Communications in Mathematical Physics - Volume 246

Commun. Math. Phys. 246, 1–18 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1033-z Communications in Mathe...

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Commun. Math. Phys. 246, 1–18 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1033-z

Communications in

Mathematical Physics

Teichmuller ¨ Groupoids, and Monodromy in Conformal Field Theory Takashi Ichikawa Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan. E-mail: [email protected] Received: 15 July 2002 / Accepted: 6 October 2003 Published online: 29 January 2004 – © Springer-Verlag 2004

Abstract: We study monodromy representations of the Teichmüller groupoid for the moduli space of pointed compact Riemann surfaces of any genus with first-order infinitesimal structure. To calculate these representations, using arithmetic Schottky-Mumford uniformization theory we construct a real orbifold in the moduli space consisting of fusing and simple moves which gives tangential base points. For a certain vector bundle on the moduli space with projectively flat connection, we show that the monodromy of each fusing move can be expressed as a connection matrix, and give the relations to the monodromy of simple moves. Furthermore, we describe the monodromy representation associated with Tsuchiya-Ueno-Yamada’s conformal field theory, and show that this representation can be expressed as the monodromy of the Wess-Zumino-Witten model. Introduction (1)

Assume that 2g + n − 2 > 0, and let Mg,n denote the moduli space classifying n-pointed compact Riemann surfaces of genus g with first-order infinitesimal struc(1) ture. Then Mg,n is known to be a complex orbifold of dimension 3g + 2n − 3, and by results of Deligne, Mumford and Knudsen [DMu, Kn], it is an open subspace of the moduli space classifying stable n-pointed complex curves of genus g with first-order (1) infinitesimal structure. The Teichmüller groupoid for Mg,n is, by definition, the funda(1) mental groupoid of Mg,n whose base points are the points at “infinity” corresponding to maximally degenerate pointed complex curves. In [MS1, MS2], Moore and Seiberg studied the monodromy representation of this Teichmüller groupoid associated with conformal field theory. The aim of this paper is to give its accurate formulation and proof (1) based on the notion of tangential base points (this notion is due to Deligne [D]) in Mg,n , and calculate the monodromy representation associated with Tsuchiya-Ueno-Yamada’s conformal field theory [TUY]. These results were obtained by Tsuchiya-Kanie [TK] and by Drinfeld [Dr] in the genus 0 case. We would claim that in higher genus case,

2

T. Ichikawa

appropriate tangential base points for calculating the monodromy can be constructed from the arithmetic Schottky-Mumford uniformization theory given in [I1]. Using these tangential base points and topological arguments, we show that the monodromy repre(1) (1) sentation is decomposed to the basic monodromy with respect to M0,4 and M1,1 , and is expressed as the monodromy of the Wess-Zumino-Witten model. Our consideration can be applicable to other conformal field theories satisfying the factorization property, and we expect that the associated monodromy representations can be described by connection matrices and by the transformation matrices of non-abelian theta functions. These tangential base points are also used in [I2] to study Grothendieck’s conjecture [Gr] on the profinite Teichmüller groupoids with Galois action which come from the moduli stack of pointed curves (not considering first-order infinitesimal structures). (1) Note that the result of [I2] can be easily extended for the natural model over Z of Mg,n as a moduli stack. See [IhN, LoNS, N1, N2, NS] for the construction of tangential base points of restricted types and its application to the study of Galois representations. This paper is organized as follows. In Sect. 1, by using the comparison theorem on deformation parameters of degenerate (1) pointed curves (see [I2], Theorem 1), we construct a real orbifold in Mg,n consisting of fusing moves (which are also called associativity moves or A-moves) and simple moves (which are also called S-moves) which gives tangential base points around each point at infinity. Our construction, which seems rather complicated, is necessary to calculate monodromy representations of the Teichmüller groupoid exactly, and will be useful if (1) one considers the rationality of vector bundles on Mg,n because we work in the framework of arithmetic geometry over Z unifying complex and formal geometry. By this result and the completeness theorem in [MS2] (see [BK1, BK2, F, FuG, HLS, NS] for its accurate formulation and proof), one can see that in the category of arithmetic geom(1) etry over Z, the Teichmüller groupoid for Mg,n can be decomposed to the fundamental generators; fusing moves, simple moves and Dehn half-twists. (1) In Sect. 2, we consider a vector bundle V with projectively flat connection ∇ over Mg,n satisfying a certain condition which is satisfied if (V, ∇) is given by Tsuchiya-UenoYamada’s theory. For such a (V, ∇), we define the associated monodromy representation with respect to the tangential base points constructed in Sect. 1, and using the comparison theorem we show that any fusing matrix (: the monodromy of a fusing move) is expressed as the connection matrix defined from the residues of ∇ at the points at infinity. Our proof is the same as given in [Dr, TK] conceptually, but is rather difficult technically because we treat projectively flat connections not necessarily flat. Combining the description of fusing matrices with the completeness theorem (we use the formulation and results in [F, NS] here), one can give a condition that the images of simple moves make a projectively linear representation of the Teichmüller groupoid. In Sect. 3, after giving a brief review of Tsuchiya-Ueno-Yamada’s theory, we apply the above results to showing that the associated monodromy representation is described by the connection matrices and by the transformation matrices of characters. From this we conclude that the representation is expressed as the monodromy of the WessZumino-Witten model written in [Ko1, Ko2]. 1. Teichmuller ¨ Groupoids 1.1. First we recall the well known correspondence between certain graphs and degenerate pointed curves, where a curve (resp. pointed curve) is called degenerate if it is a stable

Teichmüller Groupoids and Monodromy

3

curve (resp. stable pointed curve) and the normalization of its irreducible components are all projective lines (resp. pointed projective lines). A graph = (V , E, T ) means a collection of 3 finite sets V of vertices, E of edges, T of tails, and 2 boundary maps b : T → V , b : E → V ∪ {unordered pairs of elements of V } such that the geometric realization of is connected. A graph is called stable if each of its vertices has degree ≥ 3, i.e. has at least 3 branches. Denote by X the number of elements of a finite set ∼ X. Under fixing a bijection ν : T → {1, ..., T }, which we call a numbering of T , a stable graph = (V , E, T ) becomes the dual graph of a degenerate T -pointed curve of (arithmetic) genus rankZ H1 (, Z) by the correspondence: {vertices of } ←→ {irreducible components of the curve}, {edges of } ←→ {singular points of the curve}, {tails t of } ←→ {ν(t)-th marked points of the curve}, such that an edge (resp. a tail) has a vertex as its boundary if the corresponding singular (resp. marked) point belongs to the corresponding component. An orientation of = (V , E, T ) means giving an orientation of each e ∈ E. Under an orientation of , denote by ±E = {e, −e | e ∈ E} the set of oriented edges, and by vh the terminal vertex of h ∈ ±E (resp. the terminal vertex b(h) of h ∈ T ). If e is a loop, then in fact ve = v−e . Let |h| ∈ E be the edge h without orientation for each h ∈ ±E, and put |t| = t for each t ∈ T . A rigidification of a stable graph having an orientation means a collection τ = (τv )v∈V of injective maps τv : {0, 1, ∞} −→ {h ∈ ±E ∪ T | vh = v} such that τv (a) = −τv (a) if v = v and τv (a), τv (a) ∈ ±E. One can see that any stable graph has a rigidification by the induction on the number of edges and tails. For an oriented stable graph = (V , E, T ) with rigidification τ, put E = ±E ∪ T − {τv (∞) | v ∈ V }, and let A(,τ ) be the formal power series ring of variables ye (e ∈ E) over the Z-algebra A0 which is generated by xh (h ∈ E), 1/(xe − x−e ) (e, −e ∈ E − T ) and 1/(xh − xh ) (h, h ∈ E with h = h , vh = vh ), where xh = a for h = τv (a) (a ∈ {0, 1}) and xh are variables for the other h ∈ E. If is trivalent, i.e. any vertex of has just 3 branches, which is equivalent to that a (unique up to isomorphism) degenerate pointed curve with dual graph is maximally degenerate, then v∈V Im(τv ) = ±E ∪ T , and hence A0 = Z. Denote by C0 the degenerate T -pointed curve over A0 with dual graph which is obtained from the collection of Pv (v ∈ V ) by identifying the points xe ∈ Pve 1 over A having the and x−e ∈ Pv−e (e ∈ E), where Pv denotes the projective line PA 0 0 th ν(t) marked points xt if t ∈ T with vt = v, and xh denotes ∞ if h ∈ E. Then in [I2], 1.3, using the arithmetic Schottky-Mumford uniformization theory [I1] we construct a stable T -pointed curve of genus rankZ H1 (, Z) over A(,τ ) , which we denote by C(,τ ) , as a deformation by ye (e ∈ E) of C0 . Ihara and Nakamura [IhN] had already constructed this deformation when is trivalent and has no loops. For a stable curve C with marked point p over a scheme S, a first-order infinitesimal structure ξ of C at p means an OS -algebra homomorphism OS [z] (z2 ) −→ OC Ip2 ,

4

T. Ichikawa

where z is a variable and Ip is the defining ideal of p(S) in C (note that we do not require that ξ is an isomorphism). If ξ(z) = 0 in OC /Ip2 , then ξ is called trivial. For each irre∼ 1 be the isomorphism satisfying ducible component Pv (v ∈ V ) of C0 , let αv : Pv → PA 0 that for any a ∈ {0, 1, ∞}, αv−1 (a) is the singular or marked point corresponding to τv (a). Then for the ν(t)th marked point pν(t) corresponding to each tail t with terminal vertex v, (αv (u) − αv (pν(t) ))yt mod(x − αv (pν(t) ))2 if αv (pν(t) ) = ∞, yt ξyt (u) = if αv (pν(t) ) = ∞, mod(1/x)2 αv (u) 1 , gives a first-order infinitesimal structure where x denotes the natural coordinate on PA 0 (,τ ) = C(,τ ) ; ξyt (t ∈ T ) is a stable T -pointed curve with of C(,τ ) at pν(t) . Thus C (,τ ) = A(,τ ) [[yt (t ∈ T )]] . first-order infinitesimal structure over A By [I1], Theorem 3.21, taking xh (h ∈ E), ye (e ∈ E) as complex variables such that   xe = x−e for e, −e ∈ E − T , xh = xh for h, h ∈ E with h = h , vh = vh ,  x = a for h = τ (a) with a ∈ {0, 1}, h v

and that ye are sufficiently small, C(,τ ) gives rise to a family C(,τ ) of stable n-pointed complex curves of genus g. Furthermore, for each t ∈ T , ξyt gives first-order infinitesimal structures of C(,τ ) at the ν(t)th markedpoint pν(t) for sufficiently small complex numbers yt . Put C(,τ ) = C(,τ ) ; ξyt (t ∈ T ) . In particular, if is trivalent, then C(,τ ) and C(,τ ) are complex deformations of the maximally degenerate pointed complex curve C0 ⊗Z C with dual graph . 1.2. Provided that 2g + n − 2 > 0, denote by M g,n the moduli space as a complex orbifold classifying stable n-pointed complex curves of genus g, and denote by Mg,n (1)

its open subspace classifying smooth pointed curves (see [DMu, Kn]). Let M g,n be the moduli space classifying stable n-pointed complex curves (C; p1 , ..., pn ) of genus g (1) with first-order infinitesimal structure ξi at each pi . Then M g,n contains an open sub(1)

space Mg,n classifying data (C; p1 , ..., pn ; ξ1 , ..., ξn ) such that C is smooth and each ξi is an isomorphism. By adding the trivial first-order infinitesimal structure at each marked (1) point, M g,n is regarded as a subspace of M g,n , and by forgetting first-order infinitesimal (1) ∼ Cn . Using a result structures, we have a covering morphism M → M g,n with fibers = g,n

(1)

of [I2], we will construct an appropriate base set of the Teichmüller groupoid for Mg,n as a union of fusing moves and simple moves. To calculate monodromy representations of the Teichmüller groupoid exactly, we will also give the associated tangential base points. First, we consider fusing moves which, topologically, connect different sewing processes of two 3-holed spheres to one 4-holed sphere. Let = (V , E, T ) be a stable graph such that rankZ H1 (, Z) = g, T = n, and that only one vertex, which we denote by v0 , has 4 branches and that the other vertices have 3 branches. Take a numbering of T and an orientation of , and denote by h1 h2 , h3 and h4 the mutually different elements of ±E ∪ T with terminal vertex v0 . Then one can take a rigidification τ = (τv )v∈V of such that τv0 (0) = h2 , τv0 (1) = h3 , τv0 (∞) = h4 , and hence x = xh1 gives


5

(,τ ) is a stable T -pointed curve with first-order the coordinate on Pv0 ∼ = P1 . Hence C infinitesimal structure over

(,τ ) = Z x, 1 , 1 A [[ye (e ∈ E ∪ T )]] . x 1−x Let = (V , E , T ) (resp.

= (V

, E

, T

)) be the trivalent graph obtained by replacing v0 with an edge e0 (resp. e0

) having two boundary vertices one of which is the terminal vertex of h1 , h2 (resp. h1 , h3 ) and another is the terminal vertex of h3 , h4 (resp. h2 , h4 ). Then one can identify T , T

with T naturally, and see that according as x → 0 (resp. x → 1), the degenerate pointed curve corresponding to x becomes the maximally degenerate pointed curve with dual graph (resp.

). Let (resp.

) without e0

(resp. e0

) have the orientation naturally induced from that of , and let h 0 (resp. h

0 ) be the edge e0 (resp. e0

) with orientation. For 1 ≤ i ≤ 4, we denote by h i (resp. h

i ) the oriented edge in (resp.

) corresponding to hi , and identify the invariant part (E ∪ T )inv = E ∪ T − {|hi | | 1 ≤ i ≤ 4} of E ∪T with that of E ∪T and of E

∪T

. Assume that all branches starting from v0 are not loops. We put u0 = x, yi = y|hi | and attach variables ui (1 ≤ i ≤ G := 3g +2n−4) to elements of E ∪ T such that  y /x(1 − x) or −y1 /x(1 − x) (i = 1),   1 y2 /x or −y2 /x (i = 2), ui =   y3 /(1 − x) or −y3 /(1 − x) (i = 3), y4 or −y4 (i = 4), and that ui (i ≥ 5) are obtained by specifying one of ye and −ye for each e ∈ (E ∪T )inv . Moreover, we assume that for any closed path π = h(1) · h(2) · · · h(l) in , the product of the signs of these variables attached to h(j ) (1 ≤ j ≤ l) is +1 under, regarding the signs of yi ,

yi yi yi , , (1 ≤ i ≤ 4), ye (e ∈ (E ∪ T )inv ) x 1 − x x(1 − x)

as +1 (there are 22g+2n−4 ways of choosing such variables). Then by [I2], Theorem 1, under u0 → 0 (resp. 1), the variables u0 (resp. 1 − u0 ) and (ui )1≤i≤G are deformation parameters of the maximally degenerate pointed curve C0 (resp. C0

) with dual graph (resp.

), and hence these variables give the basis of the tangent space at the point (1) P (resp. P

) of M g,n corresponding to C0 (resp. C0

). In the case that there are loops with boundary vertex v0 , one can take such deformation parameters, and it is easy to see that the following argument can be applicable similarly. Denote by (a, b) the real open interval between two real numbers a, b with a < b. Then for each choice of these (,τ ) variables ui , there exists a (sufficiently small) positive real number ε such that C becomes a proper and smooth n-pointed curve of genus g with first-order infinitesimal structure over R for any u0 ∈ (0, 1) and ui ∈ (0, ε) (1 ≤ i ≤ G). Hence one can define a (1) (,τ ) with u0 ∈ (0, 1) and ui ∈ (0, ε) (i ≥ 1). fusing move ϕ(ε) in Mg,n induced from C Second, we treat simple moves which, topologically, connect different sewing processes of two 3-holed spheres to one 1-holed real surface of genus 1. We consider a trivalent graph = (V , E, T ) such that rankZ H1 (, Z) = g, T = n, and take a numbering of T and a rigidification τ of the graph with orientation. Assume that there

6

T. Ichikawa

(,τ ) is a stable n-pointed curve of genus g is a loop in E, and denote this by e0 . Then C with first-order infinitesimal structure over Z [[ye (e ∈ E ∪ T )]] . We put u0 = ye0 and attach variables ui (1 ≤ i ≤ G) to elements of (E ∪ T ) − {e0 } which are obtained by specifying one of ye and −ye such that for any closed path π = h(1) · h(2) · · · h(l) in , the product of the signs of these variables attached to h(j ) (1 ≤ j ≤ l) is +1 under, regarding the sign of each yh(j ) as +1 (there are 22g+2n−3 ways of choosing such (,τ ) variables). For each choice of these variables ui , there exists ε > 0 such that C becomes a proper and smooth n-pointed curve of genus g with first-order infinitesimal structure over R for any u0 ∈ (0, e−π ) and ui ∈ (0, ε) (1 ≤ i ≤ G). Hence we have an etale morphism from (u0 , u1 , ..., uG ) 0 < u0 < e−π , 0 < ui < ε (i ≥ 1)

(1)

into Mg,n , and denote this image by σ1 (ε). Similarly, we obtain σ2 (ε) as the image of (u0 , u1 , ..., uG ) e−4π < u0 < 1, 0 < ui < ε (i ≥ 1)

by the composite of the morphism which is obtained from replacing u0 by u 0 , and the transformation u0 −→ u 0 = exp 4π 2 / log(u0 ) which corresponds to the transformation τ → −1/τ (τ ∈ the Poincaré upper half-plane) of periods of elliptic √ curves over C because u0 can be regarded as multiplicative periods given by exp(2π −1τ ). Then we define the simple move σe0 = σ (ε) corresponding to e0 as the union σ1 (ε) σ2 (ε). (1) From the above, we have the subloci ϕ(ε) and σ (ε) of Mg,n associated with each fusing move and simple move respectively. Then by [I2], Theorem 1, moving graphs , variables ui , and taking the union of all ϕ(ε) and σ (ε) for sufficiently small ε > 0, we (1) obtain a sublocus L of Mg,n which consists of fusing moves and simple moves. Note that a 4-pointed projective line degenerates maximally in 3 ways and that these degenerations are connected by fusing moves. From this and the above consideration, we have the following result which is an extension of [I2], Theorem 2 considering first-order infinitesimal structure on pointed curves. Theorem 1. L becomes a real orbifold of dimension 3g + 2n − 3, and for an etale (1) neighborhood π : U → M g,n (U : a simply connected complex manifold) of each point corresponding to a maximally degenerate pointed curve, π −1 (L) ∩ U decomposes into 22g+2n−3 simply connected pieces, which are regarded as tangential base points of the point at infinity. (1)

(1)

We define the Teichmu¨ ller groupoid for Mg,n as the fundamental groupoid of Mg,n with base set L, which is seen to be generated by fusing moves, simple moves in L and by Dehn half-twists around points at infinity.


7

2. Connections and Fusing Matrices 2.1. In what follows, we consider curves (with additional data) in the category of complex geometry. Let (V, ∇) be a vector bundle (whose fibers are finite dimensional C-vector (1) spaces) with projectively flat connection over Mg,n , and assume that (V, ∇) has an (1)

extension (V, ∇) as a vector bundle with connection over M g,n such that ∇ has a (1)

(1)

logarithmic pole along M g,n − Mg,n . Let be a trivalent graph having n tails such that rankZ H1 (, Z) = g, and let be a subgraph of which is a trivalent graph having no loop, i.e. is the dual graph of a maximally degenerate pointed projective line. We give a numbering of the tails of from 1 to m := {tails of }. For a (not necessarily connected) maximally degenerate pointed complex curve C with dual graph − ( without tails), attaching C gives a morphism (1)

ι, : M 0,m −→ M g,n ⊂ M g,n . (1)

Then in what follows, we assume that the vector bundle V over M g,n satisfies the following condition: () The pullback (ι, )∗ (V) of V by any morphism ι, is isomorphic to a trivial bundle on M 0,m . Let C(,τ ) be as in 1.1, and take variables ze (e ∈ E ∪ T ) which are obtained by specifying one of ye and −ye . Furthermore, assume that the stable graph is trivalent, i.e. the associated pointed complex curve C0 is maximally degenerate. Denote by P the point of M g,n corresponding to C0 . Then C(,τ ) gives a deformation of C0 with (sufficiently small) complex parameters ze (e ∈ E ∪ T ) as a family of stable n-pointed complex curves of genus g with first-order infinitesimal structure. Hence by trivializing V with fiber V (: a finite dimensional C-vector space) over this family, the connection form ω of ∇ is defined by ∇v = ωv (v : local sections of V), and is expressed as ω =

Fe

e∈E∪T

dze at P , ze

where Fe are EndC (V )-valued holomorphic functions of ze . We define the residue Ae of ω for ze as the value of Fe at P which is independent of rigidifications of , and define the principal part of ω for ze (e ∈ E ∪ T ) as the EndC (V )-valued 1-form ω =

e∈E∪T

Ae

dze . ze

Since ∇ is projectively flat, [Ae , Ae ] := Ae Ae − Ae Ae is a scalar matrix in EndC (V ) for any e, e ∈ E ∪ T , and hence ω also defines a projectively flat connection on the trivial bundle with fiber V . For a piecewise smooth path γ (ζ ) ⊂ {(ze )e∈E∪T | 0 < ze ≤ 1 (e ∈ E ∪ T )}

8

T. Ichikawa (1)

from the point defined by ze = 1 (e ∈ E ∪ T ) to ζ ∈ Mg,n , let

ω · · ω

· γ (ζ )

i

denote the identity matrix in EndC (V ) if i = 0, and the i-fold iterated line integral of ω on γ (ζ ) if i ≥ 1. Then the infinite sum

ω

exp γ (ζ )

=

∞ i=0

γ (ζ )

ω · · ω

· i

of these integrals gives the transport function of γ (ζ ) with respect to the connection defined by ω . Since ω defines a projectively flat connection,

exp ω mod(C× ) ∈ P GL(V ) := GL(V )/C× γ (ζ )

is independent of paths to ζ, and we denote this element of P GL(V ) by Tω (ζ ). Denote by P(V) the projective space bundle induced from V, and by P GL(V) the associated princi(1) pal bundle over M g,n . Then ∇ defines a flat connection on P(V) and on P GL(V), which we denote by ∇ again. Let σP ,(ze ) (ζ ) be the horizontal section of (P GL(V), ∇) which is normalized with respect to the variables ze in the meaning that Tω (ζ )−1 · σP ,(ze ) (ζ ) (1)

tends to the identity element IdP GL(V ) of P GL(V ) if ζ → P in M g,n . Then one can see that σP ,(ze ) (ζ ) = lim exp ω mod(C× ) · Tω (ξ ) , ξ →P

γ (ξ,ζ )

where exp γ (ξ,ζ ) ω denotes the infinite sum of the iterated line integrals of ω on a piecewise smooth path γ (ξ, ζ ) ⊂ {(ze )e∈E∪T | 0 < ze ≤ 1 (e ∈ E ∪ T )} from ξ to ζ, and that σP ,(ze ) (ζ ) depends only on the first-order infinitesimal structure at P , i.e. σP ,(ze ) (ζ ) = σP ,(we ) (ζ ) if for any e ∈ E ∪ T , we /ze is holomorphic around P and becomes 1 at P . We define the (projectively linear) monodromy representation ρ of the Teichmüller groupoid with respect to ∇ as the map which sends each element of this groupoid to its connection matrix (modulo scalars) between normalized horizontal sections of ∇. 1/2

Proposition 1. For each e ∈ E ∪ T , denote by δe the corresponding Dehn half-twist. Then we have √ 1/2 ρ(δe ) = exp π −1Ae mod(C× ). Proof. Since ∇ is projectively flat, π √ √ 1/2 −1θ −1θ ρ(δe ) = lim exp Ae d(εe ) (εe ) mod(C× ). ε→0

0


9

2.2. Let g,n be the set of trivalent graphs with n tails having the property that rankZ H1 (, Z) = g. We define a 1-dimensional complex Xg,n whose vertices and edges are elements of g,n and fusing moves between them respectively. Then it is shown in [Ko1], Lemma 1.2 that Xg,n is connected and that Xg,n combined with 2-cells corresponding to the pentagon relation becomes a simply connected 2-dimensional complex. By the correspondence between maximally degenerate n-pointed complex curves of genus g and their dual graphs, we have a map µ : Xg,n → M g,n which sends each edge to the fusing move constructed in 1.2. Therefore, one can obtain a neighborhood of Xg,n as the pullback of a tubular neighborhood of µ(Xg,n ) in M g,n . For each fusing move ϕ, take a morphism ι : M 0,4 → M g,n and a coordinate x on M 0,4 ∼ = P1 (C) which gives M0,4 ∼ = P1 (C) − {0, 1, ∞} such that the image of {x ∈ R | 0 < x < 1} by ι represents ϕ. Then by the condition () in the case when m = 4, the pullback ι∗ (V) becomes a trivial bundle on P1 (C) whose fiber is denoted by V . This trivialization is unique up to the action of AutC (V ) because any holomorphic map P1 (C) → AutC (V ) becomes a constant map. Since the pentagon relation connects 5 points of M 0,5 corresponding maximally degenerate pointed complex curves, by the condition () in the case when m = 5 and the property of Xg,n mentioned above, one can trivialize V with fiber V on a neighborhood of Xg,n , and hence obtain Ae ∈ EndC (V ) as above for each edge of any ∈ g,n . Under this trivialization, we will express the monodromy ρ(ϕ) of ϕ (: the fusing matrix along ϕ) by the connection matrix (X, Y ) which is defined for X, Y ∈ EndC (V ) as (X, Y ) = G−1 2 · G1 ∈ EndC (V ), where Gi are the EndC (V )-valued solutions of the differential equation

G (t) =

Y X + G(t) (0 < t < 1) t t −1

X Y suchthat if we put t = exp (log(t)· X) and (1 − t) = exp (log(1 − t) · Y ) , then X Y and lim G2 /(1 − t) become the identity element of EndC (V ). Note lim G1 /t

t→0

t→1

that if X commutes with Y, then G1 = G2 = t X (1−t)Y , and hence (X, Y ) is the identity. The following assertion extends results in [Dr, TK] for projectively flat connections (not necessarily flat) in the higher genus case.

Theorem 2. Let 1 = (V1 , E1 , T1 ) and 2 = (V2 , E2 , T2 ) be elements of g,n which are connected by a fusing move, and denote by ϕ the fusing move from 1 to 2 . Identify T1 with T2 naturally, and for each i = 1, 2, decompose Ei as Ei = Ei

{i },

where ϕ replaces 1 with 2 and does not change Ei . Then we have (1) If e1 ∈ E1 ∪ T1 corresponds to e2 ∈ E2 ∪ T2 , then Ae1 = Ae2 . (2) The image ρ(ϕ) ∈ P GL(V ) of ϕ satisfies that ρ(ϕ) = (A1 , A2 ) mod(C× ).

10

T. Ichikawa

Proof. Let the notation be as above, and take an open neighborhood U0,4 of ι(M 0,4 ) in (1) M g,n , and let y1 , ..., yG (G := 3g + 2n − 4) be coordinates on U0,4 induced from the arithmetic Schottky-Mumford uniformization theory such that M 0,4 is defined by y1 = · · · = yG = 0. Then letting U0,4 be sufficiently small if necessary, the connection form ω of ∇ can be represented as G dyj B1 B2 Hj on U0,4 , + + F dx + x x−1 yj j =1

where B1 , B2 ∈ EndC (V ) and F, Hj are EndC (V )-valued holomorphic functions on U0,4 such that F = 0 if y1 = · · · = yG = 0 and that Hj |y1 =···=yG =0 is holomorphic on P1 (C) and hence is constant. We will prove the theorem in the case that 1 and 2 have 4 (distinct) incident edges because one can prove the theorem in the other cases by the same method. Put P1 = P , P2 = P

for P , P

considered in 1.2, and take y1 , y2 , y3 , y4 , u0 = x, ui (1 ≤ i ≤ G) as in 1.2 such that {(u0 , u1 , ..., uG ) | 0 < u0 < 1, 0 < ui < ε (i ≥ 1)} represents the fusing move ϕ for sufficiently small ε > 0. Since F, Hj become holomorphic for sufficiently small u0 , u1 , ..., uG and for sufficiently small 1 − u0 , u1 , ..., uG , if we put Dj = Hj |y1 =···=yG =0 ∈ EndC (V ), then the principal parts of ω for u0 , u1 , ..., uG and 1 − u0 , u1 , ..., uG are expressed as ω1 = (B1 + D1 + D2 )

duj du0 + Dj u0 uj G

j =1

and ω2 = (B2 + D1 + D3 )

duj d(1 − u0 ) + Dj 1 − u0 uj G

j =1

respectively. This implies (1) and that A1 = B1 + D1 + D2 , A2 = B2 + D1 + D3 . Furthermore, by considering the transport functions of the 2 paths: {(1, s, ..., s) | ε ≤ s ≤ 1} ∪ {(t, ε, ..., ε) | ε ≤ t ≤ 1}, {(0, s, ..., s) | ε ≤ s ≤ 1} ∪ {(1 − t, ε, ..., ε) | ε ≤ t ≤ 1} in the (u0 , u1 , ..., uG )-space for positive ε tending to 0, we have −1

ρ(ϕ) = lim exp ω · (A1 , A2 ) · exp ω ε→0

γ

mod(C× ),

γ

where γ is the path from (1, ..., 1) to (ε, ..., ε) in the (u1 , ..., uG )-space given by {(s, ..., s) | ε ≤ s ≤ 1}, and G duj ω

= Dj . uj j =1


11

Since the connections defined by ω1 , ω2 and duj du0 d(1 − u0 ) + A2 + Dj u0 1 − u0 uj G

A1

j =1

are projectively flat, we have ρ(ϕ) = (A1 , A2 ) mod(C× ), which completes the proof. 2.3. By the completeness theorem (see [BK1, BK2, F, FuG, HLS, MS2, NS]), the (1) Teichmüller groupoid for Mg,n considered in Sect. 1 is known to be generated by fusing (1) (1) moves, simple moves and Dehn half-twists with relations induced from M0,4 , M0,5 , (1)

(1)

M1,1 and M1,2 . Using this result we will give a condition that the images by ρ of Dehn half-twists, fusing moves described as above, and the images of simple moves make a projectively linear representation of the Teichmüller groupoid. Here we use the formulation and results in [F, NS] mainly. (1) First, we consider relations induced from M0,4 . Take the dual graph of a degenerate n-pointed curve of genus g such that only one vertex, which we denote by v0 , has 4 branches denoted by 1 , 2 , 3 , 4 and that the other vertices have 3 branches. Let i (i = 1, 2, 3) be the 3 elements of g,n obtained from this graph by replacing v0 with an edge, which we denote by ei respectively, having two boundary vertices. Denote by ϕ1 , ϕ2 and ϕ3 the fusing moves 1 → 2 , 2 → 3 and 3 → 1 respectively, and 1/2 1/2 by δei (i = 1, 2, 3), δj (j = 1, 2, 3, 4) the Dehn half-twists corresponding to ei , j ! 1/2 1/2 1/2 1/2 respectively. Then we have δe1 · ϕ1 · δe2 · ϕ2 · δe3 · ϕ3 = 4j =1 δj (see [NS], the proof of Claim 8.3), and hence 4 " 1/2 1/2 1/2 = . ρ δ1/2 ρ (ϕ3 ) ρ δe3 ρ (ϕ2 ) ρ δe2 ρ (ϕ1 ) ρ δe1 j j =1

Therefore, by Proposition 1 and Theorem 2 (2), we have √ √ Ae3 , Ae1 exp π −1Ae3 Ae2 , Ae3 exp π −1Ae2 √ · Ae1 , Ae2 exp π −1Ae1 mod(C× ) =

4 "

√ exp π −1Aj mod(C× ).

(2.1)

j =1 (1)

We consider relations induced from M0,5 . Let 1 , ..., 5 be 5 elements of g,n , and denote by Ei (1 ≤ i ≤ 5) the set of edges of i such that there are fusing moves ϕi (1 ≤ i ≤ 4) from i to i+1 and ϕ5 from 5 to 1 . We assume that the sequence ϕ1 · ϕ2 · ϕ3 · ϕ4 · ϕ5 of fusing moves satisfies the pentagon relation, and hence it becomes

the identity. Let ϕi replace ei ∈ Ei with ei+1 ∈ Ei+1 for 1 ≤ i ≤ 4, and let ϕ5 replace

e5 ∈ E5 with e1 ∈ E1 . Then by Theorem 2 (1), Aei = Aei+1 for 1 ≤ i ≤ 4, and Ae = Ae1 , and by Theorem 2 (2), we have 5 Ae5 , Ae1 Ae4 , Ae Ae3 , Ae4 Ae2 , Ae3 Ae1 , Ae2 ∈ C× , 5

12

T. Ichikawa

which is equivalent to the 5-cycle relation: Ae5 , Ae2 Ae4 , Ae1 Ae3 , Ae5 Ae2 , Ae4 Ae1 , Ae3 ∈ C× . (2.2) (1)

We consider relations induced from M1,1 . Let be an element of g,n having a loop which we denote by e0 , and denote by e1 the tail or the edge of incident to e0 when (g, n) = (1, 1) or not respectively. Then by [BK2], (5.2.10) and (5.2.11), the 1/2 simple move σe0 corresponding to e0 and the Dehn half-twists δei corresponding to 3 1/2 −1/2 ei (i = 0, 1) satisfy the relations (δe0 )2 · σe0 = σe20 , σe20 = δe1 , where the latter ˜ (see also the proof of [NS], Lemma relation is presented in the formulation of [F], 2S 7.3). Therefore, by Proposition 1, we have √ √ 3 ρ(σe0 ) exp 2π −1Ae0 = ρ(σe0 )2 = exp −π −1Ae1 mod(C× ). (2.3) (1)

Lastly, we consider relations induced from M1,2 . Take an element of g,n having a (unique up to isomorphism) subgraph which is the dual graph of a maximally degenerate 2-pointed curve of genus 1 having smooth irreducible components. Denote by e1 , e2 the two edges, and by e3 , e4 the two tails of this subgraph. Then for i = 1, 2, the composition of a fusing move ϕi replacing ei with i , a simple move σ3−i replacing

e3−i with e3−i and a fusing move ϕi replacing i with ei gives rise to the same graph obtained from by replacing e1 , e2 with e1 , e2 respectively. Note that simple moves act trivially on g,n , however, those constructed in 1.2 change degeneration process, and hence Aei = Aei in general. By [MS2], (4.16) and [BK2], (5.2.12), 1/2 −2 1/2 2 1/2 1/2 −1 · δe

· (ϕ2 )−1 · σ1−1 = δe3 · δ1/2 · δ · (ϕ1 )−1 · ϕ2 , σ2 · ϕ1 · δe

e 4 1 1

2

1/2 δ∗

where denotes the Dehn half-twist corresponding to ∗, and hence by Proposition 1 and Theorem 2 (2), we have √ (2.4) ρ(σ1 )−1 Ae2 , A2 exp 2π −1 Ae2 − Ae1 A1 , Ae1 ρ(σ2 ) √ = Ae2 , A2 A1 , Ae1 exp −π −1 Ae3 + A1 + Ae4 mod(C× ). Theorem 3. The system consisting of the relations (2.1) – (2.4) and the commutativity relation (C): (C) the images of α, β (α, β ∈ {fusing moves, simple moves, Dehn half-twists}) by ρ commute if α, β are supported on mutually disjoint real subsurfaces gives a necessary and sufficient condition that the images by ρ of simple moves, combined with the images of Dehn half-twists and fusing moves (described in Proposition 1 and Theorem 2 respectively), make a projectively linear representation ρ of the Teichmu¨ ller (1) groupoid for Mg,n with base set L. Proof. The necessity of the conditions is clear from the above. For an orientable real surface of genus g with n boundary components, the extended Hatcher complex is defined in [F] as a 2-dimensional complex whose 0-cells are quilt decompositions (see [NS], Sect. 7) of this real surface, 1-cells are fusing moves, simple moves and Dehn half-twists,


13

2-cells are the above relations corresponding to (2.1)–(2.4) and to (C). Then it is shown in [F], Theorem 5 (see also [NS], Sects. 7–8) that the extended Hatcher complex is connected and simply connected by using [BK1], Proposition 6.2 and the fact (see [HLS], Theorem 2) that the maximal multicurve complex is connected and simply connected. Since moves of quilt decompositions correspond to moves of the tangential base points constructed in 1.2, these relations generate all the relations in the Teichmüller groupoid. Theorem 3 follows from this fact. 3. Conformal Blocks and Monodromy 3.1. First, we review the conformal field theory by Tsuchiya, Ueno and Yamada [TUY] (see also [T, U]). Let g be a finite dimensional complex simple Lie algebra, and we fix a positive integer K called a level. Denote by P+ the set of dominant integral weights of g, and we put P+ (K) = {λ ∈ P+ | 0 ≤ (λ, θ ) ≤ K} , where ( , ) is the Cartan-Killing form which is normalized such that (θ, θ ) = 2 for the highest root θ. We define the conformal weight λ attached to λ ∈ P+ (K) by λ =

(λ, λ + 2ρ) , 2(K + g ∗ )

where ρ is the half-sum of the positive roots of g, and g ∗ is the dual Coxeter number of g. For each λ ∈ P+ (K), denote by Vλ the (finite dimensional) irreducible g-module of highest weight λ (in particular, V0 is the trivial g-module C), and define an involution λ → λ† on P+ (K) which is characterized by the fact that −λ† is the lowest weight of the g-module Vλ . Then it is known that λ = λ† and that there is a (unique up to constant multiple) nonzero g-invariant C-bilinear map βλ : Vλ ⊗C Vλ† −→ C. = (λ1 , ..., λn ) ∈ (P+ (K))n , define a g-module For each λ Vλ = Vλ1 ⊗C · · · ⊗C Vλn . Using βµi (1 ≤ i ≤ n), Homg(Vλ , Vµ ) can be identified with Homg(Vλ ⊗C Vµ † , C), where µ † = (µ†1 , ..., µ†m ) for µ = (µ1 , ..., µm ) ∈ (P+ (K))m . Let F = (π : C → B; s1 , ..., sn ; η1 , ..., ηn ) be a family over B of stable n-pointed complex curves of genus g with first-order infin ∈ (P+ (K))n , the sheaf V † (F) over B of conformal itesimal structure. Then for each λ λ blocks attached to F is constructed in [TUY]. Since this construction is rather complicated, we only recall important properties of V† (F) in the following: λ

(I)

V† (F) λ

becomes a locally free sheaf of finite rank (i.e. a vector bundle with finite dimensional fiber) over B, and if F is obtained by the base change for a morphism B → B, then there exists a canonical isomorphism † V† (F) ×B B ∼ = Vλ (F ). λ

14

T. Ichikawa

(II) V† (F) has a projectively flat connection ∇ λ (F) with logarithmic pole along the λ discriminant locus which consists of x ∈ B such that π −1 (x) is singular or ηi is trivial for some 1 ≤ i ≤ n, and the connection is canonical, i.e. compatible with respect to the isomorphism in (I). (III) Assume that C has a section s : B → C of double points, let ν : C → C be the simultaneous normalization of C, and give first-order infinitesimal structures η , η

at s (B), s

(B) ⊂ ν −1 (s(B)) respectively. Then there exists a canonical isomorphism # ∼ † V † † (F) = V (F) ×B B, (λ,µ,µ )

λ

µ∈P+ (K)

where = (C; s1 , ..., sn , s , s

; η1 , ..., ηn , η , η

) F of stable (n + 2)-pointed complex (not necessarily connected) is the family over B curves with first-order infinitesimal structure. (IV) Let C be P1 (C) with natural coordinate x, and put F = (π : C → B; s1 , ..., sn ; η1 , ..., ηn ), where n ≥ 3, si are distinct points on P1 (C) − {∞} and ηi = x − si mod(x − si )2 . Then V† (F) becomes a trivial bundle having fiber identified with a certain subλ

space, which we denote by Wλ , of Homg(Vλ , C), and the connection on V† (F) λ is given by the Knizhnik-Zamolodchikov equation. If n = 3, then Wλ consists of φ ∈ Homg(Vλ , C) satisfying that φ Vλ1 ,i ⊗C Vλ2 ,j ⊗C Vλ3 ,k = {0} if i + j + k > K,

where k denotes the principal 3-dimensional subalgebra of g, and Vλ =

K #

Vλ, k/2

k=0

denotes the decomposition to the (k + 1)-dimensional spin k/2 components as k-modules. ∈ (P+ (K))n , the sheaves V † (F) of conformal blocks By (I) and (II), for a fixed λ λ

(1)

define a vector bundle V† over M g,n with projectively flat connection ∇ λ which has λ

(1)

(1)

the logarithmic pole along M g,n − Mg,n . Then as is explained in 2.1, we obtain the monodromy representation ρλ of the Teichmüller groupoid with respect to (V† , ∇ λ ). λ From the completeness theorem reviewed in 2.3, one can see that ρλ is determined by the image of Dehn half-twists, fusing moves and simple moves.


15

3.2. We apply the results in Sect. 2 to describing ρλ . Let C(,τ ) be as in 1.1 which gives a deformation of a degenerate n-pointed complex curve. Then by (III) and (IV), over (1) the associated locus in M g,n , V† becomes a trivial bundle whose fiber is canonically λ isomorphic to the subspace $ # Wλ (a,v) W(λ ;) = ;) a∈α(λ

v∈V

) denotes the set of maps a : ±E ∪ T → P+ (K) satisof Homg(Vλ , C), where α(λ; † v) = (a(h)) fying that a(−h) = a(h) (h ∈ ±E), a(h) = λν(h) (h ∈ T ), and that λ(a, † (h ∈ ±E ∪ T such that vh = v). Therefore, V satisfies the condition () in 2.1. λ First assume that the stable graph is trivalent, i.e. the corresponding complex curve C0 is maximally degenerate, and for each e ∈ E ∪ T , define Ae ∈ EndC (W(λ ;) ) as % multiplication by a(e) on v∈V Wλ (a,v) ⊂ W(λ ;) . Then by the sewing process (see [T], Theorem III and [TUY], 6.2) which constructs a canonical base of flat sections of (V† , ∇ λ ) from the data on the boundary, one can see that for a deformation parameter λ

of C0 associated with e, the residue of the connection form of ∇ λ is Ae . Hence by Proposition 1, we have: 1/2

be the Dehn half-twist corresponding to e. Then we have √ 1/2 ρλ (δe ) = exp π −1Ae mod(C× ).

Proposition 2. Let δe

Second assume that there is only one vertex of , which we denote by v0 , having 4 branches, and that the other vertices of have 3 branches. Let hi (1 ≤ i ≤ 4) be oriented edges of with terminal vertex v0 , and take trivalent graphs 1 , 2 replacing v0 with oriented edges 1 , 2 respectively such that h1 and h2 are incident in 1 and that h1 and h3 are incident in 2 . Then by (IV), we have the following identification between subspaces of Homg(V(a(hi ))1≤i≤4 , C) : # Wλ (a,v0 ) = W(a(h1 ),a(h2 ),µ) ⊗C W(a(h3 ),a(h4 ),µ† ) µ∈P+ (K)

=

#

W(a(h1 ),a(h3 ),η) ⊗C W(a(h2 ),a(h4 ),η† )

η∈P+ (K)

which gives the identification W(λ ;) = W(λ ;1 ) = W(λ ;2 ) . Under these identifications, A1 ∈ EndC (W(λ ;1 ) ), A2 ∈ EndC (W(λ ;2 ) ) induce   $ # µ ⊗ id. on Wλ (a,v0 ) ⊗  Wλ (a,v)  , µ

# η

 η ⊗ id. on Wλ (a,v0 ) ⊗ 

v =v0

$

 Wλ (a,v) 

v =v0

), where id. denotes the identity map on respectively for any a ∈ α(λ; Then by Theorem 2 (2), we have:

%

v =v0

Wλ (a,v) .

16

T. Ichikawa

Proposition 3. The image ρλ (ϕ) ∈ P GL(W(λ ;) ) of the fusing move ϕ from 1 to 2 satisfies that ρλ (ϕ) = (A1 , A2 ) mod(C× )   # # # µ , η ⊗ id. mod(C× ). = ;) a∈α(λ

µ

η

3.3. We consider the monodromy for simple moves with respect to (V† , ∇ λ ). Assume λ that is trivalent and that there is a loop in E which we denote by e0 . Let C0 be the maximally degenerate (n+1)-pointed complex curve of genus (g −1) whose dual graph is obtained from by replacing e0 with an (n + 1)th marked point. Then by (III), we have # W(λ ;) = Wµ ⊗C W((λ ,µ† ); ) , where Wµ =

*

µ∈P+ (K)

,µ† ); ) denotes the fiber of the vector bunη∈P+ (K) W(µ,η,η† ) , and W((λ (1) over M g−1,n+1 around the point corresponding to C0 . It is known that Wµ dle (1) is the fiber of the vector bundle Vµ† over M 1,1 around the point corresponding to the degenerate elliptic curve (see (III)), and that the connection ∇ µ on Vµ† is flat (see [U],

V † † (λ,µ )

1.4 b)). Hence one can obtain the associated monodromy representation (1)

ρµ : π1 (M1,1 ) −→ GL(Wµ ) (1)

as the map sending each element of π1 (M1,1 ) to its connection matrix between normalized horizontal sections of (Vµ† , ∇ µ ). When µ = 0, it is known that the horizontal sections of ∇ 0 can be expressed as the characters {χλ (τ )}λ∈P+ (K) (τ ∈ the Poincaré upper half-plane) of the affine Lie algebra associated with g (see [U], 1.4 b)), and hence by results of Kac and Peterson [KP], the image of the simple move by ρ0 : π1 (M1,1 ) → GL(W0 ) is the transformation matrix of the vector consisting of {χλ (τ )}λ∈P+ (K) . (1) We denote by ρµ (σ ) the image by ρµ of the simple move σ in M1,1 constructed in * 1.2 for each µ ∈ P+ (K), and we define µ ρµ (σ ) ∈ GL W(λ ;) as the direct sum of ρµ (σ ) (µ ∈ P+ (K)). Since the construction of the projectively flat connection ∇ λ given in [TUY] is functorial, we have: (1)

Proposition 4. Let σe0 be the simple move in Mg,n corresponding to e0 which is constructed in 1.2. Then the image ρλ (σe0 ) of σe0 satisfies that   # ρµ (σ ) mod(C× ). ρλ (σe0 ) =  µ∈P+ (K)

3.4. Using the above results, we show: Theorem 4. ρλ becomes the projectively linear representation given in [Ko2] of the (1) Teichmu¨ ller groupoid for Mg,n .


17

Proof. Let ρ denote the representation constructed in [Ko2], Sects. 1–3. Then its repλ resentation space is identified with Wλ , and by Proposition 2, the image of each Dehn half-twist by ρ is that by ρλ . Furthermore, the image of each fusing move by ρ is the λ λ holonomy of the Knizhnik-Zamolodchikov equation, and hence is seen to be the image by ρλ from Proposition 3. The same as ρ0 , the representation ρ0 : π1 (M1,1 ) → P GL(W0 ) is described by the transformation matrix of {χλ (τ )}λ∈P+ (K) , and it is shown in [LY], Sect. 3 by a topological argument that for any µ ∈ P+ (K), ρµ , ρµ : π1 (M1,1 ) −→ P GL(Wµ ) (1)

are described by ρ0 , ρ0 respectively and by braiding matrices (which are known to be described by fusing matrices). Hence by Proposition 4, any simple move has the same image by ρλ and ρ . Thus by the completeness theorem, we have ρλ = ρ . λ

λ

Acknowledgements. I wish to thank deeply the referee whose valuable remarks helped to improve the paper, and also thank Professor Susumu Hirose who instructed me in results and references containing [F] on Teichmüller groupoids.

References [BK1]

Bakalov, B., Kirillov, A., Jr.: On the Lego-Teichmüller game. Transform. Groups 5, 207–244 (2000) [BK2] Bakalov, B., Kirillov, A., Jr.: Lectures on Tensor categories and modular functors. University Lecture Series 21, Providence RI: Am. Math. Soc., 2001 [D] Deligne, P.: Le groupe fondamental de la droite projective moins trois points. In: Galois Groups Over Q. Publ. MSRI 16, Berlin-Heidelberg-New York: Springer-Verlag, 1989, pp. 79–298 [DMu] Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Inst. Hautes ´ Etudes Sci. Publ. Math. 36, 75–109 (1969) [Dr] Drinfeld, V.G.: On quasi-triangular quasi-Hopf algebras and a group closely connected with Gal(Q/Q). Algebra i Analiz 2, 114–148 (1990) English transl. Leningrad Math. J. 2, 829–860 (1991) [F] Fuchizawa, H.: Quilt decompositions of surfaces and Torelli group action on extended Hatcher complex. Preprint [FuG] Funar, L., Gelca, R.: On the groupoid of transformations of rigid structures on surfaces. J. Math. Sci. Univ. Tokyo 6, 599–646 (1999) [Gr] Grothendieck, A.: Esquisse d’un programme, Mimeographed Note (1984). In: Geometric Galois Actions I. London Math. Soc. Lecture Note Series 242, Cambridge: Cambridge Univ. Press, 1997, pp. 5–48 [HLS] Hatcher, A., Lochak, P., Schneps, L.: On the Teichmüller tower of mapping class groups. J. Reine Angew. Math. 521, 1–24 (2000) [I1] Ichikawa, T.: Generalized Tate curve and integral Teichmüller modular forms. Am. J. Math. 122, 1139–1174 (2000) [I2] Ichikawa, T.: Teichmüller groupoids and Galois action. J. Reine Angew. Math. 559, 95–114 (2003) [IhN] Ihara, Y., Nakamura, H.: On deformation of maximally degenerate stable marked curves and Oda’s problem. J. Reine Angew. Math. 487, 125–151 (1997) [KP] Kac, V.G., Peterson, D.H.: Infinite dimensional Lie algebras, theta functions and modular forms. Adv. in Math. 53, 125–264 (1984) [Kn] Knudsen, F.: The projectivity of the moduli space of stable curves II. Math. Scand. 52, 161–199 (1983) [Ko1] Kohno, T.: Topological invariants for 3-manifolds using representations of mapping class groups I. Topology 31, 203–230 (1992) [Ko2] Kohno, T.: Topological invariants for 3-manifolds using representations of mapping class groups II: Estimating tunnel number of knots. In: Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups. Contemp. Math. 175, Providence RI: Am. Math. Soc., 1994, pp. 193–217

18 [LY] [LoNS] [MS1] [MS2] [N1] [N2]

[NS] [T] [TK] [TUY] [U]

T. Ichikawa Li, M.,Yu, M.: Brading matrices, modular transformations and topological field theories in 2+1 dimensions. Commun. Math. Phys. 127, 195–224 (1990) Lochak, P., Nakamura, H., Schneps, L.: On a new version of the Grothendieck-Teichmüller group. C. R. Acad. Sci. Paris, Série I 325, 11–16 (1997) Moore, G., Seiberg, N.: Polynomial equations for rational conformal field theories. Phys. Lett. B212, 451–460 (1988) Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123, 177–254 (1989) Nakamura, H.: Galois representations in the profinite Teichmüller modular groups. In: Geometric Galois Actions I. London Math. Soc. Lecture Note Series 242, Cambridge: Cambridge Univ. Press, 1997, pp. 159–173 Nakamura, H.: Limits of Galois representations in fundamental groups along maximal degeneration of marked curves. I. Am. J. Math. 121, 315-358 (1999); II, In: Arithmetic Fundamental Groups and Noncommutative Algebra. Proc. Symp. Pure Math. 70, Providence RI: Am. Math. Soc., 2002, pp. 43–78 Nakamura, H., Schneps, L.: On a subgroup of Grothendieck-Teichmüller group acting on the tower of profinite Teichmüller modular groups. Invent. Math. 141, 503–560 (2000) Tsuchiya, A.: Moduli of stable curves, conformal field theory and affine Lie algebras. In: Proceedings of the International Congress of Mathematicians, Kyoto 1990, Berlin-Heidelberg-New York: Springer-Verlag, 1991, pp. 1409–1419 Tsuchiya, A., Kanie, Y.: Vertex operators in conformal field theory on P1 and monodromy representations of the braid group. Adv. Stud. in Pure Math. 16, 297–372 (1988) Tsuchiya, A., Ueno, K., Yamada, Y.: Conformal field theory on universal family of stable curves with gauge symmetries. Adv. Stud. in Pure Math. 19, 459–566 (1989) Ueno, K.: On conformal field theory. In: Vector Bundles in Algebraic Geometry. Lecture Note Series 208, London: London Math. Soc., 1995, pp. 283–345

Communicated by L. Takhtajan

Commun. Math. Phys. 246, 19–42 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1043-5

Communications in


Discrete Dynamical Systems Associated with the Configuration Space of 8 Points in P3 (C) Tomoyuki Takenawa Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8914, Japan Received: 10 March 2003 / Accepted: 27 October 2003 Published online: 13 February 2004 – © Springer-Verlag 2004

Abstract: A 3 dimensional analogue of Sakai’s theory concerning the relation between rational surfaces and discrete Painlevé equations is studied. For a family of rational varieties obtained by blow-ups at 8 points in general position in P3 , we define its symmetry group using the inner product that is associated with the intersection numbers and (1) show that the group is isomorphic to the Weyl group of type E7 . By parametrizing the configuration space by means of elliptic curves, the action of the Weyl group and the dynamical system associated with a translation are explicitly described. As a result, it is found that the action of the Weyl group on P3 preserves a one parameter family of quadratic surfaces and that it can therefore be reduced to the action on P1 × P1 . 1. Introduction Relations between Painlevé equations and rational surfaces were first studied by Okamoto [12]. He showed that for each Painlevé equation, by elimination of the singularity of the equation, the solutions can be regularly extended into a family of rational surfaces. Such a family of rational surfaces is called the space of initial conditions for the Painlevé equation. Conversely, it is clarified by Saito and Takano et al. [16, 14] that for a given space of initial conditions the Hamilton system of a Painlevé equation can be determined. Since the singularity confinement method was introduced by Grammaticos et al. [6], the discrete Painlevé equations have been studied extensively ([13] for example). Emphasizing the fact that each discrete Painlevé equation preserves a family of rational surfaces, Sakai constructed the discrete Painlevé equations from families of rational surfaces (called generalized Halphen surfaces) and subsequently classified them. Such a family of rational surfaces is called the space of initial conditions for that discrete Painlevé equation. A generalized Halphen surface can be seen to be isomorphic to a surface obtained by 9 blow-ups from P2 . Sakai’s classification also shows that the spaces of initial conditions for the discrete Painlevé equations include those for continuous ones.

20

T. Takenawa

Furthermore, the largest symmetry arises when the 9 points are in general position; all other symmetries are in the case where the points are in some special position. Each Painlevé equation is obtained as a translation associated with the corresponding affine Weyl group (extended by the automorphisms of the associated Dynkin diagram). Moreover, if the space of initial conditions is that of a continuous one, its Weyl group coincides with the group of that equation’s Bäcklund transformations. The aforementioned results all concern non-autonomous dynamical systems. There also exist some studies that deal with autonomous ones. For example, in the continuous case Adler and van Moerbeke have studied Painlevé manifolds [1] and in the discrete case various authors have studied the relations between dynamical systems and the automorphism groups of manifolds ([4, 5] for example). Sakai’s procedure for describing discrete Painlevé equations is closely related to the studies on the Cremona isometry carried out by Coble et al. [2, 3] and these two approaches coincide in the case of 9 points in general position. Whereas in the case of points in general position the Weyl group is generated by the standard Cremona transformation and exchanges of the points, in the degenerate case its generators can be constructed by changing the blow-down structures. Concerning this point one has to cite the pioneering research by Looijenga [10]. Dolgachev and Ortland also studied the case of 3 (or higher) -dimensional rational (1) varieties [3]: here (for example) the affine Weyl group of type E7 appears in the case 3 where 8 generic points in P are blown-up. If the number of points is less than 8 the Weyl group is finite and it is indefinite if the number of points is larger than 8. However, in the 3-dimensional case the action of each element of the Weyl group cannot be lifted to an isomorphism between rational varieties, obtained by blow-ups at some points. Dolgachev and Ortland call such a map a pseudo-isomorphism. In this paper we study the symmetry, parametrization of the configuration space and the associated discrete dynamical systems for the family of rational varieties obtained by blow-ups at 8 ordered points in general position in P3 . In Sect. 2, we reconstruct the argument of Dolgachev and Ortland. We consider birational automorphisms of the family of varieties such that (i) each of them acts as an automorphism on the configuration space (ii) for rational varieties on the configuration space it preserves the “inner product” of the Picard group Pic(X). Here, the inner product is defined by using the intersection numbers and the canonical divisor KX as (D, D ) := D ·D ·(− 21 KX ) for D, D ∈ Pic(X). It is shown that the resulting symmetry (1) group is the Weyl group of type E7 . This group coincides with that of Dolgachev and Ortland. In Sect. 3, parametrization of the configuration space is discussed. Although there is a straightforward parametrization, it is difficult to describe the action of the Weyl group by using this and to see the properties of the resulting dynamical systems. In this paper we therefore use a parametrization in terms of elliptic curves. Quadratic surfaces passing through the 8 points we consider form at least one parameter family. Here the 8 points are on the intersection curve of the pencil of surfaces. Normalizing the pencil, one obtains a parametrization of the configuration space. In Sect. 4, we describe the action of the Weyl group obtained in Sect. 2 in normalized coordinates. In order to calculate the concrete action we apply a 3-dimensional analogue of the period map which is introduced by Looijenga [10] and Sakai [15] for surfaces. In Sect. 5, we construct a birational dynamical system in P3 by using the action obtained in Sect. 4. Such systems are obtained corresponding to translations associated to the Weyl group. We describe one of them explicitly.

Discrete Dynamical Systems with Configuration Space of 8 Points in P3 (C)

21

In Sect. 6, it is shown that the action of the Weyl group preserves each member of the pencil of quadratic surfaces and that it can therefore be reduced to an action on P1 × P1 . (1) The reduced action of the Weyl group of type E7 on P1 × P1 coincides with the action (1) of a sub-group of the Weyl group of type E8 , which is the symmetry of the family of (the most) general Halphen surfaces. Section 7 is devoted to conclusions and discussions. 2. Symmetry Let X(4, 8) denote the configuration space of ordered 8 points in P3 (C) such that all 4 points are not on the same plane:     x1 x1 · · · x8   every 4 × 4   y1 y2 · · · y8  4 8 P GL(4, C) (C× )8 , (1)  z z · · · z  ∈ (C ) minor determinant   1 3 8   is nonzero w w ··· w 1

4

8

where two configurations are identified if one can be transformed to the other by a projective transformation. We also denote the 3-dimensional rational variety obtained by successive blowing-up at distinct 8 points Pi (xi : yi : zi : wi ) by XP1 ,··· ,P8 (or simply by X) and the family of all XP1 ,··· ,P8 ’s, where {P1 , · · · , P8 } ∈ X(4, 8), by {XP1 ,··· ,P8 }. X(4, 8) is called the parameter space. Let Pic(X) be the Picard group of the variety X = XP1 ,··· ,P8 (the additive group of isomorphism classes of invertible sheaves the additive group of linear equivalence classes of divisors). We have Pic(X) = ZE ⊕ ZE1 ⊕ ZE2 ⊕ ZE3 ⊕ ZE4 ⊕ ZE5 ⊕ ZE6 ⊕ ZE7 ⊕ ZE8 , KX = −4E + 2(E1 + E2 + E3 + E4 + E5 + E6 + E7 + E8 ),

(2)

where E denotes the total transform of the divisor class of the plane in P3 and Ei denotes the total transform of the exceptional divisor generated by the blow-up at Pi . We often identify the lattices Pic(XP1 ,··· ,P8 )’s by (2). For the following argument we define δ as 1 δ := − KX , 2

(3)

and the inner product of Pic(X) as 1 (D, D ) := D · D · (− KX ) 2 for D, D ∈ Pic(X), where for D, D , D ∈ Pic(X), D · D · D denotes the intersection number Pic(X) × Pic(X) × Pic(X) → Z. We consider the group (written as Gr({X})) of birational transformations on the family {XP1 ,··· ,P8 } such that i) ϕ : X(4, 8) → X(4, 8) is an automorphism; here, we denote ϕ({P1 , · · · , P8 }) by {P1 , · · · , P8 }; ii) for any {P1 , · · · , P8 } ∈ X(4, 8) the map ϕ : XP1 ,··· ,P8 XP1 ,··· ,P8 is a birational map preserving the inner product of Pic(X), i.e. (ϕ∗ (D), ϕ∗ (D )) = (D, D ) for any D, D ∈ Pic(XP1 ,··· ,P8 ), where ϕ∗ : Pic(XP1 ,··· ,P8 ) → Pic(XP1 ,··· ,P8 ) is the push-forward action of ϕ.

22

T. Takenawa (1)

Theorem 2.1. Gr({X}) is the affine Weyl group of type E7 . Remark 2.1. Dolgachev and Ortland [3] have shown that the affine Weyl group of type (1) E7 acts on {XP1 ,··· ,P8 } and satisfies i) and ii)’: for any {P1 , · · · , P8 } ∈ X(4, 8) the map ϕ : XP1 ,··· ,P8 XP1 ,··· ,P8 is a pseudo-isomorphism, i.e. isomorphism in co-dimension one. Moreover, those maps act on Pic(X) and its Poincaré dual as the root lattice Q and ˇ respectively. Hence we have an isomorphism ν : Q → Q ˇ and the the coroot lattice Q corresponding biliniear form on Q. However, to the author’s knowledge, it has not been proved that there are no maps satisfying i) and ii)’ other than the affine Weyl group of (1) type E7 . Moreover, without concrete birational maps, it is difficult to find root and coroot bases and the isomorphism ν. Although we also do not prove the uniqueness in the category of pseudo-isomorphisms, our method makes it possible to find the symmetries and the bilinear forms on Pic(X) ⊃ Q from some families of rational varieties themselves. Before the proof we describe some formulae for the intersection numbers. For elements in Pic(X) the intersection numbers are given by E · E · E = 1, E · E · Ei = 0, E · Ei · Ei = 0, Ei · Ei · Ei = 1, Ei · Ej · D = 0 (i = j, ∀D ∈ Pic(X)) . Hence the inner product is given by (E, E) = 2, (E, Ei ) = 0, (Ei , Ej ) = −δi,j .

(4)

Lemma 2.1. If a birational map ϕ : XP1 ,··· ,P8 XP1 ,··· ,P8 preserves the inner product, then ϕ∗ : Pic(XP1 ,··· ,P8 ) → Pic(XP1 ,··· ,P8 ) is an isomorphism of lattices. Proof. Note that ϕ∗ (= (ϕ −1 )∗ ) can be considered to be a linear transformation to itself. Assume that there exists a nonzero divisor D ∈ Pic(XP1 ,··· ,P8 ) such that ϕ∗ (D) = 0. Since D is nonzero, there exists a divisor D ∈ Pic(XP1 ,··· ,P8 ) such that (D, D ) = 0. On the other hand, ϕ∗ (D) = 0 and hence (ϕ∗ (D), ϕ∗ (D )) = 0, which contradicts the assumption that ϕ preserves the inner product.

By this lemma, each ϕ∗ : Pic(XP1 ,··· ,P8 ) → Pic(XP1 ,··· ,P8 ) is an isomorphism and preserves a) the inner product; b) the anti-canonical divisor −KX , i.e. ϕ∗ (−KXP1 ,··· ,P8 ) = −KXP ,··· ,P , since ϕ is a 1 8 pseudo-isomorphism; c) the effectiveness of divisors. We call an automorphism of Pic(X) which preserves a), b), c) a (3-dimensional) Cremona isometry. Proof of Theorem 2.1. As in the 2-dimensional case, Theorem 2.1 is proved by investigating the group of Cremona isometries and realizing the corresponding birational transformations.


23

Let ϕ ∈ Gr({X}). Since ϕ∗ preserves KX and the inner product, it also preserves the orthogonal complement of KX , Q(α) :=< α0 , α1 , · · · , α7 >Q , where α0 = E1 − E2 , α1 = E2 − E3 , · · · , α6 = E7 − E8 , α7 = E − E1 − E2 − E3 − E4 .

(5)

Claim 2.1. The basis < α0 , α1 , · · · , α7 > of the linear space KX⊥ generates the lattice < α0 , α1 , · · · , α7 >Z , i.e. < α0 , α1 , · · · , α7 >Q ∩Pic(X) =< α0 , α1 , · · · , α7 >Z holds. Proof. We show that the left-hand side includes the right-hand side. Let a0 α0 + · · · + a7 α7 ∈ Pic(X). Since the coefficient of E8 is an integer, a6 is also an integer. Since the coefficient of E7 is an integer, −a5 + a6 is also an integer, and so is a5 . Along the same line all ai ’s are integers.

From this claim, ϕ∗ is an automorphism of the sub-lattice Q(α) and preserves the inner product. The matrix defined by using the inner product as (ci,j )i,j := 2

(αi , αj ) (αi , αi )

(6)

(1)

is the affine Cartan matrix of type E7 . We denote the affine Weyl group generated by rj (α) := rαj (α) = α − 2

(αj , α) αj (αj , αj )

(α ∈ Q(α))

(7)

(1)

(i = 0, 1, · · · , 7) by W (E7 ). By the following proposition of Kac we have the fact that the group of isometries of Q(α) each of which preserves the inner product is (1) ). ±Aut(Dynkin) W (E7 ) (which is written as ±W d α7 d

d

d

d

d

d

d

α0

α1

α2

α3

α4

α5

α6

(1)

Fig. 1. The Dynkin diagram of type E7

Proposition 2.1 ([8] §5.10)). If the generalized Cartan Matrix cij is a symmetric matrix of finite, affine, or hyperbolic type, then the group of all automorphisms of Q(α) pre. serving the bilinear form is ±W

24

T. Takenawa

Note that 1 δ = − KX = α0 + 2α1 + 3α2 + 4α3 + 3α4 + 2α5 + α6 + 2α7 2

(8)

(1)

is preserved by W (E7 ) and the automorphism of the Dynkin diagram. In our case, since (1) ϕ∗ preserves the anti-canonical divisor −KX , each element of −Aut(Dynkin)W (E7 ) is not a Cremona isometry. Claim 2.2. The automorphism of the Dynkin diagram is not a Cremona isometry. Proof. There exists only one automorphism of the Dynkin diagram, which is the involution exchanging α0 , α1 , α2 and α6 , α5 , α4 respectively. Denoting the actions of these involutions on Pic(X) by D → D, we have E = E + 3E1 − E2 − E3 − E4 − E5 − E6 − E7 − E8 + 4E8 , E1 = E1 − E 8 + E8 , E2 = E1 − E 7 + E8 , .. . E7 = E1 − E 2 + E8 . Set E8 = eE + e1 E1 + · · · + e8 E8 . From (E, E8 ) = (E, E8 ) = 2e − 3e1 + e2 + e3 + e4 + e5 + e6 + e7 + e8 − 4 = 0, (E1 , E8 ) = (E1 , E8 ) = −e1 + e8 − 1 = 0, .. . (E7 , E8 ) = (E7 , E8 ) = −e1 + e2 − 1 = 0, we have e = −2e1 − 3/2, which contradicts the assumption that e is an integer.

Now we have the fact that the actions of Cremona isometries on Q(α) are included (1) by W (E7 ). (1)

Claim 2.3. The action of W (E7 ) is uniquely extended onto Pic(X) as rj (D) = D − 2

(αj , D) αj (αj , αj )

(D ∈ Pic(X)).

(9)

Proof. Let s and s be Cremona isometries such that the action of s is identical to that of s on Q(α). We show s ◦ s −1 = Identity on Pic(X). Since {E1 , α0 , α1 , · · · , α7 } is a basis of Pic(X), we can set s ◦ s −1 (E1 ) = e1 E1 + a0 α0 + a1 α1 + · · · + a7 α7 . From (s ◦ s −1 (E1 ), δ) = (E1 , δ) = 1 and (αi , δ) = 0 (for ∀i), we have e1 = 1. Since (s ◦ s −1 (E1 ), αi ) = (E1 , αi ) ⇐⇒ (s ◦ s −1 (E1 ) − E1 , αi ) = 0 holds,

−(ci,j )0≤i,j ≤7 a = 0

holds, where (ci,j )0≤i,j ≤7 is the Cartan matrix of type E7 (6). Hence we have s ◦ s −1 (E1 ) = E1 +zδ (z ∈ Z). Finally, from (s ◦s −1 (E1 ), s ◦s −1 (E1 )) = −1+2z = −1, we have z = 0. Hence it has been shown that s ◦s −1 does not change the basis of Pic(X) and therefore s ◦ s −1 = Identity.

(1)


25

By this claim, the actions of simple reflections on Pic(X) are given by ri : Ei+1 → Ei+2 , Ei+2 → Ei+1 (for 0 ≤ i ≤ 6), r7 : E → 3E − 2E1 − 2E2 − 2E3 − 2E4 , Ek → E − E1 − E2 − E3 − E4 + Ek (for 1 ≤ k ≤ 4), where preserved elements are omitted. Finally, we have the claim of the theorem by the following claim. (1)

Claim 2.4. The action of each element of W (E7 ) on Pic(X) is uniquely realized as an element of Gr({X}). Proof. It is enough to show for the simple reflections ri . i) The case of 0 ≤ i ≤ 6. Since E → E, the action on P3 is linear. This is nothing but the projective transformation P GL(4). Hence, without loss of generality, we can assume that the action is the identity. Since ri exchanges Ei+1 and Ei+2 , we have ri : ((P1 , · · · , Pi+1 , Pi+2 , · · · , P8 ); x) ∈ X(4.8) × P3 → ((P1 , · · · , Pi+2 , Pi+1 , · · · , P8 ); x).

(10)

Here, we have described the action ri : XP1 ,...,P8 → XP1 ,...,P8 in terms of the coordinate x of P3 . ii) The case of r7 . Using P GL(4), we may assume P1 = P1 = (1 : 0 : 0 : 0), · · · , P4 = P4 = (0 : 0 : 0 : 1)

(11)

without loss of generality. Since E → 3E − 2E1 − 2E2 − 2E3 − 2E4 , the action on P3 : (x : y : z : w) → (x : y : z : w ) is in the 3rd degree. Moreover, from E1 ↔ E − E2 − E3 − E4 , we have (1 : 0 : 0 : 0) ↔ x = 0. Since similar facts hold for the case of E2 , E3 and E4 , considering the degree with respect to x, y, z, w, we have (x : y : z : w ) = (ayzw : bzwx : cwxy : dxyz), where a, b, c, d ∈ C× . Furthermore, we can normalize it as a = b = c = d = 1 preserving (11). This is nothing but the standard Cremona transformation of P3 . Hence the action of r7 on {X} is given by the composition of maps: ((P1 , P2 , P3 , P4 ), (P5 , P6 , P7 , P8 ); x) ∈ X(4.8) × P3     x 1000 y    0 1 0 0  −1 −1 →  ,P (P , P , P , P ); P1234  z  0 0 1 0  1234 5 6 7 8 w 0001 := ((P1 , · · · , P8 ); x )    x x x 1000 2,5 3,5 4,5 x x  0 1 0 0   x3,5 1,5   →  ,  4,5  x 0010 x4,5 x1,5 2,5 x x 0001 x1,5 2,5 3,5

··· ··· ··· ···

x x    x2,8 yzx 3,8 4,8 x x   x3,8 4,8 1,8  ;  z w x  , x x   w x y  x4,8 1,8 2,8 x x x y z x1,8 2,8 3,8

where P1234 denotes the square matrix (P1 , P2 , P3 , P4 ).

(12)

26

T. Takenawa

3. Parametrization of the Configuration Space In this section we discuss parametrization of the configuration space X(4, 8) by P GL(4), i.e. how to choose representative elements. Notice that without a good parametrization it is difficult to see the concrete action of the group and properties of associated dynamical systems. For example, although X(4, 8) is easily parametrized as 

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

1 1 1 y6 1 z6 1 w6

1 y7 z7 w7

 1 y8  , z8  w8

(13)

the action of the Weyl group on this coordinate becomes complicated. We parametrize it in terms of elliptic curves. Note the following lemma. Lemma 3.1. Let P1 , P2 , · · · , P8 be 8 points in P3 . Quadratic surfaces passing through P1 , P2 , · · · , P8 form at least one parameter family. Proof. It is clear from the fact that quadratic surfaces passing through P1 , P2 , · · · , P8 and an arbitrary point P9 in P3 is given by the equation 2 2 2 2 x y z w xy yz zw wx xz yw 2 2 2 2 x y z w x1 y1 y1 z1 z1 w1 w1 x1 x1 z1 y1 w1 1 1 1 1 = 0, .. . x 2 y 2 z2 w 2 x9 y9 y9 z9 z9 w9 w9 x9 x9 z9 y9 w9 9 9 9 9 where Pi = (xi : yi : zi : wi ) (if the left-hand side is identically zero, exchange one of the points for a generic point).

The pencil of quadratic surfaces passing through P1 , P2 , · · · , P8 can be written as xt (αA + βB)x = 0 (x ∈ P3 )

(14)

by 4 × 4 complex symmetric matrices A, B and (α : β) ∈ P1 . Normalizing (14) by P GL(4) (cf. [18]), we have the following theorem, which provides a parametrization of X(4, 8). The proof of this theorem will be given in the last part of this section. Theorem 3.1. Each element of X(4, 8) can be parametrized so that P1 , P2 , · · · , P8 are on the intersection curve(s) of one of the following 3 types of pencils of quadratic surfaces: (i) (E) x 2 − zw = 0, (F ) y 2 − 4xw + g2 xz + g3 z2 = 0, (ii) xy − zw = 0, xw − z2 = 0,


27

(iii) xy − zw = 0, 4x 2 − 2zw + w 2 = 0 . Moreover, (i-1) In the case where the intersection curve is non-singular ( = 27g32 − g23 = 0), Pi can be parameterized as Pi = (℘ (ui ) : ℘ (ui ) : 1 : ℘ 2 (ui )),

ui ∈ C/(Z + Zτ ),

(15)

where ℘ (u) is a Wierstrass ℘ function with the periods (1, τ ). (i-2) In the case where = 0 and g2 = 0, the intersection curve can be renormalized to the intersection curve of x 2 − zw = 0, y 2 − 4w(x + hz) = 0, and Pi can be parameterized as 4hui −8h3/2 ui (1 + ui ) (4hui )2 Pi = , ui ∈ P1 \ {0, ∞} . (16) : :1: (1 − ui )2 (1 − ui )3 (1 − ui )4 (i-3) In the case where g2 = g3 = 0, the intersection curve can be renormalized to the intersection curve of x 2 − zw = 0, y 2 − 4xw = 0, and Pi can be parameterized as −3 −4 Pi = u−2 , i : −2ui : 1 : ui

ui ∈ P1 \ {∞} .

(17)

(ii) The intersection consists of 2 curves {(0 : s : 0 : t) | s : t ∈ P1 } and {(s 3 : t 3 : s 2 t : st 2 ) | s : t ∈ P1 }. (iii) The intersection consists of 2 curves {(0 : s : 0 : t) | s : t ∈ P1 } and {(2st 2 : s(s 2 + 4t 2 ) : t (s 2 + 4t 2 ) : 2s 2 t) | s : t ∈ P1 }. Remark 3.1. The parameterizations of (i-2) and (i-3) of Theorem 3.1 are chosen so that the period map becomes simple. From the proof in case (i-1) of this theorem we have the following corollary. Corollary 3.1. If the intersection curve of surfaces passing through P1 , P2 , · · · , P8 is non-singular, X(4, 8) can be parametrized as P1 , P2 , · · · , P8 are on the intersection curve of 2 surfaces 2 2 2 2 2 2 θ00 x − θ01 y − θ10 z = 0, 2 2 2 2 2 2 θ10 y − θ01 z − θ00 w = 0,

(18)

where we write θij := θij (0) for the theta function θij (u) with the fundamental periods (1, τ ) and therefore Pi can be parameterized as Pi = (θ00 (2ui ) : θ01 (2ui ) : θ10 (2ui ) : θ11 (2ui )).

(19)

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T. Takenawa

Corollary 3.2. X(4, 8) restricted to case (i-1),(i-2) or (i-3) of Theorem 3.1 is isomorphic to (i-1) {(u1 , · · · , u8 , τ ) ∈ (C/(Z + Zτ ))8 ×(H/SL(2, Z)) | ui +uj +uk +ul ≡ 0},

(20)

where 1 ≤ i, j, k, l ≤ 8 are different from each other and H is the upper half of the complex plane; (i-2) {(u1 , · · · , u8 ) ∈ (P1 \ {0, ∞})8 | ui uj uk ul = 1};

(21)

{(u1 , · · · , u8 ) ∈ (P1 \ {∞})8 | ui + uj + uk + ul = 0}.

(22)

(i-3)

Proof. It is enough to show that Pi , Pj , Pk , Pl are on the same plane if and only if ui +uj +uk +ul = 0 holds (ui uj uk ul = 1 holds in case (i-2)). Notice that Pi , Pj , Pk , Pl are on the same plane if and only if xi xj x k x l

yi yj yk yl

zi zj zk zl

wi wj =0 wk wl

(23)

holds. • In case (i-1). When (23) is considered to be a rational function of ui , the origin is the unique pole of order 4. By Abel’s Theorem the sum of zero points are 0. Here, ui = uj , uk , ul are zero points and hence the other zero point is ui = −uj − uk − ul . • In case (i-2) or (i-3). When (23) is considered to be a rational function of ui , ui = 1 (the origin in case (i-3)) is the unique pole of order 4 and therefore there are 4 zero points. It is easily shown that ui = uj , uk , ul , (uj uk ul )−1 , (ui = uj , uk , ul , −uj − uk − ul in case (i-3)) are those 4 points by substitution.

A similar argument leads to the following theorem concerning the dimension of a linear system | − 21 KX | of rational variety X = XP1 ,··· ,P8 . Theorem 3.2. In case (i) of Theorem 3.1, dim(| − 21 KX |) is 2 if and only if, in case (i-1) or (i-2) u 1 + u 2 + · · · + u8 = 0

(24)

u1 u2 · · · u8 = 1

(25)

holds and in case (i-1)

holds. Since each surface in |− 21 KX | is a rational projective elliptic surface, dim |− 21 KX | ≤ 2 holds and if the equivalent conditions of Theorem 3.2 is satisfied, X is an elliptic variety.


29

Proof. Case (i-1). (⇒). If dim(|− 21 KX |) ≥ 2, there exists a surface D ∈ |− 21 KX | which does not include the intersection curve of (i-1). Let P9 and P10 be generic points on D. D is described as 2 2 2 2 x y z w yz zw wx xz yw 2 2 2 2 xy x y z w x1 y1 y1 z1 z1 w1 w1 x1 x1 z1 y1 w1 1 1 1 1 .. . = 0. (26) x 2 y 2 z2 w 2 x7 y7 y7 z7 z7 w7 w7 x7 x7 z7 y7 w7 7 7 7 7 x 2 y 2 z2 w 2 x9 y9 y9 z9 z9 w9 w9 x9 x9 z9 y9 w9 9 9 9 9 x 2 y 2 z2 w 2 x10 y10 y10 z10 z10 w10 w10 x10 x10 z10 y10 w10 10

10

10

10

By Bézout’s theorem the intersection of D and the curve of (i-1) is the 8 points P1 , P2 , · · · , P8 , which are given by the zero points of (26) with (x : y : z : w) = (℘ (u) : ℘ (u) : 1 : ℘ (u)2 ). The left-hand side of the equation is an elliptic function of u and has the unique pole u = 0 of order 8. Since the 8 points u = u1 , u2 , · · · , u8 are zero points, by Abel’s theorem we have u1 + u2 + · · · + u8 = 0. (⇐) Assume u1 + u2 + · · · + u8 = 0. By considering the zero points of the same elliptic function, it is shown that the intersection of a generic quadratic surface D passing through 7 points P1 , P2 , · · · , P7 and the curve of (i-1) is the 8 points P1 , P2 , · · · , P8 . Hence D is an element of | − 21 KX |. The dimension to choose such D is 2 or higher. In case (i-2) or (i-3), it is enough to change the argument about Abel’s theorem as the proof of Corollary 3.2.

Proof of Theorem 3.1. Since the latter part is easy, we show only the former part. We consider normalization of (14) by P GL(4). Note that P −1 in P GL(4) acts on the pencil as xt (αA + βB)x = 0 → xt (αP t AP + βP t BP )x = 0. Note also the following facts. Fact 1. Any complex symmetric matrix can be diagonalized to the form diag(1, · · · , 1, 0, · · · , 0) by P GL. Fact 2. The n × n identity matrix is not changed by orthonormal matrices. Fact 3. Since “two complex symmetric matrices are similar if and only if they are similar via a complex orthonormal similarity,” if two matrices have the same Jordan normal form, they are mapped to each other by some complex orthonormal matrix (pp. 212 in [7]). Assume that there exists (s : t) such that rank(sA + tB) = 2 by Fact 1 the matrix is normalized to diag(1, 1, 0, 0) and the defining equation can be factorized. Hence, 4 or more points in the 8 points on xt (αA + βB)x = 0 are on the same plane, which contradicts the assumption of the configuration space. So we have rank(sA + tB) ≥ 3 for all (s : t) ∈ P1 . Lemma 3.2. If rank(sA + tB) ≥ 3 for all (s : t) ∈ P1 , there exists (s : t) ∈ P1 such that rank(sE + tF ) = 4.

30

T. Takenawa

Proof. Without loss of generality we can assume A = diag(1, 1, 1, 0). We normalize B by 3 × 3 matrix. Since the sizes of Jordan blocks of the 3 × 3 submatrix of B should be (1, 1, 1), (2, 1) or (3), by Fact 3, we can set B as 

a 0 0 e

0 b 0 f

0 0 c g

 e f , g d



√ a√+ 1 −1  −1 a − 1   0 0 e f

0 0 b g

 e f , g d



a  1 √  −1 e

1 a 0 f

√

−1 0 a g

 e f . g d

Assume rank(sA + tB) = 3 for all (s, t) ∈ P1 . In the first case, we have d = e = f = g = 0 and therefore the rank becomes 2 or less for some t, which is a contradiction. Along the same line, in the second or the third case it can be shown that the rank becomes 2 or less, which is a contradiction. Hence there exists (s : t) ∈ P1 such that rank(sA + tB) = 4.

From the above lemma, we may assume A = Identity and rankB = 3. The Jordan normal form of B has the following 5 possibilities:  (i − 1)

(i − 2)

(i − 3)

(ii)

(iii)

 a000 0 b 0 0 0 0 c 0, 0000    0 0 a100 √0 0 0 a√+ 1 −1 0  0 a 0 0   0 0 b 0 ∼   0 −1 a − 1 0  , 0000 0 0 0 1     a100 0000 0 a 1 0 0 1 1 0 0 0 a 0 ∼ 0 0 1 1, 0000 0001 √     0 0 √1 −1 0100  0 0 −1 −1 0 0 1 0  √ , √ 0 0 0 1 ∼   1  −1 −1 −1 √ √ 0000 −1 −1 −1 1     1 √−1 0 0 0100 √ −1 −1 0 √0  0 0 0 0  , 0 0 a 1 ∼   0 0 √2 −1  000a 0 0 −1 0

where the right-hand side matrices are similar to B except the proportional constants. We replace B by the right-hand side matrices. • Case (i-1). Since rank(sA + tB) ≥ 3, a, b, c are not zero and different from each other. By replacing the basis of pencil xt (αA + βB)x = 0, we may assume A = diag(0, a, b, c) and B = diag(d, e, f, 0). Moreover, by the actions of diagonal matrices, we can set A = diag(0, 1, 1, 1) and B = diag(a, a, b, 0). Finally, by multiplying a constant to B, we can set A = diag(0, 1, 1, 1), B = diag(1, 1, a, 0) (a = 0, 1). On the


31

other hand if = g23 − 27g32 is not zero, using e1 = ℘ (w1 /2), e2 = ℘ (w2 /2), e3 = ℘ ((w1 + w2 )/2), (F ) can be written as y 2 − 4xw + 4(e1 + e2 + e3 )x 2 − 4(e1 e2 + e1 e3 + e2 e3 )xz + 4e1 e2 e3 z2 = 0. The matrices



1 0 E= 0 0 

0 0 0 0

0 0 0 − 21

 0 0  , − 21  0

 4(e1 + e2 + e3 ) 0 −2(e1 e2 + e1 e3 + e2 e3 ) −2 0 1 0 0   F = 4e1 e2 e3 0  −2(e1 e2 + e1 e3 + e2 e3 ) 0 −2 0 0 0 are normalized to E = diag(0, 1, 1, 1), F = diag(1, 1, λ, 0) via   01 0 0 1 0 0 0  √  P =  0 0 −1 −1  , √ 0 0 −1 1 where λ=

e2 − e3 e1 − e 3

√ √ √ √ 31/3 (3 −1 + 3)g2 − (−3 −1 + 3)(9g3 + −3g23 + 81g32 )2/3 = √ √ √ √ −31/3 (−3 −1 + 3)g2 + (3 −1 + 3)(9g3 + −3g23 + 81g32 )2/3

(27)

is the λ function. By suitably replacing P , λ can be changed to 1/λ, 1−λ, 1/(1−λ), λ/(λ − 1), (λ − 1)/λ (by using the fact that E and F are simultaneously decomposed to the eigenspaces, it can also be shown that λ cannot be other than these). Note that λ is invariant under the action (w1 , w2 ) → (sw1 , sw2 ) ⇔ (g2 , g3 ) → (g2 /s 4 , g3 /s 6 ). We show that there exist corresponding g2 and g3 when λ = 0, 1, ∞ ( = 0 if λ = 0, 1 or ∞). Setting √ √ λ − 1/2 − 3 −1/2 , y= λ−1 we have

√ √ 3 −1(1/2 − 3/2)g2 y= . √ g2 − (3 3g3 + −g23 + 27g32 )(2/3)

√ √ We show that there exist corresponding g2 and g3 when y = 1/2 + 3 −1/2, 1, ∞. √ Setting g2 = a, (3 3g3 + −g23 + 27g32 )(2/3) = 3b2 , we have √ √ 3 −1(1/2 − 3/2)a y= . a − 3b2

32

T. Takenawa

Hence, there exist corresponding a, b ∈ C for arbitrary y ∈ C. Since g2 = a, g3 = a 3 /(54b3 ) + b3 /2, we can find the corresponding g2 and g3 when b = 0. If b = 0, we have g2 = g3 = 0, which is not in the case considered here. • Case (i-2). Set   1 √0 0 0   1000 −1√  0 −√−1 0    a (a−1) 0 1 0 0 √ √a ,  , P = P1 =   −1 a 2   0020 2(1−a) 0  0 0 0001 −a 0 0 0 a−1 A = P2t (P1 AP1 − P1 BP1 )P2 , 1−a t B = P2 P1 BP1 P2 , a Then A = diag(1, 1, 1, 0) and the Jordan normal form of B is   0000 0 1 1 0 0 0 1 0. 0001 On the other hand, if = 0, setting   01 0 0 1 0 0 0  √  P1 =   0 0 −1 −1  , √ 0 0 −1 1

 √ 2 3g2  0  P2 =  0 0

(28)

0 1 0 0

0 0 1 0

 0 0 , 0 1

E = P2 P1t AP1 P2 ,

√ 3 2 t F = − P2 (P1 BP1 √ − P1t AP1 )P2 , 3 4 g2

we have E = diag(1, 1, 1, 0) and (28) as the Jordan normal form of F . Hence it has been shown that the two pencils are equivalent modulo P GL(4). • Case (i-3). If g2 = g3 = 0, setting √   0 1 2 √−1 √2 5 5   0   1 0 √0   P = −9  , √−1 √ 0 0 √ 5 5 0 0 √−1 √1 5

5

E = P t (A + B)P , we have E = Identity and



0 0 0 0

0 1 0 0

0 1 1 0

F = P t AP ,  0 0 1 1

as the Jordan normal form of F . Hence it has been shown that the two pencils are equivalent modulo P GL(4).


• Case (ii). Setting

33



 0 0 √1 √1/2  −1 −1/2  0 0 , P =  0  0 −1 √ √1/2 0 0 − −1 − −1/2 A = P t AP , B = P t BP ,

we have



0 1  A =  0 0

1 0 0 0

0 0 0 −1

  0 00 0  0 0 ,B = −1  00 0 20

• Case (iii). Setting P , A and B as in case (ii), we have    01 0 0 40 1 0 0 0  0 0 ,B = A = 0 0 0 −1  00 0 0 −1 0 00

0 0 −4 0

 2 0 . 0 0

 0 0 0 0  . 0 −1  −1 1

4. Period Map and the Action of the Weyl Group (1)

In this section we describe the action of W (E7 ) in case (i) of Theorem 3.1 in the normalized coordinate system. For this purpose, it is enough to normalize the simple reflections obtained in Claim 2.4, but it is not easy calculus. Thus, first, we define a linear map χX from the lattice Q(α) to C (which has ambiguity corresponding to the periods as discussed later). Next, we compute the action by using the fact that χ is invariant under (1) (1) the action of W (E7 ), i.e. χX (α) = χw(X) (w(α)) for α ∈ Q(α) and w ∈ W (E7 ). This method is an analogue of that of the period map essentially introduced by Looijenga for surfaces.

Period map and the action on the intersection curve. In the following, we shall discuss only case (i-1) of Theorem 3.1. For case (i-2) and (i-3), we shall write the results only. Replace x/z, y/z, w/z by x, y, w. Let D1 , D2 ∈ − 21 KX be the divisors determined by the proper tranforms of (E), (F ) in Theorem 3.1. We denote the set of piece-wise smooth singular 3-chains in X − D1 − D2 by S(X − D1 − D2 ). Using the holomorphic 3-form on X \ (D1 ∪ D2 ), ω=

c dx ∧ dy ∧ dw (x 2 − w)(y 2 − 4xw + g2 x + g3 )

(29)

(the constant c ∈ C× is determined later), we define the map χX : S(X −D1 −D2 ) → C by the pairing ω, ( ∈ S(X − D1 − D2 )). Let C denote the elliptic curve D1 ∩ D2 . We define a map Q(α) → S(X − D1 − D2 )/H1 (C, Z). For this purpose, it is enough to define the map for the basis αi ’s.

34

T. Takenawa

• In the case of 0 ≤ i ≤ 6. We have αi = Ei+1 − Ei+2 . Let Cire be a real curve on C from C ∩ Ei+1 (which is expressed as u = ui+1 in the coordinate u) to C ∩ Ei+2 (u = ui+2 ). Here, C has ambiguity of H1 (C, Z) Z + Zτ . Let ε > 0 be a sufficiently small number and i ∈ S(X − D1 − D2 ) be the set of points such that |x 2 − w| = ε, |y 2 − 4xw + g2 x + g3 | = ε and x is in the projection of Cire to the x coordinate. • The case of i = 7. We have α7 = (E − E1 − E2 − E3 ) − E4 . Let C7re be a real curve on C from C ∩ (E − E1 − E2 − E3 ) (u = −u1 − u2 − u3 , cf. Lemma 3.2) to C ∩ E4 (u = u4 ). Let 7 ∈ S(X − D1 − D2 ) be the set of points such that |x 2 − w| = ε, |y 2 − 4xw + g2 x + g3 | = ε and x is in the projection of C7re to the x coordinate. Remark 4.1. As in the 2-dimensional case, we can take i from H3 (X − D1 − D2 , Z). Let F1 , F2 be divisors such that αi is written as αi = F1 − F2 as above. For our purpose, it is enough to add singular 3-chains in F1 and F2 to the above i . Here, the extended part is included by a 2-dimensional algebraic subvariety and hence the effect for the integration is zero. By the composition Q(α) → S(X − D1 − D2 )/H1 (C, Z) → C, the map χX : Q(α) → C is determined modulo the image of H1 (C, Z). Let πx denote the projection to the x coordinate. By the residue theorem, we have χX (αi ) =

ω

i

dx ∧ dy ∧ dw |x 2 − w| = ε (x 2 − w)(y 2 − 4xw + g2 x + g3 ) |y 2 − 4xw + g2 x + g3 | = ε x ∈ πx (Cire ) dx ∧ dy = c |y 2 − 4xw + g x + g | = ε 2 2 3 (y − 4x 3 + g2 x + g3 ) x ∈ πx (Cire ) dx = c x∈πx (Cire ) y du (x = ℘ (u), y = ℘ (u)) = c

=c

=

Cire ∗

(0 ≤ i ≤ 6) c (ui+1 − ui+2 ) , c (−u1 − u2 − u3 − u4 ) (i = 7)

where Cire ∗ denotes the pullback of Cire to the space of u and the last result should be considered modulo c (Z + Zτ ). Since the constant c ∈ C× has been arbitrary, we can determine it so that c = 1. By further blow-up along lines, the simple reflection ri can be considered to be an exchange of the blow-down structure of X = XP1 ,··· ,P8 and that of ri (X) = Xri (P1 ,··· ,P8 ) , i.e. it just changes how to blow-down corresponding to the change of basis of Pic(X) (cf. Remark 4.2). Let u = u0 denote the intersection point of a effective divisor D and the curve C. Since the curve C is preserved by ri (because the modulus of C is not changed), ri (D) and C also intersect at u = u0 . Since D is arbitrary, we have χX (α) = χri (X) (ri (α))

α ∈ Q(α).


35

Considering the composition, we have χX (α) = χw(X) (w(α))

(30)

(1)

for all w ∈ W (E7 ). Remark 4.2. In the terminology of [3], this fact means that ri is a pseudo-isomorphism from X to ri (X) and determines an exchange of the points of the blow-ups. From (7),(30), we have (for 0 ≤ i ≤ 6), ri : (ui+1 , ui+2 ) → (ui+2 , ui+1 ) r7 : (u1 , · · · , u8 ) → (u1 − λ1 , · · · , u4 − λ1 , u5 + λ1 , · · · , u8 + λ1 ),

(31)

where λ1 = 21 (u1 + u2 + u3 + u4 ) (preserved elements are omitted). Moreover, since ri acts on the elliptic curve C birationally and therefore it is a translation for the points on C except Pj and ri (Pj ) (1 ≤ j ≤ 8), we have ri : u = 0 → u = 0 (for 0 ≤ i ≤ 6), r7 : u = 0 → u = λ1 .

(32)

Case (i-2). We have χX (αi ) = c

=

Cire ∗

du u

i+1 (0 ≤ i ≤ 6) c log uui+2 , −c log(u1 u2 u3 u4 ) (i = 7)

ri : (ui+1 , ui+2 ) → (ui+2 , ui+1 )

(for 0 ≤ i ≤ 6),

−1 r7 : (u1 , · · · , u8 ) → (u1 λ−1 1 , · · · , u4 λ1 , u5 λ1 , · · · , u8 λ1 ),

where λ1 = (u1 u2 u3 u4 )1/2 , and we have ri : u = 1 → u = 1 (for 0 ≤ i ≤ 6), r7 : u = 1 → u = λ1 . Case (i-3). It is the same with case (i-1). The action on P3 . We investigate the action to generic points in XP1 ,··· ,P8 . The action of ri (0 ≤ i ≤ 6). Since the simple reflection ri just exchanges the blow-up points, it acts on P3 as the identical map ri : x → x.

(33)

36

T. Takenawa

The action of r7 . We write Pi = (fx (ui ), fy (ui ), 1, fw (ui ))t by the parametric representation of C: (15),(16) or (17). Let ∗ denote the image of ∗ by r7 and set ((P1 , P2 , P3 , P4 ), (P5 , P6 , P7 , P8 )) = (A, B). For a matrix P , let 1/P denote a matrix whose elements are the reciprocal number of corresponding elements of P . The action of r7 is given by A−1

−−−−→ SCT −−−−→ diag∈P GL(4) −−−−−−−→ A −−−−→

((A, B), x) ((Id, A−1 B), A−1 x) ((Id, 1/(A−1 B)), 1/(A−1 x)) diag(b1 , b2 , b3 , b4 )((Id, 1/(A−1 B)), 1/(A−1 x)) ((A, B), x),

where SCT denotes the standard Cremona transformation. Setting x = diag(b1 , b2 , b3 , b4 )(1/(A−1 x)), we have b1 −b2 b3 −b4 x = , , , |x, P2 , P3 , P4 | |x, P3 , P4 , P1 | |x, P4 , P1 , P2 | |x, P1 , P2 , P3 | and x = A

−1

x = x, P2 , P3 , P4 , − x, P3 , P4 , P1 , x, P4 , P1 , P2 , − x, P1 , P2 , P3 .

On the other hand, from (32), in case (i-1) and case (i-2) we have p = (fx (λ1 ) : fy (λ1 ) : 1 : fw (λ1 ))t

(34)

for p = (fx (0) : fy (0) : 1 : fw (0))t = (0 : 0 : 0 : 1)t (in case (i-3) we have p = (fx (λ1 ) : fy (λ1 ) : 1 : fw (λ1 ))t for p = (fx (1) : fy (1) : 1 : fw (1))t = (0 : 0 : 0 : 1)t ). Using these, we can obtain bi explicitly. Consequently, we have 

  fx (uˇ1 ) fx (uˇ2 ) fx (uˇ3 ) fx (uˇ4 ) l2,3,4 (x) (x)   f (uˇ ) f (uˇ ) f (uˇ ) f (uˇ )   −l x =  y 1 y 2 y 3 y 4   3,4,1  , 1 1 1 1 l4,1,2 (x) −l1,2,3 (x) fw (uˇ1 ) fw (uˇ2 ) fw (uˇ3 ) fw (uˇ4 )

(35)

where uˇk := uk = uk − λ1 (1 ≤ k ≤ 4) (uˇk := uk = uk λ−1 1 (1 ≤ k ≤ 4) in case (i-2)) and fx (λ1 ) fx (uˇi ) fx (uˇj ) fx (uˇk ) 0 fx (ui ) fx (uj ) fx (uk ) fy (λ1 ) fy (uˇi ) fy (uˇj ) fy (uˇk ) 0 fy (ui ) fy (uj ) fy (uk ) 1 1 1 1 0 1 1 1 f (λ ) f (uˇ ) f (uˇ ) f (uˇ ) 1 f (u ) f (u ) f (u ) w 1 w i w j w k w i w j w k li,j,k (x) = . (36) x fx (ui ) fx (uj ) fx (uk ) y fy (ui ) fy (uj ) fy (uk ) z 1 1 1 w f (u ) f (u ) f (u ) w

i

w

j

w

k


37

5. Dynamical Systems In this section we consider a dynamical system corresponding to a translation of the (1) Weyl group W (E7 ). Note that although one can consider dynamical systems for all translations, many of them are generated by birational conjugates of one of them. Notice that rw(αi ) (β) := β − 2

(w(αi ), β) w(αi ) = w−1 ◦ ri ◦ w(β) (αi , αi )

(1)

holds for w ∈ W (E7 ), a simple reflection αi and β ∈ Q(α). Since the map T := rE−E5 −E6 −E7 −E8 ◦ rE−E1 −E2 −E3 −E4

(37)

acts on the root basis as (α0 , α1 , α2 , α3 , α4 , α5 , α6 , α7 ) → (α0 , α1 , α2 , α3 + δ, α4 , α5 , α6 , α7 − 2δ), T is a translation. Therefore T n defines a birational dynamical system on X(4, 8) × P3 . It can also be considered to be a dynamical system on P3 with the parameters ui ’s (or Pi ’s). In case (i-1) or (i-3), similar to the above section, we have the action of T on the parameter space {ui } as T : (u1 , · · · , u8 ) → (u1 + λ, u2 + λ, u3 + λ, u4 + λ, u5 − λ, · · · , u8 − λ), where λ = 21 8i=1 ui . Since the explicit action of the transformation on P3 is complicated, we give it using a decomposition.Although it is enough to compose rE−E5 −E6 −E7 −E8 and rE−E1 −E2 −E3 −E4 of course, here we use the fact T can be written as T = S 2 by S := rE4 −E8 ◦ rE3 −E7 ◦ rE2 −E6 ◦ rE1 −E5 ◦ rE−E1 −E2 −E3 −E4 ,

(38)

and describe the action of S. S acts on {ui } as S : (u1 , · · · , u8 ) → (u5 + λ1 , u6 + λ1 , u7 +λ1 , u8 + λ1 , u1 − λ1 , · · · , u4 − λ1 ),

(39)

and the action on P3 is given by (35) and (36), where uˇk := uk − λ1 (1 ≤ k ≤ 4). Along the same line, in case (i-2) we have −1 S : (u1 , · · · , u8 ) → (u5 λ1 , u6 λ1 , u7 λ1 , u8 λ1 , u1 λ−1 1 , · · · , u4 λ1 ),

and the action of S on P3 is given by (35) and (36), where uˇk := uk = uk λ−1 1 (1 ≤ k ≤ 4).

38

T. Takenawa

6. Conservation Law In this section we prove the following theorem. Theorem 6.1. In case (i) of Theorem 3.1, the action of the Weyl group preserves each member of the pencil of quadratic surfaces in Theorem 3.1. Remark 6.1. The pencil of Theorem 3.1 is given by the linear system |− 21 KX | in generic terms. By Theorem 6.1, if the dimension of | − 21 KX | is 2 or more (δ = 0 by Theorem (1) 3.2), each fiber is preserved by translations associated with the Weyl group W (E7 ), because (i) each surface is an elliptic surface (ii) each map is birational (hence continuous except at the indefinite points) and preserves the fibration (iii) the modulus of elliptic curve is preserved (iv) at least the intersection curve of the pencil is preserved. Since for every discrete Painlevé equation the polynomial degree of the nth iterate is in the order n2 as n → ∞ [17], by Theorem 6.1 the corollary below follows. (1)

Corollary 6.1. Let ϕ be a map on P3 associated with a translation of W (E7 ). The degree of ϕ n is in the order n2 as n → ∞. Since − 21 KX is preserved and therefore the pencil itself is preserved, in order to prove Theorem 6.1 it is enough to show that the automorphism of P1 defined by this correspondence is the identity. In case (i-2) and (i-3) it is easily shown by direct computation. Moreover, the simple reflection ri (0 ≤ i ≤ 6) acts on P3 as the identity. Hence it is enough to show for r7 in case (i-1). To prove r7 by direct calculation seems to be beyond our computational ability. We prove the theorem in this case by means of a birational (1) representation of W (E8 ) on P1 × P1 . Notice that a smooth quadratic surface is isomorphic to P1 × P1 via the Segré map P1 × P1 (x : 1, y : 1) → (x : y : 1 : xy) ∈ {(x : y : z : w) ∈ P3 | xy − zw = 0}. Here we reparametrize the parameter space so that the 8 points are on the intersection curve of the pencil spanned by the 2 quadratic surfaces xy − zw = 0 (G) 2 , (40) g3 g2 (x + y + z)(4w − ℘ 3 (2t) z) = w + x + y + 4℘ 2 (2t) z (H ) where t ∈ (C/(Z + Zτ )) \ {0} is an arbitrary extra-parameter. Remark 6.2. The parameter τ = w2 /w1 is invariant with respect to t. By this parametrization, Pi can be parameterized as ℘ (t + ui ) ℘ (t − ui ) ℘ (t + ui )℘ (t − ui ) : :1: . Pi ℘ (2t) ℘ (2t) ℘ 2 (2t) The action of r7 on (u, t) and on P3 are also given by (31) and (35) respectively, where we set fx (u) =

℘ (t + u) , ℘ (2t)

fy (u) =

℘ (t − u) , ℘ (2t)

fw (u) =

℘ (t + u)℘ (t − u) , ℘ 2 (2t)


39

uˇk := uk − λ1 (1 ≤ k ≤ 4) and fx (λ1 ) fx (uˇi ) fx (uˇj ) fx (uˇk ) 1 fx (ui ) fx (uj ) fx (uk ) fy (λ1 ) fy (uˇi ) fy (uˇj ) fy (uˇk ) 1 fy (ui ) fy (uj ) fy (uk ) 1 1 1 1 1 1 1 1 f (λ ) f (uˇ ) f (uˇ ) f (uˇ ) 1 f (u ) f (u ) f (u ) w 1 w i w j w k w i w j w k li,j,k (x) = . x fx (ui ) fx (uj ) fx (uk ) y fy (ui ) fy (uj ) fy (uk ) z 1 1 1 w f (u ) f (u ) f (u ) w

i

w

j

w

k

Next, we list the necessary results by Murata et al. [11] concerning birational maps on P1 × P1 whose space of initial conditions S = SP1 ,··· ,P8 ’s are given by blow-ups of P1 × P1 at generic 8 points P1 , P2 , · · · , P8 on the smooth curve of degree (2, 2), 2 g2 g3 . (41) ) = xy + x + y + (x + y + 1)(4xy − 3 ℘ (2t) 4℘ 2 (2t) Let x and y be the usual coordinates of P1 × P1 . Let H0 , H1 and Ei denote the total transform of x = c ∈ P1 , that of y = c ∈ P1 and that of the exceptional divisor generated by the blow-up at the point Pi respectively. The Picard group and the canonical divisor of the surface S are Pic(S) = ZH0 ⊕ ZH1 ⊕ ZE1 ⊕ ZE2 ⊕ · · · ⊕ ZE8 , KS = −2H0 − 2H1 + E1 + E2 + E3 + E4 + E5 + E6 + E7 + E8 , and the intersection numbers are given by Hi · Hj = 1 − δi,j ,

Hi · Ej = 0,

(Ei , Ej ) = −δi,j .

The root basis is given by αi = E7−i − E8−i (i = 0, 1, · · · , 5), α6 = H1 − E1 − E2 , α7 = H0 − H1 ,

α8 = E1 − E2 ,

and each action on the parameter space (u, t) becomes ri : (u7−i , u8−i ) → (u8−i , u7−i ), (i = 0, 1, · · · , 5), u1 u2 u3 u4 ,t r6 : u5 u6 u7 u8 u1 − 3λ1,2 u2 − 3λ1,2 u3 + λ1,2 u4 + λ1,2 → , t − λ1,2 , u5 + λ1,2 u6 + λ1,2 u7 + λ1,2 u8 + λ1,2 r7 : t → −t, r8 : (u1 , u2 ) → (u2 , u1 ), where λ1,2 =

1 4 (2t

+ u1 + u2 ). d α8 d

d

d

d

d

d

d

d

α7

α6

α5

α4

α3

α2

α1

α0

(1)

Fig. 2. E8 Dynkin diagram

(42)

40

T. Takenawa

Proof of Theorem 6.1. As mentioned above, it is sufficient to show r7 = rE−E1 −E2 −E3 −E4 in case (i-1). Let G and H denote the matrices corresponding to (G) and (H ) of the pencil (40). We write (x : y : z : w) := r7 (x : y : z : w). Since the image of a member of the pencil (x : y : z : w)|xt (α0 G+α1 H )x=0 ((α0 : α1 ) ∈ P1 ) is again an element of | − 21 KX | and therefore is again a member of the pencil (the intersection curve does not move). Hence, there exists (β0 : β1 ) ∈ P1 such that xt (β0 G + β1 H )x = 0. Since this correspondence defines an automorphism of the base space P1 of the pencil, it is enough to show that 3 members of the pencil are preserved. Hence we may assume rank(α0 G + α1 H ) = rank(β0 G + β1 H ) = 4. In the following, we assume (α0 : α1 ) = (β0 : β1 ) and lead a contradiction. Lemma 6.1. Assume rank(α0 G + α1 H ) = 4. There exist P ∈ P GL(4), v ∈ C× and t ∈ (C/(Z + Zτ )) \ {0} such that P t (α0 G + α1 H )P = G,

P t H P = vH (℘ (2t )),

and moreover there exist Q1 , Q2 ∈ P GL(4), v1 , v2 ∈ C× and c, d ∈ C \ {0, 1} such that Qt1 (α0 G + α1 H )Q1 = diag(1, 1, 1, 1), Qt1 H Q1 = v1 diag(0, 1, c, d), Qt2 GQ2 = diag(1, 1, 1, 1),

Qt2 H (℘ (2t ))Q2 = v2 diag(0, 1, c, d).

Proof. Since rankH = 3, similar to the proof of Theorem 3.1, G and H can be transformed to G = Id.,

H = vdiag(0, a, b, 1) a, b, 1 are different from each other

by P GL(4). Here, a and b are functions of g2 , g3 and ℘ (2t). Conversely, if a, b, 1 are different from each other, there exist corresponding g2 , g3 and ℘ (2t). On the other hand, they are also transformed to (α0 G + α1 H ) = Id.

H = v1 diag(0, c, d, 1)

c, d, 1 are different from each other

by P GL(4). Hence, there exist corresponding g2 , g3 , ℘ (2t ) and P ∈ P GL(4) such that P : α0 G + α1 H, H → G, H (g2 , g3 , ℘ (2t )). Here, r7 defines a birational map between the intersection curves and therefore the parameter τ = w2 /w1 does not change. As Remark 6.2, if we fix w1 as w1 = 1, only t can change, which shows that H is a function depending only on ℘ (2t ).

For (α0 : α1 ) and (β0 : β1 ) we denote t determined by Lemma 6.1 by ta and tb respectively. First, we show ta = tb . Assume ta = tb . Since only P GL(4) is needed for diag(0, 1, c, d) when one normalizes the pair diag(1, 1, 1, 1), diag(0, 1, c, d) to the pair diag(0, 1, 1, 1), diag(1, 1, λ, 0), there exist members J and J of the pencil, where J = J , rankJ = rankJ = 3, such that both pairs J, H and J , H can be transformed to the form diag(0, 1, 1, 1), diag(1, 1, λ, 0) by P GL(4). Hence there exist members K = diag(0, 1, 1, 1) and K of the pencil {s0 diag(0, 1, 1, 1) + s1 diag(1, 1, λ, 0)}, where K = K , rankK = 3, such that the pair K, H can be transformed to the pair K , H by P GL(4). The members of this pencil in rank 3 are K = diag(0, 1, 1, 1), diag(1, 1, λ, 0), K1 := diag(1, 0, λ−1, −1) and K2 := diag(−1, λ−1, 0, λ) only. Therefore, K should be K1 or K2 . Normalizing each by P GL(4), we have that λ is λ/(λ−1), (λ−1)/λ, 1−λ


41

or 1/(1 − λ). If λ = 2, 21 , 1±3I 2 , it does not coincide with λ. Hence, if τ does not correspond to these λ, we have ta = tb (as shown by (27), λ does not depend on ℘ (2t)). Let Pa , Pb denote the elements of P GL(4) which give ta , tb in Lemma 6.1. We consider the map defined by Pb ◦ r7 ◦ Pa−1 . Since the period map χX is conserved by P GL(4), we have ui = ui − λ1 + c (1 ≤ i ≤ 4), ui = ui + λ1 + c (5 ≤ i ≤ 8), where c is a constant. On the other hand, the map Pa−1

r7

Pb

xt Gx = 0−−−−→xt (α0 G + α1 H )x = 0−−−−→xt (β0 G + β1 H )x = 0−−−−→xt Gx = 0 and Segré map define a birational map on P1 × P1 . By blow-up P1 ×P1 at {Pi } and {Pi } the map is lifted to an isomorphism and included (1) by the Weyl group of type E8 . Since the Weyl group conserves χS (δ) = − 8i=1 ui , we have c = 0. Hence r7 can be written as r7 = rH0 +H1 −E1 −E2 −E3 −E4 = r6 ◦ r7 ◦ r5 ◦ r8 ◦ r4 ◦ r5 ◦ r6 ◦ r5 ◦ r4 ◦ r8 ◦ r5 ◦ r7 ◦ r6 (1)

by the root system of E8 , and we have ta = tb from (42), which contradicts ta = tb . We have shown (α0 : α1 ) = (β0 : β1 ) if τ does not correspond to λ = 2, 21 or 1±3I 2 . When λ = 2, 21 , 1±3I it can be shown by continuity of ℘ with respect to τ .

2 7. Conclusions and Discussions In this paper, we defined the inner product for the Picard group of varieties obtained by blow-ups at 8 points in P3 by means of the intersection numbers and the anti-canonical divisor and showed that the symmetry group defined by means of the inner product is (1) the Weyl group of type E7 . As in the 2-dimensional case [15], if the configuration of points is special, the symmetry may become smaller. This method can be applied to other families of 3-dimensional rational varieties. Example 7.1. Let X be a variety obtained by blow-ups at generic 6 points in P1 ×P1 ×P1 and let Hi and Ei denote the total transform of the divisor class of a plane such that one of its coordinates of P1 × P1 × P1 is constant and that of the exceptional divisor is generated by a blow-up respectively. The symmetry group Gr({X}) becomes the Weyl (1) group of type E7 defined by the root system α0 = H0 − H2 , α1 = H1 − H2 , α3 = H2 − E1 − E2 , αi = Ei−1 − Ei (3 ≤ i ≤ 6), α7 = E1 − E2 , where the inner product is given by (Hi , Hj ) = 1 − δi,j , (Hi , Ej ) = 0, (Ei , Ej ) = −δi,j . This X and the variety obtained of blow-ups at generic 8 points in P3 are not isomorphic. Thus, the relation between these 2 Weyl groups is not trivial. It may be worth commenting that the space of initial conditions for the Kajiwara(1) (1) Noumi-Yamada birational representation of the Weyl group of type A1 × A2 [9] can be obtained when the points of blow-ups are in a special position in the above example. These examples are 3-dimensional but it is expected that our method can be applied to 4 (or higher)-dimensional cases.

42

T. Takenawa

The other results are summarized as follows. • We parametrized the configuration space by normalizing the pencil of quadratic surfaces passing through 8 points (∈ | − 21 KX |) by P GL(4). The intersection curve is an elliptic curve in generic terms. • In order to obtain concrete expression of the action of the Weyl group, we introduced a 3-dimensional analogue of period map. • We showed the action of the Weyl group preserves each element of | − 21 KX | and therefore it reduces to the action on P1 × P1 . The reduced action of the Weyl group (1) (1) of type E7 is included by the action of the Weyl group of type E8 , which is the symmetry of the family of generic Halphen surfaces. Acknowledgement. The author would like to thank K. Okamoto, H. Sakai, J. Satsuma, R. Willox, M. Eguchi, M. Murata and T. Tsuda for discussions and advice. The author also appreciates a Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists.

References 1. Adler, M., van Moerbeke, P.: The complex geometry of the Kowalewski-Painleve analysis. Invent. Math. 97, 3–51 (1989) 2. Coble, A.B.: Algebraic geometry and theta functions. American Mathematical Society Colloquium Publications, vol. X. Providence, RI: American Mathematical Society, 1929 3. Dolgachev, I., Ortland, D.: Point sets in projective spaces and theta functions.Astérisque Soc.Math.de France 165, 1988 4. Diller, J., Favre, C.: Dynamics of bimeromorphic maps of surfaces. Amer. J. Math. 123, 1135–1169 (2001) 5. Gizatullin, M.H.: Rational G-surfaces. Izv. Akad. Nauk SSSR Ser. Mat. 44(239), 110–144 (1980) 6. Grammaticos, B., Ramani, A., Papageorgiou, V.: Do integrable mappings have the Painlevé property? Phys. Rev. Lett. 67, 1825–1827 (1991) 7. Horn, A., Hohnson, R.: Matrix analysis. Cambridge: Cambridge University Press, 1985 8. Kac, V.: Infinite dimensional lie algebras, 3rd ed.. Cambridge: Cambridge University Press, 1990 (1) (1) 9. Kajiwara, K., Noumi, M., Yamada, Y.: Discrete dynamical systems with W (Am−1 × An−1 ) symmetry. Lett. Math. Phys. 60, 211–219 (2002) 10. Looijenga, E.: Rational surfaces with an anti-canonical cycle. Ann. Math. 114, 267–322 (1981) 11. Murata, M., Sakai, H., Yoneda, J.: Riccati solutions of discrete Painlevé equations with Weyl group (1) symmetry of type E8 . J. Math. Phys. 44, 1396–1414 (2003) 12. Okamoto, K.: Sur les feuilletages associés aux e´ quations du second ordre a` points critiques fixes de P. Painlevé. Japan J. Math. 5, 1–79 (1979) 13. Ramani, A., Grammaticos, B., Hietarinta, J.: Discrete versions of the Painlevé equations. Phys. Rev. Lett. 67, 1829–1832 (1991) 14. Saito, M.H., Takebe, T., Terajima, H.: Deformation of Okamoto–Painlevé pairs and Painlevé equations. J. Algebraic Geom. 11, 311–362 (2002) 15. Sakai, H.: Rational surfaces associated with affine root systems and geometry of the Painlevé equations. Commun. Math. Phys. 220, 165–229 (2001) 16. Shioda, T., Takano, K.: On some Hamiltonian structures of Painlevé systems, I. Funkcial. Ekvac. 40, 271–291 (1997) 17. Takenawa, T.: Algebraic entropy and the space of initial values for discrete dynamical systems. J. Phys. A: Math. Gen. 34, 10533–10545 (2001) 18. Yoshida, M.: Hypergeometric functions, my love. Modular interpretations of configuration spaces. Aspects of Mathematics, E32. Braunschweig: Friedr. Vieweg Sohn, 1997 Communicated by L. Takhtajan


Communications in


Kähler Reduction of Metrics with Holonomy G2 Vestislav Apostolov1 , Simon Salamon2 1 2

Département de Mathématiques, UQAM, C.P. 8888, Succ. Centre-ville, Montréal (Québec), H3C 3P8, Canada. E-mail: [email protected] Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. E-mail: [email protected]

Received: 26 March 2003 / Accepted: 14 September 2003 Published online: 13 February 2004 – © Springer-Verlag 2004

Abstract: A torsion-free G2 structure admitting an infinitesimal isometry such that the quotient is a Kähler manifold is shown to give rise to a 4-manifold equipped with a complex symplectic structure and a 1-parameter family of functions and 2-forms linked by second order equations. Reversing the process in various special cases leads to the construction of explicit metrics with holonomy equal to G2 . Introduction There are now many explicitly known examples of metrics with holonomy group equal to G2 , the simplest of which admit an isometry group with orbits of codimension one. A metric with holonomy G2 on a smooth 7-manifold Y is characterized by a 3-form and a 4-form that are interrelated and both closed. If this structure is preserved by a circle group S 1 acting on Y (which cannot then be compact), the quotient Y /S 1 has a natural symplectic structure. A geometrical description of such quotients was carried out by Atiyah and Witten [8] when Y is one of three original manifolds with a complete metric of holonomy G2 described in [14]. The symplectic structure is also defined on the image of the fixed points, which in each of these cases is a Lagrangian submanifold L. The embedding of L in a neighborhood of Y /S 1 is believed to approximate the geometry of a special Lagrangian submanifold of a Calabi-Yau manifold, and this is consistent with models of special Lagrangian submanifolds of R6 described for example in [24]. This work motivates a general investigation of the quotient N = Y /S 1 of a manifold with holonomy G2 by an S 1 action. Initial results in this direction can be found in [15, 16], and in this paper we pursue the theory under the simplifying assumption that the S 1 action is free. It is an elementary fact that N has, in addition to its symplectic 2-form, a natural reduction to SU (3). Whilst this structure cannot be torsion-free (i.e. Calabi-Yau) if Y is irreducible, there are non-trivial examples in which the associated almost-complex structure is integrable, so that N is Kähler. This paper is devoted to an investigation of such a situation, which turns out to be surprisingly rich. Our study began

44

V. Apostolov, S. Salamon

with the realization that in this case, a parameter measuring the size of the fibers of Y generates both a Killing and a Ricci potential for N (Proposition 1 and Corollary 2), exhibiting a link with the theory of the so-called Hamiltonian 2-forms [4]. We first describe the induced SU (3) structure on N in §1, and then pursue consequences of the integrability condition. In particular, we prove that when N is Kähler, an infinitesimal isometry U of the SU (3) structure is inherently defined. The situation is reminiscent of the study of Einstein-Hermitian 4-manifolds in which Killing vector fields appear automatically [6, 17, 27]. We explain in §2 that U can be used to obtain a Kähler quotient of N , consisting of a 4-manifold equipped with a 1-parameter family of smooth functions and 2-forms, satisfying a coupled second-order evolution equation (Theorem 1). The procedure can be reversed so as to construct metrics with holonomy (generically equal to) G2 from a 4-manifold M with the appropriate structure. A fundamental construction of a holonomy G2 metric starting from a T2 bundle over a hyperkähler 4-manifold was discovered by Gibbons, Lü, Pope and Stelle [20], and towards the end of §3 we exhibit our reverse procedure as a generalization of the above. It can be further improved by introducing an anti-self-dual 2-form with the opposite orientation to the Kähler one (Theorem 2). This leads to the construction of new examples of G2 metrics based on the examples of Ricci-flat almost-Kähler metrics [5, 7, 28]. We point out that there also exist constructions of holonomy G2 metrics from higher dimensional hyperkähler manifolds (see for example [3]). When M is T4 , the basic examples are modelled on nilpotent Lie groups and fall into three types, special cases of which were also mentioned in [15]. We explain how to show that these are irreducible and therefore have holonomy equal to G2 . In §4, we analyse the holonomy of these metrics by restricting them to hypersurfaces so as to induce an SU (3) structure that evolves according to equations studied by Hitchin [21]. Our examples provide perhaps the simplest instance of this phenomenon other than the case of nearly-Kähler manifolds. The different SU (3) structures obtained in this way, an integrable one on a quotient and non-integrable ones on hypersurfaces, are naturally linked via the common 7-manifold. In the final section, we study a more general class of solutions of our system, formulated in terms of the complex Monge-Ampère equation. This leads to both an abstract existence theorem, some special solutions, and an explicit final example. Our assumption that the symplectic manifold Y /S 1 be Kähler leads to a (local) action on Y not just by S 1 but by the torus T2 (Corollary 3), though examples in §4 exhibit T2 actions for which the Kähler assumption fails (see also [11, 30]). We hope nonetheless that similar methods will lead to the classification of metrics with holonomy G2 admitting a T2 action. 1. The First Reduction In this section we study general properties of a 7-dimensional Riemannian manifold (Y, k) endowed with a torsion-free G2 structure and an infinitesimal isometry V . The G2 structure on Y is defined by an admissible 3-form ϕ, which itself determines the Riemannian metric k, an orientation and Hodge operator ∗. The torsion-free condition is equivalent to the closure of both ϕ and the 4-form ∗ϕ, and amounts to asserting that the holonomy group of k is contained in G2 . This theory is elaborated in the standard references [13, 19, 25, 29]. ˇ the Riemannian quotient of (Y, k), so that N is a 6-dimensional We denote by (N, k) manifold formed from the orbits of the Killing vector field V . The considerations are

Kähler Reduction of Metrics with Holonomy G2

45

local and hold in a suitable neighborhood of any point of Y where the vector field V is non-zero. The Riemannian metric kˇ is induced by the formula k = kˇ + t −2 η ⊗ η, −1/2 and η = k( · , t 2 V ) is the 1-form dual to t 2 V so that where t = V −1 k = k(V , V ) η(V ) = 1. In the case when Y can be realized as a principal S 1 bundle over N (the fibers being the closed orbits of V ), η is nothing but a connection 1-form. Since V is a Killing vector field, dt (V ) = 0 and t descends to a function on N . The 2-form

σ = ıV ϕ is horizontal in the sense that ıV σ = 0, and since V preserves the G2 structure we have dσ = d(ıV ϕ) = LV ϕ = 0, LV σ = d(ıV σ ) = 0. Thus, σ is closed and V -invariant, and therefore the pullback of a closed 2-form on N , again denoted by σ . (We identify functions and forms on N with their pullbacks to Y throughout, and L and ı stand for the Lie derivative and the interior product.) Since σ is non-degenerate transverse to the fibers of Y , it defines a symplectic form on N . We now choose to express the forms characterizing the G2 structure on Y as ϕ = σ ∧ η + t +, ∗ϕ = − ∧ η + 21 t 2 σ ∧ σ,

(1)

where − = −ıV (∗ϕ). If η were replaced by the unit form t −1 η then both σ and − would need to be multiplied by a compensatory factor of t if the left-hand sides of (1) are to remain the same. This explains why σ ∧ σ appears in (1) with a coefficient of (one half) t 2 , and + appears with a coefficient of t so as to have the same norm as − . The rescaled Riemannian metric h = t −1 kˇ

(2)

is compatible in the sense that the skew-symmetric endomorphism J defined by h(J · ,

· ) = σ( · , · )

(3)

is an almost-complex structure on N. The triple (h, σ, J ) then defines an almost-Ka¨ hler structure on N, though the qualification ‘almost’ can be deleted when J is integrable. Just as for σ , the 3-form − is closed and basic (meaning ıV − = 0, LV − = 0). Thus, it too is the pullback of a real form on N . This has type (3, 0)+(0, 3) with respect to J , so that − (J · , J · , · ) = − − ( · , · , · ). The theory of G2 structures then implies that +( · ,

· , · ) = − (J · , · , · ),

(4)

and the two real 3-forms combine to define a complex form = + + i − of type (3, 0) with respect to J .

(5)

46


Unlike − , the 3-form + is not in general closed; indeed, d + can be identified with the Nijenhius tensor of J , the obstruction to the integrability of J : Lemma 1. + is closed if and only if the almost-complex structure J is integrable. Proof. Since − is already closed, the exterior derivative d + of (5) is real. But if J is integrable so that (N, J ) is a complex manifold, this real 4-form has type (3, 1) and therefore vanishes. Conversely, if d + = d − = 0 then writing (5) locally as a wedge product of (1, 0)-forms α i shows that (dα i )0,2 = 0 for i = 1, 2, 3, and J is integrable. The almost-Kähler structure corresponds to a reduction to U (3) at each point of N . Specification of the non-zero element (5) in the space 3,0 is precisely what is needed to reduce the structure to SU (3), but one should first rescale given that ± h is not in general constant. The 3-forms ψ ± = t −1/2 ± have norm equal to 2. They are subject to the compatibility equations σ ∧ ψ ± = 0,

(6)

ψ + ∧ ψ − = 23 σ 3 = 4volh ,

(7)

and

consistent with (3), and give rise to an SU (3) structure with underlying metric h. Notation for the 3-forms is now consistent with that of [15]. Since ψ + + iψ − ∈ 3,0 , we have (γ + iJ γ ) ∧ (ψ + + iψ − ) = 0 for any 1-form γ , where J acts on 1-forms by J γ ( · ) = −γ (J · ). Whence γ ∧ ψ + = J γ ∧ ψ −.

(8)

This last equation will be useful for calculations. The real 3-forms ψ + , ψ − have a common stabilizer group SL(3, C) at each point of N , and either one determines the almost-complex structure J [22]. From this point of view, the further reduction to SU (3) is achieved by a 2-form σ ∈ 1,1 such that σ ( · , J · ) = h( · , · ) is positive definite. The SU (3) structure is therefore fully determined by (say) ψ + and σ ; in the new notation Lemma 1 reads as Corollary 1. The almost-complex structure J is integrable if and only if ∇ψ + = − 21 (d c log t) ⊗ ψ − ,

∇ψ − = 21 (d c log t) ⊗ ψ + ,

where ∇ denotes the Levi-Civita connection of h and d c f = J df . Proof. Given the above equations, d + = d(t 1/2 ψ + ) = 21 t 1/2 (d log t ∧ ψ + − d c log t ∧ ψ − ), but the right-hand side is zero by (8). Thus, J is integrable by Lemma 1.


47

Conversely, if J is integrable then (h, J ) is Kähler and ∇U ψ ± is a form of type (3, 0)+(0, 3) for any vector field U . Since ψ + and ψ − are mutually orthogonal and of constant norm 2, ∇ψ + = γ ⊗ ψ − ,

∇ψ − = −γ ⊗ ψ +

for some 1-form γ . Using (8) again, we obtain d + = d(t 1/2 ψ + ) = 21 t 1/2 (d log t ∧ ψ + + 2γ ∧ ψ − ) = 21 t 1/2 (−2J γ + d log t) ∧ ψ + . The claim follows by Lemma 1 and the injectivity of the linear map 1 → 4 defined at each point by wedging with ψ + . Corollary 2. If J is integrable, the Ricci form κ of the Ka¨ hler manifold (N, h, J ) is given by κ = 21 dd c log t. Proof. This is again an immediate consequence of the integrability criterion, which implies that the (3, 0)-form = + + i − is closed. It follows that is a holomorphic section of the canonical bundle 3,0 and the Ricci form is given by κ = i∂∂ log(2h ) = 21 dd c log(2h ), and the result follows from (7).

In the case when J is integrable, it is now possible to formulate a complete system of conditions for the SU (3) structure on N to arise from the quotient of a torsion-free G2 structure. Proposition 1. Let (N, h, J, σ ) be a Ka¨ hler manifold of real dimension 6, endowed with a compatible SU (3) structure defined by a (3, 0)-form ψ + + iψ − . Then this structure is obtained as the quotient of a 7-dimensional manifold Y with a torsion-free G2 structure by a nontrivial infinitesimal isometry V if and only if (i) dψ + = − 21 (d c log t) ∧ ψ − and dψ − = 21 (d c log t) ∧ ψ + for a smooth positive function t, and (ii) the Hamiltonian vector field U on (N, σ ), corresponding to −t, is an infinitesimal isometry of the SU (3) structure. In this case, the corresponding 7-manifold Y is locally R × N and the metric k, infinitesimal isometry V and G2 invariant forms ϕ, ∗ϕ are given by k = th + t −2 η ⊗ η, ϕ = σ ∧ η + t 3/2 ψ + ,

V =

∂ , ∂y

∗ϕ = t 1/2 ψ − ∧ η + 21 t 2 σ ∧ σ,

(9)

where y is a variable for the R factor, and η = dy + ηN is a 1-form on R × N for which dηN = −t 1/2 (ıU ψ + ).

(10)

48


Proof. Condition (i) is necessary by Corollary 1, and the equalities (9) reflect the earlier definitions of h, σ, ψ ± . Moreover, ∂/∂y is identified with the Killing field V (so that k(V , V ) = t −2 ) and η is the corresponding connection form. Using (i) and (8) in that order, we obtain 0 = dϕ = σ ∧ dη + 23 t 1/2 dt ∧ ψ + − 21 t 1/2 J dt ∧ ψ − = σ ∧ dη + t 1/2 dt ∧ ψ + . Consequently, (dη)1,1 = 0 and (10) follows from the fact that ıU σ = −dt. We now have 0 = −d(dη) = d(t 1/2 ıU ψ + ) = d(ıU + ) = LU + = t 1/2 LU ψ + . Since U is Hamiltonian, we also have LU σ = 0, i.e. U is an infinitesimal isometry for the pair (σ, ψ + ), and therefore for the SU (3) structure (using [22]). Reversing the above arguments, one can check directly that (9) and (10) define a torsion-free G2 structure on Y = R × N . Corollary 3. Under the hypothesis of Proposition 1, the horizontal lift of U to (Y, k) is an infinitesimal isometry of the G2 structure ϕ, which commutes with V . Proof. This is an immediate consequence from Proposition 1.

Remark 1. It follows from Proposition 1 that when t is constant, (Y, k) is locally the Riemannian product of a Calabi-Yau 6-manifold with R. In this case the holonomy group of h lies in SU (3). In general, the failure of the holonomy to reduce to SU (3) is measured by a torsion tensor τ ∈ Tn∗ N ⊗

so(6) su(3)

determined by dσ, dψ + , dψ − . Whereas τ has a total of 42 components, exactly two thirds of these will always vanish on N and τ is determined by the remaining 14 tracefree components of dψ + , or 6 in the Kähler case [15]. 2. A Second Reduction Proposition 1 is the key ingredient for performing a further quotient via the infinitesimal isometry U . To carry this out, we shall assume from now on that t is not constant and that J is integrable. Thus, U is a non-trivial infinitesimal isometry of the SU (3) structure on N determined by the pair (h, ψ + ). Denote by (M, J1 ) the “stable” or holomorphic quotient of (N, J ), defined at least locally as the complex two-dimensional manifold of holomorphic leaves of the foliation generated by = U − iJ U . Here we exploit the fact that is a holomorphic vector field on (N, J ). Since J is integrable, = + + i − is a closed (3, 0)-form, and we set

= 21 ı .


49

Since ı = 0, it follows from (4) that is closed and the pullback of a holomorphic symplectic form on M, again denoted by . The real closed 2-forms ω2 = Re ,

ω3 = Im

on M (that pull back to ıU + , ıU − respectively on N ) satisfy ω2 ∧ ω2 = ω3 ∧ ω3 ,

ω2 ∧ ω3 = 0.

(11)

The complex structure J1 can now be determined by the formula ω2 ( · ,

· ) = ω3 (J1 · , · ),

and Eq. (10) in Proposition 1 reads dη = −ω2 .

(12)

Set u = U −2 h and let ξ be the 1-form h-dual to the vector field uU , so that ıU ξ = 1 and ξ = uJ dt. The 3-forms ± are completely determined by (10) which forces them to be the real and imaginary components of (ω2 + iω3 ) ∧ (ξ + iJ ξ ). Thus ψ + = t −1/2 (ω2 ∧ ξ + u ω3 ∧ dt), ψ − = t −1/2 (ω3 ∧ ξ − u ω2 ∧ dt).

(13)

They are U -invariant in accordance with (ii) in Proposition 1. For any regular value of the momentum map −t, the stable quotient (M, J1 ) of (N, J ) can be identified with the symplectic quotient (M, ω˜ 1 (t)) of (N, σ ) generated by the vector field U . In this way, we obtain the Ka¨ hler quotient (M, g(t), ω˜ 1 (t), J1 ). In this correspondence, σ = ω˜ 1 (t) + dt ∧ ξ, h = g(t) + u−1 ξ ⊗ ξ + u dt ⊗ dt,

(14)

t ω˜ 1 (t) ∧ ω˜ 1 (t) = 21 u ∧ = u ω2 ∧ ω2 = u ω3 ∧ ω3 .

(15)

so that Eq. (7) reduces to

To ease the notation, we shall below omit the explicit dependence of ω˜ 1 = ω˜ 1 (t) on t except on occasion for emphasis. We now denote by P the (locally defined) space of orbits of U , so that N can be thought of as an R bundle over P with connection 1-form ξ . Locally, N = R × P , and introducing a variable x for the R factor, we may write ξ = dx + ξP ,

U=

∂ ∂x

for some 1-form ξP on P . The space of orbits of the vector field J U on P is the stable quotient of (N, J ), whereas the symplectic quotients of (N, σ ) are identified with the level sets of t in P . Using a local description P = R+ × M in which the R+ -factor

50


corresponds to t, we may regard u as a function on M for each value of t. In these terms, we have dξ = dP ξP = αM ∧ dt + βM , dP u = u dt + dM u, d ω˜ 1 = ω˜ 1 ∧ dt, where αM = αM (t), βM = βM (t) are 1-parameter families of forms on M, ∂/∂t and d, dP , dM denote the exterior derivative on N, P , M. Differentiating the first relation gives dM βM = 0,

denotes

βM = −dM αM .

Using the formula (14) for the symplectic form σ yields ω˜ 1 = βM , and it follows that βM has type (1, 1) relative to J1 . Using (13) gives dψ + = − 21 t −3/2 dt ∧ ω2 ∧ ξ + t −1/2 ω2 ∧ dξ + t −1/2 dM u ∧ ω3 ∧ dt = − 21 t −1 J dt ∧ ψ − + t −1/2 (ω2 ∧ αM + dM u ∧ ω3 ) ∧ dt, since ω2 ∧ βM = 0. In view of Proposition 1(i), ω2 ∧ αM + dM u ∧ ω3 = 0, whence c αM = J1 du = dM u,

and everything can be expressed in terms of ω˜ 1 and u. In summary, Theorem 1. Let (Y, ϕ) be a 7-manifold with a torsion-free G2 structure, admitting an infinitesimal isometry V . Suppose that the norm of V is not constant, and let y ∈ Y be a point where V does not vanish. Let V be a neighborhood of y, such that the space of orbits of V in V is a manifold N and suppose that the almost-Ka¨ hler structure on N is in fact Ka¨ hler. Then, there exists a 4-dimensional manifold M endowed with a complex structure J1 , a complex symplectic form = ω2 + iω3 , and 1-parameter families of Ka¨ hler 2-forms ω˜ 1 = ω˜ 1 (t) and positive functions u = u(t) on M, satisfying the relations c ω˜ 1

= −dM dM u,

(16)

t ω˜ 1 ∧ ω˜ 1 =

(17)

1 2u

∧ .

On a sufficiently small neighborhood of y, (ϕ, V ) is equivariantly isometric to the torsion-free G2 structure ϕ = ω˜ 1 ∧ (dy + ηN ) + dt ∧ (dx + ξP ) ∧ (dy + ηN ) + t ω2 ∧ (dx + ξP ) + u ω3 ∧ dt , 2 × M endowed with the infinitesimal isometry V = ∂/∂y, where on Rt+ × Rx,y c =J ◦d ; dM denotes the differential on M, and dM 1 M + t > 0 is the variable on the Rt -factor; 2 =R ×R ; (x, y) are standard coordinates on Rx,y x y

(18)


51

ηN is a 1-form on N = Rt+ × Rx × M with dηN = −ω2 ; c u) ∧ dt + ω ξP is a 1-form on P = Rt+ × M with dξP = (dM ˜ 1 . Remark 2. By redefining the local coordinate x, one can assume (without loss) that ηN ∂ is in fact a 1-form on M. In this case the G2 structure ϕ is invariant under ∂x as well, ∂ and ∂x is identified with the Killing vector field defined in Corollary 3. It is not difficult to see that for generic data on M, the holonomy group of the G2 structure (18) is equal to G2 . Indeed, the general theory of holonomy groups (see e.g. [29]) implies that if the holonomy group of (Y, k) were strictly less than G2 , then ∂ there would exist a non-trivial parallel vector field X on (Y, k) commuting with V = ∂y ; it would therefore come from an infinitesimal isometry of the SU (3) structure on N ∂ (still denoted by X), which preserves the level sets of t and commutes with U = ∂x . The ∂ ∂ } equations for dηN and dξP imply that there is no parallel vector field in span{ ∂x , ∂y

unless dM u = 0 and ω˜ = 0 (a situation which we shall exclude below), thus showing that X is in general different from U . Therefore, X must come from a (real) holomorphic vector field on (M, J ) which preserves the Kähler metrics ω(t) for each t. In particular, if we assume that (M, J1 , , ω˜ 1 (t), u(t)) does not admit any infinitesimal isometry and either dM u = 0 or ω˜ = 0, then we know that the holonomy group of (Y, ϕ) must equal G2 . It is important to note that the triple (ω˜ 1 (t), ω2 , ω3 ) appearing in Theorem 1 does not in general constitute a hyperkähler structure. Indeed, by (17), the holomorphic section

of the canonical bundle 2,0 M satisfies 2g = tu−1 , and the Ricci form κ of the Kähler metric ω˜ 1 (t) is given by c c κ = 21 dM dM (log t − log u) = − 21 dM dM log u.

(19)

In the next section, we shall however explain how the simplest case does correspond to a hyperkähler situation. 3. Constant Solutions A careful inspection of the proof of Theorem 1 shows that the process can be inverted so as to construct a torsion-free G2 structure from a 4-manifold M with a complex symplectic structure (J1 , ) together with a 1-parameter family (ω˜ 1 (t), u(t)) of Kähler forms and smooth functions satisfying (16) and (17). In this section, we shall carry out this inverse construction explicitly in the case in which u really is just a function of t, so that dM u = 0. The above assumption reduces (16) to ω˜ 1 = ω + tω

for some closed (1,1)-forms ω, ω on M. Consider the real symmetric bilinear form B defined by α ∧ β = 21 B(α, β) ∧

(20)

52


on the space of 2-forms. Restricting B to the subspace ω, ω and diagonalizing, we may write ω˜ 1 = (p + qt)ω0 + (r + st)ω1 ,

(21)

where ω0 ∧ ω0 = − 21 ε ∧ ,

ω1 ∧ ω1 = 21 ∧ ,

ω0 ∧ ω1 = 0,

(22)

ε is 0 or 1, and p, q, r, s are constants satisfying r + st > |p + qt| to ensure the overall positivity of ω˜ 1 . From (21) and (17) we have u = t (r + st)2 − (p + qt)2 .

(23)

(24)

Note that ε = 0 in (22) if and only if ω0 = 0; to avoid redundancy we declare that p = q = 0 in this case. The real symplectic forms ω1 , ω2 , ω3 satisfy the usual compatibility relations ωi ∧ ωi = ωj ∧ ωj , ωi ∧ ωj = 0, i = j,

(25)

extending (11). It is well known ([23, 29]) that they then determine a hyperkähler structure, consisting of (i) complex structures J1 , J2 , J3 satisfying Ji ◦ Jj + Jj ◦ Ji = −2δij Id; (ii) a Riemannian metric g0 and associated Levi-Civita connection relative to which the Ji are all orthogonal and parallel. When p 2 + q 2 > 0, in addition to the hyperkähler structure, the (1, 1) form ω0 defines an almost-complex structure I on M, such that ω0 ( · , · ) = g0 (I · , · ) as in (3). It follows that (g0 , I, ω0 ) is a Ricci-flat almost-Ka¨ hler metric on M, compatible with the opposite orientation to the one induced on M by the hyperkähler structure (ω1 , ω2 , ω3 ). The integrability of the almost-complex structure I is equivalent to the flatness of the metric g0 , and this is the only possibility when M is compact, see [31]. However, completely explicit local examples of 4-dimensional hyperkähler manifolds admitting a non-integrable almost-Kähler structure I are now known [5, 7, 28]. Theorem 2. Let (M, g0 , ω1 , ω2 , ω3 ) be a hyperka¨ hler 4-manifold. Let r, s be real constants with r + st > 0 for any t ∈ (a, b) with a > 0, and set ω˜ 1 = (r + st)ω1 ,

u = t (r + st)2 .

2 × M , where Then the 3-form (18) defines a torsion-free G2 structure on (a, b) × Rx,y

M is a suitable open subset of M. Suppose, furthermore, that (M, g0 ) admits an almost-Ka¨ hler structure (ω0 , I ) compatible with the opposite orientation to the one induced by (ω1 , ω2 , ω3 ). Let p, q, r, s be real constants satisfying (23) for t ∈ (a, b), a > 0, and set ω˜ 1 = (p + qt)ω0 + (r + st)ω1 , u = t (r + st)2 − (p + qt)2 .

Then (18) again defines a torsion-free G2 structure on a manifold of the form (a, b) × 2 × M . Rx,y


53

Remark 3. It suffices to take M to be any contractible open subset of M, in which case the 1-forms ξP and ηN of Theorem 1 can always be defined on P = Rt+ × M and N = Rt+ × Rx × M respectively. However, as we shall see below, we can alternatively keep the 4-manifold M fixed and think of the 1-forms (ξ, η) in Theorem 1 as connection forms of a principal T2 bundle W over M. Since (dξ, dη) = (qω0 + sω1 , −ω2 )

(26)

is the curvature of the principal connection, we obtain integrality constraints for the 1 1 (qω0 + sω1 )] and [ 2π ω2 ] of M, in the sense that they must cohomological classes [ 2π be contained in the image of the universal morphism H 2 (M, Z) → H 2 (M, R). Note also that in general only two of the four real parameters (p, q, r, s) are effective in the sense that one can fix two of them, by rescaling ϕ and V on Y . Moreover, if q = s = 0, (18) corresponds to the product of a Calabi-Yau 6-manifold with Rx , while generically the corresponding metric has holonomy equal to G2 in accordance with Remark 2. By way of proving Theorem 2, we shall describe the reverse construction in the case (p, q, r, s) = (0, 0, 0, 1) for simplicity, so that u = t 3 . This is also the case in which the family g(t) consists of homothetic hyperkähler metrics on M. The resulting metrics with holonomy G2 were first described in [20] in the context of T2 bundles. To duplicate this 1 1 situation, we assume that the 2-forms 2π ω1 and 2π ω2 are integral. This is always true locally, though if M is compact (i.e. a K3 surface or a torus) the assumptions imply that (M, J3 ) is exceptional – its Picard number is maximal, see e.g. [9]. On any exceptional K3 surface, Yau’s theorem implies the existence of hyperkähler metrics satisfying these integrality assumptions. Let P be the total space of the principal S 1 bundle over the hyperkähler 4-manifold M 1 classified by [ 2π ω1 ], and ξ a connection 1-form on P such that dξ = ω1 in accordance with (26). Attached to P is also a principal C∗ bundle over M, whose total space N is the real 6-manifold manifold N ∼ = Rt+ × P . Pulling back forms to this new total space, we define on N an almost-Hermitian structure (h, σ, J ) by σ = tω1 + dt ∧ ξ, h = tg0 + t −3 ξ ⊗ ξ + t 3 dt ⊗ dt,

(27)

and J α = J1 α, α ∈ 1 M,

J dt = t −3 ξ.

The 2-form σ is closed, satisfies (3), and it is easy to check that J is integrable. Thus, (h, σ, J ) defines a Kähler structure on N . Now let U be a vector field which is h-dual to t −3 ξ so that U is tangent to the fibers of P and ξ(U ) = 1; it follows that U is the generator of the natural S 1 action on N , acting as a rotation on each fibre of P . Moreover, U is a Hamiltonian isometry of the Kähler structure (27) and the corresponding momentum map is −t. The integrality assumption for ω2 implies that there exists a principal S 1 bundle over 1 N, classified by [− 2π ω2 ] ∈ H 2 (N, Z). We denote by Y the corresponding 7-dimensional total space and take η to be a connection 1-form satisfying (12).

54


Corollary 4. With the above assumptions, the 3-form ϕ = t ω1 ∧ η + dt ∧ ξ ∧ η + t 4 ω3 ∧ dt + t ω2 ∧ ξ defines a torsion-free G2 structure on Y . The corresponding 4-form is ∗ϕ = ω3 ∧ ξ ∧ η − t 3 ω2 ∧ dt ∧ η + t 3 ω1 ∧ dt ∧ ξ + 21 t 4 ω1 ∧ ω1 . Remark 4. In the above situation, the Kähler structure (h, σ, J ) on N belongs to the classes of metrics recently studied in [4] and [18]. In the terminology of the former, (h, J ) arises from a Hamiltonian 2-form of order 1 with one non-constant eigenvalue (equal to t) and one zero constant eigenvalue of multiplicity 2; the structure function F (t) of h is t −1 . Moreover, σ = tω1 + dt ∧ ξ = d(tξ ) = d(t 4 J dt) = dd c ( 15 t 5 ), using the definition of ξ after (12), and (26),(27). Not only then is f (t) = 21 log t a Ricci potential for h (see Corollary 2), but 25 t 5 = 25 e5f acts as a Kähler potential for the same metric. We now return to the general case of Theorem 2, when p and q are not both zero. Now the hyperkähler metrics g(t) are not homothetic, though for the examples in [5, 7, 28] the family g(t) is an isotopy (i.e. g(t) are all isometric to an initial hyperkähler metric, under the flow of a vector field on M). 1 1 If the cohomology classes of 2π (qω0 + sω1 ) and 2π ω2 are integral, the G2 structure is defined on the product of an interval and the principal T2 bundle W over M, classified 1 1 by [ 2π (qω0 + sω1 )] and [− 2π ω2 ]. The action of the 2-torus T2 is generated by the commuting vector fields U = k( · , t −2 ξ ) and V = k( · , t −2 η), which preserve the G2 structure. Take M = T4 to be the 4-torus with a flat hyperkähler metric. There are three cases according to the signature of the bilinear form B of (20) restricted to the 2-dimensional subspace generated by (26): (i) s > q, so B is positive definite and there is a flat metric g0 on T4 with dξ, dη ∈ 2+ . Thus, there exists a basis (ei ) of 1-forms on T4 such that dξ = ω1 and dη = −ω2 with ω1 = e14 + e23 ,

ω2 = e13 + e42 ,

ω3 = −(e12 + e34 ),

reflecting the structure of the real Lie algebra h associated to the complex Heisenberg group H . Then W = \H is the Iwasawa manifold [1, 2]. (ii) s = q so that dξ is a simple 2-form. In this case, we may assume that dξ = e24 and either dη = e14 + e23 or dη = e14 . Then W is a T2 bundle over T4 corresponding to one of two other nilmanifolds. The simplest holonomy G2 metric in the first case is obtained by setting (p, q, r, s) = (−1, 1, 1, 1) so that u = 4t 2 . The resulting metric coincides with that described in Example 2 of [15, §4] (except that t 2 there has now become t > 0). (iii) s < q and there is a basis with dξ = e14 − e23 and dη = e14 + e23 , so dξ, dη =

d ξ˜ , d η ˜ with d ξ˜ = e14 and d η˜ = e23 . In this case, we may take W to be a discrete quotient of H3 × H3 , where H3 is the real Heisenberg group.


55

Example 1. To make (i) more explicit, we may take coordinates λ, µ, , m on T4 and fibre coordinates x, y on W so that e5 = −η = dx − λd + µdm,

e6 = ξ = dy − µd − λdm

are the corresponding connection 1-forms. The resulting G2 metric k = t 2 (dλ2 + dµ2 + d2 + dm2 ) + t −2 (dx − λd + µdm)2 +t −2 (dy − µd − λdm)2 + t 4 dt 2 is defined on (0, ∞) × W . It is shown in [20] to arise from an SO(5) invariant G2 metric on the total space of 2+ → S 4 by a contraction of the isometry group. The explicit form above makes it easy to compute the Riemann tensor Rij kl of k. The package GRTensor (available from http://grtensor.phy.queensu.ca) was in fact used to verify that (a) the Ricci tensor R i j il is zero, (b) the matrix R i j kl with rows labelled by i has rank 7 everywhere, and (c) the matrix R ij kl with rows labelled by (i, j ) has rank 14 everywhere. Point (b) confirms that k is irreducible and therefore has holonomy equal to G2 , which is consistent with (c). The same technique can be used to analyse metrics arising from (ii) and (iii). 4. Hypersurface Structures We shall now investigate the metrics with holonomy G2 constructed in Theorem 2 by restricting them to hypersurfaces on which t = ηk is constant before taking an S 1 quotient. These hypersurfaces correspond to the total spaces W of the T2 bundles of Example 1. We can write the 3-form (18) as ϕ = (ω˜ 1 + dt ∧ ξ ) ∧ η + t (u ω3 ∧ dt + ω2 ∧ ξ ), where ξ, η are the corresponding connection 1-forms on N, Y respectively. Since dt2h = u−1 , it follows from (2) that dt2k = z−1 , where z = ut, so z = t 2 (r + st)2 − (p + qt)2 . (28) The 1-form z1/2 dt therefore has unit norm relative to the G2 metric k, and we may write z1/2 dt = dτ, where

(29)

1/2 τ = t (r + st)2 − (p + qt)2 dt.

The important point here is that u is constant as a function on M, and so z is really just a function of t. As a consequence, ϕ = ρ ∧ dτ + φ + , ∗ϕ = φ − ∧ dτ + 21 ρ ∧ ρ,

(30)

56


where ρ = z1/2 ω3 + z−1/2 ξ ∧ η, φ + = ω˜ 1 ∧ η + tω2 ∧ ξ, φ − = t −1 z1/2 ω2 ∧ η − t 2 z−1/2 ω˜ 1 ∧ ξ.

(31)

Restricting to an interval where the function t → τ is bijective, let Yτ denote the hypersurface of Y for which τ has the constant value “τ ” (excusing the abuse of notation). Whereas σ, ψ + , ψ − characterize the SU (3) structure on N , we are using ρ, φ + , φ − (a lexicographic shift) for the corresponding objects on Yτ . The closure of the forms (30) is equivalent to asserting that dφ + = 0,

d(ρ 2 ) = 0,

(32)

(ρ 2 denotes ρ ∧ ρ) for every fixed value of τ , and that the SU (3) structures on Yτ satisfy the equations ∂φ + = dρ, ∂τ

∂ 1 2 ( ρ ) = −dφ − . ∂τ 2

(33)

An SU (3) structure satisfying (32) is called half-integrable or half-flat in the formalism of [15]. To verify (33) directly in our situation, first observe that d(ξ ∧ η) = (qω0 + sω1 ) ∧ η + ξ ∧ ω2 (recall (26)), and φ + = (pω0 + rω1 ) ∧ η + t d(ξ ∧ η), 1 2 1 2 ρ = ξ ∧ η ∧ ω3 + 2 z ω3 ∧ ω3 .

(34)

These equations explain the significance of the coordinates (t, z) – the above forms are linear in t and z respectively. The reader may now check that (33) implies both (28) and (29); a constant of integration may be absorbed into the term r 2 − p 2 . Hitchin discovered that (33) leads to a Hamiltonian system in the symplectic vector space V × V∗ , where V is the space of exact 3-forms on the compact manifold Yτ , whose dual V∗ can be identified with the space of exact 4-forms. The Hamiltonian function H is derived from integrating volume forms determined algebraically by φ + and ρ 2 , and this also enables φ − to be determined from φ + in (31). Elements of V represent deformations of φ + in a fixed cohomology class, and those of V∗ deformations of ρ 2 . Given a solution of (32) for τ = a, a solution of (33) can then be found on some interval (a, b) [21]. This approach also underlies some of the newly constructed metrics (such as [12]) with reduced holonomy. The function H is already implicit in our calculations above, which may be summarized in the following way: Proposition 2. The solution (31) can be expressed in the form H = 0, where the function 1/2 H = 2t (r + st)2 − (p + qt)2 − 2z1/2 satisfies dt ∂H =− , dτ ∂z

dz ∂H = . dτ ∂t


57

It is an important consequence of the Kähler assumption that, for each choice (p, q, r, s) ∈ R4 , there is only one valid solution curve in the (t, z) plane. Example 2. In the light of Proposition 2, a variant of (31) and Example 1 is provided by setting ρ = ±z1/2 (e12 + e34 ) + z−1/2 e56 , φ + = φ0+ + t d(e56 ), φ − = φ0− ± t (e5 ∧ de5 + e6 ∧ de6 ), with z = (t 2 − 21 H )2 .

(35)

We assume that z1/2 > 0, so that the orientation ρ 3 remains fixed. The notation is consistent with that of [1], with de5 = e13 +e42 and de6 = e14 +e23 . Any value of the constant H gives a valid solution and so a holonomy G2 metric on the product (a, b) × W of some interval with the Iwasawa manifold W , though φ0± and the signs must be chosen to ensure compatibility (essentially (7)) and positive definiteness of the resulting metric. The figure plots the quartic curves (35) in the (t, z) plane for various values of H ; two pass through each point because of the sign ambiguity implicit in the definition of H. (i) H = 2 gives z = (1 − t 2 )2 , including the bell-shaped segment. To satisfy (33) for |t| < 1 we need to choose the plus sign in φ − . We may then define the 3-forms φ0± by setting φ + + iφ − = i((e5 + ie6 ) + t (e5 − ie6 )) ∧ d(e5 + ie6 ). It follows that φ0+ + iφ0− is a form of type (3, 0) relative to the standard complex structure on W , and taking z = 1 and the plus sign in ρ determines a compatible Hermitian metric via (3). As one flows away from the point (0, 1), the almost-complex structure induced on the hypersurface becomes non-integrable; ρ, φ ± degenerate simultaneously when one reaches t = ±1. Furthermore, we may take τ = 13 t 3 + t, a cubic equation with solution t = α 1/3 − α −1/3 ,

2α = 3τ + (9τ 2 − 4)1/2 ,

in contrast with the Kähler scheme in which τ = c + 21 rt 2 + 13 st 3 in the case p = q = 0. (ii) H = −2 gives z = (t 2 + 1)2 (the curve above and touching the bell). This requires the minus sign in φ − and provides a different deformation of the standard Hermitian structure on W . Indeed, φ0+ +iφ0− is modified by the addition of a form of type (1, 2) rather than (2, 1), and the resulting almost-complex structure is undefined when |t| reaches 1. However, ρ remains non-degenerate for all t, and this corresponds to a different singular behaviour of the G2 metric. (iii) H = 0 and z = t 4 (with the flattened base) requires φ0± = 0 and the minus signs in φ − , ρ, reproducing exactly the solution of Example 1. The almost-complex structure induced on W by φ + is constant and was first singled out for study in [2], where it is called J3 . The 2-form ρ degenerates only for t = 0, and the resulting metric has the advantage of being “half-complete”.

58

V. Apostolov, S. Salamon 5

z

4

3

2

1

t -1.5

-1

-0.5

0.5

1

1.5

We conclude this section by providing an explanation of why the system (33) often reduces to one variable. Let φ + be an invariant closed 3-form with stabilizer SL(3, C) on either of the four nilmanifolds \H described just before Example 1. Let φ − be the unique 3-form for which φ + + iφ − has type (3,0) relative to the almost-complex structure J determined by φ + . Let ker d denote the space of invariant closed 1-forms 2 h∗ ). (equivalently, the kernel of the natural mapping d : h∗ → Lemma 2. In the above situation, dψ+ = 0 implies that J (ker d) = ker d. This can be proved by generalizing an argument in the proof of [26, Theorem 1.1], which draws the same conclusion if J is integrable. In view of the structure equations for the Lie algebra h, the condition that J leave ker d invariant is equivalent to asserting that dφ − belong to the 1-dimensional space

e1234 =

4

but it is this fact that simplifies the equations.

(ker d),


59

5. Solutions Varying on M As another ramification of the evolution equations for (ω˜ 1 = ω˜ 1 (t), u = u(t)), we consider the case when (M, g0 , ω1 , ω2 , ω3 ) is a hyperkähler manifold, and c ω˜ 1 = ω1 − 21 dM dM G,

2u = G

for a smooth function G on (a, b) × M, where continues to denote ∂/∂t. We assume here that ω˜ 1 is a positive definite (1,1)-form on (M, J1 ) for each fixed t in the interval (a, b). Whilst (16) is automatically satisfied, (17) becomes 2t M(G) = G

,

(36)

where M denotes the complex Monge-Ampère operator on (M, g0 , J1 ), defined by ∧2 c f = M(f )ω1 ∧ ω1 , ω1 − 21 dM dM for all f ∈ C ∞ (M). Note that the 1-form ξP of Theorem 1 is automatically defined on P = Rt+ × M by c ξP = − 21 dM (G ).

(37)

Since the hyperkähler metric g0 is Ricci-flat and the ωi ’s are parallel 2-forms, there exists a real-analytic structure on M, compatible with (g0 , J1 , ω1 , ω2 , ω3 ). Thus, applying the Cauchy-Kowalewski theorem, one obtains a general existence result. Corollary 5. Let (M, g0 , ω1 , ω2 , ω3 ) be a hyperka¨ hler, real-analytic 4-manifold and J1 be the Ka¨ hler structure compatible with ω1 . Suppose that G0 , G1 are real-analytic funcc G0 positive definite with respect to J . Then there exist a tions on M with ω1 − 21 dM dM 1 real number a > 0 and a real-analytic solution G(t, · ) of (36) defined on (0, a) × M c G is positive definite for with G(0) = G0 , G (0) = G1 , and such that ω1 − 21 dM dM 1 1

c any t ∈ (0, a). Thus, ω˜ 1 = ω1 − 2 dM dM G and u = 2 G define, via Theorem 1, a 2 × M , where M is a torsion-free G2 structure on a manifold of the form (0, a) × Rx,y suitable open subset of M. In the above construction M should be taken so as to solve dηN = −ω2 for a 1-form ηN on N = Rt+ × Rx × M (see Theorem 1 and (37)). Alternatively, we may assume 1 ω2 ] of M is integral and η is a principal connection of the that the cohomology class [ 2π 1 1 principal S bundle Q over M, classified by [− 2π ω2 ]. In this case Corollary 5 produces examples of torsion-free G2 structures with a R×S 1 symmetry on Y = (0, a)×Rx ×Q. One has no control over the real number a. It is tempting to spot some special solutions to (36), by reducing the problem to a linear (elliptic) equation. This can be done by assuming that for each t > 0 the function G generates a complex Monge-Ampe` re foliation on (M, J1 ) (see e.g. [10]), meaning that c c dM dM G ∧ dM dM G=0 c G has constant rank. (The integral curves of d d c G then foliate M by and dM dM M M complex submanifolds.) The point is that in this case we have

M(G) = 1 + 21 G, where is the Riemannian Laplacian of the hyperkähler metric g0 .

60


The above situation appears in particular when (M, J1 ) admits a holomorphic C action and we look for equivariant solutions of (36), or when (M, J1 ) admits a holomorphic fibration p : M → C over a complex curve C and we look for solutions of the form G ◦ p, where for each t, G is a function on C. Consider finally the flat hyperkähler metric g0 on (M, J1 ) = C2 ∼ = R4 , determined by the 2-forms ω1 = 21 i(dz1 ∧ dz1 + dz2 ∧ dz2 ),

ω2 = Re(dz1 ∧ dz2 ),

ω3 = Im(dz1 ∧ dz2 ),

where z1 = λ+iµ and z2 = +im are the canonical coordinates of C2 . Letting G(t, z1 ) be a function on C2 which does not depend on z2 , Eq. (36) reduces to

G +t

∂ 2G ∂ 2G + ∂λ2 ∂µ2

= 2t.

Separable solutions are given by G(t, λ, µ) = 13 t 3 + H (λ, µ)K(t), where ∂ 2H ∂ 2H + + cH = 0, 2 ∂λ ∂µ2 c is a constant and K a solution of the Airy equation K

= ctK (equivalently, L = K −1 K satisfies the Riccati equation L + L2 = ct). Example 3. Taking H to be periodic in λ, µ (which requires c > 0) yields solutions of (36) defined on (0, a) × T4 . One such example is obtained by taking H = sin λ (so c = 1), and K = Ai(t) = 13 t 1/2 (J1/3 (ζ ) + J−1/3 (ζ )),

ζ = 23 it 3/2 .

Setting f = 1 + Ai(t) sin λ, the resulting G2 metric t (f dλ2 + f dµ2 + d2 + dm2 ) + f −1 (dx − Ai (t) cos λ dµ)2 +t −2 (dy − λd + µdm)2 + t 2 f dt 2 is Ricci-flat and irreducible. Since Ai(t) → 0 as t → ±∞, the above metric is asymptotic to a constant solution with u = t and holonomy equal to SU (3) (see Remark 3). However, the above construction can be easily modified to provide explicit deformations of the non-trivial metrics constructed in §3. Acknowledgements. This material was first developed at the joint AMS–UMI conference in Pisa in 2002. The first author was supported in part by NSERC grant OGP0023879 and by NSF grant INT-9903302. The second author is a member of EDGE, Research Training Network HPRN-CT-2000-00101, supported by the European Human Potential Programme, and a preliminary version of the results was presented at its mid-term conference. The authors are grateful to the referee for useful remarks, and to D. Conti for subsequent checking of some curvature computations.


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References 1. Abbena, E., Garbiero, S., Salamon, S.: Almost Hermitian geometry of 6-dimensional nilmanifolds. Ann. Scuola Norm. Sup. Pisa (4) 30, 147–170 (2001) 2. Abbena, E., Garbiero, S., Salamon, S.: Hermitian geometry on the Iwasawa manifold. Boll. Un. Mat. Ital. 11-B, 231–249 (1997) 3. Acharya, B., Witten, E.: Chiral fermions from manifolds of G2 holonomy. Available at arXiv: hep-th/0109152 4. Apostolov, V., Calderbank, D.M.J., Gauduchon, P.: Hamiltonian 2-forms in Kähler geometry, I. Available at arXiv:math.DG/0202280 5. Apostolov, V., Calderbank, D.M.J., Gauduchon, P.: The geometry of weakly self-dual Kähler surfaces. Compositio Math. 135, 279–322 (2003) 6. Apostolov, V., Gauduchon, P.: The Riemannian Goldberg-Sachs Theorem. Internat. J. Math. 8, 421– 439 (1997) 7. Armstrong, J.: An ansatz for Almost-Kähler, Einstein 4-manifolds. J. reine angew. Math. 542, 53–84 (2002) 8. Atiyah, M., Witten, E.: M-theory dynamics on a manifold of G2 holonomy. Adv. Theor. Math. Phys. 6, 1–106 (2003) 9. Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces. Berlin-Heidelberg-New YorkTokyo: Springer-Verlag, 1984 10. Bedford, E., Kalka, M.: Foliations and the complex Monge-Ampère equation. Commun. Pure Appl. Math. 30, 543–571 (1977) 11. Behrndt, K., Dall’Agata, G., Lüst, D., Mahapatra, S.: Intersecting 6-branes from new 7-manifolds with G2 -holonomy. JHEP 8, no. (27) (2002), 24 pp. 12. Brandhuber, A., Gomis, J., Gubser, S.S., Gukov, S.: Gauge theory at large N and new G2 holonomy metrics. Nucl. Phys. B 611, 179–204 (2001) 13. Bryant, R.L.: Metrics with exceptional holonomy. Ann. Math. 126, 525–576 (1987) 14. Bryant, R.L., Salamon, S.: On the construction of some complete metrics with exceptional holonomy. Duke Math. J. 58, 829–850 (1989) 15. Chiossi, S., Salamon, S.: The intrinsic torsion of SU(3) and G2 structures. In: Differential Geometry, Valencia 2001. River Edge, NJ: World Sci. Publishing, 2002, pp. 115–133 16. Cvetic, M., Gibbons, G.W., Lü, H., Pope, C.N.: Almost special holonomy in type IIA and M-theory. Nucl. Phys. B 638, 186–206 (2002) 17. Derdziński, A.: Self-dual Kähler manifolds and Einstein manifolds of dimension four. Compositio Math. 49, 405–433 (1983) 18. Derdziński, A., Maschler, G.: Local classification of conformally-Einstein Kähler metrics in higher dimensions. Proc. London Math. Soc. (3) 87, 779–819 (2003) 19. Fernández, M., Gray, A.: Riemannian manifolds with structure group G2 . Ann. Mat. Pura Appl. 32, 19–45 (1982) 20. Gibbons, G.W., Lü, H., Pope, C.N., Stelle, K.S.: Supersymmetric domain walls from metrics of special holonomy. Nucl. Phys. B 623, 3–46 (2002) 21. Hitchin, N.J.: Stable forms and special metrics. In: Global Differential Geometry: The Mathematical Legacy of Alfred Gray, Contemp. Math. 288, Providence RI: Am. Math. Soc., 2001, pp. 70–89 22. Hitchin, N.J.: The geometry of three-forms in six dimensions. J. Diff. Geom. 55, 547–576 (2000) 23. Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. London Math. Soc. 55, 59–126 (1989) 24. Joyce, D.D.: U (1)-invariant special Lagrangian 3-folds in C3 and special Lagrangian fibrations. Turkish J. Math. 27, 99–114 (2003) 25. Joyce, D.D.: Compact Manifolds with Special Holonomy. Oxford: Oxford University Press, 2000 26. Ketsetzis, G., Salamon, S.: Complex structures on the Iwasawa manifold. Adv. Geom. 4, 165–179 (2004) 27. LeBrun, C.: Einstein metrics on complex surfaces. In: Geometry and Physics (Aarhus, 1995), Lect. Notes Pure Appl. Math. Vol. 184, New York: Dekker, 1997, pp. 167–176 28. Nurowski, P., Przanowski, M.: A four-dimensional example of Ricci flat metric admitting almost Kähler non-Kähler structure. Classical Quant. Grav. 16, L9–L13 (1999) 29. Salamon, S.: Riemannian geometry and holonomy groups. Pitman Research Notes in Mathematics Vol. 201, London-New York: Longman, 1989 30. Santillan, O.P.: A construction of G2 holonomy spaces with torus symmetry. Nucl. Phys. B 660, 169–193 (2003) 31. Sekigawa, K.: On some compact Einstein almost-Kähler manifolds. J. Math. Soc. Japan 36, 677–684 (1987) Communicated by G.W. Gibbons


Communications in


Critical Thresholds and the Limit Distribution in the Bak-Sneppen Model Ronald Meester, Dmitri Znamenski Divisie Wiskunde, Faculteit der Exacte Wetenschappen, Vrije Universiteit Amsterdam, de Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. E-mail: [email protected]; [email protected] Received: 10 April 2003 / Accepted: 6 October 2003 Published online: 13 February 2004 – © Springer-Verlag 2004

Abstract: One of the key problems related to the Bak-Sneppen evolution model is to compute the limit distribution of the fitnesses in the stationary regime, as the size of the system tends to infinity. Simulations in [3, 1, 4] suggest that the one-dimensional limit marginal distribution is uniform on (pc , 1), for some pc ∼ 0.667. In this paper we define three critical thresholds related to avalanche characteristics. We prove that if these critical thresholds are the same and equal to some pc (we can only prove that two of them are the same) then the limit distribution is the product of uniform distributions on (pc , 1), and moreover pc < 0.75. Our proofs are based on a self-similar graphical representation of the avalanches.

1. Introduction The Bak-Sneppen model was introduced in [2], as a simple model of evolution, and has received a lot of attention in the literature: recently an Internet search engine found about 900 links by the keyword Bak-Sneppen. The model is defined as follows. Consider a system with N species. These species are represented by N vertices on a circle, evenly spaced, say. Now each of these species is assigned a so-called fitness, a number between 0 and 1. The higher the fitness, the better chance of surviving the species has. The dynamics of evolution is modelled as follows. Every discrete time step, we choose the vertex with minimal fitness, and we think of the corresponding species as disappearing completely. This species is then replaced by a new one, with a fresh and independent fitness, uniformly distributed on [0, 1]. So far, the dynamics does not have any interaction between the species, and does not result in an interesting process. Indeed, if we only replace the species with the lowest fitness, then it is easy to see that the system converges to a situation with all fitnesses equal to 1. Interaction is introduced by also replacing the two neighbours of the vertex with lowest fitness by new species with independent fitnesses. This interaction

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R. Meester, D. Znamenski

represents co-evolution of related species. The neighbour interaction makes the model very attractive and highly non-trivial from a mathematical point of view. The Bak-Sneppen model turns out to be also of some practical interest. For instance, in [16–18], variations of the Bak-Sneppen model related to real evolution of bacteria populations are considered and in [19, 20], variations related to macro-economical processes are studied. To avoid the mathematical difficulties, some simplified versions of the model have also been proposed, for example, the mean field version of the model ([9–11]) and discrete versions of the model ([12–14]). It is simple to run the model on a computer. Simulations then suggest the following behaviour, for large N ([3, 1, 4]). It appears that the one-dimensional marginals are uniform (in the limit for N → ∞) on (pc , 1) for some pc whose numerical value is close to 0.667, see [4]. In this article, we make significant progress towards this conjecture. The fitnesses of the vertices are random variables with values in [0, 1] and we update them according to the uniform distribution on [0, 1]. For computational reasons however, it is convenient for the fitnesses to have values in [0, ∞] and to update them according to the exponential distribution with parameter 1, say (see for example [5]). In this new exponential setup (denoted in the sequal by the BS-process) a threshold b corresponds to the threshold q(b) = 1 − e−b in the original uniform setup. The conjecture in the exponential setup says that the one-dimensional stationary marginals (in the limit for N → ∞) are exponentially distributed above bc , for some bc whose numerical value is close to 1.0996 q −1 (0.667). To prepare for the main results of this article, in Sect. 2 we define the so-called locking thresholds representation of the process, where we ignore all information irrelevant for the distribution of the process. This representation captures the essential characteristics of the BS-process, but at the same time, it has a much simpler spatial structure. Thus the representation has a potential to improve, for example, the simulation results. It is natural to define avalanches of fitnesses from threshold b ∈ [0, ∞]: start counting at the moment that all fitnesses are above b and finish the counting at the first next moment that all fitnesses are above b again. In Sect. 3 we will give a more precise definition of an avalanche, and relate to it three characteristics: the mean range RN (b), mean duration DN (b) and the probability PN (b) that an avalanche is of range N . In Sect. 4 we prove that the above avalanche characteristics have limits, as N → ∞, to be denoted by R∞ (b), D∞ (b) and P∞ (b) respectively, and that these limits are non-decreasing in b. p We associate to them three critical thresholds bcr , bcd and bc respectively. These critical p thresholds trivially satisfy bcr ≤ bcd ≤ bc . p The current paper is a sequel to [15]. The main result in [15] is that bc < ∞, thereby establishing the non-triviality of the limit distrbution as the size of the system tends to infinity. In the current paper, we provide a more detailed analysis of the limit distribution. For this, we use the critical threshold defined above. The central results of this article are stated in Sect. 3 and consist of three parts. First we relate mean duration DN (b) and mean range RN (b) via a differential equation, and prove that their critical thresholds are the same, i.e. bcr = bcd . Secondly, we prove that if all critical thresholds are the same and equal to bc , say, then the limit distribution is the product of exponential distributions above bc (or in terms of the uniform setup, the limit distribution is the product of the uniform distributions on [qc , 1], where qc = q(bc )). Finally we prove that bcr < 2 log 2 (in the uniform setup the upper bound is 0.75) with the help of a differential inequality for R∞ (b). Most of our proofs are based on the so-called graphical representation of an avalanche which is defined in Sect. 4. The graphical representation and some of the monotonicity

Critical Thresholds and the Limit Distribution in the Bak-Sneppen Model

65

properties of avalanches are borrowed from our previous work [15]. We include them here for convenience. 2. The Locking Thresholds Representation Let N ≥ 3, and let (N ) = {−N + 1, . . . , −1, 0} index the set of N vertices on the circle, so that 0 and −N + 1 are neighbours. We use negative indices to have a notation compatible with [15]. Later we will add and subtract the indices (N ) as integers, identifying them modulo N . For example, sometimes we denote vertex −N + 1 by vertex 1. To simplify expressions, we will omit (mod N ) in the algebraic operations on (N ) as long as it does not lead to confusion. For any N ≥ 3 consider a BS-process on (N ), and for any n ∈ N, let XN (n) = {XN,x (n)}x∈(N) be the collection of fitnesses at time n ≥ 0. We assume that at the initial time n = 0 all the fitnesses are i.i.d. and exponentially distributed. For any x1 < · · · < xk ∈ (N ), let FN,x1 ,...,xk (n, ·) denote the joint distribution function of XN,x1 (n), . . . , XN,xk (n) at time n ≥ 0, i.e. for any b1 , . . . , bk ∈ R+ , we define FN,x1 ,...,xk (n, b1 , . . . , bk ) = P XN,x1 (n) ≤ b1 , . . . , XN,xk (n) ≤ bk . We will now inductively define the so called locking thresholds YN (n) = {YN,x (n)}x∈(N) and prove that, given YN (n), the fitnesses XN (n) are independent and exponentially distributed above their locking thresholds. More precisely, we will show that for any x1 < · · · < xk ∈ (N ), and b1 , . . . , bk ∈ R+ we have the representation ∞ FN,x1 ,...,xk (n, b1 , . . . , bk ) =

··· 0

∞ k 0

s1 (bi ) GN,x1 ,...,xk (n, ds1 , . . . , dsk ),

i=1

(2.1) where, s (·) is the exponential distribution function above s, i.e. 0, if b < s, s (b) = 1 − exp(−b + s), if b ≥ s, and GN,x1 ,...,xk (n, ·) is the joint distribution of YN,x1 (n), . . . , YN,xk (n). Let us define YN,x (0) = 0, for all x ∈ (N ). Since the collection XN (0) is independent and exponentially distributed, we have the basis of the induction. Suppose that, for some n ≥ 0, the locking thresholds YN (n) are defined, and that XN (n), given YN (n), is an independent and exponentially distributed collection above their lock∗ (n) be the vertex with minimal fitness at time n. Then, ing thresholds. Let x ∗ = xN given YN (n), and given XN,x ∗ (n), the fitnesses XN,y (n) y∈(N)\{x ∗ } are independent

and exponentially distributed above thresholds max YN,y (n), XN,x ∗ (n) y∈(N)\{x ∗ } . ∗ According to the update rules, we update at time n the neighbourhood N xN (n) =

∗ ∗ (n), x ∗ (n) + 1 of x ∗ (n) and replace their fitnesses by three new indexN (n) − 1, xN N N pendent and exponentially distributed random variables. Hence, we define 

∗  max YN,y (n), XN,x ∗ (n) , for y ∈ (N ) \ N xN (n) , YN,y (n + 1) = ∗  (n) , 0, for y ∈ N xN

66


and observe that XN (n + 1), given YN (n + 1), is an independent and exponentially distributed collection above YN (n + 1), i.e. we have the step of induction and (2.1) is proved. The locking thresholds are useful for the study of the limit behaviour of the process as follows. For any x1 < · · · < xk ∈ Z, b1 , . . . , bk ∈ R+ , n ≥ 1 and N sufficiently large, we have x1 < · · · < xk ∈ (N ) ( mod N ), and thus FN,x1 ,...,xk (n, b1 , . . . , bk ) is well-defined. For any N ≥ 3 and n ∈ N, let GN (n, ·) denote the distribution function of YN,0 (n). Lemma 2.1. Suppose there exists 0 < bc < ∞, such that for any b < bc , lim sup lim sup GN (n, b) = 0, N→∞

n→∞

and for any b > bc , lim inf lim inf GN (n, b ) = 1. N→∞ n→∞

Then the limit distribution in the BS-process exists and is equal to the product of bc (·), i.e. for any x1 < · · · < xk ∈ Z and b1 , . . . , bk ∈ R+ , lim lim FN,x1 ,...,xk (n, b1 , . . . , bk ) =

N→∞ n→∞

k

bc (bi ).

i=1

Proof. The proof is a simple consequence of the representation (2.1), and for simplicity we give it for the one-dimensional marginals FN,0 (n, b) only. Let b ≥ 0, and observe that for any y ≥ 0, we have |y (b) − bc (b)| ≤ 2.

(2.2)

Since y (b) is a continuous function in y, for any 0 < ε < bc there exists 0 < δ < ε such that |y (b) − bc (b)| ≤ ε, for any y ∈ [bc − δ, bc + δ]. Hence, according to (2.1) and (2.2), we have |FN,0 (n, b) − bc (b)| ≤ 2GN (n, bc − δ) + ε GN (n, bc + δ) − GN (n, bc − δ) +2(1 − GN (n, bc + δ)) ≤ 2GN (n, bc − δ) + ε + 2(1 − GN (n, bc + δ)), and thus, due to the conditions of the lemma, lim lim |FN,0 (n, b) − bc (b)| < ε.

N→∞ n→∞

Since ε > 0 is arbitrary, this proves the lemma.


67

3. Critical Thresholds and Main Results In this section we define avalanches on (N ) and, in the second half of the section, on Z. After that we relate them to a number of critical thresholds and show that the distribution of a locking threshold, in the limit (first the time n → ∞, and then the size of the system N → ∞) is concentrated between the smallest and the largest critical threshold. We say that in the time interval [n, n + d), an avalanche from threshold b ∈ [0, ∞) (also referred to as a b-avalanche) with origin at x ∈ (N ) and duration d ≥ 1 occurs, if at time n, x is the vertex with minimal fitness, above threshold b, and n + d is the first moment after n with all fitnesses again above b. The range set of the b-avalanche is the collection of vertices updated during the avalanche, and the range of the b-avalanche is the number of different vertices in the range set. Note that, according to this definition, if at times n and n + 1 all the fitnesses are above b, then in the time interval [n, n + 1] an avalanche of range 3 and duration 1 occurs, even though there were no fitnesses below b. For any b > 0, the BS-process can be considered as a sequence of b-avalanches. Since during a b-avalanche the dynamics is completely determined by the position of the origin and the vertices with fitness below b, the range set of the b-avalanche [n, n+d] depends only on the position of the origin x, but not on the details of the initial configuration at time n. If we shift the whole range set by −x vertices, we obtain a set whose distribution is independent of x and of the initial configuration of fitnesses. The origin x itself naturally does depend on the configuration of fitnesses at time n, because x is the vertex with the minimal fitness at time n. The duration of the avalanche is clearly independent of the initial configuration of fitnesses. Thus, for any b > 0, the BS-process is a sequence of b-avalanches with i.i.d. durations and i.i.d. (shifted) range sets. These facts suggest that we should study a single b-avalanche in some detail. Consider a b-avalanche on (N ), with the origin at x. Its principal characteristics are the range set ξN (x, b) and the duration ηN (x, b). Two useful functions of the principal characteristics are the range of the avalanche rN (x, b) = |ξN (x, b)|, and the indicator function to have a spanning avalanche, 1{rN (x, b) = N }. We define the corresponding mean values, independent of x, as follows: RN (b) = E(rN (x, b)), DN (b) = E(ηN (x, b)), PN (b) = P (rN (x, b) = N ). From now on, we omit x in the notation of the avalanche characteristics if x = 0, i.e., for any b ≥ 0, ξN (b) = ξN (0, b), ηN (b) = ηN (0, b), rN (b) = rN (0, b). The following lemma is intuitively obvious. Its proof will be given in Sect. 4. Lemma 3.1. RN (b), DN (b) and PN (b) are non-decreasing in b. We now look at the limit behavior of the above functions, as N tends to infinity. The notion of an avalanche on (N ) can be naturally extended to an avalanche on Z, as follows. Assume that every vertex x ∈ Z accommodates a fitness, a random variable with value in [0, ∞]. Choose an arbitrary threshold b > 0. If the number of vertices with fitness below b is finite and positive, then the update rules of the BS-model are still well-defined. Suppose, therefore, that at time 0 we have a configuration with all fitnesses above b, and that we choose an arbitrary x ∈ Z, regardless of the values of the fitnesses, as the origin. We start by updating x and its two neighbours, x − 1 and x + 1.If, after

68


that, among x − 1, x, x + 1 there are vertices with fitnesses below b, we choose the one with minimal fitness, and update it together with its two neighbours, and so on. As soon as we get a configuration with all fitnesses above b, we stop the procedure. Then we define the duration η(x, b), and the range set ξ(x, b) of the avalanche in the obvious way. Note, however, that it is perhaps possible that we never stop and that there always is at least one fitness below the threshold. In this case, we say that an infinite avalanche occurs. For an infinite avalanche, we set η(x, b) = ∞, and define ξ(x, b) as the union of the vertices updated during [0, m], m ∈ N. It is easy to see that for an infinite avalanche, |ξ(x, b)| = ∞ a.s. The pair (ξ(x, b) − x, η(x, b)) is clearly independent of the configuration at time 0, for the same reasons as before. Similar to the avalanches on (N ), we consider the range r(x, b) = |ξ(x, b)|, and the indicator function to have an infinite avalanche 1{r(x, b) = ∞}. Define the corresponding mean values, independent of x: R∞ (b) = E(r(x, b)), D∞ (b) = E(η(x, b)), P∞ (b) = P (r(x, b) = ∞). As before, we omit x in the notation of the avalanche characteristics if x = 0, i.e., for any b ≥ 0, ξ(b) = ξ(0, b), η(b) = η(0, b), r(b) = r(0, b). The following theorem relates the avalanches on (N ) to these on Z. Theorem 3.2. For any b > 0, and any N ≥ 3, RN (b) ≤ R∞ (b), PN (b) ≥ P∞ (b), DN (b) ≤ D∞ (b),

lim RN (b) = R∞ (b),

N→∞

lim PN (b) = P∞ (b),

N→∞

lim DN (b) = D∞ (b).

N→∞

The proof of this theorem will be given in Sect. 5, with a coupling argument. The above theorem, together with Lemma 3.1, shows that R∞ (b), D∞ (b) and P∞ (b) are non-decreasing in b. Now we are ready for the definitions of the critical thresholds. According to Lemma 3.1 in [15], P∞ (68) > 0. Hence there exists a finite critical threshold p bc = inf{b > 0 : P∞ (b) > 0}. Since for any b > 0 such that P∞ (b) > 0, we also have R∞ (b) = ∞, there exists a finite critical threshold bcr = inf{b > 0 : R∞ (b) = ∞}, p

and bcr ≤ bc . Since, for any b > 0, D∞ (b) ≥ R∞ (b) − 2, infinite range implies infinite duration, and hence there exists a finite critical threshold bcd = inf{b > 0 : D∞ (b) = ∞}, with bcd ≤ bcr . Thus the three critical thresholds are ordered by definition as p

bcd ≤ bcr ≤ bc . It is not hard to see that bcd > 0, so all critical values are non-trivial.


69

Theorem 3.3. We have bcd = bcr ,

(3.1)

and as long as R∞ (b) is finite, D∞ (b) is differentiable, and d db D∞ (b)

= D∞ (b)R∞ (b).

(3.2)

The proof of this theorem is given in Sect. 6. The following two results are proved in Sect. 7 and Sect. 8. p

Theorem 3.4. If bcd = bcr = bc = bc say, then the limit distribution in the BS-process is the product of exponential distributions above bc . Theorem 3.5. It is the case that bcr ≤ 2 log 2. 4. The Self-Similar Graphical Representation Let N ≥ 3 and consider a b-avalanche on (N ), starting at time 0 at the vertex 0. So at time 0, the vertex 0 and its two neighbours are updated. Since this corresponds to a minimal avalanche from threshold 0, we can write this in terms of the range set and the duration as ξN (0) = {−1, 0, 1}, ηN (0) = 1. We can now graphically illustrate the continuation of this b-avalanche on (N ) × R+ (space × fitness) as follows. Look for the vertex with minimal fitness, and call this vertex x. Suppose that the fitness of x is equal to s < b. (Note that x must be the vertex 0 or one of its two neighbours.) Due to the lack of memory property of the exponential distribution, the fitnesses of the other two vertices in ξN (0) are independent and exponentially distributed on [s, ∞). We continue updating according to the appropriate rules, and wait until all fitnesses are above the threshold s. This in itself constitutes an s-avalanche, starting at x. We denote by x + ξˆN (s) the range set of this s-avalanche. In the graphical representation, we draw an arrow from the space-fitness point (x, s) to the space-fitness points (y, s), for all y ∈ x + ξˆN (s). In terms of the range set we write this as ξN (s) = ξN (0) ∪ {x + ξˆN (s)}, where ξN (s) is the set of vertices updated to the end of the s-avalanche. After the s-avalanche has ended, the fitnesses of all vertices in ξN (s) are independent and exponentially distributed on [s, ∞), due to the lack of memory property of the exponential distribution. We now look for the minimal fitness among all vertices in ξN (s). If this minimal fitness is above b, then the b-avalanche has stopped. If this minimal fitness is equal to t, where s < t < b, and is associated with the vertex y, say, then we start, as before, a t-avalanche with origin y. We continue updating until all fitnesses are above t. If y + ξˆN (t) denotes the range set of this t-avalanche, then we draw an arrow in the graphical representation from the space-fitness point (y, t) to all space-fitness points (z, t), for z ∈ y + ξˆN (t). In terms of the range set, we write this as ξN (t) = ξN (s) ∪ {y + ξˆN (t)}, where ξN (t) is the set of vertices updated at the end of the t-avalanche. We continue in the obvious way: this process will stop a.s. as soon as all fitnesses are above b. The idea of avalanches forming a hierarchical structure of subavalanches is also mentioned in [5], in a slightly different context. So, we have defined a random graph on (N )×R+ . This random graph is a subgraph of a graphical representation GRN defined formally as follows.

70


k- 2

b+s τk+2,1 τk-2,1 τk+1,1

^ ^ ξ (τ k+2,1) η ( τk+2,1) N

N

^ ^ ξ (τk-2,1 ) η (τk-2,1 ) N

N

^ ξ^ (τk+1,1 ) η (τk+1,1) N

N

τ k,1

^ ξ ( τ k,1) ^η ( τ k,1)

b

N

Π N,k-3

ΠN,k-2 ΠN,k-1

N

k

ΠN,k

ΠN,k+1

ΠN,k+2

ΠN,k+3

Fig. 1. The graphical representation GRN , where, for instance, (k, b)(k − 2, b + s) in GRN

Let (N) = {N,k }k∈(N) be a collection of independent homogeneous Poisson At the j th arrival processes. For each process N,k we performthe following procedure. τk,j of N,k , we draw independently the pair ξˆN (τk,j ), ηˆ N (τk,j ) , the (range set, dura tion) of a typical τk,j -avalanche with origin at 0. Then the pair k + ξˆN (τk,j ), ηˆ N (τk,j ) is distributed as the (range set, duration) of a typical τk,j -avalanche, with origin at k. We draw arrows in (N ) × R+ from (k, τk,j ) to (y, τk,j ), for all y ∈ k + ξˆN (τk,j ). For any b1 < b2 we say that (x, b1 ) is connected to (x, b2 ) by a time segment. A path in GRN is a sequence (x0 , s0 ), . . . , (xn , sn ) of points in (N ) × R+ , such that every pair (xj , sj ), (xj +1 , sj +1 ) is connected by either a time segment or an arrow. For any A, B ⊆ (N ), and b1 ≤ b2 ∈ R, we write (A, b1 )(B, b2 ) in GRN , if there exists at least one path in GRN from (x, b1 ) to (y, b2 ), for some x ∈ A and y ∈ B. See Fig. 1 for an illustration. Then, for any b > 0, the range set ξN (b) of a b-avalanche with origin at 0 is the collection of all x ∈ (N ) such that ({−1, 0, 1}, 0)(x, b) in GRN . For any b ≥ 0 ˆ and (k, j ) ∈ (N ) × N, we call the pair ξN (τk,j ), ηˆ N (τk,j ) , a subavalanche in GRN , if ({−1, 0, 1}, 0)(k, τk,j ) in GRN , and a b-subavalanche in GRN , if, in addition, τk,j ≤ b. Then, for any b > 0, the duration ηN (b) of a b-avalanche with origin at 0, is one plus the total duration of all b-subavalanches in GRN . The graphical representation provides us with the following monotonicity proper(A,b) ties. For any A ⊂ (N ) and b, s ≥ 0 we denote by ξN (s) the collection of all (A,b) x ∈ (N ) such that (A, b)(x, b + s) in GRN , and we denote by ηN (s) the sum of ηˆ N (τk,j ) over all b < τk,j ≤ b + s, such that (A, b)(k, τk,j ) in GRN . Then for any A ⊆ B ⊆ (N ), 0 ≤ s1 ≤ s2 , and b ≥ 0, (A,b)

ξN

(B,b)

(s1 ) ⊆ ξN

(A,b) ηN (s1 )

≤

(s2 ),

(B,b) ηN (s2 ).

(4.1)

In the particular cases of the range set ξN (x, b) = ξ ({x−1,x,x+1},0) (b) and the duration ηN (x, b) = η({x−1,x,x+1},0) (b) of an avalanche from threshold b and with origin at x, (4.1) gives us


ξN (x, b1 ) ⊆ ξN (x, b2 ), ηN (x, b1 ) ≤ ηN (x, b2 ),

if b1 ≤ b2 .

71

(4.2)

Proof of Lemma 3.1. It follows from (4.2) that for any b1 ≤ b2 , E(|ξN (b1 )|) ≤ E(|ξN (b2 )|), E(ηN (b1 )) ≤ E(ηN (b2 )), E(1{|ξN (b1 )| = N}) ≤ E(1{|ξN (b2 )| = N }).

We define GR, a graphical representation for the process on Z, in almost the same way as GRN ; the only thing we need to take care of are the infinite avalanches. We do p this by restricting GR to the fitnesses less than bc , i.e. GR is a random graph on the p space-fitness diagram Z × [0, bc ). We do the formal definition of GR and of the related things just without subscript N . So, let (Z) = {k }k∈Z be the collection of independent p homogeneous Poisson processes restricted to the interval [0, bc ). For any τk,j ∈ k , we denote by ξˆ (τk,j ), η(τ ˆ k,j ) , the (range set, duration) of the τk,j -avalanche. The notions of (A, b1 )(B, b2 ) in GR, subavalanche in GR and b-subavalanche in GR p are defined the same way. Again, for any b < bc , the range set ξ(b) of a b-avalanche with origin at 0 is the collection of all x ∈ Z such that ({−1, 0, 1}, 0)(x, b) in GR, and the duration η(b) of a b-avalanche with origin at 0, is one plus the total duration of all b-subavalanches in GR. The graphical representation GR provides us the same monotonicity properties as (4.1) for GRN . Thus, in particular, the range set ξ(b) and the duration η(b) are monotone in b. 5. Proof of Theorem 3.2 N , for avaIn this section we define a slightly different graphical representation, GR lanches on (N ). The reason to do this is that we want to couple avalanches on (N ) with avalanches on Z, and our current graphical representation GRN is not so suitable for avalanches on Z, for this purpose. We will also need a graphical representation GR see below. N is the specification of the Poisson The main difference between GRN and GR processes (N). In GR N we specify these during the construction of s-avalanches, by selecting the required number of Poisson processes from an infinite sequence of Poisson ˆ ˆ processes (∞). An avalanche of range k uses the first k processes of (∞). We use the ˆ same sequence (∞) for the construction of avalanches on (N ), for any N ≥ 3, and for avalanches on Z. This gives a transparent coupling of avalanches on (N ) and Z. ˆ We define the new graphical representation as follows. Let (∞) = 1 , 2 , . . . be a sequence of independent homogeneous Poisson processes. We will use the first N ˆ processes of (∞) for the new graphical representation of s-avalanches on (N ), with origin at 0. We define ξN (0) = {−1, 0, 1}, ηN (0) = 1, and associate with ξN (0) the first three Poisson processes. We write this as N,−1 = 1 , N,0 = 2 and N,1 = 3 , where N,k denotes the process associated with vertex k ∈ (N ). Let τ1N be the first arrival

72


in the superposition of N,k , k ∈ {−1, 0, 1}, and let κ1N be the position of the corresponding process: τ1N ∈ κ N . Then we have ξN (s) = {−1, 0, 1}, ηN (s) = 1, for 1 0 < s < τ1N . At s = τ1N we draw independently a pair ξˆN (τ1N ), ηˆ N (τ1N ) , distributed N as the (range set, duration) of a typical τ1 -avalanche with origin at 0. Then the pair κ1N + ξˆN (τ1N ), ηˆ N (τ1N ) is distributed as the (range set, duration) of a typical τ1N N N avalanche, with origin at κ1N .As in GRN , we draw arrows in (N )×R+ from (κ1 , τ1 ) to (y, τ N ), for all y ∈ κ N + ξˆN (τ N ). We define ξN (τ N ) = {−1, 0, 1} ∪ κ N + ξˆN (τ N ) 1

1

1

1

1

1

and ηN (τ1N ) = 1 + ηˆ N (τ1N ). If ξN (τ1N ) is larger than {−1, 0, 1} we associate with ξN (τ1N ) \ {−1, 0, 1} the next rN (τ1N ) − rN (0) Poisson processes from π(∞), and assign them to (N,k )k∈ξN (τ N )\{−1,0,1} . In this case we first add the required number of pro1 cesses to the left of {−1, 0, 1} and then to the right of {−1, 0, 1}. N We define τ2 as the first arrival after τ1N in the superposition of N,k , k ∈ ξN (τ1N ), and continue in the obvious way, each time adding a certain (random) number of vertices and the corresponding Poisson process to our collection, until we have a collection of size at least N. The first time that we have N vertices, these N vertices form (N ) and the corresponding Poisson processes define (N ). It is clear that this happens (with probability one) after a finite number of steps, and this means that we have defined (ξN (s), ηN (s)) for all s ≥ 0. We denote the collection of arrivals below threshold b in the newly formed (N ) by T (N ). for an avalanche on Z. We proceed We now define the graphical representation GR, N ; the only thing we need to take care of are the infinite avalanches. We do as for GR to the fitnesses less than bcp , i.e. GR is a random graph on the this by restricting GR p ˆ space-fitness diagram Z × [0, bc ). We again use the sequence (∞) to specify the required number of the Poisson processes. We denote by T the collection of arrivals of all specified Poisson processes in GR. The coupling just described will be used in the second part of the proof of Theorem 3.2 below. Proof of Lemma 3.2. Fix b > 0. We distinguish between two cases, P∞ (b) > 0 and P∞ (b) = 0. First, suppose that P∞ (b) > 0. In this case we have R∞ (b) = ∞ and D∞ (b) = ∞. We will couple a b-avalanche on Z, with origin at 0, with a sequence of b-avalanches on (N), N ≥ 3, with origin at 0, in such a way that |ξ(b)| = ∞ implies rN (b) = N, for all N ≥ 3. |ξ(b)| < ∞ implies rN (b) = r(b), for N large enough.

