Communications in Mathematical Physics - Volume 192

Commun. Math. Phys. 192, 1 – 7 (1998) Communications in Mathematical Physics c Springer-Verlag 1998 Equilibrium Clas...

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Commun. Math. Phys. 192, 1 – 7 (1998)

Communications in

Mathematical Physics c Springer-Verlag 1998

Equilibrium Classical Statistical Mechanics of Continuous Systems in the Kirkwood–Salsburg Approach ´ Marek Gorzelanczyk Institute of Theoretical Physics, University of Wrocław, pl. M. Borna 9, 50-204 Wrocław, Poland. E-mail: [email protected] Received: 22 July 1996 / Accepted: 9 June 1997

Abstract: We propose a new scheme for the analysis of the Kirkwood–Salsburg equation. The Kirkwood–Salsburg operator is considered on a space constructed in such a way that the spectral analysis of this operator is essentially simplified. We find a new region free of phase transitions of any order.

1. Introduction To describe the equilibrium states in classical statistical mechanics the correlation functions can be employed. The purpose of this paper is to investigate the analytic properties of the correlation functions in the thermodynamical limit. These functions fulfill the sequence of integral equations (e.g.: Kirkwood–Salsburg (K–S), Mayer–Montroll) on the suitable Banach space. The exploration of integral equations yields some information about the thermodynamical limit of correlation functions. To reach the subject we use the freedom to choose both integral equations and the Banach space. The K–S equation has caught our attention for the accompanying K–S operator is proved to be linear, bounded and independent of the parameter (i.e. chemical activity). Thus, the very point is how the K–S operator spectrum performs while approaching the thermodynamical limit. The only existing general result describing behaviour of the spectrum is the estimation of the K–S operator spectral radius by means of its norm. In the finite volume case Pastur [2] has shown that the spectrum coincides with the zeroes of the grand canonical partition function. Other results concern the properties of the spectrum but without its localization [6]. In that approach, the Banach space on which the K–S operator acts is so wide that the analysis of the spectrum is very complicated. In contrast, in the present paper we take the smallest possible Banach space which contains all the correlation functions. In addition, this space is cyclic for the K–S operator (Theorem 1) and obviously the K–S operator restricted to the cyclic space is still bounded with a norm not greater than the previous one. In Chapter 3, we show that the above restrictions

2

M. Gorzelańczyk

do simplify the spectral analysis of the K–S operator. Thus, from Theorem 2 it follows that we can investigate the spectral radius of the operator by means of its action on the cyclic vector. The latter takes particularly simple form for the K–S operator. These allow us to prove the following result (see Sect. 3 for notation and all the details): The sequence of the correlation functions tend to the thermodynamical limit uniformly on compact subsets for z −1 ∈ G1 ∪ G2 . So, the limit is the analytic function of z in the set G1 ∪ G2 . Since the set {z : 0 ≤ z ≤ [(e − 1)C]−1 } (where C is the constant defined by Eq. (3)) is contained in G1 ∪ G2 , from this result we conclude that the phase transition does not occur for the nonnegative chemical activity z less than [(e−1)C]−1 . According to [5], Sect. 4.5, we get that for the density ρ ≤ [eC]−1 there does not exist any phase transition either. It is worth stressing that the obtained region of analyticity is essentially larger than the previously known ρ ≤ [(e + 1)C]−1 (see [5]). Moreover, we have shown that all correlation functions are analytic in that region, whereas the result stated in [5] concerns only one particle correlation function (or a density). 2. Cyclic space of the K–S operator Let xi ∈ Rν for i = 1, . . . , n, (x)n = (x1 , . . . , xn ) and ϕ(x)n be a measurable complex-valued function on Rnν . Let Eξ (3) (ξ > 0) denote the Banach space of ∞ sequences ϕ = {ϕ(x)n }n=1 with the norm k ϕ k= sup ξ −n ess sup

(x)n ∈3n

n≥1

| ϕ(x)n |

(1)

where 3 is a bounded Lebesgue measureble set in Rν . Let K3 denote the KirkwoodSalsburg operator Z ∞ X 1 K(x1 , (y)m ) ϕ(y)m d(y)m , (K3 ϕ)(x)1 = χ(x)1 m! 3m m=1 ∞ 0 X 1 (2) (K3 ϕ)(x)n = χ(x)n e−βW (x1 ,(x)n ) m! m=0 Z × K(x1 , (y)m ) ϕ((x)0n , (y)m ) d(y)m for n ≥ 2 , 3m

where (x)0n = (x2 , . . . , xn ) , n X W (x1 , (x)0n ) = 8(x1 − xi ) , i=2

K(x1 , (y)m ) = χ(x)n =

m Y j=1 n Y

(e−β8(x1 −yj ) − 1) , χ3 (xi ) .

i=1

χ3 is the characteristic function of the 3 ⊂ Rν . We assume that the potential 8 is stable and regular so

Equilibrium Classical Statistical Mechanics of Continuous Systems

Z C=

3

| e−β8(x) − 1 | dx < ∞.

(3)

Under these assumptions the operator K3 is bounded [5]. Definition 1. Let Dξ (3) denote the subspace of Eξ (3) consisting of the following functions Z ∞ X al+p ∞ e−βU ((x)p ,(y)l ) d(y)l , (4) ρ3 {al }l=1 (x)p = l! 3l l=0

where U ((x)p , (y)l ) =

p X

8(xi − xj ) +

i<j

X

8(xi − yj ) +

l X

8(yi − yj ),

i<j

1≤i≤p 1≤j≤l

and the coefficients al are complex numbers such that (4) converges for each (x)p , p = 1, 2, ... . R Remark 1. If χ3 (x)p 3l e−βU ((x)p ,(y)l ) d(y)l = 0, we put al+p = 0 . According to [2] let us introduce two bounded operators defined as follows Z ∞ X 1 ϕ((x)p , (y)m ) d(y)m , (A3 ϕ)(x)p = m! 3m m=0 Z ∞ X (−1)m (B3 ϕ)(x)p = ϕ((x)p , (y)m ) d(y)m , m! 3m

(5)

A3 B3 = B3 A3 = I .

(6)

m=0

then we have Lemma 1. Dξ (3) is a Banach space.

∞ ∞ Proof. It suffices to prove the completeness of Dξ (3) . Let a sequence ρ3 {aql }l=1 q=1 be Cauchy. By the definition of the operators A3 , B3 we can write ∞ ∞ ∞ k e−βU (x)l (aql − apl ) l=1 k = k B3 (ρ3 {aql }l=1 − ρ3 {apl }l=1 ) k≤ (7) ∞ ∞ ≤ k B3 k k ρ3 {aql }l=1 − ρ3 {apl }l=1 k . Remark 1 guarantees that there exist points (x)l ∈ 3 such that e−βU (x)l > 0, so the ∞ ∞ sequences {aql }q=1 are Cauchy for any l . Let bl = limq→∞ aql , then ρ3 {bl }l=1 belongs ∞ to Eξ (3) since Eξ (3) is complete, clearly ρ3 {bl }l=1 belongs also to Dξ (3). ∞

The operator K3 acts in the space of the coefficients {al }l=1 in the following way: ∞

∞

K3 {al }l=1 = {al−1 }l=1 , where a0 = −

Z ∞ X am m=1

m!

3m

e−βU (x)m d(x)m .

(8)

4

M. Gorzelańczyk L

L−1

Lemma 2. Let the sequence {al }l=1 be finite. Then there exist a vector ρ3 {bl }l=1 and complex number β such that L

L−1

ρ3 {al }l=1 = K3 ρ3 {bl }l=1 + β 11,

(9)

where 11 = (1, 0, . . .) . Proof. Equation (9) is equivalent to Z ∞ X bm a1 = β − e−βU (x)m d(x)m , m! 3m m=1

al = bl−1 Thus we can find

L−1 {bl }l=1

for l = 2, . . . , L. L−1

and β such that ρ {bl }l=1

belongs to Dξ (3).

Theorem 1. Vector 11 is a cyclic vector of the space Dξ (3). Proof. The vectors with finite numbers of coefficients are dense in Dξ (3) . Using L Lemma 2 (L − 1) times the vector ρ3 {al }l=1 we get L

ρ3 {al }l=1 = P (K3 ) 11 ,

(10)

where P (K3 ) is the polynomial function of K3 .qed Remark 2. For the potentials with a hard core each vector belonging to Dξ (3) has the form (10). Proposition 1. The spectrum of the operator K3 on Dξ (3) coincides with the set of zeroes of the grand canonical partition function Q(z). Proof. It follows from the result of Pastur [2] and Lemma 1.

3. The Spectral Radius of the K–S Operator In this section we shall study the spectral properties of the operator K3 . Since, for z −1 ∈ ρ(K3 ), where ρ(K3 ) is the resolvent set ofK3 , we have ∞ zl = (z −1 I − K3 )−1 11 , (11) ρ3 Q3 (z) l=1 so we can find analytic properties of the correlation functions from the spectral analysis of K3 . Let T be a bounded, linear operator on a Banach space E with the cyclic vector 11. Let Pk (T ) denote a polynomial function of T , f a rational complex function and r(f (T )) be the spectral radius of the operator f (T ) . Theorem 2. We assume that for each vector ψ of a space E there exists a bounded sequence of operators {Pk (T )} such that ψ = lim Pk (T ) 11. k→∞

Then we have

r(f (T )) = lim sup k f (T )n 11 k1/n . n→∞

(12)


5

Proof. Let ψ ∈ E, lim sup k f (T )n 11 k1/n = r and k Pk (T ) k≤ N , so n→∞

lim supn→∞ k f (T )n ψ k1/n = lim supn→∞ limk→∞ k Pk (T )f (T )n 11 k1/n ≤ ≤ limn→∞ N 1/n lim supn→∞ k f (T )n 11 k1/n = r . Let us consider the following sequence: ∞ f (T )n , (r + )n n=1

(13)

(14)

where is an arbitrary positive number. By the Banach-Steinhaus theorem [4] the family of operators (14) is bounded or there exists ψ ∈ E such that f (T )n = ∞. sup ψ (15) (r + )n n But (13) implies that (15) is impossible. Thus there exists M > 0 such that f (T )n (r + )n ≤ M for any n . Hence

(16)

lim k f (T )n k1/n ≤ r + .

(17)

lim k f (T )n k1/n ≤ lim sup k f (T )n 11 k1/n ,

(18)

lim sup k f (T )n 11 k1/n ≤ lim k f (T )n k1/n lim k 11 k1/n

(19)

n→∞

Since is an arbitrary number n→∞

n→∞

we also have n→∞

n→∞

which completes the proof.

n→∞

By Remark 2 we see that the hard core potentials fulfill the assumption of Theorem 2 so we shall estimate the spectral radius of K3 for nonnegative potentials with the hard core. Using the following inequalities R | 3n K(x1 , (y)n ) d(y)n | ≤ C n , (20) e−βW (x1 ,(y)n ) ≤ 1, we observe that

k K3n 11 k≤ sup ξ −p C n−p+1 (An 11)p , p

where A is the following infinite matrix:  1 1/2! 1 1 0 1 A= 0 0  .. .. . .

1/3! 1/2! 1 1 .. .

1/4! 1/3! 1/2! 1 .. .

 ··· ··· ··· . ···  .. .

(21)

(22)

6

M. Gorzelańczyk

(An 11)p denotes a p coordinate of An 11. The norm of the matrix {ajk } generated by the vector norm sup | vp | is p

k {ajk } k= sup j

∞ X

| ajk | .

(23)

k=1

Norm (23) of the matrix A is equal to e . Therefore k K3n 11 k1/n ≤ e sup (ξ −p C n−p+1 )1/n . p≤n+1

(24)

When C ≥ ξ −1 , we get by Theorem 2 and the inequality (24) that

Otherwise, when C ≤ ξ −1 ,

r(K3 ) ≤ eC .

(25)

r(K3 ) ≤ eξ −1 .

(26)

Remark 3. As an application we derive interesting properties of the correlation functions for the values of chemical activity z on a real, positive axis. The correlation functions belong to Eξ (3) for λ = z −1 ≥ ξ −1 in the nonnegative potentials case. Thus without losing the generality we can assume that ξ −1 < C. Then the following set G1 = {λ : | λ |> eC} .

(27)

belongs to ρ(K3 ), where ρ(K3 ) is the resolvent set of K3 . If | λ − λ0 |> r(K3 − λ0 I), then ∞ X 1 (K3 − λ0 I)n . (28) (λI − K3 )−1 = (λ − λ0 )n+1 n=0

Let us assume λ0 = −C . Taking into account the inequality Z 0 | e−βW (x1 ,(x)n ) K(x1 , y1 ) dy1 + C |≤ C ,

(29)

3

we conclude in the same manner as above that the set G2 = {λ : | λ + C |> eC}

(30)

also belongs to the resolvent set of the operator K3 for ξ ≥ C −1 . Now we can formulate the technical lemma. Lemma 3. Let D be the compact subset of the set G1 ∪ G2 . If z −1 ∈ D then zl zl − ρ3j ] k= 0 , lim k χ30 [ρ3k k,j→∞ Q3j (z) l=1 Q3j (z) l=1

(31)

S∞ 0 ν where 31 ⊂ 32 ⊂ . . . , k=1 3k = R are Lebesgue measurable and 3 is an arbitrary Lebesgue mesurable compact set contained in Rν .


7

Proof. Assume that z −1 belongs to the set G2 . Since the series (28) is absolutely convergent and the estimation on the spectral radius does not depend on 3, then for any ε > 0 there exists a number N such that for k, j > N , n l o PN z zl 1 0 − ρ3j Q3 (z) ]k≤ k χ3 [ρ3k Q3 (z) n=1 |λ−λ0 |n+1 j k l=1 (32) l=1 × k χ30 [(K3k + CI)n 11 − (K3j + CI)n 11] k +ε, where λ = z −1 . It is easy to see that the expression k χ30 [(K3k + CI)n 11 − (K3j + CI)n 11] k

(33)

tends to zero for k, j → ∞ for any n , so we can estimate (32) by 2ε for sufficently large k, j. When z −1 belongs to G1 we put λ0 = 0 and complete the proof in the same way as above. Applying Lemma 3, we obtain Theorem 3. The sequence of the correlation functions tends to the thermodynamical limit uniformly on compact subsets for z −1 ∈ G1 ∪ G2 . So, the limit is the analytic function of z in the set G1 ∪ G2 . Remark 4. For a pure hard core interaction we can assume C = 1. Then the density means a ratio between the total volume occupied by spheres and the volume of the domain where the spheres are contained. We note that the estimations on the density ρ are consistent with the conjecture that the phase transition occurs for ρ = 1/2 (see [1]) in the hard core gas in dimension greater than 1. References 1. Gorzelańczyk, M.: Phase trasitions in the gas of hard core spheres. Commun. Math. Phys. 136, 43–52 (1991) 2. Pastur, L.A.: Spectral theory of the Kirkwood-Salsburg equation in finite volume. Theor. Math. Phys. 18, 233 (1974) 3. Petrina, D.Ya., Gerasimenko, V.I., Malyshev, P.V.: Mathematical fundation of classical statistical mechanics. Kiev (1985), English transl. Adv. Stud. Contemp. Math. Vol 6, New York: Gordon and Breach, 1989 4. Rudin, W.: Functional analysis. New York, Mac Graw-Hill, 1973 5. Ruelle, D.: Statistical mechanics. New York, Benjamin, 1969 6. Zagrebnov, V.A.: Spectral properties of Kirkwood–Salsburg and Kirkwood–Ruelle operators. J. Stat. Phys. 27, N 3, 577–591 (1982) Communicated by D. Brydges

Commun. Math. Phys. 192, 9 – 28 (1998)

Communications in


Chaotic Expansions of Elements of the Universal Enveloping Superalgebra Associated with a Z2 -graded Quantum Stochastic Calculus T. M. W. Eyre? Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom. E-mail: [email protected] Received: 28 January 1997 / Accepted: 10 June 1997

Abstract: Given a polynomial function f of classical stochastic integrator processes 3 = (31 , . . . , 3N ) whose differentials satisfy a closed Ito multiplication table, we can express the stochastic derivative of f as df (3) = f (3 + d3) − f (3). We establish an analogue of this formula in the form of a chaotic decomposition for Z2 graded theories of quantum stochastic calculus based on the natural coalgebra structure of the universal enveloping superalgebra.

1. Introduction In [HP1] a theory of quantum stochastic calculus was devised. The Bosonic and Fermionic versions of this calculus were unified for the one-dimensional case in [HP2] by means of the formula (1) dB = (−1)3 dA. The full Ito algebra of one-dimensional Fermionic quantum stochastic calculus consists of the integrators d3, dT, dB, dB † , these being the gauge, time, Fermionic annihilation and Fermionic creation integrators respectively. Of these, d3, dT commute about integrands whereas dB, dB † commute or anticommute about an integrand X depending on whether (−1)3 X(−1)3 = X or −X respectively. The Fermionic quantum Ito table [HP2] shows that the conditions for a Z2 -graded algebra are satisfied by this Ito algebra when the obvious assignments dB, dB † odd, dT, d3 even are made. The general theory of Z2 -graded algebras is described in [C]. ?

T. M. W. E. is supported by an EPSRC studentship.

10

T. M. W. Eyre

In [EH], an extension of (1) gave rise to a Z2 -graded theory of multidimensional quantum stochastic calculus. Here we have a mixture of multidimensional Bosonic and Fermionic creation and annihilation integrators along with graded multidimensional gauge integrators. The commutation and anticommutation properties of these integrators provide representations of Lie superalgebras, a class of structures which has received considerable attention since its introduction in 1977 by V. Kac [K]. In particular, all Lie superalgebras of the form gl(N, r) can be represented in this Z2 -graded theory by means of the Z2 -graded pure gauge processes. The purpose of this paper is to give an explicit formula for the chaotic expansion of elements of the universal enveloping superalgebra of the Lie superalgebra associated with the Z2 -graded multidimensional quantum stochastic calculus. This has been done for the ungraded case in [HPu]. Section 2 gives an outline of Z2 -graded multidimensional quantum stochastic calculus and gives some elementary results to be used further on. In Sect. 3 we work with an abstract superalgebra-with-involution, I, called the Ito superalgebra. In the (Chevalley) tensor superalgebra of I denoted by T (I) and defined as T (I) := C ⊕ I ⊕ (I⊗I) ⊕ (I⊗I⊗I) ⊕ · · · we construct a product denoted ?. Some results concerning ? are proved, including associativity. Sect. 4 takes I to be the superalgebra of Z2 -graded quantum stochastic differentials under Ito multiplication. A map I is defined on T (I) that corresponds to iterated integration against the integrators of an element of T (I) and it is shown that, in a certain sense, for a, b ∈ T (I), I(a)I(b) = I(a ? b). In Sect. 5 it is shown that the map I is injective. This is a crucial result for Sect. 6 which gives the main result of the paper, this being the existence of a chaos expansion χ from the universal enveloping superalgebra U associated with I to T (I). An explicit formula is given for this expansion. Some of the arguments given in this paper are straightforward adaptations of those given in [HPu] but there are significant deviations, most notably in Sect. 6. 2. Preliminaries 2.1. Definitions and elementary results. Let M0 (N ) be the space of all complex (N + 1) × (N + 1) matrices, indexed from 0 to N , with multiplication defined by A.B = A1B, where 1 ∈ M0 (N ) has entries that are all zero except for the 1, . . . , N th diagonal entries: 0 0 0 ··· 0 0 . 1 =  ..  0 0

1 .. .

··· ···

0 .. . 0 0

··· .. . 1 0

0 ..  . .  0 1

This space may be used to index the integrators of ungraded quantum stochastic calculus [HP1]. Denote by Eβα the element of M0 (N ) that has 1 in the (α, β)th entry and is zero elsewhere. Note that Greek letters indicate indices varying from 0 to N and Roman

Chaotic Expansions of Universal Enveloping Superalgebra

11

letters indicate indices varying from 1 to N . The Einstein repeated suffix summation convention is also in force throughout this paper. To the element Eβα of M0 (N ) there corresponds the quantum stochastic process 3Eβα which will be denoted by 3α β and which will now be defined. Let the canonical basis element (0, . . . , 0, 1, 0, . . . , 0) of CN be denoted ei , where the 1 appears in the ith position. We define 30j (t) for j ≥ 1 and t ≥ 0 by its action on an arbitrary exponential vector e(f ) in the Boson Fock space 0(L2 (R+ ; CN )): 30j (t)e(f ) =

d e(f + χ[0,t] zej )|z=0 . dz

(2)

For i, j such that 1 ≤ i, j ≤ N we define 3ij (t) for all t ≥ 0 by 3ij (t)e(f ) =

d e(eχ[0,t] z|ej ><ei | f )|z=0 , dz

(3)

where |ej >< ei | is the Dirac dyad acting in L2 (R+ ; CN ) that maps a function f to the function that has zero for all components except the j th component which is equal to the ith component of f . The process 3i0 (t), where i ≥ 1 is defined by Z t i f i (s) ds e(f ). (4) 30 (t)e(f ) = 0

Throughout this paper the notation f α indicates the αth component of f = (f 1 , . . . , f N ) ∈ L2 (R+ ; CN ). We will denote by fα the complex conjugate f¯α of f α . For α = 0 we define f 0 = f0 = 1. In the remaining case where α = β = 0 we define 300 (t) by 300 (t)e(f ) = te(f ).

(5)

The vectors {e(f ) : f ∈ L2 (R+ ; CN )} form a total set in 0(L2 (R+ ; CN )). The space of all finite linear combinations of the exponential vectors is known as the exponential domain and denoted E. All the 3α β are defined on E. The processes defined in (2) are † known as the N -dimensional creation processes and may also be written as Ai (t). The processes defined in (3) are called the N -dimensional gauge processes. The processes defined in (4) are the N -dimensional annihilation processes and may also be written Ai (t). The process (5) is known as the time process. For an arbitrary element A of M0 (N ) which decomposes as λβα Eβα the process (3A (t))t≥0 is defined to be the process (λβα 3α β (t))t≥0 . The algebra M0 (N ) is associative and therefore if it is equipped with a Lie bracket defined as [A, B] := A.B − B.A = A1B − B1A we obtain a Lie algebra M0 (N )Lie which we write as gl0 (N ). The processes and integrators of quantum stochastic calculus form a representation of gl0 (N ), that is to say [3A (t), 3B (t)] = 3[A,B] (t), [d3A (t), d3B (t)] = d3[A,B] (t).

(6)

Eq. (6) involves unbounded operators and so the composition that might be considered implicit in the bracket must be interpreted in terms of adjoints and the inner product. When doing this it is convenient to restrict our attention to the processes {3α β : 0 ≤

12

T. M. W. Eyre

α, β ≤ N }. It is straightforward to show that for all α, β the restriction to E of the α β † ˆα adjoint 3α β (t) of 3β (t) is 3α (t). We define the Evans delta δβ to be zero for all α, β except where α = β 6= 0 in which case it is equal to 1. Note that 1 = {δˆβα }. With this notation, (6) is to be interpreted rigorously as < 3βα (t)e(f ), 3γδ (t)e(g) > − < 3δγ (t)e(f ), 3α β (t)e(g) > γ γ = < e(f ), (δˆδα 3β (t) − δˆβ 3α δ (t))e(g) > .

(7)

The right hand side of (7) may be expressed as < e(f ), 3[Eβα ,Eδγ ] e(g) > . In [EH] it was shown that the one-dimensional Fermionic theory of quantum stochastic calculus may be extended to an arbitrary number of dimensions. This extension provides representations of a broad category of Lie superalgebras. For convenience we give a brief overview of Lie superalgebras here. A superalgebra or Z2 -graded algebra is an associative algebra A that may be decomposed as an internal direct sum A = A0 + A1 , where A0 and A1 satisfy the inclusions A0 A0 , A1 A1 ⊂ A0 ,

A0 A1 , A1 A0 ⊂ A1 .

(8)

We say that A0 is the even subspace and A1 is the odd subspace. If an element a ∈ A is an element of A0 then we say that a is even or of parity 0. Similarly, if a is an element of A1 then we say a is odd or of parity 1. If a is odd or even then it is said to be homogeneous or of definite parity. An element of A is not, in general, of definite parity but any element of A may be expressed uniquely as the sum of two such elements. On elements of A that are of definite parity we define the parity function σ that maps a to its parity, i.e., σ(a) = 0 if a is even and σ(a) = 1 if a is odd. Note that σ is only defined on A0 ∪ A1 , not on all of A. It is well known that an arbitrary associative algebra may be equipped with the standard commutator bracket to give a Lie algebra. An arbitrary associative superalgebra A may be similarly equipped with a superbracket to give a Lie superalgebra. The bracket will be denoted { . , . } and is defined by a bi-linear extension of the following rule on a, b ∈ A with a, b of definite parity: {a, b} = ab − (−1)σ(a)σ(b) ba.

(9)

The bracket acts as a commutator when at least one of a, b is even and as an anticommutator when a and b are both odd. When the bracket is applied to arbitrary elements of A we can see that it acts as part commutator and part anticommutator. A more detailed description of Lie superalgebras, including the full definition, may be found in [K, S]. If an integer r with 0 ≤ r < N is fixed then the algebra M0 (N ) described above may be graded as M0 (N )0 + M1 (N )1 , where M0 (N )0 is defined to be the subspace consisting of all matrices of the form r   ∗ ∗ ··· ∗ 0 ··· 0 ∗ ∗ ··· ∗ 0 ··· 0  . . .  . . ... ... . . . ...   .. ..    r  ∗ ∗ · · · ∗ 0 · · · 0 , 0 0 ··· 0 ∗ ··· ∗  . . .  . . . . ... ... . . . ...  . . 0 0 ··· 0 ∗ ··· ∗

Chaotic Expansions of Universal Enveloping Superalgebra

13

the entries marked ∗ taking any value in C. Similarly we define M0 (N )1 to be the subspace of all matrices of the form 

0 0 .  ..  r 0 ∗ . . .

∗

0 0 .. . 0 ∗ .. .

∗

··· ··· .. . ··· ··· .. . ···

r 0 0 .. . 0 ∗ .. .

∗

∗ ∗ .. .

∗ 0 .. . 0

··· ··· .. . ··· ··· .. . ···

∗ ∗ .. .

∗ 0 .. .

     .    

0

It is easy to see that M0 (N )0 and M0 (N )1 satisfy the inclusions (8). If the superbracket is defined using the established multiplication for M0 (N ) in (9) then we have derived a Lie superalgebra from M0 (N ) which we denote gl0 (N, r). This Lie superalgebra is the foundation of Z2 -graded quantum stochastic calculus, a theory introduced in [EH]. An outline of this theory will now be given. Let the value of r used in gl0 (N, r) be fixed once and for all. We define the grading process G on 0(L2 (R+ ; CN )) by its action on an arbitrary exponential vector e(f ). For time t ≥ 0 and f = (f 1 , . . . , f N ) ∈ L2 (R+ ; CN ) we set G(t)e(f ) = e(χ[0,t] (f 1 , . . . , f r , −f r+1 , . . . , −f N ) + χ(t,∞) f ). The totality of the e(f ) in 0(L2 (R+ ; CN )) and the evident isometry of each G(t) means that G is defined on the whole of that space. Note that for each t the operator G(t) is self-adjoint, involutive and leaves E invariant. It is clear that for arbitrary α, β the element Eβα is of definite parity. Thus we define α σβ to be 0 or 1 depending on whether Eβα is even or odd respectively. Note that for all α, β we have σβα = σαβ . We may now define the integrator processes of Z2 -graded quantum stochastic calculus. The integrators are denoted dΞβα (t), where 0 ≤ α, β ≤ N and are defined by α dΞβα (t) = Gσβ (t) d3α β (t). This definition is an N -dimensional extension of the Boson-Fermion unification formula dB = (−1)3 dA of [HP2]. From this definition we may construct the processes {Ξβα : 0 ≤ α, β ≤ N } by defining for all α, β and for all t ≥ 0, Z t Ξβα (t) = dΞβα (s). 0

α We see that when σβα = 0 the differential dΞβα is equal to d3α β and the process Ξβ is α α α α α equal to 3β . When σβ = 1 the differential dΞβ is equal to G d3β and the process Ξβ (t) Rt is equal to 0 G(s) d3α β (s). For an arbitrary element A of the Lie superalgebra gl0 (N, r) we may write A = λβα Eβα . This enables us to define ΞA = λβα Ξβα . In [EH] it was shown that the ΞA and their differentials form representations of the Lie superalgebra gl0 (N, r), that is to say that for arbitrary A, B ∈ gl0 (N ) we have

{ΞA , ΞB } = Ξ{A,B} , {dΞA , dΞB } = dΞ{A,B} .

(10)

14

T. M. W. Eyre

As with the ungraded case, the left-hand side of (10) must be interpreted in terms of adjoints and inner products. The self-adjointness of each G(t) provides us with Ξβα (t)† = Ξαβ (t) for each t ≥ 0 so (10) is to be interpreted rigorously as α

γ

< Ξαβ (t)e(f ), Ξδγ (t)e(g) > −(−1)σβ σδ < Ξγδ (t)e(f ), Ξβα (t)e(g) > γ

α = < e(f ), (δˆδα Ξβγ (t) − (−1)σβ σδ δˆβγ Ξδα (t))e(g) >,

where t ≥ 0 is arbitrary and f, g are arbitrary elements of L2 (R+ ; CN ). In what follows we will need to make use of the First and Second Fundamental Formulae of [HP1]. In the Z2 -graded case the First Fundamental Formula clearly takes the form Z t Z t α P (s) dΞβα (s)e(g) >= fβ (s)g α (s) < e(f ), P (s)Gσβ (s)e(g) > ds, < e(f ), 0

0

(11) where t ≥ 0, 0 ≤ α, β ≤ N , f, g ∈ L2 (R+ ; CN ) and P is an integrable process. If Q is also an arbitrary integrable process then the Z2 -graded version of the Second Fundamental Formula is easily seen to be Z t Z t β P (s) dΞα (s)e(f ), Q(s) dΞδγ (s)e(g) >= < 0

Z

t

= Z

Z

fδ (s)g γ (s)
ds β σα

α

fβ (s)g (s) < P (s)G

+ 0

+ δˆδα

Z

s

(s)e(f ), 0

Z

t

β

Q(r) dΞδγ (r)e(g) > ds γ

fβ (s)g γ (s) < P (s)Gσα e(f ), Q(s)Gσδ (s)e(g) > ds.

(12)

0

Eq. (12) may be summarised as dEF = E dF + (dE)F + dE.dF,

(13)

dΞA .dΞB = dΞA.B = dΞA1B ,

(14)

where this being the quantum Ito multiplication. Details concerning (13) and (14) can be found in [HP1]. Quantum stochastic integrals are themselves integrable processes and therefore we may form iterated quantum stochastic integrals of the form Z dΞA1 (t1 ) · · · ΞAn (tn ). 0 0, for n > 0, R(1) = 0 and R(0) > 0. If one admits that R(q n ) = 0, for some n ∈ N, then the algebra AK will be realized by endomorphisms of a finite dimensional vector space. We shall not consider this case in our paper. Once R is given we define the K--derivative according to ∂R : =∂q

1−q 1−q R(Q) = R(qQ)∂q . 1−Q 1 − qQ

(104)

The K-derivative is a natural generalization of K-derivative (95) described in the Subsect. C of the previous section. Likewise in that case one can carry out the program of finding out explicit formulae for the other objectives of the theory, i.e. the exponential function the relation between the annihilation and creation operators, the reproducing measure and the ∗-product. The reason for this is the fact that (104) gives an explicit functional realization of the K-derivative. The main results of this section are: a generalization of the q-binomial theorem (see Proposition 1); the explicit realization (see (136)) of the ∗-product formula (41) for “R-class” of physical systems and at last – the normal ordering for the operator AA† . Applying the derivative (104) to monomials z n one finds qn = R(q n ).

(105)

Then, from (14) and (21) one has ExpR (v, z) =

∞ X n=0

1 (vz)n , R(q) · · · R(q n )

(106)

where we put c0 = 1, which is equivalent to the normalization ExpR (0) = 1. By definition we assume that for n = 0 the coefficient in (106) is equal 1. The convergence radius for the series (106) is equal to R(0). Thus it is infinite, for the case when N > 0. Since, ExpR depends on the product vz only, we shall put v = 1. Subsequently the substitution ExpR and ∂R into Eq (15) gives: [

1−q R(qQ)∂q ExpR ](z) = ExpR (z). 1 − qQ

(107)

After simple calculation one can then reduce (107) to the following q-difference equation R(Q)ExpR (z) = zExpR (z), (108) of infinite rank in general.

200

A. Odzijewicz

Proposition 1. The following generalization of the q-binomial theorem is true: ExpR (z) =

∞ X n=0

∞

Y 1 n z = (1 − F (zq n )G(Q)) · 1, R(q) · · · R(q n )

(109)

n=0

where z , z − R(0) (Q − 1)R(qQ) + (1 − qQ)R(0) G(Q) = QR(qQ) F (z) =

(110)

for N = 0, and F (z) = z, qQ − 1 G(Q) = QR(qQ)

(111)

for N > 0. Proof. From the proceeding considerations it follows that the left-hand side of formula (109) is a solution of Eq. (108). However let us solve it again by the iteration method. In order to do so, we reformulate (108) ending up with the following form: ExpR (z) = [(1 − F (z)G(Q))ExpR ](qz),

(112)

where functions F and G are given by (110) and (111), in dependence whether the function R is regular (N = 0) or singular (N > 0) for z = 0. Since 1 (1 − Q), (113) ∂q = z(1 − q) one can rewrite (108) as follows: 1 1 1 1 − qQ − ExpR (qz) + z ExpR (z) [(1 − Q)ExpR ](z) = z R(qQ) R(0) Q R(0) (114) for N = 0, or 1 − qQ ExpR (qz) (115) [(1 − Q)ExpR ](z) = z R(qQ)Q for N > 0. The above expressions can be transformed into 1 1 − qQ z ExpR (z) = 1 − 1+ R(0) −1 ExpR (qz) z − R(0) Q R(qQ) or

ExpR (z) =

1−z

qQ − 1 QR(qQ)

(116)

ExpR ](qz)

(117)

respectively. This proves the formula (112). We now arrive at the right-hand side of the identity (109) via iteration of (112).

Quantum Algebras and q-Special Functions

201

Substituting the function 1−x 1−q

R(x) =

(118)

into (109) one obtains the Euler formula ∞

X 1 1 = zn. (z; q)∞ (q; q)n

(119)

n=0

If one puts x−1 x into (109), one obtains another Euler formula R(x) =

(z; q)∞ =

n ∞ X (−1)n q ( 2 )

(q; q)n

n=0

For R(x) =

(120)

zn.

1−x 1 − aq x

(121)

(122)

the identity (109) reduces to the q-binomial theorem ∞ X (a; q)n n=0

(q; q)n

zn =

(az; q)∞ . (z; q)∞

(123)

Therefore, the identity (109) is a generalization of classical identities well known in the theory of basic hypergeometric series (see [2, 7]), as expected. Let us now restrict our attention to the subcase when R is a rational function regular at infinity and without a pole at zero. If it possesses (s + 1) poles and r singularities one can represent it in the following form R(x) = rRs (x): =

(1 − x)(1 − b1 q −1 x) · · · (1 − bs q −1 x) , (−x)s−r+1 (1 − a1 q −1 x) · · · (1 − ar q −1 x)

(124)

where a1 , . . . , ar , b1 , . . . , bs are such numbers that rRs (q n ) > 0 for n ∈ N and s−r+1 ≥ 0. Applying the derivative ∂r,s : =∂rRs to the monomial z n one finds the q-factors (1 − q n )(1 − b1 q n−1 ) · · · (1 − bs q n−1 ) , (1 − a1 q n−1 ) · · · (1 − ar q n−1 )(−q)1+s−r

(125)

i1+s−r h n 1 (a1 ; q)n · · · (ar ; q)n (−1)n q ( 2 ) = , q1 · · · qn (q; q)n (b1 ; q)n · · · (bs ; q)n

(126)

qn = and thus c2n =

Hence, we find that the K-exponential function for this subcase is given by the basic hypergeometric series r8s

202

A. Odzijewicz

Expr,s (v, z) = hKr,s (v) |Kr,s (z) i = ∞ i1+s−r h X n (a1 ; q)n · · · (ar ; q)n (vz)n = (−1)n q ( 2 ) = (q; q)n (b1 ; q)n · · · (bs ; q)n n=0 a1 , · · · , a r ; q, vz . = r8s b1 , · · · , b s

(127)

(See the book by G. Gasper and M. Rahman [7] for the definition of r8s and for its fundamential properties.) Then, for r = s + 1 the coherent states map Kr,s and thus the exponential function Expr,s are defined on the unit disc D1 . If r < s + 1, they are defined on the whole complex plane. From the above it follows that the exponential function ExpR is a natural generalization of the basic hypergeometric series. Therefore we shall call it also the generalized basic hypergeometric series. The basic hypergeometric series r8s , being a K-exponential function, satisfies Eq. (108), which in that case assumes the form (−1)r

s X

r+n r+1−n (−1)n σn (b1 , . . . , bs )[8 z) − q8 z)] = r s (q r s (q

n=0

= (−1)s+1

r X

n+s+1 σn (a1 , . . . , ar )q n+s+1 z8 z), r s (q

(128)

n=0

where σn (y1 , . . . , ym ) is the symmetric polynomial defined by the identity (1 − y1 x) · · · (1 − ym x) =

m X

(−1)n σn (y1 , . . . , ym )xn .

(129)

n=0

Let us now discuss briefly the relation between A, A† and Q, where Q is defined by QK(z): =K(qz). From (19) and (105) one finds the defining relations AA† = R(qQ), A† A = R(Q)

(130)

for the quantum algebra AKR . 2 Using the above identities and ||A† A|| = ||A|| one obtains, for N = 0, the following expression for the norms r (131) ||A|| = ||A† || = sup R(q n ). n∈N

Due to (130) one finds also that the annihilation and creation operators generating the quantum algebra AKR are unbounded if N > 0. Considering A and A† as a mutually conjugate non-commuting set of coordinates on the quantum curve (23) we could interpret (130) as a parametrization of the last one by the operator Q. In such a way the meromorphic function R “uniformizes” the considered curve. So, by analogy with algebraic geometry, where algebraic curves are “coordinatized” by the field of rational functions on them (see [31]), we will interpret the generalized Heisenberg algebra AK as a “coordinatization” of the quantum curve to which it is related.


203

In [17], the authors call the algebra Aµ,r (from the subcase B. in Sect. 4) – the “quantum disc”. However, we think that it is more suitable to reserve this term for the set of coherent states K(z), z ∈ DR , and to interpret the map K : DR → H \ {0} as a quantization of the classical states z ∈ DR , (see [25] for an exhaustive disscussion). Let us remark here that K(z), z ∈ DR , are the eigenvectors of the commutative subalgebra of AK and are generated by the annihilation operator A. The ∗-product formula (41) for the quantum algebra AKR can be expressed in the functional way in terms of the q-derivative ∂q and the operator Q. In order to do this we expand the function g in a Taylor series with respect to the anti-holomorphic coordinate g(z, z) =

∞ X

gn (z)z n ,

(132)

n=0

and use the generalized Leibniz formula for the derivative ∂R , n X n (qQ; q)n−k Qk n−k n n ∂R (g · f )(x) = R(qQ) · · · R(q Q) ∂ g (x) × k q R(qQ) · · · R(q n−k Q) R k=0 (qQ; q)k k ∂ f (x). (133) × R(qQ) · · · R(q k Q) R Since

n−k ∂R ExpR (z, z) = z n−k ExpR (z, z)

1 k n n (1 − q)k ∂R z = R(qQ) · · · R(q k Q)z n−k , k!q k q (qQ; q)k after a simple calculation one comes to the following ∗-product formula:

(134)

and

∞

(135)

∞

X 1 X k 1 (1 − q)k ∂q (gn (z)z n ) × ExpR (z, z) k!q (qQ; q)n n=0 k=0 (qQ; q)n−k Qk k Exp (∂ f ) (z, z), (136) ×R(qQ) · · · R(q n Q) q R R(qQ) · · · R(q n−k Q)

(f ∗ g)(z, z) =

where ∂q and Q act on the coordinate z and ∂q acts on the coordinate z. In the special case when f (z) = z and g(z) = z, one finds from (41) the normal ordering for the operator AA† , i.e. AA† = : F (A† , A) :

(137)

where the function F is given by F (z, z) =

∞ X 1 ExpR (q k z, z) ∂R (zExpR (z, z)) = . rk q k ExpR (z, z) ExpR (z, z)

(138)

k=−N

For the definition of rk see formula (103). In Sect. 6 we will find reproducing measure (see (182) and (187)) for R which satisfies: R(0) = 1, R(q n ) > 0 and the function R(x; q)∞ := R(x)R(qx) · · · R(q n x) · · ·

(139)

is analytic without the poles in the disc D% for some % > q. So in this case one can recalculate the star product (136) using (51) with the measure dµS given by (187).

204

A. Odzijewicz

6. Generalized Basic Hypergeometric Series In this section we shall study some properties of the generalized basic hypergeometric series 8R , i.e. the exponential function ExpR defined in Sect. 5. We shall restrict our attention to the subcase R = S, where S satisfies conditions (167). These conditions are very natural and not very restrictive. Hence, they admit an ample class of meromorphic functions under consideration. We will prove the identities (177) and (179) which, respectively, provide us with the meromorphic continuation on the whole complex plane and Mittag–Leffler decomposition of generalized basic hypergeometric series ExpS . In order to achieve this we solve the infinite system of linear Eqs. (140). As a result one obtains additionally new identities (173) and (174) with the S function involved. With help of these one may then express the function ExpS in terms of the integral formula (188) and the operator formula (189). Finally, due to the identity (180) we find the reproducing measure (187) for the kernel ExpS (zv). A key role in the subsequent considerations will be played by the equation ∞ X

q (n+1)(k+1) αk = βn ,

∀n∈N∪{0}

(140)

k=0 ∞ for the complex sequence {αn }∞ n=0 , when the sequence {βn }n=0 is fixed. Below we shall assume that 0 < q < 1. In that case the infinite matrix (n+1)(k+1) ∞ (141) q n,k=0

defines a self-adjoint Hilbert-Schmidt operator in the Hilbert space l2 with the kernel equal to {0}. However, we will look for the solutions from a class of sequences which contains all convergent sequences and thus the sequences belonging to l2 . The sequences ∞ {αn }∞ n=0 , which solve Eq. (140) for some sequence {βn }n=0 form a vector space A. From the Cauchy criterion one obtains the necessary lim sup k→∞

p 1 k |αk | ≤ q

(142)

p 1 k |αk | < q

(143)

and sufficient lim sup k→∞

∞ conditions for a sequence {αk }∞ k=0 to be a solution of (140) for a given {βn }n=0 . In accordance with this, let A0 ⊂ A denotes a set of sequences satisfying the condition (143). ∞ The set A0 is a star-shaped set (i.e. {cαn }∞ n=0 ∈ A0 if {αn }n=0 ∈ A0 ) and ∞ ∞ {|α|n }n=0 ∈ A0 if {αn }n=0 ∈ A0 . The vector space of all convergent sequences is contained in A0 .

Proposition 2. Equation (140) has solution from the A0 if and only if the power series β(z) =

∞ X n=0

βn z n

(144)


205

admits the meromorphic extension on the whole complex plane with the Mittag-Leffler decomposition β(z) =

∞ X

αk

k=0

1 q −(k+1) − z

(145)

∞ ∞ such that {αn }∞ n=0 ∈ A0 . The sequence {αn }n=0 ∈ A0 solves Eq. (140) with {βn }n=0 given by (144).

Proof. From the Cauchy criterion one obtains lim sup

p k

k→∞

|αk |
1. In particular we see from (160) that the function α0 = Qr α solves the equation α0 (q n+1 ) = βr+n

(161)

α(q n+1 ) = βn .

(162)

if α solves the equation

Thus, instead of the full system (162) one can solve a reduced system (161) and one obtains α from α = Q−r α0 . Since R = q r R0 , where R0 is the convergence radius for α0 , the function α0 belongs to the domain of the operator Q−r . If βni = 0 for some infinite subset of indices ni , then βn = 0 for all n ∈ N ∪ {0}. So, in the nontrivial case, there exists r such that βn 6= 0 for n ≥ r. From the above consideration it follows that the case βn 6= 0, for n ∈ N ∪ {0}, is the crucial one. Therefore, we will consider that case below. If α0 = · · · = αN −1 = 0 and αN 6= 0, then the function S defined by α(z) =

1 S(z)α(qz) q N +1

(163)

is meromorphic on the disc D% , where R > % > q, and has inside D% a finite number of singularities. It is so, since α ∈ O(D% ) and % < R. In addition S(0) = 1, and from βn 6= 0 it follows that S(q n ) 6= 0 for n ∈ N ∪ {0}. The function S(·; q)n defined for n ≥ 1 by S(z; q)n = S(z)S(qz) · · · S(q n−1 z), and for n = 0 by S(z; q)0 = 1, is holomorphic on the D% for sufficiently large n. Iterating (163) we find that

(164)

208

A. Odzijewicz

α(z) = S(z; q)∞ z N +1 .

(165)

Thus from (162) and (165) one obtains βn =

S(q; q)∞ n+1 N +1 (q ) . S(q; q)n

(166)

Concluding, we have the following statement. Proposition 6. Equation (163) for the holomorphic function α, with S being a fixed meromorphic function on DR , R > q, which satisfies the conditions S(·; q)∞ ∈ O(DR ), S(q n ) 6= 0 and S(0) = 1,

(167)

is equivalent to Eq. (140) with βn 6= 0. The formula (166) gives the condition for {βn } which is a solvability condition for the system (140). Hence, Eq. (140) is solved by the coefficients of the Taylor expansion of S(z; q)∞ in z = 0. Since S(0) = 1, the function S is holomorphic in some neighbourhood of z = 0. Taking its expansion S(z) =

∞ X

sk z k ,

(168)

k=0

one finds from (163) the recurrence αN +n =

n−1 1 X sn−k q k αN +k 1 − qn

(169)

k=0

for n ≥ 1. Let us recall here that α0 = · · · = αN −1 = 0. This reccurence is solved by X αN +n = αN xn q −n σi1 ...in (x1 , . . . , xn−1 )si11 . . . sinn , (170) i1 +2i2 +···+nin =n

where xk =

qk 1−q k

and the polynomial σi1 ...in (x1 , . . . , xn−1 ) is defined as a linear j

n−1 combination of the monomials xj11 . . . xn−1 with coefficients equal to 0 or 1 and j1 + j2 + · · · + jn−1 = i1 + i2 + · · · + in − 1. The explicit form of σi1 ...in (x1 , . . . , xn−1 ) can be calculated from X σi1 ...in (x1 , . . . , xn−1 )si11 . . . sinn =

i1 +2i2 +···+nin =n

=

n−1 X

(

X

σi1 ...ik (x1 , . . . , xk−1 )si11 . . . sikk )sn−k .

(171)

k=0 i1 +2i2 +···+kik =k

One finds the factor αN from α(q) = β0 = S(q; q)∞ q N +1 .

(172)

Now, because of the equivalence of Eqs. (140) and (163) one obtains – as a result of the comparison of (152) with (170) – the following identities:


209

n k ∞ l S(q; q)∞ (−1)n q ( 2 ) X X (−1)k q ( 2 ) ( )q −ln = (q; q)∞ (q; q)n S(q; q)l−k (q; q)k

l=0 k=0

q = σ0 1 − qn

X

σi1 ...in (

i1 +2i2 +···+nin =n

q n−1 q ,..., )si1 . . . sinn . (173) 1−q 1 − q n−1 1

Using (165) and (152) we find other identities S(z; q)∞ =

n k ∞ ∞ l S(q; q)∞ X (−1)n q ( 2 ) X X (−1)k q ( 2 ) ( )(q −l z)n = (q; q)∞ (q; q)n S(q; q)l−k (q; q)k

n=0

=

S(q; q)∞ (q; q)∞

∞ X l X l=0 k=0

l=0 k=0

k 2

(−1)k q ( ) (q −l z; q)∞ . S(q; q)l−k (q; q)k

(174)

Because of (166) the power series β assumes the form β(z) = S(q; q)∞ q N +1 ExpS (q N +1 z),

(175)

where ExpS is the exponential function defined by (106). The exponential functions ExpS for which S satisfies the conditions (167) form an ample subclass of the class of all generalized exponential functions defined in Sect 5. For example, the conditions (167) are satisfied by the rational function rSs

(z) =

(1 − z)(1 − b1 z) · · · (1 − bs z) (1 − a1 z) · · · (1 − ar z)

(176)

if bi q n 6= 1, for i = 1, . . . , s and n ∈ N ∪ {0}, and min |a1i | =: % > q. Let us remark that after setting some parameters a1 , . . . , ar and b1 , . . . , bs to zero in (124) one can consider rSs as a special case of the rational function . If S ∈ O(DR ) then s+1Rs S(·; q)∞ ∈ O(DR ). Therefore, functions holomorphic on DR , where R > q, such that S(0) = 1 and S(q n ) 6= 0 also satisfy (167). The other properties of ExpS follow from Proposition 2. Namely, from (148) and (149) we obtain for |z| < 1 ExpS (z) =

γS (z) , (z; q)∞

(177)

where γS (z) =

∞ X n X ( n=0 k=0

k

(−1)k q ( 2 ) )z n (q; q)k S(q; q)n−k

(178)

is analytic on the whole complex plane. Hence the right-hand side of the identity gives the analytic continuation of ExpS to the meromorphic function defined on C. This continuation (that is natural to be called the generalized basic hypergeometric function) has the Mittag-Leffler decomposition given by ExpS (z) =

k ∞ X (−1)k q ( 2 )

k=0

see the identity (145).

(q; q)∞

γS (q −(k+1) )

1 , q −k − z

(179)

210

A. Odzijewicz

After the substitution z = q n+1 the identity (174) leads to the following two identities: k ∞ 1 X (−1)k q ( 2 ) 1 = γS (q −k )q (n+1)k S(q; q)n (q; q)∞ (q; q)k

(180)

k l k X X 1 (−1)k q ( 2 ) = . S(q; q)n S(q; q)l−k (q; q)k (q; q)n−l

(181)

k=0

and

l=0 k=0

Taking into account the equality k

a−k (a; q)∞ (−1)k q ( 2 ) = , a→∞ (aq k ; q)∞ (q; q)k

(182)

lim

one can rewrite (180) in the following way: 1 = S(q; q)n

Z

1

log x

1 1 a− log q (a; q)∞ x γS ( ) lim dq x, (1 − q)(q; q)∞ x a→∞ (ax; q)∞ n

0

(183)

where the dq x := (1 − q)

∞ X

xδ(x − q k )dx

(184)

k=0

is the so called q-measure, see [7, 18]. In the case when the function S(·; q)∞ has no poles in the disc D% with % > q, the identity (183) is also valid for the function S1 . Hence in that case the measure log x

1 1 a− log q (a; q)∞ dνS = γ 1 ( ) lim dq x (1 − q)(q; q)∞ S x a→∞ (ax; q)∞

(185)

solves the moment problem (55) for ExpS . Therefore, the kernel ExpS (zv) acquires the reproducing property Z ExpS (zw)ExpS (wv)dµS (w, w), (186) ExpS (zv) = D1

with respect to the measure dµS (w, w) =

1 1 dνS (x)dϕ, 2π ExpS (x)

(187)

√ where w = xeiϕ and D1 is the unit disc. In the rational case, i.e. when S = rSs (see (176)) Eq. (186) gives the reproducing formula for the kernel given by the basic hypergeometric series r8s (zv). After the substitution of (165) and (175) into the integral equation (156) and the operator equation (158), respectively, we obtain the following two expressions for the generalized exponential function: Z 1 Kq (z, ξ)S(ξ, q)∞ ξ N +1 dξ (188) ExpS (zq N +1 ) = 2πiS(q; q)∞ q N +1 ∂ D%


211

and ExpS (z) =

1 1 , S(qQ; q)∞ S(q; q)∞ 1−z

(189)

where |z| < 1 and q < % < R. The last formula may be considered as a generalization of the q-binomial theorem. 7. Examples and Applications In the case of qn = R(q n ) the quantum algebra AK =: AR is “uniformized” by the relations (130), where the operator Q plays the role of a “coordinate”. If R is invertible on the interval [0, 1] which contains the spectrum of Q then one can express Q by A and A† . Therefore in that case Q belongs to the algebra AR . Thus the algebra AR,q generated by A, A† and Q is a subalgebra of AR . In the general case one can consider AR,q as an algebra naturally related to AR . Taking into account (130) we find that AR,q is determined by the relations AA† = R(qQ),

(190)

A† A = R(Q),

(191)

QA† = qA† Q,

(192)

AQ = qQA.

(193)

It is easy to obtain the last two relations (192) and (193) if one passes to the holomorphic representation in which Q is given by (32) and A† acts as the multiplication by z. Now if one replaces in (190) the operator Q by the occupation number operator N en = nen , which is related to Q by Q = q N , one finds the equivalent relations AA† = R(q N +1 ),

(194)

A† A = R(q N ),

(195)

[N, A] = −A,

(196)

[N, A† ] = A† .

(197)

The above equivalence is valid due to 0 < q < 1, which is not the case in general, as an example of q being a root of unity shows. As is readily seen, the relations (196) and (197) do not depend on R. Therefore, in what follows, we take these relations for granted and we shall not rewrite them for each subcase separately. Let us now consider the following two subcases described by the simplest rational functions:

212

A. Odzijewicz

q 1 1 ( + x) + , (1 − q)(1 − q 2 ) x (1 − q)2 1−x R(x) = r2 . 1 − rr21 x R(x) = −

(198) (199)

In the subcase (198) the relations (194) give rise to the quantum algebra Uq (sl(2)), which is a q-deformation of U (sl(2)), see [6, 19]. Let us mention also that quantum groups SUq (2) and SUq (1, 1) can be considered as special subcases of the algebra (190)–(193). Hence we suppose that identities which are satisfied by the generalizated exponential (see Sect. 6) may by useful in the representation theory of these quantum groups, see [18, 13] It is easy to see that the relations (190)–(193) for R given by (199) take the form 1 − q N +1 , AA† = r2 1 − rr21 q N +1 A† A = r 2

1 − qN . 1 − rr21 q N (200)

The algebra (200) is the q-deformation of the algebra N +1 , N +k+1 N A† A = r2 , N +k AA† = r2

(201) where we put rr21 = q k . The above two quantum algebras are related to hyperbolic and parabolic algebras, respectively, which are investigated in Sect. 4. A natural interpretation of the algebra AR,q arises if one studies the physical system with Schrödinger operator H = H(Q) being a function of the operator Q. Then the algebra AR,q can be considered as a symmetry algebra of that system. Since Q is diagonal in the holomorphic realization such systems are integrable. The energy operator of the form H = H( 1−Q 1−q ), where H does not depend on the parameter q, corresponds to the operator H = H(N ) for q → 1. Therefore systems described by the H = H(Q) are q-deformed versions of the ones described by H = H(N ). In the physical literature there exist many examples of systems of the above type, see e.g. [21, 32]. The problem now is how to find out whether the energy operators defined in the Schrödinger representation are of the form H = H(Q). One can formulate that problem in the inverse way. Namely one may ask how to construct a Schrödinger representation of the algebra (190)–(193). Following V.Spiridonov, see [32, 33], we shall do that for the case when R is a polynomial of degree N R(x) = (x − λ1 ) · · · (x − λN ),

(202)

where λ1 , . . . , λN ∈ R. The algebra (190) with R given by (202) appeared in [32, 33] as the symmetry algebra for the one-dimensional Schrödinger operator. In order to find its representation in L2 (R, dx) let us introduce the mutually conjugated operators


213

d + fj (x), dx d † Aj = − + fj (x), dx

Aj =

(203)

where the functions fj yield the chain of differential equations 0 2 fj0 (x) + fj+1 (x) + fj2 (x) − fj+1 (x) = λj+1 − λj = µj .

(204)

This chain is equivalent to the following intertwining relations: † † Lj Aj = Aj Lj+1 , Aj Lj = Lj+1 Aj ,

(205)

where † Lj = Aj Aj + λj ,

(206)

see [33]. If we restrict ourselves to the class of solutions invariant under the transformation √ √ fj+N (x) = qfj ( qx) =: (T fj )(x), (207) µj+N = qµj , then the operators A := T −1 AN −1+j · · · Aj , † † A† := Aj · · · AN −1+j T

(208)

satisfy the relations (190)–(193). Since in that case the operator Q is – up to additive constant – a Schrödinger operator Q + const = −

d2 + fj2 (x) − fj0 (x) + λj = Lj , dx2

(209)

we discover that the algebra AR,q is its symmetry algebra. The spectral problem of the operator (209) generates a certain solution of the system given by (204) and (207), for details we refer to [34]. From the above considerations we conclude that the holomorphic representation of the algebra AR,q leads to the explicit description of coherent states of the system given by the Hamiltonian (209). If R(x) = 1−x 1−q , the physical system is a q-deformed harmonic oscillator. The standard harmonic oscillator is obtained as the limit case for q → 1. Other examples of such physical systems may be found in [30, 11, 12]. The interesting question is what kind of structures one obtains in the limit q → 1. We investigate this problem for the case when R =rSs is a rational function given by (176). Let us assume that ai = q αi −1 , bj = q βj −1 . Then the generalized derivative will correspond to the operator ∂r,s =

d d (β1 + z dz ) · · · (βs + z dz ) d = lim (1 − q)r−s−1 ∂r Ss , d d dz q→1 (α1 + z dz ) · · · (αr + z dz )

(210)

214

A. Odzijewicz

where r ≤ 1 + s, and boundary exponential function is given by the hypergeometric series lim Expr Ss [(1 − q)r−s−1 z] =rFs (z) =

q→1

=

∞ X (α1 )n · · · (αr )n n z . n!(β1 )n · · · (βs )n

(211)

n=0

Equation (15) or (108) takes the form of the differential equation s Y i=1

Y d d d (βi + z ) rFs = (αj + z )F r s , dz dz dz r

(212)

j=1

see [20]. In the considered case the algebra AR,q determined by the relations (190)– (193) corresponds to the algebra Ar,s generated by the operators A, A† and N which satisfy the relations (N + 1)(β1 + N ) · · · (βs + N ) , AA† = (α1 + N ) · · · (αr + N ) N (β1 − 1 + N ) · · · (βs − 1 + N ) A† A = . (α1 − 1 + N ) · · · (αr − 1 + N )

(213)

Algebras of this sort (when the right hand side of (213) reduces to a polynomial ) appear in problems of quantum optics as well as in quantum many–bodies physics. For the detailed description of these interrelations the reader is referred to papers of V.P. Karassiov, see [14, 15]. At last let us remark that the correspondence between quantum algebras and the theory of special functions survives if one considers a general complex manifold, e.g. noncompact Riemann surface or bounded domain in CN instead of the disc. The general case is to be investigated in a subsequent publication. Acknowledgement. The part of the article was prepared during the stay of the author at Institut für Reine Mathematik der Humboldt Universität in Berlin. The hospitality of Professor T.Friedrich at the Institute is gratefully acknowledged. The author highly appreciates all suggestions and remarks of both Referees due to whom the paper acquired a more desirable form.

References [1] [2] [3] [4] [5] [6] [7] [8]

Akhiezer, N.I.: Klassicheskaja Problema Momentov i Nekotorye Voprosy Analiza Sviazannye s Neju. Moskva: Gosudarstvennoe Izdatelstvo Fizik. Matematicheskoj Literatury, 1961 Bailey, W.N.: Generalized Hypergeometric Series. Cambridge: Cambridge University Press, 1935 Berezin, F.A.: Commun. Math. Phys. 40, 153 (1975) Berezin, F.A.: Commun. Math. Phys. 63, 131 (1978) Flato, M., Lichnerowicz, A., Sternheimer, D.:Deformation of Poisson brackets, Dirac brackets and application. J. Math. Phys. 17, 1762–1794 (1976) Drinfeld, V.G.: Quantum groups. Proceedings of the International Congress of Mathematicians (Berkeley), Providence, RI: Am. Math. Soc. 1987, pp. 789–820 Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge: Cambridge University Press, 1990 Gong, Ren-Shan: A completeness relation for the coherent states of the (p, q)-oscillator by (p, q)integration. J. Phys. A.: Math. Gen. 27, 375–379 (1994)


[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

[33] [34] [35] [36] [37]

215

Jimbo, M.: A q-difference analogue of Uq (G) and the Yang-Baxter equation. Lett. Math. Phys. 10, 63–69 (1985) Jimbo, M.: A q-analogue of U (gl(N + 1)), Hecke algebra and the Yang-Baxter equation. Lett. Math. Phys. 11, 247–252 (1986) Jorgensen, P.E.T., Werner, R.F.: Coherent states of the q-canonical commutation relations. Preprint Osnabrück, (1993) Jorgensen, P.E.T., Schmit, R.F., Werner, R.F.: q-Relations and Stability of C*-Isomorphism Classes. Algebraic Methods in Operator Theory. Basel: Birkhäuser-Verlag, 1993 Kakehi, T., Masuda, T.,Ueno, K.: Spectral analysis of a Q-difference Operator which Arises from the Quantum SU (1, 1) Group. J. Operator Theory 33, 159–196 (1995) Karassiov, V. P.: Polynomial Deformations of the Lie Algebra sl(2) in Problems of Quantum Optics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 1, 3–19, April, (1993) Karassiov, V. P.: G-invariant polynomial extensions of Lie algebras in quantum many-bodies physics. J. Phys. A: Math. Gen. 27, 153–165 (1994) Klauder, R, Skagerstam, Bo-Sture: Coherent States – Applications in Physics and Mathematical Physics. Singapore: World Scientific, (1985) Klimek, S., Lesniewski, A.: Quantum Riemann Surfaces I. The Unit Disc. Commun. Math. Phys. 146, 103–122 (1992) Klimyk, A.V., Vilenkin ,N.I.: Representations of Lie groups and Special Functions. London: KAP, 1992 Kulish, P.P., Reshetikhin, N.Yu.: J. Soviet. Math. 23, 2435 (1983) Matematicheskaja Enciklopedia T. 1. Moskva: 1977 pp. 1004–1005 Maximov, V.M., Odzijewicz, A.: The q-deformation of quantum mechanics of one degree of freedom. J. Math. Phys. 36(4), April (1995) Nikolski, N. K.: Lekcii ob operatore sdviga. Moskva: Nauka, 1980 Odzijewicz, A.: Covariant and Contravariant Berezin Symbols of Bounded Operators. Quantization and Infinite-Dimensional Systems. New York–London: Plenum Press, 1994, pp. 99–108 Odzijewicz, A.: On reproducing kernels and quantization of states. Commun. Math. Phys. 114, 577–597 (1988) Odzijewicz, A.: Coherent States and Geometric Quantization. Commun. Math. Phys. 150, 385–413 (1992) O’Raifeartaigh, L., Ryan, C.: On generalised commutation relation. Proceedings of the Royal Irish Academy. vol. 62, Sect. A Ostrovski, V.L., Samoilenko, Yu.S.: Infinite dimensional representations of quadratic and polynomial ∗-algebras. Academy of Science of the Ukrainian SSR, Institute of Mathematics, Preprint 91.4 Prugoveˇcki, E.: Stochastic Quantum Mechanics and Quantum Spacetime. Dordrecht: Reidel, 1986 Prugoveˇcki, E. and Ali, S. T.: Nuovo. Cim. A63, 171 (1981) Pusz, W., Woronowicz, L.S.: Rep.Math.Phys. 27, 231 (1989) Shafarevich, I.R.: Osnovnye poniatia algebry. Sovremennye problemy matematiki T. 11. Moskva: 1986 Spiridonov, V.: Deformation of Supersymmetric and Conformal Quantum Mechanics Through Affine Transformations. In: Proc. of the Intern. Workshop on Harmonic Oscillators, College Park, USA, March 1992. Eds. D. Han, Y.S. Kim, and W.W. Zachary, NASA Conf. Publ. 3197, 1993, pp. 93–108 Spiridonov, V.: Universal Superposition of Coherent States and Self-Similar Potentials. Phys. Rev. A, 1909–1935, (1995) Veselov, A.P., Shabat, A.B.: Dressing chain and spectral theory of Schrödinger operator. Funk. Anal. i ego Dril. 27, n.2, 1–21 (1993) Woronowicz, L.S.: Unbounded Elements Affiliated with C ∗ -Algebras and Non-Compact Quantum Groups. Commun. Math. Phys. 136, 399–432 (1991) Woronowicz, L.S.: Lett. Math. Phys. 23, 251 (1991) Woronowicz, L.S.: Commun. Math. Phys. 114, 417 (1992)

Communicated by T. Miwa

Commun. Math. Phys. 192, 217 – 244 (1998)

Communications in


Extensions of Conformal Nets and Superselection Structures D. Guido1,? , R. Longo1 , H.-W. Wiesbrock2,?? 1

Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, I–00133 Roma, Italy. E-mail: [email protected], [email protected] 2 Freie Universit¨ at Berlin, Institut für Theoretische Physik, Arnimallee 14, D-14195 Berlin, Germany. E-mail: [email protected] Received: 19 March 1997 / Accepted: 1 July 1997

Abstract: Starting with a conformal Quantum Field Theory on the real line, we show that the dual net is still conformal with respect to a new representation of the Möbius group. We infer from this that every conformal net is normal and conormal, namely the local von Neumann algebra associated with an interval coincides with its double relative commutant inside the local von Neumann algebra associated with any larger interval. The net and the dual net give together rise to an infinite dimensional symmetry group, of which we study a class of positive energy irreducible representations. We mention how superselection sectors extend to the dual net and we illustrate by examples how, in general, this process generates solitonic sectors. We describe the free theories associated with the lowest weight n representations of PSL(2, R), showing that they violate 3regularity for n > 2. When n ≥ 2, we obtain examples of non Möbius-covariant sectors of a 3-regular (non 4-regular) net. 0. Introduction Haag duality is one of the most important properties in Quantum Field Theory for the analysis of the superselection structure. It basically says that the locality principle holds maximally. Concerning Quantum Field Theory on the usual Minkowski spacetime, duality may be always assumed, in a Wightman theory, because wedge duality automatically holds and one can enlarge the net to a dual net without affecting the superselection structure [3, 26]. Nevertheless there might be good reasons in lower dimensional theories for Haag duality not to be satisfied, [24]. An important case occurs in Conformal QFT: as such a theory naturally lives on a larger spacetime, duality may fail on the original spacetime ? ??

Supported in part by Ministero della Pubblica Istruzione and CNR-GNAFA. Supported by the DFG, SFB 288 “Differentialgeometrie und Quantenphysik”.

218

D. Guido, R. Longo, H.W. Wiesbrock

because contributions at infinity are possibly not detectable there. Moreover in low spacetime dimensions, the superselection structure of the dual net may change due to the occurrence of soliton sectors [31, 26], and new information being contained in the inclusion of the two nets. This paper is devoted to an analysis of conformal QFT on the real line, namely one-dimensional components of a two-dimensional chiral conformal QFT. The first aspect that we discuss concerns the symmetries of the dual net. Starting with a conformal net A on R, the Bisognano–Wichmann property holds automatically true [5, 14], thus the dual net . Ad (a, b) = A(−∞, b) ∩ A(a, ∞) a < b on R is local and obviously translation-dilation covariant with respect to the same translation-dilation unitary representation. We shall however prove that Ad is even conformally covariant with respect to a new unitary representation of PSL(2, R). The construction of the new symmetries is achieved by a new characterization of local conformal precosheaves on the circle in terms of what we call a +hsm (halfsided modular) factorization, namely a quadruple (Ni , i ∈ Z3 ; ), where the Ni ’s are mutually commuting von Neumann algebras with a joint cyclic separating vector such that Ni ⊂ Ni+1 is a +hsm modular inclusion in the sense of [35] for all i ∈ Z3 . This characterization makes only use of the modular operators and not of the modular conjugations as in [36]. As a second result we shall deduce a structural property for any conformal net A: A is automatically normal and conormal, namely if I ⊂ I˜ is an inclusion of proper intervals then lclA(I)cc = A(I), ˜ = A(I) ∨ A(I)c , A(I)

(1) (2)

˜ denotes the relative commutant in A(I) ˜ and X cc = (X c )c . This where X c = X 0 ∩ A(I) property is useful in the analysis of the superselection structure, as discussed below. The next issue will be to compare the net A with the dual net Ad . We shall give a detailed study of the inclusion A ⊂ Ad in a particular model, namely when A is the net associated with the nth derivative of the U (1)-current algebra, the latter turning out to coincide with Ad . This is a first quantization analysis and to this end we shall give formulas relating two irreducible lowest weight representations of PSL(2, R) that agree on the upper triangular matrix subgroup P0 . In other words we are studying a class of representations of a certain infinite dimensional Lie group, the amalgamated free product PSL(2, R) ∗P0 PSL(2, R), where a classification is simply obtained. For example an explicit formula for the unitary γ whose second quantization implements the canonical endomorphism associated with the inclusion A(−1, 1) ⊂ Ad (−1, 1) (the product of the ray inversion unitaries of the two nets) will be given as a function of the skew-adjoint generator E of the dilation one-parameter subgroup γ=

(E − 1)(E − 2) · · · (E − n) . (E + 1)(E + 2) · · · (E + n)

We shall show that the net A is not 4-regular if n ≥ 2 and not 3-regular if n ≥ 3, where A is said to be k-regular if A remains irreducible after removing k − 1 points from R. The 4-regularity property played a role in the covariance analysis in [16] where the problem of its general validity remained open.

Extensions of Conformal Nets and Superselection Structures

219

We generalize the construction of the Buchholz, Mack and Todorov sectors for the current algebra ([7]) to the nth derivative of the current algebra, showing that they form a group isomorphic to R2n+1 , and that none of them is covariant w.r.t. the conformal group (if n 6= 0). In particular this shows results in [16] to be optimal, at least on the real line. Finally we shall illustrate by examples how sectors of A localized in a bounded interval may have extension to Ad with soliton localization. 1. Structural Properties of Conformal Local Precosheaves on S 1 In the algebraic approach, see [18], chiral conformal field theories are described as conformally covariant local precosheaves A of von Neumann algebras on proper intervals of the circle S 1 . We start by reviewing some aspects of this framework. An open interval I of S 1 is called proper if I and the interior I 0 of its complement are not empty. The circle will be explicitly described either as the points with modulus one in C or as the one-point compactification of R, these two description being related by the Cayley transform: C : S 1 → R ∪ {∞} given by z → i(z + 1)(z − 1)−1 . The group PSL(2, R) acts on S 1 via its action on R ∪ {∞} as fractional transformations. Intervals are labeled either by the coordinates on R or by complex coordinates of the endpoints in S 1 ⊂ C, where in the later case intervals are represented in positive cyclic order. A precosheaf A is a covariant functor from the category J of proper intervals with inclusions as arrows to the category of von Neumann algebras on a Hilbert space H with inclusions as arrows, i.e., a map I → A(I) that satisfies: A. Isotony. If I1 ⊂ I2 are proper intervals, then A(I1 ) ⊂ A(I2 ). The precosheaf A will be a (local) conformal precosheaf if in addition it satisfies the following properties: B. Locality. If I1 and I2 are disjoint proper intervals, then A(I1 ) ⊂ A(I2 )0 , C. Conformal invariance. There exists a strongly continuous unitary representation U of PSL(2, R) on H such that U (g)A(I)U (g)∗ = A(gI),

g ∈ PSL(2, R), I ∈ J .

D. Positivity of the energy. The generator of the rotation subgroup U (R(·)) (conformal Hamiltonian) is positive. Here R(ϑ) denotes the rotation of angle ϑ on S 1 . E. Existence of the vacuum. There exists a unit vector ∈ H (vacuum vector) which is U (PSL(2, R))-invariant and cyclic for ∨I∈J A(I). The Reeh-Schlieder Theorem now states that the vacuum vector is cyclic and separating for any local algebra A(I). Let us recall that uniqueness of the vacuum is equivalent both to the irreducibility of the precosheaf or to the factoriality property for local algebras. We shall denote by Mob

220


the Möbius group, namely the group of conformal transformations in C that leave the unit circle globally invariant. The group PSL(2, R) is then identified with the subgroup of orientation preserving transformations and Mob is generated by PSL(2, R) and an involution. Let I1 be the upper semi-circle parameterized as (0, +∞); we associate to I1 the following two one-parameter subgroups of Mob : First the dilations (relative to I1 ), −t/2 0 e 3I1 (t) = t/2 , 0 e leaving I1 globally stable, second the translations 1 s TI1 (s) = . 0 1 mapping I1 into itself for positive s ≥ 0. In general, if I is any interval in S 1 , there exists a g ∈ PSL(2, R) such that I = gI1 and we set 3I = g3I1 g −1 ,

TI = gTI1 g −1 .

The definition of the dilations does not depend on g, while the translations of I are defined up to a rescaling of the parameter, that however does not play any role in the following, because we are only interested in the subgroups generated by them. The subgroup generated by 3I1 (·) and TI1 (·), denoted by P0 , is the subgroup of upper triangular matrices of PSL(2, R) and plays an important role in the following, especially in the next section. We shall associate with any proper interval I a diffeomorphism rI of S 1 , the reflection mapping I onto the causal complement I 0 , i.e. fixing the boundary points of I. In the case of I1 = (0, +∞), rI1 x = −x, and one can extend this definition to a generic I as before. Notice that rI1 is orientation reversing. By a (anti-)representation U of Mob we shall mean the obvious generalization of the notion of unitary representation where U (rI ) is anti-unitary. For a general conformal precosheaf the Bisognano–Wichmann Property holds [5, 14]: U extends to a unitary (anti-)representation of Mob such that, for any I ∈ J , U (3I (2πt)) = 1it I , U (rI ) = JI ,

(3) (4)

where 1I and JI are the Tomita-Takesaki modular operator and modular conjugation associated with (A(I), ). This implies Haag duality: A(I)0 = A(I 0 ),

I ∈ I.

Let now (N ⊂ M, ) be a triple where N ⊂ M is an inclusion of von Neumann algebras acting on a Hilbert space H and ∈ H is a cyclic and separating vector for N and M. 1. (N ⊂ M, ) is said to be standard if is cyclic also for the relative commutant . N c = M ∩ N 0 of N in M, see [10].


221

2. If σtM denotes the modular automorphism associated to (M, ), then the triple M (N ) ⊂ N for, (N ⊂ M, ) is said to be ± half-sided modular (± hsm) if σ−t respectively, all t ≥ 0 or all t ≤ 0. 3. A ±hsm factorization of von Neumann algebras is a quadruple (N0 , N1 , N2 , ), where {Ni , i ∈ Z3 } is a set of pairwise commuting von Neumann algebras, is a 0 , ) is a ±hsm inclusion for cyclic separating vector for each Ni and (Ni ⊂ Ni+1 each i ∈ Z3 . In the work [37], local conformal precosheaves have been characterized in terms of ±hsm standard inclusions of von Neumann algebras and the adjoint action of the modular conjugations. (This work is based on a statement about hsm modular inclusions [34], whose correct proof is contained in [2].) We shall give here below an alternative characterization in terms of a ±hsm factorization, that has the advantage of using only the modular groups and not the modular conjugations. Lemma 1.1. Let G be the universal group (algebraically) generated by 3 one-parameter subgroups 3i (·), i ∈ Z3 , such that 3i and 3i+1 have the same commutation relations of 3Ii and 3Ii+1 for each i ∈ Z3 , where I0 , I1 , I2 are intervals forming a partition of S 1 . Then G is isomorphic to PSL(2, R), the universal covering group of PSL(2, R), and the 3i ’s are continuous one parameter subgroups naturally corresponding to 3Ii . Proof. Obviously G has a quotient isomorphic to PSL(2, R), and we denote by q the quotient map. As the exponential map is a local diffeomorphism of the Lie algebra of a Lie group and the Lie group itself, there exists a neighbourhood U of the origin R3 such that the map (t0 , t1 , t2 ) → 3I0 (2πt0 )3I1 (2πt1 )3I2 (2πt2 ) is a diffeomorphism of U with a neighbourhood of the identity of PSL(2, R). Therefore the map 8 : (t0 , t1 , t2 ) ∈ U → 30 (2πt0 )31 (2πt1 )32 (2πt2 ) ∈ G is still one-to-one. It is easily checked that the maps g8 : U → G, g ∈ G, form an atlas on G, thus G is a manifold. In fact G is a Lie group since the group operations are smooth, as they are locally smooth. Now G is connected by construction and q is a local diffeomorphism of G with PSL(2, R), hence a covering map, that has to be an isomorphism because of the universality property of PSL(2, R). Theorem 1.2. Let (N0 , N1 , N2 , ) be a +hsm factorization of von Neumann algebras and let I0 , I1 , I2 be intervals forming a partition of S 1 in counter-clockwise order. There exists a unique local conformal precosheaf A on S 1 such that A(Ii ) = Ni , i ∈ Z3 , with the vacuum vector. The (unique) positive energy unitary representation U of PSL(2, R) is determined by the modular prescription U (3Ii (2πt)) = 1it Ii . Notice that every +hsm factorization of von Neumann algebras arises by considering the von Neumann algebras associated to 3 intervals of S 1 as in the above theorem, due to the geometric property of the modular group (3). Proof. The subgroup of PSL(2, R) generated by the one-parameter subgroups 3Ii (2πt) and 3Ii+1 (2πs), i ∈ Z3 , is a two-dimensional Lie group Pi isomorphic to the translation0 , ) is a +hsm standard inclusion, by a result first stated dilation group P0 . As (Ni , Ni+1 in [34] with an erroneous proof and whose correct proof is given in [2], the unitary is group generated by 1it Ii and 1Ii+1 is isomorphic to Pi , indeed there exists a unitary representation Ui of Pi determined by Ui (3Ii (2πt)) = 1it Ii and Ui (3Ii+1 (−2πs)) = 1is , therefore by Lemma 1.1, there exists a unitary representation U of PSL(2, R), Ii+1 such that U |Pi = Ui .

222


Let t0 =

1 2π

ln 2. Then we have ([34, 2], see the remarks above) it0 0 0 Ad 1it I0 1I1 (N0 ) = N1 ,

(5)

it0 it0 it0 it0 it0 0 0 Ad 1it I2 1I0 1I1 1I2 1I0 1I1 (N0 ) = N0 .

(6)

and similarly The element 3I2 (2πt0 )3I0 (2πt0 )3I1 (2πt0 )3I2 (2πt0 )3I0 (2πt0 )3I1 (2πt0 ) is easily seen to be conjugate to the rotation by π in PSL(2, R), hence Eq. (6) entails U (2π) to implement an automorphism on N0 . Set A(I0 ) := N0 . If I is an interval of S 1 , then I = gI0 for some g ∈ PSL(2, R), and we set A(I) = U (g)A(I0 )U (g)∗ . Since the group GI0 of all g ∈ PSL(2, R) such that gI0 = I0 is generated by 3I0 (t), t ∈ R and by rotations of 2kπ, k ∈ Z, then U (g)A(I0 )U (g)∗ = A(I0 ) for all g ∈ GI0 and the von Neumann algebra A(I) is well defined. The isotony of A follows if we show that gI0 ⊂ I0 implies A(gI0 ) ⊂ A(I0 ). Indeed any such g is a product of an element in GI0 and translations TI0 (·) and TI00 (·) mapping I0 into itself, hence the isotony follows by the half-sided modular conditions. By (5) we have it0 it0 it0 0 Ad 1it I1 1I2 1I0 1I1 (N0 ) = N2 , and since the corresponding element in PSL(2, R) maps I0 onto I2 , we get N2 = A(I2 ) and analogously N1 = A(I1 ). The locality of A now follows by the factorization property. Finally U is a true representation of PSL(2, R) by the vacuum conformal spinstatistics theorem [17], and the positivity of the energy follows by the Bisognano– Wichmann property (3), see [36, 37]. Although a conformal precosheaf satisfies Haag duality on S 1 , duality on R does not necessarily hold. Lemma 1.3. Let A be a local conformal precosheaf on S 1 . The following are equivalent: (i) The restriction of A to R satisfies Haag duality: A(I) = A(R\I)0 . rm(ii) A is strongly additive: If I1 , I2 are the connected components of the interval I with one internal point removed, then A(I) = A(I1 ) ∨ A(I2 ). (iii) If I, I1 , I2 are intervals as above A(I1 )0 ∩ A(I) = A(I2 ). Proof. Note that by Möbius covariance we may suppose that the point removed in (i) and (ii) is the point ∞. Now (i) ⇔ (ii) because R\I consists of two contiguous intervals in S 1 whose union has closure equal I 0 , and by Haag duality A(I) = A(I 0 )0 . Similarly (ii) ⇔ (iii) because, after taking commutants and renaming the intervals, one relation becomes equivalent to the other one.


223

Examples of conformal precosheaf on S 1 that are not strongly additive, i.e. not Haag dual on the line, were first given in [19, 8] and [38]. We will look in some detail at an example of [38] in Sect. 3. Haag duality on S 1 entails duality for half-lines on R hence essential duality, namely the dual net of the restriction A0 to R is local: . I 7→ Ad0 (I) = A(R\I)0 , I ⊂ R. Due to locality the net Ad0 is larger than the original one, namely A0 (I) ⊂ Ad0 (I),

I ⊂ R.

Ad0 is usually called the dual net, see [3, 8] and its main feature is that it obeys Haag duality on R. The dual net does not in general transform covariantly under the covariance representation of the starting net. Theorem 1.4. Let A be a local net of von Neumann algebras on the intervals of R, a cyclic and separating vector for the von Neumann algebra A(I) associated with each interval I ⊂ R and U a -fixing unitary representation of the translation-dilation group acting covariantly on A. The following are equivalent: (i) A extends to a conformal precosheaf on S 1 . (ii) The Bisognano–Wichmann property holds for A, namely 1it R+ = U (3R+ (2πt)).

(7)

Proof. (i) ⇒ (ii): See [5, 14]. (ii) ⇒ (i): Note first that, by translation covariance, 1it (a,∞) = U (3(a,∞) (2πt)) for all a ∈ R. Hence A(−∞, a) is a von Neumann subalgebra of A(a, ∞)0 that is cyclic on and globally invariant under the modular group of A(a, ∞)0 with respect to , hence, by the Tomita-Takesaki theory, duality for half-lines holds A(a, ∞)0 = A(−∞, a). Recall now that if N ⊂ M is an inclusion of von Neumann algebras and is a cyclic and separating vector for both N and M, then (N ⊂ M, ) is +hsm iff (M0 ⊂ N 0 , ) is −hsm [34, 2]. Then it is immediate to check (A(−∞, −1), A(−1, 1), A(1, ∞), ) to be a +hsm factorization of von Neumann algebras, so we get a conformal precosheaf from Theorem 1.2. Due to Bisognano–Wichmann property this is indeed an extension to S1 of the original net. Note as a consequence that a local net on R as above with property (7) automatically has a PCT symmetry, namely JR+ A(I)JR+ = A(−I),

∀ interval I ⊂ R.

Now, if A is a local conformal precosheaf on S 1 , its restriction A0 to R does not depend, up to isomorphism, on the point we cut S 1 , because of Möbius covariance. The local precosheaf on S 1 extending Ad0 is thus well defined up to isomorphism. We call it the dual precosheaf of A and denote it by Ad . Corollary 1.5. The dual precosheaf of a conformal precosheaf on S 1 is a strongly additive conformal precosheaf on S 1 .

224


Proof. By construction, the dual net satisfies Haag duality on R, hence strong additivity by Lemma 1.3. Remark. Let us compare the precosheaves A and Ad on S 1 . First we observe that equality holds if and only if the conformal precosheaf A is strongly additive. As mentioned above locality implies / I, A(I) ⊂ Ad (I) if − 1 ∈ i.e. if I does not contain the point infinity, while Haag duality on S 1 implies Ad (I) ⊂ A(I)

if − 1 ∈ I.

Therefore the observable algebras associated with bounded intervals of the real line are enlarged while the others, associated with intervals containing the point at infinity, decrease. The observable algebras associated with half-lines, i.e. with intervals having −1 as a boundary point, remain fixed. Due to the Bisognano–Wichmann property, which holds for all conformal precosheaves, altering the algebras implies a change in the representation of the conformal group PSL(2, R). Moreover, since the algebras associated with half-lines coincide, both representations agree on the isotropy group of the point at infinity, i.e. on the subgroup P0 of PSL(2, R) generated by translations and dilations. An inclusion N ⊂ M of von Neumann algebras is said to be normal if N = N cc , where X c = X 0 ∩ M denotes the relative commutant of X in M, and conormal if M is generated by N and its relative commutant w.r.t. M, i.e., M = N ∨ N c (i.e. M0 ⊂ N 0 is normal). We shall then say that a local conformal precosheaf A is (co-)normal if the inclusion A(I1 ) ⊂ A(I2 ) is (co-)normal for any pair I1 ⊂ I2 of proper intervals of S 1 . By Haag duality, normality and conormality are equivalent properties of conformal precosheaves. Theorem 1.6. Any conformal precosheaf on S 1 is normal and conormal. Proof. Let us consider first an inclusion of two proper intervals I1 ⊂ I2 with a common boundary point. If A is strongly additive, the inclusion of von Neumann algebras A(I1 ) ⊂ A(I2 ) is conormal as in this case A(I1 )0 ∩ A(I2 ) = A(I2 \I1 ). In the general case, by conformal invariance we may assume that I1 and I2 are respectively the intervals of the real line (1, +∞) and (0, +∞). By definition then A(I1 ) = Ad (I1 ), A(I2 ) = Ad (I2 ), with Ad the dual net, hence the inclusion A(I1 ) ⊂ A(I2 ) is conormal by Corollary 1.5 and the above argument. As A(I2 )0 ⊂ A(I1 )0 is conormal, A(I1 ) ⊂ A(I2 ) is also normal. It remains to show the normality of A(I1 ) ⊂ A(I2 ) when I1 ⊂ I2 are intervals with no common boundary point, e.g. I1 = (b, c) and I2 = (a, d), with a < b < c < d. Then we set I3 = (a, c) and I4 = (b, d), therefore I1 = I3 ∩ I4 and both I3 and I4 are subintervals of I2 with a common boundary point. Then the double relative commutant of A(I1 ) in A(I2 ) is given by A(I1 )cc ⊂ A(I3 )cc ∩ A(I4 )cc = A(I3 ) ∩ A(I4 ) = A(I1 ),

(8)

where the last equality is a consequence of duality and additivity and implies the first inclusion; the opposite inclusion is elementary. Corollary 1.7. Let (N ⊂ M, ) be a +hsm standard inclusion of von Neumann algebras. In this case: • The inclusion N ⊂ M is normal and conormal.


225

• There exists a unique strongly additive local conformal precosheaf A of von Neumann algebras on S 1 with M = A(0, +∞), N = A(1, +∞), and the vacuum vector. • There exists a bijection between local conformal precosheaves A of von Neumann algebras on S 1 with M = A(0, +∞), N = A(1, +∞), the vacuum vector, and von Neumann subalgebras N0 of N 0 ∩ M cyclic on such that (N0 ⊂ M, ) is a −hsm inclusion and (N0 ⊂ N 0 , ) is a +hsm inclusion. Proof. Starting with the last point, notice that (M0 , N0 , N , ) is a +hsm factorization of von Neumann algebras, and clearly any +hsm factorization arises in this way, therefore the thesis is a consequence of Theorem 1.2. In the special case N0 = N 0 ∩ M we then obtain the second statement by Lemma 1.3(ii)⇐⇒ (iii). The first point is then a consequence of Theorem 1.6.qed Note as a consequence that a hsm modular standard inclusion (N ⊂ M, ) is also pseudonormal: N ∨ JN J = M ∩ JMJ, where J is the modular conjugation of (N 0 ∩M, ) and has the continuous interpolation property (see [10]). The irreducibility of A in Corollary 1.7 is equivalent in particular to the factoriality of N and M, see [17]. This is also equivalent to the center of N and M to have trivial intersection. Thus we have the following. Corollary 1.8. Let (N ⊂ M, ) be +hsm and standard. Then N and M have R ⊕ the same center Z and (N ⊂ M, ) has a direct integral decomposition N = Z Nλ dµ(λ), R⊕ R⊕ M = Z Mλ dµ(λ), = Z λ dµ(λ), where each (Nλ ⊂ Mλ , λ ) is either a +hsm standard inclusion of III1 factors or trivial (N = M = C). Proof. The modular group acts trivially on the center, so that 0 0 Ad 1it M (N ∩ M ) = N ∩ M , ∀t ∈ R. 0 0 0 Since Ad 1it M N = M for a suitable t0 , we immediately obtain N ∩ M = M ∩ M and M ∩ M0 = N ∩ M0 ⊂ N ∩ N 0 ,

i.e. Z(M) ⊂ Z(N ). Using the commutants we obtain the equality and the direct integral decomposition as stated. Applying [34], Theorem 12, and [2], we finish the proof. The following corollary summarizes part of the above discussion, based on results in [4, 34, 2]. Corollary 1.9. There exists a one-to-one correspondence between: • Isomorphism classes of standard +half-sided modular inclusions (N ⊂ M, ). • Isomorphism classes of Borchers triples (M, U, ), (i.e. M is a von Neumann algebra with a cyclic separating unit vector and U is a one-parameter -fixing unitary group with positive generator s. t. U (t)MU (−t) ⊂ M, t > 0) such that is cyclic for U (t)M0 U (−t) ∩ M for some, hence for all, t > 0. • Isomorphism classes of translation-dilation covariant, Haag dual nets on R with the Bisognano–Wichmann property 1it R+ = U (3(2πt)).

