Communications in Mathematical Physics - Volume 216

Commun. Math. Phys. 216, 1 – 15 (2001) Communications in Mathematical Physics © Springer-Verlag 2001 Construction of...

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Commun. Math. Phys. 216, 1 – 15 (2001)

Communications in

Mathematical Physics

© Springer-Verlag 2001

Construction of Doubly-Periodic Instantons Marcos Jardim Yale University, Department of Mathematics, 10 Hillhouse Avenue, New Haven, CT 06520-8283, USA Received: 20 September 1999 / Accepted: 15 April 2000

Abstract: We construct finite-energy instanton connections over R4 which are periodic in two directions via an analogue of the Nahm transform for certain singular solutions of Hitchin’s equations defined over a 2-torus. 1. Introduction Since the appearance of the Yang–Mills equation on the mathematical scene in the late 70’s, its anti-self-dual (ASD) solutions have been intensively studied. The first major result in the field was the ADHM construction of instantons on R4 [1]. Soon after that, W. Nahm adapted the ADHM construction to obtain the time-invariant ASD solutions of the Yang–Mills equations, the so-called monopoles [18]. It turns out that these constructions are two examples of a much more general framework. The Nahm transform can be defined in general for anti-self-dual connections on R4 , which are invariant under some subgroup of translations ⊂ R4 (see [19]). In these generalised situations, the Nahm transform gives rise to dual instantons on (R4 )∗ , which are invariant under ∗ = {α ∈ (R4 )∗ | α(λ) ∈ Z ∀λ ∈ }. There are plenty of examples of such constructions available in the literature, namely: • The trivial case = {0} is closely related to the celebrated ADHM construction of instantons, as described by Donaldson & Kronheimer [7]; in this case, ∗ = (R4 )∗ and an instanton on R4 corresponds to some algebraic data. • If = Z4 , this is the Nahm transform of Braam & van Baal [5] and Donaldson & Kronheimer [7], defining a hyperkähler isometry of the moduli space of instantons over two dual 4-tori. • = R gives rise to monopoles, extensively studied by Hitchin [10], Donaldson [6], Hurtubise & Murray [12] and Nakajima [19], among several others; here, ∗ = R3 , and the transformed object is, for SU(2) monoples, an analytic solution of certain

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M. Jardim

matrix-valued ODE’s (the so-called Nahm’s equations), defined over the open interval (0, 2) and with simple poles at the end-points. • = Z correspond to the so-called calorons, studied by Nahm [18], Garland & Murray [8] and others; the transformed object is the solution of certain nonlinear Nahm-type equations on a circle. The purpose of this paper fits well into this larger mathematical programme. Our goal is to construct finite-energy instantons over R4 provided with the Euclidean metric, which are periodic in two directions (∗ = Z2 ), so-called doubly-periodic instantons, from solutions of Hitchin’s equations [11] defined on a 2-torus, i.e. instantons over R4 which are invariant under = Z2 × R2 . The latter object is now very well studied, and their existence is determined by certain holomorphic data. One might also ask if all doubly-periodic instantons can be produced in this way. In the sequel [14] of this paper, we will show that the construction here presented is invertible by describing the Nahm transform for instantons over T 2 × R2 , which produce singular solutions of Hitchin’s equations. Indeed, Hitchin’s equations admit very few smooth solutions over elliptic curves (see [11]). Therefore, by analogy with Hitchin’s construction of monopoles [10], we will consider a certain class of singular solutions, for which existence is guaranteed [16, 21]. The singularity data is converted into the asymptotic behaviour of the Nahm transformed doubly-periodic instanton; such a picture is again familiar from the construction of monopoles. A string-theoretical version of the Nahm transform here presented was given by Kapustin & Sethi [15]. In fact, the other examples of Nahm transforms mentioned above also have string-theoretical interpretations. The ADHM construction and the Fourier transform of instantons over 4-tori were discussed in these terms by Witten [22], while Kapustin & Sethi [15] also treated the case of calorons. Let us now outline the contents of this paper. Section 2 is dedicated to a brief review of Hitchin’s self-duality equations, and the precise description of the particular type of solutions we will be interested in. The main topic of the paper is contained in Sects. 3 and 4, when we will show how to construct doubly-periodic instantons and explore some of the properties of the instantons obtained. We conclude with a few remarks and raising some questions for future investigation. 2. Singular Higgs Pairs In [11] Hitchin studied the dimensional reduction of the usual Yang–Mills anti-self-dual equations from four to two dimensions. More precisely, let V → R4 be a rank k vector bundle with a connection B˜ which does not depend on two coordinates. Pick up a global trivialisation of V and write down B˜ as a 1-form: B˜ = B1 (x, y)dx + B2 (x, y)dy + φ1 (x, y)dz + φ2 (x, y)dw. Hitchin then defined a Higgs field = (φ1 − iφ2 )dξ , where dξ = dx + idy. So is a section of 1,0 EndV , where V is now seen as a bundle over R2 with a connection B = B1 dx + B2 dy. The ASD equations for B˜ over R4 can then be rewitten as a pair of equations on (B, ) over R2 : FB + [, ∗ ] = 0 . (1) ∂B = 0

Construction of Doubly-Periodic Instantons

3

These equations are also conformally invariant, so they make sense over any Riemann surface. Solutions (B, ) are often called Higgs pairs. As we mentioned in the introduction, we are interested in singular Higgs pairs over a 2-torus Tˆ defined on an U (k)-bundle V → Tˆ . Since we want to think of Tˆ as a quotient of R4 by = Z2 × R2 , the natural choice of metric for Tˆ is the flat, Euclidean metric. Let us also fix a complex structure on Tˆ coming from a choice of complex structure on R4 . Singular Higgs bundles were widely studied by many authors ([21, 17] and [16] among others) and are closely related to the so-called parabolic Higgs bundles. Adopting this point of view, we will consider a holomorphic vector bundle V → Tˆ of degree −2 with the following quasi-parabolic structure over two points ±ξ0 ∈ Tˆ (regarding now Tˆ as an elliptic curve): V±ξ0 = F1 V±ξ0 ⊃ F2 V±ξ0 ⊃ F3 V±ξ0 = {0} order(ξ0 ) = 2, dim = 1 Vξ0 = F1 Vξ0 ⊃ F2 Vξ0 ⊃ F3 Vξ0 ⊃ F4 Vξ0 = {0} order(ξ0 ) = 2. dim = 2 dim = 1 To complete the parabolic structure we need to assign weights α1 (±ξ0 ) to F1 V±ξ0 and α2 (±ξ0 ) to F2 V±ξ0 if ξ0 = −ξ0 or α1 (ξ0 ) to F1 Vξ0 , α2 (ξ0 ) to F2 Vξ0 and α3 (ξ0 ) to F3 Vξ0 if ξ0 = −ξ0 . We assume that α1 = 0 in both cases; if ξ0 is not of order two, we fix that α2 (ξ0 ) = 1 + α and α2 (−ξ0 ) = 1 − α; if ξ0 has order two, we fix that α2 (ξ0 ) = 1 − α and α3 (ξ0 ) = 1 + α for some 0 ≤ α < 21 . Note in particular that V with this parabolic structure has zero parabolic degree. From the point of view of the Higgs pair (B, ), this means that the bundle V is defined away from ±ξ0 , and satisfies, holomorphically: V|Tˆ \{±ξ0 } (V , ∂ B ). The Higgs field has simple poles at the parabolic points ±ξ0 ∈ Tˆ such that the residues φ0 (±ξ0 ) of are k × k matrices of rank 1. If ξ0 is one of the four elements of order 2 in Tˆ , then the residue φ0 (ξ0 ) is assumed to be a k × k matrix of rank 2. Moreover, the harmonic metric h associated with the Higgs pair (B, ) is assumed to be compatible with the parabolic structure. This means that, in a holomorphic trivialisation of V over a sufficiently small neighbourhood around ±ξ0 , h is non-degenerate along the kernel of the residues of , and h ∼ O(r 1±α ) along the image of the residues of . Such metric is clearly not a hermitian metric on the extended bundle V (since it degenerates at ±ξ0 ). Let h be a hermitian metric on V bounding above the harmonic metric on V . If (V, ) is α-stable in the sense of parabolic Higgs bundles, then the existence of a meromorphic Higgs pair as above is guaranteed [21] for any rank k and any choice of ±ξ0 . Moreover, one usually fixes the eigenvalues of the residues of as well. In our situation, this amounts to choosing only one complex number that we denote by . We assume that = 0, i.e. the residues of are semi-simple. However, in this paper, these parameters (the weights αi and the eigenvalue of the residues ) will be allowed to vary; see [4] for a complete discussion. It is reassuring

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M. Jardim

to know that if two sets of parameters (α, ) and (α , ) are chosen in generic position, then α-stability and α -stability are in fact equivalent conditions [20]. In particular, the case k = 1 is very simple: once the parameters (α, ) are fixed and for any choice of ±ξ0 , the moduli space of meromorphic Higgs pairs is just the cotangent bundle of T , that is a copy of T × C. We will study solutions of (1) over Tˆ with the singularities ±ξ0 removed. Due to the non-compactness of Tˆ \ {±ξ0 }, the choice of metric on the base space is a delicate issue. From the point of view of the Nahm transform, it is important to consider the Euclidean, incomplete metric on the punctured torus, as it is well-known from the examples mentioned above. However, such a choice of metric is not a good one from the analytical point of view. For instance, one cannot expect, on general grounds, to have a finite dimensional moduli space of Higgs pairs. Fortunately, as we mentioned before, Hitchin’s equations are conformally invariant, so that we are allowed to make conformal changes in the Euclidean metric localised around the punctures to obtain a complete metric on Tˆ \ {±ξ0 }. Thus, our strategy is to obtain results concerning the Euclidean metric from known statements about complete metrics. In [2], Biquard considered the so-called Poincaré metric, which is defined as follows. We perform a conformal change on the incomplete metric over the punctured torus localised on small punctured neighbourhoods D0 of ±ξ0 , so that if ξ = (r, θ ) is a local coordinate on D0 , we have the metric: dsP2 =

dθ 2 dξ dξ dr 2 + . = 2 2 |ξ |2 log |ξ |2 r 2 log r 4 log2 r

(2)

We denote the complete metric so obtained by gP . The Euclidean metric is denoted by gE . Whenever necessary, we will denote by L2E and L2P the Sobolev norms in $(∗ V ) with respect to gE and gP , respectively, together with the hermitian metric in V . Model solutions of (1) in a neighbourhood of the singularities were described by Biquard [3]: dξ dξ + b∗ , ξ ξ dξ = φ0 , ξ B=b

where b, φ0 ∈ sl(k). Every meromorphic Higgs pair with a simple pole approaches this model close enough to the singularities. Finally, a Higgs pair (B, ) is said to be admissible if V has no covariantly constant sections. 3. Construction of Doubly-Periodic Instantons Our task now is to construct a SU (2) vector bundle over T × C, with an instanton connection on it, starting from a suitable singular Higgs pair as described in the previous section. The key feature of Nahm transforms is to try to solve a Dirac equation, and then use its L2 -solutions to form a vector bundle over the dual lattice; see the references in the introduction.


5

So let S + = 0 ⊕ 1,1 and S − = 1,0 ⊕ 0,1 , as vector bundles over Tˆ . The idea is to study the following elliptic operators: D : $(V ⊗ S + ) → $(V ⊗ S − ) D∗ : $(V ⊗ S − ) → $(V ⊗ S + ), D = (∂ B + ) − (∂ B + )∗ D∗ = (∂ B + )∗ − (∂ B + ),

(3)

where (B, ) is a Higgs pair. Note that the operators in (3) are just the Dirac operators obtained by lifting the Higgs pair (B, ) to an invariant coupled to the connection B, ASD connection on R4 , as above. The next step is to prove that the admissibility condition implies the vanishing of the L2 -kernel of D: Proposition 1. The Higgs pair (B, ) is admissible if and only if L2E −kerD = {0}. Proof. Given a section s ∈ L22 (V ⊗ S + ), the Weitzenböck formula with respect to the Euclidean metric on the punctured torus is given by: ∗

∗

(∂ B ∂ B + ∂ B ∂ B )s = ∇B∗ ∇B s + FB s = ∇B∗ ∇B s − [, ∗ ]s ⇒

∗

∗

∇B∗ ∇B s = (∂ B ∂ B + ∂ B ∂ B + ∗ + ∗ )s ∗ ∗ = (∂ B + )(∂ B + ∗ ) + (∂ B + ∗ )(∂ B + ) s = D∗ Ds,

and integrating by parts, we get: ||Ds||2L2 = ||∇B s||2L2 . E

Thus, if B is admissible, then the is also clear.

L2E -kernel

E

of D must vanish. The converse statement

In other words, the above proposition implies that the L2E -cohomology of orders 0 and 2 of the complex: +∂ B

∂ B +

C : 0 → 0 V −→ 1,0 V ⊕ 0,1 V −→ 1,1 V → 0

(4)

must vanish. On the other hand, since the L2 -norm for 1-forms is conformally invariant, the L2 -cohomology H 1 (C) does not depend on the metric itself, only on its conformal class. Motivated by a result of Biquard (Theorem 12.1 in [2]) we will see how one can identify H 1 (C) in terms of a certain hypercohomology vector space which we now introduce. Let V → Tˆ be the extended holomorphic vector bundle mentioned above. Recall that if ξ0 is not an element of order 2 then the residue of the Higgs field at ±ξ0 is a k × k matrix of rank 1. Therefore, if s is a local holomorphic section on a neighbourhood of ±ξ0 , (s) has at most a simple pole at ±ξ0 and its residue has the form (∗, 0, . . . , 0) on some suitable trivialisation. Similarly, if ξ0 is an element of order 2, (s) has at most a simple pole at ±ξ0 and its residue has the form (∗, ∗, 0, . . . , 0) on some suitable trivialisation. This local discussion motivates the definition of a sheaf P±ξ0 such that, given an open cover {Uα } of Tˆ :

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M. Jardim

• P±ξ0 (Uα ) = OTˆ (V)(Uα ), if ±ξ0 ∈ / Uα ; • P±ξ0 (Uα ) = {meromorphic sections of Uα → Uα × Ck which have at most a simple pole at ±ξ0 with residue lying either along a 2-dimensional subspace of Ck if ξ0 has order 2, or along a 1-dimensional subspace of Ck otherwise}, if ±ξ0 ∈ Uα . It is easy to see that such P±ξ0 is a coherent sheaf. To simplify notation, we drop the subscript ±ξ0 out. Hence, can be regarded as the map of sheaves: : V → P ⊗ KTˆ .

(5)

Seen as a two-term complex of sheaves, the map (5) induces an exact sequence of hypercohomology vector spaces: 0 → H0 (Tˆ , ) → H 0 (Tˆ , V) → H 0 (Tˆ , P ⊗ KTˆ ) → H1 (Tˆ , ) → H 1 (Tˆ , V) → H 1 (Tˆ , P ⊗ KTˆ ) → H2 (Tˆ , ) → 0.

(6)

It is easy to see that: H0 (Tˆ , ) = ker H 0 (Tˆ , V) → H 0 (Tˆ , P ⊗ KTˆ ) , H2 (Tˆ , ) = coker H 1 (Tˆ , V) → H 1 (Tˆ , P ⊗ KTˆ ) , and admissibility implies that the right-hand sides must vanish: restricted to Tˆ \ {±ξ0 }, a section there would give a section in the kernel of D (or, equivalently, a class in H 0 (C) and H 1 (C)). Therefore, the dimension of H1 (Tˆ , ) is equal to χ (P ⊗ KTˆ ) − χ (V) = χ (P) − χ (V). ι To compute this number, note that there is also a natural map V → P defined as the local inclusion of holomorphic local sections (elements of OTˆ (V)(Uα )), into the meromorphic ones (elements of P(Uα )). It fits into the following sequence of sheaves: ι

resξ0

0 → V → P −→ Rξ0 → 0 if ξ0 has order 2, ι

res±ξ0

0 → V → P −→ R±ξ0 → 0 otherwise,

(7) (8)

where Rξ0 is the skyscraper sheaf supported at ξ0 and stalk isomorphic to C2 and R±ξ0 is the skyscraper sheaf supported at ±ξ0 and stalks isomorphic to C. Since χ (R±ξ0 ) = χ (Rξ0 ) = 2, we conclude that H1 (Tˆ , ) is a 2-dimensional complex vector space. Proposition 2. The hypercohomology induced by the map of sheaves (5) coincides with the L2P -cohomology of the complex (4). In particular, we have identifications: H1 (Tˆ , ) ≡ L2P −cohomology H 1 (C) ≡ L2E −cohomology H 1 (C). Furthermore, note also that the L2E -cohomology of 1-forms with respect to the Euclidean metric is a 2-dimensional complex vector space.


7

Proof. The hypercohomology defined by the map (5) is given by the total cohomology of the double complex:

0 V → 1,0 P ∂ ↓ ↓ ∂

0,1 V → 1,0 P which in turn is just the cohomology of the complex: +∂

∂+

0 → 0 V → 1,0 P ⊕ 0,1 V → 1,0 P → 0. Now restricting the complex above to the punctured torus Tˆ \ {±ξ0 }, we get: +∂ B

∂ B +

0 → 0 V → 1 V → 2 V → 0 which is, of course, the complex C. So, let s be a section of 1,0 P ⊕0,1 V defining a class in H1 (Tˆ , ). Thus, restricting s to Tˆ \ {±ξ0 } yields a section sr of L2 (1 V ) defining a class in H 1 (C). Such restriction map is clearly a well-defined map: R : H1 (Tˆ , ) → H 1 (C), < s > → < sr > . We claim that it is also injective. Indeed, suppose that sr represents the zero class, i.e. there is t ∈ L22 (0 V ) such that sr = (∂ B + )t. However, L22 /→ C 0 is a bounded inclusion in real dimension 2. Thus, h(t, t) must be bounded at the punctures ±ξ0 , and t must be itself bounded along the kernel of the residues of . On the other hand, the hermitian metric degenerates along the image of the residues of , so t might be singular on this direction. Indeed, h ∼ O(r 1±α ) in a holomorphic trivialisation, so that 1 t ∼ O(r − 2 (1±α) ). But then the derivatives of t will not be square integrable, contradicting our hypothesis that t belongs to L22 . So t must be bounded at ±ξ0 . This implies that t ∈ L22 (0 V) also with respect to the h metric, so that sr is indeed the restriction of a section representing the zero class in H1 (Tˆ , ). Finally, to show that R is an isomorphism, it is enough by admissibility to argue that the L2 index of the complex C is −2. It was shown by Biquard (Theorem 5.1 in [2]) the laplacian associated to the complex C is Fredholm when acting between L2P sections. This implies that D is also Fredholm. Its index can be computed via Gromov-Lawson’s relative index theorem, and it coincides with the index of the Dirac operator on V: ∗

index(D) = index(∂ B − ∂ B ) = degV = −2 as desired.

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M. Jardim

Constructing the transformed bundle. We are finally in a position to construct a vector bundle with connection over T × C out of a Higgs pair (B, ). Recall that J (Tˆ ) = T , the Jacobian of Tˆ , is defined as the set of flat holomorphic line bundles over Tˆ . Each z ∈ T corresponds to a flat holomorphic line bundle Lz → Tˆ . Moreover, T and Tˆ are isomorphic as elliptic curves. These line bundles can be given a natural constant connection compatible with the holomorphic structure. This follows from the differential-geometric definition of T : T = {z ∈ (R2 )∗ | z(ξ ) ∈ Z, ∀ξ ∈ }, where ⊂ R4 is the two-dimensional lattice generating Tˆ . Hence each z ∈ T can be regarded as a constant, real 1-form over Tˆ , so that ωz = i · z is a connection on a topologically trivial line bundle L → Tˆ . Each such connection defines a different holomorphic structure on L, which we denote by Lz . Conversely, Tˆ parametrises the set of holomorphic flat line bundles with connection over T . Each point ξ ∈ Tˆ corresponds to the line bundle Lξ → T with a connection ωξ . Now consider the restrictions Lz → Tˆ \ {±ξ0 }, with its natural connection ωz , and form the tensor product V (z) = V ⊗ Lz . The connection B can be tensored with ωz to obtain another connection that we denote by Bz . Let i : V (z) → V (z) be the identity bundle automorphism and define w = −w·i, where w is a complex number. It is easy to see that (Bz , w ) is still an admissible Higgs pair, for all (z, w) ∈ T × C. Therefore, we get the following continuous family of Dirac-type operators: D(z,w) = (∂ Bz + w ) − (∂ Bz + w )∗ .

(9)

From Proposition 1, we have that L2E −kerD(z,w) vanishes for all (z, w) ∈ T × C. Since its index remains invariant under this continuous deformation, we conclude that ∗ L2E −kerD(z,w) has constant dimension equal to 2. Define a trivial Hilbert bundle H → T × C with fibres given by L2 (V (z) ⊗ S − ). i

∗ It follows that E(z,w) = kerD(z,w) forms a vector sub-bundle E /→ H of rank 2. Furthermore [7], E is also equipped with an hermitian metric, induced from the L2 metric on H , and an unitary connection A, defined as follows:

∇A = P ◦ d ◦ i,

(10)

where d means differentiation with respect to (z, w) on the trivial Hilbert bundle (i.e. the trivial product connection) and P is the fibrewise orthogonal projection P : L2 (V (z) ⊗ ∗ . Clearly, A defined on (10) is unitary. S − ) → kerD(z,w) Note also that the hermitian metric in H is actually conformally invariant with respect to the choice of metric in Tˆ \ {±ξ0 }, since the inner product in L2 (V (z) ⊗ S − ) is. Therefore, the induced hermitian metric in E is also conformally invariant. Monad description. The transformed bundle E also admits a monad-type description. ∗ More precisely, once a metric is chosen, the family of Dirac operators kerD(z,w) can be unfolded into the following family of elliptic complexes C(z, w): w +∂ Bz

0 → L22,E (0 V (z)) −→ L21,E (1,0 V (z) ⊕ 0,1 V (z)) ∂ Bz +w

−→ L2E (1,1 V (z)) → 0.

(11)


9

Admissibility implies that H 0 (C(z, w)) and H 2 (C(z, w)) must vanish, and ∗ coincides with L2E −kerD(z,w) . As (z, w) sweeps out T × C, H 1 (C(z, w)) forms a rank 2 holomorphic vector bundle with a natural hermitian metric and a compatible unitary connection A, equivalent to the ones defined as above; see [7].

H 1 (C(z, w))

3.1. Anti-self-duality and curvature decay. The next proposition fulfills the first goal of this paper, i.e. to show that the connection A defined above is in fact a finite-energy anti-self-dual instanton on the rank 2 bundle E → T × C. We say f ∼ O(|w|n ) if the complex function f : C → C satisfies: |f (w)| < ∞. |w|→∞ |w|n lim

(12)

Theorem 3. The transformed connection A is anti-self-dual with respect to the Euclidean metric. Furthermore, its curvature satisfies |FA | ∼ O(|w|−2 ). Proof. Since A is an unitary connection, we only have to verify that the component of FA along the Kähler class κ of T × C vanishes. Let {ψ1 , ψ2 } be a local holomorphic frame for E, orthonormal with respect to the hermitian metric induced from H . Fix some (z, w) ∈ T × C so that, as a section of ∗ V(z) ⊗ S − → Tˆ , we have ψi = ψi (ξ ; z, w) ∈ kerD(z,w) . In this trivialisation, the matrix elements of the curvature FA can then be written as follows: (FA )ij = 'ψj , ∇A ∇A ψi ( = 'ψj , d ◦ P ◦ dψi ( ∗ ∗ = 'D(z,w) (dψj ), G(z,w) D(z,w) (dψj )(,

(13)

where the inner product is taken in L2 (V (z) ⊗ S − ), integrating out the ξ coordinate; the finiteness of the integral is guaranteed by the fact that ψj ∈ L21 (V (z) ⊗ S − ). Note also that the inner product is conformally invariant with respect to the choice of metric on Tˆ \ {±ξ0 }. Hence, the expression for the curvature above is the same for both the Euclidean and Poincaré metrics. ∗ Moreover, G(z,w) is the Green’s operator for D(z,w) D(z,w) . Note that ∗ [D(z,w) , d]ψi = 9 · ψi ,

where 9 = (idz1 + dw1 ) ∧ dξ1 + (idz2 + dw2 ) ∧ dξ2 and “·” denotes Clifford multiplication. So, κ(FA )ij = 'ψj , κ(9 ∧ 9 ) ·G(z,w) ψi ( = 0, =0

(14)

and this proves the first statement. It is easy to see from (14) that the asymptotic behaviour of |(FA )ij | depends only on the behaviour of the operator norm ||G(z,w) || for large |w|. We can estimate ||G(z,w) || by looking for a lower bound for the eigenvalues of the associated laplacian acting on V ⊗ S − : ∗ D(z,w) D(z,w) = Dz Dz∗ − wφ ∗ − wφ + |w|2 ,

(15)

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M. Jardim

where Dz = D(z,w=0) and = φdξ , with φ ∈ EndV ; φ ∗ denotes the adjoint (conjugate transpose) endomorphism. In other words, we want to find a lower bound for the following expression: '(Dz D∗ + |w|2 )s, s( − '(wφ ∗ + wφ)s, s( z ≥ '(Dz D∗ + |w|2 )s, s( − |'(wφ ∗ + wφ)s, s(| (16) z

L2 (V

⊗ S−)

of unit norm. for s ∈ For the first term in the second line, it is easy to see that |'(Dz Dz∗ + |w|2 )s, s(| = ||Dz∗ s||2 + |w|2 · ||s||2 = c1 + |w|2

(17)

for some non-zero constant c1 = ||Dz∗ ||2 depending only on z ∈ T . The second term in (16) is more problematic; first note that

|'(wφ ∗ + wφ)s, s(| ≤ |w| · |'φ(s), s(| + |'φ ∗ (s), s(| . In a small neighbourhood D0 of each singularity ±ξ0 , we have

φ0 (s) regular 'φ(s), s(L2 (D0 ) = ' , s(rdrdθ + terms ξ D

0 |φ0 | regular 2 ∼ . · |s| rdrdθ + terms D0 r Let 1 < p < 2; using Hölder inequality, we obtain: 1/q 1/p |φ0 | p |φ0 | 2 2q rdrdθ |s| · |s| ≤ r D0 ξ D0 D0 ≤ c · ||s||2L2q , p where q = p−1 , and for some real constant c depending only on φ0 and on the choice of p. Since 2q > 4, the Sobolev embedding theorem tells us that L21 /→ L2q is a bounded inclusion (in real dimension 2). In other words, there is a constant C depending only on q such that ||s||L2q ≤ C · ||s||L2 . Thus, arguing similarly for the 'φ ∗ (s), s( term, we 1 conclude that |'(wφ ∗ + wφ)s, s(| ≤ c2 · |w|,

where c2 is a real constant depending neither on z nor on w, but only on the Higgs field itself and on the choice of p. Putting everything together, we have: '(Dz Dz∗ − wφ ∗ − wφ + |w|2 )s, s( ≥ |w|2 − c2 |w| + c1 so that

lim |w|2 · ||G(z,w) || < 1

|w|→∞

and the statement follows.

Remark 1. Note in particular that FA ∈ L2 (2 ⊗E) with respect to the Euclidean metric on T × C, coming from the quotient (R4 )∗ /∗ . This concludes our first task.


11

Remark 2. It is also not difficult to see that gauge equivalent Higgs pairs (B, ) and (B , ) will produce gauge equivalent instantons A and A . The dependence of A on the Higgs pair (B, ) is contained on the L2 -projection operator P , that is on the ∗ two linearly independent solutions of D(z,w) ψ = 0. Gauge equivalence of (B, ) and (B , ) gives an automorphism of the transformed bundle E, in other words, a gauge equivalence between A and A . Remark 3. The instanton connection A induces a holomorphic structure ∂ A on the the transformed bundle E → T × C. In order to further understand the asymptotic behaviour of the transformed connection, we must now pass to an equivalent holomorphic description of the above transform.

4. Holomorphic Version and Extensibility Motivated by curvature decay established above, one can expect to find a holomorphic vector bundle E → T × P1 which extends (E, ∂ A ). The idea is to find a suitable perturbation of the Higgs field for which w = ∞ makes sense. As above, the torus parameter z ∈ T simply twists the holomorphic bundle V → Tˆ . We denote: V(z) = V ⊗ Lz ,

P(z) = P ⊗ Lz .

(18)

Since ∈ H 0 (Tˆ , Hom(V, P) ⊗ KTˆ ), tensoring both sides of (5) by the line bundle Lz does not alter the sheaf homomorphism , so we have the family of maps: : V(z) → P(z) ⊗ KTˆ parametrised by z ∈ T . To define the perturbation w , recall that, regarding P1 = C ∪ {∞}, we can fix two holomorphic sections s0 , s∞ ∈ H 0 (P1 , OP1 (1)) such that s0 vanishes at 0 ∈ C and s∞ vanishes at the point added at infinity. In homogeneous coordinates {(w1 , w2 ) ∈ C2 |w2 = 0} and {(w1 , w2 ) ∈ C2 |w1 = 0}, we have that, respectively (w = w1 /w2 ): s0 (w) = w, s∞ (w) = 1,

s0 (w) = 1, 1 s∞ (w) = . w

Consider now the map of sheaves parametrised by pairs (z, w) ∈ T × P1 : w : V(z) → P(z) ⊗ KTˆ , w = s∞ (w) · − s0 (w) · ι · dξ.

(19)

Clearly, on P1 \ {∞} = C this is just w = − w · ι, the same perturbation we defined before. Moreover, if w = ∞, then ∞ = ι · dξ . The hypercohomology vector spaces H0 (Tˆ , w ) and H2 (Tˆ , w ) of the two-term complex (19) must vanish by admissibility. On the other hand, H1 (Tˆ , w ) also makes sense for ∞ ∈ P1 , and we can define a SU (2) holomorphic vector bundle E → T ×

12

M. Jardim

P1 with fibres given by E(z,w) = H1 (Tˆ , w ). Moreover, E is actually a holomorphic extension of (E, ∂ A ), in the sense that, holomorphically: E|T ×(P1 \{∞}) (E, ∂ A ).

(20)

Equivalently, E can be seen as the hermitian holomorphic vector bundle induced by the monad +∂

∂+

0 → 0 V → 1,0 P ⊕ 0,1 V → 1,0 P → 0.

(21)

Consider the metric H induced from the monad (21) above, while H is induced from the monad (11). Now, H is bounded above by H because the hermitian metric h on the bundle V in (11) is bounded above by the metric h on the bundle V in (21). We now show that the position of the singularities of the Higgs pair determines the holomorphic type of the restriction of the extended transformed bundle over the added divisor at infinity. First, recall that there is an unique line bundle P → T × Tˆ , the so-called Poincaré line bundle, satisfying: P|T ×{ξ } Lξ

P|{z}×Tˆ L−z .

It can be constructed as follows. Identifying T and Tˆ as before, let = be the diagonal inside T × Tˆ , and consider the divisor D = = − T × eˆ − e × Tˆ . Then P = OT ×Tˆ (D); it is easy to see that the sheaf so defined restricts as wanted. Note that although the two restrictions above are flat line bundles over T and Tˆ respectively, the Poincaré bundle itself is not topologically trivial; in fact, c1 (P) ∈ H 1 (T ) ⊗ H 1 (Tˆ ) ⊂ H 2 (T × Tˆ ). More precisely, the unitary connection and its corresponding curvature are given by: ω(z, ξ ) = iπ ·

2

2 dξµ ∧ dzµ . ξµ dzµ − zµ dξµ and 9(z, ξ ) = 2iπ ·

µ=1

µ=1

Restricting to each T × {ξ }, the line bundles Lξ → T are given flat connections ωξ = iπ · 2µ=1 ξµ dzµ , with constant coefficients. Similarly, the line bundles Lz → Tˆ are given the flat connections ωz = −iπ · 2µ=1 zµ dξµ as described in the previous section. Finally, note that: c1 (P) =

i 9 ⇒ c1 (P)2 = −2 · t ∧ tˆ, 2π

where t and tˆ are the generators of H 2 (T ) and H 2 (Tˆ ), respectively. Lemma 4. E|T∞ ≡ Lξ0 ⊕ L−ξ0 Proof. Substituting w = ∞ ∈ P1 , we get from (19) that ∞ = ι · dξ . Therefore, the induced hypercohomology sequence (23) coincides with the long exact sequence of cohomology induced by the sheaf sequences (7) and (8), which is given by: ∞ 0 → H 0 (Tˆ , V(z)) → H 0 (Tˆ , P(z) ⊗ KTˆ ) → H 0 (Tˆ , R±ξ0 (z)) ∞ → H 1 (Tˆ , V(z)) → H 1 (Tˆ , P(z) ⊗ KTˆ ) → 0.

(22)


13

Hence, H1 (Tˆ , (z, ∞)) = H 0 (Tˆ , R±ξ0 (z)). The right hand side is canonically identified with (Lz )ξ0 ⊕ (Lz )−ξ0 , where by (Lz )ξ0 we mean the fibre of Lz → Tˆ over the point ξ0 ∈ Tˆ . On the other hand, (Lz )ξ0 = P(z,ξ0 ) = (Lξ0 )z , where P → T × Tˆ is the Poincaré line bundle. Thus, the bundle over T∞ with fibres given by H 0 (Tˆ , R±ξ0 (z)) is isomorphic to Lξ0 ⊕ L−ξ0 , as we wished to prove. The topological type of E is also fixed from the initial data: the rank of the bundle V is translated into the second Chern class of the extended transformed bundle E. In the next lemma, we denote the generator of H 2 (P1 , Z) by p. Lemma 5. ch(E) = 2 − k · t ∧ p. Proof. The exact sequence: w

0 → H 0 (Tˆ , V(z)) → H 0 (Tˆ , P(z) ⊗ KTˆ ) → H1 (Tˆ , (z, w)) w → H 1 (Tˆ , V(z)) → H 1 (Tˆ , P(z) ⊗ K ˆ ) → 0

(23)

T

induces a sequence of coherent sheaves over T × C, with stalks over (z, w) given by the above cohomology groups: w 0 → H0 (Tˆ , V(z)) → H0 (Tˆ , P(z) ⊗ KTˆ ) → Eˇ w → H1 (Tˆ , V(z)) → H1 (Tˆ , P(z) ⊗ K ˆ ) → 0.

(24)

T

In this way, the Chern character of Eˇ will then be given by the alternating sum of the Chern characters of these sheaves, which can be computed via the usual Grothendieck– Riemann–Roch for families. ∗ P. Clearly, Consider the bundle G1 → T × P1 × Tˆ given by G1 = p3∗ V ⊗ p13 G1 |(z,w)×Tˆ = V(z), so that: ch(H0 (Tˆ , V(z))) − ch(H1 (Tˆ , V(z))) = ch(G1 )td(Tˆ )/[Tˆ ].

(25)

∗ P ⊗ p ∗ O (1). The twisting by O (1) Now consider the sheaf: G2 = p3∗ P ⊗ p13 P1 2 P1 accounts for the multiplication by the section s0 ∈ H 0 (P1 , OP1 (1)) contained in w . As above, G1 |(z,w)×Tˆ = P(z), and we have:

ch(H0 (Tˆ , P(z) ⊗ KTˆ )) − ch(H1 (Tˆ , P(z) ⊗ KTˆ )) = ch(G2 )td(Tˆ )/[Tˆ ]. Therefore: ch(E) = (26) − (25) =

k 2 = c1 (P) − c1 (V) + c1 (P) ∧ p + c1 (P) ∧ p /[Tˆ ] = 2 = χ (P) − degV + χ (P) · p − k · t ∧ p = 2 − k · t ∧ p as desired.

(26)

14

M. Jardim

Finally, we argue that the determinant bundle of E is trivial, so that A is indeed a SU (2) instanton. Note that detE is a line bundle with vanishing first Chern class, so it must be the pull back of a flat line bundle Lξ → T . But detE|T∞ = C, hence detE must be holomorphically trivial, as desired. We call ξ0 ∈ J (T ) the asymptotic state associated to the doubly-periodic instanton connection A, and the integer k its instanton number. The Nahm transform constructed above guarantees the existence of doubly-periodic instantons of any given charge and asymptotic state.

4.1. Extensible doubly-periodic instanton connections. Motivated by the properties established above, we say that a doubly-periodic instanton connection A on a bundle E → T × C is extensible if the following hypothesis holds: 1. |FA | ∼ O(|w|−2 ); 2. there is a holomorphic vector bundle E → T × P1 with trivial determinant such that E|T ×(P1 \{∞}) (E, ∂ A ), where ∂ A is the holomorphic structure on E induced by the instanton connection A; This definition will be our starting point in [14], where we shall present the Nahm transform of doubly-periodic instantons, i.e. the inverse of the construction shown here. 5. Conclusion In this paper we have shown how finite energy, doubly-periodic instantons can be produced by performing a Nahm transform on certain singular Higgs pairs. The rank of the Higgs bundle is translated into the instanton number; the number of singularities of the Higgs field (i.e. the degree of the holomorphic Higgs bundle V) gives the rank of the transformed instanton, and its positions determine how the instanton connection “splits at infinity”. Indeed, it is easy to generalise the above construction by allowing more than two singularities on the original Higgs field, so that higher rank doubly-periodic instantons are obtained; see [14]. Moreover, one would also like to understand how the parabolic parameters (α, ) are translated into the doubly-periodic instantons produced via the Nahm transform as above. On general grounds, we expect these parameters to be translated into more detailed information on the asymptotic behaviour of A. From the more analytical point of view, it is also interesting to ask if the curvature decay (Proposition 3) is enough to ensure extensibility. More precisely, one can expect to be able to prove the following result: Conjecture 6. If A is anti-self-dual and |FA | ∼ O(|w|−2 ), then there is a holomorphic vector bundle E → T × P1 such that E|T ×(P1 \{∞}) (E, ∂ A ). In other words, A is extensible. Such conjecture motivates other questions: • Do all anti-self-dual connections on E → T × C with finite energy with respect to the Euclidean metric satisfy |FA | ∼ O(|w|−2 )?


15

• Does the converse hold, i.e. if A is extensible then |FA | ∼ O(|w|−2 )? If not, what are the necessary and sufficient analytical conditions for extensibility (in terms of the Euclidean metric)? • Given a holomorphic bundle E → T × P1 , is there a connection A on E|T ×(P1 \{∞}) such that A is anti-self-dual and |FA | ∼ O(|w|−2 ) with respect to the Euclidean metric? We hope to address these issues in a future paper [4]. Acknowledgements. This work is part of my Ph.D. project [13], which was funded by CNPq, Brazil. I am grateful to my supervisors, Simon Donaldson and Nigel Hitchin, for their constant support and guidance. I also thank Olivier Biquard, Alexei Kovalev and Brian Steer for invaluable help in the later stages of the project.

References 1. Atiyah, M., Drinfel’d, V., Hitchin, N., Manin, Y.: Construction of instantons. Phys. Lett. A 65, 185–187 (1978) 2. Biquard, O.: Fibrés de Higgs et connexions intégrables: Le cas logarithmique (diviseur lisse). Ann. Scient. Éc. Norm. Sup. (4) 30, 41–96 (1997) 3. Biquard, O.: Sur les équations de Nahm et la structure de Poisson es algébres de Lie semi-simple complexes. Math. Ann. 304, 253–276 (1996) 4. Biquard, O., Jardim, M.: Asymptotic behaviour and the moduli space of doubly-periodic instantons. In preparation 5. Braam, P., van Baal, P., Nahm’s transform for instantons. Commun. Math. Phys. 122, 267–280 (1989) 6. Donaldson, S.: Nahm’s equations and classification of monopoles. Commun. Math Phys. 96, 387–207 (1984) 7. Donaldson, S., Kronheimer, P.: Geometry of four-manifolds. Oxford: Clarendon Press 1990 8. Garland, H., Murray, M.: Kac–Moody monopoles and periodic instantons. Commun. Math. Phys. 120, 335–351 (1988) 9. Gromov, M., Lawson, H.: Positive scalar curvature and the index of the Dirac operator on complete Riemannian manifolds. Inst. des Hautes Études Scientifiques Publ. Math. 58, 295–408 (1983) 10. Hitchin, N.: Construction of monopoles. Commun. Math. Phys. 89, 145–190 (1983) 11. Hitchin, N.: The self-duality equations on a Riemann surface. Proc. London Math. Soc. 55, 59–126 (1987) 12. Hurtubise, J., Murray, M.: On the construction of monopoles for the classical groups. Commun. Math. Phys. 122, 35–89 (1989) 13. Jardim, M.: Nahm transform for doubly-periodic instantons. Ph.D. thesis, Oxford (1999); Preprint math.DG/9912028 14. Jardim, M.: Nahm transform for doubly-periodic instantons. Preprint math.DG/9910120 15. Kapustin, A., Sethi, S.: Higgs branch of impurity theories. Adv. Theor. Math. Phys. 2, 571–592 (1998) 16. Konno, H.: Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface. J. Math. Soc. Japan 45, 253–276 (1993) 17. Kovalev, A.: The geometry of dimensionally reduced anti-self-duality equations. Ph.D. thesis, Oxford (1995) 18. Nahm, W.: Self-dual monopoles and calorons. In: Denardo, G., Ghirardi, G., Weber, T. (eds.) Group theoretical methods in physics. Proceedings, Trieste 1980, Berlin, New York: Springer-Verlag, 1984, pp. 189–200 19. H. Nakajima. Monopoles and Nahm’s equations. In: Mabuchi, T., Mukai, S. (eds.) Einstein metrics and Yang–Mills connections. Proceedings, Sanda 1990, New York: Marcel Dekker, 1993, pp. 193–211 20. Nakajima, H.: Hyperkähler structures on the moduli spaces of parabolic Higgs bundles on Riemann surfaces. In: Maruyama, M. (ed.) Moduli of vector bundles. Proceedings, Sanda 1994; Kyoto 1994, New York: Marcel Dekker, 1996, pp. 199–208 21. Simpson C.: Harmonic bundles on noncompact curves. J. of Am. Math. Soc. 3, 713–770 (1990) 22. Witten, E.: Talk presented at the meeting “Interfaces in Mathematics”, in honor of the 70th birthday of Michael Atiyah, London, April 1999 Communicated by R. H. Dijkgraaf

Commun. Math. Phys. 216, 17 – 22 (2001)

Communications in



Existence of the Ginzburg–Landau Vortex Number Mats Aigner Department of Mathematics, Linköping University, 581 83 Linköping, Sweden. E-mail: [email protected] Received: 16 November 1999 / Accepted: 7 July 2000

Abstract: The existence of the Ginzburg–Landau vortex number is established for any configuration with finite action.As a consequence, Bogomol’nyi’s formula for the critical action is valid for any finite action configuration.

1. Introduction In the Ginzburg–Landau theory, as formulated in Jaffe and Taubes [2], a topological invariant called the vortex number arises as the winding number at infinity of the scalar field. Given smoothness and some decay conditions at infinity, the vortex number exists and can easily be shown to be equal to an integral of the curvature. In this paper we show that, as conjectured in [2], the vortex number can be defined under the assumption of finite action only. The Ginzburg–Landau theory we are concerned with is the following. Let A be a connection form, a real-valued 2-form in R2 , and let φ be a scalar field, a complexvalued function in R2 . Then the covariant derivative of φ is dA φ = dφ − iAφ and the curvature of A is FA = dA. In this situation, the Ginzburg–Landau action of the configuration (A, φ) is A(A, φ) =

1 2π

R2

2 |FA |2 + |dA φ|2 + λ 21 1 − |φ|2 dx dy,

where λ > 0 is a constant. The three cases λ < 1, λ = 1 (called the critical case), and λ > 1 give qualitatively different theories. A gauge transformation g, a function with values in U (1), acts on connection forms and scalar fields as g.A = A − ig −1 dg and g.φ = gφ. So dg.A (g.φ) = g dA φ and Fg.A = FA , from which it follows that the Ginzburg–Landau action is gauge invariant.

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M. Aigner

The variational equations of A, ∗ d FA = Im φ dA φ A φ = λ2 1 − |φ|2 φ, where the star denotes the formal adjoint and A φ = dA∗ dA φ, are called the vortex equations. In [4] and [5], Taubes gave a complete classification of the finite action solutions to the vortex equations in the critical case. 2. The Vortex Number If the configuration (A, φ) has finite action, then 1 − |φ|2 ∈ L2 (R2 ), making it plausible that the winding number of φ at infinity, i.e. on a circle with large radius, exists in some sense. This is confirmed by our main theorem, which also shows that the space of finite action configurations is divided into components indexed by the integers. 2 Theorem. Let λ > 0 be arbitrary and let (A, φ) ∈ L1,2 loc (R ) be a finite action configuration. Then the limit 1 χε FA (2.1) δ(A, φ) = lim ε→0+ 2π R2

exists, and it is called the vortex number of (A, φ). The vortex number is an integer and 2 is invariant under gauge transformations in L2,2 loc (R ). If φ is smooth and |φ| → 1 at infinity, then the vortex number is the winding number of φ at infinity. Here Lk,p denotes the Sobolev space whose elements have derivatives up to order k in Lp , and χε is a smooth cut-off function defined as follows. For ε > 0 we let χε (x, y) = χ (ε|(x, y)|), where χ ∈ C ∞ (R) is such that 0 χ 1, χ (t) = 1 if t 1, and χ (t) = 0 if t 2. 2 Proof. Let (A, φ) ∈ L1,2 loc (R ) be a finite action configuration. We first prove the theorem in the case that φ is smooth and |φ| → 1 at infinity. Then the winding number of φ at infinity, N , exists. Since supp χε is compact, a partial integration gives 1 1 χε FA = − dχε ∧ A. (2.2) 2π R2 2π R2

Now subtract φ dA φ = φ dφ − iA |φ|2 from φ dA φ = φ dφ + iA |φ|2 and solve for A. Since |φ| → 1 at infinity, we can divide by |φ| if |(x, y)| is large. The result is A = 2i1 |φ|−2 φ dφ − φ dφ − 2i1 |φ|−2 φ dA φ − φ dA φ = d(arg φ) − 2i1 |φ|−2 φ dA φ − φ dA φ . For small ε, we substitute this expression for A into (2.2) and get 1 1 χε FA = N + dχε ∧ 2i1 |φ|−2 φ dA φ − φ dA φ . 2π R2 2π R2

Existence of the Ginzburg–Landau Vortex Number

19

The last integral is estimated as follows: 1 −2 1 φ dA φ − φ dA φ c ε |dA φ| dx dy 2π 2 dχε ∧ 2i |φ| R 1/ε|(x,y)|2/ε c dA φ L2 (1/ε|(x,y)|2/ε) , where c stands for a constant. Since A(A, φ) < ∞, so that dA φ ∈ L2 (R2 ), the estimate shows that 1 lim χε FA = N, ε→0+ 2π R2 so the theorem is proved in this case. 2 In general we have only (A, φ) ∈ L1,2 loc (R ) and A(A, φ) < ∞, but using an idea in Taubes [6], we will now construct a scalar field φ such that – – –

A(A, φ ) < ∞ 2,4/3 φ is sufficiently smooth (we will obtain φ ∈ Lloc , which is enough) φ → 1 at infinity.

The above calculations then go through with φ instead of φ and the proof is then complete. Of course, the gauge invariance of the vortex number is a direct consequence of Fg.A = FA . 2 The first step is to construct a scalar field φ ∈ L1,2 A(A, φ ) < ∞ loc (R ) such that λ and such that (A, φ ) satisfies one of the vortex equations: A φ = 2 1 − |φ |2 φ . We accomplish this by the “direct method”, as in Sedlacek [3]. 1,2 2 2 2 2 2 2 Let L1,2 A (R ) = ψ ∈ L (R ) ; dA ψ ∈ L (R ) . Then LA (R ) is a Hilbert −1,2 space of scalar fields, with dual space LA (R2 ), and a straightforward bootstrapping 1,2 2 2 argument shows that L1,2 A (R ) ⊂ Lloc (R ). We now define an affine Hilbert space 2 H = (A, φ + ψ) ; ψ ∈ L1,2 A (R ) , and choose a sequence (A, φn ) in H such that A(A, φn ) → inf H A. 2 The action A is differentiable on H and it is not hard to verify that for ψ ∈ L1,2 A (R ) and t ∈ R, 1 A(A, φn + tψ) = A(A, φn ) + t Re A φn − λ2 1 − |φn |2 φn ψ dx dy π R2 + t 2 R(A, φn , ψ, t), where a bound on A(A, φn ), ψ L1,2 (R2 ) and t gives a bound on R. This implies that A

A φn − λ2 1 − |φn |2 φn

2 L−1,2 A (R )

→ 0,

because otherwise we could not have A(A, φn ) → inf H A. be a disc strictly containing D. Since Let D ⊂ R2 be an open disc and let D 1,2 A ∈ L (D), the norms · L1,2 (D) are equivalent, so and · L1,2 (D) A

A φn − λ2 1 − |φn |2 φn

L−1,2 (D)

→ 0.

(2.3)

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M. Aigner

λ 2 Let αn ∈ L1,2 0 (D) be the solution of αn = A φn − 2 1 − |φn | φn , and define βn by and φn = αn + βn . Then αn → 0 in L1,2 (D) βn = φn − A φn + λ2 1 − |φn |2 φn = iφ d ∗A − 2i(A, dφ ) − φ |A|2 + λ 1 − |φ |2 φ . n

n

n

2

n

n

By estimating each term, using that φn L4 (D) is controlled by the action A(A, φn ) 1,2 4 and that L (D) ⊂ L (D), we get that βn L4/3 (D) c, where c in independent of n. We also have βn L1,2 (D) c, so by elliptic regularity theory, βn is a bounded sequence in L2,4/3 (D). The embedding L2,4/3 (D) → L1,2 (D) is compact, so there , exists a subsequence of φn which converges in L1,2 (D). The limit, which we call φD λ 2 satisfies A φD = 2 1 − |φD | φD because of (2.3). To summarize the above: Given a disc there exists a subsequence of φn which converges, in L1,2 on the disc, to a limit satisfying one of the vortex equations. We now cover R2 with discs and use Cantor’s diagonal process to get a subsequence of φn which con 1,2 λ 2 2 2 verges in L1,2 loc (R ) to a scalar field φ ∈ Lloc (R ), which satisfies A φ = 2 1−|φ | φ . Since the convergence is local, A(A, φ ) inf H A < ∞. 2,4/3 It remains to prove that φ ∈ Lloc (R2 ) and that φ → 1 at infinity. Let D ⊂ R2 be a disc of radius 1. We will show that

|φ | 1/2 c, (2.4) C (D) where the norm is the Hölder norm with exponent 1/2, and c is a constant which does 2 not depend on position of D. Since A(A, φ ) < ∞, so that 1 − φ ∈ L2 (R2 ), this the implies that φ → 1 at infinity. The function φ is gauge invariant, so we use a “good gauge” for the estimates. be a disc of radius 2 concentric with D, and let B ∈ L1,2 (D) be a real-valued Let D 0 2-form such that B = FA . Then d(d ∗B −A) = 0, so there exists a real-valued function such that dψ = d ∗B − A. Then g = eiψ is a gauge transformation and ψ ∈ L2,2 (D) ∗ g.A = d B. It is clear that d ∗(g.A) = 0 and that g.A L1,2 (D) c FA L2 (D) , and one

g.A

can verify that in fact both and g.φ L1,2 (D) are controlled by A(A, φ ). L1,2 (D) So

g.A L1,2 (D) , g.φ L1,2 (D) c,

where c is independent of the D. position of λ 2 The equation A φ = 2 1 − |φ

| φ is gauge invariant, so by expanding g.A (g.φ )

and estimating the terms we get (g.φ ) L4/3 (D) c. Now (2.4) follows from elliptic regularity theory and the Sobolev embedding theorem. Finally, the same estimates as above, but without the use of gauge transformations, 2,4/3 give φ ∈ Lloc (R2 ). 3. Bogomol’nyi’s Formula for the Critical Action In this section λ = 1, the critical case. Following Bogomol’nyi [1], we rewrite the action density. With R2 and C identified in the usual way we have dA φ = ∂A φ + ∂A φ, the

Existence of the Ginzburg–Landau Vortex Number

21

covariant derivative separated into its holomorphic and anti-holomorphic parts. A short computation leads to

2 |FA |2 + |dA φ|2 + 21 1 − |φ|2 dx dy = 2 2 = 2∂A φ + ∗FA − 1 1 − |φ|2 dx dy + FA − d iφ dA φ , 2

(3.1)

where ∗ is the Hodge star. If we ignore the boundary term d iφ dA φ and the fact that we may not have FA ∈ L1 (R2 ), then 1 A(A, φ) = 2π

R2

2 2 1 2 1 2 ∂A φ + ∗FA − 2 1 − |φ| dx dy + FA , (3.2) 2π R2

which motivates the introduction of the self-dual vortex equations

∂A φ = 0 ∗FA =

1 2

1 − |φ|2 .

Similarly, by using ∂A φ instead of ∂A φ, we are lead to the anti-self-dual equations. The following theorem justifies the formal result (3.2), and shows that the minima of A, for a given vortex number δ, are the finite action solutions of the (anti-)self-dual vortex equations. 2 Theorem. Let λ = 1 and let (A, φ) ∈ L1,2 loc (R ) be a finite action configuration. Then

2 2 1 2 1 2 ∂A φ + ∗FA − 2 1 − |φ| dx dy + δ(A, φ), A(A, φ) = 2π R2 and 2 1 A(A, φ) = 2|∂A φ|2 + ∗FA + 21 1 − |φ|2 dx dy − δ(A, φ). 2π R2

(3.3)

As a consequence we get the inequality A(A, φ) |δ(A, φ)| , where the equality A = δ holds if and only if (A, φ) is a self-dual vortex, and the equality A = −δ holds if and only if (A, φ) is an anti-self-dual vortex. We only sketch the proof, because the theorem is almost a corollary of the existence of the vortex number. In short, the formula (3.3) is first proved for smooth finite action configurations (A, φ) such that φ is bounded, by multiplying (3.1) with the cut-off function χε and letting ε go to zero. The general case of a finite action configuration in 2 L1,2 loc (R ) then follows by approximation. Acknowledgement. The author wishes to thank Johan Råde for useful discussions.

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References 1. Bogomol’nyi, E.B.: The stability of classical solutions. Soviet J. Nuclear Phys. 24, no. 4, 449–454 (1976) 2. Jaffe, A. and Taubes, C.H.: Vortices and Monopoles. Boston: Birkhäuser, 1980 3. Sedlacek, S.: A direct method for minimizing the Yang-Mills functional over 4-manifolds. Commun. Math. Phys. 86, no. 4, 515–527 (1982) 4. Taubes, C.H.: Arbitrary N -vortex solutions to the first order Ginzburg–Landau equations. Commun. Math. Phys. 72, no. 3, 277–292 (1980) 5. Taubes, C.H.: On the equivalence of the first and second order equations for gauge theories. Commun. Math. Phys. 75, no. 3, 207–227 (1980) 6. Taubes, C.H.: Monopoles and maps from S 2 to S 2 ; the topology of the configuration space. Commun. Math. Phys. 95, no. 3, 345–391 (1984) Communicated by A. Jaffe

Commun. Math. Phys. 216, 23 – 57 (2001)

Communications in



Combinatorics of q-Characters of Finite-Dimensional Representations of Quantum Affine Algebras Edward Frenkel, Evgeny Mukhin Department of Mathematics, University of California, Berkeley, CA 94720, USA Received: 16 December 1999 / Accepted: 12 July 2000

Abstract: We study finite-dimensional representations of quantum affine algebras using q-characters. We prove the conjectures from [FR2] and derive some of their corollaries. In particular, we prove that the tensor product of fundamental representations is reducible if and only if at least one of the pairwise normalized R-matrices has a pole. Introduction The intricate structure of the finite-dimensional representations of quantum affine algebras has been extensively studied from different points of view, see, e.g., [CP1, CP2, CP3, CP4, GV,V, KS,AK, FR2]. While a lot of progress has been made, many basic questions remained unanswered. In order to tackle those questions, E. Frenkel and N. Reshetikhin introduced in [FR2] a theory of q-characters for these representations. One of the motivations was the theory of deformed W-algebras developed in [FR1]: the representation ring of a quantum affine algebra should be viewed as a deformed W-algebra, while the q-character homomorphism should be viewed as its free field realization. The study of q-characters in [FR2] was based on two main conjectures. One of the goals of the present paper is to prove these conjectures and to derive some of their corollaries. Let us describe our results in more detail. Let g be a simple Lie algebra, g be the corresponding non-twisted affine Kac-Moody algebra, and Uq g be its quantized universal enveloping algebra (quantum affine algebra for short). Denote by I the set of vertices of g be the Grothendieck ring of Uq g. The q-character the Dynkin diagram of g. Let Rep Uq homomorphism is an injective homomorphism χq from Rep Uq g to the ring of Laurent ±1 polynomials in infinitely many variables Y = Z[Yi,a ]i∈I ;a∈C× . This homomorphism should be viewed as a q-analogue of the ordinary character homomorphism. Indeed, let G be the connected simply-connected algebraic group corresponding to g, and let T be its maximal torus. We have a homomorphism χ : Rep G → Fun T (where Fun T stands for the ring of regular functions on T ), defined by the formula (χ (V ))(t) = Tr V t, for all t ∈ T . Upon the identification of Rep G with Rep Uq g and of

24

E. Frenkel, E. Mukhin

Fun T with Z[yi±1 ]i∈I , where yi is the function on T corresponding to the fundamental weight ωi , we obtain a homomorphism χ : Rep Uq g → Z[yi±1 ]i∈I . One of the properties ±1 by yi±1 in χq (V ), where V is a Uq g-module, then of χq is that if we replace each Yi,a we obtain χ (V |Uq g ). The two conjectures from [FR2] that we prove in this paper may be viewed as qanalogues of the well-known properties of the ordinary characters. The first of them, Theorem 4.1, is the analogue of the statement that the character of any irreducible Uq g-module W equals the sum of terms which correspond to the weights of the form λ − i∈I ni αi , ni ∈ Z+ , where λ = i∈I li ωi , li ∈ Z+ , is the highest weight of V , and αi , i ∈ I , are the simple roots. In other words, we have: χ (W ) = m+ (1 + p Mp ), where m+ = i∈I yili , and each Mp is a product of factors aj−1 , j ∈ I , corresponding to the negative simple roots. Theorem 4.1 says that for any irreducible Uq g-module V , χq (V ) = m+ (1 + p Mp ), where m+ is a monomial in Yi,a , i ∈ I, a ∈ C× , with positive powers only (the highest weight monomial), and each Mp is a product of factors × A−1 j,c , j ∈ I, c ∈ C , which are the q-analogues of the negative simple roots of g. The second statement, Theorem 5.1, gives an explicit description of the image of the q-character homomorphism χq . This is a generalization of the well-known fact that the image of the ordinary character homomorphism χ is equal to the subring of invariants of Z[yi±1 ]i∈I under the action of the Weyl group W of g. Recall that the Weyl group is generated by the simple reflections si , i ∈ I . The subring of invariants of si in Z[yi±1 ]i∈I is equal to Ki = Z[yj±1 ]j =i ⊗ Z[yi + yi ai−1 ], Ki . and hence we obtain a ring isomorphism Rep Uq g i∈I

In Theorem 5.1 (see also Corollary 5.7) we establish a q-analogue of this isomorphism. Instead of the simple reflections we have the screening operators Si , i ∈ I , introduced in [FR2]. We show that Im χq equals Ker Si . Moreover, Ker Si is equal to i∈I

⊗ Z[Yi,b + Yi,b A−1 i,bqi ]b∈C× . Thus, we obtain a ring isomorphism Rep Uq g Ki . Ki =

±1 Z[Yj,a ]j =i;a∈C×

i∈I

These results allow us to construct in a purely combinatorial way the q-characters of the fundamental representations of Uq g, see Sect. 5.5. We derive several corollaries of these results. Here is one of them (see Theorem 6.7 and Proposition 6.15). For each fundamental weight ωi , there exists a family of Uq gmodules, Vωi (a), a ∈ C× (see Sect. 1.3 for the precise definition). These are irreducible g, which have highest weight ωi if restricted to finite-dimensional representations of Uq Uq g. They are called the fundamental representations of Uq g (of level 0). According to a theorem of Chari-Pressley [CP1, CP3] (see Corollary 1.4 below), any irreducible representation of Uq g can be realized as a subquotient of a tensor product of the fundamental representations. The following theorem, which was conjectured, e.g., in [AK], describes under what conditions such a tensor product is reducible. Denote by h∨ the dual Coxeter number of g, and by r ∨ the maximal number of edges connecting two vertices of the Dynkin diagram of g. For the definition of the normalized R-matrix, see Sect. 2.3.

Combinatorics of the q-Characters

25

Theorem. Let {Vk }k=1,...,n , where Vk = Vωs(k) (ak ), be a set of fundamental representations of Uq g. The tensor product V1 ⊗. . .⊗Vn is reducible if and only if for some i, j ∈ {1, . . . , n}, i = j , the normalized R-matrix R Vi ,Vj (z) has a pole at z = aj /ai . In that case aj /ai is necessarily equal to q k , where k is an integer, such that 2 ≤ |k| ≤ r ∨ h∨ . The paper is organized as follows. In Sect. 1 we recall the main definitions and results on quantum affine algebras and their finite-dimensional representations. In Sect. 2 we give the definition of the q-character homomorphism and list some of its properties. In Sect. 3 we develop our main technical tool: the restriction homomorphisms τJ . Sections 4 and 5 contain the proofs of Conjectures 1 and 2 from [FR2], respectively. In Sect. 6 we use these results to describe the structure of the q-characters of the fundamental representations and to prove the above Theorem. The results of this paper can be generalized to the case of the twisted quantum affine algebras. In the course of writing this paper we were informed by H. Nakajima that he obtained an independent proof of Conjecture 1 from [FR2] in the ADE case using a geometric approach. 1. Preliminaries on Finite-Dimensional Representations of Uq g 1.1. Root data. Let g be a simple Lie algebra of rank +. Let h∨ be the dual Coxeter number of g. Let ·, · be the invariant inner product on g, normalized as in [K], so that the square of the length of the maximal root equals 2 with respect to the induced inner product on the dual space to the Cartan subalgebra h of g (also denoted by ·, ·). Denote by I the set {1, . . . , +}. Let {αi }i∈I and {ωi }i∈I be the sets of simple roots and of fundamental weights of g, respectively. We have: αi , ωj =

αi , αi δij . 2

Let r ∨ be the maximal number of edges connecting two vertices of the Dynkin diagram of g. Thus, r ∨ = 1 for simply-laced g, r ∨ = 2 for B+ , C+ , F4 , and r ∨ = 3 for G2 . In this paper we will use the rescaled inner product (·, ·) = r ∨ ·, · on h∗ . Set

D = diag(r1 , . . . , r+ ),

where ri =

αi , αi (αi , αi ) = r∨ . 2 2

(1.1)

The ri ’s are relatively prime integers. For simply-laced g, all ri ’s are equal to 1 and D is the identity matrix. Now let C = (Cij )1≤i,j ≤+ be the Cartan matrix of g, Cij =

2(αi , αj ) . (αi , αi )

26


Let B = (Bij )1≤i,j ≤+ be the symmetric matrix B = DC, i.e., Bij = (αi , αj ) = r ∨ αi , αj . Let q ∈ C× be such that |q| < 1. Set qi = q ri , and [n]q =

q n − q −n . q − q −1

Following [FR1, FR2], define the + × + matrices B(q), C(q), D(q) by the formulas Bij (q) = [Bij ]q , Cij (q) = (qi + qi−1 )δij + (1 − δij )[Cij ]q , Dij (q) = [Dij ]q = δij [ri ]q . We have: B(q) = D(q)C(q). Let C(q) be the inverse of the Cartan matrix C(q), C(q)C(q) = Id. We will need the following property of matrix C(q). Lemma 1.1. All coefficients of the matrix C(q) can be written in the form ij (q) = C

(q) C ij d(q)

,

i, j ∈ I,

(1.2)

(q), d(q) are Laurent polynomials in q with non-negative integral coefficients, where C ij symmetric with respect to the substitution q → q −1 . Moreover, ij (q) < deg d(q), deg C

i, j ∈ I.

Proof. We write here the minimal choice of d(q), which we use in Sect. 3.2: A+ B+ C+ D+ E6 E7

: : : : : :

d(q) d(q) d(q) d(q) d(q) d(q)

= = = = = =

q + + q +−2 + · · · + q −+ , q 2+−1 + q 2+−3 + · · · + q −2+−1 , q ++1 + q −+−1 , (q + q −1 )(q +−1 + q −++1 ), (q 2 + 1 + q −2 )(q 6 + q −6 ), (q + q −1 )(q 9 + q −9 ),

E8 : d(q) = (q + q −1 )(q 15 + q −15 ), F4 : d(q) = q 9 + q −9 , G2 : d(q) = q 6 + q −6 . For Lie algebras of classical series, the statement of the lemma with the above d(q) ij (q) of the matrix C(q) given in follows from the explicit formulas for the entries C Appendix C of [FR1]. For exceptional types, the lemma follows from a case by case inspection of the matrix C(q).


27

g in the Drinfeld–Jimbo 1.2. Quantum affine algebras. The quantum affine algebra Uq realization [Dr1, J] is an associative algebra over C with generators xi± , ki±1 (i = 0, . . . , +), and relations: ki ki−1 = ki−1 ki = 1,

ki kj = kj ki ,

ki xj± ki−1 = q ±Bij xj± , [xi+ , xj− ] = δij 1−Cij

r=0

(−1)r

1 − Cij r

qi

ki − ki−1

qi − qi−1

,

(xi± )r xj± (xi± )1−Cij −r = 0,

i = j.

g. Here (Cij )0≤i,j ≤+ denotes the Cartan matrix of The algebra Uq g has a structure of a Hopf algebra with the comultiplication 3 and the antipode S given on the generators by the formulas: 3(ki ) = ki ⊗ ki , 3(xi+ ) = xi+ ⊗ 1 + ki ⊗ xi+ ,

3(xi− ) = xi− ⊗ ki−1 + 1 ⊗ xi− ,

S(xi+ ) = −xi+ ki ,

S(xi− ) = −ki−1 xi− ,

S(ki±1 ) = ki∓1 .

We define a Z-gradation on Uq g by setting: deg x0± = ±1, deg xi± = deg ki = 0, i ∈ I = {1, . . . , +}. Denote the subalgebra of Uq g generated by ki±1 , xi+ (resp., ki±1 , xi− ), i = 0, . . . , +, by Uq b+ (resp., Uq b− ). The algebra Uq g is defined as the subalgebra of Uq g with generators xi± , ki±1 , where i ∈ I. We will use Drinfeld’s “new” realization of Uq g, see [Dr2], described by the following theorem. g has another realization as the algeTheorem 1.2 ([Dr2, KT, LSS, B]). The algebra Uq ± bra with generators xi,n (i ∈ I , n ∈ Z), ki±1 (i ∈ I ), hi,n (i ∈ I , n ∈ Z\0) and central elements c±1/2 , with the following relations: ki kj = kj ki , ki hj,n = hj,n ki , ± −1 ± ki = q ±Bij xj,n , ki xj,n

1 ± ± [hi,n , xj,m ] = ± [nBij ]q c∓|n|/2 xj,n+m , n ± ± ± ± ± ± ± ± xi,n+1 xj,m − q ±Bij xj,m xi,n+1 = q ±Bij xi,n xj,m+1 − xj,m+1 xi,n , 1 cn − c−n [hi,n , hj,m ] = δn,−m [nBij ]q , n q − q −1 + − c(n−m)/2 φi,n+m − c−(n−m)/2 φi,n+m + − [xi,n , xj,m ] = δij , qi − qi−1

28

E. Frenkel, E. Mukhin s π∈6s k=0

(−1)k

s ± ± x± . . . xi,n x± x± . . . xi,n = 0, π(k) j,m i,nπ(k+1) π(s) k q i,nπ(1) i

s = 1 − Cij , for all sequences of integers n1 , . . . , ns , and i = j , where 6s is the symmetric group ± on s letters, and φi,n ’s are determined by the formula

∞ ∞ ± 7± (1.3) φi,±n u±n = ki±1 exp ±(q − q −1 ) hi,±m u±m . i (u) := n=0

m=1

C× ,

there is a Hopf algebra automorphism τa of Uq g defined on the For any a ∈ generators by the following formulas: ± ± τa (xi,n ) = a n xi,n ,

τa (c

1/2

) =

c1/2 ,

± ± τa (φi,n ) = a n φi,n ,

(1.4)

τa (ki ) = ki ,

for all i ∈ I, n ∈ Z. Given a Uq g-module V and a ∈ C × , we denote by V (a) the pull-back of V under τa . Define new variables ki±1 , i ∈ I , such that Cij kj = ki , kj = ki . ki kj (1.5) i∈I

Thus, while ki corresponds to the simple root αi , ki corresponds to the fundamental weight ωi . We extend the algebra Uq g by replacing the generators ki±1 , i ∈ I with g will stand for the extended algebra. ki±1 , i ∈ I . From now on Uq Let q 2ρ = k12 . . . k+2 . The square of the antipode acts as follows (see [Dr3]): S 2 (x) = τq −2r ∨ h∨ (q −2ρ xq 2ρ ),

∀x ∈ Uq g.

(1.6)

Let w0 be the longest element of the Weyl group of g. Let i → i¯ be the bijection I → I , such that w0 (αi ) = −αi¯ . Define the algebra automorphism w0 : Uq g → Uq g by w0 ( ki ) = ki¯ ,

w0 (hi,n ) = hi,n ¯ ,

± ± w0 (xi,n ) = xi,n ¯ .

(1.7)

We have: w02 = Id. Actually, w0 is a Hopf algebra automorphism, but we will not use this fact. g. In this section we recall some of the 1.3. Finite-dimensional representations of Uq results of Chari and Pressley [CP1, CP2, CP3, CP4] on the structure of finite-dimensional representations of Uq g. Let P be the weight lattice of g. It is equipped with the standard partial order: the weight λ is higher than the weight µ if λ − µ can be written as a combination of the simple roots with positive integral coefficients. A vector w in a Uq g-module W is called a vector of weight λ ∈ P , if ki · w = q (λ,αi ) w,

i ∈ I.

(1.8)


29

A representation W of Uq g is said to be of type 1 if it is the direct sum of its weight spaces W = ⊕λ∈P Wλ , where Wλ = {w ∈ W |ki · w = q (λ,αi ) w}. If Wλ = 0, then λ is called a weight of W . A representation V of Uq g is called of type 1 if c1/2 acts as the identity on V , and if V is of type 1 as a representation of Uq g. According to [CP1], every finite-dimensional irreducible representation of Uq g can be obtained from a type 1 representation by twisting with an automorphism of Uq g. Because of that, we will only consider type 1 representations in this paper. A vector v ∈ V is called a highest weight vector if + · v = 0, xi,n

± ± φi,n · v = ψi,n v,

c1/2 v = v,

∀i ∈ I, n ∈ Z,

(1.9)

± . A type 1 representation V is a highest weight represenfor some complex numbers ψi,n tation if V = Uq g · v, for some highest weight vector v. In that case the set of generating functions ∞ ± ?i± (u) = ψi,±n u±n , i ∈ I, n=0

is called the highest weight of V . Warning. The above notions of highest weight vector and highest weight representation are different from standard. Sometimes they are called pseudo-highest weight vector and pseudo-highest weight representation. Let P be the set of all I -tuples (Pi )i∈I of polynomials Pi ∈ C[u], with constant term 1. Theorem 1.3 ([CP1, CP3]). (1) Every finite-dimensional irreducible representation of Uq g of type 1 is a highest weight representation. (2) Let V be a finite-dimensional irreducible representation of Uq g of type 1 and highest weight (?i± (u))i∈I . Then, there exists P = (Pi )i∈I ∈ P such that −1 deg(Pi ) Pi (uqi )

?i± (u) = qi

Pi (uqi )

,

(1.10)

as an element of C[[u±1 ]]. Assigning to V the I -tuple P ∈ P defines a bijection between P and the set of isomorphism classes of finite-dimensional irreducible representations of Uq g of type 1. The irreducible representation associated to P will be denoted by V (P). (3) The highest weight of V (P) considered as a Uq g-module is λ = i∈I deg Pi ·ωi , the lowest weight of V (P) is λ = − i∈I deg Pi · ωi¯ , and each of them has multiplicity 1. (4) If P = (Pi )i∈I ∈ P, a ∈ C× , and if τa∗ (V (P)) denotes the pull-back of V (P) by the automorphism τa , we have τa∗ (V (P)) ∼ g, where = V (Pa ) as representations of Uq Pa = (Pia )i∈I and Pia (u) = Pi (ua). (5) For P, Q ∈ P denote by P ⊗ Q ∈ P the I -tuple (Pi Qi )i∈I . Then V (P ⊗ Q) is isomorphic to a quotient of the subrepresentation of V (P) ⊗ V (Q) generated by the tensor product of the highest weight vectors.

30


An analogous classification result for Yangians has been obtained earlier by Drinfeld [Dr2]. Because of that, the polynomials Pi (u) are called Drinfeld polynomials. Note that in our notation the polynomials Pi (u) correspond to the polynomials Pi (uqi−1 ) in the notation of [CP1, CP3]. (i) For each i ∈ I and a ∈ C× , define the irreducible representation Vωi (a) as V (Pa ), (i) where Pa is the I -tuple of polynomials, such that Pi (u) = 1 − ua and Pj (u) = 1, ∀j = i. We call Vωi (a) the i th fundamental representation of Uq g. Note that in general Vωi (a) is reducible as a Uq g-module. Theorem 1.3 implies the following g ocCorollary 1.4 ([CP3]). Any irreducible finite-dimensional representation V of Uq curs as a quotient of the submodule of the tensor product Vωi1 (a1 ) ⊗ . . . ⊗ Vωin (an ), generated by the tensor product of the highest weight vectors. The parameters (ωik , ak ), k = 1, . . . , n, are uniquely determined by V up to permutation. 2. Definition and First Properties of q-Characters 2.1. Definition of q-characters. Let us recall the definition of the q-characters of finiteg from [FR2]. dimensional representations of Uq Uq The completed tensor product Uq g ⊗ g contains a special element R called Uq b− and satisfies the the universal R-matrix (at level 0). It actually lies in Uq b+ ⊗ following identities: 3 (x) = R3(x)R−1 , (3 ⊗ id)R = R R , 13

23

∀x ∈ Uq g, (id ⊗3)R = R13 R12 .

For more details, see [Dr3, EFK]. Now let (V , πV ) be a finite-dimensional representation of Uq g. Define the transfermatrix corresponding to V by tV = tV (z) = Tr V (πV (z) ⊗ id)(R).

(2.1)

Thus we obtain a map νq : Rep Uq g → Uq b− [[z]], sending V to tV (z). Remark 2.1. Note that in [FR2] there was an extra factor q 2ρ in formula (2.1). This factor is inessential for the purposes of this paper, and therefore can be dropped. ± , ki , hi,r , n ≤ 0, r < 0, i ∈ I . g the subalgebra of Uq g generated by xi,n Denote by Uq It follows from the proof of Theorem 1.2 that Uq b− ⊂ Uq g. As a vector space, Uq g can be decomposed as follows: Uq g = Uq n− ⊗ Uq n+ , where Uq n± (resp., Uq h ⊗ Uq h) is ± generated by xi,n , i ∈ I, n ≤ 0 (resp., ki , hi,n , i ∈ I, n < 0). Hence

Uq g = Uq g · (Uq n+ )0 + (Uq n− )0 · Uq g , h ⊕ Uq

where (Uq n± )0 stands for the augmentation ideal of Uq n± . Denote by hq the projection Uq g → Uq h along the last two summands (this is an analogue of the Harish-Chandra homomorphism). We denote by the same letter its restriction to Uq b− . Now we define the map χq : Rep Uq g → Uq h[[z]] as the composition of νq : Rep Uq g → Uq b− [[z]] and hq [[z]] : Uq b− [[z]] → Uq h[[z]].


31

To describe the image of χq we need to introduce some more notation. Let j i (q m )hj,m , C hi,m =

(2.2)

j ∈I

where C(q) is the inverse matrix to C(q) defined in Sect. 1.1. Set Yi,a

−1 −1 n n hi,−n z a , = ki exp −(q − q )

a ∈ C× .

(2.3)

n>0

±1 the weight ±ωi . We assign to Yi,a

We have the ordinary character homomorphism χ : Rep Uq g → Z[yi±1 ]i∈I : if V = µ ⊕µ Vµ is the weight decomposition of V , then χ (V ) = µ dim Vµ · y , where for mi µ µ = i∈I mi ωi we set y = i∈I yi . Define the homomorphism ±1 β : Z[Yi,a ]i∈I ;a∈C× → Z[yi±1 ]i∈I

±1 to yi±1 , and denote by sending Yi,a

g → Rep Uq g res : Rep Uq the restriction homomorphism. Given a polynomial ring Z[xα±1 ]α∈A , we denote by Z+ [xα±1 ]α∈A its subset consisting of all linear combinations of monomials in xα±1 with positive integral coefficients. Theorem 2.2 ([FR2]). ±1 (1) χq is an injective homomorphism from Rep Uq ]i∈I ;a∈C× ⊂ Uq g to Z[Yi,a h[[z]].

±1 ]i∈I ;a∈C× . g, χq (V ) ∈ Z+ [Yi,a (2) For any finite-dimensional representation V of Uq (3) The diagram χq

±1 Rep Uq g −−−−→ Z[Yi,a ]i∈I ;a∈C×   res β χ

Rep Uq g −−−−→

Z[yi±1 ]i∈I

is commutative. g is a commutative ring that is isomorphic to Z[ti,a ]i∈I ;a∈C× , where ti,a is (4) Rep Uq the class of Vωi (a). The homomorphism ±1 χq : Rep Uq ]i∈I ;a∈C× g → Z[Yi,a

is called the q-character homomorphism. For a finite-dimensional representation V of Uq g, χq (V ) is called the q-character of V .

32


2.2. Spectra of 7± (u). According to Theorem 2.2(1), the q-character of any finite±1 with dimensional representation V of Uq g is a linear combination of monomials in Yi,a positive integral coefficients. The proof of Theorem 2.2 from [FR2] allows us to relate the monomials appearing in χq (V ) to the spectra of the operators 7± i (u) on V as follows. ± commute with each other. It follows from the defining relations that the operators φi,n Hence we can decompose any representation V of Uq g into a direct sum V = ⊕V(γ ± ) i,n of generalized eigenspaces ± ± p V(γ ± ) = {x ∈ V | there exists p, such that (φi,n − γi,n ) · x = 0, ∀i ∈ I, n ∈ Z}. i,n

Since φ0± = ki±1 , all vectors in V(γ ± ) have the same weight (see formula (1.8) for the i,n definition of weight). Therefore the decomposition of V into a direct sum of subspaces V(γ ± ) is a refinement of its weight decomposition. i,n

± Given a collection (γi,n ) of generalized eigenvalues, we form the generating functions ± Ei± (u) = γi,±n u±n . n≥0

We will refer to each collection {Ei± (u)}i∈I occurring on a given representation V as ± as the the common (generalized) eigenvalues of 7± ) i (u), i ∈ I , on V , and to dim V(γi,n multiplicity of this eigenvalue. ± Let BV be a Jordan basis of φi,n , i ∈ I, n ∈ Z. Consider the module V (z) = τz∗ (V ), see formula (1.4). Then V (z) = V as a vector space. Moreover, the decomposition in the ± direct sum of generalized eigenspaces of operators φi,n does not depend on z, because ± the action of φi,n on V and on V (z) differs only by scalar factors zn . In particular, ± BV is also a Jordan basis for φi,n acting on V (z) for all z ∈ C× . If v ∈ BV is a generalized eigenvector with common eigenvalues {Ei± (u)}i∈I , then the corresponding common eigenvalues on v in V (z) are {Ei± (zu)}i∈I The following result is a generalization of Theorem 1.3. Proposition 2.3 ([FR2]). The eigenvalues Ei± (u) of 7± i (u) on any finite-dimensional g have the form: representation of Uq deg Qi −deg Ri

Ei± (u) = qi

Qi (uqi−1 )Ri (uqi ) Qi (uqi )Ri (uqi−1 )

,

(2.4)

as elements of C[[u±1 ]], where Qi (u), Ri (u) are polynomials in u with constant term 1. Now we can relate the monomials appearing in χq (V ) to the common eigenvalues of 7± i (u) on V . Proposition 2.4. Let V be a finite-dimensional Uq g-module. There is a one-to-one correspondence between the monomials occurring in χq (V ) and the common eigenvalues of 7± i (u), i ∈ I , on V . Namely, the monomial   ki li −1   Yi,air (2.5) Yi,b is i∈I

r=1

s=1


33

corresponds to the common eigenvalues (2.4), where Qi (z) =

ki

(1 − zair ),

Ri (z) =

r=1

li

(1 − zbis ),

i ∈ I.

(2.6)

s=1

The weight of each monomial equals the weight of the corresponding generalized eigenspace. Moreover, the coefficient of each monomial in χq (V ) equals the multiplicity of the corresponding common eigenvalue. ± Proof. Denote by Uq , i ∈ I, n ∈ Z. Let B(q) n± the subalgebra of Uq g generated by xi,n be the inverse matrix to B(q) from Sect. 1.1. The following formula for the universal R-matrix has been proved in [KT, LSS, Da]:

R = R+ R0 R− T ,

(2.7)

where R = exp − 0

n(q − q −1 )2 n>0 i∈I

qin − qi−n

hi,n ⊗ hi,−n z

n

(2.8)

(here we use the notation (2.2)), R± ∈ Uq n± ⊗ Uq n∓ , and T acts as follows: if x, y satisfy ki · x = q (λ,αi ) x, ki · y = q (µ,αi ) y, then T · x ⊗ y = q −(λ,µ) x ⊗ y.

(2.9)

By definition, χq (V ) is obtained by taking the trace of (πV (z) ⊗ id)(R) over V and then projecting it on Uq h[[z]] using the projection operator hq . This projection eliminates the factor R− , and then taking the trace eliminates R+ (recall that Uq n+ acts nilpotently on V ). Hence we obtain:

χq (V ) = Tr V exp −

n(q − q −1 )2 n>0 i∈I

qin − qi−n

n πV (hi,n ) ⊗ hi,−n z (πV ⊗ 1)T . (2.10)

The trace can be written as the sum of terms mv corresponding to the (generalized) ± eigenvalues of hi,n on the vectors v of the Jordan basis BV of V for the operators φi,n (and hence for hi,n ). The eigenvalues of 7± i (u) on each vector v ∈ BV are given by formula (2.4). Suppose that Qi (u) and Ri (u) are given by formula (2.6). Then the eigenvalue of hi,n on v equals   ki li qin − qi−n  (air )n − (bis )n  , n(q − q −1 ) r=1

n > 0.

(2.11)

s=1

Substituting into formula (2.10) and recalling the definition (2.3) of Yi,a we obtain that the corresponding term mv in χq (V ) is the monomial (2.5).

34


Let V = V (P), where Pi (u) =

ni

(i)

(1 − uak ),

i ∈ I.

(2.12)

k=1

Then by Theorem 1.3(3), the module V has highest weight λ = i∈I deg Pi · ωi , which has multiplicity 1. Proposition 2.4 implies that χq (V ) contains a unique monomial of weight λ. This monomial equals ni i∈I k=1

Yi,a (i) .

(2.13)

k

We call it the highest weight monomial of V . All other monomials in χq (V ) have lower weight than λ. ±1 −1 A monomial in Z[Yi,a ]i∈I,a∈C× is called dominant if it does not contain factors Yi,a (i.e., if it is a product of Yi,a ’s in positive powers only). The highest weight monomial is dominant, but in general the highest weight monomial is not the only dominant monomial occurring in χq (V ). Nevertheless, we prove below in Corollary 4.5 that the only dominant monomial contained in the q-character of a fundamental representation Vωi (a) is its highest weight monomial Yi,a . Note that a dominant monomial has dominant weight but not all monomials of dominant weight are dominant. ±1 Similarly, a monomial in Z[Yi,a ]i∈I,a∈C× is called antidominant if it does not contain −1 factors Yi,a (i.e., if it is a product of Yi,a ’s in negative powers only). The roles of dominant and antidominant monomials are similar, see, e.g., Remark 6.19. By Corollary 6.9, the lowest weight monomial is antidominant. Remark 2.5. The statement analogous to Proposition 2.3 in the case of the Yangians has been proved by Knight [Kn]. Using this statement, he introduced the notion of character of a representation of Yangian. 2.3. Connection with the entries of the R-matrix. We already described the q-character of Uq g module V in terms of universal R-matrix and in terms of generalized eigenvalues ± of operators φi,n . It allows us to describe the q-character of V in terms of diagonal entries of R-matrices acting on the tensor products V ⊗Vωi (a) with fundamental representations. We will use this description in Sect. 6. Define

−1 −1 n n Ai,a = ki exp −(q − q ) a ∈ C× . hi,−n z a , (2.14) n>0

Using formula (2.2), we can express Ai,a in terms of Yj,b ’s: −1 −1 −1 −1 −1 −1 Yj,a Yj,aq Yj,aq −1 Yj,aq Ai,a = Yi,aqi Yi,aq −1 2 Yj,a Yj,aq −2 . (2.15) i

Cj i =−1

Cj i =−2

Cj i =−3

±1 Thus, Ai,a ∈ Z[Yj,b ]j ∈I ;b∈C× , and the weight of Ai,a equals αi .


35

g with highest Let V and W be irreducible finite-dimensional representations of Uq weight vectors v and w. Let R V W (z) ∈ End(V ⊗ W ) be the normalized R-matrix, R V W (z) = fV−1 W (z)(πV (z) ⊗ πW )(R), where fV W (z) is the scalar function, such that R V W (z)(v ⊗ w) = w ⊗ v.

(2.16)

In what follows we always consider the normalized R-matrix R V W (z) written in the basis BV ⊗ BW . Recall the definition of the fundamental representation Vωi (a) from Sect. 1.3. Denote its highest weight vector by vωi . Lemma 2.6. Let v ∈ BV and suppose that the corresponding monomial mv in χq (V ) is given by mv = m+ M A−1 (2.17) i,ak , k

× where M is a product of factors A−1 j,b , b ∈ C , j ∈ I , j = i. Then the diagonal entry of the normalized R-matrix R V ,Vωi (b) (z) corresponding to the vector v ⊗ vωi is

R V ,Vωi (b) (z)

v⊗vωi ,v⊗vωi

=

k

qi

1 − ak zb−1 qi−1 . 1 − ak zb−1 qi

(2.18)

Proof. Recall formula (2.7) for R. We have: R− (v ⊗ vωi ) = 0; v ⊗ vωi is a generalized eigenvector of R0 ; and R+ (v ⊗ vωi ) is a linear combination of tensor products x ⊗ y ∈ BV ⊗ BVωi (b) , where y has a lower weight than vωi . Therefore the diagonal matrix element of R on v ⊗ vωi ∈ V (z) ⊗ Vωi (b) equals the generalized eigenvalue of (πV (z) ⊗ πVωi (b) )(R0 ) on v ⊗ vωi . On the other hand, as explained in the proof of Proposition 2.4, the monomial mv is equal to the diagonal matrix element of (πV (z) ⊗ 1)(R0 ) corresponding to v. Therefore the diagonal matrix element of R corresponding to v ⊗ vωi equals the eigenvalue of mv (considered as an element of Uq h[[z]]) on vωi . In particular, if v is the highest weight vector, then the corresponding monomial mv is the highest weight monomial m+ . Therefore we find that the diagonal matrix element of the non-normalized R-matrix corresponding to v ⊗ vωi equals the eigenvalue of m+ on vωi . By formula (2.16) the diagonal matrix element of the normalized R-matrix equals 1. Therefore the eigenvalue of m+ on vωi equals the scalar function fV ,Vωi (b) (z). Therefore we obtain that the diagonal matrix element of the normalized R-matrix R V ,Vωi (b) (z) corresponding to the vector v⊗vωi is equal to the eigenvalue of mv m−1 + on vωi .According −1 a −1 ). Therefore, if m is given by formula (2.17), we (z to formula (2.14), Ai,a = 7− v i obtain from formula (1.10) that this matrix element is given by formula (2.18). Note that by Theorem 4.1 below every monomial occurring in the q-character of an irreducible representation V can be written in the form (2.17).

36


3. The Homomorphisms τJ and Restrictions gJ . Given a subset J of I , we denote by Uq gJ the subalgebra of 3.1. Restriction to Uq ± ±1 Uq , ki , hi,r , i ∈ J, n ∈ Z, r ∈ Z\0. Let g generated by xi,n g → Rep Uq gJ resJ : Rep Uq be the restriction map and βJ be the homomorphism ±1 ±1 ]i∈I ;a∈C× → Z[Yi,a ]i∈J ;a∈C× , Z[Yi,a ±1 to itself for i ∈ J and to 1 for i ∈J . sending Yi,a According to Theorem 3(3) of [FR2], the diagram χq

±1 Rep Uq g −−−−→ Z[Yi,a ]i∈I ;a∈C×   res β J J χq,J

±1 Rep Uq gJ −−−−→ Z[Yi,a ]i∈J ;a∈C×

is commutative. We will now refine the homomorphisms βJ and resJ . 3.2. The homomorphism τJ . Consider the elements hi,n defined by formula (2.2) and ki±1 defined by formula (1.5). Lemma 3.1. ± −1 ± ki xj,n ki = q ±ri δij xj,n ,

[nri ]q ∓|n|/2 ± xj,n+m , c n [nri ]q cn − c−n [ hi,n , hj,m ] = δi,j δn,−m . n q − q −1 ± [ hi,n , xj,m ] = ±δij

hi,n , i ∈ J , n ∈ Z\0, where J = I −J , commute with the subalgebra In particular, ki±1 , Uq gJ of Uq g. Proof. These formulas follow from the relations given in Theorem 1.2 and the formula B(q)C(q) = D(q). Denote by Uq g generated by ki±1 , h⊥ hi,n , i ∈ J , n ∈ Z\0. Then J the subalgebra of Uq is naturally a subalgebra of U gJ ⊗ Uq g. We can therefore refine the restriction Uq h⊥ q J from Uq g-modules to Uq gJ -modules by considering the restriction from Uq g-modules to Uq -modules. gJ ⊗ Uq h⊥ J Thus, we look at the common (generalized) eigenvalues of the operators ki±1 , hi,n , i ∈ J , and ki±1 , hi,n , i ∈ J . We know that the eigenvalues of hi,n have the form (2.11). The corresponding eigenvalue of hi,n equals   kj lj [n]q j i (q n )[rj ]q n  (aj r )n − n > 0. (3.1) (bj s )n  , C n j ∈I

r=1

s=1


37

j i (x) = C (x)/d(x), where C (x) and d(x) are certain According to Lemma 1.1, C ji ji polynomials with positive integral coefficients (we fix a choice of such d(x) once and for all). Therefore formula (3.1) can be rewritten as   ui ti [n]q  (cim )n − (dip )n  , nd(q n ) m=1

(3.2)

p=1

where cim and dip are certain complex numbers (they are obtained by multiplying aj r (q)[rj ]q ). and bj s with all monomials appearing in C ji According to Proposition 2.4, to each monomial (2.5) in χq (V ) corresponds a generalized eigenspace of hi,n , i ∈ I, n ∈ Z \ 0, with the common eigenvalues given by formula (2.11) (note that the eigenvalues of ki , i ∈ I , can be read off from the weight of the monomial). Using formula (3.1) we find the corresponding eigenvalues of hi,n , i ∈ J in the form (3.2). Now we attach to these common eigenvalues the following monomial ±1 ±1 in the letters Yi,a , i ∈ J , and Zj,c ,j ∈ J: 

ki



i∈J r=1

Yi,air

li s=1

  −1   · Yi,b is

uk

Zk,ckm

k∈J m=1

tk p=1

 −1  . Zk,d kp

The above procedure can be interpreted as follows. Introduce the notation ±1 Y = Z[Yi,a ]i∈I,a∈C× ,

(3.3)

±1 ±1 Y(J ) = Z[Yi,a ]i∈J,a∈C× ⊗ Z[Zk,c ]k∈J ,c∈C× .

(3.4)

Write (q))ij = (D(q)C

pij (k)q k .

k∈Z

Definition 3.2. The homomorphism τJ : Y → Y(J ) is defined by the formulas τJ (Yi,a ) = Yi,a ·

j ∈J k∈Z

τJ (Yi,a ) =

j ∈J k∈Z

p (k)

ij Zj,aq k ,

p (k)

ij Zj,aq k ,

i ∈ J,

i ∈ J.

(3.5) (3.6)

Observe that the homomorphism βJ can be represented as the composition of τJ and ±1 ]i∈J,a∈C× sending all Zk,c , k ∈ J , to 1. Therefore the homomorphism Y(J ) → Z[Yi,a τJ is indeed a refinement of τJ , and so the restriction of τJ to the image of Rep Uq g in Y is a refinement of the restriction homomorphism resJ .

38


3.3. Properties of τJ . The main advantage of τJ over βJ is the following. Lemma 3.3. The homomorphism τJ is injective. (q) is nonProof. The statement of the lemma follows from the fact that the matrix C degenerate. ±1 Lemma 3.4. Let us write χq (V ) as the sum k Pk Qk , where Pk ∈ Z[Yi,a ]i∈J,a∈C× , ±1 Qk is a monomial in Z[Zj,c ]j ∈J ,c∈C× , and all monomials Qk are distinct. Then the restriction of V to Uq gJ is isomorphic to ⊕k Vk , where Vk ’s are Uq gJ -modules with χqJ (Vk ) = Pk . In particular, there are no extensions between different Vk ’s in V . Proof. The monomials in χq (V ) ∈ Y encode the common eigenvalues of hi,n , i ∈ I on V . It follows from Sect. 3.2 that the monomials in τJ (χq (V )) encode the common eigenvalues of hi,n , i ∈ J , and hj,n , j ∈ J , on V . gJ ⊗ Uq h⊥ Therefore we obtain that the restriction of V to Uq J has a filtration with the gJ -module with χqJ (Vk ) = Pk , and associated graded factors Vk ⊗ Wk , where Vk is a Uq Wk is a one-dimensional Uq h⊥ J -module, which corresponds to Qk . By our assumption, ⊥ gJ , the modules Wk over Uq hJ are pairwise distinct. Because Uq h⊥ J commutes with Uq gJ ⊗ Uq there are no extensions between Vk ⊗ Wk and Vl ⊗ Wl for k = l, as Uq h⊥ Jmodules. Hence the restriction of V to Uq gJ is isomorphic to ⊕k Vk . Write d(q)[ri ]q =

si (k)q k .

k∈Z

Set Bi,a =

k∈Z

s (k)

i Zi,aq k.

Lemma 3.5. We have: τJ (Ai,a ) = βJ (Ai,a ),

i ∈ J,

τJ (Ai,a ) = βJ (Ai,a )Bi,a ,

(3.7)

i ∈ J.

(q)C(q) = D(q)d(q). Proof. This follows from the formula D(q)C

(3.8)

In the case when J consists of a single element j ∈ I , we will write Y(J ) , τJ and βJ simply as Y(j ) , τj and βj . Consider the diagram (we use the notation (3.3), (3.4)): τj

Y −→ ↓

Y(j ) −1

τj

Y −→

↓ Aj,x

(3.9)

Y(j )

where the map corresponding to the right vertical row is the multiplication by βj (Aj,x )−1 ⊗ 1. 2 . The following result will allow us to reduce various statements to the case of Uq sl


39

Lemma 3.6. There exists a unique map Y → Y, which makes the diagram (3.9) commutative. This map is the multiplication by A−1 j,x . Proof. The fact that multiplication by A−1 j,x makes the diagram commutative follows from formula (3.7). The uniqueness follows from the fact that τj and the multiplication by βj (Aj,x )−1 ⊗ 1 are injective maps. 4. The Structure of q-Characters In this section we prove Conjecture 1 from [FR2]. Let V be an irreducible finite-dimensional Uq g module V generated by highest weight vector v. Then by Proposition 3 in [FR2], χq (V ) = m+ (1 + Mp ), (4.1) p × where each Mp is a monomial in A±1 i,c , c ∈ C and m+ is the highest weight monomial. ±1 In what follows, by a monomial in Z[xα ]α∈A we will always understand a monomial in reduced form, i.e., one that does not contain factors of the form xα xα−1 . Thus, in particular, if we say that a monomial M contains xα , it means that there is a factor xα in M which can not be cancelled.

Theorem 4.1. The q-character of an irreducible finite-dimensional Uq g module V has × (i.e., it does not the form (4.1), where each Mp is a monomial in A−1 , i ∈ I , c ∈ C i,c contain any factors Ai,c ). Proof. The proof follows from a combination of Lemmas 3.3, 3.6 and 1.1. First, we observe that it suffices to prove the statement of Theorem 4.1 for fundamental representations Vωi (a). Indeed, then Theorem 4.1 will be true for any tensor product of the fundamental representations. By Corollary 1.4, any irreducible representation V can be represented as a quotient of a submodule of a tensor product W of fundamental representations, which is generated by the highest weight vector. Therefore each monomial in a q-character of V is also a monomial in the q-character of W . In addition, the highest weight monomials of the q-characters of V and W coincide. This implies that Theorem 4.1 holds for V . 2 . Indeed, by the argument above, it suffices Second, Theorem 4.1 is true for g = Uq sl to check the statement for the fundamental representation V1 (a). But its q-character is known explicitly (see [FR2], formula (4.3)): −1 −1 χq (V1 (a)) = Ya + Yaq 2 = Ya (1 + Aaq ),

(4.2)

and it satisfies the required property. g, we will prove Theorem 4.1 (for the case of For general quantum affine algebra Uq the fundamental representations) by contradiction. Suppose that the theorem fails for some fundamental representation Vωi0 (a0 ) = V and denote by χ its q-character χq (V ). Denote by m+ the highest weight monomial Yi0 ,a of χ . Recall from Sect. 1.3 that we have a partial order on the weight lattice. It induces a partial order on the monomials occurring in χ . Let m be the highest weight monomial

40


in χ , such that m can not be written as a product of m+ with a monomial in A−1 i,c , i ∈ I , × c ∈ C . This means that any monomial m in χ , such that m > m, is a product of m+ and A−1 i,c ’s.

(4.3)

In Lemmas 4.2 and 4.3 we will establish certain properties of m and in Lemma 4.4 we will prove that these properties can not be satisfied simultaneously. ±1 ]i∈I,a∈C× is called dominant if does not contain Recall that a monomial in Z[Yi,a −1 factors Yi,a (i.e., if it is a product of Yi,a ’s in positive powers only). Lemma 4.2. The monomial m is dominant. −1 Proof. Suppose m is not dominant. Then it contains a factor of the form Yi,a , for some i ∈ I . Consider τi (χ ). By Lemma 3.4, we have χqi (Vp ) · Np , τi (χ ) = p

±1 2 = Uq where Vp ’s are representation of Uqi sl , j = i. g{i} and Np ’s are monomials in Zj,a We have already shown that Theorem 4.1 holds for Uqi sl2 , so

mp (1 + τi (χ ) = M r,p ) · Np , (4.4) p

r

where each mp is a product of Yi,b ’s (in positive powers only), and each M r,p is a product −1

−1 −1 of several factors Ai,c = Yi,cq −1 Yi,cq (note that M r,p = τi (Mr,p ).

−1 by our assumption, the monomial τi (m) is not among the Since m contains Yi,a monomials {mp · Np }. Hence

τi (m) = mp0 M r0 ,p0 · Np0 , for some p0 , r0 and M r0 ,p0 = 1. There exists a monomial m in χ , such that τi (m ) = mp0 · Np0 . Therefore using Lemma 3.6 we obtain that m = m Mr0 ,p0 , −1

where Mr0 ,p0 is obtained from M r0 ,p0 by replacing all Ai,c by A−1 i,c . In particular, m > m and by our assumption (4.3) it can be written as m = m+ M , where M is a product of A−1 k,c . But then m = m Mr0 ,p0 = m+ M Mr0 ,p0 , and so m can be written as a product of m+ and a product of factors A−1 k,c . This is a contradiction. Therefore m has to be dominant.

Lemma 4.3. The monomial m can be written in the form m = m+ M Aj0 ,ap ,

(4.5)

p

× where M is a product of factors A−1 i,c , i ∈ I , c ∈ C . In other words, if m contains factors Aj,a , then all such Aj,a have the same index j = j0 .


41

Proof. Suppose that m = m+ M, where M contains a factor Ai,c . Let Vm be the generalized eigenspace of the operators kj±1 , hj,n , j ∈ I , corresponding to the monomial m. We claim that for all v ∈ Vm we have: + · v = 0, xj,n

j ∈ I, j = i,

n ∈ Z.

(4.6)

Indeed, let τj (m) = βj (m) · N (recall that βj (m) is obtained from m by erasing all Ys,c ±1 , s ∈ I , s = j ). By Lemma 3.4, x + · v belongs with s = j and N is a monomial in Zs,c j,n to the direct sum of the generalized eigenspaces Vmp , corresponding to the monomials mp in χ such that τj (mp ) = βj (mp ) · N (with the same N as in τj (m) = βj (m) · N ). By formula (3.8), A±1 βj (Aik ,ck )±1 Bi±1 . τj m+ ik ,ck = τj (m+ ) k ,ck ik =j

In particular, N contains a factor Bi,c , and therefore all monomials mp with the above property must contain a factor Ai,c . By our assumption (4.3), the weight of each mp can + not be higher than the weight of m. But the weight of xj,n · v should be greater than the weight of m. Therefore we obtain formula (4.6). Now, if M contained factors Ai,c and Aj,d with i = j , then any non-zero eigenvector (not generalized) in the generalized eigenspace Vm corresponding to m would be a highest weight vector (see formula (1.9)). Such vectors do not exist in V , because V is irreducible. The statement of the lemma now follows. Lemma 4.4. Let m be any monomial in the q-character of a fundamental representation that can be written in the form (4.5). Then m is not dominant. Proof. We say a monomial M ∈ Y (see (3.3)) has lattice support with base a0 ∈ C× if ±1 ] . M ∈ Z[Yi,a q k i∈I,k∈Z 0

Any monomial m ∈ Y can be uniquely written as a product m = m(1) . . . m(s) , where each monomial m(i) has lattice support with a base ai , and ai /aj ∈ q Z for i = j . Note that a non-constant monomial in A±1 , i ∈ I, k ∈ Z, can not be equal to a monomial in i,bq k

, i ∈ I, k ∈ Z if b/c ∈ q Z . Therefore if m can be written in the form (4.5), then A±1 i,cq k

each m(i) can be written in the form (4.5), where m+ = Yi0 ,a if ai = a, and m+ = 1 if a/ai ∈ q Z (note that the product over p in (4.5) may be empty for some m(i) ). We will prove that none of m(i) ’s is dominant unless m(i) = m+ or m(i) = 1. Consider first the case of m(1) , which has lattice support with base a. Then p (n) i Yi,aq m(1) = n . i∈I n∈Z

Define Laurent polynomials Pi (x), i ∈ I by pi (n)x n . Pi (x) = n∈Z

If m(1) can be written in the form (4.5), then Cij (x)Rj (x) + δi,i0 , Pi (x) = − j ∈I

∀i ∈ I,

(4.7)

42


where Rj (x)’s are some polynomials with integral coefficients. All of these coefficients are non-negative if j = j0 . Now suppose that m(1) is a dominant monomial. Then each Pi (x) is a polynomial with non-negative coefficients. We claim that this is possible only if all Ri (x) = 0. Indeed, according to Lemma 1.1, the coefficients of the inverse matrix to C(x), C(x), (x), d(x) are polynomials with non-negative can be written in the form (1.2), where C jk (x), we obtain coefficients. Multiplying (4.7) by C j ∈I

j k (x) + d(x)Rk (x) = C i ,k (x), Pj (x)C 0

∀k ∈ I.

(4.8)

Given a Laurent polynomial p(x) =

pi x i ,

p−r = 0, ps = 0,

−r≤i≤s

we will say that the length of p(x) equals r + s. Clearly, the length of the sum and of the product of two polynomials with non-negative coefficients is greater than or equal to the length of each of them. Therefore if k = j0 , and if Rk (x) = 0, then the length of the LHS is greater than or equal to the length of d(x), which is greater than the length by Lemma 1.1. This implies that Rk (x) = 0 for k = j0 . of C i0 ,k Hence m(1) can be written in the form

m(1) = Yi,a

n∈Z

Acj0n,aq n .

But such a monomial can not be dominant because its weight is ωi − nαj0 , where n > 0, and such a weight is not dominant. This proves the required statement for the factor m(1) of m (which has lattice support with base a). Now consider a factor m(i) with lattice support with base b, such that b/a ∈ q Z . In this case we obtain the following equation: the LHS of formula (4.8) = 0. The previous discussion immediately implies that there are no solutions of this equation with nonzero polynomials Rk (x) satisfying the above conditions. This completes the proof of the lemma. Theorem 4.1 now follows from Lemmas 4.2, 4.3 and 4.4.

Corollary 4.5. The only dominant monomial in χq (Vωi (a)) is the highest weight monomial Yi,a . Proof. This follows from the proof of Lemma 4.4.

5. A Characterization of q-Characters in Terms of the Screening Operators In this section we prove Conjecture 2 from [FR2].


43

5.1. Definition of the screening operators. First we recall the definition of the screening ±1 ]i∈I ;a∈C× from [FR2] and state the main result. operators on Y = Z[Yi,a Consider the free Y-module with generators Si,x , x ∈ C× , Yi = ⊕ Y · Si,x . x∈C×

Let Yi be the quotient of Yi by the relations Si,xq 2 = Ai,xqi Si,x . i

Clearly,

Yi

⊕

x∈(C× /qi2Z )

(5.1)

Y · Si,x ,

and so Yi is also a free Y-module. Define a linear operator Si : Y → Yi by the formula Si (Yj,a ) = δij Yi,a Si,a and the Leibniz rule: Si (ab) = b Si (a) + a Si (b). In particular, −1 −1 ) = −δij Yi,a Si,a . Si (Yj,a

Finally, let

Si : Y → Yi be the composition of Si and the projection Yi → Yi . We call Si the i th screening operator. The following statement was conjectured in [FR2] (Conjecture 2). Theorem 5.1. The image of the homomorphism χq equals the intersection of the kernels of the operators Si , i ∈ I . 2 . In the rest of this section we In [FR2] this theorem was proved in the case of Uq sl prove it for an arbitrary Uq g. 5.2. Description of Ker Si . First, we describe the kernel of Si on Y. The following result was announced in [FR2], Proposition 6. Proposition 5.2. The kernel of Si : Y → Yi equals ±1 ]j =i;a∈C× ⊗ Z[Yi,b + Yi,b A−1 Ki = Z[Yj,a i,bqi ]b∈C× .

(5.2)

Proof. A simple computation shows that Ki ⊂ Ker Y Si . Let us show that Ker Y Si ⊂ Ki . ±1 For x ∈ C× , denote by Y(x) the subring Z[Yj,xq n ]j ∈I,n∈Z of Y. We have: Y Lemma 5.3.

Ker Y Si =

⊗

Y(x).

⊗

Ker Y(x) Si .

x∈(C× /q Z )

x∈(C× /q Z )

44


±1 Proof. Let P ∈ Y, and suppose it contains Yj,a for some a ∈ C× and j ∈ I . Then we can write P as the sum k Rk Qk , where Qk ’s are distinct monomials, which are ±1 products of the factors Ys,aq n , s ∈ I, n ∈ Z (in particular, one of the Qk ’s could be equal ±1 to 1), and Rk ’s are polynomials which do not contain Ys,aq n , s ∈ I, n ∈ Z. Then

Si (P ) =

(Qk · Si (Rk ) + Rk · Si (Qk )). k

By definition of Si , Si (Qk ) belongs to Y · Si,a , while Si (Rk ) belongs to the direct sum of Y · Si,b , where b ∈ aq Z . Therefore if P ∈ Ker Y Si , then k Q k ·Si (Rk ) = 0. Since Qk ’s are distinct, we obtain that Rk ∈ Ker Y Si . But then Si (P ) = k Rk · Sk (Qk ). Therefore P can be written as l Rl Ql , where each Ql is a linear combination of the Qk ’s, such that Ql ∈ Ker Y Si . This proves that P ∈ Ker Y( =a) Si ⊗ Ker Y(a) Si , ±1 where Y( =a) = Z[Yj,b ]j ∈I,b ∈aq Z . By repeating this procedure we obtain the lemma

±1 (because each polynomial contains a finite number of variables Yj,a , we need to apply this procedure finitely many times).

According to Lemma 5.3, it suffices to show that Ker Y(x) Si ⊂ Ki (x), where ±1 −1 Ki (x) = Z[Yj,xq n ]j =i;n∈Z ⊗ Z[Yi,xq n + Yi,xq n Ai,xq n q ]n∈Z . i −1 −1 Denote Yj,xq n by yj,n , Aj,xq n by aj,n , and Aj,xq n Yj,xq nq Y j

j,xq n qj−1

by a j,n . Note that

±1 a j,n does not contain factors yj,m , m ∈ Z. Let T be the shift operator on Y(x) sending yj,n to yj,n+1 for all j ∈ I . It follows from the definition of Si that P ∈ Ker Y(x) Si if and only if T (P ) ∈ Ker Y(x) Si . Therefore (applying T m with large enough m to P ) we can assume without loss of generality that −1 ±1 ] ⊗ Z[yj,n ]j =i,n≥0 . P ∈ Z[yi,n , yi,n+2r i n≥0 We find from the definition of Si :

Si (yj,n ) = 0, Si (yi,2ri n+I ) = yi,I

j = i, n k=1

2 yi,2r a · Si,xq I , i k+I i,ri (2k−1)+I

(5.3)

whereI ∈ {0, 1, . . . , 2ri − 1}. Therefore each P ∈ Ker Y(x) Si can be written as a sum P = PI , where each PI ∈ Ker Y(x) Si and −1 ±1 ]n≥0 ⊗ Z[yj,n ]j =i,n≥0 . PI ∈ Z[yi,2ri n+I , yi,2r i (n+1)+I

It suffices to consider the case I = 0. Thus, we show that if −1 ±1 P ∈ Y≥0 i (x) = Z[yi,2ri n , yi,2ri (n+1) ]n≥0 ⊗ Z[yj,n ]j =i,n≥0 ,

then

±1 P ∈ Ki≥0 (x) = Z[tn ]n≥0 ⊗ Z[yj,n ]j =i,n≥0 ,


where

45

−1 −1 tn = yi,2ri n + yi,2ri n ai,r = yi,2ri n + yi,2r a −1 . i (2n+1) i (n+1) i,ri (2n+1)

±1 Consider a homomorphism Ki≥0 (x) ⊗ Z[yi,2ri n ]n≥0 → Y≥0 i (x) sending yj,n , j = i

±1 −1 to yj,n , yi,2ri n to yi,2ri n , and tn to yi,2ri n + yi,2r a −1 . This homomorphism is i (n+1) i,ri (2n+1) surjective, and its kernel is generated by the elements

(tn − yi,2ri n )a i,ri (2n+1) yi,2ri (n+1) − 1.

(5.4)

≥0 Therefore we identify Y≥0 i (x) with the quotient of Ki (x) ⊗ Z[yi,2ri n ]n≥0 by the ideal generated by elements of the form (5.4). Consider the set of monomials ±1 tn1 . . . tnk yi,2ri m1 . . . yi,2ri ml yj,p , j j =i,pj ≥0

where all n1 ≥ n2 ≥ . . . nk ≥ 0, m1 ≥ m2 ≥ . . . ml ≥ 0, and also mj = ni + 1 for all i and j . We call these monomials reduced. It is easy to see that the set of reduced monomials is a basis of Y≥0 i (x). Now let P be an element of the kernel of Si on Y≥0 i (x). Let us write it as a linear a combination of the reduced monomials. We represent P as yi,2r Q + R. Here N is iN the largest integer, such that yi,2ri N is present in at least one of the basis monomials appearing in its decomposition; a > 0 is the largest power of yi,2ri N in P ; Q = 0 does a a not contain yi,2ri N , and R is not divisible by yi,2r . Recall that here both yi,2r Q and iN iN R are linear combinations of reduced monomials. ±1 Recall that Si (tn ) = 0, Si (yj,n ) = 0, j = i, and Si (yi,2ri n ) is given by formula (5.3). Suppose that N > 0. According to formula (5.3), a+1 Si (P ) = ayi,2r iN

N−1 k=1

yi,2ri k

N

a i,ri (2l−1) yi,0 Q · Si,x + . . . ,

(5.5)

l=1

a+1 . Note that the where the dots represent the sum of terms that are not divisible by yi,2r iN

first term in (5.5) is non-zero because the ring Y≥0 i (x) has no divisors of zero. The monomials appearing in (5.5) are not necessarily reduced. However, by construca tion, Q does not contain tN−1 , for otherwise yi,2r Q would not be a linear combination iN of reduced monomials. Therefore when we rewrite (5.5) as a linear combination of reduced monomials, each reduced monomial occurring in this linear combination is still a+1 divisible by yi,2r . On the other hand, no reduced monomials occurring in the other iN a+1 terms of Si (P ) (represented by dots) are divisible by yi,2r . Hence for P to be in the iN kernel, the first term of (5.5) has to vanish, which is impossible. Therefore P does not contain yi,2ri m ’s with m > 0. p p −1 But then P = k yi,0k Rk , where Rk ∈ Ki≥0 (x), and Si (P ) = k pk yi,0k Rk · Si,x . Such P is in the kernel of Si if and only if all pk = 0 and so P ∈ Ki≥0 (x). This completes the proof of Proposition 5.2. Set K=

i∈I

Ki =

i∈I

±1 ]j =i;a∈C× ⊗ Z[Yi,b + Yi,b A−1 Z[Yj,a i,bqi ]b∈C× .

(5.6)

Now we will prove that the image of the q-character homomorphism χq equals K.

46


g in Y 5.3. The image of χq is a subspace of K. First we show that the image of Rep Uq under the q-character homomorphism belongs to the kernel of Si . ±1 ±1 ]a∈C× ⊗ Z[Zj,c ]j =i,c∈C× and the homomorphism τi : Recall the ring Y(i) = Z[Yi,a (i) Y → Y from Sect. 3.2. ±1 Let Yi be the quotient of ⊕ Z[Yi,a ]a∈C× · Si,x by the submodule generated by the x∈C×

elements of the form Si,xq 2 − Ai,xqi Si,x , where Ai,xqi = Yi,x Yi,xq 2 . Define a derivation i

i

±1 S i : Z[Yi,a ]a∈C× → Yi by the formula S i (Yi,a ) = Yi,a Si,a . Thus, Yi coincides with the 2 and S i is the corresponding screening operator. module Yi in the case of Uqi sl Set (i) ±1 Yi = Z[Zj,c ]j =i,c∈C× ⊗ Yi . (i)

The map S i can be extended uniquely to a map Y(i) → Yi by S i (Zj,c ) = 0 for all j = i, c ∈ C× and the Leibniz rule. We will also denote it by S i . The embedding τi (i) gives rise to an embedding Yi → Yi which we also denote by τi . Lemma 5.4. The following diagram is commutative τi

Y(i)   S i

τi

(i)

Y −−−−→  S i

Yi −−−−→ Yi

Proof. Since τi is a ring homomorphism and both Si , S i are derivations, it suffices to check commutativity on the generators. Let us choose a representative x in each qi2Z coset of C× . Then we can write: Yi =

⊕

x∈C× /qi2Z

(i)

Y · Si,x ,

Yi =

By definition, Si (Yj,xq 2n ) = δij Yi,x

i

S i (Yi,xq 2n ) = Yi,x i

S i (Zj,c ) = 0,

m

⊕

x∈C× /qi2Z

A±1

i,xqi2m+1

m

Y(i) · Si,x .

Si,x ,

±1

Ai,xq 2m+1 Si,x , i

∀j = i.

±1 , j = i, and Recall from formula (3.5) that τi (Yi,x ) equals Yi,x times a monomial in Zj,c ±1

from formula (3.8) that τi (A±1 i,b ) = Ai,b . Using these formulas we obtain: ±1 Ai,xq 2m+1 Si,x . (τi ◦ Si )(Yi,xq 2n ) = (S i ◦ τi )(Yi,xq 2n ) = τi (Yi,x ) i

On the other hand, when j = i, τi (Yj,x ) is a monomial in formula (3.6). Therefore (τi ◦ Si )(Yj,x ) = (S i ◦ τi )(Yj,x ) = 0, This proves the lemma.

i

±1 Zk,c ,k

j = i.

= i, according to


47

g → Y is Corollary 5.5. The image of the q-character homomorphism χq : Rep Uq contained in the kernel of Si on Y. Proof. Let V be a finite-dimensional representation of Uq g. We need to show that Si (χq (V )) = 0. By Lemma 3.4, we can write χq (V ) as the sum k Pk Qk , where (i) ±1 2 → ]a∈C× is in the image of the homomorphism χq : Rep Uqi sl each Pk ∈ Z[Yi,a ±1 ±1 Z[Yi,a ]a∈C× , and Qk is a monomial in Zj,c , j = i. (i)

The image of χq lies in the kernel of the operator S i (in fact, they are equal, but 2 we will not use this now). This immediately follows from the fact that Rep Uq sl Z[χq (V1 (a))] and S i (χq (V1 (a))) = 0, which is obtained by a straightforward calculation. We also have: S i (Zj,c ) = 0, ∀j = i. Therefore (S i ◦ τi )(χq (V )) = 0. By Lemma 5.4, (τi ◦ Si )(χq (V )) = 0. Since τi is injective by Lemma 3.3, we obtain: Si (χq (V )) = 0. 5.4. K is a subspace of the image of χq . Let P ∈ K. We want to show that P ∈ Im χq . A monomial m contained in P ∈ Y is called highest monomial (resp., lowest monomial), if its weight is not lower (resp., not higher) than the weight of any other monomial contained in P . Lemma 5.6. Let P ∈ K. Then any highest monomial in P is dominant and any lowest weight monomial in P is antidominant. Proof. First we prove that the highest monomials are dominant. By Proposition 5.2, ±1 ]j =i;a∈C× ⊗ Z[Yi,b + Yi,b A−1 P ∈ Ki = Z[Yj,a i,bqi ]b∈C× .

The statement of the lemma will follow if we show that a highest weight monomial −1 contained in any element of Ki does not contain factors Yi,a . Indeed, the weight of Yi,a is ωi , and the weight of Yi,b A−1 i,bqi is ωi − αi . Denote

tb = Z[Yi,b + Yi,b A−1 i,bqi ]b∈C× . Given a polynomial Q ∈ Z[tb ]b∈C× , let m1 , . . . , mk be its monomials (in tb ) of highest degree. Clearly, the monomials of highest weight in Q ±1 (considered as a polynomial in Yj,a ) are m1 , . . . , mk , in which we substitute each tb by −1 Yi,b . These monomials do not contain factors Yi,a . The statement about the lowest weight monomials is proved similarly, once we observe that ±1 −1 ]j =i;a∈C× ⊗ Z[Yi,b + Yi,bq −2 Ai,bq −1 ]b∈C× . Ki = Z[Yj,a i

i

Let m be a highest monomial in P , and suppose that it enters P with the coefficient νm ∈ Z \ 0. Then m is dominant by Lemma 5.2. According to Theorem 1.3(2) and g, such that m is the formula (2.13), there exists an irreducible representation V1 of Uq highest weight monomial in χq (V1 ). Since χq (V1 ) ∈ K by Corollary 5.5, we obtain that P1 = P − νm · χq (V1 ) ∈ K. For P ∈ Y, denote by J(P ) the (finite) set of dominant weights λ, such that P contains a monomial of weight greater than or equal to λ. By Proposition 5.2, if P ∈ K and J(P ) is empty, then P is necessarily equal to 0.

48


g of highest weight µ, J(χq (V )) Note that for any irreducible representation V of Uq is the set of all dominant weights which are less than or equal to µ. Therefore J(P1 ) is properly contained in J(P ). By applying the above subtraction procedure finitely many k times, we obtain an element Pk = P − χq (Vi ), for which J(Pk ) is empty. But then i=1

Pk = 0. This shows that K ⊂ Im χq . Together with Lemma 5.5, this gives us Theorem 5.1 and the following corollary. Corollary 5.7. The q-character homomorphism, χq : Rep Uq g → K, where K is given by (5.6), is a ring isomorphism.

5.5. Application: Algorithm for constructing q-characters. Consider the following problem: Give an algorithm which for any dominant monomial m+ constructs the qg-module whose highest weight monomial is m+ . In this character of the irreducible Uq section we propose such an algorithm. We prove that our algorithm produces the qcharacters of the fundamental representations (in this case m+ = Yi,a ). We conjecture that the algorithm works for any irreducible module. Roughly speaking, in our algorithm we start from m+ and gradually expand it in all 2 directions. (Here we use the explicit formulas for q-characters of Uq sl 2 possible Uqi sl and Lemma 3.6.) In the process of expansion some monomials may come from different directions. We identify them in the maximal possible way. First we introduce some terminology. ±1 Let χ ∈ Z≥0 [Yi,a ]i∈I,a∈C× be a polynomial and m a monomial in χ occurring with coefficient s ∈ Z>0 . By definition, a coloring of m is a set {si }i∈I of non-negative integers such that si ≤ s. A polynomial χ in which all monomials are colored is called a colored polynomial. We think of si as the number of monomials of type m which have come from direction 2 ). i (or by expanding with respect to the i th subalgebra Uqi sl −1 , a ∈ C× . A monomial m is called i-dominant if it does not contain variables Yi,a A monomial m occurring in a colored polynomial χ with coefficient s is called admissible if m is j -dominant for all j such that sj < s. A colored polynomial is called admissible if all of its monomials are admissible. Given an admissible monomial m occurring with coefficient s in a colored polynomial χ , we define a new colored polynomial im (χ ), called the i-expansion of χ with respect to m, as follows. If si = s, then im (χ ) = χ . Suppose that si < s and let m be obtained from m by ±1 = 1, for all j = i. Since m is admissible, m is a dominant monomial. setting Yj,a 2 module V , such that the highest weight Therefore there exists an irreducible Uqi sl monomial of V is m. We have explicit formulas for the q-characters of all irreducible 2 -modules (see, e.g., [FR2, Sect. 4.1]). We write χqi (V ) = m(1 + Uq sl p M p ), where −1

M p is a product of Ai,a . Let


49

µ = m(1 +

Mp ),

(5.7)

p −1

where Mp is obtained from M p by replacing all Ai,a by A−1 i,a . The colored polynomial im (χ ) is obtained from χ by adding monomials occurring in µ by the following rule. Let monomial n occur in µ with coefficient t ∈ Z>0 . If n does not occur in χ then it is added with the coefficient t (s − si ) and we set the i th coloring of n to be t (s − si ), and the other colorings to be 0. If n occurs in χ with coefficient r and coloring {ri }i∈I , then the new coefficient of n in im (χ ) is max{r, ri + t (s − si )}. In this case the i th coloring is changed to ri + t (s − si ) and other colorings are not changed. Obviously, the i-expansions of χ with respect to m commute for different i. To expand a monomial m in all directions means to compute +m (. . . 2m (1m (χ )) . . . ), where + = rk(g). Now we describe the algorithm. We start with the colored polynomial m+ with all colorings set equal zero. Let the Uq g-weight of m+ be λ. The set of weights of the form λ − i ai αi , ai ∈ Z≥0 has a natural partial order. Choose any total order compatible with this partial order, so we have λ = λ1 > λ2 > λ3 > . . . . At the first step we expand m+ in all directions. Then we expand in all directions all monomials of weight λ1 obtained at the first step. Then we expand in all directions all monomials of weight λ2 obtained at the previous steps, and so on. Since the monomials obtained in the expansion of a monomial of Uq g-weight µ have weights less than µ, the result does not depend on the choice of the total order. Note that for any monomial m except for m+ occurring with coefficient s at any step, we have maxi {si } = s. This property means that we identify the monomials coming from different directions in the maximal possible way. The algorithm stops if all monomials have been expanded. We say that the algorithm fails at a monomial m if m is the first non-admissible monomial to be expanded. Let m+ be a dominant monomial and V the corresponding irreducible module. Conjecture 5.8. The algorithm never fails and stops after finitely many steps. Moreover, the final result of the algorithm is the q-character of V . Theorem 5.9. Suppose that χq (V ) does not contain dominant monomials other than m+ . Then Conjecture 5.8 is true. In particular, Conjecture 5.8 is true in the case of fundamental representations. Proof. For i ∈ I , let Di be a decomposition of the set of monomials in χq (V ) with multiplicities into a disjoint union of subsets such that each subset forms the q-character of 2 module. We refer to this decompostion Di as the i th decomposition an irreducible Uqi sl of χq (V ). Denote D the collection of Di , i ∈ I . Consider the following colored oriented graph KV (D). The vertices are monomials in χq (V ) with multiplicities. We draw an arrow of color i from a monomial m1 to a monomial m2 if and only if m1 and m2 are in the same subset of the i th decomposition × and m2 = A−1 i,a m1 for some a ∈ C . We call an oriented graph a tree (with one root) if there exists a vertex v (called root), such that there is an oriented path from v to any other vertex. The graph KW (D), where 2 -module is always a tree and its root corresponds to the highest W is an irreducible Uq sl weight monomial. Consider the full subgraph of KV (D) whose vertices correspond to monomials from a given subset of the i th decomposition of χq (V ). All arrows of this subgraph are of color

50


i. By Lemma 3.6, this subgraph is a tree isomorphic to the graph of the corresponding 2 -module. Moreover, its root corresponds to an i-dominant monomial. irreducible Uqi sl Therefore if a vertex of KV (D) has no incoming arrows of color i, then it corresponds to an i-dominant monomial. In particular, if m has no incoming arrows in KV (D), then m is dominant. Since by our assumption χq (V ) does not contain any dominant monomials except for m+ , the graph KV (D) is a tree with root m+ . Choose a sequence of weights λ1 > λ2 > . . . as above. We prove by induction on r the following statement Sr : The algorithm does not fail during the first r steps. Let χr be the resulting polynomial after these steps. Then the coefficient of each monomial m in χr is not greater than that in χq (V ) and the coefficients of monomials of weights λ1 , . . . , λr in χr and χq (V ) are equal. Furthermore, there exists a decomposition D of χq (V ), such that monomials in χr can be identified with vertices in KV (D) in such a way that all outgoing arrows from vertices with Uq g-weights λ1 , . . . , λr go to vertices of χr . Finally, the j th coloring of a monomial m in χr is just the number of vertices of type m in χr which have incoming arrows of color j in KV (D). The statement S0 is obviously true. Assume that the statement Sr is true for some r ≥ 0. Recall that at the (r + 1)st step we expand all monomials of χr of weight λr+1 . Let m be a monomial of weight λr+1 in χr , which enters with coefficient s and coloring {si }i∈I . Then the monomial m enters χq (V ) with coefficient s as well. Indeed, KV (D) is a tree, so all vertices m have incoming arrows from vertices of larger weight. By the statement Sr theses arrows go to vertices corresponding to monomials in χr . Suppose that sj < s for some j ∈ I . Then m is j -dominant. Indeed, otherwise each vertex of type m in KV (D) has an incoming arrow of color j coming from a vertex of higher weight. Then by the last part of the statement Sr , sj = s. Therefore the monomial m is admissible, and the algorithm does not fail at m. Consider the expansion jm (χr ). Let µ be as in (5.7). In the j th decomposition of χq (V ), m corresponds to a root of a tree whose vertices can be identified with monomials in µ. We fix such an identification. Then monomials in µ get identified with vertices in KV (D). Let v be the vertex in KV (D), corresponding to a monomial n in µ. Denote the coefficient of n in χr by p and the coloring by {pi }i∈I . We have two cases: a) pj = p. Then the last part of the statement Sr implies that the vertex v does not belong to χr . We add the monomial n to χr and increase pj by one (we have already identified it with v). b) pj < p. Then by Sr there exists a vertex w in χr of type n with no incoming arrows of color j . We change the decomposition Dj by switching the vertices v and w and identify n with the new v. We also increase pj by one. (Thus, in this case we do not add n to χr .) In both cases, the statement Sr+1 follows. Since the set of weights of monomials occurring in χq (V ) is contained in a finite set λ1 , λ2 , . . . , λN , the statement SN proves the first part of the theorem. Corollary 4.5 then implies the second part of the theorem. We plan to use the above algorithm to compute explicitly the q-characters of the g and to obtain their decompositions under Uq g. fundamental representations of Uq


51

Remark 5.10. There is a similar algorithm for computing the ordinary characters of finitedimensional g-modules (equivalently, Uq g-modules). That algorithm works for those representations (called miniscule) whose characters do not contain dominant weights other than the highest weight (for other representations the algorthim does not work). However, there are very few miniscule representations for a general simple Lie algebra g. In contrast, in the case of quantum affine algebras there are many representations whose characters do not contain any dominant monomials except for the highest weight monomials (for example, all fundamental representations), and our algorithm may be applied to them. 6. The Fundamental Representations In this section we prove several theorems about the irreducibility of tensor products of fundamental representations. 6.1. Reducible tensor products of fundamental representations and poles of R-matrices. In this section we prove that the reducibility of a tensor product of the fundamental representations is always caused by a pole in the R-matrix. We say that a monomial m has positive lattice support with base a if m is a product ±1 Yi,aq n with n ≥ 0. Lemma 6.1. All monomials in χq (Vωi (a)) have positive lattice support with base a. 2 , the statement follows from the explicit formula (4.2) for χq (V1 (a)). Proof. For Uq sl 2 is a subsum of a product The q-character of any irreducible representation V of Uq sl of the q-characters of V1 (b)’s. Moreover, this subsum includes the highest monomial. Hence if the highest weight monomial of χq (V ) has positive lattice support with base a, then so do all monomials in χq (V ). Now consider the case of general Uq g. Suppose there exists a monomial in χ = χq (Vωi (a)), which does not have positive lattice support with base a. Let m be a highest among such monomials (with respect to the partial ordering by weights). By Corollary 4.5, the monomial m is not dominant. In other words, if we rewrite m ±1 as a product of Yi,b , we will have at least one generator in negative power, say Yi−1 . 0 ,b0 Write τi0 (χ ) in the form (4.4). The monomial τi0 (m) can not be among the monomials {mp Np }, since m contains Yi−1 . Therefore τi0 (m) = mp0 Np0 M r0 ,p0 for some M r0 ,p0 = 0 ,b0 −1

1, which is a product of factors Ai,c . Let m1 be a monomial in χ , such that τi0 (m1 ) = mp0 Np0 . Then by Lemma 3.6, m = m1 Mr0 ,p0 , where Mr0 ,p0 is obtained from M r0 ,p0 −1

by replacing all Ai,c with A−1 i,c . By construction, the weight of m1 is higher than the weight of m, so by our assumption, m1 has positive lattice support with base a. But then mp0 also has positive lattice support with base a. Therefore all monomials in mp0 (1 + r M r,p ) have positive lattice support with base a. This implies that Mr0 ,p0 , and hence m = m1 Mr0 ,p0 , has positive lattice support with base a. This is a contradiction, so the lemma is proved. Remark 6.2. From the proof of Lemma 6.1 is clear that the only monomial in χq (Vωi (a)) ±1 which contains Yj,aq n with n = 0 is the highest weight monomial Yi,a .

52


g-module with the q-character χq (V ). Define the oriented graph EV as Let V be a Uq follows. The vertices of EV are monomials in χq (V ) with multiplicities. Thus, there are dim V vertices. We denote the monomial corresponding to a vertex α by mα . We draw an arrow from the vertex α to the vertex β if and only if mβ = mα A−1 i,x for some i ∈ I , x ∈ C× . 2 -module, then the graph EV is connected. Indeed, every If V is an irreducible Uq sl 2 -module is isomorphic to a tensor product of evaluation modules. The irreducible Uq sl graph associated to each evaluation module is connected according to the explicit formulas for the corresponding q-characters (see formula (4.3) in [FR2]). Clearly, a tensor product of two modules with connected graphs also has a connected graph. Lemma 6.3. Let α ∈ EV be a vertex with no incoming arrows. Then mα is a dominant monomial. −1 Proof. Let α contain Yi,b for some i ∈ I , b ∈ C× . We write the restricted q-character τi (χq (V )) in the form (4.4), where each mp (1 + r M r,p ) is a q-character of an irre2 module. ducible Uqi sl −1 The monomial τi (m) contains Yi,b and therefore can not be among the monomials 2 -modules are connected. So we obtain that {mp Np }. But the graphs of irreducible Uq sl −1 τi (m) = τi (Ai,c )τi (m ) for some monomial m in χq (V ), and some c ∈ C× . By Lemma 3.6, we have m = A−1 i,c m which is a contradiction.

Now Corollary 4.5 implies: Corollary 6.4. The graphs of all fundamental representations are connected. Let a monomial m have lattice support with base a. We call m right negative if the factors Yi,aq k appearing in m, for which k is maximal, have negative powers. Lemma 6.5. All monomials in the q-character of the fundamental representation Vωi (a), except for the highest weight monomial, are right negative. Proof. Let us show first that from the highest weight monomial m+ there is only one outgoing arrow to the monomial m1 = m+ A−1 i,aqi . Indeed, the weight of a monomial that is connected to m+ by an arrow has to be equal to ωi −αj for some j ∈ I . The restriction of Vωi (a) to Uq g is isomorphic to the direct some of its i th fundamental representation Vωi and possibly some other irreducible representations with dominant weights less than ωi . However, the weight ωi − αj is not dominant for any i and j . Therefore this weight has to belong to the set of weights of Vωi , and the multiplicity of this weight in Vωi (a) has to be the same as that in Vωi . It is clear that the only weight of the form ωi − αj that occurs in Vωi is ωi − αi , and it has multiplicity one. By Theorem 4.1, this monomial must have the form m1 = m+ A−1 i,aqi . Now, the graph EVωi (a) is connected. Therefore each monomial m in χq (Vωi (a)) is −1 a product of m1 and factors A−1 j,b . Note that m1 is right negative and all Aj,b are right negative (this follows from the explicit formula (2.15)). The product of two right negative monomials is right negative. This implies the lemma. Remark 6.6. It follows from the proof of the lemma that the rightmost factor of each −1 non-highest weight monomial occurring in χq (Vωi (a)) equals Yj,aq n , where n ≥ 2ri . Moreover, the equality holds only for the above monomial m1 (in that case j = i).


53

Recall the definition of the normalized R-matrix R V ,W (z) from Sect. 2.3. The following theorem was conjectured, e.g., in [AK]. Theorem 6.7. Let {Vk }k=1,...,n , where Vk = Vωs(k) (ak ), be a set of fundamental repreg. The tensor product V1 ⊗ . . . ⊗ Vn is reducible if and only if for some sentations of Uq i, j ∈ {1, . . . , n}, i = j , the normalized R-matrix R Vi ,Vj (z) has a pole at z = aj /ai . Proof. The “if” part of the theorem is obvious. Let us explain the case when n = 2. Let σ : V1 ⊗ V2 → V2 ⊗ V1 be the transposition. By definition of R V1 ,V2 (z), the linear map σ ◦ R V1 ,V2 (z) is a homomorphism of Uq g-modules V1 ⊗ V2 → V2 ⊗ V1 . Therefore if R V1 ,V2 (z) has a pole at z = a2 /a1 , then V1 ⊗ V2 is reducible. It is easy to generalize this argument to general n. Now we prove the “only if” part. nIf the product V1 ⊗ · · · ⊗ Vn is reducible, then the product of the q-characters i=1 χq (Vi ) contains a dominant monomial m that is different from the product of the highest weight monomials. Therefore m is not right negative and m is a product of some monomials mi from χq (Vi ). Hence at least one of the factors mi = mi must be −1 appearing in, the highest weight monomial and it has to cancel with the rightmost Yi,b say, mj . According to Lemma 6.1, mj = mj M where M is a product of A−1 s,aj q n . By our

assumption, the maximal n0 occurring among n is such that aj q n0 = ai qi−1 . Using Lemma 2.6 we obtain that one of the diagonal entries of R Vi ,Vj has a factor 1/(1 − ai aj−1 z), which can not be cancelled. Therefore R Vi ,Vj has a pole at z = aj /ai . This proves the “only if” part. Moreover, we see that the pole necessarily occurs in a diagonal entry.

6.2. The lowest weight monomial. Our next goal is to describe (see Proposition 6.15 below) the possible values of the spectral parameters of the fundamental representations for which the tensor product is reducible. First we develop an analogue of the formalism of Sect. 4 from the point of view of the lowest weight monomials. Recall the involution I → I, i → i¯ from Sect. 1.2. According to Theorem 1.3(3), there is a unique lowest weight monomial m− in χq (Vωi (a)), and its weight is −ωi¯ . Lemma 6.8. The lowest weight monomial of χq (Vωi (a)) equals Y¯−1 r ∨ h∨ . i,aq

−1 Proof. By Lemma 5.6, m− must be antidominant. Thus, by Lemma 6.1, m− = Yi,aq ¯ ni for some ni > 0. Recall the automorphism w0 defined in (1.7). The module Vωi¯ (a) is obtained from Vωi (a) by pull-back with respect to w0 . From the interpretation of the q-character in terms of the eigenvalues of 7± i (u), it is clear that the q-character of Vωi¯ (a) is obtained ±1 from the q-character of Vωi (a) by replacing each Yj,b by Yj±1 ¯,b . Therefore we obtain: ni = ni¯ . Consider the dual module Vωi (a)∗ . By Theorem 1.3(3), its highest weight equals ωi¯ . Hence Vωi (a)∗ is isomorphic to Vωi¯ (b) for some b ∈ C× . Since Uq g is a Hopf algebra, the module Vωi (a)⊗ Vωi (a)∗ contains a one–dimensional trivial submodule. Therefore

54


the product of the corresponding q-characters contains the monomial m = 1. According to Lemma 6.5, it can be obtained only as a product of the highest weight monomial in one q-character and the lowest monomial in another. Therefore, b = aq ±ni . In the same way we obtain that Vωi¯ (a)∗ is isomorphic to Vωi (aq ±ni ). From formula (1.6) for the square of the antipode, we obtain that the double dual, ∨ ∨ Vωi (a)∗∗ , is isomorphic to Vωi (aq −2r h ). Since ni > 0, we obtain that ni = r ∨ h∨ . Having found the lowest weight monomial in the q-characters of the fundamental representations, we obtain using Theorem 1.3 the lowest weight monomial in the qcharacter of any irreducible module. Corollary 6.9. Let V be an irreducible Uq g-module. Let the highest weight monomial in χq (V ) be m+ =

sk i∈I k=1

Yi,a (i) . k

Then the lowest weight monomial in χq (V ) is given by m− =

sk i∈I k=1

Y¯−1(i)

i,ak q r

∨ h∨

.

We also obtain a new proof of the following corollary, which has been previously proved in [CP1], Proposition 5.1(b): Corollary 6.10. Vωi (a)∗ Vωi¯ (aq −r

∨ h∨

).

Now we are in position to develop the theory of q-characters based on the lowest weight and antidominant monomials as opposed to the highest weight and dominant ones. Proposition 6.11. The q-character of an irreducible finite-dimensional Uq g module V has the form χq (V ) = m− (1 + Np ), where m− is the lowest weight monomial and each Np is a monomial in Ai,c , i ∈ I , c ∈ C× (i.e., it does not contain any factors A−1 i,c ). Proof. First we prove the following analogue of formula (4.1): χq (V ) = m− (1 + Np ), p × where each Np is a monomial in A±1 i,c , c ∈ C . The proof of this formula is exactly the same as the proof of Proposition 3 in [FR2]. The rest of the proof is completely parallel to the proof of Theorem 4.1.


55

Lemma 6.12. The only antidominant monomial of q-character of a fundamental representation is the lowest weight monomial. Proof. The proof is completely parallel to the proof of Lemma 4.5.

Lemma 6.13. All monomials in a q-character of a fundamental representation are prod±1 ∨ ∨ ucts Yi,aq n with n ≤ r h . Proof. The proof is completely parallel to the proof of Lemma 6.1.

The combination of Lemmas 6.1 and 6.13 yields the following result. Corollary 6.14. Let the highest weight monomial m+ of the q-character of an irreducible (i) Uq g-module V be a product of monomials m+ which have positive lattice support with (i) bases ai . Let si be the maximal integer s, such that Yk,ai q s is present in m+ for some k ∈ I . Then any monomial m in χq (V ) can be written as a product of monomials m(i) , where each m(i) is a product of Yj,ai q n with n ∈ Z, 0 ≤ n ≤ si + r ∨ h∨ 6.3. Restrictions on the values of spectral parameters of reducible tensor products of fundamental representations. It was proved in [KS] that Vωi (a) ⊗ Vωj (b) is irreducible if a/b does not belong to a countable set. As M. Kashiwara explained to us, one can show that this set is then necessarily finite. The following proposition, which was conjectured, e.g., in [AK], gives a more precise description of this set. Proposition 6.15. Let ai ∈ C, i = 1, . . . , n, and suppose that the tensor product of fundamental representations Vωi1 (a1 ) ⊗ . . . ⊗ Vωin (an ) is reducible. Then there exist m = j such that am /aj = q k , where k ∈ Z and 2 ≤ k ≤ r ∨ h∨ . Proof. If Vωi1 (a1 ) ⊗ . . . ⊗ Vωin (an ) is reducible, then χq (Vωi1 (a1 )) . . . χq (Vωin (an )) should contain a dominant term other than the product of the highest weight terms. But for that to happen, for some m and j , there have to be cancellations between some −1 Yp,a n appearing in χq (Vωim (am )) and some Yp,aj q l appearing in χq (Vωi (aj )). These mq j cancellations may only occur if am /aj = q ±k , k ∈ Z, and 0 ≤ k ≤ r ∨ h∨ , by Lemmas 6.1 and 6.13. Moreover, k ≥ 2 according to Remark 6.6. Note that combining Theorem 6.7, Proposition 6.15 and Remark 6.6 we obtain: Corollary 6.16. The set of poles of the normalized R-matrix R Vωi (a),Vωj (a) (z) is a subset

of the set {q k |k ∈ Z, 2ri ≤ |k| ≤ r ∨ h∨ }, if i = j ; {q k |k ∈ Z, 2ri < k ≤ r ∨ h∨ or 2rj < −k ≤ r ∨ h∨ }.

6.4. The q-characters of the dual representations. In this subsection we show a simple way to obtain the q-character of the dual representation. Recall that K is given by (5.6). Lemma 6.17. Let χ1 , χ2 ∈ K. Assume that all dominant monomials in χ1 are the same as in χ2 (counted with multiplicities). Then χ1 = χ2 .

56


Proof. Consider χ = χ1 − χ2 . We have χ ∈ K and χ has no dominant monomials. Then χ = 0 by Lemma 5.6. Note that the similar statement is true for antidominant monomials. Proposition 6.18. Let Vωi (a) be a fundamental representation. Then the q-character of ∨ ∨ the dual representation Vωi (a)∗ Vωi¯ (aq −r h ) is obtained from the q-character of ±1 ∓1 Vωi (a) by replacing each Yi,aq n by Yi,aq −n . ∨ ∨

Proof. Let χ1 = χq (Vωi¯ (aq −r h )) and χ2 is obtained from χ (Vωi (a)) by replacing ±1 ∓1 Yi,aq n by Yi,aq −n . Then χ1 and χ2 are elements in K with the only dominant monomial Yi,aq ¯ −r ∨ h∨ by Corollary 4.5 and Lemma 6.12. Therefore χ1 = χ2 by Lemma 6.17. Remark 6.19. One can define a similar procedure for obtaining the q-character of the g-module V . Namely, by Theorem 1.3, χq (V ) is a subsum in dual to any irreducible Uq the product of q-characters of fundamental representations. In particular, any monomial m in χq (V ) is a product of monomials m(i) from the q-characters of these fundamental representations and Proposition 6.18 tells us what to do with each m(i) . This procedure is consistent because χq ((V ⊗ W )∗ ) = χq (V ∗ ) · χq (W ∗ ). Note that under this procedure the dominant monomials go to the antidominant monomials and vice versa. Acknowledgements. We thank N. Reshetikhin for useful discussions. The research of both authors was supported through E. Frenkel’s Fellowship from the Packard Foundation.

References [AK]

Akasaka, T., Kashiwara, M.: Finite-dimensional representations of quantum affine algebras. Publ. Res. Inst. Math. Sci. 33, no. 5, 839–867 (1997) [B] Beck, J.: Braid group action and quantum affine algebras. Commun. Math. Phys. 165, no. 3, 555–568 (1994) [CP1] Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge: Cambridge University Press, 1994 [CP2] Chari, V., Pressley, A.: Quantum affine algebras. Commun. Math. Phys. 142, no. 2, 261–283 (1991) [CP3] Chari, V., Pressley, A.: Quantum affine algebras and their representations. In: Representations of groups (Banff, AB, 1994), 59–78, CMS Conf. Proc. 16, Providence, RI: Am. Math. Soc., 1995 [CP4] Chari, V., Pressley, A.: Minimal affinizations of representations of quantum groups: The simply laced case. J. Algebra 184, no. 1, 1–30 (1996) [CP5] Chari, V., Pressley, A.: Yangians: their representations and characters. Representations of Lie groups, Lie algebras and their quantum analogues. Acta Appl. Math. 44, no. 1–2, 39–58 (1996) [Da] Damiani, I.: La R-matrice pour les algebres quantiques de type affine non tordu. Ann. Sci. Ecole Norm. Sup. 31, no. 4, 493–523 (1998) [Dr1] Drinfeld, V.G.: Hopf algebras and the quantum Yang–Baxter equation. Sov. Math. Dokl. 32, 254–258 (1985) [Dr2] Drinfeld, V.G.: A new realization of Yangians and of quantum affine algebras. Sov. Math. Dokl. 36, 212–216 (1987) [Dr3] Drinfeld, V.G.: On almost cocommutative Hopf algebras. Leningrad Math. J. 1, 1419–1457 (1990) [EFK] Etingof, P.I., Frenkel, I.B., Kirillov, A.A. Jr., Lectures on Representation Theory and Knizhnik– Zamolodchikov Equations. Providence, RI: AMS, 1998 [FR1] Frenkel, E., Reshetikhin, N.: Deformations of W-algebras associated to simple Lie algebras. Commun. Math. Phys. 197, no. 1, 1–32 (1998) [FR2] Frenkel, E., Reshetikhin, N.: The q-characters of representations of quantum affine agebras and deformations of W-algebras. Preprint math.QA/9810055; in Contemporary Math 248, 163–205, AMS 2000

Combinatorics of the q-Characters [GV] [J] [K] [KS] [KT] [Kn] [LSS] [V]

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Ginzburg, V., Vasserot, E.: Langlands reciprocity for affine quantum groups of type An . Int. Math. Res. Not. no. 3, 67–85 (1993) Jimbo, M.: A q-difference analogue of U (g) and the Yang–Baxter equation. Lett. Math. Phys. 10 no. 1, 63–69 (1985) Kac, V.G.: Infinite-dimensional Lie Algebras. 3rd Edition, Cambridge: Cambridge University Press, 1990 Kazhdan, D., Soibelman, Y.: Representations of quantum affine algebras. Selecta Math. (N.S.) 1, 537–595 (1995) Khoroshkin, S., Tolstoy, V.: Twisting of quantum (super)algebras. Connection of Drinfeld’s and Cartan-Weyl realizations for quantum affine algebras. In: Generalized symmetries in physics (Clausthal, 1993), River Edge, NJ: World Sci. Publishing, 1994, pp. 42–54 Knight, H.: Spectra of tensor products of finite-dimensional representations of Yangians. J. Algebra 174, 187–196 (1995) Levendorsky, S., Soibelman, Ya., Stukopin, V.: The quantum Weyl group and the universal quantum (1) R-matrix for affine Lie algebra A1 . Lett. Math. Phys. 27, no. 4, 253–264 (1993) Vasserot, E.: Affine quantum groups and equivariant K-theory. Transform. Groups. 3, no. 3, 269–299 (1998)

Communicated by T. Miwa

Commun. Math. Phys. 216, 59 – 83 (2001)

Communications in



Reconstructing the Thermal Green Functions at Real Times from Those at Imaginary Times Giovanni Cuniberti1 , Enrico De Micheli2 , Giovanni Alberto Viano3 1 Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany.

E-mail: [email protected]

2 Istituto di Cibernetica e Biofisica – Consiglio Nazionale delle Ricerche, Via De Marini 6, 16149 Genova,

Italy. E-mail: [email protected]

3 Dipartimento di Fisica – Università di Genova and Istituto Nazionale di Fisica Nucleare,

Via Dodecaneso 33, 16146 Genova, Italy. E-mail: [email protected] Received: 17 February 2000 / Accepted: 12 July 2000

Abstract: By exploiting the analyticity and boundary value properties of the thermal Green functions that result from the KMS condition in both time and energy complex variables, we treat the general (non-perturbative) problem of recovering the thermal functions at real times from the corresponding functions at imaginary times, introduced as primary objects in the Matsubara formalism. The key property on which we rely is the fact that the Fourier transforms of the retarded and advanced functions in the energy variable have to be the “unique Carlsonian analytic interpolations” of the Fourier coefficients of the imaginary-time correlator, the latter being taken at the discrete Matsubara imaginary energies, respectively in the upper and lower half-planes. Starting from the Fourier coefficients regarded as “data set”, we then develop a method based on the Pollaczek polynomials for constructing explicitly their analytic interpolations. 1. Introduction In the standard imaginary-time formalism of quantum statistical mechanics (tracing back to Matsubara [15]) and, later on, of quantum field theory at finite temperature (see e.g. [14] and references therein), there arises the a-priori non-trivial problem of recovering the “physical” correlations at real times starting from data at imaginary times. More specifically, the correlations at imaginary-time of observables (or, more generally, of boson or fermion fields) in a thermal equilibrium state at temperature T = β −1 are defined as periodic (or antiperiodic) functions of period β, and therefore they are equivalently characterized by their discrete mode expansion β1 n G n exp(−iζn τ ) in terms of the so-called “Matsubara energies” iζn , where ζn = 2nπ/β (or (2n + 1)π/β). The problem of recovering the correlations at real time, or equivalently the retarded and advanced Green’s functions at real energies, from the previous sequence of Fourier coefficients {Gn } admits a unique and well-defined theoretical solution in terms of the notion of “Carlsonian analytic interpolation of this sequence”. This can be achieved [5], and will be recalled below in Sect. 2, if the imaginary-time formalism is embedded in

60

G. Cuniberti, E. De Micheli, G. A. Viano

the (conceptually more satisfactory) general description of quantum thermal states as KMS states [11]. However, as suggested by the lattice approach of the imaginary-time formalism, it may be interesting to have a concrete procedure for constructing satisfactory approximate solutions of this problem when one starts from incomplete data sets. In this paper we give a precise algorithm for the previous reconstruction problem; this mathematical method is presented in Sect. 3. Moreover, in the subsequent Sect. 4, the method is applied to the case when the data are finite in number and affected by noise. Let us consider the algebra A generated by the observables of a quantum system. Denoting by A, B, . . . arbitrary elements of A and by A → A(t) (A = A(0)) the action of the (time-evolution) group of automorphisms on this algebra, we now recall the KMS analytic structure of two-point correlation functions < A(t1 )B(t2 ) >β , in a thermal equilibrium state β of the system at temperature T = β −1 . By time-translation invariance, these quantities only depend on t = t1 − t2 , and we shall put WAB (t) =< A(t) B >β ,

(1)

=< B A(t) >β .

(2)

(t) WAB

In finite volume approximations, the time-evolution is represented by a unitary group eiH t , so that

A(t) = eiH t A e−iH t ,

(3)

H

where = H − µN , H being the Hamiltonian, µ the chemical potential, and N the particle number; under general conditions, the operators e−βH have finite traces for all β > 0 (see e.g. [11]). Then the correlation functions are given, correspondingly, by the formulae 1 WAB (t) = Tr e−βH A(t) B , (4) Zβ 1 WAB (t) = Tr e−βH BA(t) , (5) Zβ

where Zβ = Tre−βH . One then introduces the following holomorphic functions of the complex time variable t + iγ : 1 Tr e−(β+γ )H A(t) eγ H B , (6) GAB (t + iγ ) = Zβ analytic in the strip {t + iγ ; t ∈ R, −β < γ < 0}, and 1 GAB (t + iγ ) = Tr e−(β−γ )H B e−γ H A(t) , Zβ

(7)

analytic in the strip {t + iγ ; t ∈ R, 0 < γ < β}, which are such that: lim GAB (t + iγ ) = WAB (t),

(8)

lim GAB (t + iγ ) = WAB (t).

(9)

γ →0 γ 0

Reconstructing Thermal Green Functions

61

From (6), (7) and the cyclic property of Tr, we then obtain the KMS relation

WAB (t) = Tre−βH A(t) B = TrB e−βH A(t) = GAB (t + iβ),

(10)

which implies the identity of holomorphic functions (in the strip 0 < γ < β) GAB (t + i(γ − β)) = GAB (t + iγ ).

(11)

According to the analysis of [11] in the Quantum Mechanical framework and of [8] in the Field-theoretical framework, this KMS analytic structure is preserved by the thermodynamic limit under rather general conditions. In the case when the algebra A is generated by smeared-out bosonic or fermionic field operators (field theory at finite temperature), the principle of relativistic causality of the theory implies additional relations for the corresponding pairs of analytic functions (G, G ). In fact, this principle of relativistic causality is expressed by the commutativity (resp. anticommutativity) relations for the boson field (x) (resp. fermion field (x)) at space-like separation: (t, x), (t , x ) = 0 (resp. (t, x), (t , x ) = 0) (12) for (t − t )2 < (x − x )2 . In this field-theoretical case, we can choose as suitable operators A the “smeared-out field operators” of the form A = (y0 , y)f (y0 , y)dy0 dy (resp. (y0 , y)f (y0 , y)dy0 dy), where f is any smooth test-function with (arbitrary small) compact support around the origin in space-time variables. For the observable B, we can then choose any operator Ax obtained from A by the action of the space-translation group (which amounts to replace the test-function f (y0 , y) by f (y0 , y) = f (y0 , y − x)). It then follows from (12) that the corresponding analytic functions GAAx (t +iγ ) and GAAx (t +iγ ) (satisfying (11)) have real boundary values WAAx (t) and WAA (t) which satisfy, on some interval x |t| < t (x, f ), coincidence relations of the following form: WAAx (t) = WAA (t) x

in the boson case,

(13)

WAAx (t) =

in the fermion case.

(14)

−WAA (t) x

Then, in view of identity (11), the coincidence relations (13) and (14) imply the existence of a single analytic function GAAx (t + iγ ) which is such that: a) in the boson case:

b) in the fermion case:

GAAx = GAAx

for − β < γ < 0,

(15)

GAAx =

for 0 < γ < β;

(16)

GAAx

GAAx = GAAx

GAAx = −GAAx

for − β < γ < 0,

(17)

for 0 < γ < β.

(18)

62


Correspondingly, it follows that GAAx is either periodic or antiperiodic with period iβ in the full complex plane minus periodic cuts along the half-lines {t +iγ ; t > t (x, f ), γ = kβ, k ∈ Z} and {t + iγ ; t < −t (x, f ), γ = kβ, k ∈ Z}. These analytic functions GAAx (t + iγ ) are smeared-out forms (corresponding to various test-functions f ) of the thermal two-point function of the fields (or ) in the complex time variable. In other words, this thermal two-point function can be fully characterized in terms of an analytic function G(t + iγ , x) (with regular dependence in the space variables) enjoying the following properties: a) G(t + iγ , x) = G(t + i(γ − β), x), where = + for a boson field, and = − for a fermion field; b) for each x, the domain of G in the complex variable t is C \ {t + iγ ; |t| > |x|; γ = kβ, k ∈ Z}; c) the boundary values of G at real times are the thermal correlations of the field, namely: lim G(t + iγ , x) = W(t, x),

(19)

lim G(t + iγ , x) = W (t, x),

(20)

γ →0 γ 0

where in finite volume regions, W and W can be formally expressed as follows (a rigorous justification of the trace-operator formalism in the appropriate Hilbert space being given in [8]): 1 Tre−βH (t, x) (0, 0), Zβ 1 W (t, x) = Tre−βH (0, 0) (t, x), Zβ W(t, x) =

(21) (22)

for the boson case, and similarly in terms of (t, x) for the fermion case. In this analytic structure, we shall distinguish two quantities that play an important role: i) the restriction G(iγ , x) of the function G to the imaginary axis is a β-periodic (or antiperiodic) function of γ which must be identified with the “time-ordered product at imaginary times”, considered in the Matsubara approach of imaginary-time formalism. In the latter, this quantity or its set of Fourier coefficients plays the role of initial data. ii) The “retarded” and “advanced” two-point functions R(t, x) = i θ (t)[W(t, x) − W (t, x)], A(t, x) = −i θ(−t)[W(t, x) − W (t, x)],

(23) (24)

which are respectively the “jumps” of the function G across the real cuts {t; t ≥ |x|} and {t; t < −|x|}. These kernels have an important causal interpretation; in particular, R describes the “response of the system” to small perturbations of the equilibrium state. The knowledge of R and A and, consequently, of W − W = −i (R − A) allows one to reconstruct W and W by the application of the Bose–

(ω) (this procedure Einstein factor 1/(1−e∓βω ) to their Fourier transforms W(ω), W being an implementation of the KMS property in the energy variable ω).