(5.1)

Then the theorem follows from(5.1), Fatou’s lemma and the relationηN (b) ≥ rN (b)−2. (1) (2) (3) (i) be a sequence of triples, where Gj are indepenLet G = Gj , Gj , Gj i∈N dent and exponentially distributed with parameter 1. We can use the sequence G as a sequence of updates to define a b-avalanche on Z. Indeed, consider at the initial moment an arbitrary configuration of fitnesses above threshold b, and replace the fitnesses of (1) (2) (3) {−1, 0, 1} by G1 , G1 , G1 . If within {−1, 0, 1} there are vertices with fitnesses below the threshold b, we choose fitness, y say, and replace the the one with minimal (1) (2) (3) fitnesses in {y − 1, y, y + 1} by G2 , G2 , G2 , and so on. We can also, at the same


73

time, use the sequence G to define a b-avalanche on (N ). Since the avalanche on (N ) is identical to the avalanche on Z until it spans the whole system, we have (5.1). Next, suppose that P∞ (b) = 0. The above coupling via the sequence G still gives us the first two lines of the theorem, but not the third line, because we might have ηN (b) > η(b), if r(b) ≥ N . We use the more complicated coupling described before this proof. Since P∞ (b) = 0, we can construct (ξ(s), η(s)), s ≤ b with the graphical representa restricted to the thresholds [0, b]. Then we will couple this GR with a sequence tion GR of GR N , N ≥ 3, restricted to the thresholds [0, b], in such a way that (ξN (s), ηN (s)), N satisfies, for any s ≤ b, s ≤ b, corresponding to GR ξN (s) = ξ(s) ( mod N ), ηN (s) = η(s), on |ξ(s)| < N, rN (s) = N, ηN (s) ≤ η(s), on |ξ(s)| ≥ N.

(5.2)

The theorem then follows from Fatou’s lemma, because for all N big enough, we have |ξ(b)| < N. N requires the specification of (∞) ˆ A construction of the above GR and the b-sub ˆ avalanches. We will borrow (∞) from GR and then couple the b-subavalanches with these on Z in a proper way. the Poisson processes −1 , 0 Suppose first that η(b) = 1. This means that in GR and 1 have no arrivals at the time interval [0, b], and hence ξ(b) = {−1, 0, 1}. Since we N we also have that in GR N the Poisson ˆ use the same (∞) for the construction of GR processes N,−1 , N,0 and N,1 have no arrivals at the time interval [0, b], and hence ξN (b) = {−1, 0, 1}, ηN (b) = 1 and we have constructed the coupling satisfying (5.2), on the event η(b) = 1. Suppose now in an inductive fashion that for some n ≥ 1 we can construct the cou N , for all pling satisfying (5.2), for all N ≥ 3, on η(b) ≤ n. We now construct GR N ≥ 3, on η(b) ≤ n + 1. has duration at most n + 1 then all b-subavalanches on GR If a b-avalanche on GR have duration at most n, and we can couple them by induction, i.e. for any τi ∈ T and any N ≥ 3 we can define (ξˆN (τi ), ηN (τi )) satisfying: ξˆN (τi ) = ξˆ (τi ) ( mod N ), ηˆ N (τi ) = η(τ ˆ i ), on |ξˆ (τi )| < N, ˆ i ), on |ξˆ (τi )| ≥ N. |ξˆ (τi )| = N, ηˆ N (τi ) ≤ η(τ

(5.3)

If |ξ(b)| < N, then |ξˆ (τi )| < N, for every τi ∈ T and due to the first line of (5.3) the N and GR are identical and we have the first line in (5.2). b-avalanches on GR If |ξ(b)| ≥ N, we define τ (N, b) as the first arrival in T such that |ξ(τ (N, b))| ≥ N in For any s ∈ [0, τ (N, b)), we have |ξ(s)| < N, in GR and, by the same reasoning GR. as above, the first line in (5.2). Since τ (N, b) ∈ T (N ) we have, by the second line N uses the first N Poisof (5.3), that |ξN (τ (N, b)| = N . Hence, for any t ∈ (s, b], GR uses at least these first N processes. Thus any arrival son processes of π(∞), while GR in T (N ) is in T , and we have the second line of (5.2).

6. Proof of Theorem 3.3 via Differential Equations for DN (b) and RN (b) Theorem 6.1. For any N ≥ 3DN (b) is differentiable with respect to b, and d DN (b) = DN (b)RN (b). db

74


The proof of the theorem is a simple corollary of the following two lemmas. Lemma 6.2. For any b > 0 we have DN (b+ε)−DN (b) ε

≥ DN (b)RN (b) 1 + o(1) , as ε → 0.

(6.1)

Lemma 6.3. For any b > 0 we have DN (b+ε)−DN (b) ε

RN (b) ≤ DN (b) 1−εR 1 + o(1) , as ε → 0. N (b)

(6.2)

Proof of Theorem 3.3. Suppose that R∞ (b) < ∞. Since RN (b) ≤ R∞ (b), and DN (0) = 1, for any N ≥ 3, Theorem 6.1 gives us DN (b) ≤ exp(RN (b)b) ≤ exp(R∞ (b)b), and hence D∞ (b) < ∞. Thus R∞ (b) < ∞ implies D∞ (b) < ∞, that is, bcd = bcr . To obtain (3.2), one can simply take the limit N → ∞ in (6.2) and (6.1), and then the limit ε → ∞.

It remains to prove Lemma 6.2 and Lemma 6.3. The ideas involved are actually quite simple, but it requires some care to carry these out. We need to estimate DN (b + ε) − DN (b) for small ε. For ε small, it is unlikely that more than one subavalanche occurs at any vertex in the range set at threshold b. Hence this difference is roughly DN (τ ) times the probability that such a subavalanche occurs, where b ≤ τ ≤ b + ε. This last probability is about εRN (b), and then the result follows after division by ε and taking the limit of ε → 0. Now this intuitive estimate is in terms of expectations; a certain amount of care is required to estimate the difference pointwise from above and below. This analysis is carried out in the following two proofs. Proof of Lemma 6.2. It follows from the properties of GRN that DN (b) = E ηN (b) = E

ηˆ N (τx,i ) .

x∈(N),τx,i ∈N,x ∩(0,b]

s.t. ({−1,0,1},0)(x,τx,i ) in GRN Hence

DN (b + ε) − DN (b) = E =E

ηˆ N (τx,i )

x∈(N),τx,i ∈N,x ∩[b,b+ε) s.t. ({−1,0,1},0)(x,τx,i ) in GRN

(6.3)

ηˆ N (τx,i ) .

x∈(N),τx,i ∈N,x ∩[b,b+ε)

s.t. (ξN (b),b)(x,τx,i ) in GRN

For every y ∈ (N ), define τ (y) as the first arrival in N,y after time b. Since, for any y ∈ ξN (b), we have always (ξN (b), b)(y, τ (y)) in GRN , (6.3) is at least E ηˆ N (τ (y)) = E sN (y, b) , (6.4) y∈ξN (b) s.t. τ (y) 0 such that 0 < ε < (RN (b))−1 , the above inequality is equivalent to (6.6) DN (b + ε) − DN (b) ≤ RN (b)DN (b + ε) ε + o(ε) . To prove (6.6) we again use the decomposition (6.3), DN (b + ε) − DN (b) = E

ηˆ N (τx,i ) .

x∈(N),τx,i ∈N,x ∩[b,b+ε)

s.t. (ξN (b),b)(x,τx,i ) in GRN

For every y ∈ (N ), define τ (y) as the first arrival in N,y after time b. Observe that if (ξN (b), b)(x, τx,i ) in GRN , for some x ∈ (N ), τx,i ∈ N,x ∩[b, b+ε), then there exists at least one y ∈ ξN (b) such that τ (y) < b+ε, and (ξN (b), b)(y, τ (y))(x, τx,j ) in GRN . See Fig. 2 for an illustration. Hence we can continue the estimate as ηˆ N (τ (y)) + ηˆ N (τx,j ) , ≤E y∈ξN (b),τ (y) 0, uniformly on If b > b > N ≥ 3. This gives the following corollary of Lemma 7.2. p bc

Corollary 1. For any b > bc we have p

lim inf lim inf GN (n, b ) = 1. N→∞ n→∞

78

R. Meester, D. Znamenski p

Proof of Theorem 3.4. Suppose bcd = bcr = bc and equal to bc . Then, by Lemma 7.1 and Corollary 1, for any b > bc , lim inf lim inf GN (n, b ) = 1, N→∞ n→∞

and for any b < bc ,

lim sup lim sup GN (n, b ) = 0. N→∞

n→∞

Hence, by Lemma 2.1, the limit distribution in the BS-process exists and is equal to the product of exponential distributions above bc .

Note that the condition of Lemma 7.2 is weaker than that of Corollary 1, and to have the statement of Theorem 3.4 it would be sufficient to prove that for any b > b > bcr , lim N PN (b)DN (b )/DN (b) = ∞.

N→∞

It remains to prove Lemma 7.1 and Lemma 7.2. The strategy for Lemma 7.1 is that the mean duration of b-avalanches is uniformly bounded in N . At the same time, if the range of such an avalanche is k say, then during any moment of the avalanche, the probability that the locking threshold at a given vertex is below b is at most k/N . Hence when N grows, only a few vertices will have locking thresholds below b. For Lemma 7.2, we use a decomposition of avalanches, and show that the locking thresholds are high in precisely those parts of the decomposition which have high probability in large systems. Proof of Lemma 7.1. Let PN (b, k) denote the probability that a b-avalanche has range k. Let DN (b|k) denote the mean duration of a b-avalanche, given the avalanche has range k. Decompose the Bak-Sneppen process into the sequence of b-avalanches. Since the b-avalanches have i.i.d. (range, duration), the probability that at time n we are in a b-avalanche of range k converges, as n → ∞, to DN (b|k)PN (b, k) N

DN (b|l)PN (b, l)

=

DN (b|k)PN (b, k) . DN (b)

l=3

For any k ≥ 3, given that we are in a b-avalanche of range k, we have at most k locking thresholds below b. Thus for any N ≥ 3 and n ≥ 3, we have

lim GN,0 (n, b) = lim P (YN,0 (n) ≤ b) ≤

n→∞

N k DN (b|k)PN (b, k) N k=3

n→∞

DN (b)

.

(7.1)

It follows from the coupling of the avalanches on (N ) and Z, introduced in the proof of Theorem 3.2, that for k < N, the values of DN (b|k) and PN (b, k) are independent of N, DN (b|k) = D(b|k) = E η(b) |ξ(b)| = k , PN (b, k) = P (b, k) = P (|ξ(b)| = k).


Hence DN (b) = DN (b|N )PN (b, N ) +

N

79

D(b|k)P (b, k).

k=3

Since, for any b < bcd , D∞ (b) < ∞, we have, according to Lemma 3.2, lim DN (b) = E(η(b)) =

N→∞

∞

D(b|k)P (b, k) < ∞.

k=3

Hence for any ε > 0, there exists K(ε) ≥ 3 such that for any N > K(ε), N

DN (b|k)PN (b, k)

k=K(ε)+1 N−1

≤

D(b|k)P (b, k) + DN (b|N )PN (b, N ) < ε.

k=K(ε)+1

Hence the r.h.s. of (7.1) is at most K(ε) k=3

K(ε) DN (b|k)PN (b, k) + N N

≤

N

DN (b|k)PN (b, k)

k=K(ε)+1

DN (b|k)PN (b, k)

k=3 K(ε) D (b) + ε N N

≤ 2ε,

DN (b) for N > K(ε)/ε.

(7.2)

Combining (7.1) and (7.2) we have lim sup lim GN,0 (n, b) ≤ 2ε. N→∞ n→∞

Since ε > 0 was arbitrary, we have the lemma.

Proof of Lemma 7.2. We will modify the proof of Theorem 1.1 in [15]. Let 0 < b < b < ∞ be fixed and satisfy the condition of the lemma. Fix some arbitrary b > b . For any N ≥ 3 consider a BS-process on (N ), such that at the initial time all the fitnesses are i.i.d. and exponentially distributed. Define a sequence (τj,A , τj,R )j ∈N of stopping times, with respect to the natural filtration, as follows: τ0,A = τ0,R and they are equal to the first moment that all the locking thresholds are above b. For any j ∈ N, τj +1,A is the end of the first b-avalanche of range N after τj,R , and τj +1,R is the first moment after time τj +1,A such that all the fitnesses are above threshold b . See Fig. 4 for an illustration. For any j ∈ N, we call the time interval IjN (b, b ) = [τj,R , τj +1,R ) the j th period, and within the period IjN (b, b ) we distinguish between the avalanche N (b) = [τ N part Ij,A j,R , τj +1,A ), and the recovery part Ij,R (b, b ) = [τj +1,A , τj +1,R ). Observe that the recovery part can be empty, if at time τj +1,A the minimal fitness is

80


N (b) = Fig. 4. IjN (b, b ) = [τj,R , τj +1,R ) is the j th period. It consists of the avalanche part Ij,A N [τj,R , τj +1,A ), and the recovery part Ij,R (b, b ) = [τj +1,A , τj +1,R )

larger than b . For any time n, we denote by j (n) the number of the period containing n, i.e. by definition n ∈ IjN(n) (b, b ). Suppose n is in the recovery part of its period, i.e. n ∈ IjN(n),R (b, b ). Then, we claim that at time n, all the locking thresholds are below b . Indeed, during the b-avalanche of range N at the end of IjN(n)+1,A (b) every locking threshold has been updated by a new value, below b. Hence at time τj (n)+1,A − 1, the end of this b-avalanche, all the locking thresholds are below b < b , and at time interval [τj (n)+1,A , n], we assign to the locking thresholds only values below b . Hence at any n ∈ IjN(n),R (b, b ), all the locking thresholds are below b . Therefore we have GN (n, b ) = P YN,0 (n) < b ≥ P for all x ∈ (N ), YN,x (n) < b ≥ P n ∈ IjN(n),R (b, b ) . Thus to prove the lemma it suffices to show that lim inf lim inf P n ∈ IjN(n),R (b, b ) = 1. N→∞ n→∞

It is clear that at every τj,R the fitnesses are i.i.d. and exponentially distributed above the threshold b , and at every τj +1,A the fitnesses are distributed i.i.d. and exponentially N (b)| above the threshold b. Thus the sequences of lengths |Ij,A

j ∈N

N (b, b )| and |Ij,R

j ∈N

N (b)| and are independent, and each consists of i.i.d. random variables. Since both |Ij,A N |Ij,R (b, b )| have non-lattice distributions, in the stationary regime with N vertices we can write (using standard alternating renewal process theory), N (b,b )| E |I0,R . lim P n ∈ IjN(n),R (b, b ) = (7.3) n→∞

N (b)| +E |I N (b,b )| E |I0,A 0,R


81

N (b)| can be decomposed into two parts: the duraThe duration of the avalanche part |I0,A tion of the b-avalanche of range N , and the waiting time until this avalanche. We denote by WN a typical waiting time before the b-avalanche of range N , and by AN the duration of this avalanche. We already found in [15] that

E(WN ) =

1 − 1 E ηN (b) ξN (b) < N . PN (b)

Thus N (b)| = E(W ) + E(A ) E |I0,A N N =

1 PN (b)

− 1 E ηN (b) ξN (b) < N + E ηN (b) ξN (b) = N

= DN (b)/PN (b).

(7.4)

Furthermore ((N),b) N (b, b )| = E ηN (b − b) E |I0,R ((N),b ) ≥ E ηN (b − b ) ≥ (b − b + o(b − b ))N D(b ),

(7.5)

where the last inequality is obtained in the same way as (6.5) in the proof of Lemma 6.2. Equation (7.4) together with inequality (7.5) shows that the r.h.s. of (7.3) is at least (b − b + o(b − b ))N D(b ) , DN (b)/PN (b) + (b − b + o(b − b ))N D(b ) and the last expression tends to 1, as N tends to infinity, according to the condition of our lemma.

8. The Upper Bound for bcr For any b ≥ 0, let (b) denote the leftmost vertex of ξ(b), i.e. (b) = min k ∈ Z : k ∈ ξ(b) . Let L∞ (b) denote the expectation of (b). It is clear that L∞ (b) is decreasing in b, and it becomes −∞ at the same point as the function R∞ (b). Lemma 8.1. If R∞ (b) < ∞, we have ∞ (b) ≤ − 21 L2∞ (b) + 21 L∞ (b). lim sup L∞ (b+ε)−L ε

ε→0

(8.1)

82


Proof of Theorem 3.5. By definition we have, (0) = −1, and hence L∞ (0) = −1. Thus L∞ (b) decays at least as fast as the solution of 2 y (b) = − y 2(b) + y(b) 2 y(0) = −1. The above differential equation can be solved analytically: y(b) =

−1 2e−b/2

−1

,

and the solution blows up at b = 2 log 2. Thus bcr < 2 log 2.

Proof of Lemma 8.1. Fix any b > 0, such that R∞ (b) < ∞. Since L∞ (·) and R∞ (·) have the same critical thresholds, we automatically have L∞ (b) > −∞. Consider the range set as a function of threshold b ≤ b on GR. Fix ε > 0. We will estimate the mean of the following increment (b, ε) = (b + ε) − (b). Let x(b), τ (b) be the position and the moment of the first arrival in the superposition of the Poisson processes x , x∈[(b),0]

of GR after time (threshold) b. It follows from the definition of x(b) and τ (b) that P (τ (b) < b + ε | (b) = l) = −(l + 1)(ε + o(ε)), as ε → 0, P (x(b) = x | (b) = l) =

−1 l+1 ,

x ∈ [l, 0].

(8.2)

Moreover, x(b) and τ (b) are conditionally independent given (b). In the graphical representation GR, at the moment τ (b) we have the subavalanche ξˆ (τ (b)), η(τ (b)) . If τ (b) < b + ε we have (b, ε) ≤ min m(b) + x(b), (b) − (b), (8.3) where m(b) is the leftmost point of ξˆ (τ (b)), i.e. m(b) = min k : k ∈ ξˆ (τ (b)) . Observe that the conditional distribution of m(b) given (t) depends on (b), but only through the value of τ (b). Since any τ (b)-avalanche contains a b-avalanche, we can ˆ couple m(b) with (b), a random variable independent of τ (b) and (b), and distributed ˆ as the leftmost point of a b-avalanche with origin at 0, and therefore E((b)) = L∞ (b). So the r.h.s. of (8.3) is at most ˆ + x(b), (b) − (b). min (b) (8.4)


83

ˆ and (b) have finite expectations, but we have no information The random variables (b) about their second moments. To deal with this, we use the truncations aM (b) = max (b), −M , ˆ −M , M ≥ 1. aˆ M (b) = max (b), Since,

lim aM (b) = (b),

M→∞

ˆ lim aˆ M (b) = (b),

M→∞

we also have by Fatou’s lemma lim E(aM (b)) = E((b)) = L∞ (b),

M→∞

(8.5)

ˆ = L∞ (b). lim E(aˆ M (b)) = E((b))

M→∞

The expression in (8.4) is at most ≤ min aˆ M (b) + x(b), (b) − (b) 1 x(b) ≤ (b) − aM (b) =

aˆ M (b) + x − (b) 1 x(b) = x

(b)−max(a M (b),aˆ M (b)) x=(b)

=

− max(aM (b),aˆ M (b))

j + aˆ M (b) 1 x(b) = (b) + j .

(8.6)

j =0

Combining the above estimates with (8.3) and taking expectations on both sides, we get


E( (b, ε)) ≤ E 1{τ (b) < b + ε}

j + aˆ M (b) 1{x(b) = (b) + j }

j =0

=

−1

− max(aM (b),aˆ M (b)) j + aˆ M (b) E 1{τ (b) < b + ε} j =0

l=−∞

×1{x(b) = (b) + j } (b) = l P ((b) = l) .

(8.7) Since τ (b) and x(b) are conditionally independent given (b), and since aˆ M (b) and (b) are independent, the r.h.s. is equal to −1 l=−∞

(b),aˆ M (b)) − max(a M P τ (b) < b + ε (b) = l E j + aˆ M (b) (b) = l j =0

×P x(b) = (b) + j (b) = l) P ((b) = l) .

(8.8)

84


After substituting (8.2) in the above expression, we get

−1

E (ε + o(ε))

− max(aM (b),aˆ M (b)) j =0

l=−∞

=E


aˆ M (b) + j

aˆ M (b) + j (b) = l P ((b) = l)

(ε + o(ε))

j =0

aˆ M (b)(aˆ M (b) − 1) =E − 1{aˆ M (b) > aM (b)} (ε + o(ε)) 2 aM (b)(aM (b) − 1) +E −(aM (b) − 1)aˆ M (b) + 1{aˆ M (b) ≤ aM (b)} (ε + o(ε)) 2 =E

aˆ M (b)(aˆ M (b) − 1) aM (b)(aM (b) − 1) − + (aM (b) − 1)aˆ M (b) − 2 2

×1{aˆ M (b) > aM (b)} (ε + o(ε)) aM (b)(aM (b) − 1) +E −(aM (b) − 1)aˆ M (b) + (ε + o(ε)) 2 =E

−

(aˆ M (b) − aM (b))2 aˆ M (b) − aM (b)) − 1{aˆ M (b) > aM (b)} (ε + o(ε)) 2 2

(E(aM (b)))2 E(aM (b)) + −E(aM (b))E(aˆ M (b)) + E(aˆ M (b)) + − (ε + o(ε)) 2 2 ≤ E

M (b)) − (aˆ M (b)−a 2

2

1{aˆ M (b) > aM (b)} (ε + o(ε))

+ −E(aM (b))E(aˆ M (b)) + E(aˆ M (b)) +

(E(aM (b)))2 2

−

E(aM (b)) 2

(ε + o(ε)).

(8.9) Since aM (b) and aˆ M (b) are i.i.d., the difference aM (b) − aˆ M (b) has a symmetric distribution. Hence 2 2 1 E aM (b) − aˆ M (b) 1{aM (b) > aˆ M (b)} = E aM (b) − aˆ M (b) , 2


85

and we can continue (8.9) as (aM (b) − aˆ M (b))2 (aM (b))2 aM (b) = −E − (EaM (b))2 + E +E 4 2 2 (EaM (b))2 EaM (b) = − + (ε + o(ε)). 2 2 Substituting the above estimates in (8.7) results in −(Ea (b))2 EaM (b) M E N (b, ε) ≤ + (ε + o(ε)). 2 2 The lemma is now straightforward by (8.5). Acknowledgement. We would like to thank Natalia Davydova for helpful discussions.

References 1. Bak, P.: How Nature Works. New-York: Springer-Verlag, 1996 2. Bak, P., Sneppen, K.: Punctuated equilibrium and criticality in a simple model of evolution. Phys. Rev. Let. 74, 4083–4086 (1993) 3. Jensen, H.J.: Self-Organized Criticality. Cambridge: Cambridge Lecture Notes in Physics, 1998 4. Grassberger, P.: The Bak-Sneppen model for punctuated evolution. Phys. Lett. A 200(3), 277–282 (1995) 5. Maslov, S.: Infinite hierarchy of exact equations in the Bak-Sneppen model. Phys. Rev. Lett. 77, 1182–1186 (1996) 6. Moreno, Y., Vazquez: The Bak-Sneppen model on scale-free networks. Europhys. Lett. 57, 765–771 (2002) 7. Cafiero, R., De los Rios, P., Valleriani, A., Vega, J.L.: Levy-nearest-neighbors Bak-Sneppen model. Phys. Rev. E.60 (2 Part A), R1111-R1114 (1999) 8. De Los Rios, P., Marsili, M., Vendruscolo, M.: High dimensional Bak-Sneppen model. Phys. Rev. Lett. 80(26), 5746–5749 (2001) 9. Pis‘mak, Yu.M.: Self-organized criticality in simple model of evolution: exact description of scaling laws. Acta Physica Slovatica 52(6), 525–532 (2002) 10. de Boer, J., Derrida, B., Flyvbjerg, H., Jackson, A.D., Wettig, T.: Simple model of self-organized biological evolution. Phys. Rev. Lett. 73, 906–909 (1994) 11. Marsili, M., De Los Rios, P., Maslov, S.: Expansion around the mean-field solution of the Bak-Sneppen model. Phys. Rev. Lett. 80(7), 1457–1460 (1998) 12. Barbay, J., Kenyon, C.: On the discrete Bak-Sneppen model of self-organized criticality. In: Proceedings of the Twelth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Washington, DC, January 2001 13. Meester, R., Znamenski, D.: Non-triviality of a discrete Bak-Sneppen evolution model. J. Stat. Phys. 109(516), 987–1004 (2002) 14. Jovanovic, B., Buldyrev, S.V., Havlin, S., Stanley, H.E.: Punctuated Equilibrium and ‘History Dependent’ percolation. Phys. Rev. E 50, R2403-R2406 (1994) 15. Meester, R., Znamenski, D.: On the limit behaviour of the Bak-Sneppen evolution model. To appear in Annals of Probability (2002) 16. Bose, I., Chaudhuri, I.: Bacterial evolution and the Bak-Sneppen model. Int. J. Mod. Phy. C 12(5), 675–683 (2001) 17. Kovalev, O.V., Pis’mak, Yu. M., Vechernin, V.V.: Self-organized criticality in the model of biological evolution describing interaction of “Coenophilous” and “Coenophobous” species. Europhys. Lett. 40, 471–476 (1997) 18. Donangelo, R., Fort, H.: A model for mutation in bacterial populations. To appear in Phys. Rev. Lett. (2002)

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19. Cuniberti,G., Valleriani, A., Vega, J. L.: Effects of regulation on a self-organized market. Quantitative Finance 1, 332–338 (2001) 20. Yamano, T.: Regulation effects on market with Bak-Sneppen model in high dimensions. Int. J. Mod. Phys. C 12(9), 1329–1333 (2001) Communicated by H. Spohn


Communications in


Some Stability and Instability Criteria for Ideal Plane Flows Zhiwu Lin Department of Mathematics, Brown University, Providence, RI 02912, USA Received: 9 May 2003 / Accepted: 18 September 2003 Published online: 13 February 2004 – © Springer-Verlag 2004

Abstract: We investigate stability and instability of steady ideal plane flows for an arbitrary bounded domain. First, we obtain some general criteria for linear and nonlinear stability. Second, we find a sufficient condition for the existence of a growing mode to the linearized equation. Third, we construct a steady flow which is nonlinearly and linearly stable in the L2 norm of vorticity but linearly unstable in the L2 norm of velocity. 1. Introduction We consider an incompressible inviscid flow satisfying the Euler equation ∂t u + (u · ∇u) + ∇p = 0,

(1a)

∇ · u = 0,

(1b)

in a bounded domain ⊂ R2 with smooth boundary ∂ composed of a finite number of connected components i . The boundary condition is u·n=0

on ∂,

where n stands for the unit outer normal of ∂. The vorticity form of (1) is given by (2) ∂t ω − ψy ∂x ω + ψx ∂y ω = 0, where ψ is the stream function, and ω ≡ −ψ = − ∂x2 + ∂y2 ψ is the vorticity. The boundary conditions associated with (2) are given by ψ|i = i ,

(3a)

88

Z. Lin

and i

∂ψ = Ai , ∂n

(3b)

with i depending on time only, and Ai being some constants. A steady flow satisfying (2), (3) has a stream function ψ0 satisfying −ψ0y ∂x ω0 + ψ0x ∂y ω0 = 0,

(4)

where ω0 ≡ −ψ0 is the associated vorticity. Consider ψ0 satisfying the following elliptic equation: −ψ = g (ψ)

(5)

with boundary conditions (3) and g being some differentiable function. Then ω0 ≡ −ψ0 = g (ψ0 ) is a steady solution of (2). In this paper we study stability and instability of these steady solutions. If g > 0, Wolansky and Ghil [17] derived some linear and nonlinear stability criteria, using the energy-Casimir method and a supporting functional method. However their conditions involve some unspecified finite dimensional function spaces and are not easy to check. Our first theorem is a refinement of their results. We state the theorem only for a simply connected domain. For this case, the boundary conditions (3) can be simplified to be ψ = 0 on ∂. First we introduce some notations. We call a real number ρ a critical value of ψ0 if ψ0 takes the value ρ at a critical point. The set of all critical values of ψ0 has zero measure by Sard’s Theorem. For any ρ which is not a critical value, the level set {ψ0 = ρ} consists of a finite number of disjoint closed curves, which we denote by 1 (ρ) , 2 (ρ) , · · · , nρ (ρ). Let X = H01 () ∩ H2 () and Y = H10 () , with ψ2X =

|ψ|2 dxdy, ψ2Y =

|∇ψ|2 dxdy.

Note that ψ2X , ψ2Y are the enstrophy and energy of the flow with the stream function ψ. The linearized equation of (2) around the steady state (ψ0 , ω0 ) is ∂t ω˜ − ψ0y ∂x ω˜ + ψ0x ∂y ω˜ = ψ˜ y ∂x ω0 − ψ˜ x ∂y ω0 , with ω˜ = −ψ˜ and ψ˜ = 0 on ∂. We have the following result on nonlinear and linear stability.

(6)

Some Stability and Instability Criteria for Ideal Plane Flows

89

Theorem 1.1. Suppose ω0 = g (ψ0 ) g ∈ C1 is a steady flow with g > 0. (i)We define the functional a (φ) for φ ∈ Y by

g (ψ0 ) φ 2 dxdy

|∇φ|2 dxdy −

a (φ) =

+

nρ max ψ0 i=1 i (ρ) g (ρ) nρ

2 φ |∇ψ0 |

1 i=1 i (ρ) |∇ψ0 |

min ψ0

dρ .

If inf a (φ) > 0,

(7)

φ2 =1

then the flow is nonlinearly stable in the following sense: for any ε > 0, there exists δ > 0 such that ψ (., 0) − ψ0 X < δ ⇒ sup ψ (., t) − ψ0 X < ε, t>0

where ψ (., t) is the solution to (2) with the initial state ψ (., 0) ∈ X. Condition (7) is equivalent to the operator B (defined by (14)) being positive. (ii)We define the functional b (φ) for φ ∈ Y by |∇φ|2 dxdy − b (φ) = g (ψ0 ) φ 2 dxdy

nρ max ψ0 i (ρ)

g (ρ) + min ψ0

i=1

2 φ |∇ψ0 | dρ 1 i (ρ) |∇ψ0 |

.

If inf b (φ) > 0,

φ2 =1

(8)

then the flow is linearly stable in the following sense : for any ε > 0, there exists δ > 0 such that ψ (., 0) − ψ0 X < δ ⇒ sup ψ (., t) − ψ0 X < ε, t>0

where ψ (., t) is the solution to the linearized Euler equation (6) with the initial state ψ (., 0) ∈ X. Note that we have b (φ) ≥ a (φ) for any φ ∈ Y and the equality only holds in the case when each level set {ψ0 = ρ} consists of only a single curve. Theorem 1.1(ii) also gives a criterion for spectral stability. That is, if b (φ) is positive definite then there is no exponentially growing solution to the linearized Euler equation (6). So the linearized operator has no unstable discrete eigenvalue. For shear flows and rotating flows, it was proved in [12] that this criterion is also necessary, namely if b (φ) is negative then we can find a growing mode. Now we give a new criterion for the existence of a growing mode, for steady flows satisfying (5) on a bounded domain and without the assumption that g > 0. We state the result only for the simply connected case. For

90

Z. Lin

the non-simply connected case, see Sect. 5 for some modifications. Define the operator A0 : X → L2 () as follows: A0 φ := −φ − g (ψ0 ) φ + g (ψ0 ) P˜ φ,

(9)

where P˜ is the orthogonal projection operator of L2 () onto S = ker L and L = −∂y ψ0 ∂x + ∂x ψ0 ∂y is defined on H01 (). We prove later that b (φ) = (A0 φ, φ) .

(10)

So the fact that b (φ) is negative for some φ is equivalent to the existence of a negative eigenvalue of A0 , and condition (8) is equivalent to the operator A0 being positive. Theorem 1.2. If A0 has an odd number of negative eigenvalues and no kernel, then there exists a purely growing mode eλt ω (with λ > 0 and ω ∈ X) to the linearized Euler equation (6) . It is hard to prove the existence of unstable discrete eigenvalues for the linearized Euler operator since it is degenerate and non-elliptic. Even for the simplest shear flow case, little has been known about sufficient conditions for the existence of unstable discrete eigenvalues ([2, 13]). We can modify the proof of Theorem 1.2 to get an instability criterion for the case when A0 has a nontrivial kernel. For the case when A0 has an even number of negative eigenvalues and no kernel, we cannot expect to find purely growing modes. Some new methods to find non-purely growing modes were developed in [13] for shear flows and rotating flows. λtNow λtwe sketch the main idea for the proof of Theorem 1.2. For a growing mode e ω, e ψ (with λ > 0) to the linearized Euler equation (6), (ω, φ) satisfies the following equations λω − ψ0y ∂x ω + ψ0x ∂y ω = ψy ∂x ω0 − ψx ∂y ω0 ,

(11)

ω = −ψ,

(12)

ψ = 0 on ∂. Using the strategy in [12], we represent ω in terms of ψ by integrating (11) along the fluid trajectory, then plug it into the Poisson equation (12). The resulting equation can be written as Aλ ψ = 0. The operator Aλ is the minus Laplacian plus a bounded operator. For the existence of a purely growing mode, it suffices to show that for some λ0 > 0, Aλ0 has a nontrivial kernel. The main difference from the case in [12] is that here Aλ is not self-adjoint. This makes the analysis more difficult. We use the infinite determinant method developed in [13]. We study the infinite determinant d (λ) of I d − exp(−Aλ ) as λ → 0+ and λ → ∞. It turns out that d (λ) is nonnegative when λ is sufficiently large. It can be shown that d (λ) is negative as λ → 0+ under the conditions of Theorem 1.2. These two facts imply that for some λ0 > 0 d (λ0 ) = 0, which implies that Aλ0 has a nontrivial kernel. The stability result in Theorem 1.1 is proved in the vorticity norm. This norm was also used in [1] to prove nonlinear instability from the existence of an unstable discrete eigenvalue of the linearized Euler operator. However, the stability problem in the L2 norm of velocity (energy norm) is quite different. So far there is no general method to prove nonlinear stability and instability in the energy norm. In the last part of this paper,


91

we construct a steady flow which is nonlinearly and linearly stable in the enstrophy norm || · ||X but linearly unstable in the energy norm || · ||Y . This example illustrates the importance of the norm to adopt when studying stability of incompressible inviscid flows. 2. Stability Criteria In this section, we prove Theorem 1.1. We need the following result from Wolansky and Gill [17]. We state it in the following form. Lemma 2.1 ([17]). The steady flow as in Theorem 1.1 is nonlinearly stable in X if for some integer m, the following functional em (φ) is positive definite in Y:

g (ψ0 ) φ 2 dxdy +

|∇φ|2 dxdy −

em (φ) :=

m

< ξi0 , φ >2 ,

i=1

where < ., . > is the g weighted L2 inner product, and ξ10 , · · · , ξm0 is a g weighted orthogonal basis of some m-dimensional subspace Wm of

W = {ψ ∈ Y |ψ = h (ψ0 ) , h being measurable on the range of ψ0 } . Denote P (Pm ) the orthogonal projection operators of L2 () onto W (Wm ) and define the operators B (Bm ) : X → L2 () ,

(13)

Bφ (Bm φ) = −φ − g (ψ0 ) φ + g (ψ0 ) P φ(Pm φ).

(14)

Then we readily see that em (φ) = (Bm φ, φ). Thus to show that em (φ) is positive definite, it is equivalent to show that Bm is positive. Lemma 2.2. If the operator B is positive, then for some integer m there exists a m-dimensional subspace Wm ⊂ W such that the operator Bm is positive. Proof. Let ζ1 , ζ2 , · · · be a complete orthogonal basis of L2 (), and denote Wn the space spanned by P ζ1 , P ζ2 , · · · , P ζn . Let Pn be the corresponding orthogonal projection. Then it is readily seen that Pn → P strongly in the sense that for any φ ∈ X, Pn φ → P φ strongly in L2 (). If the conclusion of the lemma is not true, then for each n ∈ N we can find λn ≤ 0 and φn 2 = 1 such that Bn φn = λn φn .

(15)

Let λn → λ0 ≤ 0. Then it is obvious that φn H2 () ≤ C (independent of n). So after taking some subsequence, we have φn → φ0 strongly in L2 () with φ0 2 = 1. Since Bn → B strongly, we have Bn φn → Bφ0 weakly. So taking the limit in (15), we have Bφ0 = λ0 φ0 which is a contradiction to the positivity of B.

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Lemma 2.3. For any regular point (x, y) ∈ of the function ψ0 , let ρ = ψ0 (x, y) nρ i (ρ). For any φ ∈ Y, we define two and (x, y) ∈ i (ρ) , where {ψ0 = ρ} = ∪i=1 2 functions φ1 , φ2 ∈ L () in the following way: nρ φ i=1 i (ρ) |∇ψ0 | 1 i=1 i (ρ) |∇ψ0 |

φ1 (x, y) = nρ and

(16)

φ2 (x, y) =

φ i (ρ) |∇ψ0 | 1 i (ρ) |∇ψ0 |

(17)

.

Then we have P φ = φ1 and P˜ φ = φ2 in the L2 sense. Here P˜ is the projection operator of L2 () onto S = ker −∂y ψ0 ∂x + ∂x ψ0 ∂y . Proof. To show P φ = φ1 , we take any ξ = h (ψ0 ) ∈ W . Then (φ − φ1 , ξ ) = (φ − φ1 ) h (ψ0 ) dxdy

max ψ0 φ − φ1 = h (ρ) dρ (by the co-area formula) {ψ0 =ρ} |∇ψ0 | min ψ0 nρ nρ max ψ0

φ 1 = h (ρ) − φ1 (ρ) |ψ0 =ρ dρ min ψ0 i (ρ) |∇ψ0 | i (ρ) |∇ψ0 | i=1

i=1

= 0. So φ − φ1 ∈ W ⊥ . Since clearly φ1 ∈ W, we have P φ = φ1 . To show that P˜ φ = φ2 , we take any η ∈ S. Then (φ − φ2 , η) = (φ − φ2 ) ηdxdy nρ max ψ0

=

min ψ0

=

i=1 nρ max ψ0

min ψ0

i=1

i (ρ)

(φ − φ2 ) η dρ |∇ψ0 |

η| i (ρ)

i (ρ)

φ − φ2 | i (ρ) |∇ψ0 |

i (ρ)

1 dρ |∇ψ0 |

(since η, φ2 take constant values on each i (ρ) ) = 0. So φ − φ2 ∈ S⊥ . Since φ2 ∈ S, we have P˜ φ = φ2 .

Now Theorem 1.1(i) follows from the above three lemmas. By Lemmas 2.2 and 2.3, if the condition (7) is satisfied, then there exists some integer m such that Bm is positive. Then by Lemma 2.1, the steady flow is nonlinearly stable. Theorem 1.1(ii) can be proved in the same way.


93

Corollary 2.1. Under assumption (8), there is no unstable discrete eigenvalue for the linearized Euler operator. Proof. This is a consequence of Theorem 1.1(ii). Here we give a direct proof of it. First we notice that if (ω, ψ) is a solution to (6), then for any ξ ∈ S = ker −∂y ψ0 ∂x + ∂x ψ0 ∂y , the following two functionals |ω|2 2 |∇ψ| − dxdy E (ω, ψ) = g (ψ0 ) Mξ (ω) = ωξ dxdy are conserved. This can be checked by a straightforward computation. If there exists a growing mode eλt ω (x, y) , eλt ψ (x, y) to (6) with Re λ > 0, then E eλt ω, eλt ψ = e2 Re λt E (ω, ψ) , Mg eλt ω = eλt Mg (ω) are independent of t. Thus it follows that E (ω, ψ) = Mξ (ω) = 0 for any ξ ∈ S. Noticing that

|∇ψ|2 dxdy =

ψω∗ dxdy,

where ω∗ is the complex conjugate of ω, we have |ω|2 ∗ 2 − 2ψω + |∇ψ| dxdy 0 = E (ω, ψ) = g (ψ0 )   2   ω − ψ g (ψ0 ) − g (ψ0 ) |ψ|2 + |∇ψ|2 dxdy =   g (ψ0 ) 2 ω − 1 − P˜ ψ g (ψ0 ) − g (ψ0 ) |ψ|2 + |∇ψ|2 = g (ψ ) 0 2 + g (ψ0 ) P˜ ψ dxdy 2 |∇ψ|2 − g (ψ0 ) |ψ|2 + g (ψ0 ) P˜ ψ dxdy. ≥

So if the last quadratic form in the above is positive, we get a contradiction. This proves the conclusion. Here for the equality in the third line above, we use the fact that

ω g (ψ0 )

∈ S⊥ .

Remark 2.1. The two conditions (7) and (8) are the same if and only if {ψ0 = ρ} consists of only one closed curve for any ρ. We shall prove that in this case the linearized Euler operator has no growing modes.

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Z. Lin

We consider the steady flow with the stream function ψ0 defined on a simply connected domain. Here ψ0 satisfies the following elliptic equation: −ψ0 = g (ψ0 ) , in ψ0 = 0, on ∂. Lemma 2.4. If {ψ0 = ρ} consists of only one closed curve for any ρ ∈ (min ψ0 , max ψ0 ), then there are no exponentially growing modes to the linearized Euler equation (6). Proof. First we define the generalized polar coordinates (r, θ) in the following way. Let l be the arc length variable on the stream line {ψ0 = ρ} and define l 2π 1 dl r = ψ0 (x, y) , , θ = v (r) . = v (r) {ψ0 =r} |∇ψ0 | 0 |∇ψ0 | Then min ψ0 ≤ r ≤ max ψ0 , 0 ≤ θ ≤ 2π . If eλt ψ (Re λ > 0) is a growing mode to (6), then ψ satisfies (see Lemma 3.1) 0 eλs ψ (X (s; x, y) , Y (s; x, y)) ds = 0. (18) −ψ − g (ψ0 ) ψ + g (ψ0 ) λ −∞

Here (X (s; x, y) , Y (s; x, y)) is the solution of the characteristic equation as defined in (22). In the polar coordinates (r, θ) , the characteristic equation becomes r˙ = 0, θ˙ = −v (r) . Let ψ (r, θ) =

+∞

ψk (r) eikθ ,

−∞

then (18) becomes −ψ − g (ψ0 ) ψ + g (ψ0 )

+∞

−∞

λ ψk (r) eikθ = 0. λ − ikv (r)

(19)

Taking the inner product of (19) with ψ ∗ , we have max ψ0 +∞

1 ikv (r) 2 |ψk (r)|2 dr = 0, |∇ψ| dxdy + g (r) a + bi − ikv (r) min ψ0 v (r) −∞ (20) where λ = a + bi (a > 0) . Taking the imaginary part of (20), we get max ψ0 +∞

1 akv (r) dr = 0. g (r) 2 2 min ψ0 v (r) −∞ a + (b − kv (r)) So a = 0 which is a contradiction. Thus there are no growing modes to the linearized Euler equation (6).


95

3. Instability Criterion We divide the proof of Theorem 1.2 into several steps. 3.1. Dispersion operators. In this part, we introduce dispersion operators Aλ and study their basic properties. Definition 3.1. The dispersion operators are a family of operators Aλ λ ∈ R+ : X → L2 (). Here 0 Aλ ψ := −ψ − g (ψ0 ) ψ + g (ψ0 ) λ eλs ψ (X (s; x, y) , Y (s; x, y)) ds, (21) −∞

where (X (s; x, y) , Y (s; x, y)) is the solution to the characteristic equation X˙ (s) = −∂y ψ0 (X (s) , Y (s)) Y˙ (s) = ∂x ψ0 (X (s) , Y (s)),

(22)

with the initial value X (0) = x, Y (0) = y. Remark 3.1. Aλ is well-defined. Denoting 0 eλs ψ (X (s; x, y) , Y (s; x, y)) ds, Kλ ψ := −g (ψ0 ) ψ + g (ψ0 ) λ

(23)

−∞

then we have

Kλ ψ2 ≤ 2 g (ψ0 )∞ ψ2 .

(24)

Indeed, for any function φ ∈ L2 () , we have 0 λs λe g (ψ0 ) φ(x, y)ψ (X (s; x, y) , Y (s; x, y)) dsdxdy −∞

≤

0

−∞

· ≤

0

0

2

λe g (ψ0 )∞

|ψ| dxds 2

λs

−∞

2

λe |φ| g (ψ0 ) dsdxdy λs

−∞

λe |ψ| g (ψ0 ) (X (s; x, y) , Y (s; x, y))dsdxdy λs

· λe g (ψ0 )∞ −∞ = g (ψ0 )∞ φ2 ψ2 . 0

21

21

|φ| dxds 2

λs

21

Thus (24) follows. Here we used the fact that the Jacobian of the mapping (x, y) → (X (s; x, y) , Y (s; x, y)) is one.

21

96

Z. Lin

The following lemma indicates the reason why we introduce Aλ . Lemma 3.1. Let λ > 0, then there exists a nontrivial solution λt e ω (x, y) , eλt ψ (x, y) to (6) with ω ∈ C 1 and ψ ∈ X, if and only if there exists some ψ ∈ X such that Aλ ψ = 0. In this case 0 ω = g (ψ0 ) ψ − g (ψ0 ) λ eλs ψ (X (s; x, y) , Y (s; x, y)) ds. (25)

eλt ω (x, y) , eλt ψ

−∞

Proof. If (x, y) is a solution to (6), then (ω, ψ) satisfies (11). We can rewrite (11) at (X (s) , Y (s)) as d λs (e ω ((X (s) , Y (s)))) = eλs ψy ∂x ω0 − ψx ∂y ω0 (X (s) , Y (s)) ds = eλs g (ψ0 ) −∂y ψ0 ψx + ∂x ψ0 ψy (X (s) , Y (s)) dψ = eλs g (ψ0 ) (X (s) , Y (s)) . ds Integrating above from −∞ to 0, we get 0 ω (x, y) = g (ψ0 ) −∞ eλs dψ ds (X (s; x, y) , Y (s; x, y)) ds 0 = g (ψ0 ) ψ − g (ψ0 ) λ −∞ eλs ψ (X (s; x, y) , Y (s; x, y)) ds. Plugging the above equality into the Poisson equation, we get 0 eλs ψ (X (s; x, y) , Y (s; x, y)) ds, −ψ = g (ψ0 ) ψ − g (ψ0 ) λ −∞

which is exactly Aλ ψ = 0. Conversely, if ω ∈ C 1 and satisfies (25), we can show that it satisfies (11) by the same argument as in [12]. In the following, we show that (ω, ψ) is a weak solution to (11). Moreover ω is differentiable almost everywhere. Lemma 3.2. Given ψ ∈ Y satisfying Aλ ψ = 0 and ω (x, y) defined by (25), then (ψ, ω) is a weak solution of (11). Moreover ψ is differentiable besides the critical set, which is the set of all points (x, y) such that ψ0 (x, y) is equal to the critical value at a saddle point. So according to Lemma 3.1 (ψ, ω) satisfies (11) almost everywhere in the classical sense. Proof. To show that (ψ, ω) is a weak solution of (11), we take any φ ∈ C01 (), then ψ0y ∂x φ − ψ0x ∂y φ ωdxdy

=−

+

= I + I I.

ψ0y ∂x φ − ψ0x ∂y φ g (ψ0 )

0 −∞

λeλs ψ(X(s),Y (s) )ds dxdy

ψ0y ∂x φ − ψ0x ∂y φ g (ψ0 ) ψ (x, y) dxdy


97

For the first term, we have 0 λs I =− λe g (ψ0 ) ψ0y ∂x φ − ψ0x ∂y φ (X(−s),Y (−s)) ψ (x, y) dxdyds −∞

0

d λeλs − φ (X(−s),Y (−s)) dsψ (x, y) dxdy ds −∞

0 = g (ψ0 ) −λφ (x, y) + λ2 eλs φ (X(−s),Y (−s)) ds ψ (x, y) dxdy −∞ g (ψ0 ) λφ (x, y) ψ (x, y) dxdy =−

g (ψ0 )

=

+

0

−∞

λ2 eλs

=λ

g (ψ0 ) φ (X(−s),Y (−s)) ψ (x, y) dxdy

−g (ψ0 ) ψ + g (ψ0 ) λ

× φ (x, y) dxdy ωφdxdy. = −λ

0 −∞

λs

e ψ (X (s; x, y) , Y (s; x, y)) ds

Here in the first and fourth equality we change the variable (x, y) → (X (s; x, y) , Y (s; x, y)) . By integration by parts ψ0y ∂x φ − ψ0x ∂y φ g (ψ0 ) ψ (x, y) dxdy II = = φ −ψ0y ∂x + ψ0x ∂y g (ψ0 ) ψ (x, y) dxdy = g (ψ0 ) −ψ0y ∂x ψ + ψ0x ∂y ψ φdxdy = ψy ∂x ω0 − ψx ∂y ω0 φdxdy.

So

ψ0y ∂x φ − ψ0x ∂y φ ωdxdy = I + I I = −λω + ψy ∂x ω0 − ψx ∂y ω0 φdxdy

which means that (ψ, ω) is a weak solution of (11). Taking the derivative ∂x on the right hand side of (25), we get the expression 0 ∂x g (ψ0 ) ψ (x, y) − ∂x g (ψ0 ) λeλs ψ (X (s) , Y (s)) ds − g (ψ0 ) ×

0

λeλs

−∞

−∞

∂X(s; x, y) ∂Y (s; x, y) ∂x ψ (X (s) , Y (s)) + ∂y ψ (X (s) , Y (s)) ds. ∂x ∂x

98

Z. Lin

If (x, y) is not in the critical set, then the fluid particle with the initial position (x, y) has a periodic trajectory. So ∂X(s;x,y) and ∂Y (s;x,y) can only have linear growth and the ∂x ∂x third term above is finite. Since the other terms are also finite, we prove that ω defined by (25) is differentiable if (x, y) is not in the critical set. Note that in [1], it was shown that if the growth rate Re λ is greater than the Liapunov exponent of the steady flow, then the growing mode ω ∈ H1 () . Next we study some properties of Aλ . Lemma 3.3. Aλ is a densely defined closed operator and for any ξ in its resolvent set ρ (Aλ ) , (ξ − Aλ )−1 is a trace class operator. The eigenvalues of Aλ appear in complex conjugate pairs and they are all discrete with finite multiplicity. Proof. Denote A = − with D (A) = X. Then clearly (ξ − A)−1 is a trace class operator for any ξ ∈ ρ (A). We have 1 (A + l)−1 ≤ l for any l > 0. By Remark 3.1, Aλ − A = Kλ are uniformly bounded operators with Kλ ≤ 2 g (ψ0 )∞ . We have Aλ + l = A + l + Kλ = 1 + Kλ (A + l)−1 (A + l) . So if 2 g (ψ0 )∞ < l, then −l ∈ ρ (Aλ ) and −1 . (Aλ + l)−1 = (A + l)−1 1 + Kλ (A + l)−1 This is the multiplication of a bounded operator with a trace class operator, so it is also in trace class. For any ξ ∈ ρ (Aλ ) , from formula (ξ − Aλ )−1 = (−l − Aλ )−1 + (ξ + l) (ξ − Aλ )−1 (−l − Aλ )−1 , we can see that (ξ − Aλ )−1 is in trace class. Now the conclusions about the eigenvalues of Aλ follow from the trace class property just proved and the fact that Aλ commutes with complex conjugation. Lemma 3.4. There exists 0 > 0 such that if λ > 0 then Aλ has no negative eigenvalues. Proof. First we show that for all ψ ∈ H01 () , Kλ ψ2 ≤

|ω0 |C 1 ∇ψ2 . λ


99

In fact, for any φ ∈ L2 (), 0 |(Kλ ψ, φ)| ≤ eλs ψy ∂x ω0 − ψx ∂y ω0 (X (s) , Y (s)) . |φ| dxdyds −∞ 0

≤ |ω0 |C 1

≤ |ω0 |C 1 = |ω0 |C 1

−∞ 0 −∞

|∇ψ| (X (s) , Y (s)) |φ| dxdyds

eλs

eλs

1 1 2 2 |∇ψ|2 (X(s) , Y (s)) dxdy |φ|2 dxdy ds

1 ∇ψ2 φ2 . λ

Suppose there exists some negative eigenvalue for Aλ , that is −ψ + Kλ ψ = kψ,

(26)

for some k < 0, 0 = ψ ∈ H01 () . By Poincaré’s inequality, ψ2 ≤ c0 ∇ψ2 for some constant c0 . So taking the inner product of (26) with φ, we have 0 > (kψ, ψ) = ∇ψ22 + (Kλ ψ, ψ) ≥ ∇ψ22 − Kλ ψ2 ψ2 c0 |ω0 |C 1 ∇ψ22 ≥ ∇ψ22 − λ > 0, which is a contradiction if λ > 0 = c0 |ω0 |C 1 .

3.2. Infinite determinant and an abstract theorem. In this part, we prove the following abstract result using the infinite determinant method developed in [13]. Theorem 3.1. Consider a continuous family of operators Aλ : H → L, Aλ = A + Bλ , λ ∈ R+ . We assume that: (I) Bλ are uniformly bounded operators and Aλ commute with complex conjugation. (II) The self-adjoint operator −A generates a generalized parabolic semigroup, that is, exp (−tA) is in the trace class and A exp (−tA) is bounded. Furthermore, the embedding i : (H, .A ) → (L, .) is compact. Here .A is the graph norm of operator A and . is the norm in L. (III) When λ is sufficiently large, Aλ has no negative eigenvalue. (IV) When λ tends to 0, Aλ tends to A0 strongly in the sense that: for any u ∈ H, Aλ u → A0 u and A∗λ u → A∗0 u strongly, as λ → 0 +

(27)

for any function u ∈ H. Then if A0 has an odd number of negative eigenvalues and no kernel, there must exist some λ0 > 0 such that Aλ0 has a nontrivial kernel.

100

Z. Lin

Note that here the condition (IV) is weaker than that in [13], where it is required that (Aλ − A0 ) φ ≤ c (λ) (Aφ + φ) ,

(28)

for some function c (λ) approaching 0 as λ → 0+ . Condition (28) can not be proved for the case of Theorem 1.2, and only (27) is available. Proof. The line of proof follows that of in [13]. So we just sketch it and indicate some differences caused by the weaker assumption (IV). Denote all the distinct eigenvalues of Aλ (arranged with non-decreasing real parts) by µ1 (λ) , µ2 (λ) , · · · , µk (λ) , · · · , with multiplicities n1 , n2 , · · · , nk , · · · . We define

d (λ) =

∞

(1 − exp (−µk (λ)))nk ,

k=1

which is the infinite determinant of the operator exp (−Aλ ) . By assumptions (I) (II), exp (−Aλ ) is a trace class operator. Since µk (λ) appears in complex conjugate pairs, d (λ) is a finite real number. From the definition of d (λ), we know that the sign of d (λ) is determined by the number of negative eigenvalues of Aλ. If this number is odd, then d (λ) is negative and d (λ) is positive if this number is even. Here we assume that Aλ has no kernel, since otherwise we have already found the growing mode. The main idea is to keep track of the sign change of d (λ), especially as λ tends to zero and infinity. By assumption (III) , d (λ) is nonnegative when λ is large. So if d (λ) is negative for small λ, by the continuity argument as that of [13] we conclude that there exists some λ0 such that d (λ0 ) = 0, which implies the singularity of Aλ0 . We show that when λ is small enough, the sign of d (λ) can be determined by the number of negative eigenvalues of A0 . If the number is odd as assumed in the theorem, then d (λ) is negative for small λ. So the key issue is to show that the negative spectrum of A0 is stable when perturbed to Aλ . Since only the weaker convergence (27) is available, the regular perturbation theory as that of in [13] is not applicable. We deal with this issue by using ideas from the asymptotic perturbation theory for Schrödinger operators (see [7]). First we show the following: (i) For any eigenvalue µ (λ) of Aλ , we have |Im µ (λ)| < M (here M is such that Bλ < M). (ii) Let b > 0 be such that there are no eigenvalues of A0 with real part b. There exists positive ε1 , δ1 such that if λ < δ1 , then for any eigenvalue µ (λ) of Aλ , we have |Re µ (λ) − b| > ε1 . (iii) Define P (A) = z|Rk (z) = (z − Aλ )−1 exists and is uniformly bounded for small λ and P (A∗ ) is defined in a similar way. Then we have ρ (A0 ) ⊂ P (A) , (ρ (A0 ))∗ ⊂ P A∗ .


101

The proof of (i) is obvious by our assumptions. Now we prove (ii): Supposing it were false, we could find a sequence λn → 0, µn being an eigenvalue of Aλn , and Re µn → b. Let un be the corresponding eigenfunction and un = 1. By (i), {µn } is a bounded sequence. We can find a subsequence µnk → µ0 and Re µ0 = b. For convenience, we still denote the subsequence by {µn } . It is easy to see that un A ≤ C (independent of n). So by assumption (II), there exists some u0 with un → u0 strongly in L and u0 = 1. Moreover, Aλn un → A0 u0 weakly in L. To see that, we take any function v ∈ L, then lim Aλn un , v = lim un , A∗λn v = u0 , A∗0 v (by assumption (IV)) n→∞

n→∞

= (A0 u0 , v) . Combining the above with Aλn un = µn u0 , we get A0 u0 = µ0 u0 in the limit n → ∞. This is a contradiction since Re µ0 = b. To prove (iii), we note that z ∈ P (A) is equivalent to the following: there exists some ε > 0 such that (z − Aλ ) u ≥ ε, for small λ and any u ∈ H.

(29)

Indeed assuming z ∈ / P (A), we have z − Aλ uk → 0 k for some sequence {λk } → 0 and uk ∈ H with uk = 1. By the same argument as in the proof of (ii), we have (z − A0 ) u0 = 0 for some nontrivial function u0 . This is a contradiction. The proof for P (A∗ ) is the same since (ρ (A0 ))∗ = ρ A∗0 . This proves (i)–(iii). Let be the minimum of the real part of eigenvalues of Aλ . The number is finite since Aλ are uniformly bounded from below. Define ε1 D = (x, y) | − 1 < x < − + b, − M < y < M 2 and = ∂D. By taking M, large, we can assume ⊂ ρ (A0 ). By claim (ii) just proved, if λ < δ1 then all eigenvalues of Aλ with negative real part lie in D. Define the Riesz projection as 1 Rλ (k) dk Pλ = (30) 2πi and R (Pλ ) its range. Similarly we define P0 =

1 2πi

R0 (k) dk.

Here the -integral is in the counterclockwise sense.

102

Z. Lin

Now we show that dim (R(Pλ )) = dim (R(P0 )) , if λ is small enough,

(31)

which together with (34) implies that Pλ − P0 → 0 as λ → 0

(32)

(see [8, Lemma 1.21 of Chapter VIII]). Let us prove (31). Since ⊂ ρ (A0 ) is compact, by claim (iii) above there exists δ > 0 such that if λ < δ and k ∈ , then Rλ (k) , Rλ∗ (k) ≤ C1 (33) for some constant C1 . It follows that Rλ (k) , Rλ∗ (k) are strongly continuous at λ = 0. Indeed for any u ∈ H, we have (Rλ (k) − R0 (k)) u = Rλ (k) (Aλ − A0 ) u ≤ C1 (Aλ − A0 ) u → 0 (by (27)). The strong continuity of Rλ∗ (k) can be shown in the same way. So we have Pλ → P0 and Pλ∗ → P0∗ strongly.

(34)

Therefore dim Pλ ≥ dim P0 for small λ. To show (31), we only need to prove dim Pλ ≤ dim P0 , for small λ.

(35)

Supposing otherwise, then we can find a sequence {λn } → 0 and {un } ⊂ H with un = 1, such that Pλn un = un and P0 un = 0.

(36)

By passing to a subsequence we can assume that un → u0 weakly. By (27), we have Pλn un → P0 u0 weakly. So passing to the limit in (36), we have P0 u0 = u0 and P0 u0 = 0, which implies that u0 = 0 and un → 0 weakly. But by (33) and the definition of Pλ (30), Aλ un = Aλ Pλ un ≤ Aλ Pλ ≤ const, for small λn . n

n

n

n

n

This implies the bound un A ≤ const, from which we deduce that un → 0 strongly in L by assumption (II). This is a contradiction and ends the proof of (31). Let µ1 , µ2, · · · , µN be all the distinct eigenvalues of A0 in D. Let mk be the multiplicity of µk . For each µk , we can pick a small ball Bk = B (µk ; rk ), inside which µk is the isolated eigenvalue of A0 . And by taking rk small enough we can ensure that Bk does not intersect with the imaginary axis if Re µk = 0, and Bk does not intersect with the real axis if Re µk = 0. We also assume {Bk } does intersect with . The disks {Bk } are disjoint and for the conjugate of µk we take the disk with the same radius. Then if λ is small enough, by the same proof as that of (31), there are exactly mk eigenvalues (counting multiplicity) of Aλ in each Bk . Since dim (R (Pλ )) = dim R(Pλ0 ) , these are all the eigenvalues of Aλ in D. By our construction of Bk , if we multiply all the eigenvalues of Aλ contained in them, the sign is the same as that of A0 . Thus in the definition of d (λ) the product corresponding to all the eigenvalue of Aλ with real part smaller than b has the same sign as that of A0 . Thus it is negative if λ is small. Since the other part of the product is always positive, we have proved that d (λ) is negative when λ is small. This finishes the proof of the theorem.


103

Remark 3.2. If A0 has a kernel, we denote by e (A0 ) the number of null vectors in ker A0 perturbed to be negative eigenfunctions of Aλ as λ > 0 is small and n (A0 ) the number of negative eigenvalues of A0 . Then the conclusion of the theorem still holds if n (A0 ) + e (A0 ) is odd. The proof is the same as above. 3.3. Proof of Theorem 1.2. Now we use the abstract theorem above to prove Theorem 1.2. The operator Aλ is defined by (21) with A = −, Bλ = Kλ , H = X, L = L2 () . Now we check the assumptions in Theorem 3.1. Assumption (I) is proved in Remark 3.1. Assumption (II) is standard for the Laplacian defined in a bounded domain. Assumption (III) is proved in Lemma 3.4. Moreover A0 has an odd number of negative eigenvalues and no kernel as assumed in Theorem 1.2. So we only need to prove assumption (IV). This is in the following lemma. Lemma 3.5. For any φ ∈ X, Aλ φ → A0 φ and A∗λ φ → A0 φ strongly in L2 () , as λ → 0 + . Proof. It is easy to show that A∗λ ψ := −ψ − g (ψ0 ) ψ + g (ψ0 ) λ

0 −∞

eλs ψ (X (−s; x, y) , Y (−s; x, y)) ds,

where (X (s; x, y) , Y (s; x, y)) is the solution to the characteristic equation (22). So we only need to show the strong convergence of Aλ since the proof for A∗λ is the same. For any φ ∈ X, denote 0 eλs φ (X (s; x, y) , Y (s; x, y)) ds φλ = g (ψ0 ) λ −∞

and φ0 = g (ψ0 ) P˜ φ, where P˜ is defined in the introduction and given by the formula (17). We have Aλ φ − A0 φ22 = (Aλ φ − A0 φ, Aλ φ − A0 φ) = (φλ − φ0 , φλ − φ0 ) = (φλ , φλ ) − 2 (φλ , φ0 ) + (φ0 , φ0 ) . We analyze the first term 2 g (ψ0 ) λ2 (φλ , φλ ) =

0

0

−∞ −∞

eλs eλt

·φ (X (s; x, y) , Y (s; x, y)) φ (X (t; x, y) , Y (t; x, y)) dsdtdxdy = fλ (x, y) dxdy,

where 2 fλ (x, y) = g (ψ0 ) λ2

0

0

−∞ −∞

eλs eλt φ (X (s; x, y) , Y (s; x, y))

·φ (X (t; x, y) , Y (t; x, y)) dsdt.

104

Z. Lin

We claim the following: (i) As λ → 0+, f λ (x, y) → φ02 almost everywhere. (ii) {fλ } λ ∈ R+ are uniformly integrable. Proof of claim (i). First we study the characteristic equation (22) for the fluid particle. For each initial position (x, y) not on the critical level sets, we know that the trajectory (X (s; x, y) , Y (s; x, y)) always lies in some component of {ψ0 = ψ0 (x, y)} which is a closed curve denoted by . If we denote by γ the arc length variable of and by L ( ) the length, then the particle has a periodic trajectory according to the law dγ (s) = |∇ψ0 | (X (s; x, y) , Y (s; x, y)) ds with the period

L( )

T ( ) = 0

dγ . |∇ψ0 |

In the following we identify the point (X (s; x, y) , Y (s; x, y)) with its arc length variable γ (s). We recall the following fact proved in [12]: For any T - periodic function A (x) ∈ L1 (0, T ) ,

0

es A

lim

λ→0+ −∞

s λ

ds =

1 T

T

A (s) ds . 0

Using this, we have lim λ

λ→0+

0

−∞

= lim

λ→0+

eλs φ (X (s; x, y) , Y (s; x, y)) ds

s s ; x, y , Y ; x, y ds es φ X λ λ −∞ T ( ) φ (X (s; x, y) , Y (s; x, y)) ds 0

1 T ( ) 0 L( ) 1 φ (γ ) dγ = |∇ψ0 | T ( ) 0 φ

=

=

|∇ψ0 | 1 |∇ψ0 |

.

The last expression is exactly the formula (17) for P˜ φ, so claim (i) is proved. Proof of claim (ii). For any δ > 0, there exists ε0 > 0, such that for any set B ⊂ with |B| < ε0, we have |ψ|2 dxdy < δ. B


105

Then |fλ (x, y)| dxdy B

0 0 2 λ2 eλs eλt ≤ g (ψ0 ) ∞ −∞ −∞ |φ (X (s; x, y) , Y (s; x, y)) φ (X (t; x, y) , Y (t; x, y))| dxdydsdt · B

2 ≤ g (ψ0 )∞

0

0

−∞ −∞

1 2 |φ (X (s; x, y) , Y (s; x, y))|2 dxdy

λ2 eλs eλt B

1

|φ (X (t; x, y) , Y (t; x, y))|2 dxdy

· B

2 = g (ψ0 )∞

0

0

−∞ −∞

t (B)

2 ≤ g (ψ0 )∞

0

0

−∞ −∞

dsdt

1 |φ (x, y)|2 dxdy

λ2 eλs eλt 1

|φ (x, y)|2 dxdy

·

2

2

s (B)

2

dsdt

2 λ2 eλs eλt δ 2 dsdt = g (ψ0 )∞ δ 2 .

We thus prove claim (ii). Here s denotes the mapping (x, y) → (X (s; x, y) , Y (s; x, y)) and we use the fact |s (B)| = |B| < ε0 . Now by claims (i), (ii) and the Dominant Convergence Theorem, we have lim (φλ , φλ ) = lim fλ (x, y) dxdy λ→0+ λ→0+ = lim fλ (x, y) dxdy λ→0+ = φ02 = (φ0 , φ0 ) .

By the same proof we have lim (φλ , φ0 ) = (φ0 , φ0 ) .

λ→0+

So lim Aλ φ − A0 φ22 = lim (φλ , φλ ) − 2 (φλ , φ0 ) + (φ0 , φ0 )

λ→0+

λ→0+

= (φ0 , φ0 ) − 2 (φ0 , φ0 ) + (φ0 , φ0 ) = 0.

106

Z. Lin

4. An Enstrophy-Stable but Energy-Unstable Steady Flow In this section, we consider a bounded simply connected domain ⊂ R2 with a smooth boundary ∂. Let λ0 < λ1 ≤ λ2 ≤ · · · ≤ λn ≤ · · · be all the eigenvalues of − with Dirichlet boundary condition. Denote φ0 , φ1 , · · · , φn , · · · the corresponding normalized orthogonal eigenfunctions. We show that φ0 is a nonlinearly stable steady state of the 2-D incompressible Euler equation (2) in the L2 -norm of the vorticity (enstrophy). This can be deduced from Theorem 1.1 (i). But in the following we give a direct proof. Denote ω0 = −λ0 φ0 . Theorem 4.1. The steady flow with the stream function φ0 is nonlinearly stable in the following sense: for any ε > 0, there exists some δ > 0 such that ω (., 0) − ω0 L2 < δ ⇒ sup ω (., t) − ω0 L2 < ε. t>0

Proof. For ψ ∈ X, we define the following energy-Casimir functional: 1 1 |ω|2 dxdy. − |∇ψ|2 + H (ψ) = 2 λ0 Then

1 − |∇ (ψ − φ0 )|2 − Re ∇ (ψ − φ0 ) .∇φ0 2 1 1 |ω − ω0 |2 + + Re (ω − ω0 ) ω0 2λ λ0 0 1 1 |ω − ω0 |2 = − |∇ (ψ − φ0 )|2 + 2 2λ 0

1 + Re (ω − ω0 ) ω0 + φ 0 λ0 1 1 |ω − ω0 |2 = − |∇ (ψ − φ0 )|2 + 2λ0 2 1 1 |∇ (ψ − φ0 )|2 − |∇ (ψ − φ0 )|2 + |ω − ω0 |2 = 2 2λ 0 1 1 |∇ (ψ − φ0 )|2 + (ω − ω0 ) (ψ − φ0 )∗ + |ω − ω0 |2 = 2λ0 2 1 1 |(ω − ω0 ) + λ0 (ψ − φ0 )|2 |∇ (ψ − φ0 )|2 + = 2λ0 2 1 − λ0 |ψ − φ0 |2 . 2

H (ψ) − H (φ0 ) =

So if we denote ψ − φ0 =

∞

ai φi ,

i=0

then ω − ω0 = (ψ − φ0 ) = −

∞

i=0

λ i ai φi


107

and 1 |(ω − ω0 ) + λ0 (ψ − φ0 )|2 2λ0 ∞ 1

= (λi − λ0 )2 |ai |2 2λ0 i=1

∞ λ0 2 2 1 1− ≥ λi |ai |2 . 2λ0 λ1

H (ψ) − H (φ0 ) ≥

i=1

Thus if H (ψ) − H (φ0 ) < b, then ∞

2bλ0

λ2i |ai |2 <

1−

i=1

Let

λ0 λ1

2 .

(37)

|ω|2 dxdy,

C (ω) =

then

|ω − ω0 |2 + 2 Re (ω − ω0 ) ω0

C (ω) − C (ω0 ) =

(38)

∞

= λ20 |a0 |2 + 2 Re a0 + λ2i |ai |2 . i=1

Notice that H (ψ) and C (ω) are both invariants of (2). Now a0 (t) = (ψ − φ0 )φ0 dxdy

is a continuous function of t, so for ε > 0 small we can find some d (ε) > 0 such that ε ε |a0 (0)| < √ , |a0 (t)|2 + 2 Re a0 (t) < d (ε) ⇒ |a0 (t)| < √ . (39) 2λ0 2λ0 Choose δ > 0 such that ω (., 0) − ω0 L2 < δ to satisfy |C (ω (0)) − c (ω0 )|
0, ω (., t) − ω0 2L2 = λ20 |a0 |2 +

∞

λ2i |ai |2 < ε2 .

i=1

This finishes the stability proof.

By Theorem 1.1(ii), the steady flow with the stream function φ0 is also linearly stable in the norm ·X . However, in the following we will show that for some domain , this flow is linearly unstable in the L2 −norm of the velocity (energy norm). Starting with Echhoff in the 1970s (see [3]), there are lots of papers using geometric optics or the WKB asymptotic method to treat the local instability in fluid dynamics. Typically, it allows one to estimate from below the growth rate of the solutions of the initial value problem for the linearized equation in terms of the growth rate of solutions of an ODE system. For Euler equation of 3D inviscid incompressible fluid, the ODE system (see e.g. [11]) is  ˙ X=−U (X) ,   0 T  ∂U0 ˙ K = ∂X K,    a˙ = − ∂U0 a + 2 ∂U0 a · K K 2 , ∂X

∂X

|K|

with initial conditions at t = 0, X = X0 , K = K0 , a = a0 , where K0 · a0 = 0. Here U0 (X) is the steady flow velocity field and the matrix ∂U0 /∂X has components ∂U0i /∂Xj , i, j = 1, 2, 3. Theorem 4.2. [11, 4]. If sup

lim |a (t; X0 , K0 , a0 )| = ∞,

t→∞ X0 ,K0 ,a0 |K0 |=|a0 |=1,K0 ·a0 =0

(40)

then the steady flow U0 (X) is linearly unstable in the sense that for suitable initial data, the L2 −norm of the velocity of the corresponding solution of the linearized Euler equation is not bounded in time. We said a point x0 is a hyperbolic stagnation point of the flow U0 (X), if U0 (x0 ) = 0 and the matrix ∂U0 /∂X has at least one positive real eigenvalue. It is shown by Friedlander and Vishik (see [5]) that Lemma 4.1. Let the 3-D flow dX dt = U0 (X) have a hyperbolic stagnation point at some point x0 . Then U0 (X) is linearly unstable in the L2 −norm of the velocity, as a steady flow of an ideal fluid. For the 2D case, let the corresponding stream function of U0 (X) be φ0 (X). Then for U0 (X) to have a hyperbolic stagnation point, it is equivalent that φ0 has a saddle point. In the following, we construct a stable flow with a hyperbolic stagnation point. Lemma 4.2. There exists some domain with smooth boundary such that the eigenfunction with the lowest eigenvalue of − on H01 () has a saddle point.


109

Proof. First we consider a smooth domain 0 composed of two circular disks of radius R smoothly connected by a thin channel of width 2ε (see the graph). We also make 0 symmetric with respect to the middle vertical axis. Let φ0 be the eigenfunction with the lowest eigenvalue λ0 of − on H01 (0 ) . From the standard theory, we know that φ0 is positive and is symmetric with respect to the axis. We shall prove the following estimate:

2 sup |∇φ0 | ≤ k0 + k0 + λ0 sup φ0 . (41) ¯0

0

Here k0 is a positive number such that the curvature k ≥ −k0 at each point of ∂0 . The proof of (41) follows from an idea in [14], see also [16, Chapter 5]. Let τ = sup¯ 0 |∇φ0 | , M = sup0 φ0 and α = 2k0 τ. Define P = |∇φ0 |2 + λ20 φ02 + αφ0 . Then by a direct computation P +

L i Pi = (λ0 φ0 + α) α > 0, |∇φ0 |2

here Li = −Pi − 2∂i φ0 (λ0 φ0 + α) . So by the Maximum principle, the maximum of P is either obtained on the boundary ∂0 or at some point in 0 , where ∇φ0 = 0. For the first case, supposing the maximum of P is obtained at x0 ∈ ∂0 , we have ∂P |x > 0 ∂n 0

(42)

by Hopf’s principle. Here n is the outward normal direction. But (see [16, p. 76]) ∂P ∂φ0 ∂ 2 φ0 ∂φ0 |x0 = 2 +α 2 ∂n ∂n ∂n ∂n = −2 |∇φ0 | (α + 2k (x0 ) |∇φ0 |) . Since α + 2k (x0 ) |∇φ0 | ≥ α − 2k0 sup |∇φ0 | ≥ 0, ¯0

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we get a contradiction to (42). Thus P can obtain the maximum only at a point in 0 where ∇φ0 = 0. So we get τ 2 ≤ λ20 M 2 + αM = λ20 M 2 + 2k0 Mτ from which (41) follows. For any point D in the channel, the distance to the nearest boundary point is at most ε. So by the mean value theorem

2 φ0 (D) ≤ ετ ≤ ε k0 + k0 + λ0 sup φ0 . (43) 0

Since a disk BR/2 with radius R2 is contained in 0 , by the monotonicity of the eigen values of − with respect to the domain, we have λ0 < λ BR/2 (the lowest eigenvalue of − in BR/2 ). We can make k0 bounded (with a bound independent of ε) as long as the domain 0 is smooth. So from (43), we can see that if ε is small then φ0 can not obtain its maximum in the channel. But since φ0 is symmetric with respect to the middle vertical axis, φ0 must have at least two maximum points, one on each disk. We cannot ensure the two maximum points we get are non-degenerate. But we can deform 0 to make them non-degenerate in the following way. We quote a result of K. Uhlenbeck ([18]). First we introduce some notations as in [18]. Let N be a compact n-manifold with boundary which can be embedded in Rn and B = Embk (N, Rn ) be the set of Ck embedding of N in Rn . We associate with the embedding F : N → Rn the Laplace operator on the image of F with Dirichlet boundary condition, which we denote by Im(F ) . Consider the following properties of Im(F ) : A. One dimensional eigenspaces. B. Zero is not a critical value of the eigenfunction restricted to the interior of the domain of the operator. C. The eigenfunctions are Morse functions on the interior of the domain of the operator. Theorem 9 in [18] is Lemma 4.3. Let k > n + 2. Then the set F ∈ Embk N, Rn : properties A, B, and C hold for Im(F ) is residual in Embk (N, Rn ) . Using this result, we can deform 0 slightly to get a new domain . This domain is still symmetric to its middle vertical axis with the curvature condition k ≥ −2k0 on ∂, and the first eigenfunction φ of − on is a Morse function. Then by the same argument as above for the domain 0 , φ still has at least two maximum points. By the strong maximum principle, the normal derivative ∂φ/∂n is negative everywhere on ∂. So the vector field U0 (x, y) = −∂y φ, ∂x φ is always nonzero on ∂. It defines a non-degenerate vector field on , tangential to ∂. Denote the number of equilibria of U0 with index +1(−1) by n+1 (n−1 ). We have n+1 − n−1 = 1.

(44)


111

˜ of , which is obtained from For the proof of (44), we introduce the double manifold by attaching a second copy of along ∂. By doing so we identify each point on ∂ with its copy in the boundary of the second copy. In this way we get a 2-dimensional ˜ without boundary, which is clearly diffeomorphic to S2 . The vector field U0 manifold ˜ 0 on ˜ in the natural way. And we have that the number of equilibria of is extended to U ˜ U0 with index +1(−1) is n˜ +1 (n˜ −1 ) = 2n+1 (n−1 ). By Hopf’s Theorem, ˜ = 2, n˜ +1 − n˜ −1 = χ ˜ is the Euler characteristic of . ˜ So (44) follows. Noticing that n+1 is the where χ number of maximum and minimum points of φ on which is at least 2 and n−1 is the number of saddle points of U0 , we conclude that there exists some non-degenerate saddle point of φ. Combining the above results in this section, we get the following theorem. 1 Theorem 4.3. Let φ be the eigenfunction with the lowest eigenvalue of − on H0 (), where is constructed in Lemma 4.2. Then the steady flow U0 (x, y) = −∂y φ, ∂x φ is nonlinearly stable in the L2 −norm of the vorticity, but linearly unstable in the L2 −norm of the velocity.

We note that it was proved in [10] that a steady flow with a saddle point is nonlinearly unstable in the C1,α norm of the velocity. However the nonlinearly stability or instability in the energy space is unknown. 5. Remarks on the Case of a Non-Simply Connected Domain The results in Theorem 1.1, 1.2 can be generalized to the non-simply connected case. For this case, the boundary conditions for the vorticity equation is now (3a), (3b ). So we have to change the function space for the stability and instability results. Define ∂ψ X := ψ ∈ H2 () |ψ = i on i , ψ =0 , (45) = 0 and i ∂n Y := ψ ∈ H1 () |ψ = i on i , i

L02 := with ψX =

ψ =0 ,

(46)

ψ ∈ L2 () |

∂ψ = 0 and ∂n

ψ =0

|ψ|2 dxdy, ψY =

|∇ψ|2 dxdy, ψL2 = 0

|ψ|2 dxdy.

Here i are unspecified constants. We define the functionals a, b on Y and the operators Aλ , A0 : X → L20 in the same way as in the simply connected case. Then the conclusions

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of Theorem 1.1, 1.2 still hold true. The proofs are similar, so we skip them here. Now we explain why the spaces defined by (45) and (46) are natural for non-simply connected domains. The first condition (ψ = i on i ) is the requirement of (3a). The zero integral condition is used to get rid of the arbitrary constant which might add to the stream function ψ. For the zero circulation condition, we recall that for the Euler equation the circulation is invariant, and this is also true for the linearized Euler equation. So for a growing mode eλt φ satisfying the linearized Euler equation, we must have ∂φ = 0. i ∂n So we only need consider function spaces defined by (45) and (46) when studying the existence of growing modes. For the stability study, we can decompose any function ψ satisfying the boundary conditions (3a),(3b) as ψ = ψ + ψ0 , where ψ is in X or Y and ψ0 is a harmonic function with i − circulation ∂ψ . i ∂n When ψ is the stream function for a solution of Euler equation, the circulation is fixed and thus ψ0 is independent of time. We can use the energy-Casimir method as in [17] to control ψ under the vorticity norm. Acknowledgements. I thank Walter Strauss for many stimulating discussions and great help during the preparation of this paper. And I would like to thank Yan Guo for his useful comments.

References 1. Bardos, C., Guo, Y., Strauss, W.: Stable and unstable ideal plane flows. Chinese Annals Math. 23B, 149–164 (2002) 2. Drazin, P. G., Howard, L. N.: Hydrodynamic stability of parallel flow of inviscid fluid. Adv. Appl. Math. 9, 1–89 (1966) 3. Echhoff, K. S.: On stability for symmetric hyperbolic systems I. J. Differ. Eq. 40, 94–115 (1981) 4. Friedlander, S., Vishik, M. M.: Instability criteria for the flow of an inviscid incompressible fluid. Phys. Rev. Letts. 66(17), 2204–2206 (1991) 5. Friedlander, S.: Lectures on stability and instability of an ideal fluid. In: Hyperbolic equations and frequency interactions (Park City, UT, 1995), IAS/Park City Math. Ser., 5, Providence, RI: Amer. Math. Soc., 1999, pp. 227–304 6. Friedlander, S., Shnirelman, A.: Instability of steady flows of an ideal incompressible fluid. 2001 7. Hunziker, W.: Notes on asymptotic perturbation theory for Schrödinger eigenvalue problems. Helv. Phys. Acta. 61, 257–304 (1988) 8. Kato, T.: Perturbation theory for linear operators. Berlin, New York: Springer Verlag, 1966 9. Kato, T.: On classical solutions of the two-dimensional non-stationary Euler equation.Arch. Rational. Mech. Anal. 25, 188–200 (1967) 10. Koch, H. Transport and instability for perfect fluids, Math. Ann. 323, 491–523 (2002) 11. Lifschitz, A., Hameiri, E.: Local stability conditions in fluid dynamics. Phys. Fluids A 3(11), 2644– 2651 (1991) 12. Lin, Z.: Instability of periodic BGK waves. Math. Res. Letts. 8, 521–534 (2001) 13. Lin, Z.: Instability of some ideal plane flows. SIAM J. Math. Anal. 35(2), 318–356 (2003) 14. Payne, L. E.: Bounds for the Maximum stress in the Saint Venant torsion problem. Indian J. Mech. and Math. Special issue, 51–59 (1968) 15. Sperb, R.P.: Extensions of two theorems of Payne to some nonlinear Dirichlet problems. Z. Angew. Math. Phys. 26, 721–726 (1975) 16. Sperb, R.P.: Maximum principles and their applications. New York: Academic Press, 1981 17. Wolansky, G., Ghil, M.: Nonlinear stability for saddle solutions of ideal flows and symmetry breaking. Commun. Math. Phys. 193, 713–736 (1998) 18. Uhlenbeck, K.: Generic properties of eigenfunctions. Am. J. Math. 98(4), 1059–1078 (1976) Communicated by P. Constantin


Communications in


Conditional Expectations Relative to a Product State and the Corresponding Standard Potentials Huzihiro Araki Research Institute for Mathematical Sciences, Kyoto University, Sakyoku, Kyoto 606-8502, Japan E-mail: [email protected] Received: 1 June 2003 / Accepted: 27 August 2003 Published online: 13 February 2004 – © Springer-Verlag 2004

Abstract: For a lattice system with a finite number of Fermions and spins on each lattice point, conditional expectations relative to an even product state (such as Fermion Fock vacuum) are introduced and the corresponding standard potential for any given dynamics, or more generally for any given time derivative (at time 0) of strictly local operators, is defined, with the case of the tracial state previously treated as a special case. The standard potentials of a given time derivative relative to different product states are necessarily different but they are shown to give the same set of equilibrium states, where one can compare states satisfying the variational principle (for translation invariant states) or the local thermodynamical stability or the Gibbs condition, all in terms of the standard potential relative to different even product states. 1. Introduction The subject of this paper is the equilibrium statistical mechanics of infinitely extended lattice systems. We focus on the equivalence of the following alternative characterizations of equilibrium states: The KMS condition, the variational principle (only for translation invariant states), the Gibbs condition, the local thermodynamical stability (LTS) condition, and another known condition to which the name dKMS (differential KMS) condition is given in [2]. Research on this subject for an infinite system started in 1960’s. Details of its history as well as the definition and significance of the above-mentioned characterizations of equilibrium states can be found in standard textbooks [6, 9]. Also see [7]. In collaboration with H. Moriya, we have recently developed a new approach to this subject in [2] and [3], which differs from the traditional approach in the above-mentioned textbooks mainly in the following 5 aspects: (1) We start from the dynamics (the change of operators with time) and associate with it a unique “standard” potential, in contrast to the traditional approach which starts

114

(2)

(3)

(4) (5)

H. Araki

from a potential, and the dynamics is derived from the potential (where a difficult problem arises). The general potential used in the traditional approach is indirectly treated by comparing it with the unique standard potential which has the same dynamics (or more precisely the same time derivatives of local operators at time 0) as the given general potential (the adjective “general” indicating not necessarily standard) and by showing that both potentials have the same set of equilibrium states for each characterization. We make some minimal assumptions (assumptions necessary for the formulation of the above-mentioned characterizations) and the (convergence) property of the potential needed for the definition of the time derivative of local operators automatically follow without any further assumption in contrast to the traditional approach where some convergence assumption in terms of the norm of the individual potential operators is made. (The natural convergence property in our approach does not use the norm of individual potential operators.) We obtain the equivalence results under a kind of minimal assumption on dynamics and no assumptions on potentials, which is a vast improvement over the traditional approach. We can deal with Fermion systems equally well as the quantum spin systems despite the non-commutativity of local algebras at mutually disjoint regions.

The crucial tool which made our new approach in [2 and 3] work (especially with regard to the points (1) and (3) above) is the C ∗ -algebra conditional expectations with respect to the tracial state. (For the sake of non-experts, we add some digressions on the above points at the end of this section.) In the present work, we generalize the definitions and results in [2] by introducing ω-conditional expectations relative to an even product state ω. A mathematical difference from the slice map treated in mathematical literatures is the mutual non-commutativity of factor subalgebras, relative to which the state ω has the product property. The algebra under consideration is somewhat generalized from the one in [2] to a graded C ∗ algebra with a graded commutation relations, which may include simultaneously both Fermion creation and annihilation operators and spin operators at each lattice point, as long as the local algebra at each lattice point is a full matrix algebra (i.e. a finite dimensional factor), excluding the possibility for Boson creation and annihilation operators. All results in [2] as well as those in [3] hold also for ω-standard potentials relative to a product state ω. For different choices of ω and a fixed dynamics, they provide examples of equivalent potentials. Each of characterizations of equilibrium states in terms of the ω-standard potential, such as the variational principle, the Gibbs condition and the LTS condition, gives the same set of equilibrium state for any different choice of the product state ω (Theorems 5.1 and 7.2). As an immediate consequence, mutual equivalence of the KMS condition, the dKMS condition, the Gibbs condition, the LTS condition, and the variational principle (the last one only for translation invariant states), which is derived in [2] and [3] (Theorems A, B, 7.5, 7.6, and Proposition 12.1 of [2] and Theorems 1, 2, 3 and Corollary 4 of [3]) for (τ -) standard potentials under (minimal) assumptions on dynamics, holds also for the general ω-standard potentials. The paper is organized as follows. The graded algebra and its graded commutation relations along with results on commutants (Theorem 2.4 and 2.5) and intersections (Theorem 2.2) are described and proved in Sect. 2. The ω-conditional expectations relative to a product state ω along with their basic properties (Theorems 3.1 and 3.2) are given

Conditional Expectations Relative to a Product State

115

in Sect. 3. The ω-standard potentials for a given dynamics are introduced in Sect. 4 with a use of the ω-conditional expectations. The Gibbs and LTS conditions are described in terms of the ω-standard potentials in Sect. 5. Translation invariance is introduced in Sect. 6 and the variational principle is discussed in Sect. 7. All results and their proofs in [2] and [3] can be carried over to the present generalized situation (Theorems 4.1, 4.2, 5.1, 6.1, 6.2, 6.3, and 7.1). Comparison of the ω-standard potentials for different choices of ω (the tracial state and the vacuum state of a Fermion lattice system) are made for one-body and two-body potentials in Sect. 8. The ω-conditional expectations for a non-even product state is discussed in Sect. 9. A necessary and sufficient condition for a subset I of the lattice is given for the existence of the ω-conditional expectation onto the subalgebra for the subset I in the case of non-even ω (Theorem 9.1). We now add some digressions on some of the points mentioned earlier. The dynamics and potential are mutually related √ by the formula for the time derivative (at time 0) of a local operator which is i = −1 times the commutator with the “Hamiltonian”. For a finite system, the Hamiltonian (which is the total energy of the system) is the sum of all potentials. For an infinite system, such a sum does not converge. By omitting those potentials commuting with the local operator (of which the time derivative is being computed), one obtains a converging sum of potentials, which gives a local Hamiltonian for each finite support region of the local operator under consideration. (Note that the local operators are taken in computation of time derivatives in order to obtain the local Hamiltonian as a converging sum of potentials, which has an interpretation of the interaction energy of the local system in the specified region as an open system, i.e. including the interaction energy with the outside region in addition to the internal interaction energy.) This is how the time derivatives at time 0 for local operators (i.e. differential dynamics) are introduced in terms of the potentials in the traditional approach. The dynamics is then defined from the time derivative, the existence and uniqueness of dynamics depending on the potential. The same formula is used to define the potential from a given time derivative of local operators in our approach. At this point the following ambiguity arises as to the region to which an operator is to be associated as a local potential. A potential for a local region may be viewed as (a part of) the potential for a bigger region containing the originally assigned region. Thus different potentials may give rise to the same time derivatives at time 0 for all local operators. Such potentials are said to be mutually equivalent. In order to select a unique potential for a given time derivative of local operators out of many equivalent potentials, we have to establish a criterion for those operators which are to be taken as the potential for a given region and not for any other regions. Such a criterion is the standardness of the potential formulated in terms of the conditional expectation with respect to a fixed even product state. In our new approach, we start from the dynamics and make the basic assumption that the local operators have the time derivatives at time 0 (i.e. they are differentiable at time 0). Then the time derivatives of local operators are associated with the given dynamics and give rise to a unique standard potential, for which the local Hamiltonian is definable as a convergent sum of potentials (the convergence being in a specific technical sense) and the traditional formula, giving the time derivatives of local operators in terms of the commutator with a sum of potentials, holds. The conditional expectation strictly distinguishes the supporting region of any potential operator, giving rise to the unique standard potential for each local region. Different even product states (and the associated conditional expectations) yield different criteria and define different (but equivalent) standard potentials for each given set of time derivatives.

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In associating the standard potential with a set of time derivatives of local operators (at time 0), we need one more assumption that it preserves the even-oddness of the local operators. In terms of dynamics, this is the assumption that the dynamics is even. This assumption is shown in Proposition 8.1 of [2] to be implied by the translation invariance of dynamics, an assumption needed for the formulation of the variational principle. The conditional expectation for von Neumann algebras is explained in standard textbooks [8 and 10]. We will be using the conditional expectations for C ∗ -algebras. In the case of the tensor product systems (e.g. spin lattice systems), the conditional expectation with respect to a product state is called a slice map in mathematics literature and a partial state (partial trace in the case of the tracial state) in physics literature (e.g. [7]). The use of the conditional expectations for associating a unique standard potential to a given set of time derivatives of local operators (at time 0) was developed for a quantum spin system in [4] and for the classical spin lattice systems in [5]. For the latter, the conditional expectation in the probabilistic sense is used. 2. Algebra We consider a C ∗ -algebra A equipped with the following structure, modeled after Fermion and spin lattice systems. (a) Local structure. For each point i of a lattice L = Zν , there corresponds a subalgebra Ai of A, which is isomorphic to a full matrix algebra of d × d matrices, d independent of i (independence needed for lattice translation automorphisms). For each subset I of L, A(I ) denotes the C ∗ -subalgebra of A generated by Ai , i ∈ I . A(L) is assumed to be A. In most part of this work except Sects. 2 and 9, we assume the existence of a representation of the group L by automorphisms τk of A, k ∈ L, such that τk (Ai ) = Ai+k . Then τk (A(I )) = A(I + k),

I + k = {i + k; i ∈ I }.

(b) Graded structure. There exists an involutive

C ∗ -automorphism

(2.1)

of A such that

(A(I )) = A(I ), τk = τk , (k ∈ L).

(2.2) (2.3)

Then any A ∈ A splits uniquely as a sum of even and odd elements A+ and A− : A = A+ + A− , A± = (1/2)(A ± (A)),

(A± ) = ±A± .

(2.4) (2.5)

Accordingly, A and the subalgebras A(I ) split as a sum of even and odd parts which have a trivial (i.e. zero) intersection: A = A+ + A− , A± = {A ∈ A; (A) = ±A}, A(I ) = A(I )+ + A(I )− , A(I )± = A(I ) ∩ A± .

(2.6) (2.7)

The following graded commutation relations hold: if I ∩ J = ∅, Aσ ∈ A(I )σ , and Bσ ∈ A(J )σ (σ, σ = ±), then Aσ Bσ = (σ, σ )Bσ Aσ ,

(2.8)


117

(σ, σ ) =

−1, if σ = σ = −, +1, otherwise.

(2.9)

Namely, odd elements of disjoint regions anticommute, while other pairs of even and odd elements commute. The graded commutation relations hold for any pair of disjoint I and J if they hold for a pair of disjoint one-point sets because A(I ) is generated by Ai , i ∈ I . We use the notation I ⊂⊂ L to mean that I is a finite subset of L. Then |I | denotes the number of points in I , sometimes called the volume of I . We denote A0 = ∪I ⊂⊂L A(I ).

(2.10)

It is a dense ∗-subalgebra of A. Lemma 2.1. For I ⊂⊂ L, A(I ) is isomorphic to a full matrix algebra of d |I | × d |I | matrices. For an infinite subset I of L, A(I ) is a UHF algebra of type d ∞ . In particular, A(I ) is simple for all I . As a special case, A is simple. Proof. First we prove the first assertion inductively for increasing |I |. For this, it is enough to consider disjoint finite subsets I and J of L and to prove that A(I ∪ J ) satisfies the first assertion if A(I ) and A(J ) do. Since any ∗-automorphism of a type I factor is inner, there exists a unitary u ∈ A(I ) satisfying Ad u = on A(I ). By adjusting a constant multiple of modulus 1, we may assume u2 = 1 due to 2 = id. (Then ±u are the only selfadjoint unitaries in A(I ) which implement on A(I ).) By (u) = u3 = u, we have u ∈ A(I )+ and hence u ∈ A(J ) . Consider the mapping π:

A ∈ A(J )

−→

π(A) = A+ + uA− .

It is readily seen that π is a unital ∗-homomorphism. Since a full matrix algebra is simple, π is an isomorphism. Furthermore, π(A(J )) ∈ A(I ) due to Ad u = on A(I ). Therefore the C ∗ -subalgebra of A generated by A(I ) and π(A(J )), which is the same as the C ∗ -subalgebra of A generated by A(I ) and A(J ), i.e. A(I ∪ J ), is isomorphic to a full matrix algebra of d |I | d |J | × d |I | d |J | matrices. This proves the first assertion. Suppose I is infinite. For an increasing sequence of finite subsets Li of I tending to I , the union of A(Li ) generates A(I ). Hence A(I ) is the UHF algebra of type d ∞ . Consequently, A(I ) is simple for any I . We need, in Sect. 4 and later, the following results. Theorem 2.2. For any countable family {In } of subsets of L, ∞ ∩∞ n=1 A(In ) = A(∩n=1 In ).

(2.11)

The proof is the same as that of Corollary 4.12 of [2], where we use Theorem 3.2 of Sect. 3. Definition 2.1. For a subset I of L, I− = {i ∈ I ; (Ai )− = 0}, namely I− is the set of all i ∈ I for which Ai has non-zero odd elements.

(2.12)

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Note that if the lattice translation automorphisms τk exist, and is non-trivial, then (Ai )− = 0 for all i and I− = I . The above notation is used in Sects. 2 and 9, where some results depend delicately on I− and so they are stated in the situation with the translation uniformity assumption tentatively dropped in order to draw attention to the delicate situation. A state ωi of Ai is said to be even if ωi = ωi . Lemma 2.3. (1) For each i ∈ L− , there exists a self-adjoint unitary ui ∈ (Ai )+ implementing on Ai . It is unique up to ± . (2) If I− is finite , ui (2.13) uI = i∈I−

is a self-adjoint unitary in A(I )+ implementing on A(I ), where the product is taken to be 1 if I− is empty. Such uI is unique up to ±. (3) For each i ∈ I− , there exists an even state ωi of Ai satisfying ωi (ui ) = 0 Proof. (1) This follows from the beginning part of the proof of Lemma 2.1. (2) The first part follows from (1). The second part is due to the triviality of the center of A(I ) given in Lemma 2.1. (3) Since ui is a non-trivial self-adjoint unitary, ui = Ei+ − Ei− for mutually orthogonal non-trivial projections Ei± with sum 1. Set 1 (τ (Ei+ )−1 Ei+ + τ (Ei− )−1 Ei− ), 2 ωi (A) = τ (ρi A), (A ∈ Ai ). ρi =

Since ui is even, Ei± are even. Hence ωi is even and satisfies ωi (ui ) = 0.

(2.14) (2.15)

Theorem 2.4. (1) If I− is finite, A(I ) ∩ A = A(I c )+ + uI A(I c )− ,

(2.16)

where I c denotes the complement of I in L and uI is a self-adjoint unitary in A(I ) implementing on A(I ), which exists. (2) If I− is infinite, A(I ) ∩ A = A(I c )+ .

(2.17)

The proof is the same as that of Theorem 4.17 of [2], except for two modifications. First we use a self-adjoint unitary uI given in Lemma 2.3. Second we apply the proof of Lemma 4.16 of [2] to the case of an infinite I− , by using, instead of EI in [2], the conditional expectations EIω in Sect. 3 for an even product state ω of the tracial state of Ai for i ∈ / L− and the state ωi given by Lemma 2.3 (3) for i ∈ L− . Theorem 2.5. (1) If I− is finite, (A(I )+ ) ∩ A = A(I c ) + uI A(I c ).

(2.18)

(A(I )+ ) ∩ A = A(I c ).

(2.19)

(2) If I− is infinite,


119

The proof is the same as that of Theorem 4.19 of [2] with the same modification as the proof of the preceding theorem. Lemma 2.6. Assume that I− is infinite. Then any u ∈ A satisfying uA = (A)u for all A ∈ A(I ) is 0. In particular, on A(I ) is outer. Proof is the same as that of Lemma 4.20 of [2]. 3. Conditional Expectations Let ω be a state of A possessing the following property. Product Property. : For any disjoint subsets I1 , . . . , Ik of L and for any Ai ∈ A(Ii ) (i = 1, . . . , k), ω(A1 , . . . , Ak ) = ω(A1 ) . . . ω(Ak ).

(3.1)

This property for an arbitrary pair of two disjoint one-point subsets (k = 2, |I1 | = |I2 | = 1) implies (3.1) for the general case because each A(Ii ) is generated by Al , l ∈ Ii . Such a state is called a product state and is denoted as ω= ωi , (3.2) i∈L

where ωi is the restriction of ω to Ai . It is uniquely determined by ωi . It is known ([1],Theorem 1) that such a product state for given ωi , i ∈ L, exists if and only if all ωi with at most one exception are even, i.e. ωi ((Ai )) = ω(Ai )

for Ai ∈ Ai

(3.3)

or equivalently ωi (Ai ) = 0

for Ai ∈ (Ai )−

(3.4)

for all but one i ∈ L. The product state (3.2) is even if and only if all ωi are even ([1],Theorem 1). Throughout this paper, except in Sect. 9, ω is assumed to be an even product state. A typical even product state is the tracial state τ which can be characterized by the following tracial property (see Proposition 8.1): τ (AB) = τ (BA)

for all A, B ∈ A.

(3.5)

Another example is the Fock vacuum in the case of Fermion lattice systems (see Proposition 8.2). Theorem 3.1. Let ω be an even product state. (1) For any subset I of L and any A ∈ A, there exists a unique EIω (A) ∈ A(I ) satisfying ω(B1 AB2 ) = ω(B1 EIω (A)B2 ) for all B1 , B2 ∈ A(I ).

(3.6)

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(2) The map EIω from A to EIω (A) is a conditional expectation from A to A(I ), namely the following holds: (2-1) It is linear, ∗-preserving, positive and unital. (2-2) For B1 , B2 ∈ A(I ), EIω (B1 AB2 ) = B1 EIω (A)B2 .

(3.7)

(2-3) It is a projection of norm 1. (2-4) EIω = EIω . (2-5) If ω is translation invariant, then τk EIω = EIω+k τk ,

(n ∈ L).

(3) The following relation holds: EIω EJω = EJω EIω = EIω∩J .

(3.8)

Namely, the following diagram is a commuting square. EIω

A(I ∪ J ) −−−−→   EJω A(J )

A(I )  E ω J

(3.9)

−−−−→ A(I ∩ J ) EIω

Before presenting the proof of this theorem, we give a result on continuity of EIω on I . For any net Iα of subsets of L, Iα → I means I = ∩β (∪α≥β Iα ) = ∪β (∩α≥β Iα ),

(3.10)

the second equality being the condition for the convergence of the net {Iα }. In particular, if Iα is monotone increasing, its limit is I = ∪α Iα and if Iα is monotone decreasing, its limit is I = ∩α Iα , the convergence being automatic in both cases. Theorem 3.2. If Iα → I , then lim EIωα (A) − EIω (A) = 0 α

(3.11)

for any A ∈ A . In particular, if Iα → L, then lim EIωα (A) − A = 0.

(3.12)

lim EIω = 1.

(3.13)

α

In other words, I →L

The proof of this theorem is exactly the same as that of Theorem 4.11 in [2]. The rest of this section is devoted to the proof of the first theorem. Lemma 3.3. If EIω (A) satisfying (3.6) exists, then it is unique and EIω (A) ≤ A.

(3.14)


121

Proof. Consider the (GNS) triplet consisting of a Hilbert space HωI , a representation πωI of A(I ) and a cyclic unit vector Iω ∈ HωI giving rise to the restriction of the state ω to A(I ). For B1 , B, B2 ∈ A(I ), ω(B1 BB2 ) = ( 1 , πωI (B) 2 ), 1 =

πωI (B1 )∗ Iω ,

2 =

πωI (B2 )Iω ,

(3.15) (3.16)

where { 1 ; B1 ∈ A(I )} and { 2 ; B2 ∈ A(I )} are dense in HωI . If B and B in A(I ) satisfy ω(B1 BB2 ) = ω(B1 B B2 )

(3.17)

for all B1 and B2 in A(I ), then (3.15) implies πωI (B) = πωI (B ).

(3.18)

Hence B = B due to the simplicity of A(I ). This proves the uniqueness. Since 1 2 = ω(B1 B1∗ ) and 2 2 = ω(B2∗ B2 ), we obtain πωI (B) = sup{|ω(B1 BB2 )|/(ω(B1 B1∗ )ω(B2∗ B2 ))1/2 },

(3.19)

where the sup is taken over all B1 and B2 in A(I ) satisfying ω(B1 B1∗ ) = 0 and ω(B2∗ B2 ) = 0. The same formula as (3.15) and (3.16) for I = L imply |ω(B1 AB2 )/(ω(B1 B1∗ )ω(B2∗ B2 ))1/2 | ≤ πωL (A).

(3.20)

Since A(I ) and A are simple, we have πωI (B) = B,

πωL (A) = A.

Hence the above two relations imply (for B = EIω (A)) EIω (A) ≤ A.

(3.21)

The following lemma obviously holds. Lemma 3.4. If EIω (A1 ) and EIω (A2 ) satisfying (3.6) exist, then EIω (c1 A1 + c2 A2 ) satisfying (3.6) exists and is given by EIω (c1 A1 + c2 A2 ) = c1 EIω (A1 ) + c2 EIω (A2 ).

(3.22)

Proof of Theorem 3.1. (1) First we consider A = BC,

B ∈ A(I ),

C ∈ A(I c ),

(3.23)

where I c denotes the complement of I in L. We claim that EIω (A) = ω(C)B

(3.24)

satisfies (3.6). It is enough to check (3.6) for B1 ∈ A(I )σ1 and B2 ∈ A(I )σ2 for all choices of σ1 = ± and σ2 = ±. By the decompositon (2.4), we have C = C+ + C− with Cσ ∈ A(I c )σ (σ = ±) and it is enough to check (3.6) for Cσ , σ = ± instead of C.

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We have ω(B1 AB2 ) = ω(B1 BCσ B2 ) = (σ, σ2 )ω(B1 BB2 Cσ ) = (σ, σ2 )ω(B1 BB2 )ω(Cσ ) = (σ, σ2 )ω(B1 EIω (A)B2 ). If σ = −, then ω(Cσ ) = 0 because ω is even. Hence, (3.6) holds. If σ = +, then (σ, σ2 ) = 1 irrespective of σ2 and (3.6) holds. So (3.24) satisfies (3.6). By the graded commutation relations between elements of A(I ) and A(I c ), any polynomial of a finite number of elements in A(I ) and A(I c ) can be written as a linear combination of the product (3.23). By Lemma 3.4, EωI (A) satisfying (3.6) exists for any element A in the algebraic span of A(I ) and A(I c ). If An is a Cauchy sequence tending to A, and EIω (An ) exists for all n, then EIω (An ) is a Cauchy sequence by Lemma 3.4 and Lemma 3.3. Hence the limit EIω (A) = lim EIω (An ) ∈ A(I ) n

(3.25)

exists in A(I ) and satisfies (3.6). This proves the existence of EIω for all A ∈ A. The uniqueness of EIω (A) is already given by Lemma 3.3. (2) (2-1) The linearity is given by Lemma 3.4. By ω(B1 EIω (A)∗ B2 ) = ω(B2∗ EIω (A)B1∗ ) = ω(B2∗ AB1∗ ) = ω(B1 A∗ B2 ) = ω(B1 EIω (A∗ )B2 ), we obtain EIω (A)∗ = EIω (A∗ ). By (3.15) and (3.16) with B1 = B2∗ , we have ( 2 , πωI (EIω (A∗ A)) 2 ) = ω(B2∗ EIω (A∗ A)B2 ) = ω(B2∗ A∗ AB2 ) ≥ 0. This implies πω (EIω (A∗ A)) ≥ 0 and hence EIω (A∗ A) ≥ 0 by the faithfulness of πω (due to the simplicity of A(I )). Finally EIω (1) = 1 ∈ A(I ) satisfies (3.6) and hence EIω is unital. (2-2) If B1 , B2 , B1 , B2 ∈ A(I ), then ω(B1 B1 EIω (A)B2 B2 ) = ω(B1 B1 AB2 B2 ) = ω(B1 EIω (B1 AB2 )B2 ). The uniqueness and B1 EIω (A)B2 ∈ A(I ) implies (2-2). (2-3) Since EIω is unital, (2-2) implies EIω (B) = B if B ∈ A(I ). Hence EIω (EIω (A)) = EIω (A), namely EIω is a projection. Lemma 3.3 and EIω (1) = 1 imply EIω = 1. (2-4) Since ω is even, we have ω(B1 EIω ((A))B2 ) = ω(B1 (A)B2 ) = ω((B1 (A)B2 )) = ω((B1 )A(B2 )) = ω((B1 )EIω (A)(B2 )) = ω(((B1 )EIω (A)(B2 ))) = ω(B1 (EIω (A))B2 ).


123

By uniqueness, we have EIω ((A)) = (EIω (A)).

(3.26)

(2-5) Due to τk (A(I )) = A(I + k), we have for B1 , B2 ∈ A(I + k) ω(B1 EIω+k (τk (A))B2 ) = ω(B1 τk (A)B2 ) = ω(τ−k (B1 τk (A)B2 )) = ω(τ−k (B1 )Aτ−k (B2 )) = ω(τ−k (B1 )EIω (A)τ−k (B2 )) = ω(τk (τ−k (B1 )EIω (A)τ−k (B2 ))) = ω(B1 τk (EIω (A))B2 ), where τ−k (B1 ), τ−k (B2 ) ∈ A(I ). Hence EIω+k (τk (A)) = τk (EIω (A)). (3) If A ∈ A(K) in the proof of (1), it is enough to take B ∈ A(I ∩K) and C ∈ A(I c ∩K) due to A(K) = A((I ∩ K) ∪ (I c ∩ K)). Hence we have EIω (A) ∈ A(I ∩ K)

(3.27)

if A ∈ A(K). On the other hand, for B1 , B2 ∈ A(I ∩ J ), ω(B1 EIω∩J (A)B2 ) = ω(B1 AB2 ) = ω(B1 EJω (A)B2 ) = ω(B1 EIω (EJω (A))B2 ),

(3.28) (3.29)

where the second equality is due to B1 , B2 ∈ A(J ) and the third due to B1 , B2 ∈ A(I ). Hence EIω∩J = EIω EJω by uniqueness. By interchanging the role of I and J, we also obtain EIω∩J = EJω EIω . 4. ω-Standard Potential We use notation in [2]. We start with the real vector space (A0 ) of all ∗-derivations δ with domain A0 and commuting with . If a dynamics αt of A (i.e. a continuous one-parameter group of automorphisms) satisfies Assumption I : αt = αt . Assumption II : The domain of the generator δα of αt contains A0 , then the restriction of δα to A0 is in (A0 ). We consider the real vector space Hω of functions H ω of finite subsets I of L with values H ω (I ) in A satisfying the following properties. (The vector space structure is taken to be that of a function space with values in a vector space.) (H-1)ω H ω (I )∗ = H ω (I ) ∈ A, (H-2)ω (H ω (I )) = H ω (I ) (i.e. H ω (I ) ∈ A+ ), (H-4)ω EIωc (H ω (I )) = 0, (H-5)ω H ω (I ) = H ω (J ) − EIωc (H ω (J )) for I ⊂ J ⊂⊂ L. Theorem 4.1. The following relation between H ω ∈ Hω and δ ∈ (A0 ) gives a bijective, real linear map from Hω to (A0 ). (H-3)ω δA = i[H ω (I ), A] (A ∈ A(I )).

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The proof is the same as that of Theorem 5.7 in [2]. The operator H ω (I ) will be called the ω-standard local Hamiltonian for the region I (for a given δ). The internal energy is defined by U ω (I ) = EIω (H ω (I ))

(A ∈ A(I )).

(4.1)

The local Hamiltonians H ω (I ) are recovered from the family { U ω (I )} as follows. H ω (I ) = lim {U ω (J ) − EIωc (U ω (J ))}. J L

(4.2)

Definition 4.1. A function ω of a finite subset of L with values in A is called an ω-standard potential if it satisfies the following conditions. (-a) ω (I ) ∈ A(I ), ω (∅) = 0. (-b) ω (I )∗ = ω (I ). (-c) (ω (I )) = ω (I ). (-d) EJω (ω (I )) = 0 if J ⊂ I and J = I . (-e) For each I ⊂⊂ L, the net HJω (I ) =

{ω (K); K ∩ I = ∅, K ⊂ J }

(4.3)

K

is a Cauchy net in the norm topology of A for J → L. (The index set of the net is the family of all finite subsets J of L partially ordered by the set inclusion.) The vector space of all ω-standard potentials is denoted by P ω . Remark 4.1. The condition (-d) is equivalent to the following condition due to (-a):

(-d)

EJω (ω (I ))

=

ω (I ), if I ⊂ J , 0, otherwise.

(See the Remark to Definition 5.10 in [2].) Theorem 4.2. (1) The following equation gives a bijective, real linear map from ω ∈ P ω to H ω ∈ Hω . H ω (I ) = lim {ω (K); K ∩ I = ∅, K ⊂ J }. (4.4) J L

K

(2) The relations (H-3)ω and (4.4) give a bijective, real linear map from ω ∈ P ω to δ ∈ (A0 ). Remark 4.2. The following relations hold: U ω (I ) = K⊂I ω (K), ω (I ) = K⊂I (−1)|I |−|K| U ω (K).

(4.5) (4.6)

The proof of these theorem and remarks are the same as those of Theorems 5.12 and 5.13 in [2].


125

5. Gibbs and LTS Conditions We call a function of a finite subset I of L with values (I ) in A a general potential if the conditions (-a), (-b), (-c) and (-e) of Definition 4.1 are satisfied (where ω there is replaced by ). Then the relations H (I ) = limJ L {(K); K ∩ I = ∅, K ⊂ J }, (5.1) δ (A) = i[H (I ), A], (A ∈ A(I )) (5.2) defines δ ∈ (A0 ) consistently. Consistency means [H (I ), A] = [H (J ), A]

(5.3)

if A ∈ A(I ) ∩ A(J ) (= A(I ∩ J )). If δ1 = δ2 , then 1 and 2 are said to be equivalent. For a given δ ∈ (A0 ), the corresponding ω-standard potential ω satisfies δω = δ, and hence ω for different ω’s and for a fixed δ are equivalent potentials. We shall now check that each of Gibbs and LTS conditions, which are possible characterization of equilibrium states, are mutually equivalent for equivalent potentials. Definition 5.1. A state ϕ of A satisfies (, β)-Gibbs condition for a general potential and β ∈ R if the following two conditions hold: (1) It is modular (i.e. its extention to the weak closure πϕ (A) of the (GNS) cyclic representation πϕ of A is separating). (2) The (GNS) representing operators πϕ (A(I )) is in the centralizer of the perturbed functional ϕ h for the perturbation h = βH (I ), i.e. they are elementwise invariant under the modular automorphism group of ϕ h . Equivalently, ϕ h (AB) = ϕ h (BA)

(5.4)

for any A ∈ A(I ) and any B ∈ A. (See Definition 7.1 with conditions (D-1) and (D-2)’ in [2].) If 1 and 2 are equivalent general potentials, we have [H1 (I ), A] = [H2 (I ), A]

(5.5)

H = βH1 (I ) − βH2 (I ) ∈ A(I ) .

(5.6)

for all A ∈ A(I ) and hence For ϕi = ϕ Hi where Hi = βHi (I ), we have ϕ2 = (ϕ1 )− H ,

ϕ1 = (ϕ2 ) H .

(5.7)

ϕ

Hence, for modular automorphisms σt i , we have d ϕ d ϕ1 σt (πϕ (A)) = σt 2 (πϕ (A)) + iπϕ ([ H, A]) dt dt d ϕ = σt 2 (πϕ (A)) dt ϕ

(5.8) (5.9)

d at t = 0 for A ∈ A(I ). Therefore the vanishing of dt σt i (πϕ (A)), which is necesary and sufficient for the validity of the condition (2) of the Gibbs condition for ϕi , is equivalent for i = 1 and i = 2. This proves that a state satisfies the (1 , β)-Gibbs condition if and only if it satisfies the (2 , β)-Gibbs condition.

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Definition 5.2. (1) A state ϕ satisfies the (, β)-LTSM condition if S˜IM (ϕ) − βϕ(H (I )) ≥ S˜IM (ψ) − βψ(H (I ))

(5.10)

for each finite subset I and for all states ψ with the same restriction to A(I ) as the state ϕ. (2) A state ϕ satisfies the (, β)-LTSP condition if the above condition (5.10) with M replaced by P holds for all states ψ which have the same restriction to A(I c ) as ϕ. Here S˜IM and S˜IP are conditional entropy, independent of the potential. (See [3]; M and P refer to mathematical and physical.) The equivalence of LTS conditions for equivalent general potentials is already obtained in Corollary 5 of [3] with its proof in Sect. 4 of [3]. Thus we have the following. Theorem 5.1. Let 1 and 2 be equivalent general potentials. (1) The (i , β)-Gibbs conditions for i = 1 and i = 2 are equivalent. (2) Each of (i , β)-LTSM conditions and (i , β)-LTSP conditions for i = 1 and i = 2 are equivalent.

6. Translation Invariance A dynamics αt is said to be translation invariant if the following holds. Asumption IV : τk αt = αt τk for any t ∈ R and k ∈ L. This assumption implies Assumption I in Sect. 4. (See Proposition 8.1 of [2].) A ∗-derivation δ ∈ (A0 ) is said to be translation invariant if τk δ = δτk for all k ∈ L. The real vector subspace of (A0 ) consisting of all translation invariant δ ∈ (A0 ) will be denoted by τ (A0 ). A potential ∈ P ω is said to be translation covariant if (-f ) τk ((I )) = (I + k) for all I ⊂⊂ L and k ∈ L. The real linear subspace of P ω consisting of all translation covariant ∈ P ω is denoted by Pτω . Theorem 6.1. The bijection of Theorem 4.2(2) maps τ (A0 ) onto Pτω . The proof is by a straightforward computation. (See Corollary 8.5 of [2].) A potential is said to be of finite range if there is a positive d ∈ R such that (I ) = 0 whenever the maximum distance of two points in I exceeds d. Theorem 6.2. With respect to := ({n}),

∈ Pτω ,

(6.1)

which is independent of a point n ∈ L and is a norm, Pτω is a separable Banach space, in which the subspace of all finite range potentials is dense.


127

The proof is the same as those of Proposition 8.8, Proposition 8.12 and Corollary 8.13 of [2]. The following energy estimates can be shown by the same proof as those of Lemmas 8.6 and 9.1 of [2], where W ω (I ) = H ω (I ) − U ω (I ) = lim {ω (K); K ∩ I = ∅, K ∩ I c = ∅, K ⊂ J }. J →∞

(6.2) (6.3)

Theorem 6.3. U ω (I ) ≤ H ω (I ) ≤ |I |ω , v.H. limI →∞ |I1| W ω (I ) = 0.

(6.4) (6.5)

Here, v.H. limI →∞ denotes the van Hove limit. (See Appendix of [2].) 7. Variational Principle Theorem 7.1. For a translation invariant state ϕ of A and ω ∈ Pτω , the following limits exist. p(ω ) = v.H. lim |I |−1 log τ (eH

ω (I )

I →∞

eω (ϕ) = v.H. lim ϕ(H ω (I ))/|I | = I →∞

) = v.H. lim |I |−1 log τ (eU

ω (I )

I →∞ v.H. lim ϕ(U ω (I ))/|I |, I →∞

sˆ (ϕ) = v.H. lim SÎ (ϕ)/|I |, I →∞

), (7.1) (7.2) (7.3)

where SÎ (ϕ) = −τ (ρˆϕI log ρˆϕI ) for the adjusted density matrix ρˆϕI of the restriction of ϕ to A(I ), characterized by ρˆϕI ∈ A(I ) and ϕ(A) = τ (ρˆϕI A) for all A ∈ A(I ). The proof is the same as those of Theorems 9.3, 9.5 and 10.3 in [2]. A translation invariant state ϕ satisfies the (, β)- variational principle if p(β) = sˆ (ϕ) − βe (ϕ).

(7.4)

By Proposition 14.1 of [2], ϕ is a solution of (7.4) for a general potential if and only if it is a solution of (7.4) for the τ -standard potential τ equivalent to , under the condition (14.2) and (14.3) of [2] for . For ω-standard potential ω , the condition (14.2) of [2] is fulfilled due to (6.5) and the limit (14.3) of [2], being eω (τ ), converges. Therefore the solutions of the (ω , β)-variational principle coincide with those of the (τ , β)-variational principle and hence the solution set is independent of ω. Thus we have established the following result. Theorem 7.2. For any pair of even product states ω and ω , and for the ω- and ω standard potentials ω and ω corresponding to the same δ ∈ (A0 ), a translation invariant state ϕ is a solution of the (ω , β)-variational principle if and only if it is a solution of the (ω , β)-variational principle.

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8. Examples of Equivalent ω-Standard Potentials for Different ω A potential (I ) belongs to A(J ) if J ⊃ I . Hence a part of (I ) may be taken out and included in the potential (J ) without changing the dynamics (more specifically, without changing the corresponding derivation). This is the origin of the existence of equivalent potentials. In this section, we illustrate this by taking two different even product states ω and comparing the potentials ω for the same derivation δ. The following two propositions provide examples of even product states. Proposition 8.1. The tracial state τ of A is an even product state. Proof. The tracial state of A is unique and hence invariant under any automorphism. In particular it is even and τ (A) = 0 for any odd A. Let i = j and Aσ ∈ (Ai )σ , Bσ ∈ (Aj )σ (σ, σ = ±). It is enough to show τ (Aσ Bσ ) = τ (Aσ )τ (Bσ )

(8.1)

for all pairs i, j and all combinations of σ = ± and σ = ±. Consider the case σ = − first. Then the right-hand side of (8.1) vanishes. If σ = +, then the left-hand side also vanishes because A+ B− is odd. If σ = −, then τ (A− B− ) = τ (B− A− ) = −τ (A− B− ) = 0

(8.2)

due to the tracial property of τ and the anti-commutativity of A− and B− . Therefore, (8.1) holds when σ = −. Consider the case σ = +. For any A1 , A2 ∈ Ai , τ (A1 A2 B+ ) = τ (A2 B+ A1 ) = τ (A2 A1 B+ )

(8.3)

due to the tracial property of τ and the commutativity of A1 and B+ . Hence τ ([A1 , A2 ]B+ ) = 0.

(8.4)

Since Ai is isomorphic to a full matrix algebra, its element A is a sum of τ (A)1 and commutators of elements of Ai . Hence (8.1) holds for the present case too. Proposition 8.2. For Fermion algebras (the case where each Ai is generated by a finite ∗ and a ), the vacuum state number of Fermion creation and annihilation operators aiα iα ω0 (uniquely characterized by ∗ aiα ) = 0 ω0 (aiα

(8.5)

for all i and α) is an even product state. Proof. Let the restriction of ω0 to Ai be ω0i . Since (8.5) is invariant under the transposed action ω → ω of on states ω, ω0 as well as all ω0i are even. Then i ω0i is a state of A satisfying (8.5) and hence coincides with ω0 . Therefore, ω0 is a product state.


129

Example of Equivalent Potentials. Consider the algebra A generated by Fermion creation and annihilation operators ai∗ and ai , i ∈ L, which is studied in [2]. We give below the ω-standard (one-body and two body) potentials for the same dynamics and two different even product states: ω = τ (the tracial state) and ω = ω0 (the Fermion vacuum state). They give examples of equivalent potentials caused by different choices of ω. (1) One-body ω-standard potentials. For ω = τ : τ ({i}) = c(ai∗ ai − ai ai∗ ),

(c = 0).

(8.6)

For ω = ω0 : ω0 ({i}) = 2cai∗ ai ,

(c = 0).

(8.7)

They are related by τ ({i}) − ω0 ({i}) = −2c1 ∈ A(∅) = C 1.

(8.8)

Since a multiple of the identity operator does not give any contribution to its commutator and hence to the corresponding derivation, the above τ -standard and ω0 -standard one-body potentials are equivalent. (2) Two-body ω-standard potentials. For ω = τ : Let i = j . τ ({i, j }) = c1 (ai aj − ai∗ aj∗ ) + c2 (ai aj∗ − ai∗ aj ) + c3 (ai ai∗

− ai∗ ai )(aj aj∗

− aj∗ aj ).

(8.9) (8.10)

For ω = ω0 : Let i = j . ω0 ({i, j }) = c1 (ai aj − ai∗ aj∗ ) + c2 (ai aj∗ − ai∗ aj ) + 4c3 ai∗ ai aj∗ aj .

(8.11)

They are related by τ ({i, j }) − ω0 ({i, j }) = −2c3 aj∗ aj − 2c3 ai∗ ai + c3 1 =

−c3 (aj∗ aj

− aj aj∗ ) − c3 (ai∗ ai

(8.12) − ai ai∗ ) − c3 1.

(8.13)

Namely the difference is expressed as a sum of ω0 -standard one-body potentials at lattice sites i and j , and also as a sum of τ -standard one-body potentials at lattice sites i and j , both modulo multiples of the identity operator. Therefore, the above τ -standard two-body potential is equivalent to the above ω0 -standard two-body potential combined with ω0 -standard one-body potentials at lattice sites i and j , and conversely the ω0 -standard two-body potential is equivalent to the τ -standard two-body potential combined with (−1) times τ -standard one-body potentials at lattice sites i and j . In the case of translation covariant potentials, we will have (covariantly related) τ -standard two-body potentials at all shifted pairs {i + n, j + n} of lattice sites, n ∈ L. They are then equivalent to (covariantly related) ω0 -standard two-body potentials at all shifted pairs {i + n, j + n} of lattice sites, n ∈ L combined with twice ω0 -standard one-body potentials at all lattice sites (twice because any site will appear as i + n once and as j + n another time). Similarly, (covariantly related) ω0 -standard two-body potentials at all shifted pairs {i + n, j + n} of lattice sites are equivalent to (covariantly related) τ -standard two-body potentials at all shifted pairs {i +n, j +n} of lattice sites combined with (−2) times τ -standard one-body potentials at all lattice sites.

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9. Non-Even Product State and Conditional Expectations We give a necessary and sufficient condition for the extistence of ω-conditional expectation when one of the factor states of ω is not even. A state ωI of A(I ) will be called an eigenstate of u ∈ A(I ) belonging to an eigenvalue λ if ωI (Au) = λωI (A)

(9.1)

for all A ∈ A(I ). We use the notation (I2 )− defined by Definition 2.1. Theorem 9.1. Let I1 and I2 be mutually disjoint non-empty subsets of L and I = I1 ∪I2 . Let ωi be a state of A(Ii )(i = 1, 2) and a state ω of A(I ) be a product state of ω1 and ω2 . Assume that ω1 is not even. (1) There exists the unique ω-conditional expectation EIω1 from A(I ) onto A(I1 ) in the sense of Theorem 3.1. (2) No ω-conditional expectation EIω2 from A(I ) onto A(I2 ) in the sense of Theorem 3.1 exists if (I2 )− is infinite. (3) Assume that (I2 )− is finite. Let u2 ∈ A(I2 ) be a selfadjoint unitary implementing restricted to A(I2 ) (which exists). An ω-conditional expectation EIω2 from A(I ) onto A(I2 ) in the sense of Theorem 3.1 exists if and only if ω2 is an eigenstate of u2 . It is unique if it exists. Proof. By Theorem 1 of [1], ω2 must be even (in order that a product state of a non-even ω1 and ω2 exists). (1) The proof of Theorem 3.1 goes through without any change. (2) Let A be an odd element of A(I1 ) such that ω1 (A) = 0. (Such an A exists because ω1 is assumed to be not even.) Assuming that the ω-conditional expectation EIω2 from A(I ) onto A(I2 ) exists, we show a contradiction. Set x = EIω2 (A). It has the following properties: (α) x = 0 because ω(x) = ω(A) = ω1 (A) = 0.

(9.2)

(The first equality is due to (3.6) with B1 = B2 = 1.) (β) x ∈ A(I2 ) by the defining property of EIω2 . (γ ) By the property (2-2) of EIω2 in Theorem 3.1, we have the following relation for any B ∈ A(I2 ). xB = EIω2 (A)B = EIω2 (AB) = EIω2 ((B)A) = (B)EIω2 (A) = (B)x, where the third equality is by the graded commutation relations. By Lemma 2.6, x = 0. This contradicts with (α). (3) For sufficiency proof, assume that ω is an eigenstate of a selfadjoint unitary u2 ∈ A(I2 ) which implements on A(I2 ). (Due to Lemma 2.3, the existence of such a u2 follows from the assumption that (I2 )− is finite.) Then the eigenvalue is either 1 or -1 due to (u2 )2 = 1. By choosing u2 from ±u2 , we may assume that the eigenvalue is 1. For Aσ = BCσ with B ∈ A(I2 ), Cσ ∈ A(I1 )σ , we set EIω2 (A+ ) = ω1 (C+ )B,

EIω2 (A− ) = ω1 (C− )Bu2 ,

(9.3)


131

and show that they satisfy (3.6) by the following computations, thereby showing that their linear combination gives an ω-conditional expectation from A(I ) onto A(I2 ) due to Theorem 3.1, ω(B1 A+ B2 ) = ω(B1 BB2 C+ ) = ω2 (B1 BB2 )ω1 (C+) = ω2 (B1 (ω1 (C+ )B)B2 ), ω(B1 A− B2 ) = ω(B1 BC− B2 ) = ω(B1 B(B2 )C− ) = ω2 (B1 Bu2 B2 u2 )ω1 (C− ) = ω2 (B1 (ω1 (C− )Bu2 )B2 ) where the last equality is due to the assumption that ω2 is an eigenstate of u2 belonging to an eigenvalue 1. For necessity proof, assume that EIω2 exists. Let C− ∈ A(I1 )− be such that ω1 (C− ) = 0 and set x = EIω2 (C− ). It satisfies the properties (α), (β), and (γ ) (except for the conclusion x = 0 of (γ )) in the proof of (2). In particular, (γ ) implies that u2 x ∈ A(I2 ) commutes with all B ∈ A(I2 ) and hence belongs to the center of A(I2 ), which is trivial by Lemma 2.1. Hence u2 x = c1 and x = cu2 for some scalar c. By the same computation as in the sufficiency proof, we obtain the following relation for any B1 , B2 ∈ A(I2 ). ω2 (B1 xB2 ) = ω(B1 C− B2 ) = ω(B1 (B2 )C− ) = ω2 (B1 u2 B2 u2 )ω1 (C− ) = ω2 (B1 ω1 (C− )u2 B2 u2 ).

(9.4) (9.5) (9.6)

Since x = cu2 , we have ω1 (C− ) = ω(C− ) = ω2 (x) = cω2 (u2 ).

(9.7)

By ω1 (C− ) = 0, c = 0. Hence Eq. (9.4) = (9.5) with B2 = 1 and Eq. (9.7) give ω2 (B1 u2 ) = ω2 (B1 )ω2 (u2 ) for all B1 ∈ A(I2 ). Hence ω2 is an eigenstate of u2 .

(9.8)

Acknowledgement. This work has been completed during a visit to the University of Florida, made possible through the financial support of the Institute of Fundamental Theory. The author gratefully acknowledge hospitality by members of the Department of Physics and Department of Mathematics, in particular by Professor John Klauder and by Professor Gerald Emch.

References 1. Araki, H., Moriya, H.: Joint extension of states of subsystems for a CAR system. Commun. Math. Phys. 237, 105–122 (2003) 2. Araki, H., Moriya, H.: Equilibrium statistical mechanics of Fermion lattice systems. Rev. Math. Phys. 15, 93–198 (2003) 3. Araki, H., Moriya, H.: Local thermodynamical stability of Fermion lattice systems. Lett. Math. Phys. 62, 33–45 (2002) 4. Araki, H.: On KMS states of a C ∗ -dynamical system. Lecture Notes in Math, 650, Berlin-HeidelbergNew York: Springer-Verlag, 1978 5. Araki, H.: Toukeirikigaku no suuri, Iwanami (in Japanese), 1994 6. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics 2. 2nd edition. Berlin-Heidelberg-New York: Springer-Verlag, 1996 7. Israel, R.B.: Convexity in the Theory of Lattice Gases. Princeton, NJ: Princeton University Press, 1979

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8. Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras 2. London-New York: Academic Press, 1986 9. Simon, B.: The Statistical Mechanics of Lattice Gases. Princeton, NJ: Princeton University Press, 1993 10. Takesaki, M.: Theory of Operator Algebras 1. Berlin-Heidelberg-New York: Springer-Verlag, 2003 Communicated by M.B. Ruskai


Communications in


Boltzmann Equation: Micro-Macro Decompositions and Positivity of Shock Profiles Tai-Ping Liu1,2 , Shih-Hsien Yu3 1 2

Institute of Mathematics, Academia Sinica, Taipei, Taiwan R.O.C. Department of Mathematics, Stanford University Stanford, CA 94305, USA. E-mail: [email protected] 3 Department of Mathematics, City University of Hong Kong, Hong Kong, P. R. China. E-mail: [email protected]

Received: 24 July 2003 / Accepted: 29 August 2003 Published online: 29 January 2004 – © Springer-Verlag 2004

Abstract: We introduce an elementary energy method for the Boltzmann equation based on a decomposition of the equation into macroscopic and microscopic components. The decomposition is useful for the study of time-asymptotic stability of nonlinear waves. The wave location is determined by the macroscopic equation. The microscopic component has an equilibrating property. The coupling of macroscopic and microscopic components gives rise naturally to the dissipations similar to those obtained by the Chapman-Enskog expansion. Our main result is the establishment of the positivity of shock profiles for the Boltzmann equation. This is shown by the time-asymptotic approach and the maximal principle for the collision operator.

Contents 1. 2. 3. 4. 5.

Introduction . . . . . . . . . . . . . . . . . . Macro-Micro Decomposition . . . . . . . . . Representation of the Macroscopic Variables . Energy Estimates for Linear Equation . . . . Nonlinear Stability of Planar Waves . . . . . 5.1 Lower Order Energy Estimates . . . . . 5.2 Higher Order Energy Estimates . . . . . 6. Shock Profiles . . . . . . . . . . . . . . . . . 7. Local Macroscopic and Microscopic Variables 8. Basic Matrix Representations . . . . . . . . . 9. Nonlinear Stability of Shock Profiles . . . . . 10. Positivity of Shock Profiles . . . . . . . . . .

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The research of the first author was supported by the Institute of Mathematics, Academia Sinica, Taipei and NSC #91-2115-M-001-004. The research of the second author was supported by the SRG of City University of Hong Kong Grant #7001426.

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Appendix A. Chapman-Enskog Expansion . . . . . . . Appendix B. Estimates on Collision Operators . . . . . Appendix C. Construction of Boltzmann Shock Profile References . . . . . . . . . . . . . . . . . . . . . . . .

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168 172 175 178

1. Introduction Consider the Boltzmann equation, [2], Ft + ξ · ∇x F = Q(F, F ).

(1.1)

The goal of the present paper is, first, to introduce decompositions of the Boltzmann equation into microscopic and macroscopic parts. We then devise an energy method for the study of time-asymptotic stability of nonlinear waves. Our main result is the establishment of the positivity of shock profiles for the Boltzmann equation using the time-asymptotic approach. As usual, the macro-micro decomposition of a solution to the Boltzmann equation is made with respect to local Maxwellian states, [14]: −

e ω(ξ ; u0 , T0 ) ≡

|ξ −u0 |2 2T0

(2πT0 )3

for u0 , ξ ∈ R 3 ; and T0 ∈ R + ,

where the macroscopic velocity u0 , and temperature T0 of the local thermal equilibrium state ω may be varying. Consider the perturbation, as customarily expressed, [8, 4, 7]: 1

F (x, t, ξ ) ≡ ω(ξ ; u0 , T0 ) + ω 2 (ξ ; u0 , T0 ) f (x, t, ξ ).

(1.2)

Here ρ0 is the macroscopic density. The collision invariants:  χ0 (ξ ) ≡ 1,     ξ i − ui0    χ (ξ ) ≡ for i = 1, 2, 3, √ i   T0 1 |ξ − u0 |2  χ4 (ξ ) ≡ √ −3 ,    T0 6      χi (ξ )Q(F, F )(ξ )dξ = 0, i = 0, · · · , 4, R3

are normalized with respect to the Maxwellian ω(ξ ; u0 , T0 ); ω(ξ ; u0 , T0 )χi (ξ )χj (ξ )dξ = δij , i, j = 0, · · · , 4. R3

The perturbation is decomposed into the macroscopic and microscopic parts. The macroscopic part f0 is in the range of the projection operator P 0 on the space spanned by χi ω1/2 , i = 0, · · · , 4,; and the microscopic part is in the range of the orthogonal projection P 1 :   f (x, t, ξ ) = f0 (x, t, ξ ) + f1 (x, t, ξ ),    f0 = P 0 f, f1 = P 1 f, 3

1  i   m (x, t)χi (ξ ) + e(x, t)χ4 (ξ ) ω 2 (ξ ; u0 , T0 ).  f0 (x, t, ξ ) ≡ ρ(x, t) + i=1

Boltzmann Equation: Micro-Macro Decompositions and Positivity of Shock Profiles

135

The linearized collision operator 1

1

1

L(h) ≡ ω− 2 [Q(ω, ω 2 h) + Q(ω 2 h, ω)], has two basic properties: The macroscopic part is in the null space: Lf0 = 0. Another basic property, studied by [4] for the hard sphere and the refinement of cutoff potentials by [7], is that the linearized collision operator is negative definite on the microscopic part: f1 Lf1 dξ < −ν0 (f1 )2 dξ (1.3) R3

R3

for some positive constant ν0 . The Boltzmann equation is decomposed into macroscopic and microscopic equations, cf. (2.10):

f0t + P 0 ξ · ∇x (f0 + f1 ) = 0, f1t + P 1 ξ · ∇x (f0 + f1 ) = (1 + ρ)Lf1 + N (f ).

(1.4)

Here, (1.3) is a dissipative mechanism driving the non-equilibrium state towards the equilibrium state. It resembles the Boltzmann H-theorem. We will apply the basic energy method to both equations. Our energy method will yield naturally the following expression of the dissipation for the macroscopic part: −P 0 ξ · ∇x L−1 P 1 (ξ · ∇x f0 ) 3

= −P 0 ξ · ∇x L−1 P 1 (ξ · ∇x mi ) χi + P 1 (ξ · ∇x e) χ4 .

(1.5)

i=1

The right-hand side of this identity represents the dissipation of the momentum and the energy, but not the mass. This is consistent with the compressible Navier-Stokes equations. The Navier-Stokes equations are an approximation to the Boltzmann equation through the Chapman-Enskog expansion, [5]. The above is an exact dissipation expression. The Equilibrating Property (1.3) is used for the microscopic equation, while the dissipation expression (1.5) arises when we apply the energy method to the macroscopic equation. In the study of nonlinear waves, we make the decomposition around the local Maxwellian states corresponding to the waves. The macroscopic component determines the wave propagation. For instance, it dictates the phase location of the traveling waves. The microscopic component is the faster decaying part. The geometry of the wave dictates its stability and comes up naturally in the convection term of the macroscopic equation. These are the basic understandings when carrying out the energy estimates. In Sect. 2, we first briefly recall the basics of the Boltzmann equation and introduce the macro-micro decomposition. The elementary relationship between the macroscopic and microscopic components, particularly the dissipation expression (1.5) are presented in Sect. 3. Basic energy estimates are carried out in Sect. 4 for the linear equation. The nonlinear stability of the Maxwellian states is shown in Sect. 5. The time-asymptotic stability of the Maxwellian states has been shown previously using a different energy method based on the Fourier transform and spectral analysis,

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[17, 18, 15, 10]. These studies yield stronger results with rates of convergence. Our thinking is that the energy method and the more quantitative method based on the spectral analysis, or equivalently, the Green functions, should be viewed as separate approaches. The present energy method is an elementary and basic approach. Our method is also being applied to the study of supersonic boundary layers, [19]. We then study the positivity of shock profiles. In [3] the profiles are constructed by solving the equation −sF + ξ1 F = Q(F, F ). Here s is the speed of the shock and the space variable is now taken to be one dimensional, x ∈ R 1 . In that approach, the Boltzmann shock profiles are approximated by the Navier-Stokes shock profiles. The approach does not yield the positivity of the shocks due to the polynomial perturbation to the exponentially small Gaussian tail in the momentum variable ξ . Our idea is to show the positivity using the time-asymptotic stability analysis. The positivity property is necessary for the profile to be physical, both for the boundedness of the entropy density f log f and for f to be the distribution function. We take as initial data a positive approximation of the exact profile with the same total macroscopic variables, Sect. 6. In Sect. 7 we present the local macroscopic and microscopic decomposition based on local Maxwellians. The decomposition is based on the Navier-Stokes profile through the Chapman-Enskog expansion. The fluid speeds are studied in Sect. 8 and the stability of shocks shown in Sect. 9. It is well-known that positive initial data for the Boltzmann equation yield positive solutions. Thus the time-asymptotic stability of the exact profile yields its non-negativity. The positivity is then shown by a maximum principle for the full Boltzmann equation satisfied by the profile, Sect. 10. We choose the perturbation to have zero total macroscopic variables. This choice allows us to study the positivity of the shock profiles using only the energy method. For the energy method for viscous conservation laws, see [6], [11], [16]. Without the zero macroscopic variables condition, there should be interaction of fluid waves of distinct families. For the study of the interactions for viscous conservation laws using pointwise estimates see [13]. It would be interesting to study the nonlinear stability of shock profiles with general perturbation. This is, however, left to the future. The stability analysis for the shock profile demands a sufficient accuracy of the approximate profile. The approximate profile is obtained by Chapman-Enskog expansion. Instead of using the Lyapunov-Schmidt process as in [3], we apply the weighted energy estimate method to macro-micro equations for the construction of the shock profiles. The method requires the detailed study of the tail behavior of the approximate profiles. For this, we make good uses of the exact expression of the dissipation parameters and collision frequency ,[5]. The analysis is carried out in the Appendix for the hard spheres and yields the needed stronger accuracy estimate. 2. Macro-Micro Decomposition Consider the Boltzmann equation Ft + ξ · ∇x F = Q(F, F ), and a Maxwellian state −

e ω(ξ ) ≡

|ξ −u0 |2 2T0

(2πT0 )3

for ξ ∈ R 3 .

(2.1)


137

The collision invariants χi (ξ ), i = 0, · · · , 4, are normalized with respect to the Maxwellian state: χi Q(g, h) dξ = 0 for i = 0, · · · 4, (2.2) R3

 χ0 (ξ ) ≡ 1,     ξ i − ui  χi (ξ ) ≡ √ 0 for i = 1, 2, 3, T0    1 |ξ − u0 |2  χ4 (ξ ) ≡ √ −3 , T0 6 1

1

ω 2 χi , ω 2 χj = δij . Here , is the inner product of the Hilbert space, H ≡ L2 (R 3 ), for the momentum variable ξ : g, h ≡

R3

g h dξ for g, h ∈ H.

The Maxwellian ω satisfies Q(ω, ω) = 0, and the Boltzmann solution around ω satisfies 1

F (x, t, ξ ) ≡ ω(ξ ; u0 , T0 ) + ω 2 (ξ ; u0 , T0 ) f (x, t, ξ ) for x = (x 1 , x 2 , x 3 ) ∈ R 3 , ft + ξ · ∇x f − Lf = N (f ), 1

1

1

L(h) ≡ ω− 2 [Q(ω, ω 2 h) + Q(ω 2 h, ω)], N (f ) ≡ ω

− 21

1 2

(2.3)

1 2

Q(ω f, ω f ).

The linearized collision operator L is self-adjoint, non-positive, and has the same collision invariants as Q, [7]: Lg, h = g, Lh, Lg, g ≤ 0, 1 2

L(ω χi ) = 0, i = 0, · · · , 4. 1

(2.4) (2.5)

In fact, ω 2 χi , i = 0, · · · , 4 span the null space of L. We denote by P 0 the projection on this null space and P 1 the complementary projection:  f0 ≡ P 0 f,   

1 1   P 0 f ≡ 4i=0ω 2 χi , f ω 2 χi ,   1 

  f0 (x, t, ξ ) ≡ ρ(x, t) + 3i=1 mi (x, t) χi (ξ ) + e(x, t) χ4 (ξ ) ω 2 ,       f = P 1 f, P 1 f ≡ f − f0 ,   1 P0 ≡ Range(P 0 ),   P1 ≡ P0⊥ ,    1  ρ(x, t) ≡ ω 2 χ0 , f ,      i (x, t) ≡ ω 21 χ , f for i = 1, 2, 3,  m i     e(x, t) ≡ ω 21 χ , f . 4

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T.-P. Liu, S.-H. Yu

It follows that LP 0 f = Lf0 = 0, Lf = LP 1 f = Lf1 . Here, we introduce new norms for f (x, t, ξ ):   f 2 (x, t) ≡ f, f 21 , L ξ

 f L∞ (L2 ) ≡ sup(x,t)∈R×R + f L2 (x, t). x,t

ξ

ξ

Remark 2.1. The variables f0 and f1 , respectively, are called the macroscopic and microscopic components of f . The macroscopic component is often also called the fluid component; and the microscopic component the non-fluid component. The Hilbert space H is decomposed into the macroscopic space P0 ≡ ker(L) and its orthogonal complement P1 . Thus f0 ∈ P0 and f1 ∈ P1 . The decomposition is standard. The main interest here is the decomposition of the Boltzmann equation and the energy estimates to be followed. For hard sphere, [4], and Grad’s cutoff potentials, [7], L is the sum of a multiplication operator −ν(ξ ), the collision frequency, and a compact operator K: Lh(ξ ) = −ν(ξ )h(ξ ) + K(h)(ξ ). The function ν(ξ ) has a positive lower bound. These facts, (2.4) and (2.5) imply that there exists ν0 > 0 such that any h ∈ P1 , h, Lh ≤ −ν0 h, h.

(2.6)

We will write the negative operator in the space P1 as L¯ ≡ L|P ≤ −ν0 . 1

The linearized Boltzmann equation ft + ξ · ∇x f = Lf

(2.7)

is decomposed into macroscopic and microscopic equations: f0t + P 0 ξ · ∇x f0 = −P 0 ξ · ∇x f1 , f1t + P 1 ξ · ∇x f1 − Lf1 = −P 1 ξ · ∇x f0 ,

(2.8)

It will be useful later to have from the second equation the following expression for the function f1 : f1 = L¯ −1 (f1t + P 1 ξ · ∇x (f0 + f1 )) , and rewrite the first equation as f0t + P 0 ξ · ∇x f0 + P 0 ξ · ∇x L¯ −1 (f1t + P 1 ξ · ∇x (f0 + f1 )) = 0. Similarly, the full Boltzmann equation (2.3) is decomposed as: f0t + P 0 ξ · ∇x (f0 + f1 ) = 0, f1t + P 1 ξ · ∇x (f0 + f1 ) = (1 + ρ)Lf1 + N (f ).

(2.9)

(2.10)

The nonlinear term N(f ) is expressed as follows: 1

1

1

N (f ) ≡ ω− 2 Q(ρ ω + (h + f1 )ω 2 , ρ ω + (h + f1 )ω 2 ) = ρ(x, t) Lf1 + N(f ), 1

1

1

N (f ) ≡ ω− 2 Q(ω 2 (h + f1 ), ω 2 (h + f1 )),

(2.11) 1 2

where the perturbation has been written as f = (ρ(x, t) + h(x, t, ξ ))ω + f1 (x, t, ξ ).


139

3. Representation of the Macroscopic Variables The macroscopic dissipation, for both Boltzmann and Navier-Stokes equations, is for momentum and energy, but not for mass. In other words, it applies to the following function h(x, t, ξ ):

1 2

h(x, t, ξ ) ≡ f0 − ρ(x, t) ω =

3

m ψi + e ψ4 i

1

ω2.

i=1

The following lemma describes an expression of dissipation resulting from the energy estimate in the next section from the macroscopic equation, cf. (4.2), (4.3), Lemma 3.1. For any macroscopic function f0 ≡ (ρ(x, t)+mi (x, t)χi (ξ )+e(x, t)χ4 (ξ )) 1 ω 2 (ξ ) and η ∈ R 3 , there exists C > 0 such that P 1 η · ξf0 , P 1 η · ξf0 (x, t)

(η · m)2 + 3i=1 (mi ηi )2 2 2 2 2 ≤ η · m + + 4 η e (x, t); 3

(η · m)2 + 3i=1 (mi ηi )2 −1 2 2 2 2 C

η · m + + 4 η e 3 ≤ L¯ −1 P 1 η · ξf0 , L¯ −1 P 1 η · ξf0

(η · m)2 + 3i=1 (mi ηi )2 2 2 2 2 ≤ C η · m + + 4 η e ; 3 C

−1

3

∂x i h, ∂x i h ≤ P 1 ξ · ∇x f0 , P 1 ξ · ∇x f0 ≤ C

i=1

R3

3

(3.2)

∂x i h, ∂x i h;

i=1

P 1 ξ · ∇x f0 , P 1 ξ · ∇x f0 dx

=

(3.1)

(3.3)

3 3

(div · m)2 (∂x i mi )2 + + (∂x i mj )2 + 4 (∂x i e)2 dx, 3 3 R3 1≤i,j ≤3

i=1

i=1

where · is a standard Euclidean norm in R 3 . Proof. We have, by straightforward computations, P 1 ξ i χ0 ω1/2 = 0,

P 1 ξ i χj ω1/2 = ξ i χj − 13 δ ij 3k=1 (ξ k )2 for 1 ≤ i, j ≤ 3, P 1 ξ i χ4 ω1/2 = ξ i χ4 − √2 ξ i for 1 ≤ i ≤ 3, 6

140

T.-P. Liu, S.-H. Yu

1

1

P 1 ξ i χj ω 2 , P 1 ξ i χj ω 2

1 ≤ i, i , j, j ≤ 3

1

(i = j ) & (i = j ) & (i = i )& (j = j )

1

(i = j ) & (i = j ) & (i = j )& (j = i )

0

(i = j ) & (i = j ) & {i, j } = {i , j }

0

(i = j ) & (i = j )

0

(i = j ) & (i = j )

− 23

(i = j ) & (i = j )&(i = i )

4 3

(i = j ) & (i = j )&(i = i )

1

1

P 1 ξ i χ4 ω 2 , P 1 ξ i χj ω 2 = 0, 1 1 P 1 ξ i χ4 ω 2 , P 1 ξ i χ4 ω 2 = 4δ ii . The above table yields

1 1 ηi ηi bj bj P 1 ξ i χj ω 2 , P 1 ξ i χj ω 2 1≤i,i ≤3 1≤j,j ≤4

=

3

i=1

(ηj )

2

3

j =1

2 3 3 3

1

2 j 2 (b ) + (ηj ) (b ) + η i bi + 4 (ηi )2 (b4 )2 . 3 j 2

i=1

i=1

(3.4)

i=1

This equality yields (3.1) as well as (3.3). Consider the following diagram: P 1 ξ ·η

P0 −−−−−−−−→

L¯ −1

P1 ↑

−−−−−−−→ L¯ −1 |

P1 ↑

P 1 ξ ·η(P0 ) P 1 ξ · η(P0 ) −−−−−−−−−−−−−→ L¯ −1 (P 1 ξ · η(P0 )).

Since the operator L : P1 −→ P1 is injective and the dimension of P 1 ξ · η(P0 ) is finite, the operator L¯ −1 |P 1 ξ ·η(P0 ) is bijective, bounded and invertible. Thus there exists C > 0 such that 1 P 1 ξ · ηf0 , P 1 ξ · ηf0 ≤ L¯ −1 (P 1 ξ · ηf0 ), L¯ −1 (P 1 ξ · ηf0 ) C ≤ CP 1 ξ · ηf0 , P 1 ξ · ηf0 for all η = 1. This proves (3.2).

The proof of the next lemma is straightforward and is omitted. Lemma 3.2. There exists C > 0 such that the following holds for all η = 1 and P1 -valued L2 -function f1 : P 0 ξ · ηf1 , P 0 ξ · ηf1 dx ≤ C f1 , f1 dx. R3

R3


141

4. Energy Estimates for Linear Equation The preliminary energy estimate is obtained directly from (2.7), making use of (2.6): 0=

τ

fx i , ∂x i (ft + ξ · ∇x f − Lf ) dxdt

0 τ R

3

1 ∂t fx i , fx i + fx i , ξ · ∇x fx i − fx i , Lfx i dxdt R3 2 0 τ t=τ 1 = − fx i , fx i dx fx i , Lfx i dxdt 2 R3 R3 0 t=0 t=τ τ 1 ≥ fx i , fx i dx f1x i , f1x i dxdt + ν0 3 3 2 R R 0 t=0

=

for i = 1, 2, 3. This yields

1 2

R3

t=τ t fx i , fx i dx + ν0 0

t=0

R3

f1x i , f1x i dxdt ≤ 0 for i = 1, 2, 3.

(4.1)

Note that we have obtained only partial dissipation, that is, only the microscopic dissipation f1x . Next we perform the energy estimates for the macroscopic component by integrating f0 times (2.9): 1 2

R3

t=τ 3

f0 , f0 (x, t)dx − t=0

τ

+

R3

0

i=1

τ 0

R3

f0x i , P 0 ξ i L¯ −1 P 1 ξ · ∇x f0 (x, t)dxdt

f0 , P 0 ξ · ∇x L¯ −1 (f1t + P 1 ξ · ∇x f1 )(x, t)dxdt = 0. (4.2)

From (2.6) and Lemma 3.1, there exist C and C1 > 0 such that −

3

i=1

τ

R3

0

τ

=− 0

τ

≥ ν0 0

R3

> C1 ν0 > Cν0

f0x i , P 0 ξ i L¯ −1 P 1 ξ · ∇x f0 (x, t)dxdt (L¯ · L¯ −1 )P 1 ξ · ∇x f¯0 , L¯ −1 P 1 ξ · ∇x f0 (x, t)dxdt

L¯ −1 P 1 ξ · ∇x f 0 , L¯ −1 P 1 ξ · ∇x f0 (x, t)dxdt P 1 ξ · ∇x f 0 , P 1 ξ · ∇x f0 (x, t)dxdt

R3

τ

R3 0 3

τ i=0

0

R3

∂x i h, ∂x i h(x, t)dxdt.

(4.3)

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T.-P. Liu, S.-H. Yu

Combine (4.2) and (4.3) to yield 1 2

t=τ 3 τ

f0 , f0 (x, t)dx + ν0 C ∂x i h, ∂x i h(x, t)dxdt R3 R3 t=0 i=1 0 τ + f0 , P 0 ξ · ∇x L¯ −1 (f1t + P 1 ξ · ∇x f1 )(x, t)dxdt ≤ 0.

R3

0

(4.4)

By the Schwartz inequality and (3.3), there exist C, C1 > 0 such that t τ f0 , P 0 ξ · ∇x L¯ −1 f1t dxdt = − P 1 ξ · ∇x f0 , L¯ −1 f1t dxdt R3

0

≤

τ

R3

0

3 τ

ν0 C 4

R3

0

i=1

≤

i=1

Cν0 8

τ 0

∂x i h, ∂x i hdxdt + C1

f0 , ξ · ∇x L¯ −1 ξ · ∇x f1 dxdt = 3

R3

0

3

τ

R3

j =1 0

R3

τ

R3

0

f1t , f1t dxdt;

P 1 ξ j L¯ −1 P 1 ξ · ∇x f0 , f1x j dxdt

∂x i h, ∂x i hdxdt + C1

τ

R3

0

f1x i , f1x i dxdt ;

and

τ

R3

0

P 0 ξ i ∂x i f1 , P 0 ξ j ∂x j L¯ −1 f1 dxdt ≤ C1

3

τ

R3

0

i=1

f1x i , f1x i dxdt.

This, (4.4) and (4.5) yield the macroscopic dissipation: t=τ 1 C 1 ν0 t f0 , f0 dx + P 1 ξ · ∇x f0 , P 1 ξ · ∇x f0 dxdt 4 R3 2 0 R3 t=0 3 τ t

≤ 3C0 f1t , f1t dxdt + f1x i , f1x i dxdt . 0

R3

i=1

0

R3

(4.6)

From (4.6) and (4.1), we can choose γ > 0 sufficiently small to result in

R3

+

f0 , f0 + γ ν0 C 4

3 τ

i=1

0

−1

R3

3

i=0

t=τ fx i , fx i (x, t)dx

t=0

∂x i h, ∂x i h dxdt + γ −1 ν0

3

i=1

τ 0

R3

f1x i , f1x i dxdt < 0. (4.7)

This basic energy estimate yields the stability of the global Maxwellian states for the linearized Boltzmann equation.


143

5. Nonlinear Stability of Planar Waves In preparation for later study of the positivity of shock profiles, we will only consider the planar wave propagation, x ∈ R 1 , Ft + ξ 1 Fx = Q(F, F );

(5.1)

F (x, t, ξ ) ∈ R, (x, t, ξ ) ∈ R × R + × R 3 , and assume that the perturbation has zero total macroscopic variables: 1 2 χi ω , f dx = 0 for i = 0, · · · , 4, i.e. f0 dx = 0. R

R

t=0

The macroscopic component plays a different role from the microscopic component and the above zero mass assumption simplifies the analysis, cf. Remark 5.2. From (2.2), χi (Ft + ξ 1 Fx − Q(F, F )) dξ dx = χi Ft dξ dx; 0= R R3 R R3 1 1 χi ω 2 , f dx = χi ω 2 , f dx = 0 for i = 0, · · · , 4 and t ≥ 0. R

t

This yields

R

R

0

f0 dx ≡ 0 for all t ≥ 0, t

and prompts us to introduce the anti-derivative of the macroscopic component: x W0 (x, t, ξ ) ≡ f0 (y, t, ξ ) dy. −∞

Integrate the first equation in (2.10) to obtain W0t + P 0 ξ 1 W0x + P 0 ξ 1 f1 = 0, f1t + P 1 ξ 1 (f1x + W0xx ) = Lf1 + N (f, f ) = (1 + ρ) Lf1 + N (f ), W0x = f0 .

(5.2a) (5.2b)

Write 1

W0 (x, t, ξ ) ≡ {R(x, t) χ0 (ξ ) + M(x, t) χ1 (ξ ) + E(x, t) χ4 (ξ )} ω 2 .   Rx = ρ, Mx = m,  E = e; x and from (5.2a),

  Rt + Mx = 0,    1 2  Mt + Rx + √ Ex + χ1 ω 2 , ξ 1 f1 = 0, 6   1 2   Et + √ Mx + χ4 ω 2 , ξ 1 f1 = 0. 6

(5.3)

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T.-P. Liu, S.-H. Yu

Denote the sup norm for W0 : |||W0 |||∞ ≡

sup

(x,t)∈R×R +

(|R| + |M| + |E|) (x, t).

For the variables W0 (x, t), we make an a priori smallness assumption: 1

(1 + |ξ |) 2 f1 L∞ (L2 ) + |||W0 |||∞ 1. x,t

ξ

(5.4)

For these new variables (R, M, E), we have that 0

τ

t=τ τ Rx Mt dxdt = Rx Mdx + Rt Mx dxdt R R R 0 t=0 τ t=τ = Rx Mdx − Mx Mx dxdt.

R

R

t=0

0

This results in

1 2 Rx (Mt + Rx + √ Ex + χ1 ω 2 , ξ 1 f1 dxdt 6 R 0 t=τ τ 1 = Rx Mdx + −|Mx |2 + |Rx |2 + Rx Ex + Rx χ1 ω 2 , ξ 1 f1 dxdt.

0=

τ

R

t=0

R

0

By the Schwartz inequality, one can conclude

τ 0

t=τ + O(1) ρ dxdt = O(1) ρMdx

τ

2

R

R

t=0

0

R

m2 + e2 + f1 , f1 dxdt. (5.5)

Remark 5.1. The procedure for the estimate in [9].

τ 0

R

ρ 2 dxdt is motivated by the method

Lemma 5.2. For any i ≥ 0,   ∂xi h 2 2 = O(1)(|∂xi m|2 + |∂xi e|2 ), L ξ

 ∂xi ∂t h 2 2 = O(1)(|∂xi ∂t m|2 + |∂xi ∂t e|2 ). L ξ

For the nonlinear term N (f ), we have the following lemma:


145

Lemma 5.3. For any i ≥ 0 it follows    

1

(1 + |ξ |) 2 ∂xi−β f1 2L2 + |∂xi−β m|2 + |∂xi−β e|2  ∂xi f1 , ∂xi N (f ) ≤ O(1)   

 

·   0≤β≤

i 2

ξ

i 2

0≤β≤





∂xβ f1 L∞ (L2 ) + ∂xβ m ∞ + ∂xβ e ∞   x,t

ξ







+ O(1)  





∂xi−β f1 2L2 + |∂xi−β m|2 + |∂xi−β e|2   ξ

i 2

0≤β≤



· 

  1

(1 + |ξ |) 2 ∂xβ f1 L∞ (L2 ) + ∂xβ m ∞ + ∂xβ e ∞  ; x,t

ξ

(5.6)

i 2

0≤β≤

∂xi ∂t f1 , ∂xi ∂t N (f ) 

(5.7)



≤ O(1)  

1

(1 + |ξ |) 2 ∂xi−β ∂t f1 2L2 + |∂xi−β ∂t m|2 + |∂xi−β ∂t e|2 ξ

i 2

0≤β≤



 1 + (1 + |ξ |) 2 ∂xi−β f1 2L2 + |∂xi−β m|2 + |∂xi−β e|2   ξ

 

· 

0≤β≤

∂xβ ∂t f1 L∞ (L2 ) + ∂xβ ∂t m ∞ + ∂xβ ∂t e ∞ x,t

ξ

i 2



 + ∂xβ f1 L∞ (L2 ) + ∂xβ m ∞ + ∂xβ e ∞   x,t

ξ

 

+O(1)  

0≤β≤

i 2

∂xi−β ∂t f1 2L2 + |∂xi−β ∂t m|2 + |∂xi−β ∂t e|2 ξ



 + ∂xi−β f1 2L2 + |∂xi−β m|2 + |∂xi−β e|2   ξ

146

T.-P. Liu, S.-H. Yu

 

· 

1

0≤β≤

+

x,t

ξ

i 2

0≤β≤

(1 + |ξ |) 2 ∂xβ ∂t f1 L∞ (L2 ) + ∂xβ ∂t m ∞ + ∂xβ ∂t e ∞ 

 1

(1 + |ξ |) 2 ∂xβ f1 L∞ (L2 ) + ∂xβ m ∞ + ∂xβ e ∞  . x,t

ξ

i 2

This lemma is a consequence of Lemma 5.2, (2.11), Lemma B.1, and the following 1 1 f1 , N (f ) = (1 + |ξ |) 2 f1 , (1 + |ξ |)− 2 N (f ) .

5.1. Lower Order Energy Estimates. Similar to (2.9), we have W0t + P 0 ξ 1 W0x + P 0 ξ 1 L¯ −1 (P 1 ξ 1 W0xx + f1t + P 1 ξf1x − ρLf1 − N (f )) = 0. Multiply the above equation by W0 and integrate it over [0, τ ] × R: 0

τ

∞

−∞

W0 , W0t + P 0 ξ 1 W0x

1 ¯ −1

+ P 0ξ L

(P 1 ξ W0xx + f1t + P 1 ξf1x − ρLf1 − N (f )) dxdt = 0. (5.8) 1

Since (5.2a) differs from (2.8) in the differentiation order of the variable f1 , we need to modify the procedure for (4.7) in order to obtain the nonlinear stability for the variable W0 . By (3.1) in Lemma 3.1, there exists C > 0 such that C h, h ≤ P 0 ξ 1 P 1 ξ 1 W0x , P 0 ξ 1 P 1 ξ 1 W0x . Thus, similar to (4.2), we have 1 2

t=τ τ τ W0 , W0 dx + Cν0 h, h dxdt + P 1 ξ 1 W0 , L¯ −1 f1t dxdt R R R 0 0 t=0 τ + P 1 ξ 1 W0 , L¯ −1 P 1 ξf1x dxdt R 0 τ − P 1 ξ 1 W0 , ρf1 dxdt R 0 τ − P 1 ξ 1 W0 , L¯ −1 N (f ) dxdt ≤ 0. (5.9)

0

R


147

The second double integral in (5.9) can be estimated as follows: For any γ1 > 0, t=τ −1 1 −1 ¯ ¯ P 1 ξ W0 , L f1t dxdt − P 1 ξ W0 , L f1 dx R R 0 t=0 τ =− P 1 ξ 1 W0t , L¯ −1 f1 dxdt R 0 τ = P 1 ξ 1 P 0 (ξ 1 W0x + ξ 1 f1 ), L¯ −1 f1 dxdt 0 τ R τ ≤ f1 , f1 dxdt P 1 ξ 1 P 0 ξ 1 W0x , L¯ −1 f1 dxdt + O(1) 0 R τ 0τ R h, h + ρ 2 dxdt + O(1)(1 + γ1−1 ) ≤ Cγ1 f1 , f1 dxdt.

τ

1

R

0

0

R

(5.10)

By integration by parts and Schwartz’s inequality, there exists C > 0 such that 0

τ

P 1 ξ 1 W0 , L¯ −1 P 1 ξf1x dxdt τ ∞ P 1 ξ 1 L¯ −1 P 1 ξ 1 W0x , f1 dxdt =− 0 −∞ τ 1 ≤2 γ1 P 1 ξ 1 L¯ −1 P 1 ξ 1 W0x , P 1 ξ 1 L¯ −1 P 1 ξ 1 W0x + f1 , f1 dxdt γ 1 0 τ R 1 ≤ 2C γ1 h, h + f1 , f1 dxdt for all γ1 > 0. (5.11) γ1 R 0 R

From the structure of N (f ) in (2.11) and Lemma B.1 we have from the Schwartz inequality 1 1

(1 + |ξ |)− 2 N (f ) L2 ≤ O(1) m2 + e2 + (1 + |ξ |) 2 f1 2L2 . ξ

ξ

From this, there exists C 0

τ

P 1 ξ 1 W0 , L¯ −1 N (f )dxdt τ 1 1 = (1 + |ξ |) 2 P 1 ξ 1 W0 , (1 + |ξ |)− 2 L¯ −1 N (f )dxdt R 0 τ 1 ≤ C|||W0 |||∞ m2 + e2 + (1 + |ξ |) 2 f1 2L2 dxdt. R

0

R

ξ

(5.12)

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T.-P. Liu, S.-H. Yu

By the Schwartz inequality and (5.5),

τ

P 1 ξ 1 W0 , ρf1 dxdt (5.13) R 0 τ 1 ≤ P 1 ξ 1 W0 L∞ (L2 ) ρ 2 + f1 , f1 dxdt x,t ξ 2 R 0 t=τ τ 1 ≤ O(1)|||W0 |||∞ ρMdx + m2 + e2 + (1 + |ξ |) 2 f1 2L2 dxdt . R

t=0

0

R

ξ

From the above, for a small γ1 > 0 satisfying 2γ1 C
0, 1 ¯ −1 −1 ¯ P 1 ξ W0 , L f1 dx ≤ L f1 , L¯ −1 f1 dx γ1 P 1 ξ 1 W 0 , P 1 ξ 1 W 0 + γ1 R 1 f1 , f1 dx. ≤C (5.15) γ1 W0 , W0 + γ1 R

R

1

t Combine (5.14) and (5.15) with the estimate in (5.5) to bound the term 0 R ρ 2 dxdt in (5.10). Then, under the smallness condition (5.4), it follows that there exists γ1 ∈ (0, 1) such that 1 C ν0 τ f1 , f1 dx + h, hdxdt W0 , W0 + γ1 8 0 R R 4 t=τ 1 2C 2 f1 , f1 + O(1)|||W0 |||∞ ρ dx ≤ W0 , W0 + γ1 R 2 t=0 + O(1) |||W0 |||∞ ρ 2 dx R t=τ τ 1 + O(1)

(1 + |ξ |) 2 f1 2L2 dxdt. (5.16) 0

R

ξ

Similar to (4.1), we consider the following τ ∞ energy estimate using the original Boltzmann equation (2.10) to handle O(1) 0 −∞ f1 , f1 dxdt in the RHS of (5.16): t 0

∞ −∞

f, ft + ξ 1 fx − (1 + ρ)Lf − N (f ) dxdt = 0.


149

This and Lemma 5.3 results in the following energy estimate: t=τ τ ∞ f, f dx − (1 + ρ) f1 , Lf1 dxdt −∞ 0 −∞ t=0 τ ∞ f1 , N(f ) dxdt = 0 −∞ τ ∞ 1 1 ≤ O(1)

(1+|ξ |) 2 f1 L∞ (L2 ) (m2 +e2 )+ f1 L∞ (L2 ) (1+|ξ |) 2 f1 2L2 dxdt. 1 2

∞

x,t

−∞

0

ξ

x,t

ξ

ξ

(5.17)

Under the smallness assumption (5.4), that is, for some γ ∈ (0, 1), 1

(1 + |ξ |) 2 f1 L∞ (L2 ) + |||W0 |||∞ γ γ1 < 1, x,t

ξ

the combination of (5.16) + γ2 (5.17) results in, for some C > 1, 1 W0 , W0 + f, f dx 2Cγ −∞ t=τ τ 1 ν0 1 + h, h + (1 + |ξ |) 2 f1 2L2 dxdt ξ 8C 0 R γ ∞ 1 f, f dx . ≤ 2 W0 , W0 + Cγ −∞ t=0

∞

(5.18)

This concludes the lower order energy estimates for W0 (x, t) under the smallness assumption (5.4). Remark 5.4. We have derived this lower energy estimate as a consequence of the following combination:

W0 , W0t + ξ 1 W0x + P 0 ξ 1 f1 dxdt 0 −∞ τ ∞ ¯ 1 − N (f )) dxdt + W0 , L¯ −1 (f1t + P 1 ξ 1 f1x − ρ Lf 0 −∞ 1 τ ∞ ¯ − ρ Lf ¯ − N (f ) dxdt = 0. + f, ft + ξ 1 fx − Lf γ 0 −∞

τ

∞

(5.19)

1 f, f dx Remark 5.5. In the lower order energy estimate (5.18), the integral R 2Cγ t=τ 1 1 2 2 does not contain a component (1 + |ξ |) f1 , (1 + |ξ |) f1 , which will be used to close the smallness assumption boundary integral, we use τ (5.4). Instead of using the time τ the double integral 0 R (1 + |ξ |)f1 , f1 dxdt and 0 R (1 + |ξ |)∂t f1 , ∂t f1 dxdt to resolve the smallness assumption.

150

T.-P. Liu, S.-H. Yu

5.2. Higher Order Energy Estimates. In order to complete the nonlinear stability analysis, we need to extend the lower energy estimates to the high order derivatives of W0 (x, t, ξ ). For this, we need to estimate: 1 O(1) W0 H 6 (L2 ) + (1 + |ξ |) 2 f1 H 6 (L2 ) x

x

ξ

ξ

+ ∂t W0 H 6 (L2 ) + (1 + |ξ |) ∂t f1 H 6 (L2 ) , 1 2

x

x

ξ

ξ

here · H β (L2 ) is the Sobolev norm, x

ξ

g 2 β 2 Hx (Lξ )

≡

0≤|α|≤β R

∂xα g, ∂xα g dx.

The order of the Sobolev norm is chosen to guarantee the following smallness a priori assumption:

! |||∂xα W0 |||∞ + |||∂xα ∂t W0 |||∞ +

∂xα f1 L∞ (L2 ) + ∂xα ∂t f1 L∞ (L2 )

|α|≤4

|α|≤3

x,t

x,t

ξ

ξ

(5.20) ≤ ς0 for some given ς0 1, needed to take care of the nonlinear term. This assumption is consistent with the smallness condition (5.4). Lemma 5.6. For any gi satisfying P 0 gi ≡ 0, | g1 , Lg2 | ≤ −

# 1" δ g1 , Lg1 + δ −1 g2 , Lg2 , 2

for any constant δ > 0. Proof. Since both g1 and g2 are in P1 , √

√ 1 1 δg1 + √ g2 , δg1 − √ g2 ∈ P1 . δ δ

Thus, we have √ √ √ √ 1 1 1 1 δg1 + √ g2 , L δg1 + √ g2 ≤ 0, δg1 − √ g2 , L δg1 − √ g2 ≤ 0. δ δ δ δ This and the self-adjoint property of L ,(2.4), result in the lemma.


151

Let γ0 be a given positive number satisfying ς γ0 1. Consider the following combination:

γ0−i

0≤i≤6

+

τ

R

0

∂xi W0 , ∂xi (W0t + P 0 ξ 1 W0x + P 0 ξ 1 f1 dxdt

γ0−i

τ

0≤i≤6

∂xi W0 , ∂xi P 0 ξ 1 L¯ −1 ((f1t + P 1 ξ 1 f1x

R

0

¯ 1 − N(f ))) dxdt +P 1 ξ 1 f0x − ρ Lf τ

+ γ0−i−1 ∂xi f, ∂xi (ft + ξfx − Lf − N (f )) dxdt 0≤i≤6

+

γ0−i

0≤i≤6

+

γ0−i

τ

τ

R

0

R

0

0≤i≤6

R

0

∂xi ∂t W0 , ∂xi ∂t (W0t + P 0 ξ 1 W0x + P 0 ξ 1 f1 dxdt ∂xi ∂t W0 , ∂xi ∂t P 0 ξ 1 L¯ −1 ((f1t + P 1 ξ 1 f1x

¯ +P 1 ξ f0x − ρ Lf1 − N(f ))) dxdt τ

−i−1 + γ0 ∂xi ∂t f, ∂xi ∂t (ft + ξfx − Lf − N (f )) dxdt = 0. 1

R

0

0≤i≤6

(5.21)

This generalizes (5.19), which was used to obtain the lower order energy estimate (5.18). Finally,

6

γ0−i

i=0

+ + ≤

6

i=0 6

i=0 6

i=0

∞ −∞

∂xi W0 2L2 + ∂xi ∂t W0 2L2 + ξ

γ0−i

ν0 8C

γ0−i

ν0 8Cγ

γ0−i

τ

R

0

ξ

2Cγ

ξ

0

ξ dx

∂xi f 2L2 + ∂xi ∂t f 2L2

τ

t=τ

∂xi h 2L2 + ∂xi ∂t h 2L2 dxdt ξ

ξ

1

R

1

(1 + |ξ |) 2 ∂xi f1 2L2 + (1 + |ξ |) 2 ∂xi ∂t f1 2L2 dxdt ξ

ξ

1 2 ∂xi W0 2L2 + ∂xi ∂t W0 2L2 +

∂xi f 2L2 + ∂xi ∂t f 2L2 dx . ξ ξ ξ ξ Cγ −∞ t=0 ∞

(5.22)

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T.-P. Liu, S.-H. Yu

By the Sobolev inequality with (5.22) and the following inequality: 1 i 2

(1 + |ξ |) 2 ∂x f1 L2 dx ξ R t=τ 1 i 2 ≤

(1 + |ξ |) 2 ∂x f1 L2 dx ξ R t=0 τ 1 1 +

(1 + |ξ |) 2 ∂xi f1 L2 (1 + |ξ |) 2 ∂xi ∂t f1 L2 dxdt ξ ξ 0 R τ 1 1 ≤2

(1 + |ξ |) 2 ∂xi f1 2L2 dx +

(1 + |ξ |) 2 ∂xi f1 2L2 R

+ (1 + |ξ |)

ξ

1 2

t=0

0

R

ξ

∂xi ∂t f1 2L2 dxdt

(5.23)

ξ

the smallness assumption (5.20) is justified when the initial data is sufficiently small compared to ς0 . Thus, (5.22) concludes the nonlinear stability. 6. Shock Profiles Let φ(x − st, ξ ) be a travelling wave solution of the Boltzmann equation (5.1) −sφ + ξ 1 φ = Q(φ, φ).

(6.1)

Denote by φT (x − st, ξ ) and ωT (x − st, ξ ) the corresponding profiles of local thermal equilibrium distributions:  |ξ −u|2 − 2T  e  φT (x − st) ≡ ρ(x − st) √ ; (2πT )3 (6.2) 2 |ξ −u|  − 2T  e ωT (x − st, ξ ) ≡ √ , 3 (2πT )

where (ρ, u, T ) are the macroscopic variables of the travelling wave solution:    ρ(x − st) ≡ φ(x − st, ξ ) dξ,    R 3    1   m(x − st) ≡ 3 ξ φ(x − st, ξ ) dξ, R |ξ |2   E(x − st) ≡ φ(x − st, ξ ) dξ,    R3 2 2    m2 u   +ρ T ≡E ≡ρ +e .  2ρ 2 The states (ρ± , m± , E± ) ≡ limx→±∞ (ρ, m, E)(x) satisfy the Rankine-Hugoniot condition:   s(ρ− − ρ+ ) = m− − m+ , (R-H) s(m− − m+ ) = (u− m− + p− ) − (u+ m+ + p+ ),  s(E − E ) = (u (E + p ) − u (E + p )). − + − − − + + +


153

We consider a weak shock with the following normalization:  R = 1 (Gas Constant)     ρ− = 1,      u − = 0,    T− = 1,

p− − p+ > 0, (Entropy Condition)     |ρ− − ρ+ | 1,      − ρ+ |,  ≡ |ρ  √−   − √5 1. s

(6.3)

3

Let (ρNS , uNS , ENS )(x − st) be the Navier-Stokes shock profile (A.9) obtained through the Chapman-Enskog expansion. This profile connects the same end states (ρ± , u± , E± ). We denote the corresponding local Maxwellians by  |ξ −u |2 − 2T N S   NS e  φtr (x − st, ξ ) ≡ ρNS √ 3 (2πTN S )

|ξ −u |  − 2T N S  NS  e ωtr (x − st, ξ ) ≡ √ . 3 2

(6.4)

(2πTN S )

7. Local Macroscopic and Microscopic Variables Denote by J (x, t; ξ ) the perturbation of the travelling wave φ(x − st, ξ ) with local Maxwellian initial state: F (x, t, ξ ) ≡ φ(x − st, ξ ) + J (x, t, ξ ), (7.1) F (x, 0, ξ ) ≡ φT (x, ξ ). The equation for J (x, t, ξ ) is

∂t J + ξ 1 Jx = Q(φ + J, φ + J ) − Q(φ, φ), J (x, 0, ξ ) ≡ φT (x, ξ ) − φ(x, ξ ).

(7.2)

With the change of coordinates:

x → x − st, t → t,

the shock profile becomes stationary, and (5.1) and (7.2) become Ft + (ξ 1 − s)Fx = Q(F, F ), Jt + (ξ 1 − s)Jx = Q(J + φ, J + φ) − Q(φ, φ).

(7.3)

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T.-P. Liu, S.-H. Yu

Since a shock profile is orbitally stable, to properly locate the shock, we use the conservation laws for the macroscopic variables. Our choice of the initial state yields

∞ J (x, 0, ξ ) dξ dx = J (x, t, ξ ) dξ dx = 0, −∞ R 3 −∞ R 3 ∞ ∞ ξ i J (x, 0, ξ ) dξ dx = ξ i J (x, t, ξ ) dξ dx = 0 for i = 1, · · · , 3, 3 3 −∞ R −∞ R ∞ ∞ 2 |ξ | J (x, 0, ξ ) dξ dx = |ξ |2 J (x, t, ξ ) dξ dx = 0. ∞

−∞ R 3

−∞ R 3

This implies that the perturbation does not cause the time asymptotic phase shift of the shock, [12]. Consider the anti-derivative: W (x, t, ξ ) ≡ Wt + (ξ 1 − s)Wx −

x −∞

x −∞

J (y, t, ξ ) dy;

Q(J + φ, J + φ) − Q(φ, φ)dy = 0.

(7.4)

We use the local Maxwellian profiles φtr (x, ξ ) and ωtr (x, ξ ) for the macro-micro decomposition. Let L and L , respectively, be the linearized collision operator around φtr and φ:

LJ ≡ Q(φtr , J ) + Q(J, φtr ), L J ≡ Q(φ, J ) + Q(J, φ).

We also introduce the following deviations:

DJ ≡ L J − LJ, N (J ) ≡ Q(J + φ, J + φ) − Q(φ, φ) − LJ − DJ.

The functions ψi (x, ξ )ωtr (ξ ), 1 = 0, · · · , 4, span the kernel of L: L(ψi ωtr ) = 0 for i = 0, · · · , 4,  ψ0 (x, ξ ) ≡ 1,     (ξ i − ui )  ψi (x, ξ ) ≡ √ NS for i = 1, 2, 3 TNS    1 |ξ − uNS |2  ψ4 (x, ξ ) ≡ √ −3 . TNS 6 Remark 7.1. These invariants ψi ωtr are orthogonal with respect to the weight ωtr in (6.2).


155

For h(x, t, ξ ) and j (x, t, ξ ) functions on R × R + × R 3 ; and A an operator on × R + × R 3 ), we set    h|g (x, t) ≡ hg/ωtr dξ,   3  R  h|A|g (x, t) ≡ h Ag /ωtr dξ,  R3     (h, j )(x, t) ≡ h j dξ.

L2 (R

R3

Remark 7.2. The inner product , of Sect. 2 differs from the present one. The is so because the perturbations are expressed differently in (1.2) and (7.1). The above collision invariants are orthogonal: For 0 ≤ i, j ≤ 4, ψi ωtr |ψj ωtr = δji . The following lemma follows from the definition (6.2) of φT by direct calculations. Lemma 7.3. The profile φT in (6.2) satisfies φ − φT dξ ≡ 0, R3

R3

ξi

− ui

(φ − φT )dξ ≡ 0 for x ∈ R and i = 0, · · · , 3, T 1 |ξ − u|2 − 3 (φ − φT )dξ = 0. √ T 6 R3

√

The macroscopic and microscopic variables, for W (x, t, ξ ) and J (x, t, ξ ) are defined by  4 4 

 W (x, t, ξ ) ≡  W0i ψi ωtr ; J0 (x, t, ξ ) ≡ J0i ψi ωtr ,  0 i=0

i=0

  W0i ≡ W |ψi ωtr , J0i ≡ J |ψi ωtr ,    P 0 W ≡ W0 ; P 0 J ≡ J0 , W1 (x, t, ξ ) ≡ W (x, t, ξ ) − W0 (x, t, ξ ); J1 (x, t, ξ ) ≡ J (x, t, ξ ) − J0 (x, t, ξ ), P 1 W ≡ W1 ; P 1 J ≡ J1 . Remark 7.4. The function W0 is the macroscopic component of the anti-derivative W , and not the anti-derivative of the macroscopic component J0 , see Lemma 7.6 and the remark following it. The two are the same when the underline Maxwellian state is global, as in Sect. 5. Definition 7.5. M (x, t, ξ ) is a purely microscopic function iff M0 (x, t, ξ ) ≡ 0 for all x and t. M (x, t, ξ ) is a purely macroscopic function iff M1 (x, t, ξ ) ≡ 0 for all x and t, where

M0 ≡ 4i=0 ψi ω|M ψi ω, M1 ≡ M − M 0 . .

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Lemma 7.6. If M (x, t, ξ ) is purely microscopic, then Mt (x, t, ξ ) and Mx (x, t, ξ ) are purely microscopic. Proof. Since M is purely microscopic and χi is independent of (x, t), 0 = (χi , M ); 0 = (χi , M )t = (χi , Mt ) for i = 0, . . . , 4. Similarly, Mx is also purely microscopic.

Remark 7.7. Since the base equilibrium state ωtr is local and depends on (x, t), Mt is not necessarily purely macroscopic even when M is. We have from Lemma 7.6, P 0 Wt = W0t , P 0 Wx = W0x and since J = Wx , J0 = P 0 W0x ; J1 = P 1 W0x + W1x .

(7.5)

From (7.3) and Lemma 7.6, we have the decomposition ∂t W0 + P 0 (ξ 1 − s)P 0 ∂x W0 + P 0 (ξ 1 − s)J1 = 0, J1t + P 1 (ξ 1 − s)(J1x + J0x ) − LJ1 = DJ + N (J ).

(7.6a) (7.6b)

From (7.6b), J1 = L−1 (J1t + P 1 (ξ 1 − s)(J1x + ∂x P 0 W0x )) − L−1 DJ − L−1 N (J ), L−1 DJ = L−1 D(P 0 ∂x W0 + J1 ).

(7.7)

Substitute (7.7) into (7.6a) to result in: ∂t W0 + P 0 (ξ 1 − s)P 0 ∂x W0 + P 0 (ξ 1 − s)L−1 P 1 (ξ 1 − s)∂x P 0 ∂x W0 + P 0 (ξ 1 − s)L−1 ∂t J1 + P 1 (ξ 1 − s)∂x J1 − ρLJ1 − N (J ) − DJ = 0.

(7.8)

Reference Macro-Micro Decomposition. Relative to the local macro-micro decomposition in the above, we introduce a reference macro-micro decomposition. Since the fluid variable (ρ, u, T ) for the left states of the shock wave have been normalized in (6.3), we can use the absolute Maxwellian state M[1,0,1] (ξ ), |ξ |2

e− 2 M[1,0,1] (ξ ) ≡ (2π)3 ref

to define the following reference macro-micro decomposition P 0  H1 H 2  H1 , H2 ref ≡ dξ,   3  R M[1,0,1]   4

ref P H ≡ χi M[1,0,1] , H ref χi M[1,0,1] ,  0    i=0   ref ref P 1 H ≡ H − P 0 H, j

ref

+ P1

where χi is defined in (2.2) with u0 = 0 for j = 1, 2, 3 and with T0 = 1.

:


157

Notations of Norms.  1

H L2 ≡ H |H 2 ,   ξ   1   2  H, ≡ H

H

2  ref,Lξ ref ,     H 2 2 ≡ H 2 dx, Lx (Lξ ) R L2ξ  

H ref,L2 (L2 ) ≡ R H 2ref,L2 dx,  x ξ  ξ     + ≡ sup

H L2 (x, t),

H

∞ 2 (x,t)∈R×R  Lx,t (Lξ ) ξ    H

+ H

∞ 2 ≡ sup 2 (x, t). (x,t)∈R×R

ref,Lx,t (Lξ )

ref,Lξ

Remark 7.8. For the macroscopic component P 0 H , the two norms, P 0 H L2 and ξ

P 0 H ref,L2 , are equivalent, that is, there exists K > 1 such that ξ

K−1 P 0 H L2 ≤ P 0 H ref,L2 ≤ K P 0 H L2 . ξ

ξ

ξ

Remark 7.9. Local Existence in Time. With Lemmas B.1 and B.2, one can have the local existence of J in the norm · ref,L2 (L2 ) by rewriting the equation of J as follows: x

ξ

∂t J + ξ 1 ∂x J = LJ + ((L − L)J + Q(J, J )), $ %& ' SMALL

and applying the standard Picard iteration to the above equation. Here, LJ ≡ Q(M, J ) + Q(J, M).

4 About the variable W0 = i=0 W0i φi ω, it can be represented by a finite dimensional hyperbolic system with a given source terms as follows:  0 1 0 1  ∂t (χ0 , W ) + ∂x (χ0 , ξ W ) + (χ0 , P 0 ξ J1 ) = 0, ∂t (χ1 , W 0 ) + ∂x (χ1 , ξ 1 W 0 ) + (χ1 , P 0 ξ 1 J1 ) = 0,  ∂ (χ , W 0 ) + ∂ (χ , ξ 1 W 0 ) + (χ , P ξ 1 J ) = 0, t 4 x 4 4 0 1 j

where W0 = 0 for j = 2, 3 due to the planar wave assumption. Thus from this finite dimensional hyperbolic system, the local existence of J in

· ref,L2 (L2 ) also results in the local existence of W0 in · ref,L2 (L2 ) . x

x

ξ

ξ

8. Basic Matrix Representations We denote by [ξ 1 − s] the matrix representation of the multiplication operator ξ 1 − s: [ξ 1 − s]ij ≡ ψi ωtr |ξ 1 − s|ψj ωtr . The system vt + [ξ 1 − s]vx = 0, v ∈ R 3 ,

(8.1)

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T.-P. Liu, S.-H. Yu

defines a strictly hyperbolic system. We denote by λj , rj ≡ (rj1 , rj2 , rj3 )t , and lj ≡ (lj1 , lj2 , lj3 ) the j th eigenvalue, and the corresponding right and left eigenvectors of [ξ 1 − s]:  λ1 < λ2 < λ3 ,      λ1 < λ2 < 0,      (|λ3 | )1, ξ 1 − s rj = λj rj , (8.2) ( 1 )    l ξ − s = λ l , j j j    3 (r k )2 = 1,   j   k=1 j  3 k k k=1 li rj = δi . The eigenvalues λi are *  5T 1−  λ = u  1  3 − s, λ2 = u1 − * s,   λ = u1 + 5T − s. 3 3 From (8.1), the matrix [ξ 1 −s] is symmetric. Combine this with the normalized condition

3 i 2 i=1 (rj ) = 1 to yield 3

lik ∂x rik = 0 for i = 1, 2, 3.

k=1

From the vectors rj , we define r j and l j as follows: r j ≡ (rj1 ψ0 + rj2 ψ1 + rj3 ψ4 ) ωtr , l j ≡ (lj1 ψ0 + lj2 ψ1 + lj3 ψ4 ) ωtr ,

(8.3)

so that, for any v,

P 0 (ξ 1 − s)r j = λj r j , l j (ξ 1 − s)|P 0 v = λj l j |v..

(8.4)

The macro-micro components satisfy, cf. (7.5),  3

   ≡ wj r j , W 0     j =1   h ≡ J − ω |J ω , 0 tr 0 tr ! h ≡ h1 ψ1 (ξ ) + h2 ψ4 (ξ ) ωtr ,    3 3 

   w j r j , J 1 = P 1 ∂x wj r j + ∂ x W1 .  J0 = P 0 ∂x j =1

j =1

(8.5)


159

Theorem 8.1. There exists C0 , C1 , C2 1, and C3 , C4 ∈ (0, 1) so that the profile φ(x − st, ξ ) satisfies 1

e

|φ(x, ξ ) − φtr (x, ξ )| ≤ C0 2 e−C3 |x| ρ 1

e

2

T (x)3

(8.6)

,

2

−u (x)| − |ξ2C T (x)

|∂xk φ(x, ξ )| ≤ C0 1+k e−C3 |x| ρ C4 2 e−2C3 |x|

2

−u (x)| − |ξ2C T (x)

2

for k = 1, · · · , 6, T (x)3 2 −C1 |x| ≤ −∂x l 3 (ξ 1 − s)|r 3 = −∂x λ3 ≤ e . C4

(8.7)

Remark 8.2. The above rate of the profile converging to the Maxwellian equilibrium states at |ξ | = ∞ improves that of [3]. The inequality (8.7) is the compressibility of sound speed across a shock profile; and (8.7) is a fact on the traveling wave solution of the compressible Navier-Stokes equations obtained by Chapman-Enskog expansion. The proof of this theorem will be given in Appendix C for the hard sphere.

9. Nonlinear Stability of Shock Profiles We impose an a priori smallness assumption:

∂xα W0 L∞ (L2 ) + ∂xα ∂t W0 L∞ (L2 ) x,t

|α|≤4

+

x,t

ξ

ξ

∂xα J1 ref,L∞ (L2 ) + ∂xα ∂t J1 ref,L∞ (L2 )

|α|≤3

x,t

ξ

x,t

ξ

≤ ς0 for some given ς0 1.

(9.1)

Similar to (5.8), we consider the energy estimate by integrating W0 times (7.8): τ

∞ −∞

0

+ 0

τ

W0 |∂t W0 + P 0 (ξ 1 − s)P 0 ∂x W0 + P 0 (ξ 1 − s)L−1 P 1 (ξ 1 − s)∂x P 0 ∂x W0

dxdt ∞ −∞

W0 |P 0 (ξ 1 − s)L−1 ∂t J1 + P 1 (ξ 1 − s)∂x J1 − DJ − N

dxdt = 0.