226


Isomorphism classes of strongly additive local conformal precosheaves of von Neumann algebras on S 1 . The notion of isomorphism in the above setting has an obvious meaning. Note however that an isomorphism between local conformal precosheaves can be defined as an isomorphism of precosheaves relating the vacuum states, as in this case it will automatically intertwine the Möbius representations as these are unique, being fixed by the modular prescriptions. 2. Representations of PSL(2, R) and Derivatives of the U (1)-Current 2.1. A class of representations of PSL(2, R) ∗P0 PSL(2, R). In Sect. 1 we have seen that we may associate with any conformal precosheaf on S 1 another conformal precosheaf on S 1 which is its Haag dual net on R. This amounts to “cut the circle,” namely to fix a special point (“∞”) and to redefine the local algebras associated to intervals which are relatively compact in S 1 \ {∞} in such a way that Haag duality holds on S 1 \ {∞}. The representations of PSL(2, R) associated with the two nets coincide when restricted to the group P0 generated by translations and dilations, therefore give a representation of PSL(2, R) ∗P0 PSL(2, R), i.e., the free product of two copies of PSL(2, R) amalgamated by the subgroup P0 . Let us denote by i1 , resp. i2 , the embeddings of PSL(2, R) into the first, resp. the second, component of the free product, and by i the immersion of P0 in the amalgamated free product. Then we shall consider on PSL(2, R)∗P0 PSL(2, R) the topology generated by the maps i1 i2 , namely a unitary representation U of PSL(2, R) ∗P0 PSL(2, R) is strongly continuous if and only if U ◦ i1 and U ◦ i2 are strongly continuous. We shall classify the class of strongly continuous unitary positive energy irreducible representations of PSL(2, R)∗P0 PSL(2, R) whose restrictions U ◦ik are still irreducible or, equivalently, such that U ((H1 − H2 )T ) is a scalar, where T is the generator of the translations belonging to p0 , the Lie algebra of P0 and Hk = ik (H) are the generators of the rotation subgroup. This amounts to classify the unitary positive energy representation with U ((H1 − H2 )T ) central, as these decompose into a direct integral of irreducible representations in the previous class. As we shall see, this is the general case in a free theory. Theorem 2.1. Let U be an irreducible unitary representation of PSL(2, R)∗P0 PSL(2, R) with positive energy, namely −iU (T ) is positive. Then U ◦ ik is irreducible for some k = 1, 2 if and only if both the U ◦ ik are irreducible, and if and only if U ((i1 (H) − i2 (H))i(T )) ∈ C, where H resp. T generate rotations resp. translations in PSL(2, R). Moreover, such representations are classified by pairs of natural numbers (n1 , n2 ), where nk is the lowest weight of the representation U ◦ ik of PSL(2, R). As is known the matrices 1 1 E= 2 0

0 , −1

T =

0 0

1 , 0

S=

0 −1

0 0

form a basis for the Lie algebra sl(2, R) and verify the commutation relations [E, T ] = T,

[E, S] = −S,

[T, S] = −2E.

(9)


227

Let us recall that the conformal Hamiltonian is H = 2i (T + S) and that the lowest weight of the representation is its lowest eigenvalue. The Casimir operator λ = E(E − 1) − T S

(10)

is a central element of the universal enveloping Lie algebra, thus its value in an irreducible unitary representation is a scalar. If U is a unitary irreducible non-trivial lowest weight representation of PSL(2, R), then the selfadjoint generator −iU (T ) of U (etT ) is positive and non-singular, therefore U (etE ) and eit log(−iU (T )) give a representation of the Weyl commutation relations, namely U restricts to a unitary representation of P0 , that has to be irreducible because any bounded operator commuting with E and T also commutes with S due to the formula (10). The von Neumann uniqueness theorem then implies that the restriction of U to P0 is unitarily equivalent to the Schrödinger representation, therefore E 7→ d/dx, T 7→ −iex on L2 (R). We now describe all lowest weight representations of PSL(2, R) (or its universal covering group PSL(2, R)) as extensions of the representation of P0 . Let us fix now the unitary irreducible representation of PSL(2, R) with lowest weight 1 and denote by E0 , T0 and S0 the image in this representation of the above Lie algebra generators E, T and S. Proposition 2.2. • Each non-trivial irreducible unitary representation U of PSL(2, R) with lowest weight ≥ 1 is unitarily equivalent to the representation obtained by exponentiation of the operators Tλ = T0 , Eλ = E0 , Sλ = S0 − λT0−1 , for some λ > 01 . • All λ > 0 appear and λ = α(α − 1) if U has lowest weight α. m • λ may be written as λ = m n ( n − 1), m, n ∈ N, if and only if U is a representation of the nth covering of PSL(2, R). Proof. The first two statements follow from the above discussion since the value of the Casimir operator in the unitary representation with lowest weight α is equal to λ = α(α − 1), see [20], and one gets the formula for Sλ by multiplying both sides of (10) by T −1 . To check the last point, first observe that when λ ≥ 0, λ = α(α − 1), α ≥ 1, we get an orthonormal set of eigenvectors for the (self-adjoint) conformal Hamiltonian 1 x d −x d −x e − e + λe . Hλ = 2 dx dx x

In fact, set φα = eαx e−e and define the following operators aα ± = 2Eλ ± i(Tλ + Sλ ). We . also set for simplicity of notation H = Hα(α−1) . Since Haα = aα ± ± (H ± 1), Hφα = αφα α n . α n and a− φα = 0 then φα = (a+ ) φα is an orthogonal set of eigenvectors of H with eigenvalues α + n. An application of the Stone-Weierstrass theorem shows that it is actually a basis, and the generated vector space is a G˚arding domain for Hλ , T , E. The rest of the statement follows easily. Proof of Theorem 2.1. If U ((i1 (H) − i2 (H))i(T )) is a scalar, U ◦ i2 (etH ) belongs to (U ◦ i1 (PSL(2, R) ∗P0 PSL(2, R)))00 and U ◦ i1 (etH ) belongs to (U ◦ i2 (PSL(2, R) ∗P0 PSL(2, R)))00 , therefore, since U is irreducible, U ◦ ik is irreducible too, k = 1, 2. On the other hand, if say U ◦ i1 is irreducible, we may identify it with one of the 1

If A and B are linear operators with closable sum, the closure of their sum is denoted simply by A + B.

228


representations described in Proposition 2.2 for some α ∈ R. Then, since U is irreducible and U ◦ i1 |P0 = U ◦ i2 |P0 , U ◦ i2 too has to be of the form described in Proposition 2.2, hence U ((i1 (H) − i2 (H))i(T )) is a scalar. The rest of the statement is now obvious. Corollary 2.3. Let I → A(I) be a second quantization conformal precosheaf on S 1 as described in the following subsection, I → Ad (I) be its dual net and U be the above representation of PSL(2, R)∗P0 PSL(2, R), then the irreducible components of U belong to the family described in Theorem 2.1. . Proof. Since the dual net may be described in terms of the local algebras Ad (a, b) = A(−∞, b) ∩ A(a, ∞) and the map which associates a local algebra with a local subspace is an isomorphism of complemented lattices (cf. [1]), the representation U is indeed a second quantization. On the one-particle space, the construction of the dual net may be done on any irreducible component, and the result follows. Corollary 2.4. Every irreducible lowest weight representation of PSL(2, R) extends to a (anti-)representation of Mob in a unique (up to a phase) way. Proof. Let Eλ , Tλ and Sλ be the generators of the representation of lowest weight −1 0 α as above. Since Mob is generated by PSL(2, R) and e.g. the matrix , which 0 1 correspond to the change of sign on R, we need an antiunitary C which satisfies CEλ C = Eλ , CTλ C = −Tλ and CSλ C = −Sλ . (Because Eλ , Tλ and Sλ are generators, C is then uniquely defined up to a phase.) Since in the Schrödinger representation the complex conjugation C satisfies the mentioned commutation relations with T0 , S0 and E0 , it trivially has the prescribed commutation relations with Tλ and Eλ , and the last relation follows by the formula Sλ = S0 − λT0−1 . 2.2. A modular construction of free conformal fields on S 1 . For a certain class of pairs (M, G), where M is a homogeneous space for the symmetry group G, modular theory may be used to construct a net of local algebras on M starting from a suitable (anti-) representation of the symmetry group G [6]. For related works, pointing also to other directions, the interested reader should consult [27–29]. We sketch here the case of the action of the Möbius group on S 1 . We recall that a real subspace K of a complex Hilbert space H is called standard if K ∩ iK = {0} and K + iK is dense in H, and Tomita operators j, δ are canonically associated with any standard space (cf. [25]). One may easily show that the subspaces K0 , iK and iK0 are standard subspaces if K is such, where the symplectic complement K0 is defined by K0 = {h ∈ H ; Im(h, g) = 0 ∀g ∈ K}. As shown in [6], with any positive energy representation of Mob on a Hilbert space H we may uniquely associate a family I → K(I) of standard subspaces attached to proper intervals I in S 1 satisfying the following properties: 1) 2) 3) 4)

I1 ⊂ I2 ⇒ K(I1 ) ⊂ K(I2 ) cK(I)0 = K(I 0 ) U (g)K(I) = K(gI) δIit = U (3I (t)), jI = U (rI )

(isotony), (duality), (conformal covariance), (Bisognano–Wichmann property),

that is to say, I → K(I) is a local conformal precosheaf of standard subspaces of H on the proper intervals of S 1 . The subspaces K(I) are defined as


229

1 . K(I) = {h ∈ H | jI δI2 h = h}.

We notice that the precosheaf is irreducible, i.e. ∨K(I) is dense in H, if and only if U does not contain the trivial representation. Applying the second quantization functor, we then get a local conformal precosheaf of von Neumann algebras acting on the Fock space eH . Now we observe that we may extend the lowest weight representations (with integral α = n) described in Proposition 2.2, to an (anti-)representation of Mob (cf. Corollary 2.4) in a coherent way, e.g. choosing the complex conjugation C for any α as in the proof of Corollary 2.4, and we get a family of conformal precosheaves I → Kn (I) of standard spaces on S 1 . The groups etE and etT have a unique fixed point, namely {∞}, in S 1 and we may therefore identify S 1 \ {∞} with R. Then, since the modular groups of the half-lines do not depend on n by construction, we get Kn (I) = K1 (I) when I is a half-line, namely {∞} is one of its edges. Theorem 2.5. With the above notations, let I → K(I) be a conformal precosheaf of standard subspaces of H on S 1 such that K(I) = K1 (I) for any half-line I. Then, there exists n ∈ N such that K(I) = Kn (I) for any interval I. Proof. By the Bisognano–Wichmann Theorem for conformal precosheaves on S 1 (cf. [5, 14]) we get that U|P0 coincides with the restriction to P0 of the n = 1 lowest weight representation of PSL(2, R). Therefore, since U is a positive energy representation, it should be of the form described in Proposition 2.2. Suppose now we start with the unique irreducible positive energy unitary representation U of the translation-dilation group, with non-trivial restriction to the translation subgroup, on a Hilbert space H. According to [6], we may then consider the associated precosheaf of standard subspaces I → K(I) on the half-lines I ⊂ R. The following corollary summarizes some properties discussed in Sect. 1 and some results of the previous subsection. Corollary 2.6. Let I → K(I) the above described precosheaf on the half-lines of R. Then there exists a bijective correspondence between • Extensions of K to a local conformal precosheaf on the intervals of S 1 . • Real standard subspaces of K(−∞, 1)0 ∩ K(0, ∞) -halfsided invariant w. r. t. the subgroup of dilations centered in 0 and +halfsided invariant w.r.t. the subgroup of dilations centered in 1. • The real linear spaces Kn (0, 1), n ∈ N. 2.3. Multiplicative perturbations: a formula for the canonical endomorphism. We now give an alternative way to pass from the representation of lowest weight 1 to the representation with lowest weight α ≥ 1. In this subsection we denote by E, T, S the Lie algebra generators in the lowest weight 1 representation, and with E, T, Sα the corresponding generators in the lowest weight α case. Instead of defining the generator Sα as S − λT −1 , λ = α(α − 1), we will define the unitary Rα corresponding to the ray inversion or, equivalently, the unitary γ = γα = Rα R = Jα J,

(11)

where J, resp. Jα is the modular conjugation of K(−1, 1), resp. Kα (−1, 1), as J = CR and Jα = CRα with the same anti-unitary conjugation commuting with them. In the

230


examples below the second quantization of γ will implement the canonical endomorphism of the inclusion of algebras Aα (−1, 1) ⊂ A(−1, 1) given by (α − 1)-derivative of the current algebra (in case α is an integer). We now make some formal motivation calculations, that may however be given a rigorous meaning. First note that γ commutes with E, because both J and Jα commute with E, hence γ must be a bounded Borel function of E γ = fα (E)

(12)

because the bounded Borel functions of E form a maximal abelian von Neumann algebra. In order to determine f = fα , note that the formulas RT R = S, Rα T Rα = Sα = S − λT −1

(13) (14)

imply γ ∗ T γ = T − λRT −1 R, hence γ ∗ T γT −1 = 1 − λRT −1 RT −1 = 1 − λ(T RT R)−1 . On the other hand by (10), and since T ET

T RT R = T S = E(E − 1) −1

(15) (16)

= E − 1, thus T f (E)T −1 = f (E − 1),

(17)

formula (15) implies f to satisfy the functional equation λ f (z − 1) =1− f (z) z(z − 1)

(18)

and |f (z)| = 1 for all z ∈ iR. Proposition 2.7. If α = n is an integer, then γ=

(E − 1)(E − 2) · · · (E − n + 1) . (E + 1)(E + 2) · · · (E + n − 1)

(19)

In the general case, γ = fα (E) with fα (z) =

0(z + 1)0(z) , 0(z + α)0(z − α + 1)

(20)

where 0 is the Euler Gamma-function. Proof. Let γα be given by the formula (19). In order to check that γα gives (up to a phase) the unitary (11) it is enough to check that γα Eγα∗ = Rα ERα = E, γα Sγα∗ = Rα T Rα = Sα

(21) (22)

because the representation generated by E and S is irreducible, see (10) and the remarks below it. The first equation is obvious because γα is a function of E. To verify the second equation we notice that from S = RT R we get S positive and non-singular and (10)


231

shows that this also holds for E(E + 1) = T S. Since SES −1 = E + 1 the functional equation for fα implies fα (E)Sfα (E)∗ = fα (E)fα (E + 1)∗ S = (1 −

λ )S = S − λT −1 = Sα . E(E + 1)

2.4. Lowest weight representations of PSL(2, R) and derivatives of the U (1)-current. On the space C ∞ (S 1 , R) of real valued smooth functions on the circle S 1 , we consider the seminorm ∞ X k|φˆ k |2 kφk2 = k=1

c = −isign(k)φˆ k , where the φˆ k ’s denote the Fourier coefficients and the operator I: Iφ k of φ. Since I 2 = −1 and I is an isometry w.r.t. k · k, (C ∞ (S 1 , R), I, k · k) becomes a complex vector space with a positive bilinear form, defined by polarization. Thus, taking the quotient by constant functions and completing, we get a complex Hilbert space H. We note that the symplectic form ω may be written as Z 1 −i X ˆ k f−k gˆ k = gdf. ω(f, g) = Im(f, g) = 2 2 S1 k∈Z

One might recognize this form as coming from the commutation relations for U (1)− currents. The natural action of PSL(2, R) on S 1 gives rise to a unitary representation on H: U (g)φ(t) = φ(g −1 t). Then, observing that I cos kt = sin kt for k ≥ 1, it is easy to see that cos kt is an eigenvector of the rotation subgroup U (θ): U (θ) cos kt = cos k(t − θ) = (cos kθ + sin kθI) cos kt = eikθ cos kt,

k ≥ 1,

and that all the eigenvectors have this form. Therefore the representation has lowest weight 1. We need another description of the Hilbert space H which is more suitable to be generalized. First we choose another coordinate on S 1 , namely x = tan(t/2), x ∈ R, ˙ R ˙ being the one-point compactification and therefore identify C ∞ (S 1 , R) with C ∞ (R), of R. Since the symplectic form is the integral of a differential form it does not depend on the coordinate: Z 1 g(x)df (x). ω(f, g) = 2 R A computation shows that the anti-unitary I applied to a function f coincides up to an additive constant with the convolution of f with the distribution 1/(x + i0) on R, therefore, since the symplectic form is trivial on the constants, the (real) scalar product may be written as

232


Z 1 1 ∗ g(x) f 0 (x)dx hf, gi = ω(f, Ig) = 2 x + i0 Z Z ∞ 1 1 f (x)g(y) dxdy = const pfˆ(−p)g(p)dp ˆ = 4π |x − y + i0|2 0

(23)

˙ w.r.t. this norm. and H may be identified with the completion of C ∞ (R) Note that since If = −if if suppfˆ ⊂ [0, +∞), H is also the completion of C ∞ (R, C) R∞ modulo {f |fˆ|(−∞,0] = 0} with scalar product (f, g) = 0 pfˆ(p)g(p)dp. ˆ . ˙ + R2(n−1) [x], n ≥ 1, where Rp [x] Let us now consider the space X n = C ∞ (R) denotes the space of real polynomials of degree p, and the bilinear form on it given by Z 1 1 hf, gin = f (x)g(y) dxdy. 4π |x − y + i0|2n It turns out that h · , · in is a well defined positive semi-definite bilinear form on X n which degenerates exactly on R2(n−1) [x]. On this space one may define also a symplectic form by Z 1 f (x)g(y)δ0(2n−1) (x − y)dxdy. ωn (f, g) = 2 This form might be read as the restriction of ω1 to the nth derivatives. Therefore we can recognize this symplectic form as coming from the commutation relations for the nth derivatives of U (1)−currents. This form again degenerates exactly on R2(n−1) [x], and the operator I defined before connects the positive form with the symplectic form for . any n in such a way that (·, ·)n = h·, ·in + iωn (·, ·) becomes a complex bilinear form on (X n , I). We shall denote by Hn the complex Hilbert space obtained by completing the quotient X n /R2(n−1) [x]. a b With any matrix g = in SL(2, R) we may associate the rational transforc d ax+b mation x → gx = cx+d and then, for any n ≥ 1, the operators U n (g) on X n : U n (g)f (x) = (cx − a)2(n−1) f (g −1 x). It turns out that g → U n (g) is a representation of PSL(2, R), n ≥ 1, and that the positive form is preserved (cf. [38]) as well as the symplectic form and the operator I, therefore U n extends to a unitary representation of PSL(2, R) on Hn . We remark that while X n and R2(n−1) [x] are globally preserved by U n , the space ˙ is not, and that explains why the space X n had to be introduced. C ∞ (R) By definition the space H1 coincides with the space H and the representation U 1 with the representation U , which we proved to be lowest weight 1. We observe that, for ˙ one gets functions in C ∞ (R), Z 1 ˆ |p|2n−1 fˆ(−p)g(p), hf, gin = 2 R Z 1 ˆ ω(f, g)n = p2n−1 fˆ(−p)g(p), 2 R hence (f, g)n = (Dn−1 f, Dn−1 g)1 , i.e. Dn−1 is a unitary between Hn and H1 ≡ H, where D is the derivative operator. The following holds:


233

Theorem 2.8. The representation U n has lowest weight n. Proof. Making use of the results of Proposition 2.7, we have to show that Rn R =

n−1 Y k=1

E−k E+k

,

n ≥ 1,

where Rn = Dn−1 U n (r)(Dn−1 )∗ with r the ray inversion, R = R1 . This amounts to prove n−1 Y E − k n−1 n U (r)Dn−1 . U (r) = (24) D E+k k=1

Now we take Eq. (24) as an inductive hypothesis. Then, Eq. (24) for n+1 can be rewritten, using the inductive hypothesis and the relation U n+1 (r) = x2 U n (r), as E−n n 2 n Dn−1 U n (r)D. (25) D (x U (r)) = E+n Finally we observe that U n (r)D = x2 DU n − 2(n − 1)xU n , hence Eq. (25) is equivalent to (26) (E + n)D n (x2 ·) = (E − n)Dn−1 (x2 D · −2(n − 1)x·). Since E = −xD, Eq. (26) follows by a straightforward computation.

Proposition 2.9. The unitary representations of PSL(2, R) on H given by Dn−1 U n (Dn−1 )∗ ,

n≥1

coincide when restricted to the subgroup of translations and dilations on R. Proof. We have to prove that Dn−1 U n (g) = U (g)Dn−1 when g is a translation or a dilation. Fortranslations, Un (t)f (x) = f (x − t), and the equality is obvious; for 0 eλ/2 dilations, g = , U n (λ)f (x) = eλn f (e−λ x), hence Dn−1 U n (λ)f (x) = 0 e−λ/2 f (n) (e−λ x) = U (λ)Dn f (x). The family of representations Dn−1 U n (Dn−1 )∗ on the Hilbert space H constitute a concrete realization of the family of (integral) lowest weight representations described in Proposition 2.2, therefore we may construct a family of local conformal precosheaves of standard subspaces of H as explained in Subsect. 2. In the next subsection we shall give another description of these precosheaves, showing that they coincide with the ones described in [38]. 2.5. Relations among local spaces. Let us fix an n ≥ 1 and, for any proper interval I of ˙ let us set R, X n (I) = {f ∈ X n : f |I 0 ≡ 0}. It is easy to check that these spaces satisfy the properties 1. I1 ⊂ I2 =⇒ X n (I1 ) ⊂ X n (I2 ) (isotony), 2. I1 ∩ I2 = ∅ =⇒ X n (I1 ) ⊂ X n (I2 )0 (locality), 3. U n (g)X n (I) = X n (gI), ∀g ∈ PSL(2, R) (covariance),

234


. and that the immersion iIn : X n (I) → Hn is injective. Therefore the spaces Kn (I) = I n − (in X (I)) , where the closure is taken w.r.t. k · kn , form a local conformal precosheaf of subspaces of Hn , and the following property obviously holds: 4.

_

Kn (I) = Hn

(irreducibility).

˙ I⊂R

Therefore, by the first quantization version of results mentioned in Sect. 1, these spaces are standard, the Bisognano–Wichmann property and duality on the circle hold. . Now we identify Hn with H via the unitary Dn−1 , and set Kn (I) = Dn−1 Kn (I). Then, if I is compact in R and f ∈ Kn (I), f may be integrated n − 1 times, giving a function which still has support in I, therefore Z tj f = 0, j = 0, . . . , n − 2}, I ⊂⊂ R, (a) Kn (I) = {[f ] ∈ H : f |I 0 = 0, where [f ] denotes the equivalence class of f modulo polynomials. If I is a half line in R, Kn (I) is an invariant subspace of the dilation subgroup, which is the modular group of K(I). Using Takesaki’s result, see [32], this implies that Kn (I) = K(I),

I a half-line,

(b)

(For an alternative proof of this fact, see [38].) Then, by duality and the formula for the compact case, we obtain Kn (I) = {[f ] ∈ H : f |I 0 = pf,I ∈ Rn−1 [x]}

I 0 ⊂⊂ R.

(c)

Finally we observe that, since the Bisognano–Wichmann property holds, these precosheaves coincide with those abstractly constructed in Subsect. 2. . Now, we fix a bounded interval in R, e.g. (−1, 1), and consider the family Kn = Kn ((−1, 1)). The concrete characterization of Kn given in the preceding subsection shows that Km ⊆ Kn if m ≥ n. Now, we may show Theorem 2.10. The following dimensional relations hold: codim(Km ⊂ Kn ) = m − n, m ≥ n, 0 ∩ Kn ) = max((m − n − 1), 0). dim(Km Before proving Theorem 2.10, we discuss some of its consequences. Definition 1. A precosheaf W K is said n−regular if, for any partition of S 1 into n intervals n I1 , . . . , In , the linear space j=1 K(Ij ) is dense in H. We recall that irreducible conformal precosheaves are 2-regular, because duality holds and local algebras are factors. Corollary 2.11. The conformal precosheaf K1 is n-regular for any n. The conformal precosheaf K2 is 3-regular but it is not 4-regular. The conformal precosheaves Kn , n ≥ 3, are not 3-regular. Moreover, strong additivity and duality on the line hold for the precosheaf I → K1 (I) only, therefore it is the dual precosheaf of I → Kn (I) for any n.


235

Proof. First we recall that a precosheaf is strongly additive if and only if it coincides with its dual precosheaf. Then, the precosheaf K ≡ K1 is strongly additive because its dual net should be of the form Kn (cf. Corollary 2.6) and should satisfy Kd (−1, 1) ⊇ K(−1, 1). As a consequence, K is n-regular for any n. Then, since the spaces for the half-lines do not depend on n, the dual net of Kn does not depend on n either, hence coincides with K. Since PSL(2, R) acts transitively on the triples of distinct points, we may study 3-regularity for the special triple (−1, 1, ∞) in R ∪ {∞}. Then, (Kn (∞, −1) ∨ Kn (−1, 1) ∨ Kn (1, ∞))0 = (K1 (∞, −1) ∨ Kn (−1, 1) ∨ K1 (1, ∞))0 = (K1 (−1, 1)0 ∨ Kn (−1, 1))0 = Kn (−1, 1)0 ∧ K1 (−1, 1), where we used strong additivity and duality for K1 . By Theorem 2.10, 3-regularity holds if and only if n = 1, 2. Violation of 4-regularity for K2 may be proved by exhibiting a function which is localized in the complement of any of the intervals (∞, −1), (−1, 0), (0, 1), (1, ∞), i.e. belongs to K2 (−1, 0)0 ∩ K2 (0, 1)0 ∩ K1 (−1, 1):   1 + x if − 1 ≥ x ≥ 0, φ(x) = 1 − x if 0 ≥ x ≥ 1, 0 if |x| ≥ 1. In the same way we may construct a function which violates 3-regularity for K3 , namely 2 x − 1 if |x| < 1, φ(x) = 0 if |x| ≥ 1 Clearly, φ ∈ K3 (∞, −1)0 ∩ K3 (−1, 1)0 ∩ K3 (1, ∞)0 = K30 ∩ K1 .

Lemma 2.12. codim(Km+1 ⊂ Km ) = 1. R m−1 Proof. Since Km+1 = {φ ∈ Km φ(x)dx = 0}, and we may find a funcx 0 tion ψm−2 ∈ C0∞ (R) : ψm−1 (x) = xm−1 , x ∈ (−1, 1), we get Km+1 = {φ ∈ Km ω(ψm−1 , φ) = 0}. Because the functional φ −→ ω(ψm , φ) is continuous and non zero on Km , the thesis follows. Proof of Theorem 2.10. The first statement of the theorem easily follows from 0 ∩ Km . We observe Lemma 2.12. Now, let us consider the relative commutants Km+p that, by the Poincaré inequality, the norm on K1 is equivalent to the Sobolev norm for the space H 1/2 , i.e., we may identify K1 ' H 1/2 (−1, 1) as real Hilbert spaces. We also recall that the Dirac measure δ does not belong to H −1/2 , but belongs to H −1/2− for each > 0 (see, e.g., [33]). Then 0 ∩ Km = {φ ∈ Km hφ0 , ψi = 0 ∀ψ ∈ Km+p } Km+p Z tj φ = 0, j = 0, . . . , m − 2, = {φ ∈ H 1/2 (−1, 1)

hφ(m+p) , ψi = 0, ∀ψ ∈ H m+p−1/2 (−1, 1)}. . Then f = φ(m+p) ∈ H 1/2−m−p {−1, 1}, i.e., f should be a combination of Dirac’s δ measures with supports in {−1, 1} and their derivatives. Since f ∈ H 1/2−m−p , it has Pm+p−2 (j) (j) the form f = j=0 (cj δ(−1) + dj δ(1) ). The condition φ ∈ Km may be written as

236


hf, tq i = 0,

q = 0, . . . , 2m + p − 2.

(27)

0 ∩ Km will be the difference between the dimension of the space The dimension of Km+p . Pm+p−2 (j) (j) ) cj , dj ∈ R}, which is 2(m + p − 1), and the number of 1 = { j=0 (cj δ(−1) + dj δ(1) independent conditions in Eq. (27). We may also write the conditions in Eq. (27) as

hf, P i = 0

where P is a polynomial of degree 2m + p − 2.

They are independent if the only polynomial P of degree ≤ 2m + p − 2 satisfying hf, P i = 0 for any f ∈ 1 is the null polynomial. Indeed, such polynomial should have zeroes with multiplicities greater than m + p − 1 for the points −1 and 1, therefore, either p = 0, and then there exists exactly one non trivial such polynomial, or p > 0, and the null polynomial is the unique solution. In conclusion, if p > 0, the conditions 0 ∩ Km is in Eq. (27) are independent, and the dimension of Km+p 2(m + p − 1) − (2m + p − 1) = p − 1. If p = 0 the independent conditions in Eq. (27) are 2m − 2, and the dimension is 2(m − 1) − (2m − 2) = 0, which corresponds to the general fact (see, [17]) that local algebras of irreducible conformal theories are factors.

3. Examples of Superselection Sectors for the First Derivative of the U (1)-current In this section we shall discuss examples of superselection sectors of the first-derivative theory. All these sectors are abelian, i.e. are equivalence classes of automorphisms, and we will see that they are non covariant under the conformal group. In particular, recalling that the first-derivative precosheaf is 3-regular but not 4-regular, this shows that the assumption of 4-regularity in [16] cannot be avoided in general in order to obtain the automatic covariance of superselection sectors. As we shall see, all these sectors will be obtained by generalizing methods of the Buchholz-Mack-Todorov approach to sectors (see [7]). On the one hand the conformal net on R associated with the current algebra contains as a subnet the one associated with the first (nth ) derivative of the current algebra (cf. Corollary 2.11 and the Remark after Corollary 1.5), therefore BMT sectors may be restricted to the conformal net on R associated with the first (nth ) derivative of the current algebra. The sectors described here will be extensions (of such restrictions) to the conformal precosheaf on S 1 associated with the first (nth ) derivative of the current algebra. On the other hand they may be seen as sectors on a suitable global algebra in a way which is formally identical to the BMT procedure. Now let A be a local conformal precosheaf on R and Ad its Bisognano–Wichmann dual net. Consider the unitary 0 = JJd , where J and Jd are the modular conjugations of A(−1, 1) and Ad (−1, 1) with respect to the vacuum vector , in other words 0 is the product of the two ray inversion unitaries of the nets A and Ad . The unitary 0 implements the canonical endomorphism γ of Ad (−1, 1) into A(−1, 1). Let now ρ be a morphism of A. We define the “extension” of ρ to Ad by ρ˜ = Ad0∗ ργ . If I contains the origin (possibly at the boundary), γ sends Ad (I) into A(I):


ˆ 0 J ⊂ JA(I) ˆ 0 J = A(I), Ad0(Ad (I)) = JAd (I)

237

(28)

where Iˆ is the image of I under the ray inversion map. Therefore, if {ρI } is the family of representations defining ρ and I is an interval containing the origin, then ρ˜I = γ −1 ρI γ gives a representation of A(I) and these representations are coherent. However if I does not contain the origin, there are two minimal intervals containing both I and the origin, one in which they are in clockwise order and the other in which they are in the counterclockwise one, and the two corresponding representations do not necessarily agree on the algebra of the intersection. If they do not, ρ˜ is not a representation of the dual precosheaf, nevertheless, if the point at infinity is removed and ρ is localized in a compact interval, only one choice remains, and we get a representation of the net Ad0 on the line. Clearly equivalent endomorphisms give rise to equivalent representations and if we choose the localization region I0 not containing the origin, say I0 = (a, b), b > a > 0, then ρ˜ is localized in (a, ∞). We have therefore shown that any transportable sector on A(I) gives rise to a (possibly solitonic) sector on Ad0 (I). (In this lower dimensional theory one might also interpret these sectors as coming from order variables, [30], Chapter 3.8.) In Subsect. 3 we show examples of this phenomenon. Conversely, if we assume that the two above mentioned representations agree, we get a representation of the precosheaf Ad , and assuming again that the localization interval I0 does not contain the origin, ρ˜ is localized in I0 . If we further assume that ρ is covariant and finite statistics, we obtain that ρ˜ is finite statistics too, because the index may be computed by looking at the endomorphisms of the von Neumann algebra Ad (0, ∞) = A(0, ∞). Hence ρ˜ is covariant by the strong additivity of the dual net (see [16]) and this implies that ρ and ρ˜ determine equivalent representations of the net A0 on the line. In fact, by the construction of the dual net, the product JJd of the modular conjugations for the interval (−1, 1) relative to the two theories coincides with the product of the two unitaries implementing the conformal transformation t → −1/t. Then, denoting by r, rd the corresponding automorphisms and by ud and u the unitaries ˜ d and Ad(u)ρ = rρr we have such that Ad(ud )ρ˜ = rd ρr Ad(ud )ρ˜ = rd ρr ˜ d = rρr = Ad(u)ρ. 3.1. Buchholz, Mack and Todorov approach to sectors of the current algebra . We defined the one-particle space for the current algebra as the completion of the space ˙ modulo constant functions w.r.t. the norm given in (23). We may X = X 1 = C ∞ (R) 1 then define A(S ) as the ∗ - algebra generated by W (h), h ∈ X with the relations W (h)W (h)∗ = 1 (unitarity) and W (h)W (k) = exp(i/2ω(h, k))W (h + k) (CCR). BMT automorphisms of A(S 1 ) are then given in terms of differential forms φ on S 1 . Setting R . αφ (W (h)) = ei hφ W (h), it is easy to see that α extends to an automorphism of A(S 1 ). By CCR, it follows that αφ is inner if and only if the form φ is exact, i.e. there exists a function R f ∈ RX s.t. φ = df , and that two automorphisms αφ , αψ are equivalent if and only if φ = ψ, i.e. R if the two forms give the same cohomology class in H 1 (S 1 ). The constant Q(αφ ) := φ will be called the charge of αφ . For any open interval I in S 1 we set A(I) to be the subalgebra of A(S 1 ) generated by Weyl unitaries W (h) such that the support of h is contained in I. Clearly the algebras . . associated with disjoint intervals commute and βg A(I) = A(gI), where βg (W (f )) = W (U (g)f ).

238


We observe that BMT automorphisms are locally internal, i.e. for any interval I and any form φ there exists a function f with support in some larger interval Iˆ such that df |I ≡ φ|I , therefore αφ |A(I) ≡ adW (f )|A(I) . Also, the superselection sectors corresponding to a given charge are conformally covariant w.r.t. the adjoint action of the conformal group on X, i.e. the automorphisms αφ and βg · αφ · βg−1 are in the same class for any conformal transformation g. Indeed, since the class of inner automorphisms is globally stable under the action of the conformal group and the charge is additive, namely Q(αφ ◦ αψ ) = Q(αφ ) + Q(αψ ), the action of PSL(2, R) on BMT automorphisms gives a linear action on BMT charges, i.e. a one dimensional linear representation of PSL(2, R). Any such representation being trivial, BMT sectors are covariant. Now we give a local description for these sectors. We observe that the second quantization algebra associated with the standard space K(I) coincides with π(A(I))00 = R(I), where π is the vacuum representation of A(S 1 ) on the Fock space eH . Moreover, the ˆ map π|A(I) is faithful, and the restriction of αφ to A(I), being implemented in A(I), uniquely extends to a normal automorphism of R(I). As a consequence, αφ gives rise to a representation I → αφI of the precosheaf A in the sense of [17], where R i hI φ I W (h) αφ (π(W (h))) = e and, recalling that h ∈ H is localized in I if it is equal to a constant cI in I 0 , we have . set hI = h − cI . We described BMT locally normal representations via automorphisms of A(S 1 ). Conversely, the global algebra A(S 1 ) plays the role of the universal algebra w.r.t. the family of the locally normal BMT representations, in the sense that the classes of such representations modulo unitary equivalence appear as classes of global automorphisms of A(S 1 ) up to inners. 3.2. Restriction of localized sectors. As we have already seen, local algebras associated with compact intervals on the line for the first-derivative net may be described as Z . R2 (I) = {W (h) ∈ R(I) : h(x)dx = 0}00 . Of course these algebras form a net of local algebras on the real line which is covariant with respect to the action of translations and dilations, but Haag duality does not hold on R. The quasi-local algebra A2 (R) generated by the algebras of compact intervals is a subalgebra of the quasi-local algebra A(R) of the current algebra on the line, therefore any BMT automorphism of A localized in some compact interval I gives a representation of A2 (R) which is equivalent to the vacuum representation if and only if it has zero charge, but, due to the failure of Haag duality, the intertwining unitary is not necessarily localized in I. Such a unitary exhibits instead a solitonic localization, i.e. it necessarily belongs to the von Neumann algebra of any half line containing the localization region. The restrictions of BMT sectors are then translation and dilation covariant. On the contrary, appears, if we consider classes of automorphisms of A2 (R) modulo inners, a new charge R R i.e. R twoRautomorphisms αφ and αψ are equivalent if and only if both φ = ψ and tφ = tψ are equal. As a consequence, such sectors are no longer translation covariant. 3.3. Conformal solitonic sectors. In the first-derivative theory, the automorphism αφ is localized in a compact interval I of R when φ is constant outside I, therefore solitonic


239

sectors on A(R) may become localized when restricted to A2 (R). This shows that, conversely, sectors on A2 may become solitonic when extended to A as described at the beginning of this section. Here we shall consider φ as a function on R rather than as a differential form, identifying φ(t) with φ(t)dt. If φ is constant outside I, αφ Ris equivalent to the vacuum (as +∞ a representation) when both φ(+∞) = φ(−∞) (= 0) and −∞ φ = 0. As a consequence, superselection sectors are described by two charges: Z Z Q0 = φ(+∞) − φ(−∞) = φ0 (t)dt, Q1 = tφ0 (t)dt. These sectors are clearly transportable, but a simple computation shows that they are covariant under translations and dilations if and only if Q0 = 0, i.e. only if they are restrictions of BMT sectors. Restrictions to A2 (R) of solitonic sectors on A(R) give then an example of localized non covariant sectors on the line. As we shall see, these sectors may be extended to transportable sectors on the circle. 3.4. Generalized BMT approch to sectors: Local description. We recall that the conformal precosheaf R2 of the first-derivative theory may be described as second quantization algebras on the same Fock space as the current algebra, R2 (I) = {W (h) : h ∈ K2 (I)}00 . We have seen that Z K2 (I) = {[f ] ∈ H : f |I 0 ≡ pf,I , pf,I ∈ R0 [x], (f (x) − pf,I )dx = 0} I ⊂⊂ R, K2 (I) = {[f ] ∈ H : f |I 0 ≡ pf,I pf,I ∈ R0 [x]}

I half line,

K2 (I) = {[f ] ∈ H : f |I 0 ≡ pf,I , pf,I ∈ R1 [x]}

I 0 ⊂⊂ R.

In order to extend a BMT automorphism αφ to the first-derivative theory on the circle we have to choose a real number λ and then set I (W (f )) = eihφ,f iI,λ W (f ), αφ,λ

where f belongs to K2 (I), with

Z

hφ, f iI,λ = Taking φ, ψ such that Q =

R

φ=

R

φ(x)(f (x) − pf,I (λ))dx. −1 ψ we may compute αφ,λ · αψ,µ :

−1 I (αφ,λ · αψ,µ ) (W (f )) = ei(hφ,f iI,λ −hψ,f iI,µ ) W (f ) 0

= e−iQ(λ−µ)hδx ,f i ad W (h)(W (f )),

(29)

with h0 = φ − ψ, where x is any point in I 0 . Therefore we have proved that two BMT automorphisms with the same (non zero) charge extend to equivalent automorphisms if and only if λ = µ. Since a simple calculation shows that the translated automorphism βT (t) αφ,λ βT (−t) is equal to αφ( · +t),λ−t , we conclude that non trivial BMT sectors give rise to a one parameter family of non covariant sectors on the circle. −1 is equivalent to a new autoMoreover, when λ 6= µ, the automorphism αφ,λ · αψ,µ morphism θc , c = −Q(λ − µ): 0

θcI (W (f )) = eichδx ,f i W (f ),

x ∈ I 0.

(30)

240


Since pf,I is constant whenever I ⊂ R and hence hδx0 , f i vanishes, we conclude that θc is localized in only one point, the point at infinity. We may easily show that θc is invariant under translations and that dilations act on the charge c. By conjugating θc with a conformal transformation we will get a family of automorphisms localized in different points of the real line. In particular, requiring the automorphisms to be localized in zero, we get a family ζc : R . ic x (f (y)−pf,I (y))dy W (f ) x ∈ I 0 . (31) ζc (W (f )) = e 0 Rx Indeed it is easy to see that, if f ∈ K2 (I), then 0 (f (y)−pf,I (y))dy does not depend 0 / I. Now let 0 be in I, and suppose for simplicity on x ∈ I , and is equal to zero when 0 ∈ that I ⊂ R. Then formula (31) may be obtained by formula (30) using R(π), the rotation by π, which in the real line coordinates is t → −1/t. As in Proposition 2.9, such rotation is implemented by DU (2) (R(π))D∗ on the Hilbert space H, and, since f has compact support, we may set x = ∞ in formula (31). Hence the correspondence between (31) and (30) follows by the equality Z

∞

2 0

Z

− x1

(f (y) − pf,I )(y) = hδ0 , 2x 0

1 i. − x

(f − pf,I )(y)dy + f

When f is localized in R and we choose x = ∞ as before, the automorphisms ζc furnish extensions to the circle of the restriction to A2 (R) of solitonic sectors on A(R). It is not difficult to see that dilations act on these automorphisms dilating the charge, therefore these sectors are non covariant too. In the following subsection we shall see that the Weyl algebra on the symplectic space (X 2 , ω2 ) is a global algebra for all these sectors, i.e. the given automorphisms of the precosheaf modulo unitaries on the Fock space are described by automorphisms of this global algebra modulo inners. In doing that we shall see that the described sectors form a group isomorphic to R3 . 3.5. Generalized BMT approach to sectors: Global description. Now we describe some ˙ on (X 2 , ω2 ). If φ is a measure on R ˙ natural automorphisms of the Weyl algebra A2 (R) R 2 such that (1 + t )d|φ|(t) < ∞, we set R αφ2 (W (h)) = ei hdφ W (h) h ∈ X 2 , ˙ This automorphism is inner if and only and αφ2 extends to an automorphism of A2 (R). 000 if it is of the form adW (h), where h = φ, i.e. if and only if the first three moments (charges) of φ vanish: Z . Qk (φ) = tk dφ(t) = 0, k = 0, 1, 2. As a consequence, two such automorphisms are equivalent if all their charges coincide. We now consider the corresponding sectors, i.e. classes of automorphisms modulo inners. Proposition 3.1. The only covariant sector on A2 (R) in the above class is the identity sector.


241

Proof. First we see the behavior of the automorphisms under translations: Q0 (βT (t) αφ2 βT (−t) ) = Q0 (αφ2 ), Z 2 Q1 (βT (t) αφ βT (−t) ) = (x − t)dφ(x) = Q1 (αφ2 ) − tQ0 (αφ2 ), Z Q2 (βT (t) αφ2 βT (−t) ) = (x − t)2 dφ(x) = Q2 (αφ2 ) − 2tQ1 (αφ2 ) + t2 Q0 (αφ2 ). Then we compute the charges of automorphisms transformed with the ray inversion r : x → −1/x: Z Q0 (βr αφ2 βr ) = x2 dφ(x) = Q2 (αφ2 ), Z 2 Q1 (βr αφ βr ) = x2 (−1/x)dφ(x) = −Q1 (αφ2 ), Z Q2 (βr αφ2 βr ) = x2 (−1/x)2 dφ(x) = Q0 (αφ2 ). From the first equations we derive that a translation covariant sector has Q0 = Q1 = 0, while covariance under ray inversion amounts to Q0 = Q2 and Q1 = 0, from which the thesis easily follows. Remark. If we generalize the preceding construction to the case of n derivatives, thus obtaining sectors parameterized by 2n + 1 charges, the preceding proof generalizes as well, then showing that the identity sector is the only covariant sector in that case also. Remark. As BMT sectors, also the sectors described above are additive in the sense that the vector charge (Q0 , Q1 , Q2 ) of the composition of two sectors is just the sum of the two charges. Then the action of PSL(2, R) on these sectors gives a linear representation of this group on R3 . The absence of covariant sectors means that the action is free and therefore has no one-dimensional representations, i.e. it is irreducible. ˙ are the sub-algebras generated by the Weyl unitaries The local subalgebras for A2 (R) whose test functions are zero outside I. Then the representation of A2 (R) on the Fock 2 space eH is faithful when restricted to local algebras, i.e. A2 (I) may be seen as a weakly dense subalgebra of the second quantization algebra of the space K2 (I). By a classical Sobolev embedding argument, the functions in K2 (I) are continuous, therefore the automorphisms αφ2 |A2 (I) uniquely extend to normal automorphisms of R2 (I), so that αφ2 gives rise to an automorphism of the precosheaf I → R2 (I). Proposition 3.2. All sectors described above may be localized in two points. Proof. First we observe that some of them may be localized even in one point, in fact the multiples of the δ0 function give sectors with Q1 = Q2 = 0, while for the measures cx−2 δ∞ (x) we have Q0 = 0, Q1 = 0, Q2 = c, therefore we may restrict to the case (Q0 , Q1 ) 6= (0, 0). Then we have to show that for any triple Qi , i = 0, 1, 2, (Q0 , Q1 ) 6= (0, 0), we may find a measure φ = λδa + µδb with the given momenta for some λ, µ, a, b ∈ R, or equivalently solve the system ( =λ+µ Q0 Q1 = λa + µb , Q2 = λa2 + µb2

242


whose solutions are obtained choosing a b for which Q1 −Q0 b 6= 0 and Q0 b2 −2Q1 b+Q2 6= 0 and then setting a = (Q2 − Q1 b)(Q1 − Q0 b)−1 ,λ = (Q1 − Q0 b)2 (Q0 b2 − 2Q1 b + Q2 )−1 , µ = Q0 − λ. The preceding proposition constitutes indeed another proof that these sectors are non covariant, since the following theorem holds: Proposition 3.3. Let I → A(I) be a local conformal precosheaf on S 1 . Then a covariant sector ρ with finite index which may be localized in two points is trivial. Proof. We may suppose that the two points are {0, ∞}. We first observe that ρ is indeed an automorphism because, if j is the antiunitary modular conjugation for R(0, ∞), jρj is still localized in the same two points and then any intertwiner between ρjρjand the identity (which exists e.g. by [17]) is localized in these two points and is therefore a number by two-regularity. The same argument shows that ρ commutes with the dilations, because the cocycle in the covariance equation for the dilation group is then trivial. Now the state ω0 · ρ−1 is dilation invariant, and therefore, by a cluster argument, coincides with the vacuum state, which ends the proof. In this last part of the subsection we show the relation between the local and the global picture of the superselection sectors of the first-derivative theory or, more precisely, we show that all the sectors described in Subsect refsubsec:locsect are (normal extensions ˙ described here. of) the sectors of A2 (R) Proposition 3.4. The sectors [αφ,λ ], [θc ] [ζc ] are of the form [αµ2 ], µ measure. Proof. Given a non trivial sector [αφ,λ ], we may choose a representative s.t. supp φ ⊂ I0 , where I0 is a given compact interval in R. In order [αφ,λ ] to be localized in I0 , we should have hφ, f iI0 ,λ = 0 for any f localized inR I00 , i.e., since on I00 f coincides with pf,I0 , we R xφ(x) get φ(x)(x − λ)dx = 0, i.e. λ = λ0 = R (the denominator does not vanish since φ(x)

[αφ,λ ] is non trivial). Then, according to formula (3), one has [αφ,λ ] = [θ−Q(λ−λ0 ) ] · [αφ,λ ], hence, since the class {αµ , µ measure} is closed under composition, it is enough to prove the statement for [αφ,λ0 ], [θc ] and R[ζc ]. R we have φ(x)pf,I (λ0 )dx = φ(x)pf,I (x)dx, therefore As far as αφ,λ0 is concerned, R hφ, f iI,λ and therefore, integrating by parts, R R 0 Rcoincides with φ(x)(f (x) − pf,I (x))dx with − ( (f − pf,I ))dφ(x). Observing that (f − pf,I ) is exactly the representative in X 2 of D−1 [f ] which vanishes outside I we conclude that the automorphism αφ,λ0 2 ˙ and the relation among the charges is comes from the automorphism α−dφ onA2 (R), 2 2 2 Q0 (α−dφ ) = 0, Q1 (α−dφ ) = Q(αφ,λ ), Q2 (α−dφ ) = 2λ0 Q(αφ,λ ). In the same way we may show that the “solitonic” automorphisms ζc in Eq. (31) come from the automorphisms αµ2 on A2 (R) with µ = −cδ0 . Conjugating ζc with the ray inversion we see that the automorphisms θc in Eq. (30) localized at infinity come ˙ with µ = cx−2 δ∞ . from the automorphisms αµ2 on A2 (R) Remark. In [9] it is shown that in any diffeomorphism covariant theory on S 1 , in physics terms a theory with a stress-energy tensor, superselection sectors are covariant. It is well known that the usual way to associate a stress-energy tensor to the derivative of the


243

U (1)-current formally leads to a conformal charge c = ∞. Our result then shows that the U (1)-current derivative theories indeed do not have a stress energy tensor. Acknowledgement. H.-W. W. wishes to thank the University of Rome II for the kind hospitality and the CNR for financial support during a visit in Rome where this collaboration started. He also wants to thank Bert Schroer warmly for various helpful discussions.

References 1. Araki, H.: A lattice of von Neumann algebras associated with the quantum field theory of a free Bose field. J. Math. Phys. 4, 1343 (1963) 2. Araki, H., Zsido, L.: Extension of the structure theorem of Borchers and its application to half-sided modular inclusions. Manuscript, preliminary version (1995), to appear 3. Bisognano, J., Wichmann, E.: On the duality condition for a Hermitean scalar field. J. Math. Phys. 16, 985 (1975) 4. Borchers, H.-J.: The CPT Theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315 (1992) 5. Brunetti, R., Guido, D., Longo, R.: Modular structure and duality in conformal Quantum Field Theory. Commun. Math. Phys. 156, 201–219 (1993) 6. Brunetti, R., Guido, D., Longo, R.: In preparation 7. Buchholz, D., Mack, G., Todorov, I.: The current algebra on the circle as a germ of local field theories. Nucl. Phys. B, Proc. Suppl. 56, 20 (1988) 8. Buchholz, D., Schulz-Mirbach, H.: Haag duality in conformal quantum field theory. Rev. Math. Phys. 2, 105 (1990) 9. D’Antoni, C., Fredenhagen, K.: In preparation 10. Doplicher, S., Longo, R.: Standard and split inclusions of von-Neumann-algebras. Inv. Math. 75, 493 (1984) 11. Fredenhagen, K.: Generalization of the theory of superselection sectors. In: The algebraic theory of superselection sectors. Introduction and recent results. D. Kastler, ed. Singapore: World Scientific, 1990 p. 379 12. Fredenhagen, K., Jörss, M.: Conformal Haag-Kastler nets, pointlike localized fields and the existence of operator product expansion. Commun. Math. Phys.176, 541 (1996) 13. Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebra II: Geometric aspects and conformal covariance. Rev. Math. Phys. Special Issue, 113 (1992) 14. Fröhlich, J., Gabbiani, F.: Operator algebras and conformal field theory. Commun. Math. Phys. 155, 569 (1993) 15. Guido, D.: Modular covariance, PCT, Spin and Statistics. Ann. Ist. H. Poincarè, 63, 383 (1995) 16. Guido, D., Longo, R.: Relativistic Invariance and Charge Conjugation in Quantum Field Theory. Commun. Math. Phys. 148, 521 (1992) 17. Guido, D., Longo, R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181, 11 (1996) 18. Haag, R.:Local Quantum Physics. Berlin–Heidelberg–New York: Springer-Verlag, 1996 19. Hislop, P.D., Longo, R.: Modular structure of the local algebras associated with the free massless scalar field theory. Commun. Math. Phys. 84, 71 (1982) 20. Lang, S.: SL(2, R). Berlin–Heidelberg–New York: Springer-Verlag, 1985 21. Longo, R.: Solution of the factorial Stone-Weierstrass Conjecture. Invent. Math. 76 145 (1984) 22. Longo, R.: An analogue of the Kac-Wakimoto formula and black hole conditional entropy. Commun. Math. Phys. 186, 451 (1997) 23. Longo, R., Rehren, R.: Net of subfactors. Rev. Math. Phys. 7, 567 (1995) 24. Müger, M.: Superselection Structure of massive Quantum Field Theories in 1+1 Dimensions. hepth/9705019 25. Rieffel, M., Van Daele, A.: A bounded operator approach to Tomita-Takesaki theory. Pacific J. Math. 69, 187–221 (1976) 26. Roberts, J.E. Spontaneously broken gauge symmetry and superselection rules. Proc. Int. School of Math. Physics, Camerino (1974), G. Gallavotti, ed., Univ. di Camerino, 1976

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27. Schroer, B.: Motivations and Physical Aims of Algebraic QFT” Ann. of Phys. Vol. 255 No. 2, 270 (1997) 28. Schroer, B.: Wigner Representation Theory of the Poincare Group, Localization, Statistics and the SMatrix. Nucl. Phys. B 499, 519 (1997) 29. Schroer, B.: Modular Localization and the Bootstrap-Formfactor Program. Nucl. Phys. B 499, 547 (1997) 30. Schroer, B.: A Course on: Quantum Field Theory and Local Observables. Manuscript, based on Lectures held in Rio de Janeiro and Berlin 31. Streater, R.F., Wilde, I.F.: Fermion states of a Boson field” Nucl. Phys. B24, 561 (1970) 32. Stratila, S., Zsido, L.: Lectures on von Neumann algebra. Turnbridge Wells: Abacus Press, 1979 33. Tréves, F.: Topological vector spaces, distributions and kernels. New York: Academic Press, 1967 34. Wiesbrock, H.-W.: Half-Sided Modular Inclusions of von Neumann algebras. Commun. Math. Phys. 157, 83 (1993); Erratum. Commun. Math. Phys. 184, 683–685 (1997) 35. Wiesbrock, H.-W.: Symmetries and Half-Sided Modular Inclusions of von Neumann algebras. Lett. Math. Phys. 29, 107 (1993) 36. Wiesbrock, H.-W.: A comment on a recent work of Borchers. Lett. Math. Phys. 25, 157-159 (1992) 37. Wiesbrock, H.-W.: Conformal Quantum Field Theory and Half-Sided Modular Inclusions of von Neumann algebras. Commun. Math. Phys. 158, 537 (1993) 38. Yngvason, J.: A Note on Essential Duality. Lett. Math. Phys. 31, 127 (1995) Communicated by H. Araki

Commun. Math. Phys. 192, 245 – 260 (1998)

Communications in


A Trinomial Analogue of Bailey’s Lemma and N = 2 Superconformal Invariance? G. E. Andrews, A. Berkovich Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA. E-mail: [email protected], berkov [email protected] Received: 8 March 1997 / Accepted: 29 June 1997

Dedicated to Dora Bitman on her 70th birthday

Abstract: We propose and prove a trinomial version of the celebrated Bailey’s lemma. As an application we obtain new fermionic representations for characters of some unitary as well as nonunitary models of N = 2 superconformal field theory (SCFT). We also establish interesting relations between N = 1 and N = 2 models of SCFT with 2 2 and 3 1 − 4ν . A number of new mock theta function central charges 23 1 − 2(2−4ν) 2(4ν) identities are derived.

1. Brief Review of Bailey’s Method and its Generalizations It may come as a surprise that Manchester, England was an ideal setting for pure mathematics during the height of World War II. However, a number of historical coincidences conspired to make this the case. In particular, mathematics that would later prove extremely valuable in the development of statistical mechanics and conformal field theory (CFT) flourished there. Essentially, Bailey, extending the original ideas of Rogers, came up with a variety of new Rogers-Ramanujan type identities during the winter 1943–44 [1]. Hardy who was then editor for the Journal of London Mathematical Society got Freeman Dyson to referee the paper. Realizing that Bailey and Dyson were the only people in England interested in the subject, Hardy put Bailey in contact with Dyson. The resulting correspondence lead to [2], the first paper which, implicitly at least, contained Bailey’s lemma and the first hint at the “Bailey chain”. A charming account of the Dyson-Bailey collaboration appears in Dyson’s article, “A Walk Through Ramanujan’s Garden” [3]. A few years later, Slater, in a study building on Bailey’s work, systematically derived 130 identities of the Rogers-Ramanujan type [4,5]. In the last decade, Bailey’s technique ?

Partially supported by National Science Foundation Grant: DMS-9501101.

246

G. E. Andrews, A. Berkovich

was streamlined and generalized by Andrews [6] and further extended by Agarwal, Andrews and Bressoud [7,8]. Bailey’s method may be summarized as follows. Let α = {αr }r≥0 , β = {βL }L≥0 be sequences related by the identities βL =

∞ X

αr (q)L−r (aq)L+r

r=0

,

L ∈ Z≥0 ,

  (1−aq−1 )(1−aq1−2 )···(1−aq−n ) , (a)n = 1 ,  (1 − a)(1 − aq) · · · (1 − aq n−1 ) ,

(1.1)

n ∈ Z0

(1.2)

and let γ = {γL }L≥0 , δ = {δr }r≥0 be another pair of sequences related by γL =

∞ X r=L

Then the new identity

∞ X

δr . (q)r−L (aq)r+L

αL γL =

L=0

∞ X

(1.3)

β L δL

(1.4)

L=0

holds. A pair of sequences (α, β) that satisfies (1.1) is called a Bailey pair relative to a. Analogously, a pair of sequences (γ, δ) subject to (1.3) is referred to as the conjugate Bailey pair relative to a. In [2], Bailey proved that (ρ1 , ρ2 )L (aq/ρ1 ρ2 )L 1 , (aq/ρ1 , aq/ρ2 )L (q)M −L (aq)M +L (ρ1 , ρ2 )L (aq/ρ1 ρ2 )L (aq/ρ1 ρ2 )M −L δL = (aq/ρ1 , aq/ρ2 )M (q)M −L

γL =

(1.5) (1.6)

with (a1 , a2 )L ≡ (a1 )L (a2 )L , L ≤ M ∈ Z≥0 satisfy (1.3) for any choice of parameters ρ1 , ρ2 . Combining (1.4, 1.5, 1.6) yields L ∞ X (ρ1 , ρ2 )L (aq/ρ1 ρ2 )M −L aq βL = (aq/ρ1 , aq/ρ2 )M ρ1 ρ2 (q)M −L L=0 (1.7) ∞ X (ρ1 , ρ2 )L (aq/ρ1 ρ2 )L αL . (aq/ρ1 , aq/ρ2 )L (q)M −L (aq)M +L L=0

From the last equation, we deduce immediately: Bailey’s Lemma. Sequences (α0 , β 0 ) defined by 0 αL =

βL0 =

(ρ1 , ρ2 )L (aq/ρ1 ρ2 )L αL , (aq/ρ1 , aq/ρ2 )L

∞ X (ρ1 , ρ2 )r (aq/ρ1 ρ2 )r (aq/ρ1 ρ2 )L−r r=0

(aq/ρ1 , aq/ρ2 )L (q)L−r

form again a Bailey pair relative to a.

(1.8) βr

(1.9)

A Trinomial Analogue of Bailey’s Lemma

247

Obviously, Bailey’s lemma can be iterated ad infinitum leading to a Bailey chain [6,9] of new identities (α, β) → (α0 , β 0 ) → (α00 , β 00 ) → (α000 , β 000 ) → . . .

(1.10)

with the parameter a remaining unchanged throughout the chain. P. Paule [10] independently discovered the essentials of the iterative process formalized by the Bailey chain. The notion of a Bailey chain was upgraded to a “Bailey lattice” in [7,8] where it was shown how to pass from a Bailey pair with given parameter a to another pair with an arbitrary new parameter. Further important developments have taken place in the last few years. In [11], Milne and Lilly found higher-rank generalizations of Bailey’s lemma. Many new polynomial identities of Rogers-Ramanujan type were discovered in [12-19] as the result of recent progress in CFT and Statistical Mechanics initiated by the Stony Brook group [20-22]. Following an observation made by Foda-Quano [23], these identities were recognized as new (α, β) pairs. New (γ, δ) pairs were discovered in [24,25]. Intriguing connections between Bailey’s lemma and the so-called renormalization group flows connecting different models at CFT were discussed in [18,26–28]. This paper is intended as the first step towards a multinomial (or higher-spin) generalization of Bailey’s lemma. Here we concentrate on the trinomial case. Our main assertion is Theorem 1 (Trinomial analogue of Bailey’s lemma). If for L ≥ 0, a = 0, 1, βea (L) =

L X

α ea (r)

Ta (L, r, q) , (q)L

α e0 (r)

(−1)M +1 r r q 2 + q− 2

r=0

(1.11)

then for M ∈ Z≥0 M X

L (−1)L q 2 βe0 (L) =

∞ X r=0

L=0

T1 (M, r, q) (q)M

(1.12)

and M X L=0

(−q −1 )L q L βe1 (L) =

∞ X r=0

α e1 (r)

(−1)M T1 (M, r, q) (q)M (1 − q M ) T1 (M − 1, r + 1, q) 1 + q −1−r (1 − q M ) T (M − 1, r − 1, q) , − 1 1 + q r−1

−

(1.13)

where Ta (L, r, q) are q-trinomial coefficients [28] to be defined in the next section. The pair of sequences (e αa , βea ) that satisfies identities (1.11) will be called a trinomial Bailey pair. The rest of this paper is organized as follows. In Sect. 2 we shall collect the necessary background on q-trinomials and then prove Theorem 1. In Sect. 3 we shall exploit this theorem to derive a number of new q-series identities related to characters of N = 2 SCFT. We conclude with a brief discussion of the physical significance of our results and some comments about possible generalizations.

248


2. q-Trinomial Coefficients and a Trinomial Analogue of Bailey’s Lemma 2.1. Preliminaries. Before turning our attention to the q-trinomial coefficients, let us L briefly recall the ordinary trinomials A [29], defined by 2 x+1+

1 x

L =

L = 0, A 2

L X L xA , A 2

(2.1)

A=−L

|A| > L.

(2.2)

By applying the binomial theorem twice to (2.1) we find by coefficient comparison that X L! L . (2.3) = A 2 j!(j + A)! (L − 2j − A)! j≥0

Furthermore, it is easy to deduce from (2.1) the following recurrences: L L−1 L−1 L−1 = + + , A 2 A−1 2 A A+1 2 2 which along with (2.2) and

0 =1 0 2

(2.4)

(2.5)

specify trinomials uniquely. Equations (2.4) and (2.5) lead to the Pascal-like triangle for L numbers A 2

−

− −

1 − −

1 3 − −

1 2 6 − −

1 1 3 7 − −

1 2 6 − −

1 3 − −

1 − −

− −

− .

q-Analogues of trinomials were forced into existence in the work of Andrews and Baxter on the generalized Hard Hexagon model [29]. These analogues proved to play an important role in Partition Theory [30-32] and Statistical mechanics [33-35]. Unlike binomials, trinomials admit not one but many q-analogues which we now proceed to describe. 2.2. Definitions and properties of q-trinomials. The straightforward q-deformation of (2.3) is as follows X q j(j+B) (q)L L; B; q = . (2.6) (q)j (q)j+A (q)L−2j−A A 2 j≥0

Let us further define another useful q-analogue of (2.3): L(L−n)−A(A−n) L; A − n; q −1 2 , Tn (L, A, q) = q A 2

n ∈ Z.

(2.7)


249

The polynomials Tn (L, A, q) are symmetric under A → −A, Tn (L, A, q) = Tn (L, −A, q),

(2.8)

and vanish for |A| > L: Tn (L, A, q) = 0

if

|A| > L .

(2.9)

The generalization of Pascal-triangle type recurrences (2.4) found in [32, 34] is Tn (L, A, q) = Tn (L − 1, A − 1, q) + Tn (L − 1, A + 1, q)+ q L−

1+n 2

Tn (L − 1, A, q) + (q L−1 − 1)Tn (L − 2, A, q) .

(2.10)

Additionally, there are four more identities needed Tn (L, A, q) = Tn+2 (L, A, q) + (q L − 1)q − q

L−A 2

1+n 2

Tn (L − 1, A, q),

L

Tn+1 (L, A, q) = Tn (L, A, q) + (q − 1)Tn (L − 1, A + 1, q), T1 (L, A, q) − T1 (L − 1, A, q) =

(2.11) (2.12) (2.13)

L−A

L+A

= q 2 T0 (L − 1, A + 1, q) + q 2 T0 (L − 1, A − 1, q), (2.14) T1 (L, A, q) + T1 (L − 1, A, q) = = T−1 (L − 1, A + 1, q) + T−1 (L − 1, A − 1, q) + 2T−1 (L − 1, A, q). Identities (2.11), (2.12), (2.13) follow from Eqs. (2.24), (2.23), (2.16) of [29] and identity (2.14) is Eq. (4.5) of [34]. Next we shall require the limiting formula lim T1 (L, A, q) =

L→∞

(−q)∞ , (q)∞

(2.15)

which is equation (2.51) of [29]. Let us combine (2.11) and (2.12) with n = −1 to obtain q

L−A 2

T0 (L, A, q) − T1 (L, A, q) = = (q L − 1){T−1 (L − 1, A + 1, q) + T−1 (L − 1, A, q)}.

(2.16)

We now replace A by −A in the above equation to get, with the help of (2.8), q

L+A 2

T0 (L, A, q) − T1 (L, A, q) = = (q L − 1){T−1 (L − 1, A − 1, q) + T−1 (L − 1, A, q)}.

If we add (2.16) and (2.17) and use (2.14), the result is A L A q 2 q 2 + q − 2 T0 (L, A, q) − 2T1 (L, A, q) = = (q L − 1){T1 (L, A, q) + T1 (L − 1, A, q)}, which may be conveniently rewritten as

(2.17)

(2.18)

250

G. E. Andrews, A. Berkovich L

q− 2

T0 (L, A, q) =

A

A

q 2 + q− 2

(1 + q L )T1 (L, A, q) (2.19)

L

q − 2 (1 − q L )

−

A

A

q 2 + q− 2

T1 (L − 1, A, q).

We’re now ready to prove Theorem 1. 2.3. Proof of Theorem 1. We shall prove Theorem 1 in two steps. First, we find a trinomial analogue of a conjugate Bailey pair and then use a standard Bailey Transform argument. To this end let us introduce an auxiliary function φ(L, q) L

φ(L, q) = q 2

(−1)L (q)L

(2.20)

with the easily verifiable property φ(L + 1, q) =

√

q

1 + qL φ(L, q). 1 − q L+1

(2.21)

Next we multiply both sides of (2.19) by φ(L, q) and sum both extremes of the result on L from A to M to obtain with the aid of (2.21) the following M X L=A

L

q2

(−1)M +1 T1 (M, A, q) (−1)L T0 (L, A, q) = A A , (q)L (q)M q 2 + q− 2

(2.22)

which can be restated as a trinomial analogue of the conjugate relation (1.3), ∞ X

γ e0 (A, M ) =

L=A

T0 (L, A, q) , δe0 (L, M ) (q)L

(2.23)

with conjugate pair (e γ0 , δe0 ) in this case being γ e0 (A, M ) =

(−1)M +1 T1 (M, A, q) , A A (q)M q 2 + q− 2

L δe0 (L, M ) = θ(L ≤ M )q 2 (−1)L ,

where

(2.24) (2.25)

n

1 if a ≤ b (2.26) 0 otherwise . The proof of the first statement of Theorem 1 (1.12) now easily follows by a Bailey Transform argument θ(a ≤ b) =

∞ X

α e0 (r)e γ0 (r, M ) =

r=0

=

∞ X L=0

∞ X r=0

δe0 (L, M )

L X r=0

α e0 (r)

∞ X L=r

T0 (L, r, q) δe0 (L, M ) (q)L ∞

T0 (L, r, q) X e α e0 (r) = δ0 (L, M )βe0 (L). (q)L

(2.27)

L=0

Substituting (2.24) and (2.25) into (2.27) we arrive at the desired result (1.12).


251

Similar to the binomial case, (1.12) can be interpreted as a defining relation identity e for a new trinomial Bailey pair α e1 , β1 . However, unlike the binomial case, the second analogue of (1.3) ∞ X T1 (L, A, q) (2.28) δe1 (L, M ) γ e1 (A, M ) = (q)L L=A

is now needed to iterate further. A A e To find a γ e1 , δ1 pair we multiply Eq. (2.22) by q 2 q − 2 and then replace A by A + 1(A − 1) to get M X (−1)M +1 T1 (M, A + 1, q) (−1)L L+A+1 q 2 T0 (L, A + 1, q) = , (q)L (q)M 1 + q −1−A

(2.29)

M X (−1)M +1 T1 (M, A − 1, q) (−1)L L−A+1 q 2 T0 (L, A − 1, q) = . (q)L (q)M 1 + q −1+A

(2.30)

L=A+1

L=A−1

Adding (2.29) and (2.30) and using (2.9), (2.13) gives M X (−1)L {T1 (L + 1, A, q) − T1 (L, A, q)} = (q)L L=A−1 (−1)M +1 T1 (M, A + 1, q) T1 (M, A − 1, q) + . = (q)M 1 + q −1−A 1 + q −1+A

(2.31)

Next we treat the sum in (2.31) as follows M X (−1)L {T1 (L + 1, A, q) − T1 (L, A, q)} = (q)L

L=A−1

M X (−1)L+1 (−1)L − = T1 (L + 1, A, q)+ (q)L (q)L+1 L=A−1

M X (−1)L (−1)L+1 + T1 (L + 1, A, q) − T1 (L, A, q) (q)L+1 (q)L

(2.32)

L=A−1

=−

M +1 X

−q −1

L=A

L

qL

T1 (L, A, q) (−1)M +1 + T1 (M + 1, A, q). (q)L (q)M +1

Combining (2.31), (2.32) and replacing M by M − 1 yields M X L=A

−q

−1

L

q

L T1 (L, A, q)

(q)L

(−1)M T1 (M, A, q) = (q)M (1 − q M ) T1 (M − 1, A + 1, q) 1 + q −1−A (1 − q M ) T1 (M − 1, A − 1, q) , − 1 + q −1+A

−

(2.33)

252


which is nothing else but (2.28) with 1 δe1 (L, M ) = θ(L ≤ M ) − qL , q L (−1)M γ e1 (L, M ) = T1 (M, A, q) (q)M

(2.34) (2.35)

(1 − q M ) T1 (M − 1, A + 1, q) 1 + q −1−A (1 − q M ) T (M − 1, A − 1, q) . − 1 1 + q −1+A

−

The proof of the second statement of Theorem 1 (1.13) follows again by the Bailey Transform argument (2.27) (with subindex 0 replaced by 1, everywhere). Unlike (1.12), Eq. (1.13) does not appear to be a defining relation for the new Bailey pair and therefore can not be iterated further. Finally, letting M tend to infinity in (1.12), (1.13) and using the limiting formula (2.15) we find Theorem 2. If a pair of sequences (e αa=0,1 , βea=0,1 ) is subject to identities (1.11) then ∞ X

(−1)L q 2 βe0 (L) = L

L=0

∞ (−1)∞ (−q)∞ X α e0 (r) r r 2 + q− 2 (q)2∞ q r=0

(2.36)

and ∞ X L=0

−q −1

q L βe1 (L) = L

∞ (−1)∞ (−q)∞ X 1 1 α e (r) − 1 (q)2∞ 1 + q r+1 1 + q r−1

(2.37)

r=0

hold.

3. Applications 3.1. Preliminaries. Recently it was shown [26] that Bailey’s lemma “connects” M (p, p+ 1) models of CFT with N = 1 SM (p+1, p+3) and N = 2 SM (p+1, 1) models of SCFT1 . In this section we shall demonstrate that Theorem 2 leads to very different relations between these models. We begin by collecting h A inecessary definitions and formulas. are defined as For A, B ∈ Z q-binomial coefficients B q hAi (q)A for 0 ≤ B ≤ A = (q)B (q)A−B (3.1) B q 0 otherwise . 1 Throughout this paper notations M (p, p0 ), N = 1 SM (p, p0 ) N = 2 SM (p, p0 ) stand for models of CFT and SCFT with central charges

1−

6(p0 − p)2 3 , pp0 2

1−

2(p − p0 )2 pp0

, 3 1−

2p0 p

respectively.


253

The following properties of q-binomials hAi lim

B 1/q hAi B

A→∞

q

= q B(B−A)

hAi B

q

,

1 (q)B

=

(3.2) (3.3)

are well known. Next we state some bosonic character formulas 0

M (p, p0 )[36 − 38] : χp,p r,s (q) =

∞ 0 1 X j(jpp0 +rp0 −sp) {q − q (jp+r)(jp +s) }, (q)∞ j=−∞

(3.4)

where p0 > p ≥ 2 are positive coprime integers and r ∈ {1, 2, . . . , p − 1}, s ∈ {1, 2, . . . , p0 − 1} are labels of irreducible highest weight representations ∈ (−q (br−bs) )∞ b p ,b p0 b (q) = N = 1 SM (b p , pb )[38 − 41] : χ b r ,b s (q)∞ ∞ X j(jb pb p 0 +b rb p 0 −b sb p) (jb p +b r )(jb p 0 +b s) 2 2 q , −q 0

(3.5)

j=−∞

where ∈a =

1/2 1

for a = 0 for a = 1

(mod 2) . (mod 2)

(3.6)

0

and p being coprime integers) and rb ∈ pb 0 > pb ≥ 2 are positive integers (with p −p 2 {1, 2, . . . , pb − 1}, sb ∈ {1, 2, . . . , pb 0 − 1}, ∈ ∈ −1 (−qe y)∞ (−qe y )∞ ∗ 2 (q)∞ ∞ p j+e r +e s X 2 1 − q 2e p +j(e r +e s) qj e , p j+e r )(1 + yq e p j+e s) (1 + y −1 qe j=−∞

N = 2 SM (e p , 1)†) [42 − 44] :

p ,1 χ ee (q, y) = e r ,e s

(3.7)

where pe ≥ 2 is a positive integer and e = 1/2, (1) in the A sector, re, se are half integers with 0 < re, se, re + se ≤ pe − 1; ∈ e = 1. 2 (2) in the P sector, re, se are integers with 0 < re − 1, se, re + se ≤ pe − 1; ∈ All N = 2 SM (e p , pe 0 > 1) characters were calculated by Ahn et al [46] in terms of fractional level string functions. However, for the vacuum sector for the N = 2 SM (e p , pe > 1) model (with pe > pe 0 ≥ 2; pe , pe 0 coprime) character formula similar to (3.7) (−q 1/2 y)∞ (−q 1/2 y −1 )∞ p ,e p0 χ ee (q, y) = ∗ e r ,e s (q)2∞ (3.8) 2 ∞ pe p 0 +j(e r +e s)e p0 p j+e r +e s X qj e (1 − q 2e ) ; re = se = 1/2 −1 q e p j+e r )(1 + yq e p j+e s) j=−∞ (1 + y 2

See [45] for the latest discussion regarding (3.7).

254


was recently found in [47]. Presumably, (3.8) also holds for sufficiently small re, se ∈ Z+ 21 , such that the embedding diagram is the same as in vacuum case re = se = 1/2. There are many important differences between N = 2 SM (p, 1) and N = 2 SM (p, p0 > 1) models. In particular, in contrast to the N = 2 SM (p, 1) case, N = 2 SM (p, p0 > 1) models are neither unitary nor rational [46, 47]. Moreover, while characters (3.7) have nice modular properties [48], those of N = 2 SM (p, p0 > 1) do not. Nevertheless, one can show that the characters (3.8) are, in fact, mock theta functions3 , i.e. they exhibit sharp asymptotic behaviour when q(|q| < 1) tends to a rational point of the unit circle. While bosonic characters (3.4, 3.5, 3.7) were known for quite some time, new fermionic expressions for these characters became available only in the last few years. Existence of the fermionic representations suggests that the Hilbert space of (S)CFT can be described in terms of quasi-particles obeying Pauli’s exclusion principle. The equivalence of the bosonic and fermionic character formulas gives rise to many new q-series identities of Rogers-Ramanujan type. Remarkably, in many known cases these identities admit polynomial analogues which can be written as defining relations (1.11) for trinomial Bailey pairs. 3.2. Polynomial analogues of generalized Göllnitz-Gordon identities. N = 1 SM (2, 4ν), N = 2 SM (4ν, 1) relation. Many polynomial Fermi-Bose character identities for N = 1 SM (2, 4ν), ν ≥ 2 were derived in [34, 35]. Not to overburden our narrative with cumbersome notations we shall consider here only the simplest of these identities ∞ X

q

n2 1 2

ν P

−n1 N2 +

Nj2

j=2

n1 ,...,nν =0

=

∞ X

(− )j q νj

" # i P ν hN i Y ni + L + n 1 − 2 N j 2 = j=2 n1 q ni i=2 q 2

+j/2

(3.9)

{T0 (L, 2νj, q) + T0 (L, 2νj + 1, q)},

j=−∞

where

Nj = nj + nj+1 + . . . + nν .

(3.10)

Letting L in (3.9) tend to infinity yields ∞ X

q

n2 1 2

ν P

−n1 N2 +

Nj2

j=2

n1 ,...,nν =0

hN i 2

n1

q

1 = (q)n2 · · · (q)nν

∞ 2 (−q 1/2 )∞ X = (− )j q νj +j/2 = χ b2,4ν 1,2ν−1 (q), (q)∞ j=−∞

(3.11)

where we used (3.3) and a limiting formula lim {T0 (L, A, q) + T0 (L, A + 1, q)} =

L→∞

(−q 1/2 )∞ (q)∞

(3.12)

proven in [29]. Identity (3.11) is nothing else but Andrews generalization of GöllnitzGordon identities [49]. A moment’s reflection shows that (3.9) is in the form (1.11) with 3 Notion of mock theta function was introduced by Ramanujan in his last letter to Hardy, dated January 1920.


255

 j νj 2 j/2 −j/2  )   (−1) q (q + q 1 2 α e0 (r) = (−1)j q νj +j/2    2 (−1)j q νj −j/2

1 βe0 (L) = (q)L

∞ X

q

n2 1 2

−n1 N2 +

ν P

for r = 2νj, j > 0 for r = 0 . for r = 2νj + 1, j ≥ 0 for r = 2νj − 1, j ≥ 1

Nj2

j=2

n1 ,...,nν =0

(3.13)

" # i P ν hN i Y ni + L + n 1 − 2 N j 2 . j=2 n1 q ni j=2 q (3.14)

Substituting (3.13), (3.14) into (2.36) gives ∞ X L,n1 ,...,nν =0

(−1)L q (q)L =

L+n2 1 2

ν P

" # i P ν hN i Y ni + L + n 1 − 2 N j 2 = j=2 n1 q ni i=2 q ∞ (3.15) X q νj+1/2 q νj j νj 2 +j/2 (− ) q + = 1 + q 2νj 1 + q 2νj+1 j=−∞

−n1 N2 +

Nj2

j=2

(−1)∞ (−q)∞ (q)2∞

1/2 1/2 ) + q 1/2 χ e 4ν,1 ), =χ e 4ν,1 1/2,2ν+1/2 (q, q 3/2,2ν−1/2 (q, q

which establishes an advertised relation between N = 1 SM (2, 4ν) and N = 2 SM (4ν, 1) models of SCFT. Moreover, the left-hand side of Eq. (3.15) provides new fermionic companion form for N = 2 SM (4ν, 1) characters. This form is quite different from the known fermionic representation [22,26] given in terms of D4ν -Cartan matrix. 3.3. Trinomial Bailey flow from M (3, 4) (Ising) model to N = 2 SM (6, 1) model of SCFT. In [30] the following polynomial identity L h i ∞ X X 2 2 L q j /2 = q 6j +j (T0 (L, 6j, q) + T0 (L, 6j + 1, q)) j q j=0

−

j=−∞ ∞ X

(3.16) q

6j 2 +5j+1

(T0 (L, 6j + 2, q) + T0 (L, 6j + 3, q))

j=−∞

was proven. One may check that in the limit L → ∞ this identity reduces to Fermi-Bose character identity for M (3, 4) (Ising) model X q j 2 /2 j≥0

(q)j

∞ 2 (−q 1/2 )∞ X 6j 2 +j = (q − q 6j +5j+1 ) (q)∞ j=−∞

=

χ3,4 1,1 (q)

+q

1/2

(3.17)

χ3,4 2,1 (q).

The middle expression in (3.17) is remarkably similar to (3.5) with pb = 3, pb 0 = 4. This similarity suggests an interpretation of (3.17) as a character of some extended Virasoro algebra. It is straightforward to verify that (3.16) is the defining relation (1.11) for the trinomial pair

256


α e0 (r) =

 2  q 6j (q j + q −j )    1    q 6j 2 +j

= 6j, j > 0 =0 = 6j + 1, j ≥ 0 , = 6j − 1, j > 0 = 6j + 2 and r = 6j + 3 , j ≥ 0 = 6j − 2 and r = 6j − 3 , j > 0 1 X j2 h L i q2 . βe0 (L) = j q (q)L

q 6j −j    2   −q 6j +5j+1   6j 2 −5j+1 −q 2

for r for r for r for r for r for r

(3.18)

(3.19)

j≥0

Next we apply Theorem 2 to the pair (3.18, 3.19), the result is  3j ∞ X L+j2 (−1)L h L i (−1)∞ (−q)∞  X 6j 2 +j q 3j+1/2 q q 2 = q +  (q)L j q (q)2∞ 1 + q 6j 1 + q 6j+1 j=−∞ L,j≥0  3j+1  ∞ 3j+3/2 X 2 q q − q 6j +5j+1 + 6j+2 1 + q 1 + q 6j+3  j=−∞ 1 3 − 21 6,1 2 2 =χ e 6,1 q, q + q q, q . χ e 1 3 1 3 2,2 2,2 (3.20) Recently, Warnaar proposed polynomial identities similar to (3.16) for all models M (p, p + 1), p ≥ 3 [17]. The p = 3 case is the one treated above. We have checked that his conjecture implies the following identities:  p−2 P L+m2 1 −Lm + m1 m2 + 1  mi (2mi −mi−1 −mi+1 ) 1 X 2 4 4 i=2 q  L,m1 ,...,mp−2 ≥0 ) p−2 (−1)L h L i Y h mi−12+mi+1 i = mi (q)L m1 q q i=2

=q

∞ (−1)∞ (−q)∞ X j 2 p(p−1)+j(p−1) 1 − q 4pj+2 , q 2 (q)∞ (1 + q 2pj )(1 + q 2pj+2 ) j=−∞

where mp−1 ≡ 0. Therefore the Trinomial Bailey flow for p = 1 (mod 2) is p−1 M (p, p + 1) ←→ N = 2 SM 2p, . 2

(3.21)

(3.22)

This is to be contrasted with the Bailey flow discussed in [26] where one has M (p, p + 1) ←→ N = 2 SM (p + 1, 1).

(3.23)

3.4. Results related to Rogers-Ramanujan identities. It is well known that the RogersRamanujan identities X q j(j+a) j≥0

(q)j

=

∞ 1 X j(10j+1+2a) − q (2j+1)(5j+2−a) q (q)∞ j=−∞

= χ2,5 1,2−a (q) ; a = 0, 1

(3.24)


257

admit the polynomial analogues X

q j(j+a)

h 2L − j − a i j

j≥0

q

∞ X

q j(10j+1+2a)

=

j=−∞

−q

h

i 2L L − 5j − a q

(2j+1)(5j+2−a)

h

i 2L , L − 5j − 2 q

(3.25)

which reduce to (3.24) as L → ∞. It is rather surprising that polynomials appearing in (3.25) have a q-trinomial representation as well [30]. In particular, for a = 0, one has X

q j2

h 2L − j i j

j≥0

∞ X

q 60j

=

q

2

−4j

L, 10j; q 2

j=−∞

− q 60j +q

2

+44j+8

60j 2 +16j+1

− q 60j

2

10j

2

L, 10j + 4; q 2 10j + 4 L, 10j + 1; q 2 2

+64j+17

(3.26)

10j + 1 2 L, 10j + 5; q 2 10j + 5

2

Identity (3.26) is not of the form (1.11). However, if we replace q by multiply the result by q X

q

j2 2

hL + j i

j≥0

2j

√

q

L /2 2

√1 q

in (3.26) and

we obtain with the help of (2.7), (3.2),

∞ X

=

.

q 20j

2

+2j

(T0 (L, 10j, q) + T0 (L, 10j + 1, q))

j=−∞ ∞ X

−

(3.27) q

20j 2 +18j+4

(T0 (L, 10j + 4, q) + T0 (L, 10j + 5, q)),

j=−∞

which gives rise to trinomial Bailey pair βe0 (L) =

1 X j2 h L + j i q2 , 2j √q (q)L

(3.28)

j≥0

α e0 (r) =

 2  q 20j (q 2j + q −2j )     1    20j 2 +2j q        

q 20j 2

2

−2j

−q 20j +18j+4 2 −q 20j −18j+4

for r for r for r for r for r for r

Next we apply Theorem 2 to derive

= 10j, j > 0 =0 = 10j + 1, j ≥ 0 . = 10j − 1, j > 0 = 10j + 4 and r = 10j + 5 , j ≥ 0 = 10j − 4 and r = 10j − 5 , j > 0

(3.29)

258


X

q

L+j 2 2

L,j≥0

(−1)L h L + j i = 2j √q (q)L  ∞ (−1)∞ (−q)∞  X 20j 2 +2j q =  (q)2∞ j=−∞

−

∞ X j=−∞

q 20j

2

+18j+4

q 5j+2 1 + q 10j+4

1

q 5j+ 2 q 5j + 1 + q 10j 1 + q 10j+1 ! 5  q 5j+ 2 . + 1 + q 10j+5 

! (3.30)

We note that the expression on the right-hand side of (3.30) bears a strong resemblance to formula (3.8) with pb = 10, pb 0 = 2. It is also similar in form to the 8(q) and 9(q) considered by Ramanujan in his development of the fifth-order mock theta functions [50]. Expressions in (3.30) are not modular functions. Nevertheless, using a Poisson summation formula one can show that asymptotic behaviour of (3.30) can be neatly expressed in terms of exponential forms. For instance, when q = e−t and t → 0 we have for (3.30) r π 9π2 2 cos e 20t . (3.31) 5πt 10 Proof of (3.31) along with asymptotic analysis of nonunitary characters (3.8) will be given elsewhere. 4. Discussion It is widely believed that different fermionic expressions for the (super) conformal character are related to different integrable pertubations of the same (super) conformal model. Thus, it would be interesting to identify perturbations which correspond to the new fermionic representations for N = 2 SM (4ν, 1) characters found in Sect. 3.2. Furthermore, following [26,27], it is tempting to interpret the relations N = 1 SM (2, 4ν) ←→ N = 2 SM (4ν, 1) p−1 , p = 1 (mod 2) M (p, p + 1) ←→ N = 2 SM 2p, 2

(4.1)

established here as massless renormalization group flows. If such an interpretation is indeed correct, then one should be able to carry out Thermodynamic Bethe Ansatz (TBA) analysis of these flows along the lines of [51]. We expect that related TBA systems will have the same incidence structure as that of fermionic forms discussed in Sect. 3. Also, we would like to point out that a “folding in half” relation between N = 2 SM (4ν, 1) and N = 1 SM (2, 4ν) models has already been noticed in [52,53]. Partition theoretical interpretation of our results will undoubtably lead to construction of subtractionless bases for N = 2 super Virasoro modules. From the mathematical point of view it is highly desirable to find an appropriate qhypergeometric background for the Trinomial analogue of Bailey’s lemma. Recall that the classical Bailey’s lemma is intimately related to the q-Pfaff-Saalschütz formula n X (a)n (b)n (q −n )j (c/a)j (c/b)j j q = , (q)j (c)j (cq 1−n /ab)j (c)n (ab/c)n j=0

(4.2)


259

which was first derived by Jackson [54]. In this direction we have already determined that ∞ X

(−q −n )L q

(1+2n)L 2

L=0 ∞ X

(−q −n )L q nL βe1 (L) =

L=0

βe0 (L) =

1+2n ∞ (−q n+1 )2∞ X q 2 r (−q −n )r α e0 (r), (q)∞ (q 2n+1 )∞ (−q n+1 )r

(4.3)

r=0

∞ X (−q n+1 )2∞ q nr (1 + q r )(−q −n )r α e1 (r) (4.4) (1 − q n )(q)∞ (q 2n+1 )∞ (−q n+1 )r r=0

with nonnegative integral n in (4.3) and positive integral n in (4.4). Identities (4.3, 4.4) can be derived from (2.12, 2.13, 2.15, 2.22) after a bit of labour. It immediately follows that q n in (4.3, 4.4) may be replaced by an arbitrary parameter, say ρ. Details will be given elsewhere [55]. Building on a proposal made in [32], Schilling and Warnaar defined and extensively studied q-multinomials [56–58]. One may wonder if these new objects will lead to additional generalizations of Bailey’s lemma. We strongly believe that the answer is “yes” and hope to say more about it in a subsequent paper. Note Added. Soon after this paper was completed, Warnaar [59] provided a simple and elegant proof of the conjecture from [17] used in deriving (3.21). Moreover, he has shown that each ordinary Bailey pair gives rise to a trinomial Bailey pair. In particular, he demonstrated that the trinomial Bailey pair (3.18–19) is a “descendant” of the A(1) and A(2) Bailey pairs of Slater’s list [4]. Acknowledgement. We would like to thank O. Foda, B.M. McCoy, Z. Reti, K. Voss, and S.O. Warnaar for interesting discussions and helpful comments.