63

The rest of the paper is devoted to the problem of recovering the “real-time quantities” R and A, starting from the “time-ordered product at imaginary times” as initial data. This will require the conjoint use of the analytic structure of G in complex time and of its Fourier–Laplace transform in the complex energy variable. In fact, the key property on which our reconstruction of real-time quantities relies is the following one: of the functions R and A, which are defined the Fourier–Laplace transforms R and A and analytic respectively in the upper and lower half-planes of the energy variable ω, are analytic interpolations of the set of Fourier coefficients {Gn } of the function G at imaginary times, the latter being taken at the Matsubara energies ω = iζn . Moreover, according to the uniqueness of this interpolation is ensured by global bounds on R and A, n ) to a standard theorem by Carlson [3]. The basic equalities that relate R(iζn ) and A(iζ the corresponding coefficients Gn will be called “Froissart–Gribov-type equalities” for the following historical reason. A general n-dimensional mathematical study of the type of double-analytic structure encountered here has been performed in [6] in connection with the theory of complex angular momentum, where the original Froissart–Gribov equalities had been first discovered (in the old framework of S-matrix theory). The fact that this structure is relevant (in its simplest one-dimensional form) in the analysis of thermal quantum states has been already presented in [5] in the framework of Quantum Field Theory at finite temperature. 2. Double Analytic Structure of the Thermal Green Function and Froissart–Gribov-type Equalities In the following mathematical study we replace the complex time variable t + iγ of the introduction by τ = i(t +iγ ) in such a way that, in our “reconstruction problem” treated in Sects. 3 and 4, the initial data of the function G(τ, ·) considered below correspond to real values of τ . Up to this change of notation, this general analytic function G(τ, ·) can play the role of the previously described two-point function of a boson or fermion field at fixed x. However, since the only variables involved in the forthcoming study are τ and its Fourier-conjugate variable ζ , the extra “spectator variables”, denoted by the point (·), may as well represent a fixed momentum (after Fourier transformation with respect to the space variables) or the action on a test-function f (as for the correlations of field observables A = A(f ) described in the introduction). Let us summarize the analytic structure that we want to study. Hypotheses. The function G(τ, ·), (τ = u + iv, u, v ∈ R), satisfies the following properties: a) it is analytic in the open strips kβ < u < (k + 1)β (v ∈ R, k ∈ Z, β = 1/T ) and continuous at the boundaries; a) it is periodic (antiperiodic) for bosons (fermions) with period β, i.e. G(τ, ·) for bosons, (τ ∈ C), G(τ + β, ·) = (25) −G(τ, ·) for fermions, (τ ∈ C); c)

sup−kβ0

(34)

(v ≤ 0, k ∈ Z). Let us note that these definitions are well-posed and appropriate because, as we shall see in the following theorem, G (+) (τ, ·) and G (−) (τ, ·) are holomorphic in the cut-domains ◦ ◦ I− ∪ [I+ \ (+ ] and I+ ∪ [I− \ (− ], respectively. Moreover, we suppose hereafter that (±) the slow-growth condition (26) extends to the discontinuities J(kβ) (v, ·), that turn out


65

to be “tempered functions” [4]. Finally, in view of the periodicity properties of G(τ, ·), it is sufficient to consider only the strip, in the τ -plane, defined by −a ≤ u ≤ 2β − a (0 < a < β), v ∈ R (see Fig. 1). u 2β

γa

2β−a

β

ε

γβ

0

γ0 γa

-a

v ε

−β

Fig. 1. Integration paths used in the proof of Theorem 1

We then introduce the Laplace transforms of the jump functions across the cuts located at Reτ = 0, and at Reτ = β; i.e. +∞ (+) (+) J˜(0) (ζ, ·) = J(0) (v, ·) e−ζ v dv, (ζ = ξ + iη, Reζ > 0), (35) 0

(−) J˜(0) (ζ, ·) = (+) J˜(β) (ζ, ·) = (−) J˜(β) (ζ, ·) =

0

(−)

J(0) (v, ·) e−ζ v dv,

−∞ +∞ 0

0

−∞

(Reζ < 0),

(36)

J(β) (v, ·) e−ζ v dv, (Reζ > 0),

(37)

(+)

(−)

J(β) (v, ·) e−ζ v dv,

(Reζ < 0).

(38)

We can state the following theorem. (±)

Theorem 1. If the functions G(τ, ·) and J(kβ) (v, ·) satisfy the slow-growth condition (26) ◦

◦

◦

uniformly in )τ = C \ ((+ ∪ (− ) up to the closure, the following properties hold true: i) The function G (+) (τ, ·) (respectively G (−) (τ, ·)) is holomorphic in the cut-domain ◦ ◦ I− ∪ [I+ \ (+ ] (respectively I+ ∪ [I− \ (− ]). (+) (+) ii-a) The Laplace transforms J˜(0) (ζ, ·) and J˜(β) (ζ, ·) are holomorphic in the half-plane (−) (−) Reζ > 0. The Laplace transforms J˜(0) (ζ, ·) and J˜(β) (ζ, ·) are holomorphic in the half-plane Reζ < 0.

66


(+) (+) (+) (+) ii-b) J˜(0) (ζ, ·) and J˜(β) (ζ, ·) belong to the Hardy space H2 C(δ) , where C(δ) =

(−) (−) {ζ ∈ C | Reζ > δ, δ ≥ > 0}. J˜(0) (ζ, ·) and J˜(β) (ζ, ·) belong to the Hardy

(−) (−) space H2 C(δ) , where C(δ) = {ζ ∈ C | Reζ < δ, δ ≥ > 0}. iii-a) In the case of the boson statistics the symmetric combinations G˜ (+,b) (ζ, ·) :≡ (+) (+) (−) (−) J˜ (ζ, ·) + J˜ (ζ, ·) and G˜ (−,b) (ζ, ·) :≡ J˜ (ζ, ·) + J˜ (ζ, ·) interpolate (0)

(β)

(+)

(0) (−)

(β)

uniquely the Fourier coefficients G2m (·) and G2m (·) respectively (hereafter the superscript (b) stands for the boson statistics). Let ζm = 2mπ/β, then the following Froissart–Gribov-type equalities hold: (+) (+) (+) G˜ (+,b) (ζm , ·) = J˜(0) (ζm , ·) + J˜(β) (ζm , ·) = G2m (·),

(39)

(m = 1, 2, 3, . . . ), (−) (−) (−) G˜ (−,b) (ζm , ·) = J˜(0) (ζm , ·) + J˜(β) (ζm , ·) = G2m (·),

(40)

(m = −1, −2, −3, . . . ). ii-b) In the case of the fermion statistics the antisymmetric combinations G˜ (+,f ) (ζ, ·) :≡ (+) (+) (−) (−) J˜(0) (ζ, ·) − J˜(β) (ζ, ·) and G˜ (−,f ) (ζ, ·) :≡ J˜(0) (ζ, ·) − J˜(β) (ζ, ·) interpolate (+)

(−)

uniquely the Fourier coefficients G2m+1 (·) and G2m+1 (·) respectively (hereafter the superscript (f ) stands for the fermion statistics). Let ζm = (2m + 1)π/β, then the following Froissart–Gribov-type equalities hold: (+) (+) (+) G˜ (+,f ) (ζm , ·) = J˜(0) (ζm , ·) − J˜(β) (ζm , ·) = G2m+1 (·),

(41)

(m = 0, 1, 2, 3, . . . ), ˜ (−,f )

G

(−)

(−)

(−)

(ζm , ·) = J˜(0) (ζm , ·) − J˜(β) (ζm , ·) = G2m+1 (·),

(42)

(m = −1, −2, −3, . . . ). Proof. (i) In view of the Riemann–Lebesgue theorem, and since G (+) (τ, ·) ∈ L1 [−β, β], (+) the Fourier coefficients Gn (·) tend to zero as n → ∞. From expansion (29) we have for all τ = u + iv, with v < 0: +∞ 1 (+) (+) −iζn τ |G (τ, ·)| = Gn (·)e eζn v , (43) ≤K 2β n=0

n≥0

β ζn v converges uniformly in any domain where K = −β |G(τ, ·)| dτ . The series +∞ n≥0 e compactly contained in the half-plane Imτ < 0. In view of the Weierstrass theorem on the uniformly convergent series of analytic functions, we can conclude that G (+) (τ, ·) is holomorphic in the half-plane Imτ < 0. By using analogous arguments we can prove that G (−) (τ, ·) is holomorphic in the half-plane Imτ > 0. Furthermore, we know from Hypothesis a) that G(τ, ·) = G (+) (τ, ·) + G (−) (τ, ·) is holomorphic in the strips kβ < u < (k + 1)β (k ∈ Z, v ∈ R), and continuous at the boundaries of the strips. ◦ We can conclude that G (+) (τ, ·) is holomorphic in the cut-domain I− ∪ [I+ \ (+ ], and ◦ G (−) (τ, ·) is holomorphic in the cut-domain I+ ∪ [I− \ (− ].


67

(ii) Property (ii-a) follows easily from the assumption of “temperateness” of the jump (+) functions [4]. For what concerns property (ii-b) we limit ourselves to prove that J˜(0) (ζ, ·)

(+) belongs to the Hardy space H2 C(δ) , since the remaining part of the statement can be proved analogously. To this purpose, we rewrite the Laplace transform (35) in the following form: +∞ (+) (+) (Reζ > 0), (44) J(0) (v, ·)e−δv e−ζ v dv :≡ J˜(0)(δ) (ζ , ·), 0

where Reζ

= Reζ −δ (δ ≥

(+)

> 0). In view of the slow-growth property of J(0) (v, ·), we (+)

can then say that the function J(0) (v, ·) exp(−δv) belongs to the intersection L1 [0, +∞)∩ L2 [0, +∞). Then, thanks to the Paley–Wiener theorem, we can conclude (returning to (+) (+) 2 ˜ the variable ζ ) that J(0) (ζ, ·) belongs to the Hardy space H C(δ) (see Ref. [12]).

(+) Accordingly, J˜(0) (ζ, ·) tends uniformly to zero as ζ tends to infinity inside any fixed (+) half-plane Reζ ≥ δ > δ. In particular, J˜(0) (ζn , ·), with ζn = nπ/β (n = 1, 2, . . . ), tends to zero as n → ∞. (+) (iii) We introduce the integral Iγ defined as follows (this method has been introduced by B.B. [5], and will be developed in a more detailed form in [7] within the general framework of Q.F.T.): (+) Iγ (ζ, ·) = G (+) (τ, ·) eiζ τ dτ, (45) γ

where the path γ encloses both the cuts located at u = 0, v ≥ 0 and at u = β, v ≥ 0 (see Fig. 1). In view of the slow-growth condition (26), this integral is well-defined. By choosing as integration path a pair of contours (γ0 , γβ ) enclosing respectively the cuts at u = 0, v ≥ 0 and at u = β, v ≥ 0, and then flattening them (in a folded way) onto the cuts (see Fig. 1), we obtain: +∞ +∞ (+) (+) (+) −ζ v iζβ I(γ0 ∪γβ ) (ζ, ·) = J(0) (v, ·) e dv + e J(β) (v, ·) e−ζ v dv 0 0 (46) (+) (+) = J˜(0) (ζ, ·) + eiζβ J˜(β) (ζ, ·). (0)

Next, we choose the path γa , whose support is: ] − a + i∞, −a] ∪ [−a, − ] ∪ [γ ] ∪ (β) (β) (0) [ , β − ] ∪ [γ ] ∪ [β + , 2β − a] ∪ [2β − a, 2β − a + i∞[, where γ and γ are half-circles turning around the points τ = 0 and τ = β, respectively (see Fig. 1). By taking into account the 2β-periodicity of G (+) (τ, ·), we get, for ζ = ζn = nπ/β, (n = 1, 2, . . . ): 2β−a lim Iγ(+) (ζ , ·) = G (+) (τ, ·) eiζn τ dτ = Gn(+) (·). (47) n a →0

−a

(+)

Then, from the Cauchy distorsion argument, we have Iγ0 ∪γβ (ζn , ·) = lim that is (+) (+) J˜(0) (ζn , ·) + eiζn β J˜(β) (ζn , ·) = Gn(+) (·).

(+) →0 Iγa (ζn , ·),

(48)

68


We now distinguish two cases: 1) n even: n = 2m, ζm = 2mπ/β (m = 1, 2, . . . ); then from (48) we obtain equalities (39). 2) n odd: n = 2m + 1, ζm = (2m + 1)π/β (m = 0, 1, 2, . . . ); then from (48) we obtain equalities (41). We have thus obtained two combinations (symmetric and antisymmetric, respectively) (+) that interpolate the Fourier coefficients Gn (·). The uniqueness of the interpolation is (+) (+) guaranteed by the Carlson theorem [3] that can be applied since J˜(0) (ζ, ·) and J˜(β) (ζ, ·)

(+) belong to the Hardy space H2 C(δ) . Proceeding with analogous arguments applied to G (−) (τ, ·) equalities (40) and (42) are obtained.

In conclusion, we can say that the thermal Green functions present a double analytic structure involving the analyticity properties in the τ = u + iv and ζ = ξ + iη planes. The 2β-periodic function G (+) (τ, ·) (resp. G (−) (τ, ·)) is analytic in the cut-domain I− ∪ ◦ ◦ [I+ \ (+ ] (resp. I+ ∪[I− \ (− ]); its Fourier coefficients can be uniquely interpolated (in the sense of the Carlson theorem), and are the restriction to the appropriate Matsubara energies of a function G˜ (+,b−f ) (ζ, ·) (resp. G˜ (−,b−f ) (ζ, ·)), analytic in the half-plane (+) Reζ > 0 (resp. Reζ < 0). It is straightforward to verify that the jump function J(0) (v, ·) (−)

coincides with the retarded Green function, and J(0) (v, ·) coincides with the advanced one; analogously, putting iζ = ω, we can identify G˜ (+,b−f ) (ζ, ·) and G˜ (−,b−f ) (ζ, ·) respectively with the retarded and advanced Green functions in the energy variable ω conjugate to the real time t. 3. Representation of the Jump Function in Terms of an Infinite Set of Fourier Coefficients First let us consider a system of bosons; since n is even, i.e. n = 2m, ζm = (2mπ )/β, (m = 0, 1, 2, . . . ), we have: β i 2mπ τ (+,b) 2mπ ˜ G ,· = 2 G (+) (τ, ·) e β dτ . (49) β 0 Next, recalling that G (+) (τ, ·) is β-periodic, we can write also the following Fourier expansion: ∞ 1 ˜ (+,b) 2mπ −i 2mπ τ (+) G(β) G (τ, ·) = ,· e β , (50) β β m=0 β 1 ˜ (+,b) 2mπ i 2mπ τ (+,b) 2mπ (+) β ˜ G(β) G (τ, ·) e dτ = G ,· = ,· . (51) β 2 β 0 Finally, putting β = 2π, formulae (50), (51) can be rewritten in the more convenient form: ∞ 1 ˜ (+,b) G (+) (τ, ·) = G(2π) (m, ·)e−imτ , (52) 2π m=0 2π 1 (+,b) G˜(2π) (m, ·) = G (+) (τ, ·) eimτ dτ = G˜ (+,b) (m, ·). (53) 2 0


69

Recalling once again the β-periodicity of the function G (+) (τ, ·), we write now the Froissart–Gribov equalities (39) as (+,b) (+,b) G˜ (+,b) (m, ·) = J˜(0) (m, ·) + J˜(2π) (m, ·) (+,b) (+,b) = 2J˜(0) (m, ·) = 2G˜(2π) (m, ·),

(m = 1, 2, 3, . . . ).

(54)

(b)

It is now convenient to introduce an auxiliary function J∗ (v, ·), defined as follows: (+,b)

J∗(b) (v, ·) = e−v J(0)

(v, ·),

(v ∈ R+ ),

(55)

and the corresponding Laplace transform: J˜∗(b) (ζ, ·) =

0

+∞

J∗(b) (v, ·) e−ζ v dv,

(ζ = ξ + iη, Reζ > −1 + δ, δ ≥

> 0). (56)

(b) It is straightforward to prove,via the Paley–Wiener theorem, that J˜∗ (ζ, ·) belongs to (+) (+) the Hardy space H2 C(−1+δ) , where C(−1+δ) = {ζ ∈ C | Reζ > −1 + δ, δ ≥ > 0}. Next, the Froissart–Gribov equalities (54) can be rewritten as (+,b) J˜∗(b) (m, ·) = G˜(2π) (m + 1, ·) ,

(m = 0, 1, 2, . . . ).

(57)

Then we can prove the following lemma. (b) Lemma 1. The function J˜∗ (−1/2+iη, ·), (η ∈ R) can be represented by the following series, that converges in the sense of the L2 -norm:

∞ 1 c3 ψ3 (η), J˜∗(b) − + iη, · = 2

(58)

3=0

ψ3 (η) denoting the Pollaczek functions defined by 1 1 ψ3 (η) = √ 6 + iη P3 (η), 2 π

(59)

6 being the Euler gamma function, and P3 the Pollaczek polynomials [2, 16]. The coefficients c3 are given by: ∞ √ (−1)m ˜ (+,b) 1 G(2π) (m + 1, ·)P3 −i m + . c3 = 2 π m! 2

(60)

m=0

(α)

Proof. The Pollaczek polynomials P3 (η), (η ∈ R), are orthogonal in L2 (−∞, +∞) with weight function (see refs. [2, 16]): w(η) =

1 (2α−1) |6(α + iη)|2 . 2 π

(61)

70


For α = 1/2, the orthogonality property reads:

+∞ −∞

(1/2) (1/2) w(η)P3 (η)P3 (η) dη

= δ3,3 ,

1 w(η) = π

2 6 1 + iη , 2

(62)

(in the following, when α = 1/2, we omit the index α in the notation). Next, we introduce the following functions, that will be called Pollaczek functions (of index α = 1/2): 1 1 ψ3 (η) = √ 6 + iη P3 (η), (63) 2 π (b)

which form a complete basis in L2 (−∞, +∞) [13]. Since J˜∗ (ζ, ·) belongs to the

(+) (b) Hardy space H2 C(−1+δ) , then J˜∗ (−1/2 + iη, ·) (η ∈ R) belongs to L2 (−∞, +∞). (b)

Therefore, we may expand J˜∗ (−1/2+iη, ·) in terms of Pollaczek functions as follows: ∞ 1 (b) ˜ − + iη, · = c3 ψ3 (η), J∗ 2

(64)

3=0

(b) where the series at the r.h.s. of (64) converges to J˜∗ (−1/2 + iη, ·) in the sense of the 2 L -norm. From (64) we get +∞ 1 1 1 J˜∗(b) − + iη, · 6 − iη P3 (η) dη. (65) c3 = √ 2 2 π −∞

The integral at the r.h.s. of (65) can be evaluated by the contour integration method along the path shown in Fig. 2, and taking into account the asymptotic behaviour of the gamma function given by the Stirling formula. We obtain: ∞ √ (−1)m ˜(b) 1 J∗ (m, ·)P3 −i m + . c3 = 2 π m! 2

(66)

m=0

Finally, from (57), (64) and (66) the proof of the lemma follows.

From (56), when ζ = −1/2 + iη (η ∈ R), we have: +∞ 1 (b) ˜ J∗ − + iη, · = J∗(b) (v, ·) ev/2 e−iηv dv. 2 0

(67)

(b) (b) The r.h.s. of (67) is the Fourier transform of J∗ (v, ·)ev/2 . Noting that J˜∗ (−1/2+iη, ·) 2 1 belongs to L (−∞, +∞), but not necessarily to L (−∞, +∞), the inversion of the Fourier transform (67) holds only as a limit in the mean order two, and can be written as follows: η0 1 1 J∗(b) (v, ·) ev/2 = l.i.m. J˜∗(b) − + iη, · eiηv dη , (v ∈ R+ ). (68) η0 →+∞ 2π −η 2 0

Then, we can prove the following lemma.


71

Im ζ ζ − plane

0

1 2

1

2

3

Re ζ

Fig. 2. Integration path for the evaluation of integral (65) (b)

Lemma 2. The function J∗ (v, ·)ev/2 can be represented by the following expansion that converges in the sense of the L2 -norm: ev/2 J∗(b) (v, ·) =

∞

a3 3 (v),

(v ∈ R+ ),

(69)

3=0

where the coefficients a3 are given by:

∞ √ (−1)m ˜ (+,b) 1 G(2π) (m + 1, ·) P3 −i m + , a3 = 2 m! 2

(70)

m=0

P3 being the Pollaczek polynomials, and the functions 3 (v) are given by √ −v 3 (v) = i 3 2 L3 (2e−v ) e−e e−v/2 ,

(71)

L3 being the Laguerre polynomials. Proof. Let us observe that +∞ +∞ 1 −v e−t t (iη−1/2) dt = e−e e−v/2 e−iηv dv + iη = 6 2 0 −∞ −v −e −v/2 , e =F e

(72)

where F denotes the Fourier integral operator. Let us note that the function exp(−e−v )e−v/2 belongs to S ∞ (R), i.e. the Schwartz space of the C ∞ (R) functions that, together with all their derivatives, tend to zero, for |v| tending to +∞, faster than any negative power of |v|. Therefore, we can write (see formula (63)): d −e−v −v/2 1 ψ3 (η) = √ F P3 −i . (73) e e dv π

72


Substituting in expansion (58) to the Pollaczek functions their representation (73), we obtain: ∞ 1 d −e−v −v/2 1 c3 √ F P3 −i e . (74) e J˜∗(b) (− + iη, ·) = 2 dv π 3=0

Let us now apply the operator F −1 to the r.h.s. of (74). If we exchange the integral operator F −1 with the sum, and this is legitimate within the L2 -norm convergence, we obtain: F

−1

∞ 3=0

c3

1 d −e−v −v/2 e e √ F P3 −i dv π ∞ 1 d −e−v −v/2 = c3 √ e . e P3 −i dv π

(75)

3=0

Finally, recalling formula (68), we obtain the following expansion for the function (b) J∗ (v, ·)ev/2 : ev/2 J∗(b) (v, ·) =

∞ c3 d −e−v −v/2 e , e √ P3 −i dv π

(76)

3=0

whose convergence is in the sense of the L2 -norm. It can be easily verified that [9] √ √ d −e−v −v/2 −v 2P3 −i = i 3 2 L3 2e−v e−e e−v/2 , e e (77) dv where L3 denotes the Laguerre polynomials. √ It can be checked that the polynomials L3 (v) = i 3 2 L3 (2e−v ) are a set of polynomials orthonormal on the real line with weight function w(v) = exp(−v) exp(−2e−v ), and, consequently, the set of functions 3 (v), defined by formula (71), forms an orthonormal basis in L2 (−∞, +∞). Finally, from (76) we obtain: ev/2 J∗(b) (v, ·) =

∞

∞ √ −v a3 i 3 2 L3 (2e−v ) e−e e−v/2 = a3 3 (v),

3=0

(v ∈ R+ ),

3=0

(78) √

where a3 = c3 / 2π, and the functions 3 (v) are given by formula (71).

We now introduce the weighted L2 -space L2(w) [0, +∞), whose norm is defined by: f L2

(w) [0,+∞)

+∞

=

1/2 w(v) |f (v)| dv 2

,

(79)

0

w(v) being a weight function which will be specified in the following. Then we can prove the following result.


73 (+,b)

Theorem 2. The jump function J(0) sion: (+,b)

J(0)

(v, ·) can be represented by the following expan-

(v, ·) = ev/2

∞

(v ∈ R+ ),

a3 3 (v),

(80)

3=0

which converges in the sense of the L2(w) [0, +∞)-norm, with weight function w(v) = e−v , (v ∈ R+ ). Proof. We can write: L (+,b) v/2 a3 3 (v) J(0) (v, ·) − e 3=0



+∞

=

0



L2(w) [0,+∞)

2 1/2 L (+,b) e−v J(0) (v, ·) − ev/2 a3 3 (v) dv  3=0

2 1/2 L v/2 (b) a3 3 (v) dv  . e J∗ (v, ·) −

+∞

=

0

(81)

3=0

In view of Lemma 2 we can thus state that: L (+,b) lim J(0) (v, ·) − ev/2 a3 3 (v) L→∞ 3=0

= 0,

(82)

L2(w) [0,+∞)

that proves the statement. Consider now a system of fermions. In this case the function G (+) (τ, ·) is antiperiodic with period β. Then, if we put ζm = (2m + 1)π/β (m = 0, 1, 2, . . . ) and β = 2π , we have the following expansion: ∞ 1 1 ˜ (+,f ) G(2π) m + , · e−i(m+1/2)τ , (83) 2π 2 m=0 2π 1 1 1 (+,f ) G˜(2π) m + , · = G (+) (τ, ·) ei(m+1/2)τ dτ = G˜ (+,f ) m + , · . (84) 2 2 2 0 G (+) (τ, ·) =

Recalling once again the antiperiodicity of G (+) (τ, ·), we write the Froissart–Gribov equalities (41) in the following form: 1 1 1 (+,f ) (+,f ) G˜ (+,f ) m + , · = J˜(0) m + , · − J˜(2π) m + , · 2 2 2 1 1 (+,f ) (+,f ) = 2 J˜(0) m + , · = 2 G˜(2π) m + , · , (85) 2 2 (m = 0, 1, 2, . . . ).

74


We can now proceed in a way strictly analogous to that followed in the case of bosons. We +∞ (f ) (f ) (+,f ) (f ) put: J∗ (v, ·) = e−v J(0) (v, ·) and, accordingly, J˜∗ (ζ, ·) = 0 J∗ (v, ·)e−ζ v dv (ζ = ξ + iη, Reζ ≡ ξ > −1 + δ, δ ≥ > 0). Then, the Froissart–Gribov equalities (85) now read: 1 3 (f ) (+,f ) J˜∗ m + , · = G˜(2π) m + , · , (m = 0, 1, 2, . . . ). (86) 2 2 We can now state the following theorem. (f ) Theorem 3. i) The function J˜∗ (iη, ·), (η ∈ R) can be represented by the following series, that converges in the sense of the L2 -norm: (f ) J˜∗ (iη, ·) =

∞

d3 ψ3 (η),

(87)

3=0

where ψ3 (η) are the Pollaczek functions defined by formula (59), and the coefficients d3 are given by: ∞ √ (−1)m ˜ (+,f ) 3 1 G(2π) m + , · P3 −i m + , (88) d3 = 2 π m! 2 2 m=0

P3 denoting the Pollaczek polynomials. (f ) ii) The function J∗ (v, ·) can be represented by the following expansion that converges in the sense of L2 -norm: (f ) J∗ (v, ·)

=

∞

b3 3 (v),

(v ∈ R+ ),

(89)

3=0

√ where the coefficients b3 are given by b3 = d3 / 2π , and the functions 3 (v) are defined by formula (71). (+,f ) iii) The function J(0) (v, ·) can be represented by the following expansion: (+,f )

J(0)

(v, ·) = ev

∞

b3 3 (v),

(v ∈ R+ ),

(90)

3=0

that converges in the sense of the L2(w) [0, +∞)-norm with weight function w(v) = e−2v , (v ∈ R+ ). Proof. The proof runs exactly as in the case of the boson statistics, with the only remarkable difference that we use the Froissart–Gribov equalities (86) instead of (57). (f ) We can reconstruct, by the use of this method, the function J˜∗ (iη, ·) but not the (+,f ) function J˜(0) (iη, ·), which is much more interesting from the physical viewpoint. In (+,f ) order to recover the function J˜(0) (iη, ·) we must introduce a more restrictive assump +∞ (+,f ) (+,f ) tion, requiring the function J˜(0) (ζ, ·) = 0 J(0) (v, ·)e−ζ v dv to be holomorphic


75

in the half-plane Reζ > −γ (γ > 0). Accordingly, in place of the temperateness con(+,f ) dition (26) we assume that J(0) (v, ·) belongs to L1 [0, +∞) ∩ L2 [0, +∞). Here, for the sake of simplicity, we treat only the case of fermions; analogous considerations hold true also in the case of the boson statistics. We can thus suppose that the singularities of (+,f ) J˜(0) (ζ, ·), corresponding to the excited states, all lie in the half-plane Reζ < −γ , γ being the smallest damping factor of the spectrum (see refs. [1, 10]). If this is the case, (+,f ) J˜(0) (iη, ·) is analytic, and, moreover, belongs also to L2 (−∞, +∞). We can thus state the following result. (+,f )

Theorem 4. Let us assume that J˜(0) (ζ, ·) is a function holomorphic in the half-plane (+,f ) (iη, ·) can be represented by the following expansion Reζ > −γ (γ > 0); then J˜ (0)

that converges in the sense of the L2 -norm: (+,f ) J˜(0) (iη, ·) =

∞ 3=0

d3 ψ3 (η),

(91)

where ψ3 (η) are the Pollaczek functions defined by formula (59), and the coefficients d3 are given by: ∞ √ (−1)m ˜ (+,f ) 1 1 d3 = 2 π G(2π) m + , · P3 −i m + , m! 2 2

(92)

m=0

P3 denoting the Pollaczek polynomials. Proof. The proof is strictly analogous to the one followed for proving equality (58), and successively adapted to the fermion statistics in order to obtain expansion (87). The only remarkable difference is that now in the expression of the coefficients d3 we have (+,f ) (+,f ) the terms G˜(2π) m + 21 , · instead of G˜(2π) m + 23 , · ; therefore all the coefficients corresponding to m = 0, 1, 2, . . . , are involved in the determination of the function (+,f ) J˜(0) (iη, ·). (−,f ) Analogous methods and results can be worked out for the function J˜(0) (iη, ·), (−,f ) assuming that J˜(0) (ζ, ·) is holomorphic in the half-plane Reζ < γ (γ > 0). We are (+,f ) (−,f ) then able to reconstruct the difference J˜(0) (iη, ·) − J˜(0) (iη, ·) which leads to the determination of the “spectral density” [17].

4. Reconstruction of the Jump Function in Terms of a Finite Number of Fourier Coefficients Up to now we have assumed that all the Fourier coefficients are known, and, in addition, that they are noiseless; but this assumption is clearly unrealistic. We now suppose that only a finite number of coefficients are known within a certain degree of approximation. We focus our attention on the case of the boson statistics, and specifically on the results contained in Lemmas 1 and 2, and Theorem 2. The case of the fermion statistics can be (+,b) treated similarly. We can simplify the notation, without ambiguity, by putting: G˜(2π) (m+

76

G. Cuniberti, E. De Micheli, G. A. Viano (b)

(+,b)

( )

1, ·) = gm , ev/2 J∗ (v, ·) = F∗ (v), and J(0) (v, ·) = F (v). Then, we denote by gm (+,b) the Fourier coefficients G˜(2π) (m + 1, ·) when they are perturbed by noise. We now assume that only (N + 1) Fourier coefficients are known within an approximation error ( ) of order : i.e. |gm − gm | ≤ (m = 0, 1, 2, . . . , N). We consider the following finite sums: ( ,N)

a3

=

N √ 1 (−1) ( ) gm P3 −i m + . 2 m! 2

(0,∞)

Accordingly, we have a3

= a3 (see (70)). We can then prove the following lemma.

Lemma 3. The following statements hold true: ∞ (0,∞) 2 i) = F∗ 2L2 [0,∞) = C, 3=0 a3 ∞ ( ,N) 2 ii) = +∞. 3=0 a3 ( ,N) limN→∞ a3 →0

iii)

(93)

m=0

(0,∞) a3

=

= a3 ,

(C = constant).

(94) (95)

(3 = 0, 1, 2, . . . ).

(96)

iv) If k0 ( , N ) is defined as

k0 ( , N ) = max k ∈ N :

k 3=0

i.e. it is the largest integer such that

( 3=0 a3

k

" ( ,N) 2 |a3 |

≤C ,

(97)

,N) 2

≤ C, then

lim k0 ( , N ) = +∞.

(98)

N→∞ →0

v) The sum ( ,N)

Mk

=

k ( a3

,N) 2

,

(k ∈ N),

(99)

3=0

satisfies the following properties: a) it increases for increasing values of k; b) the following relationships hold true: ( ,N) 2 ( ,N) ≥ ak Mk ∼

k→∞

1 (2k)2N , (N !)2

(N fixed ).

(100)

Proof. (i) Equality (94) follows from the Parseval theorem applied to expansion (69), and recalling that F∗ (v) belongs to L2 (−∞, +∞). ( ,N)

(ii) Let us rewrite the sums a3 ( ,N)

a3

=

as follows:

N m=0

1 ( ) bm P3 −i m + , 2

(101)


77

√ ( ) 2(−1)m gm /m!. Now, we can write the following inequality: N 1 ( ,N) ( ) bm P3 −i m + a3 = 2 m=0 N−1 ( ) m=0 bm P3 −i m + 21 () 1 . (102) · 1 − ≥ bN P3 −i N + () 1 2 bN P3 −i N + 2 ( )

where bm =

Let us now recall that in the Appendix of Ref. [9] the asymptotic behaviour of the Pollaczek polynomials P3 [−i(m + 1/2)] for large values of l (at fixed m) is proved to be: (−1)3 i 3 1 P3 −i m + (103) ∼ (23)m . 3→∞ 2 m! Therefore, we have: N−1 ( ) N−1 ( ) 1 P −i m + m=0 bm P3 −i m + 21 b 3 m m=0 2 ≤ () () bN P3 −i N + 21 bN P3 −i N + 21 N−1 ( ) bm N! ∼ (23)m−N −→ 0. ( ) b m! 3→∞ 3→∞ m=0

N

From (102), (103) and (104) it follows that for 3 sufficiently large: ( ) b N ( ,N) (23)N . a3 ∼ 3→∞ N ! ( ,N) Therefore, lim3→∞ a3 = +∞, and statement (ii) follows. (0,∞)

(iii) We can write the difference a3 (0,∞) a3

( ,N) − a3

= +

( ,N)

− a3

(105)

as follows:

N (−1)m 1 ( ) )P3 −i m + (gm − gm m! 2 m=0 " ∞ (−1)m 1 gm P3 −i m + . m! 2

√

(104)

2

(106)

m=N+1

√ (0,∞) (−1)m 1 In view of the fact that the series 2 ∞ , m=0 m! gm P3 [−i(m+ 2 )] converges to a3 it follows that the second term in bracket (106) tends to zero as N → ∞. Concerning the first term, we may write the inequality: N N (−1)m 1 1 1 ( ) −i m + (gm − gm )P3 −i m + P , ≤ 3 m! 2 m! 2 m=0

m=0

(107)

78


( ) where the inequalities gm − gm ≤ , (m = 0, 1, 2, . . . , N ) have been used. Next, by rewriting the Pollaczek polynomials P3 [−i(m + 1/2)] as P3

1 −i m + 2

=

3 j =0

(3) pj

1 m+ 2

j

,

(108)

and, substituting this expression in inequality (107), we obtain:   j N l 1  1  (3) . pj m + m! 2 m=0

(109)

j =0

(3) Next, we perform the limit for N → ∞. In view of the fact that lj =0 pj (m + 1/2)j ∞ is finite, and the series m=0 (m + 1/2)j /m! converges, we can exchange the order of the sums and write: l ∞ 1 j (3) 1 . (110) m+ pj m! 2 m=0

j =0

Finally, performing the limit for → 0, and recalling equality (106), statement (iii) is obtained. (iv) From definition (97) it follows, for k1 = k0 + 1, that: k1 ( a3

,N) 2

> C.

(111)

3=0

Statement (iv) (formula (98)) is proved if we can show that limN→∞ k1 ( , N ) = +∞. →0

Let us suppose that limN→∞ k1 ( , N ) is finite. Then there should exist a finite number →0

K (independent of and N ) such that, for N tending to ∞ and k1 ( , N ) ≤ K. Then, from inequality (111) we have: C
0). B

(127)

0

Moreover, from (123) we have: B(

,N)

(v) −→ 0 ,

v ∈ [0, +∞).

N→∞ →0

(128)

The family of functions {B ( ,N) (v)} is bounded in L2 [0, +∞), therefore it has a subsequence which is weakly convergent in L2 [0, +∞). The limit of this subsequence is zero. In fact, let us observe that |B ( ,N) (v)| ≤ 2C; then we consider the function B ( ,N) (v)φ(v), where φ is an arbitrary element of the class of functions Cc∞ (R+ ). We then have |B ( ,N) (v)φ(v)| ≤ 2C|φ(v)|, and this inequality does not depend on N and . In view of the Lebesgue dominated convergence theorem we can then write (see also limit (128)): +∞ ( ,N) lim sup B (v)φ(v) dv = 0. (129) N→∞ →0

0

Since the set of functions Cc∞ (R+ ) is everywhere dense in L2 [0, +∞), given an arbitrary function ψ ∈ L2 [0, +∞) and an arbitrary number η > 0, there exists a function φk ∈ Cc∞ (R+ ) such that ψ − φk L2 [0,+∞) < η. Furthermore, through the Schwarz inequality we have: +∞ ( ,N) (v)[φk (v) − ψ(v)] dv B 0

+∞

≤ 0

( B

,N)

2 1/2 (v) dv

√

+∞ 0

1/2 |φk (v) − ψ(v)| dv

≤ 2 C η.

2

(130)

From (129) and (130) we can conclude that +∞ ( ,N) B (v)ψ(v) dv = 0, lim sup N→∞ →0

(131)

0

for any ψ ∈ L2 [0, +∞). Next, by using the same type of arguments, we can state that if there is an arbitrary subsequence belonging to the family {B ( ,N) } that weakly converges in L2 [0, +∞), then the weak limit of this subsequence is necessarily zero. Finally, from the uniqueness of the (weak) limit point, it follows that the whole family {B ( ,N) } converges weakly to zero in L2 [0, +∞). We can thus write: +∞ lim H (0,∞) (v)B ( ,N) (v) dv = 0, (132) N→∞ 0 →0

82


and from equality (124) we have

V ( ,N)

H(

lim

N→∞ 0 →0

Since

k0 ( 3=0 a3

,N)

(0,∞) 2

− a3

≤

,N)

V(

2

(v) − H (0,∞) (v)

,N)

H(

0

k0 ( lim a3

,N)

N→∞ →0 3=0

dv = 0.

,N) (v) − H (0,∞) (v) 2

(133)

dv, we have:

= 0,

(0,∞) 2

− a3

(134)

and, in view of equality (116), the theorem is proved. We can then prove the following corollary. Corollary 1. The following equality holds true: k0 ( ,N) ( v/2 a3 lim F (v) − e N→∞ 3=0 →0

,N)

3 (v)

= 0,

(135)

L2(w) [0,+∞)

L2(w) [0, +∞) being the weighted L2 -space with weight function w(v) = e−v , (v ∈ R+ ), and the functions 3 (v) are defined by formula (71). Proof. The statement follows immediately from Theorem 5 by noting that:

2 k0 ( ,N) ( ,N) F∗ (v) − a3 3 (v) dv 3=0 2 +∞ k0 ( ,N) ( ,N) e−v F (v) − ev/2 a3 3 (v) dv = 0 3=0 2 k0 ( ,N) ( ,N) v/2 a3 3 (v) . = F (v) − e 2 3=0

+∞ 0

(136)

L(w) [0,+∞)

(+,b)

We can thus conclude that the jump function J(0) by the truncated expansion (+,b)

J(0)

(v, ·) ∼ ev/2

k0 ( ,N) 3=0

( ,N)

a3

3 (v),

(v, ·) = F (v) can be approximated

(v ∈ R+ ).

✷

(137)


83

References 1. Abrikosov, A.A., Gorkov, L.P. and Dryaloshinski, I.E.: Methods of Quantum Field Theory in Statistical Physics. Englewood Cliffs: Prentice–Hall, 1963 2. Bateman Manuscript Project: Higher Trascendental Functions. A. Erdelyi, Director. Vol. 2, New York: Krieger, 1953 3. Boas, R.P.: Entire Functions. New York: Academic Press, 1954 4. Bremermann, H.: Distributions, Complex Variables, and Fourier Transforms. Reading: Addison-Wesley, 1965 5. Bros, J. and Buchholz, D.: Axiomatic Analyticity Properties and Representations of Particles in Thermal Quantum Field Theory. Ann. Inst. H. Poincaré – Physique Theorique 64, 495–521 (1996) 6. Bros, J. and Viano, G.A.: Connection Between the Harmonic Analysis on the Sphere and the Harmonic Analysis on the One–sheeted Hyperboloid: An Analytic Continuaton Viewpoint. I Forum Mathematicum 8, 621–658 (1996); II Forum Mathematicum 8, 659–722 (1996); III Forum Mathematicum 9, 165–191 (1997) 7. Bros, J. and Buchholz, D.: Fields at finite temperature: A general theory of the two–point functions. In preparation 8. Buchholz, D. and Junglas, P: On the Existence of Equilibrium States in Local Quantum Field Theory. Commun. Math. Phys. 121, 255–270 (1989) 9. De Micheli, E. and Viano, G.A.: On the Solution of a Class of Cauchy Integral Equations. J. Math. Anal. Appl. 246, 520–543 (2000) 10. Fetter, A.L. and Walecka, J.D.: Quantum Theory of Many–Particle Systems. New York: McGraw–Hill, 1971 11. Haag, R., Hugenholtz, N.M. and Winnink, M.: On the Equilibrium States in Quantum Statistical Mechanics. Commun. Math. Phys. 5, 215–236 (1967) 12. Hoffman, K.: Banach Spaces of Analytic Functions. Englewood Cliffs: Prentice–Hall, 1962 13. Itzykson, C.: Group Representation in a Continuous Basis: An Example. J. Math. Phys. 10, 1109–1114 (1969) 14. Le Bellac, M.: Thermal Field Theory. Cambridge: Cambridge Univ. Press, 1996 15. Matsubara, T.: A new approach to quantum–statistical mechanics. Prog. Theor. Phys. 14, 351–378 (1955) 16. Szegö, G.: Orthogonal Polynomials. New York: Academic Press, 1954 17. Yukalov, V.I.: Statistical Green’s Functions. Kingston: Queen’s University Press, 1998 Communicated by D. C. Brydges

Commun. Math. Phys. 216, 85 – 138 (2001)

Communications in



A Hiker’s Guide to K3. Aspects of N = (4, 4) Superconformal Field Theory with Central Charge c = 6 Werner Nahm, Katrin Wendland Physikalisches Institut, Universität Bonn, Nussallee 12, 53115 Bonn, Germany. E-mail: [email protected]; [email protected] Received: 1 March 2000 / Accepted: 17 July 2000

Abstract: We study the moduli space M of N = (4, 4) superconformal field theories with central charge c = 6. After a slight emendation of its global description we find the locations of various known models in the component of M associated to K3 surfaces. Among them are the Z2 and Z4 orbifold theories obtained from the torus component of M. Here, SO(4, 4) triality is found to play a dominant role. We obtain the B-field values in direction of the exceptional divisors which arise from orbifolding. We prove T-duality for the Z2 orbifolds and use it to derive the form of M purely within conformal field theory. For the Gepner model (2)4 and some of its orbifolds we find the locations in M and prove isomorphisms to nonlinear σ models. In particular we prove that the Gepner model (2)4 has a geometric interpretation with Fermat quartic target space. Introduction This paper aims to make a contribution to a better understanding of the N = (4, 4) superconformal field theories with left and right central charge c = 6. Ultimately, one would like to know their moduli space M as an algebraic space, their partition functions as functions on M and modular functions on the upper half plane, and an algorithm for the calculation of all operator product coefficients, depending again on M. This would constitute a good basis for the understanding of quantum supergravity in six dimensions, and presumably for an investigation of the more complicated physics in four dimensions. The moduli space M has been identified with a high degree of plausibility, though a number of details remain to be clarified. It has two components, Mtori and MK3 , one 16-dimensional associated to the four-torus and one 80-dimensional associated to K3. The superconformal field theories in Mtori are well understood. One also understands some varieties of theories which belong to MK3 , including about 30 isolated Gepner type models and varieties which contain orbifolds of theories in Mtori . In the literature one can find statements concerning intersections of these subvarieties, but not all of them are correct. Indeed, their precise positions in M had not been studied up to now. One

86

W. Nahm, K. Wendland

difficulty is due to the fact that the standard description of Mtori is based on the odd cohomology of the torus, which does not survive the orbifolding. As varieties of superconformal theories Mtori and MK3 cannot intersect for trivial reasons. As ordinary conformal theories without Z2 grading intersections are possible and will be shown to occur. The plan of our paper is as follows. In Sect. 1 we will review known results following [A-M,As2]. We correct some of the details and add proofs for well-known conjectural features. In Sect. 1.1 we explain the connection between our description of Mtori in terms of the even cohomology and the one given by Narain much earlier by odd cohomology [C-E-N-T, Na]. Both are eight-dimensional, and they are related by SO(4, 4) triality. Section 2 deals with Z2 and Z4 orbifold conformal field theories. We arrive at a description for the subvarieties of these theories within MK3 . In particular, we present a proof for the well-known conjecture that orbifold conformal field theories tend to give the value B = 21 [As2, Sect. 4] to the B-field in direction of the exceptional divisors gained from the orbifold procedure and determine the correct B-field values for Z4 orbifolds. Our results are in agreement with those of [Do, B-I], that were obtained in a different context. We calculate the conjugate of torus T-duality under the Z2 orbifolding map to MK3 and find that it is a kind of squareroot of the Fourier-Mukai T-duality on K3. This yields a proof of the latter and allows us to determine the form of MK3 purely within conformal field theory, without having recourse to Landau–Ginzburg arguments. We disprove the conjecture that Z2 and Z4 orbifold moduli spaces meet in the Gepner model (2)4 [E-O-T-Y]. We show that the Z4 orbifold of the nonlinear σ model on the torus with lattice = Z4 has a geometric interpretation on the Fermat quartic hypersurface. Section 3 is devoted to the study of special points with higher discrete symmetry groups in the moduli space, namely Gepner models (actually (2)4 and some of its orbifolds by phase symmetries). We stress that our approach is different from the one advocated in [F-K-S-S, F-K-S] where massless spectra and symmetries of all Gepner models and their orbifolds were matched to those of algebraic manifolds corresponding to these models. The correspondence there was understood in terms of Landau–Ginzburg models, a limit which we do not make use of at all. We instead explicitly prove equivalence of the Gepner models under investigation to nonlinear σ models. This also enables us to give the precise location of the respective models within the moduli space MK3 . We prove that the Gepner model (2)4 is isomorphic to the Z4 orbifold and therefore to the Fermat quartic model studied in the previous section. We moreover find two meeting points of MK3 and Mtori generalizing earlier results for bosonic theories [K-S] to the corresponding N = (4, 4) supersymmetric models. We find a meeting point of the moduli spaces of Z2 and Z4 orbifold conformal field theories different from the one conjectured in [E-O-T-Y]. In Sect. 4 we conclude by gathering the results and joining them to a panoramic view of part of the moduli space (Fig. 4.1). In the context of σ models we must fix our α conventions. For ease of notation we use the rather unusual α = 1, so T-duality for a bosonic string compactified on a circle √ of radius R reads R → R1 . We hoped to save us a lot of factors of 2 this way. Often, the left–right transformed analogue of some statement will not be mentioned explicitly, in order to avoid tedious repetitions. Fourier components of holomorphic fields are labeled by the energy, not by its negative.

A Hiker’s Guide to K3

87

1. The Moduli Space of N = (4, 4) Superconformal Field Theories with Central Charge c = 6 We consider unitary two dimensional superconformal quantum field theories. They can be described as Minkowskian theories on the circle or equivalently as euclidean theories on tori with parameter τ in the upper complex halfplane. The worldsheet coordinates are called σ0 , σ1 . The space of states H of a quantum field theory has a real structure given by CPT. For any N = (4, 4) superconformal theory H contains four-dimensional vector spaces Ql and Qr of real left and right supercharges. Since we consider left and right central charge c = 6, we use the extension of the N = (2, 2) superconformal algebra by an su(2) ⊕ su(2) current algebra of level 1 [A-B-D+]. The (3+3)-dimensional Lie group susy susy generated by the corresponding charges will be denoted by SU (2)l × SU (2)r and susy susy its {(1, 1), (−1, −1)} quotient by SO(4) . The commutant of SU (2)l in SO(Ql ) will be called SU (2)l . Here and in the following we use the notation SO(W ) for the special orthogonal group of a real vector space W with given scalar product. susy One can identify SU (2)l with SU (2)l by selecting one vector in Ql . The subgroup of SO(Ql ) which fixes this vector is an SO(3) group with surjective projections to the two SU (2) groups modulo their centers and allows an identification of the images. Such an identification seems to be implicit in many discussions in the literature, but will not be used in this section. We will consider canonical subspaces of H spanned by the states with specified conformal dimensions (h; h) which belong to some irreducible representation of susy susy SU (2)l × SU (2)r . The latter are labeled by the charges (Q; Q) with respect to susy susy a Cartan torus of SU (2)l × SU (2)r . Since any two Cartan tori are related by a conjugation, the spectrum does not depend on the choice of this torus. Charges are normalized to integral values, as has become conventional in the context of extended supersymmetry. We assume the existence of a quartet of spectral flow fields with (h, Q; h, Q) = ( 41 , ε1 ; 41 , ε2 ), εi ∈ {±1}. Operator products with each of them yield a combined left+right spectral flow. Instead of using N = (4, 4) supersymmetry it suffices to start with N = (2, 2) and this quartet. Indeed, the operator product of a pair of quartet fields yields lefthanded flow operators with (h, Q; h, Q) = (1, ±2; 0, 0), and analogously on susy susy the righthanded side for another pair. These enhance the u(1)l ⊕ u(1)r subalgebra (1) (1) of the N = (2, 2) superconformal algebra to an A1 × A1 Kac–Moody algebra. Thus the N = (2, 2) superconformal algebra is enhanced to N = (4, 4) [E-O-T-Y]. Our assumptions are natural in the context of superstring compactification. There, unbroken extended spacetime supersymmetry is obtained from N = (2, 2) worldsheet supersymmetry with spectral flow operators [Se1, Se2]. Thus our superconformal theories may be used as a background for N = 4 supergravity in six dimensions. Here, however, we concentrate on the internal conformal field theory. External degrees of freedom are not taken into account. Let us give a brief summary on what is known about the moduli space M so far. The spaces of states of the conformal theories form a bundle with local grading by finite dimensional subbundles over M. They can be decomposed into irreducible representations of the left and right N = 4 supersymmetries. The irreducible representations are determined by their lowest weight values of (h, Q). These representations can be deformed continuously with respect to the value of h, except for the representations of non-zero Witten index, also called massless representations [E-T1, E-T2,Ta]. Apart

88


from the vacuum representation with (h, Q) = (0, 0), the lowest weight states of massless representations are labeled by (h, Q) = ( 21 , ±1) in the Neveu–Schwarz sector and by (h, Q) = ( 41 , ±1) or (h, Q) = ( 41 , 0) in the Ramond sector. Let us enumerate the representations which are massless with respect to both the left and the right handed side. Apart from the vacuum we already mentioned the spectral flow operators with (h, Q; h, Q) = ( 41 , ε1 ; 41 , ε2 ), εi ∈ {±1}. They form a vector multiplet under SO(4)susy . Since the vacuum is unique, there is exactly one multiplet of such fields. On the other hand, the dimension of the vector space of real ( 41 , 0; 41 , 0) fields is not fixed a priori. We shall denote it by 4 + δ. With a slight abuse of notation, the orthogonal group of this vector space will be called O(4 + δ). These are all the possibilities of massless representations in the Ramond sector. The corresponding ground state fields describe the entire cohomology of Landau–Ginzburg or σ model descriptions of our theories [L-V-W]. If in a given model there is a field with (h, Q; h, Q) = ( 21 , ±1; 0, 0), application of su(2)l and supersymmetry operators yields four lefthanded Majorana fermions and the corresponding abelian currents. As we shall see below, this suffices to show that the model has an interpretation as nonlinear σ model on a torus, with the currents as generators of translation and the fermions as parallel sections of a flat spin bundle. Such models have δ = 0 and constitute the component Mtori of M. The vector space F1/2 spanned by the fields with (h, Q; h, Q) = ( 21 , ε1 ; 21 , ε2 ), εi ∈ {±1} is obtained from the ( 41 , 0; 41 , 0) Ramond fields by spectral flow. Thus it gives susy susy an irreducible 4(4 +δ)-dimensional representation of su(2)l ⊕su(2)r ⊕o(4 +δ). It determines the supersymmetric deformations of the theory, as will be considered below. The massless representations cannot be deformed, so δ is constant over the generic points of a connected component of M and can only increase over nongeneric ones. Tensor products of a massive lefthanded representation with a righthanded massless representation cannot be deformed either, since h − h must remain integral. The span of such tensor products in the space of states yields a string theoretic generalization E of the elliptic genus [S-W1, S-W2], which is constant for all theories within a connected component of M. Since for c = 6 and theories with merely integer charges E is a theta function of degree n = 2 and characteristic (0, 0; −2π in, b), eb = q −1 , by its properties under modular transformations one can show that E is a multiple of the elliptic genus EK3 of a K3 surface. According to their charges, the numbers of ( 41 , 41 ) fields can be arranged into a Hodge diamond nl 1

nr

1 4+δ

nr nl

1

1 where by the above nl ∈ {0, 2} also yields the number of lefthanded Dirac fermions. The uniqueness of the left and right elliptic genera shows nl = nr and δ = 16 − 8nl . Moreover, left and righthanded elliptic genera have the same power series expression. They vanish over Mtori . In particular, as was anticipated above, the existence of one field with (h, Q; h, Q) = ( 21 , ±1; 0, 0) suffices to show that the theory is toroidal. The elliptic genus on M is interpreted as index of a supercharge acting on the loop space of K3 [Wi1,Wi2]. We call one of our conformal field theories associated to torus or K3, depending on the elliptic genus. For the theories associated to K3 one has δ = 16.


89

To understand the local structure of the moduli space M we must determine the tangent space H1 in a given point of M, i.e. describe the deformation moduli of a given theory. This space consists of real fields of dimensions h = h = 1 in the space of states H over the chosen point. The Zamolodchikov metric [Za] on the space of such fields establishes on M the structure of a Riemannian manifold, with holonomy group contained in O(H1 ). To preserve the supersymmetry algebra, H1 must consist of SO(4)susy invariant fields in the image of F1/2 under (Ql )1/2 ⊗ (Qr )1/2 , where the latter subscripts denote Fourier components. Accordingly, F1/2 ⊕ H1 yields a wellsusy known representation of the osp(2, 2) superalgebra spanned by (Ql )±1/2 , su(2)l and the Virasoro operator L0 . In particular, H1 should be 4(4 + δ)-dimensional and form an irreducible representation of su(2)l ⊕ su(2)r ⊕ o(4 + δ). We shall assume that all elements of H1 really give integrable deformations, as has been shown to all orders in perturbation theory [Di]. Note, however, that there is no complete proof yet. The holonomy group of M projects to an O(4 + δ) action on the uncharged massless Ramond representations and to an SO(4) action on Ql ⊗ Qr . Thus its Lie algebra is contained in su(2)l ⊕ su(2)r ⊕ o(4 + δ). The two Lie algebras are equal for Mtori and one expects the same for δ = 16. Below we shall find an isometry from Mtori to a subvariety of MK3 , such that the holonomy Lie algebra of the latter space is at least su(2)l ⊕ su(2)r ⊕ so(4). Moreover, this isometry shows that MK3 is not compact. Since one has the inclusion su(2) ⊕ su(2) ⊕ o(4 + δ) ∼ = sp(1) ⊕ sp(1) ⊕ o(4 + δ) $→ sp(1) ⊕ sp(4 + δ), the moduli space of N = (4, 4) superconformal field theories with c = 6 associated to torus or K3 is a quaternionic Kähler manifold of real dimension 4(4 + δ). To determine its local structure, recall that we are looking for a noncompact space. By Berger’s classification of quaternionic Kähler manifolds [Be] it can only be reducible or quaternionic symmetric [Si, Th. 9]. Because non-Ricci flat quaternionic Kähler manifolds are (even locally) de Rham irreducible [Wo], this means that it can only be Ricci flat or quaternionic symmetric. The former is excluded because geodesic submanifolds on which all holomorphic sectional curvatures are negative and bounded away from zero have been found [P-S, C-F-G, Ce1]. Hence the moduli space must locally be the Wolf space T 4,4+δ = O + (4, 4 + δ; R) SO(4) × O(4 + δ) ∼ = SO + (4, 4 + δ; R) SO(4) × SO(4 + δ),

(1.1)

i.e. one component of the Grassmannian of oriented spacelike four-planes x ⊂ R4,4+δ [Ce2], reproducing Narain’s and Seiberg’s previous results [C-E-N-T, Na, Sei]. Here SO + (W ) denotes the identity component of the special orthogonal group SO(W ) of a vector space W with given scalar product. The space of maximal positive definite subspaces of W has two components, and O + (W ) denotes the subgroup of elements of O(W ) which do not interchange these components. Note that for positive definite W we have SO(W ) = O + (W ). The Zamolodchikov metric on T 4,4+δ is the group invariant one. From the preceding discussion, x can be interpreted as the SO(4)susy invariant part of the tensor product of Ql ⊗ Qr with the four-dimensional space of charged Ramond ground states. Note that the action of so(4) = su(2)l ⊕su(2)r discussed above generates orthogonal transformations of the four-plane x ∈ T 4,4+δ corresponding to the theory under inspection, whereas o(4 + δ) acts on its orthogonal complement.

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We repeatedly used the splitting so(4) = su(2)l ⊕su(2)r . Consider the antisymmetric product 2 x of the above four-plane x. We choose the orientation of x such that su(2)l fixes the anti-selfdual part (2 x)− of 2 x with respect to the group invariant metric on O + (4, 4 + δ; R). When the theory has a parity operation which interchanges Ql and Qr , this induces a change of orientation of x. The choice of an N = (2, 2) subalgebra within the N = (4, 4) superconformal algebra corresponds to the selection of a Cartan torus u(1)l ⊕ u(1)r of su(2)l ⊕ su(2)r . This induces the choice of an oriented two-plane in x. The rotations of x in this two-plane are generated by u(1)l+r , those perpendicular to the plane by u(1)l−r . Thus the moduli space of N = (2, 2) superconformal field theories with central charge c = 6 is given by a Grassmann bundle over M, with fibre SO(4)/(SO(2)l+r × SO(2)l−r ) ∼ = S2 × S2 . Generic examples for our conformal theories are the nonlinear σ models with the oriented four-torus or the K3 surface as target space X. In the K3 case, the existence of these quantum field theories has not been proven yet, but their conformal dimensions and operator product coefficients have a well defined perturbation theory in terms of inverse powers of the volume. We tacitly make the assumption that a rigorous treatment is possible and warn the reader that many of our statements depend on this assumption. A nonlinear σ model on X assigns an action to any twocycle on X. This action is the sum of the area of the cycle for a given Ricci flat metric plus the image of the cycle under a cohomology element B ∈ H 2 (X, R). Since integer shifts of the action are irrelevant, the physically relevant B-field is the projection of B to H 2 (X, R)/H 2 (X, Z). Thus the parameter space of nonlinear σ models has the form {Ricci flat metrics} × {B − fields}. The corresponding Teichmüller space is T 3,3+δ × R+ × H 2 (X, R).

(1.2)

Its elements will be denoted by (), V , B). The first factor of the product is the Teichmüller space of Ricci flat metrics of volume 1 on X, the second parametrizes the volume, and the last one represents the B-field. The Zamolodchikov metric gives a warped product structure to this space. Worldsheet parity transformations (σ0 , σ1 ) → (−σ0 , σ1 ) change the sign of the cycles, or equivalently the sign of B, which yields an automorphism of the parameter space. Target space parity for B = 0 yields a specific worldsheet parity transformation and thus an identification of su(2)l with su(2)r . The corresponding diagonal Lie algebra su(2)l+r generates an SO(3) subgroup of SO(4). Under the action of this subgroup x decomposes into a line and its orthogonal three-plane ) ⊂ x. The S2 × S2 bundle over M now has a diagonal S2 subbundle. Each point in the fibre corresponds to the choice of an SO(2) subgroup of SO(3) or a subalgebra u(1)l+r of su(2)l+r . Geometrically this yields a complex structure in the target space. Thus the S2 bundle over the B = 0 subspace of M is the bundle of complex structures over the moduli space of Ricci flat metrics on the target space. Recall some basic facts about the Teichmüller space T 3,3+δ of Einstein metrics on an oriented four-torus or K3 surface X. We consider the vector space H 2 (X, R) together with its intersection product, such that H 2 (X, R) ∼ = R3,3+δ . In other words, positive definite subspaces have at most dimension three, negative definite ones at most dimension 3 + δ. On K3 this choice of sign determines a canonical orientation. When one wants to study Mtori by itself, the choice of a torus orientation is superfluous. Our main interest, however, is the study of torus orbifolds. For a canonical blow-up of the resulting singularities one needs an orientation. The effect of a change of orientation on the torus will be considered below.


91

Metric and orientation on X define a Hodge star operator, which on H 2 (X, R) has eigenvalues +1 and -1. The corresponding eigenspaces of dimensions three and 3 + δ are positive and negative definite, respectively. Let ) ⊂ H 2 (X, R) be the positive definite three-plane obtained in this way. The orientation on X induces an orientation on ). One can show that Ricci flat metrics are locally uniquely specified by ), apart from a scale factor given by the volume. Since the Hodge star operator in the middle dimension does not change under a rescaling of the metric, the volume V must be specified separately. It follows that T 3,3+δ × R+ is the Teichmüller space of Einstein metrics on X. Explicitly, we have (1.3) T 3,3+δ = O + (H 2 (X, R)) SO(3) × O(3 + δ). The SO(3) group in the denominator is to be interpreted as SO()0 ) for some positive definite reference three-plane in H 2 (X, R), while O(3 + δ) is the corresponding group for the orthogonal complement of )0 . Equivalently, T 3,3+δ could have been written + 2 as SO (H (X, R)) SO(3) × SO(3 + δ). We choose the description (1.3) for later convenience in the construction of the entire moduli space. For higher dimensional Calabi–Yau spaces the σ model description works only for large volume due to instanton corrections. In our case, however, the metric on the moduli space does not receive corrections [N-S]. Therefore the Teichmüller space (1.2) of σ models on X should be a covering of a component of M, thus isomorphic to the Teichmüller space T 4,4+δ obtained in (1.1). Indeed, for δ = 16 a natural isomorphism T 4,4+δ ∼ = T 3,3+δ × R+ × H 2 (X, R)

(1.4)

was given in [A-M,As2], with a correction and clarification by [R-W, Di]. The same construction actually works for δ = 0, too. It uses the identification T 4,4+δ = O + (H even (X, R)) SO(4) × O(4 + δ), where SO(4) is to be interpreted as SO(x0 ) for some positive definite reference fourplane in H even (X, R), while O(4 + δ) is the corresponding group for the orthogonal complement of x0 . In other words, the elements of T 4,4+δ are interpreted as positive definite oriented four-planes x ⊂ H even (X, R) by H even (X, R) ∼ = R4,4+δ . Note that all odd 4,4 ∼ the cohomology of K3 is even, whereas H (X, R) = R when X is a four-torus. To explicitly realize the isomorphism (1.4) one also needs the positive generators υ of H 4 (X, Z) and υ 0 of H 0 (X, Z), which are Poincaré dual to points and to the whole oriented cycle X, respectively. They are nullvectors in H even (X, R) and satisfy υ, υ 0 = 1. Thus over Z they span an even, unimodular lattice isomorphic to the standard hyperbolic lattice U with bilinear form 01 . 10 Now consider a triple (), V , B) in the righthand side of (1.4). Define ξ : ) → H even (X, R), ξ(σ ) := σ − B, σ υ, 2 υ . x := spanR ξ ()) , ξ4 := υ 0 + B + V − B 2

(1.5)

= ξ()) is a positive definite oriented three-plane in H even (X, R), and the vector Then ) . Since ξ4 2 = 2V , it has positive square. Together, ) and ξ4 ξ4 is orthogonal to )

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span an oriented four-plane x ⊂ H even (X, R). Obviously, the map (), V , B) → x is invertible, once υ and υ 0 are given. To describe the projection from Teichmüller space to M we need to consider the lattices H 2 (X, Z) and H even (X, Z). They are even, unimodular, and have signature (p, p + δ) with p = 3 and p = 4, respectively. Such lattices are isometric to - p,p+δ = U p ⊕ (E8 (−1))δ/8 . Here each summand is a free Z module, E8 has as bilinear form the Cartan matrix of E8 , and for any lattice - we denote by -(n) the same Z module - with quadratic form scaled by n. We now consider the projection from Teichmüller space to M. First we have to identify all points in T 3,3+δ which yield the same Ricci flat metric. This means that we have to quotient the Teichmüller space (1.3) by the so-called classical symmetries. The projection is given by (1.6) O + (H 2 (X, Z)) T 3,3+δ [K-T]. Here we use the notation O + (-) for the intersection of O + (W ) with the automorphism group of a lattice - ⊂ W . The interpretation of the quotient space (1.6) as moduli space of Einstein metrics of volume 1 on X is straightforward in the torus case, but for X = K3 one has to include orbifold limits (see Sect. 2). The corresponding σ models are not expected to exist for all values of B [Wi3]. To simplify the discussion we include such conifold points in M. On T 4,4+δ the group of classical symmetries lifts by (1.5) to the subgroup of O + (H even (X, Z)) which fixes both lattice vectors υ and υ 0 . Next we consider the shifts of B by elements λ ∈ H 2 (X, Z), which neither change the physical content. One easily calculates that this also yields a left action on T 4,4+δ by a lattice automorphism in O + (H even (X, Z)), generated by w → w − w, λυ for 2 0 w, υ = 0 and υ 0 → υ 0 + λ − λ 2 υ. These transformations fix υ and shift υ to 0 arbitrary nullvectors dual to υ. Thus the choice of υ is physically irrelevant. We shall argue that the projection from Teichmüller space to M is given by (1.7) T 4,4+δ −→ O + (H even (X, Z)) T 4,4+δ . The group O + (H even (X, Z)) acts transitively on pairs of primitive lattice vectors of equal length [L-P, Ni3]. Thus (1.7) would imply that different choices of υ, υ 0 are equivalent. Anticipating this result in general, we call the choice of an arbitrary primitive nullvector υ ∈ H even (X, Z) a geometric interpretation of a positive oriented four-plane x ⊂ H even (X, Z). Such a choice yields a family of σ models with physically equivalent data (), V , B). A conformal field theory has various different geometric interpretations, and the choice of υ is comparable to a choice of a chart of M. Aspinwall and Morrison also identify theories which are related by the worldsheet parity transformation [A-M]. We regard the latter as a symmetry of M. It is given by change of orientation of the four-plane x or equivalently by a conjugation of O + (H even (X, R)) with an element of O(H even (X, R)) − O + (H even (X, R)) which transforms the lattice H even (X, Z) and the reference four-plane x0 into themselves. To stay in the classical context, one may choose an element which fixes υ and υ 0 . More canonically, parity corresponds to (υ, υ 0 ) → (−υ, −υ 0 ). The latter induces ξ4 → −ξ4 and (), V , B) → (), V , −B). Let us consider the general pattern of identifications. When two points in Teichmüller space are identified the same is true for their tangent spaces. Higher derivatives can be treated by perturbation theory in terms of tensor products of the tangent spaces H1 . Assuming the convergence of the perturbation expansion in conformal field theory,


93

any such isomorphism can be transported to all points of T 4,4+δ . Therefore σ model isomorphisms are given by the action of a group G (δ) on this space. In the previous considerations we have found a subgroup of G (δ) . Below we shall prove that the interchange of υ and υ 0 , which is the Fourier-Mukai transform [R-W], also belongs to G (δ) . When B = 0, this yields the map (), V , 0) → (), V −1 , 0). In the torus case, it is known as T-duality and it seems natural to extend this name to X = K3. We will not use the name mirror symmetry for this transformation. It is obvious that classical symmetries, integral B-field shifts, and T-duality generate all of O + (H even (X, Z)). Thus G (δ) contains all of this group. As argued in [A-M, As2], it cannot be larger, since otherwise the quotient of T 4,4+δ by G (δ) plus the parity automorphism would not be Hausdorff [Al]. For a proof of the Hausdorff property of M one will need some features of the superconformal field theories, which should be easy to verify once they are somewhat better understood. First, one has to check that all fields are generated by the iterated operator products of a finite dimensional subspace of basic fields. Next one has to show that the operator product coefficients are determined in terms of a finite number of basic coefficients, and that the latter are constrained by algebraic equations only. This would show that M is an algebraic space. In particular, every point has a neighborhood which contains no isomorphic point. All of these features are true in the known examples of conformal field theories with finite effective central charge, in particular for the unitary theories. They certainly should be true in our case. In the context of σ models it often is useful to choose a complex structure on X. When such a structure is given, the real and imaginary parts of any generator of H 2,0 (X, C) span an oriented two-plane 1 ⊂ ). Conversely, any such subspace 1 defines a complex structure. This means that the choice of an Einstein metric is nothing but the choice of an S2 of complex structures on X, in other words a hyperkähler structure. In terms of cohomology, 1 specifies H 2,0 (X, C) ⊕ H 0,2 (X, C). The orthogonal complement of 1 in H 2 (X, R) yields H 1,1 (X, R). Any vector ω ∈ H 1,1 (X, R) of positive norm yields a Kähler class compatible with the complex structure and the hyperkähler structure ) spanned by 1 and ω. Since H 2 (X, Z) is torsionfree for tori and K3 surfaces, the Néron-Severi group N S(X) can be identified with P ic(X) := H 2 (X, Z) ∩ H 1,1 (X, R), the Picard lattice of X. By a result of Kodaira’s, X is algebraic, if N S(X) contains an element ρ of positive length squared [Ko]. Given a hyperkähler structure ) we can always find 1 ⊂ ) such that X becomes an algebraic surface. It suffices to choose ω as the projection of ρ on ) and 1 as the corresponding orthogonal complement. The projection is non-vanishing, since the orthogonal complement of ) in H 2 (X, R) is negative definite. Varying ρ one obtains a countable infinity of algebraic structures on X. Thus the occasionally encountered interpretation of moduli of conformal field theories as corresponding to nonalgebraic deformations of K3 surfaces does not make sense (this was already pointed out in [Ce2] by different arguments). ⊂ x. As The choice of 1 ⊂ ) lifts to a corresponding choice of a two-plane 1 discussed above this selects a (2, 2) subalgebra of the (4, 4) superalgebra. We will refer to the choice of such a two-plane as fixing a complex structure. More precisely, the two specifies a complex structure in every geometric interpretation of the conformal plane 1 field theory.

1.1. Moduli space of theories associated to tori. Originally, Narain determined the moduli space Mtori of superconformal field theories associated to tori by explicit construction

94


of nonlinear σ models [C-E-N-T, Na]. With the above formalism we can reproduce his description as follows. Let us consider tori of arbitrary dimension d. We change the notation by transposing the group elements, which exchanges left and right group actions. This yields MNarain = O(d) × O(d) O(d, d)/O(- d,d ). This moduli space has a symmetry given by worldsheet parity. We shall see that its action on O(d, d) exchanges the two O(d) factors. For later convenience we are going to use the cover SO(d) × SO(d) SO + (d, d)/SO + (- d,d ) of MNarain . For even d this is a four-fold cover, for odd d a two-fold one. The R-span of - d,d is naturally isomorphic to Rd ⊕ (Rd )∗ , where Rd is considered as an isotropic subspace and W ∗ denotes the dual of a vector space W , and analogously for lattices. Thus O(d, d) can be considered as the orthogonal group of a vector space with elements (α, β), α, β ∈ Rd and scalar product (α, β) · (α , β ) = α · β + α · β. There is a canonical maximal positive definite d-plane given by α = β in Rd ⊕ (Rd )∗ = Rd,d . The group SO(d) × SO(d) is supposed to describe rotations in this d-plane and in its orthogonal complement. In this description, the parity transformation consists of interchanging these two orthogonal d-planes, plus a sign change of the bilinear form on Rd,d . Now we use the isometry V : SO(d) GL+ (d) × Skew(d × d, R) −→ SO(d) × SO(d) SO + (d, d) ∼ = T d,d (1.8) given by V (, B) =

(T )−1 0 0

1 −B 0 1

.

(1.9)

+ d We identify ∈ GL √ (d) with the image√of Z under . Finally we change coordinates by pl := (α + β)/ 2, pr := (α − β)/ 2, such that the scalar product becomes

(pl ; pr ) · (pl ; pr ) := pl pl − pr pr .

(1.10)

This means that the positive definite d-plane is given by pr = 0 and its orthogonal := (T )−1 B−1 a point in Mtori is now complement by pl = 0. Altogether, with B described by the lattice -(, B) = (pl (λ, µ); pr (λ, µ)) (1.11)

+ λ; µ − Bλ − λ (λ, µ) ∈ ⊕ ∗ . := √1 µ − Bλ 2

The corresponding σ model has the real torus T = Rd / as target space and B ∈ H 2 (T , R) ∼ = Skew(d ×d, R) as B-field. Introducing d Majorana fermions ψ1 , . . . , ψd as superpartners of the abelian currents j1 , . . . , jd on the torus one constructs an N = (2, 2) superconformal field theory with central charge c = 3d/2 which will be denoted by T (, B). From Eq. (1.9) it is clear that integral shifts of B and lattice automorphisms yield isomorphic theories.


95

The theory is specified by its charge lattice -(, B). Namely, to any pair (λ, µ) ∈ ⊕∗ there corresponds a vertex operator Vλ,µ with charge (pl (λ, µ); pr (λ, µ)) with respect to (j1 , . . . , jd ;  1 , . . . ,  d ) and with dimensions (h; h) = ( 21 pl2 ; 21 pr2 ). Thus h and −h are the squares of the projections of (pl ; pr ) to the positive definite d-plane and its orthogonal complement, respectively. In this description, the parity operation is represented by the interchange of the latter two planes plus a sign change in the quadratic form on Rd,d . The transformations which exchange the sheets of our covering of Narain’s moduli space MNarain are given by target space orientation change and T-duality, as can be read off from Eq. (1.11). The partition function of this theory is Z(τ, z) = Z,B (τ ) · Z,B (τ ) =

1 2

4

ϑi (τ, z) d η(τ ) , i=1

1

|η(τ )|2d

(λ,µ)∈⊕∗

(1.12)

q 2 (pl (λ,µ)) q¯ 2 (pr (λ,µ)) , 1

2

1

2

where q = exp(2πiτ ) and analogously for q. The functions ϑj (τ, z), j = 1, . . . , 4 are the classical theta functions and η(τ ) is the Dedekind eta function. For ease of notation we will write η = η(τ ), ϑj (z) = ϑj (τ, z), and ϑj = ϑj (τ, 0) in the following. By considering H1 one easily checks that all theories in Mtori are described by some even unimodular lattice -. We want to show that every such lattice has a σ model interpretation - = -(, B) (see also [As2]). Choose a maximal nullplane Y ⊂ Rd,d = Rd ⊕ (Rd )∗ such that Y ∩ - ⊂ - is a primitive sublattice. Apply an SO(d) × O(d) transformation such that the equation of this plane becomes β = 0. Put Y ∩- = (∗ , 0). Next choose a dual nullplane Y 0 such that Y ⊕ Y 0 = Rd,d and Y 0 ∩ - ⊂ - is a primitive lattice, too. Existence of Y 0 can be shown by a Gram type algorithm. Then Y 0 = {(−Bβ, β) | β ∈ Rd } for some skew matrix B, and - = -(, B). Note that different choices of Y 0 merely correspond to translations of B by integral matrices. So the geometric interpretation is actually fixed by the choice of Y alone as soon as B is viewed as an element of Skew(d)/ Skew(d × d; Z). In this interpretation, Rd is identified with the cohomology group H 1 (Rd /Zd , R) of the reference torus T = Rd /Zd . In addition to its defining representation, the double cover of the group SO + (d, d) also has half-spinor representations, namely its images in SO + (H odd (T , R)) and in SO + (H even (T , R)). For d = 4 one has the obvious isomorphism SO + (4, 4) ∼ = SO + (H odd (T , R)), which together with SO + (4, 4) ∼ = SO + (H even (T , R)) yields the celebrated D4 triality [L-M, I.8]. It is the latter automorphism which we will need in this paper, since the odd cohomology of X does not survive orbifold maps. Note that for Spin(4, 4) representations on R4,4 there is the same triality relation as for Spin(8) representations on R8 , i.e. an S3 permuting the vector representation, the chiral and the antichiral Weyl spinor representation. The role of triality is already visible upon comparison of the geometric interpretations, where the analogy between choices of nullplanes Y, Y 0 as described above and nullvectors υ, υ 0 in (1.5) is apparent. Indeed, part of the triality manifests itself in a one to one correspondence between maximal isotropic subspaces Y ⊂ R4,4 and null Weyl spinors υ such that Y = {y ∈ Rd,d | c(y)(υ) = 0}, where c denotes Clifford multiplication on the spinor bundle [B-T]. One can regard this as further justification for the interpretation of υ as a volume form

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which generates H 4 (T , Z) in our geometric interpretation. Recall also that in both cases different choices of Y 0 , υ 0 correspond to B-field shifts by integral forms. We now explicitly describe the isomorphism (1.8) to show that it is a triality automorphism. First compare (1.8) to (1.4) and notice that Skew(4) ∼ = R3,3 which will 3,3 simply be written Skew(4) B → b ∈ R in the following. Moreover, because |det | is the volume of the torus T = Rd /, we can decompose SO(4)\GL+ (4) ∼ = SO(4)\SL(4) × R+ . Now let T0 = R4 /0 , where 0 is a lattice of determinant 1 and is viewed as an element of SL(4). Consider the induced representation ρ of SL(4) on the exterior product 2 (R4 ) which defines an isomorphism 2 (0 ) ∼ = H2 (T0 , Z) for every 0 ∈ SL(4). Because ρ commutes with the action of the Hodge star operator ∗ and ∗2 = 1 on twoforms, SL(4) is actually represented by SO + (3, 3). In terms of coordinates as in (1.9) and with = V 1/4 0 = (λ1 , . . . , λ4 ), V = |det |, we can write ρ (0 ) = V −1/2 (λ1 ∧ λ2 , λ1 ∧ λ3 , λ1 ∧ λ4 , λ3 ∧ λ4 , λ4 ∧ λ2 , λ2 ∧ λ3 ) (1.13) ∈ SO + (H2 (T , R)) ∼ = SO + (3, 3). + ∼ ∼ Because SO (3, 3)3,3= SL(4)/Z2 and SO(3) × SO(3)/Z2 = SO(4) we find SO(4) SL(4) ∼ T and all in all have = (1.8) (1.4) ∼ = T 4,4 ∼ = SO(4) GL+ (4) × Skew(4) −→ T 3,3 × R+ × R3,3 ∼ = T 4,4 .

(1.14)

By (1.14) the geometric interpretation of a superconformal field theory is translated from a description in terms of the lattice of the underlying torus, i.e. in terms of ∼ = H1 (T , Z), to a description in terms of H2 (T , Z) ∼ = 2 (). This translation is essential for understanding the relation between the moduli spaces Mtori and MK3 . To actually arrive at the description (1.4) in terms of hyperkähler structures, i.e. in terms of H 2 (T , Z), we have to apply Poincaré duality or use the dual lattice ∗ instead of . This distinction will no longer be relevant after theories related by T-duality have been identified. We insert the coordinate expressions in (1.9) and (1.5) into (1.14), write = V 1/4 0 , V = |det | as before and arrive at   1/2  1 0 0 V 0 0      b 1 0 0 ρ( ) 0 V (, B) −→ S(, B) =  (1.15)  . 0    2 0 0 V −1/2 − B −bT 1 2

Observe that (1.15) is a homomorphism T 4,4 → T 4,4 and thus gives a natural explanation for the quadratic dependence on B in (1.5). Moreover, (1.15) reveals the structure of the warped product (1.4) alluded to before. But above all on the Lie algebra level one can now easily read off that (1.15) is the triality automorphism exchanging the two half spinor representations V and S. Namely, let h1 , . . . , h4 denote generators of the Cartan subalgebra of so(4, 4). Here hi generates dilations of the radius Ri of our torus in direction λi . Since exp(ϑhi ) scales V ±1/2 by e±ϑ/2 and with (1.13) one then finds that (1.15) indeed is induced by the triality automorphism which acts on the Cartan subalgebra by t✙ ❥ th1 −h3 ❅ t h2 −h❅ 3

h2 +h4

h1 → 21 (h1 + h2 + h3 + h4 ), h2 → 21 (h1 + h2 − h3 − h4 ), h3 → 21 (h1 − h2 + h3 − h4 ), h4 → 21 (h1 − h2 − h3 + h4 ) :

h2 −h4

t


97

Note that triality interchanges the outer automorphisms of SO + (4, 4) related to worldsheet parity and target space orientation. Triality considerations have a long history in superstring and supergravity theories, see for example [Sha, Cu, G-O]. Concerning recent work, as communicated to us by N. Obers, SO(4, 4) is crucial in the conjectured duality between heterotic strings on the fourtorus and type IIA on K3 [O-P1, K-O-P]. In connection with the calculation of G(Z) invariant string theory amplitudes one can use triality to write down new identities for Eisenstein series [O-P1, O-P2]. We now come to a concept which is of major importance in the context of Calabi– Yau compactification and nonlinear σ models, namely the idea of large volume limit. A precise notion is necessary of how to associate a unique geometric interpretation to a theory described by an even self dual lattice - when parameters of volume go to infinity. Intuitively, because of the uniqueness condition, this should describe the limit where all the radii of the torus in this particular geometric interpretation are large. Because in the charge lattice (1.11) λ ∈ and µ ∈ ∗ are interpreted as winding and momentum modes, the corresponding nullplane Y should have the property Y ∩ - = spanZ √1 (µ; µ) ∈ - µ2 1 2 (1.16) -. ⊂ spanZ (pl ; pr ) ∈ - pl 2 1, pr 2 1 =: Because pl 2 − pr 2 ∈ Z, for (pl ; pr ) ∈ - we have pl 2 = pr 2 . This shows Y ∩- = - because any (pl ; pr ) ∈ Y ⊥ = Y must have large components. Moreover, if a maximal isotropic plane Y as in (1.16) exists, then it is uniquely defined, thus yielding a sensible notion of large volume limit. Large volume and small volume limits are exchanged by T-duality. For our embedding of torus orbifold theories into the K3 moduli space MK3 we have to keep target space orientation. We also want to keep the left-right distinction in the conformal field theory. Torus T-duality just yields a reparametrization of the theory and should be divided out of the moduli space. Thus for us the relevant moduli space of torus theories is given by Mtori = SO(d) × O(d) O + (d, d)/O + (- d,d ). (1.17) Notice that this is a double cover of MNarain . 1.2. Moduli space of theories associated to K3 surfaces. We now give some more details about the moduli space of conformal field theories associated to K3 which we will concentrate on for the rest of the paper, namely MK3 = O + (H even (X, Z)) T 4,20 (1.18) by (1.7). For other presentations see [A-M, R-W, Di]. In the decomposition (1.4) we determine the product metric such that it becomes an isometry. In particular, it faithfully relates moduli of the conformal field theory to deformations of geometric objects. Recall that the structure of the tangent space H1 of MK3 in a given superconformal field theory is best understood by examining the ( 21 , 21 )susy susy fields in F1/2 . In our case we have related it to the su(2)l ⊕su(2)r invariant subspace (4) (0) (4) (0) of the tensor product Ql ⊗ Qr ⊗ H1/4 ⊗ H1/4 , where H1/4 denotes the charged and H1/4

98

W. Nahm, K. Wendland (4)

the uncharged Ramond ground states. The invariant subspace of Ql ⊗ Qr ⊗ H1/4 yields a four-plane with an orthogonal group generated by su(2)l ⊕ su(2)r . When a frame in Ql ⊗ Qr is chosen, the latter tensor product factor can be omitted. The description of (4) (0) M implies that H1/4 ⊕ H1/4 has a natural non-degenerate indefinite metric and remains invariant under deformations, but it has not been understood how this comes about. In terms of the four-plane x ∈ T 4,20 giving the location of our theory in moduli space, specific vectors in the tangent space Tx T 4,20 are described by infinitesimal deformations of one generator ξ ∈ x in direction x ⊥ that leaves ξ ⊥ ∩ x invariant. To formulate this in terms of a geometric interpretation (), V , B) specified by (1.5), pick a basis η1 , . . . , η19 of ) ⊥ ⊂ H 2 (X, R) ∼ = R3,19 . Then x ⊥ is spanned by {ηi − 2 ηi , B υ; i = 1, . . . , 19} and η20 := υ 0 + B − ( B 2 + V )υ. In each of the SO(4) fibres of H1 over ηi − ηi , B υ, i = 1, . . . , 19 we find a three dimensional subspace deforming generators of ) by ηi , as well as the deformation of B in direction of ηi . The fibre over η20 contains B-field deformations in direction of ) and the deformation of volume. All in all, a 3 · 19 = 57 dimensional subspace of H1 = Tx MK3 is mapped onto deformations of ) by (1, 1)-forms η ∈ ) ⊥ ∩ H 2 (X, R) ⊂ H 1,1 (X, R), no matter what complex structure we pick in ). The 23 dimensional complement of this subspace is given by 19 + 3 deformations of the B-field by forms η ∈ H 2 (X, R) and the volume deformation. One of the most valuable tools for understanding the structure of the moduli space is the study of symmetries. So the next question to be answered is how to translate symmetries of our superconformal field theory to its geometric interpretations. Those symmetries which commute with the su(2)l ⊕ su(2)r action leave the four-plane x invariant and are called algebraic symmetries. When the N = (4, 4) supersymmetric theories are constructed in terms of (2, 2) supersymmetric theories one has a natural framing. In this context, algebraic symmetries are those which leave the entire vector space Ql ⊗ Qr of supercharges invariant. More generally, any abelian symmetry group of our theory projects to a u(1)l ⊕ u(1)r subgroup of su(2)l ⊕ su(2)r and fixes the corresponding N = (2, 2) subalgebra. When corresponding supercharges are fixed, the ± susy abelian symmetry group acts diagonally on the charge generators J ± , J of su(2)l ⊕ susy su(2)r . The algebraic subgroup of this symmetry group is the one which fixes these charges. If the primitive nullvector υ specifying our geometric interpretation (), V , B) is invariant upon the induced action of an algebraic symmetry we call the latter a classical symmetry of the geometric interpretation (), V , B). Because a classical symmetry α ∗ fixes x by definition we get an induced automorphism of H 2 (X, R) which leaves ) ⊂ H 2 (X, R) and B ∈ H 2 (X, R)/H 2 (X, Z) invariant. Moreover, because ξ4 in (1.5) is 2 ∗ invariant as well, η20 = υ 0 + B − ( B 2 + V )υ is fixed. Thus α acts trivially on moduli of volume and B-field deformation in direction of ). Because α ∗ acts as automorphism on H 1,1 (X, R) = 1⊥ ∩ H 2 (X, R) for any choice of complex structure 1 ⊂ ) on X leaving the one dimensional H 1,1 (X, R) ∩ ) invariant, all in all, x → (), V , B) maps the action of α ∗ to an automorphism of H 2 (X, R) which on H 1,1 (X, R) has exactly the same spectrum as α ∗ on ( 21 , 21 )-fields with charge, say, Q = Q = 1. If the integral action of α ∗ on H 2 (X, C) is induced by an automorphism α ∈ Aut (X) of finite order of the K3 surface X, then by definition, because α ∗ acts trivially on H 2,0 (X, C), α is an algebraic automorphism [Ni2]. This notion of course only makes sense after a choice of complex structure, or in conformal field theory language an N = (2, 2) subalgebra of the N = (4, 4) superconformal algebra fixing generators


99

±

J, J ± , J , J of su(2)l ⊕ su(2)r . Still, because we always assume the metric to be ∗ invariant under α ∗ as well, i.e. ) ⊂ H 2 (X, R)α , this is no further restriction. On the other hand, given an algebraic automorphism α of X which induces an automorphism of H 2 (X, R) that leaves the B-field invariant, α induces a symmetry of our conformal ± field theory which leaves J, J ± , J , J invariant. This gives a precise notion of how to continue such an algebraic automorphism to the conformal field theory level. We are thus naturally led to a discussion of algebraic automorphisms of K3 surfaces, which are mathematically well understood thanks to the work of Nikulin [Ni2] for the abelian and Mukai [Mu] for the general case. The first to explicitly take advantage of their special properties in the context of conformal field theory was P.S. Aspinwall [As1]. From [Ni2, Th. 4.3,4.7,4.15] one can deduce the following consequence of the global Torelli theorem: Theorem 1.1. Let g denote an automorphism of H 2 (X, C) of finite order which maps forms corresponding to effective divisors of self intersection number −2 in P ic(X) to forms corresponding to effective divisors. Then g is induced by an algebraic automor ⊥ phism of X iff H 2 (X, Z)g ∩ H 2 (X, Z) ⊂ P ic(X) is negative definite with respect to the intersection form and does not contain elements of length squared −2. If for a geometric interpretation (), V , B) of x ∈ O + (H even (X, Z))\T 4,20 we have classical symmetries which act effectively on what we read off as H 2 (X, C) but are not induced by an algebraic automorphism of the K3 surface X by Theorem 1.1, then our interpretation of x as giving a superconformal field theory breaks down. Such points should be conifold points of the moduli space MK3 , characterized by too high an amount of symmetry. One can regard Nikulin’s Theorem 1.1 as harbinger of Witten’s result that in points of enhanced symmetry on the moduli space of type IIA string theories compactified on K3 the conformal field theory description breaks down [Wi3]. By abuse of notation we will often renounce to distinguish between an algebraic automorphism on K3 and its induced action on cohomology. From Mukai’s work [Mu, Th. 1.4] one may learn that the induced action of any algebraic automorphism group G on the total rational cohomology H ∗ (X, Q) is a Mathieu representation of G over Q, i.e. a representation with character χ (g) = µ(ord(g)), where for n ∈ N : µ(n) :=

n

24 (1 + p1 )

.

(1.19)

p prime, p|n

It follows that dimQ H ∗ (X, Q)G = µ(G) :=

1 µ(ord(g)) |G|

(1.20)

g∈G

[Mu, Prop. 3.4]. We remark that because G acts algebraically, we have dimQ H ∗ (X, Q)G = dimR H ∗ (X, R)G = dimC H ∗ (X, C)G . By definition of algebraic automorphisms H ∗ (X, C)G ⊃ H 0 (X, C) ⊕ H 2,0 (X, C) ⊕ H 0,2 (X, C) ⊕ H 2,2 (X, C), so µ(G) − 4 = dimR H 1,1 (X, R)G . (1.21)

2 ⊥ Moreover, from Theorem 1.1 we know that H (X, R)G ⊂ H 1,1 (X, R) is negative definite, and because H 1,1 (X, R) has signature (1, 19), we may conclude that it contains

100


an invariant element with positive length squared. Thus µ(G) ≥ 5 for every algebraic automorphism group G [Mu, Th. 1.4]. Moreover [Mu, Cor. 3.5, Prop. 3.6], G = {1} #⇒ µ(G) ≤ 16.

(1.22)

Finally let us consider the special case of an algebraic automorphism α of order 4, which will be useful in due course. By nk we denote the multiplicity of the eigenvalue i k of the induced action α ∗ on H 1,1 (X, C). Because by (1.19) and (1.20) µ(Z4 ) = 10 and µ(Z2 ) = 16, using (1.21) we find n0 = 10 − 4 = 6, n2 = 16 − 4 − n0 = 6. The automorphism α ∗ acts on the lattice H 2 (X, Z), so it must have integer trace. On the other hand 20 = dimC H 1,1 (X, C) = n0 + n1 + n2 + n3 , hence n0 = n2 = 6,

n1 = n3 = 4.

(1.23)

2. Special Subspaces of the Moduli Space: Orbifold Theories This section is devoted to the study of theories which have a geometrical interpretation on an orbifold limit of K3. We begin by giving a short account on the relevant features of the orbifold construction, for details the reader is referred to the vast literature, e.g. [D-H-V-W, Di]. On the geometric side, the Zl orbifold construction of K3 can be described as follows [Wa]: Consider a four torus T , where T = T 2 × T2 with two Zl symmetric two tori which need not be orthogonal. Let ζ ∈ Zl act algebraically on T 2 = C/L, T2 = C/L 2 2 (z1 , z2 ) ∈ T × T by (z1 , z2 ) → (ζ z1 , ζ −1 z2 ). Mod out this symmetry and blow up the resulting singularities; that is, replace each singular point by a chain of exceptional divisors, which in the case of Zl -fixed points have as intersection matrix the Cartan matrix of Al−1 . In particular, the exceptional divisors themselves are rational curves, i.e. holomorphically embedded spheres with self intersection number −2. In terms of the homology of the resulting surface X these rational curves are elements of H2 (X, Z) ∩ H1,1 (X, C). To translate to cohomology we work with their Poincaré duals, which now are elements of P ic(X) with length squared −2. One may check that for l ∈ {2, 3, 4, 6} this procedure changes the Hodge diamond by 1 2 1

1 2

4 2

2 1

1 −→ 1

0

0 20

0

1 0

1

and indeed produces a K3 surface X, because the automorphism we modded out was algebraic. We also obtain a rational map π : T → X of degree l by this procedure. To fix a hyperkähler structure we additionally need to pick the class of a Kähler metric on X. We will consider orbifold limits of K3 surfaces, that is use the orbifold singular metric on X which is induced from the flat metric on T and assigns volume zero to all the exceptional divisors. The corresponding Einstein metric is constructed by excising a sphere around each singular point of T /Zl and gluing in an Eguchi Hanson sphere E2 instead for l = 2, or a generalized version El with boundary ∂El = S3 /Zl at infinity and nonvanishing Betti numbers b0 (El ) = 1, b2 (El ) = b2− (El ) = l − 1, i.e. χ (El ) = l. The orbifold limit is the limit these Eguchi Hanson type spheres have shrunk to zero size in. The description (1.6) of the moduli space of Einstein metrics of volume 1 on K3


101

includes orbifold limits [K-T], and as was shown by Anderson [An] one can define an extrinsic L2 -metric on the space E of regular Einstein metrics of volume 1 on K3 such that the completion of E is contained in the set of regular and orbifold singular Einstein metrics. On the conformal field theory side the orbifold construction is in total analogy to the geometric one described above. Assume we know the action of Zl on the space of states H of a conformal field theory with geometric interpretation on the torus T we had above. To construct the orbifold conformal field theory, keep all the invariant states in H and then – for the sake of modular invariance, if we argue on the level of partition functions – add twisted sectors. For ζ ∈ Zl , the ζ -twisted sector consists of states corresponding to fields ϕ which are only well defined up to ζ -action on the world sheet of the original theory, that is ϕ : Z → T , ϕ(σ0 + 1, σ1 ) = ζ ϕ(σ0 , σ1 ). Z denotes the configuration space as mentioned in the introduction and coordinates (σ0 , σ1 ), σ0 ∼ σ0 + 1, (σ0 , σ1 ) ∼ (σ0 + τ0 , σ1 + τ1 ) are chosen such that ϕ(0, 0) is a fixed point. In other words, the constant mode in the Fourier expansion of ϕ is a fixed point pζ of ζ . The other modes are of non-integral level, so the ground state energy in the twisted sector is shifted away from zero. More precisely, the ground state |)ζ,pζ of the ζ -twisted sector Hζ,pζ belongs to the Ramond sector and has dimensions c h = h = 24 = 41 . The corresponding field )ζ,pζ introduces a cut in Z from (0, 0) to (τ0 , τ1 ) ∼ (0, 0) to establish the transformation property ϕ(σ0 + 1, σ1 ) = ζ ϕ(σ0 , σ1 ) for |ϕ ∈ Hζ,pζ , often referred to as boundary condition. The field )ζ,pζ is called a twist field. For explicit formulae of partition functions for Zl orbifold conformal field theories see [E-O-T-Y], for the special cases l = 2 and l = 4 we are studying here see (2.3) and (2.14). To summarize, we stress the analogy between orbifolds in the geometric and the conformal field theory sense once again; in particular, the introduction of a twist field for each fixed point and boundary condition corresponds to the introduction of an exceptional divisor in the course of blowing up the quotient singularity, if we use the metric which assigns volume zero to all the exceptional divisors. By construction orbifold conformal field theories have a preferred geometric interpretation in the sense of Sect. 1.2. We will now investigate this geometric interpretation for Z2 and Z4 orbifolds, particularly taking advantage of their specific algebraic automorphisms. A program for finding a stratification of the moduli space could even be formulated as follows: Find all subspaces of theories having a geometric interpretation (), V , B) with given algebraic automorphism group G. Relations between such subspaces may be described by the modding out of algebraic automorphisms. Any infinitesimal deformation of ) by an element of H 1,1 (X, R)G will preserve the symmetries in G, as well as volume deformations and B-field deformations by elements in H 2 (X, R)G . The subspace of theories with given classical symmetry group G in a geometric interpretation therefore can maximally have real dimension 3(µ(G)−5)+1+µ(G)−2 = 4(µ(G)−4) in accord with (1.21). In particular, for the minimal value µ(G) = 5, the only deformations preserving the entire symmetry are deformations of volume and those of the B-field by elements of ). Of course, the above program is far from utterly realizable, even in the pure geometric context, but it might serve as a useful line of thought. Z2 Orbifolds actually yield the first item of this program: We can map the entire torus moduli space into the K3 moduli space by modding out the symmetry z → −z. The description is straightforward if we make use of the geometric interpretation of torus theories given by the triality automorphism

102


(1.14), because the geometric data then turn out to translate in a simple way into the corresponding data on K3. 2.1. Z2 Orbifolds in the moduli space. Some comments on Z2 orbifold conformal field theories as described at the beginning of the section are due, before we can show where they are located within the moduli space MK3 . We denote the Z2 orbifold obtained from the nonlinear σ model T (, BT ) by K(, BT ). If the theory on the torus has an enhanced symmetry G we frequently simply write G/Z2 , e.g. SU (2)14 /Z2 for K(Z4 , 0). In the nonlinear σ model on the torus T = R4 / as described in Sect. 1.1 the current jk generates translations in direction of coordinate xk . This induces a natural correspondence between tangent vectors of T and fields of the nonlinear σ model which is compatible with the so(4) action on the tangent spaces of T and the moduli space, susy respectively. After selection of an appropriate framing of Ql ⊗ Qr to identify su(2)l,r with su(2)l,r as described in Sect. 1 the ψk are the superpartners of the jk . Hence the choice of complex coordinates z1 := √1 (x1 + ix2 ), z2 := √1 (x3 + ix4 ) corresponds to 2 2 setting (1)

ψ± :=

√1 (ψ1 2

± iψ2 ),

(2)

ψ± :=

√1 (ψ3 2

± iψ4 ).

(2.1)

The holomorphic W -algebra of our theory has an su(2)12 -subalgebra generated by (1)

(1)

(2)

(2)

(1)

(2)

(1)

(1)

(2)

(2)

(1)

(2)

J := ψ+ ψ− + ψ+ ψ− , J + := ψ+ ψ+ ,

(2)

(1)

(2)

(1)

J − := ψ− ψ− ;

A := ψ+ ψ− − ψ+ ψ− , A+ := ψ+ ψ− , A− := ψ+ ψ− .

(2.2)

Its geometric counterpart on the torus is the Clifford algebra generated by the two forms dz1 ∧dz1 +dz2 ∧dz2 , dz1 ∧dz2 , dz1 ∧dz2 ; dz1 ∧dz1 −dz2 ∧dz2 , dz1 ∧dz2 , dz2 ∧dz1 upon Clifford multiplication. The nonlinear σ model on the Kummer surface K() is the “ordinary” Z2 orbifold of the above, where Z2 acts by jk → −jk , ψk → −ψk , k = 1, . . . , 4. Note that the entire su(2)12 -algebra (2.2) is invariant under this action, thus any nonlinear σ model on a Kummer surface possesses an su(2)12 -current algebra. The N S-part of its partition function is ϑ3 (z) 4 ϑ3 ϑ4 4 ϑ4 (z) 4 1 ZNS (τ, z) = 2 Z,B (τ ) + 2 η η η (2.3) ϑ2 ϑ3 4 ϑ2 (z) 4 ϑ2 ϑ4 4 ϑ1 (z) 4 + + 2 η2 η . η η Here and in the following we decompose partition functions into four parts corresponding i.e. with y = exp(2π iz), y = exp(−2π iz), to the four sectors N S, N S, R, R,

Z = 21 ZNS + ZR + ZR , + ZcNS c ZNS (τ, z) = tr NS q L0 − 24 q L0 − 24 y J0 y J 0 , c c F L0 − 24 ZNS q L0 − 24 y J0 y J 0 = ZNS (τ, z + 21 ), (τ, z) = tr NS (−1) q c c c c ZR (τ, z) = tr R q L0 − 24 q L0 − 24 y J0 y J 0 = (qq) 24 (yy) 6 ZNS (τ, z + τ2 ), c c ZR(τ, z) = tr R (−1)F q L0 − 24 q L0 − 24 y J0 y J 0 = ZR (τ, z + 21 ). (2.4)


103

Given ZNS the entire partition function can be determined by using the above flows to find ZNS , ZR and ZR . This orbifold model has an N = (16, 16) supersymmetry. We are interested in deformations which conserve N = (4, 4) subalgebras. As explained in Sect. 1, the latter are given by chiral and antichiral ( 21 , 21 )-fields. Generically, the Neveu–Schwarz sector contains 144 fields with dimensions (h, h) = ( 21 , 21 ). Their quantum numbers under (J, A; J , A) are (ε1 , ε2 ; ε3 , ε4 ), εi ∈ {±1} (16 fields), (ε1 , 0; ε3 , 0) (64 fields), and (0, ε2 ; 0, ε4 ) (64 fields). The 80 fields which are charged under (J ; J ) yield the N = (4, 4) supersymmetric deformations which conserve the superalgebra that contains the J currents. The 80 fields which are charged under (A; A) yield deformations conserving a different N = (4, 4) superalgebra. The latter corresponds to the opposite torus orientation. Let us now focus on the description of the resulting geometric objects, namely Kummer surfaces denoted by K() if obtained by the Z2 orbifold procedure from the four torus T = R4 /. Generators of the lattice are denoted by λ1 , . . . , λ4 . From (1.14) we obtain an associated three-plane )T ⊂ H 2 (T , R), i.e. an Einstein metric on T , and we must describe how the Teichmüller space T 3,3 of Einstein metrics of volume 1 on the torus is mapped into the corresponding space T 3,19 for K3. This is best understood in terms of the lattices H 2 (T , Z) ∼ = - 3,3 and H 2 (X, Z) ∼ = 3,19 2 - , X = K(). In our notation H (T , Z) is generated by µj ∧ µk , j, k ∈ {1, . . . , 4} if (µ1 , . . . , µ4 ) is the basis dual to (λ1 , . . . , λ4 ). )T is defined by its relative position to a reference lattice - 3,3 ∼ = H 2 (T , Z) ⊂ H 2 (T , R). Note that in order to simplify the following argumentation we rather regard )T ⊂ H 2 (T , Z) as giving the position of the lattice H 2 (T , Z) = spanZ (µj ∧ µk ) relative to a fixed three–plane spanR (e1 ∧ e2 + e3 ∧ e4 , e1 ∧ e3 + e4 ∧ e2 , e1 ∧ e4 + e2 ∧ e3 ) with respect to the standard basis (e1 , . . . , e4 ) of R4 . To make contact with the theory of Kummer surfaces we pick a complex structure 1T ⊂ )T . The Z2 action on T has 16 fixed points 21 4k=1 εk λk , ε ∈ F42 . We can therefore choose indices in F42 to label the fixed points1 . Note that this is not only a labeling but the torus geometry indeed induces a natural affine F42 -structure on the set I of fixed points [Ni1, Cor. 5]. The two forms corresponding to the 16 exceptional divisors obtained from blowing up the fixed points are denoted by {Ei | i ∈ I }. They are elements of P ic(X) no matter what complex structure we choose, because we are working in the orbifold limit, i.e. Ei ⊥ ) ∀ i ∈ I . Let N ⊂ P ic(X) denote the primitive sublattice of the Picard lattice that contains {Ei | i ∈ I }. It is called Kummer lattice and by [Ni1, Th. 3]: Theorem 2.1. The Kummer lattice N is spanned by the exceptional divisors {Ei | i ∈ I } and { 21 i∈H Ei | H ⊂ I is a hyperplane}. On the other hand, a K3 surface X is a Kummer surface iff P ic(X) contains a primitive sublattice isomorphic to N. Let π : T → X be the degree two map from the torus to the orbifold singular Kummer surface. Using Poincaré duality, one gets maps π∗ from the homology and cohomology groups of T to those of X, and π ∗ in the other direction. In particular, this gives the natural embedding π∗ : H 2 (T , Z)(2) $→ H 2 (X, Z) (here -(2) denotes - with quadratic form scaled by 2). The image lattice will be called K. We prefer to work with metric √ 2 (T , Z) by 2a. isomorphisms and therefore denote the image in K of an element a ∈ H √ In particular, we write 2µj ∧ µk , j, k = 1, . . . , 4 for the generators of K. The lattice 1 F denotes the unique finite field with two elements. 2

104


H 2 (X, Z) contains K ⊕ N and is contained in the dual lattice K ∗ ⊕ N∗ . The three-plane ) ⊂ H 2 (X, R) which describes the location of the singular Kummer surface within the moduli space (1.6) of Einstein metrics of volume 1 on K3 is given by ) = π∗ )T . A description of how the lattices K and N are embedded in H 2 (X, Z) can be found in [Ni1]. First notice K ∗ /K ∼ = (Z2 )6 ∼ = N∗ /N, where N∗ /N is generated by { 21 i∈P Ei | P ⊂ I is a plane}. The isomorphism γ : K ∗ /K −→ N∗ /N is most easily understood in terms of homology by assigning the image in X of a twocycle through four fixed points in a plane P ⊂ I to 21 i∈P Ei . For example, γ ( √1 µj ∧ µk ) = 21 i∈Pj k Ei , 2

Pj k = spanF2 (fj , fk ) ⊂ F42 , fj ∈ F42 the j th standard basis vector. Note that Pj k may be exchanged by any of its translates l + Pj k , l ∈ F42 . Next check that the discriminant forms of K ∗ /K and N∗ /N, i.e. the induced Q/2Z valued quadratic forms, agree up to a sign. Then H 2 (X, Z) ∼ = (x, y) ∈ K ∗ ⊕ N∗ | γ (x) = y ,

(2.5)

x, y denoting the images of x, y under projection to K ∗ /K, N∗ /N. The isomorphism (2.5) provides a natural primitive embedding K ⊥ N $→ H 2 (X, Z), which is unique up to isomorphism [Ni1, Lemma 7]. Here, H 2 (X, Z) ∼ = - 3,19 is generated by

M := √1 µj ∧ µk + 21 Ei+l , l ∈ I and spanZ (Ei , i ∈ I ) . (2.6) 2

i∈Pj k

∼ Hence = H 2 (T , Z)(2) $→ H 2 (X, Z) ∼ = - 3,19 is naturally embedded, and in 2 2 ∼ particular ) ⊂ H (X, R) = H (X, Z) ⊗ R is obtained directly by regarding )T ⊂ H 2 (T , R) ∼ = H 2 (T , Z) ⊗ R $→ H 2 (X, Z) ⊗ R as three-plane in H 2 (X, R). To describe where the image K(, BT ) of a superconformal field theory T (, BT ) under Z2 orbifold is located in MK3 we now generalize the above construction to the quantum level. We have to lift π∗ to an embedding π∗ : H even (T , Z)(2) $→ H even (X, Z). The image will be denoted by K. Apart from µj ∧ µk the lattice H even (T , Z) has cannot be embedded as a primitive generators υ, υ 0 as defined in (1.5). Note that K ⊥ N because K ∗ /K ∼ sublattice in - 4,20 such that K = (Z2 )8 ∼ = (Z2 )6 ∼ = N∗ /N. This means that the B-field of the orbifold theory must have components in the Picard lattice. The torus model is given by a four-plane xT ⊂ H even (T , R), the corresponding orbifold model by its image x = π∗ xT in H even (X, Z) ⊗ R. To arrive at a complete ⊗ R + H 2 (X, R). Since description, we must find the embedding of H even (X, Z)√ in K must be integral and 2υ 0 ∈ K, every a ∈ N must scalar products with elements of K 1 have a lift √ υ + a or 0 + a in H even (X, Z). Those elements for which the lift has the - 3,3 (2)

2

form 0 + a must form an O + (H even (T , Z)) invariant sublattice of N. One may easily check that this sublattice cannot contain the exceptional divisors Ei , i ∈ I . Moreover, as unimodular lattice H even (X, Z) must contain an element of the form √1 υ 0 + a with 2 a ∈ N∗ . One finds that H even (X, Z) must contain the set of elements

:= M ∪ √1 υ 0 − 1 M Ei ; − √1 υ + Ei , i ∈ I . (2.7) 4 2

i∈I

2

In analogy to Nikulin’s description (2.5) and (2.6) of H 2 (X, Z) ∼ = - 3,19 we now find and {π ∈ N | ∀ m ∈ M : π, m ∈ Z} is Lemma 2.2. The lattice - spanned by M isomorphic to - 4,20 .


Proof. Define √ υ := 2υ,

105

υ 0 :=

√1 υ 0 2

−

1 4

Ei +

√

2υ,

i := − √1 υ + Ei . E 2

i∈I

(2.8)

Then - is generated by υ, υ 0 and the lattice

i+l , l ∈ I ; E i , i ∈ I . - := spanZ √1 µj ∧ µk + 21 E 2

i∈Pj k

i , E j = −2δij and upon comparison to (2.6) it is now easy to see that Because E 3,19 ∼ - = - . Moreover, υ, υ0 ⊥ - and spanZ ( υ, υ0) ∼ ) = U completes the proof. ( In particular, Lemma 2.2 gives a natural embedding - 4,4 (2) ∼ = H even (T , Z)(2) $→ H even (X, Z) ∼ = - 4,20 . As in the case of embedding the Teichmüller spaces T 3,3 $→ T 3,19 this enables us to directly locate the image under the Z2 orbifold of a conformal field theory corresponding to a four-plane x ⊂ H even (T , R) ∼ = - 4,4 ⊗ R within MK3 even 4,20 ∼ by regarding x as four-plane in H (X, R) = ⊗ R. Note that in this geometric interpretation υ, υ 0 are the generators of H 4 (X, Z) and H 0 (X, Z). Theorem 2.3. Let ()T , VT , BT ) denote a geometric interpretation of the nonlinear σ model T (, BT ) as given by (1.14). Then the corresponding orbifold conformal field theory K(, BT ) associated to the Kummer surface X = K() has geometric interpretation (), V , B), where ) ∈ T 3,19 as described after Theorem 2.1, V = V2T and (2) (2) i ∈ H even (X, Z) of i ∈ H even (X, Z) with E B = √1 BT + 21 BZ , BZ = 21 i∈I E 2 length squared -2 given in (2.8). In particular, the Z2 orbifold procedure induces an embedding Mtori $→ MK3 as quaternionic submanifold. Proof. Pick a basis σi , i ∈ {1, 2, 3} of )T . Then by (1.5) the nonlinear σ model T (, BT ) is given bythe four-plane x with generators ξi = σi −σi , BT υ, i ∈ {1, 2, 3} 2 and ξ4 = υ 0 + BT + VT − B2T υ. By the embedding - 4,4 ⊗ R ∼ = H even (T , R) $→ ∼ - 4,20 ⊗ R given in Lemma 2.2 it is now a simple task to reexpress the H even (X, R) = generators of x using the generators υ, υ 0 of H 4 (X, Z) and H 0 (X, Z): √ √ √ 2 (σi − σi , BT υ) = 2σi − 2σi , √1 BT υ, 2 2 (2) √1 υ 0 + BT + VT − B2T υ = υ 0 + √1 BT + 21 BZ 2 2 2 VT 1 √1 1 (2) υ. + 2 − 2 BT + 2 B Z 2

Comparison with (1.5) directly gives the assertion of the theorem.

) (

Theorem 2.3 makes precise how the statement that orbifold conformal field theories tend to give value B = 21 to the B-field in direction of exceptional divisors [As2, Sect. 4] is to be understood. Note that x ⊥ ∩ - 4,20 does not contain vectors of length squared −2, namely Ei ∈ x ⊥ , Ei 2 = −2 but Ei ∈ H even (X, Z). In the context of compactifactions of the type IIA string on K3 this proves that Z2 orbifold conformal field theories do not have enhanced gauge symmetry. A similar statement was made in [As1] and widely spread in the literature, but we were unable to follow the argument up to our result of Theorem 2.3.

106


2.2. T-duality and Fourier–Mukai transform. By Theorem 2.3 any automorphism on the Teichmüller space T 4,4 of Mtori is conjugate to an automorphism on the Teichmüller space T 4,20 of MK3 . In particular, nonlinear σ models on tori related by T-duality must give isomorphic theories on K3 under Z2 orbifolding. To show this explicitly and discuss the duality transformation on MK3 obtained this way is the object of this subsection. For simplicity first assume that our σ model on the torus T = R4 / has vanishing B-field, where we have chosen a geometric interpretation ()T , VT , 0). Then T-duality acts by ()T , VT , 0) → ()T , 1/VT , 0). By Theorem 2.3 the corresponding Z2 orbifold theories have geometric interpretations (), VT /2, B) and (), 1/2VT , B), respectively, where ) is obtained as image of the embedding )T ⊂ H 2 (T , R) $→ H 2 (X, R) and (2) i . We will now construct an automorphism T of the lattice B = 21 BZ = 41 i∈I E even H (X, Z) which fixes the four-plane x corresponding to the model with geometric interpretation (), VT /2, B) and acts by VT /2 → 1/2VT . In other words, we will explicitly construct the duality transformation induced by torus T-duality on MK3 . Our transformation T below was already given in [R-W] but not with complete proof. Within the context of boundary conformal field theories, in [B-E-R] it was shown that T induces an isomorphism on the corresponding conformal field theories. The relation to the Fourier–Mukai transform which we will show in Theorem 2.4 has not been clarified up to now. = ξ()) and the vector By (1.5), the four-plane x ⊂ H even (X, Z) is spanned by ) ξ4 = υ 0 + B + ( V2T + 1) υ (notations as in Theorem 2.3). Because by the above T fixes x √ pointwise, the unit vector ξ4 / VT ∈ ) ⊥ ∩ x must be invariant, too, i.e. invariant and ) under the transformation VT → 1/VT . Hence 0 0 + √1 + VT √1 √1 B + 1 VT + √1 υ + V υ + V υ = υ B T T 2 V V V 2 V T

T

T

T

for any value of VT . We set υ := T( υ ), υ 0 := T( υ 0 ) etc. and deduce + υ0 + B υ = 21 υ,

υ0 + B + υ = 21 υ.

(2.9)

2 = −2 implies B, The first equation together with B, υ = B, υ 0 = 0, B υ = −4 and justifies the ansatz

= −4 i + a B υ0 − αi E υ #⇒ (αi − 1)2 = 1, αi = 8 − 2a. i∈I

i∈I

i∈I

i ∈ H even (X, Z), which must be true by (2.9), The only solutions satisfying i∈I αi E are αi0 ∈ {0, 2} for some i0 ∈ I and αi = 1 for i = i0 , correspondingly a ∈ {− 27 , − 29 }. We conclude that if the automorphism T exists, then it is already uniquely determined up to the choice of a and of one point i0 ∈ I . The two possible choices of a turn out to → B − 2B = −B and yield equivalent results. In the be related by the B-field shift B 7 following we pick a = − 2 and find

i0 . i , i − E υ = 2( υ + υ 0 ) + 21 υ 0 = 2( υ + υ 0 ) + 21 (2.10) E E i∈I

i∈I

:= spanZ ( we denote the orthogonal One easily checks that U υ, υ0) ∼ = U . By N 0 ∼ complement of U in spanZ ( υ, υ ) ⊥ N = U ⊥ N, where N is the Kummer lattice of X as introduced in Theorem 2.1. Note that in I there are 15 hyperplanes Hi , i ∈ I0 =


107

I − {i0 } which do not contain i0 . The label i ∈ I0 is understood as the vector dual to the hyperplane Hi . Since the choice of i0 can be seen as the choice of an origin in the affine space F42 , the latter can be regarded as a vector space, and we have a unique natural i , i ∈ I , with is spanned by E isomorphism (F42 )∗ ∼ = F42 . One now checks that N

i := − 1 j − !i0 := υ − υ0, E υ − υ 0 (i = i0 ) (2.11) E E 2 j ∈Hi

1

i have been chosen i for any hyperplane H ⊂ I . The signs of the E as well as 2 i∈H E (2) 1 = 1B such that B i∈I Ei . 2 Z = 4 ! i ) = E i is a continuation of ∼ Since Ei , Ej = −2δij , one has N = N. Hence T(E ∼ ⊥N , and we find T2 = 1. Note that (2.10) to an automorphism of lattices U ⊥ N = U the action of T can be viewed as a duality transformation exchanging vectors i ∈ I with hyperplanes Hi , i ∈ I . Two-planes P ⊂ I are exchanged with their duals P ∗ which shows that T can be continued to a map on the entire lattice H ∗ (X, Z) consistently with (2.5). The induced action on K = π∗ H 2 (T , Z) leaves ) invariant. We also see that the above procedure is easily generalized to arbitrary nonlinear σ models T (, BT ). i0 and leaves the Let S denote the classical symmetry which changes the sign of E 0 i , i = i0 , other lattice generators E υ, υ , µj ∧ µk invariant. By (2.10) and (2.11) one has TS = TF M T, where TF M is the Fourier-Mukai transformation which exchanges υ with υ 0 . Since TF M = TST, all in all we have Theorem 2.4. Torus T-duality induces a duality transformation T as given by (2.10) and (2.11) on the subspace of MK3 of theories associated to Kummer surfaces in the orbifold limit (see also [R-W]). The Fourier–Mukai transform TF M which exchanges υ with υ 0 is conjugate to a classical symmetry S by the image T of the T-duality map on theories associated to the torus. Note that by Theorem 2.4 we can prove Aspinwall’s and Morrison’s description (1.7) of the moduli space MK3 purely within conformal field theory without recourse to Landau–Ginzburg arguments. Namely, as explained in Sect. 1, the group G (16) needed to project from the Teichmüller space (1.1) to the component MK3 of the moduli space contains the group O + (H 2 (X, Z)) of classical symmetries which fix the vectors υ, υ0 determining our geometric interpretation. Moreover, for any primitive nullvector υ0 0 (16) 0 with υ, υ = 1 there exists an element g∈G such that g υ = υ and g υ = υ0. By Theorem 2.4 the symmetry TF M ∈ O + (H even (X, Z)) which exchanges υ and υ0 + even (16) (16) and leaves x invariant also is an element of G , thus O (H (X, Z)) ⊂ G and O + (H even (X, Z)) = G (16) under the assumption that MK3 is Hausdorff, as argued in Sect. 1. 2.3. Algebraic automorphisms of Kummer surfaces. To describe strata of the moduli space MK3 we will study subspaces of the Kummer stratum found above which consist of theories with enhanced classical symmetry groups in the geometric interpretation given there. Concentrating on the geometric objects first, in this subsection we investigate algebraic automorphisms of Kummer surfaces which fix the orbifold singular metric. Such an automorphism induces an automorphism of the Kummer lattice N because by K ∼ = H 2 (T , Z)(2) and (2.5) all the lattice vectors of length squared −2 in ) ⊥ belong to N, and N ⊗ R by Theorem 2.1 is spanned by the lattice vectors Ei , i ∈ I of length squared −2. Vice versa,

108


Lemma 2.5. The action of an algebraic automorphism α which fixes the orbifold singular metric on a Kummer surface X is uniquely determined by its action on the set {Ei | i ∈ I } of forms corresponding to exceptional divisors, i.e. by an affine transformation Aα ∈ Aff(I ). Proof. Let α ∗ denote the induced automorphism on the Kummer lattice N. By Theorem 2.1 and (2.5) the intersection form on N is negative definite and the ±Ei , i ∈ I are the only lattice vectors of length squared −2. Therefore, α ∗ is uniquely determined by α ∗ (Ei ) = εi (α)eAα (i) for i ∈ I , where εi (α) ∈ {±1} and Aα ∈ Aff(I ). Actually, ∗ εi (α) = εi (Aα ), because Aα (i) = 1 #⇒ εi (α) = 1 for otherwise Ei ∈ (H 2 (X, Z)α )⊥ with length squared −2 contradicting Theorem 1.1. Assume Aα = Aα for another algebraic automorphism α fixing the metric. Then g := (α −1 ◦ α )∗ acts trivially on N, and because ) is fixed by g as well, for the group G generated by α −1 ◦ α we find ) µ(G) ≥ 2 + 3 + 16 = 21. Now (1.22) shows that G is trivial, proving α = α . ( By abuse of language in the following we will frequently use the induced action of an algebraic automorphism on N or in Aff(I ) as a shorthand for the entire action. Theorem 2.6. For every Kummer surface X the group of algebraic automorphisms fixing the orbifold singular metric contains F42 ⊂ Aff(I ), which acts by translations on I . Proof. Any translation ti ∈ Aff(I ) by i ∈ I acts trivially on N∗ /N. Thus ti can be continued trivially to H 2 (X, Z) by (2.5). One now easily checks that the resulting automorphism of H 2 (X, C) satisfies the criteria of Theorem 1.1. ( ) Next we will determine the group of algebraic automorphisms for the Kummer surface associated to a torus with enhanced symmetry: Theorem 2.7. The group of algebraic automorphisms fixing the orbifold singular metric + of X = K(), ∼ Z4 is GKummer = Z22 F42 . Here, Z22 F42 ⊂ GL(F42 ) F42 = Aff(I ) = K( ), where is generated is equipped with the standard semidirect product. For X by i ∼ = Ri Z2 , Ri ∈ R, i = 1, 2, the group of algebraic automorphisms fixing the 4 + orbifold singular metric generically is G Kummer = Z2 F2 . + Proof. To demonstrate Z22 F42 ⊂ GKummer we will show that certain algebraic automorphisms on the underlying torus T = R4 / can be pushed to X and generate an additional group of automorphisms Z22 ⊂ GL(F42 ) on N. Namely, in terms of standard coordinates (x1 , . . . , x4 ) on T , we are looking for automorphisms which leave the forms

dx1 ∧ dx3 + dx4 ∧ dx2 ,

dx1 ∧ dx4 + dx2 ∧ dx3 ,

dx1 ∧ dx2 + dx3 ∧ dx4 (2.12)

invariant. This is true for

r14

r12 :(x1 , x2 , x3 , x4 ) → (−x2 , x1 , x4 , −x3 ), r13 :(x1 , x2 , x3 , x4 ) → (−x3 , −x4 , x1 , x2 ), = r12 ◦ r13 :(x1 , x2 , x3 , x4 ) → (x4 , −x3 , x2 , −x1 ).

(2.13)

The induced action on N is described by permutations Akl ∈ Aff(I ) of the F42 -coordinates, namely r12 = A12 = (12)(34), r13 = A13 = (13)(24). To visualize this action we introduce the following helpful pictures first used by H. Inose [In]: The vertical line labeled by j ∈ F22 symbolizes the image of the twocycle {x ∈ T | (x1 , x2 ) = 21 j } in X,


✻ ❄

00

109

✻ ❄

11

10

✲ ❅ I ✎ ❅ ❘

01

✛ ✲

11

✛ ✲

00

01

01

✐

❅ I ❘ ❅ ❅ ❅ I ❅ I ❅ ❅ ❘ ❅ ❘ ❅ ❅ ❅ ❅ I ❅ ❅ ❘ ❅ ❘ ❅ 

10

00

❅ I ❅

11

10

10 11 00

01

Fig. 2.1. Action of the algebraic automorphisms r12 (left) and r13 (right) on N

and analogously for the horizontal line labeled by j ∈ F22 we have {x ∈ T | (x3 , x4 ) = 1 2 j }. Then the diagonal lines from cycle j to cycle j symbolize the exceptional divisor obtained from blowing up the fixed point labeled (j, j ) ∈ I . Fat diagonal lines mark those exceptional divisors which are fixed by the respective automorphism. One may now easily check that the automorphisms (2.13), viewed as automorphisms on H 2 (X, C), satisfy the criteria of Theorem 1.1 and thus indeed are induced by algebraic automorphisms of X. + To see that GKummer does not contain any further elements, by Lemma 2.5 it will suffice to show that no other element of Aut(N) can be continued to H 2 (X, Z) consistently such that it satisfies the criteria of Theorem 1.1. Because all the translations of I are + already contained in GKummer we can restrict our investigation to those elements A ∈ 4 GL(F2 ) ⊂ Aff(I ) which can be continued to H 2 (X, Z) preserving the symplectic forms on F42 that correspond to (2.12). After some calculation one finds that A must commute with all the transformations listed in (2.13). This means that A acts on I by + Akl (i) = Akl (i) + |i|(1, 1, 1, 1), |i| = k ik ∈ F2 . But if any such Akl ∈ GKummer , then + also A ∈ GKummer , where A (i) = i + |i|(1, 1, 1, 1). A leaves invariant a sublattice of N of rank 12. But then, because of (1.22) and from (1.21) A cannot be induced by an algebraic automorphism fixing the orbifold singular metric of X. + The result for G Kummer follows from the above proof. Namely, if (x1 , x2 ) are standard coordinates on 1 ⊗ R and (x3 , x4 ) on 2 ⊗ R, then among the automorphisms (2.13) . ( only r12 is generically defined on ) 2.4. Z4 Orbifolds in the moduli space. This subsection is devoted to the study of Z4 orbifolds in the moduli space MK3 . We first turn to some features of the Z4 orbifold construction on the conformal field theory side which need further discussion. From what was said at the beginning of the section, in terms of complex coordinates (2.1) (1) (2) on T = R4 / the Z4 action on the nonlinear σ model is given by (ψ± , ψ± ) → (1) (2) (±iψ± , ∓iψ± ). From (2.2) we readily read off that there always is a surviving su(2)1 ⊕ u(1) subalgebra of the holomorphic W-algebra generated by J, J ± , A. To have a Z4 symmetry on the entire space of states of the torus theory, the charge lattice (1.11) must obey this symmetry. So in addition to picking a Z4 symmetric torus, i.e. a lattice generated by i ∼ = Ri Z2 , Ri ∈ R, i = 1, 2, we must have an appropriate B-field BT in the nonlinear σ model on T which preserves this symmetry. In terms of cohomology we need BT ∈ H 2 (T , R)Z4 = spanR (µ1 ∧ µ2 , µ3 ∧ µ4 , µ1 ∧ µ3 + µ4 ∧ µ2 , µ1 ∧ µ4 + µ2 ∧ µ3 ). As in Sect. 2.1 (µ1 , . . . , µ4 ) denotes a basis dual to (λ1 , . . . , λ4 ), λi being genera-

110


tors of and )T ⊂ H 2 (T , R) is regarded as giving the position of H 2 (T , Z) relative to a fixed three-plane spanR (e1 ∧ e2 + e3 ∧ e4 , e1 ∧ e3 + e4 ∧ e2 , e1 ∧ e4 + e2 ∧ e3 ). To determine the partition function, a lengthy but straightforward calculation using [E-O-T-Y, (5.2)–(5.5)] shows ZNS (τ, z) "# =

1 2

$ ϑ3 ϑ4 4 1 ϑ2 ϑ3 4 1 ϑ2 ϑ4 4 ϑ3 (z) 4 + + η2 η 2 η2 2 η2 % ϑ3 ϑ4 4 ϑ4 (z) 4 ϑ2 ϑ3 4 ϑ2 (z) 4 ϑ2 ϑ4 4 ϑ1 (z) 4 + + 2 η2 η + η2 η , η η (2.14) 1 1 2 Z,BT (τ ) + 2

where for Z,BT (τ ) one has to insert the expression for the specific torus T as obtained from (1.12). Comparing to (2.3) the partition function (2.14) coincides with that of the Z2 orbifold of a theory whose NS-partition function is the expression in curly brackets in (2.14). Indeed, the partition function of SU (2)14 /Z4 , i.e. of the Z4 orbifold of T = R4 /Z4 with BT = 0, agrees with that of the Z2 orbifold K(D4 , 0) [E-O-T-Y]. In Sect. 2.1 we showed that every Z2 orbifold conformal field theory has an su(2)12 subalgebra of the holomorphic W-algebra. On the other hand, as demonstrated above, the Z4 orbifold generically only possesses an su(2)1 ⊕ u(1) current algebra. For SU (2)14 /Z4 this is enhanced to su(2)1 ⊕u(1)3 which still does not agree with the one for Kummer surfaces. Hence although the theories have the same partition function, they are not isomorphic. Similarly, the partition function of the Z4 orbifold of the torus model with SO(8)1 symmetry agrees with that of K(Z4 , 0) as can be seen from (3.7). In this case the theories indeed are the same as will be shown in Theorem 3.9. To have a better understanding of their location within the moduli space and their geometric properties we now construct Z4 orbifolds by applying another orbifold procedure to theories with enhanced symmetries which have already been located in moduli space. denote a lattice generated by i ∼ Theorem 2.8. Let = Ri Z2 , Ri ∈ R, i = 1, 2. ) by modding out Consider the K3 surface X obtained from the Kummer surface K( + the algebraic automorphism r12 ∈ GKummer , blowing up the singularities and using the . induced orbifold singular metric. Then X is the Z4 orbifold of T = R4 / Proof. By construction (2.13), r12 is induced by the automorphism (x1 , x2 , x3 , x4 ) → (−x2 , x1 , x4 , −x3 ) with respect to standard coordinates on T . In terms of complex coordinates as in (2.1) this is just the action ρ : (z1 , z2 ) → (iz1 , −iz2 ), and because ) K() = T /ρ 2 , the assertion is clear. ( Remark. Study Fig. 2.1 to see how the structure A61 ⊕ A43 of the exceptional divisors in ) are identified pairwise the Z4 orbifold comes about: Twelve of the fixed points in K( to yield six Z2 fixed points in the Z4 orbifold, that is A61 . The four points labeled i ∈ {(0, 0, 0, 0), (1, 1, 0, 0), (0, 0, 1, 1), (1, 1, 1, 1)} are true Z4 fixed points. The induced action of r12 on the corresponding exceptional divisor CP1 ∼ = S2 is just a 180◦ rotation about the north-south axis, and north and south poles are fixed points. Blow up the resulting singularities in K()/r12 to see how an A3 arises from the A1 over each true Z4 fixed point.


111

For a Z4 orbifold X there is an analog of the Kummer lattice N described in Theorem 2.1, the primitive sublattice of P ic(X) containing all the two forms which correspond to exceptional divisors by Poincaré duality. We will give an analogous description of as for N in Lemma 2.9 below. The embedding of the moduli space of Z4 orbifolds in MK3 then works analogously to that of Z2 orbifold conformal field theories as described in Subsect. 2.1. Let us fix some notations. Let π : T → X denote the rational map of degree four. Then K := π∗ H 2 (T , R)Z4 ∩H 2 (X, Z) = spanZ (2µ1 ∧ µ2 , 2µ3 ∧ µ4 , µ1 ∧ µ3+ µ4 ∧ µ2 , µ1 ∧ µ4 + µ2 ∧ µ3 ). For the two forms corresponding to the exceptional divisors of the Z4 orbifold we adopt the labeling of fixed points by I ∼ = F42 as used in the Z2 orbifold case. Here, we have six Z2 fixed points labeled by i ∈ I (2) := {(j1 , j2 , 1, 0), (1, 0, j3 , j4 ) | jk ∈ F2 }. The four true Z4 fixed points are labeled by i ∈ I (4) := {(j, j, k, k) | j, k ∈ F2 }. The corresponding two forms are denoted by Ei for i ∈ I (2) , and for each Z4 fixed point i ∈ I (4) we have three irre(±) (0) ducible components of each exceptional divisor Poincaré dual to Ei , Ei such that (±) (0) (+) (−) Ei , Ei = 1, Ei , Ei = 0. For ease of notation we also use the combination (+) (0) (−) Ei := 3Ei + 2Ei + Ei if i ∈ I (4) . As a first step we determine the analogs of (2.5) and (2.6) in order to describe the primitive embedding K ⊥ $→ H 2 (X, Z). By (2.5) images κ ∈ K ∗ of forms corresponding to torus cycles do not necessarily correspond to cycles in H2 (X, Z). Namely, the Poincaré dual of a representative κ of κ ∈ K ∗ /K built from combinations of 21 µj ∧ µk can be interpreted as the π∗ image of a torus cycle which contains Z4 fixed points. It is not a cycle on X, since it has boundaries where the exceptional divisors were glued in instead of the fixed points by the blow up procedure. Since the discriminant forms ∗ ∗ / agree up to a sign, there is a representative η of η ∈ / of K ∗ /K and in the image of κ whose Poincaré dual has the same boundary as that of κ but orientation reversed. We can glue a part of a rational sphere corresponding to η into the boundary of the Poincaré dual of κ to obtain a cocycle κ + η ∈ H 2 (X, Z), where up to a sign κ (•) has the same intersection number as η with every Ei . We again adopt the notation Pj k = spanF2 (fj , fk ) used in Subsect. 2.1. Remember to count Z2 fixed points only once, e.g. P12 = {(0, 0, 0, 0), (1, 0, 0, 0), (1, 1, 0, 0)}. We then have Lemma 2.9. The lattice generated by the set M which consists of

1 1 1 Ei+ε(0,0,1,1) , 2 µ1 ∧ µ2 − 2 E(0,1,0,0)+ε(0,0,1,1) − 4

ε ∈ {0, 1};

i∈P12 ∩I (4) 1 2 µ3

∧ µ4 + 21 E(0,0,0,1)+ε(1,1,0,0) +

1 2

(µ1 ∧ µ3 + µ4 ∧ µ2 ) −

1 2

1 2

(µ1 ∧ µ4 + µ2 ∧ µ3 ) −

1 2

(±)

(0)

1 4

i∈P34

Ei+ε(1,1,0,0) ,

ε ∈ {0, 1};

∩I (4)

Ei+j ,

j ∈ I (4) ;

Ei+j ,

j ∈ I (4) ;

i∈P13

i∈P14

and by E := {Ei , Ei , i ∈ I (4) ; Ei , i ∈ I (2) } is isomorphic to - 3,19 . In particular, is generated by E and

1

1 4 E(0,0,0,0) + E(1,1,1,1) − E(0,0,1,1) − E(1,1,0,0) + 2 E(0,1,0,1) + E(0,1,1,0) ,

112

W. Nahm, K. Wendland 1 2 1 2

E(0,0,0,0) + E(0,0,1,1) + E(0,1,0,0) + E(0,1,1,1) + E(0,1,0,1) + E(0,1,1,0) ,

E(1,1,0,0) + E(0,0,1,1) + E(0,0,0,1) + E(0,1,0,0) + E(1,1,0,1) + E(0,1,1,1) .

$→ H 2 (X, Z), and (H 2 (T , Z))Z4 $→ This gives a natural embedding K ⊥ 2 3,19 ∼ H (X, Z) = - . Given a Kähler–Einstein metric in T 3,3 defined by )T ⊂ H 2 (T , R)Z4 , its image ) under the Z4 orbifold procedure is read off from )T ⊂ H 2 (T , R)Z4 ∼ = H 2 (X, R). = (H 2 (T , Z))Z4 ⊗ R $→ H 2 (X, Z) ⊗ R ∼ In order to prove Lemma 2.9 one has to show that the lattice under inspection has signature (3, 19) and is self dual. We omit the tedious calculation. The construction will be described in more detail in [We]. To give the location in MK3 of the image of T (, BT ) under the Z4 orbifold we have to lift the above picture to the quantum level. As before, H even (T , Z) ∼ = - 4,4 is 0 generated by µj ∧ µk and υ, υ defined in (1.5). As in (2.7) we extend the set M of := M ∪ { Lemma 2.9 to M υ, υ 0 } by

(+) (0) (−) υ := 2υ, υ 0 := 21 υ 0 − 41 3Ei + 4Ei + 3Ei + 2υ. Ei − 18 i∈I (2)

i∈I (4)

Defining (±) := − 1 υ + E (±) , for i ∈ I (4) : E i i 2 (2) for i ∈ I :Ei := −υ + Ei

(0) := − 1 υ + E (0) , E i i 2

(2.15)

one now checks in exactly the same fashion as in Lemma 2.2, and {π ∈ spanZ {E (±) , E (0) , i ∈ I (4) ; E i , i ∈ Lemma 2.10. The lattice generated by M i i (2) 4,20 I } | ∀ m ∈ M : π, m ∈ Z} is isomorphic to - . The embedding H even (T , Z)Z4 $→ H even (X, Z) that is now established actually is the unique one up to lattice automorphisms (see [We], where also the other ZM orbifold conformal field theories, M ∈ {3, 6}, will be treated). Now use

(+)

(4) (0) + 3E (−) ∈ H even (X, Z) + 4E i + 1 BZ := (2.16) 3 E E i i i 2 i∈I (2)

i∈I (4)

to find

1 2

υ, 2 (σi − σi , BT υ) = 2σi − 2σi , 21 BT 2 (4) υ 0 + BT + V − B2T υ = υ 0 + 21 BT + 41 BZ (4) 2 υ, + V4T − 21 21 BT + 41 BZ

hence Theorem 2.11. Let ()T , VT , BT ) denote a geometric interpretation of the nonlinear σ model T (, BT ) as given by (1.14). Assume that is generated by i ∼ = Ri Z2 , Ri ∈ 2 Z 4 R, i = 1, 2, and BT ∈ H (T , Z) such that a Z4 action is well defined on T (, BT ). Then the image x ∈ T 4,20 under the Z4 orbifold procedure has geometric interpretation (), V , B), where ) ∈ T 3,19 is found as described in Lemma 2.9, V = V4T , and B =

A Hiker’s Guide to K3 1 2 BT

(4)

113

(4)

+ 41 BZ , BZ ∈ H even (X, Z) as in (2.16). In particular, the moduli space of superconformal field theories admitting an interpretation as Z4 orbifold is a quaternionic submanifold of MK3 . Moreover, x ⊥ ∩ H even (X, Z) does not contain vectors of length squared −2. Note that from (2.16) it is easy to read off the flow of the B-field obtained from the orbifold procedure through an A3 divisor over one of the true Z4 fixed points of X: On integration over any of the divisors that correspond to a Zm fixed point, we get B-field flux m1 . This is also true for the other ZM orbifold conformal field theories and confirms earlier results [Do, B-I] obtained in the context of brane physics. Theorem 2.11 proves that Z4 orbifold conformal field theories do not correspond to string compactifications of the type IIA string on K3 with enhanced gauge symmetry. Concerning the algebraic automorphism group of Z4 orbifolds we can prove Theorem 2.12. Let X denote the Z4 orbifold of T = R4 /. Then the group G of algebraic automorphisms fixing the orbifold singular metric of X consists of all the residual symmetries induced by algebraic automorphisms of K() which commute with r12 . Thus, generically G ∼ = F22 is generated by the induced actions of t1100 and t0011 . If ∼ Z4 , G∼ = D4 is generated by the induced actions of t1100 and r13 . If we want invariance of the conformal field theory under the entire group G ∼ = D4 of algebraic automorphisms found in Theorem 2.12 we must restrict BT to values such that BT ∈ H 2 (T , R)Z4 ∩ H 2 (X, R)D4 = ), where we regard H 2 (T , R)Z4 $→ H 2 (X, R) as described in Lemma 2.9. If BT is viewed as element of Skew(4) acting on R4 this condition is equivalent to BT commuting with the automorphisms listed in (2.13). 2.5. Application: Fermat’s description for SU (2)14 /Z4 . Theorem 2.13. The Z4 orbifold of T (Z4 , 0) admits a geometric interpretation on the Fermat quartic 3 Q = (x0 , x1 , x2 , x3 ) ∈ CP3 xi4 = 0 (2.17) i=0

(Q)

in CP3 with volume VQ = 21 and B-field BQ = − 21 σ1 (Q) where σ1 denotes the Kähler class of Q.

up to a shift in H 2 (X, Z),

Proof. Let e1 , . . . , e4 denote the standard basis of Z4 . Then µi = ei , and by Theorem (4) 2.11 with BZ 2 = −32 the Z4 orbifold of T (Z4 , 0) is described by the four-plane x ∈ T 4,20 spanned by ξ1 = µ1 ∧ µ3 + µ4 ∧ µ2 ,

ξ 2 = µ1 ∧ µ 4 + µ 2 ∧ µ 3 ,

ξ3 = 2(µ1 ∧ µ2 + µ3 ∧ µ4 ), ξ4

(4)

= 4 υ 0 + BZ + 5 υ.

To read off a different geometric interpretation, we define υQ :=

(µ1 ∧ µ3 + µ4 ∧ µ2 − µ1 ∧ µ4 − µ2 ∧ µ3 )

(0,1,1,0) − E (1,0,1,0) , +1 E

1 2

2

υQ := µ1 ∧ µ3 + µ4 ∧ µ2 + µ1 ∧ µ2

(0,0,0,1) + E (1,1,0,1) − E (0,1,1,0) − E (1,0,1,0) . +1 E 0

2

(2.18)

114


0 ∈ H even (X, Z) as given in Lemma 2.10, υ 2 = υ 0 2 = 0 and One checks υQ , υQ Q Q 0 = 1 to show that υ , υ 0 is an admissible choice for nullvectors in (1.5). For υQ , υQ Q Q the corresponding geometric interpretation ()Q , VQ , BQ ) we find that )Q is spanned by (Q)

σ1

= µ1 ∧ µ3 + µ4 ∧ µ2 + µ1 ∧ µ4 + µ2 ∧ µ3 − 2υQ ,

(Q) σ2 (Q) σ3

= 2(µ1 ∧ µ2 + µ3 ∧ µ4 ) − 2υQ , (4)

= 4 υ 0 + BZ + 5 υ. (Q)

(Q)

As complex structure 1Q ⊂ )Q we pick the two-plane spanned by σ2 and σ3 . Note that this plane is generated by lattice vectors, so the Picard number ρ(X) := rk P ic(X) = rk (1⊥ ∩H 2 (X, Z)) of the corresponding geometric interpretation X is 20, the maximal possible value. K3 surfaces with Picard number 20 are called singular and are classified by the quadratic form on their transcendental lattice P ic(X)⊥ ∩H 2 (X, Z). In other words there is a one to one correspondence between singular K3 surfaces and even quadratic positive definite forms modulo SL(2, Z) equivalence [Shi]2 . Because (Q) (Q) σ2 , σ3 are primitive lattice vectors, one now easily checks that X equipped with the complex structure given by 1Q has quadratic form diag(8, 8) on the transcendental lattice. By [In] this means that our variety indeed is the Fermat quartic (2.17) in CP3 . Volume and B-field can now be read off using (1.5) and noting that in our geometric interpretation 2 (Q) 0 µ1 ∧ µ3 +µ4 ∧ µ2 − µ1 ∧ µ4 − µ2 ∧ µ3 = ξ4 ∼ υQ +BQ + VQ − 21 BQ υQ . ) ( 3. Special Points in Moduli Space: Gepner and Gepner Type Models Finally we discuss the probably best understood models of superconformal field theories associated to K3 surfaces, namely Gepner models [Ge1, Ge2]. The latter are rational conformal field theories and thus exactly solvable. For a short account on the Gepner construction and its most important features in the context of our investigations see Appendix A. In this section, we explicitly locate the Gepner model (2)4 and some of its orbifolds within the moduli space MK3 . This is achieved by giving σ model descriptions of these models in terms of Z2 and Z4 orbifolds which we know how to locate in moduli space by the results of Sect. 2. 3.1. Discrete symmetries of Gepner models and algebraic automorphisms of K3 surfaces. As argued before, a basic tool to characterize a given conformal field theory is the study of its discrete symmetry group. We will first discuss the abelian group given by phase symmetries of a Gepner model rj =1 (kj ) with central charge c = 6 and r even [Ge1]. Recall that this theory is obtained from the fermionic tensor product of the N = 2 superconformal minimal models (kj ), j = 1, . . . , r, by modding out a cyclic group Z ∼ = Zn , n = lcm {2; ki + 2, i = 1, . . . , r}. The model therefore inherits a Zkj +2 2 We thank Noriko Yui and Yasuhiro Goto for drawing our attention to the relevant literature concerning singular K3 surfaces.


115

symmetry from the parafermionic subtheories of each minimal model factor (kj ) whose generator in the bosonic sector acts by 2π i

l

Vmj j ,sj ;mj ,s j −→ e 2(kj +2)

(mj +mj )

l

Vmj j ,sj ;mj ,s j

(3.1)

on the j th factor. The resulting abelian symmetry group of rj =1 (kj ) is Z2 × Gab , where Z2 denotes charge conjugation and Gab = ( rj =1 Zkj +2 )/Zm , m = lcm {ki + 2, i = 1, . . . , r}. Here, Zm acts by r & j =1

Zkj +2 −→

r &

Zkj +2 ,

[a1 , . . . , ar ] −→ [a1 + 1, . . . , ar + 1]

j =1

(see also [G-P]). Note that only elements of the subgroup ' ( r aj alg Gab := [a1 , . . . , ar ] ∈ Gab ∈ Z ⊂ Gab kj + 2

(3.2)

j =1

alg

commute with spacetime supersymmetry, elements of Gab −Gab describe R-symmetries [Ge1]. Assume we can locate our Gepner model within MK3 , that is we explicitly know the corresponding four-plane x ⊂ H even (X, R) as described in Sect. 1. Furthermore assume that by picking a primitive nullvector υ ∈ H even (X, Z) we have chosen a specific geometric interpretation (), V , B). By construction, a Gepner model comes with a specific choice of the N = (2, 2) subalgebra corresponding to a specific two plane 1 ⊂ ). We stress that this is true for any geometric interpretation of j (kj ): The choice of the N = (2, 2) subalgebra does not fix a complex structure a priori, it fixes a choice of complex structure in every geometric interpretation of our model, as was explained in Sect. 1. Still, we now assume our K3 surface X to be equipped with complex structure and Kähler metric. By our discussion in Sect. 1.2 we know that susy susy any symmetry of the Gepner model which leaves the su(2)l ⊕ su(2)r currents ± J, J ± , J , J and the vector υ invariant may act as an algebraic automorphism on X. ⊗r ⊗r ± and J = V00,0;∓2,2 (see Appendix A) we conclude Because J ± = V0∓2,2;0,0 alg

from (3.2) that elements of Gab can act as algebraic automorphisms on X fixing the Bfield B ∈ H 2 (X, R), and vice versa. More explicitly by what was said in Sect. 1.2, the

action of such a Gepner-symmetry on the 21 , 21 -fields with charges, say, Q = Q = 1 should be identified with the induced action of an algebraic automorphism of X on alg H 1,1 (X, R). With reference to its possible geometric interpretation we call Gab the abelian algebraic symmetry group of the Gepner model. In the following subsections we will investigate where in the moduli space of superconformal field theories associated to K3 surfaces to locate the Gepner model (2)4 and alg some of its orbifolds by elements of Gab ∼ = (Z4 )2 . From the above discussion it is clear that given a definite geometric interpretation for (2)4 the geometric interpretation of its orbifold models is obtained by modding out the corresponding algebraic automorphisms. Apart from symmetries in Z2 × Gab our Gepner model will possess permutation symmetries involving identical factor theories. Their discussion is a bit more subtle, because as noted in [F-K-S] permuting fermionic fields will involve additional signs

116


⊗r (A.9). This in particular applies to J ± = V0∓2,2;0,0 , meaning that odd permutations can only act algebraically when accompanied by a phase symmetry [a1 , . . . , ar ] ∈ Gab :

r

j =1

aj ∈ Z + 21 . kj + 2

(3.3)

We will discuss this phenomenon in detail for the example of prime interest to us, namely alg the Gepner model (2)4 . Here Gab ∼ = (Z4 )2 , and the entire algebraic symmetry group is alg generally believed to be G ∼ = (Z4 )2 S4 [As1]. Moreover, based on Landau–Ginzburg computations and comparison of symmetries [G-V-W, G-P, F-K-S-S,As1] it is generally believed that (2)4 has a geometric interpretation ()Q , VQ , BQ ) given by the Fermat quartic (2.17) in CP3 . Indeed, Q is a K3 surface with algebraic automorphism group (Z4 )2 S4 [Mu], and arguments in favour of the viewpoint that it yields a geometric interpretation of (2)4 will arise from the following discussion. It is proved in Corollary 3.6. alg To give the action of the two generators [1, 3, 0, 0] and [1, 0, 3, 0] of Gab ∼ = (Z4 )2 1 1

on the 2 , 2 -fields with charges Q = Q = 1 we use the shorthand notation X := (V11,0;−3,2 )⊗4 ,

Y (n1 , n2 , n3 , n4 ) := Vnn11 ,0;n1 ,0 ⊗ Vnn22 ,0;n2 ,0 ⊗ Vnn33 ,0;n3 ,0 ⊗ Vnn44 ,0;n4 ,0

(3.4)

(ni ∈ N) and find [1, 3, 0, 0] →

1

−1

i

−i

↓ [1, 0, 3, 0] 1

Y (1, 1, 1, 1), X Y (0, 2, 0, 2), Y (2, 0, 2, 0) Y (1, 0, 1, 2) Y (1, 2, 1, 0)

−1

Y (2, 2, 0, 0), Y (0, 0, 2, 2)

Y (2, 0, 0, 2), Y (0, 2, 2, 0) Y (2, 1, 0, 1) Y (0, 1, 2, 1)

i

Y (1, 1, 0, 2)

Y (2, 0, 1, 1) Y (2, 1, 1, 0) Y (1, 2, 0, 1)

−i

Y (1, 1, 2, 0)

Y (0, 2, 1, 1) Y (1, 0, 2, 1) Y (0, 1, 1, 2)

(3.5)

Note first that by (1.20) we have µ(Z4 × Z4 ) = 6, in accordance with (1.21) and 2 = 6 − 4 invariant fields in the above table. One moreover easily checks that the alg spectrum of every element g ∈ Gab of order four agrees with the one computed in (1.23) for algebraic automorphisms of order four on K3 surfaces. This is a strong and highly non-trivial evidence for the fact that one possible geometric interpretation of (2)4 is given by a K3 surface whose algebraic automorphism group contains (Z4 )2 . As stated above, further discussion is due concerning the action of S4 because transpositions of fermionic modes introduce sign flips (A.9). In particular, odd elements of S4 do not leave J ± invariant. To have an algebraic action of the entire group S4 we must therefore accompany σ ∈ S4 by a phase symmetry aσ = [a1 (σ ), a2 (σ ), a3 (σ ), a4 (σ )] ∈ Gab which for odd σ satisfies (3.3). Thus a transposition (α, ω) ∈ S4 must be represented by ρ((α, ω)) = (α, ω) ◦ a(α,ω) = a(α,ω) ◦ (α, ω) in order to have ρ((α, ω))2 = 1. With any such choice of ρ on generators (αj , ωj ) of S4 one may then check explicitly that ρ

defines an algebraic action of S4 , i.e. its spectrum on the 21 , 21 -fields coincides with the


117

spectrum of the algebraic automorphism group S4 . Namely, any element of order two (or three, four) in S4 leaves µ(Z2 ) − 4 = 12 (or µ(Z3 ) − 4 = 8, µ(Z4 ) − 4 = 6) states invariant, and elements of order four have the spectrum given in (1.23). Note in particular that by (3.3) with any consistent choice of σ → aσ the group S4 acts by σ → sign(σ ) on Y (1, 1, 1, 1) and trivially on X. This leaves X = (V11,0;−3,2 )⊗4 as the unique invariant state upon the action of (Z4 )2 S4 in accordance with µ((Z4 )2 S4 ) = 5 and (1.21). Summarizing, we have shown that the action of the entire algebraic symmetry group G alg = (Z4 )2 S4 of (2)4 as described above exhibits a spectrum consistent with its interpretation as group of algebraic automorphisms of a K3 surface, e.g. the Fermat quartic with geometric interpretation ()Q , VQ , BQ ). Remember that µ G alg = 5 is the minimal possible value of µ by the discussion in Sect. 1.2. Thus by what was said in Sect. 2 the only four invariant ( 21 , 21 )-fields (V1±1,0;∓3,2 )⊗4 , (V1±1,0;∓1,0 )⊗4 are those corresponding to moduli of volume deformation and of B-field deformation in direction of )Q . 3.2. Ideas of proof: An example with c = 3. In this subsection we give a survey on the steps of proof we will perform to show equivalences between Gepner or Gepner type models and nonlinear σ models. As an illustration we then prove the well known fact that Gepner’s model (2)2 admits a nonlinear σ model description on the torus associated to the Z2 lattice. Given two N = 2 superconformal field theories C 1 , C 2 with central charge c = 3d/2 (d = 2 or d = 4) and spaces of states H1 , H2 , to prove their equivalence we show the following: (i)

The partition functions of the two theories agree sector by sector in the sense of (2.4). (ii) The fields of dimensions (h, h) = (1, 0) in the two theories generate the same algebra A = Af ⊕ Ab , where Af = u(1) for d = 2, Af = su(2)12 for d = 4, and u(1)d ⊂ Ab . In particular, u(1)c ⊂ A. Af contains the U (1)-current J (1) = J of the N = 2 superconformal algebra, and a second U (1)-generator J (2) if d = 4. Furthermore, the fields of dimensions (h, h) = (0, 1) in both theories generate algebras isomorphic to A as well, such that each of the left moving U (1)-currents j has a right moving partner  . (iii) For i = 1, 2 define Hbi := |ϕ ∈ Hi J (k) |ϕ = 0, k ∈ {1, d2 } and denote the U (1)-currents in u(1)d ⊂ Ab by j 1 , . . . , j d . We normalize them to j k (z) j l (w) ∼

δkl . (z − w)2

(3.6)

Let j d+k ∼ J (k) , k ∈ {1, d2 } denote the remaining U (1)-currents when normalized to (3.6), too, and set J := (j 1 , . . . , j d ;  1 , . . . ,  d ). The charge lattices -bi := γ ∈ Rd;d ∃ |ϕ ∈ Hbi : J |ϕ = γ |ϕ of Hb1 and Hb2 with respect to J are isomorphic to the same self dual lattice -b ⊂ Rd;d ; because the states in Hbi are pairwise local, in order to prove this it suffices to

118


show agreement of the J -action on a set of states whose charge vectors generate a self dual lattice -b . Theorem 3.1. If (i)–(iii) are true then theories C 1 and C 2 are isomorphic (the converse generically is wrong, of course). ∼ H2 =: Hb . Denote by V i [γ ] the primary Proof. Using (i)–(iii) we first show Hb1 = b field corresponding to a state in Hbi with charge γ = (γl ; γr ) ∈ -b . Notice that in both theories every charge γ ∈ -b must appear with multiplicity one, because otherwise by fusing [Vki [γ ]] × [Vki [−γ ]] = [1ik ] we find two states 1i1 , 1i2 ∈ Hbi with vanishing charges under a total u(1)c ⊂ A in contradiction to uniqueness of the vacuum. Now for any α = (αl ; αr ), β = (βl ; βr ) ∈ -b we have i V i [α](z) V i [β](w) ∼ cα,β (z − w)αl βl (z − w)αr βr V i [α + β](w) + · · · , 1 = c2 for all α, β ∈ - by normalizso it remains to be shown that we can arrange cα,β b α,β ing the primary fields appropriately. In other words, we must find constants dγ ∈ R for 2 = d d c1 . This is possible, because having any γ ∈ -b such that ∀ α, β ∈ -b : cα,β α β α,β fixed dα , dβ , dγ , dδ ∈ R such that 2 1 2 1 2 1 2 1 = dα dβ cα,β , cα,γ = dα dγ cα,γ , cα,δ = dα dδ cα,δ , cβ,γ = dβ dγ cβ,γ cα,β i , ci , ci , ci , by the crossing symmetries for four nonzero two point functions cα,β α,γ α,δ β,δ 1 c1 cα,β γ ,δ 1 c1 cα,γ β,δ

=

2 c2 cα,β γ ,δ 2 c2 cα,γ β,δ

and

1 c1 cα,γ β,δ 1 c1 cα,δ β,γ

=

2 c2 cα,γ β,δ 2 c2 cα,δ β,γ

2 = d d c1 . If more than two of the etc. we automatically have cγ2 ,δ = dγ dδ cγ1 ,δ and cβ,δ β δ β,δ six two point functions vanish, then by similar arguments the normalization of one of the primaries is independent of the three others and a consistent choice of dα , dβ , dγ , dδ ∈ R is therefore possible, too. The proof of Hb1 ∼ = Hb2 ∼ = Hb is now complete. Because -b is self dual, for any state |ϕ ∈ Hi carrying charge γ with respect to J we have γ ∈ -b and thus find vertex operators V i [±γ ] ∈ Hbi . By ii. and iii. T := 21 ck=1 (j k )2 acts as Virasoro field T i on each of the theories (check that T − T i has dimensions h = h = 0 with respect to T i ). Thus the restriction of the Virasoro field T i to Hbi is given by Tbi := 21 dk=1 (j k )2 , and by picking suitable combinations k and P of ascendants jnk , n ≥ 0, k ∈ {1, . . . , d}, we find |ψ := P of descendants j−n i P V [−γ ]|ϕ such that |0b and |ψ ∈ Hfi := |χ ∈ Hi | Tbi |χ = 0 . |ϕ = |ψ ⊗ V i [γ ] P

This shows Hi ∼ = Hfi ⊗ Hb for i = 1, 2. Hf1 and Hf2 are representations of Af = u(1) (for d = 2) or Af = su(2)12 (for d = 4) which are completely determined by charge and dimension of the lowest weight states. Because by ii. Af contains the U (1)-current J of the total N = 2 superconformal algebra, the partition functions of our theories agree by (i), and we already know Hi ∼ = Hfi ⊗ Hb for i = 1, 2, we may conclude Hf1 ∼ = Hf2 . ) (


119

Let’s see how the procedure described above works: Theorem 3.2. Gepner’s model C 1 = (2)2 has a nonlinear σ model description C 2 on the two dimensional torus TSU (2) 2 with SU (2)12 lattice = Z2 and B-field B = 0. 1

Proof. If we can prove (i)–(iii) in the above list, by Theorem 3.1 we are done. (i) Using (A.10) for computing the partition function of (2)2 on one hand and (1.12) for the partition function of the σ model on TSU (2) 2 with B = 0 on the other, we find 1 " 4 4 % 2 4 ϑ3 1 ϑ2 + ϑ4 ϑ3 (z) + ZNS (τ, z) = 2 η η η η for both theories. (ii) The nonlinear σ model on TSU (2) 2 has two rightmoving abelian currents j1 , j2 which 1 we normalize to 1 δαβ jα (z) jβ (w) ∼ 2 . (z − w)2 Their superpartners are free Majorana fermions ψ1 , ψ2 with coupled boundary conditions. By e1 , e2 we denote the generators of the lattice = ∗ = Z2 which defines our torus. Then the (1, 0)-fields in the nonlinear σ model are given by the three abelian currents J = iψ2 ψ1 (the U (1) current of the N = 2 superconformal algebra), Q = j1 + j2 , R = j1 − j2 , and the four vertex operators V±ei ,±ei , i = 1, 2. In the Gepner model (2)2 we have an abelian current j, j from each minimal model factor along with Majorana fermions ψ, ψ , where by (A.8) ψψ = V04,2;0,0 ⊗ V04,2;0,0 . The U (1) current of the total N = 2 superconformal algebra is J = j + j , and comparing J, Q, R-charges we can make the following identifications: iψ2 ψ1 = J = j + j ,

j1 + j2 = Q = j − j ,

j1 − j2 = R = iψψ ,

Ve1 ,e1 = V02,0;0,0 ⊗ V02,2;0,0 + V0−2,0;0,0 ⊗ V0−2,2;0,0 , Ve2 ,e2 = V02,0;0,0 ⊗ V02,2;0,0 − V0−2,0;0,0 ⊗ V0−2,2;0,0 , V−e1 ,−e1 = V02,2;0,0 ⊗ V02,0;0,0 + V0−2,2;0,0 ⊗ V0−2,0;0,0 , V−e2 ,−e2 = V02,2;0,0 ⊗ V02,0;0,0 − V0−2,2;0,0 ⊗ V0−2,0;0,0 . Thus the (1, 0)-fields in the two theories generate the same algebra A = u(1)⊕su(2)12 = Af ⊕ Ab . Obviously, the same structure arises on the right handed sides. (iii) The space Hb1 for the σ model is just the bosonic part of the theory. The charge lattice -b with respect to the currents J := (Q, R; Q, R) = (j1 +j2 , j1 −j2 ;  1 + 2 ,  1 − 2 ) thus contains the charges M := 21 (ε; ±ε), ε ∈ {±1}2 , carried by vertex operators V±ei ,0 , V0,±ei , i = 1, 2. M generates the self dual lattice 21 (a + b; a − b) | a, b ∈ Z2 , 2k=1 ak ≡ 2k=1 bk ≡ 0 (2) = -b . To complete the proof of (iii) we observe that in the Gepner model the fields V1n,0;n,0 ⊗ V1−n,0;−n,0 ± V1−3n,2;n,0 ⊗ V13n,2;−n,0 , n ∈ {±1}, and V1n,0;−n,0 ⊗ V13n,2;n,0 ± V1−3n,2;−n,0 ⊗ V1−n,0;n,0 , n ∈ {±1},

are uncharged with respect to 1 J and carry J = (j − j , iψψ ;  −  , iψ ψ )-charges 2 M = 2 (ε; ±ε), ε ∈ {±1} generating -b . ( )

120


3.3. Gepner type description of SU (2)14 /Z2 . 2)4 denote the Gepner type model which is obtained as orbifold Theorem 3.3. Let C 1 = ( alg 4 ∼ of (2) by the group Z2 = [2, 2, 0, 0] ⊂ Gab . Then C 1 admits a nonlinear σ model 2 4 description C = K(Z , 0) on the Kummer surface K() associated to the torus TSU (2) 4 1

with SU (2)14 lattice = Z4 and vanishing B-field. Proof. We prove conditions (i)–(iii) of Sect. 3.2 and then use Theorem 3.1. (i) From (1.12) one finds " # $%2 1 ϑ2 4 ϑ3 4 ϑ4 4 Z=Z4 ,BT =0 (τ ) = + + . 2 η η η

(3.7)

alg

Applying the orbifold procedure for the Z2 -action of [2, 2, 0, 0] ∈ Gab to the partition function (A.10) of the Gepner model (2)4 [F-K-S-S] one checks that C 1 and C 2 have the same partition function obtained by inserting (3.7) into (2.3). (ii) In the nonlinear σ model C 2 the current algebra (2.2) is enhanced to u(1)4 ⊕ su(2)12 . The additional U (1)-currents are Ui := Vei ,ei + V−ei ,−ei , i = 1, . . . , 4, where the ei are the standard generators of = ∗ = Z4 . In the Gepner type model C 1 = ( 2)4 , apart from the U (1)-currents J1 , . . . , J4 from the factor theories, where J = J1 + · · · + J4 , we find four additional fields with dimensions (h, h) = (1, 0); comparing the respective operator product expansions the following identifications can be made: ⊗4 J = J1 + J2 + J3 + J4 , J ± = V0∓2,2;0,0 ; ⊗2 ⊗2 ⊗ V0±2,2;0,0 ; A = J1 + J2 − J3 − J4 , A± = V0∓2,2;0,0 1 2 1 2

(U1 + U2 ) = P = J1 − J2 ;

(U3 + U4 ) = Q = J3 − J4 ; ⊗2 ⊗2 0 0 1 − U ⊗ V ; = R = i V (U ) 1 2 4,2;0,0 0,0;0,0 2 ⊗2 ⊗2 0 1 ⊗ V04,2;0,0 . 2 (U3 − U4 ) = S = i V0,0;0,0 (3.8) Thus the (1, 0)-fields in the two theories generate the same algebra A = su(2)12 ⊕ u(1)4 = Af ⊕ Ab . Obviously, the same structure arises on the right handed sides. (iii) We show that Hb1 and Hb2 both have self dual J := (P , Q, R, S; P , Q, R, S)-charge lattice3 -b = (x + y; x − y) x ∈ 21 D4 , y ∈ D4∗ , (3.9) 3 In our coordinates D = {x ∈ Z4 | 4 x ≡ 0 (2)} and D ∗ = Z4 + (Z + 1/2)4 . 4 i=1 i 4


121

generated by Mtw := and Minv

1 2 (x; x)

∈ R4,4 x ∈ {(ε1 , ε2 , 0, 0), (0, 0, ε1 , ε2 ),

(0, ε1 , ε2 , 0), (ε1 , 0, 0, ε2 ), εi ∈ {±1}} := (ε; 0) ε ∈ {±1}4 .