(9.2)

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This results in I 1 + I2 + I 3 + I 4 + I 5 + I 6 + I 7 + I 8 t=τ τ ∞ 1 ∞ W0 |W0 dx W0 |P 0 (ξ 1 − s)P 0 ∂x W0 dxdt ≡ + 2 −∞ 0 −∞ t=0 τ ∞ + W0 |P 0 (ξ 1 − s)L−1 P 1 (ξ 1 − s)∂x P 0 ∂x W0 dxdt 0 −∞ τ ∞ + W0 |P 0 (ξ 1 − s)L−1 ∂t J1 dxdt 0 −∞ τ ∞ + W0 |P 0 (ξ 1 − s)L−1 P 1 (ξ 1 − s)∂x J1 dxdt 0 −∞ τ ∞ − W0 |P 0 (ξ 1 − s)ρJ1 dxdt 0 −∞ τ ∞ − W0 |P 0 (ξ 1 − s)L−1 N (J ) dxdt 0 −∞ τ ∞ − W0 |P 0 (ξ 1 − s)L−1 DJ dxdt = 0. −∞

0

We now estimate Ii , i = 2, . . . , 8 using (8.5). First, I2 ≡

τ

τ

0

=

0

τ

= 0

∞ −∞ ∞

W0 |P 0 (ξ 1 − s)P 0 ∂x W0 dxdt

3

−∞ j =1 ∞

3

+ λj w j w j x +

,

wk l j |∂x r k

dxdt

k=1

−∞ j =1

3

3

∂ x λj − wk l j |∂x r k dxdt. (wj )2 + λj wj 2

(9.3a)

k=1

Remark 9.1. In the above last double integral, the term − ∂x2λ3 (w3 )2 is a positive term due to the compressibility of the profile (8.7). It is a good term for the energy estimates. ∂ λ The other two transversal terms − x2 j (wj )2 , j = 1, 2, are not necessarily positive and will be estimated later. By (8.2) and (8.7), the term

3

2 i=1 wi .

1≤j,k≤3 λj wj wk

l j |∂x r k can bounded by O(1)|∂x λ3 |

Thus there exists C > 0 such that I2 satisfies 0

τ

∞ −∞

" # |∂x λ3 | (w3 )2 − C {(w1 )2 + (w2 )2 } dxdt ≤ I2 .


The term I3 is defined and estimated as follows: τ ∞ I3 ≡ W0 |P 0 (ξ 1 − s)L−1 P 1 (ξ 1 − s)∂x P 0 ∂x W0 dxdt 0 −∞ τ ∞ = P 1 (ξ 1 − s)W0 |L−1 P 1 ∂x (ξ 1 − s)J0 dxdt 0 −∞ τ ∞ =− P 1 (ξ 1 − s)J0 |L−1 |P 1 (ξ 1 − s)J0 dxdt 0 −∞ τ ∞ − P 1 (ξ 1 − s)W0 |L−1 |P 1x (ξ 1 − s)J0 0 −∞ + P 1 (ξ 1 − s)W0 |(L−1 )x |P 1 (ξ 1 − s)J0 dxdt τ ∞ P 1 (ξ 1 − s)P 1 W0x |L−1 |P 1 (ξ 1 − s)J0 + 0 −∞ (ωtr )x 1 (ξ − s)W0 |L−1 |P 1 (ξ 1 − s)J0 dxdt. + P1 ωtr

161

(9.3b)

The x-derivatives in the above eventually will be applied to the profile φtr to generate = O(1) 2 e−C3 |x| . This and (9.3b) yield that there exists C > 0 such that a factor φtr τ ∞ 1 −1 1 I3 + P 1 (ξ − s)J0 |L |P 1 (ξ − s)J0 dxdt 0 −∞ τ ∞ 2 −2C3 |x| 1 1 W0 |W0 e + P 1 (ξ − s)J0 |P 1 (ξ − s)J0 dxdt . ≤ C 0

−∞

Apply the estimates in (5.10), (5.11), (5.12), and (5.13) to I4 , I5 , I6 , and I7 , respectively to yield that there exists C > 0 such that for any γ ∈ (0, 1), ∞ t=τ 1 −1 I4 + I5 + I6 + I7 − (ξ − s)W0 |L J1 dx −∞ t=0 τ ∞ P 1 ξ 1 P 0 ∂x W0 |L−1 |P 1 ξ 1 P 0 ∂x W0 + 0 −∞ τ ∞ ≤C γ 2 e−2C3 |x| W0 |W0 dxdt 0 −∞ 1 τ +

J1 2ref,L2 dxdt. ξ γ 0 R Here, the first term in the RHS of the above inequality is a correction to the energy estimate in Sect. 5.1 because of the presence of the shock profile. The second term in the RHS results from the inner product Macro|Mirco in (5.10), (5.11), (5.12), and (5.13): √ M Macro|Mirco ≤ O(1) √ Macro L2 · Mirco ref,L2 ξ ξ ω = O(1) Macro L2 · Mirco ref,L2 (9.3c) ξ

as long as

M3 R 3 ω2 dξ

< ∞ uniformly in x.

ξ

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T.-P. Liu, S.-H. Yu

The remaining term I8 is estimated as follows: τ ∞ I8 ≡ − W0 |P 0 (ξ 1 − s)L−1 DJ dxdt 0 −∞ τ ∞ =− P 1 (ξ 1 − s)W0 |L−1 D(P 0 ∂x W0 + J1 ) dxdt 0 −∞ τ ∞ =− P 1 (ξ 1 − s)W0 |L−1 DP 0 ∂x W0 dxdt 0 −∞ τ ∞ − P 1 (ξ 1 − s)W0 |L−1 DJ1 dxdt 0 −∞ τ ∞ = O(1) h|L−1 DW0 + O(1)|∂λ3 | D ref,L2 W0 |W0 dxdt ξ 0 −∞ τ ∞ − P 1 (ξ 1 − s)W0 |L−1 DJ1 dxdt. −∞

0

From (8.6) and Lemma B.2, the operator in ξ is of following order:

L−1 D ref,L2 ≤ O(1) 2 e−C3 |x| . ξ

Then, by the Schwartz inequality C > 1 exists such that for any γ1 ∈ (0, 1), τ ∞ J1 |J1 + h|h I8 ≤ C dxdt. γ1 W0 |W0 |∂x λ3 | + γ1 0 −∞ Finally, from the above estimates on Ii ’s we conclude that there exists C > 0 such that for any γ ∈ (0, 1), t=τ 1 ∞ 1 −1 W0 |W0 + (ξ − s)W0 |L J1 dx 2 −∞ t=0 τ ∞ 2 2 2 + |∂x λ3 |{(w3 ) − C(w1 + w2 )}dxdt 0 τ −∞ ∞ − P 1 ξ 1 P 0 ∂x W0 |L−1 |P 1 ξ 1 P 0 ∂x W0 dxdt τ 0 ∞−∞ C ≤ Cγ |∂x λ3 | W0 2L2 + J1 2ref,L2 dxdt. (9.3d) ξ ξ γ 0 −∞ τ ∞ Transversal Estimates. In (9.3d), the non-positive double integral 0 −∞ |∂x λ3 | (−Cw12 − Cw22 )dxdt remains to be estimated. For this, we consider the following double integration: τ ∞ G ≡ ∂x λ3 W0 |(ξ 1 − s)W0 dxdt. 0

−∞

Since the shock wave is a 3-shock wave (8.2), there exists C > 0 such that τ ∞ 3

G = ∂x λ 3 λj (wj )2 dxdt 0

≥

0

τ

−∞ ∞ −∞

j =1

|∂x λ3 | λ2 ∞ ((w1 )2 + (w2 )2 ) − C(w3 )2 dxdt.


163

Remark 9.2. The consideration of a transversal wave estimate originates from the energy estimate for the stability analysis of a viscous shock profile, [6]. Here, we have rewritten it in a compact format in order to be able to apply it to the Boltzmann equation. Use integration by parts to yield G =−

τ

∞

τ

−∞ ∞

−∞ ∞

0

=− 0

λ3 ∂x W0 |(ξ 1 − s)W0 dx

(9.4)

λ3 2 W0 |(ξ 1 − s)P 0 ∂x W0 + O(1) 3 e−C1 |x| W0 |W0 dxdt

λ3 2 W0 | − ∂t W0 − (ξ 1 − s)J1 + O(1) 3 e−C1 |x| W0 |W0 dxdt 0 −∞ t=τ ∞ λ3 W0 |W0 dx = −∞ t=0 τ ∞ + 2λ3 W0 |(ξ 1 − s)J1 + O(1) 3 e−C1 |x| W0 |W0 dxdt,

=−

0

τ

−∞

and so ∞

t=τ τ ∞ −λ3 W0 |W0 dx + −∂x λ3 ((w1 )2 + (w2 )2 − C(w3 )2 )dxdt −∞ 0 −∞ t=0 τ ∞ ≤ 2λ3 W0 |J1 + O(1) 3 e−C1 |x| W0 |W0 dxdt. 0

−∞

τ ∞ In the above estimates, the term 0 −∞ 2λ3 W0 |J1 dxdt can be bounded by O(1) (I3 + · · · + I8 ). It follows that τ ∞ − ∂x λ3 ((w1 )2 + (w2 )2 − C(w3 )2 )dxdt 0 −∞ ∞

W0 2L2 + J1 2ref,L2 ≤ O(1) ξ ξ −∞ t=0 2 2 + W0 L2 + J1 ref,L2 dx ξ ξ t=τ τ ∞ e−C1 |x| W0 2L2 dxdt +O(1) 3 γ ξ 0 −∞ τ ∞ +O(1)

J1 2ref,L2 dxdt. (9.5) ξ γ 0 −∞ Take γ to satisfy γ 1; and consider the combination (9.2) +

1 (9.5) + γ

τ 0

∞ −∞

J, (7.2)ref dxdt.

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T.-P. Liu, S.-H. Yu

τ ∞ (Here, the estimate for 0 −∞ J, (7.2)ref dxdt is almost identical to (5.17) and is omitted.) This results in the first order energy estimate . 1 1 + O(1) W0 2L2 + (1 + O(1)) J 2ref,L2 dx ξ ξ γC −∞ 2 t=τ τ ∞ 1 −1 1 P 1 (ξ − s)J0 |L |P 1 (ξ − s)J0 dxdt − 0 −∞ τ ∞ |∂x λ3 |

λ2 ∞ + (w3 )2 + ((w1 )2 + (w2 )2 ) dxdt 2 2γ 0 −∞ . ∞ 1 1 + O(1) W0 2L2 + (1 + O(1)) J 2ref,L2 dx . ≤2 ξ ξ γC −∞ 2 t=0

-

∞

This corresponds to the energy estimate of (5.14) with corrections due to the presence of the shock layer. With the shock correcting terms estimated in (9.5), we can apply the same analysis leading to (5.22) to conclude that, for given σ and γ sufficiently small,

γ

0≤i≤6

+

−i

τ

R

0

γ

−i

+ +

γ

−i

0≤i≤6

+

R

∂xi W0 |∂xi P 0 (ξ 1 − s)L¯ −1 ((J1t + P 1 (ξ 1 − s)J1x

¯ − N (J )))dxdt − s)J0x − ρ LJ τ 1 γ −i−1 ∂xi J, ∂xi (Jt + ξ Jx − LJ − N (J ))

0≤i≤6

τ 0

0≤i≤6 +P 1 (ξ 1

∂xi W0 |∂xi (W0t + P 0 (ξ 1 − s)W0x + P 0 (ξ 1 − s)J1 dxdt

γ −i

τ

τ

0

R

R

0

0≤i≤6

R

0

ref

dxdt

∂xi ∂t W0 |∂xi ∂t (W0t + P 0 (ξ 1 − s)W0x + P 0 (ξ 1 − s)J1 dxdt

∂xi ∂t W0 |∂xi ∂t P 0 (ξ 1 − s)L¯ −1

¯ 1 − N (J )))dxdt × ((J1t + P 1 (ξ 1 − s)J1x + P 1 (ξ 1 − s)J0x − ρ LJ τ

+ γ −i−1 dxdt ∂xi ∂t J, ∂xi ∂t (Jt + ξ Jx − LJ − N (J )) +

0

0≤i≤6 1 τ

γ

−∞ 3 −C1 |x|

0

∞

+O(1) e

R

ref

∂x λ3 W0 |(ξ 1 − s)W0 dxdt − 2λ3 W0 |Wt + (ξ 1 − s)J1 W0 |W0 dxdt = 0.

Finally, this combination yields the higher order energy estimate: There exists C > 0 such that


165

1

C 0≤i≤6 R γ −i ∂xi W0 2L2 + ∂xi ∂t W0 2L2 + γ −i−1 ∂xi J 2ref,L2 + ∂xi ∂t J 2ref,L2 ξ

+ +

2

τ

+

dx t=τ

W0 2L2 e−C1 |x| dxdt ξ −i i 2 i 2

∂x h L2 + ∂x ∂t h L2 dxdt γ R

τ

0≤i≤6 0

≤C

ξ

R τ

0≤i≤6 0

ξ

(9.6)

C 0

ξ

γ

ξ

−i−1

(1 + |ξ |)

ξ

1 2

∂xi J1 2ref,L2 ξ

+ (1 + |ξ |)

1 2

∂xi ∂t J1 2ref,L2

dxdt

ξ

0≤i≤6 R

γ −i ∂xi W0 2L2 + ∂xi ∂t W0 2L2 + γ −i−1 ∂xi J 2ref,L2 ξ

+ ∂xi ∂t J 2ref,L2 ξ

ξ

ξ

dx.

t=0

With this, (5.23), and the Sobolev inequality, the smallness assumption (9.1) is justified when ς0 is sufficiently small. Thus, nonlinear stability of the shock profile follows. 10. Positivity of Shock Profiles We consider the initial value problem Ft + (ξ 1 − s)Fx = Q(F, F ), F (x, 0) ≡ φT (x) > 0. From Lemma 7.3, J ≡ F − φ satisfies ∞ χi J (x, 0, ξ ) d 3 ξ dx = 0, for i = 0, · · · , 4. −∞ R 3

The stability theory yields that lim J (x, t, ξ ) = 0 or equivalently lim F (x, t, ξ ) = φ(x, ξ ).

t→∞

t→∞

(10.1)

We break the collision operator into lost-gain parts: Ft + (ξ 1 − s)Fx + Q− (F, F ) = Q+ (F, F ),

(10.2a)

Q− (F, F ) ≡

R 3 ×S 2

F (x, t, ξ∗ ) C(, ξ − ξ∗ ) dξ∗ d F (x, t, ξ ),

≡ (F ) · F, Q+ (F, F ) ≡ F (x, t, ξ )F (x, t, ξ∗ ) C(, ξ − ξ∗ ) dξ∗ d, R 3 ×S 2

(10.2b)

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T.-P. Liu, S.-H. Yu

where

ξ ξ∗

ξ

  ξ = ξ + ( · (ξ∗ − ξ )) , ξ∗ = ξ∗ − ( · (ξ∗ − ξ )) ,   ∈ S 2 .

ξ∗

ξ∗ ξ ξ∗

ξ and C(, ξ −ξ∗ ) is the angular cut-off function by [7]. In our case, the function C(, ξ − ξ∗ ) for a hard sphere collision is C(, ξ − ξ∗ ) ≡ | · (ξ − ξ∗ )|.

(10.3)

One can rewrite (10.2a) as follows: Ft + (ξ 1 − s)Fx + (F ) F = Q+ (F, F ).

(10.4)

This yields the representation of the solution: F (x, t, ξ ) =

t

e−

t τ

(F )(x−(ξ 1 −s)(t−τ ),τ ,ξ ) dτ

0

× Q+ (F, F )(x − (ξ 1 − s)(t − τ ), τ, ξ ) dτ + e−

t 0

(F )(x−(ξ 1 −s)(t−τ ),τ ,ξ )dτ

F (x − (ξ 1 − s)t, 0, ξ ). (10.5)

Here, the functions (F ) and Q+ (F, F ) are positive when the function F is positive and integrable in ξ . Since the initial data of F is positive, the global existence of F and (10.5) yield that F > 0. This and (10.1) result in φ(x, ξ ) = lim F (x, ξ, t) ≥ 0. t→∞

From the stability analysis, φ is a small perturbation of a local Maxwellian profile. Thus, we conclude that there exists a sufficiently large R > 0 such that


φ(x, ξ ) > 0 for all |ξ | < R.

167

(10.6)

We now show that φ is strictly positive for all ξ ∈ R 3 . This is proved by contradiction. Suppose that φ(x0 , ξ0 ) = 0 for some (x0 , ξ0 ) ∈ R × R 3 .

(10.7)

Since φ(x, ξ ) is a stationary solution of the evolution equation, φt (x, ξ ) + (ξ 1 − s)φx + Q− (φ, φ) = Q+ (φ, φ). We have the integral representation of φ(x, ξ ): φ(x, ξ ) =

t

e−

0

+ e−

t

t 0

τ

(φ)(x−(ξ 1 −s)(t−τ ),ξ ) dτ

(φ)(x−(ξ 1 −s)(t−τ ),ξ )dτ

Q+ (φ, φ)(x − (ξ 1 − s)(t − τ ), ξ ) dτ

φ(x − (ξ 1 − s)t, ξ ).

Since φ ≥ 0, the above double integrals are non-negative. With the assumption (10.7), both of the above double integrals are zero for (x, ξ ) = (x0 , ξ0 ). Thus

φ(x0 − (ξ01 − s)t, ξ0 ) ≡ 0, Q+ (φ, φ)|(x,ξ )=(x0 −(ξ 1 −s)t,ξ0 ) = 0. 0

Set t ≡ 0 to yield 0=

R 3 ×S 2

φ(x0 , ξ0 )φ(x0 , ξ∗ )C(, ξ0 − ξ∗ )dξ∗ d.

Here, the function C(, ξ − ξ∗ ) defined in the operator Q+ in (10.2b) is a positive function, except for the case when ξ0 − ξ∗ and ξ0 − ξ0 are orthogonal, φ(x0 , ξ0 )φ(x0 , ξ∗ ) = 0 for (ξ0 − ξ∗ ) · (ξ0 − ξ0 ) = 0.

(10.8)

We now set up a reduction procedure to find a sequence ξ0 , ξ1 , · · · with limi→∞ ξi = 0 and φ(ξ0 , ξi ) = 0. This would lead to contradiction to (10.6).

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ξ∗

ξi

ξi

π/4 ξ∗ = 0

ξ∗ ξi

Case 1. φ(x0 , ξ∗ ) = 0 Case 2. φ(x0 , ξ0 ) = 0

ξ∗ = 0 ξ∗ ξi+1 π/4

ξ∗

ξi+1

ξi+1

ξi π/4

ξ∗ = 0 ξ∗ = 0

ξi+1

Case 2 Case 1 Suppose that (x0 , ξi ) is given and φ(x0 , ξi ) = 0. Consider the collision of a particle with velocity ξi to a stationary particle with velocity ξ∗ = 0 with π/4 scattering angle, the angle between ξi − ξ∗ and ξi : (ξi − ξi ) · (ξi − ξ∗ ) = cos

π

ξ − ξi · ξi − ξ∗ = 0. 4 i

Apply (10.8) to either φ(ξi , x0 ) = 0 or φ(ξ∗ , x0 ) = 0. Take ξi+1 to be ξi or ξ∗ , √ φ(ξi+1 , x0 ) = 0. By the choice of the angle, we have |ξi+1 | = |ξi |/ 2 = 2−(i+1)/2 |ξ0 |. This leads to a contradiction to (10.6) and we have thus proved the positivity of the shock profile φ(x, ξ ) > 0 for all (x, ξ ) ∈ R × R 3 .

Appendix A. Chapman-Enskog Expansion Consider the Boltzmann equation ∂t F + ξ · ∇x F = where κ > 0 is the mean free path. Expand the solution F and the operator ∂t :

1 Q(F, F ), κ

(A.1)


169

F = F0 + κF1 + κ 2 F2 + . . . , ∂ ∂2 ∂0 ∂1 = + κ + κ2 + . . . , ∂t ∂t ∂t ∂t ∂i are differential operators of x on the fluid variable ρ, ui , T , . . . . where ∂t The leading term F0 : |ξ −u|2

e− 2RT F0 (ξ, x, t) ≡ ρ (2πRT )3 ∂ 0  ρ ≡ −div m,    ∂t   3 

 ∂ i j ∂  ∂0 mi ≡ − m u − i p, ∂t ∂x j ∂x j =1    3 

 ∂0 ∂   E ≡ − [E + p] uj ,   ∂t ∂x j j =1 2 |u| 3 mi ≡ ρui , E ≡ ρ + E , E ≡ RT , p ≡ ρRT . (R ≡ 1). 2 2 F1 is uniquely solved by: ∂0 Q(F0 , F1 ) + Q(F1 , F0 ) = F0 + ξ · ∇x F0 , ∂t F1 dξ = 0, 3 R ξ i F1 dξ = 0 for i = 1, 2, 3, 3 R |ξ |2 F1 dξ = 0.

(A.2)

R3

Notice that F1 is purely microscopic; and |∂xki F1 | ≤ O(1) sup |∂xki ∂x j F0 |. 1≤j ≤k

The operator

∂1 ∂t

(A.3)

is defined by

∂1 ρ≡− ∂t ∂1 i uρ≡− ∂t

R

3

ξ · ∇x F1 dξ = 0,

ξ i ξ · ∇x F1 dξ + , 3 3

∂ui ∂ ∂uj 2 ∂uk i = µ(T ) for i = 1, 2, 3, + − δ ∂x j ∂x j ∂x i 3 ∂x k j j =1

R3

k=1

170

∂1 ρ ∂t

T.-P. Liu, S.-H. Yu

|u|2 +E 2

≡− =

R3

|ξ |2 ξ · ∇x F1 dξ

1≤j,k≤3

+

. - k 2 j ∂uk ∂ ∂uj k ∂u µ(T ) u + k − u ∂x j ∂x j ∂x 3 ∂x j

3

∂ ∂T λ(T ) j , j ∂x ∂x j =1

where µ(T ) > 0 is the viscosity coefficient and λ(T ) > 0 is the heat conductivity coefficient. These dissipation coefficients can be estimated for the hard sphere, (B.3). From this definition of ∂∂t1 , ∂1 F0 + ξ · ∇x F1 is purely microscopic. ∂t

(A.4)

From (A.2), (A.3), and (A.4), . κ∂1 ∂0 + F0 + ξ · ∇x (F0 + κF1 ) ∂t ∂t 1 − (Q(F0 + κF1 , F0 + κF1 ) − Q(κF1 , κF1 )) ≡ S , κ S : Purelymicroscopic.

(A.5) (A.6)

The first two terms in the expansion of ∂t∂  ∂ ∂0 ∂1   − − κ ρ = 0,    ∂t ∂t ∂t   ∂ ∂0 ∂1 − −κ ρui = 0  ∂t ∂t ∂t  2   ∂ ∂0 ∂1 |u|    − −κ ρ + E = 0, ∂t ∂t ∂t 2 give the compressible Navier-Stokes equation:   ρt + div m = 0,  + ,   3 3 3 

 2 k i j  i i j i mt + (m u )x j +px i = κ µ(T ) ux j +ux i − ux k δ j , i = 1, 2, 3,   3   j j =1 j =1 k=1  x  - . 3 2

j j k k j k Et + µ(T ) u ux j + ux k − u ux j u [E + p] j = κ    x 3  xj j =1 1≤j,k≤3    3 

 !    +κ λ(T ) Tx j x j .   j =1

(A.7) Let (ρ, m, E) be the solution of (A.7). From this, ∂1 ∂0 ∂ F0 − +κ F0 = 0. ∂t ∂t ∂t


171

From this and (A.5), ∂ 1 (F0 + κF1 ) + ξ · ∇x (F0 + κF1 ) − Q(F0 + κF1 , F0 + κF1 ) = S, (A.8) ∂t κ ∂ S ≡ κ F1 + κQ(F1 , F1 ) + S , ∂t S : Purely microscopic , √ 9 λ |S| ≤ O(1) 3 e− 10 (µ+ 5 )|u+ −s+ 5T+ /3| |x|. ¯ 1 − st) be a travelling wave solution of (A.7) Let (ρ, m, E)(x, t) = (ρ, ¯ m, ¯ E)(x connecting the two end states (ρ± , m± , E± ) of a planar entropy shock wave in the x 1 -direction; and let the length of the mean free path κ: κ ≡ 1; and the equation of the shock profile is  −s ρ¯x 1 + m ¯ 1x 1 = 0,     4 ¯ 1 ¯ 1 u¯ 1 )x 1 + p¯ x 1 = µ(T )u¯ x 1 1 , −s m ¯ 1x 1 + (m x 3  (  ) ! 4  −s E¯ x 1 + u¯ 1 E¯ + p¯ ¯ )u¯ 1 u¯ 1 1 = + λ(T¯ ) T¯x 1 x 1 , µ( T x 1 1 x x 3 ¯ lim (ρ, ¯ m, ¯ E)(x) = (ρ± , m± , E± ).

(A.9)

x→±∞

We will view A = F0 + F1 as the approximate travelling wave for Boltzmann equation. The truncation error is then S ≡ (ξ 1 − s)(F0 + F1 )x 1 − Q(F0 + F1 , F0 + F1 ), S : Purely Microscopic, √ 9 λ |S | ≤ O(1) 3 e− 10 (µ+ 5 )|u+ −s+ 5T+ /3| |x| , A ≡ F0 + F1 .

(A.10) (A.11) (A.12) (A.13)

When the strength |ρ− − ρ+ | + |m− − m+ | + |E− − E+ | of the shock wave is sufficiently small, the acoustic speed of the travelling solution is monotone: (Here, the shock is a 3-shock.)  /  5T¯  ∂x u¯ 1 + < 0. (A.14) 3 Under the same setting as in (6.3), there exist Cu , cl , and cu satisfying for x 1 ≥ 0 :

for x 1 ≤ 0 :

 *  2 c e− 109 ex 1 ≤ −∂ u¯ 1 + 5T¯ ≤ 2 c e− 109 ex 1 , l x u 3 9 1  |S| ≤ 3 CU e− 10 ex ,

(A.15a)

 *  2 c e 109 ex 1 ≤ −∂ u¯ 1 + 5T¯ ≤ 2 c e 109 ex 1 , l x u 3 9 1  ex 3 |S| ≤ CU e 10 ,

(A.15b)

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T.-P. Liu, S.-H. Yu

where

√ √ λ |u+ − s + 5T+ /3| 1 105 3 |u+ − s + 5T+ /3| µ+ ∼√ lim . →0+ 4 5 π 256 →0+ (A.15c)

e ≡ lim

The last estimate is for the hard sphere, see (B.3) of the next section. These facts about travelling wave solutions of the compressible Navier-Stokes equation can be easily verified. Appendix B. Estimates on Collision Operators From now on we will consider only the hard spheres. The collision operator can be written as follows see [8, 7]: Lh(ξ ) ≡ −ν(ξ )h(ξ ) + (−k1 (ξ, η) + k2 (ξ, η))h(η) dη, (B.1) R3

Lh ≡ −νh − K1 h + K2 h. (B.2)  1 2 2 |ξ |2 √ |ξ | u   ν(ξ ) ≡ 2πb0 e− 2 + |ξ |2 e− 2 du ,    0   |ξ |2 +|η|2 b0 k1 (ξ, η) = √ |ξ − η|e− 4 ,  2 2π   2 2 2 2  2b 1  − |ξ −η| − 18 (|ξ | −|η|2 )  |ξ −η| k2 (ξ, η) = √ 0 , e 8 2π |ξ − η| here b0 is a positive constant. (We set b0 = 1.) Under the normalized condition (6.3), the coefficient of viscosity, heat conductivity and minimum of collision frequency are, ([5, 7]):  5   µ∼ √ ,   16 π  75 (B.3) λ∼ √ ,   64 π  √   ν0 ∼ 2π. From (B.3) and (A.15c), we obtain the crucial estimate on the strength of the collision frequency needed for the energy estimate in the next section: 8e ν0 < . (B.4) 12 15|λ± | 3 Lemma B.1. Let the function C(, ξ − ξ∗ ) be the collision operator Q(F, G) for the hard sphere model, (see (10.3)). There exists K > 0 such that the following inequality 1 1 holds for any g and h satisfying (1 + |ξ |) 2 g L2 (R 3 ) , (1 + |ξ |) 2 h L2 (R 3 ) < ∞: Q(ω0 g, ω0 h)2 + Q(ω0 h, ω0 g)2 2 dξ ≤ K (|ξ | + 1)g(ξ ) dξ · h(ξ )2 dξ (|ξ | + 1)ω0 (ξ )2 R3

R3

+K R3

where ω0 (ξ ) is a given Maxwellian distribution.

g(ξ )2 dξ ·

R3

(B.5)

(|ξ | + 1)h(ξ )2 dξ, R3


173

Proof. Since (ξ , ξ∗ ) given in (10.2b) is constructed to conserve the kinetic energy and momentum after collision, |ξ |2 + |ξ∗ |2 = |ξ |2 + |ξ∗ |2 . From this,

ω0 (ξ )ω0 (ξ∗ ) = ω0 (ξ )ω0 (ξ∗ ).

This results in

g(ξ )h(ξ∗ )ω0 (ξ )ω0 (ξ∗ )C(, ξ − ξ∗ ) ddξ∗ √ ω0 (ξ ) 1 + |ξ | R 3 S2 g(ξ )h(ξ∗ )ω0 (ξ∗ )C(ξ − ξ ∗, ) = ddξ∗ . √ 1 + |ξ | R 3 S2

We also have 1 Q(ω0 g, ω0 h) √ ω0 (ξ ) 1 + |ξ | 1 =√ −g(ξ )h(ξ∗ )ω0 (ξ∗ )C(, ξ − ξ∗ )ddξ∗ 1 + |ξ | R 3 S2 1 g(ξ )h(ξ∗ )ω0 (ξ∗ )C(, ξ − ξ∗ )ddξ∗ . +√ 1 + |ξ | R 3 S2

(B.6)

By Hölder’s inequality and by ω0 (ξ∗ )C(, ξ − ξ∗ )2 dξ∗ = O(1)(1 + |ξ |)2 , R3

it follows R3



1   |ξ | + 1

2  g(ξ )h(ξ∗ )ω0 (ξ∗ )C(ξ − ξ∗ , )dξ∗ d dξ

R 3 ×S 2

≤ O(1) R3



1   |ξ | + 1

≤ O(1)

  g(ξ )2 h(ξ∗ )2 ω0 (ξ∗ )C(ξ − ξ∗ , )2 dξ∗ d dξ

R 3 ×S 2

g(ξ )2 h(ξ∗ )2 (1 + |ξ |)dξ ddξ∗

R 3 ×S 2 ×R 3 1

= O(1) (1 + |ξ |) 2 g 2L2 (R 3 ) h 2L2 (R 3 ) .

(B.7)

By Hörder’s inequality and by the following three properties of (ξ , ξ∗ ):  The six dimensional volume element is invariant: |dξ dξ∗ | = |dξ dξ∗ | with fixed,     C(, ξ − ξ∗ ) = C(, ξ − ξ∗ ), C(ξ − ξ∗ , )2 ω0 (ξ∗ )    = O(1),  sup R 3 ×S 2 ×R 3 (1 + |ξ∗ |)(1 + |ξ |)

174

T.-P. Liu, S.-H. Yu

it follows R3

 1   |ξ | + 1

2

 g(ξ )h(ξ∗ )ω0 (ξ∗ )C(ξ − ξ∗ , )dξ∗ d dξ

R 3 ×S 2



1   |ξ | + 1

≤ O(1) R3

  g(ξ )2 h(ξ∗ )2 ω0 (ξ∗ )C(ξ − ξ∗ , )2 dξ∗ d dξ

R 3 ×S 2 g(ξ )2 h(ξ∗ )2 ω0 (ξ∗ )C(ξ

= O(1)

|ξ | + 1

R 3 ×S 2 ×R 3

= O(1) R 3 ×S 2 ×R 3

− ξ∗ , )2

dξ∗ ddξ

g(ξ )2 h(ξ∗ )2 ω0 (ξ∗ )C(ξ − ξ∗ , )2 dξ dξ∗ d |ξ | + 1 g(ξ )2 (1 + |ξ |)h(ξ∗ )2 dξ dξ∗ d

= O(1) R 3 ×S 2 ×R 3

g(ξ )2 (1 + |ξ |)h(ξ∗ )2 dξ dξ∗ d

= O(1) R 3 ×S 2 ×R 3

1

= O(1) (1 + |ξ |) 2 g 2L2 (R 3 ) h 2L2 (R 3 ) .

(B.8)

We can conclude that R3

1 Q(ω0 g, ω0 h)2 dξ = O(1) (1 + |ξ |) 2 g 2L2 (R 3 ) h 2L2 (R 3 ) . (1 + |ξ |)ω0 (ξ )2

Similarly, we have R3

1 Q(ω0 h, ω0 g)2 dξ = O(1) (1 + |ξ |) 2 h 2L2 (R 3 ) g 2L2 (R 3 ) . (1 + |ξ |)ω0 (ξ )2

The above two estimates conclude this lemma.

Lemma B.2. For a given Maxwellian ω0 and a given function d satisfying d(ξ ) ≤ D0 ω0 (ξ )α with α ∈ [1/6, 1/3], a linear operator Dh ≡ ω10 [Q(ω0 d, ω0 h)+Q(ω0 h, ω0 d)]. There exists a constant K0 satisfying

h Dhdξ ≤ K0 D0

R3

R3

h(ξ )2 (1 + |ξ |) dξ.


175

Proof. By the Schwartz inequality and Lemma B.1, it follows 1 1 Dh h Dh dξ = D0 (1 + |ξ |) 2 h (1 + |ξ |)− 2 dξ D0 R3

R3

D0 ≤ 2

(1 + |ξ |)h(ξ ) + (1 + |ξ |) 2

R3

−1

(B.9) Dh D0

2 dξ

= O(1)D0

(1 + |ξ |)h(ξ )2 dξ. R3

This yields the lemma.

Appendix C. Construction of Boltzmann Shock Profile We now prove Theorem 8.1 on the accuracy of the Navier-Stokes shock profiles as an approximation of the Boltzmann shock profiles . We will use the weighted energy method. For other weighted energy methods for the study of boundary layers, see [1, 19] and references therein. We view the shock profile φ(x − st, ξ ), x ∈ R 1 of (6.1) as a perturbation of the Navier-Stokes profile A defined in (A.13): V ≡ φ − A. The equation for V is (ξ 1 − s)Vx − [Q(A, V) + Q(V, A)] = Q(V, V) − S.

(C.1)

− + + Let (P − 0 , P 1 ) and (P 0 , P 0 ). They are the macro-micro decompositions corresponding to the end Maxwellian states ω− ≡ ω(ξ ; u− , T− ) and ω+ ≡ ω(ξ ; u+ , T+ ), respectively. We have therefore two sets of variables   V = V0+ + V1+ ,   V = V− + V− , 0 1  V0± ≡ P ± V,  0  V± ≡ P ± V; 1 1

and the operator Q(V, A) + Q(A, V) can be written two different ways:  Q(V, A) + Q(A, V) = L+ V + R+ V,     − −  Q(V, A) + Q(A, V) = L V + R V, + + + L V ≡ Q(ω , V) + Q(V, ω ),    L− V ≡ Q(ω− , V) + Q(V, ω− ),    ± R V ≡ Q(V, A − ω± ) + Q(A − ω± , V).

(C.2)

Substitute the previous decomposition A = F0 +F1 into R± V to obtain, for a constant C > 0, R± V = [Q(F0 − ω± , V1± ) + Q(V1± , F0 − ω± )] + [Q(F1 , V) + Q(V, F1 )] ≤ C |V1± | + 2 |V| . (C.3)

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T.-P. Liu, S.-H. Yu

The equations for V0± and V1 are + + 1 + 1 P+ 0 (ξ − s)V0 + P 0 (ξ − s)V1 = 0, 1 P+ 1 (ξ

+ − s)V0x

1 + P+ 1 (ξ

+ − s)V1x

− L+ V1+

(C.4a) +

= R V + Q(V, V) − S, (C.4b)

and − − 1 − 1 P− 0 (ξ − s)V0 + P 0 (ξ − s)V1 = 0, 1 P− 1 (ξ

− − s)V0x

1 + P− 1 (ξ

− − s)V1x

− L− V1−

(C.5a) −

= R V + Q(V, V) − S. (C.5b)

Equations (C.4a) and (C.5a) give algebraic relationships between V0± and V1± . Similar to the matrix representations in (8.1) and the corresponding diagonalizations in (8.3) and (8.4), for V0± we have

i;± ∈ R, V0± = 3i=1 v0i;± r ± i , v ± 1 ± ± ± P 0 (ξ − s) r i = λi r i , λ± i ∈ R.

(C.6)

From the entropy condition in (8.2), there exists c > 0 such that   λ+  i < 0, for i = 1, 2, 3,  −λ− < 0, 3  |λ± | ≥ c , (because P 0 (ξ 1 − s) is a non-singular matrix),  3  c−1 > −λ± ≥ c for i = 1, 2. i

(C.7)

From (C.6), (C.7), (C.4a) and (C.5a), there exists C > 0 such that *  ± i;±  |v V1± |V1± for i = 1, 2, | ≤ C   0 * ± √ 4/3 V1± |V1±  3;±  . |v0 | ≤ λ± 3

(C.8)

From (C.4) and (C.5), we have the following L2 -estimates:

± 4 ± ± 1 ± 1 e± 5 ex dx V0± |P ± 0 (ξ − s)V0x + P 0 (ξ − s)V1x R ± ± ± 1 ± 1 ± ± ± V1± |P ± + 1 (ξ − s)V0x + P 1 (ξ − s)V1x − L V1 − R V − Q(V, V) − S R

4

× e± 5 ex dx = 0.

(C.9)

Here, the inner product g|h± are defined as in Sect. 7: g|h± ≡

R3

g(ξ )h(ξ ) dξ. ω± (ξ )


177

From (C.8) and (C.9), ± 4 ± ± 1 ± 1 V0± |P ± e± 5 ex dx 0 (ξ − s)V0x + P 0 (ξ − s)V1x R ± 4 ± ± 1 ± 1 V1± |P ± + (ξ − s)V + P (ξ − s)V e± 5 ex dx 1 0x 1 1x R ± ± 4 4 1 ± 1 = ± e e± 5 ex dx V0 |(ξ − s)V0± − V1± |(ξ 1 − s)V1± 5 R 2   3 ±

4 2e 4 j ;± ± ± ± 2 1  =± λj (v0 ) − e V1 |(ξ − s)V1  e± 5 ex dx 5 5 R j =1 ± 4 8e |V1± |2 ± 4 ex ± 2 ± ± 1 5 ≤ |V e± 5 ex dx. e dx +O(1) | + V |(ξ − s)V 1 1 1 ± 15 |λ | R R 3 (C.10) From (C.10) and (B.4), ± 4 ± ± 1 ± 1 e± 5 ex dx V0± |P ± 0 (ξ − s)V0x + P 0 (ξ − s)V1x R ± 4 ± ± 1 ± 1 + (ξ − s)V + P (ξ − s)V e± 5 ex dx V1± |P ± 1 0x 1 1x R ± ± ± ± ± 1 ± ± ± ± 4 ex ν0 ± 5 ≤ V1 |V1 + V1 |(ξ − s)V1 e dx + O(1) V |V 12 R 1 1 R 4

× e± 5 ex dx.

(C.11)

From (B.1), there exists C > 0, ± ± ξ 1 V1± |V1± ≤ −C V1± |L± V1± . And so O(1)

R

(ξ 1 − s)V1± |V1±

±

e±ex dx ≤ O(1)

R

(C.12)

± − V1± |L± V1± e±ex dx. (C.13)

From (C.3), (C.11), (C.12), and (C.13) together with Schwartz’s inequality, for sufficiently small, 4 ± ± ± ± 4 ex ν0 1 V1 |V1 e 5 dx ≤ − V1± |L± V1± e± 5 ex dx 2 R 2 R ± 4 ± ± 1 ± 1 e± 5 ex dx V0± |P ± ≤ 0 (ξ − s)V0x + P 0 (ξ − s)V1x R ± 4 ± ± 1 ± 1 ± ± ± + e± 5 ex dx V1± |P ± 1 (ξ − s)V0x + P 1 (ξ − s)V1x − L V1 − R V R ± 4 ν0 ± ± ± 8 S|S± + V1± |Q(V, V) ≤ e± 5 ex dx. V1 |V1 + 8 ν0 R

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And so R

ν0 ± ± ± ± 4 ex e 5 dx ≤ V |V 4 1 1

± 4 1 − V1± |L± V1± e± 5 ex dx 4 R ± ± 4 8 ± S|S + V1 |Q(V, V) ≤ e± 5 ex dx. ν 0 R (C.14)

From (A.15a) and (A.15b), R

S|S± e± 5 ex dx ≤ O(1) 5 . 4

(C.15)

With (C.14) and (C.15), we make an a priori assumption, −1

V1± ω± 2 ∞ ≤ O(1) 2 . Substitute (C.16) and (C.15) into (C.14). We have ± 4 ν0 ± ± ± ± 4 ex 1 dx ≤ − V1± |L± V1± e± 5 ex dx V1 |V1 e 5 8 R 8 R 4 8 S|S± e± 5 ex dx ≤ O(1) 5 . ≤ ν R 0

(C.16)

(C.17)

To obtain the desired higher order estimates in Theorem 8.1 and close the energy estimates, we increase the differentiation order in the a priori estimates in (C.16) up to C 10 in ξ -x space and perform the energy estimate up to H 20 . This proves the existence of the shock profile with the property in Theorem 8.1. The details are similar to the energy estimates before, so it is omitted. References 1. Bardos, C., Caflisch, R. E., Nicolaenko, B.: The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas, Commun. Pure Appl. Math. 49, 323–352 (1986) 2. Ludwig Boltzmann: (Translated by Stephen G. Brush): Lectures on Gas Theory. New York: Dover Publications, Inc. 1964 3. Caflish, R. E., Nicolaenko, B.: Shock Profile Solutions of the Boltzmann Equation, Commun. Math. Phys. 86, 161–194 (1982) ´ 4. Carleman, T.: Sur La Théorie de l’Equation Intégrodifférentielle de Boltzmann. Acta Mathematica 60, 91–142 5. Chapman, S., Cowling, T. G.: The Mathematical Theory of Non-Uniform Gases. Cambridge: Cambridge University Press, 1990, 3rd edition 6. Goodman, J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rational Mech. Anal. 95 (4), 325–344 (1986) 7. Grad, H.: Asymptotic Theory of the Boltzmann Equation. In: Rarefied Gas Dynamics, J. A. Laurmann, ed., Vol. 1, New York: Academic Press, pp. 26–59 1963 8. Hilbert, D.: Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen. Leipzig: Teubner, Chap. 22 9. Kawashima, S.: Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications. Proc. Roy. Soc. Edinburgh Sect. A 106(1–2), 169–194 (1987) 10. Kawashima, S., Matsumura, A., Nishida, T.: On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation. Commun. Math. Phys. 70(2), 97–124 (1979) 11. Kawashima, S., Matsumura, A.: Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Commun. Math. Phys. 101(1), 97–127 (1985)


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12. Liu, T.-P.: Nonlinear Stability of Shock Waves for Viscous Conservation Laws. Mem. Amer. Math. Soc. 56, 329 (1985) 13. Liu, T.-P.: Pointwise Convergence to Shock Waves for Viscous Conservation Laws. Commun. Pure and Appl. Math. Vol. 11, 1113–1182 (1997) 14. Maxwell, J. C.: The Scientific Papers of James Clerk Maxwell. Cambridge: Cambridge University University Press, 1990: (a) On the Dynamical Theory of Gases, Vol. II, p. 26. (b) On Stresses in Rarefied Gases Arising from Inequalities of Temperature, Vol. p.681 15. Nishida, T.: Takaaki Fluid Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation. Commun. Math. Phys. 61(2), 119–148 (1978) 16. Matsumura, A., Nishihara, K.: On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 2(1), 17–25 (1985) 17. Ukai, S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Japan Acad. 50, 179–184 (1974) 18. Ukai, S.: Les solutions globales de l’équation de Boltzmann dans l’espace tout entier et dans le demi-espace. C.R.Acad. Sci. Paris Ser. A-B 282(6), Ai, A317–A320 (1976) 19. Ukai, S., Yang, T., Yu, S.-H.: Existence and Stability of A SuperSonic Boundary Layer for Boltzmann Equation. To appear Communicated by P. Sarnak


Communications in


Rational Conformal Field Theories and Complex Multiplication Sergei Gukov, Cumrun Vafa Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA Received: 18 July 2003 / Accepted: 26 August 2003 Published online: 23 January 2004 – © Springer-Verlag 2004

Abstract: We study the geometric interpretation of two dimensional rational conformal field theories, corresponding to sigma models on Calabi-Yau manifolds. We perform a detailed study of RCFT’s corresponding to the T 2 target and identify the Cardy branes with geometric branes. The T 2 ’s leading to RCFT’s admit “complex multiplication” which characterizes Cardy branes as specific D0-branes. We propose a condition for the conformal sigma model to be RCFT for arbitrary Calabi-Yau n-folds, which agrees with the known cases. Together with recent conjectures by mathematicians it appears that rational conformal theories are not dense in the space of all conformal theories, and sometimes appear to be finite in number for Calabi-Yau n-folds for n > 2. RCFT’s on K3 may be dense. We speculate about the meaning of these special points in the moduli spaces of Calabi-Yau n-folds in connection with freezing geometric moduli. 1. Introduction Two dimensional “Rational Conformal Field Theories” (RCFT) were introduced in [1] as a particularly nice class of conformal field theories which have more structure and could be potentially classified. Moreover it was suggested that perhaps they may be dense in the space of all conformal theories, and so in this way one can potentially get a handle on all conformal theories. RCFT’s are characterized by having a symmetry algebra extending the Virasoro algebra (chiral algebra), in terms of which the Hilbert space can be decomposed into finite irreducible representations. Thus RCFT’s generalize the notion of minimal models introduced in [2]. Their conjecture motivated a great deal of work on RCFT’s leading in particular to Verlinde algebra [3] and the rich structure they encode [4]. Moreover it was shown in [5, 6] that RCFT’s naturally lead to boundary states, which in modern terminology we call D-brane states. In the post duality era, we have learned the importance of D-branes in uncovering non-perturbative aspects of string theory. Thus it is natural to ask the following question: We consider strings propagating on a Calabi-Yau background and we vary the moduli

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of Calabi-Yau. At points on the moduli which correspond to RCFT’s we naturally have some finite number of “special” D-branes. What do these D-branes correspond to? How do we interpret their preferred role at those moduli among the infinitely many allowed D-branes? Interesting as these questions appear, the modern discovery of S-duality raises a further question: Is the notion of RCFT an S-duality invariant concept? The answer to this is no [7]. In fact it could hardly be an S-duality invariant concept because the S-dual theory may not even correspond to string theory so there is no notion of 2d conformal theory on the S-dual side. Even if the S-dual theory is a string theory, one can easily see that the RCFT’s on one side do not correspond to RCFT’s on the dual side. For example type IIB is self-dual, with the roles of fundamental and D-strings exchanged; the rationality of its toroidal compactifications strongly depends on the BNS but is independent of BR . However on the S-dual side the roles of BNS and BR are exchanged. Thus rationality is not an S-duality invariant concept. One might thus consider this concept as not being a fundamental concept. However, it turns out that at least in some cases the concept of special points on moduli space makes sense even non-perturbatively. For example the same considerations that apply to rational points on compactification of strings on T 2 lead to singling out special points on the moduli space of the type IIB string coupling constant τ which could potentially have some significance in the full non-perturbative theory. Another way such a concept may remain relevant non-perturbatively is exemplified by compactification on Calabi-Yau 3-folds. In this case the string coupling constant combines with other hypermultiplets which come from Kähler moduli (in the type IIB case) and so the question of rationality, which seems to split between Kähler and Complex moduli, picks out, in a non-perturbative sense, some special complex moduli on the Calabi-Yau. More generally, at the very least the moduli corresponding to RCFT’s must be somehow special at weak string coupling and thus must teach us some extra symmetries about the target space physics. In this paper we take up the question of RCFT’s for sigma models on Calabi-Yau n-folds. More specifically we consider the case of complex dimension 1 in detail and use it to advance a conjecture about rational points for the more complicated case of sigma models on Calabi-Yau n-folds. For the case of (complex dimension one) T 2 target space we uncover the extra symmetry principle for the target space for it to correspond to a RCFT. These correspond to tori which admit Complex Multiplication. This means that there is a complex number λ (not real) for which z → λz maps T 2 to itself. This is not necessarily an isomorphism, and is in general a many to one map. Moreover the corresponding Kähler class has to be a complex multiplier λ for this to correspond to a RCFT with a diagonal modular invariant. Moreover for each Kähler class there exists a canonical complex multiplication which has the significance that gives the corresponding Cardy states as D0 branes localized at preimages of a given point on T 2 under this complex multiplication. In higher dimensions, mathematicians have a generalized notion of complex multiplication [8–10], which is natural to conjecture is related to the notion of rationality of conformal theory. This is basically the statement that the mid-dimensional cohomology and the associated variation of Hodge structure, leads naturally to the period matrix of a higher dimensional torus and one asks whether the associated torus admits a complex multiplication. In the case of Calabi-Yau threefolds this leads to a complex torus corresponding to a coupling constant matrix of the gauge fields. In the geometric engineering of N = 2 theories, this is the associated complex torus encoding the BPS masses of N = 2 electric and magnetic charge states. Translated in this way, there is a mathematical conjecture which suggests that in many cases there are only a finite number of points

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183

on moduli where the theory is rational. This is in sharp contrast to the case of T 2 where the rational points are dense in the space of all conformal theories. The organization of this paper is as follows: In the next section we review a simple example of c = 1 RCFT based on a circle, which will help us to introduce the notations and relevant concepts. In Sect. 3 we discuss two families of RCFT based on two-dimensional tori with extra symmetries: (a) direct product of two circles, and (b) a torus with Z3 symmetry. Both examples have been extensively studied in the literature, and we use them to illustrate general features of rational conformal field theories. Section 4 gives a friendly introduction into basics of imaginary quadratic number fields, which play a central role in the characterization of rational CFT’s. Following these introductory sections, in Sect. 5 we proceed to the general case of c = 2 CFT based on elliptic curve, and formulate the criteria for CFT to be rational and, further, to be diagonal. Our results allow to classify such rational conformal field theories. In Sect. 6 we discuss geometric and arithmetic interpretation of Cardy states in RCFT based on the elliptic curve. Finally, in Sect. 7 we conjecture a generalization of these results to Calabi-Yau manifolds of higher dimension. In different contexts, the relation between string theory and number theory has been discussed previously in [11, 7, 12–16]. 2. Review of a Compact Free Boson We start with a review of c = 1 CFT associated with a free bosonic field on a circle of radius R. Since this theory was extensively studied in the literature, and has been nicely reviewed in a number of recent papers, see e.g. [17–20], here we briefly describe only some aspects that will be relevant in the following sections. In our conventions α = 1, so that the self-dual radius is Rs.d = 1. For generic values of the radius R, the torus partition function has the following form: 1 2 1 2 1 Z(q, q; R) = q 2p q 2p , (2.1) 2 |η| 1,1 (p,p)∈

where q = exp(2π iτ ) and Dedekind’s η-function is defined as η = q 1/24

∞

(1 − q n ).

n=1

The partition function Z(q, q; R) is given by a sum over the even, self-dual momentum lattice 1,1 . Explicitly, the left and the right momenta read: 1 n 1 n p=√ + mR , p = √ − mR . (2.2) 2 R 2 R By definition, the theory is rational if one can represent the partition function Z(q, q) as a finite sum of the form: Z(q, q) = Mj j χj (q)χ j (q), (2.3) j,j

where Mj j ∈ Z≥0 and χi (resp. χ j ) are holomorphic (resp. anti-holomorphic) characters: χj (q) = tr Vj q L0 −c/24 . For the toroidal examples χi and χ j are generalized θ -functions with characteristics.

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Modular invariance and existence of the unique vacuum state impose further restrictions on the matrix M. Namely, uniqueness of the vacuum implies: M00 = 1.

(2.4)

On the other hand, modular invariance of the torus partition function requires: [M, T ] = 0,

[M, S] = 0,

(2.5)

where matrices S and T determine transformation of the characters under SL(2, Z): Sij χj (τ ), χi (−1/τ ) = χi (τ + 1) =

j

Tij χj (τ ).

(2.6)

j

The matrix T is diagonal and can be written in terms of the conformal dimensions i and the central charge c: Tij = δij e2iπ( i −c/24) .

(2.7)

There is no such simple general expression for the matrix S. However, in RCFT’s where the Verlinde algebra is an abelian group algebra Sij are proportional to roots of unity, which follows from the fact that S diagonalizes the Verlinde algebra. For example, in the rational c = 1 CFT of a compact boson, we have: Sjj = √

1 2N

e−iπjj /N .

(2.8)

For a given RCFT, it is an interesting problem to classify integer matrices M, which satisfy the relations (2.4) and (2.5), see [21] for a review. Now, let us explain a geometric interpretation of the rationality condition in a theory of a free compact boson. In other words, we want to analyze when the exponent set I = {i} becomes finite. From the explicit form of the left and right momenta (2.2) it is clear that this happens when R 2 is a rational number: R2 =

k , l

k, l ∈ Z,

where k and l are relatively prime integer numbers. The partition function in this case reads: χi (q)χ j (q). Z(q, q) = i+j =0 mod 2k i−j =0 mod 2l

It is manifestly invariant under T-duality symmetry, which among other things inverts the radius R and exchanges the winding and momentum modes: Z2

:

R↔

1 , R

m ↔ n.

The chiral primaries in this theory are labeled by index j ∈ Z mod 2kl. All theories with the same value of N = kl have the same fusion ring, which in this simple case is just the group algebra of: Z2N .


185

Therefore, theories with the same chiral algebra, but different modular invariants correspond to different ways one can decompose N into a product of two integers. In particular, R 2 = integer (or R 2 = 1/integer) correspond to diagonal modular invariants Mj j = δj j (or charge conjugation modular invariants): Z(q, q) =

χj (q) χ j (q).

(2.9)

j ∈I

2.1. Cardy States. Let us now discuss D-branes in rational conformal field theories. Among all consistent boundary states, there is a special (finite) subset of states, which are invariant under the full extended chiral algebra. This distinguished set of states, called Cardy states, can be systematically constructed in a diagonal RCFT following the original work of Cardy [5]. Let us recall that Cardy states are linear combinations of the Ishibashi states obtained by imposing a modular invariance on the string world-sheet: |i =

Sij |j S0j i

(2.10)

In particular, the number of the Cardy states is equal to the number of the Ishibashi states. The Ishibashi states, in turn, are defined as generalized solutions to the gluing conditions [6]: (2.11) Wn − (−1) W (W −n ) |B = 0, where is a gluing automorphism, and W is the conformal dimension of the chiral algebra operator W . For example, in the case √ of a free compact boson, the boundary state should preserve the U (1) current J = i 2N ∂X. Since the Virasoro generators are quadratic in the oscillator modes, the Ishibashi boundary condition (2.11) is solved by: ν α−n )|B = 0. (αnµ − Rνµ

(2.12)

We can also write this condition as: ∂Xµ (z) = Rνµ ∂X ν (z).

(2.13)

Here we restored space-time indices µ, ν = 1, . . . , d, and R is an automorphism of the chiral algebra, such that R ∈ O(d) and |det(R)| = 1. In particular, +1 eigenvalues of the automorphism correspond to Neumann boundary conditions, whereas −1 eigenvalues correspond to Dirichlet boundary conditions [17]. For this reason, a boundary state corresponding to an automorphism with p eigenvalues +1 is referred to as a Dp-brane. For a given automorphism R, the explicit form of the boundary state |B satisfying (2.12) is given by: ∞ 1 µ ν |B = exp − α−n × |p, p. α−n Rµν n n=1

(2.14)

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R

2N D0−branes

Fig. 1. Cardy states in rational c = 1 CFT can be identified with equally spaced D0-branes (black dots) on a circle

The sum over momenta in this expression goes over all the elements in the momentum lattice, which satisfy the condition (2.12). Let us now come back to√the case of the diagonal RCFT corresponding to a free boson on a circle of radius N Rs.d. . In this case, there are only two choices for the automorphism, R = ±1, corresponding to D1-branes and D0-branes, respectively. In the latter case, we obtain 2N Ishibashi states, the A-states in the notations of [22, 18]: ∞ n √ √ 1 n |An, n = exp + α−n × | √ + m N , √ + m N . α−n n N N m

(2.15)

n=1

where n ∈ Z mod 2N. For the other automorphism one has only two Ishibashi B-states: ∞ n √ √ 1 n |Bn, −n = exp − α−n × | √ + m N , − √ − m N , α−n n N N n=1

(2.16) where n = 0 or n = N . Using Cardy’s formula (2.10) and the explicit form for the elements of matrix S, in this model one finds two boundary states corresponding to D1-branes with different values (±1) of the Wilson line, and 2N D0-branes located at the equidistant positions on the circle, see Fig. 1: 2N p = 0 #(Dp − branes) = 2 p = 1. These Cardy states will be our basic building blocks in a specific example of c = 2 RCFT discussed next. 3. Simple Examples of c = 2 RCFT Starting with examples in this section, we proceed to our main subject, namely c = 2 RCFT based on the elliptic curve E. Specifically, we analyze the conditions on the complex structure parameter τ and the complexified Kähler modulus ρ under which the theory becomes rational and has a diagonal partition function (2.9). These results will help us build some intuition about what should happen in the general case that will be discussed in the following sections.


187

3.1. Basic Notations. To begin, we summarize our conventions and recall the discrete symmetries of a general RCFT based on the elliptic curve (see also [23] for a nice exposition). Throughout the paper we use the following notations for the real and imaginary parts of τ and ρ: √ detG G12 τ = τ1 + iτ2 = +i , G11 √ G11 ρ = ρ1 + iρ2 = B + i detG,

(3.1)

where Gij denote components of the (flat) metric on E. Indeed, it is straightforward to check that ρ2 |dx + τ dy|2 τ2 = G11 dx 2 + 2G12 dxdy + G22 dy 2 .

ds 2 =

(3.2)

Since we are interested in geometric interpretation of RCFT, we further assume that: τ2 > 0,

ρ2 > 0.

(3.3)

RCFT based on the elliptic curve E enjoys a large group of discrete symmetries: P SL(2, Z)τ × P SL(2, Z)ρ × Z2 × Z2 × Z2 .

(3.4)

Apart from the last factor, this symmetry group may be viewed as a group of T-dualities. In particular, the first two factors in this group act via modular transformations on τ and ρ, respectively. For example: P SL(2, Z)τ : τ →

aτ + b . cτ + d

The Z2 factors, on the other hand, interchange τ , ρ, and their complex conjugates. Specifically, the first Z2 acts as: Z2 : (τ, ρ) → (ρ, τ ).

(3.5)

By analogy with the corresponding symmetry of higher dimensional varieties, we refer to this Z2 as to the mirror transform. The second Z2 factor in (3.4) is space-time parity transformation acting as follows: Z2 : (τ, ρ) → (−τ , −ρ).

(3.6)

Finally, the last Z2 factor in (3.4) reverses world-sheet orientation: Z2 : (τ, ρ) → (τ, −ρ).

(3.7)

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3.2. A Product of Two Circles: E = S1 × S1 . Our first simple example of c = 2 RCFT will be a product of two c = 1 rational CFT’s corresponding to a product of two circles of radii:

k1 k2 R1 = and R2 = , l1 l2 where ki and li are some (pairwise co-prime) integers. In these models both τ and ρ are pure imaginary (modulo real integer part, which can be set to zero by P SL(2, Z) transformations):

k1 l2 k 1 k2 τ =i , ρ=i . k2 l 1 l1 l2 Note that both τ and ρ satisfy quadratic equations with integer coefficients (and the same discriminant). Using the analogous result for a single compact boson, we find that RCFT based S1 × S1 has a diagonal modular invariant if l1 = 1 and l2 = 1, i.e. up to modular transformations:

k1 τ =i , ρ = i k1 k2 . k2 Another way to describe these τ and ρ is to say that τ is a solution to the quadratic equation: k 2 τ 2 + k1 = 0

(3.8)

while ρ is an integer multiple of τ : ρ = k2 τ. This (strange) relation between τ and ρ is a precursor of the general property of the elliptic curve, called complex multiplication. As we will see in the following sections, all elliptic curves with this property correspond to a diagonal RCFT, and the converse is also true. Since in the present example, the RCFT is a product of two theories, the Verlinde algebra is just a product: Z2k1 × Z2k2 . Note that the total dimension of the chiral ring is equal to 4k1 k2 . This also gives the number of D0-branes in this theory. Indeed, boundary states corresponding to D-branes in the c = 2 RCFT in consideration can be obtained by tensoring the suitable Cardy states in the two copies of c = 1 RCFT. Specifically, a product of Dp1 boundary state with Dp2 boundary state gives a D(p1 + p2 )-brane boundary state: |D(p1 + p2 ) = |Dp1 ⊗ |Dp2 . If we tensor two A-type Cardy states, we obtain boundary states corresponding to D0-branes, 4k1 k2 in number. The D0-branes are distributed on a torus in a regular lattice of 2k1 rows and 2k2 columns, as shown in Fig. 2. On the other hand, if we tensor two B-type boundary states, we get four D2-branes, which cover the entire torus and differ in the values of Wilson lines they carry. Specifically, since each D1-brane on a circle carries ±1 Wilson line, by tensoring two of them we get four boundary states, labeled by (±1, ±1).


189

τ

1 D0−branes

D1−branes

Fig. 2. Cardy states in rational CFT based on a torus E = S1 × S1

Finally, tensoring A-type Cardy states with B-type Cardy states gives D1-branes parallel to the sides of the torus. Namely, if we tensor the D0-brane state on the first circle with a D1-brane state on the second circle, we obtain D1-branes wrapped on the second circle inside E = S1 × S1 . Since we could take 2k1 boundary states corresponding to D0-branes and there are 2 possible choices for the Wilson line on the D1-brane, in total we get 2 × 2k1 = 4k1 parallel D1-branes. Similar arguments show that there are 4k2 parallel D1-branes wrapped on the other basic cycle of the torus, see Fig. 2. Hence, there are 4(k1 + k2 ) D1-branes in total. Summarizing, we have:   p=0 4k1 k2 #(Dp − branes) = 4(k1 + k2 ) p = 1 . (3.9)  4 p=2

3.3. SU (3) Torus. There is another simple example of c = 2 RCFT based on the torus that has been extensively studied in the literature [22, 24, 25]. It is SU (3) WZW theory at level 1 corresponding to the elliptic curve E with extra Z3 symmetry [26]: τ = ρ = exp(2πi/3). As in the previous example, in this case both τ and ρ also satisfy a quadratic equation: τ 2 + τ + 1 = 0.

(3.10)

The dimension of the chiral ring of this theory is equal to 3, and the Verlinde algebra is just a group algebra of: Z3 . Comparing this with the previous example, one might expect that in general the fusion rules are given by (a product of) cyclic groups. In what follows we will show that this is indeed always the case; specifically, the Verlinde algebra is a group algebra of: Zn1 × Zn2 .

(3.11)

This guess includes the special case of a single cyclic group (as in the present example) when one of the factors is trivial, ni = 1.

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τ 2π/ 3

1

D0−branes

D1−branes

Fig. 3. D-branes on the SU (3) torus

Cardy states corresponding to various D-branes in this model have been studied in a number of papers [22, 24, 25, 27]. Here, we summarize the result: #(D0 − branes) = 3, #(D2 − branes) = 1.

(3.12)

The number of D0-branes is expected to be 3 on general grounds. In fact, in all theories the number of Cardy states corresponding to D0-branes should be equal to the dimension of the chiral ring, which is indeed 3 in the present case. Since the generic elliptic curve E does not reduce to a product of two circles, at the moment we have nothing to say about D1-branes. We can just briefly mention that certain boundary states in this model, studied in [22, 24, 25, 27], can be identified with D1-branes wrapped on the shortest cycles of the torus rotated by π/3, as illustrated on Fig. 3. In the general discussion below we will find the complete set of boundary states in this theory. A little bit more interesting is a result for D2-branes. Combining it with the result of the previous example, one might conclude that the number of D2-branes can be at least 1 or 4. Quite surprisingly, we will show that this a general answer: in all c = 2 RCFT’s based on elliptic curve, there is either one or four D2-branes. The number depends on certain arithmetic properties of E. 4. Imaginary Quadratic Number Fields Before we discuss the general case of (rational) conformal field theory based on the elliptic curve E we need to introduce a few basic notions from number theory that will naturally appear in our discussion. In fact, some of these objects have already entered our discussion in implicit form. For example, in two special cases discussed in the previous section we have noticed that the complex parameter τ satisfies a quadratic equation of the form, cf. (3.8) and (3.10): aτ 2 + bτ + c = 0

(4.1)

with relatively prime integer coefficients a, b, and c. We shall call this quadratic equation a minimal polynomial for τ , and denote by D its discriminant: D = b2 − 4ac. In all cases relevant to physics D < 0, so that τ has a non-zero imaginary part: √ −b + D . τ= 2a

(4.2)

(4.3)


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Mathematically, it means that τ is valued in the imaginary quadratic number field: √ K = Q( D). This particular way of writing the number field K indicates that √ it can be obtained from the familiar field√of all rational numbers, Q, by introducing D. In other words, every number x in Q( D) can be written in the form: √ x = α + β D, where α√ and β are rational numbers, α, β ∈ Q. Notice, the way we construct the number field Q( D) from rational numbers is very similar to how one usually defines √ the field of complex numbers, C, supplementing the field of real numbers, R, with −1. In our applications, τ is not just a number – it is a modulus of the elliptic curve: E = C/(Z ⊕ τ Z).

(4.4)

√ It turns out that elliptic curves with modular parameter τ ∈ Q( D) have a nice property called complex multiplication [28–30]. Elliptic curves with this property enjoy a lot of wonderful arithmetic and geometric properties. To explain what complex multiplication means, let us consider endomorphisms of the elliptic curve E, i.e. holomorphic maps from E to itself: ϕ

:

E → E.

Note ϕ is a finite degree map (not necessarily degree one). To describe such maps more explicitly, we can view the elliptic curve E as a quotient of a complex plane (parameterized by z) by a lattice Z ⊕ τ Z, cf. (4.4). Then, an endomorphism ϕ simply acts as z → ϕz. Since Z ⊕ τ Z is a two-dimensional lattice, we have only have to verify that ϕ maps its generators to some other other elements in this lattice: ϕ · 1 = m1 + n1 τ, ϕ · τ = m2 + n2 τ.

(4.5)

Clearly, any elliptic curve has many trivial endomorphisms corresponding to multiplication by an integer, ϕ ∈ Z. In order to see if there exist any non-trivial endomorphisms, one can take ϕ from the first equation and substitute it to the second equation. As a result, one finds a quadratic equation with integer coefficients of the form (4.1). This simple calculation illustrates that elliptic curves with non-trivial endomorphisms have τ in some imaginary quadratic field. In fact, it turns out that an elliptic curve E has a non-trivial endomorphism if and only if τ obeys a quadratic equation (4.1) with integer coefficients. In this case, E is said to have complex multiplication (or to be of CM-type). Summarizing, the endomorphism ring of a general elliptic curve can be one of the following: Z, no CM √ End(E) = (4.6) D Z + Zaτ, CM-type, τ = −b+ . 2a Thus, complex multiplication gives another way to characterize elliptic curves with such “special” values of τ .

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There is a close relation between imaginary quadratic τ , which obey (4.1), and binary quadratic forms: 2a b . b 2c In our discussion, such forms will be associated with the intersection form of lattices. Notice the form above naturally defines a two-dimensional even lattice. However, this form is not unique. Namely, for any S ∈ SL(2, Z), the intersection form: 2a b 2a b = S S tr b 2c b 2c defines the same lattice. The invariant associated with such a lattice is the discriminant D = b2 − 4ac. For a given value of D, the equivalence classes of the integral binary quadratic forms form a finite abelian group, the so-called class group [31, 28, 32]: Cl(D). The order of this group |Cl(D)| = h(D) is called the class number. It is naturally identified√with the number of ideal classes, h(K), of the imaginary quadratic field K = Q( D). The last object we need to introduce is an order, Of , Of = Z ⊕ Z[f aτ ].

(4.7)

Here, a is a leading coefficient in the quadratic polynomial (4.1) for τ , and f is a positive integer number, called a conductor of Of .√ We can view an order Of as a two-dimensional lattice in the number field K = Q( D), generated by 1 and f aτ . The reason Of will appear in our discussion is that any element ϕ ∈ Of obviously gives a complex multiplication, cf. (4.5). Note also that the endomorphism ring itself, End (K) is also an order with1 f = 1. 5. General c = 2 RCFT Based on the Elliptic Curve E In this section we are going to study general c = 2 conformal field theory based on the elliptic curve with arbitrary parameters τ and ρ. We will show that CFT is rational if (and only if) both τ and ρ take values in the same imaginary quadratic field. The discriminant of this field gives the dimension of the chiral ring. We also show that a condition for a diagonal modular invariant implies a further relation between τ and ρ. Namely, one has to be a complex multiplication for the other. 5.1. Momentum Lattices. In general, the partition function of c = 2 CFT based on the elliptic curve E = C/(Z ⊕ τ Z): Z(q, q) = 1

1 2 η η2

(p,p)∈ 2,2

This special order in K is called the ring of integers.

1

2

1

q 2p q 2p

2

(5.1)


is given by the sum over an even, self-dual momentum lattice: i p 1 ρ τ ρτ 2,2 Z ⊕Z ⊕Z ⊕Z . ∈ = √ 1 ρ τ ρτ p 2τ2 ρ2

193

(5.2)

This lattice will be one of the central objects in our discussion, and, as we shall see in a moment, properties of the CFT, like rationality etc., can be formulated and analyzed in terms of momentum lattices. For this reason, it is convenient to introduce a few more objects associated with the momentum lattice 2,2 . First, we can define the following sublattices in 2,2 . Let 0 be the lattice of leftmoving momenta p for a fixed value of p (say, for p = 0) and a similar lattice 0 : p 0 = { p | ∈ 2,2 }, 0 0 0 = { p | (5.3) ∈ 2,2 }. p 0 . The rank of these Since 2,2 is an even integer lattice, the same is true about 0 and lattices is not greater than 2, and generically it is zero. By simply forgetting the right (or left) momentum, we can also define the following projections: p L = { p | ∈ 2,2 }, ∗ ∗ R = { p | (5.4) ∈ 2,2 }. p Both L and R can be characterized as sets where p (or p) take their values. In general, unlike (5.3), L,R are not lattices. Of course, in some special cases it may happen that the rank of L,R is less than 4. As we shall see below, these are precisely the occasions relevant to rational theories. Note, from the above definitions we obtain straightforward relations: 0 ⊆ L , 0 ⊆ R , 0 ⊕ 0 ⊆ 2,2 .

(5.5)

5.2. A Criterion for Rationality. There are various ways of defining rational CFT’s. In our examples associated with tori, it is convenient to formulate the condition for rationality in terms of momentum lattices: a CFT is rational if and only if the left momentum lattice 0 is a finite index sublattice in L . In such cases, both 0 and L are rank two sublattices in 2,2 . Moreover, it is easy to see that they are dual lattices: 0 ∼ = L∗ ,

0 ∼ = R∗ .

(5.6)

Indeed, since 2,2 is an even, self-dual integer lattice, for any vector (q, q) ∈ 2,2 and a given vector (p, 0) ∈ 0 we have a pairing: (p, 0) · (q, q) = pq ∈ Z.

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Therefore, any vector in 0 also belongs to L∗ . Conversely, to show L∗ ⊆ 0 let us take a vector p ∈ L∗ . Then, using the above equation and self-duality of 2,2 , one finds that (p, 0) is a vector in 2,2 . Hence, L∗ ∼ = 0 . Similar arguments give the second isomorphism in (5.6). Therefore, we conclude that the study of rational conformal field theories based on the elliptic curve E is related to the study of integer even two-dimensional lattices. In particular, it gives a classification of such RCFT’s. We will come back to this later. Now, let us discuss the geometric properties of the elliptic curve corresponding to rational conformal field theory. Suppose that a CFT associated with E is rational, i.e. 0 is a finite index sublattice in L . Using the explicit expression (5.2) for the right momenta p, we find that the elements of 0 correspond to integer numbers (m1 , m2 , n1 , n2 ) ∈ Z4 , which obey two independent linear relations [7]: m1 + m2 ρ + n1 τ + n2 τρ = 0, m1 + m2 ρ + n1 τ + n2 τρ = 0.

(5.7)

If we solve, for example, for ρ from the second equation and substitute the result to the first equation, we find a quadratic equation for τ , with integer coefficients2 : aτ 2 + bτ + c = 0,

gcd(a, b, c) = 1

(5.8)

with discriminant D = b2 − 4ac. Since the imaginary part of τ has to be strictly positive, cf. (3.3), √ we conclude that τ has to belong to the imaginary quadratic number field K = Q( D): √ −b + D τ= . 2a

(5.9)

√ If we now substitute this τ back into (5.7), we find that ρ is linear in D over Q. Hence, both τ and ρ are elements in K. In order to show that the converse is also true, one can take τ, ρ ∈ K, and construct, for example, √ τ and τρ in terms of 1 and ρ. Since both τ and ρ are assumed to be linear functions of D with rational coefficients (with τ2 > 0 and ρ2 > 0), one can always write τ and τρ as linear functions of ρ with rational coefficients. Multiplying the resulting relations by a suitable integer, one finds two equations of the form (5.7) with integer coefficients, where n2 = 0 and n1 = 0. Therefore, by construction these relations are independent and define a lattice 0 . Summarizing, we obtain an effective criterion for rationality of c = 2 conformal field theory based on the elliptic curve E: √ RCFT ⇐⇒ τ, ρ ∈ Q( D). In other words, we found that in order for CFT to be rational, both the target space torus and its dual should have complex multiplication relative to the same quadratic imaginary field. 2 Alternatively, we could use the fact that any three momentum vectors p ∈ (or p ∈ ) satisfy a L R linear relation over Q, since 0 is a Z ⊕ Z-module.


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5.3. Dimension Of The Chiral Ring And Verlinde Algebra. As we found in the previous subsection, rational conformal field theories based on the elliptic curve E are naturally attached to even integer lattices. Specifically, let vi , i = 1, 2, be the generators of the momentum lattice 0 . Since the intersection form is even, we can write it as: 2a b (5.10) vi · vj = f b 2c for some integer numbers a, b, c, and f , such that gcd(a, b, c) = 1. By definition, the dual lattice L = 0∗ is generated by vectors vi∗ with intersection form: 1 −2a b ∗ ∗ , (5.11) vi · vj = b −2c fD where D = b2 − 4ac is the discriminant. It is clear from (5.10) and (5.11) that the dimension of the chiral ring, given by the index [L : 0 ], is equal to: [L : 0 ] = f 2 |D|. Notice that the right hand side is expressed in terms of invariant quantities. Furthermore, the Verlinde algebra is the group algebra of: D(0 ) = 0∗ / 0 .

(5.12)

In mathematics literature this group is usually called the discriminant group [33, 34], see also [32]. It is a finite abelian group of order f 2 |D|. Since 0 is a lattice of rank two, in general, the discriminant group is a product of two cyclic groups, in agreement with what we found in the specific examples of tori with extra symmetries, cf. (3.11). Specifically, D(0 ) is generated by two elements, g and h, such that: g 2af hbf = 1

and

g bf h2cf = 1.

The structure of this group depends in a crucial way on the arithmetic. Specifically,   D = 1mod4 Zf × Zf D , D(0 ) = Z2f × Z2f D , D = 0mod4(D = D/4), b = 0 .  Z 2f a × Z2f c , b = 0 This gives a general characterization of the Verlinde algebra in the rational conformal field theory based on the elliptic curve E. 5.4. Diagonal Modular Invariants. Now we turn to the main problem, namely, analysis of the conditions under which the partition function (5.1) takes the diagonal form (2.9). As we explained in the previous subsections, curve E associated with a rational CFT has complex multiplication. However, the ring End(E) itself did not enter our discussion so far. Here, we show that RCFT has a diagonal modular invariant iff either ρ or τ (or modular transformations thereof) belong to End(E), i.e. iff ρ is a complex multiplication for a given τ , up to discrete symmetries (3.4). A diagonal modular invariant essentially implies identification of left and right momentum lattices. Namely, given an even integer lattice 0 , one can canonically reconstruct

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the whole momentum lattice 2,2 , which is even and self-dual. Specifically, we take two copies of L = 0∗ : 2,2 = (L , L )

(5.13)

L − L = 0 .

(5.14)

with the equivalence relation:

To see that the lattice (5.13) constructed in this way is even, let us take a vector (p, p) ∈ 2,2 . By the equivalence relation (5.14) we have p = p + v, where v ∈ 0 , and p ∈ 0∗ . Therefore, (p, p) · (p, p) = p 2 − p 2 = (p + p)(p − p) = −v(2p + v) = −2vp − v 2 .

(5.15)

Since 0 is taken even, both terms here are even and the claim follows. Furthermore, to show that 2,2 constructed in (5.13) is self-dual, one can provide a basis, (pi , pi ) = {(vi∗ , vi∗ ), (vj , 0)}, where vj are the generators of 0 , and vi∗ ∈ 0∗ are the dual generators. Then, the bilinear form looks like 01 (pi , pi ) · (pj , p j ) = . 1∗ The determinant of this matrix is clearly equal to 1. Therefore, 2,2 is self-dual. Now we want to compare a general lattice of the form (5.11) with the momentum lattice (5.2). Up to discrete symmetries (3.4) we can choose τ and ρ, such that the left momentum lattice is generated by (vectors proportional to) 1 and τ : i iτ ∗ 0 = L = √ . ,√ 2τ2 ρ2 2τ2 ρ2 It is easy to compute the intersection form of this lattice: 1 τ vi∗ · vj∗ =

1

2τ2 ρ2 2τ2 ρ2 |τ |2 τ1 2τ2 ρ2 2τ2 ρ2

.

Since 0 has to be an even integer lattice, one should be able to write this intersection as (5.11). Comparing the individual entries, we find that in diagonal RCFT τ and ρ look like (up to discrete symmetries (3.4)): √ −b + D τ= , ρ = f aτ . (5.16) 2a Hence, ρ is a complex multiplication for an imaginary quadratic τ . More precisely, ρ should√be associated with a generator of an order in the imaginary quadratic field K = Q( D): ρ ∈ Of with the conductor f . It is easy to verify that the converse is also true. Namely, given τ and ρ of the form (5.16), the corresponding modular invariant is diagonal. Indeed, substituting (5.16) in


197

the formulas for the momentum vectors (5.2), we find that the momentum lattice 2,2 is of the form (5.13). Specifically, we have: −i 2af τ2 p = m1 + m2 ρ + n1 τ + n2 ρτ = (m1 + f cn2 ) + (n1 − af m2 + bf n2 )τ. (5.17) It immediately follows that: L = {(p, ∗)T ∈ 2,2 } = √

i (Z ⊕ Zτ ). 2af τ2

(5.18)

In order to find elements of 0 , we have to solve p = 0. For the right momenta we find: −i 2af τ2 p = m1 + m2 ρ + n1 τ + n2 ρτ = (m1 − f cn2 ) + (n1 + af m2 − bf n2 )τ. (5.19) Hence, p = 0 gives two conditions: m1 = f cn2 n1 = −af m2 + bf n2 . Substituting this into (5.17) yields: if p|p=0 = √ (2aτ + b)(−m2 ) + (2c + bτ )n2 ∈ 0 . 2af τ2 Therefore, we explicitly constructed 0 : 0 = {(p, 0)T ∈ 2,2 } = √

if (Z[2aτ + b] ⊕ Z[2c + bτ ]). 2af τ2

(5.20)

The resulting lattices L , 0 , and 2,2 are related as in (5.13) – (5.14), so that the corresponding CFT is diagonal. It is straightforward to check this directly computing the partition function (5.1) for these momentum lattices. As a result, one finds a diagonal modular invariant: Z(q, q) = χω (q)χ ω (q), ω∈I

where the exponent ω ∈ I that labels representations of the chiral algebra can be identified with left momenta, cf. (5.12): ω ∈ L / 0 ,

(5.21)

and the characters χω (q) have the form: χω (q) =

1 1 (v+ω)2 q2 . η2

(5.22)

v∈0

Finally, we can combine all the results in this section to conclude that diagonal c = 2 rational conformal field theories are classified by the following data: (i) discriminant D = 0, 1 mod 4 (a negative integer, such that (−D) is square-free); (ii) conductor f (a positive integer); (iii) an element of the class group Cl(D). In terms of this data, the CFT has chiral ring of dimension |D|f 2 .

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Suppose, for example, we want to know how many diagonal RCFT’s have the chiral ring of dimension 163. Using the above results, we can immediately answer this question. Since h(163) = 1, the answer is surprisingly simple: there is a unique RCFT with this property. On the other hand, if we asked a similar question about RCFT’s with chiral ring of dimension 159, we would find h(159) = 10 such theories. This simple example illustrates sporadic pattern of RCFT’s, which nevertheless can be completely explained by the methods of number theory. Note, it also follows from our analysis that the set of c = 2 rational conformal field theories on tori is dense.

5.5. A Digression: The First Main Theorem of Complex Multiplication. In this subsection (which can be skipped, especially in a single reading) we digress on another remarkable property of the elliptic curve with complex multiplication. Throughout the paper we mainly viewed the elliptic curve as a quotient of C by a twodimensional lattice. However, we could also view E as an algebraic curve defines, say, by a Weierstrass polynomial: E : y 2 = 4x 3 − g2 x − g3 ,

(5.23)

where g2 and g3 are related to the modular parameter τ via invariant j -function: j (τ ) =

1728g23

θE8 (τ )3 η(τ )24 + 744 + 196844q + 21493760q 2 + 864299970q 3 + . . . .

g23 − 27g32 −1

=q

=

(5.24)

Notice that the coefficients in the power series expansion of the j -function are integer numbers. Even though j (τ ) is √ a very non-trivial function of τ , there is a nice characterization of its values for τ ∈ Q( D), known as the first main theorem of complex multiplication. Namely, suppose the elliptic curve E has a complex multiplication, i.e. τ obeys the quadratic equation (4.1). Then, j (τ ) also obeys a polynomial equation with √ integer coefficients of degree h (where h = h(D) is the class number of the field Q( D)): P (z) = zh + a1 zh−1 + . . . + ah = 0,

ai ∈ Z.

(5.25)

A solution to such an equation is called an algebraic integer 3 . Therefore, the j -invariant of the elliptic curve E with complex multiplication is an algebraic integer and E is naturally defined over the number field K(j (τ )). Motivated by this nice result, one might expect that a proper criterion for CFT to be rational should be formulated as a condition on the algebraic variety to be defined over the algebraic closure Q, obtained from the field Q by adjoining the roots of all irreducible polynomials like (5.25). It is easy to see, however, that this criterion would be wrong. Indeed, it would predict “too many” RCFT’s. For example, in the case of Calabi-Yau manifolds it would predict existence of infinitely many points (which are dense) in the moduli space, whereas in the later sections we will argue to the contrary. 3 The first main theorem of complex multiplication further says that if z = j (τ ) is one of these numbers, √ then K(j (τ )) is the maximal abelian extension of K = Q( D), with Gal(K(j (τ ))/K) = Cl(End(E)) acting transitively on the set of numbers j (τ ) [28, 31].


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More elementary is to see that the above criterion fails already in the case of the elliptic curve E. Indeed, in general, the converse of the first main theorem of complex multiplication is not true, so it can not be formulated as an “if and only if” condition. On the other hand, from the analysis of the previous sections, we know that CFT is rational if and only if E has complex multiplication. This demonstrates that the right signature of the rational CFT is complex multiplication, rather than a possibility to define the target space variety over Q. 6. Geometric Interpretation of Cardy States In the previous sections we have established a relation between RCFT data and the arithmetic of the elliptic curve E. Motivated by such a relation, one may wonder if it can be extended to string theory, including D-branes and other non-perturbative objects. Due to their geometric nature, D-branes seem to be especially promising. In the weak coupling limit they can be viewed as submanifolds of E of various (co)dimension. From the CFT point of view, there are some “special” D0-branes, which preserve the full chiral algebra. The corresponding boundary states were explicitly constructed by Cardy [5]. Therefore, one could ask: “What is arithmetic/geometric interpretation of the Cardy states?” There are several natural candidates for the answer to this question. For example, once we deal with arithmetic of E, one might consider rational points of E, i.e. solutions of (5.23) with rational values of the coordinates x, y ∈ Q. However, these can not correspond to Cardy states. Indeed, the number of rational points on E may be infinite, whereas the number of Cardy states is always finite (and equal to the dimension of the chiral algebra): #(Cardy states) = |D|f 2 , where D is the discriminant of the quadratic polynomial (5.8) for τ , and f is the conductor. In this section we study D-branes on the elliptic curve E with diagonal modular invariant: Z(q, q) = χj (q) χ j (q), (6.1) j ∈I

where the exponent j can be identified with momentum, cf. (5.21): j ∈ L / 0 ,

(6.2)

and the characters have the form (5.22). Following Cardy [5], we show that there are always |D|f 2 D0-branes in this theory, which correspond to the regular points of L / 0 . On the other hand, the number of D2- and D1-branes depends on the arithmetic of the elliptic curve in a very interesting way. For example, #(D2 − branes) =

1 4

Df odd Df even.

Note the simple examples studied in Sect. 3 agree with this general result, see (3.9) and (3.12). Below we explain these results in more detail.

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Before we start, let us recall that Cardy states are linear combinations of Ishibashi states (2.14): ∞ 1 µ ν |B = exp − × |p, p α−n Rµν α−n n

(6.3)

n=1

which satisfy boundary conditions (2.12). It is convenient to write this boundary condition in the following form: p = −Rp.

(6.4)

Since the labels j in (6.2) are identified with momenta, we can say that for a given R the Ishibashi states are labeled by: (j, −Rj ),

j ∈ L / 0 .

However, since we assume that RCFT is diagonal (6.1), the Ishibashi state (j, j ) appears in the closed string spectrum only if j = j : (j, j ) = (j, −Rj ). It follows that for a given gluing condition (6.4) the Ishibashi states are labeled by fixed points of R in the exponent set I: j = −Rj,

j ∈ L / 0 .

(6.5)

In particular, the number of solutions to this condition gives us the number of the corresponding Dp-branes. In the following subsections we solve (6.5) for each value of p. Note that j = 0 is always a solution to (6.5) for any allowed R, i.e. there is always at least one corresponding D-brane.

6.1. D0-branes. In order to get a D0-brane, we have to impose Dirichlet boundary conditions in both spatial directions, so that 2 × 2 matrix R must have two eigenvalues −1, i.e.: R = −1. In this case, the boundary condition (6.4) has p = p. It does not impose any further constraints on the momentum, so that and p ∈ L / 0 is a solution. Hence, the number of D0-branes is given by the following universal formula for all models: #(D0 − branes) = |D|f 2 .

(6.6)

This number is the same as the dimension of the chiral ring, and suggests interpretation of D0-branes as special points on the torus E. Indeed, we can think of D0-branes as


201

2aτ+b τ

2c+bτ 1

Fig. 4. D0-branes in rational CFT can be identified with points (black dots) in the lattice 1, τ modulo ρ, ρ τ . In this figure we illustrate this for a specific model where τ satisfies the quadratic equation: τ 2 + 2τ + 2 = 0. In this example, a = 1, b = 2, c = 2, and D = −4

regular points in the quotient L / 0 or else, as preimages of a marked point on a torus E under a specific complex multiplication 4 (see Fig. 4): z → ρ · z, where, cf. (5.16): ρ = f (2aτ + b).

(6.7)

Indeed, one can easily check that ρ · 1 = f (2aτ + b) and ρ · τ = −f (2c + bτ ) are the two generators of the lattice 0 , cf. (5.20). Geometrically, it is convenient to visualize the set L / 0 as a parallelogram with edges f (2aτ + b) and f (2c + bτ ) in the lattice L . Note that ρ = f (2aτ + b) is closely related to the Kähler structure of the torus, which is ρ = f aτ . In fact, if bf is even ρ can be viewed as twice the Kähler class (with a suitable shift of ρ by bf/2. Thus roughly speaking the Kähler class defines the relevant endomorphism of the torus which defines the Cardy states by its preimage. 6.2. D2-branes. The next simplest case is when we impose Neumann boundary conditions in all the directions: R = 1. 4 The complex multiplication ρ has a number of special properties. First, note that it can be obtained by taking a derivative of the defining quadratic polynomial for τ :

ρ = f

dQ(τ ) , dτ

Q(τ ) = aτ 2 + bτ + c.

Another distinguished property of ρ is that it corresponds to a complex multiplication whose square is multiplication by an integer. Moreover, it is the only complex element (up to integer multiples) in the End(E) with this property. Explicitly, the square of the element 2aτ + b is: (2aτ + b)2 = 4a 2 τ 2 + 4abτ + b2 = 4a(aτ 2 + bτ ) + b2 = 4a(−c) + b2 which is the discriminant, D.

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R f(2aτ+b) f(2c+bτ) Fig. 5. In the case of D2-brane boundary conditions, the involution R flips the parallelogram made by vectors f (2aτ + b) and f (2c + bτ ). The origin (black dot) is always fixed under this involution. The other three potential fixed points are denoted by empty circles

The involution R inverts the exponent set I = L / 0 : p → −p

:

R

(6.8)

and flips the parallelogram subtended by vectors f (2aτ + b) and f (2c + bτ ), as shown in Fig. 5. According to (6.5), the Ishibashi states are in one-to-one correspondence with the fixed points of this involution, modulo the lattice 0 . Geometrically, it is clear that there are either one or four fixed points on the parallelogram (see Fig. 5), depending on whether its edges have odd or even coordinates in the lattice L ∼ = {1, τ }. Indeed, the origin is always a fixed point of R. Let us see when there are extra fixed points. The potential candidates are middle points on the edges of the parallelogram and a point in the middle (denoted by empty circles in Fig. 5). The explicit coordinates of these points in the lattice L ∼ = {1, τ } are the following: (f b/2, f a),

(f c, f b/2),

(f c + f b/2, f a + f b/2).

It is clear that all of these points are in the lattice 0 if and only if both f b is even. Hence, we arrive to the following general result: #(D2 − branes) =

1 4

bf odd bf even.

(6.9)

Note that in the case of bf even, the four inequivalent D2 branes differ by the value of Z2 Wilson lines along the two cycles.

6.3. D1-branes. Finally we consider the case of D1-branes. In this case we have one Neumann and one Dirichlet direction, which correspond to +1 and −1 eigenvalues of R, respectively. Allowing for D1-branes of arbitrary orientation, we can write the corresponding involution R as: R

:

p → αp ∗ ,

(6.10)


203

where α is some phase, |α| = 1. Note that this is an order 2 operation. Since p takes values in the lattice L ∼ = {1, τ }, the involution R must respect this lattice. In particular it should map a basis of L into another basis: 1 → −A − Bτ, τ → C + Dτ,

(6.11)

where A, B, C, D are integer numbers, such that AD − BC = 1. Therefore, we get two conditions: α = −A − Bτ, ατ = C + Dτ,

(6.12)

from which we can eliminate α: −τ =

A + Bτ . C + Dτ

(6.13)

Simply put, the last condition says that −τ should be an involution of P SL(2; Z) acting on τ , for otherwise we wouldn’t have any D1-branes. Then, for every τ , which satisfy a relation of the form (6.13), there might be different involutions corresponding to different α’s. Therefore, it is natural to split the question in two parts, and classify first all τ which solve (6.13), and then classify all possible α’s. It is easy to see that the only τ (in the fundamental domain), which solve (6.13), are (i) either pure imaginary, τ1 = 0, or (ii) lie on a unit circle, |τ | = 1, see Fig. 6. To see that these are the only solutions, with no loss of generality we can assume τ is in the fundamental domain of the upper half plane. We are looking for involutions of SL(2, Z) acting on it which give −τ . On the other hand −τ is also in the fundamental domain of the upper half plane. This can only be consistent with the notion of the fundamental

Fig. 6. Two families of the solution for τ lie either on a unit circle or on the imaginary line. There are also two special cases, τ = i and τ = exp(2πi/3), represented by black dots. A number near every point indicates the total number of D1-brane boundary states in the corresponding RCFT

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domain if the involution is ± the identity matrix, or it is ±S in which case it maps the boundary of the fundamental domain to itself. There is one extra case corresponding to the involution T ST −1 which fixes τ = exp(2πi/3). In the first case we have τ = −τ which states that τ is pure imaginary. In the latter case it implies that τ is on the unit circle. The two different families of the solutions for τ correspond to elliptic curves with different symmetries, simple examples of which were discussed in Sect. 3. Let us now consider each of the two cases in turn: (i) A product of two circles. In the first case b = 0 and the solutions to (6.13) look like: c τ= − . a In fact, this is precisely the case discussed in Subsect. 3.1, where the torus is a product of two circles: E = S1 × S1 . All possible D1-branes in this case were classified in (3.9), with k1 = f a and k2 = f c: #(D1 − branes) = 4f (a + c)

(6.14)

and come from α = 1 or α = −1. They correspond to 2f a equally spaced D1 branes along one direction and 2f c equally spaced D1 branes along the other direction of the torus. Moreover each of these one branes can have a Z2 Wilson line on them. (ii) A torus symmetric relative to the diagonal. In this case |τ | = 1 (which implies a = c) and the torus E has extra symmetry corresponding to the reflections relative to the diagonals. In particular z → αz is a symmetry when α = ±τ , which thus generically yields two involutions R. For the case of τ = i we have 4 involutions given by z → i k z. For τ = e2πi/3 we have 6 involutions given by z → ωk z, where ω is a 6th root of unity. The corresponding fixed characters under √ this involution will correspond to equally spaced D1 branes in the direction given by α. The corresponding values of τ are: √ −b + D τ= , D = b2 − 4a 2 . 2a For the generic case, the total number of D1-branes, which is just the number of the fixed lattice points under the two reflections is #(D1 − branes) = 4f a.

(6.15)

For the case τ = i the D1 branes which make angles that are multiples of π/4. For D1 branes in the direction 2nπ/4 (n = 0, 1) we have 2f equally spaced branes each of which can have an extra Z2 Wilson line. For D1 branes in the directions π/4, 3π/4 we have 2f equally spaced branes (all branes passing through the origin) without any extra possibility of Wilson lines. The D1 branes corresponding to the latter angles do not come from tensoring Ishibashi states of the two decoupled circles, but rather they correspond to using the extra Z2 exchanging the two circles. For τ = e2πi/3 we have D1 branes which make angles of 2πn/12, where n = 0, ..., 5. One can check that for n


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even there are 3f equally spaced ones and for n odd there are f equally spaced ones. In both special cases (τ = i and τ = exp(2πi/3)) the total number of D1-branes is 12f . We have thus seen that the classification of Dp branes for p > 0 is much more sporadic than those for the D0 branes. This is to be expected because the D0 branes are precisely the ones naturally picked out by the diagonal modular invariant.

7. RCFT and Higher Dimensional Calabi-Yau So far we have talked about CFT’s based on the simplest Calabi-Yau sigma model, namely the target being T 2 . More precisely, we focused only on the bosonic sigma model, but in this case the incorporation of fermions does not modify our discussion, as the fermion partition function is independent of T 2 moduli. It is natural then to raise the question of RCFT’s corresponding to supersymmetric sigma models propagating on higher dimensional Calabi-Yau manifolds. Unfortunately not much is known about the exact solutions in such cases, and we only have existence proof for such CFT’s. The only general classes we know are tensor products of minimal N = 2 supersymmetric conformal theories (Gepner models) and toroidal orbifolds. It turns out that both classes are RCFT’s or have moduli for which RCFT’s appear in a dense subspace. It is the existence of this class of examples which motivates the belief that CFT’s are “exactly solvable” if they are rational or near one. Consider the quintic threefold, for example. There is only one point in its moduli of Kähler and complex deformation where the CFT is exactly known and that is the RCFT corresponding to the Gepner point. One wonders whether there are other points on the moduli space where they are rational and therefore, perhaps solvable. In this section we propose a criterion for rationality of conformal theory on Calabi-Yau manifolds which agrees with all the known examples of rational points discussed above (see [35] for further evidence in the case of toroidal conformal field theories). However, given mathematical conjectures a la André and Oort [36, 37] our proposal for rationality suggests that RCFT’s are not dense in the generic case of Calabi-Yau sigma models! Our criterion for rationality is motivated by generalization of the notion of complex multiplication to higher dimensional varieties. Indeed, it was pointed out to us by Kazhdan and Mazur that there already exists a suitable notion of complex multiplication5 for higher dimensional varieties introduced in 1969 by D. Mumford [8]. In particular, complex multiplication was studied in the context of a K3 surface by Piateckii-Shapiro and Shafarevich [9], and more generally, in the context of Calabi-Yau manifolds by Borcea [10]. The idea is rather simple: One first defines what it means for an abelian variety (i.e. complex tori) to admit complex multiplication. Then one asks if the variation of the Hodge structure of the Calabi-Yau M and its mirror W , whose period matrices lead to a pair of associated abelian varieties admit complex multiplication (of a “compatible” type).

7.1. Complex Multiplication for Complex Tori. Consider a complex n-dimensional torus. T 2n ∼ = Cn /Z2n . 5 Mathematically, it says that manifold M has complex multiplication when its Hodge group, Hg(M), is commutative.

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This is defined by identifications zi ∼ zi + δij

zi ∼ zi + Tij ,

where T is an n×n complex symmetric period matrix. Then we say that the torus admits complex multiplication if there exists a non-trivial endomorphism z → Az

(7.1)

which implies that A = M + NT , T A = M + N T for some integer matrices M, N, M , N . In other words we have a second order matrix equation T (M + N T ) = M + N T ⇒ T N T + T M − N T − M = 0.

(7.2)

Moreover one requires that N has rank n.6

7.2. Calabi-Yau and the Intermediate Jacobian. The notion of considering mid dimensional cohomology elements and integrating over a mid dimensional integral lattice of cycles to define an abelian variety is well known. For example for the case of a genus g Riemann surface with a symplectic pairing of 1-cycles: (Ai , Bj ) = δij

(Ai , Aj ) = (Bi , Bj ) = 0,

(7.3)

one considers g holomorphic 1-forms ωi normalized relative to the A-cycles ωi = δij Aj

and defines the Jacobian by ωi = Tij .

(7.4)

Bj

A similar idea works for arbitrary complex varieties and in particular for Calabi-Yau 3-folds. In the case of the Calabi-Yau threefold Tij can be identified with the complex torus defining the coupling constants of the associated U (1)n gauge fields and is related to the prepotential F (in the homogeneous coordinates) by Tij = ∂i ∂j F. We say that the Calabi-Yau admits complex multiplication if the corresponding intermediate Jacobian associated with T admits complex multiplication. 6 This rules out examples like M = E CM × E , where ECM is an elliptic curve with complex multiplication and E is another arbitrary elliptic curve without CM.


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7.3. A Criterion for RCFT for Calabi-Yau Sigma Models. A Calabi-Yau sigma model is completely characterized by its complex and Kähler moduli. Since the notion of complex multiplication is natural only for complex structure of the variety, it is natural to associate to a given Calabi-Yau M with a given complex and Kähler structure, a mirror pair of Calabi-Yau (M, W ) with fixed complex structures (where we have traded the Kähler structure of the original Calabi-Yau with complex moduli of its mirror). This is effectively how we studied the case of the elliptic curve, by viewing τ and ρ as defining pairs of elliptic curves. We propose the following criterion of the Calabi-Yau sigma model to correspond to a RCFT: Sigma model on Calabi-Yau corresponding to the pair (M, W ) is RCFT if and only if M and W admit complex multiplication over the same number field. For example, for a Calabi-Yau threefold M satisfying complex multiplication one gets an equation (7.2) of order two in the (h2,1 (M) + 1) × (h2,1 (M) + 1) matrix T . In this case, it has been shown by Borcea [10] that existence of complex multiplication is equivalent to the condition that elements of the endomorphism matrix A generate imaginary number field K: K∼ = End(H 3 (M, Q)) ⊗ Q

(7.5)

of degree: [K : Q] = 2(h2,1 (M) + 1). On the other hand, the mirror W admitting complex multiplication gives elements which are in an algebraic number field of degree 2(h2,1 (W ) + 1) = 2(h1,1 (M) + 1), and the criterion we are imposing for RCFT is that they are elements of the same number field7 . 7.4. Application of the Criterion. To check the criterion, we have to make sure it agrees with the known cases of RCFT’s for Calabi-Yau sigma models. Indeed it does. Toroidal orbifolds corresponding to RCFT’s obviously admit complex multiplication inherited from the fact that the underlying torus admits complex multiplication (extending our discussion from the elliptic case – the simplest case being orbifolds of the product of elliptic curves). Much more non-trivial are the Gepner points, corresponding to Fermat polynomials. It is also known that these also do admit complex multiplication [38–40, 10]. Below we show how this works for the quintic threefold with one complex moduli. This is already impressive evidence for the criterion we have proposed for rationality. We now wish to study how frequently one would encounter rational conformal theory in the moduli of a given Calabi-Yau sigma model, assuming the criterion we have proposed holds. To get a feel for this, consider Riemann surfaces. As discussed above we can identify with it an associated Jacobian. However the moduli space of genus g curves is 3g − 3 complex dimensional whereas the moduli space of the abelian varieties of dimension g has dimension g(g + 1)/2. Thus for g > 4 the Riemann surfaces are not dense in moduli of the corresponding tori. The Schottky problem is to identify which abelian varieties can arise for Riemann surfaces. Similarly, one could ask which Riemann surfaces admit complex multiplication. Even though there is a dense set of points in the moduli of complex structure of the tori admitting complex multiplication this may not hold true for the measure zero subspace of it corresponding to those coming from Riemann surfaces. Unlikely as this sounds, indeed there is evidence and a standing mathematical conjecture by Coleman [41] (see also [42] 7

Clearly, the degree of this number field is bounded by min(2(h2,1 (M) + 1), 2(h1,1 (M) + 1)).

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for recent developments) that for sufficiently large g there are only a finite number of Riemann surfaces admitting CM! Indeed a similar conjecture exists for arbitrary varieties8 and it is believed that the number of CM points are dense only if the relevant moduli space itself is of the form G/H (i.e. a submoduli of the full toroidal moduli defined by some linear algebraic constraint). In particular, in the case of complex tori (of complex dimension n) and K3 surfaces this conjecture predicts a dense set of CM/RCFT points. Indeed, in both cases the moduli space turns out to be a coset space: SO(2n, 2n; Z)\SO(2n, 2n)/SO(2n) × SO(2n) and

(7.6)

SO(20, 4; Z)\SO(20, 4)/SO(20) × SO(4)

respectively. On the other hand, for the case of the one parameter family of quintic three-folds, complex multiplication is conjectured to occur at most at a finite number of points. It would be very interesting to test this conjecture as it seems to be at odds with the common lore for RCFT’s. This of course might be a blessing in disguise as it seems to point to the existence of some finite number of interesting points on the moduli of Calabi-Yau compactifications. These may end up being interesting points when the moduli of Calabi-Yau manifolds get frozen by some mechanism. 7.5. The Example of Fermat Quintic. Finally, a non-trivial test of our criterion can be obtained by considering a one-dimensional family of quintic three-folds M:

z15 + z25 + z35 + z45 + z55 − 5ψz1 z2 z3 z4 z5 = 0.

At the Fermat point, ψ = 0, the corresponding sigma-model becomes rational, namely it is the (k = 3)5 Gepner model. On the other hand, the Calabi-Yau manifold M has complex multiplication at ψ = 0 [10, 38, 39]. This is related to the fact that the automorphism group is bigger for the Fermat quintic than for any other generic member in this family. Moreover, ψ = 0 is the only known non-trivial CM-point in the whole moduli space of M. In this sense, there is the same amount of physical and mathematical data on this question, which therefore provides at least one non-trivial check of our proposal. In order to see explicitly that the Fermat quintic has sufficiently many holomorphic endomorphisms (and, therefore, admits complex multiplication) let us evaluate the period matrix (7.4) at ψ = 0. In a particular basis of A and B cycles (7.3), the standard calculation gives [13, 43]: α − 1 α + α3 T = , α + α 3 −α 4 where α is a (non-trivial) 5th root of unity, α 5 = 1. Note that α is a solution to the degree 4 polynomial with integer coefficients: x 4 + x 3 + x 2 + x + 1 = 0. 8 Mathematically, a basic version of this conjecture is known as André-Oort conjecture [36, 37], and we thank F. Oort and B. Mazur for explaining to us the general philosophy behind it. Roughly, the André-Oort conjecture says that in order for a (sub)family of algebraic varieties to contain a dense set of CM-points, the corresponding moduli space has to be a “Shimura (sub)variety”. For example, the moduli spaces of elliptic curves and K3 surfaces are of this type, however the moduli space of a Calabi-Yau manifold in general is not.


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It is straightforward to check that the matrix T satisfies the quadratic matrix equation of the form (7.2): T N T + T M − N T − M = 0, where

N=

1 −1 , 0 1

M=

00 , 00

N =

−1 0 , −1 0

M =

−1 0 . −1 −1

The corresponding endomorphism is given by the matrix: α − 1 α + α3 . A= 1 + α + α 3 −α 4 Notice that elements of T and A take values in a degree 4 number field K, cf. (7.5): K = Q(α) which can be obtained from the field of rational numbers, Q, by adjoining the fifth root of unity. Since in the present example h2,1 (M) = 1, this is in complete agreement with the general formula for the degree, [K : Q] = 2(h2,1 + 1). Acknowledgements. We would like to thank D. Kazhdan and B. Mazur for many illuminating discussions on complex multiplication. We are also grateful to J. de Jong, J. Maldacena, K. Oguiso, H. Ooguri, F. Oort, A. Recknagel, S. Shenker, F. Rodriguez-Villegas, and E. Witten for valuable discussions. This research was partially conducted during the period S.G. served as a Clay Mathematics Institute Long-Term Prize Fellow. The work of S.G. is also supported in part by grant RFBR No. 01-02-17488, and the Russian President’s grant No. 00-15-99296. The work of C.V. is supported in part by NSF grants PHY-9802709 and DMS 0074329.

References 1. Friedan, D., Qiu, Z., Shenker, S.: Conformal Invariance, Unitarity, and Two-Dimensional Critical Exponents. Phys. Rev. Lett. 52, 1575 (1984) 2. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory. Nucl. Phys. B241, 333 (1984) 3. Verlinde, E.: Fusion Rules and Modular Transformations in 2D Conformal Field Theory. Nucl. Phys. B300, 360 (1988) 4. Moore, G., Seiberg, N.: Polynomial Equations for Rational Conformal Field Theories. Phys. Lett. B212, 451 (1988); Classical and Quantum Conformal Field Theory. Commun. Math. Phys. 123, 177 (1989); Taming the Conformal Zoo. Phys. Lett. 220, 422 (1989) 5. Cardy, J.L.: Boundary Conditions, Fusion Rules and the Verlinde Formula. Nucl. Phys. B 324, 581 (1989) 6. Ishibashi, N.: The Boundary And Crosscap States In Conformal Field Theories. Mod. Phys. Lett. A4, 161 (1989) 7. Moore, G.: Arithmetic and attractors. arXiv:hep-th/9807087 8. Mumford, D.: A note on Shimura’s paper Discontinuous Groups and Abelian Varieties. Math. Ann. 181, 345 (1969) 9. Pjateckii-Shapiro, I., Shafarevich, I.R.: The Arithmetic of K3 Surfaces. Proc. Steklov Inst. Math. 132, 45 (1973) 10. Borcea, C.: Calabi-Yau Threefolds and Complex Multiplication. In: Essays on Mirror Manifolds, S.-T. Yau, (ed.), Cambridge, MA: International Press, 1992 11. Lian, B.H.,Yau, S.-T.: Arithmetic Properties of Mirror Map and Quantum Coupling. Commun. Math. Phys. 176, 163 (1996) 12. Miller, S.D., Moore, G.: Landau-Siegel zeroes and black hole entropy. arXiv:hep-th/9903267 13. Candelas, P., de la Ossa, X., Rodriguez-Villegas, F.: Calabi-Yau manifolds over finite fields, I. arXiv:hep-th/0012233

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14. Schimmrigk, R.: Arithmetic of Calabi-Yau varieties and rational conformal field theory. arXiv:hepth/0111226 15. Kachru, S., Schulz, M., Trivedi, S.: Moduli Stabilization from Fluxes in a Simple IIB Orientifold. hep-th/0201028; see also a talk of S. Kachru at Strings 2002 Conference, http://www.damtp.cam.ac.uk/strings02/avt/kachru 16. Manin, Y., Marcolli, M.: Holography Principle and Arithmetic of Algebraic Curves. hep-th/0201036 17. Fuchs, J., Schweigert, C.: Branes: From free fields to general backgrounds. Nucl. Phys. B 530, 99 (1998) [arXiv:hep-th/9712257] 18. Maldacena, J., Moore, G., Seiberg, N.: Geometrical interpretation of D-branes in gauged WZW models. JHEP 0107, 046 (2001) 19. Cappelli, A., D’Appollonio, G.: Boundary States of c = 1 and c = 3/2 Rational Conformal Field Theories. JHEP 0202, 039 (2002) hep-th/0201173 20. Gaberdiel, M.R., Recknagel, A.: Conformal boundary states for free bosons and fermions. JHEP 0111, 016 (2001) [arXiv:hep-th/0108238] 21. Gannon, T.: Monstrous Moonshine and the Classification of CFT. math.QA/ 9909080 22. Ooguri, H., Oz, Y., Yin, Z.: D-Branes on Calabi-Yau Spaces and Their Mirrors. Nucl. Phys. B477, 407 (1996) 23. Dijkgraaf, R., Verlinde, E., Verlinde, H.: On Moduli Spaces of Conformal Field Theories with c ≥ 1. Proc. of 1987 Copenhagen Conference Perspectives in String Theory 24. Recknagel, A., Schomerus, V.: D-branes in Gepner models. Nucl. Phys. B 531, 185 (1998) [arXiv:hep-th/9712186] 25. Gutperle, M., Satoh, Y.: D-branes in Gepner models and supersymmetry. Nucl. Phys. B 543, 73 (1999) [arXiv:hep-th/9808080] 26. Vafa, C.: Quantum Symmetries of String Vacua. Mod. Phys. Lett. A4, 1615 (1989) 27. Mizoguchi, S., Tani, T.: Wound D-Branes in Gepner Models. Nucl. Phys. B611, 253 (2001) 28. Parshin, A.N., Shafarevich, I.R., (eds.): Number Theory II. Berlin-Heidelberg-New York: SpringerVerlag, 1992 29. Shimura, G., Taniyama, Y: Complex Multiplication of Abelian Varieties and its Applications to Number Theory. Japan Math. Soc. 1961 30. Lang, S., (ed.): Number Theory III. Berlin-Heidelberg-New York: Springer-Verlag, 1991 31. Borevic, Z., Shafarevich, I.: Number theory. 1985 32. Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices, and Codes. Berlin-Heidelberg-New York: Springer-Verlag, 1993 33. Nikulin, V.: Integral symmetric bilinear forms and some of their Applications. Math. Izv. 14, 103 (1980) 34. Dolgachev, I.: Integral quadratic forms: Applications to algebraic geometry. Sem. Bourbaki 611, 251 (1982) 35. Wendland, W.: Moduli Spaces of Unitary Conformal Field Theories. PhD Thesis, September 2000 36. André, Y.: G-functions and Geometry. Aspects of Mathematics, Vol. E13, Braunshweig: Vieweg, 1989; André, Y.: Distribution des points CM sur les sous-variétés de modules de variétés abéliennes, 1997 37. Oort, F.: Canonical Lifts and Dense Sets of CM-points. Arithmetic Geometry. Proc. Cortona Symposium 1994, F. Catanese, (ed.), Symposia Math., Vol. XXXVII, Cambridge: Cambridge Univ. Press, 1997, p. 228 38. Shioda, T.: What is known about the Hodge conjecture? In: Algebraic Varieties and Analytic Varieties. Adv. Studies Pure Math 1, 55 (1983); Shioda, T.: Geometry of Fermat Varieties. In: Number Theory Related to Fermat’s Last Theorem, Progress in Math. 26, 45 (1982) 39. Deligne, P.: Local Behavior of Hodge Structures at Infinity. AMS/IP Studies Adv. Math. 1, 683 (1997) 40. Deligne, P. (Notes by Milne, J.): Hodge Cycles on Abelian Varieties. Lecture Notes in Math 900, 9 (1982) 41. Coleman, R.: Torsion Points on Curves. In Galois representations and arithmetic algebraic geometry, Y. Ihara, ed., Adv. Studies Pure Math. 12, 235 (1987) 42. de Jong, J., Noot, R.: Jacobians with Complex Multiplication. In: Arithmetic Algebraic Geometry, G. van der Geer, F. Oort, J. Steenbrink, (eds.), Basel-Boston: Birkhäuser, 1991 43. Hori, K., Iqbal, A., Vafa, C.: D-Branes and Mirror Symmetry. hep-th/0005247. Communicated by Y. Kawahigashi


Communications in


Noncommutative Rigidity Eli Hawkins Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 4, 34014 Trieste, Italy. E-mail: [email protected] Received: 27 December 2002 / Accepted: 29 September 2003 Published online: 2 March 2004 – © Springer-Verlag 2004

Abstract: Using very weak criteria for what may constitute a noncommutative geometry, I show that a pseudo-Riemannian manifold can only be smoothly deformed into noncommutative geometries if certain geometric obstructions vanish. These obstructions can be expressed as a system of partial differential equations relating the metric and the Poisson structure that describes the noncommutativity. I illustrate this by computing the obstructions for well known examples of noncommutative geometries and quantum groups. These rigid conditions may cast doubt on the idea of noncommutatively deformed space-time. 1. Introduction One plausible way to try and construct examples of noncommutative geometry is to start with an ordinary, commutative manifold and deform it. One can try to construct noncommutative algebras that in some sense approximate the algebra of smooth functions on the manifold, and then to construct noncommutative geometries which approximate the geometry of the original manifold. There has been considerable success with the first step. Techniques of geometric quantization can be applied in many cases to construct a sequence of algebras which approximate the algebra of functions in a very strong sense. In a much weaker sense, the formal deformation quantization constructions of Fedosov [16] and Kontsevich [21] give noncommutative approximations to any manifold. Another motive for considering deformations is physical. There are many reasons to suspect that pseudo-Riemannian geometry might not accurately describe the small scale structure of space-time. Noncommutative geometry is a plausible route toward a better description. However, the fact that pseudo-Riemannian geometry is a sufficient description of space-time for most purposes, suggests that noncommutativity might be treated as a perturbation. If so, then this noncommutativity would be described in the leading order by a Poisson structure. Much optimism about this direction was generated by Kontsevich’s remarkable

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proof [21] that there exists a deformation quantization corresponding to any Poisson structure. Kontsevich’s proof was partly inspired by string theory, and this contributed to interest in possible connections between noncommutative geometry and string theory. In [7] Connes, Douglas, and Schwarz argued that string theory compactified on a torus with a nonzero 3-form potential is equivalent to string theory compactified on a noncommutative torus. Various authors have argued that a limit of string theory on flat space-time with a constant 2-form potential (B-field) is described by a noncommutative Yang-Mills theory, with the B-field providing the Poisson structure (see [25, 20] and refences therein). Cornalba and Schiappo [11] argued that the Poisson field is more generally equal to B + F , although they went on to advocate using nonassociative “algebras”. Although there exist many examples of noncommutative deformations of the algebra of functions on a manifold, there are very few examples in which this is extended to a deformation of the geometry. The only examples for which the complete axioms of noncommutative geometry are satisfied are the noncommutative torus and a generalization of this constructed by Connes and Landi [10] for any compact Riemannian manifold on which T2 acts by isometries (but see also [8, 9]). Of course, the axioms of noncommutative geometry are not set in stone, and Da˛browski and Sitarz have constructed an example which only satisfies some of these axioms. On the other hand, Chakraborty and Pal [4] have constructed geometries on SUq (2) which satisfy the axioms but do not correspond to the classical geometry of SU(2) = S 3 . All this suggests that deforming geometry is not easy and may not be possible generically. As I explain here, the structures of geometry cannot be deformed in general. Specifically, integration can only be deformed nicely if the divergence of the Poisson field ij vanishes, π |j = 0. The other conditions involve a type of generalized connection called a “contravariant connection” (see Sect. 2.2). The first order differential calculus of 1-forms and the gradient operator d : Cc∞ (M) → 1c (M) can only be deformed if there exists a flat, torsion free contravariant connection. A (pseudo)Riemannian metric can only be deformed if 0 = Ki

j kl

,

where K is the curvature of a certain contravariant connection constructed from the Poisson structure and metric. The first two conditions are completely independent of any specific formulation of noncommutative geometry. The derivation of the last condition is motivated by Connes’ formulation of noncommutative geometry, but as I explain, it appears to be much more general. These conditions are necessary, but not sufficient. For one thing, they only depend upon the description of the deformation to leading order; they do not ask whether something may appear in higher order to obstruct the deformation of geometry. This is not even the most complete set of necessary conditions of this kind, but that is for a future paper. I anticipate three likely interpretations of this result: (1) Noncommutative deformations have no relevance to the geometry of the universe. (2) The geometry of space-time is described by a noncommutative deformation at some level, and my conditions are physical equations of motion of the Poisson field (or whatever fields determine the Poisson structure). (3) Noncommutative deformations (in some sense) are relevant, but my assumptions about their properties are too restrictive.

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In the last case, my results can be taken as a guide to what one should not expect or assume when trying to construct noncommutative deformations. In Sect. 2, I give the necessary background and define what I mean by a deformation. I review how a Poisson structure is derived from a deformation and review the definition of a contravariant connection. In Sect. 3, I give the obstruction to deforming integration. In Sect. 5, I give the obstruction to deforming 1-forms and the gradient operator. In Sect. 6, I give the obstruction to deforming a metric. In Sects. 7 and 8, I discuss the structure of solutions of these conditions and show how some examples of noncommutative geometry and Poisson manifolds do or do not satisfy these conditions. 1.1. Notation. Throughout, M will denote a locally compact, smooth manifold. C ∞ (M) will denote the space of smooth (infinitely differentiable), C-valued functions on M. (M, V ) will denote the space of smooth sections of a vector bundle V over M. p (M) := (M, p T ∗M) will denote the space of smooth differential p-forms. Cc∞ , c , and ∗c will denote the spaces of compactly supported smooth functions, sections, and forms. Lower case Latin characters f , g, and h will mostly denote smooth functions, except in Sect. 6 after which g will denote a metric. Lower case Latin characters a, b, and c will denote elements of a (possibly noncommutative) algebra. Lower case Greek characters σ , ρ, α, β, γ will mostly denote 1-forms. However, π will denote the Poisson bivector field, ω will denote a symplectic 2-form, and will denote a volume form. Multivectors (vectors, bivectors, et cetera) are sections of the exterior powers ∗ T M of the tangent bundle. I will use the symbol to denote not only the contraction of a vector into a differential form, but also the contraction of a multivector into a differential form. This is such that, for instance, (X ∧ Y ) = Y (X ) = (X, Y, . . . ). LX denotes the Lie derivative with respect to a vector field X ∈ (M, T M). A covariant connection will be denoted as usual by ∇. In index notation this will also be i := ∇ v i . A contravariant connection (see denoted with a vertical stroke as, e. g., v|j j Sect. 2.2) will be denoted as D to distinguish it from a Dirac-type operator which will be denoted D in Sect. 6. The symbol # will denote a certain map, # : T ∗M → T M, determined by the Poisson structure (Sect. 2.1). Square brackets will denote the Lie bracket of vector fields or the bracket of a given Lie algebra. Curly brackets will denote the Poisson bracket and its generalizations. [a, b]− := ab − ba will denote a commutator. Finally, [ · , · ]π will denote the Koszul bracket of 1-forms (Sect. 2.1)

2. Deformations Let M be a smooth manifold. I am interested in a smooth, noncommutative deformation of M. What does this mean? Suppose that there exists a one-parameter family of algebras Aκ with A0 = Cc∞ (M). We don’t need to assume that κ takes a continuous range of values, only that Aκ is defined for all κ in some subset of R that is dense at 0. Because deformations of this kind are often discussed in the context of the classical limit of a quantum system, the deformation parameter is often called . However, in general, the deformation parameter is not necessarily Planck’s constant and I denote it as κ here to maintain this distinction.

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Simply having a collection of algebras is not enough. We must tie them together somehow. Thinking of these Aκ ’s as algebras of smooth functions on noncommutative spaces, the spaces should fit together smoothly into a larger noncommutative space, a sort of noncommutative cylinder. In this way, we have another algebra A which is thought of as the algebra of compactly supported, smooth functions on this larger noncommutative space. The algebras Aκ form something like a bundle over the set of values of κ, although it may not be locally trivial. Let k be the algebra of smooth functions of κ. The algebra A should be a k-algebra; that is, A is a k-module and multiplication in A is k-linear. We need just a little local triviality. We should be able to (non-canonically) expand any element of A as a power series in κ with coefficients in Cc∞ (M). Algebraically, this means that κ m A/κ m+1 A ∼ = Cc∞ (M) for any positive integer m. It is also possible to consider formal deformations in which κ is only a formal parameter. In that case we should take k = C[[κ]], the algebra of formal power series in κ. In fact, we will only need series expansions to second order in κ. At a minimum, we can take for k the algebra C[κ]/κ 3 and for A the free k-module Cc∞ (M) ⊗ k with some noncommutative product. The trivial case provides some guidance here. Consider the cylindrical space X ∼ = M × R and the commutative algebras A := Cc∞ (X) and k = C ∞ (R); identify M with M × {0}. There are many ways of identifying X with M × R. If we choose one, then this gives a canonical way of expanding any function f ∈ A = Cc∞ (X) as a power series in κ (the coordinate on R) with coefficients in Cc∞ (M). However, if we do not choose such an identification, then such an expansion is not canonical. The “operator ordering” ambiguity in a noncommutative deformation is thus equivalent to a coordinate freedom in this commutative case. Nevertheless, any function f ∈ A = Cc∞ (X) that vanishes along M can be uniquely written as a multiple of κ, any function that vanishes to second order along M can be written uniquely as a multiple of κ 2 , and so on. The following definition is limited to the properties that I will actually need here. Definition 1. A smooth deformation (to second order) of a manifold M is given by a commutative C-algebra k, a k-algebra A, an element κ ∈ k, and an isomorphism Cc∞ (M) ∼ = A/κA, such that A/κ 3 A is a free C[κ]/κ 3 -module. The isomorphism Cc∞ (M) ∼ = A/κA is a generalization of the requirement that A0 = Cc∞ (M). The last condition (freeness) expresses the requirement that any element of A can be expanded to second order in powers of κ with coefficients in Cc∞ (M). Consequently, A/κ 3 A is isomorphic to Cc∞ (M) ⊗ C[κ]/κ 3 as a C[κ]/κ 3 -module. The isomorphism is not canonical. Nevertheless, κA/κ 2 A is canonically isomorphic to Cc∞ (M); that is, the class of any κa ∈ A modulo κ 2 is naturally identified with κf for some unique f ∈ Cc∞ (M). We might also require that A be a ∗-algebra. That is, that there exists an antilinear involution ∗ : A → A such that (ab)∗ = b∗ a ∗ . This would ensure that the Poisson field is real. However, this will not affect the form of my constraints at all, so I will not bother discussing it further. Definition 1 encompasses both concrete deformations in which the algebras Aκ actually exist, and formal deformations in which κ is only a formal parameter. The concrete case will be important in motivating my definitions.


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2.1. Poisson Structure. From any smooth deformation, we can construct a Poisson bracket. Use the commutator notation, [a, b]− := ab − ba, and use modκ to denote the equivalence class modulo κA. For f, g ∈ Cc∞ (M), let a, b ∈ A such that f = a mod κ and g = b mod κ and define the Poisson bracket {f, g} ∈ Cc∞ (M) by κ{f, g} = i[a, b]− mod κ 2 . The right-hand side is a multiple of κ because [a, b]− mod κ = [f, g]− = 0. This defines {f, g} uniquely because of the freeness assumption in Definition 1. It only depends on f and g, because if we add a multiple of κ to a or b, this only changes [a, b]− by a multiple of κ 2 . This is explicitly C-bilinear and antisymmetric in f and g. The properties of the Poisson bracket derive from associativity in A to first and second order in κ. The identity [ab, c]− = [a, c]− b + a[b, c]− for a, b, c ∈ A, modulo κ 2 gives the Leibniz rule (derivation property), {f g, h} = {f, h}g + f {g, h}. The Jacobi identity for the commutator, modulo κ 3 , gives the Jacobi identity for the Poisson bracket, {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0. The fact that the Poisson bracket is bilinear, antisymmetric, and a derivation on both arguments implies that it is given by a Poisson bivector field π ∈ (M, 2 T M). That is, {f, g} = π(df, dg) := π ij dfi dgj . The Jacobi identity is equivalent to a condition on π which is expressed most succinctly (and cryptically) in terms of the Schouten-Nijenhuis bracket (see e. g., [28]) as 0 = [π, π ],

(2.1a)

or more explicitly as jk

ij

ka 0 = π ia π |a + π j a π ki |a + π π |a .

(2.1b)

This is a diffeomorphism invariant differential equation; it does not depend on the choice of torsion-free connection. The Poisson structure determines a vector bundle homomorphism # : T ∗M → T M, defined in index notation by, (#σ )j := π ij σi . In general, # is not an isomorphism, but if it is, then its inverse is given by a symplectic structure. That is, there exists a 2-form ω ∈ 2 (M) which is closed, dω = 0, and inverse to π , π ai ωaj = δji .

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In general, the image of # gives preferred directions of the tangent bundle which determine a symplectic foliation, so-called because the restriction of the Poisson structure to a leaf is symplectic. The Koszul bracket [ · , · ]π of 1-forms is defined by the properties [df, dg]π = d{f, g}

(2.2)

and [σ, fρ]π = #σ (f )ρ + f [σ, ρ]π , where #σ acts on f by a directional derivative. This satisfies the Jacobi identity and is related to the Lie bracket by #[σ, ρ]π = [#σ, #ρ].

(2.3)

The cotangent bundle T ∗M with the map # : T ∗M → T M and the Koszul bracket is an example of a more general structure called a Lie algebroid over M. A more elementary example of a Lie algebroid is the tangent bundle itself with the identity map and the Lie bracket. 2.2. Contravariant Connections. I will use the concept of contravariant connection extensively here. This is actually just a case of the natural concept of a connection with respect to a Lie algebroid. Contravariant connections (or “contravariant derivatives”) were defined by Vaisman [28] and analyzed in detail by Fernandes [18]; the concept also occurs in [24] under a different name. Definition 2. Given a vector bundle V → M, a contravariant connection D is a linear map from 1-forms to first order differential operators on V such that for any σ ∈ 1c (M), f ∈ Cc∞ (M), and v ∈ c (M, V ), Df σ v = f Dσ v and Dσ (f v) = #σ (f ) v + f Dσ v.

(2.4)

Here, #σ (f ) means the directional derivative of f by the vector field #σ . This is very similar to the definition of an ordinary (covariant) connection, except that cotangent vectors have taken the place of tangent vectors. This can also be written using index notation such that Dσ = σi Di . The contravariant index on Di is the reason for the name. In many ways, contravariant connections behave much like covariant connections. Many standard definitions, identities, and proofs for covariant connections can be translated to contravariant connections simply by exchanging the roles of tangent and cotangent vectors (or covariant and contravariant indices) and replacing Lie brackets with Koszul brackets. A contravariant connection on a vector bundle V is equivalent to one on the dual bundle V ∗ and determines a contravariant connection on the matrix bundle End V = V ⊗ V ∗ . We can mimic the standard definition of curvature and define K(σ, ρ)v := Dσ Dρ v − Dρ Dσ v − D[σ,ρ]π v.

(2.5)


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Note that the Koszul bracket has taken the place of the Lie bracket. A simple calculation shows that K(σ, ρ) is a Cc∞ (M)-linear operator on c (M, V ) and thus all derivatives of v have cancelled. Similarly, K(σ, ρ) is Cc∞ (M)-linear in σ and ρ. So, K is really an End(V )-valued bivector – just as the curvature of a covariant connection is an End(V )valued 2-form. I will use the term “flat” to mean that a contravariant connection has K = 0. In the special case that V = T ∗M, we can mimic the definition of torsion and define, T (σ, ρ) := Dσ ρ − Dρ σ − [ρ, σ ]π .

(2.6)

In the same way, T turns out to be a type (2, 1)-tensor – just as the torsion of a covariant connection is a type (1, 2)-tensor. The most significant difference between a covariant and a contravariant connection is that Dσ does not necessarily involve any derivatives. If the 1-form σ is such that #σ = 0, then Dσ is simply an operator of multiplication by some section of the bundle of matrices End V . When Dσ does involve derivatives, it only takes derivatives parallel to the symplectic foliation determined by π . The simplest examples of contravariant connections are derived from covariant connections. If ∇ is a covariant connection, then Dσ := ∇#σ

(2.7)

defines a contravariant connection. However, this clearly has the property that #σ = 0 implies Dσ = 0. As this property does not hold in general, not every contravariant connection is of this form. 3. Integration If an n-dimensional manifold, M, has a volume form ∈ n (M) (e. g., a Riemannian volume form) then integration of functions defines a linear map, τ0 : Cc∞ (M) →C f →

M

f .

This is trivially a trace, τ0 (f g) = τ0 (gf ), because Cc∞ (M) is commutative. When generalizing to noncommutative algebras, it is natural to require integration to be a trace, that is such that τ (ab) = τ (ba). In Connes’ formulation of noncommutative geometry, the generalized integration constructed from a spectral triple is automatically a trace. Given a smooth deformation of M, we can try to smoothly deform integration to a trace. For a concrete deformation, this means that we should have a trace map τκ : Aκ → C for each value of κ, such that these together give a smooth function of κ for every a ∈ A. More generally, and abstractly, we would like a k-linear trace τ : A → k. Use the notation for the contraction of a vector or multivector into a differential form. In particular (X ∧ Y ) = (X, Y, . . . ) and π is the differential (n − 2)-form (π )ij ··· = 21 π ab abij ··· . Theorem 1. Let A be a smooth deformation of M. If there exists a k-linear trace τ : A → k such that f = τ (a) mod κ M

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for f = a mod κ, then the Poisson field must satisfy, 0 = d(π ),

(3.1a)

0 = ∇j π ij

(3.1b)

or equivalently, it has 0 divergence

for any torsion-free connection such that ∇ = 0. Proof. The trace property can be equivalently stated as, 0 = τ [a, b]− for all a, b ∈ A. Commutativity of Cc∞ (M) already implies that 0 = τ [a, b]− mod κ. The condition for it to vanish to first order in κ is 0 = τ [a, b]− mod κ 2 = −iκ {f, g} M

(3.2)

for any a, b ∈ A and f = a mod κ, g = b mod κ. However, we can rewrite the integrand as {f, g} = df ∧ dg ∧ (π ). Because f and g are compactly supported, there are no boundary terms when we rewrite the condition as 0=

M

df ∧ dg ∧ (π ) =

M

f dg ∧ d(π ).

Since this must hold for any f, g ∈ Cc∞ (M), this implies Eq. (3.1a). In index notation, d(π ) is d(π )ij ··· = So, Eq. (3.1b) follows.

n−1 ab 2 π |[i j ··· ]ab

= π ab |b aij ··· .

Equation (3.2) means that integration gives a “Poisson trace”. In these terms, this result was already given by Weinstein in [29]. This condition was applied to quantization in [1, 17]. Interestingly, this same conclusion can also be reached from a completely different hypothesis. The orientation axiom for a real spectral triple [6] requires the existence of a Hochschild homology class which generalizes the volume form of a commutative manifold. If can be smoothly deformed into a Hochschild homology class for A, then (3.1) must be satisfied. This is indicative of the interplay among Connes’ axioms for noncommutative geometry.


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4. Bimodules Let V be a smooth vector bundle over M. The product of a smooth function with a smooth section of V is again a smooth section of V . So, as is well known, the space of sections, c (M, V ) is a module of Cc∞ (M). It is also a bimodule, if we define left and right multiplication to be the same. Because Cc∞ (M) is commutative, this satisfies the bimodule condition of associativity, f (vg) = (f v)g for any f, g ∈ Cc∞ (M) and v ∈ c (M, V ). In Sects. 5 and 6 we will be interested in deforming c (M, V ) into a bimodule of a deformed algebra. In the case of a concrete deformation, for each value of κ we want an Aκ -bimodule, and these should fit together nicely into an A-bimodule, V, such that V/κV ∼ = c (M, V ). Definition 3. Given a smooth deformation A of M, a bimodule deformation of a smooth vector bundle V over M is an A-bimodule V with a Cc∞ (M)-bimodule isomorphism c (M, V ) ∼ = V/κV such that V/κ 3 V is a free C[κ]/κ 3 -module. Since V is a bimodule, we can define the commutator of a ∈ A and vˆ ∈ V in the same way as in an algebra, [a, v] ˆ − := a vˆ − va. ˆ From this, we can construct the (generalized) Poisson bracket of functions and sections, { · , · } : Cc∞ (M) × c (M, V ) → c (M, V ). The following result is present in different terms in [24]; see also [3]. Theorem 2. A bimodule deformation of V determines a flat contravariant connection D on V such that, Ddf v = {f, v} for any f ∈

Cc∞ (M)

(4.1)

and v ∈ c (M, V ).

Proof. This bracket satisfies the same properties as the ordinary Poisson bracket (except for antisymmetry, which is just a matter of notation). It is a derivation on both arguments and satisfies the Jacobi identity. The proof is identical to that for the ordinary Poisson bracket. The Leibniz identity on the first argument is, {f g, v} = {f, v}g + f {g, v}. This implies that {f, v} depends locally and linearly on df . So, we can define D by Ddf v := {f, v}. The Leibniz identity on the second argument is, {f, gv} = {f, g}v + g{f, v}. In terms of D this is, Ddf (gv) = {f, g}v + g Ddf v = (#df )g v + g Ddf v. By Cc∞ (M)-linearity, this property is still true if we replace df with an arbitrary 1-form σ ∈ 1c (M), Dσ (f v) = #σ (g) v + g Dσ v. This is precisely the condition for D to be a contravariant connection.

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Using Eqs. (4.1), (2.2), and (2.5), the Jacobi identity can be written as 0 = {f, {g, v}} − {g, {f, v}} − {{f, g}, v} = Ddf Ddg v − Ddg Ddf v − Dd{f,g} v = Ddf Ddg v − Ddg Ddf v − D[df,dg]π v = K(df, dg)v, and therefore K = 0. That is, D is a flat contravariant connection.

Unless M is a symplectic manifold, this flatness condition is not nearly so restrictive as flatness of an ordinary connection. The equation 0 = Dv cannot be solved locally for a general flat contravariant connection. Nevertheless, this condition is still quite restrictive. Using a contravariant connection, one can construct characteristic classes in Poisson cohomology [18] from the curvature. These are simply the images of the conventional characteristic classes by the natural map, #∗ : H ∗ (M) → Hπ∗ (M). The existence of a flat contravariant connection thus implies that any (rational) characteristic class of V must be contained in the kernel of #∗ , except in degree 0. In particular, any characteristic class of the restriction of V to a symplectic leaf must be trivial. 5. One-Forms Most notions of a noncommutative space involve some generalization of differential forms. In Connes’ formulation, a complex of differential forms can be constructed from the algebra (of “functions”) and the Dirac operator. Other authors (e. g., [14]) simply posit such a complex as a fundamental structure. A common feature of these is that the exterior derivative should be a derivation (satisfying the Leibniz rule). In the case of a concrete deformation, suppose that for each value of κ there exists 1κ and a derivation d : Aκ → 1κ . For κ = 0 these should be 10 = 1c (M) and the gradient operator. The Leibniz rule is simply, d(ab) = da b + a db.

(5.1)

This is only meaningful if 1κ is an Aκ -bimodule. We should then require the bimodules and derivations to fit together smoothly. This suggests the following definition. Definition 4. Given a smooth deformation of M, a deformation of 1-forms is a bimodule deformation 1 of the cotangent bundle T ∗M along with a derivation d : A → 1 that reduces to the gradient operator d : Cc∞ (M) → 1c (M) modulo κ. Theorem 3. A deformation of 1-forms determines a contravariant connection on T ∗M which is not only flat, but also torsion-free. Proof. By Theorem 2, the bimodule deformation of T ∗M determines a flat contravariant connection D on T ∗M. The Leibniz rule (5.1) implies that d[a, b]− = [da, b]− + [a, db]− = [a, db]− − [b, da]− ,


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and the Poisson bracket inherits this as, d{f, g} = {f, dg} − {g, df }. In terms of the contravariant connection, 0 = Ddf dg − Ddg df − d{f, g} = Ddf dg − Ddg df − [df, dg]π = T (df, dg). Therefore T = 0.

This is philosophically unsurprising if we consider that torsion of a covariant connection is what happens if we forget that the tangent bundle is the tangent bundle. In this case, the gradient operator establishes that the cotangent bundle really is the cotangent bundle. In index notation, the property that a contravariant connection is flat and torsion-free can be stated simply as Di Dj = Dj Di . This is not the only condition of this kind. In fact, the existence of a consistent deformation of 2-forms places an additional condition on this same contravariant connection. However, the geometric interpretation of this condition is much more complicated and I will postpone discussion to a future paper. Corollary 4. If M is a symplectic manifold and there exists a deformation of 1-forms, then M admits a flat, torsion-free covariant connection. Proof. Symplectic means that the bundle homomorphism # : T ∗M → T M is an isomorphism. In this case, the contravariant connection D is equivalent to a covariant connection ∇ related by the formula ∇#σ (#ρ) = #(Dσ ρ).

(5.2)

Note that this differs from Eq. (2.7). As noted in [18] (Remark 2.3.2) when a covariant and a contravariant connection are intertwined in this way, their curvatures and torsions are intertwined by #. So, the fact that D is flat and torsion-free implies that ∇ is flat and torsion-free.

6. Metric Structure The most restrictive (and concretely defined) notion of noncommutative geometry is given by Connes’ axioms for a real spectral triple [6]. These describe a noncommutative generalization of a Riemannian Spin manifold. A real spectral triple involves the following structures: a ∗-algebra A, a Hilbert space H on which A is faithfully represented, an unbounded self-adjoint operator D on H, an antiunitary operator J on H, and (for even dimensions) a Z2 -grading on H. For a compact Spin manifold, the algebra is A = C ∞ (M), the Hilbert space is that of square-integrable sections of the spinor bundle, D is the Dirac operator, J is the charge-conjugation operator, and the grading is that into left and right handed spinors. The axioms for a real spectral triple give the following properties, among others: H is an A-bimodule with the left and right multiplications intertwined by J . The common domain, H∞ :=

∞ m=1

dom D m

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of all powers of D is a projective right (and left) A-module. For any a ∈ A, the commutator [D, a]− is a bounded operator on H (or H∞ ) and commutes with right-multiplication by any b ∈ A. In the case of a compact Spin manifold, H∞ is the space of smooth sections of the spinor bundle and [D, f ]− = iγ i dfi , in terms of the Dirac matrix-vector. For a spectral triple, there is a construction of noncommutative differential forms which is quite simple in degree 1. The A-bimodule 1D ⊂ L(H) is generated by all commutators [D, a]− for a ∈ A and the differential map d : A → 1D is given by da = −i[D, a]− . The properties that we will need here are more general than Spin manifolds and real spectral triples. We do not need the operator J , only the bimodule structure. We do not need the Hilbert space H, only the bimodule H∞ . Indeed, H could even be a Krein space, as in the approach to noncommutative space-time advocated by Strohmaier [26]. We do not need spinors, only some Spin bundle; that is, a bundle carrying a representation of the bundle of Clifford algebras. The Spin bundle ∗ T ∗M, which exists for any Riemannian or pseudo-Riemannian manifold, is sufficient. Definition 5. A deformed spectral triple (A, V, D) for a (pseudo)Riemannian manifold M consists of a smooth deformation A of M, a bimodule deformation V of some Spin bundle V , and a k-linear operator D : V → V such that (1) D mod κ is the Dirac-type operator on sections of V . (2) For any a ∈ A, [D, a]− commutes with the right multiplication of A on V. (3) The construction of noncommutative differential forms from A and D gives a deformation of 1-forms on M. Let 1 := 1D be the space of noncommutative 1-forms constructed with A and D. Modulo κ, this reduces to the classical space of 1-forms, 1c (M). The isomorphism c : 1c (M) −→ 1 /κ1 is the Clifford representation on V ; if V is the spinor bundle, this is given by the Dirac matrix-vector as c(σ ) = γ j σj . The only part of the definition of a deformation of 1-forms which is not automatically satisfied is the condition that 1 /κ 3 1 be a free C[κ]/κ 3 -module. This is effectively a condition on the behavior of D. This condition is automatically satisfied for an “isospectral” deformation in which V can be represented as a free k-module such that D is independent of κ. Theorem 5. A deformed spectral triple determines a flat, torsion-free contravariant connection on T ∗M which is compatible with the metric, 0 = Di g j k .

(6.1)

Proof. Since by hypothesis V and 1 are bimodule deformations of V and T ∗M, we have flat contravariant connections on V and T ∗M by Theorem 2. By Theorem 3, the contravariant connection on T ∗M is also torsion-free. For, a ∈ A, σˆ ∈ 1 , and vˆ ∈ V, associativity gives the elementary identity, ˆ −. [a, σˆ v] ˆ − = [a, σˆ ]− vˆ + σˆ [a, v]

(6.2)

This is automatically true to 0th order in κ. At first order, (6.2) gives an identity for generalized Poisson brackets; let f ∈ Cc∞ (M), σ ∈ 1c (M), and v ∈ c (M, V ), {f, c(σ )v} = c({f, σ })v + c(σ ){f, v}.


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In terms of the contravariant connections for T ∗M and V , Ddf [c(σ )v] = c(Ddf σ )v + c(σ )Ddf v, and so c is “parallel” with respect to the contravariant connections. The Clifford identity, g ij σi ρj =

1 2

[c(σ )c(ρ) + c(ρ)c(σ )]

then shows that the metric pairing is also parallel, which is equivalent to (6.1).

Other arguments are possible for this same metric-compatibility condition (6.1). For instance, given a deformation of 1-forms, we might try to define a deformed noncommutative metric as an A-bimodule homomorphism · , · : 1 ⊗A 1 → A which reduces modulo κ to the classical metric (in contravariant form). By bimodule linearity, ˆ γˆ ∈ 1 , for a ∈ A and β, ˆ γˆ ]− = [a, β] ˆ − , γˆ + β, ˆ [a, γˆ ]− . [a, β, Modulo κ, this gives (#df ) β, γ = Ddf β, γ + β, Ddf γ , which implies the metric-compatibility property (6.1). Alternatively, suppose that S2 ⊂ 1 ⊗A 1 is a bimodule deformation of the symmetric 2-tensor bundle S 2 T ∗M. For instance, this might be the kernel of a deformed exterior product map ∧ : 1 ⊗A 1 → 2 . We might define a deformed noncommutative metric to be a bimodule homomorphism S2 → A which reduces to the classical metric modulo κ. The deformation determines a contravariant connection on S 2 T ∗M which coincides with that induced by the connection for T ∗M. Again, this implies metric compatibility. Corollary 6. If a Riemannian manifold admits a deformed spectral triple such that the Poisson structure is symplectic, then it admits a flat Riemannian metric defined by gj k := ωj a ωkb g ab .

(6.3)

Proof. By Corollary 4, the contravariant connection D for T ∗M is equivalent to a flat, torsion-free covariant connection ∇. This is related by Eq. (5.2). In terms of index notation, we can translate between expressions in terms of D and ∇ by raising and lowering all indices with the Poisson field π and the symplectic form, respectively. Because D is compatible with the metric g, we have 0 = ωia ωj b ωkc Da g bc = ∇i (ωj b ωkc g bc ) = ∇i gj k . Therefore, ∇ is the Levi-Civita connection for g and its curvature, 0, is the Riemannian curvature of g .

Note that if g is Riemannian, then so is g . In general, g has the same signature as g. A priori, the condition presented by Thm. 5 is existential. One might despair of proving the nonexistence of a suitable connection. However, the situation is actually much better. There is a strong formal analogy between formulas in Riemannian geometry and in the present setting; the roles of covariant and contravariant indices are simply interchanged. In particular, there is an analogue of the Levi-Civita connection.

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Theorem 7. If a manifold M has both a metric g and a Poisson field π , then there exists a unique torsion-free, metric-compatible, contravariant connection D given by,

Dα β, γ = 21 (#α) β, γ − (#γ ) α, β + (#β) γ , α (6.4) + [γ , α]π , β − [β, γ ]π , α + [α, β]π , γ , for α, β, γ ∈ 1 (M). Here, α, β := g ij αi βj is the metric pairing and the vectors #α, et cetera act by directional derivatives. Definition 6. This D is the metric contravariant connection. Proof. The construction is precisely analogous to that of the Levi-Civita connection. Suppose that such a connection exists. The metric compatibility condition implies that (#α) β, γ = Dα β, γ + β, Dα γ , and the torsion-free condition is Dα γ = Dγ α − [γ , α]π . Putting these facts together gives

Dα β, γ = (#α) β, γ + [γ , α]π , β − Dγ α, β. By iterating this three times, we can solve for Dα β, γ and get Eq. (6.4). This construction proves uniqueness. To prove that this defines a connection, one can multiply the arguments by functions to show that all derivatives of α and γ cancel and verify that this satisfies the correct Leibniz rule in the argument β.

The formula (6.4) is the natural analogue of a formula for the Levi-Civita connection. In fact (6.4) has already appeared, although without derivation, in [2]. Corollary 8. If a (pseudo)Riemannian manifold M admits a deformed spectral triple, then the metric contravariant connection must be flat. Proof. This is immediate from Theorems 5 and 7.

This means that the necessary condition for compatibility of Riemannian and Poisson structures may be expressed as a differential equation, albeit a very cumbersome one when written out explicitly. We can also construct the metric contravariant connection in terms of the LeviCivita (covariant) connection, ∇. The contravariant connection ∇# defined by Eq. (2.7) is metric-compatible, but has torsion. Using Eqs. (2.2) and (2.6), the torsion is T ∇# (df, dh) = ∇#df dh − ∇#dh df − d{f, h}. Using index notation (with a vertical stroke for ∇) this is, ij

ij

T k f|i h|j = π ij f|i h|j k + π ij f|ik h|j − (π ij f|i h|j )|k = −π |k f|i h|j . ij

ij

Thus T k = −π |k or equivalently, T ∇# = −∇π.


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The metric contravariant connection can be constructed by correcting this; its action on a 1-form is ij

Di σk = π ij ∇j σk + A k σj , where ij

A k :=

1 2

ij

j |i

π |k − π k

i |j

−πk

(6.5)

,

and indices are raised and lowered with the metric. Using this, we can explicitly construct the curvature of D. It is jk

jk

ja

ij

K ij k l = π ia π j b R k lab − π ia A l |a + π j a Aikl |a − Aial A a + A l Aika + π |a Aakl . (6.6) Although it is not at all apparent from this expression, this tensor actually has precisely the symmetries of the Riemann tensor. Setting this to 0 gives a differential equation relating π and the metric. 7. Some Solutions If a commutative manifold is deformed into noncommutative geometry, it ought to have both a metric structure and integration. If a manifold can be deformed with respect to a given Poisson structure, then we should expect the conclusions of both Theorem 1 and Corollary 8 to hold. That is, it should satisfy both the conditions of 0 divergence and flatness of the metric contravariant connection. There may indeed be other necessary conditions, but just considering the conditions at hand is illuminating. There is one case in which a general solution to the second condition can be written down explicitly. Suppose that the manifold M is R2n and the Poisson field is symplectic. Because R2n is contractible, the symplectic form must be exact and can be written in terms of a potential 1-form as ω = dθ. According to Cor. 6, a flat metric g can be constructed from ω and the original metric g. We can spend our coordinate freedom and fix g to be some constant metric on R2n . The metric g and symplectic form are now both determined by θ as gij = ωia ωj b g , ab

(7.1)

and ω = dθ. Let and be the volume 2n-forms of the metrics g and g , respectively. Since all volume forms are proportional, the symplectic volume form can be written as ωn = h n!

(7.2)

in terms of some smooth function h ∈ C ∞ (R2n ). In coordinates the volume forms are given by the determinants of g, g , and ω. Equation (7.1) gives det g = (det ω)2 (det g )−1 and Eq. (7.2) gives det ω = h2 det g . Thus (det ω)2 = det g det g = h4 (det g )2

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E. Hawkins n

n−1

and = h2 = h ωn! . We can therefore write, π = h ωn! and the divergence condition is 0 = d(π ) = dh ∧

ωn−1 ; n!

that is, that h is constant. So, the general solution of the conditions for symplectic R2n is given by a constant metric g on R2n and a 1-form θ ∈ 1 (R2n ) such that det dθ is constant. There is a gauge freedom to add any gradient to θ , thus there are 2n − 2 local degrees of freedom to this solution. This is in contrast to an arbitrary Riemannian metric on R2n . That has n(2n + 1) components, but there are 2n degrees of diffeomorphism freedom per point. Therefore an arbitrary Riemannian metric has n(2n − 1) degrees of freedom per point. Of course, an arbitrary symplectic manifold resembles this locally, so the counting of degrees of freedom is not limited to R2n . In two dimensions this is particularly striking. There are no local degrees of freedom. In this way, only a flat 2-torus can be deformed into a noncommutative geometry. From this it is clear that only a very restricted class of Riemannian geometries is compatible with any symplectic structure. The requirement of compatibility with a deformation leaves geometry quite rigid. The condition of contravariant flatness given by Corollary 8 can be considered as a second order nonlinear partial differential equation in g and π . It is homogeneously quadratic in π . Notably, it only involves derivatives in directions in which π is nonvanishing; that is, along the symplectic foliation determined by π . This gives a sort of causal structure. We should expect solutions to propagate only along the symplectic leaves. To get a greater sense of the structure of solutions of these conditions, we can consider perturbations of a given solution. However, perturbations of π are in general badly behaved. If the rank of π (in some neighborhood) is less than the dimension of M, then an arbitrarily small perturbation can increase the rank. This would increase the dimension of the symplectic leaves and thus drastically change the causal structure of the contravariant flatness condition. It is thus much easier to consider perturbations of the metric relative to a fixed Poisson structure. The curvature of the metric contravariant connection is in some ways analogous to the curvature of the Levi-Civita connection, so consider the analogous problem: perturbing a flat metric to other flat metrics. If we are given a flat metric on some manifold, then locally any other flat metric is equivalent via a diffeomorphism. So, locally, any perturbation of the flat metric to another flat metric is given by an infinitesimal diffeomorphism; that is, the perturbation is the Lie derivative with respect to some vector field ξ , δgij = Lξ gij = ∇i ξj + ∇j ξi ,

(7.3)

where the index on ξi is of course lowered with the metric. The linearized contravariant flatness condition is formally very similar. Let hij = δg ij be a perturbation of the contravariant form of the metric, and suppose that the metric contravariant connection of g is flat. The condition that the perturbation preserves flatness is, 0 = Dj Dl hik + Di Dk hj l − Dk Dl hij − Di Dj hkl .

(7.4)


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Note that the order of the D’s is irrelevant because D is flat and torsion-free. Using the Koszul bracket in place of the Lie bracket, we can define a sort of Lie derivative with respect to a 1-form. This can be expressed in terms of D, and the analogue of (7.3) is, for some α ∈ 1 (M), hij = Di α j + Dj α i .

(7.5)

It is easy to check that this is a solution of Eq. (7.4). In fact, the computation is formally identical to checking that (7.3) is a solution of the linearized Riemann flatness condition. The formula (7.5) gives enough solutions locally in the symplectic case, but not generically. For example, if π = 0, then any metric satisfies the contravariant flatness condition; so, any perturbation will satisfy Eq. (7.4). Yet (7.5) is identically 0. Clearly, there are other solutions in general. If the metric is perturbed by hij , then the change in the volume form is δ = − 21 hk k . The linearization of Eq. (3.1a) is thus, 0 = d(hk k π ) = d(hk k ) ∧ (π ) or simply, 0 = π ij ∇j hk k . That is, hk k must be constant along the symplectic leaves. Inserting the expression (7.5) into this equation gives the condition, 0 = π ij ∇j (Dk αk ) = #d(Dk αk ).

(7.6)

That is, Dk αk should be constant along symplectic leaves. So, a class of perturbations preserving the conditions for deformation is given by (7.5) for α satisfying Eq. (7.6).

8. Examples and Counterexamples 8.1. The Noncommutative Torus. The oldest and best known example of a noncommutative geometry obtained by deforming a commutative manifold is the noncommutative 2-torus. The real spectral triple which describes the geometry was given in [6]. The classical Riemannian manifold is a flat 2-torus. The classical geometry and the deformed algebra are both invariant under the action of the 2-torus group T2 , therefore the Poisson field must be invariant. The Poisson structure is thus symplectic with symplectic form a multiple of the volume form. The Levi-Civita connection is just the trivial connection, so 0 = ∇k π ij , and in particular, ∇j π ij = 0 which is precisely what we should expect from Thm. 1. The metric contravariant connection must be T2 invariant, so the simplest possibility is the connection ∇# derived from the Levi-Civita connection. A simple computation shows that this is flat, torsion-free, and metric-compatible; thus the conclusion of Thm. 5 also holds.

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8.2. Isospectral Deformations. The noncommutative torus was generalized considerably by Connes and Landi [10]. The construction applies to a compact Riemannian Spin manifold M on which a torus group Tm (m ≥ 2) acts isometrically. We can assume without loss of generality that Tm acts faithfully. Given any constant (i. e., Tm -invariant) Poisson field on Tm , we can deform the algebra of smooth functions to a noncommutative algebra C ∞ (Tm κ ) which is a direct generalization of the noncommutative 2-torus. The algebra C ∞ (M) is then twisted by the noncommutative Tm . The deformed algebra is the Tm -invariant subalgebra of C ∞ (Tm C ∞ (M)⊗ κ ). The construction of the spectral triple is quite simple. The deformed algebra is represented on the same Hilbert space of square-integrable sections of the classical spinor bundle. The classical Dirac operator is used with the deformed algebras as well. Since the operators are identified, their spectra are the same and this gives the name “isospectral” to this construction. This family of spectral triples is very well behaved. They satisfy the axioms [6] for real spectral triples. In particular, a trace on each algebra can be constructed using the Dixmier trace and the Dirac operator. We should certainly expect that these examples will fit my definitions and satisfy the conditions I have formulated. Let {Xa } be a basis of infinitesimal generators of the Tm -action on M and let AB be the components of the (constant) Poisson field on Tm . The Poisson field on M is simply π = 21 AB XA ∧ XB . The vectors XA are Killing vectors, so in particular 0 = LXA = d(XA ). Because Tm is abelian, [XA , XB ] = 0, and thus 0 = [XA , XB ] = LXA (XB ) − XB (LXA ) = XA d(XB ) + d [XA (XB )] = −d [(XA ∧ XB ) ] . This implies that d(π ) = 0, so this indeed satisfies the conclusion of Thm. 1. The map # : T ∗M → T M is given by, #σ = AB (XA σ )XB . Define a contravariant connection by Dσ = AB (XA σ )LXB . Because the XA ’s are Killing vectors, we have immediately metric-compatibility, Dg = 0. It is sufficient to compute the torsion with exact 1-forms. So for f, h ∈ C ∞ (M) we find the derivative Ddf dh = AB XA (f ) d(XB h). This gives T (df, dh) = Ddf dh − Ddh df − d{f, h} = AB [XA f d(XB h) + d(XA f ) XB h − d(XB f XB h)] = 0, and hence T = 0. Therefore, by the uniqueness result of Thm. 7, D is the metric contravariant connection. If σ ∈ 1 (M) is Tm -invariant, then Dσ = 0, so clearly Kσ = 0. This shows that K = 0 at any point that is not fixed by any proper subgroup of Tm , but such points are dense and therefore K = 0 identically. So, with this contravariant connection, M satisfies the conclusion of Thm. 5.


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8.3. The Fuzzy Sphere. The “fuzzy sphere” [23] is another popular noncommutative space. It is a concrete, SU(2)-equivariant deformation of the sphere S 2 . Algebras Aκ exist for κ = 0 and κ = N −1 (N a positive integer). The algebra A1/N is simply the algebra of N by N matrices. The equivariance provides a way of simulating geometric constructions on the fuzzy sphere, but a spectral triple that describes its geometry has not been found. Because the deformation is equivariant, the Poisson field π must be SU(2)-invariant. The only possibility is symplectic, with the symplectic form some multiple of the volume form. The contraction π is a constant function on S 2 , so d(π ) = 0 and the conclusion of Thm. 1 is satisfied. Indeed, integration on S 2 can be smoothly deformed to traces. On the algebra A1/N , this is simply τ1/N = 4π N tr. By Cor. 6, there can be no deformed spectral triple for the fuzzy sphere, because S 2 does not admit a flat metric. By Cor. 4, it is not even possible to deform 1-forms to the fuzzy sphere, because T ∗ S 2 does not admit any flat connection. The fuzzy sphere generalizes nicely to other coadjoint orbits. A coadjoint orbit of a compact, semisimple Lie group G is deformed to a sequence of G-equivariant matrix algebras. The Poisson structures are symplectic and G-invariant. None of these admit deformed spectral triples, because none of these coadjoint orbits admit a flat metric. 8.4. Two Dimensions. Suppose that M is a 2-dimensional, connected Poisson manifold with a volume form ∈ 2 (M), and let ij be the inverse of the volume form. Because the bundle of bivectors is rank 1, we must have π ij = h ij for some smooth function h ∈ C ∞ (M). This ansatz gives that π = h. So, the condition from Thm. 1 that 0 = d(π ) = dh implies that h is constant. In other words, either π = 0 or M is symplectic with symplectic form a constant multiple of . If M is also Riemannian and admits a deformed spectral triple, Corollary 8 then shows that if π = 0 then M admits a flat metric. That is, it can only be a torus T2 , a cylinder S 1 × R, or the plane R2 . Suppose instead that M admits a deformed spectral triple, but disregard the hypothesis of Thm. 1 (that integration is deformed to a trace). The open submanifold := {x ∈ M | π(x) = 0} is symplectic with symplectic form ω = h−1 . If M admits a deformed spectral triple, then by Cor. 6 there exists a flat metric on given by, gij = ωia ωj b g ab = h−2 gij . In other words, gij must be flattened by the conformal factor h−1 . Of course, any 2-dimensional manifold is locally conformally flat, and so such an h always exists locally. Consider the unit sphere S 2 . This is not globally conformally flat, but if we remove a point, then = S 2 ∗ is. We can write the metric in an explicitly conformally flat form using a complex coordinate ζ , ds 2 = 4(ζ ζ¯ + 1)−2 dζ d ζ¯ .

(8.1)

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This shows that the condition will be satisfied by h = 2c(ζ ζ¯ + 1)−1 for c constant. In terms of the standard embedding coordinates, h = c(1 + z), so this is a smooth function on S 2 and vanishes at the deleted point. The volume form for the metric (8.1) is = 2i(ζ ζ¯ + 1)−2 dζ ∧ d ζ¯ , and so the symplectic form will be ω = h−1 = −ic−1 (ζ ζ¯ + 1)−1 dζ ∧ d ζ¯ . The Poisson bracket is thus given by, {ζ, ζ¯ } = −ci(ζ ζ¯ + 1).

(8.2)

The sphere minus two points is also globally conformally flat. However, the conformal factor is such that h would not be differentiable on S 2 . This would mean that the Poisson bracket of two smooth functions would not always be differentiable. This does not correspond to a smooth deformation. The Podle´s standard sphere is a (noncommutative) homogeneous space of the quantum group SUq (2). For q = 1 it is the commutative sphere, S 2 . Da˛browski and Sitarz have constructed an unusual spectral triple for the Podle´s sphere which is SUq (2)-equivariant and reduces to the spectral triple for a sphere of radius 1 when q = 1; it satisfies some – but not all – of Connes’ axioms for a real spectral triple. The Podle´s standard sphere algebra A(Sq2 ) (for q ∈ R) is the ∗-algebra generated by A and B such that, A = A∗ ,

AB = q 2 BA,

BB ∗ = q −2 A(1 − A),

B ∗ B = A(1 − q 2 A).

This is a smooth deformation of S 2 with q = 1 + κ. When q = 1, A(S12 ) is the algebra of polynomial functions on the unit sphere and we can write the generators in terms of the standard embedding coordinates as A = 21 (1 + z), B = 21 (x + iy). The commutator in A(Sq2 ) gives the Poisson bracket in the commutative algebra A(S12 ), [B, B ∗ ]

−

⇒ {A, B} = 2iAB, [A, B]− = (q 2 − 1)BA = (q 2 − q −2 )A2 − (1 − q −2 )A ⇒ {B, B ∗ } = 2iA(2A − 1).

The complex coordinate ζ is related to these generators by ζ = A−1 B, and so we can compute the Poisson bracket {ζ, ζ¯ } = 2i(ζ ζ¯ + 1). This agrees with (8.2) if c = −2. So, we see that the Podle´s standard sphere does satisfy the condition for the existence of a deformed spectral triple. It seems likely that the Da˛browski-Sitarz example is indeed a deformed spectral triple, but this has not yet been checked explicitly. 8.5. The Dual of a Lie Algebra. Let g∗ be the linear dual of a semisimple Lie algebra with a constant, g-invariant Riemannian metric (e. g., the Cartan-Killing form). The natij ij ural Poisson structure is given explicitly by π ij = Ck x k , where Ck are the structure


231

coefficients of g and x k are the coordinates. The Levi-Civita connection is simply the trivial connection given by partial derivatives on the vector space g∗ . ij The covariant derivative of the Poisson field is simply ∇k π ij = Ck . So we have, ij

∇j π ij = Cj = 0. That is, g∗ satisfies the conclusion of Thm. 1. The possibility of smoothly deforming integration into a trace is not ruled out. This is discussed more extensively in [1]. In order to compute the curvature of the metric contravariant connection, it is sufficient to work with constant 1-forms. The space of constant 1-forms is naturally identified with g. Let α, β, γ ∈ g. Let x·α = x i αi be the linear function on g∗ with derivative α. The definition of the Poisson structure on g∗ can be stated succinctly as {x·α, x·β} = x·[α, β]. With this, we see that the Koszul bracket of constant 1-forms is simply the g-bracket, [α, β]π = [d(x · α), d(x · β)]π = d{x · α, x · β} = [α, β]. We can now construct the metric contravariant connection. Because the metric inner products of α, β and γ are constant over g∗ , Eq. (6.4) simplifies to

Dα β, γ = 21 [γ , α], β − [β, γ ], α + [α, β], γ = 21 [α, β], γ , using that [α, β], γ = [β, γ ], α = [γ , α], β. So, the connection is given by Dα β = 21 [α, β]. The definition (2.5) of curvature now gives, K(α, β)γ = Dα Dβ γ − Dβ Dα γ − D[α,β]π γ = 41 [α, [β, γ ]] − 41 [β, [α, γ ]] − 21 [[α, β].γ ] = − 41 [[α, β], γ ]. We see that K = 0, and so g∗ does not admit a deformed spectral triple. 8.6. Quantum Groups. Suppose that G is a semisimple Lie group with Lie algebra g and a left and right invariant Riemannian metric. Consider a deformation of G into quantum groups. Because of semisimplicity, the Poisson field for such a deformation is of the form, π = rR − rL ,

(8.3)

where rR and rL are respectively the right and left invariant bivector fields which are both equal to some r ∈ 2 g at the identity e ∈ G. The Levi-Civita connection is left and right invariant. The divergence ∇j π ij is a vector field on G which decomposes as a sum of left and right invariant vector fields. If we identify g with left invariant vector fields on G, then the g-bracket is identified with the Lie bracket. For X, Y ∈ g, the Levi-Civita connection is given by ∇X Y = 21 [X, Y ]. Using index notation and the structure coefficients C ij k , the covariant derivative of rL is ij

j

∇k rL = 21 r aj C ika + 21 r ia C ka ,

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and the divergence is ij

j

∇j rL = 21 r kj C ij k + 21 r ik C j k = − 21 r j k C ij k .

(8.4)

This amounts to applying the Lie bracket to 2 g; for example, if r = X ∧ Y , then (8.4) would be −[X, Y ]. This is the left-invariant part of ∇j π ij . The right invariant part is essentially the same. The condition that ∇j π ij = 0 is only satisfied if (8.4) vanishes. So, integration over G can only be smoothly deformed to a trace on quantum groups if (8.4) vanishes. For the Drinfel’d-Jimbo r-matrix (see, e. g., [19]), (8.4) is a nonzero element of the Cartan subalgebra and thus the condition is not satisfied. The Koszul bracket of left-invariant 1-forms is left-invariant (see [29, 22, 28]). If we identify left-invariant 1-forms with the dual, g∗ , of the Lie algebra, then this bracket is given by, [α, β]π = ad∗rα β − ad∗rβ α, where ad∗ is the coadjoint representation of g on g∗ and (rα)j := r ij αi . If α, β, γ ∈ 1 (G) are left-invariant, the metric inner products α, β et cetera are constant functions on G. This simplifies Eq. (6.4) for the metric contravariant connection to

Dα β, γ = 21 ad∗rγ α, β + ad∗rγ β, α + ad∗rα β, γ − ad∗rα γ , β − ad∗rβ γ , α − ad∗rβ α, γ . Invariance of the metric shows that ad∗rα β, γ + ad∗rα γ , β = 0, et cetera. So, the metric contravariant connection on left-invariant 1-forms is simply, Dα β = ad∗rα β.

(8.5)

Note that the metric has factored out of this formula. The connection is determined by the invariance of the metric. We can now compute the curvature, K(α, β)γ = ad∗[rα,rβ]−r[α,β]π γ = ad∗[r,r](α,β) γ , where [r, r] ∈ 3 g is treated as a map [r, r] : g∗ ⊗ g∗ → g. This only vanishes if [r, r] is constructed from the center of g, but G is semisimple, so the curvature only vanishes if [r, r] = 0.

(8.6)

This means that if a quantum group deformation of G admits a deformed spectral triple, then the corresponding classical r-matrix must satisfy (8.6), which is the well-known “classical” Yang-Baxter equation. 9. Conclusions What are the implications of these results? Of course this depends on what, if anything, one wants to do with noncommutative geometry.


233

Suppose that space-time noncommutativity exists and is relevant to the classical physics of general relativity. We know by observation that noncommutativity can be ignored to a very good approximation, therefore we ought to be able to treat noncommutativity as a perturbation of commutative space-time. One should therefore consider general relativity with first order noncommutative corrections described by a Poisson field. However, the result is not general relativity to leading order! As I explained in Sect. 7, consistency with a noncommutative deformation restricts the geometry of space-time. This makes space-time rigid, probably too rigid for the gravitational field to propagate in the way it is observed to. In 4 dimensions, compatibility with a symplectic deformation leaves the metric with at most 2 degrees of freedom per point of space-time, rather than the usual 6 degrees of freedom. This sort of noncommutativity in the space-time of classical physics thus appears to be ruled out. This leaves the possibility of noncommutative space-time at some quantum level. In such a domain, there is less reason to assume that space-time is approximately commutative at all, but one can still consider the possibility. If a noncommutative deformation describes the geometry of space-time in some model “before quantization” or the geometry of space-time in some exotic limit of physics, then the Poisson field can be taken as a physical field and my obstructions can be taken as physical conditions (perhaps equations of motion) on this field and the metric. One might also imagine that the rigidity engendered by noncommutativity is the mechanism responsible for freezing out physically unobserved extra dimensions, such as are found in Kaluza-Klein, supergravity, and string theories. A similar idea has been suggested by Dubois-Violette, Kerner, Madore, Doplicher, Fredenhagen, and Roberts [13, 15]; with the motivation of preserving Poincaré symmetry, they propose a noncommutative space-time which is a deformation of a higher-dimensional manifold. Another possibility is that my hypotheses in Theorems 1, 2, and 5 are too restrictive. There are several ways in which this might be so. Perhaps the assumptions of smoothness in the deformation of geometry are too restrictive. This would be surprising given how well smoothness works in the deformation of the algebra alone. Perhaps deformations should be considered in which noncommutativity is only “nonperturbative”. That is, it does not appear at finite order in κ. This is not an altogether unreasonable possibility. Consider a lattice approximation. That is, let M be a compact Riemannian manifold and Xj ⊂ M a sequence of finite subsets which grow uniformly dense at every point as j → ∞. Construct the subset X := ({0} × M) ∪

∞

{j −1 } × Xj

⊂ [0, 1] × M.

j =1

Now let A ⊂ C(X) be the algebra of functions on X which are restrictions of smooth functions on [0, 1] × M and let κ ∈ A be the coordinate on the interval [0, 1]. This is a smooth commutative deformation of M, but the algebra is actually unchanged to all orders in κ. The algebra A/κ m A is simply the tensor product of C ∞ (M) with the algebra C[κ]/κ m . As this can happen with a commutative deformation, it can certainly happen with a noncommutative deformation. Of course, in this case there may not be anything as convenient as a Poisson field to describe the deformation. Perhaps the structures I chose to deform are too restrictive. If one accepts the idea of a smooth deformation characterized by a Poisson field, then my results here can be taken as a guide to the structures one should not in general expect to be able to deform. If π has

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E. Hawkins

nonzero divergence, then don’t expect to deform integration by traces. If there does not exist a flat, torsion-free contravariant connection, then don’t expect to deform 1-forms with a gradient operator that is a derivation. If the metric contravariant connection is not flat, then don’t expect to construct a well-behaved family of spectral triples. Another possibility is the sort of noncommutativity that occurs in the almost commutative spaces of Connes-Lott models. In this case, the noncommutative space is the product of a commutative manifold and a small noncommutative space. In such a scenario, the noncommutativity is not given by a deformation. Such a space appears to be commutative not because of a scaling limit, but because the noncommutativity isn’t interpreted as geometry, but as gauge and Higgs fields. Acknowledgements. I wish to thank Ludwik Da˛browski, Lee Smolin, Ted Jacobson, and Alan Weinstein ´ for their comments on earlier versions of this paper. This work was initiated while visiting the IHES.

References 1. Bieliavsky, P., Bordemann, M., Gutt, S., Waldmann, S.: Traces for star products on the dual of a Lie algebra. math.QA/0202126 2. Boucetta, M.: Compatabilité des structures pseudo-riemanniennes et des structures de Poisson. C. R. Acad. Sci. Paris 333, Série I, 763–768 (2001) 3. Bursztyn, H.: Poisson Vector Bundles, Contravariant Connections and Deformations. Prog. Theor. Phys. Supp. 144 (2001) 4. Chakraborty, P.S., Pal, A.: Equivariant Spectral triple on the Quantum SU(2)-group. K-Theory 28, 107–126 (2003) math.KT/0201004 5. Connes, A.: Noncommutative Geometry. London-New York: Academic Press, 1994 6. Connes, A.: Gravity Coupled with Matter and the Foundation of Non-commutative Geometry. Commun. Math. Phys. 182, 155–176 (1996) hep-th/9603053 7. Connes, A., Douglas, M., Schwarz, A.: Noncommutative Geometry and Matrix Theory: Compactification on Tori. JHEP 9802, 003 (1998) hep-th/9711162 8. Connes, A., Dubois-Violette, M.: Noncommutative Finite-Dimensional Manifolds. I. Spherical manifolds and related examples. math.QA/0107070 9. Connes, A., Dubois-Violette, M.: Moduli Space and Structure of Noncommutative 3-Spheres. math.QA/0308275 10. Connes, A., Landi, G.: Noncommutative Manifolds, the Instanton Algebra and Isospectral Deformations. Commun. Math. Phys. 221, 141–159 (2001) math.QA/0011194 11. Cornalba, L., Schiappa, R.: Nonassociative Star Product Deformations for D-brane Worldvolumes in Curved Backgrounds. Commun. Math. Phys. 225, 33–66 (2002) hep-th/0101219 12. Da˛browski, L., Sitarz, A.: Dirac Operator on the Standard Podle´s Quantum Sphere. To appear in Banach Center Publications. math.QA/0209048 13. Doplicher, S., Fredenhagen, K., Roberts, J.E.: The quantum structure of spacetime at the Planck scale and quantum fields. Commun. Math. Phys. 172, 187–220 (1995) 14. Dubois-Violette, M., Kerner, R., Madore, J.: Noncommutative Differential Geometry of Matrix Algebras. J. Math. Phys. 31(2), 316–322 (1990) 15. Dubois-Violette, M., Kerner, R., Madore, J.: Shadow of noncommutativity. J. Math. Phys. 39, 730– 738 (1998) q-alg/9702030 16. Fedosov, B.: Deformation Quantization and Index Theory. Mathematical Topics. Vol. 9, Berlin: Akademie Verlag, 1995 17. Felder, G., Shoikhet, B.: Deformation Quantization with Traces. Lett. Math. Phys. 53, 73–86 (2000) math.QA/0002057 18. Fernandes, R.L.: Connections in Poisson Geometry. I. Holonomy and Invariants. J. Diff. Geom. 54(2), 303–365 (2000) math.DG/0001129 19. Fuchs, J.: Affine Lie Algebras and Quantum Groups. Cambridge: Cambridge University Press, 1995 20. Konechny, A., Schwarz, A.: Introduction to M(atrix) Theory and Noncommutative Geometry. Phys. Rept. 360, 353–465 (2002) hep-th/0012145 21. Kontsevich, M.: Deformation Quantization of Poisson Manifolds, I. q-alg/9709040 22. Lu, J.-H., Weinstein, A.: Poisson-Lie Groups, Dressing Transformations and Bruhat Decompositions J. Differ. Geom. 31, 501–526 (1990)


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23. Madore, J.: The Fuzzy Sphere. Classical Quantum Gravity 9(1), 69–87 (1992) 24. Reshetikhin, N., Voronov, A., Weinstein, A.: Semiquantum Geometry. Algebraic geometry, 5. J. Math. Sci. 82(1), 3255–3267 (1996) q-alg/9606007 25. Seiberg, N., Witten, E.: String Theory and Noncommutative Geometry. JHEP 9909 032 (1999) hep-th/9908142 26. Strohmaier, A.: On Noncommutative and Semi-Riemannian Geometry. math-ph/0110001 27. Vaisman, I.: On the Geometric Quantization of Poisson Manifolds. J. Math. Phys. 32, 3339–3345 (1991) 28. Vaisman, I.: Lectures on the Geometry of Poisson Manifolds. Basel-Boston: Birkhauser, 1994 29. Weinstein, A.: Some Remarks on Dressing Transformations. J. Fac. Sci. Univ. Tokyo. Sect. 1A Math. 36, 163–167 (1988) 30. Weinstein, A.: The modular automorphism group of a Poisson manifold. J. Geom. Phys. 23, 379–394 (1997) Communicated by A. Connes


Communications in


On a Kinetic Equation for Coalescing Particles Miguel Escobedo1 , Philippe Lauren¸cot2 , Stéphane Mischler3,4 1

Departamento de Matemáticas, Universidad del Pa´ıs Vasco, Apartado 644, 48080 Bilbao, Spain. E-mail: [email protected] 2 Mathématiques pour l’Industrie et la Physique, CNRS UMR 5640, Université Paul Sabatier– Toulouse 3, 118 route de Narbonne, 31062 Toulouse cedex 4, France. E-mail: [email protected] 3 Laboratoire de Mathématiques Appliquées, Université de Versailles – Saint Quentin, 45 avenue des Etats-Unis, 78035 Versailles, France 4 Projet BANG, INRIA Rocquencourt, B.P.105, 78153 Le Chesnay Cedex, France. E-mail: [email protected] Received: 21 January 2003 / Accepted: 6 October 2003 Published online: 2 March 2004 – © Springer-Verlag 2004

Abstract: Existence of global weak solutions to a spatially inhomogeneous kinetic model for coalescing particles is proved, each particle being identified by its mass, momentum and position. The large time convergence to zero is also shown. The cornestone of our analysis is that, for any nonnegative and convex function, the associated Orlicz norm is a Liapunov functional. Existence and asymptotic behaviour then rely on weak and strong compactness methods in L1 in the spirit of the DiPerna-Lions theory for the Boltzmann equation.

1. Introduction We consider the Cauchy problem for a kinetic equation modelling (at a mesoscopic level) the dynamic of a system of particles undergoing a coalescence (or sticky) process. More precisely, describing the gas of particles by the density f (t, x, m, p) ≥ 0 of particles with mass m ∈ R+ := (0, +∞) and momentum p ∈ R3 at time t ≥ 0 and position x ∈ ⊂ R3 , we study the existence and long time behaviour of solutions to the equation ∂t f + v · ∇x f = Q(f ) in (0, +∞) × × Y, f (0) = f in in × Y.

(1.1) (1.2)

Here and below, in order to shorten the notations, we introduce the mass-momentum variable y := (m, p) ∈ Y := R+ × R3 and the velocity variable v = p/m. The collision operator Q(f ) is given by Q(f ) = Q1 (f ) − Q2 (f ), where m 1 Q1 (f )(y) = a(y , y − y ) f (y ) f (y − y ) dm dp , (1.3) 2 R3 0 ∞ a(y, y ) f (y) f (y ) dm dp . (1.4) Q2 (f )(y) = R3 0

238

M. Escobedo, P. Lauren¸cot, S. Mischler

The meaning of these terms is the following. Denoting by {y} = (m, p) a particle of mass-momentum y, Q1 (f )(y) accounts for the formation of particles {y} by coalescence of smaller ones, i.e., by the reaction {y } + {y − y }

a(y ,y−y )

−→

{y} ,

y = (m , p ) ∈ (0, m) × R3 ,

and Q2 (f )(y) describes the depletion of particles {y} by coagulation with other particles, i.e., by the reaction a(y,y )

{y} + {y } −→ {y + y } ,

y ∈ Y.

At a microscopic level, the collision of two particles {y} and {y } leads to the formation (by coalescence) of a single particle {y } with rate a(y, y ), the mass and momentum being conserved during the collision. In other words, a(y,y )

{y} + {y } −→ {y }

with y = (m , p ) = (m + m , p + p ).

In contrast to the Boltzmann equation for elastic collisions, this is an irreversible microscopic process. Observe, in particular, that it does not preserve the kinetic energy: ke (y, y ) :=

|p |2 |p|2 |p |2 m m + |v − v |2 ≥ 0. − = 2m 2 m 2 m 2(m + m )

Particles evolving according to these rules are met, for instance, in dense sprays involved in combustion reactions, where liquid droplets are carried by a gaseous phase and undergo coalescence processes (due to collisions). The dynamics of the density of liquid droplets are then described by Eq. (1.1) [3, 36–38], see also [10, 12, 27]. In this context, the coalescence kernel a is given by a(y, y ) = aH S (y, y ) := (r + r )2 |v − v |,

(1.5)

where r = m1/3 , r = (m )1/3 denote the radii of the particles and v = p/m, v = p /m , their velocities. Note that this kernel corresponds to the well-known cross section for hard spheres in the Boltzmann theory. In fact, Eq. (1.1) only takes into account coalescence and neglects the fragmentation of the droplets due to the action of the gas, as well as the condensation/evaporation of the droplets and the elastic (grazing) collisions of Boltzmann type. However, all these effects may be met together, for instance, in combustion theory [39]. In this case, one usually considers the following equation [21]: ∂t g + ∇x (v g) + ∇v (β g) + ∂r (ω g) = Qcoll (g) + Qbr (g), for the density g(t, x, r, v) of droplets which, at time t ≥ 0, are at the position x ∈ with a radius r > 0 (droplets are supposed to be spheres), and velocity v ∈ R3 . In this equation, β and ω denote the droplet acceleration and evaporation rate, respectively, the term Qbr (g) represents the effects of the break-up of the droplets due to the action of the gas, and the term Qcoll (g) represents the effects of binary collisions between particles. It includes both elastic collisions and collisions giving rise to the coalescence of the colliding particles. Similar equations may also be found in aerosol theory [24]. Another physical situation in which coalescence occurs may be found in stellar dynamics in the modelling of clouds of particles (galaxies!) interacting by an attractive

Kinetic Equation for Coalescing Particles

239

Manev pair potential −α/r − ε/r 2 (see [6, 25] and the references therein). In particular, for α = 0 and ε = 1, the associated cross-section is a(y, y ) = aNP (y, y ) :=

m + m 1 . mm |v − v |2

(1.6)

In both models described here, the coalescence rate a corresponds to the collision frequency. But, collisions may not always result in a coagulation event. This fact can be accounted for by the introduction of a coalescence efficiency E (representing the probability that the two colliding particles do really stick). Then a(y, y ) = E(y, y ) a0 (y, y ), where a0 is the frequency of collisions and may be given, for instance, by (1.5) or (1.6), see [37]. Let us now describe the coalescence rate a that we will consider in this paper. We assume that a fulfills the symmetry and positivity conditions 0 < a(y, y ) = a(y , y)

a.e. on Y 2 ,

(1.7)

and the structure condition a(y, y ) ≤ a(y, y + y ) + a(y , y + y ) ,

y, y ∈ Y.

(1.8)

We also require the growth conditions

a(y, y ) 1a(y,y )≥M dy −→ 0,

aδ,R (M) := sup

y ∈Yδ,R

ωδ,R (R ) :=

Yδ,R

sup p ∈R3 ,m ≥R Yδ,R

M→∞

a(y, y ) 0, dy −→ |y | R →∞

(1.9) (1.10)

for any R > δ, with Yδ,R := (δ, R) × BR . In particular, the coalescence kernel aH S and aNP fulfill the above assumptions, as it is shown in Appendix B. As for , we may consider the case of the whole space = R3 , the case of the 3-dimensional torus = T or the case of a bounded domain of R3 . In the latter cases, one has to supplement (1.1) with either periodic boundary conditions or, for instance, no-incoming flux conditions γf = 0

on

{x ∈ ∂ , n(x) · v < 0} ,

(1.11)

where γf stands for the trace of f on the boundary and n(x) denotes the outward normal unit vector field. In order to simplify the presentation, we only consider the case = R3 in the sequel. However, we keep the notation to differentiate between the space of positions and the space of momenta (or velocities). We finally require that the initial datum has finite total number of particles, finite total mass and finite mean momentum, that is, 0 ≤ f in ∈ L1 ( × Y, (1 + m + |p|) dydx) .

(1.12)

240


Our main result is the following. Theorem 1.1. Assume that the coalescence kernel a fulfills the assumptions (1.7)–(1.10) and that the initial datum satisfies (1.12). There exists at least a solution f ≥ 0 to the coalescence Eq. (1.1)-(1.2) which satisfies f ∈ C([0, +∞); L1 ( × Y )),

Q(f ) ∈ L1 (T × Y )

for all T > 0, where T := (0, T ) × . Moreover, ∂t ρ + divx j ≤ 0

in

D (T ),

with ρ =

m f dy, j =

Y

p f dy, Y

(1.13) and, in particular, t −→ m f (t, x, y) dydx is a non-increasing function of time.

(1.14)

Y

Finally, the solution satisfies f (t) → 0 in L1 ( × Y ) as t → +∞.

(1.15)

Furthermore, if f in ∈ Lp ( × Y ) for some p ∈ [1, ∞], then t −→ f (t) Lp (×Y ) is a non-increasing function of time.

(1.16)

Finally, if supp f in ⊂ VR = {(x, y) ∈ × Y, |v| ≤ R} for some R > 0, then supp f (t) ⊂ VR for any t ≥ 0. If f in m |v|2 ∈ L1 ( × Y ) (that is, f in has finite kinetic energy), then t −→ f (t) m |v|2 L1 (×Y ) is a non-increasing function of time.

(1.17)

It is worth mentioning that (1.15) somehow means that the mean mass of the particles grows without bound as time goes to infinity. But, the total mass of the system f (t, y) m dydx Y

is expected to remain constant through time evolution, at least for coalescence rates with a moderate growth with respect to m. Indeed, we recall that, for the classical Smoluchowski coagulation equation which corresponds to (1.1) when f = f (t, m), a loss of mass takes place in finite time for coalescence rates such that a(m, m ) ≥ (m m )α , α > 1/2. This phenomenon is usually referred to as the occurrence of gelation, see [1, 22] and the references therein. In our general setting, the total mass conservation is still an open problem. In the spatially homogeneous case f = f (t, m, p), the total conservation is proved in [23] for the coalescence rate aH S . To our knowledge, the existence of solutions to the coalescence equation (1.1) has not been studied yet when the distribution function f depends on the four variables (t, x, m, p). In the spatially homogeneous case f = f (t, m, p), existence and uniqueness of solutions are established in [37], while a numerical scheme is developed in [38]. Let us point out that, in [37, 38], the formulation of (1.1) is different and involves the variables (r, v) (with r = m1/3 , v = p/m) instead of (m, p). Nevertheless, the two formulations are equivalent as it is shown in Appendix A. When the distribution function


241

f = f (t, m) does not depend on (x, p), Eq. (1.1) reduces to the classical Smoluchowski coagulation equation which has been extensively studied since the pioneering work of Melzak [32]. We refer to the survey by Aldous [1] and the book by Dubovski [20] for a more detailed description. When the distribution function f = f (t, x, m) does not depend on the momentum, a related equation, the diffusive coagulation equation, has received much attention recently. In this equation, the evolution of f with respect to the position x ∈ is modelled by a diffusion term −d(m)x f instead of the advection term v · ∇x f . Since the works by Bénilan & Wrzosek [5] and Collet & Poupaud [11] concerning the discrete diffusive coagulation equation, existence of solutions to the continuous diffusive coagulation equation has been investigated in [2, 13, 29, 33]. Let us point out here that the key estimates used in Theorem 1.1 are in the spirit of [28, 29, 33]. We finally mention that, very recently, a kinetic equation (with velocity variable) for particles undergoing linear fragmentation has been studied in [21, 26]. Remark 1.2. On the one hand, it is likely that our existence result stated in Theorem 1.1 extends to the case where the droplets of acceleration and evaporation are taken into account (that is, ∇v (βf ) + ∂m (ωf ) is added to the left-hand side of (1.1)), under suitable assumptions on the acceleration and evaporation rates. Following [29], the linear fragmentation kernel should also be added to Eq. (1.1) under, for instance, an assumption of weakness of the fragmentation mechanism with respect to the coalescence mechanism. On the other hand, the present analysis does not carry over to the case where elastic collisions are also included and a collision term of Boltzmann type is added to the righthand side of (1.1). In that case, the Lp -norms are no longer Liapunov functionals and we do not know how to remedy this fact. Let us now give some comments on Theorem 1.1. It turns out that Theorem 1.1 is a consequence of a stability result (see Sect. 3) which asserts that a sequence (fn ) of solutions to (1.1) satisfying natural bounds converges weakly in L1 , up to a subsequence, to a solution to (1.1). Roughly speaking, the main mathematical difficulty is to prevent the formation of a Dirac mass at some point of the phase space × Y . The structure assumption (1.8) allows us to prove that, for any nonnegative and convex function, the associated Orlicz norm is a Liapunov functional, which prevents concentration by the Dunford-Pettis theorem. In fact, we can prove the weak compactness in L1 of both (fn ) and (Q(fn )). Strong compactness in L1 of y-averages of fn then follows by the velocity averaging lemma of solutions to the transport equation. The remainder of the proof is then performed in the spirit of the DiPerna-Lions theory for the Boltzmann equation [15]. Let us finally remark that, for the kernel aH S , stationary solutions to (1.1) are S(m, p) = µ(m) δv=u

(1.18)

where µ ∈ M 1 (0, +∞) is a bounded measure and u ∈ R3 . Theorem 1.1 implies that the zero solution is the only stationary state which is reached in the long time when starting from an L1 initial data. We thus identify more accurately the asymptotic state than in [37]. Let us also mention that, if St,x is a solution to (1.1) which is a stationary solution for each (t, x), that is, St,x is given by (1.18) with µ = µt,x and u = u(t, x), it satisfies ∂t St,x + v · ∇x St,x = 0. Introducing (t, x) :=< m, µt,x >, we realize that ( , u) satisfies the pressureless gases system ∂t + ∂x ( u) = 0, see [8, 9, 40] and the references therein.

∂t ( u) + ∂x ( u2 ) = 0,

(1.19)

242


We now outline the contents of this paper. In the next section, we collect some qualitative and formal information on the solutions. They lead to the natural bounds that one can expect on the solutions. In Sect. 3 we specify the notion of solution we deal with, and state the key stability theorem (Theorem 3.2) which is proved in Sect. 4. Section 5 is devoted to the proof of the convergence of the solutions to zero for large times. We then briefly explain in Sect. 6 how the stability result adapts to prove Theorem 1.1. We finally establish the equivalence between the two formulations radius-velocity (r, v) and mass-momentum (m, p) of (1.1) in Appendix A and then check that the coalescence kernels aH S and aNP given by (1.5) and (1.6) do satisfy (1.8)–(1.10) in Appendix B. 2. Conservation Laws and Liapunov Functionals In this section we derive some (formal) conserved quantities and non-increasing ones as well. Let us start with the following fundamental (and formal) identity: for any φ : Y → R+ there holds 1 Q(f ) φ dy = a(y, y ) f f φ − φ − φ dy dy. (2.1) 2 Y Y Y This identity is obtained after changing variables and applying (without justification) the Fubini theorem to Q1 (f ). Here and below, we put g = g(y), g = g(y ) and g = g(y + y ) to shorten notations. Suitable choices of functions φ in (2.1) lead to several qualitative information on the solution f to the coalescence equation (1.1), and on the reaction term Q(f ) as well. We list some of them now. • Mass conservation. With the choice φ(y) = m, the term φ − φ − φ vanishes and we deduce the total mass conservation: m f (t, x, y) dydx = m f in (x, y) dydx , t ≥ 0. (2.2) Y

Y

• Momentum conservation. Similarly, with the choice φ(y) = p, the term φ − φ − φ also vanishes and we deduce the mean momentum conservation: p f (t, x, y) dydx = p f in (x, y) dydx , t ≥ 0. (2.3) Y

Y

It is next clear from (2.1) that, if φ : Y → R is subadditive, that is, φ(y + y ) ≤ φ(y) + φ(y ) for all (y, y ) ∈ Y 2 , the following map

(2.4)

t →

f (t, x, y) φ(y) dydx Y

is a non-increasing function of time. We now identify several classes of functions satisfying (2.4): we first consider functions depending solely on m, then functions depending solely on v, and finally functions being the product of a function of m and a function of v.


243

• Typical examples of subadditive functions is φ(y) = mα for α ∈ (−∞, 1]. In particular, the choice φ(y) = 1 shows that the so-called total number of particles decreases with time (as expected) t 1 f (t, x, y) dydx + a f f dy dydxdt = f in (x, y) dydx. 2 Y Y 0 Y Y (2.5) Another consequence is that, if supp f in (x, y) ⊂ Mδ := {(x, y) ∈ × Y , m > δ}, then supp f (t) ⊂ Mδ for any t ≥ 0. Indeed, it is sufficient to notice that φ(y) = 1[0,δ] (m) satisfies (2.4). That means that no particle of size smaller than δ can be created if there are none initially. A weaker version of this fact is the following: for any non-increasing function φ : R+ → R+ (such as φ(m) = mα , α < 0), we have t 1 Yφ (f (τ )) dxdτ ≤ Yφ (f in ) , (2.6) Yφ (f (t, .)) + 2 0 with

Yφ (f ) :=

f (x, y) φ(m) dydx, Y

Yφ (f ) := Y

a φ(m) f f dydy .

Y

• For any non-decreasing and nonnegative function φ : R+ → R+ , we notice that |p + p | ≤ |p| + |p | ≤ (m + m ) max(|v|, |v |), and thus |v | ≤ max(|v|, |v |). The monotonicity of φ then implies that the map y → φ(|v|) fulfills (2.4). Consequently, f (t, x, y) φ(|v|) dydx t −→ Y

is a non-increasing function of time. As a first consequence of this fact, we realize that, if supp f in ⊂ VR := {(x, y) ∈ × Y, |v| ≤ R}, then supp f (t) ⊂ VR for any t ≥ 0 (apply the previous result with the choice φ = 1[R,+∞) ). • Consider now a nonnegative and non-increasing function φ1 : R+ → R and a nonnegative and convex function φ2 : R3 → R. Then f (t, x, y) m φ1 (m) φ2 (v) dydx t −→ Y

is a non-increasing function of time. Indeed, putting φ(y) = m φ1 (m) φ2 (p/m), y ∈ Y , the convexity of φ2 and the monotonicity of φ1 ensure that m m φ(y + y ) = (m + m ) φ1 (m + m ) φ2 v+ v m + m m + m ≤ φ1 (m + m ) m φ2 (v) + m φ2 (v ) ≤ m φ1 (m) φ2 (v) + m φ1 (m ) φ2 (v ) , and the function φ satisfies (2.4).