References 1. Bailey, W.N.: Proc. London Math. Soc. (2) 49, 421 (1947) 2. Bailey, W.W.: Proc. London Math. Soc (2) 50, 1 (1949) 3. Dyson, F.J.: In: Ramanujan Revisited. Ed. by G. E. Andrews et al., London–New York: Academic Press, 1988 p. 7 4. Slater, L.J.: Proc. London Math. Soc. (2) 53, 460 (1951) 5. Slater, L.J.: Proc. London Math. Soc. (2) 54, 147 (1952) 6. Andrews, G.E.: Pac. Journ. Math. 114, 267 (1984) 7. Agarwal, A.K., Andrews, G.E., Bressoud, D.M.: J. Indian Math. Soc. 51, 57 (1987) 8. Bressoud, D.M.: In: Ramanujan Revisited. Ed. by G. E. Andrews et al., London–New York: Academic Press, 1988 p. 57 9. Andrews, G.E.: q-series: Their development and application in analysis, number theory, combinatorics, physics and computer algebra. Providence, Rhode Island: American Math. Society, 1986 10. Paule, P.: J. Math. Anal. Appl. 107, 225 (1985) 11. Milne, S.C., Lilly, G.M.: Bull. Amer. Math. Soc. 26, 258 (1992) 12. Melzer, E.: Int. J. Mod. Phys. A9, 1115 (1994) 13. Berkovich, A.: Nucl. Phys. B431, 315 (1994) 14. Foda, O., Quano, Y.-H.: Int. J. Mod. Phys. A10, 2291 (1995) 15. Kirillov, A.N.: Prog. Theor. Phys. Suppl. 118, 61 (1995) 16. Warnaar, S.O.: J. Stat. Phys. 82, 657 (1996) 17. Warnaar, S.O.: J. Stat. Phys. 84, 49 (1996) 18. Berkovich, A., McCoy, B.M.: Lett. Math. Phys. 37, 49 (1996) 19. Berkovich, A., McCoy, B.M., Schilling, A.: Rogers-Schur-Ramanujan type identities for M (p, p0 ) minimal models of conformal field theory. q-alg/9607020, Commun. Math. Phys. (to appear)

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20. Kedem, R., Klassen, T.R., McCoy, B.M., Melzer, E.: Phys. Lett. B304, 263 (1993) 21. Kedem, R., Klassen, T.R., McCoy, B.M., Melzer, E.: Phys. Lett. B307, 68 (1993) 22. Dasmahapatra, S., Kedem, R., Klassen, T.R., McCoy, B.M., Melzer, E.: Int. J. Mod. Phys. B7, 3677 (1993) 23. Foda, O., Quano, Y.-H.: Int. J. of Mod. Phys. A12, 1651 (1997) 24. Schilling, A., Warnaar, S.O.: Int. J. Mod. Phys. B11, 189 (1997) 25. Schilling, A., Warnaar, S.O.: A higher level Bailey lemma: Proof and Applications. q-alg/9607019, Ramanujan J. (to appear) 26. Berkovich, A., McCoy, B.M., Schilling, A.: Physica A228, 33 (1996) 27. Chin, L.: Central charge and the Andrews-Bailey Constructions hep-th/9607168 28. Berkovich, A., McCoy, B.M., Schilling, A., Warnaar, S.O.: Bailey flows and Bose-Fermi identities for the (1) (1) conformal coset models (A(1) 1 )N × (A1 )N 0 /(A1 )N +N 0 . hep-th/9702026, Nucl. Phys. B (to appear) 29. Andrews, G.E., Baxter, R.J.: J. Stat. Phys. 47, 297 (1987) 30. Andrews, G.E.: J. Amer. Math. Soc. 3, 653 (1990) 31. Andrews, G.E.: In: Analytic Number Theory. B. Berndt et al eds., Boston: Birkhäuser, 1990, pp. 1–11 32. Andrews, G.E. Contemp. Math.166, 141 (1994) 33. Warnaar, S.O., Pearce, P.A.: J. Phys. A27, L891 (1994) 34. Berkovich, A., McCoy, B.M., Orrick, W.P.: J. Stat. Phys. 83, 795 (1996) 35. Berkovich, A., McCoy, B.M.: Generalizations of Andrews-Bressoud Identities for N = 1 Superconformal Model SM (2, 4ν). hep-th/9508110, Int. J. of Math. and Comp. Modelling (to appear) 36. Feigin, B.L., Fuchs, D.B.: Funct. Anal. Appl. 17, 241 (1983) 37. Rocha-Caridi, A.: In: Vertex Operators in Mathematics and Physics. ed. J. Lepowsky et al. Berlin: Springer, 1985 38. Dobrev, V.K.: Suppl. Rendiconti Circolo Matematici di Palermo, Serie II, Numero 14, 25 (1987) 39. Feigin, B.L., Fuchs, D.B.: Func. Anal. Appl. 16, 114 (1982) 40. Goddard, P., Kent, A., Olive, D.: Commun. Math. Phys. 103, 105 (1986) 41. Meurman, A., Rocha-Caridi, A.: Commun. Math. Phys. 107, 263 (1986) 42. Dobrev,V.K.: Phys. Lett. B186, 43 (1987) 43. Matsuo, Y.: Prog. Theor. Phys. 77, 793 (1987) 44. Kiritsis, E.B.: Int. J. Mod. Phys. A3, 1871 (1988) 45. Dörrzapf, M.. Commun. Math. Phys. 180, 195 (1996) 46. Ahn, C., Chung, S., Tye, S.-H.: Nucl. Phys. B365, 191 (1991) 47. Eholzer, W., Gaberdiel, M.R.: Unitarity of rational N = 2 superconformal theories. hep-th/9601163 48. Ravanini, F., Yang, S.-K.: Phys. Lett. B195, 202 (1987) 49. Andrews, G.E.: The theory of partitions. London: Addison-Wesley, 1967 50. Andrews, G.E., Garvan, F.G.: Adv. in Math. 73, 242 (1989) 51. Zamolodchikov, A.: Nucl. Phys. B358, 524 (1991) 52. Melzer, E.: Supersymmetric Analogs of the Gordon-Andrews Identities, and related TBA systems. hep-th/9412154 53. Moriconi, M., Schoutens, K.: Nucl. Phys. B464, 472 (1996) 54. Jackson, F.H.: Messenger of Math. 39, 745 (1910) 55. Andrews, G.E., Berkovich, A.: In preparation 56. Schilling, A.: Nucl. Phys. B467, 247 (1996) 57. Warnaar, S.O.: Commun. Math. Phys. 184, 203 (1997) 58. Schilling, A., Warnaar, S.O.: Supernomial coefficients, polynomial identities and q-series. q-alg/9701007 59. Warnaar, S.O.: A note on the trinomial analogue of Bailey’s lemma. info q-alg/9702021 Communicated by T. Miwa

Commun. Math. Phys. 192, 261 – 285 (1998)

Communications in


Operator of Fractional Derivative in the Complex Plane Petr Závada Institute of Physics, Academy of Sciences of Czech Republic, Na Slovance 2, CZ-180 40 Prague 8, Czech Republic. E-mail: [email protected] Received: 31 July 1996 / Accepted: 30 June 1997

Abstract: The paper deals with a fractional derivative introduced by means of the Fourier transform. The explicit form of the kernel of the general derivative operator acting on the functions analytic on a curve in the complex plane is deduced and the correspondence with some well known approaches is shown. In particular, it is shown how the uniqueness of the operation depends on the derivative order type (integer, rational, irrational, complex) and the number of poles of the considered function in the complex plane. 1. Introduction The fractional differentiation and integration (also called fractional calculus) is a notion almost more as old as the ordinary differential and integral calculus. Naturally, each slightly gifted student who has just understood what is the first and second derivative can ask the question: well, but what is, for example, a 1.5-fold derivative? There are several ways to answer such a question. An excellent review on the theory of arbitrary order differentiation (generally of complex order) including also interesting historical notes and a comprehensive list of references to original papers (more than a thousand items) is given in a recently published monograph [8]. Some new results were presented also in a recent conference [5] dedicated to this topic. The fractional calculus has also plenty of applications, see also e.g. [6, 4, 9, 10] and citations therein. Possible use in quantum mechanics and the field theory is discussed in [11]. Recently, the fractional derivative was mentioned also in [2] as a particular case of pseudo-differential operators applied in non-local field theory. Apparently, the general prescription for a definition of fractional derivative is using some representation of an ordinary n-fold derivative (primitive function) which can be in some natural way interpolated to an n-non integer. Actually, following the mentioned monograph, all the known approaches are always somehow connected with some of the following relations.

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1) The well known formula for an n-fold integral Z x1 Z x1 Z x2 Z xn 1 dx2 dx3 ... f (t)dt = (x1 − t)n−1 f (t)dt 0(n) a a a a

(1.1)

allows the substitution of n by some real α > 0. In this way fractional integration is introduced. Then fractional derivatives can be obtained by ordinary differentiation of fractional integrals. This is the basis of the construction known as the Riemann–Liouville fractional calculus. Let us note that in this approach the resulting function even in the case of fractional derivatives in general depends on fixing the integration limit a on the right hand side of (1.1). 2) The Cauchy formula for analytic functions in some region of the complex plane Z 0(n + 1) f (z)dz (1.2) f (n) (z0 ) = 2iπ (z − z0 )n+1 C in principle enables generalization to fractional derivatives, nevertheless the direct extension to non-integer values of n leads to difficulties arising from multivaluedness of the term (z − z0 )α+1 , and the result also depends on the choice of the cut and integration curve. 3) Analytic continuation of the derivative (integral) of the exponential and power function dα exp(cz) = cα exp(cz), dz α

dα 0(β + 1) (z − c)β−α . (z − c)β = α dz 0(β − α + 1)

(1.3)

Obviously these relations allow to define fractional derivatives of the functions which can be expressed as linear combinations of power and exponential functions. Also this approach is not completely consistent, as can be illustrated by fractional derivative of exponential function expanded to the power series exp(cz) =

∞ X (cz)k , 0(k + 1)

(1.4)

k=0

but for α-non integer, ∞

X (cz)k−α dα dα α α = exp(cz) = c exp(cz) = 6 c dz α 0(k − α + 1) dz α k=0

∞ X (cz)k 0(k + 1)

! .

(1.5)

k=0

So the task of extrapolation of the integer derivative order to an arbitrary one has no unique solution. The approach proposed in this paper is based on the fractional derivative of the exponential function (1.3) entering the Fourier transform of a given function. On this basis in Sect. 2 the explicit form of the kernel of the fractional derivative operator is deduced. In Sect. 3 the composition relation for the derivative operator is proved. The generalization of the case of the functions on the real axis to the case of the function on the complex plane is done in Sect. 4, which is concluded by the theorem summarizing the results. The last section is devoted to the discussion of some consequences following from the theorem and to a comparison with known approaches as well.

Operator of Fractional Derivative in the Complex Plane

263

2. Definition of the Fractional Derivative by Means of Fourier Transform Let f (x) be a function having the Fourier picture fe(k) : Z +∞ Z +∞ 1 e f (x) exp(ikx)dx, f (x) = fe(k) exp(−ikx)dk. f (k) = 2π −∞ −∞ Then let us create the function Z +∞ 1 α (−ik)α fe(k) exp(−ikx)dk, f (x) = 2π −∞ and define Dα (w) =

1 2π

Z

α > −1,

(2.1)

(2.2)

+∞

(−ik)α exp(−ikw)dk.

(2.3)

The function f α (x) can be formally expressed Z +∞ Dα (x − y)f (y)dy. f α (x) = Dα f =

(2.4)

−∞

−∞

Now let us calculate the integral (2.3), which depends on the way of passing about the singularity k = 0 and the choice of the branch and cut orientation of the function k α . To begin let us assume the cut is given by the half line either (0, −∞) or (0, +∞). For complex functions ξ α = (ξ1 + iξ2 )α we shall accept the phase convention lim (ξ1 + iξ2 )α =| ξ1α |

ξ2 →0+

ξ1 , ξ2 ≥ 0,

(2.5)

i.e. it holds ξ ≥ 0 cut orientation ξ 0,

(2.9)

which is evident from the corresponding integrals having paths closed in infinity – for α ) in the upper (lower) half-plane. Now we shall calculate the integrals (2.7) in D+α (D− the remaining regions of w. Let us split them into two parts:

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P. Závada

α D± (w)

(−1)α = 2π

"Z

Z

0±i0

α

(ik) exp(−ikw)dk ,

(ik) exp(−ikw)dk + −∞±i0

#

+∞±i0

α

0±i0

(2.10) and substitute the real parameter w for the complex one z1 = w + i z2 = w − i

for k < 0 for k > 0,

(2.11)

where > 0. This substitution ensures absolute convergence of both integrals in (2.10). Next let us make in (2.10) the further substitution ξ = ikz1 ξ = ikz2

for k < 0 for k > 0.

(2.12)

α (w) we get functions depending also on : In this way instead of D± Z Z (−1)α 1 1 α α α E± (w, ) = ξ exp(−ξ)dξ + ξ exp(−ξ)dξ . 2πi z1α+1 K1 z2α+1 K2

(2.13)

We assume a) The cut of the complex function k α is given by half-line (0,+∞). b) Function values z1α , z2α are considered values of one complex function z α in two different points. According to the cut orientation of z α one can get either z2α = (z1α )∗ or z2α = (z1α )∗ exp(2iπα). Cut of the function z α is assumed (0, +∞) (0, −∞)

for E+α α for E− .

(2.14)

Later on we shall come back to these assumptions and judge how they affected our result. Assumptions concerning the cuts of k α , z α correspond to phases or to the intervals of phases of variables k, z and correspondingly to phases of the variable ξ. All considered possibilities are summarized in Table 1. Table 1. The phase correspondence of variables ξ, z, k depending on the cut orientation integral E+α

α E−

variable

k0

k

π

0

z

(0,π)

(π,2π)

ξ

(3π/2,5π/2)

(3π/2,5π/2)

k

π

2π

z

(0,π)

(−π,0)

ξ

(3π/2,5π/2)

(3π/2,5π/2)

The corresponding integration paths are shown in Fig. 1. Instead of the interval (3π/2, 5π/2) for arg ξ we took the interval (−π/2, π/2) since the corresponding phase shift is the same for both integrals in (2.13) and can be included in the factor (−1)α ahead of the integrals. Obviously, it holds Z Z Z ∞ ξ α exp(−ξ)dξ = − ξ α exp(−ξ)dξ = ξ α exp(−ξ)dξ = 0(α + 1). (2.15) K2

K1

0


265

Im ξ K2′

K2

←K2″ K1″→

Re ξ K1′

K1

Fig. 1. Integration paths in Eqs. (2.13), (2.15)

After inserting into (2.13) we get α E± (w, )

(−1)α+1 0(α + 1) = 2iπ

1 1 − (w + i)α+1 (w − i)α+1

,

> 0.

(2.16)

We shall regard the integrals (2.7) as the limits α α D± (w) = lim E± (w, ), →0+

(2.17)

hence the kernel of the operator (2.3) which can be also considered the generalized function is symbolically written (−1)α+1 0(α + 1) 1 1 α , (2.18) − D± (w) = 2iπ (w + i0)α+1 (w − i0)α+1 where the two modes correspond to different cuts of (w + i)α+1 , D+α (w) α (w) D−

for cut (0, +∞) for cut (0, −∞).

(2.19)

Let us note the function (2.18) is well defined for any complex α = α1 + iα2 6= −1, −2, . . ., but for the beginning we assume α2 = 0. Now let us go back to the assumptions a), b) which imply result (2.16). a)Assuming the opposite cut orientation (0,−∞) for k α in Eq. (2.10) and repeating the corresponding sequence of steps does not change anything for E+α (w, ) whereas in α (w, ) the phase arg ξ will shift by −2π in both integrals in (2.13), but the case of E− this change can be included in the factor (−1)α ahead of the integral. b) Let us assume the cuts in z α having the opposite orientation than in (2.14). Let us take e.g., function E+α (w, ), then the phase of ξ complies with 3π/2 < arg ξ < 5π/2 −π/2 < arg ξ < π/2

for k > 0 for k < 0,

(2.20)

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P. Závada

i.e. the phase of the second integral in (2.13) is now shifted by −2πα, and instead of (2.16) we get (−1)α+1 0(α + 1) exp(−2iπα) 1 α − , > 0. (2.21) E+ (w, ) = 2iπ (w + i)α+1 (w − i)α+1 Obviously this function for w ∈ (−∞, +∞) with an assumed cut of z α on (0, −∞) is identically equal to the function (2.16) with the cut on (0, +∞). The analogous result α . can be obtained also for E− So we have shown that the result (2.16) of the integration (2.7) does not depend on the choice of cut orientation of the functions k α and wα . The cut orientation of wα in (2.16) and (2.18) is dictated only by the way of passing around the singularity k = 0 in initial integrals (2.7) and the correspondence (2.19) always holds. Next let us notice the α α , E± . Obviously it holds important property of the functions D± d dw d dw

α α+1 (w, ) = E± (w, ), E±

(2.22) α (w) D±

=

α+1 D± (w).

Now let us go back to Eq. (2.4) and consider how to calculate this integral. It is possible either a) First to find the difference of both terms in (2.16), then take the integration and limit for → 0, or b) First to calculate both integrals independently, then take the limit of their difference. Let us discuss both ways and compare the results. a) The action of operator Dα ± as the integral of difference. We make a calculation separately for the two cases: a1) α = n ≥ 0 is an integer number. In that case the complex function wα has no cuts (we can omit the subscript ±) and for n = 0 we can write E 0 (w, ) =

1 · , π w 2 + 2

(2.23)

i.e. for → 0 we get the known representation of the δ-function (see e.g. [3], p.35) δ(w) = lim

→0+

1 · π w 2 + 2

which acts on the function f , Z Z +∞ 1 +∞ δ(x − y)f (y)dy = lim f (y)dy = f (x). 2 + 2 →0+ π (x − y) −∞ −∞ Using Eqs. (2.22)−(2.25) one can easily show Z +∞ dn f (x) = lim E n (x − y, )f (y)dy →0+ −∞ dxn for any n ≥ 0 for which the integral converges. So it is possible to identify

(2.24)

(2.25)

(2.26)


Dn (w) = dn f (x) = dxn

Z

267

dn δ(w), dwn

+∞ −∞

(2.27)

Dn (x − y)f (y)dy,

n

i.e. action of the operator D corresponds to an n-fold derivative. a2) α is a real, non integer number. Taking into account the cut orientations (2.19), calculation of limits (2.17) gives α (w) = 0, D− α D+ (w) = 0,

w > 0, w < 0,

α (w) = − D−

w < 0,

0(α + 1) sin([α + 1]π) , πwα+1 0(α + 1) sin([α + 1]π) , D+α (w) = + πwα+1

(2.28)

w > 0.

The first two equations corresponds with Eqs. (2.9), therefore again it confirms the correct correspondence of cuts in (2.19). After inserting (2.28) into Eq. (2.4) we get Z 0(α + 1) sin([α + 1]π) ∓∞ f (y)dy α (x) = − . (2.29) f± π (x − y)α+1 x For y = x and α ≥ 0 the integral has a singularity. The method for regularization of this integral will become apparent in the next part. b) The action of operator Dα ± as a difference of two integrals. After inserting of (2.18) into (2.4) we get (−1)α+1 0(α + 1) α (x) = lim f± →0+ 2iπ (2.30) Z +∞ Z +∞ f (y)dy f (y)dy − . α+1 α+1 −∞ (x − y + i) −∞ (x − y − i) We assume for the present the function f (y) is analytic on the whole real axis. The last equation can be rewritten Z +∞−i Z +∞+i f (z)dz f (z)dz (−1)α+1 0(α + 1) α . − f± (x) = lim α+1 α+1 →0+ 2iπ −∞−i (x − z) −∞+i (x − z) (2.31) Again let us separate two cases: b1) α = n ≥ 0 is an integer number. In Eq. (2.31) we can link up both integration paths in z = ±∞ and write Z dn f (−1)n+1 0(n + 1) f (z)dz = , (2.32) f n (x) = lim n+1 →0+ 2iπ dxn C (x − z) where C is any closed curve enclosing the singular point. Therefore for n ≥ 0 the operator Dn can be again identified with an ordinary n-fold derivative. b2) α is a real, non integer number. Similarly, as in case b1) we get the integrals Z (−1)α+1 0(α + 1) f (z)dz α (x) = lim , (2.33) f± α+1 →0+ 2iπ C± (x − z)

268

P. Závada

where the paths C± can pass about the corresponding cut as shown in Fig. 2. The integrals converge provided that f (z) = 0. (2.34) lim z→±∞ z α The integration paths can be decomposed into three parts L1± , L2± , L3± and after evaluation of the corresponding integrals one gets Z ∓∞ (±1)α f (x) 0(α + 1) sin([α + 1]π) f (z)dz α + , (2.35) f± (x) = − lim α+1 →0+ π αα x∓ (x − z) or using the relation (A.10) from the Appendix,

a

Im z

← L3 +

L1 + →

x

L2-

ε L2 +

b

Im z

x

Re z

← L1-

ε

L3- →

Re z

.

(2.36)

Fig. 2. Integration paths in Eq. (2.23): a) C+ , b)C−

1 →0+ 0(−α)

α f± (x) = − lim

Z

∓∞ x∓

(±1)α f (x) f (z)dz + α+1 (x − z) αα

For α < 0 the integrals are finite and the second terms vanish. In this case Eq. (2.35) is identical with Eq. (2.29). On the other hand we have assumed the initial integral (2.33) is finite, therefore the sum of both terms in (2.35) is also finite even for any α > 0. In this sense by addition of the second term the integral in (2.29) can be regularized. Note that in contradistinction to (2.18) the relation (2.36) is well defined also for α = −1, −2, −3, . . ., but α 6= 0, 1, 2, 3, . . .. The last equation for the α negative integer Z x 1 −n (x) = (x − z)n−1 f (z)dz (2.37) f± 0(n) ∓∞ is a special case of the n-fold integral formula (1.1). Now, let us assume 0 < α < 1 and calculate integral (2.33) by parts. Obviously ∓∞± i0 Z Z 1 f (z)dz f 0 (z)dz f (z) = − + (2.38) α+1 α C± (x − z)α (x − z)α ∓∞∓ i0 C± (x − z) and the integral on the right side is finite. Any α can be written as the sum of an integer and fractional part, α = n + 1α,

n = [α],

0 ≤ 1α < 1.

(2.39)

For n ≥ 0 we can repeat integration by parts n + 1 times and if the function f meets the requirements,


269

f p (z) =0 z→±∞ (x − z)α−p

for p = 0, 1, ...n,

lim

(2.40)

then instead of (2.35) we get (−1)1α 0(1α) →0+ 2iπ

α f± (x) = lim

=−

0(1α) sin(1απ) π

Z

Z

f n+1 (z)dz = (x − z)1α

C±

∓∞ x

f n+1 (z)dz . (x − z)1α

(2.41)

Let us note that the integrals (2.33) can be modified also in another way. We assume that all integrals Z f (z)dz , γ = 1α, 1α + 1, ...α + 1 (2.42) I± (γ, x) = (x − z)γ C± converge, then the recurrent relation holds: d I± (γ, x) = −γI± (γ + 1, x). dx

(2.43)

Application of this relation for integral (2.33) and α = n + 1α gives (−1)1α 0(1α) →0+ 2iπ

α (x) = lim f±

=−

0(1α) sin(1απ) π

d dx

d dx

n+1 Z

n+1 Z

C± ∓∞ x

f (z)dz = (x − z)1α

f (z)dz . (x − z)1α

(2.44)

on n+1 - fold derivative of the Actually in Eq. (2.41) we apply the operation D1α−1 ± function f whereas in Eq. (2.44) both operations are interchanged. Using relation (A.10), obviously we can modify both equations: Z

x

1 f n+1 (z)dz = (x − z)1α 0(1 − 1α)

n+1 Z

x

f (z)dz . (x − z)1α ∓∞ ∓∞ (2.45) So we have shown the operator Dα with the kernel (2.18) can be considered a continuous interpolation of the ordinary n-fold derivative (integral) of the functions analytic on the real axis and fulfilling the condition (2.34). α (x) = f±

1 0(1 − 1α)

d dx

3. Composition of Fractional Derivatives In this section we shall investigate how the composition of the operators Dα is realized in the representation given by Eq. (2.18). Therefore we shall deal with the integrals Z +∞ Dα (x − y)Dβ (y − z)dy. (3.1) I= −∞

Let us denote

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P. Závada

1 , (w + iτ )γ 1 h• (γ, w) = − , (w − iτ )γ h• (γ, w) =

and I1 I2 I3 I4 then I=

a

τ >0

(3.2)

R +∞ = −∞ h• (α + 1, x − y)h• (β + 1, y − z)dy, R +∞ = −∞ h• (α + 1, x − y)h• (β + 1, y − z)dy, R +∞ = −∞ h• (α + 1, x − y)h• (β + 1, y − z)dy, R +∞ = −∞ h• (α + 1, x − y)h• (β + 1, y − z)dy,

(3.3)

(−1)α+β+1 0(α + 1)0(β + 1) (I1 + I2 + I3 + I4 ). 4π 2

(3.4)

b

Im z

Im z

z1

K'

K z1

K

L ϕ

Re z

Re z

z2 z2

Fig. 3. Integration paths in Eq. (3.5): a) case when the integral vanishes, b) case leading to the result (3.8)

Now let us calculate the more general integral Z dz , J= α+1 (z − z )β+1 (z − z) 2 1 K

(3.5)

where K is an arbitrary line in the complex plane, z1 , z2 any two (diverse) points and let α + β > −1. We also assume the cuts of the function in the integral do not intersect the line K. Then there are two possibilities: a) K is not passing between the points z1 , z2 , see Fig. 3a. Then obviously J = 0, since the line K can be closed in infinity by the arc in the half plane which does not contain singularities z1 , z2 . b) K is passing between the points z1 , z2 , see Fig. 3b. K 0 denotes line crossing the segment hz1 , z2 i perpendicularly at its center. If we assume that the cuts do not intersect any of both lines K, K 0 , then in the integral (3.5) path K can be substituted by K 0 . Further, if we denote z0 = (z1 + z2 )/2,

r exp(iϕ) = (z2 − z1 )/2,

(3.6)

and substitute z = z0 + it exp(iϕ), then J=

i exp[iϕ(α + β + 1)]

Z

+∞

−∞

dt . (r − it)α+1 (r + it)β+1

(3.7)


271

The last integral can be found in tables (see e.g. [7], p.301), using (3.6) we obtain 2iπ 0(α + β + 1) . α+β+1 (z2 − z1 ) 0(α + 1)0(β + 1)

J=

(3.8)

Let us note that the opposite orientation of the line K (i.e. point z2 on the left side with respect to the direction of K, as the line L in Fig. 3b) should give result (3.8) with opposite sign. Obviously the integrals I3 , I4 vanish and for I1 , I2 we get I1 =

−2iπ 0(α + β + 1) , (x − z + 2iτ )α+β+1 0(α + 1)0(β + 1)

(3.9)

I2 =

+2iπ 0(α + β + 1) , α+β+1 (x − z − 2iτ ) 0(α + 1)0(β + 1)

(3.10)

and inserting into (3.4) gives I=

(−1)α+β+1 0(α + β + 1) 2iπ

1 1 − (x − z + 2iτ )α+β+1 (x − z − 2iτ )α+β+1

. (3.11)

For τ → 0 this expression corresponds to Dα+β (x − z) in (2.18). This result is formally correct, nevertheless its drawback is that it does not reflect the correspondence of the cut orientations in the initial expressions (3.1) and the final (3.11). More rigorous discussion about the cuts we postpone to the Appendix, and here give only the result. The following composition relation holds for operators (2.18) with equally oriented cuts: Z α+β (x − z) = D±

+∞ −∞

β α D± (x − y)D± (y − z)dy

α, β 6= −1, −2, −3...;

α + β > −1.

(3.12) Let us note that validity of relation (3.12) can be verified in the initial representation α as a generalized function, then the condition (2.7) as well. Further, considering D± α + β > −1 can be omitted and the composition relation has the form Z

+∞ −∞

Z α+β D± (x−ξ)f (ξ)dξ

+∞

Z

+∞

= −∞

−∞

β α D± (x−y)D± (y−ξ)f (ξ)dξdy,

α, β 6= −1, −2, .. (3.13)

for all functions analytic on the real axis and fulfilling lim

z→±∞

f (z) = 0. z α+β

(3.14)

γ−1 γ d Equation (3.13) follows from (3.12) and relation dw D± (w) = D± (w). Repetitional integration by parts on both sides can reduce the sum α + β by any natural number, so in this way validity of (3.13) is proved. All our previous considerations concerned the action of the operator Dα on the real axis. Next we shall try to enlarge them on the whole complex plane.

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P. Závada

4. Fractional Derivative in Complex Plane First, let us illustrate the notion of fractional derivative introduced in the previous part with a particular example. Take the function 1 1 1 1 − , (4.1) = − f (x) = 1 + x2 2i x + i x − i for which (2.33) gives (−1)α+1 0(α + 1) →0+ 2iπ

Z

α (x) = lim f±

C±

dz . (x − z)α+1 (1 + z 2 )

(4.2)

Obviously for α > −2, the integration paths in Fig. 2 closed by arcs in infinity as shown in Fig. 4, are possible and

a

Im z

b

C(∞)

+i

Im z

C(∞)

+i C-

C+

x x

Re z

Re z

-i

-i

Fig. 4. Integration paths in Eq. (4.2) closed in infinity: a) C+ , b) C−

α f± (x)

(−1)α+1 0(α + 1) = 2i

1 1 − α+1 (x + i) (x − i)α+1

.

(4.3)

Even though the result formally does not depend on a given subscript (+ or −), it is α . In accordance with necessary to take into account a different cut orientation for f+α , f− the phase convention (2.6), we take ) (x + i)α+1 = Rα+1 exp[iϕ(α + 1)] for f+α (x − i)α+1 = Rα+1 exp[i(2π − ϕ)(α + 1)] ) , (4.4) (x + i)α+1 = Rα+1 exp[iϕ(α + 1)] α for f− , (x − i)α+1 = Rα+1 exp[−iϕ(α + 1)] where R=

√ 1 + x2 ,

ϕ=

x π − arcsin . 2 R

After inserting into (4.3) and a simple rearrangement, we obtain

(4.5)


273

Im z

C1

+i

C2

x

Re z

Fig. 5. Tentative modification of integration paths in Eq. (4.2) and Fig. 4

α f± (x) =

√

±1 1 + x2

α+1 0(α + 1) sin

arcsin √

x 1 + x2

±

π 2

(α + 1) .

(4.6)

Remark. From the last relation it can be easily shown that e.g. n (x) ≡ f n (x), where f n is an ordinary a) for α integer, α = n > −1, it holds f+n (x) = f− n-fold derivative, α (x) = (−1)α f+α (−x), b) f− α (x) → arctan(x) ± π/2. The c) for α → −1 we get a primitive function of f : f± same results can be obtained also from (2.45) for n = −1, 1α = 0.

Using formula (2.33) we shall try to generalize the operator of the fractional derivative on real axis to the whole complex plane. For this operator we shall demand again: Dα f (z) =

dn f dz n

for α = n ≥ 0,

(4.7)

(4.8) Dα ◦ Dβ = Dα+β . For making the generalization more transparent, we shall do it in several steps. 1) Let us go back to Fig. 4 and ask a question what the results (4.3)−−(4.6) will change, if the cut (and integration path correspondingly) is oriented otherwise than along real axis, but e.g. as shown in Fig. 5. Obviously for Eq. (4.3) nothing will change, but the form of Eqs. (4.4),(4.6) will depend on the mutual position of the cut and both poles. Because we have accepted phase convention (2.6) only for cuts on the real axis, it is now necessary to make a more general consideration to determine the phases of both terms in (4.3). Instead of (4.1) let us take a complex function a2 a1 + (4.9) f (z) = z − z1 z − z2 and integration paths C displayed in Fig. 6a. The corresponding integral will be a2 a1 . (4.10) + f α (z0 ) = −(−1)α+1 0(α + 1) (z0 − z1 )α+1 (z0 − z2 )α+1 The path C passes about the cut of function 1/wα+1 = 1/(z0 − z)α+1 in variable z, i.e. the cut of the function 1/wα+1 is oriented in the opposite direction, see Fig. 6b. The phases of w1α+1 = (z0 − z1 )α+1 and w2α+1 = (z0 − z2 )α+1 must be fixed with respect to this cut. Fig. 6b prompts the following rule.

274

P. Závada Im z

z1

C

a

Im w

b

ϕ1 ϕ2

w2

z0

ϕ2

z2

ϕ1

Re z

Re w

w1

Fig. 6. a) Integration paths in Eq. (2.33) for the function (4.9). ϕ1 , ϕ2 are phases corresponding to terms wkα+1 = (z0 − zk )α+1 in the result of integration (4.11). b) The same phases represented for variable w

Rule 1. The phase of the complex variable w is given by the angle ϕ of the arc leading from the positive real half axis and measured in the positive direction (against the clockwise sense) to the point w, and if the arc intersects the cut, ϕ is reduced by 2π. This rule can be applied also directly for the situation in Fig. 6a for fixing the phase of (z0 −z)α+1 . The only modification is that ϕ is measured from the half line (z0 , z0 −∞). Therefore angles ϕ1 , ϕ2 in Fig. 6 fix phases in (4.10), a1 exp(−iϕ1 [α + 1]) a2 exp(−iϕ2 [α + 1]) f α (z0 ) = (−1)α 0(α + 1) + . (4.11) | (z0 − z1 )α+1 | | (z0 − z2 )α+1 | Now the introduced rule can be applied also for integration on paths C1 (C2 ) in Fig. 5. Doing this, we shall get the same result as in the case of integration on paths C+ (C− ) in Fig. 4. Therefore depending on the position of the chosen derivative cut in respect to the both poles, the function ( 4.1) has (up to factor (2.8)) in a given point x two different values of fractional derivative given by Eq. (4.6). 2) We shall now generalize the prescription for calculating the fractional derivative of function having the form (4.9) for functions having a finite number of poles h(z) =

N X k=1

ak , (z − zk )nk +1

nk ≥ 0.

(4.12)

Let the derivative cut be given by the half line L ≡ [z0 , z0 + exp(iθ)∞] which does not go through any of poles zk . Then (−1)α+1 0(α + 1) X h (z0 ) = lim →0+ 2iπ N

Z

α

k=1

= (−1)α

N X 0(α + nk + 1) k=1

0(nk + 1)

C(L)

ak dz (z0 − z)α+1 (z − zk )nk +1

ak exp(−iϕk [α + 1]) , | z0 − zk |α+1 (z0 − zk )nk

(4.13)

the integration path C(L) is shown in Fig. 7. Angles ϕk are calculated using Rule 1, therefore it is obvious that the function hα (z0 ) in (4.13) can have, according to the cut orientation, as many values, as many different poles zk the function (4.12) has (number of values = number of poles of the function (4.12)).


z1

275

Im z

zN

C(∞)

ϕ1

C(L) ϕ2

z0

z2

Re z

z3

Fig. 7. Integration paths in Eq. (2.33) with the function (4.12) and phases ϕk appearing in the result of integration (4.13)

3) Now let us consider functions (4.12), (4.13) in the case when the cut is not a line, but some general curve connecting the points z0 and exp(iθ)∞. Then the result (4.13) will be formally the same, only the prescription for fixing angles ϕk must be modified. General cases are illustrated in Fig. 8, where it is apparent that the arc, which “measures” the angle, may intersect the cut several times. The consistent assigning of angles ϕk corresponding to points zk in Eq. (4.13) can be ensured by the following prescription. Rule 2. Let us choose on the half line (z0 , z0 − ∞) any reference point zR . The angle ϕk is given by angle zR z0 zk measured in the positive sense and then for each intersection with the cut is corrected as follows. Superpose the palm of the right- or left-hand at an intersection in such a way that fingers lead from zR towards zk and thumb leads in the direction of the cut from the branching point z0 . If this condition is met by the right (left) hand, ϕk will be enhanced (reduced) by 2π. Remark. It is substantial for all zk to choose one common reference point zR . A shift of this point results at most in only equal shifts of all angles ϕk → ϕk + 2nπ, which do not change Eq. (4.13).

a

Im z

zR

z0

ϕ2

b

z1

Im z zR

ϕ1

Re z

ϕ2

z1 ϕ1

z0 z2

Re z

z2

Fig. 8. a), b) Examples of the curvilinear cuts generating integration paths for Eq. (2.33) with the function (4.12). Other symbols are defined in Rule 2

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P. Závada

Therefore for any curvilinear cut the relation (4.13) can be written hα (z0 ) = (−1)α

N X 0(α + nk + 1) ak exp(−i[ϕk + 2mk π][α + 1]) k=1

0(nk + 1)

| z0 − zk |α+1 (z0 − zk )nk

.

(4.14)

The set of integer numbers (or more exactly their differences) characterizes the way the curvilinear cut passes among the poles zk . If we accept curvilinear cuts for the derivative of the function (4.1), then using (4.14) one can obtain (−1)k(α+1) 0(α + 1) x π α √ (α + 1) , (4.15) + (2k + 1) f (x) = √ α+1 sin arcsin 2 1 + x2 1 + x2 where k = m2 − m1 . Let us note that π lim f α (x) = arctan(x) + (2k + 1) , 2

α→−1

(4.16)

i.e. we obtain the infinite (but countable) set of primitive functions for the function (4.1). 4) So far we have considered only analytic functions of the form (4.12), for which the corresponding integral on the whole circle C(∞) vanishes. Now we are going to consider the general case, when this integral vanishes only on a part of the circle. But first, let us go back to the operator (2.18) acting on analytic functions on the real axis (or part of this axis) and try to generalize it for analytic functions on a curve in complex plane connecting a pair of points on the circle C(∞), see Fig. 9a. Let this curve be given as a continuous complex function of the real parameter t ∈ (−∞, +∞) z = x(t) + iy(t) ≡ ψ(t),

ψ(−∞) = ∞1 ,

ψ(+∞) = ∞2 ,

(4.17)

then its derivative

dy dx +i (4.18) dt dt determines in the complex plane the vector tangent to the curve ψ and oriented in direction of increasing t. The function ψ 0 (t) =

ν(t) = √

ψ 0 (t) = exp[iω(t)] · ψ ∗0 (t)

ψ 0 (t)

(4.19)

represents this vector in its normalized value (ω(t) is phase of this vector) and the function iν(t) normalized vector perpendicular to ψ and oriented left with respect to the course of ψ. Now let us define the function of the complex variable z ∈ ψ with complex parameters z0 ∈ ψ and α 6= −1, −2, −3, . . ., 1 (−1)α+1 0(α + 1) 1 α Dψ (z0 − z) = lim − →0+ 2iπ (z0 − z + iν)α+1 (z0 − z − iν)α+1 (4.20) having the curvilinear cuts (for α 6= 0, 1, 2, ..) corresponding to both terms coming out of the points z0 ± iν and going jointly along the curve ψ to points ∞1 or ∞2 , see Fig. 9b. So, in contradistinction to the linear cuts the form of the cut of wα+1 = (z0 − z)α+1 does depend on the position of point z0 on ψ. Next, using this function we define the operator


a

Im z

b

C∞

∞1

277

Im z

∞1 ∞2

c

C∞

z0+iεν

z0

Im z

C(ψ,ε)

∞2

∞2

z0-iεν Re z

Re z

Re z

K-

Fig. 9. a) Integration paths for the operator (4.20) in complex plane. b) Corresponding curvilinear cuts in Eq. (4.20). c) Corresponding integration path in Eq. (4.23)

Z Dα ψ

f=

α fψ± (z0 )

Z

= ψ

α Dψ (z0

− z)f (z)dz =

+∞ −∞

α Dψ (ψ(t0 ) − ψ(t))f (ψ(t))ψ 0 (t)dt

(4.21)

acting on the functions analytic on the curve ψ for which the integral converges. Remark. Let us note the integral (4.21) depends on the choice of the cut end-point (∞1 or ∞2 ) but does not depend on in which the direction (∞1 → ∞2 or ∞1 ← ∞2 ) the integration is done. That is a consequence of the fact that the change of integration direction ψ(t) → ψ(−t), dz → −dz implies the change ν(t) → −ν(t), which implies α α (z0 − z) → −Dψ (z0 − z), therefore the integral does not change. the change Dψ If we calculate (4.21) as the difference Z Z (−1)α+1 0(α + 1) f (z)dz f (z)dz α − , Dψ f = lim α+1 α+1 →0+ 2iπ ψ (z0 − z + iν) ψ (z0 − z − iν) (4.22) then this difference can be expressed as Z Z (−1)α+1 0(α + 1) f (z)dz α Dψ (z0 − z)f (z)dz = lim , (4.23) α+1 →0+ 2iπ ψ C(ψ,) (z0 − z) where the path in “distance” passes about the cut coming out the branching point z0 on the curve ψ to infinity, see Fig. 9c. To label the integration cuts ending either at ∞1 or ∞2 we can accept the following convention. Let ∞1 = exp(iθ1 )∞ and ∞2 = exp(iθ2 )∞, then for ( ) ( ) cos θ1 < cos θ2 ψ+ = (z0 , ∞1 ), ψ− = (z0 , ∞2 ) we define cos θ1 6= cos θ2 cos θ1 > cos θ2 ψ = (z0 , ∞2 ), ψ− = (z0 , ∞1 ) ( ) ( + ) sin θ1 < sin θ2 ψ+ = (z0 , ∞1 ), ψ− = (z0 , ∞2 ) we define cos θ1 = cos θ2 sin θ1 > sin θ2 ψ+ = (z0 , ∞2 ), ψ− = (z0 , ∞1 ) (4.24) α α , D , f , C (ψ, ). and correspondingly we index related symbols Dα ± ψ± ψ± ψ± Let us note the definition of the operator Dα ψ by Eqs. (4.20), (4.21) ensures that the both curves C± (ψ, ) are oriented in such a way that after their closing by the circle

278

P. Závada

Im z

C1

ψ+

z0

ψ−

Re z C0

Fig. 10. Domain G (grey area) and the curves defined in the assumptions of the Theorem

C(∞), see Fig. 9c, there arises a closed curve having always clockwise orientation. We denote these curves as K± (ψ, ). Now, after all these preparation steps, we can come to the formulation and proof of the following theorem. Theorem. Assumptions: i) G is the domain in complex plane containing the part (or parts) C1 ≡ G ∩ C(∞) of the circle C(∞). The curve C1 forms one part, the curve C0 is the second part and closed curve CG ≡ C0 ∪ C1 constitutes the complete domain boundary (Fig. 10). ii) ψ is a curve defined in (4.17) and ψ ⊂ G, ψ ∩ C0 = ∅. / ψ, zk ∈ / CG , k = 1 . . . N iii) The function f (z) is analytic in G except for the poles zk ∈ and for a given α ≡ α1 + iα2 6= −1, −2, −3... and any z1 ∈ C1 meets the condition lim

z→z1

f (z) = 0. z α1

(4.25)

Statements: 1. Operation

Z α Dα ψ± f = fψ± (z0 ) =

ψ

α Dψ± (z0 − z)f (z)dz,

z0 ∈ ψ

(4.26)

is a fractional derivative satisfying the conditions (4.7), (4.8). In particular that means the operation does not depend on ψ if α = n ≥ 0 is integer. α is given also equivalently by 2. fψ± Z (−1)α+1 0(α + 1) f (z)dz α fψ± (z0 ) = lim . (4.27) α+1 →0+ 2iπ C± (ψ,) (z0 − z) α 3. For N = 0 the value fψ± (z0 ) does not depend on ψ, i.e. up to the factor (−1)α is uniquely defined. 4. For N ≥ 1 and α non integer there remain cases: α (z0 ) does depend on ψ, but the set of values 4.1 α1 is rational and α2 = 0, then fψ± is finite.


279

α 4.2 α1 is irrational or α2 6= 0, then fψ± (z0 ) does depend on ψ as well and in general the set of values is infinite, but countable.

Proof. We shall start from Statement 2. Obviously, its validity follows from Eqs. (4.22), (4.23). Now let us consider Statement 4. If we express the function f (z) as the sum f (z) = g(z) + h(z), where g(z) is analytic in G and h(z) has the form (4.12), then obviously Z (−1)α+1 0(α + 1) f (z)dz α fψ± (z0 ) = lim α+1 →0+ 2iπ (z 0 − z) C± (ψ,) Z Z (−1)α+1 0(α + 1) (−1)α+1 0(α + 1) f (z)dz f (z)dz = lim − α+1 α+1 →0+ 2iπ (z − z) 2iπ (z 0 0 − z) K± (ψ,) C0 (4.28) ! N X 0(α + n + 1) exp(−i[ϕ + 2m π][α + 1]) a k k k k = (−1)α + I(z0 ) , 0(nk + 1) | z0 − zk |α+1 (z0 − zk )nk k=1

where we have denoted I(z0 ) =

0(α + 1) 2iπ

Z C0

f (z)dz . (z0 − z)α+1

(4.29)

The angles ϕk + 2mk π corresponding to the poles zk are evaluated according to Rule 2. In the function I(z0 ) the orientation of the path C0 is accordant with K± and the phase of (z0 − z)α+1 must be also according to the Rule 2 related to the same reference point zR . Now, if α1 = p/q (p, q are not commensurable) and α2 = 0, then each term in the last sum has the same value also for m0k = mk + q, i.e. depending on the form of the cut and the corresponding set {mk , 1 ≤ k ≤ N } the expression (4.28) has only a finite number of values for some z0 . On the other hand for α irrational different sets {mk } give different values of the sum, therefore in general the number of values is infinite. For α2 6= 0 multi-value factors in the sum (4.28) are expanded 1α = exp(2iπmα1 ) · exp(−2πmα2 ),

(4.30)

i.e. mα1 , mα2 determine the phase and scale of individual terms in the sum. Obviously for a complex α the set of values f α (z0 ) depending on ψ is in general infinite, similarly as for α irrational. Therefore Statements 4.1, 4.2 are proved. For N = 0, Eq. (4.28) is simplified: α (z0 ) = (−1)α I(z0 ), fψ±

(4.31)

and the trueness of Statement 3 is evident. Finally, let us consider Statement 1. Validity of the condition (4.7) follows from the statement 2. For α integer the path C± (ψ, ) can be closed around the pole z0 (having positive orientation) and we get the Cauchy integral Z f (z)dz n! = f n (z0 ). (4.32) 2iπ C(z0 ) (z − z0 )n+1 The condition (4.8) requires validity Z Z Z α+β β α Dψ± (x − ξ)f (ξ)dξ = Dψ± (x − y)Dψ± (y − ξ)f (ξ)dξdy. ψ

ψ

(4.33)

ψ

This relation follows from (4.21) and (3.13) in which the substitutions ξ → ψ(ξ) and y → ψ(y) are applied. So the whole proof is completed.

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P. Závada

5. Discussion 5.1. Some remarks regarding the theorem. Now, let us look into the statements of the Theorem more comprehensively. The function I(z0 ) in (4.28) can be expressed Z 0(α + 1) f (z)dz = I(z0 ) = 2iπ (z − z)α+1 0 C0 Z 0(α + 1) exp(−i2mπ[α + 1]) f (z) exp[−iϕ(z)(α + 1)]dz . (5.1) = 2iπ |(z0 − z)α+1 | C0 Since the cut ψ does not intersect the curve C0 the phases of all points on C0 are “corrected” by the same factor standing ahead of the integral. Now it is obvious that for α = p/q the function (4.28) can have at most q N +1 different values, including a multi-value factor (−1)α . Actually, the function (4.28) can be written as ! Z N X 0(α + 1) 0(α + nk + 1) f (ξ)dξ ak α α + f (z) = (−1) α+1 0(nk + 1) (z − zk )nk +α+1 2iπ C0 (z − ξ) k=1 (5.2) and considered a multi-value function with the value at point z depending on the choice of cut ψ in (4.26). At a given point z there are equivalent any two cuts for which the region closed by them and the curve C1 does not contain any pole zk . Obviously, except for the points zk the function f α (z) is (as an original function f ) analytic in the domain G. Moreover, in the region of α in which f α = Dα f exists, this function is apparently analytic also with respect to α. A special case takes place when α is a negative integer. Then due to the singularity 0(−n) the kernel (4.20) loses the sense. Nevertheless, assuming in (4.20) initially α 6= −n one can proceed to representation (4.27), then making the limit → 0 (like Eq. (2.35)) gives Z f (z0 ) 0(α + 1) sin([α + 1]π) f (z + ν0 )dz α + , fψ± (z0 ) = − lim α+1 →0+ π α(−ν0 )α ψ± (z0 − z − ν0 ) (5.3) where ν0 is given by Eq. (4.19) and represents the direction of the cut at z0 (orientation is assumed from z0 to infinity). This formula already makes sense for α = −n (if condition (4.25) holds). Using relation (A.10) gives Z 1 −n (z0 ) = − (z0 − z)n−1 f (z)dz, (5.4) fψ± (n − 1)! ψ± which is a modification of the known formula (1.1) for an n-fold primitive function. The set of values f −n (z0 ) depends on ψ is as follows. If f (z) has no poles in the domain G, then any two integration paths ψ1 , ψ2 in (5.4) can be connected by a fragment of C1 , in this way there arises closed curve C and it holds Z 1 (z0 − z)n−1 f (z)dz = fψ−n (z0 ) − fψ−n (z0 ), (5.5) 0=− 2 1 (n − 1)! C i.e. f −n (z0 ) is determined uniquely. Now, let us suppose f (z) has inside curve C just one pole, for z → zp ,


f (z) →

ap . (z − zp )np +1

281

(5.6)

Then obviously any curve C in the integral (5.5) can be always substituted by a couple of curves K0 ; each of them is closed in the phase range h0, 2πi and having the pole zp inside (e.g. circles centered at zp ). Instead of (5.5) one gets the difference Z map (z0 − z)n−1 −n −n dz, (5.7) 1p ≡ fψ2 (z0 ) − fψ1 (z0 ) = (n − 1)! K0 (z − zp )np +1 where m is the integer depending on the shape of the curve C (m represents the number of “twists” on C). Using the Cauchy formula, the last equation gives  0 np ≥ n   2iπmap . (5.8) 1p =  (z0 − zp )n−np −1 np < n  (n − np − 1)!np ! For more poles all the corresponding terms (5.8) are simply added. For example integration constants in (4.16) representing the case n = 1, np = 0 fulfill (5.8). The composition relation for α = −n, β < 0 and ψ ≡real axis is proved in the Appendix (Eq. (A.17)) and apparently can be transformed to any curvilinear path ψ. Let us note, for α negative integer only the representation of Dα given by Eq. (5.3) (or (5.4)) makes sense and conversely for α non-negative integer only the representation given by Eq. (4.26) with kernel (4.20) (or equivalently by Eq. (4.27)) is well defined. For any α complex but non-integer both representations are well defined. These two representations differ in the corresponding integration paths: i) Eq. (4.27) – integration on some curve enveloping the cut, the path can be closed. ii) Eq. (5.3) – integration on the cut itself, the path cannot be closed. For α integer it is specific that the cuts disappear. Perhaps the most restrictive assumption in the Theorem is the condition (4.25). The question if one can in some consistent way apply Dα to the functions not obeying this condition in any part of C(∞) (and simultaneously having the ordinary derivatives – primitive functions) requires further study. Obviously one possible way is to consider such functions as generalized functions as well. 5.2. Concluding remarks. Have we said something new? First let us show what is not new. Apparently: a) The content of Eq. (2.45) is almost identical with the Liouville definition of right and left -handed fractional derivatives in [8], p. 95. The only distinction is in the phase of α since in the cited definition for the real functions the real value of the derivatives f− is ad hoc postulated. b) Equation (4.27) is the Cauchy type integral which ordinarily serves as one of the possible starting points for the definition of the fractional derivative in the complex plane, see [8],p. 415. In our approach the integration path is uniquely defined by the chosen cut. On the other hand, what is new seems be the following: 1) The general form of the kernel (4.20) from which both above mentioned formulae follow.

282

P. Závada

2) The construction based on the integration paths enveloping curvilinear cuts, which in the result allows one to identify the fractional derivative-integral with the multivalued function and to determine how the number of values depends on the derivative order type and the number of poles which the given function has in the considered region.

6. Appendix: The Correspondence of Cuts in the Composition Relation Similarly as in (3.2) we define 1 h± (γ, w) = (w+iτ )γ 1 h± (γ, w) = − (w−iτ )γ

τ > 0,

(A.1)

where indices ± denote the cut orientations (0,±∞). Let us consider the integrals Z +∞ h(a, x − y)h(b, y − z)dy (A.2) I = lim τ →0+

−∞

for various combinations of the cut orientations and their locations above or below the real axis. First let us assume a < 1, b < 1, a + b > 1.

(A.3)

For the more general integral (3.5) we have shown the integral vanishes, when the integration line does not separate off both singularities, which is the case of the eight integrals (A.2) involving combinations h± h± , h± h∓ , h± h± , h∓ h± . Let us calculate (A.2) when e.g. x < z and both singularities are above the real axis. Then |I| = |exp(iaπ)I1 + I2 + exp(−ibπ)I3 | = 0, where

Z

Ik = Lk

dy |(x − y)a (y − z)b |

L1 ≡ (−∞, x), L2 ≡ (x, z), L3 ≡ (z, +∞).

Using simple substitutions in the known relation Z 1 0(λ)0(µ) , xλ−1 (1 − x)µ−1 dx = 0(λ + µ) 0 and denoting da+b−1 ≡ (x − z)a+b−1 one can get I1 =

(A.4)

(A.5)

(A.6)

1 0(1 − a)0(a + b − 1) , · da+b−1 0(b)

1 0(1 − a)0(1 − b) , · da+b−1 0(2 − a − b) 1 0(1 − b)0(a + b − 1) . I3 = a+b−1 · d 0(a) After inserting into (A.4) we get I2 =

(A.7)


cos(aπ)

283

0(1 − b)0(a + b − 1) 0(1 − a)0(a + b − 1) 0(1 − a)0(1 − b) + + cos(bπ) =0 0(b) 0(2 − a − b) 0(a) (A.8) sin(aπ)0(1 − a) sin(bπ)0(1 − b) = . (A.9) 0(b) 0(a)

The last identity also follows from the known formula (see e.g.[1] ,p. 256) 0(γ)0(1 − γ) sin(γπ) = π.

(A.10)

Now let us calculate (A.2) for the remaining combinations h± h± , h± h∓ , h± h± , h∓ h± . For τ → 0 it holds exp(iπc) lim h(γ, w) = , (A.11) τ →0+ |wγ | where the phases c of functions h entering the integral (A.2) are in accordance with the convention (2.6) summarized in Table 2. This table enables one to obtain the phases ck of their products which are listed in the first three columns of Table 3 The integrals of Table 2. The phases c depending on the cut location h(a, x − y) y<x h+

y>x

0

−a

h−

0

−a

h+

−2a

−a

h−

0

h(b, x − z) yz 0 0 − 2b 0

all the combinations summarized in the table can be expressed like (A.4), I=

3 X

exp(ick π)Ik .

(A.12)

k=1

If we denote G≡

0(a + b − 1) 2iπ , · da+b−1 0(a)0(b)

(A.13)

then using identities (A.8),(A.10) the sum (A.12) can be evaluated. The results are given in the last column of Table 3. Let us compare the corresponding rows in the upper and lower part of the table. Obviously for any x, z one can write Z +∞ 2iπ0(a + b − 1) ± h (a + b − 1, x − z) h± (a, x − y)h± (b, y − z)dy = lim lim τ →0+ −∞ τ →0+ 0(a)0(b) (A.14) and equally for h± . So far we have assumed (A.3), however using the identity d h(γ, w) = −γh(γ + 1, w), dw

(A.15)

we can enlarge the validity of (A.14) to any a, b, a + b > 1,

a, b 6= 0, −1, −2, −3, . . . .

(A.16)

284

P. Závada

Table 3. The phases ck of products h(a, x − y)h(b, y − z) and the resulting integral I (last column) for the case x < z (upper part) and x > z (lower part) y<x

x 0. Then, for every δ > 0, i h X 2 tN − v > δ =0, (1.5) lim PµN t κ0 (·) N →∞ N Z

where

∞

n

vt =

o F (u) − ρ(t, u) du

(1.6)

0

and ρ is the solution of equation (1.3). Theorem 1.2. Assume p < 1. For α < 1 define ψα (u) = α1{u < 0}+(qα/p)1{u > 0}. Fix a profile κ0 : R → [0, 1] such that ψα ≤ κ0 ≤ 1 − σ for some σ > 0, 0 < α < 1. Then, for every δ > 0, (1.5) holds provided vt is given by (1.6) and ρ is the solution of Eq. (1.4). The integral defining vt in (1.6) must be understood in the following sense: consider the sequence {Hn , n ≥ 1} of real functions defined by (1.7) Hn (u) = (1 − un−1 )+ . R +∞ It follows from the equation satisfied by ρ that 0 Hn (u){F (u)−ρ(t, u)}du converges as n ↑ ∞. This limit defines the right-hand side of (1.6). In the case where the initial state is a Bernoulli product measure µα with a fixed density α, we can make more explicit computations: Theorem 1.3. If the initial state is µα , then (1 − α) vt = p−q→0 p − q α lim

r

2t . π

Theorems 1.1 and 1.2 are proven in Sect. 5. Theorem 1.3 and more asymptotic results are proven in Sect. 6. We √ now explain why in Theorems 1.1 and 1.2 the asymmetric tagged particle moves at scale t and why the displacement is related to the solution of the differential equations (1.3), (1.4). We start labeling all particles. The tagged asymmetric particle is labeled 0. For j ≥ 1, we label the j th particle at the right (left) of the tagged particle by j (−j). For x in Z, denote by η(x) the number of holes between particle x and particle x + 1. In this way we transform a configuration of {0, 1}Z with a particle at some site X into a configuration {η(x), x ∈ Z} ∈ NZ . Denote by T : Z × {0, 1}Z → NZ the transformation

Driven Tracer Particle

291

just described. T induces a transformation on the space of functions (resp. probability measures) of Z × {0, 1}Z to the space of functions (resp. probability measures) of NZ still denoted by T . The dynamics of the process (Xt , ξt ) induces a dynamics for ηt that can be informally described as follows. For every x 6= −1, if there is at least one particle at site x, at rate 1/2 one of them jumps to site x + 1 and, symmetrically, if there is at least one particle at site x + 1, at rate 1/2 one of them jumps to site x. The picture is slightly different between sites −1 and 0 due to the behavior of the asymmetric tagged particle. A particle jumps at rate q from site −1 to site 0 if there is a particle at −1 and a particle jumps at rate p from site 0 to site −1 if there is a particle at the origin. This process is the so-called zero range process, with an asymmetry at the origin. The position at time t of the asymmetric tagged particle corresponds in the zero range model to the total number of jumps between 0 and t from 0 to −1 minus the total number of jumps in the same interval from −1 to 0: X {η0 (x) − ηt (x)}. (1.8) Xt = x≥0

The right-hand side is to be understood in the same sense as the righ-hand side of (1.6) by the use of the functions (1.7) (with the limit in the L2 sense) (cf. [RV]). Since in the zero range process the jumps of particles over all bonds, except the bond {−1, 0}, are symmetric, we expect the process to have a diffusive hydrodynamic behavior, i.e., that for a large class of initial profiles, the process accelerated by N 2 is such that for all continuous functions with compact support G, X G(x/N )ηtN 2 (x) (1.9) N −1 R

x

converges in probability to R G(u)ρ(t, u)du, where ρ is the solution of a nonlinear heat equation. In particular, approximating 1{u > 0} by the sequence defined in (1.7), it follows from (1.8) and (1.9) that X XtN 2 = N −1 {η0 (x) − ηtN 2 (x)} N x≥0

converges in probability to vt given by (1.6). 2. The Case p=1 In the case where the asymmetric tagged particle jumps only to the right, the evolution of the medium on its left is irrelevant for its motion. For the corresponding zero range dynamics, p = 1 means that at rate 1 a particle at the origin jumps to −1 and no particle jumps from −1 to 0. We may therefore assume that there is at −1 an infinite reservoir or an absorption point to which particles from the origin jump at rate 1 and from which no particle jumps. Moreover, the position of the tagged particle at time t corresponds in the zero range process to the number of particles that left the system before time t. Consider the zero–range process on N whose generator acts on cylinder functions as o Xn Lx,x+1 + Lx+1,x , (2.1) L = Lb + x≥0

292

C. Landim, S. Olla, S. B. Volchan

where

Lx,y f (η) = (1/2)g(η(x))[f (σ x,y η) − f (η)]

and

Lb f (η) = g(η(0))[f (η − d0 ) − f (η)]. Here, for a site x, dx stands for the configuration with no particles but one at x, summation is performed site by site, σ x,y η is the configuration η with one particle less at x and one more at y: σ x,y η = η − dx + dy and g(k) = 1{k ≥ 1}. For α > 0, denote by να+ the product measure on NN with marginals given by 1 α k . (2.2) να+ {η, η(x) = k} = 1+α 1+α It is easy to check that these measures are reversible with respect to the generators Lx,x+1 + Lx+1,x defined above and that Eνα+ [η(x)] = α. We need to introduce some terminology on weak solutions of non linear parabolic equations. Fix a bounded initial profile ρ0 : R+ → R. A bounded function ρ: [0, T ] × R+ → R is said to be a weak solution of the partial differential equation (1.3) with initial condition ρ0 in the layer [0, T ] × R+ if (a) 8(ρ(t, u)) is absolutely continuous in the space variable and Z T Z 2 ds du e−u {∂u 8(ρ(s, u))} < ∞ , 0

R+

(b) ρ(t, 0) = 0 for almost every 0 ≤ t ≤ T , and (c) For every smooth function with compact support G: R+ → R vanishing at the origin and for every 0 ≤ t ≤ T , Z t Z Z Z ds du G0 (u)∂u 8(ρ(s, u)). du ρ(t, u)G(u) − du ρ0 (u)G(u) = − (1/2) 0

R+

Uniqueness of weak solutions of (1.3) can be proved with similar methods to the ones presented in [ELS], we outline the argument in the appendix. The existence for special initial conditions ρ0 follows from the tightness of the sequence QµN defined below in Theorem 2.2. We now describe the initial states considered in this section. Fix a sequence of probability measures {µN , N ≥ 1} on NN .We assume that (H1) The sequence µN is bounded above (resp. below) by να+ (resp. νλ+ ) for some 0 < λ < α. (H2) There exists a bounded function ρ0 : R+ → R+ such that for each continuous function G: R+ → R with compact support and each δ > 0, Z i h X G(x/N )η(x) − du G(u)ρ0 (u) ≥ δ = 0. lim µN N −1 N →∞

x

The first assumption is needed in order to prove the two block estimates for zero range processes with bounded jump rate (cf. [KL]). The second one just imposes a hydrodynamic behavior at time 0. For each probability measure µ on NN , denote by PN µ the probability measure on the path space D(R+ , NN ) induced by the Markov process with generator (2.1) accelerated N by N 2 and the initial measure µ. Expectation with respect to PN µ is denoted by Eµ .


293

Theorem 2.1. Fix a sequence of initial measures satisfying assumptions (H1) and (H2). For any continuous function G: R+ → R with compact support and any δ > 0, " # Z −1 X N G(x/N )ηt (x) − du G(u)ρ(t, u) ≥ δ = 0, lim PµN N N →∞

x

where ρ is the unique solution of (1.3). For each positive integer N and each configuration η, define the empirical distribution Radon measure on R+ obtained by assigning a mass N −1 π N = π N (η) as the positiveP N −1 N N to each particle: π = N z≥0 η(z)δz/N and set πt = π (ηt ). Fix T > 0. Theorem 2.1 follows from the convergence in distribution of the process {πtN , 0 ≤ t ≤ T }, stated below in Theorem 2.2, and some standard topology arguments (cf. Chap. IV of [KL]). To state the convergence in distribution of the empirical measure we need some notation. Denote by M+ = M+ (R+ ) the space of positive Radon measures on R+ endowed with the vague topology, a metrizable topology. For each probability measure µ on NN , denote N by QN µ the probability measure on the path space D([0, T ], M+ ) induced by Pµ and the N empirical measure π . Theorem 2.2. The sequence QN µN converges to the probability measure concentrated on the absolutely continuous path π(t, du) = ρ(t, u)du whose density is the solution of (1.3). Guo, Papanicolaou and Varadhan introduced in [GPV] a method, well known by now, to prove Theorem 2.2 provided one has a bound on the entropy and on the Dirichlet form of the system with respect to some invariant measure. These bounds are usually obtained computing the time derivative of the entropy of the distribution of particles at time t relative to the equilibrium distribution. In the present context, however, there is only one invariant measure: the trivial one δ0 concentrated on the configuration 0 with no particles. Since all other probability measures on NN are orthogonal with respect to this one, the entropy of any reasonable measure with respect to δ0 is infinite and the entropy method does not apply straightforwardly. To overcome this problem, we compute the relative entropy with respect to an inhomogeneous product measure that is not invariant but close to the invariant measure. To obtain an estimate on the entropy and on the Dirichlet form, we first assume that there exists a parameter β > 0 for which the relative entropy H(µN |νβ+ ) is bounded by C0 N for some finite constant C0 . Coupling arguments permit to remove this assumption. This is explained at the end of this section. To deduce an estimate on the entropy of the system, we need to introduce a class of inhomogeneous product measures. For x ≥ 0, define γx by γx = β(1 + x)/N for N the product measure on NN 0 ≤ x ≤ N − 1 and γx = β for x ≥ N . Denote by νγ(·) with marginals given by N νγ(·) {η, η(x) = k} = (1 − γx )γxk

(2.3)

for all x ≥ 0 and k ≥ 0. A simple computation relying on the entropy inequality shows that the entropy of N is bounded by C1 N for some finite constant C1 depending only µN with respect to νγ(·) N N ) ≤ C1 N (cf. Remark V.1.2 in [KL]). on C0 , α and β: H(µ | νγ(·) N , define the Dirichlet form Dγ (f ) For each probability density f with respect to νγ(·) by

294


Dγ (f ) = Dγ,b (f ) + Dγ,i (f ) = Dγ,b (f ) +

X

Dx,x+1 (f ) ,

x≥0

Z

hp i2 p N Dγ,b (f ) = (1/2) g(η(0)) f (η − d0 ) − f (η) dνγ(·) , Z hp i p 2 N f (η + dx+1 − dx ) − f (η) dνγ(·) . Dx,x+1 (f ) = (1/2) g(η(x)) where

(2.4)

Proposition 2.3. Let StN be the semigroup associated to the generator L introduced in (2.1) accelerated by N 2 . Denote by ft = ftN the Radon–Nikodym derivative of µN StN N with respect to νγ(·) . There exists a finite constant C = C(β) such that N ) ≤ −N 2 Dγ (ft ) + CN. ∂t H(µN StN |νγ(·) N . It is easy to check Proof. Denote by L∗γ the adjoint operator of L with respect to νγ(·) that ft is the solution of the forward equation

(

∂t ft = N 2 L∗γ ft

(2.5)

N f0 = (dµN )/(dνγ(·) ).

Then by explicit calculation Z Z N N N ) = N 2 L∗γ ft log ft dνγ(·) + N 2 L∗γ ft dνγ(·) ∂t H(µN StN |νγ(·) Z Z Z Lft N N N = ft N 2 L log ft dνγ(·) = N 2 ft (L log ft − ) dνγ(·) + N 2 Lft dνγ(·) . ft (2.6) N were an invariant measure. Notice that the last term would vanish if νγ(·) √ √ Since for every a, b > 0, a log(b/a) − (b − a) is less than or equal to −( b − a)2 , for every x, y ≥ 0, we have that hp i2 p ft Lx,y log ft − Lx,y ft ≤ −(1/2)g(η(x)) ft (η + dy − dx ) − ft (η) hp i2 p ft Lb log ft − Lb ft ≤ −g(η(0)) ft (η − d0 ) − ft (η) . Recall the definition of the Dirichlet form Dγ (·) introduced in (2.4). The previous estimate shows that the first term on the rightmost expression of (2.6) is bounded above by −2N 2 Dγ (ft ). R N , which corresponds to the price we are paying To estimate the term N 2 Lft dνγ(·) for not using an invariant distribution as a reference measure, let us write it explicitly: Z N2

N Lft dνγ(·) = N2

XZ

N Lx,x+1 ft + Lx+1,x ft dνγ(·) +N2

Z

N Lb ft dνγ(·) . (2.7)

x≥0

Performing the change of variables ξ = η − dx + dy , the measures change as N N dνγ(·) (η)/dνγ(·) (ξ) = γx g(ξ(y))/γy g(η(x)). In particular, we have that


295

Z

N Lx,x+1 ft + Lx+1,x ft dνγ(·) γ Z x N = (1/2) −1 g(η(x + 1))ft (η) dνγ(·) γx+1 γ Z x+1 N + (1/2) −1 g(η(x))ft (η) dνγ(·) . γx

We may thus rewrite the right-hand side of (2.7) as X (1N γ)(x) Z N g(η(x))ft (η) dνγ(·) (1/2) γx x≥1 γ Z 1 N + (N 2 /2) −1 g(η(0))ft (η) dνγ(·) γ0 Z N + N 2 g(η(0))[ft (η − d0 ) − ft (η)] dνγ(·) .

(2.8)

In this formula, (1N γ)(x) stands for N 2 {γx+1 + γx−1 − 2γx }. By definition of γ, (1N γ)(x) = 0 for all x except at x = N −1, where (1N γ)(N −1) = N 2 (γN −2 −γN −1 ), which is negative because γ is non decreasing. The first line of (2.8) is therefore negative. A change of variables ξ = η − d0 permits to write the second term of the second line as Z 2 N N [γ0 − g(η(0))]ft (η) dνγ(·) . The second line of (2.8) is therefore equal to γ Z 1 N βN + (1/2)N 2 −3 g(η(0))ft (η) dνγ(·) ≤ βN γ0 because γ0 = β/N , f is a density and γ1 /γ0 = 2. This concludes the proof of the proposition. With the previous estimate on the entropy and on the Dirichlet form, we are in a position to apply the classical entropy method to prove the hydrodynamic behavior of the system (cf. Chapter V in [KL]). We just point out here the main difference coming from the absorption point at the origin. Lemma 2.4. For every 0 ≤ t ≤ T , lim EµN

N →∞

hZ

t

i g(ηs (0))ds = 0.

0

Proof. Recall that we denote by ft the Radon–Nikodym derivative of µN StN with respect Rt N to νγ(·) . Set f¯t = t−1 0 fs ds. With this notation, the expectation in the statement writes Z N t f¯t (η)g(η(0))dνγ(·) . Adding and subtracting f¯t (η − d0 ) and changing variables, we obtain that this integral is equal to

296


Z t

N g(η(0))[f¯t (η) − f¯t (η − d0 )]dνγ(·) + tγ0 .

The second term vanishes as N ↑ ∞ because γ0 = β/N . The first one, by Schwarz inequality and a change of variables, is bounded above by t {kgk∞ + γ0 } + tADγ,b (f¯t ) A √ for every A > 0. Choosing A = N , we conclude the proof of the lemma by virtue of Proposition 2.3 and the convexity of the Dirichlet form. Lemma 2.5. For every 0 ≤ t ≤ T , lim sup lim sup EµN →0

N →∞

hZ

t 0

i ds 8(2ηsN (0)) = 0.

Notice that in this last expression we multiply ηtN (0) by 2 to obtain the density of particles on the box [0, N ]. The proof of Lemma 2.5 is performed in three steps. We first show that we may replace the cylinder function g(η(0)) by an average over a small macroscopic box around the origin. We then replace this average by 8(2η N (0)) and recall Lemma 2.4 to conclude. Lemma 2.6. For each 0 ≤ t ≤ T and smooth G: R+ → R, lim sup lim sup EµN →0

hZ

N →∞

t

N n oi X ds G(s) g(ηs (0)) − (N )−1 g(ηs (y)) = 0.

0

y=0

Proof. Denote by V (ηs ) the expression inside braces in the previous formula: V (η) = g(η(0)) − (N )−1

N X

g(η(y)).

(2.9)

y=0

Since f¯t = t−1 as

Rt 0

fs ds, we may rewrite the expectation in the statement of the lemma Z N (dη). t V (η)f¯t (η)νγ(·)

A change of variables ξ = η − dx gives that (N )

−1

x−1 N X X

Z n γy

R

N V (η)f¯t (η)νγ(·) (dη) is equal to

o N f¯t (η + dy ) − f¯t (η + dy+1 ) νγ(·) (dη)

x=0 y=0

Z + [γy − γy+1 ]

N ¯ ft (η + dy+1 )νγ(·) (dη) .

Since γx is increasing in x, the second term is negative. On the hand, rewriting the difference {a − b} = {f¯t (η + dy ) − f¯t (η + dy+1 )} as √ other √ √ √ { a − b}{ a + b} and applying the Schwarz inequality, we bound the first term by


297

Z nq N x−1 q o2 A XX N ¯ ft (η + dy ) − f¯t (η + dy+1 ) νγ(·) γy (dη) 2N x=0 y=0

Z n N x−1 o 1 XX N f¯t (η + dy ) + f¯t (η + dy+1 ) νγ(·) + γy (dη) AN x=0 y=0

for every A > 0. Changing variables back, keeping in mind that γx is a non decreasing function and inverting the order of summation, we show that this expression is bounded above by N −1 2kgk∞ A X N Dy,y+1 (f¯t ) + 2 A y=0

for every positive A. Recalling Proposition 2.3 and taking A = proof of the lemma.

√

N , we conclude the

In view of Lemma 2.4, to conclude the proof of Lemma 2.5, it remains to replace the average of the cylinder function g(η(x)) by 8(2η N (0)) but this is the classical two blocks estimate. This concludes the proof of Theorem 2.2 under the assumption that the entropy of the initial state with respect to the product measure νβ+ is bounded above by C0 N for some finite constant C0 . A coupling argument permits to remove Assumption (H2’). Consider a sequence µN satisfying assumptions (H1) and (H2). Fix A > 0 and let µN,A be the probability measure on NN defined by µN,A = µN

3AN

⊗ νβ+

3cAN

,

where 3AN = {0, . . . , AN } and ν3 is the marginal of the probability measure ν on 3. Since νλ+ ≤ µN ≤ να+ and since all cylinder functions can be decomposed as the difference of two monotone functions (cf. [KL]), a simple computation and the explicit formula for the relative entropy give that H(µN,A |νβ+ ) ≤

o 1n H(να+,AN |νβ+,AN ) + H(νλ+,AN |νβ+,AN ) , 2

where νγ+,m is the marginal of νγ+ on {0, . . . , m}. In particular, the entropy H(µN,A |νβ+ ) is bounded above by C0 N for some finite constant C0 depending only on A, α and λ. Let ρA (t, u) denote the solution of (1.3) with initial condition ρRA 0 (u) = ρ0 (u)1{u ≤ A} + β1{u > A}. Investigating the time evolution of the integral R+ due−u ρA (t, u)2 we obtain uniform in A a priori estimates that show that ρA converges to the unique solution of (1.3) with initial condition ρ0 . Since the jump rate g is non decreasing, we may couple a zero range starting from µN with another one starting from µN,A and show that as A ↑ ∞ both behave exactly in the same way on compact sets. This coupling, the hydrodynamic behavior of the empirical measure for a process starting from µN,A and the convergence of ρA to ρ, permit to extend Theorem 2.2 to the sequence of measures satisfying assumptions (H1) and (H2).

298


3. The Case p < 1 We turn in this section to the case where the asymmetric tagged particle jumps at rate p to the right and at rate q to the left. The corresponding zero range process has jumps at rate (1/2) over all bonds but {−1, 0}. From the origin, particles jump at rate p to −1 and from −1 particles jump at rate q to 0. Recall that to fix ideas we assumed p > q. The purpose of this section is to deduce the hydrodynamic behavior of the just described space inhomogeneous process. Consider the zero–range process on Z with generator given by X {Lx,x+1 + Lx+1,x } + 2pL0,−1 + 2qL−1,0 , (3.1) L= x6=−1

where Lx,y is the generator defined just after (2.1). In contrast with the previous section, this system possesses a one parameter family of invariant measures. For each ϕ < p−1 , denote by ν¯ ϕi the product measure on NZ with marginals given by ν¯ ϕi {η, η(x) = k} =

1 ϕkx , Z(ϕx ) g(k)!

(3.2)

where ϕx = pϕ for x ≤ −1 and ϕx = qϕ for x ≥ 0. A direct computation shows that the Markov process with generator given by (3.1) is reversible with respect to these product measures. Before stating the main result of this section, we introduce some terminology on weak solutions of non-linear parabolic equations. Fix a bounded function ρ0 : R → R. A bounded function ρ: R+ × R → R is said to be a weak solution of the partial differential equation (1.4) with initial condition ρ0 if (a) 8(ρ(t, u)) is absolutely continuous in the space variable and for every t > 0, Z

Z

t

du e−|u| {∂u 8(ρ(s, u))} < ∞ , 2

ds R

0

(b) p8(ρ(t, 0+)) = q8(ρ(t, 0−)) for almost every t ≥ 0 and (c) For every smooth function with compact support G: R → R and for every t > 0, Z

Z R

du ρ(t, u)G(u) −

Z R

Z

t

du ρ0 (u)G(u) = −

ds 0

R

du G0 (u)∂u 8(ρ(s, u)).

Since ρ(t, u) is only a measurable function, requirement (b) must be understood as Z

t

lim

→0

0

n 1 Z 1 Z 0 o h(s) p8 ρ(s, u)du − q8 ρ(s, u)du ds = 0 0 −

(3.3)

for every t ≥ 0 and any continuous function h(t). The third property in (1.4) just states that there is conservation of the total mass at the origin. Uniqueness of weak solutions of (1.4) is proved with similar techniques to the ones presented in [ELS](cf. Appendix). The existence for special initial conditions ρ0 follows from the tightness of the sequence QN µN defined below.


299

For each probability measure µ on NZ , denote by PN µ the probability measure on the Z path space D(R+ , N ) induced by the Markov process with generator (3.1) accelerated N by N 2 and the initial measure µ. Expectation with respect to PN µ is denoted by Eµ . We now define the initial states considered in the first main theorem of this section. Fix a sequence of initial measures µN on NZ , we assume that (IS1) The sequence µN is bounded above (resp. below) by some invariant state ν¯ αi (resp. ν¯ λi ) for some 0 < λ < α. (IS2) There exists a function ρ0 : R → R+ such that for each continuous function G: R → R+ and each δ > 0, Z h i X G(x/N )η(x) − du G(u)ρ0 (u) ≥ δ = 0. lim µN N −1 N →∞

x

Notice that it follows from assumption (IS1) that the function ρ0 in (IS2) is necessarily bounded. Theorem 3.1. Consider a sequence of initial states µN satisfying assumptions (IS1), (IS2). For any continuous function G: R → R with compact support and any δ > 0, " # Z −1 X N G(x/N )ηt (x) − du G(u)ρ(t, u) ≥ δ = 0, lim PµN N N →∞

x

where ρ is the unique solution of (1.4). Like in Sect. 3 (cf. also Chap. IV of [KL]), we deduce this result from the convergence in distribution of the empirical measure π N = π N (η) defined as the −1 to each particle: positive Radon P measure on R obtained by assigning a mass N π N = N −1 z∈Z η(z)δz/N . Set πtN = π N (ηt ) and denote by M+ = M+ (R) the space of positive Radon measures on R endowed with the vague topology, a metrizable topology. Fix T > 0. For each probability measure µ on NZ , denote by QN µ the probability and the empirical measure measure on the path space D([0, T ], M+ ) induced by PN µ N π . Theorem 3.2. The sequence QN µN converges to the probability measure concentrated on the absolutely continuous path π(t, du) = ρ(t, u)du whose density is the solution of (1.4). Coupling arguments similar to the ones presented at the end of the previous section show that it is enough to prove Theorem 3.2 under the assumption that there exist a density β > 0 and a finite constant C0 such that the entropy of µN with respect to ν¯ βi is bounded by C0 N : H(µN |ν¯ βi ) ≤ C0 N for every N ≥ 1. We therefore assume until the end of this section the existence of such constants β and C0 . The main difference in the proof of the hydrodynamic limit of this model and the classical proof for space homogeneous systems resides in the behavior at the boundary u = 0. The next four lemmas solve this question. For a site x, a configuration η and a positive integer `, denote by M`± (x, η) the density of particles for the configuration η on a box of size ` at the right (left) of x: x+`

M`+ (x, η) =

1 X η(y) , ` + 1 y=x

M`− (x, η) =

x 1 X η(y). `+1 y=x−`

300


Lemma 3.3. For every continuous function H: [0, T ] → R, h Z T n o i + lim sup lim sup EN dt H(t) g(η (0)) − 8(M (0, η )) = 0. N t t N µ →0

N →∞

0

+ (0, ηt ) by The same result holds if g(ηt (0)) is replaced by g(ηt (−1)) and MN − MN (−1, ηt ).

This result follows from the next lemma and the two blocks estimate. Lemma 3.4. For every continuous function H: [0, T ] → R, lim sup lim sup EN µN →0 N →∞

h Z

T

N n o i X dt H(t) g(ηt (0)) − (N )−1 g(ηt (x)) = 0.

0

x=0

The same result holds if g(ηt (0)) is replaced by g(ηt (−1)) and the average over {0, . . . , N } is replaced by the average over {−N, . . . , 0}. Proof. Recall from (2.9) the definition of V (ηt ). By the entropy inequality, i h Z T E µN ds H(s)V (ηs ) 0

h n H(µN |ν¯ βi ) 1 + log Eν¯ βi exp ≤ NA AN

Z

T

oi ds G(s)AN V (ηs )

0

for every A > 0. By assumption, the first term on the right-hand side is bounded by CA−1 . To prove the lemma it is therefore enough to show that the limit of the second one is less than or equal to 0 for every A > 0. Since e|x| ≤ ex + e−x and lim supN N −1 log{aN +bN } ≤ max{lim supN N −1 log aN , lim supN N −1 log bN }, replacing H by −H we deduce that we only need to prove the previous statement without the absolute value in the exponent. By the Feynman–Kac formula and the variational formula for the largest eigenvalue of an operator, h nZ T oi 1 log Eν¯ βi exp ds G(s)AN V (ηs ) lim sup N →∞ AN 0 (3.4) Z T nZ o H(t)V (η)f (η)ν¯ βi (dη) + A−1 N D(f ) . dt sup ≤ 0

f

In this formula, the supremum is taken over all densities f with respect to ν¯βi and D(f ) is the Dirichlet form Z p p f L f dνβi . D(f ) = We are now ready to integrate by parts the cylinder function V . The rest of the proof is similar to the proof of Lemma 2.6 and omitted for this reason. The same argument permits to deduce the following result. Lemma 3.5. For every continuous function H: [0, T ] → R, i h Z T dt H(t){pg(ηt (0)) − qg(ηt (−1))} = 0. lim sup EN µN N →∞

0


301

The next result follows from Lemma 3.3 and Lemma 3.5. Corollary 3.6. For every continuous function H: [0, T ] → R, h Z T n o i − + dt H(t) p8(M (0, η )) − q8(M (−1, η )) lim sup EN = 0. N t t N µ N N →∞

0

These technical lemmas permit to adapt the classical proof of the hydrodynamic behavior of reversible systems to the present context. Details are left to the reader. Remark 3.7. In Sects 2 and 3 only the monotonicity and the boundness of the jump rate g(·) were used. The same arguments permit therefore to deduce the hydrodynamic behavior of a more general class of processes. 4. The Asymmetric Tagged Particle We prove in this section Theorems 1.1 and 1.2 through the hydrodynamic behavior of the inhomogeneous zero range processes considered in the previous two sections. We have seen in the first section that the displacement of the asymmetric tagged particle corresponds in the zero range process to the total flux of particles through the origin. For this reason, we start deducing the total flux through the origin from the hydrodynamic limit proved in the previous two sections. Proposition 4.1. In the case p = 1, consider a sequence of probability measures µN satisfying assumptions (H1) and (H2). Then, for every t ≥ 0 and δ > 0, Z ∞ h i −1 X {η (x) − η (x)} − du {ρ(t, u) − ρ (u)} > δ = 0 , (4.1) lim PN N N t 0 0 µ N →∞

x≥0

0

where ρ is the solution of (1.3). In the case p < 1, consider a sequence of probability measures µN satisfying assumptions (IS1), (IS2). Then, for every t ≥ 0 and δ > 0 (4.1) holds, where ρ is now the solution of (1.4). Proposition 4.1 follows from the hydrodynamic behavior of the inhomogeneous processes considered in Sects. 3, 4 and from the definition of the infinite sums appearing in (4.1). Theorem 1.1 and 1.2 follow from Proposition 4.1 if we prove the following proposition: Proposition 4.2. Fix a sequence of initial states µN ρ0 (·) satisfying the assumptions of N Theorem 1.1 or 1.2. The sequence T µρ0 (·) satisfy assumptions (H1), (H2) in the case p = 1 or (IS1), (IS2) in the case p < 1, where T is the transformation defined in Sect. 1. Proof. We start with the case p = 1. A simple computation shows that T transforms + defined by (2.2). Fix the Bernoulli product measure µρ in the product measure ν(1−ρ)/ρ a profile ρ0 : R+ → [0, 1] for which there exists σ > 0 such that σ ≤ ρ0 ≤ 1 − σ. Recall that we denote by µN ρ0 (·) the inhomogeneous product measure associated to ρ0 . N = ν . We shall now show that νρN0 (·) fulfills assumptions (H1 ), (H2). Let T µN ρ0 (·) ρ0 (·) We first claim that if µ is a product measure on {0, 1}N∗ bounded above (resp. below) + . Here by µ+ρ for some 0 < ρ < 1, then T µ is bounded below (resp. above) by ν(1−ρ)/ρ

302


µ+ρ stands for the restriction on N of the measures µρ . Notice that the inequalities are reversed by the application T . To fix ideas assume that µ ≤ µ+ρ . For x ≥ 1, denote by γx the probability of finding a particle at x for the probability µ so that γx ≤ ρ. For j ≥ 1, denote by Nj the position of the j th particle at the right of the origin. Since γx ≤ ρ for every x and µ, µρ are product measures, it is possible to couple µ and µρ in such a way µ ρ −Njµ ≥ Nj+1 −Njρ for all j ≥ 1. In this formula, Njµ (resp. Njρ ) that N1µ ≥ N1ρ and Nj+1 stands for the position of the j th particle under the distribution µ (resp. µρ ). Applying the transformation T to this coupling measure, we construct a measure on NN × NN with + and concentrated first marginal equal to T µ, second marginal equal to T µ+ρ = ν(1−ρ)/ρ 1 2 + , what on configurations (η , η ) below the diagonal. This shows that T µ ≥ ν(1−ρ)/ρ + N + concludes the proof of the claim. In particular, νσ/(1−σ) ≤ νρ0 (·) ≤ ν(1−σ)/σ for every N ≥ 1 and assumption (H1) is verified. Notice, however, that the claim “µ1 ≤ µ2 implies T µ1 ≥ T µ2 ” is not correct. Consider, for instance, the configuration ξ 1 , ξ 2 such that ξ 1 (x) = 1

if and only if x 6= 1, 2, 3 and ξ 2 (x) = 1

if and only if x 6= 1, 3.

In this case the deterministic measures δξi are such that δξ1 ≤ δξ2 but it is not correct that δT ξ1 is above δT ξ2 . We turn now to the second assumption (H2). It follows from (1.2) that Z B F(u) du = H−1 (B) − B (4.2) 0

for every B > 0. In order to check (H2), we just need to show that under νρN0 (·) , RB P[BN ] N −1 x=0 η(x) converges in probability to 0 F (u)du for every B > 0. Fix a positive integer n. The following inequalities state that for the exclusion process the total number of sites in 3n = {0, . . . , n} is equal to the total number of particles plus the total number of holes (that corresponds to the total number of particles for the zero range process): Pn Pn −1+ ξ(x) ξ(x) n n x=0 x=0 X X X X ξ(x) + η(y) ≤ n + 1 ≤ ξ(x) + η(y). x=0

y=0

x=0

y=0

P[BN ] The convergence of N −1 x=0 η(x) follows from theseRinequalities, the fact that under P n −1 the measure µN ρ0 (·) , N 0≤x≤[nN ] ξ(x) converges to 0 ρ0 (u) du and identity (4.2). Details are left to the reader. In exactly the same way, assumptions (IS1), (IS2) can be checked in the case p < 1. The only difference is that we assume in (IS1) that the sequence of initial measures is bounded below by an invariant measure ν¯ϕi which is inhomogeneous in space. This forces the initial profile ρ0 to be bounded below by the function ψα (u) = (1 − α)1{u < 0} + [1 − (q/p)α]1{u > 0} for some 0 < α < 1. 5. Einstein Relation We consider in this section initial profiles for which the solution of equation (1.4) is selfscaling. For two fixed densities ρ− and ρ+ consider, for instance, the initial condition ρ0 (·) given by


303

ρ0 (u) = ρ+ 1{u ≥ 0} + ρ− 1{y < 0}. √ The solution of (1.4) takes the form ρ(t, u) = ϕ(u/ t), where ϕ(·) is the solution of  − zϕ0 (z) = ∂z2 8(ϕ(z)) ,     ϕ0 (0−) ϕ0 (0+) (5.1) = ϕ(±∞) = ρ± .  (1 + ϕ(0+))2 (1 + ϕ(0−))2    p8(ϕ(0+)) = q8(ϕ(0−)). √ It easy to see that in this case vt = v t, where v is given by Z +∞ {ρ+ − ϕ(y)} dy. v = 0

Moreover, since ρ+ = ϕ(∞), we may write the expression inside braces as R ∂ ϕ(z)dz. Performing an integration by parts and keeping in mind that ϕ is the z [y,∞) solution of (5.1), we obtain that v =

ϕ0 (0+) · (1 + ϕ(0+))2

We now transform (5.1) in a linear equation through the following Lagrangian change of coordinates: Z z 1 · (1 + ϕ(y)) dy m(x) = x(z) = 1 + ϕ(z(x)) 0 We leave to the reader to check that this transformation is in fact the inverse of the transformation T described in (1.2). Moreover, a simple computation shows that m(x) is the solution of the linear equation  00 m (x) = −(x + v)m0 (x) ,      m0 (0+) m0 (0−)   = ,  −v = m(0+) m(0−) (5.2)  p(1 − m(0+)) = q(1 − m(0−)) ,      1   m(±∞) = α± = . 1 + ρ± In fact (5.2) describes the selfscaling solution of the Stefan problem:  1  ∂t m∗ (x, t) = ∂xx m∗ (x, t) ,    2    ∂x m∗ (vt +, t) ∂x m∗ (vt −, t) − vt = = , (5.3) m∗ (vt +, t) m∗ (vt −, t)    ∗ ∗    p{1 − m (vt +, t)} = q{1 − m (vt −, t)} ,  ∗ m (x, 0) = α+ 1{x ≥ 0} + α− 1{x < 0} . √ In other words, m(x/ t) is the macroscopic profile of density as seen from the tagged asymmetric particle.