In the σ model C 2 we denote by )δ , δ ∈ F42 the twist field corresponding to the fixed point pδ = 21 4i=1 δi ei of the Z2 orbifold. To determine the action of Ui on twist fields notice that by definition, )δ introduces a cut on the configuration space Z to establish the boundary condition ϕ(σ0 + 1, σ1 ) = −ϕ(σ0 , σ1 ) for fields ϕ in the corresponding twisted sector, i.e. ϕ(0, 0) = pδ (see Sect. 2). Action of a vertex operator with winding mode λ will shift the constant mode pδ of each twisted field by λ2 [H-V]. Hence, Ui (z) )δ (w) ∼

1/2 )δ+ei (w), z−w

(3.10)

where the factor 21 is determined up to phases by observing Tf2 |)δ = 0, Tb2 = 1 4 1 2 i=1 (Ui ) , and h = h = 4 for twist fields. The phases are fixed by appropriately 4 normalizing the twist fields. One now checks that ∀ ε ∈ {±1}4 :

sε :=

4

&

(εi )δi )δ

δ∈F42 i=1

are uncharged under (J ; J ) and (A; A) and carry J -charges Mtw . For ε, δ ∈ {±1} and k, l ∈ {1, . . . , 4} we define

εδ Ekl := jk − 2δ Vek ,ek − V−ek ,−ek jl − 2ε Vel ,el − V−el ,−el . εδ , E εδ , E εδ , E εδ are (J, A; J , A)-uncharged and carry J -charges M . Then E13 inv 14 23 24 ⊗2 1 , P(n2 ) := V0n2 ,n2 ;n2 ,n2 In the Gepner model, introducing O(n1 ) := V2,1;2n1 ,n1

⊗V0−n2 ,−n2 ;−n2 ,−n2 (ni ∈ {±1}) as shorthand notation we find (J, A; J , A)-uncharged fields O(n1 ) ⊗ O(n2 ), O(n1 ) ⊗ P(n2 ), P(n1 ) ⊗ O(n2 ), P(n1 ) ⊗ P(n2 ) which after diagonalization with respect to the J -action carry charges Mtw . Similarly, setting Q(n, s) := V02n,s;0,0 ⊗V02n,s+2;0,0 , the fields Q(n1 , s1 )⊗Q(n2 , s2 ), ni ∈ {±1}, si ∈ {0, 2} after diagonalization have charges Minv . For later reference we note that by what was said in Sect. 1 there are eight more fields in the Ramond sector with dimensions h = h = 41 . Each of them is uncharged under J and either (A; A) or (J ; J ). We denote by WεJ1 ,ε2 , WεA1 ,ε2 , εi ∈ {±1} the fields corresponding to the lowest weight states of su(2)1 ∼ = J, J ± or su(2)1 ∼ = A, A± , with (J ; J ) or (A; A)-charge (ε1 ; ε2 ) respectively and identify ⊗4 WεJ1 ,ε2 = V0−ε1 ,−ε1 ;−ε2 ,−ε2 , (3.11) ⊗2 ⊗2 ⊗ V0ε1 ,ε1 ;ε2 ,ε2 . WεA1 ,ε2 = V0−ε1 ,−ε1 ;−ε2 ,−ε2

122


In σ model language and by the discussion in Sect. 1, by applying left and right handed spectral flow to the J -uncharged WεA1 ,ε2 we obtain ( 21 , 21 )-fields in F1/2 , the real and imaginary parts of whose (1, 1)-superpartners describe infinitesimal deformations of the torus TSU (2) 4 our Kummer surface is associated to. 1

Summarizing, we can now obtain a list of all fields needed to generate H1 and H2 as well as a complete field by field identification by comparison of charges; for the resulting list of ( 41 , 41 )-fields see Appendix B. ( ) √ Note that because D4 ∼ = 2D4∗ for the J -charge lattice (3.9) -b ∼ = √12 (µ + λ, µ − λ) µ ∈ D4∗ , λ ∈ D4 . Thus -b is the charge lattice of the bosonic part of the σ model C 3 = T (D4 , 0). Theory C 1 was obtained by taking the ordinary Z2 orbifold of the torus model on TSU (2) 4 , but as 1 pointed out in [K-S], for the bosonic part of the theory this is equivalent to taking the Z2 1 orbifold associated to a shift δ = √ (µ0 ; µ0 ), µ0 = i ei ∈ ∗ on the charge lattice 2 2

of TSU (2) 4 . Under this shift orbifold, the lattices = ∗ = Z4 are transformed by 1

∗ → ∗ + ∗ + 21 µ0 = D4∗ ,

→ {λ ∈ |µ0 , λ ≡ 0 (2) } = D4 ,

so the bosonic part of the resulting theory indeed is that of C 3 . The entire bosonic sector of C 1 = C 2 agrees with that of C 3 , because the shift acts trivially on fermions, and the ordinary Z2 orbifold just interchanges twisted and untwisted boundary conditions of the fermions in the time direction. The difference between the theories merely amounts in opposite assignments of Ramond and Neveu–Schwarz sector on the twisted states resulting in different elliptic genera for the K3-model C 1 = C 2 and the torus model C 3 . The fact that the partition functions actually do not agree before projection onto even fermion numbers is not relevant here because locality is violated before the projection is carried out. So, on the level of conformal field theory: 2)4 viewed as a nonlinear σ model C 2 on the Remark 3.4. The Gepner type model C 1 = ( Kummer surface K(Z4 , 0) is located at a meeting point of the moduli spaces of theories associated to K3 surfaces and tori, respectively. Namely, its bosonic sector is identical with that of the nonlinear σ model C 3 = T (D4 , 0). This property does not translate to the stringy interpretation of our conformal field theories, though. When we take external degrees of freedom into account, the spin statistics theorem dictates in which representations of SO(4) the external free fields may couple to internal Neveu–Schwarz or Ramond fields, respectively. The theories C 1 = C 2 and C 3 therefore correspond to different compactifications of the type IIA string. 3.4. Gepner’s description for SU (2)14 /Z4 . Theorem 3.5. The Gepner model C I = (2)4 admits a nonlinear σ model description C II on the Z4 orbifold of the torus TSU (2) 4 with SU (2)14 -lattice = Z4 and vanishing 1 B-field.


123

Proof. It is clear that C I = (2)4 can be obtained from C 1 = ( 2)4 , for which we already have a σ model description by Theorem 3.3, by the Z2 orbifold procedure which revokes the orbifold used to construct C 1 . The corresponding action is multiplication by −1 on [2, 2, 0, 0]-twisted states, i.e. )

* 2 , 2 , 0, 0 :

4 + i=1

Vlmi i ,si ;mi ,s i −→ e

2π i 8 (m1 −m1 −m3 +m3 )

4 + i=1

Vlmi i ,si ;mi ,s i .

(3.12)

Among the (1, 0)-fields the following are invariant under [2 , 2 , 0, 0] (use (2.2) and (3.8)): (1)

(1)

(2)

(2)

(1)

(1)

(2)

(2)

J = ψ+ ψ− + ψ+ ψ− , A = ψ+ ψ− − ψ+ ψ− ;

(1)

(2)

J + = ψ+ ψ + , P =

1 2

(U1 + U2 ) ;

(2)

(1)

J − = ψ− ψ − ; Q=

1 2

(U3 + U4 ) .

(3.13)

Hence we have a surviving su(2)1 ⊕ u(1)3 subalgebra of our holomorphic W-algebra. In Appendix B we give a list of all ( 41 , 41 )-fields in C 1 = ( 2)4 together with their description 2 4 in the σ model C on the Z2 orbifold K(Z , 0). A similar list can be obtained for the (2, 0)-fields as discussed in the proof of Theorem 3.3. From these lists and (3.13) one readily reads off that the states invariant under (3.12) coincide with those invariant under the automorphism r12 on K(Z4 , 0) (see Theorem 2.7) which is induced by the (1) (2) (1) (2) Z4 action (j1 , j2 , j3 , j4 ) → (−j2 , j1 , j4 , −j3 ), i.e. (ψ± , ψ± ) → (±iψ± , ∓iψ± ) on the underlying torus TSU (2) 4 . The appertaining permutation of exceptional divisors 1 in the Z2 fixed points is depicted in Fig. 2.1. The action of r12 and that induced by (3.12) agree on the algebra A of (1, 0)-fields and a set of states generating the entire space of states, thus they are the same. Because of C 1 = C 2 (Theorem 3.3) and the fact that C I = (2)4 is obtained from C 1 by modding out (3.12), it is clear that modding out K(Z4 , 0) by the algebraic automorphism r12 will lead to a σ model description of (2)4 . As shown in Theorem 2.8 the result is the Z4 orbifold C II of TSU (2) 4 . ( ) 1

Theorem 3.5 has been conjectured in [E-O-T-Y] because of agreement of the partition functions of C I and C II . This of course is only part of the proof as can be seen from our argumentation in Sect. 2.4. There we showed that SU (2)14 /Z4 does not admit a σ model description on a Kummer surface although its partition function by [E-O-T-Y] agrees with that of K(D4 , 0), too. From Theorem 2.13 and Theorem 3.5 we conclude: Corollary 3.6. The Gepner model (2)4 admits a geometric interpretation on the Fermat quartic (2.17) in CP3 with volume VQ = 21 . Let (), V , B) denote the geometric interpretation of (2)4 we gain from Theorem 3.5. By ± ± ± ± the proof of Theorem 3.3 we know the moduli Vδ,ε +V−δ,−ε and i(Vδ,ε −V−δ,−ε ), δ, ε ∈ {±1} for volume and B-field deformation in direction of ) of the underlying torus TSU (2) 4 1

A , WA of our Z4 orbifold: We apply left and right handed spectral flows to W1,1 −1,−1 as given in (3.11) and then compute the corresponding (1, 1)-superpartners. In terms of

124


Gepner fields this means ⊗2 + Vδ,ε = V22δ,2;2ε,2 ⊗ V22δ,0;2ε,0 ⊗ V00,0;0,0 ⊗2 + V22δ,0;2ε,0 ⊗ V22δ,2;2ε,2 ⊗ V00,0;0,0 , ⊗2 − = V00,0;0,0 ⊗ V22δ,2;2ε,2 ⊗ V22δ,0;2ε,0 Vδ,ε ⊗2 + V00,0;0,0 ⊗ V22δ,0;2ε,0 ⊗ V22δ,2;2ε,2 .

(3.14)

± are uncharged under J and A as they should, because both U (1)-currents Indeed, Vδ,ε must survive deformations within the moduli space of Z4 orbifold conformal field theories. On the other hand by our discussion in Sect. 3.1 the (1, 1)-superpartners of (V1±1,0;∓3,2 )⊗4 , (V1±1,0;∓1,0 )⊗4 , which carry (A; A)-charges ∓(1; 1), give the moduli of volume and corresponding B-field deformation if we choose the quartic hypersurface (2.17) as geometric interpretation of Gepner’s model (2)4 . Hence along the “quartic line” we generically only have an su(2)1 -algebra of (1, 0)-fields. This agrees with the analogous picture for c = 9 and the Gepner model (3)5 where all additional U (1)-currents vanish upon deformation along the quintic line [D-G].

Symmetries and algebraic automorphisms revised: (2)4 and ( 2)4 . Among the algebraic 2 4 symmetries Z4 S4 of the Gepner model (2) all the phase symmetries Z24 commute with the action of [2, 2, 0, 0] which we mod out to obtain ( 2)4 . The residual Z2 × Z4 has 4 a straightforward continuation to (2) (i.e. to the twisted states). Moreover, [2 , 2 , 0, 0] as given in (3.12) which reverts the orbifold with respect to [2, 2, 0, 0] must belong to alg of ( the algebraic symmetry group G 2)4 . Nevertheless, one notices that Z2 × Z2 ∼ = [2 , 2 , 0, 0], [1, 3, 0, 0] leaves 6 = 8 = µ(Z2 × Z2 ) − 4 states invariant and thus does not act algebraically by (1.21). We temporarily leave the symmetry [1, 3, 0, 0] out of the discussion, because then by the methods described in Sect. 3.1 we find a consistent algebraic action of (Z2 × Z4 ) D4 on ( 2)4 , where Z2 × Z4 = [2 , 2 , 0, 0], [1, 0, 3, 0] and D4 = (12), (13)(24) ⊂ S4 is the commutant of [2, 2, 0, 0]. 2)4 : In Theorem 2.7 the Let us compare to the σ model description K(Z4 , 0) of ( 4 group of algebraic automorphisms of K(Z , 0) which leave the orbifold singular metric + invariant was determined to be GKummer = Z22 F42 . Although it is isomorphic to the + algebraic symmetry group (Z2 × Z4 ) D4 of ( 2)4 found so far, GKummer must act 4 differently on ( 2) . Namely, from the proof of Theorem 3.5 we know that the σ model + + equivalent of [2 , 2 , 0, 0] is r12 ∈ GKummer . Thus only the commutant H ⊂ GKummer of r12 can comprise residual symmetries descending from the Z4 orbifold description on (2)4 . This is no contradiction, because by the discussion in Sect. 1.2 different subgroups of the entire algebraic symmetry group of ( 2)4 may leave the respective nullvector υ invariant which defines the geometric interpretation. By what was said in Sect. 1 it is actually no surprise to find symmetries of conformal field theories which do not descend to classical symmetries of a given geometric interpretation. The Gepner type model ( 2)4 is an example where the existence of such symmetries can be checked explicitly. By the results of Sect. 2.3 we find H = Z2 × D4 = r12 , r13 , t1100 (see also Theorem 2.12). We now use our state by state identification obtained in the proof of alg and Theorem 3.3 (see Appendix B) to determine the corresponding elements of G


125

find r13 = (13)(24)

∈ S4 ,

(3.15)

t1100 = ξ ◦ [1, 3, 0, 0] =: [1 , 3 , 0, 0].

Here ξ acts by multiplication with −1 on those Gepner states corresponding to the 16 twist fields )δ of the Kummer surface and trivially on all the other generating fields of the space of states we discussed in the proof of Theorem 3.3. Note that ξ is a symmetry of the theory because by the selection rules for amplitudes of twist fields any n-point function containing an odd number of twist fields will vanish. The geometric interpretation tells us that modding out ( 2)4 by ξ will revoke the ordinary Z2 orbifold procedure, i.e. produce 4 T (Z , 0). We conclude remarking that by the modification (3.15) of the [1, 3, 0, 0]alg = (Z2 × Z4 ) D4 acts algebraically on ( action the full group G 2)4 . The subgroup 2 H consists of all the residual symmetries of (2)4 surviving both deformations along the quartic and the Z4 orbifold line and acting classically in both geometric interpretations of (2)4 known so far, the Z4 orbifold and the quartic one. 3.5. Gepner type description of SO(8)1 /Z2 . Theorem 3.7. Let C1 = ( 2)4 denote the Gepner type model which is obtained as an alg 4 orbifold of (2) by the group Z2 × Z2 ∼ = [2, 2, 0, 0], [2, 0, 2, 0] ⊂ Gab . This model admits a nonlinear σ model description C2 on the Kummer surface K( √1 D4 , B ∗ ) as2

sociated to the torus TSO(8)1 with SO(8)1 -lattice = √1 D4 and B-field value B ∗ for 2 which the theory has enhanced symmetry by the Frenkel–Kac mechanism. Proof. Let e1 , . . . , e4 denote the standard basis of Z4 . With respect to this basis the B-field which leads to a full SO(8)1 symmetry for the σ model on TSO(8)1 is   0 1 0  −1 0  (3.16) : ⊗ R −→ ∗ ⊗ R , B∗ =  0 1 0 −1 0 a two torsion point in H 2 (TSO(8)1 , R)/H 2 (TSO(8)1 , Z). We are now ready to use Theorem 3.1 if we can prove (i)–(iii) of Sect. 3.2. (i) From (1.12) we find # $ 1 ϑ2 8 ϑ3 8 ϑ4 8 Z √1 D4 ,B ∗ (τ ) = + + . 2 η η η 2

(3.17) alg

Applying the orbifold procedure for the Z2 ×Z2 action of [2, 2, 0, 0], [2, 0, 2, 0] ⊂ Gab to the partition function (A.10) of the Gepner model (2)4 [F-K-S-S] one checks that C1 and C2 have the same partition function obtained by inserting (3.17) into (2.3). 2 (ii) We have an enhancement of the current algebra (2.2) of the nonlinear σ model

C to 1 6 su(2)1 . The 12 additional (1, 0)-fields are Uα := √ Vα,α+B ∗ α +V−α,−α−B ∗ α , where 2

α belongs to the D4 rootsystem {± √1 ei ± √1 ej }. We set 2 2 ± Wi,j := 21 U √1 (ei +ej ) ± U √1 (ei −ej ) 2

2

126


to see that upon a consistent choice of cocycle factors for the vertex operators these fields indeed comprise an extra su(2)14 : + + W1,2 ± + W3,4 + + + + Q := W1,2 − W3,4 , Q± := √1 W1,3 − W2,4 ± 2 − − − − R := iW2,4 − iW1,3 , R ± := √1 W1,4 − W2,3 ± 2 − − − − ± S := W1,4 + W2,3 , S ± := √1 W1,2 + W3,4 + + P := W1,4 + W2,3 ,

P ± :=

√1 2

2

+ + W2,4 , + W1,3 + + √1 − W2,3 W1,4 , 2 − − √1 − W3,4 W1,2 , 2 − − √1 W2,4 . + W1,3 √1 2

(3.18)

2

2)4 we use Xij as a shorthand notation for the field For the Gepner type model C2 = ( having factors V04,2;0,0 in the i th and j th position and factors V00,0;0,0 otherwise, and Yij for the field having factors V0−2,2;0,0 in the i th and j th position and factors V02,2;0,0 otherwise. By comparison of operator product expansions one then checks that the following identifications can be made: J = J1 + J 2 + J 3 + J4 , A = J1 + J2 − J3 − J4 ,

⊗4 J ± = V0∓2,2;0,0 ; A+ = Y12 , A− = Y34 ;

P =

(J1 − J2 + J3 − J4 ) ,

P + = Y13 , P − = Y24 ;

(J1 − J2 − J3 + J4 ) ,

Q+ = Y14 , Q− = Y23 ;

(X13 − X24 ) ,

R ± = ∓ 21 (X12 + X34 ) +

Q= R= S=

√1 2 √1 2 √i 2 √i 2

S±

(X13 + X24 ) ,

=

± 21

(X12 − X34 ) +

i 2 i 2

(X14 + X23 ) ; (X14 − X23 ) .

Thus the (1, 0)-fields in the two theories generate the same algebra A = su(2)12 ⊕ su(2)14 = Af ⊕ Ab . Obviously, the same structure arises on the right hand sides. 2 of C1 and C2 both have self dual 1 and H (iii) We will show that the spaces of states H b b J := (P , Q, R, S; P , Q, R, S)-charge lattice, -b = √1 (x + y; x − y) x, y ∈ Z4 . (3.19) 2

In the Gepner type model C1 = ( 2)4 we find 16 fields with dimensions h = h = 41 which are uncharged under (J, A; J , A); diagonalizing them with respect to the J -action for j ∈ {P , Q, R, S} we obtain fields Ej± , Fj± uncharged under all U (1)-currents apart from j and with (j,  )-charge √1 (±1, ±1) and √1 (±1, ∓1), respectively. Namely, 2

2

EP± = V0∓1,∓1;∓1,∓1 ⊗ V0±1,±1;±1,±1 ⊗ V0∓1,∓1;∓1,∓1 ⊗ V0±1,±1;±1,±1 ,

FP± = V0∓1,∓1;±1,±1 ⊗ V0±1,±1;∓1,∓1 ⊗ V0∓1,∓1;±1,±1 ⊗ V0±1,±1;∓1,∓1 , ± EQ = V0∓1,∓1;∓1,∓1 ⊗ V0±1,±1;±1,±1 ⊗ V0±1,±1;±1,±1 ⊗ V0∓1,∓1;∓1,∓1 ,

± = V0∓1,∓1;±1,±1 ⊗ V0±1,±1;∓1,∓1 ⊗ V0±1,±1;∓1,∓1 ⊗ V0∓1,∓1;±1,±1 , FQ


127

and with εR := −1, εS := 1 for j ∈ {R, S}, ⊗4 ⊗4 + εj V12,1;−2,−1 Ej± = V12,1;2,1 ± V12,1;−2,−1 ⊗ V12,1;2,1 ⊗ V12,1;−2,−1 ⊗ V12,1;2,1

Fj±

+εj V12,1;2,1 ⊗ V12,1;−2,−1 ⊗ V12,1;2,1 ⊗ V12,1;−2,−1 , ⊗2 ⊗2 ⊗2 ⊗2 = V12,1;2,1 ⊗ V12,1;−2,−1 + εj V12,1;−2,−1 ⊗ V12,1;2,1 ± V12,1;−2,−1 ⊗ V12,1;2,1 ⊗ V12,1;2,1 ⊗ V12,1;−2,−1 +εj V12,1;2,1 ⊗ V12,1;−2,−1 ⊗ V12,1;−2,−1 ⊗ V12,1;2,1 .

Among the corresponding charges under J we find In the sigma model C1 we set α1 := α3 :=

√1 2 √1 2

(e1 + e2 ) ,

α2 :=

(e1 + e3 ) ,

α4 :=

√1 (ei ; ±ei ) 2 √1 2 √1 2

generating -b .

(e2 − e1 ) , (e4 − e2 ) .

Let )δ with δ ∈ F42 denote the twist field corresponding to the fixed point 21 4i=1 δi αi . The action of P , Q, R, S and their right handed partners is determined as in (3.10). Then by normalizing appropriately and matching (J , J )-charges we find that the following identifications can be made (sums run over δ ∈ F42 with the indicated restrictions): EP± = FP± = ± = EQ ± FQ

ER±

=

FS±

δ1 =δ2 ,δ3 =δ4

)δ ±

=

=

(−1) )δ ±

=

δ1 =δ2 ,δ3 =δ4

(−1)δ3 )δ ,

δ1 =δ2 ,δ3 =δ4

δ1

(−1) )δ ±

(−1)δ1 )δ ,

δ1 =δ2 ,δ3 =δ4

(−1)δ2 )δ ±

(−1)δ2 )δ ,

δ1 =δ2 ,δ3 =δ4

(−1)

δ2 +δ3

)δ ±

δ1 =δ2 ,δ3 =δ4

(−1)δ4 )δ ,

δ1 =δ2 ,δ3 =δ4 δ3

δ1 =δ2 ,δ3 =δ4

(−1)δ4 )δ ±

δ1 =δ2 ,δ3 =δ4

)δ ,

δ1 =δ2 ,δ3 =δ4

δ1 =δ2 ,δ3 =δ4

)δ ,

δ1 =δ2 ,δ3 =δ4

δ1 =δ2 ,δ3 =δ4

)δ ±

δ1 =δ2 ,δ3 =δ4

FR± = ES±

(−1)δ2 +δ3 )δ ,

δ1 =δ2 ,δ3 =δ4

(−1)

δ2 +δ3

)δ ±

(−1)δ2 +δ3 )δ .

δ1 =δ2 ,δ3 =δ4

In particular, the corresponding (J , J )-charges generate -b .

) (

128


Recall the Greene–Plesser construction for mirror symmetry [G-P] to observe that the Z2 × Z2 orbifold ( 2)4 of (2)4 is invariant under mirror symmetry. This can be regarded as an explanation for the high degree of symmetry found for ( 2)4 = C1 . In view of (3.19) it is clear that the same phenomenon as described in Remark 3.4 appears for the theory discussed above: Remark 3.8. The Gepner type model C1 = ( 2)4 , or equivalently the nonlinear σ model C2 = K( √1 D4 , B ∗ ), B ∗ given by (3.16), is located at a meeting point of the moduli 2 spaces of theories associated to K3 surfaces and tori, respectively. Namely, its bosonic sector is identical with that of the nonlinear σ model C3 on the SU (2)14 -torus with vanishing B-field. This again can be deduced from the results in [K-S] once one observes that the lattice denoted by O(n)×O(n) there in the case n = 4 is isomorphic to -b as defined in (3.19). 4 1 2 ∼ 2)4 = C1 = The relation between the two meeting points (2) = C = C = C 3 and ( C2 ∼ 2)4 = C3 of the moduli spaces found so far is best understood by observing that C1 = ( ab 1 4 can be constructed from C = ( 2) by modding out Z2 ∼ = [2, 0, 2, 0] ⊂ Galg . If we formulate the orbifold procedure in terms of the charge lattice -b of C 1 = ( 2)4 as described 1 in [G-P], this amounts to a shift orbifold by the vector δ = 2 (−1, 1, 0, 0; 1, −1, 0, 0) on -b . Indeed, this shift simply reverts the shift we used to explain Remark 3.4 and brings us back onto the torus TSU (2) 4 . But as for C 1 = C 2 and C 3 , C1 = C2 and C3 will 1 correspond to different compactifications of the type IIA string. From (3.15) we are able to determine the geometric counterpart of [2, 0, 2, 0] on K(Z4 , 0): It is the unique nontrivial central element t1111 of the algebraic automorphism + + group GKummer depicted in Fig. 3.1. Hence the commutant of t1111 is the entire GKummer ,

00

✲ I ❅ ✎ ❅ ❘

✲ I ❅ ✎ ❅ ❘

01

✲ ❅ I ✎ ❅ ❘

✲ ❅ I ✎ ❅ ❘

11

11

10

10

00

01

Fig. 3.1. Action of the algebraic automorphism t1111 on the Kummer lattice N + but it is not clear so far how to continue the residual GKummer /Z2 algebraically to the 4 twisted sectors in ( 2) with respect to the t1111 orbifold. We remark that conformal field theory also helps us to draw conclusions on the geometry of the Kummer surfaces under inspection: K( √1 D4 , B ∗ ) is obtained from 2

K(Z4 , 0) by modding out the classical symmetry t1111 , so in terms of the decomposition (1.4) we stay in the same “chart” of MK3 , i.e. choose the same nullvector υ for both theories. This means that we can explicitly relate the respective geometric data. For both Kummer surfaces we choose the complex structures induced by the N = (2, 2) algebra in the corresponding Gepner models ( 2)4 and ( 2)4 . Thus we identify J ± = ⊗4 in both theories with the two forms π∗ (dz1 ∧ dz2 ), π∗ (dz1 ∧ dz2 ) V0∓2,2;∓0,0


129

defining the complex structure of K(). Here π : T → K() is the rational map of degree two, = Z4 or = √1 D4 , respectively. Then both K() are singular 2 K3 surfaces (see Sect. 2.5). Given the lattices of the underlying tori one can compute the intersection form for real and imaginary part of the above two forms defining the complex structure. One finds that they span sublattices of the transcendental lattices with forms diag(4, 4) for K(Z4 ) and diag(8, 8) for K( √1 D4 ), respectively. The factor 2 of two difference was to be expected, because t1111 has degree two. Nevertheless, one may check that the transcendental lattices themselves for both surfaces have quadratic form diag(4, 4). Note that for a given algebraic automorphism in general it is hard to decide how the transcendental lattices transform under modding out [In, Cor. 1.3.3]. In our case, we could read it off thanks to the Gepner type descriptions of our conformal field theories. 3.6. Gepner type description of SO(8)1 /Z4 . Theorem 3.9. The Gepner type model C 1 = ( 2)4 which agrees with C 2 = K(Z4 , 0) by Theorem 3.3 admits a nonlinear σ model description as Z4 orbifold of the torus model T ( √1 D4 , B ∗ ) with SO(8)1 symmetry. 2

Proof. The proof works analogously to that of Theorem 3.5. From Theorem 2.8 it follows that the Z4 orbifold of T ( √1 D4 , B ∗ ) with B ∗ defined by (3.16) is obtained from 2 C2 = K( √1 D4 , B ∗ ) by modding out the automorphism r12 as depicted in Fig. 2.1. 2

Thus we should work with the models C1 = ( 2)4 and C2 = K( √1 D4 , B ∗ ) which are 2 isomorphic by Theorem 3.7. We use the notations introduced there. Then r12 is induced by e1 → e2 , e2 → −e1 , e3 → −e4 , e4 → e3 . Of the su(2)16 current algebra of C2 we find a surviving su(2)12 ⊕ u(1)4 current algebra on the Z4 orbifold generated by J, J ± , A; P , P ± , Q, R, S (see Eqs. (2.2) and (3.18)). The action on the generators Ej± , Fj± ; j ∈ {P , Q, R, S} is already diagonalized. All the Ej± are invariant as well as FP± . On the fermionic part of the space of states of C2 the identifications (3.11) hold. The fields WεJ1 ,ε2 and WεA1 ,ε1 , εi ∈ {±1} are those invariant under the Z4 action. Our field by field identifications of Theorem 3.7 now allow us to read off the induced action on the Gepner type model C1 = ( 2)4 . One checks that it agrees with the symmetry [2 , 2 , 0, 0] defined in (3.12) which revokes the orbifold by the Z2 action of [2, 2, 0, 0]. Because C1 = ( 2)4 was constructed from the Gepner model (2)4 by modalg ding out Z2 × Z2 ∼ = [2, 2, 0, 0], [2, 0, 2, 0] ⊂ Gab , it follows that the Z4 orbifold of T ( √1 D4 , B ∗ ) agrees with the Gepner type model obtained from (2)4 by modding out 2 Z2 ∼ 2)4 by a permutation of the minimal = [2, 0, 2, 0]. This clearly is isomorphic to ( model factors. ( ) 4. Conclusions: A Panoramic Picture of the Moduli Space We conclude by joining the information we gathered so far to a panoramic picture of those strata of the moduli space we have fully under control now (Fig. 4.1). The rest of this section is devoted to a summary of what we have learned about the various components depicted in Fig. 4.1. All the strata are defined as quaternionic submanifolds of the moduli space MK3 consisting of theories which admit certain

130


Z4 Orbifold-line

✚ ✚ ✻ ✚ ✚

Quartic line

❍❍ = Z4 , BT = 0

SO(8)1 /Z4 ∼ = K(Z4 , 0) Tori T (, BT ), T = R4 /

Z4 Orbifold-plane

✚ ✚ ✚ ✚ ✻ Z2 Orbifolds K(, BT ),

✚ ❍ (2)4 ✚ r12 ✚ r12 ❍❍r✚ T = R/, B = ✚ ✚I ❍ ✚ ❅ γ ✚ ✻ ✚ 4 ∼ D4 ✚ α ❅ ✚ ∼Z ✚ ✏ t1111✲ ✏ ✚ ❅✏✏ r✏ ω z✏ ✚✏✏ 4 ✏✏ K(D4 , 0) ( 2)4 ❄✏ ( 2)✏ β ❅ ❘ ✚ r ✛ ✲✏ r ✏ ✏✏ ✏✏ T (D4 , 0)

❅ T (Z4 , 0)

❅ ❅

√1 BT + 1 B (2) 2 Z 2

K( √1 D4 , B ∗ ) 2

Fig. 4.1. Strata of the moduli space

restricted geometric interpretations. In other words, a suitable choice of υ as described in Sect. 1 yields (), V , B) such that ), B have the respective properties. In the following we will always tacitly assume that an appropriate choice of υ has been performed already. Figure 4.1 contains two strata of real dimension 16, depicted as a horizontal plane and a mexican hat like object, respectively. The horizontal plane is the Kummer stratum, the subspace of the moduli space consisting of all theories which admit a geometric interpretation on a Kummer surface X in the orbifold limit. In other words, it is the 16 dimensional moduli space of all theories K(, BT ) obtained from a nonlinear σ model on a torus T = R4 / by applying the ordinary Z2 orbifold procedure; the B-field takes (2) values B = √1 BT + 21 BZ , where BT ∈ H 2 (T , R) $→ H 2 (X, R) (see the explanation 2

(2)

after Theorem 2.1), and BZ ∈ H even (X, Z) as described in Theorem 2.3. We have an embedding Mtori $→ MK3 as quaternionic submanifold, and we know how to locate this stratum within MK3 . Kummer surfaces in the orbifold limit have a generic group F42 of algebraic automorphisms which leave the metric invariant. Any conformal field theory associated to such a Kummer surface possesses an su(2)12 subalgebra (2.2) of the holomorphic W-algebra. The mexican hat like object in Fig. 4.1 depicts the moduli space (1.17) of theories associated to tori. Two meeting points with the Kummer stratum have been determined so far, namely ( 2)4 and ( 2)4 (see Remarks 3.4 and 3.8). We found ( 2)4 = K(Z4 , 0) = 1 4 ∗ 4 ∗ √ T (D4 , 0) and (2) = K( D4 , B ) = T (Z , 0), where B was defined in (3.16). 2 The vertical plane in Fig. 4.1 depicts a stratum of real dimension 8, namely the moduli space of theories admitting a geometric interpretation as Z4 orbifold of a nonlinear σ model on T = R4 /. In order for the orbifold procedure to be well defined we assume to be generated by i ∼ = Ri Z2 , Ri ∈ R, i = 1, 2 (1 is not necessarily orthogonal 2 to 2 ) and BT ∈ H (T , R)Z4 $→ H 2 (X, R) (see Lemma 2.9). The B-field then takes (4) values B = 21 BT + 41 BZ as described in Theorem 2.11, where the embedding of this K3 stratum in M is also explained. The generic group of algebraic automorphisms for Z4 orbifolds is Z2 F42 . By Theorem 3.9 there is a meeting point with the Kummer stratum in the Z4 orbifold of T ( √1 D4 , B ∗ ), where B ∗ is given by (3.16), which agrees 2 with K(Z4 , 0) = ( 2)4 .


131

The four lines in Fig. 4.1 are strata of real dimension 4 which are defined by restriction to theories admitting a geometric interpretation (), V , B) with fixed ) and allowed Bfield values B ∈ ). Thus the volume is the only geometric parameter along the lines and we can associate a fixed hyperkähler structure on K3 to each of them. For all four lines it turns out that one can choose a complex structure such that the respective K3 surface is singular. Hence ) can be described by giving the quadratic form on the transcendental lattice and the Kähler class for this choice of complex structure. Specifically we have: – Z4 -line: The subspace of the Kummer stratum given by theories K(, BT ) with ∼ Z4 and BT ∈ ), which is marked by ∼ Z4 in Fig. 4.1. – Z4 Orbifold-line: The moduli space of all theories which admit a geometric interpretation on a K3 surface obtained from the nonlinear σ model on a torus T = R4 /, ∼ Z4 with B-field BT commuting with the automorphisms listed in (2.13). – Quartic line: Though well established in the context of Landau–Ginzburg theories, this stratum has been somewhat conjectural up to now. We describe it as the moduli space of theories admitting a geometric interpretation ()Q , VQ , BQ ) on the Fermat quartic (2.17) equipped with a Kähler metric in the class of the Fubini-Study metric, in order for )Q to be invariant under the algebraic automorphism group G = Z24 S4 . The B-field is restricted to values BQ ∈ )Q , because µ(G) = 5 and therefore H 2 (X, R)G = )Q . – D4 -line: The moduli space of theories K(, BT ), ∼ D4 admitting as geometric interpretation a Kummer surface K() and BT ∈ ). This line is labeled by ∼ D4 in Fig. 4.1. The four lines are characterized by the following data4 : name of line

Z4 -line Z4 orbifold-line

quartic line D4 -line

associated form on the transcendental lattice # $ 4 0 0 4 # $ 2 0 0 2 # $ 8 0 0 8 # $ 4 0 0 4

group of algebraic automorphisms leaving the metric invariant

generic (1, 0)-current algebra

+ GKummer = Z22 F42 ∼ (Z2 × Z4 ) D4 =

su(2)12

D4

su(2)1 ⊕ u(1)

(Z4 × Z4 ) S4

su(2)1

Z2 F42

su(2)12

In Fig. 4.1 we have two different shortdashed arrows indicating relations between lines. Consider the Kummer surface K(Z4 ) associated to the Z4 -line. As demonstrated in + Theorem 2.8, the group GKummer of algebraic automorphisms of K(Z4 ) which leave the metric invariant contains the automorphism r12 of order two (see Fig. 2.1) which upon modding out produces the Z4 orbifold-line. The entire moduli space of Z4 orbifold conformal field theories is obtained this way from Z2 orbifold theories K(, BT ), where is generated by i ∼ = Ri Z2 , Ri ∈ R, i = 1, 2 and BT ∈ H 2 (T , R)Z4 . + Modding out t1111 ∈ GKummer (see Fig. 3.1) on the Z4 -line produces the D4 -line, as argued at the end of Sect. 3.5. Note that the K3 surfaces associated to Z4 - and D4 -lines 4 The quadratic form for the transcendental lattice of quartic and the Z orbifold of T = R4 /Z4 can be 4 found in [In, Shi].

132


have the same quadratic form on their transcendental lattices and hence are identical as algebraic varieties. Still, the corresponding lines in moduli space are different because different Kähler classes are fixed. In our terminology this is expressed by the change of lattices of the underlying tori on transition from one line to the other. The D4 -line can also be viewed as the image of the Z4 -line upon shift orbifold on the underlying torus. Finally, we list the zero dimensional strata shown in Fig. 4.1. To construct K(D4 , 0) on the D4 -line, we may as well apply the ordinary Z2 orbifold procedure to the D4 -torus theory in the meeting point ( 2)4 (the arrow with label ω in Fig. 4.1). We stress that in contrast to what was conjectured in [E-O-T-Y] this is not a meeting point with the Z4 orbifold-plane. As demonstrated in Theorem 3.5 and also conjectured in [E-O-T-Y], Gepner’s model (2)4 is the point of enhanced symmetry = Z4 , BT = 0 on the Z4 orbifold-line. In Sect. 3.1 we have studied the algebraic symmetry group of (2)4 and in Corollary 3.6 proved that it admits a geometric interpretation with Fermat quartic target space, too. In terms of the Gepner model, the moduli of infinitesimal deformation along the Z4 orbifold and ± the quartic line are real and imaginary parts of Vδ,ε (δ, ε ∈ {±1}) as in (3.14) and of the 1 1 ⊗4 (1, 1)-superpartners of (V±1,0;∓3,2 ) , (V±1,0;∓1,0 )⊗4 , respectively (see Sect. 3.4). The Gepner type models ( 2)4 and ( 2)4 which are meeting points of torus and K3 moduli spaces have been mentioned above. For all the longdash arrowed correspondences γ β α (2)4 ←→ ( 2)4 ←→ ( 2)4 ←→ (2)4 in Fig. 4.1 we explicitly know the symmetries to be modded out from the Gepner (type) model as well as the corresponding algebraic r12 r12 automorphisms on the geometric interpretations. For instance, ( 2)4 −→ ( 2)4 −→ (2)4 . Hence for these examples we know precisely how to continue geometric symmetries to the quantum level. Acknowledgements. The authors would like to thank A. Taormina for very helpful discussions on N = 4 superconformal field theory and V. Nikulin for his explanations concerning the geometry of Kummer surfaces. K.W. thanks F. Rohsiepe for valuable discussions and his most efficient crash course in C ++ . We thank M. Rösgen and F. Rohsiepe for proof reading. Work on this paper was supported by TMR.

A. Minimal Models and Gepner Models The N = 2 minimal superconformal models form the discrete series (k), k ∈ N of unitary representations of the N = 2 superconformal algebra with central charges c = 3k/(k + 2). For constructing the model (k) we may start from a Zk parafermion theory and add a free bosonic field. More precisely, (k) is the coset model SU (2)k ⊗ U (1)2 . U (1)k+2,diag

(A.1)

The primary fields are denoted by Vlm,s;m,s (z, z), where l ∈ {0, . . . , k} is twice the spin of the corresponding field in the affine SU (2)k and we have tacitly specialized to the diagonal invariant by imposing l = l. The remaining quantum numbers m, m ∈ Z2(k+2) and s, s ∈ Z4 label the representations of U (1)k+2,diag and U (1)2 in the decomposition (A.1), respectively, and must obey l ≡ m + s ≡ m + s (2). Here, the fields with even (odd) s create states in the lefthanded Neveu–Schwarz (Ramond) sector, and analogously for s and the righthanded sectors. Moreover the identification Vlm,s;m,s (z, z) ∼ Vk−l m+2+k,s+2;m+2+k,s+2 (z, z)

(A.2)


133

l holds. By (A.1), the corresponding characters Xm,s;m,s can be obtained from the level l k string functions cj , l ∈ {0, . . . , k}, j ∈ Z2k of SU (2)k and classical theta functions Ta,b , a ∈ Z2b of level b = 2k(k + 2) by [Ge2, R-Y, Qi] l l l Xm,s;m,s (τ, z) = χm,s (τ, z) · χm,s (τ , z), l χm,s (τ, z)

=

k

j =1

l c4j +s−m (τ )T2m−(k+2)(4j +s),2k(k+2) (τ,

z ). k+2

(A.3)

Modular transformations act by , s2 c l(l + 2) − m2 l l χm,s (τ + 1, z) = exp 2π i + − χm,s (τ, z), 4(k + 2) 8 24

1 z

mm ss π(l + 1)(l + 1) πi (k+2) l χm,s sin e−πi 2 χml ,s (τ, z), − τ , τ = κ(k) e k+2 l ,m ,s

(A.4) where κ(k) is a constant depending only on k and the summation runs over l ∈ {0, . . . , k}, m ∈ {−k − 1, . . . , k + 2}, s ∈ {−1, . . . , 2}, l + m + s ≡ 0 (2). l Let ψm,s denote a lowest weight state in the irreducible representation of the N = 2 l . Conformal dimension and charge of ψ l superconformal algebra with character χm,s m,s then are hlm,s =

l(l + 2) − m2 s2 + 4(k + 2) 8

mod 1,

Qlm,s =

m s − k+2 2

mod 2.

(A.5)

The fusion-algebra is

l ψm,s

,2k−l−l ) min (l+l l l × ψm ,s = ψm+m ,s+s .

(A.6)

l=|l−l |, l≡l+l (2)

Note that by (A.5) and (A.6) the operators of left and right handed spectral flow are 0 0 and V00,0;−1,−1 = ψ−1,−1 , respectively. associated to the fields V0−1,−1;0,0 = ψ−1,−1 The NS-part of our modular invariant partition function is now given by

ZNS (τ, z) = 21 (A.7) χml,0 (τ, z) + χml,2 (τ, z) χml,0 (τ , z) + χml,2 (τ , z) , l=0,...,k, m=−k−1,...,k+2,

l+m≡0(2)

and expressions for the other three parts ZNS are obtained by flows as described , ZR , ZR in (2.4). In the case k = 2 which we employ in this paper, the parafermion algebra is nothing but the algebra satisfied by the Majorana fermion ψ of the Ising model. By inspection of the charge lattice one may confirm that the minimal model (2) can readily be constructed by tensoring the Ising model with the one dimensional free theory which describes a

134


bosonic field ϕ compactified on a circle of radius R = 2. The primary fields decompose as Vlm,s;m,s (z, z) = Xlm−s;m−s (z, z) e 2 X0j ; (z, z)

=

X2j ±2; ±2 (z, z)

=

i √

2

(−m+2s)ϕ

(z) e 2

ξj0 (z)ξ0 (z),

ξ00

i √ (−m+2s)ϕ 2

= 1,

ξ20

(z),

= ψ,

(A.8)

1 and X11,1 = X1−1,−1 , X11,−1 = X1−1,1 denote the ground states of the two h = h = 16 representations of the Ising model. Indeed, the level 2 string functions are obtained from the characters of lowest weight representations in the Ising model by dividing by the Dedekind eta function. To construct a Gepner model with central charge c = 3d/2, d ∈ {2, 4, 6}, one r first takes the (fermionic) tensor r product of r minimal models ⊗i=1 (ki ) such that the central charges add up to i=1 3ki /(ki + 2) = 3d/2. The bosonic modes acting on different theories commute and the fermionic modes anticommute. More concretely [F-K-S, (4.5)],

Vlm1 1 ,s1 ;m1 ,s 1 ⊗ Vlm2 2 ,s2 ;m2 ,s 2 = (−1) 4 (s1 −s 1 )(s2 −s 2 ) Vlm2 2 ,s2 ;m2 ,s 2 ⊗ Vlm1 1 ,s1 ;m1 ,s 1 . (A.9) 1

The diagonal sums T , J, G± of the fields which generate the N = 2 algebras of the factor theories (ki ) then comprise a total N = 2 superconformal algebra of central 2πiJ0 , then Z ∼ Z charge c = 3d/2. Denote by Z the cyclic group generated by = n er (k ) is the orbifold with n = lcm {2; ki + 2, i = 1, . . . , r}. Now the Gepner model i i=1 of ⊗ri=1 (ki ) with respect to Z. Effectively this means that ri=1 (ki ) is obtained from S)-sector, onto ⊗ri=1 (ki ) by projecting onto integer left and right charges in the (N S + N integer or half integer left and right charges in the (R + R)-sector according to c being even or odd, and adding twisted sectors for the sake of modular invariance. In particular, the so constructed model describes an N = (2, 2) superconformal field theory with central charge c = 3d/2 and (half) integer charges. For d = 4 the Gepner model is thus associated to a K3 surface or a torus, as discussed in the introduction. We again decompose the partition function as in (2.4) and find ZNS (τ, z) =

n & r

b=0 (l,m) j =1

·

1 2

l ,0 l ,2 χmjj (τ, z) + χmjj (τ, z) ·

l ,0 l ,2 χmjj +2b (τ , z) + χmjj +2b (τ , z)

(A.10) ,

denotes the sum over all values (l, m) ∈ Z2r with lj ∈ {0, . . . , kj }, mj mj mj ∈ {−kj − 1, . . . , kj + 2}, lj + mj ≡ 0 (2) and rj =1 kj +2 , rj =1 kj +2 ∈ Z. r li We note that the field j =1 Vmj ,sj ;mj ,s j of the resulting Gepner model belongs to the where

(l,m)

bth twisted sector with respect to the orbifold by Z iff 2b ≡ (mj − mj ) mod n for j = 1, . . . , r. This means that the (b+1)st twisted sector is obtained from the bth twisted sector by applying the twofold right handed spectral flow which itself is associated to ⊗r of our theory. We explicitly see that for c = 6 the the primary field V00,0;−2,2 ⊗r belonging to the operators of twofold lefthanded spectral flow are fields V0∓2,2;0,0

nothing but the SU (2)-currents J ± which extend the N = 2 superconformal algebra


135

to an N = 4 superconformal algebra, and analogously for the righthanded algebra. Moreover, to calculate ZNS (τ, z; τ , z) instead of using the closed formula (A.10) one may proceed as follows: Start by multiplying the NS-parts of the partition functions of the minimal models (ki ), i = 1, . . . , r. Keep only the Z-invariant i.e. integrally charged part of this function; let us denote the result by F (τ, z; τ , z). Add the bth twisted sectors, b = 1, . . . , n − 1, by performing a 2b-fold righthanded spectral flow, i.e. by adding 2 q db /4 y db/2 F (τ, z; τ , z + bτ ). This way calculations get extremely simple as soon as the characters of the minimal models are written out in terms of classical theta functions. We further note that to accomplish Gepner’s actual construction of a consistent theory of superstrings in 10 − d dimensions we would first have to take into account 8 − d additional free superfields representing flat (10-d)-dimensional Minkowski space in light-cone gauge, second, perform the GSO projection onto odd integer left and right charges, and thirdly, convert the resulting theory into a heterotic one. However, at the stage described above we have constructed a consistent conformal field theory with central charge c = 3d/2 which for d = 4 is associated to a K3 surface or a torus, so we may and will omit these last three steps of Gepner’s construction. B. Explicit Field Identifications: ( 2)4 = K(Z4 , 0) 2)4 (see Theorem 3.3) In this appendix, we give a complete list of ( 41 , 41 )-fields in ( together with their equivalents in the nonlinear σ model on K(Z4 , 0). As usual, ε, εi ∈ {±1} and we use notations as in (3.10) and (3.11). Untwisted ( 41 , 41 )-fields with respect to the [2, 2, 0, 0]-orbifold. ⊗4 V0−ε1 ,−ε1 ;−ε2 ,−ε2 ⊗2 ⊗2 ⊗ V0ε,ε;ε,ε V0−ε,−ε;−ε,−ε ⊗4 V12,1;2,1 ⊗4 V12,1;−2,−1

V12,1;2,1

⊗2

= WεJ1 ,ε2 , A = Wε,ε ,

= )0000 − )1100 + )1111 − )0011 , = )1010 + )0101 − )0110 − )1001 ,

⊗ V0−1,−1;−1,−1 ⊗ V01,1;1,1

= )0000 − )1100 − )1111 + )0011 + )0010 + )0001 − )1101 − )1110 , ⊗2 V12,1;2,1 ⊗ V01,1;1,1 ⊗ V0−1,−1;−1,−1

= )0000 − )1100 − )1111 + )0011 − )0010 − )0001 + )1101 + )1110 , ⊗2 V0−1,−1;−1,−1 ⊗ V01,1;1,1 ⊗ V12,1;2,1 = )0000 + )1100 − )1111 − )0011 + )1000 + )0100 − )1011 − )0111 , ⊗2 V01,1;1,1 ⊗ V0−1,−1;−1,−1 ⊗ V12,1;2,1 = )0000 + )1100 − )1111 − )0011 − )1000 − )0100 0 V−1,−1;−1,−1 ⊗ V01,1;1,1 ⊗ V0−1,−1;−1,−1 ⊗ V01,1;1,1

+ )1011 + )0111 ,

136


= ()0000 + )1100 + )1111 + )0011 ) + ()1000 + )0100 + )0111 + )1011 ) + ()0010 + )0001 + )1101 + )1110 ) + ()1010 + )0101 + )0110 + )1001 ) , V0−1,−1;−1,−1 ⊗ V01,1;1,1 ⊗ V01,1;1,1 ⊗ V0−1,−1;−1,−1 = ()0000 + )1100 + )1111 + )0011 ) + ()1000 + )0100 + )0111 + )1011 ) − ()0010 + )0001 + )1101 + )1110 ) − ()1010 + )0101 + )0110 + )1001 ) , V01,1;1,1 ⊗ V0−1,−1;−1,−1 ⊗ V01,1;1,1 ⊗ V0−1,−1;−1,−1 = ()0000 + )1100 + )1111 + )0011 ) − ()1000 + )0100 + )0111 + )1011 ) − ()0010 + )0001 + )1101 + )1110 ) + ()1010 + )0101 + )0110 + )1001 ) , V01,1;1,1 ⊗ V0−1,−1;−1,−1 ⊗ V0−1,−1;−1,−1 ⊗ V01,1;1,1 = ()0000 + )1100 + )1111 + )0011 ) − ()1000 + )0100 + )0111 + )1011 ) + ()0010 + )0001 + )1101 + )1110 ) − ()1010 + )0101 + )0110 + )1001 ) . Twisted ( 41 , 41 )-fields with respect to the [2, 2, 0, 0]-orbifold. ⊗2 ⊗2 A ⊗ V0ε,ε;−ε,−ε = Wε,−ε , V0−ε,−ε;ε,ε ⊗2 ⊗2 ⊗ V12,1;2,1 = )1000 − )0100 + )0111 − )1011 , V12,1;−2,−1 ⊗2 ⊗2 ⊗ V12,1;−2,−1 = )0010 − )0001 + )1101 − )1110 , V12,1;2,1

V12,1;−2,−1

⊗2

⊗ V0−1,−1;−1,−1 ⊗ V01,1;1,1

= )1000 − )0100 + )1011 − )0111 + )1010 − )0101 + )1001 − )0110 , ⊗2 V12,1;−2,−1 ⊗ V01,1;1,1 ⊗ V0−1,−1;−1,−1

= )1000 − )0100 + )1011 + )0111 − )1010 + )0101 − )1001 + )0110 , ⊗2 V0−1,−1;−1,−1 ⊗ V01,1;1,1 ⊗ V12,1;−2,−1 = )0010 − )0001 − )1101 + )1110 + )1010 − )0101 − )1001 + )0110 , ⊗2 V01,1;1,1 ⊗ V0−1,−1;−1,−1 ⊗ V12,1;−2,−1 = )0010 − )0001 − )1101 + )1110 − )1010 + )0101 + )1001 − )0110 .

References [A-B-D+] [Al] [An]

Ademollo, M., Brink, L., D’Adda, A., D’Auria, R., Napolitano, E., Sciuto, S., Del Giudice, E., Di Vecchia, P., Ferrara, S., Gliozzi, F., Musto, R., Pettorino, R.: Supersymmetric strings and color confinement. Phys. Lett. B 62, 105–110 (1976) Allan, N.: Maximality of some arithmetic groups. Ann. Acad. Brasil. Ci. 38, 223–227 and 243–244 (1966) Anderson, M.T.: The L2 structure of moduli spaces of Einstein metrics on 4-manifolds. Geom. Funct. Anal. 2, 29–89 (1992)


[A-M]

137

Aspinwall, P.S., Morrison, D.R.: String theory on K3 surfaces. In: Greene, B., Yau, S.T. (eds), Mirror symmetry, vol. II, 1994, pp. 703–716; hep-th/9404151 [As1] Aspinwall, P.: Enhanced gauge symmetries and K3 surfaces. Phys. Lett. B357, 329–334 (1995); hep-th/9507012 [As2] Aspinwall, P.S.: K3 surfaces and string duality. In: Fields, strings and duality (Boulder, CO, 1996). River Edge, NJ: World Sci. Publishing, 1997, pp. 421–540; hep-th/9611137 [Be] Berger, M.: Sur les groupes d’holonomie des variétés à connexion affine et des variétés riemmannienes. Bull. Soc. Math. Fr. 83, 279–330 (1955) [B-I] Blum, J.D., Intriligator, K.: Consistency conditions for branes at orbifold singularities. Nucl. Phys. B 506, 223–235 (1997); hep-th/9705030 [B-E-R] Brunner, I., Entin, R., Römelsberger, Ch.: D-branes on T 4 /Z2 and T-Duality. JHEP 9906:016, (1999); hep-th/9905078 [B-T] Budinich, P., Trautmann, A.: The Spinorial Chessboard. Berlin–Heidelberg–New York– London–Paris–Tokyo: Springer-Verlag, 1988 [Ce1] Cecotti, S.: N = 2 supergravity, type IIB superstrings and algebraic geometry. Commun. Math. Phys. 131, 517–536 (1990) [Ce2] Cecotti, S.: N = 2 Landau–Ginzburg vs. Calabi–Yau σ -models: Non-perturbative aspects. Int. J. Mod. Phys. A 6, 1749–1813 (1991) [C-E-N-T] Casher, A., Englert, F., Nicolai, H., Taormina, A.: Consistent superstrings as solutions of the D = 26 bosonic string theory. Phys. Lett. B 162, 121–126 (1985) [C-F-G] Cecotti, S., Ferrara, S., Girardello, L.: Geometry of type II superstrings and the moduli of superconformal field theories. Int. J. Mod. Phys. A 4, 2475–2529 (1989) [Cu] Curtright, T.L.: Indices, triality and ultraviolet divergences for supersymmetric theories. Phys. Rev. Lett. 48, 1704–1708 (1982) [D-G] Distler, J., Greene, B.: Some exact results on the superpotential from Calabi–Yau compactifications. Nucl. Phys. B 309, 295–316 (1988) [Di] Dijkgraaf, R.: Instanton strings and hyperkaehler geometry. Nucl. Phys. B 543, 545–571 (1999); hep-th/9810210 [D-H-V-W] Dixon, L.J., Harvey, J., Vafa, C., Witten, E.: Strings on orbifolds II. Nucl. Phys. B 274, 285–314 (1986) [Di] Dixon, L.J.: Some world-sheet properties of superstring compactifications, on orbifolds and otherwise. Lectures given at the 1987 ICTP Summer Workshop in High Energy Physics and Cosmology (Trieste, June 29–August 7) [Do] Douglas, M.R.: Enhanced gauge symmetry in M(atrix) theory. JHEP 9707:004 (1997); hep-th/9612126 [E-O-T-Y] Eguchi, T., Ooguri, H., Taormina, A., Yang, S.-K.: Superconformal Algebras and String Compactification on Manifolds with SU(n) Holonomy. Nucl. Phys. B 315, 193–221 (1989) [E-T1] Eguchi, T., Taormina, A.: Extended superconformal algebras and string compactifications. Trieste School 1988: Superstrings, pp. 167–188 [E-T2] Eguchi, T., Taormina, A.: On the unitary representations of N = 2 and N = 4 superconformal algebras. Phys. Lett. 210, 125–132 (1988) [F-K-S] Fuchs, J., Klemm, A., Schmidt, M.G.: Orbifolds by cyclic permutations in Gepner type superstrings and in the corresponding Calabi–Yau manifolds. Ann. Phys. 214, 221–257 (1992) [F-K-S-S] Fuchs, J., Klemm, A., Scheich, Ch., Schmidt, M.G.: Spectra and symmetries of Gepner models compared to Calabi–Yau compactifications. Ann. Phys. 204, 1–51 (1990) [Ge1] Gepner, D.: Exactly solvable string compactifications on manifolds of SU (N ) holonomy. Phys. Lett. B 199, 380–388 (1987) [Ge2] Gepner, D.: Space-time supersymmetry in compactified string theory and superconformal models. Nucl. Phys. B 296, 757–778 (1988) [G-O] Goddard, P., Olive, D.: Kac–Moody and Virasoro algebras in relation to quantum physics. Int. J. Mod. Phys. A 2, 303–414 (1986) [G-P] Greene, B.R., Plesser, M.R.: Duality in Calabi–Yau moduli space. Nucl. Phys. B 338, 15–37 (1990) [G-V-W] Greene, B.R., Vafa, C., Warner, N.: Calabi–Yau manifolds and renormalization group flows. Nucl. Phys. B 324, 371–390 (1989) [H-V] Hamidi, S., Vafa, C.: Interactions on orbifolds. Nucl. Phys. B 279, 465–513 (1987) [In] Inose, H.: On certain Kummer surfaces which can be realized as non-singular quartic surfaces in P3 . J. Fac. Sci. Univ. Tokyo Sec. IA 23, 545–560 (1976) [Ko] Kodaira, K.: On the structure of compact complex analytic surfaces. I. Am. J. Math. 86, 751–798 (1964) [K-O-P] Kiritsis, E., Obers, N., Pioline, B.: Heterotic/type II triality and instantons on K(3). JHEP 0001:029, (2000); hep-th/0001083

138

[K-S] [K-T] [L-M] [L-P] [L-V-W] [Mu] [Na] [Ni1] [Ni2] [Ni3] [N-S] [O-P1] [O-P2] [P-S] [Qi] [R-W] [R-Y] [Sei] [Se1] [Se2] [Sha] [Shi] [Si] [S-W1] [S-W2] [Ta] [Wa] [We] [Wi1] [Wi2] [Wi3] [Wo] [Za]


Kobayashi, K.-I., Sakamoto, M.: Orbifold-compactified models in torus-compactified string theories. Z. Phys. C 41, 55–63 (1988) Kobayashi, R., Todorov, A.N.: Polarized period map for generalized K3 surfaces and the moduli of Einstein metrics. Tôhoku Math. J. 39, 341–363 (1987) Lawson, H.B., Michelsohn, M.-L.: Spin Geometry. Princeton, NJ: Princeton University Press, 1989 Looijenga, E., Peters, C.: Torelli theorems for K3-surfaces. Compos. Math. 42, 145–186 (1981) Lerche, W., Vafa, C., Warner, N.P.: Chiral rings in N = 2 superconformal theories. Nucl. Phys. B 324, 427–474 (1989) Mukai, S.: Finite groups of automorphisms of K3 surfaces and the Mathieu group. Invent. Math. 94, 183–221 (1988) Narain, K.S.: New heterotic string theories in uncompactified dimensions < 10. Phys. Lett. B 169, 41–46 (1986) Nikulin, V.V.: On Kummer Surfaces. Math. USSR Isv. 9, 261–275 (1975) Nikulin, V.V.: Finite automorphism groups of Kaehler K3 surfaces. Trans. Mosc. Math. Soc. 38, 71–135 (1980) Nikulin, V.V.: Integral symmetric bilinear forms and some of their applications. Math. USSR Isv. 14, 103–167 (1980) Nagura, M., Sugiyama, K.: Mirror symmetry of K3 and torus. Int. J. Mod. Phys. A 10, 233–252 (1995); hep-th/9312159 Obers, N., Pioline, B.: Eisenstein series and string thresholds. Commun. Math. Phys. 209, 275– 324 (2000); hep-th/9903113 Obers, N., Pioline, B.: Eisenstein series in string theory. Classical Quantum Gravity 17, 1215– 1224 (2000); hep-th/9910115 Periwal, V., Strominger, A.: Kähler geometry of the space of N = 2 superconformal field theories. Phys. Lett. B 235, 261–267 (1990) Qiu, Z.: Modular invariant partition functions for N = 2 superconformal field theories. Phys. Lett. B 198, 497–502 (1987) Ramgoolam, S., Waldram, D.: Zero branes on a compact orbifold. JHEP 9807:009 (1998); hep-th/9805191 Ravanini, F., Yang, S.-K.: Modular invariance in N = 2 superconformal field theories. Phys. Lett. B 195, 202–208 (1987) Seiberg, N.: Observations on the moduli space of superconformal field theories. Nucl. Phys. B 303, 286–304 (1988) Sen, A.: (2, 0) supersymmetry and space-time supersymmetry in the heterotic string theory. Nucl. Phys. B 278, 289–308 (1986) Sen, A.: Heterotic string theory on Calabi–Yau manifolds in the Green–Schwarz formalism. Nucl. Phys. B 284, 423–448 (1987) Shankar, R.: Some novel features of the Gross–Neveu model. Phys. Lett. 92B, 333–336 (1980) Shioda, T., Inose, H.: On singular K3 surfaces. In: Bailey, W.L., Shioda, T. (eds), Complex Analysis and Algebraic Geometry. Cambridge: Cambridge Univ. Press, 1977, pp. 119–136 Simons, J.: On transitivity on holonomy systems. Ann. Math. 76, 213–234 (1962) Schellekens, A., Warner, N.: Anomalies and modular invariance in string theory. Phys. Lett. B 177, 317–323 (1986) Schellekens, A., Warner, N.: Anomaly cancellation and selfdual lattices. Phys. Lett. B 181, 339–343 (1986) Taormina, A.: The N = 2 and N = 4 superconformal algebras and string compactifications. Islamabad 1989, Proc. Mathematical Physics. Walton, M.: The Heterotic string on the simplest Calabi–Yau manifold and its orbifold limits. Phys. Rev. D 37, 377–390 (1988) Wendland, K.: Consistency of orbifold conformal field theories on K3. hep-th/0010281 Witten, E.: Elliptic genera and quantum field theory. Commun. Math. Phys. 109, 525–536 (1987) Witten, E.: The index of the Dirac operator in loop space. In: Landweber, P. (ed), Elliptic curves and modular forms in algebraic geometry. Berlin–Heidelberg–New York: SpringerVerlag, 1988, pp. 161–181 Witten, E.: String theory dynamics in various dimensions. Nucl. Phys. B 443, 85–126 (1995); hep-th/9503124 Wolf, J.A.: Complex homogeneous contact manifolds and quaternionic symmetric spaces. J. Math. Mech. 14, 1033–1047 (1965) Zamolodchikov, A.B.: “Irreversibility” of the flux of the renormalization group in a 2-D field theory. JETP Lett. 43, 730–732 (1986)

Communicated by R.H. Dijkgraaf

Commun. Math. Phys. 216, 139 – 177 (2001)

Communications in



On the Critical Capacity of the Hopfield Model Jianfeng Feng1 , Mariya Shcherbina2 , Brunello Tirozzi3 1 Laboratory of Neurocomputation, The Babraham Institute, Cambridge, CB2 4AT, UK.


2 Institute for Low Temperature Physics, Ukr. Ac. Sci., 47 Lenin ave., Kharkov, Ukraine.


3 Department of Physics of Rome University “La Sapienza”, 5, p-za A. Moro, Rome, Italy.

E-mail: [email protected] Received: 1 December 1999 / Accepted: 21 July 2000

Abstract: We estimate the critical capacity of the zero-temperature Hopfield model by using a novel and rigorous method. The probability of having a stable fixed point is one when α ≤ 0.113 for a large number of neurons. This result is an advance on all rigorous results in the literature and the relationship between the capacity α and retrieval errors obtained here for small α coincides with replica calculation results. 1. Introduction and Main Results The Hopfield model is one of the most important models in the theory of spin glasses and neural networks [H, M-P-V]. It has been intensively investigated in the past few years (see e.g. book [M-P-V] and references therein). One of the main problems is the critical capacity which has been studied by means of the replica trick [A,A-G-S]. Here the value αc = 0.138 . . . (coinciding also with numerical experiments) was found. But this result is nonrigorous from the mathematical point of view. There are few rigorous approaches in the literature to estimate the critical capacity of the Hopfield model [N, L, T]. Here we introduce a novel approach based upon analysis of the Fourier transform of the joint distribution of the effective fields. It enables us to obtain a new bound for the critical capacity and also allows us to prove rigorously, for small α, the results obtained in terms of the extreme value theory [F-T]. Consider the sequential dynamics of the Hopfield model in the form N ˜ σk (t + 1) = sign Jkj σj (t) , (1.1) j =1,j =k

where, as usual, p+1 1 µ µ J˜j k = ξ˜j ξ˜k , N µ=1

p → α, N

as

N → ∞,

(1.2)

140

J. Feng, M. Shcherbina, B. Tirozzi µ

and ξ˜k (j = 1, . . . , N ), (µ = 1, . . . , p + 1) are i.i.d. random variables assuming values ±1 with probability 21 . This dynamical system is determined by the energy function N

H(σ ) = −

1 ˜ Jj k σj σk , 2

(1.3)

j =k

where we denote σ ≡ (σ1 , . . . , σN ). It is easily seen that the function H(σ ) does not increase in the process of evolution. Thus, the dynamics of the model depends on the “energy landscape” of the function H(σ ) and the local minima of the function are the fixed points of dynamics (1.1). Newman [N] was the first, who proved, that for α ≤ 0.056., an “energy barrier” exists with probability 1 around every point σ µ = µ µ ξ µ ≡ (ξ˜1 , . . . , ξÑ ), i.e. there exist some positive numbers δ and ε, such that for any σ , belonging to µ δ ≡ {σ : ||σ − ξ µ ||2 = 2[δN ]}, the following inequality holds: H(σ ) − H(ξ µ ) ≥ εN (here and below the norm || . . . || corresponds to the usual scalar product (. . . , . . . ) in RN ). In other words, it means that minµ H(σ ) − H(ξ µ ) ≥ ε2 N.

(1.4)

σ ∈δ

This result was improved by Loukianova [L], who proved the existence of the “energy barriers” for α ≤ 0.071 and then by Talagrand [T]. One can show, that if such a “barrier” exists, then inside each open ball µ

Bδ ≡ {σ : ||σ − ξ µ ||2 < 2[δN ]} there exists a point of local minimum of the function H(σ ), which, as it was mentioned above, is the fixed point of dynamics (1.1). Thus, it is clear that the point σ ∗ in which H(σ ∗ ) = minσ ∈µ H(σ ) plays an important δ

role in dynamics (1.1). We shall study the probability of the event, that the point σ (1,δ) ∈ 1δ with (1,δ)

σk

= −ξ˜k1 , (k = 1, . . . , [δN]),

(1,δ)

σk

= ξ˜k1 , (k = 1 + [δN ], . . . , N )

(1.5)

is a local minimum of the function H(σ ) on 1δ . This means that H(σ (1,δ) ) must be less than the value of H(σ ) for any σ ∈ 1δ which is the “nearest neighbor” of σ (1,δ) in 1δ . It is easy to see that, it is so if and only if for any k = 1, . . . , [δN ] and j = [δN ]+1, . . . , N, (1,δ) (1,δ) σk

−2J˜kj σj

(1,δ)

+ σk

N i=1,i=k

(1,δ) (1,δ) + σj J˜ki σi

N i=1,i=j

(1,δ) ≥ 0. J˜j i σi

It is useful to introduce at this point the definition of “effective fields”.

(1.6)

On the Critical Capacity of the Hopfield Model

141

Definition 1. The effective fields generated by the configuration σ on the neuron k is zk ≡ σk

N

J˜ki σi .

i=1,i=k

Our approach is based on the analysis of the joint probability distribution of the variables zk (k = 1, . . . , N). 2 Since with probability larger than 1 − e−N const ε˜ all matrix elements J˜kj satisfy the inequality |J˜kj | ≤

ε˜ 2

(k, j = 1, . . . , N ),

(1.7)

one can derive from (1.6) that, if we denote by x˜k0 the effective fields, generated by the configuration σ (1,δ) (1,δ)

x˜k0 = σk

N i=1,i=k

(1,δ) , J˜ki σi

(1.8)

the necessary condition for σ (1,δ) to be a local minimum point is min

k=1,...,[δN]

x˜k0 +

min

j =[δN]+1,...,N

x˜j0 ≥ −˜ε ,

(1.9)

and the sufficient condition has the same form with +˜ε in the r.h.s. Thus, if we consider the events A0k (q) = {x˜k0 ≥ q},

(1.10)

then the event M that σ (1,δ) is a local minimum point satisfies the relations 0 N 0 ∪q+q ≥˜ε ∩[δN] k=1 Ak (q) ∩k=[δN]+1 Ak (q ) ⊂ M 0 N 0 ⊂ ∪q+q ≥−˜ε ∩[δN] . A (q) ∩ A (q k k=[δN]+1 k k=1

(1.11)

So we should study the behaviour of 0 N 0 A (q) ∩ A (q ) . PN (q, q ) ≡ Prob ∩[δN] k k=[δN]+1 k k=1

(1.12)

Observe that, in particular, PN (0, 0) is the probability to have a fixed point of dynamics (1.1) at the point σ (1,δ) . Now let us introduce the new notation: µ

(1,δ) ˜ µ+1 ξk ,

ξ k ≡ σk

(µ = 1, . . . p, k = 1, . . . N ).

(1.13)

µ

Then ξk (k = 1, . . . , N), (µ = 1, . . . , p) are also i.i.d. random variables assuming the values ±1 with probability 21 . Denote x˜k =

p N 1 µ µ ξk ξj = x˜k0 + αN ± (1 − 2δN ), N µ=1 j =1

αN =

p+1 , N

δN =

[δN ] . N (1.14)

142

J. Feng, M. Shcherbina, B. Tirozzi

Here αN appears because we include in the summation the term with j = k, the term ±(1 − 2δN ) is due to the term N −1 (ξ 1 , σ (1,δ) ), and the sign here depends on k: it is plus for k = 1, . . . , [δN] and minus for k = [δN ] + 1, . . . N. To simplify formulae we introduce also a1 ≡ αN + 1 − 2δN + q → a1∗ ,

a1∗ ≡ α + 1 − 2δ + q,

(1.15)

a2 ≡ αN − 1 + 2δN + q → a2∗ , a2∗ ≡ α − 1 + 2δ + q , which yield PN (q, q ) ≡

[δN] k=1

θ(x˜k − a2 ) .

N

θ(x˜k − a1 )

(1.16)

k=1+[δN] µ

Here and below the symbol . . . denotes averaging with respect to all {ξk } (k = 1, · · · , N, µ = 1, · · · , p + 1). In order to formulate the main results of the paper we need some other definitions. Consider the function F0 (U, V ; α, δ, q, q ) of the form a∗

a∗ − V + (1 − δ) log H 2 − V U U 1 2 − U V + V + α log U, 2

F0 (U, V ; α, δ, q, q ) ≡ δ log H

1

where 1 H (x) ≡ √ 2π

x

∞

e−t

2 /2

dt.

(1.17)

(1.18)

Define also e−x /2 d log H (x) = √ , dx 2π H (x) ∗ 1 a1,2 −V , A1,2 (U, V ) ≡ A U U 1 D(U, V ) ≡ − δA1 (U, V ) − (1 − δ)A2 (U, V ) 2 1 − δ(1 − δ)(A1 (U, V ) − A2 (U, V ))2 , 2 2

A(x) ≡ −

and

(1.19)

(1.20)

 F (U, V ; α, δ, q, q ), if D(U, V ) ≥ 0   0 a∗   1   δ log H 1 − V  U 1 − 2D(U, V ) F0D (U, V ; α, δ, q, q ) ≡ a∗ V2  2   +(1 − δ) log H − V − UV + + α log U,   U 2   if D(U, V ) < 0. (1.21)


143

Theorem 1.

1 θ (x˜k − a1 ) log N [δN]

lim sup N→∞

k=1

N

θ(x˜k − a2 )

k=1+[δN]

≤ max min F0D (U, V ; α, δ, q, q ) − U >0 V

α α log α + . 2 2

(1.22)

Remark 1. Note that in all interesting cases (see Theorems 2 and 3 below) max min F0D (U, V ; α, δ, q, q ) = max min F0 (U, V ; α, δ, q, q ) U >0 V

U >0 V

and one can substitute F0D by F0 in the r.h.s. of (1.22). Remark 2. The proof of Theorem 1 can be generalized almost literally to the case ( cf. (1.16)) PN,[δ1 N] (q, q ) ≡

[δN]

θ(x˜k − a1 )

k=1+[δ1 N]

N

θ(x˜k − a2 ) .

(1.23)

k=1+[δN]

We obtain lim sup N→∞

1 log PN,[δ1 N] (q, q ) N ≤ max min F1D (U, V ; α, δ, δ1 , q, q ) − U >0 V

α α log α + , 2 2

(1.24)

with (cf. (1.17)–(1.21))  F1 (U, V ; α, δ, δ if D 1 (U, V ) ≥ 0; 1 , q, q ),    a1∗  1   −V ) log H (δ − δ 1  1 1 − 2D (U, V ) U F1D (U, V ; α, δ, δ1 , q, q ) ≡ a2∗ 1    +(1 − δ) log H − V − U V + V 2 + α log U,   U 2   if D 1 (U, V ) ≤ 0; (1.25) where a1∗

−V (1.26) U ∗ a 1 + (1 − δ) log H 2 − V − U V + V 2 + α log U, U 2

F1 (U, V ; α, δ, δ1 , q, q ) ≡ (δ − δ1 ) log H

and D 1 (U, V ) ≡ (1 − δ1 )−1

1

− (δ − δ1 )A1 (U, V ) − (1 − δ)A2 (U, V ) 1 − (δ − δ1 )(1 − δ)(A1 (U, V ) − A2 (U, V ))2 ) 2 2

with A1,2 (U, V ) defined in (1.20).

(1.27)

144


Theorem 2. If α is small enough, δ 0 and some point σ 0 ∈ Bδ1 , such that minσ ∈1 H(σ ) − H(σ 0 ) > ε2 N . δ Then if for some α and δ max max min{F0D (U, V ; α, δ, q, −q)} − 0≤q

U

V

α α log α + + C ∗ (δ) < 0, 2 2

(1.29)

then there exists some C(α) > 0 such that Prob{A} ≤ e−NC(α) .

(1.30)

C ∗ (δ) ≡ −δ log δ − (1 − δ) log(1 − δ).

(1.31)

Here and below

Numerical calculations show that condition (1.29) is fulfilled for any α ≤ αc = 0.113 . . . . The paper is organized as follows. In Sect. 2 we prove Theorems 1, 2 and 3. In the process of the proof we shall need some auxiliary facts which we formulate there as Lemmas 1–4 and Propositions 1–4. Section 3 is devoted to the proof of the auxiliary results.


145

2. Proof of Main Results Proof of Theorem 1. To make the idea of the proof more understandable we first carry µ out all computations when {ξj } are Gaussian random variables. Since this part has no connection with the rigorous proof of Theorem 1, we just sketch the proof, without going into details. g To find PN which corresponds to PN (see (1.16)) in the Gaussian case, we study the Fourier transform of the joint probability distribution of the variables x˜k , p x˜k ζk F (ζ1 , . . . , ζN ) ≡ (2π)−N/2 exp i

= (2π)−N/2 exp i = (2π)−N/2

p

k=1 p

N

µ=1

k=1

N −1/2

µ v˜ µ

ei u˜

µ

ξ k ζk

N −1/2

N j =1

µ

ξj

(2.1)

,

µ=1

where we use notations u˜ µ ≡ N −1/2

N k=1

µ

ξ k ζk ,

v˜ µ ≡ N −1/2

N j =1

µ

ξj .

(2.2)

It is easy to see that µ v˜ µ

ei u˜

= (2π )−1

µu ˜ µ +v µ v˜ µ )

duµ dv µ ei(u

µ vµ

e−iu

.

(2.3)

Thus, using the inverse Fourier transform for the function F (ζ1 , . . . , ζN ), we get g

PN = =

1 (2π)N/2

N

N

k=1

1 (2π)(N+p)

θ (xk − ak )dxk

p

N dζj exp − i xk ζk F (ζ1 , . . . , ζN )

j =1 µ vµ

e−iu

duµ dv µ

µ=1

N

k=1

dxk θ(xk − ak )

k=1

p dζk exp − iζk xk + i(uµ u˜ µ + v µ v˜ µ )

1 = (2π)N+p

p µ=1

µ=1

e

−iuµ v µ

µ

du dv

µ

N k=1

dxk θ(xk − ak )

dζk e−iζk xk

µ 2

p p p e−(ξk ) /2 µ µ uµ ξk ζk + N −1/2 v µ ξk exp i N −1/2 √ 2π µ=1 µ=1 µ=1

146


1 = (2π)(N+p)

=

p

dζk · e−iζk xk

e

µ=1 p

µ=1 p

−iuµ v µ

µ

du dv

µ

N

dxk θ(xk − ak )

k=1

(uµ ζ + v µ )2 k exp − 2N

(2.4)

(v µ )2 µ µ µ µ exp − iu v − du dv N 2 (2π)( 2 +p) µ=1 p N (ixk + N −1 µ=1 uµ v µ )2 θ (xk − ak ) , dxk exp U 2U 2 1

k=1

where U ≡ (N −1 g PN

p

µ=1 (u

−p

= (2π) N

µ )2 )1/2 .

p

H

µ

du dv

µ=1

ak

Therefore we have

− iN −1

p

µ

µ=1

exp

µ=1

uµ v µ

p 1 µ 2 u v − (v ) 2 µ µ

µ=1

(2.5)

.

U

k=1

−i

p

p

Now let us fix u = {uµ }µ=1 and change variables in the integral with respect to v = p {v µ }µ=1 , 1 v1 = √ (e1 , v), v2 = (e2 , v), . . . , vp = (ep , v), N

(2.6)

√ p µ where {ei }i=1 is the orthonormal system of vectors in Rp such that e1 = (U N )−1 uµ . Then, integrating with respect v2 , . . . , vp , we obtain g

p

N dv1 exp − iN U v1 − (v1 )2 2 µ=1 a1 a2 + [N δ] log H ( − iv1 ) + (N − [N δ]) log H ( − iv1 ) . U U

PN = (2π)−(p−1)/2

duµ

(2.7)

Using the spherical coordinates in the integral with respect to u and integrating with respect to angular variables, we get g PN

= /(p) 0

∞

dU

dv1 exp{(p − 1) log U − iN U v1 −

N (v1 )2 2

a1 a2 + [N δ] log H ( − iv1 ) + (N − [N δ]) log H ( − iv1 )}. U U

(2.8)

Let V (U ) be the point of minimum with respect to V of the function F0 (U, V ) defined by (1.17). Let us change the path of integration with respect to v1 in (2.8) from the real


147

axis to the line L which is parallel to it, but contains the point z = −iV (U ). Then, following the saddle point method, we divide the integral into two parts

∞

g dU + dt exp (p − 1) log U PN = /(p) |t|>N −1/3

0

|t|≤N −1/3

N N (V (U ))2 − iN U t − t 2 2 2 a1 a2 + [N δ] log H ( − V (U ) − it) + (N − [N δ]) log H ( − V (U ) − it) . U U (2.9) − N U V (U ) +

Due to the simple inequality |H (a + ic)| ≤ H (a)ec

2 /2

,

(2.10)

valid for any real numbers a and c, we conclude, that the second integral is o(1) exp{N F0 (U, V ; α, δ, q, q )}. Replacing in the first integral F0 (U, V (U ) − it) by its Taylor expansion up to the second order term (the first order term is zero due to the choice V (U )) and then performing the Gaussian integration, we see that

∞ g dU exp{N (F0 (U, V (U ); δ, q, q ) + o(1))}. (2.11) PN ≤ /(p) 0

Applying the standard Laplace method, we conclude that for the Gaussian random variµ ables ξk Eq. (1.22) can be replaced by the following stronger statement: lim sup N→∞

1 α α g log PN = max F0 (U, V (U ); δ, q, q ) − log α + . U >0 N 2 2

The difference of non-Gaussian case from the Gaussian one is that we have, in the p p µ ζ +v µ µ +v µ )2 k k instead of µ=1 exp − (u ζ2N . To replace sixth line of (2.4), µ=1 cos u √ N the former term by the latter one we have to estimate the difference between them for different u, v and ζ . To this end we introduce some smoothing factors in the integration (2.4). Lemma 1. N

k=1

∗ −1/2 2 p/2 θ (x˜k − ak ) ≤ PN1 lN (1 − e−h /2λ )−N eNo(1) + e−constN(εN ) ,

where PN1 ≡

1 ∗ (u, u) ∗ (v, v) dudv exp − ilN (u, v) − εN − εN N+p (2π) 2N 2N

p N uµ ζk + v µ 2 · dζk χˆ N,h (ζk )e−λζk /2−iak ζk cos , √ N µ=1 k=1

∗ = (log log N )−1 , l ≡ εN N

1 2

χˆ N,h (ζ ) =

+

1 2

∗ )2 , λ is a fixed positive number and 1 − 4(εN

2 N 1/2+d + 2h N 1/2+d sin ζ exp − iζ ζ 2 2

(2.12)

148


is the complex conjugate of the Fourier transform of χN,h (x) – the characteristic function 2 of the interval (−h, N 1/2+d + h) with some positive d and h > ( 2λ π ) . Here and below p p 1 p 1 p µ v = (v , . . . , v ), u = (u , . . . , u ), dv = µ=1 dv and du = µ=1 duµ . ∗ → 0 as slowly as we want, we can even fix ε ∗ = ε Remark 4. In fact we can take εN N with ε being small enough. However, in this case we have to be more careful to control the constants which will appear in our estimates.

Now we start to prove Theorem 1. Denote FN,k (u, v) = F˜ (u, v) =

1 2π

dζk χˆ N,h (ζk )e−λζk /2−iak ζk 2

p

cos

µ=1

uµ ζk + v µ ; √ N

(2.13)

FN,k (u, v).

k

To simplify formulae in the places where it is not important, we confine ourselves to the case ak = a. Since in this case all FN,k (u, v) are identical, we could omit the index k. To replace the product term of cos in Eq. (2.13) by the exponent we modify a method originally proposed by Lyapunov. He employed it to prove that the distribution of the sum of independent variables uniformly converges to the normal distribution (see [Lo]). To ensure the method to work, the second and the third moments of the random variables µ must be bounded. Since in our setting the random variables have the form uµ ξk and µ µ µ 2,3 µ 2,3 v ξk and their moments coincide with |u | and |v | , we need to remove large |uµ | and |v ν | in the integrals. For this purpose we take εN = (log N )−1 and denote √ √ 2 χεN (uµ , v µ ) = θ (εN N − |uµ |)θ (εN N − |v µ |). (2.14) Note that the different powers of εN in the θ -functions for u and v are necessary in our estimates below. Rewrite

∗ ε∗ εN 1 N 1 −ilN (u,v) ˜ F (u, e v) exp − v) − u) dudv (v, (u, PN = (2π)p 2 2

p p m = Cpm dudv (1 − χεN (uµ , v µ )) χεN (uν , v ν ) (2.15) m=0

·e

µ=1

ε∗ − 2N

(u,u)

e

ε∗ − 2N

(v,v) −ilN (u,v)

e

ν=m+1 p

F˜ (u, v) ≡

m=0

Cpm Im .

Let us first estimate Im in the above equation 1 |Im | ≤ (2π)p ·

p ν=m+1

dudv

m

(1 − χεN (uµ , v µ ))

µ=1

χεN (uν , v ν )e

ε∗ − 2N

(u,u)

e

ε∗ − 2N

(v,v)

N k=1

(2.16) dζk |χˆ N,h (ζk )|e

−λζk2 /2

.


149

Now, using the bound

2 N 1/2+d −λζk2 /2 −λζk2 /2 = dζk sin ζk dζk |χˆ N,h (ζk )|e e ζk 2 ≤ const log N, we arrive at

(2.17)

∗ 4

∗ −p −mNεN εN /2 ) e . |Im | ≤ e const N log log N (εN

Thus, p Cpm Im ≤ e− const N log log N ,

(2.18)

m=m0

∗ )−1 ε −4 log log N . In the following it would be more where m0 = [(log N )5 ] >> (εN N convenient to have the integration with respect to u1 , . . . , um and v 1 , . . . , v m in the 0 m whole R. Therefore, we perform the first product in (2.15) and rewrite m m=0 Cp Im in the form m0 m=0

Cpm Im =

m0

C˜ m I˜m ,

(2.19)

m=0

where I˜m ≡

1 (2π)p

dudv

p

χεN (uµ , v µ )e−

∗ εN 2

(u,u) −

e

∗ εN 2

(v,v) −ilN (u,v)

e

F˜ (u, v) (2.20)

ν=m+1

and C˜ m are some combinatorial coefficients. These coefficients are not important, because for our choice of m (m ≤ m0 = o(N )) all of them are of the order eo(N) and after taking the logarithm and dividing by N give us o(1)-terms. Thus, we have PN1 =

m0

C˜ m I˜m + O(e− const N log log N ).

(2.21)

m=0

To proceed further we define (v , v ) 2 2 F (m) (u1 , v 1 ; u2 , v 2 ) ≡ exp − 2N a − h − i (u2 ,v 2 ) − (u√1 ,ξ 1 ) (v 1 , ξ ) N N · HN,h,U˜ exp i √ 1 N U˜ 2 + λ

m uµ ζ + v µ 2 cos √ = dζ χˆ N,h (ζ )e−λζ /2−iaζ (2.22) N µ=1 1 1 1 exp − (v 2 , v 2 ) − (u2 , v 2 )ζ − (u2 , u2 )ζ 2 , 2N N 2N where

∞ 1/2+d 1 + 2h 1 N (2.23) θ − t exp − (t + x)2 dt. HN,h,U˜ (x) = √ 2 2π 0 U˜ 2 + λ

150


Here and below u1 ≡ (u1 , . . . , um ) and v 1 ≡ (v 1 , . . . , v m ), u2 ≡ (um+1 , . . . , up ), v 2 ≡ (v m+1 , . . . , v p ), so that u = {u1 , u2 }, v = {u1 , u2 }, ξ 1 ≡ (ξ11 , . . . , ξ1m ) is the random vector with independent components, assuming values ±1 with probability 21 , . . . means the average with respect to ξ 1 and U˜ ≡ [ N1 (u2 , u2 )]1/2 . Expression (2.22) p is obtained from (2.13) by changing cos in the product µ=m+1 by the correspondent exponent and then by integration with respect to ζk . The main technical tool at this step is a lemma, which is a modification of the Lyapunov theorem. √ Lemma 2. For any u2 , v 2 , λ2 such that |uν |, |v ν |, |λν | ≤ εN N and any u1 , v 1 , λ1 the function R (m) (u1 , w 1 ; u2 , w 2 ) ≡ FN (u1 , w 1 ; u2 , w 2 ) − F (m) (u1 , w 1 ; u2 , w 2 ) admits the bound (λ2 , λ2 ) ˜ 2 2 |R (m) (u1 , w 1 ; u2 , w 2 )| ≤ const εN 1+ (U + λ)1/2 N λ(v 2 , v 2 ) (λ, λ) (λ, λ) −4 · exp − + + exp − const εN . + N N 4N (U˜ 2 + λ)

(2.24)

Here and below w ≡ v + iλ. This lemma allows us to replace in our formulae FN by F (m) in the following sense. Let us write I˜m ≡

1 (2π)p

p

dudv

χεN (uµ , v µ ) exp{−ilN (u, v)}

ν=m+1

· (F (m) (u1 , v 1 , u, v) + R (m) (u1 , v 1 , u2 , v 2 ))N e− ≡

N k=0

∗ εN 2

(v,v) −

e

∗ εN 2

(u,u)

(2.25)

k CN Im,k ,

where Im,k

1 ≡ (2π)p

dudv

p

χεN (uµ , v µ )e−ilN (u,v) (F (m) (u1 , v 1 , u, v))N−k

ν=m+1

· (R (m) (u1 , v 1 , u2 , v 2 ))k e−

∗ εN 2

(v,v) −

e

∗ εN 2

(u,u)

.

−1 ] Lemma 3. For k > k0 ≡ [N log−1/2 εN −1 ∗ −2p ) exp{−kconst log εN }. |Im,k | ≤ eNconst (εN )2k (εN


151 1/2 −1

Thus, we get that for k > k0 Im,k have the order e−N const log εN and so we can neglect these terms in (2.25). Now we shall study the leading terms in the r.h.s. of Eq. (2.25) (Im,k with k < k0 ). In fact, the next step is a version of the saddle point method (cf.(2.8)–(2.11)). Let us take any real fixed√V and √ change the path of integration w.r. to v 2 from the product of intervals (−εN N , εN N ) to the product of the paths Lν1 ∪ Lν2 , with √ √ √ √ ν ν ν Lν1 = (−εN N − iV˜u , εN N − iV˜u ) and Lν2 = (−εN N , −εN N − iV˜u ) ∪ U U U √ √ ν (εN N − iV˜u , εN N ) (ν = m + 1, . . . N). It can be done, since all our functions are U analytical w.r.to v ν , Then take any real λµ , such that (λ1 , λ1 ) ≤ N const and choose the paths of integration with respect to v 1 as Lµ = {w µ = t µ − iλµ , t µ ∈ R}. Finally, we get Im,k

p−m 1 n = Cp−m du1 dw 1 p−n dw 3 m ν µ (2π)p µ=1 L ν=m+1 L1 n=1

ε 2 √N

N ∗ (w,w)/2 −ε ∗ (u,u)/2 −εN · p dw 4 e N √ du3 du4 e ν ν=p−n+1 L2

·e ≡

−ilN (u,w)

p−m n=1

(F

2 N −εN

(m)

(u, w))

N−k

(R

(m)

(u, w))

(2.26)

k

n Cp−m Im,k,n .

Here and below u = {u1 , u3 , u4 }, w = {w 1 , w 3 , w 4 }, where u1 , w 1 are the same as before and we divide vectors w 2 and u2 in two sub-vectors u2 = {u3 , u4 }, w 2 = {w 3 , w4 } in such a way that u4 , w 4 include the last n components of u2 and w2 respectively. Now let us get rid of Im,k,n with sufficiently large n. Similarly to the proof of Lemma 3 on the basis of Lemma 2, we get ∗ −p − const nNεN |Im,k,n | ≤ eN const (εN ) e exp{(λ1 , λ1 ) + N V 2 }. 2

(2.27)

−5/2

So, taking n > n0 = [εN ], on the basis of (2.27) one can conclude that we need to study only the first n0 terms in (2.26). We remark that starting from this moment, we shall distinguish the terms with a1 and a2 . Denote

a − h − V U − N −1/2 (u , ξ ) (λ , ξ ) δ 1 1 1 1 G∗m (U, V , u1 , λ1 ) ≡ H exp √ √ 1 N U2 + λ

a − h − V U − N −1/2 (u , ξ ) (λ , ξ ) 1−δ 2 1 1 1 · H( exp √ √ 1 N U2 + λ l 1 N · exp − (u1 , λ1 ) − lN U V + V 2 . N 2 (2.28) Lemma 4. Let Gm,k,n (V , u1 , λ1 , u3 ) be the function which we get, if in (2.26) integrate with respect to w 1 , w 3 , u4 and w 4 . Then

152


|Gm,k,n (V , u1 , λ1 , u3 )| ≤ (2π)−p/2 (G∗m (U, V , u1 , λ1 ))N e−

∗ εN 2

(u1 ,u1 )+No(1)

.

(2.29)

Here and below U = [N −1 (u3 , u3 )]1/2 , so that U˜ 2 = U 2 + N −1 (u4 , u4 ). Once we have an upper bound for Gm,k,n we can estimate all the I˜m in (2.21). Let us study first the term with m = 0. Consider the function Fλ,h (U, V ) ≡ δ log H

a∗ − h − V U a∗ − h − V U 1 + (1 − δ) log H 2√ √ U2 + λ U2 + λ 1 2 −U V + V . 2

(2.30)

Let V (U ) be chosen from the condition F0 (U, V (U ); α, δ, q, q ) = min F0 (U, V ; αδ, q, q ). V

(2.31)

The function Fλ,h (U, V (U )) and the functions which appear in the exponent of (2.29) for m = 0 satisfy the inequalities of the type Fλ,h (U, V (U )) ≤ α log U −

U2 2

∗ and l → 1 (it follows from log H (x) ≤ 0 and V (U ) ≤ U ). Thus, since a1,2 → a1,2 N as N → ∞, on the basis of (2.29) for m = 0, we get

−p/2 ˜ |I0 | ≤ (2π) du3 exp{N [Fλ,h (U, V (U )) + o(1)]},

where I˜0 is defined by formula (2.20) for m = 0. Remark 5. Let us note that here we have use the following simple statement. If the continuous functions φ(U ), φN (U ) (N = 1, 2, . . . ) (U ∈ R+ ) satisfy the inequalities φ(U ), φN (U ) ≤ −C1 U 2 , U ≥ L, φ(U ), φN (U ) ≤ C2 log U, U ≤ ε,

(2.32)

with some positive C1 and ! C2 and φN (U ) → φ(U ),!as N → ∞, uniformly in each compact set in R+ , then exp{N φN (U )}dU = eo(N) exp{N φ(U )}dU . The proof of this statement is very simple, and we omit it. Below we shall use this remark without additional comments. Performing the spherical change of variables and using the Laplace method, we get now α α |I˜0 | ≤ exp{N [max Fλ,h (U, V (U )) + α log U − log α + + o(1)]}. (2.33) U 2 2 To study the terms with m = 0 we chose λ1 (U, V , u1 ) in such a way that G∗m (U, V , u1 , λ1 (U, V , u1 )) = min G∗m (U, V , u1 , λ1 ), λ1 ∈Rm

(2.34)


153

where the function G∗m is defined by (2.28). Then we use the inequality, which follows from the fact that (log H (x)) ≤ 0, H (x + y) ≤ H (x)e−A(x)y

(2.35)

with the function A(x) defined by (1.19). On the basis of this inequality we get m µ

a1,2 − h − V U − N −1/2 m uµ ξ µ ξ µ=1 1 λµ √1 exp H √ N U2 + λ µ=1 m µ

a −h−VU ξ 1,2 (λ,h) (A1,2 uµ + λµ ) √1 ) exp ≤ H( √ N U2 + λ µ=1

=H ≤H

(2.36)

(λ,h)

a

m A1,2 uµ + λµ −h−VU cosh √ √ N U2 + λ µ=1

a

m 1 −h−VU (λ,h) (A1,2 uµ + λµ )2 , exp √ 2N U2 + λ µ=1

1,2

1,2

where (λ,h)

A1,2

a

= (U 2 + λ)−1/2 A

−h−VU . √ U2 + λ

1,2

(2.37)

Thus, G∗m (U, V , u1 , λ1 (U, V , u1 ))|

a − h − UV 1 √ U2 + λ a − h − UV 1 2 + (1 − δ) log H − lN U V + V 2 √ 2 2 U +λ m δ (λ,h) + min (A1 uµ + λµ )2 (2.38) λµ 2N µ=1 m m lN µ µ 1−δ (λ,h) . (A2 uµ + λµ )2 − λ u + 2N N

≤ exp δ log H

µ=1

(λ,h)

µ=1

(λ,h)

Taking λµ = (1 − A1 δ − A2 (1 − δ))uµ , which give us the minimum of the expression in the r.h.s. of (2.38), we get a − h − UV 1 |G∗m (U, V , u1 , λ1 (U, V , u1 ))| ≤ exp N δ log H √ U2 + λ a − h − UV (2.39) 1 2 2 (λ,h) +(1 − δ) log H (U, V )(u1 , u1 ) , − UV + V − D √ 2 U2 + λ where D (λ,h) (U, V ) is defined by (1.20) if we substitute there A1,2 (U, V ) by (λ,h) A1,2 (U, V ). From (2.39) it is easy to see that if D (λ,h) (U, V ) ≥ 0, then

154


ε∗ du1 G∗m (U, V , u1 , λ1 (U, V , u1 )) exp − N (u1 , u1 ) 2 a∗ − h − U V a∗ − h − U V ≤ eNo(1) exp N δ log H 1√ + (1 − δ) log H 2√ U2 + λ U2 + λ 1 2 . (2.40) − UV + V 2 If D (λ,h) (U, V ) is negative, we use Proposition 1. If D (λ,h) (U, V ) < 0, λ and h are small enough, then ∗ du1 G∗ (U, V , u1 , λ1 (U, V , u1 )) exp − εN (u1 , u1 ) m 2 a∗ − h − U V δ 1 ≤ exp N log H √ 1 − 2D (λ,h) (U, V ) U2 + λ ∗ a2 − h − U V 1 2 1−δ + o(1) . − U V + V + log H √ 2 1 − 2D (λ,h) (U, V ) U2 + λ (2.41) Thus, on the basis of (2.39) and (2.41), we have got that for any n-independent finite V , ε∗ du1 G∗m (U, V , u1 , λ1 (U, V , u1 )) exp − N ((u1 , u1 ) + (u3 , u3 )) 2 D ≤ exp{N [Fλ,h (U, V ) + o(1)]}, D (U, V ) is defined by the expression in the exponent where for D (λ,h) (U, V ) < 0, Fλ,h in the r.h.s. of (2.41) and for D (λ,h) (U, V ) ≥ 0, it coincides with Fλ,h (U, V ). Then, choosing V to minimise this estimate for any U , we get

ε∗ du3 du1 G∗m (U, V , u1 , λ1 (U, V , u1 )) exp − N ((u1 , u1 ) + (u3 , u3 )) 2

α α D ≤ dU exp N[min Fλ,h (U, V ) + α log U − log α + + o(1)] . (2.42) V 2 2

Thus, for any m ≤ m0 = o(N ), D (U, V ) + α log U } − |I˜m | ≤ exp{N [max{min Fλ,h U

V

α α log α + + o(1)]}. 2 2

Hence, D (U, V ) + α log U } − PN ≤ exp{N [max min{Fλ,h U

V

α α log α + + o(1)]}. 2 2


155

Therefore, on the basis of Lemma 1, we have

1 log θ (x˜k − a1 ) N [δN]

lim sup N→∞

k=1

N

θ(x˜k − a2 )

k=1+[δN]

D ≤ max min{Fλ,h (U, V )} + α log U } − U

V

α α log α + + o(1). 2 2

We get the conclusions of Theorem 1, after taking the limits λ → 0 and then h → 0. Proof of Theorem 2. To prove Theorem 2 let us show that if α is small enough to satisfy the condition e− 2α < α 4 , 1

(2.43)

then a∗ − α a∗ − α 1 + (1 − δ) log H 2√ √ α α α α 2 −3 −1/α ) + log α − + O(δ α ) + O(e 2 2 √ √ = F0 ( α, α; α, δ, 0, 0) + O(δ 2 α −3 ) + O(e−1/α ). (2.44)

max min F0D (U, V ; α, δ, 0, 0) ≤ log H U

V

By virtue of the condition δ 2 α. √ √ Proposition 2. If U > 2 α, and V (U ) is defined by condition (2.31), then α ≤ V (U ) ≤ U . On the basis of Proposition 2, we get 1 F0D (U, V (U ); α, δ, 0, 0) ≤ α log U − V (U )U + (V (U ))2 2 (2.45) √ √ α α ≤ α log U − αU + ≤ α log 2 α − 2α + . 2 2 Here the first inequality is due to log H (x) ≤ 0, while the second and the third follow from Proposition 2. But, using the asymptotic formulae 1 2 H (x) = √ e−x /2 (1 + O(1/x 2 )) (x >> 1), x 2π 1 2 H (x) = 1 + √ e−x /2 (1 + O(1/x 2 )) (x 2 α.

(2.46)

(2.47)

156


√ Now let us check (2.44) for U < 0.5 α. To this end let us write an equation for V (U ) which follows from (2.31), α + 1 − 2δ α − (1 − 2δ) U = V + δA − V + (1 − δ)A −V , (2.48) U U where the function A(x) is defined by (1.19). By using asymptotic formulae

A(x) = x 1 + O

1 x

1 e−x /2 (x >> 1), A(x) = √ (x 0 and therefore for 0.5 α ≤ U ≤ 2 α, F0D (U, V (U ); α, δ, 0, 0) = min F0 (U, V ; α, δ, 0, 0). V

But max min F0 (U, V ; α, δ, 0, 0) ≤ max F0 (U, U ; α, δ, 0, 0) U

V

= max α log U − U

U

U2 2

+ δ log H

a∗

a∗ − U + (1 − δ) log H 2 − U . U U 1

(2.51)

Taking the derivative of the r.h.s. of (2.51) with respect to U we get: ∂ F0 (U, U ; α, δ, 0, 0) ∂U a∗ a∗ a∗ a∗ α = − U + δ 12 + 1 A 1 − U + (1 − δ) 22 + 1 A 2 − U . U U U U U

(2.52)

Using asymptotic formulae (2.49) we get the equation for U ∗ which is the maximum point of the r.h.s. of (2.51): δ α ∗ + O(e−1/2α ) = 0, − U + O U∗ α 3/2


157

so

δ √ α + O 3/2 + O(e−1/2α ). α d 1 2 But since α log U − U √ = 0, the Taylor expansion for this function starts dU 2 U= α √ from the term (U − α)2 and we get √ √ F0 (U ∗ , U ∗ ; α, δ, 0, 0) = F0 ( α, α; α, δ, 0, 0) + O(δ 2 α −3 ) + O(e−1/α ). U∗ =

Hence, we have proved (2.44) and so (1.28) is proven. Now one can easily derive the estimate for PN∗ (δ, α) from the inequality [δN] PN (0, 0), PN∗ (δ, α) ≤ CN

where PN (q, q ) is defined by (1.12). Thus, we have finished the proof of Theorem 2. Proof of Theorem 3. It is easy to see that if for some ε > 0 for any local minimum point σ ∗ in 1δ , we can find a point σ ∗∗ inside the ball Bδ1 , such that H(σ ∗ ) − H(σ ∗∗ ) ≥ ε2 N,

(2.53)

then the event A takes place. Let {xk∗ }N k=1 be the effective field generated by the configuration σ ∗ . Consider I (σ ∗ ) ⊂ {1, 2, . . . , N} - the set of indexes i1 , . . . , i[Nδ] such that σi∗ ξ˜i1 = −1.Assume that the number Nε of indexes i ∈ I (σ ∗ ) for which xk∗ ≤ −( 21 +α)ε, is larger than εN (we denote the set of these indexes by Iε (σ ∗ )). Then consider the point σ ∗∗ , which differ from σ ∗ in the components with [εN ] + 1 first indexes i ∈ Iε (σ ∗ ), and coincides with σ ∗ in all the other components. Since we have changed only the components of σ ∗ with indexes i ∈ Iε (σ ∗ ) ⊂ I (σ ∗ ), σ ∗∗ ∈ Bδ1 . On the other hand, 1 ˜ 0 ∗∗ (J (σ − σ ∗ ), (σ ∗∗ + σ ∗ )) 2 1 xi∗ + (J˜ 0 (σ ∗∗ − σ ∗ ), (σ ∗∗ − σ ∗ )) = −2 2 ∗

H(σ ∗ ) − H(σ ∗∗ ) =

i∈Iε (σ )

(2.54)

α ((σ ∗∗ − σ ∗ ), (σ ∗∗ − σ ∗ )) 2 ≥ (1 + 2α)ε2 N − 2αε 2 N ≥ ε2 N,

≥ (1 + 2α)ε2 N −

where J˜ 0 is defined by (1.2) with zero diagonal elements and we have used the inequality J˜ 0 + αI = J˜ ≥ 0. So, we have proved that A ⊃ ∪ε>0 Bε ,

(2.55)

where Bε denotes the event, that for any extreme point σ ∗ ∈ 1δ , the number Nε of indexes in the set Iε (σ ∗ ) is larger than εN . Hence, A ⊂ ∩ε>0 B ε ,

Prob(A) ≤ inf Prob(B ε ∩ Kε˜ ) + Prob{Kε˜ }, ε>0

(2.56)

158


where the event Kε˜ means that inequalities (1.7) hold. Let us note now that B ε corresponds to the event, that there exists a local minimal point σ ∗ ∈ 1δ , such that Nε ≤ N ε. Thus, Prob(B ε ∩ Kε˜ ) ≤

[εN] k=0

[δN] k 0 CN C[δN] Prob(Bε,k ∩ Kε˜ ),

(2.57)

0 denotes the event, that the point σ (1,δ) of the form (1.14) is a local minimal where Bε,k point in 1δ , and x˜i0 ≤ −( 21 + α)ε for i = 1, . . . , k. Taking into account that under condition (1.7) the necessary condition for σ (1,δ) to be a minimum point is (1.9), we obtain that for k = 0,

1 0 ∩ Kε˜ ) ≤ Prob{x˜i0 ≥ − + α ε, i = k + 1, . . . , [δN ]; Prob(Bε,k 2 (2.58) 1 0 x˜j ≥ −˜ε , j = [δN ] + 1, . . . , N} = PN,k − ( + α ε, −˜ε ). 2 And for k = 0, 0 ∩ Kε˜ ⊂ Bε,0

1 0 0 ∩[δN] ε ) ∪ (∪q>−ε(0.5+α) C(q)), ˜ + α ε ∩N j =[δN]+1 Aj (−˜ i=1 Ai − 2 (2.59)

where A0j (q) ˜ is defined by (1.10) and C(q) ≡

min

i=1,...,[δN]

x˜i0 ≥ q,

min

j =[δN]+1,...,N

x˜j0 = −q − ε˜ .

(2.60)

But it is easy to see that for any ? > 0, if we denote 0 N 0 A(q, −q − ?) ≡ ∩[δN] i=1 Ai (q) ∩j =[δN]+1 Aj (−q − ? − ε˜ ),

then ∪0≤t≤1 C(q + t?) ⊂ A(q, −q − ? − ε˜ ) ⇒Prob{∪0≤t≤1 C(q + t?)} ≤ PN (q, −q − ? − ε˜ ).

(2.61)

To have an upper bound for the value of q which we need to consider we use Proposition 3. For any positive α ≤ 0.113 and δ ≤ 0.6α 2 there exists q0 (α, δ), such that for any d˜ > 0, Prob{∪q>q0 +d˜ C(q)} ≤ exp{−N Cd˜ }, where Cd˜ > C ∗ (δ) with C ∗ (δ) defined in (1.31). For α ≤ 0.113, δ ≤ 0.00645 and δ ≤ 0.6α 2 q0 (α, δ) ≤ 0.13.


159

On the basis of this proposition, we can restrict ourselves by 0 ≤ q ≤ q0 + d˜ and, using (2.59)–(2.61), write M 1 Prob{B ∩ Kε˜ } ≤ PN − PN (l?, −˜ε − (l + 1)?) + α ε, −˜ε + 2 l=1 1 + α ε, ε˜ + M max PÑ (q, −q − ? − ε˜ ) + e−NCd˜ , ≤ PN − 2 0≤q≤q0 +d˜ (2.62)

where M =

˜ q0 +d+ε(0.5+α] . Now, using Theorem 1, we get from (2.56), (2.57) and (2.62), ?

Prob(A ∩ Kε˜ ) ≤ exp{−N Cd˜ }

[δN] [εN] (M + 1)CN C[δN] (exp{N [C(α, δ, ε˜ , ε, ?) + o(1)]},

(2.63)

where 1

α α + α ε, −˜ε ) − log α + ; 0≤δ1 ≤ε U 2 2 2 1 α α max min F0D (U, V ; α, δ, − + α)ε, −˜ε − log α + ; U V 2 2 2 α α D . max max min F0 (U, V ; α, δ, q, −q − ? − ε˜ ) − log α + V q>ε(0.5+α) U 2 2

C(α,δ, ε˜ , ε, ? = max

max max F1D (U ; α, δ, δ1 , −

Since F0D and F1D are continuous with respect to q, q , δ1 , we get for ?, ε → 0, ˜ + o(1)]} + exp{−N (C ˜ − C ∗ (δ))}, Prob(A ∩ Kε˜ ) ≤ exp{N [C(α, δ, ε˜ , d) d

(2.64)

where ˜ ε˜ ) = C(α, δ, d,

max

max min{F0D (U, V ; α, δ, q, −q − ε˜ )

0≤q≤q0 +d˜ U

V

α α − log α + + C ∗ (δ)}, 2 2

(2.65)

and therefore ˜ + o(1)]} + exp{−N (C ˜ − C ∗ (δ))} + Prob{Kε˜ } Prob(A) ≤ exp{N [C(α, δ, ε˜ , d) d ˜ + o(1)]} ≤ exp{N [C(α, δ, ε˜ , d) + exp{−N (Cd˜ − C ∗ (δ))} + exp{− const N ε˜ 2 }.

(2.66)

Since (Cd˜ − C ∗ (δ)) > 0 for all d˜ > 0, we conclude, that if for some δ > 0, C(α, δ, 0, 0) < 0, then we always can choose d˜ and ε˜ small enough to provide that all the exponents in the r.h.s. of (2.66) are negative. Thus, we obtain the statement of Theorem 3.

160


Proposition 4. Consider the functions α α A(U, q, α, δ) ≡ min F0 (U, V ; α, δ, q, −q) − log α + + C ∗ (δ) , V 2 2 A0 (q, α, δ) ≡ max A(U, q, α, δ) ≡ A(U (q, α, δ), q, α, δ).

(2.67)

U

If for some 0.071 ≤ α1 ≤ α2 ≤ αc , 0.0035 ≤ δ ≤ δc = 0.00778, A0 (0, α2 , δ) < 0,

∂A (U2 , 0, α2 , δ) < 0, ∂q

∂A (U1 , 0, α2 , δ) > 0, ∂α

(2.68)

then A0 (q, α, δ) < 0 for any α1 ≤ α ≤ α2 and 0 ≤ q ≤ q0 . Here U1 = U (0, α1 , δ) < U2 = U (q0 , α2 , δ). If also δ ≤ kc α 2 (kc ≡ αδc2 ) and c

D ∗ max √ min F0 (U, V ; α, δ) + C (δ) −

U≤ α V

α α log α + < 0, 2 2

(2.69)

then C(α, δ, 0, 0) defined by (2.65) is negative. From (1.29) it is easy to see that to find αc and δc we should study the field of parameters α, δ where A0 (0, α, δ) < 0. Let us fix for the moment α and study the behaviour of the function A0 (0, α, δ) as a function of δ. We find, that it is negative for 0 ≤ δ ≤ δ1 (α) and δ2 (α) ≤ δ ≤ δ3 (α). But for 0 ≤ δ ≤ δ1 (α) C(α, δ, 0, 0) defined by (2.65) cannot be negative, because if it is so, then according to Theorem 3, there exists a minimum point inside the ball Bδ11 . But by the virtue of Theorem 1, the probability to have the minimum point in 1δ (δ < δ1 ) vanishes, as N → ∞, because A0 (0, α, δ) < 0. Thus we should study δ2 (α) ≤ δ ≤ δ3 (α). When α increases, |δ3 (α) − δ2 (α)| decreases and for α = αc δ3 (αc ) = δ2 (αc ) = δc . Then evidently A0 (0, αc , δc ) = 0,

∂A0 (0, αc , δc ) = 0. ∂δ

So we find from these equations, that αc = 0.11326 . . . , δc = 0.00777 . . . Unfortunately, for this (αc , δc ) condition (2.69) is not fulfilled. So we take a bit smaller α = 0.113 and δ = 0.00645, for which (2.69) is fulfilled. Then, using (2.68), we obtain the statement of Theorem 3 for all 0.071 ≤ α ≤ 0.113 in three steps: (1) 0.1105 ≤ α ≤ 0.113, δ = 0.00645; (2) 0.095 ≤ α ≤ 0.1105, δ = 0.0042; (3) 0.071 ≤ α ≤ 0.095, δ = 0.0035. For α ≤ 0.071 the statement of Theorem 3 follows from the result of [L].

3. Auxiliary Results Proof of Lemma 1. At the first step we check that, if x˜k are defined by relations (1.14), then " # 1+2d θ x˜k − ak + N 1/2+d ≤ e− const N .


161

To this end we use the Chebyshev inequality, according to which θ (x˜k − (ak + N 1/2+d ) ≤ minexp{τ x˜k − τ (ak + N 1/2+d )} τ >0

= min e−τ (ak +N

1/2+d )

τ >0

p

exp

N τ

N

µ=1

j =1

µ µ

ξk ξj

τ (pN) 1/2+d ) cosh = min e−τ (ak +N τ >0 N τ2 1+2d ≤ e− const N ≤ min exp − τ (ak + N 1/2+d ) + α . τ >0 2

Thus, N

N θ (x˜k − ak ) = θ(x˜k − ak )(θ (ak + N 1/2+d − x˜k )

k=1

k=1

+ θ (x˜k − (ak + N 1/2+d ))) ≤

N

θ(x˜k − ak )θ (ak + N 1/2+d − x˜k )

(3.1)

k=1

2N

N

θ(x˜k − (ak + N 1/2+d ))

k=1

≤

N

1+2d θ(x˜k − ak )θ (ak + N 1/2+d − x˜k ) + e− const N .

k=1

Consider

exp −

Dλ,εN∗ (x 1 , . . . , xN ) ≡

1 N 2

∗ −1 −1 j,k=1 (λI + εN lN J)j k xj xk −

1 2

N

∗ −1 j,k=1 εN lN Jj k

p/2

∗ l −1 J} lN (2π )N/2 det1/2 {λI + εN N

,

where I is a unit matrix and J is a matrix with entries Jj k

p 1 µ µ = ξj ξk . N µ=1

We study the composition Dλ,εN∗ ∗ χN,h of this function with the product of χN,h (xk ) ! (recall that (f ∗ g)(x) ≡ f (x − x )g(x )dx ). Let us check that for 0 ≤ xk ≤ N 1/2+d , N

θ (xk )θ (N 1/2+d − xk ) ≤ (1 − e−h

2 /2λ

k=1

· Dλ,εN∗ ∗

χN,h (x1 , . . . , xN )det

1/2

p/2

)−N lN

∗ N ∗ εN εN J exp Jj k . I+ λlN 2lN

j,k=1

(3.2)

162


Indeed, by definition of composition,

Dλ,εN∗ ∗ =

N ε∗ ε∗ p/2 χN,h )(x1 , . . . , xN det1/2 λI + N J exp N J j k lN lN 2lN

1 (2π)N/2

j,k=1

exp −

1 2

N

(λI +

j,k=1

∗ εN

lN

N

J)−1 j k (xj − xj )(xk − xk )

k=1

χN,h (xk )dxk

N

1 1 exp − (xk − xk )2 χN,h (xk )dxk (2π)N/2 2λ k=1 k=1 1 (x − x )2 N ≥ √ χN,h (x ) . dx exp − 2λ 2π N

≥

(3.3)

But for x ∈ (0, N 1/2+d ),

(x − x )2 I1 = dx exp − (1 − χN,h (x )) 2λ

−h

∞ (x − x )2 (x − x )2 = dx + dx exp − exp − 1/2+d 2λ 2λ +h −∞ N

−h

∞ (x )2 (x )2 2λ − h2 ≤ exp − exp − dx + dx ≤ e 2λ . 2λ 2λ h −∞ h 1/2 , So for h > ( 2λ π )

(x − x )2 √ 1 I1 √ 2 λ− √ χN,h (x ) = dx exp − ≥ λ(1 − e−h /2λ ). √ 2λ 2π 2π

Thus, we have proved (3.2) for xk ∈ (0, N 1/2+d ). Besides, using the inequality log(1 + x) ≤ x, we get ∗ 1 εN ε ∗ 1/2 det I+ J = exp log 1 + N λi λlN 2 λlN λi ∈σ (J) ∗ 1 εN (3.4) ≤ exp λi 2 λlN λi ∈σ (J)

ε∗ α ε∗ N TrJ = exp N N . = exp 2λlN 2λlN Here σ (J) is a spectrum of the matrix J. Therefore, it follows from (3.2) and (3.4) that for xk ∈ (0, N 1/2+d ), N

θ (xk )θ (N 1/2+d − xk ) ≤ (1 − e−h

2 /2λ

k=1

· exp

ε∗ αN N

2λlN

Dλ,εN∗ ∗

p/2

)−N lN

χN,h )(x1 , . . . , xN

N ∗ εN Jj k . exp 2lN

j,k=1

(3.5)


163

But for all the other values of {xk } the l.h.s. of this inequality is zero, while the r.h.s. is positive, so we can extend (3.5) to all {xk } ∈ RN . Besides, according to the Chebyshev inequality, ∗ −1/2 ∗ −1/2 Prob Jj k ≤ N (εN ≤ min e−τ (εN ) N E{eτ Jj k } ) τ >O 1 1 1 ∗ )−1/2 N p −τ (εN E exp τ ξ ξ = min e τ >O N j k p ∗ −1/2 ≤ min exp − τ (εN ) N − log(1 − τ ) 1>τ >O 2 ∗ −1/2 ≤ exp{− const (εN ) N }.

(3.6)

Here we have used the standard trick, valid for τ < 1, √ 1 1 x2 1 1 1 = (2π )−1/2 E dx exp − τ x √ ξj ξk E exp τ ξi − N 2 N √ N

x τ 2 = (2π)−1/2 dx cosh √ e−x /2 = (1 − τ )−1/2 (1 + O(N −1 )). N Therefore finally, on the basis (3.1), (3.5) and (3.6), we get N

∗ −1/2 const θ (x˜k − ak ) ≤ e−N(εN )

k=1

+

∗ 1/2 p/2 e const N(εN ) lN

(1 − e−h

2 /2λ

Dλ,εN∗ ∗

)N

N

(3.7)

χN,h (x˜1 − a1 , . . . , xÑ − aN ) .

k=1

Now to finish the proof of Lemma 1 we are left to find the Fourier transform Dˆ λ,εN∗ of the function Dλ,εN∗ ,

Dˆ λ,εN∗ (ζ ) = (2π)−N/2

−p/2

dxei(x,ζ ) Dλ,εN∗ (x) = lN

exp

−

λ (ζ , ζ ) 2 2

∗ µ 2 εN ε∗ µ ξ k ζk − N ξk 2lN N µ 2lN N µ k k ∗ εN λ −p/2 µ 2 µ 2 = lN exp − (ζ , ζ ) − ((u˜ ) + (v˜ ) ) , 2 2lN µ

−

164


where u˜ µ and v˜ µ are defined by (2.2). Then

Dλ,εN∗ ∗

N

χN,h (x˜1 − a1 , . . . , xÑ − aN )

k=1 −N

= (2π)

N k=1

−p/2

= lN ·

(2π)−N

µ

N

dζk χˆ N,h (ζk ) exp{−iak ζk } · Dˆ λ,εN∗ (ζ ) exp i ζk x˜k

N

dζk χˆ N,h (ζk ) exp

− iak ζk −

k=1

λ 2 ζ 2 k

k=1

ε∗ exp − N (u˜ µ )2 + (v˜ µ )2 + i u˜ v˜ . 2lN (3.8)

Let us use the representation (cf. (2.3) )

ε∗ exp − N ((u˜ µ )2 + (v˜ µ )2 ) + i u˜ µ v˜ µ 2lN 1/2

ε∗ l = N duµ dv µ exp − N ((uµ )2 + (v µ )2 ) − ilN uµ v µ + iuµ u˜ µ + iv µ v˜ µ , 2π 2

2 + (ε ∗ )2 . where we have taken into account, that by definition (see Lemma 1) lN = lN N Substituting this representation into (3.8), we get

Dλ,εN∗ ∗

N

χN,h (x˜1 − a1 , . . . , xÑ − aN )

k=1 −N−p

= (2π)

N k=1

λ dζk χˆ N,h (ζk ) exp − ζk2 − iak ζk 2 µ

· dv µ exp − iuµ v µ −

duµ

∗ εN ε∗ uµ ζk + v µ cos (uµ )2 − N (v µ )2 = PN1 . √ 2 2 N N

k=1

(3.9)

Inequality (3.7) and this representation prove Lemma 1.

On the Critical Capacity of the Hopfield Model π 2 6εN

Proof of Lemma 2. Take L =

(m)

FcL (u1 , v 1 , u2 , v 2 ) ≡

L

and consider an intermediate functions:

dζk χˆ N,h (ζk )e−λζk /2−iaζk 2

−L

·

cos

µ≤m

−

FNL (u1 , v 1 , u2 , v 2 ) ≡ (m)

Denote also Fc Then

165

uµ ζk + w µ 1 exp − (u2 , w 2 )ζk √ N N

p 1 wν (u2 , u2 )ζk2 cos √ ; 2N N ν>m L

dζk χˆ N,h (ζk )e−λζk /2−iaζk 2

−L

(3.10)

p

cos

µ=1

uµ ζk + w µ . √ N

(m)

by the same formula as FcL with L = ∞. (m)

R (m) ≡ FN − F (m) = (FN − FNL ) + (FNL − FcL ) (m)

+ (FcL − Fc(m) ) + (Fc(m) − F (m) ).

(3.11)

One could easily estimate (FN − FNL ) by using the simple inequalities |(FN − FNL )(u, w)| ≤ (m)

Let us estimate R∗

f (ζk ) =

e

(λ2 ,λ2 ) N

2π

|ζk |>L

e−ζk /2λ dζk ≤ e 2

(λ,λ) N

−4

e− const εN .

(3.12)

(m)

≡ FNL − FcL . To this end we consider ν>m

log cos

ζ2 u ν ζk + w ν ζk + k U˜ 2 + (u2 , w 2 ) √ 2 N N

and use the inequality |ef (ζk ) − ef (0) | ≤ |f (ζk ) − f (0)|(|ef (ζk ) | + |ef (0) |). √ ν ξ uν 2 ≤ π and |uν |, |v ν |, |λν | ≤ ε |, | √v | ≤ LεN Then, since | √ N N , we get 6 N

N

uν ξ u ν + w ν uν w ν (uν )2 |f (ζk ) − f (0)| ≤ |ζk ||f (ξ )| = |ζk | − √ tg √ + +ξ N N N N ν>m uν ξ uν + w ν 3 ≤ |ζk | const (3.13) √ √ N N ν>m 1 ν2 2 ≤ εN |ζk | const U˜ 2 |ζk |3 + (|v | + |λν |2 ) . N ν>m To estimate |ef (ζk ) | we use the inequality, valid for |#z| ≤

π 2,

1 1 # log cos z + z2 ≤ ($z)2 . 2 2

(3.14)

166


(The proof of this inequality is given at the end of the proof of Lemma 2.) It follows from (3.14) that p

#f (ζk ) = #

nu=m+1

ζk u ν + w ν (w ν )2 (ζk uν + w ν )2 log cos − + √ 2N 2N N

(${ζk uν + w ν })2 #{(w ν )2 } (v 2 , v 2 ) (λ2 , λ2 ) − =− + . ≤ 2N 2N 2N 2N ν>m ν>m

(3.15)

Therefore we derive from (3.13) and (3.15) that (u , v )ζ p (u2 , u2 )ζk2 uν ζk + w ν w ν 2 2 k − cos cos √ − exp − √ N 2N N N ν>m ν>m 1 1 (u2 , u2 )ζk2 ef (ζk ) − ef (0) = exp − (u2 , v 2 )ζk − N 2N (v 2 , v 2 ) + (λ2 , λ2 ) 2 ≤ const εN |ζk | U˜ 2 |ζk |3 + N ζ 2 U˜ 2 (u2 , v 2 ) (v 2 , v 2 ) (λ2 , λ2 ) · exp − k − ζk − + 2 N N N ζ 2 U˜ 2 ν v + iλν (u2 , v 2 ) + exp − k − ζk cos √ . 2 N N ν>m ν

(3.16)

ν

Using inequality (3.14) for | cos v √+iλ | (ν > m), we get N

|R∗(m) (u1 , v 1 , u2 , v 2 + iλ2 )|

(v 2 , v 2 ) + (λ2 , λ2 ) −λζ 2 /2 2 e k ≤ εN dζk U˜ 2 |ζk |3 + N ζk2 U˜ 2 uµ ζk + w µ (u2 , v 2 ) (v 2 , v 2 ) (λ2 , λ2 ) − ζk − + · √ cos exp − 2 N 2N N N µ≤m (v 2 , v 2 ) + (λ2 , λ2 ) (u2 , v 2 )2 (λ, λ) (v 2 , v 2 ) 2 ≤ εN const 1 + + . + exp − 2N N N 2 (U˜ 2 + λ) N U˜ 2 + λ (3.17) Now to obtain the estimate of the form (2.24) we use (3.23) and the inequality (v 2 , v 2 ) 2(U˜ 2 + λ) λ(v 2 , v 2 ) ≤ exp . 2N λ 4N (U˜ 2 + λ) Combining them with (3.17), we get (m) R (u1 , v 1 , u2 , v 2 + iλ2 ) ≤ ε 2 const (U˜ 2 + λ)1/2 (3.18) ∗ N (λ2 , λ2 ) λ(v 2 , v 2 ) (λ, λ) · 1+ exp − . + N N 4N (U˜ 2 + λ)

On the Critical Capacity of the Hopfield Model (m)

(m)

To estimate (FcL − Fc

167 ν

ν

) we use again the inequality (3.14) for | cos v √+iλ | (ν > m), N

(m)

|FcL (u, w) − Fc(m) (u, w)| ≤ e ·

(λ,λ) N

e−

(v 2 ,v 2 ) 2N

dζk |χˆ N,h (ζk )|e−λζk /2 2

|ζk |≥L

1 1 (u2 , u2 )ζk2 exp − (u2 , v 2 )ζk − 2N

N (λ,λ) (λ,λ) −4 2 /2 −λζ ≤e N dζk e k ≤ const e N e− const εN . |ζk |≥L

(3.19) Thus, we are left to estimate the difference Fc(m) (u1 , w 1 , u2 , w 2 ) − F (m) (u1 , w 1 , u2 , w 2 ) a − i(u , w ) − (u√1 ,ξ 1 ) i(v ,ξ ) 2 2 1 1 (w2 ,w2 ) wν √ N = HN,h,U˜ cos √ − e− 2N . e N N U˜ 2 + λ µ>m

(3.20)

The last multiplier here can be estimated by the same way as in (3.10)–(3.16). Then we get (w2 ,w2 ) wν cos √ − e− 2N N µ>m 2 ≤ const εN

|(w 2 , w 2 )| exp N

−

(v 2 , v 2 ) (λ2 , λ2 ) + . 2N N

To estimate the first multiplier we use the bound |HN,h,U˜ (a + ic)| ≤ ec

2 /2

(3.21)

. Thus,

) a − i(u , w ) − (u,ξ √ 1 i(v 1 ,ξ 1 ) 2 2 √ N HN,h,U˜ e N 2 U˜ + λ (λ ,ξ ) 1 1 (u2 , v 2 )2 (λ1 , λ1 ) (u2 , v 2 )2 √ ≤ exp + e N ≤ exp . N 2N 2 (U˜ 2 + λ) 2N 2 (U˜ 2 + λ)

By the same way as in (3.16)-(3.18) we can obtain now from (3.20) and (3.21) the bound of the form (2.24). Now to finish the proof of Lemma 2 we are left to prove inequality (3.14). For z = x + iy (x, y ∈ R) by the simple algebraic transformations we get that (3.14) is equivalent to the inequality 1 2 2 (cosh 2y + cos 2x) ≤ e2y −x . 2 2

Since cosh 2y ≤ e2y , to prove (3.22) it is enough to prove that cos 2x ≤ e2y (2e−x − 1), 2

2

(3.22)

168


which evidently follows from cos 2x ≤ (2e−x − 1) 2

⇐⇒

Since the last inequality is valid for |x| ≤ Lemma 2 is proven.

π 2,

cos x ≤ e−x

2 /2

.

we have proved (3.22) and so (3.14).

Proof of Lemma 3. We use (2.24) to estimate the integral Im,k

≡

√ εN N √ −εN N (m)

dv 2 e−ilN (u2 ,v 2 ) e−

∗ εN 2

(v 2 ,v 2 )

(u1 , v 1 , u2 , v 2 ))N−k (R (m) (u1 , v 1 , u2 , v 2 ))k .

· (F

By using (2.10), which is evidently valid also for HN,h,U˜ we get |F

(m)

(u2 , v 2 )2 (v 2 , v 2 ) (u1 , v 1 , u2 , v 2 )| ≤ exp − 2N 2N 2 (U˜ 2 + λ) (3.23) λ(v 2 , v 2 ) λ(v 2 , v 2 ) ≤ exp − . ≤ exp − 2N (U˜ 2 + λ) 4N (U˜ 2 + λ)

The second inequality here can be obtained if we observe that (u2 , v 2 )2 U˜ 2 = (Pu v 2 , v 2 ), N 2 (U˜ 2 + λ) U˜ 2 + λ where Pu is the orthogonal projection operator on the unit vector (U˜ )−1 N −1/2 u2 , and ˜2 use the trivial inequality I − ˜U2 Pu ≥ ˜ 2λ I. Note also, that we replace in (3.23) 2 in U +λ U +λ the denominator by 4 in order to have the same factor as in (2.24). Hence, on the basis of Lemma 2, we have | |Im,k

≤ ≤

√ εN N √

−εN N

dv 2 |(F (m) (u1 , v 1 , u2 , v 2 ))N−k (R (m) (u1 , v 1 , u2 , v 2 ))k |

√ εN N

λ(v 2 , v 2 ) ∗ (v 2 , v 2 ) − εN dv 2 exp − 2 4(U˜ 2 + λ) N −ε

N λ(N − k)(v , v ) −4 2 2 ∗ (v 2 , v 2 ) + ek const e−k const εN dv 2 exp − − εN 2 4N (U˜ 2 + λ)

2k ˜ 2 (U ek const εN

+ λ)

k/2

√

−4

2k ∗ −p/2 −k const εN + ek const (εN ) e . ≤ eN const (U˜ 2 + λ)p/2 εN

(3.24)

Substituting estimate (3.24) in the expression for Im,k integrating over u1 , v 1 , and U˜ we get finally ∗ 2 |Im,k | ≤ (U˜ 2 + λ)p/2 U˜ p−m e−NεN U /2 d U˜ eN const (εN )2k −4

∗ −p −k const εN + ek const (εN ) e .


169

Using the Laplace method for the integration with respect to U˜ and taking into account that the second term in the r.h.s. here for k > k0 is much smaller than the first one, we obtain the statement of Lemma 3. Proof of Lemma 4. To prove (2.29) we use the variables w µ = −iλµ + t µ , (t µ ∈ R) ν (µ = 1, . . . , m) and wν = −i u˜ V + t ν , (t ν ∈ R) (ν = m0 + 1, . . . , p − n) defined in U (2.26) and estimate |Gm,k,n (V , u1 , λ1 , u3 )|

≤

k1 +k2 =k

Ckk1 (2π)−p

√ 2 N εN

√ 2 N −εN

(λ ,ξ ) a − V 1 1 1 e √N H N,h,U˜

U2 U˜

du4

Lν2

|dw 4 |

dt 1

− i (u3N,t 3 ) − i (u4N,w4 ) −

√ εN N

√ −εN N (u1 ,ξ 1 ) √ N

U 2 + λ + N −1 (u4 , u4 )

dt 3

[Nδ]−k1

(λ ,ξ ) a − V U 2 − i (u3 ,t 3 ) − i (u4 ,w4 ) − (u√1 ,ξ 1 ) N−[Nδ]−k2 2 1 1 √ N N N U˜ · e N HN,h,U˜ U 2 + λ + N −1 (u4 , u4 ) · |Rm (u, w)|k exp − lN ((u1 , λ1 ) + N V U˜ − $(u4 , w 4 )) 1 U2 1 − (N − k) + (3.25) (t 3 , t 3 ) − V 2 #(w 4 , w 4 ) 2N 2N 2U˜ 2 ε∗ U2 − N (u, u) + (t 1 , t 1 ) − (λ1 , λ1 ) + (t 3 , t 3 ) − N V 2 + #(w 4 , w 4 ) . 2 U˜ 2 Here we consider Im,k as the sum of terms, in which k1 remainder functions R (m) come from the first [δN ] factors in (2.25) and k2 of R (m) come from the last N − [δN ] ones. Since k = o(N) we have that k1,2 = o(N ) and Ckk1 = eo(N) . Now we use (2.10) for HN,h,U˜ and the inequalities |N

−1

(u4 , w 4 )| ≤ N

−1

√

εN N

p

ν

|u | +

ν=p−n+1

V

p

N U˜

ν=p−n+1

|uν |2

V 4 1/2 3 ≤ nεN + n εN ≤ const εN ; U˜ 3/2 4 0 ≤ N −1 (u4 , u4 ) = U˜ 2 − U 2 ≤ nεN ≤ εN ,

(3.26)

√ −5/2 2 N (see formula (2.14)) and |w ν | < which are valid since n ≤ εN , |uν | ≤ εN √ 2 ε∗ εN N + V˜ |uν | (ν = p − n + 1, . . . , p). Besides, exp{ 2N [(λ1 , λ1 ) + N V 2 U˜ 2 ]} ≤ ∗

U

U

eN const εN = eo(N) because of the chosen bounds on λ1 and V . Then, using the inequality HN,h,U˜ (x) ≤ H (x),

(3.27)

170


and the fact that k1,2 = o(N ), we get from (3.25), √ 2 U 2 + λ)k ( const εN 2 |Gm,k,n (V , u1 , λ1 , u3 )| ≤ e−nNεN /4 (2π )p ε∗ ε∗ · (G∗m (U, V , u1 , λ1 ))N exp − N (u1 , u1 ) − N N U 2 + N o(1) 2 2

ε2 √N

(3.28) N N −k−n (u3 , t 3 )2 (t du exp − , t ) − · dt 1 dt 3 4 3 3 √ 2 N 2N N (U 2 + λ) −εN ε∗ − N ((t 1 , t 1 ) + (t 3 , t 3 ) + (u4 , u4 )) . 2 √ 2 U 2 + λ)k is due to Lemma 2 and the last line of (3.26), and the Here the term ( const εN 2 term e−nNεN /4 is due to the integration with respect to w 4 . On the other hand, we should ∗ )−(m+n) note that in fact integrals with respect t 1 and u4 can give us only ( const )m+n (εN ∗ −1 as a multiplier. Since m, n = o(N | log εN | ), we take it into account as eo(N) . Our main problem is to estimate the integral with respect t 3 , because it contains almost p integrations. To perform this integration let us note that it is of the Gaussian type with the ˜2 matrix of the form A = (I − ˜U2 Pu ), where I is a unit matrix and Pu is the orthogonal U +λ

projector on the normalized vector

√u3 . NU

Since such a matrix A has (p − m − n − 1)

eigenvalues equal to 1 and only one eigenvalue equal to 1− with respect to t 3 gives us (2π)

p−n−m 2

U˜ 2 U˜ 2 +λ

=

λ , the integration U˜ 2 +λ

const . Thus we obtain (2.29).

Proof of Proposition 1. It follows from (2.39) that log |G∗m (U, V , u1 , λ1 (U, V , u1 ))| 1 ≤ N o(1) − U V + V 2 + C(U, V ) − D(U, V )(u1 , u1 ), 2

(3.29)

where a − h − UV a1 − h − U V 2 C(U, V ) = N δ log H ( √ ) + N (1 − δ) log H . √ U2 + λ U2 + λ On the other hand, using that H (x) < 1, we get a − h − V U − 1,2 H √ U2 + λ

(u1 ,ξ 1 ) √ N

e

(λ1 ,ξ 1 ) √ N

(λ√1 ,ξ 1 ) (λ1 ,λ1 ) ≤ e N ≤ e 2N .

Therefore, taking in (2.28) λµ = uµ we obtain 1 1 log |G∗m (U, u1 , λ1 (U, V , u1 ))| ≤ N − U V + V 2 − (u1 , u1 ). 2 2

(3.30)


171

Inequalities (3.29) and (3.30) give us log |G∗m (U, V , u1 , λ1 (U, V , u1 ))| 1 1 ≤ N o(1) − U V + V 2 + min C(U, V ) − D(U, V )(u1 , u1 ); − (u1 , u1 ) . 2 2 (3.31) Now, applying the Laplace method, we get

m ε∗ N du1 |G∗m (U, V , u1 , λ1 (U, V , u1 ))| exp − N (uµ )2 2 µ=1 1 ≤ exp N − U V + V 2 + o(1) 2 1 + max min C(U, V ) − D(U, V )(u1 , u1 ); − (u1 , u1 ) . (u1 ,u1 ) 2

(3.32)

But since both functions in the r.h.s. of (3.32) are linear ones with respect to (u1 , u1 ), one can find the maximum value explicitly. It is just the intersection point of two functions y = − 21 x and y = C(U, V ) − D(U, V )x. It is easy to see that xint = −

C(U, V ) , 0.5 − D(U, V )

yint =

C(U, V ) . 1 − 2D(U, V )

Substituting yint in (3.32) we get the statement of Proposition 1.

Proof of Proposition 2. The inequality V (U ) < U follows easily from (2.48), if we take √ into account, that A(x) > 0. To prove that V (U ) ≥ α we use the inequalities: 0 < A (x) < 1,

A(x + y) < A(x) + y < 1 + y (x < 0, y > 0).

(3.33)

From the relations 2 √ e−x /2 −x 2 /2 − x 2π H (x) , e 2π H 2 (x)

∞ √ 2 2 2π H (x)x ≤ te−t /2 dt = e−x /2 ,

A (x) =

x

it is easy to derive that A (x) > 0. To get the upper bound for A (x) let us introduce the 2 function φ(x) ≡ log H (x) + x2 . Using the identities

∞ dt 2 φ(x) = log √ e−tx−t /2 , φ (x) = (t − tx )2 x ≥ 0, 2π 0 !∞ −tx−t 2 /2 dt 0 (. . . )e where . . .x ≡ ! ∞ , we obtain that A (x) ≡ 1 − φ (x) < 1. −tx−t 2 /2 dt e 0 The last bound in (3.33) can be obtained as A(x + y) ≤ A(x) + y

max |A (s)| < A(x) + y.

x≤s≤x+y

172


Taking into account, that A(x) < π2 < 1 for x < 0, we get the last inequality in (3.33). Now from the bound A (x) < 1 we get that the r.h.s. of (2.48) is an increasing function with respect to V . Thus, to prove Proposition 2 it is enough to prove that U>

√

α + p

α + δA

U

−

α − p √ √ α + (1 − δ)A − α , U

√ for U ≥ 2 α. Here and below we denote p = 1 − 2δ. √ Using the last inequality in (3.33) with x = − α and y = A, we get α + p

δA

α+p U

(3.34)

to estimate the first

α − p √ √ α + (1 − δ)A − α U U α + p U (1 − δ)

√ α + 0.3U (1 + O(α)) + o(α 2 ),

and (3.34) is valid. Thus, we have finished the proof of Proposition 2.

(3.36)

0 Proof of Proposition 3. Since for any q˜ > q C(q) ˜ ⊂ ∩[δN] j =1 {x˜ j ≥ q}, on the basis of Theorem 1, we have got

Prob ∪q>q C(q) ˜ ˜

α α ≤ exp N max min F0D (U, V ; α, δ, q, −∞) − log α + . U >0 V 2 2

Let us denote f0 (U, V ; q, α, δ) ≡ F0 (U, V ; α, δ, q, −∞) + f D (U, V ; q, α, δ) ≡

α α log α + + C ∗ (δ) 2 2

α α log α + + C ∗ (δ) + α log U − U V 2 2 log H (a1∗ U −1 − V ) V2 + +δ , 2 1 − 2D(U, V )

(3.37)


173

and consider max min f0 (U, V ; q, α, δ) ≤ max f0 (U, U ; q, α, δ) U

V

U

α δ a∗ ≤ max α log U − U 2 /2 − ( 1 − U )2 + log α U 2 U 2 α ∗ + + C (δ) → −∞, 2 (3.38)

as a1∗ → ∞. Here we have used the inequality log H (x) ≤ − f D (U, V ; q, α, δ), when D(U, V ) < 0 we have the bound

x2 (x > 0). Similarly, for 2

max minf D (U, V ; q, α, δ) ≤ max f D (U, U ; q, α, δ) U V U A(a1∗ U −1 − U ) U2 2 ≤ max α log U − U /2 − U 2 2U + (1 − δ)A(a1∗ U −1 − U ) α α − log α + + C ∗ (δ) ≤ max α log U − U 2 /2 (3.39) U 2 2 U2 α α α p−U 2 − − log α + + C ∗ (δ) → − log 2 + C ∗ (δ). 2 2 p(1 − δ) + U (1 + δ) 2 2 2 Here we have used the inequalities log H (x) ≤ −A(x)2 /2 (x > 0) and A(x) ≥ x. Thus, inequalities (3.38) and (3.39) under conditions δ ≤ 0.6α 2 , α ≤ 0.113 prove the first statement of Proposition 3. Besides, (3.39) shows that it is enough to study only f0 . Since maxU minV f0 (U, V ; q, α, δ) for fixed p increases with α and δ, to prove the second statement of Proposition 3 it is enough to check that for α = 0.113, δ = δmax = 0.00645 and q = q0 + 2δmin − 2δmax = 0.126 maxU minV f0 (U, V ; q, α, δ) < 0. We do this numerically. Thus, we obtain the statement of Proposition 3. Proof of Proposition 4. Let I = IU × Iα × Iq ⊂ R3 with IU = [U1 , U2 ], Iα = [α1 , α2 ] and Iq = [0, q0 ]. Denote by V (U, q, α) the point of minimum of F0 (U, V ; α, δ, q, −q) and by U (q, α) the point of maximum of A(U, q, α). Let us note that during the proof of Proposition 4 the variable δ is fixed. So here and below we omit δ as an argument of the functions A and A0 . The first statement follows from the relations: U (q, α) ∈ IU (q ∈ Iq , α ∈ Iα ), A(U, q, α) ≤ A(U, 0, α) ≤ A(U, 0, α2 ) ≤ A(U (0, α2 ), 0, α2 ) ≤ 0.

(3.40)

To prove the first line of (3.40) it is enough to check that in I ∂ 2A ∂ 2A ≥ 0, ≥ 0, ∂U ∂α ∂U ∂q

(0 ≤ q ≤ q0 , 0.071 ≤ α ≤ 0.113),

(3.41)

174


because in this case we have for any q ∈ Iq , α ∈ Iα , ∂A ∂A ∂A (U1 , 0, α1 ) < (U1 , q, α1 ) < (U1 , q, α), ∂U ∂U ∂U ∂A ∂A ∂A 0= (U2 , q0 , α2 ) > (U2 , q, α2 ) > (U2 , q, α), ∂U ∂U ∂U and thus U1 ≡ U (0, α1 ) ≤ U (q, α) ≤ U (q0 , α2 ) ≡ U2 . Note, that for our choice of 0.0035 ≤ δ ≤ 0.00778, 0.71 ≤ α ≤ 0.1133 and 0 ≤ q ≤ q0 ≤ 0.13 we get, that 0.25 < U1 < U2 < 0.41. Let us prove (3.41). To this end we write 0=

∂ 2A ∂ 2 F˜ 0 ∂ 2 F˜ 0 = + Vα ; ∂U ∂α ∂U ∂α ∂U ∂V ∂ 2A ∂ 2 F˜ 0 ∂ 2 F˜ 0 = + Vq , ∂U ∂q ∂U ∂q ∂U ∂V

(3.42)

are the derivawhere F˜ 0 (U, V ; α, δ, q) ≡ F0 (U, V ; α, δ, q, −q)− α2 log α+ α2 and Vq,α tives with respect to q and α of the function V (U, q, α) defined above. By the standard ∂ F˜ 0 method, from the equation (U, V (q, α)) = 0 we get ∂V ∂ 2 F˜ −1 ∂ 2 F˜ ∂ 2 F˜ −1 ∂ 2 F˜ 0 0 0 0 Vα = − = − , V . (3.43) q 2 2 ∂V ∂V ∂α ∂V ∂V ∂q

Now let us find the expressions for the derivatives of the function F˜ 0 , ∂ 2 F˜ 0 ∂ 2 F˜ 0 2 2 = 1 − δU A − (1 − δ)U A > 0; = −δA1 − (1 − δ)A2 < 0; 1 2 ∂V 2 ∂q 2 ∂ 2 F˜ 0 1 ∂ 2 F˜ 0 − δA = δU A1 + (1 − δ)U A2 > 0; = − − (1 − δ)A < 0; 1 2 ∂α 2 2α ∂V ∂α ∂ 2 F˜ 0 1 δ (1 − δ) = + A1 + A2 + δa1∗ A1 + (1 − δ)a2∗ A2 ; ∂U ∂α U U U ∂ 2 F˜ 0 (1 − δ) δ (1 − δ) ∗ δ = A1 − A2 + a1∗ A1 − a 2 A2 ; ∂U ∂q U U U U ∂ 2 F˜ 0 (3.44) = −1 − δa1∗ A1 − (1 − δ)a2∗ A2 ; ∂U ∂V ∂ 2 F˜ 0 = δU A1 − (1 − δ)U A2 ; ∂V ∂q where A1,2 are defined in (1.20) and ∗ ∗ a1,2 1 a1,2 V − V = A1,2 A1,2 − 2 + , A1,2 ≡ 2 A U U U U with function A(x) defined by (1.19). We recall here, that from definition (1.15), it follows that 1 < a1∗ < 1.25,

−1.1 < a2∗ < −0.85.

(3.45)


175

Let us note also, that for U ≤ U2 < 0.41, 0 < A2 =

1 a2∗ 1 a2∗ < 0.7. ≤ A A U2 U U2 U22

(3.46)

Thus, ∂ 2 F˜ 0 > 0, ∂U ∂α

∂ 2 F˜ 0 < 0, ∂U ∂V

(3.47)

∂ 2A > 0. To obtain the second and using (3.42)–(3.47), we can see immediately that ∂U ∂α inequality in (3.41) we write, using (3.44)–(3.47), ˜

∂ F0 − ∂U ∂V 2

0
[0.5A2 − A2 ] U U a∗ (1 − δ)A2 V = 0.5 A2 − 22 + − 2 > 0. U U U +

Thus, we have finished the proof of the first line of (3.40). To prove the second line we use the simple statement Remark 6. If f0 (x) = miny g(x, y) and

∂ 2g ∂ 2 f0 ≤ 0, then also ≤ 0. ∂x 2 ∂x 2

This statement can be easily proved on the basis of the characteristic property of the x + x f (x1 ) + f (x2 ) 1 2 ≤f concave functions . 2 2 ∂ 2A Then on the basis of the second line of (3.44) we get automatically that ≤ 0. ∂α 2 Therefore, using (2.68) and (3.41), we get 0
0 it is enough to prove (3.49) only for ∂q∂U

U = U2 . The second inequality in (2.68) implies that

δ A2 (U2 , 0, α2 ) < . A1 (U2 , 0, α2 ) 1−δ But

A A 1 A d A2 2 2 − Vα = − 1 dα A1 U A1 A2 A1 1 A 2 = ((A(x2 ) − x2 ) − (A(x1 ) − x1 )), − Vα U A1

where x1,2 = and

(3.50)

U −1

∗ a1,2

− Vα

− V (U, q, α). Since A(x) − x is a decreasing function (see (3.33)) U > 0 (see (3.43) and (3.44)), we have got that

A2 (U2 , 0, α2 ) δ ∂A A2 (U2 , 0, α) < < ⇔ (U2 , 0, α) < 0. A1 (U2 , 0, α) A1 (U2 , 0, α2 ) 1−δ ∂q Thus we have proved the first statement of Proposition 4. Now we are left to prove that inequalities (2.68) √ and (2.69) implies (1.29). To this end it is enough to check that for δ ≤ kc α 2 and U > α, D(U,√V (U )) ≥ 0, because in this case we have that F (D) (U, V (U )) = F0 (U, V (U )) (U > α) and so (D) max (U, V (U ); q, −q, α, δ) + C ∗ (δ) − √ F

U≥ α

α α log α + = max √ A(U, q, α, δ). 2 2 U≥ α

For U > 0.5 evidently D(U, V (U ); δ) > 0. For 0.5 > U >

√ α we have

√ √ D(U, V (U ); δ) > D( α, V ( α); δ) √ √ √ √ ≥ D( α, V ( α); kc α 2 ) ≥ D( αc , V ( αc ); δc ). √ √ So, checking numerically that D( α c , V ( α c ); δc ) > 0 we finish the proof of Proposition 4. Acknowledgements. This work has been done with the support of Royal Society and with the help of a scientific agreement between the Institute for Low Temperature Physics Ukr. Ac. Sci and the University “La Sapienza” of Rome.