244


A first interesting consequence of this result is the decay of the kinetic energy (with the choice φ(y) = |p|2 /m, that is, φ1 (m) = 1 and φ2 (v) = |v|2 ): |p|2 1 t f (t, x, y) Eke (f (t, x, .)) dxdt dydx + m 2 0 Y |p|2 = f in (x, y) dydx, (2.7) m Y where

Eke (f ) :=

a Y

Y

m m |v − v |2 f f dy dy . m + m

(2.8)

A more general property is actually valid. Consider a nonnegative and non-decreasing convex function λ ∈ C 1 ([0, +∞)) such that λ(0) = 0. With the choice φ(y) = m λ(|p|/m) (that is, φ1 (m) = 1 and φ2 (v) = λ(|v|)), the function φ is subadditive and we have 1 t f (t, x, y) m λ(|v|) dydx + Eλ (f (t, x)) dxdt 2 0 Y ≤ f in (x, y) m λ(|v|) dydx, (2.9) Y

where

|p| |p| + |p | Eλ (f ) := am , m m + m Y Y B (A, B) := λ (B) − λ (z) dz .

f f dy dy , (2.10)

A

Indeed, putting w = (|p| + |p |)/(m + m ), the monotonicity of λ entails that φ(y) + φ(y ) − φ(y + y ) ≥ m (λ(|v |) − λ(w)) + m (λ(|v|) − λ(w)). The convexity of λ and the identity m (|v | − w) = m (w − |v|) then imply w φ(y) + φ(y ) − φ(y + y ) ≥ m (|v | − w) λ (w) − m λ (z) dz |v| w ≥m (λ (w) − λ (z)) dz = (|v|, w) . |v|

Another consequence is the following. For any convex function ω : R3 → R+ , the function t −→ f (t, x, y) ω(x − v t) dydx Y

is a non-increasing function of time. Indeed, for each (t, x) ∈ R+ × R3 , φ : y → ω(x − v t) is a convex function of v and is thus subadditive. Therefore, after multiplying (1.1) by ω(x − v t) and integration, the contribution of the coalescence term is nonnegative, while the terms resulting from the free transport cancels. In particular, taking ω(z) = |z|θ , θ ≥ 1, z ∈ R3 , we obtain f (t, x, y) |x − v t|θ dydx ≤ f in (x, y) |x|θ dydx , t ≥ 0 . (2.11) Y

Y


245

• Finally, let ∈ C 1 ([0, +∞)) be a nonnegative and convex function satisfying (0) = 0. Then t −→ (f (t, x, y)) dydx Y

is a non-increasing function of time. More precisely, we have the following result Lemma 2.1. Any solution f to (1.1)-(1.2) satisfies (f (t, x, y)) dydx +

t 0

Y

D (f (τ, x)) dxdτ ≤

(f in (x, y)) dydx , Y

(2.12) 1 (f ) + D 2 (f ), where D (f ) := D 1 D (f ) := 2 D (f ) :=

1 2

Y2

Y2

a(y, y ) (f ∨ f ) (f ∧ f ) dydy ≥ 0 , (2.13)

a(y, y ) (f ) f 1(m,+∞)×R3 (y ) dy dy ≥ 0 ,

and (u) := u φ(u) − (u) ≥ 0, φ(u) := d/du(u) for u ≥ 0, . Here and below, we use the notations f ∨ f = max {f, f } and f ∧ f = min {f, f }. When (u) = up , p > 1, this result has been proved in [33] for the Smoluchowski equation. Proof. We first recall that the convex conjugate ∗ of is a nonnegative convex function given by ∗ (u) := sup {u w − (w)} w≥0

for u ≥ 0. In addition, we have the Young inequality u w ≤ (u) + ∗ (w) ,

u, w ≥ 0 ,

(2.14)

u ≥ 0.

(2.15)

and the equality ∗ (φ(u)) = (u) ,

By (2.14) and (2.15) we have a f f φ(f ) dy dy = a (f ∧ f ) (f ∨ f ) φ(f ) dy dy 2 2 Y Y ≤ a (f ∧ f ) (f ∨ f ) + ∗ (φ(f )) dy dy 2 Y ≤ a (f ∧ f ) (f ∨ f ) dy dy Y2 + a (f ∧ f ) (f ) dy dy. Y2

246


We now use (1.8) to bound the second term of the right-hand side of the above inequality and deduce that a f f φ(f ) dy dy ≤ a (f ∧ f ) (f ∨ f ) dy dy Y2 Y2 + (a(y, y ) + a(y , y )) (f ∧ f ) (f ) dy dy Y2 ≤ a (f ∧ f ) (f ∨ f ) dy dy Y2 +2 a(y , y ) f (f ) dy dy Y2 ≤ a (f ∧ f ) (f ∨ f ) dy dy 2 Y +2 a f (f ) 1(0,m)×R3 (y ) dy dy . Y2

Consequently, Q(f ) φ(f ) dy Y 1 = a f f φ(f ) − φ(f ) − φ(f ) dy dy 2 2 Y 1 ≤ a (f ∧ f ) (f ∨ f ) dy dy + a f (f ) 1(0,m)×R3 (y ) dy dy 2 2 Y2 Y 1 − a (f ) f dy dy − a (f ) f + (f ) f dy dy , 2 Y2 Y2 whence (2.12).

We summarize, in the next result, the a priori estimates on the solutions to (1.1) obtained in this section. Theorem 2.2. Assume that f in satisfies |p| in in in (f (x, y)) + f (x, y) r(m) + 1 + m + |p| + m λ K(f ) := m Y dxdy < ∞, (2.16) for some nonnegative and non-decreasing convex functions ∈ C 1 ([0, +∞)) and λ ∈ C 1 ([0, +∞)) such that (0) = 0 and λ(0) = 0, and some nonnegative and nonincreasing function r ∈ C((0, +∞)). Then a solution f to (1.1)-(1.2) formally satisfies |p| (f (t, x, y)) + f (t, x, y) r(m) + 1 + m + |p| + m λ sup m t≥0 Y dxdy ≤ K(f in ), (2.17)


∞ 0

247

(D (f (τ, x)) + Y1 (f (τ, x)) + Yr (f (τ, x)) + Eλ (f (τ, x))) dxdτ ≤ 2 K(f in ),

(2.18) where Y1 , Yr , Eλ and D are defined in (2.6) (with φ = 1 and φ = r, respectively), (2.10) and (2.12), respectively.

3. Stability Result Definition 3.1. Let f in be a nonnegative function satisfying (1.12). A weak solution to (1.1)-(1.2) is a nonnegative function f ∈ C([0, +∞); L1 ( × Y )) ,

f (0) = f in ,

satisfying (2.17), (2.18) for some nonnegative and non-decreasing convex functions ∈ C 1 ([0, +∞)) and λ ∈ C 1 ([0, +∞)), and some nonnegative, convex and nonincreasing function r ∈ C 1 ((0, +∞)) such that (0) = 0, λ(0) = 0, lim λ (u) = lim

u→+∞

u→+∞

(u) = +∞ u

and

lim r(m) = +∞ ,

m→0

(3.1)

together with Eq. (1.1) in the sense of distributions. It must also satisfy the qualitative properties (1.13)–(1.17). Notice that the bounds (2.17), (2.18) imply that Eq. (1.1) makes sense since, for any T ∈ R+ , Qi (f ) ∈ L1 (T × Y ),

i = 1, 2.

(3.2)

We now state a weak stability principle for weak solutions to (1.1)-(1.2). Theorem 3.2. For n ≥ 1, let fn be a weak solution to (1.1)-(1.2) with initial datum fn (0). Assume further that (2.17) and (2.18) hold uniformly with respect to n ≥ 1 with K(fn (0)) ≤ K0 for some K0 > 0 and functions (, λ, r) satisfying (3.1). Then there exist a subsequence (fnk ) of (fn ) and a function f such that f is a weak solution to (1.1)-(1.2),   fnk −→ f in C([0, T ); w − L1 (BR × Y )), 

(3.3) Qi (fnk ) Qi (f ) in L1 ((0, T ) × BR × YR )

for any T , R ∈ R+ and i ∈ {1, 2}, with YR := (0, R) × BR , and

Y

ψ(y) fnk dy −→

ψ(y) f dy in L1 ((0, T ) × BR )

(3.4)

Y

for ψ ∈ D(Y ). Here D(Y ) denotes the space of C ∞ -smooth and compactly supported functions in Y and C([0, T ); w − L1 ( × Y )) the space of weakly continuous functions from [0, T ) in L1 ( × Y ).

248


4. Proof of Theorem 3.2 In this section, we consider a sequence (fn ) of solutions to (1.1) satisfying the requirements of Theorem 3.2. We first prove that (2.17) and (2.18) guarantee the weak compactness in L1 of (fn ) and (Qi (fn )), i = 1, 2. The next step is to use the properties of the linear transport equation to obtain the strong L1 -compactness of y-averages in a way similar to the Boltzmann equation [15]. Thanks to the strong compactness thus obtained, we may identify the limit of the nonlinear coagulation term (Q(fn )). 4.1. Weak compactness of the coalescence term. The aim of this subsection is to prove that (2.17) and (2.18) imply the weak L1 -compactness of (fn ) and (Q(fn )). While that of (fn ) will follow from Lemma 2.1, the reaction terms are more difficult to handle and will be split in several parts: an integral where either m or m are small, for which a might be singular, and which will be controlled by (2.6), an integral where p is large, for which a is unbounded, and which will be controlled by (2.9), and the remaining integral is over a compact subset of Y 2 and will be controlled by (1.9), where a is large and by Lemma 2.1 elsewhere. We first prove the claim for (fn ) and (Q1 (fn )). Lemma 4.1. For each T > 0 and R > 0, the sequence (fn ) is weakly compact in L1 ((0, T ) × BR × Y ) and the sequence (Q1 (fn )) is weakly compact in L1 ((0, T ) × BR × YR ), where YR is defined in (3.3). Proof. We fix T > 0 and R > 0. Observe first that the first assertion of Lemma 4.1 is a straightforward consequence of (2.17), (3.1) and the Dunford-Pettis theorem. We next study the weak compactness properties of (Q1 (fn )). Let E be a measurable subset of (0, T ) × BR × YR and let M, N , δ and R1 be positive real numbers such that R1 ≥ R + 2R 2 /δ. Putting ϕ = 1E and A = An (t, x, y, y ) := a(y, y ) fn (t, x, y) fn (t, x, y ) , and performing the change of variables (y, y ) → (y , y − y ), we have Q1 (fn ) ϕ dy = I1 + J1 YR

with

δ

2I1 = 0

R

R3 0 R

+

δ R

+ δ

B(−p ,R) R

{|p |>R1 } δ

{|p |≤R1 }

and 1 J1 = 2

δ

R

R δ

A ϕ dydy +

R

B(−p ,R)

{|p |≤R1 } δ

R

δ

R3 0

δ

B(−p ,R)

B(−p ,R)

A ϕ dydy

A ϕ dydy A ϕ 1a≥N dydy B(−p ,R)

A ϕ 1a≤N dydy .

(4.1)


249

On the one hand, observing that |p| R R1 |p | + |p| ≤ ≤ ≤ , m δ 2R m + m so that

|p| |p | + |p| m , m m + m

R R1 ≥ δ,R (R1 ) := δ , δ 2R

for y ∈ (δ, R) × B(−p , R) and y ∈ (δ, R) × BRc 1 , and A = a (fn ∨ fn ) (fn ∧ fn ) M a (fn ∨ fn ) (fn ∧ fn ) 1fn ∧fn ≥M (M)

≤ a (fn ∨ fn ) M 1fn ∧fn ≤M +

M a (fn ∨ fn ) (fn ∧ fn ), (M)

≤ a (fn + fn ) M +

(4.2)

we get 1 2I1 ≤ r(δ) +

δ

0

1 r(δ)

0

R

δ

B(−p ,R) δ

r(m ) A dydy

R

{|p |>R1 } δ

δ

2 Yδ,R+R

r(m) A dydy

B(−p ,R)

0 R

1 + δ,R (R1 ) +M M + (M)

R

|p| |p| + |p | m , m m + m B(−p ,R)

A dydy

a 1a≥N (fn + fn ) dydy 1

2 Yδ,R+R

a (fn ∨ fn ) (fn ∧ fn ) dydy 1

2 1 ≤ Yr (fn (t, x)) + Eλ (fn (t, x)) r(δ) δ,R (R1 ) M +2 M aδ,R+R1 (N ) fn (t, x) dy + 2 D 1 (fn (t, x)) . (M) Y On the other hand, by (1.9) and (4.2), we have

A 1a≤N ϕ dydy

J1 ≤ Yδ,R+R1

Yδ,R+R1

≤MN Yδ,R+R1

≤ 2M N

Y2

Yδ,R+R1

(fn + fn ) ϕ dydy +

fn ϕ dy dy +

2M D 1 (fn (t, x)) (M)

2M D 1 (fn (t, x)) . (M)

250


We now gather the previous estimates and integrate over (0, T ) × BR . Owing to (2.17) and (2.18), we end up with T 2 K0 K0 4M 2 Q1 (fn ) ϕ dydxdt ≤ + + K0 r(δ) δ,R (R1 ) (M) BR YR 0 T +2 M aδ,R+R1 (N )K0 + M N fn ϕ dy dydxdt . 0

BR

Y2

Using the already established weak compactness of (fn ) in L1 ((0, T ) × BR × Y ) and (1.9), we may pass to the limit first as |E| → 0, then as N → +∞, R1 → +∞, and finally as M → +∞ and δ → 0 to conclude that T lim sup Q1 (fn ) 1E dydxdt = 0 , |E|→0 n≥1

0

BR

YR

after noticing that (3.1) implies that δ,R (R1 ) → +∞ as R1 → +∞. Therefore, (Q1 (fn )) is weakly compact in L1 ((0, T ) × BR × YR ) by the Dunford-Pettis theorem. Owing to the structure of the coalescence equation (1.1), it turns out that the weak L1 -compactness of (Q1 (fn )) provides some additional information on (Q2 (fn )). This fact was first observed in [28] for the discrete diffusive coagulation equation and a result in the same spirit is also available for (1.1). Lemma 4.2. For any R, T > 0, there exists a nonnegative and non-decreasing function = R,T ∈ C 1 ([0, +∞)) such that θ (u) :=

d (u) → +∞ when u → +∞ du

and

T

0

θ (fn ) Q2 (fn ) dydxdt ≤ C(δ, R, T )

BR

(4.3)

Yδ,R

for any δ ∈ (0, 1). Proof. From Lemma 4.1, we know that (fn (0)), (fn ) and (Q1 (fn )) belong to weakly compact subsets of L1 ( × Y ) and L1 ((0, T ) × B2R × Y2R ), respectively. The DunfordPettis theorem and a refined version of the de la Vallée-Poussin theorem [30, Prop. I.1.1] imply that there is a nonnegative, convex and non-decreasing function = R,T ∈ C 1 ([0, +∞)) such that θ(u) :=

d → +∞ du

as

u → +∞

for some constant C > 1 and K1 := sup (fn (0)) dydx + n≥1

B2R Y2R

T 0

and

u θ (u) ≤ C (u) ,

u ≥ 0,

((fn ) + (Q1 (fn ))) dydxdt B2R Y2R

< +∞.


251

We fix χ ∈ D(R3 × Y ), 0 ≤ χ ≤ 1, such that χ ≡ 1 on BR × Yδ,R and supp χ ⊂ B2R × Yδ/2,2R . We infer from (1.1) and the Young inequality that (fn (T )) χ dydx + θ (fn ) Q2 (fn ) χ dydxdt Y T Y = θ (fn ) Q1 (fn ) χ dydxdt T Y + (fn ) v · ∇x χ dydxdt + (fn (0)) χ dydx T

T

≤

Y

0

B2R T

2 + δ

0

Y

{∗ (θ (fn )) + (Q1 (fn ))} dydxdt

Y2R

(fn ) p · ∇x χ dydxdt + K1 .

B2R

Yδ/2,2R

Since and θ are nonnegative, ∗ (θ (u)) ≤ u θ (u) − (u) ≤ C (u) ,

u ≥ 0,

and since 1Yδ,R ≤ χ , we deduce from the above estimate that θ (fn ) Q2 (fn ) dydxdt T

Y

δ,R

≤

θ (fn ) Q2 (fn ) χ dydxdt T

Y

4R

∇x χ L∞ ≤ 2 K1 + C + δ whence (4.3).

T

(fn ) dydxdt ,

0

B2R

Y2R

Lemma 4.3. For each R, T > 0, the sequence (Q2 (fn )) is weakly compact in L1 ((0, T )× BR × YR ). Proof. We fix T > 0 and R > 0. Let E be a measurable subset of (0, T ) × BR × YR and M, N, δ, R1 and R2 be positive real numbers such that R1 ≥ R and R2 ≥ 2R12 /δ. Putting ϕ = 1E and A = An (t, x, y, y ) := a(y, y ) fn (t, x, y) fn (t, x, y ) , we have

Q2 (fn ) ϕ dy = I2 + J2

(4.4)

YR

with

δ

I2 = 0

BR

+ Yδ,R

+

Yδ,R

∞

R3 0 ∞ R3 R1 R1 δ

A ϕ dy dy +

Yδ,R

A ϕ dy dy +

BR2

Yδ,R

A ϕ 1a≥N dy dy

δ

R3 0 R1 δ

A ϕ dy dy

{|p |>R2 }

A ϕ dy dy (4.5)

252


and

R1

J2 = Yδ,R

δ

A ϕ 1a≤N dy dy.

BR2

On the one hand, observe again that |p| |p| + |p | R R2 ≤ , ≤ ≤ m δ 2R1 m + m so that

m

|p| |p| + |p | , m m + m

≥ δ,R (R R2 /R1 ) := δ

R R2 , δ 2R1

for y ∈ Yδ,R and y ∈ (δ, R1 ) × BRc 2 , and that A = A (1fn ≥M + 1fn <M ) ≤ A

θ (fn ) a(y, y ) +M fn |y | θ (M) |y |

on Yδ,R × (R1 , ∞) × R3 . Using the growth conditions (1.9) and (1.10), a similar computation to the one performed to estimate I1 leads to I2 ≤

2 1 Yr (fn (t, x)) + Eλ (fn (t, x)) r(δ) δ,R (R R2 /R1 ) 1 + θ (fn ) Q2 (fn ) dy + M ωδ,R (R1 ) |y| fn (t, x) dy θ (M) Yδ,R Y M 1 +2 M aδ,R2 (N ) fn dy + 2 D (fn (t, x)). (M) Y

On the other hand, performing exactly the same computations as for J1 , we obtain J2 ≤ 2 M N

Y2

fn ϕ dy dy +

2M D 1 (fn (t, x)) . (M)

We integrate (4.4) over (0, T ) × BR and use the estimates for I2 and J2 and Lemma 4.2 to obtain, thanks to (2.17) and (2.18), 0

T

BR YR

Q2 (fn ) ϕ dydxdt ≤

2 K0 K(δ, R) + + M ωδ,R (R1 ) K0 r(δ) θ (M) K0 4M + 2 M aδ,R2 (N ) K0 + K0 + δ,R (R R2 /R1 ) (M) T +2M N fn ϕ dy dydxdt . 0

BR

Y2

Using the already established weak compactness of (fn ) in L1 ((0, T ) × BR × Y ), we may pass to the limit first as |E| → 0, then successively as N → +∞, R2 → +∞ and R1 → +∞, and finally as M → +∞ and δ → 0, to conclude.


253

Thanks to Lemma 4.1 and Lemma 4.3, there are a subsequence of (fn ) (not relabeled) ¯ 1, Q ¯ 2 such that and nonnegative functions f , Q fn f

weakly in

L1 ((0, T ) × BR × Y ) ,

(4.6)

and ¯i Qi (fn ) Q

weakly in

L1 ((0, T ) × BR × YR )

(4.7)

for i = 1, 2, T > 0 and R > 0. ¯ i , i = 1, 2, in terms of f . Since Q(fn ) involves quadratic It remains to identify Q terms, some strong compactness is needed and is the subject of the next section. 4.2. Strong compactness of y-averages. Theorem 4.4. Let T > 0 and consider two bounded sequences (gn ) and (Gn ) of L1 (T × Y ) such that ∂t gn + v · ∇x gn = Gn in T × Y. Assume further that (gn ) is weakly compact in L1 ((0, T ) × BR × YR ) for each R > 0. Then, for any ψ = ψ(y, y ) ∈ L∞ (Y 2 ) with compact support, there holds gn (t, x, y) ψ(y, y ) dy belongs to a strongly compact subset of Y

L1 ((0, T ) × BR × YR )

(4.8)

for each R > 0. This is a particular case of the velocity averaging lemma [7, 18, 31], noticing that v = v(y) = p/m satisfies the nondegeneracy condition meas {y ∈ Y ,

σ · v(y) = u} = 0

for every σ ∈ S2 and u ∈ R. Nevertheless, it can be seen as a consequence of the classical averaging lemma [15, 18] as we show below. Proof. First step. We assume further that (gn ) is bounded in L2 ((0, T ) × BR × YR ). We fix α = α(m) ∈ Cc (R+ ) and observe that ∞ g˜ n (t, x, v) := gn (t, x, m, mv) α(m) dm, 0 ∞ ˜ n (t, x, v) := Gn (t, x, m, mv) α(m) dm G 0

belong respectively to a bounded subset of L2loc (T × R3 ) and L1loc (T × R3 ) and satisfies ∂ g˜ n ˜ n in (0, T ) × × R3 . + v · ∇x g˜ n = G ∂t Using the classical velocity averaging lemma [18], we get that ∞ gn (t, x, m, mv) α(m) dmβ(v) dv R3 0

belongs to a strongly compact subset of L1loc (T ) for every β ∈ Cc (R3 ).

254


Second step. Performing the change of variables (m, v) → (m, p = mv), the previous step and the compactness of the support of α imply that (4.8) holds true with ψ(y, y ) = α(m) β(p/m) m−3 . We now fix χ ∈ Cc (R+ ) and ζ ∈ Cc (R3 ). Consider k ∈ N and ∈ N3 . We deduce from the previous step with α(m) = mk χ (m) and β(v) = v ζ (v) that (4.8) is valid for ψ(y, y ) = mk−||−3 p χ (m)ζ (p/m). Therefore, (4.8) holds for ψ(y, y ) = ϕ(y) χ (m)ζ (p/m) for any ϕ ∈ Cc (Y¯ ) by the Stone-Weierstrass theorem. Choosing ζ ∈ Cc (R3 ) such that ζ ≡ 1 on Bρ for a sufficiently large ρ (depending on the support of ϕ and χ ), we realize that (4.8) is actually true for ψ(y, y ) = ϕ(y) χ (m) for any ϕ ∈ Cc (Y¯ ). Finally, using the weak compactness of (gn ), we may let χ → 1 and deduce that (4.8) is valid for ψ(y, y ) = ϕ(y) for any ϕ ∈ Cc (Y¯ ). We then proceed as in [15, Cor. IV.2] to remove the additional assumption that (gn ) is bounded in L2 (derenormalization technique) and to extend the result to the class of functions ψ ∈ L∞ (Y 2 ) (density argument), thus completing the proof of Theorem 4.4. ¯ i = Qi (f ) for 4.3. Passing to the limit in the coalescence term. We aim to show that Q ¯ ¯ i = 1, 2, where f , Q1 and Q2 are defined in (4.6) and (4.7). Step 1. For n ≥ 1, we put

ρn :=

ρ :=

fn dy, Y

f dy, Y

and claim that ρn −→ ρ

in

L1 ((0, T ) × BR )

and a.e. in

(0, T ) × BR

(4.9)

for each R > 0. Indeed, fix R > 0. We have ρn = ρnM + σnM with M ρn := fn dy YM

for n ≥ 1 and M > 0. By Theorem 4.4, (ρnM ) belongs to a strongly compact subset of L1 ((0, T ) × BR ) for each M > 0, while σnM satisfies 1 M σn := fn dy ≤ fn |y| dy c M Y YM and thus converges to zero in L1 ((0, T ) × BR ) as M → +∞, the convergence being uniform with respect to n ≥ 1 by (2.17). The claim (4.9) then readily follows. Step 2. We next consider ϕ ∈ D(Y ) and R > 0 such that supp ϕ ⊂ YR . Let δ, N , R1 and R2 be positive real numbers such that R1 ≥ R + 2R 2 /δ and R2 ≥ 2R12 /δ, and put b1 =

1 a ϕ 1a≤N 1X1 , 2

b2 = a ϕ 1a≤N 1X2 ,

with X1 = {(y, y ) ∈ Y 2 ; m, m ∈ [δ, R], p ∈ BR1 , p ∈ BR (−p )}, X2 = {(y, y ) ∈ Y 2 ; y ∈ Yδ,R , m ∈ [δ, R1 ], p ∈ BR2 }.


255

We first claim that, for i = 1, 2, η ∈ (0, 1) and R0 > 0, there holds 1 1 fn bi fn dy dy f bi f dy dy 1 + η ρn Y 1 + η ρ Y Y Y

(4.10)

weakly in L1 ((0, T ) × BR0 ). Indeed, notice that, since bi ∈ L∞ (Y 2 ) with compact support, Theorem 4.4 and (4.9) imply that 1 1 bi fn dy → bi f dy a.e. in (0, T ) × BR0 × YR , 1 + η ρn Y 1+ηρ Y and it is bounded in L∞ (T × Y ) by N ϕ L∞ /η. Combining these properties with the weak convergence (4.6) of (fn ) yields (4.10) by standard integration arguments, see [29, Lemma A.2] for instance. We now aim at passing to the limit as η → 0. As a first consequence of Lemma 4.3 and (4.10), we obtain that T T bi f f bi fn fn dy dy dx dt ≤ lim inf dy dy dx dt n→+∞ 0 BR0 Y 2 1 + η ρ BR0 Y 2 1 + η ρn 0 ≤ sup Qi (fn ) dy dx dt n≥1 T

Y

for each η ∈ (0, 1). The parameters R0 , R, N , δ, R1 , R2 and η being arbitrary, we deduce from the Fatou lemma that T a ϕ f f dy dy dx dt ≤ sup Q2 (fn ) dy dx dt < ∞ . (4.11) 0

BR0

Y

Y

n≥1 T

Y

We next observe that, for M ≥ 1 and η ∈ (0, 1), we have ηρn ηρn = 1ρn <M + 1ρn ≥M ≤ η M + 1ρn ≥M . 1 + η ρn 1 + η ρn Consequently, T ηρn sup bi fn fn dy dy ≤ η M sup Q2 (fn ) dy dx dt n≥1 0 BR0 1 + η ρn Y 2 n≥1 T YR T 1ρn ≥M Q2 (fn ) dy dx dt . + sup n≥1 0

BR0 YR

We first use (2.18) to pass to the limit as η → 0 and then use Lemma 4.3 and (4.9) to let M → +∞ and conclude that T ηρn lim sup bi fn fn dy dy dx dt = 0 . η→0 n≥1 0 BR0 1 + η ρn Y 2 Also, it follows from (4.10) that T ηρ lim bi f f dy dy dx dt = 0 . 2 η→0 0 1 + η ρ BR0 Y

256


Combining the previous two identities with (4.10), we end up with fn bi fn dy dy f bi f dy dy Y

Y

Y

(4.12)

Y

weakly in L1 ((0, T ) × BR0 ). ¯ i = Qi (f ) for i = 1, 2. For i = 1, 2, we write Step 3. We now show that Q n Qi (fn ) ϕ dy = Ii + fn bi fn dy dy , Y

where

I1n

and

I2n

Y

Y

are given by (4.1) and (4.5) respectively, with A = a fn fn . Also, Qi (f ) ϕ dy = Ii + f bi f dy dy , Y

Y

Y

where I1 and I2 are given by (4.1) and (4.5) with A = a f f , respectively. By (4.11) and the analysis of Sect. 3, we have Iin + Ii → 0

in

L1 (T × Y )

uniformly with respect to n ≥ 1 as N → +∞, R2 → +∞, R1 → +∞ and δ → 0. This last fact and (4.12) then imply that Qi (fn ) ϕ dy Qi (f ) ϕ dy Y

Y

¯ i = Qi (f ) for i = 1, 2. weakly in L1 ((0, T ) × BR0 ), from which we conclude that Q Owing to (4.6) and (4.7), it is now a standard matter to pass to the limit in the equation satisfied by fn and obtain that f satisfies (1.1) in the sense of distributions. 4.4. Further properties of f . In fact, owing to (2.17), (3.1) and the Dunford-Pettis theorem, the first assertion of Lemma 4.1 can be strengthened. Lemma 4.5. For T > 0 and R > 0, there is a weakly compact subset KR of L1 (BR ×Y ) such that fn (t) ∈ KR for each t ∈ [0, T ] and n ≥ 1. We now improve the convergence (4.6) of (fn ) to (3.3). Consider T > 0 and R > 0. On the one hand, it follows from Lemma 4.5 that {fn (t) , n ≥ 1} belongs to a weakly compact subset of L1 (BR × Y ) for each t ∈ [0, T ]. On the other hand, we infer from (1.1) and (2.18) that

(fn (t + h) − fn (t)) ψ dydx

≤ C(ψ) K0 h

Y

for every every ψ ∈ D( × Y¯ ), t ∈ [0, T ) and h ∈ (0, T − t), whence

=0 lim sup

(t + h) − f (t)) ψ dydx (fn n

h→0 n≥1

Y

(4.13)


257

for every ψ ∈ D( × Y¯ ) and t ∈ [0, T ). A density argument and Lemma 4.1 then imply that (4.13) actually holds true for every ψ ∈ L∞ (BR × Y ) (recall that a function in L∞ (BR × Y ) is the pointwise limit of a sequence of functions in D( × Y¯ ) which is bounded in L∞ (BR × Y )). Therefore, a variant of the Arzelà-Ascoli theorem entails that, up to the extraction of a subsequence, fn −→ f

C([0, T ]; w − L1 (BR × Y )) .

in

We next check that f enjoys the properties listed in Definition 3.1. Owing to (4.6) and the convexity of , standard weak compactness arguments entail that f satisfies (2.17). Next, proceeding as in Sect. 4.3, we obtain that Y1 (f ), Yr (f ) and Eλ (f ) belong to L1 (R+ × ) with the help of the bound (2.18). It does not seem possible to use a similar argument to check that D (f ) belongs to L1 (R+ × ) and we thus proceed directly on the equation satisfied by f . More precisely, we approximate by a sequence of convex functions growing at most linearly at infinity and employ approximation arguments as in [16, Sect. 2] to show that D (f ) belongs to L1 (R+ × ). We finally prove the strong continuity of f as in [16, Cor. II.2] with the help of the L1 -bound on f to control the behaviour of f for small m and the following lemma to control the behaviour of f for large x.

Lemma 4.6. lim

sup

R→+∞ t∈[0,T ]

{|x|≥R} Y

f (t, x, y) dydx = 0 .

Proof. Let A > 0. We multiply (1.1) by m ∧ A and integrate over Y , which is allowed since Q(f ) and f belong to L1 (T × Y ) for every T > 0. The contribution of the collision term being nonnegative by (2.1), we get ∂ (m ∧ A) f (t, x, y) dy + divx (m ∧ A) v f (t, x, y) dy ≤ 0 in D (T ), ∂t Y Y ∀ A > 0. We obtain (1.13) by passing to the limit A → +∞. Next, let ξ ∈ C ∞ (R3 ) be such that 0 ≤ ξ ≤ 1, ξ(x) = 0 if |x| ≤ 1/2 and ξ(x) = 1 if |x| ≥ 1. For R ≥ 1 and x ∈ R3 , we put ξR (x) = ξ(x/R). Let R ≥ 1. We now multiply (1.13) by ξR (x) and integrate over t to obtain t

∇ξ L∞ in ξR (x) ρ(t, x) dx ≤ ξR (x) ρ (x) dx + j (τ, x) dxdτ R 0

∇ξ L∞ ≤ K(f in ) t , ξR (x) ρ in (x) dx + R thanks to (2.17). Now, for t ∈ [0, T ] and δ ∈ (0, 1), it follows from (2.17) and the above inequality that δ 1 f (t, x, y) dydx ≤ r(m) f (t, x, y) dydx r(δ) {|x|≥R} 0 R3 {|x|≥R} Y ∞ 1 ξR (x) m f (t, x, y) dydx + δ {|x|≥R} R3 δ K(f in ) 1

∇ξ L∞ ≤ ρ in (x) dx + + K(f in )T . r(δ) δ {|x|≥R/2} δR

258


Lemma 4.6 then follows from (1.12) and (3.1) by letting first R → +∞ and then δ → 0 in the above inequality. Let us finally briefly explain how it is possible to get the additional qualitative properties (1.14)–(1.17). Mass decreasing (1.14) (it should be a mass conservation but we do not know how to prove it) is straightforwardly obtained integrating (1.13). For (1.16), we may consider a sequence (k ) of convex functions growing at most linearly at infinity such that k (s) s p as k → ∞ for any s > 0 and we first establish that

k (f (t, .)) L1 is a non-increasing function of time by a direct (and allowed) computation as it has been (formally) done in Lemma 2.1. Then we let k → ∞ and we conclude by the monotonous convergence theorem. Concerning (1.17), we may argue as follows. We assume that the sequence of solutions (fn ) satisfies t −→ fn (t, x, y) m φ(m) (|v|) dydx is a non-increasing function of time, Y

(4.14) for any decreasing function φ ∈ L∞ (R+ ) and any convex function which grows at most quadratically at infinity. In particular, fn (t, .) m |v|2 L1 is uniformly bounded. That is not a restrictive assumption, at least for an approximated solution, because (4.14) has been (formally) derived in Sect. 2. We then consider φ(m) = φR (m) = 10≤m≤R and (s) = R (s) = s 2 10≤s≤R + (2 R s − R 2 ) 1s≥R and for A ≥ R we write fn (t, x, y) m φR (m) R (|v|) dydx = fn (t, x, y) m φR (m) R (|v|) Y

Y

1|v|≤A dydx + fn (t, x, y) m φR (m) R (|v|) Y

1|v|≤A dydx. Since m φR (m) R (|v|) 1|v|≤A ∈ L∞ (Y ) and 2R fn (t, x, y) m φR (m) R (|v|) 1|v|≤A dydx ≤ fn (t, x, y) m |v|2 dydx A Y Y 2R ≤ C0 , A we may pass to the limit n → ∞ and we get fn (t, x, y) m φR (m) R (|v|) dydx → f (t, x, y) m φR (m) R (|v|) dydx. Y

Y

As a consequence, we have yet proved that for any R ≥ 0, t −→ f (t, x, y) m φR (m) R (|v|) dydx is a non-increasing function of time. Y

(4.15) The fact that the energy is non-increasing (1.17) then follows from (4.15), leting R → ∞ and using the monotonous convergence theorem.


259

5. Large Time Behaviour and Pressureless Gases System We first prove the last assertion of Theorem 1.1. Let f be a weak solution to (1.1)-(1.2) in the sense of Definition 3.1. We consider an increasing sequence (tn )n≥1 , tn → +∞, of positive real numbers and put fn (t, .) := f (tn + t, .) for t ≥ 0 and n ≥ 1. Owing to Definition 3.1, we realize that the sequence (fn ) satisfies the assumptions of Theorem 3.2. Therefore, fixing T > 0, it follows from Theorem 3.2 that, up to the extraction of a subsequence, fn (t) F weakly in L1 (T × Y ). Moreover, on the one hand, we infer from (4.12) with b2 = a 1a≤N 1Y 2 that δ,R

2 Yδ,R

a 1a≤N fn fn dy dy

2 Yδ,R

a 1a≤N F F dy dy

On the other hand, the bound (2.18) yields a fn fn dy dydxdt ≤ T

Therefore,

Y2

T 0

whence

BR

2 Yδ,R

a F F = 0

∞ tn

weakly in

L1 ((0, T ) × BR ).

Y1 (f ) dxdt −→ 0.

a 1a≤N F F dy dydxdt = 0,

a.e. on

(0, T ) × R3 × Y 2 .

Consequently, F ≡ 0 a.e., since 0 ≤ F ∈ L1 and a > 0 a.e. on Y 2 by (1.7). Since f is nonnegative with a vanishing weak limit, we actually have a strong convergence in L1 . Remark 5.1. When = R3 and |x| f in ∈ L1 ( × Y ), an alternative proof of the convergence to zero may be performed with the help of (2.11) and (2.12). We refer to [4, 34] for more details about this dispersion argument. Remark 5.2. It is mathematically challenging to have a better understanding of the relationship between the coalescence equation (1.1) and the pressureless gases system (1.19). Since colliding particles always stick together in the pressureless gases system, one can expect that it is the natural hydrodynamic limit of the coalescence equation: ∂t fε + v · ∇x fε =

1 Q(fε ) . ε

This is not true if fε (0) = f in satisfies (1.12). Indeed, for i = 1, 2, we have 1 2 Qi (fε ) dy dx dt ≤ Y1 (fε ) dx dt ≤ 4 f in L1 , ε T Y ε T and (fε (t)) L1 ≤ (f in ) L1 for some as in Definition 3.1. We may then argue as above to prove that fε → 0 as ε → 0. It would be interesting to figure out whether, if we start from a sequence of “well-prepared” initial data, the sequence (fε ) converges to a non-trivial solution to (1.1) of the form (1.18).

260


6. Proof of Theorem 1.1 The proof is essentially the same as that in [29, Sect. 5] and proceeds in two steps: we first consider an approximated problem and then use the stability result (Theorem 3.2) in order to pass to the limit. For the sake of completeness, we sketch below the main ingredients of these two steps. Step 1. Approximation. We introduce the modified coalescence term ˜ δ (g) + λ g, Qδ,λ (g) := Q with ˜ δ (g) := Q

1 Qaδ (g), 1 + δ ρg

|g| dy,

ρg := Y

and where Qaδ is defined by (1.3), (1.4) with the coalescence kernel aδ (y, y ) := min{a(y, y ), 1/δ}. For λ ≥ δ −2 we obviously get that Qδ,λ maps X+ in X+ , where the Banach space (X, . X ) is defined by X := {g ∈ L∞ (Y );

(1 + |y|)5 g ∈ L∞ (Y )} ,

with g X := (1 + |y|)5 g L∞ and X+ denotes the positive cone of X. Moreover, ∀g ∈ X, Qδ,λ (g) X ≤ C g X ∀g1 , g2 ∈ X, gi X ≤ R ⇒ ≤ CR g2 − g1 L1 (Y )

(6.1)

Qδ,λ (g2 ) − Qδ,λ (g1 ) L1 (Y ) (6.2)

for some constants depending only on δ, λ (and R). We next consider fin ∈ L∞ (; X+ ) and T > 0. Putting B := f in L∞ (;X) , we define the metric space (XT , d) by XT := g ∈ C([0, T ]; L1 ( × Y ))∩L∞ ((0, T ) × ; X),

sup g(t) L∞ (;X) ≤ 2B , t∈[0,T ]

d(g1 , g2 ) := sup (g1 − g2 )(t) L1 (×Y ) ,

g1 , g2 ∈ XT ,

t∈[0,T ]

and the map : XT → XT in the following way: for any g ∈ XT , h = (g) is the unique solution to ∂t h + v · ∇x h + λh = Qδ,λ (g),

h(0) = f in ,

with periodic boundary conditions when is the torus and with the null flux boundary conditions when is bounded. It is straightforward to verify that is well-defined and satisfies g ≥ 0 ⇒ g ≥ 0, by the maximum principle for λ > δ −2 . In addition, we infer from (6.1) and (6.2) that there exist T > 0 and k ∈ (0, 1) such that g ∈ XT

⇒

(g) ∈ XT


261

and d ((g1 ), (g2 )) ≤ k d(g1 , g2 ) for

g1 , g2 ∈ XT .

By the contracting map theorem, has a unique fixed point fδ in XT , which is actually a solution to the modified coalescence equation ˜ δ (fδ ) in (0, T ) × × Y, ∂t fδ + v · ∇x fδ = Q fδ (0) = f in in × Y,

(6.3) (6.4)

with periodic boundary conditions when is the torus and with the null flux boundary conditions when is bounded. By standard iterative arguments, (6.1) ensures that fδ is in fact a global solution (with T = ∞) and satisfies

fδ (t) L∞ (;X) ≤ exp(C(1 + t))

t ≥ 0.

Step 2. The limit δ → 0. Given now an initial data f in satisfying the condition (1.12), we introduce the approximation fδin given by 1 in in fδ (x, y) := min f (x, y), 1δ (x) 1Bδ−1 (y) δ with δ = when is either the torus or a bounded domain, and δ := B 1 when δ

= R3 . Clearly, fδin ∈ X+ and we denote by fδ the corresponding solution to the modified coalescence equation (6.3), (6.4). We first show that, under the assumption (1.12), the initial data f in and fδin satisfy (2.16) uniformly with respect to δ. For that purpose, we collect some properties of f in in the next lemma. Lemma 6.1. Assume that f in satisfies (1.12). There exist two nonnegative and nondecreasing convex functions ∈ C 1 ([0, +∞)) and λ ∈ C 1 ([0, +∞)) and a nonnegative, convex and non-increasing function r ∈ C 1 ((0, +∞)) depending only on f in satisfying (0) = 0, λ(0) = 0, (3.1) and such that |p| f in (x, y) + f in (x, y) r(m) + λ dydx < +∞ . (6.5) m Y Proof. Since f in ∈ L1 ( × Y ), the de la Vallée-Poussin theorem [14, 35] ensures that there is a nonnegative and non-decreasing convex function ∈ C 1 ([0, +∞)) satisfying (0) = 0, (3.1) and (f in ) ∈ L1 ( × Y ). Similarly, it follows from (1.12) that (x, y) → p/m belongs to L1 ( × Y ; mf in (x, y)dxdy) and the de la Vallée-Poussin theorem [14, 35] ensures that there is a nonnegative and non-decreasing convex function λ ∈ C 1 ([0, +∞)) satisfying λ(0) = 0, (3.1) and |p| mλ f in (x, y) dydx < +∞ . m Y We next notice that f in ∈ L1 ( × Y ) implies that (x, y) −→

1 ∈ L1 ( × Y ; mf in (x, y)dxdy) . m

262


Using again the de la Vallée-Poussin theorem, we conclude that there is a nonnegative, non-increasing and convex function σ ∈ C 1 ([0, +∞)) satisfying σ (0) = 0, σ (u)/u → +∞ as u → +∞ and 1 (x, y) −→ σ ∈ L1 ( × Y ; m f in (x, y)dxdy) . m Putting r(m) := m σ (1/m) for m > 0, this last fact implies that f in ∈ L1 ( × Y, r(m)dxdy), while the behaviour of σ for large u entails that r fulfills (3.1). Next, since σ is a convex function with σ (0) = 0, m → σ (m)/m is a non-decreasing function and r is thus a non-increasing function. The other properties of r finally follows from the properties of σ . Summarizing, we have constructed three functions , λ and r with the properties stated in Lemma 6.1 such that (6.5) holds true. Since fδin ≤ f in , we readily obtain that (6.5) holds for (fδin ) uniformly with respect to δ. Owing to the properties of fδ , we may justify the computations performed in Sect. 2 and deduce that the family {fδ } satisfies the bounds (2.17) and (2.18) uniformly with respect to δ (notice also that the modified coalescence kernel aδ /(1 + δ ρfδ ) satisfies (1.7)-(1.10) uniformly with respect to δ). Arguing as in the proof of Theorem 3.2, in the same way as it is done in [29], we deduce that there are a subsequence (fδk ) of (fδ ) and a function f such that   fδk −→ f in C([0, T ); w − L1 (BR × Y )),  ˜ Qδ (fδk ) Q(f ) in L1 ((0, T ) × BR × YR ) for any T > 0 and R > 0. We therefore conclude that f is a weak solution to (1.1)-(1.2) enjoying the properties stated in Theorem 1.1. A. From (r, v) to (m, p) We first show in this appendix that Eq. (1.1) with the collision term Q(f ) given by (1.3) and (1.4) is deduced from the usual droplet equation as it is presented for instance in [38], through a simple change of variables. Let us consider a function g = g(t, x, r, v) which solves the equation: ∂t g + v · ∇x g = Q(g)

in

(0, +∞) × × R+ × R3 ,

(A.1)

where Q(g) := Q1 (g) − Q2 (g) is given by r 11 r 1 Q1 (g)(r, v) = B(r1 , r ∗ , v1 , v ∗ ) g(r1 , v1 ) g(r ∗ , v ∗ ) dr ∗ dv ∗ , (A.2) 2 R3 0 r111 ∞ B(r, r ∗ , v, v ∗ ) g(r, v) g(r ∗ , v ∗ )dr ∗ dv ∗ , (A.3) Q2 (g)(r, v) = R3 0

and 1/3 r1 = r 3 − r ∗3 ,

r 3 v − r ∗3 v ∗ , r 3 − r ∗3 B(r, r ∗ , v, v ∗ ) = E(r, r ∗ , v, v ∗ ) π (r + r ∗ )2 |v − v ∗ |, v1 =

(A.4) (A.5)


263

the function E being a positive function, called the coalescence efficiency. We perform the following change of variables: m := r 3 , p := r 3 v, and f (t, x, m, p) :=

m−11/3 p g t, x, m1/3 , . 3 m

It then follows from (A.1) that m−11/3 p ∂t g t, x, m1/3 , 3 m m−11/3 p m−11/3 p = Q1 (g) t, x, m1/3 , − Q2 (g) t, x, m1/3 , . 3 m 3 m

∂t f (t, x, m, p) =

We now compute the two terms on the right-hand side. Starting with Q2 , we obtain p Q2 (g) t, x, m1/3 , m ∞ p p B m1/3 , r ∗ , , v ∗ g(t, x, r ∗ v ∗ ) dr ∗ , dv ∗ . = g t, x, m1/3 , m R3 0 m We change variables under the integral m∗ = r ∗3 ,

p∗ = r ∗3 v ∗ ,

dr ∗ dv ∗ =

1 ∗ −11/3 dm∗ dp ∗ , m 3

and obtain,

p Q2 (g) t, x, m1/3 , m =3m

11/3

f (t, x, m, p)

R3 0

with ∗

∗

∞

a(m, p, m∗ , p∗ ) f (t, x, m∗ , p∗ )dm∗ dp ∗

a(m, p, m , p ) = B m

1/3

∗ 1/3

,m

p p∗ , , ∗ m m

.

On the other hand, using the same change of variables and (A.4), we obtain p Q1 (g) t, x, m1/3 , m m1/3 11/3 1 m = B(r1 , r ∗ , v1 , v ∗ ) g(t, x, r1 , v1 ) g(t, x, r ∗ , v ∗ ) dr ∗ dv ∗ 2 R3 0 r111 m 3 = m11/3 a(m − m∗ , p − p ∗ , m∗ , p∗ ) f (t, x, m − m∗ , p − p ∗ ) 3 2 R 0 f (t, x, m∗ , p∗ ) dm∗ dp ∗ . Using the notation y = (m, p) introduced in Sect. 1 and (A.5), we may write a as follows 1/3 1/3 p p a(y, y ) = B m , (m ) , , m m 2 = E(m1/3 , (m )1/3 , v, v ) m1/3 + (m )1/3 |v − v |.

264


B. Properties of aH S and aNP We check here that Theorem 1.1 guarantees the existence of a weak solution to (1.1)– (1.2) when the coalescence kernels are given by either (1.5) or (1.6). For that purpose, we only have to show that these kernels enjoy the properties (1.8)–(1.10). We first consider aH S and prove the following result. Lemma B.1. For any α ≥ 0 the function α 2 a(y, y ) = mα + m |v − v | satisfies (1.8) and (1.9). Moreover, if 0 ≤ α < 1/2, it also satisfies (1.10). Proof. Let us start with (1.8). To this end, we notice that

v (m + m ) mv + m v

= m |v − v |,

|v − v | = −

m m m

and similarly, |v − v | =

m |v − v |. m

Therefore, m α α 2 m |v − v | + m + m m m α α 2 m α 2 m ≥ mα + m |v − v | + m + m m m α 2 ≥ mα + m |v − v | = a(y, y ).

a(y, y ) + a(y , y ) =

mα + m

α 2

2 , so that (1.9) is obviously true. On the other hand, it is clear that a is bounded on Yδ,R Finally, if y ∈ Y is such that m ≥ R ≥ 1 + R and y ∈ Yδ,R , we have

2 R a(y, y ) ≤ R α + m α + |v | ≤ C(R, δ) m 2α + m 2α−1 |p | δ C(R, δ) ≤ 1−2α |y |, R and (1.10) follows since α ∈ [0, 1/2).

Lemma B.2. For α ∈ [0, 1] and γ ∈ R, we put a(y, y ) =

m + m m m

α

v − v γ .

If −3 < γ ≤ 0, the function a satisfies (1.8)–(1.10).


265

Proof. Since α ≤ 1, it follows from the formulae for |v − v | and |v − v | computed in the proof of Lemma B.1 and the subadditivity of u → uα that α

m + m m γ /α m + m m γ /α

v − v γ a(y, y ) + a(y , y ) ≥ + mm m m m m m (α+γ )/α α (α+γ )/α m ≥ a(y, y ) + . m m Since (α + γ )/α ≤ 1, the function u → u(α+γ )/α is subadditive, from which we deduce that m m γ +α a(y, y ) + a(y , y ) ≥ a(y, y ) + = a(y, y ) . m m We next turn to (1.9). Since γ > −3, there exists q > 1 such that γ q > −3. For y ∈ Yδ,R , we have

αq

m p

γ q −γ q 2 q

a(y, y ) dy ≤ dy

p − m m δ Yδ,R Yδ,R R −γ q |p|γ q dy ≤CR BR (−mp /m ) δ ≤C |p|γ q dp, BR+R 2 /δ

which is finite since γ q > −3 and does not depend on y ∈ Yδ,R . The property (1.9) then follows from the inequality 1 a(y, y ) 1a(y,y )≥M dy ≤ q−1 a(y, y )q dy . M Yδ,R Yδ,R Finally, we have

α

p 2 p

γ

a(y, y ) dy ≤

− m dy δ Yδ,R Yδ,R m

for m ≥ R ≥ R and p ∈ R3 . Now, if |p |/m ≥ 2R/δ, we have

p

− p ≥ |p | − |p| ≥ |p |

m m m m 2 m

and

γ

p

dy ≤ C(δ, R), a(y, y ) dy ≤ C(δ, R)

Yδ,R Yδ,R m

since γ ≤ 0. On the other hand, if |p |/m < 2R/δ, we have BR/m (−p /m ) ⊂ B3R/δ and R a(y, y ) dy ≤ C(δ, R) m3 |p|γ dy ≤ C(δ, R), Yδ,R

δ

BR/m (−p /m )

since γ > −3. The assertion (1.10) then readily follows.

266


Acknowledgements. We gratefully acknowledge the partial support of the European Research Training Network HYKE HPRN-CT-2002-00282 during this work. The first and third authors were partially supported by CNRS and UPV/EHU through a PICS between the Universidad del Pa´ıs Vasco and the Ecole Normale Supérieure. We also thank Pierre Degond for fruitful discussions which motivate us to extend Theorem 1.1 to initial data with possibly infinite kinetic energy.

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Communications in


Exponential Distribution for the Occurrence of Rare Patterns in Gibbsian Random Fields M. Abadi1, , J.-R. Chazottes2 , F. Redig3 , E. Verbitskiy4 1 2

IME-USP, cp 66281, 05508-090, São Paulo, SP, Brasil. E-mail: [email protected] CPhT, CNRS-Ecole Polytechnique, 91128 Palaiseau Cedex, France. E-mail: [email protected] 3 Faculteit Wiskunde en Informatica, Technische Universiteit Eindhoven, Postbus 513, 5600 MB Eindhoven, The Netherlands. E-mail: [email protected] 4 Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands. E-mail: [email protected] Received: 17 February 2003 / Accepted: 22 October 2003 Published online: 24 February 2004 – © Springer-Verlag 2004

Abstract: We study the distribution of the occurrence of rare patterns in sufficiently mixing Gibbs random fields on the lattice Zd , d ≥ 2. A typical example is the high temperature Ising model. This distribution is shown to converge to an exponential law as the size of the pattern diverges. Our analysis not only provides this convergence but also establishes a precise estimate of the distance between the exponential law and the distribution of the occurrence of finite patterns. A similar result holds for the repetition of a rare pattern. We apply these results to the fluctuation properties of occurrence and repetition of patterns: We prove a central limit theorem and a large deviation principle. 1. Introduction In the last decade there has been an intensive study of exponential laws for rare events in the context of dynamical systems and stochastic processes, see e.g. the review paper [2]. In general, these laws are derived under the assumption of sufficiently strong mixing conditions, which basically ensures the possibility of writing the rare event as an intersection of almost independent events. The basic example of a rare event is the occurrence or return of a large cylindrical event. Other relevant examples are approximate cylindrical events (approximate matching in the sense of Hamming distance, see e.g. [8]), or large deviation events in certain interacting particle systems, see e.g. [3, 4]. The mixing conditions appearing in the context of dynamical systems or stochastic processes are typical for Z-actions, e.g., the ψ-mixing condition is very naturally satisfied in the context of Bowen-Gibbs measures [5]. In turning to the context of random fields or Zd -actions, the ψ-mixing property is very restrictive and in many natural examples such as Gibbsian random fields, this property does not hold except (trivially) in the i.i.d. case and in non-interacting copies of one-dimensional Gibbs measures. Gibbsian random fields have an obvious relevance to various applications, e.g., statistical physics, image processing, etc. Many interesting fluctuation properties such as

Supported by FAPESP at University of São Paulo

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M. Abadi, J.-R. Chazottes, F. Redig, E. Verbitskiy

large deviations principle and central limit theorems have been derived for them, and by now Gibbs measures constitute a well-established field of research, see e.g. [14, 16, 17]. The study of exponential laws for the occurrence or repetition of rare events in random fields has been initiated by A.J. Wyner [29] for the ψ-mixing case, using the Chen-Stein method. Because of the mixing condition, the results of that paper are not applicable to Gibbsian random fields like the Ising model in the high mixing regime (such as Dobrushin uniqueness, or analyticity regime). As an example, consider the d-dimensional Ising model in the high-temperature regime and fix a pattern in a cubic box of size n: what is the size of the “observation window” in which we see this pattern for the first time? This is clearly a rare event when the size of the pattern increases, and hence one expects in the “high mixing regime” that the size of this observation window is approximately exponentially distributed with parameter proportional to the probability of the pattern. The main difficulty in making this intuition into a mathematical statement is caused by the typical non-uniform mixing of Gibbsian random fields: the influence of an event A on an event B is not only dependent on their distance but also on their size. More precisely, the difference between the conditional probabilities P(A|B) and P(A) can be estimated in the optimal situation of the Dobrushin uniqueness regime as something of the form |A| exp(−αdist(A, B)). On a technical level, this “non-uniform mixing” implies that the rare event under consideration should be written as an intersection of events which at the same time are separated by a large distance and do not have an “excessive” size. In this paper we concentrate on Gibbsian random fields in the Dobrushin uniqueness regime (e.g. high temperature case). This has to be considered as the first non-trivial test case for random fields, with a broad variety of examples. The regime of phase coexistence (such as in the low-temperature Ising model) poses an even larger non-uniformity in the mixing conditions, i.e., the difference between P(A|B) and P(A) will in that case also depend on which events B we are conditioning on. Recent techniques such as disagreement percolation constitute a powerful tool to tackle this situation. This is however not the subject of the present paper, where we want to deal with the basic non-uniformity in the mixing appearing in all non-trivial Gibbsian random fields. Besides the mere derivation of exponential laws for the occurrence and repetition of rare events, we obtain a precise and uniform estimate of the error (i.e., the difference between the law and its exponential approximation). We show that obtaining this precise control of the error has many useful non-trivial applications in studying fluctuations of both “waiting times” and repetitions of rare patterns. The derivation of the exponential law is not via the Chen-Stein method. Via a direct use of the (non-uniform) mixing we obtain more detailed information on the error term. The reason for that is that in the Chen-Stein method one gives an estimate of the variational distance between the “real counting process” and the Poisson process, whereas we only need one particular event. The precise estimation of the error turns out to be crucial in the study of large deviations. The problem of “waiting times” is to ask for the P-typical size of the “observation window” in which a Q-typical pattern occurs, where P is Gibbsian, and Q is any ergodic field. The logarithm of the size of this observation window properly normalized converges to the sum of the entropy of Q and the relative entropy density s(Q|P). To this “law of large numbers” we add precise large deviation estimates and a central limit theorem as a corollary of the exponential law with its precise error. The main point is that the exponential law provides an approximation of the logarithm of the waiting time by minus the logarithm of the probability of the corresponding pattern. For the cumulant generating function of the waiting times, we give an explicit expression in terms

Occurrence of Rare Patterns in Gibbsian Random Fields

271

of the pressure. It coincides with the cumulant generating function of the probability of patterns in the interval (−1, ∞) and is constant on (−∞, −1]. A similar phenomenon was observed numerically for the cumulant generating function of the return times (that is in dimension one), see [18]. For repetition of patterns, we prove a similar exponential law with precise error bound. However, in that case we have to exclude “badly self-repeating” patterns, which have exponentially small probability for any Gibbs measure. As a corollary, we obtain a law of large numbers and a central limit theorem for repetitions. The large deviations are more subtle due to the presence of the bad patterns. We prove a full large deviation principle for the measure conditioned on good patterns, and a restricted large deviation principle for the full measure. Our paper is organized as follows. In Sect. 2 we give basic notations and definitions and state our main result and its corollaries. In Sect. 3 we review basic properties of high-temperature Gibbs measures. Section 4 contains the proof of the exponential law for the occurrence of patterns, and Sect. 5 is devoted to the derivation of its corollaries. 2. Definitions and Results We consider a random field {σ (x) : x ∈ Zd } on the lattice Zd , d ≥ 2, where σ (x) takes values in a finite set A. The joint distribution of {σ (x) : x ∈ Zd } is denoted by d P. The configuration space = AZ is endowed with the product topology (making it into a compact metric space). The set of finite subsets of Zd is denoted by S. For A, B ∈ S we put d(A, B) = min{|x − y| : x ∈ A, y ∈ B}, where |x| = di=1 |xi | (x = (x1 , x2 , ..., xd )). For A ∈ S, FA is the sigma-field generated by {σ (x) : x ∈ A}. For V ∈ S we put V = AV . For σ ∈ , and V ∈ S, σV ∈ V denotes the restriction of σ to V . For x ∈ Zd and σ ∈ , τx σ denotes the translation of σ by x: τx σ (y) = σ (x + y). For an event E ⊆ the dependence set of E is the minimal A ∈ S such that E is FA measurable. For any n ∈ N let Cn = [0, n]d ∩ Zd . An element An ∈ Cn is called a n-pattern or a pattern of size n. Definition 2.1 (First occurrence of a pattern). For every configuration σ ∈ we define tAn (σ ) to be the first occurrence of an n-pattern An in that configuration, that is the minimal k ∈ N such that there exists a non-negative vector x = (x1 , . . . , xd ) ∈ Zd+ with xi ≤ k, i = 1, ..., d, |x| > 0, satisfying (τx σ )Cn = An .

(1)

If such a vector x does not exist then we put tAn (σ ) = ∞. We now come to the mixing hypothesis we make on our random fields. For m > 0 define 1 | P EA1 |EA2 − P EA1 | , (2) ϕ(m) = sup |A1 | where the supremum is taken over all finite subsets A1 , A2 of Zd , with d(A1 , A2 ) ≥ m and EAi ∈ FAi , with P(EA2 ) > 0. Note that this ϕ(m) differs from the usual ϕ-mixing function since we divide by the size of the dependence set of the event EA1 . Definition 2.2. A random field is non-uniformly exponentially ϕ-mixing if there exist constants C1 , C2 > 0 such that ϕ(m) ≤ C1 e−C2 m for all m > 0.

(3)

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The examples that motivate this definition are Gibbsian random fields in the Dobrushin uniqueness regime (see Definition 3.2 below and examples thereafter). We leave their definition and properties till the next section. For a pattern An ∈ Cn we define the corresponding cylinder C(An ) as C(An ) = {σ ∈ : σCn = An } . Our main result reads: Theorem 2.1. For a translation-invariant Gibbs random field satisfying (3), there exist strictly positive constants C, c, ρ, 1 , 2 , 1 ≤ 2 , such that for any n and any n-pattern An , there exists λAn ∈ [ 1 , 2 ], such that P tAn >

1/d t − e−t λAn P (C(An ))

≤ C P (C(An ))ρ e−ct

(4)

for any t > 0. Notice that P(C(An )) in the “error term” in (4) is bounded above by exp(−c nd ), with c > 0, by the Gibbs property, see (31). The proof of this theorem is given in Sect. 4. Remark 2.1. The only results we are aware of in the context of random fields appeared in [29]. The results of that paper are valid under the assumption of a much stronger mixing condition than ours, namely ψ-mixing. Most Gibbs random fields (including the Ising model at high temperature) cannot satisfy such a property. As an examples of ψ-mixing Gibbsian random fields (in the sense of Wyner) on Z2 , one can consider independent copies of a one-dimensional Markov chain. This gives a two-dimensional Gibbsian random field, but without interaction in the y-direction. From the technical point of view, Wyner uses the Chen-Stein method. This leads to an estimate which for fixed pattern size does not converge to zero as t → ∞. Here we use a different approach allowing us to get a control in t in (4). This feature will turn out to be fundamental when we prove large deviations for waiting times, see below. From the proof of Theorem 2.1 it will be clear that we can generalize it to (An )n ’s that are finite patterns supported on a van Hove sequence of subsets of Zd . We will show elsewhere how to prove an analog of Theorem 2.1 in order to obtain the same kind of result for the low temperature “plus phase” of the Ising model, where the mixing condition of Definition 2.2 is no longer satisfied. We now state a number of corollaries of the previous theorem. We first consider the repetition of patterns. Definition 2.3 (First repetition of the initial pattern). For every configuration σ ∈ and for all n ∈ N, we define the first repetition, denoted by rn (σ ), as the minimal k ∈ N such that there exist a vector x = (x1 , . . . , xd ) ∈ Zd+ , with 0 ≤ xi ≤ k and |x| > 0, satisfying (τx σ )Cn = σCn .

(5)

To obtain a similar result for the repetition times we have to exclude certain patterns with “too quick repetitions”. We will make this notion precise later. The following result is established in Subsect. 5.1.


273

Theorem 2.2. For a translation-invariant Gibbs random field satisfying (3), there exist (i) a set Gn , which is a union of cylinders; (ii) strictly positive constants B, b, C, c, ρ, such that for any n ≥ 1, P(Gcn ) ≤ Be−bn , d

and for each An with C(An ) ⊆ Gn , 1/d t P rn > C(An ) − e−t ≤ C P(C(An ))ρ e−ct λAn P(C(An ))

(6)

(7)

for all t > 0 and where λAn is given in Theorem 2.1. Notice that the constants appearing in the previous theorems may be different. Nevertheless we used the same notations for the sake of simplicity. We denote by s(P) the entropy of P (see the next section for the definition). The next result (proved in Subsect. 5.2) shows how the repetition of typical patterns allows to compute the entropy using a single “typical” configuration. Theorem 2.3. For a translation-invariant Gibbs random field satisfying (3), there exists 0 > 0 such that for all > 0 ,

− log n ≤ log (rn (σ ))d P(C(σCn )) ≤ log log n eventually P−almost surely. (8) In particular, d log rn (σ ) = s(P) P − a.s . n→∞ nd lim

(9)

Note that (9) is a particular case of the result by Ornstein and Weiss in [23] where P is only assumed to be ergodic. Under our assumptions, we get the more precise result (8). We now consider the occurrence of an n-pattern drawn from some ergodic random field in the configuration drawn from a possibly different Gibbsian random field. This is the natural d-dimensional analog of the waiting-time [26, 29]. Definition 2.4 (“Waiting time”). For all configurations ξ, σ ∈ and for all n ∈ N, we define the “waiting time”, denoted by wn (ξ, σ ), as the minimal k ∈ N such that there exist a non-negative vector x = (x1 , . . . , xd ) ∈ Zd+ , with 0 ≤ xi ≤ k and |x| > 0, satisfying (τx σ )Cn = ξCn .

(10)

Notice that wn (ξ, σ ) = tξCn (σ ). We are going to consider the situation when ξ is “randomly chosen” according to an ergodic random field Q and σ is “randomly chosen” according to a non-uniformly exponentially ϕ-mixing Gibbs random field P, i.e. (ξ, σ ) is drawn with respect to the product measure Q × P. We denote by s(Q|P) the relative entropy of Q with respect to P; see Sect. 3 for the definition and a more explicit form. We have the following result (proved in Subsect. 5.3):

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Theorem 2.4. For a translation-invariant Gibbs random field P satisfying (3), and an ergodic random field Q, there exists 0 > 0 such that for all > 0 ,

(11) − log n ≤ log (wn (ξ, σ ))d P(C(ξCn )) ≤ log log n for Q × P-eventually almost every (ξ, σ ). In particular d log wn (ξ, σ ) = s(Q) + s(Q|P) Q × P − a.s . (12) nd Statement (12) is the d-dimensional generalization of a result obtained in [7] in the case of Bowen-Gibbs measures. Using Theorem 2.3 we can rewrite (12), for a “typical” pair (ξ, σ ), as follows: lim

n→∞

wn (ξ, σ ) ≈ rn (ξ ) exp((nd /d)s(Q|P)) . (The measure Q is supposed to be Gibbsian or only ergodic if we invoke the OrnsteinWeiss theorem alluded to above.) This gives an interpretation of relative entropy in terms of repetition and waiting times. We now turn to the analysis of fluctuations of occurrence and repetitions of patterns. In the sequel, U is the interaction defining the Gibbs measure P (see Sect. 3 below). The following two theorems are proved in Subsect. 5.4. Theorem 2.5. Let U be a finite range, translation-invariant interaction, and for β small enough let Pβ be the unique Gibbs measure with interaction βU . There exists β0 > 0 such that for all β < β0 there exists θ = θβ such that log wn − E(log wn ) d

n2

→ N (0, θ 2 ) , as n → ∞, in Pβ × Pβ distribution,

(13)

where N (0, θ 2 ) denotes the normal law with mean zero and variance θ 2 , which is equal to d2 (14) (P ((1 − q)βU )) q=0 . 2 dq Theorem 2.6. Let U be a finite range, translation-invariant interaction, and for β small enough let Pβ be the unique Gibbs measure with interaction βU . There exists β0 > 0 such that for all β < β0 there exists θ = θβ (the same as in the previous theorem) such that log rn − E log rn → N (0, θ 2 ) , a.s. n → ∞, in Pβ distribution . (15) d n2 Remark 2.2. From the proof of the previous theorem it follows that one can replace the measure Pβ × Pβ by the measure Q × Pβ , where Q is any ergodic random field, and s(Pβ ) by s(Q) + s(Q|Pβ ). Remark 2.3. The β0 of Theorems 2.6 and 2.5 determines the analyticity regime of the pressure. This is related to the regime where the high-temperature expansion is convergent. The restriction to finite range interactions is here for convenience only, and can be replaced by the requirement that the norm

U =

U (A, ·) exp (α(diam(A))) A0

is finite for some α > 0, see [27].


275

We end our corollaries with large deviation estimates. In the context of Gibbs measures, it is well-known that the sequence {− n1d log P(C(σCn )) : n ∈ N} satisfies a large deviation principle see e.g., [10, 22]. Here we shall apply the more specific large deviation result of [24] that was already used in [9] to establish large deviations for log rn (in dimension one). The following theorem is proved in Subsect. 5.5. Theorem 2.7. Let P be a translation-invariant Gibbs random field satisfying (3). Then for all q ∈ R the limit 1 qd W(q) = lim d log wn dP×P (16) n→∞ n exists. Moreover, W(q) =

P ((1 − q)U ) + (q − 1)P (U ), P (2U ) − 2P (U ),

for q ≥ −1, for q < −1,

(17)

where P is the pressure defined in (30) below. The following theorem gives the precise consequence of Theorem 2.7 for the large deviations of log wn provided P ((1 − q)U ) is C 1 for all q ≥ −1. For this we can apply the result of [24]. The pressure function is C 1 for example in the Ising model. In the case P ((1 − q)U ) is not differentiable everywhere on [−1, ∞), the result of [24] will give us Large Deviations for u in some bounded interval. Theorem 2.8. Suppose U is a finite range translation-invariant interaction. Then there exists β1 > 0 such that for β ≤ β1 there exists a unique Gibbs measure Pβ with interaction βU , and for all u ≥ 0 we have 1 log wnd ≥ s(P ) + u = inf −(s(Pβ ) + u)q + W(q) , lim d log (Pβ × Pβ ) β d n→∞ n q>−1 n (18) and for all u ∈ (0, u0 ), u0 = | limq↓−1 W (q) − s(Pβ )|, 1 log wnd log (Pβ × Pβ ) ≤ s(Pβ ) − u = inf −(s(Pβ ) − u)q + W(q) lim n→∞ nd q>−1 nd (19) Remark 2.4. A more general version of Theorem 2.7 can be easily deduced by following the same lines as its proof: The measure P × P can be replaced by the measure Q × P, where Q is any Gibbsian random field (without any mixing assumption). Of course formula 17 has to be modified: Now W(q) = P (V − qU ) − P (V ) + qP (V ) for q ≥ −1, where V is the interaction of the Gibbs measure Q. Accordingly, a version of Theorem 2.8 can be obtained under a differentiability condition on W. Remark 2.5. Under the assumption of Theorem 2.7, the sequence ndd log wn satisfies a Large Deviation Principle in the sense of [12] (Theorem 4.5.20 p. 157). The following theorem derives from Theorem 2.2. Since its derivation follows verbatim along the lines of [9], we omit the proof.

276


Theorem 2.9. Suppose U is a finite range, translation-invariant interaction. There exists β1 > 0 be such that for β ≤ β1 there exists a unique Gibbs measure Pβ with interaction βU and there exists u˜ > 0 such that for all u ∈ [0, u) ˜ we have 1 log rnd ≥ s(P ) + u = I(s(Pβ ) + u), (20) lim − d log Pβ β n→∞ n nd and lim −

n→∞

where

1 log Pβ nd

log rnd ≤ s(P ) − u = I(s(Pβ ) − u), β nd

(21)

I(u) = sup (uq − P ((1 − q)U ) − (q − 1)P (U )) . q∈R

Remark 2.6. It follows from the proof of Theorem 2.8 that we have the analogue theorem for repetition times, if we condition the measure Pβ on good patterns, that is, patterns which are not “badly self-repeating”, see Lemma 5.3 below. Remark 2.7. β1 in Theorems 2.8 and 2.9 does not necessarily coincide with the critical inverse temperature βc (below which there is a unique Gibbs measure), and is in general strictly larger than β0 of Theorem 2.6 and 2.5, see [15]. 3. Gibbsian Random Fields and Dobrushin Uniqueness For the sake of convenience the present and next subsections are devoted to the notion of Gibbsian random fields and their mixing properties. More details on this subject can be found in [16, 17]. Definition 3.1. A translation-invariant interaction is a function U : S × → R,

(22)

such that the following conditions are satisfied: 1. U (A, σ ) depends on σ (x), with x ∈ A only. 2. Translation invariance: U (A + x, τ−x σ ) = U (A, σ ) 3. Uniform summability:

∀A ∈ S, x ∈ Zd , σ ∈ .

sup |U (A, σ )| < ∞ .

A0 σ ∈

(23)

(24)

An interaction U is called finite-range if there exists an R > 0 such that U (A, σ ) = 0 for all A ∈ S with diam(A) > R. The set of all such interactions is denoted by U. Mostly we will give examples of Gibbs measures satisfying our mixing conditions with interactions U ∈ U. This can be generalized easily to interactions such that

U α =

U (A, ·) exp (α(diam(A))) A0

is finite for some α > 0.


277

For U ∈ U, ζ ∈ , ∈ S, we define the finite-volume Hamiltonian with boundary condition ζ as ζ U (A, σ ζ c ) . (25) H (σ ) = A∩ =∅

Corresponding to the Hamiltonian in (25) we have the finite-volume Gibbs measures U,ζ P , ∈ S, defined on by ζ U,ζ ζ f (σ ζ c ) e−H (σ ) /Z , (26) f (ξ ) dP (ξ ) = σ ∈ ζ

where f is any continuous function and Z denotes the partition function normalizing U,ζ P to a probability measure. Because of the uniform summability condition, (24) the ζ U,ζ objects H and P are continuous as functions of the boundary condition ζ . ζ For a probability measure P on , we denote by P the conditional probability distribution of σ (x), x ∈ , given σ c = ζ c . Of course, this object is only defined on a set of P-measure one. For ∈ S, ∈ S and ⊆ , we denote by P (σ |ζ ) the conditional probability to find σ inside , given that ζ occurs in \ . For U ∈ U, we call P a Gibbs measure with interaction U if its conditional probabilities coincide with the ones prescribed by (26), i.e., if ζ

U,ζ

P = P

P − a.s.

∈ S, ζ ∈ .

(27)

We denote by G(U ) the set of all translation invariant Gibbs measures with interaction U . For any U ∈ U, G(U ) is a non-empty compact convex set. In this paper we will in fact restrict ourselves to interactions with a unique Gibbs measure. A basic example is the ferromagnetic Ising model, where U ({x, y}, σ ) = −βJ σ (x) σ (y) if |x − y| = 1, U ({x}, σ ) = −hβσ (x). Here β ∈ (0, ∞) represents the inverse temperature, J > 0 the coupling strength, and h the external magnetic field. We turn to the mixing properties of Gibbs random fields. For an interaction U ∈ U, the Dobrushin matrix is given by 1 U,ζ U,ξ γxy (U ) = sup | P{x} (α) − P{x} (α)| : ζ, ξ ∈ , ζZd \{y} = ξZd \{y} , α ∈ A . 2 The matrix γ measures the dependence of changing the spin at site y on the conditional probability at site x. Definition 3.2. The interaction U is said to satisfy the Dobrushin uniqueness condition if γxy (U ) < 1 . (28) sup x∈Zd y∈Zd

The following result is proved in [16], see also [17], Theorem 2.1.3, p. 52. Theorem 3.1. Let U ∈ U be a finite range interaction. Under the condition (28), there is a unique Gibbs measure P ∈ G(U ), and this P is non-uniformly exponentially ϕ-mixing, i.e., it satisfies the mixing property (3).

278


Examples for which (28) is satisfied are: 1. The so-called high-temperature region where U ∈ U is such that sup (|A| − 1) sup |U (A, σ ) − U (A, σ )| < 2. σ,σ ∈

x∈Zd Ax

(29)

Inequality (29) implies the Dobrushin uniqueness condition (28) (see [16], p. 143, Proposition 8.8). In particular, it implies that |G(U )| = 1 (i.e., no phase transition). Note that it is independent of the “single-site part” of the interaction, i.e., of the interactions U ({x}, σ ). For any finite range potential U there exists βc such that βU satisfies (29) for all β < βc . For the Ising model in Z2 , much more is known: the mixing property (3) holds for any β < βc (see e.g. [13]). 2. Low temperature regime for an interaction with unique ground state, e.g., the Ising model in a homogeneous magnetic field and sufficiently large β. See [17] Example (2.1.5) 3. Interactions in a large external field. See [17], Example (2.1.4). For the Ising model in two dimensions this means that the field h should satisfy |h| > 4β + log(8β) . Remark 3.1. The Dobrushin uniqueness condition is not a necessary condition for the mixing property, Definition 2.2. More general versions, known as “Dobrushin-Shlosman” conditions exist, see e.g., [21] for more details on general finite size conditions ensuring NUEM. We now recall some basic facts on entropy and relative entropy (or Kullback-Leibler information). We use the following shorthand to ease notation : = . Cn

C (An ):An ∈Cn

The entropy s(P) of P is defined as s(P) = lim − n→∞

1 P(Cn ) log P(Cn ) . nd Cn

The relative entropy s(Q|P) of a stationary random field Q with respect to a Gibbsian random field P is 1 Q(Cn ) s(Q|P) = lim d Q(Cn ) log . n→∞ n P(Cn ) Cn

In terms of the interaction U of P the relative entropy is s(Q|P) = P (U ) + fU dQ − s(Q), where fU (σ ) =

U (A, σ ) A0

|A|


279

and P (U ) is the pressure of U , which is defined as follows 1 log ZCn , n→∞ nd

P (U ) = lim where

ZCn =

exp(−

σCn ∈Cn

(30)

U (A, σ ))

A⊆Cn

is the partition function with the free boundary conditions. Proposition 3.1. Let P be a Gibbs random field and Q be an ergodic random field. Then lim

n→∞

1 Q(C(σCn )) log = s(Q|P) nd P(C(σCn ))

for Q-almost every σ . Proof. The proof is simple, but since we did not find it in the literature, we give it here for the sake of completeness. Write gn (σ ) ∼ hn (σ ) if lim

n→∞

1 sup |gn (σ ) − hn (σ )| = 0 . nd σ

Let U be the potential of the Gibbsian field P. Then we have τi fU (σ ) − log ZCn . log P(C(σCn )) ∼ − i∈Cn

Therefore, by ergodicity of Q, 1 log P(C(σCn )) nd converges Q-a.s. to

−

fU dQ − P (U ) .

By the Shannon-Mc Millan-Breiman theorem [20, 28] 1 log Q(C(σCn )) nd converges Q-a.s. to −s(Q). Hence the difference 1 (log Q(C(σCn )) − log P(C(σCn ))) nd converges Q-a.s. to P (U ) − s(Q) −

fU dQ

which is equal to s(Q|P) by the Gibbs variational principle, see [16].

280


A standard property of Gibbs measures which we will use often is the following: there exist positive constants C, c, C , c such that d

C e−c n ≤ P(Cn ) ≤ C e−cn

d

(31)

for every cylinder Cn supported on Cn . 4. Proof of Theorem 2.1 To ease notation, we will write P(A) instead of P(C(A)), where A = An is an n-pattern. 4.1. Preliminary results. In this section we prove Theorem 2.1. We follow the approach of [1]. For V ∈ S, σ ∈ and A = An an n-pattern we say that “A is present in V ”, and write A ≺ V , for the configuration σ if there exists x = (x1 , x2 , ..., xd ) ∈ Zd such that W := x + Cn ⊆ V and (τx σ )W = A. By abusing notation, we will write P(A ≺ V ) for the probability of that event. Lemma 4.1. Let V be a finite subset of Zd , and let A = An be a n-pattern. Then P(A ≺ V ) ≤ |V | P(A). Proof. P(A ≺ V ) ≤ P σ : σ |x+Cn = A = P(A) = |V | P(A). x∈V

x∈V

For every k ∈ N define NkA (σ ) =

?{τx (σ )Cn = A}.

x∈Zd

0≤xi ≤k

Then the following events coincide: {tA ≤ k} = {NkA ≥ 1}.

(32)

Moreover, ENkA = (k + 1)d P(A). Lemma 4.2 (Second moment estimate). Consider a non-uniformly exponentially ϕmixing Gibbsian random field. Then there exists δ > 0 such that for every n, k ∈ N, and every > 2n one has E(NkA )2 ≤ (k + 1)d P(A) 1 + e−δn d + (k + 1)d P(A) + (k + 1)d nd ϕ( − 2n) . Proof. Define C(x, n) = x + Cn . We have to estimate the following expression: E(NkA )2 = P(σC(x,n) = σC(y,n) = A). (33) x∈Zd y∈Zd 0≤xi ≤k 0≤yi ≤k


281

We split the above double sum into the three following sums I1 = , I2 = , I3 = . x=y |x−y|≤

x=y

x=y |x−y|>

Let us proceed with each of the sums separately. For I1 one obviously has I1 = P(A) = (k + 1)d P(A). x∈Zd 0≤xi ≤k

To estimate I2 we use that for any Gibbsian random field there exists a constant δ > 0 such that for any finite volume V, and any configuration σ and η, the conditional probability of observing σ on V , given η outside of V , can be estimated as follows [16]: P(σV |ηV c ) ≤ exp(−δ|V |). Therefore I2 = P(σC(x,n) = σC(y,n) = A) = x=y |x−y|≤

≤

P σC(x,n) = A σC(y,n) = A P(A)

x=y |x−y|≤

P(A) exp −δ |C(x, n) \ C(y, n)|) .

x=y |x−y|≤

To complete the estimate, it is sufficient to observe that since x = y, the volume of the set C(x, n) \ C(y, n) is at least n. Hence I2 ≤ (k + 1)d d exp(−δn) P(A). Finally, using the mixing condition (3), for I3 we obtain I3 = P(A) + nd ϕ( − 2n) P(A) P(σC(x,n) = σC(y,n) = A) ≤ x=y |x−y|>

x=y |x−y|>

≤ (k + 1)2d P(A) P(A) + nd ϕ( − 2n) .

Combining all the estimates together we obtain the statement of the lemma.

Lemma 4.3 (The parameter). There exist strictly positive constants 1 , 2 such that for any integer t with tP(A) ≤ 1/2, one has 1 ≤ λA,t := −

log P(tA > t 1/d ) ≤ 2 . tP(A)

Proof. Taking into account (32) and the Cauchy-Schwartz inequality we obtain P(tA ≤ k) ≥

(ENkA )2 E(NkA )2

.

(34)

282


We apply the basic inequalities κ ≤ 1 − e−κ ≤ κ , 2

(35)

where the left inequality is valid for all κ ∈ [0, 1], and the right inequality is true for κ ≥ 0. Let now κ = − log P(tA > t 1/d ). Then, using Lemma 4.2 and (34), we conclude P(tA ≤ t 1/d ) − log P(tA > t 1/d ) ≥ tP(A) tP(A) 1 1 + e−δn d + (t + 1)P(A) + (t + 1)nd ϕ( − 2n) 1 ≥ =: 1 , 1 + c1 + 3/2 + c2

≥

where we have chosen = nd+1 , c1 =

e−δn nd(d+1) < ∞, and c2 = sup (t + 1) nd ϕ( − 2n) .

n∈N

n∈N

We have to show that c2 is finite. Indeed, since for a Gibbs random field P there exist c , C > 0 such that P(A) ≥ C exp(−c nd ) for every n-pattern A; t has been chosen such that tP(A) < 1/2, we have c2 ≤ sup

n∈N

1 d d d+1 exp(c n ) + 1 n C exp −C (n − 2n) < ∞, 1 2 2C

where we have used the mixing condition (3). For the upper bound, we use (35) again, but first we have to check that κ = −log P(tA > t 1/d ) ∈ [0, 1]. Indeed, since t < (2P(A))−1 , by Lemma 4.1 we have 1 P(tA > t 1/d ) ≥ P tA > (2P(A))1/d P(A) 1 1 = 1 − P tA ≤ ≥1− = . 1/d (2P(A)) 2P(A) 2 Hence, κ ≤ log(2) < 1, therefore κ ≤ 2(1 − e−κ ), which means −log P(tA > t 1/d ) ≤ 2P(tA ≤ t 1/d ) ≤ 2t P(A) ≤ 1, where we have used Lemma 4.1 for the second inequality. Hence, we can choose 2 = 2. This finishes the proof.


283

For positive numbers xAn , yAn depending on the n-pattern An we write xAn ∼ yAn if lim

n→∞

xAn = 1. yAn

For a positive integer tA we set C(tA ) = [0, tA ]d ∩ Zd . For a subset V ⊆ Zd let A ⊀ V be the event that the n-pattern A cannot be found in V . (See above for the definition of A ≺ V .) The following lemma is crucial and gives the factorization property of the exponential distribution, i.e., the fact that asymptotically P(tAn > (t + s)/P(An )) ∼ P(tAn > t/P(An ))P(tAn > s/P(An )), where the accuracy of the approximation marked ∼ is spelled out in detail. The idea is that the event of non-occurrence of the pattern in a cube of size O(1/P(An )) can be viewed as the non-occurrence of the pattern in many sub-cubes of volume kn , where nd 0. Now, recall from Lemma 4.3 that λA = −

log P(tA > (fA )1/d ) ∈ [ 1 , 2 ] fA P(A)

for some positive constants 1 , 2 . We also define log P(tA > (fA )1/d ) + P(A) ˜λA = − . fA P(A) It is not difficult to see that λ˜ A ∈ [ 1 /2, 2 ], for n large enough. Since tA ≤ t ≤ tA , one obviously has P(tA > t 1/d ) − exp(−λA P(A) t) ≥ P(tA > (tA )1/d ) − exp(−λA P(A) tA ), and P(tA > t 1/d ) − exp(−λA P(A)t) ≤ P(tA > (tA )1/d ) − exp(−λA P(A) tA ).

(39)

286


Now, |P(tA > (tA )1/d ) − exp(−λA P(A)tA )| ≤ |P(tA > (tA )1/d ) − P(tA > (fA )1/d )k | +|P(tA > (fA )1/d )k − exp(−λA P(A) tA )| +| exp(−λA P(A) tA ) − exp(−λA P(A) tA )|. By Lemma 4.4, |P(tA > (tA )1/d ) − P(tA > (fA )1/d )k | k ≤ P(A)γ (1−δ)/d k P(tA > (fA )1/d ) + P(A)1−γ . = P(A)γ (1−δ)/d t P(A) exp(−λ˜ A P(A) tA ) ≤ P(A)γ (1−δ)/d t P(A) exp(−C1 P(A) t). By the choice of λA (39) , and since tA = kfA , P(tA > (fA )1/d )k = exp(−λA kfA P(A)) = exp(−λA tA P(A)). Finally, | exp(−λA P(A) tA ) − exp(−λA P(A) tA )| ≤ λA P(A)(tA − tA ) exp(−λA P(A) tA ) ≤ 2 P(A)fA exp(− 1 P(A) tA ) ≤ C2 P(A)1−γ exp(−C3 P(A) t). The lower estimate is obtained in a similar way. This finishes the proof. Remark 4.1. Notice that in the iteration lemma it is not used that An is a pattern. Therefore, this lemma can be generalized to arbitrary measurable events En ∈ FCkn , where kn ≪ 1/P(En ). The second moment estimate however uses that An is a pattern. Therefore Theorem 2.1 can be generalized as follows. Let En ∈ FCkn , where |Ckn | = O(nα ), and P(En ) = O(e−cn ). Suppose furthermore that d

lim sup n→∞

0n/2

≤ (n + )d

n

η

n

n

n

sup exp(−c |Cn + x\Cn |)

|x|>n/2

≤ (n + )d exp(−c nd ), where c, c are positive constants. We now use the mixing property (2) to get, for any good pattern A : P A ⊀ C(tA )\Cn+ A ≺ Cn − P (A ⊀ C(tA )\Cn+ ) ≤ |C(tA )| ϕ() . Putting together the above estimates, with the choice = nd+1 and using (3), yields P A ⊀ C(tA )\Cn A ≺ Cn − P (A ⊀ C(tA )) ≤ (n + nd+1 )d e−c This gives the desired result.

nd

+ C1 ec"n e−C2 n d

d+1

+ (n + nd+1 )d P(A) .

We also need the following lemma. Lemma 5.2 (Iteration Lemma for pattern repetitions). Let tA be such that tAd ∼ P(A)−ϑ , where ϑ ∈ (0, 1). For i = 2, . . . k, let Ci (tA ) denote any collection of k disjoint cubes of the form xi + C(tA ). Assume also that C1 (tA ) = x1 + C(tA )\{0} is disjoint from Ci (tA ), i = 2, . . . , k. Then we have the following inequality for all k: k Ci (tA ) | A ≺ Cn − P (A ⊀ C(tA ))k P A ⊀ i=1

≤ C1 exp{−C2 nd } (P (A ⊀ C(tA )) + C1 exp{−C2 n})k .

288


Proof. Proceeding as in the proof of Lemma 4.4 we have: k Ci (tA )) P (A ≺ Cn ) ∩ (A ⊀ i=1

− P((A ≺ Cn ) ∩ A ⊀ C(tA )\Cn ) P (A ⊀ C(tA ))k−1 k ≤ P(A)1−ϑ P (A ⊀ C(tA )) + P(A)1−ϑ . On the other hand, Lemma 5.1 tells us that d P(A ⊀ C(tA )\Cn |A ≺ Cn ) − P(A ⊀ C(tA )) ≤ b1 e−b2 n . The proof of (7) in Theorem 2.2 is now the same as that of Theorem 2.1. It remains to prove (6): Lemma 5.3 (Probability of badly self-repeating patterns). There exist c, C > 0 such that P(Bn ) ≤ Be−bn . d

(42)

Proof. Put Cn+ = Cn ∩ (Cn + x) and Cn+ = Cn ∩ (Cn − x) . By definition of Bn , we have the inequality: P(Bn ) ≤ P ∃x : |x| ≤ n/2 : σCn+ (x) = σCn− (x) .

(43)

Define the event Ex = {σ : σCn+ (x) = σCn− (x) }. If σ ∈ Ex , then there exist disjoint sets Sn+ (x) and Sn− (x) such that σSn+ (x) = σSn− (x) and |Sn+ (x)|, |Sn− (x)| > δnd for some positive δ. Therefore, we have P(Ex ) ≤ P σSn+ (x) = σSn− (x) ≤ sup P σSn+ (x) = η|σ(Sn+ (x))c = ξ : η ∈ Sn+ (x) , ξ ∈ (Sn+ (x))c ≤ exp(−c nd ),

(44)

where in the last inequality we used the Gibbs property (31). Finally, P(Bn ) ≤

x:|x| 0, d

P{σ : log(rn (σ )d P(C(σCn ))) ≥ log t} ≤ (C e−c n + e− 1 t ) + Ce−cn . d

Take t = tn = log(n ), > −1 1 , to get d

P{σ : log(rn (σ )d P(C(σCn ))) ≥ log log(n )} ≤ C e−c n + An application of the Borel-Cantelli lemma leads to

log (rn (σ ))d P(C(σCn )) ≤ log log(n )

1 d + Ce−cn . n 1

eventually a.s.

For the lower bound first observe that Theorem 2.2 gives, for all t > 0, d

P{σ : log(rn (σ )d P(C(σCn ))) ≤ log t} ≤ C e−c n + (1 − exp(− 2 t)) + Ce−cn . d

Choose t = tn = n− , > 1, to get, proceeding as before,

log (rn (σ ))d P(C(σCn )) ≥ − log n eventually a.s. Finally, let 0 = max( −1 1 , 1). 5.3. Proof of Theorem 2.4. We first show that the strong approximation formula (8) holds with wn in place of rn with respect to the measure Q × P. We have the following identity:

1/d t dQ(ξ ) P σ : tξCn (σ ) > P(C(ξCn ))

1/d t = (Q × P) (ξ, σ ) : wn (ξ, σ ) > . P(C(ξCn )) This shows immediately that Theorem 2.1 is valid with wn (ξ, σ ) in place of tξCn (σ ) and Q × P in place of P, hence so is Theorem 2.3. Therefore for large enough, we obtain

− log n ≤ log (wn (ξ, σ ))d P(C(ξCn )) ≤ log log n (46) for Q × P-eventually almost every (ξ, σ ). Write

Q(C(ξCn )) log (wn (ξ, σ ))d P(C(σCn )) = d log wn (ξ, σ ) + log Q(C(ξCn )) − log P(C(ξCn )) and use (46). After division by nd , we obtain (12) since limn→∞ n1d log Q(C(σCn )) = −s(Q), Q-a.s. by the Shannon-Mc Millan-Breiman theorem and limn→∞ n1d Q(C (ξ

))

log P(C (ξCCn )) = s(Q|P), Q-a.s. (Prop. 3.1 in Sect. 3). n

290


5.4. Proof of Theorem 2.6 and Theorem 2.5. We use the strong approximation formula (8) from Theorem 2.3 to get d log rn (σ ) + log Pβ (C(σCn )) d

n2

→0

when n → ∞, for Pβ − almost all σ .

(47)

Therefore, it suffices to see that in the high-temperature regime we have a central limit theorem for {− n1d log Pβ (C(σCn ))}. By a standard argument presented below (51), one has 1 lim log Pβ (C(ξCn ))−q dP(ξ ) = P ((1 − q)βU ) + (q − 1)P (βU ) , (48) n→∞ nd for all q ∈ [0, ∞). There exists β1 > 0 such that for |z| ≤ β1 the maps z → P (zU ) and 1 : z → lim d log Pβ (C(ξCn ))−z dP(ξ ) n→∞ n are analytic see e.g. [27], and [13]. Therefore, if |(q − 1)|β ≤ β1 , the map q → P ((1 − q)βU ) + (q − 1)P (βU ) is analytic, and equality holds for all q ∈ C. By Bryc’s theorem [6], this implies the CLT for {− n1d log Pβ (C(σCn ))} with variance θ 2 given by θ2 =

d2 (P ((1 − q)βU )) q=0 . 2 dq

(49)

The proof of Theorem 2.5 is the same once we observe that d log wn (σ ) + log Pβ (C(ξCn ))

→0 d n2 for Pβ × Pβ − almost all (ξ, σ )

when n → ∞, (50)

by using (46).

5.5. Proof of Theorem 2.7. Recall that for any Gibbs measure − log P(σCn ) ∼ τi fU (σ ) + log ZCn , i∈Cn

and hence we have the identity 1 log P(Cn )1−q = P ((1 − q)U ) − (1 − q)P (U ). n→∞ nd lim

(51)

Cn

In the sequel, we are going to show that qd wn dP × P ≈ P(Cn )1−q , Cn

(52)


for q > −1, and

qd

wn dP × P ≈

291

P(Cn )2 ,

(53)

Cn

for q ≤ −1. Here an ≈ bn means that max{an /bn , bn /an } is bounded from above. Clearly (52) and (53) imply (17). Let q > 0. Then qd qd wn dP × P = P(Cn ) tCn (σ )dP(σ ) (54) Cn

=q

P(Cn )1−q

Cn

∞

P(Cn )

t q−1 P tCd n ≥

t dt. P(Cn )

(55)

By Theorem 2.1, there exist positive constants A, B such that for any t > 0 one has t P tCd n ≥ ≤ Ae−Bt . P(Cn ) Theorem 2.1 also easily gives the lower bound : ∞ t dt ≥ K − C exp(−cn) K , t q−1 P tCd n > P(C n P(Cn ) ∞ q−1 − t ∞ 2 where 0 < K := 1 t e dt < ∞ and 0 < K := 0 t q−1 e− 1 t dt < ∞. For n large enough, K − C exp(−cn)K is strictly positive. Therefore we obtain qd 1−q K1 P(Cn ) ≤ wn dP × P ≤ K2 P(Cn )1−q , Cn

Cn

where K1 := q (K − C exp(−cn0 ) K ) ,

∞

K2 := qA

t q−1 e−Bt dt.

0

This establishes (51) for q ≥ 0. The case q = 0 is trivial. Let now q ∈ (−1, 0). −|q|d −|q|d wn dP × P = P(Cn ) tCn (σ ) dP(σ ) Cn

=

Cn

= |q|

P(Cn )

1

0

Cn

−|q|d P tCn ≥ t dt

P(Cn )

(56)

1+|q|

∞

P(Cn )

t

−|q|−1

(57) t dt. P tCn ≤ P(Cn )

d

(58)

The last integral is bounded from above by the integral where P(Cn ) is replaced by 1 in the integration domain. From Theorem 2.1, we get the following lower bound, for every t > 0: t P tCd n ≤ ≥ 1 − e− 1 t − C P(Cn )ρ e−C t . P(Cn )

292


The number K1 := |q|

∞

t −|q|−1 1 − e− 1 t − C P(Cn )ρ e−C t dt

1

is finite and strictly positive for n large enough. Now, putting 0 instead of P(Cn ) gives an upper bound to the integral upon condideration. We use Lemma 4.3 to get immediately t ≤ 1 − e− 2 t , P tCd n ≤ P(Cn ) provided that t ≤ 21 . We have ∞ t −|q|−1 P tCd n ≤ 0

≤

1 2

t

−|q|−1

0

≤

t dt P(Cn )

∞ t P tCn ≤ dt + t −|q|−1 dt 1 P(Cn ) 2 d

2 21−|q| 2−|q| + =: K2 < ∞ . 1 − |q| |q|

Hence, we conclude that for n large enough, −|q|d K1 P(Cn )1+|q| ≤ wn dP × P ≤ |q| K2 P(Cn )1+|q| . Cn

Cn

Therefore we obtain (52) for q ∈ (−1, 0). Finally, let us consider the remaining case q ≤ −1. Then for sufficiently large n (such that P(Cn ) < 1/2) one has ∞ t −|q|d dt dP × P = |q| P(Cn )1+|q| t −|q|−1 P tCd n ≤ wn P(Cn ) P(Cn ) Cn

= |q|

P(Cn )1+|q|

Cn

= |q|

1 2

P(Cn )

+

∞

1 2

t −|q|−1 P tCd n ≤

t dt P(Cn )

P(Cn )1+|q| [ I1 (n, Cn ) + I2 (n, Cn ) ] .

Cn

Clearly the second integral I2 (n, Cn ) is uniformly bounded in n. Indeed, ∞ 1 I2 (n, Cn ) ≤ dt < +∞. 1+|q| 1 t 2 However, the first integral I1 (n, Cn ) is diverging in the limit n → ∞. Therefore the limiting behavior as n → ∞ is determined by |q|

Cn

P(Cn )

1+|q|

I1 (n, Cn ) = |q|

Cn

P(Cn )

1+|q|

1 2

P(Cn )

t

−1−|q|

t P tCn ≤ P(Cn ) d

dt.


293

We again use Lemma 4.3 to get 1−e

− 1 t

t ≤ P tCn ≤ P(Cn ) d

≤ 1 − e− 2 t ,

provided that t ≤ 21 . Hence, using the Gibbs property (31), we have I1 (n, Cn ) ≤ 2

1 2

P(Cn )

t −|q| dt ≤

d

2 (1 − 2|q|−1 C e−c n ) P(Cn )−|q|+1 , |q| − 1

where we used the fact that for all κ ∈ R, 1 − e−κ ≤ κ. Notice that for n large enough, the term between parentheses is strictly positive. Now, using the fact that 1 − e−κ ≥ κ/2 for any κ ∈ [0, 1], and remembering that 1 /2 ≤ 1 (1 ), and using again the Gibbs property (31), we obtain 1 (1 − 2|q|−1 Ce−cn ) P(Cn )−|q|+1 , 2(|q| − 1) d

I1 (n, Cn ) ≥

where the term between parentheses is strictly positive provided that n is sufficiently large. Therefore, for n large enough, we end up with |q| 1 (1−2|q|−1 Ce−cn ) P(Cn )2 ≤ 2(|q| − 1) d

−|q|d

wn

dP × P

Cn

d

2|q| 2 (1−2|q|−1 C e−c n ) ≤ P(Cn )2 . |q| − 1 Cn

(Notice that L’Hôpital’s rule shows that there is no problem at q = −1.) Thus, we obtain (53), which finishes the proof. References 1. Abadi, M.: Exponential approximation for hitting times in mixing processes. Math. Phys. Electron. J. 7 (2001) 2. Abadi, M., Galves, A.: Inequalities for the occurrence of rare events in mixing processes. The state of the art. In: ‘Inhomogeneous random systems’ (Cergy-Pontoise, 2000), Markov Process. Related Fields 7(1), 97–112 (2001) 3. Asselah, A., Dai Pra, P.: Sharp estimates for the occurrence time of rare events for symmetric simple exclusion. Stochastic Process. Appl. 71(2), 259–273 (1997) 4. Asselah, A., Dai Pra, P.: Occurrence of rare events in ergodic interacting spin systems. Ann. Inst. H. Poincaré Probab. Statist. 33(6), 727–751 (1997) 5. Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Math. 470, Berlin-Heidelberg-New York: Springer, 1975 6. Bryc, W.: A remark on the connection between the large deviation principle and the central limit theorem. Statist. & Probab. Lett. 18, 253–256 (1993) 7. Chazottes, J.-R.: Dimensions and waiting time for Gibbs measures. J. Stat. Phys. 98(3/4), 305–320 (2000) 8. Chi, Z.: The first-order asymptotic of waiting times with distortion between stationary processes. IEEE Trans. Inform. Theory 47(1), 338–347 (2001) 9. Collet, P., Galves, A., Schmitt, B.: Fluctuations of repetition times for gibbsian sources. Nonlinearity 12, 1225–1237 (1999) 1

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294


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Communications in


Locally Conformal Dirac Structures and Infinitesimal Automorphisms A¨ıssa Wade Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA. E-mail: [email protected] Received: 18 February 2003 / Accepted: 30 October 2003 Published online: 19 February 2004 – © Springer-Verlag 2004

Abstract: We study the main properties of locally conformal Dirac bundles, which include Dirac structures on a manifold and locally conformal symplectic manifolds. It is proven that certain locally conformal Dirac bundles induce Jacobi structures on quotient manifolds. Furthermore we show that, given a locally conformal Dirac bundle over a smooth manifold M, there is a Lie homomorphism between a subalgebra of the Lie algebra of infinitesimal automorphisms and the Lie algebra of admissible functions. We also show that Dirac manifolds can be obtained from locally conformal Dirac bundles by using an appropriate covering map. Finally, we extend locally conformal Dirac bundles to the context of Lie algebroids.

1. Introduction Dirac structures generalize foliations, Poisson structures, and pre-symplectic 2-forms. They were originally studied by Courant and Weinstein (see [C] and [CW]), who showed their connection with constraint Hamiltonian systems. On the other hand, locally conformal symplectic manifolds play an important role in the theory of Jacobi manifolds and in physics. Indeed, locally conformal symplectic manifolds may be considered as phase spaces of Hamiltonian systems (see [V]). They occur, for instance, in the study of Gaussian isokinetic dynamics (see [WL]). Motivated by these facts, we introduce the notion of a locally conformal Dirac bundle which extends that of a Dirac structure on a manifold. The purpose of this paper is to investigate the basic properties of locally conformal Dirac bundles. A locally conformal Dirac bundle over M is a sub-bundle of the vector bundle T M ⊕ T ∗ M → M which is maximally isotropic with respect to the canonical symmetric bilinear operation on T M ⊕ T ∗ M and such that there exist an open cover U = (Ui )i∈I of M and smooth positive functions fi : Ui → R for which the sub-bundle Li = {(XUi , fi α|Ui ) | (X, α) ∈ L} ⊂ T Ui ⊕ T ∗ Ui is integrable in the sense that the space

296

A. Wade

(Li ) of smooth sections of Li is closed under the Courant bracket for all i ∈ I (see Sect. 2 for the definition of the Courant bracket). To any locally conformal Dirac bundle L ⊂ T M ⊕ T ∗ M, there corresponds a skewsymmetric bilinear map L on L which induces a 2-form on the distribution ρ((L)) = {X ∈ χ(M) | (X, α) ∈ (L)}: L (X, Y ) = α(Y ),

for any (X, α), (Y, β) ∈ (L).

Any locally conformal Dirac bundle L over M induces a singular foliation whose leaves are locally conformal pre-symplectic manifolds. Precisely, the foliation is determined by the distribution ρ((L)). If L is a locally conformal Dirac bundle L such that the rank of L is greater than 2 at every point x ∈ M then there exists a unique 1-cocycle ωL satisfying dL = −ωL ∧ L , where d denotes the derivative along the foliation defined by ρ((L)). The 1-cocycle ωL is called the Lee 1-form of L. There are two fundamental Lie algebras associated with a locally conformal Dirac bundle L whose 2-form L has a rank r(x) > 2 at every point: (i) The Lie algebra of L-admissible functions. (ii) The Lie algebra A(L) of infinitesimal automorphisms. Given a locally conformal Dirac bundle L with Lee 1-form ωL , we show that if there exists a vector field Z such that (Z, ωL ) ∈ (L), then there is a Lie homomorphism between a subalgebra of A(L) and the Lie algebra of L-admissible functions. We then recover results given in [B1, GL and V]. Theorem 5.5 shows that certain locally conformal Dirac bundles induce Jacobi structures on quotient manifolds. Two locally conformal Dirac bundles L1 , L2 ⊂ T M ⊕ T ∗ M are conformally equivalent if ρ((L1 )) = ρ((L2 )) and there exists a nowhere vanishing function f such that L 1 = f L 2 , where ρ : (Li ) → χ (M) is the canonical projection for i = 1, 2. An equivalence class of such locally conformal Dirac bundles is called a locally conformal Dirac structure on M. We show that, for a large family of locally conformal Dirac structures D, the extended Lee homomorphism does not depend on the locally conformal Dirac bundle representing D. One can extend the concept of a locally conformal Dirac bundle to more general Lie algebroids. This idea was already suggested in [Wa], where we study conformal E 1 (M)Dirac structures in connection with Jacobi manifolds. The last section of this paper is devoted to the generalization of locally conformal Dirac bundles to the context of Lie algebroids. The paper is organized as follows. In Sect. 2, we recall some known results concerning Dirac structures on manifolds. We give, in Sect. 3, the characterizations of locally conformal Dirac bundles over a smooth manifold M (see Theorem 3.5). We present some examples of locally conformal Dirac bundles in Sect. 4. In Sect. 5, we introduce the concept of an admissible function relative to a given locally conformal Dirac bundle L. Then, we show that the space of admissible functions is a Lie algebra. Moreover, we prove that a reducible locally conformal Dirac bundle induces a Jacobi bracket on a quotient manifold. In Sect. 6, we develop the properties of the infinitesimal automorphisms of a locally conformal Dirac bundle L. We also establish Theorems 6.5 and 6.6. Finally, in Sect. 7, we consider the generalized locally conformal Dirac structures.

Locally Conformal Dirac Structures and Infinitesimal Automorphisms

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2. Dirac Structures Let M be a smooth manifold. We denote by ., .+ the canonical symmetric bilinear operation on the vector bundle T M ⊕ T ∗ M → M. This induces a symmetric C ∞ -bilinear operation on the space of sections of T M ⊕ T ∗ M given by: (X1 , α1 ), (X2 , α2 )+ =

1 (iX α1 + iX1 α2 ), 2 2

for any (X1 , α1 ), (X2 , α2 ) ∈ (T M ⊕ T ∗ M). Definition 2.1 ([C]). An almost Dirac structure on M is a vector sub-bundle of T M ⊕ T ∗ M → M which is maximally isotropic with respect to ., .+ . Every almost Dirac structure L on M induces a skew-symmetric C ∞ -bilinear map L : ρ((L)) × ρ((L)) → C ∞ (M), where ρ is the canonical projection of (L) onto the Lie algebra of vector fields χ (M). Precisely, L is given by L (X, Y ) = α(Y ), for any (X, α), (Y, β) ∈ (L).

(1)

In fact L is well-defined since L is isotropic. Consider the Courant bracket on (T M ⊕ T ∗ M), that is, the skew-symmetric R-bilinear operation defined by: 1 [(X1 , α1 ), (X2 , α2 )]C = ([X1 , X2 ], LX1 α2 − LX2 α1 + d(iX2 α1 − iX1 α2 )), 2 where LX = d ◦ iX + iX ◦ d is the Lie derivation by X. This bracket does not satisfy the Jacobi identity. Definition 2.2 ([C]). A Dirac structure L on M is an almost Dirac structure which is integrable in the sense that the space of all sections of L is closed under the Courant bracket. In this case, the pair (M, L) is called a Dirac manifold. Basic examples of Dirac manifolds are foliations, pre-symplectic and Poisson manifolds. Now, we review some properties of Dirac manifolds that will be used later. Proposition 2.3 ([C]). Let L be an almost Dirac structure on M. If L is integrable then dL = 0, where L is the skew-symmetric C ∞ -bilinear map defined as in (1). This proposition is an immediate consequence of the following lemma. Lemma 2.4 ([C]). Let L be a Dirac structure on M. For any e1 = (X, α), e2 = (Y, β), e3 = (Z, γ ) in (L), we have 2[e1 , e2 ]C , e3 + = dL (X, Y, Z). The proof of 2.4 is straightforward. It can be found in [C]. Remark 1. This lemma is still valid if one weakens the hypothesis of Lemma 2.4 by assuming that L is an almost Dirac structure on M such that the canonical projection of (L) onto χ (M) defines an involutive distribution. Corollary 2.5 ([C]). A Dirac structure on M gives rise to a singular foliation by presymplectic leaves. Proof. Clearly, if L is a Dirac structure on M then the canonical projection of (L) onto χ (M) defines an involutive distribution. In other words, we have a singular foliation in the sense of Stefan and Sussmann. Moreover, L is closed on each leaf by Proposition 2.3. Hence, each leaf is a pre-symplectic manifold.

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3. Locally Conformal Dirac Bundles Definition 3.1. An almost Dirac structure L on M is a locally conformal Dirac bundle over M if there exist an open cover (Ui )i∈I of M and smooth positive functions fi : Ui → R such that Li = {(X|Ui , fi α|Ui ) | (X, α) ∈ L} is a Dirac structure on Ui . We say that L is a globally conformal Dirac bundle if there exists a smooth positive function f : M → R such that L = {(X, f α) | (X, α) ∈ L} is a Dirac structure on M. Remark 2. a. An immediate consequence of Definition 3.1 is that if L is a locally conformal Dirac bundle then the canonical projection of (L) onto χ (M) defines an involutive distribution. Thus, a locally conformal Dirac bundle over M induces a singular foliation. However, the leaves are not, in general, pre-symplectic manifolds. b. If L is an almost Dirac structure whose 2-form L vanishes then (X, f α) ∈ L, for any (X, α) ∈ L and for any f ∈ C ∞ (M). In other words, one obtains again L by scaling the second component. It follows from Remark 1 that L is a Dirac structure if and only if the canonical projection of (L) onto χ (M) defines an involutive distribution. This kind of Dirac structure is called a null Dirac structure. We will not consider null Dirac structures here. Proposition 3.2. Suppose that L together with the open cover U = (Ui )i∈I of M and the smooth positive functions fi : Ui → R defines a locally conformal Dirac bundle over M. Denote by ρ the canonical projection of (L) onto χ (M) and L the skewsymmetric 2-form on the C ∞ (M)-module ρ((L)) defined as in (1). If the rank of L is greater than 2 at every point x ∈ M then there exists a unique C ∞ -linear map ωL : ρ((L)) → C ∞ (M) such that dL = −ωL ∧ L and dωL = 0. The proof of this proposition is based on the following lemma: Lemma 3.3. Consider an almost Dirac structure L on M. Assume the rank of the corresponding 2-form L is greater than 2 at every point. If ξ : ρ((L)) → C ∞ (M) is a C ∞ -linear map such that (ξ ∧ L )(X, Y, Z) = 0,

∀X, Y, Z ∈ ρ((L)),

then ξ = 0. Proof. Pick a point x in M. If ξ does not vanish at x, then we can choose u1 , · · · , uk such that {ξ(x), u1 , · · · , uk } is a basis for the dual of the vector space ρ(Lx ) ⊂ Tx M. Thus, we have L (x) = ξ(x) ∧

n i=1

di ui +

i<j

cij ui ∧ uj = ξ(x) ∧ θ +

cij ui ∧ uj .

i<j

Using the fact that ξ ∧L = 0, we get cij = 0, for all i, j . This implies L (x) = ξ(x)∧θ, which is not possible since the rank of L (x) is greater than 2. Therefore, ξ = 0 on M.


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Proof of Proposition 3.2. Existence of ωL . Set ωL = (d ln fi )|ρ((L)) on each Ui . Here (d ln fi )|ρ((L)) denotes the restriction of d ln fi to the leaves of the foliation associated with L. We have to show that d(ln fi − ln fj )|ρ((L)) = 0 on Ui ∩ Uj when this intersection is not empty. By definition Li = {(X|Ui , fi α|Ui ) | (X, α) ∈ L} is a Dirac structure on Ui . Therefore Proposition 2.3 ensures that dLi = 0. But, Li = fi L on Ui . This gives 0 = d(Li ) = fi (d ln fi )|ρ((L)) ∧ L + dL on Ui , ∀i ∈ I. It follows (d ln fi )|ρ((L)) ∧ L + dL = 0 on Ui , ∀i ∈ I. Consequently, for two different open sets Ui and Uj such that Ui ∩ Uj is not empty, one has (d ln fi − d ln fj )|ρ((L)) ∧ L = 0, on Ui ∩ Uj . Using Lemma 3.3, we obtain (d ln fi − d ln fj )|ρ((L)) = 0 on Ui ∩ Uj . This shows that ωL is well-defined. Uniqueness of ωL . Assume that dL = −ω1 ∧ L = −ω2 ∧ L . Then (ω1 − ω2 ) ∧ L = 0. By Lemma 3.3, we obtain ω1 = ω2 at every point x ∈ M.

Definition 3.4. If L is a locally conformal Dirac bundle over M such that the rank of L is greater than 2 at every point x ∈ M then its corresponding 1-form ωL : ρ((L)) → C ∞ (M) is called the Lee 1-form of L. Let L be an almost Dirac structure on M and let ω : ρ((L)) → C ∞ (M) be a map. On (L), we define:

C ∞ -linear

1 ω [(X1 , α1 ), (X2 , α2 )]C = ([X1 , X2 ], LX1 α2 − LX2 α1 + d(iX2 α1 − iX1 α2 ) 2 1 +(iX1 ω)α2 − (iX2 ω)α1 + (iX2 α1 − iX1 α2 )ω). 2 Define the 3-tensor ω

T (e1 , e2 , e3 ) = 2[e1 , e2 ]C , e3 + ,

e1 , e2 , e3 ∈ (L).

Now, we can state our first main result: Theorem 3.5. Let η be a closed 1-form on a smooth manifold M. Let L be an almost Dirac structure on M such that the distribution ρ((L)) is involutive. Assume that the rank of L is greater than 2 at every point. Then the following statements are equivalent: 1. L is a locally conformal Dirac bundle with Lee 1-form ω = η|ρ((L)) . 2. (dL + ω ∧ L )(X, Y, Z) = 0 for any vector fields X, Y, Z ∈ ρ((L)). 3. The set of all sections of L is closed under [ , ]ωC . 4. T (e1 , e2 , e3 ) = 0 for any e1 , e2 , e3 ∈ (L).

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Proof. Proposition 3.2 shows 1⇒ 2. Now assume that Statement 2 is true. By the Poincaré Lemma, there exist an open cover (Ui )i∈I of M and smooth positive functions fi : Ui → R such that η = d ln fi on the open set Ui . Let ω = (d ln fi )|ρ((L)) on Ui . Define Li = {(X|Ui , fi α|Ui ) | (X, α) ∈ L}. It is clear that Li is an almost Dirac bundle. Moreover, the 2-form corresponding to Li is Li = fi L|Ui on Ui . We have: dLi = (dfi )|ρ((L)) ∧ L|Ui + fi ∧ dL|Ui = fi (ω ∧ L|Ui + dL|Ui ) = 0. Using Remark 1, one deduces that Li is integrable. This proves that 2 ⇒ 1. Clearly, Statements 3 and 4 are equivalent. We are going to show that 2 ⇔ 4. Consider three smooth sections ei = (Xi , αi ), i = 1, 2, 3 ∈ (L). Then ω

T (e1 , e2 , e3 ) = 2[e1 , e2 ]C , e3 +

1 = [X1 , X2 ], α3 + X3 , LX1 α2 − LX2 α1 + d(iX2 α1 − iX1 α2 ) 2 1 +X3 , (iX1 ω)α2 − (iX2 ω)α1 + (iX2 α1 − iX1 α2 )ω. 2

This can be written in the form T (e1 , e2 , e3 ) = L (X3 , [X1 , X2 ])+X1 ·L (X2 , X3 )+c.p +(ω∧L )(X1 , X2 , X3 ). Thus, this is equivalent to the following equation: T (e1 , e2 , e3 ) = (dL + ω ∧ L )(X1 , X2 , X3 ). We deduce that 2 ⇔ 4.

Remark 3. If one weakens the hypothesis of Theorem 3.5 by removing the fact that the Lee 1-form ω comes from a differential 1-form η ∈ (T ∗ M) then one gets 1⇒ 2 ⇔ 3 ⇔ 4. In fact, the Poincaré Lemma applies to regular foliations but it is not always valid for singular foliations. 4. Examples and Applications 4.1. Locally conformal pre-symplectic manifolds. A locally conformal pre-symplectic manifold is a manifold M endowed with a pair (, ω), where is a differential 2-form and ω is a closed 1-form such that d = −ω ∧ . When is non-degenerate, this definition is equivalent to that of a locally conformal symplectic manifold (see [B1, Li1, V]). It is easy to check that (M, , ω) is a locally conformal pre-symplectic manifold if and only if the graph of is a locally conformal Dirac bundle over M having ω as a Lee 1-form. As an immediate consequence of Proposition 3.2, one has: Corollary 4.1. Consider a locally conformal Dirac bundle L ⊂ T M ⊕ T ∗ M such that the rank of L is greater than 2 at every point. Then L induces a foliation by locally conformal pre-symplectic manifolds.


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4.2. Examples related to Jacobi manifolds. A Jacobi structure on a manifold M is given by a pair (π, E) formed by a bivector field π and a vector field E such that ( [Li1]) [E, π ]SN = 0,

[π, π ]SN = 2E ∧ π,

where [ , ]SN is the Schouten-Nijenhuis bracket on the space of multi-vector fields. A manifold endowed with a Jacobi structure is said to be a Jacobi manifold. When E is zero, we get a Poisson structure. After recalling the definition of a Jacobi manifold, we are going to give examples of locally conformal Dirac bundles which are related to Jacobi manifolds. Proposition 4.2. Let π be a bivector field on M whose rank is greater than 2 at every point. Let ω be a closed 1-form on M. Define the vector field E = −π(ω) and Lπ = {(π α, α), | α ∈ T ∗ M}. Then L is a locally conformal Dirac bundle over M with Lee 1-form ω|ρ((L)) if and only if (π, E) is a Jacobi structure on M. To prove this proposition, we will use the following lemmas: Lemma 4.3. Let π be a bivector field on M. We denote by {·, ·}π the skew-symmetric operation 1-forms defined by: {α, β}π = Lπα β − Lπβ α − d(π(α, β)), for any 1-forms α and β on M. Then 1 [π, π]SN (α, β) = π{α, β}π − [π α, πβ]. 2 This lemma is proven for instance in [KM]. Lemma 4.4. If the distribution generated by the vector fields of the form π α is involutive then we have 1 (dL + ω|ρ((L)) ∧ L )(π α, πβ, π γ ) = (− [π, π ]SN + E ∧ π )(α, β, γ ), 2 where Lπ is the skew-symmetric 2-form associated with Lπ and ρ : (Lπ ) → χ (M) is the canonical projection. Proof. Let eα = (π α, α), eβ = (πβ, β), and eγ = (π γ , γ ) be in (L). On the one hand, by Remark 1, one gets dL (π α, πβ, π γ ) = 2[eα , eβ , ], eγ + = [πα, πβ], γ + π γ , {α, β}π = [πα, πβ] − π {α, β}π , γ 1 = − [π, π](α, β, γ ). 2 On the other hand, ω|ρ((L)) ∧ L ((π α, πβ, π γ ) = πα, ω L (πβ, π γ ) + c.p = iE α π(β, γ ) + c.p. = (E ∧ π )(α, β, γ ).

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Therefore, 1 (dL + ω|ρ((L)) ∧ L )(π α, πβ, π γ ) = (− [π, π ]SN + E ∧ π )(α, β, γ ). 2 Proposition 4.2 is an immediate consequence of Lemma 4.4 and Theorem 3.5. From now on, we will only consider locally conformal Dirac bundles L such that the rank of L (x) is greater than 2 at every point x ∈ M. 5. Admissible Functions Definition 5.1. Let L be a locally conformal Dirac bundle over M with Lee 1-form ω. We say that a smooth function f on M is L-admissible if there exists a vector field Xf ω ω on M such that (Xf , d f ) is a smooth section of L, where d f = df + f ω. On the space of all L-admissible functions, we define the bracket { , } : {f, g} = Xf · g + gω(Xf ), ω

(2)

ω

for any (Xf , d f ), (Xg , d g) ∈ (L). This is well-defined and skew-symmetric since L is isotropic. We have the following result: Lemma 5.2. ωLet L be a locally conformal Dirac bundle over M with Lee 1-form ω. If ω ef = (Xf , d f ) and eg = (Xg , d g) are in (L) then ω

ω

[ef , eg ]C = ([Xf , Xg ], d {f, g}). Proof. To simplify the proof, we use the deformed Lie differential ω

ω

ω

LX = iX d + d iX ,

ω

where d µ = dµ + ω ∧ µ. ω

We get a simpler expression for the bracket [ , ]C : 1 ω ω ω ω [(X1 , α1 ), (X2 , α2 )]C = ([X1 , X2 ], LX1 α2 − LX2 α1 + d (iX2 α1 − iX1 α2 )), 2 for any (X1 , α1 ), (X2 , α2 ) ∈ (L). ω ω Assume that ef = (Xf , d f ) and eg = (Xg , d g) are in (L). Using the fact that ω ω ω ω LX d = d LX , we get ω

ω

ω

ω

ω

ω

[ef , eg ]C = ([Xf , Xg ], LXf d g − LXg d f + d ({f, g})) ω

ω

= ([Xf , Xg ], d LXf g) ω

= ([Xf , Xg ], d {f, g}). Theorem 5.3. Let L be a locally conformal Dirac bundle over M with Lee 1-form ω. Then, the space of all L-admissible functions is a Lie algebra.

Locally Conformal Dirac Structures and Infinitesimal Automorphisms ω

303

ω

ω

Proof. For any ef = (Xf , d f ), eg = (Xg , d g), and eh = (Xh , d h) in (L), we have ω

ω

ω

2[ef , eg ]C , eh + = [Xf , Xg ], d h + Xh , d {f, g} ω = L[Xf ,Xg ] h + {h, {f, g}}. ω

ω

ω

ω

ω

Applying the relation L[Xf ,Xg ] = LXf LXg − LXg LXf , we obtain ω

ω

ω

ω

ω

2[ef , eg ]C , eh + = LXf LXg h − LXg LXf h + {h, {f, g}} = {f, {g, h}} − {g, {f, h}} + {h, {f, g}}. By Theorem 3.5, we have ω

0 = [ef , eg ]C , eh + =

1 {f, {g, h}} − {g, {f, h}} + {h, {f, g}} . 2

Thus, the Jacobi identity is satisfied. Furthermore, the bracket { , } is R-bilinear and skew-symmetric. Hence, the set of all L-admissible functions is a Lie algebra. Lemma 5.4. Let L be a locally conformal Dirac bundle over M with Lee 1-form ω. Suppose there exists a vector field Z such that (Z, ω) ∈ (L) then the product f g of two L-admissible functions is L-admissible. Moreover the bracket (2) gives {f g, h} = f {g, h} + g{f, h} + f gω(Xh ), ω

ω

ω

for any ef = (Xf , d f ), eg = (Xg , d g), and eh = (Xh , d h) in (L). Proof. The condition (Z, ω) ∈ (L) means that the constant functions are L-admissible. Furthermore, ω

f eg + gef − (f gZ, f gω) = (f Xg + gXf − f gZ, d (f g)) ∈ (L). Hence f g is L-admissible. We have: ω

{f g, h} = d h(f Xg + gXf − f gZ) ω = f {g, h} + g{f, h} − f g(d h(Z)) = f {g, h} + g{f, h} + f gω(Xh ). There follows the lemma.

Remark 4. From Lemma 5.4, one gets the following relation: {f g, 1} = f {g, 1} + g{f, 1},

∀ ef , eg ∈ (L).

This shows that there exists a unique vector field E such that {·, 1} = E.

Let L be a locally conformal Dirac bundle over M. The distribution (L ∩ T M) is called the characteristic distribution of L. It is clear that (L ∩ T M) is involutive. Therefore, it determines a singular foliation FL on M. Following [LWX], we say that L is reducible if FL is simple, i.e. M/FL is a smooth manifold and the projection π : M → M/FL is a submersion. Here is our second main result:

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Theorem 5.5. Let L be a reducible locally conformal Dirac bundle over M with Lee 1-form ω. Assume that there exists a vector field Z such that (Z, ω) ∈ (L). Then L induces a Jacobi structure on M/FL . Proof. Since (Z, ω) ∈ (L), one has ω

d f (X) = df (X) + f ω(X) = df (X),

∀X ∈ L ∩ T M.

Hence f is L-admissible if and only if f is constant along FL . So, we can identify the set of all L-admissible functions with C ∞ (M/FL ). Lemma 5.4 ensures that the bracket is local. Applying Theorem 5.3, we conclude that (2) defines a Jacobi bracket on C ∞ (M/FL ). Let pr : T M → T M/L ∩ T M be the canonical projection. We set E = pr(Z). Define a bivector field M/FL as follows: (df, ·) = pr(Xf ) − f E

ω

for all (Xf , d f ) ∈ (L).

Then (, E) defines the Jacobi structure on M/FL .

Remark 5. Theorem 5.5 is a generalization of a result given in [C] (see also [LWX]). Precisely, Courant proved that a Dirac structure L on a smooth manifold M induces a Poisson bracket on a quotient space of the type M/F, where F is a foliation associated with L. It may be valuable to recall the standard description of the Dirac reduced phase space in relation with Courant’s result. Let M be a finite-dimensional smooth manifold. Its cotangent bundle P = T ∗ M together with the canonical symplectic form = dη is the phase space of a mechanical system. Consider k functions H1 , . . . , Hk ∈ C ∞ (P ). The constraint hypersurface is the subset N of P defined by the equations Hu = 0, u = 1, 2, . . . , k. We assume that N is a smooth submanifold of P of codimension k. The symplectic form induces a closed 2-form N = i ∗ , where i : N → P is the inclusion. If, in addition, N has a constant rank then the characteristic distribution C = ker(N ) defines a regular foliation on N. Suppose that the foliation is simple, then there is a symplectic form on N/C induced by N . The quotient N/C is called the space of physical states. One may replace the symplectic 2-form by a locally conformal symplectic 2-form on the phase space. In this case, the induced 2-form on N/C is a locally conformal symplectic 2-form (see for instance [MR] for more details on the reduction of Poisson manifolds). Here the quotient space obtained in Theorem 5.5 plays the role of the space of physical states. The converse of Theorem 5.5 will be studied elsewhere. We close this section by recalling that conformal symplectic 2-forms are also relevant to statistical mechanics (see [BFLS and Li2]. 6. The Extended Lee Homomorphism We start this section by introducing the notion of a pull-back of an almost Dirac structure by a surjective submersion (see [LWX]). This notion allows to better understand the Lie algebra of infinitesimal automorphisms of a locally conformal Dirac structure. Let V and W be two vector spaces over R. Given a surjective linear map : V → W , its dual ∗ : W ∗ → V ∗ is injective. Let LW be a maximally isotropic subspace of W ⊕ W ∗ . Then LV = {(x, ∗ ξ ) | x ∈ V , ξ ∈ W ∗ , ((x), ξ ) ∈ LW }


305

is a maximally isotropic subspace of V ⊕ V ∗ , which is called the pull-back of LW by the map . This definition can be extended to the context of vector bundles. Namely, if E → M and F → N are two vector bundles, and : E → F a surjective bundle map over : M → N. Then, one defines the pull-back LE of an almost Dirac structure LF by as its fiberwise pull-back. Lemma 6.1. Let ϕ : M → N be a surjective submersion and let L be a locally conformal Dirac bundle over N . Denote by Lϕ the pull-back of L by ϕ. Then Lϕ = ϕ ∗ L . Proof. For any x ∈ M, (X, ϕ ∗ α) and (Y, ϕ ∗ β) ∈ (Lϕ ), we have Lϕ (X, Y )(x) = X, ϕ ∗ β(x) = ϕ∗ X, β(ϕ(x)) = L (ϕ∗ X, ϕ∗ Y )(ϕ(x)) = (ϕ ∗ L )(X, Y )(x). This shows that Lϕ = ϕ ∗ L .

Definition 6.2. Two locally conformal Dirac bundles L1 and L2 over M with respective Lee 1-forms ω1 and ω2 are conformally equivalent if there exits a smooth function a on M such that a(x) = 0 at every point x ∈ M and L2 = {(X, aα) | (X, α) ∈ L1 }. This gives L2 = aL1 and ω2 = ω1 − d(ln |a|)|ρ((L)) . The equivalence class of a locally conformal Dirac bundle L is called a locally conformal Dirac structure on M and denoted by D = [L]. The pair (M, D) is called a locally conformal Dirac manifold. One can notice that if two locally conformal Dirac bundles are conformally equivalent then they have the same foliation. Let D = [L] be a locally conformal Dirac structure on M. A diffeomorphism ϕ is in Diff ∞ (M, D) if the pull-back of L by ϕ is conformally equivalent to L. Therefore, there exists a smooth function aL,ϕ : M → R∗ such that ϕ ∗ L = aL,ϕ L . Clearly, this definition does not depend on the locally conformal Dirac bundle L representing D. Moreover, any ϕ ∈ Diff ∞ (M, [L]) preserves the foliation associated with L. We have the following definition: Definition 6.3. An infinitesimal automorphism of D = [L] is a smooth vector field X ∈ χ (M) satisfying the conditions 1. [X, ρ((L))] ⊂ ρ((L)); 2. LX L = uX L , where uX is a smooth function on M. Here we use the formula (LX L )(Y1 , Y2 ) = X · L (Y1 , Y2 ) − L ([X, Y1 ], Y2 ) − L (Y1 , [X, Y2 ]), for (Yi , αi ) ∈ (L). We denote by χ (M, D) the space of all infinitesimal automorphisms of D. Let T (M, D) be the subspace formed by elements of χ (M, D) which are contained in ρ((L)). It is easy to check that χ (M, D) is a Lie algebra. Moreover, T (M, D) is a subalgebra of χ(M, D). One can observe that, when M is compact, the flow φtX of any infinitesimal automorphism X preserves D, i.e. φtX ∈ Diff ∞ (M, D) for all t ∈ R.

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Proposition 6.4. Let L be a locally conformal Dirac bundle with Lee 1-form ω. Denote χ0 (M, L) = {X ∈ χ (M) | LX L = 0 and [X, ρ((L))] ⊂ ρ((L))}. Then X is in χ0 (M, L) if and only if ([X, Y ], LX β) is in (L) for any (Y, β) ∈ (L). Proof. Suppose [X, ρ((L))] ⊂ ρ((L)). Pick (Y, β), (Z, γ ) ∈ (L), then 2([X, Y ], LX β), (Z, γ )+ = γ ([X, Y ]) + LX β(Z) = γ ([X, Y ]) + X · β(Z) − β([X, Z]) = (Z, [X, Y ]) + X · (Y, Z) − (Y, [X, Z]) = (LX )(Y, Z). There follows the proposition.

Now, we can state our third main theorem, which is a generalization of results proven in [GL, V and B1]. Theorem 6.5. Let L be a locally conformal Dirac bundle with Lee 1-form ω. Assume that there exists a vector field Z such that (Z, ω) ∈ (L). If X is an infinitesimal automorphism of D = [L] which belongs to ρ((L)), then there exists a smooth function cX on M which is constant along the leaves of the foliation determined by ρ((L)) and satisfies ω d θ = cX L , where θ = iX L . Furthermore the mapping X → cX is a Lie homomorphism C : T (M, D) → A∞ (M), where A∞ (M) is the Lie algebra of L-admissible functions. L L This mapping C is called the extended Lie homomorphism of L. (M) = C ∞ (M) the extended Lie homomorphism is an invariant of D = When A∞ L [L], i.e. C does not depend on the vector bundle L representing D. Proof. We have ω

d θ = diX L + ω ∧ iX L = LX L − iX dL + ω ∧ iX L = uX L + iX (ω ∧ L ) + ω ∧ iX L = (uX + ω(X))L . Set

cX = uX + ω(X). ω

Then d θ = cX L . Moreover ω

ω

ω

0 = d ◦ d θ = d (cX L ) = dcX |ρ((L)) ∧ L + cX (dL + ω ∧ L ) = dcX |ρ((L)) ∧ L . Applying Lemma 3.3, we get dcX |ρ((L)) = 0. This shows that cX is constant along the leaves of the foliation associated with L. Now we are going to prove that X → cX is a Lie homomorphism. By hypothesis, there exists a vector field Z such that (Z, ω) ∈ (L). Therefore, if X, Y ∈ ρ((L)) ω ω are two infinitesimal automorphisms of D = [L] then (cX Z, d cX ) and (cY Z, d cY ) are sections of L. In other words, cX and cY are L-admissible functions. Furthermore, {cX , cY } = 0. We also have:


307

L[X,Y ] L = LX (uY L ) − LY (uX L ) = (LX uY − LY uX )L . Consequently u[X,Y ] = LX uY − LY uX = LX (cY − ω(Y )) − LY (cX − ω(X)) = −X · (ω(Y )) + Y · (ω(X)). Therefore, c[X,Y ] = u[X,Y ] + ω([X, Y ]) = −X · (ω(Y )) + Y · (ω(X)) + ω([X, Y ]) = −dω(X, Y ) = 0. Hence C is a Lie homomorphism. Assume that A∞ (M) = C ∞ (M). One can easily show that if L is conformalL ly equivalent to L then A∞ (M) = C ∞ (M). Moreover, we have L = f L and L ω = ω − d ln |f |, for some smooth function f . Since (Z, ω) ∈ (L), we obtain that there exists a vector field Z such that (Z , ω − d ln |f |) ∈ (L ). Let X ∈ ρ((L)) be an infinitesimal automorphism of D. Then LX L = LX (f L ) = (uX + d ln |f |(X))L . It follows that uX = uX + d ln |f |(X). Hence, cX = uX + ω (X) = uX + d ln |f |(X) + ω(X) − d ln |f |(X) = uX + ω(X) = cX . This completes the proof of Theorem 6.5.

Remark 6. Examples where the situation A∞ (M) = C ∞ (M) occurs are given in PropL osition 4.2. The following theorem slightly generalizes a result obtained in [B1 and B2]: Theorem 6.6. Let ω be a closed 1-form on a smooth manifold M and let L be a locally → M be the Galois conformal Dirac bundle with Lee 1-form ωL = ω|ρ((L)) . Let p : M →R covering which resolves ω, that is, there exists a positive smooth function f : M is endowed with a Dirac structure L. Precisely, if L satisfying p ∗ ω = d ln f . Then, M is the pull-back of L by the derivative = T p : T M → T M then = {(X, f η) | (X, η) ∈ L} L is independent of f and the is a Dirac structure. Furthermore, the conformal class of L choice of L in D = [L].

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that is, ˆ = p ∗ L . Proof. Let Lˆ be the skew-symmetric 2-form associated with L, L Then L˜ (X, Y ) = f Lˆ (X, Y ) = f (p∗ L )(X, Y ). It follows that L˜ = fp ∗ L .

(3)

Therefore ∗ ∗ ∗ dL˜ = f (d ln f|ρ((L)) ˜ ∧ p L + p dL ) = fp (ωL ∧ L + dL ) = 0.

is indepenMoreover, it follows immediately from Eq. (3) that the conformal class of L dent of f and the choice of L in D = [L]. This completes the proof of Theorem 6.6. 7. Generalized Locally Conformal Dirac Structures A Lie algebroid over a smooth manifold M is a vector bundle A → M together with a Lie algebra structure on the space (A) of smooth sections of A, and a bundle map

: A → T M such that • induces a Lie homomorphism from (A) to χ (M). • [X1 , f X2 ] = f [X1 , X2 ] + ( (X1 )f )X2 . The map is called the anchor of the Lie algebroid. There is a coboundary operator dA associated with A. It is defined as follows: (dA ξ )(X1 , . . . , Xk ) =

k

(−1)i+1 (Xi )(ξ(X1 , · · · , Xˆ i , . . . , Xk ))

i=1

(−1)i+j ξ([Xi , Xj ], · · · , Xˆ i , . . . , Xˆ j , . . . , Xk ),

+

i<j

for ξ ∈ (k A∗ ), X1 , . . . , Xk ∈ (A). The corresponding cohomology is called the Lie algebroid cohomology. When A = T M, one gets the de Rham cohomology. Let A be a Lie algebroid with anchor and let η ∈ (A∗ ) be a 1-cocycle. On (A ⊕ A∗ ), we define 1 η η η η [(X1 , ξ1 ), (X2 , ξ2 )]C = ([X1 , X2 ], LX1 ξ2 − LX2 ξ1 + dA (iX2 ξ1 − iX1 ξ2 )), 2 where

η

dA ξ = dA ξ + η ∧ ξ

and

η

η

η

LX1 = iX1 dA + dA iX1 .

Consider the bilinear 2-form defined by (X1 , ξ1 ), (X2 , ξ2 )+ = for (X1 , ξ1 ), (X2 , ξ2 ) ∈ A ⊕ A∗ .

1 (iX ξ1 + iX1 ξ2 ), 2 2


309

Definition 7.1. A sub-bundle L of the vector bundle A ⊕ A∗ → M is said to be an (A, η)-Dirac bundle over M if it is maximally isotropic with respect to , + , and η (L) is closed under the bracket [ , ]C . In particular, Dirac structures on M are just (T M, 0)-Dirac bundles. A maximally isotropic sub-bundle L ⊂ A ⊕ A∗ is called a locally conformal (A, η)Dirac bundle if there exist an open cover (Ui )i∈I of M and smooth positive functions fi : Ui → R such that Li = {(X|Ui , fi ξ|Ui ) | (X, ξ ) ∈ L} is an (A, η)-Dirac bundle over Ui for all i ∈ I . Thus, locally conformal Dirac bundles coincide with locally conformal (T M, 0)-Dirac bundles. Of course, any locally conformal (A, η)-Dirac bundle induces a foliation on M which is given by the distribution (pr1 (L))), where is the anchor of the Lie algebroid A and pr1 : L → A is the projection of the first component. Furthermore, there is a skewsymmetric C ∞ -bilinear map L : 2 P → C ∞ (M) associated with L and defined as in (1), where P = pr1 ((L)). By an argument similar to the one used in Proposition 3.3, one may show that if the rank of L is greater than 2 at every point x ∈ M then there exists a unique map ωL : P → C ∞ (M) such that dA L = −ωL ∧ L . Here we restrict dA to the space of all the n-forms θ : n (P ) → C ∞ (M). We also can extend Definition 6.2. Specifically, Definition 7.2. Two locally conformal (A, η)-Dirac bundles L1 and L2 over M with respective Lee 1-forms ω1 and ω2 are conformally equivalent if there exists a smooth function f on M such that f (x) = 0 at every point x ∈ M and L2 = {(X, f α) | (X, α) ∈ L1 }. One may state results analogous to Theorem 3.5, 5.3, and 6.6. We will end this paper by giving an example of generalized locally conformal Dirac structure. Example. Let A = T M ⊕ R, with the Lie algebroid bracket and anchor given by: [(X, f ), (Y, g)] = ([X, Y ], X · g − Y · f ),

and

(X, f ) = X.

(T ∗ M

Consider the 1-cocycle η =η (0, 1) ∈ ⊕ R). Denote E 1 (M) = A ⊕ A∗ . It was observed in [GM] that [ , ]C coincides with the bracket we used in [Wa] to define the E 1 (M)-Dirac structures. Namely, η

[ (X1 , f1 ) + (α1 , g1 ), (X2 , f2 ) + (α2 , g2 )]C = [X1 , X2 ], X1 · f2 − X2 · f1 1 + LX1 α2 − LX2 α1 + d(iX2 α1 − iX1 α2 ) 2 1 +f1 α2 − f2 α1 + (g2 df1 − g1 df2 − f1 dg2 + f2 dg1 ), 2 1 × X1 · g2 − X2 · g1 + (iX2 α1 − iX1 α2 − f1 g2 + f2 g1 ) , 2 for any (Xi , fi ) + (αi , gi ) ∈ (E 1 (M)). More precisely, E 1 (M)-Dirac structures are exactly (T M ⊕ R, (0, 1))-Dirac bundles over M. Furthermore, the concept of a conformal equivalence class considered in this paper is slightly different from the one introduced in [Wa]. We defined an equivalence relation among (T M ⊕ R, (0, 1))-Dirac bundles in [Wa], while the locally conformal Dirac bundles are considered in Definition 7.2.

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Acknowledgement. I would like to thank A. Banyaga, P. Brassler, P. Foth, Y. Kosmann-Schwarzbach, C.-M. Marle, A. Weinstein, P. Xu, and N-T. Zung for discussions and for their interest in this work. Thanks go also to the Institut de Mathématiques de Jussieu for the hospitality while I was working on the revised version of this paper. Many thanks to the referee for helpful remarks.

References [B1]

Banyaga, A.: On the geometry of locally conformal symplectic manifolds. In: Infinite dimensional Lie groups in geometry and representation theory (Washington, DC, 2000), River Edge, NJ: World Sci. Publishing, 2002, pp. 79–91 [B2] Banyaga, A.: A geometric integration of the extended Lee homomorphism. J. Geom. Phys. 39, 30–44 (2001) [BFLS] Basart, H., Flato, M., Lichnerowicz, A., Sternheimer, D.: Deformation theory applied to quantization and statistical mechanics. Lett. Math. Phys. 8, 483–494 (1984) [C] Courant, T.: Dirac structures. Trans. A.M.S. 319, 631–661 (1990) [CW] Courant, T., Weinstein, A.: Beyond Poisson structures. In: Séminaire Sud-Rhodanien de Geométrie, Travaux en cours 27, Paris: Hermann, 1988, pp. 39–49 [DLM] Dazord, P., Lichnerowicz, A., Marle, C.-M.: Structure locale des variétés de Jacobi. J. Math. Pures Appl. 70, 101–152 (1991) [GL] Guédira, F., Lichnerowicz, A.: Géométrie des algèbres de Lie locales de Kirillov. J. Math. Pures et Appl. 63, 407–484 (1984) [GM] Grabowski, J., Marmo, G.: The graded Jacobi algebras and (co)homology. Preprint arXiv:math.DG/0207017 [KM] Kosmann-Schwarzbach, Y., Magri, F.: Poisson-Nijenhuis structures. Ann. Inst. H. Poincaré Phys. Théor. 53(1), 35–81 (1990) [Li1] Lichnerowicz, A.: Les variétés de Jacobi et leurs algèbres de Lie associés. J. Math. Pures et Appl. 57, 453–488 (1978) [Li2] Lichnerowicz, A.: Deformations and geometric (KMS)-conditions. In: Quantum theories and geometry. Math. Phys. Studies 10, 127–143 (1987) [LWX] Liu, Z.-J., Weinstein, A., Xu, P.: Dirac structures and Poisson homogeneous spaces. Commun. Math. Phys. 192, 121–144 (1998) [MR] Marsden, J., Ratiu, T.: Reduction of Poisson manifolds. Lett. Math. Phys. 11, 161–169 (1986) [V] Vaisman, I.: Locally conformal symplectic manifolds. Internat. J. Math. Math. Sci. 8, 521–536 (1985) [Wa] Wade, A.: Conformal Dirac structures. Lett. Math. Phys. 53, 331–348 (2000) [WL] Wojtkowski, M., Liverani, C.: Conformally symplectic dynamics and symmetry of the Lyapunov spectrum. Commun. Math. Phys. 194, 47–60 (1998) Communicated by M. Aizenman


Communications in


A Particular Bit of Universality: Scaling Limits of Some Dependent Percolation Models Federico Camia1, , Charles M. Newman2, , Vladas Sidoravicius3, 1

EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: [email protected] Courant Inst. of Mathematical Sciences, New York University, New York, NY 10012, USA. E-mail: [email protected] 3 Instituto de Matematica Pura e Aplicada, Rio de Janeiro, RJ, Brazil. E-mail: [email protected]

2

Received: 27 February 2003 / Accepted: 3 November 2003 Published online: 24 February 2004 – © Springer-Verlag 2004

Abstract: We study families of dependent site percolation models on the triangular lattice T and hexagonal lattice H that arise by applying certain cellular automata to independent percolation configurations. We analyze the scaling limit of such models and show that the distance between macroscopic portions of cluster boundaries of any two percolation models within one of our families goes to zero almost surely in the scaling limit. It follows that each of these cellular automaton generated dependent percolation models has the same scaling limit (in the sense of Aizenman-Burchard [3]) as independent site percolation on T. 1. Introduction The phase diagrams of many physical systems have a “critical region” where all traditional approximation methods, such as mean-field theory and its generalizations, fail completely to provide an accurate description of the system’s behavior. This is due to the existence of a “critical point,” approaching which, some statistical-mechanical quantities diverge, while others stay finite but have divergent derivatives. Experimentally, it is found that these quantities usually behave in the critical region as a power law. The exponents that appear in those power laws, called critical exponents, describe the nature of the singularities at the critical point. The theory of critical phenomena based on the renormalization group suggests that statistical-mechanical systems fall into “universality classes” such that systems belonging to the same universality class have the same critical exponents. The work was conducted while this author was at Department of Physics, New York University, New York, NY 10003, USA. Research partially supported by the U.S. NSF under grants DMS-98-02310 and DMS-01-02587. Research partially supported by the U.S. NSF under grants DMS-98-03267 and DMS-01-04278. Research partially supported by FAPERJ grant E-26/151.905/2000 and CNPq.

312

F. Camia, C. M. Newman, V. Sidoravicius

It should be emphasized that the phenomenon of universality was discovered at least twice prior to the introduction of the renormalization group. In fact, in what Alan Sokal [50] calls the Dark Ages of the theory of critical phenomena, before the 1940’s, physicists generally believed that all systems have the same critical exponents, namely those of the mean-field theory of Weiss [52] (1907) or its analogue for fluids, the van der Waals theory [51] (1873). The failure of this type of universality became apparent as early as 1900, following experiments on fluid systems, but it began to be taken seriously only after Onsager’s exact solution [42] (explicitly displaying non-mean-field exponents) in 1944 and the rediscovery of the experimental evidence of non-mean-field values for critical exponents by Guggenheim in 1945 [24]. Nonetheless, although meanfield theory is clearly incorrect for short range models, some universality does seem to hold. Many, if not all, different fluids, for instance, seem to have the same value for the critical exponent β (related to density fluctuations), and it is believed that, for example, carbon dioxide, xenon and the three dimensional Ising model should all have the same critical exponents. Maybe even more surprisingly, it was soon realized that some (but not all) magnetic systems have the same critical exponents as do the fluids. This remarkable phenomenon seems to suggest the existence of a mechanism that makes the details of the interaction irrelevant in the critical region. Nevertheless, the critical exponents should depend on the dimensionality of the system and on any symmetries in the Hamiltonian. Despite being a very plausible and appealing heuristic idea, backed up by renormalization group arguments and empirical evidence, only very few cases are known in which some form of universality has actually been proved, especially below the upper critical dimension, where the values of the critical exponents are expected, and in some cases proved, to be different from those predicted by mean-field theory. There are, however, some more examples even in low dimensions – see, e.g., [8, 9, 11, 43]. In particular, in [8, 9] methods analogous to the ones developed here are applied, but to different models and on different lattices. Percolation, with its simplicity and important physical applications, is a natural candidate for studying universality. This is especially so after the ground-breaking work of Schramm [45], who identified the only possible conformally invariant scaling limit of critical percolation, and that of Smirnov [47, 48] (see also the closely related work of Lawler, Schramm, and Werner [28–34]). The combination of those results made it possible [49] to verify the values of the critical exponents predicted in the physics literature in the case of critical site percolation on the triangular lattice (and to derive also some results that had not appeared in the physics literature, such as an analogue of Cardy’s formula “in the bulk” [46], or the description of the so-called backbone exponent [32]). It is generally accepted that the lattice should play no role in the scaling limit, and that there should be no difference between bond and site percolation. In other words, two-dimensional critical (independent) percolation models, both site and bond, should belong to the same universality class, regardless of the lattice (at least for periodic lattices like the square, triangular or hexagonal lattice). Once again, though, despite being a very natural and plausible conjecture, such universality has not yet been proved. There is however another natural direction in which to study universality, which consists in analyzing critical percolation models on a given lattice that differ in their dependence structures. It is this direction that we pursue in this paper. The cellular automata that we use to generate our families of dependent percolation models arise naturally in the study of the zero-temperature limit of Glauber dynamics or as coarsening or agreement-inducing dynamics. The action of such cellular automata

Scaling Limits of Some Dependent Percolation Models

313

can be viewed as a sort of “small (local) perturbation” of the original, independent percolation model, and our main corollary can be viewed as proving a form of universality for two dimensional percolation. Therefore, we provide an explicit example of the principle of universality, in the strong form concerning scaling limits. To be more precise, there are at least two, a priori different, notions of universality, one concerning the critical exponents discussed above, and a second one concerning the continuum scaling limit. The two concepts are closely related, but in this paper we are only concerned with the second type of universality. The first type is considered in [11], using methods related to, but easier than, those in this paper; it appears that universality in terms of the scaling limit is a stronger notion than that in terms of exponents. In [49], some knowledge of the scaling limit is used to determine critical exponents in the case of two-dimensional independent site percolation on the triangular lattice, but there is no general result in this direction. We remark that the continuum limit used in this paper corresponds to the “full scaling limit” (for independent percolation) discussed in [12] (see also Theorem 4 and Subsect. 3.3 of [47]). A complete proof of the existence and properties of the “full scaling limit” has, to our knowledge, not yet appeared in the literature, although there has been recent progress in that direction [12, 13]. We consider a family of dependent percolation models that arise through a (discrete time) deterministic cellular automaton T acting on site percolation configurations σ on the set of sites of the triangular lattice T. Each configuration σ corresponds to an assignment of −1 or +1 to the vertices of T. The variable σx , corresponding to the value of σ at x, is commonly called a spin variable. At discrete times n = 1, 2, . . . , each spin σx is updated according to the following rules (later, in Sect. 4, we will introduce other cellular automata, both on T and H, generated by different rules): • if x has three or more neighbors whose spin is the same as σx , then the latter does not change value, • if x has only two neighbors y1 and y2 that agree with x, and y1 and y2 are not neighbors, then σx does not change value, • otherwise, σx changes value: σx → −σx . The starting configuration σ 0 , at time n = 0, of our cellular automaton is chosen from a Bernoulli product measure corresponding to independent critical percolation, and the distributions at times n ≥ 1 (including the final state as n → ∞) of the discrete time deterministic dynamical process σ n are the other members of our family of dependent percolation models. We show that all those dependent percolation models have the same scaling limit, thus providing an explicit example in which universality can be proven. This comes as a corollary of our main result, Theorem 1. To explain the main result, we first need some terminology. In the scaling limit, the microscopic scale of the system (i.e., the lattice spacing δ) is sent to zero, while focus is kept on features manifested on a macroscopic scale. In the case of percolation, it is far from obvious how to describe such a limit and we only do so briefly here; for more details, see [1–3]. We will make use of the approach introduced by Aizenman and Burchard [3] (see also [4]) applied to portions of the boundaries between clusters of opposite sign, and present our results in terms of closed collections of curves ˙ 2 of R2 , which we identify (via the stereographic in the one-point compactification R projection) with the two-dimensional unit sphere. For each fixed δ > 0, the curves are, before compactification, polygonal paths of step size δ (i.e., polygonal paths between sites of the dual lattice). The distance between curves is defined so that two curves are

314


close if they shadow each other in a metric which shrinks at infinity (for the details, see Sect. 2.2 and 2.3). The dynamics allows one to construct the whole family of percolation models for all n ∈ {0, 1, . . . , ∞} on the same probability space, i.e., there is a natural coupling ν, realized through the dynamics, between any two percolation models in the family. In terms of this coupling, Theorem 1 states, roughly speaking, that the ν-probability that the distance between the two collections of curves corresponding to two distinct percolation models in the family is bounded away from zero vanishes as δ → 0. This means that, in the limit δ → 0, given any two models in the family, for every curve in one of them, there exists a curve in the other one that shadows the first curve and vice-versa. The rest of the paper is organized as follows. In Sect. 2, we describe the behavior of the cellular automaton T, define the family of dependent percolation models dynamically generated by T, and state the main results. The proofs of the main results are contained in Sect. 3. The dynamics described in Sect. 2 is chosen as a prototypical example, but is not the only one for which our results apply. In Sect. 4, we introduce other such dynamics (both on T and H), which can be obtained as suitable zero-temperature limits of stochastic Ising models. 2. Definitions and Results We start this section by giving a more detailed definition of one of the families of dependent percolation models that are the object of investigation of this paper. The models in that family are defined on the triangular lattice T, embedded in R2 by identifying the sites of T with the elementary cells (i.e., regular hexagons) of the hexagonal (or honeycomb) lattice H (see Fig. 1). We will use those models as a paradigm and will give for them explicit and detailed proofs of the results. Later on, we will point out how to modify the proofs to adapt them to the other models discussed in the paper. In the rest of the paper, points of R2 will be denoted by u and v, while for the sites of T we will use the Latin letters x, y, z and the Greek letters ζ and ξ . An edge of T

Fig. 1. Portion of the hexagonal (or honeycomb) lattice


315

∗ we denote the dual edge incident on sites x and y will be denoted by ηx,y , while by ηx,y (in H) perpendicular bisector of ηx,y .

2.1. The dynamics and the percolation models. We construct a family of dependent percolation models by means of a cellular automaton T acting on site percolation configurations on the triangular lattice T, i.e., T : → , where is the set of configurations {−1, +1}T . This family is parametrized by n ∈ {0, 1, . . . } ∪ {∞} representing (discrete) time. The initial configuration σ 0 consists of an assignment of −1 or +1 to the sites of T. At times, we will identify the spin variable σx with the corresponding site (or hexagon) x. We choose σ 0 according to a probability measure µ0 corresponding to independent identically distributed σx0 ’s with µ0 (σx0 = +1) = λ ∈ [0, 1]. With the exception of Propositions 2.1 and 2.2, we will set λ = 1/2, so that µ0 is the distribution corresponding to critical independent site percolation. We denote by µn the distribution n of σ n and write µn = T˜ µ0 , where T˜ is a map in the space of measures on . The action of the cellular automaton T can be described as follows:  n  σx if x has at least two neighbors y1 and y2 such that σyn1 = σyn2 = σxn , n+1 σx = and y1 and y2 are not neighbors  −σ n otherwise. x (1) Once the initial percolation configuration σ 0 is chosen, the dynamics is completely deterministic, that is, T is a deterministic cellular automaton with random initial state (see Figs. 2 and 3). Certain configurations are stable for the dynamics, in other words, they are absorbing states for the cellular automaton. To see this, let us consider a loop in T expressed as a sequence of sites (ζ0 , . . . , ζk ) which are distinct except that ζ0 = ζk and suppose moreover that k ≥ 6 and that ζi−1 and ζi+1 are not neighbors; we will call such a

1 1

2 3

2

2 2

1

1

2 1

1

Fig. 2. Example of local configuration with unstable spins. The numbered hexagons correspond to spins that will flip and the numbers indicate at what time step the spin flips occur

316


1 1

2 3

2

2 2

1

1

2 1

1

Fig. 3. Same local configuration after all the unstable spins have flipped

sequence an m-loop (these are the “minimal” loops that can be “stable” – see Definitions 3.1 and 3.3). If σζ0 = σζ1 = . . . = σζk−1 then every site ζi in such an m-loop has two neighbors, ζi−1 and ζi+1 , such that ζi−1 and ζi+1 are not neighbors of each other and σζi−1 = σζi = σζi+1 . According to the rules of the dynamics, σζi is therefore stable, that is, retains the same sign at all future times. Other stable configurations are “barbells,” where a barbell consists of two disjoint such m-loops connected by a stable m-path. The stability of certain loops under the action of T will be a key ingredient in the proof of the main theorem. A more precise definition of m-paths and m-loops is given in Sect. 3. An important feature of this dynamics is that almost surely every spin flips only a finite number of times and every local configuration gets fixated in finite time. To show this (and also to make more explicit the connection with the models discussed in Sect. 4), we introduce a formal Hamiltonian H(σ ) = −

1 Hx (σ ), 2 x

σx σ y =

x,y

where and

x,y

(2)

denotes the sum over all pairs of neighbor sites, each pair counted once, Hx (σ ) = −

σx σy ,

(3)

y∈N (x)

where N (x) is the set of six (nearest) neighbors of x. We also introduce a “local energy” σx σy − σx σz , (4) H (σ ) = − x,y

x,y∈

z∈∂

x∈

x∈N (z)

where is a subset of T and ∂ is the outer boundary of , i.e., {ζ ∈ / : x ∈ N (ζ ) for some x ∈ }. Notice that although the total energy H(σ ) is almost surely


317

infinite and is therefore only defined formally, we will only use local energies of finite subsets of T. The notion of the energy change caused by a spin flip is somewhat ambiguous in a cellular automaton because of the synchronous dynamics and hence multiple simultaneous spin flips. Nevertheless, it is easy to show (see the proof of Proposition 2.1) that each step of the dynamics either lowers or leaves unchanged the energy – both locally and globally. In this sense, our cellular automaton can be considered a zero-temperature dynamics (see, for example, [37] and references therein). Proposition 2.1. For all values of λ, almost surely, every spin flips only a finite number of times. Proof. By the translation invariance and ergodicity of the model, it is enough to prove the claim for the origin. At time zero, and for all values of λ, the origin is almost surely surrounded by an m-loop of spins of constant sign. (For λ = 1/2, there are infinitely many loops of both signs surrounding the origin.) Such an m-loop is stable for the dynamics and its spins retain the same sign at all times. Call the (a.s. finite) region surrounded by . The energy H of the finite region is bounded below. Consequently, if we can show that no step of the dynamics ever raises H , it would follow that there can only be a finite number of steps that strictly lower the energy H . Call an edge ηx,y satisfied if σx = σy and unsatisfied otherwise; then the change in energy H (σ n+1 )−H (σ n ) is twice the difference between the number of satisfied edges at time n that become unsatisfied at time n+1 and the number of unsatisfied edges at time n that become satisfied at time n + 1. The edges that change from satisfied to unsatisfied or vice-versa are those between spins that flip and their neighbors that do not, so H (σ n+1 ) − H (σ n ) = (σxn σyn − σxn+1 σyn+1 ) x∈ : σxn+1 =σxn y∈N (x)

=

n Hx ,

(5)

x∈ : σxn+1 =σxn

where n Hx = Hx (σ n+1 ) − Hx (σ n ).

(6)

Notice that the only nonzero contributions in the first sum of (5) come from those sites y ∈ that do not flip at time n. We want to show that n Hx ≤ 0 for all x ∈ and find some y with n Hy < 0 (assuming there was at least one spin flip inside at time n). Call Dxn the number of disagreeing neighbors of x at time n and notice that a necessary condition for the spin at site x to flip at that time is Dxn ≥ 4. Let us first consider the case Dxn ≥ 5 and assume, without loss of generality, that σxn = −1 and σxn+1 = +1. Then, at time n, site x has at least five plus-neighbors, and at least three of them have plus-spins at time n + 1 (those having at time n two plus-neighbors that are not neighbors of each other). This implies that the number of edges incident on x that change from unsatisfied to satisfied is at least three, while the number of edges that change from satisfied to unsatisfied is at most one. Then, n Hx = (σxn σyn − σxn+1 σyn+1 ) < 0. (7) y∈N (x)

318


~ y1 ~ y2

z1

~ y1

y1

z1 y1 x

x

~ y2

y2

y2 z2

z2

Time n

Time n+1

Fig. 4. Example of a step of the dynamics acting on a local configuration that leaves the energy of some sites (including x) unchanged and decreases that of other sites (the three spins at the top and the three at the bottom are stable, as determined also by spins that do not appear in the figure)

Next, we consider the case Dxn = 4. Again, we can assume that σxn = −1 = −σxn+1 . In this case, one can have two types of spin flips, one with Hx (σ n+1 ) − Hx (σ n ) < 0 and one with Hx (σ n+1 ) − Hx (σ n ) = 0. The second type occurs when, of the four neighbors of x that are plus at time n, two remain plus at time n + 1 (we call them y1 and y2 ) and the other two flip to minus (we call them z1 and z2 ), while the two neighbors that are minus at time n remain minus at time n + 1 (we call them y˜1 and y˜2 ). In this situation, site x disagrees with four of its six neighbors both at time n and at time n + 1 (see Fig. 4) and therefore Hx (σ n+1 ) − Hx (σ n ) = 0. The spins at z1 and z2 flip together with the spin at x. Each energy, Hz1 and Hz2 , is lowered or left unchanged. If either is lowered, then the energy H is lowered. If neither is lowered, then z1 and z2 must be in the same situation as x (but with minus and plus interchanged), which requires a configuration that looks locally like the one in the left part of Fig. 4 (or one equivalent to it under some lattice symmetry). However, such a local configuration cannot extend forever; it must be finite and contained in . This implies that we will necessarily find at least one spin y (in fact, at least two spins) that flip together with the spin at x and such that n Hy < 0. Thus, if at time n some site x ∈ flips, H (σ n+1 ) − H (σ n ) =

n Hx < 0.

(8)

x∈ : σxn+1 =σxn

It follows that there can only be a finite number of times n at which spins in (in particular, the origin) flip. This completes the proof. Let us now give two results that are analogous to results proved in [14] for a related cellular automaton that will be discussed below in Sect. 4. Proposition 2.2. If λ > 1/2 (respectively, < 1/2), then for almost every σ 0 , there is percolation of +1 (respectively, −1) spins in σ n for any n ∈ [0, ∞].


319

Proof. We only give the proof for λ > 1/2, since the case λ < 1/2 is the same by symmetry. If λ > 1/2, since the critical value for independent (Bernoulli) percolation on the triangular lattice is exactly 1/2, there is at time zero percolation of +1 spins. This implies the existence, at time 0, of doubly-infinite plus-paths (i.e., plus-paths that can be split into two disjoint infinite paths) that are stable. Therefore, there is percolation of +1 spins for all n ≥ 0. We denote by Cx the cluster at x (for a configuration σ ), i.e., the maximal connected set B ∈ T such that x ∈ B and σy = σx for all y ∈ B. We write Cx (n) to indicate the cluster at x for σ n . Cx (n) is random and its distribution is denoted µn . Eµn denotes expectation with respect to µn . Proposition 2.3. For λ = 1/2, the following two properties are valid. 1. For almost every σ 0 , there is no percolation in σ n of either +1 or −1 spins for any n ∈ [0, ∞]. 2. The mean cluster size in σ n is infinite for any n ∈ [0, ∞] : for any x ∈ T, Eµn (|Cx |) = Eµ0 (|Cx (n)|) = ∞.

(9)

Proof. To prove the first claim, notice that at time zero the origin is almost surely surrounded by both a plus and a minus m-loop. Those loops are stable and prevent the cluster at the origin, be it a plus or a minus cluster, from percolating at all subsequent times. Therefore the probability that the origin belongs to an infinite cluster is zero for all n ≥ 0. To prove the second claim, we note that because of the absence of percolation for either sign, it follows from a theorem of Russo [44] applied to the triangular lattice, that the mean cluster size of both plus and minus clusters diverges. Before stating our main theorem, we need some more definitions to formulate the continuum scaling limit. We adopt the approach of [3] (see also [4]). 2.2. Compactification of R2 . The scaling limit δ → 0 can be taken by focusing on fixed finite regions, ⊂ R2 , or by treating the whole R2 . The second option is more convenient, because it avoids technical issues that arise near the boundary of . A convenient way of dealing with the whole R2 is to replace the Euclidean metric with a distance function d(·, ·) defined on R2 × R2 by d(u, v) = inf (1 + |φ|2 )−1 ds, (10) φ

where the infimum is over all smooth curves φ(s) joining u with v, parametrized by arclength s, and | · | denotes the Euclidean norm. This metric is equivalent to the Euclidean metric in bounded regions, but it has the advantage of making R2 precompact. Adding a ˙ 2 which is isometric, via stereographic single point at infinity yields the compact space R projection, to the two-dimensional sphere. 2.3. The space of curves. Denote by S the complete, separable metric space of con˙ 2 with a distance D(·, ·) based on the metric defined by Eq. (10) as tinuous curves in R follows. Curves are regarded as equivalence classes of continuous functions γ (t) from

320


˙ 2 , modulo monotonic reparametrizations. C will represent a particthe unit interval to R ular curve and γ (t), t ∈ [0, 1], a particular parametrization of C, while F will represent a set of curves. The distance D between two curves, C1 and C2 , is defined by D(C1 , C2 ) ≡ inf sup d(γ1 (f1 (t)), γ2 (f2 (t))), f1 ,f2 t∈[0,1]

(11)

where γ1 and γ2 are particular parametrizations of C1 and C2 , and the infimum is over the set of all monotone (increasing or decreasing) continuous functions from the unit interval onto itself. The distance between two closed sets of curves is defined by the induced Hausdorff metric as follows: dist(F, F ) ≤ ε ⇔ ∀ C ∈ F, ∃ C ∈ F with D(C, C ) ≤ ε, and vice-versa.

(12)

For each fixed δ > 0, the random curves that we consider are polygonal paths in the hexagonal lattice δH, dual of δT, consisting of connected portions of the boundaries between plus and minus clusters in δT. A subscript δ may then be added to indicate that the curves correspond to a model with a “short distance cutoff.” The probability measure µnδ denotes the distribution of the random set of curves Fδn consisting of the polygonal paths on δH generated by Tn acting on σ 0 .

2.4. Main results. Since the cellular automaton is deterministic, all the percolation models σ n for n ≥ 0 are automatically coupled, once they are all constructed on the single probability space ( , , µ0 ) on which σ 0 is defined. The following theorem is valid for any λ, but λ = 1/2 is the only interesting case, therefore we restrict attention to it. Theorem 1. For λ = 1/2, the Hausdorff distance between the system of random curves Fδ0 at time 0 and the corresponding system of curves Fδn at time n goes to zero almost surely as δ → 0, for each n ∈ [1, ∞]; i.e., for µ0 -almost every σ 0 , lim dist(Fδ0 , Fδn ) = 0, for any n ∈ [1, ∞].

δ→0

(13)

The main application of the theorem is that the scaling limits of our family F1n of percolation models, if they exist, must be the same for all n ∈ [0, ∞]: Corollary 2.1. Suppose that critical site percolation on the triangular lattice has a unique scaling limit in the Aizenman-Burchard sense [3], i.e., Fδ0 converges in distribution as δ → 0 (for λ = 1/2). Then, for every n ∈ [1, ∞], Fδn converges in distribution to the same limit. Remark 2.1. A complete proof of the convergence in distribution of Fδ0 as δ → 0 (i.e., of the existence of the “full scaling limit” of independent site percolation on the triangular lattice as discussed in [47, 12]) has not yet appeared in the literature. A paper by two of the authors with a proof of this fact and of other properties of the limit, starting from Smirnov’s results [48, 49], is in preparation [13]. We also note that the proofs below show that the convergence in Theorem 1 and Corollary 2.1 is uniform in n.


321

3. Proofs In this section, we give the proofs of the main results. We start by reminding the reader of the definitions of m-path and m-loop and by giving some new definitions and two lemmas which will be used later. Definition 3.1. A path between x and y in δT (embedded in R2 ) is an ordered sequence of sites (ζ0 = x, . . . , ζk = y) with ζi = ζj for i = j and ζi+1 ∈ N (ζi ). A loop is a sequence (ζ0 , . . . , ζk+1 = ζ0 ) with k ≥ 3 such that (ζ0 , . . . , ζk ) and (ζ1 , . . . , ζk+1 ) are paths. We call a path (ζ0 , . . . , ζk ) (resp., a loop (ζ0 , . . . , ζk+1 = ζ0 )) an m-path (resp., an m-loop) if ζi−1 and ζi+1 are not neighbors for i = 1, . . . , k − 1 (resp., for i = 1, . . . , k + 1, where ζk+2 = ζ1 ). Notice that for every path between x and y, there always exists at least one m-path

˜ between the same sites, that is contained in . Definition 3.2. A boundary path (b-path) ∗ is an ordered sequence of distinct dual edges (η0∗ = ηζ∗0 ,ξ0 , . . . , ηk∗ = ηζ∗k ,ξk ) such that either ζi+1 = ζi and ξi+1 ∈ N (ξi ) or ξi+1 = ξi and ζi+1 ∈ N (ζi ), and σζi = −σξi for all i = 0, . . . , k. We call a maximal boundary path simply a boundary; it can either be a doubly-infinite path or a finite loop. b-paths represent the (random) curves introduced in the previous section for a fixed ˙2 value of the short distance cutoff δ. A collection of such curves in the compact space R is indicated by Fδ . b-paths ∗ are parametrized by functions γ (t), with t ∈ [0, 1]. When we write that, ∗ for t in some interval [t1 , t2 ] (the interval could as well be open or half-open), γ (t) ∈ ηx,y we mean that the parametrization γ (t) for t between t1 and t2 is irrelevant and can be chosen, for example, so that | dγdt(t) | is constant for t ∈ [t1 , t2 ]. The notation ∗ (u, v), where u and v can be dual sites or generic points of R2 ∩ ∗ , stands for the portion of

∗ between u and v. Definition 3.3. A stable loop l (for some σ ) is an m-loop (ζ0 = x, . . . , ζk = x) such that σζ0 = σζ1 = . . . = σζk . ∗ is stable if x and y belong to stable loops Definition 3.4. We say that a dual edge ηx,y of opposite sign.

Lemma 3.1. In general, a constant sign m-path (ζ0 , . . . , ζk ) is “fixated” (i.e., retains that same sign in σ n for all 0 ≤ n ≤ ∞) if ζ0 and ζk are fixated Proof. The claim of the lemma is straightforward. Lemma 3.2. For λ = 1/2, there is a one to one mapping from boundaries in Fδn+1 to “parent” boundaries in Fδn . Proof. Let n∗ ∈ Fδn be a boundary at time n and int( n∗ ) be the set of sites of T (i.e., hexagons) “surrounded” by n∗ . Let C be the (unique) constant sign cluster contained in int( n∗ ) that has sites next to n∗ . Suppose that sites x and y in C do not change sign at time n + 1, then, at that time, they must belong to the same cluster C . (This means that a cluster cannot split in two or more pieces under the effect of the dynamics.) The reason

322


is that x and y are connected by an m-path at time n because they are in the same cluster, and since they have not flipped between time n and time n + 1, Lemma 3.1 (or more accurately, the single time step analogue of Lemma 3.1) implies that the sites in

have not flipped either, so that x and y are still connected at time n + 1 and therefore belong to the same cluster. If at least one site x in C does not change sign at time n + 1, we say that C has survived (and evolved into a new cluster C that contains x at time n + 1). From what we just said, the sites of C that retain the same sign form a unique cluster C . We call C the parent cluster of C and the external boundary n∗ ∈ Fδn of C the parent of the ∗ ∈ Fδn+1 of C (note that C is a.s. finite – see Proposition 2.3). external boundary n+1 To prove that there is a one to one mapping between boundaries in Fδn+1 and those in Fδn , it remains to show that the parent boundary of each element of Fδn+1 is unique. (This means that clusters cannot merge under the effect of the dynamics.) If this were not the case, then a cluster C could have two or more distinct parent clusters, C1 , C2 , . . . . Notice that each of the parent clusters at time n is surrounded by a constant sign m-loop i , i = 1, 2, . . . , which is stable. Suppose that x1 ∈ C1 and x2 ∈ C2 retain their sign at time n + 1. Without loss of generality, we shall assume that this sign is plus. Assuming that C1 and C2 are both parents of C , x1 and x2 should both belong to C at time n + 1 and therefore be connected by a plus-path. But this contradicts the fact that, at time n, x1 ∈ int( 1 ) and x2 ∈ int( 2 ), that is, they belong to the interiors of two disjoint stable minus-loops. Notice that boundaries cannot be “created,” but can “disappear,” as complete clusters are “eaten” by the dynamics. 3.1. Proof of Theorem 1. Let us start, for simplicity, with the case of a single b-path. For any ε > 0, given a b-path 0∗ at time 0, parametrized by some γ (t), we will find a path n∗ at time n with parametrization γ (t) such that, for δ small enough, sup d(γ (t), γ (t)) ≤ ε + 2δ

(14)

t∈[0,1]

and vice-versa (i.e., given n∗ and γ , we need to find 0∗ and γ so that (14), in which the dependence on the scale factor δ has been suppressed, is valid). Later we will require that this holds simultaneously for all the curves in Fδ0 and Fδn , as required by Eq. (13). Let B 1 (R) be the ball of radius R, B 1 (R) = {u ∈ R2 : |u| ≤ R} in the Euclidean metric, and B 2 (R) = {u ∈ R2 : d(u) ≤ R} the ball of radius R in the metric (10). For a given ε > 0, we divide R2 into two regions: B 1 (6/ε) and R2 \ B 1 (6/ε). We start by showing that, thanks to the choice of the metric (10), one only has to worry about curves (or polygonal paths) that intersect B 1 (6/ε). In fact, the distance between any two points ˙ 2 \ B 1 (6/ε) satisfies the following bound: u, v ∈ R ∞ d(u, v) ≤ d(u, ∞) + d(v, ∞) ≤ 2 [1 + (s + 6/ε)2 ]−1 ds < ε/3. (15) 0

˙ 2 \ B 1 (6/ε), it can be approxiThus, given any curve in Fδ0 contained completely in R n 2 1 ˙ \ B (6/ε), and vice-versa. The existence mated by any curve in Fδ also contained in R ˙ 2 \ B 1 (6/ε) contains of such curves in Fδ0 is clearly not a problem, since the region R an infinite subset of δT and therefore there is zero probability that it doesn’t contain any


323

b-path at time zero. There is also zero probability that it contains no stable b-path at time zero, but any such b-path also belongs to Fδn . Before we can proceed, we need the following lemma, which is a consequence of the the fact that at time zero we are dealing with a Bernoulli product measure. In this lemma (and elsewhere), the diameter diam(·) of a subset of R2 is defined by using the Euclidean metric. Lemma 3.3. Let η0∗ be any (deterministic) dual edge; then for some constant c > 0, µ0 ∃ ∗ η0∗ : diam ( ∗ ) ≥ M and ∗ does not contain at least one stable edge ≤ e−cM .

(16)

Proof. To prove the lemma, we partition the hexagonal lattice into regions Qi as in Fig. 5. We then do an algorithmic construction of ∗ , starting from η0∗ , as a percolation exploration process, but with the additional rule that, when the exploration process hits the boundary of any Qi for the first time, all the hexagons in Qi are checked next (according to some deterministic order). From every entrance point of Qi , there is a choice of the values of the spins of the outermost layer of Qi that forces ∗ to enter Qi . Therefore, when ∗ hits Qi , it always has a positive probability of entering the region. We call Fi the event that a (dual) stable edge is found inside Qi , belonging to ∗ . It is easy to see that such an event has positive probability, bounded away from zero by a constant that does not depend on how the exploration process enters the region Qi . In fact, from any entrance point, there is clearly a choice of the values of the spins in Qi that forces ∗ to cut Qi in two symmetric parts, containing spins of opposite sign. Now, if diam( ∗ ) ≥ M, then ∗ must clearly visit at least O(M) different regions Qi . The conclusion of the proof should now be clear (see, for example, [19]). With this lemma, we can now proceed to the proof of the theorem. As explained before, we restrict attention to paths that intersect B 1 (6/ε). Given a b-path 0∗ =

Fig. 5. Elementary cell for the partition of H used in Lemma 3.3. Notice that the cell is made out of seven smaller cells, each of them formed by seven hexagons

324


(η0∗ = ηζ∗0 ,ξ0 , . . . , ηk∗ = ηζ∗k ,ξk ) in Fδ0 with parametrization γ (t), call u0 the point in R2 where the first (dual) edge η0∗ begins. The following algorithmic construction produces a sequence u0 , . . . , uN of points in 0∗ . 1. Start with u0 . 2. Once u0 , . . . , ui have been constructed, if ui ∈ B 1 (6/ε) take the ball Bu1i (ε/3) centered at ui and of radius ε/3 and let ui+1 be the first intersection of 0∗ \ 0∗ (u0 , ui ) with ∂Bu1i (ε/3), if ui ∈ / B 1 (6/ε) take the ball Bu2i (ε/3) centered at ui and of radius ε/3 and let ui+1 be the first intersection of 0∗ \ 0∗ (u0 , ui ) with ∂Bu2i (ε/3). 3. Terminate when there is no next ui . During the construction of the sequence u0 , . . . , uN , 0∗ is split in N + 1 pieces, the first N having diameter at least ε/3. The construction also produces a sequence of balls j j Bu00 (ε/3), . . . , BuNN (ε/3), with ji = 1 or 2. Notice that no two successive ui ’s can lie outside of B 1 (6/ε). In fact, if for some i, ui lies outside of B 1 (6/ε), ui+1 belongs to ∂Bu2i (ε/3), which is contained inside B 1 (6/ε), due to the choice of the metric. Each ui contained in B 1 (6/ε) lies on an edge of δH, but no more than one ui can lie on the same edge since 0∗ is self-avoiding and cannot use the same edge or site more than once. Also, the number of ui ’s lying outside of B 1 (6/ε) cannot be larger than (one more than) the number of the ui ’s lying inside B 1 (6/ε). Therefore, N ≤ const × (εδ)−2 . j j ji For any two successive balls, Buii (ε/3) and Bui+1 i+1 (ε/3), let Oi = Bui (ε/3 + δ) ∪ ji+1 ∗ Bui+1 (ε/3 + δ). Now assume that there exists a sequence η¯ 0∗ , . . . η¯ N −1 of stable (dual) ∗ ) is contained in O (for fixed edges of 0∗ , with η¯ i∗ contained in 0∗ (ui , ui+1 ). 0∗ (η¯ i∗ , η¯ i+1 i ε and small enough δ). Also contained in Oi are the two paths (i.e., paths in δT which ∗ ), may be thought of as sequences of hexagons) whose hexagons are next to 0∗ (η¯ i∗ , η¯ i+1 one on each side. From those two paths one can extract two subsets that are m-paths which are stable since the first and last hexagons of each one of them is stable (such hexa∗ ). The two m-paths so constructed gons must be stable since they are next to η¯ i∗ and η¯ i+1 ∗ ) is constitute a “barrier” that limits the movements of the boundary, so that n∗ (η¯ i∗ , η¯ i+1 in fact confined to lie within those two m-paths and thus within Oi . To parametrize n∗ , we use any parametrization γ (t) such that γ (t) = γ (t) whenever γ (t) ∈ η¯ i∗ . Using this ∗ ) parametrization and the previous fact, it is clear that the distance between 0∗ (η¯ i∗ , η¯ i+1 ∗ ) does not exceed ε + 2δ, the diameter of O . Therefore, conditioning and n∗ (η¯ i∗ , η¯ i+1 i ∗ on the existence of the above sequence η¯ 0∗ , . . . , η¯ N−1 of stable (dual) edges of 0∗ , we can conclude that sup d(γ (t), γ (t)) ≤ ε + 2δ.

(17)

t∈[0,1] ∗ It remains to prove the existence of the sequence η¯ 0∗ , . . . , η¯ N −1 of stable (dual) edges. ∗ To do that, let us call Ai the event that 0 (ui , ui+1 ) does not contain at least one stable ∗ edge, and let A = ∪N−1 i=0 Ai be the event that at least one of the first N pieces of 0 does not have any stable edge. Then, considering that the total number of edges contained in B 1 (6/ε) is bounded by const × (εδ)−2 and using Lemma 3.3, we have

µ0δ (A) ≤ (εδ)−2 e−c (ε/δ)

(18)

for some c > 0. Equation (18) means that the probability of not finding at least one stable edge in each of the first N pieces of 0∗ is very small and goes to 0, for fixed ε,


325

as δ → 0. This is enough to conclude that, with high probability (going to 1 as δ → 0), Eq. (17) holds. This proves one direction of the claim, in the case of a single curve. To obtain the other direction, notice that a large b-path at time n is part of a complete boundary n∗ which must come from a line of “ancestors” (see Lemma 3.2) that starts with some 0∗ at time 0. Therefore, one can apply the above arguments to 0∗ , provided that the latter is large enough. Although the proof of this last fact is very simple, it is convenient to state it as a separate lemma. Lemma 3.4. Set δ = 1 for simplicity; then for any boundary n∗ at time n, there is an ancestor 0∗ , with diam( 0∗ ) ≥ diam( n∗ ) − 1. Proof. The existence of an ancestor 0∗ comes from Lemma 3.2, so we just have to show that diam( 0∗ ) ≥ diam( n∗ ) − 1. 0∗ is surrounded by a connected set of hexagons that touch 0∗ and whose spins are all the same (this is the “external boundary” of the set of hexagons that are in the interior of 0∗ ). From this set, one can extract an m-path of constant sign whose diameter is bounded above by diam( 0∗ ) + 1. Since such a constant sign m-path is stable for the dynamics, n∗ must lie within its interior. This concludes the proof. At this point, we need to show that the above argument can be repeated and the construction done simultaneously for all curves in Fδ0 and Fδn (for each n). First of all notice that, for a fixed ε, any b-path ∗ of diameter less than ε/2 can be approximated by a closest stable edge, provided that one is found within the ball of radius ε/2 that contains the ∗ , with the probability of this last event clearly going to 1 as δ → 0, when we restrict attention to B 1 (6/ε). For a b-path outside B 1 (6/ε), we already noticed that it can be approximated by any other b-path also outside B 1 (6/ε). As for the remaining b-paths, notice that the total number of boundaries that intersect the ball B 1 (6/ε) cannot exceed const × (εδ)−2 (in fact, the total number of pieces in which the boundaries that intersect B 1 (6/ε) are divided cannot exceed const × (εδ)−2 ). So, we can carry out the above construction simultaneously for all the boundaries that touch B 1 (6/ε), having to deal with at most const × (εδ)−2 segments of b-paths of diameter of order at least ε. Therefore, letting Yδn = dist(Fδ0 , Fδn ), we can apply once again Lemma 3.3 and conclude that µ0 (Yδn > ε) ≤ (εδ)−2 e−c

(ε/δ)

.

(19)

To show that Yδn → 0 as, δ → 0, µ0 -almost surely and thus conclude the proof, it suffices to show that, ∀ε > 0, µ0 (lim supδ→0 Yδn > ε) = 0. To that end, first take a sequence δk = 1/2k and notice that ∞ k=0

µ0 (Yδnk > ε) ≤

∞ k 4 k=0

ε2

e−c

2k ε

< ∞,

(20)

where we have made use of (19). Equation (20) implies that we can apply the BorelCantelli lemma and deduce that µ0 (lim supk→∞ Yδnk > ε) = 0, ∀ε > 0. In order to handle the values of δ not in the sequence δk , that is for those δ such that δk+1 < δ < δk for some k, we use the following double bound, valid for any 0 < α < 1, αd(u, v) ≤ d(αu, αv) ≤

1 d(u, v), α

(21)

326


n ≤ 1 Y n . The two bounds in Eq. (21) come from writing which implies that αYδnk ≤ Yαδ k α δk d(αu, αv) as d(αu, αv) = inf φ (1 + |φ |2 )−1 ds = α inf φ (1 + α 2 |φ|2 )−1 ds, where φ (s ) are smooth curves joining αu with αv, while φ(s) are smooth curves joining u with v. The proof of the theorem is now complete.

3.2. Proof of Corollary 2.1. The corollary is an immediate consequence of Theorem 1 and of the following general fact, of which we include the proof for completeness. Lemma 3.5. If {Xδ }, {Yδ } (for δ > 0), and X are random variables taking values in a complete, separable metric space S (whose σ -algebra is the Borel algebra) with {Xδ } and {Yδ } all defined on the same probability space, then if Xδ converges in distribution to X and the metric distance between Xδ and Yδ tends to zero almost surely as δ → 0, Yδ also converges in distribution to X. Proof. Since Xδ converges to X in distribution, the family {Xδ } is relatively compact and therefore tight by an application of Prohorov’s Theorem (using the fact that S is a complete, separable metric space – see, e.g., [7]). Then, for any bounded, continuous, real function f on S, and for any ε > 0, there exists a compact set K such that |f (Xδ )|I{Xδ ∈K} / dP < ε and |f (Yδ )|I{Xδ ∈K} / dP < ε for all δ, where I{·} is the indicator function and P the probability measure of the probability space of {Xδ } and {Yδ }. Thus, for small enough δ, | f (Xδ )dP − f (Yδ )dP | < |f (Xδ ) − f (Yδ )|I{Xδ ∈K} dP + 2ε < 3ε, (22) where in the last inequality we use the uniform continuity of f when restricted to the compact set K and the fact that the metric distance between Xδ and Yδ goes to 0 as δ → 0. To conclude the proof of the corollary, it is enough to apply Lemma 3.5 to {µ0δ }, µsl (or, to be more precise, to the random variables of which those are the distributions), for each n ∈ [1, ∞], where µsl is the unique scaling limit of critical site percolation on the triangular lattice. {µnδ },

4. Dependent Site Percolation Models on the Hexagonal and Triangular Lattice The model that we have presented and discussed in Sect. 2 has been chosen as a sort of paradigm, but is not the only one for which such results can be proved. In fact, it is not the original model for which such results were obtained. In this section we describe some percolation models on the hexagonal lattice and prove that they have the same scaling limit as critical (independent) site percolation on the triangular lattice. None are independent percolation models, but nonetheless, they represent explicit examples of critical percolation models on different lattices with the same scaling limit. Besides, the construction of the models on the hexagonal lattice can be seen as a simple and natural way of producing percolation models for which all the sites of the external (site) boundary of any constant sign cluster C belong to a unique cluster C of opposite sign. In other words, this implies that the boundaries between clusters of opposite sign form a nested collection of loops, a property that site percolation


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on the triangular lattice possesses automatically because of the self-matching property of T (which is crucial in Smirnov’s proof of the existence and uniqueness of the scaling limit of crossing probabilities and Cardy’s formula). 4.1. The models. The percolation models that we briefly describe here can be constructed by means of a natural zero-temperature Glauber dynamics which is the zerotemperature case of Domany’s stochastic Ising ferromagnet on the hexagonal lattice [18]. The cellular automaton (i.e., Domany’s stochastic Ising ferromagnet at zero temperature) that gives rise to those percolation models can also be realized on the triangular lattice with flips when a site disagrees with six, five and sometimes four of its six neighbors. The initial state σ 0 consists of an assignment of −1 or +1 with equal probability to each site of the hexagonal or triangular lattice (depending on which version of the cellular automaton we are referring to). In the first version, H, as a bipartite graph, is partitioned into two subsets A and B which are alternately updated so that each σx is forced to agree with a majority of its three neighbors (which are in the other subset). In the second version, all sites are updated simultaneously according to a rule based on a deterministic pairing of the six neighbors of every site into three pairs. The rule is that σx flips if and only if it disagrees with both sites in two or more of its three neighbor pairs; thus there is (resp., is not) a flip if the number Dx of disagreeing neighbors is ≥ 5 (resp., ≤ 3) and there is also a flip for some cases of Dx = 4. These percolation models on H and T are investigated in [15], where Cardy’s formula for rectangular crossing probabilities is proved to hold in the scaling limit. The discrete time cellular automaton corresponding to the zero-temperature case of Domany’s stochastic Ising ferromagnet on the hexagonal lattice can be considered as a simplified version of a continuous time Markov process where an independent (rate 1) Poisson clock is assigned to each site x ∈ H, and the spin at site x is updated (with the same rule as in our discrete time dynamics) when the corresponding clock rings. (In particular, they have the same stable configurations – see Fig. 6.) The percolation properties of the final state σ ∞ of that process were studied, both rigorously and numerically, in [27]; the results there (about critical exponents rather than the continuum scaling limit) strongly suggest that that dependent percolation model is also in the same universality class as independent percolation. Similar stochastic processes on different types of lattices have been studied in various papers. See, for example, [10, 20, 22, 37–40] for models on Zd and [26] for a model on the homogeneous tree of degree three. Such models are also discussed extensively in the physics literature, usually on Zd (see, for example, [18] and [35]). Numerical simulations have been done by Nienhius [41] and rigorous results for both the continuous and discrete dynamics have been obtained in [14], including a detailed analysis of the discrete time (synchronous) case. Let us now describe in more detail the two deterministic cellular automata (on H and T). Later, we will prove the equivalence of the percolation models generated by those cellular automata. Zero-temperature Domany model. Consider the homogeneous ferromagnet on the hexagonal lattice H with states denoted by σ = {σx }x∈H , σx = ±1, and with (formal) Hamiltonian σx σy , (23) H(σ ) = − x,y

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Fig. 6. Example of a (local) stable configuration for the zero-temperature Domany dynamics. Heavy lines on edges of H connect, say, plus spins, while heavy broken lines connect minus spins. The dotted lines drawn on edges of the dual lattice are the perpendicular bisectors of unsatisfied edges, indicating the boundaries between plus and minus clusters. Every spin has at least two neighbors of the same sign, since only loops and barbells are stable under the effect of the dynamics

where x,y denotes the sum over all pairs of neighbor sites, each pair counted once. We write N H (x) for the set of three neighbors of x, and indicate with σx σ y (24) x H(σ ) = 2 y∈N H (x)

the change in the Hamiltonian when the spin σx at site x is flipped (i.e., changes sign). The hexagonal lattice H is partitioned into two subsets A and B in such a way that all three neighbors of a site x in A (resp., B) are in B (resp., A). By joining two sites of A whenever they are next-nearest neighbors in the hexagonal lattice (two steps away from each other), we get a triangular lattice (the same with B) (see Fig. 7). The synchronous dynamics is such that all the sites in the sublattice A (resp., B) are updated simultaneously. We now define the discrete time Markov process σ n , n ∈ N, with state space SH = {−1, +1}H , which is the zero temperature limit of a model of Domany [18], as follows: • The initial state σ 0 is chosen from a symmetric Bernoulli product measure. • At odd times n = 1, 3, . . . , the spins in the sublattice A are updated according to the following rule: σx , x ∈ A, is flipped if and only if x H(σ ) < 0. • At even times n = 2, 4, . . . , the spins in the sublattice B are updated according to the same rule as for those of the sublattice A. Cellular automaton on T. We define here a deterministic cellular automaton Q on the triangular lattice T, with random initial state chosen by assigning value +1 or −1 independently, with equal probability, to each site of T.