304


The solution of (5.2) can be written as Z x  2   e−(1/2)y −vy dy for x > 0 ,  A+ + B+ m(x) = Z0 x  2   A− + B − e−(1/2)y −vy dy for x < 0 , 0

where the parameters are related by the equations p(1 − A+ ) = q(1 − A− ) ;

−v =

B+ B− = ; A+ A−

α± = A± J(±v),

R +∞ 2 where J(v) = 1 − v 0 e−(1/2)y −vy dy. It follows from the previous identities that the parameters p, α+ , α− and v satisfy the equation α− α+ =q 1− . (5.4) p 1− J(v) J(−v) This equation was obtained heuristically by [BDMO]. In particular, we cannot write v as an explicit function of p, α+ , α− , but we can study some asymptotic relations. We consider three distinct asymptotics. We first investigate the case of a constant initial profile: α+ = α− = α. In this case elementary computations give the identity (p − q)

1 − α pJ(−v) − qJ(v) − (p − q)J(v)J(−v) = · α J(v)J(−v)

(5.5)

For small asymmetry p − q, we have a small displacement v. Replacing in (5.5) J(v) by its expansion for v small gives, for fixed α and small p − q, that r v = (p − q)

2 1−α + o(p − q). π α

This proves the validity of Einstein relation for small drifts. In the case α+ 6= α− one can expand around the equilibrium, i.e., for small p(1 − α+ ) − q(1 − α− ). The same expansions show that p(1 − α+ ) − q(1 − α− ) v= pα+ − qα−

r

2 + o p(1 − α+ ) − q(1 − α− ) . π

A third possible asymptotics is given when the initial profile is constant and the density α = α+ = α− is small. In this case, for a fixed drift p − q, the displacement v is very large. Asymptotically, for |v| close to ∞, a simple computation shows that √ 2 J(v) ∼ v −2 , J(−v) ∼ vev /2 2π. Using these expansions in (5.4) one obtains that r v∼

p−q + o α+

1 √ α+

.


305

6. Appendix: Uniqueness Case p = 1. This is an extension to infinite volume of an argument presented in [ELS2]. Fix a weak solution ρ(t, u) of the differential Eq. (1.3). Since ρ(t, ·) is in L1loc (R+ ), we may define Rt : R2+ → R by Z v ρ(t, w) dw. (6.1) Rt (u, v) = u

Denote by [·, ·] the inner product in L2 (R2+ ). Fix a smooth function H: R2+ → R with compact support. Changing the order of summations we obtain that Z du ρ(t, w)h(w) , (6.2) [Rt , H] = R+

Z

where

Z

w

∞

du

h(w) =

w

0

Z dv H(u, v) −

Z

∞

w

du w

dv H(u, v). 0

Notice that h is a smooth function with compact support that vanishes at the origin. Moreover, its derivative is given by Z ∞ 0 h (w) = du {H(w, u) − H(u, w)}. 0

Therefore, in the virtue of (6.2), property (c) of weak solutions and a change of variables, for every smooth function H with compact support, Z t Z Z ds du dv H(u, v){∂v 8(ρ(s, v)) − ∂u 8(ρ(s, u))}. [Rt , H] = [R0 , H] + 0

R+

R+

In particular, we have that Z

t

Rt (u, v) − R0 (u, v) =

ds {∂v 8(ρ(s, v)) − ∂u 8(ρ(s, u))}

(6.3)

0

for almost all (u, v) in R2+ . Consider now two solutions ρ1 , ρ2 of Eq. (1.3), denote by Rt1 , Rt2 the respective functions associated to ρ1 , ρ2 , through (6.1) and set Wt = Rt1 − Rt2 , ρ¯t = ρ1t − ρ2t . Denote by [·, ·]e the inner product on L2 (R2+ ) associated to the measure e−(u+v) dudv. In view of property (a) of weak solutions and identity (6.3), R: [0, T ] → L2 (R2+ , e−(u+v) dudv) is almost everywhere differentiable. Therefore, Z Z d ¯ t (v) − ∂u 8 ¯ t (u)} , [Wt , Wt ]e = 2 du dv e−(u+v) Wt (u, v){∂v 8 dt ¯ t (v) stands for 8(ρ1 (t, v)) − 8(ρ2 (t, v)). An integration by parts gives that the where 8 right-hand side is equal to Z Z Z ¯ t (u)ρ(t, ¯ t (v) −2 e−u 8 ¯ u) + 2 du dv e−(u+v) Wt (u, v)8 (6.4) R+

306

because by


R

du exp{−u} = 1. By Schwarz inequality, the second term is bounded above Z 1 ¯ t (u) 2 du e−u 8 k80 k∞ [Wt , Wt ]e + 0 k8 k∞ R+ Z ¯ t (u)ρ(t, ≤ k80 k∞ [Wt , Wt ]e + du e−u 8 ¯ u) R+

because 8 is an increasing function with a bounded first derivative. Adding this expression to the first term of (6.4), we obtain that the time derivative of [Wt , Wt ]e is bounded above by Z 0 ¯ t (u)ρ(t, du e−u 8 ¯ u) ≤ k80 k∞ [Wt , Wt ]e k8 k∞ [Wt , Wt ]e − R+

because 8 is non decreasing. By the Gronwall inequality, we deduce that [Wt , Wt ]e is bounded above by [W0 , W0 ]e exp{k80 k∞ t}, which concludes the proof of the uniqueness of weak solutions of Eq. (1.3). The case p < 1. The argument is similar to the one presented for p = 1. For t ≥ 0, define Rt : R2 → R+ as in (6.1). It can be shown that Z t ds {∂v 8(ρ(s, v)) − ∂u 8(ρ(s, u))} Rt (u, v) − R0 (u, v) = 0

for almost all (u, v) in R . Consider two solutions of Eq. (1.4). Denote by m(du) = m(u)du the absolutely continuous measure with density m(u) = p1{u < 0}+q1{u > 0} enough and Rand fix a smooth function θ: R → R+ such that θ(0) = 0, θ(u) = |u| for u2 large m(du) exp{−θ(u)} = 1. Let [·, ·]m stand for the inner product in L (R2 ) with respect to the measure m(du)m(dv) exp{−θ(u) − θ(v)}. Fix two solutions ρ1 , ρ2 of Eq. (1.4), denote by Rt1 , Rt2 the respective functions associated to ρ1 , ρ2 , through (6.1) and set Wt = Rt1 − Rt2 . With the same arguments presented above one can show that [Wt , Wt ]m is bounded above by [W0 , W0 ]m exp{C(θ, k80 k∞ )t}. In this deduction the use of the measure m(du) instead of the Lebesgue measure is fundamental in the integration by parts performed in (6.4) for the boundary term to cancel. 2

Acknowledgement. The authors would like to thank H. Rost and H. Spohn for fruitful discussions on the Einstein relation and Pott’s models. We also thank G. S. Oshanin for pointing out a computational mistake in Sect. 5 of a previous version of this paper.

References [A] [Ar]

Andjel, E.; Invariant measures for the zero-range processes. Ann. Probab. 10, 525–547 (1982) Arratia, R: The motion of a tagged particle in the simple symmetric exclusion system in Z. Ann.Prob. 11, 362–373 (1983) [BDMO] Burlatsky, S.F., De Coninck, J., Moreau, M., Oshanin, G. S: Dynamics of the shock front propagation in a one-dimensional hard core lattice gas. Preprint (1997) [BMMO] Burlatsky, S. F., Mogutov, A. V., Moreau, M., Oshanin, G. S: Directed walk in a one–dimensional gas. Phys. Lett. A 166, 230–234 (1992) [BMOR] Burlatsky, S.F., Moreau, M., Oshanin, G. , Reinhardt, W.P: Motion of a driven tracer particle in a one-dimensional symmetric lattice gas. Phys.Rev. E, 54, no. 4, 3165–3172 (1996) [ELS] Eyink, G., Lebowitz, J.L. and Spohn, H.: Lattice gas models in contact with stochastic reservoirs: local equilibrium and relaxation to steady states. Commun. Math. Phys. 140, 119–131 (1991)


[GPV] [H] [KL] [LM] [LR] [RV] [Spo]

307

Guo, M.Z., Papanicolaou, G.C. and Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Comm. Math. Phys. 118, 31–59 (1988) Harris, T.E: Diffusions with collisions between particles. J. Appl. Prob. 2, 323–338 (1965) Kipnis, C., Landim, C: Hydrodynamical Limit of Interacting Particle Systems., Preprint 1996 Landim, C., Mourragui, M: Hydrodynamic limit of mean zero asymmetric zero range processes in infinite volume. Ann. Inst. H. Poincaré, Prob. et Stat. 33, 65–82 (1997) Lebowitz, J.L., Rost, H.: The Einstein relation for the displacement of a test particle in a random environment. Stoc. Proc. Appl. 54, 183–196 (1994) Rost, H. and Vares, M.E: Hydrodynamics of a one dimensional nearest neighbor model. Contemp. Math. 41, 329–342 (1985) Spohn, H: Large Scale Dynamics of Interacting Particles Text and Monographs in Physics, New York: Springer Verlag, 1991

Communicated by J. L. Lebowitz

Commun. Math. Phys. 192, 309 – 347 (1998)

Communications in


Relative Zeta Functions, Relative Determinants and Scattering Theory ¨ Werner Muller Universität Bonn, Mathematisches Institut, Beringstrasse 1, D-53115 Bonn, Germany. E-mail: [email protected] Received: 15 January 1997 / Accepted: 2 July 1997

Abstract: We use the method of zeta function regularization to regularize the ratio det A/ det A0 of the determinants of two elliptic self-adjoint operators A, A0 satisfying certain natural assumptions. This is of interest, especially, if the regularized determinants of the individual operators don’t exist as, for example, in the case of elliptic operators on a noncompact manifold. 0. Introduction In the present paper we use the method of zeta function regularization to regularize the ratio det A/ det A0 of the determinants of two elliptic self-adjoint operators A and A0 , acting in the space of smooth sections of a vector bundle over a C ∞ manifold X. If X is closed, then the zeta function technique can be used to regularize the determinants of the individual operators A and A0 , and the above ratio is simply the ratio of these regularized determinants. However, if X is noncompact then, in general, an elliptic self-adjoint operator A on X has continuous spectrum and therefore, the zeta function of A cannot be defined. Since regularized determinants of elliptic operators play an important role in various fields of mathematics and physics (see [Mu6] for some references concerning applications), it is interesting to see how this technique can be extended to the case of elliptic operators on noncompact manifolds. First, we recall some facts about the zeta function regularization of determinants of elliptic operators on compact manifolds. Let E be a complex vector bundle over a closed n-dimensional C ∞ manifold M and let A : C ∞ (E) → C ∞ (E) be an elliptic pseudodifferential operator of order m > 0. Suppose that A is symmetric and nonnegative (with respect to an inner product in C ∞ (E) induced by the choice of a metric on M and a fibre metric in E). Then the zeta function of A is defined by X λ−s (0.1) ζA (s) = j , λj >0

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where the λj ’s run over the eigenvalues of A, and each eigenvalue is counted with its multiplicity. The series (0.1) is absolutely convergent in the half-plane Re(s) > n/m and admits a meromorphic extension to the entire complex plane [Se]. Moreover, s = 0 is not a pole of ζA (s). We note that the zeta function can also be expressed in terms of the heat operator by Z ∞ 1 ts−1 Tr(e−tA ) − dim ker A dt, Re(s) > n/m, (0.2) ζA (s) = 0(s) 0 and the well known heat expansion can be used to obtain the analytic continuation of ζA (s) (see e.g. [Gi]). Using the zeta function, the regularized determinant of A is defined by d . (0.3) det A = exp − ζA (s) ds s=0 The zeta function regularization was first introduced by Ray and Singer [RS1] to define a regularized determinant for the Laplacian on differential forms twisted by a flat bundle. Hawking [H] has used the same method to regularize quadratic path integrals on a curved background spacetime. In some cases it is also possible to define a determinant for self-adjoint elliptic operators that are unbounded from below. For example, let P : C ∞ (E) → C ∞ (E) be an elliptic self-adjoint differential operator of order m > 0. For such operators, Atiyah, Patodi and Singer [APS1, APS3] introduced the so-called η-function which in the half-plane Re(s) > n/m is given by ηP (s) =

X signλj |λj |s

λj 6=0

1 = 0 (s + 1)/2

Z

∞

t(s−1)/2 Tr P e−tP

2

dt.

0

(0.4) Here λj runs over the nonzero eigenvalues of P . The analytic function ηP (s) has a meromorphic extension to C with isolated simple poles. The residues at all the poles are locally computable, and s = 0 is not a pole of ηP (s) [APS3, Gi]. The fact that ηP (s) is regular at s = 0 is less obvious than in the case of the zeta function. Using ηP (0), one can define a regularized determinant for P by πi ηP (0) − ζ|P | (0) (0.5) det P = det |P | · exp 2 [Si]. As explained above, for elliptic self-adjoint operators on a noncompact manifold X, in general, the zeta function (0.1) cannot be defined. To introduce an appropriate zeta function, we consider pairs (A, A0 ) of elliptic self-adjoint operators such that A is a compactly supported perturbation of A0 . Then A0 serves as a reference operator and our goal is to regularize det A/ det A0 . In some special cases, this question has been studied in [Br, GMS, Lu, JL] and [Mu2] (see §4). Assume, in addition, that A and A0 are nonnegative. Then, under some natural assumptions, we can expect that e−tA − e−tA0 is a trace class operator for each t > 0, and that Tr e−tA − e−tA0

Relative Zeta Functions, Relative Determinants and Scattering Theory

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has an asymptotic expansion as t → 0. This holds, for example, for spinor Laplace operators on complete Riemannian manifolds [Bu]. In order to define the relative zeta is analogous to (0.2), we have to investigate function ζ(s; A, A0 ) by a formula which the behaviour of Tr e−tA − e−tA0 as t → ∞. For this purpose we use Krein’s theory of the spectral shift function [Kr, BY, Y], which implies that the spectral shift function ξ(λ) = ξ(λ; A, A0 ) of (A, A0 ) exists and the following trace formula holds: Z ∞ Tr e−tA − e−tA0 = −t e−tλ ξ(λ) dλ. (0.6) 0

The spectral shift function is locally integrable and locally constant on the complement of σ(A)∪σ(A0 ). Moreover, e−tλ ξ(λ) is absolutely integrable on R+ . Thus, if the essential spectrum of A0 has a positive lower bound, then the same holds for A and it follows from (0.6) that there exists c > 0 such that (0.7) Tr e−tA − e−tA0 = dim ker A − dim ker A0 + O(e−ct ) as t → ∞. This resembles the behaviour of the trace of the heat operator in the compact case and we can define a relative zeta function by Z ∞ 1 ts−1 Tr e−tA − e−tA0 − b dt, Re(s) 0, ζ(s; A, A0 ) = 0(s) 0 where b = dim ker A − dim ker A0 . However, if the continuous spectrum of A0 extends down to zero, (0.7) does not hold. Then the large time behaviour of Tr e−tA − e−tA0 is more complicated and this is where scattering theory comes into play. First of all, by (0.6), it follows that the large time behaviour of Tr e−tA − e−tA0 is related to the behaviour of the spectral shift function ξ(λ) near λ = 0. To study ξ(λ) near λ = 0 we shall use scattering theory. Since e−tA − e−tA0 is a trace class operator, the Birman-Krein invarinace principle for wave operators [K1, K2] implies that the Wave operators W± (A, A0 ) exist and are complete and therefore, the scattering matrix S(λ) = S(λ; A, A0 ), λ ∈ σac (A0 ), exists. Furthermore, the determinant of the scattering matrix is related to the spectral shift function by det S(λ) = e−2πiξ(λ) ,

λ ∈ σac (A0 ),

[BK, BY, Y]. Thus, the investigation of the large time behaviour of Tr e−tA − e−tA0 can be reduced to the study of log det S(λ) near λ = 0. For typical examples (see Sect. 4), det S(λ) is differentiable near zero and (0.6) specializes to a more explicit trace d log det S(λ). This formula involving the eigenvalues and the scattering phase shift dλ trace formula can be used to determine the asymptotic behaviour of Tr e−tA − e−tA0 as t → ∞. Then, by a modification of (0.2), we can define a relative zeta function ζ(s; A, A0 ) which is a meromorphic function of s ∈ C. This zeta function has some new features that do not occur in the compact case. For example, there are poles whose residues are related to the derivatives of det S(λ) at λ = 0 and therefore, these residues are not locally computable as in the compact case. If ζ(s; A, A0 ) is holomorphic at s = 0, we can introduce the regularized ratio det(A, A0 ) =

det A det A0

by the analogous formula (0.3) with ζA (s) replaced by ζ(s; A, A0 ).

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We note that we may think of A0 as being a reference operator and regard det(A, A0 ) as a function of A. In particular, we can use det(A, A0 ) in much the same way as det A is used in the case of elliptic operators on a compact manifold. Next we briefly describe the content of this paper. In Sect. 1, we introduce relative zeta functions for pairs (H, H0 ) of self-adjoint operators in a Hilbert space. In Sect. 2, we use Krein’s theory of the spectral shift function to relate the large time behaviour of Tr e−tA − e−tA0 to properties of the scattering matrix near zero. Then relative determinants are introduced and studied in Sect. 3. Finally, in Sect. 4, we discuss several natural examples: the Schrödinger operator in Rn , perturbations of the euclidean Laplacian, manifolds with cylindrical ends and surfaces with hyperbolic ends. 1. Relative Zeta Functions In this section we study relative zeta functions in the abstract setting. Let H be a separable Hilbert space and let H, H0 be two self-adjoint nonnegative linear operators in H. We assume that the following conditions hold: 1) Let e−tH and e−tH0 be the heat semigroups associated to H and H0 , respectively. Then e−tH − e−tH0 is a trace class operator for t > 0.

(1.1)

2) As t → 0, there exists an asymptotic expansion of the form Tr e

−tH

−e

−tH0

∼

k(j) ∞ X X

ajk tαj logk t,

(1.2)

j=0 k=0

where −∞ < α0 < α1 < . . . and αk → ∞. Moreover, if αj = 0 we assume that ajk = 0 for k > 0. 3) As t → ∞, there exists an asymptotic expansion of the form ∞ X bk t−βk Tr e−tH − e−tH0 ∼

(1.3)

k=0

where 0 = β0 < β1 < · · · . It follows from (1.2) that the integral Z 1 ts−1 Tr (e−tH − e−tH0 ) dt 0

is absolutely convergent in the half-plane Re(s) > −α0 and has a meromorphic continuation to the whole complex plane. The possible poles occur at s = −αj , j ∈ N, and at a given pole s = −αj , the Laurent expansion is given by k(j) X k=0

Set

ajk

k! + O(1). (s + αj )k+1

(1.4)


ζ1 (s; H, H0 ) =

1 0(s)

Z

1

ts−1 Tr(e−tH − e−tH0 ) dt.

313

(1.5)

0

Then ζ1 (s; H, H0 ) is a meromorphic function of s ∈ C. Poles may occur at s = −αj , j ∈ N, and the corresponding Laurent expansion is determined by (1.2). By our assumption about the asymptotic expansion (1.2), ζ1 (s; H; H0 ) is holomorphic at s = 0. Next consider the integral Z ∞ ts−1 Tr (e−tH − e−tH0 ) dt. 1

By (1.3), this integral is absolutely convergent in the half-plane Re(s) < β0 and admits a meromorphic continuation to C. Set Z ∞ 1 ts−1 Tr e−tH − e−tH0 dt. (1.6) ζ2 (s; H, H0 ) = 0(s) 1 Then ζ2 (s; H, H0 ) is a meromorphic function of s ∈ C with possible poles at s = βk , k ∈ N+ . All poles are simple and we have Res ζ2 (s; H, H0 ) =

s=βk

bk , k > 0. 0(βk )

Furthermore, ζ2 (s; H, H0 ) is holomorphic at s = 0. Put ζ (s; H, H0 ) = ζ1 (s; H, H0 ) + ζ2 (s; H, H0 ) .

(1.7)

We call ζ (s; H, H0 ) the relative zeta function of (H, H0 ) . Summarizing, we have proved Proposition 1.1. Assume that the pair (H, H0 ) of nonnegative self-adjoint operators satisfies conditions (1.1)–(1.3). Then the relative zeta function ζ (s; H, H0 ) is a meromorphic function of s ∈ C. The set of poles of ζ (s; H, H0 ) is contained in the set {−αj | j ∈ N} ∪ {βk | k ∈ N+ } and s = 0 is a regular point. Example 1.1. Let M be a closed Riemannian manifold of dimension n, let E be a Hermitian vector bundle over M and let Di : C ∞ (M, E) −→ C ∞ (M, E), i = 0, 1, be an elliptic self-adjoint differential operator of order mi > 0 which is nonnegative. Let Hi , i = 0, 1, be the unique self-adjoint extension of Di in L2 (M, E). It is wellknown that exp(−tHi ) is a trace class operator for t > 0 and, as t → 0, there exists an asymptotic expansion X aij tj . (1.8) Tr e−tHi ∼ t−n/mi j≥0

Let hi = dim Ker Di , i = 0, 1. Then, as t → ∞, we have Tr (e−tHi ) = hi + O(e−ct ).

(1.9)

Thus, (1.2) and (1.3) are a consequence of (1.8) and (1.9). In the present case, the zeta function ζHi (s) of Hi is defined by (0.1) and the relative zeta function of (H1 , H0 ) can be expressed in terms of the absolute zeta functions by

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ζ(s; H1 , H0 ) = ζH1 (s) − ζH0 (s),

(1.10)

which justifies the name relative zeta function. We note that (1.8) is the general form of the short time asymptotic expansion of the trace of the heat operator of an elliptic operator on a compact manifold. In particular, there occur no logarithmic terms in this expansion. However, if we pass to the noncompact setting, then, in general, logarithmic terms naturally arise in the short time asymptotic expansion of the trace of the difference of two heat semigroups. Examples are hyperbolic manifolds or, more generally, manifolds with cusps. Furthermore, the large time behaviour (1.9) of the trace of the heat semigroup has also a very simple form. In the noncompact case, (1.3) is related to the behaviour of the continuous spectrum near zero. If the essential spectrum of H0 has a positive lower bound, then the expansion (1.3) is similar to (1.9) (see Lemma 2.2). Finally, we observe that a relative version of the η-function (0.4) can be defined in the same way. Suppose that D and D0 are self-adjoint operators in H such that the following three conditions hold: 1) Let e−tD and e−tD0 be the heat semigroups of the nonnegative selfadjoint operators D2 and D02 , respectively. Then 2

2

De−tD − D0 e−tD0 is a trace class operator for t > 0. 2

2

(1.11)

2) As t → 0, there exists an asymptotic expansion of the form Tr De

−tD 2

− D0 e

−tD02

∼

k(j) ∞ X X

Ajk tκj logk t,

(1.12)

j=0 k=0

where −∞ < κ0 < κ1 < · · · → ∞. 3) As t → ∞, there exists an asymptotic expansion of the form ∞ X 2 2 Tr De−tD − D0 e−tD0 ∼ Bk t−θk ,

(1.13)

k=0

where 0 < θ0 < θ1 < · · · → ∞. Then we can introduce partial η-functions by Z 1 2 2 1 t(s−1)/2 Tr De−tD − D0 e−tD0 dt, η1 (s; D, D0 ) = 0 (s + 1)/2 0 Re(s) > −κ0 ; 1 η2 (s; D, D0 ) = 0 (s + 1)/2

Z

∞

(1.14)

2 2 t(s−1)/2 Tr De−tD − D0 e−tD0 dt,

1

Re(s) < θ0 .

(1.15)

Using the asymptotic expansions (1.12) and (1.13), the partial η-functions can be continued analytically to the whole complex plane. Set η(s; D, D0 ) = η1 (s; D, D0 ) + η2 (s; D, D0 ).

(1.16)

This is the relative η-function associated to (D, D0 ). If κj 6= −1/2 for all j ∈ N, then η(s; D, D0 ) is regular at s = 0.


315

2. The Spectral Shift Function and the Large Time Asymptotic Expansion In this section, we study the spectral shift function of (H, H0 ) and we investigate the connection between its behaviour at 0 (resp. ∞) and the asymptotic expansion (1.3) (resp. (1.2)). First, we introduce some notation. Let A be a self-adjoint operator in a Hilbert space. Then we denote by σ(A) the spectrum of A. Moreover, by σpp (A), σsc (A), σac (A) and σess (A), respectively, we shall denote the pure point spectrum, the singular continuous, the absolutely continuous and the essential spectrum of A, respectively. Next we recall some basic results of Krein [Kr] and Birman–Krein [BK] concerning the spectral shift function. See also [BY, Y]. Let A and A0 be bounded self-adjoint operators in H and suppose that V = A − A0 is a trace class operator. Let R0 (z) = (A0 − z)−1 . Then the spectral shift function ξ(λ) = ξ(λ; A, A0 ) = π −1 lim arg det(1 + V R0 (λ + i)) →0

(2.1)

exists for a.e. λ ∈ R. Note that arg det(1 + V R0 (λ + i)) is well defined by the condition that it should tend to zero as → ∞. The spectral shift function ξ(λ) is real valued, belongs to L1 (R) and satifies Z ξ(λ) dλ, ||ξ||L1 ≤ ||A − A0 ||1 , (2.2) Tr(A − A0 ) = R

where || · ||1 denotes the trace norm. Moreover, let I(A, A0 ) be the smallest interval containing σ(A) ∪ σ(A0 ). Then ξ(λ) = 0 Let

for

G=

λ∈ / I(A, A0 ). Z

f : R → R | f ∈ L1 and

R

(2.3)

|fˆ(p)|(1 + |p|) dp < ∞ .

Then for every ϕ ∈ G, ϕ(A) − ϕ(A0 ) is a trace class operator and Z ϕ0 (λ)ξ(λ) dλ. Tr(ϕ(A) − ϕ(A0 )) =

(2.4)

R

Note that (2.4) determines the spectral shift function up to an additive constant and this constant is fixed by (2.3). Applied to our setting, we obtain Proposition 2.1. Let H, H0 be two nonnegative self-adjoint operators in H and assume that e−tH − e−tH0 is a trace class operator for t > 0. Then there exists a unique real valued locally integrable function ξ(λ) = ξ(λ; H, H0 ) on R such that for each t > 0, e−tλ ξ(λ) ∈ L1 (R) and the following conditions hold R∞ 1) Tr(e−tH − e−tH0 ) = −t 0 e−tλ ξ(λ) dλ. 2) For every ϕ ∈ G, ϕ(H) − ϕ(H0 ) is a trace class operator and Z ϕ0 (λ)ξ(λ) dλ. Tr(ϕ(H) − ϕ(H0 )) = R

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3) ξ(λ) = 0 for λ < 0. Proof. First, we note that 2) and 3) determine ξ(λ) uniquely. Indeed, if ξ1 and ξ2 are any locally integrable functions on R satisfying 2) and 3), then ξ1 − ξ2 is a constant which must be zero by 3). To prove existence, let ξt (λ) = ξ(λ; e−tH , e−tH0 ) be the spectral / [0, 1]. shift function of (e−tH , e−tH0 ). Then ξt ∈ L1 (R) and by (2.3), ξt (λ) = 0 for λ ∈ Let ϕ ∈ C0∞ (R) and set ( ϕ(− 1t ln µ), if 0 < µ; f (µ) = 0, if µ ≤ 0. Then f ∈ C0∞ (R) with supp f ⊂ (0, ∞). Changing variables by µ = e−tλ and using (2.4), we obtain Z ϕ0 (λ)ξt (e−tλ ) dλ, Tr(ϕ(H) − ϕ(H0 )) = R

which holds for every ϕ ∈ C0∞ (R). Furthermore, we note that ξt (e−tλ ) is locally integrable and ξt (e−tλ ) = 0 for λ < 0. Hence, by uniqueness, ξt (e−tλ ) is independent of t. We denote this function by ξ(λ) = ξ(λ; H, H0 ). It satisfies 2) and 3). Moreover, using (2.2) we get 1) and e−tλ ξ(λ) is integrable on R. The extension to functions in G is straight forward. The function ξ(λ; H, H0 ) is called the spectral shift function of H, H0 . From Proposition 2.1, it follows that, in order to determine the asymptotic behaviour of Tr(e−tH − e−tH0 ) as t → ∞, we have to investigate the behaviour of the spectral shift function ξ(λ) near λ = 0. As a first consequence of Proposition 2.1 we obtain Lemma 2.2. Suppose that σess (H0 ) ⊂ [c, ∞], where c > 0. Then ker H and ker H0 are both finite dimensional and there exists c1 > 0 such that Tr(e−tH − e−tH0 ) = dim ker H − dim ker H0 + O(e−c1 t )

(2.5)

as t → ∞. Proof. First, we observe that by (1.1) and the invariance of the essential spectrum under compact perturbations [RS], it follows that σess (H) = σess (H0 ) ⊂ [c, ∞) , c > 0. Therefore, H0 and H have pure point spectrum in [0, c). In particular, ker H0 and ker H e1 ) be the infimum of the nonzero spectrum of H are finite dimensional. Let λ1 (resp. λ e1 ). By assumption, we have 0 < µ ≤ c and it (resp. H0 ) and set µ = 1/2 min(λ1 , λ follows from Proposition 2.1, 2), that ξ(λ) = dim ker H0 − dim ker H,

0 ≤ λ ≤ µ.

Put e = ξ(λ) + dim ker H − dim ker H0 , ξ(λ)

λ ∈ R.

−tλ e = 0 for λ < µ. Moreover, ξ(λ)e e Then we have ξ(λ) is integrable as a function of λ ∈ R. Hence, we obtain Z ∞ Z ∞ e dλ. e−tλ ξ(λ) = dim ker H − dim ker H0 − t e−tλ ξ(λ) −t 0

µ


317

The integral on the right hand side can be estimated by Z ∞ −tλ/2 e e−tµ/2 |ξ(λ)|e dλ ≤ Ce−tµ/2 µ

for t ≥ 1. Using Proposition 2.1, 1), we obtain (2.5).

Thus, if the essential spectrum of H0 , and consequently also the essential spectrum of H, has a positive lower bound, then as t → ∞, Tr(e−tH −e−tH0 ) behaves in the same way as the trace of the heat operator of an elliptic nonnegative self-adjoint differential operator on a compact manifold. Let h = dim ker H − dim ker H0 . If follows from (2.5) that under the assumption of Lemma 2.2, the relative zeta function can be defined by Z ∞ 1 ts−1 Tr(e−tH − e−tH0 ) − h dt ζ(s; H, H0 ) = 0(s) 0 for Re(s) > −α0 . This is the analogue of (0.2). A special case is that H, H0 > 0. Then (1.2) implies that for Re(s) > −α0 , Z ∞ 1 H −s − H0−s = ts−1 e−tH − e−tH0 dt, 0(s) 0 and H −s − H0−s is a trace class operator for Re(s) > −α0 . Hence, ζ(s; H, H0 ) = Tr H −s − H0−s , Re(s) > −α0 . Now assume that the continuous spectrum of H0 extends down to zero. Then the behaviour of the spectral shift function at λ = 0, in general, can be very complicated. We assume that near λ = 0, ξ(λ) admits an asymptotic expansion in powers of λ, λ ≥ 0. This implies (1.3). More precisely, we have Lemma 2.3. Suppose there exist > 0 and a sequence 0 ≤ γ0 < γ1 < γ2 < · · · with γi → ∞ such that for every N ∈ N, ξ(λ) =

N X

cj λγj + O(λγN +1 )

j=0

uniformly for λ ∈ [0, ]. Then there exists an asymptotic expansion of the form Z ∞ b1 b0 ξ(λ)e−tλ dλ ∼ t−1 γ0 + γ1 + · · · t t 0 as t → ∞.

(2.6)

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W. Müller

Proof. We split the integral as Z

∞

R R∞ + . Since ξ(λ)e−tλ is integrable we obtain 0

ξ(λ)e−tλ dλ ≤ Ce−t/2 ,

t ≥ 1.

Furthermore, note that for γ ≥ 0 and t ≥ 1, we have Z Z ∞ Z γ −tλ γ −tλ λ e dλ = λ e dλ − 0

∞

λγ e−tλ dλ

0

C(γ) + O(e−t/2 ). tγ+1 Replacing ξ(λ) by (2.6), we get the desired result. =

To investigate the asymptotic behaviour of the spectral shift function near λ = 0, one may use the relation between the spectral shift function and the scattering phase [BK, BY]. Let Hac (resp. H0,ac ) be the absolutely continuous subspace for H (resp. H0 ) and let Hac (resp. H0,ac ) be the restriction of H (resp. H0 ) to Hac (resp. H0,ac ). Furthermore, let P0 be the orthogonal projection of H onto H0,ac . It follows from (1.1) and the Birman–Krein invariance principle [BK, K1] that the wave operators W± = W± (H, H0 ) = s − lim eitH e−itH0 P0 t→±∞

(2.7)

exist and are complete. This means that the strong limit (2.7) exists and the operators W± define isometries of H0,ac onto Hac , intertwining H0,ac and Hac . In this context, the scattering operator S is defined by S = W−∗ W+ .

(2.8)

This is a unitary operator on H0,ac that commutes with H0,ac. Let σ0 = σac (H0 ) and let {E0 (λ)}λ∈σ0 be the spectral family of H0,ac . Since S commutes with H0,ac we have Z S(λ) dE0 (λ), S= σ0

where S(λ) = S(λ; H, H0 ) acts in the Hilbert space H(λ). The operator S(λ) is called the on-shell scattering matrix. We observe that S(λ) is a unitary operator in H(λ). Let Iλ be the identity of H(λ). Since S(λ; H, H0 ) = S(e−tλ ; e−tH , e−tH0 ) and exp(−tH) − exp(−tH0 ) is a trace class operator, it follows that T (λ) = S(λ) − Iλ

(2.9)

is a trace class operator (cf. [BK, BY]). The opeator T (λ) is called the scattering amplitude. Thus, the Fredholm determinant det S(λ) of S(λ) exists. Since S(λ) is unitary, we 1 log det S(λ) have | det S(λ)| = 1, λ ∈ σac (H0 ). Hence, the scattering phase s(λ) = 2πi is defined mod Z. It was proved in [BK] that there is a distinguished choice for the scattering phase, namely the spectral shift function ξ(λ). More precisely, the following identity holds det S(λ) = e−2πiξ(λ) ,

a.e. λ ∈ σac (H0 ),

(2.10)

(see [BY, Y]). This equality can be used to derive an expansion of ξ(λ) near λ = 0. For b ,m → D the ramified covering of > 0, let D = {z ∈ C | |z| < } and denote by D m order m, defined by z 7→ z . Then, as a consequence of (2.10), we obtain


319

Proposition 2.4. Let > 0 and suppose that (0, ) ⊂ σac (H0 ). Furthermore, assume that 1) ξ(λ) is continuous on (0, ); 2) There exists m ∈ N such that det S(λ), λ ∈ (0, ), extends to a holomorphic function b ,m . on D Let 0 < 1 < . Then there exists an expansion of the form ξ(λ) =

∞ X

ck λk/m ,

λ ∈ [0, 1 ),

(2.11)

k=0

for some 0 < 1 < 1/m . Proof. According to our assumption, det S(λ) is smooth on (0, ). Since | det S(λ)| = 1, λ ∈ σac (H0 ), it follows that there exists δ ∈ C ∞ ((0, )) such that det S(λ) = exp(−2πiδ(λ)), λ ∈ (0, ). On the other hand, ξ is continuous on (0, ) and therefore, (2.10) implies that there exists k ∈ Z such that ξ(λ) = δ(λ) + k, λ ∈ (0, ). Hence, ξ is smooth on (0, ). Differentiating (2.10) and using that S(λ) is unitary, we get 1 dS(λ) dξ(λ) =− Tr S ∗ (λ) , λ ∈ (0, ). (2.12) dλ 2πi dλ Furthermore, since δ(λ) has a finite limit as λ → 0, we get Z λ dS(u) 1 du Tr S ∗ (u) ξ(λ) = ξ(0+) − 2πi 0 du Z λ1/m m 1 ∗ m dS(u ) = ξ(0+) − du Tr S (u ) 2πi 0 du (2.13) for λ ∈ [0, ). By assumption, Tr S ∗ (z m ) dS(z m )/dz is analytic on the disc |z| < 1 ≤ 1/m . Inserting the Taylor expansion on the right hand side of (2.13) we obtain the desired expansion (2.11). Let

dS(z m ) Tr S (z ) dz ∗

m

=

∞ X

dk z k , |z| < 1 ,

k=0

be the Taylor expansion at z = 0. Then by (2.13), the coefficients of the expansion (2.11) are related to the dk ’s by ck = dk−1 /k, k ≥ 1. Thus, under the assumptions of Proposition 2.4, the asymptotic expansion of Tr(e−tH − e−tH0 ) for t → ∞ determines det S(λ) near λ = 0, up to an additive constant. This means that the coefficients bk in (1.3) and therefore, the residues of the poles of ζ(s; H, H0 ) at s = k/m, k ∈ N+ , are nonlocal. If we insert ξ(λ) in the trace formula of Proposition 2.1 and integrate by parts, we obtain

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W. Müller

Tr(e−tH − e−tH0 )

Z dS(λ) 1 −tλ ∗ dλ + O(e−t/2 ) e Tr S (λ) = −ξ(0+) + 2πi 0 dλ Z 1 m 1 −tλm ∗ m dS(λ ) = −ξ(0+) + dλ e Tr S (λ ) 2πi 0 dλ + O(e−t/2 ) (2.14)

as t → ∞. This formula shows more explicitly how the scattering matrix determines the asymptotic behaviour of Tr(e−tH − e−tH0 ) as t → ∞. Furthermore, in special cases, ξ(0+) can be expressed in terms of spectral data. It follows from (2.14) that Tr(e−tH − e−tH0 ) = −ξ(0+) + O(t−1/m ) as t → ∞. Inserting this equality in (1.6), we obtain the analytic extension of ζ2 (s; H, H0 ) to the half-plane Re(s) < 1/m. Now, in order to define the relative determinant of H, H0 by zeta function regularization, it is sufficient to know that ζ(s; H, H0 ) is analytic in some half-plane Re(s) < ε, ε > 0. Therefore, we may replace (1.3) by the following weaker assumption: There exist b0 ∈ C and ρ > 0 such that Tr e−tH − e−tH0 = b0 + O(t−ρ ) (2.15) as t → ∞. As above, this condition can be restated in terms of the behaviour of the spectral shift function near λ = 0. In fact, by Proposition 2.1, 1), (2.15) is equivalent to ξ(λ) = −b0 + O(λρ )

(2.16)

as λ → 0+. For this condition to hold, it is not necessary that the scattering matrix extends to an analytic function on some covering of a neighborhood of the origin. Much less stringent assumptions are sufficient. Now, suppose that (2.16) holds. Then ζ2 (s; H, H0 ) has a meromorphic extension to the half-plane Re(s) < ρ which is given by b0 0(s + 1) Z ∞ 1 ts−1 Tr(e−tH − e−tH0 ) − b0 dt. + 0(s) 1

ζ2 (s; H, H0 ) = −

(2.17)

This formula implies that ζ2 (s; H, H0 ) is regular at s = 0. Analogous results can be obtained for the small time asymptotic expansion (1.2). If we impose smoothness assumptions at λ = ∞ similar to those at λ = 0, then it follows that the asymptotic expansion (1.2) corresponds to an asymptotic expansion of the scattering phase shift d/dλ(log det S(λ)) as λ → ∞. Next, we briefly discuss relative η-functions. Details will appear elsewhere. Let D, D0 be self-adjoint operators in H, satisfying (1.11) and (1.12). Moreover, suppose that the spectra of D and D0 have a common gap, i.e. there exists an interval [a, b], a < b, such that (σ(D) ∪ σ(D0 )) ∩ [a, b] = ∅.

(2.18)


321

Then, by a modification of the proof of Proposition 2.1, one can introduce a spectral shift function ξ(λ) = ξ(λ; D, D0 ). This function has the following properties: 1) ξ ∈ L1loc (R) and ξ(λ) = 0 for λ ∈ [a, b]; 2) for all ϕ ∈ C0∞ (R), ϕ(D) − ϕ(D0 ) is a trace class operator and Z Tr (ϕ(D) − ϕ(D0 )) = ϕ0 (λ)ξ(λ) dλ;

(2.19)

R

3) Tr De

−tD 2

− D0 e

−tD02

Z

=

R

2 d (λe−tλ )ξ(λ) dλ. dλ

(2.20)

We note that 1) and 2) determine ξ uniquely. If, however, (2.18) is not satisfied, i.e. if σ(D) = σ(D0 ) = R, then the construction of the spectral shift function is slightly more complicated. In particular, ξ cannot be normalized by fixing its values outside the spectrum and therefore, ξ is determined by (2.19) constant. only up2 to an additive −tD −tD02 as t → ∞ can − D0 e By (2.20), the study of the behaviour of Tr De be reduced to the investigation of the spectral shift function near λ = 0. To handle this problem, one can use, in the same way as above, the connection between the spectral 2 2 shift function and the scattering phase. Indeed, since De−tD − D0 e−tD0 is a trace class operator for t > 0, it follows from the Birman–Kato invariance principle that the wave operators W± (D, D0 ) exist and are complete [K2]. Hence, the scattering matrix S(λ; D, D0 ) also exists and it follows from [BK] that det S(λ; D, D0 ) = e−2πiξ(λ;D,D0 ) ,

λ ∈ σac (D0 ).

As above, this equality can be used to get an expansion of ξ(λ; D, D0 ) near λ = 0, provided that the scattering matrix extends to a holomorphic function on some finite covering of a neighborhood of the origin. But, in order to define the η-invariant, we only need that there exists > 0 such that 2 2 (2.21) Tr De−tD − D0 e−tD0 = O(t−(1/2+) ) as t → ∞. Then (1.15) is absolutely convergent in the half-plane Re(s) < 2. Suppose, for example, that D and D0 are invertible. Then there exists > 0 such that (2.18) holds with respect to the interval [−, ]. Thus, the spectral shift function vanishes on [−, ] and by (2.20), we obtain 2 2 2 (2.22) Tr De−tD − D0 e−tD0 = O(t− /2 ). Hence, in this case, the relative η-function can be defined by a single Mellin transform: η(s; D, D0 ) Z ∞ 2 2 1 t(s−1)/2 Tr De−tD − D0 e−tD0 dt, = 0 (s + 1)/2 0 Re(s) > −κ0 .

(2.23)

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3. Relative Determinants Suppose that (H, H0 ) are nonnegative self-adjoint operators which satisfy (1.1), (1.2) and (2.15). Then the relative zeta function ζ(s; H, H0 ) is well defined as a meromorphic function in the half-plane Re(s) < ρ, ρ > 0, and s = 0 is not a pole of ζ(s; H, H0 ). Therefore, generalizing (0.3), we may define the regularized relative determinant of (H, H0 ) by d . (3.1) det(H, H0 ) = exp − ζ(s; H, H0 ) ds s=0 To justify the name “relative determinant”, consider two self-adjoint operators H0 and H1 such that exp(−tHi ) is trace class for t > 0 and assume that (1.8) holds for both operators. Then the absolute determinants det Hi are defined and it follows from (1.10) that det(H1 , H0 ) =

det H1 . det H0

(3.2)

Using (2.17), the logarithm of the relative determinant can be written as d ζ1 (s; H, H0 )|s=0 − b0 00 (1) ds Z ∞ t−1 Tr e−tH − e−tH0 − b0 dt. +

− log det(H, H0 ) =

1

(3.3) Furthermore, we may use the asymptotic expansion (1.2) to replace the derivative of ζ1 (s; H, H0 ) at s = 0 by a more explicit formula. Namely, let N ∈ N be such that the exponent αN in the expansion (1.2) satisfies αN > 0. Then it follows that − log det(H, H0 ) =

j=0 k=0



Z

1

+

k(j) N X X

ajk

k! − aj0 00 (1) αjk

t−1 Tr(e−tH − e−tH0 ) −

0

− b0 00 (1) +

k(j) N X X

 ajk tαj log t dt k

j=0 k=0

Z

∞

t−1 Tr(e−tH − e−tH0 ) − b0 dt.

1

(3.4) We shall now summmarize some of the elementary properties of the determinant. As an immediate consequence of the definition we get Lemma 3.1. Suppose that H0 , H1 and H2 are self-adjoint operators in H such that both (H1 , H0 ) and (H2 , H1 ) satisfy (1.1), (1.2) and (2.15). Then the following holds: 1) det(H0 , H1 ) = det(H1 , H0 )−1 . 2) The operators (H2 , H0 ) also satisfy (1.1), (1.2) and (2.15) and det(H2 , H0 ) = det(H2 , H1 ) · det(H1 , H0 ).

(3.5)


323

Next, we consider some special cases. Let H be a nonnegative self-adjoint operator in H and suppose that H has eigenvalues 0 < λ1 ≤ λ2 ≤ · · · ≤ λm . Let Pm denote the orthogonal projection of H onto the subspace of H spanned by the eigenvectors that correspond to λ1 , . . . , λm . Put H0 = (I − Pm )H. Then exp(−tH) is a finite-rank perturbation of exp(−tH0 ). In particular, (H, H0 ) satisfies (1.1)–(1.3) and we have det(H, H0 ) =

m Y

λi .

(3.6)

i=1

More generally, if the pair (H, H0 ) satisfies (1.1)– (1.3), then the pair ((I − Pm )H, H0 ) satisfies the same conditions and by (3.5), we get det(H, H0 ) =

m Y

λi · det((I − Pm )H, H0 ).

(3.7)

i=1

This means that we may split off any finite number of eigenvalues from the determinant. Next let H = Hp ⊕ Hc be the orthogonal decomposition into the subspaces that correspond to the point spectrum and the continuous spectrum of H, respectively. Let Hp and Hc denote the restriction of H to Hp and Hc , respectively. Suppose that for each t > 0, exp(−tHp ) is a trace class operator and as t → 0, Tr(exp(−tHp )) admits an asymptotic expansion of the form (1.2). Then det Hp can be defined by zeta function regularization and using (3.5), one has det(H, H0 ) = det Hp · det(Hc , H0 ).

(3.8)

Now let a > 0. Then (H + a, H0 + a) also satisfies (1.1)–(1.3) and the relative zeta function is given by Z ∞ 1 ts−1 e−ta Tr(e−tH − e−tH0 ) dt ζ(s; H + a, H0 + a) = 0(s) 0 for Re(s) > −α0 . The right hand side also makes sense if we replace a by any complex number z with Re(z) > 0. We denote the corresponding analytic function by ζ(s, z; H, H0 ). Using (1.2) it follows that for every z ∈ C with Re(z) > 0, ζ(s, z; H, H0 ), regarded as function of s, admits a meromorphic continuation to C which is holomorphic at s = 0. Thus, generalizing (3.1) we set ∂ . (3.9) det(H + z, H0 + z) = exp − ζ(s, z; H, H0 ) ∂s s=0 Again, if the operators H0 and H1 are as in (3.2), then det(H1 + z, H0 + z) =

det(H1 + z) . det(H0 + z)

(3.10)

We note that if H is a nonnegative self-adjoint operator such that exp(−tH) is trace class for t > 0 and (1.8) holds, then H has pure discrete spectrum consisting of a sequence of eigenvalues 0 ≤ λ0 < λ1 < λ2 < · · · → ∞ of finite multiplicities and det(H + z) is an entire function of z whose set of zeros equals {−λi }i∈N and the multiplicities coincide. In particular, this shows that (3.10) is the quotient of two entire functions. We don’t know yet, if this holds in general. Here we prove a weaker result.

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Proposition 3.2. Suppose that (H, H0 ) satisfies (1.2). Then the relative determinant det(H + z, H0 + z) is a holomorphic function of z ∈ C − (−∞, 0]. Proof. Let ξ(λ) be the spectral shift function associated to (H, H0 ). Let c > 0. By Proposition 2.1, 1), we have Z 2c Z ∞ e−tλ ξ(λ) dλ − t e−tλ ξ(λ) dλ. Tr(e−tH − e−tH0 ) = −t 0 −tλ

2c

Since e ξ(λ) ∈ L (R) for t > 0, the second integral is O(e−tc ) as t → ∞. Furthermore, for every N ∈ N, we have 1

Z

2c

e−tλ ξ(λ) dλ =

0

N X

ck tk + O(tN +1 )

k=0

R∞ as t → 0. Together with (1.2) it follows that as t → 0, 2c e−tλ ξ(λ) dλ has a complete asymptotic expansion similar to (1.2). This implies that the double integral Z ∞ Z ∞ 1 s −tz F (s, z) = t e e−tλ ξ(λ) dλ dt 0(s) 0 2c is absolutely convergent in the half-planes Re(s) > −α0 and Re(z) > −c. Moreover, as a function of s, it admits a meromorphic continuation to C which is holomorphic at s = 0. Next consider the first integral. Note that ξ is absolutely integrable on [0, 2c]. Hence for Re(z) > 0, we get Z ∞ Z 2c 1 ts e−tz e−tλ ξ(λ) dλ dt − 0(s) 0 0 Z 2c Z ∞ 1 ξ(λ) ts e−t(z+λ) dt dλ =− 0(s) 0 0 Z 2c −(s+1) (z + λ) ξ(λ) dλ. = −s 0

Thus, for Re(z) > 0, we have det(H + z, H0 + z) (Z 2c

(z + λ)

= exp 0

−1

) ∂F (0, z) . ξ(λ) dλ + ∂s

(3.11)

The integral has an obvious extension to an analytic function of z ∈ C − (−∞, 0]. Moreover, ∂F ∂s (0, z) is holomorphic in Re(z) > −c. Since c is arbitrary, (3.11) gives the analytic continuation to C − (−∞, 0]. We now shall study the variation of the relative determinant. So, we consider a differentiable 1-parameter family Hu , u ∈ (−, ), of nonnegative self-adjoint operators in H and a nonnegative self-adjoint operator H in H such that for each u ∈ (−, ), the pair (Hu , H) satisfies (1.1), (1.2) and (2.15). By (3.5) we have det(Hu , H) = det(Hu , H0 ) · det(H0 , H),


325

d so that we may take H = H0 . Let H˙ u = du Hu . By Duhamel’s principle, exp(−tHu ) is differentiable in u and Z t d −tHu e e−vHu H˙ u e−(t−v)Hu dv. (3.12) = du 0

Now assume that H˙ u e−tHu is a trace class operator for t > 0 with trace norm uniformly d bounded on compact subsets of (0, ∞). Then it follows from (3.12) that du exp(−tHu ) is also a trace class operator for t > 0 and d Tr e−tHu − e−tH0 = −t Tr(H˙ u e−tHu ). du

(3.13)

We shall now make the following two assumptions about the family Hu : i) Hu is invertible for each u ∈ (−, ), ii) As t → 0, there exists an asymptotic expansion of the form Tr(H˙ u Hu−1 e−tHu ) ∼

k(j) ∞ X X

Cjk (u)tγj logk t

j=0 k=0

+ C1 (u) + C2 (u) log t

(3.14)

and the exponents γj are such that −∞ < γ1 < γ2 < · · · , γj 6= 0, j ∈ N, and γj → ∞. Note that by i), there exists c > 0 such that Tr(H˙ u Hu−1 e−tHu ) = O(e−ct )

(3.15)

as t → ∞. Using i) and (3.13) - (3.15), it follows that for Re(s) 0, Z ∞ 1 d ∂ ζ(s; Hu , H0 ) = ts Tr(H˙ u Hu−1 e−tHu ) dt du 0(s) 0 ∂t Z ∞ s ts−1 Tr(H˙ u Hu−1 e−tHu ) dt. =− 0(s) 0 (3.16) By (3.14) and (3.15), Z

∞

ts−1 Tr(H˙ u Hu−1 e−tHu ) dt

0

has an analytic continuation to a meromorphic function of s ∈ C. The location of the poles is determined by the asymptotic expansion (3.14). In particular, the pole at s = 0 has order ≤ 2 and the Laurent expansion is given by −

C2 (u) C1 (u) + ··· , + s2 s

where C1 (u) and C2 (u) are the corresponding coefficients in (3.14). Using (3.16), we obtain

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W. Müller

Proposition 3.3. Suppose that the family Hu , u ∈ (−, ), satisfies assumptions i) and ii). Then det(Hu , H0 ) is differentiable and d log det(Hu , H0 ) = C1 (u) + 00 (1)C2 (u), du

(3.17)

where C1 (u) denotes the constant term and C2 (u) the coefficient of log t in the asymptotic expansion (3.14). Corollary 3.4. Let Hu , u ∈ [0, 1], be a differentiable family of self-adjoint operators in H such that for all u ∈ [0, 1], (1.1) and (1.2) holds for (Hu , H0 ) and Hu satisfies conditions i) and ii). Then (Z ) 1

det(H1 , H0 ) = exp

(C1 (u) + 00 (1)C2 (u)) du .

(3.18)

0

Remark. A similar formula was proved in [GMS] for a certain class of elliptic invertible pseudodifferential operators on a compact manifold. For each of these operators the determinant can be defined via zeta function regularization and the relative determinant of any two such operators is the fraction of the determinants of the individual operators. The assumption i) is, of course, rather restrictive and excludes many cases that are of interest for applications. However, using (3.4), differentiability can be established under more general assumptions. For example, we have Proposition 3.5. Let Hu , u ∈ (−ε, ε), be a differentiable family of self-adjoint operators which satisfy (1.1), (1.2), (2.15) and in addition, the following conditions hold: a) The constant b0 and the exponent ρ in (2.15) are independent of u, and the coefficients of the expansion (1.2) are differentiable in u. b) For t > 0, H˙ u e−tHu is a trace class operator and there exist C > 0 and δ > 0 such that Tr H˙ u e−tHu ≤ Ct−(1+δ) for t ≥ 1 and u ∈ (−ε, ε). c) As t → 0, there exists an asymptotic expansion of the form k(j) ∞ X X Tr H˙ u e−tHu ∼ Cjk (u)tνj logk t, j=0 k=0

where the exponents νj are such that −∞ < ν1 < ν2 < · · · , νj 6= 1, j ∈ N, and νj → ∞. R1 ˙ −tHu dt is absolutely convergent in the half-plane Then the integral 0 ts Tr He Re(s) > −ν1 − 1 and has a meromorphic extension to the half-plane Re(s) < δ which is holomorphic at s = 0. Moreover, det(Hu , H0 ) is a differentiable function of u and d log det(Hu , H0 ) = du

Z

1

˙ −tHu dt t Tr He s

0

s=0

Z

∞

+ 1

Tr H˙ u e−tHu dt.


327

Proof. We use formula (3.3). Since by b), Tr(H˙ u e−tHu ) is absolutely integrable on [1, ∞), it follows from a) and (3.13) that d du

Z

∞

t

−1

Tr(e

−tHu

−e

−tH0

Z

∞

) − b0 dt = −

1

Tr H˙ u e−tHu dt.

1

R1 Furthermore, by c), 0 ts Tr H˙ u e−tHu dt has a meromorphic extension to C which is holomorphic at s = 0, and 1 ∂ ζ1 (s; D, D0 ) = − ∂u 0(s)

Z

1

ts Tr H˙ u e−tHu dt.

0

Taking derivatives at s = 0 gives the desired result.

To conclude this section, we consider the case of two self-adjoint operators D, D0 which are not necessarily bounded from below. A typical example are Dirac type operators on a complete manifold. Suppose that (1.11) and (1.12) hold for (D, D0 ). For simplicity, we also assume that D and D0 are invertible, i.e. 0 is not in the spectrum of D and D0 . Then the relative η-function η(s; D, D0 ) is defined by (2.23) and has a meromorphic extension to C. If −1/2 does not occur among the exponents of the asymptotic expansion (1.12), then η(s; D, D0 ) is regular at s = 0 and therefore, we may define the relative η-invariant as η(0; D, D0 ). For example, if D and D0 are Dirac type operators, then the asymptotic expansion (1.12) contains no negative exponents so that the relative η-invariant is given by 1 η(0; D, D0 ) = √ π

Z

∞

2 2 t−1/2 Tr De−tD − D0 e−tD0 dt.

0

Suppose also that D2 , D02 satisfy (1.1) and (1.2). Since D and D0 are invertible, Lemma 2.2 implies that (1.3) holds and the relative zeta function is given by ζ(s; D2 , D02 ) = Tr (D2 )−s − (D02 )−s

= Tr |D|−2s − |D0 |−2s ,

Re(s) > −κ0 .

Hence, the relative zeta function ζ(s; |D|, |D0 |) also exists as a meromorphic function on C, and we have ζ(s; D2 , D02 ) = ζ(2s; |D|, |D0 |). This formula implies that ζ(s; |D|, |D0 |) is regular at s = 0 and therefore, det(|D|, |D0 |) can be defined by (3.1). Generalizing (0.5), we introduce the relative determinant for D, D0 by det(D, D0 ) = det(|D|, |D0 |) πi (η(0; D, D0 ) − ζ(0; |D|, |D0 |) . · exp 2

(3.19)

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4. Examples In this section we discuss a number of typical cases of operators (H, H0 ) which satisfy the assumptions (1.1) - (1.3). Such operators arise very naturally both in geometry and physics. In particular, we will show how the general trace formula of Proposition 2.1 specializes in all these cases to a rather explicit trace formula involving the eigenvalues of the operator H and the scattering phase shift. 4.1. The Schrödinger operator in Rn . A typical example is the Schrödinger operator 1 + V in Rn , where V ∈ C0∞ (Rn ) and 1 is the negative of the usual Laplacian of Rn . I shall limit myself to this simple case, where V is both smooth and has compact support, although most of the results have generalizations to potentials with less regularity assumptions and the support property replaced by growth conditions. For details we refer the reader to [CV, Gu1, Gu2]. Regarded as an operator in L2 (Rn ) with domain C0∞ (Rn ), 1 + V is essentially self-adjoint. Let H be the unique self-adjoint extension of 1 + V and let H0 = 1. The spectrum of H consists of a finite number of eigenvalues λ1 < λ2 ≤ λ3 ≤ · · · ≤ λN < λN +1 = · · · = λN 0 = 0, where each eigenvalue is repeated according to its multiplicity, and an absolutely continuous spectrum which has multiplicity 2, if n = 1, and infinite multiplicity, if n > 1. Using Duhamel’s formula e−tH − e−tH0 =

Z

t

e−sH V e−(t−s)H0 ds,

0

it is easy to see that e−tH − e−tH0 is a trace class operator for t > 0 (see also [CV, Gu1, Gu2]). Hence, the wave operators W± (H, H0 ) exist and are complete, and the scattering operator S is defined by (2.8). Let L2 (Rn ) ' L2 R+ , L2 (S n−1 ); λn−1 dλ be the spectral resolution of H0 defined by the Fourier transform Z −n/2 b f (λ)(ω) = (2π) e−iλ f (x) dx. Rn

Then the “on-shell scattering matrix” S(λ) = S(λ; H, H0 ) is given by c (λ) = S(λ2 )fb(λ), λ > 0. Sf Furthermore, S(λ) is a unitary operator in L2 (S n−1 ) which is of the form S(λ) = Id + T (λ), where T (λ) is an integral operator with a smooth kernel T (λ; ω, ω 0 ). Since V ∈ C0∞ (Rn ), it follows that for odd n, S(λ2 ) has an analytic continuation to a meromorphic operator valued function of λ ∈ C. In the even-dimensional case a similar result is valid, except that S(λ) extends to a meromorphic function on the logarithmic covering of C, i.e. as a function of the variable log λ. In particular, this implies Proposition 4.1. The Fredholm determinant det S(λ) of S(λ) is real analytic on (0, ∞).


329

Furthermore, since e−tH − e−tH0 , t > 0, is a trace class operator, the spectral shift function ξ(λ; H, H0 ) exists. In the present case, it can also be defined using the resolvents. Let R0 (λ) and R(λ) denote the resolvent of H0 and H, respectively. Let µ ∈ R − σ(H) and let k > n/2. Then R(µ)k − R0 (µ)k is of the trace class. Hence, there exists the spectral shift function ξµ (λ) = ξ(λ; R(µ)k , R0 (µ)k ) defined by (2.1) and, by the uniqueness of the spectral shift function, we get ξ(λ; H, H0 ) = ξµ (λ − µ)−k ; R(µ)k , R0 (µ)k . Using Proposition 1 in [Gu1] (or Theorem III.2 in [Gu2]), it follows that ξ(λ) is locally constant on the complement of σ(H) and continuous on (0, ∞). If n 6= 2, 4, then the discontinuity of ξ(λ) at 0 is given by ξ(0−) − ξ(0+) = m(0, H) + ,

(4.1)

where m(0, H) is the multiplicity of the eigenvalue 0 of H and = 0, if there is no resonance at 0 and = 1/2 otherwise. If n = 2, 4, the description is more difficult. Let det S(0) = lim det S(λ). λ→0+

We note that ξ(0−) ∈ Z. Hence, using (2.10), we obtain det S(0) = (−1)2 . This implies that we may pick the branch of log det S(λ) such that ξ(0−) − ξ(0+) = m(0, H) +

1 log det S(0). 2πi

Then by Proposition 2.1 we get the following trace formula for n 6= 2, 4 : 0

Tr(e

−tH

−e

−tH0

)=

N X i=1

+

1 2πi

e−tλi + Z

∞ 0

log det S(0) 2πi

e−tλ

d log det S(λ) dλ, dλ

(4.2)

[Gu1, Gu2]. If n = 2, 4, (2πi)−1 log det S(0) has to be replaced by a different constant. Let K(t, x, y) be the kernel of exp(−tH). Using the heat equation method [Gi], it follows that as t → 0, there exists an asymptotic expansion Tr(e−tH − e−tH0 ) ∼ (4πt)−n/2

∞ X

aj,n tj .

(4.3)

j=1

The first coefficients can be determined explicitly: R R a1,n = − Rn V (x) dx, a2,n = 21 Rn V 2 (x) dx, R a3,n = − 16 Rn V 3 (x) + 21 ||DV (x)||2 dx. The asymptotic expansion (4.3) corresponds to an asymptotic expansion of the scattering phase

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W. Müller

s(λ) =

1 log det S(λ) 2πi

for high energy. Namely, for λ → ∞ one has ∞

X ds (λ) ∼ λn/2−1 αj (V )λ−j , dλ

(4.4)

j=1

and the coefficients αj (V ) are related to the coefficients in (4.3). We observe that the asymptotic expansion (4.4) holds for more general potentials (cf. [R]). It remains to investigate the large time behaviour of Tr(e−tH − e−tH0 ). For this purpose we have to assume that n is odd. Then det S(λ2 ) is real analytic on (−c, ∞) for some c > 0 and therefore it has a power series expansion at λ = 0 which implies that there exists > 0 such that the series X ∞ 1 dS(λ) ∗ ds (λ) = Tr S (λ) = bj λj/2 (4.5) dλ 2πi dλ j=0

is convergent for 0 ≤ λ ≤ . Now let P1 be the orthogonal projection of L2 (Rn ) onto the orthogonal complement of the subspace spanned by the eigenfunctions of H that correspond to the negative eigenvalues. Let H1 be the restriction of H to RanP1 . Then H1 ≥ 0 and H −H1 has finite rank. Let n be odd. Using (4.2), (4.3) and (4.5), it follows that (H1 , H0 ) satisfies (1.1)– (1.3). Hence, the relative zeta function ζ(s; H1 , H0 ) is well defined. Poles may occur at the points s = k/2, k ∈ Z. All poles are simple and the residues can be determined from the expansions (4.3) and (4.5). The relative determinant det(H1 , H0 ) is then given by (3.1). Using this determinant, we define the relative determinant det(1 + V, 1) by det(1 + V, 1) =

N Y

λi · det(H1 , H0 ).

(4.6)

i=1

Remark. We recall from Sect. 2, that in order to define the determinant, the full expansion (4.5) is not needed and may be replaced by weaker assumptions like (2.16). For example, there exist asymptotic expansions of the scattering matrix S(λ) as λ → 0 under more general assumptions about the potential (see [J1, JK]). If W ∈ C0∞ (Rn ) is another potential, we may define the relative determinant det(1+ V + W, 1 + V ) in the same way, and by Lemma 3.1, we have det(1 + V + W, 1 + V ) = det(1 + V + W, 1) · det(1 + V, 1)−1 .

(4.7)

If n is even, there is no expansion of the scattering phase near 0 in powers of λ. This problem can be eliminated if we shift the spectra of H and H0 by a sufficiently large positive constant m. Let n be arbitrary. Pick m > 0 such that H + m > 0. Then there exists c > 0 such that as t → ∞, one has Tr e−t(H+m) − e−t(H0 +m) = (e−tc ). If t → 0, this trace has an asymptotic expansion similar to (4.3). Hence, the relative zeta function can be defined in the half-plane Re(s) > n/2 in the usual way by


1 ζ(s; H + m, H0 + m) = 0(s)

Z

∞

331

ts−1 Tr e−t(H+m) − e−t(H0 +m dt

0

= Tr (H + m)−s − (H0 + m)−s .

(4.8)

This zeta function was first introduced and studied by Guillopé [Gu1]. Using the trace formula (2.4), it follows that for Re(s) > n/2, 0

ζ(s; H, H0 ) =

N X

(λj + m)−s + αm−s

j=1

1 + 2πi

Z

∞

(λ + m)−s

0

d log detS(λ) dλ, dz

(4.9)

where α is determined by the singularity of ξ(λ) at λ = 0. If n 6= 2, 4, then α = (2πi)−1 log detS(0). The analytic continuation of ζ(s; H + m, H0 + m) is holomorphic at s = 0 and we can define the relative determinant by (3.1). Again, we denote this relative determinant by det(1 + V + m, 1 + m). We now consider the variation of det(1 + V + m, 1 + m) with respect to V which can be computed using (3.17). First we note that Z ∞ e−s(H+m) ds (H + m)−1 e−t(H+m) = t

= (H + m)−1 e−(H+m) +

Z

1

e−s(H+m) ds.

t

Let W ∈ C0∞ (Rn ). Using this formula together with the local heat expansion, it follows that as t → 0, there is an asymptotic expansion of the form ∞ X dj (W ) t−n/2+j Tr W (H + m)−1 e−t(H+m) ∼ j=1 j6=n/2

+c1 (W ) + c2 (W ) log t, where 1 dj (W ) = n/2 − j

Z W (x)aj−1,n (x) dx,

Rn

c1 (W ) = Tr W (H + m)−1 e−(H+m) Z p X (j + 1 − n/2)−1 +

Rn

j=0 j6=n/2−1

Z

Z W (x)

+ Rn

0

W (x)aj,n (x) dx

 1

K(s, x, x) −

p X j=0

 aj,n (x)s−n/2+j  ds dx,

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W. Müller

Z c2 (W ) = −

Rn

W (x)an/2−1,n (x) dx.

Thus, by (3.17) we have δ log det(1 + V + m, 1 + m) = c1 (δV ) + 00 (1)c2 (δV ). δV Now assume that n is odd and consider det(1+V, 1). Since the continuous spectrum of 1 + V contains 0, det(1 + V, 1) may not be differentiable in all directions. But we can use Proposition 3.5 to determine under what assumptions about the potential and the variation, det(1u , 10 ) will be differentiable. Suppose, for example, that V ≥ 0, V 6= 0, and let W ∈ C0∞ (Rn ) be such that V + uW ≥ 0 for |u| ≤ . Let 1u = 1 + V + uW and let Ku (x, y, t) be the kernel of e−t1u . Then 1u has no L2 -eigenfunctions for |u| ≤ and by Duhamel’s principle, it follows that Ku (x, y, t) − (4πt)−n/2 e−kx−yk /4t Z t Z = −(4π)−n/2 s−n/2 Ku (x, z, t − s) 2

0

Rn

× (V + uW )(z)e−kz−yk

2

/4s

dz ds.

(4.10)

Hence, if |u| ≤ , we get Ku (x, y, t) ≤ (4πt)−n/2 e−kx−yk

2

/4t

.

(4.11)

Inserting this estimate into (4.10), we obtain Tr e−t1u − e−t1 = O(t−n/2+1 )

(4.12)

as t → ∞ and |u| ≤ . Using (4.11) and (4.12), it is easy to verify that the assumptions of Proposition 3.5 are satisfied. Thus the relative determinant det(1 + V + uW, 1) is differentiable for |u| ≤ . 4.2. Perturbations of the Euclidean Laplacian. Let g be a complete Riemannian metric on Rn and let gij (x), 1 ≤ i, j ≤ n, be the components of g(x) with respect to the standard basis dxi of Tx∗ Rn . Let G = det(g) and let g kl (x) denote the components of (gij (x))−1 . Furthermore, let W be a complex vector space of dimension m with Hermitian inner product h·, ·i. Let A = (A1 , . . . , An ) be a smooth Yang–Mills potential, i.e. the Aj are C ∞ functions on Rn with values in the Hermitian operators on W . Then we consider the following differential operator: 1 = 1g,A = −G−1/4

n X √ jk ∂ ∂ + iAj Gg + iAk G−1/4 , ∂xj ∂xk

(4.13)

j,k=1

acting in the Hilbert space H = L2 (Rn ) ⊗ W. Note that this class of operators contains, for example, all spinor Laplacians on Rn , i.e. the squares of twisted Dirac operators. For p > 0, we make the following assumption: For every α ∈ Nn , there exists Cα > 0 such that


X ij

|∂ α (gij (x) − δij )| +

X

|∂ α Ai (x)| ≤ Cα (1 + ||x||2 )−p−|α| .

333

(4.14)

i

In [CKS] it was proved that 1g,A , regarded as operator in H with domain C0∞ (Rn ) ⊗ W, is essentially self-adjoint. Let H = Hg,A be the corresponding self-adjoint extension. Furthermore let 10 = −

n X ∂2 ⊗ IdW , ∂x2i i=1

and let H0 be the self-adjoint extension of 10 . H and H0 are nonnegative self-adjoint operators in H. The following results, which we summarize as a proposition, were established in [CKS]. Proposition 4.2. Suppose that p > 1. Then a) The wave operators W± (H, H0 ) exist and are complete. b) H has no point spectrum. c) The singular continuous spectrum of H is empty. If p > n, it follows from [Ro, Br] that for t > 0, e−tH − e−tH0 is a trace class operator and as t → 0, there is an asymptotic expansion of the form X aj t−n/2+j . (4.15) Tr(e−tH − e−tH0 ) ∼ j≥0

Thus for m > 0, the pair of operators (H + m, H0 + m) satisfies (1.1)– (1.3) and the relative zeta function ζ(s; H + m, H0 + m) can be defined in Re(s) > n/2 by the usual formula (4.16) ζ(s; H + m, H0 + m) = Tr (H + m)−s − (H0 + m)−s . Remark. In a more general context, such zeta functions have been studied by Bruneau [Br] in his thesis. Bruneau considers a certain class of elliptic operators in Rn which, for example, includes the Laplacians for asymptotically equal metrics and also the operators of the type (4.13). He then studies relative zeta functions like (4.16). In order to define the relative zeta function for (H, H0 ) itself, we need some information about the behaviour of the scattering matrix at λ = 0. This requires more stringent assumptions. For example, we may assume that gij (x) = δij , i, j = 1, . . . , n, Aj (x) = 0, j = 1, . . . , n for k x k≥ R. Then 1g,A = 10 outside a compact set. If n is odd, it follows from Theorem 1.1 of [SZ] that (H − λ2 )−1 , Im(λ) > 0 has a continuation to a meromorphic function of λ ∈ C with values in the operators from L2comp (Rn ) ⊗ W to L2loc (Rn ) ⊗ W . Let S(λ) = S(λ; H, H0 ) be the scattering matrix. Then, using the analytic continuation of the resolvent, it follows that S(λ2 ) extends to a meromorphic function of λ ∈ C. We now can proceed in essentially the same way as in the case of the Schrödinger operator in odd dimensions. In particular, the relative determinant det(1g,A , 10 ) exists and we can study it as a function of the metric g and the gauge field A. In the same way, one can study Dirac operators on Rn coupled to a gauge field. For example, consider the Dirac operator on R3 :

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W. Müller

DA = c

3 X j=1

γj

∂ + iAj (x) + mc2 γ4 + V (x), ∂xj

x ∈ R3 ,

where c > 0 is the speed of light, γ1 , ...γ4 are the Dirac matrices, V is the electric potential, m is the mass and A = (A1 , A2 , A3 ) is the magnetic field. We assume that the potential V and the magnetic field A are C ∞ and satisfy the following assumption: For each α ∈ N3 , there exists Cα > 0 such that |∂ α A(x)| + |∂ α V (x)| ≤ Cα (1 + |x|2 )−p−|α| ,

x ∈ R3 ,

where p > 0 is independent of α. 4 Then DA is essentially self-adjoint in L2 (R3 ) . Let D0 be the free Dirac operator, i.e. the Dirac operator with magnetic field A = 0. If p > 1, then the wave operators W± (DA , D0 ) exist and are complete. The singularly continuous spectrum of DA is empty and the essential spectrum of DA equals (−∞, −mc2 ]∪[mc2 , ∞) (cf. [Th]). In particular, if m > 0, then the continuous spectrum of DA has a gap at zero. It is proved in [Br, Br1] Due to the gap of the continuous that the operators (DA , D0 ) satisfy (1.11) and (1.12). 2 ) − D0 exp(−tD02 ) is exponentially decreasing for t → spectrum, Tr DA exp(−tDA ∞. Thus we can define the relative eta invariant η(DA , D0 ). Moreover, det(|DA |, |D0 |) also exists and therefore, det(DA , D0 ) exists too and is given by (3.19). 4.3. Manifolds with cylindrical ends. Let M be a compact n-dimensional C ∞ manifold with smooth boundary Y . Let X be the non-compact manifold obtained from M by gluing the bottom of the half-cylinder R+ × Y to the boundary of M , i.e. X = M ∪Y (R+ × Y ). We equip X with a Riemannian metric g which is a product on R+ × Y, i.e. on R+ × Y , g takes the form g = du2 + h, where h is the pull-back of a metric on Y. Then X equipped with g is called a “manifold with a cylindrical end”. In this way, we obtain a natural class of complete Riemannian metrics on X. We also may consider larger classes of metrics which are obtained by perturbations of cylindrical end metrics. For example, we may require that the perturbation together with all its derivatives is exponentially decreasing if u → ∞, where u is the radial variable on the cylinder. If we change coordinates by v = e−u , then we get a metric on the compact manifold M. Let x be a defining function of ∂M . Then near the boundary, the metric g has the form dx2 + h, x2 where h is a semi-positive metric on M which restricts to a non-degenerate metric on ∂M. These are the exact b-metrics studied by Melrose [Me1, Me2]. Another possible condition would be to require that the perturbation and a finite number of its derivatives decay like (1 + u2 )−p , for some p > 1. Here we shall restrict attention to metrics which are exact products on the cylinder. But most of the results extend to larger classes of metrics; for example, to the class of exact b-metrics. Let X be equipped with a cylindrical end metric and let 1 be the


335

corresponding Laplacian. Then 1 is essentially self-adjoint and we continue to denote its unique self-adjoint extension by 1. Now, we shall briefly describe some facts concerning the spectral decomposition of 1. For details see [Me1, Mu3]. The spectrum of 1 is the union of a pure point spectrum σpp (1) and the absolutely continuous spectrum σac (1). The point spectrum σpp (1) consists of eigenvalues of finite multiplicity with no finite point of accumulation [Do]. Let 0 < λ1 ≤ λ2 ≤ λ3 ≤ · · · → ∞ be the sequence of eigenvalues where each eigenvalue is repeated according to its mulitplicity. Let Npp (λ) = # {j | λj ≤ λ} . It was proved by Christiansen and Zworski [CZ] that there exists C > 0 such that Npp (λ) ≤ C(1 + λ)n/2 , λ ≥ 0,

(4.17)

n = dim X. The first result of this type is due to Donnelly [Do], who proved a similar bound with n replaced by 2n − 1. We note that (4.17) is the optimal bound [CZ]. The estimate (4.17) implies that for t > 0, X e−tλi < ∞. (4.18) i

Let L2p (X) ⊂ L2 (X) be the subspace spanned by the L2 -eigenfunctions of 1. Then it follows from (4.18) that e−t1 |L2p (X) is a trace class operator and X −tλj Tr e−t1 |L2p (X) = e . (4.19) j

Now note that on R × Y , one has +

1=−

∂2 + 1Y . ∂u2

(4.20)

Therefore, standard perturbation theory implies that the continuous spectrum of 1 is governed by the Laplacian 1Y of Y . We regard the right-hand side of (4.20) as an operator in L2 (R+ × Y ) with domain C0∞ (R+ × Y ) and impose Dirichlet boundary conditions at the bottom of the half-cylinder. Let 10 be the corresponding self-adjoint extension. Then e−t1 − e−t10 is a trace class operator for t > 0. Hence, by the Birman–Kato invariance principle, the wave operators W± (1, 10 ) exist and are complete. Therefore, 1ac is unitarily equivalent to 10 . Let 0 = µ 0 < µ1 < µ2 < · · · → ∞ be the sequence of eigenvalues of 1Y . Then it follows that σac (1) =

∞ [

[µj , ∞),

j=0

i.e. the µj ’s are the thresholds of the continuous spectrum. At each threshold there starts a new branch of the continuous spectrum with multiplicity equal to the rank of the corresponding eigenspace.

336

W. Müller

A more explicit description of the continuous spectrum can be given in terms of generalized eigenfunctions. Let E(µk ) be the eigenspace for the eigenvalue µk of 1Y . Let φ ∈ E(µk ) and let λ > µk be different from all thresholds. Then there exists a unique eigenfunction E(φ, λ) ∈ C ∞ (X) of 1 with eigenvalue λ which on R+ × Y has the following expansion: √ E(φ, λ, (u, y)) =eiu λ−µk φ(y) √ X + e−iu λ−µl (Tkl (λ)φ) (y) + 9, µl 0. Then the resolvent (1 − λ)−1/2 , as anoperator from L2δ (X) to L2−δ (X), extends to a holomorphic function of λ ∈ Σ. For the proof see [Me1]. One also can adapt the method described in [Mu4]. Let 1/4 λ − µk Skl (λ) = Tkl (λ), (4.22) λ − µl and put S(λ) =

M

Skl (λ).

µk ,µl n, we have X X ζ(s; 1 + z, 10 + z) = (λj + z)−s + (µk + z)−s

+

1 2πi

j

Z

k ∞

(λ + z)−s

0

d log det S(λ) dλ. dλ

(4.24)

The series and the integral are absolutely convergent for s in the given half-plane. There is a similar formula for z = 0, except that the integral has to be defined by analytic continuation using the expansion log det S(λ) =

∞ X

dj λj/2 ,

|λ| < ε.

j=0

This expansion is a consequence of the analyticity of S(λ2 ) in a neighborhood of λ = 0. Everything that has been said for the Laplacian 1 on functions extends without any difficulties to generalized Dirac operators. Suppose, for example, that X is a spin manifold and let F → X be a Hermitian vector bundle over X with a Hermitian


339

connection. We assume that all structures are adapted to the product structure on R+ ×Y . Let S be the spinor bundle of X and let D : C ∞ (S ⊗ F ) → C ∞ (S ⊗ F ) be the associated twisted Dirac operator. Then, on R+ × Y , D has the following form: ∂ D=γ + DY , (4.25) ∂u where γ denotes the Clifford multiplication by the exterior normal vector field to Y and DY is a twisted Dirac operator on Y . We note that D is essentially self-adjoint in L2 (S ⊗ F ) and we shall denote its unique self-adjoint extension also by D. Let H = D2 and let H0 be the closure of −

∂2 + DY2 : C0∞ (R+ × Y, S ⊗ F ) → L2 (R+ × Y, S ⊗ F ) ∂u2

obtained by imposing Dirichlet boundary conditions. Then, as above, H, H0 satisfy the conditions (1.1) - (1.3) and therefore, ζ(s; H, H0 ) and det(H, H0 ) are well defined. Since, by (4.25), D determines uniquely the operator H0 , det(H, H0 ) depends only on D and we shall denote it simply by det(D2 ). Now assume that dim X is even. Then the spinor bundle splits as S = S+ ⊕ S− and, with respect to this splitting, D takes the form 0 D− D= . D+ 0 Then D− D+ and D+ D− have unique self-adjoint extensions H + and H − , respectively. Note that H = H + ⊕ H − . Similarly, we get H0± such that H0 = H0+ ⊕ H0− . Then the relative determinants det(H ± , H0± ) can be defined as above and we shall denote them by det(D− D+ ) and det(D+ D− ), respectively. Next we consider families of Dirac operators on manifolds with cylindrical ends. Let M and B be connected Riemannian manifolds and assume that B is compact. Let π : M → B be a Riemannian submersion whose fibres Zb = π −1 (b) are 2k-dimensional manifolds with cylindrical ends. We assume that Zb is an oriented spin manifold. Such families can be constructed from fibrations of manifolds with boundary along the lines of [BC]. Let F → M be a Hermitian vector bundle which is adapted to the product structure and let Db be the twisted Dirac operator on the fibre Zb . Let R+ × Yb be the cylindrical end of Zb . Then on R+ × Yb , Db has the form ∂ D b = γb + D Yb . ∂u Let D denote the corresponding family of Dirac operators. Assume that 0 ∈ / Spec(DYb ) for all b ∈ B. Then for each b ∈ B, Db has discrete spectrum near zero, and we may construct the determinant line bundle L = det(kerD+ )∗ ⊗ det(kerD− )

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W. Müller

in the same way as in [BF1]. Using the determinant det(D−,b D+,b ) defined as above, we can equip L with the Quillen metric. More generally, we may assume that dim ker DYb is constant. This means that the orthogonal projections Pb of L2 (S⊗F |Yb ) onto ker DYb are a smooth family, i.e. ker DYb , b ∈ B, gives rise to a smooth vector bundle ker DY over B. Then we modify the Dirac operator as follows. Let f ∈ C ∞ (R) be such that f (u) = 0 for u ≤ 1 and f (u) = 1 for u ≥ 2. Put e b = Db − γb f (u)Pb . D By our assumption, we get an operator with smooth coefficients e : C ∞ (M, S ⊗ F ) → C ∞ (M, S ⊗ F ). D e is not a differential operator, it is easy to see that D e has similar spectral Although D properties as D. Furthermore, note that on [2, ∞) × Yb , we have e b = γb ∂ + DY − Pb D b ∂u e Y = DY − Pb is invertible for all b ∈ B. This means that the and by definition, D b b e b has a gap at zero. Thus, we may proceed as above and continuous spectrum of D e and equip Le with the Quillen construct the determinant line bundle Le for the family D metric. The following lemma implies that the two line bundles are in fact the same. e b and Db , e b and ker Db denote the spaces of L2 -solutions of D Lemma 4.5. Let ker D respectively. Then we have e b = ker Db , ker D

b ∈ B.

Proof. To simplify notation, we skip the subscript b. Let φj , j ∈ Z, be an orthonormal basis of eigenfunctions of DY and let λj be the corresponding eigenvalues. If ϕ ∈ ker D, then on R+ × Y , ϕ has the following expansion: X ϕ(u, y) = aj e−λj u φj (y). λj >0

e Now let ψ ∈ ker D. e The restriction of ψ to Thus P ϕ = 0 and therefore, ϕ ∈ ker D. R+ × Y is given by Z u X X −λj u ψ(u, y) = bj e φj (y) + ck exp f (u) du φk (y). λj >0

λk =0

0

Since f (u) = 1 for u > 2 and ψ is square integrable, it follows that ck = 0 for all k. Thus P ψ = 0 and hence, Dψ = 0. e coincides with L. Since for Thus, the determinant line bundle Le associated to D e each b ∈ B, the spectrum of Db has a gap at zero, we can construct the Quillen metric as above. 4.4. Surfaces with hyperbolic ends. Let (X, g) be a complete surface of finite area such that the Gaussian curvature K of (X, g) satisfies K ≡ −1 in the complement of a


341

compact set. We call (X, g) a surface with hyperbolic ends. Any such surface X admits a decomposition of the form X = X0 ∪ Y1 ∪ · · · ∪ Ym , where X0 is a compact surface with smooth boundary and Yi ∼ = [1, ∞) × S 1 , i = 1, . . . , m, and the metric on Yi equals ds2 =

dy 2 + dx2 , y2

where (y, x) ∈ [1, ∞) × S 1 . Each end Yi is called a cusp of X. Special cases are hyperbolic surfaces of finite area, i.e. surfaces with K ≡ −1 everywhere and Area(X) < ∞. Then there exists a discrete torsion free subgroup 0 ⊂ SL(2, R) of finite co-volume such that X = 0\H where H denotes the upper half-plane equipped with the Poincaré metric. Examples are the well known surfaces 0(N )\H, where 0(N ) ⊂ SL(2, Z) is the principal convergence subgroup of level N. Let 1 be the Laplacian of X. Then 1 is essentially self-adjoint in L2 (X). The structure of the spectrum σ(1) of 1 is well known (see e.g. [Mu2]). Namely, σ(1) is the union of the absolutely continuous spectrum σac (1), and the pure point spectrum σpp (H). The point spectrum consists of a sequence of eigenvalues 0 = λ0 < λ1 ≤ λ2 ≤ · · · of finite multiplicity. The only possible point of accumulation of the eigenvalue sequence is ∞ and, as proved by Colin de Verdiere [CV2], for a generic metric there exist only finitely many eigenvalues which are all contained in [0,1/4). The absolutely continuous spectrum equals [1/4, ∞) with multiplicity equal to the number of cusps m. Let 10 be the self-adjoint extension of the operator −

m X i=1

yi2

m m M d2 M ∞ : C ([1, ∞)) → L2 [1, ∞), yi−2 dyi 0 2 dyi i=1 i=1

with respect to Dirichlet boundary conditions. If we regard a function on [1, ∞) as a function on [1, ∞) × S 1 ∼ = Yi which is independent of the second variable, then we get a canonical inclusion m M

L2 [1, ∞), yi−2 dyi ⊂ L2 (X).

(4.26)

i=1

Hence, we may consider e−t10 as an operator in L2 (X) which is equal to zero in the orthogonal complement of the image of the inclusion (4.26). With this identification, it was proved in [Mu1] that e−t1 − e−t10 is a trace class operator for every t > 0. Thus 1ac and 10 are unitarily equivalent. Moreover, there exists a complete set of generalized eigenfunctions Ei (z, s), i = 1, ..., m, which are meromorphic functions of s ∈ C satisfying 1Ei (z, s) = s(1 − s)Ei (z, s), s ∈ C. Furthermore, each Ei (z, s) is regular on the line Re(s) = 1/2. The restriction of Ei (z, s) to the cusp Yj can be expanded in a Fourier series and the zeroth Fourier coefficient has the form δij yjs + Cij (s)yj1−s .

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Put C(s) = Cij (s) . Then C(s) is a meromorphic matrix valued function which satisfies the following functional equation: C(s)C(1 − s) = Id. Moreover, the scattering matrix S(λ) = S(λ; 1, 10 ) is related to C(s) by 1 1 + λ2 = C + iλ , λ ∈ R. S 4 2

(4.27)

Hence S(λ) has an extension to a meromorphic function on the double covering of C defined by λ = s(1 − s) and we have S(s(1 − s)) = C(s). Using the generalized eigenfunctions, the spectral shift function ξ(λ) = ξ(λ; 1, 10 ) can be computed explicitly. Together with Proposition 2.1, this leads to the following trace formula: X −tλj 1 e + Tr S(1/4) e−t/4 Tr e−t1 − e−t10 = 4 j Z ∞ 1 d log det S(λ) dλ − e−tλ 2πi 1/4 dλ X 1 e−tλj + Tr C(1/2) e−t/4 = 4 j −

1 4πi

Z

∞

e−(1/4+λ

−∞

2

)t

d log det C(1/2 + iλ) dλ, dλ

(4.28)

where we have used (4.27).

√ √ Remark. There is a similar trace formula for cos t 1 −cos t 10 . If X is hyperbolic, i.e. X = 0\H, then the left-hand side of this trace formula can be rewritten as a sum of orbital integrals where the sum runs over the conjugacy classes in 0. This is then the Selberg trace formula. It follows from (4.28) that as t → ∞, we have Tr e−t1 − e−t10 = 1 + O(e−ct ) for some c > 0. Furthermore, using Theorem 8.20 of [Mu1], we obtain the following asymptotic expansion for t → 0: Area m log t + √ Tr e−t1 − e−t10 = 4πt 2 4πt +

√ 3γm 1 χ(X) √ + O( t). + 2 6 4πt

(4.29)

Here γ is Euler’s constant and χ(X) is the Euler characteristic of X. The higher order terms are similar. This is an example where logarithmic terms arise naturally. Summarizing, it follows that 1, 10 satisfy (1.1) - (1.3) and therefore, the relative zeta function ζ(s; 1, 10 ) exists.