177

References [A] Amit, D.: Modeling Brain Function. Cambridge: Cambridge University Press, 1989 [A-G-S] Amit, D., Gutfreund, H. and Sompolinsky, H.: Statistical Mechanics of Neural Networks. Annals of Physics 173, 30–47 (1987) [F-T] Feng, J., Tirozzi, B.: Capacity of the Hopfield model. J. Phys. A: Math.Gen. 30, 3383–3391 (1997) [H] Hopfield, J.: Neural Networks and Physical Systems with Emergent Collective Computational Abilities. Proc. Nat. Ac. Sci. 79, 2554–2558 (1982) [L] Loukianova, D.: Lower bounds on the restitution error of the Hopfield model. Prob. Theor. Relat. Fields, 107, 161–176 (1997) [Lo] Loeve, M.: Probability Theory. Amsterdam: D.Van Nostrand Comp. Inc., 1960 [M-P-V] Mezard, M., Parisi, G., Virasoro, M.A.: Spin Glass Theory and Beyond. Singapore: World Scientific, 1987 [N] Newman, C.: Memory capacity in neural network models: Rigorous lower bounds. Neural Networks I, 223–238 (1988) [T] Talagrand, M.: Rigorous Results for the Hopfield Models with Many Patterns. Prob. Theor. Rel. Fields, 110, 109–176 (1998) Communicated by Ya. G. Sinai

Commun. Math. Phys. 216, 179 – 193 (2001)

Communications in



On the Gribov Copy Problem for the Coulomb Gauge J. F. Grotowski1 , P. P. Schirmer2 1 Mathematisches Institut der Universität Erlangen–Nürnberg, Bismarckstr. 1 1/2, 91054 Erlangen, Germany.


2 Universidade de São Paulo, Instituto de Matemática e Estatística, Rua do Matão 1010,

São Paulo SP 05508-900, Brazil. E-mail: [email protected] Received: 9 November 1998 / Accepted: 10 August 2000

Abstract: We consider the problem of gauge degeneracy. In particular, for connections on three-dimensional Euclidean space with the structure group SU(2), we show that a large class of spherically symmetric connections in the Coulomb gauge have distinct gauge copies in the Coulomb gauge. 1. Introduction In abelian gauge theories such as electrodynamics, the Coulomb gauge condition is sufficient to fix all gauge degrees of freedom. It was first pointed out by Gribov in [Gr] that in non-abelian gauge theories one may be confronted with the problem of degeneracy of the Coulomb gauge, i.e. the gauge orbits can intersect the Coulomb gauge hypersurface at more than one point. Later Singer [Si] showed that this problem occurs whenever one tries to choose a global section of the configuration space of Yang-Mills potentials over a compact manifold with compact gauge group. In order to discuss further the problem of gauge degeneracy, we fix some general notation; this will be made more precise in Sect. 2. We consider a smooth, n-dimensional Riemannian manifold M with structure group G and associated Lie algebra G; we further consider connections A ∈ 1 (M, G), the space of G-valued 1-forms over M, and gauge copies of A: g ∗ A = gAg −1 − dg · g −1 with g ∈ 0 (M n , G) a gauge transformation. There are two basic questions which arise, namely that of determining when gauge degeneracy occurs, and that of determining whether there is a reasonable way of choosing a distinguished representative in some “nice” gauge, in particular in the Coulomb gauge, for each gauge orbit. When the structure group is abelian, the gauge orbits are linear and the problem of finding a representative in the Coulomb gauge reduces to solving the linear Laplace equation. If, on

180

J. F. Grotowski, P. P. Schirmer

the other hand, the structure group is nonabelian then the gauge orbits are complicated nonlinear objects and the question of finding Coulomb gauge representatives reduces to a non-linear equation of harmonic map type, for which no regularity theory is directly available. One of the fundamental difficulties in dealing with these questions is to decide on an appropriate class of connections A and gauge transformations g. For example, Uhlenbeck considers in [Uh] Sobolev spaces of connections, and gauge transformations having one more weak derivative than the connections. In this setting, she is able to show ([Uh, Theorem 1.3]) that the orbit of any connection A ∈ W 1,p (Bn , G), p ≥ n2 , defined on the unit ball Bn and whose curvature FA has sufficiently small Ln/2 -norm has a representative g ∗ A ∈ W 1,p (Bn , G) in the Coulomb gauge with g ∈ W 2,p (Bn , G). On the other hand, by considering a weaker notion of connections and gauge transformations, Dell’Antonio and Zwanziger [DZ2] were able to show the existence of gauge copies in the Coulomb gauge for any A ∈ L2 (Rn ) defined on the non-compact space Rn . Note that this result will also apply to configurations on Rn which decay so fast at infinity that they induce regular connections on S n , i.e. their result is also applicable in the noncompact setting. Due to the methods they employ, any additional regularity of the initial configuration is not a priori possessed by the gauge copy, in contrast to Uhlenbeck’s result. In the current paper we will produce smooth distinct gauge copies of smooth connections which can have arbitrarily large L2 -norms, for a restricted class of radially symmetric connections on R3 . Concerning the question of finding distinguished representatives of gauge orbits, the original hope of Gribov was that imposing the positivity of the so-called Faddeev–Popov operator F (A) = −∂i (Di (A)·) (see Sect. 4) would be sufficient to uniquely specify the gauge orbit representative inside the Coulomb gauge. Although this hope turned out not to be justified – see [He, DZ1] – this set of potentials nevertheless provides a natural domain, as it can be characterized as the set of local minima of the L2 -norm along the gauge orbit: I (g) =

1 ∗ 2 g AL2 . 2

Minimizing this functional over the the gauge orbits was precisely the technique used by Dell’Antonio and Zwanziger in [DZ2]. As a consequence then, the copy they produce always lies inside the so-called Gribov horizon and in particular, they produce distinct gauge copies provided the initial connection lies outside the horizon. That one can find an initial connection outside the horizon follows from the fact that the horizon is bounded in L2 in any given direction (see [Zw, Theorem 3]). In our setting the variational methods we employ ensure that the gauge copy we produce is always distinct from the original. We are also able to show that, for a large subclass of the potentials considered, the copy produced does in fact lie inside the Gribov horizon (see Sect. 4). The paper is organized as follows. In Sect. 2 we discuss notation, explain the symmetry being considered and show how the existence of a gauge copy reduces to the existence of a nontrivial solution to a particular second-order boundary value problem with degenerate coefficients. The symmetry considered is similar to that used in [Gr]. However, it is the topological triviality implied by our boundary conditions (see Sect. 3) α(0) = 0, α(∞) = 0 (a property essential to the functioning of our minimizing arguments) which is a distinguishing aspect of the current paper.

On the Gribov Copy Problem for the Coulomb Gauge

181

Our main theorem, the existence of a distinct gauge copy in the Coulomb gauge for a large class of symmetric potentials can now be stated (the Pauli matrices {Ta }a=3 a=1 are defined in (2.1)). Theorem 1.1. Consider connections on R3 with structure group SU (2) with components Ai = Aai Ta of the form Aai (x) =

xb f (r) − 1 iab , r r

where r = |x| (such connections lie in the Coulomb gauge hypersurface). For a large class of potentials f ∈ C ∞ ((0, +∞)) we can find a nontrivial gauge transformation g(x) = exp(θ(r)

xa Ta ) ∈ C ∞ (R3 , SU (2)) r

such that the gauge copy g ∗ A = gAg −1 − dg · g −1 is smooth, square-integrable, and also lies in the Coulomb gauge. In Sect. 3 we prove the required existence result by employing the direct method of the calculus of variations. We are required to work in particular weighted Sobolev spaces to obtain existence. We also need to obtain a rather detailed picture of the qualitative behaviour of our minimizer in order to understand its behaviour at the origin (the point where the coefficients degenerate), and hence be able to show that the solution obtained is in fact smooth. As mentioned above, in Sect. 4 we discuss the relation of our solution to the so-called Gribov horizon. 2. Spherically Symmetric Gauge Fields on R Let E denote a principal fiber bundle over an n-dimensional manifold M n with structural group G, a compact semi-simple Lie Group. We denote by G the Lie algebra of G and by [·, ·] its Poisson bracket. We shall assume in what follows that G is a matrix group. A Yang–Mills potential A is a G-valued 1-form on E compatible with the action of the structure group on E. We denote by C ∞ (M, Ad E ⊗ T ∗ M) the set of all smooth connections on E and by C ∞ (M, Aut(E)) the set of all smooth gauge transformations: these are sections of the bundle Aut (E) acting on the connections by conjugation. If A ∈ C ∞ (M, Ad E ⊗ T ∗ M) and g ∈ C ∞ (M, Aut(E)), then we define the gauge transformed connection g ∗ A as: g ∗ A = gAg −1 − dg · g −1 . The curvature FA = DA A ∈ C ∞ (M, Ad E ⊗ 2 T ∗ M), the 2-form obtained by taking the covariant derivative DA = d + [A, ·] of the connection A, changes accordingly to: Fg ∗ A = gFA g −1 . In this paper we shall consider the case when M = Rn and E is the trivial bundle E = Rn ×G. In this case we can identify the spaces of connections and gauge transformations with the space C ∞ (Rn , 1 G) and C ∞ (Rn , G) of smooth G-valued 1-forms A = Ai dx i and smooth mappings g : Rn → G respectively. On Rn , the curvature FA = DA A is a G-valued 2-form FA = Fij dx i dx j given by the components Fij = ∂i Aj − ∂j Ai + [Ai , Aj ].

182


More generally, one can define the Sobolev spaces of connections and gauge transformations as: W s,p (Rn , 1 G) = {A : Ai ∈ W s,p (Rn )} , W s+1,p (Rn , G) = {g : g ∈ W s+1,p (Rn ), g(x) ∈ G, a.e.}. If p ≥ n/2, then the Sobolev embedding theorem shows that FA ∈ Lp in the case that A ∈ W 1,p , and that if p > n/2, then g ∗ A ∈ W 1,p provided A ∈ W 1,p and g ∈ W 2,p ; see [Uh, Lemma 1.2] for details. We consider the action of the rotation group SO(n) on the base manifold Rn , and all possible lifts of this action to the bundle E = Rn × G. Fixing one of these liftings, it is possible to say that a connection A = Ai dx i is spherically symmetric if it is invariant under the action of the rotation group SO(n) on the bundle. It has been established that all possible lifts of the action of SO(n) on Rn to the bundle E are in 1-1 correspondence with all possible endomorphisms λ : SO(n) → G (see e.g [KN, Theorem 11.5]). When n = 3 and G = SU(2), then, apart from the trivial map (which induces abelian connections), there is only one type of spherical symmetry. We fix a basis {Ta }3a=1 of the Lie algebra su(2): 1 01 1 0 −i 1 1 0 T1 = , T2 = , T3 = , (2.1) 2i 1 0 2i i 0 2i 0 −1 and write the components of A as Ai = Aai Ta . A connection A ∈ C ∞ (R3 , 1 su(2)) is spherically symmetric when the Lie derivatives of A along the infinitesimal generators Oa = abc xb ∂c of SO(3) verify the relations: LOa A + [Ta , A] = 0,

a = 1, 2, 3.

(2.2)

The general solution of (2.2) is a 3-parameter family of connections described by three radial functions f1 , f2 , f3 of r = |x|: xb f1 (r) − 1 x i xa x i xa f2 (r) iab + (δia − 2 ) + f3 (r) 2 . r r r r r The curvature FA = dA + [A, A] is given by: f1 f3 − f2 f + f 2 f3 xb xk xa Fija (x) = ij k kab + 1 (δka − 2 ) r r r r 1 2 x k xa 2 + 2 (f1 + f2 − 1) 2 . r r Aai (x) =

(2.3)

(2.4)

The class of spherically symmetric connections (2.3) is preserved by a residual gauge group consisting of all gauge transformations g ∈ C ∞ (R3 , SU (2)) of the kind: g(x) = exp(θ(r)

xa Ta ) , r

(2.5)

where θ is a smooth function on R+ . The action of such an element g on such connections is described by (g ∗ A)ai =

xb f1 cos θ + f2 sin θ − 1 xi xa f1 sin θ − f2 cos θ iab − (δia − 2 ) r r r r x i xa + (f3 − θ (r)) 2 . (2.6) r


183

From the expressions (2.3) and (2.4) it is easy to compute the L2 -norms of A and FA : A2L2 (R3 )

∞ = 4π [2(f1 (r) − 1)2 + 2f2 (r)2 + f3 (r)2 ]dr,

(2.7)

0

∞ FA 2L2 (R3 ) = 4π

2(f1 f3 − f2 )2 + 2(f1 + f2 f3 )2 +

0

(f12 + f22 − 1)2 dr. (2.8) r2

In this work, we shall restrict ourselves to the subfamily of connections of the form Aai (x) =

xb f (r) − 1 iab r r

(2.9)

with f ∈ C ∞ ((0, +∞)). These connections already satisfy the Coulomb gauge condition ∂i Ai = 0. If g is of the form (2.5) with θ ∈ C ∞ ((0, +∞)), then g ∗ A ∈ C ∞ ((0, +∞)). It is natural to ask for assumptions on the gauge transformation g which will ensure that the connection g ∗ A will lie in L2 (R3 ), provided A ∈ L2 (R3 ) initially. We claim that this will be the case when θ ∈ H 1 (R3 ) and f approaches 1 so fast that FA has finite L2 -norm on the sphere S 3 . Note that this means that, in particular, ∞

(f 2 − 1)2 r2

0

+ f (r)2 (1 + r 2 ) dr < ∞ .

(2.10)

This inequality implies that f is bounded on (0, +∞) and the claim then follows from considering (2.7) in the light of (2.6) and (2.10). Such properties will play a role in fixing our class of admissable potentials S in Sect. 3. Computing the divergence of the connection g ∗ A, we have:

xa 2 2 sin θ f (r) (2.11) Ta ∂i (g ∗ Ai ) = − θrr + θr − r r2 r and then, by setting θ (r) = 2α(r), we see that g ∗ A will also be in the Coulomb gauge if α(r) is a solution to the ordinary differential equation: 2 sin 2α f (r) = 0 αrr + αr − r r2

(2.12)

which satisfies the conditions α(0) = α(∞) = 0. We shall study this problem in the next section by using the direct method of the calculus of variations. 3. The Variational Problem In this section we are concerned with finding particular solutions of (2.12) on [0, ∞). Here f is a given function in a subclass of C ∞ [0, ∞) which we will describe shortly. Equation (2.12) is the Euler–Lagrange equation associated with the functional ∞ 1 sin2 α Jf (α) = ( αr2 + f )r 2 dr, 2 r2 0

(3.1)

184


where there is no risk of confusion we will omit the subscript f . We will look for solutions of (2.12) in a particular weighted Sobolev space, which we now define. We define ||u||L2;δ by 1 (1 + |x|2 )δ |β(x)|2 dx 2 , ||u||L2;δ := R3

and ||u||H 1;δ by ||u||H 1;δ :=

(1 + |x|2 )δ |β(x)|2 + (1 + |x|2 )1+δ |∇β(x)|2 ]dx

1 2

.

R3

The space L2;δ is then defined to be the space of functions for which ||u||L2;δ is finite, and H 1;δ is the space of functions for which ||u||H 1;δ is finite (the latter can of course be characterized as the subspace of functions in L2;δ with weak derivatives in L2;1+δ ; we refer the reader to [Ba, Sect. 1] for a description of such spaces and a summary of their properties. 1;δ , by which we denote radial We will look for solutions of (2.12) in the space HR 1;δ functions in H . We will further consider a fixed δ satisfying 21 < δ < 1 (as will be seen in the proof, any fixed δ in this range allows us to show the desired continuity properties of our functional J ). We now specify the class of potentials which we will consider. We denote by S those functions f in C ∞ [0, ∞) for which (2.10) holds, and for which there further hold: ∞ (f − 1)2 (1 +

1 ) dr r2

< ∞, and

(3.2)

0

inf Jf (β) < 0.

1;δ β∈HR

(3.3)

In the first instance, these conditions should be regarded as analytic constraints on f , imposed in order to obtain our desired existence result. As we discussed in Sect. 2, they are at least partially geometrically motivated: (2.10) can be viewed as the condition that FA be in L2 (S 3 ). The condition (3.2) can be thought of additionally requiring that A be 1,2 in Hloc near the origin in R3 . We of course need to verify that the class S is nonempty; this follows directly by a perturbation argument, once we establish: Lemma 3.1. There exists f , uniformly Lipschitz continuous on [0, ∞), which satisfies (2.10), (3.2) and (3.3). Proof. We consider a potential f (= fε,ρ,d,κ ) defined by  0≤r 1. Direct calculation shows that f satisfies 1;δ function defined by (2.10) and (3.2). We now consider β(= βµ,ρ,ε,κ ) to be the HR  0     µε (r − ρ + ε) β(r) := µ µ     ε (κρ + ε − r) 0

0≤r rn , ∞ 1/2 rm 2 rn 1/2 f −1 1 dr 2 2 · dr + (f ) r dr |f (rn ) − f (rm )| ≤ 2 r r r2 2

2

rn

0

rm

1/4 ∞ 1/2 ∞ 2 1/4 (f − 1)2 1 1 2 2 ≤2 (f ) r dr dr − r2 rn rm 0

+

1 1 − rn rn

0

1/2 .

Allowing first rm then rn to tend to ∞ and noting (2.10), we conclude that lim f 2 (r) r→∞ exists and by (2.10) this must be equal to 1. By (3.2) we in fact have lim f (r) = 1. r→∞

For convenience, we consider r1 = inf{r|f (r) < 0}, r2 = sup{r|f (r) < 0} in (3.5). We now describe the properties of the functional J , in order to be able to realize our 1;δ . desired solution as a minimizer of J over HR 1;δ , bounded from below. Lemma 3.3. J is a C 1 -functional on HR

Proof. From (3.5) we see that J is bounded from below by −f ∞ (r2 − r1 ). 1;δ , we begin by noting that, by continuity To establish that J is continuous on HR 1 2 2 of the Dirichlet functional α → 2 αr r dr, we only need to show continuity of the ∞ 1;δ nonlinear functional N (α) = f (r) sin2 α(r)dr. We have, for β and γ in HR and for arbitrary positive ε:

0

|N (β) − N (γ )| ≤ 2f ∞

ε

∞ | sin β − sin γ |dr +

| sin β − sin γ |dr .

ε

0

We estimate, using Hölder’s inequality ∞ | sin β − sin γ |dr ≤ ε

∞

|β − γ | r dr 2 2δ

ε

√ 2 π ε 1/2−δ = √ 1 − 2δ

1/2 ∞

ε

{x∈R3 | |x|>ε}

≤

ε 1/2+δ

1 √

1 − 2δ

r −2δ dr

1/2

|β − γ |2 2δ 1/2 |x| dx |x|2

β − γ H 1;δ .

In the light of (3.6) this establishes the continuity of N , and hence of J . Formally, for ϕ ∈ C0∞ (R3 ) radial we have sin 2α 2αr J (α)ϕ = − (αrr + f )ϕ dx. 3 r r2 R

(3.6)


187

Since α → 9 α is continuous, we only need to check the continuity of N (α), where sin 2α(|x|) N (α)ϕ = f (|x|)ϕ(|x|) dx. |x|2 R3

1;δ we have, using Hölder’s and Sobolev’s inequalities: For β, γ ∈ HR

N (β) − N (γ ) = sup 1;δ ϕ∈HR

|N (β) − N (γ ), ϕ| ϕH 1;δ (R3 )

ϕL6 (R3 ) sin 2β − sin 2γ 6/5 3 L (R ) r2 ϕH 1;δ (R3 ) sin 2β − sin 2γ ≤ f ∞ 6/5 3 . 2 L (R ) r ≤ f ∞

(3.7)

We have, for arbitrary real ε, ε ∞ sin 2β − sin 2γ 6/5 | sin 2β − sin 2γ |6/5 | sin 2β − sin 2γ |6/5 = dr+ dr. 6/5 3 2 L (R ) r r 2/5 r 2/5 ε

0

The first of these integrals is bounded above by 5ε3/5 and the second, using Hölder’s 6/5 inequality, by (3δ)−2/5 ε 6(δ−1)/5 β −γ H 1;δ . Since ε is arbitrary, we conclude from (3.7) the desired continuity of N (α). By taking γ ≡ 0 in (3.6) we obtain the estimate

1 |N (β)| ≤ 2||f ||∞ 2ε + 1 √ ||β||H 1;δ , ε 2 +δ 1 − 2δ whence J (β) ≥

1 ||β||2H 1;δ − c1 − c2 ||β||H 1;δ 2

for constants c1 , c2 depending only on ||f ||∞ and δ. Thus for a sequence {αn } minimizing 1;δ J in HR we see that {αn } is bounded in H 1;δ , and hence possesses a subsequence (which we shall also denote by {αn }) which is weakly convergent in H 1;δ , strongly convergent in 1;δ L2;δ and pointwise convergent almost everywhere to a function α ∈ HR . In particular this means that N (αn ) → N (α). Hence, due to the weak lower semicontinuity of the norm on H 1;δ , we have J (α) ≤ lim J (αn ) ; n→∞

this implies equality, since {αn } is a minimizing sequence. Thus the convergence is in fact strong in H 1;δ , and α is a minimizer. Since (2.12) is the Euler–Lagrange equation associated with J , Lemma 3.2 (i) shows that α satisfies (2.12) on (0, ∞). Further by (3.3) we see that α is not identically zero. 1;δ satisfies either 0 < α < π/2 on (0, ∞) or Lemma 3.4. A minimizer α of J over HR −π/2 < α < 0 on (0, ∞).

188


Proof. We first establish that both lim inf r→0+ α(r) and lim inf r→∞ α(r) exist and are finite. If the first of these quantities is not finite, we can find r ∗ ∈ (0, r1 ) with α(r ∗ ) = :π for some : ∈ Z. Defining α1 by α1 (r) =

:π α(r)

r < r∗ r ≥ r∗

1;δ we see α1 ∈ HR , J (α1 ) < J (α) (recall f > 0 on [0, r ∗ ]) which is a contradiction. Arguing analogously we also show the finiteness of lim inf r→∞ α(r). We next show that α is bounded on [0, ∞). From the regularity of f and the above, α is bounded on the compact set [0, r2 ]. We define

 α(r)   pπ α2 (r) =   α(r) (p + 1)π

r r r r

≤ r2 > r2 , > r2 , > r2 ,

α(r) < pπ pπ ≤ α(r) ≤ (p + 1)π α(r) > (p + 1)π,

1;δ , where p ∈ Z is specified by the condition pπ ≤ α(r2 ) < (p + 1)π . We see α2 ∈ HR and J (α2 ) < J (α) with strict inequality unless α2 ≡ α. Thus pπ ≤ α < (p + 1)π on [r2 , ∞), i.e. α is bounded on [0, ∞). Note that we must have p = 0 or p = −1 since 1;δ . α ∈ HR We argue as above to show the existence of q ∈ Z such that qπ ≤ α(r) < (q + 1)π on (0, r1 ]. Lemma 3.2 (ii) in conjunction with a straightforward reflection argument shows that we must have p = q. A similar argument shows that we must have either 0 < α(r) < π/2 (in the case that p = 0) or −π/2 < α(r) < 0 (in the case that p = −1) on (0, ∞), which completes the proof.

We can use Lemma 3.4 to give a precise description of the qualitative behaviour of our minimizer α; this result, and the subsequent comparison argument (Lemma 3.6) are needed to establish the desired boundary condition α(0) = 0. 1;δ Lemma 3.5. Let α be a minimizer of J over HR . Then α is monotone increasing on [0, r1 ) and monotone decreasing on (r2 , ∞) or vice versa.

Proof. We consider the case α > 0 on (0, ∞): the case α < 0 follows analogously. Straightforward maximum principle arguments applied to (2.12) show that the only possible local extrema of α are minima in (0, r1 ) and (r2 , ∞), and maxima in (r1 , r2 ) (note in particular that we can apply the Hopf maximum principle to rule out critical points at r1 and r2 ). This also establishes the existence of lim α(r). r→0

If we have a local minimum at r¯ ∈ (0, r1 ) we consider α8 defined by α3 (r) =

α(¯r ) α(r)

0 ≤ r < r¯ r ≥ r¯ ,

1;δ and see α3 ∈ HR ; J (α3 ) < J (α) (note that there can be at most one such r¯ ), contradicting the minimality of α. Hence α is monotone on (0, r1 ]. We argue similarly to rule out α being monotone decreasing on this interval. Thus α is monotone increasing on (0, r1 ).


189

1;δ Since α is smooth on [r2 , ∞), in HR and only permitted local minima in this region, we see immediately that α is in fact monotone decreasing to 0 as r tends to infinity. We are now in a position to prove the desired behaviour of α at zero (obviously, we define α(0) = lim α(r)). r→0

1;δ , we have α(0) = 0. Lemma 3.6. For α a minimizer of J over HR

Proof. As usual, we restrict to the case α > 0, the case of negative α following analogously. We consider α(0) = 0; by Lemma 3.5, we see that 0 < α(0) < π/2. We will 1;δ argue by contradiction, constructing β ∈ HR with J (β) < J (α) and 0 < β(0) < α(0). We write α0 for α(0). Noting sin 2α0 − sin(2α0 − 2γ ) = 2γ cos 2α0 + 2γ 2 sin 2α0 + O(γ 3 ), we can find γ0 (depending on α0 ) such that sin 2α0 − sin(2α0 − 2γ ) > 2γ cos 2α0 for all γ ∈ (0, γ0 ). We fix γ positive, γ < min{γ0 , 21 sin 2α0 , 1}, and then fix ε such that 1 2 0 < ε < 30 γ sin 2α0 . The continuity of f and α at 0 ensures the existence of ε˜ > 0 such that, for r ∈ [0, ε˜ ], there holds: |f (r) − 1| < ε,

and

0 ≤ α(r) − α0 < ε/2 .

For a comparison function we define α0 − γ + 1ε˜ (α(˜ε ) − α0 + γ )r β(r) = α(r)

0 ≤ r ≤ ε˜ r > ε˜ ;

note that β ∈ H 1;δ . Given our choices of γ , ε and ε˜ , elementary calculations show that J (β) < J (α), which is the desired contradiction. The final step in the proof of Theorem 3.1 is to establish the smoothness of the minimizer α at 0. In order to establish C 1 regularity we use arguments similar to those of the proof of [Sh, Lemma 4.1]. 1;δ Lemma 3.7. Any minimizer of J over HR is smooth at 0.

Proof. From Lemmas 3.5 and 3.6 we know that, for such a minimizer α, we have α(0) = 0, and we can restrict to the case that α is monotone increasing on [0, r1 ). Multiplying (2.12) by α and integrating by parts we have, for 0 < ε < ε small, ε

αr2 −

ε

1 α sin 2α 2ααr dr = ∂r (α 2 )|εε . + 2 r r 2

(3.8)

2 )| ≥ 0 from Lemma 3.6, we have that (3.8) is uniformly Noting that lim inf (α ∂ r ε ε →0

bounded for ε small. From Lemma 3.6 we see, for ε sufficiently small: 0≤

ε

αr2

0

α2 + 2 dr ≤ 2 r

ε

αr2 −

0

α sin 2α αr α dr, + r r2

(3.9)

190


which is uniformly bounded by the uniform boundedness of (3.8). We now rewrite (2.12) as 2 2α sin 2α 2α αrr + αr − 2 = − 2. r r r2 r

(3.10)

Since α(0) = 0 and α ∈ C γ for any γ < 21 , we have that the right-hand side of (3.10) 2α is integrable on (0, ε); from (3.10) then, so is ∂r (αr + 2α r ). Thus αr + r is absolutely continuous on [0, ε), and in particular, there exists a finite limit

lim

r→0+

αr +

2α . r

Since both these terms are nonnegative, αr must also have a finite limit as r → 0+ , i.e. α ∈ C 1 [0, ∞). Higher regularity then follows from the fact that solutions to (2.12) solve the equation 9α =

sin 2α f r2

in R3 (considering r := |x|). Since the right-hand side of this equation belongs to p Lloc (R3 ) for any p > 1, we can apply standard elliptic regularity theory (see e.g. [GT, Theorem 8.15]) to conclude that α ∈ W 2,p (R3 ), and hence in C 1,γ for some γ ; a standard bootstrap argument then yields that α is everywhere smooth. By the above remarks, this completes the proof of Theorem 3.1. 4. The Faddeev–Popov Operator Given a connection A ∈ C ∞ (Rn , 1 G), the so-called Faddeev–Popov operator is defined by DA = −∂i (Di (A)·); here DA : dom(DA ) ⊆ L2 (Rn , G) → L2 (Rn , G). This operator is the second variation of the functional F(g) =

1 ∗ 2 g AL2 (Rn ) , 2

that is to say, if we compute the Hessian matrix of F at the identity, along two directions gi = exp(ψi ), with ψi ∈ G, then: D 2 F(id) · (ψ1 , ψ2 ) = DA ψ1 (x) · ψ2 (x) dx, Rn

and therefore, if F attains its minimum along a gauge orbit, then DA is a positive-definite operator. When A is in the Coulomb gauge, DA takes the form DA = −9 − [Ai , ∂i ·]

(4.1)


191

Restricting to configurations satisfying (2.9), it is not difficult to see that (4.1) becomes 2f , r2

(4.2)

xa Ta , r

(4.3)

DA = −9 + when acting on vectors of the form: ψ(x) = ψ(r)

corresponding to the allowable gauge transformations (2.5). Noting that F(g) =

1 ∗ 2 1 g AL2 = A2L2 + 2J (α), 2 2

we conclude that the operator (4.2) will be positive definite when acting on radial vectors of the form (4.3). It is our goal to show that our minima are not only minima with respect to radial variations, but in fact global minima in H 2 (R3 , G). This can be seen by demonstrating that the terms with higher angular momentum will not effect the positivity of the restricted Faddeev–Popov operator. More precisely, we will prove: Proposition 4.1. For f ∈ S satisfying −1 ≤ f ≤ 3, the full Faddeev–Popov operator (4.1) is a positive-definite operator on H 2 (R3 , G). Proof. We shall consider an expansion of ψ ∈ H 2 (R3 , su(2)) in vector spherical harmonics (see e.g. [DM]): ψ(x) = ψj :m (r)Yja:m (ω)Ta , (4.4) j,:,m: j ≥0 |j −:|≤1 |m|≤:

where {Yja:m (ω)}3a=1 form an orthonormal basis of L2 (S 2 , su(2)) (we will assume that the restrictions on j ,: and m given in (4.4) hold in any summations considered for the remainder of this section). The radial vectors (4.3) correspond to the vector spherical harmonic j = 0 for which the only surviving modes are: a Y010 =

xa , r

Using the standard properties of the vector spherical harmonics we find: :(: + 1) −9ψ(x) = −9+ ψj :m (r)Yja:m (ω)Ta . r2

(4.5)

j,:,m

The non-abelian term [Ai , ∂i ψ] can be calculated using the fact that A is radial and that [Ta , Tb ] = abc Tc : [Ai , ∂i ψ] = abc Abi ∂i ψ c Ta =

f (r) − 1 (T · O)ψ, r2

where T · O = Ta Oa is the so-called spin-orbit coupling operator (see [DM]). Therefore [Ai , ∂i ψ] =

f (r) − 1 (j (j + 1) − :(: + 1) − 2)ψj :m (r)Yja:m (ω)Ta , 2r 2 j,:,m

192


and so, using the orthogonality on S 2 , we have

DA ψ · ψdx =

j,:,m

R3

∞ :(: + 1) 4π (−9 + )ψj :m (r)ψj :m (r)r 2 dr r2

∞ −

0

f (r) − 1 (j (j + 1) − :(: + 1) − 2)ψj2:m (r)dr 2

0

= I + II + III + IV, where I is the term corresponding to j = 0, and II, III and IV correspond to the terms with j ≥ 1 for : = j , : = j − 1 and : = j + 1 respectively. We calculate (keeping in mind (4.5)) ∞ 2f xa xa 2 I = 4π (−9 + 2 )ψ010 (r)ψ010 (r)r dr = DA (ψ010 (r) Ta ) · (ψ010 (r) Ta )dx r r r R3

0

which is nonnegative by the positivity of DA when acting on radial vectors. The second term, where j = :, is given by II = 4π

∞

r 2 (∂r ψ::m )2 + [:(: + 1) + (f (r) − 1)]ψ::m (r)2 dr,

:=j,m 0

since : ≥ 1 and f (r) ≥ −1, this term is nonnegative. The third term is III = 4π

∞

:=j −1,m 0

r 2 (∂r ψj :m )2 + :[: − f (r) + 2)]ψj2:m dr,

since f ≤ 3, for any j ≥ 2 the summands are nonnegative, and for j = 1 the last term is zero. Finally, we observe that the last term is even easier to estimate:

IV = 4π

∞

:=j +1,m 0

r 2 (∂r ψj :m )2 + (: + 1)(f (r) + : − 1)ψj2:m dr,

which is nonnegative by inspection; this concludes the proof. Acknowledgements. The second author’s work was supported by grant CNPQ 301228/96-5. Much of this work was completed during a visit of the first author to the Universidade de São Paulo, supported by Pronex grant: geometric differential equations, and Sonderforschungsbereich 288: Differentialgeometrie und Quantenphysik of the Deutsche Forschungsgemeinschaft. He wishes to thank the staff and faculty of the Instituto de Matemática e Estatística, in particular the first author, for their hospitality.


193

References [Ba]

Bartnik, R.: The Mass of an asymptotically flat manifold. Comm. Pure Appl. Math. 39, 661–693 (1986) [DM] Daumens, M., Minnaert, P.: Tensor spherical harmonics and tensor multipoles. I. J. Math. Phys. 17, 1903–1909 (1976) [DZ1] Dell’Antonio, G., Zwanziger, D.:All gauge orbits and some Gribov copies encompassed by the Gribov Horizon. In: Daamgard, P. et al. (eds.) NATO Adv. Research Workshop in Probabilistic Methods in Quantum Field Theory and Quantum Gravity. Proceedings, 1989, New York: Plenum Press, 1990, pp. 107–130 [DZ2] Dell’Antonio, G., Zwanziger, D.: Every Gauge Orbit Passes Inside the Gribov Horizon. Commun. Math. Phys. 138, 291–299 (1991) [GT] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order (2nd Edition). Berlin–Heidelberg–New York–Tokyo: Springer–Verlag, 1983 [Gr] Gribov, V.N.: Quantization of non-Abelian gauge theories. Nucl. Phys. B 139, 1–19 (1978) [He] Henyey, F.S.: Gribov Ambiguity without topological charge. Phys. Rev. D 20, 1460–1462 (1979) [KN] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 1. New York: Interscience Publishers, 1963 [Sh] Shatah, J.: Weak Solutions and Development of Singularities of the SU (2) σ -Model. Comm. Pure Appl. Math. 41, 459–469 (1988) [Si] Singer, I.M.: Some remarks on the Gribov ambiguity, Commun. Math. Phys. 60, 7–12 (1978) [Uh] Uhlenbeck, K.K.: Connections with Lp Bounds on Curvature. Commun. Math. Phys. 83, 31–42 (1982) [Zw] Zwanziger, D.:Non-perturbative modification of the Faddeev–Popov formula and banishment of the naive vacuum. Nucl. Phys. B 209, 336–348 (1982) Communicated by A. Jaffe

Commun. Math. Phys. 216, 195 – 213 (2001)

Communications in



First KdV Integrals and Absolutely Continuous Spectrum for 1-D Schrödinger Operator S. Molchanov1 , M. Novitskii2 , B. Vainberg1 1 Mathematics, UNCC, Charlotte, NC 28223, USA.

E-mail: [email protected]; [email protected]

2 Mathematics, ILT, Kharkov, 310164, Ukraine.

E-mail: [email protected] Received: 16 June 2000 / Accepted: 11 August 2000

Abstract: We consider 1-D Schrödinger operators on L2 (R+ ) with slowly decaying potentials. Under some conditions on the potential, related to the first integrals of the KdV equation, we prove that the a.c. spectrum of the operator coincides with the positive semiaxis and the singular spectrum is unstable. Examples show that for special classes of sparse potentials these results can not be improved.

Introduction Let H = H α be the Hamiltonian on L2 (R+ ) given by H ψ = −ψ + V (x)ψ,

ψ(0) cos α − ψ (0) sin α = 0,

x ≥ 0,

(1)

α ∈ S = [0, π ).

(2)

1

It is assumed that the potential V vanishes at infinity in some sense (see (4)), and from this decay it follows that the Birman condition: lim

x+1

x→∞ x

|V (x)|dx = 0

(3)

holds. Then (see the original paper [2] or the monograph [9]) the operator H α is essentially self-adjoint, and: a) the negative part of the spectrum of H α is bounded from below and discrete; b) the essential spectrum of H α coincides with the positive semiaxis: ess = [0, ∞).

196

S. Molchanov, M. Novitskii, B. Vainberg

Let µα (dλ) be the spectral measure of H α , i.e. H α is unitarily equivalent to the operator of multiplication by λ in the space L2 (R, µα (dλ)). A direct method of construction of the measure will be recalled in the next section. The paper concerns the positive part of the spectrum of the operator H α . Let H p , p = 0, 1, 2, . . . , be the space of functions on R+ whose derivative of order p belongs to L2 . The simplest form of our result is given by Theorem 1. Let V ∈ H p−1 ∩ Lp+1 for some integer p ≥ 1, i.e ∞ Jp (V ) = (|V (p−1) (x)|2 + |V (x)|p+1 ) dx < ∞.

(4)

0

Then: 1) for all α ∈ [0, π), the essential support of the absolutely continuous component of the spectral measure µα (dλ) of the operator H α coincides with [0, ∞), i.e. µαac () > 0 for any Borel set ⊂ [0, ∞), || > 0; 2) for a.e. α ∈ [0, π), the spectral measure µα (dλ) of the operator H α is absolutely continuous on [0, ∞). For p = 1, the first part of this theorem is identical to the main result of the paper [5] by P. Deift and R. Killip. The paper [5] by itself is a culmination of a long sequence of publications (see for instance [3, 4, 18]) on spectral L2 -conjecture. The central point of the proof in [5] was a relationship between the transmission coefficients of the operators with fast decaying potentials and L2 -norm of their potentials. The L2 -norm of a solution is a first integral of the KdV equation Vt = 6V Vx − Vxxx . The idea to use the high order first integrals: ∞ Ik (V ) = Pk (V , V , V , . . . , ) dx (5) 0

in a similar setting is sitting on the surface. Here Pk are certain polynomials (see [4]). 2 − 5 V 2V + 5 V 4. In particular, P1 = V 2 , P2 = (V )2 /2 + V 3 , P3 = 21 Vxx xx 2 2 Our main goal here is an implementation of this idea and its development in several directions: a) clarification of the proof by separation of the measure theory and KdV ideas; b) simplification of the result (instead of very complicated expressions for Ik , k = 1, 2, . . . , we use very simple functionals Jk , k = 1, 2, . . . ); c) construction of several classes of examples where the spectral bifurcation ac ⇒ pp is related to the transition Jk < ∞

⇒

Jk = ∞,

k = 1, 2, . . . .

The outline of the paper is the following. First we obtain a new test for the existence of the a.c. spectrum of the Schrödinger operator based on the boundedness of the transfer operator in average (with respect to the spectral parameter). Then a formula is given for the norm of the transfer operator in terms of the transmission coefficient. This provides the existence of the a.c. spectrum if there is a sequence Ln → ∞ such that the transmission coefficients of the operators with potentials Vn are bounded in average. Here Vn is the restriction of V to the segment [0, Ln ]. In Sect. 2, the transmission coefficients are estimated in terms of the first integrals of the KdV equation. Inequalities of the Kolmogorov type allow us to estimate the first integrals by the functionals Jp . A short

First KdV Integrals and AC Spectrum for 1-D Schrödinger Operator

197

Sect. 3 contains the end of the proof of Theorem 1. The last section provides examples showing the “exactness” of Theorem 1. In the spirit of general functional analysis one could expect that the first term of the integrand in condition (4) dominates and the second term can be omitted. However, the ∞ statement that 0 |V (p−1) (x)|2 dx < ∞ for some p ≥ 2 implies the existence of the a.c. spectrum on [0, ∞), is obviously wrong. If, for instance , p = 3, V1 is an infinitely smooth, compactly supported (or fast decaying) potential and V (x) = V0 (x) + V1 (x),

V0 (x) = αx,

α > 0,

then V ∈ L2 (R+ ), but the spectrum of H = −d 2 /dx 2 + V (x) is discrete for any α ∈ [0, π). Using piecewise smooth (and mainly linear) functions V0 (x) instead of αx, one can construct examples when V ∈ L2 (R+ ), but the spectrum is a.c. above an arbitrary fixed level h0 = sup V0 (x) > 0, and is singular below this level. We need some additional conditions on the potential V (x) toguarantee that ac = ∞ [0, ∞). We expect that the condition |V (x)| → 0 together with 0 |V (p−1) (x)|2 dx < ∞ for some p ≥ 1 is sufficient for ac (H ) = [0, ∞) in many interesting cases. Even the more general statement is probably true: If ||V ||∞ < ∞ , h0 = lim supx→∞ V (x) and V (p) ∈ L2 (R+ ) for some p ≥ 1, then the spectrum of H = −d 2 /dx 2 + V (x) is singular for λ ≤ h0 and contains the a.c. component which is essentially supported on [h0 , ∞). We shall return to the analysis of this conjecture in a different publication. 1. Tests for the A.C. Spectrum 1.1. Spectral measure . Let ψ = ψλ (x) be the solution of the following initial value problem on R+ : H ψ = λψ,

λ > 0,

ψ(0) = sin α,

ψ (0) = cos α,

α ∈ [0, π ).

(6)

The spectral measure µα (dλ) of the operator H = H α can be defined in terms of the generalized Fourier transform and the Parseval equality [12]: ∞ L2 (R+ ) ϕ(x) → ϕ(λ) ˆ = ϕ(x)ψ ˆ (7) λ (x)dx, 0 α ϕ(x) = ϕ(λ)ψ ˆ (8) λ (x)µ (dλ), (H α )

ˆ ϕ(λ)) ˆ (ϕ(x), ϕ(x))L2 (R+ ) = (ϕ(λ), L2 ((H α ),µα ) .

(9)

Here (H α ) is the spectrum of H α . We denote by Tλ (x1 , x2 ) the transfer operator on the interval [x1 , x2 ]:

Tλ (x1 , x2 ) : (ψλ (x1 ), ψλ (x1 )) → (ψλ (x2 ), ψλ (x2 )).

(10)

The following formula (attributed to M. Krein’s school, see, for example, [1]) gives an expression for the spectral measure, and is very important for our analysis: µα (dλ) = w − lim µαL (dλ), L→∞

(11)

198


where µαL (dλ) =

dλ ρλ2 (L)

(12)

and ρλ2 (x) = ||Tλ (0, x)eα ||2 ,

eα = (sin α, cos α).

(13)

1.2. The a.c. spectrum and the transfer operator. Let % ⊂ (0, ∞) be a closed interval. It is well-known that the spectrum of H α is pure a.c. on % if the transfer operator Tλ (0, L) is bounded uniformly in L ∈ (0, ∞) and λ ∈ % (see [20, 11] and references there). A part of this statement, namely, the existence of the a.c. spectrum can be justified easily using (11) and (12). In fact, assume that ||Tλ (0, Ln )|| ≤ c0 ,

λ∈%

(14)

for some sequence {Ln }, Ln → ∞, and a fixed interval % ⊂ (0, ∞), |%| > 0. Then µαL (dλ) 1 ≥ 2, dλ c0

λ ∈ %,

(15)

and therefore, the same estimate is valid for the limit measure µα (dλ). One of our central observations is that the uniform (14) or even a pointwise estimate can be replaced by boundedness in average. Theorem 2. Let % ⊂ (0, ∞) be a closed interval. If there is a sequence {Ln }, ∞, such that ln (||Tλ (0, Ln )||) dλ ≤ C(%), %

Ln → (16)

then i) for all α ∈ [0, π), the essential support of the measure µαac (dλ) contains %; ii) for a.e. α ∈ [0, π), the spectral measure µα (dλ) of the operator H α is absolutely continuous on %. Remark. Theorem 2 remains valid if the logarithm function in (16) is replaced by any monotonically increasing on [1, ∞) function '(λ). In order to prove the theorem we need Lemma 1. Let νn (dλ), n = 1, 2, . . . , be a sequence of measures on an interval %, absolutely continuous with respect to dλ with the densities νn (dλ) = hI%\n (λ)dλ, dλ

n = 1, 2, . . . .

Here h is a constant, IB (λ) is the characteristic function of a Borel set B, and |n | ≤ ε. Then there is a subsequence νnl (dλ) which converges weakly to a measure ν(dλ), and any such weak limit ν(dλ) has the following properties:


199

1) the measure ν(dλ) is absolutely continuous with respect to dλ, and ν(dλ)/dλ ≤ h; 2) there exists a set such that || ≤ 2ε and ν(dλ) h ≥ dλ 2

on % \ .

λ Proof. Let Fn (λ) = −∞ νn (ds) be the distribution function of the measure νn (dλ). By the second Helly theorem there exists a subsequence νnl (dλ) which converges weakly to a measure ν(dλ), and for any such subsequence the distribution functions Fnl (λ) converge to the distribution function F (λ) of the limit measure at all points λ, where F (λ) is continuous. The set where F is discontinuous is at most countable. Consider an interval [α, β) ⊂ % with α, β outside of this set. Then β β νnl (α, β) = Fnl (β) − Fnl (α) = νnl (dλ) ≤ hdλ = h(β − α). α

α

Passing to the limit l → ∞, we get ν(α, β) ≤ h(β − α). This estimate implies that the measure ν(dλ) is absolutely continuous on % and ν(dλ)/dλ ≤ h. To prove the second statement of the lemma we note that νnl (%) ≥ h(|%| − ε) and therefore, ν(%) ≥ h(|%| − ε). On the other hand, if

(17)

ν(dλ) h = λ: ≤ , dλ 2

then ν(%) ≤

h || + h(|%| − ||). 2

(18)

Inequalities (17) and (18) imply the second statement of the lemma. This completes the proof of the Lemma. Proof of Theorem 2. We shall prove Theorem 2 and the Remark simulteneously, i.e. we shall assume that the logarithm function in (16) is replaced by a positive monotonically increasing function '. Fix ε > 0. Let Rε be the solution of the equation '(Rε ) =

C(%) , ε

where C(%) is defined in (16). Let n (ε) = {λ : λ ∈ %, Since '(t) >

C(%) ε

||Tλ (0, Ln )|| ≥ Rε }.

when t > Rε , we have

%

'(||Tλ (0, Ln )||)dλ ≥ '(Rε )|n | =

C(%) |n |. ε

200


From here and (16), it immediately follows that |n (ε)| ≤ ε. Consider a sequence of measures νn,ε (dλ) on % defined by the formula νn,ε (dλ) =

I%\n (ε) (λ) dλ. Rε

µLn (dλ) =

dλ ||Tλ (0, Ln )eα ||2

Let

be the measure defined in (12) with L = Ln . Then µLn (dλ) ≥ νn,ε (dλ),

n = 1, 2, . . . ,

λ ∈ %.

Let νε (dλ) be a weak limit of some subsequence νnl ,ε (dλ), follows from Lemma 1. From (11) and (19) it follows that µα (dλ) ≥ νε (dλ),

(19)

l → ∞. Its existence

λ ∈ %.

Lemma 1 applied to the sequence νn,ε (dλ) leads to the estimate I%\ε νε (dλ) ≥ , dλ Rε

λ ∈ %,

with some ε such that |ε | ≤ 2ε. Thus, the same estimate holds for µα (dλ). Since ε > 0 is an arbitrary small number, the essential support of µαac (dλ) contains the interval %. Part i) of Theorem 2 is proved. Now we shall prove Part ii) of Theorem 2. Let us consider first a specific Hamiltonian H π/2 (Neumann boundary conditions) and let µπ/2 (dλ) be its spectral measure. Under Birman’s condition (3) on the potential,

∞

λ0

µπ/2 (dλ) < ∞, 1 + |λ|

(20)

where λ0 is the bottom of the spectrum of H π/2 . This estimate can be found in [12], (Chapter 4) for much more general potentials and for boundary condition (2) with α = 0. Estimate (20) fails if α = 0. One could also prove a stronger estimate for µα (dλ):

λ

λ0

µα (dλ) ≤ Cα 1 + |λ| ,

λ ≥ λ0 (α), α = 0.

(21)

We shall not prove (21) here, since (20) is enough for our purpose. Estimate (20) allows us to define the Borel transform mπ/2 (z) of the spectral measure π/2 µ (dλ): ∞ π/2 µ (dλ) , z > 0. mπ/2 (z) = z−λ λ0 Function mπ/2 (z) is analytic in the half-plane z > 0. Due to the general theory of Borel transforms (see [19], Chapter I), mπ/2 (z) has the following properties:


a) for a.e.

201

λ ∈ R, 0 ≤ mπ/2 (λ + i0) < ∞;

|mπ/2 (λ + i0)| < ∞.

(22)

b) the absolutely continuous component of µπ/2 (dλ) is equal to π −1 mπ/2 (λ + i0)dλ; c) the singular component of the measure µπ/2 (dλ) is supported on the set {λ : mπ/2 (λ + i0) = ∞}. Consider the set = {λ : 0 < mπ/2 (λ + i0) < ∞}, and let

(23)

2 = [0, ∞) \ .

Part i) of Theorem 2 and the properties a) - c) above imply that |2| = 0 and the singular component of µπ/2 (dλ) is supported on 2. We shall show that the singular components of all measures µα (dλ), α ∈ (0, π ), are supported on the same set 2. Recall (see [12]) that the Weyl function mα (z) of the operator H α is defined by the relation ψ1 (x, z) + mα (z)ψ2 (x, z) ∈ L2 [0, ∞),

z > 0,

(24)

−ψ

+ V (x)ψ = zψ with where ψ1 (x, z) and ψ2 (x, z) are solutions of the equation initial conditions ψ1 (0) = sin α, ψ1 (0) = cos α, and ψ2 (0) = − cos α, ψ2 (0) = sin α. The following representation is valid (see [12]): µα (dλ) , z > 0, α ∈ (0, π ). mα (z) = cot α + (H α ) z − λ The properties a) – c) of the Borel transform of a measure are valid for mα (z). Direct calculations based on (24) give mα (z) =

cos α + mπ/2 (z) sin α . sin α − mπ/2 (z) cos α

(25)

Using (25), it is easy to check that 0 < mα (λ + i0) < ∞ at points λ where 0 < mπ/2 (λ + i0) < ∞. Hence, µαsing is supported on the αindependent set 2 of Lebesgue measure zero. The spectral averaging of µα (dλ) over α ∈ (0, π ) gives (see [1] or [19]) 1 π α µ (dλ) = dλ. π 0 This formula applied to the set 2 leads to 1 π α µ (2)dα = |2| = 0. π 0 Then µα (2) = 0 for a.e. α , i.e. µαsing (dλ) = 0 for a.e. α. Theorem 2 is proved.

202


1.3. The a.c. spectrum and the transmission coefficient . First of all we need a simple proposition which provides a connection between the transfer matrix and the scattering theory. In this section, it is convenient to replace the spectral parameter λ√> 0 by k 2 . Let φk = ψk 2 = ψλ , and let a new transfer operator Pk (x1 , x2 ), k = λ, (Pruffer transformation) be given by φk (x1 ) φk (x2 ) → φk (x2 ), . (26) Pk (x1 , x2 ) : φk (x1 ), k k It follows from the definition of Pk that, for λ = k 2 and D = diag(1, k), Tλ = D(k)Pk D −1 (k).

(27)

Lemma 2. Let us consider a compactly supported on (0, L) continuous potential V (x), and let ψ(x, k) be the Jost solution of the equation −ψ (x, k) + V (x)ψ(x, k) = k 2 ψ(x, k), x ∈ R, such that

exp(−ikx), x ≤ 0, ψ(x, k) = a(k) exp(−ikx) + b(k) exp(ikx), x ≥ L. Then the norm of the transfer operator Pk is given by ||Pk (0, L)|| = |a(k)| + |b(k)|.

(28)

Remark. Everywhere below, we shall refer to a(k) as the transmission coefficient, although this term is more often used for 1/a(k), which is the corresponding coefficient in the asymptotic expansion of the solution ψ/a of the scattering problem. Proof. Denote a(k) = a1 + ia2 ; b(k) = b1 + ib2 . Vector (ψ(x, k), ψ(x, k)) is equal to (sin kx, cos kx) if x ≤ 0, and it is equal to ((a1 + b1 ) cos kx + (a2 − b2 ) sin kx, (a2 + b2 ) cos kx + (b1 − a1 ) sin kx) if x ≥ L. Hence, the matrix of the transfer operator Pk (0, x), x ≥ L is equal to (a1 + b1 ) cos kx + (a2 − b2 ) sin kx (a2 + b2 ) cos kx + (b1 − a1 ) sin kx . (29) (a2 − b2 ) cos kx − (a1 + b1 ) sin kx (b1 − a1 ) cos kx + (a2 − b2 ) sin kx Then (30) Tr(Pk (0, x)Pk∗ (0, x)) = 2(|a|2 + |b|2 ). √ The norm ||Pk || is equal to µ, where µ is the larger eigenvalue of the matrix Pk Pk∗ . Since det Pk = 1, the characteristic equation for µ is µ2 − 2(|a|2 + |b|2 )µ + 1 = 0.

(31)

The identity |a|2 = 1 + |b|2 holds for coefficients a and b; and therefore, the roots of (31) are µ1,2 = (|a| ± |b|)2 . Then √ ||Pk (0, x)|| = µ1 = |a| + |b|. The lemma is proved.


203

The identity |a|2 = 1+|b|2 and formulas (28), (27) allow us to reformulate Theorem 2 as follows. Define by I[a,b] (x) the characteristic function of the interval [a, b]. Theorem 3. Let % ⊂ (0, ∞) be a closed interval. Assume that there exists a sequence Ln → ∞ such that the transmission coefficients an (k) of the operators H = −d 2 /dx 2 + Vn (x), satisfy the condition

where Vn (x) = V (x)I[0,Ln ] (x),

%

ln |an (k)|dk ≤ C(%),

n = 1, 2, . . . .

(32)

Then i) for all α ∈ [0, π), the essential support of the measure µαac (dλ) contains %; ii) for a.e. α ∈ [0, π), the spectral measure µα (dλ) of the operator H α is absolutely continuous on %. 2. The Estimates for the First Integrals of KdV Equation Our next goal is to further simplify Theorem 2 by rewriting the estimate (32) in terms of the potential V and its derivatives. The central point here is the estimation of % ln |a(k)|dk for compactly supported or fast decaying potentials. Let V (t, x) be a solution of the Cauchy problem for the KdV-equation Vt = 6V Vx − Vxxx ,

V (0, x) = V (x),

in the class of the fast decaying functions. Let λj = −(κj )2 , j = 1, 2, . . . , q, be negative eigenvalues and let a(k) be the transmission coefficient for the Hamiltonian H = −d 2 /dx 2 + V (t, x). Then (as was discovered by Gardner, Green, Kruskal and Miura, see details in [8] or [6]) the scattering data {λj , j = 1, 2, . . . .q, a(k), k ∈ R} do not depend on t. They are invariants of the KdV-dynamics. It is also well known that the function a(k) admits analytic continuation in the half plane z > 0, and i ln a(z) has the asymptotic expansion i ln a(z) ∼ −

∞

Im (V ) , z2m+1

z > 0,

z → ∞.

(33)

m=0

Coefficients Im are invariants of the KdV equation also. They have two different representations. First, they can be written in the form Im = (−1)m+1

q

2κj2m+1 j =0

2m + 1

+

1 π

+∞

−∞

k 2m ln |a(k)|dk,

m = 0, 1, 2, . . . .

(34)

This representation can be found in [6] and can be proved in the following simple way. q Let Bq (z) be the Blashke product with the same zeros {iκk }k=1 as the zeros of the function a(z) in the upper half plane: q z − iκj Bq (z) = . z + iκj j =1

(35)

204


For real z, we have |B(z)| = 1. Hence, [i ln

a(z) ] = ln |a(z)|, Bq (z)

z ∈ R.

(36)

ln |a(k)| d k. k−z

(37)

Using the Herglotz formula, we get 1 π

i ln a(z) = i ln Bq (z) +

+∞ −∞

It is clear that i ln

∞

2κ 2m+1 (−1)m z − iκ , ∼ z + iκ 2m + 1 z2m+1

z → ∞.

(38)

m=0

Using (37) and (38), we get (34). Denote by r = (r0 , r1 , . . . . . . , r2m ) a vector with nonnegative integer coordinates, and let d(r) = 2r0 + 3r1 + · · · + (2m + 2)r2m

.

(39)

The second representation for Im is given by Lemma 3. The following formulas hold: ∞ Im = Pm (V , V , . . . .)dx, −∞

m = 0, 1, 2, 3, . . . ,

(40)

where Pm are the polynomials in the function V and its derivatives: Pm (V , V , . . . .) =

cm (r)

2m

rl

[V (x)(l) ] .

(41)

l=0

{ r: d(r)=2m+2}

Remark 1. The coefficients cm (r) can be calculated by the formula cm (r) = E

m ξ rl l

l=0

Here

ξl (x) =

1

rl !

.

ωl (τ )dτ

(42)

0

and w(τ ) is the “Wiener bridge”, i.e., conditional Wiener process determined by the condition w(0) = w(1) = 0 with zero means and the correlation function B(s, t) = (s ∧ t)(1 − s ∨ t). Remark 2. The structure (41) of the polynomials Pm is the consequence of the invariance of the KdV equation under scaling V (t, x) → ε 2 V (ε 3 t, εx). This invariance implies that the polynomial Pm (V , V , . . . .) must be homogeneous of order 2m + 2 under the transformation V (x) → ε 2 V (εx). Specific values of the coefficients cm (r) in the “Minakshisundaram type” expansion (41) are not important for our analysis. However, the estimates for cm (r),m → ∞, can be fundamental for some applications.


205

Proof of this lemma uses the formula ([6], p. 169): tr[(H − z2 I )−1 − (H0 − z2 I )−1 ] = −

1 d ln a(z) 2z dz

(43)

and the Kac–Feynman formula (for complete proof see [16]). The following inequalities are crucial. Theorem 4. The functional Im has the estimates: a) Im ≤ cm

∞

(|V (m−1) (x)|2 + |V (x)|m+1 )dx,

m = 1, 2, . . . ,

(44)

0

b) 1 π

+∞

−∞

k 4m+2 ln |a(k)|dk ≤ I2m+1 ,

m = 0, 1, 2, . . . ,

(45)

c) 1 π

+∞

−∞

k

4m

ln |a(k)|dk ≤ I2m + dm

∞

|V (x)|2m+1 dx,

m = 1, 2, . . . . (46)

0

Here cm and dm are positive constants which do not depend on V . Corollary 1. For any m = 1, 2, . . . , and any closed interval % ∈ (0, ∞), |%| < ∞, ∞ ln |a(k)|dk ≤ cm,% (|V (m−1) (x)|2 + |V (x)|m+1 )dx, m = 1, 2, . . . . . (47) %

0

Proof. Inequalities b) and c) follow from representation (34). If m is odd, then both terms in (34) are nonnegative which implies b). If m is even, we can use the one-dimensional version of the Lieb–Thirring estimate (see [13] or [17], p. 368): for any number γ > 21 , q

|λj |γ ≤ Cγ

j =1

+∞ −∞

|V− (x)| 2 +γ dx. 1

(48)

Here V− (x) is the negative part of the potential V (x). This estimate with γ = 2m + 21 together with (34) implies c). To prove a) we use the second representation for Im . Let us denote by || · ||s the norm in the space Ls (R) and by Am (r) the integral Am (r) =

2m ∞

−∞ k=0

rk

[V (k) (x)] dx,

r = (r0 , r1 , . . . . . . , r2m ).

(49)

From (40 ) and (41 ) it follows that the estimate (m−1) 2 ||2 ), |Am (r)| ≤ C(||V ||m+1 m+1 + ||V

C = C(r, m),

(50)

206


implies a). It is sufficient to prove (50) for integrals Am (r) with an additional condition rm = rm+1 = . . . . = r2m = 0,

(51)

since an arbitrary Am (r) can be reduced via integration by parts to a linear combination of similar integrals for which d(r) is the same and condition (51) holds. Applying the generalized Holder inequality,

∞ m−1 −∞ k=0

m−1 fk (x)dx ≤ k=0

∞ −∞

|fk (x)|λk dx

1/λk

m−1

,

1/λk = 1,

(52)

k=0 r

with λk = (2m + 2)/(k + 2)rk to functions fk = [V (k) (x)] k , we get |Am (r)| ≤

m−1 k=0

rk

||V (k) || 2m+2 .

(53)

k+2

We also need a Kolmogorov type inequality (see, for example, [7]): ||f (k) ||q ≤ C(q, p, r, n, k)||f ||p 1−α ||f (n) ||r

α

(54)

which is valid if 1 n−k k = + , q np nr

α=

k − 1/q + 1/p , n − 1/r + 1/p

0 ≤ k ≤ n.

(55)

Parameters q = (2m + 2)/(k + 2), p = m + 1, r = 2, n = m − 1 satisfy the first condition of (55). With these values of the parameters, α = k/(m − 1), and (54) implies m−k−1

k

m−1 ||V (k) ||(2m+2)/(k+2) ≤ C(m, k)||V ||m+1 ||V (m−1) ||2 m−1 .

(56)

κ = r0 + r1 + . . . · · · + rm−1 ,

(57)

κ−2 m−κ+1 m−1 m−1 |Am (r)| ≤ C ||V ||m+1 . ||V (m−1) ||22 m+1

(58)

If

then (53) and (56) imply

Holder inequality ab ≤ a p /p + a q /q with parameters p = (m − 1)/(κ − 2) and q = (m − 1)/(m − κ + 1) applied to (58) gives (50). Theorem 4 is proved.


207

3. Theorem 1 and Its Generalization In order to prove Theorem 1 we need Lemma 4. If V (x) ∈ H p−1 (R+ ) ∩ Lp+1 (R+ ) then it can be represented in the form V = VB + VR ,

(59)

where: a) a “bump” potential VB has the form VB =

∞

vk .

(60)

k=0

Here potentials vk , k ≥ 1, have compact supports without intersection, and the support of vk+1 is located to the right of the support of vk ; b) VB continued by zero for x ≤ 0 belongs to H p−1 (R) ∩ Lp+1 (R); c) the remainder potential VR is “small” in the following sense: ∞ |VR (x)|dx < ∞. (61) 0

Lp+1

and, as a result, satisfies the Birman condition (3). Proof. The potential V ∈ Hence, there exists a sequence {Lm }∞ 0 such that L0 = 0, Lm+1 − Lm ≥ 2 and Lm +1 Lm +1 1 1 |V (x)|dx < m ; (|V (x)|p+1 + |V (p−1) |2 )dx < m , m ≥ 1. 2 2 Lm −1 Lm −1 (62) ∞ ∞ Consider the following partition of unity 1 = k=0 ϕk on R. Let ϕ ∈ C0 (R), |ϕ| ≤ 1, ϕ = 1 in a neighborhood of x = 0, and ϕ = 0 when |x| > 1. Then ϕ2k (x) = ϕ(x − Lk ), ϕ2k+1 (x) = 1 − ϕ2k (x) − ϕ2k+2 (x) if x ∈ [Lk , Lk+1 ], and ϕ2k+1 (x) = 0 (n) if x ∈ / [Lk , Lk+1 ]. Obviously ||ϕk (x)||∞ < C, uniformly in k = 1, 2, . . . and n = 0, 1, . . . ., p. Define vk = V ϕ2k+1 ,

VR (x) = V (x)

∞

ϕ2k (x).

k=0

Then the statement a) is obvious. According to (62), the potential VR (x) belongs to L1 (0, ∞). It remains to prove that p+1

(p−1) 2 ||2

||VB ||p+1 + ||VB

< ∞.

Since V ∈ H p−1 (R+ ) ∩ Lp+1 (R+ ), it is sufficient to show that Lm +1 1 C |V (k) |dx ≤ C, |V (k) |2 dx < m , m ≥ 1 2 0 Lm −1

(63)

when 0 ≤ k ≤ p − 1. Let S = Sm , m ≥ 0, be the interval of integration in (63), and let || · ||l,S be the norm in Ll (S). Then (63) follows from (62) and the estimate (see, for example [7]) ||V (k) ||2,S ≤ C1 ||V ||2,S + C2 ||V (p−1) ||2,S , Lemma 4 is proved.

0 ≤ k ≤ p − 1.

(64)

208


Proof of Theorem 1. We need to prove only the first statement of the theorem, since it implies the second one (see the proof of Theorem 2). Since the essential support of the spectral measure is invariant under L1 -perturbations of the potentials (see [10]), it is enough to prove the first statement of the theorem only for V = VB . Let Lk be an arbitrary point in the gap between supports of the potentials vk and vk+1 and let m

vk , x > 0; Vm = 0, x < 0. Vm = k=1

Then Jp (Vm ) ≤ Jp (V ) = C < ∞.

(65)

According to Theorem 3, it is sufficient to prove that for any closed interval % ⊂ (0, ∞) there is a constant C(%) such that ln(|am (k)|)dk ≤ C(%) (66) %

for any m = 1, 2, . . . . Here am (k) is the transmission coefficient associated with operator H = −d 2 /dx 2 + Vm (x). Inequality (66) follows immediately from (65), (47), and Theorem 3. Theorem 1 is proved. The following statement can be proved similarly to Theorem 1. Theorem 5. Let V = VB + VR ,

(67)

where VB =

∞

vk ,

vk ∈ H pk −1 (R+ ) ∩ Lpk +1 (R+ ),

VR ∈ L1 (R+ ),

(68)

k=1

and the supports of vk do not intersect each other. Let 0 ≤ pk ≤ p, k = 1, 2, . . . , and ∞

Jpk (vk ) < ∞.

(69)

k=1

Then: 1) for all α ∈ [0, π), the essential support of the absolutely continuous component of the spectral measure µα (dλ) of the operator H α coincides with [0, ∞), i.e. µαac () > 0 for any Borel set ⊂ [0, ∞), || > 0. 2) for a.e. α ∈ [0, π), the spectral measure µα (dλ) of the operator H α is absolutely continuous on [0, ∞).


209

4. The Spectral Bifurcation. Examples The examples below show the “exactness” of the statement of Theorem 1. We shall construct a family of potentials V = V (x, ε; m), m is integer, m ≥ 2, with the following properties: a) for each m, the potential V depends continuously on parameter ε (in the uniform norm); b) Jp (V ) = ∞ for all ε if p = m, and therefore neither Theorem 1 with p = m, nor the results of [5] are applicable to such potentials; c) if ε < 0 then Jm (V ) < ∞, and Theorem 1 with p = m can be applied. Hence, the spectrum of H α on the semiaxis λ ≥ 0 is a.c. for a.e. α. If ε ≥ 0 then Jm (V ) = ∞, and the spectrum of H α is singular for a.e. α. In order to make the idea of the examples transparent we shall discuss in detail the case m = 2. Then, we shall briefly describe the changes which are needed to construct V for arbitrary m. Let us consider a sparse potential V of the form V =

∞

hn (ε)vn (x − xn ),

x ∈ R,

(70)

n=1

where “bumps” vn (x) have compact supports and xn → ∞ as n → ∞. We assume that the functions vn = vn (x) have the following properties: vn ∈ C ∞ (R), vn are even (for simplicity), |vn (x)| ≤ 1, vn (x) = 1 when |x| ≤ Ln , and vn (x) = 0 when |x| ≥ Ln + 1. Let the behavior of all the functions vn (x) be the same when Ln ≥ |x| ≥ Ln + 1, i.e. ω(x) = vn (Ln + 21 + x) does not depend on n when |x| ≤ 21 . Moreover, we assume that ω (x) is even when |x| ≤ 21 . We choose Ln = nβ ,

hn = n−1/2+ε ,

|ε| < ε0 ,

xn = exp(cn ln n),

c>1

(71)

β with arbitrary β ∈ (0, 1/4) and ε0 = min( 1−4β 6 , 2 ).

Theorem 6. 1) Potential V with vn described above has the following properties: ∞ |V (x)|2 dx = ∞, 0 ∞ (p−1) 2 (|V (x)| + |V (x)|p+1 )dx < ∞ if ε < 0, p > 1, (72) 0 ∞ (|V (p−1) (x)|2 + |V (x)|p+1 )dx = ∞ if 0 ≤ ε < ε0 , p > 1. (73) 0

2) For a.e. α, the spectrum of the operator H α with the potential V is pure singular if ε ≥ 0 and is a.c. on the semiaxis λ > 0 if ε < 0. Remark. Theorem 1 with any p > 1 can be applied to potential (70) if ε < 0. Hence, this potential provides an example when Theorem 1 guarantees the existence of the a.c. spectrum in spite of the fact that V ∈ / L2 . Then, V (x, ε, m) with m = 2 can be realized as V˜ = V + V1 ,

210


where V is given by (70), V1 is a compactly supported function such that |V1 | < C, V1 ∈ / L2 . Obviously, Jp (V˜ ) < ∞ only if p = 2, ε < 0. On the other hand, the a.c. spectrum of operators H α with potentials V and V˜ coincide, since the perturbation by V1 is of the trace class. Proof. First of all note that

∞

−∞

|V (x)|2 dx ≥

∞

n=1

Further, ∞ ∞

p+1 p+1 |V (x)| dx = 2 hn (Ln + a), −∞

h2n Ln = ∞.

1

a=

where

0

n=1

|V (Ln + x)|p+1 dx. (74)

Thus, if p > 1, then

∞

−∞

Similarly, if p > 1, then ∞ ∞

|V (p−1) (x)|2 dx = b h2n , −∞

|V (x)|p+1 dx < ∞. where

b=2 0

n=1

(75)

1

|V (p−1) (Ln + x)|2 dx. (76)

Together with (71) and (75) this implies (72) and (73). From (72) and Theorem 1 it follows that the spectrum of H α on the semiaxis λ > 0 is a.c. for a.e. α if ε < 0. Since V vanishes at infinity, the spectrum of H α on the semiaxis λ > 0 is discrete. It only remains to show that (H α ) on the semiaxis λ > 0 is pure singular for a.e. α if 0 ≤ ε < ε0 . Let ψ be the solution of the problem −ψ (x) + hn vn ψ(x) = k 2 ψ(x),

x ∈ R,

ψ(x) = e−ikx ,

x < −Ln − 1. (77)

Then ψ(x) = an e−ikx + bn e+ikx ,

x > Ln + 1.

(78)

The function ψ(x)/an describes the scattering of the plane wave, coming from x = ∞, on the “one bump” potential vn . The spectrum of one-dimensional Schrödinger operators with sparse potentials was studied in [14, 15], where it was proved that under an assumption on xn , ∞

xn 0. We fix a closed interval % ∈ (0, ∞). We will show that α(k) bn (k) = hn sin(2k(Ln + 1/2)) + O(h2n L2n ), n → ∞, (81) k where α(k) is an entire function and the remainder decays uniformly in k 2 ∈ %, i.e. |O(h2n L2n )| ≤ C(%)h2n L2n ,

n = 1, 2, 3, . . . . (82) ∞ 3 2 ∞ 4 4 Here C(%) does not depend on n and ∈ %. Since n=1 hn Ln < ∞, n=1 hn Ln < ∞, and α(k) = 0 only at a discrete set of points, the divergence of (80) when 0 ≤ ε < ε0 follows from the divergence of the series k2

∞

n=1

h2n | sin(2k(Ln + 1/2))|2 =

∞

n=1

h2n [1 − cos(4k(Ln + 1/2))],

The divergence of the last series is obvious, since ∞

∞

n−1+2ε cos(4k(nβ + 1/2)) < ∞,

2 n=1 hn

0 ≤ ε < ε0 . (83)

= ∞ if ε > 0 and

2ε < β,

k = 0.

(84)

n=1

Finally, it is only left to justify (81). The solution ψ of (77) satisfies the integral equation x sin k(x − ξ ) −ikx + hn vn (ξ )ψ(ξ )dξ. ψ(x) = e k −Ln −1

(85)

The norm of the integral operator in the right-hand side of (85) in the space C(R) does not exceed hn (Ln + 2)/|k| ≤ C1 (%)hn Ln . It is less than 1 if n is large enough. Thus, x sin k(x − ξ ) −ikx ψ(x) = e hn vn (ξ )e−ikξ dξ + O(h2n L2n ), + (86) k −Ln −1 and therefore, hn bn (k) = 2k

∞

−∞

vn (ξ )e−2ikξ dξ + O(h2n L2n ),

k 2 ∈ %,

n → ∞.

(87)

Since the function vn is supported on the set {ξ : Ln ≤ |ξ | ≤ Ln + 1} and is an odd function, after integration by parts, (87) takes the form −ihn ∞ bn (k) = v (ξ )e−2ikξ dξ + O(h2n L2n ) (2k)2 −∞ n Ln +1 hn =− 2 vn (ξ ) sin 2ikξ dξ + O(h2n L2n ) 2k Ln 1/2 1 hn ω (ξ ) sin[2k(ξ + Ln + )]dξ + O(h2n L2n ) =− 2 2k −1/2 2 1/2 1 hn ω (ξ ) cos 2kξ dξ sin 2k(Ln + ) + O(h2n L2n ). (88) =− 2 2k −1/2 2

212


The last equality is the consequence of the fact that ω (ξ ) is even on the interval |ξ | < 1/2. Equality (88) implies (81). The proof of Theorem 6 is complete. In order to construct potentials V (x, ε; m) for m > 2 one can consider first the family of potentials: ∞

W = hn (ε)vn (x − xn )ϕ(x − xn , n), x ∈ R, n=1

where hn , vn are the same as in (70), and ϕ is a C ∞ -function such that |ϕ (j ) | < Cj if j ≥ m, and ϕ(x − xn , n) = (Ln − |x − xn |)m−2 , |x − xn | ≥ 1. We assume that (71) holds with small enough β > 0 and ε0 = ε0 (β) > 0. Then one can easily check that Jp (W ) = ∞, Jp (W ) < ∞,

if p < m or p ≥ m, if p ≥ m, ε < 0.

ε ≥ 0;

Thus, Theorem 1 with any p ≥ m can be applied to the operator H α with potential W when ε < 0. Arguments similar to those used in the proof of Theorem 6 can be applied to prove that operator H α with potential W , has singular spectrum for a.e α if ε ≥ 0. Then, V (x, ε; m), m > 2, can be realized as V = W + V1 where V1 is a compactly supported (m−1) (m) | < C, V1 ∈ / L2 (see the Remark following Theorem 6). function such that |V1 Acknowledgements. We would like to thank V. Jaksic (University of Ottawa) and B. Simon (Caltech) for useful discussions. M.N. would like to thank the Department of Mathematics of the University of North Carolina for hospitality. The work of the first and third authors was supported in part by NSF Grant # DMS-9971592.

References 1. Atkinson, F.V.: Discrete and continuous boundary problems. Mathematics in Science and Engineering, 8, New–York-London: Academic Press, 1964 2. Birman, M.: Perturbations of quadratic forms and the spectrum of singular boundary value problems. Dokl. Akad. Nauk SSSR (Russian) 125, 471–474 (1959) 3. Christ, M., Kiselev,A., Remling, C.: The absolutely continuous spectrum for one-dimensional Schrödinger operators with decaying potentials. Math. Res. Lett. 4, no. 5, 719–723 (1997) 4. Christ, M., Kiselev, A.: The absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results. J. Am. Math. Soc. 193, no. 1, 151–170 (1998) 5. Deift, P., Killip, R.: On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials, Comm. Math. Phys. 203, 341- 347 (1999). 6. Dodd, R.K., Eilenbeck, J.C., Gibbon, J.D., Morris, H.C.: Solitons and nonlinear wave equations. London– New York: Academic Press, 1982 7. Gabushin, V.: Inequalities for norms of the functions and their derivatives in the Lp metrics. Math. Notes 1, 194–198 (1967) 8. Gardner, C., Greene, J., Kruskal, R., Miura, R.: Korteweg–de Vries equation and generalizations, VI, Methods for exact solution. Comm. Pure Appl. Math. 27, 97–133 (1974) 9. Glazman, I.: Direct methods of qualitative spectral analysis of singular differential operators. Israeli Progr. Scient. Transl., Jerusalem, 1965 10. Kato, T.: Perturbation theory for linear operators. Reprint of the 1980 edition, Berlin: Springer, 1995 11. Last, Y., Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectra of onedimensional Schrödinger operators. Invent. Math. 135, no. 2, 329–367 (1999) 12. Levitan, B., Sargsyan, I.: Introduction to spectral theory. Translation Math. Monographs, 39, Providence, RI: AMS, 1976 13. Lieb, E., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. Studies in Mathematical Physics: Essay in Honor of Valentine Bargmann, Princeton: Princeton Univ. Press, 1976, pp. 269–303


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14. Molchanov, S.: Multiscattering on sparse bumps. Contemp. Math. 217, 157–181 (1998) 15. Molchanov, S.: Multiscale averaging for ordinary differential equations. In: Homogenization, Series on advances in mathematics for applied sciences, 50, Kozlov, S., Berdichevskii, V., Papanicolaou, G., eds., Singapore: World Scientific, 1999, pp. 316–397 16. Novitskii, M.: Spectral invariants of the Schrödinger operator families, inverse problems and related fuctionals. Doctoral theses, Kharkov, 1997 17. Reed, M., Simon, B.: Methods of modern mathematical physics, IV. Analysis of operators. New York– London: Academic Press, 1972 18. Remling, C.: The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials. Commun. Math. Phys. 203, 341–347 (1999) 19. Simon, B.: Spectral analysis of rank one perturbations and applications. CRM Lecture Notes, 8, Feldman, J., Froese, R., Rosen, L., eds., Providence, RI: AMS, 1995 20. Simon, B.: Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators. Proc. Amer. Math. Soc. 124, no. 11, 3361–3369 (1996) Communicated by B. Simon

Commun. Math. Phys. 216, 215 – 241 (2001)

Communications in



Renormalization in Quantum Field Theory and the Riemann–Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group Alain Connes, Dirk Kreimer Institut des Hautes Études Scientifiques, 35 Route de Chartres, 91440 Bures-sur-Yvette, France. E-mail: [email protected]; [email protected] Received: 21 March 2000 / Accepted: 3 October 2000

Abstract: We showed in Part I that the Hopf algebra H of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group G and that the renormalized theory is obtained from the unrenormalized one by evaluating at ε = 0 the holomorphic part γ+ (ε) of the Riemann–Hilbert decomposition γ− (ε)−1 γ+ (ε) of the loop γ (ε) ∈ G provided by dimensional regularization. We show in this paper that the group G acts naturally on the complex space X of dimensionless coupling constants of the −3/2 theory. More precisely, the formula g0 = gZ1 Z3 for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra H. This allows first of all to read off directly, without using the group G, the bare coupling constant and the renormalized one from the Riemann–Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter ε. It also allows to lift both the renormalization group and the β-function as the asymptotic scaling in the group G. This exploits the full power of the Riemann–Hilbert decomposition together with the invariance of γ− (ε) under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group G for the full higher pole structure of minimal subtracted counterterms in terms of the residue. 1. Introduction We showed in Part I of this paper [1] that perturbative renormalization is a special case of a general mathematical procedure of extraction of finite values based on the Riemann– Hilbert problem. More specifically we associated to any given renormalizable quantum IHES/M/00/22, hep-th/0003188

216

A. Connes, D. Kreimer

field theory an (infinite dimensional) complex Lie group G. We then showed that passing from the unrenormalized theory to the renormalized one was exactly the replacement of the loop d → γ (d) ∈ G of elements of G obtained from dimensional regularization (for d = D = dimension of space-time) by the value γ+ (D) of its Birkhoff decomposition, γ (d) = γ− (d)−1 γ+ (d). The original loop d → γ (d) not only depends upon the parameters of the theory but also on the additional “unit of mass” µ required by dimensional analysis. We shall show in this paper that the mathematical concepts developed in Part I provide very powerful tools to lift the usual concepts of the β-function and renormalization group from the space of coupling constants of the theory to the complex Lie group G. We first observe, taking ϕ63 as an illustrative example to fix ideas and notations, that even though the loop γ (d) does depend on the additional parameter µ, µ → γµ (d),

(1)

the negative part γµ− in the Birkhoff decomposition, γµ (d) = γµ− (d)−1 γµ+ (d)

(2)

∂ γµ− (d) = 0. ∂µ

(3)

is actually independent of µ,

This is a restatement of a well known fact and follows immediately from dimensional analysis. Moreover, by construction, the Lie group G turns out to be graded, with grading, θt ∈ Aut G ,

t ∈ R,

(4)

inherited from the grading of the Hopf algebra H of Feynman graphs given by the loop number, L() = loop number of (5) for any 1PI graph . The straightforward equality, γet µ (d) = θtε (γµ (d))

∀ t ∈ R, ε = D − d

(6)

shows that the loops γµ associated to the unrenormalized theory satisfy the striking property that the negative part of their Birkhoff decomposition is unaltered by the operation, γ (ε) → θtε (γ (ε)).

(7)

In other words, if we replace γ (ε) by θtε (γ (ε)) we do not change the negative part of its Birkhoff decomposition. We settle now for the variable, ε = D − d ∈ C\{0}.

(8)

Our first result (Sect. 2) is a complete characterization of the loops γ (ε) ∈ G fulfilling the above striking invariance. This characterization only involves the negative part γ− (ε) of their Birkhoff decomposition which by hypothesis fulfills, γ− (ε) θtε (γ− (ε)−1 ) is convergent for ε → 0.

(9)

Renormalization in Quantum Field Theory and Riemann–Hilbert Problem II

217

It is easy to see that the limit of (9) for ε → 0 defines a one parameter subgroup,

and that the generator β = residue of γ ,

∂

∂t

(10) Ft ∈ G, t ∈ R Ft t=0 of this one parameter group is related to the

Res γ = − ε=0

∂ γ− ∂u

1 , u u=0

(11)

by the simple equation, ∂

β = Y Res γ ,

(12)

where Y = ∂t θt t=0 is the grading. This is straightforward, but our result is the following formula (14) which gives γ− (ε) in closed form as a function of β. We shall for convenience introduce an additional generator in the Lie algebra of G (i.e. primitive elements of H∗ ) such that, [Z0 , X] = Y (X)

∀ X ∈ Lie G.

(13)

The scattering formula for γ− (ε) is then, γ− (ε) = lim e

−t

β ε +Z0

t→∞

etZ0 .

(14)

Both factors in the right-hand side belong to the semi-direct product, = G > R G θ

(15)

of the group G by the grading, but of course the ratio (14) belongs to the group G. This shows (Sect. 3) that the higher pole structure of the divergences is uniquely determined by the residue and gives a strong form of the t’Hooft relations, which will come as an immediate corollary. In Sect. 4 we show, specializing to the massless case, that the formula for the bare coupling constant, −3/2 g0 = g Z1 Z3 , (16) where both g Z1 = g +δg and the field strength renormalization constant Z3 are thought of as power series (in g) of elements of the Hopf algebra H, does define a Hopf algebra homomorphism, g0

HCM −→ H,

(17)

from the Hopf algebra HCM of coordinates on the group of formal diffeomorphisms of C such that, ϕ(0) = 0, ϕ (0) = id (18) to the Hopf algebra H of the massless theory. We had already constructed in [2] a Hopf algebra homomorphism from HCM to the Hopf algebra of rooted trees, but the physical significance of this construction was unclear. The homomorphism (17) is quite different in that for instance the transposed group homomorphism, ρ

G −→ Diff(C)

(19)

218


lands in the subgroup of odd diffeomorphisms, ϕ(−z) = −ϕ(z)

∀ z.

(20)

Moreover its physical significance will be transparent. We shall show in particular that the image by ρ of β = Y Res γ is the usual β-function of the coupling constant g. We discovered the homomorphism (17) by lengthy concrete computations. We have chosen to include them in an appendix besides our conceptual proof given in Sect. 4. The main reason for this choice is that the explicit computation allows to validate the concrete ways of handling the coproduct, coassociativity, symmetry factors. . . that underly the theory. As a corollary of the construction of ρ we get an action by (formal) diffeomorphisms of the group G on the space X of (dimensionless) coupling constants of the theory. We can then in particular formulate the Birkhoff decomposition directly in the group, Diff (X)

(21)

of formal diffeomorphisms of the space of coupling constants. The unrenormalized theory delivers a loop δ(ε) ∈ Diff (X),

ε = 0

(22)

whose value at ε is simply the unrenormalized effective coupling constant. The Birkhoff decomposition, δ(ε) = δ+ (ε) δ− (ε)−1

(23)

δ− (ε) = bare coupling constant

(24)

δ+ (D) = renormalized effective coupling constant.

(25)

of this loop gives directly then,

and,

This result now, in its statement, no longer depends upon our group G or the Hopf algebra H. But of course the proof makes heavy use of the above ingredients. Now the Birkhoff decomposition of a loop, δ(ε) ∈ Diff (X),

(26)

admits a beautiful geometric interpretation. If we let X be a complex manifold and pass from formal diffeomorphisms to actual ones, the data (26) is the initial data to perform, by the clutching operation, the construction of a complex bundle, P = (S + × X) ∪δ (S − × X)

(27)

over the sphere S = P1 (C) = S + ∪ S − , and with fiber X, π

X −→ P −→ S.

(28)

The meaning of the Birkhoff decomposition (23), δ(ε) = δ+ (ε) δ− (ε)−1 is then exactly captured by an isomorphism of the bundle P with the trivial bundle, S × X.

(29)


219

2. Asymptotic Scaling in Graded Complex Lie Groups We shall first prove the formula (14) of the introduction in the general context of graded Hopf algebras and then apply it to the Birkhoff decomposition of the loop associated in Part I to the unrenormalized theory. We let H be a connected commutative graded Hopf algebra (connected means that H(0) = C) and let θt , t ∈ R be the one parameter group of automorphisms of H associated with the grading so that for x ∈ H of degree n, θt (x) = etn x

∀ t ∈ R.

(1)

By construction θt is a Hopf algebra automorphism, ∂

θt ∈ Aut(H).

(2)

We also let Y = ∂t θt t=0 be the generator which is a derivation of H. We let G be the group of characters of H, vp : H → C,

(3)

i.e. of homomorphisms from the algebra H to C. The product in G is given by, (ϕ1 ϕ2 )(x) = ϕ1 ⊗ ϕ2 , (x,

(4)

where ( is the coproduct in H. The augmentation e of H is the unit of G and the inverse of ϕ ∈ G is given by, ϕ −1 , x = ϕ, Sx, (5) where S is the antipode in H. We let L be the Lie algebra of derivations, δ : H → C,

(6)

i.e. of linear maps on H such that δ(xy) = δ(x) e(y) + e(x) δ(y)

∀ x, y ∈ H.

(7)

Even if H is of finite type so that H(n) is finite dimensional for any n ∈ N, there are more elements in L than in the Lie algebra P of primitive elements, (Z = Z ⊗ 1 + 1 ⊗ Z

(8)

∗ of H. But one passes from P to L by completion in the graded dual Hopf algebra Hgr ∗. relative to the I -adic topology, I being the augmentation ideal of Hgr ∗ The linear dual H is in general an algebra (with product given by (4)) but not a Hopf algebra since the coproduct is not necessarily well defined. It is however well defined for characters ϕ or derivations δ which satisfy respectively (ϕ = ϕ ⊗ ϕ, and (δ = δ ⊗ 1 + 1 ⊗ δ. For δ ∈ L the expression, ϕ = exp δ (9)

makes sense in the algebra H∗ since when evaluated on x ∈ H one has x, δ n = 0 for n large enough (since x, δ n = ((n−1) x, δ ⊗ · · · ⊗ δ vanishes for n > deg x). Moreover ϕ is a group-like element of H∗ , i.e. a character of H. Thus ϕ ∈ G.

220


The one parameter group θt ∈ Aut (H) acts by automorphisms on the group G, θt (ϕ), x = ϕ, θt (x)

∀x ∈ H

(10)

and the derivation Y of H acts on L by Y (δ), x = δ, Y (x),

(11)

and defines a derivation of the Lie algebra L where we recall that the Lie bracket in L is given by, ∀ x ∈ H. (12) [δ1 , δ2 ], x = δ1 ⊗ δ2 − δ2 ⊗ δ1 , (x Let us now consider a map,

ε ∈ C\{0} → ϕε ∈ G

(13)

such that for any x ∈ H, e(x) = 0 one has, ε → ϕε , x is a polynomial in

1 without constant term. ε

(14)

Thus ε → ϕε extends to a map from P1 (C)\{0} to G, such that ϕ∞ = 1.

(15)

For such a map we define its residue as the derivative at ∞, i.e. as, Res ϕ = lim ε(ϕε − 1).

(16)

ε→∞

By construction Res ϕ ∈ L is a derivation H → C. When evaluated on x ∈ H, Res ϕ is just the residue at ε = 0 of the function ε → ϕε , x. We shall now assume that for any t ∈ R the following limit exists for any x ∈ H, lim ϕε−1 θtε (ϕε ), x.

(17)

ε→0

Using (10), (4) and (5) we have, (18) ϕε−1 θtε (ϕε ), x = ϕε ⊗ ϕε , (S ⊗ θtε ) (x, so that with (x = x(1) ⊗ x(2) we get a sum of terms ϕε , S x(1) ϕε , θtε (x(2) ) = P1 1ε ektε P2 1ε . Thus (17) just means that the sum of these terms is holomorphic at ε = 0. It is clear that the value at ε = 0 is then a polynomial in t, Ft , x = lim ϕε−1 θtε (ϕε ), x. ε→0

(19)

Let us check that t → Ft ∈ G is a one parameter group, Ft1 +t2 = Ft1 Ft2

∀ ti ∈ R.

(20)

The group G is a topological group for the topology of simple convergence, i.e., ϕn → ϕ

iff

ϕn , x → ϕ, x

∀ x ∈ H.

(21)

Moreover, using (10) one checks that θt1 ε (ϕε−1 θt2 ε (ϕε )) → Ft2

when ε → 0.

(22)


221

We then have Ft1 +t2 = lim ϕε−1 θ(t1 +t2 )ε (ϕε ) = lim ϕε−1 θt1 ε (ϕε ) θt1 ε (ϕε−1 θt2 ε (ϕε )) = ε→0

ε→0

Ft1 Ft2 . This proves (20) and we let, β=

∂ Ft ∂t

(23)

t=0

which defines an element of L such that, Ft = exp(tβ)

∀ t ∈ R.

(24)

As above, we view H∗ as an algebra on which Y acts as a derivation by (11). Let us prove, Lemma 1. Let ε → ϕε ∈ G satisfy (17) with ϕε = 1 +

∞ n=1

Y (d1 ) = β

Y dn+1 = dn β

dn ε n , dn

∈ H∗ . One then has

∀ n ≥ 1.

Proof. Let x ∈ H and let us show that β, x = lim εϕε ⊗ ϕε , (S ⊗ Y ) ((x). ε→0

(25)

Using (18) we know by hypothesis that, ϕε ⊗ ϕε , (S ⊗ θtε ) ((x) → Ft , x,

(26)

where the convergence holds in the space of holomorphic functions of t in say |t| ≤ 1 so that the derivatives of both sides at t = 0 are also convergent, thus yielding (25). Now the function ε → ε ϕε ⊗ ϕε , (S ⊗ Y ) (x is holomorphic for ε ∈ C\{0} and also at ε = ∞ ∈ P1 (C) since ϕ∞ = 1. Moreover by (25) it is also holomorphic at ε = 0 and is thus a constant, which gives, ϕε ⊗ ϕε , (S ⊗ Y ) ((x) =

1 β, x. ε

(27)

Using the product in H∗ this means that ϕε−1 Y (ϕε ) =

1 β, ε

(28)

and multiplying by ϕε on the left, that, Y (ϕε ) = One has Y (ϕε ) = lemma.

∞ n=1

Y (dn ) εn

and

1 ε

ϕε β =

1 ϕε β. ε 1 ε

β+

∞ n=1

(29) 1 εn+1

dn β. Thus (29) gives the

222


In particular we get Y (d1 ) = β and since d1 is the residue, Res ϕ, this gives, β = Y (Res ϕ),

(30)

which shows that β is uniquely determined by the residue of ϕε . We shall now write a formula for ϕε in terms of β. This is made possible by Lemma 1 which shows that β uniquely determines ϕε . What is not transparent from Lemma 1 is that for β ∈ L the elements ϕε ∈ H∗ are group-like, so that ϕε ∈ G. In order to obtain a nice formula we take the semi direct product of G by R acting on G by the grading θt , = G > R, G

(31)

θ

be the Lie algebra and similarly we let L = L ⊕ C Z0 , L

(32)

where the Lie bracket is given by [Z0 , α] = Y (α)

∀α ∈ L

(33)

and extends the Lie bracket of L. as the Lie algebra of G in a way which will become clear in the proof of We view L the following, Theorem 2. Let ε → ϕε ∈ G satisfy (17) as above. Then with β = Y (Res ϕ) one has, ϕε = lim e

−tZ0 t

e

t→∞

e

The limit t βε +Z0

β ε +Z0

.

holds in the topology of simple convergence in G. Both terms e−tZ0 and

but their product belongs to G. belong to G

Proof. We endow H∗ with the topology of simple convergence on H and let θt act by automorphisms of the topological algebra H∗ by (10). Let us first show, with, ∞ dn , εn

dn ∈ H∗ ,

(34)

θ−s1 (β) θ−s2 (β) . . . θ−sn (β) / dsi .

(35)

ϕε = 1 +

n=1

that the following holds,

dn =

s1 ≥s2 ≥···≥sn ≥0

For n = 1, this just means that,

∞

d1 = 0

θ−s (β) ds,

which follows from (30) and the equality

∞ Y −1 (x) = θ−s (x) ds 0

∀ x ∈ H, e(x) = 0.

(36)

(37)


223

We see from (37) that for α, α ∈ H∗ such that Y (α) = α , α, 1 = α , 1 = 0 one has,

α=

∞

0

θ−s (α ) ds.

(38) (39)

Combining this equality with Lemma 1 and the fact that θs ∈ Aut H∗ is an automorphism, gives an inductive proof of (35). The meaning of this formula should be clear; we pair both sides with x ∈ H, and let ((n−1) x = x(1) ⊗ x(2) ⊗ · · · ⊗ x(n) . (40) Then the right-hand side of (35) is just,

β ⊗ · · · ⊗ β , θ−s1 (x(1) ) ⊗ θ−s2 (x(2) ) · · · ⊗ θ−sn (x(n) )/ dsi , s1 ≥···≥sn ≥0

(41)

and the convergence of the multiple integral is exponential since, β, θ−s (x(i) ) = O (e−s )

for

s → +∞.

(42)

We see moreover that if x is homogeneous of degree deg(x), and if n > deg(x), at least one of the x(i) has degree 0 so that β, θ−s (x(i) ) = 0 and (41) gives 0. This shows that the pairing of ϕε with x ∈ H only involves finitely many non-zero terms in the formula, ϕε , x = e(x) +

∞ 1 dn , x. εn

(43)

n=1

With all convergence problems out of the way we can now proceed to prove the formula of Theorem 2 without care for convergence. Let us first recall the expansional formula [3], ∞

(A+B) e = eu0 A Beu1 A . . . Beun A / duj (44) n=0

uj =1, uj ≥0

(cf. [3] for the exact range of validity of (44)). We apply this with A = tZ0 , B = tβ, t > 0 and get, ∞

ev0 Z0 βev1 Z0 β . . . βevn Z0 / dvj . et (β+Z0 ) = n=0

vj =t, vj ≥0

(45)

Thus, with s1 = t − v0 , s1 − s2 = v1 , . . . , sn−1 − sn = vn−1 , sn = vn and replacing β by 1ε β, we obtain, et (β/ε+Z0 ) =

∞ 1 etZ0 θ−s1 (β) . . . θ−sn (β) / dsi . ε n t≥s1 ≥s2 ≥···≥sn ≥0

(46)

n=0

Multiplying by e−tZ0 on the left and using (41) thus gives, ϕε = lim e−tZ0 et (β/ε+Z0 ) . t→∞

(47)

224


It is obvious conversely that this formula defines a family ε → ϕε of group-like elements of H∗ associated to any preassigned element β ∈ L. Corollary 3. For any β ∈ L there exists a (unique) map ε → ϕε ∈ G satisfying (17) and (34). 3. The Renormalization Group Flow Let us now apply the above results to the group G associated in Part I to the Hopf algebra H of 1PI Feynman graphs of a quantum field theory. We choose ϕ63 for simplicity. As explained in Part I the group G is a semi-direct product, G = G0 > Gc

(1)

of an abelian group G0 by the group Gc associated to the Hopf subalgebra Hc constructed on 1PI graphs with two or three external legs and fixed external structure. Passing from Gc to G is a trivial step and we shall thus concentrate on the group Gc . The unrenormalized theory delivers, using dimensional regularization with the unit of mass µ, a loop, ε → γµ (ε) ∈ Gc ,

(2)

and we first need to see the exact µ dependence of this loop. We consider the grading of Hc and Gc given by the loop number of a graph, L() = I − V + 1, where I is the number of internal lines and V the number of vertices. One has, γet µ (ε) = θtε (γµ (ε)) ∀ t ∈ R.

(3)

(4)

Let us check this using the formulas of Sect. 3 of Part I. For N = 2 external legs the dimension B of σ, U is equal to 0 by (12) of loc.cit. Thus the µ dependence is given by ε µ 2 V3 , (5) where V3 is the number of 3-point vertices of . One checks that 21 V3 = L as required. Similarly if N = 3 the dimension B of σ, U is equal to 1 − 23 d + 3, d = 6 − ε by (12) of loc.cit. so that the µ-dependence is, ε

µ 2 V3 µ−ε/2 .

(6)

But this time, V3 = 2L + 1 and we get µεL

(7)

as required. We now reformulate a well known result, the fact that counterterms, once appropriately normalized, are independent of m2 and µ2 . Lemma 4. Let γµ = (γµ− )−1 (γµ+ ) be the Birkhoff decomposition of γµ . Then γµ− is independent of µ. As in Part I we perform the Birkhoff decomposition with respect to a small circle C with center D = 6 and radius < 1.


225

Proof. The proof of the lemma follows immediately from [4]. Indeed the dependence in m2 has in the minimal subtraction scheme the same origin as the dependence in p 2 and we have chosen the external structure of graphs (Eq. (41) of Part I) so that no m2 dependence is left 1 . But then, since µ2 is a dimensionful parameter, it cannot be involved any longer. Corollary 5. Let ϕε = (γµ− )−1 (ε), then for any t ∈ R the following limit exists in Gc : lim ϕε−1 θtε (ϕε ).

ε→0

In other words ε → ϕε ∈ Gc fulfills condition (17) of Sect. 2. Proof. The product ϕε−1 γµ (ε) is holomorphic at ε = 0 for any value of µ. Thus by (4), for any t ∈ R, both ϕε−1 γµ (ε) and ϕε−1 θεt (γµ (ε)) are holomorphic at ε = 0. The same holds for θ−εt (ϕε−1 ) γµ (ε) and hence for the ratio ϕε−1 γµ (ε) (θ−εt (ϕε−1 ) γµ (ε))−1 = ϕε−1 θ−εt (ϕε ).

We let γ− (ε) = ϕε−1 and translate the results of Sect. 2. Corollary 6. Let Ft = lim γ− (ε) θtε (γ− (ε)−1 ). Then Ft is a one parameter subgroup ε→0

of Gc and Ft = exp(tβ), where β = Y Res ϕε is the grading operator Y applied to the residue of the loop γ (ε). In general, given a loop ε → γ (ε) ∈ G it is natural to define its residue at ε = 0 by first performing the Birkhoff decomposition on a small circle C around ε = 0 and then taking Resε=0 γ =

∂ (ϕ1/u )u=0 , ∂u

(8)

where ϕε = γ− (ε)−1 and γ (ε) = γ− (ε)−1 γ+ (ε) is the Birkhoff decomposition. As shown in Sect. 2, the residue or equivalently β = Y Res uniquely determines ϕε = γ− (ε)−1 and we thus get, from Theorem 2, Corollary 7. The negative part γ− (ε) of the Birkhoff decomposition of γµ (ε) is independent of µ and given by, γ− (ε) = lim e t→∞

−t

β ε +Z0

etZ0 .

As above we adjoined the primitive element Z0 to implement the grading Y (cf. Sect. 2). Our choice of the letter β is of course not innocent and we shall see in Sect. 5 the relation with the β-function. 1 This can be easily achieved by maintaining non-vanishing fixed external momenta. γ µ− is independent on such external structures by construction [1].

226


4. The Action of Gc on the Coupling Constants We shall show in this section that the formula for the bare coupling constant g0 in terms of 1PI graphs, i.e. the generating function, g0 = (g Z1 ) (Z3 )−3/2 ,

(1)

where we consider the right-hand side as a formal power series with values in Hc given explicitly by, (with ; = L() the loop number of the graphs),  g0 = x +

 −3/2  1 −  x 2l+1 x 2l S() S()

(2)

does define a Hopf algebra homomorphism, = : HCM → H

(3)

from the Hopf algebra HCM of coordinates on the group of formal diffeomorphisms of C with ϕ(0) = 0, ϕ (0) = 1, (4) to the Hopf algebra Hc of 1PI graphs. This result is only valid if we perform on Hc the simplification that pertains to the massless case, m = 0, but because of the m-independence of the counterterms all the corollaries will be valid in general. The desired simplification comes because in the case m = 0 there is no need to indicate by a cross left on an internal line the removal of a self energy subgraph. Indeed and with the notations of Part I we can first of all ignore all 2 2 the (1) yield a k term which exactly cancels out (0) since m = 0; moreover the k with the additional propagator when we remove the subgraph and replace it by (1). This shows that we can simply ignore all these crosses and write coproducts in the simplest possible way. To get familiar with this coproduct and with the meaning of the Hopf algebra morphism (3) we urge the reader to begin by the concrete computation done in the appendix, which checks its validity up to order six in the coupling constant. Let us now be more explicit on the meaning of formula (2). We first expand g0 as a power series in x and get a series of the form, g0 = x +

∞

αn x n ,

(5)

2

where the even coefficients α2n are zero and the coefficients α2n+1 are finite linear combinations of products of graphs, so that, α2n+1 ∈ H

∀ n ≥ 1.

(6)

We let HCM be the Hopf algebra of the group of formal diffeomorphisms such that (4) holds. We take the generators an of HCM given by the equality ϕ(x) = x + an (ϕ) x n , (7) n≥2


227

and define the coproduct in HCM by the equality (an , ϕ1 ⊗ ϕ2 = an (ϕ2 ◦ ϕ1 ).

(8)

We then define uniquely the algebra homomorphism = : HCM → H by the condition,

=(an ) = αn .

(9)

By construction = is a morphism of algebras. We shall show that it is comultiplicative, i.e. (= ⊗ =) ( x = ( =(x) ∀ x ∈ HCM (10) and comes from a group morphism, ρ : Gc → G2 ,

(11)

where G2 is the group of characters of HCM which is by construction the opposite of the group of formal diffeomorphisms. In fact we shall first describe the corresponding Lie algebra morphism, ρ. ∗ Let us first recall from Part I that a 1PI graph defines a primitive element () of Hgr which only pairs nontrivially with the monomial of H and satisfies (), = 1. We take the following natural basis = S()() for the Lie algebra of primitive elements ∗ , labelled by 1PI graphs with two or three external legs. By Part I, Theorem 2, of Hgr their Lie bracket is given by [, ] = ◦v − ◦v , (12) v

v

where ◦v is the graph obtained by grafting at the vertex v of . (Our basis differs from the one used in loc. cit. by an overall – sign, but the present choice will be more convenient.) In our context of the simplified Hopf algebra the places where a given graph can be inserted in another graph are no longer always labelled by vertices of . They are when is a vertex graph but when is a self energy graph such places are just labelled by the internal lines of , as we could discard the use of external structures and two-point vertices for self energy graphs. ∗ We also let Zn be the natural basis of the Lie algebra of primitive elements of HCM ∂ which corresponds to the vector fields x n+1 ∂x . More precisely, Zn is given as the linear form on HCM which only pairs with the monomial an+1 , Zn , an+1 = 1,

(13)

[Zn , Zm ] = (m − n) Zn+m .

(14)

and the Lie bracket is given by,

We then first prove, Lemma 8. Let ρ = 23 for 2-point graphs and ρ = 1 for 3-point graphs. The equality , where ; = L() is the loop number, defines a Lie algebra homomorρ() = ρ Z2; phism.

228


Proof. We just need to show that ρ preserves the Lie bracket. Let us first assume that 1 , 2 are vertex graphs and let Vi be the vertex number of i . One has, V = 2L + 1

(15)

for any vertex graph . Thus the Lie bracket ρ [ 1 , 2 ] provides V2 − V1 = 2(L2 − L1 ) vertex graph contributions all equal to Z2(L so that 1 +L2 ) ρ ([ 1 , 2 ]) = 2(L2 − L1 ) Z2(L 1 +L2 )

(16)

which is exactly [ρ ( 1 ), ρ ( 2 )] by (14). Let then 1 and 2 be 2-point graphs. For any such graph one has I = 3L − 1,

(17)

where I is the number of internal lines of . Thus ρ ([ 1 , 2 ]) gives I2 −I1 = 3(L2 −L1 ) 2-point graph contributions, each equal to 23 Z2(L . Thus, 1 +L2 ) ρ ([ 1 , 2 ]) =

3 , 3(L2 − L1 ) Z2(L 1 +L2 ) 2

(18)

but the right-hand side is ρ1 ρ2 2(L2 − L1 ) Z2(L so that, 1 +L2 )

ρ ([ 1 , 2 ]) = [ρ ( 1 ), ρ ( 2 )],

(19)

as required. Finally if say 1 is a 3-point graph and 2 a 2-point graph, we get from [ 1 , 2 ] a set of V2 2-point graphs minus I1 3-point graphs which gives, 3 V2 − I1 Z2(L . (20) 1 +L2 ) 2 One has V2 = 2 L2 , I1 = 3 L1 so that which gives (19) as required.

3 2

V2 − I1 = 3(L2 − L1 ) = ρ1 ρ2 2(L2 − L1 )

We now have the Lie algebra morphism ρ and the algebra morphism =. To ρ corresponds a morphism of groups, (21) ρ : Gc → G2 , and we just need to check that the algebra morphism = is the transposed of ρ on the coordinate algebras, =(a) = a ◦ ρ ∀ a ∈ HCM . (22) To prove (22) it is enough to show that = is equivariant with respect to the action of the ∗ . More precisely, given a primitive element Lie algebra L of primitive elements of Hgr ∗ Z ∈ Hgr , (Z = Z ⊗ 1 + 1 ⊗ Z,

(23)

we let ∂Z be the derivation of the algebra H given by, ∂Z (y) = Z ⊗ id, (y ∈ H

∀ y ∈ H.

What we need to check is the following: Lemma 9. For any a ∈ HCM , Z ∈ L one has ∂Z =(a) = =(∂(ρZ) (a)).

(24)


229

Proof. It is enough to check the equality when Z is of the form = S()() with the above notations. Thus we let be a 1PI graph and let ∂ be the corresponding derivation of H given by (24) with Z = S()(). Now by definition of the primitive element () one has (cf. (48) Sect. 2 of Part I) () ⊗ id, ( = n(, ; ) , (25) where the integer n(, ; ) is the number of subgraphs of which are isomorphic to while / ∼ = . By Theorem 2 of Part I we have S() S( ) n(, ; ) = i(, ; ) S( ),

(26)

where i(, ; ) is the number of times appears in ◦ . We thus get ∂

; ) = i(, , S( ) S( )

(27)

which shows that ∂ admits a very simple definition in the generators S( ) of H. The derivation ∂(ρZ) of HCM is also very easy to compute. One has by construction (Lemma 8), ρ(Z) = ρ Z2; , ; = L(), (28) ∗ is simply and the derivation dk of HCM associated to the primitive element Zk of HCM given, in the basis an ∈ HCM by

dk (an ) = (n − k) an−k . We thus get

∂(ρZ) = ρ d2; ,

; = L().

(29) (30)

Now by construction both ∂Z = and = ◦ ∂(ρZ) are derivations from the algebra HCM to H viewed as a bimodule over HCM , i.e. satisfy δ(ab) = δ(a) =(b) + =(a) δ(b).

(31)

Thus, to prove the lemma we just need to check the equality ∂ =(an ) = ρ =(d2; (an )),

; = L(),

or equivalently using the generating function g0 = x + =(an ) x n , that ∂ g0 = ρ x 2;+1

(32)

(33)

∂ g0 . ∂x

(34)

Z3 = 1 − δZ,

(35)

Now by construction of = we have g0 = (x Z1 )(Z3 )−3/2 , where Z1 = 1 +

x 2l

x 2l , δZ = . S() S()

(36)

230


∂ Thus, since both ∂ and ∂x are derivations we can eliminate the denominators in (34) and rewrite the desired equality, after multiplying both sides by (1 − δZ)5/2 as

∂ 3 ∂ x Z1 (1 − δZ) + δZ (x Z1 ) ∂ 2 ∂ ∂ ∂ 3 = ρ x 2;+1 (x Z1 ) (1 − δZ) + (x Z1 ) x 2;+1 δZ . ∂x 2 ∂x

(37)

Both sides of this formula are bilinear expressions in the 1PI graphs. We first need to ∂ ∂ compute ∂ Z1 and ∂ δZ. One has

and

∂ x 2l+2l c(, ) Z1 = ∂ S( )

(38)

∂ x 2l+2l c(, ) δZ = , ∂ S( )

(39)

where ; = L(), ; = L( ) are the loop numbers and the integral coefficient c(, ) is given by c(, ) = V if ρ = 1

and

c(, ) = I if ρ = 3/2

(40)

(where V and I are respectively the number of vertices and of internal lines of ). To prove (38) and (39) we use (27) and we get in both cases expressions like (38), (39) with i(, ; ). (41) c(, ) =

But this is exactly the number of ways we can insert inside and is thus the same as (40). Let now 1 be a 3-point graph and 2 a 2-point graph. The coefficient of the bilinear term, 2 1 , (42) S(1 ) S(2 ) in the left-hand side of (37) is given by 3 c(, 2 ) − c(, 1 ) x 2;+2;1 +2;2 +1 . 2

(43)

Its coefficient in the right-hand side of (37) is coming from the terms, x

2;+1

∂ ∂x

x

2;1 +1

1 S(1 )

−2 S(2 )

x 2;2 + 3 2;1 +1 1 ∂ x x 2;+1 2 S(1 ) ∂x

which gives

(3;2 − 2;1 − 1) x 1+2;+2;1 +2;2 .

x

2;2

2 S(2 )

(44)

(45)


231

We thus only need to check the equality 3 c(, 2 ) − c(, 1 ) = (3;2 − 2;1 − 1) ρ . 2

(46)

Note that in general, for any graph with N external legs we have V = 2(L − 1) + N, I = 3(L − 1) + N.

(47)

Let us first take for a 3-point graph so that ρ = 1. Then the left-hand side of (46) gives 23 V2 − V1 = 23 (2;2 ) − (2(;1 − 1) + 3) = 3;2 − 2;1 − 1. Let then be a 2-point graph, i.e. ρ = 23 . Then the left-hand side of (46) gives 3 3 2 I2 − I1 = 2 (3;2 − 1) − 3;1 = ρ (3;2 − 2;1 − 1), which gives the desired equality. Finally we also need to check the scalar terms and the terms linear in 1 or in 2 . ∂ ∂ The only scalar terms in the left hand side of (37) are coming from x ∂ Z1 + 23 x ∂ δZ and this gives, x 2;+1 ρ . (48) The only scalar term in the right-hand side of (37) comes from x 2;+1 thus they fulfill (37). ∂ The terms linear in 1 in the left-hand side of (37) come only from x ∂ Z1 if is a 3-point graph and the coefficient of 1 /S(1 ) is thus, c(, 1 ) x 1+2;1 +2; .

(49)

In the right-hand side of (37) we just get (2;1 + 1) x 1+2;1 +2; .

(50)

We thus need to check that c(, 1 ) = 2;1 + 1 which follows from (40) and (47) since V1 = 2;1 + 1. ∂ Similarly, if is a 2-point graph, the left side of (37) only contributes by x ∂ Z1 + 3 2;+1 x Z , so that the coefficient of /S( ) is 1 1 1 2 3 (51) x 1+2;+2;1 . c(, 1 ) + 2 In the right-hand side of (37) we get just as above (2;1 + 1) x 1+2;+2;1 ,

(52)

multiplied by ρ = 3/2. Now here, since is a 2-point graph, we have c(, 1 ) = I1 = 3(;1 − 1) + 3 = 3;1 so that 3 3 c(, 1 ) + = (2;1 + 1) = ρ (2;1 + 1) 2 2 as required. The check for terms linear in 2 is similar. We can now state the main result of this section: Theorem 10. The map = = HCM → H given by the effective coupling is a Hopf algebra homomorphism. The transposed Lie group morphism is ρ : Gc → G2 .

232


The proof follows from Lemma 9 which shows that the map from Gc to G2 given by the transpose of the algebra morphism = is the Lie group morphism ρ. By construction the morphism = is compatible with the grading @ of H and α of HCM given by deg(an ) = n − 1 (cf. [5]), one has indeed, = ◦ αt = @2t ◦ =, ∀t ∈ R.

(51)

Finally we remark that our proof of Theorem 10 is similar to the proof of the equality Fφ1 φ2 = Fφ2 ◦ Fφ1

(52)

for the Butcher series used in the numerical integration of differential equations, but that the presence of the Z3 factor makes it much more involved in our case. 5. The β-Function and the Birkhoff Decomposition of the Unrenormalized Effective Coupling in the Diffeomorphism Group Let us first recall our notations from Part I concerning the effective action. We work in the Euclidean signature of space time and in order to minimize the number of minus signs we write the functional integrals in the form,

N eS(ϕ) P (ϕ) [Dϕ] (1) so that the Euclidean action is2 1 1 g S(ϕ) = − (∂µ ϕ)2 − m2 ϕ 2 + ϕ 3 . (2) 2 2 6 The effective action, which when used at tree level in (1) gives the same answer as the full computation using (2), is then given in dimension d = 6 − ε by (µε/2 g)n−2

(p1 , . . . , pn )(3) Seff (ϕ) = S(ϕ) + n! S() (3) 1PI × ϕ(p1 ) . . . ϕ(pn ) / dpi , where, as in Part I, we do not consider tree graphs as 1PI and the integral is performed on the hyperplane pi = 0. To be more precise one should view the right-hand side of (3) as a formal power series with values in the Hopf algebra H. The theory provides us with a loop γµ (ε) = γ− (ε)−1 γµ+ (ε) (4) of characters of H. When we evaluate γµ (ε) (resp. γ− , γµ+ ) on the right-hand side of (3) we get respectively the unrenormalized effective action, the bare action and the renormalized effective action (in the MS scheme). Our notation is hiding the g-dependence of γµ (ε), but this dependence is entirely governed by the grading. Indeed with t = log(g) one has, with obvious notations, γµ,g (ε) = @2t (γµ,1 (ε)).

(5)

Since @t is an automorphism the same equality holds for both γµ+ and γ− . As in Sect. 4 we restrict ourselves to the massless case and let γµ (ε) = γ− (ε)−1 γµ+ (ε) be the Birkhoff decomposition of γµ (ε) = γµ,1 (ε). 2 We know of course that the usual sign convention is better to display the positivity of the action functional.


233

Lemma 11. Let ρ : Gc → G2 be given by Theorem 10. Then ρ(γµ (ε))(g) is the unrenormalized effective coupling constant, ρ(γµ+ (0))(g) is the renormalized effective coupling constant and ρ(γ− (ε))(g) is the bare coupling constant g0 . This follows from (3) and Theorem 10. It is now straightforward to translate the results of the previous sections in terms of diffeomorphisms. The only subtle point to remember is that the group G2 is the opposite of the group of diffeomorphisms so that if we view ρ as a map to diffeomorphisms it is an antihomomorphism, ρ(γ1 γ2 ) = ρ(γ2 ) ◦ ρ(γ1 ).

(6)

Theorem 12. The renormalization group flow is the image ρ(Ft ) by ρ : Gc → Diff of the one parameter group Ft ∈ Gc . Proof. The bare coupling constant g0 governs the bare action, g0 3 1 ϕ Sbare (ϕ0 ) = − (∂µ ϕ0 )2 + µε/2 2 6 0 in terms of the bare field ϕ0 . Now when we replace µ by

µ = e t µ

(7)

(8)

we can keep the bare action, and hence the physical theory, unchanged provided we replace the renormalized coupling constant g by g , where t

g0 (ε, g ) = e−ε 2 g0 (ε, g). By construction we have

t

g = ψε−1 (e−ε 2 ψε (g)),

(9) (10)

where ψε is the formal diffeomorphism given by ψε = ρ (γ− (ε)).

(11)

Now the behaviour for ε → 0 of g given by (10) is the same as for ψε−1 αεt/2 (ψε ),

(12)

where αs is the grading of Diff given as above by αs (ψ)(x) = e−s ψ(es x).

(13)

Thus, since the map ρ preserves the grading, ρ (θt (γ )) = αt/2 ρ(γ )

(14)

(by (51) of Sect. 4), we see by Corollary 6 of Sect. 3 that g → ρ(Ft )g As a corollary we get of course,

when ε → 0.

(15)

234


Corollary 13. The image by ρ of β ∈ L is the β-function of the theory. In fact all the results of Sect. 3 now translate to the group G2 . We get the formula for the bare coupling constant in terms of the β-function, namely, ψε = lim e t→∞

−tZ0 t

e

β ε +Z0

,

(16)

∂ where Z0 = x ∂x is the generator of scaling. But we can also express the main result of Part I independently of the group G or of its Hopf algebra H. Indeed the group homomorphism ρ : G → G2 maps the Birkhoff decomposition of γµ (ε) to the Birkhoff decomposition of ρ(γµ (ε)). But we saw above that ρ (γµ (ε)) is just the unrenormalized effective coupling constant. We can thus state

Theorem 14. Let the unrenormalized effective coupling constant geff (ε) viewed as a formal power series in g be considered as a loop of formal diffeomorphisms and let geff (ε) = geff + (ε) (geff − )−1 (ε) be its Birkhoff decomposition in the group of formal diffeomorphisms. Then the loop geff − (ε) is the bare coupling constant and geff + (0) is the renormalized effective coupling. Note that G2 is naturally isomorphic to the opposite group of Diff so we used the opposite order in the Birkhoff decomposition. This result is very striking since it no longer involves the Hopf algebra H or the group G but only the idea of thinking of the effective coupling constant as a formal diffeomorphism. The proof is immediate, by combining Lemma 11, Theorem 10 of Sect. 4 with Theorem 4 of Part I. Now in the same way as the Riemann–Hilbert problem and the Birkhoff decomposition for the group G = GL(n, C) are intimately related to the classification of holomorphic n-dimensional vector bundles on P1 (C) = C+ ∪ C− , the Birkhoff decomposition for the group G2 = Diff 0 is related to the classification of one dimensional complex (non linear) bundles P = (C+ × X) ∪geff (C− × X).

(17)

Here X stands for a formal one dimensional fiber and C± are, as in Part I, the components of the complement in P1 (C) of a small circle around D. The total space P should be thought of as a 2-dimensional complex manifold which blends together the ε = D − d and the coupling constant of the theory. 6. Conclusions We showed in this paper that the group G of characters of the Hopf algebra H of Feynman graphs plays a key role in the geometric understanding of the basic ideas of renormalization including the renormalization group and the β-function. We showed in particular that the group G acts naturally on the complex space X of dimensionless coupling constants of the theory. Thus, elements of G are a refined form of diffeomorphisms of X and as such should be called diffeographisms. The action of these diffeographisms on the space of coupling constants allowed us first of all to read off directly the bare coupling constant and the renormalized one from the Riemann–Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This showed that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of


235

the dimensional regularization parameter ε. It also allowed us to lift both the renormalization group and the β-function as the asymptotic scaling in the group of diffeographisms. This used the full power of the Riemann–Hilbert decomposition together with the invariance of γ− (ε) under a change of unit of mass. This gave us a completely streamlined proof of the existence of the renormalization group and more importantly a closed formula of scattering nature, delivering the full higher pole structure of minimal subtracted counterterms in terms of the residue. In the light of the predominant role of the residue in NCG we expect this type of formula to help us to decipher the message on space-time geometry buried in the need for renormalization. Moreover, thanks to [6] the previous results no longer depend upon dimensional regularization but can be formulated in any regularization or renormalization scheme. Also, we could discard a detailed discussion of anomalous dimensions, since it is an easy corollary [7] of the knowledge of the β-function. For reasons of simplicity our analysis was limited to the case of one coupling constant. The generalization to a higher dimensional space X of coupling constants is expected to involve the same ingredients as those which appear in higher dimensional diffeomorphism groups and Gelfand-Fuchs cohomology [5]. We left aside the detailed study of the Lie algebra of diffeographisms and its many similarities with the Lie algebra of formal vector fields. This, together with the interplay between Hopf algebras, rational homotopy theory, BRST cohomology, rooted trees and shuffle identities will be topics of future joint work. 7. Appendix: Up to Three Loops We now want to check the Hopf algebra homomorphism HCM → H up to three loops as an example. We regard g0 as a series in a variable x (which can be thought of as a −3/2 physical coupling) up to order x 6 , making use of g0 = xZ1 Z3 and the expression of the Z-factors in terms of the 1PI Feynman graph. The challenge is then to confirm that the coordinates δn on G2 , implicitly defined by [5] (n) log g0 (x) commute with the Hopf algebra homomorphism: calculating the coproduct (CM of δn and expressing the result in Feynman graphs must equal the application of the coproduct ( applied to δn expressed in Feynman graphs. −3/2 By (2) of Sect. 4 we write g0 = xZ1 Z3 , Z1 = 1 +

∞

z1,2k x 2k ,

k=1

Z3 = 1 −

∞

z3,2k x 2k ,

k=1

and

−3/2

Zg = Z1 Z3

as formal series in x 2 . Using

, zi,2k ∈ Hc , i = 1, 3,

236


log

∂ xZg ∂x

=

∞ δ2k 2k x , (2k)! k=1

which defines δ2k as the previous generators an (φ) of coordinates of G2 , we find 9 1 δ2 ≡ δ˜2 = 3z1,2 + z3,2 , 2! 2 1 3 9 2 3 2 − 6z1,2 z3,2 − z3,2 , δ4 ≡ δ˜4 = 5[z1,4 + z3,4 ] − z1,2 4! 2 2 4 1 3 3 2 δ6 ≡ δ˜6 = 9z1,2 + 18z1,2 z3,2 − 5[3z1,2 z1,4 + z3,2 z3,4 ] 6! 2

(1) (2) (3)

1 3 3 2 + 12[z1,2 z3,2 − z1,2 z3,4 − z1,4 z3,2 ] + 7[z1,6 + z3,2 + z3,6 ]. 2 2 The algebra homomorphism Hc → H of Sect. 4 is effected by expressing the zi,2k in Feynman graphs, with 1PI graphs with three external legs contributing to Z1 , and 1PI graphs with two external legs, self-energies, contributing to Z3 . Explicitly, we have z1,2 = z3,2

,

1 = 2

,

z1,4 = z3,4 =

+ 1 2

1 + 2 .

+

+

The symmetry factor

+

+

+

1 2

,

2=S is most obvious if we redraw =

.

Further, we have z1,6 = +

1 2

+

+

+

+

+

+

+

+

+

+

1 2

+

+

+ +

+

+

+

+

+

+

+

+

+

+

+

+

+

+


+ + + + +

1 4 1 2 1 4 1 2 1 2

237

+

+

+

+

+

+

+

+

+

+

+ +

+

+

+

+

+

+

+

+

+ primitive terms,

and z3,6 =

1 2 +

+ 1 2

1 8 +

+ 1 4

1 4 +

+ 1 2

1 2 +

+ 1 2

1 4 +

.

Here, primitive terms refer to 1PI three-loop vertex graphs without subdivergences. They fulfill all desired identities below trivially, and are thus not explicitly given. On the level of diffeomorphisms, we have the coproducts (CM [δ4 ] = δ4 ⊗ 1 + 1 ⊗ δ4 + 4δ2 ⊗ δ2 , (CM [δ6 ] = δ6 ⊗ 1 + 1 ⊗ δ6 + 20δ2 ⊗ δ4 + 6δ4 ⊗ δ2 + 28δ22 ⊗ δ2 ,

(4) (5)

where we skip odd gradings. We have to check that the coproduct ( of Feynman graphs reproduces these results. Applying ( to the rhs of (2) gives, using the expressions for zi,k in terms of Feynman graphs, 9 ⊗ + ⊗ + ⊗ ((δ˜4 ) = 6 2 27 + ⊗ + δ˜4 ⊗ 1 + 1 ⊗ δ˜4 . 8 ˜ ˜ This has to be compared with δ˜4 ⊗ 1 + 1 ⊗ δ˜4 + 2!2! 4! 4δ2 ⊗ δ2 , which matches perfectly, as 27 ˜ ˜ ⊗ + ⊗ + ⊗ δ2 ⊗ δ2 = 9 4 81 + ⊗ . 16 After this warming up, let us do the check at order g 6 , which will be much more demanding, as the coproduct will be noncocommutative now.

238


We need (CM (δ6 ) in (5) to be equivalent to (CM (δ˜6 ) = δ˜6 ⊗ 1 + 1 ⊗ δ˜6 + 20 + 28

2!4! 2!4! δ˜2 ⊗ δ˜4 + 6 δ˜4 ⊗ δ˜2 6! 6!

2!2!2! 2 δ˜2 ⊗ δ˜2 . 6!

(6)

Applying the Hopf algebra homomorphism to Feynman graphs on both sides of the tensor product delivers ((δ˜6 ) = δ˜6 ⊗ 1 + 1 ⊗ δ˜6 2 28 9 9 + + ⊗ 3 + 3 90 4 4 6 1 + 5 + + + + + 15 2 3 1 1 + + 2 2 2 3 1 −3 + + 2 16 9 ⊗ 3 + 4 20 9 + + 3 15 4 1 ⊗5 + + + + + + 2 3 1 1 + + 2 2 2 3 1 −3 . + + 2 16 Multiplying this out, we find the following result: ((δ˜6 ) = δ˜6 ⊗ 1 +1 ⊗ δ˜6

+

9 4

+

+10

⊗

+3 9 + 4 15 + 2

⊗

+

⊗

⊗ ⊗

+

(7)


+ +

9 2

15 2

⊗

+

+

+

+

+

⊗

+15

⊗

9 2

⊗

+15 9 + 2

⊗

+15 27 + 8 45 + 4 27 + 8 45 + 4

⊗

+

239

+

+3

+

⊗

⊗ ⊗ ⊗

⊗ ⊗

+

+10 +6

⊗

+

+

+

⊗

+

+20

⊗

−9 −

⊗

9 2

⊗ −

3 4

9 16 27 − 2 −

⊗ ⊗ ⊗

−12

⊗

−18

⊗

+

+

240


+

27 4

⊗

+

⊗

+9

⊗ 9 4

⊗

+3

⊗

+

.

(8)

Now we have to compare with ((δ˜6 ), so we first apply the homomorphism to graphs and use the coproduct ( on them. For this, we need ([z1,2 ] =

⊗1+1⊗

([z3,2 ] =

,

(9)

⊗1+1⊗

,

(10)

([z1,4 ] = z1,4 ⊗ 1 + 1 ⊗ z1,4 + 3

⊗

1 2

([z3,2 ] = z3,4 ⊗ 1 + 1 ⊗ z3,4 +

([z1,6 ] = z1,6 ⊗ 1 + 1 ⊗ z1,6 + 3 +3 + + +2 ⊗ +

+

3 2

⊗

⊗ +

⊗

+

,

⊗

,

3 2 3 + 2 1 + 2 3 + 2

⊗

+

(13)

⊗ +

+ 9 2 3 + 2 5 + 2

+

⊗

⊗

⊗

+

⊗

+

+

+

+

⊗

+

+ ⊗

+

3 + 2 3 + 2

⊗ ⊗

⊗

+

+

(12)

+

+3

(11)

+

+

⊗

3 + 2 ⊗

+ +

3 2

⊗

⊗ + ,

+ 3 2

⊗

+


1 2

([z3,6 ] = z3,6 ⊗ 1 + 1 ⊗ z3,6 + 1 2 1 8 1 2 1 4 1 2 1 2

+ + + + + + +

1 2 1 + 2

+

⊗

+

1 4

(14) ⊗

⊗ ⊗ ⊗ ⊗

+ +

⊗

⊗

⊗ ⊗

+

+

⊗

241

⊗ + ⊗ +

1 4

⊗

+ ⊗

⊗

+

+

⊗ ⊗

+

.

It is now only a matter of using the rhs of (3) for δ˜6 to confirm that we reproduce the ⊗ in ((δ˜6 ) we find result (8). For example, for the contribution to −

5×3 2

⊗

+7×

5 2

⊗

= 10

⊗

,

as desired. Similarly, one checks all of the 32 tensorproducts of (8). Acknowledgements. Both authors thank the IHES for generous support during this collaboration. D.K. is grateful to the DFG for a Heisenberg Fellowship. Both authors thank T. Krajenski for discussions and careful reading of the manuscript.

References 1. 2. 3. 4. 5. 6. 7.

Connes, A., Kreimer, D.: Commun. Math. Phys. 210, 249 (2000); hep-th/9912092 Connes, A., Kreimer, D.: Commun. Math. Phys. 199, 203 (1998); hep-th/9808042 Araki, H.: Ann. Sci. École Norm. Sup. (4) 6, 67 (1973) Collins, J.: Renormalization. Cambridge: Cambridge University Press, 1984 Connes, A., Moscovici, H.: Commun. Math. Phys. 198, 199 (1998); math.dg/9806109 Kreimer, D.: Adv. Theor. Math. Phys. 3.3, (1999); hep-th/9901099 Broadhurst, D.J., Kreimer, D.: Phys. Lett. B 475, 63 (2000); hep-th/9912093

Communicated by A. Jaffe

Commun. Math. Phys. 216, 243 – 253 (2001)

Communications in



Groups of Loops and Hoops Pablo Spallanzani Centro de Matemática, Facultad de Ciencias, Igua 4225, Montevideo CP11400, Uruguay. E-mail: [email protected] Received: 4 November 1999 / Accepted: 3 May 2000

Abstract: The approaches to quantum field theories based in the so-called loop representation deserved much attention recently. In it, closed curves and holonomies around them play a central role. In this framework the group of loops and the group of hoops have been defined, the first one consisting in closed curves quotient with the equivalence relation that identifies curves differing in retraced segments, and the second one consisting in closed curves quotient with the equivalence relation that identifies curves having the same holonomy for every connection in a fiber bundle. The purpose of this paper is to clarify the relation between hoops and loops, or in other words, to give a description of the class of holonomy equivalent curves.

1. Introduction An important step in the construction of quantum field theories is the definition of the space of states L2 (A/G). This is done in [1, 2] by first constructing generalized measures on A/G. In these constructions the notions of group of loops, holonomy around loop and group of hoops play a central role (the precise definitions are stated below). Given a differentiable manifold M and a point o of M we construct the space of closed curves in M, , as the set of piecewise regular curves α : [0, 1] → M. A curve β : [a, b] → M is regular if there exists > 0 and a differentiable (or analytic) curve γ : (a − , b + ) → M such that β and γ coincide in [a, b], and we say that a curve α : [0, 1] → M is piecewise regular if a partition of [0, 1], 0 = t0 < t1 < · · · < tn = 1 exists such that α restricted to each of the intervals [ti−1 , ti ] is regular. In we can define the inverse of a curve α −1 (t) = α(1 − t), and the composition of curves αβ(t) =

α(2t) if t < 1/2 . β(2t − 1) if t ≥ 1/2

244

P. Spallanzani

If we identify curves that only differ in a reparameterization the composition is an associative operation, but in general αα −1 = c (c being the constant curve). To make in a group we need to introduce a further equivalence relation. A possibility is to identify curves differing in retraced segments, that is we identify αβ with αρρ −1 β. The group obtained is called the group of loops and is denoted by LG or LG ω if we work with analytic curves. Other possible identification is, given a principal bundle (E, M, G, π ), G a Lie group, identify two curves α and β if they have the same holonomy for every connection in the bundle. The group obtained this way is called the group of hoops and is denoted by HG or HG ω in the analytic case. The purpose of this paper is to clarify the relation between LG and HG and how HG depends on the Lie group G. In particular we obtain results for piecewise differentiable loops without making any assumptions on the Lie group. Consider the infinite set of symbols e1 , e2 , . . . and e1−1 , e2−1 , . . . , and let E be the set of words in these symbols including the null word. A word is a finite ordered list of symbols (ex. e3 e1−1 e2 is a word). If we define the product of a word as the concatenation and identify words that differ by “canceling opposite symbols”, that is w1 ei ei−1 w2 ∼ w1 w2 , then E is a free group. Let us define EG , the group of identities of G, as the subgroup of E consisting in words, say e2 e3 e1−1 , such that if we assign to each ei a element gi of G and multiply these in the way specified by the word (as g2 g3 g1−1 ), the result is the identity of G no matter what choice of gi (ex. if G is abelian e1 e2 e1−1 e2−1 is an identity). In other words, EG is the intersection of the kernels of every homomorphism of groups from E to G, ker f. EG = f ∈ hom (E,G)

Now we define EG (LG) as the subgroup of LG generated by the loops obtained in the following way: for every word in EG (such as e2 e1 e3−1 ) and every assignment of a loop αi to each of the symbols ei take the product of αi in the same way as the word (ex. α2 α1 α3−1 ). Or equivalently EG (LG) =

f (EG ). f ∈ hom (E,LG )

Now we can state the main results, first in the analytic case. Theorem 1. For G a connected Lie group, HG ω = LG ω /EG (LG ω ). This result is complemented with results about EG of Sect. 3. Theorem 2. If G is abelian then EG is generated by elements of the form ei ej ei−1 ej−1 . Theorem 3. If G is connected and non-solvable then has no non-trivial identities. Then we have the following corollaries (see [1]). Corollary 1. If G is abelian then HG ω = LG ω /[LG ω , LG ω ]. Corollary 2. If G is connected and non-solvable then HG ω = LG ω .

Groups of Loops and Hoops

245

In Sect. 5 we show through a example that Theorem 1 is not valid in the differentiable case. However we have: Theorem 4. For G a connected Lie group, HG = LG/EG (LG). EG (LG) is the closure of EG (LG) in the quotient topology arising from the C N topology of curves for any N . The topology of the loop space is discussed in Sect. 4 where we also show that the topology introduced by Barrett [4] coincides with the usual C N topology. For a nonsolvable group a result anologous to Corollary 2 can be obtained as showed in [7]. Theorem 5. If G is connected and non-solvable then HG = LG. 2. Analytic Loops First we consider the case of analytic loops, this case is simpler because of the way in which analytic curves intersect, they intersect either in finitely many points or in a segment. From this we obtain a decomposition of a loop in independent loops. Let us define what we mean by independent loops. We say that a loop α has a segment ρ that is traced once if there exist curves β and γ such that α = βργ and β and γ do not intersect ρ except at the endpoints. A set of loops α1 , . . . , αn is independent if each loop αi has a segment ρi traced once and the segments ρi do not intersect. Theorem 6. Every loop can be decomposed into a product of independent loops. Proof. The loop γ is piecewise analytic. Thus it can be written as a product of analytic curves γ = ρ1 . . . ρn . The curves ρi intersect each other in finitely many points or in a common segment. Thus each ρi can be decomposed in segments that intersect only at the endpoints or coincide, then γ = αis11 . . . αiskk , where sj is either 1 or −1. Let us denote by e− (αi ) the initial point of αi and e+ (αi ) the final point. Let E denote the set of all the endpoints of all αi . For each point p ∈ E choose a curve β(p) from o to p that does not intersect the segments αi , and let γi = β(e− (αi ))αi β(e+ (αi )), then γ = γie11 . . . γiekk . Lemma 1. Let (E, M, G, π) be a principal bundle with G a connected Lie group and α a loop in M with a segment traced once, chose any element g of G, then there is a connection θ in E such that Hθ (α) = g. Proof. The loop α has a segment traced once, thus we can find a local parameterization of M such that 1. Its domain contains I = [0, 1]n , in what follows we identify points in I with its images in M and we fix a trivialization of the bundle over I . 2. α = βγ ξ such that β and ξ do not have points in I except their endpoints. 3. γ is the segment from a = (0, 1/2, . . . , 1/2) to b = (1, 1/2, . . . , 1/2) in I . Let A be any connection, take a small deformation of A such that A is flat over I . Take oˆ and pˆ in the fiber over o such that oˆ = pg, ˆ let βˆ be the horizontal lift of β that starts in oˆ and ξˆ be the horizontal lift of ξ that ends in p. ˆ Next fix a trivialization of the bundle over I , thus elements in fibers over points of I can be identified with elements of G. Let g1 be the endpoint of βˆ in the fiber over a and g2 the endpoint of ξˆ in the fiber over

246

P. Spallanzani

g2 ξˆ

g1

pˆ βˆ

a

β

γ H

b

I oˆ o

ξ Fig. 1.

b, see Fig. 1. G is connected then there exists a curve s : [0, 1] → G such that s(t) = g1 for t ∈ [0, ) and s(t) = g2 for t in (1 − , 1] and take ρ : [0, 1] → R differentiable such that ρ(t) = 1 for t ∈ [ , 1 − ] and ρ(t) = 0 for t ∈ [0, /2) ∪ (1 − /2, 1]. Take the connection Ax = Ls(x1 )−1 s˙ (x1 )ρ(x2 ) . . . ρ(xn )dx1 , let B be the connection defined as A outside I and as A in I (note that this is a smooth connection because A is flat in I and the way in which s and ρ were chosen), then HB (α) = g. Note that the proof of this lemma requires to change a given connection only in a small neighborhood of a point in the segment traced once, then we can use it to prove the next proposition. Proposition 1. Let (E, M, G, -) be a principal bundle with G a connected Lie group and α1 , . . . , αn be independent loops. Then for every (g1 , . . . , gn ) in Gn there is a connection A such that HA (αi ) = gi for i = 1, . . . , n. Now we state and prove the main theorem of the section. Theorem 7. For G a connected Lie group, HG ω = LG ω /EG (LG ω ). Proof. Let α be a loop in EG (LG ω ). Then there exist a word in EG , for example e1 e2 e1−1 e3 e2−1 , such that α = α1 α2 α1−1 α3 α2−1 , then for every connection A, if we define gi = HA (αi ), the holonomy of α is HA (α) = g1 g2 g1−1 g3 g2−1 = e. Conversely if α is a loop not in EG (LG ω ) then by Theorem 6 there exists independent loops α1 , . . . , αn such that α = αis11 . . . αiskk . Note that eis11 . . . eiskk is not an identity of G because α ∈ EG (LG ω ), then there exist g1 , . . . , gn in G such that gis11 . . . giskk = e; by Proposition 1 there exists a connection A such that HA (αi ) = gi then HA (α) = e. 3. Identities in Lie Groups In this section we prove the following theorems:


247

Theorem 8. If G is abelian then EG is generated by elements of the form ei ej ei−1 ej−1 . Theorem 9. If G is connected and non-solvable then it has no non-trivial identities. Proof of Theorem 8. If g is abelian then EG contains all words of the form ei ej ei−1 ej−1 . Conversely, if eis11 . . . eiskk is an identity all the words formed by reordering terms are also identities because G is abelian, thus is sufficient to prove that e1a1 . . . enan is an identity iff ai = 0 for i = 1, . . . , n. If aj = m = 0 then take g an element of G such that g m = e (for example if v is a vector in g such that exp v = e take g = exp v/m), define gi = e if i = j and gj = g. Then g1a1 . . . gnan = g m = e thus e1a1 . . . enan is not an identity of G.

To prove Theorem 9 we show that if G is non-solvable then for every n it has a free subgroup with n generators. We will use the following theorem due to Tits [9]. Theorem 10. Let G ⊂ GL(V ) be a subgroup, V a finite-dimensional vector space over a field of characteristic 0. Then G has a free subgroup with n generators for every n or G has a solvable subgroup of finite index. First we recall the definition of solvable groups. Let G be a group. The derived group G is the subgroup of G generated by elements of the form xyx −1 y −1 , x, y ∈ G, then define by induction G(1) = G

G(n+1) = G(n) ,

then G is solvable if G(n) is the trivial group for some n. Next we prove the following theorem. Theorem 11. If G is a subgroup of GL(n, R) connected and non-solvable then for every n it has a free subgroup with n generators. Proof. Suppose that G does not contain a free subgroup with n generators, then by Theorem 10 G has a solvable subgroup of finite index H . Let H be the closure of H in G; then H is a solvable subgroup of finite index of G. Then either H = G which is absurd because G is nonsolvable, or the index of H in G is greater than 1. Then G is union of finitely many disjoint closed subsets (the cosets of H ) which is absurd because G is connected. To prove Theorem 9 we use the adjoint representation of a Lie group, Ad : G → Aut(g), Ad(g)v = dag v, where ag : G → G, ag (x) = gxg −1 . Proof of Theorem 9. We need to show that Ad(G) is a connected nonsolvable subgroup of Aut(g). Clearly Ad(G) is connected because Ad is continuous. Suppose that Ad(G) is solvable, that is, there exist n such that Ad(G)(n) = {e}, but Ad(G)(n) = Ad(G(n) ). Then G(n) ⊂ Ker Ad = Z(G) (Z(G) is the set of all elements in G that commute with every other element of G). Then G(n+1) = {e} which is absurd because G is nonsolvable. Then Ad(G) is a connected nonsolvable subgroup of Aut(g) and by Theorem 11, Ad(G) has no non-trivial identities thus G has no non-trivial identities.

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P. Spallanzani

4. Topology of the Loop Space In this section we discuss several ways to give a topology to the loop space. We work in the space of parameterized paths in M. Let P N = {γ : [0, 1] → M : γ is piecewise C N } and we define P ∞ = N>0 P N . We define the C N topology in P N giving a subbase of open sets. Let φ : U ⊂ M → Rd be a coordinate system in M, a < b ∈ [0, 1] and γ a curve such that γ |[a,b] ⊂ U . Then we define N (γ , ) = {α ∈ P N : α|[a,b] ⊂ U and |α (n) (x) − γ (n) (x)| < , Uφ,a,b

∀n ≤ N, x ∈ [a, b]}, where we identify α with φ ◦ α and we denote by α (n) the nth derivative of α. We take this family of sets with N fixed as a subbase of the C N topology in P N and take the family of these sets for all N as a subbase of the C ∞ topology in P ∞ . We now give another characterization of this topology. Define a C N homotopy, where possibly N = ∞, as function 6 : U → P N that is obtained from a C N function φ : U × [0, 1] → M with U a open set of Rn , the finest topology in which all the C N homotopies are continuous is the Barrett [4] topology. We claim that the Barrett topology coincides with the C N topology. We give a proof for the C ∞ case; the C N case is similar. In what follows we consider that all the curves are contained in the domain of a coordinate system (if not we can divide paths in smaller pieces) and thus we identify them with paths in Rd . As obviously all the C ∞ homotopies are continuous in the C ∞ topology, closed sets in the C ∞ topology are closed in the topology generated by C ∞ homotopies. The converse follows from the following lemma. Lemma 2. If γn is a sequence of C ∞ curves in Rn converging in the C ∞ topology to γ then there is a homotopy 6 : (−1, 1) → P ∞ such that 6(0) = γ and αn = 6(2−n ) is a subsequence of γn . Proof. Take ρ : [0, 1] → [0, 1] a C ∞ function such that ρ(0) = 0, ρ(1) = 1, ρ (n) (0) = ρ (n) (1) = 0 for all n ≥ 1 and let an = maxx∈[0,1],k≤n |ρ (k) (x)|. Take αn a subsequence 2 −1 of γn such that αn ∈ U N (γ , 2−N −N−1 aN ) for all n ≥ N − 1. Define φ : (−1, 1) × d [0, 1] → R as γ (t) if s ≤ 0 φ(s, t) = . (1 − ρ(2n s − 1))αn (t) + ρ(2n s − 1)αn−1 (t) if 2−n < s ≤ 2−n+1 Then obviously φ(s, t) is C ∞ when s = 0. We have to show that ∂ k+l φ →0 ∂s k ∂t l when s → 0. Take n > k + l, then for 2−n < s ≤ 2−n+1 we have k+l ∂ φ (l) nk (k) n (l) nk (k) n ∂s k ∂t l = | − 2 ρ (2 s − 1)αn (t) + 2 ρ (2 s − 1)αn−1 (t)| (l)

≤ 2n an |αn(l) (t) − αn−1 (t)| ≤ 2n an 2−n 2

2

2 −n

an−1 ≤ 2−n .

Then φ(s, t) is a C ∞ function thus 6 is a C ∞ holonomy such that αn = 6(2−n ) and γ = 6(0).


249

5. Differentiable Loops The key of the proof of Theorem 1 for analytic loops was the theorem of decomposition of a loop in a product of independent loops (Theorem 6). In the case of differentiable loops this theorem is not valid because differentiable curves can intersect in complicated ways. For example let ρ : [0, 1] → [0, 1] be a differentiable function such that ρ (n) (0) = (n) ρ (1) = 0 for all n ≥ 0 and let an = maxx∈[0,1],n>0 |ρ (n) (x)|. Define f1 , f2 : [0, 1] → R as fi (x) = (3 − 2i)n

1 ρ(2n x − 1) if 2−n < x ≤ 2−n+1 , 2 n an

i = 1, 2

and f3 = −f1 , f4 = −f2 , see Fig. 2.

f1

f3

f2

f4

Fig. 2.

Take the curves ci (x) = (x, fi (x)) and c = c1 c2−1 c3 c4−1 . Then c has trivial holonomy for any connection in a bundle with abelian structure group G but it is not in EG (LG). However we have another decomposition available [2]. Let us begin with some definitions, let T be a family of curves α1 , . . . , αn in M, αi : [0, 1] → M. We define range(T ) as the union of the images of the curves in M, a point p of range(T ) is a regular point if there is a neighborhood U of p such that U ∩ range(T ) is an embedded segment in M. Definition 1. Let p be a regular point, the type of p is the subgroup of Gn generated by the elements in which the i th component is e if αi does not pass through p and the other components are all equal. Definition 2. We say that the family T is a tassel based at b if 1. range(T ) is contained in a contractible open subset of M. 2. αi (0) = b for i = 1, . . . , n. 3. There is a parameterization (x1 , . . . , xd ) → M such that b = (0, . . . , 0) and αi can be written as a graph αi (t) = (t, fi (t)), where fi : [0, ti ] → Rd−1 .

250

P. Spallanzani

4. If there is a regular point in range(T ) with a certain type, then there are points with the same type in every neighborhood of b. 5. All the curves in the family are different. Definition 3. A family of curves α1,1 , . . . , α1,n1 , . . . , αk,1 , . . . , αk,nk is a web if the curves αj,1 . . . αj,nj form a tassel Tj for j = 1, . . . , k and curves in two different tassels do not intersect (except, possibly, at their endpoints). In [2] the following proposition is proven. Proposition 2. For every family of curves F in M there is a web W such that all curves in F are products of curves in W (or their inverses). Applying this proposition to a loop γ we obtain a web W formed by a family of curves α1 , . . . , αn such that γ can be obtained as product of curves in W . Then repeating the same construction as in proof of Theorem 6 we can construct loops γi = β(e− (αi ))αi β(e+ (αi )). Then γ is a product of loops γi . Definition 4. Let T be a tassel, GT is the closed subgroup of Gn generated by all the types of regular points in T . Now let us see what are the possible holonomies for curves in a tassel. Let T be a tassel composed by curves α1 , . . . , αn ; range(T ) is contained in a contractible open set U . Then we can fix a trivialization of the bundle over U and associate to each connection a element of G over each curve. Then we can identify the set of possible values of holonomies for the curves αi with a subset of Gn . In [2] the following proposition is also proved: Proposition 3. For a tassel T the set of possible values for the holonomies is GT . To prove Theorem 5 we need this lemma which was probed in [7]. Lemma 3. If G is semisimple then GT = Gn . Proof. It is sufficient to show that all the elements of Gn of the form E(g, i) = (e, . . . , g, . . . , e) (the element of Gn that has g in the i th component and e in the others) are in GT . Now we say that an element of Gn is of the form E(g, i, i1 , . . . , ik ) if its i th component is g and the components i1 , . . . , ik are e. We will show that GT has elements of the form E(g, i, i1 , . . . , ik ) for every g, i, i1 , . . . , ik . (When k = n − 1 this implies that GT has all the elements of the form E(g, i), then GT = Gn .) We proceed by induction in k. When k = 1, given g, i, i1 , we can find a regular point of T such that αi passes through p and αi1 does not, then the type of p is a element of GT of the form E(g, i, i1 ). To proceed with the inductive step assume that we are given g = g1 g2 g1−1 g2−1 , i, i1 , . . . , ik+1 . Then take elements of GT gˆ 1 of the form E(g1 , i, i1 , . . . , ik ) and gˆ 2 of the form E(g2 , i, i2 , . . . , ik+1 ) (they exist by induction hypothesis), then gˆ 1 gˆ 2 gˆ 1−1 gˆ 2−1 is a element of GT of the form E(g, i, i1 , . . . , ik+1 ). Since G is semisimple elements of the form g1 g2 g1−1 g2−1 generate G, then for every g ∈ G GT has elements of the form E(g, i, i1 , . . . , ik+1 ). Now we can give a proof of Theorem 5 Proof of Theorem 5. Let (E, G, M, -) be a principal bundle with G a non solvable group. ˆ be the quotient of G by its radical, then G ˆ is semisimple. Let (E, ˆ G, ˆ M, -) ˆ be Let G


251

ˆ Let γ be a loop in M not null in LG, then as the extension of the bundle to the group G. remarked before we can decompose γ in a product of loops γi = β(e− (αi ))αi β(e+ (αi ), ˆ is semisimple we can choose where the curves α1 , . . . , αn form a web. Then because G holonomies independently for the loops γi . Then we can proceed as in the proof of Theorem 1 and find a connection A in Eˆ such that HA (γ ) = e. Then we can pull-back this connection to E and obtain a connection in E such that the holonomy of γ is not null. Now let us prove Theorem 4. It follows from the following proposition. Proposition 4. Let γ be a C N loop such that HA (γ ) = e for all connections in the bundle (E, G, M, -). Then there are loops in EG (LG) arbitrarily C N close to γ . Proof. We first decompose γ in curves forming a web with tassels T1 , . . . , Tn . We will do small deformations to these curves to obtain a family of curves that intersect each other only in a finite number of segments or isolated points. We need to do such deformation in a way that the holonomy of the deformed loop γˆ is e for every connection, then the same argument as in proof of Theorem 1 shows that γˆ ∈ EG (LG). To accomplish this it is sufficient that the group of possible holonomies of the deformed tassel GTî be included in GTi , and for this is sufficient that the deformation does not take apart curves that intersect. Fix > 0, take α1 , . . . , αc curves in a tassel T in the decomposition of γ and take the parameterization in Definition 2 such that α1 (t) = (t, 0, . . . , 0), t ∈ [0, 1]. We identify points in M with the corresponding points in Rd in the parameterizations and identify points t in [0, 1] with points (t, 0, . . . , 0). Also take a C ∞ function ρ : R → [0, 1] such that ρ(x) = 1 if x ≤ −1 or x ≥ 1 and ρ(x) = 0 if −1/2 ≤ x ≤ 1/2. Let (n) an = maxx∈R |ρ (n) (x)| and let ρp,δ (x) = ρ((x − p)/δ). Note that |ρp,δ (x)| ≤ an /δ n . We say that a point in the intersection of two curves is singular if it is not in the interior of a common interval of both curves. Let A be the set of singular points of intersection between α1 and the other curves, because of the way in which the coordinate system was chosen points in A are of the form (t, 0, . . . , 0) and thus we identify them with points of [0, 1] which are the values of the parameter of the point taking the parameterization αi (t) = (t, fi (t)) (where fi is as in Definition 2). Let A be the set of accumulation points of A. Thus A is a compact subset of [0, 1]. Let p ∈ A , and we define C N (p) as the set of curves αi that have a contact of order N with α1 in p (that is the first N derivatives of fi in p are null). Note that if p is an accumulation point of intersection points of α1 and αi then αi ∈ C N (p). And define C · · · > δk . Let 0 fi = fi , j fi = j −1 fi ρqj ,δj (n) then f¯i (x) = k fi (x). We will prove by induction |j f (x)| < 2j rqk (f (n) , δj )|x − i

qk |N−n , for j = 0 is clear, for j > 0, n (n) (n−@) n (@) C@ ρqj ,δj (x)j −1 fi (x) |j fi (x)| = @=0 n (n−@) n a@ j −1 N−n+@ C@ @ 2 rqk (fi , δj )|x − qk | ≤ δj @=0

(n)

< 2j rqk (fi , δj )|x − qk |N−n , where it is used that |x − qk | < δk < δj . Then c+1 (n) (n) |f¯i (x)| < 2k r(fi , δk )|x − qk |N−n < 2k < 22 . c+1

This shows that the distance of αi and α¯ i is less than 22 in the C N topology, and this construction removed all the accumulation points of singular intersection points between α1 and α¯ 2 . Then α1 and α¯ 2 intersect in a finite number of intervals or isolated points. We have to see that GT¯ , the set of possible values for holonomies of the curves α¯ i , is not larger than GT . To show this it is sufficient to show that if αi (t) = αj (t) then α¯ i (t) = α¯ j (t). Let Qi (x) = {q ∈ Qi (x) : x ∈ (q − δq , q + δq )}, then ρq,δq (x) ρi (x) = q∈Qi (x)

if q ∈ Qi (x) and q ∈ Qj (x). Then αi ∈ C N (q) and αj ∈ C 0, there is a constant C = C(L) such that L ρ(x, t)dx ≤ C(L) (1.7) −L

for all t ∈ [0, T ]. Now, in the existence theory of [8] (which deals only with the barotropic case P = P (ρ)), smooth reference functions ρ(x) ¯ and u(x) ¯ are defined which are constant for x ≤ −1, constant for x ≥ 1, and monotone for −1 ≤ x ≤ 1. The constructed solutions are then shown to satisfy a number of regularity conditions and estimates, among which the following are particularly important: R

¯ (x, t)dx + ρu2 − ρ¯ u¯ 2 + G(ρ, ρ)

T

0

R

u2x dxdt < ∞.

(Thus ux (·, t) ∈ L2 (R) for almost all t ∈ [0, T ].) Here G is the potential energy density relative to the reference state ρ, ¯ defined by ρ P (s) − P (ρ(x)) ¯ G(ρ, x) = ρ ds. 2 s ρ(x) ¯ Thus G is a smooth, nonnegative function. It was also assumed in [8] that lim inf ρ→0 G(ρ, x) ≥ C −1 for some constant C, independent of x. It is easily seen that this condition is satisfied in the representative case that P = P (ρ) = Kρ γ , γ ≥ 1. In the present paper we shall deal with weak solutions which are assumed to satisfy analogous, but somewhat weaker conditions. These conditions are formulated to be the minimum required for the proof of our theorem, and are consequently slightly technical. It is easy to see, however, that they are indeed weaker than the conditions described above, which are known to be satisfied by the solutions constructed in [8]. We thus assume that (A2 ) ux ∈ L1 [0, T ]; L2loc (R) . In particular, ux (·, t) ∈ L2loc (R) for almost all t ∈ [0, T ]. Next we assume that (A3 ) there is a function γ (t) ∈ L1 ([0, T ]) such that, for all L > 0 and almost all t ∈ [0, T ],

L

−L

1/2 (ρu )(x, t)dx 2

≤ γ (t)(1 + L),

(1.8)

≤ γ (t)(1 + L).