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Fig. 7. A star-triangle transformation

Given some site x¯ ∈ T, group its six T-neighbors y in three disjoint pairs {y1x¯ , y2x¯ }, {y5x¯ , y6x¯ }, so that y1x¯ and y2x¯ are T-neighbors, and so on for the other two pairs. Translate this construction to all sites x ∈ T, thus producing three pairs of sites {y1x , y2x }, {y3x , y4x }, {y5x , y6x } associated to each site x ∈ T. (Note that this construction does not need to specify how T is embedded in R2 .) Site x is updated at times m = 1, 2, . . . according to the following rule: the spin at site x is changed from σx to −σx if and only if at least two of its pairs of neighbors have the same sign and this sign is −σx . {y3x¯ , y4x¯ },

4.2. Equivalence between the models on H and T. We show here how the models on the hexagonal and on the triangular lattice are related through a star-triangle transformation. More precisely, we will show that the dynamics on the triangular lattice T is equivalent to the alternating sublattice dynamics on the hexagonal lattice H when restricted to the sublattice B for even times n = 2m. To see this, start with T and construct an hexagonal lattice H by means of a star-triangle transformation (see, for example, p. 335 of [23]) such that a site is added at the center of each of the triangles (x, y1x , y2x ), (x, y3x , y4x ), and (x, y5x , y6x ) (the sites yix are defined in the previous subsection). H may be partitioned into two triangular sublattices A and B with B = T. One can now see that the dynamics on T for m = 1, 2, . . . and the alternating sublattice dynamics on H restricted to B for even times n = 2m are the same. An immediate consequence of this equivalence between the two cellular automata is that the two families of percolation models that they produce are also equivalent in an obvious way through a star-triangle transformation. To be more precise, the percolation models defined on T by Q for times m = 1, 2, . . . are the same as those defined on B by the zero-temperature Domany model for even times n = 2m. 4.3. Results for the zero-temperature Domany model. In this section we explain how the results obtained for the percolation models µnδ generated by the cellular automaton T are

330


also valid for the percolation models defined by the zero-temperature Domany model. Because of the results of the previous section, we can actually consider the percolation models µ˜ nδ on δT generated by Q, for which we have results analogous to Theorem 1 and Corollary 2.1. Results equivalent to Propositions 2.1, 2.2 and 2.3 are contained in [14]. The proof of the main theorem (i.e., the analogue of Theorem 1) is basically the same as for the models generated by T, so we just point out the differences. We follow here the setup and notation of [14], but give all the relevant definitions in order to make this paper self-contained. Let us consider a loop in the triangular sublattice B, written as an ordered sequence of sites (y0 , y1 , . . . , yk ) with k ≥ 3, which are distinct except that yk = y0 . For i = 1, . . . , k, let ζi be the unique site in A that is an H-neighbor of both yi−1 and yi . We call an s-loop if ζ1 , . . . , ζk are all distinct. Similarly, a (site-self avoiding) path (y0 , y1 , . . . , yk ) in B, between y0 and yk , is called an s-path if ζ1 , . . . , ζk are all distinct. Notice that any path in B between y and y (seen as a collection of sites) contains an s-path between y and y . An s-loop of constant sign is stable for the dynamics since at the next update of A the presence of the constant sign s-loop in B will produce a stable loop of that sign in the hexagonal lattice. Similarly an s-path of constant sign between y and y will be stable if y and y are stable — e.g., if they each belong to an s-loop. A triangular loop x1 , x2 , x3 ∈ B with a common H-neighbor ζ ∈ A is called a star; it is not an s-loop. A triangular loop in B that is not a star is an s-loop and will be called an antistar, while any loop in B that contains more than three sites contains an s-loop. With these definitions, the proof of the main theorem for the zero-temperature Domany model is the same as that presented in Sect. 3 for our prototypical model, with the role of m-loops in the original proof played here by s-loops (in particular antistars), and that of m-paths by s-paths (see Sect. 3 above). 4.4. An amusing further example: Totally synchronous dynamics on H. The model that we consider here corresponds to the zero-temperature Glauber dynamics on the hexagonal lattice with all sites updated simultaneously at discrete times, without alternating between two subsets (contrary to the case of the zero-temperature Domany model), with initial configuration σ 0 chosen according to a symmetric Bernoulli product measure. Let us partition H in two subsets A and B as before and define the family of perco2m (at even times) lation models {µ¯ m ¯m A , m = 0, 1, . . . }, where µ A is the distribution of σ restricted to the subset A (naturally endowed with a triangular lattice structure, so that µ¯ 0A is the distribution of critical site percolation). The main difference consists in the fact that σ n does not fixate as n → ∞ since there is a positive density of spins that flip infinitely many times. To see this, consider a loop in H containing an even number of sites and such that at time zero its spins are alternately plus and minus. At any time n, every spin in has two neighbors of opposite sign and will therefore flip at the next update. Thus, the spins in never stop flipping. However, there is a simple observation that tremendously simplifies the analysis of this totally synchronous dynamics. Namely, that if we restrict attention to sublattice A at odd times n = 1, 3, 5, . . . and sublattice B at even times n = 0, 2, 4, . . . , the dynamics is identical to the zero-temperature Domany dynamics discussed above (let us call this σan ). On the other hand, if we instead observe A at even times and B at odd times, this is identical to an alternative zero-temperature Domany type dynamics σbn but with the n ∞ first (and third and . . . ) update on B rather than A. Furthermore, (σan )∞ n=0 and (σb )n=0 are completely independent of each other. We conclude that there are two distinct limits σa∞ and σb∞ (independent of each other) and that the scaling limit of σ n restricted to


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either A or B is the same as for independent critical percolation on T. But it appears that for any n, σ n on all of H should be subcritical and thus have a trivial scaling limit. References 1. Aizenman, M.: The geometry of critical percolation and conformal invariance. In Stat. Phys. 19, H. Balin, ed., Singapore: World Scientific, 1995 2. Aizenman, M.: Scaling limit for the incipient spanning clusters. In: Mathematics of Multiscale Materials; the IMA Volumes in Mathematics and its Applications, K. Golden, G. Grimmett, R. James, G. Milton, P. Sen, eds., Berlin-Heidelberg-New York: Springer, 1998 3. Aizenman, M., Burchard, A.: Holder regularity and dimension bounds for random curves. Duke Math. J. 99, 419–453 (1999) 4. Aizenman, M., Burchard, A., Newman, C.M., Wilson, D.B.: Scaling Limits for Minimal and Random Spanning Trees in Two Dimensions. Random Struct. Alg. 15, 319–367 (1999) 5. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry of critical fluctuations in two dimensions. J. Stat. Phys. 34, 763–774 (1984) 6. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B241, 333–380 (1984) 7. Billingsley, P.: Convergence of Probability Measures. New York: John Wiley & Sons, Inc., 1968 8. Camia, F.: Scaling Limit and Critical Exponents for 2D Bootstrap Percolation. Preprint 9. Camia, F.: Universality in Two-Dimensional Enhancement Percolation. In preparation 10. Camia, F., De Santis, E., Newman, C. M.: Clusters and recurrence in the two-dimensional zero-temperature stochastic Ising model. Ann. Appl. Probab. 12, 565–580 (2002) 11. Camia, F., Newman, C.M.: The Percolation Transition in the Zero-Temperature Domany Model. J. Stat. Phys., to appear 12. Camia, F., Newman, C.M.: Continuum Nonsimple Loops and 2D Critical Percolation. J. Stat. Phys., to appear 13. Camia, F., Newman, C.M.: In preparation 14. Camia, F., Newman, C.M., Sidoravicius, V.: Approach to fixation for zero-temperature stochastic Ising models on the hexagonal lattice. In: In and out of equilibrium: Probability with a Physics Flavor V. Sidoravicius, ed., Progress in Probability 51, Basel-Boston: Birkhäuser, 2002, pp. 163–183 15. Camia, F., Newman, C.M., Sidoravicius, V.: Cardy’s Formula for some Dependent Percolation Models. Bull. Brazilian Math. Soc. 33, 147–156 (2002) 16. Cardy, J.L.: Critical percolation in finite geometries. J. Phys. A 25, L201–L206 (1992) 17. Cardy, J.: Lectures on Conformal Invariance and Percolation. Preprint arXiv:math-ph/0103018 18. Domany, E.: Exact results for two- and three-dimensional Ising and Potts models. Phys. Rev. Lett. 52, 871–874 (1984) 19. Fontes, L.R., Newman, C.M.: First passage percolation for random colorings of Zd . Ann. Appl. Probab. 3, 746–762 (1993) 20. Fontes, L.R., Schonmann, R.H., Sidoravicius, V.: Stretched exponential fixation in stochastic Ising models at zero temperature. Commun. Math. Phys. 228, 495–518 (2002) 21. Gandolfi, A., Keane, M., Russo, L.: On the uniqueness of the infinite occupied cluster in dependent two-dimensional site percolation. Ann. Probab. 16, 1147–1157 (1988) 22. Gandolfi, A., Newman, C.M., Stein, D.L.: Zero-temperature dynamics of ±J spin glasses and related models. Commun. Math. Phys. 214, 373–387 (2000) 23. Grimmett, G.R.: Percolation. Second edition. Berlin: Springer, 1999 24. Guggenheim, E.A.: The Principle of Corresponding States. J. Chem. Phys. 13, 253–261 (1945) 25. Harris, T.E.: A correlation inequality for Markov processes in partially ordered state spaces. Ann. Probab. 5, 451–454 (1977) 26. Howard, C.D.: Zero-temperature Ising spin dynamics on the homogeneous tree of degree three. J. Appl. Probab. 37, 736–747 (2000) 27. Howard, C.D., Newman, C.M.: The percolation transition for the zero-temperature stochastic Ising model on the hexagonal lattice. J. Stat. Phys. 111, 57–72 (2003) 28. Lawler, G., Schramm, O., Werner, W.: Values of Brownian intersection exponents I: Half-plane exponents. Acta Math. 187, 237–273 (2001) 29. Lawler, G., Schramm, O., Werner, W.: Values of Brownian intersection exponents II: Plane exponents. Acta Math. 187, 275–308 (2001) 30. Lawler, G., Schramm, O., Werner, W.: Values of Brownian intersection exponents III: Two-sided exponents. Ann. Inst. Henri Poincaré 38, 109–123 (2002) 31. Lawler, G., Schramm, O., Werner, W.: Analyticity of intersection exponents for planar Brownian motion. Acta Math. 189, 179–201 (2002)

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Communications in


Short Distance Expansion from the Dual Representation of Infinite Dimensional Lie Algebras S. James Gates, Jr.1 , W.D. Linch, III1 , Joseph Phillips1 , V.G.J. Rodgers2 1

Center for String and Particle Theory, Department of Physics, University of Maryland, College Park, MD 20742-4111, USA. E-mail: [email protected]; [email protected]; [email protected] 2 Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242–1479 USA. E-mail: [email protected] Received: 1 April 2003 / Accepted: 1 November 2003 Published online: 19 February 2004 – © Springer-Verlag 2004

Abstract: We develop a method for computing the short distance expansion of fields or operators that live in the coadjoint representation of an infinite dimensional Lie algebra by using only properties of the adjoint representation and its dual. We explicitly implement this method by computing the short distance expansion for the duals of the Virasoro algebra, affine Lie algebras and the geometrically realized N -extended supersymmetric GR Virasoro algebra. This method can also be used to compute short distance expansions between fields that transform in the adjoint and those that transform in the coadjoint representations. 1. Introduction The Virasoro algebra is at the heart of understanding string theory and low dimensional gravitational theories. In string theory and conformal field theories it is often thought of as a derived quantity that comes from the mode expansion of an energy-momentum tensor. For mathematicians it also has meaning in its own right as the one dimensional algebra of centrally extended Lie derivatives. Representations of the Virasoro algebra are used to classify conformal field theories and also provide important clues as to the nature of string field theories. One representation, the coadjoint representation, has been the focus of investigations for two distinct reasons. One is that the orbits of the coadjoint representation under the action of the Virasoro group have a relationship with unitary irreducible representations of the Virasoro algebra [34, 25, 41, 37]. For the Virasoro algebra and also affine Lie algebras in one dimension these orbits can then be directly related to two dimensional field theories that are conformal field theories. These are the geometric actions [33, 2, 13]. Another reason for studying these representations comes when one studies the elements of the coadjoint representation and adjoint representation as conjugate variables

Supported in part by National Science Foundation Grant PHY-0099544 and PHY-0244377

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S.J. Gates, Jr., W.D. Linch, III, J. Phillips, V.G.J. Rodgers

of a field theory [26]. The adjoint elements in these constructions are the conjugate momenta if they generate the isotropy algebra of the coadjoint elements. Field theories constructed in this fashion are called transverse actions [9, 10] (with respect to the coadjoint representation) since the geometric actions constructed on the orbits are transverse to these transverse actions. One example of the distinction of these two types of actions for the SU(N) affine Lie algebra or what physicists sometimes call an SU(N) Kac-Moody algebra is its geometric action, an SU(N) WZNW model, and its transverse action, the two dimensional SU(N)Yang-Mills action. For the Virasoro algebra the geometric action is given by the two dimensional Polyakov action for gravity [32] and its corresponding transverse action given by the N = 0 “affirmative action” [22, 8]. Besides these constructions, other areas of interest for the coadjoint representation are the BTZ black holes [4, 5] that appear in the asymptotic Brown-Henneaux symmetry [11] on AdS3 [42, 30, 4]. On the other hand, the short distance expansion between fields and operators appears in a variety of settings which includes regulating fields that appear in path integrals, computing correlation functions in statistical field theories [6] and operator product expansions in quantum field theories [7]. Quite often the operator product expansions rely on a Lagrangian where one may extract a propagator to describe the expansion. For free field theories, this is usually accompanied with a normal ordering procedure. In this note we would like to take a primordial view of the short distance expansion which is along the lines of Wilson’s non-Lagrangian approach [7] but where in our case the algebra (its adjoint representation) and its dual (the coadjoint representation) describe a notion of the short distance expansion. This will be accomplished by describing a phase space where the coadjoint representation provides coordinates of a phase space and the adjoint representation are the conjugate momentum variables. We will construct Noether charges on the phase space which will correspond to the adjoint action on the variables. From there it is straightforward to write the short distance expansion for the elements of the dual representation with themselves or with elements of the adjoint representation. The algebras used as examples in this note will be the pure Virasoro algebras, the semi-direct product of the Virasoro algebra with an affine Lie Algebra, and a particular supersymmetric extension of the Virasoro algebra for an arbitrary number of supersymmetries where the adjoint representation is realized through bosonic and fermionic derivative operators.

1.1. The role of the coadjoint representation for the Virasoro and affine Lie alegbras. In any space-time dimension the Lie algebra of coordinate transformations can be written as Lξ ηa = −ξ b ∂b ηa + ηb ∂b ξ a = (ξ ◦ η)a ,

(1)

and the algebra of Lie derivatives satisfies, [Lξ , Lη ] = Lξ ◦η .

(2)

In one dimension, we can centrally extend this algebra by including a two cocycle which is coordinate invariant and satisfies the Jacobi identity. We write [(Lξ , a), (Lη , b)] = (Lξ ◦η , (ξ, η)),

(3)

Short Distance Expansion of Infinite Dimensional Lie Algebras

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where the two cocycle depends on the D-dimensional metric gab (used to define the connection) and a rank two tensor1 Dab , c h (ξ, η) = 2π (ξ a ∇a ∇b ∇c ηc ) dx b + 2π (ξ a Dab ∇c ηc ) dx b − (ξ ↔ η). (4) Here the index structure is left intact in order to show the invariance of the two cocycle. Since this is a one dimensional structure, one may ignore the indices as long as one is mindful of the tensor structure and write, c h (ξ, η) = 2π (ξ η − ξ η) dx + 2π (ξ η − ξ η)(D + hc ( + 21 2 )) dx, (5) where is the derivative with respect to the coordinate. In this form, it is easy to see that (5) contains the familiar “anomaly” term that arises in string theory. The two cocycle then can be reduced to c h (ξ, η) = 2π (ξ η − ξ η) dx + 2π (ξ η − ξ η) B dx, (6) dependent upon a pseudo-tensor, B = (hD + c ( + 21 2 )), which transforms as δB = −2ξ B − ξ B − c ξ

(7)

under infinitesimal coordinate transformations. This pseudo tensor absorbs the metric contribution in the central extension as well as the tensor D. B is said to transform in the coadjoint representation of the Virasoro algebra. Different choices of B will give different centrally extended algebras. In string theory, central extensions are commonly chosen so that an SL(2,R) subalgebra is centerless. Thus the coadjoint representation might be thought of as endemic to the central extension of the algebra. For example, B = 0 is the choice commonly used when the metric is fixed to gab = 1 and Dab = 0. The vector fields ξ for the one dimensional line are moded by ξ = CN x N+1 and where the integration is defined through the contour shown in Fig. 2 for w = 0. The realization for the Virasoro algebra is then [LN , LM ] = (N − M) LN+M + (cN 3 − cN ) δN+M,0 .

(8)

One can get the same central extension for the algebra on the circle in two distinct ways: either by making a change of coordinates where x = exp (iω), giving the complex metric g(ω) = exp (2iω), or by choosing g(τ ) = 1 and Dab = gab on the circle with h = −c. In this case the vector fields are ξ = CN exp (iN ω). As a parenthetic remark, one may wish to view the two cocycle (ξ, η) as a functional of the metric. Upon variation of (ξ, η) with respect to the metric gab and assuming that Dab is independent of the metric, one finds δ (ξ, η) = J + J , δg

(9)

where the current is J = η ξ − η ξ . For constant J, one recognizes this variation as the anomalous one dimensional “energy-momentum” tensor. Examples of higher dimensional versions of non-central extensions may be found in [28, 29, 14]. 1

While it is possible to choose Dab = gab , at this stage we do not impose this condition.

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Another way in which the coadjoint representation appears is directly through the construction of a representation that is dual to the adjoint representation. In this case one starts with the centrally extended algebra. Then using a suitable pairing between the algebra and its dual, one extracts the coadjoint action of the algebra. As an example consider the semi-direct product of the Virasoro algebra with an affine Lie algebra. Then the Virasoro algebra [LN , LM ] = (N − M) LN+M + cN 3 δN+M,0

(10)

β β JNα , JM = i f αβγ JN+M + N k δN+M,0 δ αβ

(11)

α α = −M JN+M , LN , JM

(12)

is augmented with and

where [τ α , τ β ] = if αβγ τ γ . The algebra is realized by LN = ξNa ∂a = ieiNθ ∂θ ,

JNα = τ α eiNθ ,

(13)

so that the centrally extended basis can be thought of as the three-tuple, β LA , JB , ρ .

(14)

The adjoint representation acts on itself as β α LA , JB , ρ ∗ LN , JM , µ = (Lnew , Jnew , λ) ,

(15)

where Lnew = (A − N ) LA+N , β α Jnew = −M JA+M + BJB+N + if

βα λ λ JB+M ,

λ = (cA3 )δA+N ,0 + Bkδ α β δB+M ,0 .

(16) α µ

A typical basis for the dual of the algebra can be written as the three-tuple L N , JM ,

.

Using the pairing,

α µ L , J β, ρ

= δN,A + δ αβ δM,B + ρ µ L A B N , JM ,

(17)

and requiring it to be invariant, one defines the coadjoint representation through the action of the adjoint on this dual as [27] β α α , µ˜ = L˜ new , J˜M ,0 with, (18) LA , JB , ρ ∗ L˜ N , J˜M 3 ˜ L˜ new = (2A − N )L˜ N−A −Bδ αβ L˜ M−B − µ(cA ˜ )L−A α α J˜M = (M − A)J˜M−A − if

βνα ˜ν JM−B

− µB ˜ k J˜−B . β

and

(19) (20)


337

Since we are interested in how we can build physical fields from these algebras it is more instructive to use explicit tensors instead of the mode decomposition. The adjoint representation may be thought of as a three-tuple, F = (ξ (θ) , (θ) , a)

(21)

ξ a,

containing a vector field coming from the Virasoro algebra, a gauge parameter coming from the affine Lie algebra and a central extension a. The coadjoint element is the three-tuple, B = (D (θ ) , A (θ) , µ) ,

(22)

which consists of a rank two pseudo tensor Dab , a gauge field Aa and a corresponding central element µ. In this way the coadjoint action can be written as

δB F = (ξ (θ) , (θ) , a) ∗ (D (θ) , A (θ ) , µ) = (δD (θ ) , δA (θ) , 0) ,

cµ ξ + δD (θ) = 2ξ D + D ξ + 2π

− T r A ,

hµ 2π ξ

coordinate transformation

(23) (24)

gauge trans

and δA(θ ) = A ξ + ξ A − [ A − A ] + k µ . coord trans

(25)

gauge transformation

Again denotes derivative with respect to the argument. 1.2. Geometric actions. There are two important features that can be extracted from the above transformation laws. The first notion is the space of all coadjoint elements that can be reached by making a finite transformation on B generated from the algebra. This space is a coadjoint orbit [25, 40]. The collections of all orbits foliates the dual space of the algebra. One of the most interesting features of these orbits is that each one admits a symplectic two form, i.e. a Poisson bracket structure, whose integration over a suitable two manifold gives a natural physical action. These actions are called Geometric Actions. The geometric actions have been constructed for the Kac-Moody and Virasoro cases [33, 13, 2] separately and correspond to the WZNW model and two dimensional Polyakov gravitation and supergravity when the coadjoint elements are set to zero. For the case of the semi-direct product of the Virasoro and affine Lie algebras the geometric action is [27, 26] 1 S = 2π dτ dθ D ∂∂τθ ss 1 + 2π dτ dθ dλ Tr A (θ ) ∂∂λθ ss ∂θ g −1 ∂τ g − ∂∂τθ ss ∂θ g −1 ∂λ g (∂ 2 s)2 ∂2s βc + g −1 ∂λ g, g −1 ∂τ g dτ dθ (∂ θs)2 ∂τ ∂θ s − θ 3 ∂τ s − 48π (∂θ s) θ βk − 4π dτ dθ Tr g −1 ∂θ gg −1 ∂τ g βk + 4π dλdτ dθ Tr g −1 ∂θ g, g −1 ∂λ g g −1 ∂τ g . (26)

338


In the above the field s(θ, τ, λ) is a two parameter family of coordinate transformations that are generated by ξ , and similarly g(θ, τ, λ) is a two parameter family of gauge group transformations generated by . 1.3. Transverse actions. The second type of physical action that can be constructed arises when one tries to describe the dynamics of the field Dab . If we look in the KacMoody sector alone we see that the dynamics for the gauge field Aa can come from Yang-Mills theory. In calculating the two dimensional anomaly, which is precisely the geometric action, the gauge field serves as a background field that is used in regulating the theory [3]. Since Yang-Mills is a theory that describes the dynamics of the gauge field, Yang Mills theory may be called a Transverse Action to the geometric action. The distinction between the geometric action and the transverse action is that for the geometric action, the coadjoint element is not dynamical and the symplectic structure lives on coadjoint orbits. There the physical fields in the geometric action are just the group elements, s(θ ) and g(θ ). In the case of transverse actions, the coadjoint elements become dynamical variables and are able to describe moving from one coadjoint orbit to another. The group transformations are gauge and coordinate transformations. The requirement for the proper description of the transverse action so that its Hamiltonian does not propagate gauge degrees of freedom is that the momentum conjugate to the coadjoint elements live in the isotropy algebra for the coadjoint elements. This is just a statement of the Gauss Law constraint. This Gauss Law constraint comes from setting Eq.(25) and Eq.(24) to zero and treating the corresponding algebraic elements as the conjugate momenta. For the pure Kac-Moody case, i.e. setting Eq.(25) to zero, the Gauss Law constraint is [E(x) A(x) − A(x) E(x)] + k µ E(x) = 0

(27)

which is found in two dimensional Yang-Mills theory. Recall that the initial data in two dimensional Yang-Mills transforms as Ai (x) → U (x)Ai U −1 (x) − g1 ∂i U (x)U −1 (x), Ei (x) → U (x)Ei U −1 (x),

(28)

where Ai (x) is an element of the coadjoint representation of an SU(N) affine Lie algebra on the line and Ei (x) is in the adjoint representation. Figure 1 shows how the dual of the algebra is foliated into parts relevant to the geometric actions and transverse actions. The transverse actions for two dimensions were constructed in [9, 10] for the Virasoro algebra, and for the super Virasoro algebra [22, 8] where the actions where dubbed “affirmative actions” since the supersymmetry put the bosons and fermions (the two genders) on equal footing. In the purely bosonic case the action is √ Sdiff = − d 2 x g q1 X lmr Da r Xmla + 2X lmr Dla X a rm (29) 2 √ 1 ab − d x g 4 X b ∇l ∇m X lm a + β2 X bga Xbga , where Xmnr = ∇ r D mn . Here β and q are constants. If one varies this action with respect to the space-time component D10 and setting D10 = 0, one recovers the expression Xlm0 ∂1 Dlm − ∂m (X ml0 Dl1 ) − ∂l (X ml0 Dm1 ) − q ∂1 ∂l ∂m X lm0 = 0.

(30)


339

Geometric and Transverse Actions

Geometric Actions

Transverse Actions

Fixed Coadjoint Orbit Here A and D are background Fields

Geometric Actions: action describes collective coordinates about fixed A and D Transverse Actions: Field equations yield constraints for A and D and dynamics

Fig. 1. The foliation of the dual of the algebra

In 1 + 1 this corresponds to the isotropy equation (i.e. setting Eq.[24] to zero in the absence of a gauge field) found on the coadjoint orbit, where D corresponds to the quadratic differential, i.e. XD + 2X D + q X = 0 → ξ D + 2ξ D + q ξ = 0.

(31)

The adjoint element ξ corresponds to the conjugate momentum, X ≡ X110 , of D = D11 . 2. As Phase Space Variables In order to build the transverse actions above new structures such as general coordinate covariance, gauge invariance, lagrange multipliers and/or tensors with components in higher dimensions have to be introduced. What we would like to do in this work is to show that irrespective of any action that described the dynamics of the coadjoint elements, there is a natural notion of the “short distance expansion" that exists for the elements of the coadjoint representation of the affine Lie algebras, Virasoro algebras and extensions of the Virasoro algebra. To show this one takes advantage of the fact that adjoint and coadjoint elements in these particular algebras can be used as phase space elements.

340


To see how one may use the adjoint and coadjoint representations as phase space elements we will use the familiar Yang-Mills example in two space-time dimensions. After gauge fixing the time component of the two-vector potential, the remaining spatial component of the vector potential is a canonical coordinate, Aai (x) and the electric field Eia (x) its conjugate momentum as realized through the equal time Poisson bracket relations, {Aai (x), Abj (y)} = 0, {Eia (x), Ejb (y)} = 0, {Aai (x), Ejb (y)} = δ ab δij δ(x, y).

(32)

We have left the index structure on the fields even though they are only the one dimensional spatial components. Algebraically, the equal time commutation relations for a quantum field theory may be written directly through the correspondence principle especially since the central extension is already present in the Poisson bracket algebra. The transformation laws for these phase space variables under the one dimensional gauge transformations is given by Ai (x) → U (x)Ai U −1 (x) − g1 ∂i U (x)U −1 (x), Ei (x) → U (x)Ei U −1 (x),

(33)

where one recognizes that Ai (x) is the second element of the three-tuple in Eq. (22) of the element of the coadjoint representation of an SU(N) affine Lie algebra on the line and Ei (x) is in the adjoint representation (the second element in Eq. (21). In this way one sees that these representations which are dual to each other are indeed phase space elements. If we were to consider the generating function for spatial gauge transformations, G(x)a = ∂i Eia + [Ei , Ai ]a ,

(34)

then one can construct the charge Q =

dx Ga a (x)

with

(35)

{Q , E(x)} = [(x), E(x)], {Q , A(x)} = [(x), A(x)] − g1 ∂(x).

(36)

For a generic function on the phase space, say F (A, E), {Q , F (A, E)} = (x)

δ ˆ δ (x)

F (A, E),

(37)

where ˆδ is the functional variation in the direction of (x). δ (x) Similarly, the Virasoro algebra and its dual produce a set of phase space variables. Let the one dimensional pseudo tensor, Dij , correspond to the first element in Eq.(22) with conjugate momentum given by the rank two tensor density of weight one, X ij . The generator corresponding to the one dimensional coordinate transformation is given by Ga (x) = Xlm ∂a Dlm − ∂l (X lm Dam ) − ∂m (Dla X lm ) − c ∂a ∂l ∂m X lm .

(38)


341

From the Poisson brackets, one can recover the transformation laws of X ab and Dab . We have Qξ = dx Ga ξ a , (39) where ξ a corresponds to ξ in the algebra. One has {Qξ , Dlm (x)} = −ξ a (x) ∂a Dlm (x) − Dam (x) ∂l ξ a (x) − Dla (x) ∂m ξ a (x) − c ∂a ∂l ∂m ξ a (x) = −2ξ (x) D(x) − ξ(x) D (x) − c ξ(x) , {Qξ , X lm (x)} = ξ a (x)∂a X lm (x) − (∂a ξ l (x)) X am (x) − (∂a ξ m (x)) Xla (x) + (∂a ξ a (x))X lm (x) = ξ(x) X (x) − ξ (x) X(x).

(40)

Again for a generic function on the phase space, say F (D, X), {Qξ , F (D, X)} = ξ(x) where

δ δ ξˆ (x)

δ F (D, X), ˆ δ ξ (x)

(41)

is the functional variation in the direction of ξ(x).

3. Short Distance Expansions for Coadjoint Representation We can now define a procedure for writing the short distance expansion [39] for elements of the coadjoint representation of affine Lie algebras and Virasoro type algebras. For concreteness we will take the coadjoint elements of the Virasoro and affine Lie algebras to be real and analytic tensors in the variable x or y (or τ in the graded Lie algebra cases). These representations will later be graded to incorporate supersymmetry. This property is preserved under smooth infinitesimal coordinate transformations and gauge transformations corresponding to smooth analytic vector fields satisfying the algebra of Eqs.(3,4). Equations (37) and (41) are the prototype expressions that we need to proceed. For any element of the adjoint representation, say α, we can construct Qα . Now α is dual to coadjoint elements, say A since (α, A) = α(x)A(x) dx = some constant. (42) We treat the above expression as the integral form of the operation with any function of the phase space F (x), {Qα , F (x)} = α(x) =

δ F (x) δ α(x) ˆ

α(y) (A(y)G(x)) dy.

δ . δ αˆ

Then for

(43) (44)

The distribution fA (x, y) = A(y)F (x) when compared to the left-hand side of Eq. (44) gives the short distance expansion between A(x) and F (x). In the following we derive short distance expansions for some simple cases and then move on to the N-extended super Virasoro algebra.

342


3.1. Case: Virasoro and affine Lie algebras. As our first example we consider the short distance expansion for the semi-direct product of the Virasoro algebra and an affine Lie algebra. We are interested in the short distance expansion of the coadjoint elements D(x) and A(x). Equations 24 and 25 allow us to construct the charge icµ Qξ = X(y) 2ξ (y) D(y) + D(y) ξ + ξ 12 + E(y) ξ (y)A(y) + A (y)ξ(y) . (45) Then from {Qξ , A(x)} we have that {Qξ , Ab (x)} = ξ(x)

δ δ ξˆ (x)

Ab (x) =

ξ(y) D(y) Ab (x) dy .

(46)

This implies that D(y) Ab (x) = (∂y δ(x, y)) Ab (x) − (∂x Ab (x)) δ(x, y).

(47)

We write the delta function as, δ(x, w) =

1 , 2πi(x − w)

(48)

with the understanding that this is defined through the contour integral in Fig. 2 in the limit that R → ∞ and r → 0. For functions f (x) whose analytic continuation f (ζ ) vanishes when |ζ | → ∞ in the upper half plane and which remains analytic on the real axis we have ∞ ∞ f (x) 1 f (w) = f (x) δ(x, w) dx ≡ dx. (49) 2πi (x − w) −∞ −∞

y

R

w r

Fig. 2. The contour for defining the delta function

x


343

With this we have that D(y) Ab (x) =

−1 1 Ab (x) − ∂x Ab (x). 2 2πi(y − x) 2π i(y − x)

(50)

Similarly we can construct a charge Q via Q = −X(y)Ab (y) b (y) − if bac E b(y) a (y)Ac (y) + kµ E b (y) b (y) dy, (51) then together QL and Qξ will give −1 1 c D(x) − , ∂x D(x) − 2πi(y − x) πi(y − x)2 2π i(y − x)4 −1 1 D(y) Ab (x) = Ab (x) − ∂x Ab (x), 2 2πi(y − x) 2πi(y − x) 1 Ab (y) D(x) = Ab (x), 2πi(y − x)2 1 ikµ Ab (y) Aa (x) = δ ba − i f bac Ac (x). 2 2πi(y − x) 2π i(y − x) D(y) D(x) =

(52) (53) (54) (55)

3.2. Case: N=1 super Virasoro algebras. Supersymmetric algebras can be treated in a similar way. From the discussion found in Siegel [35], let M represent an element of the phase space and let MN be a supersymplectic two form with an inverse MN , i.e. MN P N = δPM . The index M = {m, µ}, where the Latin indices are bosonic and the Greek indices fermionic. Then [M , N } = MN ,

(56)

and {MN ] = 0. This means that (mn) = [µν] = mν + νm = 0. The bracket is defined by ←

[A, B} = −A

∂B ∂ NM . ∂M ∂N

(57)

For the N=1 super Virasoro algebra, the adjoint representation is built from the vector field ξ , a spinor field and and a central extension so that we might write an adjoint element as F = (ξ, , a). In the same way the coadjoint representation is given by a

344


three-tuple of fields, B = (D, ψ, α). The transformation law for the coadjoint representation is F ∗ B = −ξ D − 21 ψ − 23 ψ − c8β ξ , −ξ ψ − 21 ψ − 23 ξ ψ − 4iβc , 0 . (58) The charges are Qξ = X(y) −ξ(y) D(y) −

cβ 8

ξ dy + φ(y) −ξ ψ(y) − 23 ξ ψ(y) dy, (59)

and Q = X(y) − 21 (x)u(y) − 23 (y) ψ(y) dy + φ(y) − 21 (y) − 4iβc dy.

(60)

Here the fields X(y) and φ(y) correspond respectively to the spin −1 and spin − 21 conjugate momenta for D and ψ, which gives the short distance expansion 1 3cβ −1 D(x) − , ∂x D(x) − 2 2πi(y − x) πi(y − x) 8π i(y − x)4 −3 1 D(y) ψ(x) = ∂x ψ(x), ψ(x) − 2 4πi(y − x) 2πi(y − x) 3 1 ψ(y) D(x) = ψ(x) − ψ(x), 4πi(y − x)2 4πi(y − x) −2 4βc ψ(y) ψ(x) = . D(x) − π(y − x) π(y − x)3

D(y) D(x) =

(61) (62) (63) (64)

4. N -Extended GR Super Virasoro Algebra In the pioneering work of Ref.[1], the Virasoro algebra was extended to an arbitrary number of supersymmetries. Since then several other ways of extending the Virasoro algebra were constructed from two dimensional super conformal field theories [18, 19]. There N-extended current algebras are constructed from bosonic and fermionic ghosts from a conformal field theory and the operator product expansions are computed by use of an energy-momentum tensor that is derived from the conformal field theory. As our last example, we focus on an extension of the super Virasoro algebra that are graded Lie algebras and which are realized explicitly through bosonic and fermionic first order derivatives. The version of the N Extended super Virasoro algebra that we will use as our last example is the one found in [20, 21]. The generators of this algebra provide an (almost [12]) primary basis to the K(1 | N ) contact superalgebra2 . This is the subalgebra of vect(1 | N) vector fields that preserves the contact one form σ = dτ + δI J ζ I dζ J . 2

We thank Thomas Larsson for pointing this out to us.


345

The generators are I ···Ip

QA1

I ···I = τ A ζ I1 · · · ζ Ip ∂τ , PA1 p+1 = τ A ζ I1 · · · ζ Ip ∂ Ip+1 ,

where A can be integer or half-integer moded. This algebra along with a complete classification of Lie superalgebras used in string theory are reviewed and discussed in [23]. In the “almost” primary basis the generators may be written as3 : the following: 1 1 A+ 2 A− 2 I K GA I ≡ i τ ζ ζ ∂K , (65) ∂ I − i 2 ζ I ∂τ + 2( A + 21 )τ LA ≡ − τ A+1 ∂τ + 21 (A + 1) τ A ζ I ∂I , (66) I ···Iq

UA1

I ···Ip

RA1

q

(q−2)

[ ] (A− 2 ≡ i (i) 2 τ

≡

p p [ ] (A− 2 ) I1 (i) 2 τ ζ

) I1

ζ · · · ζ Iq−1 ∂ Iq , q = 1, . . . , N + 1,

· · · ζ Ip τ ∂τ

, p = 2, . . . , N

(67) (68)

[I J] for any number N of supersymmetries. The antisymmetric part of the UA generator will generate an SO(N) subalgebra so we will denote it as [I J] . TAI J ≡ −UA

Note that the primary generators LA and GA I are used in lieu of RA and RIA1 respectively while RIA1 I2 is the only generator that is not primary. The elements of the algebra can be realized as fields whose tensor properties can be determined by the way they transform under one dimensional coordinate transformations. How each field transforms under a Lie derivative with respect to ξ is summarized below. In conformal field theory these transformation laws are generalized for two copies of the diffeomorphism algebra and characterize the fields by weight and spin. Here we treat the algebraic elements as one dimensional tensors. Table 1. Tensors associated with the algebra Element of algebra LA → η GIA T RS

→

χI

→

t RS

U V1 ···Vn → wV1 ···Vn T ···Tn

RA1

→ r T1 ···Tn

Transformation rule η →

− ξ η + ξ η

1 → + 2 ξ χ I RS RS t → − ξ (t ) 1 wV1 ···Vn → −ξ(wV1 ···Vn ) − 2 (n − 2)ξ wV1 ···Vn 1 r T1 ···Tn → − (r T1 ···Tn ) ξ − 2 (n − 2)ξ r T1 ···Tn

χI

−ξ(χ I )

Tensor structure ηa χ I;α t RS V ···V ; a w 1 α1n···αn T ···T ;a r 1 α1n···αn

In the above table we have used capital Latin letters, such as I, J, K to represent SO(N ) indices, small Latin letters to represent tensor indices, and small Greek letters for spinor indices. Spinors with their indices up transform as scalar tensor densities of weight one (1) while those with their indices down transform as scalar densities of weight minus one (−1). For example, the generator U V1 V2 V3 has a tensor density realization of contravariant tensor with rank one and weight − 23 living in the N × N × N representation of SO(N ), i.e. ωαV11αV22αV3 3 ;a . 3

We are grateful to Thomas Larsson for pointing out errors in the algebra in previous publications.

346


Table 2. Tensors associated with the dual of the algebra Dual element of algebra

Transformation rule

Tensor structure

L A → D

D → − 2ξ D − ξ D 3 ψ I → −ξ(ψ I ) − 2 ξ ψ I

Dab

G IA RS

T

U V1 ···Vn T ···Tr

R A1

→

ψI

→

ARS

→

ωV1 ···Vn

→ ρ T1 ···Tr

ARS

→

− (ξ ) ARS

−ξ

(ARS )

n → − (2 − 2 )ξ ωV1 ···Vn r ρ T1 ···Tr → − (ρ T1 ···Tr ) ξ − (2 − 2 )ξ ρ T1 ···Tr

ωV1 ···Vn

− ξ(ωV1 ···Vn )

I ψaα

ARS a V1 ···Vn ; α1 ···αn ωab T ···T ;α ···α ρab1 r 1 r

Thus for N supersymmetries there is one rank two tensor Dab , N spin- 23 fields ψ I , a spin-1 covariant tensor ARS that serves as the N (N − 1)/2 SO(N ) vector potentials (that are gauge potentials for N = 2) associated with the supersymmetries, N ( 2N ) fields for the ωV1 ···Vp and N ( 2N − N − 1) ρ T1 ···Tp fields. The entries in the third column of each of these tables are the one dimensional tensors or tensor densities corresponding to the ones that appear in the second column. 4.1. Central extensions of the N -extended GR algebra. In [12, 8] it was mistakenly reported that the N -Extended GR algebra admits a central extension for arbitrary values (I I ) I1 and the symmetric combination for UA1 2 generators were of N and further the UA omitted. It is easy to see that not only are these required to close the algebra but their existence will eliminate the central extensions for N > 2. (For N = 4 there is a subalgebra of the GR algebra that will admit a central extension [20].) Although the algebra is I fields form an Abelian extension to the algebra the obstruction not simple since the UA J1 J2 . This can be seen by looking at the comes from the required symmetric part of UA I J I K (GA , UB , GC ) contribution to the Jacobi identity. There one finds that J1 J2 J1 J2 K I I K [UBJ1 J2 , {GIA , GK C }] − {GC , [UB , GA ]} + {GA , [GC , UB ]} = 0

implies that 2 1 J K IJ IK 1 δ 2 −(A2 − 1 )δ J2 K δ Ij1 δ −2i(A−C) [UBJ1 J2 , UA A+B+C = 0. 4 +C ] − i c˜ (C − 4 )δ It is clear that the symmetric combination cannot satisfy the Jacobi identity. As we [I1 I2 ] are needed to close the algebra will show below, only the antisymmetric fields UA and a central extension exists. The constraint above forces the central extension in the SO(2) affine Kac-Moody algebra to be related to the central extension in the superdiffeomorphism algebra even before unitarity issues are considered. In what follows we will separate the N = 2 algebra from the generic case. 4.2. N = 2. For N = 2 the GR Super Virasoro Algebra contains twelve generators [21]. The usual N = 2 super Virasoro algebra is contained as a proper subgroup that admits a central extension. This subalgebra is [ LA , LB } = ( A − B ) LA+B + 18 c (A3 − A) δA+B,0 ,

(69)

IJK [ GA I , GB J } = − i 4 δ I J LA+B − i2(A − B) [ TAI J+B + 2(A + B) UA +B K ] 1 2 IJ −ic(A − 4 ) δA+B,0 δ , (70)


347

[ LA , GB I } = ( 21 A − B ) GA+B I ,

(71)

J [LA , TBI J } = − B TAI +B ,

(72)

[ TAI J , TBK L } = −2c (A − B)(δ IK δ JL − δ IL δ JK )δA+B,0 ,

(73)

[ TAI J , GB K } = 2 (δ JK GA+B I − δ IK GA+B J ),

(74)

where [I J] TAI J ≡ −UA

serves as the SO(2) generator. The remaining eight generators from the full algebra that IJ generator, two U I generators, three generators coming from the symmetric are the RA A (IJ) IJK . The role of these extra generators as combination, UA , and the two generators UA auxiliary fields in the N = 2 transverse action for this algebra is being investigated. 4.3. N > 2. The N > 2 the central extension is absent and new generators must be I . The commutation relations are included, including the “low order” generators UA [ LA , LB } = ( A − B ) LA+B ,

(75)

I1 ···Im [ LA , UBI1 ···Im } = − [ B + 21 (m − 2) A ] UA +B ,

(76)

[ GA I , GB J } = − i 4 δ I J LA+B − i2(A − B) IJK [ TAI J+B + 2(A + B) UA +B K ] ,

(77)

[ LA , GB I } = ( 21 A − B ) GA+B I ,

(78)

I1 ···Im I1 ···Im [ LA , RB } = − [ B + 21 (m − 2) A ] RA +B I1 ···Im J − [ 21 A (A + 1) ] UA +B J ,

(79)

J1 ···Jm I J1 ···Jm [ GA I , RB } = 2 (i)σ (m) [ B + (m − 1) A + 21 ] RA +B m J ···J J ···J − (i)σ (m) (−1)r−1 δ I Jr RA1 +Br−1 r+1 m r=1

J1 ···Jm I − (−i)σ (m) [ A + 21 ] UA +B I J1 ···Jm K + 2 (i)σ (m) [ A2 − 41 ] UA K , +B

m = 2,

(80)

J1 J2 IJ1 J2 J1 J2 I IJ1 J2 K 2 1 1 [ GA I , RB } = 2(A + B + 21 )RA +B − (A − 2 ) UA+B + 2(A − 4 ) UA+B K J2 J2 K 1 − δ I J1 21 GJA2 +B − UA +B + 2(A + B + 2 )UA+B K J1 J1 K 1 (81) + δ I J2 21 GJA1 +B − UA +B + 2(A + B + 2 )UA+B K ,

348

S.J. Gates, Jr., W.D. Linch, III, J. Phillips, V.G.J. Rodgers I J1 ···Jm [ GA I , UBJ1 ···Jm } = 2 (i)σ (m) [ B + (m − 2) A ] UA +B J ···J K − 2 (−i)σ (m) [ A + 21 ] δ I Jm UA1+Bm−1 K m−1 J ···J J ···J − (i)σ (m) (−1)r−1 δ I Jr UA1+Br−1 r+1 m r=1

J ···J

+ 2 (−i)σ (m) δ I Jm RA1 +Bm−1 ,

m = 2,

(82)

K IJ [ GA I , UBJ } = −2i δ IJ (LA+B + 21 (B − A) UA +B K ) − 2i (A + B + 1) UA+B, (83)

J1 J2 IJ1 J2 J1 J1 K IJ1 J2 I [ GA I , UBJ1 J2 } = −UA + 2B UA GA+B − UA +B δ +B + δ +B + 2B UA+B K , (84) I1 ···Im J1 ···Jn I1 ···Im J1 n···Jn [ RA , RB } = − (i)σ (mn) [ A − B − 21 (m − n) ] RA , (85) +B

I1 ···Im [ RA , UBJ1 ···Jn } = (−i)σ (mn)

+ i(i)

m

J ···J

(−1)r−1 δ Ir Jn RA1 +Bn−1

I1 ···Ir−1 Ir+1 ···Im

r=1 σ (mn)

I1 ···Im J1 ···Jn [ B − 21 (n − 2) ] UA , +B

m = 2, n = 1, (86)

I1 I2 I1 I2 J [ RA , UBJ } = (B + 21 )UA +B JI I2 I2 K + 21 UA δ11 − GIA2 +B + 2(A + B + 21 ) UA +B +B K I1 JI I2 I1 K 1 1 − 2 UA+B − GA+B + 2(A + B + 2 ) UA+B K δ 1 2

(87)

I1 ···Im [ UA , UBJ1 ···Jn } m I ···I J ···J J ···J J σ (mn) (−1)r−1 δ Im Jr UA1+Bm−1 1 r−1 r+1 n−1 n = − (i) r=1

−(−1)

mn

m

J ···J

(−1)r−1 δ Ir Jn UA1+Bn−1

I1 ···Ir−1 Ir+1 ···Im−1 Im

,

n = 1,

(88)

r=1

I1 [ UA , UBJ1 ···Jm } = −(i)σ (m)

n

J1 ···Jr ···Jn (−1)r−1 δ I1 Jr UA , +B ˆ

(89)

r=1

where the function σ (m) = 0 if m is even and −1 if m is odd. The central extensions c and c˜ are unrelated since we have only imposed the Jacobi identity. 4.4. Short distance expansion for D(y) O(x). The transformation laws for ξ(y) on the dual of the algebra allow us to extract the short distance expansion rules for the field D(y)


349

and any other element in the coadjoint representation O(x). Using the transformation rules ¯ = L¯ ˜ , D˜ = − 2 ξ D − ξ D − cβ¯ ξ δN,2 , Lξ ∗ (L¯ D , β) 8 D ¯ Lξ ∗ G

¯ Q Q¯

¯ Q¯ = G ˜¯

Q

¯ ¯ , ˜Q¯ = − ( 23 ξ ψ Q + ξ(ψ Q ) ) ,

¯¯ ¯¯ ¯¯ ¯¯ ¯¯ Lξ ∗ T¯τRR¯SS¯ = T¯τ˜RR¯SS¯ , τ˜ RS = −ξ τ RS − ξ (τ RS ) ,

n−2

¯ ¯ ¯ ¯ [ Lξ ∗ U¯ VV¯11······VV¯ nn = U¯ VV¯11······VV¯ nn + 2i (i) 2 ω ω˜

where

n

]−[ 2 ]

¯ ···V ¯ [V Vn−1 ]Vn , R¯ 1 V¯ 1 ···n−2 ¯n δ V ξ ω

(91) (92) (93)

ω˜ V1 ···Vn = ( n2 − 2) ξ ωV1 ···Vn − ξ(ωV1 ···Vn ) , ¯

¯

¯

¯

¯

¯

¯ ¯ ¯ ¯ Lξ ∗ R¯ TT1¯ 1······TT¯mm = R¯ TT1¯ 1······TT¯mm ,

(94)

ρ˜

ρ

where

(90)

T1 ···Tm − ξ (ρ T1 ···Tm ) , ρ˜ T1 ···Tm = ( m 2 − 2) ξ ρ ¯

¯

¯

¯

¯

¯

and [x] is the greatest integer in x. By constructing the generators we have the short distance expansion laws, D(y)D(x) = D(y) ψ Q (x) = D(y) ARS (x) = D(y) ωV1 ···Vn (x) = D(y) ρ V1 ···Vn (x) =

−1 1 3 c β δN,2 D(x) − , ∂x D(x) − 2 2πi(y − x) πi(y − x) 4π i(y − x)4 −3 1 ψ Q (x) − ∂x ψ Q (x), 4πi(y − x)2 2πi(y − x) −1 1 ∂x ARS (x), ARS (x) − 2πi(y − x)2 2πi(y − x) n−4 1 ωV1 ···Vn (x) − ∂x ωV1 ···Vn (x), 4πi(y − x)2 2π i(y − x) n−4 1 ∂x ρ V1 ···Vn (x) ρ V1 ···Vn (x) − 2 4πi(y − x) 2π i(y − x) n

(95) (96) (97) (98)

n−2

(i)[ 2 ]−[ 2 ] I1 ···In A + ω (x). 2π(y − x)3 A

(99)

It is worth commenting on the last summand that appears in Eq.(99). From Eq.(93), we notice that the transformation of ωV1 ···Vn (x) contributes to ρ I1 ···Im (x). This implies that the charge Qξ contains the summand q q−2 I ···I [ 2 ]−[ 2 ] 1 Qξ = · · · + i(i) ρ1 q−2 (x) ξ (x)ωI1 ···Iq−2 Iq−1 Iq (x) δq−1,q dx, 2 (100) I ···Iq−2

where ρ1

(x) is the conjugate momentum for ρ I1 ···Iq−2 (x). So {Qξ , ρ I1 ···Iq−2 (x)] = ξ(x)

δ ρ I1 ···Iq−2 (x), δξ(x)

(101)

350


will lead to n

D(y) ρ I1 ···In (x) = 2i i [ 2 ]−[

n−2 2 ] ∂ 2 δ(y, x) ωI1 ···In A (x) + (plus ρ I1 ···In y A

dependent terms). (102)

The existence of contributions like ξ as seen in Eq.(93) is due to the transformation of the connection . Although the connection does not appear in the algebra since gab = 1, its presence is felt in the coordinate transformations. 4.5. Short distance expansion for ψ I (y) O(x). The next sections follow in the same way as the D(y) O(x) expansions. One constructs the generators dedicated to the symmetry transformation of the particular algebraic element and then identifies the short distance expansion or operator product expansion from the variation. From [12] we can write the transformation laws due to the spin 21 fields as ¯ Q¯ ¯ = δ IQ¯ L¯ ˜ + T¯ IQ¯¯ + (1 − δ2N ) R¯ IQ¯ GIχ I ∗ G ξ Q I ˜ IQ ψ

where

χ ψ Q¯

A

¯ IQ − U¯ I

χ ψ Q¯

,

¯ ¯ ¯ ¯ ¯ ξ˜ = 21 (ψ Q ) χ I − 23 (χ I ) ψ Q , A˜ IQ = (2χ I ψ Q − 2χ Q ψ I ), (103)

¯ I I − 2i U˜ I I , ¯ = 4i G GIχ I ∗ (L¯ D , β) χ˜ χ D where

¯ δ2N (χ I ) ), χ˜ I = (−χ I D − βc

(104)

¯ ¯ ¯ S S δ RI ¯ R¯ R¯ δ IS¯ ) , χ R¯ = χ S = 2(χ I ) τ R¯ S + χ I (τ R¯ S ) , GIχ I ∗ T¯τRR¯ SS = 2i (G −G χ χ (105)

GIχ I ∗ U¯ ωVV11VV22 = −i U¯ I

(−2χ I ωV1 V2 −ωV1 V2 χ I )

δ V1 V2 + 2i U¯ V2V1 V2 (ω

χ I)

δ V1 I ,

N = 2, (106)

ILM GIχ I ∗ R¯ LM = −3i R¯ (χ I ρ LM ) ,

(107)

m+2

m ]−[ 2 ] ¯ [T¯ 1 ···T¯ m ] U I T¯ ···T¯ m m = 2 ) (χ ρ 1 m−2 [ m−1 ]−[ 2 ] I [T¯ 1 ¯ T¯ 2 ···T¯ m ] R I T¯ ···T¯ m δ − 2i 2 ¯ ¯ −(χ I )(ρ T1 ···Tm ) ) ((χ ) ρ 1 m+1 m+1 m [ ]−[ ] r−1 T¯ 1 ···T¯ r−1 I T¯ r+1 ···T¯ m

¯ ¯ [ GIχ I ∗ R¯ TT1¯ ······TT¯mm = 2i(i)m+1 (i) 2 ρ

1

− (i)(i) 2

2

(−1)

R¯

r=1

GIχ I ∗ U¯ VV¯11······VV¯ nn = −2i ¯

¯

n

¯ [ n−1 2 ]−[ 2 ] I [V1

δ

ω

¯ ¯ U¯ V2 ···Vn ] I

n−1

[ + 2(−1)n−1 (i) 2

n−1

[ + (i)(i) 2

n

]−[ 2 ]

¯

¯

(χ I ρ T1 ···Tm )

((n−4)(χ ) ωV1 ···Vn −(χ I )(ωV1 ···Vn ) )

n

¯n ]−[ 2 ] I[V

δ

n r=1

¯

¯

]K V¯ ···V¯ U¯ 1 I n−1 ¯ ···V ¯n V 1 ((χ ) ω

¯

)

K

¯ r+1 ···V ¯ ···V ¯ ¯n V ¯ V [V U¯ I1 V¯ 1 ···r−1 δ r ]I ¯n V (χ ω

)

¯

,

(108)

Short Distance Expansion of Infinite Dimensional Lie Algebras ¯

¯ [V2 +G

351 ¯ ¯

(−4i (χ I ) (ωV1 ···Vn ) −2i (χ I )(ωV1 ···Vn ) ) ¯ 1 ···V ¯ n−1 ] ¯ V n−1 I[V ¯

¯

R¯

¯

− (−1)

δ

− 2 δ I[V1 R¯

¯ n−2 ¯ 2 ···V V δ Vn−1 ]Vn ¯ ¯ ((χ I ) ωV1 ···Vn )

¯

n

¯

¯

δ V3 V4 δ V1 ]I δ n4

((χ I ) ωV1 ···Vn ) ¯

¯

n = 1, 2,

¯ IV1 ¯ IK ¯ KI GIχ I ∗ U¯ VV¯11 = −R¯ (χ I ωK ) − U(χ I ωK ) + U(ωK χ I ) . ω

(109) (110)

The short distance expansions that follow from here are: i 3 ψ I (x), ∂x (ψ I (x)) − 4π(y − x) 4π i(y − x)2 4βc −2 ψ I (y) ψ Q (x) = δ I Q D(x) − δ2N δI Q, π(y − x) π(y − x)2 π ψ A (y) ARS (x) = δ AR δ LS − δ AS δ LR ψ L (x), i(y − x) ψ I (y) D(x) =

(111) (112) (113)

A ···Aq−1

ψ J (y) ωAq1

(x) q−1 q+1 1 q J = (1 − δ2 ) 2(i)(i)q (i)[ 2 ]−[ 2 ] δA ρ A1 ···Aq−1 (x) q 2πi(y − x) q+1 q i [ 2 ]−[ 2 ] (q − 3) J A1 ···Aq−1 J A ···A − (x) − ∂x ωAq 1 q−1 (x) ωAq π i(y − x) (y − x) q−2

q

q−1

+ (1 − δ2 )(−1)(q−2) (−i)[ 2 ]−[ 2 1 A ] ω[A1 ···Aq−2 J δAqq−1 (x) 2 π i(y − x) q−1

q

]

(i)[ 2 ]−[ 2 ] A1 A Ar−1 I + δ[V1 · · · δVr−1 δVr · · · δVqq−1 δVq+1 ]Aq ωV1 ···Vq+1 (x) 2π i(y − x) q

r=1

q

q

iδ (1 − δ2N ) J[A1 A2 ] δ (1 − δ2N ) J[A1 A2 ] − 3 δ δ ψ (x) − i 2 ω (x), 2π(y − x) 2π(y − x) ψ J (y) ρ A1 ···Aq (x) =

(114)

q −(−1)(q+1) [ q+1 (i) 2 ]−[ 2 ] ωA1 ···Aq J (x) 2π(y − x) q iδ (1 − δ2N ) J[A1 A2 ] + 2 δ ψ (x) 2π(y − x)

q+1 q −(−1)(q−1) [ q+1 Aq Ar−1 Ar Ar+1 q A1 ]−[ 2 ] I J V1 ···Vq−1 2 (i) δ[V · · · δVr−1 δI δVr · · · δVq−1 (x)(1−δ2 ) ]δ ρ 1 2π(y − x) r=1

q−2 q−1 i [ 2 ]−[ 2 ] 1 J A1 ···Aq J A1 ···Aq − ρ − ∂x ρ πi(y − x) (y − x) q δ (1 − δ2N ) J[A1 A2 ] − i 2 ω (x). (115) δ 2π(y − x)

352


4.6. Short distance expansion for AJ K (y) O(x). As a generator of SO(N) transformations, the operator AJ K (y) takes on its own significance from the symmetric part of U J K (y). Here are some of the expansions for operators O(x) paired with AJ K (y): ¯ TtJJ KK ∗ G TtJJ KK

¯ Q ψ Q¯

¯ KJ K Q¯ δ QJ − G ¯ J J K Q¯ δ QK ) , = −2(G (t ψ ) (t ψ ) ¯

¯ ¯ ∗ U¯ VV¯11······VV¯ nn = − ω

n−1

¯

(116)

¯ ···V ¯ ¯ r+1 ··· ]V ¯n ¯ V | K |V (−1)r+1 (δ J [ V1 U¯ J2K V¯r−1 ¯ 1 ···Vn (t

r=1 ¯1 K[V

ω

)

¯ ···V ¯ ¯ r+1 ··· ]V ¯n V | J |V ) U¯ J2K V¯r−1 ¯ 1 ···Vn

−δ

(t

ω

)

¯ ···V ¯ ¯ [V ]J V δ nK + U¯ J K1 V¯ n−1 ¯ (t ω 1 ···Vn )

− U¯

¯ 1 ···V ¯ n−1 ] K [V ¯ ¯ (t J K ωV1 ···Vn )

¯

¯

δ Vn J

−i(−1)n−2 (δ K Vn δ J [V1 − δ J Vn δ K [V1 )R¯ ¯

TtJJ KK ∗ R¯ TT1¯ 1······TT¯mm = ¯

¯

ρ

m

¯

¯

¯ 2 ···V ¯ n−1 ] V ¯ ¯ J K ((t ) ωV1 ···Vn )

,

(117)

T¯ 2 ···T¯ r−1 | K | T¯ r+1 ···T¯ m ] ¯ ¯ (t J K ρ T1 ···Tm )

(−1)r+1 (δ [T1 | J | R¯ ¯

r=1 [T¯ 1 | K |

−δ

T¯ ···T¯ | J | T¯ r+1 ···T¯ m ] R¯ 2J K r−1 ) , T¯ 1 ···T¯ m (t

ρ

(118)

)

¯¯ ¯ ¯ ¯ ¯ ¯ JKR¯ S¯ TtJJ KK ∗ (T¯τRR¯SS¯ , β) = 21 (δ RJ δ SK −δ RK δ SJ ) L¯ ((t JK ) τ R¯ S¯ ) + 21 T¯(tAJ BK τ R¯ S¯ ) δAB +4β¯ T¯(τJKR¯ S¯ ) , ¯¯

¯ ¯

¯ ¯

¯

¯

JKRS δAB ≡ ( δ AK δ BS δ RJ − δ AK δ BR δ SJ + δ AS δ BJ δ RK

where

¯

¯

¯

¯

¯

¯

¯

¯

¯

− δ AR δ JS δ SK + δ AS δ BK δ RJ − δ AR δ KB δ SJ ¯ ¯

¯

+ δ AJ δ BS δ RK − δ JA δ RB δ SK + δ AS δ BK δ RJ ).

(119)

We get the AJ K (y) D(x) short distance expansion as: 1 RS J K RK SJ δ − δ δ δ ARS (x), 4πi(y − x)2 −1 AAB (y) ψ C (x) = δ AC ψ B (x) − δ BC ψ A (x) , πi(y − x) AJ K (y) D(x) =

AJ K (y) ARS (x) =

(120) (121)

1 β J KRS AB J R KS RK SJ A (x) + δ − δ δ δ , δAB 4πi(y − x) πi(y − x)2 (122)

AJ K (y) ωA1 ···An (x) =

n−1 (−1)r+1 KAr J A1 ···Aˆ r An−1 An ˆ δ ω (x) − δ J Ar ωKA1 ···Ar An−1 An (x) 2π(y − x) r=1 1 + (123) δ An J ωA1 ···An−1 K (x) − δ An K ωA1 ···An−1 J (x) , 2π i(y − x)


353

AJ K (y) ρ A1 ···An (x) n (−1)r+1 KAr J A1 ···Aˆ r An−1 An ˆ = ρ (x) − δ J Ar ρ KA1 ···Ar An−1 An (x) δ 2π(y − x) r=1 i(−i)n−1 J A1 ···An K KA1 ···An J + (x) − ω (x) . (124) ω 2π i(y − x)2 4.7. Short distance expansion for ωI1 ···Iq (y) O(x). The next two sections provide the short distance expansions for the fields ωI1 ···Iq and ρ P1 ···Pq . These fields appear in the GR realization as natural Grassmann extensions of the SO(N) gauge field and the one dimensional diffeomorphism field. The relevant transformation laws are : ¯J J , UµJ J ∗ L¯ D = −2 i G (µ D)

U

I1 ···Iq µ{Iq }

(125)

JK UµJ J ∗ U¯ ωKK , = R¯ (µ J ωK ) ,

(126)

¯ K K = −R¯ JKJ K , UµJ J ∗ G ψ (µ ψ )

(127)

¯ LM JN + 4R¯ JL δ MN , UµJ J ∗ U¯ ωLMN LMN = R(µ ω) δ (µω )

(128)

¯ J1 J J K δ K J2 − G ¯ J2 J J K δ KJ1 , UµJ1JJ12J2 ∗ U¯ ωKK = G (ω 1 2 ω ) (µ 1 2 ω )

(129)

¯ J2 J J K , ¯ K K = −δ J1 K G UµJ1J1J2J2 ∗ G ψ (υ 1 2 ψ )

(130)

¯ 1 ···V ¯m ∗ U¯ ωV{V m}

[ I ···I ] = −δ m q δ[V¯1 ···Vq¯ ] L¯ 4−q q 1

Vm Iq [ I1 ···Iq−1 ] q − 2 T¯(µω) δ[V¯ ···V¯ ] δm ¯

)µ ω−(

q−2 2 )µω )

1

(( 2 q q+1 ¯ 2 ···V ¯1 ¯m] ([ ]−[ 2 ]) m,(q+1) ¯ [V V G(−(q−2)µω − 2i 2 δ +(3−q)µ ω) δ[I ···Iq ] 1

q

q−1

q+1

¯ ···V ¯ ¯ m−1 ],V ¯m [V [ ]−[ 2 ] ¯ Iq V G(−µω) δ m,q+1 δ[I1 1···Iq−1m−2 + 2(−1)q (i) 2 ] δ

q

q−1 q−1 ]

[ ]−[ 2 − i(i) 2

m ¯ [Ir δ¯I1 · · · δ I¯r−1 δ I¯r+1 · · · δ I¯q ] G (−1)r−1 δq−1 (µω) [V V ] V V 1

r

r−1

m

r=1

+

q−1

¯ 1 V¯2 ¯r V¯r+1 ]V¯ Ir [V 2(−1)r+1 (T¯(µω) δ[I1 · · · δIVr−1 δIr+1 · · · δ]Iq m δ q,m )

r=1

q

m−q

{[ ]+[ 2 + i(i) 2

m+2

+2]−[ 2 ]}

m−q

[V ···Vq−1 ¯ Vq ···Vq+r−1 Iq Vq+r ···]Vm (−1)r−1 δ[ I11···Iq−1 ] Uµω ¯

¯

¯

¯

r=1

− (−1)q(m−q+2)

q

¯ ···V ¯ m−q+1 [ Ir I1 ···Ir−1 Ir+1 ··· Iq ] V δV¯ ¯ m−q+2+r ···V ¯m m−q+2 ···V

(−1)r−1 U¯ µω1

¯

¯

r=1

− (i)

q m−q q+m−4 2 +

2

−

2

[V1 ···Vm−q Vm−q+1 ··· ]Vm δ[ I1 ··· ]Iq , R¯ (µω) ¯

¯

¯

¯

(131)

354


U

I1 ···Iq µ{Iq }

m−q q m ¯ 1 ···T¯ m q(m−q+2) 2 +2 + 2 − 2 ∗ R¯ ρT{T = −i(−1) (i) m} ×

m−q+2

[ I ···I ] Tq ···T Iq T ···Tm (−1)r−1 δ[T¯ 1 ···T¯q−1 ] R¯ µρ q+r−1 q+r+1 . (132) ¯

1

¯

¯

¯

q−1

r=1

For q > 2 the short distance expansions are: 4−q 1 I1 ···Iq I1 ···Iq I1 ···Iq ω ω (y) D(x) = − (x) − ∂x ω (x) , 2πi(y − x) 2(y − x) q q+1 (−2i) 2 − 2 ωI1 ···Iq (y) ψ C (x) = 2πi(y − x) (3 − q) CI1 ···Iq CI1 ···Iq −(q − 2)∂x ω (x) + (x) ω (y − x) q q+1 (−1)q (i) 2 − 2 − δ CIq πi(y − x) 1 I ···I I I ···I I ∂x ωI1 q−1 (x) + ωI1 q (x) (y − x) q q−1 q−1 i(i) 2 − 2 ˆ − δ c[Ir ωI1 ···Ir ···Iq ] (x), (133) 2πi(y − x) r=1

ωI1 ···Iq (y) AAB (x) =

q−1

1 ˆ ˆ {ωAI1 ···Ir ···Iq (x) δ Ir B − ωBI1 ···Ir ···Iq (x) δ Ir A } 2πi(y − x) r=1 1 − ωI1 ···Iq−1 A (x) δ Iq B + ωI1 ···Iq−1 B (x) δ Iq A ] , 2πi(y − x) (134) p−2

q

ω

A1 ···Aq

(y) ω

B1 ···Bp

(i)[ 2 ]+[ 2 +2]−[ (x) = 2π(y − x)

p+q q 2 ]

ωV1 ···Vq+r−1 BVq+r ···vq+p−1 (x)

r=1

A

A

B

B

B

q−1 B1 p−1 p A−1 r+1 r × δ[V · · · δVq−1 δVq · · · δVBq+r−1 δB q δVq+r · · · δVq+p−2] δVq+p−1 1

q −(−1)q(p+q−1) V1 ···Vp+q−3 + ω (x) × 2πi(y − x) r=1

B

A

A

A ]

p−1 Bp [Ar A1 q r−1 r+1 × δVB11 · · · δVp−1 δ δVp · · · δVp+r−1 δVp+r · · · δVp+q−3 ,

q p 2q+p−4

(135)

2 + 2 −

2 ωA1 ···Aq (y) ρ B1 ···Bp (x) = −(i) B1 ···Bp A1 ···Aq (x) ∂x ωB1 ···Bp A1 ···Aq (x) ω + 2πi(y − x)2 2π i(y − x)

+

p+2 r=1

(−1)r−1 ρ A1 ···Aq−1 B1 ···Br Aq Br+1 ···Bp (x). 2πi(y − x) (136)


355

4.8. Short distance expansion for ρ A1 ···Aq (y) O(x). Our last contribution to the expansions for the coadjoint representation is given below. Special cases are explicitly written and supersede the more general expressions: ¯ J1 J J M δ MJ2 − G ¯ J2 J J M δ J1 M , RrJ1J1JJ22 ∗ U¯ ωMM = G (r 1 2 ω ) (r 1 2 ω )

(137)

J1 J2 1 J1 L J2 M ¯ N ¯ J1 + 2 δ J1 L δ MN G ¯ J2 , ¯ LMN R(r G(r ω+rω ) − 2δ J2 L δ MN G δ J1 J2 ) ∗ UωLMN = 2 δ (rω ) (r ω ) (138) J1 I ] I1 I2 J 1 K K[I J L J L 1 1 2 2 2 1 ¯ K = δ ¯ JJ L δ ¯ JJ L δ R I I ∗G −G , (139) U¯ I I K − 2 G r

J1 ···Jp r {Jp }

R

J1 ···Jp r {Jp }

R

ψ

1 2

2

(r

1 2ψ

)

(r

1 2

ψ )

(r

1 2ψ

)

p p+2 ¯ ···V ¯ ¯ V ¯m ¯ ¯ [V 1 2 − 2 ∗ U¯ ωV{V1 ··· = − i(i) δ[J¯ 1···J¯ ]m−2 δ Vm−1 ],Vm δ m,p+2 L¯ (rω) } 2 m p 1 p p+1 − 2 ¯ V¯ m δ [V¯ 1 ···V¯ m−1 ] +i 2 δ p+1,m G (rω) [J1 ···Jp ] p p+2 ¯ V − 2 ¯ 1 δ V¯ 2 ···V¯ m−2 δ V¯ m−1 , V¯ m G + 2i(−1)p+1 δ p+3,m (i) 2 (rω) [J1 ···Jp ]

1 ···Tm ∗ R¯ ρT{T m}

¯

¯

¯ 2 ···V ¯ m−1 ] p+2,m ¯ 1 [V ¯m V V + i(−1)p T¯(rω) δ δ[J ···Jp ] 1 m p m+p−2 ¯ p +1···V ¯ ···V ¯ m−p V ¯ V + 2 − 2 + (−1)pm (i) 2 δ[J11···Jpp] , U¯ (rω) (140) [T¯ 1 ···T¯ m ] p m ¯ p = δ δ L p [J1 ···Jp ]

(−( 2 −2)r ρ−( 2 −1) rρ ) T¯ 2 ···T¯ m ] m ¯ [T¯ 1 + (−1)p {2(i)G ((2−p)r ρ−(p−1)rρ ) δ[J1 ···Jp ] δp+1

p p−1 p [J ···J J ···Jp ] − 2 m ¯ Jr δp−1 (−1)r G δ[T¯1 ···T¯r−1 r+1 } + (i)(i) 2 (rρ) T¯ ...T¯ ] 1

r−1 r

m

r=1

+

p

p [T¯ |J | T¯ 2 ···T¯ m ] (−1)r+1 2T¯(rρ)1 r δ[J δ 1 ···Jr−1 Jr+1 ···Jp ] m

r=1

−

p r=1

p

[T ···T |Jr | T ···Tm ] (−1)r−1 U¯ (rρ)1 m−p+1 δJ1m−p+2 ···Jr−1 Jr+1 ···Jp ¯

m−p

{[ ]+[ 2 +i 2

¯

¯

¯

m ]−[ 2 ]} ¯ [T¯ p+1 ···T¯ m T¯ 1 ···T¯ p ] R(2r ρ+rρ ) δ[J1 ···Jp ] .

(141)

Typical operator product expansions are then (q > 2): ρ A1 ···Aq (y) D(x) A1 ···Aq ( p2 − 2) (x) ρ p A1 ···Aq = (x) − ( 2 − 1)∂x ρ 2π i(y − x) (y − x) p p+2 A ···A A A ···A A ωA1 q (x) ∂x ωA1 q (x) 1 2 A1 ···Aq A (i) 2 − 2 − + 2 ∂x ωA + (x) , 2π(y − x) (y − x)2 (y − x)

(142)

356


q q+1 2 − 2 i 1 ρ A1 ···Aq (y)ψ B (x) = ∂x ωA1 ···Aq B (x) − ωA1 ···Aq B (x) 2πi(y − x) (y − x) q q+2 − 2 2(−1)q+1 (i) 2 + π(y − x) BA1 ···Aq A BA ···A A ωA (x) ωA 1 q (x) 1 BA1 ···Aq A + 2 ωA + (x) (y − x)2 (y − x) + (−1)q (2 − p)ρ BA1 ···Aq (x) (p − 1)∂x ρ BA1 ···Aq (x) − π(y − x)2 π(y − x) q q−1 q − 2 (i) 2 ˆ + δ Ar B ρ A1 ···Ar ···Aq (x), (143) 2π(y − x) r=1 [B1 |A1 ···Aq |B2 ] ∂x ω[B1 |A1 ···Aq |B2 ] A1 ···Aq B1 B2 q ω ρ (y) A (x) = (i) + 2π(y − x)2 2π(y − x) p −1 (−1)r+1 ρ B2 A1 ···Ar−1 B1 Ar+1 ···Aq , (144) + πi(y − x) r=1

ρ A1 ···Aq (y) ωB1 ···Bn (x) n

p

= (−1)pn (i)[ 2 ]+[ 2 ]−[ −

p r=1

(−1)r−1

n+p−2 ] 2

ωA1 ···Aq B1 ···Bn (x) ∂x ωA1 ···Aq B1 ···Bn (x) + 2πi(y − x)2 2π i(y − x)

1 ˆ ρ B1 ···Bn−r+1 Ar Bn−r+2 ···Bn A1 ···Ar ···Aq(x), 2πi(y −x)

(145)

q n−2q n−p ρ A1 ···Aq (y) ρ B1 ···Bn (x) = i 2 + 2 − 2 A1 ···Aq B1 ···Bn ρ (x) ∂x ρ A1 ···Aq B1 ···Bn (x) + . (146) πi(y − x)2 π i(y − x) 5. Conclusions This work shows that the dual representation of affine Lie algebras and Virasoro-type algebras provide a natural definition of a short distance expansion or operator product expansion. The operators are not built from an enveloping algebra as in the Sugawara constructions [36] but instead exist in their own right as fundamental fields. The fields D(x) should not be confused with the energy-momentum tensor that comes from conformal field theories. However the fields D(x) and energy-momentum tensors do share the same transformation laws modulo their interpretation. Other similarities with other approaches are seen in Sect. 3.1, where the operators D(x), satisfy the same operator product expansions as one has in Virasoro theories [16] which is built from a lattice. One may also notice the A(x) that appears in our discussion of affine Lie algebras as compared to conformal field theory treatments of WZNW [38, 41, 31] where an


357

energy-momentum tensor is constructed from the current algebra. Outside of algebraic and convergent properties, our construction is otherwise model independent and can be implemented for other infinite dimensional Lie algebras that admit a dual. Underlying free-field theories are not necessary to define these expansions. Thus, the roles of symmetry groups in defining these expansions is moved firmly to the foreground. The origin of this model independence is that we exploit the natural bifurcation of the initial data where the algebraic elements serve as conjugate momenta and the dual of the algebra serves as the conjugate coordinates. This feature is shared by the symplectic structures of actions like Yang-Mills theory and N -extended affirmative actions [22, 8]. The coadjoint representation enforces a short distance expansion already at the level of the Poisson bracket of the elements with no Hamiltonian required. Another implication that we wish to note is that since we never relied on any specific model in our discussion, this implies that at no point was it necessary to invoke Wick rotations to a Euclideanized formulation to justify the form of the short-distance expansions. We note that just as in the case of Virasoro theories and some other conformal field theory based operators, the central extension can be further restricted by demanding unitarity. Choices of the central extension can be determined through the Kac determinant [24, 17, 15]. This viewpoint has been applied to the model-independent N -extended supersymmetric GR Virasoro algebra [21] to obtain for the first time its representation in terms of short distance expansions. The complete set of such expansions has been presented in the fifth section of this work. This success is expected to open up further avenues of study. Since these short distance expansions are now known for arbitrary values of N , this means that our new results may be used to study the possible existence of 1D, N = 16 or N = 32 supersymmetric NSR-type models. The question of whether such an approach can lead to a new manner for probing M-theory is now a step closer to being answered. Acknowledgements. VGJR thanks the Center for String and Particle Theory at the Physics Department of the University of Maryland for support and hospitality.

References 1. Ademollo, M. et al.: Phys. Lett. B 62, 105 (1976) 2. Alekseev, A., Shatashvili, S.: Path Integral Quantization of the Coadjoint Orbits of the Virasoro Group and 2-D Gravity. Nucl. Phys. B 323, 719 (1989) 3. Balachandran, A. P., Marmo, G., Nair, V. P., Trahern, C. G.: Phys. Rev. D 25, 2713 (1982) 4. Banados, M., Chandia, O., Ritz, A.: Holography and the Polyakov action. Phys. Rev. D 65, 126008 (2002) [arXiv:hep-th/0203021] 5. Banados, M., Henneaux, M., Teitelboim, C., Zanelli, J.: Phys. Rev. D 48, 1506 (1993) [arXiv: gr-qc/9302012] 6. Kadanoff, L. P.: Phys. Rev. Lett. 23, 1430 (1969) 7. Wilson, K. G.: Phys. Rev. 179, 1499 (1969) 8. Boveia, A., Larson, B. A., Rodgers, V. G. J., Gates, S. J., Linch, W. D., Phillips, J. A., Kimberly, D. M.: Chiral supergravitons interacting with a 0-brane N-extended NSR super-Virasoro group. Phys. Lett. B 529, 222 (2002) [arXiv:hep-th/0201094] 9. Branson, T., Lano, R. P., Rodgers, V. G. J.: Yang-Mills, gravity, and string symmetries. Phys. Lett. B 412, 253 (1997) [arXiv:hep-th/9610023] 10. Branson, T. P., Rodgers, V. G. J., Yasuda, T.: Interaction of a string inspired graviton. [arXiv:hepth/9812098] 11. Brown, J. D., Henneaux, M.: Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity. Commun. Math. Phys. 104, 207 (1986) 12. Curto, C., Gates, S. J., Rodgers, V. G. J.: Superspace geometrical realization of the N-extended super Virasoro algebra and its dual. Phys. Lett. B 480, 337 (2000) [hep-th/0002010]

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13. Delius, G. W., van Nieuwenhuizen, P., Rodgers, V. G. J.: The Method of Coadjoint Orbits: An Algorithm for the Construction of Invariant Actions. Int. J. Mod. Phys. A 5, 3943 (1990) 14. Dzhumadildaev, A.: Z. Phys. C 72, 509 (1996) 15. Feigin, B. L., Fuchs, D. B.: Funct. Anal. Appl. 16, 114 (1982) 16. Frenkel, I.B., Zhu, Y.: Duke Math. J. 66, 123 (1992) 17. Friedan, D., Qiu, Z. a., Shenker, S. H.: Phys. Rev. Lett. 52, 1575 (1984) 18. Bershadsky, M. A.: Phys. Lett. B 174, 285 (1986) 19. Bershadsky, M., Ooguri, H.: Phys. Lett. B 229, 374 (1989) 20. Gates, S. J., Rana, L.: A Theory of spinning particles for large N extended supersymmetry. Phys. Lett. B 352, 50 (1995) [hep-th/9504025] 21. Gates, S. J., Rana, L.: Superspace geometrical representations of extended super Virasoro algebras. Phys. Lett. B 438, 80 (1998) [hep-th/9806038] 22. Gates, S. J., Rodgers, V. G. J.: Super gravitons interacting with the super Virasoro group. Phys. Lett. B 512, 189 (2001) [hep-th/0105161] 23. Grozman, P.: Dimitry Leites, Irina Shchepochkina Lie superalgebras of string theories arXiv:hepth/9702120 24. Kac, V. G.: In: Austin, 1978, Proceedings, Group Theoretical Methods In Physics, Berlin-New York: Springer-Verlag, 1979, pp. 441–445 25. Kirillov, A.A.: Elements of the Theory of Representations. Berlin-Heidelberg-New York: Springer Verlag, 1975 26. Lano, R.P., Rodgers, V.G.J.: A Study of fermions coupled to gauge and gravitational fields on a cylinder. Nucl. Phys. B 437, 45 (1995) [arXiv:hep-th/9401039] 27. Lano, R.P., Rodgers, V.G.J.: Applications of W algebras to BF Theories, QCD And 4-D Gravity. Mod. Phys. Lett. A 7, 1725 (1992) [arXiv:hep-th/9203067] 28. Larsson, T. A.: Phys. Lett. B 231, 94 (1989) 29. Larsson, T. A.: J. Phys. A 25, 1177 (1992) 30. Nakatsu, T., Umetsu, H., Yokoi, N.: Prog. Theor. Phys. 102, 867 (1999) [arXiv:hep-th/9903259] 31. Novikov, S. P.: Usp. Mat. Nauk 37N5, 3 (1982) 32. Polyakov, A. M.: Quantum Gravity in Two-Dimensions. Mod. Phys. Lett. A 2, 893 (1987) 33. Rai, B., Rodgers, V. G. J.: From Coadjoint Orbits to Scale Invariant WZNW Type Actions and 2-D Quantum Gravity Action. Nucl. Phys. B 341, 119 (1990) 34. Segal, G.: Unitarity Representations of Some Infinite Dimensional Groups. Commun. Math. Phys. 80, 301 (1981) 35. Siegel, W.: Fields. [arXiv:hep-th/9912205] 36. Sugawara, H.: Phys. Rev. 170, 1659 (1968) 37. Taylor, W. I.: Coadjoint Orbits And Conformal Field Theory. hep-th/9310040 38. Wess, J., Zumino, B.: Phys. Lett. B 37, 95 (1971) 39. Wilson, K.G., Zimmermann, W.: Commun. Math. Phys. 24, 87 (1972) 40. Witten, E.: Coadjoint Orbits of the Virasoro Group. Commun. Math. Phys. 114, 1 (1988) 41. Witten, E.: Commun. Math. Phys. 92, 455 (1984) 42. Yokoi, N., Nakatsu, T.: Three-dimensional extremal black holes and the Maldacena duality. Prog. Theor. Phys. 104, 439 (2000) [arXiv:hep-th/9912096] Communicated by Y. Kawahigashi


Communications in


Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality Patrick Hayden1 , Richard Jozsa2 , Dénes Petz3 , Andreas Winter2,4 1

Institute for Quantum Information, Caltech 107–81, Pasadena, CA 91125, USA. E-mail: [email protected] 2 Department of Computer Science, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol BS8 1UB, U.K. E-mail: [email protected] 3 Department for Mathematical Analysis, Mathematical Institute, Budapest University of Technology and Economics, Egry József utca 2, 1111 Budapest, Hungary. E-mail: [email protected] 4 Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1 TW, U.K. E-mail: [email protected] Received: 10 April 2003 / Accepted: 8 October 2003 Published online: 24 February 2004 – © Springer-Verlag 2004

Abstract: We give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von Neumann entropy. By combining a result of Petz characterising the equality case for the monotonicity of relative entropy with a recent theorem by Koashi and Imoto, we show that such states will have the form of a so–called short quantum Markov chain, which in turn implies that two of the systems are independent conditioned on the third, in a physically meaningful sense. This characterisation simultaneously generalises known necessary and sufficient entropic conditions for quantum error correction as well as the conditions for the achievability of the Holevo bound on accessible information. I. Introduction The von Neumann entropy [13] S(ρ) = −Trρ log ρ, of a density operator ρ on a finite dimensional Hilbert space H shares many properties with its classical counterpart, the Shannon entropy H (P ) = − P (x) log P (x) x∈X

of a probability distribution P on a discrete set X . (All logarithms in this work are understood to be to base 2. Also, we will use the terms “state” and “density operator” interchangeably.) For example, both are nonnegative, and equal to 0 if and only if the state (distribution) is an extreme point in the set of all states (distributions), i.e. if ρ is pure (P is a point mass). Both are concave and, moreover, both are subadditive: for a state ρAB on a composite system HA ⊗ HB with reduced states ρA = Tr B ρAB , ρB = Tr A ρAB ,

360

it holds that

P. Hayden, R. Jozsa, D. Petz, A. Winter

S(ρAB ) ≤ S(ρA ) + S(ρB ).

A directly analogous inequality holds for a distribution over a product set and its marginals. (Many more properties of S are collected in the review by Wehrl [25] and in the monograph [14].) We shall view von Neumann entropy as a generalisation of Shannon entropy [19] in the following precise way: if the set X labels an orthonormal basis |x : x ∈ X of HX we can construct the state ρP = P (x)|xx| x

corresponding to the distribution P . This clearly defines an affine linear map from distributions into states. It is then straightforward to check that S(ρP ) = H (P ), so all properties of von Neumann entropy of a single system also hold for Shannon entropy of a single distribution. Similarly, for a distribution P on a cartesian product X ×Y, we use the tensor product basis |xy = |x ⊗ |y : x ∈ X , y ∈ Y to define the state ρP on HX ⊗ HY . Again, it is straightforward to check that reduced states correspond to taking marginals: Tr Y ρP = ρP |X , Tr X ρP = ρP |Y . Hence all entropy relations for bipartite states also hold for bipartite distributions. In [9] Lieb and Ruskai proved the remarkable relation S(ρAB ) + S(ρBC ) ≥ S(ρABC ) + S(ρB ),

(1)

with a tripartite state ρABC on the system HA ⊗ HB ⊗ HC . It clearly generalises the previous subadditivity relation, which is recovered for a trivial system B: HB = C. In fact, this inequality plays a crucial role in nearly every nontrivial insight in quantum information theory, from the famous Holevo bound [5] and the properties of the coherent information [3, 18] to the recently proved additivity of capacity for entanglement–breaking channels [20]. The present investigation aims to resolve the problem of characterising the states which satisfy this relation with equality: the main result is Theorem 6. Roughly speaking, the strong subadditivity inequality expresses the fact that discarding a subsystem of a quantum system is a dissipative operation, in the sense that it can only destroy correlations with the rest of the world. Our work, therefore, can be interpreted as providing a detailed description of the conditions under which the act of discarding a quantum system can be locally reversed on a particular input. We restrict ourselves to finite dimensional systems in this paper. The question of whether a similar result holds in infinite dimension is left open. The rest of the paper is organised as follows. In Sect. II we will review the case of probability distributions: there the solution to our problem is easy to obtain, and in fact well–known. This will provide the intuitive basis for understanding our main result. After that, in Sect. III we review quantum relative entropy and the relation of its monotonicity

Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality

361

property to the strong subadditivity inequality. Section IV presents a condition given by Petz for equality in the monotonicity of relative entropy, while Sect. V presents and proves our main result, a structure theorem for states which satisfy strong subadditivity with equality. An essential step is the application of a recent result of Koashi and Imoto [7] for which we give a short but non–constructive algebraic proof in the appendix. In Sec. VI, we show how the entropic conditions for quantum error correction as well as the conditions for saturation in the Holevo bound follow as easy corollaries from our structure theorem. The question of characterising the equality case of strong subadditivity as well as of the monotonicity of relative entropy was considered in earlier work by Petz [15], where it was related to the existence of quantum operations with certain properties. Ruskai [17] has given a characterisation in terms of an operator equality, which can be used to show that the states described in our main Theorem 6 are indeed equality cases (as she has informed us, this was pointed out to her by M. A. Nielsen after [17] appeared). Neither of these results is as explicit as one could wish for, however, because while both give algebraic criteria which one can check on any given state, they do not yield a simple description of all the states that satisfy equality. This simple description is exactly what our Theorem 6 provides. II. The Classical Case Let us first look at the classical case of probability distributions and their Shannon entropies. The exposition is most conveniently phrased in terms of random variables denoted A, B, C, taking values in A, B, C, respectively, with a joint distribution PABC (a, b, c) = Pr{A = a, B = b, C = c}. The distribution of A is the marginal PA = PABC |A of the joint distribution to A and similarly for the other variables. Shannon [19] defined the mutual information I (A : B) = H (A) + H (B) − H (AB), with H (A) = H (PA ) and so on. It is not hard to show that I (A : B) ≥ 0 with equality if and only if A and B are independent. Conditional mutual information is defined as I (A : C|B) = PB (b)I (A : C|B = b), b∈B

where I (A : C|B = b) is the mutual information between the variables A and C conditional on the event “B = b”, i.e. I (A|B=b : C|B=b ), with PABC (a, b, c) Pr A|B=b = a, C|B=b = c = PB (b) =: PAC|B (a, c|b). It is straightforward to check that with these definitions one has the chain rule I (A : BC) = I (A : B) + I (A : C|B).

(2)

362


This implies the formula I (A : C|B) = H (AB) + H (BC) − H (ABC) − H (B). Because the left hand side is by definition a convex combination of mutual information, each of which is always nonnegative, we obtain strong subadditivity for classical distributions. Theorem 1. I (A : C|B) = 0 if and only if A and C are conditionally independent given B, meaning ∀b s.t. PB (b) = 0 A|B=b , C|B=b are independent. This is the case if and only if PABC (a, b, c) = PB (b)PA|B (a|b)PC|B (c|b) = PA (a)PB|A (b|a)PC|B (c|b),

(3)

i.e. iff A—B—C is a Markov chain in this order. Proof. Clearly, the conditions are sufficient. Assume conversely that I (A : C|B) = 0. By definition of the latter quantity, this implies that for all b with PB (b) = 0, I (A : C|B = b) = 0. But this implies independence of A|B=b and C|B=b . Hence, Eq. (3) follows: PABC (a, b, c) = PB (b)PAC|B (a, c|b) = PB (b)PA|B (a|b)PC|B (c|b) = PA (a)PB|A (b|a)PC|B (c|b).

The remainder of the paper is devoted to describing the quantum mechanical generalisation of this equivalence between zero conditional mutual information, conditional independence, the Markov property and the factorization of the joint distribution given in Eq. (3). III. Relative Entropy Our approach to saturation of the strong subaddivity inequality will be via the quantum relative entropy; this quantity was defined by Umegaki [24] for two quantum states ρ and σ as S(ρσ ) = Tr ρ(log ρ − log σ ) if the support of ρ is contained in the support of σ , and +∞ otherwise. We note that this definition generalises the familiar Kullback–Leibler divergence [8] of two probability distributions, just as von Neumann entropy generalises Shannon entropy. For a bipartite state ρAB it is straightforward to check that S(ρAB ρA ⊗ ρB ) = S(ρA ) + S(ρB ) − S(ρAB ),

(4)

and the latter quantity is abbreviated I (A : B), in formal extension of the definition of Shannon’s mutual information [19] to quantum states.


363

Example 2. Let {p(x), ρx } be an ensemble of quantum states on H. The Holevo quantity χ is defined as χ {p(x), ρx } = S p(x)ρx − p(x)S(ρx ). x

x

Holevo [5] showed that this quantity is an upper bound to the mutual information between x and the outcomes y of any particular measurement performed on the states ρx . It is easily seen that χ {p(x), ρx } = I (A : B), with the bipartite state ρAB =

x

p(x)|xx|A ⊗ (ρx )B .

For a tripartite state ρABC we can also consider the information I (A : BC), which can be written as S(ρABC ρA ⊗ ρBC ) = S(ρA ) + S(ρBC ) − S(ρABC ).

(5)

The difference between Eqs. (5) and (4), which by virtue of the classical chain rule Eq. (2) we might call the quantum conditional mutual information I (A : C|B) is, therefore, S(ρABC ρA ⊗ ρBC ) − S(ρAB ρA ⊗ ρB ) = S(ρAB ) + S(ρBC ) − S(ρABC ) − S(ρB ).

(6)

The right-hand side here is nonnegative by strong subadditivity. (Note that this is an important theorem in the quantum case despite being an almost trivial observation classically.) The left-hand side, however, can be rewritten as S(ρσ ) − S(T ρT σ ), with the states ρ = ρABC and σ = ρA ⊗ ρBC , and the quantum operation T = Tr C , the partial trace over HC , which as a linear map can be written as T = idAB ⊗ Tr. Now a theorem of Uhlmann [23] (proved earlier by Lindblad [10] for the finite– dimensional case of interest here) says that for all states ρ and σ on a space H, and all quantum operations T : B(H) → B(K), S(ρσ ) ≥ S(T ρT σ ),

(7)

so Uhlmann’s theorem implies strong subadditivity and we have equality in the latter if and only if there is equality in the former. IV. The Equality Condition for Relative Entropy The formulation in the previous section of strong subadditivity as a relative entropy monotonicity under a partial trace operation transforms the question for the equality conditions for the former into the same question for the latter. Note that by the very monotonicity relation, there is a “trivial” case of equality in Eq. (7), namely if there exists a quantum operation T mapping T ρ to ρ and T σ to σ . In fact, this is the only case of equality:

364


Theorem 3 (Petz [16]). For states ρ and σ , S(ρσ ) = S(T ρT σ ) if and only if there exists a quantum operation T such that TT ρ = ρ,

TT σ = σ.

Furthermore, on the support of T σ , T can be given explicitly by the formula

1 1 1 1 Tα = σ 2 T ∗ (T σ )− 2 α(T σ )− 2 σ 2 ,

(8)

with the adjoint map T ∗ of T : A∗i XAi , if T (α) = Ai αA∗i .

T ∗ (X) = i

i

Observe that the definition of T in Eq. (8) depends on σ , thereby automatically ensuring that TT σ = σ . Sometimes, we add the subscript σ to T to emphasize the dependence. Example 4. As in Example 2, let {p(x), ρx } be an ensemble of states on H and define p(x)|xx|A ⊗ (ρx )B . ρAB = x

There we observed that, with σ = ρA ⊗ ρB , χ {p(x), ρx } = S(ρσ ). Now, let ϕ be a quantum operation on H (which could be a measurement), and form T = idA ⊗ ϕB . Then χ {p(x), ρx } = S(ρAB ρA ⊗ ρB ) ≥ S T ρAB T (ρA ⊗ ρB ) (9) = χ {p(x), ϕρx } , which is (a generalistion of) the famous Holevo bound [5] in the form of a data processing relation. Equality holds, according to Theorem 3, if and only if T of Eq. (8) maps T ρ to ρ. Note that we may assume without loss of generality that σ = ρA ⊗ρB is strictly positive. But it is straightforward to check that ρA ⊗ Tσ = id ϕρB = id ⊗ ϕ, hence we have equality in Eq. (9) if and only if for all x, ϕ ϕρx = ρx . Remark 5. In [2] the “transpose channel” T of Eq. (8), as it is called in [14], makes an appearance in a slightly different context: there a set of states is subjected to a quantum channel, and the problem is to find the best recovery map which maximises a fidelity criterion for the original states and the images of the channel output states. It was shown that the error using T is always at most twice the minimum error under the optimal recovery map.


365

V. Structure of States with Equality Let ρABC be a state on HA ⊗ HB ⊗ HC . As we observed earlier, Uhlmann’s theorem specialized to the states ρABC and σABC = ρA ⊗ ρBC , along with the map T = Tr C , states that S(ρABC ρA ⊗ ρBC ) ≥ S(ρAB ρA ⊗ ρB ). Consequently, Theorem 3 provides the condition for equality here: TT ρ = ρ. Now, because T = idA ⊗ RBC , with the restriction map RBC = idB ⊗ Tr C , and σ is a tensor product, we obtain (compare Example 4): T = idA ⊗ R,

(10)

= R ρBC . with R Summarising, in the above monotonicity and hence in strong subadditivity we have equality if and only if AB . ρABC = (id ⊗ R)ρ

(11)

We are now in a position to prove our main result: Theorem 6. A state ρABC on HA ⊗ HB ⊗ HC satisfies strong subadditivity (Eq. (1)) with equality if and only if there is a decomposition of system B as HB = HbL ⊗ HbR j

j

j

into a direct (orthogonal) sum of tensor products, such that ρABC = qj ρAbL ⊗ ρbR C , j

j

j

with states ρAbL on HA ⊗ HbL and ρbR C on HbR ⊗ HC , and a probability distribution j j j j {qj }. Proof. The sufficiency of the condition is immediate. The proof of necessity will come from analysing the quantum Markov chain condition, Eq. (11). the Markov condition gives us After defining the quantum operation ϕ = Tr C ◦ R, (id ⊗ ϕ)ρAB = ρAB .

(12)

Consider an operator M on HA with 0 ≤ M ≤ , and define a state µ by pµ = Tr A ρAB (M ⊗ ) , p = Tr ρAB (M ⊗ ) . Then, if p = 0, Eq. (12) implies that ϕ(µ) = µ. Varying the operator M we obtain a family M of states on HA invariant under ϕ. To this we can apply Theorem 9 from the appendix. We obtain a decomposition HB = HbL ⊗ HbR , (13) j

j

j

366


such that every µ ∈ M can be written µ= qj (µ)ρj (µ) ⊗ ωj , j

with states ρj (µ) on HbL and ωj on HbR . This, in turn, easily implies the following j j structure for ρAB : ρAB = qj ρAbL ⊗ ωbR . (14) j

j

j

To see this, introduce the quantum operation P0 (ξ ) = Tr bR j ξ j ⊗ ωj , j

j

on HB , where j is the orthogonal projector onto the subspace HbL ⊗ HbR in Eq. (13). j

j

(Its dual P0∗ is the subalgebra projection from the appendix, where it is denoted the same way.) Then it is easy to calculate, for arbitrary operators M and N bounded between 0 and : Tr ρAB (M ⊗ N ) = pTr(µN ) = pTr P0 (µ)N = pTr µP0∗ (N )

= Tr ρAB M ⊗ P0∗ (N ) = Tr (id ⊗ P0 )ρAB (M ⊗ N ) . By linearity, this holds for all operators in place of M ⊗ N , so ρAB = (id ⊗ P0 )ρAB , implying Eq. (14), because (id ⊗ P0 )ξ =

j

Tr bR ( ⊗ j )ξ( ⊗ ∗j ) ⊗ ωj . j

But Theorem 9 also gives information about ϕ: introduce an environment HE in state ε and a unitary U on HB ⊗ HC ⊗ HE such that R(α) = Tr E U (α ⊗ |00| ⊗ ε)U ∗ , with a standard state |0 ∈ HC . Because a further trace over C gives us ϕ, we obtain the following form for U (with E = HC ⊗ HE ): U= H L ⊗ Uj , (15) j

with Uj a unitary on HbR ⊗ E. j

bj


367

Putting Eqs. (14) and (15) together, we finally get: AB ρABC = (idA ⊗ R)ρ = Tr E (A ⊗ U )ρAB (A ⊗ U ∗ ) = qj ρAbL ⊗ Tr E U (ωbR ⊗ |00|)U ∗ j

j

=

j

j

qj ρAbL ⊗ ρbR C , j

which is what we wanted to prove.

j

Quantum Markov states on the infinite tensor product of matrix algebras ∞ i=−∞ Mn (C)(i) were introduced by Accardi and Frigerio [1]. Let Am be the subproduct of the factors with superscript i ≤ m. Then Am ⊂ Am+1 . A state ρ of the infinite tensorproduct is called Markovian if for every integer m there exists a unital completely positive mapping Tm,m+1 : Am+1 → Am which leaves the state ρ (restricted to Am ) invariant and the subalgebra Am−1 fixed. Accardi and Frigerio call the mapping Em,m+1 quasi–conditional expectation; its dual is the quantum analogue of the Markov kernel in classical probability theory. Assume that Am−1 = B(HA ), Mn (C)(m) = B(HB ) and Mn (C)(m+1) = B(HC ). If the equality in strong subadditivity is satisfied in this setting, then we have Eq. (10) and the dual of T is a quasi–conditional expectation. Therefore the equality in strong subadditivity for every m yields a quantum Markov state on the infinite system. This property characterises quantum Markov states, see e.g. [14], p. 201. We propose to call a state as in Eq. (11) a short quantum Markov chain (as opposed to the infinite chains introduced in [1]), since we require the existence of the quasi–conditional expectation only for B(HA ⊗HB ⊗HC ) → B(HA ⊗HB ); note that the analogous quasi–conditional expectation B(HA ⊗ HB ) → B(HA ) exists trivially because the subalgebra to be left invariant is C. VI. Applications Theorem 6 provides a convenient framework for synthesizing many previously known facts in quantum information theory. To illustrate the method of its application, we present a couple of special cases from the literature. Example 7. The fundamental problem in quantum error correction is to determine when the effect of a quantum operation ϕ acting on half of a pure entangled state can be perfectly reversed. Define the coherent information Ic (σ, ϕ) = S(ϕσ ) − S (idA ⊗ ϕ) σ , where σ is any purification of σ to system A. In [18] it was shown that there exists a quantum operation ϕˆ such that ˆ (idA ⊗ ϕϕ)

σ = σ if and only if Ic (σ, ϕ) = S(σ ).

(16)

368


By the Stinespring dilatation theorem [21], we may assume that ∗ ϕσ = Tr C UBC (σ ⊗ ψ)UBC for a unitary operator UBC and pure state ancilla ψ on system C. If we let |ω = (A ⊗ UBC )(| σ ⊗ |ψ) then, taking mutual information with respect to the state ω = |ωω|, S(σ ) = I (A : BC) − S(A) and Ic (σ, ϕ) = I (A : B) − S(A). Therefore, Eq. (16) holds iff I (A : BC) = I (A : B). By Theorem 6, we can conclude that ω= qj ωAbL ⊗ ωbR C . j

j

j

The recovery procedure ϕˆ given the state (idA ⊗ ϕ) σ = Tr C ω is then obvious: first measure j before preparing the state ωbR C on HbR ⊗ HC . Next, j

j

∗ and discard the fixed ancilla state ψ. The output is exactly . As an perform UBC σ aside, the reason that the solution to this problem was accessible without the results of the present paper is that a closer examination of the situation reveals that, because σ is pure, only the equality conditions for the usual subadditivity inequality are required in the construction of the reversal map. Strong subadditivity, in this case, is superfluous.

Example 8. Returning to our investigation of the Holevo

bound from Example 4, let {p(x), ρx } be an ensemble of states on H, ρAB = x p(x)|xx|A ⊗ (ρx )B and ϕ a quantum operation on the B system. Again by the Stinespring dilation theorem, after possibly adjoining a fixed ancilla and performing a common unitary operation to all the ρx , we may assume without loss of generality that B = B˜ C˜ and ϕ = Tr C˜ . An application of Theorem 6 then gives the conditions under which χ ({p(x), ρx }) = χ ({p(x), ϕρx }). Namely, ρAB˜ C˜ =

j

qj

x

p(x|j )|xx|A ⊗ (ρx )b˜ L ⊗ ωb˜ R C˜ , j

j

where p(x|j )qj is the joint probability distribution for (x, j ) and ωb˜ R C˜ does not depend j on x. In the special case where ϕ corresponds to a measurement operation, the additional constraint [ϕρx , ϕρx ] = 0 must hold because the output system is classical. Given the form of ρAB˜ C˜ , this implies [(ρx )b˜ L , (ρx )b˜ L ] = 0 and, in turn, that the states {ρx } all j j commute. In the language of quantum information, we have found that the accessible information of an ensemble is equal to its Holevo quantity if and only if all the states in the ensemble commute. This condition for equality actually appeared in Holevo’s original paper [5].


369

VII. Discussion We have exhibited the explicit structure of the tripartite states ρABC which satisfy strong subadditivity with equality. Not only are they short quantum Markov chains in the sense of [1], it is even the case that the A and C systems are conditionally independent given B, in a physically meaningful sense: there is information in the B system which can be obtained by a non–demolition measurement, conditioned upon which the quantum state factorises. By specialising our result to particular types of states, we can easily recover the entropic conditions for quantum error correction and the conditions for saturation in the Holevo bound. In the general case, our theorem characterises exactly when a quantum operation preserves correlations, whether they be classical, in the form of pure entanglement, or more exotic, such as combinations of the two or even bound entanglement [6]. We left open the problem of a similar characterisation in infinite dimension (of system B — infinite A and C are covered by our result): we only note that our method will certainly not work, as it relies ultimately on the classification of finite dimensional operator algebras. A further interesting problem could be to address the approximate case: if a state almost satisfies strong subadditivity, does it mean that its structure is close in some sense to the form of Theorem 6? There might be a relation to [2] (see Remark 5), where an approximate fidelity condition was studied. Acknowledgement. We would like to thank Chris Fuchs, Mary Beth Ruskai and Ben Schumacher for helpful discussions. PH acknowledges the support of the Sherman Fairchild Foundation and the U.S. National Science Foundation through grant no. EIA-0086038. DP was partially supported by Hungarian OTKA T032662. The work of RJ and AW was supported by the U.K. Engineering and Physical Sciences Research Council.

Appendix A. An Operator Algebraic Derivation of the Koashi–Imoto Theorem Let ρ1 , . . . , ρK be density operators on the finite–dimensional Hilbert space H. We are interested in the quantum operations (completely positive, trace–preserving linear maps) T : B(H) → B(H) which leave these states invariant: ∀k

T ρk = ρk .

(A1)

By possibly shrinking H to the

minimum joint supporting subspace of the ρk , we may assume that the support of K1 k ρk is H, which we shall do in the following. From the Stinespring dilation theorem [21] it follows that every such T can be represented as ∗ (A2) T σ = Tr E UHE (σ ⊗ ε)UH E , with another Hilbert space E, a state ε on it, and a unitary acting on H ⊗ E. In [7] the following result is proved by an explicit algorithmic construction: Theorem 9 (Koashi, Imoto [7]). Associated to the states ρ1 , . . . , ρK there exists a decomposition of H as Jj ⊗ Kj (A3) H= j

into a direct (orthogonal) sum of tensor products, such that:

370


1. The states ρk decompose as ρk =

qj |k ρj |k ⊗ ωj ,

j

where ρj |k is a state on Jj , ωj is a state on Kj (which is independent of k), and (qj |k )j is a probability distribution over j ’s. 2. For every T which leaves the ρk invariant, every associated unitary from Eq. (A2) has the form UHE = Jj ⊗ UKj E , k

with unitaries UKj E on Kj ⊗ E that satisfy ∗ = ωj . ∀j Tr E UKj E (ωj ⊗ ε)UK jE The purpose of this appendix is to present a short (but non–constructive) proof of this theorem, based on the theory of operator algebras. Property 1 of the theorem has previously appeared in a paper of Lindblad [12] and our approach closely follows the one taken there. We begin with a slight reformulation of the result, avoiding the environment system E: Proposition 10. In the above theorem, Property 2 is equivalent to: 2 . For every T which leaves the ρk invariant, T |B(Jj ⊗Kj ) = id ⊗ Tj ,

∀j

with id on Jj and Tj on Kj such that Tj (ωj ) = ωj . Proof. Clearly, 2 implies 2 . In the other direction, consider any U implementing T . Clearly, because of 2 , U |(Jj ⊗Kj )E = Jj ⊗ UKj E , which yields the form 2 for U .

Proof of Theorem 9. Consider the set of quantum operations F = F : ∀k Fρk = ρk , which is obviously non–empty since it contains T and id. With each F ∈ F we associate the set AF = X ∈ B(H) : F ∗ (X) = X of operators left invariant by the adjoint map F ∗ . By Lemma 11 this is a ∗–subalgebra

of B(H) and, in fact, if F ∗ (X) = i Bi∗ XBi , AF = {Bi , Bi∗ } = {X : ∀i XBi = Bi X, XBi∗ = Bi∗ X} is the commutator of the Kraus operators of F ∗ . By the same lemma, this algebra furthermore is the image of B(H) under the projection map N 1 ∗ n (F ) , N→∞ N

P ∗ = lim

n=1


whose adjoint is

371

N 1 n P = lim F . N→∞ N n=1

Clearly, P ∈ F. Next, define A0 =

AF ,

F ∈F

which clearly is a ∗–subalgebra itself. Because all dimensions are finite, it can actually be presented as a finite intersection A0 = AF1 ∩ . . . ∩ AFM , and in fact there is F0 ∈ F such that A0 = AF0 . We may take, for example, F0 =

M 1 Fµ M µ=1

and use Lemma 11. Denote the projection onto A0 derived from F0∗ by P0∗ . Lemma 12 gives us the form of A0 : B(HbL ) ⊗ bR , A0 = j

j

and, likewise, of P0∗ : P0 (ξ ) =

j

j

Tr bR j ξ j ⊗ ωj . j

Thus, we obtain the advertised form of the states: ρk = P0 (ρk ) = qj ρj |k ⊗ ωj . j

As for the properties of T , because AT ⊃ A0 , we have T ∗ |A0 = idA0 . More explicitly, for A ∈ B(Jj ) and ∈ B(Kj ), T ∗ (A ⊗ ) = A ⊗ . Now assume 0 ≤ A ≤ , and consider B ∈ B(Kj ) such that 0 ≤ B ≤ . Then 0 ≤ T ∗ (A ⊗ B) ≤ T ∗ (A ⊗ ) = A ⊗ ≤ ⊗ .

(A4)

T∗

maps B(Jj ⊗ Kj ) into itself for all j , and hence the same applies This implies that to T . Now, Eq. (A4) applied with the rank–one projector A = |ψψ|, yields that T ∗ (|ψψ| ⊗ B) = |ψψ| ⊗ B , with B depending linearly on B. Dependence on |ψψ| quickly leads to contradiction, so T ∗ (A ⊗ B) = A ⊗ Tj∗ (B), which gives the desired form of T : T (ρ ⊗ σ ) = ρ ⊗ Tj (σ ), and application to the ρk yields the invariance of the ωj under Tj .

372


Here follow the general lemmas about unital completely positive maps which were used in the Troof of theorem 9. The first one is a mean ergodic theorem for the dual of a quantum operation. (The statement is essentially the Kovács-Szücs theorem — see e.g. [4], Proposition 4.3.8 — but we give a proof in our setting.) Lemma 11. For a quantum operation F , the map N 1 ∗ n (F ) N→∞ N

P ∗ = lim

n=1

is a conditional expectation onto the ∗–subalgebra AF = {X : F ∗ (X) = X} = {Bi , Bi∗ } . subspace and Proof. First of all, we want to see that AF is a ∗–subalgebra. It is a linear ∗ the Kraus representation shows that if F ∗ (X) = X, then F ∗ (X ∗ ) = F ∗ (X) = X∗ . With this, the Schwarz inequality (see e.g. [4]) gives that for invariant X, F ∗ (X ∗ X) ≥ F ∗ (X ∗ )F ∗ (X) = X∗ X. However, applying a faithful (i.e., non–degenenerate) invariant state, such as leaves only the possibility of equality:

1 K

k

ρk ,

F ∗ (X ∗ X) = X∗ X, from which it follows straightforwardly that the product of invariant operators is again invariant. For X ∈ AF , one can confirm by direct calculation that

[X, Bi ]∗ [X, Bi ] = F ∗ (X ∗ X) − X ∗ X = 0,

i

the latter by the previous observation that X∗ X is also invariant. But since the left-hand side is a sum of positive terms, all of them must be 0, hence [X, Bi ] = 0 for all i. Similarly, [X, Bi∗ ] = 0 for all i. These facts together say that AF ⊂ {Bi , Bi∗ } , while the opposite containment is trivial. Another application of the Schwarz inequality gives that F ∗ is a contraction. Hence the mean ergodic theorem for a contraction implies that the limit in the statement exists. (Due to the finite dimensional situation all the relevant topologies coincide.) To see P ∗ (X) ∈ AF , we compute

1 ∗ N+1 (F ) (X) − F ∗ (X) , N→∞ N

(F ∗ P ∗ )(X) − P ∗ (X) = lim

which is clearly 0 so the image of P ∗ is contained in AF . Since P ∗ (X) = X when X ∈ AF , it is also onto and a projection.


373

Lemma 12. Let A be a ∗–subalgebra of B(H), with a finite dimensional H. Then there is a direct sum decomposition H= HbL ⊗ HbR , j

such that A=

j

j

B HbL ⊗ bR . j

j

j

Any completely positive and unital projection P ∗ of B(H) onto A is of the form Tr bR j Xj (bL ⊗ ωj ) ⊗ bR , P ∗ (X) = j

j

j

j

with the projections j onto the subspaces HbL ⊗ HbR , and states ωj on HbR . j

Proof. See [22], Sec. I.11.

j

j

References 1. Accardi, L., Frigerio, A.: Markovian cocycles. Proc. Proc. Roy. Irish Acad. 83A(2), 251–263 (1983) 2. Barnum, H., Knill, E.: Reversing quantum dynamics with near–optimal quantum and classical fidelity. J. Math. Phys. 43(5), 2097–2106 (2002) 3. Barnum, H., Nielsen, M. A., Schumacher, B.: Information transmission through a noisy quantum channel. Phys. Rev. A 57(6), 4153–4175 (1998) 4. Bratteli, O., Robinson, D. W.: Operator algebras and quantum statistical mechanics. 1. C∗ - and W∗ –algebras, symmetry groups, decomposition of states. 2nd ed., Texts and Monographs in Physics, New York: Springer Verlag, 1987 5. Holevo, A. S.: Bounds for the quantity of information transmitted by a quantum channel. Probl. Inf. Transm. 9(3), 177–183 (1973) 6. Horodecki, M., Horodecki, P., Horodecki, R.: Mixed-state entanglement and distillation: Is there a ‘bound’ entanglement in nature?. Phys. Rev. Lett. 80, 5239–5242 (1998) 7. Koashi, M., Imoto, N.: Operations that do not disturb partially known quantum states. Phys. Rev. A 66(2), 022318, (2002) 8. Kullback, S., Leibler, R. A.: On information and sufficiency. Ann. Math. Statistics, 1951 9. Lieb, E. H., Ruskai, M. B.: Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys. 14, 1938–1941 (1973) 10. Lindblad, G.: Completely positive maps and entropy inequalities. Commun. Math. Phys.40, 147–151 (1975) 11. Lindblad, G.: Quantum entropy and quantum measurements. In: C. Bendjaballah, O. Hirota, S. Reynaud (eds.), Quantum Aspects of Optical Communications, Lecture Notes in Physics, Vol. 378, Berlin: Springer Verlag, 1991, pp. 71–80 12. Lindblad, G.: A general no–cloning theorem. Lett. Math. Phys. 47, 189–196 (1999) 13. von Neumann, J.: Thermodynamik quantenmechanischer Gesamtheiten. Nachr. der Gesellschaft der Wiss. Gött., 273–291, (1927). (see also J. von Neumann, Mathematical Foundations of Quantum Mechanics. Princeton, NJ: Princeton University Press, 1996 14. Ohya, M., Petz, D.: Quantum Entropy and Its Use. Texts and Monographs in Physics, Berlin-Heidelberg: Springer Verlag, 1993 15. Petz, D.: Sufficient subalgebras and the relative entropy of states of a von Neumann algebra. Commun. Math. Phys. 105, no. 1, 123–131, 1986. Sufficiency of channels over von Neumann algebras. Quart. J. Math. Oxford Ser. 39(153), 97–108 (1988) 16. Petz, D.: Monotonicity of quantum relative entropy revisited. Rev. Math. Phys. 15, 79–91 (2003) 17. Ruskai, M. B.: Inequalities for Quantum Entropy: A Review with Conditions for Equality. J. Math. Phys. 43, 4358–4375 (2002) 18. Schumacher, B., Nielsen, M. A.: Quantum data processing and error correction. Phys. Rev. A. 54, 2629–2635 (1996)

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19. Shannon, C. E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948) 20. Shor, P. W.: Additivity of the Classical Capacity of Entanglement–Breaking Quantum Channels. J. Math. Phys. 43, 4334–4340 (2002) 21. Stinespring, W. F.: Positive functions on C∗ –algebras. Proc. Am. Math. Soc. 6, 211–216 (1955) 22. Takesaki, M.: Theory of Operator Algebras I. New York–Heidelberg–Berlin: Springer–Verlag, 1979 23. Uhlmann, A.: Relative entropy and the Wigner–Yanase–Dyson–Lieb concavity in an interpolation theory. Commun. Math. Phys. 54(1), 21–32 (1977) 24. Umegaki, H.: Conditional expectation in an operator algebra IV. Entropy and information. K¯odai Math. Sem. Rep. 14, 59–85 (1962) 25. Wehrl, A.: General properties of entropy. Rev. Mod. Phys. 50(2), 221–260 (1978) Communicated by M.B. Ruskai


Communications in


Microlocalization of Resonant States and Estimates of the Residue of the Scattering Amplitude Jean-Fran¸cois Bony, Laurent Michel Mathématiques Appliquées de Bordeaux, UMR 5466 du CNRS, Université de Bordeaux 1, 33405 Talence cedex, France. E-mail: [email protected]; [email protected] Received: 30 April 2003 / Accepted: 26 September 2003 Published online: 24 February 2004 – © Springer-Verlag 2004

Abstract: We obtain some microlocal estimates of the resonant states associated to a resonance z0 of an h-differential operator. More precisely, we show that the normalized √ resonant states are O( |Im z0 |/ h +h∞ ) outside the set of trapped trajectories and are O(h∞ ) in the incoming area of the phase space. As an application, we show that the residue of the scattering amplitude of a Schrödinger operator is small in some directions under an estimate of the norm of the spectral projector. Finally we prove such a bound in some examples. 1. Introduction The original motivation of this paper is the study of the residue of the scattering amplitude associated to a Schrödinger operator P (h) = −h2 + V (x) on Rn . The first works treating this question are due to Lahmar-Benbernou [17] and Lahmar-Benbernou and Martinez [18]. In these papers, they consider the case where the potential V (x) is a “well in an island” with non-degenerate local minimum. In this situation, the form of the resonances is given by the work of Helffer and Sjöstrand [13]. Near a resonance z0 simple, isolated and close to the energy of this local minimum, the scattering amplitude can be written f (ω, ω , z, h) =

f res (ω, ω , h) + f hol (ω, ω , z, h), z − z0

with f hol holomorphic near z0 . Using the form of the resonant states associated to z0 , Lahmar-Benbernou and Martinez proved that |f res (ω, ω , h)| = g(h)|Im z0 |, where g(h) has an asymptotic expansion with respect to h. Moreover, they showed that for some directions (ω, ω ), determined by the Agmon distance to the well, the residue

376

J.-F. Bony, L. Michel

is O(h∞ ) while for some other ones, they obtained an explicit non-vanishing principal term for g(h). Their proof is based on the knowledge of the resonant states given by [13]. In [30], Stefanov generalized some parts of this result and proved that for V ∈ C0∞ (Rn ) and z0 a resonance which is simple and isolated in a sense made precise in [30], one has |f res (ω, ω , h)| ≤ Ch−

n−1 2

|Im z0 |.

(1.1)

This result was next improved by the second author in [23] where estimate (1.1) is established for general long-range potentials under a weaker separation condition on resonances. In [30] and [23], the method employed stands on the semiclassical maximum principle of Tang and Zworski [32] and a resolvent estimate of Burq [5]. In the case where |Im z0 | ≤ ChM for M 1 and |Im z0 | = O(h∞ ), it shows only that |f res | = O(hN ) for N ∈ R, whereas it is proven in [18] that the decay of the residue may depend on the direction considered. In particular, one can think that there exists some couple of directions (ω, ω ) such that the associated residue is O(h∞ ). One of our motivations is to show the existence of such directions for resonances “far” from the real axis. In the case where the potential V is compactly supported, we have a nice representation formula for the scattering amplitude, so that one can easily see the link between the problem of the residue and the estimate of the resonant states announced in the title. Indeed, as is proven in [24], one has √ √ f (ω, ω , z, h) = c(z; h) e−i zx,ω / h [h2 , χ1 ]R(z, h)[h2 , χ2 ]ei zx,ω / h dx, Rn

(1.2)

2 denotes the meromorphic continuation of the resolvent where R(z, h) : L2comp → Hloc of P to a conic neighborhood of the real axis and

c(z; h) =

(n−3)π n+1 1 n−3 z 4 (2πh)− 2 e−i 4 . 2

(1.3)

Moreover, if one denotes by Pθ the operator obtained from P by analytic dilatation (see [28]), and if one assumes that the dilatation is performed sufficiently far, one gets √ √ f (ω, ω , z, h) = c(z; h) e−i zx,ω / h [h2 , χ1 ](Pθ − z)−1 [h2 , χ2 ]ei zx,ω / h dx. Rn

(1.4)

Assume that z0 is a simple resonance and that there is no other resonance in a disk D centered in z0 , then the residue is given by the formula f res (ω, ω , h) = c(z0 ; h)[h2 , χ1 ]θ [h2 , χ2 ]ei

√ z0 x,ω / h

, ei

√ z0 x,ω / h

,

(1.5)

where θ is the spectral projector associated to z0 . Moreover, as θ is a rank one operator, there exist uθ , vθ ∈ L2 such that θ = ., vθ uθ and one can show that (Pθ − z0 )uθ = 0 and (P−θ − z0 )vθ = 0. It follows that f res (ω, ω , h) = −c(z0 ; h)uθ , [h2 , χ1 ]ei

√ z0 x,ω / h

[h2 , χ2 ]ei

√ z0 x,ω / h

, vθ .

Microlocalization of Resonant States and Residue of Scattering Amplitude

377 √

∗

On the other hand, it is easy to see that the functions [h2 , χ∗ ]ei z0 x,ω / h are microlocalized near {(x, ξ ); R1 < |x| < R2 , ξ/|ξ | ∼ ω∗ } for ω∗ = ω, ω . Our approach consists to show that for suitable directions, the resonant state uθ is microlocalized out of this set. In fact, the microlocal estimate that we will prove holds for more general operators than Schrödinger ones. To state precisely our results, we need to introduce the following class of symbol (see ∗) the book of Dimassi and Sjöstrand [7] for more details). We say that g ∈ C ∞ (Rd ; R+ d α d is an order function if ∀α ∈ N , ∂x g(x) = O(g) uniformly on R . A function a(x; h) defined on Rd ×]0, h0 ] for some h0 > 0 is said to be a symbol in the class Sd (g) if a(x; h) depends smoothly on x and ∀α ∈ Nd ,

∂xα a(x; h) = O(g),

uniformly with respect to (x, h) ∈ Rd ×]0, h0 ]. We will say that a(x; h) belongs to Sdcl (g) if there exists a sequence aj (x) ∈ Sd (g) such that for all N ∈ N, a(x; h) −

N

aj (x)hj ∈ hN+1 Sd (g),

j =0

uniformly with respect to h. For a(x, ξ ; h) ∈ S2n (g), one can define the h-pseudodifferential operator (in the Weyl quantization) A = Opw h (a) = a(x, hDx ) associated with a. For f ∈ C0∞ (Rn ), x + y 1 ix−y.ξ / h e (a)f )(x) = a , ξ ; h f (y) dξ dy. (Opw h (2πh)n 2 In this case, we say that a is the Weyl symbol of A. In this paper, we consider P (h) an h-differential operator on Rn , having the form P (h) = aα (x; h)(hDx )α , (1.6) |α|≤2

where aα (x; h) ∈ Sncl (1) and aα (x; h) does not depend on h for |α| = 2. We assume that P is formally self-adjoint on L2 (Rn ), that is ∀u, v ∈ C0∞ (Rn ) (P u)v dx = u(P v) dx. (1.7) We suppose also that P is elliptic, that is, aα (x)ξ α ≥ |ξ |2 /C.

(1.8)

|α|=2

To define the resonances, we assume that the coefficients aα (x; h) extend holomorphically in x in the domain ϒ = {x ∈ Cn ; |Im x| ≤ δ0 Re x and |x| ≥ R0 },

(1.9)

R0 > 0, δ0 ∈]0, 1[ and that P converge to −h2 at infinity in the following sense: aα (x; h)ξ α −→ ξ 2 , (1.10) |α|≤2

378


as |x| → +∞, x ∈ , uniformly with respect to h. Under these assumptions, it is clear that P is a self-adjoint operator with domain H 2 (Rn ) and one can define the resonances associated to P by the method of analytic distortions (see Aguilar–Combes [1], Hunziker [14] and Sjöstrand–Zworski [28]). Let F : Rn → Rn be a smooth vector field such that F (x) = 0 if |x| ≤ R0 and F (x) = |x| for |x| large enough. For ν ∈ R small enough, we consider the unitary operator Uν on L2 (Rn ) defined by: 1

Uν ϕ(x) = det(1 + νdF (x))− 2 ϕ(x + νF (x)). Then, the operator Uν P (h)Uν−1 has coefficients which are analytic with respect to ν near 0 and can be continued to complex values of ν. For ν = iθ , with θ > 0 small enough, we get a differential operator denoted by Pθ . It is well-known that the spectrum of Pθ is discrete in the sector Sθ = {z ∈ C; Re z > 0 and − 2θ < arg z ≤ 0} (see [28] and [26]) and by definition, the resonances of P are the eigenvalues of Pθ . cl (ξ 2 ) the Weyl symbol of P and p (x, ξ ) = p(x, ξ ; h) ∈ S2n 0 We denote by α is its principal symbol. The Hamilton vector field associated with a (x)ξ |α|≤2 α,0 p0 is Hp0 = ∂ξ p0 .∂x − ∂x p0 .∂ξ and exp(tHp0 ), t ∈ R is the corresponding Hamiltonian flow. We define the outgoing tail and the incoming tail at the energy E by ± (E) = {(x, ξ ) ∈ p0−1 (E); exp(tHp0 )(x, ξ ) ∞, t → ∓∞}. Hence, the set of trapped trajectories is T (E) = + (E) ∩ − (E) = {(x, ξ ) ∈ p0−1 (E); t → exp(tHp0 )(x, ξ ) is bounded on R}. For E > 0, T (E) is a compact set (see the appendix of the paper of C. Gérard–Sjöstrand [11]). Setting T ([a, b]) = E∈[a,b] T (E), we give another proof of a result of Stefanov on the localisation of the resonant states: Theorem 1. Let E0 > 0 be a fixed energy level, > 0 small enough, θ = h/C with C > 0, let z ∈ C be a resonance of P with Re z ∈ [E0 − , E0 + ], |Im z| < θ , and let uθ ∈ L2 (Rn ) be a resonant state associated to z: (Pθ − z)uθ = 0. If w(x, ξ ) ∈ S2n (1) with supp w ∩ T ([E0 − , E0 + ]) = ∅, then |Im z| w ∞ +h uθ . Oph (w)uθ = O h

(1.11)

(1.12)

Remark 1.1. For compactly supported perturbations of the Laplacian, this is a straightforward consequence of the estimate given in Proposition 3 of [30] and propagation of singularities given in Lemma 4.1 of [31]. As remarked by Stefanov, the same arguments can be adapted for long-range perturbations of the Laplacian in view of Sect. 8 of [31]. Remark 1.2. It seems also possible to obtain such type of results using the semi-classical measures introduced by P. Gérard [12] and Lions–Paul [19]. Assume that θ = o(h) and that z → E0 (as h → 0) is a resonance with |Im z| ≤ θ . Let uθ satisfying


379

(Pθ − z)uθ = 0 and uθ = 1. Following the works of Burq [6] and Jecko [16], one can perhaps show that any semiclassical measure µ of the sequence (uθ )h verifies

supp µ ⊂ T (E0 ), (1.13) Hp0 µ = 0. Then it is enough to write (P − z)uθ = (P − Pθ )uθ with σ P − Pθ ∈ S2n (θξ 2 ) and, as uθ H 2 = O(1), we deduce (P − z)uθ = o(1) and one can apply the proof of Burq or Jecko. Before we state our second result, let us introduce the following subspaces of the phase space. For R > 0, > 0 and σ ∈ [−1, 1], set ± (R, , σ ) = {(x, ξ ) ∈ T∗ (Rn ); |x| > R, |p0 (x, ξ ) − E0 | < and ± x, ξ > ±σ |x||ξ |}. We have the following theorem which says that a resonant state is outgoing. Theorem 2. Let E0 > 0 and uθ be a resonant state associated to a resonance z as in (1.11). We assume that Re z ∈ [E0 −, E0 +], |Im z| < θ and h/C < θ < Ch ln(1/ h) with , C > 0. Let w(x, ξ ) ∈ S2n (1) and suppose that there exists T > 0 such that exp(−T Hp0 )(supp(w)) ⊂ − (R, , σ ) with R 1 and σ < 0. Then for h > 0 sufficiently small, one has w(x, hDx )uθ = O(h∞ )uθ .

(1.14)

In particular w ∈ C0∞ (T∗ (Rn )) such that supp(w) ⊂ + ([E0 − , E0 + ])C =

C E∈[E0 −,E0 +] + (E)

satisfies the hypothesis of Theorem 2. Because, for each point ρ ∈ + ([E0 − , E0 + ])C , exp(−tHp0 )(ρ) is in a set − (R, , σ ) if t is large enough.

Remark 1.3. It is possible to generalize this result to the black-box setting (see [28] and [27] for a precise formulation). Assume that the black-box is contained in D(0, R0 ), let χ ∈ C0∞ (Rn ) with χ = 1 near D(0, R0 ) and let w be supported in {|x| > R0 } and satisfying the assumptions of the above theorem. If uθ is a resonant state, then w(x, hDx )(1 − χ (x))uθ = O(h∞ )uθ . Remark 1.4. Another possible generalization concerns the case of multiple resonances. Assume that z is a resonance whose multiplicity N = N (h) is bounded uniformly with respect to h, then the conclusion of the theorem remains valid for all generalized resonant states (i.e. the functions uθ ∈ L2 (Rn ) such that (Pθ − z)N uθ = 0). We will give the idea of the proof of this generalizations at the end of Sect. 3. The plan of the paper is the following. In Sect. 2, we make precise the action of the FBI transform on pseudodifferential operators. In particular, we give the form of the term of order h in the expansion of the transformed symbol. Section 3 is devoted to the proof of Theorem 2. The demonstration is based on the construction of a suitable escape function and an application of the result of Sect. 2. The main idea consists to choose a weight G which permits to gain ellipticity near 0, whereas the dilatation F gives ellipticity at infinity. In Sect. 4, we prove Theorem 1 using again the results of Sect. 2. Applying Theorem 2, we obtain in Sect. 5 an estimate of the residue of the scattering amplitude associated to a Schrödinger operator. We treat the case of resonances whose

380


imaginary part is bounded by O(hln(1/ h)). This estimate involves the norm of the associated spectral projector on the space of resonant states. In Sect. 6, we give some examples where the spectral projector above satisfies nice estimates. These bounds on the projector permit to show that the associated residue is O(h∞ ) for some particular directions. 2. Microlocal Exponential Estimate In this section, we give a microlocal exponential weighted estimate for C ∞ symbols using a Fourier–Bros–Iagolnitzer (in short FBI) transform, widely studied by Sjöstrand [25]. The result is a slight modification of Proposition 3.1 of Martinez [21] (see also the book of Martinez [20] for a related presentation). For u ∈ S (Rn ), the FBI transform of u is given by 2 (2.1) T u(x, ξ ; h) = αn (h) ei(x−y)ξ/ h−(x−y) /2h u(y)dy, with αn (h) = 2−n/2 (π h)−3n/4 . As proved in [20], we know that T u ∈ C ∞ (R2n ) and that 2 eξ /2h T u(x, ξ ; h) is an holomorphic function of z = x − iξ . Moreover, if u ∈ L2 (Rn ) then T uL2 (R2n ) = uL2 (Rn ) . Let A be a h-differential operator of Weyl symbol a(x, ξ ; h) ∼ j ≥0 aj (x, ξ )hj ∈ cl (ξ d ). As a is polynomial with respect to ξ with coefficients in S (1), one can S2n n cl (ξ d ) of a in a D × Cn , where find an almost analytic extension a (x, ξ ; h) ∈ S2n D = {x ∈ Cn ; |Im x| < }, which satisfies

a|R2n = a,

(2.2) ∞

∂x a = O(|Im x| )ξ . d

(2.3)

Theorem 3 (Martinez). Let f (x, ξ ) ∈ S2n (1) and G(x, ξ ) ∈ C0∞ (R2n ). Then there cl (ξ d ) uniformly with respect exists a symbol q(x, ξ ; t, h) ∼ j ≥0 qj (x, ξ ; t)hj ∈ S2n ∞ to t and an operator R(t, h) such that for all u, v ∈ C0 (Rn ), one has −tG/ h −tG/ h T Opw T v L2 (R2n ) fe h (a)u, e = q(x, ξ ; t, h) + R(t, h) e−tG/ h T u, e−tG/ h T v L2 (R2n ) , (2.4) where supp qj ⊂ supp f for all j ∈ N. Here, we have with the notation ∂z = (∂x +i∂ξ )/2, a0 x + 2t∂zG(x, ξ ), ξ − 2it∂zG(x, ξ ) , (2.5) q0 (x, ξ ; t) =f (x, ξ ) 2 2 q1 (x, ξ ; t) = f a1 − f ∂xx a0 /4 − f ∂ξ ξ a0 /4 − ∂x f ∂x a0 /2 − ∂ξ f ∂ξ a0 /2 (x, ξ ) i ∂ξ a0 ∂x f − ∂x a0 ∂ξ f (x, ξ ) + O(t), + (2.6) 2 and σ ξ R(t, h)ξ −d−σ 2 n = O(h∞ + h−3n/2 |t|∞ e2 sup |G||t|/ h ), L(L (R )) for all σ ∈ R, uniformly with respect to t and h small enough.

(2.7)


381

cl Remark 2.1. Theorem 3 holds also for A = Opw h (a) with a(x, ξ ; h) ∈ S2n (1), since one can find an almost analytic extension of a which satisfies (2.2) and (2.3) with d = 0.

Proof. This theorem is a slight adaptation of Proposition 3.1 of Martinez [20] and we follow his proof. Let T0 = f e−tG/ h T Au, e−tG/ h T v y + z αn (h)2 / h = e f (x, ξ )a (2.8) , η; h u(z)v(y ) dx dξ dy dη dy , (2π h)n 2 where = −2tG(x, ξ ) + i(x − y)ξ − i(x − y )ξ − (x − y)2 /2 − (x − y )2 /2 + i(y − z)η. (2.9) We have, for Y = (y, η) ∈ R2n and X = x + 2t∂zG(x, ξ ), ξ − 2i∂zG(x, ξ ) ∈ C2n , a(Y ; h) − a (X; h) 1 ∂ a ∂ a = (Y − Re X) sY + (1 − s)X − Im X sY + (1 − s)X ds ∂Re X ∂Im X 0 1 ∂ a ∂ a (Y − X) sY + (1 − s)X ds sY + (1 − s)X + 2iIm X = ∂Re X ∂X 0 = (Y − X)b(x, ξ, y, η; t, h) + r(x, ξ, y, η; t, h), (2.10) cl (ξ, η 2 ) and r ∈ S (|t|∞ ξ, η 2 ) uniformly with respect to t. In addition with b ∈ S4n 4n

1

b0 (x, ξ, y, η; t) =

(∂x a0 , ∂ξ a0 ) sy + (1 − s)x, sη + (1 − s)ξ ds + O(tξ, η d ).

0

(2.11) So, we have a (X; h)e−tG/ h T u, e−tG/ h T v + T1 + R1 T0 = f (x, ξ ) with y+z αn (h)2 e/ h ((y + z)/2, η) − X f (x, ξ )b x, ξ, ,η n (2π h) 2 × u(z)v(y ) dx dξ dy dη dy , (2.12) αn (h)2 R1 = e/ h f (x, ξ )r(x, ξ, (y + z)/2, η)u(z)v(y ) dx dξ dy dη dy . (2.13) (2π h)n T1 =

We have ∂x + i∂ξ + i∂η /2 = (y + z)/2 − x − 2t∂zG(x, ξ ), −i∂y − i ∂x + i∂ξ /2 = η − ξ + 2it∂zG(x, ξ ).

382


Thus there exists a constant vector-field L(∂x , ∂ξ , ∂y , ∂η ) such that L() = ((y + z)/2, η) − X . Making an integration by part with L in (2.12) and using a 1 (x, ξ, (y + z)/2, η; t, h) = t L f (x, ξ )b(x, ξ, (y + z)/2, η; t, h) , cl (ξ, η 2 ), supp 1 which satisfies a 1 ∈ S4n (x,ξ ) a ⊂ supp f and a01 (x, ξ, y, η; t) = − ∂x /2 − i∂ξ /2 − i∂η /2, i∂y /2 + i∂x /2 − ∂ξ /2 . f (x, ξ )b0 (x, ξ, y, η; t) ,

(2.14)

we get T1 = h e−tG/ h Ta 1 u, e−tG/ h T v , with

Ta 1 u(x, ξ ) = αn (h)

2 /2h

ei(x−y)ξ/ h−(x−y)

(2.15)

1 Opw h (a (x, ξ, ., .; t, h))u(y) dy. (2.16)

We repeat the same work for T1 as this done for T0 and, by induction, we can find, for j = 0, 1, . . . , N, symbols qj (x, ξ ; t) ∈ S2n (1) uniformly with respect to t. Moreover, supp qj ⊂ supp f and (2.17) q0 (x, ξ ; t) = f (x, ξ )a0 x + 2t∂zG(x, ξ ), ξ − 2it∂zG(x, ξ ) , q1 (x, ξ ; t) = f (x, ξ )a1 x + 2t∂zG(x, ξ ), ξ − 2it∂zG(x, ξ ) +a01 x, ξ, x + 2t∂zG(x, ξ ), ξ − 2it∂zG(x, ξ ); t 2 = f a1 − f ∂xx a0 /4 − f ∂ξ2ξ a0 /4 − ∂x f ∂x a0 /2 − ∂ξ f ∂ξ a0 /2 (x, ξ ) i + ∂ξ a0 ∂x f − ∂x a0 ∂ξ f (x, ξ ) + O(t) (2.18) 2 such that, for each N ∈ N, T0 =

N−1

qj (x, ξ ; t)hj etG/ h T u, etG/ h T v + hN e−tG/ h Ta N u, e−tG/ h T v + RN ,

j =0

(2.19) where a N and RN satisfy the same properties as a 1 and R1 . Using what T is an isometry on L2 (R2n ), we write Ta N = e−tG/ h Ta N T ∗ T u, e−tG/ h T v , RN = e−tG/ h TrN T ∗ T u, e−tG/ h T v , (2.20) where a N , rN ∈ S4n (ξ, η 2 ) have their support inside supp f . Applying Lemma 6.1 of [21] with t = 0 and l = 0, we get ξ σ TpN T ∗ ξ −d−σ = O(1), ξ σ TrN T ∗ ξ −d−σ = O(|t|∞ ), which give Theorem 3 since |e−tG/ h | ≤ esup |G||t|/ h .

(2.21)


383

3. Proof of Theorem 2 We begin the proof with some geometric results. Lemma 3.1 (C. Gérard–Sjöstrand). Assume that K ⊂ p0−1 ([E0 − , E0 + ]) is compact and satisfies K ∩ T ([E0 − , E0 + ]) = ∅. Then, one can find a function f (x, ξ ) ∈ Cb∞ (T∗ (Rn )) such that Hp0 f ≥ 0 on p0−1 ([E0 − , E0 + ]) and Hp0 f > 1 on K. Proof. We follow the proof of Proposition A.6 of C. Gérard and Sjöstrand [11]. We give the proof for a reason of completeness and we use their notation. Let HT = {(x, ξ ) ∈ T∗ (Rn ); p0 (x,ξ ) ∈ [E0 − , E0 + ] and x.ξ = T } with T large enough. Let T > 0 and 0 < f+ ∈ C ∞ p0−1 ([E0 −, E0 +])\ − ([E0 −, E0 +]) be equal to χ (x.ξ )Hp0 (x.ξ ) outside a compact with χ ∈ C0∞ ([−T − T − 1, T + T + 1]; [0, 1]) equal to 1 near [−T − T , T + T ]. As in [11], we can solve Hp0 (G+ ) = f+ in p0−1 ([E0 − , E0 + ])\ − ([E0 − , E0 + ]) with G+ = T on HT . We have G+ ≤ T + T + 1 and, if f+ is large enough in a compact, lim sup G+ ≤ −T .

− ∩H−T

(3.1)

We construct G− with analogous properties. Let χ± ∈ C ∞ (R; R± ) with supp χ± ⊂ [∓T , ±∞[ and with χ+ (t) + χ− (t) = t. Put

∈ C∞,

= χ+ (G+ ) + χ− (G− ). By (3.1), we have, near p −1 ([E0 − , E0 + ]), G G 0 b

≥ 0 and Hp0 (G)

> c > 0 for (x, ξ ) ∈ {−T − T < x.ξ < −T } ∪ {T < x.ξ < Hp0 (G) T + T }. As K is compact and K ∩ T ([E0 − , E0 + ]) = ∅, there is s > 0 and T > 0 such that (x, ξ ) ∈ K implies exp(sHp )(x, ξ ) ∈ {T < x.ξ < T + T } or exp(−sHp )(x, ξ ) ∈ {−T − T < x.ξ < −T }. Then, we can take f ∈ C ∞ (T∗ (Rn )) with f (x, ξ ) = −1

G(exp(sH p )(x, ξ ))/c + G(exp(−sH p )(x, ξ ))/c near p ([E0 − , E0 + ]). Let σ ∈] − 1, 0[ and α ∈ C0∞ (R; [0, 1]) be a decreasing function such that α(x) = 1 if x < σ and α(x) = 0 if x > σ/2. We define x.ξ , (3.2) w

(x, ξ ) = ρ(|x|)f (p0 (x, ξ ))α |x||ξ | where f ∈ C0∞ ([E0 − , E0 + ]) and ρ ∈ C ∞ (R; [0, 1]) is growing with ρ(x) = 1 for x > R and ρ(x) = 0 for x < R − 1. It is obvious that w

∈ S2n (1). We have the following lemma for w

, that will be useful later.

≤ 0. Lemma 3.2. For σ > 0 small enough and R large enough we have Hp0 w Proof. Using (1.10), one can show that, for p0 (x, ξ ) ∈ [E0 − , E0 + ], 2ξ + o(1) , Hp0 = o(1/x)

(3.3)

where o(1) is a function which tends to 0 as |x| → +∞. Then x.ξ x.ξ + f (p(x, ξ ))α Hp0 ρ(|x|)

(x, ξ ) = f (p0 (x, ξ ))ρ(|x|)Hp0 α Hp0 w |x||ξ | |x||ξ | |ξ |2 ≥ E0 − + o(1).

384


For (x, ξ ) ∈ supp(Hp0 w

), we have Hp0 α

x.ξ 2|ξ |2 x.ξ o(1) 2(x.ξ )2 + = α − |x||ξ | |x| |x| |x|3 |x| 1 x.ξ ≤ α 2E0 − 2 + o(1) − 2σ 2 , |x| |x||ξ |

and, on the other hand, x.ξ Hp0 ρ(|x|) = ρ (|x|) + o(1) |x| ≤ ρ (|x|)(σ/2 + o(1)). If we fix σ > 0 small enough and after R large enough, we have Hp0 w

≤ 0.

Now, we can begin the proof of Theorem 2. Consider z ∈ C, uθ ∈ H 2 (Rn ) and w as in Theorem 2 such that (Pθ − z)uθ = 0. For N ∈ N, let w˜ j (x, ξ ), j = 1, . . . , N, ∞ be of the form (3.2) with w ≺ w1 ≺ · · · ≺ wN ≺ w∞ , where the ωj are defined by ωj (x, ξ ) = ω˜ 1 (exp(tHp0 )(x, ξ )). Here, the notation g1 ≺ g2 means that g2 = 1 near the support of g1 and one can easily see that Hp0 ωj ≤ 0, ∀j . Denoting by ., . the scalar product on L2 (R2n ), we have 0 = w12 (x, ξ )e−tG/ h T (Pθ − z)uθ , e−tG/ h T uθ . (3.4) We can apply Theorem 3 with Pθ to get (qθ (x, ξ ; t, h) − z)e−tG/ h T uθ , e−tG/ h T uθ 2 = O (h∞ + t ∞ h−3n/2 )esup |G||t|/ h e−tG/ h T uθ ,

(3.5)

with qθ (x, ξ ; t, h) = qθ,0 (x, ξ ; t) + hqθ,1 (x, ξ ; 0) + (h|t| + h2 )rθ (x, ξ ; t, h),

(3.6)

where qθ,0 and qθ,1 are given by (2.5) and (2.6) and rθ ∈ S2n (1) uniformly with respect to t, θ and supp rθ ⊂ supp w1 . Lemma 3.3. We have the following expansions Im qθ,0 (x, ξ ; t) = −w12 (x, ξ )Hp0 (tG(x, ξ ) + θ F (x)ξ ) + w12 (x, ξ )O(θ 2 + t 2 ), (3.7) and Im qθ,1 (x, ξ ; 0) =

1 Hp w 2 (x, ξ ) + w22 (x, ξ )O(θ ). 2 0 1

(3.8)

Proof. First, we recall that P being formally self-adjoint, the symbols p0 and p1 are real valued. We will denote by pθ (x, ξ ; h) the symbol of Pθ , and by definition, we have pθ,0 (x, ξ ) = p0 x + iθ F (x), (1 + iθ ∂x F (x))−1 ξ . (3.9)


385

Notice that one has pθ,0 (x, ξ ) = p0 (x + iθ F (x), ξ − iθ ∂x F (x)ξ ) + O(θ 2 )ξ 2 = p0 (x, ξ ) + iθ F (x)∂x p0 (x, ξ ) − ∂x F (x)ξ ∂ξ p0 (x, ξ ) + O(θ 2 )ξ 2 = p0 (x, ξ ) − iθ Hp0 (F (x)ξ ) + O(θ 2 )ξ 2 .

(3.10)

Combining Eqs. (3.10) and (2.5) one gets qθ,0 (x, ξ ; t) =w12 (x, ξ )p θ,0 x + 2t∂z G(x, ξ ), ξ − 2it∂z G(x, ξ ) =w12 (x, ξ ) p 0 x + 2t∂z G(x, ξ ), ξ − 2it∂z G(x, ξ ) − iθ Hp0 (F (x)ξ ) x + 2t∂z G(x, ξ ), ξ − 2it∂z G(x, ξ ) + O(θ 2 ) . By Taylor expansion, we obtain qθ,0 (x, ξ ; t) = w12 (x, ξ ) p0 (x, ξ ) + t ∂x p0 ∂x G + ∂ξ p0 ∂ξ G (x, ξ )

−itHp0 G(x, ξ ) − iθ Hp0 (F (x)ξ )(x, ξ ) + O(θ 2 + t 2 ) .

(3.11)

Taking the imaginary part, we obtain the announced expansion for qθ,0 . Now, let us prove the formula on qθ,1 . From formula (2.6), we know that 2 qθ,1 (x, ξ ; 0) = w12 pθ,1 − w12 ∂xx pθ,0 /4 − w12 ∂ξ2ξ pθ,0 /4 − ∂x w12 ∂x pθ,0 /2 i −∂ξ w12 ∂ξ pθ,0 /2 (x, ξ ) + Hpθ,0 w12 (x, ξ ). (3.12) 2 By Taylor expansion, we get pθ,1 (x, ξ ) = p1 (x, ξ ) + O(θ ξ 2 ), and Hpθ,0 w12 (x, ξ ) = Hp0 w12 (x, ξ ) + O(θ )w22 (x, ξ ). The symbols p1 , p0 and w1 being real valued, the result comes directly by taking the imaginary part of (3.12). As xξ is an escape function and F (x) = x for x large enough, we have, on p0−1 ([E0 − , E0 + ]),

c > 0 for |x| ≥ R, Hp0 (F (x)ξ ) ≥ (3.13) − M for |x| ≤ R, with R > 0 large enough. We fix K = supp w∞ ∩ B(0, R0 ) ⊂ p0−1 ([E0 − , E0 + ]) ∩ T ([E0 − , E0 + ])c and we denote f (x, ξ ) the function given by Lemma 3.1. Let χ1 ∈ C0∞ (Rn , [0; 1]) such that χ2 (x) = 1 for |x| ≤ R + 1 and χ2 ∈ C0∞ (R; [0, 1]) with χ2 (E) = 1 on [E0 − , E0 + ]). As in [21], we set G(x, ξ ) = χ1 (x)χ2 (p0 (x, ξ ))f (x, ξ ) ∈ C0∞ (R2n ).

(3.14)

386


Since χ1 can be chosen arbitrarily flat, the quantities µ = sup |χ2 (p0 (x, ξ ))f (x, ξ )Hp0 χ1 (x)|,

(3.15)

can be chosen arbitrarily small. We take t = Lθ , and we have, on the support on w∞ ,

−Lµ + c for |x| ≥ R0 , Hp0 (tG(x, ξ ) + θ F (x)ξ ) ≥ θ (3.16) L − Lµ − M for |x| ≤ R0 . We fix L ≥ 2M and µ small enough so that (3.16) becomes Hp0 (tG(x, ξ ) + θ F (x)ξ ) ≥ θc/2, on supp w∞ . Equations (3.6), (3.17) and Lemma 3.3 imply −Im (qθ (x, ξ ; t, h) − z)e−tG/ h T uθ , e−tG/ h T uθ 2 ≥ θ c/4w1 e−tG/ h T uθ − h w1 Hp0 w1 e−tG/ h T uθ , e−tG/ h T uθ 2 +O(θ 2 + h2 )w2 e−tG/ h T uθ .

(3.17)

(3.18)

Since w1 Hp0 w1 ≤ 0, by Lemma 3.2, −Im (qθ (x, ξ ; t, h) − z)e−tG/ h T uθ , e−tG/ h T uθ 2 2 ≥ θ c/4w1 e−tG/ h T uθ + O(θ 2 + h2 )w2 e−tG/ h T uθ .

(3.19)

Using (3.5), we get w1 e−tG/ h T uθ 2 ≤ O(θ )w2 e−tG/ h T uθ 2 + O(h∞ )T uθ 2 , and by induction, w1 e−tG/ h T uθ 2 ≤ O(θ N−1 )wN e−tG/ h T uθ 2 + O(h∞ )T uθ 2 2 ≤ O θ N−1 h−2CL sup |G| uθ ,

(3.20)

which implies w1 T uθ = O(h∞ )uθ .

(3.21)

Now, choose w0 ∈ S2n (1) such that w ≺ w0 ≺ w1 . One can write w w T Opw h (w)uθ ≤ w0 T Oph (w)uθ + (1 − w0 )T Oph (w)uθ .

Using two times Theorem 3 with A = Opw h (w) and inequality (3.21), we have 2 w w w0 T Opw h (w)uθ = w0 T Oph (w)uθ , w0 T Oph (w)uθ

= q(x, ξ ; h)T uθ , T uθ + O(h∞ )uθ 2 = O(h∞ )uθ 2 ,

(3.22)


387

since q(x, ξ ; h) ∈ S2n (1) satisfy q ≺ w1 . On the other hand, (1 − w0 )T Opw (w)uθ (x, ξ ) h αn (h) = e/ h (1 − w0 )(x, ξ )w (y + z)/2, η; h uθ (z) dz dy dη, (3.23) n (2π h) with (x, ξ, y, η, z) = i(x − y)ξ − (x − y)2 /2 + i(y − z)η.

(3.24)

We notice that ∂y − i/2∂η = i(η − ξ ) + (x − (y + z)/2). So, making integrations by parts in (3.23) with L=

(x − (y + z)/2) − i(η − ξ ) (∂y − i/2∂η ), (x − (y + z)/2)2 + (η − ξ )2

(3.25)

and using the fact that supp w ∩ supp(1 − w0 ) = ∅, we find, for each N ∈ N, N αn (h) (1−w0 )T Opw e/ h sN x, ξ, (y +z)/2, η; h uθ (z) dx dξ dy dη (w)u = h θ h n (2πh) N = h TsN T ∗ T uθ , ∞ with sN ∈ S4n (1). So (2.21) implies (1 − w0 )T Opw h (w)uθ = O(h )uθ and we get ∞ Opw h (w)uθ = O(h )uθ ,

which gives Theorem 2.

(3.26)

Let us explain briefly how to generalize Theorem 2 to the black-box setting and to multiple resonances. Assume that χ , w and uθ are as in Remark 1.3. We have (Pθ − z) (1−χ ) = (Qθ −z)(1−χ ), where Qθ is a differential operator satisfying the assumptions of Theorem 2. Following the proof above we get Im w1 (x, ξ )2 e−tG/ h T (Qθ − z)(1 − χ )uθ , e−tG/ h T (1 − χ )uθ 2 2 ≥ Cθ w1 e−tG/ h T (1 − χ )uθ − O(θ 2 )w2 e−tG/ h T (1 − χ )uθ . On the other hand, we can always assume that supp χ ∩ suppx w1 = ∅ and one deduces from Theorem 3 that w1 (x, ξ )2 e−tG/ h T (Qθ − z)(1 − χ )uθ , e−tG/ h T (1 − χ )uθ = w1 (x, ξ )2 e−tG/ h T [Qθ , χ ]uθ , e−tG/ h T (1 − χ )uθ = O(h∞ )uθ . It follows that w1 (x, ξ )T (1 − χ )uθ = O(h∞ )uθ + O(h)w2 (x, ξ )T (1 − χ )uθ and working as in the proof of Theorem 2 we show that w(x, hDx )(1 − χ )uθ = O(h∞ )uθ . Now, as in Remark 1.4, assume that z0 is a resonance whose multiplicity N = N (h) is bounded uniformly with respect to h and that uθ is a generalized resonant state associated to z0 . By definition, (Pθ − z0 )N uθ = 0 and we deduce from Theorem 2, that wN−1 (x, hDx )(Pθ − z)N−1 uθ = O(h∞ )(Pθ − z)N−1 uθ = O(h∞ )uθ . (3.27)

388


Using this estimate and the proof of Remark 1.3, we obtain wN−2 (x, hDx )(Pθ − z)N−2 uθ = O(h∞ )uθ , and repeating this argument N − 2 times (here we use that N is bounded with respect to h), we deduce that if w satisfies the assumptions of Theorem 2 and uθ is a generalized resonant state associated to z0 , then w(x, hDx )uθ = O(h∞ )uθ . 4. Proof of Theorem 1 The proof uses essentially the same arguments as in the proof of Theorem 2. For N ∈ N, let w ≺ w0 ≺ · · · ≺ wN ≺ w∞ ∈ S2n (1) with supp w∞ ∩ = ∅ and let g0 ≺ · · · ≺ gN ≺ g∞ ∈ C0∞ ([E0 −3, E0 +3]) with g0 = 1 near [E0 −2 +E0 +2].

we get Applying Theorem 3 with f = g02 (p0 (x, ξ )) and t = Ch, 0 = Im g0 (p0 (x, ξ ))e−tG/ h T (Pθ − z)uθ , e−tG/ h T uθ

h) + O(h∞ ) − z e−tG/ h T uθ , e−tG/ h T uθ , = Im qθ (x, ξ ; Ch,

(4.1)

with

h) = qθ (x, ξ ; Ch,

∞

j. qθ,j (x, ξ ; Ch)h

j =0

Following Lemma 3.3, one can choose G(x, ξ ) ∈ C0∞ (supp w2 ) as in (3.14) such that:

≤ −g02 (p0 )w02 h + O(h2 )g12 (p0 )w12 . Im qθ,0 (x, ξ ; Ch)

(4.2)

Using Remark 2.1 and the fact that Im qj (x, ξ ; 0) = 0, we get that

= O(h2 )g12 (p0 )w12 Im qθ,j (x, ξ ; Ch)

(4.3)

for j ≥ 1. So (4.1) implies g0 (p0 )w0 e−tG/ h T uθ 2 = O(h)g1 (p0 )w1 e−tG/ h T uθ 2 |Im z| + h∞ e−tG/ h T uθ 2 . +O h

(4.4)

By induction, g0 (p0 )w0 e−tG/ h T uθ 2 = O(hN )gN (p0 )wN e−tG/ h T uθ 2 |Im z| ∞ e−tG/ h T uθ 2 , +O +h h and since e−tG/ h = O(1) and etG/ h = O(1), |Im z| 2 ∞ T uθ 2 . g0 (p0 )w0 T uθ = O +h h

(4.5)


389

Now, let g−∞ ≺ g−N ≺ · · · ≺ g−1 ≺ g0 ∈ C0∞ (R) with g−∞ = 1 on [E0 − , E0 + ]. Applying Theorem 3 with t = 0 and f = (1 − g0 (p0 ))2 sign(p0 − E0 ) ∈ S2n (1), we have 0 = Re (1 − g0 (p0 ))2 T (Pθ − z)uθ , T uθ = Re qθ (x, ξ ; 0, h) − z + O(h∞ )ξ 2 T uθ , T uθ . (4.6) We have Re (qθ (x, ξ ; 0) − z) = (1−g0 (p0 ))2 sign(p0 −E0 )(p0 −Re z) + O(h)(1 − g−1 (p0 ))2 ≥ (1 − g0 (p0 ))2 ξ 2 /C + O(h)(1 − g−1 (p0 ))2 ,

(4.7)

with C > 0 large enough. On the other hand, we know that, for j ≥ 1, qθ,j (x, ξ ; 0) ∈ S2n (ξ 2 ) satisfies supp qθ,j ⊂ supp(1 − g0 (p0 )). So (4.6) proves that (1 − g0 (p0 ))ξ 2 T uθ 2 = O(h)(1 − g−1 (p0 ))ξ 2 T uθ 2 + O(h∞ )ξ 2 T uθ 2 , (4.8) and by induction (1 − g0 (p0 ))ξ 2 T uθ 2 = O(hN )(1 − g−N (p0 ))ξ 2 T uθ 2 + O(h∞ )ξ 2 T uθ 2 = O(hN )ξ 2 T uθ 2 .

(4.9)

As Pθ is elliptic in the classical sense, (1.11) implies, uθ H 2 (Rn ) = O(1)uθ L2 (Rn ) , which, in view of (2.21), implies ξ 2 T uθ = O(1)uθ .

(4.10)

(1 − g0 (p0 ))T uθ 2 = O(h∞ )uθ ,

(4.11)

|Im z| ∞ w0 T uθ = O uθ 2 . +h h

(4.12)

So (4.9) becomes

which gives, with (4.5),

2

We conclude as at the end of the proof of Theorem 2.

5. Residue Estimate of the Scattering Amplitude In this section, we assume that P is a Schrödinger operator P = −h2 + V (x),

(5.1)

390


where V (x) ∈ Sn (1) extends holomorphically to the domain ϒ defined in (1.9). To define the scattering amplitude, we make a long-range assumption on V (x): |V (x)| ≤ C|x|−ρ .

∃ρ > 0 ∃C > 0 ∀x ∈ ,

(5.2)

In particular, P satisfies the assumptions of Sect. 1. We can define the scattering matrix ∗ , related to P = −h2 and P , as a unitary operator: S(z; h), z ∈ R+ 0 S(z; h) : L2 (Sn−1 ) −→ L2 (Sn−1 ). Next, introduce the operator T (z; h) defined by S(z; h) = I d − 2iπ T (z; h). It is wellknown (see [15]) that T (z; h) has a kernel T (ω, ω , z; h), smooth in (ω, ω ) ∈ Sn−1 × Sn−1 \ {ω = ω } and the scattering amplitude is given by f (ω, ω , z; h) = c1 (z; h)T (ω, ω , z; h), with c1 (z; h) = −2π(2z)−

n−1 4

(2πh)

n−1 2

e−i

(n−3)π 4

.

(5.3)

ω

In [10], C. Gérard and Martinez have shown that for ω = fixed, the scattering ampli∗ , whose poles are tude has a meromorphic continuation to a conic neighborhood of R+ the resonances of P . Moreover, the multiplicity of each pole is exactly the multiplicity of the resonance. In this section, we still assume that z0 (h) is a simple resonance of P such that Re z0 ∈ [E0 − , E0 + ] and 0 < − Im z0 < Ch ln(1/ h). Under this condition the scattering amplitude takes the form f (ω, ω , z; h) =

f res (ω, ω ; h) + f hol (ω, ω , z; h), z − z0

(5.4)

where f hol (ω, ω , z; h) is holomorphic near z0 . Our aim is to give an estimate of the residue f res in some special directions: Definition 5.1. We say that ω ∈ Sn−1 is an incoming direction (resp. outgoing direction) for the energy E0 iff there is , R > 0 and W ⊂ Sn−1 , a neighborhood of ω, such that, for all (x, ξ ) ∈ p−1 ([E0 − , E0 + ]) |x| ≥ R and

ξ ∈ W ⇒ lim exp(tHp0 )(x, ξ ) = ∞. t→−∞ |ξ |

(5.5)

(resp. lim exp(tHp0 )(x, ξ ) = ∞ as t → +∞). Remark 5.2. If ρ > 1, ω is an incoming direction iff there is R > 0 such that p(x, ξ ) = E0 , |x| ≥ R and

ξ = ω ⇒ lim exp(tHp0 )(x, ξ ) = ∞. t→−∞ |ξ |

This is a consequence of Proposition 6.1 of [22]. For θ ≥ C|Im z| with C > 0 sufficiently large, we denote by θ the spectral projector associated to the resonance z0 : 1 (z − Pθ )−1 dz, (5.6) θ = 2iπ ∂D where D = D(z0 , r(h)) ⊂ C is a small disk such that z0 is the only resonance in D.


391

Theorem 4. Let E0 > 0 and ω, ω ∈ S n−1 with ω = ω . If ω is an outgoing direction or if ω is an incoming direction, then there exists , C > 0 such that for all simple resonance z0 ∈ [E0 − , E0 + ] − i[0, θ/C ] with h/C < θ < Ch ln(1/ h), C > 0 one has f res (ω, ω , h) = O(h∞ )θ .

(5.7)

Remark 5.3. As for Theorem 2, the assumption that z0 is simple is not necessary to estimate the corresponding residue f res . If we suppose only that z0 is a resonance whose multiplicity N is bounded with respect to h, then it is possible to show that f res (ω, ω , h) = O(h∞ ) where Aj =

N

Aj ,

(5.8)

j =0

(z − z0 )j (z − Pθ )−1 dz is a finite rank operator. We will give the proof

∂D

of this result at the end of Sect. 5.2. For the proof of Theorem 4, we need a representation formula of the scattering amplitude. This is the object of the next section. 5.1. Representation formula. In this section, we recall some results due to C. Gérard and Martinez [10]. We just have to be careful with the fact that in our case, the dilatation angle θ may depend on h. Moreover, we recall only how to continue the meromorphic part of the scattering amplitude. The main idea consists to extend Isozaki-Kitada’s formula to complex energies. For this purpose, C. Gérard and Martinez show that the symbols and the phases involved in that formula can be chosen to be analytic in a complex neighborhood of R2n . For R > 0 large enough, d > 0, > 0 and σ ∈]0, 1[, we denote ± C (R, d, ν, σ ) = (x, ξ ) ∈ C2n ; | Re x| > R, d −1 < | Re ξ | < d, | Im x| ≤ Re x , |Im ξ | ≤ Re ξ and ± Re x, Re ξ ≥ ±σ |x||ξ | . Let > 0, d > 1, −1 < σa− < σa+ < 0 < σb− < σb+ < 1 and R0 > 0 be suf+ − (R0 , d, , σ∗+ ) ∪ C (R0 , d, , σ∗− ). ficiently large. For ∗ = a, b, we denote ∗ = C ∞ 2n C. Gérard and Martinez construct some phases ∗ ∈ C (C ) and some symbols k∗ ∈ C ∞ (C2n ) ∩ S2n (1) satisfying the general assumptions of Isozaki-Kitada [15], and such that the following properties hold: The phases ∗ have an holomorphic extension to ∗ and satisfy (∇x ∗ (x, ξ ))2 + V (x) = ξ 2 , (5.9) β ∂xα ∂ξ ∗ (x, ξ ) − x, ξ = O x 1−ρ−|α| , uniformly in ∗ . There exists 0 < δ 1 and 1 > 0 such that k∗ are supported in ∗ , extend holomor+ − phically in the variables |x|, |ξ | to C (2R0 , d/2, , σ∗+ + δ) ∪ C (2R0 , d/2, , σ∗− − δ) and k∗ (x, ξ ; h) = O e−1 x / h , (5.10)

392


+ − uniformly with respect to h ∈]0, 1] and (x, ξ ) ∈ C (2R0 , d/2, , σ∗+ + δ) ∪ C (2R0 , − d/2, , σ∗ − δ). With this construction, one can show that for real energies, the scattering amplitude takes the form

f (ω, ω , z; h) = f1 (ω, ω , z; h) + f2 (ω, ω , z; h),

(5.11)

where f1 (ω, ω , z; h) has an holomorphic continuation with respect to z ∈ {|Im z| ≤ 0 |Re z|} for 0 sufficiently small independent on h, and f2 (ω, ω , z; h) √ √ −1 √ √ = c2 (z; h) P − (z + i0) kb (., zω)eib (., zω)/ h , ka (., zω )eia (., zω )/ h , (5.12) with c2 (z; h) = −2πz

n−3 4

(2πh)−

n+1 2

e−i

(n−3)π 4

.

The function f2 can be continued meromorphically by the following process. For µ > 0 small enough, let Uµ be defined as in Sect. 1 with F (x) = xχ (|x|), χ ∈ C ∞ (R), χ = 1 outside a big interval and χ = 0 near 0, then one has −1 f2 (ω, ω , z; h) = c2 (z; h) Uµ P − (z + i0) Uµ−1 Uµ kb eib / h , Uµ ka eia / h . (5.13) −1

Using the above properties on k∗ and ∗ it is easy to see that Uµ (k∗ eih ∗ ) is well defined for µ complex and | Im z| | Im µ| 1. A simple calculus shows that, for |Im z| ≤ 0 | Re z| and µ = iθ , one has √ √ −1

Uiθ kb (., zω)eih b (., zω) = kb (x, ω, z; θ, h)ei b (x,ω,z;θ,h)/ h ,

(5.14)

with

kb (x, ω, z; θ, h) √ √ √ √ 1−ρ = Jiθ (x)kb x + iθ F (x), zω e−Re zθ F (x)+Im zx,ω / h+O((θ+Im z)x )/ h , and

b (x, ω, z; θ, h) = b (x, Re

√ √ zω) − θ Im zF (x), ω .

From estimate (5.10), one deduces that there exists 2 > 0 such that kb (x, ω, z; θ, h) = θ(C−x )/ h uniformly on Cn and O e

kb (x, ω, z; θ, h) = O e−2 x / h , (5.15) uniformly with respect to x ∈ {|x| ≥ 2R0 } ∩ {x, ω ≥ (σb+ + δ)|x|} ∪ {x, ω ≤ (σb− − δ)|x|} and h ∈]0, 1]. Similarly, one can write √ √

U−iθ ka (., zω )eia (., zω )/ h = ka (x, ω , z; θ, h)ei a (x,ω ,z;θ,h)/ h ,


393

with

a (x, ω , z; θ, h) = a (x, Re

√ √ zω ) + θ Im zF (x), ω ,

ka (x, ω, z; θ, h) = O(eθ(C−x )/ h ) uniformly on Cn and

ka (x, ω, z; θ, h) = O e−2 x / h , uniformly with respect to x ∈ {|x| ≥ 2R0 } ∩ {x, ω ≥ (σa+ + δ)|x|} ∪ {x, ω ≤ (σa− − δ)|x|} and h ∈]0, 1]. It follows that f2 can be written

f2 (ω, ω , z; h) = c2 (z; h) (Pθ − z)−1 ka ei a / h , kb ei b / h ,

(5.16)

for θ > 0 and | Im z| θ| Re z|. From this formula, one deduces easily the form of the residue of f at a simple pole z0 :

f res (ω, ω , h) = c2 (z0 ; h) θ ka ei a / h , kb ei b / h , (5.17)

∗ are evaluated in z = z0 (h). where the functions k∗ , 5.2. Proof of Theorem 4. Before going further, let us discuss the properties of θ when z0 is simple. If one denotes by uθ a resonant state associated to z0 , the rank one operator θ can be written θ = ., u−θ uθ ,

(5.18)

where u−θ satisfies (P−θ − z0 )u−θ = 0. In particular, one has θ = uθ u−θ . Let R > 0, d > 0, σ > 0 and w± ∈ S2n (1) such that exp(±T Hp )(supp w± ) ⊂ ± (R, , ±σ ) for some T > 0. It follows from (5.18) and Theorem 2 that w− (x, hDx )θ = O(h∞ )θ , θ w+ (x, hDx ) = O(h∞ )θ .

(5.19)

The inequality (4.11) implies that for ρ ∈ C0∞ (R), ρ = 1 near [E0 − , E0 + ]), θ ρ(P ) = O(h∞ )θ .

(5.20)

Now, we consider χ ∈ C0∞ (Rn ) such that 1|x|≤2R0 ≺ χ ≺ 1|x|≤3R0 . Then, one has the following Lemma 5.4. For R0 > 0 large enough and | Im z| θ , one has

f res (ω, ω ; h) = c(z0 ; h) θ χ kb ei b / h , χ ka ei a / h + O(h∞ )θ .

(5.21)

Proof. First we prove that

θ (1 − χ ) kb ei b / h , ka ei a / h = O(h∞ )θ .

(5.22)

Let us denote g1 = θ (1 − χ ) kb ei b / h . Using (5.20), we have, for ρ ∈ C0∞ ([E0 − , E0 + ]) with ρ = 1 near E0 ,

g1 = θ (1 − χ )ρ(P ) kb ei b + O(h∞ )θ .

394


Next, we introduce w+ ∈ S2n (1) such that supp w+ ⊂ + (R0 , , σb− − 2δ) and that w+ = 1 on + (2R0 , /2, σb− − δ). It follows immediately from (5.19) that θ (1 − χ )ρ(P )w+ (x, hDx ) = O(h∞ )θ , so that

kb ei b / h + O(h∞ )θ , g1 = θ w− (x, hDx )

with w− (x, hDx ) = (1−χ )ρ(P )(1−w+ )(x, hDx ). In particular, supp w− ⊂ − (2R0 , , σb− − δ). Moreover, the stationary phase method gives |α| i

b ), kb,α ei b / h ∂ξα w− (x, ∇x kb e b / h ∼ h Cα w− (x, hDx )

(5.23)

α

β

b ) ∈ supp w− , we have with kb,α = β≤α O(∂x kb ). Now, if we assume that Re (x, ∇x − −

b ) ∈ (2R0 , , σ − δ). By (5.9), ∇x

b = ω + O(x −ρ ) and assuming that (x, ∇x b C i

R0 is sufficiently large, we get x, ω ≤ (σb− −δ)|x|. By (5.15), w− (x, hDx ) kb e b / h = O(h∞ ) comes and (5.22) follows. Using the same arguments, one proves that (1 −

χ) ka ei a = O(h∞ )θ . It follows that

kb ei b / h , (1 − χ ) ka ei a / h = O(h∞ )θ , θ χ

and the proof is complete.

Now, we are in position to prove Theorem 4 and we assume that ω is outgoing. Let w ∈ S2n (1) with supp w ⊂ p −1 ([E0 − , E0 + ]) ∩ { |ξξ | ∈ W } ∩ {R0 < |x| < 4R0 } and

w = 1 on p−1 ([E0 − /2, E0 + /2]) ∩ { |ξξ | ∈ W } ∩ {2R0 < |x| < 3R0 }, where W is given by Definition 5.1 and ω ∈ W ⊂⊂ W . Using the fact that ∇x b = ω + O(x −ρ ) and an argument similar to (5.23), one can prove that

kb ei b / h , χ ka ei a / h + O(h∞ )θ . f res (ω, ω ; h) = c2 (z0 ; h) θ χ w(x, hDx )

As χ ∈ C0∞ (Rn ) and ω is outgoing, there exists T > 0 such that exp(T Hp0 )(supp χ w) ⊂ + (R, , σ ), with R 1 and 0 < σ < 1. It follows from (5.19) and from the estimates

∗ that on k∗ and

kb ei b / h , χ ka ei a / h f res (ω, ω ; h) = c2 (z0 ; h) θ χ w(x, hDx ) +O(h∞ )θ = O(h∞ )θ , and the proof of Theorem 4 is complete.

Here, we give the arguments to show Remark 5.3 concerning multiple resonances. In the situation that we deal with, we have (Pθ − z0 )

−1

=

N j =1

Aj + Ahol (z), (z − z0 )j


395

where A0 = θ , Ahol (z) is holomorphic near z0 and the Aj are finite rank operators with Im Aj ⊂ Im θ . For j = 0, . . . , N fixed, one can write Aj =

Nj

j

j

., vθ,k uθ,k

k=1 j

j

with (Pθ −z0 )N uθ,k = 0 and Nj ≤ N . In particular, one can choose the sequence (uθ,k )k j

j

j

or (vθ,k )k orthogonal and one has vθ,k uθ,k ≤ Aj for all k. Moreover, it follows from Remark 1.3 and the discussion following (5.18), that for supp ω± ⊂ ± (R, , ±σ ), one gets ω− (x, hDx )Aj = O(h∞ )Aj , Aj ω+ (x, hDx ) = O(h∞ )Aj .

(5.24)

Following the proof of Theorem 4, one can show that f res (ω, ω , h) =

N

−1 b

Aj kb,β (z0 , h)eih

−1 a

, ka,α (z0 , h)eih

,

j =1 α+β=j

where the functions kb,β , ka,α have the same properties as ka , kb . Hence, one can work as in the proof of Theorem 4, to get |f res (ω, ω , h)| = O(h∞ ) Aj , j

which proves Remark 5.3.

6. Estimate on the Spectral Projector In this section, we give some examples where the spectral projector θ is bounded by O(h−M ). 6.1. Case of resonances at distance hM . In this section, we consider the case where the resonance z0 satisfies | Im z0 | = O(hM ) with M >> 1. In that case, it is possible to obtain some a priori estimates of the spectral projector by using the semiclassical maximum principle [32, 33, 29]. For this purpose, we need some exponential estimate of the modified resolvent (Pθ − z)−1 in a suitable complex neighborhood of E0 . This was done by Tang and Zworski in [32, 33] in the case where θ is fixed. Here θ depends on h so that we have to check that this estimate is still available in our case. Lemma 6.1 (Tang–Zworski). Assume that Ch < θ < Mh log(1/ h), with C > 0 large enough, and let θ = E + θ , where E ∈ [E0 − , E0 + ] and ⊂ C is a fixed simply connected and relatively compact domain. Let g(h) be a strictly positive function such that g(h) θ, then there exists C = C() such that θ Ch−n log g(h) D(zj , g(h)), (Pθ − z)−1 ≤ Ce . ∀z ∈ θ \ zj ∈Res(P )∩θ

396


Proof. The demonstration follows closely [32] and we just sketch it. The only difference is that θ (and so θ ) depends on h, so that we have to be careful with the constants appearing in the proof. The main steps are the following. As in [26] one can find K ∈ L(L2 , L2 ) with K = O(1) and rank(K) = O(h−n ) such that (Pθ + θ K − z) is invertible for z ∈ θ and (Pθ + θ K − z)−1 L(L2 ,H 2 ) = O(1/θ ). Using K as in [26, 32], we can construct, for z ∈ θ , an invertible operator P(z) =

Pθ − z R− R+ 0

: H 2 ⊕ CN → L2 ⊕ CN ,

(6.1)

where N = rank(K) = O(h−n ). Using the fact that (Pθ + θK − z)−1 (Pθ − z) = O(1), one shows that its inverse E(z) E+ (z) E(z) = (6.2) : L2 ⊕ CN → H 2 ⊕ CN , E− (z) E−+ (z) −1 ) and E (z), E satisfies E(z), E− (z) = O(θ + −+ (z) = O(1). −1 −1 −2 (z)CN ,CN , for z ∈ θ \ Res P , we obtain As (Pθ − z) = O(θ ) 1 + E−+

−n (Pθ − z)−1 CN ,CN = O θ −2 h−n eCh | det(E−+ (z))|−1 ,

(6.3)

and it remains to estimate | det(E−+ (z))| from below. For this purpose, we set

Dθ (z, h) =

zj ∈θ ∩Res P

z − zj θ

and det(E−+ (z)) = Gθ (z, h)Dθ (z, h). Using the change of variable θ z → (z − E)/θ ∈ we work on a domain independent of θ. Following the arguments of [26], −n one can show that |Gθ (z, h)| ≥ e−Ch uniformly with z ∈ θ , which implies ∀z ∈ θ \

D(zj , g(h)),

| det(E−+ (z))| ≥ e

−Ch−n

zj ∈Res(P )∩θ

≥ Ce

−n g(h) O(h ) θ

θ −C log( g(h) )h−n

Combining estimates (6.3) and (6.4), one gets the announced result.

.

(6.4)

Proposition 6.2. Assume that V is compactly supported and let E0 > 0. Let z0 be a simple resonance of P such that Res(P ) ∩ D(z0 , hM1 ) = {z0 }, for M1 sufficiently large and |Im z0 | ≤ ChM2 with M2 ≥ M1 + 2n + 2. Then θ = O(1) uniformly with respect to h/C < θ < Ch log(1/ h).


397

Proof. We can copy the proof of Proposition 3.1 of Stefanov [31] with θ0 = h ln(1/ h) to get (Pθ − z)−1 ≤

2 , Im z

(6.5)

−1/3

for all z satisfying Im z > 2e−h . Let us denote z˜ 0 = z0 + 2ihM2 . Following [31], we want to apply the semiclassical maximum principle as it is presented in Stefanov [29] −1 which is holomorphic on 0 to the function F (z, h) = z−z z−˜z0 (Pθ − z) (h) = {z ∈ C; | Re z − Re z0 | < 2hM1 , −hM1 −n−2 < Im z < hM1 }. From (6.5), it follows that F (z, h) ≤ Ch−M1 on Im z = hM1 . On the other hand, we deduce from the exponential estimate of the resolvent proved in Lemma 6.1 below, that −n ln(1/ h) on (h). By the semiclassical maximum principle, it follows F (z, h) ≤ CeCh that ˜ F (z, h) ≤ Ch−M1 on (h), ˜ with (h) = {z ∈ C; | Re z − Re z0 | < hM1 , −2hM1 < Im z < hM1 }. In particular, 1 −M1 and the proof is complete. (2h)−M1 θ ≤ |z0 −˜ z0 | θ = F (z0 , h) ≤ Ch From Theorem 4 and Proposition 6.2, one deduces immediately the following. Corollary 6.3. Assume that V is compactly supported and let E0 > 0. Suppose that ω is outgoing or ω is incoming and that ω = ω . Let z0 be a simple resonance of P such that Res(P ) ∩ D(z0 , hM1 ) = {z0 }, for M1 large enough, Re z0 ∈ [E0 − , E0 + ] and |Im z0 | ≤ ChM2 with M2 ≥ M1 + 2n + 2. Then f res (ω, ω , h) = O(h∞ ).

(6.6)

Remark 6.4. Let us notice that this result is not a consequence of the works [30] and [23]. Indeed, if one applies the theorems of [30] and [23] to this situation, one can only n−1 show that f res (ω, ω , h) = O hM2 − 2 . 6.2. Estimate in dimension one. Lemma 6.5. We assume that n = 1 and that the critical points of p0 (x, ξ ) on the energy level are non-degenerate (i.e. the points (x, ξ ) ∈ p0−1 ({E0 }) such that ∇p0 (x, ξ ) = 0 satisfy Hess p0 (x, ξ ) is invertible). Then there exists M, > 0 such that, for E ∈ [E0 − , E0 + ] and θ = N h with N > 0 large enough, (Pθ − z)−1 = O(h−M )

zj ∈Res(P )∩E,θ

θ , |z − zj |

(6.7)

where z ∈ E,θ/2 , E,δ = E + D(0, δ) and h is small enough. Proof. As for Lemma 6.1, the proof is a slight modification of Lemma 1 of Tang–Zworski [32]. It is shown in [2], that for a A − E K = χ (x)g(A)f g(A)χ (x), θ

398


where χ ∈ C0∞ (Rn ), f ∈ S(R; R+ ) with f ∈ C0∞ (R) (f is the Fourier transform of cl 2 f ), g ∈ C0∞ (R) with g = 1 near E0 and A = Opw h (a) with a ∈ S2n (ξ ) elliptic in the sense of (1.8), we have (Pθ − iθ K − z)−1 ≤ O(θ −1 ),

(6.8)

for |Re z − E| ≤ θ, Im z ≥ −θ and (Pθ − z)−1 ≤ O(θ −1 ),

(6.9)

for |Re z − E| ≤ θ , Im z ≥ Cθ . In addition, the critical points of a0 (x, ξ ) in [E0 − , E0 + ] are non degenerate. cl (x 2 + ξ 2 ; R) be such that b = a for |x| ≤ R, b(x, ξ ; h) ≥ Let b(x, ξ ; h) ∈ S2n (x 2 + ξ 2 )/C for |x| ≥ 2R and the critical points of b0 (x, ξ ) in [E0 − , E0 + ] are non-degenerate. We note B = Opw h (b) which is self-adjoint and has only pure spectrum near [E0 − , E0 + ]. Since the symbol of g(A) and g(B) coincide modulo O(h∞ ) near the support of χ (x), we get K = χ (x)g(B)f

A − E θ

g(B)χ (x) + O(h∞ ),

(6.10)

and we have A − E B − E χ(x)g(B) f −f g(B)χ (x) θ θ 1 f(t)χ(x)g(B) eitA/θ − eitB/θ g(B)χ (x) dt = 2π 1 it (A − B) 1 f(t)χ (x)g(B)eitsA/θ g(B)eit (1−s)B/θ χ (x) ds dt. = 2π θ 0

(6.11)

cl (1), vanishes Here (A − B)g(B) is a h-pseudodifferential operator whose symbol, in S2n ∞ for |x| ≤ R. On the other hand, we have |st| ≤ C since f ∈ C0 (R) and the symbol of χ(x)g(B), in S2n (1), has compact support independent of R (modulo h∞ ). If we fix R large enough, the theorem of Egorov implies

χ (x)g(B)eitsA/θ (A − B)g(B) = O(h∞ ).

(6.12)

So (6.10), (6.11) and (6.12) imply K = χ (x)g(B)f

B − E θ

g(B)χ (x) + O(h∞ ).

(6.13)

Let k ∈ C0∞ (R; [0, 1]) with k = 1 near 0. As f ∈ S(R), the functional calculus implies B − E B − E f g(B)χ (x) χ (x)g(B) 1 − k θ θ t − E t − E ≤ O(1) sup g(t) 1 − k f g(t) θ θ t∈R ≤ O(−∞ ).

(6.14)


399

So (6.13) shows that B − E B − E f g(B)χ (x) + O(−∞ + h∞ ) θ θ

+ O(−∞ + h∞ ). =K

K =χ(x)g(B)k

(6.15)

Using (6.8), we get for |Re z − E| ≤ θ and Im z ≥ −θ,

− z =Pθ − iθ K − z + θ O(−∞ + h∞ ) Pθ − iθ K =(Pθ − iθ K − z)(1 + O(−∞ + h∞ )). Now we fix large enough and we have Pθ − iθ K

− z −1 ≤ O(θ −1 ),

(6.16)

(6.17)

for |Re z − E| ≤ θ, Im z ≥ −θ and h small enough. As b0 (x, ξ ) has only non-degenerate critical point in the energy level E0 , the work of Brummelhuis–Paul–Uribe [4] or [3] shows that the number of eigenvalues of B in [E − Cθ, E + Cθ] is O(ln(1/θ ), so

≤ rank k B − E ≤ #sp(B) ∩ [E − Cθ, E + Cθ] ≤ O(ln(1/θ )), (6.18) rank K θ and

L(L2 ,L2 ) = O(1). K

(6.19)

Now the end of the proof is a repetition of the proof of Lemma 6.1: we put a Grushin problem like (6.1) which is well posed and we note E(z) is inverse as in 6.2. We −1 have (Pθ − z)−1 = O(θ −2 )(1 + E−+ (z)) with E−+ (z) = O(1). As the minor

−+ = O(h−C ), we get E (Pθ − z)−1 = O(h−C )| det(E−+ (z))|−1 .

(6.20)

As usual, we set Dθ (z, h) =

zj ∈θ ∩Res(P )

z − zj θ

and det(E−+ (z)) = Gθ (z, h)Dθ (z, h). Using the change of variable E,θ z → (z − E)/θ ∈ 0, we work on a domain independent on θ . The majoration of the number of resonances #Res(P ) ∩ E,θ = O(ln(1/ h)),

(6.21)

uniproved in [3] and the arguments of Sjöstrand [26] show that |Gθ (z, h)| ≥ hC /C formly with z ∈ E,θ/2 . The lemma follows from (6.20). Corollary 6.6. Under the hypotheses of Lemma 6.5, if #Res(P ) ∩ D(E0 , θ ) = O(1) and z0 ∈ Res(P ) is separated by hC from the other resonances of P , then

θ = O(h−C ).

(6.22)

400


Now we give an example in the 1 dimensional case where we can bound the projector θ . Consider a short range potential V (x) which is holomorphic in {x ∈ C; |Im z| ≤ Re z /C}, and has the following form: V (x) xc

E0

x

At xc , V (x) has a non-degenerate maximum. Such type of potential have been studied by Fujjié and Ramond [8] and [9]. In particular, the formula (41) of [9] implies that the resonances in E0 ,θ are of the form zj = E 0 +

S0 − (2j + 1)π h + ih ln(2) + O(h/ ln(h)2 ), K ln(h)

(6.23)

with j ∈ Z and S0 , K are some fixed constants. Let j0 ∈ Z fixed and z ∈ D(zj0 , h/ ln(1/ h)C) with C > 0 large enough. Using (6.23), we get |z−zj | ≥ (|j0 −j |)h/ ln(1/ h)C for j = j0 and we have zk ∈E0 ,θ and j ≥j0

θ ≤ |z − zj |

zj ∈E0 ,θ and j ≥j0

C ln(1/ h) (C ln(1/ h))N+ ≤ , |j − j0 | N+ !

where N+ = #{j ≥ j0 ; zj ∈ E0 ,θ }. A similar formula can be obtained for the product over j ≤ j0 and we get zk ∈E0 ,θ

θ (C ln(1/ h))N+ (C ln(1/ h))N− ≤ . |z − zj | N+ !N− !

(6.24)

Equation (6.23) implies that the number of resonances in E0 ,θ , noted N = N+ + N− , satisfy N ∼ α ln(1/ h), with α > 0 and, as j0 is fixed, |N+ − N/2| ≤ C and |N− − N/2| ≤ C,

(6.25)

so zk ∈E0 ,θ

θ (C ln(1/ h))N (CN )N ≤ NC ≤ NC , 2 |z − zj | ((N/2)!) ((N/2)!)2

(6.26)


401

√ The Stirling formula N ! ∼ N N e−N 2πN implies zk ∈E0 ,θ

θ NN ≤ C N = O(h−C ). ≤ CN |z − zj | ((N/2)N/2 )2

(6.27)

Using Lemma 6.5, we have proved Corollary 6.7. Under the previous hypotheses, the projector associated to a resonance zj satisfies, for h small enough, θ = O(h−C ).

(6.28)

In this case +1 ∈ S0 is an incoming direction and −1 is an outgoing direction. Acknowledgements. The authors thank V. Petkov for the discussions and comments and P. Stefanov for judicious remarks on this article.

References 1. Aguilar, J., Combes, J.M.: A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun. Math. Phys. 22, 269–279 (1971) 2. Bony, J.-F.: Résonances dans des domaines de taille h. Int. Math. Res. Notices (16), 817–847 (2001) 3. Bony, J.-F.: Résonances dans des petits domaines près d’une e´ nergie critique. Ann. Henri Poincaré 3(4), 693–710 (2002) 4. Brummelhuis, R., Paul, T., Uribe, A.: Spectral estimates around a critical level. Duke Math. J. 78(3), 477–530 (1995) 5. Burq, N.: Lower bounds for shape resonances widths of long range Schrödinger operators. Am. J. Math. 124(4), 677–735 (2002) 6. Burq, N.: Semi-classical estimates for the resolvent in nontrapping geometries. Int. Math. Res. Not. (5), 221–241 (2002) 7. Dimassi, M., Sjöstrand, J.: Spectral asymptotics in the semi-classical limit. Cambridge: Cambridge University Press, 1999 8. Fujiié, S., Ramond, T.: Matrice de scattering et résonances associées a` une orbite hétérocline. Ann. Inst. H. Poincaré Phys. Théor. 69(1), 31–82 (1998) 9. Fujiié, S., Ramond, T.: Breit-wigner formula at barrier tops. Preprint 2002 10. Gérard, C., Martinez, A.: Prolongement méromorphe de la matrice de scattering pour des problèmes a` deux corps a` longue portée. Ann. Inst. H. Poincaré Phys. Théor. 51(1), 81–110 (1989) 11. Gérard, C., Sjöstrand, J.: Semiclassical resonances generated by a closed trajectory of hyperbolic type. Commun. Math. Phys. 108(3), 391–421 (1987) ´ 12. Gérard, P.: Mesures semi-classiques et ondes de Bloch. Séminaire sur les Equations aux Dérivées ´ Partielles, 1990–1991, Ecole Polytech., Palaiseau, 1991, pp. Exp. No. XVI, 19 13. Helffer, B., Sjöstrand, J.: Résonances en limite semi-classique. Mém. Soc. Math. France (N.S.), no. 24-25, (1986) 14. Hunziker, W.: Distortion analyticity and molecular resonance curves. Ann. Inst. H. Poincaré Phys. Théor. 45(4), 339–358 (1986) 15. Isozaki, H., Kitada, H.: Scattering matrices for two-body schrödinger operators. Sci. Papers College Arts Sci. Univ Tokyo 35(1), 81–107 (1985) 16. Jecko, T.: From classical to semiclassical non-trapping behaviour: a new proof. Preprint Univ. Rennes 1 2002 17. Lahmar-Benbernou, A.: Estimation des résidus de la matrice de diffusion associés a` des résonances de forme. I. Ann. Inst. H. Poincaré Phys. Théor. 71(3), 303–338 (1999) 18. Lahmar-Benbernou, A., Martinez, A.: Semiclassical asymptotics of the residues of the scattering matrix for shape resonances. Asymptot. Anal. 20(1), 13–38 (1999) 19. Lions, P.-L., Paul, T.: Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9(3), 553–618 (1993) 20. Martinez, A.: An introduction to semiclassical and microlocal analysis. New York: Springer-Verlag, 2002

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21. Martinez, A.: Resonance free domains for non-globally analytic potentials. Ann. Henri Poincaré 3(4), 739–756 (2002) 22. Michel, L.: Semi-classical behavior of the scattering amplitude for trapping perturbations at fixed energy. Can. J. Math., to appear 23. Michel, L.: Semi-classical estimate of the residue of the scattering amplitude for long-range potentials. J. Phys. A 36, 4375–4393 (2003) 24. Petkov, V., Zworski, M.: Semi-classical estimates on the scattering determinant. Ann. Henri Poincaré 2(4), 675–711 (2001) 25. Sjöstrand, J.: Singularités analytiques microlocales. Astérisque, Vol. 95, Paris: Soc. Math. France, 1982, pp. 1–166 26. Sjöstrand, J.: A trace formula and review of some estimates for resonances. In: Microlocal analysis and spectral theory (Lucca, 1996), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 490, Dordrecht: Kluwer Acad. Publ., 1997, pp. 377–437 27. Sjöstrand, J.: Resonances for bottles and trace formulae. Math. Nachr. 221, 95–149 (2001) 28. Sjöstrand, J., Zworski, M.: Complex scaling and the distribution of scattering poles. J. Am. Math. Soc. 4(4), 729–769 (1991) 29. Stefanov, P.: Resonance expansions and Rayleigh waves. Math. Res. Lett. 8(1-2), 107–124 (2001) 30. Stefanov, P.: Estimates on the residue of the scattering amplitude. Asympt. Anal. 32(3,4), 317–333 (2002) 31. Stefanov, P.: Sharp upper bounds on the number of resonances near the real axis for trapped systems. Am. J. Math. 125(1), 183–224 (2003) 32. Tang, S-H., Zworski, M.: From quasimodes to reasonances. Math. Res. Lett. 5(3), 261–272 (1998) 33. Tang, S-H., Zworski, M.: Resonance expansions of scattered waves. Comm. Pure Appl. Math. 53(10), 1305–1334 (2000) Communicated by B. Simon


Communications in


Quantum Invariant for Torus Link and Modular Forms Kazuhiro Hikami Department of Physics, Graduate School of Science, University of Tokyo, Hongo 7–3–1, Bunkyo, Tokyo 113–0033, Japan. E-mail: [email protected] Received: 16 May 2003 / Accepted: 15 October 2003 Published online: 24 February 2004 – © Springer-Verlag 2004

Abstract: We consider an asymptotic expansion of Kashaev’s invariant or of the colored Jones function for the torus link T (2, 2 m). We shall give q-series identity related to these invariants, and show that the invariant is regarded as a limit of q being N -th root of unity of the Eichler integral of a modular form of weight 3/2 which is related to the su(2) m−2 character. 1. Introduction Recent studies reveal an intimate connection between the quantum knot invariant and “nearly modular forms” especially with half integral weight. In Ref. 9 Lawrence and Zagier studied an asymptotic expansion of the Witten–Reshetikhin–Turaev invariant of the Poincaré homology sphere, and they showed that the invariant can be regarded as the Eichler integral of a modular form of weight 3/2. In Ref. 19, Zagier further studied a “strange identity” related to the half-derivatives of the Dedekind η-function, and clarified a role of the Eichler integral with half-integral weight. From the viewpoint of the quantum invariant, Zagier’s q-series was originally connected with a generating function of an upper bound of the number of linearly independent Vassiliev invariants [17], and later it was found that Zagier’s q-series with q being the N th root of unity coincides with Kashaev’s invariant [5, 6], which was shown [14] to coincide with a specific value of the colored Jones function, for the trefoil knot. This correspondence was further investigated for the torus knot, and it was shown [3] that Kashaev’s invariant for the torus knot T (2, 2 m + 1) also has a nearly modular property; it can be regarded as a limit q being the root of unity of the Eichler integral of the Andrews–Gordon q-series, which is a theta series with weight 1/2 spanning m-dimensional space. As the torus knot is not hyperbolic, studies of the torus knot may not be attractive for the “Volume Conjecture” [5, 14] which states that an asymptotic limit of Kashaev’s invariant coincides with the hyperbolic volume of the knot complement, but they are rather absorbing from the point of view of the number theory, q-series and modular forms.

404

K. Hikami

Fig. 1. Hopf link T (2, 2) and torus link T (2, 4)

Motivated by our previous result on the torus knot T (2, 2 m+1), we study Kashaev’s invariant for the torus link T (2, 2 m) (see Fig. 1) in this article. We shall show that the invariant is now regarded as the half-integration or the Eichler integral of a modular form of weight 3/2. Remarkable is that this modular form is related to the su(2) m−2 character. It is noted that recent studies [20,21] reveal a relation with Ramanujan’s mock theta functions. We also propose a q-series identity, which is new as far as we know, and study an asymptotic expansion thereof. This paper is organized as follows. In Sect. 2 we construct the colored Jones polynomial for the torus link T (2, 2 m). Using the Jones–Wenzl idempotent, we give an explicit formula of the invariant. It is known [14] that Kashaev’s invariant coincides with a specific value of the colored Jones polynomial. This correspondence enables us to give an integral form of Kashaev’s invariant for the torus link T (2, 2 m) in Sect. 3. We further give an asymptotic expansion of the invariant, and see that the invariant for T (2, 2 m) also has a nearly modular property. We give an explicit form of Kashaev’s invariant for this torus link using the enhanced Yang–Baxter operator. Combining these results we obtain an asymptotic expansion of a certain ω-series. In Sect. 4 we introduce the q-series related to Kashaev’s invariant for the torus link, and prove a new q-series identity. We study the modular property of these q-series, and discuss how Kashaev’s invariant for T (2, 2 m) may be regarded as the Eichler integral of a modular form with weight 3/2 which is the affine su(2) m−2 character, in Sect. 5. In the last section, we collect some examples. 2. Colored Jones Polynomial for Torus Link (2, 2 m) The N -colored Jones polynomial for torus knot T (m, p) was studied in Refs. [12, 15]. Following these methods, we compute the colored Jones polynomial for torus link T (2, 2 m) in this section. We use the Jones–Wenzl idempotent, and use following formulae (see, e.g., Ref. 10);

= (−1)(a+b−c)/2 Aa+b−c+

a 2 +b2 −c2 2

,

(2.1)

Quantum Invariant for Torus Link and Modular Forms

=

c: (a, b, c) is admissible

405

c θ (a, b, c)

,

(2.2)

where each label denotes a color, and we mean that A2(n+1) − A−2(n+1) , A2 − A−2   a ≤ b + c, (a, b, c) is admissible ⇔ a + b + c is even, and b ≤ c + a,  c ≤ a + b. n = (−1)n

We have a θ-net

θ (a, b, c) =

(2.3)

,

which is given as θ (a, b, c) =

x+y+z ! x−1 ! y−1 ! z−1 ! , y+z−1 ! z+x−1 ! x+y−1 !

with a = y + z,

b = z + x,

c = x + y.

Proposition 1. The N -colored Jones polynomial JN (h; K) for the torus link K = T (2, 2 m) is given by

Nh 2 sh 2

N−1 1 1 JN (h; K) 2 2 = e− 2 m(N −1)h ε emhj +(m+ε) h j + 2 hε , JN (h; O) ε=±1 j =0

where a parameter q is set to be q = A4 = eh , and O denotes an unknot whose invariant is given by JN (h; O) =

sh(N h/2) . sh(h/2)

(2.4)

406

K. Hikami

Fig. 2. We apply Eq. (2.2) to the torus link T (2, 4). We set n = N − 1

Proof. We first apply Eq. (2.2) in the torus link T (2, 2 m) (see Fig. 2), and untangle crossings recursively using Eq. (2.1). We see that θ (a, b, c) vanishes at the end. We have

JN (h; K) =

c 2 1 2 2m c (−1)N−1− 2 A−2(N−1)+c−(N−1) + 2 c

c:(N − 1, N − 1, c) is admissible

=

2 N−1

A−2m(N −1) 4mj (j +1) 2(2j +1) −2(2j +1) A − A A . A2 − A−2

j =0

This proves Eq. (2.4).