343

Using (4.28), it follows that for Re(s) > 1, the relative zeta function is given by P −s s−1 Tr(C(1/2)) ζ(s; 1, 10 ) = λj >0 λj + 4 R ∞ 1 d λ−s dλ log det S(λ) dλ. − 2πi 1/4 By (4.29), ζ(s; 1, 10 ) has a meromorphic continuation to C which is holomorphic at s = 0. In particular, we can define the relative determinant det(1, 10 ) by (3.1). Since 10 is uniquely determined by 1, we shall denote the determinant by det 1. Remark. A different regularization of a relative determiant for surfaces with hyperbolic ends was introduced by Lundelius [Lu]. We recall his definition. Let Xi , i = 1, 2, be two surfaces with hyperbolic ends and assume that the number of ends is the same for both surfaces. Let 1i , i = 1, 2, be the Laplacian of Xi . By assumption there exist open relatively compact subsets Ki ⊂ Xi such that X1 − K1 = X2 − K2 and 11 |(X1 − K1 ) = 12 |(X2 − K2 ). Let H = L2 (X1 ) ⊕ L2 (K2 ) ∼ = L2 (X2 ) ⊕ L2 (K1 ). Then we may regard 1i as an unbounded operator in H which is zero in L2 (Kj ), i 6= j, i, j = 1, 2. Moreover, e−t11 − e−t12 is a trace class operator for t > 0 and (1.2), (1.3) hold for (11 , 12 ). Thus, one can define det(11 , 12 ). This is the relative determinant used by Lundelius. It is closely related to our definition. In fact, it follows from the definitions that det 11 . det(11 , 12 ) = det 12 Next consider the differentiability of the determinant. It follows from Proposition 3.5 that det 1 is a differentiable function on the space of all metrics with hyperbolic ends. For example, let g0 be any metric with hyperbolic ends and let f ∈ C0∞ (X). Put gu = euf g0 ,

u ∈ R.

Then gu is also a metric with hyperbolic ends. Let 1gu be the Laplacian with respect to gu . Then 1gu = e−uf 1g0 , and it follows as in [OPS] that −

d log det 1gu |u=0 = du

Z f (x) X

1 K0 (x) − 12π A0

dµ0 (x),

where K0 (x), A0 and dµ0 are the Gaussian curvature, the area and the volume element of (X, g0 ), respectively. Thus one can study critical points of det 1 for finite area surfaces with hyperbolic ends in the same way as in [OPS]. In the present case, det 1 can be expressed in a different way using the eigenvalues of 1 and the resonances. Here resonances are defined in terms of an analytic continuation of the resolvent of 1 to a meromorphic function with values in the linear operators from L2comp (X) to L2loc (X) (see [Mu4]). A resonance is then, by definition, a pole of the analytic continuation of the resolvent which does not correspond to an eigenvalue in the sense that the pole is either not an eigenvalue or its multiplicity (defined in a proper sense) is bigger than the dimension of the eigenspace. Let R(1) be the set of all poles of the analytic continuation of the resolvent. For each η ∈ R(1) one can define an algebraic

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multiplicity m(η). In fact, there is a nonself-adjoint operator B – the generator of the Lax–Phillips semigroup – such that R(1) equals the set of generalized eigenvalues of B and m(η) is then the algebraic multiplicity of this eigenvalue [Mu2]. We note that resonances can also be defined as poles of the scattering matrix which we regard as a function on the double covering of C defined by λ = s(1 − s). Now the resonance set R(1) can be used as a discrete set of spectral parameters in the same way as the eigenvalues are used in the case of a compact Riemannian manifold. For example, there is an analogue of Weyl’s formula [Mu2, Pa], and also a trace formula [Mu2]. In particular, one can introduce the resonance zeta function which is defined by X

ζRes (s) =

η∈R(1) η6=1

m(η) . (1 − η)s

(4.30)

This series is absolutely convergent for Re(s) > 2 and admits a meromorphic continuation to C which is holomorphic at s = 0. Using this zeta function, we define a second regularized determinant by d . det Res 1 = exp − ζRes (s) ds s=0 Then formally, one has Y

det Res 1 =

|1 − η|2m(η) .

η∈R(1) η6=1

Furthermore, the two determinants are closely related. Namely, we have Area(X) 3πγ − m detRes 1, det 1 = exp 8π 2 where γ is Euler’s constant [Mu2]. More generally, for Re(z) > 1, we can define X

ζRes (s; z) =

η∈R(1) η6=1

m(η) , Re(s) > 2. (z − η)s

(4.31)

Then, as a function of s, ζRes (s; z) has a meromorphic continuation to C and s = 0 is not a pole. Put ∂ . det Res (1 + z − 1) = exp − ζRes (s; z) ∂s s=0 For a hyperbolic surface 0\H of finite area, detRes (1 + z − 1) can be expressed in terms of the Selberg zeta function Z0 (s) and the determinant of the scattering matrix S(λ). First recall that Z0 (s) =

∞ YY γ k=0

1 − e−(s+k)`(γ) , Re(s) > 1,


345

where γ runs through the primitive closed geodesics and `(γ) is the length of γ. Furthermore, we write the spectral parameter λ as usual as λ = z(1 − z), z ∈ C, and regard the scattering matrix as a function of z. Then det 2Res (1 + z − 1) = detS(z)Z0 (z)2 Z∞ (z)2 0(z + 1/2)−2m ec(2z−1)+d ,

(4.32)

where c and d are certain constants that depend on Area(0\H) and m, and Area(0\H)/2π Z∞ (s) = (2π)s 02 (s)2 /0(s) with 02 (s) being the double Gamma function (see [Mu2]). It follows from (4.32) that detRes (1 + z − 1) extends to an analytic function on C with the set of zeros equal to R(1). We claim that this holds in general. If 0 = SL(2, Z), then the scattering matrix is a function S(z) which is given in terms of the Riemannian zeta function ζ(s) by S(z) =

√ 0(z − 1/2) ζ(2z − 1) π 0(z) ζ(2z)

[He]. Thus in this case, the resonances are precisely the numbers ρ/2, where ρ runs over the nontrivial zeros of the Riemann zeta function. Furthermore, detRes (1 + z − 1) can be factorized in the product of two determinants, where the first one is defined in terms of the eigenvalues similar to (0.3) and the second one is defined in terms of the nontrivial zeros ρ of ζ(s). We describe the second determinant. For Re(z) > 1 consider the Dirichlet series X (z − ρ)−s , (4.33) ζ(s, z) = (2π)s ρ

where arg(z − ρ) ∈ (−π/2, π/2). It was proved in [De] that (4.33) converges absolutely for Re(s) > 1 and for fixed z, it has an analytic continuation to a holomorphic function of s ∈ C\{1}. Moreover, one has 2−1/2 (2π)−2 π −z/2 0(z/2)ζ(z)z(z − 1) ∂ . = exp − ξ(s, z) ∂s s=0

(4.34)

The right hand side is then the determinant associated with the resonances. Note that (4.33) resembles (4.31). This analogy becomes even closer if we recall Colin de Verdiere’s result that for a generic metric, the number of eigenvalues is finite [CV2]. Remark. In conclusion, one can say that for a surface with hyperbolic ends, the resonances can be used as a substitute for the eigenvalues of the Laplacian of a compact surface. It would be very interesting to see if in other cases, the resonances play a similar role. Everything that we described here for surfaces can be extended to the case of manifolds with ends of hyperbolic type as studied in [Mu1]. Furthermore, the Laplacian on forms can be treated in the same way. In particular, we can introduce the L2 analytic torsion for hyperbolic manifolds of finite volume. Another interesting problem is the investigation of the finer structure of the distribution of resonances in the case of surfaces with hyperbolic ends. This should be seen in connection with quantum chaos [Sa].

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References [APS1] [APS3] [BC] [BF1] [BK] [Br] [Br1] [BY] [Bu] [Ch] [CZ] [CKS] [CV] [CV2] [De] [Do] [Gi] [GMS] [Gu1] [Gu2] [H] [He] [J1] [JK] [JK2] [JL] [K1] [K2]

Atiyah, M.F., Patodi, V.K. and I.M. Singer: Spectral asymmetry and Riemannian geometry, I. Math. Proc. Camb. Phil. Soc. 77, 43–69 (1975) Atiyah, M.F., Patodi, V.K. and Singer, I.M.: Spectral asymmetry and Riemannian geometry, III. Math. Proc. Camb. Phil. Soc. 79, 71–99 (1976) Bismut, J.-M. and Cheeger, J.: Families index for manifolds with boundary, super connections, and cones, I. J. Funct. Anal. 90, 306–354 (1990) Bismut, J.-M. and Freed, D.S.: The analysis of elliptic families, I. Metrics and connections on determinant bundles. Commun. Math. Phys. 106, 159–176 (1986) Birman, M.Sh. and Krein, M.G.: On the theory of wave operators and scattering operators. Dokl. Akad. Nauk SSSR 144, 475–478 (1962); English transl. in Soviet Math. Dokl. 3 (1962) Bruneau, V.: Propriétés asymptotiques du spectre continu d’opérateurs de Dirac. These de Doctorat, Université de Nantes, 1995 Bruneau, V.: Sur le spectre continu de l’opérateur de Dirac: formule de Weyl, limite non-relativiste. C.R. Acad. Sci. Paris 322, 43–48 (1996) Birman, M.Sh. and Yafaev, D.R.: The spectral shift function. The work of M.G. Krein and its further development. St. Petersburg Math. J. 4, 833–870 (1993) Bunke, U.: Relative index theory. J. Funct. Anal. 105, 63–76 (1992) Christiansen, T.: Scattering theory for manifolds with asymptotically cylindrical ends. J. Funct. Anal. 131, 499–530 (1995) Christiansen, T.and Zworski, M.: Spectral asymptotics for manifolds with cylindrical ends. Ann. Inst. Fourier, Grenoble 45, 251–263 (1995) Cotta-Ramusino, P.,Krüger, W. and Schrader, R.: Quantum scattering by external metrics and Yang-Mills potentials. Ann. Inst. Henri Poincaré 31, 43–71 (1979) Colin de Verdiere, Y.: Une formule de traces pour l’opérateur de Schrödinger dans R3 . Ann. scient. ´ Norm. Sup. 4e série, 14, 27–39 (1981) Ec. Colin de Verdiere, Y.: Pseudo-Laplaciens II. Ann. Inst. Fourier, Grenoble 33, 87–113 (1983) Deninger, C.: Local L-factors of motives and regularized determinants. Invent. math. 107, 135–151 (1992) Donnelly, H.: Eigenvalue estimates for certain noncompact manifolds. Michigan Math. J. 31, 349– 357 (1984) Gilkey, P.B.: Invariance theory, the heat equation, and the Atiyah-Singer index theorem. Second edition, Boca Raton, Ann Arbor: CRC Press, 1995 Gamboa Saravi, R.E., Muschietti, M.A., Schaposnik, F.A. and Solomin, J.E.: ζ-function method and the evaluation of fermion currents. J. Math. Phys. 26, 2045–2049 (1985) Guillopé, L.: Une formule de trace pour l’opérateur de Schrödinger dans Rn . These de 3eme cycle, Grenoble, 1981 Guillopé, L.: Asymptotique de la phase de diffusion pour l’opérateur de Schrödinger avec potentiel. C.R. Acad. Sci. Paris 293, 601–603 (1981) Hawking, S.W.: Zeta function regularization of path integrals in curved space time. Commun. Math. Phys. 55, 133–148 (1977) Hejhal, D.A.: The Selberg trace formula and the Riemann zeta function. Duke Math. J. 43, 441–482 (1976) Jensen, A.: Spectral properties of Schrödinger operators and time-decay of the wave functions, results in L2 (Rm ), m ≥ 5. Duke Math. J. 47, 57–80 (1980) Jensen, A. and Kato, T.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J. 46, 583–611 (1979) Jensen, A. and Kato, T.: Asymptotic behaviour of the scattering phase for exterior domains. Commun. Partial Diff. Equations 3 , 1165–1195 (1978) Jorgenson, J. and Lundelius, R.: Continuity of relative hyperbolic spectral theory through metric degeneration. Duke Math. J. 84, 47–81 (1996) Kato, T.: Perturbation theory for linear operators. Berlin: Springer-Verlag, 1966 Kato, T.: Wave operators and unitary equivalence. Pacific J. Math. 15, 171–180 (1965)


[Kr] [Lu] [Me1] [Me2] [Mu1] [Mu2] [Mu3] [Mu4] [Mu5] [Mu6] [OPS] [Pa] [RS1] [RS] [R] [Ro] [Sa] [Se] [Si] [SZ] [Th] [Y]

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Krein, M.G.: On the trace formula in perturbation theory. Mat. Sbornik 33(75), 597–626 (1953) (Russian) Lundelius, R.: Asymptotics of the determinant of the Laplacian on hyperbolic surfaces of finite volume. Duke Math. J. 71, 211–242 (1993) Melrose, R.B.: The Atiyah-Patodi-Singer index theorem. Boston: A.K. Peters, 1993 Melrose, R.B.: Geometric scattering theory. Cambridge: Cambridge University Press, 1995 Müller, W.. Spectral theory for Riemannian manifolds with cusps and a related trace formula. Math. Nachrichten 111, 197–288 (1983) Müller, W.: Spectral theory and scattering theory for certain complete surfaces of finite volume. Invent. math. 109, 265–305 (1992) Müller, W.: Eta invariants and manifolds with boundary. J. Diff. Geom. 40, 311–377 (1994) Müller, W. On the analytic continuation of rank one Eisenstein series. Geom. Funct. Anal. 6, 572–586 (1996) Müller, W. Manifolds with cusps of rank one. Lecture Notes in Math. 1244, Berlin: Springer-Verlag, 1987 Müller, W.: Relative determinants of elliptic operators and scattering theory. Journees “Equations aux derivees partielles”, Saint-Jean-de Monts 1996, Ecole Polyt. Osgood, B., Phillips, R. and Sarnak, P.: Extremals of determiants of Laplacians. J. Funct. Anal. 80, 148–211 (1988) Parnovski, L.B.: Spectral asymptotics of the Laplace operator on surfaces with cusps. Math. Annalen 303, 281–296 (1995) Ray, D.B. and Singer, I.M.: R-torsion and the Laplacian on Riemannian manifolds. Adv. Math. 7, 145–210 (1971) Reed, M. and Simon, B.: Methods of mathematical physics. IV, London: Academic Press, 1978 Robert, D.: Asymptotique a grande e´ nergie de la phase de diffusion pour un potentiel. Asymptotic Anal. 3, 301–320 (1991) Robert, D.: Asymptotique de la phase de diffusion a haute e´ nergie pour des perturbations du second ´ Norm. Sup. 4e série, 25, 107–134 (1992) ordre du Laplacien. Ann. scient. Ec. Sarnak, P.: Arithmetic quantum chaos. Israel Math. Conf. Proc. 8, 183–236 (1995) Seeley, R.T.: Complex powers of an elliptic operator. Proc. Symp. Pure Math. 10, 288–307 (1967) ´ Cartan et les Singer, I.M.: Families of Dirac operators with applications to physics. In: Elie Mathématiques d’aujourd’hui, Astérisque 1985, Numéro Hors Série. Sjöstrand, J. and Zworski, M.. Complex scaling and the distribution of scattering poles. J. Am. Math. Soc. 4, 729–769 (1991) Thaller, B.: The Dirac equation, Texts and Monographs in Physics, Berlin–Heidelberg–New York: Springer-Verlag, 1992 Yafaev, D.R.: Mathematical scattering theory. Transl. Math. Monographs, Vol. 105, Princeton, NJ: AMS, 1992

Communicated by P. Sarnak

Commun. Math. Phys. 192, 349 – 403 (1998)

Communications in


New Braided Endomorphisms from Conformal Inclusions Feng Xu Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA Received: 12 September 1996 / Accepted: 3 July 1997

Abstract: Various puzzles about subfactors and integrable lattice models associated with conformal inclusions are resolved in the framework of constructive quantum field theory in two dimensions. In particular, a new class of braided endomorphisms are obtained for a general class of conformal inclusions and their properties are analyzed. The existence of subfactors with principal graphs E6 or E8 follows from a rather simple argument in our construction. The fusion graphs of many new examples are given. 0. Introduction Let G ⊂ H be a conformal inclusion and π 0 be the vacuum representation of Loop group LH (see Sect. 1.6). Denote by e the identity element in H and I an interval on S 1 . Let LI G = {f ∈ LG | f = e on I c }, and LI H = {f ∈ LH | f = e on I c }. Then π 0 (LI G)00 ⊂ π 0 (LI H)00 is called the subfactor from conformal inclusions. Both π 0 (LI G)00 and π 0 (LI H)00 are hyperfinite III1 factors and when G = SU (N ), π 0 (LI G)00 ⊂ π 0 (LI H)00 is of finite depth (see Sect. 2). In [X3] and [X4] attempts are made to construct subfactors from conformal inclusions from some known integrable lattice models in [PZ]. The advantage of such an approach is that one can determine the principal graphs of the subfactors completely from an explicit recursive formula. However, one has to check some local conditions. The calculations which are needed are tedious but straightforward (unlike global conditions such as the flatness condition [OCN1] which is impossible to check directly) when the rank of G is small. But when the rank of G is big, the corresponding integrable lattice models become harder to construct. In fact, the main motivation of this paper is to explain and generalize some of the observations in [X3] and [PZ]. On the other hand, in [Ka1] it is suggested that one can construct certain “quotient” subfactors by subfactors from conformal inclusions. In particular, the role of intertwining YBE is emphasized in [Ka1]. However, it is clear that one can only hope to be able to check intertwining YBE in very limited cases.

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In this paper we will show the existence of “quotient” subfactors for all maximal conformal inclusions G ⊂ H when G = SU (N ) 1 and H is a simple group. In fact, we will show the existence of a new class of braided endomorphisms for all maximal conformal inclusions SU (N ) ⊂ H with H being simple. There are two main ingredients which make such a general statement possible. The first one is the work of A. Wassermann in [W1] which clarifies the fusion ring structure arising from the positive energy representations of LSU (N ) (cf. Sect. 1.6). The second one is the work of R. Longo and K.-H.Rehren in [LR] which applies to the study of conformal inclusions. Let us describe in more detail the content of the paper. Section 1 is a preliminary section of the general theory of sectors, correspondences and constructive conformal field theories. Sections 1.2 and 1.3 are contained in [GL1] and we have included them to set up the notations and concepts. In Sect. 1.4, we proved the Yang–Baxter Equation (YBE) and Braiding–Fusion Equation (BFE). We also give a proof of monodromyequations. These results are scattered in the literature and their proof is not new. We have included them for future references. Our treatment follows that of [X6]. In Sect. 1.5 we use diagrams to represent YBE and BFE proved in Sect. 1.4. These diagrammatic representations make many algebraic relations more transparent. Part of the results in [W1] and [FG] are sketched in Sect. 1.6. In Sect. 2 we study the subfactors from conformal inclusions, following the ideas of [W2] and [LR]. Section 2.1 is contained in [LR]. We have included this section to introduce concepts and notations compatible with the previous sections. In Sect. 2.2, the general theory of [LR] is applied to the subfactors associated with the conformal inclusions. Proposition 2.5 shows that such subfactors naturally give rise to a standard net. Proposition 2.6 enables us to study certain endomorphism associated with conformal inclusions by using the theory sketched in Sect. 1.6. The local property established in Proposition 2.7 and Proposition 2.8 plays a crucial role in proving Theorem 3.3 and Theorem 3.9. The idea of the proof of Proposition 2.7 and Proposition 2.8 is contained in [LR]. Section 3 is the main part of this paper. We use the results of previous sections to prove six main theorems. Theorem 3.1 and Corollary 3.2 shows the existence of “quotient” endomorphisms. Theorem 3.3 implies that these endomorphisms are braided and gives more information on the higher relative commutants of these braided endomorphisms. Theorem 3.6 shows some commutation relations which imply, for example, that for the subfactors associated to these endomorphisms, the principal graphs and the dual principal graphs are isomorphic as abstract graphs (Corollary 3.7). Connections with [X3] and [PZ] are made in Theorem 3.8, which implies that certain graphs “support” (in the sense of [PZ]) representations of Hecke algebras. Two explicit examples are given in the end of Sect. 3. The existence of subfactors with principal graphs E6 , E8 follows from Corollary 3.7 and a rather simple counting argument. In Sect. 3.2 we prove some further properties of our braided endomorphisms and establish their relations to solvable lattice models in [PZ]. Among them, Theorem 3.9 determines the spectrum of the fusion graphs completely, and Theorem 3.10 proves most of the assumptions of [PZ] in generality. In fact, the work of [PZ] has been an important source of inspiration for our work. The main part of the proof in Sect. 3 consists of computations of algebraic relations established in Sects. 1 and 2. We have used diagrammatic representations which may help to “visualize” the algebraic identities. 1

For other simple G case, see comments in Sect. 5.

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351

By using the results of Sect. 3, especially Theorem 3.3 and Lemma 3.5, the fusion graphs of the braided endomorphism associated to G = SU (3) are obtained in Sect. 4. \ \ (15) One particular interesting example associated to conformal inclusion SU (4)4 ⊂ SU is also given. It shows how commutativity among the subsectors of our braided endomorphisms may fail. It should be mentioned that all the graphs first appear in [PZ] with different meaning and are constructed by a certain hypothesis. Our results provide a rigorous foundation to the insights of [PZ]. In Sect. 5, we present our conclusions and some questions which arise naturally from our approach. 1. Preliminaries 1.1. Sectors and correspondences. Let M , N be von Neumann algebras, that we always assume to have separable preduals, and H a M − N correspondence, namely H is a (separable) Hilbert space, where M acts on the left, N acts on the right and the actions are normal. We denote by xξy, x ∈ M , y ∈ N , ξ ∈ H the relative actions. The trivial M − M correspondence is the Hilbert space L2 (M ) with the standard actions given by the modular theory xξy = xJy ∗ Jξ ,

x, y ∈ M ,

ξ ∈ L2 (M ) ,

where J is the modular conjugation of M ; the unitary correspondence is well defined modulo unitary equivalence. If ρ is a normal homomorphism of M into M we let Hρ be the Hilbert space L2 (M ) with actions: x · ξ · y ≡ ρ(x)ξ · y, x ∈ M , y ∈ M , ξ ∈ L2 (M ). Denote by End(M ) the semigroup of the endomorphism of M and Corr(M ) the set of all M − M correspondences. The following proposition is proved in [L4] (Corollary 2.2 in [L4]). Proposition 1.1.1. Let M be an infinite factor. There exists a bijection between the unitary equivalence classes of End(M ) and the unitary equivalence classes of Corr(M ), i.e., given ρ, ρ0 ∈ End(M ), Hρ is unitarily equivalent to Hρ0 iff there exists a unitary u ∈ M with ρ0 (x) = uρ(x)u∗ . Let Sect(M ) denote the quotient of End(M ) modulo unitary equivalence in M as in Proposition 1.1. We call sectors the elements of the semigroup Sect(M ); if ρ ∈ End(M ) we denote by [ρ] its class in Sect(M ). By Proposition 2.2 Sect(M ) may be naturally identified with Corr(M )∼ the quotient of Corr(M ) modulo unitary equivalence. It follows from [L3] and [L4] that Sect(M ), with M a properly infinite (on Hilbert space H) von Neumann algebra, is endowed with a natural involution θ → θ¯ that commutes with all natural operations of direct sum, tensor product and other (the tensor product of correspondences correspond to the composition of sectors). Suppose ρ ∈ End(M ) is given together with a normal faithful conditional expectation : M → ρ(M ). We define a number d (possibly ∞) such that: d−2 := Max{λ ∈ [0, +∞)|(m+ ) ≥ λm+ , ∀m+ ∈ M+ }. Now assume ρ ∈ End(M ) is given together with a normal faithful conditional expectation : M → ρ(M ), and assume d < +∞. We define d = Min {d }.

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d is called the statistical dimension of ρ. It is clear from the definition that the statistical dimension of ρ depends only on the unitary equivalence classes of ρ. The properties of the statistical dimension can be found in [L1, L3 and L4]. Denote by Sect0 (M ) those elements of Sect(M ) with finite statistical dimensions. For λ, µ ∈ Sect0 (M ), let Hom(λ, µ) denote the space of intertwiners from λ to µ, i.e. a ∈ Hom(λ, µ) iff aλ(x) = µ(x)a for any x ∈ M . Hom(λ, µ) is a finite dimensional vector space and we use hλ, µi to denote the dimension of this space. hλ, µi depends only on [λ] and [µ]. Moreover we have hνλ, µi = hλ, νµi, ¯ hνλ, µi = hλ, µλi which follows from Frobenius duality (see [L2] or [Y]). We will also use the following notations: if µ is a subsector of λ, we will write it as µ ≺ λ or λ µ. 1.2. General properties of conformal precosheaves on S 1 . In this section we recall the basic properties enjoyed by the family of the von Neumann algebras associated with a conformal Quantum Field Theory on S 1 . All the propositions in this section and Sect. 1.3 are proved in [GL1]. Our goal in this section is to give the definition of a conformal precosheaf. Let us first do some preparations. By an interval in this section only we shall always mean an open connected subset I of S 1 such that I and the interior I 0 of its complement are non-empty. We shall denote by I the set of intervals in S 1 . We shall denote by P SL(2, R) the group of conformal transformations on the complex plane that preserve the orientation and leave the unit circle S 1 globally invariant. Denote by G the universal covering group of P SL(2, R). Notice that G is a simple Lie group and has a natural action on the unit circle S 1 . Denote by R(ϑ) the (lifting to G of the) rotation by an angle ϑ. We may associate two one-parameter groups with any interval I in the following way. Let L1 be the upper semi-circle, i.e. the interval {eiϑ , ϑ ∈ (0, π)}. By using the Cayley transform C : S 1 → R ∪ {∞} given by z → −i(z − 1)(z + 1)−1 , we may identify L1 with the positive real line R+ . Then we consider the one-parameter groups 3I1 (s) and TI1 (t) of diffeomorphisms of S 1 (cf. Appendix B of [GL1]) such that C3I1 (s)C −1 x = es x ,

CTI1 (t)C −1 x = x + t ,

t, s, x ∈ R .

We also associate with I1 the reflection rI1 given by ¯ rI1 z = z, where z¯ is the complex conjugate of z. It follows from the definition that 3I1 restricts to an orientation preserving diffeomorphism of I1 , rI1 restricts to an orientation reversing diffeomorphism of I1 onto I10 and TI1 (t) is an orientation preserving diffeomorphism of I1 into itself if t ≥ 0. Then, if I is an interval and we choose g ∈ G such that I = gI1 , we may set 3I = g3I1 g −1 ,

rI = grI1 g −1 ,

TI = gTI1 g −1 .

The elements 3I (s), s ∈ R and rI are well defined, while the one parameter group TI is defined up to a scaling of the parameter. However, such a scaling plays no role in this paper. We note also that TI 0 (t) is an orientation preserving diffeomorphism of I into itself if t ≤ 0. Let r be an orientation reversing isometry of S 1 with r2 = 1 (e.g. rI1 ). The action of r on P SL(2, R) by conjugation lifts to an action σr on G, therefore we may consider the


353

semidirect product of G ×σr Z2 . Since G ×σr Z2 is a covering of the group generated by P SL(2, R) and r, G ×σr Z2 acts on S 1 . We call (anti-)unitary a representation U of G ×σr Z2 by operators on H such that U (g) is unitary, resp. antiunitary, when g is orientation preserving, resp. orientation reversing. Now we are ready to define a conformal precosheaf. A conformal precosheaf A of von Neumann algebras on the intervals of S 1 is a map I → A(I) from I to the von Neumann algebras on a Hilbert space H that verifies the following property: A. Isotony. If I1 , I2 are intervals and I1 ⊂ I2 , then A(I1 ) ⊂ A(I2 ) . B. Conformal invariance. There is a unitary representation U of G (the universal covering group of P SL(2, R)) on H such that U (g)A(I)U (g)∗ = A(gI) ,

g ∈ G,

I ∈I.

C. Positivity of the energy. The generator of the rotation subgroup U (R)(·) is positive. D. Locality. If I0 , I are disjoint intervals then A(I0 ) and A(I) commute. The lattice symbol ∨ will denote “the von Neumann algebra generated by”. E. Existence of the vacuum. There exists a unit vector (vacuum vector) which is U (G)-invariant and cyclic for ∨I∈I A(I). We have the following (cf. Proposition 1.1 of [GL1]): Proposition 1.2.1. Let A be a conformal precosheaf. The following hold: (a) Reeh-Schlieder theorem: is cyclic and separating for each von Neumann algebra A(I), I ∈ I. (b) Bisognano-Wichmann property: U extends to an (anti-)unitary representation of G ×σr Z2 such that, for any I ∈ I, U (3I (2πt)) = 1it I , U (rI ) = JI , where 1I , JI are the modular operator and the modular conjugation associated with (A(I), ) [29]. For each g ∈ G ×σr Z2 , U (g)A(I)U (g)∗ = A(gI). (c) Additivity: if a family of intervals Ii covers the interval I, then A(I) ⊂ ∨i A(Ii ) . (d) Spin and statistics for the vacuum sector [16]: U is indeed a representation of P SL(2, R), i.e. U (2π) = 1. (e) Haag duality: A(I)0 = A(I 0 ) A conformal precosheaf is called irreducible if it also satisfies the following: F. Uniqueness of the vacuum (or irreducibility). The only U (G)-invariant vectors are the scalar multiples of . The term irreducibility is due to the following (See Proposition 1.2 of [GL1]):

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Proposition 1.2.2. The following are equivalent: C are the only U (G)-invariant vectors. The algebras A(I), I ∈ I, are factors. In this case they are type III1 factors. If a family of intervals Ii intersects at only one point ζ, then ∩i A(Ii ) = C. The von Neumann algebra ∨A(I) generated by the local algebra coincides with B(H) (A is irreducible). Let I be an interval and denote by I¯ its closure on the circle. The conformal precosheaf A constructed in Sect. 1.6 has the following additonal property: the map (i) (ii) (iii) (iv)

I → A(I) extends to all the connected subsets of S 1 such that the interior of I and the interior I 0 of its complement I c are non-empty and satisfies the Isotony property; moreover, ¯ A(I) = A(I). We shall only consider a conformal precosheaf A with the above property in this paper. 1.3. Superselection structure. In this section A is an irreducible conformal precosheaf of von Neumann algebras as defined in Section 1.2. Our goal is to define the covariant representation of A and the associated concepts which will be important later on. Again all the definitions and propositions are given in [GL1] and we safely refer the reader to [GL1] for further details and unexplained notations. A covariant representation π of A is a family of representations πI of the von Neumann algebras A(I), I ∈ I, on a Hilbert space Hπ and a unitary representation Uπ of the covering group G of P SL(2, R), with positive energy, i.e. the generator of the rotation unitary subgroup has positive generator, such that the following properties hold: I ⊃ I¯ ⇒ πI¯ |A(I) = πI (isotony), adUπ (g) · πI = πgI · adU (g)(covariance) . A unitary equivalence class of representations of A is called superselection sector. Assuming Hπ to be separable, the representations πI are normal because the A(I)’s are factors . Therefore for any given I0 , πI00 is unitarily equivalent idA(I00 ) because A(I00 ) is a type III factor. By identifying Hπ and H, we can thus assume that π is localized in a given interval I0 ∈ I, i.e. πI00 = idA(I00 ) (cf. [Fro]). By Haag duality we then have πI (A(I)) ⊂ A(I) if I ⊃ I0 . In other words, given I0 ∈ I we can choose in the same sector of π a localized endomorphism with localization support in I0 , namely a representation ρ equivalent to π such that I ∈ I, I ⊃ I0 ⇒ ρI ∈ End A(I) ,

ρI00 = idI00 .

To capture the global point of view we may consider the universal algebra C ∗ (A). Recall that C ∗ (A) is a C ∗ -algebra canonically associated with the precosheaf A (see [Fre]). C ∗ (A) has the following properties: there are injective embeddings ιI : A(I) → C ∗ (A) so that the local von Neumann algebras A(I), I ∈ I, are identified with subalgebras of C ∗ (A) and generate all together a dense ∗-subalgebra of C ∗ (A), and every representation of the precosheaf A factors through a representation of C ∗ (A). Conversely any representation of C ∗ (A) restricts to a representation of A. The vacuum representation π0


355

of C ∗ (A) corresponds to the identity representation of A on H, thus π0 acts identically on the local von Neumann algebras. We shall often drop the symbols ιI and π0 when no confusion arises. By the universality property, for each g ∈ P SL(2, R) the isomorphism adU (g) : A(I) → A(gI), I ∈ I lifts to an automorphism αg of C ∗ (A). We shall lift the map g → αg to a representation, still denoted by α, of the universal covering group G of P SL(2, R) by automorphisms of C ∗ (A). The covariance property for an endomorphism ρ of C ∗ (A) localized in I0 means that αg · ρ · αg−1 is adzρ (g)∗ · ρ = αg · ρ · αg−1 g ∈ G for a suitable unitary zρ (g) ∈ C ∗ (A). We define ρg = αg · ρ · αg−1

, g ∈ G.

ρg,J is the restriction of ρg to A(J). The map g → zρ (g) can be chosen to be a localized α-cocycle, i.e. zρ (g) ∈ A(I0 ∪ gI0 ) ∀g ∈ G : I0 ∪ gI0 ∈ I zρ (gh) = zρ (g)αg (zρ (h)) , g, h ∈ G . To compare with the result of [FG], let us define: 0ρ (g) = π0 (zρ (g)∗ ). This notation will be used in Sect. 1.4. An endomorphism of C ∗ (A) localized in an interval I0 is said to have finite index if ρI (= ρ|A(I) ) has finite index, I0 ⊂ I (see [L2, L3]). The index is independent of I due to the following (See Proposition 2.1 of [GL1]) Proposition 1.3.1. Let ρ be an endomorphism localized in the interval I0 . Then the index Ind(ρ) := Ind(ρI ), the minimal index of ρI , does not depend on the interval I ⊃ I0 . The following Proposition is Proposition 2.2 of [GL1]: Proposition 1.3.2. Let ρ be a covariant (not necessarily irreducible) endomorphism with finite index. Then the representation Uρ described before is unique. In particular, any irreducible component of ρ is a covariant endomorphism. By the above proposition the univalence of an endomorphism ρ is well defined by Sρ = Uρ (2π) . When Sρ is a complex number of modulus one, since Uρ0 (g) := π0 (u)Uρ (g)π0 (u)∗ , where ρ0 (·) := uρ(·)u∗ , u ∈ C ∗ (A), Sρ depends only on the superselection class of ρ. When ρ is irreducible, Sρ is a complex number of modulus one since by definition Sρ belongs to π(C ∗ (A))0 and ρ is irreducible, and we have Sρ = e2πi1ρ . with 1ρ the lowest weight of Uρ . 1ρ is also referred to as Conformal dimension. Examples of calculations of 1ρ can be found at the beginning of Sect. 3.2. Let ρ1 , ρ2 be endomorphisms of an algebra B. Recall from Sect. 1.1 that their intertwiner space is defined by Hom(ρ1 , ρ2 ) = {T ∈ B : ρ2 (x)T = T ρ1 (x),

x ∈ B}.

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In the case B = C ∗ (A), ρi localized in the interval Ii and T ∈ (ρ1 , ρ2 ), then π0 (T ) is an intertwiner between the representations π0 · ρi . If I ⊃ I1 ∪ I2 , then by Haag duality its embedding ιI · π0 (T ) is still an intertwiner in (ρ1 , ρ2 ) and a local operator. We shall denote by (ρ1 , ρ2 )I the space of such local intertwiners (ρ1 , ρ2 )I = (ρ1 , ρ2 ) ∩ A(I) . If I1 and I2 are disjoint, we may cover I1 ∪ I2 by an interval I in two ways: we adopt the convention that, unless otherwise specified, a local intertwiner is an element of (ρ1 , ρ2 )I , where I2 follows I1 inside I in the clockwise sense. We now define the statistics. Given the endomorphism ρ of A localized in I ∈ I, choose an equivalent endomorphism ρ0 localized in an interval I0 ∈ I with I¯0 ∩ I¯ = ∅ and let u be a local intertwiner in (ρ, ρ0 ) as above, namely u ∈ (ρ, ρ0 )I˜ with I0 following ˜ clockwise I inside I. The statistics operator σ := u∗ ρ(u) = u∗ ρI˜ (u) belongs to (ρ2I˜ , ρ2I˜ ). Recall that if ρ is an endomorphism of a C ∗ -algebra B, a left inverse of ρ is a completely positive map 8 from B to itself such that 8 · ρ = id. It follows from Cor.2.12 of [GL1] that if ρ is irreducible there exists a unique left inverse 8 of ρ and that the statistics parameter λρ := 8(σ) depends only on the sector of ρ. The statistical dimension d(ρ) and the statistics phase κρ are then defined by λρ . d(ρ) = |λρ |−1 , κρ = |λρ | In [GL1], the following remarkable theorem is proved: Theorem 1.1. κρ = Sρ if ρ has finite statistics. 1.4. Coherence equations. In this section, we assume 1 is a set of localized covariant endomorphism of A with localization support in I0 . Let h, g be elements of G. We assume hI0 ∩ I0 = ∅, gI0 ∩ I0 = ∅, hI0 ∩ gI0 = ∅. Choose J1 , J2 ∈ I such that J1 ∪ J2 ( S 1 , J1 ⊃ I0 ∪ g.I0 , J2 ⊃ I0 ∪ h.I0 , J1 ∩ h.I0 = ∅, J2 ∩ g.I0 = ∅ and J1 ∩ J2 = I0 . We assume in J1 (resp. J2 ), g.I0 (resp. h.I0 ) lies a clockwise (resp. anti clockwise) from I0 . Lemma 1.1. For any J ⊃ J1 ∪ J2 , J ∈ I, γ, λ ∈ 1 and x ∈ A(J), we have (0) (1) (2) (3)

0λ (g) ∈ A(J1 ). 0λ (g)∗ γJ1 (0λ (g))γJ · λJ (x) = λJ · γJ (x)0λ (g)∗ γJ1 (0λ (g). 0λ (g)∗ γJ1 (0λ (g)) = λJ2 (0γ (h)∗ )0γ (h). 0γ (g)∗ γJ1 (0λ (g)) ∈ A(I0 ).

Proof. Recall λJ (x) = 0γ (g)∗ λg,J (x)0λ (g) for any x ∈ A(J), S 1 ) J ⊃ J1 . Since λJ (resp. λg,J ) is localized on I0 (resp. g.I0 ), it follows that 0λ (g) ∈ A(J ∩ J1c )0 for any S 1 ) J ⊃ J1 . Let us choose J4 ⊃ J1 , J3 ⊃ J1 so that I2 = J4 ∩ J1c , I3 = J3 ∩ J1c are closed intervals and I2c ∩ I3c = J1 . Then we have: 0λ (g) ∈ A(I2c ) ∩ A(I3c ).


357

We claim that

A(I2c ) ∩ A(I3c ) = A(J1 ). In fact it is clear that A(I2c ) ∩ A(I3c ) ⊃ A(J1 ). By Haag duality, it is sufficient to prove A(I2 ) ∨ A(I3 ) ⊃ A(J1c ). But I2 ∪ I3 = J1c and the inclusion above follows by (c) of Proposition 1.2.1. So we have 0λ (g) ∈ A(J1 ). By (0), γJ1 (0λ (g)) is well defined. To prove (1), we can calculate the left hand side as follows: 0λ (g)∗ γJ1 (0λ (g))γJ · λJ (x) = 0λ (g)∗ γJ (0λ (g)λJ (x)) = 0λ (g)∗ γJ (λg,J (x)0λ (g)) = 0λ (g)∗ γJ (λg,J (x))γJ1 (0λ (g) = 0λ (g)∗ λg,J (γJ (x))γJ1 (0λ (g)) = λJ · γJ (x)0λ (g)∗ γJ1 (0λ (g)), where in the first “=” we used γJ (x) = γJ1 (x) if x ∈ A(J)A(J1 ) and J ⊃ J1 . In the fourth “=” we used λg,J (γJ (x)) = γJ (λg,J (x)) for x ∈ A(J) since λg,J and γJ have disjoint support. To prove (2), it is sufficient to prove: 0γ (h)∗ 0γ (h)γJ1 (0λ (g))0γ (h)∗ = 0λ (g)λJ2 (0γ (h)∗ )0λ (g)∗ 0λ (g), i.e.

0γ (h)∗ γh,J1 (0λ (g)) = λg,J2 (0γ (h)∗ )0γ (g) .

This follows from

γh,J1 (0λ (g)) = 0λ (g) λg,J2 (0λ (h)∗ ) = 0γ (h)∗ . Since 0γ (g) (resp. 0γ (h)∗ ) is in A(J1 ) (resp. A(J2 )) and J1 (resp. J2 ) is disjoint from the support h.I0 (resp. g.I0 ) of γh,J1 (resp. λg,J2 ). It follows from (1) and the proof of (0) that 0λ (g)∗ γJ1 (0λ (g)) ∈ A(J1 ). Similarly, λJ2 (0γ (h)∗ )0γ (h) ∈ A(J2 ). From (2) we deduce that 0λ (g)∗ γJ1 (0λ (g)) ∈ A(J1 ) ∩ A(J2 ) = A(J0 ) where the last "=” follows as in the proof of (0). Because of the property (1) of Lemma 1.4.1, 0λ (g)∗ γJ1 (0λ (g)) is called the braiding operator. We shall use σγ,λ to denote 0λ (g)∗ γJ1 (0λ (g)). We are now ready to prove the following equations. For simplicity we will drop the subscript I0 and write µI0 as µ for any µ ∈ 1 in the following. Proposition 1.4.1. (1) Yang-Baxter-Equation (YBE): σµ,γ µ(σλ,γ )σλ,µ = γ(σλ,µ )σλ,γ λ(σµ,γ ) . (2) Braiding-Fusion-Equation (BFE): For any w ∈ Hom(µγ, δ), σλ,δ λ(w) = wµ(σλ,γ )σλ,µ , σδ,λ w = λ(w)σµ,λ µ(σγ,λ ), ∗ ∗ ∗ λ(w) = wµ(σγ,λ )σµ,λ , σδ,λ ∗ ∗ ∗ σλ,δ λ(w) = wµ(σγ,λ )σλ,µ .

(a) (b) (c) (d)

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Proof. To prove (1), let us first calculate the left-hand side of (1) as follows: σµ,γ µ(σλ,γ )σλ,µ = σµ,γ µ(γJ2 (0λ (h)∗ )0λ (h))µJ2 (0λ (h)∗ )0γ (h) = σµ,γ µ(γJ2 (0λ (h)∗ ))0γ (h) . For the right-hand side of (1), we have: γ(σλ,µ )σλ,γ λ(σµ,γ ) = γ(µJ2 (0λ (h)∗ )0λ (h)) · γJ2 (0λ (h)∗ )0λ (h) · λ(σµ·γ ) = γ(µJ2 (0λ (h)∗ ))λh (σµ,γ ) · 0λ (h) = γ(µJ2 (0λ (h)∗ ))σµ,γ 0λ (h) = σµ,γ µ(γJ2 (0λ (h)∗ )), where in the second “=” we have used 0λ (h)λ(σµ,γ ) = λh (σµ,γ )0λ (h); In the third “=” we have used λh (σµ,γ ) = σµ,γ , since σµ,γ ∈ A(I0 ) and λh has support on h.I0 which is disjoint from I0 . To prove (a) of (2), let us calculate, starting from the right-hand side of (a) as follows: wµ(σλ,γ )σλ,µ = wµ(γJ2 (0λ (h)∗ )0λ (h)) · µ(0λ (h)∗ )0λ (h) = wµ(γJ2 (0λ (h)∗ ))0λ (h) = δ(0λ (h)∗ )w0λ (h) = σλ,δ λ(w) . To prove (b), we make use of (2) in Lemma 1.4.1 to calculate, starting from the right-hand side of (b) in the following: λ(w)σµ,λ µ(σγ,λ ) = λ(w)0λ (g)∗ µ(0λ (g)) · µ(0λ (g)∗ γ(0λ (g))) = λ(w)0λ (g)∗ µ(γ(0λ (g))) = 0λ (g)∗ wµ(γ(0λ (g))) = 0λ (g)∗ δ(0λ (g))w = σδ,λ w. (c) (resp. (d)) is proved in exactly the same way as (a) (resp. (b)) with h replaced by g (resp. g replaced by h). Suppose ξ1 ∈ Iξ1 ⊂ J1 , Iξ1 ∩ gI1 ∩ I1 = ∅, ξ2 ∈ Iξ2 ⊂ J2 , and Iξ2 ∩ I1 ∩ h.I1 = ∅. Here g, h, J1 , J2 are defined as the beginning of this section. It follows from (2) of Lemma 1.4.1 that: γIξc (0λ (g))∗ 0λ (g) = γJ2 (0λ (h)∗ )0λ (h) = σλ,γ . 1

Hence σλ,γ σγ,λ = γIξc (0λ (g)∗ )γIξc (0λ (g)). 1 2 σλ,γ σγ,λ is called the monodromy operator. Let Te : δ → γλ be an intertwiner. Recall Sρ = Uρ (2π) is the univalence of a covariant endomorphism. When ρ is irreducable, Sρ is a complex number.


359

Proposition 1.4.2 (Monodromy Equation)). If Sδ , Sγ , Sλ are complex numbers, then Te∗ γIξc (0λ (g)∗ )γIξc (0λ (g))Te = Te∗ σλ,γ σγ,λ Te = 1

2

Sδ . Sλ Sγ

Proof. The proof is essentially contained in [FRS]. We define W = {g ∈ G | I0 ∪ gI0

is a proper interval contained in

S 1 \Iξ1 } .

For g ∈ W , we define Uγλ (g) = γI0 ∪g.I0 (0λ (g))Uγ (g). Then it is easy to check that if g1 ∈ W , g2 ∈ W and g1 g2 ∈ W , then we have: Uγλ (g1 g2 ) = Uγλ (g1 )Uγλ (g2 ) . For any g ∈ G, since G is connected, g can be decomposed as g = g1 · · · gn with gi ∈ W . Define Uγλ (g) = Uγλ (g1 ) · · · Uγλ (gn ) . By a standard deformation argument, using the fact that G is simply connected (see Proposition 8.2 in [GL2]), Uγλ (g) is independent of the decomposition of g. It follows from the proof of (v) of Lemma 4.8 in [Fro] that: Uγλ (g)(γ · λ)J (x)Uγλ (g) = (γ · λ)gJ (αg .x) for any x ∈ AJ . Since Te∗ Uγλ (g)Te is a representation of G associated with δ, it follows from Proposition 1.3.2 that Uδ (g) = Te∗ Uγλ (g)Te . We may assume, for simplicity, that I0 is so small that I0 ∩ π.I0 = ∅. Notice that in particular Uδ (2π) = Sδ = Te∗ Uγλ (2π)Te . Choose Iξ1 , Iξ2 such that Iξ1 , Iξ2 , I0 , π.I0 don’t intersect and anti-clockwise on the circle the order of the intervals are I0 , Iξ2 , π.I0 , Iξ1 . We have: Uγλ (2π) = Uγλ (π) · Uγλ (−π)∗ h i∗ = γIξc (0λ (π)∗ )Uγ (π) · γIξc (Uλ (−π)U0 (−π)∗ )Uγ (−π) 1

2

= γIξc (0λ (π)∗ )Uγ (2π) · γIξc U (π)Uλ (π)) 1

1

= γIξc (0λ (π)∗ )Sγ · Sλ · γIξc (0λ (π)). 1

2

So we have: Sδ = Te∗ · γIξc (0λ (π)∗ )γIξc (0λ (π))Te . 1 2 Sγ Sλ It is clear, by Lemma 1.4.1, that as long as g.I0 ∩ I0 = ∅, γIξc (0λ (g)∗ )γIξc (0λ (g)) = γIξc (0λ (π)∗ )γIξc (0λ (π)) . 1

2

1

The proof of the proposition is now completed.

2

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1.5. Diagrammatic representations. Let γ, α, β ∈ 1, where 1 is as the beginning of Sect. 1.4. A choice of basis of Hom(γ, αβ) which consists of isometries is called a gauge choice. The elements of the basis are called spin variables. We will denote elements in γ (x) ∈ Hom(γ, αβ) where x runs over a finite index set a basis of Hom(γ, αβ) by Pαβ which consists of n elements, and n = Hom(γ, αβ). γ γ It is instructive to represent Pαβ (x) (resp. Pαβ (x)∗ ) by the following diagrams: γ

γ Pαβ (x)

-

α

γ Pαβ (x)∗

x

@ @ α

-

β

@ % @% x

γ

β

∗ We represent σα,β (resp. σβ,α ) by an over crossing (resp. under crossing) diagram as follows:

α

β

σα,β -

β

α

∗ σβα

β

-

α

The diagram of the compositions of two operators ba with diagrammatic representations are defined to be a diagram which is obtained by putting the diagram of a on the top of the diagram of b. It will be extremely convenient to represent YBE and BFE relations in terms of diagrams. For similar treatment, see [MS or X2]. For the reader’s convenience, we have used dotted horizontal lines in the following diagrams. Between each two adjacent horizontal lines, there is only one crossing or trivalent graph indicating the corresponding operators. All the diagrams should be read from the top to the bottom, meaning the compositions of the corresponding operators from the top to the bottom. We will omit the spin variables x in the following for simplicity. However, it should be clear that it is straightforward to include the spin variables at each stage. Now the Yang-Baxter-Equation (YBE) as in Proposition 1.4.2 can be represented as:


λ

µ

361

γ

λ

µ γ

=

The Braiding-Fusion-Equation (BFE) corresponding to (a) and (b) of Proposition 1.4.2 can be represented as: λ

µ

γ

λ

µ

γ

δ

λ

=

δ

λ

µ γ

λ

µ

γ

λ

λ δ λ δ The diagrams corresponding to (c) and (d) of Proposition 1.4.2 is the same as above except the over-crossing is replaced by under-crossing. We will use these diagrammatic identities to prove most of the theorems in Sect. 3. 1.6. Conformal precosheaf from representation of Loop groups. Let G = SU (N ). We denote LG the group of smooth maps f : S 1 7→ G under pointwise multiplication. The diffeomorphism group of the circle DiffS 1 is naturally a subgroup of Aut(LG) with the action given by reparametrization. In particular the group of rotations RotS 1 ' U (1) acts on LG. We will be interested in the projective unitary representation π : LG → U (H) that are both irreducible and have positive energy. This means that π should extend to LG n Rot S 1 so that H = ⊕n≥0 H(n), where the H(n) are the eigenspace for the action of RotS 1 , i.e., rθ ξ = expinθ for θ ∈ H(n) and dim H(n) < ∞ with H(0) 6= 0. It follows

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from [PS] that for fixed level K which is a positive integer, there are only a finite number of such irreducible representations indexed by the finite set X X K λi 3i , λi ≥ 0 , λi ≤ K , P++ = λ ∈ P | λ = i=1,···,N −1

i=1,···,n−1

where P is the weight lattice of SU (N ) and 3i are the fundamental weights. We will K ν , define Nλµ = use 30 to denote the trivial representation of SU (N ). For λ, µ, ν ∈ P++ P (δ) (δ) (δ∗) (δ (δ) /S30 ), where Sλ is given by the Kac-Peterson formula: K Sλ Sµ Sν δ∈P++ X εw exp(iw(δ) · λ2π/n), Sλ(δ) = c w∈SN

where εw = det(w) and c is a normalization constant fixed by the requirement that Sµ(δ) is ν are non-negative integers. an orthonormal system. It is shown in [Kac2], p. 288 that Nλµ K with structure Moreover, define Gr(CK ) to be the ring whose basis are elements of P++ ν K ∗ constants Nλµ . The natural involution ∗ on P++ is defined by λ 7→ λ = the conjugate of λ as a representation of SU (N ). We shall use f to denote the positive energy representation of LSU (N ) at level K corresponding to the vector representation of SU (N ). The vacuum representation of a loop group LG is the positive energy representation of LG corresponding to the trivial representation of G. by S1(3) . Define dλ = We shall also denote S3(3) 0

S1(λ)

(30 )

S1

. We shall call (Sν(δ) ) the S-matrix

of LSU (N ). We shall encounter the ZN group of automorphisms of this set of weights, generated by σ : λ = (λ1 , λ2 , · · · , λN −1 ) → σ(λ) = (K − 1 − λ1 − · · · λN −1 , λ1 , · · · , λN −2 ). Define color τ (λ) ≡ Σi (λi − 1)i mod(N ). The central element ω = exp 2πi N of SU (N ) (λ) . We also make use of acts on the representation of SU (N ) labeled by λ as exp 2πiτ N the N (linearly dependent) vectors ei ˆ 1 , ei = 3 î−3 ˆ i−1 , i = 1, · · · N − 1, eN = −3 ˆ N −1 . e1 = 3 The irreducible positive energy representations of LSU (N ) at level K give rise to an irreducible conformal precosheaf A (see Sect. 2) and its covariant representations in the following way: First note if πλ is a representation of central extension of Diff+ (S 1 ) on Hλ , then πλ induces an action of G as follows: Let us denote the induced central extension of ] P SL(2, R) ⊂ Diff+ (S 1 ) from that of Diff+ (S 1 ) by P SL(2, R) which is a circle bundle, ] denoted by L, over P SL(2, R) (cf. Sect. 4.5 of [PS]). Let π2 : P SL(2, R) → P SL(2, R) and π1 : G → P SL(2, R) be the natural covering maps. Since G is a simply connected simple Lie group, the pull back circle bundle π1∗ (L) on G is homemorphic to G × S 1 . It ] follows that there exists a homomorphism ϕ from G to P SL(2, R) such that π2 .ϕ = π1 . We shall fix ϕ and denote by πλ (g) the operator πλ (ϕ(g)) for any g ∈ G in the following. The conformal precosheaf is defined by A(I) = π0 (LI G)00 . In fact, by the results in Chapter 2, Theorem A, B, C, E and F of [W2] or Theorem 3.2. of [FG] that A(I) satisfies A to F of Sect. 1.2 and therefore is indeed an irreducible conformal precosheaf.


363

Let U (λ, I) be a unitary operator from Hλ to H0 such that: πλ (x) = U (λ, I)∗ π0 (x)U (λ, I) for any x ∈ LI G. Fix I0 ⊂ S 1 . We define a collection of maps as follows. For any interval J ⊂ S 1 , x ∈ A(J), λJ (x) = U (λ, I0c )U ∗ (λ, J)xU (λ, J)U (λ, I0c )∗ . It follows that if J ⊃ I0 , then λJ (x) commutes with A(J c ) for any x ∈ A(J). By Haag-duality, if J ⊃ I0 , λJ (AJ ) ⊂ A(J). Define: Uλ (g) = U (λ, I c )πλ (g)U (λ, I c )∗ . It is easy to check that {λJ } gives a covariant representation of conformal precosheaf A. Let us note that the intervals in Sect. 1.2 are defined to be open intervals. We can actually choose the interval to be closed since we shall be concerned with the conformal precosheaves from positive energy representations of LG and by Theorem E of [W2], π(LI G)00 = π(LI¯ G)00 where I¯ is the closure of I. The collection of maps {λJ } define an endomorphism λ of C ∗ (A) (see [GL2], Sect. 8). The relation between λ and λJ is given by π0 (λ(iJ (x))) = λJ (x)

for any

x ∈ A(J) .

Here iJ : A(J) → C ∗ (A) is the embedding of A(J) in C ∗ (A), and π0 is the vacuum representation of C ∗ (A). This makes the definition of composition λ · µ of two covariant representations {λJ } and {µJ } straightforward. One simply defines λ · µ as the composition of λ, µ as endomorphisms of C ∗ (A). It is easy to check that if J ⊃ I1 , π0 ((λ · µ)(iJ (x))) = λJ · µJ (x) . An equivalent definition can be found in Sect. 4.2 of [?]. We shall also be concerned with the positive energy representations at level K of LSU (N ) which is not irreducible but may be decomposed into a direct sum of finitely many irreducible representations. We will denote the set of such representations by CK . By abuse of notation we shall use λ to denote an element of CK . It is easy to see that the previous construction will give us a covariant (not necessarily irreducible) representation of A for any λ ∈ CK . We shall also use λ to denote the localized endomorphism of A(I0 ). All the sectors [λ] with λ irreducible generate a ring (see Sect. 1.1). We will call such a ring the fusion ring. For λ irreducible, the univalence Sλ is given by an explicit formula (cf. 9.4 of [PS]). c2 (λ) , where c2 (λ) is the value of the Casimir operator on the Let us first define 1λ = K+N representation of SU (N ) labeled by the dominant weight λ. 1λ is usually called the conformal dimension. Then we have: Sλ = exp(2πi1λ ). The following remarkable result is proved in [W2] (See corollary 1 of Chapter V in [W2]). (K) has a finite index with index value d2λ . The fusion ring Theorem 1.2. Each λ ∈ P++ (K) generated by all λ ∈ P++ is isomorphic to Gr(CK ).

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Proof. It is easy to check that the tensor product of bimodules defined in Sect. 30 of [W1] correspond precisely to the compositions of the corresponding endomorphisms defined in this section. By Sect. 1.1 the fusion ring generated by λ’s is exactly the same as the fusion ring defined in Sect. 33 of [W1], and its structure is given by Corollary 1 of Chapter V in [W2]. The equivalence between the ring structure described in Corollary 1 of Chapter V in [W2] and Gr(CK ) described above is proved as an exercise on p. 288 of [?]. In Sect. 3, we will also need a similar but much simpler results as Theorem 1.6 for positive energy representations of loop group LH at level 1 where H = E6 , E7 , E8 , Spin(N ). We shall use σi to denote such representations, and Sσi σj to denote the S matrix of LH at level 1, where Sσi ,σj are defined on p. 264 of [Kac2] similar as above. For each σi we can define a covariant representation of an irreducible conformal precosheaf from the vacuum representation of LH exactly to that above, using Theorem 3.2 of [FG]. By abuse of notation, we shall use the same σi to denote the endomorphism which is localized on a fixed interval I0 . The ring structure associated to the H = Spin(N ) case is determined by Theorem 7.1 of [JB]. When H = E8 , there is only one representation, the vacuum representation at level 1, the fusion ring is generated by the identity element. When H = E6 (resp. H = E7 ), the center of E6 is Z3 (resp. Z2 ). The irreducible level 1 representations are in one-to-one correspondence with the elements of the center of H (cf. p. 181 of [PS]) . Denote by π0 , πσ the vacuum representation and the representation corresponding to the generator σ in the center with order n. Then (cf. p. 181 of [PS]): πσ ∼ = π0 · Adασ , where ασ is a smooth path from the identity element e of H to σ along S 1 , and Adασ is the adjoint action of ασ on LH. Choose ασ in such a way that ασ |I0c ≡ e. Then the sector [Adασ ] generates the fusion ring, and since ασn ∈ LI0 H, it follows that [Adnασ ] = id and the fusion ring is isomorphic to the group ring of Z. In all the cases above, i.e., when H = E6 , E7 , E8 , Spin(N ), the fusion rings determined above have also been computed by using the formula on p. 288 of [Kac2] in [Wal] on P.784-788. In particular this shows that the irreducible representations of the fusion ring are given by (cf. p. 288 of [Kac2]): σi →

Sσi σj . S1σj

Here the subscript 1 in S1σj is used to denote the vacuum representation of LH. 2. Subfactors from Conformal Inclusions 2.1. Nets of subfactors. In this section we sketch some of the results of [LR] which will be used in this paper. For the details of all the proofs and unexplained terminology , we safely refer the reader to [LR]. We have changed some of the notations in [LR] since they have been used to denote different objects in this paper. Let N ⊂ M be an inclusion of type III von Neumann algebras on a Hilbert space H. Let φ ∈ H be a joint cyclic and separating vector (which always exists if N is type


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III and H is separable). Let jN = AdJN and jM = AdJM be the modular conjugations w.r.t.φ and the respective algebra. Then α = jN jM |M ∈ End(M ) maps M into a subalgebra of N . We call α the canonical endomorphism associated with the subfactor, and denote by N1 := jN jM (M ) ⊂ N, M ⊂ M1 := jM jN (N ) the canonical extension resp. restriction. φ is again joint cyclic and separating for the new inclusions above, giving rise to new canonical endomorphisms β = jN1 jN ∈ End(N ) and α1 = jM jM1 ∈ End(M1 ). We have the following formula for canonical endomorphism: (cf. Proposition 2.9 of [LR]): Proposition 2.1. Let N ⊂ M be an inclusion of properly infinite factors, : M → N a faithful normal conditional expectation, and eN ∈ N 0 the associated Jones projection. The canonical endomorphism α : M → N is given by γ = 9−1 · 8, where

8(m) = U mU ∗ (m ∈ M )

is the isomorphism of M into N eN implemented by an isometry U ∈ M1 = hM, eN i with U U ∗ = eN , and 8 is the isomorphism of N with N eN given by 8(n) = neN (n ∈ N ). Every canonical endomorphism of M into N arises in this way. Definition. A net of von Neumann algebras over a partially ordered set J is an assignment M : i → Mi of von Neumann algebras on a Hilbert space to i ∈ J which preserves the order relation, i.e., Mi ⊂ Mk if i ≤ k. A net of subfactors consists of two nets N and M such that for every i ∈ J , Ni ⊂ Mi is an inclusion of subfactors. We simply write N ⊂ M. The net M is called standard if there is a vector ∈ H which is cyclic and separating for every Mi . The net of subfactors N ⊂ M is called standard if M is standard and N is standard on a subspace H0 ⊂ H with the same cyclic and separating vector ∈ H0 . For a net of subfactors N ⊂ M, let be a consistent assignment i → i of normal conditional expectations. Consistency means that i = k |Mi whenever i ≤ k. Then we call a normal conditional expectation from M onto N . is called standard, if it preserves the vector state ω = (, ·). If the index set J is directed, i.e., for j, k ∈ J there is m ∈ J with j, k ≤ m, we with a net M of von Neumann algebras the inductive limit C ∗ algabra S associate ( i∈J Mi )− and denote it by the same symbol M. Then we have (cf. Cor.3.3 of [LR]): Proposition 2.2. Let N ⊂ M be a directed standard net of subfactors (w.r.t. the vector ∈ H) over a directed set J , and a standard conditional expectation. For every i ∈ J there is an endomorphism αi of the C ∗ algebra M into N such that α|Mj is a canonical endomorphism of Mj into Nj whenever i ≤ j. Furthermore, αi acts trivially on Mi0 ∩ N . As i ∈ J varies to k, the corresponding endomorphisms αi and αk are inner equivalent by a unitary in Nl whenever i, k ≤ l.

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If the index set J is directed, by Cor.4.2 of [LR] the index is constant in a directed standard net of subfactors with a standard conditional expectation. We have (cf. Cor.4.3 of [LR]): Proposition 2.3. Let N ⊂ M be a directed standard net of subfactors (w.r.t. the vector ∈ H) over a directed set J , and a standard conditional expectation. If the index d2 = Ind() is finite, then for any i ∈ J , there is an isomorphic intertwiner v1 : id → α in Mi which satisfies the following identity with the isometric intertwiner w0 : id → β (β := α|N ) in Ni : w0∗ v1 = d−1 id = w0∗ α(v1 ). α is given on M by the formula α(m) = d2 (v1 mv1∗ )(m ∈ M). Furthermore, every element in M is of the form nv1 with n ∈ N , namely m = d2 (mv1∗ )v1 = d2 v1∗ (v1 m). 2.2. Subfactors from Conformal Inclusions. Let G ⊂ H be inclusions of compact simple Lie groups. LG ⊂ LH is called a conformal inclusion if the level 1 projective positive energy representations of LH decompose as a finite number of irreducible projective representations of LG. LG ⊂ LH is called a maximal conformal inclusion if there is no proper subgroup G0 of H containing G such that LG ⊂ LG0 is also a conformal inclusion. A list of maximal conformal inclusions can be found in [GNO]. Let H 0 be the vacuum representation of LH, i.e., the representation of LH associated with the trivial representation of H. Then H 0 decomposes as a direct sum of irreducible projective representation of LG at level K. K is called the Dynkin index of the conformal inclusion. Assume H 0 = ⊕λ∈P0 mλ Hλ where P0 ⊂ CK is finite and mλ ˆ is the multiplicity of Hλ in H 0 . We shall write the conformal inclusion as Gˆ K ⊂ H. We shall limit our consideration to the following conformal inclusions though most of d 1 , SU d(2)28 ⊂ d(2)10 ⊂ SO(5) the arguments apply to other cases as well. For SU (2) : SU d d d d ˆ ˆ ˆ ˆ G2 , For SU (3) : SU (3)5 ⊂ SU (6)1 , SU (3)9 ⊂ E6 , SU (3)21 ⊂ E7 ; (A8 )1 ⊂ (Eˆ 8 )1 ; and four infinite series: d N (N − 1) , N ≥ 4; d(N )N −2 ⊂ SU (a) SU 2 1 N (N + 1) d d ; (b) SU (N )N +2 ⊂ SU 2 1 2 d(N )2N ⊂ Spin(4N [ − 1)1 , N ≥ 2; SU d [ SU (2N + 1)2N +1 ⊂ Spin(4N (N + 1))1 .

(c) (d)

These cover all the maximal conformal inclusions of the form SU (N ) ⊂ H with H being a simple group. Let G = SU (N ) and I be an interval of S 1 . Set I c = S 1 \I. Let LI G = {f ∈ LG | f = e on I c } and LI H = {f ∈ LH | f = e on I c }, where e is the identity element of H. Let π 0 be the vacuum representation of LH on Hilbert space H 0 with vacuum vector . By Theorem 3.3 of [FG] π 0 (LI H)00 is isomorphic to the unique hyperfinite III1 factor M . Proposition 2.4. π 0 (LI G)00 ⊂ π 0 (LI H)00 is an irreducible inclusion with finite index.


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Proof. Since G ⊂ H is an conformal inclusion, we can assume H 0 = ⊕λ∈P0 mλ Hλ , where P0 ⊂ CK is finite and mλ is the multiplicity of Hλ . Let H0 be the vacuum representation of LG. Denote by qλ the projections from H 0 to Hλ . By Theorem 3.3 of [FG] is the unique U (G) invariant vector in H 0 of lowest weight, so must be in H0 with multiplicity m0 , and therefore m0 = 1. Assume q ∈ π 0 (LI G)0 ∩ π 0 (LI H)00 . Then q ∈ π 0 (LI G)0 ∩ π 0 (LI c G)0 . By Theorem F in [W1], π 0 (LI G)0 ∩ π 0 (LI c G)0 = π 0 (LG)0 . Notice q0 is a minimal abelian projection in π 0 (LG)0 , so we have: q = qq0 = a for some constant complex number a. Since q ∈ π 0 (LI H)00 and is separating for π 0 (LI H)00 , we must have q = a · id. To prove the inclusion is of finite index, consider the following inclusions: π 0 (LI G)00 ⊂ π 0 (LI H)00 = π 0 (LI c H)0 ⊂ π 0 (LI c G)0 , where the = in the middle follows from Haag duality (cf. Theorem 3.3 of [FG]). By [L1], the statistical dimension of π 0 (LI G)00 ⊂ π 0 (LI c G)0

P is λ∈P mλ dλ which is finite by the definition of conformal inclusion and Theorem 1.6. Hence π 0 (LI G)00 ⊂ π 0 (LI H)00 is of finite index. The subfactor in Proposition 2.4 is called the subfactor from conformal inclusions. Now fix intervals I0 , I1 , J such that J¯ = I¯0 ∪ I¯1 , I0 ∩ I1 = ∅ and I1 lies in the anticlockwise direction from I0 in J. We define a directed set J to be the set which consists ¯ For any two elements I 0 , I 00 ∈ J , I 0 ≤ I 00 iff of all the intervals I 0 such that I 0 ⊂ J. I 0 ⊂ I 00 . For simplicity, we shall use the following notations in the rest of the paper: M (I) := π 0 (LI H)00 , N (I) := π 0 (LI G)00 , M1 (I) := π 0 (LI c G)0 , A(I) := π0 (LI G)00 , M := A(I0 ). Let eN := q0 be the unique projection from H 0 to H0 as in the proof of Proposition 2.4. Proposition 2.5. (1) N (I) ⊂ M (I) is a directed standard net of subfactors w.r.t. the vector ∈ H 0 over the set J for any I ∈ J , and there exists a unique standard conditional expectation from M (I) to N (I). (2) M1 (I) = hM (I), eN i. Proof. By Theorem 3.2 of [FG], is cyclic and separating for M (I). Similarly is cyclic and separating for N (I) restricted to H0 . Since the modular conjugation 1it of M (I) associated to the state ω = (, ·) is geometric and is given by (b) of Proposition 1.2.1, it follows that 1it N (I)1−it = N (I) and by Takesaki’s theorem (cf. Sect. 12. of [W1]) there exists a normal conditional expectation I from M (I) to N (I) and preserves the state ω. In fact such an I is unique and faithful since N (I) ⊂ M (I) is irreducible and of finite index. Recall eN is the projection from H 0 to H0 . Then

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I (m) = eN m. It follows that if I ⊂ I 0 and m ∈ M (I), then I (m) = I 0 (m). So we have I (m) = I 0 (m), 0

since is separating for M (I ). Thus I is a standard conditional expectation from M (I) to N (I). To prove (2), notice that hM (I), eN i is the basic construction associated with N (I) ⊂ M (I) and the unique conditional expectation I , so the statistical dimension of N (I) ⊂ hM (I), eN i is d2 , where d is the statistical dimension of N (I) ⊂ M (I). It is obvious that hM (I), eN i ⊂ M1 (I), so we just have to show that the statistical dimension of N (I) ⊂ M1 (I) is also d2 . The subfactor M (I) ⊂ M1 (I) is anti-conjugate to N (I c ) ⊂ M (I c ) which has the same statistical dimension d as that of N (I) ⊂ M1 (I) since N (I c ) ⊂ M (I c ) is conjugate to N (I) ⊂ M1 (I), using congugation by U (g), where U (g) is the representation of G on H 0 and g.I = I c . It follows that the statistical dimension of N (I) ⊂ M1 (I) is also λ2 and hM (I), eN i = M1 (I). By Proposition 2.4 and Proposition 2.3, we have a canonical endomorphism αI0 : M (I) → N (I) for any I ⊃ I0 . Moreover, we have v1 ∈ M (I0 ) which satisfy the properties stated in Proposition 2.3. Let w1 = αI0 (v1 ). Recall β I0 := αI0 |N (I). For simplicity, we shall drop the labels and write αI0 , β I0 simply as α, β in the following. Define U (γ, I) : H 0 → H0 to be a unitary operator which commutes with the action of LI G as in Sect. 1.6. Notice that such an operator always exists since π 0 |LI G is equivalent to that of π0 . We shall think of U (γ, I) as an element of B(H 0 ) by identifying H0 as a subspace of H 0 . Define: φI : B(H 0 ) → B(H0 ) by φI (m) = U (γ, I)mU (γ, I)∗ , m ∈ B(H 0 ). We claim α can be chosen to be c α(m) = φ−1 I (φI0 (m)).

(1)

In fact by Proposition 2.1 and the proof of Proposition 2.2 in [LR], we just have to show c c φ−1 I (φI0 (m))eN = φI0 (m)

(2)

for any I ⊃ I0 and m ∈ M (I). Notice for any m ∈ M (I) and I ⊃ I0 , φI0c (m) ∈ A(I) as operators on H0 by definition of φI0c and Haag duality for A(I). So it is sufficient to check (2) for the case φI0c (m) = π0 (x) with x ∈ LI G since π0 (LI G) generates A(I). From the definitions we have


369

∗ φ−1 I (π0 (x))eN = U (γ, I) π0 (x)U (γ, I)eN

= U (γ, I)∗ U (γ, I)π 0 (x)eN = π0 (x)eN , and hence (2) and (1) hold. For any a ∈ A(I), define: γI (a) := φI · β · φ−1 I (a). Notice since φ−1 I (A(I)) = N (I), γ is well defined. By using (1), we have: γI (a) = φI0c (φ−1 I (a)).

(3)

which is precisely the definition of the localized (localized on I0 ) covariant endomorphism associated with the representation H 0 of LG as in Sect. 1.6. We shall denote γI0 for the fixed interval I0 by γ. Notice γ ∈ End(M ). Since β|N (I0 ) is a canonical endomorphism, we can find ρ1 ∈ End(N (I0 )) such that: β|N (I0 ) = ρ1 ρ¯1 with ρ¯1 (N (I0 )) ⊂ N (I0 ) conjugate to N (I0 ) ⊂ M (I0 ). Moreover, ρ1 (N (I0 )) = α(M (I0 )). Define ρ ∈ End(M ) and w ∈ M by ρ := φI0 · ρ1 · φ−1 I0 , w := φI0 (W1 ). Proposition 2.6. (1) γ = ρρ¯ ; (2) The subfactor ρ(M ¯ ) ⊂ M is conjugate to the subfactor N (I0 ) ⊂ M (I0 ); (3) There exists isometries v, v2 in M with w = ρ(v2 ) such that: ¯ vx = γ(x)v, v2 x = ρρ(x)v 2, wγ(x) = γ 2 (x)w and

γ(ρ(x))w = wρ(x),

ww∗ = γ(w∗ )w, ∗

w v=

d−1 ρ 1

for any

x∈M

ww = γ(w)w, ∗

= w γ(v).

Moreover, for any element x ∈ ρ(M ), x can be written as x = x1 w with x1 ∈ γ(M ). ¯ ). Similarly, for any element y ∈ M, y = y1 v2 with y1 ∈ ρ(M Proof. (1) and (2) follows from the definitions. (3) follows from Theorem 5.1 of [L2] and Cor.5.6 of [L4]. Let us choose h ∈ G with h.I0 = I1 . Recall from Sect. 1.3 that the braiding operator σγ,γ ∈ M is defined by: σγ,γ = γJ (0γ (h)∗ )0γ (h). Proposition 2.7. σγ,γ w = w.

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Proof. The proof is contained in the proof of Cor.4.4 of [LR]. Let us explain the proof in our notations. Notice that ∗ φ−1 J σγ,γ φJ = β(u0 )u0 −1 with u0 = φ−1 J (0γ (h)), and φJ (w) = α(v1 ). It is sufficient to show that

β(u∗0 )u0 α(v1 ) = α(v1 ). It follows from the intertwining property (see Sect. 1.3) that u0 β(n) = nu0 (∀n ∈ N (I0 )). Let u = λ(v10 v1∗ ), where v10 ∈ M (I1 ) is the choice of isometry similar to v1 but corresponds to the interval I1 in Proposition 2.3. By Proposition 2.3, uv1 = v10 , u0 β(n) = nu0 (∀n ∈ N (I0 )). It follows that

uu∗0 ∈ N (I0 )0 ∩ N (J).

So φJ (uu∗0 ) ∈ A(I0 )0 ∩ A(J) = A(I1 ), where A(I0 )0 ∩ A(J) = A(I1 ) follows from the proof of (0) in Lemma 1.4.1. By applying φ−1 J to both sides, we have proved that: uu∗0 ∈ N (I1 ). Since β is localized on I0 , we conclude that β(uu∗0 ) = uu∗0 , i.e., β(u∗0 )u0 = β(u∗ )u. Since uv1 = v10 ∈ M (I1 ) and I0 ∩ I1 = ∅, by locality we have uv1 v1 = v1 uv1 = β(u)v1 v1 . Using α(v1 )v1 = v1 v1 we have uα(v1 )v1 = β(u)α(v1 )v1 . Now multiply v1∗ from the right on both sides and apply , using (v1 v1∗ ) = λ−1 , we have β(u∗ )uα(v1 ) = β(u∗0 )u0 α(v1 ) = α(v1 ).

Recall σj is the localized endomorphism (localized on I0 ) as in the end of Sect. 1.6. Let us recall how σj is defined. Let U (j, I) : Hσj → H 0 be a unitary map which commutes with the action of LI H. Define: ψI : B(Hσj ) → B(H 0 ) by ψI (x) = U (j, I)xU (j, I)∗ . Then σj is given by: σj (m) = ψI0c (ψI−1 (m)) (∀m ∈ M (I)). Let γj be the reducible representation of LG on Hσj . Then the localized endomorphism, denoted by the same γj , is given by: γj = φI0c · ψI0c ψI−1 · φ−1 I . Define σj0 ∈ End(M ) by: σj0 = ρ−1 · φI0c · σj · φ−1 I c · ρ. 0

c Notice by the definition of ρ1 , ρ1 (N (I0 )) = α(M (I0 )) = φ−1 I0 φI0 (M (I0 )), and it follows 0 that φ−1 ρ(M ) = M (I ), so σ as above is well defined. 0 j Ic 0


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Proposition 2.8. (1) As elements in End(M ), we have ρσj0 ρ¯ = γj ; (2) σγ,γj w = ρσj0 ρ−1 (w). Proof. (1) follows directly from the definitions and the formula γ = ρρ. ¯ −1 ∗ 0 −1 (σ w) = α(σ(u ))u α(v ) and φ To prove (2), notice φ−1 γ,γ 0 1 j 0 J J (ρσj ρ (w)) = α(σj (v1 )), it is sufficient to show that α(σj (u∗0 ))u0 α(v1 ) = α(σj (v1 )). Here u0 = φ−1 J (0γ (h)) as in the proof of Proposition 2.7. Recall from the proof of Proposition 2.7 that u = λ(v10 v1∗ ), where v10 ∈ M (I1 ) is the choice of isometry similar to v1 but corresponds to the interval I1 in Proposition 2.3. Since ασj as an endomorphism of N (I) is localized on I0 , exactly the same argument as in the proof of Proposition 2.6 shows that α(σj (u∗0 ))u0 = α(σj (u∗ ))u. So we just have to show α(σj (u∗ ))uα(v1 ) = α(σj (v1 )). By Proposition 2.7 we have α(u∗ )uα(v1 ) = α(v1 ), i.e., uα(v1 ) = α(u)α(v1 ), so α(σj (u∗ ))uα(v1 ) = α(σj (u∗ ))α(u)α(v1 ) = α(σj (u∗ )uv1 ) = α(σj (u∗ )σj (uv1 )) = α(σj (v1 )), where in the third = we have used that uv1 ∈ M (I1 ) and σj is localized on I0 .qe In the next chapter all the endomorphisms will be in End(M ). For this reason we shall write σj0 in Proposition 2.8 simply as σj .

3. New Braided endomorphisms In this chapter all the endomorphisms will be in End(M ). P By formula (3) of Sect. 2, [γ] = λ∈P0 mλ [λ] ∈ Gr(CK ). By Proposition 2.6, γ = ρρ. ¯ So Proposition 2.6 will allow us to study ρ by γ which is an element in Gr(CK ). By Proposition 2.6, w ∈ Hom(γ, γ 2 ). Hence we can represent w and w∗ as:

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γ

w

-

γ

@ @ γ

w∗

-

γ

γ @ % @%

γ

Let σλ,µ : λµ → µλ be the braiding operator as in Sect. 1.4. We will drop the subscript of σλ,µ and write σλ,µ simply as σ. In any case the subscript can always be recovered by tracing which spaces it acts on. 3.1. New braided endomorphisms. Fix λ ∈ CK (λ is not necessarily an irreducible representation). Let σ : γ · λ → λ · γ, be the braiding operator as defined in Sect. 1.4. Denote by λσ ∈ End(M ) which is defined by λσ (x) = σ ∗ λ(x)σ for any x ∈ M . Let : M → ρ(M ) be the minimal conditional expectation (see [L4]) given by (x) = w∗ γ(x)w for any x ∈ M . Theorem 3.1. (1) λσ (ρ(x)) ∈ ρ(M ) for any x ∈ M . (2) (λσ (x)) = λσ ((x)) for any x ∈ M . Proof. By the definitions of λσ we have λσ γ(x) = γ(λ(x)) ⊂ ρ(M ). Since by Proposition 3.1, every element x ∈ ρ(M ) can be written as x = x1 w with x1 ∈ γ(M ). It is sufficient to show λσ (w) ⊂ ρ(M ). We claim λσ (w) = γ(σ)w i.e. σ ∗ λ(w)σ = γ(σ)w. It is equivalent to λ(w)σ = σγ(σ)w which follows from (c) of Proposition 1.4.2. Hence we have proved λσ (w) = γ(σ)w. Since w ∈ ρ(M ), γ(σ) ∈ ρ(M ), therefore λσ (w) ∈ ρ(M ) and the proof of (1) is complete. As for (2), By using λσ (w) = γ(σ)w we have λσ ((x)) = λσ (w∗ )λσ (γ(x))λσ (w) = w∗ γ(σ)∗ γ(λ(x))γ(σ)w = (λσ (x)) Now let v be the isometry in M with vx = ρρ(x)v ¯ for any x ∈ M . Define aλ (x) = v ∗ ρ¯ · λσ · ρ(x)v. Corollary 3.2. aλ ∈ End(M ) and ρ · aλ = λσ · ρ, aλ · ρ¯ = ρ¯ · λ. If we denote by daλ , dλ the statistical dimension of aλ and λ, then daλ = dλ . Proof. Let x, ∈ M be an arbitrary element. We have ρ · aλ (x) = ρ(v ∗ ρ¯ · λσ · ρ(x)v) = ρ(v ∗ ρ¯ · ρ(z)v) = ρ(z) = λσ ρ(x). And

aλ · ρ(x) ¯ = v ∗ ρ¯ · λσ ρ · ρ(x)v ¯ ¯ = ρ¯ · λ(x). = v ∗ ρ¯ · ρ · ρ(λ(x))v


373

Since ρ · aλ = λσ · ρ and ρ is one-to-one, it makes sense to write aλ = ρ−1 · λσ · ρ. It follows immediately that aλ ∈ End(M ). By the multiplicative property of the statistical dimension and aλ · ρ¯ = ρ¯ · λ, we have daλ dρ¯ = dρ¯ dλ . Since dρ¯ > 1, it follows that daλ = dλ . Remark. (1) As in the proof above, since ρ · aλ = λσ · ρ and ρ is one-to-one, it makes sense to write aλ = ρ−1 ·λσ ·ρ. One can think of aλ as measuring the non-commutativity of ρ and λσ : in fact, it follows from Theorem 3.3 that aλ is in general different from λσ as sectors. Following [Ka1] and (2) of Theorem 3.1, aλ is referred to as quotient endomorphisms as it can be thought as obtained from λ “divided” by the symmetry provided by ρ. Remark. (2): There are two kinds of braidings one may use in theorem 3.1. The one we used corresponds to over crossing (see Sect. 2). If we choose to use under crossing braiding, then we get a˜ λ which is in general different from aλ (See the beginning of Sect. 3.2). However, it is clear from the proof that all the results in this section about aλ hold true for a˜ λ . Next we claim that aλ is a braided endomorphism of M . (See [L1]), i.e., there exists a unitary σ in a2λ (M )0 ∩ M that satisfies the braiding relation σaλ (σ)σ = aλ (σ)σaλ (σ). Let us first prove the following: Theorem 3.3. Let µ, λ ∈ CK (µ, λ are not necessarily irreducible). Then: ¯ aλ ρ) ¯ = Hom(aµ , aλ ) . (1) Hom(aµ ρ, (2) Hom(ρaµ , ρaλ ) = ρ(Hom(aµ , aλ )). ¯ aλ ρ) ¯ ⊃ Hom(aµ , aλ ). We need to show Proof. It is obvious that Hom(aµ ρ, ¯ aλ ρ) ¯ ⊂ Hom(aµ , aλ ). Notice y ∈ Hom(aµ ρ, ¯ aλ ρ) ¯ iff: Hom(aµ ρ, ∀x ∈ M,

¯ = yaµ · ρ(x). ¯ aλ · ρ(x)y

Since by Corollary 3.2, aλ · ρ(x) ¯ = ρ¯ · λ(x), aµ · ρ(x) ¯ = ρ¯ · µ(x), by applying ρ to both sides of the equation the above is equivalent to: γ · λ(x)ρ(y) = ρ(y)γ · µ(x). Recall γ · λ = λσ · γ, γ · µ = µσ · γ (we have omitted the subscript of σ, but it ¯ aλ ρ) ¯ iff: should be clear from the context what they should be), hence y ∈ Hom(aµ ρ, λσ · γ(x)ρ(y) = ρ(y)µσ · γ(x) for any x ∈ M . Similarly it follows from Corollary 3.2 that z ∈ Hom(aµ , aλ ) iff λσ · ρ(x) ρ(z) = ¯ aλ ρ) ¯ ⊂ Hom(aµ , aλ ), it is ρ(z) µσ · ρ(x) for any x ∈ M . Hence to show Hom(aµ ρ, sufficient to show if ρ(z) satisfies λσ · γ(x) ρ(z) = ρ(z) µσ · γ(x) for any x ∈ M , then λσ · ρ(x) ρ(z) = ρ(z) λµ · ρ(x) for any x ∈ M . By Proposition 3.1, all we need to show is if ρ(z) ∈ Hom(µσ · γ, λσ · γ), then λσ (w)ρ(z) = ρ(z)µσ (w). Notice from the proof of Theorem 3.1 we have λσ (w) = γ(σ)w, µσ (w) = γ(σ)w. In terms of diagrams, ρ(z)µσ (w) = ρ(z)γ(σ)w can be represented as:

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γ

µ w

γ(σ) aa a ρ(z) ,@ , , @ γ λ

µ

By using YBE and BFE, the above diagram is equal to: γ

µ w σ∗ γ (ρ(z)) σ γ(σ)

γ

λ

µ

∗ Using the fact σγ,γ w = w, we see the above diagram corresponds to:

γ(σ)σγ,γ γ(ρ(z))w. Since γ(ρ(z))w = wρ(z), we have γ(σ)σγ,γ γ(ρ(z))w = γ(σ)σγ,γ wρ(z) = γ(σ)wρ(z) = λσ (w)ρ(z). As for (2), it is clear that Hom(ρaµ , ρaλ ) ⊃ ρ(Hom(aµ , aλ )). To prove the equality, we just have to show the dimension of ρ(Hom(aµ , aλ )) is equal to that of Hom(ρaµ , ρaλ ). Since ρ is one to one, all we need to show is: haµ , aλ i = hρaµ , ρaλ i. By (1) Frobenius duality: ¯ aλ ρi ¯ haµ , aλ i = haµ ρ, = hρµ, ¯ ρλi ¯ = hγµ, λi. On the other hand, from Corollary 3.2 we have


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hρaµ , ρaλ i = hµσ ρ, λσ ρi = hµρ, λρi = hµγ, λi = hγµ, λi. So we have haµ , aλ i = hρaµ , ρaλ i = hγµ, λi.

Remark. It follows immediately from the above theorem that every irreducible subsector of aµ remains irreducible when multipy from the right (resp. from the left) by ρ¯ (resp. ρ). Corollary 3.4. (1) For any λ, µ ∈ CK , aλµ = aλ aµ ; (2) aλ is a braided endomorphism. Proof. (1) Apply ρ to both sides, we just have to show ρaλµ = ρaλ aµ . By using Cor.3.2, it is sufficient to show: λ(σγ,µ )σγ,λ = σγ,λµ , which follows directly from the definitions of the braiding operator before Proposition 1.4.2 in Sect. 1.4. (2) Denote by τ = σλ,λ the braiding unitaries in λ20 (M ) ∩ M . It follows from YBE that τ λ(τ )τ = λ(τ )τ λ(τ ). Let σ = ρ(τ ¯ ) ∈ ρλ ¯ 20 (M ) ∩ M = a2λ ρ¯0 (M ) ∩ M . By Theorem 3.3, 2 0 20 σ ∈ aλ ρ¯ (M ) ∩ M = aλ (M ) ∩ M . By applying ρ¯ to the equation τ λ(τ )τ = λ(τ )τ λ(τ ) and using aλ · ρ¯ = ρ¯ · λ, we get: τ aλ (τ )τ = aλ (τ )τ aλ (τ ). Hence aλ is a braided endomorphism. Recall from Sect. 2.1 that f denotes the vector representation of SU (N ). Corollary 3.5. (1) af is irreducible, i.e. a0f (M ) ∩ M = C1. (2) [aλ¯ ] = [¯aλ ]. ¯ 0 (M ) ∩ M it is sufficient to show Proof. (1) Since, a0f (M ) ∩ M ⊂ af ρ¯0 (M ) ∩ M = ρf 0 ρ¯ (M ) ∩ M = C1. By Frobenius duality (see [L2] or [Y]), ¯ af ρi ¯ = hρf, ¯ ρf ¯ i haf ρ, = hρρ, ¯ f f¯i X mλ λ, 1 + adi, =h λ∈P0

where ad denotes the adjoint representation of SU (N ). ¯ af ρi ¯ = 1, it is sufficient to show ad ∈ / P0 . By [Kac1], Since m1 = 1, to show haf ρ, Cλ any λ ∈ P0 has the property that K+N ∈ Z, where Cλ is the eigenvalue of the Casimir operator. Cad N Since 0 < P / P0 . K+N = K+N < 1, (recall K > 0) it follows that ad ∈ (2) Let [aλ ] = i mi xi , where xi are irreducible subsectors of [aλ ] and mi are positive integers. ¯ ρi ¯ = haλ aλ¯ , 1i, ¯ ρ, ¯ by (1) of Cor.3.4. By Theorem 3.3, haλ aλ λλ P Notice aλ aλ¯ = aP ¯ ρi ¯ = i mi hxi aλ¯ , 1i. Since hxi aλ¯ ρ, ¯ ρi ¯ ≥ hxi aλ¯ , 1i, it follows that i.e., i mi hxi aλ¯ ρ, hxi aλ¯ ρ, ¯ ρi ¯ = hxi aλ¯ , 1i. On the other hand,

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¯ ρi hxi aλ¯ ρ, ¯ ρi ¯ = hxi ρ¯λ, ¯ ¯ = hxi , ρλρi ¯ λi = hxi , ρρa ≥ hxi , aλ i = mi .