(1.9)

and

L

−L

1/2 ux (x, t)2 dx

258

D. Hoff, J. Smoller

We assume also that (A4 ) for every L > 0 there is a constant C = C(L) such that L (ρ|u|)(x, t)dx ≤ C(L) −L

(1.10)

for all t ∈ [0, T ]. (We note, however, that, if (A3 ) were strengthened slightly by replacing the right side of (1.8) by C(1 + L) for some constant C and requiring (1.8) to hold for all t, then (A4 ) would be a consequence of (1.7) and (1.8); that is, finite local mass and kinetic energy would imply finite local momentum.) Finally we assume that (A5 ) there is a “potential energy density” function G(ρ, x, t), which is nonnegative and continuous on R≥0 × R × [0, T ], and for which: a) there exist positive constants C0 > 0 and ρ > 0 such that, for all x ∈ R, t ∈ [0, T ], and ρ ∈ [0, ρ], G(ρ, x, t) ≥ C0−1 ;

(1.11)

b) there exist constants C1 > 0 and θ ∈ [0, 1) such that, for all x0 , L ∈ R and all t ∈ [0, T ], x0 +L G(ρ(x, t), x, t)dx ≤ C1 + θC0−1 L. (1.12) x0

We remark that, for solutions of the nonbarotropic system alluded to earlier, in which P = P (ρ, e) = (γ − 1)ρe, the negative of the entropy density, that is, S(ρ, e) ≡ ρ(e − 1 − log e) + (γ − 1)(1 − ρ + ρ log ρ), has locally finite spatial integral at all times, at least in all known constructed solutions which could be regarded as physical (see [5], for example). The hypothesis (A5 ) above may therefore be met by taking G = (γ − 1)(1 − ρ + ρ log ρ). The results of the present paper are thus seen to apply as well to the equations of nonbarotropic flow for an ideal fluid. The following theorem is the main result of this paper. Theorem. Assume that P and f satisfy conditions (1.3)–(1.4) above, and let (ρ, u) be a solution of (1.1)–(1.2) on [0, T ] satisfying assumptions (A1 )–(A5 ). If ρ(x, 0)dx > 0 (1.13) E

for every open set E ⊂ R, then

E

ρ(x, t)dx > 0

for every open subset E ⊂ R and for every t ∈ [0, T ].

(1.14)

Non-Formation of Vacuum States for Compressible N–S Equations

259

We now give a brief, heuristic overview of the proof and explain some of the underlying physical motivations. The rigorous proof is detailed in a sequence of lemmas in Sect. 2. We first show that u ∈ L1 ([0, T ]; L∞ ([−L, L])) for every L, and that the norm in the latter space grows at most linearly in L. These facts would be immediate from (1.8) and (1.9) if we knew that ρ were bounded below away from 0. We instead apply the hypotheses (1.11) and (1.12), which imply the weaker fact that ρ cannot be close to zero on too a large set. This turns out to be sufficient for the required estimate, which is given T in Lemma 2.1 below. Observe that u(·, t)L∞ ([−L,L]) dt dominates the distance that 0

a fluid particle travels between times 0 and T , provided that it remains within [−L, L]. The fact that this integral grows at most linearly in L therefore shows, at least at the heuristic level, that a fluid particle can travel at most a finite distance in finite time. Now suppose that ρ(x, t1 ) = 0 a.e. on (a, b), where a is minimal and b is maximal. Our observation above concerning finite average convection speeds then implies that there must be nearby vacuum states at nearby times. Specifically, we construct curves y(t) and z(t) starting from a and b respectively, such that ρ(·, t) = 0 a.e. on (y(t), z(t)), and such that y(t) is minimal and z(t) is maximal. By comparing with the time-antiderivative of u(·, t)L∞ , we are able to prove that these curves are in fact absolutely continuous, and can be extended backward to a minimal time t0 ≥ 0, and that y(t0 ) = z(t0 ). Thus a vacuum exists in the wedge–shaped region V given by V = {(x, t) : y(t) ≤ x ≤ z(t), t0 ≤ t ≤ t1 } . Since ρ = 0 in V , u is evidently linear in V , say u(x, t) = α(t)x + β(t), in a suitable sense. Now, in what is the most difficult part of the analysis, we show that integral curves of u which start in V must remain in V on [t0 , t1 ]. This result depends in a crucial way on the linearity of u in V and on the absolute continuity of the boundary curves y and z, and is given in Lemma 2.6 below. This invariance of V for the fluid flow thus implies that any two integral curves of u in V , proceeding backward in time, must come together at time t0 . It therefore follows that α cannot be integrable on [t0 , t1 ]. We now apply this fact to derive a contradiction, motivated by the following physical intuition. First recall that, in the Navier–Stokes model, the term µux represents the viscous force applied at the surface of a fluid particle by an adjacent fluid particle. (The second derivative µuxx in (1.2) results from an application of the divergence theorem.) Recall also from elementary mechanics that the time-integral of a given force, which is called the impulse, equals the corresponding change in momentum of the system. Now, in the situation described above, µux = µα is therefore the viscous force applied by the “massless fluid particles" in V at the boundary of the fluid to the right of V . The nonintegrability of α on [t0 , t1 ] therefore implies that the change in momentum from time t to time t1 becomes infinite as t → t0 . But this contradicts the fact (1.8) that the momentum is locally finite, thus completing the proof. The initial-value problem for the Navier–Stokes equations (1.1)–(1.2) has been studied by many authors. See for example Kanel [12], Hoff [4, 5], and [8], Kazhikov and Shelukhin [13], and Serre [19] for existence of solutions with constant time-asymptotic states, as well as Liu [14], Hoff and Liu [9], Liu and Xin [15], Szepessy and Xin [21], and Matsumura and Nishihara [16] and [17] for cases in which the time-asymptotic state contains a viscous shock or rarefaction wave, usually of small strength. There are a number of results concerning solutions of (1.1)–(1.2) on a finite interval with suitable boundary conditions, among which we mention those of Amosov and Zlotnick [1], Chen,

260

D. Hoff, J. Smoller

Hoff and Trivisa [2], Fujita-Yashima et. al. [3], Hoff and Ziane [11], Matsumura and Yanagi [18], and Shelukhin [20]. See also Hoff and Zarnowski [10], Hoff [5] and [7], and Hoff and Ziane [11] for uniqueness and continuous dependence results for solutions with strictly positive densities. Finally we call attention to the result of Hoff [6], in which solutions are obtained for the multidimensional, spherically symmetric version of (1.1)–(1.2) with large, possibly discontinuous data. The density is assumed to be strictly positive at t = 0, but the existence theory allows for the possibility that a vacuum state forms in a ball centered at the origin in positive time. It is not known whether there are in fact solutions with such vacuum states, or whether such solutions can be precluded. Indeed, the question of the spontaneous formation of vacuum states in solutions of the Navier–Stokes equations in several space variables remains an important open question. 2. Proof of the Theorem In this section we give the details of the proof outlined above. The hypotheses (1.3), (1.4), and (A1 )–(A5 ) will be in force throughout this section, the constants C0 , C1 and θ defined in (1.11) and (1.12) will be fixed, and, unless otherwise stated, C will denote a generic positive constant whose precise meaning will be clear from the context. Lemma 2.1. u ∈ L1 ([0, T ]; L∞ loc (R)); in fact, there is a constant C > 0 such that for any L > 0, u(·, t)L∞ (−L,L) ≤ Cγ (t)(1 + L) for almost all t ∈ [0, T ], where γ is as in (1.8) and (1.9). 1 , for almost all t ∈ [0, T ]; pick such a t. If Proof. From hypothesis (A3 ), u(·, t) ∈ Hloc % > 0 is given, and x0 ∈ [−L, L], let

A% = {x ∈ [x0 , x0 + %] : ρ(x, t) ≤ ρ}. Since (1.12) implies that C0 G(ρ, x, t) ≥ 1 if 0 ≤ ρ ≤ ρ, we have, using (1.12), x0 +% meas(A% ) ≤ C0 G(ρ(x, t), x, t)dx ≤ C0 C1 + θ%. (2.1) x0

Now choose %0 such that C0 C1 + θ%0 ≤ so that meas(A%0 ) ≤

1+θ 2

1+θ %0 , 2

(2.2)

%0 . Thus if B%0 = [x0 , x0 + %0 ] − A%0 , then

1+θ 1−θ %0 = %0 ; (2.3) 2 2 that is, meas(B%0 ) is strictly positive. Now if x1 ∈ B%0 , then ρ(x1 , t) ≥ ρ, and therefore x1 |u(x0 , t)| ≤ |u(x1 , t)| + |ux |dx meas(B%0 ) ≥ %0 −

x0

≤ ρ − 2 ρ 1/2 |u| (x1 , t) + 1

x0 +%0 x0

1/2 u2x dx

1

%02 .


261

Integrating with respect to x1 over the set B%0 gives meas(B%0 )|u(x0 , t)| ≤ ρ

− 21

1/2 (ρu )(x1 , t)dx1 )

B%0

1/2 + %0 meas(B%0 )

x0 +%0

x0

1/2

meas(B%0 )

2

1/2 u2x dx

,

so − 1

|u(x0 , t)| ≤ ρ meas(B%0 )

x0 +%0

2

− 1

≤ ρ meas(B%0 )

2

x0

1/2 ρu dx 2

1 2

+ %0

x0 +%0

x0

1/2 u2x dx

1 2

(1 + L + %0 )γ (t) + %0 (1 + L + %0 )γ (t)

≤ C (1 + L + %0 )γ (t) ≤ C(1 + L)γ (t). This proves the lemma since (A3 ) implies that γ ∈ L1 [0, T ]. We shall show that the hypothesis ρ(·, t) = 0 a.e. on some open subset of R1 leads to a contradiction. In preparation for this, we first make a remark. Remark. If ρ(·, t) = 0 on some open interval (a, b), then b − a is bounded above by a constant depending only on the parameters C0 , C1 , and θ appearing in (1.11) and (1.12). Indeed, it follows from (1.11) and (1.12), that C1 + θ C0−1 (b − a) ≥

b a

G(ρ(x, t), x, t)dx ≥ C0−1 (b − a),

and as 0 ≤ θ < 1, we see that b − a is bounded, as required. The following lemma shows that if ρ(·, t) is zero a.e. on some interval, then, if t is near t, ρ(·, t ) is zero a.e. on a nearby, but possibly smaller interval. Lemma 2.2. Let t1 < T and suppose that ρ(·, t1 ) = 0 a.e. on an open interval (a, b). Let t1 1 ∞ t0 = inf t ∈ [0, t1 ] : u(·, s)L (a,b) < (b − a) 2 t and t 1 t2 = sup t ∈ [t1 , T ] : u(·, s)L∞ (a,b) < (b − a) . 2 t1 Then t0 < t1 < t2 , and for any t ∈ (t0 , t2 ), ρ(·, t) = 0 on the interval t t a + u(·, s)L∞ (a,b) ds , b − u(·, s)L∞ (a,b) ds .

t1

t1

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D. Hoff, J. Smoller

||uε||∞

Wεδ(x,t) a+b 2

a+b – δ

2

2

–||uε||∞

x

a+b + δ

Fig. 1.

δ

ψ

1

x a

a+δ

a+2δ

b–2δ b–δ

b

Fig. 2.

Proof. It is clear that t0 ≤ t1 ≤ t2 , and Lemma 2.1 shows that strict inequalities must hold because γ is integrable. Now suppose t > t1 ; the proof for t < t1 is similar, and will be omitted. Fix δ > 0 ε satisfying δ < b−a 6 , and for small ε > 0, let u denote the usual spatial regularization of u. Then for almost all t, T > t ≥ t1 , ε u (·, t) ∞ ≤ u(·, t)L∞ (a,b) . L (a+δ,b−δ) For ease in notation, let uε ∞ = uε (·, t)L∞ (a+δ,b−δ) and u∞ = u(·, t)L∞ (a,b) . Now define the smooth function wεδ (·, t) by uε (·, t)∞ , if x < εδ w (x, t) = −uε (·, t)∞ , if x >

a+b 2 a+b 2

−δ + δ,

a+b and wεδ is decreasing on a+b 2 − δ, 2 + δ ; cf. Fig. 1 (where we take a > 0). Next, define the smooth function + δ (x) by   if x < a + δ 0, + δ (x) = 1, if a + 2δ ≤ x ≤ b − 2δ  0, if x > b − δ and + δ is increasing on the interval (a + δ, a + 2δ), and decreasing on (b − 2δ, b − δ); cf. Fig. 2. Now let φ εδ be the solution to the problem φt + w εδ φx = 0, t > t1 , φ(·, t1 ) = + δ .

(2.4)


263

t εδ

Φ

Tεδ

t1

εδ

Φ

0 0 εδ > Φx

I

0

Φ εδ x 0, then x < a+b 2 −δ, a+b εδ ε εδ so from Fig. 1, w (x, t) = u ∞ . If for such t, φx (x, t) < 0, then x > 2 + δ, and wεδ (x, t) = −uε ∞ . It follows that (cf. (2.5)), b−δ t ρ(uε − w εδ )φxεδ ≤ 0. (2.9) a+δ

t1

b−δ

Next, we claim that lim

ε→0 a+δ

t t1

ρ(u − uε )φxεδ = 0.

(2.10)

Granting this for the moment, we complete the proof of Lemma 2.2 as follows. First, from (2.5), (2.9), and (2.10), we get b−δ lim (ρφ)εδ (x, t)dx ≤ 0, t ∈ [t1 , T δ ]. (2.11) ε→0 a+δ

Then from Fig. 3, we see that the support of φ εδ is the region bounded by the characteristics I and IV. As before, the x-distance traversed by these characteristics is bounded from above by t t ε u ∞ ≤ u∞ , t1

so that the interval

a+δ+

t t1

t1

uε ∞ , b − δ −

t t1

uε ∞

≡ Iδ

(2.12)

is contained in the support of φ εδ (·, t). Hence (2.11) gives that for all t ∈ [t1 , T δ ], ρ(·, t) = 0 a.e. on Iδ . If now t < t2 (cf. the statement of Lemma 2.2), then t 1 u∞ < (b − a), 2 t1 and thus there is a δ0 > 0 such that if δ ≤ δ0 , then t 1 u∞ < (b − a) − 4δ. 2 t1

(2.13)


265

For such δ, (2.7) implies that t ≤ T δ . Thus for such t and δ, ρ(·, t) = 0 a.e. on Iδ . Taking a sequence δi 0, we get that ρ(·, t) = 0 on the interval t t u∞ , b − u∞ a+ t1

t1

for all t ∈ [t1 , t2 ], and this completes the proof of the lemma. It remains to prove (2.10). To this end, we first differentiate (2.4) with respect to x to obtain εδ εδ + w εδ φxx = −wxεδ φxεδ , φxt

so that along the characteristics x = x(t), φxεδ (x(t), t)

=

+xδ (x(t1 )) exp

t εδ − wx (x(s), s)ds . t1

(2.14)

But from Fig. 1, we see that |wxεδ (·, s)| ≤ C(δ)uε (·, s)∞ ≤ C(δ)u(·, s)∞ , (where |C(δ)| → ∞ as δ → 0), and thus from (2.14), φxεδ ∞ ≤ C (δ), where C (δ) is a constant depending only on δ. Hence b−δ t t ε εδ ≤ C ρ(u − u )φ (δ) ρu − ρuε L1 (a+δ,b−δ) dt x a+δ t1 t1 T u(·, t) − uε (·, t)L∞ (a+δ,b−δ) ρ(·, t)L1 (a+δ,b−δ) dt. ≤ C (δ)

(2.15)

t1

1 and But from hypotheses (A4 ), we have that for almost all t ∈ [t1 , T ], u(·, t) ∈ Hloc from (1.7) ρ(·, t)L1 (a+δ,b−δ) is bounded; thus for each fixed t the integrand on the right-hand side of (2.15) tends to zero as ε 0. Since

u(·, t) − uε (·, t)L∞ (a+δ,b−δ) ρ(·, t)L1 (a+δ,b−δ) ≤ C(a, b)u(·, t)L∞ (a+δ,b−δ) and u(·, t)L∞ (a+δ,b−δ) is integrable (by Lemma 2.1), the Lebesgue dominated convergence theorem applies to the right-hand side of (2.15) and shows that (2.10) holds. Now suppose that ρ(·, t1 ) = 0 a.e. on (a, b), where, without loss of generality, a is minimal and b is maximal (cf. the remark following the proof of Lemma 2.1). The interval (a, b) and the time t1 will be fixed for the remainder of the argument. Let t0 be as in the statement of Lemma 2.2, and define for t ∈ (t0 , t1 ), a+b , (2.16) y(t) = inf x : ρ(·, t) = 0 a.e. on x, 2 a+b z(t) = sup x : ρ(·, t) = 0 a.e. on ,x . (2.17) 2 Clearly, y(t1 ) = a and z(t1 ) = b. In the following lemma we prove an important regularity property for the curves y and z.

266

D. Hoff, J. Smoller

a+b 2

t1 z(t)

t

t x s

z(s) Fig. 4.

Lemma 2.3. There exists a constant h = h(a, b) > 0 such that y and z are absolutely continuous functions on [t1 − h, t1 ]. Proof. First, it follows from the remark preceding Lemma 2.2 that there exists an L > 0 such that, for all t ∈ (t0 , t2 ), −L ≤ y(t), z(t) ≤ L . Next, choose h > 0 such that

t1

t1 −h

uL∞ (−L,L) dt
0 be given; then Lemma 2.1 implies that we can find δ > 0 such that if meas(E) < δ, then uL∞ (−L,L) dt ≤ ε. (2.23) E

Thus given points {sj }k1 and {τj }k1 satisfying t1 − h ≤ s1 < τ1 < s2 < τ2 < · · · < sk < τk < t1 , with (τj − sj ) ≤ δ, (2.22) and (2.23) give j

|z(τj ) − z(sj )| ≤

j

j

=

τj sj

uL∞ (−L,L)

∪[sj , τj ]

uL∞ (−L,L) ≤ ε.

This proves that z is AC on [t1 − h, t1 ]; similarly, y is AC on the same interval.

In the next lemma, we obtain further results concerning the functions y(t) and z(t). To this end, let S be defined as the set of all t ≥ 0 such that there are extensions of y and z to [t, t1 ] such that the following three properties hold: (i) y and z are absolutely continuous on [t, t1 ], (ii) y < z on [t, t1 ], z(s)+ε z(s) ρ(x, s)ds and ρ(x, s)dx are both positive for all ε > 0 and all (iii) y(s)−ε z(s) y(s) ρ(x, s)dx > 0. s ∈ [t, t1 ], and y(s)

Notice that the last lemma implies that S is nonempty; thus let τ = inf S.

(2.24)

Concerning τ we have the following result. Lemma 2.4. y and z have AC extensions to time τ , y(τ ) = z(τ ), and there is an L > 0 such that for all t ∈ [τ, t1 ], −L ≤ y(t) ≤ z(t) ≤ L. Proof. We prove the last assertion first. Let τ < c < d < f < g < t1 , and for t ∈ (τ, t1 ), let w(t) = max{z(t), −y(t)} ≥ 0.

(2.25)

Let t ∈ [c, g]; then by definition ρ(·, t) = 0 a.e. on (y(t), z(t)), and since y(t) < z(t), Lemma 2.2 shows that there is an h = h(t) > 0 such that if |t − s| ≤ h, then ρ(·, s) = 0 a.e. on the interval s s y(t) + uL∞ (−w(t),w(t)) , z(t) − uL∞ (−w(t),w(t)) , t

t

268

D. Hoff, J. Smoller

and

C

t+h

γ (s)ds ≤ 21 ,

t−h

(2.26)

where C is as in Lemma 2.1. Thus

t z(s) ≥ z(t) − uL∞ (−w(t),w(t)) , s t y(s) ≤ y(t) + uL∞ (−w(t),w(t)) , s

so that using Lemma 2.1, we get

t ∞ w(s) ≥ w(t) − uL (−w(t),w(t)) s t ≥ w(t) − C(1 + w(t)) γ (σ )dσ s t t = 1 − C γ w(t) − C γ . s

s

Thus for |t − s| ≤ h(t), (2.26) gives t −1 t

w(t) ≤ 1 − C γ w(s) + C γ s t t s

≤ 1 + C γ w(s) + C γ , s

(2.27)

s

for some positive constant C. Now choose constants A < B (depending on t, which is fixed), such that −w(t) < A < B < w(t). If h(t) is further reduced, and if |t − s| ≤ h(t), then y(s) < A < B < z(s), as follows from the continuity of y and z (Lemma 2.3). For such s, using Lemma 2.2, we find that there is a σ , depending on B−A 2 , (so σ = σ (t)), such that if s ≤ s˜ ≤ s + σ , then ρ(·, s˜ ) = 0 on

s˜ s˜ y(s) + uL∞ (−w(t),w(t)) , z(s) − uL∞ (−w(t),w(t)) . s

It follows that y(˜s ) ≤ y(s) +

s˜ s

s

uL∞ (−w(t),w(t)) and z(˜s ) ≥ z(s) −

s˜ s

uL∞ (−w(t),w(t)) .

We can further reduce h(t) so that h(t) ≤ σ (t). Thus if t − h(t) ≤ s ≤ t, then s ≤ t ≤ s + σ (t), and we may take s˜ = t, to obtain t w(t) ≥ w(s) − uL∞ (−w(t),w(t)) s t ≥ w(s) − C(1 + w(t)) γ, s


269

where we have used Lemma 2.1. Thus if t − h(t) ≤ s ≤ t, then t t

w(s) ≤ 1 + C γ w(t) + C γ . s

(2.28)

s

We now cover the interval [d, f ] by a finite number of intervals Bhj (sj ), where s1 > s2 > · · · > sp and hj = h(sj ). If τj ∈ Bhj +1 (sj +1 ) ∩ Bhj (sj ), then by (2.27)

τj

w(sj +1 ) ≤ 1 + C

sj +1

γ

w(τj ) + C

τj

sj +1

γ .

Also, from (2.28)

w(τj ) ≤ 1 + C

sj

τj

γ

w(sj ) + C

sj τj

γ .

If we set wp = w(sp ), and w1 = w(s1 ), then iterating these inequalities gives

τj sj t

wp ≤ 1+C γ 1+C γ w1 + C γ . sj +1

τj

s

(2.29)

Now if ε1 + · · · + εq = ε, and each εi > 0, then ε q ≤ eε . (1 + εj ) ≤ 1 + q Thus applying this to (2.29) gives wp ≤ eC

T 0

γ

T

w1 + C

γ ≤ C (w1 + 1)

(2.30)

0

for some constant C . As w1 = w(s1 ), it follows that for s1 near t1 , then as noted in (2.18), we can bound w1 independent of t, and so (2.30) and (2.27) bound w on [d, t1 ], for all d > τ , independent of t. Thus we have proved that there is an L > 0 such that −L ≤ y(t) ≤ z(t) < L,

t ∈ (τ, t1 ].

(2.31)

We now show that z and y are uniformly continuous on the interval (τ, t1 ]. Once this is shown then the first and third assertions of Lemma 2.4 will be proved. Thus to prove the uniform continuity of z on (τ, t1 ], let ε > 0 be given. Choose δ > 0 such that if 0 ≤ s < t ≤ T , and |s − t| ≤ δ, then t uL∞ (−L,L) ≤ ε. s

Now just as earlier in this proof, if t ∈ (τ, t1 ], we can find h(t) > 0 such that if |t − s| ≤ h(t), then t |z(s) − z(t)| ≤ uL∞ (−L,L) . (2.32) s

270

D. Hoff, J. Smoller

Now fix s < t with |s − t| ≤ δ and s, t ∈ (τ, t1 ]; then the interval [s, t] is covered by h h q ∪1 B hk (sk ), s1 < s2 < · · · < sq , where sj + 2j > sj +1 − j2+1 , and hj < δ for each j . 2

Then |sj +1 − sj | ≤

hj +hj +1 2

≤ max{hj , hj +1 } < δ. Thus by (2.32), sj +1 uL∞ (−L,L) . |z(sj ) − z(sj +1 ) ≤ sj

Now for some j and k, s ∈ B hk (sk ), t ∈ B hj (sj ), and we have 2

2

|z(t) − z(s)| ≤ |z(s) − z(sj )| + |z(sj ) − z(sj −1 )| + · · · + |z(sk ) − z(t)| sj t t ≤ +··· + = uL∞ (−L,L) s

sk

≤ ε.

s

To complete the proof, we have to show that y(τ ) = z(τ ). But this is clear, since otherwise y(τ ) < z(τ ), and if τ > 0, then τ would not be minimal, whereas if τ = 0, z(τ ) ρ(x, 0) dx > 0 would be violated. then the hypothesis that y(τ )

We next study the function u in the vacuum region. To this end, we define the set V by V = {(x, t) : y(t) < x < z(t), τ < t ≤ t1 }. Note that for τ < t ≤ t1 , ρ(·, t) = 0 a.e. on (y(t), z(t)). Lemma 2.5. There exist functions α, β ∈ L1loc ((τ, t1 ]) such that u = α(t)x + β(t) in D (V ) and u(x, t) = α(t)x + β(t) for all x and almost all t in V . Proof. From (1.2), we see that uxx = 0 in D (V ), and thus uεxx = 0 in D (V ), where uεxx is the standard regularization of uxx . Thus uε (x, t) = α ε (t)x + β ε (t). Now from (1.10), 0=

lim

ε1 ,ε2 →0 τ

=

t1

lim

ε1 ,ε2 →0 τ

z(t) y(t)

t1

uεx1

2 − uεx2 dx

1/2 dt

|α ε1 (t) − α ε2 (t)|[z(t) − y(t)]1/2 dt,

and thus {α ε } is a Cauchy sequence in L1 ([τ + δ, t1 ]) for every δ > 0; that is, {α ε } is a Cauchy sequence in L1loc ((τ, t1 ]). Also, if I is a compact set in (τ, t1 ], and t ∈ I , |β ε1 (t) − β ε2 (t)| ≤ Cuε1 (·, t) − uε2 (·, t)L∞ (y(t),z(t)) for some constant C. Since uε → u in L1 ({(x, t) : t ∈ I, y(t) ≤ x ≤ z(t)}), we see that {β ε } is a Cauchy sequence in L1 (I ) so β ε → β in L1 (I ); thus β ε → β in L1loc ((τ, t1 ]). Since uε → u in D (V ), and α ε x + β ε → αx + β in L1loc ((τ, t1 ]; L∞ ) we obtain that u = αx + β in V .


271

The last lemma which we need is Lemma 2.6. Fix w1 ∈ (a, b) and for τ < t ≤ t1 define w(t) by t1 t1 t1 α(s)ds − exp − α β(s)ds. w(t) = w1 exp − t

t

s

Then y(t) < w(t) < z(t) for τ < t ≤ t1 . Proof. We claim that dz ≤ αz + β dt

(2.33)

for almost all t ∈ (τ, t1 ]. If this holds, then since dw = αw + β, dt w(t1 ) = w1 < b = z(t1 ), we find d (z − w) ≤ α(z − w), a.e. dt so that

t d α (z − w) ≤ 0 a.e. exp − dt t1

Integrating from t to t1 and using Lemma 2.3 gives t α [z(t) − w(t)] ≥ z(t1 ) − w(t1 ) > 0, exp − t1

so that z(t) > w(t); similarly, w(t) > y(t). We now prove (2.33). For this, we define the following sets of zero measure: A = {t ∈ (τ, t1 ] : ux (·, t) ∈ L2 (y(z), z(t))}, D = {(x, t) ∈ V : u(x, t) = α(t)x + β(t)}, E = {t ∈ (τ, t1 ] : z is not differentiable at t}. Let {rk } be the set of rational numbers, and let Bj k = {x : |x −rk | < j1 }, j, k = 1, 2, . . . . From Lemma 2.1, we have u(·, t)L∞ (Bj k ) ∈ L1 ([0, T ]). Let Fj k = {t ∈ (τ, t1 ] : t is not a Lebesgue point of u(·, t)L∞ (Bj k ) } and set F = ∪Fj k ; then meas(F ) = 0, and if t¯ ∈ F , t 1 u(·, t)L∞ (Bj k ) = u(·, t¯)L∞ (Bj k ) , lim tt¯ t − t¯ t¯ for every j and k.

(2.34)

272

D. Hoff, J. Smoller

y(t) z(t)

∆t

y( t ) c

z–h

∆t

t=t e

z

d

z+h

Bjk

y(t)

z(t) Fig. 5.

Let t¯ ∈ A ∪ D ∪ E ∪ F ; we will prove that (2.33) holds at t¯. Suppose not; then there is an ε > 0 such that for t near t¯ and t > t¯, z(t) − z¯ ≥ α(t¯)¯z + β(t¯) + ε ≡ u¯ + ε, t − t¯ where z¯ = z(t¯); that is, for t near t¯, z(t) ≥ z¯ + (t − t¯)(u¯ + ε).

(2.35)

1 , we can find h > 0 such that if |x − z¯ | ≤ h, Because u(·, t) is in Hloc

|u(x, t¯) − u| ¯ ≤

ε , 2

(2.36)

and y(t¯) < z¯ − h.

(2.37)

Then choose Bj k such that z¯ ∈ Bj k ⊂ [¯z − h, z¯ + h]. Let Bj k = (c, d) and choose e such that z¯ − h < c < e < z¯ < d < z¯ + h. We can thus can find :t > 0 such that |t − t¯| < :t ⇒ y(t) < c,

e ≤ z(t) ≤ d;

(this can be done since y and z are continuous functions); cf. Fig. 5. Then if |t − t¯| < :t, ρ(·, t) = 0 a.e. on (y(t), z(t)) ⊃ (c, e), so by Lemma 2.2, there is a σ > 0 such that ρ(·, s) = 0 on s s c + uL∞ (c,e) , z(t) − uL∞ (c,e) t

t


273

if |t − s| ≤ σ, |t − t¯| < :t. Thus for these s and t, s z(s) ≥ z(t) − uL∞ [c,e] t s ≥ z(t) − uL∞ (Bj k ) . t

Let s = t¯, and take t within σ of t¯, t > t¯, to get t z(t¯) ≥ z(t) − uL∞ (Bj k ) . t¯

Thus using (2.35), we have z¯ + (t − t¯)(u¯ + ε) ≤ z(t) ≤ z¯ + so that u¯ + ε ≤

1 t − t¯

t t¯

t t¯

uL∞ (Bj k ) ,

uL∞ (Bj k ) .

If we let t t¯ in this last inequality, we get u¯ + ε ≤ uL∞ (Bj k ) . Since Bj k ⊂ [¯z − h, z¯ + h], this contradicts (2.36). This proves (2.33) and completes the proof of Lemma 2.6. t1 Corollary 2.1. lim α(s)ds = ∞. tτ

t

Proof. With w1 < w2 , wi ∈ (a, b), i = 1, 2, and wi (t) the corresponding functions w as in the last lemma, we have t1 w1 (t) − w2 (t) = (w1 − w2 ) exp − α(s)ds . t

From Lemma 2.6 lim (w1 (t) − w2 (t)) = 0,

tτ

and the last equation gives the result.

We now complete the proof of the theorem as follows. Let c(t) ≡ w1 (t) < w2 (t) ≡ d(t) be two curves as in Lemma 2.6, corresponding to points w1 , w2 respectively; then from Lemma 2.7, 0 ≤ d(t) − c(t) −→ 0 as t τ. Define functions ψ(x) and χ (x) as in Fig. 6, and define for t ∈ (τ, t1 ], wε (x, t) = α ε (t)x + β ε (t) χ (x),

274

D. Hoff, J. Smoller

1

left of vacuum

χ

ψ

c(t1)

d(t1)

ψ

χ

right of e(t ) f(t ) 1 1 vacuum Fig. 6.

χ 1 t1 t

τ

c(t1)

χ 0 d(t1)

c(t)

e(t1)

f(t1)

d(t)

(I)

(II) Fig. 7.

where α ε and β ε are regularizations of α and β. Consider the initial-value problem φtε + w ε φxε = 0, φ ε (x, t1 ) = ψ(x).

(2.38)

Using Fig. 6, we see that φ ε is a smooth compactly supported function. Thus from (1.2), we have, for τ < t < t1 , t1 ρuφ ε dx = ρu(φtε + uφxε ) + (P − µux )φxε + ρf φ ε t (2.39) = ρu(u − w ε )φxε + (P − µux )φxε + ρf φ ε . Now φ ε is constant along the characteristics of (2.38) so that the support of φxε , in the region [t, t1 ], consists of two disjoint “strip-like" regions as depicted in Fig. 7. That is, the characteristics of (2.38) which start on (spt ψx ) ∩ [c(t1 ), d(t1 )] are given by x˙ = α ε x + β ε , so for small ε (depending on t) they stay between the curves c(t) and d(t); the corresponding support of φxε is the shaded region (I) in Fig. 7. Similarly the characteristics of (2.38) outside of the vacuum, which start on (spt ψx ) ∩ [e(t1 ), f (t1 )] are given by x˙ = 0; the corresponding support of φxε is depicted in Fig. 7 as the shaded region II. We now consider (2.39). First, the left-hand side is bounded independent of t, for τ < t < t1 by virtue of (1.10). Similarly, the term ρf φ ε is bounded because of (1.4). Also I

ρu(u − w ε )φxε = 0,


275

since ρ = 0 here. In II, w ε = 0 and φxε = ψx , so that in view of hypothesis (A3 ), ε ε 2 ρu(u − w )φx = ρu ψx II II ≤C ρu2 ≤ C. Next

ε (P − µux )φx ≤ II

II

(|P | + µ|ux |)|ψx | ≤ C

because of (A1 ), (A2 ), and (1.9). Since P (0, x, t) = 0 (by (1.3)) we have ε (P − µux )φx = −µux φxε I

=−

I t1

t

=− =−

t

d(s) c(s)

t1

µux φxε

µα(s) φ ε (d(s), s) − φ ε (c(s), s) ds

t1

µα(s)ds,

t

because φ ε (d(s), s) = 1 and φ ε (c(s), s) = 0. Thus from (2.41), we obtain that t1 α(s)ds t

is bounded, independent of t. Letting t τ contradicts Corollary 2.7. This completes the proof of the theorem. References 1. Amosov, A.A. and Zlotnick, A.A.: Solvability “in the large” of a system of equations of the onedimensional motion of an inhomogeneous viscous heat-conducting gas. Mat. Zametki 52, no. 2, 3–16 (1992) 2. Chen, G.-Q., Hoff, D. and Trivisa, K.: Global Solutions of the Compressible Navier–Stokes Equations with Large Discontinuous Initial Data. To appear in Comm. PDE 3. Fujita-Yashima, H., Padula, M., Novotny, A.: Equation monodimensionnelle d’un gaz visqueux et calorifere avec des conditions initiales moins restrictives. Richerche di Matematica XLII, no. 2, 199–248 (1993) 4. Hoff, D.: Global existence for 1D, compressible, isentropic Navier–Stokes equations with large initial data. Trans. AMS 303, no. 11, 169–181 (1987) 5. Hoff, D.: Global well-posedness of the Cauchy problem for the Navier–Stokes equations of nonisentropic flow with discontinuous initial data. J. Diff. Eqns. 95, 33–74 (1992) 6. Hoff, D.: Spherically symmetric solutions of the Navier–Stokes equations for compressible, isothermal flow with large, discontinuous initial data. Indiana Univ. Math. J. 41, 1–79 (1992) 7. Hoff, D.: Continuous dependence on initial data for discontinuous solutions of the Navier–Stokes equations for one-dimensional, compressible flow. SIAM J. Math. Ana. 27, no. 5, 1193–1211 (1996) 8. Hoff, D.: Global solutions of the equations of one-dimensional compressible flow with large data and differing end states. ZAMP 49, 774–785 (1998) 9. Hoff, D. and Liu, T.-P.: The inviscid limit for the Navier–Stokes equations of compressible, isentropic flow with shock data. Indiana Univ. Math. J. 38, 861–915 (1989)

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D. Hoff, J. Smoller

10. Hoff, D. and Zarnowski, R.: Continuous dependence in L2 for discontinuous solutions of the viscous p−system. Analyse Nonlineaire 11, 159–187 (1994) 11. Hoff, D. and Ziane, M.: Finite determining modes for the uniform attractor of the Navier–Stokes equations of one-dimensional, compressible flow in a space of discontinuous solutions. Submitted to Indiana Univ. Math. J. 12. Kanel, Ya.I.: On a model system of equations of one-dimensional gas motion. Differentsial’nye Uravneniya 4, 721–734 (1968) 13. Kazhikov, A. and Shelukhin, V.: Unique global solutions in time of initial boundary value problems for one-dimensional equations of a viscous gas. PMMJ Appl. Math. Mech. 41, 273–283 (1977) 14. Liu, T.-P.: Shock waves for compressible Navier–Stokes equations are nonlinearly stable. Comm. Pure Appl. Math. 35, 565–594 (1986) 15. Liu, T-P. and Xin, Z.: Nonlinear stability of rarefaction waves for compressible Navier–Stokes equations. Commun. Math. Phys. 118, no. 3, 451–465 (1988) 16. Matsumura, A. and Nishihara, K.: On the stability of travelling wave solutions of a one dimensional model system for compressible viscous gas. Japan J. Appl. Math. 2, 17–25 (1985) 17. Matsumura, A. and Nishihara, K.: Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas. Preprint 18. Matsumura, A. and Yanagi, S.: Uniform boundedness of the solutions for a one-dimensional isentropic model system of compressible viscous gas. Commun. Math. Phys. 175, 259–274 (1996) 19. Serre, D: Sur l‘équation monodimensionnelle d‘un fluide visqueux, compressible et conducteur de chaleur. C.R. Acad. Sc. Paris 303, 703–706 (1986) 20. Shelukhin, V.V.: On the structure of generalized solutions of the one-dimensional equations of a polytropic viscous gas. PMM USSR 48, 665–672 (1984) 21. Szepessy, A. and Xin, Z.: Nonlinear stability of viscous shock waves. Archive Rational Mech. Anal. 122 no. 1, 53–103 (1993) Communicated by A. Kupiainen

Commun. Math. Phys. 216, 277 – 312 (2001)

Communications in



Multifractal Analysis of Conformal Axiom A Flows Ya. B. Pesin , V. Sadovskaya Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA. E-mail: [email protected]; [email protected] Received: 18 November 1999 / Accepted: 24 July 2000

Abstract: We develop the multifractal analysis of conformal axiom A flows. This includes the study of the Hausdorff dimension of basic sets of the flow, the description of the dimension spectra for pointwise dimension and for Lyapunov exponents and the multifractal decomposition associated with these spectra. The main tool of study is the thermodynamic formalism for hyperbolic flows by Bowen and Ruelle. Examples include suspensions over axiom A conformal diffeomorphisms, Anosov flows, and in particular, geodesic flows on compact smooth surfaces of negative curvature. 1. Introduction The multifractal analysis of dynamical systems has recently become a popular topic in the dimension theory of dynamical systems. By now only conformal dynamical systems with discrete time have been subjects of study. They include conformal expanding maps and conformal axiom A diffeomorphisms (see [10] for the definition of conformal axiom A diffeomorphisms, related results, and further references). In this paper we extend the study to include conformal dynamical systems with continuous time, more precisely, conformal axiom A flows. Our first result is the formula for the Hausdorff dimension of basic sets of axiom A flows (see Sect. 4). It is an extension to the continuous time case of the famous Bowen pressure formula for the Hausdorff dimension of hyperbolic sets. We then consider the two dimension spectra: the dimension spectrum for pointwise dimensions generated by Gibbs measures and the dimension spectrum for Lyapunov exponents. Using the symbolic representation of axiom A flows by suspensions over subshifts of finite type and the associated thermodynamic formalism of Bowen and Ruelle ([4]), we obtain a complete description of these spectra. The statements of our The authors were partially supported by the National Science Foundation grant #DMS9403723.Ya. P. was partly supported by the NATO grant CRG970161.

278

Ya. B. Pesin, V. Sadovskaya

results are similar in spirit to those in the discrete time case but proofs require some substantial technical modifications. We stress that we handle only axiom A flows which are conformal and we introduce and study this notion in Sect. 3. Examples include suspensions over conformal axiom A diffeomorphisms and two-dimensional Anosov flows. Our results provide, in particular, a formula for the dimension and a description of the dimension spectra for pointwise dimensions and for Lyapunov exponents for the time-one map of the flow. This is the first example of a partially hyperbolic diffeomorphism for which such results are now known. Let us emphasize that, in general, both dimension spectra are non-trivial. More precisely, as we show in Sect. 5, the dimension spectrum for pointwise dimension is trivial (i.e., is a δ-function) if and only if the Gibbs measure is the measure of full dimension. For an Anosov flow it holds if it preserves a smooth measure. Furthermore, the dimension spectrum for Lyapunov exponents is trivial if and only if the measure of full dimension coincides with the measure of maximal entropy. We apply this statement to geodesic flows on compact n-dimensional Riemannian manifolds of negative curvature. For n = 2 we have that the spectrum is trivial if and only if the topological entropy of the flow coincide with the metric entropy (see Sect. 3). This provides a new insight into the famous Katok’s entropy conjecture (see [8]). For n > 2, the requirement that the flow is conformal implies that the curvature is constant. In particular, the dimension spectrum for Lyapunov exponents is trivial. Finally we describe multifractal decomposition associated with the two spectra. More detailed description can be found in [2].

2. Preliminaries Let M be a smooth finite-dimensional Riemannian manifold. Throughout this paper f t : M → M is a flow on M without fixed points generated by a C r -vector field t V , r ≥ 1, i.e., dfdt(x) = V (x) for every x ∈ M. A compact f t -invariant set ⊂ M is said to be hyperbolic if there exist a continuous splitting of the tangent bundle T M = E (s) ⊕ E (u) ⊕ X and constants C > 0 and 0 < λ such that for every x ∈ and t ∈ R, 1. df t E (s) (x) = E (s) (f t (x)), df t E (u) (x) = E (u) (f t (x)), and X = {αV : α ∈ R} is a one-dimensional subbundle; 2. for all t ≥ 0,

df t v ≤ Ce−λt v

if v ∈ E (s) (x),

df −t v ≤ Ce−λt v

if v ∈ E (u) (x).

The subspaces E (s) (x) and E (u) (x) are called stable and unstable subspaces at x respectively and they depend Hölder continuously on x. It is well-known (see, for example, [8]) (s) that for every x ∈ one can construct stable and unstable local manifolds, Wloc (x) (u) and Wloc (x). They have the following properties: (s)

(u)

3. x ∈ Wloc (x), x ∈ Wloc (x); (s) (u) 4. Tx Wloc (x) = E (s) (x), Tx Wloc (x) = E (u) (x); (s) (s) (u) (u) 5. f t (Wloc (x)) ⊂ Wloc (f t (x)), f −t (Wloc (x)) ⊂ Wloc (f −t (x));

Multifractal Analysis of Conformal Axiom A Flows

279

6. there exist K > 0 and 0 < µ such that for every t ≥ 0, (s)

ρ(f t (y), f t (x)) ≤ Ke−µt ρ(y, x) for all y ∈ Wloc (x) and

(u)

ρ(f −t (y), f −t (x)) ≤ Ke−µt ρ(y, x) for all y ∈ Wloc (x),

where ρ is the distance in M induced by the Riemannian metric. A hyperbolic set is called locally maximal if there exists a neighborhood U of such that = f t (U ). −∞ 0 such that for any two points x, y ∈ with ρ(x, y) ≤ δ one can find a number t = t (x, y), |t| ≤ #, for which the intersection (s)

(u)

Wloc (f t (x)) ∩ Wloc (y) consists of a single point z ∈ . We denote this point by z = [x, y]; moreover, the maps t (x, y) and [x, y] are continuous. We define stable and unstable global manifolds at x ∈ by (u) (s) f −t Wloc (f t (x)) , W (u) (x) = f t Wloc (f −t (x)) . W (s) (x) = t≥0

t≥0

They can be characterized as follows: W (s) (x) = { y ∈ : ρ(f t (y), f t (x)) → 0 as t → ∞ }, W (u) (x) = { y ∈ : ρ(f −t (y), f −t (x)) → 0 as t → ∞ }. A flow f t is called an axiom A flow if its set of non-wandering points is hyperbolic. Let us remark that we deal only with flows without fixed points. If this assumption is dropped one should assume in the above definition that the flow has finitely many hyperbolic fixed points. The Smale Spectral Decomposition Theorem claims (see [8]) that in this case the hyperbolic set can be decomposed into finitely many disjoint closed f t -invariant locally maximal hyperbolic sets on each of which f t is topologically transitive. These sets are called basic sets. From now on we will assume that f t is topologically transitive on a locally maximal hyperbolic set . One can show that periodic orbits are dense in . In [3], Bowen constructed Markov partitions of basic sets (see also [13] for the case of Anosov flows). We provide here a concise description of his results. Given a point x ∈ , consider a small compact disk D containing x of co-dimension one which is transversal to the flow f t . This disk is a local section of the flow, i.e., there exists τ > 0 such that the map (y, t) → f t (y) is a diffeomorphism of the direct product D × [−τ, τ ] onto a neighborhood Uτ (D). The projection PD : Uτ (D) → D is a differentiable map. Consider now a closed set ( ⊂ ∩ D which does not intersect the boundary ∂D. For any two points y, z ∈ ( let {y, z} = PD [y, z]. The set ( is said to be a rectangle if ( = int( (where the interior of ( is considered with respect to the induced topology

280


of ∩ D) and {y, z} ∈ ( for any y, z ∈ (. If ( is a rectangle then for every x ∈ ( we set (s) (s) Wloc (x, () = { {x, y} : y ∈ ( } = ( ∩ PD Uτ (D) ∩ Wloc (x) , (2.1) (u) (u) Wloc (x, () = { {z, x} : z ∈ ( } = ( ∩ PD Uτ (D) ∩ Wloc (x) (we assume that diam ( is much smaller than the size of local stable and unstable manifolds). A collection of rectangles T = {(1 , . . . , (n } is called regular of size r0 if there exist small compact co-dimension one disks D1 , . . . , Dn , which are transversal to the flow f t , such that 1. diamDi < r0 and (i ⊂ intDi ; 2. for i = j at least one of the sets Di ∩ f [0,r0 ] Dj or Dj ∩ f [0,r0 ] Di is empty; in particular, Di ∩ Dj = ∅; 3. = f [−r0 ,0] ,(T ), where ,(T ) = (1 ∪ · · · ∪ (n . Let T = {(1 , . . . , (n } be a regular collection of rectangles of size r0 . For every x ∈ ,(T ) one can find the smallest positive number t (x) ≤ r0 such that f t (x) (x) ∈ ,(T ). Since the disks Di are disjoint there exists a number β > 0 such that t (x) ≥ β for all x. The map HT : ,(T ) → ,(T ) given by HT (x) = f t (x) (x)

(2.2)

is one-to-one. Note that the maps t (x) and HT are not continuous on ,(T ) but on n k , (T ) = x ∈ ,(T ) : (HT ) (x) ∈ int (i for all k ∈ Z . (2.3) i=1

, (T )

The set is dense in ,(T ) and the set ∪t∈R f t (, (T )) is dense in . Given two rectangles (i and (j we denote by U ((i , (j ) = {w ∈ , (T ) : w ∈ (i , HT (w) ∈ (j },

(2.4) V ((i , (j ) = {w ∈ , (T ) : w ∈ (j , HT−1 (w) ∈ (i }. A Markov collection of size r0 (for a basic set ) is a regular collection T = {(1 , . . . , (n } of rectangles of size r0 which satisfies the following conditions: for any 1 ≤ i, j ≤ n, (s)

1. if x ∈ U ((i , (j ) then Wloc (x, () ⊂ U ((i , (j ); (u) 2. if y ∈ V ((i , (j ) then Wloc (y, () ⊂ V ((i , (j ) (see (2.1)). In [3], Bowen proved that for any sufficiently small r0 there exist a Markov collection of size r0 . Given a rectangle (i ∈ T , we call the set Ri = f t (x) ⊂ (2.5) x∈(i 0≤t≤t (x)

a Markov set (corresponding to the Markov collection T ). Note that Ri = intRi and intRi ∩ intRj = ∅ for any i = j . Using Markov collections one can obtain symbolic representations of Axiom A flows by symbolic suspension flows (see Appendix; see also [4]).


281

Proposition 2.1. Let be a basic set for an axiom A flow f t generated by a C 1 -vector field V . Then there exists a topologically mixing subshift of finite type (2A , σ ) (see Appendix), a positive Hölder continuous function ψ (in the metric dβ for some β > 1, (see (A.22)), and a continuous projection map χ : (A, ψ) → such that the following diagram St

(A, ψ) −−−−→ (A, ψ)    χ χ

is commutative with

St

ft

−−−−→

a symbolic suspension flow (see (A.24)).

The map χ is called the coding map. The transfer matrix A = (ai,j ) is uniquely determined by a Markov collection for . Namely, if T = {(1 , . . . , (n } is such a collection then ai,j = 1 if and only if there exists a point x ∈ , (T ) such that x ∈ (i and HT (x) ∈ (j (see (2.2) and (2.3)). As an immediate consequence of Proposition 2.1 we obtain the following statement. Proposition 2.2. Let be a basic set for an axiom A flow f t and ϕ : → R a Hölder continuous function. Then there exists a unique equilibrium measure νϕ corresponding to ϕ (see (A.21)). Moreover, the measure νϕ is ergodic and positive on open sets. We describe the local structure of an equilibrium measure ν corresponding to a Hölder continuous function (see part 3 of the Appendix). Let R1 , . . . , Rn be the Markov sets corresponding to a Markov collection T for . Let us fix a set Ri and consider the partitions ξ (u) and ξ (s) of Ri by local stable and unstable manifolds. Denote by ν (u) (x) and ν (s) (x) the corresponding conditional (u) (s) measures on Wloc (x) ∩ Ri and Wloc (x) ∩ Ri (where x ∈ Ri ) generated by ν. The following statement shows that equilibrium measures have local product structure. Its proof follows from Proposition A.5 (see Appendix) and local product structure of Gibbs measures for subshifts of finite type (see [10]). Proposition 2.3. There are positive constants A1 and A2 such that for some point x ∈ Ri and any Borel set E ⊂ Ri A1 χE (y, z, t) dν (u) (y) dν (s) (z) dt ≤ ν(E) Ri (2.6) ≤ A2 χE (y, z, t) dν (u) (y) dν (s) (z) dt, Ri

where y ∈

(u) Wloc (x)

(s)

and z ∈ Wloc (x).

3. Conformal Axiom A Flows Let F = {f t } be a C 2 -flow on a locally maximal hyperbolic set . We say that F is u-conformal (respectively, s-conformal) if there exists a continuous function A(u) (respectively, A(s) ) on × R such that for every x ∈ and t ∈ R, df t |E (u) (x) = A(u) (x, t) I (u) (x, t),

282


respectively,

df t |E (s) (x) = A(s) (x, t) I (s) (x, t),

where I (u) (x, t) : E (u) (x) → E (u) (f t x) and I (s) (x, t) : E (s) (x) → E (s) (f t x) are isometries. We define functions a (u) (x) and a (s) (x) by a (u) (x) =

log df t |E (u) (x)

∂ log A(u) (x, t) |t=0 = lim , t→0 ∂t t

a (s) (x) =

log df t |E (s) (x)

∂ log A(s) (x, t) |t=0 = lim . t→0 ∂t t

Since the subspaces E (u) (x) and E (s) (x) depend Hölder continuously on x the functions a (u) (x) and a (s) (x) are also Hölder continuous. Note that a (u) (x) > 0 and a (s) (x) < 0 for every x ∈ . For any x ∈ and any t ∈ R, we have t a (u) (f τ (x)) dτ for any v ∈ E (u) (x), (3.1)

df t (v) = v exp 0

and t

t

df (w) = w exp

a (s) (f τ (x)) dτ

for any w ∈ E (s) (x).

(3.2)

0

A flow F = {f t } on is called conformal if it is u-conformal and s-conformal as well. It is easy to see that a three-dimensional flow on a locally maximal hyperbolic set is conformal. If F = {f t } is a conformal flow then for every x ∈ the Lyapunov exponent at x takes on two values which are given by log dfxt |E (u) (x)

1 t (u) τ λ+ (x) = lim a (f (x)) dτ > 0, (3.3) = lim t→∞ t→∞ t 0 t log dfxt |E (s) (x)

1 t (s) τ λ− (x) = lim a (f (x)) dτ < 0 (3.4) = lim t→∞ t→∞ t 0 t (provided the limit exists). If ν is an f -invariant measure then by the Birkhoff ergodic theorem, the above limits exist ν-almost everywhere, and if ν is ergodic then they are − constant almost everywhere. We denote the corresponding values by λ+ ν > 0 and λν < 0. We describe some examples of conformal axiom A flows. 1. A suspension flow over a conformal axiom A diffeomorphism is a conformal axiom A flow. Note that if the roof function of a suspension flow is not cohomologous to a constant then the corresponding suspension flow is mixing. 2. Consider a conformal Anosov flow F . Let be a closed locally maximal hyperbolic set for F . Then the restriction of F | is a conformal axiom A flow. 3. Consider the geodesic flow on a compact Riemannian manifold M of negative curvature. The flow acts on the space SM = {(x, v) : x ∈ M, v ∈ Tx M, v = 1} of unit tangent vectors. We endow the second tangent bundle T T M with a Riemannian metric whose projection to T M is the given metric. If dim M = 2 then the geodesic flow is conformal since stable and unstable subspaces are one dimensional, and our results apply.


283

If dim M ≥ 3 the result in [7] shows that conformality of the geodesic flow implies that M is of constant curvature (regardless of the metric on the second tangent bundle). We thank M. Kanai for informing us on his result. On the other hand, if the curvature of M is constant then the geodesic flow is conformal provided the second tangent bundle is endowed with the canonical metric. Remark 3.1. Our main results (Theorems 4.1, 4.2, 5.1, 5.2, and 5.3) can be easily generalized to the case when the flow is not conformal, but has bounded distortion. By this we mean that there exist Hölder continuous functions a (u) and a (s) on , and constants K1 , K2 > 0 such that for any x ∈ , v ∈ E (u) (x), w ∈ E (s) (x), and t ∈ R, t t (u) τ t a (f (x)) dτ ≤ df (v) ≤ K2 v exp a (u) (f τ (x)) dτ , K1 v exp 0

and

0

t

K1 w exp

a (s) (f τ (x)) dτ ≤ df t (w) ≤ K2 w exp

0

t

a (s) (f τ (x)) dτ

0

(compare to (3.1) and (3.2) ). We thank A. Katok for providing us with this remark. 4. Hausdorff and Box Dimension of Basic Sets for Conformal Axiom A Flows Let be a basic set for a u-conformal axiom A flow F = {f t }. Consider the function −t (u) a (u) (x)

(4.1)

on , where t (u) is a unique root of Bowen’s equation P (F, −t a (u) ) = 0

(4.2)

(see (A.16)-(A.18)). The function −t (u) a (u) is Hölder continuous and therefore, there exists a unique equilibrium measure corresponding to it. We denote this measure by κ (u) . Let T = {(1 , . . . , (n } be a Markov collection for and R1 , . . . , Rn the Markov sets corresponding T . Given x ∈ denote by R(x) a Markov set containing x. Consider (u) the conditional measures η(u) (y) on Wloc (y) ∩ R(x) (where y ∈ R(x)) generated by the measure κ (u) . We now state the result which describes the Hausdorff dimension of subsets of unstable manifolds. (u)

Theorem 4.1. For any x ∈ and any open set U ⊂ Wloc (x) such that U ∩ = ∅ the following statements hold: 1. dimH (U ∩ ) = dimB (U ∩ ) = dimB (U ∩ ) = t (u) ; 2. h (u) (f 1 ) t (u) = (u)κ , (u) a (y) dκ (y)

(4.3)

where hκ (u) (f 1 ) is the measure-theoretic entropy of the time-one map f 1 with respect to the measure κ (u) ; (u) 3. dη(u) (x) (y) = t (u) for all y ∈ Wloc (x) ∩ R(x);

284


4. t (u) = dimH η(u) (x), i.e., the measure η(u) (x) is the measure of full dimension (see Appendix); 5. the t (u) -Hausdorff measure of U ∩ is positive and finite; moreover, it is equivalent to the measure η(u) (x)|U . Remark 4.1. Consider a u-conformal diffeomorphism f on a locally maximal hyperbolic set X. This means that there exists a continuous function b(u) on X such that for any x ∈ X, df |E (u) (x) = b(u) (x) I (u) (x), where I (u) (x) : E (u) (x) → E (u) (f (x)) is an isometry (see [10]). It is known that for (u) any x ∈ X and any open set U ⊂ Wloc (x) such that U ∩ X = ∅, dimH (U ∩ X) = dimB (U ∩ X) = dimB (U ∩ X) = t (u) , where t (u) is the unique root of Bowen’s equation PX (f, −t log b(u) ) = 0 (see [10]). Consider a u-conformal flow F = {f t } and the corresponding time-one map f 1 . It is a partially hyperbolic diffeomorphism and the local strong unstable manifold for f 1 (su) (u) at a point x ∈ , Wloc (x), coincides with Wloc (x) for the flow F . Note that df 1 |E (su) (x) = A(u) (x, 1) I (u) (x, 1). In view of (A.19),

P (F, −ta (u) ) = P f 1 , −t

1

a (u) (f τ x) dτ

0

= P f 1 , −t log A(u) (x, 1) .

Therefore, the first statement of Theorem 4.1 and (4.2) imply that for any x ∈ and for (su) any open set U ⊂ Wloc (x) such that U ∩ = ∅, dimH (U ∩ ) = dimB (U ∩ ) = dimB (U ∩ ) = t (u) , where t (u) is the unique root of Bowen’s equation P (f 1 , −t log A(u) (x, 1)) = 0. (su)

This gives a formula for the dimension of Wloc (x) ∩ for the partially hyperbolic time-one diffeomorphism f 1 . This formula is the same as the one for a u-conformal diffeomorphism. (su) It is not known in general how to compute the dimension of Wloc (x) ∩ for an arbitrary partially hyperbolic diffeomorphism. We now consider a basic set for an s-conformal axiom A flow F = {f t }. Similarly to (4.1) and (4.2) define the function t (s) a (s) (x)

(4.4)


285

on where t (s) is a unique root of Bowen’s equation P (F, ta (s) ) = 0

(4.5)

(see (A.16)–(A.18)). The function t (s) a (s) is Hölder continuous and therefore, there exists a unique equilibrium measure corresponding to it. We denote this measure by κ (s) . (s) Given x ∈ consider the conditional measures η(s) (y) on Wloc (y) ∩ R(x) (where (s) y ∈ R(x)) generated by the measure κ on a Markov set R(x) containing x. Similarly to Theorem 4.1, one can prove that for any x ∈ and any open set (s) U ⊂ Wloc (x), dimH (U ∩ ) = dimB (U ∩ ) = dimB (U ∩ ) = t (s) . Moreover, t (s) = −

hκ (s) (f 1 ) , (s) (s) a (y) dκ (y)

(4.6)

where hκ (s) (f 1 ) is the measure-theoretic entropy of the time-one map f 1 with respect to the measure κ (s) . The t (s) -Hausdorff measure of U ∩ is positive and finite. In addition, dη(s) (x) (y) = (s)

t (s) for all y ∈ Wloc (x) ∩ R(x), and therefore dimH η(s) (x) = t (s) , i.e. the measure η(s) (x) is the measure of full dimension. We now consider the case when is a basic set for an axiom A flow F = {f t } which is both s- and u-conformal. Using Proposition 7.1 we compute the Hausdorff dimension and box dimension of . Theorem 4.2. We have dimH = dimB = dimB = t (u) + t (s) + 1, where t (u) and t (s) are unique roots of Bowen’s equations (4.2) and (4.5) and can be computed by the formulae (4.3) and (4.6). This result applies and produces a formula for the Hausdorff dimension and box dimension of a basic set of an Axiom A flow on a surface which is clearly seen to be both s- and u-conformal. Consider the measures κ (u) and κ (s) on , which are equilibrium measures for the functions −t (u) a (u) and t (s) a (s) respectively. It is easy to see that dimH κ (u) ≤ t (u) + t (s) + 1,

dimH κ (s) ≤ t (u) + t (s) + 1.

Moreover, the equalities hold if and only if κ (u) = κ (s) = κ. def

(4.7)

In this case, κ is the measure of full dimension. Condition (4.7) is a “rigidity” type condition. It holds if and only if the functions −t (u) a (u) (x) and t (s) a (s) (x) are cohomologous (see [8]). One can show that this is the case if and only if for any periodic point x ∈ of period p, p p t (u) a (u) (f τ (x)) dτ = −t (s) a (s) (f τ (x)) dτ. 0

0

286


5. Multifractal Analysis of Conformal Axiom A Flows on Basic Sets We undertake the complete multifractal analysis of equilibrium measures on a locally maximal hyperbolic set of a flow F = {f t } assuming that the flow is both s- and u-conformal. We follow the approach suggested by Pesin and Weiss in [11] (see also [10]). Let ϕ be a Hölder continuous function on and ν = νϕ a unique equilibrium measure for ϕ. Recall that a measure ν on a metric space is called Federer if there exists a constant K > 0 such that for any point x and any r > 0, ν(B(x, 2r)) ≤ Kν(B(x, r)). Theorem 5.1. The measure ν is Federer. For α ≥ 0 consider the sets α defined by α = { x ∈ : dν (x) = α } and the fν (α)-spectrum for dimensions fν (α) = dimH α (see (A.6)). Theorem 5.2. 1. The pointwise dimension dν (x) exists for ν-almost every x ∈ and

1 1 + 1, − dν (x) = hν (f 1 ) λ+ λ− ν ν − λ+ ν , λν are positive and negative values of the Lyapunov exponent of ν (see (3.3), (3.4)). 2. If ν is not the measure of full dimension then the function fν (α) is defined on an interval [α1 , α2 ] (i.e., the spectrum is complete, see [14]); it is real analytic and strictly convex. 3. If ν is not the measure of full dimension then there exists a strictly convex function T (q) such that the functions fν (α) and T (q) form a Legendre transform pair (see (A.26)) and for any q ∈ R we have log inf Br B∈Br ν(B)q , T (q) = − lim r→0 log r

where the infimum is taken over all finite covers Br of by open balls of radius r; in particular, for every q > 1, T (q) = H Pq (ν) = Rq (ν) 1−q (see (A.7), (A.8), (A.9)). 4. If ν is the measure of full dimension then T (q) = (1 − q) dimH is a linear function; in addition, fν (dimH ) = dimH and fν (α) = 0 for all α = dimH . In other words fν (α) is a δ-function if and only if ν is the measure of full dimension.


287

Remark 5.1. Consider the case when ν is not the measure of full dimension. Note that fν (α) ≤ dimH for any α ∈ [α1 , α2 ]. Since fν (α(q)) = T (q) + qα(q) (see Appendix) we obtain that f (α(0)) = T (0) = dimB = dimB = dimH (see (A.1) and Theorem 4.2). Therefore, fν attains its maximum value dimH at the point α(0). Differentiating the equality fν (α(q)) = T (q) + qα(q) with respect to q and using d the fact that α(q) = −T (q) we find that dα fν (α(q)) = q for every real q. This implies that lim

α→α1

d d d fν (α(q)) = +∞, lim fν (α(q)) = −∞, and fν (α(1)) = 1. α→α2 dα dα dα

Since T (1) = 0 we have that f (α(1)) = α(1). It follows that the graph of the function fν (α) is tangent to the line with slope 1 at the point α(1). One can show that α(1) is the information dimension of ν (see [10]). It easily follows from the above observations that dimH ∈ (α1 , α2 ). Another consequence of Theorem 5.2 is the following multifractal decomposition of a basic set associated with the pointwise dimension of an equilibrium measure ν corresponding to a Hölder continuous function. Namely, ˆ ∪ α , = α

where α is the set of points for which the pointwise dimension takes on the value α ˆ is the set of points with no pointwise dimension. One can and the irregular part ˆ = ∅; moreover, it is everywhere dense in and dimH ˆ = dimH (oral show that communication by L. Barreira). We also have that each set α is everywhere dense in . An important manifestation of Theorem 5.2 is multifractal decomposition of the basic set associated with the Lyapunov exponent λ+ (x) and λ− (x) (see (3.3), (3.4)). We consider only the positive Lyapunov exponent λ+ (x); similar statements hold true for the negative Lyapunov exponent λ− (x) at points x ∈ . We can write   , = Lˆ + ∪  L+ β β∈R

where

Lˆ + = {x ∈ : the limit in (3.3) does not exist}

is the irregular part, and + L+ β = {x ∈ : λ (x) = β}. (u)

If ν is an ergodic measure for f t we obtain that λ+ (x) = λν for ν-almost every x ∈ . Thus, the set L+(u) = ∅. λν

288


We introduce the dimension spectrum for (positive) Lyapunov exponents by E+ (β) = dimH L+ β. Let ϕ be a Hölder continuous function on and ν the unique equilibrium measure for ϕ. Let also R be a Markov set. For any y ∈ R we define a measure ν˜ (u) (y) on (u) Wloc (y) ∩ R as follows. Let ϕ˜ be the pull back of ϕ to (A, ψ) by the coding map χ . The unique equilibrium measure corresponding to ϕ˜ is λµ = ((µ × m)(Yψ ))−1 (µ × m)|Yψ , where m is the Lebesgue measure on R and µ is the unique equilibrium measure on 2A corresponding to the Hölder continuous function

ψ(ω)

H(ω) = 0

ϕ(ω, ˜ t) dt − P(A,ψ) (S, ϕ) ˜ ψ(ω)

(see Proposition A.5). + We define the measure µ(u) on 2A such that for any cylinder Ci0 ...in in 2A and its + + projection Ci0 ...in to 2A , µ(u) (Ci+0 ...in ) = µ(Ci0 ...in ).

(5.1)

− Similarly, we define the measure µ(s) on 2A such that for any cylinder Ci−n ...i0 in 2A − − and its projection Ci−n ...i0 to 2A ,

µ(s) (Ci−−n ...i0 ) = µ(Ci−n ...i0 ).

(5.2)

There exist constants K1 , K2 > 0 such that for every integers m, n ≥ 0, and any (. . . i−1 i0 i1 . . . ) ∈ 2A , K1 ≤

µ(Ci−m ...in ) − (s) µ (Ci−m ...i0 ) × µ(u) (Ci+0 ...in )

≤ K2

(see [10]). Let ( be a rectangle corresponding to R (see (2.5)), and x ∈ (. Denote by ν (u) (x) (u) the push forward of µ(u) to Wloc (x, () by the coding map χ . Let y ∈ R(x), then (u) (u) Wloc (y) ∩ R(x) is naturally diffeomorphic to Wloc (x , () for some x ∈ (. Denote (u) by ν˜ (u) (y) the push forward of ν (u) (x ) to Wloc (y) ∩ R(x). Note that ν˜ (u) (y) is defined for every y ∈ R, and it is equivalent to the conditional measure generated by ν on (u) Wloc (y) ∩ R for ν-almost every y ∈ R. There is a relation between the positive Lyapunov exponent λ+ (x) and the pointwise dimension dν (u) (x) (x), where νmax is the measure of maximal entropy. For dynamical max systems with discrete time this relation was first described by Weiss (see [15]). Notice that the measure of maximal entropy is a unique equilibrium measure corresponding to the function ϕ = 0.


289

Proposition 5.1. L+ β =

x ∈ : dν˜ (u) (x) (x) = max

h (f 1 ) β

,

(u)

where ν˜ max (x) is defined as above. Recall that we denoted by κ (u) the unique equilibrium measure corresponding to the function −t (u) a (u) , where t (u) is defined by (4.2). Let η˜ (u) (x) be the measure on (u) Wloc (x) ∩ R(x) defined as above. Theorem 4.1 implies that η˜ (u) (x) is the measure of full dimension. This together with Theorem 5.2 and Proposition 5.1 implies the following result. Theorem 5.3. (u)

1. If ν˜ max (x) is not equivalent to the measure η˜ (u) (x) for some x ∈ then the Lyapunov spectrum E+ (β) is a real analytic strictly convex function on an interval [β1 , β2 ] containing the point (u)

β = h (f 1 )/ dimH ( ∩ Wloc (x)). (u)

2. If ν˜ max (x) is equivalent to η˜ (u) (x) for some x ∈ then the Lyapunov spectrum is a delta function, i.e., (u) dimH , for β = h (f 1 )/ dimH ( ∩ Wloc (x)) + E (β) = (u) 0, for β = h (f 1 )/ dimH ( ∩ Wloc (x)). (u)

Remark 5.2. One can show that if the measures ν˜ max (x) and η˜ (u) (x) are equivalent for some x ∈ then they are equivalent for all x ∈ . As an immediate consequence of Theorem 5.3 we obtain the following statement. (u)

Corollary 5.1. Assume that the measure ν˜ max is not equivalent to the measure η˜ (u) (x) for some x ∈ then the range of the function λ+ (x) is an interval [β1 , β2 ] and for any β outside this interval the set L+ β is empty (i.e., the spectrum is complete, see [14]); in particular, the Lyapunov exponent attains uncountably many distinct values. One can also show that the set Lˆ + is not empty and has full Hausdorff dimension (oral communication by L. Barreira). Consider the geodesic flow on compact surface on negative curvature. Since the flow is conformal (see Sect. 3) the above results apply and give a complete description of the dimension spectrum for Lyapunov exponents. In particular, this spectrum is a δ-function if and only if the Liouville measure is the measure of maximal entropy and hence, the topological entropy of the flow coincide with its metric entropy (with respect to the Liouville measure). This implies that the curvature is constant. Remark 5.3. The above results provide a complete description of the dimension spectra for pointwise dimensions and Lyapunov exponents for the time-one map of a conformal axiom A flow.

290


6. Moran Covers Let x ∈ and ( be a rectangle containing x. We construct a special cover of the set (u) Wloc (x, () which will be an “optimal” cover in computing the Hausdorff dimension and box dimensions. Let x ∈ , (T ) and t > 0 be a number such that f t x ∈ T . Let also (f t x be the (u) rectangle containing f t x. For any point y ∈ Wloc (f t x, (f t x ) there exists a unique (u) number τ (y) > 0 such that f −τ (y) y ∈ Wloc (x, () and the points f −τ y (0 ≤ τ ≤ τ (y)) and f −τ (f t x) (0 ≤ τ ≤ t) visit the same rectangles in the same order. Define (u)

(u)

Q(x, t) = { f −τ (y) y, y ∈ Wloc (f t x, (f t x ) } ⊂ Wloc (x, (). (u)

Lemma 6.1. 1. Q(x, t) contains a ball in Wloc (x, () of radius r(x, t) and is contained (u) in a ball in Wloc (x, () of radius r(x, t). 2. There exist positive constants K1 and K2 independent of x and t such that for any y ∈ Q(x, t), −1 τ (y)

K1 exp

a (u) (f τ y) dτ

≤ r(x, t) ≤ r(x, t)

0

≤ K2 exp

τ (y)

−1 a

(u)

τ

(f y) dτ

.

0

We assume that the rectangles ( are small so that K2 < 1. (u) Fix a number r > 0. For any y ∈ Wloc (x, () ∩ , (T ) let t (y) be the smallest number such that f t (y) y ∈ T and −1 t (y)

exp

a (u) (f τ y) dτ

≤ r.

(6.1)

0

Among all points z such that z ∈ Q(y, t (y)) choose a point z0 for which t (z0 ) is minimal. Let Q(y) = Q(z0 , t (z0 )). The properties of the Markov collection T imply that the sets Q(y) for different y ∈ (u) Wloc (x, () ∩ , (T ) either coincide or overlap only along their boundaries. These sets (u) (u) comprise a cover of Wloc (x, () which we call a Moran cover of Wloc (x, () of size r. (u)

We can also construct a Moran cover of Wloc (x, () using the symbolic representation of the flow (see Proposition 2.1 and Appendix). Recall that any x ∈ is the image under the coding map χ of a point (ω, t) ∈ (A, ψ). If y ∈ T then y = χ (ω, 0) for some ω ∈ 2A . If a number t > 0 is such that f τ y ∈ T for 0 < τ ≤ t then f t y = χ (ω, t). Let a˜ (u) and a˜ (s) be the pull back of the functions a (u) and a (s) to (A, ψ) by the coding map χ . Let also a(s) and a(u) be the Hölder continuous function on 2A defined by ψ(ω) ψ(ω) a(s) (ω) = exp a˜ (s) (ω, t) dt, a(u) (ω) = exp a˜ (u) (ω, t) dt. (6.2) 0

0


291

Choose ωˆ = (. . . i−1 i0 i1 . . . ) ∈ 2A such that x = χ (ω). ˆ We identify the set of points + in 2A having the same past as ωˆ with the cylinder Ci+0 ⊂ 2A . + Given r > 0 and a point ω ∈ Ci0 choose the number n(ω) such that n(ω)−1

(a

(u)

k

(σ ω))

−1

n(ω)

(a(u) (σ k ω))−1 ≤ r

> r,

k=0

(6.3)

k=0

(compare to (6.1)). It is easy to see that n(ω) → ∞ as r → 0 uniformly in ω. For any ω ∈ Ci+0 consider the cylinder Ci+0 ...in(ω) . Let C(ω) ⊂ Ci+0 be the largest

cylinder set containing ω with the property that C(ω) = Ci+0 ...i and

Ci+0 ...i n(ω )

n(ω )

for some ω ∈ C(ω)

⊂ C(ω) for any ω ∈ C(ω). The sets corresponding to different ω ∈ Ci+0

either coincide or are disjoint. Thus, we obtain a cover Ur (Ci+0 ) of Ci+0 of size r which we also call a Moran cover. Similarly one can construct a Moran cover Ur (Ci−0 ) of Ci−0 of size r. The sets Q = χ (C), C ∈ Ur (Ci+0 ) (u)

(u)

comprise a cover of Wloc (x, () which is a Moran cover of Wloc (x, () of size r. These sets may overlap only along their boundaries. Lemma 6.1 implies that a Moran cover has the following properties: (6.4). Any element of the cover is contained in a ball of radius r and contains a ball of (u) radius K1 r in Wloc (x, (), where K1 is a constant independent of r. (u)

(6.5). The number of elements of the cover which intersects a ball B(x, r) ⊂ Wloc (x, () is bounded from above by a constant M independent of x and r. The number M is called the Moran multiplicity factor. (u) Let x be a point in a rectangle (. Starting with a Moran cover of Wloc (x, () we will obtain a cover of the rectangle ( by the sets

Q(y) =

(s)

Wloc (z, ().

z∈Q(y)

We call this cover the extended Moran cover corresponding to a given Moran cover. It follows from Lemma 6.1 and the construction of the sets Q(y) that sup

z∈Q(y)

exp

t (z)

−1 a

(u)

τ

(f z) dτ

≤ K3 r,

(6.6)

0

where t (z) is defined by (6.1) and K3 > 0 is a constant. 7. Proofs (u)

Proof of Theorem 4.1. We first show that t (u) ≤ d := dimH Wloc (x, () for any x ∈ , (T ).

292


Fix ε > 0. By the definition of the Hausdorff dimension there exists a number r > 0 (u) and a cover of Wloc (x, () by balls Bl , l = 1, 2, . . . of radius rl ≤ r such that rld+ε ≤ 1. l

(u)

For every l > 0 consider a Moran cover of Wloc (x, () of size rl and the corresponding extended Moran cover of (. Choose those sets from the extended cover that intersect (j ) (1) (m(l)) Bl . Denote them by Ql , . . . , Ql . The collection of sets {Ql }l=1,2,... j =1,...m(l) forms a cover of ( which we denote by G. By (6.5), m(l) ≤ M, l = 1, 2, . . . , where M is a Moran multiplicity factor. Using (6.6) we conclude that −(d+ε) t (z) (u) τ sup exp a (f z) dτ ≤M (K3 rl )d+ε ≤ K4 , (j )

0

(j ) Ql ∈G z∈Ql

l (j )

where K4 > 0 is a constant. The cylinders Cl of Ci0 = χ −1 (() for which (j ) Cl ∈G˜

 sup exp (j ) ω∈Cl

n(ω)−1 ψ(σ k ω)

(j )

(j )

= χ −1 Ql , Ql

∈ G form a cover G˜

−(d+ε) a˜ (u) (σ k ω, τ ) dτ 

≤ K4 ,

0

k=0

where n(ω) is defined by (6.3). Let

ψ(ω)

ϕ(ω) = −(d + ε)

a˜ (u) (ω, τ ) dτ.

0 (j )

Note that the cylinders Cl are of the form Ci+0 ...in(ω(l,j )) . Given a number N > 0 choose r so small that n(ω) ≥ N for any ω ∈ 2A . Then   n(ω)−1 M(Ci0 , 0, ϕ, U(0) , N) ≤ sup exp ϕ(σ k ω) ≤ K4 , (j )

(j ) Cl ∈G˜ ω∈Cl

k=0

where U(0) is the cover of 2A by cylinders Ci = {ω ∈ 2A : ω0 = i} (see (A.13)). Let U(k) be the cover of 2A by cylinders Ci−k ...ik . It follows from the definition of M that M(Ci0 , 0, ϕ, U(k) , N ) ≤ |A|k M(Ci0 , 0, ϕ, U(0) , N + k) ≤ K5 , where |A| is the number of elements in the alphabet A and K5 > 0 is a constant. This implies that mc (Ci0 , 0, ϕ, U(k) ) ≤ K5 and P˜Ci0 (ϕ, U(k) ) ≤ 0 (see (A.14), (A.15)). Hence, P˜Ci0 (σ, ϕ) ≤ 0 and P2A (σ, ϕ) = P˜2A (σ, ϕ) = max P˜Ci (σ, ϕ) ≤ 0. 1≤i≤n


293

We now estimate the topological pressure of the function −(d + ε)a˜ (u) on (A, ψ) with respect to the suspension flow S. It is known (see [9]) that P(A,H) (S, −(d + ε)a˜ (u) ) is the unique real number c such that P2A (σ, ϕ − cψ) = 0, and P2A (σ, ϕ − cψ) is a decreasing function over c. This implies that P(A,H) (S, −(d + ε)a˜ (u) ) = c ≤ 0. It follows that t (u) ≤ d + ε. Since the inequality holds true for any ε > 0, we conclude that t (u) ≤ d. This easily implies that t (u) ≤ dimH (U ∩ ) for any x ∈ and any open (u) set U ⊂ Wloc (x). (u) We prove that d = dimB (U ) ≤ t (u) , where U is an open set in Wloc (x) ∩ . Recall that log N (U, ε) d = lim sup , log (1/ε) ε→0 where N (U, ε) is the maximal cardinality of an ε-separated set in U . For any δ > 0 there exists a sequence {εk }, εk → 0, such that N (U, εk ) ≥ (1/εk )d−δ for any k > 0. Fix ε > 0. Take εk < ε and let Xεk be an εk -separated set in U . For any y ∈ Xεk let τ (y) be the number for which τ (y) 2ε exp a (u) (f τ y)dτ = . εk 0 We have that τ (y) min a (u) ≤ log

It follows that

2ε ≤ τ (y) max a (u) . εk

1 1 τ (y) ∈ K6 log , K7 log . εk εk

This implies that there exists a number tk such that (1/εk )d−δ . K8 log (1/εk )

card { y ∈ Xεk : τ (y) ∈ [tk − 1, tk ] } ≥

Let Ek = { y ∈ Xεk : τ (y) ∈ [tk − 1, tk ] }. If εk is sufficiently small we obtain card Ek ≥ (1/εk )d−2δ . By construction, Ek is an (ε, tk )-separated set in . Hence, tk (u) exp −(d − 2δ)a (u) (f τ y)dτ Ztk (F, −(d − 2δ)a , ε) ≥ 0

y∈Ek

≥ K9

(see (A.16), (A.17), (A.18)).

1 εk

τ (y)

exp

y∈Ek

≥ K9

−(d−2δ) a

(u)

τ

(f y)dτ

0

d−2δ

2ε εk

−(d−2δ)

≥ K10

294


Note that tk → ∞ as k → ∞. Therefore, P (F, −(d − 2δ)a (u) , ε) ≥ 0

and

P (F, −(d − 2δ)a (u) ) ≥ 0.

It follows that d − 2δ ≤ t (u) . Since the inequality holds true for any δ > 0, we conclude that d ≤ t (u) and complete the proof of the first statement. Since κ (u) is the unique equilibrium measure corresponding to the Hölder continuous function −t (u) a (u) (x), we have (u) (u) 1 (u) a (u) (y) dκ (u) (y) 0 = P (−t a ) = hκ (u) (f ) − t

(see (4.1), (4.2), (A.21)), and the second statement follows. We will prove the last three statements of the theorem. Consider the function −t (u) a˜ (u) which is the pull back of the function −t (u) a (u) (x) to (A, ψ) by the coding map χ. The unique equilibrium measure corresponding to −t (u) a˜ (u) is equal to λϑ = ((ϑ × m)(Yψ ))−1 (ϑ × m)|Yψ , where ϑ is the unique equilibrium measure corresponding to the Hölder continuous function ψ(ω) a˜ (u) (ω, t) dt = −t (u) log a(u) (ω) −t (u) 0

on 2A and m is the Lebesgue measure on R (see (A.25), (A.26), Proposition A.5). Since P(A,ψ) (S, −t (u) a˜ (u) ) = 0, we obtain that P2A (σ, −t (u) log a(u) ) = 0. Therefore, there exist constants K11 , K12 > 0 such that for any ω ∈ 2A and any n > 0, K11 ≤

ϑ{ω : ωi = ωi , i = 0, . . . , n} ≤ K12 n −t (u) a(u) (σ k ω)

(7.1)

k=0

(see Proposition A.4). Let ( be a rectangle, and x ∈ (. Let also Ci0 be the cylinder such that Ci0 = χ −1 ((). + We introduce the measure ϑ (u) on 2A such that for any cylinder Ci0 ...in ⊂ 2A and its + + projection Ci0 ...in to 2A , ϑ (u) (Ci+0 ...in ) = ϑ(Ci0 ...in ). (u)

Let ξ (u) (x) be the push forward of ϑ (u) to Wloc (x, () by the coding map. Then ξ (u) (x) (u) is equivalent to the conditional measure on Wloc (x, () generated by the measure κ (u) . (u) (u) Let B(y, r) be a ball in Wloc (x, () of radius r. Consider a Moran cover of Wloc (x, () of size r. Let Q1 , . . . , Qm be the elements of this cover which intersect the ball B(y, r). + Recall that Qj = C(ω(j ) ) for some ω(j ) ∈ 2A (see Sect. 6). We have ξ (u) (B(y, r)) ≤

m

ξ (u) (Qj ) =

j =1

≤ K12

m j =1

m j =1



(j ) n(ω )



k=0

ϑ (u) (C(ω(j ) )) (7.2)

−t (u)

a(u) (ω(j ) )

≤ K12 M r

t (u)

,


295

where M is the Moran multiplicity factor, that does not depend on r (see (6.3), (6.5), (7.1)). Let ω = (. . . i−1 i0 i1 . . . ) ∈ 2A be such that y = χ (ω). Consider the cylinder Ci+0 ...in(ω) , where n(ω) is defined by (6.3). Then χ (Ci+0 ...in(ω) ) is contained in B(y, r). Thus, by (7.1), ξ (u) (B(y, r)) ≥ ϑ (u) (Ci+0 ...in(ω) )  −t (u) n(ω) (u) ≥ K11  a(u) (ω) ≥ K13 r t .

(7.3)

k=0 (u)

It follows from (7.2) and (7.3) that dξ (u) (x) (y) = t (u) for all y ∈ Wloc (x, (). This together with Proposition A.2 implies that dim H ξ (u) (x) = t (u) . (u) Let G be a finite or countable cover of an open set U ⊂ Wloc (x, () by open sets V with diamV ≤ ε. For any V ∈ G there exists a ball B such that V ⊂ B and diamB ≤ 2 diamV . Such balls comprise a cover B of U . By (7.2),

(diamV )

t (u)

V ∈G

≥

diamB t (u) 2

B∈B

≥

1 K12 M

ξ (u) (B) ≥ K14 ξ (u) (U ),

B∈B

and hence, mH (U, t (u) ) ≥ K14 ξ (u) (U ) (see (A.2)). Given δ > 0 there exists ε > 0 such that for any cover G of U with diamG ≤ ε, (u) mH (U, t (u) ) ≤ (diamV )t + δ. V ∈G

Let B be a finite or countable cover of U by balls of diameter at most ε such that ξ (u) (B) ≤ ξ (u) (U ) + δ. B∈B

Using (7.3) we conclude that 1 (u) (u) mH (U, t (u) ) ≤ (diamB)t + δ ≤ ξ (B) + δ K13 B∈B B∈B

ξ (u) (U ) 1 ≤ + + 1 δ. K13 K13 Since δ can be chosen arbitrarily, it follows that mH (U, t (u) ) ≤ (u)

(u)

1 (u) (U ). K13 ξ

Note that Wloc (x, () is diffeomorphic to Wloc (x) ∩ R(x), and the push forward of (u) (u) ξ (x) to Wloc (x) ∩ R(x) is equivalent to η(u) (x). Statements 3, 4 and 5 of the theorem follow. Proof of Theorem 4.2. The following statement is a corollary of results by Hasselblatt [6]. Lemma 7.1. Let F be a conformal axiom A flow on a basic set . Then the weak unstable distribution E (u) ⊕ X and the weak stable distribution E (s) ⊕ X are Lipschitz.

296


Recall that any rectangle ( lies in a small disk of co-dimension one which is transversal to the flow. The lemma implies that ( has a Lipschitz continuous local product structure. Since (u)

(u)

(s)

(s)

dimH (Wloc (x, ()) =dimB (Wloc (x, ()) = t (u) , and dimH (Wloc (x, ()) =dimB (Wloc (x, ()) = t (s) for any x ∈ (, the Proposition A.1 implies that dimH ( = dimB ( = dimB ( = t (u) + t (s) . The theorem follows since is locally diffeomorphic to the product of a rectangle and an interval. Proof of Theorem 5.1. We begin with the following observation. Let ϕ˜ be the pull back of ϕ to (A, ψ) by the coding map χ . The unique equilibrium measure corresponding to ϕ˜ is equal to λµ = ((µ × m)(Yψ ))−1 (µ × m)|Yψ , where µ is the unique equilibrium measure corresponding to the Hölder continuous function log R on 2A such that ψ(ω) ϕ(ω, ˜ t) dt − cψ(ω), log R(ω) = 0

and c = P(A,ψ) (S, ϕ). ˜ Note that P (σ, log R) = 0. (See (A.25), (A.26), Proposition A.5.) Let us introduce the functions µ(Ci1 ...in ) , µ(Ci0 ...in ) µ(Ci−n ...i−1 ) log R(s) (ω− ) = − lim log , n→∞ µ(Ci−n ...i0 )

log R(u) (ω+ ) = − lim log n→∞

+ − where ω+ = (i0 i1 . . . in . . . ) ∈ 2A and ω− = (. . . i−n . . . i−1 i0 ) ∈ 2A . (u) One can show that the above limits exist, the functions log R and log R(s) are + − Hölder continuous, and they are projections to 2A and 2A respectively of functions on 2A which are strictly cohomologous to log R (see [10]). In particular,

P2 + (log R(u) ) = P2 − (log R(s) ) = 0. A

A

− on and µ(s) on 2A as in (5.1) and (5.2). The We introduce the measures measures µ(u) and µ(s) are unique equilibrium measures corresponding to the Hölder continuous function log R(u) and log R(s) respectively (see [10]). It follows from the definition of the equilibrium measure (see (A.12)) that log R(u) (ω+ ) dµ(u) = log R(s) (ω− ) dµ(s) = log R(ω) dµ + − 2A 2A 2A (7.4) = −hµ(u) σ |2 + = −hµ(s) σ |2 − = −hµ (σ ).

µ(u)

A

+ 2A

A


297

Starting with the functions a(s) and a(u) one can similarly define functions a(ss) on + and a(uu) on 2A which are projections of functions strictly cohomologous to a(s) (u) and a respectively. − 2A

We proceed with the proof of Theorem 5.1. Consider a rectangle ( and a point (u) x ∈ int (. Let ν (u) be the push forward of the measure µ(u) to Wloc (x, () by the coding (u) map χ. Then ν (u) is equivalent to the conditional measure on Wloc (x, () generated by ν. We will show that the measure ν (u) is Federer. Since P2 + (log R(u) ) = 0 we conclude that there exist constants K1 and K2 such that A

+ for any ω ∈ 2A ,

K1 ≤

µ(u) {ω : ωi = ωi , i = 0, . . . , n} ≤ K2 n R(u) (σ k (ω))

(7.5)

k=0

(see Proposition A.4). (u) Given a number r > 0 consider a Moran cover of Wloc (x, () of size r. Fix a point (u) y ∈ Wloc (x, (). Let Q0 be an element of the Moran cover that contains y. Let also Q0 , . . . Qm be the elements of the Moran cover that intersect B(y, 2r). Recall that + Qj = χ (C(ω(j ) )) for some ω(j ) ∈ 2A (see Sect. 6). By the property (6.5) of the Moran ˜ where M˜ is a constant independent of y and r. Since cover, we have that m ≤ M, diamQ0 < r, we obtain Q0 ⊂ B(y, r) ⊂ B(y, 2r) ⊂

m

Qj .

j =0

Since a(u) is a Hölder continuous function on 2A , it is easy to show that there exist positive constants L1 and L2 such that (0) n(ω )

L1 ≤

−1

k=0

a(u) (σ k (ω(0) ))

n(ω (0) ) k=0

−1 a(u) (σ k (ω(j ) ))

≤ L2 ,

where n(ω) is defined by (6.3). This implies that |n(ω(0) ) − n(ω(j ) )| ≤ K3 , where K3 is a constant independent of j and r. So we conclude that (0) n(ω )

K4 ≤

k=0 n(ω (j ) ) k=0

R(u) (σ k (ω(0) )) ≤ K5 . R(u) (σ k (ω(j ) ))

(7.6)

298


It follows from (7.5) and (7.6) that ν

(u)

(B(y, 2r)) ≤

m

ν

(u)

(Qj ) =

j =1 (j )

≤ K2

m n(ω )

µ(u) (C(ω(j ) ))

j =1 (0)

R

(u)

k

(σ (ω

j =1 k=0

≤ K2 M˜

m

(j )

n(ω ) 1 (u) k (0) )) ≤ K2 M˜ R (σ (ω )) K4 k=0

1 1 (u) µ (C(ω(0) )) = K6 ν (u) (Q0 ) ≤ K6 ν (u) (B(y, r)). K 4 K1 (s)

Let ν (s) be the push forward of µ(s) to Wloc (x, (). Arguing similarly one can prove that ν (s) is Federer. Since the measure ν is locally equivalent to the product ν (u) × ν (s) × m (where m is the Lebesgue measure), it is also Federer. Proof of Theorem 5.2. First we define the “symbolic” level set. Given 0 < r < 1 and ω ∈ 2A , choose n− = n− (ω, r) and n+ = n+ (ω, r) such that 0

|a(ss) (σ k (ω− ))| > r,

k=1−n− + −1 n

0

|a(ss) (σ k (ω− ))| ≤ r,

k=−n−

(7.7)

+

|a

(uu)

k

+

(σ (ω ))|

k=0

−1

> r,

n

|a

(uu)

k

+

(σ (ω ))|

−1

≤ r.

k=0

Fix a number α˜ ≥ 0 and let J˜α˜ be the set of points ω in 2A for which the limit   0 n+ (s) k − (u) k + log R (σ (ω )) log R (σ (ω ))     k=−n− k=0 lim  − +  0 n r→0   log |a(ss) (σ k (ω− ))| log |a(uu) (σ k (ω+ ))| k=−n−

k=0

exists and is equal to α. ˜ ˜ α = { (ω, t) ∈ (A, ψ) : ω ∈ J˜α−1 }. Then χ ( ˜ α ) = α . Lemma 7.2. Let Proof. Let Jα−1 = {x ∈ T : d ν (u) ×ν (s) (x) = α − 1} and B (u) (x, r) be a ball in (u) Wloc (x, () centered at x ∈ Jα−1 . Fix x and choose ω = (. . . i−1 i0 i1 . . . ) ∈ 2A such that x = χ (ω). Consider the cylinder Ci+0 ...in(ω) , where n(ω) is defined by (6.3).

Let Q(u) (x, r) = χ (Ci+0 ...in(ω) ). We have x ∈ Q(u) (x, r) and diam Q(u) (x, r) < r.

Therefore, Q(u) (x, r) ⊂ B (u) (x, r). Since Q(u) (x, r) contains a ball of radius K1 r and ν (u) is Federer, we obtain µ(u) (Ci+0 ...in(ω) ) = ν (u) (Q(u) (x, r)) ≤ ν (u) (B (u) (x, r)) ≤ K7 ν (u) (Q(u) (x, r)) = K7 µ(u) (Ci+0 ...in(ω) ).


It follows from (7.5) and (7.7) that 

299

 (uu) (σ k (ω+ ))|−1 log |a  log ν (u) (B (u) (x, r))    k=0 × lim   = 1. + n r→0   log r (u) k + log R (σ (ω )) n+

k=0

Arguing similarly one can show that   log ν (s) (B (s) (x, r))  lim  × r→0  log r

0 k=−n− 0

log |a(ss) (σ k (ω− ))|

k=−n−

log R(s) (σ k (ω− ))

    = 1, 

(s)

where B (s) (x, r) are balls in Wloc (x, (). This implies that Jα−1 = χ (J˜α−1 ). Since locally α is a direct product of Jα−1 and ˜ α ). The lemma is proven. an interval, α = χ ( We proceed with the proof of Theorem 5.2. Consider the one-parameter families of functions on 2A , ϕq(u) (ω) = −T˜ (u) (q) log |a(u) (ω)| + q log R(ω),

(7.8)

ϕq(s) (ω) = T˜ (s) (q) log |a(s) (ω)| + q log R(ω), where T˜ (u) (q) and T˜ (s) (q) are chosen such that P2A (ϕq(u) ) = 0 and P2A (ϕq(s) ) = 0.

(7.9)

It is known that that the functions T˜ (u) and T˜ (s) are real analytic (see [10]). We introduce the functions ϕq(uu) (ω+ ) = −T˜ (u) (q) log |a(uu) (ω+ )| + q log R(u) (ω+ ), ϕq(ss) (ω− ) = T˜ (s) (q) log |a(ss) (ω− )| + q log R(s) (ω− ), (u)

(s)

+ − and 2A of functions strictly cohomologous to ϕq and ϕq which are projections to 2A respectively. (u) (s) Let µq and µq be the equilibrium measures corresponding to the Hölder continuous (uu) + − and ϕ (ss) on 2A respectively. functions ϕq on 2A For each real q define (u) (ω+ ) dµ(u) (s) (ω− ) dµ(s) + log R − log R q q 2A 2A (u) (s) α˜ (q) = − , α˜ (q) = . (u) (uu) (ω+ )| dµ (ss) (ω− )| dµ(s) q q 2 + log |a 2 − log |a

Note that

+ 2A

A

A

(u) log |a(uu) (ω+ )| dµq

+ 2A

> 0. The variational principle implies that

(u) log R(u) (ω+ ) dµ(u) q ≤ P2 + (log R ) = 0 A

300


(see (A.11)), and hence α˜ (u) (q) > 0 for all q ∈ R. Similarly, α˜ (s) (q) > 0 for all q ∈ R. It is known that α˜ (u) (q) = −(T˜ (u) ) (q) and α˜ (s) (q) = −(T˜ (s) ) (q) (see [10]), in particular, (T˜ (u) ) (q) < 0 and (T˜ (s) ) (q) < 0 for all q ∈ R. Lemma 7.3. 1. If ν (u) is the measure of full dimension then (u) T˜ (u) (q) = (1 − q) dimH Wloc (x, (), and (u)

dν (u) (y) = t (u) for all y ∈ Wloc (x, (),

(7.10)

where t (u) is defined by (4.2). 2. If ν (u) is not the measure of full dimension, then (T˜ (u) ) (q) > 0 for all q ∈ R. (u)

Proof. Recall that the conditional measure on Wloc (x, () generated by the measure κ (u) is the measure of full dimension, where κ (u) is the unique equilibrium measure on for the function −t (u) a (u) . 1. If ν (u) is the measure of full dimension, then µ(u) is the equilibrium measure for the function −t (u) log |a(uu) |, and therefore the functions log R(u) and −t (u) log |a(uu) | are cohomologous (see Appendix). Since P2 + log R(u) = P2 + −t (u) log |a(uu) | = 0, A

A

the functions are strictly cohomologous. It follows that 0 = P2A ϕq(u) = P2 + ϕq(uu) = P2 + (−T˜ (u) (q) − qt (u) ) log |a(uu) | . A

A

By the definition of t (u) (see (4.2)), −T˜ (u) (q) − q t (u) = −t (u) , and hence T˜ (u) (q) = (u) (1 − q)t (u) = (1 − q) dimH Wloc (x, (). The third statement of Theorem 4.1 implies that if ν (u) is the measure of full dimen(u) sion, then dν (u) (y) = t (u) for all y ∈ Wloc (x, (). 2. It is known that (T˜ (u) ) (q) > 0 for some q if the functions log R(u) and −(T˜ (u) ) (q) log |a(uu) | are not cohomologous (see [10]). Assume that the functions are cohomologous for some q. Since (T˜ (u) ) (q) = −α˜ (u) (q), it is easy to see that log R(u) (ω+ ) + (T˜ (u) ) (q) log |a(uu) (ω+ )| dµ(u) q = 0. + 2A

This implies that the functions log R(u) and −(T˜ (u) ) (q) log |a(uu) | are strictly cohomologous, and hence P2 + −(T˜ (u) ) (q) log |a(uu) | = P2 + log R(u) = 0 A

A

(see Appendix). It follows that (T˜ (u) ) (q) = t (u) , and ν (u) is the measure of full dimension.


301

Similarly to Lemma 7.3, one can prove that 1. If ν (s) is the measure of full dimension then (s) T˜ (s) (q) = (1 − q) dimH Wloc (x, (), and (s)

dν (s) (y) = t (s) for all y ∈ Wloc (x, (),

(7.11)

where t (s) is defined by (4.5). 2. If ν (s) is not the measure of full dimension, then (T˜ (s) ) (q) > 0 for all q ∈ R. Set T˜ (q) = T˜ (u) (q) + T˜ (s) (q), and α(q) ˜ = α˜ (s) (q) + α˜ (u) (q). We can conclude that ˜ ˜ ˜ α(q) ˜ = −T (q), in particular, T < 0, T ≥ 0, and T˜ > 0 if and only if either ν (s) or (u) ν is not the measure of full dimension. Assume that ν is not the measure of full dimension and hence, ν (s) or ν (u) is not the measure of full dimension. (u) (s) (u) (s) We define the measure µq = µq × µq . Since the measures µq and µq are ergodic, it follows from the Birkhoff ergodic theorem that for µq -a.e. ω ∈ 2A ,    lim  r→0 

 log R(u) (σ k (ω+ ))  k=−n−  k=0 ˜ − +  = α(q). 0 n  (ss) k − (uu) k + log |a (σ (ω ))| log |a (σ (ω ))| 0

n+


k=−n−

(7.12)

k=0

Lemma 7.4. For all ω = (. . . i−1 i0 i1 . . . ) ∈ J˜α(q) , ˜ lim

log µq (Ci−n− ...in+ ) log r

r→0

= T˜ (q) + q α(q), ˜

where n− = n− (ω, r) and n+ = n+ (ω, r) are defined by (7.7). (s)

(u)

Proof. Since µq and µq are equilibrium measures corresponding to the functions (ss) ϕq and ϕ (uu) , Proposition A.4 implies that the ratios (s)

µq (C−in− ...i0 )

0 k=−n−

a(ss) (σ k (ω− ))T˜

(s) (q)

R(s) (σ k (ω− ))q

and (u)

µq (Ci0 ...in+ ) n+ k=0

a(uu) (σ k (ω+ ))T˜

(u) (q)

R(u) (σ k (ω+ ))q

302


are bounded from below and from above by constants independent of ω and r. Hence, for all ω ∈ J˜α(q) , ˜ lim


r→0

T˜ (s) (q) log = lim

r→0



0 k=−n−

|a(ss) (σ k (ω− ))| + T˜ (u) (q) log

n+ k=0

|a(uu) (σ k (ω+ ))|−1

log r 0


n+

log R(u) (σ k (ω+ ))



    k=−n− k=0 + q lim  − +  0 n r→0   log |a(ss) (σ k (ω− ))| log |a(uu) (σ k (ω+ ))| ˜ (s)

=T

k=−n− (u) ˜

(q) + T

k=0

(q) + q α(q). ˜

The lemma is proven. We proceed with the proof of the theorem. Consider the measure λµq = ((µq × m)(Yψ ))−1 (µq × m)|Yψ on (A, ψ). Let νq be its push forward. It follows from (7.12) that ) = 1. νq (α(q)+1 ˜

(7.13)

Similarly to the proof of Lemma 7.2 one can show that dνq (x) = lim


r→0

+ 1.

Lemma 7.4 implies that ˜ + 1 for all x ∈ α(q)+1 . dνq (x) = T˜ (q) + q α(q) ˜

(7.14)

˜ + 1) = dimH α(q)+1 = T˜ (q) + q α(q) ˜ +1 fν (α(q) ˜

(7.15)

It follows that

(see [10]). Recall that α(q) ˜ = −T˜ (q), T˜ < 0 and T˜ > 0. Let us introduce the functions α(q) = α(q) ˜ +1

and

T (q) = T˜ (q) − q + 1.

We have fν (α(q)) = T (q) + αq , where α(q) = −T (q). Therefore, the functions fν and T form a Legendre transform pair (see Appendix). Clearly, the function T is real analytic, T < 0, and T > 0. Therefore, fν is also real analytic and fν < 0. The function fν (α) is defined on an interval [α1 , α2 ], where α1 = − lim T (q), q→+∞

α2 = − lim T (q). q→−∞


303

Since P2A (R) = 0, we have that T (s) (1) = T (u) (1) = 0, and ϕ (u) (ω) = ϕ (s) (ω) = (s) (u) log R(ω) (see (7.8), (7.9)). Therefore, µ1 = µ(s) , and µ1 = µ(u) . It follows from the definition of α˜ and (7.4) that 2A log R(ω) dµ 2 log R(ω) dµ α(1) ˜ = − A (s) (u) 2A log a (ω) dµ 2A log a (ω) dµ ψ(ω) = ϕ(ω, ˜ t) dt − cψ(ω) dω 2A

×

0

1

ψ(ω) a˜ (s) dt 2A 0

dω

−

1

ψ(ω) a˜ (u) dt 2A 0

dω

= K8 ϕ(ω, t) dt − c ψ(ω) (A,ψ) 2A 1 1 × − K8 (A,ψ) a˜ (s) dt dω K8 (A,ψ) a˜ (u) dt dω

1 1 = ϕ(x) dν − c × (s) − (u) a dν a dν

1 1 = hν (f 1 ) − − , λ+ λν ν where K8 = (µ × m)(Yψ ) and c = P(A,ψ) (S, ϕ). It follows from (5.3) that µ is equivalent to µ1 , and hence ν is equivalent to ν1 . By (7.13) and (7.14), ν(α(1) ) = 1. Moreover, dν (x) = α(1) for all x ∈ α(1) . This implies that

1 1 1 dν (x) = hν (f ) − − +1 λ+ λν ν for ν-a.e. x ∈ . This completes the proof of the first statement. Let Ur (Ci+0 ) and Ur (Ci−0 ) be Moran covers of Ci+0 and Ci−0 of size r. Then Cr = Ur (Ci−0 ) × Ur (Ci+0 ), i0 ∈ A is a cover of 2A . It is known that T˜ (q) = − lim

log

r→0

C∈Cr (µ(C))

log r

q

.

˜ r be the cover of (A, ψ) which consists of the elements Let D D˜ = C × [kr, (k + 1)r), We have that − lim

r→0

log

where C ∈ Cr , and 0 ≤ k
0 (provided the limit exists).

308


Another dimension spectrum is the Hentschel–Procaccia spectrum for dimensions. It is a one-parameter family of characteristics q log inf G B(xi ,r)∈G µ(B(xi , r)) 1 H Pq (µ) = lim , (A.8) q − 1 r→0 log r where G is a finite or countable cover of the support of µ by balls of radius r and q ≥ 0, q = 1 (provided the limit exists). One can show that for q > 1, log µ(B(x, r))q−1 dµ(x) 1 H P q (µ) = lim . (A.9) q − 1 r→0 log(1/r) Moreover, Rq (µ) = H P q (µ). 3. Facts from thermodynamic formalism [3, 4, 10, 9, 13]. Let X be a compact metric space, f : X → X a continuous map, and ϕ a continuous function on X (called the potential function). For every ε > 0 and n > 0 a set E ⊂ X is called (ε, n)-separated if x, y ∈ E, x = y implies that ρ(f k (x), f k (y)) > ε for some k ∈ [0, n]. Set n−1 k exp ϕ(f (x)) , Zn (f, ϕ, ε) = sup x∈E

k=0

where the supremum is taken over all (ε, n)-separated sets E ⊂ X. Set further 1 log Zn (f, ϕ, ε), n→∞ n PX (f, ϕ) = lim PX (f, ϕ, ε).

PX (f, ϕ, ε) = lim sup ε→0

(A.10)

We call PX (f, ϕ) the topological pressure of the function ϕ on X (with respect to f ). The following result is a variational characterization of the topological pressure. Let M(f ) denote the space of all f -invariant Borel probability measures on X. Then

PX (f, ϕ) = sup hµ (f ) + ϕ dµ , (A.11) X

µ∈M(f )

where hµ (f ) is the measure-theoretic entropy of µ. Measures that realize the variational principle for topological pressure play crucial roles in ergodic theory. A measure µ ∈ M(f ) is called an equilibrium measure for the function ϕ if PX (f, ϕ) = hµ (f ) + ϕ dµ. (A.12) X

We also need the “dimensional” definition of topological pressure for the case of a symbolic dynamical system (2A , σ ) (see [10]): Let U(k) be the open cover of 2A by cylinders Ci−k ...ik . (Notice that diam U(k) → 0 as k → ∞.) Let Z be a subset of 2A , and α be a real number. Let   m M(Z, α, ϕ, U(k) , N) = inf exp −α(m + 1) + sup ϕ(σ j (ω)) , (A.13) G˜

C∈G˜

ω∈C j =0


309

where the infimum is taken over all finite or countable collections G of cylinders C = Ci−k ...ik+m with m ≥ N > k which cover Z. Define mc (Z, α, ϕ, U(k) ) = lim M(Z, α, ϕ, U(k) , N ), N→∞

(A.14)

PZ (ϕ, U(k) ) = inf { α : mc (Z, α, ϕ, U(k) ) = 0 } = sup { α : mc (Z, α, ϕ, U(k) ) = ∞}, P˜Z (f, ϕ) = lim PZ (U(k) , ϕ).

(A.15)

k→∞

If Z is a compact invariant subset of 2A then P˜Z (f, ϕ) = PZ (f, ϕ). We now describe the thermodynamic formalism for dynamical systems with continuous time. Let F = {f t } : X → X be a continuous flow (i.e., a one-parameter group of continuous maps on X which depend continuously on t) and ϕ a continuous function on X. For every ε > 0 and t > 0 a set E ⊂ X is called (ε, t)-separated if x, y ∈ E, x = y implies that ρ(f τ (x), f τ (y)) > ε for some τ ∈ [0, t]. Set t exp ϕ(f τ (x)) dτ , (A.16) Zt (F, ϕ, ε) = sup x∈E

0

where the supremum is taken over all (ε, t)-separated sets E ⊂ X. Define PX (F, ϕ, ε) = lim sup t→∞

1 log Zt (F, ϕ, ε), t

PX (F, ϕ) = lim PX (F, ϕ, ε).

(A.17) (A.18)

ε→0

We call PX (F, ϕ) the topological pressure of the function ϕ on X (with respect to the flow F = {f t }). One can show that (A.19) PX (F, ϕ) = PX (f 1 , ϕ 1 ), 1 where f 1 is a time-one map and ϕ 1 = 0 ϕ(f t (x)) dt. Moreover, one can express the variational principle for the topological pressure in the case of flows as follows:

1 1 hµ (f ) + ϕ dµ , (A.20) PX (F, ϕ) = sup µ∈M(F )

X

where M(F ) is the set of Borel probability measures on X. Note that for all1 F -invariant any such measure µ ϕ dµ = ϕ dµ. A measure µ ∈ M(F ) is called an equilibrium measure for the function ϕ if 1 1 1 ϕ dµ = hµ (f ) + ϕ dµ. (A.21) PX (F, ϕ) = hµ (f ) + X

X

4. Symbolic dynamical systems [10, 3, 4, 9]. Given a p × p matrix A of 0s and 1s (called transfer matrix), consider the subshift of finite type (2A , σ ), where 2A is the space of two-sided infinite sequences of p symbols which are admissible by the matrix

310


A (a sequence ω = (ωi ), i ∈ Z is admissible if aωi ,ωi+1 = 1 for all i ∈ Z) and σ is the shift map. The space 2A has a natural family of metrics ∞ |ωi − ωi | , dβ (ω, ω ) = β |i|

(A.22)

i=−∞

where β > 1. The set 2A is compact with respect to the topology induced by dβ and the shift map σ is a homeomorphism. If the matrix A is transitive (i.e., for every 0 ≤ i, j ≤ p there exists k > 0 such that the (i, j )-entry of the matrix Ak is strictly positive) then the shift σ is topologically transitive (i.e., for every open set U and V there exists k > 0 such that σ k (U ) ∩ V = ∅). If the matrix A is irreducible (i.e., there exists k > 0 such that Ak > 0) then the shift σ is topologically mixing (i.e., for every open set U and V there exists k > 0 such that σ n (U ) ∩ V = ∅ for every n ≥ k). Let ϕ be a Hölder continuous function on 2A . The following statement describes equilibrium measures for subshifts of finite type. Proposition A.4. Assume that the transfer matrix A is irreducible. Then 1. there exists a unique equilibrium measure µ = µϕ which is mixing and is positive on open sets; 2. there exist constants D1 , D2 > 0 such that for any ω = (ωi ) and any m, n ≥ 0, D1 ≤

µ{ω : ωi = ωi , i = −m, . . . , n} ≤ D2 . exp −(m + n + 1)P2A (σ, ϕ) + nk=−m ϕ(σ k (ω))

(A.23)

A measure µ on 2A which satisfies (A.23) is called a Gibbs measure. We describe a symbolic suspension flow over a subshift of finite type (2A , σ ). Let ψ be positive continuous function on 2A and Yψ = {(ω, s) : s ∈ [0, ψ(ω)], ω ∈ 2A } ⊂ 2A × R. If for every ω ∈ 2A we identify the points (ω, ψ(ω)) and (σ (ω), 0) we obtain a compact topological space (A, ψ). We define the symbolic suspension flow S = {S t } on (A, ψ) by S t (ω, s) = (ω, s + t)

if s + t ∈ [0, ψ(ω)],

(A.24)

taking identification into account. There is a canonical identification between the spaces of invariant measures for symbolic suspension flows and subshifts of finite type. Namely, for any measure µ ∈ M(σ ) and the Lebesgue measure m on R the measure µ × m has the property that the identifications Yψ → (A, ψ) are held on a set of measure zero. Therefore the measure λµ = ((µ × m)(Yψ ))−1 (µ × m)|Yψ

(A.25)

is a probability measure on (A, ψ). Moreover, λµ ∈ M(S) and the map µ → λµ is one-to-one. Let ϕ˜ be a continuous function on (A, ψ). Set ψ(ω) ϕ(ω, ˜ t) dt, H( ω) = H0 (ω) − cψ(ω), (A.26) H0 (ω) = 0


311

where c = P(A,ψ) (S, ϕ) ˜ is the topological pressure of the function ϕ˜ on (A, ψ) with respect to the symbolic suspension flow S. P2A (σ, H) = 0, since P(A,ψ) (S, ϕ) ˜ is the unique real number c such that P2A (σ, H0 − cψ) = 0 (see [9]). The following statement describes equilibrium measures for symbolic suspension flows. Proposition A.5. Assume that the function H(ω) is Hölder continuous on 2A with respect to the dβ -metric for some β > 1. Then 1. there exists a unique equilibrium measure µϕ˜ for the function ϕ˜ for the symbolic suspension flow S = {S t }; the measure µϕ˜ is ergodic and positive on open sets; 2. µϕ˜ = λµH where µH is a unique equilibrium measure for the function H and the measure λµH is defined by (A.25). 5. Legendre Transform. We remind the reader of the notion of a Legendre transform pair of functions. Let h be a C 2 -function on an interval I such that h (x) > 0 for all x ∈ I . The Legendre transform of h is the differentiable function g of a new variable p defined by g(p) = min(px + h(x)). (A.27) x∈I

One can show that: 1. g < 0; 2. the Legendre transform is involutive; 3. strictly convex functions h and g form a Legendre transform pair if and only if g(α) = h(q) + qα, where α(q) = −h (q) and q = g (α). 6. Cohomologous Functions [13]. Let X be a compact metric space, and f : X → X a continuous map. Two functions ϕ1 and ϕ2 on X are called cohomologous if there exists a Hölder continuous function g : X → R and a constant K such that ϕ1 − ϕ2 = g − g ◦ f + K. If the above equality holds with K = 0 the functions are called strictly cohomologous. We recall some properties of cohomologous functions: 1. The functions ϕ1 and ϕ2 are cohomologous if and only if equilibrium measures of ϕ1 and ϕ2 ; on X coincide. 2. If ϕ1 and ϕ2 are strictly cohomologous then PX (ϕ1 ) = PX (ϕ2 ). Acknowledgement. The authors would like to thank D. Burago, A. Katok, and M. Kanai for valuable comments and discussions. We also would like to thank the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) for hospitality and support during our stay in June–July 2000.

References 1. Barreira, L., Pesin, Ya., Schmeling, J.: Dimension and Product Structure of Hyperbolic Measures. Annals of Math. 149, 3, 755–783 (1999) 2. Barreira, L., Saussol, B.: Multifractal Analysis of Hyperbolic Flows. Preprint (1999) 3. Bowen, R.: Symbolic Dynamics for Hyperbolic Flows. Am. J. Math. 95, 429–460 (1973) 4. Bowen, R., Ruelle, D.: The Ergodic Theory of Axiom A Flows. Invent. Math. 29, 181–202 (1975) 5. Falconer, K.: Fractal Geometry, Mathematical Foundations and Applications. New York–London– Sydney: John Wiley & Sons, 1990

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6. Hasselblatt, B.: Regularity of the Anosov Splitting and of Horospheric Foliations. Ergod. Theory and Dyn. Syst. 14, 645–666 (1994) 7. Kanai, M.: Differential-geometric Studies on Dynamics of Geodesic and Frame Flows. Japanese J. Math. 19, 1–30 (1993) 8. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and its Applications, vol. 54. Cambridge: Cambridge University Press, 1995 9. Parry, W., Pollicott, M: Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics. Astérisque, Vol. 187–188 (1990) 10. Pesin, Ya.: Dimension Theory in Dynamical Systems: Contemporary Views and Applications. Chicago Lectures in Mathematics. Chicago: Chicago University Press, 1997 11. Pesin,Ya., Weiss, H.: The MultifractalAnalysis of Gibbs Measures: Motivation, Mathematical Foundation, and Examples. Chaos 7, 1, 89–106 (1997) 12. Ratner, M.: Markov Partitions for Anosov Flows on n-dimensional Manifolds. Israel J. Math. 15, 92–114 (1973) 13. Ruelle, D.: Thermodynamic Formalism. Reading, MA: Addison-Wesley, 1978 14. Schmeling, J.: On the Completeness of Multifractal Spectra. Ergod. Theory and Dyn. Syst. to appear 15. Weiss, H.: The Lyapunov Spectrum of Equilibrium Measures for Conformal Expanding Maps and AxiomA Surface Diffeomorphisms. J. Stat. Physics 95 (1999) 16. Young, L.-S.: Dimension, Entropy, and Lyapunov Exponents. Ergod. Theory and Dyn. Syst. 2, 109–124 (1982) Communicated by J. L. Lebowitz

Commun. Math. Phys. 216, 313 – 323 (2001)

Communications in



The Modulus of Continuity for 0 (m)\H Semi-Classical Limits Scott A. Wolpert Department of Mathematics, University of Maryland, College Park, MD 20742, USA Received: 10 March 2000 / Accepted: 26 July 2000

Abstract: We study the behavior of a large-eigenvalue limit of eigenfunctions for the hyperbolic Laplacian for the modular quotient SL(2; Z)\H. Féjer summation and results of S. Zelditch are used to show that the microlocal lifts of eigenfunctions have largeeigenvalue limit a geodesic flow invariant measure for the modular unit cotangent bundle. The limit is studied for Hecke–Maass forms, joint eigenfunctions of the Hecke operators and the hyperbolic Laplacian. The first modulus of continuity result is presented for the limit. The singular concentration set of the limit cannot be a compact union of closed geodesics and measured geodesic laminations. 1. Introduction Let be the Laplace–Beltrami operator for a finite volume Riemannian manifold M. The large-eigenvalue limit of eigenfunctions of presents a model for the transition √ it − between quantum and classical mechanics [8]. The operator e represents time evolution for the quantum mechanical system; geodesic flow represents time evolution for the classical mechanical system. In the large-eigenvalue limit the eigenfunctions (quantum states) give rise to a geodesic flow invariant measure (a classical state) on the unit cotangent bundle of M. The quantum ergodicity question is to understand the limit in the presence of a classical ergodic flow [2–4, 7, 17, 20, 21, 31, 32, 34]. The limit for finite area quotients of the hyperbolic plane and in particular modular quotients presents a setting where an explicit understanding is developing [11, 10, 13, 17, 20–23, 29–35]. The quantum ergodicity question for hyperbolic quotients involves modular functions, coefficient sums and the structure of SL(2; R). A basic construction is the microlocal lift of a Laplace–Beltrami eigenfunction. The lift is an almost measure (a distribution) on the unit cotangent bundle S ∗ (M); the first term of the lift is the eigenfunction square. For large eigenvalue the lift is almost invariant Research supported in part by NSF Grant DMS-9800701.

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under geodesic flow. A. Schnirelman [23], Y. Colin de Verdière [7], and S. Zelditch [32] showed for compact manifolds with ergodic geodesic flow that the spectral average of the microlocal lifts is the uniform distribution on S ∗ (M); a corollary provides for a full spectral density subsequence that the microlocal lifts converge to the uniform density on S ∗ (M). S. Zelditch first considered non compact hyperbolic quotients. The corresponding spectral decomposition for the hyperbolic Laplacian consists of the continuous span of the Eisenstein series and the span of the square integrable eigenfunctions, [5, 24, 28]. S. Zelditch found the appropriate renormalization for the Eisenstein series and showed that the spectral average again is the uniform distribution, [35]. For SL(2; Z) the Eisenstein series contribution in fact has smaller order of magnitude and does not contribute to the spectral average, [35]. W. Luo and P. Sarnak were able to directly analyze the modular Eisenstein series [17]. They found that the absolute square of the Eisenstein series weak∗ converges to 48π −1 for large-eigenvalues; their analysis involved the subconvexity bounds for the Riemann zeta function and the L-functions for Maass cusp forms. D. Jakobson extended the considerations to include the microlocal lift of the Eisenstein series [13]. In [30] the author found that the microlocal lift to SL(2; R) ≈ S ∗ (H)1/2 of automorphic eigenfunctions can be obtained directly from their twisted Fourier coefficient sums. The Luo–Sarnak and Jakobson result is equivalent to a limit-sum formula combining the Riemann zeta values ζ (1 + it) and the elementary divisor values σ2it . Z. Rudnick and P. Sarnak considered arithmetic compact hyperbolic quotients [20]. An Eichler order in a quaternion algebra over Q gives rise to a cocompact subgroup ⊂ SL(2; R) with a commutative ring of self-adjoint operators, Hecke operators, acting on L2 (\H) and commuting with the hyperbolic Laplacian. Closed geodesics for such a are associated with binary quadratic forms. There is a computational scheme for determining the action of the Hecke operators on closed geodesics. The authors show that a closed geodesic can be separated from any finite set of closed geodesics by a Hecke operator. The result provides for joint eigenfunctions of the hyperbolic Laplacian and the Hecke operators that a large-eigenvalue limit cannot have singular support a finite union of points a closed geodesics, [20, Theorem 1.1]. S. Zelditch introduced a microlocal lift to SL(2; R) based on Helgason’s Fourier transform [12, 34]. He found that the lift satisfies an exact differential equation; see Lemma 2 below. Properties of the large-eigenvalue limit of SL(2; R) microlocal lifts can be obtained directly from Fejér summation and integration by parts: see Proposition 4 for the basic properties and Proposition 5 for Cauchy–Schwartz and Minkowski type inequalities. We consider the Hecke operators for SL(2; Z) and the congruence subgroups 0 (m). We describe a sub-tiling for the Hecke operators Tp , p ≤ q and a basic set of diameter q −2 . We combine the sub-tiling for the Hecke operators, the structure of the microlocal lift and the partial-sums for p −1 to study limits of the lifts. The measure of a set is estimated after tiling a region with translates of the set. We find in particular for joint eigenfunctions of the hyperbolic Laplacian and the Hecke operators that a large-eigenvalue limit of microlocal lifts with compact singular support vanishes on each closed geodesic and on each geodesic lamination for a finite index subgroup. Our results and approach have similarities to the work of D. Jakobson and S. Zelditch on semi-classical limits for eigenfunctions of Hecke operators for the sphere S2 [14, Sect. 4.3]. In comparison to the considerations of Z. Rudnick and P. Sarnak the present result provides that even more general limit measures will be null on closed geodesics and geodesic laminations. A limit measure is geodesic flow invariant and hence determines a measure on the leaf space for the flow, the space of geodesics for the hyperbolic

Modulus of Continuity for 0 (m)\H Semi-Classical Limits

315

plane. We present the first explicit modulus of continuity bound for such measures. The mass in a ball of radius is bounded by (log log −1 )−2 ; see Proposition 10 below. 2. Background We recall the formalism for SL(2; R), [16], as well as the construction of S. Zelditch for the microlocal lift [31, 32, 34]. An element B ∈ SL(2; R) has the unique Iwasawa decomposition B=

ab cd

=

1x 01

y 1/2 0 0 y −1/2

cos θ sin θ − sin θ cos θ

which provides for an equivalence of SL(2; R) with S ∗ (H)1/2 , the square root of the unit cotangent bundle to the upper half plane, by the rule x + iy = y 1/2 eiθ (ai + b),

y −1/2 eiθ = d − ic

for z = x + iy ∈ H and θ the argument for the root cotangent vector measured from the positive vertical. The equivalence will play a basic role throughout. The bi-invariant volume form (Haar measure) for SL(2; R) is dV = y −2 dxdydθ. The Lie algebra acts on the right of SL(2; R) with E ± = H ± iV for

1 0 01 0 1 01 H = , V = , W = , and X = . 0 −1 10 −1 0 00 The infinitesimal generator of geodesic-flow is H = 21 (E + +E − ); W is the infinitesimal generator of K, the fiber rotations of S ∗ (H)1/2 . In terms of the coordinates (x, y, θ) for ∂ ∂ SL(2; R) the operator E + is simply E + = 4iye2iθ ∂z − ie2iθ ∂θ and the operator X is ∂ ∂ ∂ simply y cos 2θ ∂x + y sin 2θ ∂y + y sin2 θ ∂θ , [16]. A function u on H satisfying the differential equation Du + ( 41 + r 2 )u = 0, D the hyperbolic Laplacian, lifts to a K-invariant function on SL(2; R) satisfying Cu = (2ir + 1)(2ir − 1)u for the Casimir operator C = E − E + − W 2 − 2iW . The Casimir operator is in the center of the enveloping algebra. A ladder of functions, the raisings and lowerings of u, is determined by the scheme u0 = u, (2ir + 2m + 1)u2m+2 = E + u2m , (2ir − 2m + 1)u2m−2 = E − u2m

(1)

representation for K for m integral. The function u2m is in the weight 2m irreducible ∞ as demonstrated by W u2m = i2m u2m . The sum u = u2m is a distribution that is m

N -invariant as well as an eigendistribution of H [32, Prop. 2.2]; [34, p. 44]. Elements of the Lie algebra sl(2; R) preserve the volume form and can be integrated by parts. In particular the integral BκdV vanishes for B in the Lie algebra and Q

Q = SL(2; R) or Q = \SL(2; R), a discrete subgroup, with κ a smooth compactly

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Q-supported function. Consider solutions u, v of the equation Cu = (2ir + 1)(2ir − 1)u and a smooth function χ . Provided χ is smooth with compact support there is the relation ((E + u2j )v2k χ + u2j E − v2k )χ + u2j v2k E + χ dV. (2) 0= Q

We are ready to consider the microlocal lifts of automorphic eigenfunctions. Let ⊂ SL(2; R) be a cofinite subgroup and ϕ an L2 (\H) eigenfunction with unit-norm. The function ϕ lifts to a K-invariant function on SL(2; R) satisfying Cϕ = (2ir + 1)(2ir − 1)ϕ. We consider the ladder {ϕ2m } of raisings and lowerings, as well as the quantity ϕ∞ = ϕ2m . The ladder {ϕ2m } is an orthogonal basis for an irreducible principal m

continuous series representation of SL(2; R), [16]. For the L2 (\SL(2; R)) Hermitian product ϕ2m , ϕ2m = 2π is satisfied and from integration by parts E + ϕ2j , ϕ2k χ + ϕ2j , E − (ϕ2k χ ) = 0 for a -invariant test function χ . A test function χ ∈ Cc2 (\SL(2; R)) has a K Fourier expansion χ = χm with m

|χm | ≤ Cχ (1 + |m|)−2 . For L2 (\H) eigenfunctions ϕ and ψ from Parseval’s relation the pairing of χ with ϕψ ∞ is the sum χ2m ϕψ2m dV. m

\SL(2;R)

The sum is bounded by Cχ ϕψ. In consequence the quantity ϕψ ∞ is a distribu2 (SL(2; R)), the tion for Cc2 (\SL(2; R)). Equivalently ϕψ ∞ is a distribution for Cc, 2 Cc (SL(2; R)) subspace of -invariant functions. Relatedly the operator 5 : Cc2 (SL(2; R)) → Cc2 (\SL(2; R)) defined by 5χ = χ | γ is a continuγ ∈

ous surjection relative to the Fréchet topologies; furthermore for functions with support contained in a -fundamental domain χ = 5χ . In consequence a distribution for Cc2 (\SL(2; R)) has a natural extension, the formal adjoint of 5, to a distribution for Cc2 (SL(2; R)); furthermore convergence of extensions is equivalent to convergence of the original distributions. In the following considerations we will use all three settings for the distribution ϕψ ∞ . Definition 1. For L2 (\H) eigenfunctions ϕ and ψ set 2 Q(ϕ, ψ) = ϕψ ∞ + ψϕ ∞ and Q(ϕ) = Q(ϕ, ϕ). The microlocal lift Q(ϕ) is a basic quantity for the 8DO-calculus based on Helgason’s Fourier transform [12, 34]. For σ ∈ C ∞ (SL(2; R) × R), a complete symbol for a 8DO compactly supported on SL(2; R) (σ (A, τ ) is asymptotically a sum of homogeneous terms in the frequency τ with bounded left invariant derivatives in A) the associated matrix element is σr Q(u, v)dV 2πOp(σ )v, u = SL(2;R)

for σr the symbol evaluated at τ = r and − 41 − r 2 the eigenvalue for u, v [31, 34]. S. Zelditch discovered that the essential properties of the microlocal lift are given by an exact differential equation [34].

Modulus of Continuity for 0 (m)\H Semi-Classical Limits

317

Lemma 2. For ϕ and ψ weight zero eigenfunctions of the Casimir operator with eigenvalue −(4r 2 + 1) then (H 2 + 4X 2 + 4irH )Q(ϕ, ψ) = 0. A Lie algebraic proof of the lemma is presented in [30]. We are interested in the large-eigenvalue limit of a sequence of automorphic eigenfunctions for a cofinite subgroup. As noted above for a sequence {ϕn } of L2 (\H) unitnorm eigenfunctions the sequence {Q(ϕ)} of Cc2 (SL(2; R)) (and thus Cc2 (\SL(2; R)) ) distributions is precompact. Provided the eigenvalues tend to infinity then from Lemma 2 [34] the limit of a convergent subsequence is a geodesic flow invariant distribution for Cc2 (SL(2; R)). Definition 3. A sequence of normalized L2 (\H) real-valued eigenfunctions {ϕn } with eigenvalues tending to infinity has semi-classical limit µϕ provided µϕ = lim Q(ϕn ) in n

the sense of Cc2 (SL(2; R)) distributions.

We now consider an alternate construction for the microlocal lift in terms of Fejér summation of the ladder of SL(2; R) raisings and lowerings, [30]. For an eigenfunction ϕ and a positive integer M introduce the sum M

QM (ϕ) = (2M + 1)−1 |

ϕ4m |2 .

m=−M

The basic properties of the semi-classical limit are in fact simple consequences of the Fejér summation and integration by parts. Proposition 4. Notation as above. Let {ϕn } be a sequence with semi-classical limit µϕ . The limit satisfies µϕ = lim lim QM (ϕn ), is a positive real measure on SL(2; R) and is M n 0 1 invariant). Let {(ϕ , ψ )} be a sequence of pairs of time-reversal invariant (right −1 n n 0 eigenfunctions, ϕn and ψn with common eigenvalue, eigenvalues tending to infinity, such that {Q(ϕn )}, {Q(ψn )} and {Q(ϕn , ψn )} converge in the sense of Cc2 (SL(2; R)) distributions. The limit lim Q(ϕn , ψn ) is a real time-reversal invariant measure on SL(2; R). n

Proof. A semi-classical limit is determined on the subspace Cc∞ of Cc2 ; furthermore the K expansions are convergent for a convergent sequence of distributions. First we show that the terms ϕϕ2m with m odd do not contribute to the limit. From (1) we have that u−2m = (−1)m (u) ¯ 2m + O(r −1 |u2m |) in the sense of distributions. Since ϕ is real we have that ϕϕ−2m = −ϕϕ2m + O(r −1 ). Now by a repeated application of (2) we have for m = 2q + 1 that 4iϕϕ2m = (−1)q r −1 E + ((ϕ2q )2 ) + O(r −1 ). The leading-term E + ((ϕ2q )2 ) is itself a bounded distribution; in consequence for m odd ϕϕ2m and ϕϕ−2m have magnitude O(r −1 ) and thus do not contribute to a limit. In particular the limits µϕ and µϕ+ψ have K expansions with non trivial terms only for weights congruent to zero modulo 4; the limits are time-reversal invariant. From (2) we have the additional relation ϕ2j +2 ψ2k = ϕ2j ψ2k−2 + O(r −1 ) in the sense of distributions. It follows for pairs of eigenfunctions that lim Q(ϕn , ψn ) = lim Q(ψn , ϕn ) and consequently that a n

n

limit is real. It further follows that lim QM (ϕn ) = lim

2M

n

from the above result on the K

n m=−2M expansion that lim lim QM (ϕn ) M n

is positive it follows that µϕ is a positive real measure.

(1 −

|m| 2M+1 )ϕn ϕn,4m

and

= µϕ . Finally since QM

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S. A. Wolpert

We consider further properties for the semi-classical limit of tuples of eigenfunctions. Consider a sequence {(ϕn , ψn )} of pairs of eigenfunctions, ϕn and ψn with common eigenvalue λn , such that for eigenvalues tending to infinity the microlocal lifts converge to measures on SL(2; R) µϕ = lim Q(ϕn ), n

µψ = lim Q(ψn ), n

µϕ±ψ = lim Q(ϕn ± ψn ) and µϕ,ψ = lim Q(ϕn , ψn ). n

n

Proposition 5. Notation as above. The measures satisfy 2|µϕ,ψ | ≤ µϕ + µψ and ( χ µϕ,ψ )2 ≤ χ µϕ χ µψ for each positive χ ∈ Cc (SL(2; R)). In particular µϕ,ψ is absolutely continuous with respect to µϕ and to µψ . Proof. The pair of inequalities ±2µϕ,ψ ≤ µϕ + µψ are a consequence of the positivity of the measures µϕ±ψ . The first assertion is now a consequence of the Jordan decomposition of µϕ,ψ as a difference of mutually singular positive measures [19]. For the second assertion consider a non negative test function χ and the quadratic polynomial χ (α 2 µϕ + 2αµϕ,ψ + µψ ) in the real parameter α. Since µαϕ+ψ is a positive measure for each α the second assertion now follows. The measures µ∗ are outer regular: for a compact Borel set S then µ∗ (S) = inf χ

χ µ∗ for χ ∈ Cc (SL(2; R)) with χ = 1 on S,

[19]. In particular for compact Borel sets we find (µϕ,ψ (S))2 ≤ µϕ (S)µψ (S) and thus that µϕ,ψ is absolutely continuous with respect to µϕ and µψ . Corollary 6. Notation as above. For a sequence of q-tuples of eigenfunctions {(ϕ1,n , . . . , ϕq,n )}, ϕj,n with common eigenvalues, and all pairs Q(ϕj,n , ϕk,n ) converq q gent for eigenvalues tending to infinity then µ@ ≤ q( µϕj ) for @ = ϕj . j =1

Proof. The positive measure µ@ is given as a sum µ@ =

j =1

q j =1

The result follows from the inequality 2|µϕ,ψ | ≤ µϕ + µψ .

µϕj + 2

1≤j 1. Moreover, we shall see that the instability can be of two different types: either the Kuramoto phase instability, or a “period-doubling” instability, corresponding to a real eigenvalue of λ becoming smaller than −1. This result was conjectured on the basis of numerical observations by Médéric Argentina and Pierre Coullet ([1]), and their observations, conjectures, and questions were the starting point of this work. The reader interested in these observations, in the physical interpretations and implications of these results, and in the nonlinear development of these instabilities is invited to consult the references [2] and [5]. Let us also mention that this generic instability result extends to the case where space isotropy is broken ([5]). 1.1. Statement of the results. We give ourselves and fix a C 1 -vector field f0 : Rd → Rd , d ≥ 2, and we make the following hypotheses (see Fig. 1): • f0 (0) = 0 (0 denotes the origin (0, . . . , 0) of Rd ); • Df0 (0) has a simple real eigenvalue b+ > 0; if d = 2, then the second eigenvalue is not larger than −b+ ; if d ≥ 3, then the real part of any other eigenvalue is strictly smaller than −b+ ; • one of the following statements holds: (a) the differential equation du dt = f0 (u) admits a solution t → u0 (t) which is homoclinic to the fixed point 0 (i.e. u0 (.) ≡ 0 and u0 (t) → 0 when t → ±∞); (b) the differential equation du dt = f0 (u) admits two solutions t → u0 (t) and t → u˜ 0 (t) (with distinct trajectories) which are homoclinic to the fixed point 0. Let us consider any C 1 -vector field f1 : Rd → Rd , with the following properties: • f1 (.) is close to f0 (.) in the C 1 -topology (this hypothesis will be formulated more precisely below); • the differential equation du dt = f1 (u) admits a periodic solution t → u1 (t) whose trajectory is, in case (a), close to the trajectory of u0 (.), and, in case (b), close to the union of the trajectories of u0 (.) and u˜ 0 (.) (again, this hypothesis will be formulated more precisely below); • if d = 2, then the periodic orbit u1 (.) is not linearly unstable. Here the hypothesis on the closeness of the trajectories holds in the sense of the Hausdorff distance between two sets (recall that this distance can be defined the following way: dist(A, B) = inf{δ > 0 A ⊂ Neighbδ (B) and B ⊂ Neighbδ (A)}). Remark. In the case d ≥ 3, these hypotheses (in particular the ones on Df0 (0)) imply that the periodic orbit u1 (.) is linearly stable; the same is true in the case d = 2 if the second eigenvalue of Df0 (0) is strictly smaller than −b+ . On the other hand, the hypotheses on Df0 (0) are almost necessary if we want u1 (.) not to be linearly unstable. Indeed, in the case d = 2, if the second eigenvalue of Df0 (0) was strictly larger than −b+ , then the hypotheses would imply that u1 (.) is linearly unstable; the same would generically be true in the case d ≥ 3 if Df0 (0) had an eigenvalue different from b+ with a real part strictly larger than −b+ .

328

E. Risler u1

case (a)

u1

case (b)

u0

u0 T0

E s (0)

T0

E s (0) T1 0

E u (0)

E u (0)

0

u0 0 T

1 T

Fig. 1.

Now let us define the coupling terms to be added to the two previous vectors fields. We give ourselves and fix a C 0 -map C0 : Rd × R → L(Rd ) and we consider any C 0 -map C1 : Rd × R → L(Rd ) close to C0 in the C 0 -topology (this hypothesis will be formulated more precisely below). Denote by L+ (Rd ) the subset of L(Rd ) consisting of linear maps having no eigenvalue with a strictly negative real part. We will suppose that the maps C0 and C1 take their values in L+ (Rd ). This hypothesis is natural, since, as λ ≤ 0, it excludes the existence of instabilities uniquely due to the coupling. However, the results are to a large extent independent of this hypothesis (which will be necessary only in dimension d = 2, and mainly for the phase stability results in case 2 of Theorem 2 and case 2 of Theorem 3 below). For λ ≤ 0, denote by λ the (linear) flow over one period of u1 of the differential equation du = Df1 (u1 (t)) + λC1 (u1 (t), λ) u, dt

(7)

and denote by ρ(λ ) the spectral radius of λ . Let || . . . ||C 1 denote a uniform C 1 -norm on C 1 (Rd , Rd ) and let || . . . ||C 0 denote a uniform C 0 -norm on C 0 (Rd × R, L(Rd )); let T0 , T1 , and, in case (b), T˜ 0 denote the respective trajectories of u0 (.), u1 (.), and u˜ 0 (.). Our result is the following. Theorem 1. Let f0 (.) and C0 (., .) be as above. Then, if a generic condition (which will be detailed below) on f0 (.) and C0 (., .) is satisfied, there exists ε0 > 0 (small) such that, for any f1 (.) and C1 (., .) as above, if ||f1 (.)−f0 (.)||C 1 ≤ ε0 and ||C1 (., .)−C0 (., .)||C 0 ≤ ε0 and if, in case (a), dist(T0 , T1 ) < ε0 , and in case (b), dist(T0 ∪T˜ 0 , T1 ) < ε0 , one can find λ < 0 (arbitrarily close to 0 if ε0 is small enough) such that ρ(λ ) > 1. We are going to be more precise. Let f0 (.), C0 (., .), f1 (.), and C1 (., .) be as above. Up to conjugating f1 (.) by a (small) translation of Rd , we will suppose that f1 (0) = 0. Fix δ0 > 0 small, let B0 = {x ∈ Rd

Spatial Unfoldings of Almost Homoclinic Periodic Orbits

329

||x|| ≤ δ0 }, and let W1s,loc (0) denote the local stable manifold of 0 for f1 (.), i.e. say the set of points of B0 whose forward trajectory by f1 (.) remains in B0 . According to the hypotheses (for ε0 sufficiently small), the set T1 ∩∂B0 contains, in case (a), exactly two points, and, in case (b), exactly four points; in dimension d = 2, this is due to an elementary plane topology argument, and in dimension d ≥ 3, this is due to the hypotheses on Df0 (0) (and related to the fact that u1 (.) is linearly attractive). In case (a) (resp. in case (b)), denote by ζ1 (resp. by ζ1 and ζ˜1 ) the point(s) of T1 ∩∂B0 as shown on Fig. 2.

T1

W1s,loc (0)

ζ1

T1

W1s,loc (0) ζ1

0

0

ζ1

case (a)

case (b)

Fig. 2. Definition of ζ1 and ζ˜1

Let µ = dist(ζ1 , W1s,loc (0))

and, in case (b),

µ˜ = dist(ζ˜1 , W1s,loc (0))

(these quantities can be considered as bifurcation parameters: they measure the proximity to the homoclinic orbit or to the homoclinic bifurcation). In the following (Sect. 2), we will show how to associate to each triplet (f0 , u0 , C0 ) as above an index σ (f0 , u0 , C0 ) in {−1, 0, 1}, which vanishes if, for each (t, λ), the map C0 (uh (t), λ) is positively proportional to IdRd , but which is generically different from 0 for a general C0 (., .), and whose sign governs the nature of the instability. With this index, we can formulate the following more precise results (for sake of clarity, we distinguish cases (a) and (b)). Theorem 2. Let f0 (.), u0 (.), and C0 be as above, in case (a). Then, if σ (f0 , u0 , C0 ) = 0, there exists ε0 > 0 (small) such that, for any f1 (.) as above, if ||f1 (.) − f0 (.)||C 1 ≤ ε0 , dist(T0 , T1 ) < ε0 , and ||C1 (.) − C0 (.)||C 0 ≤ ε0 , then, 1. if σ (f0 , u0 , C0 ) = 1, then for any λ ∈] − ε0 ; 0[, λ has an eigenvalue which is real and strictly larger than 1 (phase instability);

330

E. Risler

λ

λ

µ

0

0

µ stable

unstable “+1”

unstable “−1” case (a), 1

case (a), 2 Fig. 3. Illustration of Theorem 2 (case (a))

2. if σ (f0 , u0 , C0 ) = −1, then there are constants K > K > 0, depending only on f0 (.) and C0 , such that, for any λ ∈] − Kµ; 0[, ρ(λ ) ≤ 1 (no phase instability), and for any λ ∈] − ε0 ; −K µ[, λ has an eigenvalue which is real and strictly smaller than −1 (“period-doubling” instability). Theorem 3. Let f0 (.), u0 (.), u˜ 0 (.), and C0 be as above, in case (b). Then, if σ (f0 , u0 , C0 ) = 0 and σ (f0 , u˜ 0 , C0 ) = 0, there exists ε0 > 0 (small) such that, for any f1 (.) as above, if ||f1 (.) − f0 (.)||C 1 ≤ ε0 , dist(T0 ∪T˜0 , T1 ) < ε0 , and ||C1 (.) − C0 (.)||C 0 ≤ ε0 , then, 1. if σ (f0 , u0 , C0 ) = −1 and σ (f0 , u˜ 0 , C0 ) = −1, then for any λ ∈] − ε0 ; 0[, λ has an eigenvalue which is real and strictly larger than 1 (combination of two phase instabilities); 2. if σ (f0 , u0 , C0 ) = +1 and σ (f0 , u˜ 0 , C0 ) = +1, then there are constants K > K > 0, depending only on f0 (.) and C0 , such that, for any λ ∈] − K min(µ, µ); ˜ 0[, ρ(λ ) ≤ 1 (no phase instability), and for any λ ∈] − ε0 ; −K max(µ, µ)[, ˜ λ has an eigenvalue which is real and strictly larger than 1 (combination of two “perioddoubling” instabilities); 3. if σ (f0 , u0 , C0 ) and σ (f0 , u˜ 0 , C0 ) have opposite signs, then there is a constant K > 0 such that, for any λ ∈] − ε0 ; −K max(µ, µ)[, ˜ λ has an eigenvalue which is real and strictly smaller than −1 (combination of a phase and a “period-doubling” instability).

λ 0

λ

µ, µ 0

λ

µ, µ stable

0

unstable “+1” unstable “+1” case (b), 1

case (b), 2

Fig. 4. Illustration of Theorem 3 (case (b))

unstable “−1” case (b), 3

µ, µ


331

In case 2 of this last result, the instability is of the same nature as the period-doubling instability (it can be viewed as the composition of two period-doubling instabilities). Case 3 is a bit more involved, but occurs less frequently than cases 1 and 2. For instance, it never occurs when the two homoclinic orbits u0 (.) and u˜ 0 (.) are symmetric. 1.2. Examples. The hypotheses of Theorems 1, 2, and 3 cover essentially two kinds of situations: homoclinic bifurcations of attractive periodic orbits in one-parameter families of ordinary differential equations on one hand, and families of periodic orbits bounded by homoclinic orbits in two-dimensional conservative ordinary differential equations on the other hand (this second case corresponds to f1 = f0 ). Moreover, these hypotheses take into account cases where, because of the presence of a symmetry or of a conserved quantity, the limit of the periodic orbits consists of two (instead of one) homoclinic orbits. We now give some examples (for other examples and references, see [2]). 1. Consider the following nonlinear wave equation: utt + (ν + u)ut + u − u2 = #x u parametrized by ν ∈ R. This is the equation governing a chain of coupled second order oscillators in the potential V (u) = 21 u2 − 13 u3 , submitted to the nonlinear damping −(ν + u)ut . This equation can be rewritten v u 0 0 #x u + = , v t 1 0 #x v −(ν + u)v − u + u2 and thus can be viewed as a spatial extension of the ordinary differential equation v u = v t −(ν + u)v − u + u2 0 0 with respect to the “coupling” matrix (here the map C(. , .) is constant and equal 1 0 to this matrix). This family of differential equations appears in the universal unfolding of the Bogdanov–Takens bifurcation ([7]). Its dynamics displays the following features. For ν > 0, the fixed point (0, 0) is linearly stable. At ν = 0, it undergoes a supercritical Hopf bifurcation and becomes unstable for ν < 0. The bifurcation gives rise to an attractive periodic orbit around (0, 0) for ν < 0 close to 0. At a certain value ν = νc < 0 of the parameter, this attractive periodic orbit disappears through homoclinic bifurcation (see Fig. 5), the limiting orbit being homoclinic to the hyperbolic fixed point (1, 0). For ν < νc , forward orbits generically go to infinity. Theorem 1 claims that, for ν > νc , ν close to νc , the attractive periodic orbit is unstable with respect to inhomogeneous perturbations. More generally, a possible physical interpretation of our results is the following: for a spatially extended dynamical system, it is impossible to cross a potential barrier in a synchronous way. According to Theorem 2, it is possible to predict the nature of the instability. We use the definitions and notations of Subsect. 2.2. On one hand, the homoclinic orbit is backward oriented, thus σor = −1. On the other hand, we can see from the expression of C(., .) that c3,0 (t) will be negative for all times, which shows that Y− (.) > 0, that Y+ (.) < 0, and thus that Y− (.) − Y+ (.) > 0. Thus, σY = +1, and, according to

332

E. Risler V

u 0

1

ut

0

1

u

Fig. 5. Phase portrait when ν = νc

Theorem 2, the instability is a phase instability (for more details on the links between the expression of C(. , .) and the nature of the instability, see [10]). 2. Consider the following partial differential equation: utt + V (u) = uxx , where V (u) = − 21 u2 + 13 u3 + 41 u4 . It represents a chain of coupled conservative oscillators in the bistable potential V (.). It can be viewed as a spatial extension of an ordinary differential equation with respect to the same coupling matrix as above. The phase space of the differential equation is as follows. It is foliated by periodic orbits, bounded by the fixed points and by two orbits homoclinic to (0, 0) and having an energy 1 2 2 ut + V (u) equal to 0. According to Theorem 1, any periodic orbit having an energy E close enough to 0 is unstable with respect to inhomogeneous perturbations; moreover, it is phase unstable (case 1 of Theorem 2) if E < 0, and not phase unstable (but “perioddoubling-like” unstable, case 2 of Theorem 3) if E > 0. 3. We end up our series of examples with the celebrated sine-Gordon equation utt + sin u = uxx . The phase space of the corresponding ordinary differential equation on (R/2π Z) × R is foliated by periodic orbits, bounded by the fixed points and by two orbits homoclinic to (π, 0) and having an energy 21 u2t − cos u equal to 1. We can easily deal with the fact that the phase space is 2π -periodic on the horizontal variable. According to Theorem 1, any periodic orbit having an energy E close enough to 1 is unstable with respect to inhomogeneous perturbations; moreover, it is phase unstable (case 1 of Theorem 3) if E < 1, and period-doubling unstable (case 2 of Theorem 2) if E > 1.


333

V u

ut

u

Fig. 6.

1.3. Sketch of the proof and organization of the paper. Let us describe rapidly how the proof goes. To simplify, we suppose that we are in case (a) and that the dimension d equals 2. We shall take a small parameter δ > 0 and cut the trajectory of u1 (.) into two parts, as shown on the Fig 7. Consider the local frame (e1 (t), e2 (t)) = (f1 (u1 (t)), Rot π2 f1 (u1 (t))) along this trajectory. Denote by ψλ (resp. by φλ ) the flow of the differ-

φλ

e2 e1

δ -λ

0

δ

Fig. 7.

334

E. Risler

ential Eq. (6), expressed in this local frame, along the part of the trajectory which lies inside (resp. outside) the box of size δ around 0. The flow ψλ ◦ φλ is conjugated to λ , and we want to study its spectral radius. The differential equation du dt = Df1 (u1 (t))u, expressed in the local frame, takes the du ˆ form dt = M1 (t)u, where the first column of Mˆ 1 (t) vanishes; this shows that ψ0 and 1 ∗ . When the distances between f0 and f1 and between T0 and φ0 are of the form 0 ∗ T1 go to 0, the flow φ0 converges to a limit, while ψ0 becomes singular. Indeed, writing 1 η ψ0 = , we will see that η goes to +∞ (or to −∞ if the orbits have the converse 0 ζ orientation) while ζ remains bounded (if b− = −b+ ) or goes to 0 (if b− < −b+ ). More precisely, we will see that η is of the order of µ−1 . w x , we The flow φλ is a non-singular perturbation of φ0 . Writing φλ = φ0 + λ y z can see that the trace of ψλ ◦ φλ reads tr ψλ ◦ φλ = tr ψ0 ◦ φ0 + ληy + . . . . We will show that, when δ is small, y is large and has a definite sign (actually, when u1 (.) is oriented as on the figure above, the index σ (f0 , u0 , C0 ) will be equal to ±1 according to this sign). Thus, we can already see on this expression of the trace what we will actually prove: for |λη| ≥ 1 (which corresponds to |λ| being at least of the order of µ), this trace is large and its sign is governed by the sign of y (i.e. by the sign of σ (f0 , u0 , C0 )). This already proves the instability. For the case |λη| < 1, we will need slightly more precise estimates, either to prove the phase instability (if y is positive) or to prove some stability (for |λη| small, if y is negative). The proofs in dimension d = 2 and in dimension d ≥ 3 differ noticeably at this point: in case d = 2, we will simply estimate the determinant of ψλ ◦ φλ , while in case d ≥ 3, we will have to construct an invariant cone for this map (none of these two strategies seems to be convenient for the other case: in dimension d ≥ 3, estimates on the trace and the determinant are not sufficient to control the eigenvalues, while the construction of an invariant cone seems to be delicate in dimension 2 in case b− = −b+ ). The paper is organized as follows. Section 2 is devoted to some notations and to the definition of the index σ (., ., .). This definition is very simple when d = 2, and slightly more involved when d ≥ 3, thus we distinguish theses two cases (Sects. 2.2 and 2.3). The proof of the results in case (a) is given in Sect. 3. After a preliminary setup (Sect. 3.1), we again distinguish the cases d = 2 (Sect. 3.2) and d ≥ 3 (Sect. 3.3). Finally, we explain in Sect. 4 how to adapt the previous arguments in order to prove the results in case (b). Notations. For n ∈ N, we will denote by Bcan (Rn ) the canonical basis of Rn and by 31 , . . . , εn the vectors forming this canonical basis. We will denote by || . . . || the usual euclidean norm on Rn , by Mn (R) the space of n × n real matrices, and by ||| . . . ||| the usual norm on Mn (R). 2. Definition of the Index σ 2.1. Notations related to the local frames. Throughout the proofs, we will have to work in local frames along the solutions u0 (.) (or u˜ 0 (.)) and u1 (.). Here we introduce some notations related to these local frames.


335

For k ∈ {0, 1} and t ∈ R, write Mk (t) = Dfk (uk (t)) and e1,k (t) = fk (uk (t)). In dimension d = 2, write

e2,k (t) = Rot π2 e1,k (t).

In dimension d ≥ 3, the local frame is not canonical, but we will define vectors e2,k (t), . . . , ed,k (t), C 1 and periodic (of the same period as u1 (.)) with respect to t, such that the family (e1,k (t), . . . , ed,k (t)) defines for each t a basis of Rd . Then, • let Pk (t) denote the matrix whose columns are the coordinates of e1,k (t) and e2,k (t), −1 −1 k ˆ • let Mˆ k (t) = −Pk (t)−1 dP dt (t) + Pk (t) Mk (t)Pk (t) and Ck (t, λ) = Pk (t) C(uk (t), λ)Pk (t). The change of variables u = Pk (t)v transforms the differential equation du = (Mk (t) + λC(uk (t), λ))u dt

(8)

dv = (Mˆ k (t) + λCˆ k (t, λ))v. dt

(9)

into

The definition of e1,k (t) ensures that the first column of Mˆ k (t) vanishes. Let us write 0 ak (t) c (t) c2,k (t) Mˆ k (t) = and Cˆ k (t, 0) = 1,k , 0 bk (t) c3,k (t) c4,k (t) where c1,k (t) is a number, ak (t) and c2,k (t) 1 × (d − 1)-matrices, c3,k (t) is a (d − 1) × 1matrix, and bk (t) and c4,k (t) are (d − 1) × (d − 1)-matrices. 2.2. Definition of σ in dimension two. We suppose that the dimension d equals 2, and we give ourselves a vector field f0 (.) and a map C0 (., .) as in Subsect. 1.1. Up to a linear change of coordinates preserving the orientation, we can suppose that E u (0) and E s (0) (the unstable and stable spaces of Df0 (0)) are respectively equal to R × {0} and {0} × R. We will say that u0 (.) (or u˜ 0 (.)) is forward oriented or backward oriented according to the orientation of its trajectory in R2 (see Fig. 8). Remark that, in case (b), u0 (.) and u˜ 0 (.) necessarily have the same orientation. We are going to define the index σ(f0 , u0 ,C0 ) (in case (b), σ (f0 , u˜ 0 , C0 ) would be defined similarly). Write b 0 Df0 (0) = + . With the notations of the preceding paragraph, we have 0 b− b0 (t) → b− − b+ < 0 when t → −∞

and

b0 (t) → b+ − b− > 0 when t → +∞ (10)

336

E. Risler u0

u0

0

0

u0

u0

“forward oriented”

“backward oriented” Fig. 8.