3. Kashaev Invariant and Asymptotic Expansion It is known [14] that Kashaev’s invariant is given from the colored Jones polynomial at a specific value h → 2 π i/N . By use of a result of Prop. 1 we obtain an integral form of the invariant as follows. Proposition 2. The Kashaev invariant for torus link T (2, 2 m) is given by an integral form as √ πi 1 3 T (2, 2 m)N = e 2N (m− m ) 8 2 (m N ) 2 e−π i/4 2 sh(4 m π N w) sh(4 π w) 2 × dw w2 e8mπ iNw +4mπNw , sh(4 m π w)

(3.1)

C

where C denotes a path passing through the origin in the steepest descent direction. Proof. A proof is essentially the same as those given in Refs. [2, 7, 8, 16] for studies in asymptotic behavior of the quantum invariants.


407

To rewrite Eq. (2.4) into an integral form, we use an integral formula z2 1 2 eh w = √ dz e− h +2wz , πh C

where an integration path C is an infinite line passing through the origin in the steepest descent direction. Then we get N h JN (h; K) 2 sh 2 JN (h; O) 1

= e− 2 mh(N

2 −1)− m2 +1 h 4m

N−1 ε=±1 j =0

=√ =√

1

2

π mh 1 π mh

e

e

− 21 mh(N 2 −1)− m4m+1 h

h − 21 mhN 2 + m 4 h− 4m

ε emh(j +

m+ε 2 2m )

N−1 z2 m+ε ε dz e− mh +2(j + 2m )z ε=±1 j =0

C

2

dz e

z − mh +Nz

C

2 sh(N z) sh(z/m) . sh(z)

Now we take a limit h→

2π i , N

to compute Kashaev’s invariant. As LHS vanishes in this limit, we take a derivative of both sides to have iz2 N 3/2 π i (m− 1 )− π i 2 sh(N z) sh(z/m) 1 m 4 T (2, 2 m)N = √ . e 2N dz z2eN( 2mπ +z) sh(z) 4 2 π3 m C

Rescaling an integral variable, we obtain Eq. (3.1).

Theorem 3. Kashaev’s invariant KN for the torus link K = T (2, 2 m) has an asymptotic expansion in N → ∞ as m−1 (m−1)2 3 k2 2 k πi− πi 3/2 2mN N (−1)k (k − m) sin T (2, 2 m)N e 4 π e− 2m πiN m m k=1 ∞ (m;0) (m−1)2 Ek πi k +e− 2mN πi N , (3.2) k! 2mN k=0

(m;0)

where Ek

is defined from a generating function (see also Eq. (3.16)) as ∞

E m sh(z) k = z2k sh(m z) (2 k)! (m;0)

k=0

= 1+

1 − m2 2 (3 − 7 m2 ) (1 − m2 ) 4 z + z + ··· . 6 360

(3.3)

408

K. Hikami

Proof. When we decompose sh(4 m π N w) in the integrand (3.1) into (e4mπNw − e−4mπNw )/2, the integral reduces to I1 − I2 up to a constant where I1 and I2 are sh(4 π w) 2 , I1 = dw w2 e8mπN(iw +w) sh(4 m π w) C

I2 =

dw w2 e8mπNiw C

2

sh(4 π w) . sh(4 m π w)

In I1 we deform an integration path C → C+ 2i . In this deformation we have contributions from residues at w = 4km i for k = 1, 2, . . . , 2 m − 1. Then we get sh(4 π w) 2 dw w2 e8mπN(iw +w) I1 = 2 π i · (Residues) + sh(4 m π w) C + 2i

2 m−1 k2 k 1 k 2 − 2m πiN π e (−1) k sin = 32 m3 m k=1 i 2 8mπ iNz2 sh(4 π z) + dz z + e 2 sh(4 m π z) C

=

m−1 k2 1 k k (−1) (k − m) sin π e− 2m πiN 2 8m m k=1 1 2 sh(4 π z) +I2 − dz e8mπ iNz . 4 sh(4 m π z) C

In the last equality, we have used a symmetry z ↔ −z of the integrand. Substituting a series expansion (3.3) into the above expression, we obtain an assertion of the theorem. Asymptotic expansion for the torus knot T (2, 2 m + 1) was studied in Ref. 3. In view of these results, a tail of asymptotic expansion of Kashaev’s invariant is given in an infinite series of N −1 with coefficients, whose generating function seems to be related to s 1/2 − s −1/2 . K (s) Here K (s) is the Alexander polynomial for knot K, and in a case of the torus link K = T (2, 2 m) we have K (s) =

s m − s −m . s 1/2 + s 1/2

We have shown that the volume conjecture [5,14] is fulfilled for the torus link T (2, 2 m),

2π log T (2, 2 m)N = 0, N→∞ N lim


409

because the torus link T (2, 2 m) is not hyperbolic. Rather we have interests in an asymptotic expansion of the q-series. To this aim, we compute the quantum invariant by another method; the explicit form of Kashaev’s invariants can be directly computed from a set of the enhanced Yang–Baxter operators [5] (see also Refs. 13, 14, 18); N ω1−(k−j +1)(−i) i j ij Rk = ·θ , (3.4a) ∗ k (ω)[−k−1] (ω)[j∗−] (ω)[i−j ] (ω)[k−i] ij (R −1 )k

N ω−1+(−i−1)(k−j ) i j = ·θ , ∗ ∗ k (ω)[−k−1] (ω)[j −] (ω)[i−j ] (ω)[k−i] µk = −δk,+1 ω1/2 ,

(3.4b) (3.4c)

where we have defined the N th root of unity as 2π i ω = exp , N

(3.5)

and ∗ means a complex conjugation. We have also used [x] ∈ {0, 1, . . . , N − 1} modulo N, and   i ≤ k < ≤ j,    j ≤ i ≤ k < , i j θ = 1, if and only if k  ≤ j ≤ i ≤ k (with < k),   k < ≤ j ≤ i. Remark that the standard notation of the q-product and the q-binomial coefficient is used; (ω)n =

n

(1 − ωk ),

(3.6)

k=1

 (ω)m   , m = (ω)n (ω)m−n n  0,

if m ≥ n ≥ 0, others.

These operators are assigned to a projection of knot as follows;

ij

Rk =

µk =

R −1

ij

µ−1

k

k

=

=

(3.7)

410

K. Hikami

Proposition 4. Kashaev’s invariant KN for the torus link K = T (2, 2 m) is explicitly given by T (2, 2 m)N

=N

(−1)

cm−1

ω

1 2 cm−1 (cm−1 +1)

m−2

N−1≥cm−1 ≥···≥c2 ≥c1 ≥0

ω

ci (ci +1)

i=1

ci+1 ci

.

(3.8)

Proof. We get this result from a direct computation using the R-matrix (3.4) for (1, 1)tangle of (2, 2 m)-torus link. See Refs. 13, 18. We note that Kashaev’s invariant for the Hopf link T (2, 2) is given by T (2, 2)N = N. Combining Prop. 4 with Thm. 3, we obtain an asymptotic expansion of the ω-series. Corollary 5.

(−1)

cm−1

N −1≥cm−1 ≥···≥c2 ≥c1 ≥0

√

Ne +e

(m−1)2 3 4 π i− 2mN πi

2 − (m−1) 2mN πi

ω

1 2 cm−1 (cm−1 +1)

m−2

ω

ci (ci +1)

i=1

2 m

m−1

(−1) (k − m) sin k

k=1

ci+1 ci

k π m

k2

e− 2m πiN

∞ (m;0) Ek πi k . k! 2mN

(3.9)

k=0

In the rest of this paper, we shall reveal a meaning of this asymptotic expansion from the point of view of the modular form. As a generalization of ω-series defined by Kashaev’s invariant we introduce Ym(a) (ω) =

N−1

c1 ,...,cm−1 =0 m−2 c i+1

×

i=1

2 +···+c 2 +c a+1 +···+cm−2 m−2

1

(−1)cm−1 ω 2 cm−1 (cm−1 +1) ωc1 + δi,a ci

,

(3.10)

for m ≥ 2 and a = 0, 1, . . . , m − 2. See that Kashaev’s invariant for T (2, 2 m) corresponds to a case of a = 0, T (2, 2 m)N = N · Ym(0) (ω). (a)

(3.11)

It is unclear whether the ω-series Ym (ω) for a = 0 represent the quantum invariant for any three manifolds.


411

(a)

Conjecture 1. Let Ym (ω) be defined by Eq. (3.10). An asymptotic expansion of this ω-series in N → ∞ is given by πi

2

e 2mN (m−1−a) · Ym(a) (ω) m−1 √ 3 k2 2 k πi

N e4 · (−1)k (k − m) sin (a + 1) π e− 2m πiN m m k=1 ∞ (m;a) k E πi k + , (3.12) 2mN k! k=0

(m;a)

is given from a generating function as where a generalized Euler number Ek ∞ (m;a) m sh (a + 1) z Ek = z2k (3.13) sh(m z) (2 k)! k=0

1 (1 + a) (1 + a)2 − m2 z2 6 1 + (1 + a) (1 + a)2 − m2 3 (1 + a)2 − 7 m2 z4 360 1 + (1 + a) (1 + a)2 − m2 15120 × 3 (1 + a)4 − 18 (1 + a)2 m2 + 31 m4 z6 + · · · .

= (a + 1) +

The case a = 0 of this conjecture is proved in Corollary 5. Note that we have ∞ sh (a + 1)z (a) χ2m (n) e−nz , = sh(m z)

(3.14)

n=0

(a)

where the odd periodic function χ2m (n) is written as n mod 2 m m − 1 − a m + 1 + a others . (a) χ2m (n) 1 −1 0.

(3.15)

Applying the Mellin transformation to Eqs. (3.13) and (3.14), we have an expression of (a) the generalized Euler number in terms of the L-function associated to χ2m (n); (m;a)

Ek

(a)

= m · L(−2 k, χ2m ) m−1−a m+1+a (2 m)2k B2k+1 − B2k+1 , = −m 2k +1 2m 2m

(3.16)

where Bn (x) is the nth Bernoulli polynomial. It should be remarked that the colored Jones polynomial (2.4) for the torus link K = T (2, 2 m) is rewritten using the periodic function as 2mN 1 m2 +1 k2 N h JN (h; K) 2 (0) = −e− 2 m(N −1)h− 4m h χ2m (k) e 4m h . (3.17) 2 sh 2 JN (h; O) k=0

412

K. Hikami

Based on this expression, we find that Kashaev’s invariant is given by T (2, 2 m)N = −

2mN (m−1)2 k2 1 (0) k 2 χ2m (k) e 2mN πi . e− 2mN πi 4mN

(3.18)

k=0

Later we shall clarify a relationship between the above conjecture and the modular form. 4. q-Series Identity (a)

In this section we study a q-series identity, which is closely related with Ym (ω) defined in Eq. (3.10). We use standard notation as in Eqs. (3.5) – (3.7), but in this section we replace ω, the N th primitive root of unity, with generic q. We define the q-series (a) Km (x) =

∞

2 +···+c 2 +c a+1 +···+cm−2 m−2

1

(−1)cm−1 q 2 cm−1 (cm−1 +1) x c1 +···+cm−1 q c1

c1 ,...,cm−1 =0 m−2 c i+1

·

+ δi,a ci

i=1

(4.1)

,

for m ≥ 2 and a = 0, 1, 2, . . . , m − 2. We simply replace N−1 cm−1 =0 in Eq. (3.10) with ∞ an infinite sum cm−1 =0 , though we have introduced an additional variable x. (a)

Theorem 6. Let the q-series Km (x) be defined in Eq. (4.1). We have (a) (x) = Km

∞

(a)

χ2m (n) q

n2 −(m−1−a)2 4m

n−(m−1−a) 2

x

(4.2)

,

n=0 (a)

where χ2m (n) is a periodic function in Eq. (3.15). Proof. We prove this statement by showing that both sides satisfy the same q-difference equation. (a) It is easy to see from a periodicity of the function χ2m (n) that RHS of Eq. (4.2) solves a difference equation (see, e.g., Refs. 1, 3, 19), (a) Km (x) = 1 − q a+1 x a+1 +

= 1−q

a+1

x

a+1

∞

(a)

χ2m (n) q

n=2m m 2m−1−a

+x q

n2 −(m−1−a)2 4m

x

n−(m−1−a) 2

(a) 2 Km (q x).

(4.3)

We shall show that LHS of Eq. (4.2) also fulfills a same difference equation. To this aim, we introduce (a) Km (x1 , . . . , xm−1 ) ∞

=

(−1)cm−1 q

c1 ,...,cm−1 =0

·q

ca2

xaca

1 2 cm−1 (cm−1 +1)

m−2 +1

ca+1 ca

i=a+1

q

cm−1 xm−1

ci 2 +ci

a−1

xi

q ci xi

i=1

ci

2

ci+1 ci

ci

ci+1 ci

.

(4.4)


413

See that by definition we have (a) (a) (x) = Km (x, . . . , x ). Km

(4.5)

m−1 (a) Km ,

We use the same symbol but we believe there is no confusion. To prove the assertion of the theorem, we use formulae for the q-binomial coefficients; n n n+1 + (4.6a) = qc c c−1 c n n = + q n+1−c . (4.6b) c c−1 +1 c in Eq. (4.4), we get Applying Eq. (4.6a) to a+1 ca (a) (0) −1 (x1 , . . . , xm−1 ) = Km (q x1 , . . . , q −1 xa−1 , xa , . . . , xm−1 ) Km (a−1) +q xa · Km (x1 , . . . , xa−1 , q xa , xa+1 , . . . , xm−1 ).

(4.7)

On the other hand, using Eq. (4.6b), we have (a) (0) −1 (x1 , . . . , xm−1 ) = Km (q x1 , . . . , q −1 xa , xa+1 , . . . , xm−1 ) Km (a−1) +q xa · Km (x1 , . . . , xa , q xa+1 , xa+2 , . . . , xm−1 ).

(4.8)

Another difference equation is given as follows; (0) (x1 , . . . , xm−1 ) Km ∞

=1+

∞

(−1)cm−1 q

1 2 cm−1 (cm−1 +1)

cm−1 =1 c1 ,...,cm−2 =0

=1−q

cm−1 xm−1

m−2

q ci (ci +1) xi

ci

i=1

ci+1 ci

(m−2) xm−1 · Km (q x1 , . . . , q xm−1 ).

(4.9)

We can prove Eq. (4.3) by use of Eqs. (4.7)–(4.9) as follows. Recursive use of Eq. (4.7) gives (a) (x1 , . . . , xa , q xa+1 , . . . , q xm−1 ) Km

(0) −1 = Km (q x1 , . . . , q −1 xa−1 , xa , q xa+1 , . . . , q xm−1 )

(0) −1 + q xa · Km (q x1 , . . . , q −1 xa−2 , xa−1 , q xa , . . . , q xm−1 )

(0) −1 + q 2 xa−1 xa · Km (q x1 , . . . , q −1 xa−3 , xa−2 , q xa−1 , . . . , q xm−1 ) (0) + · · · + q a x1 · · · xa · Km (q x1 , . . . , q xm−1 ).

(4.10)

Substituting the above equation for a = m − 2 into Eq. (4.9), we get (0) −1 (q x1 , . . . , q −1 xm−2 , xm−1 ) Km

(0) −1 +q xm−1 · Km (q x1 , . . . , q −1 xm−3 , xm−2 , q xm−1 )

(0) −1 + q 2 xm−2 xm−1 · Km (q x1 , . . . , q −1 xm−4 , xm−3 , q xm−2 , q xm−1 ) (0) + · · · + q m−2 x2 · · · xm−1 · Km (x1 , q x2 , . . . , q xm−1 ) (0) = 1 − q m−1 x1 · · · xm−1 · Km (q x1 , . . . , q xm−1 ).

(4.11)

414

K. Hikami

In the same way, iterated use of Eq. (4.8) gives (a) Km (x1 , . . . , xa+1 , q xa+2 , . . . , q xm−1 )

(0) −1 = Km (q x1 , . . . , q −1 xa , xa+1 , q xa+2 , . . . , q xm−1 )

(0) −1 + q xa · Km (q x1 , . . . , q −1 xa−1 , xa , q xa+1 , . . . , q xm−1 )

(0) −1 + q 2 xa−1 xa · Km (q x1 , . . . , q −1 xa−2 , xa−1 , q xa , . . . , q xm−1 ) (0) + · · · + q a x1 · · · xa · Km (x1 , q x2 , . . . , q xm−1 ).

(4.12)

Substituting the above equation for a = m − 2 into Eq. (4.9), we find (0) −1 Km (q x1 , . . . , q −1 xm−2 , xm−1 )

(0) −1 + q xm−2 · Km (q x1 , . . . , q −1 xm−3 , xm−2 , q xm−1 )

(0) −1 + q 2 xm−3 xm−2 · Km (q x1 , . . . , q −1 xm−4 , xm−3 , q xm−2 , q xm−1 ) (0) + · · · + q m−2 x1 · · · xm−2 · Km (x1 , q x2 , . . . , q xm−1 )

1 (0) −1 1 − Km = (q x1 , . . . , q −1 xm−1 ) . xm−1

(4.13)

Combining the two equations (4.11) and (4.13) with x ≡ x1 = · · · = xm−1 , we obtain (0) (0) 2 Km (x) = 1 − q x + q 2m−1 x m · Km (q x).

This proves Eq. (4.2) for a = 0. Other cases can be shown by using Eqs. (4.11) and (4.13) with Eqs. (4.10) and (4.12). (a) (a) Corollary 7. We define the q-series

m (τ ) by Km (x = 1) up to constant, i.e.,

(a)

m (τ ) = m q ·

∞

(m−1−a)2 4m

m−2 c i=1

2 +···+c 2 +c a+1 +···+cm−2 m−2

1

(−1)cm−1 q 2 cm−1 (cm−1 +1) q c1

c1 ,...,cm−1 =0 i+1

+ δi,a ci

(4.14)

,

where m ≥ 2 and a = 0, 1, . . . , m − 2, and we mean q = e2πiτ . Then we have (a)

m (τ ) = m

∞

(a)

1

2

χ2m (n) q 4m n .

n=0

Factor m in the above definition is merely for our later convention.

(4.15)


415

(a)

Corollary 8. Let the q-series m (τ ) be defined by 1 2 (a) n χ2m (n) q 4m n ,

(a) m (τ ) =

(4.16)

n∈Z

for m ≥ 2 and a = 0, 1, . . . , m − 2. Then we have ∞ (m−1−a)2 m−1−a (a) 4m c1 + c2 + · · · + cm−1 +

m (τ ) = 4 q 2 c1 ,...,cm−1 =0 1 2 2 2 cm−1 (cm−1 +1) c1 +···+cm−2 +ca+1 +···+cm−2

q ×(−1) m−2 c i+1 + δi,a . · ci cm−1

q

(4.17)

i=1

Proof. We differentiate Eq. (4.2) with respect to x and substitute x → 1.

5. Modular Property (a) (a) We shall reveal the modular property of the q-series m (τ ) and

m (τ ) defined by (a) Eq. (4.16) and Eq. (4.15) respectively. The theta series m (τ ) have weight 3/2, and span (m − 1)-dimensional space; it is straightforward to get a2

(m−1−a) (τ + 1) = e 2m πi · (m−1−a) (τ ), m m and from the standard method using the Poisson summation formula we have 3/2 1 i m (τ ) = Mm · m − , τ τ where



(5.1)

(5.2)

 (τ )  ..  . ,  (1)

m (τ )  (0)

m (τ ) (m−2)

m

  m (τ ) =    and Mm is an (m − 1) × (m − 1) matrix,

Mm

1≤a,b≤m−1

=

2 sin m

ab π . m

(5.3)

Remarkable is that the theta series defined by (m−λ)

chm λ (τ ) =

m+2 (τ ) 3 , 2 η(τ )

(5.4)

where η(τ ) is the Dedekind η-function (6.3), is the affine su(2) m character (see Examples in Sect. 6) [4, 11].

416

K. Hikami

As studied in Ref. 9, we have interests in the Eichler integral of the modular form m (τ ). Generally when the q-series F (τ ) =

∞

an q n ,

n=1

is a modular form with weight k ∈ Z≥2 , the Eichler integral defined as k − 1 integrations of F (τ ) with respect to τ , or explicitly defined by (τ ) = F

∞ an n q , nk−1 n=1

satisfies (γ (τ )) − F (τ ) = Gγ (τ ), (c τ + d)k−2 · F where γ =

(5.5)

ab ∈ SL(2; Z), and Gγ (τ ) is the period polynomial cd Gγ (z) =

(2 π i)k−1 (k − 2)!

i∞ γ −1 (i ∞)

F (τ ) (z − τ )k−2 dτ.

(a)

In our case, the modular form m (τ ) in Eq. (4.16) has a half-integral weight, and the above story does not work any more. But the Eichler integral as an infinite q-series can be defined in a naive sense, and we may find that (a)

m (τ ) = m

∞

(a)

1

2

χ2m (n) q 4m n ,

n=0

which is nothing but a definition (4.15). We note that a prefactor in the above definition (a) is for our convention. We can regard

m (τ ) as the Eichler integral of the modular form (a)

m (τ ) with weight 3/2. To study a nearly modular property (see Eq. (5.5)) of this Eichler integral of the half-integral weight modular form, we first recall the following result. Proposition 9. Let Cf (n) be a periodic function with mean value 0 and modulus f . Then we have an asymptotic expansion as t 0 ∞ n=1

Cf (n) e

−n2 t

∞

L(−2 k, Cf )

k=0

(−t)k , k!

where L(k, Cf ) is the L-function associated with Cf (n), and is given by f n fk L(−k, Cf ) = − Cf (n) Bk+1 . k+1 f n=1

Proof. It is a standard result using the Mellin transformation. See, e.g., Ref. 9.


417

m (τ ) near at a root of unity as Using this property, we obtain the Eichler integral

follows. (a)

Proposition 10. The Eichler integral (4.15) for τ = M N ∈ Q (N > 0, and M, N are coprime integers) reduces to mN n n2 Mπi (a) (a) M χ2m (n) 1 − . (5.6) =m e 2mN

m N mN n=0

Proof. We have from Eq. (4.15) ∞ n2 M t (a)

C2mN (n) e− 4m t , + i = m m N 2π n=0

where (a)

Mn2

C2mN (n) = χ2m (n) e 2mN πi . We see that C2mN (n + 2 m N ) = C2mN (n) and C2mN (2 m N − n) = −C2mN (n), and we can apply Prop. 9 to get an asymptotic expansion in t 0 as ∞ M t t k L(−2 k, C2mN ) (a) + i

− .

m N 2π k! 4m k=0

Then we obtain a limiting value (a)

m

M N

= m · L(0, C2mN ).

Recalling an explicit form of the Bernoulli polynomial B1 (x) = x − C2mN (2mN − n) = −C2mN (n), we obtain Eq. (5.6).

1 2

and a property

(a) Conjecture 2. Let

m (τ ) be the Eichler integral defined by Eq. (4.15) or (4.14). When th (a) q is the N root of unity,

m (1/N ), which was computed as Eq. (5.6), coincides with an expression (3.10) up to constant, i.e., (m−1−a)2 1 (a)

= e 2mN πi · Ym(a) (ω). (5.7) m N Proof for a case of a = 0. As a case of a = 0 is related to Kashaev’s invariant for the torus link as in Eq. (3.11), this case can be directly proved by using Eq. (3.18) as follows: e

(m−1)2 2mN π i

1 (m−1)2 πi T (2, 2 m)N e 2mN N 2mN k2 1 (0) =− k 2 χ2m (k) e 2mN πi 2 4mN

Ym(0) (ω) =

k=0

=

2mN

(0)

χ2m (k)

k=0

=

mN k=0

(0)

χ2m (k)

m k − 2 2N

k2

e 2mN πi

k2 1 k (0) e 2mN πi =

. m− m N N

418

K. Hikami

In the third equality, we have summed an expression with k → 2 m N − k. As a result of Eq. (5.6) the statement of the conjecture is true for a = 0. We now discuss how Conjecture 1 follows from Conjecture 2. We first recall from Eq. (5.6) that for N ∈ Z, (a)

m (N ) = (1 + a) e

(m−1−a)2 πiN 2m

(5.8)

.

Following Ref. 9 (see also Ref. 21) we define the period function ∞ (a) m

m (τ ) (a) rm (z; α) = dτ, √ 8i α τ −z

(5.9)

where α ∈ Q. It is defined for z in the lower half plane, z ∈ H− , but it is analytically (a) continued to R = ∂H− . To see a modular property of

m (α + i y) in y 0, we further define ∞ (a) m

m (τ ) (a)

(z) = dτ, (5.10) √ m 8 i z∗ τ −z where z ∈ H− . We can find that this function is nearly modular of weight 1/2 by m−1 0 (m−1−b) m−1 m

m (−1/s) ds 1 (m−1−b) Mm a,b m Mm a,b − = √ −1 z 8 i z∗ −s + z−1 s 2 b=1 b=1

√ (m−1−a) (m−1−a) = iz

(z) − r (z; 0) . (5.11) m m On the other hand, we have for z = α + i y,

∞ n πiτ m e 2m (a) n χ2m (n) dτ √ ∗ 8i τ −z z n∈Z ∞ n2 πy (a) , χ4m (n) e 2m πiz erfc n − =m m

(a)

m (z) =

2

n=1

where erfc(x) is the complementary error function ∞ 2 2 erfc(x) = √ e−t dt. π x m (α), we get As erfc(0) = 1 and we know Eq. (5.6) from a definition of

(a)

(a) (a)

m (α) = m (α),

(5.12)

for α ∈ Q. We stress that LHS is a limit from the lower half plane H− while RHS is analytically continued from the upper half plane H. Taking a limit z → 1/N for N ∈ Z in Eq. (5.11), we obtain m−1 √ 1 1 (m−1−a) (m−1−b) (m−1−a)

m = −i N ;0 . Mm a,b m (−N ) + rm N N b=1

(5.13)


419

(a) 1 N;0

When we recall Eq. (5.8) and use an asymptotic expansion of rm we obtain the following proposition.

in N → ∞,

(a) Proposition 11. Let the Eichler integral

m (τ ) be defined by Eq. (4.15). Asymptotic expansion in N → ∞ is then given by (a)

m

1 N

m−1 √

−i N b=1

b2 (m − 1 − a) b 2 (m − b) sin π e− 2m πiN m m k ∞ (m;a) Ek πi + . k! 2mN

(5.14)

k=0

Based on this proposition, we get a conjecture (3.12) assuming a conjecture (5.7). 6. Examples 6.1. (2,4)-Torus Link: m = 2. • q-series: The modular form with weight 3/2 is 1 2 1 2 1 (0) (0) n χ4 (n) q 8 n = (−1)n (2 n + 1) q 2 (n +n+ 4 )

2 (τ ) = n∈Z

= 2q

1 8

n∈Z

1 − 3 q + 5 q − 7 q 6 + 9 q 10 − 11 q 15 + · · · , 3

(6.1)

(0)

where χ4 (n) is the primitive character modulo 4, n

mod 4 1 3 others . (0) χ4 (n) 1 −1 0

It is known by Jacobi that we can write 3 (0)

2 (τ ) = 2 η(τ ) ,

(6.2)

where η(τ ) is the Dedekind η-function, η(τ ) = q 1/24 · (q)∞ .

(6.3)

From a modular property of the η-function we see that 1

(0)

(0)

2 (τ + 1) = e 4 πi 2 (τ ), (0)

2 (−1/τ ) =

(6.4)

τ 3/2 (0)

2 (τ ). i

(6.5)

The Eichler integral is given by (0) (τ ) = 2

2

∞

(0)

n=0

= 2q

1 8

1 2

1

χ4 (n) q 8 n = 2 q 8

∞

6

− q 15 + · · · .

k=0 10

1−q +q −q +q 3

1

(−1)k q 2 k(k+1) (6.6)

420

K. Hikami

• root of unity: As a limit of q being the N th root of unity, the Eichler integral (6.6) coincides with Kashaev’s invariant for torus link, (0)

2

1 N

πi

= e 4N

N−1

1

(−1)c ω 2 c(c+1) =

c=0

1 πi e 4N T (2, 4)N . N

(6.7)

We see that the Eichler integral (6.6) with q being the N th root of unity takes the same form with the original Eichler integral up to constant, only an infinite sum reduces (0) (N ) = eπiN/4 , we have the nearly modular property (see to a finite sum. Using

2 Eq. (3.9)), (0)

2 (2;0)

where En

1 N

∞ (2;0) √ En (0) (−N ) + −i N

2 n! n=0

πi 4N

n ,

(6.8)

is the Euler number defined by En(2;0) = −

1 3 24n+1 B2n+1 − B2n+1 , 2n+1 4 4

some of which are given as ∞ (2;0) 1 En = x 2n ch(x) (2 n)! n=0

1 2 5 4 61 6 x + x − x + ··· . 2 24 720

= 1−

6.2. (2,6)-Torus Link: m = 3. • q-series: A set of the theta series is given by (0)

3 (τ ) =

(0)

n∈Z

1

= 4q3 (1)

3 (τ ) =

n∈Z

1

2

n χ6 (n) q 12 n

1 − 2 q + 4 q 5 − 5 q 8 + 7 q 16 − 8 q 21 + · · · ,

(1)

1

2

n χ6 (n) q 12 n 1

= 2 q 12

1 − 5 q 2 + 7 q 4 − 11 q 10 + 13 q 14 − 17 q 24 + · · · ,

where n

(6.9a)

mod 6 2 4 others (0) χ6 (n) 1 −1 0

n

mod 6 1 5 others . (1) χ6 (n) 1 −1 0

(6.9b)


421

We note [11] that these series can be written in terms of the Dedekind η-function as

5

2 η(τ ) η(4 τ ) , 2 = 4 η(2 τ ) η(τ + 21 ) η(2 τ )

1 12 πi

(0)

3 (τ )

= 4e

(1)

3 (τ )

5 η(2 τ ) =2 2 . η(4 τ )

(6.10a)

(6.10b)

The modular property is written as 

 (1)

3 (τ )  = , 2 (0) (0)

3 (τ + 1)

3 (τ ) 0 e 3 πi

(1)

3 (τ + 1)

1

e 6 πi 0

(1) τ 3/2 1 1 1

3 (τ ) = . ·√ · (0) (0) i 2 1 −1

3 (−1/τ )

3 (τ ) (1)

3 (−1/τ )

(6.11)

(6.12)

The Eichler integrals are then defined by (0) (τ ) = 3 q 3

3 1

=3

∞

1

(−1)a q 2 a(a+1)

a=0 ∞ (0) 1 2 χ6 (n) q 12 n n=0 1 5 3

1

∞

1

(−1)a q 2 a(a+1)

a=0

=3

∞ n=0 1

= 3 q 12

q b(b+1)

b=0

a b

1 − q + q − q 8 + q 16 − q 21 + · · · ,

= 3q

(1) (τ ) = 3 q 12

3

a

a+1 b=0

1

(1)

qb

2

a+1 b

(6.13a)

2

χ6 (n) q 12 n

1 − q 2 + q 4 − q 10 + q 14 − q 24 + · · · .

(6.13b)

We note that Zagier’s identity [19] leads us to find (1) (τ ) = 3 q 12

3 1

∞

(−1)n (−1; q 2 )n+1 .

n=0

• root of unity: From Prop. 10, the Eichler integrals (6.13) reduce in the case of q being N th root of unity to

422

K. Hikami

(a)

3

1 N

=3

3N n=0

n n2 πi (a) χ6 (n) 1 − e 6N . 3N

(6.14)

This coincides with (0)

3

1 N

2π i

= e 3N

N−1

1

(−1)a ω 2 a(a+1)+b(b+1)

a,b=0

a b

1 2π i = e 3N T (2, 6)N , N (1)

3

1 N

=e

πi 6N

N−1

a

(−1) ω

(6.15a)

1 2 2 a(a+1)+b

a,b=0

a+1 , b

(6.15b)

where the second identity remains to be proved (the first identity was established from an asymptotic expansion due to a discussion in the previous section). We have checked numerically a validity of this identity. As we have

(1) (N )

3

 =

(0) (N )

3

1

2 e 6 πiN 2

e 3 πiN

 ,

modular property and Eqs. (6.15) supports that

1

(3:1) (1) ∞ (−N )

πi n 1 En 1 1 1 3

+ −i N · , · √ n! E (3:0) 6N 2 1 −1 (0) 1 (0) (−N )

n n=0 3 3 N (6.16) (1)

3

N

√

where the generalized Euler number is defined by En(3;a) = −

3 · 62n 2n+1

B2n+1

2−a 6

− B2n+1

4+a 6

,

or some of them are as follows; (3;1) ∞ sh(2 x) Ek 3 = (3;0) sh(3 x) sh(x) k=0 E k 10 2 3 = − 8 1 3

x 2k (2 k)!

34 x 4 910 x 6 x2 + − + ··· . 2 32 24 896 720


423

6.3. (2,8)-Torus Link: m = 4. • q-series: A set of theta series is given by 1 2 (0) (0)

4 (τ ) = n χ8 (n) q 16 n n∈Z

3 3

η(τ ) τ 3 τ = − η 2 η 2 η(2 τ )

9 = 2 q 16 3 − 5 q + 11 q 7 − 13 q 10 + 19 q 22 − 21 q 27 + · · · ,

(1)

4 (τ ) =

1

(1)

(6.17a)

2

n χ8 (n) q 16 n

n∈Z

3 = 4 η(2 τ )

1 = 4 q 4 1 − 3 q 2 + 5 q 6 − 7 q 12 + 9 q 20 − 11 q 30 + · · · , (2)

4 (τ ) =

1

(2)

(6.17b)

2

n χ8 (n) q 16 n

n∈Z

3 3

η(τ ) τ 3 = + η τ η( 2 ) η(2 τ ) 2

1 = 2 q 16 1 − 7 q 3 + 9 q 5 − 15 q 14 + 17 q 18 − 23 q 33 + · · · , which are modular coinvariant;   (2)   1 πi  (2) 0 0 e8

4 (τ + 1)

4 (τ )   (1)   1   (1)   (τ + 1) =  2 πi   4 (τ )  0 0 e  4   ,  9 (0) (0)

4 (τ + 1)

4 (τ ) 0 0 e 8 πi √   (2) 

4 (τ ) 2 1 1  (1)  τ 3/2 1 √ √   (1)   (−1/τ ) =   ·  2 0 − 2  4   ·  4 (τ ) . i 2 √ (0) (0) 1 − 2 1

4 (−1/τ )

4 (τ ) 

(2)

4 (−1/τ )



(6.17c)

(6.18)



(6.19)

We then have the Eichler integral as (0) (τ ) = 4 q 16

4 9

∞

a

(−1) q

1 2 a(a+1)

a=0

=4

∞ n=0 9

= 4 q 16

(0)

a b=0

1

q

b(b+1)

b a c(c+1) b q b c c=0

2

χ8 (n) q 16 n

1 − q + q 7 − q 10 + q 22 − q 27 + · · · ,

(6.20a)

424

K. Hikami ∞

(1) (τ ) = 4 q 4

4

1

=4

1

(−1)a q 2 a(a+1)

a=0 ∞ (1) 1 2 χ8 (n) q 16 n n=0

a

b+1 2 b +1 a qc b c c=0

b=0

1 − q 2 + q 6 − q 12 + q 20 − q 30 + · · · ,

1

= 4q4

∞

(2) (τ ) = 4 q 16

4 1

a

(−1) q

1 2 a(a+1)

a=0

=4

q b(b+1)

∞ n=0 1

= 4 q 16

a+1

q

b2

b=0 1

(2)

(6.20b)

b a + 1 c2 b q b c c=0

2

χ8 (n) q 16 n

1 − q 3 + q 5 − q 14 + q 18 − q 33 + · · · .

(6.20c)

One sees that (1) (τ ) = 2

(0) (2 τ ).

4 2 • root of unity: Limiting value of the Eichler integrals when q goes to the N th primitive root of unity is given by (a)

4

1 N

=4

4N n=0

n n2 πi (a) χ8 (n) 1 − e 8N , 4N

(6.21)

which is rewritten as (0)

4

1 N

9π i

= e 8N

(2)

4

1 N 1 N

1

(−1)a ω 2 a(a+1)+b(b+1)+c(c+1)

a,b,c=0

=

(1)

4

N−1

a b b c

1 9π i e 8N T (2, 8)N , N

=e

πi 2N

N−1

(6.22a)

a

1 2 2 a(a+1)+b(b+1)+c

a

1 2 2 2 a(a+1)+b +c

(−1) ω

a,b,c=0

=e

πi 8N

N−1 a,b,c=0

(−1) ω

a b+1 , (6.22b) b c

a+1 b

b . c

(6.22c)

Though we have checked these three equalities numerically, we proved only Eq. (6.22a) in this article.


425

The nearly modular properties are written as 1 

√    (2) (−N )

2 1 1 4  (1)  √  √   (1) √ 1 1  

     4 N  −i N · 2  2 0 − 2 ·  4 (−N ) √ (0) 1 (0) (−N ) 1 − 2 1

4 4 N  (4:2)  En n ∞  1  E (4:1)  π i + , n!  n  8 N n=0 (4:0) En 

(2)

4



N

where we have 

(2) (N )

4





1

3 e 8 πiN

(6.23)



  (1)   1  

(N ) =  2 πiN  ,  2 e  4    9 (0) (N )

e 8 πiN 4

and the generalized Euler number is defined by 3−a 5+a 26n+2 (4;a) En =− B2n+1 − B2n+1 . 2n+1 8 8 Some of them are explicitly given as follows:  (4;2)    Ek sh(3 x) ∞  2k 4    E (4;1)  x sh(2 x) =   (2 k)! k sh(4 x) k=0 (4;0) sh(x) Ek         119 5587 3 7 2 4 6       x  x  x + 160 − 7808 + ··· . = 2 − 8 2 24 720 109 5465 1 5 Acknowledgements. The author would like to thank Anatol N. Kirillov for suggesting to study torus links. He also thanks Hitoshi Murakami for communications in constructing quantum knot invariants. Thanks are also due Don Zagier for comments on the nearly modular form. The author thanks George Andrews for his interest. This work is supported in part by the Sumitomo Foundation, and Grant-in-Aid for Young Scientists from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

References 1. Andrews, G. E.: The Theory of Partitions. London: Addison-Wesley, 1976 2. Hikami, K.: Volume conjecture and asymptotic expansion of q-series. Exp. Math. (2003). to appear 3. Hikami, K.: q-series and L-functions related to half-derivatives of the Andrews–Gordon identity. Ramanujan J., (2003), to appear 4. Kac, V. G.: Infinite Dimensional Lie Algebras. Third ed. Cambridge: Cambridge Univ. Press, 1990 5. Kashaev, R. M.: A link invariant from quantum dilogarithm. Mod. Phys. Lett. A 10 1409–1418 (1995)

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K. Hikami

6. Kashaev, R. M.: The hyperbolic volume of knots from quantum dilogarithm. Lett. Math. Phys. 39, 269–275 (1997) 7. Kashaev, R. M., Tirkkonen, O.: Proof of the volume conjecture for torus knots. Zap. Nauch. Sem. POMI, 269, 262–268 (2000) 8. Lawrence, R., Rozansky, L.: Witten–Reshetikhin–Turaev invariants of Seifert manifolds. Commun. Math. Phys. 205, 287–314 (1999) 9. Lawrence, R., Zagier, D.: Modular forms and quantum invariants of 3-manifolds. Asian J. Math., 3, 93–107 (1999) 10. Lickorish, R.: An Introduction to Knot Theory. New York: Springer, 1997 11. Macdonald, I. G.: Affine root systems and Dedekind’s η-function. Invent. Math. 15, 91–143 (1972) 12. Morton, H. R.: The coloured Jones function and Alexander polynomial for torus knots. Proc. Cambridge Philos. Soc. 117, 129–135 (1995) 13. Murakami, H.: Kashaev’s invariant and the volume of a hyperbolic knot after Yokota. In: “Physics and Combinatorics”, A. N. Kirillov, A. Tsuchiya, H. Umemura, eds., Singapore: World Scientific, 2001, pp. 244–272 14. Murakami, H., Murakami, J.: The colored Jones polynomials and the simplicial volume of a knot. Acta Math. 186, pp. 85–104 (2001) 15. Rosso, M., Jones, V.: On the invariants of torus knots derived from quantum groups. J. Knot Theory Ramifications 2, 97–112 (1993) 16. Rozansky, L.: Higher order terms in the Melvin–Morton expansion of the colored Jones polynomial. Commun. Math. Phys. 183, (1997) 17. Stoimenow, A.: Enumeration of chord diagrams and an upper bound for Vassiliev invariants. J. Knot Theory Ramifications 7, 93–114 (1998) 18. Yokota, Y.: On the volume conjecture for hyperbolic knots. math.QA/0009165, (2000) 19. Zagier, D.: Vassiliev invariants and a strange identity related to the Dedekind eta-function. Topology 40, 945–960 (2001) 20. Zwegers, S. P.: Mock ϑ-functions and real analytic modular forms. In “q-Series with Applications to Combinatorics, Number Theory, and Physics”, B. C. Berndt, K. Ono, eds., Providence, RI: Am. Math. Soc., 2001, pp. 269–277 21. Zwegers, S. P.: Mock Theta Functions. PhD thesis, Universiteit Utrecht, 2002 Communicated by L. Takhtajan


Communications in


Remarks on Additivity of the Holevo Channel Capacity and of the Entanglement of Formation Keiji Matsumoto1 , Toshiyuki Shimono2 , Andreas Winter3 1

ERATO Quantum Computation and Information Project, Dai–ni Hongo White Bldg. 201, Hongo 5–28–3, Bunkyo–ku, Tokyo 133–0033, Japan. E-mail: [email protected] 2 Department of Computer Science, Graduate School of Information Science and Technology, University of Tokyo, Hongo 7–3–1, Bunkyo–ku, Tokyo 113–0033, Japan. E-mail: [email protected] 3 Department of Computer Science, University of Bristol, MerchantVenturers Building, Woodland Road, Bristol BS8 1UB, UK. E-mail: [email protected] Received: 4 September 2002 / Accepted: 19 May 2003 Published online: 18 November 2003 – © Springer-Verlag 2003

Abstract: The purpose of this article is to discuss the relation between the additivity questions regarding the quantities (Holevo) capacity of a quantum channel T and entanglement of formation of a bipartite state ρ. In particular, using the Stinespring dilation theorem, we give a formula for the channel capacity involving entanglement of formation. This can be used to show that additivity of the latter for some states can be inferred from the additivity of capacity for certain channels. We demonstrate this connection for some families of channels, allowing us to calculate the entanglement cost for many states, including some where a strictly smaller upper bound on the distillable entanglement is known. Group symmetry is used for more sophisticated analysis, giving formulas valid for a class of channels. This is presented in a general framework, extending recent findings of Vidal, Dür and Cirac. We also discuss the property of superadditivity of the entanglement of formation, which would imply both the general additivity of this function under tensor products and of the Holevo capacity (with or without linear cost constraints). 1. Introduction Quantum information theory has progressed considerably over the last decade: today we understand much better the information transmission properties of quantum channels, and entanglement has turned from an oddity first into a valuable effect and then into a quantifiable resource, as shown by the many well–motivated entanglement measures that have been put forward. Almost all of them are operationally grounded as some optimal performance parameter, and can be written as solutions to various high–dimensional or even asymptotic optimisation problems. All of these capacities and entanglement measures raise the natural problem of additvity under tensor products, i.e. the question, if the independent supply of two specimens of the resource has as its performance the sum of the performances of the individual objects (be they channels or states). For some of the current measures of entanglement

428

K. Matsumoto, T. Shimono, A. Winter

additivity has been disproved by counterexamples (for the so–called relative entropy of entanglement in [36]), for others, like the distillable entanglement [7] it is claimed improbable [28]. For some, however, additivity is still widely conjectured, most notably for a bound on the distillable entanglement by Rains [24], and for the entanglement of formation [7]. The literature on the subject is vast and increasing rapidly, and in the present paper we will only make a small contribution. We shall be concerned with the entanglement of formation, and with the aforementioned classical capacity of quantum channels, pointing out a connection between the two that also relates their additivity problems. We outline briefly the content of the rest of the paper: in Sects. 2 and 3 the classical capacity of a channel and the entanglement of formation of a state are reviewed. In Sect. 4 a simple observation on the Stinespring dilation of a completely positive map provides the link between the two quantities, which is exploited in a number of examples in Sect. 6; group symmetry is introduced in Sect. 7, adding another example, and to supply formulas valid for a class of channels which includes examples discussed in Sect. 4 as special cases. And in Sect. 8 some of these results are used to demonstrate a gap between entanglement cost and distillable entanglement. In Sect. 5 we discuss superadditivity of entanglement of formation as a (conjectured) property which would unify the additivity questions considered here: it implies additivity of entanglement of formation, of channel capacity, and of channel capacity with a linear cost constraint. We conclude with a discussion of our observations and related works. 2. Holevo Capacity We consider block coding of classical information via the quantum channel T : B(H) −→ B(H2 ), where H and H2 are Hilbert spaces. If the encoding is restricted to product states it is known [17, 26] that the capacity is given by C(T ) = sup I p; T (π) : {pi , πi } pure state ensemble on H , (1) where the Holevo mutual information of an ensemble {pi , ρi } is given by I (p; ρ) = S p i ρi − pi S(ρi ). i

i

Here S(ω) = −Trω log ω is the von Neumann entropy of a state. For finite dimensional H2 the sup in eq. (1) is indeed a max, attained for an ensemble of at most (dim H2 )2 states. It is conjectured that for a product of channels making use of entangled input states does not help to increase the capacity: C(T1 ⊗ T2 ) = C(T1 ) + C(T2 ).

(2)

(The question is implicit in [16] and the above references, and made explicit in [8], where it was speculated that the answer may be negative.)

Additivity of the Holevo Channel Capacity and Entanglement of Formation

429

This would imply that C(T ) is the classical capacity of T . Observe that here the inequality “≥” follows immediately from the fact that the right hand side can be achived using product states. Without additivity, the general formula for this capacity reads 1 ⊗n . C T n→∞ n lim

Despite much recent activity on the question [1, 2], and even proofs of the additivity conjecture in some cases [9, 22, 13, 19, 20, 29, 21], it is still a wide open problem.

3. Entanglement of Formation Let ρ be a state on H1 ⊗ H2 . The entanglement of formation of ρ is defined as Ef (ρ) := inf

pi E(πi ) : {pi , πi } pure state ens. with

i

p i πi = ρ ,

(3)

i

where the (entropy of) entanglement for a pure state π on H1 ⊗ H2 is defined as E(π ) := S Tr H2 π = S Tr H1 π . If the rank of ρ is finite the inf is in fact a min, achieved for an ensemble of at most (rank ρ)2 elements. This quantity was proposed in [7] as a measure of how costly in terms of entanglement the creation of ρ is. It is conjectured (but only in a few cases proved: the only published examples are in [33]) that Ef is an additive function with respect to tensor products: Ef (ρ1 ⊗ ρ2 ) = Ef (ρ1 ) + Ef (ρ2 ).

(4)

Observe that, as in the case of the Holevo capacity, “≤” follows easily from the fact that the right hand side is achieved by product state ensembles. If this would turn out to be true, the entanglement cost Ec (ρ) of ρ, i.e. the asymptotic rate of EPR pairs to approximately create n copies of ρ is given by Ef (ρ): in [14] it was proved rigorously that 1 Ec (ρ) = lim Ef ρ ⊗n . n→∞ n Note that the function Ef has the property of being a convex roof : Ef (ρ) = inf

i

pi Ef (ρi ) : {pi , ρi } ensemble with

pi ρ i = ρ .

(5)

i

The cases in which Ef is known are arbitrary states of 2 × 2–systems [41], isotropic states in arbitrary dimension [31], Werner and OO–symmetric states [36], and some other highly symmetric states [33].

430


4. Stinespring Dilations: Linking C(T ) and Ef (ρ) Due to a theorem of Stinespring [30] the completely positive and trace preserving map T can be presented as the composition of an isometric embedding of H into a bipartite system with a partial trace: U

Tr H1

T : B(H) → B(H1 ⊗ H2 ) −→ B(H2 ).

(6)

See [25] for a discussion on how to construct this from

the so–called Kraus (operator sum) representation [23], T (ρ) = i Ai ρA∗i with i A∗i Ai = ?, of T . We shall use this construction later on in Examples 5 and 6. By embedding into larger spaces we can present U as restriction of a unitary, which often we silently assume done. Denote K := U H ⊂ H1 ⊗ H2 , the image subspace of U . Then we can say that T is equivalent to the partial trace channel, with inputs restricted to states on K. This entails: Theorem 1. C(T ) = sup{S Tr H1 ρ − Ef (ρ) : ρ state on K}.

(7)

Proof. Very simple: choosing an input ensemble for T amounts

by our above observation to choosing an ensemble {pi , πi } on K. Denoting ρ = i pi πi , the average output state of T in Eq. (1) is just Tr H1 ρ, while the individual output states are the Tr H1 πi . Hence the second term in Eq. (1), the average of output entropies, has as its infimum Ef (ρ) when we vary over ensembles with fixed ρ. Note that if we choose the dimension of H1 large enough, every channel from H to H2 corresponds to a subspace of H1 ⊗ H2 (though not uniquely) and vice versa. Remark 2. The quantity S Tr H1 ρ − Ef (ρ) in the optimisation problem in Theorem 3 equals the entropy of the subalgebra B(H2 ) in B(H1 ⊗ H2 ), as defined by Connes, Narnhofer and Thirring [11]: this was observed by Benatti, Narnhofer and Uhlmann [6]. This has interesting consequences: for each subspace K of the tensor product there is a convex set OT of states ρ supported on it which maximise Eq. (7). The reason for convexity is again very simple: let ρ, ρ ∈ S(K). Then S pTr H1 ρ + (1 − p)Tr H1 ρ ≥ pS(Tr H1 ρ) + (1 − p)S(Tr H1 ρ ), Ef pρ + (1 − p)ρ ≤ pEf (ρ) + (1 − p)Ef (ρ ), by concavity (convexity) of S (Ef ). Hence the aim function in Eq. (7) is concave, which implies that the set of ρ for which it is atleast Ris a convex set, for any real R. Observe that by this argument both S Tr H1 ρ and Ef (ρ) are constants for ρ ∈ OT . Indeed, one can show (see the discussion below, in this section) that even all Tr H1 ρ, ρ ∈ OT , are identical. For such states the additivity of Ef is implied by the additivity of C for the corresponding channels: indeed, assume that for two channels T , T that optimal input states in the sense of Eq. (7) are ρ ∈ OT , ρ ∈ OT , respectively, with reduced states ρ2 and ρ2 . Then, assuming additivity we get


S(ρ2 ) − Ef (ρ) + S(ρ2 ) − Ef (ρ ) = C(T ) + C(T ) = C(T ⊗ T ) ≥ S(ρ2 ⊗ ρ2 ) − Ef (ρ ⊗ ρ ), hence

431

(8)

Ef (ρ ⊗ ρ ) ≥ Ef (ρ) + Ef (ρ ),

which by our earlier remarks implies additivity. Thus we have proved Theorem 3. If for any two channels T and T , each with a Stinespring dilation chosen as in eq. (6), C(T ⊗ T ) = C(T ) + C(T ), then ∀ρ ∈ OT , ρ ∈ OT Ef (ρ ⊗ ρ ) = Ef (ρ) + Ef (ρ ). Most interesting is the case when we know C(T ⊗n ) = nC(T ), because then we can conclude Ef (ρ ⊗n ) = nEf (ρ), thus determining the entanglement cost of ρ (see Sect. 3). For example, King [19, 20] proved this for unital qubit–channels, Shor [29] for entanglement–breaking channels, and King [21] for arbitrary depolarising channels, giving rise to a host of states for which we thus know that the entanglement cost equals Ef . Examples are discussed in Sect. 6 below and the following two sections. It is natural to consider ways to implement an implication of additivity going the other way than Theorem 3: from entanglement of formation to Holevo capacity. Indeed, in another look at Eq. (7), let us focus on the other quantity of interest in the optimisation: this is the von Neumann entropy of the output state. In general, while there can be many ensembles maximising Eq. (1) (let us assume

for the moment that the output space is finite dimensional), and in fact many averages i pi πi (the set OT of optimal input

states introduced above), the average output state of such an optimal ensemble, ω = i pi T (πi ), is unique: the reason is the strict concavity of the von Neumann entropy, so if we had two ensembles with different output states, mixing the ensembles would strictly increase the Holevo mutual information. Let us denote this optimal output state ω(T ). It is clear that the additivity conjecture Eq. (2) implies that ω(T ⊗ T ) = ω(T ) ⊗ ω(T ),

(9)

but the reverse seems not obvious. Still, Eq. (9) might be a reasonable first step towards proving additivity of C(T ) in general. Unfortunately, even assuming additivity of the entanglement of formation, we have not been able to derive additivity of the channel capacity from Eq. (9). However, let us assume that for the product channel T ⊗ T an optimal input state in Eq. (7) is a product (due to the non–uniqueness of optimal input states there might also be entangled ones!), ρ ⊗ ρ , say. Then clearly, Ef (ρ ⊗ ρ ) = Ef (ρ) + Ef (ρ ) implies C(T ⊗ T ) = C(T ) + C(T ), in a reversal of the argument from the proof of Theorem 3. 5. Superadditivity: Unifying C(T ) and Ef (ρ) Looking at Eq. (7), and trying to find a unifying reason why both of the above discussed additivity conjectures should hold, we are led to speculate that Ef might not only be additive with respect to tensor products (Eq. (4)), but have even a superadditivity property for arbitrary states on a composition of two bipartite systems:

432


Let ρ be a state on H ⊗ H , where H = H1 ⊗ H2 and H = H1 ⊗ H2 . Then superadditivity means that Ef (ρ) ≥ Ef (Tr H ρ) + Ef (Tr H ρ),

(10)

where all entanglements of formation are understood with respect to the 1–2–partition of the respective system. (This relation was apparently first considered in [36], and called strong superadditivity there. We call it just “superadditivity” here in simple analogy to, e.g., subadditivity of the von Neumann entropy.) Note that this implies additivity of Ef when applied to ρ1 ⊗ ρ2 since we remarked in Sect. 3 that the other inequality is trivial. Note on the other hand that it also implies additivity of C(T ), by Eq. (7) in Sect. 4: by replacing a supposedly optimal ρ on K ⊗ K (for two channels T and T , and corresponding Stinespring dilations which give rise to the subspaces K and K in respective bipartite systems) by the tensor product of its marginals, we can only increase the entropy (subadditivity), and only decrease the entanglement of formation (superadditivity). As an extension, let us show that it even implies an additivity formula for the classical capacity under linear cost constraints (see [18]): in this problem, there is given a selfadjoint operator A on the input system, and a real number α, additional to the channel T . ≤ nα + o(n), As signal states we allow only such states σ on H⊗n for which Tr(σ A) with n ⊗(k−1) ⊗(n−k) = ? ⊗A⊗? A k=1

(i.e., their average cost is asymptotically bounded by α). Then it can be shown [18, 39] that the capacity C(T ; A, α) in the thus constrained system and using product states is given

by a maximisation as in Eq. (1), only that the ensembles {pi , πi } are restricted by i pi Tr(πi A) ≤ α. (The same treatment applies if there are several linear cost inequalties of this kind. It is only for simplicity of notation that we stick to the case of a single one.) Because of the linearity of this condition in the states this yields a formula for C(T ; A, α) very similar to Theorem 1: C(T ; A, α) = sup{S Tr H1 ρ − Ef (ρ) : ρ state on K, Tr(ρA) ≤ α}. (11) By the general arguments given in previous sections we can conclude that this function is concave in α. The question of course is again, if entangled inputs help to increase the capacity, or if ? nα) = nC(T ; A, α). C T ⊗n ; A,

(12)

We shall show that this indeed follows from the superadditivity, by showing the following: for channels T , T , cost operators A, A , and cost threshold α: C T ⊗ T ; A ⊗ ? + ? ⊗ A ; α = sup C(T ; A, α) + C(T ; A , α ) . α+α = α

(Then, by induction and using the concavity, the equality in Eq. (12) follows.) Indeed, “≥” is obvious by choosing, for α + α = α , optimal states ρ, ρ in the sense

of Eq. (11), and considering ρ ⊗ ρ . In the other direction, assume any optimal ω for the product system, with marginal states ρ and ρ : by definition, Tr (ρ ⊗ ρ )(A ⊗ ? + ? ⊗ A ) = Tr ω(A ⊗ ? + ? ⊗ A ) ≤ α,


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so also the product ρ ⊗ ρ is admissible, and since there exist α, α summing to α such that Tr(ρA) ≤ α, Tr(ρ A ) ≤ α , the claim follows in exactly the same way as for the unconstrained capacity. We have thus proved: Theorem 4. Superadditivity of Ef , Eq. (10), implies additivity of entanglement of formation, of the Holevo capacity and of the Holevo capacity with cost constraint under tensor products. Observe the strong intuitive appeal of the superadditivity property: it says that by measuring the entanglement via Ef , a system can only appear less entangled if judged by looking at its subsystems individually. Note that this is almost trivially true (by definition) for the distillable entanglement, while wrong for the relative entropy of entanglement [32], because this would make it an additive quantity, which we know it isn’t [36, 3, 4]. The superadditivity also bears semblance to a distributional property of the so–called tangle [10]. Superadditivity is thus a very strong property. If there is one “nice” underlying mathematical structure to the additivity of Ef , it should indeed be this. Note that it is true if one of the marginal states, say Tr H , is separable: because then its Ef is 0, and Eq. (10) simply expresses the monotonicity of Ef under local operations (in this case: partial traces). This was previously noted in [36]. Observe that it is sufficient to prove superadditivity for a pure state ρ = |ψ ψ|, as then we can apply it to an optimal decomposition of ρ, together with the convex roof property, Eq. (5). This was apparently considered by Benatti and Narnhofer [5], who even conjectured “good decompositions” of the reduced states Tr H |ψ ψ| and Tr H |ψ ψ|. This latter conjecture however was refuted by Vollbrecht and Werner [35] who constructed a counterexample. On the other hand, there is limited positive evidence in favour of superadditivity: In [33], Eq. (16), it is actually proved if the partial trace in one of the subsystems is entanglement–breaking. We observed (following [36]) that it is trivially true if one of the reduced states is separable. Some of our examples yield more cases of superadditivity. E.g. in Example 6 we constructed the subspaces Kλ : for every pure state ψ ∈ Kλ1 ⊗ · · · ⊗ Kλn , with reduced density operators ρ1 , . . . , ρn we get (using the additivity of the minimal output entropy proved in [21]) E(ψ) ≥ Smin (T1 ) + . . . + Smin (Tn ) = Ef (ρ1 ) + . . . + Ef (ρn ), the second line by the insight of Example 6 that all states supported on Kλi have the same entanglement of formation. Similarly, our other examples yield certain pure states for which we obtain superadditivity. It seems to us that this question most elegantly sums up the two most prominent additivity questions in quantum information theory, and we would like to pose it as a challenge: either to prove superadditivity (thus proving additivity of Ef and of C), or to find a counterexample. 6. Examples In this and the following two sections we want to demonstrate how Theorem 3 can be used to construct nontrivial states for which we can compute the entanglement cost, to

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reproduce some known results of this sort, and even exhibit “irreversibility of entanglement”. Example 5. Consider the generalised depolarising channels of qubits: ps σs ρσs† , T : ρ −→ s=0,x,y,z

with σ0 = ?, the familiar Pauli matrices

01 0 −i 1 0 , σy = , σz = , σx = 10 i 0 0 −1 and a probability distribution (ps )s=0,x,y,z . For these channels additivity of the capacity under tensor product with an arbitrary channel was proved in [20]. Note that up to unitary transformations on the input and output system each unital qubit channel has this form, by the classification of qubit maps of King and Ruskai [22], and Fujiwara and Algoet [12]. By this result we also can assume that p0 + pz − px − py ≥ |p0 + py − px − pz |, |p0 + px − py − pz |.

(13)

It is easy to see that for such a channel the capacity is given by C(T ) = 1 − Smin (T ), with achieved at the eigenstates |0 , |1 of σz : Smin (T ) = the minimal output entropy S T (|0 0|) = S T (|1 1|) . An optimal ensemble is the uniform distribution on these states. It is easy to construct a Stinespring dilation for this map, by an isometry U : C2 −→ C2 ⊗ C4 , in block form:  √ √ p0 σ 0  p σ  U = √ x x , p σ √ y y p z σz and the corresponding subspace K ⊂ C2 ⊗ C4 is spanned by √ √ √ √ |ψT = p0 |0 ⊗ |0 + px |1 ⊗ |x + i py |1 ⊗ |y + pz |0 ⊗ |z , √ √ √ √ |ψT⊥ = p0 |1 ⊗ |0 + px |0 ⊗ |x − i py |0 ⊗ |y − pz |1 ⊗ |z . The optimal input state corresponds to the equal mixture ρT of these two pure states. From these observations, together with Theorem 3, we obtain that Ef (ρT ) = Smin (T ) = H (p0 + pz , 1 − p0 − pz ), and Ef (ρT ⊗ σ ) = Ef (ρT ) + Ef (σ ) for any σ ∈ OT , with arbitrary channel T . In particular, Ec (ρT ) = Ef (ρT ) = H (p0 + pz , 1 − p0 − pz ). In fact, we proved that the decomposition of ρT⊗n into the 2n equally weighted tensor products of |ψT ψT | and |ψT⊥ ψT⊥ | is formation–optimal. By the convex roof property of Ef this implies that any convex combination of these states is a formation–optimal decomposition (this argument was also used in [36] to extend the domain of states with known entanglement of formation). In particular, we can conclude that any mixture ρ of |ψT ψT | and |ψT⊥ ψT⊥ | has Ec (ρ) = Ef (ρ) = H (p0 + pz , 1 − p0 − pz ).

(14)


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The case of equal px , py , pz leads to the usual unitarily covariant depolarising channel. This is contained in the following: Example 6. Consider the d–dimensional depolarising channel with parameter λ: 1 T : ρ −→ λρ + (1 − λ) ?, d with − d 21−1 ≤ λ ≤ 1 for complete positivity, to ensure that T can be represented as a mixture of generalised Pauli actions: T (ρ) = p0 ρ + (1 − p0 )

2 −1 d

i=1

1 σi ρσi† , d2 − 1

with an orthogonal set of unitaries (a “nice error basis”, see e.g. [37] for constructions) σi , i.e. σ0 = ?, Tr(σi† σj ) = dδij , and p0 = λ + (1 − λ)/d 2 . For this channel, [21] proves the additivity of C(T ) and Smin (T ), and it is quite obvious that C(T ) = log d − Smin (T ) = log d − S T (|ψ ψ|) , for arbitrary |ψ ∈ Cd , optimal input ensembles being those mixing to d1 ?. It is easy to evaluate this latter von Neumann entropy:

1−λ 1−λ 1−λ S T (|ψ ψ|) =H λ + , ,... , d d d

1 1 (1 − λ), 1 − 1 − (1 − λ) =H 1− d d

1 + 1− (1 − λ) log(d − 1). d 2

Again, it is easy to construct a Stinespring dilation U : Cd −→ Cd ⊗ Cd in block form:   √ p0 ?   1−p0 σ   d 2 −1 1   U = , ..   .   1−p0 σ 2 −1 2 d d −1 such that the subspace of interest is Kλ := U Cd , its maximally mixed state denoted ρλ . Then Theorem 3 allows us to conclude that Ef (ρλ ⊗ σ ) = Ef (ρλ ) + Ef (σ ) for any σ ∈ OT . In particular Ec (ρλ ) = Ef (ρλ ) = Smin (T ). By the argument familiar from Example 5 we can conclude even that any mixture of product states on Kλ⊗n has entanglement of formation nSmin (T ), in particular for every state ρ supported on Kλ we obtain Ec (ρ) = Ef (ρ) = Smin (T ).

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In the following section we will study some other examples, involving symmetry, which allows evaluation of the entanglement of formation in some cases, and also the entanglement cost.

7. Group Symmetry Imposing a group symmetry via representation on the involved (sub–)spaces as follows, we obtain another example, such as Vidal, Dür and Cirac [33], and formulas valid for a class of channels. Note that the symmetry is used principally for simplifying computations. Assume that a compact group G (with Haar measure dg) acts irreducibly both on K and H2 by a unitary representation (which we denote by Vg and Ug ), which commutes with the map T (partial trace): Tr H1 Vg σ Vg† = Ug Tr H1 σ Ug† .

(15)

g , such For example let there also be a unitary representation of G on H1 , denoted U that K is an irreducible subspace of the representation Vg = Ug ⊗ Ug . We call this the Product Case. In the general, non–product case of Eq. (15), it is an easy exercise to show that, with P denoting the projection onto K in H1 ⊗ H2 ,

1 C(T ) = log dim H2 − Ef P , TrP

1 Ef P = min E(ψ) : |ψ ∈ K . TrP

(16) (17)

Indeed, in the second equation, “≥” is trivially true, and for the opposite direction choose a minimum entanglement 1 pure state |ψ0 ∈ K, and consider the decomposition P (by Schur’s lemma!): all these states Vg |ψ0 ψ0 |Vg† have {Vg |ψ0 ψ0 |Vg† , dg} of TrP the same entanglement, E Vg |ψ0 = S Tr 1 Vg |ψ0 ψ0 |Vg† = S Ug Tr 1 |ψ0 ψ0 |Ug† = S Tr 1 |ψ0 ψ0 | = E(ψ0 ),

(18)

using Eq. (15). As for the capacity, in the light of Eq. (7) and using Eq. (17), the “≤” is trivial, and the argument just given proves equality. Moreover, for all states ρ spanned by {Vg |ψ0 ψ0 |Vg∗ : g ∈ G}, where |ψ0 is a pure state with E(|ψ0 ) = min E(|ψ ) : |ψ ∈ K , we can conclude that Ef (ρ) = min E(ψ) : |ψ ∈ K .


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We even obtain the entanglement cost of all the ρ spanned by {Vg ρ0 Vg∗ : g ∈ G}, in 1 1 the cases where we know that Ec ( TrP P ) = Ef ( TrP P ): consider the chain of inequalities ⊗n P Ef ≤ dn gEf Vg1 ⊗ · · · ⊗ Vgn ρ ⊗n Vg†1 ⊗ · · · ⊗ Vg†n TrP n ≤ dn g Ef Vgk ρVg†k

= nEf

k1

P TrP

= Ef

P TrP

⊗n .

Here the first inequality is due to the convexity (see the definition) of Ef , applied to of Ef and the the family Vg ρVg† with Haar measure, and the others are by subadditivity ⊗n assumption. But the right-hand side in the first line equals Ef ρ , since any decomposition of ρ ⊗n translates into a decomposition of Vg1 ⊗ · · · ⊗ Vgn ρ ⊗n Vg†1 ⊗ · · · ⊗ Vg†n of the same entanglement, and vice versa. Hence Ec (ρ) = Ef (ρ) = min E(ψ) : |ψ ∈ K . (19) (Note that in [33] this was argued by making use of being in the “product case”, in which case the group action on K is performable by LOCC; then the first inequality above was argued by nonincrease of Ef under LOCC transformations.) In particular, if in addition the action of G in K is transitive, we can conclude (19) for all the state supported on K, because (18) implies that E(|ψ ) takes the same value for any pure state |ψ in K. This group symmetry argument simplifies the analysis of unital qubit channels and generalised depolarising channels. In the former case, G is chosen to be SU (d), while in the latter, we consider the group G = {?, R, R 2 , R 3 }, with

0 −1 R= . 1 0 In both cases, we define representations Vg , Ug of G by Vg = UgU ∗ and Ug = g. They are irreducible, and satisfy the condition Eq. (15). Hence, general arguments in this section, directly imply results about these examples in the previous section. The following example is constructed using group symmetry. Example 7. Vidal, Dür and Cirac [33] consider the subspace K of C3 ⊗ C6 spanned by √ 1 |0 s = |1 |2 + |2 |1 + 2|0 |3 , 2 √ 1 |1 s = |2 |0 + |0 |2 + 2|1 |4 , 2 √ 1 |2 s = |0 |1 + |1 |0 + 2|2 |5 . 2 By using the isomorphism |j ↔ |j s between C3 and K, it is easily checked that Tr C6 implements the channel map T : ρ −→

1 ? + ρ , 4

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hence we are in the transitive covariant case, with U ∈ SU(3) and V = U . It is straightforward to check that this channel is entanglement–breaking (see [33]): hence [29] tells us that its capacity is additive, and we can apply Theorem 3. By our general observations above we can conclude that for any state ρ supported on K, Ec (ρ) = Ef (ρ) = 3/2. Following [33], we can introduce (for j = 0, 1, 2) |j t = | 3 ⊗ |j ∈ C3 ⊗ C3 ⊗ C3 , and form the superpositions |  := c|j s ⊕ s|j t ∈ C3 ⊗ C6 ⊕ C9 in the direct sum of the respective supporting spaces, with |c|2 + |s|2 = 1. This obviously retains the covariant nature, and allows us to implement the mixtures of T with the constant map onto 13 ?, so we get every channel 1 Tp : ρ −→ p ? + (1 − p)ρ , 3 for 3/4 ≤ p ≤ 1, all of which are clearly entanglement–breaking, so the same technique applies, and we find subspaces on which every state has Ec = Ef = const. ∈ [3/2, log 3]. In [33], by implementing other entanglement–breaking channels (and using Shor’s result [29] on capacity additivity), other, and more general results of this type were obtained. Example 8. The “U ⊗U ”–representation of SU(3) on C3 ⊗C3 decomposes into two irreducible parts, the symmetric subspace of dimension 6 and the antisymmetric subspace A of dimension 3. The latter has a nice basis given by 1 |0 a = √ |1 |2 − |2 |1 , 2 1 |1 a = √ |2 |0 − |0 |2 , 2 1 |2 a = √ |0 |1 − |1 |0 , 2 which we use to identify A with C3 . Notice that the partial trace over the first factor (say) implements a unital channel with symmetry (U ∈ SU(3) on C3 and V = U ⊗ U on A), which is even transitive (hence all states ρa supported on A have the same entanglement of formation Ef (ρa ) = 1), but it is neither depolarising nor entanglement–breaking: in the above identification it reads

3 1 1 TVDC : ρa −→ ? − ρ. 2 3 2 Notice that this is one of the very channels used in [38] to disprove the general multiplicativity conjecture for the maximal output p-norm of a channel. Incidentally, this property is the main tool in King’s proofs of the additivity of channel capacities [19–21].


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Denoting the maximally mixed state on A by σA , it was shown in [27] that f (σ E⊗n A⊗ σA ) = 2Ef (σA ) = 2. Subsequently, Yura [42] has shown that for all n, Ef σA = n, showing that the entanglement cost of this state is indeed 1. The above examples show that using covariance one can often evaluate the entanglement of formation. By carefully choosing the supporting subspace of the state we can use our main Theorem 3, yielding even the entanglement cost. 8. Gap Between Ec and ED Returning to Example 5, let us demonstrate that the states discussed there exhibit a gap between the entanglement cost and distillable entanglement for some of these states, by use of the log–negativity bound log ρ 1 on distillable entanglement [34]. We use the notation of Example 5, in particular we assume the channel T to be a mixture of Pauli rotations, with probability weights according to Eq. (13). The partial transpose ρT of the optimal state ρT decomposes into a direct sum of two 4×4–matrices, which turn out to have the same characteristic equation f (2z) = 0, where f (z) = z4 − z3 + 4(p0 px py + p0 px pz + p0 py pz + px py pz )z − 16p0 px py pz . Since f (2z) = 0 has only one negative root z0 and f is decreasing in a neighbourhood of it, log ρT 1 < Ec (ρT ) is equivalent to 2H (p0 +pz ,1−p0 −pz ) − 1 2Ec (ρT ) − 1 > 0, (20) =f − f − 2 2 using ρT 1 = 1 − 4z0 . That is, if p0 , px , py , pz satisfy this inequality, there is a gap between the entanglement cost of ρT , and its distillable entanglement; Fig. 1 shows a plot of the region of these (px , py , pz ). By continuity, also for a mixture of |ψT ψT | and |ψT⊥ ψT⊥ | which is sufficiently close to ρT , we observe a similar gap. for p0 = 1/2, px = py = pz = 1/6, a short calculation reveals that Especially, ρ = 5/3, so ED (ρT ) ≤ log(5/3) ≈ 0.737, which is smaller than the entanglement T 1 cost Ec (ρ1 ) = H (1/3, 2/3) ≈ 0.918. If p0 + pz = px + py = 21 and p0 = pz , px = py , we can even prove for all true mixtures ρT ,s = s|ψT ψT | + (1 − s)|ψT ψT |⊥ of |ψT ψT | and |ψT⊥ ψT⊥ |, that ED (ρT ,s ) < Ec (ρT ,s ) holds: by Eq. (14) the latter is 1 for all these ρT ,s , and the key observation is that log ρT 1 is strictly than Ec (ρT ,s ) in this case, for the smaller conditon (20) is always satisfied. Hence ρT 1 < 2. The convexity of trace norm and the observation |ψT ψT | 1 = 2 leads, for 1 2 ≤ s < 1 (which we may assume by symmetry), to ρ ≤ (2s − 1) |ψT ψT | + (2 − 2s) ρ T ,s 1 T 1 1 < (2s − 1) · 2 + (2 − 2s) · 2 = 2, and consequently we have log ρT ,s < 1 = Ec (ρT ). 1

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1 0.8 0.6 0.4 0.2 0 0 x0.2 0.4

0.3 0 0.1 y

0.5

Fig. 1. Plots in a (px , py , pz )–frame of the admissible parameters according to Eq. (13) and of the region for which Eq. (20) holds (between the two surfaces)

This fact is also proven by noting that the negativity is strictly convex for mixings of |ψT ψT | and |ψT⊥ ψT⊥ |, i.e. s|ψT ψT | + (1 − s)|ψT⊥ ψT⊥ | < s |ψT ψT | 1 + (1 − s) |ψT⊥ ψT⊥ | , 1

1

if 0 < s < 1. This is proved by finding eigenvectors with nonzero overlap of the two partial transposes such that one has a negative, the other a positive eigenvalue. 9. Conclusion We demonstrated a link between the additivity problems for classical capacity of quantum channels and entanglement of formation, resulting in the additivity of the latter for many states, by invoking recent additivity results for the former. This allows us to establish in particular a gap between distillable entanglement and entanglement cost for many of these states. By exploiting the fact that Ef is a convex roof, this additivity can be extended to even more states, though it is not clear how far this would get us, even taking the general additivity conjecture for granted. It is obvious that we only probed the scope of the method, and it is clear that other examples of the same sort can be constructed, adding to the list of states for which the entanglement cost is known. Each channel for which additivity of its capacity is established will add to this list. The method generalises part of the argument found in the recent work of Vidal, Dür and Cirac [33], but for the case of entanglement breaking channels their method is more general. On the side of general insights, the attempt to link the two additivity conjectures considered here led us to consider the superadditivity of entanglement of formation as a relation which integrates them neatly. We were even able to exhibit a few cases where it is known to hold, providing modest evidence in favour of it.


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Since completion of this work, subsequent research has further clarified the picture presented here: Ruskai (quant-ph/0303141) showed that not all bipartite states can be associated with a channel such as to make use of Theorem 3 to prove additivity. Audenaert and Braunstein (quant-ph/0303045) have re-expressed the superadditivity of entanglement of formation using tools from convex analysis, and showed that the multiplicativity conjecture for maximal output p-norms [1], for p close to 1, of filtering operations implies superadditivity. Shor (quant-ph/0305035) has complemented our Theorem 4 by showing that the general conjectures of superadditivity of Ef , additivity of Ef under tensor products, and additivity of C are in fact equivalent to each other and to the additivity of minimal output entropy of a channel. Acknowledgement. We thank K. G. Vollbrecht and R. F. Werner for conversations about the superadditivity conjecture, and A. Uhlmann for pointers to the literature. KM and TS are supported by the Japan Science and Technology Corporation, AW is supported by the U.K. Engineering and Physical Sciences Research Council, and gratefully acknowledges the hospitality of the ERATO Quantum Computation and Information project, Tokyo, on the occasion of a visit during which part of the present work was done.

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Communications in


On Strong Superadditivity of the Entanglement of Formation Koenraad M.R. Audenaert, Samuel L. Braunstein∗ University of Wales, Bangor School of Informatics, Bangor (Gwynedd) LL57 1UT, UK. E-mail: [email protected] Received: 20 June 2003 / Accepted: 19 July 2003 Published online: 25 November 2003 – © Springer-Verlag 2003

Abstract: We employ a basic formalism from convex analysis to show a simple relation between the entanglement of formation EF and the conjugate function E ∗ of the entanglement function E(ρ) = S(Tr A ρ). We then consider the conjectured strong superadditivity of the entanglement of formation EF (ρ) ≥ EF (ρI ) + EF (ρI I ), where ρI and ρI I are the reductions of ρ to the different Hilbert space copies, and prove that it is equivalent with subadditivity of E ∗ . Furthermore, we show that strong superadditivity would follow from multiplicativity of the maximal channel output purity for quantum filtering operations, when purity is measured by Schatten p-norms for p tending to 1. 1. Introduction One of the central quantities in quantum information theory is the entanglement cost of a state, defined as the number of maximally entangled pairs (singlets) required to prepare this state in an asymptotic way. Calculating the entanglement cost of a general mixed state as such is, with the present state of knowledge, a formidable task because one has to consider an infinite supply of singlets and construct a protocol using local or classical (LOCC) operations only, such that the resulting (infinite-dimensional) state approximates an infinite supply of the required state to arbitrary precision. Furthermore, the protocol must have maximal yield, the number of states produced per singlet. The entanglement cost is the inverse of this yield. An important theoretical breakthrough was achieved in [1], where the entanglement cost EC was shown to be equal to the regularised entanglement of formation: EC (ρ) = limn→∞ EF (ρ ⊗n )/n. The entanglement of formation (EoF) (defined below in (4)) is defined in a mathematical and non-operational way and is therefore much more amenable to calculation. Moreover, for 2-qubit mixed states, a closed formula for the EoF exists [2]. Nevertheless, calculating the entanglement cost still requires calculations over ∗

Current address: The University of York, Dept. of Computer Science, York YO10 5DD, UK

444

K.M.R. Audenaert, S.L. Braunstein

infinite-dimensional states. For that reason one would hope for the additivity property to hold for the EoF: EF (ρ1 ⊗ρ2 ) = EF (ρ1 )+EF (ρ2 ), because then EC = EF . Additivity of the EoF has been proven in specific instances [3–8]. Some of these additivity results are sufficiently powerful to allow calculating the entanglement cost for certain classes of mixed states [5–7]. The much sought-after general proof, however, remains elusive for the time being and, in fact, general additivity is still a conjecture. It is very easy to show that the EoF is subadditive: EF (ρ1 ⊗ ρ2 ) ≤ EF (ρ1 ) + EF (ρ2 ).

(1)

Additivity would then follow from superadditivity: EF (ρ1 ⊗ ρ2 ) ≥ EF (ρ1 ) + EF (ρ2 ).

(2)

In [4] a stronger property, which would imply (super)additivity, has been conjectured for the EoF, namely strong superadditivity: EF (ρ) ≥ EF (ρI ) + EF (ρI I ),

(3)

where ρ is a general state over a duplicated Hilbert space and ρI and ρI I are its reductions to the different copies of that space. In this paper we show that strong superadditivity of EoF is equivalent to subadditivity of a much simpler quantity, the so-called conjugate of the entanglement functional E(ρ) = S(Tr A ρ). We then exploit this equivalence to show that strong superadditivity would follow as a consequence of multiplicativity of the maximal output purity, measured by a Schatten norm, for quantum filtering operations (this quantity will also be defined in due course). The main results are stated in Theorems 1 and 2. To arrive at these results, we have made use of a basic formalism from convex analysis [9, 10] and we hope that our results will stimulate usage of this elegant theory in other areas of quantum information. 2. Notations Let us first introduce the basic notations. Let S(ρ) denote the von Neumann entropy S(ρ) = − Tr ρ ln ρ. For state vectors we will typically use lowercase Greek letters, ψ, φ. For mixed states we will use lowercase Greek letters ρ, σ , τ . The identity matrix will be denoted by I. We shall denote the set of bounded Hermitian operators over the Hilbert space H by B s (H), the set of non-negative elements in B s (H) by B + (H), and the (convex) set of all states (trace 1 positive operators) over H by S(H). We will frequently slim down expressions like maxρ∈S {. . . } to maxρ {. . . }. When the domain of, say, a maximisation over states is missing it will be implicitly understood that the whole of state space S(H) is meant. The abovementioned naming convention for states and vectors will be adhered to exactly for that reason. Any state ρ can be realised by an ensemble of pure states. An ensemble is specified by a set of pairs {(pi , ψi )}N of N state vectors ψi and associated statistii=1 , consisting cal weights pi (with pi ≥ 0 and i pi = 1). Here, N is called the cardinality of the ensemble. The entanglement of formation (EoF) of a bipartite state ρ (i.e., a state over the bi-partite Hilbert space HA ⊗ HB ), is defined by [11] EF (ρ) = min pi S(Tr A |ψi ψi |) : pi |ψi ψi | = ρ . (4) {(pi ,ψi )}

i

i

On Strong Superadditivity of the Entanglement of Formation

445

3. Convex Closures Admittedly, the definition of the EoF just mentioned is not very handy to work with. Not in the least because for generic states ρ the cardinality N of the optimal realising ensemble must be larger than R 1.5 /4, where R is the rank of ρ [12]. This is one of the reasons why no really efficient numerical algorithms have been found yet to calculate the EoF [13]. Furthermore, the mere fact that the minimisation involves ensembles at all makes a theoretical study of the EoF rather difficult. One of the first attempts at proving additivity of EoF relied on the investigation of these optimal ensembles [3]. The results in the present work depend on the following simple observation. The import of the definition (4) of the EoF, as has been shown in [14, 4], is that the EoF is the convex closure (or convex roof, as it is called in [14]) of the pure state entanglement function E(|ψ ψ|) = S(Tr A |ψ ψ|), restricted to the set of pure states. This means that the epigraph of the EoF (being the set of points (ρ, x) in S(H)×R with x ≥ EF (ρ)) on the complete state space S(H) is the convex closure of the epigraph of the function E defined over S(H), where E(ρ), ρ pure E (ρ) = +∞, ρ not pure. This follows immediately from Cor. 17.1.5 of [9] and the definition (4). Note now that E is concave over its domain. There is, therefore, no need to explicitly exclude mixed states 1 , so EF is the convex closure of E as well. In the following paragraphs we will apply the standard convex analytical formalism for convex closures to general bounded functions f whose domain is the convex set of states S(H). We will denote the convex closure of f by fˆ. One definition of the convex closure of f is pi f (ρi ) : p i ρi = ρ , (5) fˆ(ρ) = min {(pi ,ρi )}

i

i

agreeing, indeed, with the definition of the EoF. A less cumbersome formulation of the convex closure is based on Cor. 12.1.1 of [9], which states that the convex closure of a function f is the pointwise supremum of the collection of all affine functions on S(H) majorised by f . So, for all states ρ: fˆ(ρ) =

sup {Tr ρX : (∀ψ ∈ H : ψ|X|ψ ≤ f (|ψ ψ|))}.

X∈Bs (H)

(6)

The mentioned affine functions are here the functions ψ|X|ψ, where X ranges over B s (H) 2 . This dual formulation is then further simplified by defining an intermediate function f ∗ : f ∗ (X) = max Tr[ρX] − f (ρ), ρ∈S (H)

(7)

1 Of course, E(ρ) has no real physical significance for mixed states. Moreover, we must be careful to distinguish between the two possible definitions E(ρ) = S(Tr A ρ) and E (ρ) = S(Tr B ρ). On pure states, these two definitions yield the same value, but for mixed states this is not so anymore. 2 For our purposes the corollaries from [9] have to be restated with Rn replaced by S (H). This causes no problems if one extends the domain of f to the affine space of all trace 1 Hermitian operators and defines f (x) = +∞ for negative x.

446


the so-called conjugate function of f [9]. If f is continuous, then the conjugate function is just the Legendre transform of f . The conjugate function is convex in X, because it is a pointwise maximum of functions that are affine in X. The importance of the conjugate function is that the conjugate of the conjugate of f is the convex closure of f , fˆ = f ∗∗ , and the conjugate of the convex closure of f is the conjugate of f , fˆ∗ = f ∗ ([9], the remark just before its Theorem 12.2). Thus fˆ(ρ) =

max Tr[ρX] − f ∗ (X),

X∈Bs (H)

f ∗ (X) = max Tr[ρX] − fˆ(ρ). ρ∈S (H)

(8) (9)

In other words, the conjugate and convex closure determine each other completely. Because f ∗ and fˆ are convex functions, the optimal X and ρ in (8) and (9), respectively, both form convex sets (possibly singleton sets). Furthermore, there is a correspondence between the optimal X in (8) and the optimal ρ in (9). Proposition 1. (a) If X is an optimal X for τ in (8), then (i) τ is an optimal ρ for X in (9), and (ii) all members of an optimal realising ensemble for τ are optimal ρ for X in (7). (b) If ρ is an optimal ρ for Y in (9), then Y is an optimal X for ρ in (8). Proof. Statement (a)(i) is proven by inserting (9) in (8) and exploiting the premise that X is an optimal X. This gives fˆ(τ ) = Tr τ X − maxρ (Tr ρX − fˆ(ρ)). Putting ρ = τ yields an upper bound on the right-hand side because τ is not necessarily optimal in the maximisation. However, the value of the bound we obtain is fˆ(τ ), which happens to be equal to the left-hand side. Thus this choice really is an optimal one, proving optimality of τ for X in (9). Statement (b) is proven similarly, by inserting (8) in (9). Considering statement (a)(ii), let {(pi , τi )} be an optimal ensemble for τ (with

∗

pi > 0). Thus fˆ(τ ) = i p fˆ(τ ) = Tr τ X i f (τi ). By assumption, − f (X ). Inserting

(7) and expanding unity as i pi yields i pi f (τi ) = Tr τ X − i pi maxρ (Tr ρX − f (ρ)). If we now replace ρ by τi in the ith summation term we get an upper bound on

the right-hand side, with equality only if all the τi are optimal ρ for X . The bound is easily seen to be i pi f (τi ), which is actually equal to the left-hand side. We find again that the bound is sharp, and optimality of the τi follows.

4. Additivity These basic results will now prove to be a powerful tool for studying the additivity issue of the EoF. Let HI and HI I be two copies of the Hilbert space HA ⊗ HB , and define H = HI HI I . We will reserve the symbol ⊗ for tensor products with respect to the A-B subdivision, and the symbol for tensor products regarding the I-II subdivision. Strong superadditivity of the EoF [4] is the inequality EF (ρ) ≥ EF (ρI ) + EF (ρI I ), for ρ a state on H, and ρI and ρI I its reductions to HI and HI I , respectively.

(10)


447

The following Lemma is simple but crucial: Lemma 1. For any bounded function f defined on S(H), strong superadditivity of fˆ, fˆ(ρ) ≥ fˆ(ρI ) + fˆ(ρI I ),

(11)

is equivalent to subadditivity of the conjugate function f ∗ with respect to the Kronecker sum: f ∗ (X1 I + I X2 ) ≤ f ∗ (X1 ) + f ∗ (X2 ).

(12)

Proof. Set Z = X1 I + I X2 . Then, using (8) and assuming the validity of (12) yields fˆ(ρ) = sup Tr[ρX] − f ∗ (X) X

≥ sup Tr[ρZ] − f ∗ (Z) X1 ,X2

≥ sup Tr[ρI X1 + ρI I X2 ] − f ∗ (X1 ) − f ∗ (X2 ) X1 ,X2

= fˆ(ρI ) + fˆ(ρI I ), which is (11). The converse follows from (9). Assuming the validity of (11) yields f ∗ (Z) = max Tr[ρZ] − fˆ(ρ) ρ

≤ max Tr[ρI X1 + ρI I X2 ] − fˆ(ρI ) − fˆ(ρI I ) ρ

= max Tr[ρ1 X1 + ρ2 X2 ] − fˆ(ρ1 ) − fˆ(ρ2 ) ρ1 ,ρ2

= f ∗ (X) + f ∗ (Y ), which is (12).

The appearance of the Kronecker sum in Lemma 1 suggests that the consideration of the function f ∗ ◦ log is a more natural setting for studying additivity. Defining g := f ∗ ◦ log and setting Xi = log Mi , (12) becomes g(M1 M2 ) ≤ g(M1 ) + g(M2 ), for M1 , M2 ∈ B + (H). Restating (8) and (9) in terms of M, we have g(M) = max Tr[ρ log(M)] − f (ρ),

(13)

fˆ(ρ) =

(14)

ρ∈S (H)

max

M∈B+ (H)

Tr[ρ log(M)] − g(M).

Strictly speaking, these quantities are defined only for positive M. However, when M is singular, we can still make sense out of it by the usual extension Tr[ρ log(M)] = −∞ for any ρ that is not completely supported on the range of M.

448


We can now restate Lemma 1 in the form of a theorem, which is our first main result: Theorem 1. For any function f defined on S(H), and with g defined on B + (H) by (13), strong superadditivity of the convex closure fˆ, fˆ(ρ) ≥ fˆ(ρI ) + fˆ(ρI I ),

(15)

is equivalent to subadditivity of g, g(M1 M2 ) ≤ g(M1 ) + g(M2 ).

(16)

Note that the expression Tr[ρ log(M)] − g(M) is invariant under multiplication of M by a positive scalar. Hence, one could impose the restriction Tr M = 1, i.e. that M should be a state, or alternatively M ≤ I, which is what we shall do. An immediate corollary of this theorem is the equivalence of the strong superadditivity of the EoF with the subadditivity of g = E ∗ ◦ log, where E ∗ is the conjugate of the entanglement functional E(ρ) = S(Tr A ρ). We have chosen to present Theorem 1 in the more general way because it obviates the rather remarkable independence of the theorem on any property of the function f at all. Specifically, while for the sake of defining the EoF it is necessary to split up the Hilbert space into two parties A and B, this is something the theorem is completely oblivious of. The only interesting feature of E we can exploit at this level is its concavity. Concavity allows to simplify the conjugation expression by replacing the maximisation over all mixed states by a maximisation over pure states. Indeed, the argument of the maximisation in g(M) = max Tr[ρ log(M)] − E(ρ) ρ∈S (H)

is a convex function of ρ, and it is well-known [9] that a convex function achieves its maximum over a convex set always in an extreme point of that set, in this case in a pure state. Thus: g(M) = max ψ| log(M)|ψ − E(). ψ∈H

Theorem 1 reduces the additivity problem for the convex closure, originally defined as a minimisation over ensembles, to an equivalent problem for the conjugate function, defined as a maximisation over pure states. If counterexamples are found for (16), this automatically disproves strong superadditivity (15), so this simplification does not come at the cost of reduced power. Specifically, by “inverting” the proof of Lemma 1 (or Theorem 1) and employing Proposition 1, we easily get the following: Proposition 2. If ρ violates strong superadditivity of fˆ, (15), M1 is optimal for ρI in (14), and M2 is optimal for ρI I , then M1 M2 violates subadditivity of g (16). If M1 M2 violates (16) and ρ is optimal for M1 M2 in (13), then ρ violates (15). 5. Maximal Output Purity Exploiting Theorem 1, we will now show that strong superadditivity of EF would follow as a consequence of another additivity conjecture, concerning quantum channel capacities. Recollect that, since E is concave, the optimal ρ in (7) will be an extreme point of the feasible set, i.e. a pure state, so: E ∗ (X) = max ψ|X|ψ − E(|ψ ψ|). ψ∈H

(17)


449

From the additivity of E over pure states it easily follows that the corresponding function g = E ∗ ◦ log is superadditive, hence subadditivity of g implies its additivity. 5.1. Step 1. The maximisation in g can be rewritten in terms of a maximal eigenvalue λmax : Lemma 2. For any M ∈ B + (H), g(M) := max( ψ| log M|ψ − S(Tr A |ψ ψ|)) ψ

=

max λmax (log M + log(IA ⊗ τ )).

τ ∈S (HB )

(18)

Note that we will henceforth consider log M + log(IA ⊗ τ ) as an operator restricted to the range intersection ran(M) ∩ ran(I ⊗ τ ). Proof. max λmax (log M + log IA ⊗ τ ) τ

= max max Tr[|ψ ψ|(log M + log IA ⊗ τ )] τ

ψ

(19)

= max max Tr[|ψ ψ| log M] + Tr[Tr A (|ψ ψ|) log τ ] τ

ψ

= max Tr[|ψ ψ| log M] − S(Tr A |ψ ψ|). ψ

(20)

In step (19) we have used the Rayleigh-Ritz representation of a maximal eigenvalue, and in step (20) we have used the fact that relative entropy is non-negative and attains the value zero when (and only when) its arguments are equal. Specifically: 0 = min S(ρ||τ ) τ

= min −S(ρ) − Tr[ρ log τ ] τ

= −S(ρ) − max Tr[ρ log τ ]. τ

5.2. Step 2. Using the Lie-Trotter formula, the logarithm can be replaced by a limit of a power function. Lemma 3. 1/p

exp g(M) = lim hp (M), p→0

where hp (M) := max ||M p/2 (I ⊗ τ )p M p/2 || τ

and ||.|| denotes the operator norm.

450


Proof. Taking the exponential of both sides of (18) and noting exp λmax (M) = λmax exp(M), we get exp g(M) = max || exp(log M + log(I ⊗ τ ))||. τ

To make sense of this formula, we extend exp(log M + log(I ⊗ τ )) as 0 on the complement of ran(M) ∩ ran(I ⊗ τ ), as in [15]. The Lie-Trotter formula has a continuous version (see the remark after Lemma 3.3 in [15]) 1/p . exp(A + B) = lim exp(pA/2) exp(pB) exp(pA/2) p→0

In particular, this gives us

1/p exp(log M + log(I ⊗ τ )) = lim M p/2 (I ⊗ τ )p M p/2 . p→0

(21)

Define the shorthand functions f (τ ) := || exp(log M + log(I ⊗ τ ))||, fp (τ ) := ||(M p/2 (I ⊗ τ )p M p/2 )1/p || over S(H). By (21) and the triangle inequality for norms, fp converges pointwise to f . The functions fp are clearly continuous for p > 0. By Lemma 4.1 of [15], f is continuous too. From [16] (p. 118) we have that fp decreases monotonously to f as p decreases to 0. The set S(H), over which f and fp are defined, is compact. Hence, all the prerequisites are fulfilled to apply Dini’s theorem [17], and we get that the convergence of fp to f is uniform over S(H). Finally, uniform convergence is equivalent with convergence in the sup-norm. By the triangle inequality for norms, that in turn implies that the sup-norm of fp con1/p verges to the sup-norm of f . Therefore, hp (M) = maxτ fp (τ ) = ||fp ||S converges to ||f ||S = maxτ f (τ ) = exp g(M).

Additivity of g would thus follow as a consequence of multiplicativity of hp , hp (M1 M2 ) = hp (M1 )hp (M2 ), for p ↓ 0. Following [18], we say that a property holds for p ↓ a if it holds for an arbitrarily small, but finite, interval p ∈ (a, a + ], > 0. 5.3. Step 3. The quantity hp (M) is formally equal to the maximal output purity [18–20] of quantum filtering operations. Indeed, hp (M) = max Tr[|φ φ|(M p/2 (I ⊗ τ )p M p/2 )] τ,φ

= max Tr[τ p Tr A [M p/2 |φ φ|M p/2 ]] τ,φ

= max || Tr A [M p/2 |φ φ|M p/2 ]||q φ

= νq ( ), where q = 1/(1 − p) and ||.||q denotes the Schatten q-norm [21], and νq ( ) is the maximal output purity measured by the Schatten q-norm of the (non-trace preserving) operation

: ρ → (ρ) = Tr A [M p/2 ρM p/2 ]. If this operation would be trace preserving, we would call it a channel.

(22)


451

5.4. Step 4. We now claim that there is no advantage in restricting attention to operations of the form (22). It is of course true that the class of operations (22) is rather specific. They admit a Kraus representation such that the block column matrix (Ai )i obtained by stacking the Kraus element matrices Ai vertically, equals M p/2 , which is a positive matrix. Necessary conditions are that i A†i Ai = M p (which is ≤ I) and the input dimension of the operation should equal the output dimension times the number of elements. However, as regards the maximal output purity question, these structural peculiarities offer no additional mileage. To see this, consider the specific case that M is a partial isometry M = U U † , where = |1 1| ⊗ IB and U is any unitary, then νq ( ) = max || Tr A [U p/2 U † |φ φ|U p/2 U † ]||q φ∈H

= max || Tr A [U p/2 |φ φ | p/2 U † ]||q φ ∈H

= max || Tr A [U (|1 1| ⊗ |φ

φ

|)U † ]||q , φ

∈HB

which is the generic case for operations from HA to HA . Thus, the case for the “special operations” H → HA contains the generic HA → HA case and is therefore not easier to prove.

5.5. Step 5. The exponent p of M, occurring in , is coupled to q, occurring in νq , via the relation q = 1/(1 − p). To cap off our argument, we “decouple” p and q by replacing M p/2 with a general matrix 0 ≤ X ≤ I, strengthening our multiplicativity conjecture ever so slightly. This is allowed only when we first fix the interval of the values p for which multiplicativity has to hold, i.e. these values should not depend on

. Noting finally that p ↓ 0 corresponds to q ↓ 1, we get our second main result: Theorem 2. If there exists a real number q0 > 1 such that νq ( ) is multiplicative for all 1 ≤ q ≤ q0 and for any filtering operation , then the entanglement of formation is strongly superadditive. Multiplicativity of νq had been conjectured in [18] for trace preserving channels. It has been proven for entanglement breaking channels [20], unital qubit maps [22] and depolarising channels [23], but, unfortunately, was refuted in [19] for q > 4.79. Nevertheless, the conjecture might still be true for q ↓ 1. Theorem 2 has to be compared to the main technical result in [7], which states that additivity of the Holevo capacity for given channels implies additivity of the EoF for certain states. In a sense, our Theorem 2 is stronger because we get the stronger outcome of strong superadditivity. On the other hand, this comes at the price of having to consider non-trace-preserving operations. After the appearance of the first draft of this manuscript, Shor proved [24] the equivalence of four additivity conjectures: strong superadditivity of the EoF, ordinary additivity of the EoF, additivity of the maximal output purity νS of a channel as measured by the entropy, and additivity of the classical (Holevo) capacity of a channel. As multiplicativity of νq ( ) for q ↓ 1 implies additivity of νS ( ) [18], Shor’s third equivalence provides an alternative proof for our result Theorem 2.

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6. Conclusion In conclusion, we have shown how a simple convex analytical argument leads to a simpler formulation of the entanglement of formation and an especially simple equivalent condition for strong superadditivity of the EoF. Based on this we have found the second result that strong superadditivity of the EoF would follow as a consequence of the multiplicativity of the maximum output purity νq of quantum filtering operations, for q ↓ 1. Acknowledgement. We gratefully acknowledge comments by M.B. Plenio, J. Eisert, M.B. Ruskai and Ch. King. SLB currently holds a Wolfson-Royal Society Research Merit Award.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

Hayden, P.M., Horodecki, M., Terhal, B.M.: J. Phys. A 34(35), 6891–6898 (2001) Wootters, W. : Phys. Rev. Lett. 80, 2245 (1998) Benatti, F., Narnhofer, H.: Phys. Rev. A 63, 042306 (2001) Vollbrecht, KG.H., Werner, R.F.: Phys. Rev. A 64, 062307 (2001) Vidal, G., Dür, W., Cirac, J.I.: Phys. Rev. Lett. 89, 027901 (2002) Horodecki, M., Sen De, A. Sen, U.: quant-ph/0207031, 2002 Matsumoto, K., Shimono, T., Winter, A.: quant-ph/0206148, 2002 Heng Fan: quant-ph/0210169, 2002 Rockafellar, R.T.: Convex Analysis. Princeton, NJ: Princeton University Press, 1970 Boyd, S., Vandenberghe, L.: Convex Optimization. Available online at http://www.stanford.edu/∼boyd/cvxbook.html, 2002 Bennett, C.H., DiVincenzo, D.P., Smolin, J., Wootters, W.K.: Phys. Rev. A 54, 3824 (1996) Lockhart, R.B.: J. Math. Phys. 41(10), 6766–6771 (2000) Audenaert, K.M.R., Verstraete, F., DeMoor, B.: Phys. Rev. A 64, 052304 (2001) Uhlmann, A.: quant-ph/9704017, 1997 Hiai, F., Petz, D.: Lin. Alg. Appl. 181, 153–185 (1993) Ando, T., Hiai, F.: Lin. Alg. Appl. 197, 198, 113–131 (1994) Apostol, T.M.: Mathematical Analysis. Reading MA: Addison-Wesley, 1974 Amosov, G.G., Holevo, A.S., Werner, R.F.: Problems in Information Transmission 36, 25–34 (2000) and math-ph/0003002 (2000) Werner, R.F., Holevo, A.S.: J. Math. Phys. 43(9), 4353–4357 (2002) King, C.: quant-ph/0212057, 2002 Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge: Cambridge University Press, 1991 King, C.: J. Math. Phys. 43(9), 4334–4340 (2002) King, C.: quant-ph/0204172 (2002) Shor, P.W.: quant-ph/0305035 (2003)

Communicated by M.B. Ruskai


Communications in


Equivalence of Additivity Questions in Quantum Information Theory Peter W. Shor∗ AT&T Labs Research, Florham Park, NJ 07922, USA Received: 8 May 2003 / Accepted: 17 July 2003 Published online: 18 November 2003 – © Springer-Verlag 2003

Abstract: We reduce the number of open additivity problems in quantum information theory by showing that four of them are equivalent. Namely, we show that the conjectures of additivity of the minimum output entropy of a quantum channel, additivity of the Holevo expression for the classical capacity of a quantum channel, additivity of the entanglement of formation, and strong superadditivity of the entanglement of formation, are either all true or all false. 1. Introduction The study of quantum information theory has led to a number of seemingly related open questions that center around whether certain quantities are additive. We show that four of these questions are equivalent. In particular, we show that the four conjectures of i. ii. iii. iv.

additivity of the minimum entropy output of a quantum channel, additivity of the Holevo capacity of a quantum channel, additivity of the entanglement of formation, strong superadditivity of the entanglement of formation,

are either all true or all false. Two of the basic ingredients in our proofs are already known. The first is an observation of Matsumoto, Shimono and Winter [12] that the Stinespring dilation theorem relates a constrained version of the Holevo capacity formula to the entanglement of formation. The second is the realization that the entanglement of formation (or the constrained Holevo capacity) is a linear programming problem, and so there is also a dual linear formulation. This formulation was first presented by Audenaert and Braunstein [1], who expressed it in the language of convexity rather than that of linear programming. We noted this independently [16]. These two ingredients are explained in Sect. 3 and 5. ∗ Current address: Dept. of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

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The rest of this paper is organized as follows. Sect. 2 gives some background in quantum information theory, describes the additivity questions we consider, and gives brief histories of them. Sect. 3 and 5 explain the two ingredients we describe above, and are positioned immediately before the first sections in which they are used. To show that the conditions (i) to (iv) are equivalent, in Sect. 4 we prove that (ii) → (iii): additivity of the Holevo capacity implies additivity of entanglement of formation. In Sect. 6 we prove (iii) → (iv): additivity of entanglement of formation implies strong superadditivity of entanglement of formation. This implication was independently discovered by Pomeransky [13]. In Sect. 7 we prove that (i) → (iii): additivity of minimum entropy output implies additivity of entanglement of formation. In Sect. 8, we give simple proofs showing that (iv) → (i), (iv) → (ii), and (iv) → (iii). The first implication is the only one that was not in the literature, and we assume this is mainly because nobody had tried to prove it. The second of these implications was already known, but for completeness we give a proof. The third of these implications is trivial.1 In Sect. 9 we give proofs that (ii) → (i) and (iii) → (i): either additivity of the Holevo capacity or of the entanglement of formation implies additivity of the minimum entropy output. These implications complete the proof of equivalence. Strictly speaking, the only implications we need for the proof of equivalence are those in Sect. 6–9. We include the proof in Sect. 4 because it uses one of the techniques used later for Sect. 7 without introducing the extra complexity of the dual linear programming formulation. Finally, in Sect. 10 we comment on the implications of the results in our paper and give some open problems. 2. Background and Results One of the important intellectual breakthroughs of the 20th century was the discovery and development of information theory. A cornerstone of this field is Shannon’s proof that a communication channel has a well-defined information carrying capacity and his formula for calculating it. For communication channels that intrinsically incorporate quantum effects, this classical theory is no longer valid. The search for the proof of the analogous quantum formulae is a subarea of quantum information theory that has recently received much study. In the generalization of Shannon theory to the quantum realm, the definition of a stochastic communication channel generalizes to a completely positive trace-preserving linear map (CPT map). We call such a map a quantum channel. In this paper, we consider only finite-dimensional CPT maps; these take din × din Hermitian matrices to dout × dout Hermitian matrices. In particular, these maps take density matrices (trace 1 positive semidefinite matrices) to density matrices. Note that the input dimension can be different from the output dimension, and that these dimensions are both finite. Infinite dimensional quantum channels (CPT maps) are both important and interesting, but dealing with them also introduces extra complications that are beyond the scope of this paper. There are several characterizations of CPT maps. We need the characterization given by the Stinespring dilation theorem, which says that every CPT map can be described by an unitary embedding followed by a partial trace. In particular, given a finite-dimensional CPT map N, we can express it as N (ρ) = Tr B U (ρ), 1 In fact, property (iv), strong superadditivity of E , seems to be in some sense the “strongest” of these F equivalent statements, as it is fairly easy to show that strong superadditivity of entanglement of formation implies the other three additivity results whereas the reverse directions appear to require substantial work. Similarly, property (i) appears to be the “weakest” of these statements.

Equivalence of Additivity Questions in Quantum Information Theory

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where U (ρ) is a unitary embedding, i.e., there is some ancillary space HB such that U takes Hin to Hout ⊗ HB by U (ρ) = VρV † and V is a unitary matrix mapping Hin to range(V ) ⊆ Hout ⊗ HB . We also need the operator sum characterization of CPT maps. This characterization says that any finitedimensional CPT map N can be represented as † N (ρ) = Ak ρAk , k

where the Ak are complex matrices satisfying † Ak Ak = I. The Holevo information2 χ is a quantity which is associated with a probabilistic ensemble of quantum states (density matrices). If density matrix ρi occurs in the ensemble with probability qi , the Holevo information χ of the ensemble is χ =H qi ρi − qi H (ρi ), i

i

where H is the von Neumann entropy H (ρ) = −Tr ρ log ρ. This quantity was introduced in [6, 11, 8] as a bound for the amount of information extractable by measurements from this ensemble of quantum states. The first published proof of this bound was given by Holevo [8]. It was much later shown that maximizing the Holevo capacity over all probabilistic ensembles of a set of quantum states gives the information transmission capacity of this set of quantum states; more specifically, this is the amount of classical information which can be transmitted asymptotically per quantum state by using codewords that are tensor products of these quantum states, as the length of these codewords goes to infinity [9, 15]. Optimizing χ over ensembles composed of states that are potential outputs of a quantum channel gives the quantum capacity of this quantum channel over a restricted set of protocols, namely those protocols which are not allowed to send inputs entangled between different channel uses. If the channel is N, we call this quantity χN ; it is defined as χN = max H N pi |vi vi | − pi H (N (|vi vi |)), (1) {pi ,| vi }

i

i

where the maximization is over ensembles {pi , | vi }, where the input space of the channel N . The regularized Holevo capacity is lim

n→∞

i

pi = 1 and | vi ∈ Hin ,

1 χN ⊗n ; n

this gives the capacity of a quantum channel to transmit classical information when inputs entangled between different channel uses are allowed. The question of whether 2

This has also been called the Holevo bound and the Holevo χ-quantity.

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the quantum capacity is given by the single-symbol Holevo capacity χN is the question of whether the capacity χN is additive; that is, whether χN1 ⊗N2 = χN1 + χN2 . The ≥ relation is easy; the open question is the ≤ relation. The question of additivity of the minimum entropy output of a quantum channel was originally considered independently by several people, including the author, and appears to have been first considered in print in [10]. It was originally posed as a possible first step to proving additivity of the Holevo capacity χN . The question is whether min H (N1 ⊗ N2 (|φφ|)) = min H (N1 (|φφ|)) + min H (N2 (|φφ|)), | φ

| φ

| φ

where the minimization ranges over states | φ in the input space of the channel. Note that by the concavity of the von Neumann entropy, if we minimize over mixed states ρ – i.e., minρ H (N (ρ)) – there will always be a rank one ρ = |φφ| achieving the minimum. The statements (iii) and (iv) in our equivalence theorem both deal with entanglement. This is one of the stranger phenomena of quantum mechanics. Entanglement occurs when two (or more) quantum systems are non-classically correlated. The canonical example of this phenomenon is an EPR pair. This is the state of two quantum systems (called qubits, as they are each two-dimensional): 1 √ | 01 − | 10 . 2 Measurements on each of these two qubits separately can exhibit correlations which cannot be modeled by two separated classical systems [2]. A topic in quantum information theory that has recently attracted much study is that of quantifying entanglement. The entanglement of a bipartite pure state is easy to define and compute; this is the entropy of the partial trace over one of the two parts Epure (|vv|) = H (Tr B |vv|). Asymptotically, two parties sharing n copies of a bipartite pure state |vv| can use local quantum operations and classical communication (called LOCC operations) to produce nEpure (|vv|) − o(n) nearly perfect EPR pairs, and can similarly form n nearly perfect copies of |vv| from nEpure (|vv|) + o(n) EPR pairs [4]. This implies that for a pure state |vv|, the entropy of the partial trace is the natural quantitative measure of the amount of entanglement contained in |vv|. For mixed states (density matrices of rank > 1), things become more complicated. The amount of pure state entanglement asymptotically extractable from a state using LOCC operations (the distillable entanglement) is now no longer necessarily equal to the amount of pure state entanglement asymptotically required to create a state using LOCC operations (the entanglement cost) [17]. In general, the entanglement cost must be at least the distillable entanglement, as LOCC operations cannot increase the amount of entanglement. The entanglement of formation was introduced in [5]. Suppose we have a bipartite state σ on a Hilbert space HA ⊗ HB . The entanglement of formation is EF (σ ) = min pi H (Tr B |vi vi |), (2) {pi , | vi }

i


457

where the minimization is over all ensembles such that i pi |vi vi | = σ with probabilities pi satisfying i pi = 1. The entanglement of formation must be at least the entanglement cost, as the decomposition of the state σ yielding EF (σ ) can be used to create a prescription for asymptotically constructing σ ⊗n from nEF (σ ) + o(n) EPR pairs. The regularized entanglement of formation lim

n→∞

1 EF (σ ⊗n ) n

has been proven to give the entanglement cost of a quantum state [7]. As in the case of channel capacity, a proof of additivity, i.e., that EF (σ1 ⊗ σ2 ) = EF (σ1 ) + EF (σ2 ), would imply that regularization is not necessary. The question of strong superadditivity of entanglement of formation has been previously considered in [3, 17, 12, 1]. This conjecture says that for all states σ over a quadripartite system HA1 ⊗ HA2 ⊗ HB1 ⊗ HB2 , we have EF (σ ) ≥ EF (Tr 2 σ ) + EF (Tr 1 σ ), where the entanglement of formation EF is taken over the bipartite A-B division, as in (2). This question was originally considered in relation to the question of additivity of EF . The strong superadditivity of entanglement of formation is known to imply both the additivity of entanglement of formation (trivially) and the additivity of Holevo capacity of a channel [12]. A proof similar to ours that additivity of EF implies strong superadditivity of EF was discovered independently; it appears in [13]. We can now state the main result of our paper. Theorem 1. The following are equivalent. i. The additivity of the minimum entropy output of a quantum channel. Suppose we have two quantum channels (CPT maps) N1 (taking C d1,in ×d1,in to C d1,out ×d1,out ) and N2 (taking C d2,in ×d2,in to C d2,out ×d2,out ). Then min H ((N1 ⊗ N2 )(|φφ|)) = min H (N1 (|φφ|)) + min H (N2 (|φφ|)), | φ

| φ

| φ

where H is the von Neumann entropy and the minimization is taken over all vectors | φ in the input space of the channels. ii. The additivity of the Holevo capacity of a quantum channel. Assume we have two quantum channels N1 and N2 , as in (i). Then χN1 ⊗N2 = χN1 + χN2 , where χ is defined as in Eq. (1). iii. Additivity of the entanglement of formation. Suppose we have two quantum states σ1 ∈ HA1 ⊗ HB1 and σ2 ∈ HA2 ⊗ HB2 . Then EF (σ1 ⊗ σ2 ) = EF (σ1 ) + EF (σ2 ), where EF is defined as in Eq. (2). In particular, the entanglement of formation is calculated over the bipartite A–B partition.

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iv. The strong superadditivity of the entanglement of formation. Suppose we have a density matrix σ over a quadripartite system HA1 ⊗ HA2 ⊗ HB1 ⊗ HB2 . Then EF (σ ) ≥ EF (Tr 2 σ ) + EF (Tr 1 σ ), where the entanglement of formation is calculated over the bipartite A–B partition. Here, the operator Tr 1 traces out the space HA1 ⊗HB1 , and Tr 2 traces out the space HA2 ⊗ HB2 . 3. The Correspondence of Matsumoto, Shimono and Winter Recall the definition of the Holevo capacity for a channel N: χN = max H (N( pi H (N (|φi φi |)). pi |φi φi |)) − {pi , | φi }

i

i

Recall also the definition of entanglement of formation. For a bipartite state σ on HA ⊗ HB , the entanglement of formation is EF (σ ) = min pi H (Tr B |vi vi |). {pi , | vi } pi |vi vi |=σ

i

Let us define a constrained version of the Holevo capacity, which is just the Holevo capacity over ensembles whose average input is ρ, max H (N( pi |φi φi |)) − pi H (N (|φi φi |)). (3) χN (ρ) = {pi , | φi } i pi |φi φi |=ρ

i

i

The paper of Matsumoto, Shimono and Winter [12] gives a connection between this constrained version of the Holevo capacity and the entanglement of formation, which we now explain. The Stinespring dilation theorem says that any quantum channel can be realized as a unitary transformation followed by a partial trace. Suppose we have a channel N taking Hin to HA . We can find a unitary embedding U (ρ) = VρV † that takes Hin to HA ⊗ HB such that N (µ) = Tr B U (µ) for all density matrices µ ∈ Hin . Now, U maps an ensemble of input states {pi , | φi } with ρ = i pi |φi φi | to an ensemble of states {pi , | vi = V | φi } on the bipartite system HA ⊗ HB such that i pi |vi vi | = σ = U (ρ). Conversely, if we are given a bipartite state σ ∈ HA ⊗ HB , we can find an input space Hin with dim Hin = rank σ , a density matrix ρ ∈ Hin , and a unitary embedding U : Hin → Hout such that U (ρ) = σ . We can then define N by N (µ) = Tr B U (µ), establishing the same relation between N, U , ρ and σ . Note that since we chose dim Hin = rank σ = rank ρ, ρ has full rank in Hin . Since N (|φi φi |) = Tr B |vi vi |, we have χN (ρ) = H (N(ρ)) − EF (σ ).


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Now, suppose EF (σ ) is additive. I claim that χN (ρ) is as well, and vice versa. Let us take N1 (ρ) = Tr B U1 (ρ) and N2 (ρ) = Tr B U2 (ρ). If U1 (ρ1 ) = σ1 and U2 (ρ2 ) = σ2 , then we have χN1 ⊗N2 (ρ1 ⊗ ρ2 ) = H (N1 ⊗ N2 (ρ1 ⊗ ρ2 )) − EF (σ1 ⊗ σ2 ) = H (N1 (ρ1 )) + H (N2 (ρ2 )) − EF (σ1 ⊗ σ2 ). The first term on the right-hand side is additive, so the entanglement of formation EF is additive if and only if the constrained capacity χN (ρ) is. 4. Additivity of χ Implies Additivity of EF Recall the definition of the Holevo capacity for a channel N: pi |φi φi | − pi H (N (|φi φi |)), χN = max H N {pi , | φi }

i

i

where the maximization is over ensembles {pi , | φi } with i pi = 1. Recall also our definition of a constrained version of the Holevo capacity, which is just the definition of the Holevo capacity with the maximization only over ensembles whose average input is ρ, pi |φi φi | − pi H (N (|φi φi |)). max H N χN (ρ) = {pi , | φi } i pi |φi φi |=ρ

i

i

Let σ be the state whose entanglement of formation we are trying to compute. The MSW correspondence yields a channel N and an input state ρ so that N (ρ) = Tr B σ and χN (ρ) = H (N(ρ)) − EF (σ ). This is very nearly the channel capacity, the only difference being that the ρ above is not necessarily the ρ that maximizes χN . Only one element is missing for the proof that additivity of channel capacity implies additivity of entanglement of formation: namely making sure that the average density matrix for the ensemble giving the optimum channel capacity is equal to a desired matrix ρ0 . This cannot be done directly [14], but we solve the problem indirectly. We now give the intuition for our proof. Suppose we could define a new channel N which, instead of having capacity χN = max χN (ρ), ρ

has capacity χN = max χN (ρ) + Tr ρτ ρ

(4)

for some fixed Hermitian matrix τ . For a proper choice of τ , this will ensure that the maximum of this channel occurs at the desired ρ. Consider two entangled states σ1 and σ2 which we wish to show are additive. We can find the associated channels N1 and N2 ,

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with the capacity maximized when the average input density matrix is ρ1 and ρ2 , respectively. By our hypothesis of additivity of channel capacity, the tensor product channel N1 ⊗ N2 has capacity equal to the sum of the capacities of N1 and N2 . If we can now analyze the capacity of the channel N1 ⊗ N2 carefully, we might be able to show that the entanglement of formation of EF (σ1 ⊗ σ2 ) is indeed the sum of EF (σ1 ) and EF (σ2 ). We do not know how to define such a channel N satisfying (4). What we actually do is find a channel whose capacity is close to (4), or more precisely a sequence of channels approximating (4) in the asymptotic limit. It turns out that this will be adequate to prove the desired theorem. We now give the definition of our new channel N . It takes as its input, the input to the channel N , along with k additional classical bits (formally, this is actually a 2k dimensional Hilbert space on which the first action of the channel is to measure it in the canonical basis). With probability q the channel N sends the first part of its input through the channel N and discards the classical bits; with probability 1 − q the channel N makes a measurement on the first part of the input, and uses the results of this measurement to decide whether or not to send the auxiliary classical bits. When the auxiliary classical bits are not sent, an erasure symbol is sent to the receiver instead. When the auxiliary classical bits are sent, they are labeled, so the receiver knows whether he is receiving the output of the original channel or the auxiliary bits. What is the capacity of this new channel N ? Let E be the element of the POVM measurement in the case that we send the auxiliary bits (so I − E is the element of the POVM in the case that we do not send these bits). Now, we claim that for some set of vectors | vi and some associated set of probabilities pi , the optimum signal states of this new channel N will be |vi vi | ⊗ |bb| with associated probabilities pi /2k , where b ranges over all values of the classical bits.3 We now can find bounds on the capacity of N . Let | vi and pi be the optimal signal states and probabilities for χN (ρ). We compute pi |vi vi | − pi H (N (|vi vi |)) χN (ρ) = q H N i

+(1 − q)k

i

i

pi Tr E|vi vi |

+(1 − q) H2 Tr E

i

pi |vi vi | − pi H2 (Tr E|vi vi |) ,

(5)

i

where H2 is the binary entropy function H2 (x) = −x log x − (1 − x) log(1 − x). The first term is the information associated with the channel N , the second is that associated with the auxiliary classical bits, and the third is the information associated with the measurement E. Let ρ = i pi |vi vi | and let σ be the associated entangled state. We can now deduce from (5) that χN (ρ) = qχN (ρ) + (1 − q)kTr Eρ + (1 − q)δ,

(6)

3 This just says that we want to use the classical part of the channel as efficiently as possible. The formal proof is straightforward: First, we show that it doesn’t help to send superpositions of the auxiliary bits, so we can assume that the signal states are indeed of the form |vi vi | ⊗ |bb|. Next, we show that if two signals |vi vi | ⊗ |b1 b1 | and |vi vi | ⊗ |b2 b2 | do not have the same probabilities associated with them, a greater capacity can be achieved by making these probabilities equal.


where δ is defined as δ = H2 (Tr Eρ) −

461

pi H2 (vi | E | vi ).

i

Note that 0 ≤ δ ≤ 1, since δ is positive by the concavity of the entropy function H2 , and is at most 1 since H2 (p) ≤ 1 for 0 ≤ p ≤ 1. Similarly, if we use the optimal states for χN (ρ), we find that χN (ρ) ≥ χN (ρ) + (1 − q)kTr Eρ.

(7)

From Eq. (6) and (7), if we find the ρ0 that maximizes the quantity qχN (ρ) + (1 − q)kTr Eρ;

(8)

we are guaranteed to be within 1 − q of the capacity of N . We next show that we can find a measurement E such that an arbitrary density matrix ρ0 is a maximum of (8). Lemma 2. For any probability 0 < q < 1, any channel N, and any fixed positive matrix ρ0 over the input space of N , there is a sufficiently large k0 such that for k ≥ k0 we can find an E so that the maximum of (8) occurs at ρ0 . (This maximum need not be unique. If χN (ρ) is not strictly concave at ρ0 , then ρ0 will be just one of several points attaining the maximum.) Proof. It follows from the concavity of von Neumann entropy that χN (ρ) is concave in ρ. The intuition is that we must choose E so that the derivative 4 of (8) with respect to ρ at ρ0 is 0. Because we only vary over matrices with Tr ρ = 1, we can add any multiple of I to E and not change the derivative. Suppose that in the neighborhood of ρ0 , χN (ρ) ≤ χN (ρ0 ) + Tr τ (ρ − ρ0 ).

(9)

That such an expression exists follows from the concavity of χN (ρ) and the assumption that ρ0 is not on the boundary of the state space, i.e., has no zero eigenvalues. A full rank ρ0 is guaranteed by the MSW correspondence. To make ρ0 a maximum for Eq. (8), we see from Eq. (9) that we need to find E so that (1 − q) kE = λI − τ q with 0 ≤ E ≤ I . This can be done by choosing k and λ appropriately.

Now, suppose we have two entangled states σ1 and σ2 for which we want to show that the entanglement of formation is additive. We create the channels N1 and N2 as detailed above. By the additivity of channel capacity (which we’re assuming), the signal states (1) (2) of the tensor product channel can be taken to be | vi | b1 ⊗ | vj | b2 for b1 , b2 any (1) (2)

k-bit strings, with probability pi pj /22k . This gives a bound on the channel capacity of at most χN1 ⊗N2 ≤ q (H (N1 (ρ1 )) − EF (σ1 )) + (1 − q)kTr E1 ρ1 +q (H (N2 (ρ2 )) − EF (σ2 )) + (1 − q)kTr E2 ρ2 + 2(1 − q). 4

This is the intuition. This derivative need not actually exist.

(10)

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The 2(1 − q) term at the end comes from the fact that the formula (8) is within 1 − q of the capacity. Now, we want to show that we can find a larger capacity than this if there is a better decomposition of σ1 ⊗ σ2 , i.e., if the entanglement of formation of σ1 ⊗ σ2 is not additive. The central idea here is to let q go to 1; this forces k to simultaneously go to ∞. There is a contribution from entangled states, which goes as q 2 , a contribution from the auxiliary k-bit classical channel, which goes as (1 − q)k, but which is equal in both cases, and a contribution from unentangled states, which goes as q(1 − q). As q goes to 1, the contribution from the entangled states dominates the difference. Suppose there is a set of entangled states which gives a smaller entanglement of formation for σ1 ⊗ σ2 than EF σ1 + EF σ2 . By the MSW correspondence, this gives a set of signal states for the map N1 ⊗ N2 which yields a larger constrained capacity than χN1 (ρ1 ) + χN2 (ρ2 ). We define this set of signal states for N1 ⊗ N2 to be the states |φi φi |, and let the associated probabilities be πi . Now, using the | φi as signal states in N1 ⊗ N2 shows that χN1 ⊗N2 ≥ q 2 H (N1 ⊗ N2 (ρ1 ⊗ ρ2 )) − q 2 EF (σ1 ⊗ σ2 ) + (1 − q)kTr E2 ρ2 . This estimate comes from considering the information transmitted by the signal states |φi φi | in the case (occurring with probability q 2 ) when the channels operate as N1 ⊗N2 , as well as the information transmitted by the k classical bits. We now consider the difference between this lower bound (11) for the capacity of N1 ⊗ N2 and the upper bound (10) we showed for the capacity using tensor product signal states. In this difference, the terms containing (1 − q)k cancel out. The remaining terms give 0 ≥ qEF (σ1 ) + qEF (σ2 ) − q 2 EF (σ1 ⊗ σ2 ) − 2(1 − q) −q(1 − q)H (N1 (ρ1 )) − q(1 − q)H (N2 (ρ2 )). For q sufficiently close to 1, the (1 − q) terms can be made arbitrarily small, and q and q 2 are both arbitrarily close to 1. This difference can thus be made positive if the entanglement of formation is strictly subadditive, contradicting our assumption that the Holevo channel capacity is additive. 5. The Linear Programming Formulation We now give the linear programming dual formulation for the constrained capacity problem. Recall the definition of the constrained Holevo capacity χN (ρ) = max H N pi |φi φi | − pi H (N (|φi φi |)). (11) {pi , | φi } i pi |φi φi |=ρ

i

i

This is a linear program, and as such it has a formulation of a dual problem that also gives the maximum value. This dual problem is crucial to several of our proofs. For this paper, we only deal with channels having finite dimensional input and output spaces. For infinite dimensional channels, the duality theorem fails unless the maxima are replaced by suprema. We have not analyzed the effects this has on the proof of our equivalence theorem, but even if it still holds the proofs will become more complicated. By the duality theorem for linear programming there is another expression for EF (σ1 ). This was observed in [1, 16]. It is χN (ρ) = H (N(ρ)) − f (ρ),

(12)


463

where f is the linear function defined by the maximization max f (ρ) such thatf (|vv|) ≤ H (N(|vv|)) f

for all | v ∈ Hin .

(13)

Here Hin is the input space for N and the maximum is taken over all linear functions f (ρ) = Tr τρ. Equations (12) and (13) can be proved if ρ is full rank by using the duality theorem of linear programming. The duality theorem applies directly if there are only a finite number of possible signal states allowed, showing the equality of the modified version of Eqs. (11) and (12) where the constraints in (13) are limited to a finite number of possible signal states | vi , which are also the only signal states allowed in the capacity calculation (11). To extend from all finite collections of signal states |vi vi | to all |vv|, we need to show that we can find a compact set of linear functions f (ρ) = Tr τρ which suffice to satisfy Eq. (13). We can then use compactness to show that a limit of these functions exists, where in the limit Eqs. (11) and (13) must hold on a countable set of possible signal states | vi dense in the set of unit vectors, thus showing that they hold on the set of all unit vectors | v. The compactness follows from ρ being full rank, and H (N(|vv|)) ≤ log dout for all |vv|, where dout is the dimension of the output space of N. The case where ρ is not full rank can be proved by using the observation that the only values of the function f which are relevant in this case are those in the support of ρ. Equality must hold in (13) for those | v which are signal states in an optimal decomposition. This can be seen by considering the inequalities χN (ρ) = H (N(ρ)) − pi H (N (|vi vi |)) i

≤ H (N(ρ)) −

pi f (|vi vi |)

i

= H (N(ρ)) − f (ρ). For equality to hold, it must hold in all the terms in the summation, which are exactly the signal states | vi . 6. Additivity of EF Implies Strong Superadditivity of EF In this section, we will show that additivity of entanglement of formation implies strong superadditivity of entanglement of formation. Another proof was discovered independently by Pomeransky [13]; it is quite similar, although it is expressed using different terminology. We first give the statement of strong superadditivity. Assume we have a quadripartite density matrix σ whose four parts are A1, A2, B1 and B2. The statement of strong superadditivity is that EF (σ ) ≥ EF (Tr 2 σ ) + EF (Tr 1 σ ),

(14)

where EF is the entanglement of formation when the state is considered as a bipartite state where the two parts are A and B; that is, EF (σ ) = min pi H (Tr B |φi φi |). (15) {pi , | φi } i pi |φi φi |=σ

i

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First, we show that it is sufficient to prove this when σ is a pure state. Consider the opti mal decomposition of σ = i πi |φi φi |. We can apply the theorem of strong subadditiv (1) (1) (1) ity to the pure states |φi φi | to obtain decompositions Tr 1 |φi φi | = j pi,j |vi,j vi,j | (2) (2) (2) and Tr 2 |φi φi | = j pi,j |vi,j vi,j | so that H (Tr B |φi φi |) ≥

(1)

(1)

(1)

pi,j H (Tr B |vi,j vi,j |) +

j

(2)

(2)

(2)

pi,j H (Tr B |vi,j vi,j |).

j

Summing these inequalities over i gives the desired inequality. We now show that additivity of EF implies strong superadditivity of EF . Let | φ be a quadripartite pure state for which we wish to show strong superadditivity. We define σ1 = Tr 2 |φφ| and σ2 = Tr 1 |φφ|. Now, let us use the MSW correspondence to find channels N1 and N2 and density matrices ρ1 and ρ2 such that N1 (ρ1 ) = Tr B σ1

and

N2 (ρ2 ) = Tr B σ2

and χN1 (ρ1 ) = H (N1 (ρ1 )) − EF (σ1 ), χN2 (ρ2 ) = H (N2 (ρ2 )) − EF (σ2 ). We first do an easy case which illustrates how the proof works without introducing additional complexities. Let d1 and d2 be the dimensions of the input spaces of N1 and N2 . In the easy case, we assume that there are d12 linearly independent signal states in an optimal decomposition of ρ1 for χN1 (ρ1 ), and d22 linearly independent signal states in (1) (1) an optimal decomposition of ρ2 for χN2 (ρ2 ). Let these sets of signal states be |vi vi | (1) (2) (2) (2) with probabilities pi , and |vj vj | with probabilities pj , respectively. It now follows from our assumption of the additivity of entanglement of formation that an optimal (1) (2) (1) (2) ensemble of signal states for χN1 ⊗N2 (ρ1 ⊗ρ2 ) is | vi ⊗| vj with probability pi pj . Now, let us consider the dual linear function fT for the tensor product channel N1 ⊗ N2 . Since we assumed that entanglement of formation is additive, by the MSW correspondence χN (ρ) is also additive. We claim that the dual function fT must satisfy (1)

(1)

(2)

(2)

(1)

(1)

(2)

(2)

fT (|vi vi | ⊗ |vj vj |) = H (N1 (|vi vi |)) + H (N2 (|vj vj |)) (1)

(16)

(2)

for all signal states | vi | vj . This is simply because equality must hold in the inequality (13) for all signal states. However, we now have that fT is a linear function in a d12 d22 − 1 dimensional space which has been specified on d12 d22 linearly independent points; this implies that the linear function fT is uniquely defined. It is easy to see that it thus must be the case that fT (ρ) = f1 (Tr 2 ρ) + f2 (Tr 1 ρ),

(17)

as this holds for the d12 d22 signal states We now let |ψψ| be the preimage of Tr B |φφ| under the channel N1 ⊗ N2 . We have, from. Eq. (13) and (17), that f1 (Tr 2 |ψψ|) + f2 (Tr 1 |ψψ|) ≤ H (N1 ⊗ N2 (|ψψ|)).

(18)


465

But recall that f1 (Tr 2 |ψψ|) = EF (σ1 ), f2 (Tr 1 |ψψ|) = EF (σ2 ),

(19)

because (13) holds with equality for signal states, and that N1 ⊗ N2 (|ψψ|) = Tr B |φφ|. Thus, substituting into (18), we find that EF (σ1 ) + EF (σ2 ) ≤ H (Tr B |φφ|), which is the statement for the strong superadditivity of entanglement of formation of the pure state |φφ|. We now consider the case where there are fewer than di2 signal states for χNi (ρi ), i = 1, 2. We still know that the average density matrices of the signal states for N1 and N2 are ρ1 and ρ2 , and that the support of these two matrices are the entire input spaces H1,in and H2,in . The argument will go as before if we can again show that the dual function fT must be f1 (Tr 2 ρ) + f2 (Tr 1 ρ). In this case we do not know d12 d22 points of the function fT , and thus cannot use the same argument as above to show that fT is determined. However, there is more information that we have available. Namely, we (1) know that in the neighborhood of the signal states | vi , the entropy H (N1 (|vv|)) must be at least the dual function f1 = Tr τ1 |vv|, and that these two functions are equal at the signal states. If we assume that the derivative of H (N1 (|vv|)) exists at (1) (1) |vi vi |, then we can conclude that this is also the derivative of f1 = Tr τ1 |vv|. For the time being we will assume that the first derivative of this entropy function does in fact exist.5 We need a lemma. Lemma 3. Suppose that we have a set of unit vectors | vi that span a Hilbert space H. If we are given the value of f at all the vectors | vi as well as the value of the first derivative of f ,

1 lim f (|vi vi |) − f ( (1 − 2 | vi + | wi )( 1 − 2 vi | + wi |)

→0

at all the vectors | vi and for all orthogonal | w, then f is completely determined. Proof. Let us use the representation f (ρ) = Tr τρ (we do not need a constant term on the right-hand side because we need only specify f on trace 1 matrices). Suppose that vi |w = 0. We compute the derivative at | vi in the | w direction:

1 − 2 vi | + w | τ 1 − 2 | vi + | w − vi |τ | vi ≈ (vi | τ | w + w | τ | vi ) .

(20)

The derivative in the i | w direction gives i (vi | τ | w − w | τ | vi ) ,

(21)

5 In fact, I believe the function is smooth enough that these derivatives do exist. However, we find it easier to deal with the cases where N1 (|vv|) has zero eigenvalues by expressing N1 and N2 as a limit of nonsingular completely positive maps.

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so a linear combination of (20) and (21) shows that the value of vi | τ | w is determined for all | w orthogonal to | vi . We also know the value of vi | τ | vi ; it follows that the value of vi |τ | w is determined for all | w. Since the vi | span the vector space, this determines the value of u | τ | w for all u | and all | w, thus determining the matrix τ .

We now need to compute the derivative of the entropy of N1 . Let N1 (ρ) =

† Ai ρAi

i

with

i

† Ai Ai = I . Then if Tr σ = 0,

H (N1 (ρ + σ )) − H (N1 (ρ)) ≈ − Tr (I + log(N (ρ))N1 (σ ) † = − Tr σ Ak log N1 (ρ) Ak .

(22)

k

Now, if the entanglement of formation is additive, then the derivative of H (N1 ⊗ N2 ) at (1) (1) (2) (2) the tensor product signal states |vi vi | ⊗ |vj vj | must also match the derivative of the function fT at these points. We calculate: H (N1 ⊗ N2 (ρ + σ )) − H (N1 ⊗ N2 (ρ))   (1)† (2)† (1) (2) ≈ − Tr σ (Ak1 ⊗ Ak2 )(log(N1 ⊗ N2 (ρ)))(Ak1 ⊗ Ak2 ) . k1 ,k2

Now at a point ρ = ρ1 ⊗ ρ2 ,

(1)†

(Ak1

k1 ,k2

=

(2)† (1) (2) ⊗ Ak2 )(log N1 ⊗ N2 (ρ))(Ak1 ⊗ Ak2 ) (1)† (1) Ak1 log N1 (ρ1 )Ak1

k1

⊗I +I ⊗

(2)† (2) (Ak2 log N2 (ρ2 )Ak2

,

k2 (1)

(2)

showing that at the states | vi ⊗ | vj , we have not only that fT = f1 + f2 , but that the first derivatives (for directions σ with Tr σ = 0) are equal as well. Since the states (1) (2) | vi ⊗ | vj span the vector space, Lemma 3 shows that fT = f1 + f2 everywhere, giving us the last element of the proof.


467

The one thing remaining to do to show that the assumption that the first derivative of entropy exists everywhere is unnecessary. It suffices to show that there are dual functions fT = f1 + f2 such that Eq. (18) holds. We do this by taking limits. For x = 1, 2 let (q) Nx be the quantum channel 1

(q)

Nx (ρ) = Nx (ρ) + (1 − q)

dout,x

I (q)

(q)

which averages the map Nx with the maximally mixed state I /dout,x . Let NT = N1 ⊗ (q) (q) (q) (q) N2 . We need to show that some limits of the dual functions f1 , f2 and fT exist. (q) By continuity of Nx , they will be forced to have the desired properties (17), (18), (q) (q) and (19). Let ρT = ρ1 ⊗ ρ2 . Now, fT is a linear function with fT (ρT ) ≥ 0 and (q) (q) fT (ρ) ≤ log dout,T for all ρ, so the fT lie in a compact set. Thus, some subsequence (q) (q) (q) of fT has a limit as q → 1. The same argument applies to f1 and f2 , so by taking (1) these limits we find that the functions fx have the desired properties, completing our proof. 7. Additivity of min H (N ) Implies Additivity of EF Suppose that we have two bipartite states for which we wish to prove that the entanglement of formation is additive. We use the MSW correspondence to convert this problem to a question about the Holevo capacity with a constrained average signal state. We thus now have two quantum channels N1 and N2 , and two states ρ1 and ρ2 . We want to show that χN1 ⊗N2 (ρ1 ⊗ ρ2 ) = χN1 (ρ1 ) + χN2 (ρ2 ). In fact, we need only prove the ≤ direction of the inequality, as the ≥ direction is easy. (2) (1) Let | vi and | vi be optimal sets of signal states for χN1 (ρ1 ) and χN2 (ρ2 ), so that χN1 (ρ1 ) = H (N1 (ρ1 )) −

(1)

(1)

(1)

pi N (|vi vi |),

i

(1) (1) (1) where ρ1 = i pi |vi vi |, and similarly for N2 . By the linear programming dual formulation in Sect. 5, we have that there is a matrix τ1 such that χN1 (ρ1 ) = H (N1 (ρ1 )) − Tr τ1 ρ1 and Tr τ1 ρ ≤ H (N1 (ρ) (1)

(1)

for all ρ, with equality for signal states ρ = |vi vi |, and similarly for τ2 and N2 . Suppose we could find a channel N1 and N2 such that H (N1 (|vv|)) = H (N1 (|vv|)) + C1 − v | τ | v

(23)

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for all vectors | v (similarly for N2 ). We know from the linear programming duality theorem that H (N1 (ρ)) = H (N1 (ρ)) + C1 − Tr τ1 ρ ≥ C1 (1)

(1)

for all input states ρ, with equality holding for the signal states ρ = |vi vi |. Thus, the minimum entropy output of N1 is C1 and of N2 is C2 . Also, (1) (1) (1) pi H (N1 (|vi vi |)) χN1 (ρ1 ) = H (N1 (ρ1 ))) − i

= H (N1 (ρ1 ))) − C1 , and similarly for N2 . Now, if we assume the additivity of minimum entropy, we know that the minimum entropy output of N1 ⊗ N2 has entropy C1 + C2 . We have for some probability distribution πi on signal states | φi , that χN1 ⊗N2 (ρ1 ⊗ ρ2 ) = H (N1 ⊗ N2 (ρ1 ⊗ ρ2 )) − πi H (N1 ⊗ N2 (|φi φi |)) i

≤ H (N1 (ρ1 )) + H (N2 (ρ2 )) − C1 − C2 = χN1 (ρ1 ) + χN2 (ρ2 ). Now, if we can examine the construction of the channels N1 and N2 and show that the additivity of the constrained Holevo capacity for N1 and N2 implies the additivity of the constrained Holevo capacity for N1 and N2 , we will be done. We will not be able to achieve Eq. (23) exactly, but will be able to achieve this approximately, in much the same way we defined N in Sect. 4. Given a channel N, we define a new channel N . On input ρ, with probability q the channel N outputs N(ρ). With probability 1 − q the channel makes a POVM measurement with elements E and I − E. If the measurement outcome is E, N outputs the tensor product of a pure state signifying that the result was E and the maximally mixed state on k qubits. If the result is I − E the channel N outputs only a pure state signifying this fact. We have H (N (ρ)) = qH (N (ρ)) + H2 (q) + (1 − q)kTr Eρ + (1 − q)H2 (Tr Eρ). If we choose k and E such that (1 − q) kE = λI − τ, q we will have H (N (|vv|)) = qH (N (|vv|)) − q v | τ | v + qλ + H2 (q) + (1 − q)H2 (v | E | v). The minimum entropy H (N (|vv|)) is thus at least qλ + H2 (q). For signal states | vi of N , H (N (|vi vi |)) is at least qλ + H2 (q) and at most qλ + H2 (q) + 1 − q. As q goes to 0, this is approximately a constant. We thus see that H (N1 (ρ1 )) − qλ1 − H2 (q) − (1 − q) ≤ χN1 (ρ1 )


≤ H (N1 (ρ1 )) − qλ1 − H2 (q).

469

(24)

Now, given two channels N1 and N2 , we can prepare N1 and N2 as above. If we assume the additivity of minimum entropy, this implies the constrained channel capacity satisfies, for the optimal input ensembles | φi , πi , χN1 ⊗N2 (ρ1 ⊗ ρ2 ) = H (N1 (ρ1 )) + H (N2 (ρ2 )) −

πi H (N1 ⊗ N2 (|φi φi |))

i

≤ H (N1 (ρ1 )) + H (N2 (ρ2 )) − qλ1 − qλ2 − 2H2 (q) ≤ χN1 (ρ1 ) + χN2 (ρ2 ) + 2(1 − q), where the first inequality follows from the assumption of additivity of the minimum entropy output, and the second from Eq. (24). We now need to relate χN1 (ρ1 ) and χN1 (ρ1 ). Suppose we have an ensemble of signal states |vi vi | with associated probabilities pi , and such that i pi |vi vi | = ρ. Define CN1 (CN1 ) to be the information transmitted by channel N1 (N1 ) using these signal states. We then have CN1 = qCN1 + (1 − q)δ1 , where δ1 = H2 (Tr Eρ) −

pi H2 (vi | E | vi ).

i

This shows that qχN1 (ρ1 ) ≤ χN1 (ρ1 ) ≤ qχN1 (ρ1 ) + (1 − q). Also, by using the optimal set of signal states for χN1 ⊗N2 (ρ1 ⊗ ρ2 ) as signal states for the channel N1 ⊗ N2 , we find that χN1 ⊗N2 (ρ1 ⊗ ρ2 ) ≥ q 2 χN1 ⊗N2 (ρ1 ⊗ ρ2 ), since with probability q 2 , the channel N1 ⊗ N2 simulates N1 ⊗ N2 . Thus, we have that χN1 ⊗N2 (ρ1 ⊗ ρ2 ) ≤ q −2 χN1 ⊗N2 (ρ1 ⊗ ρ2 ) ≤ q −2 (χN1 (ρ1 ) + χN2 (ρ2 )) + 2(1 − q)q −2 ≤ q −1 (χN1 (ρ1 ) + χN2 (ρ2 )) + 4(1 − q)q −2 holds for all q, 0 < q < 1. Letting q go to 1, we have subadditivity of the constrained Holevo capacity, implying additivity of the entanglement of formation.

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8. Implications of Strong Superadditivity of EF All three additivity properties (i) to (iii) follow easily from the assumption of strong superadditivity of EF . The additivity of EF follows trivially from this assumption. That the additivity of χN follows is known [12]. We repeat this argument below for completeness. Recall the definition of χN : χN = max H N pi |φi φi | − pi H (N (|φi φi |)). (25) {pi , | φi }

i

i

Suppose that this maximum is attained at an ensemble pi , | φi that is not a tensor product distribution. If we replace this ensemble with the product of the marginal ensembles, the concavity of von Neumann entropy implies that the first term increases, and the superadditivity of entanglement of formation implies that the second term decreases, showing that we can do at least as well by using a tensor product distribution, and that χN is thus additive. Finally, the proof that strong superadditivity of EF implies additivity of minimum output entropy is equally easy, although I am not aware of its being in the literature. Suppose that we have a minimum entropy output χN1 ⊗N2 (|φφ|). The strong superadditivity (1) (1) (2) (2) of EF implies that there are ensembles pi , | vi and pi , | vi such that (1) (2) (1) (1) (2) (2) H (N1 ⊗ N2 (|φφ|)) ≥ pi H (N1 (|vi vi |)) + pi H (N2 (|vi vi |)). i

i

But the two sums on the right-hand side are averages, so there must be one quantum state in each of these sums which has smaller output entropy than the average output entropy; this shows additivity of the minimum entropy output. 9. Additivity of χN or of EF Implies Additivity of min H (N ) Suppose we have two channels N1 and N2 which map their input onto d-dimensional output spaces. We can assume that the two output dimensions are the same by embedding the smaller dimensional output space into a larger dimensional one.6 We will define two new channels N1 and N2 . The channel N1 will take as input the tensor product of the input space of channel N1 and an integer between 0 and d 2 − 1. Now, let X0 . . . Xd 2 −1 be the d-dimensional generalization of the Pauli matrices: Xda+b = T a R b , where T takes | j to | j + 1(mod d) and R takes | j to e2πij/d | j . Let † N1 (ρ ⊗ |ii|) = Xi N1 (ρ)Xi . Now, suppose that |v1 v1 | is the input giving the minimal entropy output N1 (|v1 v1 |). We claim that a good ensemble of signal states for the channel N1 is |v1 v1 | ⊗ |ii|, where i = 0, 1, . . . , d 2 −1, with equal probabilities. This is because for this set of signal states, the first term in the formula for Holevo capacity (1) is maximized (taking any state † ρ and averaging over all Xi ρXi gives the maximally mixed state, which has the largest possible entropy in d dimensions), and the second term is minimized. The same holds 6

This is not necessary for the proof, but it reduces the number of subscripts required to express it.


471

for the channel N2 . Now, suppose there is some state |ww| which has smaller output entropy for the channel N1 ⊗ N2 than H (N1 (|v1 v1 |)) + H (N2 (|v2 v2 |)). We can use the ensemble containing states |ww| ⊗ |i1 , i2 i1 , i2 |, for i1 , i2 = 0 . . . d 2 − 1, with equal probabilities, to obtain a larger capacity for the tensor product channel N1 ⊗ N2 . The above argument works equally well to show that additivity of entanglement of formation implies additivity of minimum entropy output. We know that to achieve the maximum capacity, the average output state must be the maximally mixed state, so we can equally well use the fact that the constrained Holevo capacity χN (ρ) is additive to show that the minimum entropy output is additive. 10. Discussion We have shown that four open additivity questions are equivalent. This makes these questions of even greater interest to quantum information theorists. Unfortunately, our techniques do not appear to be powerful enough to resolve these questions. The relative difficulty of the proofs of the implications given in this paper would seem to imply that of these equivalent conjectures, additivity of minimum entropy output is in some sense the “easiest” and strong superadditivity of EF is in some sense the “hardest.” One might thus try to prove additivity of the minimum entropy output as a means of solving all of these equivalent conjectures. One step towards solving this problem might be a proof that the tensor product of states producing locally minimum output entropy gives a local minimum of output entropy in the tensor product channel. Acknowledgement. I would like to thank Beth Ruskai for calling my attention to the papers [1, 12] and for helpful discussions, and to Beth Ruskai, Keiji Matsumoto, and an anonymous referee for useful comments on drafts of this paper.

References 1. Koenraad, M.R., Audenaert, Braunstein, S.L.: On strong superadditivity of the entanglement of formation. quant-ph/0303045 2. Bell, J.: On the Einstein Podolsky Rosen paradox. Physics 1, 195–200 (1964) 3. Benatti, F., Narnhofer, H.: Additivity of the entanglement of formation. Phys. Rev. A 63, art. 042306 (2001) 4. Bennett, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046–2052 (1996) 5. Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824–3851 (1996), quant-ph/9604024 6. Gordon, J.P.: Noise at optical frequencies: Information theory. In: Proceedings of the International School of Physics Enrico Fermi. Course XXXI: Quantum Electronics and Coherent Light, P.A. Mills, (ed.), New York: Academic Press, 1964), pp. 156–181 7. Hayden, P.M., Horodecki, M., Terhal, B.M.: The asymptotic entanglement cost of preparing a quantum state. J. Phys. A: Math. Gen. 34, 6891–6898 (2001) 8. Holevo, A.S.: Bounds for the quantity of information transmitted by a quantum communication channel. Probl. Peredachi Inf. 9(3), 3–11 (1973) [in Russian; English translation in Probl. Inf. Transm. (USSR) 9, 177–183 (1973)] 9. Holevo, A.S.: The capacity of the quantum channel with general signal states. IEEE Trans. Info. Theory 44, 269–273 (1998) 10. King, C., Ruskai, M.B.: Minimal entropy of states emerging from noisy quantum channels. IEEE Trans. Info. Theory 47, 192–209 (2001) 11. Levitin, L.B.: On the quantum measure of the amount of information. In: Proceedings of the Fourth All-Union Conference on Information Theory, Tashkent (1969), pp. 111–115, (in Russian) 12. Matsumoto, K., Shimono, T., Winter, A.: Remarks on additivity of the Holevo channel capacity and of the entanglement of formation. quant-ph/0206148

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13. Pomeransky, A.: Strong superadditivity of the entanglement of formation follows from its additivity. quant-ph/0305056 14. Ruskai, M.B.: Some bipartite states do not arise from channels. quant-ph/0303141 15. Schumacher, B., Westmoreland.: Sending classical information via a noisy quantum channel. Phys. Rev. A 56, 131–138 (1997) 16. Shor, P.W.: Capacities of quantum channels and how to find them. quant-ph/0304102 17. Vidal, G., Dür, W., Cirac, J.I.: Entanglement cost of mixed states. Phys. Rev. Lett. 89, art. 027901 (2002) Communicated by M.B. Ruskai

Commun. Math. Phys. 246, 473 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1071-1

Communications in


Erratum

Equivalence of Additivity Questions in Quantum Information Theory Peter W. Shor AT&T Labs Research, Florham Park, NJ 07922, USA c Springer-Verlag 2004 Erratum published online: 2 April 2004 – Commun. Math. Phys. 246, 453–472 (2004)

Unfortunately, an editing error occurred in the first reference of the article, which appears in this issue, when the article was published “online”. Please note that the first reference should read as follows: References 1. Audenaert, K.M.R., Braunstein, S.L.: On strong superadditivity of the entanglement of formation. http://arxiv.org/abs/quant-ph/0303045

Current address: Dept. of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Commun. Math. Phys. 246, 475–502 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1026-y

Communications in


Gravity from Lie Algebroid Morphisms T. Strobl Institut für Theoretische Physik, Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany. E-mail: [email protected] Received: 28 May 2003 / Accepted: 8 August 2003 Published online: 19 February 2004 – © Springer-Verlag 2004

Abstract: Inspired by the Poisson Sigma Model and its relation to 2d gravity, we consider models governing morphisms from T to any Lie algebroid E, where is regarded as a d-dimensional spacetime manifold. We address the question of minimal conditions to be placed on a bilinear expression in the 1-form fields, S ij (X)Ai Aj , so as to permit an interpretation as a metric on . This becomes a simple compatibility condition of the E-tensor S with the chosen Lie algebroid structure on E. For the standard Lie algebroid E = T M the additional structure is identified with a Riemannian foliation of M, in the Poisson case E = T ∗ M with a sub-Riemannian structure which is Poisson invariant with respect to its annihilator bundle. (For integrable image of S, this means that the induced Riemannian leaves should be invariant with respect to all Hamiltonian vector fields of functions which are locally constant on this foliation). This provides a huge class of new gravity models in d dimensions, embedding known 2d and 3d models as particular examples. 1. Introduction 1.1. The problem – in the 2d Poisson setting. In the present paper we want to address essentially the following question: Given a manifold M equipped with some 2-tensor T , say contravariant, and given a two-dimensional manifold , which we call worldsheet or spacetime, under what conditions can we define a “reasonable” theory of gravity on with just these data? Certainly one needs to specify what one means by reasonable, and we will do this – in a rather minimalistic but also precise way – below. Moreover, we will also address a likewise question for higher dimensional spacetime manifolds ; but the formulation of the respective higher dimensional problem, including a replacement of M by a vector bundle with structure – which then also explains the title of the present work – needs further background material, which we will present only later on. Without any further restriction, the above question is not very fruitful: Indeed, for a generic choice of T , we may split T into its symmetric and antisymmetric part, S and P,

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respectively. Then we may use the antisymmetric part to define the following canonical action functional: L[X, A] : = Ai ∧ dX i + 21 P ij Ai ∧ Aj , (1)

where parametrizes the map X from to M, which is used as a target manifold, and A = Ai ⊗ dX i ≡ Aµi (x)dx µ ⊗ dX i is a 1-form on with values in the pullback bundle X ∗ T ∗ M. The symmetric part, on the other hand, may be used to define a metric on by means of Xi (x)

g :=

1 ij 2 S Ai Aj

.

(2)

(Throughout this paper any suppressed tensor symbol will denote a symmetrized tensor product, Ai Aj ≡ Ai ⊗Aj +Aj ⊗Ai , while the antisymmetrized one, the wedge product, will be displayed explicitly.) By construction, the functional (1) is invariant with respect to diffeomorphisms of , which clearly is a necessary condition for a theory describing gravity. For generic enough choice of T (and thus also of P) there are no further local or gauge symmetries of L and there will be solutions to the field equations

dAi +

dX i + P ij (X)Aj = 0,

(3)

(X) Ak ∧ Al = 0,

(4)

1 kl 2 P ,i

such that g constructed according to (2) satisfies det g = 0 on all of . This latter condition is also necessary for regarding g as a metric tensor field. We may, however, not expect it to hold for all solutions to the field equations: For example, Ai :≡ 0, Xi :≡ const is a solution to the field equations (3), (4), leading to g ≡ 0 irrespective of whatever T is. Still, S should not be e.g. vanishing, since then there would be no gravitationally interesting or admissible solutions; for a generic enough choice of T this will not happen, however, and then there is no obstacle to regard (1) and (2) as defining some 2d theory of gravity. Note that we did not require L to be a functional of g alone or a functional of g at all. Applications, some of which we will recall in the body of the paper, show that this would be too restrictive: There may be some other “parent” action functional, which depends explicitly on g, or maybe only on some vielbeins from which g results by the standard bilinear combination, and possibly on some other more or less physical fields and only after some (possibly involved) identifications, including (2), one ends up with the functional (1). The initially posed question bears more structure1 , if we add an additional requirement, namely that the theory should be “topological”. This notion may be understood in various ways, cf. e.g. [3]. Here we imply the following with it: in addition to the diffeomorphism invariance of the action functional L, we require its moduli space of classical solutions (for reasonable enough choice of , such that its fundamental group has a finite rank) to be finite dimensional. For dim M > 2 and generic choice of P this is not the case: For fixed topology of the space of smooth solutions to (3) and (4) is infinite dimensional, even after identification of solutions differing by a gauge symmetry (here only diffeomorphisms of ). 1 This is true at least from the mathematical point of view. The condition imposed excludes propagating modes, which is not so desirable from the physical point of view. Still, many examples in two dimensions have this feature, and many more theories may be captured if the topological theory describes only one sector of a more extended one, cf. e.g. [1, 2] as well as the discussion further below.

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The theory described by (1) is topological, iff [4] the tensor P satisfies P [ij ,s P k]s = 0 ,

(5)

i.e. iff P endows M with a Poisson structure where the Poisson bracket between functions f and g on M is defined by {f, g} := f,i g,j P ij . Now there are much more gauge symmetries. Indeed, for any choice of functions εi (x) (ε corresponding to a section of X ∗ T ∗ M), one verifies easily that if the fields are transformed infinitesimally according to δε X i = εj P j i (X) , δε Ai = dεi + P kl ,i (X)Ak εl ,

(6)

L changes by a boundary term only. The diffeomorphism invariance of L now is a particular case of (6): a simple calculation shows that on-shell, i.e. by use of the field equations (3) and (4), the Lie derivative of the fields with respect to v ∈ (T ) is obtained upon the (field dependent) choice εi = Aµi v µ . In view of these additional symmetries a new question arises if one intends to identify (2) with a metric on : How do these symmetries act on g? If one is “lucky” – or, more precisely, upon a reasonable choice of S – all the symmetries (6), when acting on g obtained as a non-degenerate (det g = 0 everywhere) solution to the field equations, will boil down to diffeomorphisms of only. Otherwise there is an additional local symmetry, which will be hard to interpret in general: There may be e.g. a flat spacetime metric g on which would be “gauge equivalent” to a spacetime metric g describing some black hole. It is our intention to clarify the conditions to be placed on S which ensure that this rather unreasonable scenario is avoided. To be precise: Definition 1. A Poisson Sigma Model (PSM) [5, 6], i.e. the action functional (1) where P satisfies (5), together with an identification (2) for the metric g on is called a reasonable theory of 2d Poisson gravity, if the following two conditions hold (d = 2): • For any point X ∈ M there exist smooth solutions to the field equations on ∼ = Rd with a base map X : → M such that the image of X contains X and everywhere on det g = 0. • For any two gauge equivalent solutions (X(1) , A(1) ) and (X(2) , A(2) ) on a given manifold which yield a non-degenerate metric, det g(i) = 0, ∀x ∈ , and which are maximally extended, there exists a diffeomorphism D : → such that g(1) = D∗ g(2) . We will add remarks explaining some of the technicalities of this definition after its generalization in Subsect. 2.3 below. Requiring (2) to define a reasonable theory of gravity will pose some simple geometrical restriction on S. This will be seen to define a sub-Riemannian structure (cf. e.g. [7] or the beginning of Sect. 5.1 below) on M, which has to be “compatible” with the Poisson structure P in a particular manner (Eq. (40) or (46) below). From a certain perspective to become more transparent below this will be seen to correspond to a Poisson or Lie algebroid extension of the notion of a Riemannian foliation (cf. [8]), which may be also interesting mathematically in its own right. By the above method we will be able to define a much wider class (also of just 2d) gravity theories, the previously known ones – such as those in [5, 9, 1, 2] – arising as a relatively small subclass. As a straightforward application of this extension it will be immediate to construct a simple PSM, the gravity solution space of which coincides with the one of the exact string black hole, something excluded within the previously known models according to [10]. We also demonstrate with an example that by these means one can construct

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gravitational models with solutions for the metric that have no local Killing vector field, a feature missing in the class of matterless 2d gravity models considered hitherto (cf. e.g. [1, 2]). We also hope that by the above analysis much of the in part intricate global structure arising on in the known gravity models, as classified in [11, 12], will find some explanation from the implicitly underlying geometrical struture (M, P, S) on the target. In fact, already in the present work we want to highlight the emergence of a Killing vector field in these models as resulting from structures on M. Finally, we may apply our considerations to the case of kappa deformed gravity [13, 14], providing some coordinate independent [15] grasp on the discussion which arose in that context. 1.2. Generalizations. Given a metric on , be it of the form (2) or not, and the data specified in the previous subsection, we may also add another term to (1): 1 ij 2 S Ai

∧ ∗Aj .

(7)

Here ∗ : p () → 2−p () denotes the operation of taking the Hodge dual of a form with respect to g. In fact, upon this addition to (1) one may obtain a theory that is at least classically equivalent to standard String Theory: Assuming the matrix T ij to be invertible, we may integrate out the A–fields altogether, as they enter the action quadratically only; the result is the usual String Theory action in Polyakov form. This may be of particular interest also if P is Poisson, since then one expects P to govern the noncommutativity of the effective action induced on D-branes via the Kontsevich formula [16], which itself just results upon perturbative quantization of (1) [17]. For constant T this is verified easily [18]. We will not consider additions of the form (7) any further within the present work. Much closer to the present subject are the following modifications of the problem posed in the previous subsection: Suppose that in addition to T one is given also a covariant 2-tensor on M. We again may split it into symmetric and antisymmetric parts, say G and B, respectively. Then we may add the pullback of B = 21 Bij dX i ∧ dXj with respect to X : → M to (1) and the pullback of G = 21 Gij dX i dX j to the right hand side of (2). We then may repeat the questions of the previous subsection. Requiring the resulting action functional to be topological now yields a modified condition (5), its right hand side consisting of the 3-form H = dB with all indices raised by contraction with P [19]. This leads to the notion of a twisted Poisson, WZ-Poisson, or an H-Poisson structure on M, cf. e.g. [20]. Now the second set of field equations receives an H–dependent contribution. Likewise, there are still gauge symmetries for any εi , while the transformation on Ai in (6) receives some H–dependent additions. Despite these evident changes, much of the standard PSM can be transferred to the modified theory. Indeed, if (1 + BP) is invertible, one even may get rid of the additional term in L by some redefinition of the A–fields (as seen best in the Hamiltonian formulation) leading to a modified, P– and B–dependent effective Poisson structure P . (Such transformations were called gauge transformations in [20].) Likewise a statement holds for g: Using the first set of field equations, (3), yields a new effective tensor S = S + PGP. Thus essentially everything of what will be said on the PSM with (2) will have a straightforward generalization to the twisted WZ-, or HPSM with the modified identification for g.2 2 In fact, from the mathematical point of view, the more detailed study of the modified situation still may be interesting – and possibly also intricate (the non-degeneracy condition on g, e.g., will have quite different implications than those in the present note, cf. Lemma 1 below).


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Finally, we may also be given a (1,1)–tensor field on M. Adding corresponding terms to (2) does not change much since again by use of the field equations one may reexpress everything in terms of a bilinear combination of A–fields. If one uses such a tensor field j so as to redefine the first term in (1), on the other hand, replacing it e.g. by Ai ∧ dX j Ci , the situation changes if C has a kernel. Let us mention here parenthetically that such a modification of (1) arises if one considers 2d gravity models with additional scalar or fermionic matter fields, where in the first case there is a nontrivial kernel of C considered as a map from T M to T M and in the second case a nontrivial kernel of the transposed map from T ∗ M to itself, cf. [21, 1]. More generally we may replace T ∗ M by some vector bundle E over M. Together with a (coanchor) map C : E → T ∗ M and a covariant E-2-tensor TI J = PI J + SI J we may replace (1) by an integral over AI CI i ∧ dX i + 21 PI J AI ∧ AJ . It may be interesting to clarify the conditions on such a more general action functional to be topological and then to address the question of when g = 21 SI J AI AJ

(8)

defines a reasonable notion of a metric on . Although this generalization of the PSM seems interesting also in its own right, we will not pursue it here any further. Instead we want to permit an even more drastic, but different change – not sticking to the underlying action functional, but only to the relevant mathematical structures: We thus first want to allow for any spacetime dimension of . Moreover, the Poisson manifold (M, P) may be viewed as a particular Lie algebroid structure on the vector bundle T ∗ M over M; we will replace it by any Lie algebroid defined on a vector bundle E over M. For the metric g on we then will use (8) as a generalization of (2). First we note that the field equations (3) and (4) make perfect sense also if is replaced by a manifold of some higher dimension, and likewise so for the symmetries (6). So, the questions addressed above do make sense also in d > 2 spacetime dimensions. The generalization of the target structure to Lie algebroids needs some extra explanations; before we can approach it, we want to recall the notion and the basic features of Lie algebroids in the next section, since we cannot assume that the average physics reader is familiar with it. In the present work we will not focus on the action functional that produces the desired field equations and symmetries. Whenever an action functional which has field equations and symmetries containing the ones to be specified below, our analysis will apply. It is still comforting to know, however, that such action functionals can be constructed [22]. 1.3. Struture of the paper. In the following section we will recall the definition of Lie algebroids over M and provide the corresponding generalization of the field equations (3) and (4) as well as of the gauge symmetries (6) to this more general setting. In the subsequent section we then determine the conditions on S which ensure that g as defined in (8) provides a reasonable theory of gravity, as defined by the two marked conditions above or more precisely by Definition 3 below. This section provides the main result of the present work, summarized in Theorem 1. Section 4 contains first illustrations of the general results, applying it to various particular Lie algebroids. Standard (2+1)-gravity is among the examples of the general framework developed here. Only in Sect. 5 do we come back to the Poisson case – still

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permitting arbitrary dimension of ; in a further specialization we finally arrive back to the Poisson Sigma model. We then will show how previously known 2d gravity models fit into the present more general framework and provide some simple examples of new models. Finally we will address some more recent issues such as kappa deformation of 2d gravity theories. We complete the paper with an outlook including a list of open questions and possible further developments. 2. Lie Algebroids, Generalized Field Equations and Symmetries 2.1. Lie algebroids, basic facts and examples. A Lie algebroid is a simultaneous generalization of a Lie algebra and a tangent bundle. Let us begin with a formal definition. (For further details cf. e.g. [23].) Definition 2. A Lie algebroid (E, ρ, [·, ·]) is a vector bundle π : E → M together with a bundle map (“anchor”) ρ : E → T M and a Lie algebra structure [·, ·] : (E) × (E) → (E) satisfying the Leibniz identity [ψ, f ψ ] = f [ψ, ψ ] + (ρ(ψ)f ) ψ

(9)

∀ψ, ψ ∈ (E), ∀f ∈ C ∞ (M). Let us provide some basic examples of Lie algebroids: 1. If M is a point, ρ is trivial and E is just an ordinary Lie algebra. 2. M arbitrary, but ρ ≡ 0, mapping all elements of the fiber of E at any point X ∈ M to the zero vector in TX M, E is a bundle of Lie algebras, since then (9) provides a Lie bracket on each fiber of E; M then may be viewed as a manifold of deformations of a Lie algebra. 3. E = T M, ρ = idT M , the bracket being the ordinary Lie bracket between vector fields. This is the so-called standard Lie algebroid. 4. E = T ∗ M, (M, P) a Poisson manifold. Here ρ = P , ρ(αi dX i ) = αi P ij ∂j , and the bracket [df, dg] := d{f, g} between exact 1-forms is extended to all 1-forms by means of (9). We now add some remarks: Eq. (9) restricts ρ; using the Jacobi identity of the bracket [·, ·] between sections, it is possible to show [24] that ρ is a morphism of Lie algebras, ∀ψ, ψ ∈ (E): ρ[ψ, ψ ] = [ρ(ψ), ρ(ψ )] .

(10)

As a consequence, the image of ρ is always an integrable distribution in T M, defining the orbits of the Lie algebroid in M. In the case of Poisson manifolds, these orbits coincide with the symplectic leaves of (M, P). Generalizing the observation in the second example above, ker ρ ⊂ E always defines a bundle of Lie algebras. For different points on the same orbit, these Lie algebras are isomorphic, while they are not necessarily so for any two points in M. For Poisson manifolds ker ρ|X∈M is the conormal bundle of the respective symplectic leaf L at a given point X ∈ M. For regular points on Poisson manifolds these Lie algebras are abelian; the origin of a Lie Poisson manifold g∗ provides an example for a nonregular point. If {bI }nI=1 denotes a local basis of E, the bracket and anchor give rise to structure functions cI J K (X) and ρIi (X), respectively: [bI , bJ ] = cI J K bK ,

ρ(bI ) = ρIi ∂i .

(11)


481

The compatibility conditions above then provide differential equations for them: i + cycl(I J K) = 0 , cI J S cKS L + cI J L ,i ρK i cI J K ρ K

j − ρI ρJi ,j

j +ρJ ρIi ,j

(12)

= 0.

(13)

While ρIi behaves as a tensor with respect to a change of local basis and coordinates, cI J K certainly does not: With bI = FII bI and bI = FII bI one has

I I I K L I i I i I I cKL = FK FL FI cKL + ρK FI − ρ L FI . FL,i FK,i

(14)

In the case of ρ ≡ 0, Example 2 above, the second equation above becomes trivial and the first one reduces to the standard Jacobi identity at any point X ∈ M. For Poisson manifolds, on the other hand, bI ∼ dX i , ρJi ∼ P j i , and cI J K ∼ P ij ,k . We see that this case is particular insofar as the anchor and the structure functions are essentially one and the same object; Eq. (13) reduces to the Jacobi identity (5) and Eq. (12) to its derivative. A Lie algebroid structure on a vector bundle E permits a generalization of differenp tial geometry to E. In particular, there is a natural exterior E-derivative dE : E (M) → p+1 p E (M), where E (M) ≡ ( p E ∗ ) and 0E (M) ≡ C ∞ (M). If {bI }nI=1 denotes the ∗ local basis of E dual to {bI }nI=1 , it may be defined by means of dE f := f,i ρIi bI ,

dE bI := − 21 cJ K I bJ ∧ bK ,

(15)

p

extended to all of E (M) by a graded Leibniz rule. As a consequence of the Lie al2 = 0. (E together with a nilpotent d even provides an alternative gebroid axioms, dE E definition of a Lie algebroid, cf. [23].) p p Likewise there is an E-Lie derivative EL· : (E) × Tq (E) → Tq (E), where p ⊗q ⊗p ∗ Tq (E) ≡ E ⊗ (E ) . For p = 1, q = 0 it is defined by the ordinary bracket, EL ψ := [ψ, ψ ], while for α ∈ p (M) one uses the formula (generalizing a wellψ E known pendant in differential geometry resulting from E = T M) Lψ α = (dE ψ + ψdE ) α ;

E

(16)

p

it is extended to Tq (E) by means of the ordinary Leibniz rule. This concludes our short excursion into the mathematics of Lie algebroids.

2.2. Generalized field equations and symmetries. For a generalization we first need to interpret the field content of the PSM. The coordinates X i define a map X : → M, where is our spacetime and M the base of a vector bundle E. The fields Ai then correspond to a linear assignment of an element in E (which is T ∗ M in the PSM) in the fiber over X (x) to any element in Tx . Together they thus parametrize a vector bundle morphism ϕ: T → E ,

(17)

with X being the respective base map. Besides X such a morphism is characterized by an element A = AI ⊗ bI ∈ (X ∗ E), where bI here corresponds to a local basis in the pullback bundle X ∗ E over (induced by a likewise basis in E).

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As a generalization of the field equations (3) and (4), we then consider the following set of equations: dA + I

dX i − ρIi AI = 0 ,

(18)

∧A

(19)

I J 1 2 cJ K A

K

= 0,

where E has been given the structure of a Lie algebroid and ρ and c are the corresponding previously introduced structure functions. This clearly generalizes the field equations of the PSM. Moreover, there is also some mathematical meaning to these equations in the general case: They state that the vector bundle morphism ϕ : T M → E is also a morphism of Lie algebroids (cf. [25] for details). It is straightforward to check that on behalf of the structural compatibility equations (12) and (13), the above field equations define a free differential algebra (cf. e.g. [26, 27] for a definition). In particular, their integrability conditions are satisfied and for any choice of “initial data” Xi (x0 ), AI |x0 given at a point x0 ∈ there is a neighborhood U x0 in with smooth continuations X i (x), AI (x) solving the field equations (18) and (19). Such a continuation is however far from unique. There is a set of symmetry transformations which map solutions of the generalized field equations (18) and (19) into solutions. Indeed, one may check that the infinitesimal transformations δε X i = εI ρIi , δε AI ≈ dεI + cKJ I AK ε J ,

(20)

induced by some element ε ∈ (X ∗ E), leave the field equations invariant. Here again one needs to use the structural compatibility conditions, generalizing the Jacobi identity for the PSM. These are the symmetries we want to consider as gauge symmetries. Note that in the above we used a somewhat modified equality sign in the definition of the symmetries for the A-fields. By this we intend to imply that the respective symmetry transformations need to have this form only on-shell, i.e. upon use of the field equations (18) and (19). Indeed, examples such as the HPSM [19] show that in general there may be some addition proportional to (18) so as to leave invariant the respective action functional. Within the present setting it is, however, completely sufficient to know the gauge symmetries on-shell. The above gauge symmetries generate diffeomorphisms of . Indeed, again the image of a vector field v under the map ϕ : T M → E provides the respective infinitesimal generator ε; one finds Lv ≈ δA,v ,

(21)

where Lv denotes the standard Lie derivative with respect to v, and A, v ≡ AIµ v µ the image of v with respect to ϕ ∼ (X , A). As a further consistency check we may verify that the complete set of field equations is covariant. This indeed follows quite easily by use of Eq. (14). Likewise so for the symmetries (20), keeping in mind their infinitesimal form, however; in particular,

I K L δ(FII AI ) = δ(FII )AI + FII δAI ≈ dεI + cKL A ε ,

(22)

where εI ≡ FII ε I and FII is an arbitrary function of X(x), governing a change of basis in E. (Although not also of x alone, as it would be permitted as a change of basis in the pullback bundle X ∗ E – at least if the structure functions cI J K in (20) are understood as pullback of the structure functions on E, as we want to interpret them.)


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As mentioned already in the Introduction, action functionals yielding field equations and symmetries containing (18), (19), and (20) exist. One option, which works in any dimension d, is to multiply the left-hand sides of (18) and (19) by Bi and BI , a (d − 2)and a (d − 1)-form, respectively (cf. also [22] for more details). 2.3. Definition of a reasonable theory of E-gravity. We now are in the position to generalize Definition 1: Definition 3. A Lie algebroid E together with a symmetric section S of T02 (E) is said to define a reasonable theory of E-gravity, if the two marked items in Definition 1 hold true for g as defined in (8), where solutions refer to the set of field equations (18),(19) and the gauge symmetries are given by (20) (infinitesimally and on-shell). We now add some remarks explaining the two items in the definition: In the first one, we required the existence of a (metric) non-degenerate solution in a neighborhood of the preimage of any point X ∈ M. If this was not satisfied, but only for a submanifold U ∈ M, we could equally well take the subbundle E|U as a new Lie algebroid, defined just over U . Moreover, there would be no condition on S for points X ∈ M \ U . Thus, essentially without loss of generality, we require non-degeneracy for all points X ∈ M. Since the second item refers to a global gauge equivalence, while the symmetries (20) are given in infinitesimal form only, we need to add that within the present framework and in the context of (8) we regard two metric non-degenerate solutions (X(1) , A(1) ) and (X(2) , A(2) ) to the field equations as gauge equivalent if they may be connected by a flow of gauge transformations generated by (20) which does not enter a degenerate sector of the theory, i.e. which does not pass through a solution not satisfying the first condition on g. This last specification is necessary because otherwise the set of theories would be empty: Even if the gauge transformations (20) when acting on non-degenerate solutions correspond merely to diffeomorphisms, they allow non-degenerate solutions to connect with degenerate ones. E.g. in the 2d Poisson case it has been shown that by such transformations, even if they are contained in the component of unity of the gauge group, one may generate non-degenerate gravity solutions with nontrivial kink number from non-degenerate ones with trivial kink number (if π1 () = 0), something that can never be achieved by a diffeomorphism of – cf. [28] for an explicit example of this scenario within the Jackiw-Teitelboim model of 2d gravity [29, 30], and [31] for a generalization to arbitrary 2d dilaton gravity theories. Finally, the addition that the solution should be maximally extended results from the following typical feature of a gravity theory: the “time-evolution” – or likewise the exten ⊃ – is governed by the sion of a solution on to one defined on a bigger manifold symmetries, even if these are on-shell diffeomorphisms only. Thus evolving e.g. some in some locally defined time parameter t such that Ut ⊂ and U0 = U , region U ⊂ it may happen that for some large enough t = t1 one has U1 ∩ U = ∅, U1 ≡ Ut1 , and that g on all of U = U0 is flat while on U1 it may be nowhere flat. Then U0 and U1 may be still gauge equivalent with respect to some symmetries which only on-shell reduce to the diffeomorphism, while certainly there does not exist a diffeomorphism relating (U0 , g|U0 )) and (U1 , g|U1 )). However, by construction of this example, there exists , g) of the local solution defined on ∼ some extension ( = U0 such that (U0 , g|U0 ) may still and (U1 , g|U1 ) are part of this extended solution. (Note that, as a manifold, ∼ be diffeomorphic to ∼ U U ; thus, extending a solution defined on need not = 0 = 1 change its topology.) However, if two solutions are already maximally extended, then

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we require that they are gauge equivalent only if they (or better their respective metrics) differ by some diffeomorphism from one another. Note that by our definition it is not excluded that two not gauge equivalent solutions still have diffeomorphic metrics. This can happen, if the model carries also some other physically interesting fields, such as e.g. non-abelian gauge fields. We do not want to exclude such possibilities. Alternatively, in the second item we could have taken recourse to infinitesimal symmetries only. So, we could have required that for any metric non-degenerate solution (X , A), the infinitesimal variation of g, as defined in (8), by a gauge transformation (20) may be expressed as the Lie derivative of some vector field v ∈ (T ) acting on g. Despite the technical complications, to our mind the global description is formulated closer to the desiderata. Further illustration of the assumptions or requirements in the definition will be provided by the examples below. 3. Conditions for a Reasonable Theory of Gravity 3.1. Metric non-degeneracy. A metric g on is non-degenerate iff g(v, v )|x = 0 ∀v ∈ Tx implies v = 0. Likewise, it is non-degenerate iff the map g : T → T ∗ , v → g(v, ·) is invertible. (To be precise, we would have to restrict to the fiber over a point x ∈ T in the last sentence; this is to be understood, also analogously in what follows.) Due to (8), we have g = 21 ϕ ∗ ◦ S ◦ ϕ, where S : E → E ∗ , a = a I bI → a I SI J bJ and ϕ ∗ is the transpose map to ϕ (again, when restricted to X (x), certainly). This implies that ker ϕ has to be the zero vector. Moreover, we learn that the image of this map is not permitted to have a nontrivial intersection with the kernel of S . Thus we obtain: Lemma 1. For (metric) non-degenerate solutions to the field equations (18) and (19), one needs (d = dim ) ker ϕ = {0} ⇔ dim im ϕ = d

and

im ϕ ∩ ker S = {0}.

(23)

As a simple corollary of this, we find that d ≤ r, where r ≡ rkE denotes the rank (fiber dimension) of E, and dim ker S ≤ r − d

⇔

dim imS ≥ d.

(24)

If S has definite signature, (23) is also sufficient to ensure non-degeneracy of g; and then (24) is sufficient for the existence of a non-degenerate solution ϕ at least in some neighborhood U x of a given point x ∈ due to the local integrability of the field equations for any given choice of ϕx ∼ (X , A)x at that point. Choosing the trivial topology for , ≈ Rd , we can identify U with and then fulfill the first of the two conditions for an E-gravity to be reasonable. Proposition 1. For semi-definite S or for any S with dim imS = d the conditions (23) are sufficient to ensure that (8) is non-degenerate. In general, however, (23) is not sufficient to ensure non-degeneracy of g, as the following example may illustrate: Let d = 2, r = 3, S = b1 b1 + b2 b2 − b3 b3 , such that ker S = {0}, and let im ϕ = b1 , b2 + b3 . The conditions in (23) are satisfied, but restricting S to the two-dimensional image of ϕ, it reduces to S = b1 b1 , which is degenerate. To exclude such counter-examples, one may employ the following


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Proposition 2. For any W ⊂ E, S|W is non-deg.

⇔

S|W ∩Z is non-deg. ,

(25)

where Z (or better Z|X∈M ) denotes the set (not a vector space) of null vectors, Z|X = {V ∈ EX |S(V , V ) = 0}. Proof. One direction is trivial, namely ⇐, since for any non-null vector V , one already has S(V , V ) = 0. The other direction is an exercise in elementary linear algebra. So, in addition to (23), it suffices to check that for any vector V ∈ im ϕ ∩ Z, there exists a vector V ∈ im ϕ ∩ Z such that S(V , V ) = 0 so as to ensure that the map (or solution) ϕ ∼ (X , A) leads to a non-degenerate metric g (upon usage of the defining relation (8)). 3.2. Transformation of the metric. We now come to the derivation of the key relation(s) of the present paper. Herein we first assume that ϕ is a solution to the field equations (18), (19) providing a non-degenerate metric g. With the considerations of the previous subsection, it then will be easy to determine some minimal conditions on S such that this can be achieved at all, which was one of the two conditions placed on a reasonable theory of E-gravity. According to the second item (cf. Defintion 3), we are then left with finding conditions on S such that the symmetries (20) coincide with diffeomorphisms of on-shell. In other words, we need to ensure that for any ε ∈ (X ∗ E) one either has δε g ≈ 0 or there has to exist a v ∈ (T ) such that δε g ≈ Lv g. According to (21) we may, however, rewrite the right-hand side of the last equation as another gauge transformation, Lv g ≈ δϕ(v) g. Correspondingly, we find that ∀ε ∈ (X ∗ E) ∃ v ∈ (T ) :

!

δε−ϕ(v) g ≈ 0,

(26)

where v may also be the zero vector field. This condition is sufficient and also necessary so as to ensure that the underlying assignment for g defines a reasonable theory of E-gravity. We first compute δε g. By a straightforward calculation, using the Leibniz rule for the symmetries, one obtains δε g ≈ 21 X ∗ ELbK S ε K AI AJ + X ∗ (SI J )dε I AJ . (27) IJ

If, moreover, ε is the pull back of a section ∈ (E), εI = X ∗ I , then, using the field equations, Eq. (27) may be simplified further: δε g ≈ 21 X ∗ EL S AI AJ . IJ

(28)

(29)

For a general map ϕ we may choose ε of the form (28). On the other hand, for a given map ϕ this covers any choice of ε, if the base map X : → M is an embedding; in particular, due to the first field equation (18) and due to (23), this requires im ϕ ∩ ker ρ = {0} .

(30)

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Let us in the following assume for simplicity that S is semi-definite, i.e., e.g. S ≥ 0. (We will soon relax this condition again; however, much of the discussion simplifies in this case and may at least serve as a first orientation.) Then, due to (23), there is – for fixed, permitted ϕ and for any point x ∈ – a unique orthogonal decomposition of EX(x) according to EX(x) = Wx ⊕ Wx⊥ ,

Wx ≡ im ϕ|x ,

(31)

where a possibly nonvanishing ker S |X(x) is part of Wx⊥ certainly. Note that this decomposition depends on x and not only on X(x), which may be a decisive difference, if (30) is not satisfied. Alternatively, we may formulate it as a decomposition of the pullback bundle: X ∗E = W ⊕ W ⊥ ,

W = im ϕ ,

(32)

where here im ϕ is interpreted as a subbundle of X ∗ E. In the case that X : → M is an embedding, however, we may rewrite (31) also in the form E|im X = W ⊕ W ⊥ ,

W = im ϕ ,

(33)

where now W is a subbundle of E (restricted to the image of X ). If not otherwise stated, we will use (32). We know that (26) has to be satisfied for any ε : → E covering X : → M. We now can uniquely decompose any ε into a part inside W and W ⊥ , calling it εW and ε⊥ , respectively: ε = εW + ε⊥ . One has to be careful: if ε satisfies (28), this in general does not imply a likewise decomposition of – since the decomposition (31) or (32) depends explicitly on x and not only on X(x). This changes, however, if (30) is satisfied; then also = W + ⊥ . It is suggestive to assume that if (26) is to be satisfied, then εW needs to equal ϕ(v). This is, however, not mandatory: Although there is a unique decomposition of ε into the two orthogonal parts, it does not imply that also the respective variations of g are orthogonal (so that each of them would need to vanish separately). Instead we only know that there is some v such that εW = ϕ(v ) and then, with v − v := w, we can merely conclude from (26) that !

δε⊥ g ≈ δϕ(w) g

for some w ∈ W .

(34)

In the present paper we do not intend to explore this relation in full generality or even in examples; nor do we want to extend the general consideration to the case of an indefinite S, where part of W may be contained in W ⊥ . Here we will content ourselves with solving the above condition for w := 0, i.e. we require !

δε⊥ g ≈ 0

∀ε⊥ ∈ (W ⊥ ) .

(35)

In Subsect. 3.4 below, however, we will also permit indefinite S, but only for dim ker S = r − d. In this case, one may just replace W ⊥ by ker S in the above. We will make this more explicit later on.


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3.3. First compatibility conditions. Within this subsection we will first assume that (30) is satisfied. Note that this can be achieved only if ker ρ ⊂ E is not too big; in particular, due to the first condition in (23), it cannot be satisfied if dim ker ρ > r − d, r ≡ rankE. Thus here we may analyse (35) for the case of (28) with ε⊥ = X ∗ ⊥ . Using (29) in combination with (35), we find E (36) Lψ1 S (ψ2 , ψ3 ) = 0 ∀ψ1 ∈ (W ⊥ ), ψ2 , ψ3 ∈ (W ). This relation needs to hold for any point X ∈ M, because any X can be in the image of X , the base map of ϕ. And according to the above definition of a reasonable theory of E-gravity, by using all possible maps ϕ, we may also vary W essentially at will: Eq. (36) has to hold ∀W ⊂ E with W ∩ ker S = {0} = W ∩ ker ρ, dim W = d.

(37)

In summary: Proposition 3. In any Lie algebroid E together with a symmetric, semi-definite section S of E ∗ ⊗ E ∗ defining a reasonable theory of E-gravity (cf. Definition 3), the conditions (24) and (36) with (37) are satisfied. Condition (36) with (37) becomes empty certainly if dim ker ρ > r −d. The following observations may be helpful in the present context: Corollary 1. Any semi-definite S of a reasonable theory of E-gravity with dim(ker ρ) ≤ r − d satisfies Lψ S = 0 ∀ψ ∈ ker S .

E

(38)

Proof. Take ψ1 := ψ ∈ ker S in (36), which is possible for any choice of W , since always ker S ∈ W ⊥ . Next take ψ2 in (36) as any vector in E\ ker ρ ∩E\ ker S . Finally, choose ψ3 in such a way that the span of ψ2 and ψ3 can lie in some W compatible with (37). Since d ≥ 2, dim(ker ρ) ≤ r − d, and dim(ker S ) ≤ r − d, choices the possible of ψ2 and ψ3 lie dense in E. By continuity we thus can conclude ELψ S (ψ2 , ψ3 ) = 0 for all ψ2 , ψ3 ∈ E – and thus Eq. (38). Proposition 4. Conditions (36) and (38) (or (40) below) are C ∞ (M)-linear. Correspondingly it suffices to check them on a local basis. Proof. In (36) we only need to check it for ψ1 . Using the simple-to-verify relation LF ψ S = F ELψ S + dE F S(ψ, ·)

E

(39)

with ψ = ψ1 , we find that additional contributions vanish due to orthogonality of ψ1 with ψ2 and ψ3 . Likewise, in (38) an eventually additional term vanishes since ψ ∈ ker S . A simplification occurs, if (24) is saturated: Corollary 2. For dim imS = d and dim(ker ρ) ≤ r − d, Eq. (36) reduces to Eq. (38). Proof. In this case always W ⊥ = ker S . Thus (36) is automatically satisfied upon (38), which in turn was already found to be necessary.

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The conditions established in the present section are necessary conditions (for satisfying (35)) only, which, as remarked, in the case of ker ρ too big even become vacuous. On the other hand, when ker ρ is trivial – so that (30) is satisfied always, no matter what ϕ – the conditions in Proposition 3 are also sufficient to define a reasonable theory of E-gravity. This is quite restrictive, however: it implies that E is isomorphic to some integrable distribution, E ∼ = D ⊂ T M with [(D), (D)] ⊂ (D). So not many Lie algebroids may be covered by that proposition. We will consider the case of injective anchor map in some detail in the examples. It will be seen by explicit inspection then that in this case it is necessary that ker S has maximal dimension so as to be compatible with (24). At least in this case it must be possible, therefore, to derive dim imS = d from the conditions summarized in Proposition 3. Moreover, in the present paper we were not able to provide examples of admissible E-gravity theories which do not saturate the bound. It may be conjectured, therefore, that this is even a necessary condition in general. In any case, Conditions (36) with (37) seem very restrictive, and it is plausible that stronger results can be derived from them. 3.4. Sufficient conditions. The main complication one encounters with generalizing the previous considerations to arbitrary Lie algebroids (also permitting dim ker ρ > r − d), and to possibly necessary and sufficient conditions to ensure (35), is the fact that in general the decomposition (31) depends on x ∈ itself, and not just on its image X(x) with respect to X : → M. Correspondingly, in (27) it is not always possible to choose a basis bI in E adapted to the decomposition of ε into its two parts εW and ε⊥ . (Consider e.g. the extreme case where the image of X is just a point in M.) This then can I AJ = 0 (which is true by construction of ε ) does not have the effect that X ∗ (SI J )ε⊥ ⊥ I ∗ J imply X (SI J )dε⊥ A = 0 (the second term in Eq. (27)). And indeed, as we will also illustrate by an explicit example in Sect. 4.2 below, conditions of the form (36) turn out insufficient in general, no matter what W is permitted to be. In the following we will thus content ourselves with providing some sufficient conditions for an E-gravity theory to be reasonable. For this purpose we assume saturation of the general bound (24). This was seen to provide a simplification already in the previous subsection (cf. Corollary 2). We now state the main result of the present paper: Theorem 1. Any Lie algebroid E together with a symmetric section S of E ∗ ⊗E ∗ defines a reasonable theory of E-gravity (as defined in Definition 3), if dim imS = d and Lψ S = 0 ∀ψ ∈ ker S .

E

(40)

As before, d = dim and r = rank E and EL denotes the E-Lie derivative of the Lie algebroid as defined at the end of Sect. 2.1. Proof of Theorem 1. We remarked already previously that the symmetries are on-shell covariant, cf. Eq. (22). Correspondingly, the total expression (27) is independent of a choice of frame bI in E; the unwanted contribution from the first term cancels a likewise contribution from the second one on use of the field equations (18). Let us therefore choose a basis bI adapted to the kernel of S : E → E ∗ : {bI }rI =1 = ker }d−r , {b0 }d ker 0 {{bA M M=1 }, where {bA } spans ker S and {bM } some d-dimensional comA=1 plement W 0 . Note that this provides a fixed decomposition of E according to E = W 0 ⊕ ker S .

(41)


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This is in spirit quite different from the decomposition (31), where for the same point X in M there may be different decompositions into Wx and Wx⊥ , depending on the choice of x ∈ . In the present context, always Wx⊥ = ker SX(x) , no matter what x. 0 . Now we use the unique decomposition of On the other hand, in general Wx = WX(x) ε into its two parts corresponding to (31). For the calculation of the change of g with respect to ε = εW + ε⊥ we use the other decomposition (41), we thus need to express εW and ε⊥ in terms of the adapted basis introduced above. While the first of these two quantities in general has contributions into directions of both W0 and ker S , which, moreover, depend on x (and not just on X(x)), ε⊥ appearing in (35) has a very simple A (x)bker | 0 decomposition: ε⊥ = ε⊥ A X(x) . This has no non-vanishing components in the W I A direction, and likewise also dε⊥ (x) = (dε⊥ (x), 0) does not. Thus obviously (in the adapted basis chosen) the second term in (27) vanishes identically for ε = ε⊥ . But also the first term vanishes on behalf of (40) and the fact that in the sum over the index K ker have non-zero coefficient ε K (x). This concludes the proof. only contributions along bA ⊥

(For non-semidefinite S one merely replaces Wx⊥ in (31) by ker SX(x) so as to guarantee uniqueness of the decomposition. Cf. also Proposition 1 for what concerns nondegeneracy of g.)

If we restrict our attention to (35) (instead of the more general condition (34)) then under the assumption on the dimension of ker S the condition (40) is also necessary for defining a reasonable theory of E-gravity. This is obvious from the above proof. As mentioned in Proposition 4, the condition (40) is C ∞ -linear. In fact, this goes even further3 : Proposition 5. For any S ∈ (∨2 E ∗ ) of constant rank satisfying (40) in a Lie algebroid E, K := ker S defines a Lie subalgebroid. Proof. Due to [ELψ1 , ELψ2 ] = EL[ψ1 ,ψ2 ] also the product of two sections ψ1 , ψ2 ∈ (K) annihilate S. Assuming that [ψ1 , ψ2 ] ∈ (K), one obtains a contradiction: ∃ψ3 ∈ (E) such that 0 = S(ELψ1 ψ2 , ψ3 ) = ELψ1 (S(ψ2 , ψ3 )) − S(ψ2 , ELψ1 ψ3 ) = 0, since ψ2 ∈ ker S . As an obvious consequence we find Corollary 3. ρ(K) ⊂ T M is an integrable distribution. Under appropriate conditions, including that the foliation induced by ρ(K) is a fibration, this may be used to define quotient bundles, to which S projects as an otherwise arbitrary nondegenerate E-2-tensor. We intend to come back to this elsewhere.

4. First Examples In this section we provide some elementary examples following the list after Definition 2. 3 This observation has been inspired by discussions with A. Weinstein, in particular those about the standard Lie algebroid, discussed in Sect. 4 below, in which he pointed out to me that in this case the structure induced by a compatible S is the one of a Riemannian foliation.