P So we have hxi aλ¯ , 1i = haλ¯ , x¯ i i ≥ mi and we can write [aλ¯ ] = i mi x¯ i + y = [¯aλ ] + y. But the statistical dimension of aλ¯ is dλ¯ = dλ = the statistical dimension of [¯aλ ], therefore y = 0. ¯ Let Aρ ⊂ Sect(M ) which is generated by all the irreducible subsectors of [ρλρ], where λ ∈ CK . Since ρρ¯ ∈ CK and the elements of CK generate a finite subring of Sect(M ), it follows Aρ is a finite ring. Many examples suggest Aρ is a commutative ring but we have an example in the next section showing that is not the case. However, we have the following theorem. Theorem 3.6. (1) Let [b] be any subsector of [A(aµ )], where A is an arbitrary polynomial in aµ , µ ∈ CK , then [aλ ][b] = [b][aλ ] for any λ ∈ CK ; (2) Let [c] be any subsector of [ρρ], ¯ then [aλ ][c] = [c][aλ ] for any λ ∈ CK . Proof. (1) For simplicity we denote A(aµ ) by A. It follows from Corollary 3.2 that ˜ there exists A(µ) ∈ CK (denote by A˜ in the following) such that: Aρ¯ = A˜ ρ¯ and ˜ is the braiding operator. Since b is a subsector of ˜ ρA = Aσ ρ, where σ : γ A˜ → Aγ A, there exists partial isometry v ∈ M with b(x) = v ∗ A(x)v , ∀x ∈ M . Denote by p = vv ∗ ∈ A0 (M ) ∩ M . Then aλ b(x) = aλ (v ∗ )aλ A(x)aλ (v) baλ (x) = v ∗ Aaλ (x)v. Notice aλ · A · ρ¯ = ρ¯ · λA˜ = ρ¯ · A˜ 1 · λ = ρ(σ ¯ 1 )ρ¯ · A˜ · λ ρ(σ ¯ 1∗ ) = ρ(σ ¯ 1 )A · aλ · ρ¯ ρ(σ ¯ 1∗ ), ∗ where A˜ 1 = σ1 Aσ1 and σ1 : A˜ · λ → λ · A˜ is the under crossing braiding operator. Let y = ρ(σ ¯ 1∗ ). First we claim y aλ · A(x) = A · aλ (x)y, ∀x ∈ M . Since ρ · aλ · A = λσ · A˜ σ · ρ, ρ · A · aλ = A˜ σ · λσ · ρ, it is sufficient to show ρ(y) λσ · A˜ σ · ρ(x) = A˜ σ · λσ · ρ(x)ρ(y), ∀x ∈ M . ˜ ˜ = γ(σ1∗ λ· A(x)) = Recall ρ(y) λσ · A˜ σ ·γ(x) = γ(σ1∗ )λσ · A˜ σ ·γ(x) = γ(σ1∗ )γ ·λ· A(x) ∗ ˜ ˜ γ(A · λ(x)σ1 ) = Aσ · λσ · γ(x)ρ(y), ∀x ∈ M . By Proposition 2.6, ρ(M ) is generated by γ(M ) and w, it is sufficient to show γ(σ1∗ )λσ · A˜ σ (w) = A˜ σ · λσ (w)γ(σ1∗ ). In terms of diagrams (using δσ (w) = γ(σ)w, ∀δ as in the proof of Theorem 3.1): γ

λ

eσ (w) ←→ γ(σ1∗ ) · λσ · A

e A eσ (w) λσ A γ(σ1∗ )

γ

e λA γ(σ1∗ )

eσ · λσ (w) · γ(σ ∗ ) ←→ A 1 A˜ σ · λσ (w)


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It follows from YBE that γ(σ1∗ )λσ · A˜ σ (w) = A˜ σ · λσ (w)γ(σ1∗ ). Hence b · aλ (x) = v ∗ A · aλ (x)v = v ∗ yaλ · A(x)y ∗ v, ∀x ∈ M . Recall aλ · b(x) = aλ (v ∗ )aλ · A(x)aλ (v). To show [aλ · b] = [b · aλ ], it is sufficient to ¯ 1 )pρ(σ ¯ 1∗ ). Apply ρ to both sides and use show that aλ (vv ∗ ) = y ∗ vv ∗ y, i.e., aλ (p) = ρ(σ Corollary 3.2, it is sufficient to show: λσ (ρ(p)) = γ(σ ∗ )ρ(p)γ(σ). Recall p ∈ A0 (M ) ∩ M = Aρ¯0 (M ) ∩ M = ρ¯A˜ 0 (M ) ∩ M . Hence ρ(p) ∈ γ A˜ 0 (M ) ∩ M . In terms of diagrams: A˜

γ λ

σ

λσ (ρ(p)) ←→

λ (ρ(p))

σ∗ γ

λ

A˜ γ(σ1∗ )

γ(σ1 )ρ(p)γ(σ1∗ ) ←→

ρ(p)

γ(σ1 ) By YBE and BFE, λσ (ρ(p)) = γ(σ1 )ρ(p)γ(σ1∗ ). (2) The proof is quite similar to that of (1). Let u be the partial isometry with c(x) = u∗ ρρ(x)u ¯ for any x ∈ M and let q = uu∗ . Then aλ · c(x) = aλ (u∗ )aλ · ρ¯ · ρ(x)aλ (u) , c · aλ (x) = u∗ ρ¯ · ρ · aλ (x)u. By Corollary 3.2, ρ¯ · ρ · aλ (x) = ρ¯ · λσ · ρ(x) = ρ(σ ¯ ∗ )ρ¯ · λ · ρ(x)ρ(σ) ¯ ∗ ¯ = ρ(σ ¯ )aλ · ρ¯ · ρ(x)ρ(σ). ¯ ∗ )aλ · ρ¯ · ρ(x)ρ(σ)u. ¯ To show [aλ · c] = [c · aλ ], it is sufficient Hence c · aλ (x) = u∗ ρ(σ to show aλ (q) = ρ(σ)q ¯ ρ(σ ¯ ∗ ), which is equivalent to (applying ρ to both sides and using Corollary 3.2) λσ (ρ(q)) = γ(σ)ρ(q)γ(σ ∗ ). Notice ρ(q) ∈ γρ0 (M ) ∩ M ⊂ γ 20 (M ) ∩ M . Hence

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γ λ

γ σ

λσ (ρ(q)) ←→

λ (ρ(q))

σ∗

γ

λ

γ γ(σ ∗ )

γ(σ)ρ(q)γ(σ ∗ ) ←→

ρ(q)

γ(σ)

It follows from YBE and BFE that λσ (ρ(q)) = γ(σ)ρ(q)γ(σ ∗ ).

Notice since [af ] ∈ Aρ (see the definition after Corollary 3.5), the subfactor M ⊃ af (M ) has finite depth. Theorem 3.6 gives us more information on the higher relative commutants of this subfactor. We have: ¯ (In particular this shows [af ] Corollary 3.7. (1) h(af a¯ f )n , (af a¯ f )n i = hf 2n f¯2n , ρρi. is different from [f ] in general.) (2) h(af a¯ f )n af , (af a¯ f )n af i = hf 2n+1 f¯2n+1 , ρρi. ¯ (3) The principal graph of M ⊃ af (M ) is isomorphic to its dual principal graph as abstract graphs. Proof. h(af a¯ f )n , (af a¯ f )n i = h(af af¯ )n , (af af¯ )n i = ha(f f¯)n , a(f f¯)n i = hρ(f ¯ f¯)n , ρ(f ¯ f¯)n i ¯ = hf 2n f¯2n , ρρi, where we have used (2) of Cor.3.5, (1) of Cor.3.4, (1) of Theorem 3.3 in the first, the second and the third "=". We have also used Frobenius duality and Theorem 1.6 in the last 2 identities. (2) is proved in a similar way as in (1).


379

To prove (3), let G, G0 , G1 (resp. H, H 0 , H 1 ) denote the principal graph (resp. the dual principal graph), the even nodes and the odd nodes of the principal graphs (the dual principal graphs). Then G0 , H 0 correspond to irreducible subsectors of (af a¯ f )n , (where n > the depth of M ⊃ af (M )) and G1 , (resp. H 1 ) corresponds to irreducible subsectors of (af a¯ f )n af (resp. (af a¯ f )n a¯ f ). The isomorphism ϕ from G to H is defined to be: for any x ∈ G0 ∪ G1 , ϕ(x) = x¯ ∈ H 0 ∪ H 1 , where x¯ is the conjugate sector of x. ϕ is an isomorphism between G and H since by (1) of Theorem 3.6 and (2) of Corollary 3.5, ¯ = [af¯ x] ¯ = [xa ¯ f¯ ] and hx af , yi = h¯af x, ¯ yi ¯ = hx¯ a¯ f , yi ¯ = hϕ(x)ϕ(af ), ϕ(y)i. [¯af x] In [X3] , certain subfactors are constructed from some exceptional representations of Hecke-algebras. The theory developed in this section allow us to generalize some of the observations of [X3]. In fact, part of the motivation of this paper is to give a better explanation of some results in [X3]. Let us first introduce some notations. ¯ n (M )0 ∩ ρ(M ¯ ) and Bn = anf (M )0 ∩ M . Let n be a positive integer. Set An = ρf S S Denote by A = n An and B = n Bn . Let bk = f k−1 (σ), where σ = σf,f is the braiding operator. Let hk = ρ(b ¯ k ). Theorem 3.8. (1) An ⊂ Bn ; (2) There exists a Markov trace tr defined on B, i.e., tr(hn−1 x) = tr(hn−1 ) tr(x) for any x ∈ Bn−1 ; 2πi (3) Let q = exp( K+N ) and h0i = q 2N hi . If K + N > 4, then h0i satisfies the following type A Hecke algebra relations: N +1

h0i h0i+1 h0i = h0i+1 h0i h0i , h0i h0j = h0j h0i (|i − j| > 1), h0i = (q − 1)h0i + q; 2

(4) An as an algebra is generated by 1, h1 , ...hn−1 ; (5) The Markov trace tr as in (1) makes the following inclusions: An ⊂ An+1 ∩ ∩ Bn ⊂ Bn+1 a periodic commuting square in the sense of [GHJ] (see Chapter 4 of [GHJ]) when n is sufficiently large. Proof. (1) By Theorem 3.3 and Corollary 3.2, n 0 ¯ (M ) ∩ M = ρf ¯ n0 (M ) ∩ M ⊃ ρf ¯ n0 (M ) ∩ ρ(M ¯ ) = An . Bn = an0 f (M ) ∩ M = af ρ

(2) Let us define tr as in Corollary 2.5 of [L1]. Let εn : M → ρf ¯ n (M ) be the minimal expectations. For any b ∈ B, define tr(b) = limi→+∞ εi (b). By Corollary 3.5, a0f (M ) ∩ M = C1. It follows from Corollary 2.5 of [L1] that tr is really a trace. Let us show for any x ∈ Bn−1 , εn (hn−1 x) = tr(x) tr(hn−1 ), which implies tr is a Markov trace. Notice ¯ n−1 ) ∈ ρf ¯ n−2 (M ). Therefore hn−1 = ρ(g εn (hn−1 x) = εn (εn−2 (hn−1 x)) = εn (hn−1 εn−2 (x)).

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∀x ∈ Bn−1 , εn−2 (x) ∈ ρf ¯ n−2 (M ) ∩ ρf ¯ n−10 (M ) = ρf ¯ n−2 (M ∩ ρf ¯ 0 (M )) = ρf ¯ n−2 (M ∩ a0f (M )) = C1.

So we have εn−2 (x) = tr(x). Hence εn (hn−2 (x)) = εn (hn−1 )· tr(x) = tr(hn−1 ) tr(x). (3) The first two relations follow immediately from the definitions of h0i and YBE. We just have to show the third relation, and it is sufficient to prove it for h0 := h01 . By Theorem 1.6, f 2 (M )0 ∩ M ' M2 (C), where M2 (C) denotes the set of twoby-two matrices. So the unitary operator b := b1 ∈ f 2 (M )0 ∩ M satisfies a quadratic relation. Denote by Spec(A) the set of eigenvalue of an operator A. It follows from Proposition 1.4.3 and the formula given at the beginning of the proof of Lemma 3.1 that Spec(b2 ) = {q It follows that

N −1 N

,q

−N −1 N

}.

Spec(h0 ) = {q 2 , 1}. 2

¯ (M )). Let f (resp. ρf ¯ ) be the minimal conditional expectation from M to f (M ) (resp. ρf By Sect. 2 of [L1] and the end of Sect. 1.3, ¯ f (b)) tr(h) = ρf ¯ (h) = ρ( = f (b) = f −1 f (b) = 8f (b) = d−1 f κf , where we have also used the fact that both ρf ¯ (h) and f (g) are scalars. By Theorem 1.3.3, κf = Sf , and using the explicit formula of Sf given in Sect. 1.6, tr(h0 ) =

q−1 . 1 − q −N

We claim that if K + N > 4, then Spec(h0 ) = {q, −1}. We just have to show that Spec(h0 ) cannot be {q, 1}, {−q, 1}, {−q, −1}. If Spec(h0 ) = {q, 1}, then gi → −h0i gives a C ∗ representation of the Hecke algebra H∞ (−q) of [We1]. By the statement on p. 378 of [We1] this is possible only if −q = exp( 2πi l ) for some integer l with |l| ≥ 4. One checks easily that this is impossible if K + N > 4. Similar argument shows that Spec(h0 ) 6= {−q, −1}. If Spec(h0 ) = {−q, 1}, then gi → −h0i gives a C ∗ representation of the Hecke algebra H∞ (q) of [We1]. Since the Markov trace tr factors through such a representation, by (b) q−1 of Theorem 3.6 of [We1], there exists 1 ≤ k 0 ≤ K + N − 1 such that tr(−h0 ) = 1−q −k0 0

which leads to 2 = q −N + q −k , a contradiction. So if K + N > 4, then Spec(h0 ) = {q, −1}, and (3) is proved. When K+N ≤ 4, one checks easily from the argument above that the only possibility is N = 3, K = 1 and Spec(h0 ) = {i, −1} or Spec(h0 ) = {−i, −1}. (4) Assume K + N > 4 or N = 3, K = 1 and Spec(h0 ) = {i, −1}. Let Aˆ n be the subalgebra of An generated by 1, h1 , ...hn−1 . By the proof of (3) above and (b) of Theorem 3.6 of [We1], the simple ideals of Aˆ n are given by (N, K + N ) diagrams (cf. p. 367 of [We1]), which are in one-to-one correspondence with the irreducible descendants of f n by Theorem 1.6. The inclusion matrix of Aˆ n ⊂ Aˆ n+1 , given by (2.14) on P.369 of


381

[We1], is precisely the same as the inclusion matrix of An ⊂ An+1 by Theorem 1.6. Since Aˆ 0 = A0 ' C, (4) is proved. If K = 1, N = 3 and Spec(h0 ) = {−i, −1}, the simple ideals of Aˆ n are given by (1, 4) diagrams (cf. p. 367 of [We1]). But it is easy to check that such diagrams are in one-to-one correspondence with the irreducible descendants of f n by Theorem 1.6, and the inclusion matrix of Aˆ n ⊂ Aˆ n+1 , given by (2.14) on P.369 of [We1], is precisely the same as the inclusion matrix of An ⊂ An+1 by Theorem 1.6. So the same proof as above shows (4) in this case. (5) It follows from the proof of Proposition 3.1 in [X3] and (1) to (4) that the square in (5) of the theorem is a commuting square. ¯ has finite Finally let us prove the periodicity. For any irreducible λ ∈ CK , since [ρ·λ] statistical dimension, it decomposes as a direct sum of a finite number of irreducible P sectors. Let us write [ρ¯ · λ] = a Vρ¯λa [a], where Vρ¯λa are non-negative integers. Let P be the set of all [a]’s which appear in the above decompositions, i.e. Vρ¯λa 6= 0 for some λ ∈ CK . P is a finite set andPwe denote by ]P the number of elements in P . We claim µ µ [b] for some non-negative integers Vab . In fact for any µ ∈ CK , [a · µ] = [b]∈P Vab if [b] is any irreducible subsector of [a · µ], then [b] is also an irreducible subsector of ν ν λ [ρ¯ · λ · µ] = Σν [ρ¯ · ν]Nλµ , where Nλµ is as in Sect. 1.6. Hence [b] ∈ P . Notice λ → Vab gives a representation of Gr(CK ) by ]P × ]P matrices whose entries are non-negative integers. Recall from Sect. 1.6 that each irreducible λ ∈ CK can be assigned one of the N -colors C(λ) ∈ 0, 1, 2, · · · N − 1. We denote by Pi the set of those elements a ∈ P such that Vρ¯λa 6= 0 for some λ ∈ CK of color i. Notice for n > m (m is a fixed number which depends on N and K), all irreducible λ with color ≡ n(mod N ) appears as irreducible components of f n (recall f has color 1). Hence the minimal projections of ¯ n , are Bn , which are in one to one correspondence with the irreducible subsectors of ρf in one to one correspondence with the elements of Pi with i ≡ n(mod N ). The inclusion f , where a ∈ Pi , b ∈ Pi+1 . It is the same as that of matrix for Bn ⊂ Bn+1 is given by Vab Bn+N ⊂ Bn+N +1 for n > m. Next let us show Bn ⊂ Bn+N is primitive, i.e. given any a ∈ Pi , b ∈ Pi , there exists r such that b is contained in af N r (we use notations b ≺ af N r in the following). ¯ Therefore b ≺ ρf ¯ n+N ≺ Since ρf ¯ n ≺ a it follows from Frobenius duality af¯n ≺ ρ. n n+N Nr n n N (r−1) ¯ ¯ af f ≺ af , where we use the fact that f f ≺ f for some positive integer r. The fact that An ⊂ An+1 is periodic with period N and that An ⊂ An+N is primitive follows from the lemma on p. 369 of [We1] and (4). In the following we will give two well studied examples when G = SU (2). In this case there are two nontrivial conformal inclusions: SU (2)10 ⊂ Spin(5) and SU (2)28 ⊂ G2 . The representation of SU (2) is denoted by spin j which is a half integer. Let us consider these two cases separately. The fusion ring for SU (2) is well known. The one which will be used is: 21 · j = (j − 21 ) ⊕ (j + 21 ). Example 1. SU (2)10 ⊂ Spin(5). In this case K = 10, the subfactor M ⊃ af (M ) has π . γ = 0 ⊕ 3 (cf.[CIZ]). The principal graph can be either A11 or E6 (see index 4 cos2 12 [GHJ]). From (2) of Corollary 3.7 we have haf a¯ f af , af a¯ f af i = hf 6 , 0 ⊕ 3i = 6, while it is easy to check that if the principal graph is A11 , then haf a¯ f af , af a¯ f af i = 5. Therefore the principal graph of M ⊃ af (M ) is E6 . Example 2. SU (2)28 ⊂ G2 . In this case K = 28 and the subfactor M ⊃ af (M ) has π . γ = 0 ⊕ 5 ⊕ 9 ⊕ 14 (cf.[CIZ]). The principal graph can only be A29 , D16 index 4 cos2 30 or E8 by [GHJ].

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From (2) of Corollary 3.7 we have h(af a¯ f )2 af , (af a¯ f )2 af i = hf 10 , ρρ¯ = 0 ⊕ 5 ⊕ 9 ⊕ 14i. Notice spin 5 representation appears in the irreducible decomposition of f 10 with multiplicity 1. Hence h(af a¯ f )2 af , (af a¯ f )2 af i = hf 10 , 0i + 1 = 43. If M ⊃ af (M ) has principal graph A29 or D16 , it is easy to check that h(af a¯ f )2 af , (af a¯ f )2 af i = hf 10 , 0i = 52 + 42 + 1 = 42. Therefore the principal graph of M ⊃ af (M ) is E8 . Hence there exist hyperfinite type II1 subfactors with principal graph E8 by [Po]. The original proof of this fact (see [I2]) is completed by some tedious but straightforward calculations of about 15 matrices. 3.2. More properties. As mentioned at Sect. 3.1 (after Corollary 3.2), there are in fact two kinds of braided endomorphisms aλ and a˜ λ associated to each λ ∈ CK . In general c2 (λ) , where c2 (λ) is [aλ ] 6= [ãλ ]. (See Lemma 3.2 below). Recall from Sect. 1.6 1λ = K+N the value of the Casimir operator on the representation of SU (N ) labeled by λ. Then we have: (K) . If for any λ0 ≺ λ · f , 1λ0 − 1λ − 1f are integers, Lemma 3.1. Suppose λ ∈ P++ then λ is the trivial representation. P P Proof. Recall for λ = 1≤j≤N −1 λj 3j , where λj ≥ 0, 1≤j≤N −1 λj ≤ K (We also write such λ as [λ1 , ...λN −1 ]),

c2 (λ) =

1 2N +

and 1λ =

c2 (λ) K+N .

1 2

X

j(N − j)λ2j +

1≤j≤N −1

X

1 N

X

j(N − k)λj λk

1≤j 0), it (2) Let Gab = Vab follows Gab is irreducible (see [GHJ] for definition). By the Perron–Frobenius theorem, there exists a unique unit eigenvector with nonnegative entries corresponding to the largest eigenvalue (1) above we have:

Sf(1) S1(1)

of Gab . In our notation this eigenvector is {ψa(1) }. Using (*) of (ψ1(1) )2 S1(1)

=

X

1 S1σj V1σ = S11 . j

j

q Hence ψ1(1) = S1(1) S11 . P f f (3) From af a = b Vab b we have df da = Vab db . It follows {db }b∈V is a Perron– f (1) Frobenius eigenvector of Vab . Hence db = xψb , where x > 0 is a fixed constant. Since P P ψ (1) 1 1 1 (1) 2 d1 = 1, x = ψ1(1) , db = ψb(1) . So a∈V d2a = (ψ(1) a∈V (ψa ) = (ψ (1) )2 = S (1) S . )2 1

(4) Notice that

1

1

1

1

11

b = hcσj , bi = hσj c, bi = Nσbj c . Ncσ j

So Vσλj b =

X

V1cλ Nσbj c ,

c

i.e., X

Sλ(µ)

S (µ) µ,i,s∈(Exp) 1

X

∗

· ψσ(µ,i,s) ψb(µ,i,s) = j

= It follows immediately that ψσ(µ,i,s) j

Sλ(ν)

S (ν) ν,k,t∈(Exp) 1

Sσj σi S1σi

ψ1(µ,i,s) .

·

∗ Sσj σk · ψ1(ν,k,t) ψb(ν,k,t) . S1,σk

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F. Xu

3i Let (Gi )ab = Vab with a, b ∈ V . In [PZ], certain postulations on graphs are given based on the connections with integrable lattice models. We shall show that our Gi satisfy all the postulations i) to ix) of 2.2 in [PZ] except (2) of ix). Let {σj } denote all the level 1 representations of LH. and χσj the corresponding character. Then χσj = P P 2 (K) hλ, νj ixλ , and Z = (K) hλ, νj iχλ | is called the partition function j |Σλ∈P++ λ∈P++ associated with G ⊂ H. Recall γ is the representation of LSU (N ) obtained by restriction from the vacuum representation of LH to LSU (N ).

b [ Lemma 3.6. For all the conformal inclusions listed in Sect. 2.2 except (A 8 )1 ⊂ E8 , γ has color 0, i.e., for any λ with hλ, γi ≥ 1, τ (λ) ≡ 0 mod(N ). Proof. For SU (2), SU (3) cases, one can check the lemma is true by using the explicit form of ν given in [PZ]. For series (a) and (b), notice in both cases the generator w = exp 2πi N of the center of 2 SU (N ) is embedded as w → w in H which acts trivially on the vacuum representation of LH. Hence the center of SU (N ) acts trivially on any λ with hλ, γi ≥ 1, so τ (λ) ≡ 0(mod N ). For series (c) and (d), the generator w = exp 2πi N is embedded trivially, (i.e. w → 1 in H). Hence for any λ with hλ, γi ≥ 1, τ (λ) ≡ 0(mod N ). We are now ready to prove the following: Theorem 3.10. The following statements (i) and (ii) are true for all the conformal b [ inclusions in Sect. 2.2 except (A 8 )1 ⊂ E8 . The remaining statements are true for all the conformal inclusions in Sect. 2.2. (i)

To each a ∈ V may be attached a Z/N Z grading τ (a) such that (Gi )ab 6= 0 only if τ (b) = τ (a) + i mod N.

(ii) The complex conjugation a → a¯ satisfies τ (¯a) = −τ (a) and (Gi )ab = (Gi )b¯ a¯ . (iii) The matrices Gi are pairwise transposed of one another: τ

Gi = Gn−i .

(iv) The matrices Gi commute among themselves and are diagonalizable in an orthonormal basis common to all of them. (v) The corresponding eigenvalues of G1 , . . . , GN −1 are given by S3(λ)1

S3(λ)i

S1

S1

,..., (λ)

,..., (λ)

S3(λ)N −1 S1(λ)

;

some of these λ may occur with multiplicities larger than 1 but hλ, γj i 6= 0 for some γj . (vi) The graph defined by G1 admits one extremal vertex, i.e., a vertex on which only one edge is ending and from which only one edge is starting. Proof. (i) Let us first show if a is an irreducible subsector of anf and am f , then n ≡ m mod(N ). Notice under the assumption, 1 is a subsector of [anf a¯ m f ] = [af n f¯m ]. By ¯ af n f¯m ρi ¯ = hρ, ¯ ρf ¯ n f¯m i = hγ, f n f¯m i ≥ 1, Theorem 3.3, we have h1, af n f¯m i = hρ, m where we have also used [¯af ] = [af¯m ] (see (2) of Corollary 3.5). By Lemma


391

3.5 γ has color 0, but all descendants of f n f¯m have color n − m, it follows n ≡ m mod(N ). So it makes sense to define τ (a) ≡ n mod(N ) if a is an irreducible subsector of anf . It follows from such a definition that τ (ab) ≡ τ (a) + τ (b) mod N . Since a3i has color i, it follows (Gi )ab 6= 0 only if τ (b) = τ (a) + i mod N . (ii) τ (¯a) = −τ (a) follows from the definition of τ (a) in (i). (Gi )ab = ha3i a, bi = ¯ a¯ i = (Gi )a¯ b¯ , where we have used [b¯ a3i ] = [a3i b] ¯ which ¯ 3i , a¯ i = ha3i b, hba follows from (1) of Theorem 3.6. (iii) (Gi )ab = ha3i a, bi = ha, a¯ 3i bi = ha, a3¯ i bi = ha, a3n−i bi = ( t Gn−i )ab where we have used [¯a3i ] = [a3¯ i ] which follows from (2) of Corollary 3.5. (iv) and (v) follow trivially from the definitions of Gi Lemma 3.3 and (1) of Theorem 3.9. As for (vi), we claim the extremal vertex is the identity sector. All we need to show is af is irreducible which follows from (1) of Corollary 3.5. Remark. The only postulate in [PZ] which remains to be proved or disproved is (2) of ix) in [PZ] which says φ(λ) 1 > 0 for any λ ∈ (Exp). 4. More Examples In this section we give some examples as an application of the general theory developed in the previous sections. We will be mainly interested in the subring C which is generated by all the irreducible descendants of anf , where n is a positive integer. What we mean by the fusion diagram of a3i is the oriented graph determined by ¯ i = 3i , the graph is unoriented.) Clearly these graphs determine Gi = V 3i . (In the case 3 the multiplication of aλ with the elements of C for any λ. The following simple lemma will be used repeatedly in Sect. 4.1 and Sect. 4.2. Lemma 4.1. (1) If hλγ, λi = m ≤ 3, then aλ decomposes into a sum of m different irreducible subsectors. (2) If hλγ, µi = n and hµγ, µi = 1, then aµ is irreducible and naµ is a subsector of aλ . (3) Let σj be the special nodes (see the definition before Proposition 3.9) corresponding to Hσj of the level 1 representation of LH. Let γj denote the representation of LSU (N ) obtained by restriction of Hσj to LSU (N ). Then hσj , aλ i = hγj , λi. Proof. (1) By Theorem 3.3, ¯ aλ ρi ¯ = hρλ, ¯ ρλi ¯ haλ , aλ i = haλ ρ, = hγλ, λi = m ≤ 3. PS PS If aλ = i=1 mi xi , where mi ≥ 1 are integers, then i=1 m2i = m ≤ 3. It follows that mi = 1 and S = m. (2) It follows from (1) that aµ is irreducible if hµγ, µi = 1. Again by Theorem 3.3, haλ , aµ i = haλ ρ, ¯ aµ ρi ¯ = hρλ, ¯ ρµi ¯ = hγλ, µi = n. Therefore naµ is a subsector of aλ . (3) It follows from Lemma 3.4 that ¯ aλ ρi ¯ = hσf ρ, ¯ ρλi ¯ = hρσj ρ, ¯ λi hσj , aλ i = hσj ρ, = hγj , λi. We have also used aλ ρ¯ = ρλ ¯ and ρσj ρ¯ = γj (cf. Cor.3.2 and Proposition 2.8).

392

F. Xu

ν We will also need some of the structure constants Nµλ of Gr(Ck ). The general ν using such a formula. formula is given in Sect. 1.6 but it is quite tedious to calculate Nµλ A very efficient algorithm is known and can be found on p. 288 of [Kac2] or [Wal]. We will use aλ instead of [aλ ] to simplify notations. However it should be clear that the ring structure is about sectors rather than endomorphisms. We shall denote the weights (λ1 − λ2 )31 + (λ1 − λ2 )31 + ... + λN −1 3N −1 with λ1 ≥ λ2 ≥ ... ≥ λN −1 of SU (N ) by (λ1 , ...λN −1 ). If a sector is denoted by 0, it will always be the identity sector in the rest of this section.

4.1. E6 and E8 revisited. As in Sect. 3.2, we use a half integer j to denote the representation of SU (2). The ring structure of C is given by the following fusion diagram of a1/2 : b = a3/2 − aq/2

0

a1/2

a1 c = aq/2 d = a5

where the vertices correspond to the irreducible sectors in C. Let us explain how we obtain b = a3/2 − a9/2 . By Lemma 3.3, a1 · a1/2 = a3/2 + a1/2 = b + a1/2 + c. Hence a3/2 = b + c. That a3/2 has two irreducible sectors can be seen from ha3/2 , a3/2 i = ¯ a3/2 ρi ¯ = hρ¯ 23 , ρ¯ 23 i = hγ, 23 · 23 i = h0 ⊕ 3, 0 ⊕ 1 ⊕ 2 ⊕ 3i = 2. ha3/2 ρ, On the other hand, 3 3 3 5 7 9 3 · γ = (0 ⊕ 3) = ⊕ ⊕ ⊕ ⊕ . 2 2 2 2 2 2 2 It follows ha3/2 , a9/2 i = h 23 · γ, 29 i = 1. But ha9/2 , a9/2 i = h0 ⊕ 3, 0 ⊕ 1i = 1, which means a9/2 is irreducible. So b and c must be equal to a9/2 . However dc = d9/2 6= d6 . It follows c = a9/2 and b = a3/2 − a9/2 . It is amusing to check b2 = 0 + a5 , b · a5 = b which means (0, b, a5 ) generates a subring isomorphic to the fusion ring of the Ising model. These are the special nodes (see the definition before Theorem 3.9). Notice it is impossible to have such results in the bimodule picture since b would correspond to the N − M bimodule (see [K]). In exactly the same way we can obtain the ring structure of all irreducible subsectors of a˜ nf (n > 0) b

0

a˜ 1/2

a˜ 1

a˜ q/2

a5

where b = a3/2 − a9/2 = a˜ 3/2 − a˜ 9/2 . It follows from [K] that the principal graph and dual principal graphs of the subfactor ρ(M ) ⊂ M are given by:


3 0

%J %

S

S S% ρ

1 4 2

S

%J %

S S% ρ¯

5

# # S JJ# S S x y

α 0

393

β β1 δ

# # S JJ# S S x¯ y¯

where we have used β1 instead of γ (γ is reserved for the vacuum representation of LH) compared to [K]. By using the results of [K], we can even determine the ring structure of Aρ whose irreducible elements are given by: 0, a1/2 , a˜ 1/2 , a1 , a˜ 1 , b, a9/2 , a˜ 9/2 , a5 , α = a1/2 a˜ 1/2 , a5 α and αb. Also β = a1 , β1 = a˜ 1 , δ = a5 · α, ε = a5 . That β = a1 , β1 = a˜ 1 agree with the observation in [K]. The ring structure of Aρ is completely determined by the following formula: a1/2 a˜ 1/2 = α, a1/2 a˜ 1 = bα, a1/2 a˜ 9/2 = a5 α; a1/2 α = a˜ 1/2 + bα, a˜ 1/2 α = a1/2 + bα. Perhaps the most surprising identity is α = a1/2 a˜ 1/2 .2 Let us explain why α = a1/2 a˜ 1/2 . First of all, ha1/2 a˜ 1/2 , a1/2 a˜ 1/2 i = ha1/2 a1/2 , a˜ 1/2 a˜ 1/2 i = h0 + a1 , 0 + a˜ 1 i = 1, hence a1/2 a˜ 1/2 is irreducible. Then ha1/2 a˜ 1/2 , ρρi ¯ = hρa1/2 a˜ 1/2 , ρi = h(0 ⊕ 1)ρ, ρi = √ h0 ⊕ 1, 0 ⊕ 3i = 1. This shows ρρ ¯ = 0 + α contains a1/2 a˜ 1/2 . But dα = 2 + 3 = da1/2 · da˜ 1/2 , so we have α = a1/2 a˜ 1/2 . The rest the of irreducible sectors are obtained by a similar or simpler calculation. As for the multiplication rule, we explain how we obtain a1/2 α = a˜ 1/2 + bα which requires more computations than the others. Notice ¯ = ρ¯ · 21 · ρ = a1/2 · (0 + α) = a1/2 + a1/2 α. Since hã1/2 , ρ¯ · 21 · ρi ≥ 1, and by a1/2 ρρ Lemma 3.2, a˜ 1/2 6= a1/2 , so we must have hã1/2 , a1/2 αi ≥ 1. But hρ¯ · 21 · ρ, ρ¯ · 21 · ρi = hγ · 21 , 21 · γi = h(0 ⊕ 3) · 21 , (0 ⊕ 3) · 21 i = 3. This shows a1/2 · α = a˜ 1/2 + c where c is an irreducible sector. Since ha1/2 · α, bαi = ha1/2 · b, ααi ¯ = hβ, 0 + α + β + β1 + δi = 1 and hbα, bαi = hb2 , α2 i = h0 + a5 , 0 + α + β + β1 + δi = 1, it follows bα is irreducible and a1/2 · α = a˜ 1/2 + bα. The commutativity of Aρ , for example a˜ 1/2 b = bã1/2 , a1/2 a˜ 1/2 = a˜ 1/2 a1/2 , follows from Lemma 3.3 and Theorem 3.6. For the E8 case the structure of C is given by the fusion diagram of the a1/2 diagram c a b a2 a3/2 a1 a1/2

0

2 This identity is an indication of the relevance of a and a ˜ f to the study of the still mysterious dual f subfactor ρ(M ¯ ) ⊂ M in the general case.

394

F. Xu

with a = a3 − a2 , b = a7/2 − a3/2 and c = a5/2 + a3/2 − a7/2 . Again it is easy to check that a2 = a + 0. The structure of Aρ is more complicated and we haven’t done the calculation. 4.2. The case of SU (3) and SU (4). The weights of SU (3) are labeled by (λ1 , λ2 ) with λ1 ≥ λ2 ≥ 0. d(3)3 ⊂ Spin(8). [ The partition function is Z = |χ(0,0) + χ(3,0) + χ(3,3) |2 + Example 0. SU 2 3|χ(2,1) | . γ = (0, 0) + (3, 0) + (3, 3). The special nodes (see the definition before Proposition 3.9) corresponding to the blocks of Z from the left to the right are denoted by 0, b1 , b2 , b3 . The fusion graph G1 is given by: a¯ fb3 HH j H H @ @ / b @b1 2 j 0HH HH 6 af G1 is determined as follows. We have (1, 0)(1, 1) =(0, 0) + (2, 1), (1, 0)(1, 0) =(2, 0) + (1, 1), (1, 1)(1, 1) =(1, 0) + (2, 2), (1, 1)γ =(1, 1) + (2, 0) + (3, 2). By using Lemma 4.1, the first identity says a(1,0) a(1,1) = 0+a(2,1) = 0+b1 +b2 +b3 . (In fact we only have a(2,1) b1 + b2 + b3 but by calculating statistical dimension we conclude that a(2,1) = b1 + b2 + b3 ), the fourth identity says a(1,1) is irreducible and a(1,1) = a(2,0) . The graph G1 is then completely determined as above. d(3)5 ⊂ SU d(6). The special nodes are denoted by e0 , e1 , e2 , a(5,0) , Example 1. SU a(5,5) and a(0,0) which correspond to the blocks in Z = |χ(5,2) + χ(2,2) |2 + |χ(3,0) + χ(3,3) |2 + |χ(2,0) + χ(5,3) |2 + |χ(3,2) + χ(5,0) |2 + |χ(3,1) + χ(5,5) |2 + |χ(0,0) + χ(4,2) |2 from the left to the right. f = (1, 0) and f¯ = (1, 1) correspond to vector representation and its dual respectively. The multiplication by (1, 0) and (1, 1) is given by (λ1 , λ2 )(1, 0) = (λ1 + 1, λ2 ) + (λ1 , λ2 + 1) + (λ1 − 1, λ2 − 1), (λ1 , λ2 )(1, 1) = (λ1 − 1, λ2 ) + (λ1 , λ2 − 1) + (λ1 + 1, λ2 + 1). σ(c) c Let σ be the action of Z3 on the weights. Then Nσ(a)b = Nab (cf. p. 783 of [Wal]). First we claim the principal graphs for the subfactor M ⊃ af (M ) is given by the following: a(5,5) 0 a a f

e0

a(5,1)

(5,4)

a(4,0)

e1


395

In fact from (2, 1)γ = (2, 1) + (2, 1) + (5, 1) + (5, 4) + · · ·, (where · · · denotes the rest of sectors which are different from (2, 1), (5, 1) and (5, 4)), by Lemma 4.1, a(2,1) decompose into two irreducible sectors. Since σ(1, 0) = (5, 1), σ 2 (1, 1) = (5, 4) and af , af¯ are irreducible, it follows a(5,1) , a(5,4) are irreducible and are subsectors of a(2,1) . Hence a(2,1) = a(5,1) + a(5,4) . Since (5, 1)(1, 0) = (5, 2) + (4, 0), (5, 4)(1, 0) = (5, 5) + (4, 3), it remains to show that a(4,3) = a(1,0) + a(4,0) , a(5,2) = a(1,0) + e0 , a(3,0) = a(5,4) + e1 . One checks (5, 2) γ = σ(2, 0)γ = σ((2, 0)γ) = (5, 2) + (5, 2) + (1, 0) + · · · . Hence a(5,2) = a(1,0) + e0 (e0 is a subsector of a(5,2) follows from that (5, 2) belongs to the blocks in Z correspond to e0 and Lemma 4.1). Similarly one has (4, 3) γ = (4, 3) + (4, 3) + (1, 0) + (4, 0) + · · · , (3, 0) γ = (3, 0) + (3, 0) + (5, 4) + · · · which leads to a(4,3) = a(2,1) + a(4,0) , a(3,0) = a(5,4) + e1 . a(4,0) , a(5,4) are irreducible follows from a(4,0) · γ = σ(a(1,1) γ) = a(4,0) + · · · and a(5,4) · γ = σ 2 (a(1,1) γ) = a(5,4) + · · · . By using Z3 symmetry, we can immediately determine the fusion graph of af :

a(5,5) A T U A T

T a(5,4) (4,4) T a e1 H T T HH * jH T T HHT KT ?HT a(4,0) af THH6 T JJ T HH ^T Y H H H T HH T a(5,0) 0 T

a(5,1) T

T ] J TT J e0

e2

396

F. Xu

d(3)9 ⊂ E c6 . The three special nodes are denoted by 10 , 12 , 11 which Example 2. SU correspond to the three blocks (from the left to the right) in the partition function Z = 2|χ(4,2) + χ(7,2) + χ(7,5) |2 + |χ(0,0) + χ(9,0) + χ(9,9) + χ(8,4) + χ(5,1) + χ(5,4) |2 . d(3)9 but correspond to two The first 2 blocks are identical as representations of SU ˆ different representations of E6 at level 1. We claim the principal graph for af (M ) ⊂ M is given by 11 ,,

af , @ @ I a a a(2,2) @ ! ! " " a(2,1) 22 " A 12 A l 20 l k KA l 10 with 5 = a(2,1) , 6 = a(2,2) , 20 = 10 af , 22 = 12 af . First it is clear (2, 1)γ = (2, 1)+· · ·, so a(2,1) is irreducible and 5 = a(2,1) . (2, 1)(1, 0) = (3, 1) + (2, 2) + (1, 0). Let us show a(3,1) = a(2,2) + 10 af + 12 af . Notice (3, 1)γ = 3(3, 1) + (2, 2) + · · · and (3, 1)(1, 1) = (2, 1) + (3, 0) + (4, 2). Remember 10 + 12 are subsectors of a(4,2) by (3) of Lemma 4.1. It follows a(3,1) decomposes into 3 irreducible subsectors and ha(3,1) af¯ , 10 i = ha(3,1) af¯ , 12 i = 1. So are exactly the irreducible a(2,2) , 10 af , 12 af √ subsectors of a(3,1) . The statistical di1 mension d of af is 3 + 1, da(2,2) = 2 d − d . If a(2,2) · a(1,1) = 2a(2,1) + x, it follows dx = 0 and therefore a(2,2) a(1,1) = 2a(2,1) . (In fact it follows a(2,1) = a(3,3) .) The principal graph is completely determined as above. The fusion graph of af is determined similarly as: 11 %C % CM 21 %- C 31 C %A % AK5 C C 6 , A 4%" - , EL i + C , " L"- , C 22 E" 6 E L ,32C L , EO C 30 12 E, 20 L L

KL L L 10


397

where 30 = 10 · af¯ , 32 = 12 · af¯ , 4 = a(2,0) . d(3)21 ⊂ Eˆ 7 . The partition function is given by: Example 3. SU Z =|χ(0,0) + χ(21,0) + χ(21,21) + χ(8,4) + χ(17,12) + χ(17,4) + χ(20,10) + χ(11,1) + χ(11,9) + χ(12,5) + χ(15,9) + χ(15,5) + |2 + |χ(6,0) + χ(21,6) + χ(15,15) + χ(6,6) + χ(21,15) + χ(15,0) + χ(11,7) + χ(14,4) + χ(11,4) + χ(14,10) + χ(17,7) + χ(17,10) |2 . The special nodes are 10 and 11 . The fusion graph of af is given by: 11

31

21

51

41

61

81

91

71

101 120

110

110

122 100

70 80

90 60

40

50 20

30

10

,

where 20 = af , 30 = a(1,1) , 40 = a(2,0) , 50 = a(2,1) , 60 = a(2,2) , 70 = a(4,1) − 11 · a(3,2) , 80 = a(3,1) , 90 = a(3,2) , 100 = a(4,3) − 11 · a(3,1) , 110 = a(3,0) , 120 = 70 · a(1,0) − 110 and i1 = 11 · i0 for 1 ≤ i ≤ 12. The checking of the above formula is tedious for 70 , 100 , 120 but quite straightforward. The determination of the fusion graphs in Examples 1 to 3 prove a conjecture on p. 20 of [X3]. d(10)1 . The partition function is d(5)3 ⊂ SU Example 4. SU Z = |χ(0,0,0,0) + χ(2,2,1,0) |2 + |χ(3,1,1,0) + χ(3,3,2,2) |2 + (|χ(1,1,0,0) + χ(3,2,2,0) |2 + |χ(3,3,0,0) + χ(2,2,1,1) |2 + |χ(2,2,0,0) + χ(3,2,2,2) |2 + |χ(3,0,0,0) + χ(3,2,2,1) |2 + c.c),

398

F. Xu

where c.c indicates the conjugate of the terms inside the parenthesis (the conjugate of |χλ + χµ |2 is defined to be |χλ¯ + χµ¯ |2 ). The fusion graphs G1 and G2 are given by:

G

G

where the broken lines indicated double edges. The arrows are omitted but can be recovered from the following rule (See (i) of Proposition 3.10): the color of vertex a is a − 1 (mod 5), and in Gi , i = 1, 2, a points to b if and only if a = b + i (mod 5). The determination of G1 is rather similar to Example 1. One determines the principal graph of subfactor af (M ) ⊂ M first and then use Z5 symmetry to obtain G1 . G2 is determined by tedious but rather straightforward calculations. The special nodes are labeled from 1 to 10 in the above graphs. d(6)1 . The partition function is d(4)2 ⊂ SU Example 5. SU Z = |χ(0,0,0) + χ(2,2,0) |2 + |χ(2,2,2) + χ(2,0,0) |2 + 2|χ(1,1,0) |2 + 2|χ(2,1,1) |2 ,

γ = (0, 0, 0) + (2, 2, 0). The corresponding special vertices corresponding to the blocks in Z from the left to the right are labeled by 1,4,2,6,5,3. The fusion graphs G1 and G2 are determined from the following formula and Lemma 4.1: (1, 0, 0)(1, 0, 0) = (2, 0, 0) + (1, 1, 1), (1, 1, 1)(1, 1, 1) = (1, 1, 0) + (2, 2, 2); (1, 1, 1)γ (2, 1, 0).

Graphs G1 , G2 are the following:


399

7

7 6

6

1

5

5 4 3

2

4

1 3

2 8 8

d(4)4 ⊂ Spin(15) [ Example 6. SU 1 . The partition function Z = |χ(0,0,0) +χ(4,4,0) +χ(3,3,2) + χ(3,1,0) |2 + |χ(4,4,4) + χ(4,0,0) + χ(2,1,1) + χ(4,3,1) |2 + |2χ(3,2,1) |2 . The corresponding special nodes are denoted by 1, ε and τ which correspond to the level 1 representation of LSO(15). The fusion graphs of af and a(1,1,0) determined by G1 and G2 respectively are given by: , 9

9

9

3

5 6

5

2

7

9

8

8

1

,

4

10

3

,

6

1

4 2

10

G2

10

, 10

7

where 2 = ε, 3 = τ, 6 = a(2,0,0) = a(1,1,0) , τ ·a(1,1,0) = 4+5, τ ·af = 9+90 , τ ·af¯ = 10+100 . It is interesting to note d9 = d90 = d10 = d100 = 2 cos π8 . Here as in the previous examples, we have enumerated the nodes in the graph and we have used the number to denote the corresponding sector. Let us explain how the graphs are obtained. Notice (1, 0, 0)(1, 0, 0) = (2, 0, 0) + (1, 1, 0). One checks (2, 0, 0)γ = (2, 0, 0) + (1, 1, 0) + · · ·, which implies a(2,0,0) = a(1,1,0) = 6 by Lemma 4.1. Note (1, 0, 0)(1, 1, 1) = (2, 1, 1) + (0, 0, 0), hτ af , τ af i = h1 + ε, af af¯ i = h1, af af¯ i + hε, af af¯ i = 1 + hε, a(2,1,1) i = 2. Hence we can write τ af = 9 + 90 , similarly τ · a¯ f = 10 + 100 . Since hτ af , τ a¯ f i = h1 + ε, af af i = h1 + ε, a(2,0,0) i = 0, 9, 90 , 10, 100 are all different sectors. Notice 9¯ + 9¯ 0 = 10 + 100 , we may choose our notation such that 9¯ = 10, 9¯0 = 100 . Similarly one has τ · a(1,1,0) = 4 + 5. Let us show a(2,1,0) = τ · af + a(1,1,1) . First one checks (2, 1, 0) · γ = 3(2, 1, 0) + (1, 1, 1) + · · ·. Then ha(2,1,0) , τ · af i = ha(2,1,0) af¯ , τ i = ha(3,2,1) , τ i = 2. It follows a(2,1,0) = τ · af + a(1,1,1) . In the same way 6 · af¯ = 2af + τ · af¯ and a(2,1,1) = ε + τ · a(1,1,0) . From τ · af = 9 + 90 it follows d9 + d90 = 4 cos π8 . Without loss of generality, let us assume d9 ≤ 2 cos π8 . It follows the fusion graph of 9 is a A or D graph (see [GHJ]).

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Hence at most 9 · 9¯ = 0 + X + Y and haf · 9, af · 9i = h0 + ε + 4 + 5, 0 + X + Y i ≤ 3. On the other hand haf · 9, τ · a(1,1,0) i = hτ · 9, a(1,1,0) af¯ i = hτ · 9, 2af + τ · af¯ i = h9, 2τ · af + af¯ + af¯ i = 2 (we here used εaf¯ = af¯ which follows from hεaf¯ , af¯ i = hε, af af¯ i = 1). If af · 9 = 4 + 4 or 5 + 5, then haf · 9, af · 9i ≥ 4 which is impossible so af · 9 = 4 + 5 = τ · a(1,1,0) , and it follows d9 = d90 = 2 cos π8 . Similarly one can show af · 90 = 4 + 5, a¯ f · 10 = 4 + 5 = a¯ f · 100 . It follows af · 4 = 7 + 10 + 100 + W . However, from the explicit formula for the statistical dimensions it follows dW = 0, therefore af · 4 = 7 + 10 + 100 . The above calculations determine G1 completely. G2 is determined in a similar way. Since d9 = d10 = 2 cos π8 , the principal graph associated of the subfactor associated with 9 and 10 are given as follows: 5 4 0 0 4 5 A A J A A J A A J A A J J A A J A A a f 10 100 a¯ f 9 90 The nodes in the diagram are determined by the fact that they must be among the nodes of G1 and calculations of statistical dimensions. A bit surprising fact is that 9 · 10 = 0 + 4 but 10 · 9 = 0 + 5, in particular 9 · 10 6= 10 · 9 ! Let us explain how we derive it. ¯ = h6 + τ, 6 + τ i = 2. On the other hand, In fact, h9 · 9, af · af i = haf¯ · 9, af · 9i h9 · 9, af · af i = h9 · 9, 6 + 6i, it follows that 9 · 9 6 and by computing the statistical dimensions we conclude that 9 · 9 = 6. So h9 · 9, 9 · 9i = h9 · 10, 10 · 9i = 1. So if we choose our notation such that 9 · 10 = 0 + 4, then 10 · 9 6= 0 + 4, but the only nodes of G1 other than 4 which has the same statistical dimension as that of d4 is 5. It follows that 10 · 9 = 0 + 5. This is the first indication that the subring generated by all the subsectors of anf , where n runs over all positive integers are noncommutative! It follows from the above principal graphs that · 9 = 90 since · (9 + 90 ) = (9 + 90 ) and 9 · 9¯ = 0 + 4. In fact, one can determine the complete multiplication table among the nodes of G1 quite easily from the formula we presented above. We omit the details. All the diagrams in this section first appear in [Fran] and [PZ]. They are constructed by spectral analysis based on certain assumptions . Our theory gives explanations and generalize many observations in [Fran] and [PZ], and furthermore prove that these observations apply to a general class of conformal inclusions. In particular, all the fusion graphs G1 constructed in Examples 1 to 6 support a representation of Hecke algebras by Theorem 3.8 in the sense of [PZ]. This fact is established for Example 1 to 5 in [Fran, Sch] by explicit but rather tedious calculations and for Example 6, no such explicit calculation has been done. Finally let us note that the principal graph of subfactor M ⊃ ρ(M ) is given as the connected part (containing 1) of the graph determined by V1bλ . In fact, since ρρ¯ = γ, the principal graph of M ⊃ ρ(M ) is given by the connected part containing ρ¯ of the Bratteli diagram of γ n (M )0 ∩ M ⊂ γ n ρ0 (M ) ∩ M for n big enough. Notice the minimal projections of γ n (M )0 ∩ M and γ n ρ0 (M ) ∩ M are in one to one correspondence with [λ]’s and irreducible subsectors of [λρ]’s respectively since [λρ] = [ρaλ ] and by (2) of


401

Theorem 3.3, the irreducible subsectors of [ρaλ ] are in one to one correspondence with the irreducible subsectors of [aλ ]. (Recall the set of such sectors are denoted by V .) Under such a correspondence, the Bratteli diagram is determined by the right multiplication of λ . aλ on V which is given by Vab f . In the first four examples of Sect. 4.2, we have In Sect. 4.1 we have determined Vab ¯ f f¯ f f (1,1,0,0) (1,1,1,0) , Vab , Vab , Vab = determined Vab , Vab . In example 4, we have determined Vab f¯ (1,1,0,0) f (1,1,0) λ and in the last two examples, Vab , Vab , Vab . Since λ → V is a repreVba sentation of the fusion ring and in each case we have determined the image of the λ are completely determined for all the examples in this chapter generators, it follows Vab (see [X3] and [X4] for similar considerations). Hence the principal graphs of the subfactor M ⊃ ρ(M ) are completely determined. The principal graph of M ⊃ ρ(M ) for d(6) can be determined in exactly the same way as [X3] and is the d(3)5 ⊂ SU the case SU same as the graph given there. The following is the principal graph of M ⊃ ρ(M ) for d(6) from [X3]. d(3)5 ⊂ SU the case SU

5. Conclusions and Questions In this paper we have shown the existence of a new class of braided endomorphisms for all the maximal conformal inclusions G = SU (N ) ⊂ H with H being simple. The properties of these braided endomorphisms are analyzed and many new examples are given. Our approach suggests some natural questions. It is clear that our method can be applied to other conformal inclusions (G is not necessarily of type A) once the analog of Theorem 1.6 for LG, where G is any simple simply connected compact group is established. In fact, in [X4], certain subfactors associated to the case when G is of type B are constructed.3 Such a theory has been sketched in [W1] but details have not appeared, at least to us. Another question is the ring structure of Aρ . Such a ring structure not only encodes the structure of braided endomorphisms aλ and a˜ λ , it also contains the information about the ring structure of the subfactors from conformal inclusions. It is worthwhile to notice that the fusion graph of af (see the examples in Sect. 4) also appears in a different context, namely in the construction of some integrable N = 2 supersymmetric models (see [PZ] and [Z]). It will be very interesting to establish such connections. It is also a very interesting question to study the TQFT associated to these new braided endomorphisms along the lines of [OCN2] and [EK]. Finally, based on the relations between subfactors and quantum-groups at roots of unity, it is interesting to see if our new braided endomorphisms have any implications in the quantum group context. 3 We have checked that by assuming Theorem 1.6 for LG where G = B and G , the fusion graphs of a 2 2 f ˆ [ are those given by [X4] and [Fran] for the conformal inclusions b so(5)3 ⊂ b so(10) and (G 2 )3 ⊂ E6 by using Theorem 3.3 of Sect. 3.

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Acknowledgement. I’d like to thank Prof. Kawahigashi and Prof. A. Wassermann for useful correspondences via e-mail. I’d also like to thank the referee for helpful suggestions. This work is partially supported by NSF grant DMS-9500882.

References [L1] [L2]

Longo, R.: Minimal index and braided subfactors. J. Funct. Anal. 109, 98–112 (1992) Longo, R.: Duality for Hopf algebras and for subfactors, I. Commun. Math. Phys. 159, 133–150 (1994) [L3] Longo, R.: Index of subfactors and statistics of quantum fields, I. Commun. Math. Phys., 126, 217–247 (1989) [L4] Longo, R.: Index of subfactors and statistics of quantum fields, II. Commun. Math. Phys., 130, 285– 309 (1990) [L5] Longo, R.: Proceedings of International Congress of Mathematicians, 1281–1291 (1994) [LR] Longo, R. and Rehren, K.-H. Nets of subfactors. Rev. Math. Phys., 7, 567–597 (1995) [Po] Popa, S. Classification of subfactors and of their endomorphisms. CBMS Lecture Notes Series, 86 [W1] Wassermann, A.: Operator algebras and Conformal field theories III. To appear [W2] Wassermann, A.: Proceedings of International Congress of Mathematicians, 966–979 (1994) [PZ] Petkova, V.B. and Zuber, J.-B.: From CFT to graphs. hep-th-9510198 [Fran] Di Francesco, P. and Zuber, J.-B.: Integrable lattice models associated with SU (N ). Nucl. Phys. B, 338, 602–623 (1990) [Ka1] Kawahigashi, Y.: Classification of paragroup actions on subfactors. Publ. RIMS, Kyoto Univ., 31 481–517 (1995) [EK] Evans, D.E. and Kawahigashi, Y.: From subfactors to 3-dimensional topological quantum field theories and back – A detailed account of Ocneanu’s theory. Internat. J. Math., 6, 537–558 (1995) [OCN1] Ocneanu, A.: Quantum symmetry, differential geometry of finite graphs and classification of subfactors. University of Tokyo Seminary Notes 45, (Notes recorded by Y. Kawahigashi), 1991 [OCN2] Ocneanu, A.: An invariant couplings between 3-manifolds and subfactors. Preprint (1991) [GHJ] Goodman, F.M., de la Harpe, P. and Jones, V.: Towers of algebras and Coxeter graphs. MSRI publication, no. 14 [PS] Pressley, A. and Segal, G.: Loop groups. Oxford: Clarendon Press, 1986 [I1] Izumi, M.: Applications of fusion rules to classification of subfactors. Publ. RIMS, Kyoto Univ., 27, 953–994 (1991) [I2] Izumi, M.: On Flatness of the Coxeter graph E8 .Pacific J. Math. 166, 305–327 (1994) (1992) [JB] Bockenhauer, J.: An algebraic formulation of level 1 WZW models. Hep-th 9507047, Rev. Math. Phys. 8, 925–948 (1996) [GNO] Goddard, P., Nahm, W. and Olive, D.: Symmetric spaces, Sugawara’s energy momentum tensor in two dimensions and free fermions. Phy. Lett. 160B, 111–116 (1985) [We1] Wenzl, H.: Hecke algebras of type An and subfactors. Invent. Math., 92, 349–383 (1988) [M] Maclane, S.: Categories for the working mathematicians. Graduate Texts in Mathematics 5, Berlin: Springer 1977 [FG] Fröhlich, J. and Gabbiani, F.: Operator algebras and CFT. Commun. Math. Phys. 155, 569–640 (1993) [Kac1] Kac, V.G. and Wakimoto, M.: Modular and conformal invariance constraints in representation theory of affine algebras. Adv. in Math. 70, 156–234 (1988) [Kac2] Kac, V.G.: Infinite dimensional algebras, 3rd Edition. Cambridge: Cambridge University Press, 1990 [Sch] Sochen, N.: Integrable models from Hecke algebras. Nucl. Phys.B 360, 613–637 (1991) [MS] Moore, G. and Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123, 77–184 (1989) [X1] Xu, F.: Orbifold construction in subfactors. Commun. Math. Phys. 166, 237–253 (1994) [X2] Xu, F: A new series of subfactors. Ph.D. thesis, Berkeley, 1995 [X3] Xu, F.: Generalized Goodman-Harper-Jones construction of subfactors, I. Commun. Math. Phys. 184, 475–491 (1997) [X4] Xu, F.: Generalized Goodman-Harpe-Jones construction of subfactors, II. Commun. Math. Phys. 184, 493–508 (1997) [X5] Xu, F.: The flat part of non-flat orbifold. Pac. J. Math., 172(1), 299–306 (1996) [X6] Xu, F.: Jones-Wassermann subfactors for Disconnected Intervals. q-alg 9704003


[Wal] [CIZ] [Y] [FRS]

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Walton, M.A.: Fusion rules in WZW models. Nucl. Phys. B 340, 777–790 (1990) Cappeli, A., Itzykson, C. and Zuber, J.-B.: Commun. Math. Phys. 116, 1–23 (1987) Yamagami, S.: A note on Ocneanu’s approach to Jones index theory. Int. J. Math. 4, 859–871 (1993) Fredenhagen, K., Rehren, K.-H. and Schroer, B.: Superselection sectors with braid group statistics and exchange algebras, II. Rev. Math. Phys. Special issue (1992), 113–157 [Fre] Fredenhagen, K.: Generalizations of the theory of superselection sectors. In The algebraic theory of superselection sectors D.Kastler ed., Singapore. World Scientific, 1990 [GL1] Guido, D. and Longo, R.: The Conformal Spin and Statistics Theorem. hep-th 9505059 [GL2] Guido, D. and Longo, R.: Relativistic invariance and charge conjugation in quantum field theory. Commun. Math. Phys. 148, 521–551 (1995) [GL3] Guido, D. and Longo, R.: An Algebraic Spin and Statistics Theorem. To appear in Commun. Math. Phys. Communicated by H. Araki

Commun. Math. Phys. 192, 405 – 432 (1998)

Communications in


R-Matrix Quantization of the Elliptic Ruijsenaars– Schneider Model G.E. Arutyunov, L.O. Chekhov, S.A. Frolov Steklov Mathematical Institute, Gubkina 8, GSP-1, 117966, Moscow, Russia. E-mail: [email protected]; [email protected]; [email protected] Received: 17 March 1997 / Accepted: 8 July 1997

Abstract: It is shown that the classical L-operator algebra of the elliptic RuijsenaarsSchneider model can be realized as a subalgebra of the algebra of functions on the cotangent bundle over the centrally extended current group in two dimensions. It is governed by two dynamical r and r-matrices ¯ satisfying a closed system of equations. The corresponding quantum R and R-matrices are found as solutions to quantum analogs of these equations. We present the quantum L-operator algebra and show that the system of equations on R and R arises as the compatibility condition for this algebra. It turns out that the R-matrix is twist-equivalent to the Felder elliptic RF -matrix with R playing the role of the twist. The simplest representation of the quantum L-operator algebra corresponding to the elliptic Ruijsenaars-Schneider model is obtained. The connection of the quantum L-operator algebra to the fundamental relation RLL = LLR with Belavin’s elliptic R matrix is established. As a byproduct of our construction, we find a new N -parameter elliptic solution to the classical Yang-Baxter equation. 1. Introduction The appearance of classical dynamical r-matrices [1, 2] in the theory of integrable manybody systems raises an interesting problem of their quantization. In this way one may hope to separate the variables explicitly. At present, the classical dynamical r-matrices are known for the rational, trigonometric [2, 3] and elliptic [4, 5] Calogero–Moser (CM) systems, as well as for their relativistic generalizations – the rational, trigonometric [6, 7] and elliptic [8, 9] Ruijsenaars– Schneider (RS) systems [10]. It is recognized that dynamical systems of the Calogero type can be naturally understood in the framework of the Hamiltonian reduction procedure [11, 12]. Moreover, the reduction procedure provides an effective scheme to compute the corresponding dynamical r-matrices [3, 13]. Depending explicitly on the phase variables, the dynamical r-matrices do not satisfy a single closed equation of the Yang–Baxter type, that makes the problem of their

406

G.E. Arutyunov, L.O. Chekhov, S.A. Frolov

quantization rather nontrivial. In [14], the spin generalization of the Calogero–Sutherland system was quantized by using the particular solution [15] of the Gervais–Neveu–Felder equation [16, 15] and in [17], it was interpreted in terms of quasi-Hopf algebras. This system is not integrable, but zero-weight representations of the quantum L-operator algebra admit a proper number of commuting integrals of motion. However, it seems to be important to find such a quantum L-operator algebra for the Calogero-type systems that possesses a sufficiently large abelian subalgebra. Recently, an algebraic scheme for quantizing the rational RS model in the R-matrix formalism was proposed [18]. We introduced a special parameterization of the cotangent bundle over GL(N, C). In new variables, the standard symplectic structure was described by a classical (Frobenius) r-matrix and by a new dynamical r-matrix. ¯ The classical L-operator was introduced as a special matrix function on the cotangent bundle. The Poisson algebra of L inherited from the cotangent bundle coincided with the L-operator algebra of the rational RS model. It is this reason why we called L the L-operator. Quantizing the Poisson structure for L, we found the quantum L-operator algebra and constructed its particular representation corresponding to the rational RS system. This quantum algebra has a remarkable property, namely, it possesses a family of N mutually commuting operators directly on the algebra level. It is well-known that the elliptic RS model is the most general among the systems of the CM and RS types. In this paper, we aim to include this model in our scheme. Recall that the classical L-operator algebra for the elliptic RS model can be obtained by means of the Poissonian [19] or the Hamiltonian reduction schemes [20]. In the first scheme, the affine Heisenberg double is used as the initial phase space and in the second one, the cotangent bundle over the centrally extended current group in two dimensions is considered. Thus, the appropriate phase space we choose to deal with the elliptic RS model d )(z, z) d )(z, z) ¯ over the centrally extended group GL(N ¯ is the cotangent bundle T ? GL(N of double loops. The application of our approach [18] is not straightforward since one should work with the infinite-dimensional phase space and, therefore, the correct ded )(z, z) ¯ in the desired parameterization scription of the Poisson structure on T ? GL(N requires an intermediate regularization. We describe briefly the content of the paper and the results obtained. In the second d )(z, z) ¯ that depends on section we start by describing the Poisson structure on T ? GL(N d )(z, z) ¯ in a special way. two complex parameters k and α. Then we parametrize T ? GL(N The Poisson structure in new variables is ill defined due to the presence of singularities. To overcome this problem, we introduce an intermediate regularization. Removing the regularization we find that only for α = 1/N the resulting Poisson structure is well defined. The value α = 1/N corresponds, in fact, to the case where only the SL(N )(z, z)¯ subgroup is centrally extended. The corresponding Poisson structure is described by two matrices r and r¯ , which depend on N dynamical variables qi . It follows from the Jacoby identity (see Appendix A) that r is an N -parameter elliptic solution to the classical YangBaxter equation (CYBE). It is worthwhile to mention that the main elliptic identities (see Appendix B) follow from the fulfillment of the CYBE for r. We expect that the matrix r is related to a special Frobenius subgroup in GL(N )(z, z) ¯ as it was in the finitedimensional case [18]. The Jacoby identity also implies a closed system of equations on r and r¯ . d )(z, z). ¯ We call this function the We define a special matrix function L on T ? GL(N d )(z, z) ¯ “L-operator" since as we show the Poisson algebra of L inherited from T ? GL(N literally coincides with the one for the elliptic RS model [9, 20]. Thus, the classical L-operator algebra can be realized as a subalgebra of the algebra of functions on

R-Matrix Quantization of Elliptic Ruijsenaars–Schneider Model

407

d )(z, z). d )(z, z) T ? GL(N ¯ It turns out that the L-operator as a function on T ? GL(N ¯ admits a factorization L = W P , where pi = log Pi are the variables canonically conjugated to ¯ The Poisson bracket for the qi and W belongs to some special subgroup in GL(N )(z, z). entries of W is given by the matrix r and coincides with the Sklyanin bracket defining the structure of the Poisson–Lie group. Although the quantum analogs of equations on r and r¯ can be easily established, it is rather difficult to find the corresponding quantum R and R-matrices. The matter is that the matrices r and r¯ have the complicated structure, r = r−s⊗I +I ⊗s and r¯ = r−s⊗I, ¯ due to the presence of the s-matrix. However, we observe that the classical L-operator algebra does not depend on s and, moreover, the matrices r and r¯ also obey a closed system of equations. We show that this system arises as the compatibility condition for a d )(z, z) ¯ new Poisson algebra. This algebra contains both the Poisson algebra of T ? GL(N and the classical L-operator algebra as its subalgebras. In the third section, using this key observation, we pass to the quantization. We find the corresponding quantum R and R-matrices as solutions to quantum analogs of the equations for r and r. ¯ In particular, the R-matrix satisfies a novel triangle relation that differs from the standard quantum Yang-Baxter equation by shifting the spectral parameters in a special way. The Felder elliptic RF -matrix naturally arises in our construction. It turns out that the R-matrix is twist-equivalent to the RF -matrix with the R-matrix playing the role of the twist. Then we derive a new quadratic algebra satisfied by the “quantum" L-operator. This algebra is described by the quantum dynamical R-matrices, namely, R, RF , and R: F . R12 L1 R21 L2 = L2 R12 L1 R12

We show that the system of equations on R, RF , and R-matrices arises as the compatibility condition for this algebra. We present the simplest representation of the quantum L-operator algebra corresponding to the elliptic RS model. We note that when performing a simple canonical transformation, the quantum L-operator coincides essentially with the classical L-operator found in [10]. The quantum integrals of motion for the elliptic RS model were obtained in [10]. In [21], it was shown that any operator from the Ruijsenaars commuting family can be ˆ realized as the trace of a proper transfer matrix for the special L-operator that obeys the ˆ with Belavin’s elliptic R-matrix [26]. We note that our L-operator relation RLˆ Lˆ = Lˆ LR ˆ It follows from this observation that the determinant formula is gauge-equivalent to L. for the commuting family [21] is also valid for L. We show that any representation of our ˆ L-operator algebra is gauge equivalent to a representation of the relations RLˆ Lˆ = Lˆ LR. In the Conclusion we discuss some problems to be solved. 2. Classical L-Operator Algebra d )(z, z). ¯ Let Tτ be a torus endowed with the standard 2.1. Poisson structure of T ? GL(N complex structure and periods 1 and τ . Denote by G a group of smooth mappings from Tτ into the group GL(N, C). Then g ∈ G is a double-periodic matrix function g(z, z). ¯ The dual space to the Lie algebra of G is spanned by double-periodic functions A(z, z) ¯ with values in Mat (N, C). In what follows, we often use the concise notation b [22]. The g(z, z) ¯ = g(z) and A(z, z) ¯ = A(z). The group G admits central extensions G ∗b Poisson structure on T G with fixed central charges reads

408


1 ∂ [C, A1 (z) − A2 (w)]δ(z − w) − k(C − αI) δ(z − w), (2.1) 2 ∂ z¯ {g1 (z), g2 (w)} = 0, (2.2) (2.3) {A1 (z), g2 (w)} = g2 (w)Cδ(z − w),

{A1 (z), A2 (w)} =

where k, α are central charges and δ(z) is the two-dimensional δ-function. Here we use the standard tensor notation, and C is the permutation operator. b One can consider the following Hamiltonian action of G on T ∗ G: ¯ (z), A(z) → T −1 (z)A(z)T (z) + kT −1 (z)∂T g(z) → T −1 (z)g(z)T (z).

(2.4)

We restrict our consideration to the case of smooth elements A(z). Then the generic element A(z) can be diagonalized by the transformation (2.4) [23]: ¯ (z)T −1 (z). A(z) = T (z)DT −1 (z) − k ∂T

(2.5)

Here D is a constant diagonal matrix with entries Di , Di 6= Dj and T (z) is doubleperiodic. Matrix D is defined up to the action of the elliptic Weyl group. One can fix D by choosing the fundamental Weyl chamber. The matrix T (z) in Eq. (2.5) is not uniquely defined. Any element T˜ (z, z) ¯ = T (z, z)h(z), ¯ where a diagonal matrix h(z) is an entire function of z, also satisfies (2.5). Demanding T˜ (z, z) ¯ to be double-periodic, we obtain that h(z) is a constant matrix. We can remove this ambiguity by imposing the condition T (ε)e = e,

(2.6)

where e is a vector such that ei = 1 ∀i, and ε is an arbitrary point on Tτ . In what follows, we denote the matrix T (z) that solves Eq. (2.5) and satisfies Eq. (2.6) by T ε (z). Such matrices evidently form a group. Now we try to rewrite the Poisson structure (2.1) in terms of variables T and D. Since Di are G-invariant functions, they belong to the center of (2.1) and, therefore, it is enough to calculate the bracket {T ε (z), T ε (w)}. However, the straightforward calculation reveals that this bracket is ill defined. So, we begin with calculating the bracket {T ε (z), T η (w)}, where T ε (z) and T η (w) satisfy (2.6) at different points ε and η, η X Z δTijε (z) δTkl (w) η ε {Tij (z), Tkl (w)} = d2 z 0 d 2 w 0 {Amn (z 0 ), Aps (w0 )}. (2.7) 0 0) δA (z ) δA (w mn ps mnps To calculate the functional derivative

ε δTij (z) δAmn (z 0 ) ,

we consider the variation of (2.5):

¯ X(z) = t(z)D − Dt(z) − k ∂t(z) + d,

(2.8)

where X(z) = T −1 (z)δA(z)T (z), t(z) = T −1 (z)δT (z) and d = δD. First, from (2.8), we immediately obtain 1 δDi = T −1 ε (z)Tliε (z). δAkl (z) (τ − τ ) ik Let us introduce the function 8(z, s) of two complex variables

(2.9)


8(z, s) =

409

z¯ σ(z + s) −2ζ( 1 )zs 2πis τz− −τ . 2 e e σ(z)σ(s)

(2.10)

Here σ(z) and ζ(z) are the Weierstrass σ- and ζ-functions with periods equal to 1 and τ . The function 8(z, s) is the only double-periodic solution to the following equation: 2πis ¯ ∂8(z, s) + 8(z, s) = 2πiδ(z). τ −τ

(2.11)

It is also convenient to define 8(z, 0) as follows: 1 z − z¯ 1 8(z, 0) = lim 8(z, ε) − = ζ(z) − 2ζ( )z + 2πi . ε→0 ε 2 τ −τ This function solves the equation 2πi ¯ ∂8(z, 0) = 2πiδ(z) − . τ −τ τ −τ We introduce the notation qij ≡ qi − qj , where qi = 2πik Di . Using these functions, one can write the solution to (2.8) obeying the condition t(ε)e = 0 [20]: XZ d2 w(8(ε − w, qij )Xij (w)Eii − 8(z − w, qij )Xij (w)Eij ). (2.12) t(z) = κ i,j 1 Hereafter, we denote 2πik by κ. Performing the variation of Eq. (2.12) with respect to Amn (w) one gets

X δTijε (z) −1 ε ε = κ 8(ε − w, qjk )Tijε (z)Tjm (w)Tnk (w) δAmn (w) k ! X −1 ε ε ε − 8(z − w, qkj )Tik (z)Tkm (w)Tnj (w) . k

To compute the bracket (2.7), one needs the following relation between T ε (z) and T (z): (2.13) T ε (z) = T η (z)H ηε , η

where H ηε is a constant diagonal matrix. By direct computation, one finds 1 ε εη {T (z), T2η (w)} = T1ε (z)T2η (w)(H1εη H2ηε r12 (z, w) − αf εη (z, w)), κ 1 where εη (z, w) = r12

X

8(ε − η, qij )Eii ⊗ Ejj +

ij

−

X ij

and

X

(2.14)

8(z − w, qij )Eij ⊗ Eji (2.15)

ij

8(z − η, qij )Eij ⊗ Ejj +

X ij

8(w − ε, qij )Ejj ⊗ Eij (2.16)

410


f εη (z, w) = 8(ε − η, 0) + 8(ω − ε, 0) + 8(z − ω, 0) − 8(z − η, 0). The bracket (2.14) has the r-matrix form with the r-matrix depending not only on coordinates qi but also on the additional variables H. In the limit η → ε, one encounters the singularity. This shows that the variable T (z) is not a good candidate to describe the Poisson structure (2.1). However, one can use the freedom to multiply T (z) by any functional of A. So, we introduce a new variable Tε (z) = T ε (z)(det T ε (ε))β .

(2.17)

We use det T ε (ε) in the definition of Tε (z) in order to have the group structure for the new variables. Using the Poisson bracket (2.14) one immediately finds 1 ε εη {T (z), Tη2 (w)} = Tε1 (z)Tη2 (w) H1εη H2ηε r12 (z, w) − αf εη (z, w) κ 1 εη εη + βI ⊗ tr 3 H3εη H2ηε r32 (ε, w) + β tr 3 H1εη H3ηε r13 (z, η) ⊗ I εη ηε εη 2 (2.18) + β tr 34 H3 H4 r34 (ε, η) I ⊗ I , since f (ε, w) = f (z, η) = 0. Now we are going to pass to the limit η → ε.1 For this purpose one should take into account the following behavior of H εη when η goes to ε: H εη = 1 + (ε − η)h + o(ε − η), where h is a constant diagonal matrix being the functional of A. It turns out that there exists a unique choice for α and β, namely, α = 1/N , β = −1/N , for which the singularities cancel and there is no contribution from the matrix h. In the limit η → ε = 0, 2 for these values of α and β, one gets 1 {T1 (z), T2 (w)} = T1 (z)T2 (w)r12 (z, w). κ

(2.19)

Here the limiting r-matrix is given by r12 (z, w) = r12 (z, w) − s(z) ⊗ I + I ⊗ s(w) − where r(z, w) =

X

8(qij )Eii ⊗ Ejj +

i6=j

−

X ij

and s(z) =

X

1 8(z − w, 0)I ⊗ I, N

8(z − w, qij )Eij ⊗ Eji

ij

8(z, qij )Eij ⊗ Ejj +

X

8(w, qij )Ejj ⊗ Eij

1 X 8(qij )Eii − 8(z, qij )Eij . N ij

1 8(qij ) = ζ(qij ) − 2ζ( )qij , i 6= j, 2 8(qii ) = 0. 1

(2.21)

ij

Here we denote by the function 8(qij ) the regular part of 8(ε, qij ) for ε → 0:

2

(2.20)

Without loss of generality we assume that ε and η are real. The ε-dependence can be easily restored by shifting both z and w by ε.

(2.22)


411

Note that both r and r are skew-symmetric: r12 (z, w) = −r21 (w, z). A natural conjecture is that the r-matrix obtained satisfies the classical Yang–Baxter equation [[r, r]] ≡ [r12 (z1 , z2 ), r13 (z1 , z3 ) + r23 (z2 , z3 )] + [r13 (z1 , z3 ), r23 (z2 , z3 )] = 0. (2.23) It can be verified either by direct calculation or by considering the limiting case of the Jacoby identity for the bracket (2.18) as is done in Appendix A. Thereby, the r-matrix (2.20) is an N -parameter solution of the classical Yang–Baxter equation. Let us note that, as one could expect, the condition det T(0) = 1 is compatible with the bracket (2.19), since det T(z) is a central element of algebra (2.19). Remark that the choice α = 1/N corresponds to the case where only the sl(N )(z, z)¯ subalgebra is centrally extended. In terms of T(z) (β = −1/N ), the boundary condition looks like T(0)e = λe and det T(0) = 1. One can also check that the field A(z) defined by (2.5) with the substitution T(z) for T (z) obeys Poisson algebra (2.1). The next step is to consider the special parameterization for the field g(z). To this ˜ end, we introduce A(z): ¯ −1 . ¯ −1 + k tr ∂gg A˜ = gAg −1 − k ∂gg N

(2.24)

˜ One can check that A(z) Poisson commutes with A(w) and obeys the Poisson algebra: ∂ 1 1 {A˜ 1 (z), A˜ 2 (w)} = − [C, A˜ 1 (z) − A˜ 2 (w)]δ(z − w) + k(C − I) δ(z − w), 2 N ∂ z¯ (2.25) {A˜ 1 (z), g2 (w)} = Cg2 (w)δ(z − w). ˜ Now we factorize A(z) in the same manner as was done for A(z), −1 ¯ ˜ A(z) = U(z)DU−1 (z) − k ∂U(z)U (z),

(2.26)

where U(z) satisfies the boundary condition U(0)e = λe and det U(0) = 1. Obviously, U(z) Poisson commutes with T(w) and satisfies the Poisson algebra 1 {U1 (z), U2 (w)} = −U1 (z)U2 (w)r12 (z, w). κ

(2.27)

One can find from (2.24) and (2.26) the representation for the field g, g(z) = (det g(z)) N U(z)PT−1 (z), 1

(2.28)

where P is a constant diagonal matrix. Computing the determinants of both sides of Eq. (2.28) one gets det P = det (T(z)/U(z)). Since the l.h.s. does not depend on z and det T(0) = det U(0) = 1 we obtain that det P = 1 and det T(z) = det U(z). Calculating the Poisson brackets of P with P and Q = diag(q1 , . . . qN ) in the same manner as above one reveals that {P1 , P2 } = 0, X 1 1 {Q1 , P2 } = P2 ( Eii ⊗ Eii − I ⊗ I). κ N ii

(2.29) (2.30)

412


P In fact, it means that log Pi = pi − N1 i pi , where pi are canonically conjugated to qi . For the remaining Poisson brackets of P with the fields T, U, we have 1 {T1 (z), P2 } = T1 (z)P2 r¯ 12 (z), κ 1 {U1 (z), P2 } = U1 (z)P2 r¯ 12 (z). κ Here r¯ 12 (z) = r¯12 (z) − s(z) ⊗ I −

X 1 I⊗ 8(qij )Ejj , N ij

where we introduced the r-matrix: ¯ X X r¯12 (z) = 8(qij )Eii ⊗ Ejj − 8(z, qij )Eij ⊗ Ejj . ij

(2.31) (2.32)

(2.33)

(2.34)

ij

To complete the description of the classical Poisson structure of the cotangent bundle we present the Poisson bracket of det g with other variables: 1 {Q, det g(w)} = det g(w), κ

{P, det g(w)} = 0,

1 {T(z), det g(w)} = − det g(w)T(z)(8(z − w, 0) + 8(w, 0)), κ 1 {U(z), det g(w)} = − det g(w)U(z)(8(z − w, 0) + 8(w, 0)). κ Recall that the Jacoby identity for bracket (2.19) reduces to the classical Yang-Baxter equation for the r-matrix. As to the Poisson relations (2.31) and (2.32), one finds that the Jacoby identity is equivalent to the following quadratic in the r¯ equation: [¯r12 (z), r¯ 13 (z)] − P3−1 {¯r12 (z), P3 } + P2−1 {¯r13 (z), P2 } = 0

(2.35)

and the equation involving r and r¯ , [r12 (z, w), r¯ 13 (z) + r¯ 23 (w)] + [¯r13 (z), r¯ 23 (w)] − P3−1 {¯r12 (z, w), P3 } = 0.

(2.36)

One can check by direct calculations that the matrices r and r¯ given by (2.20) and (2.33) do solve these equations. Finally, we remark that the fields A(z) and g(z) defined by (2.5) and (2.28) obey Poisson relations (2.1–2.3). Let us note that the Poisson relation for the generator W(z) = T−1 (z)U(z) turns out to be the Sklyanin bracket: 1 {W1 (z), W2 (w)} = [r12 (z, w), W1 (z)W2 (w)], κ

(2.37)

which, therefore, defines the structure of a Poisson-Lie group. This group is an infinitedimensional analog of the Frobenius group appeared in [18], where the Poisson-Lie group structure was related to the existence of a non-degenerate two-cocycle on the corresponding Lie algebra. It would be interesting to find a similar interpretation in the infinite-dimensional case.


413

2.2. Classical L-operator. In this subsection, we define a special function on the cotangent bundle, which we call the classical L-operator. The motivation to treat this function as the L-operator is that the Poisson algebra of L is equivalent to the one found in [9] for the L-operator of the elliptic RS model. Denote by L the following function L(z) = T−1 (z)g(z)T(z) = (det g(z)) N T−1 (z)U(z)P. 1

(2.38)

By using the formulas of the previous subsection, one can easily derive the Poisson bracket of L and Q: X 1 {Q1 , L2 (z)} = L2 (z) Eii ⊗ Eii , κ i

(2.39)

and the Poisson algebra of the L-operator, 1 {L1 (z), L2 (w)} = r12 (z, w)L1 (z)L2 (w) κ + L1 (z)L2 (w)(¯r12 (z) − r¯ 21 (w) − r12 (z, w)) + L1 (z)¯r21 (w)L2 (w) − L2 (w)¯r12 (z)L1 (z).

(2.40)

Clearly, the generators Q and L form a Poisson subalgebra in the Poisson algebra of the cotangent bundle. An important feature of this subalgebra is that In (z) = tr Ln (z) form a set of mutually commuting variables. Just as in the finite-dimensional case [18], one can see from (2.39) that the L-operator admits the following factorization: L(z) = W (z)P , where Q and log P are canonically conjugated variables, the W -algebra coincides with (2.37), and the bracket of W and P is 1 {W1 (z), P2 } = −P2 [¯r12 (z), W1 (z)]. (2.41) κ In fact, everything that we need to quantize the L-operator algebra (2.40) is prepared. The problem of quantization is reduced to finding the quantum R and R-matrices satisfying the quantum analogs of Eqs. (2.23), (2.35), and (2.36), R12 (z1 , z2 )R21 (z2 , z1 ) = 1, R12 (z1 , z2 )R13 (z1 , z3 )R23 (z2 , z3 ) = R23 (z2 , z3 )R13 (z1 , z3 )R12 (z1 , z2 ), R12 (z1 , z2 )R13 (z1 )R23 (z2 ) = R23 (z2 )R13 (z1 )P3−1 R12 (z1 , z2 )P3 ,

(2.42) (2.43) (2.44)

R12 (z)P2−1 R13 (z)P2 = R13 (z)P3−1 R12 (z)P3 .

(2.45)

These matrices are assumed to have the standard behavior near ~ = 0: R = 1 + ~r + o(~),

R = 1 + ~¯r + o(~),

where ~ is a quantization parameter. The problem formulated seems to be rather complicated due to the presence of the s-matrix in the classical r and r¯ -matrices. However, the Poisson algebra (2.40) possesses an important property allowing one to avoid the problem at hand. Namely, the matrix s(z) coming both in r and r¯ drops out from the r.h.s. of (2.40). Thereby, Eq. (2.40) can be eventually rewritten as

414


1 {L1 (z), L2 (w)} = r12 (z, w)L1 (z)L2 (w) κ − L1 (z)L2 (w)(r12 (z, w) + r¯21 (w) − r¯12 (z)) + L1 (z)r¯21 (w)L2 (w) − L2 (w)r¯12 (z)L1 (z).

(2.46)

F the sum Moreover, if we denote by r12 F r12 (z, w) = r12 (z, w) + r¯21 (w) − r¯12 (z),

(2.47)

then using (2.21) and (2.34) we obtain X X F (z − w) = − 8(qij )Eii ⊗ Ejj + 8(z − w, qij )Eij ⊗ Eji . r12 ij

(2.48)

ij

In this expression one can recognize the elliptic solution to the classical Gervais–Neveu– Felder equation [16, 15]: F F F F F (z1 − z2 ), r13 (z1 − z3 ) + r23 (z2 − z3 )] + [r13 (z1 − z3 ), r23 (z2 − z3 )] [r12 F (z1 −P3−1 {r12

− z2 ), P3 } +

F P2−1 {r13 (z1

− z3 ), P2 } −

F P1−1 {r23 (z2

(2.49)

− z3 ), P1 } = 0.

In fact, rF emerges as the semiclassical limit of the quantum R-matrix found in [15]. The absence of the s-matrix in the resulting L-operator algebra and the appearance of the rF -matrix show that there may exist a closed system of equations involving only r- and r-matrices ¯ in the classical case, and R- and R-matrices in the quantum one. In the next subsection we find the desired system of equations and describe a Poisson structure for which these equations ensure the fulfillment of the Jacoby identity. Note that the algebra (2.46) literally coincides with the one obtained in [20] by using the Hamiltonian reduction procedure. A mere similarity transformation of L turns algebra (2.46) to the one previously found in [9]. In contrast to [9] where (2.46) was derived by direct calculation with the usage of the particular form of the L-operator for the RS model, our treatment does not appeal to the particular form of L. 2.3. Quadratic Poisson algebra with derivatives. In the first subsection, we obtained the matrices r and r¯ obeying system of equations (2.23), (2.35) and (2.36). Clearly, these equations are not satisfied when substituting r and r¯ for r and r¯ . However, computing the l.h.s. of these equations after this substitution we arrive at the surprisingly simple result: [r12 (z1 , z2 ), r13 (z1 , z3 ) + r23 (z2 , z3 )] + [r13 (z1 , z3 ), r23 (z2 , z3 )] = −(∂1 + ∂2 )r12 (z1 , z2 ) + (∂1 + ∂3 )r13 (z1 , z3 ) − (∂2 + ∂3 )r23 (z2 , z3 ),

(2.50)

[r¯12 (z), r¯13 (z)] − P3−1 {r¯12 (z), P3 } + P2−1 {r¯13 (z), P2 } = −∂(r¯12 (z) − r¯13 (z)), (2.51) and [r12 (z1 , z2 ), r¯13 (z1 ) + r¯23 (z2 )] + [r¯13 (z1 ), r¯23 (z2 )] − P3−1 {r12 (z1 , z2 ), P3 } (2.52) = −(∂1 + ∂2 )r12 (z1 , z2 ) + ∂1 r¯13 (z1 ) − ∂2 r¯23 (z2 ). ∂ , where x = Re z. Note that Eqs. (2.35) and (2.36) are formulated with the Here ∂ = ∂x help of P. However, since all the matrices depend only on the difference qij = qi − qj , we simply replace P by P .