(see assertion (11) below). Thus, the differential equation dY = b0 (t)Y + c3,0 (t), dt

t ∈ R,

has a unique solution Y+ (.) (resp. Y− (.)) which is bounded when t → +∞ (resp. when t → −∞). The difference Y− (.) − Y+ (.) is either identically 0, or does not vanish, and in this case its sign is constant. Let σor = +1 (resp. σor = −1) if u0 (.) is forward (resp. backward) oriented. Let σY = +1 (resp. σY = 0, σY = −1) if Y− (.) − Y+ (.) > 0 (resp. Y− (.) − Y+ (.) ≡ 0, Y− (.) − Y+ (.) < 0). Finally, let us define our index σ (f0 , u0 , C0 ) by σ (f0 , u0 , C0 ) = −σor σY . The condition Y− (.) − Y+ (.) = 0 is generic, except if the map C0 (., .) is identically proportional to the identity (in this case, we have c3,0 (.) ≡ 0, and thus Y− (.) ≡ Y+ (.) ≡ 0), and the condition σ (f0 , u0 , C0 ) = 0 is thus also generic. If C0 (., .) is constant and not proportional to the identity and if its two eigenvalues are either complex conjugated or equal, then one can check that c3,0 (.) ≡ 0 and that the sign of c3,0 (.) is constant, given by the “sense of rotation” of the flow t → exp(−tC0 ) (for more precisions on these “monotonic” matrices, see [10]); in this case, Y− (.) has the sign of c3,0 (.), and Y+ (.) has the opposite sign, and the condition Y− (.) − Y+ (.) = 0 (and σ (f0 , u0 , C0 ) = 0) is thus always fulfilled. Moreover, in this last case, the sign of σ (f0 , u0 , C0 ), and thus the nature of the instability, can be predicted geometrically, from the orientation (forward or backward) of the homoclinic orbit and the “sense of rotation” of C0 ([10]). We finish with a rapid computation which will justify the limits (10), and which will be used later. For k = 0 or 1, denote by θk (t) the angle between the vectors (1, 0) and M1,1 M1,2 . e1,k (t), and write Mk (t) = M2,1 M2,2 Claim. We have

ak (t) M1,2 + M2,1 . = Rot−2θk (t) M2,2 − M1,1 bk (t)

(11)


337

Indeed, we have (forgetting the indices k and the dependence with respect to t), dP 0 a 0 −1 ˆ − + MP . =M =P 1 b 1 dt Besides,

0 P = Rot π2 e1 1

Thus,

and thus

dP dt

0 = Rot π2 Me1 . 1

1 a −1 = P [M, Rot π2 ]P 0 b

and we have P

−1

M + M M − M 1,2 2,1 2,2 1,1 Rotθ [M, Rot π2 ]P = Rot −θ M2,2 − M1,1 −M1,2 − M2,1 M1,2 + M2,1 M2,2 − M1,1 , = Rot−2θ M2,2 − M1,1 −M1,2 − M2,1

which proves the claim. 2.3. Definition of σ in dimension higher than two. We suppose that d ≥ 3, and we give ourselves a vector field f0 (.) and a map C0 (., .) as in Subsect. 1.1. We are going to define the index σ (f0 , u0 , C0 ) (in case (b), σ (f0 , u˜ 0 , C0 ) would be defined similarly). Up to a linear change of coordinates, we can suppose that E u (0) and E s (0) (the unstable and stable spaces of Df0 (0)) are respectively equal to Vect(31 ) and {0} × Rd−1 , and that the first coordinate of u0 (t) is positive when t is large negative. b+ 0 , B− ∈ Md−1 (R). We can suppose that B− is diagonal by Write Df0 (0) = 0 B −

blocks, i.e. that it reads

 B1 0 B2   ..  . 0

  ,  Bs

each block Bj corresponding to an eigenvalue bj . We can suppose that the non-real eigenvalues of B− are bs +1 , . . . , bs , where 0 ≤ s ≤ s. For j ≥ s + 1, denote by ρj (resp. by θj ) the real part (resp. the imaginary part) of bj . We can suppose that, for j ≥ s + 1, Bj takes the form   ρj −θj ∗   θj ρ j     . .. .     ρj −θj  0 θj ρ j For t ∈ R denote by Rt the linear map of Rd whose restriction to the characteristic spaces corresponding to the eigenvalues b+ and bj , j ≤ s , is the identity, and whose

338

E. Risler

restriction to the characteristic space corresponding to any eigenvalue bj , j ≥ s + 1, reads   Rottθj 0 cos tθj − sin tθj   . . where Rot . =   tθj . sin tθj cos tθj 0 Rottθj The change of variables v = R−t u transforms the differential equation dv dt = g0 (v, t), where g0 (v, t) = R−t f0 (Rt v) +

du dt

= f0 (u) into

dR−t Rt v. dt

Write R = dRdt−t Rt ; this matrix does not depend on t and we have Dv g0 (0, t) = Df0 (0)+ R. Thus, Dv g0 (0, t) does not depend on t, and we can see that its eigenvalues are real (these eigenvalues are b+ , b1 , . . . , bs , ρs +1 , . . . , ρs ). Write v0 (t) = R−t u0 (t), t ∈ R. The following lemma is classical (see for instance [3]), and we shall omit its proof. Lemma 1. The quantity vector of Dv g0 (0, .).

v0 (t) ||v0 (t)||

has a limit when t → +∞, and this limit is an eigen-

Denote by w this eigenvector. It belongs to one of the characteristic spaces of Df0 (0), corresponding to an eigenvalue bj0 of Df0 (0). We know that ρj0 < −b+ . Remark. Generically, we have ρj0 ≥ Re bj , 1 ≤ j ≤ s, but we shall not need this in the following. Denote by 31 , . . . , 3d the canonical basis of Rd . Up to another change of coordinates, we can suppose that w = 32 , and that, if bj0 is real, then Df0 (0) reads   b+ 0 0  0 bj0 ∗  0 0 B˜ − with B˜ − ∈ Md−2 (R) (in this case, write E = Vect(32 )), and, if bj0 is non-real, then Df0 (0) reads   b+ 0 0   ρj0 −θj0 0 ∗    θj0 ρj0 0 0 B˜ − with B˜ − ∈ Md−3 (R) (in this case, write E = Vect(32 , 33 )). We can now define the moving frame (e1,0 (t) . . . , ed,0 (t)), t ∈ R. Let e1,0 (t) = f0 (u0 (t)), t ∈ R. This vector e1,0 (t) is almost parallel to 31 when t is large negative, and almost parallel to E when t is large positive. Denote by ;1 (resp. by ;E ) the orthogonal projection onto Vect(31 ) (resp. onto E) in Rd . There exists T > 0 (large) such that, for t < −T , ;1 (e1,0 (t)) = 0, and, for t > T , ;E (e1,0 (t)) = 0. For t < −T , let ej,0 (t) = ||e1,0 (t)||3j , 2 ≤ j ≤ d. For t > T , let e2,0 (t) = ||e1,0 (t)||31 , and, • if bj0 is real, then let ej,0 (t) = ||e1,0 (t)||3j , 3 ≤ j ≤ d;


339

• if bj0 is non-real, then let ej,0 (t) = ||e1,0 (t)||3j , 4 ≤ j ≤ d, and let e3,0 (t) =

||e1,0 (t)|| Rot− π2 ;E e1,0 (t) ||;E e1,0 (t)||

(here Rot− π2 denotes the rotation of angle − π2 in the subspace E equipped with the orientation of the basis (32 , 33 )). We can see that, for any t with |t| > T , the family (e1,0 (t) . . . , ed,0 (t)) defines a basis of Rd ; it depends smoothly on t, it is almost orthogonal for large |t|, it satisfies ||ej,0 (t)|| = ||e1,0 (t)||, 2 ≤ j ≤ d, and it has the direct orientation. It is thus possible to extend smoothly each map t → ej,0 (t) to the whole real line, in such a way that, for any t ∈ R, (e1,0 (t), . . . , ed,0 (t)) defines a (positively oriented) basis of Rd . We use the notations of Subsect. 2.1. We have, when t → −∞, ||e1,0 (t)||−1 P0 (t) → 0 IdRd , and, by calculus, P0 (t)−1 dP dt (t) → b+ IdRd . Thus, b0 (t) → B− − b+ IdRd−1 when t → −∞.

Suppose that bj0 0 where ς = −1 case

(12) 0 ,

ς is real. Then, when t → +∞, ||e1,0 (t)||−1 P0 (t) → 0 IdRd−2 1 0 , and, by calculus, P0 (t)−1 dP dt (t) → bj0 IdRd . We thus have in this 0

b0 (t) →

b+ − bj0 0 0 B˜ − − bj0 IdRd−2

when t → +∞.

(13)

Now suppose that bj0 is non-real. Then, when t → +∞, ||e1,0 (t)||−1 P0 (t) is close to be of the form    0 1 0 ∗ 0 ∗ 0   ,  ∗ 0 ∗  0 IdRd−3 and, by calculus, 

P0 (t)−1

dP0 (t) → ρj0 IdRd dt

  0 0 −θj0  0 0 0  0 . +  θj0 0 0  0 0

We thus have in this case   b+ − ρj0 0 0  when t → +∞ 0 0 ∗ b0 (t) →  ˜ 0 0 B− − ρj0 IdRd−3 (where the terms ∗ may depend on time). Consider the differential equation dY = b0 (t)Y + c3,0 (t), dt

Y ∈ Rd−1 ,

t ∈ R.

(14)

340

E. Risler

According to (12), this equation has a unique solution t → Y− (t) which is bounded when t → −∞. On the other hand, according to (13) and (14), this equation admits a (unique) affine hyperplane of solutions Y (of dimension d − 2) such that, for any Y (.) ∈ Y, the vector ebj0 t Y (t) is bounded when t → +∞. Let S denote the set of all solutions of the preceding differential equation, and denote by S+ (resp. by S− ) the set of solutions Y (.) such that the first coordinate of the vector ebj0 t Y (t) goes to +∞ (resp. to −∞) when t → +∞. We have S \ Y = S+ S− . Let us define our index σ (f0 , u0 , C0 ) by σ (f0 , u0 , C0 ) = + 1 (resp. σ (f0 , u0 , C0 ) = 0, σ (f0 , u0 , C0 ) = −1) if Y− ∈ S+ (resp. Y− ∈ Y, Y− ∈ S− ). The condition Y− ∈ S+ S− (and thus σ (f0 , u0 , C0 ) = 0) is again generic, except if the map C0 (., .) is identically proportional to the identity (in this case, we have c3,0 (.) ≡ 0, and thus Y− (.) ≡ 0 and Y ≡ {0} × Rd−2 ). 3. Proof in Case (a) 3.1. Setup for the proof. We give ourselves and fix a vector field f0 (.) and a map C0 (., .) as in Subsect. 1.1, in case (a). We adopt the conventions (choice of a convenient basis) and notations of Sect. 2 and we suppose that σ (f0 , u0 , C0 ) = 0. Let δ > 0 and ε0 > 0 be two constants to be chosen later. Throughout the proof, we will often have to make the hypotheses that δ or ε0 are small. The hypotheses on δ will always depend only on C0 and f0 (.) (although this will not be stated explicitly), and the ones on ε0 only on C0 , f0 (.), and δ. Thus the final convenient choices of δ and ε0 will only depend on C0 and f0 (.). Consider any vector field f1 (.) and any map C1 (., .) with the same hypotheses as in Theorems 1 and 2, in particular ||f1 (.) − f0 (.)||C 1 < ε0 ,

||C1 (.) − C0 (.)||C 0 < ε0 , and

dist(T0 , T1 ) < ε0 ,

and let ν = ||f1 (.) − f0 (.)||C 1 . Because of the continuous dependence of a local stable manifold on the vector field, we have µ → 0 when ε0 → 0 (recall, see Subsect. 1.1, that µ = dist(ζ1 , W1s,loc (0)). Let >0 be a small hypersurface crossing transversally T0 at u0 (0) (see Fig. 9). For ε0 sufficiently small, >0 ∩ T1 = ∅, and, up to reparametrizing t → u1 (t) we will suppose that u1 (0) ∈ >0 . Let > = {(x, y) x ∈ [−δ; δ] and y ∈ Rd−1 , ||y|| = δ}, > = {(x, y) x = ±δ and y ∈ Rd−1 , ||y|| ≤ δ}.


341

φ0,λ λ >0

δ

ξ0

ξ1

> > -λ

ξ1 ξ0

0 δ

Fig. 9.

For δ and ε0 sufficiently small, the intersection T0 ∩> (resp. T0 ∩> , T1 ∩>, T1 ∩> ) contains exactly one point (same reason as in Subsect. 1.1); denote it by ξ0 (resp. ξ0 , ξ1 , ξ1 ). Denote by T the period of u1 (.), define t0 , t0 , t1 , and t1 by: u0 (t0 ) = ξ0 ,

u0 (t0 ) = ξ0 ,

u1 (t1 ) = ξ1 ,

u1 (t1 ) = ξ1 ,

t1 < 0 < t1 < t1 + T ,

and write t1 = t1 + T . Let µ = dist(u1 (t1 ), W1s,loc (0))

and

ε=

µ δ

(W1s,loc (0) was defined in Subsect. 1.1, we suppose that δ < δ0 ). We remark that µ, µ , and ε are of the same order (they are equal up to multiplicative constants depending on the choice of δ). For λ ∈ R, denote by φλ (resp. ψλ ) the flow of the differential Eq. (9) with k = 1, between the times t = t1 and t = t1 (resp. between the times t = t1 and t = t1 ) (in the case d ≥ 3, the local frames will be defined in Subsect. 3.3). Denote by φ0,λ the flow of the differential Eq. (9)) with k = 0, between the times t = t0 and t = t0 . Write 1 α 1 η 1 α0 , ψ0 = , and φ0,0 = φ0 = 0 β 0 ζ 0 β0

342

E. Risler

(where α, η, and α0 are 1 × (d − 1)-matrices, and β, ζ , and β0 are (d − 1) × (d − 1)matrices) and write wλ xλ w0,λ x0,λ φλ = φ0 + λ and φ0,λ = φ0,0 + λ yλ zλ y0,λ z0,λ (with similar conventions). The quantities w0,λ , x0,λ , y0,λ , and z0,λ have limits w0,0 , x0,0 , y0,0 , and z0,0 when λ → 0; these limits can be obtained as values at time t0 of solutions of explicit differential equations involving a0 (.), b0 (.), and cj,0 (.), 1 ≤ j ≤ 4); the differential equation for y0,0 reads dy = b0 (t)y + c3,0 (t) dt

(15)

(it is the differential equation used in paragraphs 2.2 and 2.3 for the definition of σY ). According to classical results on continuous dependence with respect to parameters for solutions of ordinary differential equations, the quantities wλ , xλ , yλ , and zλ are arbitrarily close to w0,0 , x0,0 , y0,0 , and z0,0 if |λ| and ε0 are sufficiently small (depending on δ). For the remainder of the proof, we impose λ ∈] − ε0 ; 0[; moreover, we will suppose that ε0 is small enough (depending on δ) in order to have δ > ν, δ > |λ|, and δ > ε. Thus, in all the following estimates, the terms of the order of O(ν), O(λ) or O(ε) will be absorbed in the terms O(δ). 3.2. Estimates in dimension two. Estimates on ψλ . Denote by -λ the flow of the differential equation (8) with k = 1 between the times t1 and t1 . Write Q = P1 (t1 ) and Q = P1 (t1 ). We have ψλ = Q

−1

-λ Q.

A cone-invariance argument on the flow of (8) shows that -λ has two eigenvectors iλ and jλ of the form 1 (δ) iλ = and jλ = O 1 O(δ) (the terms O(ν) and O(λ) are absorbed in O(δ)). Denote by Rλ the matrix of M2 (R) whose columns are the coordinates of iλ and jλ (we have Rλ = IdR2 + O(δ)). The matrix Rλ−1 -λ Rλ is diagonal; denote it by Lλ and write Aλ 0 Lλ = . 0 aλ Let us estimate ψ0 . Write γ =

|b− | b+

≥ 1. As ε = µ /δ, we have

t1 − t1 = and thus

1 log ε −1 b+ + O(δ)

A0 = ε−1+O(δ) ! 1

and

a0 = εγ +O(δ) " 1


343

(these last estimates are not optimal but are sufficient for the moment; we will prove a more precise estimateon A0 in the following). (δ 2 ) (δ 2 ) + µ O O = , and thus We have u1 (t1 ) = σor δ σor δ 0 1 Q = σor δ|b− | + O(δ) . −1 0 δ δ We have u1 (t1 ) = , thus Q = δb+ (IdR2 + O(δ)) = O(δ 2 ) O(δ 2 ) + σor δε γ +O(δ) and 1 −1 Q = (Id 2 + O(δ)). δb+ R Finally, as ψ0 = Q

−1

R0 L0 R0−1 Q,

(16)

we get ζ = detψ0 = γ 2 (1 + O(δ))A0 a0 = γ 2 (1 + O(δ))ε γ −1+O(δ)

(17)

and, identifying in the expression (16) of ψ0 , we find η = σor γ (1 + O(δ))A0 = σor γ (1 + O(δ))ε −1+O(δ) . Now we estimate ψλ . Write qλ =

Aλ A0

and dλ =

A0 Aλ aλ

Lλ = qλ

A0 0 0 a0 + dλ

= qλ

(18)

− a0 ; then we have

0 0 L0 + 0 dλ

.

A cone-invariance argument shows that Rλ = R0 + O(λ), and we have

Aλ = A0 e(t1 −t1 ) O(λ) = A0 ε O(λ) ;

aλ = a0 e(t1 −t1 ) O(λ) = a0 ε O(λ) ; thus qλ = εO(λ) and dλ = a0 (ε O(λ) − 1). Now we have −1

ψλ = Q Rλ Lλ Rλ−1 Q 0 0 −1 (R0−1 Q)(Id + O(λ)) = qλ (Id + O(λ))(Q R0 ) L0 + 0 dλ = qλ (Id + O(λ))ψ0 (Id + O(λ)) + O(dλ ) and we obtain ψλ = qλ (ψ0 + S),

(19)

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E. Risler

where S = (Si,j )1≤i,j ≤2 satisfies Si,j = η O(λ) + O(dλ ) if (i, j ) = (2, 1); S2,1 = (1 + ζ ) O(λ) + η O(λ2 ) + O(dλ ). We remark that dλ = a0 (eO(λ) log ε − 1) = ε γ +O(δ) (log ε) O(λ) which shows that Si,j = η O(λ) if

(i, j ) = (2, 1)

and

S2,1 = (1 + ζ ) O(λ) + η O(λ2 ).

(20)

Estimates on the trace of ψλ ◦ φλ . Denote by Tλ the trace of ψλ ◦ φλ . We have T0 = 1 + ζβ and calculus yields Tλ = qλ T0 + λη(yλ + r(λ)) ,

(21)

where (forgetting the indices λ) r(λ) = η−1 (w + ζ z) + (λη)−1 S1,1 (1 + λw) + S1,2 λy + S2,1 (α + λx) + S2,2 (β + λz .

Lemma 2. The quantity β0 is bounded by a constant which does not depend on δ. Proof. We have

β0 = exp

t0

t0

b0 (s)ds.

Write u0 (t) = (x0 (t), y0 (t)). We have log |y0 (t)|−1 ∼ t|b− | when t → +∞ and log |x0 (t)|−1 ∼ |t|b+ when t → −∞. In particular, we have t0 ∼ |b− |−1 log δ −1

and

−1 t0 ∼ −b+ log δ −1

when

δ → 0.

As b0 (t) → ±(b+ − b− ) when t → ±∞, this shows that, if |b− | > b+ , then β0 → 0 when δ → 0, and this proves the lemma in this case. In the remaining case |b− | = b+ , we have to be slightly more precise. When t → +∞, we have x0 (t) = O(y0 (t)2 ) and thus, according to claim (11), b0 (t) = b− − b+ + O(y0 (t)). Similarly, when t → −∞, we have b0 (t) = b+ − b− + O(x0 (t)). Thus, β0 is equal, up to a multiplicative constant independent of δ, to the quantity e(b+ −b− )(t0 −|t0 |) . On the other hand, we have dy0 = b− y0 + O(y02 ) dt

when

t → +∞,

which shows that log |y0 (t)|−1 − t|b− | is bounded when t → +∞. Similarly, log |x0 (t)|−1 − |t|b+ is bounded when t → −∞, which shows that t0 − |t0 | is bounded independently of δ, and the lemma follows. $


345

According to estimates (17) on ζ , (18) on η, (20) on Si,j , and to the lemma above, r(λ) is bounded, for ε0 sufficiently small (depending on δ) by a constant which does not depend on δ. The quantity y0,0 is the value at time t0 of the solution of the differential Eq. (15), namely dY = b0 (t)Y + c3,0 (t) dt with initial condition Y = 0 at time t = t0 . This differential equation is precisely the one governing the functions Y− (.) and Y+ (.) of Subsect. 2.2. We know that σY = ±1, and, as b0 (t) → b+ − b− when t → +∞, we see that σY Y− (t) → +∞ when t → +∞. We thus have σY y0,0 → +∞

when

δ → 0.

Before we can conclude, we need a more precise estimate on η. Lemma 3. We have

A0 = (1 + O(δ))ε −1 .

Proof. We could use Hartman’s C 1 linearization theorem ([8]) but we will give a more elementary proof. There is a smooth map g1 , defined on a neighborhood of 0 in R2 , with values in R2 , satisfying g1 (0) = 0, and mapping W1u,loc (0) (resp. W1s,loc (0)) to the x-axis (resp. to the y-axis). We have Dg1 (0) = IdR2 + O(ν). Denote by fˆ1 the vector field obtained by conjugating f1 by g1 (i.e. fˆ1 (.) = Dg1 (g1−1 (.))f1 (g1−1 (.))), denote by fˆ1,1 the first component of fˆ1 , and let b+,1 = b+ + O(ν) and b−,1 = b− + O(ν) denote the two eigenvalues of Df1 (0). Then we have fˆ1,1 (x, y) = x b+,1 + O(||(x, y)||) . (22) Write uˆ 1 (t) = g1 (u1 (t)), t ∈ R, and denote by xˆ1 (t) the first coordinate of uˆ 1 (t). We have xˆ1 (t1 ) = µ (1 + O(δ)),

xˆ1 (t1 ) = δ(1 + O(δ)),

(23)

and, according to (22), d xˆ1 = xˆ1 (t)(b+,1 + O(||uˆ 1 (t)||)), dt

t ∈ [t1 ; t1 ].

(24)

On the other hand, the dynamics close to 0 shows that

t1

t1

O(||uˆ 1 (t)||)dt = O(δ).

(25)

Thus, we deduce from (23) and (24) that

eb+,1 (t1 −t1 ) = (1 + O(δ))ε −1 .

(26)

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E. Risler

Denote by Aˆ 0 the largest eigenvalue of the flow of the differential equation du = D fˆ1 (uˆ 1 (t))u dt

(27)

between the times t = t1 and t = t1 , and denote by v the corresponding eigenvector (with the normalization constraint that the first coordinate of v is equal to 1). We have Aˆ 0 = (1 + O(δ))A0 . Let v(t) denote the solution of the differential equation (27) with initial condition v at time t = t1 . Write v(t) = (v1 (t), v2 (t)). Then v1 (t1 ) = Aˆ 0 . A cone-invariance argument shows that, for any t ∈ [t1 ; t1 ], we have v2 (t)/v1 (t) = O(δ). Thus, according to (27), we have dv1 = v1 (t) b+,1 ) + O(||uˆ 1 (t)||) . dt The lemma thus follows from (25) and (26). $ According to this lemma and to estimate (18) on η, we have η = σor γ (1 + O(δ))ε −1 .

(28)

End of the proof. To conclude, we will distinguish two cases. (i) |λ| ≥ ε. In this case, write ε = s|λ|, 0 < s ≤ 1. We have 1 < T0 ≤ 2. Thus, according to (28), the formula (21) for Tλ yields Tλ = qλ ληyλ (1 + . . . ) = (−σY σor )γ s −1+O(λ) |yλ |(1 + · · · ), where the “ . . . ” denote terms which are arbitrarily small if δ is sufficiently small and ε0 is sufficiently small (depending on δ). Thus, for δ sufficiently small and ε0 sufficiently small (depending on δ), the quantity (−σY σor )Tλ is arbitrarily large, in particular larger than 2. On the other hand, we know, as the trace of C1 (., .) is nonnegative (according to the hypothesis that C1 (., .) ∈ L+ (Rd ), see §1.1), that the determinant of ψλ ◦ φλ is not larger than 1. Thus, (−σY σor )Tλ > 2 implies that ψλ ◦ φλ has an eigenvalue which is real and strictly larger than one in modulus, its sign being the sign of −σY σor . This proves the instability in case |λ| ≥ ε; in particular, this proves the instability in case 2 of Theorem 2 (i.e. when −σY σor = −1); indeed, as we already mentioned, the quantity ε µ is bounded from above by a constant (which depends on the choice of δ) which is convenient for the choice of the constant K appearing in the theorem. (ii) |λ| < ε. In this case, write |λ| = tε, 0 < t < 1. Write Tλ = T0 + tT λ . According to (28), we have T λ = (−σY σor )γ |yλ |(1 + . . . ). In particular, T λ is arbitrarily large, and has the sign of (−σY σor ), if δ is sufficiently small and ε0 is sufficiently small (depending on δ). Denote by Dλ the determinant of ψλ ◦ φλ . We have det ψλ = det Q and

−1

(Aλ aλ ) det Q = εO(λ) det ψ0

det φλ = det φ0 + O(λ) = (1 + O(λ)) det φ0


347

(be careful that in this last expression, the term O(λ) depends on δ !); thus Dλ = εO(λ) (1 + O(λ))D0 . Write Dλ = D0 + tDλ . As D0 ≤ 1, we see that Dλ is arbitrarily small if ε0 is sufficiently small (depending on δ). Write #λ = Tλ2 − 4Dλ and #λ = #0 + t#λ . We have #λ = 2T0 T λ + tT λ − 4Dλ . 2

If #λ ≥ 0, denote by mλ the largest eigenvalue of ψλ ◦ φλ . We have m0 = 1 and 1 tT λ + #0 + t#λ − #0 . mλ = 1 + 2 Now we can conclude. We know that #0 ≥ 0. If −σor σY = 1, we see that #λ > 0 (thus #λ > 0) and mλ > 1. This proves the instability result in case 1 of Theorem 2. If on the other hand −σor σY = −1, then we see that, for t sufficiently small (depending on δ), #λ < 0, and, if #λ ≥ 0, then the two eigenvalues of ψλ ◦ φλ are strictly between 0 and 1. Finally, if #λ < 0, then we know that Dλ ≤ 1 (according to the hypothesis that C1 (., .) ∈ L+ (Rd ), the trace of C1 (., .) is nonnegative) and the spectral radius of ψλ ◦ φλ is thus not larger than 1. This proves the stability result in case 2 of Theorem 2 (the value of t “sufficiently small” provides a convenient choice for the constant K). The proof in dimension 2 of Theorem 2 (and thus of Theorem 1 in case (a)) is now complete. $ 3.3. Estimates in dimension higher than two. For t ∈ [t1 ; t1 ], let ej,1 (t) = ej,0 (t), j = 2, . . . , d (the vectors ej,0 (t) were defined in Subsect. 2.3). If ε0 is sufficiently small, then, for any t ∈ [t1 ; t1 ], the family (e1,1 (t), . . . , ed,1 (t)) defines a basis of Rd . This enables to define P1 (t), Mˆ 1 (t), and Cˆ01 (t, λ) for t ∈ [t1 ; t1 ] as in Subsect. 2.1. We can thus define φ0,λ and φλ as in Subsect. 3.1. To define ψλ , we do not have to define explicitly the local frame between t = t1 and t = t1 ; indeed, ψλ actually depends only on the local frame at t = t1 and t = t1 . Write Q = P1 (t1 ) and Q = P1 (t1 ), and denote by -λ the flow of the differential Eq. (8)) between the times t1 and t1 . We can define ψλ by: ψλ = Q

−1

-λ Q.

Estimates on ψλ . We suppose, as in the case d = 2, that ε0 is sufficiently small (depending on δ) to have δ > ν, δ > |λ|, and δ > ε, so that the terms O(µ), O(λ), and O(ε) are absorbed by terms O(δ). A cone-invariance argument shows that -λ has two invariant subspaces Iλ and Jλ , with dim Iλ = 1 and dim Jλ = d − 1. The subspace Iλ (resp. Jλ ) is almost parallel to 31 (resp. to {0} × Rd−1 ). Denote by ;I (resp. by ;J ) the projector on Iλ along {0} × Rd−1 (resp. the projector on Jλ along Vect(31 )), and write 31,λ = ;I 31 and 3j,λ = ;J 3j , j = 2, . . . , d. These vectors define a basis of Rd , and we have 3j,λ = 3j + O(δ),

j = 1, . . . , d.

348

E. Risler

Denote by Rλ the matrix of Md (R) whose columns are the coordinates of the vectors 3j,λ , j = 1, . . . , d, and write Lλ = Rλ−1 -λ Rλ . The matrix Lλ reads Aλ 0 0 aλ with aλ ∈ Md−1 (R). Let us estimate ψ0 . Fix a real number b− < 0 satisfying maxj =1,...,s Re bj < b− < −b+ and let γ = |bb−+ | > 1. We have (as in the case d = 2) t1 − t1 = and thus

1 log ε −1 , b+ + O(δ)

A0 = ε−1+O(δ) ! 1,

and, for δ sufficiently small (according to the margin between maxj =1,...,s Re bj and b− ), |||a0 ||| < εγ " 1. Write η = (η1 , . . . , ηd−1 ). According to the estimates of Subsect. 2.3 on P0 (t), |t| > T , and computing ψ0 = Q −1 R0 L0 R0−1 Q, we get η1 =

|bj0 | A0 (1 + O(δ)) b+

and

ηj = A0 O(δ),

j = 2, . . . , d − 1.

(29)

Lemma 4. For δ sufficiently small and ε0 sufficiently small (depending on δ), we have |||ζ ||| < εγ −1 . Proof. For t ∈ [t1 ; t1 ], write u1 (t) = (x1 (t), y1 (t)), x1 (t) ∈ R, y1 (t) ∈ Rd−1 . We have d x1 b+ 0 x1 = + O(δ) 0 B y y1 1 − dt which shows that, for δ sufficiently small, there exists a unique time t1 ∈]t1 ; t1 [ such that x1 (t1 ) = ||y1 (t1 )||. For t ∈ [t1 ; t1 ], write e1,1 (t) = f1 (u1 (t)) and ej,1 (t) = 3j , 2 ≤ j ≤ d. For δ sufficiently small, these vectors define, for any t ∈ [t1 ; t1 ], a basis of Rd . Let us define the matrices P1 (t) and Mˆ 1 (t) as in Subsect. 2.1. Let Q = P1 (t1 ), and let -(1) denote the flow of the differential Eq. (8)) between the times t1 and t1 . Write ψ(1) = Q −1 -(1) Q and let ψ(2) denote the flow of the differential ˆ equation du dt = M1 (t)u between the times t1 and t1 . We have ψ0 = ψ(2) ◦ ψ(1) and we can write 1 η(1) 1 η(2) ψ(1) = and ψ(2) = , 0 ζ(1) 0 ζ(2) where ζ(1) and ζ(2) belong to Md−1 (R). Then we have ζ = ζ(2) ◦ ζ(1) .


349

We have |||Q||| = O(δ) and |||Q −1 ||| = ||e1,1 (t1 )||−1 O(1). We have t1 − t1
b+ ) and we leave it to the reader. The quantity y0,0 is the value at time t1 of the solution of the differential Eq. (15), namely dY = b0 (t)Y + c3,0 (t) dt


351

with initial condition Y = 0 at time t = t1 . Thus, we see from the definition of σ (f0 , u0 , C0 ) that σ (f0 , u0 , C0 )y0,0,1 → +∞ when δ → 0, and from this differential equation that the ratio y0,0,j /y0,0,1 goes to 0 when δ → 0. Thus, for δ sufficiently small (depending on c) and for ε0 sufficiently small (depending on δ), we have σ (f0 , u0 , C0 )yλ,1 > 0 and χ = qλ (1 + λη1 yλ,1 (1 + . . . ))

(33)

where “ . . . ” is small. In the following, σ (f0 , u0 , C0 ) will simply be denoted by σ . Let us consider ξ . We have λ−1 qλ−1 ξ = λ−1 S2,1 (1 + λwλ ) + S2,2 yλ + S2,1 (α + λxλ ) + S2,2 (β + λzλ ) ϕ + . . . , where “. . . ” denotes terms which are arbitrarily small if ε0 is sufficiently small (depending on δ and c). We thus have, according to (31), |λ|−1 qλ−1 ||ξ || = O(1) + yλ,1 η1 O(λ) + cη1 O(λ).

(34)

As in the case d = 2, we have the following more precise estimate on A0 : A0 = (1 + O(δ))ε −1 (the proof of this estimate is similar to that of Lemma 3). According to (29), this yields |bj0 | (1 + O(δ))ε −1 . (35) η1 = b+ To conclude, we have, as in the case d = 2, to distinguish two cases. (i) |λ| ≥ ε. In this case, write ε = s|λ|, 0 < s ≤ 1. We deduce from (33) and (35) that |bj | qλ−1 χ = −σ 0 s −1 |yλ,1 |(1 + . . . ) (36) b+ and thus |bj | χ = −σ 0 s −1+O(λ) |yλ,1 |(1 + . . . ), (37) b+ where the terms “ . . . ” are small. Thus, if δ is sufficiently small and c is sufficiently large, we see from (34) and (36) that |λ|−1 ||ξ || < c|χ |. This shows the existence of an eigendirection in the cone C for ψλ ◦ φλ , the corresponding eigenvalue being real. According to (37), the modulus of this eigenvalue is strictly larger than 1, and its sign is the sign of yλ,1 , i.e. the sign of −σ . In particular, this proves the instability in case 2 of Theorem 2. (ii) |λ| < ε. In this case, write |λ| = tε, 0 < t < 1. We see from the expression (32) that |bj | χ = 1 − tσ 0 |yλ,1 |(1 + . . . ), (38) b+ where “ . . . ” are small, and from (34) that |λ|−1 ||ξ || = O(1) + t (yλ,1 O(1) + c O(1)).

(39)

Let us suppose that −σ = 1 (we are proving the instability in case 1 of Theorem 2). Then, we see that χ > 1 and that, for δ sufficiently small and c sufficiently large, |λ|−1 ||ξ || < c|χ |. This shows the existence of an eigendirection in the cone C for ψλ ◦ φλ , the corresponding eigenvalue being real and strictly larger than 1. This finishes the proof of the instability results when d ≥ 3.

352

E. Risler

A stability result. Here we suppose that −σ = 1 and that |λ| < ε, and we still write |λ| = tε, 0 < t < 1. It remains to prove that, for t sufficiently small, the eigenvalues of ψλ ◦ φλ are not larger than 1 in modulus. We see from (38) and (39) that, for t sufficiently small (depending on δ) and c sufficiently large, the cone C is still invariant by ψλ ◦ φλ . Thus, this linear map admits two invariant subspaces E1 and E2 , with Rd = E1 ⊕ E2 , dim E1 = 1, dim E2 = d − 1, E1 ⊂ C , and E2 ⊂ Rd \ C . The eigenvalue corresponding to the eigenspace E1 is, according to (38) (and for t sufficiently small) 0 and 1. −1between λ xˆ , xˆ ∈ R, ϕˆ ∈ Rd−1 , and suppose that Let v be any vector of E2 . Write v = ϕˆ ||ϕ|| ˆ = 1. Then, as v ∈ / C , we have |x| ˆ ≤ c−1 . Write −1 χˆ λ xˆ . = ψ ◦ φ λ λ ϕˆ ξˆ We have ξˆ = ζyλ + λ−1 S2,1 (1 + λwλ ) + S2,2 yλ xˆ + (S2,2 (β + λzλ ))ϕˆ + . . . , where “. . . ” denotes terms which are arbitrarily small if ε0 is sufficiently small. We thus have, for δ sufficiently small (depending on c), ||ξˆ || ≤ (1 + tyλ,1 )c−1 O(1). In particular, for t sufficiently small (depending on δ) and c sufficiently large, we have ||ξˆ || < 1, which shows that all the eigenvalues of (ψλ ◦ φλ )|E2 are strictly smaller than 1 in modulus. This proves the desired stability result, and finishes the proof of Theorems 1 and 2 in dimension d ≥ 3. $ Remark. This method of construction of an invariant cone works all the same in dimension d = 2, providing that b− < −b+ . Thus, under the hypotheses of Subsect. 3.2, if b− < −b+ , then we can say that the unstable eigendirection of ψλ ◦ φλ (which was proved to exist via estimation of the trace and determinant) is actually close to the hori zontal direction (it belongs to a cone C = {(x, y) ∈ R2 |y| < c|λ| |x|}, for c sufficiently large). 4. Proof in Case (b) 4.1. Setup for the proof. We give ourselves and fix f0 (.), C0 (., .) as in Subsect. 1.1, in case (b), and we suppose that σ (f0 , u0 , C0 ) = 0 and σ (f0 , u˜ 0 , C0 ) = 0. 10). We introduce δ, ε0 , f1 (.), C1 (., .), and ν as in Subsect. 3.1 (see Fig. b 0 Up to a linear change of coordinates, we suppose that Df0 (0) reads + . Let 0 B− >0 , the parametrization of u0 (.), >, and > be as in Subsect. 3.1. For δ and ε0 sufficiently small, the intersection T0 ∩> (resp. T0 ∩> , T˜ 0 ∩>, T˜ 0 ∩> ) contains exactly one point; denote it by ξ0 (resp. ξ0 , ξ˜0 , ξ˜0 ); moreover, the intersection T1 ∩> (resp. T1 ∩> ) contains exactly two points (see Subsect. 1.1); denote them by ξ1 , ξ˜1 (resp. by ξ1 , ξ˜1 ), in such a way that ξ1 ' ξ0 , ξ1 ' ξ0 , ξ˜1 ' ξ˜0 , ξ˜1 ' ξ˜0 .


353

φ0,λ

ξ1

> ξ0

ξ0

ξ1

0 λ -

>

ξ0 ξ1

>

λ φ

>

-λ

ξ1

φλ

0,λ φ

Fig. 10.

Define t0 , t0 , t˜0 , and t˜0 by: u0 (t0 ) = ξ0 ,

u0 (t0 ) = ξ0 ,

u˜ 0 (t˜0 ) = ξ˜0 ,

and

u˜ 0 (t˜0 ) = ξ˜0 .

Denote by T the period of u1 (.), define t1 , t1 , t˜1 , and t˜1 by: u1 (t1 ) = ξ1 , u˜ 1 (t˜1 ) = ξ˜1 ,

u1 (t1 ) = ξ1 , t1

u˜ 1 (t˜1 ) = ξ˜1 ,

< 0 < t1 < t˜1 < t˜1 < t1 + T ,

and write t1 = t1 + T . Define µ , ε as in Subsect. 3.1, and define µ˜ , ε˜ similarly. Define φλ and φ0,λ as in Subsect. 3.1, and define φ˜ λ and φ˜ 0,λ similarly. Let ψλ (resp. ψ˜ λ ) denote the flow of the differential Eq. (9) with k = 1, between the times t = t1 and t = t˜1 (resp. between the times t = t˜1 and t = t1 ). We adopt the same notations as in in Subsect. 3.1 for φλ , ψλ , φ0,λ , and similar notations (with a tilde) for φ˜ λ , ψ˜ λ , φ˜ 0,λ . The map ψ˜ λ ◦ φ˜ λ ◦ ψλ ◦ φλ is conjugated to λ , and our aim is to study its spectral radius. 4.2. Estimates in dimension two. Estimates on ψλ are the same as in Subsect. 3.2 (in particular estimates (17) on ζ and (20) on Si,j ), and similar estimates hold for ψ˜ λ . Estimates on β0 and A0 are the same as in Subsect. 3.2 (Lemmas 2 and 3) and similar estimates hold for β˜0 and A˜ 0 . We deduce from the estimates on A0 and A˜ 0 that η = −σor γ (1 + O(δ))ε −1

and

η˜ = −σor γ (1 + O(δ))˜ε −1

(40)

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E. Risler

(these estimates are similar to estimate (28) of Subsect. 3.2 on η, except that σor is replaced by −σor ). Denote by Tλ the trace of ψ˜ λ ◦ φ˜ λ ◦ ψλ ◦ φλ . We have ˜ T0 = 1 + ζ˜ βζβ. According to the expression of ψλ ◦ φλ (see Subsect. 3.3) and to estimates on η, ζ , and Si,j of Subsect. 3.2, we have 1 + ληyλ (1 + . . .) η(β + . . . ) −1 qλ ψλ ◦ φλ = , λyλ ζ (1 + . . . ) + λη O(1) + λ O(1) ζβ + λη O(1) where the “O(1)” denote quantities which, for ε0 sufficiently small (depending on δ), are bounded independently of δ, and the “. . . ” denote quantities which are arbitrarily small if δ is sufficiently small and ε0 is sufficiently small (depending on δ). A similar expression holds for q˜λ−1 ψ˜ λ ◦ φ˜ λ . We thus have (qλ q˜λ )−1 Tλ = T0 + ληyλ (1 + . . . ) + λη˜ y˜λ (1 + . . . ) + ληληy ˜ λ y˜λ (1 + . . . ) ˜ O(1), ˜ β˜ + . . . )λyλ ζ + ληζ˜ β˜ O(1) + ληζβ + η(β + . . . )λy˜λ ζ˜ + η( where the “. . . ” denote quantities which are arbitrarily small if δ is sufficiently small and ε0 is sufficiently small (depending on δ). Lemma 6. We have

a0 = (1 + O(δ))ε γ .

We omit the proof which is very similar to that of Lemma 3. According to this lemma and to estimate (17) on ζ , we have ζ = γ 2 (1 + O(δ))ε γ −1 = O(1). As a consequence, in the above expression of (qλ q˜λ )−1 Tλ , the last two terms can be removed. Now, we once again distinguish several cases. (i) max(ε, ε˜ ) ≤ |λ|. In this case, write ε = s|λ|, 0 < s ≤ 1, and ε˜ = s˜ |λ|, 0 < s˜ ≤ 1. For δ sufficiently small, the dominant term in Tλ reads, according to (40), (qλ q˜λ )ληληy ˜ λ y˜λ = γ 2 (1 + O(δ))|λ|O(λ) s −1+O(λ) s˜ −1+O(λ) yλ y˜λ . If δ is small, this term is large, thus Tλ is large and has the sign of yλ y˜λ ; this proves the desired instability (in particular, this proves the instability in cases 2 and 3 of Theorem 3). (ii) min(ε, ε˜ ) ≤ |λ| < max(ε, ε˜ ). This situation has to be considered only in case 1 of Theorem 3, called “case (b),1”, namely when σ (f0 , u0 , C0 ) = σ (f0 , u˜ 0 , C0 ) = −1. In this case, σor yλ < 0 and σor y˜λ < 0, and we can see that all the terms in the above expression of (qλ q˜λ )−1 Tλ are positive. Suppose for instance that ε ≤ |λ| < ε˜ and write ε = s|λ|, 0 < s ≤ 1, and ε˜ = s˜ |λ|, 1 < s˜ . Then the term ληyλ (1 + . . . ) is large, and, as the other terms are positive, we find, according to (40), Tλ ≥ q˜λ (qλ ληyλ )(1 + . . . ) = ε˜ O(λ) γ (1 + O(δ))|λ|O(λ) s −1+O(λ) |yλ | (1 + . . . ).


355

As |λ| < ε˜ , we have ε˜ O(λ) = ε˜ O(˜ε) ' 1, thus Tλ is large positive, which proves the desired instability. (iii) |λ| < min(ε, ε˜ ). In this case, write |λ| = tε, 0 < t < 1, and |λ| = t˜ε˜ , 0 < t˜ < 1. It remains to prove the instability in case 1 of Theorem 3 and the stability result in case 2 of Theorem 3. In these two cases, σ (f0 , u0 , C0 ) and σ (f0 , u˜ 0 , C0 ) have the same sign, and, equivalently, for δ sufficiently small, yλ and y˜λ have the same sign. Write Tλ = T0 + tT λ + t˜T˜λ + t t˜T λ . We have qλ q˜λ = 1 + t O(ε) log ε + t˜ O(˜ε ) log ε˜ , and thus T λ = σor γ yλ (1 + . . . ) + ζ˜ y˜λ (1 + . . . ) , T˜λ = σor γ y˜λ (1 + . . . ) + ζyλ (1 + . . . ) ,

T λ = γ 2 yλ y˜λ (1 + . . . ), where the terms “ . . . ” are small. As yλ and y˜λ have the same sign, we see that T λ , T˜λ , and T λ are arbitrarily large in modulus if δ is sufficiently small. Denote by Dλ the determinant of ψ˜ λ ◦ φ˜ λ ◦ ψλ ◦ φλ , and write Dλ = D0 + tDλ + t˜D˜ λ . Proceeding as in Subsect. 3.2, we see that Dλ and D˜ λ are arbitrarily small if ε0 is sufficiently small (depending on δ). Write #λ = Tλ2 − 4Dλ , and, as in Subsect. 3.2, if #λ ≥ 0, let mλ = 1 +

1 Tλ − T 0 + #λ − #0 . 2

Let us conclude. If σor yλ > 0 and σor y˜λ > 0, then we see that T λ , T˜λ , and T λ are all large positive, thus Tλ > T0 and #λ > #0 ≥ 0, and finally mλ > 1. This finishes the proof of the instability result in case 1 of Theorem 3. If on the other hand σor yλ < 0 and σor y˜λ < 0, then we see that, for t and t˜ sufficiently small (depending on δ), the term t t˜T λ is dominated by tT λ + t˜T˜λ , which is negative. Thus we see Tλ < T0 and #λ < #0 , and thus that, if #λ ≥ 0, then mλ < 1. This proves the stability result in case 2 of Theorem 3. The proof in dimension 2 of Theorem 3 (and thus of Theorem 1) is complete. $ 4.3. Estimates in dimension higher than two. Estimates on ψλ (in particular on ζ , si,j , A0 ) are the same as in Subsect. 3.3, and similar estimates hold for ψ˜ λ . We deduce from the estimates on A0 and A˜ 0 that η1 = −

|bj0 | (1 + O(δ))ε −1 b+

and

η˜ 1 = −

|bj0 | (1 + O(δ))˜ε −1 . b+

As in Subsect. 3.3, let C = {(x, y) ∈ R × Rd−1 ||y|| < c|λ| |x|}, where c is a large constant to be chosen. Let us denote σ (f0 , u0 , C0 ) by σ and σ (f0 , u˜ 0 , C0 ) by σ˜ . Then, proceeding as in Subsect. 3.3, we obtain that, if c is sufficiently large, δ sufficiently small (depending on c), and ε0 sufficiently small (depending on δ), in the three following cases: (i) |λ| ≥ max(ε, ε˜ ), (ii) |λ| < max(ε, ε˜ ) and σ = 1 and σ˜ = 1, (iii) |λ| < τ min(ε, ε˜ ), where τ is a small constant (depending on δ), and σ = −1 and σ˜ = −1,

356

E. Risler

the cone C is invariant by ψλ ◦ φλ and by ψ˜ λ ◦ φ˜ λ . Thus, it is also invariant by the composition ψ˜ λ ◦ φ˜ λ ◦ψλ ◦φλ , which shows the existence of an eigendirection in the cone C for this map, the corresponding eigenvalue being real. Proceeding as in Subsect. 3.3, we obtain that, in cases (i) and (ii) above, this eigenvalue is strictly larger than 1 in modulus, and has the sign of σ σ˜ (this proves the instability results); in case (iii), if τ is small enough, this eigenvalue belongs to ]0; 1[, and we can show as in Subsect. 3.3 that the other eigenvalues are smaller than 1 in modulus (this proves the stability result). This finishes the proof of Theorem 3. $ Acknowledgements. I am grateful to Médéric Argentina and Pierre Coullet, who introduced me to spatially extended differential equations, and who conjectured, on the basis of numerical observations ([1]), the results established in this paper. This work owes much to their support through numerous discussions (in particular, Pierre Coullet helped me in considerably simplifying the proofs).

References 1. Argentina, M.: Dynamique des systèmes bistables spatialement étendus. Thèse Institut Non Linéaire de Nice, 1999 2. Argentina, M., Coullet, P., Risler, E.: Self-parametric instability in spatially extended systems. Preprint INLN 1999, to appear in Phys. Rev. Lett. 3. Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. New York: SpringerVerlag, 1983 4. Benjamin, T.B., Feir, J.E.: The disintegration of wave trains on deep water. J. Fluid Mech. 27, 417 (1967) 5. Coullet, P., Risler, E., Vandenberghe, N.: Spatial unfolding of elementary bifurcations. J. of Stat. Phys. 101, (1/2), 521 (2000) 6. Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851 (1993) 7. Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Appl. Math. Sci. 42 (1983) 8. Hartman, P.: Ordinary differential equations. New York: Wiley, 1967 9. Newell, A.C.: Envelop equation. Lect. Appl. Math. 15, 157 (1974) 10. Risler, E.: Criteria for the stability of spatial extensions of fixed points and periodic orbits of differential equations in dimension 2. Physica D 146, 121 (2000) 11. Yamada, T., Kuramoto, Y.: Pattern formation in oscillatory chemical reactions. Prog. Theor. Phys. 56, 681 (1976) Communicated by A. Kupiainen

Commun. Math. Phys. 216, 357 – 373 (2001)

Communications in



On the Fundamental Solution of Semiclassical Schrödinger Equations at Resonant Times André Martinez1, , Kenji Yajima2, 1 Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40127 Bologna, Italy 2 Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, Japan

Received: 18 April 2000 / Accepted: 31 July 2000

Abstract: We consider perturbations of the semiclassical harmonic oscillator of the form 2 2 P = − h2 + x2 + hδ W (x), x ∈ Rm , with W (x) ∼ x2−µ as |x| → +∞ and δ, µ ∈ (0, 1), and we investigate the fundamental solution E(t, x, y) of the corresponding timedependent Schrödinger equation. We prove that at resonant times t = nπ (n ∈ Z) it admits a semiclassical asymptotics of the form: E(nπ, x, y) ∼ h−m(1+ν)/2 a0 eiS(x,y)/ h with a0 = 0 and ν = δ/(1 − µ), under the conditions x = (−1)n y and ν < 1. 1. Introduction and Main Result We consider time dependent Schrödinger equation in L2 (Rm ): ih

h2 x2 ∂u = − u + u + hδ W (x)u = P h u, ∂t 2 2

(1.1)

where δ ∈ (0, 1) and W (x) ∈ C ∞ (Rm ) is real valued. We assume for some constants C > 0 and µ ∈ (0, 1):  1 x−µ ≤ D 2 W (x) ≤ Cx−µ , (1.2) C |∂ α W (x)| ≤ C x2−µ−|α| for |α| ≥ 3 α

for x ∈ Rm . In particular W is subquadratic at infinity. Under this assumption, P h on C0∞ (Rm ) admits a unique selfadjoint extension, which we denote by P h again, and the solution of (1.1) with initial data u(0, x) = φ(x) is given by u(t) = exp(−itP h / h)φ. The distribution kernel E(t, x, y) of exp(−itP h / h) is called the fundamental solution Investigation supported by University of Bologna. Funds for selected research topics

Partly supported by the Grant-in-Aid for Scientific Research, The Ministry of Education, Science, Sports

and Culture, Japan #11304006

358

A. Martinez, K. Yajima

(FDS for short) of (1.1) and we investigate the behaviour as h → 0+ of E(t, x, y) at δ the resonant times t = nπ (n ∈ Z∗ ). We set ν = and assume 0 < ν < 1. Our 1−µ main result is: Theorem 1.1. Let n ∈ Z∗ . Then, E(nπ, x, y) is a C ∞ function of (x, y) and for h small enough it can be written in the form: E(nπ, x, y) = h−m(1+ν)/2 a(x, y, h)eiS(x,y)/ h ,

(1.3)

where S(x, y) is the action integral of classical trajectory corresponding to (1.1) connecting x(0) = y and x(nπ ) = x, and for any compact subset K of R2m \ , = {(x, (−1)n x) ; x ∈ Rm }, a(x, y, h) satisfies 0 < C −1 ≤ |a(x, y, h)| ≤ C < ∞ for (x, y) ∈ K uniformly with respect to small h. The estimate (1.3) should be compared with the result at non-resonant time: If t ∈ πZ, then the FDS solution behaves as h → 0+, E(t, x, y) = h−m/2 a(x, y, h)eiS(x,y)/ h = O(h−m/2 ), and (1.3) represents the anomalous increase of the amplitude as h → 0. We should also remark that, if W is sublinear, viz. W = O(x1−ε ), then for (x, y) ∈ K, K being as above, E(nπ, x, y) = O(hN ) for any N as h → 0+. Indeed in this case, E(nπ, ·, y) has singularities at (−1)n y. These remarks can be easily obtained by applying the standard stationary phase method to (1.5) below. Motivation to this work comes from the study of the behavior at infinity x 2 +y 2 → ∞ of the FDS of Eq. (1.1) with fixed h = 1 under the condition (1.2): i

∂u 1 x2 = − u + u + W (x)u. ∂t 2 2

(1.4)

When t ∈ πZ, E(t, x, y) for (1.4) converges to the FDS solution of the harmonic oscillator as x 2 + y 2 → ∞ ([Ya-1]). At resonant times, however, we believe that E(nπ, x, y), n = 0, blows up as |x − y| → ∞. It turns out that proving the latter is a little too intricate and still out of reach, although intimately related to the semiclas2−µ the change of sical problem we investigate √ here. Indeed, in the case W (x) = |x| scale u(t, x) → u(t, x/ h) converts (1.4) to (1.1) with δ = µ/2 and the study of the solution of (1.4) as |x| → ∞ is equivalent to that of (1.1) at fixed x as h → 0. Thus, we expect in general |E(nπ, x, y)| ∼ |x − y|mν as |x − y| → ∞, with ν = µ/(2 − 2µ). The strategy for proving the theorem is as follows. First of all, one can see as in [Ya-1] (see also [Ro, KK] for a semiclassical version for short time) that E(nπ, x, y) can be written under the form of an oscillatory integral: 1 E(nπ, x, y) = ei(xξ −ψ(y,ξ ))/ h b(y, ξ, h)dξ. (1.5) (2π h)m Here b is a semiclassical symbol which is uniformly bounded together with all its derivatives, and ψ is the function: nπ ˙ x (t, y, ξ )2 − V ( x (t, y, ξ )) dt, (1.6) ψ(y, ξ ) = x (nπ, y, ξ ) · ξ − 2 0

On the Fundamental Solution of Semiclassical Schrödinger Equations at Resonant Times

359

x2 where V (x) = x (t, y, ξ ) denotes the x-projection at time t of the + hδ W (x) and 2 unique classical trajectory t → (x(t), p(t)) satisfying x(0) = y and p(nπ ) = ξ , that is the unique solution of:  x(t) ˙ = p(t)  p(t) ˙ = −∇V (x(t))  x(0) = y ; p(nπ ) = ξ (notice that since V depends on h, the same is true for x (t, y, ξ )). We then apply the stationary phase method to (1.5). It is standard to show that: x (nπ, y, ξ ) ∇ξ ψ(y, ξ ) =

(1.7)

and the point of stationary phase is given as the solution of x = x (nπ, y, ξ ). We study the properties of x (nπ, y, ξ ) as h → 0 as well as |ξ | → ∞ in Sect. 2. Section 3 is devoted to studying the phase function ψ(y, ξ ). We show there exits a unique point of stationary phase for x = (−1)n y and we estimate |x − ∇ξ ψ(y, ξ )| from below. Estimates on the symbol b is given in Sect. 4 and the proof of the theorem is completed in Sect. 5. In the Appendix an implicit function theorem for mappings in Rm with positive definite differentials outside a compact set is given. 2. Estimates on the Classical Flow The purpose of this section is to show the following proposition. Proposition 2.1. Let a compact set K ⊂ Rm be fixed and α, β ∈ Nm . Then: (1) For all t ∈ [0, nπ] one has: β

|∂yα ∂ξ ( x (t, y, ξ ) − ycost − (−1)n ξ sint)| = O(hδ ξ (1−|β|)+ −µ ) and n

δ

x (t, y, ξ ) = ycost + (−1) ξ sint + h cost

t

sins∇W (ycoss + (−1)n ξ sins)ds

0

+h2δ r(t, y, ξ ) with

β

∂yα ∂ξ r(t, y, ξ ) = O(ξ (1−|β|)+ −2µ + |sint|ξ 1−2µ )

uniformly with respect to ξ ∈ Rm , y ∈ K and h > 0 small enough. (2) For any ε > 0, there exists h0 = h0 (ε, K) such that β

|∂yα ∂ξ ( x (t, y, ξ ) − ycost − (−1)n ξ sint)| = O((hδ ξ −µ )|β| (hδ ξ 1−µ )(1−|β|)+ ) uniformly with respect to |ξ | ≥ εh−ν , 0 < h < h0 , t ∈ [0, nπ ] and y ∈ K. Proof. For (y, k) ∈ R2m , we denote (x(t, y, k), p(t, y, k)) the unique classical trajectory t → (x(t), p(t)) satisfying (x(0), p(0)) = (y, k). We also denote k(y, ξ ) the value of k for which p(nπ, y, k) = ξ (so that we have x (t, y, ξ ) = x(t, y, k(y, ξ ))).

360


We use the following lemma: Lemma 2.2. (1) For h > 0 small enough and for all α, β ∈ Nm , one has: β

|∂yα ∂ξ (k(y, ξ ) − (−1)n ξ )| = O(hδ ξ (1−|β|)+ −µ ) and k(y, ξ ) = (−1)n ξ + hδ

nπ

coss∇W (ycoss + (−1)n ξ sins)ds + h2δ r1 (y, ξ )

0

with

β

∂yα ∂ξ r1 (y, ξ ) = O(ξ 1−2µ )

uniformly with respect to ξ and h. (2) For any ε > 0, there exists h0 = h0 (ε, K) such that for all α, β ∈ Nm , β

|∂yα ∂ξ (k(y, ξ ) − (−1)n ξ )| = O((hδ ξ −µ )|β| (hδ ξ 1−µ )(1−|β|)+ ) uniformly with respect to |ξ | ≥ εh−ν , 0 < h < h0 , t ∈ [0, nπ ] and y ∈ K. Proof of the lemma. By Duhamel principle, we have for any (t, y, k) ∈ R × R2m :

t x(t, y, k) = ycost + ksint − hδ 0 sin(t − s)∇W (x(s, y, k))ds,

t (2.1) p(t, y, k) = −ysint + kcost − hδ 0 cos(t − s)∇W (x(s, y, k))ds and therefore, k = k(y, ξ ) is the unique solution of the equation: nπ (−1)n ξ = k − hδ coss∇W (x(s, y, k))ds.

(2.2)

0

Denoting by F (y, k) the right-hand-side of (2.2), we see that: nπ ∂x ∂F (s, y, k)x(s, y, k)−µ ds = I + O hδ ∂k ∂k 0

(2.3)

while, using Gronwall’s inequality iteratively, we deduce from (2.1) that for all α, β ∈ Nm one has: nπ α β ∂x −µ ∂ ∂ = O hδ x(u, y, k) du (2.4) (s, y, k) − sins y k ∂k 0 uniformly with respect to k and h (here and in the sequels we have denoted sins for (sins)I , where I is the identity matrix of Rm ). Moreover, using the same arguments as in [Ya] Lemmas 4.2–4.4, we see that: nπ x(s, y, k)−µ ds = O(k−µ ). (2.5) 0

In particular ∂x/∂k is uniformly bounded and we deduce from (2.3)–(2.5) that for any α, β ∈ Nm : α β ∂F ∂ ∂ = O(hδ k−µ ) (2.6) − I y k ∂k


361

uniformly. It follows from (2.6) that k → F (y, k) is a global diffeomorphism in Rm for all y and for h small enough, and that moreover the solution k(y, ξ ) of (2.2) satisfies: β

|∂yα ∂ξ (k(y, ξ ) − (−1)n ξ )| = O(hδ ξ (1−|β|)+ −µ )

(2.7)

for any α, β ∈ Nm . Inserting this estimate in (2.2) and using again (2.4) as well as (1.2), we get in particular: n

k(y, ξ ) = (−1) ξ + h

δ

nπ

coss∇W (x(s, y, (−1)n ξ ))ds + O h2δ ξ 1−2µ

0

and analogous estimates for the derivatives. Then the result follows by using (see (2.1)) that x(s, y, (−1)n ξ ) = ycoss + (−1)n ξ sins + O(hδ ξ 1−µ ). For proving the second statement, we need two lemmas. Lemma 2.3. Let ρ > 0 and 2 ≥ 0. Let T > 0 and a compact set K ⊂ Rm be fixed. Then, for any ε > 0, there exist h0 = h0 (ε, K) and C = C(ε, K) > 0 such that

T

x(t, y, k)−ρ |sint|2 dt ≤ C(|k|−1 (hδ |k|−µ )2 + |k|− min(1+2,ρ) )

(2.8)

0

for y ∈ K and |k| ≥ εh−ν , 0 < h < h0 . Here we have assumed ρ ∈ N for simplicity. Proof. We have δ h

T 0

sin(t − s)∇x W (x(s, y, k))ds ≤ CT hδ |k|1−µ

for |k| ≥ C0 .

Set Cε,K = ε −1/(1−µ) supy∈K |y| so that Cε,K hδ |k|1−µ ≥ sup |y| for |k| ≥ εh−ν , 0 < y∈K

h < 1. Define D1 = {t ∈ [0, T ] : |sint| ≤ 2(CT + Cε,K )hδ |k|−µ },

For t ∈ D1 , we have |x(t, y, k)| ≤ 3(CT + Cε,K )hδ |k|1−µ ≤ εh−ν and, as in [Ya] Lemma 4.2, we see that

D1

x(t)

−ρ

(sint) dt ≤ C 2

j

|t−j π|≤θ

D2 = [0, T ] \ D1 .

|k|2 + y 2 /10 if |k| ≥

|t − j π |2 (t − aj )k−ρ dt,

where we have set θ = 3(CT + Cε,K )hδ |k|−µ and where the sum over j integer is finite, and aj ∈ [−2T , 2T ] is the unique time in [j π − π/10, j π + π/10] for which |x(t)|2 is minimal. In particular, using that x(t, y, k) = ycost + ksint + O(hδ k1−µ ) we get: |aj − j π | = O(θ )

(2.9)

362


uniformly. Therefore, denoting bj = aj − j π we get by a change of variable: −ρ 2 x(t) (sint) dt ≤ C |t + bj |2 tk−ρ dt D1

|t+bj |≤θ

j

≤C

2

|bj |

2−q

j q=0

≤C

2

|t|≤Cθ

|t|q tk−ρ dt

|k|− min(q+1,ρ) θ 2−q+(q+1−2)+

q=0

≤ C(|k|−ρ θ 2+1 + |k|−1 θ 2 ). Now, if t ∈ D2 , then |x(t, y, k)| ≥ (1/2)|(sint)k| and we have T −ρ 2 x(t, y, k) (sin t) dt ≤ C |(sin t)2 |(sin t)k−ρ dt ≤ C|k|− min(2+1,ρ) . D2

0

Adding all the contributions completes the proof of the lemma.

Lemma 2.4. Let a compact set K ⊂ Rm and T > 0 be fixed. Then, for any ε > 0 we have for any |β| ≥ 1, β

|∂yα ∂k (x(t, y, k) − ycos t − ksin t)| ≤ O((hδ k −µ )|β| ), β

|∂yα ∂k (p(t, y, k) + ysin t − kcos t)| ≤ O((hδ k −µ )|β| ) for |k| ≥ εh−ν , 0 < h < h0 = h0 (ε, K) and y ∈ K. Proof. We prove the case α = 0 only. The proofs for other cases are similar. We write t sin(t − s)∂x2 W (x(s))∂k x(s)ds = sin t + X(t). (2.10) ∂k x(t) = sin t − hδ 0

Then |X(t)| ≤ Chδ |k|−µ and this proves the case |β| = 1. We prove the general case by induction on |β|. We assume that the lemma holds for |β| ≤ 2 − 1 and let |β| = 2, 2 ≥ 2. We have by Leibniz’ formula t β β sin(t − s)∂x2 W (x(s))∂k x(s)ds ∂k x(t) = hδ 0 (2.11) β

t + hδ sin(t − s)∂xκ ∂x W (x(s)) ∂k j x(s) ds, 0

where the sum is taken over βj such that j βj = β, |κ| ≥ 2 and |κ| is the number of the factors in the product. We estimate each integral under the sign of summation. β Replacing all ∂k x(s) by sins + X(s) and using the induction hypothesis for |∂k j x(s)| δ −µ with |βj | ≥ 2 and |X(s)| ≤ Ch |k| , we estimate it by hδ (hδ |k|−µ )|β|−q ·

T 0

x(s)1−µ−|κ| |sin s|q ds,


363

where q ≤ |κ| is the number of sins’s which appear from the factors ∂k x(s) = sin s + X(s). We have µ + |κ| − 1 > q + 1 unless q ≥ |κ| − 1. Hence, we have from Lemma 2.3 that, if q ≤ |κ| − 2, hδ

T

x(s)1−µ−|κ| |sin s|q ds ≤ Chδ (|k|−1 (hδ |k|−µ )q + |k|−(q+1) )

(2.12)

0

and if |κ| − 1 ≤ q ≤ |κ|, hδ

T

x(s)1−µ−|κ| |sin s|q ds ≤ Chδ (|k|−1 (hδ |k|−µ )q + |k|1−µ−q ).

(2.13)

0

Using that |k| ≥ εh−ν , we see that the right-hand sides of (2.12) and (2.13) are bounded by C(hδ |k|−µ )q and the lemma follows by applying Gronwall’s inequality to (2.11). Completion of the proof of Lemma 2.2. When |ξ | ≥ εh−ν , we improve (2.6) to α β ∂F = O((hδ k−µ )|β|+1 ) ∂ ∂ − I y k ∂k by using the argument of the proof of the previous Lemma 2.4 which leads to the following improvement of (2.7): β

|∂yα ∂ξ (k(y, ξ ) − (−1)n ξ )| = O((hδ ξ −µ )|β| (hδ ξ 1−µ )(1−|β|)+ ). This completes the proof of the lemma. Completion of the proof of Proposition 2.1. Going back to (2.1) and using the first estimate of Lemma 2.2, we first get: x (t, y, ξ ) = ycost + k(y, ξ )sint − hδ

t

sin(t − s)∇W (x(s, y, (−1)n ξ )ds

0

(2.14)

+ h r2 (t, y, ξ ) 2δ

with

β ∂yα ∂ξ r2 (t, y, ξ ) = O ξ (1−|β|)+ −2µ

and the first estimate of statement (1) follows by using also (2.5). The second estimate of statement (1) is also obtained immediately from (2.14) by using the second estimate of Lemma 2.2. Statement (2) may be proved by differentiating x(t, ˜ y, ξ ) = x(t, y, k(y, ξ )) and applying Lemma 2.4 and Lemma 2.2. Note that |ξ | ∼ |k| by virtue of (2.2).

3. Estimates on the Phase Let ψ be the phase defined in (1.6). In this section we show:

364


Proposition 3.1. For all (x, y) ∈ R2m with x = (−1)n y, there exists a unique ξc = ξc (x, y, h) ∈ Rm for 0 < h small enough such that: ∇ξ ψ(y, ξc ) = x. Moreover, if (x, y) remains in a compact set K of R2m ∩ {x = (−1)n y}, then there exists a constant CK > 0 such that: 1 −ν h ≤ |ξc (x, y, h)| ≤ CK h−ν CK with ν =

δ , and for any ξ ∈ Rm : 1−µ |x − ∇ξ ψ(y, ξ )| ≥

hδ |ξ − ξc |. CK (|ξc | + |ξ − ξc |)µ

Proof. By (1.7) and (2.1), we have to solve the equation:

nπ

sin s∇W (x(s, y, k))ds =

0

(−1)n x − y , hδ

(3.1)

where k = k(y, ξ ). Actually, since the mapping ξ → k(y, ξ ) is one-to-one on Rm (for y fixed), it is enough to solve (3.1) by taking k as the unknown variable. Denote by G(y, k) the function defined by the left-hand side of (3.1). Then computing as ∇k G(y, k) =

nπ

sin sD 2 W (x(s, y, k))

0

∂x (s, y, k)ds ∂k

and using (2.4) and (1.2), we see as in [Ya], that ∇k G(y, k) satisfies ∇k G(y, k) − P (k) ≤ Ck−2µ for some positive definite matrix P (k) such that C −1 k−µ ≤ P (k) ≤ Ck−µ . It follows from the global implicit function theorem given in the appendix that, for large enough R > 0, the mapping k → G(y, k) =

nπ

sin s∇W (x(s, y, k))ds

0

is a diffeomorphism from the exterior B>R ⊂ Rm of the ball of radius R to its image and the image contains another exterior domain B>R1 . In particular, for h small enough we get the existence of a unique solution kc = k(y, ξc ) of (3.1). Moreover, we have: Lemma 3.2. For y remaining in a fixed compact set of Rm and for |k| large enough, one has: 1 1−µ nπ ≤ sin s∇W (x(s, y, k))ds ≤ C|k|1−µ |k| C 0

for some constant C > 0 and uniformly with respect to h and k.


365

Proof of the lemma. Since the upper-bound is obvious, we concentrate on the lowerbound. We see from (2.1) that x(s, y, k) = ycos s + ksin s + O(hδ k1−µ ) and we also use (2.4) to obtain: nπ nπ ∂ sin s∇W (x(s, y, θk))ds = sin2 sD 2 W (ycos s + θksin s)k ds ∂θ 0 0

+ O(hδ ) θk1−µ |k| + k1−µ . Integrating with respect to θ from 0 to 1, this gives:

nπ

1 nπ

sin s∇W (x(s, y, k))ds =

0

0

sin2 sD 2 W (ycos s + θksin s)k dsdθ

0 δ

+ O(1 + h k

1−µ

(3.2)

).

Then we fix ρ ∈ (µ, 1), and we consider the set: Dρ = {(θ, s) ∈ [0, 1] × [0, nπ ] ; |θ sin s| ≥ k−ρ } and its complementary DρC in [0, 1] × [0, nπ ]. Since on Dρ we have |θksins| → +∞ as |k| → +∞, we can use (1.2) to get:

sin sD W (ycos s + θksin s)k dsdθ, k 2

Dρ

2

≥

Dρ

sin2 s ycos s + θksin s−µ dθds |k|2 C

and thus, by Cauchy–Schwarz inequality and since for |k| large enough Dρ contains {(θ, s) ∈ [0, 1] × [0, nπ] ; |θ sins| ≥ δ1 } (of measure ∼ 1) for any fixed δ1 > 0 small enough, we get (with some other constant C > 0): 1 2 2 sin sD W (ycos s + θksin s)k ds dθ ≥ k1−µ . (3.3) C Dρ On the other hand, since the Lebesgue measure of DρC is O(k−ρ lnkρ ) as |k| → +∞, we have: 2 2 sin sD W (ycos s + θksin s)k dsdθ = O(k1−ρ lnkρ ) = O(k1−µ−ε ) DρC

(3.4) for some ε > 0. Then the result follows from (3.2)–(3.4).

Completion of the proof of Proposition 3.1. We deduce from the lemma and from (3.1) that |kc | (and thus also |ξc |) behaves like h−ν as h → 0.

366


Now, denoting nπ G(y, ξ ) := sin s∇W (x(s, y, k(y, ξ ))) ds = 0

nπ

sin s∇W ( x (s, y, ξ ))ds,

0

(3.5) we can write: x − ∇ξ ψ(y, ξ ) = x − (−1)n y − (−1)n hδ G(y, ξ ) = (−1)n hδ (G(y, ξc ) − G(y, ξ )) 1 ∂G n δ = (−1) h (y, tξc + (1 − t)ξ ) · (ξc − ξ )dt, 0 ∂ξ that is: n δ

x − ∇ξ ψ(y, ξ ) = (−1) h

1 nπ 0

A(t, s, x, y, ξ )dsdt

(3.6)

0

with A(t, s, x, y, ξ ) = sin sD 2 W ( x (s, y, tξc + (1 − t)ξ ))

∂ x (s, y, tξc + (1 − t)ξ ) ∂ξ

· (ξc − ξ ). Now, given some constant λ > 0 large enough, we split the integral in (3.6) in two pieces by setting: B = B(x, y, ξ ) = {(s, t) ∈ [0, nπ ] × [0, 1] ; |(tξc + (1 − t)ξ )sin s| ≥ λ} and n δ

I1 (x, y, ξ ) = (−1) h

I2 (x, y, ξ ) = (−1)n hδ

B

A(t, s, x, y, ξ )dsdt,

BC

A(t, s, x, y, ξ )dsdt.

If λ is taken sufficiently large, for (s, t) ∈ B we can apply (1.2) with x = x (s, y, tξc + (1 − t)ξ ). Since also, by Proposition 2.1, we have ∂ x (s, y, ξ ) = (−1)n sin s + O(hδ ξ −µ ) ∂ξ this permits us to get: I1 , ξc − ξ ≥

hδ C0

sin2 s x (s, y, tξc + (1 − t)ξ )−µ |ξc − ξ |2 ds dt B − O(h2δ ) tξc + (1 − t)ξ −µ |ξc − ξ |2 ds dt.

(3.7)

B

Now, let us estimate the measure of B C . Since |ξc + (1 − t)(ξ − ξc )| ≥ ||ξc | − (1 − C t)|ξ −

to see that if (s, t) ∈ B , then t belongs to an interval of length ξc ||, it is easy 2λ Min 1, |(ξ −ξc )sin s| . When e.g. |ξ − ξc | ≥ |ξc |/2, this gives a set in [0, nπ ] × [0, 1] of


measure

ln|ξ −ξc | |ξ −ξc | . On the other hand, if |ξ O(|ξc |−1 ). Thus we get in any case:

measure O

Measure(B C ) = O It follows that

367

− ξc | ≤ |ξc |/2 then s belongs to a set of

ln(|ξc | + |ξ − ξc |) . |ξc | + |ξ − ξc |

(3.8)

hδ |ξ − ξc |ln(|ξc | + |ξ − ξc |) |I2 | = O (3.9) |ξc | + |ξ − ξc | and also, in view of (3.7): hδ 1 nπ 2 I1 , ξc − ξ ≥ sin s x (s, y, tξc + (1 − t)ξ )−µ |ξc − ξ |2 dsdt C0 0 0 δ 1 h |ξ − ξc |2 − O(h2δ ) tξc + (1 − t)ξ −µ |ξc − ξ |2 dt −O |ξc | + |ξ − ξc | 0 that is: hδ 1 nπ 2 I1 , ξc − ξ ≥ sin s x (s, y, tξc + (1 − t)ξ )−µ |ξc − ξ |2 ds dt C0 0 0 (3.10) δ h |ξ − ξc |2 h2δ |ξc − ξ |2 −O . + |ξc | + |ξ − ξc | (1 + |ξc | + |ξ − ξc |)µ As a consequence, since | x (s, y, ξ )| = O(|ξ |) uniformly, we get from (3.10): hδ 1 tξc + (1 − t)ξ −µ |ξc − ξ |2 dt I1 , ξc − ξ ≥ C1 0 δ h |ξ − ξc |2 h2δ |ξc − ξ |2 −O , + |ξc | + |ξ − ξc | (1 + |ξc | + |ξ − ξc |)µ where C1 > 0 is a constant, and thus (with some other constant C2 > 0):

|ξc − ξ |2 hδ C2 (1 + |ξc | + |ξ − ξc |)µ δ h |ξ − ξc |2 h2δ |ξc − ξ |2 . −O + |ξc | + |ξ − ξc | (1 + |ξc | + |ξ − ξc |)µ Putting together (3.6), (3.9) and (3.11), we get for h small enough: I1 , ξc − ξ ≥

|x − ∇ξ ψ(y, ξ )| 1 C3 ln(|ξc | + |ξ − ξc |) |ξc − ξ | − ≥ hδ C3 (1 + |ξc | + |ξ − ξc |)µ |ξc | + |ξ − ξc |

(3.11)

(3.12)

with C3 > 0 constant. Now the result follows by observing that |ξc | ∼ h−ν ∼ 1+|ξc | and, for any fixed ρ ∈ (µ, 1), (|ξc |+|ξ −ξc |)−1 ln(|ξc |+|ξ −ξc |) = O((|ξc |+|ξ −ξc |)−ρ ) = O(|ξc |µ−ρ (|ξc | + |ξ − ξc |)−µ ) = O(hν(ρ−µ) (|ξc | + |ξ − ξc |)−µ ). Remark. We deduce in particular from Proposition 3.1 that for ξ such that |ξ − ξc | = O(|ξc |), one has: hν |x − ∇ψ(y, ξ )| ≥ " |ξ − ξc | CK " with CK > 0 constant.

368


4. Estimates on the Symbol Let b(y, ξ, h) be the amplitude function in (1.5). We denote y, ξ −µ = (y, ξ )−µ . The purpose of this section is to prove: β

Proposition 4.1. As h → +0, ∂yα ∂ξ (b(y, ξ, h) − 1) = O(hδ y, ξ −µ ) + O(h). Proof. For small ε > 0 and large T > 0 fixed, we set Iε,T = {|t| < T : |t − (m + 1/2)π| > ε, ∀m ∈ Z}. For |t| < T , we have ∂p (t, y, k) − cos t ≤ CT hδ ∂k and for 0 < h small, the map k → p(t, y, k) is a diffeomorphism of Rm for every fixed t ∈ Iε,T and y ∈ Rm . It follows for such t that the phase function is globally defined by t p(s, y, k)2 ψ(t, y, ξ ) = x(t, y, k) · ξ − − V (x(s, y, k)) ds, 2 0 k being such that ξ = p(t, y, k), and that E(t, x, y), (n − 1/2)π < t < (n + 1/2)π , can be written ([Ya-1], Theorem 5.5) in the form i −n ei(xξ −ψ(t,y,ξ ))/ h b(t, y, ξ )dξ. E(t, x, y) = (2πh)m |cos t|m/2 When |t| ≤ T1 ≡ (π/2) − ε, it can be shown as in ([Ya-1]) that −1/2 b0 (t, y, ξ ) ∂p b(t, y, k) = = det + hO(1), ξ = p(t, y, k). (t, y, k) (cos t)m/2 ∂k α β ∂p (t, y, k) − cos t = O(hδ y, k−µ ) ([Ya], Lemma 4.4) and y, ξ ∼ Since ∂y ∂k ∂k y, k for |t| ≤ T1 , (4.1) holds for small t. For obtaining the proposition, it suffices via an induction argument to show the following lemma. We let t, s ∈ Iε,T be such that, for some n1 , n2 , n3 ∈ Z, |s − n1 π| < π/2, |t − n2 π | < π/2 and |s + t − n3 π | < π/2 and set i −n1 ei(xξ −ψ(t,y,ξ ))/ h (b0 (y, ξ ) + hb1 (y, ξ )), (2πh)m |cost|m/2 i −n2 G(x, y, ξ ) = ei(xξ −ψ(s,y,ξ ))/ h (c0 (y, ξ ) + hc1 (y, ξ )) (2πh)m |coss|m/2 F (x, y, ξ ) =

and define

H (y, ξ ) =

e−ixξ/ h F (x, z, η)G(z, y, ζ )dζ dzdηdx. β

Lemma 4.2. Suppose that b0 and c0 satisfy ∂yα ∂ξ (b0 (y, ξ, h) − 1) = O(hδ y, ξ −µ ) β

and ∂yα ∂ξ (c0 (y, ξ, h) − 1) = O(hδ y, ξ −µ ) and b1 , c1 = O(1). Then H (y, ξ ) = β

i −n3 e−iψ(t+s,y,ξ )/ h (d0 (y, ξ ) + hd1 (y, ξ )), |cos(t + s)|m/2

where ∂yα ∂ξ (d0 (y, ξ, h) − 1) = O(hδ y, ξ −µ ) and d1 = O(1).


369

Proof. Set C(x, z, η, y, ζ ) = −xξ + xη − ψ(t, z, η) + zζ − ψ(s, y, ζ ). The derivatives of C of order higher than one are bounded and Hess x,η,z,ζ C is given by     0 1 0 0 0 1 0 0 0 0   1 − tan t − sec t  1 −ψηη −ψzη  δ =  0 −ψ −ψ  + O(h ). 1 1   0 − sec t − tan t ηz zz 0 0 1 − tan s 0 0 1 −ψζ ζ Denote by A the matrix on the right. It is easy to see that cos(t + s) m = 0. | det A| = | tan t tan s − 1|m = cos t · cos s Thus the point of stationary phase exists uniquely for every (ξ, y) and is determined by the system of equations   ∂x C = −ξ + η = 0,   ∂ C = x − ∂ ψ(t, z, η) = 0, η η (4.1) ∂z C = −∂z ψ(t, z, η) + ζ = 0,   ∂ C = z − ∂ ψ(s, y, ζ ) = 0. ζ ζ For any k, (x, η, z, ζ ) = (x(t + s, y, k), p(t + s, y, k), x(s, y, k), p(s, y, k))

(4.2)

satisfies the last three equations of (4.1) and (y, k) → (y, p(t + s, y, k)) is a diffeomorphism on R2m . It follows that the unique stationary phase point (xc , ηc , zc , ζc ) is given by the right hand side of (4.2) with k being replaced by the solution k(y, ξ ) of ξ = p(t + s, y, k). The quadratic form defined by the matrix A can be written for x = (a, b, c, d) ∈ R4m in the form tan t tan s tan t tan s − 1 2 tan t tan s − 1 2 a − b− a Ax, x = tan t tan s tan t tan s − 1 tan t tan s 2 tan t tan s − 1 c 2 tan t sec t + , c− b + tan s d − tan s tan t tan s − 1 tan s and we see that the signature of A is given by   if tan t tan s < 1, 0 sgn(A) = −2m if tan t tan s > 1 and tan s > 0, (4.3)  2m if tan t tan s > 1 and tan s < 0. It follows by the standard stationary phase method that H (y, ξ ) is given by i −n1 −n2 eiπsgn(A)/4 i(−ψ(t,zc ,ηc )+zc ζc −ψ(s,y,ζc ))/ h e · (b0 (zc , ηc )c0 (y, ζc ) + hd1 (y, ξ )). |cos(t + s)|m/2 Notice that tan t tan s < 1 if and only if |t + s − (n1 + n2 )π | < π/2 and, tan t tan s > 1 and ± tan s > 0 if and only if |t + s − (n1 + n2 ± 1)π | < π/2 and that −ψ(t, zc , ηc ) + zc ζc − ψ(s, y, ζc ) = −ψ(t + s, y, ξ ). Moreover, because y, ξ ∼ y, k ∼ y, ζc ∼ zc , ζc , we have b0 (zc , ηc )c0 (y, ζc ) − 1 = (b0 (zc , ηc ) − 1)c0 (y, ζc ) + c0 (y, ζc ) − 1 = O(hδ y, ξ −µ ).

370


5. Completion of the Proof In what follows we fix a compact set K ⊂ Rm and always assume that x, y ∈ K. We apply the method of stationary phase to the integral on the right of (1.5). As the magnitude of the critical point ξc of the phase function ξ → xξ − ψ(y, ξ, h) is of order h−ν as was shown in Proposition 3.1, we change the variables ξ → h−ν ξ to make |ξc | ∼ 1 in the new scale. Thus, we consider i −n ν −ν 1+ν E(x, y, h) = ei(xξ −h ψ(y,h ξ,h)/ h b(y, h−ν ξ, h)dξ. (5.1) (2π)m h(1+ν)m Set D(x, y, ξ, h) = xξ − hν ψ(y, h−ν ξ, h) and denote by ξc = ξc (x, y, h) the critical point of the function ξ → D(x, y, ξ, h). By virtue of Proposition 3.1, x = ∂ξ ψ(y, h−ν ξc , h), |∇ξ D(x, y, ξ, h)| ≥

−1 CK ≤ |ξc | ≤ CK ,

|ξ − ξc | . CK (1 + |ξ − ξc |)µ

(5.2)

In view of (5.2), we split the integral (5.1) E(x,y, h) = E≤ε (x, y, h) + E≥ε (x, y, h) ξ − ξc by using the cutoff function χε (ξ ) = χ : ε i −n 1+ν E≤ε (x, y, h) = eiD(x,y,ξ )/ h χε (ξ )b(y, h−ν ξ, h)dξ, (1+ν)m (2πh) i −n 1+ν E≥ε (x, y, h) = eiD(x,y,ξ )/ h (1 − χε (ξ ))b(y, h−ν ξ, h)dξ, (1+ν)m (2πh) where χ ∈ C0∞ (Rm ) is such that χ (ξ ) = 1 for |ξ | < 1/2 and χ (ξ ) = 0 for |ξ | > 1. β

Lemma 5.1. Let ε > 0. For any N = 0, 1, . . . , ∂xα ∂y E≥ε (x, y, h) = O(hN ). Proof. We apply integration by parts by using the identitity N ∇ξ D 1+ν 1+ν · ∇ eiD/ h = eiD/ h h(1+ν)N ξ 2 i|∇ξ D| and write in the form E≥ε (x, y, h) =

i −n h(1+ν)N (2πh)(1+ν)m †N ∇ξ D 1+ν · eiD/ h · ∇ (1 − χε )b(y, h−ν ξ, h)dξ, ξ i|∇ξ D|2

where † stands for the real transpose. Since ∂ξα ∇ξ D(x, y, ξ ) = O(h−ν|α| ),

∂ξα b(y, h−ν ξ, h) = O(h−ν|α| ),

|α| ≥ 1,

we have by virtue of (5.2), †N ∇ D CN h−Nν ξ −ν · ∇ (1 − χ (ξ ))b(y, h ξ, h) , ≤ ξ ε 2 i|∇ξ D| ξ − ξc N(1−µ) and we obtain the lemma for α = β = 0 by letting N large enough. The proof for the derivatives of E≥ε is similar.


We deal with E≤ε (x, y, h) next. Assume ε > 0 is small enough and |ξ | ≥ ξ ∈ suppχε . Since ∇ξ ψ(y, ξ ) = x(nπ, ˜ y, ξ ), we have

371

1 for 2CK

Hess ξ D(x, y, ξ ) = −h−ν (∂ξ x)(nπ, ˜ y, h−ν ξ ) and by Proposition 2.1 the right-hand side can be written as nπ n+1 −ν+δ sin2 s∂ξ2 W (ycoss + (−1)n h−ν ξ sins)ds + h2δ−ν O(h−ν ξ −2µ ). (−1) h 0

(5.3) It follows by an estimate similar to the one used in the proof of Lemma 4.2 that the symmetric matrix given by the integral (5.3) is larger than Chµν on the support of χε . Thus we have for x, y ∈ K and ξ ∈ supp χε : 0 < C1 ≤ (−1)n+1 Hess ξ D(x, y, ξ ) ≤ C2 < ∞.

(5.4)

Moreover, by virtue of the second statement of Proposition 2.1, we have for x, y ∈ K and ξ ∈ supp χε : β

∂yα ∂ξ D = O(h−ν(|β|+1) (hδ |h−ν ξ |−µ )) = O(1).

(5.5)

By Taylor’s formula we have D(x, y, ξ ) = D(x, y, ξc ) + (ξ − ξc , B(x, y, ξ )(ξ − ξc ))/2, 1 B(x, y, ξ ) = 2 (1 − θ)Hess ξ D(x, y, θ ξ + (1 − θ)ξc )dθ. 0

It is obvious from (5.4) that for x, y ∈ K and ξ ∈ supp χε , 0 < C1 ≤ (−1)n+1 B(x, y, ξ ) ≤ C2 < ∞.

(5.6)

Set M(x, y, ξ ) = ((−1)n B(x, y, ξ ))1/2 and define η = M(x, y, ξ )(ξ − ξc ). Then ∂ξ η = M(x, y, ξ ) + (∂ξ M(x, y, ξ ))(ξ − ξc ) and, if we replace ε > 0 by a smaller one if necessary, we see from (5.6) and (5.5) that the map ξ → η is a diffeomorphism on the ball {ξ : |ξ − ξc | < 2ε} to its image with uniformly bounded derivatives and the same for its inverse map. We change the variables in the integral for E≤ε (x, y, h) from ξ to η: i −n eiD(x,y,ξc )/ h (2π)m h(1+ν)m ∂η −1 n+1 2 1+ν · ei(−1) η / h χε (ξ )b(y, h−ν ξ, h) det dη, ∂ξ 1+ν

E≤ε (x, y, h) =

where ξ = ξ(x, y, η) is the inverse of ξ → η(x, y, ξ ). Since 1 + ν > 2ν by our assumption, we can apply the extended form of stationary phase and, in virtue of Proposition 4.1, n+1

i −n eiD(x,y,ξc )/ h +iπ(−1) m/4 (2π )m/2 h(1+ν)m/2 −1/2 ∂ x˜ −ν · det (1 + O(hµ ) + O(h1−ν )), (nπ, y, h ξc ) ∂ξ 1+ν

E≤ε (x, y, h) =

This concludes the proof of the theorem.

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6. Appendix For R > 0, we write B>R = {x ∈ Rm : |x| > R}, B 0 and 0 < δ < 1. Then, there exists R1 such that F (x) is a diffeomorphism from B>R1 onto its image and such that the image F (B>R1 ) contains the exterior domain B>ρ for some ρ > 0. Proof. Take R2 > 0 large enough such that for a constant C4 > 0, (∂x F (x)u, u) ≥ C4 x−δ u2 ,

x ∈ B≥R2 , u ∈ Rm .

Then ∂x F (x) is non-singular and F (x) is a local diffeomorphism in B≥R2 . We suppose R1 > 10R2 and show first that F is one to one on B≥R1 . Let x, y ∈ B≥R1 and x = y. If x and y may be connected by a line segment L ⊂ B≥R2 , then we have 1 (∂x F (tx + (1 − t)y)(x − y), x − y)dt > 0 (6.1) (F (x) − F (y), x − y) = 0

and F (x) = F (y). Suppose, therefore, L ∩ B 0 such that supp b ⊂ B(0, R), where B(0, R) denotes the ball of radius R around the origin. Using the proposition above and the compact support of b we get τ|B+tb(x)| (ρt (x)) − τ|B| (ρt (x)) dx = τ|B+tb(x)| (ρt (x)) − τ|B| (ρt (x)) dx B(0,R)

=

|B+tb(x)|

B(0,R) |B| |B+tb(x)|

=

B(0,R) |B|

3 " ˜ τB˜ (ρt ) − ρt τB˜ (ρt ) d Bdx 5 5 3 " ˜ + o(t), τB˜ (ρ0 ) − ρ0 τB˜ (ρ0 ) d Bdx 5 2B˜ 5 2B˜

where we used the bounds on τ, τ " and the weak convergence in L5/3 to get the last equality. This proves the differentiability and the formula

d 5 3 " B · curl a τ (ρ) − (ρ) dx. ρτ E(N, B + t curl a, V ) = B dt t=0 2 B2 5 B If we apply the Thomas–Fermi equation (Theorem 6) we get:

5 3 B · curl a jMTF · a = τ dx. (ρ) + ρV B eff 2 B2 5 Now it only remains to notice, from the Thomas–Fermi equation, that ρ(x) = 0 if Veff (x) > 0. 3.1. A useful relation. When we calculate the limit of the current in quantum mechanics we will get a term which looks like (x − y) · (ã(x) − a˜ (y)) ρ(x) ρ(y) dxdy. |x − y|3 We will now use the minimizing property of the MTF-density ρ to obtain an equality for this term: Lemma 4. Let ρ be the minimizer in MTF-theory and let a˜ ∈ C0∞ (R3 ). Define a˜ 0 (x) = a˜ (x) − a˜ (0). Then 1 (x − y) · (ã(x) − a˜ (y)) ρ a˜ 0 · ∇V + ρ(y) dxdy = tr(D a˜ )PB (|Veff |− ). ρ(x) 2 |x − y|3 Proof. Let us define ρt (x) = ;t (x)ρ(x + t a˜ 0 (x)) for small t, where ;t (x) = | det(I + tD a˜ 0 (x))| = 1 + t tr(D a˜ 0 (x)) + O(t 2 ). Notice, that for small t we may write x = φt (x + t a˜ 0 (x)), where φt (y) = y − t a˜ 0 (y) + O(t 2 ). Now dy EMTF [ρt ] = τB (;t (φt (y))ρ(y)) + V (φt (y))ρ(y) dy ;t (φt (y)) 1 1 + ρ(x) ρ(y) dxdy. 2 |φt (x) − φt (y)|

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Using the minimizing property of ρ we get: d 0 = EMTF [ρt ] dt t=0 " = τB (ρ)ρ tr(D a˜ ) − τB (ρ) tr(D a˜ ) − ρ∇V · a˜ 0 (x − y) · (ã(x) − a˜ (y)) 1 ρ(x) − ρ(y) dxdy. 2 |x − y|3 Thus,

(x − y) · (ã(x) − a˜ (y)) 1 ρ(x) ρ(y) dxdy ρ a˜ 0 · ∇V + 2 |x − y|3 = − tr(D a˜ )(τB (ρ) + ρVeff ) = tr(D a˜ )PB (|Veff |− ).

3.2. Scaling in MTF-theory. When V (x) = −Z |x| and B = B(1, 0, 0), we may define a scaled functional −ρ(x) Eˆβ [ρ] = τˆβ (ρ(x)) dx + dx + D(ρ, ρ), |x| where τˆβ (t) = (1 + β)−8/5 τβ ((1 + β)6/5 t), β = B/Z 4/3 . The corresponding energy is (remember that λ = N/Z): ˆ E(λ, β) = inf Eˆβ [ρ]|ρ ∈ L1 ∩ L5/3 , ρ ≥ 0, ρ ≤ λ = Z −7/3 (1 + β)−2/5 E MTF (N, B, Z), and the minimizing densities ρ MTF and ρλ,β are related by MTF (x) = Z 2 (1 + β)6/5 ρλ,β (Z 1/3 (1 + β)2/5 x). ρN,B,Z

Let us state the scaled TF-equation: τˆβ" (ρλ,β ) = |veff |− ,

(8)

ρλ,β = Pˆβ" (|veff |− ).

(9)

or

Here Pˆβ (w) = (1 + β)−8/5 Pβ ((1 + β)2/5 w), and veff (x) = Notice also that

−1 |x|

−Pˆβ (|veff |− ) = τˆβ (ρλ,β ) + veff ρλ,β .

+ ρλ,β ∗

1 |x|

+ ν(λ, β). (10)

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Using the identity from Lemma 4 we can prove the Theorem 5 – assuming the validity of Theorem 4: Proof. From Theorem 4 and Theorem 3 we get that the quantum mechanical current in the limit behaves like: a˜ 0 (x) · x ρλ,β (x) dx Z |x|3 3 − Z (∂x (1) a (2) − ∂x (2) a (1) )Pˆβ (|veff |− ) dx 2 − Z (∂x (1) a (2) − ∂x (2) a (1) )veff ρλ,β dx 1 (x − y) · (ã(x) − a˜ (y)) +Z ρλ,β (x) ρλ,β (y) dxdy. 2 |x − y|3 If we use Lemma 4 above on the first and the last term we get: 5 = − Z (∂x (1) a (2) − ∂x (2) a (1) )Pˆβ (|veff |− ) dx 2 − Z (∂x (1) a (2) − ∂x (2) a (1) )veff ρλ,β dx 5 = Z (∂x (1) a (2) − ∂x (2) a (1) )τˆβ (ρλ,β ) dx 2 3 + Z (∂x (1) a (2) − ∂x (2) a (1) )veff ρλ,β dx, 2 which is exactly the result from MTF-theory. 4. MTF-Theory with a Current Term In order to calculate the current we need to introduce a perturbed MTF- functional with a current term. We want to stress that this functional is not meant to have anything to do with the so-called current density functional theories appearing in the physics literature. This functional is still a functional of the density alone, but it will enable us to calculate a part of the current. The functional is: ˜ V , a˜ ] = τB(x) (ρ(x)) dx + V (x)ρ(x) dx + D˜ t (ρ, ρ), EC-MTF,t [ρ; B, ˜ R3

where 1 D˜ t (f, g) = 2

f (x)

R3

(x − y) · (x − y + t (ã(x) − a˜ (y))) g(y) dxdy, |x − y|3

and where the other terms are as in standard MTF-theory. It is easy to see that (1 − tc)D(ρ, ρ) ≤ D˜ t (ρ, ρ) ≤ (1 + tc)D(ρ, ρ), where c only depends on D a˜ ∞ . We will assume t so small that the constants 1 − tc and 1 + tc, appearing in the inequality above, can be bounded below resp. above by 1/2 resp. 3/2. Therefore the proofs of the main theorems in MTF-theory apply to C-MTFtheory essentially without change so we only state the following conclusion:

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Theorem 7. Let B = B(0, 0, 1). For sufficiently small (depending only on D a˜ ∞ ) t we have: There is a unique minimizer ρt ∈ CB,N of EC-MTF . This ρt satisfies the Thomas–Fermi equation: τB" (ρt (x)) = |Veff,t |− ,

(11)

where Veff,t = V (x) +

(x − y) · (x − y + t (ã(x) − a˜ (y))) ρt (y) dy + ν. |x − y|3

The Thomas–Fermi equation can also be formulated: −PB (|Veff,t (x)|− ) = τB (ρt (x)) + Veff,t ρt (x).

(12)

The energy EC-MTF,t (N, B, Z) is differentiable in t at t = 0 and satisfies d 1 EC-MTF,t (N, B, Z) = dt 2

ρ0 (x)

(x − y) · (ã(x) − a˜ (y)) ρ0 (y) dxdy. |x − y|3

(13)

4.1. Scaled C-MTF-theory. Suppose now a˜ (x) = l a˜ 0 (x/ l) and define τˆβ (t) = (1 + β)−8/5 τβ ((1 + β)6/5 t) = (1 + β)2/5 τβ(1+β)−4/5 (t) with l = Z −1/3 (1 + β)−2/5 , β = B/Z 4/3 . Define furthermore, −1 ˆ Eβ,t [ρ] = τˆβ (ρ(x)) dx + ρ(x) dx + Dt (ρ, ρ), |x| where Dt (f, g) =

1 2

f (x)

(x − y) · (x − y + t (ã0 (x) − a˜ 0 (y))) g(y) dxdy. |x − y|3

Then the energy corresponding to Eˆβ,t : Eˆ C-MTF,t (λ, β) = inf Eˆβ,t [ρ]|ρ ∈ L1 ∩ L5/3 , ρ ≥ 0, ρ ≤ λ , satisfies

EC-MTF,t (N, B, Z) = Z 2 l −1 Eˆ C-MTF,t (λ, β),

with λ = N/Z and the minimizers ρ = ρC-MTF,t (N, B, Z) and ρλ,β,t of Eˆβ,t satisfy ρ(x) = Zl −3 ρλ,β,t (x/ l). Finally, we can scale the relation (13) above to get: d 1 (x − y) · (ã(x) − a˜ (y)) |t=0 Eˆ C-MTF,t (λ, β) = ρλ,β,t (x) ρλ,β,t (y) dxdy. (14) dt 2 |x − y|3

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5. The Calculation of the Quantum Current In this section we will prove Theorem 4. In the proof we will use the semiclassical lower bound from [3]: (ã(x)−ã(y))) Lemma 5. Let veff,t = −|x|−1 + (x−y)·(x−y+t ρt (y) dy + ν(λ, β), and |x−y|3 HN0 (t, h, µ, veff,t ) =

N

(pA )j · St (xj )(pA )j − µh(1 + tb(3) (xj )) + veff,t (xj ),

j =1

where St (x) = 1 + tM(x), M(x) = − D a˜ (x) + (D a˜ (x))T . Then inf Spec HN0 (t, h, µ, veff,t ) ≥ Escl (t, h, µ, veff,t ) + o(h−3 + µh−2 ), where Escl (t, h, µ, veff,t ) ∞ 3/2 µ 4bu,t = 2 (2ν + 1)µhbu,t − µh(1 + tb3 (u)) + veff,t (u) − du, dν h 6π;u,t ν=0

where bu,t = | curl x

;u,t = | det

1 + tM(u)A( 1 + tM(u)x)|,

1 + tM(u)| and d0 =

1 , 2π

dν =

1 π

for ν ≥ 1.

Remark 9. Notice that the bound above only was proved for a fixed (independent of h) v. In the present case veff,t depends on h. It is, however, easy (see [4, p. 99]) to see that the proof of the bound holds in our case as well. With the help of this lemma we can now prove Theorem 4: Proof. Please recall the definitions of JÑ,KIN and JÑ,INT from (5). Define HN (t, h, µ) = HN (h, µ) + t (JÑ,KIN (h, µ, a˜ ) + JÑ,INT (h, µ, a˜ )). We seek a lower bound on EN (t, h, µ) = inf ψ|HN (t, h, µ)|ψ. ψ=1

Remark 10. Formally, we should instead define HN (t1 , t2 , h, µ) = HN (h, µ) + t1 JÑ,KIN (h, µ, a˜ ) + t2 JÑ,INT (h, µ, a˜ ), and then analyze HN (t1 , 0, h, µ) and HN (0, t2 , h, µ) separately in order to obtain information on the two contributions to the current. For shortness, however, we will work with the two terms together since the methods of calculation are very similar.

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2 3 2 Lower bound. Let ψ ∈ ∧N j =1 L (R , C ), and let ρt = ρt,λ,β be the minimizer of scaled C-MTF theory, then

ψ|HN (t, h, µ)|ψ ≥ inf Spec HN0 (t, h, µ, veff,t ) (xj − xk ) · (xj − xk − t (ã(xj ) − a˜ (xk ))) + Z −1 ψ| |ψ |xj − xk |3 j 1, examples for T = R/Z from [12] will be the desired ones. To deal with a simple AF C ∗ -algebra A let us proceed as follows. First express A as ∪n An with An finite-dimensional, as in the proof of 2.1. With the notation there, we

One-Parameter Automorphism Groups of UHF Algebras

403

assume this time that all the multiplicities χn (i, j ) ≥ 4. We define a homomorphism ϕn,ij of Anj ⊗ C(T) into Anj ⊗ Mχn (i,j ) ⊗ C(T) by 0z 0z χn (i,j )−5 ϕn,ij (x)(z) = ⊕ x(1), ⊕ ⊕=0 10 10 and define accordingly ϕn : An ⊗ C(T) → An+1 ⊗ C(T). Then it follows [11] that the inductive limit C ∗ -algebra of (An ⊗ C(T), ϕn ) is isomorphic to the original A; we have thus expressed A as ∪n Bn where Bn = An ⊗ C(T) ⊂ Bn+1 . We define a sequence (Hn ) with self-adjoint Hn ∈ An ⊗ 1 ⊂ Bn by H1 = 0 and Hn = Hn−1 + i j hn,ij , where hn,ij ∈ 1 ⊗ Mχn−1 (i,j ) ⊗ 1 ⊂ Bn is given by 10 −1 0 hn,ij = ⊕ ⊕ 0 ⊕ · · · ⊕ 0. 01 0 −1 Then we define a one-parameter automorphism group α of A by αt |Bn = Ad eitHn |Bn . Note that Sp(Hn ) ⊂ Z and α2π = id. Then we can easily conclude that the fixed point algebra Aα is not AF; K1 (Aα ) is not trivial. What we need is this property to conclude that D(δα ) does not contain a canonical AF masa of A. Before proving this as a lemma below we shall show that D(δα ) is AF. Let z be the canonical unitary in C(T) and let zn = 1 ⊗ z ∈ An ⊗ C(T) = Bn . We have to use an estimate in the approximation of zn by a unitary with finite spectrum in Bm for m > n. Let u be the image of zn in Bm . Then the part of u(z) which is not constant in z = e2πt , t ∈ [0, 1), has an equal number of eigenvalues exp(±i2π 2n−m (t + k)) with k = 0, 1, . . . , 2m−n −1. By using this we approximate u by a unitary v ∈ Bm ∩(An ⊗1) with finite spectrum with the order u − v ≈ 2n−m (see [2]). But the norm of δα |Bm ∩ (An ⊗ 1) can be estimated as m − n, which yields δα (u) − δα (v) ≤ (m − n)u − v. Thus we can conclude that we can make the approximation in · δα , which shows that D(δα ) is AF. Lemma 3.5. If α is a periodic one-parameter automorphism group of a simple AF C ∗ algebra A and Aα is not AF, then D(δα ) does not contain a canonical AF masa of A. Proof. Let δ = δα and suppose that D(δ) contains a canonical AF masa C. Then by 3.1 we have a self-adjoint h ∈ A and an increasing sequence (An ) of finite-dimensional C ∗ -subalgebras of A with dense union such that δ + ad ih leaves An invariant and C ∩ An ∩ An−1 is a masa of An ∩ An−1 . Also δ + ad ih vanishes on C. Let β be the one-parameter automorphism group generated by δ +ad ih. Then there is an α-cocycle u such that βt = Ad ut αt . If α1 = id, then it follows that β1 = Ad u1 , i.e., u1 ∈ C. Since C is AF , we find a self-adjoint k ∈ C such that eik = u1 . Since (δ + ad ih)(k) = 0, one can conclude that δ +ad(ih−ik) generates a one-parameter automorphism group γ with γ1 = id such that γ leaves each An invariant. Since α and γ can be considered as actions of T and γ is a cocycle-perturbation of α, it follows that the crossed products A ×α T and A ×γ T are isomorphic. Since A ×γ T is AF as the inductive limit of An ⊗ C0 (Z) and Aα is a hereditary C ∗ -subalgebra of A ×α T, Aα must be AF. This contradiction shows that D(δ) cannot contain a canonical AF masa. " # If α is not periodic in the proof of 3.4, we could still use the following property for δα : Condition. For any ! > 0 there exists a ν > 0 with the following property: If u ∈ D(δα ) is a unitary with δα (u) < ν there is a continuous path (ut ) of unitaries in D(δα ) such that u0 = 1, u1 = u, and δα (ut ) < !, t ∈ [0, 1].

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A. Kishimoto

Proposition 3.6. Let α be a one-parameter automorphism group of a unital simple AF C ∗ -algebra. If D(δα ) contains a canonical AF masa, then the above Condition for δα is satisfied. Proof. First suppose that A is finite-dimensional. Then there is an h = h∗ ∈ A such that δα = ad ih. The condition δα (u) < ν reads h − uhu∗ < ν. Then the Condition follows from Theorem 4.1, which will be given later. Note that here the choice of ν does not depend on A nor δα . Let (An ) be an increasing sequence of finite-dimensional C ∗ -subalgebras of A with dense union such that αt (An ) = An . Let u ∈ D(δα ) be a unitary with δα (u) < ν. Since ∪n An is dense in D(δα ), there is a sequence (un ) in ∪n An such that u − un δα → 0. Since un u∗ ≈ 1 and δα (un u∗ ) ≈ 0, we can find a continuous path (un (t)) in D(δα ) such that un (0) = un , un (1) = u, and δα (un (t)) is of the order of δα (u). Thus we can suppose that u ∈ ∪n An and the assertion follows from the previous paragraph. If δα = δβ + ad ih with βt (An ) = An , there is a sequence (hn ) with hn = h∗n ∈ An such that h − hn → 0. Then δβ + ad ihn = δα + ad(ihn − ih) generates an AF locally representable action. Thus we may as well assume that α is AF locally representable. This completes the proof. " # If α is periodic and K1 (Aα ) is not trivial, then the Condition is not satisfied. But we note: Remark 3.7. In the above proposition the converse does not hold. In fact the example in the proof of 2.1 satisfies the above Condition. By using Proposition 3.6 we can give more examples with the property that D(δα ) contains no canonical AF masa. For example, as in the proof of Proposition 3.4, suppose that we express A as ∪n Bn with Bn = An ⊗ C(T) and that we define an α by defining hn,ij . This time we choose hn,ij to be of the form: −an 0 an 0 ⊕ ⊕ 0 ⊕ · · · ⊕ 0, hn,ij = 0 an 0 −an where (an ) is an arbitrary sequence such that a = inf an > 0. If we had a continuous path (ut ) of unitaries in Bn such that u0 = 1, u1 = z1 , and δα (ut ) < a for the canonical unitary z1 , we could reach a contradiction as follows. Let Hn = j λj Ej be the spectral decomposition with λ1 > λ2 > · · · . By the assumption we have that λ1 − λ2 ≥ a. Since [Hn , ut ] < a, we can estimate (λ1 − λj )E1 ut Ej < a, j

which shows that

E1 ut (1 − E1 ) < 1.

Since E1 ut E1 u∗t E1 + E1 ut (1 − E1 )2 E1 ≥ E1 , it follows that E1 ut E1 is invertible. Since E1 u0 E1 = E1 and E1 u1 E1 = z1 E1 is a unitary with non-trivial K1 , this is a contradiction. If we have a continuous path of unitaries in D(δα ) with the above property, we approximate the path by a path in ∪n Bn to reach the contradiction. Since ∪n Bn is


405

dense in the Banach *-algebra D(δα ), this is possible. Thus we have shown that D(δα ) contains no canonical AF masa. If lim sup an < ∞, one can also show that D(δα ) is AF. There is a standard way to construct a one-parameter automorphism group α of a certain UHF algebra through an interaction of a quantum spin system [8]. If the interaction is quantum, we expect that any inner perturbation of α is not AF locally representable. We also expect that the quasi-free one-parameter automorphism group of the CAR algebra induced by a one-particle Hamiltonian with continuous spectrum [8, 17] or any inner perturbation of it is not AF locally representable. We conclude this section by posing: Problem. Prove the above conjecture. 4. A Homotopy Lemma We prove here a technical lemma which is used in the proof of Proposition 3.6. With an additional assumption on h below (saying the norm is less than 1), this follows from Lemma 5.1 of [6]. To remove this assumption we have to replace a certain approximation argument used there by a constructive argument, which will constitute the main part of the proof. Theorem 4.1. For any ! > 0 there exists a ν > 0 satisfying the following condition: For any unital AF algebra A and u, h ∈ A such that u∗ u = uu∗ = 1, h∗ = h, and [h, u] < ν, there is a rectifiable path (ut )t∈[0,1] in the unitary group of A such that u0 = 1, u1 = u, [h, ut ] < !, and the length of (ut ) is smaller than 3π + !. Proof. We may assume that A is finite-dimensional; in particular we assume that h is diagonal. Let δ > 0 be a sufficiently small number, which will be chosen later depending on !. Let f be a C ∞ -function on R such that f ≥ 0, f (t)dt = 1, and supp fˆ ⊂ (−δ, δ). Define x = f (t)eith ue−ith dt. Then it follows that x ≤ 1, [h, x] ≤ [h, u], and ith x − u ≤ f (t)(Ad e (u) − u)dt ≤ f (t)|t|dt[h, u], where we have used that Ad e

ith

t

(u) − u =

eish [ih, u]e−ish ds ≤ |t|[h, u].

0

If we denote by Eh the spectral measure of h, then we have that for x # = x or x ∗ and t ∈ R, Eh (−∞, t)x # Eh [t + δ, ∞) = 0. We define a projection ek for each k ∈ Z by ek = Eh [2kδ, 2(k + 1)δ). Then there are only a finite number of non-zero ek . It follows that ∗

xek x ≤ Eh [(2k − 1)δ, (2k + 3)δ).

k ek

= 1 and

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We suppose that x − u < µ, where µ can be made arbitrarily small by choosing ν ∗ ∗ ∗ 2 ∗ small. Since 0 ≤ 1∗ − x x < 2µ and 0 ∗≤ xek x − (xek x ) ≤ 2µxek x , we have that 0 ≤ 1 − k xek x ≤ 2µ, and Sp(xek x ) ⊂ {0} ∪ [1 − 2µ, 1]. If x were a unitary (and so xek x ∗ were a projection), we could skip most of the arguments below. What we will do next is to construct a unitary v by using x such that v is close to u and satisfies that vek v ∗ ≤ Eh [(2k − 1)δ, (2k + 3)δ). By multiplying Fj = Eh [(2j − 1)δ, ∞) with 1 − k xek x ∗ whose norm is less than 2µ, we get that xek x ∗ + Fj xej −1 x ∗ − Fj < 2µ, k≥j

which implies that

Fj xej −1 x ∗ − xej −1 x ∗ Fj < 4µ.

Since (Fj xej −1 x ∗ Fj )2 − Fj xej −1 x ∗ Fj < 4µ + Fj ((xej −1 x ∗ )2 − xej −1 x ∗ )Fj < 6µ, Fj xej −1 x ∗ Fj is close to a projection for a small µ. If we denote by fj+−1 the support projection of Fj xej −1 x ∗ Fj , then we have that fj+−1 − Eh [(2j − 1)δ, ∞)xej −1 x ∗ Eh [(2j − 1)δ, ∞) < 6µ , √ where µ = (1 − 1 − 24µ)/12 ≈ µ, which we again denote by µ below. Note that fj+−1 − fj+−1 xej −1 x ∗ < 10µ and that fj+−1 ≤ Eh [(2j − 1)δ, (2j + 1)δ). In the same way we denote by fj− the support projection of Eh (−∞, (2j + 1)δ)xej x ∗ Eh (−∞, (2j + 1)δ) = Eh [(2j − 1)δ, (2j + 1)δ)xej x ∗ Eh [(2j − 1)δ, (2j + 1)δ); then we have that fj− − Eh [(2j − 1)δ, (2j + 1)δ)xej x ∗ Eh [(2j − 1)δ, (2j + 1)δta) < 6µ. Let fj = fj− + fj+ . Then summing up the above calculations, we obtain that fj − xej x ∗ = fj − (Eh [(2j − 1)δ, (2j + 1)) + Eh [(2j + 1)δ, (2j + 3)δ))xej x ∗ < 8µ + fj − Eh [(2j − 1)δ, (2j + 1)δ)xej x ∗ · Eh [(2j − 1)δ, (2j + 1)δ) − Eh [(2j + 1)δ, (2j + 3)δ)xej x ∗ Eh [(2j + 1)δ, (2j + 3)δ) < 14µ. Hence if µ is small, fj xej (ej x ∗ fj xej )−1/2 defines a partial isometry with initial projection ej and final projection fj . Let gj− = Eh [(2j − 1)δ, (2j + 1)δ) − fj+−1 , gj+ = Eh [(2j + 1)δ, (2j + 3)δ) − fj−+1 , and gj = gj− + gj+ . Since gj xej −1 x ∗ = gj− xej −1 x ∗ = (1 − fj+−1 )Eh [(2j − 1)δ, (2j + 1)δ)xej −1 x ∗ < 4µ etc., we obtain that gj xej x ∗ − gj < 10µ.


407

Since fj −1 xej x ∗ < 6µ + Eh [(2j − 1)δ, (2j + 1)δ)xej −1 · x ∗ Eh [(2j − 1)δ, (2j + 1)δ)xej x ∗ < 10µ + Eh [(2j − 1)δ, (2j + 1)δ)xej −1 x ∗ xej x ∗ < 12µ, we have that xej x ∗ − gj ≤ gj xej x ∗ − gj + fj+−1 xej x ∗ + fj−+1 xej x ∗ , < 34µ.

Let v=

j

f2j xe2j (e2j x ∗ f2j xe2j )−1/2 +

g2j −1 xe2j −1 (e2j −1 x ∗ g2j −1 xe2j −1 )−1/2 ,

j

which is the unitary part of the polar decomposition of y = j f2j xe2j + √ ∗ < 14µ, we have that v − y < 1/ 1 − 14µ − 1. g xe . Since 0 ≤ 1 − yy 2j −1 2j −1 j Note also that vej v ∗ ≤ Eh [(2j − 1)δ, (2j + 3)δ). Since (f2j − 1)xe2j 2 = ∗ ∗ supj (f2j − 1)xe2j x (f2j − 1) ≤ supj xe2j x − f2j , we get that (f2j − 1)xe2j < 14µ. Since (g2j −1 − 1)xe2j −1 x ∗ (g2j −1 − 1) < 34µ, we get that (g2j −1 − 1)xe2j −1 < 34µ. Since y √ − x ≤ (f2j − 1)xe2j + (g2j −1 − 1)xe2j −1 , we get y − x < √ √ 14µ + 34µ < 10 µ. Hence we get that if µ is sufficiently small, √ √ v −u < v −y+y −x+x −u < 1/ 1 − 14µ−1+10 µ+µ < 10(µ+ µ). √ We assume that the constant 10(µ + µ) is sufficiently small. Let k = j 2j δEh [2j δ, 2(j + 1)δ) = 2j δej . Then h − k < 2δ and [k, u] ≤ 2h − k + [h, u] < 4δ + [h, u] < 4δ + ν. Since vek v ∗ ≤ ek−1 + ek + ek+1 , we have that k − 2δ ≤ vkv ∗ ≤ k + 2δ. Hence it follows that vkv ∗ − k ≤ 2δ. Since [vu∗ , k] ≤ [v, k] + [u, k] < 6δ + ν and vu∗ = eia with a ∗ = a ≈ 0, we get that [a, k] ≈ 0 (up to the order of 6δ + ν). We take a continuous path t ∈ [0, 1] → wt = eita u of length a. Then since [k, wt ] = [k, eita ]u + eita [k, u] ≈ 0 (up to the order of 10δ + 2ν) and w1 = v, we may replace u by v. From now on we can proceed as in the proof of Lemma 5.1 of [6]. Let En = j ≥n ej . Then k equals 2δ n>m En +2mδ, where m is the biggest integer satisfying Em = 1, and the sequence (En )n≥m of projections decreases from 1 to 0 as n increases. Let Fn = vEn v ∗ . Then we have that En+1 ≤ Fn ≤ En−1 . Since F2n+2 ≤ F2n+1 ≤ F2n and F2n+2 ≤ E2n+1 ≤ F2n , we find a continuous path (wt ) of unitaries of ∗ length at most π such that w0 = 1, [wt , F2n −F 2n+2 ] = 0, and w1 (F2n+1 −F2n+2 )w1 = ∗ E2n+1 − F2n+2 for all n. Since vkv − 4δ 2n>m F2n − 2mδ ≤ 2δ, we have that wt vkv ∗ wt∗ − vkv ∗ ≤ 4δ, and hence [wt v, k] ≤ 6δ. Next we find a continuous path (zt ) of unitaries of length at most π such that z0 = 1, [zt , E2n−1 − E2n+1 ] =

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∗ 0, w1 vkv ∗ w1∗ = 2δ( 2n>m F2n + 2n+1 . Since and z1 (F2n − E2n+1 )z1 = E2n − E ∗ ∗ 2n+1>m E2n+1 ) + 2mδ and w1 vkv w1 − 4δ 2n+1>m E2n+1 − 2mδ ≤ 2δ, we ∗ ∗ ∗ ∗ get that zt w1 vkv w1 zt − w1 vkv w1 ≤ 4δ, and hence [zt w1 v, k] ≤ 10δ. Since z1 w1 vkv ∗ w1∗ z1∗ = k, we can find a continuous path of unitaries from z1 w1 v to 1 in the commutant of k, whose length is at most π . (Here we use the fact that the unitary group of eAe for any projection e ∈ A is connected.) Note that the path obtained by combining these three paths has length at most 3π . The above calculations show that we can choose δ just depending on !. (For example δ should be smaller than !/15 and much smaller √ than 1.) Then we choose ν independently (such that ν is smaller than !/30 and 10(µ + µ) is much smaller than !, where µ is proportional to ν as shown at the beginning of the proof). This concludes the proof. " # References 1. Blackadar, B.: Symmetries of the CAR algebras. Ann. of Math. 131, 589–623 (1990) 2. Blackadar, B., Bratteli, O., Elliott, G.A. and Kumjian, A.: Reductions of real rank in inductive limits of C ∗ -algebras. Math. Ann. 292, 111–126 (1992) 3. Bratteli, O.: Derivations, dissipations and group actions on C ∗ -algebras. Lecture Notes in Math. 1229, Berlin–Heidelberg–New York: Springer, 1986 4. Bratteli, O., Elliott, G.A., Evans, D.E. and Kishimoto, A.: On the classification of inductive limits of inner actions of a compact group. In: Current topics in operator algebras, edited by H. Araki et al., London–Hong Kong–Singapore–New Jersey: Word Scientific, 1991, pp. 13–24 5. Bratteli, O., Elliott, G.A., Evans, D.E. and Kishimoto, A.: Finite group actions on AF algebras obtained by folding the interval. K-theory 8, 443–464 (1994) 6. Bratteli, O., Elliott, G.A., Evans, D.E. and Kishimoto,A.: Homotopy of a pair of approximately commuting unitaries in a simple purely infinite unital C ∗ -algebra. J. Funct. Anal. 160, 466–523 (1998) 7. Bratteli, O. and Kishimoto, A.: Generation of semi-groups, and two-dimensional quantum lattice systems. J. Funct. Anal. 35, 344–368 (1980) 8. Bratteli, O. and Robinson, D.W.: Operator algebras and quantum statistical mechanics, I, II. Berlin– Heidelberg–New York: Springer, 1979, 1981 9. Brown, L. and Pedersen, G.K.: C ∗ -algebras of real rank zero. J. Funct. Anal. 99, 131–149 (1991) 10. Dadarlat, M. and Gong, G.: A classification result for approximately homogeneous C ∗ -algebras of real rank zero. Preprint 11. Elliott, G.A.: On the classification of C ∗ -algebras of real rank zero. J. reine angew. Math. 443, 179–219 (1993) 12. Evans, D.E. and Kishimoto, A.: Compact group actions on UHF algebras obtained by folding the interval. J. Funct. Anal. 98, 346–360 (1991) 13. Kishimoto, A.: Actions of finite groups on certain inductive limit C ∗ -algebras. Internat. J. Math. 1, 267– 292 (1990) 14. Kishimoto, A.: Locally representable one-parameter automorphism groups of AF algebras and KMS states. Rep. Math. Phys. 45, 333–356 (2000) 15. Ringrose, J.R.: Automatic continuity of derivations of operator algebras. J. London Math. Soc. 5, 432–438 (1972) 16. Sakai, S.: On one-parameter subgroups of *-automorphisms on operator algebras and the corresponding unbounded derivations. Am. J. Math. 98, 427–440 (1976) 17. Sakai, S.: Operator Algebras in Dynamical Systems. Cambridge: Cambridge Univ. Press, 1991 18. Strˇatilˇa S. and Voiculescu, D.: Representations of AF-algebras and of the group U (∞). Lecture Notes in Math. 486, Berlin–Heidelberg–New York: Springer, 1975 Communicated by H. Araki

Commun. Math. Phys. 216, 409 – 430 (2001)

Communications in



Representations of Hermitian Kernels by Means of Krein Spaces. II. Invariant Kernels T. Constantinescu1 , A. Gheondea2 1 Department of Mathematics, University of Texas at Dallas, Box 830688, Richardson, TX 75083-0688, USA.


2 Institutul de Matematic˘a al Academiei Române, C.P. 1-764, 70700 Bucure¸sti, România.

E-mail: [email protected] Received: 27 March 2000/ Accepted: 5 September 2000

Abstract: In this paper we study hermitian kernels invariant under the action of a semigroup with involution. We characterize those hermitian kernels that realize the given action by bounded operators on a Kre˘ın space. This is motivated by the GNS representation of ∗-algebras associated to hermitian functionals, the dilation theory of hermitian maps on C ∗ -algebras, as well as others. We explain the key role played by the technique of induced Kre˘ın spaces and a lifting property associated to them.

1. Introduction The Hilbert space H associated to a positive definite kernel K is an abstract version of the L2 space associated to a positive measure and the Kolmogorov decomposition of K gives a useful expansion of the elements of H in terms of a geometrical model of a stochastic process with covariance kernel K. Therefore, it is quite natural to seek similar constructions for an arbitrary kernel. While the decomposition into a real and an imaginary part can be realized without difficulties, the study of hermitian kernels is no longer straightforward. This was shown in the work of L. Schwartz [25], where a characterization of the hermitian kernels admitting a Jordan decomposition was obtained in terms of a boundedness condition that we call the Schwartz condition (the statement (1) of Theorem 2.5 in Sect. 2 below). A key difficulty of the theory was identified in [25] in the lack of uniqueness of the associated reproducing kernel spaces. It was shown in [5] that the Schwartz condition is also equivalent to the existence of a Kolmogorov decomposition, while the uniqueness of the Kolmogorov decomposition was characterized in spectral terms (Theorem 2.5 and, respectively, Theorem 2.6 in Sect. 2). The purpose of this paper is to continue these investigations by considering hermitian kernels with additional symmetries given by the action of a semigroup. The main result gives a characterization of those hermitian kernels that produce a representation of the action by bounded operators on a certain Kre˘ın space. It turns out that such a

410

T. Constantinescu, A. Gheondea

result has many applications and in this paper we discuss GNS representations on inner product spaces. The paper is organized as follows. In Sect. 2 we review the concept of induced Kre˘ın space and we show its key role in the construction of Kolmogorov decompositions as described in [5]. A new result is added here in connection with a lifting property for induced Kre˘ın spaces that is related to an important inequality of M. G. Kre˘ın. Theorem 2.3 gives an example of an induced Kre˘ın space without the lifting property, adding one more pathology to the study of hermitian kernels. Incidentally, this result answers negatively a question raised in [9]. We show the applicability of our results to questions concerning GNS representations of ∗-algebras on Kre˘ın spaces. The whole issue is motivated by the lack of positivity in some models in local quantum field theories. We relate these questions to properties of Kolmogorov decompositions so that we can characterize the existence (Theorem 2.8) and the uniqueness (Theorem 2.9) of the GNS data. This is also a motivation for considering the general case of semigroups with involution. For example, Theorem 5.1 characterizes the boundedness of the GNS data. In Sect. 3 we prove the main result of the paper. We consider the action of a semigroup on a hermitian kernel and Theorem 3.1 gives the conditions that insure the representation of this action as a semigroup of bounded operators on a Kre˘ın space. We also address the uniqueness property of such representations. While the case of the trivial semigroup with one element is settled in [5] (Theorem 2.6 in Sect. 2) and Theorem 3.4 gives another partial answer, the general case remains open. The proof used for the trivial semigroup cannot be easily extended precisely because Theorem 2.3 is true. In Sect. 4 we analyze the case when the projective representation given by Theorem 3.1 is fundamentally reducible or, equivalently, it is similar to a projective Hilbert space representation, a question closely related to other similarity problems and uniformly bounded representations. The last section contains an application related to the GNS construction. 2. Preliminaries We briefly review the concept of a Kolmogorov decomposition for hermitian kernels. The natural framework to deal with these kernels is that of Kre˘ın spaces. We recall first some definitions and a few items of notation. An indefinite inner product space (H, [·, ·]) is called Kre˘ın space provided that there exists a positive inner product ·, · on H turning (H, ·, ·) into a Hilbert space such that [ξ, η] = J ξ, η, ξ, η ∈ H, for some symmetry J (J ∗ = J −1 = J ) on H. Such a symmetry J is called a fundamental symmetry. The norm ξ 2 = ξ, ξ is called a unitary norm. The underlying Hilbert space topology of K is called the strong topology and does not depend on the choice of the fundamental symmetry. For two Kre˘ın spaces H and K we denote by L(H, K) the set of linear bounded operators from H to K. For T ∈ L(H, K) we denote by T the adjoint of T with respect to [·, ·]. We say that A ∈ L(H) is a selfadjoint operator if A = A. A possibly unbounded operator V between two Kre˘ın spaces is called isometric if [V ξ, V η] = [ξ, η] for all ξ, η in the domain of V . Also, we say that the operator U ∈ L(H) is unitary if U U = U U = I , where I denotes the identity operator on H. The notation T ∗ is used for the adjoint of T with respect to the positive inner product ·, ·. 2.1. Kre˘ın spaces induced by selfadjoint operators. Many difficulties in dealing with operators on Kre˘ın spaces are caused by the lack of a well-behaved factorization theory.

Representations of Hermitian Kernels of Krein Spaces

411

The concept of induced space turned out to be quite useful in this direction. Thus, for a selfadjoint operator A in L(H) we define a new inner product [·, ·]A on H by the formula [ξ, η]A = [Aξ, η],

ξ, η ∈ H,

(2.1)

and a pair (K, ) consisting of a Kre˘ın space K and a bounded operator ∈ L(H, K) is called a Kre˘ın space induced by A provided that has dense range and the relation [ξ, η]K = [ξ, η]A

(2.2)

holds for all ξ, η ∈ H, where [·, ·]K denotes the indefinite inner product on K. One well-known example is obtained in the following way. Example 2.1. Let J be a fundamental symmetry on H and let ·, ·J be the associated positive inner product turning H into a Hilbert space. Then J A is a selfadjoint operator on this Hilbert space and let H− and H+ be the spectral subspaces of J A corresponding to (−∞, 0) and, respectively, (0, ∞). We obtain the decomposition H = H− ⊕ ker A ⊕ H+ . Note that (H− , −[·, ·]A ) and (H+ , [·, ·]A ) are positive inner product spaces and hence they can be completed to the Hilbert spaces K− and, respectively, K+ . Let KA be the Hilbert direct sum of K− and K+ and denote by ·, ·KA the positive inner product on KA . Define JA (k− ⊕ k+ ) = −k− ⊕ k+ for k− ∈ K− and k+ ∈ K+ . We can easily check that JA is a symmetry on KA and then the inner product [k, k ]KA = JA k, k KA turns KA into a Kre˘ın space. The map A : H → KA is defined by the formula A ξ = [PH− ξ ] ⊕ [PH+ ξ ], where ξ ∈ H, PH± denotes the orthogonal projection of the Hilbert space (H, ·, ·J ) onto the subspace H± , and [PH± ξ ] denotes the class of PH± ξ in K± . Then one checks that (KA , A ) is a Kre˘ın space induced by A. In addition, if J A = SJ A |J A| is the polar decomposition of J A, then we note that JA A = A SJ A .

(2.3)

This example proved to be very useful since it is accompanied by a good property concerning the lifting of operators, as shown by a classical result of M.G. Kre˘ın, [16]. The result was rediscovered by W.J. Reid [24], P.D. Lax [17], and J. Dieudonné [6]. The indefinite version presented below was proved in [7] by using a 2×2 matrix construction that reduces the proof to the positive definite case. Theorem 2.2. Let A and B be bounded selfadjoint operators on the Kre˘ın spaces H1 and H2 . Assume that the operators T1 ∈ L(H1 , H2 ) and T2 ∈ L(H2 , H1 ) satisfy the relation T2 A = BT1 . Then there exist (unique) operators T˜1 ∈ L(KA , KB ) and T˜2 ∈ L(KB , KA ) such that T˜1 A = B T1 , T˜2 B = A T2 , and [T˜1 f, g]KB = [f, T˜2 g]KA for all f ∈ KA , g ∈ KB .

412


Theorem 2.2 will be used in an essential way in the proof of the main result of the next section and it is also related to the uniqueness property of a Kolmogorov decomposition for invariant hermitian kernels. For these reasons we discuss one more question related to this result, namely whether this lifting property holds for other induced Kre˘ın spaces. More precisely, two Kre˘ın spaces (Ki , i ), i = 1, 2, induced by the same selfadjoint operator A ∈ L(H) are unitarily equivalent if there exists a unitary operator U in L(K1 , K2 ) such that U 1 = 2 . Theorem 2.8 in [5] shows that there exist selfadjoint operators with the property that not all of their induced Kre˘ın spaces are unitarily equivalent. Let (K, ) be a Kre˘ın space induced by A. We say that (K, ) has the lifting property if for any pair of operators T , S ∈ L(H) satisfying the relation AT = SA there exist ˜ = S. From Theorem 2.2 unique operators T˜ , S˜ ∈ L(K) such that T˜ = T , S it follows that the induced Kre˘ın space (KA , A ) constructed in Example 2.1 has the lifting property, as do all the others which are unitarily equivalent to it. However, as the following result shows, this is not true for all induced Kre˘ın spaces of A. Theorem 2.3. There exists a selfadjoint operator that has an induced Kre˘ın space without the lifting property. Proof. Let H0 be an infinite dimensional Hilbert space and A0 is a bounded selfadjoint operator in H0 such that 0 ≤ A0 ≤ I , ker A0 = 0, and the spectrum of A0 accumulates to 0, equivalently, its range is not closed. Consider the Hilbert space H = H0 ⊕ H0 as well as the bounded selfadjoint operator A0 0 A= . (2.4) 0 −A0 Let K be the Hilbert space H with the indefinite inner product [·, ·] defined by the symmetry I 0 J = . 0 −I Consider the operator ∈ L(H, K), I −(I − A0 )1/2 = . (I − A0 )1/2 −I

(2.5)

It is a straightforward calculation to see that ∗ J = A and, by performing a Frobenius– Schur factorization, it follows that has dense range. Thus, (K, ) is a Kre˘ın space induced by A and we show that it does not have the lifting property. Let T be an operator in L(H) such that, with respect to its 2 × 2 block-matrix representation, all its entries Tij , i, j = 1, 2, commute with A0 . Define the operator S = J T J and note that AT = SA. Let us assume that there exists a bounded operator T ∈ L(K) such that T = T . Then, there exists the constant C = T K < ∞ such that

T ξ ≤ C ξ ,

ξ ∈ H,

or, equivalently, that T ∗ H T ≤ C 2 H,

(2.6)


where

413

2 − A0 −2(I − A0 )1/2 H = . −2(I − A0 )1/2 2 − A0

Taking into account that A0 commutes with all the other operator entries involved in (2.6), it follows that the inequality (2.6) is equivalent to I − I − T∗ T ≤ C2 , (2.7) − I − I where we denoted = 2(I − A0 )1/2 (2 − A0 )−1 . Note that, by continuous functional calculus, is an operator in H such that 0 ≤ ≤ I and its spectrum accumulates to 1. The use of the Frobenius–Schur factorization I 0 I − I − I 0 (2.8) = 0 I − I − I 0 I − 2 suggests to take

T =

I − , 0 I

and this choice is consistent with our assumption that all its entries commute with A0 . Since T is bounded invertible, from (2.7) we get I 0 I − . ≤ C2 − I 0 I − 2 Looking at the lower right corners of the matrices in the previous inequality we get I ≤ C 2 (I − 2 ) which yields a contradiction since the spectrum of the operator I − 2 accumulates to 0. Remark 2.4. (1) Incidentally, the example in Theorem 2.3 can be used to answer the following question raised in [9], Lecture 6A: let A be a selfadjoint operator on a Kre˘ın space H, and construct a factorization A = DD , where D ∈ L(D, H) is an one-to-one operator. If X ∈ L(H) and XA is selfadjoint, does there exist a (unique) selfadjoint operator Y ∈ L(D) such that XD = DY ? We show that the answer to this question is negative. Indeed, an operator D as above produces the induced Kre˘ın space (D, D ) for A. Let A be the operator defined by (2.4). Let us take I I T = . −I I One checks that AT = T ∗ A. Define X = T ∗ , then XA is selfadjoint. If Y ∈ L(D) exists such that XD = DY , then Y ∗ D ∗ = D ∗ T and a similar reasoning as in the proof of Theorem 2.3 shows that from (2.7) and (2.8) we get 2(3 + 2 − + I ) ≤ C 2 (I − 2 ), which is impossible since the spectrum of the operator from the left side is bounded away from 0.

414


(2) One might ask whether another additional assumption on the operator T that is frequently used in applications, namely that T is A-isometric, could enforce the lifting property. To see that this is not the case, let us take 2 I − 21 I . T =√ 1 3 2 I −I It is easy to prove that T ∗ AT = A, that is, T is A-isometric. Noting that T is boundedly invertible, this corresponds to S = T ∗−1 .As before, from (2.7) and (2.8) we get 43 (−3 + 15 2 5 2 2 4 − 3 + 4 I ) ≤ C (I − ). But this is again contradictory since the spectrum of the operator from the left side is bounded away from 0. 2.2. Kolmogorov decompositions of hermitian kernels. We can use the concept of induced space in order to describe the Kolmogorov decomposition of a hermitian kernel. Let X be an arbitrary set. A mapping K defined on X × X with values in L(H), where (H, [·, ·]H ) is a Kre˘ın space, is called a hermitian kernel on X if K(x, y) = K(y, x) for all x, y ∈ X. Let F0 (X, H) denote the vector space of H-valued functions on X having finite support. We associate to K an inner product on F0 (X, H) by the formula: [f, g]K = [K(x, y)f (y), g(x)]H (2.9) x,y∈X

for f, g ∈ F0 (X, H). We say that the hermitian kernel L : X × X → L(H) is positive definite if the inner product [·, ·]L associated to L by the formula (2.9) is positive. On the set of hermitian kernels on X with values in L(H) we also have a natural partial order defined as follows: if A, B are hermitian kernels, then A ≤ B means [f, f ]A ≤ [f, f ]B for all f ∈ F0 (X, H). Following L. Schwartz [25], we say that two positive definite kernels A and B are disjoint if for any positive definite kernel P such that P ≤ A and P ≤ B it follows that P = 0. A Kolmogorov decomposition of the hermitian kernel K is a pair (V ; K), where K is a Kre˘ın space and V = {V (x)}x∈X is a family of bounded operators V(x) ∈ L(H, K) such that K(x, y) = V (x) V (y) for all x, y ∈ X, and the closure of x∈X V (x)H is K ([15, 22, 10]). Note that here and throughout this paper ∨ stands for the linear manifold generated by some set, without taking any closure. The next result, obtained in [5], settles the question concerning the existence of a Kolmogorov decomposition for a given hermitian kernel. Theorem 2.5. Let K : X × X → L(H) be a hermitian kernel. The following assertions are equivalent: (1) There exists a positive definite kernel L : X × X → L(H) such that −L ≤ K ≤ L. (2) K has a Kolmogorov decomposition. The condition −L ≤ K ≤ L of the previous result appeared in the work of L. Schwartz [25] concerning the structure of hermitian kernels. We will call it the Schwartz condition. It is proved in [25] that this condition is also equivalent to the Jordan decomposition of K, which means that the kernel K is a difference of two disjoint positive definite kernels. It is convenient for our purpose to review the construction of the Kolmogorov decomposition. We assume that there exists a positive definite kernel L : X × X → L(H) such that −L ≤ K ≤ L. Let HL be the Hilbert space obtained


415

by the completion of the quotient space F0 (X, H)/NL with respect to [·, ·]L , where NL = {f ∈ F0 (X, H) | [f, f ]L = 0} is the isotropic subspace of the inner product space (F0 (X, H), [·, ·]L ). Since −L ≤ K ≤ L is equivalent to 1/2

1/2

|[f, g]K | ≤ [f, f ]L [g, g]L

(2.10)

for all f, g ∈ F0 (X, H) (see Proposition 38, [25]), it follows that NL is a subset of the isotropic subspace NK of the inner product space (F0 (X, H), [·, ·]K ). Therefore, [·, ·]K uniquely induces an inner product on HL , still denoted by [·, ·]K , such that (2.10) holds for f, g ∈ HL . By the Riesz representation theorem we obtain a selfadjoint contractive operator AL ∈ L(HL ), referred to as the Gram, or metric operator of K with respect to L, such that [f, g]K = [AL f, g]L

(2.11)

for all f, g ∈ HL . Let (KAL , AL ) be the Kre˘ın space induced by AL given by Example 2.1. For ξ ∈ H and x ∈ X, we define the element ξx ∈ F0 (X, H) by the formula: ξ, y = x (2.12) ξx (y) = 0, y = x. Then we define V (x)ξ = AL [ξx ],

(2.13)

where [ξx ] denotes the class of ξx in HL and it can be verified that (V ; KAL ) is a Kolmogorov decomposition of the kernel K. We finally review the uniqueness property of the Kolmogorov decomposition. Two Kolmogorov decompositions (V1 , K1 ) and (V2 , K2 ) of the same hermitian kernel K are unitarily equivalent if there exists a unitary operator # ∈ L(K1 , K2 ) such that for all x ∈ X we have V2 (x) = #V1 (x). The following result was obtained in [5]. Theorem 2.6. Let K be a hermitian kernel which has Kolmogorov decompositions. The following assertions are equivalent: (1) All Kolmogorov decompositions of K are unitarily equivalent. (2) For each positive definite kernel L such that −L ≤ K ≤ L, there exists $ > 0 such that either (0, $) ⊂ ρ(AL ) or (−$, 0) ⊂ ρ(AL ), where AL is the Gram operator of K with respect to L. 2.3. Motivation. In this subsection we give some motivation for the study of hermitian kernels invariant under the action on a semigroup with involution. Thus, we first discuss the GNS representation for unital ∗-algebras from the point of view of hermitian kernels, showing that considering only actions on groups is not sufficient. Our goal is to make connections with some constructions of interest in quantum field theories such as those summarized in [26]. Another important issue is that we should consider projectively invariant hermitian kernels. This is emphasized, for example, by the Fock representation of the canonical commutation relations obtained from an action of the rigid motions of a Hilbert space on the exponential vectors of a Fock space, since it is natural to consider a similar construction for other groups, the like the Poincaré group, involving an indefinite inner product. Various models involving Fock spaces associated to indefinite inner products were studied in [19, 26]. Here we emphasize that the Kolmogorov decomposition gives a simple construction of the Weyl exponentials (the related topic of the representations of the Heisenberg algebra in Kre˘ın spaces is taken up in [18]).

416


2.3.1. Representations of ∗-algebras associated to hermitian forms. Let A be a ∗algebra with identity 1 and let Z be a linear hermitian functional on A with mass 1 (Z(1) = 1). Then A is a unital multiplicative semigroup with involution acting on itself by φ(a, x) = xa ∗

(2.14)

KZ (x, y) = Z(xy ∗ )

(2.15)

for a, x ∈ A. We define

for x, y ∈ A. Then KZ is a hermitian kernel on A with scalar values and satisfies the symmetry relation KZ (x, φ(a, y)) = Z(xay ∗ ) = KZ (φ(a ∗ , x), y)

(2.16)

for a, x, y ∈ A. In order to describe the GNS construction for Z we will use the concept of unbounded representations of A. Thus, a mapping π of A into the set of closable operators defined on a common dense domain D(π ) of a Banach space K is called a closable representation if it is linear, D(π ) is invariant under all operators π(a), a ∈ A, and π(ab) = π(a)π(b) for all a, b ∈ A. If, in addition, K is a Kre˘ın space and, for all a ∈ A, the domain of π(a) contains D(π ) and π(a) |D(π ) = π(a ∗ ),

(2.17)

then π is called a hermitian closable representation on the Kre˘ın space K (or, a J representation, as introduced in [20], see also [13]). The GNS data (π, K, ,) associated to Z consists of a hermitian closable representation of A on the Kre˘ın space K and a vector , ∈ D(π ) such that

Z(a) = [π(a),, ,]K

(2.18)

for all a ∈ A and a∈A π(a), = D(π ). It was known that not every hermitian functional Z admits GNS data. Characterizations of those Z that do admit GNS data appeared in papers such as [19, 1, 13]. We first show that the GNS data associated to a hermitian form can be equivalently described in terms of Kolmogorov decompositions of the kernel KZ . The proof is straightforward and can be omitted. Proposition 2.7. Let A be a unital ∗-algebra, let Z be a linear hermitian functional on A with Z(1) = 1, and consider the kernel KZ associated to Z by (2.15). For every GNS data (π, K, ,) of Z define V (a)λ = π(a ∗ )λ,,

a ∈ A, λ ∈ C.

(2.19)

Then (V , K) is a Kolmogorov decomposition of the hermitian kernel KZ and (2.19) establishes a bijective correspondence between the set of all GNS data of Z and the set of all Kolmogorov decompositions of KZ . In particular, Z admits GNS data if and only if the hermitian kernel KZ has Kolmogorov decompositions. As a consequence, Proposition 2.7 reduces the characterization of those hermitian functionals that admit GNS data to Theorem 2.5. A different characterization was obtained in Theorem 2 in [13].


417

Theorem 2.8. Let A be a unital ∗-algebra and let Z be a linear hermitian functional on A with Z(1) = 1. Then Z admits GNS data if and only if there exists a positive definite scalar kernel L on A such that |Z(

n i,j =1

λi λj xi xj∗ )| ≤

n

λi λj L(xi , xj ),

i,j =1

n ∈ N, {λi }ni=1 ⊂ C, {xi }ni=1 ⊂ A. (2.20)

Proof. Note that (2.20) is equivalent to −L ≤ KZ ≤ L and then apply Proposition 2.7 and Theorem 2.5. We now discuss the uniqueness property of the GNS data, an issue previously addressed in [13], but not completely solved. Two GNS data (π1 , K1 , ,1 ) and (π2 , K2 , ,2 ) are unitarily equivalent if there exists a unitary operator # ∈ L(K1 , K2 ) such that #D(π1 ) = D(π2 ), π2 (a)# = #π1 (a) for all a ∈ A, and #,1 = ,2 . Theorem 2.9. Let A be a unital ∗-algebra and let Z be a linear hermitian functional on A with Z(1) = 1, admitting GNS data. The following assertions are equivalent: (1) All GNS data of Z are unitarily equivalent. (2) For each positive definite kernel L on A such that −L ≤ KZ ≤ L, there exists $ > 0 such that either (0, $) ⊂ ρ(AL ) or (−$, 0) ⊂ ρ(AL ), where AL is the Gram operator of KZ with respect to L. Proof. Let (Vi , Ki ), i = 1, 2, be two Kolmogorov decompositions of KZ that are unitarily equivalent, that is, there exists a unitary operator # ∈ L(K1 , K2 ) such that V2 (x) = #V1 (x). Let (πi , Ki , ,i ), i = 1, 2, be the corresponding GNS data for Z as in Proposition 2.7. Then, D(π2 ) = V2 (x)C = #V1 (x)C = #( V1 (x)C) = #D(π1 ). x∈A

x∈A

x∈A

Also, for a ∈ cA and λ ∈ C, π2 (a)#V1 (x)λ = π2 (a)V2 (x)λ = V2 (xa ∗ )λ = #V1 (xa ∗ )λ = #π1 (a)V1 (x)λ, which implies that π2 (a)# = #π1 (a). Finally, #,1 = #V1 (1)1 = V2 (1)1 = ,2 , therefore (π1 , K1 , ,1 ) and (π2 , K2 , ,2 ) are unitarily equivalent GNS data for Z. Conversely, let (πi , Ki , ,i ), i = 1, 2, be two unitarily equivalent GNS data for Z and let (Vi , Ki ), i = 1, 2, be the Kolmogorov decompositions of KZ associated to these GNS data by Proposition 2.7. Therefore, there exists a unitary operator # ∈ L(K1 , K2 ) such that #D(π1 ) = D(π2 ), π2 (a)# = #π1 (a) for all a ∈ A and #,1 = ,2 . It follows that V2 (x)λ = π2 (a ∗ )λ,2 = π2 (a ∗ )λ#,1 = π2 (a ∗ )#λ,1 = #π1 (a ∗ )λ,1 = #V1 (x)λ, which shows that (V1 , K1 ) and (V2 , K2 ) are unitarily equivalent Kolmogorov decompositions of the kernel KZ . Now, an application of Theorem 2.6 concludes the proof.

418


2.3.2. An example: Weyl exponentials.. Let (H, [·, ·]) be a Kre˘ın space and consider P the group of its rigid motions. This is the semidirect product of the additive group H and the group of the bounded unitary operators on H. The group law is given by (ξ, U )(ξ , U ) = (ξ + U ξ , U U ) and an action of P on H can be defined by the formula φ((ξ, U ), ξ ) = ξ + U ξ . In particular, the normal subgroup H of P acts on H by translations. For simplicity, we restrict here to this action by translations. The hermitian kernel associated to this construction is defined by the formula: K(ξ, η) = exp(

i[η, ξ ] [ξ − η, ξ − η] ) exp(− ), 2 4

(2.21)

for ξ, η ∈ H. The additive group H acts on itself by the translations φ(ξ, η) = ξ + η and we notice that K(φ(ξ, η), φ(ξ, η )) = α(ξ, η)α(ξ, η )K(η, η )

(2.22)

for all ξ, η, η ∈ H, where α(ξ, η) = exp(−

i[ξ, η] ) 2

and then σ (ξ, η) = α(ξ, η + η )−1 α(η, η )−1 α(ξ + η, η ) = exp(

i[ξ, η] ). 2

In the terminology to be introduced within the next section, it is readily verified that α is a φ-multiplier and hence that σ has the 2-cocycle property. Then (2.22) means that the (scalar) hermitian kernel K is projectively φ-invariant. We can obtain a Kolmogorov decomposition of the kernel K by adapting the Fock space construction from the positive definite case, similar to the Kolmogorov decomposition that gives the Bose-Fock space (see [10] or [22] for more details). Proposition 2.10. The kernel K defined by (2.21) has a Kolmogorov decomposition (V , K) with the property that the operators defined by the formula α(ξ, η)W (ξ )V (η) = V (ξ + η) are defined on the common dense domain commutation relations

ξ ∈H V (ξ )C

(2.23)

in K and satisfy the canonical

W (ξ )W (η) = σ (ξ, η)W (ξ + η).

(2.24)


419

3. Invariant Hermitian Kernels In this section we study properties of the Kolmogorov decompositions of hermitian kernels with additional symmetries. Let S be a unital semigroup and φ an action of S on the set X, this means that φ : S × X → X, φ(a, φ(b, x)) = φ(ab, x) for all a, b ∈ S, x ∈ X, and φ(e, x) = x, where e denotes the unit element of S. We are interested in those kernels K on X assumed to satisfy a certain invariance property with respect to the action φ because this leads to the construction of a representation of S on the space of a Kolmogorov decomposition of K. This kind of construction is well-known for a positive definite kernel (it just extends the construction of the regular representation, see for instance, [22]), but for the Kre˘ın space setting the question concerning the boundedness of the representation operators is more delicate. It is the goal of this section to deal with this matter in a more detailed way. We now introduce additional notation and terminology. Let α be a φ-multiplier, that is, a complex-valued function on S×X such that α(a, x) = 0 and subject to the following relation: α(ab, x)α(ab, y) = α(a, φ(b, x))α(a, φ(b, y))α(b, x)α(b, y)

(3.1)

for all x, y ∈ X. This implies that σ (a, b) = α(a, φ(b, x))−1 α(b, x)−1 α(ab, x) does not depend on x; moreover, |σ (a, b)| = 1, and σ has the 2-cocycle property: σ (a, b)σ (ab, c) = σ (a, bc)σ (b, c)

(3.2)

for all a, b, c ∈ S (see [22, Lemma 2.2]). For each a ∈ S we define a projective shift ψa : F0 (X, H) → F0 (X, H) by (ψa (f ))(x) = α(a, x)−1 f (φ(a, x)),

f ∈ F0 (X, H), x ∈ X.

(3.3)

In terms of the atoms of the vector space F0 (X, H), ψa acts as follows ψa0 (ξx ) = α(a, x)−1 ξφ(a,x) (= (α(a, x)−1 ξ )φ(a,x) ),

(3.4)

where ξx is defined as in (2.12). This can be used as an alternate definitionof ψa since each element h of F0 (X, H) can be uniquely written as a finite sum h = nk=1 ξxkk for vectors ξ 1 , . . . , ξ k ∈ H and distinct elements x1 , x2 , . . . , xn in X and then the projective shift ψa0 is the extension by linearity to a linear map ψa , from F0 (X, H) into F0 (X, H), ψa

n k=1

ξxkk

=

n k=1

ψa0 (ξxkk ).

We say that a positive definite kernel L is projectively φ-bounded provided that for all 1/2 a ∈ S, ψa is bounded with respect to the seminorm [·, ·]L induced by L on F0 (X, H). We denote by Bφ+ (X, H) the set of positive definite projectively φ-bounded kernels on X with values in L(H). In addition, from now on we assume that S is a unital semigroup with involution, that is, there exists a mapping I : S → S such that I(I(a)) = a and I(ab) = I(b)I(a) for

420


all a, b ∈ S. The connection between the involution I and the φ-multiplier α is given by the assumption α(aI(a), x) = 1,

a ∈ S, x ∈ X.

(3.5)

Finally, with the notation and the assumptions as before, we say that the hermitian kernel K on X is projectively φ-invariant if K(x, φ(a, y)) = α(a, φ(I(a), x))α(a, y)K(φ(I(a), x), y)

(3.6)

for all x, y ∈ X and a ∈ S. In order to keep the terminology simple, the function α and the involution I will be made each time precise, if not clear from the context. If α(a, x) = 1 for all a ∈ S and x ∈ X then the hermitian kernel K satisfying (3.6) is called simply φ-ivariant. The following is the main result of this section. Theorem 3.1. Let φ be an action of the unital semigroup S with involution I satisfying (3.5) on the set X and let K be an L(H)-valued projectively φ-invariant hermitian kernel on X. The following assertions are equivalent: (1) There exists L ∈ Bφ+ (X, H) such that −L ≤ K ≤ L. (2) K has a Kolmogorov decomposition (V ; K) with the property that there exists a projective representation U of S on K (that is, U (a)U (b) = σ (a, b)U (ab) for all a, b ∈ S) such that V (φ(a, x)) = α(a, x)U (a)V (x)

(3.7)

for all x ∈ X, a ∈ S. In addition, σ (I(a), a)U (I(a)) = U (a) for all a ∈ S. (3) K = K1 − K2 for two positive definite kernels such that K1 + K2 ∈ Bφ+ (X, H). (4) K = K+ − K− for two disjoint positive definite kernels such that K+ + K− ∈ Bφ+ (X, H). Proof. (1)⇒(2). Let HL be the Hilbert space obtained by the completion of the quotient space F0 (X, H)/NL with respect to [·, ·]L , where NL = {f ∈ F0 (X, H) | [f, f ]L = 0} is the isotropic subspace of the inner product space (F0 (X, H), [·, ·]L ). Let AL be the Gram operator of K with respect to L and let (V ; KAL ) be the Kolmogorov decomposition of the kernel K described in the previous section. Since L is φ-bounded, it follows that each ψa extends to a bounded operator F (a) on HL . We notice that [ψa (ξx ), ηy ]K = [(α(a, x)−1 ξ )φ(a,x) , ηy ]K = α(a, x)−1 [K(y, φ(a, x))ξ, η]H = α(a, φ(I(a), y))[K(φ(I(a), y), x)ξ, η]H = α(a, φ(I(a), y))α(I(a), y)[ξx , ψI(a) (ηy )]K . From the definition of σ we have that for y ∈ X, σ (a, I(a)) = α(a, φ(I(a), y))−1 α(I(a), y)−1 α(aI(a), y).


421

By our assumption (3.5), α(aI(a), y) = 1, so that σ (a, I(a)) = α(a, φ(I(a), y))−1 α(I(a), y)−1 . Since |σ (a, I(a))| = 1, we deduce that [ψa (ξx ), ηy ]K = σ (a, I(a))[ξx , ψI(a) (ηy )]K . This relation can be extended by linearity to [ψa (f ), g]K = σ (a, I(a))[f, ψI(a) (g)]K for all f, g ∈ F0 (X, H). We deduce that [AL ψa (f ), g]L = σ (a, I(a))[AL f, ψI(a) (g)]L , which implies that AL F (a) = σ (a, I(a))F (I(a))∗ AL .

(3.8)

Theorem 2.2 implies that there exists a unique operator U (a) ∈ L(KAL ) such that U (a)AL = AL F (a). Moreover, for h ∈ HL , U (a)U (b)AL h = U (a)AL F (b)h = AL F (a)F (b)h. We also notice that ψa ψb (ξx ) = ψa (α(b, x)−1 ξφ(b,x) ) = α(b, x)−1 α(a, φ(b, x))−1 ξφ(a,φ(b,x)) = σ (a, b)α(ab, x)−1 ξφ(ab,x) = σ (a, b)ψab (ξx ). We deduce that F (a)F (b) = σ (a, b)F (ab) and this relation implies that U (a)U (b)AL h = σ (a, b)U (ab)AL h. Since the set {AL h | h ∈ HL } is dense in KAL , we deduce that U is a projective representation of S on KAL . For ξ ∈ H we have V (φ(a, x))ξ = AL [ξφ(a,x) ] and U (a)V (x)ξ = U (a)AL [ξx ] = AL F (a)[ξx ]. Since ψa (ξx ) = α(a, x)−1 ξφ(a,x) , we deduce that F (a)[ξx ] = α(a, x)−1 [ξφ(a,x) ], so that (3.7) holds.

422


Finally, the relation (3.8) implies that [U (a)AL f, AL g]KAL = [AL F (a)f, AL g]KAL = [AL F (a)f, g]L = σ (a, I(a))[F (I(a))∗ AL f, g]L = σ (a, I(a))[AL f, F (I(a))g]L = σ (a, I(a))[AL f, AL F (I(a))g]KAL = σ (a, I(a))[AL f, U (I(a))AL g]KAL for all f, g ∈ HL , which implies that σ (a, I(a))U (I(a)) = U (a) . We now notice that the relation (3.6) implies that σ (a, I(a)) = σ (I(a), a), which concludes the proof of the relation σ (I(a), a)U (I(a)) = U (a) for all a ∈ S. (2)⇒(4). Let J be a fundamental symmetry on K. Then J is a selfadjoint operator with respect to the positive definite inner product h, gJ = [J h, g]K . Let J = J+ − J− be the Jordan decomposition of J and define the hermitian kernels K± (x, y) = ±V (x) J± V (y),

L(x, y) = V (x) J V (y),

x, y ∈ X.

From J+ + J− = I and ±J± = J± J J± we get K(x, y) = K+ (x, y) − K− (x, y) and L(x, y) = K+ (x, y) + K− (x, y). To prove that K+ and K− are positive definite kernels let h ∈ F0 (X, H). Then [K± (x, y)h(y), h(x)]H = [±V (x) J± V (y)h(y), h(x)]H x,y∈X

x,y∈X

=

[±J± V (y)h(y), V (x)h(x)]K

x,y∈X

=

[J± J J± V (y)h(y), V (x)h(x)]K

x,y∈X

=

J± V (y)h(y), J± V (x)h(x)J

x,y∈X

=

x∈X

J± V (x)h(x) 2J ≥ 0.

It remains to show that L is φ-bounded. If h ∈ F0 (X, H), then h = nk=1 ξxkk for some n ∈ N, vectors ξ 1 , . . . , ξ n ∈ H and distinct elements x1 , x2 , . . . , xn in X. Then [ψa (h), ψa (h)]L = = =

n j,k=1 n j,k=1 n j,k=1

j

[ψa (ξxj ), ψa (ξxkk ]L j

k α(a, xj )−1 α(a, xk )−1 [ξφ(a,xj ) , ξφ(a,x ] k) L

α(a, xj )−1 α(a, xk )−1 [L(φ(a, xk ), φ(a, xj ))ξ j , ξ k ]H


= =

n

423

α(a, xj )−1 α(a, xk )−1 V (φ(a, xj ))ξ j , V (φ(a, xk ))ξ k J

j,k=1 n

U (a)V (xj )ξ j , U (a)V (xk )ξ k J

j,k=1

= U (a)

n k=1

≤ U (a) 2J

= U (a) 2J = U (a) 2J =

V (xk )ξ k 2J n

V (xk )ξ k 2J

k=1

n

V (xj )ξ j , V (xk )ξ k J

j,k=1 n

j

[ξxj , ξxkk ]L

j,k=1

U (a) 2J [h, h]L ,

so that L is φ-bounded. We also deduce that (V , (K, ·, ·J )) is the Kolmogorov decomposition of the positive definite kernel L and (J± V , (J± K, ·, ·J )) is the Kolmogorov decomposition of K± . Since J+ J− = 0 we deduce that J+ K ∩ J− K = {0} and, by Proposition 16, in [25] we deduce that K+ and K− are disjoint kernels. Since (4)⇒(3) and (3)⇒(1) are obvious implications, the proof is complete. A Kolmogorov decomposition (V , K) of the hermitian kernel K for which there exists a projective representation U such that (3.7) holds is called a projectively invariant Kolmogorov decomposition. Also, a projective representation U satisfying the additional property U (a) = σ (I(a), a)U (I(a)) for all a ∈ S, is called symmetric projective representation. A natural question that can be raised in connection with the previous result is whether Bφ+ (X, H) is a sufficiently rich class of kernels. Proposition 3.2. Assume that S is a group and I(a) = a −1 , a ∈ S. If K is a projectively φ-invariant hermitian kernel on X then, for any a ∈ S the operator ψa is isometric with respect to the inner product [·, ·]K . In particular, any projectively φ-invariant positive definite kernel on X belongs to Bφ+ (X, H). Proof. Indeed, in this case (3.5) becomes α(e, x) = 1 for all x ∈ X, where e is the unit of the group S. Also, if K is a hermitian kernel then it is projectively φ-invariant if and only if K(φ(a, x), φ(a, y)) = α(a, x)α(a, y)K(x, y),

x, y ∈ X, a ∈ S.