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4.1. Lie algebra case. In the case of an ordinary Lie algebra, E = g, only the second field equation is nontrivial. One then regards flat connections modulo gauge transformations on a d-dimensional spacetime . In this way for d = 3 one may reobtain standard 2+1 gravity (with or without cosmological constant) in its Chern Simons formulation [32, 33], to mention the most prominent example: If there is no cosmological constant, the Lie algebra is the one of I SO(2, 1), while in the presence of a cosmological constant λ it is so(3, 1) and so(2, 2), depending on the sign of λ. So, in each of these cases, the “rank” r of the bundle (here just over a point) is six. The rank of the bilinear form S used in [32, 33] to construct the metric g on is three, on the other hand, saturating our general bound (24). Equation (40) just reduces to ad-invariance of that (degenerate) inner product on g with respect to those elements of g which are in its kernel k. According to Proposition 5, k is a Lie subalgebra of g. It is worthwhile to mention that all of these three Lie algebras permit an ad-invariant non-degenerate inner product. This is used to construct the action functional. However, this inner product cannot be used to construct the metric g – at least if we want to comply to the conditions in Theorem 1, where we need a degenerate inner product. Moreover, for non-zero λ, the degenerate bilinear form S of [32, 33] used to construct the metric is in fact not ad-invariant with respect to all of g, but only for k < g. 4.2. Bundles of Lie algebras. This is the case where E is some vector bundle over M, equipped with a smoothly varying Lie algebra on the fiber. The field equations (18) and (19) then just imply that has to be mapped to one (arbitrary) point X0 in M and that the A–field is a flat connection of the Lie algebra in the fiber above X0 . Gauge transformations are again just of the standard Yang-Mills type, where the point X0 cannot be changed (although there are solutions for any point X0 ∈ M). The type of the Lie algebra (although not its dimension, since this is fixed by the rank of the bundle E) can change upon a different choice of X0 . Let us further specialize to the case of abelian Lie algebras, i.e. all the fibers carry the same, trivial Lie algebra [·, ·] ≡ 0. Then, irrespective of the choice of X0 , one finds as local solutions for the A-fields, AI = df I , i = 1, . . . , r, for some functions f I . Gauge transformations correspond to AI ∼ AI + dg I for some arbitrary functions g I . We want to work out this example in full detail from scratch, making a general ansatz for S, S = 21 S I J (X)bI bJ ,

(42)

and then determine the conditions to be fulfilled such that the respective E-gravity becomes reasonable (cf. Definition 3). This then will be compared with the general results obtained in the previous section. Note that in the present case ker ρ = E, so that we cannot refer to the (necessary) conditions obtained in Subsect. 3.3; we can only refer to 3.4, which, however, had the drawback that it provided only sufficient conditions – due to the restriction on the rank of S. In the present example we will find this restriction to be also necessary. In the case of bundles of abelian Lie algebras, moreover, the condition (36) (or also (40)) is empty, since the E-Lie derivative vanishes identically. Let us now work out this example explicitly. With (8) we find that any solution for g is of the form g = 21 SI J (X0 ) df I df J

(43)

in the present case. First we require the existence of non-degenerate solutions. For this S(X0 ) needs to have at least rank d ≡ dim – and in the spirit of the remark at the


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end of Sect. 3.1 – for all X0 ∈ M. This coincides with what we found in (24). (For simplicity we take S to have positive semi-definite signature so that this condition is also sufficient.) To simplify the further discussion, let us in the following diagonalize the matrix S(X0 )I J : This can be done most easily by transition to new functions representing the solutions for A: f I (x) → MJI (X0 )f J (x). The net effect in (43) is that S(X0 ) = diag(1, . . . , 1, 0, . . . , 0), where m ≥ d entries are non-vanishing. Now a possible nondegenerate representative is obtained on ≈ Rd upon the choice f I = x J for I = 1, . . . , d and f I = 0 for I > d; then g = 21 dx µ dx µ ,

(44)

which is just the standard flat metric on Rd . Next we need to check the behavior of g under gauge transformations. We at once observe that with respect to gauge transformations (20), on a topologically trivial , any g is gauge equivalent to the identically vanishing tensor field, g ∼ 0. A likewise statement holds also for 2+1 gravity in its Chern Simons formulation discussed above or 2d dilaton gravity in its Poisson Sigma formulation recalled below – and we thus do not want to exclude such a feature. However, restricting the gauge transformations to the non-degenerate sector, i.e. disregarding all solutions to the field equations which somewhere on have det g = 0, we want the gauge transformations to boil down to diffeomorphisms of , and to nothing else. How or under what conditions is this realized in the present simple example? Indeed, suppose first that the rank m of S equals d. Then any permitted g is of the form g = 1 d µ µ 1 d µ=1 df df with det g = 0, which is equivalent to df ∧ . . . ∧ df = 0 in this 2 case. Clearly, such a g is always flat, and any two such non-degenerate solutions are related to one another by a diffeomorphism of . Now let us assume that m > d. We obtain the above flat solution g always, too, since we can just require all f I with I > d to vanish. But then any other non-degenerate solution g on Rd should be diffeomorphic to this flat solution (since the moduli space of solutions with respect to the symmetries (20) consists of one point here and in a reasonable theory of gravity these two notions coincide – up to the subtleties mentioned), i.e. be flat too. This, however, is not the case for m > d: A straightforward calculation shows that the curvature of gµν = δµν + f,µ f,ν is non-zero (for sufficiently general second derivatives of f ). This metric, however, is a gauge relative to (44) with respect to the symmetries (20), provided only m > d (and if f is kept sufficiently small, it never passes through a non-degenerate sector). In summary we find in this simple case: Proposition 6. For E being a bundle of abelian Lie algebras, dim imS = d (the only nontrivial condition in Theorem 1) is necessary and sufficient for defining an admissible E-gravity theory. Let us use the occasion to also illustrate the possible complications arising from the x-dependence of the decomposition (31): We choose = R2 , E = M × R3 , and SI J = δI J . Consider A1 = dx 1 , A2 = dx 2 , A3 = df with f ∈ C ∞ (R2 ). Thus Wx = ∂1 +f,1 (x)∂3 , ∂2 +f,2 (x)∂3 . Clearly, the following gauge parameter is in Wx⊥ : ε = ε⊥ = (f,1 , f,2 , −1) (the entries corresponding to the standard basis {∂i }3i=1 in the fiber of E). We now turn to (27): the first term in (27) vanishes identically, whatever ε. In this example, the second term is already on-shell covariant by itself (due to dX i ≈ 0). I AJ ≈ 0. By construction, but also obvious by direct insertion from above, X ∗ SI J ε⊥

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I ), AJ = f, (x) ≈ 0 However, dε⊥ (x) ∈ Wx⊥ and consequently e.g. δε⊥ g11 ≈ δI J (ε⊥ 1 1 11 1 (for nonvanishing second derivatives of f with respect to x ). This may be contrasted with the proof of Theorem 1, where the second term in (27) was found to vanish (upon an appropriate choice of a basis) no matter how complicated the x-dependence of the decomposition (31). In the present example the resulting solutions for g were found to be flat always – for admissible theories. This will change in general, if the Lie algeras are taken non-abelian.

4.3. The standard Lie algebroid and integrable distributions. We first consider the illuminating case E = T M. The field equations (18) reduce to dX i = Ai in this case, while the second set of field equations, (19), becomes an identity. Thus, in the present case, using (8) we find g ≈ 21 Sij dX i dX j = X ∗ S ,

(45)

g is just the pullback of the section S ∈ (∨2 T ∗ M) to by the map X . Let us first consider n = d, n ≡ dim M. Then by condition (24), S is non-degenerate and defines a metric on M and g is nothing but the pullback of this metric to . Note that this provides a simple example where the necessity for the addition of maximal extension of in the second item of Definition 1 or 3 becomes transparent (cf. also the discussion at the end of Subsect. 2.3): S may be quite different for different regions in M. After maximal extension, X : → M is a (possibly branched) covering map (thus locally a diffeomorphism). If one does not permit branched coverings (e.g. by requiring solutions to be either geodesically complete or that curvature invariants diverge towards the ideal boundary – assuming that this is satisfied for S on M), so as to exclude constructions of solutions such as those in [31], the moduli space of classical solutions in this example is zero dimensional, independently of the topology of . It may consist, however, of several representative solutions, parametrized by the homotopy (covering) classes of the map X . We now turn to d < n. It is obvious that for dim imS > d the conditions of a reasonable theory of E-gravity cannot be satisfied, since then we can embed some maximally extended in various ways such that X ∗ S will give non-equivalent (i.e. non-diffeomorphic) metrics on . Correspondingly, we find upon explicit inspection that in the case of T M we need saturation of the bound (24). The condition (38) just becomes an ordinary Lie derivative on M in the present context. S has to be invariant with respect to motions in directions of its kernel. This kernel is integrable according to Corollary 3. S then provides a metric in the normal bundle of the leaves of the respective foliation. This metric is invariant along the foliation, moreover; it thus projects to locally defined quotient manifolds (we are dealing with regular foliations, since the rank of S is constant; thus locally the foliation is a fibration and a quotient manifold, the base of the fibration, can be defined), on which it is nondegenerate, moreover. Such a structure on M is called a Riemannian foliation, cf. e.g. [8] (or correspondingly maybe pseudo-Riemannian foliation if the locally projected metric has indefinite signature). Let us provide a simple example of such a feature on M = R3 coordinatized by (X, Y, Z): S := dX2 + dY 2 . Clearly this is invariant with respect to ∂Z , LZ S = 0, which spans its kernel; according to Proposition 4 and Corollary 2 this is already sufficient for a check. The foliation in this simple example is a fibration over R2 (X, Y ) with fiber R Z. Thus the quotient manifold here exists even globally, and it is equipped with the non-degenerate metric dX2 + dY 2 (the projection of S).


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In this example we assumed d = 2, certainly. The representative metric g on is the unique flat metric in this case (if it exists); whatever embedding of , for a nondegenerate solution one needs dX = 0 and dY = 0, and then X and Y can be taken as possible coordinates on . The topology of permitting everywhere non-degenerate solutions for g is quite restricted; in fact, there is e.g. no compact Riemann surface for with this property. At least locally we can identify a solution on with the quotient manifold constructed above. Different embeddings of (provided homotopic along the Z-fibers) are just gauge equivalent; and this is also obvious from the explicit solutions since in any solution as constructed above g is independent of the function Z. In the present context one may certainly also easily obtain non-flat metrics g on ; e.g., just multiply the above S by some non-trivial function f (X, Y ). As the standard Lie algebroid E = T M in general is one of the main models for a Lie algebroid—the guideline for it, besides ordinary Lie algebras—it may be used also in the present context for further orientation in the general Lie algebroid case. Equipping a Lie algebroid E with an E-tensor S satisfying the conditions in Theorem 1 might thus be called equipping E with an E-Riemannian foliation. Qualitatively, the situation does not change much when we regard integrable distributions D ⊂ T M: First we observe that according to (18) X maps completely into a leaf L of the distribution. The moduli space of classical solutions will now consist of homotopy classes of such maps. But otherwise the discussion is essentially as before, just – for any fixed base map X – with T M replaced by D|L ∼ = T L. Note that now S is not just a covariant 2-tensor on M. Rather, it is the equivalence class of such 2-tensors, where two of them are to be identified, if they coincide upon contraction with arbitrary vectors tangent to the distribution D. As for an explicit example we may extend the previous one for T M by adding one or more further directions to M, considering the previous M as a typical leaf L. Proposition 7. For Lie algebroids with injective anchor map ρ the conditions in Theorem 1 are also necessary.

5. Specializing to Poisson Gravity 5.1. Compatibility and integrability. We first want to specialize (40) to the Poisson case. Here E = T ∗ M, M the given Poisson manifold, and S becomes a symmetric, contravariant 2-tensor on M, S = 21 S ij ∂i ∂j . Correspondingly, S : T ∗ M → T M and imS ⊂ T M defines a distribution on M. In contrast to the distrubtion of P , which induces the symplectic foliation of M, this distribution is not necessarily integrable. To provide a non-integrable example, just consider d = 2, M = R3 with the trivial Poisson bracket – so that the condition (40) becomes empty – and take S = (∂1 + X 3 ∂2 )∂3 . The structure defined on M by S (with constant rank) is called a sub-Riemannian (or in the indefinite case then sub-pseudo-Riemannian) structure, cf. e.g. [7]. According to Theorem 1 it has to satisfy a particular compatibility condition with the Poisson structure on M. We will now make this explicit under a further assumption, namely that imS is an integrable distribution. (In the context of sub-Riemannian structures this is usually assumed to not be the case; still, here we assume this for simplicity.) Then at least locally one may characterize the corresponding leaves, which we want to denote by R, by the level set of some functions f α ∈ C ∞ (M), α = 1 . . . n − d, where n ≡ dim M. (Likewise the symplectic leaves will be denoted by L or (L, L ), where L is the symplectic

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2-form induced on L by the Poisson tensor P on M; locally any leaf L can be characterized as the level set of some Casimir functions C I , I = 1 . . . n − k, where k = rankPX for any point X ∈ L.) For any symmetric or antisymmetric (say contravariant) 2-tensor T , the annihilator of its image imT ⊂ T M is its kernel, ker T ⊂ T ∗ M. In the above case we thus can span ∗ α n−d the kernel of S by df α , ker S = df α n−d α=1 . The quotient map S : T M/df α=1 → imS has an inverse. Since at any point T ∗ R may be identified with T ∗ M/df α n−d α=1 , this implies that the leaves R are equipped with some pseudo-Riemannian metric. Let us denote the latter by GR . different ways, by symplectic leaves, M = So, in this scenario, M is foliated in two (L, L ), and by Riemannian ones, M = (R, GR ). While the former foliation may be quite wild, the leaves of the latter ones all have at least the same dimension, namely d ≡ dim . (We will content ourselves here with specializing the conditions of Theorem 1 to the Poisson case – under the additional assumption of integrable imS .) The situation simplifies further if the pseudo-Riemannian foliation is even a fibration, such that in particular the quotient manifold M/R is well-defined (and leaves R lying dense in some higher dimensional submanifolds of M are excluded). The two foliations (or, more precisely, the two structures induced on M by P and S, respectively) are not independent from one another, due to condition (40): Proposition 8. Let (M, P) be a Poisson manifold and S ∈ ∨2 T M define a d-dimensional fibration into pseudo-Riemannian leaves (R, GR ) as described above. Then (M, P, S) define a reasonable theory of Poisson gravity, if Lv f S = 0

∀f ∈ C ∞ (M/R),

(46)

where vf ≡ {·, f } = P ij f,j ∂i is the Hamiltonian vector field of the function f and L denotes the usual Lie derivative. To prove this statement it suffices to check that for closed 1-forms α ∈ 1 (M) one has ELα = Lρ(α) – since ρ(df ) = vf .4 Due to Proposition 4 it is sufficient to check (46) for the set of n − d functions f α introduced above. Let us add two remarks, valid in the simplified circumstances of the above proposition. First, according to Proposition 5, ker S = df α n−d α=1 defines a Lie subalgebroid. It is isomorphic to the Poisson type Lie algebroid T ∗ (M/R), where, by construction, the projection π : M → M/R is a Poisson map (since f α can be thought of as pullback of at least locally defined functions on M/R – and since ker S is a Lie subalgebroid of T ∗ M)5 . Second, according to Corollary 3, P (ker S ) defines an integrable distribution; it is spanned by the vector fields vf appearing in Eq. (46). In general this foliation is distinct from the symplectic and from the Riemannian foliation. Clearly, its leaves always lie in symplectic leaves L, however. Its relation to the Riemannian leaves is given by Proposition 9. The leaves of P (ker S ) lie inside the intersection L ∩ R of symplectic and Riemannian leaves, iff R are coisotropic submanifolds of (M, P). Proof. vf α ∈ imS ⇔ ker S , vf α = 0 ⇔ {f α , f β } = 0.

As a rather obvious consequence of this we find 4 5

I am grateful to A. Kotov for discussions of this point. Cf. also Eq. (54) below. I am grateful to A. Weinstein for this remark.


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Proposition 10. The (pseudo-)Riemannian structures on the leaves R may be chosen independently from one another (still varying smoothly certainly) iff the leaves R are coisotropic submanifolds of the Poisson manifold M. In this case the metric on each leaf R has k independent Killing vector fields, where k = dim P (ker S ). Note that in the present Poisson case this does not imply automatically that also the metric g, defined on , has Killing vector fields. In fact, by the field equations, is mapped into the symplectic leaves, which in general are different from the Riemannian ones; moreover, g is not just an ordinary pullback. Cf. also Sect. 5.5 below for illustration. In Proposition 8 we assumed a fibration. More generally, for a foliation one has a likewise statement, if one uses for any X ∈ M sufficiently small neighborhoods UX and replaces the above functions f by functions constant along connected components of R ∩ UX for any leaf R. We conclude this subsection with an obvious remark: Corollary 4. If each leaf of the foliation induced by S consists of a union of symplectic leaves (the leaves induced by P), Condition (40) or (46) is fulfilled trivially. This just corresponds to the particular case k = 0 in Proposition 10. 5.2. Examples in two dimensions. Most of the studied 2d gravity models result from the following simple choice of (M, P, S) [5]: Take M = R3 with linear coordinates (Xa , X3 ), where the index a runs over two values, which conventionally we denote by 1 and 2 when we consider Euclidean signature gravity and by + and − for Lorentzian signature; these indices will be raised and lowered by means of the standard flat Riemannian or Lorentzian metric ηab , furthermore. The Poisson structure is then defined by {X a , Xb } = εab W (X c Xc , X3 ) and {X a , X3 } = εab Xb ,

(47)

where εab are the (contravariant) components of the antisymmetric ε-tensor and W is any smooth two-argument function. With an ansatz S := λab (X)∂a ∂b , ker S = dX 3 . Equation (47) shows that X3 generates (Lorentzian or Euclidean) rotations in the planes of constant X3 . According to Proposition 8, S has to be invariant with respect to these “rotations”. As one may easily convince oneself, requiring S to be smooth on all of R3 leaves only S = 21 γ (X c Xc , X3 ) ηab ∂a ∂b

(48)

for some nonvanishing two-argument function γ . In most applications (cf. below) either γ = 1 or γ depends on X3 only. In this example, the symplectic foliation is given by variation of the integration constant in the solution of the first order differential equation du/dv = W (u, v), replacing u X3 by Xa Xa and v by X3 [9]. E.g. for W = V (X3 ) one finds Xa Xa − V (v)dv = const. as for the symplectic leaves. They are generically two-dimensional, except at simultaneous zeros of Xa and W (0, X 3 ), where the Poisson tensor vanishes altogether. On the two-dimensional leaves the symplectic form can be written as L = dX 3 ∧ dϕ, where in the Euclidean case ϕ is the standard azimuthal angular variable around the X 3 axis and in the Lorentzian case e.g. ln X+ .

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The pseudo-Riemannian leaves are characterized by X 3 = const in the above ansatz 1 and GR = 2γ ηab dX a dX b . Note that the intersection of the symplectic and the Riemannian leaves are rotationally invariant. So, e.g. for Euclidean signature they are generically circles in the X3 -planes around the origin X a = 0. 5.3. Relation to 2d Lagrangians. We want to be rather brief here due to the rather extensive literature on this subject (cf. e.g. [2] and [1] for two recent reviews). Still it is illustrative to show how intricate the relation to some more established actions may be. of all we just mention that geometrical action functionals of the form L[g] = First 2 x √| det g|f (R) or also d

d2 x | det g| F (R, τ a τa ) (49) L[g, τ a ] ∼ L[ea , ω] =

may be covered by the choice in the previous subsection with γ ≡ 1. Here R denotes the Ricci scalar (in two dimensions this contains all information about the curvature tensor), in the first case of the torsion-free Levi-Civita connection of g and in the second case of the connection ω with torsion scalar τ a . The functions f and F are the Legendre transforms of the functions V and W of the preceding subsection, respectively. In these examples one identifies Aa with a zweibein ea and A3 with the single non-trivial component ω of the spin connection, using the action functional (1) Poisson Sigma Model. We refer to [1] for a detailed study of this relation, including a discussion of the situation when the functions f or F have Legendre transforms locally only. Secondly, also dilaton-like Lagrangians

d2 x | det g| φR + W ((∇φ)2 , φ) (50) L[g, φ] =

can be covered. At least if W in (50) is at most linear in its first argument, there are various ways of doing so. In one of these, one usually performs a dilaton-dependent conformal transformation of g and then identifies Ai in (1) with the zweibein and spin connection of the new metric) and X3 with the dilaton φ (while X a become Lagrange multipliers for torsion zero; in this case WPoisson (X a Xa , X3 ) = Wabove (0, X3 ), cf. e.g. [9] for further details). Closer to the present point of view is to regard this as a one-step-procedure, using the conformal factor as an X3 -dependent γ in (48). There is also another route to (50), which moreover works for general W (which now agrees with the function introduced in the Poisson structure within the previous subsection), cf. e.g. [2]. We believe that it is worthwhile to be a bit more explicit in this case. We first choose to identify g with g = 21 ηab Aa Ab ,

(51)

corresponding to γ = 1 in (48). Then let ω(A) denote the solution of dAa +ε ab ω∧Ab = 0 – such a solution always exists and is even unique (interpreting Aa as zweibein, this is the corresponding torsion-free connection 1-form). Next, shift A3 by ω(A) in (1), 3 := A3 + ω(A). Then, with the previous choice of P, the action functional A3 → A takes the form (again we replace X 3 by φ) ω(A) ∧ dφ + 21 ε ab Aa ∧ Ab W (X c Xc , φ) L[X, A] = 3 ∧ dφ + εab X a Ab . +A (52)


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Classically it is always permitted to eliminate fields of an action functional by their own 3 and X a : variation with respect to A 3 determines Xa field equations. We apply this to A uniquely, and vice versa. Implementing this into (52), the last term vanishes and X a is replaced by −εa b Abµ g µν ∂ν φ. After a partial integration in the first term, this is now seen to coincide with (50). Note that the elimination of X a required that Aaµ has an inverse (this is reflected by the use of g µν in the above explicit expression), which is legitimate after one identifies g as in (51); the equivalence of action functionals then certainly may be expected also only on the metric non-degenerate solutions. If in the first term of (50) φ is replaced by some function U (φ), one still may use the functional (1) along the above lines, where now X 3 = U (φ). Certainly this works only in regions of φ where U is monotonic. If U has several monotonic parts, one needs to use several action functionals to recover all the classical solutions. However, at least if W in (50) is at most linear in its first argument, W = V1 (φ) + V2 (φ)(∇φ)2 , it may be shown [34] under rather mild conditions on U , V1 , and V2 , that for any solution with φ taking values within one of the monotonic sectors of U , the classically allowed values of φ remain in this sector. 5.4. Further 2d examples. Up to now, 2d models were considered merely with the 3d Poisson manifold (M3 , P (W ) ) as provided in Subsect. 5.2 – with only one exception, namely the inclusion of Yang-Mills fields [9]. In this case the total Poisson manifold has the form (M, P) = M3 × g∗ , where the function W is permitted to depend also on the Casimir functions of the Lie Poisson manifold g∗ . If γ in (48) depends on X a Xa , X 3 and again these Casimirs only, then also the condition (46) in Proposition 8 is satisfied and (M, P, S) defines a reasonable theory of 2d Poisson gravity. But certainly many more examples or models can be constructed by the general methods of the present paper, even in the realm of two dimensions. Let us use this fact to construct a simple PSM, the local solution space of which coincides with any given parametric family of 2d metrics, say g(Cα ) = 21 gµν (Cα , x 1 , x 2 ) dx µ dx ν in some local coordinate system x µ and with Cα , α = 1 . . . N local parameters in some given Ndimensional manifold M of moduli. Just take M = M × R2 , equipping the former factor with zero Poisson bracket and R2 with the standard one: (q, p) ∈ R2 , {q, p} = 1, {q, q} = 0 = {p, p}. Now choose S as S = 21 g11 ∂q ∂q + 21 g22 ∂p ∂p − g12 ∂q ∂p , where all the coefficient functions gµν are evaluated at (Cα , q, p). Due to Corollary 4 this provides a reasonable theory of 2d gravity. With the simple field equations dq + Ap = 0, dp − Aq = 0, dC α = 0, following from (3) – and the fact that locally the second set of field equations (4) may be satisfied without the addition of any further moduli parameters when using the symmetries (6) – one obtains the desired result. In [10] it was shown that for (50) one finds the (local) solution space of the exact string black hole for no choice of W ; a model which does the job is trivially included in our framework, as just demonstrated for an arbitrary family of 2d metrics. (This can be easily adapted to also include the given parametric dependence of the dilaton field φ). Likewise constructions also work in any dimension d, moreover. In [13, 14], on the other hand, the Poisson bracket on R3 recalled above was modified, in particular breaking (or “κ-deforming”) its rotational invariance. This is used hand in hand with the unmodified notion (51) for the metric. Correspondingly, the conditions for a theory to be admissible or “reasonable” in our sense and as made precise in Defintion 3 are violated – with the corresponding features, too.

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To provide an even simpler example of what happens when these conditions are violated, consider e.g. R3 (q, p, C) with P = 21 ∂q ∧ ∂p and S = ∂q ∂p + f (q, p)∂C ∂C for some fixed, non-constant function f . Here ker S is trivial, and ker ρ = dC. Now the necessary conditions of Proposition 3 are violated: Indeed, take W = dq, dp + dC such that W ⊥ = d(p − C); now (36) is violated due to L∂q S (dp + dC, dp + dC) ∝ ∂q f = 0. The general local solution for g = 21 S ij Ai Aj , on the other hand, takes the form g = dq 2 + dp 2 + f dλ2 , where λ is an arbitrary function of q and p and pure gauge according to (6). Clearly, for λ ≡ 0 the metric is flat, while otherwise it is generically not. (Note that this example of a non-reasonable theory works also for f = 1, while then the necessary conditions found in Proposition 3 are too weak to exclude this case). As we want to stress in the present note, the action functional (1) of the PSM alone certainly does not entail any notion of a metric g on ; and the identification of g implicitly introduces or requires further structures. These structures are identified as a symmetric contravariant 2-tensor on M in the present paper. Moreover, the 2-tensor may not be chosen at will, but has to satisfy a compatibility condition with the chosen Poisson structure. The present framework is independent of coordinates on M, thus also permitting a coordinate independent discussion of compatibility (cf. [15]).

5.5. Relation of Killing vectors to target geometry. It has been observed [9] that all solutions of any gravity model resulting from the structures as defined in Subsect. 5.2 – and thus, as a consequence, of the action functionals (49) and (50) – have at least one local Killing vector field v ∈ (T ). Moreover, along these lines, X 3 is constant; correspondingly, the Killing lines are apparently related to the one-dimensional intersection of the symplectic with the pseudo-Riemannian leaves. (Recall that classically only maps from into a symplectic leaf is compatible with (3), while the pseudo-Riemannian leaves are characterized by X3 = const.) This is qualitatively different from the construction of models in Subsect. 5.4 for any given family of metrics g on , in particular also metrics without Killing symmetry: there the symplectic foliation agreed with the pseudo-Riemannian one and the intersection was two-dimensional. It is the purpose of the present subsection to relate the appearance of a Killing vector field in the models of Subsect. 5.3 to the target geometry (M, P, S) defined in Subsect. 5.2. We will make this relation explicit in the somewhat simplified scenario of Eq. (30), which, on behalf of the field equations (3), implies that locally we may identify with a symplectic leaf L and we likewise may fall back on (28) and (29). In the present case ker S = dX 3 . Let us denote a local Casimir function of the symplectic foliation by C, then ker ρ = ker P = dC. Now, for any point X ∈ imX ⊂ M there is a unique ∗ M. Thus, for any µ ∈ C ∞ (M) we λ(X) such that (λdX 3 + dC)X(x) ∈ imϕx ⊂ TX(x) 3 have σ := µ(λdX + dC) ∈ imϕ. Let us calculate the variation of g with respect to X ∗σ : δX ∗ σ g ≈ 21 X ∗

Lσ S

E

ij

Ai Aj = 21 X ∗

LµdC S

E

ij Ai Aj ,

(53)

where we made use of Eq. (40). We know that the left-hand side may be rewritten as the Lie derivative of a (for fixed σ ) unique (local) vector field v on , since σ was in the image of imϕ and we then may employ (21). Thus, we will have accomplished our task of showing the existence of a local Killing vector field provided that the right-hand side of Eq. (53) can be made to vanish. Note that in this process the ambiguity in µ ∈ C ∞ (M)


499

has to be cut down since in the space of Killing vector fields is a finite dimensional vector space (and not a module over the functions). For any α ∈ T ∗ M, one has Lα S = Lρ(α) S + P is (dα)st S tj ∂i ∂j .

E

(54)

Using α = µdC and the fact that this α is in the kernel of ρ, we obviously only need to ensure that dα = dµ ∧ dC = 0 which is satisfied, iff µ = µ(C). Since on the classical solutions C = const, this corresponds only to a constant rescaling of the Killing vector. This concludes our proof for the case of (30). To show the appearance of this Killing vector field more generally, one proceeds similarly to the proof of Theorem 1, replacing (41) by E = dC ⊕ W¯ 0 ⊕ dX 3 (which may be used at least locally on M and for dC ∝ dX3 ). 6. Conclusion In the present paper we discussed conditions which can be used to endow a large class of topological models with some gravitational interpretation. Concerning the models we only specified their field equations and local symmetries in the present paper, so as to remain as general as possible (while still action functionals can be constructed for them, at least if one permits auxiliary fields). They could be assigned to any d-dimensional spacetime manifold and to any Lie algebroid E used as target. On some more abstract level the field equations state that the maps ϕ : T → E should be Lie algebroid p morphisms (i.e. the induced map : E (M) → p (M) should be a chain map), while the symmetries correspond to a notion of homotopy of such morphisms (cf. [25] for more details). Our ansatz for the metric g on was that it should be the image of a symmetric, covariant E-2-tensor S, S ∈ (∨2 E ∗ ), with respect to the naturally extended map just introduced above. (g = (S); in explicit formulas this gives (8).) We attempted to clearly formulate the desiderata of what we want to call a reasonable theory of E-gravity (defined over ) – cf. Defintion 3. Essentially it was the requirement that on solutions the gauge transformations of the model when applied to a non-degenerate metric g should boil down to just diffeomorphisms of . In Theorem 1 we summarized our main result: If S has a kernel of maximal rank so as to be still compatible with metric non-degeneracy (cf. Eq. (24)), dim imS = d,

(55)

then it suffices to have S invariant with respect to any section in its kernel, cf. Eq. (40). This last condition was found also to be necessary (to be precise, cf. Corollary 1, this was shown only under the assumption dim ker ρ ≤ r − d and by requiring the somewhat strengthened condition (35)). We believe that Eq. (55) is necessary, too; however, in the present paper, we managed to show this only under the relatively strong additional assumption that ker ρ is trivial (cf. Proposition 7) or that it is all of E and the bracket abelian (cf. Sect. 4.2). The invariance condition (40) may be reformulated as a kind of ad-invariance of the degenerate inner product induced by S: Let (ψ1 , ψ2 ) := S, ψ1 ⊗ ψ2 , then Eq. (40) is equivalent to ρ(ψ) · (ψ1 , ψ2 ) = ([ψ, ψ1 ], ψ2 ) + (ψ1 , [ψ, ψ2 ])

(56)

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valid ∀ψ ∈ (ker S ) and for all sections ψ1 , ψ2 ; here · denotes application of the vector field to the respective function to follow. It is decisive that this condition has to hold only for the Lie subalgbroid ker S ; if e.g. it were to hold for any section of E and if the inner product was non-degenerate, then necessarily ρ ≡ 0, i.e. one would be left with a bundle of Lie algebras equipped with a fiberwise ad-invariant inner product. Moreover, as illustrated in Sect. 4.1 and 4.2, even for ρ ≡ 0 a kernel of S is/will be required (for r > d) so as to really yield a reasonable (cf. Definition 3) notion of a metric g on by means of g = (S). General ad-invariance of an inner product appears in a slight modification of the notion of a Lie algebroid, namely the Courant algebroid; modifying the bracket [·, ·] by a symmetric contribution stemming from the inner product, one can have ad-invariance even without ρ ≡ 0 and despite non-degeneracy of the inner product, cf. e.g. [35, 20]. Let us remark parenthetically that from a more recent perspective, cf. [36], it is also natural to drop the non-degeneracy condition; Lie algebroids then appear as completely degenerate Courant algebroids. In any case, it may be reasonable to generalize the considerations of the present paper to the realm of Courant algebroids. Closer to the present structure are however possible quotient constructions. The guideline may be taken from the standard Lie algebroid E = T M. Assuming ker S to define a fibration, then S obviously is just the pullback (by the projection) of a non-degenerate and unrestricted metric on the base or quotient manifold. This generalizes: As one may show and as shall be made more explicit elsewhere, one may construct a reasonable theory of E-gravity just by means of a Lie algebroid morphism from E to some Lie algebroid E¯ of rank d, where the latter is equipped with an arbitrary non-degenerate fiber metric. For E = T M the structure governing a gravity model was found to be the one of a Riemannian foliation (cf. Sect. 4.3). The general structure required then may be viewed as the corresponding Lie algebroid generalization, i.e. an E-Riemannian foliation. In the case of E = T M, the metric g was found to locally coincide just with the metric on the locally defined quotient of the Riemannian foliation. In general, the relation between S and g is more subtle, and worth further investigations. The usually considered gravity models in three spacetime dimensions result from E being a Lie algebra (cf. Sect. 4.1). For E the cotangent bundle of a Poisson manifold, finally, a reasonable choice of S boiled down to the choice of a sub-Riemannian structure, compatible with the given Poisson structure. Under the assumption that imS defines an integrable distribution, a sub-(pseudo)-Riemannian structure gives rise to a foliation of M by d-dimensional (pseudo)-Riemannian leaves. (This is not to be confused with the foliation generated by ρ(ker S ), cf. the discussion after Proposition 8.) The invariance condition (56) then was tantamount to requiring that S – or the degenerate inner product induced by it – is invariant with respect to the Hamiltonian vector fields of the (possibly only locally defined) functions characterizing the pseudo-Riemannian foliation (cf. 8). Known models of 2d gravity were seen to be a special case of the much more general construction of the present paper. Also we were able to relate the existence of a Killing vector of g in the previously known models to the invariance condition (56) together with the particularities of the two foliations defining these models. In view of Proposition 10, it may be rewarding to strive at a generalization of these results. It is worthwhile also to generalize our consideration to the existence of a vielbein ea and a spin connection ωa b – instead of just a metric g, which results from the vielbein


501

according to g = ea eb ηab , i.e. to determine the conditions such that some of the 1-form fields AI permit identification with the Einstein-Cartan variables (ea , ωa b ). The present paper is meant to open a new arena for studying models of gravity. It should be possible to extend the in part extensive studies of lower dimensional models of gravity – on the classical and on the quantum level – to a much larger set of theories, including topological gravity models in arbitrary spacetime dimensions. Finally, we mention again that within this paper we focused on the field equations and symmetries corresponding to Lie algebroids. Depending on the spacetime dimension and possibly additional structures chosen, there may be several different action functionals producing the same desired theory, at least in a subsector. To give a simple example: If E is a Lie algebra g, the field equations correspond just to flat connections and the symmetries to the standard non-abelian gauge transformations. This may be described without any further structure and restriction to spacetime by a BF -theory. For d = 3 and with g permitting a non-degenerate, ad-invariant inner product, one may also take a Chern-Simons action; this even has the advantage of getting by without additional auxiliary fields – but on the other hand requires additional structures and restriction to a specific spacetime dimension. This generalizes also to the present context. We intend to make this more explicit elsewhere [22]. Merely on the level of the field equations and symmetries, one may already address the interesting but also challenging question of observables. Since the theories constructed are topological, they are expected to capture important and interesting mathematical information–displaying an interplay between the topology of and the target Lie algebroid and, if one also involves g, the target E-Riemannian structure. As suggested by the particular case E = g, one should, among others, introduce a generalization of the notion of a Wilson loop for this purpose. Acknowledgements. First of all I need to acknowledge the profit I have drawn from a collaboration with Martin Bojowald and Alexei Kotov on closely related subjects, which we are going to publish seperately [25]. In addition, I am grateful for discussions with D. Grumiller, M. Grützmann, W. Kummer, D. Länge, S. Mignemi, D. Roytenberg, H. Urbantke, D. Vassilevich and in particular with A. Weinstein.

References 1. Strobl, T.: Gravity in Two Spacetime Dimensions. http://arXiv.org/abs/hep-th/0011240. Habilitationsschrift, 224 pages 2. Grumiller, D., Kummer, W., Vassilevich, D. V.: Dilaton Gravity in Two Dimensions. http://arXiv.org/ abs/hep-th/0204253 3. Birmingham, D., Blau, M., Rakowski, M., Thompson, G.: Topological field theory. Phys. Rept. 209, 129–340 (1991) 4. Schaller, P., Strobl, T.: Quantization of Field Theories Generalizing Gravity Yang- Mills Systems on the Cylinder. http://arXiv.org/abs/gr-qc/9406027 5. Schaller, P., Strobl, T.: Poisson structure induced (topological) field theories. Mod. Phys. Lett. A9, 3129–3136 (1994), [arXiv:hep-th/9405110] 6. Ikeda, N.: Two-dimensional gravity and nonlinear gauge theory, Ann. Phys. 235, 435–464 (1994), [arXiv:hep-th/9312059] 7. Langerock, B.: A connection theoretic approach to sub-Riemannian geometry. http://arXiv.org/ abs/math.DG/0210004 8. Molino, P.: Riemannian Foliations. Basel: Birkhäuser, 1988 9. Klösch, T., Strobl, T.: Classical and quantum gravity in (1+1)-dimensions. Part 1: A unifying approach. Class. Quant. Grav. 13, 965–984 (1996), [arXiv:gr-qc/9508020] 10. Grumiller, D., Vassilevich, D.: Non-existence of a Dilaton Gravity Action for the Exact String Black Hole. http://arXiv.org/abs/hep-th/0210060

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11. Klösch, T., Strobl, T.: Classical and quantum gravity in 1+1 dimensions. Part 2: The universal coverings, Class. Quant. Grav. 13, 2395–2422 (1996), [arXiv:gr-qc/9511081] 12. Klösch, T., Strobl, T.: Classical and quantum gravity in (1+1)-dimensions. Part 3: Solutions of arbitrary topology, Class. Quant. Grav. 14, 1689–1723 (1997) 13. Mignemi, S.: Two-dimensional Gravity with an Invariant Energy Scale. hep-th/0208062 14. Mignemi, S.: Two-dimensional Gravity with an Invariant Energy Scale and Arbitrary Dilaton Potential. hep-th/0210213 15. Grumiller, D., Kummer, W., Vassilevich, D. V.: A Note on the Triviality of Kappa-deformations of Gravity. hep-th/0301061 16. Kontsevich, M.: Deformation Quantization of Poisson Manifolds, i. http://arXiv.org/abs/q-alg/ 9709040 17. Cattaneo, A. S., Felder, G.: A path integral approach to the Kontsevich quantization formula. Commun. Math. Phys. 212, 591 (2000), [arXiv:math.qa/9902090] 18. Schomerus, V., Strobl, T.: Unpublished notes 19. Klimcik, C., Strobl, T.: WZW-Poisson manifolds. J. Geom. Phys. 43, 341–344 (2002), [math.sg/0104189] 20. Severa, P., Weinstein, A.: Poisson geometry with a 3-form background. Prog. Theor. Phys. Suppl. 144, 145–154 (2001), [math.sg/0107133] 21. Pelzer, H., Strobl, T.: Generalized 2d dilaton gravity with matter fields. Class. Quant. Grav. 15, 3803–3825 (1998), [gr-qc/9805059] 22. Strobl, T.: Algebroid Yang-Mills theories. In preparation 23. da Silva, A. C., Weinstein, A.: Geometric Models for Noncommutative Algebras, Vol. 10, of Berkeley Mathematics Lecture Notes. Providence, RI: American Mathematical Society, 1999. Available at http://www.math.berkeley.edu/ alanw/ 24. Boyom, M. N.: The Cohomology of Koszul-Vinberg Algebras. arXiv:math.DG/0202259 25. Bojowald, M., Kotov, A., Strobl, T.: Lie algebroid morphisms, Poisson sigma models, and off-shell closed symmetries. In preparation 26. Sullivan, D.: Inst. des Haut Etud. Sci. Pub. Math 47, 269 (1977) 27. Izquierdo, J. M.: Free differential algebras and generic 2D dilatonic (super) gravities. Phys. Rev. D59, 084017 (1999), [hep-th/9807007] 28. Schaller, P., Strobl, T.: Diffeomorphisms versus nonabelian gauge transformations: An example of (1+1)-dimensional gravity. Phys. Lett. B337, 266–270 (1994), [http://arXiv.org/abs/hep-th/9401110] 29. Teitelboim, C.: Gravitation and hamiltonian structure in two space-time dimensions. Phys. Lett. B126, 41 (1983) 30. Jackiw, R.: Liouville field theory: A two-dimensional model for gravity. In: Christensen, S. (eds), Quantum Theory of Gravity. Essays in Honor of the 60th Birthday of Bryce S. DeWitt, Bristol: Hilger, 1984, pp. 403–420 31. Klösch, T., Strobl, T.: A global view of kinks in 1+1 gravity, Phys. Rev. D57, 1034–1044 (1998), [arXiv:gr-qc/9707053] 32. Achucarro, A., Townsend, P. K.: A chern-simons action for three-dimensional anti-de sitter supergravity theories. Phys. Lett. B180, 89 (1986) 33. Witten, E.: (2+1)-dimensional gravity as an exactly soluble system. Nucl. Phys. B311, 46 (1988) 34. Düchting, N., Strobl, T.: Unpublished notes 35. Roytenberg, D.: Courant algebroids, derived brackets and even symplectic supermanifolds. PhD thesis, UC Berkeley, 1999, http://arXiv.org/abs/math.DG/9910078 36. Alekseev, A., Strobl, T.: Courant algebroids from current algebras. In preparation Communicated by G.W. Gibbons


Communications in


On the Boltzmann Equation for Diffusively Excited Granular Media I.M. Gamba1 , V. Panferov1, , C. Villani2 1

Department of Mathematics, The University of Texas at Austin, Austin, TX 78712-1082, USA. E-mail: [email protected] 2 UMPA, ENS Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07, France Received: 29 May 2003 / Accepted: 7 October 2003 Published online: 19 February 2004 – © Springer-Verlag 2004

Abstract: We study the Boltzmann equation for a space-homogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing. Under the assumption that the initial datum is a nonnegative L2 (RN ) function, with bounded mass and kinetic energy (second moment), we prove the existence of a solution to this model, which instantaneously becomes smooth and rapidly decaying. Under a weak additional assumption of bounded third moment, the solution is shown to be unique. We also establish the existence (but not uniqueness) of a stationary solution. In addition we show that the high-velocity tails of both the stationary and time-dependent particle distribution functions are overpopulated with respect to the Maxwellian distribution, as conjectured by previous authors, and we prove pointwise lower estimates for the solutions.

0. Introduction In recent years a significant interest has been focused on the study of kinetic models for granular flows [10, 23, 20]. Depending on the external conditions (geometry, gravity, interactions with surface of a vessel) granular systems may be in a variety of regimes, displaying typical features of solids, liquids or gases and also producing quite surprising effects [37]. Finding a systematic way to describe such systems under different conditions is a physical problem of considerable importance. At the same time, recent developments in this area gave rise to several novel mathematical models with interesting properties. In the case of rapid, dilute flows, the binary collisions between particles may be considered the main mechanism of inter-particle interactions in the system. In such cases methods of the kinetic theory of rarefied gases, based on the Boltzmann-Enskog equations have been applied [25, 24, 21]. Current address: Department of Mathematics and Statistics, University of Victoria, Victoria B.C. V8W 3P4, Canada

504

I.M. Gamba, V. Panferov, C. Villani

A very important feature of inter-particle interactions in granular flows is their inelastic character: the total kinetic energy is generally not preserved in the collisions. Therefore, in order to keep the system out of the “freezing” state, when particles cease to move and the system becomes static, a certain driving mechanism, supplying the system with energy, is required. Physically realistic driven regimes include excitation from the moving boundary, through-flow of air, fluidized beds, gravity, and other special conditions. We accept a simple model for a driving mechanism, the so-called thermal bath, in which we assume that the particles are subject to uncorrelated random accelerations between the collisions. Such a model was studied in [41] in the one-dimensional case, and in [38] in general dimension. We study the model [38] in the space-homogeneous regime, described by the following equation: ∂t f − µ v f = Q(f, f ),

v ∈ RN ,

t > 0.

(0.1)

Here f is the one-particle distribution function (particle density function in the phase space), which is a nonnegative function of the microscopic velocity v and the time t; we shall assume N ≥ 2 (dimension 1 could be treated as well but would require a few notational changes). On the right-hand side of Eq. (0.1) there is the inelastic Boltzmann-Enskog operator for hard spheres (the details of which are given below); the term −µ v f , µ = const, represents the effect of the heat bath. Without loss of generality we can set µ = 1 (see Sect. 1.5), which we will from now on assume. In the sequel, we shall often abbreviate v into just . One of the interesting features of the model (0.1) is the fact that it possesses nontrivial steady states described by the balance between the collisions and the thermal bath forcing. Such steady states are given by solutions of the equation µ v f + Q(f, f ) = 0,

v ∈ RN .

(0.2)

It is interesting to point out that such solutions may exist only in the case of inelastic collisions, at least in the class of integrable functions with finite second moment: see Sect. 2. Solutions of (0.2) have been studied in [38] by means of formal expansions. The same problem was also studied in [9] and in [6], for a different kind of interactions, namely the Maxwell pseudo-particle model [5, 26, 27], by methods of expansions and the Fourier transforms, respectively. In reference [11] the rigorous existence of radially symmetric steady solutions for the Maxwell model was established. The aim of this study is to develop a rigorous theory for the inelastic hard sphere model, and to investigate the regularity and qualitative properties of the solutions. We prove that Eq. (0.1) has a unique weak solution under basic assumptions that the initial data have bounded mass and kinetic energy, and satisfy some additional conditions (bounded entropy for existence, L2 (RN ) for regularity, and bounded third moment in |v| for uniqueness). The thermal bath (diffusion) term in (0.1) is responsible for the parabolic regularity of solutions: the weak solutions become smooth, classical solutions after an arbitrarily short time. We apply generally similar techniques, based on elliptic regularity, to treat the steady case. Finally, we establish lower bounds, for both steady and time-dependent solutions, proving that the distribution tails are “overpopulated” with respect to the Maxwellian, as was suggested in [38]. The lower bound for steady solutions is given by a “stretched exponential” A exp(−a|v|3/2 ), with a = a(α, µ). In the time-dependent case the bound holds with A = A(t), where A(t) is a generally decaying function of time.

On the Boltzmann Equation for Diffusively Excited Granular Media

505

We emphasize that the appearance of the “3/2” exponent is a specific feature of the hard sphere model with diffusion, and could be predicted by dimensional arguments (cf. [38]). On the other hand, the Maxwell model with diffusion results in a high-velocity tail with asymptotic behavior C exp(−c|v|), see [6]. As a general rule, the exponents in the tails are expected to depend on the driving and collision mechanisms [2, 16, 17, 7]. In fact, deviations of the steady states of granular systems from Maxwellian equilibria (“thickening of tails”) is one of the characteristic features of dynamics of granular systems, and has been an object of intensive study in recent years [30, 28, 36, 33]. We remark that the “3/2” bound has rather important practical implications as well. In particular, it indicates that the approximate solutions based on the truncated expansion of the deviation from the Maxwellian into Sonine polynomials [38, 9, 33] could only be valid for moderate values of |v|2 . Any conclusions about the tail behavior drawn from such an expansion should be questioned. Indeed, since the deviation function is growing rapidly for |v| large (it is in the weighted L1 space, but not in L2 !), the Sonine polynomial expansion should in general be expected to have poor approximation properties in this region. The paper is organized as follows. The first section contains the preliminaries, where we introduce the inelastic collision operator and establish several basic identities which are important in the sequel. In Sect. 2 we establish the bounds for the energy and entropy of solutions. In Sect. 3 we study the moments of the distribution function by analyzing the moment inequalities for Eqs. (0.1) and (0.2). The key point in analyzing the moments is the so-called Povzner inequalities, well-known for the classical Boltzmann equation [35, 15, 12, 40, 4, 31], which we here extend to the case of inelastic interactions and present in a general setting of polynomially increasing convex test functions. In Sect. 4 we study the estimates of the inelastic collision operator in Lp spaces with polynomial weights, extending the results in [22] to the inelastic hard sphere case. We continue by establishing apriori regularity estimates, based on the interpolation of Lp spaces and the Sobolev-type inequalities. In Sect. 5 we present a rigorous proof of the existence and regularity of the time-dependent and steady solutions. The arguments presented there also justify the formal manipulations performed in Sects. 2, 3 and 4. In Sect. 6 we show the uniqueness for the time-dependent problem using Gronwall’s lemma. Finally, in Sect. 7 we compute lower bounds for the stationary and time-dependent solutions. 1. Preliminaries 1.1. Binary inelastic collisions. We study the dynamics of inelastic identical hard balls with the following law of interactions. Let v and v∗ be the velocities of two particles before a collision, and denote by u = v − v∗ their relative velocity. Let the prime symbol denote the same quantities after the collision. Then we assume (u · n) = −α (u · n), u − (u · n) = u − (u · n),

(1.1)

where n is the unit vector in the direction of impact, and 0 < α < 1 is a constant called the coefficient of normal restitution. Setting w = v + v∗ and using the momentum conservation we can express v and v∗ as follows: v =

w u + , 2 2

v∗ =

w u − . 2 2

(1.2)

506


v

σ

n v β

λ

ω

θ

µ π −χ

1- β w

α

v *

v *

0 Fig. 1. A two-dimensional illustration of the collision mechanism: − − − − − − : possible locations of v ; −·−· − : possible locations of v∗ . All lengths shown in assumption |u|/2 = 1. Unit vectors n, σ , ω not to scale

By substituting (1.1) into (1.2) and Eqs. (1.1), the post-collisional velocities v and v∗ are uniquely determined by the pre-collisional ones, v and v∗ , and the impact parameter n (cf. [10, 38]). The geometry of the inelastic collisions defined by relations (1.1), (1.2) is shown in Fig. 1. For every v and v∗ fixed, the sets of possible outcomes for post-collisional velocities are two (distinct) spheres of diameter 1+α 2 |u|. Thus, it is convenient to parametrize the relative velocity after collision as follows: u = (1 − β) u + β |u| σ,

(1.3)

where we denoted β = 1+α 2 . The relations (1.2) and (1.3) define the post-collisional velocities in terms of v, v∗ and the angular parameter σ ∈ S N−1 . 1.2. Weak form of the collision operator. We define the collision operator by its action on test functions, or observables. Taking ψ = ψ(v, t) to be a suitably regular test function, we introduce the following weak bilinear form of the collision term:


RN

Q(g, f ) ψ dv =

RN

RN

S N −1

507

f g∗ (ψ − ψ) |u| b(u, σ ) dσ dv dv∗ .

(1.4)

Here and below we use the shorthand notations f = f (v, t), g∗ = g(v∗ , t), ψ = ψ(v , t), etc. The function b(u, σ ) in (1.6) is the product of the Enskog correlation factor k(ρ, d) (which is a constant in the space-homogeneous case) by the differential collision cross-section, expressed in the variables u, σ . In the case of hard-sphere interactions, N−1 N −3 d 1 − (ν · σ ) − 2 b(u, σ ) = k(ρ, d) , 2 2 where ν = u/|u|, and d is the diameter of the particles. Notice that the hard sphere cross-section depends only on the angle between u and σ , and is generally anisotropic, unless N = 3. Without restricting generality, by choosing the value of d accordingly, we can always assume that b(u, σ ) dσ = 1. (1.5) S N −1

Of course, to write down the Boltzmann operator we only need Q(f, f ), but later on it will be sometimes convenient to work with the bilinear form Q(g, f ). An explicit form of Q will be given later on; however for many purposes it will be easier to work with the weak formulation which is also quite natural from the physical point of view (it is analogous to the well-known Maxwell form of the Boltzmann collision operator [39, Chap. 1, Sect. 2.3]). In the case when f = g in (1.4), we can further symmetrize and write Q(f, f ) ψ dv RN 1 = ff∗ (ψ + ψ∗ − ψ − ψ∗ ) |u| b(u, σ ) dσ dv dv∗ . (1.6) 2 RN RN S N −1 Notice that the particular form of the inelastic collision laws enters (1.6) only through the test functions ψ and ψ∗ . 1.3. Equations for observables and conservation relations. Using the weak form (1.6) allows us to study equations for average values of observables given by the functionals of the form RN f ψ dv. Namely, multiplying Eq. (0.1) by a test function ψ(v, t) and integrating by parts we obtain T t=T T f ψ dv − f (∂t ψ + v ψ) dv dt = Q(f, f ) ψ dv dt. RN

t=0

0

RN

0

RN

(1.7)

With the weak form (1.6) of the collision operator, it is easy to verify formally the basic conservation relations that follow from (0.1). Namely, setting ψ = 1 and ψ = vi in (1.7) and assuming that RN f ψ dv is differentiable in t, we obtain the conservation of mass and momentum: d f {1, v1 , . . . , vN } dv = 0. (1.8) dt RN

508


Further, taking ψ = |v|2 and computing |v |2 + |v∗ |2 − |v|2 − |v∗ |2 = −

1 − α 2 1 − (ν · σ ) 2 |u| , 2 2

(1.9)

we obtain the following relation for the dissipation of kinetic energy: d 1 − α2 2 f |v| dv = 2N − N ff∗ |u|3 dv∗ dv, dt RN 4 RN RN where

N =

S N −1

(1.10)

1 − (ν · σ ) b(u, σ ) dσ = const. 2

Notice that, unlike the no-diffusion case, the kinetic energy is not necessarily a monotone function of time. However, it is not difficult to show using (1.10) (see Sect. 2) that the kinetic energy remains bounded for all times, provided the initial distribution function has finite energy. Finally, Eq. (1.7) allows us to define the concept of solutions of (0.1) which we use throughout the paper. Namely, we say that a function f is a weak solution of (0.1) if for every T > 0, f ∈ L1 ([0, T ] × RN ), Q(f, f ) ∈ L1 ([0, T ] × RN ) and (1.7) holds for every ψ ∈ C 1 ([0, ∞), C 2 (RN )) vanishing for t > T . It can be shown in the usual way that if a weak solution is sufficiently smooth (say, continuously differentiable with respect to time and twice continuously differentiable with respect to velocity) and satisfies suitable decay conditions for large |v|, then it also is a classical solution. 1.4. Entropy identity. Taking in the weak form (1.6) ψ = log f we obtain an interesting identity for the entropy RN f log f dv. First, we compute Q(f, f ) log f dv RN 1 f f∗ = ff∗ log |u| b(u, σ ) dσ dv dv∗ 2 RN RN S N −1 ff∗ f f∗ 1 f f∗ = ff∗ log − + 1 |u| b(u, σ ) dσ dv dv∗ 2 RN RN S N −1 ff∗ ff∗ 1 + (f f∗ − ff∗ ) |u| b(u, σ ) dσ dv dv∗ . (1.11) 2 RN RN S N −1 The last term vanishes in the elastic case α = 1; however, as we see below, it is generally different from zero if α < 1. To compare the integral of f f∗ to that of ff∗ we perform the transformation corresponding to the inverse collision, passing from the velocities v , v∗ to their predecessors v and v∗ . Such a transformation is more easily expressed in the variables u and n. Passing to these variables, we can write the integral of f f∗ as follows: d N−1 f f∗ |u · n| dn dv dv∗ , (1.12) RN RN

N −1 S+


509

N−1 where S+ = {n ∈ S N−1 | u · n > 0}. The “inverse collision” transformation (v, v∗ , n) → (v , v∗ , −n) has the Jacobian determinant equal to α [10]. Therefore, using the first of Eqs. (1.1), the integral (1.12) is computed as

d

N−1

1 ff∗ |u · n| dn dv dv∗ . α 2 RN RN S+N −1

(1.13)

Changing variables in the angular integral from n to σ , we rewrite (1.12) as 1 1 ff∗ |u| b(u, σ ) dσ dv dv∗ = 2 ff∗ |u| dv dv∗ . (1.14) α RN RN α 2 RN RN S N −1 In view of (1.11) and (1.14) the entropy equation becomes d 2 f log f dv + 4 ∇ f dv dt RN RN 1 f f∗ f f∗ = ff∗ log − + 1 |u| b(u, σ ) dσ dv dv∗ 2 RN RN S N −1 ff∗ ff∗

1 1 + −1 ff∗ |u| dv dv∗ . (1.15) 2 α2 RN RN In these equations as in all the sequel, the symbol ∇ will stand for the gradient operator with respect to velocity variables. Here the first term on the right-hand side is nonpositive (notice the inequality log x − x + 1 ≤ 0) and similar to the entropy dissipation in the elastic case. The last term in (1.15) is a nonnegative correction term that vanishes in the elastic limit α → 1.

1.5. Similarity in the equations and normalization of solutions. As a consequence of (1.8), the total density (mass) and momentum (mean value) of the distribution function are equal to those of the initial distribution. We can write this as follows: RN

f dv = ρ0 = const,

and

RN

f vi dv = ρ0 v0i = consti ,

i = 1, . . . , N.

In fact, we can always assume that ρ0 = 1, v0 = 0 and µ = 1 in (0.1). Indeed, if f (v, t) is such a solution to (0.1), then, for every ρ0 , v0 and µ, the function f{ρ0 ,v0 ,µ} (v, t) = ρ0 η−N f t/τ, (v − v0 )/η , where −2/3 −1/3

τ = ρ0

µ

,

−1/3 1/3

and η = ρ0

µ

is a solution corresponding to the given values of ρ0 , v0 and µ.

,

510


1.6. Strong form of the collision operator. Using the weak form (1.6) we can derive the usual strong form of the collision operator. We notice the obvious splitting into the “gain” and the “loss” terms, Q(g, f ) = Q+ (g, f ) − Q− (g, f ). Assuming that f is regular enough, setting ψ(v) = δ(v − v0 ) in the part of (1.6) corresponding to Q− (g, f ), and using (1.5) we find − Q (g, f ) = f g∗ |u| b(u, σ ) dσ dv∗ = f (g ∗ |v|). RN

S N −1

To find the explicit form of Q+ (g, f ) we invoke the inverse collision transformation, tracing the collision history back from the pair v, v∗ to their predecessors, which we denote by v and v∗ . Setting ψ(v) = δ(v − v0 ) and arguing similarly to the derivation of the entropy identity we obtain 1 Q+ (g, f ) = f g∗ 2 |u| b(u, σ ) dσ dv∗ , α RN S N −1 where f = f ( v, t), g∗ = g( v∗ , t), and the pre-collisional velocities are defined as

v=

u w + , 2 2

and γ =

v∗ =

u w − , 2 2

where

u = (1 − γ )u + γ |u|σ,

(1.16)

α+1 2α .

2. Basic Apriori Estimates: Energy and Entropy In the classical theory of the elastic Boltzmann equation, the energy conservation and the entropy decay are the most fundamental facts which provide the base for every analysis. In the present setting naturally we do not have energy conservation, and the energy inequality (expressing that collisions do not increase the energy) would by no means be sufficient to compensate for that. So the key ingredient will be to replace it by the more precise energy dissipation estimate, as follows. To study solutions of (0.1) and (0.2) we assume for simplicity that they satisfy the normalization conditions of unit mass and zero average; however the estimates we derive below will be by no means restricted to such solutions. We use the energy equation (1.10) and apply Jensen’s inequality for the last term to get 3 f∗ |u|3 dv∗ ≥ v − f (t, v) v dv = |v|3 , RN

and therefore,

RN

RN

RN

ff∗ |u|3 dv∗ dv ≥

RN

f |v|3 dv.

We then get (in the time-dependent case) the differential inequality d f |v|2 dv + k1 f |v|3 dv ≤ K1 , dt RN RN

(2.1)


511

2

where K1 = 2N and k1 = N 1−α 4 . Further, by Jensen’s inequality,

3/2 f |v|2 dv , f |v|3 dv ≥ RN

RN

and we obtain

3/2 d f |v|2 dv ≤ K1 − k1 f |v|2 dv . dt RN RN d 2 Thus, if RN f |v|2 dv > (K1 /k1 )2/3 , then dt RN f |v| dv < 0 and so,

2/3 sup f |v|2 dv ≤ max f0 |v|2 dv, K1 /k1 . t≥0

RN

RN

In the steady case the derivative term drops in (2.1), and we obtain K1 . f |v|3 dv ≤ N k1 R Let us introduce the following weighted L1 spaces: L1k (RN ) = {f | f vk ∈ L1 (RN )},

(2.2)

where k ≥ 0 and v = (1 + |v|2 )1/2 . We then define the norms in L1k as RN |f | vk dv, which for f nonnegative coincide with the moments RN f vk dv. The above argument implies apriori estimates for the steady solutions in L13 (RN ), and for the time-dependent ones in L∞ ([0, ∞), L12 (RN )) and L1loc ([0, ∞), L13 (RN )). We emphasize that the bounds depend on α and deteriorate in the elastic limit α → 1. In fact, in the case α = 1 it is easy to see formally that the kinetic energy of solutions increases linearly in time, and so, no nonnegative steady solution with finite mass and second moment is possible. Next, using the entropy equation (1.15) we show that the entropy is bounded uniformly in time, for initial data with finite mass, kinetic energy and entropy. To obtain this, we first estimate the second term in (1.15) using the Sobolev embedding inequality: assuming for simplicity here that N ≥ 3, we have |∇ f |2 dv ≥ c f Lp∗ , RN

where p∗ = N/(N − 2). Further, we have the inequality f log f dv ≤ Cε f εLp∗ ,

(2.3)

for all ε > 0. Indeed, obviously, for every δ > 0, f log f dv ≤ Cδ

(2.4)

RN

RN

RN

f 1+δ dv.

Further, by Hölder’s inequality, for δ < p∗ , f L1+δ ≤ f 1−ν f νLp∗ , L1

512

where ν =

I.M. Gamba, V. Panferov, C. Villani p∗ δ (p∗ −1)(1+δ) .

Therefore, RN

f

1+δ

p∗ δ p ∗ −1 p∗

dv ≤ f L

,

which together with (2.4) implies (2.3). Now, coming back to estimating the terms in the entropy equation (1.15), we get

1/ε d f log f dv + cε f log f dv ≤ C f 2L1 . (2.5) 1 dt RN RN The established bound in L∞ ([0, ∞), L12 (RN )) implies that for initial data with finite mass and energy, the right-hand side of (2.5) is bounded by a constant, and we obtain by Gronwall’s lemma, sup f log f dv ≤ C( RN f0 log f0 dv, f0 L1 ) . t≥0

RN

2

√ Integrating (1.15) in time, we also get f ∈ L2 ([0, T ], H 1 (RN )), for every T > 0, ∗ which implies in particular, f ∈ Lp ([0, T ] × RN ), where the constants in the estimates depend on the initial mass, energy and entropy √ of the solutions. For the steady solutions we obtain a particularly simple estimate ∇ f L2 ≤ C f L1 . As the reader will easily 1 check, our assumption that N ≥ 3 is just for convenience, and can easily be circumvented in dimension 2√ by the Moser-Trudinger inequality, or just the local control of all Lp norms of f by ∇ f L2 , together with a moment-based localization argument. 3. Moment Inequalities We further look for apriori estimates of the solutions in the spaces L1k (2.2) with k > 2. Such estimates will play a very important role in our study of regularity, which we perform in Sect. 4. The key technique for obtaining the necessary estimates is the so-called Povzner inequalities [35, 15, 12, 32, 4, 31] which we here extend to the inelastic case.

3.1. The Povzner-type inequalities. We take ψ(x), x > 0 to be a convex nondecreasing function and look for estimates of the expressions q [ψ](v, v∗ , σ ) = ψ(|v |2 ) + ψ(|v∗ |2 ) − ψ(|v|2 ) − ψ(|v∗ |2 )

(3.1)

and q¯ [ψ](v, v∗ ) =

S N −1

(ψ(|v |2 ) + ψ(|v∗ |2 ) − ψ(|v|2 ) − ψ(|v∗ |2 )) b(u, σ ) dσ, (3.2)

which appear in the weak form of the collision operator (1.6). Our aim is to treat the cases of ψ(x) = x p ,

and ψ(x) = (1 + x)p − 1,

p > 1,

(3.3)


513

and also truncated versions of such functions which will be required in the rigorous analysis of moments in Sect. 5. Thus, we will require functions ψ to satisfy the following list of conditions: ψ(x) ≥ 0, x > 0; ψ(0) = 0; ψ(x) is convex, C 1 ([0, ∞)), ψ (x) is locally bounded; ψ (ax) ≤ η1 (a) ψ (x), x > 0, a > 1; ψ (ax) ≤ η2 (a) ψ (x), x > 0 a > 1,

(3.4) (3.5) (3.6) (3.7)

where η1 (a) and η2 (a) are functions of a only, bounded on every finite interval of a > 0. The above conditions are easily verified for the functions (3.3). We will further establish the following elementary lemma. Lemma 3.1. Assume that ψ(x) satisfies (3.4)–(3.7). Then ψ(x + y) − ψ(x) − ψ(y) ≤ A ( x ψ (y) + y ψ (x) )

(3.8)

ψ(x + y) − ψ(x) − ψ(y) ≥ b xy ψ (x + y),

(3.9)

and

where A = η1 (2) and b = (2η2 (2))−1 . Proof. To establish the first of the bounds assume that x ≥ y. Then, since ψ(y) ≥ 0, ψ(x + y) − ψ(x) − ψ(y) ≤ ψ(x + y) − ψ(x) y = ψ (x + t) dt ≤ 0

y

η1 (2) ψ (x) dt = A y ψ (x).

0

By symmetry we have ψ(x + y) − ψ(x) − ψ(y) ≤ A x ψ (y), when x ≤ y. This proves the required inequality for all x and y. To prove the second of the bounds in the lemma, we can write, using (3.7) and the normalization ψ(0) = 0, y y x ψ(x + y) − ψ(x) − ψ(y) = (ψ (x + t) − ψ (t)) dt = ψ (t + τ ) dτ dt 0 0 y 0 x −1 ≥ (η2 (2)) ψ (x + y) χ{t+τ >(x+y)/2} dτ dt = (2η2 (2))−1 xy ψ (x + y). 0

This completes the proof.

0

In the sequel, we shall use some relations involving post-collisional velocities v and v∗ . It becomes more convenient to parametrize them in the center of mass–relative velocity variables. We therefore set v =

w + λ|u|ω , 2

and

v∗ =

w − λ|u|ω , 2

(3.10)

where w = v + v∗ , u = v − v∗ , and ω is a parameter vector on the sphere S N−1 (see Fig. 1). We have λω = βσ + (1 − β)ν,

514


where β =

1+α 2

and ν = u/|u|, and therefore,

λ = λ(cos χ) = (1 − β) cos χ +

(1 − β)2 (cos2 χ − 1) + β 2 ,

(3.11)

where χ is the angle between u and ω. Notice that 0 < α ≤ λ(cos χ ) ≤ 1, for all χ . With this parametrization we have |w|2 + λ2 |u|2 + 2λ|u||w| cos µ , 4 |w|2 + λ2 |u|2 − 2λ|u||w| cos µ |v∗ |2 = , 4 |v |2 =

(3.12)

where µ is the angle between the vectors w = v + v∗ and ω. Lemma 3.2. Assume that the function ψ satisfies (3.4)–(3.7). Then we have q [ψ] = −n [ψ] + p [ψ], where p [ψ] ≤ A (|v|2 ψ (|v∗ |2 ) + |v∗ |2 ψ (|v|2 ) ) and n [ψ] ≥ κ(λ, µ) (|v|2 + |v∗ |2 )2 ψ ( |v|2 + |v∗ |2 ). Here A is the constant in estimate (3.8), κ(λ, µ) =

b 2 λ (η2 (λ−2 ))−1 sin2 µ, 4

and b is the constant in estimate (3.9). Proof. We start by setting p [ψ] = ψ(|v|2 + |v∗ |2 ) − ψ(|v|2 ) − ψ(|v∗ |2 ) and n [ψ] = ψ(|v|2 + |v∗ |2 ) − ψ(|v |2 ) − ψ(|v∗ |2 ). The estimate for p [ψ] follows easily by (3.8). It remains to verify the lower bound for n [ψ]. For this we use (3.9), noticing that ψ is monotone and that |v|2 + |v∗ |2 ≥ |v |2 + |v∗ |2 . We then obtain: n [ψ] ≥ ψ(|v |2 + |v∗ |2 ) − ψ(|v |2 ) − ψ(|v∗ |2 ) ≥ b |v |2 |v∗ |2 ψ (|v |2 + |v∗ |2 )

= b ζ (v , v∗ ) (|v |2 + |v∗ |2 )2 ψ (|v |2 + |v∗ |2 ) ,


515

where ζ (v , v∗ ) =

|v∗ |2 |v |2 . |v |2 + |v∗ |2 |v |2 + |v∗ |2

Further, using (3.12), we get ζ (v , v∗ ) =

1 4λ2 |u|2 |w|2 1 1 2 1− 2 2 cos µ ≥ (1 − cos2 µ) = sin2 µ. 2 2 4 (λ |u| + |w| ) 4 4

Finally, noticing that |v |2 + |v∗ |2 =

|u|2 + |w|2 λ2 |u|2 + |w|2 ≥ λ2 = λ2 (|v|2 + |v∗ |2 ), 2 2

we obtain b 2 λ (η2 (λ−2 ))−1 sin2 µ (|v|2 + |v∗ |2 )2 ψ (|v|2 + |v∗ |2 ). 4 This completes the proof of the lemma. n [ψ] ≥

Lemma 3.2 gives us the basic formulation of the Povzner inequality for the considered class of test functions ψ. In the example ψ(x) = x p we have p [ψ] ∼ C(|v|2 |v∗ |2p−2 + |v∗ |2 |v|2p−2 ) and n[ψ] ∼ c(|v|2p + |v∗ |2p ), outside the set where κ(λ, µ) is small (which amounts to a small set of angles). This implies that the nonpositive term −n [ψ] is dominating, at least when |v| >> |v∗ | or |v∗ | >> |v|, which are the most important regions of integration from the point of view of calculation of moments (cf. also [12, 32]). We can further simplify the inequalities and get rid of the dependence on the angular variables, by integration with respect to σ ∈ S N−1 . We then obtain the following lemma. Lemma 3.3. Assume that the function ψ satisfies (3.4)–(3.7). Then q¯ [ψ] ≤ − k (|v|2 + |v∗ |2 )2 ψ (|v|2 + |v∗ |2 ) + A (|v|2 ψ (|v∗ |2 ) + |v∗ |2 ψ (|v|2 ) ), where the constant A is as in Lemma 3.2, and k > 0 is a constant that depends on the function ψ but not on α. Proof. For the proof we notice that λ(cos χ ) is pointwise decreasing as α 0 and so, λ(cos χ ) ≥ cos χ ,

for

cos χ > 0,

for all α > 0. We then denote cos θ = (ν · σ ), b0 (cos θ) = b(u, σ ), and estimate the integral κ(λ, µ) b0 (cos θ ) dσ ≥ κ(λ, µ) b0 (cos θ) dσ, S N −1

{cos χ>ε0 , sin µ>ε1 , 1−cos θ>ε2 }

(3.13) setting ε0 , ε1 and ε2 small enough. The integrand on the right-hand side of (3.13) is bounded below by a constant, and so is the area of the domain of integration. (The verification of the last statement for the condition sin µ > ε1 is somewhat tedious and is achieved by changing the variables of integration from ω to σ : we omit the technical details.) We therefore find that the integral (3.13) is bounded below by a constant k > 0, independent on α. The rest of the claim is easy to verify.

516


Finally, we present estimates for the integral expression (3.2) multiplied by the relative speed, in the cases when ψ(x) is given by one of the functions (3.3). Lemma 3.4. Take p > 1 and ψ(x) = x p . Then |u| q¯ [ψ](v, v∗ ) ≤ −kp (|v|2p+1 + |v∗ |2p+1 ) + Ap (|v||v∗ |2p + |v|2p |v∗ |). Also, take ψ(x) = (1 + x)p − 1, then |u| q¯ [ψ](v, v∗ ) ≤ −kp ( v2p+1 + v∗ 2p+1 ) + Ap ( v v∗ 2p + v2p v∗ ). Here the constants kp and Ap are independent on the restitution coefficient α. Proof. We use Lemma 3.3 and the inequalities ||v| − |v∗ || ≤ |u| = |v − v∗ | ≤ |v| + |v∗ |. Then in the case ψ(x) = x p the bounds have the form − p(p − 1)kp |u| (|v|2 + |v∗ |2 )p + pAp |u| (|v|2 |v∗ |2p−2 + |v|2p−2 |v∗ |2 ) . The terms appearing with the negative sign are estimated using the inequality |u| (|v|2 + |v∗ |2 )p ≥

1 1 (|v|2p+1 + |v|2p+1 ) − (|v||v∗ |2p + |v|2p |v∗ |) . 2 2

For the remaining terms we have |u| (|v|2 |v∗ |2p−2 + |v|2p−2 |v∗ |2 ) ≤ Cp (|v||v∗ |2p + |v|2p |v∗ |) , which completes the proof of the first part of the lemma. The case ψ(x) = (1 + x)p − 1 can be treated by arguing along the same lines, by using the inequalities (|v|2 + |v∗ |2 )2 ≥ and |v| ≥ (1 + |v|2 )1/2 − 1.

1 (1 + |v|2 + |v∗ |2 )2 − 1 2

3.2. Estimates for higher-order moments. The Povzner-type inequalities of Lemma 3.4 allow us to study the topics of propagation and appearance of moments. We find that results known for the classical Boltzmann equation with “hard-forces” interactions [15, 12] transfer to present case. We introduce the notation Ys (t) =

RN

f vs dv,

and denote by Y¯s the corresponding steady moment.


517

Lemma 3.5. Let f be a sufficiently regular and rapidly decaying solution of (0.1). Then, the following differential inequality holds: d Ys + 2ks Ys+1 ≤ Ks (Ys + Ys−2 ), dt

(3.14)

where Ks and ks are positive constants. Further, s sup Ys (t) ≤ Ys∗ = max Ys (0), Ks /ks , t>0

and for every τ > 0

τ

0

Ys+1 (t) dt ≤

Ks τ + 1/2 ∗ Ys . ks

(3.15)

Finally, for the steady equation (0.2) we obtain the apriori estimate Ks ¯ (Ys + Y¯s−2 ). Y¯s+1 ≤ 2ks Proof of Lemma 3.5. Using the weak form of Eq. (0.1) with ψ(v) = vs we find d f vs dv − f vs dv = Q(f, f ) vs dv. (3.16) N N dt RN R R Estimating the moments of the collision integral according to Lemma 3.4 we get Q(f, f ) vs dv ≤ −2ks Ys+1 + 2As Y1 Ys . RN

The moments of the Laplacian term are computed as follows: s f v dv = f ( (s(s − 2) + sN ) vs−2 − s(s − 2) vs−4 ) dv RN

RN

= (s(s − 2) + sN ) Ys−2 − s(s − 2) Ys−4 .

(3.17)

Combining (3.16) and (3.17) and neglecting the non-positive Ys−4 term, we obtain inequality (3.14) with Ks = max{2As , s(s − 2) + sN }. To obtain a uniform bound for Ys (t), we use Jensen’s inequality to write Ys+1 ≥ (Ys )(s+1)/s . Then we find, estimating the right-hand side of (3.14) by 2Ks Ys , d Ys ≤ −2ks (Ys )(s+1)/s + 2Ks Ys . dt Thus, Ys (t) < 0 if Ys > (Ks /ks )s , and so, the upper bound for supt>0 Ys (t) must hold. Further, integrating in time we obtain τ 2ks Ys+1 ≤ 2Ks τ Ys∗ − Y (s) + Y (0) ≤ (2Ks τ + 1)Ys∗ , 0

which proves (3.15). Finally, the last inequality is obtained by the same arguments as (3.14) applied to the steady equation.

518


Based on the lemma just proven we can make the following conclusions about the behavior of the moments of the solutions. First, if a moment Ys is finite initially, it propagates, that is, it remains bounded for the whole time-evolution. Further, the integral condition on Ys+1 implies the appearance of moments of order s + 1: these moments become finite after an arbitrarily short time, even if they are initially infinite (cf. [12]). Indeed, suppose that Ys+1 (0) = +∞, then for every τ > 0 there is a t0 < τ such that Ys+1 (t0 ) < +∞. Then, applying the lemma to Ys+1 , starting with t = t0 , we obtain that for every t0 > 0, sup Ys+1 (t) ≤ Ct0 ,s ,

t>t0

which implies the above statement. The last part of the lemma implies an important statement concerning the moments of the steady solution: on the formal level, every solution that has a finite moment of order s > 2 has finite moments of all positive orders. In fact, in view of the L13 (RN ) estimate of the previous section, this implies that every solution with finite mass should have this property. 4. Lp Bounds and Apriori Regularity In this section we study the apriori regularity of solutions to (0.1) and (0.2). The presence of the diffusion term in the equation makes it plausible that solutions to the steady equation should be smooth, and those for the time-dependent equation should gain smoothness after an arbitrarily short time. However, to realize this idea we need to make use of the particular structure of the collision term. As we will see below, the moment bounds of the previous section will also be of crucial importance. We start by establishing the bounds for the collision operator in the spaces Lp with a polynomial weight, extending the results well-known in the case of the classical Boltzmann equation, and first derived by Gustafsson [22]. Below, we shall establish these bounds by adapting the simple strategy that was suggested in [39, Chap. 2, Sect. 3.3] and later developed in [34] to establish improved Lp bounds in the elastic case.

4.1. Lp bounds for the collision operator. We will use the following weighted Lp spaces: p

Lk (RN ) = {f | f vk ∈ Lp (RN )}, where v = (1 + |v|2 )1/2 . The necessity to introduce a weight comes from the presence of the factor |u| in the hard sphere collision term (1.6). The collision operator is generally p unbounded on Lp : in order to control its norm we will invoke the Lk norms with higher powers of v. The precise formulation of this statement is given in next lemma. Lemma 4.1. For every 1 ≤ p ≤ ∞ and every k ≥ 0, Q(g, f ) Lp ≤ C g Lp f L1 k

k+1

k+1

+ g L1 f Lp

where C is a constant depending on p, k and N only.

k+1

k+1

,


519

Proof. We fix an exponent 1 ≤ p ≤ ∞. It is easy to estimate the “loss” part Q− (g, f ) = (g ∗ |v|)f , using the inequality | g ∗ |v| | ≤ g L1 v, 1

from which it follows Q− (g, f ) Lp ≤ g L1 f Lp . 1

k

k+1

(4.1)

We now turn to estimate the Q+ term: starting from the weak form (1.6), we find Q+ (g, f ) vk Lp = sup Q+ (g, f ) ψ vk dv ψ

=1 Lp

=

sup

ψ

=1 Lp

RN

RN

RN

f g∗ |u|

S N −1

ψ v k b(u, σ ) dσ dv dv∗ .

(4.2)

By using the inequalities |u| ≤ v + v∗ and v k ≤ ( v + v∗ )k the integral (4.2) is bounded as f g∗ ( v + v∗ )k+1 ψ b(u, σ ) dσ dv dv∗ . (4.3) RN

S N −1

RN

We now see that the problem comes down to estimating the integral S[ψ](v, v∗ ) = ψ b(u, σ ) dσ S N −1

p

in either L∞ (RvN , L (RvN∗ )) or L∞ (RvN∗ , Lp (RvN )). In fact, we split S[ψ] into two parts S+ [ψ] and S− [ψ] and prove the bounds for each of the parts in the respective spaces. We set ψ b(u, σ ) dσ S± [ψ](v, v∗ ) = {±u·σ >0}

and establish the bounds for S+ and S− in the following proposition. Proposition 4.2. The operators S+ : Lq (RN ) → L∞ (RvN , Lq (RvN∗ )), S− : Lq (RN ) → L∞ (RvN∗ , Lq (RvN )),

are bounded for every 1 ≤ q ≤ ∞. Proof. We prove the Lq bounds by interpolation between L∞ and L1 . The L∞ estimates are clear due to the boundedness of the domain of integration. To check the L1 bounds we assume without loss of generality that ψ ≥ 0 and calculate the L1 norms as follows:

β − u + |u|σ b(u, σ ) dσ du ψ v+ S− [ψ](v, v∗ ) L1 (RNv ) = ∗ 2 RN {u·σ 0} ψ(v + z) dσ dz , = |J+ (u(z, σ ), σ )| RN S N −1 where now z = v − v∗ = 21 (2 − β))u + β |u|σ , and J+ (u, σ ) =

2 − β N 2

1+

β (u · σ )

. 2 − β |u|

Then, since (u · σ ) > 0, we can argue similarly to the previous case to obtain S+ [ψ](v, v∗ ) L1 (RNv ) ≤

2 − β −N 2

ψ L1 ,

uniformly in v∗ ∈ RN . The statement of the proposition now follows by the Marcinkiewicz interpolation theorem. End of proof of Lemma 4.1. Combining the bound (4.3) with the ones proven in Proposition 4.2 we find Q(g, f ) ψ dv RN ≤ f g∗ ( v + v∗ )k+1 S+ [ψ](v, v∗ ) + S− [ψ](v, v∗ ) dv dv∗ N N R R ≤ Ck g∗ f ( vk+1 + v∗ k+1 ) S+ [ψ](v, v∗ ) dv dv∗ RN RN +Ck f g∗ ( vk+1 + v∗ k+1 ) S− [ψ](v, v∗ ) dv∗ dv RN RN ≤ C g L1 f Lp + g L1 f Lp + f L1 g Lp + f L1 g Lp , k+1

k+1

k+1

k+1

since ψ Lp = 1. From this the conclusion of the lemma follows easily.


521

4.2. H 1 regularity: Steady state equation. We start by establishing apriori estimates for solutions to the steady equation (0.2), for which the analysis is performed in a rather more direct way than for the time-dependent problem. We first show the bounds in the Sobolev spaces with the weight vk = (1 + |v|2 )k/2 : Hk1 (RN ) = {f ∈ L2k (RN ) | ∇f ∈ L2k (RN )}. The main tools are the coercivity of the diffusion part, the estimates of the collision operator in Lp , and the interpolation inequalities for Lp spaces. The constants in the estimates are expressed in terms of the L1 moments. In all this section, we shall assume for simplicity that N ≥ 3, but there is no difficulty to adapt the proofs to cover the case N = 2 as well. We begin with an estimate for the gradient in L2 . Lemma 4.3. Assume that the function f ∈ H 1 (RN ) ∩ L1r (RN ), where r = solution of (0.2). Then

N+2 4 ,

is a

∇f L2 ≤ CA B r , where A = f L1r ,

B = f L1 , 1

and C is a constant depending on the dimension. Proof. Multiplying Eq. (0.2) by f , integrating and applying Hölder’s inequality yields |∇f |2 dv = Q(f, f )f dv ≤ C f Lp Q(f, f ) Lp , (4.4) RN

RN

for all 1 ≤ p ≤ ∞. We choose p = 2∗ = 2N/(N − 2), where 2∗ is the critical Sobolev exponent, and apply the Sobolev’s embedding inequality f L2∗ ≤ C ∇f L2 .

(4.5)

(Note: for N = 3, 2∗ = 6 and (2∗ ) = 6/5.) Then, by Lemma 4.1, Q(f, f ) L(2∗ ) ≤ C f To estimate f

(2∗ )

L1

(2∗ )

L1

f L1 . 1

(4.6)

we use the following interpolation inequality for weighted Lp

norms (ϕ is any weight function), which can be easily verified using Hölder’s inequality: f ϕ k Lq ≤ f ϕ k1 νLq1 f ϕ k2 1−ν Lq2 ,

(4.7)

where 1−ν 1 ν + = q1 q2 q

and

k1 ν + k2 (1 − ν) = k.

Now, interpolating the norm in L1 for q = (2∗ ) between q1 = 2∗ and q2 = 1, we get q

f

(2∗ )

L1

≤ f νL2∗ f 1−ν , L1 r

(4.8)

522


where ν and r are determined from the following equations: ν 1 1−ν = ∗ + 2∗ 1 (2 )

and r (1 − ν) = 1,

so that ν=

N −2 N +2

and r =

1 N +2 = . 1−ν 4

(4.9)

Combining estimates (4.4)–(4.8) we obtain the inequality f 1+ν ≤ CBA1−ν ∇f 1+ν , ∇f 2L2 ≤ C f L1 f 1−ν L1 L2∗ L2 1

r

from which the conclusion of the lemma follows.

(4.10)

The result of the lemma implies a bound for the solutions in the space H 1 (RN ). Indeed, by the Sobolev embedding, f L2∗ ≤ CA B r . ∗

Interpolating between L1 and L2 using inequality (4.7) we get a bound for the L2 norm, which then implies a bound in H 1 . Since the constants in the estimates depend on the L1k norms only, and the latter are controlled by the moments bounds, we gain an apriori control of the H 1 norm by means of the mass and the energy only. We next see that the derivatives of the solutions have an appropriate decay, so even L2k norms for all k ≥ 0 are bounded. Lemma 4.4. Let f be a solution of Eq. (0.2) and assume that f ∈ Hk1 (RN ) ∩ L1(k+1)r (RN ), where k ≥ 0 and r = N+2 4 . Then ∇(f vk ) L2 ≤ C A1 Ar2 + k 2r A3 , where A1 = f L1

(k+1)r

,

A2 = f L1 , k+1

A3 = f L1

k−2/r

,

and C is a constant depending on the dimension N . Proof. Integrating Eq. (0.2) against f v2k we obtain 2k ∇f · ∇(f v ) dv = Q(f, f )f v2k dv. RN

RN

(4.11)

Using estimates from the previous lemma, the right-hand side can be bounded above as follows: Q(f, f ) vk L(2∗ ) f vk L2∗ ≤ C f vk+1 L(2∗ ) f vk+1 L1 f vk L2∗ . (4.12) Interpolating as in (4.7) we find f vk+1 L(2∗ ) ≤ f νL2∗ f 1−ν L1

(k+1)r

,

(4.13)


523

where ν and r are as defined in (4.9). Therefore, combining (4.12) with (4.13) we bound the right hand side of (4.11) by k 1+ν f L1 f vk 1+ν ≤ CA1−ν (4.14) C f 1−ν 1 A2 ∇(f v ) L2 . L2∗ L1 (k+1)r

k+1

The integral on the left-hand side of (4.11) is estimated as follows: 2k k 2 f 2 |∇ vk |2 dv ∇f · ∇(f v ) dv = |∇(f v )| dv − RN

RN

RN

≥ ∇(f v

k

) 2L2

−k

2

f 2L2 . k−1

(4.15)

∗

Further, interpolating the L2k−1 norm between L2 and L1 we get ≤ C ∇(f vk ) λL2 f 1−λ , f vk−1 L2 ≤ C f vk λL2∗ f vk2 1−λ L1 L1 k2

where λ 1 1−λ = , + ∗ 2 1 2

so that λ =

N 1+ν , = 2 N +2

(4.16)

and k − 1 = λk + (1 − λ)k2 ,

so that k2 = k −

1 = k − 2/r. 1−λ

Gathering the above inequalities and noticing that 2λ = 1 + ν we obtain: 2 1−ν ∇(f vk ) 1+ν . ∇(f vk ) 2L2 ≤ C A1−ν 1 A2 + k A3 L2 Dividing by the norm of the gradient to the power 1 + ν we get 1 2 1−ν 1−ν . ∇(f vk ) L2 ≤ CA1−ν 1 A2 + k A 3 1 Noticing that 1−ν = r and using the inequality (x + y)r ≤ Cr (x r + y r ) we arrive at the conclusion of the lemma.

Using the lemma just proven we find bounds for solutions f in Hk1 for every k ≥ 0. Indeed, using the inequality |(∇f ) vk |2 ≤ C( |∇(f vk )|2 + |f ∇ vk |2 ) ∗

and interpolating in the second term between L2 and L1k−2/r we get , ∇f 2L2 ≤ C ∇(f vk ) 2L2 + ∇(f vk ) 1+ν f 1−ν L2 L1 k

k−2/r

from which an estimate in terms of the L1 moments follows. Further, by the interpolation inequality (4.7), f L2 ≤ f λL2∗ f 1−λ L1 k

,

k/(1−λ)

and so, in view of our earlier remarks, the norm in L2k is also estimated in terms of L1 moments only. Summarizing the results obtained so far, the solutions are controlled apriori in Hk1 (RN ) for any k ≥ 0 in terms of mass and kinetic energy only.

524


4.3. Schwartz class regularity: Steady problem. Our next aim is now to establish a priori bounds for solutions to (0.2) in the spaces Hkn (RN ) = {f ∈ L2k (RN ) | ∇ m f ∈ L2k (RN ), 1 ≤ m ≤ n}, for all 1 ≤ n < ∞ and all 0 ≤ k < ∞. We use induction on n, differentiating the equation in v in each step. The base of the induction is given by Lemma 4.4. We recall the following rule for differentiating the collision integral. Proposition 4.5. Let f and g be smooth, rapidly decaying functions of v. Then ∇Q(g, f ) = Q(∇g, f ) + Q(g, ∇f ). Proof. We use the splitting into the “gain” and “loss” terms, Q(g, f ) = Q+ (g, f ) − Q− (g, f ). Since Q− (g, f ) = f (g ∗ |v|), the differentiation rule for the “loss” term is obvious. To prove the proposition for the “gain” term Q+ (g, f ) we represent it as follows, using (1.16): u − u u + u 1 Q+ (g, f ) = f v+ |u| b(u, σ ) dσ du. g v− 2 2 α2 RN S N −1 Since u is a function of u and σ only, the statement follows by differentiation under the integral sign. Remark. The above statement is in fact a corollary of the following abstract statement which can be proven very easily: Let Q be a bilinear operator commuting with translations, continuously differentiable; then ∇Q(g, f ) = Q(∇f, g) + Q(f, ∇g). Thus, the differentiation formula of Proposition 4.5 can be seen as a consequence of the translation invariance of Q. As a direct corollary of Proposition 4.5, higher-order derivatives of Q can be calculated using the following Leibniz formula: j j ∂ Q(g, f ) = Q(∂ j −l g, ∂ l f ), l 0≤l≤j

where j and l are multi-indices j = (j1 . . . jN ), and l = (l1 . . . lN ); ∂ j = ∂vj11 . . . ∂vjNN ,

and jl are the multinomial coefficients. Thus, for every multi-index j , by formal differentiation of (0.2) we obtain the following equations for higher-order derivatives: j j − ∂ f = Q(∂ j −l g, ∂ l f ). (4.17) l 0≤l≤j

By applying the methods developed in Lemmas 4.3 and 4.4 to Eq. (4.17) we arrive at the following result. n+1 Lemma 4.6. Let f be a solution to (0.2), such that f ∈ Hk+µ (RN ), with n ≥ 0, k ≥ 0 and µ > 1 + N2 . Then

∇ n+1 f L2 ≤ C (1 + k + f H n−1 ) (1 + ∇ n f L2 ), k

k+µ

where C is a constant depending on n and N only.

k+µ


525

Proof. Taking a multi-index j with |j | = n, multiplying Eq. (4.17) by ∂ j f v2k and integrating by parts we obtain: j j j 2k Q(∂ j −l f, ∂ l f ) ∂ j f v2k dv. ∇∂ f · ∇(∂ f v ) dv = l RN RN 0≤l≤j

(4.18) Similarly to (4.15), the left-hand side can be written as ∇(∂ j f vk ) 2L2 − ∂ j f ∇ vk 2L2 . Each integral on the right-hand side of (4.18) can be bounded above by using CauchySchwartz’s inequality and Lemma 4.1 as follows: Q(∂ j −l f, ∂ l f ) ∂ j f v2k dv ≤ Q(∂ j −l f, ∂ l f ) L2 ∂ j f L2 RN

k

k

≤ C ∂ f L1 ∂ l

l−j

k+1

f L2 ∇ n f 2L2 ∂ j f L2 . k+1

k

k+1

Now, the L1 norms can be estimated as follows: ∂ l f L1

k+1

as soon as µ > 1 +

N 2.

≤ v1−µ L2 ∂ l f L2

k+µ

≤ ∂

f ∇ vk 2L2

k+µ

Gathering the above estimates we obtain:

(∂ j f vk ) 2L2 j

≤ C ∂ l f L2

j ∂ l f L2 ∂ l−j f L2 . k+µ k+1 l

+ C ∂ f L2 j

k

0≤l≤j

≤ k ∂

j

≤ k ∂

j

2 2

f 2L2 k−1

+ C f L2 ∂ f 2L2

f 2L2 k+µ

+ C(1 + f 2H n−1 ) ∂ j f 2L2 k+µ k+µ

j

k+µ

k+µ

+ C ∂ j f L2 f 2H n−1 k+µ

k+µ

+ C(1 + ∂ j f 2L2 ) f 2H n−1 . k+µ

k+µ

Since ∇(∂ j f vk ) = (∇∂ j f ) vk + k ∂ j f |v| vk−2 , we obtain ∇∂ j f 2L2 ≤ C (1 + k 2 + f 2H n−1 ) (1 + ∂ j f 2L2 ). k

k+µ

k+µ

Taking the sum over all j with |j | = n implies the estimate of the lemma.

Lemma 4.6 gives us a way to estimate higher-order derivatives of solutions in terms of lower-order ones. Thus, provided we have a solution to (0.2) that has all Hk1 norms bounded in terms of mass and energy (as we assumed in the previous section), we can derive bounds in Hk2 for every k, and then proceed by induction, obtaining bounds in Hkn (RN ), for all n and all k ≥ 0. We then obtain f ∈ Hkn = S, n≥1, k≥0

where S is the Schwartz class of rapidly decaying smooth functions. Notice that the bounds in each of the spaces Hkn (RN ) can be expressed in terms of mass and energy of the solutions.

526


4.4. Regularity for the time-dependent problem. An analysis of the regularity of the time-dependent solutions can be performed in the same vein as for the steady problem. Using the estimates obtained in the previous section in combination with the Gronwall lemma will give us results for the time-dependent equation (0.1). Our first lemma is an analog of Lemma 4.3. Lemma 4.7. Let f be a sufficiently regular solution to (0.1) with the initial condition f (·, 0) = f0 ∈ L2 (RN ), such that f has a moment of order r = N+2 4 bounded uniformly in time. Then f (·, t) L2 ≤ C,

0 ≤ t < ∞,

and ∇f L2 ([0,T ]×RN ) ≤ CT , for every 0 ≤ T < ∞, where the constants C and CT depend only on N, f0 L2 and sup f (·, t) L1r . t≥0

Proof. Integrating Eq. (0.1) against f we get, arguing similarly to the case of the steady problem: 1 d , f 2L2 + ∇f 2L2 ≤ K(t) ∇f 1+ν L2 2 dt where K(t) = C f L1 f 1−ν and ν = L1r 1 embedding,

N−2 N+2

(4.19)

as in (4.9). By interpolation and Sobolev

≤ C ∇f λL2 f 1−λ , f L2 ≤ f λL2∗ f 1−λ L1 L1 where λ =

N N+2

as given by (4.16). Therefore, 2/λ

∇f 2L2 ≥ k f L2 ,

(4.20)

is a constant. Distributing the term ∇f L2 in (4.19) equally where k = C −1 f L1 between the left and the right-hand sides and using inequality (4.20) we obtain (λ−1)/λ

d 2/λ f 2L2 + k f L2 ≤ − ∇f 2L2 + 2K(t) ∇f 1+ν . L2 dt

(4.21)

The function X → −X2 + 2K(t)X 1+ν , appearing on the right-hand side of (4.21) has a global maximum (1 + ν)2r−1 (1 − ν)K(t)2r = CK(t)2r , so we obtain d 2/λ f 2L2 + k f L2 ≤ CK(t)2r ≤ C K¯ 2r , dt

(4.22)

where K¯ = sup K(t) ≤ sup f 2L1 . Applying Gronwall’s lemma argument to (4.22) we t≥0

r

t≥0

then obtain a bound of the

L2

norm of f in terms of f0 L2 and sup f L1r . Further, t≥0

integrating (4.19) over time, we get T 2 ¯ ¯ (1−ν)/2 ∇f 1+ν ∇f L2 ([0,T ]×RN ) ≤ C + K ∇f 1+ν dt ≤ C + KT , L2 L2 ([0,T ]×RN ) 0

which proves the second claim of the lemma.


527

Similar results can be established about the time-dependence of the L2k norms of the solutions. Lemma 4.8. Let f be a sufficiently regular solution to (0.1), with initial data f0 ∈ L2k (RN ), where k ≥ 0, and such that f has a moment of order r(k + 1), where r = N+2 4 , bounded uniformly in time. Then f (·, t) L2 ≤ C, k

0 ≤ t < ∞,

and ∇f L2 ([0,T ]×RN ) ≤ CT , k

for every 0 ≤ T < ∞, where the constants C and CT depend on N, f0 L2 and k sup f (·, t) L1 only. t≥0

r(k+1)

Proof. Multiplying the equation by f v2k and integrating we obtain: 1 d ∇f · ∇(f v2k ) dv = Q(f, f )f v2k dv. f 2L2 + 2 dt k RN RN Following the steps of the proof of Lemma 4.4 and distributing the term ∇(f vk ) 2 evenly between the left and right-hand sides we obtain the following differential inequality: d f vk 2L2 + ∇(f vk ) 2L2 dt

≤ − ∇(f vk ) 2L2 + CA1 (t)1−ν A2 (t) + k 2 A3 (t)1−ν ∇(f vk ) 1+ν , (4.23) L2

where A1 (t), A2 (t), and A3 (t) are the moments defined in Lemma 4.4. The uniform bounds of the moments imply that the right-hand side of (4.23) is bounded above by a constant. The left-hand side is estimated below as d 2/λ f vk 2L2 + c f vk L2 dt analogously to (4.19). Thus, by a Gronwall-type argument we obtain that the L2k -norm of f is bounded uniformly in time. Integrating (4.23) over time we also get the second claim of the lemma. Finally, we establish the following analog of Lemma 4.6 which will allow us to study the regularity of higher-order derivatives. Lemma 4.9. Let f be a solution to (0.1) with initial data f0 ∈ Hkn (RN ) where k ≥ 0 and n ≥ 0, such that f has a moment of order r ∗ = r(2n (k + µ) − 2µ + 1), where N+2 r = N+2 4 and µ > 2 , bounded uniformly in time. Then f (·, t) Hkn ≤ C,

0 ≤ t < ∞,

and f L2 ([0,T ],H n+1 (RN )) ≤ CT , k

for every 0 ≤ T < ∞, where the constants C and CT depend on N, f0 Hkn and sup f (·, t) L1∗ only. t≥0

r

528


Proof. We will use induction on n. The case n = 0 is already proven in Lemma 4.8. Assuming that the statement of the lemma holds for n − 1, we differentiate the equation in v and argue as in the proof of Lemma 4.6, obtaining the following inequality: 1 d ∇ n f 2L2 + ∇ n+1 f 2L2 ≤ C(1 + k 2 + f 2H n−1 )(1 + ∇ n f 2L2 ). 2 dt k k k+µ k+µ We estimate ∇ n f 2L2

integrating by parts and using Young’s inequality (cf. [13]):

k+µ

∇ n f 2L2

k+µ

≤ δ ∇ n+1 f 2L2 + Cδ ∇ n−1 f L2

2(k+µ)

.

n−1 we get Then, since we assumed f to be bounded in H2(k+µ)

1 d ∇ n f 2L2 ≤ − ∇ n+1 f 2L2 + C1 (n, k)δ ∇ n+1 f 2L2 + C2 (n, k, δ). 2 dt k k Choosing δ suitably small, we obtain the conclusion by Gronwall’s lemma.

Lemmas 4.7 – 4.9 allow us to make the following conclusions about the regularity of solutions to (0.1). Provided a sufficient number of moments is initially available, the H n regularity of the initial data is preserved with time. Moreover, the established bounds for the derivatives in L2 ([0, T ] × RN ) imply that after an arbitrarily short time the derivatives ∂ j f (·, t) of any order are in L2 (RN ), and then they propagate in time. Thus, on the level of apriori estimates we find that the solutions become immediately infinitely smooth in v and decay faster than any negative power for |v| large. We can also see that the solutions are infinitely differentiable in t. Indeed, in view of the established Hkn regularity we have f (·, t) ∈ S(RN ) for t > 0, and then Eq. (0.1) implies ∂t f (·, t) ∈ S(RN ), for every t > 0. Differentiating the equation in time and proceeding by induction we find also that ∂tm f (·, t) ∈ S(RN ), for every m = 1, 2, . . . , and for every t > 0. The time derivatives also remain bounded uniformly in time. 5. Existence We next proceed with a rigorous proof of existence that will also justify the formal manipulations performed in the derivation of apriori inequalities. Theorem 5.1. For every f0 ≥ 0, f0 ∈ L12 ∩ L log L(RN ) there exists a nonnegative weak solution f ∈ L∞ ([0, ∞), L12 (RN )),

f log f ∈ L∞ ([0, ∞), L1 (RN ))

to Eq. (0.1), with the initial condition f (·, 0) = f0 . Furthermore, if in addition f0 ∈ L1r ∩ L2 (RN ), where r = max{2, N+2 4 }, then for every t0 > 0, f ∈ Cb∞ ([t0 , ∞), S(RN )), where Cb∞ denotes the class of functions with bounded derivatives of any order, and S is the Schwartz class of rapidly decaying smooth functions. In particular, for t > 0, f is a classical solution of (0.1).


529

Theorem 5.2. For every ρ > 0 there exists a nonnegative solution f to (0.2), N f ∈ S(R ), satisfying f dv = ρ. RN

Furthermore, every nonnegative solution in L1r ∩ L2 (RN )), where r = max{2, N+2 4 } is in fact in S(RN ). Proof of Theorem 5.1. We assume that the initial datum f0 is in C ∞ (RN ) and has compact support (we will remove this assumption in the end of the proof). We also introduce a truncation in the collision term by replacing the factor |u| in (1.6) by |u|m,M = m + min{|u|, M},

(5.1)

and m > 0, M > 0 are truncation parameters. We then denote by Qm,M (f, f ) the corresponding collision operator. The first step of the proof will be to find approximating solutions which we define using the following truncated problem: ∂t f − v f = Qm,M (f, f ), f (0, v) = f0 (v),

v ∈ R3 ,

t ∈ [0, T ], (5.2)

where m, M and T are fixed positive parameters. We will denote by f solutions to (5.2), keeping in mind that they generally depend on m and M. The solutions will be constructed by applying a fixed point argument to the following approximation scheme: ∂t f − v f + Mf = Qm,M (g, g) + Mg, f (v, 0) = f0 (v).

v ∈ R3 ,

t ∈ [0, T ], (5.3)

Here g is a nonnegative function from L∞ ([0, T ], L12 ∩ L2 (RN )), which for every t > 0 has unit mass and zero average. Denoting by h the right hand side of Eq. (5.3) we notice that h ≥ 0, for every g ≥ 0, due to the truncation of the kernel. Indeed, h = Qm,M (g, g) + Mg ≥ −g (g ∗ |v|m,M ) + Mg ≥ 0.

(5.4)

Further, by analogy with Lemma 4.1 we can estimate Qm,M (g, g) as follows: Qm,M (g, g) Lp ≤ CM g L1 g Lp , k

k

k

1≤p≤∞

(5.5)

(there will be no loss of moments since the kernel Bm,M is bounded). Therefore, h ∈ L∞ ([0, T ], L12 ∩ L2 (RN )), as soon as g is in the same space. The unique weak solution f ∈ L∞ ([0, T ], L12 ∩ L2 (RN )) of (5.3) is then obtained from the following integral representation: t f (v, t) = f0 (v) ∗ E(v, t) + h(v, τ ) ∗ E(v, t − τ ) dτ, (5.6) 0

530


where ∗ denotes the convolution in v, and E(v, t) is the fundamental solution of (5.3): E(v, t) =

|v|2 1 e− 4t −Mt . N/2 (4πt)

The H 2 regularity of f is then guaranteed by the classical parabolic regularity result [29, Sect. 3.3], and we have the bound f H 2 ([0,T ]×RN ) ≤ CM ( h L2 ([0,T ]×RN ) + f0 H 1 (RN ) ).

(5.7)

We denote by T the operator that maps g into f . We next establish that for a certain choice of constants A1 and A2 this operator maps the set

1 N f dv = 1, f v dv = 0, B = f ∈ L ([0, T ] × R )) f ≥ 0, RN RN f 2 dv ≤ A22 , for a. a. t ∈ [0, T ] (5.8) f |v|2 dv ≤ A1 , RN

RN

into itself. Indeed, the nonnegativity of f is evident from the integral representation (5.6), since h ≥ 0. The mass and momentum normalization conditions follow easily, since for g ∈ B the collision term Qm,M (g, g) integrates to zero when multiplied by 1 or v. It remains to verify the last two conditions in (5.8). For the first of these conditions, multiplying the Eq. (5.3) by |v|2 and integrating by parts we obtain: d 2 f |v| dv + M f |v|2 dv N dt RN R ≤ 2N + M g |v|2 dv − k g g∗ |v|2 |v − v∗ |m,M dv dv∗ RN RN RN g |v|2 dv, (5.9) ≤ 2N + (M − mk) RN

where k = N (1 − α 2 )/4. Therefore, taking g so that 2N , g |v|2 dv ≤ A1 = mk RN yields the differential inequality d f |v|2 dv + M f |v|2 dv ≤ MA1 . N dt RN R Then, by Gronwall’s lemma, 2 f |v| dv ≤ max A1 , RN

Therefore, setting

RN

A1 = max A1 ,

RN

we obtain the required estimate.

f0 |v|2 dv .

f0 |v|2 dv ,


531

To obtain a bound of f in L2 we integrate the equation against f and use the inequality (5.5) to estimate Qm,M (g, g): 1 d |∇f |2 dv + M f 2 dv f 2 dv + N N 2 dt RN R R ≤ CM g L1 g L2 f L2 + M g L2 f L2 . (5.10) By Sobolev’s embedding and interpolation, 1/λ

−(1−λ)/λ

∇f L2 ≥ K f L2∗ ≥ K f L2 f L1

,

where 0 < λ < 1 is as in (4.16). Therefore, dividing (5.10) by f L2 and taking into account that g L1 = f L1 = 1, we get d 2/λ−1 f L2 + K 2 f L2 + M f L2 ≤ (CM + M) g L2 . dt Using the inequality K 2 x 2/λ−1 ≥

1 x − Kε , ε

true for all ε > 0, we find d f L2 + (1/ε + M) f L2 ≤ Kε + (CM + M) g L2 . dt We then get by Gronwall’s lemma: f L2 ≤ max f0 L2 , β + γ g L2 .

(5.11)

CM + M . 1/ε + M

(5.12)

where β=

Kε 1/ε + M

and

γ =

Choosing ε < 1/CM we get γ < 1. Therefore, we obtain the inequality f L2 ≤ A2 if we set A2 = max f0 L2 , β/(1 − γ ) . It is straightforward to verify that the set B is convex and closed in the strong topology of L1 ([0, T ]×RN ), using Fatou’s lemma and the fact that the second moment in |v| is uniformly bounded for g ∈ B. Further, the uniform in time bounds assumed in the definition of B imply the continuity of Qm,M (g, g) in L1 . We can then deduce easily that the solution operator T itself is continuous, based on the representation (5.6). Finally, the bound for the second moment and the regularity estimate (5.7) imply that the operator T maps B into its compact subset. By the Schauder theorem, this proves the existence of a fixed point for T in B, which is thereby a weak solution fm,M ∈ L∞ ([0, T ], L12 ∩ L2 (RN )) of (5.2). Our next goal is to pass to the limit as M → ∞ and then as m → 0, to recover the solutions with the “hard sphere” collision kernel. To this end, we will show that the bounds set forth in the apriori estimates hold for the fixed point solutions, and are uniform in M (and m). First of all, using the computation (5.9) it is easy to conclude that

532


the second moment is bounded uniformly in M, as soon as m > 0. Indeed, we obtain the following inequality for f = fm,M , d f |v|2 dv ≤ 2N − km f |v|2 dv, (5.13) dt RN RN so the required bound follows by Gronwall’s lemma. Further, we see that for every m > 0, M > 0 and for every T > 0, the solutions are in L∞ ([0, T ], L12p (RN )), for every p > 1. To see this we take K > 0 and introduce the truncated function p x , 0≤x 0 fixed and every p ≥ 0, the bounds of f = fm,M in L∞ ([0, T ], L12p (RN )) are independent of M.


533

Using the established L12p bounds and the fact that f ∈ H 2 ([0, T ] × RN ) we can make rigorous the arguments of Lemma 4.7 and then proceed as in Lemmas 4.8-4.9 obtaining that n f ∈ L∞ ([0, T ], H2p (RN )),

for every n = 1, 2..., and every p ≥ 0, with bounds independent on M. This will allow us to pass to the limit as M → ∞ in the weak form and to show that the limit solutions satisfy the equation with the kernel (m + |u|) b(u, σ ). We can then substitute the computation (5.15) by the argument of Lemma 3.4 and find the bounds in L∞ ([0, T ], L12p (RN )) that are independent on m and T . Arguing as above we can then pass to the limit as m → 0. The limit solution obtained in this step will then satisfy the equation with the “hard sphere” kernel. Finally, in order to treat the problem with the initial data f0 ∈ L12 ∩ L log L(RN ) we can take a sequence f n ∈ C0∞ (RN ) that converges to f0 in L1 (RN ). Then, since the constants in the bounds for the energy and entropy from Sect. 2 are independent of n, we can pass to the weak L1 -limit in the equations. The fact that the bounds of the solutions are independent of T allows us to continue the obtained solutions to [0, 2T ], and by induction, to [0, ∞). To study the regularity of solutions with L2 initial data we use the parabolic regularity of the equation [29] to find that f ∈ H 1 ([0, T ] × RN ), for any T > 0. Using this fact in combination with the bound f ∈ L∞ ([0, ∞), L1r (RN )) we can make rigorous the argument of Lemma 4.7 and then proceed as in Lemmas 4.8 and 4.9 to find the infinite differentiability of the solutions. We now turn our attention to the steady equation (0.2) and give a proof of Theorem 5.2. One of the possible approaches consists of adapting the arguments developed above for the time-dependent case. In fact, as a careful reader will easily check, practically all arguments in the above proof apply to the steady equation: the Gronwall lemma arguments will be replaced by the inequalities obtained by dropping the time-derivative terms. The only point that would need more careful attention is the moment estimate (5.15), which is not uniform in T . It can be replaced by a more elaborate argument for the moment bounds in the case of the truncated collision kernel. We will, however, take another approach, which will allow us to obtain the existence of the steady problem as a consequence of the regularization properties of the time-dependent equation. Proof of Theorem 5.2. The proof of Theorem 5.1 enables us to construct a semigroup on the convex set C made of those functions in L12 ∩ L2 (RN ) with unit mass and zero mean. Denote it by (St )t≥0 . Our bounds imply that for all t > 0, the range of St is compact in C. Therefore, for all n the equation fn = S2−n fn is solvable by Schauder’s theorem. Since fn = S1 fn , the sequence fn is contained in a fixed compact of C, namely S1 (C). We can therefore extract a subsequence which converges towards some f . Now for all k ≤ n we have fn = S2−k fn

534


(because 2−k is a multiple of 2−n ), and we can pass to the limit as n → ∞ using the continuity of the semigroup, thereby obtaining f = S2−k f,

for all

k ≥ 0.

Therefore f = St f for every t which is a sum of inverse powers of 2. Since the set of such times forms a dense subset of R+ and since the semigroup is continuous with respect to t, we conclude that f = St f, This ends the proof.

for all

t ≥ 0.

6. Uniqueness by Gronwall’s Lemma We next show that under the assumption that the initial data has the moment of order 3 finite, the solution to the time-dependent problem is unique. The proof uses an argument based on a certain cancellation property of the collision operator multiplied by sgn(f ) [1, 14] (see also [32] for discussion). We show that this property yields the desired result for the operator with inelastic collisions as well. Theorem 6.1. Assume that f0 ∈ L13 (RN ); then Eq. (0.1) with the initial condition f (·, 0) = f0 has at most one solution. Proof. Assume that f and g are solutions of (0.1), with the same initial data f0 . Set h = f − g and H = f + g. Then h satisfies the equation ∂t h − v h =

1 (Q(h, H ) + Q(H, h)), 2

with the homogeneous initial data. Now take mation of sgn(x). We can take   −1, ψε (x) = x/ε,  1,

(6.1)

a function ψε (x), a continuous approxix ≤ −ε −ε < x ≤ ε x > ε.

Multiplying Eq. (6.1) by ψε (h) (1 + |v|2 ) and integrating by parts we get d 1 2 h ψε (h)(1 + |v| ) dv + |∇h|2 (1 + |v|2 ) dv dt RN 2ε {|h|≤ε} 1 −2N φε (h) dv = (Q(h, H ) + Q(H, h)) ψε (h) (1 + |v|2 ) dv, 2 RN RN where

x

φε (x) = 0

  −x + ε/2, ψ(t) dt = x 2 /2ε,  x − ε/2,

x ≤ −ε −ε < x ≤ ε x > ε.


535

To estimate the right-hand side we can adapt the argument that is known to work in the case of the elastic Boltzmann equation (cf. [1, 14]). Passing to the weak form we get: (Q(h, H ) + Q(H, h)) ψε (h) (1 + |v|2 ) dv N R = H∗ h ψε (h ) (1 + |v |2 ) + h ψε (h∗ ) (1 + |v∗ |2 ) RN RN S N −1 − h ψε (h) (1 + |v|2 ) − h ψε (h∗ ) (1 + |v∗ |2 ) |u| b(u, σ ) dσ dv dv∗ . (6.2) Since |v |2 + |v∗ |2 ≤ |v|2 + |v∗ |2 , we can estimate the integrals of the first two terms in the braces as follows: 1 − α2 |h| H∗ |v − v∗ |2 ) (2 + |v|2 + |v∗ |2 − cN 4 RN RN ×(2 + |v|2 + |v∗ |2 ) |v − v∗ | dv dv∗ . Subtracting the third term in the integral (6.2) and noticing that hψε (h) − |h| ≤ |h| χε (h), where χε (x) is the characteristic function of the interval [−ε, ε], we obtain the estimate for the first three terms: |h| H∗ (1 + |v∗ |2 ) |v − v∗ | dv dv∗ RN RN + H∗ h (1 + |v|2 )|v − v∗ | dv dv∗ . RN

{|h|≤ε}

The fourth term in (6.2) contributes with another integral like the first one above, so we finally get d h ψε (h)(1 + |v|2 ) dv dt RN ≤ 2N φε (h) dv + |h| H∗ (1 + |v∗ |2 ) |v − v∗ | dv dv∗ RN RN RN 1 + H∗ h (1 + |v|2 )|v − v∗ | dv dv∗ . 2 RN {|h|≤ε} Passing to the limit as ε → 0, we find: d |h| (1 + |v|2 ) dv dt RN 2 1/2 ≤ 2N |h| dv + |h| (1 + |v| ) dv H (1 + |v|2 )3/2 dv N N N R R R ≤C |h| (1 + |v|2 ) dv, RN

since H is assumed to be bounded in L13 (RN ). Now since h(0, v) = 0 it follows by Gronwall’s lemma that h(t, v) = 0 for all times.

536


Remark. The uniqueness result of Theorem 6.1 is most certainly suboptimal. We believe that the uniqueness could be obtained in the class of initial conditions with finite mass and energy, with no additional assumptions, similarly to the classical Boltzmann equation [32]. The main technical obstacle for such a result is extending the Povzner inequalities in the case of inelastic collisions to the class of slowly growing piecewise linear functions ψ studied in [32]. We believe that this can be overcome with a more careful analysis of the inelastic collision mechanism. 7. Lower Bounds with Overpopulated High Energy Tails In this section we obtain pointwise lower estimates of solutions to (0.1) and (0.2) showing that the behavior of the high-energy tails of solutions is controlled from below by “stretched exponentials” A exp(−a|v|3/2 ). The bounds are established by using the comparison principle based on the parabolic (elliptic) structure of the equations. The following proposition establishes the particular role played by the “stretched exponentials”: they can be used as barrier functions in the comparison principle. Proposition 7.1. Let g(v) be a nonnegative function with finite mass ρ0 = RN g dv and moment of order one, ρ1 = RN g |v| dv. Then for every r > 0, and every K > 0, there is a constant a > 0 such that the function h(v) = Ke−a|v| , 3/2

satisfies h − Q− (g, h) ≥ 0,

for all |v| > r.

(7.1)

Further, choosing b > 0 large enough, the function h(v, t) = Ke−bt−a(1+|v|

2 )3/4

satisfies −∂t h + h − Q− (g, h) ≥ 0,

(7.2)

for all t > 0 and all v ∈ RN . Proof. To prove inequality (7.1) we fix an r > 0, compute h =

9 4

a 2 |v| −

3(2N − 1) a|v|−1/2 h, 4

and use the estimate Q− (g, h) = h (g ∗ |v|) ≤ (ρ1 + ρ0 |v|) h, to obtain h − Q(g, h) ≥

9

3(2N − 1) a 2 − ρ0 |v| − ρ1 − a|v|−1/2 h . 4 4

(7.3)


537

If 49 a 2 ≥ ρ0 , the factor on right-hand side of (7.3) attains its minimum for |v| = r. Therefore, inequality (7.1) holds for every a ≥ a ∗ , where a ∗ is the positive root of the quadratic equation 9r 2 3(2N − 1)r −1/2 a − a − (ρ0 r + ρ1 ) = 0. 4 4 1

For the time-dependent operator, setting v = (1 + |v|2 ) 2 , we obtain 3a

3N a −1/2 h − ∂t h = |v|2 v−1 3a + v−1/2 −

v + b h. 4 2 Choosing a so that 49 a 2 ≥ ρ0 and then b ≥ and complete the proof.

3Na 2

(7.4)

+ ρ0 + ρ1 we obtain inequality (7.2)

The established property of the function h(v) is used in next lemma to obtain a comparison result for the steady equation. Lemma 7.2. Let f ∈ L11 (RN ) be a nonnegative smooth solution to (0.2), with the mass ρ0 > 0. Then, there is a constant K > 0, such that f (v) ≥ Ke−2a|v| , 3/2

(7.5)

for all v ∈ RN , where a is a constant as in Proposition 7.1. Proof. Assuming the smoothness of the solution to (0.2), there is a constant c0 > 0 and a ball B(v0 , r0 ) with v0 ∈ RN and r0 > 0, such that f (v) ≥ c0 > 0,

if

v ∈ B(v0 , r0 ).

(7.6)

The value of c0 (as well as r0 and v0 ) depend on the solution f and use the fact that ρ0 > 0. Since Eq. (0.2) is translation invariant, we can take g(v) = f (v + v0 ); then g − Q− (g, g) = g − (g ∗ |v|)g ≤ 0.

(7.7)

Applying Proposition 7.1 to the function g(v) with r = r0 we find the barrier function h(v) = c0 exp(−a|v|3/2 ), for which we have h − Q− (g, h) = h − (g ∗ |v|)h ≥ 0,

for |v| > r0

(7.8)

and g(v) ≥ h(v),

for |v| ≤ r0 .

Therefore, letting U (v) = g(v) − h(v), subtracting (7.8) from (7.7) we obtain the inequality U − (g ∗ |v|) U ≤ 0,

|v| > r0 .

To prove that U (v) ≥ 0 everywhere we apply a form of a strong maximum principle (see, for example, [19]) to the operator L U = U − ν(U + h) U.

538


We can reduce the problem to proving that U ≥ 0 in a bounded domain. Indeed, the decay conditions on f imply that for every ε > 0 we can find R > 0 such that |U (v)| < ε if |v| ≥ R. Then we have L(U + ε) = L U − εν(g) ≤ 0,

r0 < |v| < R

and U + ε > 0 for |v| = r0 and |v| = R. The strong maximum principle then implies that U + ε ≥ 0 for all r0 ≤ |v| ≤ R. Letting ε go to zero we get U ≥ 0,

for all |v| ≥ r0 .

In view of the inequality (7.6) this implies g(v) ≥ c0 e−a|v| , 3/2

or, applied to the function f (v), f (v) ≥ c0 e−a|v−v0 |

3/2

≥ Ke−2a|v| , 3/2

with K = c0 e−a|v0 | . This completes the proof of the lemma. 3/2

By using a version of the maximum principle for the parabolic operator, we obtain, in a similar fashion, the pointwise lower bound for the time dependent problem. Lemma 7.3. Let f ∈ L∞ ([0, ∞), L12 (RN )) be a nonnegative smooth solution to (0.1) with the initial data f0 ≥ c0 exp(−a0 |v|3/2 ). Then, there are positive constants K, b and a, generally depending on the solution, such that f (v, t) ≥ Ke−bt−a|v| , 3/2

for all t > 0 and all v ∈ RN . Further, if there is a constant c1 and a ball B(v0 , r0 ), such that f (v, t) ≥ c1 ,

if v ∈ B(v0 , r0 ),

for all t, then the lower bound f (v, t) ≥ Ke−a|v| , 3/2

holds uniformly in time, where now K > 0, a > 0 and b > 0 will depend on c1 , v0 and r0 . Proof. To prove the first statement of the lemma we use the second part of Proposition 7.1 and repeat the comparison arguments of Lemma 7.2 taking h(v, t) = Ke−bt−a v

3/2

and using the strong maximum principle for the parabolic operator on U = f (v + v0 ) − h(v), L U = U − ν(f ) U − ∂t U. For the second part, the additional assumption made on f allows us to repeat the proof of Lemma 7.2 using the function h from (7.4).


539

Remark. It is natural to conjecture that solutions to (0.2) should satisfy a pointwise upper bound of the type K exp(−a |v|3/2 ), for certain values of a and K . A recent result by Bobylev et al. [7] shows that such a bound holds “in the L1 sense”, namely, for certain b > 0, f (v) exp(b|v|3/2 ) dv < +∞. RN

Further, an application of the maximum principle formulated in [39] allows one to reduce the problem to obtaining an estimate of the type Q+ (f, h) = o (Q− (f, h)),

|v| → ∞,

where f is a solution of (0.2) and h = K exp(−a |v|3/2 ). We will return to this problem in a forthcoming paper [18]. 8. Concluding Remarks We studied the existence, uniqueness and regularity for the time-dependent equation (0.1) and the existence, regularity for the steady equation (0.2). An important problem that remained beyond the scope of our study is the convergence of the time-dependent solutions to the steady ones as time approaches infinity. In fact, this remains a serious open problem, since no Lyapunov functional for the time-evolution is known to exist. A number of other interesting questions can be raised in connection to the obtained results. Are the steady states unique up to a normalization? Do the steady solutions necessarily have radial symmetry? (This can be expected from the rotation invariance of the equations; the existence and regularity of radial solutions can be obtained by applying our analysis to the reduced one-dimensional problem, as in [11], or just by working in spaces of radially symmetric functions.) We hope that the methods developed in the present work for the case of diffusion forcing could be useful for studying other problems involving the Boltzmann (Enskog) collision terms with other collision and driving mechanisms. In particular, a generalization to the case of a heat bath including a friction term seems to be rather straightforward. (The lower bounds in that case are expected to be Maxwellians.) It is also likely that applying the techniques of this paper should yield results for problems with the normal restitution coefficient dependent on the relative velocity [3, 8], which would allow us to study a broader range of physical phenomena. Another problem worth studying is the (quasi-)elastic limit α → 1. The steady states for the Boltzmann equation with elastic interactions (α = 1) and vanishing diffusion (µ = 0) are Maxwellians, while for every µ > 0 and every α < 1 we have a “3/2” lower bound. Obtaining quantitative information on the transition to the Maxwellian steady states would be valuable. We hope to address some of these questions in our future work. Acknowledgements. A number of people have contributed in a very important way to the development of the present paper. We would like to thank C. Bizon, S. J. Moon, J. Swift and H. Swinney for discussions concerning the physical aspects of the problem. We are also thankful to A. V. Bobylev, J. A. Carrillo, C. Cercignani and S. Rjasanow for fruitful discussions and a number of suggestions that helped us to improve the presentation. The first author has been supported by NSF under grant DMS 9971779 and by TARP under grant 003658-0459-1999; the third author has been supported by the HYKE European network, contract HPRN-CT-2002-00282. Support from the Texas Institute for Computational and Applied Mathematics/Austin is also gratefully acknowledged.

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References 1. Arkeryd, L.: On the Boltzmann equation. II. The full initial value problem. Arch. Rational Mech. Anal. 45, 17–34 (1972) 2. Benedetto, D., Caglioti, E., Carrillo, J.A., Pulvirenti, M.: A non-Maxwellian steady distribution for one-dimensional granular media. J. Statist. Phys. 91, 979–990 (1998) 3. Bizon, C., Shattuck, M.D., Swift, J.B., Swinney, H.L.: Transport coefficients for granular media from molecular dynamics simulations. Phys. Rev. E 60, 4340–4351 (1999), ArXiv:cond-mat/9904132 4. Bobylev, A.V.: Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems. J. Statist. Phys. 88, 1183–1214 (1997) 5. Bobylev, A.V., Carrillo, J.A., Gamba, I.M.: On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Statist. Phys. 98, 743–773 (2000) 6. Bobylev, A.V., Cercignani, C.: Moment equations for a granular material in a thermal bath. J. Statist. Phys. 106, 547–567 (2002) 7. Bobylev, A.V., Gamba, I.M., Panferov, V.: Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions. In preparation 8. Brilliantov, N.V., Poeschel, T.: Granular gases with impact-velocity dependent restitution coefficient. In: Granular Gases, T. Poeschel, S. Luding (eds.), Lecture Notes in Physics, Vol. 564, Berlin: Springer, 2000, pp. 100–124, ArXiv:cond-mat/0204105 9. Carrillo, J.A., Cercignani, C., Gamba, I.M.: Steady states of a Boltzmann equation for driven granular media. Phys. Rev. E (3), 62, 7700–7707 (2000) 10. Cercignani, C.: Recent developments in the mechanics of granular materials. In: Fisica matematica e ingegneria delle strutture, Bologna: Pitagora Editrice, 1995, pp. 119–132 11. Cercignani, C., Illner, R., Stoica, C.: On diffusive equilibria in generalized kinetic theory. J. Statist. Phys. 105, 337–352 (2001) 12. Desvillettes, L.: Some applications of the method of moments for the homogeneous Boltzmann and Kac equations. Arch. Rational Mech. Anal. 123, 387–404 (1993) 13. Desvillettes, L., Villani, C.: On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness. Comm. Partial Diff. Eqs. 25, 179–259 (2000) 14. Di Blasio, G.: Differentiability of spatially homogeneous solutions of the Boltzmann equation in the non Maxwellian case. Commun. Math. Phys. 38, 331–340 (1974) 15. Elmroth, T.: Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range. Arch. Ra. Mech. Anal. 82, 1–12 (1983) 16. Ernst, M.H., Brito, R.: Velocity tails for inelastic Maxwell models (2001), ArXiv:cond-mat/0111093 17. Ernst, M.H., Brito, R.: Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails. J. Stat. Phys., to appear (2002), ArXiv:cond-mat/0112417 18. Gamba, I.M., Panferov, V., Villani, C.: Work in progress 19. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Vol. 224 of Grundlehren der Mathematischen Wissenschaften, Berlin: Springer-Verlag, second ed., 1983 20. Goldhirsch, I.: Rapid granular flows: kinetics and hydrodynamics. In: Modeling in applied sciences, Model. Simul. Sci. Eng. Technol., Boston, MA: Birkhäuser Boston, 2000, pp. 21–79 21. Goldshtein, A., Shapiro, M.: Mechanics of collisional motion of granular materials. I. General hydrodynamic equations. J. Fluid Mech. 282, 75–114 (1995) 22. Gustafsson, T.: Global Lp -properties for the spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal. 103, 1–38 (1988) 23. Jenkins, J.T.: Kinetic theory for nearly elastic spheres. In: Physics of dry granular media (Cargèse, 1997), NATO Adv. Sci. Inst. Ser. E Appl. Sci., 350, Dordrecht: Kluwer Acad. Publ., 1998, pp. 353– 369 24. Jenkins, J.T., Richman, M.W.: Grad’s 13-moment system for a dense gas of inelastic spheres. Arch. Rational Mech. Anal. 87, 355–377 (1985) 25. Jenkins, J.T., Savage, S.B.: A theory for rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187–202 (1983) 26. Krapivsky, P.L., Ben-Naim, E.: Multiscaling in infinite dimensional collision processes. Phys. Rev. E 61, R5 (2000), ArXiv:cond-mat/9909176 27. Krapivsky, P.L., Ben-Naim, E.: Nontrivial velocity distributions in inelastic gases. J. Phys. A 35, L147 (2002), ArXiv:cond-mat/0111044 28. Kudrolli, A., Henry, J.: Non-Gaussian velocity distributions in excited granular matter in the absence of clustering. Phys. Rev. E 62, R1489 (2000), ArXiv:cond-mat/0001233 29. Ladyzhenskaya, O., Uraltseva, N., Solonnikov, V.: Linear and quasi-linear equations of parabolic type. Vol. 23 of AMS Translations of Mathematical Monographs, Providence, RI: Am. Math. Soc., 1988 30. Losert, W., Cooper, D., Delour, J., Kudrolli, A., Gollub, J.: Velocity statistics in excited granular media. Chaos 9, 682–690 (1999), ArXiv:cond-mat/9901203


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Communications in


Moment Maps, Scalar Curvature and Quantization of Ka¨ hler Manifolds Claudio Arezzo1 , Andrea Loi2 1

Dipartimento di Matematica, Università di Parma, Via D’Azeglio 85, Italy. E-mail: [email protected] 2 Dipartimento di Matematica, Università di Cagliari, Via Ospedale 72, Italy. E-mail: [email protected] Received: 23 June 2003 / Accepted: 3 October 2003 Published online: 2 March 2004 – © Springer-Verlag 2004

Abstract: Building on Donaldson’s work on constant scalar curvature metrics, we study the space of regular Kähler metrics Eω , i.e. those for which deformation quantization has been defined by Cahen, Gutt and Rawnsley. After giving, in Sects. 2 and 3 a review of Donaldson’s moment map approach, we study the “essential” uniqueness of balanced basis (i.e. of coherent states) in a more general setting (Theorem 2.5). We then study the space Eω in Sect. 4 and we show in Sect. 5 how all the tools needed can be defined also in the case of non-compact manifolds. 1. Introduction Let (M, ω) be a polarized Kähler manifold with polarization L, namely a compact Kähler manifold endowed with a holomorphic line bundle L whose first Chern class c1 (L) = [ω]dR . One can define two natural subspaces of Cω , the set of Kähler forms on M cohomologous to ω. The first one, denoted by Sω , is the space of constant scalar curvature metrics cohomologous to ω and the second one is the space Eω consisting of Kähler forms cohomologous to ω such that their Tian’s function is constant (see formula (15) for its definition, compare also Sect. 4). While Sω has an obvious geometric interest, Eω has been proved to be of great importance in the theory of quantization. Indeed, for a Kähler manifold with Eω non-empty Cahen, Gutt and Rawnsley [3] have shown how to generalize Berezin’s quantization procedure. In [9] Donaldson, using the concept of balanced basis of H 0 (L) studies the interplay between the two spaces above. Under the hypothesis that AutC(M,L) is discrete he ∗ shows that there exists at most one Kähler metric in Sω and in Eω (see Theorem 2.1 and Corollary 2.3 below). Moreover, he shows (see Theorem 2.2 below) that if Sω is non-empty then Lm , the mth tensor power of L, is stable, for m sufficiently large. Since Kähler-Einstein metrics have constant scalar curvature, Donaldson’s result confirms in one direction the well-known

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conjecture of Yau [24] which asserts that the existence of a Kähler-Einstein metric is equivalent to the stability in the sense of geometric invariant theory. Additional evidence for Yau’s conjecture had been provided earlier by Tian (see [21–23]), who showed that the existence of constant scalar curvature metrics implies K-stability and CM-stability. The aim of this paper is twofold. First, we study the space Eω without Donaldson’s assumption on AutC(M,L) . Our main result is Theorem 2.5 below where we generalize ∗ Donaldson’s result on balanced bases. As a consequence (cfr. Theorem 4.1 below) we have that two cohomologous Kähler forms in Eω belong to the same orbit under the action of the group of biholomorphisms of M which lift to the line bundle L. In Theorem 4.3, we also give a description of the space Eω in the case of compact coadjoint orbits. Second, by using the quantization tools developed by Cahen, Gutt and Rawnsley we show that all the spaces which come in this study can be defined also for non-compact manifolds and we explicitly compute Tian’s function in some cases. We end this paper discussing the difficulties we run into trying to study the geometry of the space Eω in this situation. The paper is organized as follows. In Sect. 2 we describe Donaldson’s results about balanced bases and we state our main Theorem 2.5 which is proved in Sect. 3. Section 4 is devoted to the study of the space Eω . Finally, in Sect. 5, we treat the non-compact case. 2. Preliminaries and Statements of the Main Results Consider the complex projective space CP N with standard homogeneous coordinates [z0 , . . . , zN ] and the matrix valued function on CP N given by: zj zk , j, k = 0, . . . , N. Bj k = 2 l |zl | Let V ⊂ CP N be a projective variety. We define M(V ) to be the skew-adjoint (N + 1) × (N + 1) matrix with entries M(V )j k = i Bj k dµV , j, k = 0, . . . N, (1) V

where dµV is the standard measure on V induced by the Fubini–Study form F S on n CP N , namely dµV = n!F S |V , where n is the complex dimension of V . Recall that i ¯ ∂ ∂ log |zl |2 2π N

F S =

(2)

l=0

for the homogeneous coordinate system [z0 , . . . , zN ] in CP N . Donaldson christened a projective variety V ⊂ CP N balanced if M(V ) is a multiple of the identity matrix. By formula (1) this multiple is a purely imaginary number iλ with λ > 0. Hence we deduce that V ⊂ CP N is balanced iff there exists a positive real number λ such that Bj k dµV = λδj k , j, k = 0, . . . N. (3) V

Moment Maps, Scalar Curvature and Quantization of Kähler Manifolds

545

The relevance of this condition in the algebro-geometric context is given by Luo’s Theorem [17], which proved that a balanced variety is stable in the sense of Hilbert– Mumford (the converse is still an open question). The previous definition can be extended to the case of polarized manifolds. So, let (M, L) be a polarized manifold with polarization L. This means that M is a compact complex n-dimensional manifold endowed with a holomorphic line bundle L whose first Chern class c1 (L) can be represented by a Kähler form ω on M, namely c1 (L) = [ω]dR . The line bundle L is called a polarization of M. For a positive integer m, let Lm = L⊗m be the mth tensor power of L and denote by H 0 (Lm ) the space of holomorphic sections of Lm . The Kodaira embedding theorem asserts that for m sufficiently large the holomorphic sections of Lm define a projective embedding im : M → P(H 0 (Lm )∗ ).

(4)

A choice of a basis (s0m , . . . , sdmm ) in H 0 (Lm ) identifies P(H 0 (Lm )∗ ) with the standard CP dm , where dm = dim H 0 (Lm ) − 1 is given with the help of the Riemann– Roch formula. We say that the pair (M, Lm ) is balanced if one can choose a basis in H 0 (Lm ) such that V = im (M) is a balanced variety in CP dm . In this case we call this basis a balanced basis of H 0 (Lm ). In order to see how this definition depends on the basis (s0m , . . . , sdmm −1 ) we write the Kodaira embedding (4) more explicitly. Thus, let σ : U → Lm be a trivializing holomorphic section of Lm on the open set U ⊂ M and define the map sdmm (x) s0m (x) dm +1 iσ : U → C \ {0} : x → ,..., . (5) σ (x) σ (x) If τ : V → Lm is another holomorphic trivialization then there exists a non-vanishing holomorphic function f on U ∩ V such that σ (x) = f (x)τ (x). Then, the Kodaira embedding written in the basis s m = (s0m , . . . , sdmm ) denoted by im (s m ), is the holomorphic embedding im (s m ) : M → P(H 0 (Lm )∗ ) ∼ = CP dm ,

(6)

whose local expression in the open set U is given by (5). Consider the Kähler form ωm on M induced by the Fubini–Study form on CP dm namely: ωm = im (s m )∗ (F S ) =

dm m sl (x) 2 i ¯ ∂ ∂ log σ (x) , 2π

(7)

l=0

and the smooth function bjmk on M with values in the set of (dm + 1) × (dm + 1) complex matrices defined by: bjmk (x)

=

sjm (x) skm (x) σ (x) σ (x)

. dm slm (x) 2 l=0 σ (x)

(8)

(The expression (8) is given in terms of the trivializing holomorphic section σ : U → Lm but it is straightforward to verify that it is defined on the whole M.) It follows from the

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definition of balanced variety (cfr. formula (3)) that s0m , . . . , sdmm is a balanced basis of H 0 (Lm ) iff there exists a positive real number λ such that: ωn bjmk m = λδj k , j, k = 0, . . . , dm . (9) n! M Consider now the Hermitian metric hm : Lm × Lm → C defined on a pair of points (q, q ) ∈ Lm × Lm by the formula: q

q

1 σ (x) σ (x) hm (q, q ) = . λ dm slm (x) 2 l=0 σ (x)

(10)

Observe that the Ricci curvature form Ric(hm ) of hm is equal to ωm (see also formula (13) in Sect. 3). Formulae (8), (9) and (10) tell us that a basis (s0m , . . . , sdmm ) is balanced iff ωn

sjm , skm hm = hm (sjm (x), skm (x)) m = δj k , n! M or equivalently, iff (s0m , . . . , sdmm ) is an orthonormal basis for (H 0 (Lm ), ·, ·hm ), where

·, ·hm is the L2 -product in H 0 (Lm ) given by:

s m , t m hm =

hm (s m (x), t m (x)) M

n ωm , ∀s m , t m ∈ H 0 (Lm ). n!

(11)

√ √ √ Observe that by replacing hm by hcm , c > 0, and by taking cs m = ( cs0m , . . . , csdmm ) we still obtain a balanced basis. The same happens by acting, in the natural way, on s m by an element U of the unitary group U (dm + 1). We denote this action by U · s m . The first important Donaldson’s result in [9] asserts that if the group AutC(M,L) is dis∗ (M,L) Aut m crete this is the only way to make the pair (M, L ) balanced. Here denotes the C∗

group biholomorphisms of M which lift to holomorphic bundles maps L → L modulo the trivial automorphism group C∗ . More precisely, using the same notations as above, the following holds (see Theorem 1 in [9]):

Theorem 2.1 (Donaldson). Assume that AutC(M,L) is discrete. Let s m and s˜ m be two ∗ balanced bases of H 0 (Lm ). Then there exist U ∈ U (dm + 1) and c ∈ R+ such that s m = c U · s˜ m . Thus, under the hypothesis that AutC(M,L) is discrete the sequence of Kähler forms ∗ ωm given by (7) is uniquely determined. Indeed if ω˜ m is the sequence determined by the balanced basis s˜ m , we get: ωm = im (s m )∗ (F S ) = im (c U · s˜ m )∗ (F S ) = im (˜s m )∗ (F S ) = ω˜ m .

(12)

Donaldson’s main results about this sequence are summarized in the following (see Theorems 2 and 3 in [9]):


547

is discrete. Theorem 2.2 (Donaldson). Assume that AutC(M,L) ∗ (i) If, for m sufficiently large, the pair (M, Lm ) is balanced and the sequence ωmm C ∞ converges to some limit ω∞ as m → ∞, then ω∞ has constant scalar curvature; (ii) if ω∞ is a Ka¨ hler form in c1 (L) with constant scalar curvature, then, for m sufficiently large the pair (M, Lm ) is balanced and the sequence ωmm C ∞ -converges to ω∞ as m → ∞. Consequently, under the assumption of (ii) in the previous theorem and by the above mentioned result of Luo , Lm is stable for m sufficiently large. From the Kähler geometry point of view Theorem 2.2 implies the following: Corollary 2.3 (Donaldson). Assume that AutC(M,L) is discrete. Then there exists at most ∗ one Ka¨ hler form ω on M with c1 (L) = [ω]dR having constant scalar curvature. It is worth mentioning that Chen [8] proved the uniqueness result of the previous corollary under the hypothesis c1 (M) < 0. Hence Corollary 2.3 is an extension of Chen’s result , being Aut(M), and hence AutC(M,L) , finite if c1 (M) < 0 (see e.g. Theorem 2.1 ∗ in [12]). In this paper we study what happens when we drop the hypothesis that AutC(M,L) is ∗ discrete. The following example shows that the hypothesis on AutC(M,L) in Theorems 2.1 and ∗ 2.2 can not be dropped. Example 2.4. Consider the complex projective space M = CP n equipped with the hyperplane bundle L. We have c1 (L) = ω, being ω the Fubini–Study form on M. In this case every automorphism lifts to a bundle map Fˆ : L → L and we obviously have AutC(M,L) = Aut(M) = PGL(N + 1, C). One can easy verify that sj = zj , j = ∗ 0, . . . , N is a balanced basis of H 0 (L), where we are identifying H 0 (L) with the space of homogeneous polynomials in the variable z0 , . . . , zn . Take any F ∈ Aut(M); an easy calculation shows that s˜j = Fˆ · sj is still a balanced basis of H 0 (L). If we take F ∈ Aut(M) \ PU(N + 1), where PU(N + 1) is the projective unitary group we immediately see that its lift Fˆ does not act unitarily on H 0 (L) and so the hypothesis on Aut(M,L) in Theorem 2.1 is necessary. More generally, for any non-negative integer m C∗ the elements of the form m! j j z 0 . . . znn , j0 + · · · jn = m, sjm = j0 ! · · · jn ! 0 where j is a multi-index is a balanced basis of H 0 (Lm ) (which we are identifying with the space of homogeneous polynomials of degree m) and the Kodaira map in this basis N +m m N(m) , N (m) = im (s ) : M → CP N satisfies im (s m )∗ (F S ) = mω, F S being the Fubini–Study form on CP N(m) . It is worth mentioning that the map im (s m ) is obtained by rescaling the Veronese embedding and has been introduced by Calabi in [7]. As before if we take F ∈ Aut(M) \ PU(N + 1) then s˜jm = Fˆ ·sj is a balanced basis of H 0 (Lm ) for any lift Fˆ ∈ Aut(Lm ) of F . Moreover,

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one can easily verify that im (˜s m )∗ (F S ) = mF ∗−1 (ω) = mω. Finally, the sequence of Kähler forms ω2m = 2mω, ω2m+1 = (2m + 1)F ∗−1 (ω), m = 1, 2 . . . , shows that the hypothesis on AutC(M,L) in Theorem 2.2 is necessary (cfr. also Conjecture 1 below). In ∗ Sect. 4 we generalize all the considerations of this example to the case of homogeneous and simply connected Kähler manifolds. The main result of this paper is the following theorem which generalizes Theorem 2.1: Theorem 2.5. Let s m and s˜ m be two balanced bases of H 0 (Lm ). Then there exist U ∈ U (dm + 1), c ∈ R+ and Fˆ ∈ Aut(Lm ) such that s m = c U · Fˆ · s˜ m , where (Fˆ · s m )j = Fˆ ◦ sjm ◦ F −1 j = 0, . . . , dm , and where F ∈ Aut(M, Lm ) is the biholomorphism underlying Fˆ . It follows from this theorem that, without Donaldson’s assumption on AutC(M,L) , ∗ there could exist different sequences ωm and ω˜ m coming from different balanced bases of H 0 (Lm ). Nevertheless, a careful reading of the proof of (i) in Theorem 2.2 shows that once we fix such a sequence, say ωm , then ωmm C ∞ -converges to a constant scalar curvature Kähler form ω∞ . About the proof of (ii) in Theorem 2.2 we point out that the assumption on AutC(M,L) is used in different places. Nevertheless, we believe the ∗ following holds true: Conjecture 1. If ω∞ is a Ka¨ hler form in c1 (L) with constant scalar curvature, then, for m sufficiently large, the pair (M, Lm ) is balanced and one can find a sequence ωm = im (s m )∗ (F S ) for a balanced basis s m of H 0 (Lm ) such that ωmm C ∞ -converges to ω∞ as m → ∞. 3. Moment Maps and the Proof of Theorem 2.5 One of the main ingredients to prove Theorem 2.5 and Donaldson’s theorems above is summarized in the following (see [9] and also [11]): Proposition 3.1. Let G be a group acting on a Ka¨ hler manifold (H, ) by preserving the Ka¨ hler form ; let Lie(G) be its Lie algebra and let µG : H → Lie(G)∗ be the corresponding moment map. Assume that GC , the complexification of G, also acts on H. Then the following hold: (i) Let k be an element in the center of Lie(G)∗ fixed by the coadjoint action. Then for every ξ ∈ µ−1 G (k) we have C µ−1 G (k) ∩ (G · ξ ) = G · ξ,

namely the GC -orbit of ξ intersects the level set µ−1 G (k) in the G-orbit of ξ .


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(ii) Let StabG (ξ ) and StabGC (ξ ) denote the stabilizers of a point ξ ∈ H under the action of G and GC respectively and let Stab0G (ξ ) and Stab0GC (ξ ) be the identity components of these groups. Then the inclusion of StabG (ξ ) in StabGC (ξ ) induces StabGC (ξ ) an isomorphism between the discrete groups StabG0 (ξ ) and . StabG (ξ ) Stab0GC (ξ ) Throughout this section we fix a natural number m and we will assume that the line bundle Lm is very ample, namely the Kodaira map defined by a basis in H 0 (Lm ) is an embedding. In order to avoid heavy notations we put L = Lm . Fix a basis s = (s0 , . . . , sN ) and let ω0 be the Kähler form on M given by i(s)∗ (), where i(s) : M → CP N is the Kodaira map with respect to the basis s, N = dim H 0 (L) − 1. Let h0 be the Hermitian metric on L such that ω0 = Ric(h0 ), where Ric(h0 ) is the Ricci curvature form, namely, the 2-form on M defined in terms of the Hermitian metric h0 by the equation: i ¯ ∂ ∂ log h0 (σ (x), σ (x)), (13) 2π for a trivializing holomorphic section σ : U ⊂ M → L \ {0} of L. (ω0 and h0 play the role of ωm and hm given by the formulae (7) and (10) respectively.) The Hermitian metric h0 is defined up to a positive constant factor. With this notation we then have that the polarized manifold (M, L) is balanced with respect to the above basis s = (s0 , . . . , sN ) iff s is orthonormal with respect to the L2 -product ωn h0 (s(x), t (x)) 0 , ∀s, t ∈ H 0 (L). (14)

s, th0 = n! M Ric(h0 ) = −

We start by giving an equivalent criterion expressing that the pair (M, L) is balanced in terms of cohomologous Kähler forms. Let ω be any Kähler form on M such that c1 (L) = [ω]dR . Define a smooth function on M by the formula

ω (x) =

N

h(sj (x), sj (x)), x ∈ M,

(15)

j =0

where s0 , . . . , sN is an orthonormal basis for (H 0 (L), ·, ·h ) and h is an Hermitian metric on L whose Ricci curvature equals ω. It is easy to verify, as the notation suggests, that the function ω depends only on the Kähler form ω and not on the Hermitian metric h with Ric(h) = ω or on the orthonormal basis chosen. The above mentioned criterion is expressed in terms of this function by the following: Proposition 3.2. Let (M, L) be a polarized manifold. The following assertions are equivalent: (i) (M, L) is balanced; (ii) there exists a Ka¨ hler form ω on M with c1 (L) = [ω]dR such that ω equals a constant. Proof. Observe that for any Kähler form ω on M with [ω]dR = c1 (L) the following formula holds: i ¯ i(s)∗ (F S ) = ω + ∂ ∂ log ω , (16) 2π where s = (s0 , . . . , sN ) is an orthonormal basis with respect to the L2 -product ·, ·h and where h is the Hermitian metric on L satisfying Ric(h) = ω. The equivalence between assertions of (i) and (ii) is now immediate.

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Remark 3.3. In the hypothesis that ω is constant its value can be easily calculated as

ω = volN+1 . Indeed (M,ω) N +1=

N M j =0

ωn = h(sj , sj ) n!

ω M

ωn = ω vol(M, ω). n!

Observe also that if ω˜ is another Kähler form in the same cohomology class of ω with

ω˜ constant then ω˜ = ω = volN+1 . (M,ω) Remark 3.4. We can express Donaldson’s Theorem 2.1 in terms of cohomology classes ˜ dR = by saying that if two cohomologous Ka¨ hler forms ω and ω˜ on M, with [ω]dR = [ω] c1 (L) are such that ω and ω˜ are constants and AutC(M,L) is discrete, then ω = ω˜ (cfr. ∗ formula (12) above). The ingenious idea of Donaldson was to relate the balanced condition to the level set of an appropriate moment map and to apply Proposition 3.1. Therefore one needs to find a Kähler manifold (H, ), a group G acting on H in such a way that a balanced basis of H 0 (L) belongs to the level set of µG : M → Lie(G)∗ , where µG is the moment map for this action. The infinite dimensional manifold H. Until now we have thought of M and of L as complex manifolds equipped with fixed complex structures, say I0 and J0 respectively. Observe that I0 belongs to Iint , the set of all ω0 -compatible complex structures on M. This set consist of all the almost complex structures I on M such that, for all vector fields X, Y on M the following hold: NI (X, Y ) = 0, ω0 (X, I X) > 0, ω0 (I X, I Y ) = ω0 (X, Y ), where NI is denoted the Nijenhius tensor on M associated to the complex structure I . In order to vary the complex structure J0 on L, we need a brief digression on holomorphic and Chern connections. Consider two complex structures I on M and J on L. Let (L) be the space of smooth sections on L and let s be in (L). Given a trivializing (I, J )holomorphic section σ : U → L, i.e. dσ ◦ I = J ◦ dσ , let f be the smooth complex valued function on U such that s = f σ . Define the map ∂¯I,J : (L) → (0,1 (M) ⊗ L), : s → ∂¯I f ⊗ σ, where ∂¯I : (T M) → (T 0,1 M) is the usual operator associated to the complex structure I . Observe that s ∈ (L) is (I, J )-holomorphic iff ∂¯I,J (s) = 0. Given a connection ∇ on L, the decomposition of 1-forms into a (1, 0)-part and a (0, 1)-part with respect to I induced a decomposition of ∇ = ∇ 0,1 + ∇ 1,0 . We say that a connection ∇ on L is (I, J )-holomorphic if its (0, 1)-part (with respect to I ) equals ∂¯I,J , i.e. ∇ 0,1 = ∂¯I,J . We refer to [13] p. 85 for the proof of the following: Proposition 3.5. Let (M, I ) be a complex manifold and let L be a smooth complex line bundle on M equipped with a connection ∇ such that its curvature is purely of type (1, 1). Then there exists a unique complex structure J on L such that ∇ is (I, J )-holomorphic. We denote the complex structure given by the previous proposition by JI,∇ . One immediately obtains the following:


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Corollary 3.6. Let (M, I ) be a complex manifold and let L be a smooth complex line bundle on M equipped with two connections ∇ and ∇˜ such that their curvatures are of type (1, 1). Suppose that there exists a diffeomorphism F : M → M which admits a lift ˜ = ∇. Then Fˆ : L → L such that Fˆ ∗ (∇) JF ·I,∇˜ = d Fˆ ◦ JI,∇ ◦ d Fˆ −1 , where F · I is the complex structure on M defined by F · I = dF ◦ I ◦ dF −1 . In other words if F is (I, F · I )-holomorphic and Fˆ preserves the connections then Fˆ is (JI,∇ , JF ·I,∇˜ )-holomorphic. We now fix the Chern connection on L associated to the triple (h0 , I0 , J0 ), namely the unique (I0 , J0 )-holomorphic connection ∇ 0 on L which is compatible with the Hermitian metric h0 (see e.g. [10] p. 73). With the notation above we obviously have J0 = JI0 ,∇ 0 . Observe also that the curvature of ∇ 0 equals −2π iω0 , hence is of type (1, 1). To simplify the notation we give the following: Definition 3.7. Given a complex structure I on the complex manifold M we say that a section s ∈ (L) is holomorphic with respect to the complex structure I if s is (I, JI,∇ 0 )holomorphic, with JI,∇ 0 being the unique complex structure on L given by Proposition 3.5. We are now in the position to define our manifold H as the subset of (L)N+1 × Iint consisting of pairs (s , I ), where s = (s0 , . . . , sN ) are C-linearly independent elements of (L) which are holomorphic with respect to the (ω0 -compatible) complex structure I . The group G, its action on H and the moment map µG . Let G be the group of Hermitian bundle maps of L which preserves the Chern connection ∇ 0 , namely the C ∞ bundle-maps Fˆ : L → L such that Fˆ ∗ (h0 ) = h0 and Fˆ ∗ (∇ 0 ) = ∇ 0 . Observe that if Fˆ belongs to G then its underlying map F : M → M is a simplectomorphism, namely F ∗ (ω0 ) = ω0 . If I is ω0 -compatible then for all vector fields X, Y on M one has: ω0 ((F · I )(X), X) = ω0 (I dF −1 (X), dF −1 (X)) > 0, ω0 ((F · I )(X), (F · I )(Y )) = ω0 (I dF −1 (X), I dF −1 (Y )) = ω0 (X, Y ), which show that F ·I is ω0 -compatible. Moreover, let s be a section in (L) holomorphic with respect to the complex structure I , then, it is easily seen, using Corollary 3.6, that Fˆ · s = Fˆ ◦ s ◦ F −1 is holomorphic with respect to F · I , for all Fˆ ∈ G. Then we have the following: Proposition 3.8. The group G acts on H by Fˆ · (s , I ) = (Fˆ · s , F · I ) = (Fˆ ◦ s ◦ F −1 , dF ◦ I ◦ dF −1 ), where, for s = (s0 , . . . , sN ), Fˆ ◦ s ◦ F −1 = (Fˆ ◦ s0 ◦ F −1 , . . . , Fˆ ◦ sN ◦ F −1 ).

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There is another group acting naturally on H. This is the finite dimensional group SU (N + 1) acting on (s , I ) ∈ H by U · (s , I ) = (U · s , I ), U ∈ SU (N + 1), and hence leaving unchanged the complex structure I . The actions of the groups G and SU (N + 1) on H commutes, hence these give rise to an action of the product group G = G × SU (N + 1) on H. We refer to Donaldson for the proof of the following: Theorem 3.9 (Donaldson). The space H admits a Ka¨ hler form invariant by the action of G. The corresponding moment map µG : H → C ∞ (M, C) ⊕ su(N + 1), is given by: µG (s , I ) =

1 + , i( sj , sk h0 − 2

N

2 l=0 sl h0

N +1

δj k ) ,

(17)

where is the smooth function on M defined by (x) = N j =0 h0 (sj (x), sj (x)), is the Laplacian with respect to the metric g0 associated to ω0 (and I0 ) and su(N + 1) denotes the space of traceless skew-Hermitian matrices. The complexification GC of G. In order to apply Proposition 3.1 to our case we need to understand what GC is for G = G × SU (N + 1). The complexification of SU (N + 1) is SL(N + 1, C), namely the set of non-singular matrices with determinant 1. So it remains to give a meaning to G C . Even if one is not able to define G C , one can formally identify the G C -orbit of a point (s , I ) ∈ H with the element in the equivalence class of (s , I ) under the following equivalence relation (see Proposition 11 in [9] for details). We declare (s , I ) ∈ H equivalent to (s , I ) ∈ H iff there exists a bundle map Fˆ : L → L (not necessarily preserving ∇ 0 or h0 ) such that s = Fˆ · s and I = F · I (F as above denotes the underlying map of Fˆ ). Roughly speaking one can think of G C as the set of C ∞ -bundle maps Fˆ : L → L such that if (s , I ) belongs to H then (Fˆ · s , F · I ) still belongs to H. Proposition 3.1 in our case reads as: Proposition 3.10. Let (C, O) be in C ∞ (M, C) ⊕ su(N + 1), where C is a constant and O is the (N + 1) × (N + 1) zero matrix, and let (s , I ) be in µ−1 G ((C, O)). Then the −1 C G -orbit of (s, I ) intersects µG ((C, O)) in the G-orbit of (s , I ) or, equivalently, up to the G-action there exists a unique element in (GC · (s , I )) ∩ µ−1 G ((C, O)). We are now in the position to prove Theorem 2.5. Observe that with the notation so far and by Proposition 3.2 we are reduced to prove the following: let (s , I0 ) and (˜s , I0 ) be two elements in H and let ω0 = i(s)∗ (F S ) and ω˜ = i(˜s )∗ (F S ) be the induced Ka¨ hler forms via the Kodaira embedding. Suppose that ω0 and ω˜ are constant functions. Then there exist U ∈ U (N + 1), c ∈ R+ and Fˆ ∈ Aut(L) such that s = c U · Fˆ · s˜ .

(18)


Let vol(M, ω0 ) =

ω0n M n! . Consider K =

N+1 vol(M,ω0 ) , O

553

as an element in C ∞ (M, C)⊕

su(N +1) and observe that from formula (17) and by the fact that ω0 = volN+1 it fol(M,ω0 ) lows that µG (s , I0 ) = K. Let A be in GL(N +1, C) such that A· s˜ = s. Let z be a N +1root of det A and define t˜ = z−1 s˜ . Then there exists T ∈ SL(N +1, C) such that T · t˜ = s. ˜ Take a lift Fˆ1 : L → L Let F1 : M → M be a diffeomorphism such that F1∗ (ω0 ) = ω. ∗ 2 ˆ ˜ ˜ = ω. of F1 such that F1 (h0 ) = λh, λ = |z| , where Ric(h0 ) = ω0 and Ric(h) ˜ Now ∗ 0 observe that the connection ∇˜ = Fˆ (∇ ) has curvature equal to −2π i ω˜ and, therefore, by Corollary 3.6 one easily deduces that (Fˆ1 · t˜ , F1 · I0 ) is an element of H, namely Fˆ1 · t˜ is holomorphic with respect to F1 · I0 . Moreover (s , I0 ) and (Fˆ1 · t˜ , F1 · I0 ) belong to the same G C -orbit since (Fˆ1 · t˜ , F1 · I0 ) is obtained by acting on (s , I0 ) with (idG C , T ) followed by (Fˆ1 , idSL(N+1,C) ). We claim that µG (Fˆ1 · t˜ , F1 · I0 ) = K. Indeed, since ˜ sj , s˜j ) = N+1 = N+1 , one gets:

ω˜ = N j =0 h(˜ vol(M,ω) ˜ vol(M,ω ) 0

N

1 ˆ∗ ˆ · t˜j , Fˆ1 · t˜j ) = N (F h0 )(˜sj ◦ F1−1 , s˜j ◦ F1−1 ) jN=0 λ 1 −1 ˜ sj ◦ F , s˜j ◦ F −1 ) = N+1 , = j =0 h(˜ 1 1 vol(M,ω )

j =0 h0 (F1

0

and

M

h0 (Fˆ1 · t˜j , Fˆ1 · t˜k )

ω0n = n!

n ˜ sj , s˜k ) ω˜ Fˆ1−1∗ h(˜ = ˜sj , s˜k h˜ = δj k , n! M

which prove our claim. Hence, by Proposition 3.10 there exists (Fˆ2 , V ) ∈ G = G × SU (N + 1) such that: (s , I0 ) = (z−1 V · (Fˆ2 ◦ Fˆ1 ) · s˜ , (F2 ◦ F1 ) · I0 ). By taking F = F2 ◦ F1 we then have that I0 = F · I0 , i.e. F and its lift Fˆ = Fˆ2 ◦ Fˆ1 are holomorphic maps. By writing z−1 in polar coordinates z−1 = ceiθ and by putting U = eiθ V ∈ U (N + 1) we obtain Eq. (18) as desired. Remark 3.11. In order to reobtain Donaldson’s Theorem 2.1 one must show that, if Aut(M,L) is discrete, then Fˆ ∈ Aut(L) in formula (18) acts on (H 0 (L), ·, · ) as a h0 C∗ unitary transformation. This can be shown as follows. First, it is immediate to show that StabGC (s , I0 ), the stabilizer of the point (s , I0 ) for the GC -action, coincides with the set of (F, Fˆ ) ∈ Aut(M, L) which acts with determinant 1 on H 0 (L). Second, if Fˆ and cFˆ , c ∈ C∗ both act on H 0 (L) with determinant 1, then c is forced to be an N + 1 root of unity. Hence, if we denote by R the group of the N + 1-roots of unity, the inclusion StabGC (s , I0 ) ⊂ Aut(M, L) induces an inclusion between the quotient spaces StabGC (s , I0 ) Aut(M, L) ⊂ . R C∗ If the latter is discrete it follows that StabGC (s , I0 ) is discrete, being R finite. By (ii) of Proposition 3.1 we then have that StabG (s , I0 ) = StabGC (s , I0 ), where StabG (s , I0 ) denotes the stabilizer of the point (s , I0 ) for the G-action. Finally, observe that Fˆ ∈ Aut(M, L) in formula (18) satisfies Fˆ · t˜ = (V −1 T ) · t˜ which implies that it acts on H 0 (L) with determinant 1 and so (F, Fˆ ) belongs to StabGC (s , I0 ) = StabG (s , I0 ). Since the group G preserves h0 , Fˆ acts on (H 0 (L), ·, ·h0 ) as a unitary transformation and this concludes the proof of Donaldson’s theorem.

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4. On the Constancy of ω in a Fixed Cohomology Class In this section we study some geometric properties of the function ω . We are mainly interested in the case when ω is constant. Therefore, let M be a complex manifold endowed with a holomorphic line bundle L, and let ω be a Kähler form on M with [ω]dR = c1 (L) and ω equals a constant. Denote by Cω the set of Kähler forms on M cohomologous to ω. We want to study the set Eω consisting of the ω˜ ∈ Cω such that ω˜ is constant. We believe that an understanding of the “topology” of this set could give interesting information about (M, ω). We start with the following theorem which shows that two elements in Eω belong to the same Aut(M, L)-orbit. Theorem 4.1. Let M be a compact complex manifold M endowed with a holomorphic ˜ line bundle L. Let ω˜ be in Eω . Then there exists F ∈ Aut(M, L) such that F ∗ (ω) = ω. If AutC(M,L) is discrete then ω = ω. ˜ ∗ Proof. For the first part, it is enough to observe that by formula (18) (which holds with our assumption with ω = ω0 ) one gets: ω = i(s)∗ (F S ) = i(c U · Fˆ · s˜ )∗ (F S ) = F −1∗ (i(s)∗ (F S )) = F −1∗ (ω). ˜ The second part follows from Remark 3.11 above. Indeed under the assumption that AutC(M,L) is discrete, we can take Fˆ in G and so its underlying map F belongs to ∗ Aut(M, L) ∩ Symp(M, ω). Corollary 4.2. Let ω be a Ka¨ hler form on CP N with ω constant. Then there exists a natural number m and F ∈ PGL(N + 1, C) = Aut(CP N ) such that F ∗ (ω) = mF S . Proof. Since the second Betti number of CP N is 1 there exists a natural number m such that ω is cohomologous to mF S and thus the conclusion by Theorem 4.1 (see also Example 2.4). Let us denote by Aut(M, L, Cω ) the set of maps in Aut(M, L) which preserves Cω . Observe that if Aut(M, L) is connected then Aut(M, L, Cω ) = Aut(M, L). Further denote by Aut(M, L, Eω ) the subset of Aut(M, L, Cω ) which preserves Eω . We have the natural inclusion Symp(M, ω) ∩ Aut(M, L) ⊂ Aut(M, L, Eω ). The map F ∈ Aut(M, L) in Theorem 4.1 belongs to Aut(M, L, Eω ). Moreover F is uniquely determined up to the action (on the left) of the group Symp(M, ω)∩Aut(M, L). We declare F and G in Aut(M, L, Eω ) equivalent if they belong to the same orbit under the action of this group and we denote by [F ] the corresponding equivalence class of F . Hence we can define a bijection: :

Aut(M, L, Eω ) → Eω : [F ] → F ∗ (ω), Symp(M, ω) ∩ Aut(M, L)

whose surjectivity follows from the first part Theorem 4.1. Observe also that if AutC(M,L) ∗ is discrete then, from the second part of Theorem 4.1, we deduce that Symp(M, ω) ∩ Aut(M, L) = Aut(M, L, Eω ) and hence Aut(M, L, Eω ) = Eω = {ω}. Symp(M, ω) ∩ Aut(M, L)


555

We want to analyze the case of homogeneous Kähler manifolds (M, ω), namely those manifolds which are acted upon transitively by the group Symp(M, ω) ∩ Aut(M). The following theorem can be considered as a generalization of Example 2.4 above. Theorem 4.3. Let (M, ω) be a homogeneous and simply-connected Ka¨ hler manifold. Then the function ω is constant and the map above defines a bijection between Eω Aut(M) and the quotient space Symp(M,ω)∩ Aut(M) . Proof. Let Aut(M, L, h) be the subgroup of Aut(M, L) consisting of (F, Fˆ ) such that Fˆ ∗ (h) = h. Being M simply-connected, we have the following equality Aut(M, L, h) = Symp(M, ω) ∩ Aut(M). It is straightforward to verify that the function ω is invariant by Aut(M, L, h) and hence, in the homogeneous case, it reduces to a constant. To prove the second assertion, simply observe that for any F ∈ Aut(M) the pair (M, F ∗ (ω)) is a simply-connected and homogeneous Kähler manifold (the group F −1 ◦ (Symp(M, ω) ∩ Aut(M)) ◦ F acts transitively on it) and hence F ∗ (ω) is constant by the proof of first part. It follows that Aut(M) = Aut(M, L, Eω ), which concludes the proof of our theorem. Remark 4.4. Note that the condition of simply-connectedness in Theorem 4.3 can not be relaxed. In fact the n-dimensional complex torus M = Cn /Z2n endowed with the flat Kähler form ω is a homogeneous Kähler manifold. On the other hand ω can not be induced by the Fubini–Study metric (see Lemma 2.2 in [19] for a proof) and hence, in particular, ω can not be constant by formula (16). (See also [14] for the calculation of

ω in this case.) 5. Applications to the Theory of Quantization and the Non-Compact Case In the quantum mechanics terminology introduced by Kostant and Souriau a holomorphic Hermitian line bundle (L, h) such that Ric(h) = ω is called a geometric quantization of the Kähler manifold (M, ω) and L is called the quantum line bundle. The function

ω of the previous section or more generally mω , for a non-negative integer m, plays a fundamental role in the theory of quantization carried out in [2, 3 and 4]. As we have already observed in [1], the function mω equals the function Tm introduced in [20] which enables Tian to solve a conjecture posed by Yau [24] by proving that the Kähler form ω can be obtained as the limit of Bergmann metrics on M. Tian’s Theorem was generalized by Zelditch [25], who, using the theory of the Szegö Kernel on the unit circle bundle L∗ over M, proved that there is a asymptotic expansion

mω (x) = mn + a1 (x)mn−1 + a2 (x)mn−2 + · · · ,

(19)

as m → ∞. Later Lu [16], by using Tian’s peak section method, gave a detailed description of the smooth coefficients aj (x). For example he proved that a1 (x) = 21 ρ, where ρ is the scalar curvature of the metric g associated to ω. It is worth mentioning that these results on mω together with the moment maps tools, are the key ingredients used by Donaldson in the proof of Theorem 2.2. The special class of quantizations (L, h) of a Kähler manifold (M, ω) having ω constant is called regular. We call a quantization m-regular if mω is constant for a non-negative integer m. The m-regular quantizations enjoy very nice properties which

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enable Cahen, Gutt and Rawnsley to generalize Berezin’s method [2] to the case of compact Kähler manifolds and to obtain a deformation quantization of the Kähler manifold (M, ω). In view of Proposition 3.2, Donaldson’s Theorem 2.1 and our Theorem 4.1 one gets the following: Corollary 5.1. Let ω and ω˜ be two cohomologous Ka¨ hler forms on a compact complex ˜ respectively. Then manifold M both defining regular quantizations (L, h) and (L, h) (M,L) Aut there exists F ∈ Aut(M, L) such that F ∗ (ω) = ω. ˜ Moreover if is discrete then C∗ ω = ω. ˜ Regarding the existence of regular quantizations we observe that, from Donaldson’s Theorem 2.2, one immediately gets: Corollary 5.2. Given a geometric quantization (L, h) of a compact Ka¨ hler manifold (M, ω), such that ω has constant scalar curvature, suppose AutC(M,L) is discrete, then ∗ there exists a sequence of Ka¨ hler forms ωm on M all defining regular quantizations such that ωmm C ∞ -converges to ω. We believe that in the previous corollary the assumption on AutC(M,L) can be dropped ∗ (compare Conjecture 1 above) and this could have interesting consequences for the quantization deformation of constant scalar curvature metrics. Disregarding the applications to the theory of quantization, we believe that the study of the Kähler metrics ω such that mω is constant for all m deserves further study. Examples of this kind of metrics are the homogeneous ones on simply-connected manifolds as follows from Theorem 4.3. Observe that such a metrics has necessarily constant scalar curvature, as follows from the above result of Lu (see also [1]). Observe also that the balanced basis s m of H 0 (Lm ) such that im (s m )∗ (F S ) = ωm = mω is very special and the proof of Theorem 2.2 is trivial for this kind of metrics. The non-compact case. We want now to show that the tools developed in the theory of quantization can be used to extend all the previous definitions to the non-compact case. In what follows, we refer to [18] (see also [5, 6 and 14]) for details and further results. Suppose that (L, h) is a geometric quantization of a (not necessarily compact) Kähler manifold. Consider the space Hh0 (L) ⊂ H 0 (L) consisting of global holomorphic sections s of L which are bounded with respect to ωn 2

s, sh = sh = h(s(x), s(x)) . n! M One can show that Hh0 (L) is a separable complex Hilbert space. Let x ∈ M and q ∈ L\{0} be a fixed point of the fiber over x. If one evaluates s ∈ Hh0 (L) at x, one gets a multiple δq (s) of q, i.e. s(x) = δq (s)q. The map δq : Hh0 (L) → C is a continuous linear functional [3], hence from Riesz’s Theorem, there exists a unique eq ∈ Hh0 (L) such that δq (s) = s, eq h , ∀s ∈ Hh0 (L), i.e. s(x) = s, eq hq . It follows, by (20), that

ecq = c−1 eq , ∀c ∈ C∗ .

(20)


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The holomorphic section eq ∈ Hh0 (L) is called the coherent state relative to the point q. Thus, one can define a smooth function on M

(x) = h(q, q)eq 2h ,

(21)

where q ∈ L \ {0} is any point on the fiber of x. If sj , j = 0, . . . N N ≤ ∞ is a ortho normal basis for (Hh0 (L), ·, ·h ) then one can easily verify that = N j =0 h(sj , sj ) and therefore, in the compact case the function equals the function ω . Suppose the following condition holds: Condition A. For every x ∈ M there exists s ∈ Hh0 (L) such that s(x) = 0. Then, for every fixed basis s of Hh0 (L) , one can define, as in formula (6) above, Kodaira’s map: i(s) : M → CP N , N ≤ ∞. One call this map the coherent states map. We say that a basis s = (s0 , . . . , sN ) is a balanced basis of Hh0 (L) iff s is an orthonormal basis of (Hh0 (L), ·, ·h ), where Ric(h) = ω and ω = i(s)∗ (F S ), where F S is the Fubini–Study form on CP N (which can be defined also if N = +∞). As for the compact case we can prove formula (16) above, namely: i ¯ i(s)∗ (F S ) = ω + ∂ ∂ log ω . 2π Therefore if ω is constant then the basis s is balanced. (The converse is not generally ¯ true since, in the non-compact case, a function f satisfying ∂∂f = 0 is not necessarily constant.) In the following two examples we show that Condition A above is satisfied and ω is constant. Example 5.3. Let ω = nj=0 dzj ∧ d z¯ j be the standard Kähler form on Cn . The trivial bundle L = Cn × C on Cn equipped with the Hermitian metric: hz (w1 , w2 ) = e−π z w1 w¯2 , ∀w1 , w2 ∈ C, 2

where z2 = |z1 |2 + · · · + |zn |2 defines a geometric quantization of (Cn , ω). One can j j easily verify that tj = z11 . . . znn is an orthogonal basis for (Hh0 (L), ·, ·h ) and 1 jl ! . n! π jl n

tj 2h =

l=1

Therefore

ω = n!e−π z

2

+∞ jl n

π |zl |2jl = n!. l! l=1 jl =0

Therefore sj = n! nl=1 is given in this case by:

π jl l! tj

is a balanced basis of Hh0 (L) and the coherent states map

 n j π l j1 j : (z1 , . . . , zn ) → . . . , z . . . znn , . . .  . l! 1 

i(s) : Cn → CP ∞

l=1

(22)

558

C. Arezzo, A. Loi

Example 5.4. Let D = {z ∈ C||z|2 < 1} be the unit disk in C equipped with the hyperdz∧d z¯ bolic form ω = πi (1−|z| 2 )2 . The trivial bundle D × C → D endowed with the Hermitian metric 1 − |z|2 hz (w1 , w2 ) = w1 w¯2 , ∀w1 , w2 ∈ C, 2 satisfies Ric(h) = ω and so the pair (L, h) is a geometric quantization of (D, ω). It √ is easily seen that the holomorphic function sj = j + 1zj , j = 0, 1, . . . defines an orthonormal basis for the Hilbert space (Hh0 (L), ·, ·h ). Furthermore s = (. . . , sj , . . . ) is a balanced basis for Hh0 (L) since +∞ 1 1 − |z|2 1 − |z|2 1 (j + 1)|z|2j = =

ω (z) = 2 2 1 − |z|2 2 j =0

is constant. The coherent states map is given by: i(s) : D → CP ∞ : z → [. . . , j + 1zj , . . . ], j = 0, 1 . . . .

(23)

Remark 5.5. Example (5.4) can be generalized to the case of bounded symmetric domains of non-compact type endowed with the Bergmann metric (see [15 and 12]). Remark 5.6. The maps (22) and (23) were considered also by Calabi [7] in the context of holomorphic and isometric immersions in the infinite dimensional complex projective space. Remark 5.7. Observe that the constancy of ω in the previous examples can also be obtained from Theorem 4.3 above whose proof extends without any change to the noncompact case. We wonder if the Kähler metrics with ω constant, as in the previous examples, enjoy nice properties similar to those in the compact case. For example if ω and ω˜ are two cohomologous Kähler forms on a complex manifold M such that ω and ω˜ are both constant, how are the Kähler forms related? One could compare this question with the analogous one posed by Yau (see Sect. 6 in [24]) for Kähler-Einstein metrics on non-compact manifolds. One of the main difficulties in answering this kind of question is that the Hermitian ˜ = ω˜ define two Hilbert spaces metrics h and h˜ on L such that Ric(h) = ω and Ric(h) 0 0 Hh (L) and H ˜ (L) which can be different. Thus, two balanced bases s and s˜ could h “live” in different Hilbert spaces and we do not know how to connect them as we did, for example, in Theorem 4.1. Acknowledgement. We wish to thank Simon Donaldson for useful discussions about his work and for his interest in ours.

References 1. Arezzo, C., Loi, A.: Quantization of Kähler manifolds and the asymptotic expansion of Tian–Yau– Zelditch. J. Geom. Phys. 47, 87–99 (2003) 2. Berezin, F.A.: Quantization. Math. USSR Izv. 8, 1109–1165 (1974) 3. Cahen, M., Gutt, S., Rawnsley, J. H.: Quantization of Kähler manifolds I: Geometric interpretation of Berezin’s quantization. J. Geom. Phys. 7, 45–62 (1990)


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4. Cahen, M., Gutt, S., Rawnsley, J. H.: Quantization of Kähler manifolds II. Trans. Am. Math. Soc. 337, 73–98 (1993) 5. Cahen, M., Gutt, S., Rawnsley, J. H.: Quantization of Kähler manifolds III. Lett. Math. Phys. 30, 291–305 (1994) 6. Cahen, M., Gutt, S., Rawnsley, J. H.: Quantization of Kähler manifolds IV. Lett. Math. Phys. 34, 159–168 (1995) 7. Calabi, E.: Isometric Imbeddings of Complex Manifolds. Ann. of Math. 58, 1–23 (1953) 8. Chen, X–X.: The space of Kähler metrics. J. Diff. Geom. 56, 189–234 (2000) 9. Donaldson, S.: Scalar Curvature and Projective Embeddings, I. J. Diff. Geom. 59, 479–522 (2001) 10. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. NewYork: John Wiley and Sons Inc., 1978 11. Hitchin, N., Karlhede, A., Lindström, U., Roˇcek, M.: Hyperkäler Metrics and Supersymmetry. Commun. Math. Phys. 108, 535–589 (1987) 12. Kobayashi, S.: Trasformation Group in Differential Geometry. Berlin-Heidelberg-New York: Springer Verlag, 1972 13. Kobayashi, S.: Differential Geometry of Complex Vector bundles. Princeton, NJ: Princeton University Press, 1987 14. Loi, A.: The function epsilon for complex tori and Riemann surfaces. Bull. Belg. Math. Soc. Simon Stevin 7(2), 229–236 (2000) 15. Loi, A.: Quantization of bounded domains. J. Geom. Phys. 29, 1–4 (1999) 16. Lu, Z.: On the lower terms of the asymptotic expansion of Tian–Yau–Zelditch. Am. J. Math. 122, 235–273 (2000) 17. Luo, H.: Geometric criterion for Gieseker–Mumford stability of polarized Kähler manifolds. J. Diff. Geom. 49, 577–599 (1998) 18. Rawnsley, J. H.: Coherent states and Kähler manifolds. The Quarterly J. Math. 403–415 (1977) 19. Takeuchi, M.: Homogeneous Kähler Manifolds in Complex Projective Space. Japan J. Math. 4, 171–219 (1978) 20. Tian, G.: On a set of polarized Kähler metrics on algebraic manifolds. J. Diff. Geom. 32, 99–130 (1990) 21. Tian, G.: The K-energy on hypersurfaces and stability. Comm. Anal. Geom. 2, 239–265 (1994) 22. Tian, G.: Kähler–Einstein metrics with positive scalar curvature. Invent. Math. 130, 1–37 (1997) 23. Tian, G.: Bott–Chern forms and geometric stability. Discrete Contin. Dynam. Syst. 6, 211–220 (2000) 24. Yau, S.-T.: Nonlinear analysis in geometry. Enseign. Math. 33, 109–158 (1987) 25. Zeldtich, S.: Szegö Kernel and a Theorem of Tian. Int. Math. Res. Notices 317–331 (1998) Communicated by M.R. Douglas


Communications in


Factoriality of Boz˙ ejko–Speicher von Neumann Algebras ∗ ´ Piotr Sniady

Institute of Mathematics, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland. E-mail: [email protected] Received: 16 July 2003 / Accepted: 27 September 2003 Published online: 23 January 2004 – © Springer-Verlag 2004

Abstract: We study the von Neumann algebra generated by q–deformed Gaussian elements li + li∗ , where operators li fulfill the q–deformed canonical commutation relations li lj∗ − qlj∗ li = δij for −1 < q < 1. We show that if the number of generators is finite, greater than some constant depending on q, it is a II1 factor which does not have the property . Our technique can be used for proving factoriality of many examples of von Neumann algebras arising from some generalized Brownian motions, both for the type II1 and type III case. 1. Introduction In 1970 Frisch and Bourret [FB70] considered the commutation relation lφ lψ∗ − q lψ∗ lφ = (φ, ψ)

(1)

fulfilled by annihilation operator l and its adjoint creation operator l ∗ for q ∈ R. Annihilation and creation operators are indexed by elements of a fixed real Hilbert space HR and they act on a complex Hilbert space F, called Fock space. There is a distinguished unital vector ∈ F, called the vacuum, such that lφ = 0 for every φ ∈ HR . It was a long time until Bo˙zejko and Speicher [BS94] showed in 1994 the existence of the operators considered by Frisch and Bourret for all −1 < q < 1. Frisch and Bourret were studying generalized Gaussian variables Lφ = lφ + lφ∗ ; it turns out that the algebra q generated by (Lφ ) can be equipped with a tracial state a → , a. The motivation for studying such Gaussian variables is that if q = 1 then (1) coincides with the canonical commutation relations and hence (Lφ ) can be identified with a family of classical Gaussian random variables; for q = −1 relation (1) coincides with the canonical anticommutation relations; furthermore it turned out much later that ∗

Research supported by State Committee for Scientific Research (KBN) grant 2 P03A 007 23.

´ P. Sniady

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for q = 0 relation (1) coincides with the free relation and hence (Lφ ) is a family of Voiculescu semicircular elements [VDN92]. In this article we will study the von Neumann algebra q generated by q–deformed Gaussian variables Lφ . Since for q = 0 these von Neumann algebras are isomorphic to the free group factors, we can consider the general case as a ‘smooth’ deformation of this eminent case. Not too much is known about q , in particular it is not clear if it is always isomorphic to the free group factors. Bo˙zejko and Speicher [BS94] showed that under certain conditions q is non–injective; recently Nou [Nou02] showed that it is enough to assume that √−1 < q < 1 and dim HR ≥ 2. Recently Shlyakhtenko [Shl03] showed that if |q| < 2 − 1 and dim HR ≥ 2 then algebras q are solid (cf. [Oza03]) and if they are factors then they do not have the property . Bo˙zejko, Kümmerer and Speicher [BKS97] showed that if the number of generators is infinite (i.e. if dim HR = ∞) then for every −1 < q < 1 the algebra q is a II1 factor. In this article we show that this result remains true if the number of generators is finite, greater than some constant depending on q and that this factor does not have the property . We also point out that the same proof can be used for proving the analogous result for many other von Neumann algebras, both finite and infinite. 2. Notations If not stated otherwise, all results presented in this section are due to Bo˙zejko and Speicher [BS94]. 2.1. Fock space. Let HR be a real Hilbert space equipped with a bilinear scalar product (·, ·); we will denote its complexification by H and the corresponding sesquilinear scalar product by ·, ·. Let furthermore −1 < q < 1 be fixed. For integer n ≥ 0 we introduce an operator P (n) : H⊗n → H⊗n given by P (n) (ψ1 ⊗ · · · ⊗ ψn ) = q inv σ ψσ (1) ⊗ · · · ⊗ ψσ (n) , (2) σ ∈Sn

where inv σ is the number of inversions in σ , i.e. the number of pairs (i, j ) such that 1 ≤ i < j ≤ n and σ (i) > σ (j ). Operator P (n) is strictly positive, therefore we can equip H⊗n with a new scalar product , q = , P (n) , for , ∈ H⊗n , where ·, · denotes the standard scalar product on H⊗n . In the following by H⊗n we will mean the n–fold tensor product equipped with the standard scalar product ·, · and by Hq⊗n the n–fold tensor product equipped with the scalar product ·, ·q . The q–Fock space F is a complex Hilbert space defined by Hq⊗n , F = F(HR ) = n≥0

where the term Hq⊗0 should be understood as one–dimensional space C for some unital vector . By H ⊗ F we will mean the tensor product of Hilbert spaces H and F

Factoriality of Bo˙zejko–Speicher von Neumann Algebras

563

equipped with the canonical product of the scalar product in H and the scalar product in F. We consider the Hilbert space F+ = Hq⊗n = ⊥ ⊂ F n≥1

and the map j : H ⊗ F → F+

(3)

given by the trivial mapping of Hilbert spaces φ ⊗ (ψ1 ⊗ · · · ⊗ ψn ) → φ ⊗ ψ1 ⊗ · · · ⊗ ψn . Please note that the map (3) is not as trivial as it might appear since the scalar products are different on the domain and on the range. Proposition 1. For every −1 < q < 1 there exist constants C1 , C2 such that for every choice of the real Hilbert space HR , j ≤ C1 ,

j −1 ≤ C2 .

Similar inequalities hold for the right trivial mapping F ⊗ H → F + given by (ψ1 ⊗ · · · ⊗ ψn ) ⊗ φ → ψ1 ⊗ · · · ⊗ ψn ⊗ φ. Proof. Bo˙zejko and Speicher showed that 1 1 ⊗ P (n) 1 − |q| and Bo˙zejko [Bo˙z98] showed that there exists a positive constant ω(q) such that for each n we have 1 1 ⊗ P (n) ≤ P (n+1) . ω(q) P (n+1) ≤

2.2. Annihilation and creation operators. For φ ∈ HR we consider left and right creation operators lφ∗ , rφ∗ : F → F defined on elementary tensors by lφ∗ (ψ1 ⊗ · · · ⊗ ψn ) = φ ⊗ ψ1 ⊗ · · · ⊗ ψn ,

(4)

rφ∗ (ψ1 ⊗ · · · ⊗ ψn ) = ψ1 ⊗ · · · ⊗ ψn ⊗ φ.

(5)

We also consider their adjoints: left and right annihilation operators i ⊗ · · · ⊗ ψn , lφ (ψ1 ⊗ · · · ⊗ ψn ) = q i−1 (φ, ψi ) ψ1 ⊗ · · · ⊗ ψ

(6)

1≤i≤n

rφ (ψ1 ⊗ · · · ⊗ ψn ) =

i ⊗ · · · ⊗ ψn , q n−i (φ, ψi )ψ1 ⊗ · · · ⊗ ψ

(7)

1≤i≤n

i denotes an omitted factor and n ≥ 1. The case n = 0 should be understood as where ψ lφ = rφ = 0. All creation and annihilation operators are bounded. Remark 1. Please note that in the definition of the annihilation operators the natural extension of the bilinear scalar product (·, ·) was used and not the sesquilinear scalar product ·, ·.

´ P. Sniady

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2.3. von Neumann algebra. Let q denote the von Neumann algebra generated by the family of selfadjoint operators Lφ : F → F given by Lφ = lφ + lφ∗ for φ ∈ HR . Algebra q is equipped with a faithful tracial state x → , x. Vector is separating and cyclic for q ; let J : F → F denote the canonical involution. Then J Lφ J = Rφ , where Rφ = rφ + rφ∗ for φ ∈ HR . Operators (Lφ ) commute with operators (Rψ ) [BX00]. 3. The Main Result ⊆ H and let e , . . . , e Let us fix some finite–dimensional real Hilbert subspace HR 1 d R . be an orthonormal basis of HR . We denote by H the complexification of HR In Sect. 4 we will define a certain operator M. It is enough to know now that |M|2 : F → F is given by

|M|2 = M ∗ M =

(Lei − Rei )2 .

(8)

i

In Sect. 4 we will show the following. Proposition 2. C belongs to the kernel of |M|. For every −1 < q < 1 there exists d0 such that if d ≥ d0 then the restriction of |M| to the space F + is strictly positive in the sense that |M| ≥ > 0 holds for some . The above result has an immediate consequence. Theorem 1. Let d0 be the constant from Proposition 2. If dim HR ≥ d0 (the case dim HR = ∞ is allowed) then q is a II1 factor which does not have the property . ⊆ H in such a way that dim H ≥ d . Proof. We choose finite–dimensional HR 0 R R Let a ∈ q be central; it follows that a commutes with |M| hence |M|a = a|M| = 0 and a belongs to the kernel of |M|. Proposition 2 implies that a ∈ C. Since is separating it follows that a ∈ C, hence q is a factor. The following observation was pointed out to me by Dimitri Shlyakhtenko. The operator |M| belongs to the C ∗ -algebra generated by q and J q J . Proposition 2 shows that the kernel of |M| is equal to C and that the zero eigenvalue is separated. It follows that the orthogonal projection F → C belongs to C ∗ (q , J q J ). By the result of Connes [Con76] it follows that q does not have the property .


565

4. Inequalities We define M : F → H ⊗ F by

M() =

ei ⊗ Lei () − Rei () .

i

It is easy to check that (8) is indeed fulfilled. We also define auxiliary maps m, m† : F → H ⊗ F given by m() =

ei ⊗ lei () − rei () ,

i

m† () =

ei ⊗ le∗i () − re∗i () .

i

Clearly M = m + m† . Please note that the definitions of the above operators do not depend on the choice of the orthonormal basis (ei ). Lemma 1. The norm of m fulfills m ≤ 2C1 , where C1 is the constant from Proposition 1. Proof. Consider the map ml : F + → H ⊗ F given by ml =

ei ⊗ (lei ).

i

Its adjoint m∗l : H ⊗ F → F + is given by m∗l (φ ⊗ ) =

(ei , φ) le∗i = j (H φ) ⊗ ,

i

where j denotes the trivial map (3) and H : H → H is the orthogonal projection. Therefore ml = m∗l ≤ j ≤ C1 . Similar inequality can be obtained for the right annihilator mr =

ei ⊗ (rei );

i

it remains to notice that m = ml + mr .

Lemma 2. The restriction of |m† | to the space F + fulfills d − C 1 C2 |m† | ≥ √ . C2 d

´ P. Sniady

566

Proof. Consider the map S : F + → F + : S(φ1 ⊗ · · · ⊗ φn ) = φ2 ⊗ · · · ⊗ φn ⊗ (H φ1 ), where H : H → H denotes the orthogonal projection. Proposition 1 implies that S ≤ C1 C2 . Consider the map f˜ : H ⊗ H ⊗ F → F given by f˜ φ ⊗ ψ1 ⊗ (ψ2 ⊗ · · · ⊗ ψn ) = ek ⊗ e k , φ ⊗ ψ 1 ψ2 ⊗ · · · ⊗ ψ n ; k

√ its norm is equal to k ek ⊗ ek = d. From Proposition 1 it follows that f : H ⊗ F + → F given by f φ ⊗ (ψ1 ⊗ · · · ⊗ ψn ) = ek ⊗ e k , φ ⊗ ψ 1 ψ 2 ⊗ · · · ⊗ ψ n √

k

fulfills f ≤ C2 d (because the only difference between the maps f and f˜ is the choice of the norm on the domain). It is easy to check that for ∈ F + , f m† = (d − S). It follows √ C2 d m† () ≥ f m† () = (d − S) ≥ (d − C1 C2 ) . Proof (of Proposition 2). It is easy to check that belongs to the kernel of |M|2 . Observe that if the restrictions of the operators to the space F + fulfill |m† | > m then the restriction of |M| = |m† + m| to F + is strictly positive. Estimates from the above lemmas finish the proof. Remark 2. For the case q = 0 one can take C1 = C2 = 1 and Theorem 1 shows factoriality of 0 for dim HR ≥ 6 while we know (from the free group construction) that it is valid also for dim HR ≥ 2, therefore our result is far from being fully satisfactory. 5. Generalizations A careful reader might easily observe that in Sects. 3 and 4 we used only very few properties of Bo˙zejko–Speicher algebras. In particular, we did not use the exact form of the symmetrizer (2) and of the annihilation operators (6), (7), nor the existence of the tracial state a → , a. It follows that our proof can be used in a more general context to proof factoriality (both in the type II1 and type III case) of some von Neumann algebras arising from some generalized Brownian motions for which an analogue of Proposition 1 holds. It is easy to find a whole zoo of examples since already Bo˙zejko and Speicher in their original article [BS94] considered symmetrizations arising from Yang–Baxter operators


567

which are much more general than (2); Królak [Kr˙o00] generalized further these results. Also Hiai [Hia03] generalized results of Shlyakhtenko [Shl97] and constructed von Neumann algebras which are q–analogues of free Araki–Woods algebras. It was pointed out to me by M˘ad˘alin Gut¸a˘ that the generalized Brownian motions considered in the work [GM02] might also provide appropriate examples. Acknowledgement. I thank Marek Bo˙zejko for introducing me into the subject and for many discussions. I thank Benoˆıt Collins for teaching me the right way of thinking about tensors. I also thank M˘ad˘alin Gut¸a˘ , Fumio Hiai, Dimitri Shlyakhtenko and the Reviewer for many helpful remarks. Research supported by State Committee for Scientific Research (KBN) grant No. 2 P03A 007 23. The research was conducted in Syddansk Universitet (Odense, Denmark) and Banach Center (Warszawa, Poland) on a grant funded by European Post–Doctoral Institute for Mathematical Sciences.