415

Comparing Eqs. (2.50–2.52) for r and r¯ with (2.23), (2.35), and (2.36) for r and r¯ , we come to the conclusion that the s(z)-matrix coming in r and r¯ effectively plays the role of the derivative with respect to the spectral parameter. It is worth mentioning that Eqs. (2.50)-(2.52) obtained for r and r¯ can be rewritten in the same form as Eqs. (2.23), (2.35), and (2.36) if we replace r and r¯ by r12 − ∂1 + ∂2 and r¯12 − ∂1 . In particular, for (2.50), we have [r12 −∂1 +∂2 , r13 −∂1 +∂3 ]+[r13 −∂1 +∂3 , r23 −∂2 +∂3 ]+[r12 −∂1 +∂2 , r23 −∂2 +∂3 ] = 0. (2.53) Thus, r12 − ∂1 + ∂2 is a matrix first-order differential operator satisfying the standard classical Yang-Baxter equation. Using this fact we write down the Poisson algebra generated by the fields T (z), U (z), Q and P , having Eqs. (2.50)–(2.52) as the consistency conditions: 1 {T1 (z), T2 (w)} = T1 (z)T2 (w)r12 (z, w) + T10 T2 − T1 T20 , κ 1 {U1 (z), U2 (w)} = −U1 (z)U2 (w)r12 (z, w) − U10 U2 + U1 U20 , κ 1 {T1 (z), P2 } = P2 T1 (z)r¯12 (z) + P2 T10 (z), κ 1 {U1 (z), P2 } = P2 U1 (z)r¯12 (z) + P2 U10 (z), κ X {Q1 , P2 } = P2 Eii ⊗ Eii , {Q1 , T2 } = {Q1 , U2 } = 0,

(2.54) (2.55) (2.56) (2.57) (2.58)

i

where T 0 = ∂T . It is worth mentioning that the Poisson structure (2.54–2.58) is not compatible with the boundary condition T (0)e = λe. Let us note that there exists a Poisson subalgebra of Poisson algebra (2.54–2.58), formed by the generators: ¯ (z)T −1 (z), g(z) = U (z)P T −1 (z) A(z) = T (z)DT −1 (z) − k ∂T that coincides with the Poisson algebra of the cotangent bundle with the central charge α = 0. Defining the L-operator as L(z) = T −1 (z)g(z)T (z) = T −1 (z)U (z)P , we get for L algebra (2.46) obtained previously. As in the previous subsection the commutativity of In (z) follows again from the one of g(z). The main advantage of Poisson algebra (2.54–2.58) is that it can be easily quantized. 3. Quantization 3.1. Quantum R-matrices. In this section, following the ideology of the Quantum Inverse Scattering Method [24, 25], we quantize the classical r and r-matrices ¯ and derive the quantum L-operator algebra. We start with quantization of the relations (2.50)-(2.52). Let T (z), U (z) be matrix generating functions being the formal Fourier series in variables x and y: X X Tmn e2πi(mx+ny) , U (z) = Umn e2πi(mx+ny) , T (z) = mn

mn

416


where z = x + τ y. Denote by A a free associative unital algebra over the field C generated by matrix elements of the Fourier modes of T (z), U (z), and by the entries of the diagonal matrices P and Q modulo the relations T1 (z)T2 (w − ~) = T2 (w)T1 (z − ~)R12 (−~, z, w), U1 (z)U2 (w + ~) = U2 (w)U1 (z + ~)R12 (~, z, w), T1 (z + ~)P2 R12 (~, z) = P2 T1 (z), U1 (z + ~)P2 R12 (~, z) = P2 U1 (z), X Eii ⊗ Eii , [Q1 , P2 ] = −~P2

(3.1) (3.2) (3.3) (3.4) (3.5)

i

[T1 (z), U2 (w)] = [T1 (z), Q2 ] = [U1 (z), Q2 ] = [P1 , P2 ] = [Q1 , Q2 ] = 0. Here R(~, z, w) and R12 (~, z) are double-periodic matrix functions of spectral parameters. These functions also depend on the coordinates qi and have the following semiclassical behavior at ~ = 0: R = 1 + ~r + o(~), R = 1 + ~r¯ + o(~).

(3.6)

The next step is to find the conditions on R and R that ensure the consistency of the defining relations for A. In the sequel we often use R(z, w) as a shorthand notation for R(~, z, w). First, we write down the compatibility condition for algebra (3.1) or (3.2), which reduces to the Quantum Yang-Baxter equation with spectral parameters shifted by ~, R12 (z, w)R13 (z − ~, s − ~)R23 (w, s) = R23 (w − ~, s − ~)R13 (z, s)R12 (z − ~, w − ~). (3.7) Analogously to the classical case, one can introduce the following matrix differential ∂ ∂ operator R(z, w) = e~ ∂w R(z, w)e−~ ∂z in terms of which Eq. (3.7) reads as the standard Quantum Yang-Baxter equation: R12 (z, w)R13 (z, s)R23 (w, s) = R23 (w, s)R13 (z, s)R12 (z, w).

(3.8)

Relation (3.1) also requires the fulfillment of the “unitarity" condition for R, R12 (z, w)R21 (w, z) = 1.

(3.9)

Analogously, we find the following compatibility conditions for (3.3): P3−1 R12 (z)P3 R13 (z − ~) = P2−1 R13 (z)P2 R12 (z − ~)

(3.10)

and P3−1 R12 (z, w)P3 R13 (z − ~)R23 (w) = R23 (w − ~)R13 (z)R12 (z − ~, w − ~). (3.11) Now taking into account (3.6) one can easily see that in the semiclassical limit 1 1 − {·, ·} = lim [·, ·], ~→0 ~ κ relations (3.1–3.5) determine Poisson structure (2.54–2.58), while Eqs. (3.7), (3.10), and (3.11) turn into (2.50), (2.51), and (2.52), respectively, in order ~2 . In the first order in ~, the unitarity condition (3.9) requires r to be skew-symmetric. Hence, the algebra A with


417

defining relations (3.1)–(3.5), where R and R are the solutions of (3.7–3.11) obeying (3.6), is a quantization of the Poisson structure (2.54–2.58). Now we are in a position to find the matrices R and R explicitly. We start with the R-matrix for which we assume the following natural ansatz: X X 8(~1 , qij + ~2 )Eii ⊗ Ejj + 8(z − w + ~3 , qij + ~4 )Eij ⊗ Eji f R(~, z, w) = −

X

ij

8(z + ~5 , qij + ~6 )Eij ⊗ Ejj +

ij

X

ij

8(w + ~7 , qij + ~8 )Ejj ⊗ Eij .

(3.12)

ij

This form is compatible with the structure of the classical r-matrix. Here ~1 , . . . , ~8 are arbitrary parameters that should be specified by Eqs. (3.7) and (3.9), and f is a scalar function that may depend only on ~i and spectral parameters. It turns out that the parameters hi are almost uniquely fixed by the unitarity condition (3.9). Substituting (3.12) into (3.9) and using the elliptic function identities we obtain ~2 = ~3 = ~4 = ~6 = ~8 = 0, ~5 = ~1 + ~7 , f 2 (z, w) = P(~1 ) − P(z − w), where P(z) is the Weierstrass P-function. Now it is a matter of direct calculation to check that Eq. (3.7) holds for ~1 = ~. The remaining parameter ~7 is inessential since it corresponds to an arbitrary common shift of the spectral parameters z and w. In the sequel, we choose ~7 = 0. Therefore, the obtained solution to (3.7) and (3.6) reads as follows: X X 8(~, qij )Eii ⊗ Ejj + 8(z − w, qij )Eij ⊗ Eji (3.13) f (z, w)R(~, z, w) = ij

−

X

ij

8(z + ~, qij )Eij ⊗ Ejj +

ij

X

8(w, qij )Ejj ⊗ Eij ,

ij

√ where f (z, w) = P(~) − P(z − w). One must be careful in the definition of R(−~, z, w). This matrix is defined by (3.13) with the replacement ~ → −~ and f → −f . Therefore, R(~, z, w) and R(−~, z, w) are related as R12 (−~, z, w) = R21 (~, w − ~, z − ~).

(3.14)

To find the R-matrix, we adopt the following ansatz: X X 1 R12 (~, z) = 8(~1 , qij +~2 )Eii ⊗Ejj − 8(z +h3 , qij +~4 +δij ~5 )Eij ⊗Ejj . σ(~) ij ij (3.15) It has almost the same matrix structure as the classical r-matrix. ¯ Since Eq. (3.10) is easier to deal with than Eq. (3.11), we first substitute (3.15) into Eq. (3.10) thus obtaining R: X X 1 R12 (~, z) = − 8(~, qij )Eii ⊗ Ejj 8(z + ~3 , qij )Eij ⊗ Ejj σ(~) i6=j i6=j X − 8(z + ~3 , −~) Eii ⊗ Eii , (3.16) i

where h3 remains unfixed.

418


Equation (3.11) involves both the R- and R-matrices and is independent on (3.7) and (3.10). One can verify by direct calculations that R and R given by Eqs. (3.13) and (3.16) also satisfy (3.11) as soon as ~3 = ~. One can easily check that in the case of real ~, the matrices R and R have the proper semiclassical behavior (3.6). −1 In what follows we also need the R -matrix, X X 1 −1 R12 (~, z) = − 8(−~, qij + ~)Eii ⊗ Ejj + 8(z, qij + ~)Eij ⊗ Ejj . (3.17) σ(~) ij ij It would be of interest to mention that just as in the rational case without the spectral parameter [18], one can introduce the formal variable W (z) = T −1 (z)U (z) with permutation relations following from (3.1–3.5): R12 (z, w)W1 (z)W2 (w + ~) = W2 (w)W1 (z + ~)R12 (z, w), W1 (z + ~)P2 R12 (z) = P2 R12 (z)W1 (z).

(3.18) (3.19)

In analogy with the rational case, it is natural to treat Eq. (3.18) as the defining relation of the quantum elliptic Frobenius group. 3.2. Quantum L-operator algebra. Just as in the classical case, we introduce a new variable: (3.20) L(z) = T −1 (z)U (z)P = W (z)P, which we call a quantum L-operator. Using the relations of the algebra A one can formally derive the following algebraic relations satisfied by the quantum L-operator: X Eii ⊗ Eii , (3.21) [Q1 , L2 (z)] = −~L2 (z) i

R12 (z, w)L1 (z)R21 (w)L2 (w) −1

= L2 (w)R12 (z)L1 (z)R21 (w − ~)R12 (z − ~, w − ~)R12 (z − ~).

(3.22)

In spite of the fact that L has the form L(z) = W (z)P , we can not reconstruct from Eqs. (3.21) and (3.22) the relations (3.18) and (3.19) for W and P . So, in the sequel, we do not assume any relations on W and P . Let us define −1 F (z, w) = R21 (w)R12 (z, w)R12 (z). (3.23) R12 Then, by using the explicit form of the R- and R-matrices and elliptic function identities, we obtain X F (z − w) = − 8(−~, qij )Eii ⊗ Ejj f R12 i6=j

+

X i6=j

8(z − w, qij )Eij ⊗ Eji + 8(z − w, ~)

X

Eii ⊗ Eii , (3.24)

i

which is nothing but the R-matrix by Felder [15], i.e., an elliptic solution to the quantum Gervais–Neveu–Felder equation [16, 15]:


419

F F F F F F P1−1 R23 (w−s)P1 R13 (z−s)P3−1 R12 (z−w)P3 = R12 (z−w)P2−1 R13 (z−s)P2 R23 (w−s). (3.25) Recall that one feature of RF is the “weight zero" condition: F (z − w)] = 0. [P1 P2 , R12

(3.26)

Developing RF in powers of ~, we have RF = 1 + ~rF + o(~), where rF is given by (2.48). Let us stress that in our consideration RF arises to account for the explicit form of R and R, and that the Gervais–Neveu–Felder equation does not follow from system (3.7–3.11). Formula (3.23) shows that the matrix R plays the role of the twist, which transforms the matrix R(z, w) – a particular solution of (3.7) – into such a solution of (3.25) which depends only on the difference z − w. Thus, the quantum L-operator algebra (3.22) can be presented in the following form: F (z − w). R12 (z, w)L1 (z)R21 (w)L2 (w) = L2 (w)R12 (z)L1 (z)R12

(3.27)

The quantum L-operator algebra seems to be automatically compatible as A is compatible. However, a simple analysis shows that A and the algebra (3.27) admit different supplies of representations. In particular, the simplest representation for L we present below does not realize the algebra (3.18), (3.19). Therefore, we find it necessary to give a direct proof of the compatibility of (3.27). In this way, we come across Eq. (3.25) and discover a new relation involving RF and R. To this end, let us multiply both sides of (3.27) by P2−1 R31 (s + ~)P2 R32 (s)L3 (s) and subsequently using Eq. (3.27) we transform the string L1 · · · L2 · · · L3 into L3 · · · L2 · · · L1 . For the l.h.s., we have R12 (z, w)L1 R21 (w)L2 P2−1 R31 (s + ~)P2 R32 (s)L3 = R12 (z, w)L1 R21 (w)R31 (s + ~)L2 R32 (s)L3 = F (w − s) = R12 (z, w)L1 R21 (w)R31 (s + ~)R32 (s, w)L3 R23 (w)L2 R23 F (w − s) = R12 (z, w)R32 (s + ~, w + ~)L1 R31 (s)L3 P3−1 R21 (w + ~)P3 R23 (w)L2 R23

R12 (z, w)R32 (s + ~, w + ~)R31 (s, z)L3 R13 (z)L1 × F F (z − s)P3−1 R21 (w + ~)P3 R23 (w)L2 R23 (w − s). R13

(3.28)

At this point, we interrupt the chain of calculations by remarking that the next step implies the possibility to push RF somehow through P3−1 R21 (w + ~)P3 R23 (w). It can be done by virtue of the following new relation involving RF and R: F F (z)P2−1 R31 (w)P2 R32 (w − ~) = P1−1 R32 (w)P1 R31 (w − ~)R12 (z), R12

(3.29)

which can be checked directly by using the explicit forms (3.24) and (3.16) of RF and R respectively. Now we pursue calculation of (3.28) with the relation (3.29) at hand: R12 (z, w)R32 (s + ~, w + ~)R31 (s, z)L3 R13 (z)R23 (w + ~) × F F (z − s)P2 R23 (w − s). L1 R21 (w)L2 P2−1 R13 As to the r.h.s., the same technique yields

420

G.E. Arutyunov, L.O. Chekhov, S.A. Frolov F L2 R12 (z)L1 R12 (z − w)P2−1 R31 (s + ~)P2 R32 (s)L3 = F (z − w)P3 = L2 R12 (z)R32 (s + ~)L1 R31 (s)L3 P3−1 R12 F F (z − s)P3−1 R12 (z − w)P3 L2 R12 (z)R32 (s + ~)R31 (s, z)L3 R13 (z)L1 R13

= R31 (s + ~, z + ~)L2 R32 (s)L3 P3−1 R12 (z + ~)P3 × F F R13 (z)L1 R13 (z − s)P3−1 R12 (z − w)P3 =

R31 (s + ~, z + ~)R32 (s, w)L3 R23 (w)R13 (z + ~)L2 R12 (z)L1 × F F F (w − s)P1 R13 (z − s)P3−1 R12 (z − w)P3 = P1−1 R23

(3.30)

R31 (s + ~, z + ~)R32 (s, w)R12 (z + ~, w + ~)L3 R13 (z)R23 (w + ~)L1 × F F F F R21 (w)L2 R21 (w − z)P1−1 R23 (w − s)P1 R13 (z − s)P3−1 R12 (z − w)P3 . Therefore, comparing the resulting expressions we conclude that the compatibility condition for the L-operator algebra (3.27) reduces to four equations (3.7), (3.11), (3.24), and (3.29). The existence of the Poisson commuting functions In (z) in the classical case implies that the commuting family should exist in the quantum case as well. It should be the intrinsic property of the algebra (3.27) itself, without referring to the explicit form of its representations. Let us demonstrate the commutativity of the simplest quantities tr L(z) and tr L−1 (z) postponing the discussion of the general case to the next section. To this end, we need one more relation involving the matrices RF , R and R. In analogy with the rational case, it is useful to introduce the variable g(z) = U (z)P T −1 (z). Calculation of the commutator [g1 (z), g2 (w)] with the help of the defining relations of A results in [g1 (z), g2 (w)] = U2 (w)U1 (z + ~) R12 (~, z, w)P1 R21 (w)P2 R12 (z − ~)R12 (−~, z, w) − P2 R12 (z)P1 R21 (w − ~) T2−1 (w − ~)T1−1 (z). When the spectral parameter is absent, the algebra A allows one to establish a connection with the quantum cotangent bundle (see [18] for details). Then, in particular, the quantity [g1 , g2 ] is equal to zero. In the case at hand, we can not construct a subalgebra of A that is isomorphic to the quantum cotangent bundle. However, one can note that in the elliptic case, the commutativity of g(z) with g(w) follows from the identity R12 (~, z, w)P1 R21 (w)P2 R12 (z − ~)R12 (−~, z, w) = P2 R12 (z)P1 R21 (w − ~). Using the definition of RF , Eq. (3.14), and the "weight zero" condition (3.26), the last formula can be written in the following elegant form: −1

F (z − w)P1 R21 (w)P1−1 . R12 (z, w) = P2 R12 (z)P2−1 R12

(3.31)

Identity (3.31) plays the primary role in proving the commutativity of the family tr L(z) . To prove the commutativity, let us multiply both sides of F (w − z) L2 (w)R12 (z)L1 (z) = R12 (z, w)L1 (z)R21 (w)L2 (w)R21 −1

by P2 R12 (z)P2−1 and take the trace in the first and the second matrix spaces. We get −1

tr 12 P2 R12 (z)P2−1 L2 (w)R12 (z)L1 (z) = −1

F tr 12 P2 R12 (z)P2−1 R12 (z, w)L1 (z)R21 (w)L2 (w)R21 (w − z) .

(3.32)


421

−1

It is useful to write R12 in the factorized form X −1 −1 R12 (z) = Rij ⊗ Ejj ,

(3.33)

ij

where

−1

Rij = −σ(~)8(−~, qij + ~)Eii + σ(~)8(z, qij + ~)Eij . Then the l.h.s. of (3.32) reads as X −1 tr 12 (Pj Rij Pj−1 ⊗ Ejj )L2 R12 (z)L1 . ij

Taking into account that R12 is diagonal in the second matrix space and using the cyclic property of the trace, we obtain X −1 tr 12 (Pj Rij Pj−1 ⊗ I)(I ⊗ LEjj )R12 (z)L1 . ij

Since L = W P , where all entries of W commute with qi , we arrive at X −1 tr 12 (Pj Rij Pj−1 ⊗ I)(I ⊗ W Pj Ejj )R12 (z)L1 = ij

X

−1

tr 12 L2 (Rij ⊗ Ejj )R12 (z)L1 = tr L(w) tr L(z) .

ij

As to the r.h.s. of (3.32), we use identity (3.31) to rewrite it as −1

F F (z − w)P1 R21 (w)P1−1 L1 (z)R21 (w)L2 (w)R21 (w − z) . tr 12 R12

Having in mind that R21 is diagonal in the first matrix space and taking into account the property (3.26) one can easily see that under the trace sign, the matrix RF can be F . Therefore, we get pushed to the right where it cancels with R21 −1

tr 12 P1 R21 (w)P1−1 L1 (z)R21 (w)L2 (w) .

(3.34)

Now applying to this expression the technique we used above for the l.h.s of (3.32) we conclude that Eq. (3.34) is equal to tr L(z) tr L(w) . Thus, we proved that tr L(z) commutes with tr L(w) . Quite analogously one can prove that tr L−1 (z) commutes with tr L−1 (w) and with tr L(w) . Now we give an example of the simplest representation of algebra (3.21) and (3.27) associated with the elliptic RS model. Namely, the following L-operator satisfies algebra (3.27): X 8(z, qij + γ)bj Pj Eij , (3.35) L(z) = ij

where bj =

Y

8(γ, qaj ).

(3.36)

a6=j

Here the parameter γ is a coupling constant of the elliptic RS model. This can be checked by straightforward calculations. Some comments are in order. The L-operator of the form

422


(3.35) already appeared in [20] as a result of the Hamiltonian reduction procedure applied d )(z, z). ¯ To guess the explicit form of bj one should note that in the rational to T ? GL(N limit algebra (3.27) tends to the one obtained in [18], where the coefficients bj are found to be Y qaj + γ . (3.37) bj = qaj a6=j

Therefore, it is natural to assume that the elliptic analog of (3.37) is given by (3.36). It is worthwhile to mention that bj are not uniquely defined since one can perform a canonical transformation of (q, p)-variables. In particular, the variables b˜ j =

Y σ(qaj + γ) σ(γ)σ(qaj )

(3.38)

a6=j

are related to bj by the canonical transformation qi → qi and P Pi → eα a qai Pi , γ¯ where α = 2ζ(1/2)γ − i γ− τ −τ¯ . We call this L the quantum L-operator of the elliptic RS model. Indeed, taking the Hamiltonian to be H = tr L(z) one can see that the quantum canonical transformation of the form: Y σ(qai + γ) 1/2 Y σ(qai ) 1/2 R Pi , (3.39) Pi = σ(qai ) σ(qai − γ) a6=i

a6=i

where PiR is the momentum in the Ruijsenaars Hamiltonian, turn H into the first integral S1 from the Ruijsenaars commuting family [10]. Moreover, after the canonical transformation (3.39), the L-operator (3.35) coincides essentially with the classical L-operator of the RS model. The generating function for the commuting family in terms of L can be written as X (−µ)N −k Ik (z), (3.40) I(z, µ) =: det (L(z) − µ) := k

where the normal ordering :: means that all momentum operators are pushed to the right. It follows from the results obtained in the next section. 4. Connection to the Fundamental Relation RLL = LLR In this section we establish a connection of the quantum L-operator algebra (3.27) with the fundamental relation RLL = LLR. In [21], the operators from the Ruijsenaars commuting family were obtained by using a special representation Lˆ of the algebra B B (z − w)Lˆ 1 (z)Lˆ 2 (w) = Lˆ 2 (w)Lˆ 1 (z)R12 (z − w), R12

(4.1)

where RB (z) is Belavin’s R-matrix being an elliptic solution to the quantum YangBaxter equation [26]. The explicit form of RB we use here can be found in [27]. For the reader’s convenience we recall a construction of Lˆ [28, 29].


423

Denote by h∗ the weight space for slN (C) that can be realized in CN with a basis PN i , < i , j >= δij , as the orthogonal complement to i=1 i . Let ¯k be the orthogonal P N projection of k : ¯k = k − N1 i=1 i . ∗ For each q ∈ h one can introduce the intertwining vectors [32, 33] z ¯k − < q, ¯k >)/iη(τ ), (4.2) φ(z)q+~ j = θj ( q N where X n2 1 θj (z) = τ exp 2πi n(z + ) + 2 2N n∈ N −j+N Z 2 Q∞ and η(τ ) = p1/24 m=1 (1 − pm ) is the Dedekind eta function with p = exp 2πiτ . ¯k j ¯k ¯ q+~ Following [28] we denote by φ(z) the entries of the matrix inverse to φ(z)q+~ j. q q Then the orthogonality relations read as follows n X

¯k j ¯k0 ¯ q+~ φ(z) φ(z)q+~ j = δkk0 q q

j=1

n X

q+~¯k j 0 ¯k ¯ φ(z)q+~ = δjj 0 . j φ(z)q q

(4.3)

k=1

In the sequel, the following formula [21] will be of intensive use n X

0

¯i ¯ q0 +~¯j m φ(z)q+~ φ(z) m q q

m=1

=

θ(z+ < q 0 , ¯j > − < q, ¯i >) Y θ(< q 0 , ¯j 0 > − < q, ¯i >) θ(z) θ(< q 0 , ¯j 0 > − < q 0 , ¯j >) 0

(4.4)

j 6=j

Here θ(z) denotes the Jacoby θ-function X 2 1 1 1 2 1 e2πi(z+ 2 )(n+ 2 )+iπτ (n+ 2 ) = θ0 (0)e−ζ( 2 )z σ(z). θ(z) = − n

ˆ It is shown in [29, 30] that the L-operator Lˆ ij (z) =

N X

¯k ¯ q+~¯k j ~¯k ∂qk φ(z + γN )q+~ e , i φ(z)q q ∂

(4.5)

k=1

acting on the space of functions on h∗ satisfies relation (4.1). This Lˆ is an N × N generalization of the 2 × 2 Sklyanin L-operator [31]. ¯k ˆ The intertwining vectors φ(z)q+~ j coming in the definition of L relate the matrix q (1) B R with the Boltzmann weights for the An−1 face model. Recall [33] that the nonzero Boltzmann weights depending on the spectral parameter z are explicitly given by   q + ~¯i ˇ  q z q + 2~¯i  = θ(z + ~) , (4.6) W θ(~) q + ~¯i   q + ~¯i ˇ  q z q + ~(¯i + ¯j )  = θ(−z + qij ) (i 6= j), W θ(qij ) q + ~¯i   q + ~¯i θ(z) θ(~ + qij ) ˇ q z q + ~(¯i + ¯j )  = W (i 6= j), θ(~) θ(qij ) q + ~¯ j

424


where qij =< q, ¯i − ¯j >. The relation between RB and the face weights is given by X i0 j 0

X

0 0

q+~(¯k +¯m ) ¯k RB (z − w)iijj φ(z)q+~ i0 φ(w)q+~¯k j0 = q

(4.7)



q+~(¯k +¯m ) ¯s φ(w)q+~ j φ(z)q+~¯s i q

s

 q + ~¯k ˇ  q z − w q + ~(¯k + ¯m )  . W q + ~¯s

In what follows we use the concise notation   q + ~¯k ˇ  q z − w q + ~(¯k + ¯m )  . Wsk [k + m] = W q + ~¯s Then the dual relation to (4.7) is X q+~(¯k +¯m ) i0 B ¯k j 0 ¯ ¯ q+~ φ(w) φ(z)q+~ R (z − w)ij q i0 j 0 ¯k i0 j 0

=

X

q+~(¯k +¯m ) j ¯s i ¯ ¯ q+~ φ(w)q+~ Wks [k + m]φ(z) . q ¯s

(4.8)

s

In [21] another L-operator L˜ appeared. It is related to Lˆ in the following way: Lˆ ij (z) → L˜ ij (z) =

X

¯i i0 ¯j ˆ ¯ q+~ φ(z) φ(z)q+~ j 0 Li0 j 0 (z) q q

(4.9)

i0 j 0

=

θ(z + γ + qij ) Y θ(γ + qnj ) ~¯j ∂q∂j e . θ(z) θ(qni ) n6=i

In what follows we need to remove from the quantum L-operator algebra (3.27) the nonholomorphic dependence on the spectral and quantization parameters. This can be achieved by considering the following transformation of the L-operator: L(z) → eα(z)Q e−βQ L(z)eβQ e−α(z)Q ,

(4.10)

where α(z) is an arbitrary function of the spectral parameter and β is a complex number. Since the transformed L-operator also has the structure W P , then the following formula is valid: (4.11) L2 (w)eα(z)Q1 = eα(z)Q1 L2 (w)e~α(z)r0 , P where the notation r0 = i Eii ⊗ Eii was used. Recalling that the L-operator (3.35) satisfies the quantum L-operator algebra (3.27) and using Eq. (4.11) one can easily establish the algebra satisfied by the transformed L: ˇ (w)L (w) = L (w)R ˇ (z)L (z)Rˇ F (z − w), Rˇ 12 (z, w)L1 (z)R 21 2 2 12 1 12 ˇ and Rˇ F are ˇ R where the matrices R,


425

Rˇ 12 (z, w) = e−α(z)Q1 −α(w)Q2 +βQ2 R12 (z, w)eα(z)Q1 +α(w)Q2 −βQ1 , ˇ (z) = e~α(z)r0 −α(z)Q1 +βQ2 R (z)eα(z)Q1 −βQ1 , R 12

12

F (z, w) = Rˇ 12 ~

(4.12) (4.13) (4.14)

~

F (z, w)eα(z)Q1 +α(w)Q2 + 2 (α(w)−α(z))r0 −βQ2 . e−α(z)Q1 −α(w)Q2 + 2 (α(w)−α(z))r0 +βQ1 R12

Since the transformation in question keeps the form of the quantum L-operator algebra ˇ and Rˇ F also satisfy all the compatibility conditions. ˇ R intact, the transformed matrices R, In particular, the transformation (4.14) defines another solution of (3.25). For β = 0 this was observed in [14]. ¯ h z−z¯ For the particular choice β = 2πi h− τ −τ¯ and α(z) = 2πi τ −τ¯ + β we find that the ˇ and Rˇ F are given by the same formulae (3.13), (3.16), (3.24) with the ˇ R matrices R, z−z¯

θ(z+s) 0 θ(z+s) θ (0)e2πi τ −τ¯ by the meromorphic function θ(z)θ(s) . The total change of 8(z, s) = θ(z)θ(s) transformation with such a choice of α and β transforms (up to an unessential multiplier) the L-operator (3.35) into

Lij (z) =

θ(z + qij + γ) Y θ(qnj + γ) Pj , θ(z)θ(qij + γ) θ(qnj )

(4.15)

n6=j

which is a quasi-periodic meromorphic matrix function of the spectral parameter: L(z + 1) = L(z), L(z + τ ) = e−2πi(γ+~) e−2πiQ L(z)e2πiQ . ~¯

∂

We assume that the L-operator is of the form W P, where Pi = e i ∂qi . The replacement of P by P preserves all the consistency conditions because the R-matrices depend only on the difference qi − qj . Thus, Eq. (3.27) with R-matrices defined via 8(z, s) = θ(z+s) θ(z)θ(s) refers to the meromorphic version of the quantum L-operator algebra while (4.15) provides its particular meromorphic representation. In what follows we use only this meromorphic version. Comparing (4.15) to (4.9) we can read off that L and L˜ are related in the following way: Q n6=j θ(qnj ) Lij (z). (4.16) L˜ ij (z) = Q n6=i θ(qni ) Since the combined transformation (4.9), (4.16) from Lˆ to L depends only on q we can conjecture that any representation L of the quantum L-operator algebra (3.27) is gauge equivalent to some representation Lˆ of (4.1) with a gauge-equivalence defined as Q X 0 θ(qnj 0 ) q+~¯j 0 j q+~¯i0 ¯ ˆ Qn6=j Li0 j 0 (z). φ(z)q (4.17) Lij (z) = i φ(z)q n6=i0 θ(qni0 ) 0 0 ij

Now we are in a position to prove this conjecture. Suppose Lˆ is an abstract L-operator satisfying algebra (4.1) and introduce L˜ by Eq. (4.9). Assume that L˜ has the structure ~¯ ∂ W P, where the entries of the diagonal matrix P are Pi = e i ∂qi and the entries of W commute with qi . Then substituting Lˆ expressed via L˜ in (4.1) and performing the ˜ straightforward calculation with the use of (4.8) one finds an algebra satisfied by L:

426


X i0 j 0

¯a +¯c ) j ¯a i0 ¯b ¯ ¯ q+~ φ(w)q+~( Wka [a + c]δa+c,i+k φ(z) φ(z)q+~ q q q+~¯a i0

0

abcd

¯ q+~(¯j +¯d ) L˜ bj (z)L˜ dl (w) = φ(w) q+~¯j j0 X ¯k +¯i ) j 0 ¯k i0 ¯d ¯ ¯ q+~ φ(z)q+~( Waj [j + l]δj+l,a+c φ(w) φ(w)q+~ q q q+~¯k i0 i0 j 0

abcd

¯ q+~(¯c +¯b ) L˜ dc (w)L˜ ba (z). φ(z) q+~¯c j0 Performing the summation in i0 and j 0 with the help of (4.4) we obtain X

Wkb [b + a]δa+b,i+k

abc

θ(w + qac + ~δab − ~δjc ) θ(w)

Y θ(qnc + ~δnb − ~δjc ) L˜ bj (z)L˜ cl (w) = θ(qna + ~δnb − ~δab )

n6=a

X

Waj [j + l]δj+l,a+c

abc

θ(z + qib + ~δik − ~δbc ) θ(z)

Y θ(qnb + ~δnk − ~δbc ) n6=i

θ(qni + ~δnk − ~δik )

L˜ kc (w)L˜ ba (z).

Let us introduce an operator L by inverting (4.16). Substituting this L in the last formula, taking into account the nonzero components of the face weights and multiplying both sides by the function Q Q θ(qni + ~δnk − ~δik ) n6=k θ(qnk ) Q Qn6=i , θ(q ) nj n6=j n6=l θ(qnl + ~δnj − ~δjl ) we finally arrive at the algebra satisfied by L: Q X n6=k θ(qns − ~δjs ) k k θ(w + qks + ~ − ~δjs ) Q Lkj (z)Lsl (w) + Wk [2k]δi θ(w) n6=s θ(qns + ~δnj − ~δjs ) s Q X θ(w + qis − ~δjs ) n6=i θ(qns + ~δnk − ~δjs ) k Q Lkj (z)Lsl (w) + Wk [k + i] θ(w) n6=s θ(qns + ~δnj − ~δjs ) s Q X θ(qki +~) θ(w+qks −~δjs ) n6=k θ(qns +~δni −~δjs ) i Q Lij (z)Lsl (w) Wk [i + k] θ(qki −~) θ(w) n6=s θ(qns +~δnj −~δjs ) s X j θ(z + qis + ~δik − ~δls ) θ(qlj + ~) = + 2δlj × Wj [j + l] θ(q − ~) θ(z) lj s Q n6=i θ(qns + ~δnk − ~δls ) Lkl (w)Lsj (z) + ×Q n6=s θ(qns + ~δnl − ~δls ) Q X j θ(z + qis + ~δik − ~δjs ) n6=i θ(qns + ~δnk − ~δjs ) Q Lkj (w)Lsl (z). Wl [j + l] θ(z) n6=s θ(qns + ~δnj − ~δjs ) s (4.18)


427

Here in the second and the third lines i 6= k. The ratio of products of theta-functions occurring in each term in (4.18) allows one to take off the sum over s, e.g., when i 6= k, we have Q θ(qns + ~δnk − ~δjs ) Qn6=i = n6=s θ(qns + ~δnj − ~δjs ) θ(qki + ~)θ(qji ) θ(qki ) |j6=k + δkj (1 − δij ) + | j6=i + δis δij θ(qki − ~) θ(qki )θ(qji + ~) j6=k θ(qjk )θ(~) θ(~) + |j6=i + δks (1 − δkj ) δij θ(qik + ~) θ(qjk + ~)θ(qik ) θ(qkj )θ(~) δjs |j6=i . θ(qkj − ~)θ(qji + ~) To compare (4.18) to (3.27) we rewrite relation (3.27) with the help of Eq. (3.31) in the following form: −1

F (z − w)P1 R21 (w)P1−1 L1 (z)R21 (w)L2 (w) = R12 −1

F P2 R12 (z)P2−1 L2 (w)R12 (z)L1 (z)R12 (z − w).

(4.19)

In the component form algebra (4.19) is presented in Appendix C. Comparing the components of (4.18) to the ones of (4.19) we establish that they coincide up to the overall multiplicative factor θ(z − w)θ(~)2 . Thus, we have shown that any representation of algebra (3.27) by the transformation (4.17) turns into a representation of (4.1). The connection established gives the right to assert that algebra (3.27) possesses a family of N -commuting integrals and that the formula of the determinant type (3.40) for the commuting family proved in [21] is also valid for the L-operator (3.35).

5. Conclusion In this paper, we described the dynamical R-matrix structure of the quantum elliptic RS model. The quantum L-operator algebra possesses a family of commuting operators. It turns out that this algebra has a surprisingly simple structure and can be analysed explicitly in the component form. Furthermore, one can hope that the problem of finding new representations of the algebra obtained is simpler than the corresponding problem for the algebra RLL = LLR. There are several interesting problems to be discussed. First, we recall that in the classical case we obtained two different Poisson algebras, which lead to the same classical L-operator algebra. Only one of them was quantized. It is desirable to quantize the second one and to show that the corresponding quantum L-operator algebra is isomorphic to the algebra obtained in the paper. The elliptic RS model we dealt with in the paper corresponds to the AN −1 root system. It seems to be possible to extend our approach to other root systems and to derive the corresponding L-operator algebras. To this end, one should find a proper parameterization of the corresponding cotangent bundle. Generalizing our approach to the cotangent bundle over a centrally extended group of smooth mappings from a higher-genus Riemann surface into a Lie group, one may expect to obtain new integrable systems.

428


It is known that the CM systems admit spin generalizations [34–36]. Recently, the spin generalization was found for the elliptic RS model [37]. However, the Hamiltonian formulation for the model is not found yet. One may hope that in our approach the spin models can arise as higher representations of the L-operator algebra. Probably, the most interesting and complicated problem is to separate variables for the quantum elliptic RS model. Up to now only the three-particle case for the trigonometric RS model was solved explicitly [38]. One could expect that the L-operator algebra obtained in the paper may shed light on the problem. Acknowledgement. The authors are grateful to M.A.Olshanetsky and N.A.Slavnov for the valuable discussions. This work is supported in part by the RFBI grants N96-01-00608, N96-02-19085 and N96-01-00551 and by the ISF grant a96-1516.

Appendix A In this Appendix, we prove that the limiting r-matrix (2.20) satisfies the classical Yang– Baxter equation. For this, let us write the equation, which follows from the Jacoby identity for the bracket (2.18): εη ερ ηρ ερ ηρ (z, w), r13 (z, s) + r23 (w, s)] + [r13 (z, s), r23 (w, s)] [r12 1 ε −1 1 ρ −1 ρ εη ηρ (w, s)] + T3 (s){T3 (s), r12 (z, w)] + T1 (z){T1ε (z), r23 κ κ 1 ερ (z, s)] = 0. − T2η −1 (w){T2η (w), r13 κ

(A.1)

Here εη εη (z, w) = H1εη H2ηε r12 (z, w) − αf εη (z, w) r12 εη εη (z, w) + β tr 3 H1εη H3ηε r13 (z, w) ⊗ I + βI ⊗ tr 3 H3εη H2ηε r32 εη (z, w) I ⊗ I. + β 2 tr 34 H3εη H4ηε r34

(A.2) ε

ηρ

To study (A.1), one needs to know the Poisson bracket of T (z) with H . This bracket can be easily derived from Eqs. (2.13), (2.14), and (2.17): X ερ ηε 1 1 ε −1 T1 (z){Tε1 (z), H2ηρ } = Hi Hj 8(ε − ρ, qij )(Eii − I) ⊗ Ejj (A.3) κ N −

X

i6=j

i6=j

−

X i6=j

1 I) ⊗ Ejj N X εη ηε ηρ + Hi Hj Hj 8(z − η, qij )Eij ⊗ Ejj

Hiεη Hjηε Hjηρ 8(ε − η, qij )(Eii − Hiερ Hjηε 8(z − ρ, qij )Eij ⊗ Ejj

X 1 ηρ H (8(ε − ρ, 0) + N j j

i6=j

1 I) ⊗ Ejj . (A.4) N Equation (A.1) holds at arbitrary values of all the parameters. Without loss of generality one can put ρ = 0 and η = aε, where a and ε are real. Let us now perform the change of variables Hiεη : +8(η − ε, 0) + 8(z − η, 0) − 8(z − ρ, 0)) (Ejj −


429

Hiεη = 1 + hεη i and consider the expansion of the l.h.s. of (A.1) in powers of ε and h. Let us note that the matrix rεη (z, w) has the following expansion in powers of h at ε − η → 0: (1) (z, w, ε)h + r(2) (z, w, ε)h2 + o(ε − η), rεη (z, w) = r(z, w) + rreg

(A.5)

(1) (z, w, ε) is regular when ε → 0 and the where r(z, w) is given by (2.20). The matrix rreg ε-dependence of r(2) (z, w, ε) is inessential. Substituting (A.5) into the bracket T−1 {T, r} one gets: (1) (ε)T−1 {T, h} + r(2) (ε)hT−1 {T, h}. (A.6) T−1 {T, r} = rreg

Clearly, Eq. (A.1) should be satisfied in any order in h and ε. Since we are interested in finding an equation for r, we consider only the terms independent of h and ε in the expansion of Eq. (A.1). The terms of zero order in h and ε occurring in (A.1) come from rεη and from the first term in (A.6). However, from the explicit expression for the bracket {T, H} one can see that {T, h}|h=0 = o(ε). Now, taking into account that rreg (ε) is regular at ε → 0, we conclude that the last three terms in (A.1) do not contribute. Thus, in the zero order in h and ε, Eq. (A.1) reduces to the CYBE for r(z, w). The remarkable thing is that the main elliptic identities (see Appendix B) follow from the Jacoby identity for the bracket (2.18) or, equivalently, from the Yang-Baxter equation for r(z, w). (1)

Appendix B Here we present the basic elliptic function identities, formulated as a set of functional relations on 8(z, w) [8]: 8(z, x)8(w, y) = 8(z, x − y)8(z + w, y) + 8(z + w, x)8(w, y − x), (B.1) 8(z, x)8(z, y) = 8(z, x + y) (8(z, 0) + 8(x, 0) + 8(y, 0) − 8(z + x + y, 0)) , (B.2) 8(z − w, a − b)8(z, x + b)8(w, y + a) − 8(z − w, x − y)8(z, y + a)8(w, x + b) = 8(z, x + a)8(w, y + b) (8(a − b, 0) + 8(x + b, 0) − 8(x − y, 0) − 8(a + y, 0)) . (B.3) Equation (B.2) is the limiting case of Eq. (B.1) where w → z, and Eq. (B.3) is a z−z¯ consequence of (B.1) and (B.2). Note that the exponent term e−2ζ(1/2)zs+2πis τ −τ in z¯ 8(z, s) as well as the linear term −2ζ(1/2)z + 2πi τz− −τ in 8(z, 0) are irrelevant since they drop out from (B.1–B.2). To establish the unitarity relation for R, one also needs the identity involving the Weierstrass P-function 8(z, s)8(z, −s) = P(z) − P(s), and to prove Eq. (2.50), the following relation between the derivatives of 8 is of use: ∂8(z, qij ) ∂8(z, qij ) = − (8(z, 0) − 8(qij ))8(z, qij ). ∂z ∂qij

430


Appendix C In this Appendix we present the quantum L-operator algebra −1

F (z − w)P1 R21 (w)P1−1 L1 (z)R21 (w)L2 (w) R12 −1

F = P2 R12 (z)P2−1 L2 (w)R12 (z)L1 (z)R12 (z − w)

(C.1)

in the component form. The l.h.s. of (C.1) has the form X 8(~, qki )8(~, qik )8(~, qkj − ~)Lij (z)Lkl (w)Eij ⊗ Ekl i6=k;j,l

X

−

8(~, qki )8(~, qij − ~)8(w, qkj − ~)Lij (z)Ljl (w)Eij ⊗ Ekl

i6=k;j,l

X

+

8(~, qki )8(w, qki )8(~, qij − ~)Lij (z)Lil (w)Eij ⊗ Ekl

i6=k;j,l

X

+

8(z − w, qik )8(~, qki )8(~, qij − ~)Lkj (z)Lil (w)Eij ⊗ Ekl

i6=k;j,l

X

−

8(z − w, qik )8(~, qkj − ~)8(w, qij − ~)Lkj (z)Ljl (w)Eij ⊗ Ekl

i6=k;j,l

X

+

8(z − w, qik )8(w, qik )8(~, qkj − ~)Lkj (z)Lkl (w)Eij ⊗ Ekl

i6=k;j,l

+

X

8(z − w, ~)8(w, ~)8(~, qkj − ~)Lkj (z)Lkl (w)Ekj ⊗ Ekl

j,k,l

−

X

8(z − w, ~)8(w, ~)8(w + ~, qkj − ~)Lkj (z)Ljl (w)Ekj ⊗ Ekl .

j,k,l

The r.h.s. of (C.1) reads X 8(~, qki )8(~, qil − ~)8(~, qlj )Lkl (w)Lij (z)Eij ⊗ Ekl i6=k;j6=l

+

X

8(~, qki )8(~, qij − ~)8(z − w, qlj + ~δlj )Lkj (w)Lil (z)Eij ⊗ Ekl

i6=k;j,l

−

X

8(~, qkl − ~)8(z, qil − ~)8(~, qlj )Lkl (w)Llj (z)Eij ⊗ Ekl

i6=k;j6=l

−

X

8(~, qkj − ~)8(z, qij − ~)8(z − w, qlj + ~δlj )Lkj (w)Ljl (z)Eij ⊗ Ekl

i6=k;j,l

+

X

8(z, qik )8(~, qkl − ~)8(~, qlj )Lkl (w)Lkj (z)Eij ⊗ Ekl

i6=k;j6=l

+

X

8(z, qik )8(~, qkj − ~)8(z − w, qlj + ~δlj )Lkj (w)Lkl (z)Eij ⊗ Ekl

i6=k;j,l

+

X

j6=l;i

8(z, ~)8(~, qil − ~)8(~, qlj )Lil (w)Lij (z)Eij ⊗ Eil


+

X

431

8(z, ~)8(~, qij − ~)8(z − w, qlj + ~δlj )Lij (w)Lil (z)Eij ⊗ Eil

i,j,l

−

X

8(z, ~)8(z + ~, qil − ~)8(~, qlj )Lil (w)Llj (z)Eij ⊗ Eil

j6=l;i

−

X

8(z, ~)8(z + ~, qij − ~)8(z − w, qlj + ~δjl )Lij (w)Ljl (z)Eij ⊗ Eil .

i,j,l

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. 27. 28. 29. 30. 31. 32. 33.

Babelon, O.and Viallet, C.M.: Phys. Lett. B237, 411 (1990) Avan, J. and Talon, M.: Phys.Lett. B303, 33–37 (1993) Avan, J., Babelon, O. and Talon, M.: Alg.Anal. 6(2), 67 (1994) Sklyanin, E.K.: Alg. Anal. 6(2), 227 (1994) Braden, H.W. and Suzuki, T.: Lett. Math. Phys. 30, 147 (1994) Avan, J. and Rollet, G.: The classical r-matrix for the relativistic Ruijsenaars-Schneider system. Preprint BROWN-HET-1014 (1995) Suris, Yu.B.: Why are the rational and hyperbolic Ruijsenaars-Schneider hierarchies governed by the same R-matrix as the Calogero-Moser ones? hep-th/9602160 Nijhoff, F.W., Kuznetsov, V.B., Sklyanin, E.K. and Ragnisco, O.: J. Phys. A: Math. Gen. 29, 333–340 (1996) Suris, Yu.B.: Elliptic Ruijsenaars-Schneider and Calogero-Moser hierarchies are governed by the same r-matrix. solv-int/9603011 Ruijsenaars, S.N.: Commun. Math. Phys. 110, 191 (1987) Olshanetsky, M.A., Perelomov, A.M.: Phys. Reps. 71, 313 (1981) Gorsky, A. and Nekrasov, N.: Nucl. Phys. B414, 213 (1994); Nucl. Phys. B436, 582 (1995); Gorsky, A.: Integrable many body systems in the field theories. Prep. UUITP-16/94, (1994) Arutyunov, G.E. and Medvedev, P.B.: Phys. Lett. A 223, 66–74 (1996) Avan, J., Babelon, O. and Billey, E.: The Gervais-Neveu-Felder equation and the quantum CalogeroMoser systems. Commun. Math. Phys. 178, 281–300 (1996) Felder, G.: Conformal field theory and integrable systems associated to elliptic curves. hep-th/9407154 Gervais, J.L. and Neveu, A.: Nucl. Phys. B238, 125 (1984) Babelon, O., Bernard, D. and Billey, E.: Phys. Lett B 375, 89–97 (1996) Arutyunov, G.E., Frolov, S.A.: Quantum dynamical R-matrices and quantum Frobenius group. To appear in Commun. Math. Phys. Arutyunov, G.E., Frolov, S.A. and Medvedev, P.B.: J. Phys. A: Math. Gen. 30, 5051–5063 (1997) Arutyunov, G.E., Frolov, S.A. and Medvedev, P.B.: J. Math. Phys. 38, 5682–5689 (1997) Hasegawa, K.: Ruijsenaars’ commuting difference operators as commuting transfer matrices, qalg/9512029 Etingof, P.I., Frenkel, I.B.: Commu. Math. Phys 165, 429–444 (1994) Falceto, F. and Gawedski, K.: Commun. Math. Phys. 159, 549 (1994) Faddeev, L.D.: Integrable models in (1+1)-dimensional quantum field theory. In: Recent advances in field theory and statistical mechanics. Eds. Zuber, J.B., Stora, R., Les Houches Summer School Proc. session XXXiX, 1982, Elsevier Sci.Publ., 1984 pp. 561 Kulish, P.P., Sklyanin, E.K.: Quantum spectral transform method. Recent developments. In: Integrable quantum field theories. Eds. Hietarinta, J., Montonen, C., Lect. Not. Phys. 51, 1982, pp. 61 Belavin, A.A.: Nucl. Phys. B180 [FS2], 189–200 (1981) Richey, M.P. and Tracy, C.A.: J. Stat. Phys. 42, 311–348 (1986) Quano, Y. and Fujii, A.: Modern Phys. Lett. A vol 8, No 17, 1585–1597 (1993) Hasegawa, K.: J. Phys. A: Math. Gen. 26, 3211–3228 (1993) Hasegawa, K.: J. Math. Phys. 35 (11), 6158–6171 (1994) Sklyanin, E.K.: Funct. Anal. and Appl. (Engl. transl.), 17, 273–284 (1983) Baxter, R.J.: I. Ann. Phys. 76, 1–24 (1973) II. ibid. 25–47, III. ibid. 48–71 Jimbo, M., Miwa T. and Okado, M.: Lett. Math. Phys. 14, 123–131 (1987); Nucl. Phys. B300, 74–108 (1988)

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34. Krichever, I.M., Babelon, O., Billey, E., Talon, M.: Spin generalization of Calogero-Moser system and the matrix KP equation. hep-th/9411160 35. Nekrasov, N.: Commun. Math. Phys. 180, 587–604 (1996) 36. Enriquez, B. and Rubtsov, V.: Hitchin systems, higher Gauden operators and r-matrices. Math. Research Lett. V3, 343–358 (1996) 37. Krichever, I.M., Zabrodin, A.V.: Spin generalizations of the Ruijsenaars-Schneider model, non-abelian 2D Toda chain and representations of Sklyanin algebra. hep-th/9505039 38. Kuznetsov, V.B. and Sklyanin, E.K.: J. Phys. A: Math. Gen. 29, 2779–2804 (1996) Communicated by G. Felder

Commun. Math. Phys. 192, 433 – 461 (1998)

Communications in


Zero Viscosity Limit for Analytic Solutions, of the Navier-Stokes Equation on a Half-Space. I. Existence for Euler and Prandtl Equations Marco Sammartino1,? , Russel E. Caflisch2,?? 1 Dipartimento di Matematica, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy. E-mail: [email protected] 2 Mathematics Department, UCLA, Los Angeles, CA 90096-1555, USA. E-Mail: [email protected]

Received: 5 September 1995 / Accepted: 14 July 1997

Abstract: This is the first of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space. In this paper we prove short time existence theorems for the Euler and Prandtl equations with analytic initial data in either two or three spatial dimensions. The main technical tool in this analysis is the abstract Cauchy-Kowalewski theorem. For the Euler equations, the projection method is used in the primitive variables, to which the Cauchy-Kowalewski theorem is directly applicable. For the Prandtl equations, Cauchy-Kowalewski is applicable once the diffusion operator in the vertical direction is inverted. 1. Introduction The zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space is a challenging problem due to the formation of a boundary layer whose thickness is proportional to the square root of the viscosity. Boundary layer separation, which is difficult to control, may cause singularities in the boundary layer equations. In this and the companion paper Part II, we overcome these difficulties by imposing analyticity on the initial data. Under this condition, we prove that, in the zero-viscosity limit and for a short time, the Navier-Stokes solution in a half-space goes to an Euler solution outside a boundary layer in either two or three spatial dimensions, and that it is close to a solution of the Prandtl equations within the boundary layer. The construction of the Navier-Stokes solution is performed as a composite asymptotic expansion involving an Euler solution, a Prandtl boundary layer solution and a correction term. It follows the earlier, unpublished analysis of Asano [1], who also restricted the data to be analytic, but our work contains a considerably simplified exposition, explicit use of the Prandtl equations, and several other technical differences: ? ??

Research supported in part by the NSF under grant #DMS-9306720. Research supported in part by DARPA under URI grant number # N00014092-J-1890.

434

M. Sammartino, R. E. Caflisch

Asano used a sup-estimate on the divergence-free projection operator, which we have been unable to verify. He also used high order derivative norms in the y (normal) variable, whereas we find it necessary to only use second derivatives. An earlier attempt to analyze this problem, without the requirement of analyticity and without explicit use of the Prandtl equations, was made by Kato [8]. It was not completely successful, since it required some unverified assumptions on the NavierStokes solution. Analysis of the zero-viscosity limit for the Navier-Stokes solution in an unbounded domain was performed in [3, 7, 13]. In this first part, we present short-time existence results for the Euler and Prandtl equations in a half-space with analytic initial data. The main significance of the Euler result is that it is stated in terms of the function spaces used in the Navier-Stokes result of Part II. For the Euler equations, of course, this is not an optimal result since analyticity is not needed for existence of a solution. Moreover, a more general existence result for analytic solutions of the incompressible Euler equations was proved earlier by Bardos and Benachour [2]. The present proof is somewhat different, since it uses the projection method on the primitive variables, rather than the vorticity formulation. For the Prandtl equations, on the other hand, our result on existence for short time and analytic initial data is the first general existence theorem for the unsteady problem. To the best of our knowledge, the only previous existence theorem for the unsteady Prandtl equations was by Oleinik [10]. For the Prandtl equations with upstream velocity prescribed at the left (x = 0) as well as at infinity and at t = 0, she proved existence for either a short time for all x > 0 or for a short distance and for all time, without the analyticity assumption. The proof required conditions that the prescribed horizontal velocities are all positive and strictly increasing, which are not required in our result. For a review of related mathematical results on both the steady and unsteady Prandtl equations, see [9]. In fact, we conjecture that the general initial value problem for the Prandtl equations is ill-posed in Sobolev space. Although ill-posedness has not been proved, there is some evidence in its favor: First, previous attempts to construct such solutions have failed. Second, there are numerical solutions of Prandtl that develop singularities associated with boundary layer separation in finite time [4–6]. Most recently, E and Engquist [14] have proved existence of Prandtl solutions with singularities. This is not enough to show ill-posedness, however, because in these computations and analysis the singularity time is not small. The main technical tool of our analysis is the abstract Cauchy-Kowalewski Theorem (ACK), the optimal form of which is due to Safonov [11]. This theorem, which is for systems that are first order in some sense, is directly applicable to the Euler equations. Since the Prandtl equations are diffusive rather than first order, the classical CauchyKowalewski Theorem cannot be applied, and it may at first seem surprising that the ACK Theorem is applicable to them. As pointed out by Asano [1], however, the ACK Theorem may be used for a nonlinear diffusion equation after inversion of the diffusion operator. We show below that this strategy works for the Prandtl equations, and in Part II, we shall also apply it to the Navier-Stokes equations. The main simplification of this analytic method over Sobolev methods is that it uses Cauchy estimates to bound derivatives rather than energy estimates. In Sect. 2 the Euler and Prandtl equations are stated and a number of function spaces and norms are defined. The abstract Cauchy-Kowalewski Theorem is formulated in Sect. 3. The existence theorem for the Euler equations is stated and proved in Sect. 4, which includes a convenient formulation and some useful bounds for the projection method. The existence theorem for the Prandtl equations is stated and proved in Sect. 5,

Zero Viscosity Limit for Analytic Solutions, of N-S Equation. I.

435

using properties of the heat operator, which are proved in an appendix. The analysis for Prandtl is completely independent of that for Euler. Some concluding remarks are made in Sect. 6. For convenience, the formulation and analysis will often be written in 2D, but the extension to 3D is straightforward. Key points in the 3D extension will be noted and the main results will be stated for 2D and 3D.

2. Statement of the Problem and Notation 2.1. Euler equations. The Euler equations for a velocity field u E = uE , v E are ∂t u E + u E · ∇u E + ∇pE = 0, ∇ · u E = 0, γn u E ≡ v E (x, y = 0, t) = 0, u E (x, y, t = 0) = u E 0 (x, y) .

(2.1) (2.2) (2.3) (2.4)

Here u E depends on the variables (x, y, t), where x is the transversal variable going from −∞ to ∞, y is the normal variable going from 0 to ∞, and t is the time. The operator γn acting on vectorial functions gives the normal component calculated at the boundary. In the rest of this paper we shall also use the trace operator γ, defined by γu = (u (x, y = 0, t) , v (x, y = 0, t)) .

(2.5)

In Sect. 3 we shall prove that under suitable hypotheses for the initial condition u E 0 (essentially analyticity in x and y ) Euler equations admit a unique solution. Although stated here for 2D, the analysis works equally well in 3D. The existence result is only for a short time. 2.2. Prandtl equations. The Euler equations are a particular case of the Navier-Stokes (N-S) equations, when the fluid has zero viscosity. Therefore, the Navier-Stokes solution for small viscosity ν is expected to be well approximated by an Euler solution, at least away from boundaries, which is confirmed by numerical and experimental observations. An analysis of the short time, spatially global behavior (in presence of a boundary) of N-S equations will be the subject of part II of this work, [12]. In the vicinity of the boundaries, on the other hand, the effect of viscosity is O(1) even as the viscosity goes to zero. The no-slip condition causes the creation of vorticity; moreover in a small layer there is an adjustment of the flow to the outer (inviscid) flow. Due to the resulting rapid variation of the√fluid velocity, the velocity depends on a scaled is of size ε. The normal variable Y = y/ε in which ε = ν. Also the vertical velocity resulting equations governing the velocity field u P = uP , εv P are Prandtl equations: (∂t − ∂Y Y ) uP + uP ∂x uP + v P ∂Y uP + ∂x pP = 0, ∂Y pP = 0, P ∂x u + ∂Y v P = 0, uP (x, Y = 0, t) = 0, P u (x, Y → ∞, t) −→ uE (x, y = 0, t) , uP (x, Y, t = 0) = uP 0 (x, Y ) .

(2.6) (2.7) (2.8) (2.9) (2.10) (2.11)

436


Equation (2.10) is the matching condition between the flow inside the boundary layer and the outer Euler flow. In Sect. 5 we shall prove that the Prandtl solution approaches the boundary value of the Euler solution at an exponential rate as Y goes to infinity. Equation (2.7) implies that the pressure is constant across the boundary layer; to match with the Euler pressure pE , it must satisfy ∂x pP = ∂x pE (x, y = 0, t) = −γ(∂t + uE ∂x )uE .

(2.12)

The normal component of the velocity v P can be found, using the incompressibility condition, to be Z Y ∂x uP (x, Y 0 , t)dY 0 . (2.13) vP = − 0 P

0

P

In 3D, ∂x u is replaced by ∇ · u in this integral, where ∇0 is the gradient with respect to the transversal variables. Therefore Eq. (2.6) can be considered as an equation for the transversal component uP , with v P given by Eq. (2.13), with boundary conditions (2.9)–(2.10), and with initial conditions (2.11). Moreover there must be compatibility between the boundary conditions and initial conditions; i.e. γuP 0 = 0,

(2.14)

E uP 0 (Y → ∞) − γu0 −→ 0.

(2.15)

In this paper we shall prove the existence and the uniqueness of the solutions for Eqs. (2.1)–(2.4) and (2.6)–(2.11). We now introduce the appropriate function spaces. 2.3. Function spaces. Let us introduce the “strip", the angular sector and the “conoid" in the complex plane D(ρ) = R × (−ρ, ρ) = {x ∈ C : =x ∈ (−ρ, ρ)} , Σ(θ) = {y ∈ C : 0.

Theorem 4.1 follows directly from Theorem 4.2, using the following proposition, which is a consequence of Lemma 4.2: 0 ,θ0 0 ,θ0 . Then Pt u ? ∈ Hβl,ρ and Proposition 4.1. Let u ? ∈ Hβl,ρ 0 ,T 0 ,T

|Pt u ? |l,ρ0 ,θ0 ,β0 ,T ≤ c|u ? |l,ρ0 ,θ0 ,β0 ,T .

(4.39)

We have also the following bound on Pt : 0 ,θ0 Proposition 4.2. Let u ? ∈ Hβl,ρ and let ρ0 < ρ0 − β0 T and θ0 < θ0 − β0 T . Then 0 ,T ? l,ρ0 ,θ 0 Pt u ∈ H for each 0 < t < T and Z t ds|u ? (·, ·, s)|l,ρ0 ,θ0 ≤ c|u ? |l,ρ0 ,θ0 ,β0 ,T . (4.40) |Pt u ? |l,ρ0 ,θ0 ≤ c

0

In the rest of this section we shall be concerned with proving Theorem 4.2. To do this we shall verify that the operator H satisfies all the hypotheses of the ACK Theorem in l,ρ,θ (as a function the function spaces Xρ,θ = H l,ρ,θ (at each fixed t) and Yρ,θ,β,T = Hβ,T of t), and with ρ replaced by the vector (ρ, θ). 4.5. The forcing term. It is obvious that H satisfies the first condition of the ACK Theorem in the norms H l,ρ,θ . In this subsection we shall prove that there exists a constant R0 such that |H (t, 0)|l,ρ0 −βt,θ0 −βt ≤ R0 (4.41) in H l,ρ,θ for 0 ≤ t ≤ T , which verifies the second assumption of the theorem. The constant R0 will of course depend on |u E 0 |l,ρ,θ and on the difference between ρ and ρ0 , θ and θ0 . From Eq. (4.38), we see that E H (t, 0) = u E 0 · ∇u 0 ,

(4.42)

E E 2 |u E 0 · ∇u 0 |l,ρ0 −βt,θ0 −βt ≤ c|u 0 |l,ρ,θ ,

(4.43)

and Lemmas 4.6 and 4.7 imply

which gives the desired bound (4.41). We now pass to the Cauchy estimate.

448


4.6. The Cauchy estimate. In this subsection we shall be concerned with proving that the operator H satisfies the last hypothesis of the ACK Theorem. We have to show that, l,ρ0 ,θ0 l ≥ 4, if ρ0 < ρ(s) ≤ ρ0 − βs, θ0 < θ(s) ≤ θ0 − βs, and if u ?1 and u ?2 are in Hβ,T with (4.44) |u ?1 |l,ρ0 ,θ0 ,β,T ≤ R, |u ?2 |l,ρ0 ,θ0 ,β,T ≤ R, then in H l,ρ,θ |H (t, u ?1 ) − H (t, u ?2 )|l,ρ0 ,θ0 Z t ?1 |u − u ?2 |l,ρ(s),θ0 |u ?1 − u ?2 |l,ρ0 ,θ(s) ≤C ds + . ρ(s) − ρ0 θ(s) − θ0 0

(4.45)

First estimate the nonlinear term of H . Using Lemma 4.8 to estimate the convective part of the operator H and then Proposition 4.2 leads to   Pt u ?1 · ∇Pt u ?1 − Pt u ?2 · ∇Pt u ?2  0 0 l,ρ ,θ Z t ?1 ?2 ?1 0 |u − u ?2 |l,ρ0 ,θ(s) |u − u |l,ρ(s),θ ≤c ds + ρ(s) − ρ0 θ(s) − θ0 0 Z t |u ?1 − u ?2 |l,ρ0 ,θ0 ,β0 ,T ds + 0 Z t ?1 |u |l,ρ(s),θ0 + |u ?2 |l,ρ(s),θ0 |u ?1 |l,ρ0 ,θ(s) + |u ?2 |l,ρ0 ,θ(s) + ds × ρ(s) − ρ0 θ(s) − θ0 0 Z t ?1 |u − u ?2 |l,ρ(s),θ0 |u ?1 − u ?2 |l,ρ0 ,θ(s) ds + (4.46) ≤C ρ(s) − ρ0 θ(s) − θ0 0 in H l,ρ,θ , using Lemma 3.1 and the bound (4.44) in the last step. The estimate of the linear part is similar. 4.7. Conclusion of the proof of Theorem 4.1. Since all of the hypotheses of the ACK Theorem have been verified, the proof of Theorem 4.2 has been achieved. There exist 0 < ρ0 < ρ, 0 < θ0 < θ, and a β0 > 0 such that Eq. (4.37) admits a unique solution in 0 ,θ0 . This also concludes the proof of Theorem 4.1 for the Euler equations. Hβl,ρ 0 ,T 5. Existence and Uniqueness for Prandtl’s Equations We want to prove that the Prandtl equations (2.6)-(2.11) admit a unique solution in an appropriate function space. The main result of this section is the following theorem: Theorem 5.1. Suppose that uP 0 satisfies the compatibility conditions (2.14) and (2.15), l+1,ρ,θ P E l+1,ρ0 ,θ0 ,µ0 that u E ∈ H , and that u l ≥ 3 (l ≥ 4 in 3D). Then there 0 0 − γu0 ∈ K P exists a unique solution u of the Prandtl equations (2.6)-(2.11). This solution can be written as: (5.1) uP (x, Y, t) = u˜ P (x, Y, t) + γuE , 1 ,θ1 ,µ1 , with 0 < ρ1 < ρ0 , 0 < θ1 < θ0 , 0 < µ1 < µ0 , β1 > β0 > 0. where u˜ P ∈ Kβl,ρ 1 ,T

1 ,θ1 ,µ1 This solution satisfies the following bound in Kβl,ρ : 1 ,T

E E |u˜ P |l,ρ1 ,θ1 ,µ1 ,β1 ,T < c |uP 0 − γu0 |l+1,ρ0 ,θ0 ,µ0 + |u 0 |l+1,ρ,θ .

(5.2)

Zero Viscosity Limit for Analytic Solutions, of N-S Equation. I.

449

In particular this shows that if the initial condition for Prandtl equations exponentially approaches the initial value of the Euler flow calculated at the boundary, the same property will be true for the Prandtl solution at least for short time. The proof of this theorem will occupy the remainder of this section. As in the proof of existence and uniqueness for the Euler equation, we shall recast the Prandtl equations in a form suitable for the use of the ACK Theorem (see Eq. (5.38) below). In Prandtl’s equations a second order operator (the heat operator) is present. The key idea is to invert this operator, taking into account boundary and initial conditions. Therefore we shall first introduce the heat operators, and prove some bounds on them. In Subsect. 5.2 we find an operator form for Prandtl equations, Eq. (5.38). The resulting operator F consists of two terms: The first is a forcing term that accounts for BC and IC. The second is the composition of a convective operator and the inverse of the heat operator with zero BC and IC. With the bounds on the heat and convective operators, it is then straightforward to get the desired bounds, which is performed in Subsects. 5.3 and 5.4. In the rest of this section we shall always suppose l ≥ 4 (l ≥ 5 in 3D), as needed for Proposition 2.3. 5.1. Estimates on heat operators. To solve Prandtl equations we introduce the heat kernel: 1 exp (−Y 2 /4t), (5.3) E0 (Y, t) = (4πt)1/2 and the heat operators acting on functions f (Y ) with 0, all independent of ε, such that the solution of the Navier-Stokes equations can be written in the form u N S = u E (x, y, t) + u P (x, Y, t) + ε [ω(x, y, t) + Ω(x, Y, t) + e (x, Y, t)]

(2.27)

in which (i)

u E ∈ H l,ρ,θ is the solution of the Euler equations (2.1)–(2.4), β,T

(ii) u = (u˜ P , εv P ) ∈ K l,ρ,θ,µ is the modified Prandtl solution as defined in (2.18) β,T and (2.24), exponentially decaying outside the boundary layer, P

(iii) ω ∈ N l,ρ,θ is the first order correction to the inviscid flow; it solves Eqs. (4.7)– β,T (4.10) below, (iv) Ω ∈ K l,ρ,θ,µ is the first order correction to the boundary layer flow; it solves β,T Eqs. (4.11)–(4.14) below,

468


(v) e ∈ Ll,ρ,θ is an overall correction; it solves Eqs. (4.15)–(4.18) below. β,T

The norms of ω, Ω and e in the above spaces are bounded by a constant that does not depend on the viscosity. 2.2. Discussion of the Theorem. Since u P is exponentially decaying for large Y = y/ε, then the expression (2.27) shows that u N S = u E + O(ε) for y outside of the boundary layer (i.e. y >> ε). For y inside the boundary layer (i.e. y ≤ ε), u E = (γuE , 0) + O(ε), so that u N S = u P + O(ε). This shows that the informal statement of the theorem follows from the rigorous statement. In this theorem the Navier-Stokes solution is represented in terms of a composite expansion of the form (2.27), which includes a regular (Euler) term u E , a boundary layer term u P and a correction term. Since the Euler solution has non-zero boundary values, the Prandtl solution must be modified so that the sum of the two is zero at the boundary and approaches the Euler solution at the outer edge of the boundary layer. The theorem says that if the initial condition is a function L2 in transversal and normal component (together with its derivatives up to order l), then the solution of the NavierStokes equations will have the composite expansion form given in Eq. (2.27), at least for a short time. There are several other ways to represent the Navier-Stokes solution for small viscosity. The most common method in perturbation theory [3] is to write the solution as a matched asymptotic expansion in which u N S = u P + O(ε) u N S = u E + O(ε)

for y small enough, for y not too small.

(2.28) (2.29)

The formal validity of this representation is usually demonstrated by showing that the O(ε) terms are small, and that there is a region of overlap for the validity of the two expansions. While this representation is more easily understood than the composite expansion, it is much more difficult to rigorously analyze due to the two spatial regimes. A second method for representing the solution, which has been used for example in [4, 8], is to introduce a cut off function m = m(y/εα ) with m(0) = 1, m(∞) = 0, and 0 < α < 1. The solution is then written as u N S = mu P + (1 − m)u E + O(εα ).

(2.30)

This method has two difficulties: It introduces an artificial length scale εα which makes the error terms artificially large. It also requires error terms in the incompressibility equation, since mu P + (1 − m)u E is not divergence-free. For these reasons we have found the composite expansion method to be the most convenient for analysis. The rest of this paper is devoted to proving Theorem 1. Unless otherwise stated, l ≥ 6 throughout. 2.3. The error equation. If we pose uN S = uE + u˜ RP + εw1 , ∞ v N S = v E + ε Y dY 0 ∂x u˜ P + εw2 = v E + εv P + εw2 , NS E w = p + εp , p

(2.31)

and use these expressions in the N-S equations, we get the following equation for the error w = (w1 , w2 ):

Zero Viscosity Limit for Analytic Solutions of N-S Equation. II.

∂t − ε2 1 w + w · ∇u 0 + u 0 · ∇w + εw · ∇w + ∇pw ∇·w γw w (t = 0)

469

= f + g · ∂y u˜ P , 0 , = 0, = (0, g) , = ω 0 + Ω0 + e 0 ,

(2.32) (2.33) (2.34) (2.35)

in which u 0 = (u0 , v 0 ) is defined by u0 = uE + u˜ P , R∞ v 0 = v E + εv P = v E + ε Y dY 0 ∂x u˜ P .

(2.36)

The forcing term is f = (f 1 , f 2 ) given by f 1 = −ε−1 u˜ P ∂x uE − ∂x γuE + (∂x u˜ P ) uE − γuE + (∂y u˜ P ) v E + y∂x γuE

− v P ∂y uE + ε1uE + ε∂x2 u˜ P , P

P

P

P

f = − ∂t v + u ∂x v + v ∂y v + v ∂y v 2

0

0

Z

and also g=

∞

E

(2.37) −ε

−1 P

E

0

u˜ ∂x v + ε1v ,

dY 0 ∂x u˜ P .

(2.38)

(2.39)

0 1 ,θ1 We want to show that the forcing term f is in Ll−2,ρ , and that in this space it has β1 ,T O(1) norm, namely that

2

˜P |f |l−2,ρ1 ,θ1 ,β1 ,T ≤ c |u E 0 |l,ρ,θ + |u 0 |l,ρ,θ,µ + 1

,

(2.40)

where the constant c does not depend on ε. Let us consider f 1 . From Theorems 4.1 and 5.1 of Part I [6], it is clear that the terms ε1uE and ε∂x2 u˜ P satisfy the estimate (2.40). Each of the remaining terms in f has a similar form: They are each ε−1 times the product of a function which is exponentially decaying (with respect to Y = y/) outside the boundary layer (terms containing u˜ P and v P ), and a function that is O(ε) inside the boundary layer (e.g. uE − γuE ). It follows that they all satisfy (2.40). In an analogous way one can see that f 2 is O(1) and satisfies the estimate (2.40). Thus Eqs. (2.32)–(2.35) for the error term w (x, Y, t) have bounded forcing terms. In Sects. 4–7 we shall prove that this system admits a solution w which can be represented in the following form: w =ω+Ω+e, (2.41) where the norms (in the appropriate function spaces) of ω, Ω and e remain bounded by a constant independent of ε. The difficulty of this proof is the presence in Eq. (2.32) of terms like ∂y u˜ P , which are O(ε−1 ) inside the boundary layer. 3. The Boundary Layer Analysis for Stokes Equations Before addressing the problem of solving Eqs. (2.32)–(2.35), it is useful to consider a somewhat simpler problem, the Stokes equations with zero initial condition and boundary data g . This problem is of intrinsic interest, and the results will be used in the analysis of the Navier-Stokes equations. The time-dependent Stokes equations are

470


(∂t − ν1) u S + ∇pS ∇·uS γu S S u (x, y, t = 0)

= 0, = 0, = g (x, t), = 0.

(3.1) (3.2) (3.3) (3.4)

Here g is a vectorial function g = (g 0 , gn ). Primed quantities denote the tangential components of a vector, while the subscript n denotes the normal component. The compatibility condition g (x, t = 0) = 0 is required for the boundary data. In this section we shall show that the solution of the above problem has a structure similar to that for the Navier-Stokes solution Eq. (2.27); i.e. it is the superposition of an inviscid (Euler) part, a boundary √ layer (Prandtl) part which exponentially decays to zero outside a region of size ε = ν, and a correction term which is size O(ε) everywhere. The Stokes problem Eqs. (3.1)–(3.4) has already been addressed by Ukai in [7], (where even the case of non-zero initial data was considered), without making the distinction between inviscid part, boundary layer part and correction term. We seek a solution of the form uS = uE + u˜ P + w1 , v S = v E + εv P + w2 , pS = pE + pw ,

(3.5)

so that (uE , v E ) represents an inviscid solution, (u˜ P , v P ) is a boundary layer solution decaying (in both components) outside the boundary layer, (w1 , w2 ) is a small correction, and the pressures pE and pw are bounded at infinity. Please note that in this section uE , u˜ P and w refer to the “Euler”, “Prandtl” and correction components of the Stokes solution; everywhere else in the paper, this notation is used for the usual Euler and Prandtl solutions and for the correction in the Navier-Stokes solution. These quantities solve the following equations: ∂t u E + ∇pE = 0, ∇ · u E = 0, γn u E = g n , u E (x, y, t = 0) = 0,

(3.6) (3.7) (3.8) (3.9)

(∂t − ν1) u˜ P = 0, ∂x u˜ P + ∂Y v P = 0, v P → 0 asY → ∞, γ u˜ P = g 0 − γuE , u˜ P (x, y, t = 0) = 0,

(3.10) (3.11) (3.12) (3.13)

(∂t − ν1) w + ∇pw = 0, ∇ · w = 0, γw = (0, −εγv P ), w (x, y, t = 0) = 0.

(3.14) (3.15) (3.16) (3.17)

Note that Eq. (3.10)–Eq. (3.13) use the fast variable Y = y/ with ν = 2 , in terms of which 1 = 2 ∂xx + ∂Y Y . Also, there is no term 1u E , since it is identically zero. We now solve explicitly these equations.


471

3.1. Convective equation. Take the divergence of (3.6) to obtain 1pE = 0. Then apply 1 to (3.6) and use the initial condition u E = 0, to obtain 1u E = 0.

(3.18)

Therefore the solution of Euler problem is u E = ∇N gn ,

(3.19)

where the operator N = −1/|ξ 0 | exp (−|ξ 0 |y) solves the Laplace equation with Neumann boundary condition; i.e. 1N gn = 0, (3.20) γ∂y N gn = gn . 3.2. Boundary Layer Problem. To solve Eqs. (3.10)–(3.13) it is useful to introduce the operator E˜ 1 acting on functions f (x, t) defined on the boundary Z t Y exp −Y 2 /4(t − s) ˜ E1 f (x, Y, t) =2 ds t − s (4π(t − s))1/2 0 (3.21) Z ∞ exp −(x − x0 )2 /4ε2 (t − s) 0 dx0 f (x , s). 1/2 −∞ 4πε2 (t − s) This operator solves the heat equation with boundary conditions ∂t − ε2 ∂xx − ∂Y Y E˜ 1 f = γ E˜ 1 f = E˜ 1 f (x, Y, t = 0) =

conditions f and zero initial 0, f, 0.

(3.22)

Note that the operator E˜ 1 differs from the operator E1 (defined in Sect. 5.1 of Part I) by the fact that it involves an integration on the transversal component x also. Define M g = g 0 + N 0 gn .

(3.23)

The solution of the boundary layer equations is written as u˜ P = E˜ 1 M g .

(3.24)

Using the incompressibility condition and the limiting condition, the normal component is Z vP =

∞

Y

dY 0 ∂x u˜ P .

(3.25)

3.3. The Correction Term. Here we shall use the Fourier transform variable with respect to x. As in Part I, the corresponding transform variable is denoted ξ 0 . As in Subsect. 3.1, 1pw = 0. Since pw is bounded at ∞, then (3.26) ∂y + |ξ 0 | pw = 0. Define τ = ∂y + |ξ 0 | w2 , so that Eqs. (3.14)–(3.16) imply

472


∂t − ε2 1 τ = 0,

(3.27)

γτ = γ ∂y + |ξ 0 | w2 ,

= γ −∇0 w1 + |ξ 0 |w2 ,

= |ξ 0 |α, Z

in which

∞

α = −ε

(3.28)

dY 0 ∂x u˜ P .

(3.29)

0

Since τ solves the heat equation with the above boundary condition, then τ = |ξ 0 |E˜ 1 α.

(3.30)

∂y w2 + |ξ 0 |w2 = |ξ 0 |E˜ 1 α,

(3.31)

0 w2 (x, Y, t) = e−|ξ |y α + U E˜ 1 α

(3.32)

From the definition of τ , w2 satisfies

which leads to in which U is defined as

U f (ξ 0 , Y ) = ε|ξ 0 |

Z

Y

0

0

e−ε|ξ |(Y −Y ) f (ξ 0 , Y 0 )dY 0 .

(3.33)

0

Notice that a similar operator occurs in Eq.(4.12) in [6]. Finally, the incompressibility condition implies that 0

w1 = −N 0 e−|ξ |y α + N 0 (1 − U )E˜ 1 α.

(3.34)

These above results can be summarized as follows: The solution of the Stokes problem Eqs. (3.1)–(3.4) is denoted by Sg , with u S = Sg = S E g + S P g + S C g 0 E˜ 1 M g −N 0 Dgn −N 0 e−|ξ |y + N 0 (1 − U )E˜ 1 R = + α. + 0 ∞ Dgn ε Y dY 0 ∂x E˜ 1 M g e−|ξ |y + U E˜ 1 (3.35) After some manipulation, this can be simplified, as in [7], to 0 −N 0 e−|ξ |y gn + N 0 (1 − U )E˜ 1 V1 g , u S = Sg = 0 e−|ξ |y gn + U E˜ 1 V1 g in which

V1 g = g n − N 0 g 0 .

(3.36)

(3.37)

3.4. Estimates. In this subsection we prove some basic simple estimates on the operators S E , S P , and S C . Propositions 3.1, 3.2 and 3.3 are presented as results on the timedependent Stokes equations, but are not used in the sequel. For analysis of the NavierStokes equations, only Proposition 3.4 and Lemma 3.2 will be used. We cannot in general give an estimate for the operator S E in a space involving the 2 L norm in y. Nevertheless it is possible to give such an estimate for a special class of boundary data.


473

Proposition 3.1. Suppose that g satisfies Z ∞ gn = |ξ 0 | dy 0 f (ξ 0 , y 0 , t)k(ξ 0 , y 0 )

(3.38)

0

R∞ l,ρ,θ l,ρ,θ . Then S E g ∈ Hβ,T , and the following with |ξ 0 | 0 dy 0 |k(ξ 0 , y 0 )| ≤ 1 and f ∈ Hβ,T estimate holds: (3.39) |S E g |l,ρ,θ,β,T ≤ c|f |l,ρ,θ,β,T . R ∞ Using Jensen’s inequality and replacing a factor of |ξ 0 | 0 dy 0 |k(ξ 0 , y 0 )| by 1, Z supθ0 ≤θ

Z

dy 0(θ 0 ) Z ≤ sup θ 0 ≤θ

Z =

2 Z ∞ 0 0 dξ 0 e2ρ|ξ | dy 0 |ξ 0 |e−|ξ |y k(ξ 0 , y 0 )f (ξ 0 , y 0 ) 0 Z ∞ Z 2 0 0 2ρ|ξ 0 | dy dξ e dy 0 |ξ 0 |e−2|ξ |y k(ξ 0 , y 0 ) f (ξ 0 , y 0 )

0(θ 0 )

0

Z

0

dξ 0 e2ρ|ξ |

∞

2 dy 0 k(ξ 0 , y 0 ) f (ξ 0 , y 0 )

0

≤ |f |20,ρ,θ,β,T .

(3.40)

Analogous bounds can be proved for the differentiated terms in the norm. Now consider the “Prandtl” part. We first state an estimate for the operator E˜ 1 . 0l,ρ with f (t = 0) = 0. Then E˜ 1 f ∈ Ll,ρ,θ Lemma 3.1. Let f ∈ Kβ,T β,T for some θ, and the

following estimate holds in Ll,ρ,θ β,T :

|E˜ 1 f |l,ρ,θ,β,T ≤ c|f |l,ρ,β,T .

(3.41)

A much stronger estimate actually holds. One can in fact prove the exponential decay of E˜ 1 f in the normal variable away from the boundary; see the proof given in the Appendix. Using Lemma 3.1, the following estimates on S P 0 and SnP (respectively the transversal and the normal components of the operator S P ) are obvious: 0l,ρ with g (t = 0) = 0. Then S P 0 g ∈ Ll,ρ,θ Proposition 3.2. Suppose g ∈ Kβ,T β,T and

SnP g ∈ Ll−1,ρ,θ for some θ, and β,T

|S P 0 g |l,ρ,θ,β,T ≤ c|g |l,ρ,β,T ,

(3.42)

|SnP g |l−1,ρ,θ,β,T ≤ c|g |l,ρ,β,T .

(3.43)

P

Again a stronger estimate could be proved, namely that S g is exponentially decaying when Y −→ ∞ (i.e. outside the boundary layer). The loss of one derivative in the normal component is due to the incompressibility condition (see e.g., Eq. (3.25)). The estimate on S C will be a consequence of the following bound on the operator U: l,ρ,θ Lemma 3.2. Let f ∈ Ll,ρ,θ β,T . Then U f ∈ Lβ,T and

|U f |l,ρ,θ,β,T ≤ c|f |l,ρ,θ,β,T .

(3.44)

474


The proof of Lemma 3.2 is like the proof of Proposition 3.1, and is based on the fact that U f can be written as a derivative with respect to the normal variable. Lemma 3.2 leads to the following proposition for S C : 0l,ρ . Then S C g ∈ Ll−1,ρ,θ for some θ, and Proposition 3.3. Suppose g ∈ Kβ,T β,T

|S C g |l−1,ρ,θ,β,T ≤ ε1/2 c|g |l,ρ,β,T .

(3.45)

This estimate on the size of the error is not optimal. In fact, a more careful analysis of S C would reveal that the error term is made up of two parts: a Eulerian part (namely 0 l−1,ρ,θ , and a e−|ξ |y α) depending on the unscaled variable y, which is of size ε in Hβ,T l−1,ρ,θ ˜ part (namely U E1 α) depending on the scaled variable Y , which is of size ε in Lβ,T . Something similar occurs in the analysis of the error for the Navier-Stokes equations (see Sect. 4); to prove that the error w is size ε we shall break it up in several parts (see Eq. (4.6) below) and estimate them in the appropriate function spaces. We now give an estimate on the Stokes operator S. Combine Lemma 3.1, 3.2 with the representation (3.36) to obtain the following bound on S: R∞ 0 0 Proposition 3.4. Suppose that g ∈ K 0 l,ρ β,T , with g (t = 0) = 0 and gn = |ξ | 0 dy R ∞ l,ρ,θ f (ξ 0 , y 0 , t)k(ξ 0 , y 0 ) with |ξ 0 | 0 dy 0 |k(ξ 0 , y 0 )| ≤ 1 and f ∈ Ll,ρ,θ β,T . Then Sg ∈ Lβ,T , and (3.46) |Sg |l,ρ,θ,β,T ≤ c |g 0 |l,ρ,β,T + |f |l,ρ,θ,β,T . 0

0

In addition, for each t ≤ T , Sg ∈ K l,ρ ,θ , and satisfies sup |Sg |l,ρ0 ,θ0 ≤ c |g 0 |l,ρ,β,T + |f |l,ρ,θ,β,T

(3.47)

0≤t≤T

in which 0 < ρ0 < ρ − βT and 0 < θ0 < θ − βt. The proof of this proposition uses Jensen’s inequality as in the proof of Proposition 3.1. Proposition 3.4 and Lemma 3.2 are the only results from this section that will be used in the rest of this paper. 4. The Error Equation The equation for the error is ∂t − ε2 1 w + w · ∇u 0 + u 0 · ∇w +εw · ∇w + ∇pw ∇·w γw w (t = 0)

= f + g · ∂y u˜ P , 0 , = 0, = (0, g) , = ω 0 + Ω0 + e 0 ,

(4.1) (4.2) (4.3) (4.4)

1 ,θ1 in which the forcing term f is in Ll−2,ρ , and is O(1) (see Eq. (2.40)). Notice that β1 ,T in Eqs. (4.1) and (4.2), and in the rest of this paper, the divergence and the gradient are taken with respect to the unscaled variable y; i.e. (4.5) ∇ = ∂x , ∂ y .

The rest of this paper is concerned with proving that equations (4.1)–(4.4) admit a unique solution, and that this solution is O(1). We shall prove the following Theorem:


475

l,ρ,θ l,ρ,θ,µ Theorem 2. Suppose that u E ∈ Hβ,T , that u˜ P ∈ Kβ,T , so that f has norm in

Ll−2,ρ,θ bounded by a constant independent of ε. Then there exist ρ2 < ρ, θ2 < θ and β,T β2 > β and µ2 > 0 such that Eqs. (4.1)–(4.4) admit a solution which can be written in the form: w =ω+Ω+e, (4.6) where 2 ,θ2 satisfies Eqs. (4.7)–(4.10); • ω ∈ Nβl−2,ρ 2 ,T

2 ,θ2 ,µ2 • Ω ∈ Kβl−2,ρ satisfies Eqs. (4.11)–(4.14), and 2 ,T

2 ,θ2 • e ∈ Ll−2,ρ satisfies Eqs. (4.15)–(4.18). β2 ,T

The quantity ω represents the first order correction to the Euler flow. It satisfies the following equations: ∂t ω + ω · ∇u E + u E · ∇ω + ∇pω = 0, ∇ · ω = 0, γn ω = g, ω(t = 0) = ω 0 .

(4.7) (4.8) (4.9) (4.10)

In addition the initial data ω 0 is required to satisfy the condition (iii) of Theorem 2.1. The quantity Ω = 1 , 2 represents the first order correction inside the boundary layer, with the convective terms omitted. It satisfies the following equations: (∂t − ∂Y Y ) 1 = 0, Z 2 =ε

(4.11) ∞ Y

dY 0 ∂x 1 ,

γ1 = −γω 1 , 1 (t = 0) = 10 . The third part of the error e satisfies the following equations: ∂t − ε2 1 e + e · ∇ u 0 + ε (ω + Ω) + u 0 + ε (ω + Ω) · ∇e + εe · ∇e + ∇pe = Ξ, ∇ · e = 0, γe = 0, −γ2 , e (t = 0) = e 0 .

(4.12) (4.13) (4.14)

(4.15) (4.16) (4.17) (4.18)

The forcing term Ξ is given by: Ξ = − u 0 · ∇Ω+ω · ∇ u P +εΩ +Ω · ∇u 0 + u P +εΩ+εω · ∇ω+εΩ · ∇Ω (4.19) +ε2 1ω + ∂xx 1 , 0 − 0, (∂t − ε2 1)2 + f + g · ∂y u˜ P , 0 . The initial data 10 and e 0 are required to satisfy conditions (iv) and (v) of Theorem 2.1. The reason for the complicated representation Eq. (4.6) for the error w is the following: To solve Eqs. (4.1)–(4.4) one has to use the projection operator due to the incompressibility condition. The natural ambient space is therefore the space of functions which are L2 in both transversal and normal components. In the right-hand side of

476


Eq. (4.1), there are terms which are rapidly varying inside the boundary layer, and thus depend on the rescaled variable Y . So, in taking the L2 norm with respect to the normal variable we are forced to use the variable Y instead of y. The boundary condition (4.3), on the other hand, gives rise to terms which depend on the variable y. Their L2 norm evaluated using the rescaled variable Y would be O(ε−1/2 ). To avoid such a catastrophic error, we use the decomposition (4.6): ω, which is L2 in y, takes care of the boundary condition (4.3) (see Eq. (4.9)); e , which is L2 in Y , takes care of the rapidly varying forcing term; Ω cancels the transversal component of ω at the boundary (see Eq. (4.13)). 5. The Correction to the Euler Flow In this section we shall prove the following theorem: 0l−1,ρ . Then there exist ρ2 < ρ, θ2 < θ and β2 > β such Theorem 3. Suppose that g ∈ Kβ,T

2 ,θ2 that Eqs. (4.7)–(4.10) admit a unique solution ω ∈ Nβl−2,ρ . The following estimate 2 ,T 2 ,θ2 in Nβl−2,ρ holds: 2 ,T

2 ,θ2 ≤ c |u E ˜P |ω|l−2,ρ 0 |l,ρ,θ + |u 0 |l,ρ,θ,µ + |ω 0 |l,ρ,θ , β2 ,T

(5.1)

l,ρ,θ ˜P , K l,ρ,θ,µ and N l,ρ,θ respectively. where the norms of u E 0 ,u 0 and ω 0 are taken in H

The structure of Eqs. (4.7)–(4.10) is somewhat similar to the structure of Euler equations and the proof of the above theorem closely follows the proof of Theorem 4.1 in [6]. The functional setting here is slightly different; in fact Theorem 3 above is stated in the space l,ρ,θ , where only the first derivative with respect to time is taken, instead of the space Nβ,T l,ρ,θ Hβ,T , where time derivatives up to order l are allowed. This is due to the presence of the boundary condition g deriving from Prandtl equations. We shall prove the above theorem using the ACK Theorem. The solution of Eqs. (4.7)–(4.10) can be written as 0 (5.2) ω = ω 0 + −N 0 , 1 e−|ξ |y (g − g0 ) + Pt ω ∗ ,

where the operator Pt is the integrated (with respect to time) half space projection operator defined in Eq.(4.35) of [6]. The first term in this expression provides the correct initial data, the second term the correct boundary data, and the third term the correct forcing terms. l,ρ,θ : The projection operator Pt satisfies the following bounds in Nβ,T l,ρ,θ l,ρ,θ Proposition 5.1. Let u ∗ ∈ Nβ,T . Then Pt u ∗ ∈ Nβ,T and

|Pt u ∗ |l,ρ,θ,β,T ≤ c|u|l,ρ,θ,β,T .