Let ξ, η ∈ H be arbitrary. Then −1

[ψa (ξx ), ψa (ηy )]K = α(a, x)−1 α(a, y)

[ξφ(a,x) , ηφ(a,y) ]K

424

T. Constantinescu, A. Gheondea −1

= α(a, x)−1 α(a, y) [K(φ(a, y), φ(a, x))ξ, η]H = [K(y, x)ξ, η]H = [ξx , ηy ]K , and hence ψa is [·, ·]K isometric.

Remark 3.3. (1) Theorem 3.1 is known when H is a Hilbert space and the kernel K is positive definite and satisfies K(φ(a, x), φ(a, y)) = α(a, x)α(a, y)K(x, y),

a ∈ S, x, y ∈ X

(3.9)

(see, for instance, [22]). In that case the proof is easily obtained by defining directly U (a)V (x)ξ = α(a, x)−1 V (φ(a, x))ξ

(3.10)

for ξ ∈ H and verify that U (a) satisfies all the required properties (we note that no involution is considered in this case). We have to emphasize that this direct approach does not work in the hermitian case since the formula (3.10) does not necessarily give a bounded operator. In order to overcome this difficulty we have to replace the symmetry condition in (3.9) by the symmetry condition in (3.6) and then use Theorem 2.2. This was the main point in the proof of Theorem 3.1. (2) The positive definite version of Theorem 3.1 has many applications, some of them mentioned for instance in [10, 11], and [22]. Such a typical application gives a Naimark dilation for Toeplitz kernels. Thus, if X = S, φ(a, x) = ax, and α(a, x) = 1 for all a, x ∈ S, then (3.9) becomes the well-known Toeplitz condition K(ab, ac) = K(b, c) for all a, b, c ∈ S. If K is a positive definite kernel on S satisfying the Toeplitz condition and K(e, e) = I , where e is the unit of S, then {U (a)}a∈S defined by (3.10) is a semigroup of isometries on a Hilbert space K containing H such that K(a, b) = PH U (a)∗ U (b)|H, for all a, b ∈ S, where PH denotes the orthogonal projection of K onto H. (3) The next example explores the fact that for positive definite kernels the representation {U (a)}a∈S given by (3.10) is unique up to unitary equivalence. Thus, consider the action of a group G on the Hilbert space H such that φ(g, ξ ), φ(g, η) = ξ, η for all g ∈ G and ξ, η ∈ H. We consider the kernel K(ξ, η) = η, ξ on H and notice that K is positive definite. Its Kolmogorov decomposition is given by V (ξ ) : C → H, V (ξ )λ = λξ, λ ∈ C, ξ ∈ H. If we use the positive definite version of Theorem 3.1, we deduce that there exists a Kolmogorov decomposition V of K and a representation U of G such that V (φ(g, ξ )) = U (g)V (ξ ) for all g ∈ G and ξ ∈ H. From the uniqueness of V up to unitary equivalence, it follows that there exists a unitary operator # such that V (φ(g, ξ )) = #U (g)#∗ V (ξ ), or φ(g, ξ ) = U (g)ξ, with U (g) = #U (g)#∗ . Therefore we obtained the well-known result that φ acts by linear unitary operators. The last example was intended to emphasize the importance of the uniqueness up to unitary equivalence of the projectively invariant Kolmogorov decompositions. This issue turns out to be rather delicate in the hermitian case. Theorem 2.6 settles this question only in the case of the trivial semigroup S with one element. It is easily seen that the spectral condition in Theorem 2.6 is also sufficient for the uniqueness of a projectively


425

invariant Kolmogorov decomposition. However, Theorem 2.3 shows that the proof in [5] of Theorem 2.6 cannot be easily adapted to the case of an arbitrary semigroup S. We conclude this section with another case when uniqueness holds. Given a hermitian kernel K, the rank rank(K) is, by definition, the supremum of rank(K ) taken over all finite subsets ⊂ X, where K is the restricted kernel (K(x, y))x,y∈ . By definition rank(K) is either a positive integer or the symbol ∞. A hermitian kernel K has κ negative squares if the inner product space (F0 (X, H), [·, ·]K ) has negative signature κ, that is, κ is the maximal dimension of all its negative subspaces. It is easy to see that this is equivalent to K = K+ − K− , where K± are disjoint positive definite kernels such that rank(K− ) = κ, see e.g. [25]. This allows us to define κ − (K) = κ, the number of negative squares of the kernel K. In particular, hermitian kernels with a finite number of negative squares always have Kolmogorov decompositions and for any Kolmogorov decomposition (V ; K) of K we have κ − (K) = κ − (K) < ∞, hence K is a Pontryagin space with negative signature κ. In Pontryagin spaces the strong topology is intrinsically characterized in terms of the indefinite inner product, e.g. see [12]. Therefore, by using Proposition 3.2 and Shmul’yan’s Theorem (e.g. see Theorem 2.10 in [9]) we get: Theorem 3.4. Let φ be an action of the group S on the set X and let K be an L(H)-valued projectively φ-invariant hermitian kernel on X with a finite number of negative squares. Then K has a projectively invariant Kolmogorov decomposition on a Pontryagin space, that is unique up to unitary equivalence. 4. Similarity The symmetric projective representation U of S obtained in Theorem 3.1 acts on a Kre˘ın space. It would be of special interest to decide whether U is at least similar to a symmetric projective representation on a Hilbert space, a property related to the well-known similarity problem for group representations, see [23] for a recent survey. The above mentioned problem is also closely related to the characterization of those φ-invariant hermitian kernels K with the property that the representation K = K+ −K− holds for two positive definite φ-invariant kernels. In this section we give an answer to these two questions in terms of fundamental reducibility. We say that the projective representation U of S on the Krein space K is fundamentally reducible if there exists a fundamental symmetry J on K such that U (a)J = J U (a) for all a ∈ S. This condition is readily equivalent to the condition U (a) = U (a)∗ for all a ∈ S, and further, equivalent to the diagonal representation of U (a) with respect to a fundamental decomposition of the Kre˘ın space K. Proposition 4.1. Let S be a semigroup with involution I and σ satisfies the 2-cocycle property (3.2) on S. Let U be a symmetric projective representation of S on the Kre˘ın space K. Then the following assertions are equivalent: (1) U is similar to a symmetric projective representation T on a Hilbert space. (2) U is fundamentally reducible. Proof. (1)⇒(2). Let # ∈ L(K, G) be the similarity such that T (a)# = #U (a) for a ∈ S. We first notice that # is also an involutory similarity (with the terminology from [14]), that is T (a)∗ = #U (a) #−1 ,

a ∈ S.

(4.1)

426


Then, we consider on K the positive inner product ξ, η# = #ξ, #η, ξ, η ∈ K. Since # is boundedly invertible, there exists a selfadjoint and boundedly invertible operator G ∈ L(K) such that [ξ, η] = Gξ, η# , ξ, η ∈ K. Therefore, for arbitrary a ∈ S and ξ, η ∈ K we have U (a)ξ, η# = #U (a)ξ, #η = T (a)#ξ, #η = #ξ, T (a)∗ #η = #ξ, #U (a) η = ξ, U (a) η# = [G−1 ξ, U (a) η] = [U (a)G−1 ξ, η] = GU (a)G−1 ξ, η# . Thus, GU (a) = U (a)G and letting J = sgn(G) it follows that J is a fundamental symmetry on the Kre˘ın space K such that J U (a) = U (a)J . (2)⇒(1). If J is a fundamental symmetry on the Kre˘ın space K such that J U (a) = U (a)J , for all a ∈ S, then U is a symmetric projective representation with respect to the Hilbert space (K, ·, ·J ). With the notation as in Proposition 4.1, if σ has the 2-cocycle property (3.2) and |σ (a, b)| = 1 for all a, b ∈ S, then it follows that U (a) U (a) = U (I(a)a),

a ∈ S.

(4.2)

Thus, in certain applications where U consists of (Kre˘ın space) isometric operators, it is interesting to know whether U is similar to a symmetric projective representation of isometric operators on a Hilbert space. Clearly, a necessary condition is that for some (equivalently for all) unitary norm · on K there exists C > 0 such that 1

ξ ≤ U (a)ξ ≤ C ξ , C

a ∈ S, ξ ∈ K.

(4.3)

As expected, the converse implication is related to the assumption of amenability of the semigroup S. More precisely, following closely the idea in the proof of Théorème 6 in [8], we get: Theorem 4.2. Let S be an amenable semigroup, σ has the 2-cocycle property (3.2), |σ (a, b)| = 1 for all a, b ∈ S, and let U be a projective representation (without any assumption of symmetry) of S on a Hilbert space K, such that (4.3) holds for some constant C > 0. Then U is similar to a projective representation T of S on a Hilbert space G such that T (a) are isometric for all a ∈ S. We come now to the problem of characterizing those hermitian invariant kernels that can be represented as a difference of two positive invariant kernels. Theorem 4.3. Let φ be an action of the unital semigroup S with involution I satisfying (3.5) on the set X and let K be an L(H)-valued φ-invariant hermitian kernel on X. The following assertions are equivalent: (1) There exists L ∈ Bφ+ (X, H) such that −L ≤ K ≤ L and L is φ-invariant. (2) K has a projectively invariant Kolmogorov decomposition (V ; K) such that the associated projective representation is fundamentally reducible. (3) K = K+ − K− for two disjoint positive definite kernels such that K+ + K− ∈ Bφ+ (X, H) and both K± are φ-invariant.


427

Proof. (1)⇒(2). We use the same notation as in the proof of Theorem 3.1. Thus, HL is the Hilbert space obtained by the completion of the quotient space F0 (X, H)/NL with respect to [·, ·]L , where NL is the isotropic subspace of the inner product space (F0 (X, H), [·, ·]L ). Let AL be the Gram operator of K with respect to L and let (V ; KAL ) be the projectively invariant Kolmogorov decomposition of the kernel K described in the proof of (1)⇒(2) in Theorem 3.1. Since L is φ-bounded, it follows that each ψa extends to a bounded operator F (a) on HL . Since L is φ-invariant, we deduce that [ψa (f ), g]L = σ (a, I(a))[f, ψI(a) (g)]L for all f, g ∈ F0 (X, H), which implies that F (a) = σ (a, I(a))F (I(a))∗ . This relation and (3.8) imply that AL F (a) = F (a)AL for all a ∈ S. Let AL = SAL |AL | be the polar decomposition of AL and let JAL be the symmetry introduced in Example 2.1. Using (2.3), we deduce that U (a)JAL AL = U (a)AL SAL = AL F (a)SAL = AL SAL F (a) = JAL AL F (a) = JAL U (a)AL , therefore the representation U is fundamentally reducible. (2)⇒(3). We consider the elements involved in the proof of (2)⇒(4) in Theorem 3.1 for a fundamental symmetry J on K for which U (a)J = J U (a), a ∈ S. Therefore U (a)J± = J± U (a) for all a ∈ S, and then K± (x, φ(a, y)) = ±V (x) J± V (φ(a, y)) = ±α(a, y)V (x) J± U (a)V (y) = ±α(a, y)V (x) U (a)J± V (y) = ±α(a, y)σ (I(a), a)V (x) U (I(a)) J± V (y) = ±α(a, φ(I(a), x))α(a, y)V (φ(I(a), x)) J± V (y) = α(a, φ(I(a), x))α(a, y)K± (φ(I(a), x), y)). (3)⇒(1). Just set L(x, y) = K+ (x, y) + K− (x, y). In case S is a group with the involution I(a) = a −1 , then some of the assumptions in the previous results simplify to a certain extent. In this case, as a consequence of (4.2), the symmetric projective representation U associated to a φ-invariant Kolmogorov decomposition consists of unitary operators. Theorem 4.4. Let S be a group and σ a 2-cocycle on S with |σ (a, b)| = 1 for all a, b ∈ S. Let U be a unitary projective representation of S on the Kre˘ın space K. Then the following assertions are equivalent:

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(1) U is similar to a unitary projective representation T on a Hilbert space, that is, T : S → L(G), G a Hilbert space, T (a)T (b) = σ (a, b)T (ab) and T (a)∗ = σ (a −1 , a)T (a −1 ) for all a ∈ S. (2) U is fundamentally reducible. Moreover, if U satisfies one (hence both) of the assumptions (1) and (2) then U is uniformly bounded, that is, sup U (a) < ∞.

(4.4)

a∈S

If, in addition, S is amenable, then (4.4) is equivalent to (any of) the conditions (1) and (2). Proof. This follows from Proposition 4.1 and Theorem 4.2.

Theorem 4.5. Let φ be an action of the group S on the set X and let K be an L(H)-valued φ-invariant hermitian kernel on X. The following assertions are equivalent: (1) There exists a φ-invariant positive definite L on X such that −L ≤ K ≤ L. (2) K has a projectively invariant Kolmogorov decomposition (V ; K) such that the associated symmetric projective representation is similar to a symmetric projective representation on a Hilbert space. (3) K = K+ − K− for two disjoint positive definite φ-invariant kernels. Proof. This follows from Proposition 3.2 and Theorem 4.3.

5. An Application: Representations of ∗-Algebras Another consequence of the Kolmogorov decomposition approach is the possibility of obtaining a characterization of those hermitian functionals Z that admit bounded GNS data, that is, the representation π is made of bounded operators. We use the same notation as in Subsect. 2.3.1. Theorem 5.1. Let A be a unital ∗-algebra and let Z be a linear hermitian functional on A with Z(1) = 1. Then Z admits bounded GNS data if and only if there exists a positive definite scalar kernel L on A having the property (2.20) and such that for every a ∈ A there exists Ca > 0 with the property that n i,j =1

λi λj L(xi a ∗ , xj a ∗ ) ≤ Ca

n

λi λj L(xi , xj ),

i,j =1

n ∈ N, {λi }ni=1 ⊂ C, {xi }ni=1 ⊂ A. Proof. This is a consequence of Theorem 3.1 and Proposition 2.7.

We conclude this section with a discussion of the Jordan decomposition of a linear hermitian functional on a ∗-algebra A, that is, the possibility of writing the hermitian functional as the difference of two positive functionals. Let us first note that a functional F : A → C is positive, that is, F (a ∗ a) ≥ 0 for all a ∈ A, if and only if the kernel KF associated to F by the formula (2.15) is positive definite. Also, if F is a positive


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functional on A, then KF is φ-bounded, with the action φ defined as in (2.14), if and only if for any a ∈ A there exists Ca > 0 such that F (xa ∗ ax ∗ ) ≤ Ca F (xx ∗ ),

x ∈ A.

(5.1)

For simplicity, we call the positive functional F φ-bounded if KF is φ-bounded. Let F1 , F2 be two positive functionals on the ∗-algebra A. Then F1 ≤ F2 , by definition, if F2 − F1 is a positive functional. It is easy to see that F1 ≤ F2 if and only if KF1 ≤ KF2 . The functionals F1 and F2 are called disjoint if their associated kernels KF1 and KF2 are disjoint. Theorem 5.2. Let A be a unital ∗-algebra, let Z be a linear hermitian functional on A with Z(1) = 1, and let φ be the action given by (2.14). The following assertions are equivalent: (1) There exists a linear positive φ-bounded functional Z0 on A such that −Z0 ≤ Z ≤ Z0 . (2) Z admits bounded GNS data (π, K, ,) such that the representation π is similar with a ∗-representation on a Hilbert space. (3) Z = Z+ − Z− for two disjoint linear positive definite functionals on A with the property that (Z+ + Z− ) is φ-bounded. Proof. The implications (1) ⇒ (2) ⇒ (3) are direct consequences of Theorem 4.3 and Proposition 2.7. For (3) ⇒ (1) we use the proof of Theorem 3.1 in order to deduce that there exists L ∈ Bφ+ (A, C) such that −L ≤ KZ ≤ L. Then Theorem 4.3 shows that L(x, φ(a, y)) = L(φ(a ∗ , x), y) for all x, y, a ∈ A. Also, in this case, L is linear in the first variable (hence, antilinear in the second variable). If we define Z0 (x) = L(x, 1) for x ∈ A, then Z0 is a linear functional on A and KZ0 (x, y) = Z0 (xy ∗ ) = L(xy ∗ , 1) = L(x, y). Now all the required properties of Z0 follow from the corresponding properties of L. Remark 5.3. It is interesting to note that under fairly general assumptions on the ∗algebra A, every positive functional F on A is φ-bounded, that is, for all a in A we have (5.1). This holds, for instance, if A is a Banach ∗-algebra, cf. Lemma 37.6 in [4], with the constant Ca equal to the spectral radius of a ∗ a. References 1. Albeverio, S., Gottschalk, H.,Wu, J.-L.: Models of local relativistic quantum fields with indefinite metric (in all dimensions). Commun. Math. Phys. 184, 509–531 (1997) 2. Antoine, J.-P., Ôta, S.: Unbounded GNS representations of a ∗-algebra in a Kre˘ın space. Lett. Math. Phys. 18, 267–274 (1989) 3. Araki, H.: Indecomposable representations with invariant inner product. A theory of the Gupta–Bleuler triplet. Commun. Math. Phys. 97, 149–159 (1985) 4. Bonsall, F.F., Duncan, J.: Complete Normed Algebras, Berlin–Heidelberg–New York: Springer-Verlag, 1973 5. Constantinescu, T., Gheondea, A.: Representations of hermitian kernels by means of Kre˘ın spaces. Publ. RIMS. Kyoto Univ. 33, 917–951 (1997)

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6. Dieudonné, J.: Quasi-hermitian operators. In: Proceedings of International Symposium on Linear Spaces, Jerusalem 1961, pp. 115–122 7. Dijksma, A., Langer, H., de Snoo, H.S.: Unitary colligations in Kre˘ın spaces and their role in extension theory of isometries and symmetric linear relations in Hilbert spaces. In: Functional Analysis. II, Lecture Notes in Math. 1242, Berlin: Springer Verlag, 1987, pp. 1–42 8. Dixmier, J.: Les moyennes invariante dans les semi-groupes et leurs applications. Acta. Sci. Math. Szeged 12, 213–227 (1950) 9. Dritschel, M.A.,Rovnyak, J.: Operators on indefinite inner product spaces. In: Lectures on Operator Theory and its Applications, Fields Institute Monographs, Vol. 3, Providence, RI: Amer. Math. Soc., 1996 10. Evans, D.E., Lewis, J.T.: Dilations of Irreducible Evolutions in Algebraic Quantum Theory. Dublin Institute for Advanced Studies, Dublin, 1977 11. Evans, D.E., Kawahigashi, Y.: Quantum Symmetries on Operator Algebras. Oxford Science Publications, 1998 12. Iokhvidov, I.S., Kre˘ın, M.G., Langer, H.: Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric. Berlin: Akademie-Verlag, 1982 13. Hofmann, G.: On GNS representations on inner product spaces I. The structure of the representation space. Commun. Math. Phys. 191, 299–323 (1998) 14. Kissin, E., Shulman, V.: Representations of Krein Spaces and Derivations of C ∗ -Algebras. Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 89, Harlow: Longman, 1997 15. Kolmogorov, A.N.: Stationary sequences in Hilbert space. Bull. Math. Univ. Moscow 2, 1–40 (1941) 16. Kre˘ın, M.G.: On linear completely continuous operators in functional spaces with two norms. [Ukrainian], Zbirnik Prak. Inst. Mat. Akad. Nauk USSR 9, 104–129 (1947) 17. Lax, P.: Symmetrizable linear transformations. Comm. Pure Appl. Math. 7, 633–647 (1954) 18. Mnatsakanova, M., Morchio, G.,Strocchi, F., Vernov, Yu.: Irreducible representations of the Heisenberg algebra in Kre˘ın spaces. J. Math. Phys. 39, 2969–2982 (1998) 19. Morchio, G., Pierotti, D., Strocchi, F.: Infrared and vacuum structure in two-dimensional local quantum field theory models. The massless scalar field. J. Math. Phys. 31, 1467–1477 (1990) 20. Ôta, S.: Unbounded representation of a ∗-algebra on indefinite metric space. Ann. Inst. Henri Poincaré 48, 333–353(1988) 21. Paulsen, V.I.: Completely bounded maps and dilations. Pitman Research Notes in Math. 146, New York: Longman, Wiley, 1986 22. Parthasaraty, K.R., Schmidt, K.: Positive-Definite Kernels, Continous Tensor Products and Central Limit Theorems of Probability Theory. Lecture Notes in Mathematics, Vol. 272, Berlin: Springer-Verlag, 1972 23. Pisier, G.: Similarity Problems and Completely Bounded Maps. Springer Lecture Notes 1618, Berlin– Heidelberg–New York: Springer, 1996. 24. Reid, W.T.: Symmetrizable completely continuous linear transformations in Hilbert space Duke Math. J. 18, 41–56 (1951) 25. Schwartz, L.: Sous espace Hilbertiens d’espaces vectoriel topologiques et noyaux associés (noyaux reproduisants). J. Analyse Math. 13, 115–256 (1964) 26. Strocchi, F.: Selected Topics on the General Properties of Quantum Field Theory. Lecture Notes Phys. 51, Singapore: World Scientific, 1993 Communicated by H. Araki

Commun. Math. Phys. 216, 431 – 459 (2001)

Communications in



BPS States of D = 4 N = 1 Supersymmetry Jerome P. Gauntlett1 , Gary W. Gibbons2,3,5 , Christopher M. Hull1,4 , Paul K. Townsend5 1 Department of Physics, Queen Mary and Westfield College, Mile End Rd., London E1 4NS, UK. 2 3 4 5

E-mail: [email protected]; [email protected] Laboratoire de Physique Théorique, Ecole Normale Supérieure, 24 Rue Lhomond, Paris 05, France Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Institute for Theoretical Physics, University of California Santa Barbara, CA 93106-4030, USA DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK. E-mail: [email protected]; [email protected]

Received: 6 April 2000/ Accepted: 10 September 2000

Abstract: We find the combinations of momentum and domain-wall charges corresponding to BPS states preserving 1/4, 1/2 or 3/4 of D = 4 N = 1 supersymmetry, and we show how the supersymmetry algebra implies their stability. These states form the boundary of the convex cone associated with the Jordan algebra of 4 × 4 real symmetric matrices, and we explore some implications of the associated geometry. For the Wess– Zumino model we derive the conditions for preservation of 1/4 supersymmetry when one of two parallel domain-walls is rotated and in addition show that this model does not admit any classical configurations with 3/4 supersymmetry. Our analysis also provides information about BPS states of N = 1 D = 4 anti-de Sitter supersymmetry. 1. Introduction Although N=1 supersymmetric field theories in 3 + 1 dimensions have been extensively investigated for more than twenty five years, most of these investigations have been based on the standard supersymmetry algebra. It has been known for some time, however, that p-brane solitons in supersymmetric theories carry p-form charges that appear as central charges in the spacetime supertranslation algebra [1]. Allowing for all such charges, the D = 4 N = 1 supertranslation algebra is spanned by a four component Majorana spinor charge Q, the 4-vector Pµ and a Lorentz 2-form charge Zµν . The only non-trivial relation is the anticommutator 1 {Q, Q} = Cγ µ Pµ + Cγ µν Zµν , 2

(1)

where C is the charge conjugation matrix and γµ = (γ0 , γi ) are the four Dirac matrices. Our metric convention is “mostly plus” so that we may choose a real representation of the Dirac matrices. In this representation the Majorana spinor charges Q are real, so {Q, Q} is a symmetric 4 × 4 matrix with a total of ten real entries. The number of components

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J. P. Gauntlett, G. W. Gibbons, C. M. Hull, P. K.Townsend

of Pµ and Zµν is also ten, so that we have indeed included all possible bosonic central charges. Note that the automorphism group of this algebra is GL(4; R). The components of Zµν can be interpreted as charges carried by domain walls [1], while Pµ is (in general) a linear combination of the momentum and a string charge. In the case of a domain wall, the tension is bounded by the charge, and saturation of this bound implies preservation of 1/2 of the N = 1 D = 4 supersymmetry. This is one example in the class of “1/2 supersymmetric” configurations allowed by the supersymmetry algebra1 . Such 1/2 supersymmetric domain walls were shown to occur in [2] in the Wess–Zumino (WZ) model, for an appropriate superpotential, and also arise in the SU (n) SQCD [3] because the low-energy effective dynamics is related to that of a WZ model with a superpotential admitting n discrete vacua [4]. More recently, it was shown that the WZ model also admits (again for an appropriate superpotential) 1/4 supersymmetric configurations that can be interpreted as intersecting domain walls [6, 7]. More precisely, it was established that such configurations must solve a certain “Bogomol’nyi” equation for which earlier mathematical studies had made the existence of appropriate solutions plausible (especially in view of the results of [8] which were recently brought to our attention). Domain wall junctions of the WZ model have since been studied further in [9–12] and an explicit 1/4 supersymmetric domain wall junction of a related model has recently been found [13]. It was pointed out in [6] that the possibility of 1/4 supersymmetric intersecting domain walls is inherent in the supersymmetry algebra. If we choose C = γ 0 and γ5 = γ 0 γ 1 γ 2 γ 3 , then (1) becomes 1 1 {Q, Q} = H + γ 0i Pi + γ 0ij Uij + γ 0ij γ5 Vij , 2 2

(2)

where H = P 0 , Uij = Zij and Vij = −εij k Z0k . One is thus led to expect “electric” type domain walls with non-zero 2-form Uij but vanishing Vij and “magnetic” type domain walls with non-zero 2-form Vij but vanishing Uij . In general, a domain wall will be specified not only by its tension and orientation but also by an angle in the electric-magnetic charge space; the domain wall is “dyonic” when this angle is not a multiple of π. It is not difficult to show that the algebra (2) allows for dyonic charge configurations preserving 1/4 supersymmetry. In this paper we determine the modelindependent restrictions on such configurations that are implied by the supersymmetry algebra. As pointed out in [6], the charge associated with the stringlike junctions of domain walls in the WZ model appears in the supersymmetry algebra in the same way as the 3-momentum, so for a static 1/4 supersymmetric configuration of the WZ model the 3-vector P must be interpreted as a string charge carried by the domain wall junction. It was supposed in [6] that this junction charge contributes positively to the energy of the 1/4 supersymmetric configuration as a whole. In contrast, the charge associated to domain wall junctions of the model considered in [13] was shown there to contribute negatively to the total energy. As we shall see, either sign is possible, depending on the central charge structure. There may therefore be more than one field theory realization of static intersecting domain walls preserving 1/4 supersymmetry, but as yet no example that exploits the most obvious possibility in which P vanishes but Uij and Vij do not. These observations underscore the importance of the model-independent analysis of 1/4 supersymmetric configurations based only on the N = 1 D = 4 supersymmetry 1 An analysis of 1/2 supersymmetric combinations of charges in N > 1 D = 4 theories, N = 2 in particular, can be found in [5].

BPS States of D = 4 N = 1 Supersymmetry

433

algebra, but our aim is to understand the implications of the supersymmetry algebra for all supersymmetric configurations, not just those preserving 1/4 supersymmetry. Since the matrix {Q, Q} is a positive definite real symmetric one, it can be brought to diagonal form with real non-negative eigenvalues. The number of zero eigenvalues is the number of supersymmetries preserved by the configuration. The “supersymmetric” configurations are those for which this number is 1, 2, 3 or 4. There is a unique “vacuum” charge configuration preserving all four supersymmetries. Configurations preserving two supersymmetries are 1/2 supersymmetric while those preserving one supersymmetry are 1/4 supersymmetric. Configurations preserving three supersymmetries are 3/4 supersymmetric, but there is no known field theoretic realization of this possibility. Indeed, we will show here that there is no classical field configuration of the WZ model that preserves 3/4 supersymmetry. However, possible string-theoretic realizations of exotic supersymmetry fractions such as 3/4 supersymmetry were recently explored in [14], and this possibility has been considered previously in a variety of other contexts [15–18]. In particular, the OSp(1|8; R)-invariant superparticle model of [16] provides a simple realization in the context of particle mechanics. The fundamental representation of OSp(1|8; R) is spanned by (ρ α , λα , ζ ), where ρ and λ are two 4-component real commuting spinors of Spin(1, 3), and ζ is a real anticommuting scalar. The action S = dt ρ α λ˙ α + ζ ζ˙ (3) is manifestly OSp(1|8) invariant; in particular, it is supersymmetric with supersymmetry charge Q = λζ . The canonical (anti)commutation relations imply that {Qα , Qβ } = λα λβ , which is a matrix of rank one, corresponding to 3/4 supersymmetry. Thus, there exist models of one kind or another in which all possible fractions of D = 4 N = 1 supersymmetry are preserved. This fact provides further motivation for the general model-independent analysis of the possibilities allowed by the supersymmetry algebra that we present here. As we shall explain, the space of supersymmetric charge configurations, or “BPS states”, is the boundary of the convex cone of 4×4 real symmetric matrices and this has an interpretation in terms of Jordan algebras. In analogy with the way that the conformal group acts on massless states on the light-front P 2 = 0, there is a group Sp(8, R) that acts on the “BPS-front” of supersymmetric configurations and which has an interpretation in this context as the Möbius group of the Jordan algebra [19]. Another purpose of this paper is to explore some of the geometrical ideas underlying this interpretation of supersymmetric charge configurations. It is generally appreciated that BPS states are stable states, this being the main reason for their importance, but some “standard” arguments for stability rely on physical intuition derived from special cases. For example, a massive charged particle that minimises the energy for given charge cannot radiate its energy away in the form of uncharged photons because this would leave behind a particle with the same charge but lower energy, contradicting the statement that the original particle minimised the energy in its charge sector. However, this heuristic argument is not conclusive. For instance, the stability against radiative relaxation to a lower energy state of the same “charge vector” assumes that the radiated energy carries away no momentum because momentum is one of the charges, and this assumption would be violated by a decay in which just one photon is emitted. It is also implicit in the heuristic argument that prior to decay one can go to the rest frame, but the supersymmetry algebra allows BPS states for which this is not possible, a massless particle being an obvious, but by no means the only, example. These considerations show that it is not quite as obvious as generally supposed that BPS states

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are stable. Here we provide a complete analysis, for the general D = 4 N = 1 supersymmetry algebra, based on a combination of the Minkowski reverse-triangle inequality for positive-definite matrices and the ordinary triangle inequality for BPS energies. The supertranslation algebra for which (1) is the only non-trivial (anti)commutator is a contraction of the superalgebra osp(1|4; R), which is the D = 4 N = 1 anti-de Sitter (adS) superalgebra. The anticommutator of the 4 real supercharges of the latter is 1 {Q, Q} = Cγ µ Pµ + Cγ µν Mµν , 2

(4)

where Mµν are the Lorentz generators. This is formally equivalent to (1), although the charges on the right-hand side are no longer central because they generate the adS group SO(3, 2). However the positivity conditions on these charges are the same, as are the conditions for preservation of supersymmetry. This fact means that much of our analysis of the centrally-extended supertranslation algebra can be immediately applied to the adS case. A related analysis has been considered previously for D = 5 in [20], where the D = 4 case was briefly mentioned, and BPS states in D = 4 adS have also been analysed by other methods in [21]. We begin with an analysis of the N = 1 D = 4 supersymmetry algebra, determining the charge configurations that preserve the various possible fractions of supersymmetry, and we show how the positivity of {Q, Q} implies the stability of BPS states carrying these charges. We also show how the supersymmetry algebra determines, in a modelindependent way, some properties of the 1/4 supersymmetric intersecting domain walls that are realized by the WZ model, but show also that 3/4 supersymmetry is not realized by classical WZ field configurations. We then turn to an exposition of the geometry associated with the supersymmetric configurations, which is that of self-dual homogeneous convex cones, and review their relation to Jordan algebras. We then discuss how our results apply to D = 4 N = 1 adS supersymmetry, and conclude with comments on implications and generalizations of our work, in particular to M-theory. 2. BPS States The anticommutator (2) can be rewritten as {Q, Q} = H + γ 0i Pi + γ5 γ i Ui + γ i Vi ,

(5)

where Ui =

1 εij k Uj k 2

Vi =

1 εij k Vj k . 2

(6)

As mentioned above, a charge configuration is supersymmetric if the matrix {Q, Q} has at least one zero eigenvalue. Thus, supersymmetric charge configurations are those for which {Q, Q} has vanishing determinant. We see from (5) that this determinant must be expressible in terms of H and the three 3-vectors P, U and V. Now det{Q, Q} is manifestly SL(4; R) invariant, but the subgroup that keeps H fixed is its maximal compact SO(4) ∼ = [SU (2) × SU (2)R ]/Z2 subgroup. Ignoring the quotient by Z2 , the first SU (2) factor can be identified with the 3-space rotation group while the SU (2)R group rotates the three 3-vectors P, U and V into each other, i.e. these three 3-vectors form a triplet of SU (2)R . The notation chosen here reflects the fact that SU (2)R ⊃ U (1)R , where U (1)R is the R-symmetry group2 rotating U into V keeping P


435

fixed (this is the automorphism group of the standard supersymmetry algebra, including Lorentz generators). We conclude from this that det{Q, Q} is a fourth-order polynomial in H with coefficients that are homogeneous polynomials in the three algebraicallyindependent SU (2) × SU (2)R invariants that can be constructed from P, U and V. These are a = U 2 + V 2 + P 2, b = P · U × V, c = |U × V|2 + |P × U|2 + |P × V|2 .

(7)

An explicit computation shows that det{Q, Q} = P (H ),

(8)

where P (H ) is the quartic polynomial P (H ) = H 4 − 2aH 2 − 8bH + a 2 − 4c.

(9)

The fact that {Q, Q} is a positive real symmetric matrix imposes a bound on H in terms of the invariants a, b, c. Specifically, H ≥ E(a, b, c),

(10)

where E(a, b, c) is the largest root of P (H ) = (H − λ1 )(H − λ2 )(H − λ3 )(H − λ4 ). Since the sum of the roots vanishes, the largest root E is necessarily non-negative. The number of supersymmetries preserved is then the number of roots equal to E. The vacuum configuration has all roots equal with E = 0. In all other cases E is strictly positive and the number of roots equal to it is 1,2 or 3, corresponding to 1/4,1/2 or 3/4 supersymmetry. Our first task, to be undertaken below, is to analyse the conditions required for the realization of each of these possibilities. We will then show how the stability of states preserving supersymmetry, alias “BPS states”, is guaranteed by the supersymmetry algebra. Although all model-independent consequences of supersymmetry are encoded in the supersymmetry algebra, the extraction of these consequences for BPS states is facilitated by methods that involve only the constraints on the Killing spinors associated with these states, and we show in the subsequent subsection how these methods can be used to learn about restrictions imposed by the preservation of supersymmetry on intersecting domain walls. We conclude with a discussion of 3/4 supersymmetry, and a proof that this fraction is not realized in the WZ model. 2.1. Supersymmetry fractions. The analysis of the conditions on the invariants a, b, c required for the preservation of the various possible fractions of supersymmetry is fairly straightforward when the polynomial P (H ) has at least two equal roots, and is especially simple when there are three equal roots. We shall therefore begin with the case of three equal roots, followed by the case of two equal roots, arriving finally at the generic case. 2 This symmetry is usually broken in D = 4 N = 1 QFTs, either by the superpotential or by anomalies. We shall comment on this fact in the conclusions, but it is not relevant to the purely algebraic analysis presented here.

436


The quartic polynomial P (H ) has three equal roots if c=

a2 , 3

a b = ∓( )3/2 , 3

(11)

and the roots are a λ1 = λ2 = λ3 ≡ λ = ±( )1/2 , 3

λ4 = −3λ.

(12)

If λ is positive then we have the BPS bound H ≥ λ, and charge configurations saturating this bound preserve 3/4 supersymmetry. If λ is negative then we instead find the BPS bound H ≥ −3λ, with only 1/4 supersymmetry being preserved by charge configurations that saturate it. Charge configurations preserving 1/2 supersymmetry can occur only when P (H ) has two equal roots. The conditions for the special case in which λ1 = λ2 and λ3 = λ4 are b = c = 0, √ λ1 = −λ3 = ± a.

(13)

In the more general case when λ1 = λ2 ≡ λ and λ3 ≡ ρ we have λ4 = −(2λ + ρ). If λ = 0 we have a 2 = 4c, b = 0 and ρ 2 = 2a, with 1/4 supersymmetry when H = |ρ|. Otherwise we find the condition 4a 3 b2 + 27b4 − 18ab2 c − a 2 c2 + 4c3 = 0

(14)

with 3λ2 = a ± 2(a 2 − 3c)1/2 ,

ρ 2 + 2λρ + 3λ2 = 2a,

(15)

with 1/2 supersymmetry possible when λ is the largest root. The general case of four unequal roots is quite complicated, unless b = 0, in which case √ √ √ √ a + 2 c , a − 2 c , − a − 2 c , − a + 2 c . (16) (λ1 , λ2 , λ3 , λ4 ) = One way to achieve b = 0 is to set P = 0. In this case the bound on H becomes H ≥ U 2 + V 2 + 2|U × V|.

(17)

Note that this becomes H ≥ |U| + |V| when U · V = 0, which is typical of 1/4 supersymmetric orthogonal intersections of branes. The four eigenvalues of {Q, Q} are, in order of increasing magnitude, √ √ √ √ H − a + 2 c, H − a − 2 c, H + a − 2 c, H + a + 2 c. (18) The first of these vanishes when the bound is saturated. The last two are never zero unless all four vanish, which is the vacuum charge sector. The second eigenvalue equals the first only when c = 0, so in this case there are two zero eigenvalues when the bound is saturated and we have 1/2 supersymmetry. Otherwise we have 1/4 supersymmetry. As emphasized earlier, static configurations need not have P = 0 because P may have an interpretation as a domain-wall junction charge, rather than 3-momentum (in general it must be interpreted as a sum of the 3-momentum and a string junction charge).


437

Nevertheless, one may still have b = 0 if U × V vanishes, which it will do if, say, V = 0. In this case, the results are exactly as in the P = 0 case just analysed but with V replaced by P. In particular, if P·U = 0 we then have H ≥ |P|+|U|, and static 1/4 supersymmetric configurations have H ≥ |P| + |U|. For this case, we can bring the charges to the form P = (0, 0, Q),

U = (u1 , u2 , 0),

V = (0, 0, 0),

(19)

where Q is a junction charge. This case is the one analysed in [6], with T = u1 +iu2 being the complex scalar charge in the D = 3 supersymmetry algebra obtained by dimensional reduction on the 3-direction. In agreement with [6] we find that H = |T | + |Q|, so the junction charge contributes positively to the energy of the whole configuration. More generally, we might have P = (0, 0, Q),

U = (u1 , u2 , 0),

V = (v1 , v2 , 0).

(20)

This case was analysed in [13], and an explicit realization of it was found in a model with several chiral superfields; in this model the charge Q is again associated with a domain wall junction. In agreement with [13] we find the four roots to be λ1 = −Q +

(u2 + v1 )2 + (u1 − v2 )2 ,

λ2 = −Q − (u2 + v1 )2 + (u1 − v2 )2 , λ3 = Q − (u2 − v1 )2 + (u1 + v2 )2 , λ4 = Q + (u2 − v1 )2 + (u1 + v2 )2 .

(21)

Note that the four roots are distinct, in general, and (in contrast to the previous case) b = 0. If Q is positive and λ1 is the largest root, the junction charge Q contributes negatively to the total energy as in [13]. The case just considered is a special case of the larger class of configurations with b = 0 for which P (H ) has four distinct roots. At this point the analysis becomes quite complicated, and we shall not pursue it further.

2.2. Stability of BPS states. Our aim in this subsection is to prove the stability of BPS states. We begin by considering the possible decay of a general state, not necessarily BPS, with energy H3 into two other states, not necessarily BPS, with energies H1 and H2 . This can be represented schematically as (state)3 → (state)1 + (state)2 .

(22)

{Q, Q} = H + K(a, b, c),

(23)

Let us write

where K is a traceless symmetric matrix, and (a, b, c) are the three SU (2) × SU (2)R invariants introduced previously. Conservation of charges and energy requires that H 3 = H1 + H 2 , K 3 = K1 + K 2 ,

(24) (25)

438


where Ki = K(ai , bi , ci ), with (ai , bi , ci ) being the values of the invariants (a, b, c) for the ith state. Since the matrices Hi + Ki are positive definite they are subject to the Minkowski reverse triangle inequality (see e.g. [22]) 1

1

1

[det(H3 + K3 )] 4 ≥ [det(H1 + K1 )] 4 + [det(H2 + K2 )] 4 .

(26)

We now want to see the consequences of supposing state 3 to be BPS. We observe that the left hand side of (26) vanishes if state 3 is BPS, but the right-hand side can vanish only if both states 1 and 2 are also BPS. The extension to more than two decay products is immediate so we conclude that any unstable BPS state would have to decay into other BPS states. To complete the proof of stability we now show that a BPS state cannot decay into other BPS states. A BPS state has an energy H = E(K) ≡ E(a, b, c), where E(K) is the largest value of H for which det(H + K) = 0. An equivalent characterization of E(K) is as the smallest eigenvalue of K. It follows that E(K) = min ζ T Kζ , where ζ is a commuting spinor normalized such that ζ T ζ = 1 but otherwise arbitrary. From this and the fact that min(a + b) ≤ min(a) + min(b), we deduce the triangle inequality E(K1 + K2 ) ≤ E(K1 ) + E(K2 ).

(27)

Generic models will have a spectrum of BPS states for which this inequality is never saturated. In such cases BPS states are absolutely stable. In those cases for which there are BPS energies saturating the inequality (27) there may be states of marginal stability3 . The inequality (27) is saturated when K1 and K2 are proportional, with positive constant of proportionality, but this is only a sufficient condition for equality. Another sufficient condition, which we believe to be necessary, is the coincidence, up to normalization, of the eigenvectors of K1 and K2 with lowest eigenvalue. It is instructive to see how the above comments apply to the special case in which H + K = Cγ µ Pµ . The Minkowski inequality becomes −(P1 + P2 )2 ≥ −P12 + −P22 . Since

(28)

√ −P 2 is the rest mass m of a particle with 4-momentum P , we learn that m3 ≥ m1 + m2 .

(29)

This is the familiar rule that the sum of the masses of the decay products cannot exceed the mass of the particle undergoing decay. Given that m3 = 0 we deduce that m1 = m2 = 0, so if a massless particle decays into two other particles those two particles must also be massless. For this special case the triangle inequality (27) reduces to |P1 + P2 | ≤ |P1 | + |P2 |,

(30)

which is saturated if and only if P1 and P2 are parallel, and in this case there is no phase space for the decay. 3 It is well known that marginal stability is the mechanism by which BPS states “decay” as one moves in the space of parameters defining certain theories, but this is a discontinuity of the BPS spectrum as a function of parameters and not a process within a given theory.


439

2.3. Domain walls at angles. Each supersymmetric configuration is associated with a set of Killing spinors 1 which span the kernel of {Q, Q}. With the exception of the vacuum configuration, these spinors are subject to constraints that reduce the dimension of the space that they span. Some properties of supersymmetric configurations follow directly from the nature of these constraints. In particular, intersecting brane configurations can be considered as configurations obtained from parallel branes by rotation of one or more of them. The constraints can be similarly obtained, and then analysed to determine the dimension of the space of Killing spinors they allow [23]. We shall apply this analysis here to intersecting domain walls of N = 1 D = 4 theories. We begin with two coincident domain walls, corresponding to the constraint γ013 1 = 1.

(31)

We then rotate one of them around the 3-axis until it makes an angle β in the 12-plane, and simultaneously rotate by some angle α in the electric-magnetic charge space. This operation is represented by the matrix R = e 2 αγ5 e 2 βγ12 ,

(32)

γ013 R −1 = Rγ013 .

(33)

1

1

which satisfies

The constraint on the Killing spinor 1 imposed by the rotated brane is Rγ013 R −1 1 = 1.

(34)

Using (33) and (31), one easily verifies that this second constraint is equivalent to

R 2 − 1 1 = 0. (35) It is not difficult to show that this equation has no non-zero solutions for 1 unless α ± β = 0. We thus have R = eα3 ,

3=

1 (γ5 ± γ12 ) . 2

(36)

Using the identity 3 3 = −3 one can establish that R 2 − 1 = (2R)(sin α 3).

(37)

Since 2R is invertible, it follows that (35) is equivalent to sin α 3 1 = 0.

(38)

This is trivially satisfied if sin α = 0. Otherwise it reduces to 31 = 0, which is equivalent to γ03 1 = ±1.

(39)

If this is combined with (31) we deduce that γ5 γ023 1 = ∓1,

(40)

440


which is the constraint associated with a purely magnetic domain wall in the 23-plane. We may take any two of these three constraints as the independent ones; the choice (31) and (40) have an obvious interpretation as the constraints associated with the orthogonal intersection of an electric wall with a magnetic one. This constitutes the special α = π/2 case of the more general configuration of rotated intersecting branes that we have been studying. But we have now derived these constraints for any angle α = 0, π . The fraction of supersymmetry preserved by the general rotated brane configuration is therefore the same as the fraction preserved in the special case of orthogonal intersection. Standard arguments can now be used to show that this fraction is 1/4. We have thus shown that starting from a 1/2 supersymmetric configuration of two parallel coincident domain walls with normal n, one of them may be rotated relative to the other by an arbitrary angle in a plane containing n, preserving 1/4 supersymmetry, provided that the charge of the rotated wall is simultaneously rotated by the same angle in the “electric-magnetic” charge space. In practice it may not be possible for the domain walls to intersect at arbitrary angles (preserving supersymmetry). For example, in the Z3 -invariant model discussed in [6], supersymmetric intersections are necessarily at 2π/3 angles. But such additional restrictions are model-dependent. What we learn from the supersymmetry algebra is the model-independent result that the angle separating 1/4 supersymmetric intersecting domain walls must equal the angle between them in the “electric/magnetic” charge space. Since the constraint (39) is associated with non-zero P3 we also learn from the above analysis that we can include this charge, provided it has the appropriate sign, which is determined by the sign in (36), without affecting the constraints imposed by 1/4 supersymmetry, although we then leave the class of configurations for which b = 0. Setting P3 = 0 might be considered as performing a boost along the 3-direction except for the previously noted fact that P3 is not necessarily to be interpreted as momentum. Nevertheless, as a terminological convenience we shall call P the “3-momentum” in what follows. Consider the charge configuration obtained by adding the charges of an electric brane in the 13-plane with a brane rotated in the 12-plane, preserving 1/4 supersymmetry, and then adding momentum in the 3 direction: U = v cos α(sin α, − cos α, 0) + (0, −u, 0), V = v sin α (sin α, − cos α, 0) , P = (0, 0, p).

(41)

a = u2 + v 2 + 2uv cos2 α + p 2 , b = puv sin2 α, c = u2 v 2 sin4 α + p 2 (u2 + v 2 + 2uv cos2 α).

(42)

We now have

One can show that the eigenvalues of {Q, Q} are H + p ± u2 + v 2 + 2uv cos 2α,

H − p ± (u + v).

(43)

For u, v, p ≥ 0, we conclude that H ≥ p + u + v and that 1/4 supersymmetry is preserved when the bound is saturated. Note that in this case

(44) {Q, Q} = u (1 − γ013 ) + v 1 − γ013 R 2 + p (1 − γ03 ) , for the upper sign in (36), confirming that the projections remain unchanged by the inclusion of momentum.


441

2.4. 3/4 Supersymmetry. Continuing the above analysis, we now turn to the case in which u, v, p are not necessarily all positive because this case includes the possibility of domain wall configurations preserving 3/4 supersymmetry [14]. Consider the case α = π/2 for an electric wall and a magnetic wall intersecting at right angles, so that the eigenvalues (43) are H + p ± (u − v),

H − p ± (u + v).

(45)

It follows that H is bounded below by each of the eigenvalues λ1 λ2 λ3 λ4

= = = =

p − u − v, v − u − p, u − v − p, u + v + p.

(46)

If only one of the charges is non-zero, u say, then we obtain the standard BPS bound, H ≥ |u|, which is saturated by the electrically charged BPS domain wall. With two charges, u and v say, we obtain H ≥ |u + v| and H ≥ |u − v|, and when the stronger of these is saturated we have the intersecting domain wall configuration preserving 1/4 supersymmetry. With all three charges, there are four bounds corresponding to the four eigenvalues and 1/4 supersymmetry is preserved, generically, when the strongest bound is saturated. There are then two subcases to consider according to whether or not λ4 is the largest eigenvalue. If λ4 is the largest eigenvalue, as happens, for example, when u, v, p are all positive, then we recover the standard 1/4 supersymmetric case considered above, unless two of the three charges u, v, p vanish in which case 1/2 supersymmetry is preserved. If λ4 is not the largest eigenvalue then one of the others is, and we may choose it to be λ1 because the other possibilities are related to this one by SU (2)R transformations. Given this, H is bounded below by p − u − v and if there is a state saturating this bound with H = p − u − v then the eigenvalues of {Q, Q} are 0,

2(p − v),

2(p − u),

−2(u + v).

(47)

It follows that 1/4 supersymmetry is preserved generically but more supersymmetry is preserved for special values of the charges. The possibility of this kind of enhancement of supersymmetry, including the possibility of 3/4 supersymmetry, was recently discussed in [14] and the case under consideration here is very similar. If p = v or p = u or u = −v, then a charge configuration saturating the BPS bound will preserve 1/2 supersymmetry and if p = u = v or u = −v = ±p then 3/4 supersymmetry will be preserved. Thus, a charge configuration saturating the bound H ≥ λ1 will preserve 1/4 supersymmetry for generic values of the charges, but 1/2 or 3/4 supersymmetry for certain special values. We should stress that the above analysis is purely algebraic and it is an open question whether there exists a physical model with domain wall configurations preserving 3/4 supersymmetry. As we now show, this possibility is not realized by the WZ model. 2.5. BPS solutions of the Wess–Zumino model. The WZ model is known to admit both 1/4 and 1/2 supersymmetric classical solutions, which (at least potentially) correspond to states in the quantum theory. We shall show here that there are no classical solutions preserving 3/4 supersymmetry. We shall begin by considering purely bosonic field configurations and then extend the result to arbitrary classical configurations.

442


The fields of the WZ model belong to a single chiral superfield, the components of which are a complex physical scalar φ = A + iB, a complex two-component spinor, which is equivalent to a 4-component Majorana spinor λ, and a complex auxiliary field F = f + ig. We will continue to use a real representation of the four Dirac matrices γ µ . For purely bosonic field configurations we need only consider fermion supersymmetry transformations. Our starting point will therefore be the (off-shell) supersymmetry transformation of λ, which takes the form δλ = M1, where 1 is a real constant spinor parameter and M is the real 4 × 4 matrix M = γ µ (∂µ A + γ5 ∂µ B) + f + γ5 g.

(48)

This transformation is valid for the spinor component of any chiral superfield. The Wess–Zumino model is characterised by the fact that the auxiliary field equation is F ≡ f + ig = W (φ),

(49)

where W (φ) is the derivative with respect to φ of the holomorphic superpotential W (φ). A bosonic field configuration of the WZ model will be supersymmetric if there is a spinor field 1 that is both annihilated by M(x), for all x, and covariantly constant with respect to a metric connection on E(1,3) . Thus, for there to be n preserved supersymmetries it is a necessary condition that M(x) has an n dimensional kernel for each x. Our strategy for showing that there are no 3/4 supersymmetric field configurations will be to analyse necessary conditions for the matrix M0 ≡ M(x0 ) at a fixed point x0 to have an n-dimensional kernel. We begin by noting that a WZ field configuration can preserve 1/4 supersymmetry only if det M0 vanishes, which is equivalent to

2

(∂A)2 + (∂B)2 − f 2 − g 2 = 4 (∂A)2 (∂B)2 − (∂A · ∂B)2 . (50) This condition is necessary for the preservation of at least 1/4 supersymmetry in any model with a single chiral superfield, and in particular in the WZ model. Configurations preserving more than 1/4 supersymmetry are characterized by additional constraints on the fields. Necessary constraints can be found very easily by making use of the fact that M0 can be brought to (real) upper-triangular form by a similarity transformation. We may therefore assume that M0 is upper triangular. If, in addition, it has a 2-dimensional kernel then it may be brought to the form   0 0 ∗ ∗  0 ∗ ∗ , (51)  ∗ ∗ ∗ where ∗ indicates an entry that is not zero (or not necessarily zero in the case of the off-diagonal entries). This matrix has the property that 2tr M03 − 3tr M0 tr M02 + (tr M0 )3 = 0, and substituting (48) we learn that

f f 2 + g 2 − (∂A)2 − (∂B)2 = 0.

(52)

(53)


443

This condition is therefore necessary for a field configuration to preserve 1/2 supersymmetry. Similarly, any upper-triangular matrix with a 3-dimensional kernel can be brought to the form   0 0 0 ∗  0 0 ∗ . (54)  0 ∗ ∗ This matrix satisfies both (52) and tr M02 = (tr M0 )2 , in addition to (50). These conditions, which are therefore necessary for 3/4 supersymmetry, are equivalent to the joint conditions f = 0, g 2 = (∂A)2 + (∂B)2 , 2 (∂A) (∂B)2 = (∂A · ∂B)2 .

(55)

We are now in a position to show that there are no 3/4 supersymmetric WZ field configurations (other than the vacuum which has 4/4 supersymmetry). The conditions (55) must be satisfied by such a field configuration. We shall analyse these conditions at a fixed point x = x0 and consider separately the cases in which g = 0 and g = 0 at that point. If g = 0 then the second condition in (55) implies that at x0 either the 4-vectors ∂A and ∂B are both null or one is spacelike and the other is timelike. The latter option contradicts the third of Eqs. (55) so both are null. It then follows from (55) that ∂A and ∂B are parallel, so that f = g = 0,

∂A = α v,

∂B = β v,

(56)

where α and β are constants and v is a null 4-vector. This field configuration is therefore a candidate for 3/4 supersymmetry, but because the conditions leading to it were not sufficient for 3/4 supersymmetry this must be checked. In fact, it is readily shown that the matrix M corresponding to the configuration (56) has only a two-dimensional kernel so that at most 1/2 supersymmetry can be preserved. The remaining candidates for 3/4 supersymmetry in the WZ model arise from field configurations in which f vanishes but g is non-zero. Then (55) implies that at x0 either ∂A and ∂B are both spacelike, or one is spacelike and the other is null. Suppose first that either ∂A or ∂B is null. In the case in which ∂B is null we have f =0

∂A = gs,

∂B = v,

(57)

where v is a null vector orthogonal to a spacelike vector s normalized such that s 2 = 1. For this configuration one can check that the matrix M generically has a one dimensional kernel, and has a two dimensional kernel when either g = 0 or β = 0. The case in which ∂A is null is similar, with the same result that at most 1/2 of the supersymmetry is preserved. If neither ∂A nor ∂B is null then they are both spacelike and we can arrange for them to take the form ∂B = β(0, 1, 0, 0), ∂A = α(sin θ, cos θ, 0, sin θ),

(58)

444


with g 2 = α 2 cos2 θ + β 2 . One then finds that the kernel of M(x0 ) is 2-dimensional if αβ sin θ = 0 and otherwise 1-dimensional. Configurations of the form (58) can therefore preserve at most 1/2 supersymmetry. We have now shown that there are no non-vacuum bosonic WZ field configurations that preserve 3/4 supersymmetry. We now wish to consider whether this remains true when we consider general configurations that are not necessarily bosonic. This question is perhaps best posed in the context of the quantum theory, which we will not consider here, but it can also be posed classically by taking all fields to be supernumbers with a “body” and a nilpotent “soul”. Any general field configuration of this kind preserving 3/4 supersymmetry must have a body preserving at least 3/4 supersymmetry and, as we have just seen, the vacuum configuration is the only candidate. It follows that the only remaining way in which a classical field configuration could be 3/4 supersymmetric is if the 4/4 supersymmetry of the bosonic vacuum configuration is broken to 3/4 by fermions. Preservation of any fraction of supersymmetry in a fermionic background requires the vanishing of the supersymmetry transformations of the bosons. For the WZ model this implies (λ¯ ≡ λT C) ¯ = 0, λ1

¯ 5 1 = 0, λγ

(59)

and for 3/4 supersymmetry there must be a three-dimensional space of parameters 1 for which this condition holds. At a given point in space we may choose, without loss of generality, a basis in spinor space such that C1 = (0, ∗, ∗, ∗)T , where an asterisk indicates an entry that may be non-zero. The first equation then implies that λT = (∗, 0, 0, 0) and the second that λT γ5 = (∗, 0, 0, 0). But since γ5 is both real and satisfies γ52 = −1 these conditions are not mutually compatible. This concludes our proof that the WZ model has no non-vacuum classical configurations, bosonic or otherwise, that preserve 3/4 supersymmetry 3. The Geometry of Supersymmetry We now turn to a discussion of the geometry associated with BPS representations of the algebra (2), which we may re-write in terms of a positive semi-definite symmetric bispinor Z as {Q, Q} = Z. The positivity of {Q, Q} implies that Z is a vector in a convex cone, with the boundary of the cone corresponding to the BPS condition det Z = 0. We shall first explain some of the geometry associated with convex cones, and how it relates to BPS states. We will then explain how this ties in with the theory of Jordan algebras. 3.1. Convex cones. Let us begin with the standard D = 4 N = 1 supersymmetry algebra, in which case Z = γ · P and the positivity of {Q, Q} implies that P lies either in the forward lightcone of D = 4 Minkowski momentum-spacetime or on its boundary, the lightfront. In the latter case, P 2 = 0 and any states with this 4-momentum are BPS, preserving 1/2 supersymmetry. The forward lightcone in momentum space and the forward lightcone in position space are both examples of convex cones. An ndimensional cone C is a subspace of an n-dimensional vector space V with the property that λx ∈ C for all x ∈ C and all real positive λ. The cone is convex if the sum of any two vectors in the cone is also in it. The dual cone is then defined as follows. Let y be a vector in the dual vector space V ∗ and let y · x be a bilinear map from V × V ∗ to R. The dual cone C ∗ is the subspace of V ∗ for which y · x > 0 for all x ∈ C.


445

Given a translation-invariant measure on V we can associate with each convex cone in V a characteristic function ω defined by −1 e−y·x d n y. (60) ω (x) = C∗

As all translation-invariant measures are multiples of any given translation-invariant measure, this formula defines ω up to a scale factor, but this ambiguity will not affect the statements to follow. The cone is foliated by hypersurfaces of constant ω, with the limiting hypersurface ω = 0 being the boundary of the cone. In the case of the forward light cone in D=4 Minkowski spacetime the vector space V is R4 and ω = N 2 , where N (x) = −ηµν x µ x ν is the quadratic form defined by the Minkowski metric η (we adopt a “mostly plus” metric convention). The hypersurfaces of constant ω are therefore hyperboloids homothetic to SO(1, 3)/SO(3). Note that this is a symmetric space; this is a general feature of self-dual homogeneous convex cones, of which the forward lightcone in Minkowski space is an example. Homogeneous convex cones that are not self-dual are foliated by homogeneous spaces that are not symmetric spaces. Because, in this example, ω is determined by a quadratic function N , the vector space V = R4 can be viewed as a metric space, with Minkowski metric η. More generally, ω is not quadratic and hence does not furnish V with a metric. Nevertheless, ω does provide a positive definite metric for C (obviously, this differs from the Minkowski metric of the “quadratic” case discussed above). Let us first note that, by the definition of a cone, the map D : x → λx is an automorphism, in that Dx ∈ C if x ∈ C. It follows immediately that ω(x) is a homogeneous function of degree n. A corollary of this is that π(x) · x = 1, where π(x) =

1 ∂ log ω . n ∂x

(61)

Thus, π ∈ C ∗ , and as x ranges over all vectors in C so π ranges over all vectors in C ∗ . One can now introduce a metric g on C with components4 1 gij = − ∂i ∂j log ω(x). n

(62)

πj = x i gij .

(63)

One may verify that

The map from C to C ∗ provided by the metric (62) has a natural interpretation in terms of Hamilton–Jacobi theory: if log ω is interpreted as a characteristic function in the sense of Hamilton, then π as defined by (61) is the conjugate momentum. A feature of the metric g is that it is invariant under automorphisms of C. For example it follows from the homogeneity of ω that the linear map D is an isometry of g. The group of automorphisms will generally be a semi-direct product of D with some group G that acts on the leaves of the foliation. The cone is homogeneous if G acts transitively. A homogeneous cone is foliated by homogeneous hypersurfaces of the form G/H for some isotropy group H . For a self-dual cone this homogeneous space is also a symmetric space. As already mentioned, the forward light cone in E(1,3) is foliated by hyperboloids 4 For the forward light-cone in Minkowski spacetime with Minkowski metric η, we have g ij = (x 2 )−2 (2xi xj − x 2 ηij ), where x 2 = ηij x i x j and xi = ηij x j , so that πi = (x 2 )−1 xi .

446


homothetic to SO(1, 3)/SO(3), so G is the (proper orthochronous) Lorentz group. The metric induced on each leaf of the foliation by the metric gij of the cone is the positivedefinite SO(1, 3)-invariant metric on SO(1, 3)/SO(3). Let us now turn to the general D = 4 N = 1 supersymmetry algebra {Q, Q} = Z. The bispinor charge Z can be interpreted as a vector in the convex cone of positivedefinite real 4 × 4 symmetric matrices. This is a cone in R10 which, since Z includes the 4-momentum, we may consider as a “momentum-space” cone C ∗ . We set aside to the following subsection consideration of the corresponding “position space” cone C. The characteristic function of C ∗ is5 5

ω(Z) = (det Z) 2 .

(64)

The cone is again a self-dual homogeneous one, and is foliated by symmetric spaces that are homothetic to SL(4; R)/[SO(4)]. Of principal interest here is the boundary of C ∗ , defined by det Z = 0, because this is the condition for preservation of supersymmetry. The geometry of this boundary is now rather more complicated than it was before. The basic observation required to understand this geometry is that the cone is a stratified space with strata Sn , n = 0, 1, 2, 3, 4, where Sn is the subspace in which at least n of the four eigenvalues vanish, corresponding to at least n supersymmetries being preserved, and Sn+1 is the boundary of Sn . The boundary of the cone is the space S1 , which is the 9-dimensional space of matrices of rank 3 or less. The boundary of this is the space S2 of matrices of rank 2 or less which make up a 7 dimensional space. To see why it is 7 dimensional recall that to specify a matrix of rank 2 it suffices to give the normalised eigenvectors with non-vanishing eigenvalues together with their eigenvalues. The two eigenvectors define a 2-plane in R4 , corresponding to an element of the 4-dimensional Grassmannian SO(4)/(SO(2) × SO(2)). Giving the orientation of the eigenvectors within the 2-plane means specifying one of the SO(2) factors. In other words the basis of 2 eigenvectors corresponds to the 5 dimensional Stiefel manifold SO(4)/SO(2). Taking into account the two eigenvalues we have a 7-dimensional space, as claimed. The boundary of this stratum is the set S3 of matrices of rank 1 or less. These span a 4-dimensional space, since a rank 1 matrix is specified by the direction, up to a sign, of its eigenvector with non-zero eigenvalue together with the eigenvalue. This is a point in RP 3 × R+ . Finally, the boundary of S3 is the stratum S4 consisting of the zero matrix, which is the vertex of the cone.

3.2. Reverse triangle inequalities. The Minkowski inequality that we used previously to establish the stability of BPS states is a special case of a reverse-triangle inequality valid for all convex cones. Let us define the “length” of a vector in an n-dimensional convex cone with characteristic function ω as L(x) = ω1/n (x).

(65)

This is a homogeneous function of degree 1. Because the hypersurfaces of constant ω are concave, this “length” satisfies the reverse triangle inequality L(x + x ) ≥ L(x) + L(x )

(66)

5 Note that ω2 is a polynomial. A theorem of Koecher states that ω2 is a polynomial for all self-dual homogeneous convex cones.


447

with equality if and only if x and x are proportional. In the case of the cone of m × m positive definite hermitian matrices we have L(x) = (det x)1/m and the reverse triangle inequality is the Minkowski inequality 1

1

1

[det(x + y)] m ≥ [det x] m + [det y] m ,

(67)

with equality if the two matrices are proportional. In the special case of diagonal m matrices, the cone becomes the positive orthant Rm + in E . The length of a vector m 1/m x = diag(x1 , . . . , xm ) in R+ is L(x) = (x1 . . . xm ) , and Minkowski’s inequality for positive definite matrices reduces to a form of Holder’s inequality (see e.g. [22]). 2 The metric g on Rm (d log x i )2 . The automorphism + is the flat metric dl = (1/m) group is the permutation group Sm , which is clearly an invariance of the length. 3.3. Conformal invariance. For the standard D = 4 N = 1 supersymmetry algebra without central charges all BPS states have P 2 = 0. This is the momentum space version of the massless wave-equation, which is invariant under the action of the conformal group SU (2, 2) on compactified Minkowski spacetime. Our aim here is to show how this generalizes when the domain wall charges are included. This will turn out to be a straightforward extension of the standard case, appropriately formulated, so we consider that first. It is convenient to identify a point in Minkowski spacetime with a matrix X = Xµ σµ , where σµ = (1, σ1 , σ2 , σ3 ) are the 2 × 2 Hermitian sigma-matrices. The conjugate momentum P is then similarly a 2 × 2 Hermitian matrix and −P 2 becomes det P . (The momentum P should not be confused with the dual variable π introduced in the previous subsection.) Let us now consider the massless particle action I = [trP dX − e det P ], (68) where e (the einbein) is a Lagrange multiplier for the mass-shell constraint det P = 0. The conformal group SU (2, 2) acts on the compactification of Minkowski space via the fractional linear transformation X → X = (AX + B)(CX + D)−1 , where the hermiticity of

X

requires that A B ∈ SU (2, 2). C D

(69)

(70)

This implies that dX (CX + D) = (A − X C)dX.

(71)

We deduce from this that the P dX part of the action I is invariant (up to a surface term) if P → P = (CX + D)P (A − X C)−1 .

(72)

This transformation implies det P → det P = G−1 det P ,

(73)

448


where G=

det(A − X C) . det(CX + D)

(74)

The action I is therefore invariant if we assign to the einbein the transformation e → e = Ge. We now wish to determine the analogous symmetry group of the more general BPS condition det Z = 0. The matrix Z can be viewed as a vector in a 10-dimensional vector space. Let X be coordinates of the dual space and consider the particle action I = [trZdX − e det Z]. (75) Special cases of actions of this type were considered previously by Cederwall [24], with a motivation derived from Jordan algebra considerations that we shall explain in the following subsection (see also [25, 16]). Now consider the fractional linear transformation X → X = (AX + B)(CX + D)−1 ,

(76)

which acts on the compactification of the space of symmetric matrices [26]. The matrix X will also be real and symmetric provided that A B ∈ Sp(8; R). (77) C D That is, AT D − C T B = 1,

AT C = C T A,

B T D = D T B.

(78)

As before, we deduce (71) and from this that the ZdX term is invariant up to a surface term if Z → Z = (CX + D)Z(A − X C)−1 .

(79)

det Z → det Z = G−1 det Z,

(80)

This implies

where G has form of (74). We may again take e → e = Ge to achieve an invariance of the action I . In this case, the invariance group is Sp(8; R). Note that this conclusion rests on an interpretation of the 4-dimensional compactified Minkowski spacetime as a subspace of a ten-dimensional vector space of the 4 × 4 real symmetric matrices X. A field theory realization of Sp(8; R) would require fields defined on this larger space. For example, the analogue of the massless wave equation on Minkowski space is the fourth-order equation det(−i∂/∂X) H = 0.

(81)

The symmetry group of this equation is Sp(8; R). By analogy with the Minkowski case, we expect this to be the maximal symmetry group of this equation.


449

3.4. Jordan algebras. The results of the previous subsections have an interpretation in terms of Jordan algebras. A Jordan algebra J of dimension n and degree ν is an ndimensional real vector space with a commutative, power associative, bilinear product, and a norm N that is a homogeneous polynomial of degree ν (see e.g. [27]). There are four infinite series of simple Jordan algebras, realizable as matrices with the Jordan product being the anticommutator: the degree 2 algebras 3(n) to be discussed below, and the series JkR , JkC , JkH , which are realized by k × k hermitian matrices over R, C or H, with norm given by the determinant, N (x) = det(x). In addition, there is one “exceptional” Jordan algebra J3O realizable by 3 × 3 hermitian matrices over the octonions. Associated with any Jordan algebra J with product x ◦ y is a self-dual homogeneous convex cone C(J ). This is the subspace of J consisting of elements ex with x ∈ J (where ex is defined by the usual power series with x n+1 = x n ◦ x). The characteristic function is ω = N n/ν ,

(82)

so the boundary of the cone corresponds to N = 0. The cone is foliated by copies of the homogeneous space Str(J )/Aut(J ), where Str(J ) is the invariance group of N (the “structure group” of the algebra) and Aut(J ) is the automorphism group of the algebra (the subgroup of Str(J ) that fixes the identity element in J ). The relation of self-dual homogeneous convex cones to Jordan algebras has similarities to the relation between Lie groups and Lie algebras. Recall that a Lie group is parallelizable but has a non-zero torsion given by the structure constants of its Lie algebra. A self-dual homogeneous convex cone C, on the other hand, is not parallelizable (in general) but its torsion-free affine connection is determined by the structure constants of a Jordan algebra. Because of homogeneity it suffices to know the connection at the “base” point c ∈ C defined by6 gij |c = δij .

(83)

Let fij k be the structure constants of J in a basis ei = (c, ea ). Then Jij k |c = fij k .

(84)

Although Jordan algebras are commutative they are nonassociative. Define the associator {a, b, c} ≡ (a ◦ b) ◦ c − a ◦ (b ◦ c).

(85)

The curvature tensor of the cone at the base point is then given by the relation {ei , ej , ek } = Rij k l |c el .

(86)

In addition to the automorphism and structure groups, there is a larger “Möbius group” associated with any Jordan algebra J , acting on elements of J by fractional linear transformations. We therefore have the sequence of groups Aut(J ) ⊂ Str(J ) ⊂ Mo(J ),

(87)

6 There is only one such point, even in those cases for which C is flat. It corresponds to the identity element in the algebra. We use the notation c to indicate both the identity element of J and the base point of the cone C(J ).

450


associated with any Jordan algebra J . These can be interpreted as generalized, rotation, Lorentz and conformal groups, respectively [19]. To motivate this interpretation, we return to the representation of a Minkowski 4-vector as the 2 × 2 Hermitian matrix X. This is an element in the degree 2 Jordan algebra J2C . The dimension is 4 and the norm is the determinant, which is the SL(2; C) invariant Minkowski norm N on R4 . The group SL(2; C) acts on 2 × 2 matrices by conjugation so the subgroup leaving invariant the identity matrix is its maximal compact SU (2) subgroup. The convex cone associated with this Jordan algebra is the forward light-cone of D = 4 Minkowski spacetime. As we saw in the previous subsection, the group of fractional linear transformations of X is SU (2, 2), so the sequence (87) is, in this case, SU (2) ⊂ SL(2; C) ⊂ SU (2, 2).

(88)

These are the standard rotation, Lorentz and conformal groups. The inclusion of domain wall charges means that we should replace J2C by J4R , the algebra of 4×4 symmetric real matrices. One can see that J2C is a subalgebra of J4R from the fact that J2C ∼ = 3(4), where 3(n) is the n-dimensional Jordan algebra with basis (1, σ1 , . . . σn−1 ) and Jordan product σa ◦ σb = 2δab ; this has a realization in which σa are sigma-matrices of an n-dimensional Minkowski spacetime, with the Jordan product being the anticommutator; it follows that the standard supersymmetry algebra in D dimensions is naturally associated with 3(D). For D = 4 one can choose the σa to be the three 2 × 2 hermitian Pauli matrices, hence the isomorphism J2C ∼ = 3(4). All simple Jordan algebras of degree 2 are isomorphic to 3(n) for some n. Having replaced J2C by J = J4R we find that the sequence (88) is generalized to [19] SU (2) × SU (2) ⊂ SL(4; R) ⊂ Sp(8; R).

(89)

We now turn to the Jordan algebraic interpretation of the boundary of the convex cone C(J ). This consists of elements λP ∈ J , where λ is a positive real number and P is an idempotent of J with less than maximal rank, i.e. its trace, defined by tr X = log N (eX ), is less than ν. An idempotent is a non-zero element P ∈ J satisfying P ◦ P = P , and two idempotents P and P are said to be orthogonal if P ◦ P = 0. The idempotents with unit trace are called the primitive idempotents, and the number of mutually orthogonal primitive idempotents equals the degree ν of the algebra. For a Jordan algebra of degree 2 all idempotents of less than maximal rank have unit trace and are therefore primitive. This is true of J2C , in particular, corresponding to the fact that the only supersymmetric states other than the vacuum permitted by the standard D = 4 N = 1 supersymmetry algebra are 1/2 supersymmetric states associated with massless particles (for which the 4-momentum lies on the positive light-front). Note that although at most two primitive idempotents of a degree 2 Jordan algebra can be orthogonal in the above sense, the space of primitive idempotents of 3(D) is (D − 1)-dimensional. The boundary of the associated convex cone is therefore (D − 1)-dimensional. For 3(4) ∼ = J2C , in particular, this boundary is the three-dimensional forward light-front of the origin of 4-dimensional Minkowski momentum space. For a Jordan algebra J of degree ν > 2, there are idempotents of less than maximal rank that are not primitive. For an algebra of degree 3, these non-primitive idempotents generate faces of the boundary of eJ which themselves have a boundary generated by the primitive idempotents. An example is the (non-simple) Jordan algebra J = R ⊕ R ⊕ R for which eJ is the positive octant in E3 ; its boundary consists of three faces that meet on the three axes generated by the three primitive idempotents (in this case there are


451

only three primitive idempotents, which are therefore orthogonal; details can be found in [28]). More generally, for Jordan algebras of higher degree, the boundary of the associated convex cone is a stratified set of faces. In particular, J4R has degree 4 so the faces of the boundary of its associated convex cone are generated by idempotents of trace 1,2 and 3, corresponding to 3/4,1/2 and 1/4 supersymmetry respectively. The primitive idempotents, of unit trace, correspond to 3/4 supersymmetry.

3.5. Entropy of BPS fusion. In a quantum field theory realization of the D = 4 N = 1 supersymmetry algebra the central charges Z are labels of quantum states. We have now seen that the set of these charges naturally carries the structure of a Jordan algebra. This algebra may itself be regarded as a finite-dimensional state space (not to be confused with infinite-dimensional space of states of the field theory that carry these charges). This interpretation is of course how Jordan algebras originally arose (see [29] for a review). The exceptional Jordan algebra provides a state space more general than conventional quantum mechanics but for all other Jordan algebras the formalism is equivalent to one in which a state is represented by a density matrix. The general state is therefore a mixed state. The pure states correspond to the primitive idempotents; these lie on the boundary of the convex cone C(J ) but do not in general exhaust it. Rather, the boundary is stratified by sets of states of successively less purity, corresponding in our application to states with successively less supersymmetry. Thus, the pure states in this sense are the charge configurations that preserve 3/4 supersymmetry, the remaining supersymmetric configurations corresponding to states on the boundary of the cone that are not pure. We previously showed that a BPS state is stable against decay into any other pair of states; in particular it cannot decay into two BPS states. Consider now the reverse process, i.e. fusion of two BPS states to form a third via the inverse of the reaction (22), i.e. (BPS)1 + (BPS)2 → (BPS)3 .

(90)

If the first two states preserve 3/4 supersymmetry then the third one will generally preserve less supersymmetry. This is like passing from a pure to a mixed state. There is also a formal resemblance here to classical thermodynamics. The Jordan algebra J , now viewed as vector space V containing the convex cone C(J ), is spanned by the extensive quantities while the dual vector space V ∗ is spanned by the intensive variables. The function S(x) = log ω(x)

(91)

of the extensive variables may be interpreted as entropy. Because it is convex S(µx + (1 − µ)x ) ≥ µS(x) + (1 − µ)S(x ),

(92)

with equality when x is proportional to x , the entropy can not decrease as a result of a fusion process such as (90). Conversely, the (marginal) stability of a single BPS state against decay into two other BPS states can now be understood as being forbidden by a version of the second law of thermodynamics.

452


4. BPS States for adS The N = 1 D = 4 adS anticommutator (4) may be written as

1 {Qα , Qβ } = MAB CJ AB , αβ 2

(93)

where J A = (γ µ , γ5 )

(94)

and MAB = −MBA are the generators of the adS group SO(3, 2) (and so are no longer central). The matrix C is the SO(3, 2) charge conjugation matrix; we can choose a representation in which C = γ0 γ5 ,

(95)

and this choice will be implicit in what follows. Note that {J A , J B } = 2ηAB ,

(96)

where η is a flat metric on E(2,4) , such that η = diag(−1, 1, 1, 1, −1) in cartesian coordinates. Although (4) is preserved by GL(4; R), the automorphism group of the adS supergroup OSp(1|4; R) is Sp(4; R) ⊂ SL(4; R) ⊂ GL(4; R). The anticommutator (4) can also be written in the form (2), with M04 = H,

Mi4 = −Pi ,

M0i = −Ui ,

Ji ≡

1 1ij k M j k = −Vi , 2

(97)

where H is the hamiltonian, P the 3-momentum, J the angular momentum while the 3-vector U generates boosts. The analysis of supersymmetric charge configurations is then exactly the same as in the super-Poincaré case considered earlier, and in particular requiring 41 , 21 or 43 supersymmetry gives exactly the same conditions on the charges H, U, V, P as were found earlier. The condition for preservation of supersymmetry can be expressed in terms of the SO(3, 2) Casimirs. We will first show how the values of these Casimirs are constrained by the physical state condition, and then turn to the supersymmetric states. 4.1. Physical states in adS. Physical states lie either in the convex cone for which AB is positive, or on its boundary, for which det Z = 0. This cone Z = 21 MAB CJαβ is a subspace of the 10-dimensional vector space spanned by 5 × 5 skew-symmetric matrices M with entries MAB . The matrix commutator turns this space into the Lie algebra so(3, 2). This algebra has rank 2, with quadratic Casimir7 c2 =

1 MAB M AB , 2

(98)

and quartic Casimir c4 = M A B M B C M C D M D A .

(99)

7 The quadratic Casimir provides a metric of signature (4, 6) on the 10-dimensional vector space, but this metric (which is inherited from the metric η on E(3,2) ) does not play a crucial role in the following analysis.


453

Since detZ is both a quartic polynomial of the charges and SO(3, 2) invariant it must be a linear combination of c4 and c22 . In fact det Z = c4 − c22 ,

(100)

c4 ≥ c22

(101)

and hence

for physical states. There is a further constraint on the Casimirs required by physical states. To see this, we begin by noting that the vacuum is the only physical state for which the energy M04 vanishes. This follows from the fact that {Q, Q} is positive semi-definite, with a trace equal to 4M04 . We next prove that M04 must vanish if the kernel of M contains a timelike 5-vector. Suppose that such a 5-vector exists. By an SO(3, 2) transformation, we can arrange for it to have only one non-vanishing component, in the 4-direction. It then follows that the only non-vanishing components of M are Mµν . In particular, the energy M04 vanishes. Thus, for any non-vacuum physical state the kernel of M contains no timelike vectors. Note that the kernel of M has dimension 1, 3 or 5, according to whether M has rank 4, 2 or 0, respectively. The vacuum is the only physical state for which M has rank 0. Now consider the Pauli–Lubanski 5-vector sA =

1 ABCDE MBC MDE . 1 8

(102)

This satisfies the identity MAB s B ≡ 0,

(103)

which shows that, unless it vanishes, s is in the kernel of M. A timelike s would therefore be in the kernel of M but, as we have just seen, the kernel of M cannot contain timelike vectors unless M vanishes, but in that case s also vanishes. Thus, s cannot be timelike. Now, s 2 ≡ ηAB sA sB =

1 2 (2c − c4 ), 4 2

(104)

so s will be non-timelike if and only if c4 ≤ 2c22 .

(105)

This bound implies (for physical states) that c4 = 0 when c2 = 0 . 4.2. Supersymmetric states. Our main interest is in BPS states, i.e. the subset of physical states that are supersymmetric. These must saturate the bound (101), so BPS states are those for which c4 = c22 .

(106)

1 2 c 4 2

(107)

Using this in (104) we see that s2 =

454


for supersymmetric states. We will organise our discussion of the supersymmetric states according to whether s is zero, spacelike or non-vanishing null. If s vanishes then M has either a 3-dimensional or a 5-dimensional kernel. M will have a 5-dimensional kernel only if it vanishes. If the kernel is 3-dimensional then, as we have seen, it cannot contain timelike vectors. It may contain null vectors but any such null vector must be orthogonal to all other vectors in the kernel, spacelike or null, because we could otherwise find a timelike linear combination. Since the maximum number of mutually orthogonal null 5-vectors is 2, a 3-dimensional kernel must contain at least one spacelike vector. There are three possible choices for the other two linearly independent 5-vectors: (i) both spacelike, (ii) one spacelike and one null, or (iii) both null. In all cases M can be brought to a form in which M04 = E ≥ 0 is its only independent entry. In case (i) M04 and M40 are the only entries, and the only supersymmetric state with this property is the vacuum, with E = 0. In case (ii) M can be brought to a form for which the only non-zero upper-triangular entries are M04 = M02 = E. It then follows from the discussion of Sect. 2.4, on which we will elaborate below, that all such states are 1/2 supersymmetric. In case (iii) M can brought to a form for which the only nonzero upper-triangular entries are M04 = −M02 = M23 = M34 ; all such states are 3/4 supersymmetric. Consider now spacelike s. In this case we may choose the only non-vanishing component of s to be its 1-component. Since s now spans the kernel of M, this 5 × 5 matrix M then reduces to a 4 × 4 matrix F acting on the 4-dimensional (0234) subspace orthogonal to s, on which η restricts to a metric η˜ of signature (2, 2). The matrix F is equivalent to a second-rank antisymmetric tensor in E(2,2) that can be written uniquely as F = F + + F − , where F + is real and self-dual while F − is real and anti-self-dual matrix. Now

c4 − c22 = tr(ηF ˜ − )2 . (108) ˜ + )2 tr(ηF We can write F as



 0 u b E  −u 0 −v c  F = −b v 0 −p  −E −c p 0

(109)

vE + bc + up = 0,

(110)

provided that since s would otherwise vanish. Now −tr(ηF ˜ ± )2 = (E ∓ v)2 − (u ± p)2 − (b ± c)2 .

(111)

Configurations with self-dual or anti-self-dual F , for which E = ∓v, u = ±p and b = ±c, are 1/2 supersymmetric. However, any configuration for which (E ∓ v)2 = (u ± p)2 + (b ± c)2 is also supersymmetric. In fact

{Q, Q} = (E ∓ v) − (b ± c)γ 012 + (u ± p)γ 013

+ v − cγ 02 + pγ 03 1 ± γ 1 .

(112)

(113)


455

Given (112), the term in square brackets is proportional to a 1/2supersymmetry pro jector that commutes with the 1/2 supersymmetry projector (1/2) 1 ± γ 1 which leads generically to 1/4 supersymmetry. The final case to consider is s null but non-zero. By means of an SO(3, 2) transformation we may choose s ∝ (1, 0, 1, 0, 0).

(114)

This choice is preserved by an SO(1, 2) “stability” subgroup, and by a transformation in the SO(2) subgroup of this group we can bring M to the standard form   0 0 −a 0 1 0 a 0 −1  0   M = E  a −a 0 t −q  . (115) 0 0 −t 0 −r  −1 1 q r 0 One then finds that c2 = E 2 (t 2 − q 2 − r 2 ), so that supersymmetric states are those with t = ± q 2 + r 2.

(116)

(117)

Actually, in arriving at the above form of M we have used only that the null 5-vector (1, 1, 0, 0, 0) is in the kernel of M. To ensure that this 5-vector is proportional to s (with non-zero constant of proportionality) we require that t + ra = 0.

(118)

This condition also ensures (as it must) that M has rank 4. When combined with (117) it implies that t = 0. For M of the form (115) we have {Q, Q} = E (1 − aγ3 ) (1 − γ01 ) − tγ1 1 − (q/t)γ012 − (r/t)γ013 . A spinor 1 is in the kernel of {Q, Q} if (q/t)γ012 + (r/t)γ013 1 = 1,

(119)

(120)

(121)

and γ01 1 = 1,

(122)

and these two constraints imply 1/4 supersymmetry. Note that when a = ±1 and q = 0 and hence t = ±r, the latter constraint can be replaced by γ3 1 = ±1, which again yields 1/4 supersymmetry.

456


4.3. Examples. Many of the possibilities for BPS configurations just noted are illustrated by the class of examples considered in Sect. 2.4. This means, in the language of this section, that the non-zero upper-triangular components of MAB are taken to be M04 = E, M34 = −p, M02 = u and M23 = −v. The Pauli–Lubanski 5-vector is then s = (0, Ev + up, 0, 0, 0),

(123)

so s is spacelike unless it vanishes. The Casimirs for this class are given by c2 = E 2 + v 2 − p 2 − u2 , c4 = 2[E 4 + u4 + v 4 + p 4 − 2(v 2 + E 2 )(u2 + p 2 ) − 4Euvp].

(124) (125)

The BPS condition c4 = c22 becomes (E − u − v − p)(E − u − v + p)(E − u + v − p)(E + u − v − p) = 0,

(126)

in agreement with (45). Let us first consider vanishing s. We have seen above that M can be brought to a standard form in which all charges are determined in terms of M04 = E. The nonvacuum BPS states occurred for cases (ii) and (iii) discussed above. An example of case (ii) within the class of configurations now under discussion is found8 by setting v = p = 0 and E = |u|. Finally, an example of case (iii), with 3/4 supersymmetry, is obtained by setting u = v = p = −E < 0, although there is no known field theoretic realization of this case. We next to turn to examples with s spacelike. Let us first consider u = p = 0 and set v = −J , where J is the spin about the 1-axis. We then have c2 = E 2 + J 2 ,

c4 = 2E 4 + 2J 4 ,

(127)

which is equivalent to

c2 1 c4 − c22 , + E= 2 2 c2 1 J =± c4 − c22 . − 2 2

(128)

The physical states satisfy E ≥ |J | and states that saturate this bound preserve 1/2 supersymmetry. For these configurations the matrix F of (109) is either self-dual or anti-self-dual. An example of states with s spacelike and F neither self-dual or anti-selfdual can be obtained by taking u, v, p to be positive and solving (112) via E = u+v +p. We then have {Q, Q} = u(1 + γ 013 ) + p(1 + γ 03 ) + v(1 + γ 1 )

(129)

and 1/4 of the supersymmetry is preserved. 8 The charge u can be interpreted as a membrane charge. To see this note that there is a static planar solution of the equations of motion of a test membrane in ad S4 at a fixed radial distance, in horospherical coordinates, from the Killing horizon [30]. This solution must preserve 1/2 supersymmetry of the ad S4 supersymmetry because ad S4 can itself be interpreted as a membrane, at the horizon, to which the test membrane is parallel. Because this test membrane remains at a fixed distance from the horizon, the worldline of a point on it is uniformly accelerated, and therefore naturally associated with a non-zero boost u.


457

5. Comments We have seen that a full analysis of the D = 4 N = 1 supersymmetry algebra not only confirms the existence of 1/2 and 1/4 supersymmetric states, realizable within the WZ model, and determines some of their properties, but it also permits states with 3/4 supersymmetry [14, 16, 18] which, as we have shown, cannot be realized by solutions of the WZ model. However, it has been argued that such ‘exotic” fractions might play a role in other contexts, and with this in mind we have provided a detailed analysis of the BPS states of D = 4 N = 1 supersymmetry. We have also seen that these states can be understood in terms of the geometry associated with the convex cone of the Jordan algebra J4R , and that this leads to a natural generalization of the rotation, Lorentz and conformal groups. In general, the U (1)R symmetry will be broken to at most a discrete subgroup. For theories with domain walls (e.g. the WZ model), the R-symmetry will be explicitly broken by the scalar potential. In theories with only massless particles, and no domain walls, the U (1)R symmetry will be generically broken to a discrete subgroup by chiral anomalies. For theories in which the domain wall charges are quantized, the U (1)R symmetry will be broken to the discrete subgroup preserving the quantization condition. An example of this is given by M-theory compactified on a 7-manifold of G2 holonomy, yielding a D = 4 N = 1 theory in which the domain walls are M2-branes and wrapped M5-branes, with the M2-brane and M5-brane charges quantized. Given that only a discrete subgroup of U (1)R survives the same is true of the larger group SU (2)R . We noted that, in the classical theory, the automorphism group of the full supertranslation algebra is GL(4, R), but it seems that any realization of this on fields, and any realisation of the generalized conformal group Sp(8, R), requires an enlargement of 3-space to include coordinates conjugate to the “domain-wall” charges U and V. Of course, the domain wall interpretation is probably no longer appropriate in this case. Other interpretations are certainly possible in the context of particle mechanics [17]. In such one-dimensional field theories it is possible to realize the SU (2)R symmetry between the three 3-vector “charges” P, U, V as an internal symmetry. For such models that arise from the toroidal compactification of some D = 4 theory with quantized U and V, the 3-momentum will also be quantized and the classical GL(4; R) symmetry will be broken to the discrete GL(4; Z) subgroup preserving the 9-dimensional charge lattice. Many of the observations made here for N = 1 D = 4 can of course be generalized to N > 1 or to D > 4. For example the general N extended D = 4 supersymmetry algebra has automorphism group GL(4N ; R) and det{Q, Q} is preserved by the subgroup SL(4N, R). This leads to the sequence SO(4N ) ⊂ SL(4N ; R) ⊂ Sp(8N ; R)

(130)

R of 4N ×4N symmetric matrices over the reals. The generalised for the Jordan algebra J4N conformal symmetry of the BPS condition is then Sp(8N ; R), as deduced from a different analysis in [20]. A D > 4 case of particular interest is the D = 11 “M-theory algebra” {Q, Q} = Z, where Q is now a 32 component real spinor of the D = 11 Lorentz group and Z is a 32×32 real symmetric matrix containing the Hamiltonian and 527 central charges carried by M-branes [31]. This supersymmetry algebra has automorphism group GL(32; R), as noted independently in [32], and Z takes values in the convex cone associated with the

458


R . The sequence (87) of groups associated with this algebra is Jordan algebra J32

SO(32) ⊂ SL(32; R) ⊂ Sp(64; R),

(131)

so that Sp(64; R) is the M-theoretic generalisation of the D = 11 conformal group. As in the D = 4 case, the realization of any of these larger “spacetime” symmetry groups, or discrete subgroups such as GL(32; Z), would seem to require consideration of an enlarged space of 527 coordinates, as considered for other reasons in [33]. Finally, we have found many possibilities for new BPS states in anti de Sitter space. It seems likely that some of these, in particular those with 1/4 supersymmetry, will have a realization in the context of N = 1 D = 4 supersymmetric field theories in an adS spacetime. Acknowledgements. We would like to thank C. Gui for bringing ref. [8] to our attention. We also thank M. Günaydin and J. Lukierski for helpful correspondence. JPG thanks the EPSRC for partial support. The work of CMH was supported in part by the National Science Foundation under Grant No. PHY94-07194. All authors are supported in part by PPARC through their SPG #613.

References 1. de Azcárraga, J.A., Gauntlett, J.P., Izquierdo, J.M. and Townsend, P.K.: Topological extensions of the supersymmetry algebra for extended objects. Phys. Rev. Lett. 63, 2443 (1989) 2. Abraham, E. and Townsend, P.K.: Intersecting extended objects in supersymmetric field theories. Nucl. Phys. B 351, 313 (1991); Cvetiˇc, M., Quevedo, F. and Rey, S-J.: Target space duality and stringy domain walls. Phys. Rev. Lett. 67, 1836 (1991) 3. Dvali, G. and Shifman, M.: Domain walls and strongly coupled theories. Phys. Lett. B 396, 64 (1997); erratum: ibid B 407, 452 (1997) 4. Veneziano, G. and Yankielowicz, S.: An effective Lagrangian for the pure N = 1 supersymmetric YangMills theory. Phys. Lett. B113, 231 (1982) 5. Kostelecky, V.A. and Perry, M.J.: Solitonic black holes in gauged N = 2 supergravity. Phys. Lett. B 371, 191 (1996); Ferrara, S. and Porrati, M.: Central extensions of Supersymmetry in Four and Three Dimensions. Phys. Lett. B 423, 255 (1998) 6. Townsend, P.K. and Gibbons, G.W.: A Bogomol’nyi equation for intersecting domain walls. Phys. Rev. Lett. 83, 1727 (1999) 7. Carroll, S.M., Hellerman, S. and Trodden, M.: Domain wall junctions are 1/4 supersymmetric. Phys. Rev. D 61, 065001 (2000) 8. Bronsard, L., Gui, C. and Schatzman, M.: A three-layered minimiser in R3 for a variational problem with a symmetric three-well potential. Commun. Pure Appl. Math. 49, 677 (1996) 9. Saffin, P.: Tiling with almost-BPS-invariant domain-wall junctions. Phys. Rev. Lett. 83, 4249 (1999) 10. Gorsky, A. and Shifman, M.: More on the tensorial central charges in N = 1 supersymmetric gauge theories (BPS wall junctions and strings). hep-th/9909015 11. Binosi, D. and ter Veldhuis, T.: Domain wall junctions in a generalized Wess–Zumino model. hepth/9912081 12. Shifman, M. and ter Veldhuis, T.: Calculating the tension of domain wall junctions and vortices in generalized Wess–Zumino models. hep-th/9912162 13. Oda, H., Ito, K. Naganuma, M. and Sakai, N.: An exact solution of BPS domain wall junction. Phys. Lett. B 471, 140 (1999) 14. Gauntlett, J.P. and Hull, C.M.: BPS states with extra supersymmetry. JHEP 0001, 004 (2000), hepth/9909098 15. Lu, H. and Pope, C.N.: Multi-scalar p-brane solitons. Int. J. Mod. Phys. A 12, 437 (1997); p-brane taxonomy. hep-th/9512153 16. Bandos, I. and Lukierski, J.: Tensorial central charges and new superparticle models with fundamental spinor coordinates. Mod. Phys. Lett. A 14, 1257 (1999); New superparticle models outside the HLS supersymmetry scheme. hep-th/9812074 17. Bandos, I., Lukierski, J. and Sorokin, D.: Superparticle models with tensorial central charges. Phys. Rev. D 61, 045002 (2000)


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18. Ueno, T.: BPS states in 10+2 dimensions. hep-th/9909007 19. Günaydin, M.: Generalized conformal and superconformal group actions and Jordan algebras. Mod. Phys. Lett. A 15, 1407 (1993) 20. Ferrara, S. and Porrati, M.: AdS Superalgebras with Brane Charges. Phys. Lett. B 458, 43 (1999) 21. Bandos, I., Lukierski, J., Preitschopf, C. and Sorokin, D.: OSp supergroup manifolds, superparticles and supertwistors. Phys. Rev. D 61, 065009 (2000); hep-th/9907113; The Osp(1|4) superparticle and exotic BPS states. hep-th/9912264. 22. Hardy, G., Littlewood, J.E. and Polya, G.: Inequalities. Cambridge: Cambridge University Press, 1952; Horn, R.A. and Johnson, C.R.: Matrix Analysis. Cambridge: Cambridge University Press, 1985 23. Berkooz, M., Douglas, M.R. and Leigh, R.G.: Branes intersecting at angles. Nucl. Phys. B 480, 265 (1996); Gauntlett, J.P., Gibbons, G.W., Papadopoulos, G. and Townsend, P.K.: Hyper-Kähler manifolds and multiply intersecting branes. Nucl. Phys. B 500, 133 (1997); Ohta, N. and Townsend, P.K.: Supersymmetry of M-branes at angles. Phys. Lett. B 418, 77 (1988) 24. Cederwall, M.: Jordan algebra dynamics. Phys. Lett. B 210, 169 (1988) 25. Rudychev, I. and Sezgin, E.: Superparticles in D > 11. Phys. Lett. B 415, 363 (1997) 26. Hua, L.-K.: Geometry of symmetric matrices over any field with characteristic other than two. Ann. Math. 50, 8 (1949); Causality and the Lorentz group. Proc. Roy. Soc. Lond. A 380, 487 (1982) 27. Schafer, R.D.: An Introduction to Nonassociative Algebras. Dover, 1995 28. Günaydin, M., Sierra, G. and Townsend, P.K.: Gauging the d = 5 Maxwell/Einstein supergravity theories: More on Jordan algebras. Nucl. Phys. B 253, 573 (1985) 29. Townsend, P.K.: The Jordan formulation of quantum mechanics: A review. In: Supersymmetry, Supergravity, and Related Topics, eds. F. del Aguila, J.A. de Azcárraga and L.E. Ibañez, Singapore: World Scientific, 1985; Gibbons, G.W.: Master equations and Majorana spinors. Class. Quantum Grav. 14, A155 (1997) 30. Claus, P., Kallosh, R., Kumar, J., Townsend, P.K. and Van Proeyen, A.: Conformal theory of M2, D3, M5 and “D1 + D5” branes. JHEP 9806, 004 (1998); hep-th/9801206 31. Townsend, P.K.: p-brane democracy. hep-th/9507048; M-theory from its superalgebra. hep-th/9712004 32. Barwald, O. and West, P.C.: Brane rotating symmetries and the fivebrane equations of motion. hepth/9912226 33. Hull, C.M.: Duality and strings, space and time. hep-th/9911080 Communicated by R. H. Dijkgraaf

Commun. Math. Phys. 216, 461 – 490 (2001)

Communications in



Goldstone Boson Normal Coordinates T. Michoel , A. Verbeure Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium. E-mail: [email protected]; [email protected] Received: 2 February 2000 / Accepted: 11 September 2000

Abstract: The phenomenon of spontaneous symmetry breaking is well known. It is known to be accompanied with the appearance of the “Goldstone boson”. In this paper we construct the canonical coordinates of the Goldstone boson, for quantum spin systems with short range as well as long range interactions. 1. Introduction As is well known, spontaneous symmetry breakdown (SSB) is one of the basic phenomena accompanying collective phenomena, such as phase transitions in statistical mechanics, or ground state excitations in field theory. SSB is a representative tool for the analysis of many phenomena in modern physics. The study of SSB goes back to the Goldstone Theorem [1], which was the subject of much analysis. This theorem refers usually to the ground state property that for short range interacting systems, SSB implies the absence of an energy gap in the excitation spectrum [2,3]. In this paper we concentrate on the non-relativistic Goldstone Theorem, and we mean by this spontaneous symmetry breaking of a continuous symmetry group in condensed matter homogeneous many particles systems, with short range as well as long range interactions. There are many different situations to consider. For short range interactions, it is typical that SSB yields a dynamics which remains symmetric in the thermodynamic limit.At temperature T = 0, one has as main characteristics the absence of an energy gap. However for equilibrium states (T > 0), SSB is better characterized by bad clustering properties [4, 5]. For long range interactions, it is typical that SSB breaks also the symmetry of the dynamics. This situation has been studied extensively in the literature. In physics the phenomenon is known as the occurrence of oscillations with energy spectrum taking a Research Assistant of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.)

462

T. Michoel, A. Verbeure

finite value (k → 0) = 0. Different approximation methods, typical here is the random phase approximation, yield the computation of these frequencies. For mean field models, such as the BCS-model [6], the Overhauser model [7], a spin density wave model [8], the anharmonic crystal model [9], and for the jellium model [10], one is able to give the rigorous mathematical status of these frequencies as elements of the spectrum of typical fluctuation operators [11, 12]. The typical operators entering in the discussion are the generator of the broken symmetry and the order parameter operator. In a physical language they are the charge density and current density operators. It is proved that their fluctuation operators form a quantum canonical pair, which decouples from the other degrees of freedom of the system. As fluctuation operators are collective operators, they describe the collective mode accompanying the SSB phenomenon. Hence for long range interacting systems, we realised mathematically rigorously in these models, the socalled Anderson theorem [13, 14] of “restoration of symmetry”, stating that there exists a spectrum of collective modes (k → 0) = 0 and that the mode in the limit k → 0 is the operator which connects the set of degenerate temperature states, i.e. “rotates” one ergodic state into another. We conjecture that our results of [6–10] can be proved for general long range two-body interacting systems as a universal theorem. However Anderson did formulate his theorem in the context of the Goldstone theorem for short range interacting systems, i.e. in the case (k → 0) = 0 of absence of an energy gap in the ground state. Of course one knows that there is no one-to-one relation between long range interactions and the presence of an energy gap for symmetry breaking systems (see e.g. [9]). The imperfect Bose gas and the weakly interacting Bose gas are examples of long range interacting systems showing SSB, but without energy gap. In [15] we realise for these boson models the above described programme of construction of the collective modes operators of condensate density and condensate current, as normal modes dynamically independent from the other degrees of freedom of the system. We consider the whole temperature range, the ground state included. In particular the ground state situation is interesting, because it yields a non-trivial quantum mechanical canonical pair of conjugate operators, giving an explicit representation of the field variables of the so-called Goldstone boson. In this paper we are able to present the analogous proof for general interacting quantum lattice systems, and hence give a model independent construction. We construct the fluctuation operators of the generator of a broken symmetry and of the order parameter and prove that they form a canonical pair. We prove that this pair is dynamically independent from the other degrees of freedom of the system. In the case of long range interactions, we prove that the appearance of a plasmon fequency is a natural phenomenon corresponding to the spectrum of the above mentioned canonical pair. Moreover these fluctuation operators are normal. Our main contribution here is the construction of a canonical order parameter. Usually there are many order parameter operators. Therefore the identification of the right one for the purpose is important. For short range interactions in the ground state, we find again the phenomenon of squeezing of the fluctuation operator of the generator of the broken symmetry. In the literature this is sometimes referred to the statement that in case of SSB, the broken symmetry behaves like an approximate symmetry. The amount of squeezing is inversely related to the anormality of the fluctuation operator of the order parameter, which itself is directly related to the degree of off-diagonal long range order. Using an appropriate volume scaling, which is determined by the long wavelength behaviour of the spectrum, we arrive at the construction of the Goldstone boson normal coordinates. We consider this

Goldstone Boson Normal Coordinates

463

result as a formal step forward, beyond the known analysis of the Goldstone phenomenon. We repeat that our construction is solely determined by the long wavelength behaviour of the microscopic energy spectrum of the system. Finally, we want to throw the attention of the reader to the direct open questions which should keep our attention. There is first of all the problem of SSB of more dimensional symmetries. One should expect a more dimensional Goldstone boson. There is also the problem of SSB of non-commutative symmetry groups. An insight in this situation would certainly contribute to information on the situation of SSB in gauge theories in relativistic field theory. 2. Canonical Coordinates 2.1. Introduction. In [11, 12] a dynamical system of macroscopic quantum fluctuations is constructed for sufficiently clustering states. We repeat the main results in order to fix the notation and refer to the original papers for more details and proofs. The main issue of this section is the construction of creation and annihilation operators for this system of macroscopic fluctuation observables. We start by formulating the systems and the technical settings. With each x ∈ Zν we associate the algebra Ax , a copy of the matrix algebra MN of N × N matrices. For each ⊂ Zν , consider the tensor product A = x∈ Ax . The algebra of all local observables is A . AL = ⊂Zν

The norm closure A of AL is again a C ∗ -algebra A = AL =

A ,

⊂Zν

and is considered the algebra of quasi-local observables of our system. The group Zν of space translations of the lattice acts as a group of *-automorphisms on A by: τx : A ∈ A → τx (A) ∈ A +x , x ∈ Zν . The dynamics of our system is determined in the usual way by the local Hamiltonians (X), ⊂ Zν H = X⊆

with self adjoint (X) ∈ AX for all X ⊂ Zν . The interaction is supposed to be translation invariant: τx (X) = (X + x). For each ⊂ Zν , the local dynamics αt is given by αt : A → A ,

αt (A) = eitH Ae−itH , A ∈ A .

464


If there exists λ > 0 such that λ ≡

|X|N 2|X| eλd(X) (X) < ∞,

(1)

0∈X

with d(X) = supx,y∈X |x − y| the diameter of the set X and |X| the number of elements in X, then the global dynamics αt is well defined as the norm limit of the local dynamics αt [16]. The state ω is an (αt , β)-KMS state which is supposed to have good spatial clustering expressed by αω (|x|) < ∞, (2) x∈Zν

with αω the following clustering function: 1 |ω(AB) − ω(A)ω(B)| d ≤ d( , ) . (3) sup αω (d) = sup , A∈A ,B∈A A B Through the GNS construction, ω defines the Gelfand triple (H, π, "), where H is a Hilbert space, π a *-representation of A as bounded operators on H and " a cyclic vector of H such that ω(A) = (", π(A)"). 2.2. Normal fluctuations. Denote by n the cube centered around the origin with edges of length 2n + 1. For any A ∈ A, the local fluctuation Fn (A) of A in the state ω is given by Fn (A) =

1 τx A − ω(A) . 1/2 | n | x∈ n

In [12] it is proved that under the condition (2), the central limits exist: for all A, B ∈ AL,sa (self-adjoint elements of AL ) 1 lim ω eiFn (A) eiFn (B) = lim ω eiFn (A+B) e− 2 ω([Fn (A),Fn (B)]) n→∞ n→∞ 1

i = exp − sω (A + B, A + B) − σω (A, B) , 2 2 where ω(A∗ τx B) − ω(A∗ )ω(B) , sω (A, B) = lim Re ω Fn (A)∗ Fn (B) = Re n→∞

∗

x∈Zν

σω (A, B) = lim 2 Im ω Fn (A) Fn (B) = −i n→∞

ω [A, τx B]). x

Now we are able to introduce the algebra of normal fluctuations of the system (A, AL , ω). Consider the symplectic space (AL,sa , σω ). Denote by W (AL,sa , σω ) the


465

CCR-algebra generated by the Weyl operators {W (A)|A ∈ AL,sa }, satisfying the product rule i

W (A)W (B) = W (A + B)e− 2 σω (A,B) . The central limit theorem fixes a representation of thisCCR-algebra in the following way. For each A ∈ AL,sa the limits limn→∞ ω eiFn (A) define a quasi-free state ω˜ of the CCR-algebra W (AL,sa , σω ) by 1 ω˜ W (A) = e− 2 sω (A,A) . Moreover if γ is a *-automorphism of A leaving AL invariant, commuting with the space translations and leaving the state ω invariant, then γ˜ given by γ˜ (W (A)) = W (γ (A))

(4)

defines a quasi-free *-automorphism of W (AL,sa , σω ). ˜ π˜ , ") ˜ and yields a von Neumann The quasi-free state ω˜ induces a GNS-triplet (H, algebra ˜ = π˜ W (AL,sa , σω ) . M This algebra will be called the algebra of normal (macroscopic) fluctuations. By the fact that the representation π˜ is regular, we can define boson fields F0 (A) given by π˜ (W (A)) = eiF0 (A) , and satisfying [F0 (A), F0 (B)] = iσω (A, B). Through the relation

lim ω eiFn (A) = ω˜ eiF0 (A) ,

n→∞

we are able to identify the macroscopic fluctuations of the system (A, ω) with the boson field F0 (·): lim Fn (A) = F0 (A).

n→∞

Let (H, π, ") be the GNS-triplet induced by the state ω and consider the sesquilinear form ·, ·0 on H with domain π(AL )" which we simply denote by AL : i ω(A∗ τx B) − ω(A∗ )ω(B) . A, B0 = sω (A, B) + σω (A, B) = 2 ν x∈Z

We call A and B in AL equivalent, denoted A ≡0 B if A − B, A − B0 = 0. The following important result holds: A ≡0 B ⇔ π˜ (W (A)) = π˜ (W (B)) .

(5)

This is the property of coarse graining: different micro observables yield the same macroscopic fluctuation operator. Denote by [AL ] the equivalence classes of AL for the equivalence relation ≡0 . The form ·, ·0 is a scalar product on [AL ]. Denote by Kω the Hilbert space obtained as the completion of [AL ]. Clearly sω and σω extend continuously to Kω . Denote by KωRe the real subspace of Kω generated by [AL,sa ]. Now one considers the CCR-algebra W KωRe , σω in the same representation induced by the state ω, ˜ and one has the following equality: ˜ = π˜ W (KωRe , σω ) . M

466


2.3. Reversible dynamics of fluctuations. Property (4) is not directly applicable with γ = αt , because with this choice it is not clear, and generally not true that αt AL ⊆ AL . Nevertheless, since αt Fn (A) = Fn (αt A) one is tempted to define the dynamics α˜ t of the fluctuations by the formula α˜ t F0 (A) = F0 (αt A). The non-trivial point in this formula is that it is unclear whether the central limit of the non-local observable αt A exists or not. Furthermore if F0 (αt A) exists it remains to prove that (α˜ t )t defines a weakly continuous group of *-automorphisms on the fluctuation ˜ algebra M. In [12] it is shown that if the interaction is of short range, i.e. if satisfies condition (1), then for all A ∈ [AL ], one has that for all t ∈ R, αt A ∈ Kω and if A ∈ [AL,sa ] then ˜ and as αt A ∈ KωRe . W (αt A) is a well defined element of M W (αt A) = eiF0 (αt A) , A ∈ [AL,sa ] the fluctuation F0 (αt A) exists for all t ∈ R. The map Ut : [AL ] → Kω , Ut A = αt A is a well defined linear operator on the Hilbert space (Kω , ·, ·0 ) extending to a unitary operator for all t ∈ R. The map t → Ut is a strongly continuous one-parameter group, and for all elements A ∈ KωRe we can define α˜ t W (A) = W (Ut A). Then α˜ t extends to a weakly continuous one-parameter group of ˜ *-automorphisms of M. Moreover it is shown that if the microsystem is in an equilibrium state, then also the macro system of fluctuations is in an equilibrium state for the dynamics constructed in the previous theorem, i.e. the notion of equilibrium is preserved under the operation of coarse graining induced by the central limit. In particular, if ω is an αt -KMS state of ˜ at the same A at β > 0, then ω˜ is an α˜ t -KMS state of the von Neumann algebra M temperature.

2.4. Canonical coordinates. Now we proceed to the explicit construction of creation ˜ For product states this and annihilation operators of fluctuations in the algebra M. construction can be found in [17]. Here we work out the construction for the most general system. From the definition of KωRe and Kω we can write Kω = KωRe + iKωRe . Let * be the operation on Kω defined by A∗ = (A1 + iA2 )∗ = A1 − iA2 ,

A1 , A2 ∈ KωRe .

Clearly for X ∈ AL one has [X]∗ = [X ∗ ] and it follows from the properties of Ut (see above) that (Ut A)∗ = Ut A∗ for all A ∈ Kω .


467

Let D denote the set of infinitely differentiable functions on R with compact support. D is dense in C0 (R), the continuous functions vanishing at ∞, for the supremum norm. If fˆ ∈ D then the inverse Fourier transform +∞ f (z) = dλfˆ(λ)eiλz −∞

is an entire analytic function. If supp fˆ ∈ [−R, R] then it follows from the theorem of Paley–Wiener [16] that for all n ∈ N there exists a constant Cn such that |f (z)| ≤ Cn (1 + |z|)−n eR|Imz| . ˜ Let Ut = eit h = eitλ d E˜ λ be the spectral resolution of the unitary group Ut and for A ∈ Kω , f ∈ L1 (R) denote +∞ +∞ ˜ A(f ) = fˆ(−λ)d E˜ λ A = fˆ(−h)A. dtf (t)Ut A = −∞

−∞

Clearly one has A(f )∗ = A∗ (f¯). Let W be an open set in R and let E˜ W = W d E˜ λ be the spectral projection onto the spectral subspace KW . It follows from the spectral theory [16, 18] that KW is generated by the set {A(f )|A ∈ Kω , f ∈ D, supp fˆ ⊂ W }. Finally for A ∈ Kω denote the associated spectral measure by d µ˜ A (λ) = A, d E˜ λ A0 and its spectral support 2A 2A = {λ ∈ R | µ˜ A ([λ − , λ + ]) > 0 ∀ > 0}.

(6)

It is easy to see that 2A is also given by 2A = {λ ∈ R | fˆ(λ) = 0, ∀fˆ ∈ D such that A(f ) = 0}. From this expression and fˆ¯(λ) = f¯ˆ(−λ) it follows that 2A∗ = −2A , and from the same argument one also has E˜ + A∗ = (E˜ − A)∗ ,

(7)

where E˜ + = E˜ (0,+∞) and E˜ − = E˜ (−∞,0) are the projections onto positive, respectively negative energy. Lemma 1. Let ω be an (αt , β)-KMS state on the algebra A. For all A ∈ Kω , fˆ ∈ D fˆ(λ)dµA (λ) = fˆ(λ)eβλ dµA∗ (−λ). Proof. Follows from the KMS-properties of ω. ˜

Re = E ˜ 0 KωRe and KRe = (E˜ + + E˜ − )KωRe . Define the operator J on KRe by Let Kω,0 ω,1 ω,1

J = i(E˜ + − E˜ − ). Re , (J A)∗ = J A∗ and thus J KRe ⊆ KRe . From (7) one has for all A ∈ Kω,1 ω,1 ω,1

(8)

468


Proposition 2. The operator J defined above is a complex structure on the symplectic Re , σ ): space (Kω,1 ω (i) J 2 = −1, Re , (ii) σω (A, J B) = −σω (J A, B), A, B ∈ Kω,1 Re (iii) σω (A, J A) > 0, 0 = A ∈ Kω,1 . Proof. From the definition of J and σω = 2 Im ·, ·0 , (i) and (ii) are trivially satisfied. Now we prove (iii). Let E be the set of real functions f such that fˆ ∈ D and 0 ∈ / supp fˆ. Re , f ∈ E} is dense in KRe . By the spectral theory, the set generated by {A(f )|A ∈ Kω,1 ω,1 Take such an element A(f ). Using the previous lemma one computes E˜ − A(f ), E˜ − A(f )0 = |fˆ(λ)|2 χ(−∞,0) (λ)dµA (λ) = |fˆ(−λ)|2 e−βλ χ(0,∞) (λ)dµA (λ) ˜

= E˜ + A(f ), e−β h E˜ + A(f )0 . ˜ Re , this Because E˜ + , E˜ − are projections and e−β h = e−βλ d E˜ λ is bounded on E˜ + Kω,1 Re . Using this property one has relation holds for all B ∈ Kω,1

σω (A, J A) = −2i Im A, J A0 = 2 E˜ + A, E˜ + A0 − E˜ − A, E˜ − A0 ∞ (1 − e−βλ )A, d E˜ λ A0 ≥ 0. =2 0

The strict inequality holds because the spectral measure d µ˜ A (λ) is regular and E˜ 0 A = 0. The existence of a complex structure J yields the existence of creation and annihilation operators a0± (A) =

F0 (A) ∓ iF0 (J A) √ 2

(9)

Re . They satisfy the property for all A ∈ Kω,1

a0± (J A) = ±ia0± (A). Re , 2.5. Normal modes. Consider a given microscopic observable A such that [A] ∈ Kω,1 i.e. such that F0 (A) evolves non-trivially under the dynamics α˜ t . For simplicity we will denote A = [A]. We will construct the normal modes corresponding to the macroscopic fluctuations of the observable A. In order to make clear the idea we will first make the simplifying assumption that the spectral measure d µ˜ A (λ) consists of two δ-peaks, at ±A , with A > 0. Afterwards we will show how to extend the construction to more general (absolutely continuous) measures d µ˜ A . Notice also that the prototype examples of systems with normal fluctuations, i.e. mean field systems, have a discrete energy spectrum and therefore obey the δ-peak assumption (see Sect. 3 for an explicit example).


469

Re , Lemma 3. For fˆ ∈ D and [A] ∈ Kω,1 ∞ fˆ(λ) + fˆ(−λ)e−βλ d µ˜ A (λ), fˆ(λ)d µ˜ A (λ) = 0

˜ ∈ KωRe (i.e. f (t) real), and for fˆ(h)A

2 ˜ ω˜ F0 fˆ(h)A = |fˆ(λ)|2 d µ˜ A (λ). Proof. This is a simple computation and application of Lemma 1.

It will turn out to be more natural to work in terms of the following measure: for λ > 0, dcA (λ) ≡ 2

1 − e−βλ d µ˜ A (λ), λ

and 0 otherwise, such that by Lemma 3, ∞ +∞ 1 − e−βλ cA ≡ d µ˜ A (λ) = β F0 (A), F0 (A) ∼ dcA (λ) = λ 0 −∞ is the well known Duhamel two point function, or canonical correlation. In the sequel, cA will act as a quantization parameter or Planck’s constant for the normal modes corresponding to the fluctuations of A. The assumption on the spectral measure of the fluctuations of A then amounts to the assumption that there exists A > 0 such that dcA (λ) = cA δ(λ − A )dλ.

(10)

The “position” operator Q0 (A) and “momentum” operator P0 (A) of the normal mode are now defined by Q0 (A) ≡ F0 (A),

P0 (A) ≡ F0 (i h˜ −1 A).

Obviously P0 (A) is well defined because of the assumption (10). The following proposition justifies the name normal mode: Proposition 4. The pair Q0 (A), P0 (A) forms a quantum canonical pair, Q0 (A), P0 (A) = icA , satisfying the equations of motion of a free quantum harmonic oscillator with frequency A : α˜ t Q0 (A) = Q0 (A) cos A t + A P0 (A) sin A t, 1 α˜ t P0 (A) = − Q0 (A) sin A t + P0 (A) cos A t. A The (α˜ t , β)-KMS property of ω˜ is expressed by

c

βA A A ω˜ Q0 (A)2 = A2 ω˜ P0 (A)2 = coth . 2 2

470


Proof. By the KMS property of ω, ˜

σ F0 (A), F0 (i h˜ −1 A) =

(1 − e−βλ )λ−1 d µ˜ A (λ) = cA .

Lemma 3 and assumption (10) yield

c

βA A A ω˜ Q0 (A)2 = A2 ω˜ P0 (A)2 = coth . 2 2 ˜ A J A−i hA ˜ A = 0, and by the equivalence A similar computation yields A J A−i hA, ˜ relation (Eq. (5)), F0 (i hA) = A F0 (J A), and by exponentiation: α˜ t F0 (A) = F0 (eA tJ A); J 2 = −1 yields α˜ t F0 (A) = F0 (A) cos A t + F0 (J A) sin A t. As above one shows that by the equivalence relation (5), F0 (i h˜ −1 A) = A−1 F0 (J A) yielding the equations of motion as stated in the proposition.

The creation and annihilation operators corresponding to this harmonic mode are simply the creation√and annihilation operators defined in (9), although it is customary to rescale them with A , i.e. 1 Q0 (A) ∓ iA P0 (A) . √ √ a0± (A) = A 2A Let us now consider how this situation can be extended to the more general case where the measure d µ˜ A (λ) has some spectral support 2A (see (6)). To avoid problems at energy λ = 0, we assume 2A to be bounded away from 0, i.e. there exists A > 0 such that + 2+ A ≡ 2A ∩ R ⊆ [A , +∞).

Remark that 2+ A is the support of the measure dcA (λ). In this case we can safely assume this measure to be absolutely continuous, i.e. dcA (λ) = cA (λ)dλ. Lemma 3 yields ω˜ F0 (A)2 =

βλ cA (λ)λ coth dλ. 2 2 It is easily seen that instead of a single mode Q0 (A), P0 (A) one can construct in this situation a continuous family of harmonic modes, i.e. two operator valued distributions Q0,A (λ), P0,A (λ) | λ ∈ 2+ A , 2+ A


471

such that

Q0,A (λ), P0,A (λ ) = icA (λ)δ(λ − λ ), cA (λ)λ βλ ω˜ Q0,A (λ)2 = λ2 ω˜ P0,A (λ)2 = coth , 2 2 α˜ t Q0,A (λ) = Q0,A (λ) cos λt + λP0,A (λ) sin λt, 1 α˜ t P0,A (λ) = − Q0,A (λ) sin λt + P0,A (λ) cos λt. λ

One identifies

F0 (A) = Q0 (A) = F0 (i h˜ −1 A) = P0 (A) =

2+ A 2+ A

Q0,A (λ)dλ P0,A (λ)dλ.

Remark that due to the spectral gap P0 (A) is well defined and that by the spectral theory [18], Q0,A (λ) can be arbitrarily well approximated [16, Proposition 3.2.40 ] by a sequence of operators F0 (A(fi )), where fî ∈ D is a sequence converging to a double δ-peak in ±λ. The content of this paper is to apply the construction of Proposition 4 to the situation of spontaneous breaking of a continuous symmetry, where we take for A the symmetry generator (i.e. the “charge” operator). The normal modes corresponding to the fluctuations of the symmetry generator as constructed above then yield a rigorous mathematical representation of the collective modes accompanying the spontaneous symmetry breaking (SSB), i.e. of the Goldstone bosons. There are two distinct situations to consider, either the system with SSB has a gap in the energy spectrum, or it has not. The former situation is typically connected with long range interactions, the latter with short range interactions. Both situations introduce specific problems that make Proposition 4 not directly applicable as such. Long range interacting systems in general do not possess a well-defined time evolution in the thermodynamic limit. Therefore one is restricted to studying specific models. In Sect. 3 we study a prototype model of a long range interacting system with a well-defined time evolution and a spectral gap, i.e. a mean field system. These systems have normal fluctuations, hence one can apply Proposition 4 directly. The presence of SSB in short range interacting systems is characterized by either bad clustering properties (for temperature T > 0) or the absence of a spectral gap (T = 0). This is the content of the Goldstone Theorem (see Sect. 4 and references [19, 20] for more details). Therefore these systems do not have normal fluctuations as defined in this section, i.e. there is off diagonal long range order in the system. For the systems we are interested in, this is a statement that applies to momentum k = 0 only, and one goes around this problem by working with the k-mode fluctuations, k = 0, Fn,k (A) =

1 τx A − ω(A) cos k.x. 1/2 | n | x∈ n

These fluctuation operators will be shown to be normal and it will also be shown that in the ground state (T = 0) one can recover the situation of Proposition 4 in a properly scaled limit k → 0. This is the content of Sect. 4.

472


3. Long Range Interactions 3.1. Introduction. In this section we study symmetry breaking systems whose Hamiltonian has a gap in the ground state. These systems typically have long range interactions, but since there is no general criterium whether a long range interacting system has a spectral gap or not, and since an infinite volume time evolution in general may not exist for these systems (see condition (1)), we restrict ourself to mean field systems which are long range interacting systems with a well defined time evolution in the thermodynamic limit and with a spectral gap. For the sake of clarity we consider an explicit example, namely the strong coupling BCS-model for superconductivity. Similar results as the ones presented here have already been obtained for different other mean field models [7, 8], and for the jellium model [10], albeit by different methods. Moreover our main contribution in this section is the construction of a canonical order parameter. The Hamiltonian for the strong coupling BCS-model is given by [21, 22] HN =

N i=−N

σiz

N 1 − σi+ σj− , 2N + 1

βc where βc is determined by the equation tanh βc = 2. These solutions λ = 0 are understood to describe the superconducting phase. Remark that if λ = 0 is a solution of (11), then for all φ ∈ [0, 2π ), λeiφ is a solution as well. There is an infinite degeneracy of the states for the superconducting phase. The degeneracy is due to the breaking of the gauge symmetry. As σ z = σ + σ − − σ − σ + it is clear that the Hamiltonian HN is invariant under the continuous gauge transformations automorphism group G = {γφ |φ ∈ [0, 2π)} of A, γφ : σi+ → γφ (σi+ ) = e−iφ σi+ . However the solutions ωλ are not invariant for this symmetry transformation, because: ωλ (γφ (σi+ )) = e−iφ ωλ (σi+ ) = ωλ (σi+ ).

(12)

The gauge symmetry of the system is spontaneously broken. Remark that hλ is no longer invariant under the symmetry transformation, this is a typical feature of long range interacting systems. From (12) it follows also that ωλ ◦ γφ = ωλeiφ , i.e. one solution ωλ is transformed into another solution ωλeiφ by the gauge transformation γφ .


473

The gauge group G is not implemented by unitaries in any of the representations induced by the solutions ωλ . Locally however, the gauge transformation γφ is implemented by unitaries: take any finite set of indices, then

∗ γφ σi− = Uφ σi− Uφ , i∈

i∈

where i

Uφ = e 2 φQ ,

Q =

j

σjz .

The operator Q is called the local charge or symmetry generator and σ z the charge density or symmetry generator density.

3.2. Canonical coordinates of the Goldstone mode. Next we introduce the algebra of fluctuations and show how the Goldstone mode operators are to be defined in a canonical way. The relation between symmetry breaking and quantum fluctuations in the strong coupling BCS model has been studied before in [6]. This analysis is here extended. Per lattice site j ∈ Z one has the local algeba of observables, the real (2×2) matrices, M2 , generated by the Pauli matrices.As state we consider a particular equilibrium state ωλ with β > βc which reduces per lattice point to the trace state ωλ (A) = tr ρλ A, A ∈ M2 . Because of the product character of the algebra, the state and the time evolution, it is sufficient to consider fluctuations of one-point observables. Locally the fluctuation of A in the state ωλ is: FN (A) =

N 1 A − ρ (A) , i λ (2N + 1)1/2

A ∈ M2 .

i=−N

The commutator of two fluctuations is a mean, indeed: FN (A), FN (B) =

N 1 [A, B] i . 2N + 1 i=−N

For A, B ∈ M2 define

sλ (A, B) = Re ρλ A − ρλ (A) B − ρλ (B) ,

σλ (A, B) = Im ρλ A − ρλ (A) B − ρλ (B) = −iρλ [A, B] . Clearly (M2,sa , σλ ) is a symplectic space and sλ is a symmetric positive bilinear form on M2,sa . Because ρλ is time invariant, ρλ ◦ αt = ρλ and because the evolution αt is local, αt : M2,sa → M2,sa , one has that αt is a symplectic operator on (M2,sa , σλ ): for all t ∈ R, σλ (αt A, αt B) = σλ (A, B).

474


The

structure (M2,sa , σλ , sλ , αt ) defines in a canonical way the CCR-dynamical system W (M2,sa , σλ ), ω˜ λ , α˜ t ; ω˜ λ is a quasi-free state on the CCR-algebra W (M2,sa , σλ ): 1 ω˜ λ W (A) = e− 2 sλ (A,A)

and

α˜ t W (A) = W αt (A)

for all A ∈ M2,sa . ˜ λ ) be the GNS triplet of ω˜ λ . As the state ω˜ λ is regular, there exists Let (H˜ λ , π˜ λ , " a real linear map, called the bose field Fλ : M2,sa → L(H˜ λ ) such that π˜ λ W (A) = eiFλ (A) and the commutation relations Fλ (A), Fλ (B) = iσλ (A, B). As in Sect. 2.2, a central limit theorem allows the identification limN→∞ FN (A) = Fλ (A). The state ω˜ λ is completely characterized by the two-point function on the algebra of fluctuations i ω˜ λ Fλ (A)Fλ (B) = lim ωλ FN (A)FN (B) = sλ (A, B) + σλ (A, B). N→∞ 2 Now we proceed to the construction of the complex structure J (see Sect. 2.4). By diagonalisation of the matrix hλ it is easily seen that hλ has eigenvalues ±µ, where µ = ( 2 + |λ|2 )1/2 . The spectral resolution of hλ is hence given by hλ = −µP− + µP+ . In order to construct J we need to know the spectral resolution of [hλ , ·] considered as operator on M2 . The spectrum of [hλ , ·] is given by {−2µ, 0, 2µ}, the corresponding spectral projections are respectively: E− = E(−2µ) = P− · P+ ,

E0 = P− · P− + P+ · P+ ,

E+ = E(2µ) = P+ · P− ,

and [hλ , A] = −2µE− (A) + 2µE+ (A). 1 On M2,sa ≡ (E+ + E− )M2,sa define J as in Sect. 2 (Eq. (8)) by J (E+ + E− )(A) = i(E+ − E− )(A). 1 , σ ), satisfying This operator J is a complex structure on the symplectic space (M2,sa λ 1 2 the properties of Proposition 2: J = −1, σλ (A, J B) = −σλ (J A, B), A, B ∈ M2,sa 1 1 and σλ (A, J A) > 0, if 0 = A ∈ M2,sa . Remark that on M2,sa , [hλ , ·] = −2iµJ (·) (cf. Proposition 4). For λ = 0, we have [hλ , σ z ] = 0. However [hλ , E0 (σ z )] = 0, and the state ωλ and the corresponding time evolution αt are still invariant under the symmetry generated by E0 (σ z ): N lim ωλ E0 (σ z )i , A = 0 N→∞

i=−N

for all local A. Symmetry breaking is only concerned with the operator σˆ z ≡ σ z − E0 (σ z ) = (E+ + E− )(σ z ); 1 and we are interested in the fluctuations of the operator σˆ z together with its σˆ z ∈ M2,sa z adjoint J σˆ . By calculating [hλ , σ z ] = 2µ(E+ − E− )(σ z ), we find

J σˆ z =

i ¯ − ). (λσ + − λσ µ


475

Similarly [hλ , J σˆ z ] = 2iµ(E+ + E− )(σ z ) yields σˆ z =

|λ|2 z ¯ − ). σ + 2 (λσ + + λσ 2 µ µ

Note that J σˆ z is the usual order parameter operator for the BCS model, but now constructed by means of σ z and the spectrum of the Hamiltonian. z Therefore it is called the canonical order parameter operator. We have also ω J σˆ = 0 and 0 = λ ωλ [hλ , J σˆ z ] = 2iµωλ (σˆ z ). The variances of the fluctuation operators are easily calculated since (E0 σ z )2 =

2 , µ2

(σˆ z )2 = (J σˆ z )2 =

|λ|2 . µ2

Note 1 = (σ z )2 = E0 (σ z )2 + (σˆ z )2 . Also ρλ (σ z ) = ρλ (E0 σ z ) = − Hence

tanh βµ = −2. µ

ω˜ λ Fλ (E0 σ z )2 = sλ (E0 σ z , E0 σ z ) 2 = ρλ (E0 σ z )2 − ρλ (E0 σ z )2 = 2 − 4 2 , µ 2 |λ| ω˜ λ Fλ (σˆ z )2 = sλ (σˆ z , σˆ z ) = ρλ (σˆ z )2 = 2 , µ |λ|2 ω˜ λ Fλ (J σˆ z )2 = sλ (J σˆ z , J σˆ z ) = ρλ (J σˆ z )2 = 2 . µ

The only non-trivial commutator is Fλ (σˆ z ), Fλ (J σˆ z ) = iσλ (σˆ z , J σˆ z ) = ωλ [σˆ z , J σˆ z ] 4|λ|2 = ωλ [σ z , J σˆ z ] = i , µ expressing the bosonic character of the fluctuations. Remark on the other hand that the microscopic observables σˆ z and J σˆ z do not satisfy canonical commutation relations, only their fluctuations do. The flucuation operator Fλ (E0 σ z ) is invariant under the dynamics α˜ t , but the operators Fλ (σˆ z ) and Fλ (J σˆ z ) satisfy the equations of motion d α˜ t Fλ (σˆ z ) = Fλ [hλ , αt (σˆ z )] = −2iµFλ αt (J σˆ z ) = −2iµα˜ t Fλ (J σˆ z ), idt (13) d (14) α˜ t Fλ (J σˆ z ) = Fλ [hλ , αt (J σˆ z )] = 2iµFλ αt (σˆ z ) = 2iµα˜ t Fλ σˆ z . idt In integrated form one gets: α˜ t Fλ (σˆ z ) = Fλ (σˆ z ) cos 2µt + Fλ (J σˆ z ) sin 2µt, α˜ t Fλ (J σˆ z ) = −Fλ (σˆ z ) sin 2µt + Fλ (J σˆ z ) cos 2µt.

476


Hence by an explicit calculation we have arrived at the results of Proposition 4, for A = σˆ z , the generator of the broken symmetry. Therefore, denoting Qλ ≡ Fλ (σˆ z ) and 1 Pλ ≡ 2µ Fλ (J σˆ z ), we defined the pair (Qλ , Pλ ) as the canonical pair of the Goldstone bosons. Writing down the previous results in terms of Qλ and Pλ (as in Proposition 4) one sees that this pair shares indeed all physical properties for Goldstone bosons. Remark that the frequency of oscillation is 2µ. This is the phenomenon of the doubling of the frequency for the inherent plasmon frequency. The formula

|λ|2 cλ (2µ) β(2µ) ω˜ λ Q2λ = (2µ)2 ω˜ λ Pλ2 = 2 = coth , µ 2 2 is a quantum mechanical expression of a virial theorem. Remark that in the normal phase (λ → 0), Qλ=0 = Pλ=0 = 0, i.e. the Goldstone boson disappears. The creation and annihilation operators of the Goldstone bosons are as usual aλ± =

Qλ ∓ i2µPλ . √ 4µ

to the gauge transformations The state ω˜ λ is gauge-invariant and quasi-free with respect of these creation and annihilation operators, i.e. ω˜ λ aλ+ aλ+ = 0 = ω˜ λ aλ+ , and the two-point function ω˜ λ aλ+ aλ− =

1 . e2βµ − 1

4. Short Range Interactions 4.1. Goldstone theorem and canonical order parameter. Let ω be an extremal translation invariant (αt , β)-KMS state, αt a dynamics generated by a translation invariant Hamiltonian H and let γs be a strongly continuous one-parameter symmetry group which is locally generated by a generator qx , Qn = x∈ n

where n = [−n, n]ν ∩ Zν and qx is the symmetry generator density, i.e. for A ∈ A n , γs (A) = eisQn A e−isQn . Denote q = qx=0 , and for convenience denote again q − ω(q) by q. For systems with short range interactions, assuming spontaneous symmetry breaking amounts to: Assumption 1. Assume that there exists an (αt , β)-KMS or ground state ω such that ω is not invariant under the symmetry transformation γ , while the dynamics αt remains invariant under γ , i.e. ∃A ∈ AL such that ω γs (A) = ω(A), (15) αt ◦ γs = γs ◦ αt .

(16)


477

The invariance of the dynamics (16) is crucial in this context (see [23] and Proposition 6 and Eq. (24) below). For a more complete discussion of the phenomenon of spontaneous symmetry breaking, see [20]. An operator A satisfying (15) is called an order parameter operator. Equation (15) is equivalent to d ω γs (A) = lim ω [Qn , A] = 0. n→∞ s=0 ds The local Hamiltonians are determined by an interaction (X) Hn = X⊆ n

and the infinite volume Hamiltonian H is defined such that for A ∈ A 0 , H A" = [(X), A]", X∩ 0 =∅

where " is the cyclic vector of the state ω. The relation between spontaneous symmetry breaking and the absence of a gap in the energy spectrum in the ground state was originally put forward by Goldstone [1]. For short range interactions in many-body systems, it is proved [2, 3] that spontaneous symmetry breaking implies the absence of an energy gap in the excitation spectrum. We refer here to [19] where the Goldstone theorem is proved rigorously for quantum lattice systems. Theorem 5 (Goldstone Theorem [19]). If is translation invariant and satisfies |X| (X) < ∞, (17) X*0

then (i) At T = 0: If the system has an energy gap then there is no spontaneous symmetry breakdown. (ii) At T > 0: If the system has L1 clustering then there is no spontaneous symmetry breakdown. The L1 clustering means here that for each observable A, one has: ω(Aτx A) − ω(A)2 < ∞. x∈Zν

The first step is to construct something like a canonical order parameter operator. See Sect. 3 for an example of this construction. Denote L(A) = [H, A]. The Duhamel two-point function becomes now: 1 − e−βL 1 β (A, B)∼ ≡ B . ω(A∗ αiu B)du = ω A∗ β 0 βL

478


The KMS-condition, ω (AB) = ω Bαiβ A , yields

ω [A, B] = ω A(1 − e−βL )B , for A, B in a dense domain of A, and hence if B ∈ Dom(L−1 ) then

β(A, B)∼ = ω A, L−1 B , and the Bogoliubov inequality [24] for KMS-states is given by: ∗ ω [A , B] 2 ≤ βω [A∗ , L(A)] (B, B)∼ . Finally denote the local 0-mode fluctuation of an observable A in the state ω by 1 τx A − ω(A). Fn,0 (A) = | n |1/2 x∈ n

Assumption 2. Assume that there are no long range correlations in the fluctuations of the symmetry generator density, i.e. assume lim ω Fn,0 (q)2 = ω(qτz q) − ω(q)2 < ∞. n→∞

z∈Zν β

Then also the uniform susceptibility c0 defined by β

c0 ≡ lim

n→∞

β Fn,0 (q), Fn,0 (q) ∼ 2

(18)

β

is finite, i.e. c0 < ∞. Proposition 6. Under Assumption 1 and 2 we have β

β Fn,0 (q), αt Fn,0 (q) ∼ > 0 n→∞ 2

c0 = lim

(19)

β

and c0 is independent of t, and given by β

1 ω Qn , L−1 (q) . n→∞ 2

c0 = lim Proof. Let β

β Fn,0 (q), αt Fn,0 (q) ∼ n→∞ 2 1 1 − e−βL itL = lim ω Fn,0 (q) e Fn,0 (q) . n→∞ 2 L

c0 (t) = lim

β

First we show c0 (t = 0) > 0. Let A be an arbitrary order parameter operator. SSB, translation invariance and the Bogoliubov inequality yield 2 0 < lim ω Fn,0 (q), Fn,0 (A) n→∞

≤ lim βω Fn,0 (A), L Fn,0 (A) Fn,0 (q), Fn,0 (q) ∼ . n→∞


479

In [19] it is shown that (17) also implies

lim ω Fn,0 (A), L Fn,0 (A) = ω τz A, L(A) < ∞ n→∞

z∈Zν

for each local observable A. Hence 0< ω τz A, L(A) lim β Fn,0 (q), Fn,0 (q) ∼ z∈Zν

n→∞

β

yielding c0 (t = 0) > 0. β The proof of the time invariance of c0 is based on [23] and goes as follows: d β β c0 (t) = lim Fn,0 (q), αt L Fn,0 (q) n→∞ 2 ∼ idt 1 = lim ω Fn,0 (q)(1 − e−βL )eitL Fn,0 (q) n→∞ 2 1 = lim ω Fn,0 (q), eitL Fn,0 (q) . n→∞ 2 Translation invariance and (16) yield: d β 1 c0 (t) = lim ω Qn , eitL q n→∞ 2 idt 1 d 1 d = ω γ (α q) = ω αt (γs q) s t 2 ids s=0 2 ids s=0 1 d = ω(q) = 0. 2 ids s=0 From the proposition it follows that if L−1 q exists, it is an order parameter operator. We call it the canonical order parameter operator, it is an order parameter constructed directly from the two given quantities, the Hamiltonian and the symmetry generator. However it can not be expected in general that q ∈ Dom(L−1 ), especially not for systems without an energy gap, because of problems at zero energy. Expressions like (1−e−βL )L−1 q on the contrary are well defined. The bulk of our efforts below consists of mastering the difficulties with the canonical order parameter by considering the k-mode fluctuations and by afterwards taking the limit k → 0. This method has already been used in [15], where the Goldstone coordinates are constructed for models of interacting Bose gases. 4.2. Fluctuations. By the Goldstone theorem, spontaneous symmetry breaking implies that the system does not have exponential or L1 clustering. In particular the variances of local fluctuations Fn,0 (A) may not be convergent in the thermodynamic limit for certain A (in particular for A an order parameter operator) because of long range order correlations. The central limit as described in Sect. 2.2 no longer holds. However one can study the k-mode fluctuations, i.e. one considers for k = (k1 , k2 , . . . , kν ) ∈ Rν , with kj = 0 for j = 1, 2, . . . , ν: 1 Fn,k (A) = τx (A) − ω(A) cos k.x. 1/2 | n | x∈ n

480


It is believed that the central limit theorem holds for the k-mode fluctuations in every extremal translation invariant state, even at criticality. This is essentially because one stays away from the singularity at k = 0. A completely rigorous proof of this statement is found in [25], for the absolute convergent case under a very mild cluster condition. Below we prove the convergence of the Fourier series for translation invariant states with singularities occuring only at zero momentum (see further on). See also [26] for a similar line of reasoning. For A ∈ AL , denote the Fourier transforms of the l-point correlation functions ω(τx1 Aτx2 A · · · τxl A) by µ(k1 , k2 , · · · , kl ) (i.e. kj are different vectors in Rν here, not the components of a particular k). In general µ is a measure. By translation invariance it can be written as a function of k1 , k1 + k2 , . . . , k1 + k2 + · · · + kl . As in [26], assume that the only singularities in µ are of the type δ(k1 + · · · + ki ) (i.e. singularities occurring only at zero momentum). We show now that the truncated correlation functions ωT Fn,k (A)l vanish for l ≥ 3 and remain finite for l = 2. Let ω(A) = 0, then

ωT Fn,k (A)l 1 = ωT (τx1 Aτx2 A · · · τxl A) cos k.x1 cos k.x2 · · · cos k.xl l/2 | n | x ,x ,... ,x 1 2 l 1 = ωT (Aτy1 A · · · τyl−1 A) cos k.x1 cos k.(y1 + x1 ) | n |l/2 x ,y ... ,y 1

1

l−1

· · · cos k.(yl−1 + x1 ). The expansion of the cosines into exponentials yields two types of terms, namely terms which do not depend on x1 and terms which do depend on x1 . The first kind of terms do not appear for l odd and for l even they are exactly the ones which are cancelled out by the truncation. The second kind of terms tend to zero because of the scaling factors. Let us illustrate this by means of an example. First let l = 2:

ωT Fn,k (A)2 = ω Fn,k (A)2 =

1 ω(Aτy−x A) cos k.x cos k.y. | n | x,y

Since µ(k) =

ω(Aτz A)e−ik.z

z

can at most have a singularity at k = 0, µ(k) < ∞ for k = 0. Also 1 ik.z e → δk,0 . | n | z Hence the only terms contributing in the two-point correlation function are the terms containing the factor e±ik.(y−x) , i.e. the terms of the first kind. In the limit we find

1 lim ωT Fn,k (A)2 = [µ(k) + µ(−k)] < ∞. n 4


481

Now let l = 4 and consider a typical term: 1 | n |2

ω Aτy1 Aτy2 Aτy3 A e−ik.(y1 −y2 +y3 ) .

x1 ,y1 ,y2 ,y3

Ignoring boundary effects in the sums, this becomes 1 ω Aτy1 Aτy2 Aτy3 A e−ik.(y1 −y2 +y3 ) | n | y ,y ,y 1

2

3

1 = ω Aτy1 Aτy2 [Aτy3 −y2 A] e−ik.(y1 −y2 +y3 ) | n | y ,y ,y 1 2 3 1 = ω Aτx A τy [Aτz A] e−ik.(x+z) . | n | y x,z

In the limit we get

ω (Aτx A) e−ik.x

x

ω (Aτz A) e−ik.z ,

z

cancelling out against two-point correlations in the 4-point truncated correlation function. Finally, take l = 3, then all terms are of the second kind and vanish, e.g. 1 ω(Aτy1 Aτy2 A)eik.(x1 +y1 −y2 ) | n |3/2 x ,y ,y 1

1

2

=

1 ik.(y1 −y2 ) ω(Aτ [Aτ A])e eik.x1 . y y −y 1 2 1 | n |3/2 y ,y x 1

The sum over x1 is bounded by

ν

2

kj −1 j =1 | sin 2 | ,

1

yielding

1 ω A τx [Aτy A] e−ik.y | n | x y which converges to ω(A)

ω Aτy A e−ik.y = 0.

y

Using the formula ∞

(iλ)l ωT Q, . . . , Q , ω eiλQ = exp l! l=1

l times

482


one arrives at the central limit theorem

1 lim ω eiFn,k (A) = e− 2 sk (A,A) , n

with sk (A, A) = limn→∞ ω Fn,k (A)2 . In [25] one can find a rigorous proof of the central limit theorem for the k-mode fluctuations, k = (kj = 0)νj =1 , for states satisfying a certain clustering condition, expressed as a condition on the function αω (see Eq. (3)). Although this condition is much weaker than for the k = 0 fluctuations, it is not clear whether it is always satisfied for any extremal translation invariant state. The arguments above however suggest that this clustering condition on the state is merely technical and that a general rigorous proof of the central limit theorem along the lines of [25] is possible for k = (kj = 0)νj =1 under even weaker conditions. We continue on the basis of the arguments above.

Theorem 7 (Central limit theorem). If the state ω has only singularities at zero momentum, for all A ∈ AL,sa and k = (kj = 0)νj =1 , then (i) limn→∞ ω Fn,k (A)2 < ∞, 1 (ii) limn→∞ ω eiFn,k (A) = e− 2 sk (A,A) with sk (A, B) = limn→∞ Re ω Fn,k (A)∗ Fn,k (B) . Because of (i), the limit

lim ω Fn,k (A)∗ Fn,k (B) ≡ A, Bk

n→∞

defines a positive sesquilinear form which satisfies the Cauchy–Schwarz inequality |A, Bk |2 ≤ A, Ak B, Bk . More explicitly A, Bk =

1 ω(A∗ τz B) − ω(A∗ )ω(B) cos k.z. 2 ν z∈Z

Let σk (A, B) = 2 Im A, Bk , then

strong − lim π [Fn,k (A), Fn,k (B)] = iσk (A, B). n→∞

The identification of the central limit with bose fields is as in Sect. 2.2, and worked out in full detail for k = 0 in [25]. The bilinear form sk determines a quasi free state ω˜ k on the CCR-algebra W(AL,sa , σk ): ω˜ k (Wk (A)) = e− 2 sk (A,A) . 1

The Wk (A), A ∈ AL,sa are the Weyl operators generating W(AL,sa , σk ). Via the central limit theorem, one shows for A1 , A2 , . . . , Al ∈ AL,sa ,

lim ω eiFn,k (A1 ) eiFn,k (A2 ) . . . eiFn,k (Al ) = ω˜ k Wk (A1 )Wk (A2 ) . . . Wk (Al ) . n→∞


483

The state ω˜ k is regular and hence for every A ∈ AL,sa there exists a self-adjoint bosonic ˜ k ) of ω˜ k such that field Fk (A) in the GNS representation (H˜ k , π˜ k , " π˜ k (Wk (A)) = eiFk (A) . This implies that in the sense of the central converge to the

limit, the local fluctuations bosonic fields associated with the system W(AL,sa , σk ), ω˜ k , lim Fn,k (A) = Fk (A).

n→∞

As in Sect. 2.2, fluctuation operators are only defined up to equivalence i.e. A ≡k B if A − B, A − Bk = 0 and A ≡k B ⇔ π˜ k Wk (A) = π˜ k Wk (B) . (20) The form ·, ·k thus becomes a scalar product on [AL ], the equivalence classes of AL for the relation ≡k . Denote by Kk the Hilbert space obtained as completion of [AL ] and by KkRe the real subspace of Kk generated by [AL,sa ]. 4.3. Goldstone modes for finite wavelengths. The finiteness of limn→∞ ω Fn,k (q)2 for all k (k = 0 included by Assumption 2) implies the finiteness of lim |fˆ(λ)|ω Fn,k (q)dEλ Fn,k (q) n→∞

for fˆ ∈ D, and hence the existence of a measure d µ˜ k (λ) = lim ω Fn,k (q)dEλ Fn,k (q) ; n→∞

dEλ is the spectral measure of the Hamiltonian H , i.e. H = λdEλ . β As in Sect. 2.5, define the measure dck (λ) with support on R+ only by β

dck (λ) = 2

1 − e−βλ d µ˜ k (λ), λ

such that for fˆ ∈ D (cf. Lemma 3) lim fˆ(λ)ω Fn,k (q)dEλ Fn,k (q) n→∞ ∞ fˆ(λ) + fˆ(−λ)e−βλ = 0

Proposition 8. For fˆ ∈ D, ∞ β lim fˆ(λ)dck (λ) = k→0 0

where

β c0

∞ 0

λ β dc (λ). 2(1 − e−βλ ) k

(21)

β β fˆ(λ)dc0 (λ) = c0 fˆ(0), β

β

β

is given by Eq. (18). In other words limk→0 dck (λ) = dc0 (λ) = c0 δ(λ)dλ.

484

T. Michoel, A. Verbeure β

β

Proof. The statement that limk→0 dck (λ) = dc0 (λ) follows from Assumption 2. The proof of the second statement is based on the time invariance of β c0 (t) = lim β Fn,0 (q), αt Fn,0 (q) ∼ (Proposition 6) n→∞

and by (21): for fˆ ∈ D,

β f (t) Fn,0 (q), αt Fn,0 (q) ∼ e−iλt dt fˆ(λ)c0 = β lim n→∞ ∞ β fˆ(λ − λ )dc0 (dλ ), = 0

β

β

i.e. dc0 (λ) = c0 δ(λ)dλ.

In order not to obscure the construction of the Goldstone boson normal coordinates by technical details, we will first consider the case that β

β

β

(22) dck (λ) = ck δ(λ − k )dλ, β β with k > 0 and ck = limn→∞ β Fn,k (q), αt Fn,k (q) ∼ . From Proposition 8 we deduce that this is a good approximation for sufficiently small |k|, and we will show later that this approximation becomes exact in a certain limit k → 0, to be specified later. From Eq. (21) and (22), it follows

lim ω Fn,k (q)fˆ(H )Fn,k (q) =

n→∞

β β β fˆ(k ) + fˆ(−k )e−βk .

β β

ck k

β

2(1 − e−βk )

(23)

In particular one has β

cβ β

β ω˜ k Fk (q)2 = lim ω Fn,k (q)2 = k k coth k . n→∞ 2 2 β

Also time invariance of c0 (t) (see above) (i.e. SSB) implies β lim k→0 k

= 0,

(24)

as can be seen from (23): β

β

β

β

c0 (t) = lim ck (t) = lim ck cos k t. k→0

For fˆ ∈ D, denote

q(f ) =

k→0

f (t)α−t q = fˆ(L)q

and consider the equivalence class [q(f )]k . For q(f ) ∈ AL,sa the fluctuation operator Fk [q(f )]k is well defined, β β β

2 β c β ω˜ k Fk [q(f )]k = [q(f )]k , [q(f )]k k = |fˆ(k )|2 k k coth k , 2 2

(25)


485

(we used that q(f ) ∈ AL,sa ifffˆ¯(λ) = fˆ(−λ) ), and obviously for these functions f , we can define elements [q]k (f ) ∈ KkRe through the relation [q]k (f ) = [q(f )]k . However since Kk is by definition closed for the ·, ·k topology, we can define elements [q]k (f ) for β a much wider class of functions F, namely all those functions for which |fˆ(k )| < ∞: β β let fi be a sequence of functions such that [q(fi )]k ∈ KkRe and limi fî (k ) = fˆ(k ), and define [q]k (f ) = strong- lim[q(fi )]k . i

In particular we have i , λ and obviously we interpret Fk [q]k (g) as “Fk iL−1 (q) ”, i.e. as the k-fluctuation operator of the canonical order parameter, even though iL−1 (q) does not exist in general. In the spirit of Proposition 4, denote [q]k (g) ∈ KkRe with g(λ) ˆ =

i Pk = Fk [q]k (g) with g(λ) ˆ = , λ

Qk = Fk (q),

and denote by B˜k the algebra generated by Qk and Pk . Also denote by C˜k the algebra generated by the operators Fk [q]k (f ) with f ∈ F. Our main result is then that the pair (Qk , Pk ), constructed directly from the generator of the broken symmetry, forms a harmonic normal mode, therefore properly called the Goldstone boson normal mode. This result is an extension of Proposition 4 to the case of k = 0 fluctuations in the presence of SSB. Theorem 9. In the presence of SSB (Assumption 1), and in the case (22), the generator of the broken symmetry determines uniquely the construction of a canonical pair of fluctuation operators (Qk , Pk ), β

[Qk , Pk ] = ick

β with ck = limn→∞ β Fn,k (q), Fn,k (q) > 0, satisfying a virial theorem:

∼

β ω˜ k Q2k = (k )2 ω˜ k Pk2 . The microscopic time evolution αt induces a time evolution α˜ tk on C˜k through the relation α˜ tk Fk [q]k (f ) ≡ Fk [q]k (Ut f ) ,

itλ (U t f )(λ) = e fˆ(λ);

α˜ tk leaves B˜k invariant and leads to the equations of motion β

β

β

α˜ tk Qk = Qk cos k t + k Pk sin k t, Qk β β α˜ tk Pk = − β sin k t + Pk cos k t. k

(26) (27)

486


The operators (Qk , Pk ) are called the Goldstone boson normal coordinates. The Goldstone boson creation and annihilation operators are defined by ak±

β

=

Qk ∓ ik Pk β

2k

β

satisfying [ak− , ak+ ] = ck . The quasi-free state ω˜ k is a β-KMS state on B˜k for the evolution α˜ tk , i.e. the Goldstone bosons have a Bose–Einstein distribution: ω˜ k ak+ ak− =

β

ck β

eβk − 1

,

which is equivalent to β

cβ β β ω˜ k Q2k = k k coth k . 2 2 + + The state ω˜ k is gauge invariant: ω˜ k ak ak = 0 = ω˜ k ak+ .

Proof. The commutator follows from σk [q]k , [q]k (g) = −i

g(λ)(1 ˆ − e−βλ )d µ˜ k (λ).

The variance of Pk is obtained from (25): β β β c 1 ω˜ k Pk2 = kβ coth k = β ω˜ k Q2k . 2 2 2k (k )

ˆ Denote h(λ) = iλ. Clearly the infinitesimal generator of α˜ tk is given by d k Fk [q]k (f ) = Fk [q]k (hf ) . α˜ dt t t=0 Hence the first relation d k α˜ Pk = −Qk dt t t=0

(28)

d k β α˜ Qk = (k )2 Pk , dt t t=0

(29)

follows trivially. The second,

follows from the equivalence relation (20): from Eq. (23) one computes straightforwardly β β [q]k (h) − (k )2 [q]k (g), [q]k (h) − (k )2 [q]k (g) k = 0, where g(λ) ˆ = iλ−1 as before. Exponentiation of (28) and (29) leads to the equations of motion. Also the remainder of the theorem follows from (23).


487

β Remark that for k → 0, ω˜ k Pk2 diverges as (k )−2 . This divergence corresponds to the well known phenomenon of long range correlations in the order parameter fluctuations. Similarly to what we did after Proposition 4, the proper generalisation of (22), is to consider the case that for k = 0, the support 2k of the measure d µ˜ k (λ) is bounded away from 0 and absolutely continuous, i.e. β

Assumption 3. By translation invariance we assume that for k = 0, there exists k > 0 β β + such that 2+ k ≡ 2k ∩ R ⊆ [k , +∞) and that there exists a function ck (λ) such that β

β

dck (λ) = ck (λ)dλ.

(30)

Equation (23) becomes

lim ω Fn,k (q)fˆ(H )Fn,k (q) =

n→∞

∞ β

k

β

ck (λ)λ fˆ(λ) + fˆ(−λ)e−βλ . β 2(1 − e λ)

It is clear that again the single mode (Qk , Pk ) gets replaced by a continuous family of modes Qk (λ), Pk (λ) | λ ∈ 2+ k , such that

β Qk (λ), Pk (λ ) = ck (λ)δ(λ − λ ),

β

c (λ)λ ω˜ k Qk (λ) = λ ω˜ k Pk (λ) = k coth βλ2, 2 α˜ tk Qk (λ) = Qk (λ) cos λt + λPk (λ) sin λt, Qk (λ) sin λt + Pk (λ) cos λt, α˜ tk Pk (λ) = − λ 2

2

2

and Qk =

∞ β

k

Qk (λ)dλ,

Pk =

∞ β

k

Pk (λ)dλ.

4.4. Goldstone mode for infinite wavelength. Next we look for the Goldstone mode operators in the limit of k tending to zero, i.e. in the long wavelength limit. We take the results of Sect. 4.3 and study the limit k → 0. Among other results, we show that the long wavelength Goldstone mode survives in this limit only in the ground state. This shows also that no long wavelength quantum Goldstone modes are present for temperatures T > 0. For T > 0, the spontaneous symmetry breakdown does not show any quantum behaviour, only classical modes are present. For simplicity we will first consider the case of a single harmonic mode (Qk , Pk ), i.e. the case (22). However we will prove afterwards that the results we obtain in the limit k → 0 are independent of this choice and are valid in general. β Let k = limβ→∞ k , the ground state spectrum. Because of the Goldstone theorem, β β β we have that limk→0 k = 0. Let c0 = limk→0 ck and ck = limβ→∞ ck . β

Assumption 4. Assume limk→0 ck = limβ→∞ c0 = c0 < ∞.

488


First let β < ∞. The variances β

cβ β

β β ω˜ k Q2k = k k coth k = (k )2 ω˜ k Pk2 2 2

behave as follows for k → 0: β

cβ c ω˜ k Q2k ≈ k → 0 (finite), β β

ω˜ k Pk2 ≈

β

ck

β

β(k )2

→ ∞.

Since observable fluctuation operators are always characterized by a finite, non-zero variance, it is clear that we have to renormalize Pk before taking a limit k → 0: β Pˇk = k Pk .