References [BKS97] Bo˙zejko, M., Kümmerer, B., Speicher R.: q-Gaussian processes: non-commutative and classical aspects. Commun. Math. Phys. 185(1), 129–154 (1997) [Bo˙z98] Bo˙zejko, M.: Completely positive maps on Coxeter groups and the ultracontractivity of the q-Ornstein-Uhlenbeck semigroup. In: Quantum probability (Gdańsk, 1997), Warsaw: Polish Acad. Sci., 1998, pp. 87–93 [BS94] Bo˙zejko, M., Speicher, R.: Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces. Math. Ann. 300(1), 97–120 (1994) [BX00] Bo˙zejko, M., Xu, Q.: Factoriality and second quantization. Unpublished, 2000 [Con76] Connes, A.: Classification of injective factors. Cases I I1 , I I∞ , I I Iλ , λ = 1. Ann. of Math. (2), 104(1), 73–115 (1976) [FB70] Frisch, U., Bourret, R.: Parastochastics. J. Math. Phys. 11, 364–390 (1970) [GM02] Gut¸a˘ , M., Maassen, H.: Generalised Brownian motion and second quantisation. J. Funct. Anal. 191(2), 241–275 (2002) [Hia03] Hiai, F.: q-deformed Araki-Woods factors. In J.-M. Combes et al. (eds.) Operator Algebras and Mathematical Physics, Bucharest: Theta, 2003, pp. 169–202 [Kr˙o00] Kr˙olak, I.: Wick product for commutation relations connected with Yang-Baxter operators and new constructions of factors. Commun. Math. Phys. 210(3), 685–701 (2000) [Nou02] Nou, A.: Non injectivity of the q–deformed von Neumann algebras. Preprint, 2002 [Oza03] Ozawa, N.: Solid von Neumann Algebras. Preprint math.OA/0302082, 2003 [Shl97] Shlyakhtenko, D.: Free quasi-free states. Pacific J. Math. 177(2), 329–368 (1997) [Shl03] Shlyakhtenko, D.: Some estimates for non-microstates free entropy dimension, with applications to q-semicircular families. Preprint math.OA/0308093, 2003 [VDN92] Voiculescu, D. V., Dykema, K. J., Nica, A.: Free random variables. Providence, RI: American Mathematical Society, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups Communicated by Y. Kawahigashi


Communications in


Moyal Planes are Spectral Triples 1,2 , J.C. V´ V. Gayral1,2 , J.M. Gracia-Bond´ıa3 , B. Iochum1,2 , T. Schucker arilly4,5 ¨ 1 2 3 4 5

Centre de Physique Théorique , CNRS–Luminy, Case 907, 13288 Marseille Cedex 9, France. E-mail: [email protected]; [email protected]; [email protected] Université de Provence, France Departamento de F´ısica, Universidad de Costa Rica, 2060 San Pedro, Costa Rica Departamento de Matemáticas, Universidad de Costa Rica, 2060 San Pedro, Costa Rica Regular Associate of the Abdus Salam ICTP, 34014 Trieste, Italy. E-mail: [email protected]

Received: 24 July 2003 / Accepted: 31 October 2003 Published online: 2 March 2004 – © Springer-Verlag 2004

Abstract: Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, are proposed. As a guide, and for the sake of their importance in noncommutative quantum field theory, the spaces R2N endowed with Moyal products are intensively investigated. Some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes–Lott functional action, are given for these noncommutative hyperplanes. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Theory of Distributions and Moyal Analysis . . . . . . . . . . . 2.1 Basic facts of Moyalology . . . . . . . . . . . . . . . . . . . 2.2 The oscillator basis . . . . . . . . . . . . . . . . . . . . . . . 2.3 Moyal multiplier algebras . . . . . . . . . . . . . . . . . . . . 2.4 Smooth test function spaces, their duals and the Moyal product 2.5 The preferred unitization of the Schwartz Moyal algebra . . . 3. Axioms for Noncompact Spin Geometries . . . . . . . . . . . . . . 3.1 Generalization of the unital case conditions . . . . . . . . . . 3.2 Modified conditions for nonunital spectral triples . . . . . . . 3.3 The commutative case . . . . . . . . . . . . . . . . . . . . . . 3.4 On the Connes–Landi spaces example . . . . . . . . . . . . . 4. The Moyal 2N-Plane as a Spectral Triple . . . . . . . . . . . . . . 4.1 The compactness condition . . . . . . . . . . . . . . . . . . . 4.2 Spectral dimension of the Moyal planes . . . . . . . . . . . . 4.3 The regularity condition . . . . . . . . . . . . . . . . . . . . . 4.4 The finiteness condition . . . . . . . . . . . . . . . . . . . . .

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4.5 The other axioms for the Moyal 2N -plane 5. Moyal–Wick Monomials . . . . . . . . . . . . 5.1 An algebraic mould . . . . . . . . . . . . 5.2 The noncommutative Wick monomials . . 6. The Functional Action . . . . . . . . . . . . . . 6.1 Connes–Terashima fermions . . . . . . . 6.2 The differential algebra . . . . . . . . . . 6.3 The action . . . . . . . . . . . . . . . . . 7. Conclusions and Outlook . . . . . . . . . . . . 8. Appendix: A Few Explicit Formulas . . . . . . 8.1 On the oscillator basis functions . . . . . 8.2 More junk . . . . . . . . . . . . . . . . .

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1. Introduction Since Seiberg and Witten conclusively confirmed [79] that the endpoints of open strings in a magnetic field background effectively live on a noncommutative space, string theory has given much impetus to noncommutative field theory (NCFT). This noncommutative space turns out to be of the Moyal type, for which there already existed a respectable body of mathematical knowledge, in connection with the phase-space formulation of quantum mechanics [65]. However, NCFT is a problematic realm. Its bane is the trouble with both unitarity and causality [39, 78]. Feynman rules for NCFT can be derived either using the canonical operator formalism for quantized fields, working with the scattering matrix in the Heisenberg picture by means of Yang–Feldman–Källén equations; or from the functional integral formalism. These two approaches clash [3], and there is the distinct possibility that both fail to make sense. The difficulties vanish if we look instead at NCFT in the Euclidean signature. Also, in spite of the tremendous influence on NCFT, direct and indirect, of the work by Connes, it is surprising that NCFT based on the Moyal product as currently practised does not appeal to the spectral triple formalism. So we may, and should, raise a basic question: namely, whether the Euclidean version of Moyal noncommutative field theory is compatible with the full strength of Connes’ formulation of noncommutative geometry, or not. The prospective benefits of such an endeavour are mutual. Those interested in applications may win a new toolkit, and Connes’ paradigm stands to gain from careful consideration of new examples. In order to speak of noncommutative spaces endowed with topological, differential and metric structures, Connes has put forward an axiomatic scheme for “noncommutative spin manifolds”, which in fact is the end product of a long process of learning how to express the concept of an ordinary spin manifold in algebraic and operatorial terms. A compact noncommutative spin manifold consists of a spectral triple (A, H, D), subject to the six or seven special conditions laid out in [19] —and reviewed below in due course. Here A is a unital algebra, represented on a Hilbert space H, together with a distinguished selfadjoint operator, the abstract Dirac operator D, whose resolvent is completely continuous, such that each operator [D, a] for a ∈ A is bounded. A spectral triple is even if it possesses a Z2 -grading operator χ commuting with A and anticommuting with D. The key result is the reconstruction theorem [19, 20] which recovers the classical geometry of a compact spin manifold M from the noncommutative setup, once the

Moyal Planes are Spectral Triples

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algebra of coordinates is assumed to be isomorphic to the space of smooth functions C ∞ (M). Details of this reconstruction are given in [45, Chapters 10 and 11] and in a different vein in [71]. Thus, for compact noncommutative spaces, the answer to our question is clearly in the affirmative. Indeed the first worked examples of noncommutative differential geometries are the noncommutative tori (NC tori), as introduced already in 1980 [14, 74]. It is a simple observation that the NC torus can be obtained as an ordinary torus endowed with a periodic version of the Moyal product. The NC tori have been thoroughly exploited in NCFT [24, 92]. The restriction to compact noncommutative spaces (“compactness” being a metaphor for the unitality of the coordinate algebra A) is essentially a technical one, and no fundamental obstacle to extending the theory of spectral triples to nonunital algebras was foreseen. However, it is fair to say that so far a complete treatment of the nonunital case has not been written down. (There have been, of course, some noteworthy partial treatments: one can mention [41, 73], which identify some of the outstanding issues.) The time has come to add a new twist to the tale. In this article we show in detail how to build noncompact noncommutative spin geometries. The indispensable commutative example of noncompact manifolds is considered first. Then the geometry associated to the Moyal product is laid out. One of the difficulties for doing this is to pin down a “natural” compactification or unitization (embedding of the coordinate algebra as an essential ideal in a unital algebra), the main idea being that the chosen Dirac operator must play a role in this choice. Since the resolvent of D is no longer compact, some adjustments need to be made; for instance, we now ask for a(D − λ)−1 to be compact for a ∈ A and λ ∈ / sp D. Then, thanks to a variation of the famous Cwikel inequality [27, 81] —often used for estimating bound states of Schrödinger operators— we prove that the spectral triple N (S(R2N ), ), L2 (R2N ) ⊗ C2 , −i∂µ ⊗ γ µ , where S denotes the space of Schwartz functions and a Moyal product, is 2N + summable and has in fact the spectral dimension 2N. The interplay between all suitable algebras containing (S(R2N ), ) must be validated by the orientation and finiteness conditions [19, 20]. In so doing, we prove that the classical background of modern-day NCFTs does fit in the framework of the rigorous Connes formalism for geometrical noncommutative spaces. This accomplished, the construction of noncommutative gauge theories, that we perform by means of the primitive form of the spectral action functional, is straightforward. The issue of understanding the fluctuations of the geometry, in order to develop “noncommutative gravity” [12] has not reached a comparable degree of mathematical maturity, and is not examined yet. As a byproduct of our analysis, and although we do not deal here with NCFT proper, a mathematically satisfactory construction of the Moyal–Wick monomials is also given. The main results in this paper have been announced and summarized in [38]. The first order of business is to review the Moyal product more carefully with due attention paid to the mathematical details. 2. The Theory of Distributions and Moyal Analysis In this first paragraph we fix the notations and recall basic definitions. For any finite dimension k, let be a real skewsymmetric k × k matrix, let s · t denote the usual scalar

572


product on Euclidean Rk and let S(Rk ) be the space of complex Schwartz (smooth, rapidly decreasing) functions on Rk . One defines, for f, h ∈ S(Rk ), the corresponding Moyal or twisted product: −k f h(x) := (2π) f (x − 21 u) h(x + t) e−iu·t d k u d k t, (2.1) where d k x is the ordinary Lebesgue measure on Rk . In Euclidean field theory, the entries of have the dimensions of an area. Because is skewsymmetric, complex conjugation reverses the product: (f h)∗ = h∗ f ∗ . Assume to be nondegenerate, that is to say, σ (s, t) := s · t to be symplectic. This implies even dimension, k = 2N. We note that −1 is also skewsymmetric; let θ > 0 be defined by θ 2N := det . Then formula (2.1) may be rewritten as −1 f h(x) = (πθ )−2N f (x + s) h(x + t) e−2is· t d 2N s d 2N t. (2.2) The latter form is very familiar from phase-space quantum mechanics [40], where R2N is parametrized by N conjugate pairs of position and momentum variables, and the entries of have the dimensions of an action; one then selects 0 1N = S := . −1N 0 Indeed, the product (or rather, its commutator) was introduced in that context by Moyal [65], using a series development in powers of whose first nontrivial term gives the Poisson bracket; later, it was rewritten in the above integral form. These are actually oscillatory integrals, of which Moyal’s series development, f g(x) =

i |α| 1 ∂f ∂g (x) (x), α α 2 α! ∂x ∂(Sx) 2N

(2.3)

α∈N

is an asymptotic expansion. The development (2.3) holds —and sometimes becomes exact— under conditions spelled out in [33]. The first integral form (2.1) of the Moyal product was exploited by Rieffel in a remarkable monograph [75], who made it the starting point for a more general deformation theory of C ∗ -algebras. Since the problems we are concerned with in this paper are of functional analytic nature, there is little point in using the most general here: we concentrate on the nondegenerate case and adopt the form = θ S with θ real. Therefore, the corresponding Moyal products are indexed by the real parameter θ ; we denote them by θ and usually omit explicit reference to N in the notation. The plan of the rest of this section is roughly as follows. The Schwartz space S(R2N ) endowed with these products is an algebra without unit and its unitization will not be unique. Below, after extending the Moyal product to large classes of distributions, we find and choose unitizations suitable for our construction of a noncompact spectral triple, and show that (S(R2N ), θ ) is a pre-C ∗ -algebra. We prove that the left Moyal product by a function f ∈ S(R2N ) is a regularizing operator on R2N . In connection with that, we examine the matter of Calderón–Vaillancourt-type theorems in Moyal analysis. We inspect as well the relation of our compactifications with NC tori.


573

2.1. Basic facts of Moyalology. With the choice = θS made, the Moyal product can also be written 2i −2N f θ g(x) := (πθ ) f (y)g(z) e θ (x−y) · S(x−z) d 2N y d 2N z. (2.4) Of course, our definitions make sense only under certain hypotheses on f and g. A good chunk of Moyal analysis can be found in [43, 90], from which we extract the following lemma. Lemma 2.1 [43]. Let f, g ∈ S(R2N ). Then (i) f θ g ∈ S(R2N ). (ii) θ is a bilinear associative product on S(R2N ). Moreover, complex conjugation of functions f → f ∗ is an involution for θ . (iii) Let j = 1, 2, . . . , 2N. The Leibniz rule is satisfied: ∂ ∂g ∂f . (f θ g) = g + f θ ∂xj ∂xj ∂xj θ

(2.5)

(iv) Pointwise multiplication by any coordinate xj obeys xj (f θ g) = f θ (xj g) +

∂g i θ ∂f iθ g = (xj f )θ g − . f θ 2 ∂(Sx)j θ 2 ∂(Sx)j

(2.6)

(v) The product has the tracial property: 1 1 2N f, g := f θ g(x) d x = gθ f (x) d 2N x (πθ )N (π θ)N 1 = f (x) g(x) d 2N x. (πθ )N (vi) Let Lθf ≡ Lθ (f ) be the left multiplication g → f θ g. Then limθ↓0 Lθf g(x) = f (x) g(x), for x ∈ R2N . Property (vi) is a consequence of the distributional identity limε↓0 ε −k eia·b/ε = (2π )k δ(a)δ(b), for a, b ∈ Rk ; convergence takes place in the standard topology [77] of S(R2N ). To simplify notation, we put S := S(R2N ) and let S := S (R2N ) be the dual space of tempered distributions. In view of (vi), we may denote by L0f the pointwise product by f . Theorem 2.2 [43]. Aθ := (S, θ ) is a nonunital associative, involutive Fre´ chet algebra with a jointly continuous product and a distinguished faithful trace. Introduce the symplectic Fourier transform F by Ff (x) := (2π)−N f (t)eix·St d 2N t.

(2.7)

It is obviously a symmetry, i.e., an involutive selfadjoint operator. Since δθ δ = (π θ)−2N , the maps f → (π θ)N δθ f and f → f θ (πθ )N δ are unitary, too; they turn out to be [(πθ )N δθ f ](y) = (2/θ )N Ff (−2y/θ ),

[f θ (π θ)N δ](y) = (2/θ )N Ff (2y/θ ).

574


This prompts us to consider the unitary dilation operators Ea given by Ea f (x) := a N/2 f (a 1/2 x), and it is immediate from (2.7) that F Ea = E1/a F . We also remark that f θ g = (θ/2)−N/2 E2/θ (Eθ/2 f 2 Eθ/2 g).

(2.8)

Nearly all formulas in this paper simplify when θ = 2. Thanks to the scaling relation (2.8), it is often enough, when studying properties of the Moyal product, to work out the case θ = 2. 2.2. The oscillator basis Definition 2.3. The algebra Aθ has a natural basis of eigentransitions fmn of the harmonic oscillator, indexed by m, n ∈ NN . As usual, for m = (m1 , . . . , mN ) ∈ NN , we write |m| := m1 + · · · + mN and m! := m1 ! . . . mN !. If 2 Hl := 21 (xl2 + xl+N ) for l = 1, . . . , N and H := H1 + H2 + · · · + HN ,

then the fmn diagonalize these harmonic oscillator Hamiltonians: Hl θ fmn = θ (ml + 21 )fmn , fmn θ Hl = θ (nl + 21 )fmn .

(2.9)

They may be defined by 1 fmn := √ (a ∗ )m θ f00 θ a n , |m|+|n| θ m!n!

(2.10)

where f00 is the Gaussian function f00 (x) := 2N e−2H /θ , and the annihilation and creation functions respectively are 1 1 al := √ (xl + ixl+N ) and al∗ := √ (xl − ixl+N ). 2 2

(2.11)

nN θ nN One finds that a n := a1n1 . . . aN = a1θ n1 θ · · · θ aN .

These Wigner eigentransitions are already found in [46] and also in [6]. (Incidentally, the “first” attributions in [36] are quite mistaken). The fmn can be expressed with the help of Laguerre functions in the variables Hl : see Subsect. 8.1 of the Appendix. The next lemma summarizes their chief properties. ∗ = f . Thus Lemma 2.4 [43]. Let m, n, k, l ∈ NN . Then fmn θ fkl = δnk fml and fmn nm fnn is an orthogonal projector and fmn is nilpotent for m = n. Moreover, fmn , fkl = 2N δmk δnl . The family { fmn : m, n ∈ NN } ⊂ S ⊂ L2 (R2N ) is an orthogonal basis. It is clear that eK := |n|≤K fnn , for K ∈ N, defines a (not uniformly bounded) approximate unit {eK } for Aθ . As a consequence of Lemma 2.4, the Moyal product has a matricial form.


575

Proposition 2.5 [43]. Let N = 1. Then Aθ has a Fre´ chet algebra isomorphism with the matrix algebra of rapidly decreasing double sequences c = (cmn ) such that, for each k ∈ N, 1/2 ∞ 2 2k 1 k 1 k rk (c) := θ (m + 2 ) (n + 2 ) |cmn | m,n=0

is finite, topologized by all the seminorms (rk ); via the decomposition f = m,n∈NN cmn fmn of S(R2 ) in the {fmn } basis. For N > 1, Aθ is isomorphic to the (projective) tensor product of N matrix algebras of this kind. Definition 2.6. We may as well introduce more Hilbert spaces Gst (for s, t ∈ R) of those f ∈ S (R2 ) for which the following sum is finite: ∞

f 2st :=

θ s+t (m + 21 )s (n + 21 )t |cmn |2 .

m,n=0

We define Gst , for s, t now in RN , as the tensor product of Hilbert spaces Gs1 t1 ⊗ · · · ⊗ GsN tN . In other words, the elements (2π)−N/2 θ −(N +s+t)/2 (m + 21 )−s/2 (n + 21 )−t/2 fmn (with an obvious multiindex notation), for m, n ∈ NN , are declared to be an orthonormal basis for Gst . N If q ≤ s and r ≤ t in

R , then S ⊂ Gst ⊆ Gqr ⊂ S with continuous dense inclusions. Moreover, S = s,t∈RN Gst topologically (i.e., the projective limit topology of the intersection induces the usual Fréchet space topology on S) and S = s,t∈RN Gst topologically (i.e., the inductive limit topology of the union induces the usual DF topology on S ). In particular, the expansion f = m,n∈NN cmn fmn of f ∈ S converges in the strong dual topology. We will use the notational convention that if F, G are spaces such that f θ g is defined whenever f ∈ F and g ∈ G, then F θ G is the linear span of the set { f θ g : f ∈ F, g ∈ G }; in many cases of interest, this set is already a vector space. It is now easy to show that Sθ S = S; more precisely, the following result holds.

Proposition 2.7 [43, p. 877]. The algebra (S, θ ) has the (nonunique) factorization property: for all h ∈ S there exist f, g ∈ S such that h = f θ g.

2.3. Moyal multiplier algebras Definition 2.8. The Moyal product can be defined, by duality, on larger sets than S. For T ∈ S , write the evaluation on g ∈ S as T , g ∈ C; then, for f ∈ S we may define T θ f and f θ T as elements of S by T θ f, g := T , f θ g and f θ T , g := T , gθ f , using the continuity of the star product on S. Also, the involution is extended to S by T ∗ , g := T , g ∗ . We shall soon argue [43] that if T ∈ S and f ∈ S, then T θ f, f θ T ∈ C ∞ (R2N ). Consider the left and right multiplier algebras: MθL := { T ∈ S (R2N ) : T θ h ∈ S(R2N ) for all h ∈ S(R2N ) }, MθR := { T ∈ S (R2N ) : hθ T ∈ S(R2N ) for all h ∈ S(R2N ) }, and set Mθ := MθL ∩ MθR .

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It is clear from Lemma 2.1(ii) that the map h → T θ h, for T ∈ MθL , is adjointable, with adjoint given by (left) multiplication by T ∗ . One can then define the Moyal products MθR θ S = S and S θ MθL = S as well. Theorem 2.9 [90]. Mθ is a complete nuclear semireflexive locally convex unital ∗-algebra with hypocontinuous multiplication and continuous involution. Moreover, in view of the previous proposition, Mθ is the maximal compactification of Aθ defined by duality (see [45, Sect. 1.3]). This maximal unitization Mθ of Aθ contains, beyond the constant functions (in particular 1 is the identity), the plane waves. By plane waves we understand all functions of the form x → exp(ik · x) for k a 2N-vector. They are important in physics. Also Mθ contains the Dirac δ and all its derivatives, and all monomials x → x α for α ∈ N2N . Clearly M2 is Fourier invariant, so more generally F Mθ = M4/θ . When θ = 0, the place of Mθ is taken by the space OM (“M” for multiplier) of smooth functions of polynomial growth on R2N in all derivatives. There is a new way of defining the Moyal lying in product for pairs of distributions the Sobolev-like spaces Gst [43]. If f = m,n cmn fmn ∈ Gst , g = m,n dmn fmn ∈ Gqr and if t +q ≥ 0, then for amn := k cmk dkn , the series h := m,n amn fmn converges in Gsr ; f θ g is defined, and f θ g = h. Furthermore, the following useful norm estimates hold: f θ gst ≤ f sq grt whenever q + r ≥ 0. In particular, Gt,−t is a Banach algebra, for all t ∈ RN . This is consistent with the previous definition.

We let G−∞,t := s∈RN Gst (with the projective limit topology) and Gs,+∞ :=

MθL = s∈RN Gs,+∞ topologt∈RN Gst (with the inductive limit topology). Then ically, and the strong (pre-)dual (MθL ) equals t∈RN G−∞,t topologically. Note in passing that (MθL ) → MθL with a continuous inclusion. Yet alternatively, we may work with another algebra of distributions including (S, θ ), to wit, the multiplier algebra of G00 = L2 (R2N ) considered in [56, 90]. We first record the analogue of Lemma 2.1. Lemma 2.10 [43, 48, 49, 90]. Let f, g ∈ L2 (R2N ). Then (i) For θ = 0, f θ g lies in L2 (R2N ). Moreover, f θ g is uniformly continuous. (ii) θ is a bilinear associative product on L2 (R2N ). The complex conjugation of functions f → f ∗ is an involution for θ . (iii) The linear functional f → f (x) dx on S extends to I00 (R2N ) := L2 (R2N )θ L2 (R2N ), and the product has the tracial property: −N 2N −N f, g := (πθ ) f θ g(x) d x = (π θ) gθ f (x) d 2N x = (πθ )−N f (x) g(x) d 2N x. We are not asserting that h = f θ g is absolutely integrable. We can nevertheless find u ∈ S with u∗ θ u = 1 and |h| ∈ I00 so that h = uθ |h| and |h| = l ∗ θ l with l ∈ G00 . Writing h00,1 := 1, |h| = l200 , we obtain a Banach space norm for I00 such that f θ g00,1 ≤ f 00 g00 . (iv) limθ↓0 Lθf g(x) = f (x) g(x) almost everywhere on R2N .


577

In Subsect. 8.1 of the Appendix it is discussed why I00 ⊂ L1 (R2N ). Since f ∈ I00 if and only if the Schrödinger representative σ θ (f ) is trace-class (see the proof of the next Proposition 2.13), one can obtain sufficient conditions for f to belong in I00 from the treatment in [29]. Definition 2.11. Let Aθ := { T ∈ S : T θ g ∈ L2 (R2N ) for all g ∈ L2 (R2N ) }, provided with the operator norm Lθ (T )op := sup{ T θ g2 /g2 : 0 = g ∈ L2 (R2N ) }. Obviously Aθ = S → Aθ . But Aθ is not dense in Aθ (see below), and we shall denote by A0θ its closure in Aθ . Note that G00 ⊂ Aθ . This is clear from the following estimate. Lemma 2.12 [43]. If f, g ∈ L2 (R2N ), then f θ g ∈ L2 (R2N ) and Lθf op ≤ (2π θ )−N/2 f 2 . Proof. Expand f = m,n cmn αmn and g = m,n dmn αmn with respect to the orthonormal basis {αnm } := (2πθ )−N/2 {fnm } of L2 (R2N ). Then

2

2 2 −2N −N cmn dnl fml cmn dnl f θ g2 = (2π θ )

= (2π θ) m,l

≤ (2π θ )−N

m,j

n

|cmj |2

2

m,l

n

|dkl |2 = (2π θ)−N f 22 g22 ,

k,l

on applying the Cauchy–Schwarz inequality. The algebra Aθ contains moreover L1 (R2N ) and its Fourier transform [57], even the bounded measures and their Fourier transforms; the plane waves; but no nonconstant polynomials, nor derivatives of δ. The algebra Aθ is selfconjugate, and it could have been defined using right Moyal multiplication instead. Proposition 2.13 [56, 90]. (Aθ , .op ) is a unital C ∗ -algebra of operators on L2 (R2N ), isomorphic to L(L2 (RN )) and including L2 (R2N ). Also, (I00 ) = Aθ . Moreover, there is a continuous injection of ∗-algebras Aθ → Aθ , but Aθ is not dense in Aθ , namely A0θ Aθ . Proof. We prove the nondensity result. The left regular representation Lθ of Aθ is a denumerable direct sum of copies of the Schrödinger representation σ θ on L2 (RN ) [66]. Indeed, there is a unitary operator, the Wigner transformation W [36, 90], from L2 (R2N ) onto L2 (RN ) ⊗ L2 (RN ), such that W Lθ (f ) W −1 = σ θ (f ) ⊗ 1. If f ∈ S, then σ θ (f ) is a compact (indeed, trace-class) operator on L2 (RN ), and so A0θ equals { W −1 (T ⊗1)W : T compact }, while Aθ itself is { W −1 (T ⊗1)W : T bounded }. Clearly the dual space is (A0θ ) = I00 . Notice as well that conjugation by W yields an explicit isomorphism between Aθ and L(L2 (RN )). Consequently, Aθ is a Fréchet algebra whose topology is finer than the .op -topology. Moreover, it is stable under holomorphic functional calculus in its C ∗ -completion A0θ , as the next proposition shows.

578


Proposition 2.14. Aθ is a (nonunital) Fre´ chet pre-C ∗ -algebra. Proof. We adapt the argument for the commutative case in [45, p. 135]. To show that Aθ is stable under the holomorphic functional calculus, we need only check that if f ∈ Aθ and 1 + f is invertible in Aθ with inverse 1 + g, then the quasiinverse g of f must lie in Aθ . From f + g + f θ g = 0, we obtain f θ f + gθ f + f θ gθ f = 0, and it is enough to show that f θ gθ f ∈ Aθ , since the previous relation then implies gθ f ∈ Aθ , and then g = −f − gθ f ∈ Aθ also. Now, Aθ ⊂ G−r,0 for any r > N [90, p. 886]. Since f ∈ Gs,p+r ∩ Gqt , for s, t arbitrary

and p, q positive, we conclude that f θ gθ f ∈ Gs,p+r θ G−r,0 θ Gqt ⊂ Gst ; as S = s,t∈R Gst , the proof is complete. The Fréchet algebras Aθ are automatically good (their sets of quasiinvertible elements are open); and by an old result of Banach [5], the quasiinversion operation is continuous in a good Fréchet algebra. Note that a good algebra with identity cannot have proper (even one-sided) dense ideals. However, the nonunital (MθL ) provides an example of a good Fréchet algebra that harbours Aθ as a proper dense left ideal [44]. We noticed already that the extensions Mθ and Aθ of Aθ are quite different. Clearly θ M is associated with smoothness; however, even though the Sobolev-like spaces Gst grow more regular with increasing s and t [90], Mθ includes none of them; in particular, L2 (R2N ) ⊂ Mθ for any θ. Be that as it may, the plane waves belong both to Mθ and Aθ . One obtains for the Moyal product of plane waves: exp(ik ·)θ exp(il ·) = e− 2 θ k·Sl exp(i(k + l)·), i

(2.12)

or, reinstalling the generic Moyal product: exp(ik ·) exp(il ·) = e− 2 k·l exp(i(k + l)·). i

(2.13)

Therefore the plane waves close to an algebra, the Weyl algebra. It represents the translation group of R2N : exp(ik ·)θ f θ exp(−ik ·) (x) = f (x + θSk), for f ∈ S or f ∈ G00 , say. 2.4. Smooth test function spaces, their duals and the Moyal product. Here there is a fascinating interplay. Recall that a pseudodifferential operator A ∈ DO on Rk is a linear operator which can be written as −k A h(x) = (2π) σ [A](x, ξ ) h(y) eiξ ·(x−y) d k ξ d k y. Let d := { A ∈ DO : σ [A] ∈ S d } be the class of DOs of order d, with S d := { σ ∈ C ∞ (Rk × Rk ) : |∂xα ∂ξ σ (x, ξ )| ≤ CKαβ (1 + |ξ |2 )(d−|β|)/2 for x ∈ K }, β

where K is any compact subset of Rk , α, β ∈ Nk , and CKαβ is some constant. Also

∞ := d∈R d and −∞ := d∈R d . Recall, too, that a DO A is called regularizing or smoothing if A ∈ −∞ , or equivalently [52, 80], if A extends to a continuous linear map from the dual of the space of smooth functions C ∞ (Rk ) to itself.


579

Lemma 2.15. If f ∈ S, then Lθf is a regularizing DO. Proof. From (2.1), one at once sees that left Moyal multiplication by f is the pseudodifferential operator on R2N with symbol f (x − θ2 Sξ ). Clearly Lθf extends to a continuous linear map from C ∞ (R2N ) → S to C ∞ (R2N ). The lemma also follows from the inequality β

|∂xα ∂ξ f (x − θ2 Sξ )| ≤ CKαβ (1 + |ξ |2 )(d−|β|)/2 , valid for all α, β ∈ N2N , any compact K ⊂ R2N , and any d ∈ R, since f ∈ S. Remark 2.16. Unlike for the case of a compact manifold, regularizing DOs are not necessarily compact operators. For instance, for each n, Lθ (fnn ) possesses the eigenvalue 1 with infinite multiplicity, so it cannot be compact. Definition 2.17. For m ∈ N, f ∈ C m (Rk ) —functions with m continuous derivatives— and γ , l ∈ R, let qγ lm (f ) := sup{ (1 + |x|2 )(−l+γ |α|)/2 |∂ α f (x)| : x ∈ Rk , |α| ≤ m }; m m k and then let V m γ ,l , respectively Vγ ,l , be the space of functions in C (R ) for which

(1 + |x|2 )(−l+γ |α|)/2 ∂ α f (x) vanishes at infinity for all |α| ≤ m, respectively is finite for all x ∈ Rk , normed by qγ lm . m [53]. We define Note that V m ´ th’s space S−2l 0,l is Horva Vγ :=

Vγm,l ,

and, more generally, Vγ ,l :=

l∈R m∈N

Vγm,l ,

m∈N

so that Vγ = l∈R Vγ ,l . Particularly interesting cases include the space K := V1 of Grossmann–Loupias–Stein functions [47], whose dual K is the space of Cesa` ro-summable distributions [34], the space OC := V0 whose dual OC is the space of convolution multipliers (Fourier transforms of OM ), and the space OT := V−1 [43]. Similarly, Kr := V1,r and Or := V0,r are defined. We see that S=

m∈N l∈R

Vm 0,l ,

OM =

Vm 0,l .

m∈N l∈R

Following Schwartz, we denote B := O0 , the space of smooth functions bounded together with all derivatives.

We shall also need B˙ := m∈N V m 0,0 , the space of smooth functions vanishing at infinity together with all derivatives, and the weighted test space DL2 , the space of elements of L2 (R2N ) all of whose (distributional) derivatives also lie in L2 [68, 77]; by Sobolev’s lemma, these are in fact smooth functions and moreover DL2 ⊂ B˙ [77]: actually if f ∈ DL2 , then the (ordinary) Fourier transform F(f ) satisfies (1 + |ξ |2n )F(f ) ∈ L2 for all integer n, and by the Cauchy–Schwarz inequality F(f ) ∈ L1 , thus f tends to zero at infinity. In the notation of [84], DL2 is H 2,∞ .

580


There are continuous inclusions D → Vγ → Vγ → OM → D for γ > γ ; these are all normal spaces of distributions, namely, locally convex spaces which include S as a dense subspace and are continuously included in S . Also DL2 (density of S in this space follows from density of the Schwartz functions in L2 and invariance of S under derivations) and MθL , MθR and Mθ [90] are normal space of distributions. By the way, there are suggestive Tauberian-type theorems for these spaces, establishing when their intersections with their respective dual spaces are included in S. Concretely, we quote the following result from [32]. Proposition 2.18. If C is a space of smooth functions on R2N which is closed under complex conjugation, and if the pointwise product space KC lies within C, then C ∩ C ⊆ S. = S and C ∞ ∩ (C ∞ ) = In particular, Vγ ∩ Vγ = S for γ ≤ 1. Also OM ∩ OM D ⊂ S. Now, what can be said about the relation of all these spaces with Mθ ? In [43] it , is included in Mθ , for all θ. Therefore by is established that OT , and a fortiori OM Fourier analysis OC is included in Mθ for all θ, and gθ f is defined as a tempered distribution whenever f, g ∈ OC . Growth estimates may be obtained as follows. It is true that OC = r∈R Or topologically. If g ∈ Or and f ∈ Os , the following crucial proposition shows that the Or spaces have similar behaviour under pointwise and Moyal products.

Proposition 2.19. The space OC is an associative ∗-algebra under the Moyal product. In fact, the Moyal product is a jointly continuous map from Or × Os into Or+s , for all r, s ∈ R. Moreover, Aθ is a two sided essential ideal in OC . Proof. For the reader’s convenience, we reproduce part of Theorem 2 of [35]. Let g ∈ Os . By the Leibniz rule for the Moyal product, ∂ α (f θ g) = f ∈ Or and α β γ β+γ =α β ∂ f θ ∂ g. Hence we need only show that there are constants Crsm such that (1 + |x|2 )−(r+s)/2 |(∂ β f θ ∂ γ g)(x)| ≤ Crsm q0rm (f ) q0sm (g)

(2.14)

for all x ∈ R2N , for large enough m ≥ |β| + |γ |. If k ∈ N (to be determined later), we can write (∂ β f θ ∂ γ g)(x) β ∂ f (x + y) ∂ γ g(x + z) −2N = (π θ ) (1 + |y|2 )k (1 + |y|2 )k (1 + |z|2 )k 2i

× (1 + |z|2 )k e θ y·Sz d 2N y d 2N z β 2i ∂ f (x + y) ∂ γ g(x + z) = (π θ )−2N Pk (∂y , ∂z ) e θ y·Sz d 2N y d 2N z k k 2 2 (1 + |y| ) (1 + |z| ) β 2i ∂ f (x + y) ∂ γ g(x + z) 2N 2N = (π θ )−2N d y d z, e θ y·Sz Pk (−∂y , −∂z ) (1 + |y|2 )k (1 + |z|2 )k


581

where Pk is a polynomial of degree 2k in both y and z variables. From the elementary estimates |∂ α ((1 + |x|2 )−k )| ≤ cα,k (1 + |x|2 )−k it follows that |∂ β f θ ∂ γ g|(x) ≤ Ck k

∂ β+k f (x + y) ∂ γ +k g(x + z) 2N 2N d yd z 2 )k 2 )k (1 + |z| (1 + |y| k ,k ≤2k (1 + |x + y|2 )r/2 (1 + |x + z|2 )s/2 2N 2N d yd z q0rm (f ) q0sm (g) ≤ Crsm (1 + |y|2 )k (1 + |z|2 )k ≤ Crsm q0rm (f ) q0sm (g) (1 + |x|2 )(r+s)/2 (1 + |y|2 )r/2−k d 2N y × (1 + |z|2 )s/2−k d 2N z,

provided m ≥ |β| + |γ | + 2k; here the Cauchy inequality 1 + |x + y|2 ≤ 2(1 + |x|2 )(1 + |y|2 ) has been used to extract the x variables. If we now choose k > N + max{r, s}/2 (and therefore take m ≥ |β| + |γ | + 2N + max{r, s}), the integrals will be finite. The joint continuity now follows directly from the estimates (2.14). That S is a two-sided ideal in OC follows from the inclusion OC ⊂ Mθ . Essentiality

= 0 for any nonzero g ∈ Os ; for the ideal S = Aθ is equivalent [45, Prop. 1.8] to gθ S but if gθ fmn = 0 for all m, n, then in the expansion g = m,n cmn fmn (as an element of S , say) all coefficients must vanish, so that g = 0. Similar results hold for Vγ when γ > 0. Indeed, the Moyal product (f, g) → f θ g is a jointly continuous map from Kr × Ks into Kr+s ; moreover, f θ g − f g ∈ Kr+s−2 , which is a bonus for semiclassical analysis (while on the contrary the similar statement for Or × Os is in general false). For γ < 0, we lose control of the estimates; indeed, Lassner and Lassner [59] gave an example of two functions in OT whose twisted product can be defined but is not a smooth function, but rather a distribution (of noncompact support). Also, in the next subsection we prove by counterexample that OT ⊂ MθL . The integral estimates on the derivatives of gθ f can be refined to show that in fact OM θ OC = OM . However, since these estimates depend on the order of the derivatives in a complicated way, it is doubtful that the twisted product can be extended to OM . The regularizing property of θ proved at the beginning of the section can be vastly improved, as follows. Proposition 2.20 [43]. If T ∈ S and f ∈ S, then T θ f and f θ T lie in OT . Moreover, these bilinear maps of S × S and S × S into OT are hypocontinuous. In fact, Sθ S equals (MθL ) , so the latter is made of smooth functions. But (MθL ) ∩ (MθL ) = (MθL ) ∩ MθL = (MθL ) S; so (MθL ) and (MθR ) do not satisfy the conclusion of Proposition 2.18. (Here of course denotes the strong bidual space, not a bicommutant.) As distributions, the elements of (MθL ) and (MθR ) belong to OC , and a fortiori they are Cesàro summable [34]. Finally, it is important to know when smooth functions give rise to elements of A0θ or Aθ . Sufficient conditions are the following (quite strong) results of the Calderón– Vaillancourt type [36, 54]. 2N+1 Theorem 2.21. The inclusion V0,0 ⊂ Aθ holds. In particular, B ⊂ Aθ . The inclusion 2N+1 2N+1 0 V 00 ⊂ Aθ also holds. In particular, B˙ ⊂ A0θ . Moreover, if b ∈ V0,0 belongs to A0θ , 2N+1 then b ∈ V 0,0 .

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We have also proved that the function space B is a ∗-algebra under the Moyal product θ for any θ, in which Aθ is a two sided essential ideal. Recall that DL2 ⊂ B˙ ⊂ Mθ . We will now show that DL2 is a ∗-algebra under the Moyal product as well. Lemma 2.22. (DL2 , θ ) is a ∗-algebra with continuous product and involution. Moreover, it is an ideal in (B, θ ). Proof. The closure under the twisted product follows from the Leibniz rule and Lemma 2.12: α α −N/2 ∂ (f θ g)2 ≤ (2πθ ) ∂ β f 2 ∂ α−β g2 . β β≤α

This also shows that the product is separately continuous, indeed jointly continuous since DL2 is a Fréchet space. The continuity of the involution f → f ∗ is immediate. The fact that DL2 is a two sided ideal in B comes directly from the stability of these spaces under partial derivations and from the inclusion B ⊂ Aθ given by the previous theorem, since then ∂ α f θ ∂ β g2 < ∞ for all f ∈ B, g ∈ DL2 and all α, β ∈ N2N . ˇ 2.5. The preferred unitization of the Schwartz Moyal algebra. As with Stone–Cech compactifications, the algebras Mθ are too vast to be of much practical use (in particular, to define noncommutative vector bundles). A more suitable unitization of Aθ is given θ := (B, θ ). This algebra possesses an intrinsic characterization as by the algebra A the smooth commutant of right Moyal multiplication (see our comments at the end of θ contains Subsect. 4.5). The inclusion of Aθ in B is not dense, but this is not needed. A the constant functions and the plane waves, but no nonconstant polynomials and no imaginary-quadratic exponentials, such as eiax1 x2 in the case N = 1 (we will see later the pertinence of this). θ is a unital Fre´ chet pre-C ∗ -algebra. Proposition 2.23. A Proof. We already know that B is a unital ∗-algebra with the Moyal product, and that θ is continuous in the topology of the Fréchet space B defined by the seminorms q00m , for m ∈ N. Its elements have all derivatives bounded, and so are uniformly continuous functions on R2N , as are their derivatives: the group of translations τy f = f (· − y), for θ (i.e., y → τy f is continuous for each f ). y ∈ R2N , acts strongly continuously on A This action preserves the seminorms q00m , and it is clear that B is a subspace of the space of smooth elements for τ , which we provisionally call A∞ θ . The latter space has its own Fréchet topology, coming from the strongly continuous action. Rieffel [75, Thm. 7.1] proves two important properties in this setting: firstly, based on a density theorem of Dixmier and Malliavin [30], that the inclusion B → A∞ θ is continuous and dense. Secondly, using a “-twisting” of C ∗ -algebras with an Rk -action which generθ , −θ ), alizes (2.1), whereby the pointwise product can be recovered as (B, 0 ) = (A one obtains the reverse inclusion; thus, B = A∞ . (Thus, the smooth subalgebra is θ independent of .) θ , as a subalgebra of the C ∗ -algebra Aθ , is stable under It is now easy to show that A the holomorphic functional calculus. Indeed, since G(τy (f )) = τy (G(f )) for any function G which is holomorphic in the neighbourhood of sp Lθ (f ) = sp Lθ (τy (f )), it is θ entails G(f ) ∈ A θ . clear that f ∈ A


583

θ properly contains A0 ; it is not known to Clearly the C ∗ -algebra completion of A θ θ ≡ B is nonseparable as it stands; there us whether it is equal to Aθ . At any rate, A is, however, another topology on B, induced by the topology of C ∞ (R2N ) [77, p. 203], under which this space is separable. That latter topology is very natural in the context of commutative and Connes–Landi spaces (see Subsects. 3.3 and 3.4). To investigate its pertinence in the context of Moyal spaces would take us too far afield. θ is that the covering relation of the noncommutative plane to An advantage of A the NC torus is made transparent. To wit, the smooth noncommutative torus algebra C ∞ (T2N ) can be embedded in B as periodic functions (with a fixed period parallelogram). In that respect, it is well to recall [76, 87] how far the algebraic structure of C ∞ (T2N ) can be obtained from the integral form (2.1) of (a periodic version of) the Moyal product. θ , Anticipating the next section, we finally note the main reason for suitability of A θ and D namely, that each [D / , Lθ (f ) ⊗ 12N ] lies in Aθ ⊗ M2N (C), for f ∈ A / the Dirac operator on R2N . The previous proposition has another useful consequence. Corollary 2.24. (DL2 , θ ) is a (nonunital) Fre´ chet pre-C ∗ -algebra, whose C ∗ -completion is A0θ . Proof. The argument of the proof of Proposition 2.14 applies, with the following modifications. Firstly, S ⊂ DL2 ⊂ A0θ with continuous inclusions, so that A0θ is indeed the C ∗ -completion of (DL2 , θ ). Indeed, for the second inclusion one can notice that if f ∈ DL2 , then W Lθ (f ) W −1 = σ θ (f ) ⊗ 1, where σ θ (f ) is a Hilbert–Schmidt operator, hence compact. The same conclusion follows from Theorem 2.21. Secondly, if f ∈ DL2 has a quasiinverse g ∈ A0θ , then the previous proposition shows θ , too. Since (DL2 , θ ) is an ideal in A θ by Lemma 2.22, we conclude that that g ∈ A f θ gθ f ∈ DL2 , which is enough to establish that g ∈ DL2 . A relevant group of inner automorphisms of the “big algebras” Mθ or Aθ is given by the metaplectic representation. Real symplectic 2N × 2N matrices act on functions by Mf (x) := f (M −1 x). We can consider inhomogeneous symplectic transformations, i.e., affine transformations leaving the symplectic structure invariant. Let (s, M) denote an element of the inhomogeneous symplectic group I Sp(2N, R), i.e., the semidirect product of the group of translations and the symplectic group, with group law (s1 , M1 )(s2 , M2 ) = (M2−1 s1 + s2 , M1 M2 ), acting by (s, M)f (x) = f (M −1 x − s).

(2.15)

The equivariance of the twisted product is readily checked: (s, M)f θ (s, M)g = (s, M)(f θ g).

(2.16)

We concentrate on the homogeneous (0, M) transformations. The symplectic action is realized by the adjoint -action of unitaries E(M, ·), belonging also to the multiplier Moyal algebra Mθ . They constitute a variant of the metaplectic representation; E(M, ·) is a distribution on the space of smooth sections of a nontrivial line bundle

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over I Sp(2N, R), that works like the exponential kernel of a noncommutative Fourier transform: E(M, ·) θ f θ E(M, ·)∗ = Mf,

(2.17)

for all f ∈ S or f ∈ L2 (R2N ) or even f ∈ S . Explicitly, for elements M of Sp(2N, R) which are “nonexceptional”, i.e., det(1 + M) = 0, there is the presentation 2N 1−M iα E(M, x) = e √ x . (2.18) exp −ix · S θ(1 + M) det(1 + M) Thus, such E(M, .) are imaginary-quadratic exponentials; the quadratic form in the exponent is actually an important symplectic invariant, solving a modified Hamilton– Jacobi equation, introduced by Poincaré [69] and nowadays all but forgotten. The phase prefactor in (2.18) reflects the ambiguity inherent in (2.17), which can be reduced to a sign, so that E(M, ·)θ E(M , ·) = ±E(MM , ·). The curious reader can directly check the last two formulas, aided by the method of the stationary phase; a look at [1, 37, 89] will help. In contradistinction to the Weyl algebra, the E(M, ·) do not belong to B, and so they θ —and of course of Aθ . yield outer automorphisms of A Note that the values exp ±2iθ −1 (x1 xN+1 + · · · + xN x2N ) are never reached by E in (2.18). For good reason: these functions do not belong to the multiplier algebras Mθ or Aθ , as the following lemma shows. Lemma 2.25. Let ha (x) := exp ia(x1 xN+1 +· · ·+xN x2N ) for a = 0. Then ha ∈ Mθ , or ha ∈ Aθ , if and only if |a| = 2/θ . Proof. We show this for N = 1, the general case follows immediately. In view of (2.8), it suffices to consider the case θ = 2. We must determine whether ha 2 fmn ∈ S; because of the multiplication rule (2.6), it is enough to check this for the Gaussian function f00 . From (2.4), 1 ha 2 f00 (x) = exp iay1 y2 − 21 z12 − 21 z22 + i(x1 − y1 )(x2 − z2 ) 2π 2 −i(x2 − y2 )(x1 − z1 ) d 2 y d 2 z. With u = (y1 , y2 , z1 , z2 ), the integral is of the type exp(− 21 u·Qu−iu·Rx ) d 4 u, where the quadratic form u·Qu, with Q ≥ 0, is degenerate if and only if det Q = a 2 −1 = 0. Thus if |a| = 1, then ha 2 f00 ∈ / S, and also ha 2 f00 ∈ / L2 , while if |a| = 1, an explicit calculation shows that ha 2 f00 ∈ S. This shows, by the way, that OT ⊂ Mθ and that the Mθ and Aθ for different θ are all distinct spaces of tempered distributions. θ . Linear functions, not belonging Next we look briefly at the derivations of Aθ , A θ either, at the infinitesimal level double as Hamiltonians for the translations; to B ≡ A quadratic functions double as Hamiltonians for linear symplectomorphisms. For h affine quadratic [h, f ]θ = iθ {h, f },


585

that is, the Moyal and Poisson brackets in this case essentially coincide. (Note that the θ corresponding to quadratic Hamiltonians are unbounded.) derivations of A On the other hand, all derivations of Mθ are inner, as it is easily proved using the Poincaré lemma in the distributional context [31]. θ , as An important task is to compute the Hochschild cohomologies of Aθ and A Connes did for the NC torus in [15]; we have already seen that they are not entirely trivial. The “big algebras” Mθ and Aθ , on the other hand, risk having uninteresting cohomology. Rennie has proposed to equip nonunital noncommutative algebras A like Aθ with a “local ideal” Ac ⊂ A [73], which would be a noncommutative generalization of the space Cc∞ (M) of smooth functions with compact support. A Fréchet algebra A is local in his sense if it has a dense ideal Ac with local units; an algebra Ac has local units when, for any finite subset of elements {a1 , . . . , ak } of Ac , there exists u ∈ Ac such that uai = ai u = ai for i = 1, . . . , k. Certainly the Moyal product θ is not “local” in the ordinary sense: the formulas (2.3) and (2.4) are two different definitions, as may be noticed in the simple example of a couple f, g with disjoint supports; then (2.3) gives zero outside the supports; while (2.4) does not. The algebras Aθ are not known to have bilateral ideals; it is very likely that they are simple, and if so, they would not be local in the sense of [73], either (thus, it is not clear if Rennie’s device can carry the full weight of noncommutative spin geometry). However, one can define a useful weaker notion of locality: Definition 2.26. A Fre´ chet algebra A is quasilocal if it has a dense ∗-subalgebra Ac with local units. Here, we choose Ac,K , where Ac,K := f ∈ S : f = cmn fmn . Ac := K∈N

0≤|m|,|n|≤K

That is, Ac is the algebra of finite linear combinations of the { fmn : m, n ∈ NN }; it possesses local units, and so Aθ is quasilocal. Rennie further argues that possession of a local ideal in his sense guarantees H -unitality [97] of the original algebra. Certainly our Ac is algebraically H -unital, as it possesses local units [61]. It would be good to know whether Aθ is topologically H -unital. 3. Axioms for Noncompact Spin Geometries 3.1. Generalization of the unital case conditions. To define and construct noncommutative spin manifolds, one starts from an operatorial version of ordinary spin geometry, that can be generalized to noncommutative manifolds. Ideally, one should prove a reconstruction theorem, allowing to recover all of the (topological, smooth, geometrical) concrete structure from the abstract geometry over a suitable commutative algebra; this has been performed to satisfaction for compact manifolds without boundary [18, 19, 45, 71]. However, to rush to that at the present stage would not do. It is better for now to patiently listen to what the possible examples have to say. As the first part of our task, therefore, we seek a collection of Connes-like axioms for not necessarily compact noncommutative manifolds. Such a list of conditions should be compatible with the previous axiomatic framework, and be fulfilled by noncompact commutative manifolds. We expect it also to encompass other interesting cases. Our main task will then be to prove that noncommutative Moyal-product algebras constitute

586


one of the examples. (To eventually reach this goal, we use heavy machinery wholesale; we do not claim to have the “best” proofs.) The discussion in this section will be relatively informal; a formal proposal is made in the next one. We set out by discussing what a real noncompact spectral triple might be. As mentioned in the Introduction, the basic data (A, H, D) —or (A, H, D, χ )— for a spectral triple consist of an algebra A represented by bounded operators on a Hilbert space H and an unbounded selfadjoint operator D on H, such that each commutator [D, a], for a ∈ A (densely defined as an operator on H) extends to a bounded operator; it is understood that a Dom D ⊆ Dom D. To get an idea of the difficulties involved in the choice of A, consider the commutative case, say of the manifold Rk . Depending on the fall-off conditions deemed suitable, the smooth nonunital algebras that can represent the manifold are numerous as the stars in the sky. The problem is compounded in the noncommutative case, say when A is a deformation of an algebra of functions. To be on the safe side, we will take a relatively small algebra at the start of our investigation of the Moyal examples; during its course, a larger candidate will emerge. ConsiderAlso, when A is not unital, we need choose a preferred unitization A. ation of the links to K-theory and K-homology makes it prudent to require that A, A be pre-C ∗ -algebras, whose K-theories then coincide with those of their respective C ∗ completions [17]. Denote by K(H) the compact operators on H, and by Lp (H) the Schatten ideal in K(H) defined by a finite norm Ap := Tr(|A|p )1/p , for p ≥ 1. For compact or unital spectral triples, it is further required that the operator D have compact resolvent, that / sp D. Consequently D must have discrete is, (D − λ)−1 must belong to K(H) for λ ∈ spectrum of finite multiplicity. Since this is clearly not the case for the Dirac operator D / on Rk , in the nonunital case we only demand [18] that a(D −λ)−1 be compact for a ∈ A. This condition ensures that the spectral triple (A, H, D) corresponds to a well-defined K-homology class [51, Chap. 10], and could be termed “Axiom 0” for an —in general noncompact— noncommutative geometry. We turn to the several conditions which spectral triples must satisfy to yield noncommutative spin geometries. To formulate the generalization to the noncompact case, we focus first on commutative geometries. First in line there is a summability condition, namely that the operators a(D 2 + ε 2 )−1 be not merely compact, but belong to the generalized Schatten class called Lk+ , with k an integer; this is a kind of k th root in the sense of operator products of the Dixmier trace class L1+ [17, 45]. More concretely, a compact operator T belongs to Lk+ for k > 1 if its singular values satisfy µm (T ) = O(m−1/k ) as m → ∞. In the compact commutative case of a k-dimensional spin manifold, choosing D to be the ordinary Dirac operator D / on a spinor space H, one finds that / |−k ) = Ck a(x) d k x, Tr + (a|D M

where Tr + denotes any Dixmier trace, for a universal constant Ck . For the noncompact commutative case, we expect Tr + (·|D / |−k ) still to exist for a suitable algebra of inte+ grable functions, and we regard Tr (·|D / |−k ) as a noncommutative integral. These two summability conditions together constitute Axiom 1. A further necessary condition was regularity or smoothness of the spectral triple. If δ(T ) := [|D|, T ] for an operator T on H, regularity means that each a ∈ A and each


587

[D, a] lies in the domain of δ n for all n ∈ N. In the commutative case, |D / | is a first-order pseudodifferential operator, and computing δ n (a) is onerous; it is somewhat easier to handle the (commuting) operations L(T ) := |D / |−1 [D / 2 , T ],

R(T ) := [D / 2 , T ] |D / |−1 ,

(3.1)

and one can show that the smooth domain of δ equals the common smooth domain of k/2 as an (ordinary) L and R [23]. If f is a Schwartz function acting on L2 (Rk ) ⊗ C2 multiplication operator, we can regard it as a pseudodifferential operator with symbol f (x) ⊗ 12k/2 , and one checks that Ln R m f is a bounded pseudodifferential operator of order (at most) zero. On the subject of regularity, the reader is advised to look at the discussions in [45, Sect. 10.3] and also in [50, 73]. There is no obvious need to modify this axiom in the noncompact case. However, at the technical level, |D / |−1 is a somewhat more problematic object than in the compact case, and one must find a substitute for it. The condition of finiteness, in the unital case, is that the smooth domain H∞ of D in H be a finitely generated projective (left) module over the unital algebra A, that is, H∞ Am p for some projector p = p ∗ = p2 in Mm (A) with a suitable m. In the case of M = Rk , under either the pointwise or the Moyal product, the module of smooth spinors is free since the spinor bundle is trivial. However, when A is nonunital, to get a projective A-module one should select the projector p in a matrix algebra over See Rennie [72] for a discussion both of this point the preferred compactification A. and if E is a left A-module, and of “pullback modules”. Concretely, if A1 is an ideal of A its pullback to A1 is the left A1 -module E1 := A1 E. The finiteness condition for the nonunital case should then demand that H∞ densely contains a pullback of a finite projective A-module to A; or, better still, that it can be m is also a unitization of A1 ), for some identified with A1 p, with A an ideal in A1 (thus A Moreover, a hermitian structure should be defined m and some projector p ∈ Mm (A). ∞ on the module H through the noncommutative integral; we shall see the details of this further on. We bring up next the axioms having an algebraic flavour. The reality condition is the existence of an antilinear conjugation operator J on H such that a → J a ∗ J −1 gives a second representation of A on H commuting with the original one, and with certain algebraic properties listed in [18, 20] and reviewed later: for the commutative case of spin manifolds, J is just the charge conjugation operator on spinors. There is no need to modify this axiom in the noncompact case. The first order condition is that [[D, a], J b∗ J −1 ] = 0,

for all a, b ∈ A.

For the commutative case this is a simple check, since D = D / is a first-order differential operator. There is no need to modify this axiom in the noncompact case. The orientability condition is that the spectral triple (A, H, D) carry an algebraic version of a “volume k-form”, where k is the integer summability exponent (k = 2N in op the nondegenerate Moyal case). Let b0 denote b0 ∈ A as an element of the opposite algebra Aop , with the product reversed; this algebraic version consists of a Hochschild op k-cycle c, that is, a sum of terms of the form (a0 ⊗ b0 ) ⊗ a1 ⊗ · · · ⊗ ak satisfying b c = 0 (cycle property), that we represent by bounded operators πD ((a0 ⊗ b0 ) ⊗ a1 ⊗ · · · ⊗ ak ) := a0 J b0∗ J −1 [D, a1 ] . . . [D, ak ], op

(3.2)

588


and on which we impose πD (c) = χ (orientation), where χ is the given Z2 -grading operator on H. We just use χ = 1 if k is odd; and, in the even case for ordinary spinors, one uses χ := (−i)m γ 1 γ 2 . . . γ 2m . For the commutative or noncommutative torus C ∞ (Tk ), with unitary generators u1 , . . . , uk satisfying uk uj = eij k uj uk ,

(3.3)

the good Hochschild cycle is known [20, 45] to be c=

(−i)k/2 (−1)σ (uσ (1) uσ (2) . . . uσ (k) )−1 ⊗ uσ (1) ⊗ uσ (2) ⊗ · · · ⊗ uσ (k) , k! σ (3.4)

where the sum is over all permutations of 1, 2, . . . , k. For nonunital algebras, we might expect something similar. However, the fact that the plane waves belong to B suggests, in the light of the NC torus example, taking the rather than A itself. This has the happy consequence of cycle over the unitization A bypassing the many difficulties of Hochschild cohomology for nonunital algebras. Poincare´ duality for a noncompact orientable manifold M is usually expressed as the isomorphism between the compactly supported de Rham cohomology and the homology of M, mediated by the fundamental class [M]. In noncommutative geometry a K-theoretic version is in order. One would expect that some kind of compactly supported K-homology of the initial nonunital algebra A be isomorphic to its K-theory, through a fundamental K-homology class of A ⊗ Aop given by the spectral triple itself. We shall actually leave aside the final condition of Poincaré duality in K-theory, since it is not central to the present form of the reconstruction theorem in the compact case [45], and the details of its reformulation in the nonunital noncommutative case are still somewhat clouded. 3.2. Modified conditions for nonunital spectral triples Definition 3.1. By a real noncompact spectral triple of dimension k, we mean the data H, D, J, χ ), (A, A, where A is an (a priori nonunital) algebra acting faithfully (via a representation some is a preferred unitization of A, acting times denoted by π) on the Hilbert space H, A on the same Hilbert space, and D is an unbounded selfadjoint operator on H such that extends to a bounded operator on H. [D, a], for each a in A, Furthermore, J and χ are respectively an antiunitary and a selfadjoint operator, such that χ = 1 when k is odd, and otherwise χ 2 = 1, χ a = aχ for a ∈ A, and Dχ = −χD, satisfying the conditions which follow. 0. Compactness: The operator a(D − λ)−1 is compact for a ∈ A and λ ∈ / sp D. 1. Spectral dimension: There is a unique nonnegative integer k, the spectral or “classical" dimension of the geometry, for which a(D 2 + ε 2 )−1/2 belongs to the generalized Schatten class Lk+ for each a ∈ A and moreover Tr + (a(|D| + ε)−k ) is finite and not identically zero, for any ε > 0. This k is even if and only if the spectral triple is even.


589

2. Regularity: lie in the smooth domain of The bounded operators a and [D, a], for each a ∈ A, the derivation δ : T → [|D|, T ]. 3. Finiteness: are pre-C ∗ -algebras. There exists an The algebra A and its preferred unitization A including A, which is also a pre-C ∗ -algebra with the same C ∗ -comideal A1 of A, pletion as A, such that the space of smooth vectors C ∞ (D) ≡ H∞ := Dom(D 1 ) 1∈N

is an A1 -pullback of a finite projective A-module. Moreover, an A1 -valued hermitian structure (· | ·) is implicitly defined on H∞ with the noncommutative integral, as follows: (3.5) Tr + (aξ | η)(|D| + ε)−k = η | aξ , and · | · denote the standard inner product on H. This is an absolute where a ∈ A continuity condition, since (· | ·) is a kind of Radon–Nikodým derivative with respect to the functional Tr + (· (|D| + ε)−k ). 4. Reality: There is an antiunitary operator J on H, such that [a, J b∗ J −1 ] = 0 for all a, b ∈ A op ∗ −1 (thus b → J b J is a commuting representation on H of the opposite algebra A ). Moreover, J 2 = ±1 and J D = ±DJ , and also J χ = ±χ J in the even case, where the signs depend only on k mod 8. Here is the table for the even case; see the full table in [45, p. 405]. N mod 4

0 1 2 3

J2 = ±1

+−−+

(3.6)

J D = ± DJ + + + + J χ = ± χJ + − + −

5. First order: The bounded operators [D, a] also commute with the opposite algebra representation: [[D, a], J b∗ J −1 ] = 0 for all a, b ∈ A. 6. Orientation: with values in A ⊗ A op . Such a k-cycle is a There is a Hochschild k-cycle c on A, op finite sum of terms like (a ⊗ b ) ⊗ a1 ⊗ · · · ⊗ ak , whose natural representative by operators on H is given by πD (c) in formula (3.2); the “volume form” πD (c) must solve the equation πD (c) = χ

(even case),

or

πD (c) = 1

(odd case).

(3.7)

Finally, a geometry is called connected or irreducible if the only operators commuting with A and D are the scalars. We are mainly interested in connected noncompact noncommutative geometries. The discussion in the previous subsection, and this proposal, are very much in the vein of [41]. We may also keep the concept in that article of “star triples”, a specialization of the spectral triple to deformations of the algebra of functions on a noncompact

590


manifold, wherein the Dirac operator (is possibly deformed, but) remains an ordinary (pseudo-)differential operator on that original manifold. However, the authors of [41] got carried away in that they confused properties of Lθf with properties of the Weyl pseudodifferential operator associated (by the Schrödinger representation) to the “symbol” f . And thus, the dreaded “dimension drop”, apparent there, does not actually take place. But before going to the Moyal case, we need to reexamine the commutative case. 3.3. The commutative case. The outcome of the discussion in Subsect. 3.1 is that the main outstanding issues, in order to obtain noncompact noncommutative spin geometries, are the analytical ones. Let A be some appropriate subalgebra of C ∞ (M) and D / be the Dirac operator, with k equal to the ordinary dimension of the spin manifold M. Let H be the space of square-integrable spinors. Then [D /,f] = D / (f ), just as in the unital case, and so the boundedness of [D, A] is unproblematic. In order to check whether (A, H, D / , χ ) is a spectral triple in our sense, one first needs to determine whether products of the form f (|D / | + ε)−k are compact operators of Dixmier trace class, whose Dixmier trace is (a standard multiple of) f (x) d k x. This compactness condition is guaranteed in the flat space case (taking A = S(Rk ), say) by celebrated estimates in scattering theory [81], that we review in Subsect. 4.1. The summability condition is a bit tougher. The Cesàro summability theory of [34] establishes that, for a positive pseudodifferential operator H of order d, acting on spinors, the spectral density asymptotically behaves as dH (x, x; λ ) ∼

2k/2 −k/d (k−d)/d wres H (λ ) + · · · , d (2π)k

in the Cesàro sense. Here wres denotes the Wodzicki residue density [45]. (If the operator is not positive, one uses the “four parts” argument.) In our case, H = a(|D / | + ε)−k is pseudodifferential of order −k, so dH (x, x; λ ) ∼ −

2k/2 k a(x) −2 (λ + · · · ), k (2π)k

as λ → ∞ in the Cesàro sense; here k is the hyperarea of the unit sphere in Rk . We independently know that H is compact, so on integrating the spectral density over x and over 0 ≤ λ ≤ λ, we get 2k/2 k a(x) d k x 1 NH (λ) ∼ as λ → ∞. k (2π)k λ This holds in the ordinary asymptotic sense, and not merely the Cesàro sense, by the “sandwich” argument used in the proof of [34, Cor. 4.1]. So finally, 2k/2 k a(x) d k x 1 λm (H ) ∼ as m → ∞, (3.8) k (2π)k m and the Dixmier traceability of a(|D / | + ε)−k , plus the value of its trace, follow at once. The rest is a long but almost trivial verification. For instance, J is the charge conjugation operator on spinors; the algebra (B, 0 ) is a suitable compactification; the domain H∞ consists of the smooth spinors; and so on. See below the parallel discussion for the Moyal case.


591

The following theorem sums it up. Theorem 3.2. The triple (S(Rk ), L2 (Rk ) ⊗ C2 commutative geometry of spectral dimension k.

k/2

,D / ) on Rk defines a noncompact

What about the nonflat case (of a spin manifold such that D / is selfadjoint)? Mainly because the previous Cesàro summability argument is purely local, everything carries over, if we choose for A the algebra of smooth and compactly supported functions. Of course, in some contexts it may be useful to demand that M also has conic exits. We want to remark that formula (3.8) for the flat case has been proved by Chakraborty et al in [11], using an ingenious reasoning involving two Laplacians on Rk . Theirs is a kind of “poor man’s argument” for ours, because what is really used is that the spectral density has the same asymptotic behaviour for the two Laplacians. Also, our inference is not confined to flat manifolds, rather it is directly valid on any decent noncompact manifold (without recourse to “lifting” devices). 3.4. On the Connes–Landi spaces example. An interesting family of compact spectral triples was constructed by Connes and Landi [25], by isospectral deformation of a commutative spectral triple wherein the Dirac operator is kept fixed (just as for our Moyal-product example) but the algebra is “twisted”. One starts with a smooth boundaryless manifold M carrying a smooth effective action of a torus Tk of dimension k ≥ 2. The orbits on which Tk acts freely determine maps C ∞ (M) → C ∞ (Tk ), and with these maps one can pull back the NC torus structure on C ∞ (Tk ) := (C ∞ (Tk ), ∗ ) to get an algebra C ∞ (M ) := (C ∞ (M), ∗ ). This algebra is given in fact by a periodic Moyal product just like (2.1), with the translations replaced by the Tk -action. See [26, 82, 87, 88] for several equivalent formulations of this construction. Now, as pointed out in [26], there is no need to assume that the manifold M be compact: we only need that the group action on M be periodic. Taking M = Rk , we get a noncompact spectral triple which is not isomorphic to the Moyal product examples considered in this article; one can regard it as intermediate between the commutative case and the full Moyal cases (with nonperiodic action). Concretely, the sphere S2N−1 = SO(2N )/SO(2N − 1) carries an effective action of N T , namely the rotations by elements of a maximal torus of SO(2N ); and this extends to a TN -action by rotations of R2N preserving the radial coordinate r. Each f ∈ S is a function of coordinates f (r, α1 , . . . , αN−1 , φ1 , . . . , φN ) where φ = (φ1 , . . . , φN ) ∈ TN . If Eq. (2.1) is interpreted as involving integration over the φj coordinates only, it defines a new twisted product on S (for each real skewsymmetric N × N matrix ). To define a spectral triple over this algebra, we need an operator D which is also TN invariant. For instance, one can construct D by extending radially the Dirac operator for (say) the round metric on S2N−1 , with its spinor bundle; it will be necessary to lift the torus action to a doubly covering action of Tn on spinors [26]. It remains to check that B is still a suitable unitization of S (note that abstract smoothness of B is proved like in Sect. 2 here [87]) in the case of the Connes–Landi twisted 2N-planes, in order to conclude that these fit into the framework developed in this paper. 4. The Moyal 2N -Plane as a Spectral Triple There is a natural star triple associated to the Moyal plane and we will see that it is part of the data for an even spectral triple fulfilling all required conditions.

592


:= (B(R2N ), θ ). The Hilbert Let A = (S(R2N ), θ ), with preferred unitization A N space will be H := L2 (R2N ) ⊗ C2 of ordinary square-integrable spinors. The representation of A is given by π θ : A → L(H) : f → Lθf ⊗ 12N , where Lθf acts on the “reduced” Hilbert space Hr := L2 (R2N ). In other words, if a ∈ A and ∈ H, to obtain π θ (a) we just left Moyal multiply by a componentwise. This operator π θ (f ) is bounded, since it acts diagonally on H and Lθf ≤ (2πθ)−N/2 f 2 was proved in Lemma 2.12. Under this action, the elements of H get the lofty name of Moyal spinors. The selfadjoint Dirac operator is not “deformed”: it will be the ordinary Euclidean Dirac operator D / := −i γ µ ∂µ , where the hermitian Dirac matrices γ 1 , . . . , γ 2N satisν µ fying {γ , γ } = +2 δ µν irreducibly represent the Clifford algebra C(R2N ) associated to (R2N , η), with η the standard Euclidean metric. As a grading operator χ we take the usual chirality associated to the Clifford algebra: χ := γ2N+1 := 1Hr ⊗ (−i)N γ 1 γ 2 . . . γ 2N . The notation γ2N+1 is a nod to physicists’ γ5 . Thus χ 2 = (−1)N (γ 1 . . . γ 2N )2 = (−1)2N = 1 and χ γ µ = −γ µ χ . The real structure J is chosen to be the usual charge conjugation operator for spinors on R2N endowed with an Euclidean metric. Here, we only assume that J 2 = ±1 according to the “sign table” (3.6) and that J (1Hr ⊗ γ µ )J −1 = −1Hr ⊗ γ µ which guarantees the other requirements of (3.6). In general, in a given representation, it can be written as J := CK,

(4.1)

where C denotes a suitable 2N × 2N unitary matrix and K means complex conjugation. An explicit form for J in a particular representation can be found in [98] where all γ µ are hermitian matrices with purely imaginary (respectively real) entries when µ is even (respectively odd). An important property of J is J (Lθ (f ∗ ) ⊗ 12N )J −1 = R θ (f ) ⊗ 12N ,

(4.2)

where R θ (f ) ≡ Rfθ is the right Moyal multiplication by f ; this follows from the antilinearity of J and the reversal of the twisted product under complex conjugation. Lemma 2.1(iii) implies that [D / , π θ (f )] = −iLθ (∂µ f ) ⊗ γ µ =: π θ (D / (f )); by θ = B(R2N ) —just as in the commutative case. Theorem 2.21 this is bounded for f ∈ A 4.1. The compactness condition. In this subsection and the next, the main tools are techniques developed some time ago for scattering theory problems, as summarized in Simon’s booklet [81, Chap. 4]. We adopt the convention that L∞ (H) := K(H), with A∞ := Aop . Let g ∈ L∞ (R2N ). We define the operator g(−i∇) on Hr as g(−i∇)ψ := F −1 (g Fψ),


593

where F is the ordinary Fourier transform. In more detail, for ψ in the correct domain, g(−i∇)ψ(x) = (2π)−2N eiξ ·(x−y) g(ξ )ψ(y) d 2N ξ d 2N y. The inequality g(−i∇)ψ2 = F −1 gFψ2 ≤ g∞ ψ2 entails that g(−i∇)∞ ≤ g∞ . /, Theorem 4.1. Let f ∈ A and λ ∈ / sp D / . Then, if RD/ (λ) is the resolvent operator of D then π θ (f ) RD/ (λ) is compact. Thanks to the first resolvent equation, RD/ (λ) = RD/ (λ ) + (λ − λ)RD/ (λ)RD/ (λ ), we may assume that λ = iµ with µ ∈ R∗ . The theorem will follow from a series of lemmas interesting in themselves. Lemma 4.2. If f ∈ S and 0 = µ ∈ R, then π θ (f )RD/ (iµ) ∈ K(H) ⇐⇒ π θ (f )|RD/ (iµ)|2 ∈ K(H). Proof. We know that Lθ (f )∗ = Lθ (f ∗ ). The “only if” part is obvious since RD/ (iµ) is a bounded normal operator. Conversely, if π θ (f )|RD/ (iµ)|2 is compact, then π θ (f ) |RD/ (iµ)|2 π θ (f ∗ ) is compact. Since an operator T is compact if and only if T T ∗ is compact, the proof is complete. The usefulness of this lemma stems from the diagonal nature of the action of π θ (f ) N |RD/ (iµ)|2 on H = Hr ⊗ C2 ; so in our arguments it is feasible to replace H by Hr , 2 π θ (f ) by Lθf , and to use the scalar Laplacian − := − 2N µ=1 ∂µ instead of the square of the Dirac operator D / 2. Lemma 4.3. When f, g ∈ Hr , Lθf g(−i∇) is a Hilbert–Schmidt operator such that, for all real θ, Lθf g(−i∇)2 = L0f g(−i∇)2 = (2π )−N f 2 g2 . Proof. To prove that an operator A with integral kernel KA is Hilbert–Schmidt, it suffices to check that |KA (x, y)|2 dx dy is finite, and this will be equal to A22 [81, Thm. 2.11]. So we compute KLθ (f ) g(−i∇) . In view of Lemma 2.15, 1 θ f (x − θ2 Sξ ) g(ξ )ψ(y) eiξ ·(x−y) d 2N ξ d 2N y. [L (f ) g(−i∇)ψ](x) = (2π)2N Thus KLθ (f ) g(−i∇) (x, y) = and

1 (2π)2N

f (x − θ2 Sξ ) g(ξ ) eiξ ·(x−y) d 2N ξ,

|KLθ (f ) g(−i∇) (x, y)|2 dx dy is given by 1 · · · f¯(x − θ2 Sξ ) g(ξ ¯ ) f (x − θ2 Sζ ) g(ζ ) ei(x−y)·(ζ −ξ ) (2π )4N ×d 2N x d 2N y d 2N ζ d 2N ξ 1 |f (x − θ2 Sξ )|2 |g(ξ )|2 d 2N x d 2N ξ = (2π )2N = (2π )−2N f 22 g22 < ∞.

594


Remark 4.4. As a consequence, we get .2 - lim Lθf g(−i∇) = L0f g(−i∇). θ→0

Lemma 4.5. If f ∈ Hr and g ∈ Lp (R2N ) with 2 ≤ p < ∞, then Lθf g(−i∇) ∈ Lp (Hr ) and Lθf g(−i∇)p ≤ (2π)−N(1/2+1/p) θ −N(1/2−1/p) f 2 gp . Proof. The case p = 2 (with equality) is just the previous lemma. For p = ∞, we estimate Lθf g(−i∇)∞ ≤ (2πθ )−N/2 f 2 g∞ : since Lθf g(−i∇)∞ ≤ Lθf ∞ g(−i∇)∞ , this follows from Lemma 2.12 and a previous remark. Now use complex interpolation for 2 < p < ∞. For that, we first note that we may suppose g ≥ 0: defining the function a with |a| = 1 and g = a|g|, we see that ¯ Lθf ∗ Lθf g(−i∇)) Lθf g(−i∇)22 = Tr(|Lθf g(−i∇)|2 ) = Tr(g(−i∇) = Tr(|g|(−i∇) a(−i∇) ¯ Lθf ∗ Lθf a(−i∇) |g|(−i∇)) = Tr(a(−i∇) ¯ |g|(−i∇) Lθf ∗ Lθf |g|(−i∇) a(−i∇)) = Tr(|Lθf |g|(−i∇)|2 ) = Lθf |g|(−i∇)22 , and Lθf g(−i∇)∞ = Lθf a(−i∇) |g|(−i∇)∞ = Lθf |g|(−i∇) a(−i∇)∞ ≤ Lθf |g|(−i∇)∞ a(−i∇)∞ = Lθf |g|(−i∇)∞ . Secondly, for any positive, bounded function g with compact support, we define the maps: Fp : z → Lθf g zp (−i∇) : S = { z ∈ C : 0 ≤ z ≤

1 2

} → L(Hr ).

For all y ∈ R, Fp (iy) = Lθf g iyp (−i∇) ∈ L∞ (Hr ) by Lemma 4.3 since g, being compactly supported, lies in Hr . Moreover, Fp (iy)∞ ≤ (2π θ)−N/2 f 2 . Also, by Lemma 4.3, Fp ( 21 +iy)∈L2 (Hr ) and Fp ( 21 +iy)2 = (2π )−N f 2 g p/2 2 . Then complex interpolation (see [70, Chap. 9] and [81]) yields F (z) ∈ L1/z (Hr ), for all z in the strip S. Moreover, Fp (z)1/z ≤ F (0)1−2z F ( 21 )2z ∞ 2 = f 2 (2πθ )− 2 (1−2z) (2π )−2Nz g p/2 2z 2 , N

and applying this result at z = 1/p, we get for such g: Lθf g(−i∇)p = F (1/p)p ≤ (2π)−N(1/2+1/p) θ −N(1/2−1/p) f 2 gp . We finish by using the density of compactly supported bounded functions in Lp(R2N ).


595

Remark 4.6. In the commutative case, if f and g are bounded on Rk , then f (x) g(−i∇)∞ ≤ f ∞ g∞ . Complex interpolation [8, 70, 81] leads then to an estimate of the form f (x) g(−i∇)p ≤ (2π)−k/p f p gp when p ≥ 2. For f ∈ S and for g(y) := 1/ |y|2 + µ2 , which lies in Lp (Rk ) for all p > k we conclude that f (x) g(−i∇) is compact and in Lp for p > k. This has already strongly pointed to compliance with Axiom 1 (verified above using Cesàro summability considerations), since Lk+ is larger than Lk , but smaller than the intersection of the Lp for p > k. Lemma 4.7. If f ∈ S and 0 = µ ∈ R, then π θ (f ) |RD/ (iµ)|2 ∈ Lp for p > N. Proof. We see that / − iµ)−1 (D / + iµ)−1 = Lθf (−∂ ν ∂ν + µ2 )−1 ⊗ 12N . π θ (f ) |RD/ (iµ)|2 = (Lθf ⊗ 12N ) (D N

So this operator acts diagonally on Hr ⊗ C2 and Lemma 4.5 implies that 1/p

θ

d 2N ξ

L (−∂ ν ∂ν +µ2 )−1 ≤ (2π)−N(1/2+1/p) θ −N(1/2−1/p) f 2 , f p (ξ ν ξν + µ2 )p which is finite for p > N.

Proof of Theorem 4.1. By Lemma 4.2, it was enough to prove that π θ (f ) |RD/ (iµ)|2 is compact for a nonzero real µ. H, D The conclusion is that (A, A, / , χ , J ) defines a noncompact spectral triple; recall are pre-C ∗ that we proved in Sect. 2 that both A and its preferred compactification A algebras. 4.2. Spectral dimension of the Moyal planes Theorem 4.8. The spectral dimension of the Moyal 2N-plane spectral triple is 2N. We shall first establish existence properties. Thanks to Lemma 4.5 and because / 2 + ε2 )−l and [D / , π θ (f )] (D /2+ [D / , π θ (f )] = −iLθ (∂µ f ) ⊗ γ µ , we see that π θ (f )(D 2 −l p ε ) lie in L (H) whenever p > N/ l (we always assume ε > 0). In the next lemma, we show that [|D / |, π θ (f )] (D / 2 + ε2 )−l has the same property of summability; this will become our main technical instrument for the subsection. Lemma 4.9. If f ∈ S and p > N/ l.

1 2

≤ l ≤ N, then [|D / |, π θ (f )] (D / 2 + ε2 )−l ∈ Lp (H) for

Proof. We use the following spectral identity for a positive operator A: dµ 1 ∞ A2 A= √ , 2 π 0 A +µ µ and another identity for any operators A, B and λ ∈ / sp A: [B, (A − λ)−1 ] = (A − λ)−1 [A, B](A − λ)−1 .

(4.3)

596


Hence, for any ρ > 0, / | + ρ, π θ (f )] = [|D / |, π θ (f )] = [|D =

1 π

∞

1−

1 π

∞

0 (|D / | + ρ)2

(|D / | + ρ)2 dµ θ (f ) , π √ 2 (|D / | + ρ) + µ µ (|D / | + ρ)2 , π θ (f )

(|D / | + ρ)2 + µ 1 dµ × √ (|D / | + ρ)2 + µ µ 1 ∞ 1 1 √ = µ dµ (|D / | + ρ)2 , π θ (f ) π 0 (|D / | + ρ)2 + µ (|D / | + ρ)2 + µ ∞ 1 1 / = −π θ (∂ µ ∂µ f ) − 2i(Lθ (∂µ f ) ⊗ γ µ )D π 0 (|D / | + ρ)2 + µ 1 √ +2ρ |D / |, π θ (f ) µ dµ. (4.4) (|D / | + ρ)2 + µ 0

This implies that

[|D / |, π θ (f )] (D / 2 + ε 2 )−l p 1 1 ∞

−π θ (∂ µ ∂µ f ) − 2i(Lθ (∂µ f ) ⊗ γ µ )D ≤ /

2 π 0 (|D / | + ρ) + µ

1 2 2 −l √ + 2ρ |D / |, π θ (f ) + ε ) µ dµ. (D /

2 (|D / | + ρ) + µ p Thus, the proof reduces to show that for any f ∈ S,

1 1 1 ∞ θ 2 2 −l √

π (D / (f )D / + ε ) µ dµ < ∞.

2 2 π 0 (|D / | + ρ) + µ (|D / | + ρ) + µ p (4.5) Since the Schatten p-norm is a symmetric norm, and since, as in the proof of Theorem 4.1, only the reduced Hilbert space is affected, expression (4.5) is majorized by

3/2

θ 1 ∞ 1 D / 1

π (f )

2 2 π 0 (|D / | + ρ) + µ (D / + ε 2 )1/2

(D / 2 + ε 2 )l−1/2

√ 1

× µ dµ 2 1/2 ((|D / | + ρ) + µ) p √

µ dµ 1 ∞

π θ (f ) (D ≤ / 2 + ε 2 )−l+1/2 ((|D / | + ρ)2 + µ)−1/2 p . π 0 (µ + ρ 2 )3/2 Thanks to Lemma 4.5, we can estimate the µ-dependence of the last p-norm:

θ

π (f )((|D / | + ρ)2 + µ)−1/2 (D / 2 + ε 2 )−l+1/2 p

≤ (2π )−N(1/2+1/p) θ −N(1/2−1/p) f 2 ((|ξ | + ρ)2 + µ)−1/2 (|ξ |2 + ε 2 )−l+1/2 p

≤ C(p, θ ) ((|ξ | + ρ)2 + µ)−1/2 q (|ξ |2 + ε 2 )−l+1/2 r ;


597

with p −1 = q −1 + r −1 appropriately chosen, these integrals are finite for q > 2N and r > 2N/(2l − 1); for l = 21 , take r = ∞ and q = p. For such values,

θ

π (f )((|D / | + ρ)2 + µ)−1/2 (D / 2 + ε 2 )−l+1/2 p ∞ 1/q R 2N−1 1/q 2 2 −l+1/2 dR ≤ C(p, θ, N; f )(|ξ | + ε ) r 2N ((R + ρ)2 + µ)q/2 0 = C(p, θ, N; f )(|ξ |2 + ε 2 )−l+1/2 r π N/q

1/q ( q2 − N ) −1/2+N/q µ 1/q ( q2 )

=: C (p, q, θ, N; f ) µ−1/2+N/q . Finally, the integral (4.5) is less than C (p, q, θ, N; f )

∞ 0

µN/q dµ, (µ + ρ 2 )3/2

which is finite for q > 2N and p > N/ l. This concludes the proof.

/ | + ε)−1 π θ (f ∗ ) ∈ L2N+ (H). Lemma 4.10. If f ∈ S, then π θ (f ) (|D Proof. This is an extension to the Moyal context of the renowned inequality by Cwikel [27, 81, 94]. As remarked before, it√is possible to replace D / 2 by −, π θ (f ) by θ −1 Lf and H by Hr . Consider g(−i∇) := ( − + ε) . Since g is positive, it can be decomposed as g = n∈Z gn , where g(x) if 2n−1 < g(x) ≤ 2n , gn (x) := 0 otherwise. For each n ∈ Z, let An and Bn be the two operators Lθf gk (−i∇) Lθf ∗ , Bn := Lθf gk (−i∇) Lθf ∗ . An := k≤n

k>n

We estimate the uniform norm of the first part:

−N 2

g (−i∇) ≤ (2π θ) f gk An ∞ ≤ Lθf 2 k 2

≤

k≤n −N (2πθ ) f 22 2n

∞

k≤n

∞

=: 2 c1 (θ, N ; f ). n

The trace norm of Bn can be computed using Lemma 4.3:

2

1/2 1/2

2

θ θ

Bn 1 = g (−i∇) L = L g (−i∇) k k f∗

f 2

k>n

2 1/2

−2N 2

= (2π) f 2 gk

2 k>n

−2N 2 −2N = (2π) f 2 gk f 22 gk 1

= (2π ) k>n

k>n

≤ (2π)

−2N

f 22

k>n

1

gk ∞ ν{supp(gk )},

k>n

2

598


where ν is the Lebesgue measure on R2N . By definition, gk ∞ ≤ 2k and ν{supp(gk )} = ν{ x ∈ R2N : 2k−1 < g(x) ≤ 2k } ≤ ν{ x ∈ R2N : (|x| + ε)−1 ≥ 2k−1 } ≤ 22N(1−k) c2 . Therefore Bn 1 ≤ (2π)−2N f 22 22N c2

2k(1−2N)

k>n

< π −2N c2 f 22 2n(1−2N) =: 2n(1−2N) c3 (N; f ), where the second inequality follows because N > 21 . th singular value µ of B (arranged in decreasing order We can now estimate the m m n with multiplicity): Bn 1 = ∞ µk (Bn ). Note that, for m = 1, 2, 3, . . . , Bn 1 ≥ k=0 m−1 −1 ≤ 2n(1−2N) c m−1 . Now 3 k=0 µk (Bn ) ≥ m µm (Bn ). Thus, µm (Bn ) ≤ Bn 1 m Fan’s inequality [81, Thm. 1.7] yields µm (Lθf g(−i∇) Lθf ∗ ) = µm (An + Bn ) ≤ µ1 (An ) + µm (Bn ) ≤ An + Bn 1 m−1 ≤ 2n c1 + 2n(1−2N) c3 m−1 . Given m, choose n ∈ Z so that 2n ≤ m−1/2N < 2n+1 . Then µm (Lθf g(−i∇) Lθf ∗ ) ≤ c1 m−1/2N + c3 m−(1−2N)/2N m−1 =: c4 (θ, N ; f ) m−1/2N . √ Therefore Lθf ( − + ε)−1 Lθf ∗ ∈ L2N+ (Hr ), and the statement of the lemma follows. Corollary 4.11. If f, g ∈ S, then π θ (f ) (|D / | + ε)−1 π θ (g) ∈ L2N+ (H). / | + ε)−1 π θ (f ∗ ± g) and π θ (f ± ig ∗ ) (|D / | + ε)−1 π θ Proof. Consider π θ (f ± g ∗ ) (|D ∗ (f ∓ ig). / | + ε)−1 ∈ L2N+ (H). Corollary 4.12. If h ∈ S, then π θ (h) (|D Proof. Let h = f θ g. Then / | + ε)−1 = π θ (f ) (|D / | + ε)−1 π θ (g) + π θ (f ) [π θ (g), (|D / | + ε)−1 ], π θ (h) (|D and we obtain from the identity (4.3) that π θ (h) (|D / | + ε)−1 = π θ (f ) (|D / | + ε)−1 π θ (g) θ +π (f ) (|D / | + ε)−1 [|D / |, π θ (g)] (|D / | + ε)−1 . By arguments similar to those of Lemmata 4.5 and 4.9, the last term belongs to Lp for p > N, and thus to L2N+ . Boundedness of (|D / | + ε)(D / 2 + ε 2 )−1/2 follows from elementary Fourier analysis. And so the last corollary means that the spectral triple is “2N + -summable”. We have taken care of the first assertion of the theorem. The next lemma is the last property of existence that we need. / |+ε)−2N and π θ (f )(D / 2 +ε 2 )−N are in L1+ (H). Lemma 4.13. If f ∈ S, then π θ (f )(|D


599

Proof. It suffices to prove that π θ (f )(|D / | + ε)−2N ∈ L1+ (H). We factorize f ∈ S according to Proposition 2.7, with the following notation: f = f1 θ f2 = f1 θ f21 θ f22 = f1 θ f21 θ f221 θ f222 = · · · = f1 θ f21 θ f221 θ · · · θ f22···21 θ f22···22 . Therefore, / | + ε)−1 π θ (f2 ) (|D / | + ε)−2N+1 / | + ε)−2N = π θ (f1 ) (|D π θ (f ) (|D +π θ (f1 ) (|D / | + ε)−1 [|D / |, π θ (f2 )] (|D / | + ε)−2N . (4.6) / | + ε)−1 ∈ Lp (H) whenever p > 2N; and by Lemma 4.9, By Lemma 4.5, π θ (f1 )(|D the term [|D / |, π θ (f2 )](|D / | + ε)−2N lies in Lq (H) for q > 1. Hence, the last term on the right hand side of Eq. (4.6) lies in L1 (H). We may write the following equivalence relation: / | + ε)−2N+1 , / | + ε)−1 π θ (f2 )(|D / | + ε)−2N ∼ π θ (f1 )(|D π θ (f )(|D where A ∼ B for A, B ∈ K(H) means that A − B is trace-class. Thus, / | + ε)−1 π θ (f2 )(|D / | + ε)−2N+1 π θ (f )(|D / | + ε)−2N ∼ π θ (f1 )(|D θ −1 θ −1 θ = π (f1 )(|D / | + ε) π (f21 )(|D / | + ε) π (f22 )(|D / | + ε)−2N+2 +π θ (f1 )(|D / | + ε)−1 π θ (f21 )(|D / | + ε)−1 [|D / |, π θ (f22 )] (|D / | + ε)−2N+1 θ −1 θ −1 θ −2N+2 ∼ π (f1 )(|D / | + ε) π (f21 )(|D / | + ε) π (f22 )(|D / | + ε) ∼ ··· θ −1 θ −1 θ −1 ∼ π (f1 )(|D / | + ε) π (f21 )(|D / | + ε) π (f221 )(|D / | + ε) . . . π θ (f22···22 )(|D / | + ε)−1 . / | + ε)−1 π θ (f21 )(|D /| + The second equivalence relation holds because π θ (f1 )(|D p θ ∈ L (H) for p > N by Lemma 4.5, and [|D / |, π (f22 )](|D / | + ε)−2N+1 ∈ Lq (H) for q > 2N/(2N − 1) by Lemma 4.9 again. The other equivalences come from similar arguments. Corollary 4.11, the Hölder inequality (see [45, Prop. 7.16]) and the inclusion L1 (H) ⊂ L1+ (H) finally yield the result.

ε)−1

Now we go for the computation of the Dixmier trace. Using the regularized trace for a DO: Tr (A) := (2π)−2N σ [A](x, ξ ) d 2N ξ d 2N x, |ξ |≤

the result can be conjectured because lim→∞ Tr (·)/ log(2N ) is heuristically linked with the Dixmier trace, and the following computation: 1 Tr π θ (f )(D / 2 + ε 2 )−N →∞ 2N log 2N = lim f (x − θ2 Sξ ) (|ξ |2 + ε 2 )−N d 2N ξ d 2N x →∞ 2N (2π)2N log |ξ |≤ 2N 2N = f (x) d 2N x. 2N (2π )2N lim

This is precisely the same result of (3.8), in the commutative case, for k = 2N. However, to establish it rigorously in the Moyal context requires a subtler strategy. We shall

600


compute the Dixmier trace of π θ (f ) (D / 2 + ε 2 )−N as the residue of the ordinary trace of a related meromorphic family of operators. For this, recent results of Carey and coworkers [10] extending Connes’ trace theorem (see [16] and [45, Chap. 7]) come in handy. In turn we are allowed to introduce the explicit symbol formula that will establish measurability [17, 45], too. In the language of [50], thus, we seek first to verify that Aθ has analytical dimension equal to 2N; that is, for f ∈ Aθ the operator π θ (f ) (D / 2 + ε2 )−z/2 is trace-class if z > 2N. / 2 + ε 2 )−z/2 is trace-class for z > 2N , and Lemma 4.14. If f ∈ S, then Lθf (D Tr[Lθf

2 −z/2

(D / +ε ) 2

−2N

] = (2π)

f (x) (|ξ |2 + ε 2 )−z/2 d 2N ξ d 2N x.

Proof. If a(x, ξ ) ∈ Kp (R2k ), for p < −k, is the symbol of a pseudodifferential operator A, then the operator is trace-class and moreover Tr A = (2π)−k a(x, ξ ) d k x d k ξ. This is easily proved by taking a ∈ S(R2k ) first and extending the resulting formula by continuity; have a look at [29, 67, 93] as well. In our case, the symbol formula for a product of DOs yields, for p > N, (−i)|α| σ Lθf (− + ε 2 )−p (x, ξ ) = ∂ξα σ [Lθf ](x, ξ ) ∂xα σ (− + ε 2 )−p (x, ξ ) α! α∈NN = σ [Lθf ](x, ξ ) σ (− + ε 2 )−p (x, ξ ) = f (x − θ2 Sξ ) (|ξ |2 + ε 2 )−p . Therefore, for p > N, Tr

Lθf (− + ε 2 )−p

= (2π)

−2N

= (2π)−2N

f (x − θ2 Sξ ) (|ξ |2 + ε 2 )−p d 2N ξ d 2N x f (x) (|ξ |2 + ε 2 )−p d 2N ξ d 2N x.

We continue with a technical lemma, in the spirit of [73]. Consider the approximate unit {eK }K∈N ⊂ Ac , where eK := 0≤|n|≤K fnn . These eK are projectors with a natural ordering: eK θ eL = eL θ eK = eK for K ≤ L, and they are local units for Ac . Lemma 4.15. Let f ∈ Ac,K . Then N π θ (f ) (D / 2 + ε 2 )−N − π θ (f ) π θ (eK )(D / 2 + ε 2 )−1 π θ (eK ) ∈ L1 (H). Proof. For simplicity we use the notation e := eK and en := eK+n . By the boundedness of π θ (f ), we may assume that f = e ∈ Ac,K . Because en θ e = eθ en = e, it is clear that π θ (e)(D / + λ)−1 1 − π θ (en ) = π θ (e) (D / + λ)−1 [D / , π θ (en )] (D / + λ)−1 . (4.7)


601

/ , π θ (e)] π θ (en ) = 0 because [D / , π θ (e)] Also, π θ (e) [D / , π θ (en )] = [D / , π θ (eθ en )] − [D θ θ π (en ) = [D / , π (e)] for n = 1 or bigger —see Eq. (8.1) of the Appendix. We obtain An := = = =

/ + λ)−1 / + λ)−1 [D / , π θ (en )](D π θ (e)(D / + λ)−1 [D / , π θ (en )](D / + λ)−1 / , π θ (e1 )](D π θ (e)(D / + λ)−1 [D −1 θ −1 θ θ θ / + λ) [D / , π (en )](D / + λ)−1 = · · · / + λ) [D / , π (e1 )]π (e2 )(D π (e)(D θ / + λ)−1 · · · / , π θ (e2 )](D π (e)(D / + λ)−1 [D / , π θ (e1 )](D / + λ)−1 [D [D / , π θ (en )](D / + λ)−1 .

Taking n = 2N here, A2N appears as a product of 2N + 1 terms in parentheses, each in L2N+1 (H) by Lemma 4.5. Hence, by Hölder’s inequality, A2N is trace-class and therefore π θ (e)(D / + λ)−1 (1 − π θ (e2N )) ∈ L1 (H). Thus, / 2 + ε 2 )−1 1 − π θ (e4N ) π θ (e) (D = π θ (e)(D / − iε)−1 1 − π θ (e2N ) + π θ (e2N ) (D / + iε)−1 1 − π θ (e4N ) = π θ (e)(D / + iε)−1 1 − π θ (e4N ) / − iε)−1 1 − π θ (e2N ) (D +π θ (e)(D / − iε)−1 π θ (e2N )(D / + iε)−1 1 − π θ (e4N ) ∈ L1 (H). (4.8) This is to say π θ (e)(D / 2 + ε 2 )−1 ∼ π θ (e)(D / 2 + ε 2 )−1 π θ (e4N ). Shifting this property, we get π θ (e)(D / 2 + ε 2 )−N ∼ π θ (e)(D / 2 + ε 2 )−1 π θ (e4N )(D / 2 + ε 2 )−N+1 θ 2 2 −1 θ ∼ π (e)(D / + ε ) π (e4N )(D / 2 + ε 2 )−1 π θ (e8N )(D / 2 + ε 2 )−N+2 ∼ · · · / 2 + ε 2 )−1 π θ (e4N )(D / 2 + ε 2 )−1 π θ (e8N ) · · · (D / 2 + ε 2 )−1 π θ (e4N 2 ). ∼ π θ (e)(D By identity (4.3), the last term on the right equals π θ (e)(D / + iε)−1 π θ (e)(D / − iε)−1 π θ (e4N )(D / 2 + ε 2 )−1 π θ (e8N ) · · · (D / 2 + ε 2 )−1 π θ (e4N 2 ) +π θ (e)(D / + iε)−1 [D / , π θ (e)](D / 2 + ε 2 )−1 π θ (e4N )(D / 2 + ε 2 )−1 π θ (e8N ) · · · 2 −1 θ 2 (D / + ε ) π (e4N 2 ). The last term is trace-class because it is a product of N terms in Lp (H) for p > N and one term in Lq (H) for q > 2N, by Lemma 4.5. Removing the second π θ (e) once again, by the ordering property of the local units eK yields / + iε)−1 π θ (e)(D / − iε)−1 π θ (e4N )(D / 2 + ε 2 )−1 π θ (e8N ) · · · π θ (e)(D (D / 2 + ε 2 )−1 π θ (e4N 2 ) = π θ (e)(D / 2 + ε 2 )−1 π θ (e8N ) · · · (D / 2 + ε 2 )−1 π θ (e4N 2 ) / 2 + ε 2 )−1 π θ (e)(D +π θ (e)(D / 2 + ε 2 )−1 [D / , π θ (e)](D / − iε)−1 π θ (e4N )(D / 2 + ε 2 )−1 π θ (e8N ) · · · 2 2 −1 θ (D / + ε ) π (e4N 2 ). The last term is still trace-class, hence π θ (e)(D / 2 + ε 2 )−N ∼ π θ (e)(D / 2 + ε 2 )−1 π θ (e)(D / 2 + ε 2 )−1 π θ (e8N ) · · · 2 2 −1 θ (D / + ε ) π (e4N 2 ).

602


This algorithm, applied another (N − 1) times, yields the result: N π θ (e)(D / 2 + ε 2 )−N ∼ π θ (e)(D / 2 + ε 2 )−1 π θ (e) . We retain the following consequence. / 2 + ε 2 )−N ] = 0 for any g ∈ S and any proCorollary 4.16. Tr + π θ (g) [π θ (f ), (D jector f ∈ Ac . Proof. This follows from Lemma 4.15 applied to π θ (f ) (D / 2 +ε2 )−N and its adjoint.

Now we are finally ready to evaluate the Dixmier traces. Proposition 4.17. For f ∈ S, any Dixmier trace Tr + of π θ (f ) (D / 2 + ε2 )−N is independent of ε, and / 2 + ε 2 )−N = Tr + π θ (f ) (D

2N 2N 2N (2π)2N

f (x) d 2N x =

1 N! (2π )N

f (x) d 2N x.

Proof. We will first prove it for f ∈ Ac . Choose e a unit for f , that is, eθ f = f θ e = f . By Lemmata 4.13 and 4.15, and because L1 (H) lies inside the kernel of the Dixmier trace, we obtain / 2 + ε 2 )−N ) = Tr + π θ (f ) (π θ (e)(D / 2 + ε 2 )−1 π θ (e))N . Tr + (π θ (f ) (D N Lemma 4.15 applied to f = e implies that π θ (e)(D / 2 + ε2 )−1 π θ (e) is a positive / 2 + ε2 )−N plus a term in L1 (H). Thus, operator in L1+ (H), since it is equal to π θ (e)(D [10, Thm. 5.6] yields (since the limit converges, any Dixmier trace will give the same result): / 2 + ε 2 )−N = lim(s − 1) Tr π θ (f ) (π θ (e)(D / 2 + ε 2 )−1 π θ (e))Ns Tr + π θ (f ) (D s↓1 = lim(s − 1) Tr π θ (f )π θ (e)(D / 2 + ε 2 )−Ns π θ (e) + ENs , s↓1

(4.9) where Ns / 2 + ε 2 )−1 π θ (e) − π θ (f )π θ (e)(D / 2 + ε 2 )−Ns π θ (e). ENs := π θ (f ) π θ (e)(D Lemma 4.15 again shows that EN ∈ L1 (H). Ns Now for s > 1, the first term π θ (f ) π θ (e)(D / 2 +ε2 )−1 π θ (e) of ENs is in L1 (H). 2 θ 2 −1 p / + ε ) ∈ L (H) for p > N, we have In effect, using Lemma 4.5 and since π (e)(D π θ (e)(D / 2 + ε 2 )−1 π θ (e) ∈ LNs (H). This operator being positive, one concludes

π θ (e)(D / 2 + ε 2 )−1 π θ (e)

Ns

∈ L1 (H).

The second term π θ (f )π θ (e)(D / 2 + ε 2 )−Ns π θ (e) lies in L1 (H) too, because / 2 + ε 2 )−Ns π θ (e)1 = (D / 2 + ε 2 )−Ns/2 π θ (e)22 = π θ (e)(D / 2 + ε 2 )−Ns/2 22 π θ (e)(D


603

is finite by Lemma 4.3. So ENs ∈ L1 (H) for s ≥ 1, and (4.9) implies Tr + π θ (f ) (D / 2 + ε 2 )−N = lim(s − 1) Tr π θ (f )π θ (e)(D / 2 + ε 2 )−Ns π θ (e) s↓1 = lim(s − 1) Tr π θ (f )(D / 2 + ε 2 )−Ns . s↓1

Applying now Lemma 4.14, we obtain Tr + π θ (f ) (D / 2 + ε 2 )−N

= lim(s − 1) Tr(12N ) Tr Lθf (− + ε 2 )−Ns s↓1 N −2N = 2 (2π) lim(s − 1) f (x) (|ξ |2 + ε 2 )−Ns d 2N ξ d 2N x s↓1 1 = f (x) d 2N x, N! (2π)N

where the identity

(|ξ |2 + ε 2 )−Ns d 2N ξ = π N

(N (s − 1)) , (N s) ε 2N(s−1)

and (N α) ∼ 1/Nα as α ↓ 0 have been used. The proposition is proved for f ∈ Ac . Finally, take f arbitrary in S, and recall that {eK } is an approximate unit for Aθ . Since f = gθ h for some g, h ∈ S, Corollary 4.16 implies + θ Tr (π (f ) − π θ (eK f eK ))(D / 2 + ε 2 )−N θ θ = Tr + (π θ (f ) − π θ (eK θ f )) (D / 2 + ε 2 )−N = Tr + (π θ (g) − π θ (eK θ g)) π θ (h)(D / 2 + ε 2 )−N ≤ π θ (g) − π θ (eK θ g)∞ Tr + π θ (h) (D / 2 + ε 2 )−N . Since π θ (g) − π θ (eK θ g)∞ ≤ (2πθ )−N/2 g − eK θ g2 tends to zero when K increases, the proof is complete because eK θ f θ eK lies in Ac and

[eK θ f θ eK ](x) d

2N

x→

f (x) d 2N x

as K ↑ ∞.

Remark 4.18. Similar arguments to those of this section (or a simple comparison argument) show that for f ∈ S, Tr + π θ (f ) (|D / | + ε)−2N = Tr + π θ (f ) (D / 2 + ε 2 )−N . In conclusion: the analytical and spectral dimension of Moyal planes coincide. Lemma 4.13, Proposition 4.17 and the previous remark have concluded the proof of Theorem 4.8.

604


4.3. The regularity condition θ , the bounded operators π θ (f ) and [D Theorem 4.19. For f ∈ A / , π θ (f )] lie in the smooth domain of the derivation δ(T ) := [|D / |, T ]. The traditional recursive proof [23, 45] does not work in its original form because the useful transformations L and R are undefined in the noncompact case (i.e., |D / |−1 is not available). However, an analogue of this proof may exist if instead of (3.1) we define Lλ and Rλ , for real λ, as Lλ (·) := (|D / | + iλ)−1 [D / 2 , ·],

Rλ (·) := [D / 2 , ·] (|D / | − iλ)−1 .

Here we prefer to prove the theorem by its north face: this approach is still valid for the commutative case, compact or not. Proof of Theorem 4.19. As before, because [D / , π θ (f )] = −iLθ (∂µ f ) ⊗ γ µ , it is sufθ ficient to prove that π (f ) lies in the smooth domain of δ. For each n ∈ N and ρ > 0, we may iterate the spectral identity (4.4) n times, to get for δ n (π θ (f )): √ ∞ ∞ n 1 λi ··· (ad(|D / | + ρ)2 )n (π θ (f )) n π 0 (|D / | + ρ)2 + λi 0 i=1

×

n i=1

1 dλn . . . dλ1 , (|D / | + ρ)2 + λi

with an obvious notation for the n-fold iterated commutators. Because [D / 2 , π θ (f )] = D / 2 (f )+2D / (f ) D / , with the notation D / (f ) := −iLθ (∂µ f )⊗ µ γ , we can check that the term with the highest power of D / in the expansion of (ad(|D / |+ ρ)2 )n (π θ (f )) is 2n D / n (f ) D / n . For the rest of the proof, we consider only such highestpower terms. As in the proof of Lemma 4.9, all commutators [|D / |, π θ (f )], which appear due to the artificial presence of ρ, will be treated as a sum of two first order operators. Hence, √ ∞ ∞ n 1 λi ··· 2n D / n (f )D /n n π 0 (|D / | + ρ)2 + λi 0 i=1

n

1 dλn . . . dλ1 (|D / | + ρ)2 + λj j =1 √ ∞ ∞ n λi 1 n n n = n ··· 2 D / (f ) D / dλn . . . dλ1 2 + λ )2 π 0 ((|D / | + ρ) i 0 i=1 ∞ ∞ n 1 1 n n + n ··· ,2 D / (f ) D /n π 0 (|D / | + ρ)2 + λi 0 i=1 √ n λi × dλn . . . dλ1 . (4.10) (|D / | + ρ)2 + λi i=1 √ ∞ Using 0 t (λ + t 2 )−2 λ dλ = π/2, the first term on the right hand side of (4.10) equals ∞ n √ |D /| + ρ 1 D /n D /n n n n 2 D / (f ) λ dλ = D / (f ) , (|D / | + ρ)n π 0 ((|D / | + ρ)2 + λ)2 (|D / | + ρ)n which is a bounded operator. ×


605

/ | + ρ)2 + λi )−1 , D / n (f ) can For the other term, notice that the commutator i ((|D be rewritten as n n n 2 −1 2 n − ((|D / | + ρ) + λi ) ((|D / | + ρ) + λj ), D / (f ) ((|D / | + ρ)2 + λk )−1 , j =1

i=1

k=1

and the highest-power term of this expression is, up to a constant: n n ((|D / | + ρ)2 + λi )−1 D / n+1 (f ) D / 2n−1 ((|D / | + ρ)2 + λk )−1 . i=1

k=1

So the proof reduces to showing the finiteness of the following norm: √

∞ ∞ n

λi

D / n+1 (f ) D / 3n−1 ···

2+λ (|D / | + ρ) i 0 i=1 0

2 n

1

dλ . . . dλ × n 1 2 (|D / | + ρ) + λj j =1

∞

≤ D / n+1 (f ) 0

D / 3−1/n

((|D 2 3/2−1/2n / | + ρ) + λ )

n ∞

··· 0

i=1

i

3/2+1/2n

1

× λi dλi (|D / | + ρ)2 + λi √ ∞ n λ ≤ D / n+1 (f ) dλ . (ρ 2 + λ)3/2+1/2n 0

θ ⊂ Aθ for This integral is finite for all n ∈ N and so is the norm since ∂ α f ∈ A |α| ≤ n + 1. The proof is complete.

4.4. The finiteness condition Lemma 4.20. The smooth vectors for D / are given by N C ∞ (D / ) ≡ H∞ := Dom(D / k ) DL2 ⊗ C2 . k∈N

Proof. Since DL2 is the common smooth domain of the partial derivatives ∂µ , for µ = 1, . . . , 2N, and since D / = −i∂µ ⊗ γ µ , the conclusion is clear. θ . Then H∞ is an A1 -pullback Take A1 := DL2 ; by Lemma 2.22, this is an ideal in A of a free left Aθ -module. On H∞ , there is a natural A1 -valued hermitian structure, given by N

(ξ | η) :=

2 j =1

ξj θ ηj∗ ,

for all ξ, η ∈ H∞ .

606


Because DL2 ⊂ Mθ , the hermitian pairing (π θ (a)ξ | η) = aθ (ξ | η) is Aθ -valued whenever a ∈ Aθ . Proposition 4.17 and Lemma 2.1(v) now imply Tr

+

2 −N

/ +ε ) π ((ξ | η) ) (D θ

2

2 2N 2N (ξj θ ηj∗ )(x) d 2N x = 2N (2π)2N N

j =1

2 1 = ηj∗ (x) ξj (x) d 2N x. N! (2π)N N

j =1

Therefore, (ξ | η) := N! (2π)N (ξ | η) is the desired hermitian structure satisfying (3.5). Its uniqueness can be checked in the same way as in [45, p. 501]. In summary: the inner product on H is tightly linked to the natural hermitian structure on H∞ (D) by means of the resolvent of D and the noncommutative integral. Remark 4.21. An obvious integral estimate makes it clear that Or ⊂ DL2 if and only if r < −N. Consider, therefore, N := r 0; so in particular it is invertible. The operator D might not be strictly positive at the outset; but a related operator will do. For instance, for the scalar case, commutative or not, / 2 + ε 2 )1/2 .

we may use D :=k (D ∞ Denote K (D) := k∈N Dom(d(D) ) ⊂ K. A typical element of K∞ (D) is a symmetrized tensor power of elements of H∞ (D); in fact the algebraic span of such vectors is dense in K∞ (D). The boson field ϕ(v) is just the selfadjoint generator of β(tv); the Segal field ϕ(v) = a(v) + a † (v) (here a(v) and a † (v) are the usual annihilation and creation operators) is essentially selfadjoint on K∞ (D), and it is easy to see that for v ∈ H∞ (D), it sends K∞ (D) continuously into itself. It is advantageous to think of ϕ(v) as a quadratic form; we recall how this comes about. Let L be a dense subspace of H, gifted with a topology stronger than that of H (in our case, H∞ (D) and K∞ (D) are given the projective Fréchet space topologies associated to the families of norms D n (·) and d(D)n (·), respectively), and let f be a continuous sesquilinear form on L. One could try to introduce a Hilbert space operator TfH through f (u, v) =: u | TfH v, defined on elements v of L for which f (u, v) ≤ cv u for all u; but that condition might only hold for, say, v = 0. However, if L is the antidual of L, then H → L with a continuous embedding, since u → u | v is an antilinear continuous functional on L, and f defines a map Tf : L → L by Tf v(u) := f (u, v). The elements of L in a concrete representation for H are distributions; and so quadratic forms are generalized operators. Often, (H∞ (D)) is denoted H−∞ (D). We refer to [2, Sec. 7.3] for the following estimate: for all v ∈ H, ∈ K∞ (D) and m ≥ 1, a(v) ≤ C D −m v d(D)m .


609

From that, and the formula | :ϕ(w1 ) . . . ϕ(wn ): | =

I ⊆{1,...,n} i∈I c

a(wi ) a(wj ) ,

(5.1)

j ∈I

with , ∈ K∞ (H) and I c = {1, . . . , n} \ I , it is immediate that one can define a Wick map from monomials in the free algebra over H−∞ (D) to quadratic forms on K∞ (D), extending the similar map in the subalgebra generated by H. To fix ideas in the following, the reader can put 2N = 4. We work on Euclidean space rather than on Minkowski spacetime, but formal passage to relativistic field theory (where however everything takes place on-shell) is quite simple. The conservative approach is to have the :ϕ r (x): living in the commutative context, that is, in the boson algebra K = (H) over the Hilbert space functions on H of∨nsquare-summable ∨n is identified to the momentum space. As already indicated, K ∞ H , where H n=0 space of complex symmetric functions , square-integrable with respect to the standard volume form d 2N p1 . . . d 2N pn in R2Nn . Precisely, the norm on H∨n is taken to be (n) 2 :=

···

n! |(n) (p1 , . . . , pn )|2

n

d 2N pi .

i=1

Then H∞ (D) is nothing other than the space DL2 ! Furthermore, it is possible to take wi (x) = δ(x −xi ) in the above (5.1), as the distribution δ(·−xi ) belongs to H−∞ (D) = H 2,−∞ = DL 2 . An outcome of the previous discussion is that the Wick products :ϕ(x1 ) . . . ϕ(xl ): := :ϕ(δ(x − x1 )) . . . ϕ(δ(x − xl )): used by physicists make perfect sense as continuous sesquilinear forms on the corresponding K∞ (D), and a fortiori on the space of Fock vectors with finitely many nonvanishing components, each one belonging to (a symmetrized tensor power of) DL2 . The function from R2Nl × (K∞ (D))2 to C given by (x1 , . . . , xn ; , ) → | :ϕ(x1 ) . . . ϕ(xl ): | , being continuous (indeed, smooth) in x1 , . . . , xl , can be restricted to the diagonal; and this defines the (ordinary) Wick monomials :ϕ l (x): for any l. That is to say, | :ϕ l (x): | = | :ϕ(x1 ) . . . ϕ(xl ):, δ(x − x1 ) . . . δ(x − xl )x1 ,...,xl | (5.2) is a well-defined expression. Thus, and more important still, we have established that manipulations with Dirac delta functions —such as the ones we are going to use later to define Moyal–Wick monomials— are justifiable. In this respect, the good behaviour of DL2 under the Moyal product, as under the ordinary one, becomes crucial. Also, for the same reason that the better algebra to represent the Moyal plane is (DL2 , θ ) rather than (S, θ ), the use of Schwartz functions and tempered distributions in the classic paper by Wightman and G˚arding [95], in which Wick products and Wick monomials were defined as operator-valued distributions, has been revealed as artificial.

610


5.2. The noncommutative Wick monomials. For ease of reference, we give here the explicit expression of the ordinary commuting Wick products (n) :ϕ(x1 ) . . . ϕ(xl ): (p1 , . . . , pn ) l 1 −Nl = (2π ) ··· j ! (l − j )! j =0 |X|=l−j i(x1 η1 +···+xj ηj −xj +1 ηj +1 −···−xl ηl ) × Pe P

×(n−l+2j ) (η1 , . . . , ηj , p1 , . . . , η j +1 , . . . , ηl , . . . , pn )

j

d 2N ηk ,

(5.3)

k=1

where P runs over all permutations of the momentum variables, and X = {ηj +1 , . . . , ηl } ranges over all subsets of l −j distinct elements of {p1 , . . . , pn }. Consequently, for good measure: (n) (n) l :ϕ :(x) (p1 , . . . , pn ) := :ϕ l (x): (p1 , . . . , pn ) l 1 = (2π )−Nl ··· P eix(η1 +···+ηj −ηj +1 −···−ηl ) j ! (l − j )! j =0

|X|=l−j

P

×(n−l+2j ) (η1 , . . . , ηj , p1 , . . . , η j +1 , . . . , ηl , . . . , pn )

j

d 2N ηk .

k=1

We have used operator rather than sesquilinear-form notation, although :ϕ l (x): for ∈ K∞ (D) is not in K, instead it is an (actually rather tame) vector-valued distribution. But it is guaranteed that K∞ (D) | :ϕ l (x): | K∞ (D) is finite. Let us now reinstate the Moyal product associated to a k×k skewsymmetric matrix ; for now, we assume to be nondegenerate. Formula (2.2) can be construed as meaning −1 t)

δ(x − s) δ(x − t) = (πθ )−2N e−2i(s·

−1 s+t·−1 x)

e−2i(x·

.

The left-hand side could of course have been written, somewhat more correctly, as (δs δt )(x). More generally, an easy two-step induction gives δ(x − x1 ) · · · δ(x − x2m )

−1 x j

= (π θ )−2Nm e2i i<j (−) xi · δ(x − x1 ) · · · δ(x − x2m+1 ) = (π θ )−2Nm e2i

i+j

i+j x ·−1 x i j i<j (−)

−1 (x −x +x −···−x ) 1 2 3 2m

e−2ix·

,

δ(x − x1 + x2 − x3 + · · · − x2m+1 ).

(5.4a) (5.4b)

These functionals of x1 , . . . , x2m or x1 , . . . , x2m+1 belong to (DL 2 )2m or respectively (DL 2 )2m+1 —recall that the space of rapidly decreasing distributions OC is a subspace of DL 2 [77]. There can be no question of making :ϕ(x1 ) . . . ϕ(xl ): “noncommutative”; so, how are we to define the Moyal–Wick products :ϕ l :(x)?


611

A “quantum Wick product” was recently introduced in [4]; but it is at variance with Moyal NCFT, and so is unsuitable for our present purposes. A different course is suggested by the older duality theory of [43, 90] and the discussion in the previous subsection. Our declared tactics are to construct :ϕ l :(x) on the very same Fock space of the real scalar field. This would seem to run against the spirit of noncommutative geometry, but is in fact demanded by our results here so far, and the treatment in the previous subsection. We posit :ϕ l :(x) := :ϕ(x1 ) . . . ϕ(xl ):, δ(x − x1 ) · · · δ(x − xl )x1 ,...,xl ,

(5.5)

to be compared with (5.2). We may also define Moyal products of Moyal–Wick monomials with suitable scalar functions or distributions on configuration space: :ϕ l : h(x) = :ϕ(x1 ) . . . ϕ(xl ):, δ(x − x1 ) · · · δ(x − xl ) h(x)x1 ,...,xl , h :ϕ l :(x) = :ϕ(x1 ) . . . ϕ(xl ):, h(x) δ(x − x1 ) · · · δ(x − xl )x1 ,...,xl . What it is required is that the functional δ(x − x1 ) · · · δ(x − xl ) h(x), in the x1 , . . . , xl variables, belong to (H−∞ (D))l . A seemingly alternative definition is given by :ϕ l : h(x), g(x) = :ϕ l :(x), h g(x), h :ϕ l :(x), g(x) = :ϕ l :(x), g h(x), in the spirit of [43, 90], for suitable spaces of functions g and distributions h. The verification that both kinds of definition coincide is immediate. Note that the identity :ϕ l :(x), h(x) = :ϕ(x1 ) . . . ϕ(xl ):, h(x1 ) δ(x1 − x2 ) · · · δ(x1 − xl )x1 ,...,xl affords a definition of the Moyal–Wick monomials a` la Wightman and G˚arding. Using now (5.4) together with (5.3) and (5.5), we obtain the completely explicit formula on the boson Fock space (n) (2π )Nl ϕ l (x) (p1 , . . . , pn ) l i 1 = ··· P eix(η1 +···+ηj −ηj +1 −···−ηl ) e∓ 2 m j , the + sign otherwise. In the simplest instance, we get

612


(n) (2π )3N ϕ 2 (x), h(x) (k1 , . . . , kn ) ˆ 1 + κ2 ) cos 1 κ1 κ2 (n+2) (κ1 , κ2 , k1 , . . . , kn ) d 2N κ1 d 2N κ2 = h(κ 2 +

n

i

i

ˆ − kj )e 2 κkj + h(k ˆ j − κ)e 2 kj κ h(κ

j =1

×(n) (κ, k1 , . . . , kj , . . . , kn ) d 2N κ (n−2) 1 ˆ + (k1 , . . . , kj , . . . , kl , . . . , kn ). h(−k j − kl ) cos 2 kj kl 1≤j =l≤n

(We underline again that the :ϕ(x1 ) . . . ϕ(xl ):, here as in (5.2), are the usual commutative boson products of fields, with all creation operators to the left of the annihilation operators —see [9, Sec. 4.1]— as for instance in (2π )3N :ϕ(x1 )ϕ(x2 )ϕ(x3 ): i(k x +k x +k x ) = ··· e 1 1 2 2 3 3 a(k1 )a(k2 )a(k3 ) + e−i(k1 x1 −k2 x2 −k3 x3 ) ×a † (k1 )a(k2 )a(k3 ) +ei(k1 x1 −k2 x2 +k3 x3 ) a † (k2 )a(k1 )a(k3 ) + ei(k1 x1 +k2 x2 −k3 x3 ) a † (k3 )a(k1 )a(k2 ) +e−i(k1 x1 +k2 x2 −k3 x3 ) a † (k1 )a † (k2 )a(k3 ) + e−i(k1 x1 −k2 x2 +k3 x3 ) a † (k1 )a † (k3 )a(k2 ) +ei(k1 x1 −k2 x2 −k3 x3 ) a † (k2 )a † (k3 )a(k1 ) + e−i(k1 x1 +k2 x2 +k3 x3 ) a † (k1 )a † (k2 )a † (k3 ) ×

3

d 2N ki .

i=1

In turn we are assured that ϕ l (x) is normally ordered. Had we tried in (5.5) to use the operator product instead of the normal product, we would have been punished by extra divergent terms of the type δ(k1 − k2 ) d 2N k1 d 2N k2 , just as in the commutative case. Thus, as anticipated, the twisted product does not help with the ordering problem.) The previous formulae have been obtained under the assumption that det > 0. For k = 2N , the set of nonsingular skewsymmetric k × k matrices is open and dense in the set of all skewsymmetric k × k matrices, and the same formulae are valid when det = 0 by continuity. We also conclude their validity in the case that k is odd, by consideration of an extra dimension with trivial commutation relations. 6. The Functional Action The functional action plays a great role in the applications to physics of noncommutative geometry, because it reproduces not only the Yang–Mills action but also the full Yang– Mills–Higgs [55, 91] and even, in its more general incarnation [13], the Einstein–Hilbert action. Here we choose, for the reasons indicated in the introduction, to compute the Connes–Lott action [21, 22, 16, 63], which notionally is Tr + (F 2 D / −2N ), for F the field strength or curvature associated to a vector potential α. Due to some ambiguity in the transition to F from α, unimportant for the general theory but crucial for physics, we need to deal with “junk”: that is, to quotient by an ideal living in the representation πD of


613

the universal differential algebra on H. Then we show that the action coincides with the noncommutative Yang–Mills action currently used in Moyal gauge theory. Of course, physicists have not waited for formal developments of this kind before forging ahead (see [42], for instance); but for our purposes this is an indispensable check. We first make some necessary remarks on the bimodule nature of the image of πD . 6.1. Connes–Terashima fermions. That bimodule nature is completely familiar to customers of Connes’ noncommutative geometry, and basically means that H∞ can sustain a bimodule action of two algebras. The reconstruction of the Standard Model Lagrangian in [18] uses actions of this type, exchanged by the charge conjugation operator. Independently, in the traditional context of Lie algebras, Terashima [83] summarized to similar effect some natural methods and restrictions, that were scattered in practice, to introduce noncommutative gauge fields. First of all, assume an infinitesimal gauge variation given by δλ Aµ (x) = ∂µ λ(x) − i[Aµ , λ]θ (x), or explicitly, δλ Aµ (x) = T a ∂µ λa (x) −

i 2

[T a , T b ] {Aaµ , λb }θ (x) + {T a , T b } [Aaµ , λb ]θ (x) ,

where the T a denote the gauge “group” generators, normalized as Tr(T a T b ) =

1 ab 2δ ,

closing to a Lie algebra [T a , T b ] = i fcab T c , and [·, ·]θ , {·, ·}θ denote the Moyal commutator and anticommutator brackets, respectively. Let us think of the Lie algebra of SU (n), to fix ideas. Then {T a , T b } =

1 ab δ + dcab T c , n

where the dcab are totally symmetric and real. This is not a linear combination of T d ’s. Therefore noncommutative gauge transformations are consistent only for unitary groups (there are some ways round this obstacle; but they are not very appealing). But then the gauge group of unitary transformations is identified to the unitary endomorphism group of a module, and we are back in Connes’ context. The second remark by Terashima is that the same closure requirement and consideration of the covariant derivative forces the representation of U (n) to be fundamental or antifundamental. It is possible, however, for a gauge group to act from the left, say in the fundamental representation, and (perhaps a different one) from the right in the antifundamental one, with gauge transformations given by ∗ . → U(1) θ θ U(2)

Again, this is completely natural in the context of algebra bimodules. We remark that already the chiral anomaly for these fermions has been calculated [62]. We want to add that, even in the context of pure group theory, the concept of bimodule is called for. We formalize this remark, in the spirit of [96]. By definition, a linear space V is a (G, H )-bimodule if it carries a left action $ of the group G (with the usual

614


continuity or smoothness conditions) and a right action % of the group H which are compatible, that is to say, g $ (v % h) = (g $ v) % h, for g ∈ G, h ∈ H , v ∈ V . A bimodule is irreducible if there is no proper subspace of V stable under both actions. If V is a G left-module and W is an H right-module, then V ⊗ W is a (G, H )-bimodule. When G = H , interesting bimodules usually have a conjugation operator that exchanges the actions. In that case, the bimodule is a very canonical object in harmonic analysis: the space of functions on a group G is a (G, G)bimodule; and if G possesses a representation on a space W , then V = End W is a (G, G)-bimodule; but, strangely enough, it does not seem to be in use. 6.2. The differential algebra. In the Connes–Lott approach one works with the tensor product of some finite dimensional Eigenschaften algebra and a spacetime algebra (that here is no longer commutative). We disregard the Eigenschaften algebra in what follows; in other words, we concentrate on the analytical details of the U (1) Moyal gauge theory. In this subsection we do not need to consider the preferred unitization of Aθ . As in [17, 21, 22], let • Aθ := p∈N p Aθ be the universal differential graded algebra over Aθ , where p Aθ := { f0 δf1 . . . δfp : fi ∈ Aθ } and the only constraint on δ is to satisfy the Leibniz rule δ(f1 θ f2 ) = δf1 f2 + f1 δf2 so δ can be extended on • Aθ . Since Aθ has no unit, we define [17, III.1.α] 0 Aθ := Aθ ⊕ C, which is the minimal unitization of Aθ , and δ(0 ⊕ 1) := 0. Moreover (δf )∗ := δf ∗ . The representation π θ of Aθ by elements of L(H) extends naturally to • Aθ , by π˜ θ : p Aθ → L(H) : f0 δf1 . . . δfp → i p π θ (f0 ) [D / , π θ (f1 )] . . . [D / , π θ (fp )]. Lemma 6.1. If fi ∈ Aθ , then π˜ θ (f0 δf1 . . . δfp ) = Lθ (f0 θ ∂µ1 f1 θ · · · θ ∂µp fp ) ⊗ γ µ1 . . . γ µp . Proof. This follows from [D / , Lθf ⊗12N ] = −iLθ (∂µ f )⊗γ µ and Lθf Lθg = Lθ (f θ g).

To overcome the unfaithfulness of π˜ θ (even if π θ is faithful), one introduces a graded p p−1 p 2-sided ideal of • Aθ , namely Junk := p∈N J p = p∈N J0 + δJ0 , J0 := { ω ∈ p θ Aθ : π˜ (ω) = 0 }, and finally D/ Aθ := π˜ θ (• Aθ )/π˜ θ (Junk). Here, the 2-junk is particularly simple since it is isomorphic to π θ (Aθ ), as we now show. Proposition 6.2. There is a natural identification π˜ θ (J 2 ) π θ (Aθ ) = Lθ (Aθ ) ⊗ 12N . Proof. Any ω ∈ π˜ θ (J 2 ) ⊂ π˜ θ (2 Aθ ) can be written as ω = j ∈I Lθ (∂µ fj )Lθ (∂ν gj )⊗ γ µ γ ν , where I is a finite set, and satisfies j ∈I Lθ (fj θ ∂µ gj )⊗γ µ = 0. By the Leibniz rule, ω= Lθ (∂µ (fj θ ∂ν gj ) − fj θ ∂µ ∂ν gj ) ⊗ γ µ γ ν = − Lθ (fj θ ∂µ ∂ν gj ) ⊗ γ µ γ ν j ∈I

=−

j ∈I

L (fj θ ∂µ ∂ν gj ) ⊗ η θ

µν

12 N .

j ∈I

Hence π˜ θ (J 2 ) ⊂ π θ (Aθ ) = Lθ (Aθ ) ⊗ 12N .


615

Consider ωmnkl := fmk δfkn − fml δfln (no summation) in 1 Aθ . In Subsect. 8.1 of the Appendix, it is shown that π˜ θ (ωmnkl ) = 0 and π˜ θ (δωmnkl ) = θ2 N j =1 (kj − θ θ lj ) L (fmn ) ⊗ 12N , which is nonzero if |l| = |k|. Thus, L (fmn ) ⊗ 12N lies in π˜ θ (J 2 ) for all m, n ∈ NN . Since {fmn } is a basis for Aθ , we conclude that π θ (Aθ ) Lθ (Aθ ) ⊗ 12N ⊂ π˜ θ (J 2 ). It is easy to generalize the above proof, to get the next corollary. Corollary 6.3. For p ≥ 2, π˜ θ (J p ) is the linear span of the elements in π˜ θ (p Aθ ) of the form Lθf ⊗ γ µ1 . . . γ µk , with k ≤ p − 2 and of the same parity as p. p be the Hilbert space obtained by completion of π˜ θ (p Aθ ) 6.3. The action. Let H under the scalar product / 2 + ε 2 )−N , π˜ θ (ω) | π˜ θ (ω )p := Tr + π˜ θ (ω)∗ π˜ θ (ω ) (D for ω, ω ∈ p Aθ . This defines a natural pre-action I (η) when p = 2 and ω = ω = δη + η2 : (6.1) / 2 + ε 2 )−N . I (η) := Tr + π˜ θ (ω)∗ π˜ θ (ω) (D p whose range is the orthogonal complement of Let P be the orthogonal projector on H p−1 p . Then P extends the quotient map from π˜ θ (p Aθ ) π˜ θ (δJ0 ), and define Hp := P H p onto D/ Aθ , which is identified with a dense subspace of Hp . The possible ambiguity in (6.1) due to the unfaithfulness of π˜ θ disappears if we define the functional action (noncommutative Yang–Mills action) as: Y M(α) :=

N! (2π)N P π˜ θ (F ) | P π˜ θ (F )2 , 8g 2

(6.2)

where 1D/ Aθ ( α = π˜ θ (η) and F = δη + η2 is the curvature of the 1-form η and g is the coupling constant. It is shown in [21, 91] that Y M(α) is equal to the infimum of the preaction on all η ∈ 1 Aθ with the same image in 1D/ Aθ : Y M(α) =

N! (2π)N inf{ I (η) : π˜ θ (η) = α }. 8g 2

This result justifies the notation Y M(α), because this positive quartic functional of η depends only on its equivalence class in 1D/ Aθ , namely α. Theorem 6.4. Let η = −η∗ ∈ 1 Aθ . Then the Yang–Mills action Y M(α) of the universal connection δ + η, with α = π˜ θ (η), is equal to 1 1 Y M(α) = − 2 F µν θ Fµν (x) d 2N x = − 2 F µν (x) Fµν (x) d 2N x, 4g 4g where Fµν := 21 (∂µ Aν − ∂ν Aµ + [Aµ , Aν ]θ ) and Aµ is defined by α = Lθ (Aµ ) ⊗ γ µ .

616


fj δgj for some fj , gj ∈ S and a finite set I , then α = j ∈I Lθfj Lθ∂µ gj ⊗ γ µ = j ∈I Lθ (fj θ ∂µ gj ) ⊗ γ µ . Thus Aµ := j ∈I fj θ ∂µ gj and, with a sum over j, k ∈ I understood, Proof. If η =

j ∈I

π˜ θ (δη + η2 ) = π˜ θ (δfj δgj + (fj δgj )(fk δgk )) = π˜ θ (δfj δgj + fj δ(gj θ fk )δgk − (fj θ gj ) δfk δgk ) = Lθ (∂µ fj θ ∂ν gj + fj θ ∂µ (gj θ fk )θ ∂ν gk −fj θ gj θ ∂µ fk θ ∂ν gk ) ⊗ γ µ γ ν = Lθ (∂µ fj θ ∂ν gk + fj θ ∂µ gj θ fk θ ∂ν gk ) ⊗ γ µ γ ν = Lθ (∂µ (fj θ ∂ν gj ) + fj θ ∂µ gj θ fk θ ∂ν gk ) ⊗ γ µ γ ν −Lθ (fj θ ∂µ ∂ν gj ) ⊗ ηµν 12N = Lθ (∂µ Aν + Aµ θ Aν ) ⊗ 21 [γ µ , γ ν ] + ηµν Lθ (∂µ Aν +Aµ θ Aν ) ⊗ 12N − ηµν Lθ (f θ ∂µ ∂ν g) ⊗ 12N . The two last terms are in π˜ θ (J 2 ). Thus, P (π˜ θ (F )) = P Lθ (∂µ Aν + Aµ θ Aν ) ⊗ 21 [γ µ , γ ν ] = P Lθ ( 21 (∂µ Aν − ∂ν Aµ + [Aµ , Aν ]θ ) ⊗ γ µ γ ν = P (Lθ (Fµν ) ⊗ γ µ γ ν ) = Lθ (Fµν ) ⊗ γ µ γ ν , where the last equality follows because the junk affects only the scalar part of π˜ θ (• Aθ ). To repeat: each ω = ωµν ⊗ γ µ γ ν ∈ π˜ θ (2 Aθ ) can be uniquely decomposed as ω = ωµν ⊗ 21 (γ µ γ ν − γ ν γ µ ) + ωµν ⊗ 21 (γ µ γ ν + γ ν γ µ ) in π˜ θ (2 Aθ )a ⊕ π˜ θ (2 Aθ )s = π˜ θ (2 Aθ )a ⊕ π˜ θ (J 2 ), the direct sum of its alternating and symmetric parts. ∗ = −F and therefore P (π Since Aµ = −A∗µ , we also find Fµν ˜ θ (F ))∗ = Lθ (Fµν )⊗ µν µ ν γ γ . Then Tr + Lθ (Fµν θ Fρσ ) (− + ε 2 )−N ⊗ γ µ γ ν γ ρ γ σ = Tr + Lθ (Fµν θ Fρσ ) (− + ε 2 )−N Tr(γ µ γ ν γ ρ γ σ ). But Tr(γ µ γ ν γ ρ γ σ ) = 2N (ηµν ηρσ − ηµρ ηνσ + ηµσ ηνρ ), so since Fµν = −Fνµ Proposition 4.17, computed with −i∇ instead of D / , yields 2 N! (4π)N + θ Tr L (Fµν θ F µν ) (− + ε 2 )−N 8g 2 1 = − 2 (Fµν θ F µν )(x) d 2N x, 4g

Y M(α) = −

and according to Lemma 2.1(v) the pointwise product can replace the Moyal product. Remark 6.5. The action as we have defined it is positive definite, since 2N 1 |Fµν (x)|2 d 2N x. Y M(α) = 2 4g µ,ν=1


617

7. Conclusions and Outlook We have shown in detail how to build noncompact noncommutative spin geometries. As a consequence, the classical background of present-day NCFTs is recast in the framework of the rigorous Connes formalism for geometrical noncommutative spaces. One can wonder about the uniqueness of the constructions presented here. Our detailed scrutiny shows that appropriate algebras for the spectral triples are to a large extent “selected” by the Dirac operator itself. The choice of A = S for the original nonunital algebra, made in the flat space cases, has much to recommend it, not least Fourier invariance and the existence of a body of tempered distribution analysis. However, an outcome of the study in this paper is that, both in the commutative and the Moyalalgebra example, a more canonical ‘arrival’point is the bigger algebra A1 := DL2 (R2N ); we found that nearly everything that works for A works also for A1 , with significant improvement of the finiteness axiom; also, A1 yields the most advantageous framework for quantization. We can accommodate an A1 -triple instead of an A-triple, provided we make a slight H, D, J, χ ), modification of the summability axiom. That is, we use the data (A1 , A, and the suggested new version of the noncompact noncommutative geometry postulates runs as follows. 1. Spectral dimension, 2nd version: There is a unique nonnegative integer k, the spectral or “classical” dimension of the geometry, for which a(|D| + ε)−1 belongs to the Schatten class Lp for p > k whenever a ∈ A1 , for any ε > 0; and moreover, for a in a dense ideal A of A1 , a(|D|+ ε)−1 lies in the generalized Schatten class Lk+ and the trace a → Tr + (a(|D| + ε)−k ) is finite and not identically zero. This k is even if and only if the spectral triple is even. 3. Finiteness, 2nd version: are pre-C ∗ -algebras. The space of The algebras A1 and its preferred unitization A ∞ smooth vectors H is the A1 -pullback of a finite projective A-module. Moreover, an A1 -valued hermitian structure (· | ·) is implicitly defined on H∞ with the noncommutative integral, as follows: Tr + (aξ | η)(|D| + ε)−k = η | aξ , and · | · denotes the standard inner product on H. where a ∈ A In the other postulates A is replaced by A1 ; they are otherwise unchanged. In our case A could be taken equal to S or larger: Tr + (a(|D| + ε)−k ) < ∞ is valid for a belonging to a larger ideal of DL2 . Support for enrollment of A1 comes from physics, on one hand, and abstract nonsense, on the other. Langmann and Mickelsson [58] found existence of the quantum scattering matrix for quantized fermions in external gauge potentials with components precisely in the sibling N of DL2 ; this is a both strong and significant result. Also, as exploited in Sect. 5, the more correct and general approach to the construction of Wick monomials makes use precisely of the smooth domain of the Dirac operator. The close relation of A1 to this smooth domain points to generalizations of the pseudodifferential calculus in the fully noncommutative context [50, 58]. The orientation condition and the required boundedness of the operators [D, a] give rather tight lower and upper bounds (so to speak) on what the preferred compactification of A1 should be. It would be good to know whether these two conditions determine such a unitization uniquely. The following conjecture is strengthened by the result of [64].

618


= B(R2N ) is the largest Moyal multiplier algebra of A1 = DL2 (R2N ) Conjecture 7.1. A such that [D / , a] is bounded for each a ∈ A. The clever argument in [11] leads one to ponder what kind of boundary conditions one would impose on D / (without presumably changing the leading term behaviour of its spectral density) in order to obtain a compact spectral triple canonically associated to the given noncompact one. This should allow the anomaly calculations in [42, 62] to be made more rigorous. The subject of noncommutative manifolds with boundary is still in its infancy, however, and we shall not elaborate the point. Apart from eventually proving a reconstruction theorem (a rather strenuous task), much remains to be done. There are probably already enough examples of noncommutative spaces around for consideration of the “category” of spectral triples to be promising. For instance, NC tori are quotients of the spaces considered in this paper. A mathematically important question is the computation of the Hochschild cohomology of θ . Another is the explicit lifting of (a central extension of) the group of (nonlinear, in A general) symplectomorphisms (or at least, of those connected to the identity) to a group of inner automorphisms of M θ (or of Aθ ), which should be irreducibly represented on H. In this context, work on the geometry of the gauge algebra in noncommutative Yang–Mills theories [60] can be pursued. 8. Appendix: A Few Explicit Formulas 8.1. On the oscillator basis functions. For N = 1 and m, n ∈ N, the basic eigentransition fmn (x1 , x2 ) is explicitly given by √ n!m! min(m,n) 2(−1) (4H1 /θ )|m−n|/2 ei(n−m) arctan(x2 /x1 ) max(m, n)! |m−n|

× exp(−2H1 /θ ) Lmin(m,n) (4H1 /θ ), with Lrj being the generalized Laguerre polynomials of order j and H1 = 21 (x12 + x22 ). In general, fmn (x1 , . . . , xN ) = fm1 n1 (x1 , x1+N ) . . . fmN nN (xN , x2N ). Also, using the coalgebra formula for the Laguerre polynomials (u + v) = Lrj (u)Lsl (v), Lr+s+1 n j +l=n

one obtains [7] eigenstates for H = H1 + · · · + HN : N H θ fM = fM θ H = θ M + fM , 2 where fM (x1 , . . . , x2N ) :=

fm1 m1 . . . fmN mN (x1 , . . . , x2N )

|m|=M

= 2N (−)M exp(−2H /θ ) LN−1 M (4H /θ ). √ It is known that |fnn (x1 , x2 )| dx1 dx2 ∼ n as n → ∞. From this, using the closed graph theorem, it easy to show that there are non-absolutely integrable functions in I00 [28].


619

8.2. More junk Lemma 8.1. For m, n, k, l ∈ NN , let ωmnkl := fmk δfkn − fml δfln ∈ 1 Aθ (no summation on k or l). Then π˜ θ (ωmnkl ) = 0 and π˜ θ (δωmnkl ) = θ2 (|k| − |l|) Lθ (fmn ) ⊗ 12N . Proof. Using the creation and annihilation functions (2.11) we may rewrite the Dirac operator as follows; we adopt the convention that j = 1, . . . , N, and write ∂aj = ∂/∂aj and ∂aj∗ = ∂/∂aj∗ : ∗ i j D / = −√ γ (∂aj + ∂aj∗ ) + iγ j +N (∂aj − ∂aj∗ ) = −i (γ aj ∂aj + γ aj ∂aj∗ ), 2 j j ∗

where γ aj := √1 (γ j + iγ j +N ) and γ aj := √1 (γ j − iγ j +N ). 2 2 Lemma 2.1(iv), applied to aj and aj∗ respectively, yields 1 1 1 1 ∂aj = − adθ aj∗ := − [aj∗ , · ]θ , ∂aj∗ = adθ aj := [aj , · ]θ , θ θ θ θ and hence i aj∗ (γ adθ aj − γ aj adθ aj∗ ). D / =− θ j

Let uj := (0, 0, . . . , 1, . . . , 0) be the j th standard basis vector of RN . From the definition (2.10) of fmn , we directly compute: ! aj∗ θ fmn = θ (mj + 1) fm+uj ,n , fmn θ aj∗ = θnj fm,n−uj , ! aj θ fmn = θ mj fm−uj ,n , fmn θ aj = θ(nj + 1) fm,n+uj . Consequently, ! i aj γ θ nj fm,n−uj − θ(mj + 1) fm+uj ,n θ j ! aj∗ +γ θ mj fm−uj ,n − θ (nj + 1) fm,n+uj .

D / (fmn ) = −

(8.1)

We are now able to compute π˜ θ (ωmnkl ) and π˜ θ (δωmnkl ). Firstly, π˜ θ (ωmnkl ) = π˜ θ (fmk δfkn − fml δfln ) = Lθ (fmk θ ∂µ fkn − fml θ ∂µ fln ) ⊗ γ µ ! 1 = θ nj Lθ (fmk θ fk,n−uj ) − θ(kj + 1) Lθ (fmk θ fk+uj ,n ) θ j ! − θ nj Lθ (fml θ fl,n−uj ) + θ (lj + 1) Lθ (fml θ fl+uj ,n ) ⊗ γ aj ! + θ kj Lθ (fmk θ fk−uj ,n ) − θ (nj + 1) Lθ (fmk θ fk,n+uj ) ! ∗ − θ lj Lθ (fml θ fl−uj ,n ) + θ (nj + 1) Lθ (fml θ fl,n+uj ) ⊗ γ aj = 0.

620


Secondly, we calculate that π˜ θ (δωmnkl ) = π˜ θ (δfmk δfkn −δfml δfln ) = Lθ (∂µ fmk θ ∂ν fkn −∂µ fml θ ∂ν fln )⊗γ µ γ ν equals

! 1 θ θ k L (f ) − θ (mj + 1) Lθ (fm+uj ,k ) ⊗ γ aj j m,k−uj 2 θ j ! ∗ + θ mj Lθ (fm−uj ,k ) − θ (kj + 1) Lθ (fm,k+uj ) ⊗ γ aj ! × θ np Lθ (fk,n−up ) − θ (kp + 1) Lθ (fk+up ,n ) ⊗ γ ap p

! ∗ θ kp Lθ (fk−up ,n ) − θ (np + 1) Lθ (fk,n+up ) ⊗ γ ap ! − θ lj Lθ (fm,l−uj ) − θ (mj + 1) Lθ (fm+uj ,l ) ⊗ γ aj

+

j

! ∗ θ mj Lθ (fm−uj ,l ) − θ (lj + 1) Lθ (fm,l+uj ) ⊗ γ aj ! θ np Lθ (fl,n−up ) − θ (lp + 1) Lθ (fl+up ,n ) ⊗ γ ap ×

+

p

+

θ lp L (fl−up ,n ) − θ

!

θ (np + 1) L (fl,n+up ) ⊗ γ θ

ap∗

.

Using the elementary properties of the fmn from Lemma 2.4, this simplifies to ∗ ∗ 1 kj Lθ (fmn ) ⊗ γ aj γ aj + (kj + 1)Lθ (fmn ) ⊗ γ aj γ aj θ j ∗ ∗ −lj Lθ (fmn ) ⊗ γ aj γ aj − (lj + 1)Lθ (fmn ) ⊗ γ aj γ aj ∗ ∗ 1 = Lθ (fmn ) ⊗ (kj − lj ) (γ aj γ aj + γ aj γ aj ) θ

π˜ θ (δωmnkl ) =

j

2 = (kj − lj ) Lθ (fmn ) ⊗ 12N . θ

j

Acknowledgements. We thank A. Connes, K. Fredenhagen, H. Grosse, F. Lizzi, C. P. Mart´ın, M. Puschnigg, M. Rieffel, A. Schwarz and A. Wassermann for suggestions and/or helpful discussions, and G. Rozenblum and T. Weidl for correspondence on matters pertaining to the subject of this paper. The work of JMGB and JCV was supported by the Vicerrector´ıa de Investigación and the Facultad de Ciencias of the Universidad de Costa Rica. VG and JCV are grateful to Vanderbilt University for providing a splendid occasion and nice surroundings for discussions. JMGB also thanks the Université de Provence, and JCV thanks the Abdus Salam ICTP, for their customarily excellent hospitality during various stages of this work.

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Communications in


An Algebraic Characterization of Vacuum States in Minkowski Space. III. Reflection Maps Detlev Buchholz1 , Stephen J. Summers2 1 2

Institut für Theoretische Physik, Universität Göttingen, 37077 Göttingen, Germany Department of Mathematics, University of Florida, Gainesville, FL 32611, USA

Received: 10 September 2003 / Accepted: 15 October 2003 Published online: 5 March 2004 – © Springer-Verlag 2004

Abstract: Employing the algebraic framework of local quantum physics,vacuum states in Minkowski space are distinguished by a property of geometric modular action. This property allows one to construct from any locally generated net of observables and corresponding state a continuous unitary representation of the proper Poincaré group which acts covariantly on the net and leaves the state invariant. The present results and methods substantially improve upon previous work. In particular, the continuity properties of the representation are shown to be a consequence of the net structure, and surmised cohomological problems in the construction of the representation are resolved by demonstrating that, for the Poincaré group, continuous reflection maps are restrictions of continuous homomorphisms. 1. Introduction A basic conceptual problem in local quantum physics [12] is the determination of the spacetime symmetry, causality and stability properties of a theory from the structure of the observable algebras associated with spacetime regions. Within this general setting, it seems unnatural to appeal from the outset to symmetry properties of a theory — such as the action of a spacetime isometry group upon the states and observables — which are absent in generic spacetimes. Instead, the pertinent notions characterizing specific physical systems ought to be based on the states and observables of the system, and possible symmetry and stability properties should be deduced, not posited. Light was shed on these matters by a condition of geometric modular action (CGMA), proposed in [4, 7] and briefly recalled in Sect. 4, which is designed to characterize those elements in the state space of a quantum system which admit an interpretation as a “vacuum”. This condition is expressed in terms of the modular conjugations associated to any given family of algebras paired with suitable subregions (wedges) of the underlying space–time and any states by the Tomita–Takesaki modular theory, cf. [2, 13]. It thereby can be applied, in principle, to theories on any space–time manifold. For a

626

D. Buchholz, S.J. Summers

motivation of this condition and applications to theories in Minkowski, de–Sitter, anti-de Sitter and a class of Robertson–Walker space–times, we refer the interested reader to [7, 5, 8, 6]. In the present article we revisit the case of Minkowski space theories and resolve some intriguing questions left open by our previous work in [7, 5]. The basic ingredients in that investigation are an isotonous map (henceforth, a net) W → R(W ) from the family of wedge-shaped regions W ⊂ R4 , bounded by two characteristic planes, to von Neumann algebras R(W ) on a Hilbert space H, and a state vector ∈ H complying with the CGMA. The modular conjugation associated to any given pair R(W ), was shown to have the geometrical meaning of a reflection λ about the edge of W . More precisely, denoting this conjugation by J (λ), one has for any wedge W0 the equation [7] J (λ)R(W0 )J (λ) = R(λW0 ) .

(1.1)

This implies, in particular, that J (λ) is also the modular conjugation of the pair R(W ), , where W = λW denotes the causal complement of W ; so the net satisfies wedge duality. Moreover, one has J (λ)J (λ0 )J (λ) = J (λλ0 λ) ,

(1.2)

in an obvious notation. As the reflections λ generate the proper Poincaré group P+ , these relations lead naturally to the question of whether products of the conjugations J (λ) generate an (anti) unitary representation of P+ which acts covariantly on the net. This question was answered in the affirmative in [7, 11] under a technical assumption of net continuity. It follows from that assumption that the map λ → J (λ)

(1.3)

from the reflections in P+ into the group of (anti)unitary operators is continuous. This information, together with relation (1.2), implies that there is a continuous projective representation of P+ on H with coefficients in the center of the group J generated by all conjugations J (λ). By an application of Moore cohomology theory, this projective representation lifted to a true representation and the center of J turned out to be trivial [7]. These latter results suggested that viewing the problem cohomologically was misleading and obscured the presence of an extremely rigid structure encoded in the modular conjugations. It was also desirable to clarify the conceptual status of the technical assumptions underlying the crucial continuity property of the map (1.3). A first step towards the clarification of these points was taken in [5]. There it was shown without any a priori continuity assumptions that a continuous unitary representation of the subgroup of translations acting covariantly on the net can be constructed from the modular conjugations. In the present investigation we want to extend this result to the full proper Poincaré group P+ . We shall restrict our attention here to nets W → R(W ) which are locally generated, a case of particular interest being the situation where each R(W ) is the inductive limit of algebras R(C) associated to double cones C ⊂ W . In fact, this condition is already satisfied if is cyclic for the algebras R(C) and satisfies, in addition to the CGMA, a modular stability condition (CMS), recalled in Sect. 4, which was proposed in [7] for the characterization of stable states. We shall show under these latter two conditions that the map (1.3) provided by the CGMA is continuous. Our second result clarifies the nature of continuous maps (1.3) of reflections λ ∈ P+ into the topological group J , which satisfy the basic relations J (λ)2 = 1

Algebraic Characterization of Vacuum States in Minkowski Space. III

627

and J (λ)J (λ0 )J (λ) = J (λλ0 λ) for any pair of reflections λ, λ0 ∈ P+ . We shall show that such reflection maps are restrictions of continuous homomorphisms from P+ into J . Phrased differently, any reflection map can be extended uniquely to a true continuous (anti)unitary representation U of P+ which, in view of (1.1), acts covariantly upon the net. Thus the outcome of the present investigation is the insight that any vector which is cyclic for the local algebras and complies with the CGMA and the CMS is a vacuum state which is invariant under a continuous (anti)unitary representation of P+ acting covariantly on the net. No further assumptions are needed for the proof of this result. That this arises in the manner shown here provides further evidence that the modular involutions are fundamental objects encoding crucial physical data, which include the causal structure of the theory, its dynamics, and the action of the isometry group upon the observables. Even the space–time itself can be found to be encoded in the modular involutions in certain cases [16]. Our paper is organized as follows. In the next section we consider continuous reflection maps from the proper Lorentz group into an arbitrary topological group and show that they are restrictions of continuous homomorphisms. The continuity of the reflection maps arising in the present context is established in Sect. 3. These results are combined in Sect. 4 with theorems we have previously established to yield the desired characterization of Poincaré covariant vacuum states in Minkowski space in terms of the modular objects. 2. Reflection Maps and Homomorphisms In this section we study continuous reflection maps from the proper Lorentz group into an arbitrary topological group. We shall show that any such map is the restriction of a continuous homomorphism. Combining this information with results obtained in [5], this feature can be established also for reflection maps on the proper Poincaré group. In fact, similar results hold for many other groups, suggesting the possibility of a general theorem about reflection maps. This would be of interest in the general context of the CGMA, where reflection maps appear naturally [7], and in the application of the CGMA to other space–times. But we shall not address the general problem here. ↑

2.1. Group theoretical considerations. Let L+ be the proper Lorentz group and L+ be its orthochronous subgroup. Fixing a Lorentz system with proper coordinates (x0 , x) ∈ R4 and metric in diagonal form g = diag(1, −1, −1, −1), one can uniquely decompose any ↑ ∈ L+ into a rotation R in the time-zero plane and a boost (velocity transformation) B, = RB.

(2.1)

Remark. This formula is simply the polar decomposition of in the space M(4, R) of real four–by–four matrices. In particular, any Lorentz transformation which is represented by a positive matrix is a boost. This well–known fact recently received some attention again, cf. [14, 17]. ↑

Thus, any ∈ L+ generically fixes two spatial directions, the axis of revolution r of R and the boost direction b of B. We adopt the convention that r, b are normalized

628


and that rotations are performed by an angle less than or equal to π about r in the counterclockwise direction. So r is fixed unless R = 1 or R is a rotation through the angle π, where r is fixed only up to a sign. Similarly, unless B = 1, the direction of b is fixed by the condition that the lightlike vector (1, b) is an eigenvector of B corresponding to its eigenvalue which is larger than 1. Making use of this convention, we want to show that any can be represented as the product of two reflections about the edges of suitable wedges. Although there are results in the mathematical literature which establish that any element of the Lorentz group can be written as the product of two involutions, the reflections we must employ are restricted to lie in a single conjugacy class of these involutions. We are therefore obliged to provide a proof of this fact here. We begin by defining for a given unit vector e ∈ R3 the wedge . We = {x ∈ R4 | x · e > |x0 |}

(2.2)

and the involution λe ∈ L+ inducing the reflection about its edge, i.e. λe (1, 0) = −(1, 0) ,

λe (0, e) = −(0, e) ,

λe (0, e⊥ ) = (0, e⊥ ) ,

(2.3)

where the latter equality holds for any e⊥ which is perpendicular to e. One finds that if b · e = 0 and B is any boost in the direction of b, then λe B = B −1 λe is a reflection about the edge of the boosted wedge B −1/2 We . Similarly, if r · e = 0 and R is any rotation about the direction of r, then R λe = λe R −1 is a reflection about the edge of R 1/2 We , where R 1/2 is the rotation about r through half the angle of R. Thus, choosing for given R, B the direction e such that r · e = b · e = 0, R λe as well as λe B are reflections about the edges of wedges and R λe λe B = RB = . In the following we shall call the involutions λ inducing reflections about the edges of wedges simply reflections, for short, and we shall denote by R the set of all such reflections. We have therefore just proved the following result which has been proven independently by Ellers [10] using a very different argument. Lemma 2.1. Every element of the proper orthochronous Lorentz group can be written ↑ as a product of two reflections, i.e. for every ∈ L+ there exist two elements λ1 , λ2 ∈ R such that = λ1 λ2 . This result is crucial in our investigation of reflection maps on the Lorentz group, and a similar result is likely to be just as important in any attempt to generalize our results to other groups. We refer the interested reader to the recent paper of Ellers [10] for a beginning of such a program. ↑ We shall discuss the ambiguities involved in this representation. Let ∈ L+ be given ↑ and let λ1 , λ2 ∈ R be reflections such that λ1 λ2 = . If ∈ L+ is any Lorentz transfor. mation commuting with , it is clear that the product of the reflections λ1 = λ1 −1 , . λ2 = λ2 −1 is equal to . Yet this may not be the only ambiguity in the choice of pairs of reflections corresponding to . Since λ1 2 = 1, one has λ2 = λ1 , and since λ2 2 = 1, one also has λ1 = −1 λ1 . Now if λ1 is another reflection satisfying the latter equation, one gets λ1 λ1 = λ1 λ1 , i.e. λ1 = λ1 , where commutes with . Moreover, as λ1 is an involution, must satisfy λ1 = −1 λ1 . We therefore consider for given and reflection λ1 as above the set of Lorentz transformations ↑

Λ = { ∈ L+ | λ1 = −1 λ1 , = } .

(2.4)


629

. . Given any ∈ Λ , the elements λ1 = λ1 and λ2 = λ1 are involutions, and their product is equal to . But these involutions are not always reflections. Nonetheless, if is such that for each ∈ Λ there is some 1/2 ∈ Λ whose square is , one has λ1 = −1/2 λ1 1/2 and λ2 = −1/2 −1/2 λ1 1/2 1/2 . So, in this case, λ1 and λ2 are both reflections of the form given above. ↑ In the following we shall focus our attention on certain specific elements ∈ L+ ↑ 2 which are of the form = 1 0 −1 1 , where 1 ∈ L+ is arbitrary and 0 (0 = 1) ↑ is an element of the stability group L0 ⊂ L+ of some fixed wedge We0 . We recall that ↑ L0 is the abelian subgroup of L+ generated by all rotations R0 about e0 and all boosts B0 in the direction of e0 . ↑

Remark. Disregarding three special cases, all conjugacy classes of L+ are of this form. ↑ This can be seen by proceeding to the covering group SL(2, C) of L+ and making use of the Jordan normal form of two–by–two matrices. One finds by explicit computation (most conveniently in the covering group) that the commutant of is equal to the abelian group 1 L0 −1 1 . Hence if e is such that e · e0 = 0 and if λ is the reflection about the edge of We , one obtains for the reflection . λ1 = 1 λ−1 1 about the edge of 1 We the equality λ1 = −1 λ1

∈ 1 L0 −1 1 .

(2.5)

Hence Λ = 1 L0 −1 1 in this case. Moreover, as L0 is stable under taking square roots, 1/2 ∈ L −1 . Hence for these there is also for each ∈ 1 L0 −1 1 0 1 1 a square root special elements we have complete control of their representation in terms of products of reflections. We summarize these results in the following lemma. Lemma 2.2. Let ∈ L0 −1 , 2 = 1, and let λ1 be the reflection about the edge of We , where e is orthogonal to e0 . Then λ2 = λ1 is a reflection and λ1 λ2 = . Moreover, any pair of reflections λ1 , λ2 with product arises from λ1 , λ2 by the adjoint action of some element of L0 −1 . 2.2. Reflection maps. Let J be a topological group. We consider maps λ → J (λ) from the set of reflections R ⊂ L+ into J . There is no loss of generality to assume that the subgroup generated by the set of elements {J (λ) | λ ∈ R} is dense in J . Definition 2.3. A map J : R → J is a reflection map if for every λ ∈ R the element J (λ) ∈ J is an involution and J (λ1 )J (λ2 )J (λ1 ) = J (λ1 λ2 λ1 ) ,

(2.6)

for all λ1 , λ2 ∈ R. We want to show that any continuous reflection map is the restriction of a continuous homomorphism V : L+ → J . ↑ For this to be true, it would be necessary to define for given ∈ L+ and corresponding pair of reflections λ1 , λ2 with λ1 λ2 = the element . V () = J (λ1 )J (λ2 ) . (2.7)

630


Yet it is a priori not clear whether this element (a) is independent of the choice of the pair of reflections into which is decomposed, (b) has the right continuity properties and (c) defines a homomorphism. We shall start by making specific choices of reflections for special and establish, step by step, properties (a)–(c) of V . In this discussion we make use of arguments and results in [7], which we shall recall here in somewhat modified form for the convenience of the reader. Let us consider first the action of V on rotations and boosts, R, B. To this end we choose a vector e which is orthogonal to the axis of revolution of R, respectively the boost direction of B; such vectors are called admissible in the following. Let λe , Rλe and Bλe be the reflections defined above. We then set . . Ve (R) = J (Rλe )J (λe ) , Ve (B) = J (Bλe )J (λe ) (2.8) and observe that Ve (R)−1 = J (λe )J (Rλe ) and Ve (B)−1 = J (λe )J (Bλe ). Note that because of relation (2.6), we also have Ve (R)J (λ)Ve (R)−1 = J (Rλe )J (λe )J (λ)J (λe )J (Rλe ) = J (RλR −1 ) ,

(2.9)

for every λ ∈ R, where we have used λe Rλe = R −1 . Similarly, we also have Ve (B)J (λ)Ve (B)−1 = J (BλB −1 ) .

(2.10)

Lemma 2.4. The elements Ve (B), Ve (R) defined above do not depend on the choice of the vector e within the above-stated limitations. Proof. Consider first the case of boosts. If B = 1, there is nothing to prove. So let B = 1, e be one of the admissible vectors for this boost, and let B1 be any boost in the same direction as that of B. Note that Ve (B1 )J (λe ) = J (B1 λe )J (λe )2 = J (B1 λe ) = J (λe )2 J (B1 λe ) = J (λe )Ve (B1 )−1 . Hence, for any n ∈ N, Ve (B1 )2n J (λe ) = Ve (B1 )n J (λe )Ve (B1 )−n = J (B1 n λe B1 −n ) = J (B1 2n λe ) , using (2.10). Consequently, one has Ve (B1 )2n = Ve (B1 )2n J (λe )2 = J (B1 2n λe )J (λe ) = Ve (B1 2n ) . Similarly, one sees that Ve (B1 )J (B1 λe ) = J (B1 λe )J (λe )J (B1 λe ) = J (B1 λe )Ve (B1 )−1 and therefore Ve (B1 )2n+1 = Ve (B1 )2n J (B1 λe )J (λe ) = Ve (B1 )n J (B1 λe )Ve (B1 )−n J (λe ) = J (B12n+1 λe )J (λe ) = Ve (B12n+1 ) . Thus, one has Ve (B1 )n = Ve (B1 n ), for all n ∈ N. Now let Rφ be a rotation by φ about the axis established by the direction of the boost B. Since Rφ and B1 commute, one obtains from relation (2.9), Ve (Rφ )Ve (B1 )Ve (Rφ )−1 = Ve (Rφ )J (B1 λe )J (λe )Ve (Rφ )−1 = J (B1 Rφ λe Rφ−1 )J (Rφ λe Rφ−1 ) = J (B1 λRφ e )J (λRφ e ) = VRφ e (B1 ) .


631

On the other hand, according to (2.10), the element VRφ e (B1 )Ve (B1 )−1 must commute with J (λ), for every λ ∈ R. Since J (R) generates J , this implies that there exists some element Zφ in the center of J such that VRφ e (B1 ) = Zφ Ve (B1 ) . Setting φ = 2mπ/n, for n ∈ N and m ∈ Z, one sees from the preceding two relations that n −n n n Ve (B1 ) = VR2mπ/n = Z2mπ/n Ve (B1 ) , e (B1 ) = Ve (R2mπ/n ) Ve (B1 )Ve (R2mπ/n ) n and consequently Z2mπ/n = 1. Hence, n VR2mπ/n e (B1n ) = VR2mπ/n e (B1 ) n = Z2mπ/n Ve (B1 )n = Ve (B1 )n = Ve (B1n ) ,

and setting B1 = B 1/n one obtains VR2mπ/n e (B) = Ve (B) , for all n ∈ N, m ∈ Z. By hypothesis, the reflection map is continuous, so the element Ve (Rφ ) depends continuously on φ for any admissible e, and the same is thus also true of VRφ e (B). It therefore follows from the preceding relation that VRφ e (B) = Ve (B) for any rotation Rφ , proving the assertion for the case of the boosts. For the rotations R, one proceeds in exactly the same way. The role of Rφ is now to be played by the rotations about the axis of revolution fixed by R.

In light of this result, we may omit the index e and set . . V (B) = Ve (B) , V (R) = Ve (R) .

(2.11)

Lemma 2.5. The elements V (B) and V (R) depend continuously on the boosts B and rotations R, respectively. Proof. Let {Bn }n∈N be a sequence of boosts converging to B. If B = 1 the distance between the unit disks parametrizing the corresponding orthogonal admissible vectors converges to 0. In particular, there exists a sequence of unit vectors en , admissible for Bn , converging to the unit vector e, admissible for B. Hence, the sequence {Bn en } converges to Be. By the assumed continuity of the reflection map, one concludes that V (Bn ) = J (Bn λen )J (λen ) converges to J (Bλe )J (λe ) = V (B) as n → ∞. If, on the other hand, the sequence {Bn } converges to 1, the corresponding unit disks need not converge. Nonetheless, due to the compactness of the unit ball in R3 , for any sequence of unit vectors en ∈ R3 there exists a subsequence {eσ (n) } which converges to some unit vector eσ . Since {Bσ (n) } converges to 1, the corresponding sequence {Bσ (n) eσ (n) } converges to eσ . One therefore has V (Bσ (n) ) = J (Bσ (n) λeσ (n) )J (λeσ (n) ) → J (λeσ )J (λeσ ) = 1 . Since the choice of sequence {en } was arbitrary, the proof of the continuity of V (B) with respect to the boosts B is complete. The argument for the rotations is analogous after the boost direction is replaced by the axis of revolution of the respective rotation.

Lemma 2.6. With the above definitions, one has the following: (1) V (R)V (B)V (R)−1 = V (RBR −1 ) for all boosts B and rotations R.

632


(2) V (·) defines a true representation of every continuous one-parameter subgroup of boosts or rotations. Proof. Statement (1) follows from relation (2.9) and Lemma 2.4, which imply (e being admissible for both the rotation R and the boost B) V (R)V (B)V (R)−1 = V (R)J (Bλe )J (λe )V (R)−1 = J (RBR −1 Rλe R −1 )J (Rλe R −1 ) = J (RBR −1 λR e )J (λR e ) = V (RBR −1 ) . The last equality follows from the fact that RBR −1 is again a boost whose direction is orthogonal to Re . ↑ Now let G : R → L+ be a continuous one-parameter group of boosts or rotations. As in the proof of Lemma 2.4, one shows by an elementary computation on the basis of relation (2.6) that V (G(u))n = V (G(u)n ) = V (G(nu)). Consequently, one finds that, for m1 , m2 , n ∈ N, V (G(m1 /n))V (G(m2 /n)) = V (G(1/n))m1 V (G(1/n))m2 = V (G(1/n))m1 +m2 = V (G((m1 + m2 )/n)) . As V (G(−u)) = V (G(u)−1 ) = V (G(u))−1 by (2.8) and Lemma 2.4, this relation extends to arbitrary m1 , m2 ∈ Z and n ∈ N. The stated assertion (2) thus follows once again from the continuity properties of V ( · ) established so far.

↑

Given any ∈ L+ we make use of its unique polar decomposition = RB and choose a direction e which is orthogonal to both the axis of revolution of R and the boost direction of B. We recall that the corresponding reflection λe satisfies both Rλe = λe R −1 and Bλe = λe B −1 . Hence Rλe = RBλe B and λe B are reflections, and we can define . V () = J (RBλe B)J (λe B) = J (Rλe )J (λe )2 J (λe B)J (λe )2 = J (Rλe )J (λe )J (Bλe )J (λe ) = V (R)V (B) , (2.12) where we made use of the properties of reflection maps. Since R, B depend continuously on , V () is continuous in as well. Moreover, V ()J (λ)V ()−1 = J (λ−1 ) ,

(2.13)

↑

for any λ ∈ R and ∈ L+ . ↑ Let L0 ⊂ L+ be the abelian stability group of any given wedge We0 . It is generated by two one-parameter subgroups: the rotations about the axis fixed by e0 and the boosts in the corresponding direction. It follows from the properties of V ( · ) established so far that for any = RB ∈ L0 one has V (R)V (B) = V (R)V (B)V (R)−1 V (R) = V (RBR −1 )V (R) = V (B)V (R) . (2.14) Hence for any 0 = R0 B0 ∈ L0 one obtains V (0 )V ()V (0 )−1 = V (R0 )V (B0 )V (R)V (B)V (B0 )−1 V (R0 )−1 = V () . (2.15) This fact puts us into the position of being able to prove that for a large set of Lo↑ rentz transformations ∈ L+ the corresponding V () do not depend on the choice of reflections in the decomposition of .


633

↑

Lemma 2.7. Let ∈ L0 −1 , 2 = 1, ∈ L+ . Then V () = J (λ1 )J (λ2 ) for any pair of reflections λ1 , λ2 satisfying λ1 λ2 = . Proof. In view of relation (2.13) it suffices to establish the statement for ∈ L0 , 2 = 1. Let λ1 , λ2 be reflections as in definition (2.12) such that = λ1 λ2 and V () = J (λ1 )J (λ2 ). If λ3 , λ4 are reflections such that λ3 λ4 = λ1 λ2 , there is by −1 Lemma 2.2 a 0 ∈ L0 such that λ3 = 0 λ1 −1 0 , λ4 = 0 λ2 0 . Hence by relations (2.13) and (2.15) one obtains J (λ3 )J (λ4 ) = V (0 )J (λ1 )J (λ2 )V (0 )−1 = V (0 )V ()V (0 )−1 = V () , proving the statement.

This result will greatly simplify the computations which will show that V ( · ) is a homomorphism. Let R1 , R2 be arbitrary rotations such that (R1 R2 )2 = 1 and let e be orthogonal to the axes of revolution of R1 , R2 . Taking into account the fact that any rotation is an element of the stability group L0 of some suitable wedge We0 , we obtain from the preceding lemma the equalities V (R1 )V (R2 ) = J (R1 λe )J (λe )J (R2 λe )J (λe ) = J (R1 λe )J (λe R2 ) = V (R1 R2 ) ,

(2.16)

and this equation extends by continuity to arbitrary pairs of rotations. Next, let B1 , B2 be arbitrary boosts. Then B1 B2 = B1 1/2 (B1 1/2 B2 B1 1/2 )B1 −1/2 , where the expression in brackets is a positive matrix and hence a boost. So B1 B2 belongs to the class of Lorentz transformations covered by the preceding lemma. Choosing e orthogonal to the boost directions of B1 , B2 , we therefore have V (B1 B2 ) = J (B1 λe )J (λe B2 ) = J (B1 λe )J (λe )2 J (λe B2 )J (λe )2 = V (B1 )V (B2 ) .

(2.17)

On the other hand, proceeding to the polar decomposition B1 B2 = RB we get by definition V (B1 B2 ) = V (R)V (B), and hence V (B1 )V (B2 ) = V (R)V (B) .

(2.18)

Now let 1 = R1 B1 , 2 = R2 B2 be arbitrary proper orthochronous Lorentz transformations. Introducing the boost B3 = R2−1 B1 R2 and making use of the polar decomposition B3 B2 = RB, we obtain from the preceding results the chain of equalities V (1 )V (2 ) = V (R1 )V (B1 )V (R2 )V (B2 ) = V (R1 )V (R2 )V (B3 )V (B2 ) = V (R1 R2 )V (R)V (B) = V (R1 R2 R)V (B) = V (R1 R2 RB) = V (1 2 ). (2.19) ↑

Thus V ( · ) is a continuous homomorphism from L+ into J . It remains to extend V ( · ) to the component of L+ which is disconnected from unity. To this end we fix a reflection λ0 ∈ L+ corresponding to some wedge We0 and note that all elements in the discon↑ nected part can be represented uniquely in the form λ0 , where ∈ L+ . We set . V (λ0 ) = J (λ0 )V () . (2.20) In view of the defining properties of reflection maps and the definition of V ( · ), we get J (λ0 )V ()J (λ0 ) = V (λ0 λ0 ) .

(2.21)

634


(Note that the sets of rotations and boosts are mapped onto themselves by the adjoint action of λ0 , and the set of distinguished wedges We is stable under the action of λ0 , as ↑ well.) Hence for any ∈ L+ , V ( )V (λ0 ) = J (λ0 )2 V ( )J (λ0 )V () = J (λ0 )V (λ0 λ0 )V () = J (λ0 )V (λ0 λ0 ) = V ( λ0 ) , (2.22) and similarly V (λ0 )V ( ) = V (λ0 ). Moreover, V (λ0 )V (λ0 ) = J (λ0 )V ()J (λ0 )V ( ) = V (λ0 λ0 )V ( ) = V (λ0 λ0 ) .

(2.23)

Thus V ( · ) is a continuous homomorphism from L+ into J . ↑ As any reflection λ can be represented in the form λ = λ0 −1 for some ∈ L+ , it follows that J (λ) = J (λ0 )2 V ()J (λ0 )V ()−1 = J (λ0 )V (λ0 λ0 −1 ) = V (λ) .

(2.24)

Thus we finally see that J ( · ) is indeed the restriction of the continuous homomorphism V ( · ) to the set of reflections R and that V () does not depend on the decomposition of into reflections for any ∈ L+ . Since R generates L+ , V is the only extension of the reflection map to a homomorphism from L+ into T . We summarize these results in the following proposition. Proposition 2.8. Let J be a continuous reflection map from the set of reflections R ⊂ L+ into an arbitrary topological group J . Then J is the restriction to R of a unique continuous homomorphism mapping L+ into J . 3. Continuity of Modular Reflection Maps In view of the preceding proposition it is of interest to clarify the continuity properties of the modular reflection maps appearing in quantum field theory. In order to reveal the pertinent structures, we discuss this problem in a setting which is slightly more general than that outlined in the introduction. Let W → R(W ) be any net of von Neumann algebras indexed by wedge regions, which satisfies the condition of wedge duality, R(W ) = R(W ), and let ∈ H be any vector which is cyclic and separating for all algebras R(W ). We denote the modular conjugation corresponding to the pair R(W ), by JW . We shall show that the map W → JW from the family of wedges W into the group of (anti)unitary operators on H ↑ is continuous under quite general conditions. Making use of the fact that P+ acts tran↑ sitively on W, we identify W, as a topological space, with the quotient space P+ /P0 , ↑ where P0 ⊂ P+ is the invariance subgroup of any given wedge W0 ∈ W; note that the topology does not depend on the choice of W0 . On the group of (anti)unitary operators we use the strong–*–topology. As we shall see, the desired result follows from the assumption that the net W → R(W ) is locally generated in the following specific sense: Let C be a family of closed regions C ⊂ R4 subject to the conditions: (a) Each C ∈ C can be approximated from the outside by wedges W ∈ W, i.e. C = W C W . Here the inclusion relation W C means that there is some open neighborhood of W in W all of whose elements contain C.


635

(b) Each wedge W ∈ W can be approximated from the inside by regions C ∈ C, i.e. W = C W C, where C W if C is contained in all wedges in some neighborhood of W . ↑ (c) The family C is stable under the action of P+ . We say in this case that C is a generating family of regions. A familiar example of such a generating family is the set of closed double cones in Minkowski space; another one is the family of closed spacelike cones considered in [3] in the context of theories with topological charges. Given a generating family C, we define corresponding algebras R(C), C ∈ C, setting . R(C) = R(W ) . (3.1) W C

Clearly, R(C) ⊂ R(W ) whenever C W . Definition. The net W → R(W ) is said to be locally generated if there is a generating family C of regions such that is cyclic for R(C), C ∈ C, and R(C), W ∈ W . (3.2) R(W ) = C W

Note that the nets affiliated with quantum field theories satisfying the Wightman axioms are locally generated [15]. We shall establish the continuity properties of the modular conjugations by first showing that any locally generated net satisfying wedge duality complies with the net continuity condition introduced in [7], see below. Let {Wδ }δ>0 be a family of wedges converging to some wedge W0 as δ converges to 0. We define corresponding von Neumann algebras . . Rε = R(Wδ ) , Rε = R(Wδ ) , (3.3) 0≤δ≤ε

0≤δ≤ε

and note that, by construction, Rε ⊂ R(Wδ ) ⊂ Rε ,

(3.4)

for any 0 ≤ δ ≤ ε. Moreover, for any ε1 ≥ ε2 ≥ 0, one has Rε1 ⊂ Rε2 and Rε1 ⊃ Rε2 . Now let C ∈ C be such that C W0 . Bearing in mind the meaning of this inclusion relation, it is apparent that C Wδ for all sufficiently small δ >0 and consequently R(C) ⊂ Rε for sufficiently small ε > 0. Hence, R(C) ⊂ ε>0 Rε . As algebra and the net is locally generated, it follows that ε>0 Rε isa von Neumann R(W0 ) = C W0 R(C) ⊂ ε>0 Rε . On the other hand, the inclusion (3.4) implies ε>0 Rε ⊂ R(W0 ), proving the equality Rε = R(W0 ) . (3.5) ε>0

In a similar manner one shows that also Rε = R(W0 ) , ε>0

(3.6)

636


since, by wedge duality,

Rε =

0≤δ≤ε

R(Wδ ) =

R(Wδ ) ,

(3.7)

0≤δ≤ε

and it is easy to see that the family of wedges {Wδ }δ>0 converges to W0 . Applying the arguments in the preceding step, one obtains ε>0 Rε = R(W0 ) = R(W0 ) . From this equality the assertion follows by taking commutants. Relations (3.5) and (3.6) comprise the net continuity condition used in [7] as a crucial technical assumption. We have therefore shown that this condition is met by any net which is locally generated and satisfies wedge duality. This fact puts us into the position to apply Proposition 4.6 in [7], giving the following unexpected result.1 Proposition 3.1. Let W → R(W ) be a locally generated net which satisfies the condition of wedge duality. Then the mapW → JW from the wedges W ∈ W to the modular conjugations JW corresponding to R(W ), is continuous. The conditions underlying this result are natural from a physical point of view and obtain in many models. We therefore believe that the continuity of the modular conjugations is a generic feature in local quantum physics. As a matter of fact, such continuity properties can be established for locally generated nets on many space–times, provided they satisfy an analogue of the wedge duality condition. It is necessary to prove a version of Proposition 3.1 formulated in more group theoretical terms. Because of wedge duality, the modular conjugations JW , JW corresponding to the pairs (R(W ), ) and (R(W ), ), respectively, coincide. We may therefore return . to the notation used in the introduction, i.e. put J (λ) = JW = JW , where λ ∈ P+ is the reflection about the common edge E of the wedges W and W . This notation is consistent, since every two–dimensional spacelike plane E determines uniquely a corresponding pair of wedges W , W having E as their edge. As we shall see, Proposition 3.1 implies that the map λ → J (λ) from the family of reflections λ ∈ P+ into the group of (anti)unitary operators is continuous. In order to prove this fact, we have to show that for any given family of reflections {λδ }δ>0 converging to some reflection λ0 there exists a corresponding convergent family of wedges whose elements Wδ have edges Eδ which are pointwise fixed under the action of λδ , δ > 0. The edges Eδ can be specified easily: Let E0 be the two–dimensional spacelike plane which is pointwise fixed under the action of λ0 . Introducing on R4 the maps x → 21 (1 + λδ λ0 )x one finds that the planes Eδ = 21 (1 + λδ λ0 )E0 are pointwise fixed under the action of the reflections λδ , δ > 0. Since λδ converges to λ0 it follows that λδ λ0 converges to the unit element of P+ , so the above maps converge to the identity, uniformly on compact subsets of R4 . Thus, for sufficiently small δ > 0, each Eδ is a two–dimensional spacelike plane and therefore constitutes the edge of a pair of wedges. Moreover, since the points on the edges Eδ converge to points on E0 in the ↑ limit of small δ, it is straightforward to exhibit Poincaré transformations υδ ∈ P+ such that υδ E0 = Eδ , δ > 0, and {υδ }δ>0 converges to the identity. Picking one of the two . wedges with edge E0 , say W0 , we take as our family of wedges {Wδ = υδ W0 }δ>0 and note that it has all of the desired properties. Since J (λδ ) = JWδ , we are now in a position to apply Proposition 3.1, entailing the following result. 1 Although the CGMA is mentioned in the statement of [7, Prop. 4.6], only wedge duality and the cyclicity properties of given above enter into its proof.


637

Corollary 3.2. Let W → R(W ) be a locally generated net which satisfies the condition of wedge duality. Then the map λ → J (λ) fromthe reflections λ ∈ P+ tothe modular conjugations J (λ) corresponding to R(W ), , hence also to R(W ), , is continuous. Here the wedges W , W are fixed by the condition λW = W . We conclude this section with a technical result pertaining to nets which satisfy the condition of geometric modular action, CGMA, and the modular stability condition, CMS; see the next section for their definitions. The latter condition says that the elements of the modular groups itW , t ∈ R, corresponding to the pairs R(W ), are contained in the group generated by all finite products of the modular conjugations JW , W ∈ W. Under these circumstances one can relax the condition that the nets are locally generated without changing the conclusions of the preceding discussion. In fact, one has the following result. Lemma 3.3. Let W → R(W ) be a net satisfying the CGMA and the CMS. If there is a generating family of regions C such that is cyclic for R(C), given any C ∈ C, then the net is locally generated. Proof. As was shown in [7], the CGMA entails relation (1.1), hence the CMS implies that, for any given W ∈ W,

itW R(W0 ) −it W = R(υW (t)W0 ) ,

W0 ∈ W , it/2

where υW (t) ∈ P+ for t ∈ R; as a matter of fact, since itW = ( W )2 , υW (t) is the square of an element of P+ and thus lies in the identity component of this group. Moreover, since the modular unitaries itW induce automorphisms of R(W ) and since, by the CGMA, the map W → R(W ) is a bijection, one finds that υW (t) must be an element of the stability group of W , t ∈ R. Now let C be a generating family of regions as hypothesized, and let C ∈ C be such that C W for a given W ∈ W. Bearing in mind the definition of the algebras R(C) and the stability of the family C under Poincaré transformations, one obtains

itW R(C) −it W = R(υW (t)C) and υW (t)C ∈ C. Since υW (t) is an element of the stability group of W , one alsofinds, after a moment’s reflection, that υW (t)C W . It follows −it that itW C W R(C) W = C W R(C) ⊂ R(W ), t ∈ R. But is cyclic for the algebras R(C), hence C W R(C) = R(W ), by a well known result of Takesaki.

Analogous results can be established for other space–times, a prominent example being de–Sitter space. 4. Modular Action and Poincaré Covariance Making use of the results obtained in the preceding two sections, we are now able to improve considerably on the analysis of theories complying with the CGMA carried out in [5, 7]. For the convenience of the reader, we recall this condition in a form appropriate for the present discussion as well as some of the important consequences established in the cited articles. A state vector and a net W → R(W ) from the family of wedge regions W in Minkowski space to von Neumann algebras are said to comply with the CGMA if the following conditions are satisfied.

638


(a) W → R(W ) is an order-preserving bijection. (b) If W1 ∩ W2 = ∅, then is cyclic and separating for R(W1 ) ∩ R(W2 ). Conversely, if is cyclic and separating for R(W1 ) ∩ R(W2 ), then W1 ∩ W2 = ∅, where the bar denotes closure. (c) For each W ∈ W, the adjoint action of the modular conjugation JW corresponding to the pair (R(W ), ) leaves the set {R(W )}W ∈W invariant. (d) The group of (anti)automorphisms generated by the adjoint action of the modular conjugations JW , W ∈ W, acts transitively on {R(W )}W ∈W . The core of this condition is part (c), which says that the modular conjugations generate a subgroup of the symmetric (permutation) group on the set {R(W )}W ∈W . So, in view of the correspondence between algebras and wedge regions, it is appropriate to say that these conjugations act geometrically. No a priori assumptions are made about the specific form of this action and the nature of the resulting group. This general formulation of the condition is also appropriate if one thinks of applications to nets labelled by other families of regions appearing, for example, in theories on curved space–times. As was shown in [7], cf. also [1], the CGMA implies that the net satisfies wedge duality. So one may consistently label these conjugations by the reflections λ ∈ P+ about . the edges of the wedges W , respectively W , setting J (λ) = JW = JW . Moreover, the modular conjugations act on the net covariantly in the sense of relation (1.1) and satisfy the fundamental relation (1.2), see [7]. We make use now of the additional assumption that the net is locally generated. Then, by Corollary 3.2, λ → J (λ) is a continuous map from the set of reflections λ ∈ P+ , equipped with the topology induced by P+ , to the group of (anti)unitary operators, equipped with the strong–*–topology. Using this information, we want to construct a continuous representation of the semidirect product L+ R4 = P+ whose elements are denoted, as usual, by (, x). Restricting attention first to the subgroup L+ , we can apply Proposition 2.8 and extend the reflection maps λ → J (λ, 0), λ ∈ L+ , to a continuous (anti)unitary representation U of the proper Lorentz group L+ , (, 0) → U (, 0). Turning to the subgroup R4 of translations, let λ ∈ L+ be any reflection and let x ∈ R4 be such that λx = −x. The set of such x constitutes a two–dimensional subgroup of R4 containing timelike translations. Moreover, (λ, x) = (1, x/2)(λ, 0)(1, −x/2), hence (λ, x) is again a reflection. We put . Uλ (x) = J (λ, x)J (λ, 0) , λx = −x , (4.1) and note that it has been shown in [7, Sect. 4.3] that Uλ defines a continuous unitary representation of the two–dimensional subgroup of translations x satisfying the stated condition.2 As a matter of fact, the continuity of this representation follows directly from the CGMA without any further assumptions [5]. We want to show next that these representations of subgroups of translations can be combined with the above representation of the Lorentz group to yield a true representation of the proper Poincaré group. Let x ∈ R4 be any timelike vector; so its stability subgroup in L+ is conjugate to the entire group of rotations. Hence, applying the arguments in the proof of Lemma 2.4 to the present situation, one finds that for all elements S ∈ L+ of the stability group of x one has USλS −1 (x) = Uλ (x). We may therefore omit the dependence of these operators on the reflections λ and set, for given timelike x, 2 This result was established in [7] only for reflections about edges of certain specific wedges; the general statement follows from the special case by an application of relation (1.2).


. U (1, x) = Uλ (x) ,

639

(4.2)

for any reflection λ such that λx = −x. Now if x, y are positive timelike, their sum is positive timelike, too, and there is a reflection λ ∈ L+ such that λx = −x, λy = −y. Hence we can compute U (1, x) U (1, y) = Uλ (x)Uλ (y) = Uλ (x + y) = U (1, x + y) ,

(4.3)

where, in the second equality, we made use of the fact established above that Uλ is a representation of the respective two–dimensional subgroup. We also note that, for timelike x and admissible λ, U (1, x)−1 = J (λ, 0) J (λ, x) = J (λ, 0) J (λ, x) J (λ, 0)2 = J (λ, −x) J (λ, 0) = U (1, −x) ,

(4.4)

where we made use of relation (1.2). The preceding two relations allow us to extend the unitary operators U (1, z) to arbitrary translations z ∈ R4 . Indeed, any z can be decomposed into z = x − y, where x, y are positive timelike, and, if z = x − y is another such decomposition, one has x + y = y + x . Hence U (1, x)U (1, y ) = U (1, x + y ) = U (1, y + x ) = U (1, y)U (1, x ). All operators appearing in this equality commute with each other according to relation (4.3). Thus, making use of relation (4.4), one gets U (1, x)U (1, −y) = U (1, x )U (1, −y ). One can therefore consistently define for z ∈ R4 , . (4.5) U (1, z) = U (1, x)U (1, −y) , x, y ∈ V+ , x − y = z . Based on the equalities established thus far, one can show that U defines a continuous unitary representation of the subgroup R4 of translations. We omit the straightforward proof of this fact. Since timelike vectors x are mapped by the elements ∈ L+ to timelike vectors, one can also compute the adjoint action of the unitary operators U (, 0) on U (1, x). Fixing x and making use of relations (4.1) and (1.2), one gets for any admissible reflection λ the equation U (, 0) U (1, x) U (, 0)−1 = U (, 0) J (λ, x) J (λ, 0) U (, 0)−1 = J (λ−1 , x) J (λ−1 , 0) = U (1, x) .

(4.6)

This equation can be extended to arbitrary x ∈ R4 by means of relation (4.5). Hence, setting . (4.7) U (, x) = U (1, x) U (, 0) , (, x) ∈ P+ , we arrive at a continuous unitary representation of P+ . Since for any reflection λ ∈ L+ and x ∈ R4 such that λx = −x we have U (λ, x) = U (1, x)U (λ, 0) = U (1, x/2)U (λ, 0)U (1, x/2)−1 = U (1, x/2)J (λ, 0)U (1, x/2)−1 = J (λ, x) ,

(4.8)

we also see that the representation U extends the reflection map J . As a consequence of relation (1.1) and the fact that the operators U (, x) are certain specific products of modular conjugations, these operators act covariantly on the net. Moreover, because of the invariance of under the action of the modular conjugations, is invariant under the action of U (, x), (, x) ∈ P+ . We summarize these results in the following theorem.

640


Theorem 4.1. Let W → R(W ) be a locally generated net and a state vector complying with the CGMA, i.e. conditions (a) to (d). Then the net satisfies wedge duality and there is a continuous (anti)unitary representation U of P+ which leaves invariant and acts covariantly on the net. Moreover, for any given wedge W and reflection λ about its edge, U (λ) is the modular conjugation corresponding to the pair (R(W ), ). Although is invariant under the action of U and as such clearly is a distinguished state, the CGMA does not imply that it is necessarily a ground state. In fact, there exist examples conforming with the hypothesis of Theorem 4.1 for which the joint spectrum sp U of the generators of the subgroup of translations is all of R4 , cf. [7, Sect. 5.3]. So, for the characterization of ground states on Minkowski space, where sp U is contained in a light cone, one has to supplement the CGMA by additional constraints. A conceptually simple and quite general requirement is the modular stability condition CMS, proposed in [7]. We recall this condition here as the last item in our list of constraints characterizing Poincaré invariant ground states describing the vacuum. (e) For any W ∈ W, the elements itW , t ∈ R, of the modular group corresponding to (R(W ), ) are contained in the group generated by all finite products of the modular involutions {JW }W ∈W . We refer the interested reader to [5, 7] for a discussion of the background of this condition and a brief account of other interesting approaches towards an algebraic characterization of ground states. Within the present context, the implications of condition (e) are twofold. On the one hand, we can apply Lemma 3.3 and replace in Theorem 4.1 the assumption that the net is locally generated by the weaker requirement that there is some generating family C of regions such that is cyclic for the algebras R(C), C ∈ C. On the other hand, we can employ the results in [5, 7], which imply that the representation U has the desired spectral properties. Theorem 4.2. Let W → R(W ) be a net and a state vector satisfying the CGMA and CMS, i.e. conditions (a) to (e), and let C be some generating family of regions such that is cyclic for the algebras R(C), C ∈ C. Then the net satisfies wedge duality and there is a representation U of P+ with properties described in the preceding theorem such that sp U ⊂ V + or sp U ⊂ − V + , where V + denotes the closed forward lightcone. The fact that both the forward and the backward lightcone appear as possible supports of the spectrum of the generators of the translations can be understood easily: Neither the CGMA nor the CMS contains any input about the arrow of time. By choosing proper coordinates, one may therefore assume without loss of generality that sp U ⊂ V + . With this convention, U is then the only continuous unitary representation of the spacetime translations which acts covariantly on the given net and leaves invariant [7, Prop. 5.2], cf. also [4, Prop. 2.4]. We have thus attained our goals, the characterization of vacuum states in Minkowski space and the construction of continuous unitary representations of the isometry group of this space, using conditions which are expressed solely in terms of the algebraically determined modular objects. All technical assumptions about continuity properties of the net have been eliminated from this analysis. Instead, they follow in a natural manner from the net structure. And Proposition 2.8 has made clear that once this continuity has been assured, the modular reflection map determines the representation of the isometry group in a canonical and unique manner. There are already strong indications from studies of nets on de Sitter [7, 11] and anti-de Sitter space–times [6], and also more general Robertson–Walker space–times


641

[8, 9] that this strategy is applicable to a large class of space–times of physical interest. This is true in spite of the fact that there is no translation subgroup in the isometry group of these spaces, and thus the standard definition of vacuum state is inapplicable. We therefore believe that the analysis of the modular data in quantum field theories on curved space–time deserves further attention. Acknowledgements. DB wishes to thank the Institute for Fundamental Theory of the University of Florida and SJS wishes to thank the Institute for Theoretical Physics of the University of Göttingen for hospitality and financial support which facilitated this research. This work was supported in part by a research grant of Deutsche Forschungsgemeinschaft (DFG)

References 1. Borchers, H.-J.: On revolutionizing quantum field theory with Tomita’s modular theory. J. Math. Phys. 41, 3604–3673 (2000) 2. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I, Berlin, Heidelberg, New York: Springer Verlag, 1979 3. Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states. Commun. Math. Phys. 84, 1–54 (1982) 4. Buchholz, D., Summers, S.J.: An algebraic characterization of vacuum states in Minkowski space. Commun. Math. Phys. 155, 449–458 (1993) 5. Buchholz, D., Florig, M., Summers, S.J.: An algebraic characterization of vacuum states in Minkowski space, II: Continuity aspects. Lett. Math. Phys. 49, 337–350 (1999) 6. Buchholz, D., Florig, M., Summers, S.J.: The second law of thermodynamics, TCP and Einstein causality in anti-de Sitter space–time. Class. Quantum Grav. 17, L31–L37 (2000) 7. Buchholz, D., Dreyer, O., Florig, M., Summers, S.J.: Geometric modular action and spacetime symmetry groups. Rev. Math. Phys. 12, 475–560 (2000) 8. Buchholz, D., Mund, J., Summers, S.J.: Transplantation of local nets and geometric modular action on Robertson–Walker space–times. In: Mathematical Physics in Mathematics and Physics (Siena), R. Longo, ed., Fields Institute Commun., 30, 65–81 (2001) 9. Buchholz, D., Mund, J., Summers, S.J.: Covariant and quasi-covariant quantum dynamics in Robertson–Walker space–times. Class. Quant. Grav. 19, 6417–6434 (2002) 10. Ellers, E.: Conjugacy classes of involutions in the Lorentz group (V ) and in SO(V ). To appear in Linear Algebra Appl., and private communication 11. Florig, M.: Geometric Modular Action. Ph.D. Dissertation, University of Florida, 1999 12. Haag, R.: Local Quantum Physics. Berlin: Springer-Verlag, 1992 13. Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, Volume II, Orlando: Academic Press, 1986 14. Moretti, V.: The interplay of the polar decomposition theorem and the Lorentz group. Preprint math-ph/0211047 15. Summers, S.J., Wichmann, E.H.: Concerning the condition of additivity in quantum field theory. Ann. Inst. Henri Poincaré 47, 113–124 (1987) 16. Summers, S.J., White, R.K.: On deriving space–time from quantum observables and states. Commun. Math. Phys. 237, 203–220 (2003) 17. Urbantke, H.K.: Lorentz transformations from reflections: Some applications. Found. Phys. Lett. 16, 111–117 (2003) Communicated by Y. Kawahigashi

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