This however implies that the commutator β β

[Qk , Pˇk ] = ick k

vanishes in the limit k → 0. In other words the quantum character and hence also the harmonic oscillation of the Goldstone mode disappears in the appropriate limit k → 0, at least at non-zero temperature. At zero temperature (β = ∞), in the ground state, the situation is completely different. The variances behave now for k → 0 as follows:

c

ck k k → ∞, → 0, ω˜ k Pk2 = ω˜ k Q2k = 2 2k but their product

c2 c2 ω˜ k Q2k ω˜ k Pk2 = k → 0 4 4 remains finite. This means that the divergence of the order parameter operator fluctuations due to long range correlations is exactly compensated by a proportional squeezing of the symmetry generator fluctuations. Therefore one can find a renormalized Qk and Pk , ˇ k and Pˇk , having both a finite, non-zero variance, with a finite non-zero denoted by Q commutator; indeed take e.g. ˇ k ≡ −1/2 Qk , Q k then

1/2 Pˇk ≡ k Pk ,

ˇ 2k = ω˜ k Pˇk2 = ck → c , ω˜ k Q 2 2

ˇ k , Pˇk ] = ick → ic. [Q

Remark that this scaling transformation has no effect on the creation and annihilation operators, in particular: ak± =

ˇ k ∓ i Pˇk Qk ∓ ik Pk Q = . √ √ 2k 2


489

On the other hand, the equations of motion (26) and (27) are transformed into ˇk = Q ˇ k cos k t + Pˇk sin k t, α˜ tk Q ˇ k sin k t + Pˇk cos k t. α˜ tk Pˇk = −Q Hence in order to retain a non-trivial time evolution in the k → 0 limit, one has to rescale time as well in the following way: t → τ = k t. ˇ 0 , Pˇ0 ), Let B˜0 be an algebra generated by a canonical pair (Q ˇ 0 , Pˇ0 = ic0 ; Q α˜ τ0 , τ ∈ R is a time evolution on B˜0 defined through the equations of motion ˇ0 = Q ˇ 0 cos τ + Pˇ0 sin τ, α˜ τ0 Q ˇ 0 sin τ + Pˇ0 cos τ, α˜ τ0 Pˇ0 = −Q and ω˜ 0 is a state on B˜0 defined through the relation ˇ 0 , Pˇ0 ) ≡ lim ω˜ k F (Q ˇ k , Pˇk ) , ω˜ 0 F (Q k→0

where F is any polynomial in two variables. Summarizing our results: Theorem 10. In the ground state (β = ∞), the dynamical system (B˜k , α˜ tk , ω˜ k ) converges in the limit k → 0 to the dynamical system (B˜0 , α˜ τ0 , ω˜ 0 ) in the sense that for any two polynomials F1 , F2 in two variables, ˇ 0 , Pˇ0 )α˜ τ0 F2 (Q ˇ 0 , Pˇ0 ) = lim ω˜ k F1 (Q ˇ k , Pˇk )α˜ kτ F2 (Q ˇ k , Pˇk ) . ω˜ 0 F1 (Q k→0

k

ˇ k and Pˇ0 = limk→0 Pˇk . Moreover ω˜ 0 is a ˇ 0 = limk→0 Q Therefore we can identify Q 0 ˜ ground state for α˜ τ , i.e. for all X ∈ B0 , d ω˜ 0 X ∗ α˜ τ0 X ≥ 0. dt t=0 ˇ 0 , Pˇ0 ) is called the canonical pair of the collective Goldstone mode. The pair (Q Proof. Due to quasi-freeness, it is sufficient to check these properties for the two-point correlation function. But in this case they follow immediately from the very definition of α˜ τ0 and ω˜ 0 . Remark that although formally, Theorem 9 and 10 are very similar, it is important to remember the rescaling that has been done. In fact the previous theorem tells us that in the ground state the long range correlations in the order parameter fluctuations are exactly compensated by a squeezing of the generator fluctuations. Both operators continue to form a harmonic oscillator pair in the limit k → 0, although the frequency becomes infinitesimally small and hence the period of oscillation infinitely (or macroscopically) large. Considering the most common case of powerlaw behaviour of the energy spectrum, i.e. k = |k|δ , this rescaling provides information about the size of the 0-mode fluctuations. In a finite box n of length L = 2n + 1, the smallest non-zero wave vector has

490

T. Michoel, A. Verbeure −1/2

length |k| ∝ L−1 . Therefore the rescaling of Qk with a factor k by Lδ/2 = | n |δ/2ν of the fluctuation, i.e. 1 qx − ω(q) , Fn,0 (q) = δ 1 | n | 2 − 2ν x∈ n

suggests a rescaling

in order that its variance is non-zero and finite. This means that the fluctuations of the 1 δ symmetry generator are of order | n | 2 − 2ν , i.e. subnormal fluctuations. Similarly the 1 δ fluctuations of the order parameter are of order | n | 2 + 2ν , i.e. abnormal fluctuations. δ 1 This requires 2ν ≤ 2 , or δ ≤ ν. This condition is undoubtly related to the condition c < ∞ (Assumption 4). Remark also that if SSB disappears, i.e. if c = 0, then the Goldstone boson disappears. Finally we remark that the results of Theorem 10 do not depend on the particular β form of the measure dck (λ), in this case given by (22). One could equally well take the more general form (30), since in the limit k → 0 this measure also reduces to a δ-peak by Proposition 8. It is a straightforward calculation to show that Theorem 10 holds in general (i.e. under Assumption 3), upon interpreting k as the gap in the support of the measure dck (λ). Therefore we find that at zero temperature, the fluctuations of the symmetry generator lead to a single harmonic mode with vanishingly small frequency in the long-wavelength limit, even though at finite wavelength, there exists a continuous family of modes associated to the fluctuations of the symmetry generator. It is hence also appropriate to consider the results of Theorem 9 as being physically valid in general, as long as one considers low enough temperatures and large enough wavelengths. 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. 25. 26.

Goldstone, J.: Il Nuovo Cim. 19, 154 (1961) Kastler, D., Robinson, D.W., and Swieca, A.: Commun. Math. Phys. 2, 108–120 (1966) Swieca, J.A.: Commun. Math. Phys. 4, 1–7 (1967) Martin, P.A.: Il Nuovo Cim. 68 B(2), 302–313 (1982) Fannes, M., Pulè, J.V., and Verbeure, A.: Lett. Math. Phys. 6, 385–389 (1982) Goderis, D., Verbeure, A., and Vets, P.: Il Nuovo Cim. 106 B(4), 375–383 (1991) Broidioi, M., Nachtergaele, B., and Verbeure, A.: J. Math. Phys. 32 (10), 2929–2935 (1991) Broidioi, M. and Verbeure, A.: Helv. Phys. Acta 64, 1093–1112 (1991) Verbeure, A. and Zagrebnov, V.A.: J. Stat. Phys. 69, 329 (1992) Broidioi, M. and Verbeure, A.: Helv. Phys. Acta 66, 155–180 (1993) Goderis, D. and Vets, P.: Commun. Math. Phys. 122, 249 (1989) Goderis, D., Verbeure, A., and Vets, P.: Commun. Math. Phys. 128, 533–549 (1990) Anderson, P.W.: Phys. Rev., 112 (6), 1900–1916 (1958) Stern, H.: Phys. Rev. 147 (1), 94–101 (1966) Michoel, T. and Verbeure, A.: J. Stat. Phys. 96 (5/6), 1125–1162 (1999) Bratteli, O. and Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics 2. Berlin– Heidelberg–New York: Springer, 1996 Goderis, D., Verbeure, A., and Vets, P.: Probability Theory and Related Fields 82, 527–544 (1989) Arveson, W.: J. Funct. Anal. 15 (3), 217–243 (1974) Landau, L., Fernando Perez, J., and Wreszinski, W.F.: J. Stat. Phys. 26 (4), 755–766 (1981) Wreszinski, W.F.: Forts. der Physik 35 (5), 379–413 (1987) Thirring, W. and Wehrl, A.: Commun. Math. Phys. 4, 303–314 (1967) Thirring, W.: Commun. Math. Phys. 7, 181–189 (1968) Requardt, M.: J. Stat. Phys. 29 (3), 117–127 (1982) Bogoliubov, N.N.: Phys. Abh. S.U. 1, 229 (1962) Michoel, T., Momont, B., and Verbeure, A.: Rep. on Math. Phys. 41 (3), 361–395 (1998) Narnhofer, H., Requardt, M., and Thirring, W.: Commun. Math. Phys. 92, 247–268 (1983)

Communicated by H. Araki

Commun. Math. Phys. 216, 491 – 513 (2001)

Communications in



Scattering Theory for Quantum Fields with Indefinite Metric Sergio Albeverio, Hanno Gottschalk Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität Bonn, Wegelerstr. 6, 53115 Bonn, Germany. E-mail: [email protected]; [email protected] Received: 13 September 1999/ Accepted: 1 August 2000

Abstract: In this work, we discuss the scattering theory of local, relativistic quantum fields with indefinite metric. Since the results of Haag–Ruelle theory do not carry over to the case of indefinite metric [4], we propose an axiomatic framework for the construction of in- and out-states, such that the LSZ asymptotic condition can be derived from the assumptions. The central mathematical object for this construction is the collection of mixed vacuum expectation values of local, in- and out-fields, called the “form factor functional”, which is required to fulfill a Hilbert space structure condition. Given a scattering matrix with polynomial transfer functions, we then construct interpolating, local, relativistic quantum fields with indefinite metric, which fit into the given scattering framework. 1. Introduction The Wightman framework of local, relativistic quantum field theory (QFT) turned out to be too narrow for theoretical physicists, who were interested in handling situations involving in particular gauge fields (like in quantum electrodynamics). For several reasons which are intimately connected with the needs of the standard procedure of the perturbative calculation of the scattering matrix (for a detailed discussion, see [32]), the concept of QFT with indefinite metric was introduced, where a probability interpretation is possible only on Hilbert subspaces singled out by a gauge condition in the sense of Gupta [18] and Bleuler [9]. On the other hand, “ghosts”, which are quantum fields with the “wrong” connection of spin and statistics, entered the physical scene in connection with the Fadeev–Popov determinant in perturbation theory [15]. As a consequence of Pauli’s spin and statistics theorem, such quantum fields can not be realized on a state space with positive metric. Mathematical foundations for QFT with indefinite metric were laid by several authors, among them Scheibe [29], Yngvason [35], Araki [6], Morchio and Strocchi [26], Mintchev [25] and more recently by G. Hoffmann, see e.g. [22]. The results obtainable

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S. Albeverio, H. Gottschalk

from the axioms of indefinite metric QFT in many aspects are less strong than the axiomatic results of positive metric QFT. As the richness of the axiomatic results can be seen as a measure for the difficulty to construct theories which fulfill such axioms [31], the construction of indefinite metric quantum fields can be expected to be simpler than that of positive metric QFTs. Up to now, however, the linkage between these mathematical foundations and scattering theory, which in the day to day use of physicists is based on the LSZ reduction formalism [23], remained open, since the only available axiomatic scattering theory (Haag–Ruelle theory[19, 28, 20]) heavily relies on the positivity of Wightman functions. One can even give explicit counterexamples of local, relativistic QFTs with indefinite metric [4, 8], such that the LSZ asymptotic condition fails and Haag–Ruelle like scattering amplitudes diverge polynomially in time [4]. A scattering theory for QFTs with indefinite metric which fits well to the LSZ formalism and the mathematically rigorous construction of models of indefinite metric quantum fields (in arbitrary space-time dimension) with nontrivial scattering behavior are the topic of this work, which is organized as follows: In the second section (and Appendix A) we set up the frame of QFT with indefinite metric and recall some GNS-like results on the representation of ∗-algebras on state spaces with indefinite inner product. In Sect. 3 we introduce a set of conditions which is tailored just in the way to imply the LSZ asymptotic condition. The main mathematical object is the collection of mixed expectation values of incoming, local and outgoing fields, called “form factor functional”, which is required to fulfill a Hilbert space structure condition (HSSC), cf. [22, 26]. The existence of the form factor functional can be understood as a restriction of the strength of mass-shell singularities in energy-momentum space which rules out the counterexamples in [4]. In Sect. 4 we construct a class of QFTs with indefinite metric and nontrivial scattering behaviour fitting into the frame of Sect. 3. The main ingredient of this section is a sequence of local, relativistic truncated Wightman functions called the “structure functions”, which have been introduced and studied in [1–5,8,17,18,25]. The non-trivial scattering behaviour of the structure functions has been observed in [3, 16, 24]. The class of such QFTs is rich enough to interpolate essentially all scattering matrices with polynomial transfer functions1 . Some technical proofs can be found in Appendix B. Section 5 is a supplement to Sect. 4, in which we discuss the approximation of any set of measurement data for energies below a maximal experimental energy Emax up to an experimental accuracy given by an error tolerance > 0 with models in the class of Sect. 4. 2. Quantum Fields with Indefinite Metric In this section we introduce our notation and we collect some known facts about quantum field theories with indefinite metric following [2, 21, 22, 26]. In order to keep notations simple we study Bosonic, chargeless QFTs2 over a d dimensional Minkowski space-time (Rd , ·) where x · y = x 0 y 0 − x · y for x = (x 0 , x) = (x 0 , x 1 , . . . , x d−1 ), y = (y 0 , y) = (y 0 , y 1 , . . . , y d−1 ) ∈ Rd . For x · x we will frequently write x 2 . The collection of all k ∈ Rd with k 2 > m2 ≥ 0 and k 0 > 0 (k 0 < 0) is 1 Schneider, Baumgärtel and Wollenberg constructed a class of weakly local interpolating QFTs with positive metric [7, 30]. These fields however are not local [7] and are not related to the models we study here. 2 All results of this article can be generalized to fields with arbitrary parameters and statistics, cf. [16].

Scattering Theory for Quantum Fields with Indefinite Metric

493

called the forward (backward) mass-cone of mass m and is denoted by the symbol Vm+ (Vm− ). By V¯m± we denote the closure of Vm± . The (topological) boundary of Vm+ (Vm− ) is called the forward (backward) mass shell. By L we denote the full Lorentz group and ↑ by P˜ + the (covering group of the) orthochronous, proper Poincaré group. Sn stands for the complex valued Schwartz functions over Rdn and we set S0 = C. The topology on the spaces Sn is induced by the Schwartz norms n f K,L = sup (1) (1 + |xl |2 )L/2 D β1 ···βn f (x1 , . . . , xn ) , x1 ,... ,xn ∈Rd 0≤|β1 |,... ,|βn |≤K

l=1

where K, L ∈ N, βl = (βl1 , . . . , βld−1 ) ∈ Nd0 , l = 1 . . . , n, are multi indices with n j β1 ...βn = |βl | /∂x βl ). |βl | = d−1 l=1 (∂ j =0 βl , D l ↑ The canonical representation α of P˜ + on Sn is given by α{,a} f (x1 , . . . , xn ) = f (−1 (x1 − a), . . . , −1 (xn − a))

(2)

↑ ∀{, a} ∈ P˜+ , f ∈ Sn . We normalise the Fourier transform F : Sn → Sn as follows −dn/2 Ff (k1 , . . . , kn ) = (2π) e−i(x1 ·k1 +···+xn ·kn ) f (x1 , . . . , xn ) dx1 · · · dxn (3) Rdn

∀f ∈ Sn . Frequently we will also use the notation fˆ instead of Ff . For the inverse ¯ . Fourier transform of f we write Ff Let S be the Borchers’ algebra over S1 , namely S = ∞ n=0 Sn . f ∈ S can be written in the form f = (f0 , f1 , . . . , fj , 0, . . . , 0, . . . ) with f0 ∈ C and fn ∈ Sn , j ∈ N. The addition and multiplication on S are defined as follows: f + h = (f0 + h0 , f1 + h1 , . . . )

(4)

and f ⊗ h = ((f ⊗ h)0 , (f ⊗ h)1 , . . . ), (f ⊗ h)n =

∞

fj ⊗ hl for n ∈ N0 .

(5)

j,l=0 j +l=n

↑ The involution ∗, the Fourier transform F and the representation α of P˜ + on S are defined through

f ∗ = (f0∗ , f1∗ , f2∗ . . . ), Ff = (f0 , Ff1 , Ff2 , . . . ), α {,a} f = (f0 , α{,a} f1 , α{,a} f2 , . . . ),

(6)

where fn∗ (x1 , . . . , xn ) = fn (xn , . . . , x1 ). We endow S with the strongest topology, such that the relative topology of Sn in S is the Schwartz topology (direct sum topology). Let S = S (Rd , C) be the topological dual space of S. Then R ∈ S is of the form R = (R0 , R1 , R2 , . . . ) with R0 ∈ C, Rn ∈

494


Sn , n ∈ N. Furthermore, any such sequence defines uniquely an element of S . As in the ↑ case of S, the involution, Fourier transform and representation of P˜ + are on S , defined by the corresponding actions on the components Sn of S . Elements of S are also called Wightman functionals. The tempered distributions Wn ∈ Sn associated to a Wightman functional W ∈ S are also called (n-point) Wightman functions. Next we introduce the modified Wightman axioms of Morchio and Strocchi for QFTs in indefinite metric. Axioms 2.1. A1) Temperedness and normalization: W ∈ S and W0 = 1. ↑ A2) Poincaré invariance: α {,a} W = W ∀{, a} ∈ P˜ + . A3) Spectral property: Let Isp be the left ideal in S generated by elements of the form (0, . . . , 0, fn , 0 . . . ) with supp fˆn ⊆ {(k1 , . . . , kn ) ∈ Rdn : nl=1 kl ∈ V¯0+ }. Then Isp ⊆ kernel W . A4) Locality: Let Iloc be the two-sided ideal in S generated by elements of the form (0, 0, [f1 , h1 ], 0, . . . ) with supp f1 and supp h1 space-like separated. Then Iloc ⊆ kernel W . A5) Hilbert space structure condition (HSSC): There exists a Hilbert seminorm p on S s.t. W (f ∗ ⊗ g) ≤ p(f )p(g)∀f , g ∈ S. A6) Cluster Property: limt→∞ W (f ⊗ α {1,ta} g) = W (f ) W (h) ∀f , g ∈ S, a ∈ Rd space like (i.e. a 2 < 0). A7) Hermiticity: W ∗ = W . All these axioms can be equivalently expressed in terms of Wightman functions in the usual way, cf. [13, 26, 31]. The significance ofAxioms 2.1 can be seen from the following GNS-like construction: A metric operator η : H → H by definition is a self adjoint operator on the separable Hilbert space (H, (., .)) with η2 = 1. Let D be a dense and linear subspace. We denote the set of (possibly unbounded) Hilbert space operators A : D → D with (restricted) η-adjoint A[∗] = ηA∗ η|D : D → D with Oη (D). Clearly, Oη (D) is an unital algebra with involution [∗]. The canonical topology on Oη (D) is generated by the seminorms A → |(*1 , ηA*2 )|, *1 , *2 ∈ D. We then have the following theorem: Theorem 2.2. Let W ∈ S be a Wightman functional which fulfills the Axioms 2.1. Then (i) There is a Hilbert space (H, (., .)) with a distinguished normalized vector *0 ∈ H called the vacuum, a metric operator η with η*0 = *0 inducing a nondegenerate inner product ., . = (., η.) and a continuous ∗-algebra representation φ : S → Oη (D) with D = φ(S)*0 which is connected to the Wightman functional W via W (f ) = *0 , φ(f )*0 ∀f ∈ S. ↑ (ii) There is a η-unitary continuous representation U : P˜ + → Oη (D) ( U[∗] = U−1 ) ↑ −1 such that U(, a)φ(f )U(, a)−1 = φ(α f ) ∀f ∈ S, {, a} ∈ P˜ + and *0 is {,a}

invariant under the action of U. (iii) φ fulfills the spectral condition φ(Isp ), = 0. (iv) φ is a local representation in the sense that Iloc ⊆ kernel φ. (v) For *1 , *2 ∈ D and a ∈ Rd space like, we get limt→∞ *1 , U(1, ta)*2 = *1 , *0 *0 , *2 .


495

A quadruple ((H, ., ., *0 ), η, U, φ) is called a local relativistic QFT with indefinite metric. Conversely, let ((H, ., ., *0 ), η, U, φ) be a local relativistic QFT with indefinite metric. Then W (f ) = *0 , φ(f )*0 ∀f ∈ S defines a Wightman functional W ∈ S which fulfills Axioms 2.1. Proof. See [11, 26]. For the fact that we can choose the metric operator in such a way that η*0 = *0 , cf. [22]. Item (v) is just a rephrasing of the cluster property (A6). It should be mentioned that the pair (W , p) uniquely determines the associated QFT with indefinite metric, but it is believed that in general the Wightman functional W admits non equivalent representations as the vacuum expectation value of a QFT with indefinite metric depending on the choice of p, cf. [6] for a related situation. See however [22] for sufficient conditions s.t. only W determines the (maximal) Hilbert space structure. We want to study sufficient topological conditions on the Wightman functionals which imply the HSSC and therefore the existence of ∗-algebra representations with indefinite metric. To this aim let γK,L be the strongest topology on S s.t. ∀n ∈ N the restriction of γK,L to Sn is induced by the norms (1). Let γ be the weakest topology on S generated by all γK,L . Then we get e.g. by Theorem 3 of [26]: Theorem 2.3. If W ∈ S fulfills the condition (A5’): W is continuous w.r.t. the topology γ , then W fulfills the HSSC. We note that F, F¯ : S → S are γ -continuous, thus there is no difference between the γ -continuity of W and Wˆ . Topological conditions of this kind obviously are “linear” in the sense that they are preserved under linear combinations. The only essentially non-linear condition in the set of Axioms 2.1 thus is the cluster property (A6). It is linearized by an algebraic transformation S W ! → W T ∈ S known as “truncation”. As we shall see, this transformation preserves (A2)–(A4), (A7) and transforms (A1) into an equivalent linear condition. The crucial observation now is that truncation also preserves the γ -continuity of W [2, 21]. Consequently we can translate the modified Wightman axioms 2.1 into a purely linear set of conditions for the truncated Wightman functional. For the technicalities we refer to Appendix A. 3. Construction of Asymptotic States In this section we develop a mathematical framework for scattering in indefinite metric relativistic local QFT. In a certain sense we go in the opposite direction as the axiomatic scattering theory with positive metric [19, 20, 28] where asymptotic fields are being constructed first and the scattering amplitudes are calculated in a second step [20, 23]. Here we postulate the existence of the mixed vacuum expectation values of in- loc- and out- fields and we then construct these fields using the GNS-like procedure of Sect. 2. Let S ext be the “extended” Borchers’ algebra over the test function space S1ext = S(Rd , C3 ), which is the space of Schwartz functions with values in C3 . For a =in/loc/out we define J a : S1 → S1ext to be the injection of S1 into the first/second/third component of S1ext , i.e. J in f = (f, 0, 0), J loc f = (0, f, 0), J out f = (0, 0, f ), f ∈ S1 . Then J a uniquely induces a continuous unital ∗-algebra homomorphism J a : S → S ext given a⊗n . by J a = ⊕∞ n=0 J

496


We also define a suitable “projection” J : S ext → S as the unique continuous unital ∗-algebra homomorphism induced by J : S1ext → S1 , J (f in , f loc , f out ) = f in + f loc + f out . For simplicity, we only consider the case of only one stable particle mass m > 0. Let and ϕ ∈ C0∞ (R, R) with support in (−, ) with 0 < < m2 and ϕ(x) = 1 if −/2 < x < /2. We define χ ± (k) = θ(±k 0 )ϕ(k 2 − m2 ) with θ the Heavyside step function and we set  0 0  χ + (k)e−i(k −ω)t + χ − (k)e−i(k +ω)t for a =in for a =loc . (7) χt (a, k) = 1  + 0 −ω)t 0 +ω)t i(k − i(k + χ (k)e for a =out χ (k)e We then define ,t : S1ext → S1ext by   χt (in, k) 0 0 . 0 χt (loc, k) 0 F,t F¯ =  0 0 χt (out, k)

(8)

Next, we introduce the multi parameter t = (t1 , t2 , . . . ), tn = (tn1 , . . . , tnn ), tnl ∈ R and we write t → +∞ if tnl → +∞ in any order, i.e. first one tnl goes to infinity, then the next, etc. We say that the limit t → +∞ of any given object exists, if it exists for tnl → +∞ in any order and it does not depend on the order. We now define the finite times wave operator ,t : S ext → S as n ,t = J ◦ ⊕ ∞ n=0 ,n,tn , ,0,t0 = 1, ,n,tn = ⊗l=1 ,tnl .

(9) in/out

Furthermore, we define the finite times in- and out- wave operators ,t :S →S as ,t ◦ J in/out . Up to changes of the time parameter which do not matter in the limit (in/out)

t → +∞, the wave operators ,t verified from the definitions.

are ∗-algebra homomorphisms, as can be easily

Definition 3.1. (i) Let W ∈ S be a Wightman functional s.t. the functionals W ◦ ,t

converge in S ext as t → +∞. We then define the form factor functional F ∈ S ext associated to W as this limit, i.e. F = lim W ◦ ,t . t→+∞

(10)

(ii) The scattering matrix S associated to W is defined by S(f , g) = F (J in f ⊗ J out g) =

lim

t,t →+∞

out W (,in t f ⊗ ,t g) ∀f , g ∈ S.

(11)

We are now in the position to state a set of conditions which allow a reasonable definition of the scattering matrix, in- and out-fields and states in indefinite metric QFT.


497

Condition 3.2. Let W ∈ S . We assume that m0 s1) W fulfills Axioms 2.1 and W is a theory with a mass gap m0 > 0, i.e. W T (Isp )= m0 0 with Isp the vector space generated by (0, . . . , 0, fn , 0, . . . ) with supp fˆn ⊆ {(k1 , . . . , kn ) ∈ Rdn : ∃j, 2 ≤ j ≤ n, such that nl=j kl ∈ V¯m+0 }. s2) The truncated two point function W2T of W is of the form ∞ − (k1 ) + δµ− (k1 )ρ(µ)dµ δ(k1 + k2 ) (12) Wˆ 2T (k1 , k2 ) = δm m0

with ρ a positive polynomially bounded locally integrable density. s3) The form factor functional F associated to W exists, is Poincaré invariant and fulfills the Hilbert space structure condition (HSSC). The following theorem shows that Condition 3.2 just implies the LSZ asymptotic condition. Theorem 3.3. We suppose that W fulfills Condition 3.2. Then (i) There exists a (in general not local) quantum field theory with indefinite metric ((H, ., ., *0 ), η, U, 9) over the Borchers algebra S ext such that the statements (i)–(iii) of Theorem 2.2 hold. (ii) There exist relativistic local quantum fields with indefinite metric φ in/loc/out = 9 ◦ J in/loc/out over S s.t. φ in/out are free fields of mass m (for d ≥ 4)and φ = φ loc fulfills the LSZ asymptotic condition, namely in/out

lim φ(,t

t→+∞

f ) = φ in/out (f ) ∀f ∈ S,

(13)

where the limit is taken in Oη (D). (iii) There exist U-invariant Hilbert spaces Hin/out ⊆ H defined as Hin/out = φ in/out (S)*0 , s.t. the restriction of ., . to Hin/out is positive semidefinite (d ≥ 4). Proof. (i) Except for the spectral property and hermiticity, this point of the theorem follows immediately from s3) and Theorem 2.2. Concerning the spectral property we note that supp F(Wn ◦ ,n,tn ) ⊆ supp Wˆ n . Thus, supp Fˆn ⊆ supp Wˆ n . Since Wˆ n has the spectral property, which is actually a restriction on the support of Wˆ n , the spectral property of Fˆn follows. 9(Isp ), = {0} now follows from Theorem 2.2. The hermiticity follows from the hermiticity of W , the fact that ,t is a ∗-algebra homomorphism (in the sense given above) and that the limit of Hermitian functionals is Hermitean itself. (ii) The existence of the fields φ in/loc/out follows immediately from point (i) of the theorem, namely from the existence of the field 9. That these fields fulfill the properties of 2.2 for φ in/out follows from the fact that they are free fields (cf. [31]) and for φ loc this statement by Theorem 2.2 follows from the assumption s1) on W . That φ in/out for d ≥ 4are free is a consequence of the fact that the mass gap assumption is fulfilled and thus the truncated Wightman functionals WnT fulfill the strong cluster in/out property Theorem XI.110 of [27] Vol. III. Consequently, limt→+∞ W T (,t f) = 0 for f ∈ S with f1 = 0, f2 = 0 follows from Theorem XI.111 in [27] Vol. III ( the negative frequency terms which occur in our framework are just the complex conjugation of some positive frequency term with the same “time direction”). The fact that the locally

498


integrable density ρ(µ)dµ does not give a contribution to the two point function of φ in/out follows from the Riemann lemma, cf. [27] Vol. II ( for the details of the argument, see the proof of Proposition 4.7 below). In order to prove the Oη (D)-convergence in Eq. (13), we have to show that in/out lim 9(f )*0 , φ(,t g)9(h)*0 = 9(f )*0 , φ in/out (g)9(h)*0 t→+∞

holds for all g ∈ S, f , h ∈ S ext . Rewriting the left-hand side and the right-hand side of this formula in terms of the quantum field 9 we can verify it using also s3) by the following calculation in/out lim 9(f )*0 , 9(J loc ,t g)9(h)*0 t→+∞ in/out = lim *0 , 9(f ∗ ⊗ J loc ,t g ⊗ h)*0 t→+∞ in/out = lim F f ∗ ⊗ J loc ,t g⊗h t→+∞ = lim lim W ,s 1 f ∗ ⊗ ,t J in/out g ⊗ ,s 2 h t→+∞ s 1 ,s 2 →+∞ = lim W ,t (f ∗ ⊗ J in/out g ⊗ h) t→+∞ = F f ∗ ⊗ J in/out g ⊗ h = *0 , 9(f ∗ ⊗ J in/out g ⊗ h)*0 = 9(f )*0 , 9(J in/out g)9(h)*0 . (iii) The U-invariance of Hin/out results from the transformation law U(, a)φ in/out (f )U(, a)−1 = φ in/out (α −1 {,a} f ) and the U-invariance of *0 . The transformation law holds by Theorem 2.2 (ii) and s3). That ., . is positive semidefinite on Hin/out follows from the fact that the dense subspaces spaces φ in/out (S)*0 are also dense subsets of Fock spaces over the one particle space S1 with positive semidefinite inner product induced in/out,T by W2 , cf. [10] p. 288. Here we do not give precise conditions for the existence of the form factor functional, but we refer to the methods of Sect. 4 and Appendix B where the form factor functional has been constructed for a special class of models. Looking into the details of the proof, one notices that what one really requires in order to get the existence of this ± (k) functional is the restriction of mass-shell singularities to singularities of the type δm 2 2 and 1/(k − m ). These are just the singularities occurring in the Feynman propagator. It therefore seems to be reasonable that the form factor functional can be defined in theories where Yang–Feldman equations [34] hold. Since for the physicists’ common sense Yang–Feldman equations are an alternative formulation of the LSZ asymptotic condition, to us it seems that our Condition 3.2 does not rule out many cases of physical interest. The assumption of the existence of the form factor functional also can not be dropped from Condition 3.2, since we have to exclude those models from [4] which have too strong mass shell singularities leading to divergent Haag–Ruelle like scattering amplitudes.


499

Finally in this section we translate Condition 3.2 into the language of truncated functionals: Proposition 3.4. Let W ∈ S be given and W T be the associated truncated Wightman functional. ˜

(i) F = limt→+∞ W ◦ ,t exists if and only if F T = limt→+∞ W T ◦ ,t exists. In this ˜

case F T = F T (we may thus omit the tilde in the following). (ii) S T (f , g) = F T (J in f ⊗ J out g) ∀f , g ∈ S. (iii) Suppose that W T fulfills s1) (transcribed to the language of truncated Wightman functionals according to Proposition A.1) and s2) of Condition 3.2 and furthermore s3T): F T ∈ S ext exists, is Poincaré invariant and γ -continuous. Then the associated Wightman functional fulfills s1)-s3) of Condition 3.2. Proof. (i) We note that up to the ordering of the time parameter t this statement follows from Lemma A.2. But the ordering of t does not matter due to the definition of the limit t → +∞. (ii) This equation follows by application of Lemma A.3 to F . (iii) is a corollary to (i), Proposition A.1 and Theorem2.3. 4. An Interpolation Theorem In this section we construct a class of quantum fields with indefinite metric which have a well defined scattering behavior in the sense of Theorem 3.3 and which interpolate a certain class of scattering matrices. This is being done by a rather explicit construction of the truncated Wightman functional and the verification of the conditions given in item (iii) of Proposition 3.4. The existence of quantum fields with indefinite metric then follows from Theorem 3.3. We first recall a well known result of scattering-(S)-matrix theory following [20, 23]: Let us for a moment consider a quantum field as a operator valued distribution φ(x) (i.e. the restriction of the homomorphism φ : S → Oη (D) to S1 ). We assume that φ in/out fulfills the LSZ-asymptotic condition φ ◦ ,t → φ in/out as t → ∞ in an appropriate sense, where the asymptotic fields φ in/out are free fields of mass m. Let φˆ in/out (k) denote the Fourier transform of φ in/out (x). Then the expectation values of states created by application of the in-fields to the vacuum *0 with states generated analogously by the out-fields have the following general shape:

φˆ in (kr ) · · · φˆ in (k1 )*0 , φˆ out (kr+1 ) · · · φˆ out (kn )*0 = 2πi Mn (−kr , . . . , −k1 , kr+1 , . . . , kn ) “transfer

function

n l=1

T

+ δm (kl ) δ(

on−shell term

n l=r+1

kl −

r l=1

kl ),

(14)

energy−momentum conservation term

where n ≥ 3, kl0 > 0, l = 1, . . . , n, i.e. all operators φˆ in/out (kl ) are creation operators. Since the in- and out-fields fulfill canonical commutation relations this is sufficient to 0 calculate also those expectation n values with the condition on the kl dropped. Here the distribution Mn (k1 , . . . , kn )δ( l=1 kl ) is given (up to a constant) by the Fourier transform

500


of the time-ordered vacuum expectation values of φ(x) multiplied by nl=1 (kl2 − m2 ) and thus is symmetric under permutation of the arguments and Poincaré invariant. By the definition of the scattering matrix in Sect. 3, we can equivalently write for Eq. (14), T (k1 , . . . , kr ; kr+1 , . . . , kn ) Sˆr,n−r

= 2πi Mn (k1 , . . . , kn )

r l=1

− δm (kl )

n l=r+1

+ δm (kl ) δ(

n

kl )

(15)

l=1

for n ≥ 3 and kl0 < 0 for l = 1, . . . , r and kl0 > 0 for l = r + 1, . . . , n. Here we used φˆ in/out (k) = φˆ in/out[∗] (−k). Given this general form of the S-matrix, one can ask whether under some conditions on the transfer functions Mn there exists an interpolating quantum field φ s.t. φ fulfills the LSZ asymptotic condition and the scattering matrix S is determined by Eq. (15). In the following we give a (partial) answer to this question for the case of quantum fields with indefinite metric. First we fix some conditions on the sequence of transfer functions Mn . Condition 4.1. We assume that M ∈ S fulfills the following conditions: I1) Mn is symmetric under permutation of arguments and Lorentz invariant (w.r.t. the entire Lorentz group L); I2) Mn is real, M2 = 1; I3) Mn is a polynomial; I4) ∃Lmax ∈ N0 s. t. ∀n ∈ N the degree of Mn (k1 , . . . , kn ) in any of the arguments k1 , . . . , kn is at most Lmax . Remark 4.2. The “essentially linear” set of conditions given above of course does not imply unitarity of the scattering matrix, which connects transfer functions of different orders, cf. [10]. Up to now it is not clear, whether in the class of transfer functions described by Condition 4.1 there are exact solutions to the unitarity condition. “Approximate” solutions however are possible due to Proposition 5.1 below. While the specific properties of the system under consideration are encoded in the transfer functions, we also need an input creating the “axiomatic structure”, namely the on-shell terms and the energy-momentum conservation term. In the following we define a sequence of “structure functions3 ” with the required properties. Definition 4.3. For n ≥ 3 we define the n-point structure function Gn as the inverse ˆ n given by Fourier transform of G   −1 n n j n   1 − + ˆ n (k1 , . . . , kn ) = δm (kl ) 2 δ (k ) δ( kl ). (16) G m l   kj − m 2 j =1 l=1

l=j +1

l=1

ˆ 2 given by Eq. (12). The structure functional G ∈ S is defined by G0 = 0, G1 = 0 and G 3 These functions have nothing to do with the “structure functions” describing inelastic scattering in the phenomenology of elementary particles.


501

The structure functions first have been defined in [5] (for m = 0), the present form given in Definition 4.3 was obtained in [1]. Further properties of the structure functions are given in [2, 3], see also [16, 17, 24]. The following proposition summarizes the results obtained in these references: Proposition 4.4. G fulfills all properties of a truncated Wightman functional of a QFT with indefinite metric with a mass gap m0 > 0 (cf. Proposition A.1, Condition 3.2 s1)). If M is a functional which fulfills Cond. 4.1, then we define the dot-product of the ˆ by (M · G) ˆ n = Mn · G ˆ n , where the multiplication on the rightfunctionals M and G ˆ n is a tempered distribution and Mn is a hand side obviously is well defined, since G polynomial. We now have collected the pieces, which are being put together in the following “interpolation theorem”. Theorem 4.5. Let G be the structure functional (cf. Definition 4.3) and let M ∈ S fulfill Condition 4.1. Then T ˆ fulfills the conditions of Proposition 3.4 (iii). (i) Wˆ = M · G (ii) The truncated S-matrix (cf. Prop. 3.4 (ii)) is determined by Eq. (15). (iii) In particular, there exists a local, relativistic quantum field φ with indefinite metric (see Theorem 2.2) which fulfills the LSZ asymptotic condition Eq. (13) w.r.t. free fields φ in/out of mass m and has scattering behavior determined by Eq. (14). The restriction of the indefinite inner product ., . to the Hilbert spaces Hin/out = φ in/out (S)*0 is positive semidefinite.

The rest of this section is devoted to the proof of Theorem 4.5. Obviously, item (iii) is a straightforward application of (i), (ii), Proposition 3.4 and Theorem 3.34 . Therefore, we only have to check statements (i) and (ii). T

Proof of statement (i). Step 1) Verification of the modified Wightman axioms for Wˆ and s1),s2): (A1T) holds by G0 = 0 and G ∈ S , cf. Prop. 4.4. Poincaré invariance (A2) follows straightforwardly from the translation invariance of G and Lorentz invariance of G and M. The (strong) spectral property (A3) (s1) can be verified by supp Wˆ nT = supp Mn · ˆ n ⊆ {(k1 , . . . , kn ) ∈ Rdn : nl=j kl ∈ V¯m+ for j = 2, . . . , n} for n ≥ 2, ˆ n ⊆ supp G G 0 where the last inclusion holds by Prop. 4.4. Locality (A4) can equivalently be expressed T in terms of the (truncated) Wightman functions via supp Wn,[,] ⊆ {(x1 , . . . , xn ) ∈ Rdn : j

T (xj − xj +1 )2 ≥ 0} for j = 1, . . . , n − 1, where Wn,[,] (x1 , . . . , xj , xj +1 , . . . , xn ) = j

WnT (x1 , . . . , xj , xj +1 , . . . , xn ) − WnT (x1 , . . . , xj +1 , xj , . . . , xn ). This follows by T = supp Mn (−i supp Wn,[,] j

∂ ∂ , . . . , −i )Gn,[,]j ⊆ supp Gn,[,]j , ∂x1 ∂xn

where in the first step we have made use of the definition of WnT and the symmetry of Mn under permutation of the arguments j, j + 1, and in the second step we used that multiplication by a polynomial in momentum space gives differentiation in position space which is a local operation. Now the assertion follows from the locality of Gn , cf. Prop. 4.4. The proof of (A5’) follows from the observation that the γc,r -continuity 4 By a direct calculation as in the proof of (ii) below one can show that the fields φ in/out are free fields also for d = 2, 3.

502


ˆ (which holds for some c, r ∈ N by Prop. 4.4) implies the γc,r+Lmax -continuity of of G T Wˆ , where Lmax is given in Condition 4.1 I4). (A6T) follows from the strong spectral property, invariance and locality, cf. Theorem XI.110 of [27] Vol. III. Hermiticity (A7) immediately follows from the hermiticity of G and the fact that Mn (−kn , . . . , −k1 ) = Mn (kn , . . . , k1 ) = Mn (k1 , . . . , kn ), where we have also used the real valuedness, reflection invariance and symmetry of Mn . But this is just the relation defining hermiticity in momentum space. Finally, s2) holds by the definition of G and M2 = 1. Step 2) Calculation of the truncated form factor functional and verification of (A5’), (s3T):. We proceed as follows: We define a functional F G and we prove that this is the form factor functional associated to G. To show this, we require two technical lemmas; their proofs can be found in Appendix B. The rest of the proof of this step is in a similar fashion as the preceding paragraph. We define the distribution ?m ∈ S1ext by the following formula for the Fourier transform of its in-, loc- and out-component:  + −  −iπ(δ 2 m (k) 2− δm (k)) for a = in ˆ m (a, k) = 1 (k − m ) (17) ? for a = loc .  + (k) − δ − (k)) iπ(δm for a = out m Here, as in the definition of the structure functions, the singularity 1/(k 2 − m2 ) has to be understood in the sense of Cauchy’s principal value. We now define the functional F G which turns out to be the form factor functional associated with G:

Definition 4.6. The functional F G ∈ S ext is defined by the following formulae for G(a ,... ,an ) , al = in/loc/out, l = 1, . . . , n: the Fourier transform of the components Fˆn 1 G(a ) 1 G ˆ ˆ (k1 ) = 0, F0 = 0, F1 ˆ 2 (k1 , k2 ) for a1 = a2 = loc G G(a ,a ) (18) Fˆ2 1 2 (k1 , k2 ) = − (k )δ(k + k ) otherwise δm 1 1 2 and FˆnG(a1 ,... ,an ) (k1 , . . . , kn ) =

 −1 n j  

j =1 l=1

− ˆ m (aj , kj ) δm (kl )?

n l=j +1

 

+ δm (kl ) δ( 

n

kl ).

l=1

(19)

That F G is in S ext , as stated in the Definition 4.6, is contained in the following Proposition 4.7. F G is the form factor functional associated to G. Furthermore, F G is Poincaré invariant and γ -continuous. ·

2,L For the proof of this proposition we introduce the test function space S1,2 = ∩∞ L=0 S 1 (the bar stands for completion) with the topology of the inductive limit. By S1,2 we denote the topological dual space. It is well-known that 1/(k 2 − m2 ) as a distribution lies in S1,2 (since the Cauchy principle value in a neighborhood of the singularity is continuous w.r.t. ˆ m (a, k) ∈ S for a = in/loc/out. The following two lemmas the C 1 -norm) and thus ? 1,2 contain the analytic part of the proof of Proposition 4.7. For the proof see Appendix B:


Lemma 4.8. limt→+∞

χt (a,k) (k 2 −m2 )

503

ˆ m (a, k) holds in S for a = in/loc/out. =? 1,2

Lemma 4.9. For f ∈ Sn , n ≥ 3, j = 1, . . . , n, let gj : Rd → C be defined as gj (kj ) =

Rd(n−1) n

× δ(

f (k1 , . . . , kn )

j −1 l=1

− δm (kl )

n l=j +1

+ δm (kl )

kl )dk1 · · · dkj −1 dkj +1 · · · dkn .

(20)

l=1

Then gj ∈ S1,2 and gj 2,L ≤ cL f 2,L for L ∈ N, L = max{d, L} and cL > 0 sufficiently large. G Proof of Proposition 4.7. We first note that Fˆ is manifestly Poincaré invariant. The γ -continuity of F G can be seen as follows: Let al = in/loc/out, l = 1, . . . , n be fixed ˆ m (aj , kj ) is continuous w.r.t. and f ∈ Sn , n ≥ 3. Then by Lemma 4.9 and the fact that ? . 2,L for L ≥ d + 1 (with continuity constant dL > 0 sufficiently large) we get the following estimate: n ˆ G(a1 ,... ,an ) ˆ (f ) = ?m (aj , kj )gj (kj )dkj Fn d j =1 R

≤ dL gj 2,L ≤ dL cL f 2,L . ˆ 2 (which is deterThus, if we choose L sufficiently large s.t. the “continuous part” of G G ˆ mined by ρ, cf. Eq. 12) is continuous w.r.t. . 0,L , we get that F is continuous w.r.t. γ2,L and hence w.r.t. γ . To finish the proof we have to show that for n ∈ N0 , n

lim

tn1 ,... ,tnn →+∞

l=1

ˆ n (k1 , . . . , kn ) = FˆnG(a1 ,... ,an ) (k1 , . . . , kn ), χtnl (al , kl )G

(21)

where tnl → +∞, l = 1, . . . , n, in arbitrary order and the limit is being taken in Sn . For n = 0, 1 this holds by definition (G0 , G1 = 0 and F0G , FG1 = 0). Let n = 2. For a1 = a2 =loc there is nothing to prove since χt (loc, k) = 1. Let e.g. a1 =out and f ∈ S2 . Then we get for the left-hand side of Eq. (21) smeared out with f for the case first t21 → +∞ and then t22 → +∞, ∞ − − δm (k) + lim δµ (k)ρ(µ)dµ . . . = lim t22 →+∞ t21 →+∞ Rd

×e =

Rd

χ − (k)χt 2 (a2 , −k)f (k, −k) dk 2 ∞ 1 − δm (k)f (k, −k) dk + lim lim ei(ω−ωµ )t2 t22 →∞ t21 →+∞ m0

×

m0

i(k 0 +ω)t21

Rd−1

f ((−µ, k), (µ, −k))ϕ(µ2 − m2 )χt 2 (a2 , (−µ, k)) 2

dk ρ(µ)dµ. 2ωµ

504


Here ωµ = |k|2 + µ2 . We want to show that the limit of the second integral vanishes. To do this, we note that the expression in the brackets [. . . ] defines a smooth and fast falling (for µ → +∞) function in µ and the change of variables µ → ξ = ω − ωµ is smooth (with polynomially bounded determinant) since m0 > 0. Thus, the second integral can be written as the Fourier transform evaluated at t21 of a L1 (R)-function in the variable ξ (which might depend on t22 ). By the lemma of Riemann-Lebesgue (cf. Theorem IX.7 [27] Vol. II), the Fourier transform of such a function vanishes at infinity. Thus, the second integral vanishes. If we first take the limit t22 → +∞ and then t21 → ∞, we can distinguish two cases: If a2 = loc, the second integral does not depend on t22 and we can thus take the limit t21 → +∞ as before. If a2 = loc we get by an argument which is analogous to the one given above, that the limit t22 → +∞ of the second integral on the r.h.s. vanishes. This proves Eq. (21) for the case n = 2. ± (k) = δ ± (k) we get for the left-hand Let thus n ≥ 3. Using the fact that χt (a, k)δm m side of (21) smeared out with f ∈ Sn : χ j (a , k ) n tn j j ... = lim gj (kj ) dkj , j d kj2 − m2 j =1 tn →+∞ R where we have used the notation introduced in Lemma 4.9. Using now that by Lemma 4.9 gj ∈ S1,2 we get by Lemma 4.8 for the right-hand side of this equation ... =

n d j =1 R

ˆ m (aj , kj )gj (kj ) dkj . ?

But this is just the right-hand side of Eq. (21) smeared out with f . G G Similar as above, we define the dot-product M · Fˆ ∈ S ext of Fˆ with M via G (a1 ,... ,an ) G(a ,... ,an ) (M · Fˆ )n = Mn · Fˆn 1 . We then get from Proposition 4.7 by a simple use of duality and the same arguments as in Step 1):

G

¯ · Fˆ ) exists, is Poincaré invariant and γ -continuous. Corollary 4.10. F T = F(M Proof of statement (ii). Let n ≥ 3, 1 ≤ r ≤ n − 1, k10 , . . . , kr0 < 0 and 0 , . . . , k 0 > 0. Then by Corollary 4.10 kr+1 n T (k1 , . . . , kr ; kr+1 , . . . , kn ) = FˆnT (in,... ,in,out,... ,out) (k1 , . . . , kn ) Sˆr,n−r

= Mn (k1 , . . . , kn )FˆnG(in,... ,in,out,... ,out) (k1 , . . . , kn ), where the “in” is being repeated r times and the “out” n − r times. Inserting (19) into this expression we get  −1 r j n  − + ˆ Mn (k1 , . . . , kn ) δm (kl )?m (in, kj ) δm (kl )  j =1 l=1 l=j +1  −1 n j n n  − + ˆ + δm (kl )?m (out, kj ) δm (kl ) δ( kl ).  j =r+1 l=1

l=j +1

l=1


505

0 , . . . , k 0 > 0 for j = r + 1, . . . , n, Using the assumption k10 , . . . , kr0 < 0 and kr+1 n we see that in the first sum only the term j = r gives a non vanishing contribution whereas in the second sum all terms vanish except for the term j = r + 1. Inserting the 0 + (k )-term in ? ˆ m (in, kr ) expression (17) and using kr0 < 0 and kr+1 > 0 we see that the δm r − ˆ gives no contribution and this is also true for the δm (kr+1 )-term in ?m (out, kr+1 ). We thus get for the above expression r−1 n − − + Mn (k1 , . . . , kn ) δm (kl )[iπ δm (kr )] δm (kl ) l=1 r

+

l=1

l=r+1

− + δm (kl )[iπ δm (kr+1 )]

= 2πi Mn (k1 , . . . , kn )

r l=1

This finishes the proof of Theorem 4.5.

n l=r+2

− δm (kl )

n + δm (kl ) δ( kl )

n l=r+1

l=1

+ δm (kl ) δ(

n

kl ).

l=1

5. Approximation of Arbitrary Scattering Amplitudes Here we want to discuss the approximation of a given (“reference”) set of transfer functions (cf. Eq. (14)) R with polynomial transfer functions M. For R we assume full Lorentz invariance (including reflections) and symmetry under permutation of the arguments, which is motivated from the LSZ formalism (see Sect. 4). Furthermore, we assume that the Rn are continuous, real functions5 . Since the models of Sect. 4 have polynomial transfer functions, which grow very fast for large energy arguments and therefore have a somehow “bad” high energy behaviour, we only consider scattering experiments with maximal energy Emax > 0, which can be chosen arbitrarily large. By Qn (Emax ), we denote the set of points in energy-momentum space which can be reached by a scattering experiment of maximal energy Emax : !

"

(k1 , . . . , kn ) ∈ Rdn : kl2 = m2 , l = 1, . . . , n; k10 , . . . , kr0 < 0,

1≤r≤n−1 0 , . . . , kn0 > 0, kr+1

n l=r+1

kl0 ≤ Emax ,

n

# kl = 0 .

(22)

l=1

It is easy to verify that for Emax < ∞, Qn (Emax ) is compact and that Qn (Emax ) = ∅ for n > Emax /m. We say that M approximates R for energies smaller than Emax up to an error > 0, if for n ∈ N |Mn (k1 , . . . , kn ) − Rn (k1 , . . . , kn )| < holds ∀(k1 , . . . , kn ) ∈ Qn (Emax ). We then get 5 In general scattering amplitudes are analytic functions on a “cut” neighborhood of the on-shell region and therefore can have discontinuities or singularities on these “cuts”, cf. [14, 33]. Therefore, we do not consider R as the transfer functions of some “real” theory, but as a set of “measurement data”. Then, the requirement for the Rn to be real can be justified by the fact that only the square modulus of Rn enters in the measurable transition probabilities and continuity can be understood in the sense that Rn was obtained by some continuous interpolation of a discrete set of measurements.

506


Proposition 5.1. Let R be a real, fully Lorentz invariant and symmetric functional consisting of continuous functions. For any error parameter > 0 arbitrarily small and any energy cut-off parameter Emax > 0, there exists a functional M which fulfills Conditions 4.1 and which approximates R for energies smaller than Emax up to an error (in the sense given above). In particular, there exists a QFT with indefinite metric in the class of QFTs given in Theorem 4.5 with scattering behavior which for energies smaller than Emax differs from the data R at most by an error . We start the proof with a technical lemma: Lemma 5.2. Let Rn : (V¯m+0 ∪ V¯m−0 ) × Rd(n−1) → R be continuous and invariant under the full Lorentz group L. Then there exists a continuous function Vn : Rn(n+1)/2 → R s.t. Rn (k1 , . . . , kn ) = Vn (k12 , k1 · k2 , k22 , . . . , k1 · kn , k2 · kn , . . . , kn2 ). Sketch of the Proof. Let π : (V¯m+0 ∪ V¯m−0 ) × Rd(n−1) /L → Rn(n+1)/2 be defined by Lk¯ = L(k1 , . . . , kn ) → (k12 , k1 · k2 , k22 , . . . , k1 · kn , k2 · kn , . . . , kn2 ) = (q1,1 , . . . , qn,n ) = q. ¯ We want to define Vn on the image of π as Rn ◦ π −1 . Hence we have to show that ¯ k¯ ∈ π −1 (q) k, ¯ are in the same orbit of L in Rdn . First, we can apply $ a Lorentz boost (possibly in connection with time reflection) which

maps k1 (k1 ) to ( k12 , 0). Then, in this new frame of reference the zero components of $ kl , l = 2, . . . , n are given by kl · k1 / k12 (for kl we proceed analogously). Since the zero components are known, also scalar products of the kl (kl ) are known in this new frame which fixes distances of “points” from the origin and “angles” of the “rigid body” spanned by the kl (kl ) in Rd−1 . But then there is an orthogonal transformation on Rd−1 moving the “rigid body” spanned by kl onto the one spanned by the kl . Hence k¯ and k¯ are in the same orbit. Furthermore, the mapping Vn is continuous on the set Ran π . This follows from the fact that one can construct a reference vector r¯ (q) ¯ ∈ Rdn corresponding to fixing the zero component and a “standard orientation” for the “rigid body” which depends smoothly on q. ¯ Thus, for q¯n → q¯ in Ran π we get r¯n → r¯ and thus Vn (q¯n ) = Rn (¯rn ) → Rn (¯r ) = Vn (q¯ ). Since Ran π is closed in Rn(n+1)/2 , there exists a continuous extension of Vn to Rn(n+1)/2 .

Proof of Proposition 5.1. We use the same notations as in the proof of Lemma 5.2. Note that π(Qn (Emax )) is compact since π is continuous and Qn (Emax ) is compact. Thus, for > 0 by the Stone–Weierstrass theorem there exists a polynomial pn such ¯ then that |pn (q) ¯ − Vn (q)| ¯ < ∀q¯ ∈ π(Qn (Emax )). Let thus Mn (k) = pn (π(k)), ¯ − Rn (k)| ¯ < ∀k¯ ∈ Qn (Emax ). Furthermore, there is no problem to assume that |Mn (k) Mn is real and symmetric under exchange of variables, since if this is not the case we can replace Mn with ReMn and symmetrize without changing the approximation properties. By construction Mn is invariant under the full Lorentz group. It remains to show that the uniform bound in the degree of Mn (k1 , . . . , kn ) can be obtained. But this follows from Qn (Emax ) = ∅ for n > Emax /m, which means that we can choose {Mn }n>Emax /n as arbitrary real, symmetric and Lorentz invariant polynomials with uniform bound. By Proposition 5.1, there is no “falsification” based on scattering experiments for the statement that the “true” theory explaining a set of measurements R is in the class of


507

models given in Theorem 4.5 (note that ., . is positive semidefinite on the asymptotic states, thus there is no problem with the probability interpretation of such experiments). Of course, we do not consider this as a serious physical statement. Instead, we think that this result emphasizes the importance of structural aspects (as e.g. a “good” high energy behavior, “exact” unitarity), which might go beyond an explicit and exact measurability. A. Truncation of (Bi-) Linear Functionals on Borchers’ Algebra j

We introduce the following notation: Let λl = (λ1l , . . . , λl ) ⊆ (1, . . . , n), where the inclusion means that λl is a subset of {1, . . . , n} and the natural order of (1, . . . , n) is preserved. Let P(1, . . . , n) denote the collection of all partitions of (1, . . . , n) into disjoint sets λl , i.e. for λ ∈ P(1, . . . , n) we have λ = {λ1 , . . . , λr } for some r, where λl ⊆ (1, . . . , n), λl ∩ λl = ∅ for l = l and ∪rl=1 λl = {1, . . . , n}. Given a Wightman j functional W ∈ S and λl = (λ1l , . . . , λl ), we set W (λl ) = Wj (xλ1 , . . . , xλj ). l l With this definition at hand we can recursively define the truncated Wightman func T tional W ∈ S associated to W ∈ S via W0T = 0 and W (1, . . . , n) =

|λ|

W T (λl ) , n ∈ N,

(23)

λ∈P (1,... ,n) l=1

where |λ| is the number of sets λl in λ. We have the following proposition on the properties of W T : Proposition A.1. W fulfills Axioms 2.1 (A1)-(A4),(A5’),(A6) and (A7) if and only if W T fulfills (A1T): W0 = 0, W T ∈ S , (A2)-(A4), (A5’),(A7) and (A6T): limt→∞ W T (f ⊗ α {1,ta} g) = 0 for a ∈ Rd space like and f , g ∈ S with f0 = g0 = 0. Proof. The equivalence of (A1)/(A2)–(A4)/(A7) for W ⇔ (A1T)/(A2)–(A4)/(A7) for W T can be found e.g. in [10] pp. 492–493. (A6) for W ⇔ (A6T) for W T is well-known, for a detailed proof cf. [1] Sect. 4. (A5’) for W ⇔ (A5’) for W T is proven in [2, 21]. For continuous operators A : S1 → S1 we define An = A⊗n , A0 = 1 and we set : S → S setting A⊗ = ⊕∞ n=0 An . We get

A⊗

Lemma A.2. Let A : S1 → S1 be linear and continuous. Then W T ◦ A⊗ = (W ◦ A⊗ )T ∀W ∈ S . Since the scattering matrix can be considered as a bilinear functional on the Borchers’ algebra, we require a definition of truncation for these objects. By the Schwartz kernel theorem it is clear that there is a one to one correspondence of the bilinear functionals S on S with sets of tempered distributions {Sn,m }n,m∈N0 , where Sn,m ∈ Sn+m and S(f , g) = ∞ q 1 r 1 n,m=0 Sn,m (fn ⊗ gm ). For λl = (λl , . . . , λl ) ⊆ (1, . . . , n), νj = (νj , . . . , νj ) ⊆ (n + 1, . . . , n + m) we define S(λl , νj ) = Sr,q (xλ1 , . . . , xλrl ; xν 1 , . . . , xν q ). With this l

j

j

notation we define recursively the truncated bilinear functional S T associated with S via S(1, . . . , n; n + 1, . . . , n + m) =

|λ|

λ∈P (1,... ,n+m) l=1

> S T (λ< l , λl ).

(24)

508


> Here λ< l = λl ∩ (1, . . . , n) and λl = λl ∩ (n + 1, . . . , n + m). The truncation of linear and bilinear functionals is related as follows: Let ı⊗ be the injection of linear functionals into the bilinear functionals on S given by ı⊗ W (f , g) = W (f ⊗ g) ∀f , g ∈ S. Then we get from these definitions:

Lemma A.3. ı⊗ W T = (ı⊗ W )T ∀W ∈ S . B. Proof of Lemma 4.8 and Lemma 4.9 Proof of Lemma 4.8. We begin the proof of Lemma 4.8 with two auxiliary lemmas (for the definition of S1,2 , S1,2 cf. Sect. 4).

Lemma B.1. The Fourier transform is a continuous mapping from L1 (R, C) to (R, C). S1,2 Proof. We prove that F : S1,2 (R, C) → L1 (R, C) is continuous. Then the statement of the lemma follows by duality. The stated continuity property is established by the following estimate: Ff L1 (R,C)

−iξ t = (2π) f (ξ ) dξ dt e R R % & dt d2 −1/2 −iξ t = (2π) 1 − 2 f (ξ ) dξ e dξ 1 + t2 R R & % 2 1− d f (ξ ) dξ ≤ c f 2,2 , ≤ π(2π)−1/2 2 dξ R −1/2

for a sufficiently large constant c > 0 (here we have used

'

R dt/(1 + t

2)

(25)

= π ).

Let 1/ξ be defined as the Cauchy principal value of the function 1/ξ and the distribution 1/(ξ ± i0) as the boundary value of 1/(ξ ± i) for → +0. 1/ξ and 1/(ξ ± i0) are related via the Sokhotsky–Plemelji formula 1 1 = ∓ iπ δ(ξ ) , ξ ± i0 ξ

(26)

(R, C), since the cf. [12] p. 45. These distributions can be understood as elements on S1,2 Cauchy principle value is defined on S1,2 (R, C) by [12] p. 44 and the delta distribution of course also is defined on this space. Furthermore, the Fourier transform (in S1 (R, C)) of the step function 1{0≤±s} is

Fs (1{0≤±s} (s))(ξ ) = (2π )−1/2

∓i , ξ ∓ i0

see [12, p. 94]. (R, C). Lemma B.2. limt→+∞ e±iξ t /ξ = ±iπ δ(ξ ) in S1,2

(27)


509

Proof. We note that 1 ±iξ t 1 = e lim t→+∞ ξ ξ t→+∞

t

lim

= ±i(2π )

1/2

0

d ±iξ s ds + 1 e ds

1 lim F¯ s (1{0≤s≤t} )(±ξ ) + . t→+∞ ξ

Since by Lemma B.1 the (inverse) Fourier transform F¯ s is continuous from L1 (R, C) (R, C) and 1 1 to S1,2 {0≤s≤t} (s) → 1{0≤s≤∞} (s) as t → +∞ in L (R, C), we get for the r.h.s. of the above equation using also the formulae (26), (27):

1 · · · = ±i(2π )1/2 F¯ s (1{0≤s} )(±ξ ) + ξ 1 1 = ∓ ± − iπ δ(ξ ) + = ±iπ δ(ξ ). ξ ξ

Now we are in the position to prove Lemma 4.8. We only prove the lemma for a =out. The case a =in is in the same manner and the case a =loc is trivial. We note that the function f in the expression χt (out, k)f (k), f ∈ S1,2 , can be written as a sum of a function f1 with supp f1 ⊆ Rd+ = (0, ∞) × Rd−1 and supp f2 ⊆ Rd− = (−∞, 0) × Rd−1 . Here we only deal with the “positive frequency part” f1 , and identify f1 with the expression χ + f1 , which does not change f1 on the mass shell. Furthermore, we omit the index 1 in the following. Let thus f ∈ S1,2 with supp f ⊆ Rd+ . Then lim

t→+∞ Rd

ei(k −ω)t f (k) dk = lim t→+∞ k 2 − m2 0

= lim

(

Rd−1

t→+∞ Rd−1

)

R

ei(k −ω)t f (k) dk 0 k 2 − m2

R

eiξ t f (ξ + ω, k) dξ ξ ξ + 2ω

0

dk dk,

where we have used the change of variables k 0 → ξ = k 0 − ω in the last step. We note that f (ξ +ω, k)/(ξ +2ω) is in S1,2 (R, C) for k ∈ Rd−1 since the denominator ' (ξ +2ω) is smooth on the support of f (ξ + ω, k). Thus, if we can interchange the Rd−1 · · · dk + = δ(k 0 −ω)/2ω integral and the limit limt→+∞ we get the formula of Lemma 4.8 by δm and application of Lemma B.2. ' iξ t Let h ∈ S1,2 (Rd , C). We define gt (k) = R e ξ h(ξ, k) dξ . Using the product formula for the inverse Fourier transform on S (R, C) we get 1 ¯ ¯ |gt (k)| = 2π Fξ ( ) ∗ Fξ (h(ξ, k)) (t) ξ ¯ = 2π i(π − 1{t−x>0} (t − x))Fξ (h(ξ, k))(x) dx R ≤ 2π(π + 1) F¯ ξ (h(ξ, k))(x) dx R

≤ c1

sup

ξ ∈R,0≤l≤2

|(1 + ξ 2 )

h 2,d dl h(ξ, k)| ≤ c2 , l dξ (1 + |k|2 )d/2

510


for some c1 , c2 > 0 sufficiently large. Here we made use of the estimate (25) and we also applied the formulae (26) and (27). But this estimate shows that there is an integrable majorant for gt , t ∈ R, namely c/(1 + |k|2 )d/2 , and we may therefore interchange the limit t → +∞ and the integral over Rd−1 by the theorem of dominated convergence. Proof of Lemma 4.9. For notational convenience we only prove the lemma for j = 1. The proof for j = 2, . . . , n − 1 can be carried out analogously. By integrating over the variables k20 , . . . , kn0 and over k2 we obtain for the right-hand side of (20)

f (k1 , (ω2 , −k1 − R(d−1)(n−2)

n

l=3 kl ), (ω3 , k3 ), . . . n l=2 ωl

, (ωn , kn ))

× δ(k10 +

n

ωl ) dk3 · · · dkn .

l=2

Here ω2 = (|k1 +

n

2 l=3 kl |

h(k1 , k3 , . . . , kn ) =

+ m2 )1/2 . We set

f (k1 , (ω2 , −k1 −

n l=3 kl ), (ω3 , k3 ), . . . , (ωn , kn )) n , l=2 2ωl

and we get that h(k1 , k3 , . . . , kn ) ∈ S1,2 (Rd+(d−1)(n−2) , C) and h 2,L ≤ cL f 2,L for some cL > 0. We thus have to show that for such h, g(k) = h(k, k3 , . . . , kn )δ(ρ(k, k3 , . . . , kn ) + k 0 ) dk3 · · · dkn (28) R(d−1)(n−2)

defines a S1,2 -function g and that g 2,L ≤ cL h 2,L for cL > 0 sufficiently large, where we have set ρ(k, k3 , . . . , kn ) = nl=2 ωl . Using a smooth partition of unity which has bounded derivatives we can write h h1 we have |k| > 1 and asa sum of functions h1 , h2 , h3 , where on the support of | nl=3 kl | > 1, on the support of h2 we have |k| < 2, | nl=3 kl | > 1 and on the support of h3 we have |k| < 2, | nl=3 kl | < 2. By the boundedness of derivatives of the partition of unity, hj 2,L ≤ cL f 2,L holds for L ∈ N0 , j = 1, 2, 3 and sufficiently large cL > 0. We denote the functions associated to hj via Eq. (28) by gj , j = 1, 2, 3. Let us first consider the right hand side of Eq. (28) for h replaced by h1 . We introduce the variables Kj = nl=j kl for j = 3, . . . , n and we set cos θ3 = K3 · k/(|K3 ||k|) and cos θj = Kj −1 · Kj /(|Kj −1 ||Kj |) for j = 4, . . . , n. We then get ρ(k, k3 , . . . , kn ) = ρ(k, (|K3 |, cos θ3 ), . . . , (|Kn |, cos θn )) = (|k|2 + |K3 |2 + 2|k||K3 | cos θ3 + m2 )1/2 +

n−1

(|Kl |2 + |Kl+1 |2 − 2|Kl ||Kl+1 | cos θl+1 + m2 )1/2

l=3

+ (|Kn |2 + m2 )1/2 . If we now change variables k2 , . . . , kn → K3 , . . . , Kn in (28) and we then pass over to spherical coordinates (|Kl |, cos θl , Fl ), l = 3, . . . , n, where (cos θl , Fl ) are coordinates on the sphere S d−2 (and the surface element on S d−2 is denoted by d cos θl dFl )


we get

511

(S d−2 ×(0,∞))×n−2

h1 (k, (|K3 |, cos θ3 , F3 ), . . . , (|Kn |, cos θn , Fn )) (29)

× δ(ρ(k, (|K3 |, cos θ3 ), . . . , (|Kn |, cos θn )) + k 0 ) × d cos θ3 dF3 |K3 |d−2 d|K3 | · · · d cos θn dFn |Kn |d−2 d|Kn |, where we have written the function h1 as a function of the new variables. Using the formula 1 δ(x − y) δ(ρ(x) − a) = |ρ (y)| y:ρ(y)=a

which holds if ρ (y) = 0 if ρ(y) = a and setting d ρ(k, (|K3 |, cos θ3 ), . . . , (|Kn |, cos θn )) d cos θ3 |K3 ||k| = , (|k|2 + |K3 |2 + 2|k||K3 | cos θ3 + m2 )1/2

ϕ(k, K3 , cos θ3 ) =

we get for (29) (S d−2 ×(0,∞))×n−2

×

h1 (k, (|K3 |, cos θ3 , F3 ), . . . , (|Kn |, cos θn , Fn ))

δ(cos θ3 − ψ(k, |K3 |, (|K4 |, cos θ4 ), . . . , (|Kn |, cos θn ))) ϕ(k, K3 , cos θ3 )

× d cos θ3 dF3 |K3 |d−2 d|K3 | · · · d cos θn dFn |Kn |d−2 d|Kn |,

(30)

where ψ(k, |K3 |, (|K4 |, cos θ4 ), . . . , (|Kn |, cos θn )) n−1 0 = −k − (|Kl |2 + |Kl+1 |2 − 2|Kl ||Kl+1 | cos θl+1 + m2 )1/2 l=4

− (|Kn | + m ) 2

2 1/2

2

* − |k| − |K3 | − m 2

2

2

(2|k||K3 |)

is a smooth function on the set of arguments which are in the support of h1 . Furthermore, since |k|, |K3 | > 1 in the support of h1 , derivatives (∂ |α| /∂k α )ψ (∂ |α| /∂k α ) cos θ also are bounded on the support of h1 for any multiindex α. We now set h˜ 1 = h1 /ϕ and we get that h˜ 1 ∈ S1,2 (Rd+(d−1)(n−2) , C) with h˜ 1 2,L ≤ cL h1 2,L for some cL > 0, l ∈ N0 . Consequently, we get for a multinindex α with |α| = 0, 1, 2, |α| ∞ 1 ∂ d−2 ˜ δ(cos θ − ψ) h d cos θ |K | d|K | 3 1 3 3 3 ∂k α 0 −1 n 1 1 ≤ c h˜ 1 2,L , 2 L/2 (1 + |k| ) (1 + |Kl |2 )d/2 l=4

512


for c sufficiently large and L = max{L, d}. If we insert this estimate into (30), we get h g1 2,L ≤ cL h˜ 1 2,L ≤ cL 2,L . If we can prove similar estimates for g2 , g3 , the proof is finished. This is simple for g2 : We consider h2 and g2 as functions of the new variable k = k+a for some a ∈ Rd−1 with |a| ≥ 3. Then function h2 in these new variables fulfills the same conditions as h1 before and we get the desired estimate. It remains to show the estimate for g3 . Let k, K3 , . . . , Kn be the coordinates introduced above. We define the vector field b = b(Kn ) = 3Kn /|Kn | and we introduce new variables k = k − b, Kl = Kl + b, l = 3, . . . , n. In the polar coordinates ||K |) we then get for |k |, |Kl |, cos θ3 = k · K3 /(|k ||K3 |), cos θl = Kl−1 · Kl /(|Kl−1 l ρ(k, k3 , . . . , kn ): (|k |2 + |K3 |2 + 2|k ||K3 | cos θ3 + m2 )1/2 +

n−1 l=3

(|Kl |2 + |Kl+1 |2 − 2|Kl ||Kl+1 | cos θl+1 + m2 )1/2

+ ((|Kn | + 3)2 + m2 )1/2 and we can proceed as before, since |k |, |K3 | > 1 on the support of h3 .

Acknowledgements. We thank C. Becker, D. Buchholz, S. Doplicher, R. Gielerak, O. W. Greenberg, K. Iwata, T. Kolsrud, G. Morchio, F. Strocchi and J.-L. Wu for interesting discussions. This work was made possible through financial support of D.F.G., SFB 237 and the “Hochschulsonderprogramm III” of the federation and lands of Germany via a D.A.A.D. scholarship for the second named author.

References 1. Albeverio, S., Gottschalk, H., Wu, J.-L.: Convoluted generalized white noise, Schwinger functions and their continuation to Wightman functions. Rev. Math. Phys. 8, No. 6, 763 (1996) 2. Albeverio, S., Gottschalk, H., Wu, J.-L.: Models of local relativistic quantum fields with indefinite metric (in all dimensions). Commun. Math. Phys. 184, 509 (1997) 3. Albeverio, S., Gottschalk, H., Wu, J.-L.: Nontrivial scattering amplitudes for some local relativistic quantum field models with indefinite metric. Phys. Lett. B 405, 243 (1997) 4. Albeverio, S., Gottschalk, H., Wu, J.-L.: Scattering behaviour of quantum vector fields obtained from Euclidean covariant SPDEs. Rep. on Math. Phys. 44, 1/2, 21–28 (1999) 5. Albeverio, S., Iwata, K., Kolsrud, T.: Random fields as solutions of the inhomogenous quaternionic Cauchy–Riemann equation. I. Invariance and analytic continuation . Commun. Math. Phys. 132, 550 (1990) 6. Araki, H.: On a pathology in indefinite inner product spaces. Commun. Math. Phys. 85, 121 (1982) 7. Baumgärtel, H., Wollenberg, M.: A class of nontrivial weakly local massive Wightman fields with interpolating properties. Commun. Math. Phys. 94, 331 (1984) 8. Becker, C., Gielerak, R., Ługiewicz, P.: Covariant SPDEs and quantum field structures. J. Phys. A 31, 231–258 (1998) 9. Bleuler, K.: Eine neue Methode zur Behandlung der longitudinalen und skalaren Photonen. Helv. Phys. Acta 23, 567 (1950) 10. Bogulubov, N.N., Logunov, A.A., Ossak, A.I., Todorov, I.T.: General principles of quantum field theories. Amsterdam: Kluwer Academic Publishers, 1990 11. Borchers, H.-J.: Algebraic aspects of Wightman field theory. In: Statistical Mechanics and Field Theory, Ed. R.N. Sen and C. Weil, Jerusalem/London: Halsted Press, New York/Israel Universities Press, 1972, pp. 31–80 12. Contantinescu, F.: Distributionen und ihre Anwendung in der Physik. Stuttgart: Teubner, 1973 13. Doplicher, S.: An algebraic spectrum condition. Commun. Math. Phys. 1, 1 (1965) 14. Epstein, H.: Some analytic properties of scattering amplitudes in quantum field theory. In: Axiomatic quantum field theory, Proc. 1965 Brandeis University Summer Scool on Theoret. Phys., Ed. M. Chretien, S. Deser, New York: Gordon and Breach, 1966


15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

513

Fadeev, L.D., Popov, V.N.: Feynman diagrams for the Yang–Mills field. Phys. Lett. B 25, 29 (1967) Gottschalk, H.: Green’s functions for scattering in local relativistic QFT. Dissertation, Bochum 1998 Gottschalk, H.: A characterization of locality in momentum space. Lett. Math. Phys.50, 259–273 (1999) Gupta, S.N.: Theory of longitudinal photons in quantum electrodynamics. Proc. Phys. Soc. A 63, 681 (1950) Haag, R.: Quantum field theories with composite particles and aymptotic condition. Phys. Rev. 112, 669 (1958) Hepp, K.: On the connection between the LSZ and Wightman quantum field theory. Commun. Math. Phys. 1, 95 (1965) Hoffmann, G.: The Hilbert space structure condition for quantum field theories with indefinite metric and transformations with linear functionals. Lett. Math. Phys. 42, 281 (1997) Hoffmann, G.: On GNS representations on inner product spaces: I. The structure of the representation space. Commun. Math. Phys. 191, 299 (1998) Lehmann, H., Symanzik, K., Zimmermann, W.: Zur Formulierung quantisierter Feldtheorien. Il Nuovo Cimento 1, 205 (1954) Johnson, G.E.: Interacting quantum fields. Rev. Math. Phys. 11 7, 881–928 (1999) with Erratum: A Comment on Interacting quantum fields. Rev. Math. Phys. 12, 687–689 (2000) Mintchev, M.: Quantization in indefinite metric. J. Phys. A 13, 1841–1859 (1980) Morchio, G., Strocchi, F.: Infrared singularities, vacuum structure and pure phases in local quantum field theory. Ann. Inst. H. Poincaré, 33, 251 (1980) Reed, M., Simon, B.: Methods of modern mathematical physics. Vol. II + III, San Diego: Academic Press, 1979 Ruelle, D.: On the asymptotic condition in quantum field theory. Helv. Phys. Acta 35, 147 (1962) Scheibe, E.: Über Feldtheorien in Zustandsräumen mit indefiniter Metrik. Max-Planck-Institut für Physik und Astrophysik, München, 1960 Schneider, W.: S-Matrix und interpolierende Felder. Helv. Phys. Acta 39, 81 (1966) Streater, R.F., Wightman, A.S.: PCT, spin and statistics, and all that. New York, Amsterdam: Benjamin, 1964 Strocchi, F.: Selected topics on the general properties of quantum field theory. Lecture Notes in Physics 51, Singapore–New York–London–Hong Kong: World Scientific, 1993 Weinberg, S.: The quantum theory of fields. Vol. I, Cambridge: Cambridge Univ. Press, 1995 Yang, C.N., Feldman, D.: The S-matrix in the Heisenberg representation. Phys. Rev. 79, 972 (1950) Yngvason, J.: On the algebra of test functions for field operators. Commun. Math. Phys. 34, 315 (1973)

Communicated by H. Araki

Commun. Math. Phys. 216, 515 – 537 (2001)

Communications in



Gaussian Random Matrix Models for q-deformed Gaussian Variables ´ Piotr Sniady Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland. E-mail: [email protected] Received: 29 March 2000 / Accepted: 1 August 2000

Abstract: We construct a family of random matrix models for the q-deformed Gaussian random variables Gµ = aµ + aµ , where the annihilation operators aµ and creation operators aν fulfill the q-deformed commutation relation aµ aν − qaν aµ = µν , µν is the covariance and 0 < q < 1 is a given number. An important feature of the considered random matrices is that the joint distribution of their entries is Gaussian. 1. Introduction 1.1. The deformed Gaussian variables. The q-deformed Gaussian random variables Gµ = aµ + aµ , where operators aµ and their adjoints aµ fulfill deformed commutation relations aµ aν − qaν aµ = νµ 1

(1)

were introduced by Bourret and Frisch [FB]. These operators act on a Hilbert space K which has a unital vector , called a vacuum, with the property that aµ = 0

(2)

for every value of the index µ. With the help of the vector one can introduce a state τ on the algebra of operators acting on K as follows: τ (X) = , X . The state τ plays the role of the non-commutative expectation value. From (1) and (2) it follows [BS1] that for any m ∈ N and any indexes µ1 , . . . , µ2m we have that τ (Gµ1 · · · Gµ2m−1 ) = 0, τ (Gµ1 · · · Gµ2m ) =

π

q i(π) c1 d1 · · · cm dm ,

(3) (4)

´ P. Sniady

516

where the sum is taken over all pair partitions π = {c1 , d1 }, . . . , {cm , dm } of the set {1, . . . , 2m} and i(π ) is the number of crossings of the partition π . For the reader’s convenience we shall recall definitions of a pair partition and of its number of crossings in Sect. 3. From the quantum probability point of view all the information about non-commutative random variables Gµ is encoded in their moments τ (Gµ1 · · · Gµm ), and therefore Eq. (3) and (4) can be treated as an alternative definition of q-deformed Gaussian variables Gµ . 1.1.1. Applications of deformed Gaussian variables. Equations (3) and (4) show that for q = 1 operators Gµ have the same moments as classical Gaussian variables with mean zero and covariance µν , which should explain why we call Gµ deformed Gaussian variables. Equation (1) for q = 1 is called the canonical (or bosonic) commutation relation. For other special choices of the deformation parameter q variables Gµ also have natural probabilistic interpretations [FB], namely as increments of a dichotomic Markov process (for q = −1) or as Wigner’s large random matrices (for q = 0). Voiculescu [V1] has made a remarkable observation that for q = 0 random variables Gµ are free semicircular elements (an analogue of independent Gaussian variables in the free probability theory of Voiculescu [V3,VDN]). Equation (1) for q = −1 is called the canonical anticommutation relation (or fermionic relation) and for q = 0 is called the free relation. Therefore it was natural to expect that the relations (3) and (4) which are a simple generalisation of the three mentioned above: bosonic, fermionic and free cases, would give rise to interesting probabilistic objects. Indeed, it was observed by Bo˙zejko and Speicher [BS1] that related to Eq. (1) Brownian motion is a one component of an n-dimensional Brownian motion which is invariant under the quantum group S√q U (n) of Woronowicz for 0 < q < 1. Another application of q-deformed Gaussian variables, this time as generalised quantum statistics, was proposed by Greenberg [Gr] and Speicher [Sp2]. The existence of operators aµ and aµ fulfilling deformed commutation relations (1) was proven by Bo˙zejko and Speicher [BS2]. Later it was proven by Bo˙zejko, Kümmerer, and Speicher [BKS] that the von Neumann algebra generated by q-deformed Gaussian variables G1 , G2 , . . . (−1 < q < 1) is a II1 factor. There are today many open questions concerning these factors, particularly if they are different from the free group factors. In this paper we present a natural probabilistic representation of the q-deformed Gaussian variables for all q ∈ [0, 1] as some random matrices, which was one of the open questions posed in the paper [FB]. A remarkable property of our model is that the joint distribution of entries of our matrices is Gaussian. Recently a related problem of finding a random matrix model for the so-called qdeformed circular system was solved by Mingo and Nica [MN]. 1.1.2. The covariance µν . Indexes µ, ν are elements of a certain set M. A necessary and sufficient condition for operators Gµ to exist is that the function µν is positive definite [BS2], i.e. αi αj µi µj ≥ 0 1≤i,j ≤n

for all α1 , . . . , αn ∈ R and µ1 , . . . , µn ∈ M. Typical examples of sets M and covariance functions are:

Gaussian Random Matrix Models for q-deformed Gaussian Variables

517

– M = N and i,j = δij . For q = 1 we have that G1 , G2 , . . . is a sequence of independent, standard Gaussian variables while for q = 0 we have that G1 , G2 , . . . is a sequence of free semicircular elements [VDN]. – M = R+ and t,s = min(t, s). For q = 1 we have that Gt is a Brownian motion, for q = 0 we have that Gt is a noncommutative stochastic process with free increments – M is a real Hilbert space and the covariance is defined by the scalar product: φψ = φ, ψ. The case M = L2 (R+ ) is often used in white noise calculus. 1.1.3. The distribution of a deformed Gaussian variable. A distribution of a random variable corresponding to the bounded selfadjoint operator G is a measure ν supported on the real line R such that τ (Gn ) = x n dν(x) for all n ∈ N. It can be shown [Sz] that the distribution νq of a q-deformed Gaussian variable with 2 2 the variance equal to 1 is given by a measure νq supported on the interval − √1−q , √1−q with a density

νq (dx) =

∞ 1 1 − q sin θ (1 − q n )|1 − q n e2iθ |2 dx, π n=1

where x=√

2 cos θ 1−q

with θ ∈ [0, π ]. 1.1.4. Canonical commutation relations and Itô’s formula. It is not merely an accident that for q = 1 there is a correspondence between the commutation relation (1) and Gaussian random variables. If we consider a probability space generated by a Brownian motion B(t) then every real random variable X with a finite second moment can be uniquely expressed as a series of iterated Itô integrals X=X

(0)

+

∞ i=1

0≤t1 ≤···≤ti j , are defined by the hermitianity condition Mij = Mj i . One can show that both definitions do not depend on the choice of the orthonormal basis of V .


523

For each A ⊆ {1, . . . , N} let us consider a family of hermitian standard random (N),A,µ ∈ End(HA ) indexed by µ ∈ M such that the entries of different matrices R1 matrices are independent. We define a family of random matrices R (N),A,µ by (N),A,µ R (N),A,µ = jÃ R1 ⊗ 1HA ) ∈ End(H(N) ,

where 1HA : HA → HA denotes the identity operator. Intuitively speaking, a matrix R (N),A,µ consists of d N−|A| copies of a d |A| × d |A| standard hermitian random matrix. As one can see, matrices R A,µ are hermitian and the joint distribution of their entries is Gaussian, but different entries need not be independent. We have: A,µ

Rij

A,µ E Rij = 0,

A,µ

= Rji ,

and from (7) it follows that A,µ B,ν A,µ B,ν (8) = E Rij Rlk E Rij Rkl [ir = lr ][jr = kr ] = δAB δµν [ir = jr ][kr = lr ] . d r∈A

r∈A

3.5. Tensors T . The formula (8) can be written shorter if we introduce for all A ⊆ A,r {1, . . . , N} and 1 ≤ r ≤ N tensors Tij,kl as follows: A,r Tij,kl

1 =

d [i

= l][j = k]

[i = j ][k = l]

if r ∈ A if r ∈ A

.

We define TA ij,kl =

r

TiA,r , r jr ,kr lr

(9)

which with a small abuse of notation can be written as TA = T A,1 ⊗ · · · ⊗ T A,N .

(10)

A,µ B,ν A,µ B,ν E Rij Rkl = δAB δµν TA = E Rij Rlk ij,kl .

(11)

Then (8) can be written as

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524

3.6. Examples. First of all note that for the trivial case d = 1 all Hilbert spaces are one dimensional and all random matrices R A,µ are in fact scalar random Gaussian variables. In general case d ≥ 2, the random matrix R (N),µ,∅ is simply a scalar real random Gaussian variable multiplied by an identity matrix. The random matrix R (N),µ,{1,...,N} is a hermitian standard random matrix from Definition 2. There is a correspondence between sequences i = (i1 , . . . , iN ) such that 0 ≤ i1 , . . . , iN ≤ d − 1 and the set of integer numbers {0, 1, . . . , d N − 1} given by the digit representation of natural numbers in the system with base d: i = (i1 , . . . , iN ) → i1 + di2 + · · · + d N−1 iN . Therefore we can introduce an orthonormal basis g0 , . . . , gd N −1 of H(N) indexed by integer numbers: gi1 +di2 +···+d N −1 iN = fi1 ⊗ · · · ⊗ fiN = e(i1 ,...,iN ) , where 0 ≤ i1 , . . . , iN ≤ d − 1. In the following, if we want to write an endomorphism M ∈ End(H(N) ) as a matrix (Mij )0≤i,j ≤d N −1 we shall do it in the basis (gi )0≤i≤d N −1 . For d = 2 and N = 2 the matrices R (N),µ,A are of the following form:   a00 a01 0 0 a 0 0   a , R {1} =  10 11 0 0 a00 a01  0 0 a10 a11   b00 0 b01 0  0 b00 0 b01  , R {2} =  b10 0 b11 0  0 b10 0 b11

a00 a01 b00 b01 where and are standard hermitian random matrices. Entries a10 a11 b10 b11 of the first matrix are by definition independent of the entries of the second matrix. The index µ was omitted; however it should be understood that for different values of µ the entries of matrices are independent. For d = 2 and N = 3 we have:   c00 c01 0 0 0 0 0 0  c10 c11 0 0 0 0 0 0     0 0 c00 c01 0 0 0 0    0 c10 c11 0 0 0 0   0 R {1} =  , 0 0 0 c00 c01 0 0    0   0 0 0 0 c10 c11 0 0    0 0 0 0 0 0 c00 c01  0 0 0 0 0 0 c10 c11   d00 0 d01 0 0 0 0 0  0 d00 0 d01 0 0 0 0     d10 0 d11 0 0 0 0 0    0 0 0   0 d10 0 d11 0 , R {2} =  0 0 0 d00 0 d01 0    0   0 0 0 0 0 d00 0 d01    0 0 0 0 d10 0 d11 0  0 0 0 0 0 d10 0 d11




R {3}

     =    

e00 0 0 0 e01 0 0 0 0 0 e01 0 0 0 e00 0 0 0 e00 0 0 0 e01 0 0 0 e01 0 0 0 e00 0 e10 0 0 0 e11 0 0 0 0 e10 0 0 0 e11 0 0 0 0 e10 0 0 0 e11 0 0 0 0 e10 0 0 0 e11

525

      ,    

where again (cpq )0≤p,q≤1 , (dpq )0≤p,q≤1 , (epq )0≤p,q≤1 are standard hermitian random matrices. Furthermore   f00 f01 f02 f03 0 0 0 0  f10 f11 f12 f13 0 0 0 0     f20 f21 f22 f23 0 0 0 0    f f f 0 0 0 0   f , R {1,2} =  30 31 32 33 0 0 0 f00 f01 f02 f03    0   0 0 0 0 f10 f11 f12 f13    0 0 0 0 f20 f21 f22 f23  0 0 0 0 f30 f31 f32 f33   g00 g01 0 0 g02 g03 0 0  g10 g11 0 0 g12 g13 0 0     0 0 g00 g01 0 0 g02 g03    0 g10 g11 0 0 g12 g13   0 R {1,3} =  , 0 g22 g23 0 0    g20 g21 0   g 0 g32 g33 0 0   30 g31 0  0 0 g20 g21 0 0 g22 g23  0 0 g30 g31 0 0 g32 g33   h00 0 h01 0 h02 0 h03 0  0 h00 0 h01 0 h02 0 h03     h10 0 h11 0 h12 0 h13 0     0 h10 0 h11 0 h12 0 h13  R {2,3} =  ,  h20 0 h21 0 h22 0 h23 0   0 h 0 h21 0 h22 0 h23  20    h30 0 h31 0 h32 0 h33 0  0 h30 0 h31 0 h32 0 h33 where (fpq )0≤p,q≤3 , (gpq )0≤p,q≤3 , (hpq )0≤p,q≤3 are standard hermitian random matrices. 3.7. The case of a general covariance µν . By a small change of definition of the matrices R we obtain a more general case. Let µν be a real positive definite function. For every A ⊆ {1, . . . , N} we consider a (N),A,µ ∈ End(H ) such that for each pair family of random (non-hermitian) matrices R0 (N),A,µ A µ ∈ M is Gaussian, of indexes i, j we have that the joint distribution of R0 ij (N),A,µ E R0 = 0, ij

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526

the covariance of real and imaginary parts are defined by the function : (N),A,ν (N),A,ν (N),A,µ (N),A,µ R ! R0 = E ! R = E R0 0 0 ij ij ij ij

1 µν , 2d |A|

and the real and imaginary parts are independent: (N),A,ν (N),A,µ ! R0 = 0. E R0 ij ij For different choices of sets A or a pair of indexes (i, j) the random variables (N),A,µ should be independent. R0 ij We define hermitian random matrices (N),A,µ (N),A,µ (N),A,µ = R0 + R0 . R1 Note that for the simplest choice of a positive definite function µν = δµν this definition (N),A,µ coincides with the definition from Subsect. 3.4. of random matrices R1 Similarly as in Subsect. 3.4 we define (N),A,µ R (N),A,µ = jÃ R1 ⊗ 1H A . The joint distribution of entries of hermitian matrices R (N),A,µ is Gaussian and E R A,µ ij = 0, A,µ B,ν A,µ B,ν E Rij Rkl = δAB µν TA = E Rij Rlk ij,kl .

(12)

4. The Main Theorem We define a family of random matrices indexed by µ ∈ M, (N) σA R (N),A,µ , S (N),µ = A⊆{1,...,N}

where σ (N) is a real-valued function on the set of all subsets of {1, . . . , N}. Matrices S (N) are hermitian and the joint distribution of their entries is Gaussian. Alternatively one can define these matrices by giving the mean and the covariance of the entries: we have µ

E[Sij ] = 0,

(13)

µ ν µ ν = µν E[Sij Skl ] = E Sij Slk (σA )2 TA ij,kl .

(14)

A

In the following theorem we show conditions which the sequence of functions (σ (N) ) needs to fulfill. Since these conditions may seem quite disgusting, we would like to give some hope to the reader by pointing to Eq. (15), which gives a simple example of a covariance function fulfilling all assumptions.


527

Theorem 1. If for each N ∈ N we have that σ (N) is a real-valued function on the set of all subsets of {1, . . . , N} such that: 1. (Normalisation) For each N ∈ N we have (N) 2 = 1. σA A⊆{1,...,N}

2. (Triple coincidations are rare) lim N→∞

A1 ,A2 ,A3 ⊆{1,...,N } A1 ∩A2 ∩A3 =∅

(N) 2 (N) 2 (N) 2 σA2 σA3

σA1

= 0.

3. (Distribution of coincidations) There exists a sequence (pi )i≥0 of nonnegative real numbers such that i≥0 pi = 1 and for any k ∈ N and any nonnegative integer numbers nij , 1 ≤ i < j ≤ k we have (N) 2 (N) 2 lim · · · σAk = pnij . σA1 N→∞

4. For each n ∈ N, lim

N→∞

Then for q =

∞

1≤i<j ≤k

A1 ,...,Ak ⊆{1,...,N } |Ai ∩Aj |=nij for any 1≤i<j ≤k

A1 ,...,An ⊆{1,...,N}

1 i=0 pi d 2i

(N) 2

σA1

(N) 2 · · · σAn

1 d 2|A1 \(A2 ∪···∪An )|

= 0.

we have

lim E[tr S (N),µ1 · · · S (N),µn ] =

N→∞

π

q i(π) µc1 µd1 · · · µcm µdm = τ [Gµ1 · · · Gµn ]

for every n ∈ N and µ1 , . . . , µn ∈ M, where Gµ1 , . . . , Gµn are q-deformed Gaussian variables with covariance . Before we prove this theorem we would like to make some remarks and state auxiliary lemmas. Remark 1. For a given function σ (N) we define a measure ρ (N) on the set of all subsets (N) of {1, . . . , N} by assigning to set A the weight (σA )2 . Then the first three assumptions of the theorem can be reformulated as follows: 1. For each N ∈ N the measure ρ (N) is probabilistic. (N) (N) (N) 2. If for each N ∈ N we have that A1 , A2 , A3 are independent random variables (N) with distribution given by the measure ρ (N) then the probability of the event: A1 ∩ (N) (N) A2 ∩ A3 = ∅ tends to 0 as N tends to infinity. (N) (N) 3. Let k be a fixed natural number. If for each N ∈ N we have that A1 , . . . , Ak are independent random variables with distribution given by the measure ρ (N) , then the joint distribution of the random variables |Ai ∩ Aj |, 1 ≤ i < j ≤ k tends to a product distribution as N tends to infinity. The limit distribution of a single variable |Ai ∩ Aj | is given by (N) (N) lim P |Ai ∩ Aj | = k = pk . N→∞

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528

Remark 2. Assumption 4 follows from other assumptions, however the proof of this is rather technical and we omit it. Remark 3. We would like to point out an interesting informal connection between stochastical properties of Gi regarded as large matrices and their entries. Let us consider µν = δµν . For q = 1 we have that Gµ is a family of independent Gaussian variables and for q = 0 we have that Gµ is a family of free non-commutative random variables. Freeness is an analogue of classical independence; we can expect therefore that in a general case −1 ≤ q ≤ 1 variables Gµ are independent in some generalised way. However, it was proven by Speicher [Sp3] that there are only three generalisations of the notion of independence of random variables to the non-commutative setup which would satisfy certain natural properties. These three generalisations are: classical independence, free independence (freeness) and boolean independence. Therefore except the cases q ∈ {0, 1} which correspond to the free and the classical situation respectively we cannot formally say that the non-commutative random variables Gµ are independent in some sense. We can of course weaken Speicher’s axioms and treat this “independence” on a very informal level. It is worth pointing out that a family of “independent” variables Gi is asymptotically represented as random matrices such that the entries of different matrices are classically independent random variables. Similarly, for the choice of the covariance function ts = min(t, s) for t, s ≥ 0 we obtain a non-commutative stochastic process Gt which can be regarded as some kind of a Brownian motion [BS1] and Gt can be asymptotically represented as a matrix valued stochastic process. Entries of this matrix are classical Brownian motions. Remark 4. The assumptions of the theorem are fulfilled for the following important examples of the functions σ : Proposition 1. For every real c > 0 the sequence of functions defined by (N) 2 = σA

c √ N

|A|

c 1− √ N

N−|A|

fulfills the assumptions of Theorem 1 with the sequence pk = Poisson distribution with parameter c2 and

q=e

− 1−

1 d2

1 2k −c2 k! c e

being the

c2

.

In this case the covariance (14) takes a beautiful form µ ν µ ν E Sij Skl = E Sij Slk (15)

c [ir = lr ][jr = kr ] c = µν + 1− √ [ir = jr ][kr = lr ] . √ d N N r Proof of this proposition will be presented in Sect. 6.


529

Proposition 2. For every real number c > 0 the sequence of functions σ (N) defined for N sufficiently large by √ 1 if |A| = "c N # (N) 2 N √ , = ("c N#) σA 0 otherwise where "x# denotes the integer part of a real number x, fulfills the assumptions of Theo2 rem 1 with pk = k!1 c2k e−c and q=e

−(1−

1 )c2 d2

.

Since proof of this proposition is similar to the proof of Proposition 1 we skip it. Lemma 1. For any pair partition π = {c1 , d1 }, . . . , {cm , dm } of the set {1, . . . , 2m} and any sets A1 , . . . , A2m ⊆ {1, . . . , N} we have Ac 1 0≤ N Ticv vicv +1 ,idv idv +1 ≤ 1. d 1 2m i ,...,i

1≤v≤m

If furthermore Aci ∩ Acj ∩ Ack = ∅ for all 1 ≤ i < j < k ≤ m, then 1 dN

i1 ,...,i2m 1≤v≤m

A

Ticvcvicv +1 ,idv idv +1 =

1≤i<j ≤m lines {ci ,di },{cj ,dj } cross

1 d

2|Aci ∩Acj |

.

The proof of this lemma will be presented in Sect. 6. The following lemma states a well known property of the Gaussian distribution. Lemma 2. If the joint distribution of random variables (Xk ) is Gaussian and E[Xk ] = 0 then E(X1 · · · X2m−1 ) = 0, E(X1 · · · X2m ) = E(Xc1 Xd1 ) · · · E(Xcm Xdm ), π

where the sum is taken over all pair partitions π = {c1 , d1 }, . . . , {cm , dm } of the set {1, . . . , 2m}. With this preparation we are able to start the proof of the main theorem. Proof of Theorem 1. In the following the sums over π are taken over all pair partitions π = {c1 , d1 }, . . . , {cm , dm } of the set {1, . . . , 2m} and sums over A1 , . . . , An are taken over all subsets of the set {1, . . . , N}. Products over v are taken over 1 ≤ v ≤ m. From Lemma 2 follows that for any m ∈ N and indexes µ1 , . . . , µ2m ∈ M we have: E tr S (N),µ1 · · · S (N),µ2m−1 = 0 and furthermore U (N) := E tr S (N),µ1 · · · S (N),µ2m  (N) (N) σA1 · · · σA2m  = dN π A1 ,...,A2m

i1 ,...,i2m v

 A δAcv Adv µcv µdv Ticvcvicv +1 ,idv idv +1  .

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530

We define V

(N)

:=

π A1 ,...,A2m

v

×

(N)

(N)

δAcv Adv µcv µdv σA1 · · · σA2m 1


d

2|Aci ∩Acj |

.

Lemma 1 shows that the corresponding summands in the definitions of U and V are equal unless there are some indexes 1 ≤ p < q < r ≤ m such that Acp ∩Acq ∩Acr = ∅. There are m3 choices of these indexes and again from Lemma 1 and Assumption 1 we have (N) 2 (N) 2 m σA1 · · · σAm |U (N) − V (N) | ≤ C 3 A ,A ,A ⊆{1,...,N } m =C 3 A

A4 ,...,Am ⊆{1,...,N}

1 2 3 A1 ∩A2 ∩A3 =∅

(N) 2 (N) 2 (N) 2 σA2 σA3 ,

σA1

1 ,A2 ,A3 ⊆{1,...,N } A1 ∩A2 ∩A3 =∅

m where C = max µp ,µq and therefore from Assumption 2 we have lim |U (N) − V (N) | = 0.

N→∞

We have V

(N)

=

π

v

×

µcv µdv

A1 ,...,Am

=

π

v

µcv µdv

(N) 2

σA1

(N) 2 · · · σAm

1


d

2|Aci ∩Acj |


1 d 2nij

dλ(N) (nij ),

where measures (λ(N) ) are defined on the set of all sequences (nij )1≤i<j ≤m , nij ∈ {0, 1, 2, . . . } by condition (N) 2 (N) 2 λ(N) {(nij )} = σA1 · · · σAN . A1 ,...,Ak ⊆{1,...,N } |Ai ∩Aj |=nij for all 1≤i<j ≤m

From Assumption 3 it follows that this sequence converges pointwise to the product measure defined on the atoms by λ {(nij )} = pij . 1≤i<j ≤m

Since measures λ(N)

and the measure λ are probabilistic, this convergence is uniform and the statement of the theorem follows. $ %


531

5. The Almost Surely Convergence As a simple corollary of Lemma 2 we have Lemma 3. If the joint distribution of random variables X1 , . . . , X2m is Gaussian and E[Xk ] = 0 for all 1 ≤ k ≤ 2m then E[Xcv Xdv ], E[X1 · · · X2m ] − E[X1 · · · Xm ] E[Xm+1 · · · X2m ] = π 1≤v≤m

where the sum is taken over pair partitions π = {c1 , d1 }, . . . , {cm , dm } of the set {1, . . . , 2m} which have the additional property that there exist x ∈ {1, . . . , m} and y ∈ {m + 1, . . . , 2m} such that {x, y} ∈ π . We define a permutation σ : {1, . . . , 2m} → {1, . . . , 2m} by σ (k) = k + 1 for k ∈ {m, 2m}, σ (m) = 1, σ (2m) = m + 1. Lemma 4. Let π = {c1 , d1 }, . . . , {cm , dm } be a pair partition as in Lemma 3, i.e. for some 1 ≤ k ≤ m we have ck ∈ {1, . . . , m}, dk ∈ {m + 1, . . . , 2m}. Then Ac 1 1 v ≤ T . c σ (c ) d σ (d ) v v v v d 2N i i ,i i 2|Ak \ j =k Aj | d 1 2m i ,...,i

1≤v≤m

This lemma follows directly from Lemmas 5 and 7 from Sect. 6. Proposition 3.

Var tr S (N),µ1 · · · S (N),µm ≤ C

A1 ,...,Am ⊆{1,...,N}

(N) 2

σA1

(N) 2 · · · σAm

1 , d 2|A1 \(A2 ∪···∪Am )|

where C = (2m)!! max |µp ,µq |m . Proof. We define µm+k = µk . In the following sums over π ´ are taken over all pair partitions π = {c1 , d1 }, . . . , {cm , dm } of the set {1, . . . , 2m} with the additional property that there exist x ∈ {1, . . . , m} and y ∈ {m + 1, . . . , 2m} such that {x, y} ∈ π . From Lemma 3 we have that Var tr S (N),µ1 · · · S (N),µm 2 (N),µ 1 · · · S (N),µm 2 − E tr S = E tr S (N),µ1 · · · S (N),µm µ 1 µm−1 µm µm−1 µ1 µm E Si1 i12 · · · Sim−1 = 2N m Sim i1 Sim+1 im+2 · · · Si2m−1 i2m Si2m im+1 i d i1 ,...,i2m

µ1 µm−1 µm µ1 µm−1 µm − E Si1 i2 · · · Sim−1 im Sim i1 E Sim+1 im+2 · · · Si2m−1 i2m Si2m im+1 = =

1 d 2N

π i1 ,...,i2m 1≤v≤m

1 d 2N

π

µ µ E Sicvcviσ (cv ) Sidvdviσ (dv )

A1 ,...,A2m ⊆{1,...,N} i1 ,...,i2m

(N)

(N)

σA1 · · · σA2m

´ P. Sniady

532

×

A

1≤v≤m

[Acv = Adv ] µcv ,µdv Ticvcviσ (cv ) ,idv iσ (dv )

≤C

A1 ,...,Am ⊆{1,...,N}

(N) 2

σA1

(N) 2 · · · σAm

1 , d 2|A1 \(A2 ∪···∪Am )|

where in the last inequality we used Lemmas 5 and 7.

% $

If (Mij )1≤i,j ≤K is a hermitian matrix with eigenvalues λ1 , . . . , λK we define a probabilistic measure νM on the real line R by the following: νM =

1 δλn . K 1≤n≤K

Theorem 2. If the assumptions of Theorem 1 are fulfilled then there exists an increasing sequence of natural numbers (Ni ) such that the sequence of measures νS (Ni ) almost surely converges weakly to the measure νq . Since for 0 ≤ q < 1 the support of the limit measure νq is compactly supported, the theorem follows from the following stronger statement. Theorem 3. If the assumptions of Theorem 1 are fulfilled then there exists a sequence (Ni ) such that the limit lim tr S (Ni ),µ1 · · · S (Ni ),µm = τ (Gµ1 · · · Gµm )

m→∞

holds almost surely. Proof. Our goal is to construct a sequence (Ni ) such that 2 E tr S (Ni ),µ1 · · · S (Ni ),µm − τ (Gµ1 · · · Gµm ) 0}, and assume that ϕ0 = id. (The formulae are slightly simpler in this case.) Then the potential A(t, z) in Loewner’s equations (1.3) and (1.7) has the form dµt (ξ ) . A(t, z) = R ξ −z Differentiating the inverse equation (1.7) with respect to z, and then integrating over [0, t], we have t [A (s, w(s))] ds, (1.19) log |ϕ (t, z)| = 0

where w(s) := w(s; t, z) is the corresponding characteristic. Similarly, we find t $[ϕ(t, z)] = M(s, w(s)) ds, (1.20) log $z 0 where

M(s, x + iy) =

dµs (ξ ) . (ξ − x)2 + y 2

(1.21)

Note that the sum K := [A ] + M of the kernels in (1.19) and (1.20) is positive: 2(ξ − x)2 dµs (ξ ). (1.22) K(s, x + iy) = [(ξ − x)2 + y 2 ]2 For a fixed δ > 0, let ρ = ρ(t, x) be such that dist(ϕt (x + iρ), ∂ t ) δ , i.e. ρ |ϕt (x + iρ)| δ,

(1.23)

Aggregation in the Plane and Loewner’s Equation

591

and let wt,x (s) denote the characteristic w = w(s; t, x + iρ(t, x)). By (1.19), (1.20), and (1.23), we obtain the following system of equations: t $[wt,x (0)] log = M(s, wt,x (s)) ds, ρ(t, x) 0 t δ $[wt,x (0)] log K(s, wt,x (s)) ds + O(1). = ρ 2 (t, x) 0

(1.24) (1.25)

The main observation is that M(s, wt,x (s)) K(s, wt,x (s)), unless the measure µs is “concentrated” near the point [wt,x (s)]. The positivity of K implies a Beurling-type bound for ρ, but we get a better estimate for non-concentrated growth measures. This argument can be used to prove results similar to the estimate (1.16), which essentially states that growth measures are not concentrated with respect to most of the phase flow trajectories. Moreover, the method applies to arbitrary evolution processes related to first order PDEs as long as the corresponding kernels have suitable localization properties.

1.5. Non-analytic chains. In the second part of the paper, we illustrate this real-variable approach by applying it to a system which is similar to (1.24)–(1.25) but has a lower dimensional phase space. Roughly speaking, we consider only the imaginary parts of characteristics assuming that the real parts remain constant. We establish an analogue of (1.16), see Theorem 3.1, and derive the following growth estimate for the corresponding version of an HS(δ)-process. Let µt be a family of positive measures on R periodic with period 1. Define the kernels M and K by the formulae (1.21) and (1.22) respectively. Given a positive function ρ = ρ(t, x), for each (t, x) let y(s) = yt,x (s) denote the solution of the initial value problem ∂s y = −yM(s, x + iy),

y(t) = ρ(t, x).

(1.26)

Define the times T0 and T by the equations

T0 0

µt [0, 1] dt =

T

T0

µt [0, 1] dt =

1 . 2

Theorem B. Suppose that for all t ≥ T0 , we have log

δyt,x (0) = ρ 2 (t, x)

t 0

K(s, x + iyt,x (s)) ds + O(1),

dµt (x) = ρt2 (t, x) dx. Then T − T0 δ −(1+c) for some absolute constant c > 0.

(1.27)

(1.28)

592

L. Carleson, N. Makarov

The system (1.26)–(1.27) is of course an analogue of (1.24)–(1.25). Together with (1.28) it describes an evolution process corresponding to the family of non-analytic maps ψt : x + iy → x + iβ(t, x, y),

ψ0 = id,

with β = β(t, x, y) satisfying the partial differential equation ∂t β = P ∂y β,

(1.29)

where P = P (t, x, y) is the Poisson integral of the function ρ(t, ·)2 such that ρ |∂y β(t, x, ρ)| δ,

ρ = ρ(t, x).

The latter is similar to the relation (1.23). The PDE (1.29) should be compared to the equation ∂t β = P ∂y β + P˜ ∂x β, (P˜ is the conjugate Poisson integral), which is equivalent to the analytic Loewner equation (take the imaginary part in (1.3) and denote β := $ϕt ). The complex analytic method of Sect. 2 does not apply to the system of Theorem B, and we don’t know if Kesten’s 3/2-result is valid in this case. 2. Deterministic Version of Kesten’s Theorem In this section we prove the bound T δ −3/2 for HS(δ)-chains (Theorem A). We also establish a stronger form of the Kesten-type estimate for Loewner chains with growth measures satisfying the equation dµt (ζ ) = |ϕt (ζ + εζ )|−2 |dζ |,

(ε > 0

fixed).

2.1. Proof of Theorem A. Let ϕt be an HS(δ)-process. Denote R(t) := max{|ϕt (ζ )| : ζ ∈ ∂ }. Assuming R(0) 1,

R(T ) − R(0) 1,

we want to show that T δ −3/2 . Let 0 ≤ t2 < t1 ≤ T , and suppose that R(t1 ) − R(t2 ) > 10δ. Let us choose a point a such that |a| = R(t1 ) + δ,

dist(a, ∂ t1 ) = δ,

and define t0 to be the first time when the cluster intersects the disc of radius 10δ centered at a. Clearly, t0 > t2 . To prove the theorem, it will be sufficient to show that t1 − t0 δ −1/2 .

(2.1)


593

We will use the inverse Loewner equation z˙ (s) z(s) + ζ =− dµs (ζ ) z(s) z(s) − ζ for the function

z(s) := ϕs−1 (a),

(t0 ≤ s ≤ t1 ).

If we denote y(s) := |z(s)| − 1, then the equation implies that y(s) ˙ − y(s)

dµs (ζ ) . |z(s) − ζ |2

Lemma. For all s ∈ (t0 , t1 ), we have √ dµs (ζ ) δ. 2 |z(s) − ζ |

(2.2)

(2.3)

Assuming this fact, we can finish the proof as follows. Observe that log

y(t0 ) 1. y(t1 )

(2.4)

Indeed, let G1 and G0 be the Green functions of the domains t1 and t0 respectively, with pole at infinity. By the comparison principle, we have G0 ≥ G1 . Consider the disc B := B(a, 5δ) , and let U be the component of t1 ∩ B containing a. Define α := ∂U ∩ ∂B, β := ∂U ∩ B. If ω denotes the harmonic measure of U evaluated at a, then we have G1 (a) = G1 dω, α

G0 (a) =

α

G0 dω +

G0 dω.

β

By Beurling’s projection theorem, we have ω(β) > 1/2, and by Harnack’s inequality, G0 (·) G0 (a) It follows that

on

β.

β

G0 dω ≥ cG0 (a)

594


with an absolute constant c ∈ (0, 1), and therefore (1 − c)G0 (a) ≥ G0 dω ≥ G1 dω = G1 (a). α

α

Thus we have log

G0 (a) 1, G1 (a)

which is equivalent to (2.4). The estimate (2.1) now follows from (2.2)–(2.4): t0 y(t0 ) y˙ 1 log = y(t1 ) t1 y t1 √ dµs (ζ ) = ds (t1 − t0 ) δ. 2 |z(s) − ζ | t0 This completes the proof of Theorem A except that we still need to prove the lemma. 2.2. Proof of Lemma. The argument is based on the estimate (1.16), which we apply in the following form: Let ψ ∈ (-) and 0 < σ < 1. Suppose that a function ρ(η) on the unit circle is such that ρ |ψ (η + ρη)| ≤ σ (ρ := ρ(η), η ∈ ∂ ). Then for some absolute constant c > 0, we have ρ 2 (η) |dη| σ 1+c .

(2.5)

Let us fix s ∈ (t0 , t1 ) and drop the index s from the notation s , ϕs , εs ; in particular, we have dµs (ζ ) = ε2 (ζ ) |dζ |. We will also write y for y(s) and we will assume that z(s) = 1 + y. Moving the point a within a disc of a fixed hyperbolic radius and applying Harnack’s inequality, we can assume without loss of generality that y ε(1). For 0 ≤ n | log ε|, we denote

yn = 2n y,

and consider the intervals In ⊂ ∂ of length yn centered at 1. Then we have ∂

dµs (ζ ) = |z(s) − ζ |2

We also denote and observe that

I0

+

n≥0 In+1 \In

rn = 1 + y n ,

1 ε 2 (ζ ) |dζ |. yn2 In n≥0

dn = yn |ϕ (rn )|,

dn dist(ϕ(rn ), ∂ ),

d0 δ.

(2.6)


595

We claim that |ϕ (1 + y)| |ϕ (rn )|, and

In

ε 2 (ζ ) |dζ | yn3

δ dn

(2.7)

1+c ,

(2.8)

where the constant c is the same as in (2.5). The statement of the lemma is a consequence of these two facts. Indeed, by (2.6) and (2.8) we have

δ 1+c δ c dµs (ζ ) y y , n |z(s) − ζ |2 dn dn

(2.9)

where the last inequality follows from the distortion theorem and (2.7): y yn . dn δ For the same reason, the last sum in (2.9) is finite, and therefore √ dµs (ζ ) y δ. 2 |z(s) − ζ | To verify (2.7) and (2.8), we denote by wn the point in such that ϕ(wn ) = (1 + dn )a. By construction, the halfline R+ a lies in a conformal Stoltz angle of the domain , and therefore the hyperbolic distance between the points rn and wn is bounded from above by a universal constant. Let us fix n, and consider the conformal automorphism τ : → given by the formula τ (w) =

wn w + 1 , w + wn

so that ∞ → wn and 1 → 1. Let ψ ∈ (-) be the corresponding Möbius transform of the function ϕ: (1 − wn2 )ϕ (wn ) ψ := = T ◦ ϕ ◦ τ, ϕ ◦ τ − ϕ(wn ) where (1 − wn2 )ϕ (wn ) T (z) = z − ϕ(wn ) ˜ := ψ . is the conformal map from onto the domain Let us prove (2.7). Denote by w the preimage of the point 1 + y under τ . Then we have |ψ (w)| = |T (a) ϕ (1 + y) τ (w)| dn |ϕ (1 + y)| yn |ϕ(wn ) − a|2 |ϕ (1 + y)| dn−1 |ϕ (1 + y)| yn = , |ϕ (rn )|

596


and so we need to show that |ψ (w)| 1. To see this, denote U := {|z| > R(t1 )}, Clearly,

U˜ := T U.

dist(a, ∂U ) dist(a, ∂ ),

and applying the map T , we have ˜ dist(T a, ∂ U˜ ) dist(a, ∂ ). By the comparison principle, we have the inequality GU˜ (·) ≤ G ˜ (·), for Green’s functions with pole at infinity. It follows that ˜ ˜ dist(ψ(w), ∂ ) dist(T a, ∂ ) |w| − 1 G ˜ (T a) dist(T a, ∂ U˜ ) 1; GU˜ (T a)

|ψ (w)|

the latter holds since ∂ U˜ is a round circle of radius 1. Finally, let us prove (2.8). Define the function ρ(η) by the equation ρ(η) = yn−1 ε(τ η),

(η ∈ ∂ ).

We will see that ˜ dist (ψ(η + ρ(η)η), ∂ )

δ . dn

(2.10)

It then follows from the estimate (2.5) with σ := dn−1 δ that 1+c δ 2 . ρ (η) |dη| dn Since

|(τ −1 ) | yn−1

on

In ,

the left hand side of the last inequality does not exceed τ −1 In

ρ 2 (η) |dη| =

1 yn2

In

ε 2 (ζ ) |(τ −1 ) (ζ )| |dζ |

1 yn3

In

ε 2 (ζ ) |dζ |,

which implies (2.8). It remains to prove (2.10). For η ∈ τ −1 In , denote ζ := τ η,

z := τ (η + ρ(η)η).

Since T ϕ(z) = ψ(η + ρ(η)η), by the distortion theorem we have ˜ |T (ϕ(z))| dist(ϕ(z), ∂ ). dist (ψ(η + ρ(η)η), ∂ )

(2.11)


597

The hyperbolic distance between the points z and ζ + ε(ζ )ζ is bounded by an absolute constant, which implies that dist(ϕ(z), ∂ ) dist (ϕ(ζ + ε(ζ )ζ ), ∂ ) δ

(2.12)

by the definition of ε(ζ ). On the other hand, we have |ϕ(wn ) − ϕ(z)| ≥ dist(ϕ(wn ), ∂ ) − O(δ) dn , and |T (ϕ(z))|

dn 1 . 2 |ϕ(wn ) − ϕ(z)| dn

The estimate (2.8) now follows from (2.11)–(2.13).

(2.13)

# "

2.3. A version of Theorem A. In the definition of an HS(δ)-process, the parameter δ was fixed, but the quantity ε depended on ζ ∈ ∂ and time t. We obtain a somewhat simpler process if we keep ε > 0 fixed and define the growth measures by the equation dµt (ζ ) = |ϕt (ζ + εζ )|−2 |dζ |.

(2.14)

For such processes, we can establish a slightly stronger form of the Kesten-type estimate by working with the direct Loewner equation (1.3). Theorem 2.1. Let ε < 1 be a positive number, and suppose that the growth measures µt of a Loewner chain {ϕt } satisfy (2.14). Denote R(t) := max {|ϕt (rζ )| : ζ ∈ ∂ }

(r := 1 + ε).

Then there is an absolute constant C such that if R(t) ≤ 1, then lim sup

t→0

R(t + t) − R(t) ≤ Cε −1 .

t

(2.15)

It is perhaps interesting to mention that a similar estimate c(t) ˙ ε −1 for the logarithmic capacity is not clear at all. By (1.4), the latter is essentially equivalent to the Brennan conjecture [4]: 1 |ψ (rζ )|−2 |dζ | , ∀ψ ∈ (-). r − 1 ∂ The inequality (2.15) is of course best possible, as the case of the function ϕ0 (z) = z + z−1 shows. The theorem implies the bound T ε for the doubling time defined by Eqs. (1.13). Proof. For each ζ ∈ ∂ , the function t → Rζ (t) := |ϕt (rζ )| is differentiable, and there is a uniform upper bound for the derivative, e.g. R˙ ζ (t) ε −3 . It follows that to prove (2.15) it is enough to show that for any fixed t, we have R˙ ζ∗ (t) ε−1 ,

598


where ζ∗ is a point at which the function ζ → Rζ (t) attains its maximum value. To this end, let us rewrite the Loewner equation in the form ϕ˙t (z) zϕt (z) z+ζ = dµt (ζ ), (2.16) ϕt (z) ϕt (z) z−ζ and note that

zϕt (z) ∈R ϕt (z) Taking the real part in (2.16), we have R˙ ζ∗ (t) |ϕt (rζ∗ )|

∂

at

z = rζ∗ .

r2 − 1 1 |ϕ (rζ )|−2 |dζ | . |rζ∗ − ζ |2 t ε

(2.17)

The second inequality in (2.17) is an analogue of (2.3) and can be established by the same method. Namely, let us assume that ζ∗ = 1 and write ϕ for ϕt . If we consider the intervals In ⊂ ∂ of length ε2n centered at 1 with n ≥ 1 and 2n 1/ε, then the middle expression in (2.17) does not exceed ε |ϕ (rζ )|−2 |dζ |. (2.18) |ϕ (r)| |In |2 In (n)

Denote rn = 1 + ε2n and observe that for all n, we have |ϕ (r)| |ϕ (rn )|, and 1 |In |

In

|ϕ (rζ )|−2 |dζ |

1 |ϕ (rn )|2

(2.19)

rn − 1 r −1

q (2.20)

with an absolute constant q < 2. It follows that

ε rn − 1 q 1 (2.18) |ϕ (r)| |In | |ϕ (rn )|2 r − 1 (n) 1 rn − 1 rn − 1 q−2 1 . |ϕ (rn )| r − 1 ε ε

(n)

The inequalities (2.19) and (2.20) can be verified as in the proof of (2.3) by considering the Möbius transform (1 − rn2 )ϕ (rn ) ψ = ∈ (-) ϕ ◦ τ − ϕ(rn ) of the function ϕ, where τ is the conformal automorphism of such that ∞ → rn , 1 → 1. The constant q in (2.20) comes from a universal estimate of integral means of the derivative in the class (-): 1.9 1 −2 |ψ (ρζ )| |dζ | , (0 < ρ < 1), ρ−1 ∂ see [18], Chapter 8.


599

3. Estimates for the Inverse Loewner Equation In this section, we discuss a real variable approach to growth estimates which is based on the study of the inverse Loewner chains, see Sect. 1.4. We will be considering the simplified, non-analytic model described in Sect. 1.5. Let us recall the equations for this model. We will use a slightly different notation. Let {µt }, 0 ≤ t ≤ T , be a family of finite positive measures on T := R/Z. Denote := [0, T ] × T. (The same symbol µt will be used for the periodic extension of µt to R.) We have two measures on – the area measure A, and the measure µ corresponding to the family {µt }. For each x ∈ T, we define the functions dµs (ξ ) Mx (s, y) := , 2 2 R (x − ξ ) + y (x − ξ )2 dµs (ξ ) Kx (s, y) := , 2 2 2 R [(x − ξ ) + y ] where 0 ≤ s ≤ T and y > 0, and we consider the following family ordinary differential equations: y˙ = − Mx (s, y). y

(Ex )

Every positive solution y = y(s) of (Ex ) is monotone decreasing. We specify the initial conditions as follows. Let ρ = ρ(ω) be a given a positive function on . Then for each ω = (t, x), we denote by yω (·) the solution of (Ex ) satisfying yω (t) = ρ(ω). This solution exists on the interval [0, t], and we define Y (ω) := yω (0). Our goal is to prove the following statement. Theorem 3.1. Let µ be a probability measure and let ρ be a positive function on = [0, T ] × T. Suppose that for some δ ∈ (0, 1), the following inequality holds for all points ω = (t, x) ∈ : t δY (ω) log 2 ≥ Kx (s, yω (s)) ds. (3.1) ρ (ω) 0 Then, for some absolute constant c > 0, either 1+c 1 µ ρ≤δ 2 > , 2

(3.2)

or there is a subset ⊂ satisfying µ( ) > c,

A( ) < δ c A( ).

(3.3)

600


Observe that for all ω ∈ , we have Y (ω) ≤ 2.

(3.4)

To see this, note that Mx (s, y) ≤ y −2 µs , and therefore if ω = (t, x), then ∂s yω2 (s) ≥ − µs ,

(0 ≤ s ≤ t).

On the other hand, since Kx ≥ 0, it follows from (3.1) that ρ 2 (ω) ≤ δY (ω).

(3.5)

Since µ is a probability measure, we have Y 2 (ω) ≤ 2 + ρ 2 (ω) ≤ 2 + δY (ω), which implies √ √ (3.4) for δ < 1. From (3.4) and (3.5), we obtain a Beurling-type estimate: ρ(ω) δ. Our theorem states that ρ can not be too close to the upper bound δ on a set of large µ-measure unless µ is “very singular” with respect to the area measure. The following two simple examples illustrate the dichotomy (3.2)–(3.3). (i) Let µ be the area measure on = [0, 1] × T. Then we have Kx ≥ and so (3.1) implies δY ≥ ρ2

1 Mx , 2

Y , ρ

(∀x ∈ T),

(∀ω ∈ ).

In particular, by (3.4) we must have ρ δ 2/3 . We can actually get ρ δ 2/3 on a set of large measure – it is easy to show that (3.1) holds for ρ(t, x) = const (δt)2/3 provided that the constant is sufficiently small. (ii) Let µ be the linear measure of the set [0, 1] × {x0 } and let ρ ≡ δ 1/2 on this set and be zero elsewhere. Then (3.1) is clearly satisfied and the bound (3.5) is attained on a set of full measure. Theorem 3.1 implies the following estimate of the “doubling” time T . Corollary. Suppose µ and ρ satisfy the hypothesis of the theorem, and suppose that they are related by the equation dµ = ρ 2 dA. Then T δ −(1+c) . Indeed, applying Theorem 3.1 we see that if (3.2) holds, then 1 ≤ δ 1+c dA ≤ T δ 1+c , 2 and if (3.3) holds, then we have ρ 2 dA ≤ δA( ) ≤ T δ 1+c . c≤

Theorem B (see Introduction) is a version of the last statement. It will be clear from the proof of Theorem 3.1 that the changes in the argument are quite obvious. We now turn to the proof of Theorem 3.1. Our construction depends on a certain numerical parameter ε which will be specified at the end of the argument. By the notation of the type a b we mean the inequality a ≤ const b with an absolute constant independent of the choice of ε; a b means that a b and a b.


601

3.1. Martingale structure. We first introduce a certain martingale structure in the set . Lemma 1. There exists a sequence of partitions {Dj }j ≥1 of satisfying the following conditions. • Each partition Dj consists of rectangles P = I × l, where l ⊂ T is a dyadic interval of length 2−j and I is a subinterval of [0, T ]. • Rectangles of the same rank j have disjoint interiors. Each P ∈ Dj is a union of rectangles of rank j + 1. • If P = I × l ∈ Dj and if xl is the center of l, then dµs (ξ ) 1 ≤ ds ≤ 4. 2 −j 4 I T (xl − ξ ) + 4 Proof. Denote j Mx (s)

Clearly,

j Mx

≤

j +1 Mx ,

:=

T

(3.6)

dµs (ξ ) . (xl − ξ )2 + 4−j

and it is easy to see that

|x − x| ≤ 2−j −1

⇒

1 j j j Mx ≤ Mx ≤ 2Mx . 2

(3.7)

We begin by setting P0 = . Suppose a rectangle Pj = I × l of rank j is already constructed and suppose that 1 j (3.8) < Mxl ≤ 1. 2 I Note that (3.8) holds for P0 . Let k > j be the least integer such that Mxkl > 2. I

We first subdivide Pj successively into rectangles of ranks j + 1, . . . , k − 1 with the same t-projection I as Pj . By (3.8) and (3.7), the new rectangles satisfy (3.6). Then we subdivide each of the rectangles of rank k − 1 already obtained into rectangles of rank k satisfying (3.8). The latter is possible by (3.7). " # We fix partitions Dj constructed in Lemma 1 for the rest of the proof. If ω ∈ , then we write Pj (ω) for the element of Dj containing ω. It is clear from (3.6) that P ∈ Dj

⇒

µ(P ) 4−j .

(3.9)

Another property of the construction is stated in the following lemma, the proof of which we leave as a simple exercise. Lemma 2. Let P = I × l ∈ Dj , x ∈ l, and let y(·) be a solution of (Ex ). Then (i) if y(s1 ) ≥ 2−j +10 for some s1 ∈ I , then y(s) ≥ 2−j for all s ∈ I ; (ii) if y(s1 ) 2−j for s1 ∈ I , then y(s) 2−j for all s ∈ I , and y(t− )/y(t+ ) − 1 1, where t− < t+ are the endpoints of I .

602


3.2. Rectangles Rj (ω). Let ω = (t, x) ∈ and let yω (·) be the corresponding solution of (Ex ). We now construct certain rectangles associated with this solution. Suppose that a positive integer j satisfies ρ(ω) ≤ 2−j −10 ≤ Y (ω). Then we define Rj (ω) as a rectangle I × l ∈ Dj such that yω (s) = 2−j +10

for some

s ∈ I.

We don’t define rectangles Rj (ω) for other values of j . Let Ij (ω) denote the projection of Rj (ω) to the t-axis. It follows from Lemma 2 that yω (s) 2−j ,

(∀s ∈ Ij (ω)),

(3.10)

and that the intervals Ij (ω) with the same ω are essentially disjoint: |k − j | > 10

Ik (ω) ∩ Ij (ω) = ∅.

⇒

(3.11)

We will also need the following fact. Lemma 3. If P ∈ Dj , then µ{ω : Rj (ω) = P } 4−j . Proof. Let P = (s− , s+ ) × l ∈ Dj , and S = {ω : Rj (ω) = P }. For each point ω = (t, x) ∈ S with x ∈ l, we denote by Cω the family of rectangles Q satisfying the following condition: ∃s,

s+ ≤ s ≤ t,

Define

yω (s) 2−k , C∗ :=

k := rank Q > j,

(s, x) ∈ Q.

{Cω : ω ∈ S},

and consider a subcollection C ⊂ C∗ constructed as follows. We first take rectangles Q ∈ C∗ of rank j + 1, then we add rectangles of rank j + 2 which are not contained in any C∗ -rectangle of rank j + 1, then we add rectangles of rank j + 3 which are not contained in any rectangle of rank j + 1 or j + 2, etc. Clearly, S⊂ {Q : Q ∈ C}, and so it is sufficient to show that

µQ 4−j .

Q∈C

One of the properties of the family C is that the number of rectangles Qk ∈ C of rank k intersecting any given dyadic strip {x1 ≤ x ≤ x2 } of width 2−k is bounded by an absolute constant. To see this, consider the rank k rectangle with the largest t-coordinate t∗ of the center, and let ω ∈ S be such that yω (t∗ ) 2−k . By construction, the solution yω must be 2−k in all other rectangles Qk , because otherwise Qk would be covered by a C-rectangle of a smaller rank. It remains to note that by Lemma 2, the function yω drops by a constant factor across every rectangle Qk . It follows that Nk := #{Q ∈ C : rank Q = k} 2k−j and therefore by (3.9), we have µQ Nk 4−k 4−j . Q∈C

k≥j

# "


603

3.3. Concentrated rectangles. We will write Kω (s) for the function s → Kx (s, yω (s)). The notation Mω (s) has similar meaning. Our next goal is to estimate the integral of Kω over Ij (ω). Note that by (3.6) and (3.10), we have Mω (s) ds ≥ 2a (3.12) Ij (ω)

for some absolute constant a > 0. Fix a small dyadic number ε > 0 to be specified later. The following definitions depend on the choice of ε. A rectangle P = I × l ∈ Dj is said to be concentrated if there is a subinterval l ⊂ l of length |l | =

ε |l| 10

such that for P := I × l we have µP > a4−j ,

µ(P \ P )
ε |l|. Then either Rj (ω) is concentrated and x ∈ λ(Rj (ω)), or Kω ≥ α, Ij (ω)

(3.14)

(3.15)

where α = α(ε) is a positive number depending on the choice of ε. Proof. Let l denote the interval of length 10−1 ε |l| centered at x. Then l ⊂ l by (3.14). We have (x − ξ )2 dµs (ξ ) Kω ≥ ds 2 2 2 Ij (ω) Ij (ω) l\l [(x − ξ ) + yω (s)] dµs (ξ ) ε2 ds . 2 2 Ij (ω) l\l (x − ξ ) + yω (s) It follows that if (3.15) does not hold with α - ε 2 , then the latter iterated integral is - 1, and therefore (3.12) implies the estimate dµs (ξ ) ds ≥ a. (3.16) (x − ξ )2 + yω2 (s) Ij (ω) l

604


Let us show that the inequalities (3.13) hold in this case, and so Rj (ω) is a concentrated rectangle. The first inequality follows from (3.16) and the fact that yω (·) > 2−j on Ij (ω). To prove the second inequality in (3.13), we observe that

µ(R \ R ) 4

−j

Ij (ω)

ds

l\l

dµs (ξ ) - 4−j . (x − ξ )2 + yω2 (s)

# "

3.4. Proof of Theorem. Let c < 1/4 be a positive number and E := {ω : ρ(ω) > δ If ω ∈ E, then Y (ω) ≥

1+c 2

}.

ρ 2 (ω) > δc , δ

√ and since ρ(ω) δ, the rectangles Rj (ω) are defined for all j ∈ [1 + N, 2N ], where 1 N log . δ For ω = (t, x) ∈ E, denote n(ω) := #{j ∈ [1 + N, 2N ] : x ∈ λ(Rj (ω))}. We also need the function N (ω) := #{j ∈ [1 + N, 2N ] : ω ∈ Pˆj (ω)}, Define the sets E := {ω ∈ E : n(ω) ≥ and

(ω ∈ ).

1 N} 4

:= {ω ∈ : N (ω) ≥ 2εN}.

Lemma 5. (i) There is a positive constant c1 = c1 (ε) such that A( ) ≤ δ c1 A( ). (ii) There is an absolute constant ε1 such that if ε ≤ ε1 , then µE ≥

1 4

⇒

µ 1.

Proof. (i) Denote by Xj the characteristic function of the set {Pˆ : P ∈ Dj }. If ε = 21−m , where m is a positive integer, then we have the submartingale property E(Xj +m − ε | Dj ) ≤ 0,


605

where E is the conditional expectation with respect to the normalized area measure. For 1 ≤ ν ≤ m, let   N   −1  m m  XN+ν+j m ≥ 2ε . ν :=    N j =0  (We can assume that m divides N .) Simple large deviation argument shows that A( ν ) ≤ q N/m A( ) for some q = q(ε) < 1. Since

⊂

ν ,

the statement follows. (ii) Suppose µE ≥ 41 . Denote f (ω) := N −1 N (ω). Since f ≤ 1, we have µ = µ{f ≥ 2ε} ≥ f dµ − f dµ ≥ f dµ − 2ε, {f 1, first use linearity to obtain 4-point functions of the form E k γα γβ γµ 0,s , then apply Lemma 4.2 to obtain 4-point functions involving E and repeat the previous step to express everything in terms of 3-point functions. • Use WDVV equation and its derivatives to exchange position of vectors in products of correlation functions and simplify expressions if necessary. • Use Lemma 4.3 to simplify expressions whenever possible. After expressing φm in terms of 3-point functions, we then simplify the expression to get the desired formula. Remark. The expression for φm in Theorem 4.4 is the same as that for E k 1,s obtained in [DZ2] (4.42) for the case where the quantum cohomology of V is semisimple. 5. A Necessary and Sufficient Condition for the Genus-1 Virasoro Conjecture The main purpose of this section is to prove the following Theorem 5.1. For any manifold V , the genus-1 Virasoro conjecture holds if and only if 2 E 1,s = φ2 . In this section, we will use {u1 , . . . , uN } to denote the coordinate on the small phase space in order to distinguish the one on the big phase space. In this coordinate, the vector field ∂u∂ α is identified with γα . Let uα = β ηαβ uβ . Then ∂u∂ α is identified with γ α . Let

Elliptic Gromov–Witten Invariants and Virasoro Conjecture

719

M be an N × N matrix whose entries are uαβ . Temporarily, we think of each uαβ as an independent variable. Define & % 1 α F1 (u1 , . . . , uN ; M) := e α uα γ + log det η−1 M . 1 24 Then ∂F1 1 % −1 & ∂F1 = γ α 0,s and = . (23) M αβ ∂uα ∂uαβ 24 The genus-1 constitutive relation says that F1 is equal to F1 after the transformation uα = γ1 γα 0 and uαβ = γ1 γα γβ 0 . (24) Taking derivative of the genus-1 constitutive relation once, we obtain ∂F1 ∂F1 τm,α 1 = γ1 τm,α γσ 0 γ1 τm,α γσ γρ 0 + ∂u ∂uσρ σ σ σ,ρ

(25)

for any m and α. On the other hand, the genus-0 constitutive relation says, in particular, that '' = γα γβ 0 . (26) γα γβ 0,s ' uσ =γ1 γσ 0

Taking derivative of this relation once, we get & % '' γα γβ γµ 0,s ' = M −1 η uσ =γ1 γσ 0

' ' ' νµ '

ν

uσρ =γ1 γσ γρ 0

γα γβ γ ν

0

.

(27)

Moreover combining formula (26) with formula (6) and Lemma 2.2 (i), we obtain '' γα Eγβ 0,s ' = γα X γβ 0 . (28) uσ =γ1 γσ 0

The following lemma will be useful in the proof of Theorem 5.1. Lemma 5.2.

%

M −1

& αβ

α,β,µ1 ,... ,µk−1

γ1

#

γα X γ µ1

=k

γµ1 X γ µ2

0

µ1 ,... ,µk−1

γµk−1 X γ µ1

0

0

$ · · · γµk−1 X γβ 0

γµ1 X γ µ2

0

· · · γµk−2 X γ µk−1 0 .

Proof. By formula (6), γ1 γα X γβ 0 = (bα + bβ ) γ1 γα γβ 0 = (bα + bβ )uαβ . Therefore

%

M −1

α,β,µ1 ,... ,µk−1

= 2k

& αβ

γ1

µ1 ,... ,µk−1

#

γα X γ µ1

0

γµ1 X γ µ2

0

$ · · · γµk−1 X γβ 0

bµ1 γµk−1 X γ µ1 0 γµ1 X γ µ2 0 · · · γµk−2 X γ µk−1 0 .

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X. Liu

In this calculation, one needs to switch the position of γ1 and that ot X by using the generalized WDVV equation so that γ1 can be pushed to the beginning or the end of the chain of the multiplications of 3-point functions. In this way we can always create entries of M which can be used to eliminate entries of M −1 . The lemma then follows from an argument similar to the proof of Lemma 4.3 (i.e., by interchanging all upper indices with the corresponding lower indices). Recall that L1 is the vector field on the big phase space which is defined to be the first derivative part of the L1 operator. The genus-1 L1 constraint is 21,1 = 0, where 21,1 = L1 1 +

# $ 1 bα (1 − bα ) γα γ α 0 + 2 γ α 0 γα 1 . 2 α

We have the following Proposition 5.3. 21,1 =

! − E2

1,s

"' ' + φ2 ''

uσ =γ1 γσ 0

.

Proof. Applying Eqs. (23) and (25) to each genus-1 1-point function in 21,1 , we obtain ( α γ1 L1 γσ 0 + 21,1 = bα (1 − bα ) γ 0 γ1 γα γσ 0 σ

!

α

" ' ' · γ 0,s ' (29) uβ =γ1 γβ 0 ( % −1 & α 1 + γ1 L1 γσ γρ 0 + bα (1 − bα ) γ 0 γ1 γα γσ γρ 0 M σρ 24 σ,ρ α +

σ

1 bα (1 − bα ) γα γ α 0 . 2 α

By the second' equation of (8) and Eq. (28), the first line of the right-hand side is equal ' to − E 2 1,s ' . Now we compute the second line. Since uσ =γ1 γσ 0

γ1 L1 γσ γρ

0

= γ1 L1 γσ γρ 0 − b1 (b1 + 1) τ1,1 γσ γρ 0 − (2b1 + 1) C1α γα γσ γρ 0 , α

by Lemma 5.2 and the second equation of (8), the second line of (29) is equal to −

1 1 % −1 & M γα γ α 0 − σρ 12 α 24 σ,ρ,α # · (bα (1 − bα ) + b1 (b1 + 1)) γ1 γα 0 γ α γσ γρ 0 $ + (2b1 + 1)C1α γα γσ γρ 0 .

(30)


721

On the other hand, by Lemma 2.2 (ii) EEγα γ α 0,s = Eγ1 γ β 0,s γβ Eγα γ α 0,s α

α,β

=

α,β

By Eqs. (27) and (28), '' EEγα γ α 0,s '

uσ =γ1 γσ 0

α

=

(bβ − b1 ) Eγ1 γ β 0,s γβ γα γ α 0,s .

β,σ,ρ

& % (bβ − b1 ) Xγ1 γ β 0 γβ γσ γρ 0 M −1

.

σρ

By formula (6), α

=

b α γ1 X γ α 0

#

$ −b1 γ1 X γ α 0 + (2b1 + 1)C1α + (bα (1 − bα ) + b1 (b1 + 1)) γ1 γ α 0 .

α

Moreover & % Xγ1 γ β 0 γβ γσ γρ 0 M −1

σρ

β,σ,ρ

=

Xγσ γ β

β,σ,ρ

=

Xγσ γ

σ

=

γσ γ σ

σ

%

0

0

γβ γ1 γρ

M −1

& σρ

σ 0

0

.

Therefore we have

EEγα γ α

α

=

uσ =γ1 γσ 0

& $ % (2b1 + 1)C1α + (bα (1 − bα ) + b1 (b1 + 1)) γ1 γ α 0 γα γσ γρ 0 M −1

#

α,σ,ρ

' ' 0,s '

σρ

− 2b1

γσ γ

σ

σ

Comparing this equation with (30) and using (26), we obtain the desired formula.

0

.

We next prove the analogue of this proposition for the genus-1 L2 constraint. We need the following Lemma 5.4. 1 β α (i) γα γ β 0 γβ γ α 0 ; α,β bα γα γ 0 γβ γ 0 = 2 α,β 3 β (ii) γβ γ α 0 = α,β − 41 + 23 bα2 γα γ β 0 γβ γ α 0 ; α,β bα γα γ 0 k β α (iii) = 0 if k is odd. α,β (bα ) Cα γβ γ 0

722

X. Liu

Proof. Interchanging in the expression theα upper indices and lower indices β αβ = 0 implies b = 1 − b , we γ γ b γ γ and using the fact that b η α β α β α α,β α 0 0 obtain b α γα γ β 0 γ β γ α 0 = (1 − bα ) γβ γ α 0 γα γ β 0 . α,β

α,β

This implies (i). Similarly we have α,β

bα3 γα γ β 0 γβ γ α 0 = (1 − bα )3 γβ γ α 0 γα γ β 0 . α,β

Together with (i), this implies (ii). Using the fact bα Cαβ = 0 implies bβ = −bα , we have (bα )k Cαβ γβ γ α 0 = (bα )k Cαβ γ β γ α 0 = (−bβ )k Cαβ γ β γ α 0 . α,β

α,β

α,β

Interchanging α with β, we have (bα )k Cαβ γβ γ α 0 = (−1)k (bα )k Cαβ γβ γ α 0 . α,β

α,β

This implies (iii). The genus-1 L2 constraint is the equation 21,2 = 0 where 21,2 = L2 1 +

α

bα (1 − bα2 )

# $ τ1,α γ α 0 + τ1,α 0 γ α 1 + τ1,α 1 γ α 0

# $ 1 2 − (3bα − 1)Cαβ γ α γβ 0 + 2 γ α 1 γβ 0 . 2 α,β

We have the following Proposition 5.5. 21,2

! = − E3

1,s

"' ' + φ3 ''

uσ =γ1 γσ 0

.

Proof. Applying Eqs. (23) and (25) to each genus-1 1-point function in 21,2 , using Eq. (8) and the fact γ1 L2 γσ γρ 0 = γ1 L2 γσ γρ 0 − b1 (b1 + 1)(b1 + 2) τ2,1 γσ γρ 0 β − (3b12 + 6b1 + 2)C1 τ1,β γσ γρ 0 β

−

β

β 3(b1 + 1)(C 2 )1 γβ γσ γρ 0 ,


723

then applying Lemma 5.2, Eq. (28) and the genus-0 topological recursion relation, we obtain (' ' 1 2 ' 3 α E γα γ − 21,2 = − E ' 1,s 1,s ' 8 α uσ =γ1 γσ 0 1 % −1 & + M γ µ γν γ β 0 µν 24 µ,ν,β (3bα2 − 1)Cαβ γ1 γ α 0 + bα (bα2 − 1) γ1 γ α 0 γα γβ 0 α

α

+ bβ (bβ2 − 1) γ1 τ1,β 0 − b1 (b1 + 1)(b1 + 2) τ1,1 γβ 0 ( 2 α 2 − (3b1 + 6b1 + 2)C1 γα γβ 0 − 3(b1 + 1)(C )1β −

β

α

bβ (bβ2

1 2 − 1) τ1,β γ β 0 − (3bα − 1)Cαβ γβ γ α 0 . 2

(31)

α,β

A simple combination of formula (6) and the genus-0 topological recursion relation gives the following (cf. [LT] formula (8) and (9)) # $ γα γ σ 0 Cσβ + (bσ + bβ ) γσ γβ 0 (1 + bα + bβ ) τ1,α γβ 0 = σ

−

σ

Cασ γσ γβ 0 .

This is a special case of the fundamental recursion relationof [EHX1]. Using this for mula, we can express 2-point correlation functions of type τ1,α γβ 0 in the right-hand side of Eq. (31) in terms of correlation functions only involving γσ , σ = 1, . . . , N. (In this procedure, first Lemma 3.2 in [LT] to shift the level of descendant in applying the term bβ (1 + bβ ) γ1 τ1,β 0 may simplify the computation.) Then a straightforward computation using formula (6) and Lemma 5.4 shows that (' ' 1 2 ' 3 α E γα γ − 21,2 = − E ' 1,s 1,s ' 8 α

uσ =γ1 γσ 0

1 % −1 & M γµ γν γ β 0 (b1 + bα + 1 − bβ ) γ1 X γ α 0 γα X γβ 0 + µν 24 µ,ν,β 3 1 1 2 − b β − bα b β γ α X γ β 0 γβ X γ α 0 . + 8 2 4 α,β

The proposition then follows from Eqs. (27), (28), and Theorem 4.4.

Now we are ready to prove Theorem 5.1. Proof of Theorem 5.1. The string equation implies that the transformation uα = γ1 γ α 0,s is an identity map when the right-hand side of this equation is restricted to the small phase space. Therefore, by Proposition 5.3, the restriction of the genus-1

724

X. Liu

L1 constraint to the small phase space is equivalent to the condition that E 2 0,s = φ2 . Hence E 2 0,s = φ2 is a necessary condition for the genus-1 Virasoro conjecture. On the other hand, if E 2 0,s = φ2 , Proposition 5.3 also implies that the genus-1 L1 constraint is true. Moreover, Theorem 4.1 and Proposition 5.5 implies that the genus-1 L2 constraint is also true. By the virasoro relation among the Ln operators, the genus-1 Virasoro conjecture holds.

6. Virasoro Type Relation for {φk }

Because of Theorem 5.1, we are interested in when the equality E 2 1,s = φ2 holds. The Virasoro relation (15) and Theorem 4.1 implies that Eq. (1) is a necessary condition for this equality to hold. In this section, we prove that this condition is always satisfied. Theorem 6.1. For any manifold V , we always have E k φm − E m φk = (m − k)φk+m−1 . We first prove a special case of Theorem 6.1. Proposition 6.2. E k φ2 − E 2 φk = (2 − k)φk+1 . Proof. We will use the formula given in Theorem 4.4 for φ2 and φk . Using formula (10) and (13), we can express E k φ2 − E 2 φk in terms of 3-point and 4-point functions involving quantum powers of the Euler vector field. As in the proof of Theorem 4.4, we can use Lemma 2.2 and Lemma 4.2 to represent all 4-point functions in this expression by 3-point functions. After simplifying the resulting expression, we obtain the desired formula. Now we are ready to prove Theorem 6.1. Proof of Theorem 6.1. We prove this theorem by induction on min{m, k}. Without loss of generality, we may assume that m ≤ k. If m = 0, Eq. (1) is equivalent to γ1 φk = kφk−1 . This equality holds trivially when k = 0 or k = 1. When k = 2, it follows from formula (10), Lemma 2.1, and the following formula (cf. [Bor]) 1 b1 + 1 1 bβ (1 − bβ ) − c1 (V ) ∪ cd−1 (V ). χ (V ) = − 2 12 12 V β

Note that this is the reason why bα is defined in terms of the holomorphic dimension of γα rather than a half of the real dimension of γα as proposed in [EHX2]. For k > 2, the equality follows from Theorem 4.4, formula (10), the fact that ∇γ1 E k = [γ1 , E k ] = kE k−1 , and Lemma 2.1. Assume that equality (1) holds for m ≤ n. We want to show that it also holds for m = n + 1. In fact for any k, by Eq. (15) and Proposition 6.2, we have & % & 1 % 2 n E n+1 φk − E k φn+1 = E E − E n E 2 φ k − E k E 2 φn − E n φ2 . n−2


725

By the induction hypothesis, E n φk = E k φn + (k − n)φn+k−1 , and by Proposition 6.2, E 2 φk = E k φ2 + (k − 2)φk+1 . Therefore, by Eq. (15), we have & % 1 E n+1 φk − E k φn+1 = (k − 2) E k+1 φn − E n φk+1 n−2 % & + (k − n) E 2 φn+k−1 − E n+k−1 φ2 . Using the induction hypothesis and Proposition 6.2 again, we have E n+1 φk − E k φn+1 = (k − n − 1)φn+k . This proves the theorem. We can use Theorem 6.1 to construct a representation of the Lie algebra spanned by {E k | k ≥ 0} in the following way. Let hk := E k − φk . (32) 1,s

By Theorem 3.4 and the definition of φk , h0 = h1 = 0 and hk =

k k−1 E h2 . 2

(33)

More generally, we have the following Lemma 6.3. For all k ≥ 0 and m > 0, Ek

hm hm+k−1 = (m − 1) . m m+k−1

Proof. Theorem 6.1 and formula (15) imply E k hm − E m hk = (m − k)hk+m−1 for all m and k. Using this formula, one can show that the equation Ek

hm hm+k−1 = (m − 1) m m+k−1

is equivalent to the equation Em

hk hm+k−1 = (k − 1) . k m+k−1

Formula (33) says that the lemma is true if min{m, k} = 2. By formulas (33) and (15), we have " ! 2 m 2 E k hm = E m−1 hk+1 + (m − k − 1) hm+k−1 . 2 k+1 m+k−1 The lemma then follows from induction on min{m, k}. Lemma 6.3 tells us that the linear span of {hk | k ≥ 2} gives a representation of the Lie algebra spanned by {E k | k ≥ 0}. Theorem 5.1 means that the genus-1 Virasoro conjecture holds if and only if h2 = 0, which is equivalent to say that this representation is trivial.

726

X. Liu

7. Further Remarks In this section we briefly describe some applications of the results obtained in previous sections. We first notice that since the small phase space is finite dimensional, the set of vectors {E k | k ≥ 0} must be linearly dependent. Consequently, in an open subset of the small phase space, there exists an integer n such that {E k | 0 ≤ k ≤ n} are linearly independent and there are functions fi , 0 ≤ i ≤ n such that E n+1 =

n

fk E k .

(34)

k=0

Due to Theorem 4.1 and Theorem 5.1, a necessary condition for the genus-1 Virasoro conjecture is that φn+1 =

n

fk φ k .

(35)

k=0

This condition only involves genus-0 data. We conjecture that this condition is always satisfied. This can be verified easily for manifolds with semisimple quantum cohomology. In fact, Eqs. (1) and (35) are equivalent to the existence of a local potential function whose derivative along E k is φk for all k. For manifolds with semisimple quantum cohomology, such a potential function exists globally and can be explicitly expressed in terms of the τ -function of the isomonodromy deformation (cf. [DZ2] proof of Proposition 4). It is also easy to verify this for all algebraic curves and K3 surfaces. Since E k+n+1 = E k • E n+1 , Eq. (34) implies E k+n+1 =

n

fi E k+i ,

(36)

i=0

for every k ≥ 0. In particular, we have

E k+n+1

1,s

=

n

fi E k+i

i=0

1,s

for every k ≥ 0. On the other hand, Theorem 3.4 tells us that each E k 1,s is equal to (k/2)E k−1 E 2 1,s plus genus-0 data. In this way, we obtain a system of equations which represent certain first order directional derivatives of E 2 1,s in terms of genus-0 data. After some simplification using the Virasoro bracket relation (15), this system of equations can be writen in the following form Zk E 2 = genus-0 data, (37) 1,s

where Zk = (n + 1)E n+k −

n−1

(i + 1)fi+1 E i+k

i=0

for k ≥ 0. If the quantum cohomology is not too degenerate, this system of equations will completely determine the function E 2 1,s in terms of genus-0 data, which together with


727

Eq. (35) actually implies that E 2 1,s = φ2 (therefore also implies the genus-1 Virasoro conjecture). In fact, because of the relation (15), all what we need is that for some m ≥ 1, E m can be expressed as a combination of E 0 and Zk ’s. This condition is satisfied for manifolds with semisimple quantum cohomology (in this case, {Zk | k ≥ 0} span entire tangent spaces of the small phase space) and also for all curves and K3 surfaces (in these cases, the span of {E k | k ≥ 0} are proper subspaces of tangent spaces of dimension less than or equal to 2). Therefore the genus-1 Virasoro conjecture holds for such manifolds. We believe that the class of manifolds satisfying this condition is much larger than the examples given here. It would be interesting to give a geometric characterization of such manifolds. The genus-1 Virasoro conjecture for manifolds with semisimple quantum cohomology was proved in [DZ2]. The genus-1 Virasoro conjecture for elliptic curves is not known before. After this work has been finished, the author was informed that F. Zahariev found a combinatorial proof to the genus-1 Virasoro conjecture for elliptic curves. It seems that there is a gap in the proof of Virasoro conjecture for K3 surfaces in [G2] since it omits to verify the genus-1 degree-0 case. Moreover, the proof contained in this paper is technically simpler since it doesn’t use deformation invariance of GW invariants. Acknowledgement. The author would like to thank V. Kac, G. Tian, and E. Witten for very helpful discussions. He is grateful to G. Tian for encouragement during this work and collaboration in the previous work. The author is partially supported by an NSF postdoctoral fellowship. He also wants to thank the Mathematics Department of MIT, where most part of this work has been done.

References [BF] [Bor] [CK] [DVV] [DW] [Du] [DZ1] [DZ2] [EHX1] [EHX2] [EJX] [G1] [G2] [HM] [H] [Ko] [LiT1]

Behrend, K., Fantechi, B.: The intrinsic normal cone. Invent. Math. 128, 45–88 (1997) Borisov, L.: On betti numbers and Chern classes of varieties with trivial odd cohomology groups. alg-geom/9703023 Cox, D. and Katz, S.: Mirror symmetry and algebraic geometry. Providence, R.I.: AMS, 1999 Dijkgraaf, R., Verlinde, H., and Verlinde, E.: Loop equations and Virasoro constraints in nonperturbative two-dimensional quantum gravity. Nucl. Phys. B 348, no. 3, 435–456 (1991) Dijkgraaf, R. and Witten, E.: Mean field theory, topological field theory, and multimatrix models. Nucl. Phys. B 342, 486–522 (1990) Dubrovin, B.: Geometry of 2D topological field theories, Integrable systems and quantum groups. Lecture Notes in Math. 1620. Berlin: Springer, 1996, pp. 120–348 Dubrovin, B., Zhang, Y.: Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation. To appear in Commun. Math. Phys.; hep-th/9712232 Dubrovin, B., Zhang, Y.: Frobenius manifolds and Virasoro constraints. math.AG/9808048 Eguchi, T., Hori, K., and Xiong, C.: Gravitational Quantum Cohomology. Int. J. Mod. Phys. A 12, 1743–1782 (1997); hep-th/9605225 Eguchi, T., Hori, K., and Xiong, C.: Quantum Cohomology and Virasoro Algebra. Phys. Lett. B 402, 71–80 (1997); hep-th/9703086 Eguchi, T., Jinzinji, M., and Xiong, C.: Quantum Cohomology and Free Field Representation. Nucl. Phys. B 510 608–622 (1998); hep-th/9709152 Getzler, E.: Intersection theory on M¯ 1,4 and elliptic Gromov–Witten Invariants. J. Am. Math. Soc. 10, 973–998 (1997) Getzler, E.: The Virasoro conjecture for Gromov–Witten invariants. In: Algebraic Geometry: Hirzebruch 70, Providence, RI: AMS, 1999, 147–176 Hertling, C., Manin, Y.; Weak Frobenius manifolds. math.QA/9810132 Hori, K.: Constraints For Topological Strings In D ≥ 1. Nucl. Phys. B 439, 395 (1995), hepth/9411135 Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix airy function. Commun. Math. Phys. 147, 1–23 (1992) Li, J. and Tian, G.: Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties. J. Am. Math. Soc. 11, 119–174 (1998)

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Li, J. and Tian, G.: Virtual moduli cycles and Gromov–Witten invariants of general symplectic manifolds. alg-geom/9608032 Liu, X. and Tian, G.: Virasoro constraints for quantum cohomology. J. Diff. Geom. 50, 537–591 (1998); math.AG/9806028 Ruan, Y. and Tian, G.: A mathematical theory of quantum cohomology. J. Diff. Geom. 42, 259–367 (1995) Ruan, Y. and Tian, G.: Higher genus symplectic invariants and sigma models coupled with gravity. Invent. Math. 130, 455–516 (1997); alg-geom/9601005 Tian, G.: Quantum cohomology and its associativity. Current developments in mathematics. Cambridge, MA: Internat. Press, 1995, pp. 360–401 Tian, G., Xu, G.: On the semi-simplicity of the quantum cohomology algebras of complete intersections Math. Res. Lett. 4, 481–488 (1997); 9611035 Witten, E.: Topological sigma models. Commun. Math. Phys. 118, 411–449 (1988) Witten, E.: Two dimensional gravity and intersection theory on Moduli space. Surveys in Diff. Geom. 1, 243–310 (1991) Witten, E.: On the Kontsevich model and other models of two dimensional gravity. In: Proceedings of the XXth international conference on differential geometric methods in theoretical physics (New York, 1991). River Edge, NJ: World Sci. Publishing, 1992, pp. 176–216

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