(5.3)

l,ρ,θ Proposition 5.2. Let u ∗ ∈ Nβ,T . Let ρ0 < ρ − βT and θ0 < θ − βT . Then Pt u ∗ ∈ 0 0 N l,ρ ,θ for all 0 ≤ t ≤ T , and Z t ∗ 0 0 ds|u(·, ·, s)|l,ρ0 ,θ0 ≤ c|u|l,ρ,θ,β,T . (5.4) |Pt u |l,ρ ,θ ≤ c 0


477

Using Eq. (5.5) one sees that (4.7)–(4.10) are equivalent to the following equation for ω ∗ : (5.5) ω ∗ + H 0 (ω ∗ , t) = 0, where

i h 0 H 0 (ω ∗ , t) = ω 0 + (−N 0 , 1)e−|ξ |y (g − g0 ) + Pt ω ∗ · ∇u E i h 0 +u E · ∇ ω 0 + (−N 0 , 1)e−|ξ |y (g − g0 ) + Pt ω ∗ .

(5.6)

Using the Cauchy estimate, and with the same procedure we used to prove existence and uniqueness for Euler equations in [6], one can see that the operator H 0 satisfies all the hypotheses of the ACK Theorem; therefore there exist ρ2 < ρ, θ2 < θ and β2 > β such 2 ,θ2 . Equation (5.2) and Proposition that Eq. (5.5) admits a unique solution ω ∗ ∈ Nβl−2,ρ 2 ,T 2 ,θ2 5.1 also imply ω ∈ Nβl−2,ρ . Theorem 3 is thus proved. 2 ,T

6. The Boundary Layer Correction We prove the following theorem: Theorem 4. Let ω be the solution of Eqs. (4.7)–(4.10) found in Theorem 5.1. Then there exist ρ02 > ρ2 , θ20 > θ2 , β20 > β2 , and µ2 > 0 such that Eqs. (4.11)–(4.14) admit a unique l−2,ρ0 ,θ 0 ,µ l−2,ρ0 ,θ 0 ,µ solution Ω ∈ Kβ 0 ,T 2 2 2 . It satisfies the following estimate in Kβ 0 ,T 2 2 2 : 2

2

˜P |Ω|l−2,ρ02 ,θ20 ,µ2 ,β20 ,T ≤ c |u E 0 |l,ρ,θ + |u 0 |l,ρ,θ,µ + |ω 0 |l,ρ,θ + |Ω0 |l,ρ,θ,µ2 ,

(6.1)

l,ρ,θ where the norms of u E ˜P , K l,ρ,θ,µ , N l,ρ,θ and K l,ρ,θ,µ 0 ,u 0 , ω 0 and Ω0 are taken in H respectively.

The proof of this theorem uses the following lemma: 0l−2,ρ02

Lemma 6.1. There exists ρ02 < ρ2 such that the boundary data γω 1 is in Kβ2 ,T following estimate holds in

. The

0l−2,ρ0 Kβ2 ,T 2 :

|γω 1 |l−2,ρ02 ,β2 ,T ≤ c|ω|l−2,ρ2 ,θ2 ,β2 ,T .

(6.2)

The above lemma can be proved using a Sobolev estimate to bound the sup with respect to y of ω 1 , and then a Cauchy estimate on the x derivative to bound the term ∂y ∂xl−2 ω 1 . The solution of Eqs. (4.11)–(4.14) can be explicitly written as (6.3) 1 = E0 (t)10 − E1 γω 1 = E0 (t) 10 + γω01 − E1 γ ω 1 − ω01 − γω01 , where the operator E0 (t) and E1 have been defined in [6]. Proposition 5.1 and Propol−2,ρ0 ,θ 0 ,µ sition 5.3 of [6], imply that 1 ∈ Kβ 0 ,T 2 2 2 . Using the expression (4.12) for 2 2 and again shrinking the domain of analyticity in x, and renaming ρ02 , we obtain also l−2,ρ0 ,θ 0 ,µ 2 ∈ Kβ 0 ,T 2 2 2 . The proof of Theorem 4 is thus complete. 2 By a redefinition of ρ2 , θ2 , β2 , we may take ρ02 = ρ2 , θ20 = θ2 , β20 = β2 .

478


7. The Navier-Stokes Operator In this section we shall prove the following theorem Theorem 5. Under the hypotheses of Theorem 2, there exist ρ2 , θ2 , β2 , such that 2 ,θ2 . This solution satisfies the Eqs. (4.15)–(4.18) admit a unique solution e ∈ Ll−2,ρ β2 ,T

2 ,θ2 following estimate in Ll−2,ρ : β2 ,T

˜P |e |l−2,ρ2 ,θ2 ,β2 ,T ≤ c |u E 0 |l,ρ,θ + |u 0 |l,ρ,θ,µ + |ω 0 |l,ρ,θ + |Ω0 |l,ρ,θ,µ + |e 0 |l,ρ,θ , (7.1) P l,ρ,θ l,ρ,θ,µ l,ρ,θ l,ρ,θ,µ , u ˜ , ω , Ω and e are taken in H , K , N , K where the norms of u E 0 0 0 0 0 and Ll,ρ,θ respectively. We shall prove this theorem using the ACK Theorem. In the same way as for the Euler and Prandtl equations, we first invert the second order heat operator, taking into account the incompressibility condition and the BC and IC. This is performed using the heat op erator,defined in Subsect. 7.1, which inverts ∂t − ∂Y Y − ε2 ∂xx . Then in Subsect. 7.2 we insert the divergence-free projection and obtain the operator N0 . Using the Stokes operator from Sect. 3 to handle the boundary data, in Subsect. 7.3 we define the operator N ∗ , which is suitable for the iterative solution of the Navier-Stokes equations (i.e. treating initial data and nonlinearities as forcing terms). Bounds on this operator are given in Propositions 7.6 and 7.7. With the use of this Navier-Stokes operator, and taking into account initial and boundary data Eq. (4.17) and Eq. (4.18), in Subsect. 7.4 we finally solve the error equation. In Subsects. 7.5 and 7.6 we prove by the ACK Theorem that this iterative procedure converges to a unique solution. 7.1. The heat operator. We have already introduced the operator E˜ 1 in (3.21) which solves the heat equation with boundary data. We now want to solve the heat equation with a source and with zero initial and boundary data on the half plane Y ≥ 0; i.e. ∂t − ε2 ∂xx − ∂Y Y u = w(x, Y, t), (7.2) u(x, Y, t = 0) = 0, γu = 0. First introduce the heat kernel E˜ 0 (x, Y, t), defined by e−x /4tε e−Y /4t √ . E˜ 0 (x, Y, t) = √ 4πt 4πtε2 2

2

2

(7.3)

We solve the problem (7.2) on the half plane with the following operator: u(x, Y, t) = E˜ 2 w Z t Z ds =

∞

Z

∞

dx0 E˜ 0 (x − x0 , Y − Y 0 , t − s) 0 0 −∞ −E˜ 0 (x − x0 , Y + Y 0 , t − s) w(x0 , Y 0 , s). dY

0

(7.4)

We now state some estimates on this operator. In these estimates w is defined for Y ≥ 0. l,ρ,θ ˜ Proposition 7.1. Let w ∈ Ll,ρ,θ β,T . Then E2 w ∈ Lβ,T and

|E˜ 2 w|l,ρ,θ,β,T ≤ c|w|l,ρ,θ,β,T .

(7.5)


479

0 0 Proposition 7.2. Suppose w ∈ Ll,ρ,θ β,T with γw = 0 and that ρ ≤ ρ − βt, θ ≤ θ − βt. 0 0 l,ρ ,θ Then E˜ 2 w(t) ∈ L and Z t ds|w(·, ·, s)|l,ρ0 ,θ0 ≤ c|w|l,ρ,θ,β,T . (7.6) |E˜ 2 w|l,ρ0 ,θ0 ≤ c 0

The proofs of these two propositions are given in the Appendix. 7.2. The projected heat operator. In [6] we introduced the divergence-free projection operator P ∞ . Here we employ a similar operator with the normal variable rescaled by a ∞ factor ε. The projection operator in the x and Y variable, P , is the pseudodifferential operator whose symbol is 1 ∞ ξn2 −εξ 0 ξn P = 2 02 , (7.7) ε ξ + ξn2 −εξ 0 ξn ε2 ξ 02 where ξ 0 and ξn denote the Fourier variables corresponding to x and Y respectively. For all w this operator satisfies ∞

∇ · P w = ∂x P

∞0

w + ε∂Y P

∞

nw

= 0.

(7.8)

In [6] to avoid Fourier transform in y we expressed P ∞ as an integration in the normal ∞ variable. For P one can similarly see that " Z Y 0 0 1 ∞ ε|ξ 0 | P nw = dY 0 e−ε|ξ |(Y −Y ) (−N 0 w1 + w2 ) 2 −∞ (7.9) Z ∞ 0 0 ε|ξ 0 |(Y −Y 0 ) 0 1 2 + ε|ξ | dY e (N w + w ) , Y

" P

∞0

1 −ε|ξ 0 | w =w + 2

Z

1

Y

0

0

dY 0 e−ε|ξ |(Y −Y ) (w1 + N 0 w2 )

−∞ 0

− ε|ξ |

Z

∞

0 ε|ξ 0 |(Y −Y 0 )

dY e Y

0

(7.10)

(w − N w ) . 1

2

Next we present estimates on the projection operator. In these estimates w is defined ∞ on Y ≥ 0, but we write P w to mean the following: First extend w oddly to Y < 0, i.e. w(x, Y ) = −w(x, −Y ) when Y ≤ 0 ; (7.11) ∞

then apply P , and finally restrict the result to Y ≥ 0 for application of the norm. The ∞ resulting expressions for P are "Z Y 0 0 0 0 1 0 ∞ P n w = ε|ξ | dY 0 e−ε|ξ |(Y −Y ) − e−ε|ξ |(Y +Y ) (−N 0 w1 + w2 ) 2 0 Z ∞ 0 0 0 0 + dY 0 eε|ξ |(Y −Y ) (N 0 w1 + w2 ) − eε|ξ |(−Y −Y ) (−N 0 w1 + w2 ) , (7.12) Y

480


P

∞0

"Z Y 0 0 0 0 1 0 w = w − ε|ξ | dY 0 e−ε|ξ |(Y −Y ) − e−ε|ξ |(Y +Y ) (w1 + N 0 w2 ) 2 0 Z ∞ 0 0 0 0 + dY 0 eε|ξ |(Y −Y ) (w1 − N 0 w2 )−eε|ξ |(−Y −Y ) (w1 + N 0 w2 ) . 1

Y

(7.13) The following estimate is easily proved ∞

Proposition 7.3. Let w ∈ Ll,ρ,θ with γw = 0. Then P w ∈ Ll,ρ,θ and ∞

|P w |l,ρ,θ ≤ c|w |l,ρ,θ .

(7.14)

We are now ready to introduce the projected heat operator N0 , acting on vectorial functions, defined as ∞ (7.15) N0 = P E˜ 2 . ∞ 2 One can easily show that P commutes with the heat operator ∂t − ∂Y Y − ε ∂xx . It then follows that for each w such that γw = 0, ∇ · N0 w = 0, ∞ ∂t − ∂Y Y − ε2 ∂xx N0 w = P w . The following estimates are a consequence of the properties of P Proposition 7.4. Suppose w ∈

Ll,ρ,θ β,T .

Then N0 w ∈

Ll,ρ,θ β,T

|N0 w |l,ρ,θ,β,T ≤ c|w |l,ρ,θ,β,T .

(7.16) (7.17) ∞

and E˜ 2 separately:

and (7.18)

0 0 Proposition 7.5. Suppose w ∈ Ll,ρ,θ β,T with γw = 0 and that ρ ≤ ρ − βt, θ ≤ θ − βt. 0 0 Then w and N0 w are in Ll,ρ ,θ for each t, and Z t ds|w (·, ·, s)|l,ρ0 ,θ0 ≤ c|w|l,ρ,θ,β,T . (7.19) |N0 w |l,ρ0 ,θ0 ≤ c 0

Note that E˜ 2 has zero boundary data; thus the conditions in Proposition 7.3 are all satisfied. 7.3. The Navier-Stokes operator. With the Stokes operator defined in Sect. 3 and the projected heat operator of the previous subsection, we now introduce the Navier-Stokes operator N ∗ defined as (7.20) N ∗ = N0 − SγN0 . This operator is used to solve the time-dependent Stokes equations with forcing, which is equivalent to the Navier-Stokes equations if the nonlinear terms are put into the forcing. In fact (7.21) w = N ∗w ? solves the system (7.22) ∂t − ∂Y Y − ε2 ∂xx w + ∇pw = w ? , ∇ · w = 0, (7.23) γw = 0, (7.24) w (t = 0) = 0, (7.25) and satisfies the following bound:


481

l,ρ,θ ∗ Proposition 7.6. Suppose w ∈ Ll,ρ,θ β,T . Then N w ∈ Lβ,T and

|N ∗ w |l,ρ,θ,β,T ≤ c|w |l,ρ,θ,β,T .

(7.26)

We already know, from Proposition 7.4, that N0 obeys an estimate like (7.26). Therefore the only part of N ∗ which has to be estimated is that involving the Stokes operator S. To bound this term it is enough to notice that γN0 w is a boundary data for which the assumptions of Proposition 3.4 hold. In fact since E˜ 2 w has been extended oddly for ∞ Y < 0, then γn N0 w = γP n E˜ 2 w is (see Eq. (7.12) ) Z ∞ 0 0 ∞ γP n E˜ 2 w = ε|ξ 0 | dY 0 e−ε|ξ |Y N 0 E˜ 2 w1 . (7.27) 0

According to Proposition 7.1, this is of the form required in Proposition 3.4 for the normal part gn of g = N0 w . The tangential part g 0 satisfies the bound

Therefore

|g 0 |l,ρ,θ,β,T ≤ c|E˜ 2 w |l,ρ,θ,β,T .

(7.28)

|SγN0 w |l,ρ,θ,β,T ≤ c|E˜ 2 w |l,ρ,θ,β,T ≤ c|w |l,ρ,θ,β,T ,

(7.29)

which concludes the proof of Proposition 7.6. We shall also use the following Proposition, which is proved in the same way as the previous result, using Proposition 7.5: 0 0 Proposition 7.7. Suppose w ∈ Ll,ρ,θ β,T with γw = 0 and that ρ ≤ ρ − βt, θ ≤ θ − βt. l,ρ0 ,θ 0 Then in L , Z t |N ∗ w |l,ρ0 ,θ0 ≤ c ds|w (·, ·, s)|l,ρ0 ,θ0 ≤ c|w|l,ρ,θ,β,T . (7.30) 0

7.4. The solution of the error equation. We can now solve Eqs. (4.15)–(4.18). If one looks at these equations one sees that they are of the form (7.22)–(7.25) (where all forcing and nonlinear terms are in w ? , see Eq. (7.37) below) plus boundary and initial data. We therefore express e as the sum of two terms: the first involving the Navier-Stokes operator and the second where all boundary and initial data are. In fact we write e = N ∗e ∗ + σ ,

(7.31)

where σ solves the following time-dependent Stokes problem with initial and boundary data: (∂t − ∂Y Y ) σ + ∇φ, = 0 ∇ · σ = 0, γσ = (0, εG), σ (t = 0) = e 0 , having denoted:

Z

∞

G=−

dY 0 ∂x 1 ,

0

and where e ∗ satisfies the following equation:

(7.32) (7.33) (7.34) (7.35)

(7.36)

482


e ∗ = Ξ − e · ∇ u 0 + ε (ω + Ω) + u 0 + ε (ω + Ω) · ∇e + εe · ∇e − ε2 ∂xx σ .

(7.37)

Equations (7.32)–(7.35) can be solved explicitly. First note that φ is harmonic, so that, imposing it to be bounded at infinity, (7.38) ∂y + |ξ 0 | φ = 0. Apply (∂y + |ξ 0 |) to the normal component of Eq. (7.32), and define τ = ∂y + |ξ 0 | σ 2

(7.39)

which satisfies (∂t − ∂Y Y ) τ = 0, γτ = ε|ξ 0 |G, τ (t = 0) = |ξ 0 |V1 e 0 ,

(7.40) (7.41) (7.42)

in which V1 e 0 = e20 − N 0 e10 . Denote G0 = G(t = 0). Then the solution of the system (7.40)–(7.42) is τ = E0 (t) |ξ 0 |V1 e 0 − ε|ξ 0 |G0 + E1 γε|ξ 0 |G − ε|ξ 0 |G0 + ε|ξ 0 |G0 = |ξ 0 |τ˜ .

(7.43) 00

00

The initial condition e 0 is in Ll,ρ,θ ; this obviously implies e 0 ∈ Ll−2,ρ2 ,θ2 . One has the following proposition: 2 Proposition 7.8. Given that e 0 ∈ Ll−2,ρ2 ,θ2 , that G ∈ Kβ0l−2,ρ , and the compatibility 2 ,T 2 ,θ2 condition γn e 0 = εG0 , then τ˜ ∈ Ll−2,ρ and β2 ,T

|τ˜ |l−2,ρ2 ,θ2 ,β2 ,T ≤ c |e 0 |l−2,ρ2 ,θ2 + |G|l−2,ρ2 ,β2 ,T .

(7.44)

The proof of this proposition is based on the estimates on the operators E0 (t) and E1 given in Propositions 5.2 and 5.3 of [6]; regarding the estimate in Proposition 5.3, we l,ρ,θ,µ it is a fortiori in Ll,ρ,θ notice in fact that if a function is in Kβ,T β,T . Now, the expression (7.43) for τ in (7.39) and the boundary condition (7.34) on σ 2 imply that 0 (7.45) σ 2 = εe−ε|ξ |Y G + U τ˜ , where U has been defined in (3.33). The incompressibility condition then leads to 0

σ 1 = −εN 0 e−ε|ξ |Y G + N 0 (1 − U )τ˜ .

(7.46)

A bound for σ is given by ˜ with G˜ ∈ K l−2,ρ2 , then σ ∈ Ll−2,ρ2 ,θ2 and Proposition 7.9. Suppose that G = |ξ 0 |G, β2 ,T β2 ,T ˜ l−2,ρ2 ,β2 ,T |σ |l−2,ρ2 ,θ2 ,β2 ,T ≤ c |e 0 |l−2,ρ2 ,θ2 + |G| ˜P ≤ c |u E 0 |l,ρ,θ + |u 0 |l,ρ,θ,µ + |ω 0 |l,ρ,θ + |Ω0 |l,ρ,θ,µ + |e 0 |l,ρ,θ . (7.47)


483

The proof of this proposition is based on Lemma 3.2 and Proposition 7.8 for the estimate 0 2 , then ε|ξ 0 |e−ε|ξ |Y G˜ ∈ of the terms involving τ˜ , and on the fact that if G˜ ∈ Kβ0l−2,ρ 2 ,T

2 ,θ2 Ll−2,ρ . β2 ,T We are now ready to prove existence and uniqueness for Eqs. (4.15)–(4.18). Use Eq. (7.31) in (7.37), interpret this equation as an equation for e ∗ , and use the abstract version of the Cauchy–Kowalewski Theorem, in the function spaces Xρ = Ll,ρ,θ and Yρ,β,T = Ll,ρ,θ β,T , to prove existence and uniqueness for the solution. This is similar to the procedure used in [6] to prove existence and uniqueness for the Euler and Prandtl equations. Rewrite Eq. (7.37) as

where F (e ∗ , t) is F (e ∗ , t) = k −

e ∗ = F (e ∗ , t),

(7.48)

u 0 + ε (ω + Ω + σ ) · ∇N ∗ e ∗ + N ∗ e ∗ · ∇ u 0 + ε (ω + Ω + σ ) +εN ∗ e ∗ · ∇N ∗ e ∗ }

(7.49)

and k is the forcing term k = Ξ − u 0 +ε (ω+Ω) · ∇σ +σ · ∇ u 0 +ε (ω+Ω) +εσ · ∇σ 0 =f − u +ε(ω+Ω+σ ) · ∇Ω+ ω · ∇u P − (g∂y u˜ P , 0) +(Ω+σ ) · ∇u P (Ω+σ ) · ∇u E + u P +ε(ω+Ω+σ ) · ∇ω+ u 0 +ε(ω+Ω+σ ) · ∇σ (7.50) +ε2 1ω+∂xx 1 , 0 +∂xx σ − 0, (∂t − ε2 1)2 . The rest of this section is concerned with proving that the operator F satisfies all the hypotheses of ACK Theorem. 7.5. The forcing term. In this subsection we shall prove the following proposition, 2 ,θ2 and O(1): asserting that the forcing term is bounded in Ll−2,ρ β2 ,T Proposition 7.10. There exists a constant R0 such that

Equation (7.49) shows that

|F (0, t)|l−2,ρ2 −β2 t,θ2 −β2 t ≤ R0 .

(7.51)

F (0, t) = k

(7.52) 2 ,θ2 Ll−2,ρ β2 ,T

(see the discussion after with k given by (7.50). We already know that f ∈ Eq. (2.40)). The terms in the first square brackets are exponentially decaying outside the boundary layer. Inside the boundary layer they can be shown to be O(1) with a Cauchy estimate on the terms where ∂y is present: this is possible because they go linearly fast to zero at the boundary. All terms inside the second square brackets are more easily handled because no O(ε−1 ) appear. Proposition 7.10 is thus proved. 7.6. The Cauchy estimate. In this subsection we shall prove that the operator F satisfies the last hypothesis of the ACK Theorem. Here and in the rest of this section ρ0 < ρ(s) ≤ ρ2 − β2 s, θ0 < θ(s) ≤ θ2 − β2 s.

484


Proposition 7.11. Suppose ρ0 < ρ(s) ≤ ρ2 − β2 s and θ0 < θ(s) ≤ θ2 − β2 s. If e ∗1 2 ,θ2 and e ∗2 are in Ll−2,ρ with β2 ,T |e ∗1 |l−2,ρ2 ,θ2 ,β2 ,T ≤ R, |e ∗2 |l−2,ρ2 ,θ2 ,β2 ,T ≤ R,

(7.53)

then |F (e ∗1 , t) − F (e ∗2 , t)|l−2,ρ0 ,θ0 Z t ∗1 |e − e ∗2 |l−2,ρ(s),θ0 |e ∗1 − e ∗2 |l−2,ρ0 ,θ(s) ≤C ds + ρ(s) − ρ0 θ(s) − θ0 0 Z t X ∗i |e |l−2,ρ(s),θ0 |e ∗i |l−2,ρ0 ,θ(s) ∗1 ∗2 ds + (7.54) +C|e − e |l−2,ρ0 ,θ0 ,β2 ,t ρ(s) − ρ0 θ(s) − θ0 0 i=1,2

in which all the norms are in Ll,ρ,θ and Ll,ρ,θ β,T . The proof of the above proposition occupies the remainder of this section. First introduce the Cauchy estimates in Ll,ρ,θ . Lemma 7.1. Let f (x, Y ) ∈ Ll,ρ,θ . Then for 0 < ρ0 < ρ and 0 < θ0 < θ, |f |l,ρ,θ , ρ − ρ0 |f |l,ρ,θ ≤c . θ − θ0

|∂x f |l,ρ0 ,θ ≤ c |χ(Y )∂Y f |l,ρ,θ0

(7.55) (7.56)

In the above proposition χ(Y ) is a monotone, bounded function, going to zero linearly fast near the origin (see e.g. Eq.(4.28) ) of [6]. The Sobolev inequality implies the following lemmas: Lemma 7.2. Let f (x, Y ) and g(x, Y ) be in Ll,ρ,θ . Then for 0 < ρ0 < ρ, |g∂x f |l,ρ0 ,θ ≤ c|g|l,ρ0 ,θ

|f |l,ρ,θ . ρ − ρ0

(7.57)

Lemma 7.3. Let f (x, Y ) and g(x, Y ) be in Ll,ρ,θ with g(x, Y = 0) = 0. Then for 0 < θ0 < θ, |f |l,ρ,θ . (7.58) |g∂Y f |l,ρ,θ0 ≤ c|g|l,ρ,θ0 θ − θ0 Lemmas 7.2 and 7.3 then imply with γn e 1 = γn e 2 = 0. Then for Lemma 7.4. Suppose e 1 and e 2 are in Ll−2,ρ,θ β,T 0 0 0 < ρ < ρ and 0 < θ < θ, |e 1 · ∇e 1 −e 2 · ∇e 2 |l−2,ρ0 ,θ0 ≤ c

|e 1 − e 2 |l−2,ρ,θ0 |e 1 − e 2 |l−2,ρ0 ,θ , (7.59) + ρ − ρ0 θ − θ0

where the constant c depends only on |e 1 |l−2,ρ,θ,β,T and |e 2 |l−2,ρ,θ,β,T .


485

We are now ready to prove Proposition 7.11. We first take into consideration the nonlinear part N ∗ e ∗ · ∇N ∗ e ∗ . From the estimates (7.26) and (7.30) on the Navier-Stokes operator, the estimate (7.59) on the convective operator and the fact that γn N ∗ e ∗ = 0, it follows that |N ∗ e ∗1 · ∇N ∗ e ∗1 − N ∗ e ∗2 · ∇N ∗ e ∗2 |l−2,ρ0 ,θ0 Z t ∗1 |e (·, ·, s) − e ∗2 (·, ·, s)|l−2,ρ(s),θ2 ≤C ds ρ(s) − ρ0 0 ∗1 ∗2 |e (·, ·, s) − e (·, ·, s)|l−2,ρ2 ,θ(s) + θ(s) − θ0 Z t X ∗i |e (·, ·, s)|l−2,ρ(s),θ2 ∗1 ∗2 ds +C|e − e |l−2,ρ2 ,θ2 ,β2 ,T ρ(s) − ρ0 0 i=1,2 |e ∗i (·, ·, s)|l−2,ρ2 ,θ(s) + θ(s) − θ0 Z t ∗1 |e (·, ·, s) − e ∗2 (·, ·, s)|l−2,ρ(s),θ2 ds ≤C ρ(s) − ρ0 0 |e ∗1 (·, ·, s) − e ∗2 (·, ·, s)|l−2,ρ2 ,θ(s) + . (7.60) θ(s) − θ0 +Ω+σ) · Since γn u 0 + ε(ω + Ω + σ ) = 0, one can estimate the term u 0 + ε(ω ∇N ∗ w ? in a similar fashion. The term N ∗ w ? · ∇ u 0 + ε(ω + Ω + σ ) is easily estimated. The proof of Proposition 7.11 is thus achieved. 7.7. Conclusion of the Proof of Theorem 5. The operator F (e ∗ , t) satisfies all the hypotheses of the ACK Theorem. Therefore, there exists a β2 > 0 such that Eq. (7.48) has 2 ,θ2 2 ,θ2 . Because of Proposition 7.6, then N ∗ e ∗ ∈ Ll−2,ρ . a unique solution e ∗ ∈ Ll−2,ρ β2 ,T β2 ,T Given the expression (7.31) for the error e and Proposition [7.9] for σ , the proof of Theorem 5 is achieved. l,ρ,θ 7.8. Conclusion of the Proof of Theorem 1. We have thus proved that u E ∈ Hβ,T l−1,ρ,θ,µ 2 ,θ2 (Theorem 4.1 of [6]), that u P ∈ Kβ,T (Theorem 3 of [6]), that ω ∈ Nβl−2,ρ 2 ,T

2 ,θ2 ,µ2 2 ,θ2 (Theorem 5.1), that Ω ∈ Kβl−2,ρ (Theorem 4), and that e ∈ Ll−2,ρ (Theorem β2 ,T 2 ,T 5). By a redefinition of the parameters, we may take (ρ2 , θ2 , β2 , µ2 ) = (ρ, θ, β, µ), and the proof of Theorem 1 is achieved.

8. Conclusions In the analysis above, we have proved existence of solutions of the Navier-Stokes equations in two and three dimensions for a time that is short but independent of the viscosity. As the viscosity goes to zero, the Navier-Stokes solution has been shown to approach an Euler solution away from the boundary and a Prandtl solution in a thin boundary layer. The initial data were assumed to be analytic: although this restriction is severe, we believe that it might be optimal. In fact separation of the boundary layer is related to development of a singularity in the solution of the time-dependent Prandtl equations, as

486


discussed in [2]. We conjecture that the time of separation (and thus the singularity time) cannot be controlled by a Sobolev bound on the initial data, unless some positivity and monotonicity is assumed as in [5]. It would be very important to verify this by an explicit singularity construction, or to refute it by an existence theorem in Sobolev spaces for Prandtl. This result suggest further work on several related problems: Analysis of the zeroviscosity limit for Navier-Stokes equations in the exterior of a ball is presented in [1]. An alternative derivation of this result may be possible by a more direct analysis of the Navier-Stokes solution. In two-dimensions, a solution is known to exist for a time that is independent of the viscosity. Thus by writing the solution as a Stokes operator times the nonlinear terms and analysis of the Stokes operator, it should be possible to recognize the regular (Euler) and boundary layer (Prandtl) parts directly. We believe that the method of the present paper could be used to prove convergence of the Navier-Stokes solution to an Euler solution with a vortex sheet, in the zero viscosity limit outside a boundary layer around the sheet. Note that the problem with a vortex sheet should be easier because the boundary layer is weaker since tangential slip is allowed, but it is more complicated since the boundary is curved and moving. Appendix A: The Estimates for the Heat Operators Proof of Lemma 3.1. To prove Lemma 3.1 it is useful to introduce the following changes of variables into the expression (3.21) for the operator E˜ 1 : ζ= One has 2 E˜ 1 f = π

Z

∞

x0 − x Y , η= . 1/2 [4(t − s)] [4(t − s)]1/2

dζe−ζ

√

2

Y / (4t)

Z

∞ −∞

2 dηe−η f x + ηY /ζ, t − Y 2 /4ζ 2 .

(A.1)

(A.2)

2 To get an estimate in Ll,ρ,θ β,T one has to bound the appropriate L norm in x and Y of j ∂xi E˜ 1 f with i ≤ l, ∂t ∂xi E˜ 1 f with i ≤ l − 2 and ∂xi ∂Y E2 f with i ≤ l − 2, j ≤ 2. We shall in fact prove a stronger estimate; we shall in fact prove that these terms are exponentially decaying in the Y variable. Let us first bound ∂xi E˜ 1 f :

sup

e(µ−βt) 0 and initial data (w(·, 0), ρ1 (·, 0)) for (1.5) satisfying w(·, 0) ∈ W 2,2 , ρ1 (·, 0) ∈ W 1,2 ,

(1.6)

where W 2,2 = W 2,2 (Rn ), etc.; we assume that the force f satisfies

f ∈ L∞ (I; L2 ) ∩ L2 (I; L2 ) ft ∈ L2 (I; L2 ), ∇f ∈ L2 (I; L∞ ),

(1.7)

where I = [0, T ) and T ≤ ∞ (T is not to be confused with the scaling time which occurred earlier); and we assume finally that (1.5) has a corresponding solution (w, ρ1 ) defined on Rn × I, satisfying the following conditions:  ρ1 ∈ L∞ (I; L∞ ∩ W 1,2 ) ∩ L2 (I; L∞ )     ρ1t , ∇ρ1 ∈ (L1 ∩ L2 )(I; L∞ ) ∩ L2 (I; L2 ) w ∈ L∞ (I; W 2,2 ∩ W 1,∞ )  2  L2 (I; L2 ∩ L∞ )   ∇w, Dx w ∈ 1 ρ1 4w ∈ L (I; L2 ∩ L∞ ).

(1.8)

When T is finite, the requirements (1.8) are simply that the solution of (1.5) be reasonably smooth. The relevant existence and regularity theory is well-known, and differs considerably in the case that n = 3, where smallness conditions are needed, from the case that n = 2. When T = ∞ on the other hand, the conditions in (1.8) require that the incompressible solution decays in a certain sense. See Schonbek [7], Wiegner [9], and the references therein for various results concerning rates of decay of solutions of (1.5). Of course, we do not exclude here the important case that the incompressible flow (w, ρ1 ) exists for all time and satisfies the conditions in (1.8) for all finite T but not for T = ∞. We now let δ-dependent initial data (ρδ (·, 0), uδ (·, 0)) be given for some or all values of δ ∈ (0, 1], and assume that the perturbations ϕδ and ψ δ defined in (1.4) satisfy kψ δ (·, 0)kL2 + δk∇ψ δ (·, 0)kL2 + δ 2 kDx2 ψ δ (·, 0)kL2 ≤ C0 δ,

(1.9)

kϕδ (·, 0)kL2 + δk∇ϕδ (·, 0)kL2 ≤ C0 δ 2 ,

(1.10)

kϕδ (·, 0)kL∞ ≤ h(δ),

(1.11)

where h is a strictly increasing, continuous function on [0, 1] satisfying h(0) = 0 and h(1) ≤ C0 . We then obtain the following convergence estimates, which are the main results of this paper:

546

D. Hoff

Theorem. Fix n = 2 or 3, let an external force f satisfying (1.7) be given, and let (ρ0 , w, c2 ρ1 ) be a corresponding solution of (1.5) as described above, satisfying the conditions in (1.8), and with all the norms indicated in (1.7) and (1.8) bounded by C1 . Let the function P = P (ρ) be given, satisfying (1.2), and let C0 > 0 be given. Then there are positive constants δ0 and C, depending on C0 , C1 , ε, λ, P, ρ0 , and, in the case that T < ∞, on T , such that: given initial data (ρδ (·, 0), uδ (·, 0)) for which the perturbations ϕδ and ψ δ defined in (1.4) satisfy (1.9)–(1.11), there is a corresponding solution (ρδ , uδ ) of (1.3) defined on Rn × [0, T ). This solution satisfies sup kρδ (·, t) − ρ0 kL2 ≤ Cδ 2 ,

(1.12)

sup kρδ (·, t) − ρ0 kL∞ ≤ C h(δ) + δ 1/4 ,

(1.13)

sup kuδ (·, t) − w(·, t)kL2 ≤ Cδ,

(1.14)

sup k∇(uδ − w)(·, t)kL2 ≤ C.

(1.15)

0≤t 0 for some µ ∈ Pcl+ , then rj w w. Proof. From hµ, w−1 hj i > 0, we see w−1 hj is a positive coroot. This is equivalent to w−1 αj ∈ Σ+ , where Σ+ is the set of positive roots. To see this is equivalent to rj w w, we refer, for example, to Proposition 4(i) in Sect. 5.2 (p.411) of [MP]. Now we define a subset P (k) (λ, B) of P(λ, B) as follows. We set P (0) (λ, B) = {p}. For k > 0, we take j and a such that k = (j − 1)d + a, j ≥ 1, 1 ≤ a ≤ d, and set · · · ⊗ B0(j+2) ⊗ B0(j+1) ⊗ Ba(j,···,1) (j < κ) (k) (2.3) P (λ, B) = · · · ⊗ B0(j+2) ⊗ B0(j+1) ⊗ Ba(j,···,j−κ+1) ⊗ B ⊗(j−κ) (j ≥ κ). Theorem 1. Under the assumptions (I-IV), we have Bw(k) (λ) ' P (k) (λ, B). 2.4. Proof of the theorem. In view of assumption (IV), we prove by induction on the length of w(k) . If k = 0, i.e., w(k) = 1, the statement is true. Next assume k > 0 and take j and a such that k = (j − 1)d + a, j ≥ 1, 1 ≤ a ≤ d. From the recursive formula (2.1) and w(k−1) ≺ w(k) , we have [ f˜in(j) Bw(k−1) (λ) \ {0}. Bw(k) (λ) = n≥0

a

On the other hand, from induction hypothesis, we have Bw(k−1) (λ) ' P (k−1) (λ, B) (j,···,1) (j < κ) · · · ⊗ B0(j+2) ⊗ B0(j+1) ⊗ Ba−1 = (j+2) (j+1) (j,···,j−κ+1) ⊗ B0 ⊗ Ba−1 ⊗ B ⊗(j−κ) (j ≥ κ). · · · ⊗ B0 Note that by definition, this is valid even for a = 1. In view of assumption (III), we see the action of f˜i(j) does not give any effect on the part · · · ⊗ B0(j+2) ⊗ B0(j+1) . Thus we a ignore this part in the following consideration. Then, in the case of j < κ, the proof ⊗κ , B2 = B ⊗(j−κ) , and take turns out to be trivial. Assume j ≥ κ, set i = i(j) a , B1 = B (j,···,j−κ+1) any elements b1 and b2 from Ba−1 ⊂ B1 and B2 . From Lemma 1, for any n ≥ 0, there exist p, q ≥ 0 such that f˜in b1 ⊗ b2 = f˜ip (b1 ⊗ e˜qi b2 ). Noting that e˜qi b2 ∈ B ⊗(j−κ) , we can conclude · · · ⊗ B0(j+1) ⊗ Ba(j,···,j−κ+1) ⊗ B ⊗(j−κ) ⊂ Bw(k) (λ). The other direction of the inclusion is clear. 2.5. Application to characters. Since we have established the weight preserving bijection between Bw(k) (λ) and P (k) (λ, B), the following proposition turns out to be an immediate corollary.

562

A. Kuniba, K. C. Misra, M. Okado, J. Uchiyama

Corollary 1. Under the assumptions (I-IV), we have X ewt p , ch Bw(k) (λ) = p∈P (k) (λ,B)

where wt p is given in (1.4). In view of the tensor product structure of P (k) (λ, B) (2.3), it would be more interesting to consider a classical character, which we define by X ]Bw (λ)µ ecl(µ) , cl ch Bw (λ) = µ∈P

Bw (λ)µ = {b ∈ Bw (λ) | wt b = µ}. Note that for B(λ), the classical character does not make sense. Corollary 2. Assume (I-IV). For k ≥ 1, take j and a such that k = (j − 1)d + a, j ≥ 1, 1 ≤ a ≤ d. Then, λ (j < κ) e j ch Ba(j,···,1) cl ch Bw(k) (λ) = eλj (ch Ba(j,···,j−κ+1) )(chB)j−κ (j ≥ κ). Proof. Let p be an element in P (k) (λ, B). From (1.4), the classical weight of p is given by ∞ X (wt p(i) − wt bi ). (2.4) λ+ i=1

Noting that p(i) = bi (i > j), wt bi = λi−1 − λi and λ0 = λ, (2.4) reads as λj +

j X

wt p(i),

i=1

which immediately implies the statement.

3.

c sl n Symmetric Tensor Case

In this section, we apply our theorem to the case of symmetric tensor representations of b n ). Uq0 (sl 3.1. Preliminaries. In what follows, ≡ always means ≡ mod n. Define δi(n) by δi(n) = 1 b n , we have hαi , hj i = 2δ (n) − δ (n) − δ (n) (i ≡ 0), = 0 (i 6≡ 0). In the case of g = sl i−j i−j−1 i−j+1 (i, j ∈ I), where I = {0, 1, · · · , n − 1}. For our purpose, it is convenient to use the notations αi , hi , 3i , ri for i ∈ Z by defining αi = αi0 , etc., if i ≡ i0 . Let B l be the classical crystal of the level l symmetric tensor representation of b n ). As a set, B l is described as B l = {(x0 , · · · , xn−1 ) ∈ Zn | Pn−1 xi = l}. The Uq0 (sl ≥0 i=0 actions of e˜i , f˜i are defined as follows.

Demazure Modules and Perfect Crystals

e˜i (x0 , · · · , xn−1 ) = f˜i (x0 , · · · , xn−1 ) =

563

(x0 , · · · , xi−1 + 1, xi − 1, · · · , xn−1 ) (i 6= 0) , (i = 0) (x0 − 1, · · · , xn−1 + 1)

(3.1)

(x0 , · · · , xi−1 − 1, xi + 1, · · · , xn−1 ) (i 6= 0) . (i = 0) (x0 + 1, · · · , xn−1 − 1)

(3.2)

If the right-hand side contains a negative component, we should understand it as 0. Following Sect. 1.2 of [KMN2], we describe the perfect crystal structure of B l . (Note that we deal with the case of (A(1) n−1 , B(l31 )) in their notation.) From (3.1) and (3.2), Pn−1 Pn−1 it is easy to see ε (x0 , · · · , xn−1 ) = i=0 xi 3i , ϕ (x0 , · · · , xn−1 ) = i=0 xi 3i+1 . Pn−1 Setting λ = i=0 mi 3i , we have b(λ) = (m1 , · · · , mn−1 , m0 ) and the automorphism Pn−1 σ is given by σλ = i=0 mi 3i−1 . Thus, the order of σ is n. It is often convenient to consider automorphisms on I and B l . By abuse of notation, we also denote them by σ. They are defined by σ(i) ≡ i − 1, 0 ≤ σ(i) ≤ n − 1 for i ∈ I, σ (x0 , · · · , xn−1 ) = (x1 , · · · , xn−1 , x0 ) for (x0 , · · · , xn−1 ) ∈ B l . It is easy to see the following properties. f˜σ(i) σ(b) = σ(f˜i b) Using σ, we have bj = σ j (m0 , · · · , mn−1 ) . e˜σ(i) σ(b) = σ(e˜i b),

for b ∈ B l .

3.2. Sequence of Weyl group elements. We define [x]+ by the largest integer not exceeding x (x ≥ 0) [x]+ = 0 (x < 0),

and set w

(k)

=

1 (k = 0) rk−1 · · · r1 r0 (k > 0).

(3.3)

We are to prove Proposition 3. The sequence of Weyl group elements w(0) , w(1) , · · · is increasing with respect to the Bruhat order. For the proof, we prepare a lemma. Pk−1 Lemma 2. w(k) 30 = 30 − i=0 i+d d + αi , with d = n − 1. Proof. i+d We prove by induction on k. Assuming the statement for k and setting γi = d + , we have w(k+1) 30 = w(k) 30 − hw(k) 30 , hk iαk = 30 −

k−1 X

γi αi

i=0

X − δk(n) + γk−1 + (γk−1−nj − 2γk−nj + γk+1−nj ) αk . j≥1

564


Therefore, it is sufficient to show X (γk−1−nj − 2γk−nj + γk+1−nj ) = γk . δk(n) + γk−1 + j≥1

Note that γa − γa−1 = θ(a), where θ(a) = 1 if a/d ∈ Z≥0 , = 0 otherwise. We have X γk − γk−1 − (γk−1−nj − 2γk−nj + γk+1−nj ) j≥1

= θ(k) + =

X

X

θ(k − nj) − θ(k + 1 − nj)

j≥1

θ(k − nj) − θ(k − nj − d)

j≥0

= δk(n) . This completes the proof.

to show Proof of the proposition. We take 30 for µ, and apply Proposition 2. It suffices hw(k) 30 , hk i > 0. The left hand side was already calculated to be k+d d + in the proof of the previous lemma. 3.3. λ = l30 case. We have seen that B l is perfect of level l. We have shown that the sequence of Weyl group elements w(k) (3.3) is increasing with respect to the Bruhat (j) order. Thus, taking d = n − 1 and i(j) a ≡ a − j (0 ≤ ia ≤ n − 1), the assumptions (I) and (IV) are already satisfied. Let us take λ = l30 . Then, we have λj = l3−j and the i

ground-state path is given by p = · · · ⊗ b2 ⊗ b1 , bj = (0, · · · , 0, l, 0, · · · , 0) with i ≡ −j (0 ≤ i ≤ n − 1). Let us describe Ba(j) . Thanks to the automorphism σ, it is sufficient to consider the case of j = n. From (2.2) and i(n) a = a, we easily have Ba(n) = {(x0 , · · · , xn−1 ) ∈ B l |

a X

xi = l}.

i=0

Checking the assumption (III) is also trivial. Therefore, we have the following proposition. (j) Proposition 4. For the case of λ = l30 , take d = n−1 and i(j) a ≡ a−j (0 ≤ ia ≤ n−1). Then, the assumptions (II) and (III) are satisfied with κ = 1.

3.4. λ arbitrary case. We show even if λ is any element in (Pcl+ )l , we have κ = 2 with Pn−1 the same choice of d and i(j) a as in the previous subsection. For λ = i=0 mi 3i , define a subset Bλl of B l by Bλl = {(x0 , · · · , xn−1 ) ∈ B l | x0 + · · · + xi−1 ≤ m0 + · · · + mi−1 for 1 ≤ i ≤ n − 1}. We now prepare a lemma. Pn−1 Pn−1 Lemma 3. Let λ = i=0 mi 3i , i=0 mi = l1 . For any b1 ∈ Bλl1 and b2 ∈ B l2 , there pn−1 exist an element bˇ 2 ∈ B l2 and integers p1 , · · · , pn−1 ≥ 0 such that fñ−1 · · · f˜1p1 (bˇ 1 ⊗ bˇ 2 ) = b1 ⊗ b2 , zi ≤ xi−1 (i = 1, · · · , n − 1), where we set bˇ 1 = (m0 , · · · , mn−1 ), bˇ 2 = (z0 , · · · , zn−1 ), b1 = (x0 , · · · , xn−1 ).


565

Proof. We prove by induction on n. If n = 1, the statement is trivial. Now assume the statement is valid when n−1, and set b2 = (y0 , · · · , yn−1 ). We divide it into two cases:(a) xn−2 > yn−1 , (b) xn−2 ≤ yn−1 . Consider the case (a). Assume (x0 , · · · , xn−1 ) ∈ Bλl1 , then we have xn−1 ≥ mn−1 . Consider elements (x0 , · · · , xn−3 , xn−2 +xn−1 −mn−1 , mn−1 ) ∈ Bλl1 , (y0 , · · · , yn−1 ) ∈ B l2 . Ignoring the last component and using the induction hypothesis, we see there exist z0 , · · · , zn−2 and p1 , · · · , pn−2 ≥ 0 such that (z0 , · · · , zn−2 , yn−1 ) ∈ B l , and pn−2 fñ−2 · · · f˜1p1 (m0 , · · · , mn−1 ) ⊗ (z0 , · · · , zn−2 , yn−1 ) = (x0 , · · · , xn−3 , xn−2 + xn−1 − mn−1 , mn−1 ) ⊗ (y0 , · · · , yn−1 ). x

−m

n−1 n−1 Apply fñ−1 further, then we get (x0 , · · · , xn−1 ) ⊗ (y0 , · · · , yn−1 ). Checking another condition is easy. We move to (b). The proof goes similarly. Take bˇ 2 = (z0 , · · · , zn−2 , xn−2 ) (z0 , · · · , zn−2 are determined from the induction hypothesis) and pn−1 = xn−1 − mn−1 + yn−1 − xn−2 .

(j) Proposition 5. For any λ ∈ (Pcl+ )l , take d = n − 1 and i(j) a ≡ a − j (0 ≤ ia ≤ n − 1). Then, the assumptions (II) and (III) are satisfied with κ = 2.

Proof. We again reduce the proof for all j to the j = n case by the automorphism σ. Set Pn−1 (n) λ = i=0 mi 3i . Note that i(n) a = a. Firstly, we claim that Ba consists of the elements l (x0 , · · · , xn−1 ) ∈ B satisfying x0 + · · · + xi−1 ≤ m0 + · · · + mi−1 xi = mi

(1 ≤ i ≤ a), (a + 1 ≤ i ≤ n − 1).

(n) ) = ma , and This is easily shown by induction on a. Thus, we have hλn , ha i = εa (Ba−1 (III) is checked. It remains to show Bd(n,n−1) = Bd(n) ⊗ B l . Noting that Bd(n) = Bλl , it is equivalent to check [ pn−1 fñ−1 · · · f˜1p1 {bn } ⊗ σ −1 (Bλl ) \ {0} = Bλl ⊗ B l . (3.4) p1 ,···,pn−1 ≥0

Take any b1 = (x0 , · · · , xn−1 ) ∈ Bλl and b2 ∈ B l . From the lemma, there exist bˇ 2 = pn−1 · · · f˜1p1 (bn ⊗ bˇ 2 ) = b1 ⊗ b2 , (z0 , · · · , zn−1 ) ∈ B l and p1 , · · · , pn−1 ≥ 0 such that fñ−1 zi ≤ xi−1 (i = 1, · · · , n−1). Noting that z1 +· · ·+zi ≤ x0 +· · ·+xi−1 ≤ m0 +· · ·+mi−1 , we see bˇ 2 ∈ σ −1 (Bλl ). Thus, we have shown the inclusion ⊃ in (3.4). cmp1 The other direction of the inclusion is clear. 4. Discussion b n , B = Bl . We retain We explain the suggestion by Kirillov. Consider the case of g = sl the notations in Sect. 3. If L is divisible by n, we already know Bw(Ld) (l30 ) ' ul30 ⊗ (B l )⊗L . Therefore, Bw(Ld) (l30 ) is invariant under the action of e˜i , f˜i (i 6= 0). On the other hand, it has been shown recently that the Kostka-Foulkes polynomial Kλ(lL ) (q) has the following expression (cf. [DF, D, NY]).

566


Kλ(lL ) (q) =

X PL−1 jH(bj+1 ⊗bj ) q j=1 ,

where the sum is over all highest weight vectors bL ⊗ · · · ⊗ b1 with respect to e˜i (i 6= 0) Pn−1 with highest weight i=1 (λi − λi+1 )3i . Now Kirillov’s suggestion reads as e−l30 ch Bw(Ld) (l30 ) (z1 , · · · , zn−1 ; q) X = q −E0 Kλ(lL ) (q)sλ (x1 , · · · , xn ) x1 ···xn =1 , λ L where λ runs over all partitions of lL of depth less than or equal to n, E0 = lL 2 ( n − 1), −δ −αi = xi+1 /xi (i = 1, · · · , n − 1), and sλ is the Schur function. Note that q = e , zi = e an expression of Kλµ (q) using Gaussian polynomials is known [Ki]. In this article, we have presented a criterion for the Demazure crystal to have a tensor b n symmetric tensor product structure. One may ask if there are other cases than the sl case which have a similar property. We claim that if g is non-exceptional and λ = l30 (plus some other λ), we can find a sequence of Weyl group elements which satisfies the assumptions (II-III) with κ = 1. We would like to report on that in near future. We are not only interested in the classical character, but also in the full character. One of the advantages to relate the Demazure crystal to a set of paths is that we can utilize the results on so-called 1D sums in solvable lattice models. In [FMO] we have seen that for b 2 case, all characters of the Demazure modules are expressed in terms of q-multinomial sl coefficients. We also would like to deal with this subject for any non-exceptional affine Lie algebra (though the level will be 1) in another publication.

Acknowledgement. K.C.M. and M.O. would like to thank Omar Foda for discussions, and collaboration in our earlier work [FMO]. We would like to thank Anatol N. Kirillov for suggesting to us that in the sbln symmetric tensor case, Demazure characters can be expressed using Kostka-Foulkes polynomials. We would also like to thank Noriaki Kawanaka for introducing the book [MP]. M.O. would like to thank Peter Littelmann for stimulating discussions. K.C.M. is supported in part by NSA/MSP Grant No. 96-1-0013. A.K. and M.O. are supported in part by Grant-in-Aid for Scientific Research on Priority Areas, the Ministry of Education, Science and Culture, Japan. M.O. is supported in part by the Australian Research Council.

References Dasmahapatra, S.: On the combinatorics of row and corner transfer matrices of the A(1) n−1 restricted models. hep-th/9512095 [DF] S. Dasmahapatra and O. Foda: Strings, paths, and standard tableaux. q-alg/9601011 [DJKMO1] Date, E., Jimbo, , Kuniba, A., Miwa, T. and Okado, M.: One dimensional configuration sums in vertex models and affine Lie algebra characters. Lett. Math. Phys. 17, 69–77 (1989) [DJKMO2] Date, E., Jimbo, , Kuniba, A., Miwa, T. and Okado, M.: Paths, Maya diagrams and representations of sbl(r, C). Adv. Stud. Pure Math. 19, 149–191 (1989) [FMO] Foda, O., Misra, K.C. and Okado, M.: Demazure modules and vertex models: the sbl(2) case. q-alg/9602018 [L] Littelmann, P.: Paths and root operators in representation theory. Ann. Math. 142, 499–525 (1995) [Ka] Kashiwara, M.: The crystal base and Littelmann’s refined Demazure character formula. Duke Math. J. 71, 839–858 (1993) [Ki] Kirillov, A.N.: Dilogarithm identities. Lectures in Mathematical Sciences 7, The University of Tokyo, 1995 [KMN1] Kang, S-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T. and Nakayashiki, A.: Affine crystals and vertex models. Int. J. Mod. Phys. A 7 (suppl. 1A), 449–484 (1992)

[D]


[KMN2] [KN] [MP] [NY] [S]

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Kang, S-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T. and Nakayashiki, A.: Perfect crystals of quantum affine Lie algebras. Duke Math. J. 68, 499–607 (1992) Kashiwara, M. and Nakashima, T.: Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra 165, 295–345 (1994) Moody, R.V. and Pianzola, A.: Lie algebras with triangular decompositions. New York: WileyInterscience, 1995 Nakayashiki, A. and Yamada, Y.: Kostka polynomials and energy functions in solvable lattice models. q-alg/9512027 Sanderson, Y.B.: On characters of Demazure modules. Ph. D. Thesis, Rutgers University (1995)

Communicated by T. Miwa

Commun. Math. Phys. 192, 569 – 604 (1998)

Communications in


Hilbert Modules in Quantum Electro Dynamics and Quantum Probability? Michael Skeide?? Centro Vito Volterra, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Rome, Italy Received: 28 October 1996 / Accepted: 21 July 1997

Abstract: A physical system of the form R ⊗ S with a distinguished state on B(R) may be described in a natural way on a Hilbert B(S)-module. Following the ideas of Accardi and Lu [1], we apply this possibility to a concrete system consisting of a boson field in the vacuum state coupled to a free electron. We show that the physical system is described adequately on a new type of Fock module: the symmetric Fock module. It turns out that a module has to fulfill an algebraic condition in order to allow for the construction of a symmetric Fock module. We prove in a central limit theorem that in the stochastic limit the moments of the collective operators (i.e. more or less the time-integrated interaction Hamiltonian) converge to the moments of free creators and annihilators on a full Fock module. In the sense of Voiculescu [22] and Speicher [20] these operators form a free white noise over the algebra B(S). 1. Introduction In elementary particle physics and statistical physics, usually, one is interested in the evolution of a system (usually one or a small number of elementary particles) described on a Hilbert space S subject to interaction with a system (usually a field or a heat bath) described on a Hilbert space R. The system R is assumed to be in a distinguished state (usually the vacuum state of the field or some temperature state). The goal of these notes is to show how the language of Hilbert modules can be used advantageously to describe physical systems of this type. As an example we consider a single free electron coupled to the electro magnetic field in the vacuum state. Like Accardi and Lu in [1] we compute the stochastic limit of this system. However, we ?

This work has been supported by the Deutsche Forschungsgemeinschaft. Present adress: Lehrstuhl für Wahrscheinlichkeitstheorie und Statistik, Brandenburgische Technische Universität Cottbus, Postfach 10 13 44, D-03013, Germany, E-mail: [email protected] ??

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do this from the beginning in terms of Hilbert modules. Along the computations we are led in a natural manner to the concepts of a symmetric and a full Fock module. These concepts are direct generalizations of the Bose and full Fock space well-known in quantum probability. The full Fock module introduced by Pimsner [14] is closely related to Voiculescu’s operator-valued free independence [22]. The symmetric Fock module is new. It is for operator-valued Bose independence what the full Fock module is for operator-valued free independence. We investigate this point in [18]. e of the field space R The interacting system is described on R ⊗ S. The vacuum determines a conditional expectation from the ∗-algebra of operators on R ⊗ S to the ∗-algebra of operators on S; see Eq. (2.6). The dynamics of the full system is determined by the interaction Hamiltonian. However, actually we are interested only in the reduced dynamics of S which is obtained by ‘projecting down’ the full dynamics from R ⊗ S to S via . Hilbert modules arise in a natural manner, namely, by GNS-construction. This GNSconstruction is, however, not based on a state, but on the completely positive mapping . The necessary facts about Hilbert modules and GNS-construction are explained in Sect. 3. In our example R is the space of the electromagnetic field, i.e. the symmetric or Bose Fock space over the one-photon sector. In Section 4 we introduce a Hilbert module which turns out to be the space of the GNS-representation of . This module has a close formal similarity to a symmetric Fock space. For this reason we call it a symmetric Fock module. In Sect. 6 we justify this name by showing that this module, indeed, arises by a systematic construction out of its one-particle sector. This construction completely parallels the construction of the symmetric Fock space. However, the possibilty for such a construction depends on an additional algebraic condition on the one-particle sector, namely, it has to be a centered Hilbert module; see Section 6. This condition turns out to be crucial for operator-valued Bose independence; see [18]. The proof that the GNSrepresentation of on the symmetric Fock module is faithful, is postponed mainly to the appendix. The methods developed in the appendix are applicable to arbitrary R and S as long as the state on R is pure. The physical model is discussed in Sect. 2. The most important objects are the collective operators defined by (2.3). When translated into the language of modules, they appear just as creators and annihilators on the symmetric Fock module. In the stochastic limit one is interested in a certain scaling limit of the moments of the collective operators in the vacuum conditional expectation . In [1] the limit has been calculated in terms of matrix elements of operators on S. Our first computational result (basically Lemma 5.6 and Corollary 5.8) going beyond [1] tells us that the limits of the matrix elements, indeed, define operators on S. In a central limit theorem we show that the moments of collective operators converge to moments of free creators and annihilators on a full Fock module. These free operators form operator-valued free white noises; see Speicher [20]. The proof of the central limit theorem is split into several steps. In the first step we compute the limit of the one-particle sector. This is done in Sect. 5. We provide a definition of convergence of Hilbert modules. All technical difficulties already arise in the limit of the one-particle sector. In Sect. 7 we generalize our techniques to the full system. The basic idea is to show in the proof of Theorem 7.3 that the limits of the collective operators fulfill the free commutation relations; see Eq. (6.1). For this aim it is indispensable to know from the general construction in Section 6 that the symmetric

Hilbert Modules in Quantum Electro Dynamics and Quantum Probability

571

Fock module is contained in the full Fock module as the submodule being spanned by symmetric tensors. Already Pimsner has shown that Relations (6.1) determine more or less the ∗-algebra generated by the free operators. This fact has the more probabilistic interpretation that the moments of the free operators are already determined by their second moments. The second moments, in turn, are known, if we know the one-particle sector. Following Speicher [20] Shlyakhtenko calls in [16] the one-particle sector a covariance matrix. He also finds the distributions of free creators as limit distributions of certain operatorvalued random variables in the totally different setup of random Gaussian band matrices (these are operator-valued generalizations of the usual random matrices leading to the Wigner semi-circle law). Our example shows that operator-valued free noise also can emerge from a purely physical setup. We remark that the appearance of free noise is due to the choice of the physical system. In particular, if the particle has discrete spectrum, it is very well possible to obtain operator-valued Bose white noise in the limit. For a detailed discussion we refer the reader to Gough [4]. He shows that the stochastic limit and the infinite-volume limit are interchangeable. This means that the limit does not change, if we first compute the stochastic limit for an electron in a finite volume and then let the volume go to infinity. Gough finds diagram rules of how to write down concrete expressions for the limit. In his diagrams the “dying out” of the non-identical permutations has a beautiful interpretation as an additional rule of energy conservation between certain vertices. According to this rule, like in the Wigner semi-circle law, only the non-crossing diagrams survive. A right Hilbert module already was constructed in [1]. However, the construction of Fock modules is based on two-sided modules. The left multiplication introduced in Sect. 4 and its limit calculated in Sect. 5 is a further computational result exceeding [1]. It turns out that the left multiplication makes the limit module much bigger than the module given in [1]. These notes are organized such that rather abstract sections where general notions are introduced change with sections where the abstract notions are applied to the concrete problem. The reader who prefers to have the general theory at the beginning can very well start reading Sections 3, 6 and the appendix and then the remaining sections. Conventions and notations. All our vector spaces are vector spaces over C. All our algebras are unital algebras. All our modules are modules over algebras (compatible with the unit of the algebra) and cary a vector space structure which is compatible with the algebra structure. All tensor products are algebraic tensor products. If V is a vector space and B an algebra, then we denote by VBf = B ⊗ V ⊗ B, VBr = V ⊗ B, VBl = V ⊗ B, and VB = V ⊗ B the free two-sided, free right, free left and free centered B-modules, respectively, generated by V equipped with their natural structures. (See Sect. 6 for the definition of a centered module.) Spaces of linear mappings between linear spaces are denoted by using the letter L. For right linear mappings between right modules we use Lr . For adjointable mappings between inner product spaces or modules we use La . Adjoint always means mutually adjoint; cf. also Remark 3.5. Notice that in the case of modules the adjointable mappings are always right linear. In the case of mappings between normed spaces we also use the letter B instead of L with the same use of the superscripts to denote spaces of bounded mappings. Spaces of continuous functions on Rd with values in a normed space B are denoted by C(Rd , B). A subscript b means bounded functions, a subscript 0 means functions vanishing at infinity and a subscript c means functions with compact support. If the

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range is not specified, we mean C. In this case a superscript n means n-fold continuously differentiable. The Schwartz functions on Rd are denoted by S(Rd ). The simple functions on R are denoted by A0 (R).

2. The Physical Model In [1] and also in [2] Accardi and Lu investigate the so-called stochastic limit or Friedrichs-van Hove scaling limit for the non-relativistic QED-Hamiltonian in d ∈ N dimensions of a single free electron coupled to the photon field without dipole approximation. Originally, the photon field has d components. However, since we neglect the possibility of polarization, we may restrict to a single component. From the mathematical point of view this is not a serious simplification. Our results can be generalized easily to d components. In addition, we forget about the fact that the electron couples to the field via the component of p into the direction of the field. The p may be reinserted after the computations easily, because we work in Coulomb gauge. Throughout these notes we assume d ≥ 3. In the sequel, we describe our simplified setup and refer to [1, 4] for a detailed description. The Hilbert space R of the field is theRsymmetric or boson Fock space 0(L2 (Rd )) over L2 (Rd ) with the Hamiltonian HR = dk |k| a+k ak . (By a+k and ak we denote the usual creator and annihilator densities which fulfill ak a+k0 − a+k0 ak = δ(k − k 0 ).) The particle space S is the representation space L2 (Rd ) of the d-dimensional Weyl algebra 2 W in momentum representation with the usual free Hamiltonian HS = p2 . The interaction is described on the compound system R ⊗ S by the interaction Hamiltonian Z HI = λ dk a+k ⊗ eik·q c(k) + h.c. . λ is a (positive) coupling constant. In the original physical model the function c is given by c(k) = √1|k| , see [1]. As in [1] we replace it by a suitable cut-off function c ∈ Cc (Rd ). Since we will identify operators on R and S, respectively, with their ampliations to R ⊗ S, we omit in the sequel the ⊗-sign in between such operators. The time-dependent interaction Hamiltonian in the interaction picture, defined by HI (t) = eit(HR +HS ) HI e−it(HR +HS ) , takes the form Z 2 1 HI (t) = λ dk a+k eik·q eitk·p eit(|k|+ 2 |k| ) c(k) + h.c. . This follows directly from the commutation relations fulfilled by a+k and ak and from the basic relation f (p)eik·q = eik·q f (p + k) for all f ∈ L∞ (Rd ). In the sequel, the special case 0

0

eik·p eik ·q = eik ·q eik·p eik·k

0

(2.1)

will be of particular interest. The wave operators U (t) defined by U (t) = eit(HR +HS ) e−it(HR +HS +HI ) are the objects of main physical interest. They fulfill the differential equation dU (t) = −iHI (t)U (t) and U (0) = 1. dt

(2.2)


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For the stochastic limit the time t is replaced by λt2 and one considers the limit λ → 0. So we define the rescaled wave operators Uλ (t) = U ( λt2 ). The problem is to give sense to U0 = lim Uλ . For this aim one usually proceeds in the following way. Let V denote λ→0

the vector space which is linearly spanned by all functions f : R × Rd → C of the form f (τ, k) = χ[t,T ] (τ )fe(k) (t < T, fe ∈ Cc (Rd )). Obviously, we have V = A0 (R) ⊗ Cc (Rd ). For f ∈ V define the collective creators Z A+λ (f ) =

Z dτ

dk a+k γλ (τ, k)f (τ, k)

(2.3)

and their adjoints Aλ (f ), the collective annihilators. Here we set γλ (τ, k) =

1 ik·q i t2 k·p i t2 ω(|k|) 1 e e λ e λ and ω(r) = r + r2 . λ 2

In view of Remark 4.3 and Lemma 5.6 we keep in mind that in some respects also more general choices for γλ and ω are possible. Obviously, we have Z A+λ (χ[t,T ] c)

+ Aλ (χ[t,T ] c) =

T λ2 t λ2

dτ HI (τ ).

(2.4)

With the definition A+λ (t) = A+λ (χ[0,t] c) Eq. (2.2) transforms into dA+ dAλ dUλ λ (t) = −i (t) + (t) Uλ (t) and Uλ (0) = 1. dt dt dt

(2.5)

Henceforth, if we are able to give sense to the limit not only of Uλ but also of A+λ (f ) for fixed f ∈ V , we may expect this differential equation to hold also in the limit λ → 0. On the other hand, if we find a limit of the A+λ (f ) and a quantum stochastic calculus in which (2.5) makes sense and has a solution U0 , we may hope that U0 is the limit of the Uλ . In these notes we will be concerned exclusively with the limit of the collective operators. However, we provide a natural language which promises to allow to describe several types of rich quantum stochastic calculi. e denote the vacuum in R. Then we define the vacuum conditional expectation Let : La (R ⊗ S) → La (S) by setting e ⊗ id)a(|i e ⊗ id). (a) = (h|

(2.6)

In [1] the limit lim hξ, (Mλ )ζi, for Mλ being an arbitrary monomial in collective λ→0

operators and ξ, ζ being Schwartz functions, has been calculated. We repeat the major ideas of the proof in a new formulation. Moreover, we show as a new result that the limit considered as a sesquilinear form in ξ and ζ, indeed, determines an element of B(S).

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3. GNS–Construction and Hilbert Modules The idea to generalize the GNS-construction based on a state to a construction based on a completely positive mapping between ∗-algebras A and B is not new. The first step in this direction is the Stinespring construction for the case when A is a C ∗ -algebra and B is an operator C ∗ -algebra over a Hilbert space G. Also in the Stinespring construction we obtain a representation of A on a Hilbert space H. However, the cyclic vector is replaced by an adjointable mapping from G into H; cf. Eq. (2.6) and also the appendix. In a generalization of the Stinespring construction due to Kasparov one may replace G by a Hilbert module; see e.g. the book [8] of Lance. A more direct generalization of the GNS-construction goes back to Paschke [12]. He also obtains a representation of A. However, this representation acts on a Hilbert B-module E rather than on a Hilbert space. One benefit from this description is the existence of a cyclic vector in E. Moreover, B may be an arbitrary C ∗ -algebra and A may be any ∗-algebra which is spanned linearly by its unitaries. The last condition yields to bounded representation operators automatically. Since the collective operators are unbounded, we have to further generalize Paschke’s GNSconstruction in order to describe the ∗-algebra Aλ , which is generated by the collective operators, and the conditional expectation (2.6) on a Hilbert module. Before we do that we recall the basic definitions which can be found in an equivalent form e.g. in [8]. Definition 3.1. A pre-Hilbert B-module over a C ∗ -algebra B is a right B-module E with a sesquilinear inner product h•, •i : E × E → B, such that hx, xi ≥ 0 for x ∈ E (positivity), that hx, ybi = hx, yib for x, y ∈ E; b ∈ B (right linearity), and that hx, xi = 0 implies x = 0 (strict positivity). If h•, •i is not necessarily strictly positive, we speak of a semi-inner product and of a semi-Hilbert B-module. We remark that sesquilinearity and positivity imply hx, yi = hy, xi∗ (symmetry), and that right linearity and symmetry imply hxb, yi = b∗ hx, yi (left anti-linearity). Let A be a ∗-algebra. A pre- (or semi-) Hilbert A-B-module is a two-sided A-Bmodule E which is also a pre- (or semi-) Hilbert B-module, such that hx, ayi = ha∗ x, yi for x, y ∈ E; a ∈ A (∗-property). P A mapping : A → B is called completely positive, if b∗i (a∗i aj )bj ≥ 0 for all i,j

ai ∈ A; bi ∈ B; i = 1, . . . , n; n ∈ N. As was noted by Paschke [12, Remark 5.1] this is equivalent to the usual definition of complete positivity. A representation of a ∗-algebra A over a C ∗ -algebra B is a pre-Hilbert A-B-module E. For B = C we recover the usual notion of a representation of A on a pre-Hilbert space. Lemma 3.2. Let E be a semi-Hilbert A-B-module and denote by N the set of all x ∈ E having square length hx, xi = 0. Then N is an A-B-submodule of E and the quotient E/N inherits a pre-Hilbert A-B-module structure by hx + N, y + Ni = hx, yi. Proof. We have to show that N is stable under all module operations (including addition) and that the definition of the inner product on E/N does not depend on the choice of a representative x + n (n ∈ N) of an element x + N. Both assertions follow easily, if we establish the equivalence hx, xi = 0 ⇐⇒ hy, xi = 0∀y ∈ E.

(3.1)

But this follows by applying an arbitrary separating family of states to (3.1) and CauchySchwartz inequality (cf. also Remark 3.4).


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Corollary 3.3. To any completely positive mapping : A → B from a ∗-algebra A into a C ∗ -algebra B there exists a pre-Hilbert A-B-module E and a cyclic vector η ∈ E (i.e. span AηB = E), such that (a) = hη, aηi. The pair (E, η) is determined up to two-sided pre-Hilbert module isomorphism (i.e. an isomorphism of two-sided modules, which is also an isometry). (E, η) is called the GNS-representation of . Proof. Consider A ⊗ B with its natural A-B-module structure. Since is completely positive, we turn A ⊗ B into a semi-Hilbert A-B-module by setting E X DX X a i ⊗ bi , a0j ⊗ b0j = b∗i (a∗i a0j )b0j (ai , a0j ∈ A; bi , b0j ∈ B). i

j

i,j

By the preceding lemma E = A ⊗ B/N is a pre-Hilbert A-B-module. Setting η = 1 ⊗ 1 + N, the pair (E, η) has the claimed properties. Uniqueness follows in the usual way. Remark 3.4. Up to now we did not use the C ∗ -algebra structure of B. The above definitions and results are formulated in a way such that they can be generalized to the much wider class of ∗-algebras which admit a separating family S0 of states (i.e. ϕ(b) = 0 for all ϕ ∈ S0 implies b = 0.) In this case an element b is called positive, if ϕ(b) ≥ 0 for all ϕ ∈ S0 . In particular, B may be only a pre-C ∗ -algebra. In this case on a semi-Hilbert Bmodule E we have the generalized Cauchy-Schwartz inequality hx, yihy, xi ≤ khy, yik hx, xi. (This follows either by the same method we proved (3.1), or by an investigation of yhy,xi when khy, yik 6= 0. We emphasize, however, that the case the length of x − khy,yik khy, yik = khx, xik = 0 requires a different argument.) As a consequence a pre-Hilbert B-module E is turned into a normedp module (i.e. a normed vector space fulfilling khx, xik. If E is complete, then it is called kxbk ≤ kxk kbk), by setting kxk = a Hilbert C ∗ -module or Hilbert B-module. A Hilbert C ∗ -module always admits an extension of the module operation to the completion of B. If E is a semi-Hilbert B-module, then by E we mean the Hilbert B-module associated with E, i.e. the completion of E after having divided out elements of length 0. Remark 3.5. The set La (E) of adjointable mappings on a pre-Hilbert B-module E (i.e. mappings T on E for which there exists a mapping T ∗ on E, such that hx, T yi = hT ∗ x, yi) is a subset of the set Lr (E) of right module mappings. Since the adjoint is unique, the elements of La (E) form a ∗-algebra. If E is a pre-Hilbert A-B-module, then A has a ∗-homomorphic image in La (E). The elements of La (E) extend closeably to the completion of E. Therefore, on a Hilbert B-module E the algebra La (E) and the algebra B a (E) of bounded adjointable operators on E coincide. However, in general B a (E) and B r (E) do not coincide; see [8, 12]. B a (E) is a C ∗ -algebra with respect to the operator norm. If A is a C ∗ - or a Banach ∗-algebra and E a Hilbert A-B-module, then kaxk ≤ kak kxk.

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Definition 3.6. The tensor product E ⊗ F of a semi-Hilbert B-module E and a semiHilbert C-module FDis turned into a semi-Hilbert E P B ⊗ C-module called exterior tensor P 0 P 0 xi ⊗yi , xj ⊗yj = hxi , x0j i⊗hyi , yj0 i (xi , x0j ∈ E; yi , yj0 ∈ product, by setting i

j

i,j

F ). We P show that if E and F are pre-Hilbert modules, then so is E ⊗ F . Indeed, let z = xi ⊗yi be an arbitrary element of E ⊗F . We may assume that the xi form a C-linearly i P independent set. If hz, zi = 0, then by (3.1) we have hu ⊗ v, zi = hu, xi i ⊗ hv, yi i = 0 i P for all u ∈ E, v ∈ F . For an arbitrary state ϕ on C we define 8v : z 7→ xi ϕ(hv, yi i). i

We have hu, 8v (z)i = (id ⊗ϕ)(hu ⊗ v, zi) = 0 for all u ∈ E, hence 8v (z) = 0. From linear independence of the xi we conclude that ϕ(hv, yi i) = 0 for all i. Since ϕ and v are arbitrary, we find yi = 0 for all i, i.e. z = 0. (Confer also the proof of this fact given in [8].) If E and F have a two-sided structure with ∗-algebras A and B, respectively, acting from the left, then E ⊗ F inherits a left structure over A ⊗ B which turns it into a semior pre-Hilbert A ⊗ B-B ⊗ C-module. Definition 3.7. The module tensor product E F of an A-B-module E and a B-Cmodule F is the vector space (E ⊗ F )/(xb ⊗ y − x ⊗ by) equipped with its natural A-C-module structure. By x y we denote the image of x ⊗ y under the quotient map. A mapping j : E × F → G into an A-C-module G is called A-C-bilinear, if it is left A-linear in the first and right C-linear in the second argument. j is called balanced, if j(xb, y) = j(x, by) for all x ∈ E, b ∈ B, y ∈ F . Obviously, the mapping i : (x, y) 7→ x y is balanced and A-C-bilinear. E F together with i is determined uniquely up to A-C-module isomorphism by the universal property: For an arbitrary balanced A-C-bilinear mapping j : E × F → G there exists a unique A-C-linear mapping e j : E F → G fulfilling j = e j ◦ i. Uniqueness follows in the usual way. We sketch this once for all other cases in these notes. Assume we have two tensor products G and G0 which have with mappings i and i0 the universal property. Denote by ie0 : G → G0 and ei : G0 → G the unique mappings determined by the universal property of G applied to i0 and conversely. We have ei ◦ ie0 ◦ i = ei ◦ i0 = i. By the universal property of G there is precisely one A-C-linear mapping j : G → G fulfilling j ◦ i = i, namely j = idG . We conclude ei ◦ ie0 = idG and, similarly, ie0 ◦ ei = idG0 . This means that G and G0 are isomorphic A-C-modules. We state some results which may be checked by using the universal property. Firstly, building the module tensor product is an associative operation. Secondly, given E and F as above, we find E VBf = E ⊗ VBr ,

E VB = E ⊗ V,

VBf

VB F, = V ⊗ F.

F =

VBl

⊗ F,

The ⊗-signs are those of the exterior tensor product. The right formulae applied to V = C show explicitly that B plays the role of the complex numbers. Given an A-B-linear mapping j : E → E 0 and a B-C-linear mapping k : F → F 0 , by the universal property there exists a unique a A-C-linear mapping j k : E F → E 0 F 0 , fulfilling (j k)(x y) = j(x) k(y). We call j k the tensor product of


577

j and k. If, for instance, E and F are submodules of E 0 and F 0 and j and k are the canonical embeddings, respectively, then j k defines a canonical embedding of E F into E 0 F 0 . However, unlike the vector space case, this embedding is, in general, not injective. This may happen, because the number of relations to be divided out in the definition of E F will usually be much smaller than the corresponding number for E0 F 0. Definition 3.8. If E and F are, in addition, two-sided semi-Hilbert modules, we may define a two-sided semi-Hilbert module structure also on the tensor product, by setting hx y, x0 y 0 i = hy, hx, x0 iy 0 i.

(3.2)

By a twofold application of the universal property we see that this is, indeed, welldefined. To see positivity it is necessary to know that B is a C ∗ -algebra. (In this case also Mn (B) is a C ∗ -algebra and we may refer to the positive square root of positive elements (hxi , xj i)i,j=1,... ,n ∈ Mn (B); see Lance [8, Chapter 4] for details.) E F is called the interior tensor product. If no confusion can arise, we say just tensor product or B-tensor product. Lance also shows that E F is a pre-Hilbert module, if E and F are Hilbert modules. We remark that the same is true (without changing a single word in the proof), if F is only a pre-Hilbert module. The same construction may be performed, when B is only a pre-C ∗ -algebra, if any b ∈ B acts as a bounded operator on F . In this case the argument to see positivity applies to the C ∗ -algebra B a F . Remark 3.9. Assume again that E and F are submodules of E 0 and F 0 , respectively. We already remarked that the canonical embedding from E F into E 0 F 0 needs not to be injective. It is, however, always an isometry. The same statement is true, if we consider the tensor product E B0 F over a subalgebra B 0 ⊂ B. Obviously, (3.2) defines a (B-valued) inner product also on E B0 F . The tensor product over a smaller algebra is always isometric to the tensor product over the bigger algebra. In the extreme case, one may choose even the tensor product of vector spaces (i.e. the tensor product over C). Often, we are interested in semi-Hilbert modules only up to isometry. Remark 3.10. Let us have another look at Corollary 3.3. Suppose that is a conditional expectation. In this case A is an example for a B-algebra; see Definition 6.2. It is easily checked that the semi-Hilbert module A ⊗ B may be replaced by the isometric semiHilbert module A B = A. This is a strong hint that a ∗-B-algebra with a positive B-B-linear mapping into B is the proper generalization of a ∗-algebra with a positive functional. See also [20, 22]. Definition 3.11. Let E denote a pre-Hilbert module over a von Neumann algebra of operators on a Hilbert space (B(S), for simplicity). By the strong Hilbert module topology on E we understand generated by the p the locally convex Hausdorff topology s family of semi-norms x 7→ hξ, hx, xiξi (ξ ∈ S). Denote by E the space obtained from E by completing the unit-ball of E in the strong Hilbert module topology. We observe that h•, •i : E × E → B(S) is jointly continuous in the strong Hilbert module topology of E and the weak topology of B(S). It follows that the inner product s extends strongly-weakly continuously to E . The strong topology is weaker than the s s norm topology, so that E is complete also in the norm topolgy. It can be shown that E

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is a self-dual Hilbert module (see the appendix for the definition of self-dual). Moreover, it can be shown that an arbitrary element of B a (E) extends uniquely to an element of s s a B E ; see [12, 19]. Therefore, if E is a pre-Hilbert A-B(S)-module then so is E . Example 3.12. The modules C0 (Rd , B(S))s and Cc (Rd , B(S))s . The examples of preHilbert B-modules most important for us are modules of functions with values in B. The inner product is given by a straightforward generalization of the usual inner product of L2 -spaces. To reduce technical difficulties it seems convenient to consider continuous functions. Unfortunately, it turns out that we need functions which are only strongly continuous. We close this section with some technical details. Although many of our statements can be generalized to other setups, we only consider the C ∗ -algebra B(S). By C0 (Rd , B(S))s = {f : Rd → B(S)|f ξ ∈ C0 (Rd , S) (ξ ∈ S)} we denote the module of all strongly continuous functions on Rd with values in B(S) vanishing strongly at infinity equipped with a B(S)-B(S)-module structure defined by the pointwise operations. Observe that C0 (Rd , S) is turned into a Banach space by the usual supremum norm. The strong topology on C0 (Rd , B(S))s is defined by the family f 7→ kf ξk (ξ ∈ S) of semi-norms. In addition, it follows by an application of the principle of uniform boundedness that the supremum norm exists also on C0 (Rd , B(S))s . In the same manner, it follows that C0 (Rd , B(S))s is an algebra (but not a ∗-algebra). We observe that the unit-ball of C0 (Rd , B(S))s is complete in both topologies in the uniform topology defined by the norm and in the strong topology. The B(S)-B(S)-module C0 (Rd ) ⊗ B(S) is contained in C0 (Rd , B(S))s , if we identify f ⊗ b with the function f b. It is a standard result that C0 (Rd ) ⊗ B(S) is dense in C0 (Rd , B(S)) with respect to the uniform topology; see e.g. Murphy [11]. Replacing in the proof uniform neighbourhoods by strong neighbourhoods, we obtain that C0 (Rd ) ⊗ B(S) is dense in C0 (Rd , B(S))s with respect to the strong topology. By Cc (Rd , B(S))s we denote the ideal of C0 (Rd , B(S))s , consisting of those functions with compact support. We turn Cc (Rd , B(S))s into a pre-Hilbert B(S)-B(S)-module by defining the inner product Z hf, gi = dk f ∗ (k)g(k). This integral is to be understood as the weak limit of Riemann sums. Observe that this inner product is jointly continuous with respect to the toplogy arising by restriction from C0 (Rd , B(S))s to functions with support in the same compact subset of Rd and the weak s topology of B(S). We denote L2 (Rd , B(S))s = Cc (Rd , B(S))s . Suppose fn n∈N is a net of elements fn ∈ C0 (Rd , B(S))s converging in the strong topology of C0 (Rd , B(S))s to an element f ∈ Cc (Rd , B(S))s . Let χ ∈ Cc (Rd ) such that χ is 1 on the support of f . Then also fn χ → f . From continuity of the inner product we conclude that fn χ → f also in the strong topolgy of Cc (Rd , B(S))s . Therefore, Cc (Rd ) ⊗ B(S) is dense in the strong Hilbert module topology of L2 (Rd , B(S))s . Actually, it can be shown that Cc (Rd ) ⊗ B(S) is sequentially dense in both strong topologies. The mapping b 7→ hf, bgi is weakly continuous on bounded subsets of B(S). This follows from the observation that if a certain Riemann sum for hξ, hf, f iξi differs from its limit at most by , then for b ≥ 0 the corresponding RiemannP sum for a∗i bai hξ, hf, bf iξi differs from its limit at most by kbk, and that, of course, b 7→ i


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is a weakly continuous mapping. (Notice also hf, bf i ≤ kbk hf, f i.)R Furthermore, if f ∈ Cc (Rd , B(S))s and g ∈ Cc (Rd+e , B(S))s , then the mapping ` 7→ dk f ∗ (k)g(k, `) is an element of Cc (Re , B(S))s . Finally, for f ∈ Cc (Rd , B(S))s and g ∈ Cc (Re , B(S))s the mapping f · g : (k, `) 7→ f (k)g(`) is an element of Cc (Rd+e , B(S))s . These remarks show that f g 7→ f · g defines an isometry from Cc (Rd , B(S))s Cc (Re , B(S))s into Cc (Rd+e , B(S))s . Moreover, given an arbitrary element of Cc (Rd+e , B(S))s the order of k- and `-integration does not matter. 4. The Module for Finite λ In this section we explicitly define a pre-Hilbert module on which the ∗-algebra Aλ of collective operators may be represented. The module does not depend on λ. In view of the appendix it may be considered (up to completion) as the GNS-representation of the ∗-algebra La (R ⊗ S) in the vacuum conditional expectation as defined in Sect. 2. In view of Sect. 6 we call this module a symmetric Fock module. For the time being we fix on the algebra B(S) which contains the Weyl algebra W as a strongly dense subalgebra. In the limit λ → 0 the algebra B(S) turns out to be too big. However, in view of Remark 3.9 it is easy to restrict to submodules as well as to subalgebras. We consider the module ∞ s M Ccsym (Rd )n , B(S) . 0B(S) Cc (Rd , B(S))s = n=0

s

Ccsym (Rd )n , B(S) are the strongly continuous symmetric functions on (Rd )n with compact support and values in B(S). (A function F (kn , . . . , k1 ) depends on n arguments in Rd . F is symmetric, if it is invariant under all exchanges ki ↔ kj .) With the pointwise multiplication and the inner product Z Z hF, Gi = dkn · · · dk1 F ∗ (kn , . . . , k1 )G(kn , . . . , k1 ), (4.1) 0B(S) Cc (Rd , B(S))s is turned into a pre-Hilbert B(S)-B(S)-module; see Example 3.12 for details. We call this module the symmetric Fock module over Cc (Rd , B(S))s . Forany function f ∈ Cc (Rd , B(S))s we define the creator a+ (f ) on 0B(S) Cc (Rd , B(S))s , by setting X 1 f (ki )F (kn+1 , . . . , kbi , . . . , k1 ) n + 1 i=1 s for F ∈ Ccsym (Rd )n , B(S) . The creator has an adjoint in La 0B(S) Cc (Rd , B(S))s , namely, the annihilator a(f ), which is defined by setting Z √ [a(f )F ](kn−1 , . . . , k1 ) = n dk f ∗ (k)F (k, kn−1 , . . . , k1 ). n+1

[a+ (f )F ](kn+1 , . . . , k1 ) = √

Under the condition [f ∗ (k), g(k)] = 0 for all k one easily checks the relations a(f )a+ (g) − a+ (g)a(f ) = hf, gi

(4.2)

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M. Skeide

which parallel the relations fulfilled by the creators and annihilators on the usual symmetric Fock space. The condition on f and g is fulfilled, if, for instance, one of them takes values only in the complex multiples of 1. Such a function commutes with all algebra elements. The vector subspace of a B-B-module E consisting of all x ∈ E which commute with all elements of B is called the B-center of Eand denoted by CB (E). We remark that both Cc (Rd , B(S))s and 0B(S) Cc (Rd , B(S))s are toplogically generated by their B(S)-center (i.e. the submodule generated by the B(S)-center is dense); see Example 3.12. In Sect. 6 this property shows to be essential for a systematic construction of a symmetric Fock module. The left-hand side of (4.2) is an operator. Consequently, also the algebra element hf, gi on the right-hand side is an operator, namely, multiplication of a module element by hf, gi from the left. Henceforth, it is impossible to understand Relations (4.2) without explicit reference to the left module structure of 0B(S) Cc (Rd , B(S))s . Notice also that a+ (bf b0 ) = ba+ (f )b0 for all b, b0 ∈ B(S). This means f 7→ a+ (f ) is a B(S)-B(S)d s linear mapping and f 7→a(f ) is a B(S)-B(S)-anti-linear mapping Cc (R , B(S)) → a d s L 0B(S) Cc (R , B(S)) . From these remarks it follows that the ∗-algebra generated by a+ Cc (Rd , B(S))s is an example for a ∗-B(S)-algebra; see Definition 6.2. Now our aim is to represent the collective operators as suitable creators and annihi lators on the symmetric Fock module 0B(S) Cc (Rd , B(S))s . Let f be an element in V (see Sect. 2). We define the mapping ϕλ : V → Cc (Rd , B(S))s , by setting Z [ϕλ (f )](k) = dτ γλ (τ, k)f (τ, k). (Since γλ is only strongly continuous, ϕλ maps, indeed, into Cc (Rd , B(S))s and not into Cc (Rd , B(S)).) Having a look at (2.3), we get the impression, as if A+λ (f ) wants to create the function ϕλ (f ). This impression is fully reconfirmed by the language of modules. Theorem 4.1. The equation α(A+λ (f )) = a+ (ϕλ (f )) defines a ∗-algebra monomorphism α : Aλ → La 0B(S) Cc (Rd , B(S))s . Moreover, s identifying 1 ∈ B(S) = Ccsym (Rd )0 , B(S) ⊂ 0B(S) Cc (Rd , B(S))s , we have (Mλ ) = h1, α(Mλ )1i

(4.3)

for any monomial Mλ in collective operators. Proof. We could check this directly by applying a state of the form hξ, •ξi to (4.3) and realizing that the left-hand side is 0, if and only if the right-hand side is 0. However, we prefer to show how our methods from the appendix work. ∞ L L2 (Rd )⊗sym n ⊗ S . The Denote by H a partial completion of R ⊗ S = n=0

completion is partial in the sense that each of the direct summands is completed, but the direct sum remains algebraic. H is a common invariant domain for the collective operators. a According to Theorem A.5 the ∗-algebra L (H) of adjointable operators on H is isomorphic to the ∗-algebra La La (S, H) of adjointable operators on the La (H)La (S)-module La (S, H). The isomorphism, denoted by α, ¯ maps the element a ∈ La (H) a to the map e a : L 7→ aL for L ∈ L (S, H).


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An element F ∈ 0B(S) Cc (Rd , B(S))s may be considered as an element of of S. It is easily checked that α(A ¯ +λ (f )) La (S, H) by letting F act pointwise on elements d s + restricted to 0B(S) Cc (R , B(S)) is α(Aλ (f )). This implies that α is a homomorphism. a a By Corollary A.6 an element of L L (S, H) is already determined by its restriction to R ⊗ La (S). 0B(S) Cc (Rd , B(S))s contains R ⊗ La (S). Therefore, α is injective. a e Eq. (4.3) follows from (A.1) and from the fact that ( ⊗ id) ∈ L (S, H) corresponds d s to 1 ∈ 0B(S) Cc (R , B(S)) . 1 is not yet necessarily a cyclic vector for the range of α. However, if we denote by d V (B(S)) the module spanned by functions f : R × R → B(S) of the form f (τ, k) = d s ˘ ˘ χ[t,T ] (τ )f (τ, k) f ∈ Cc (R × R , B(S)) , then it is possible to extend the definitions of the collective operators and of ϕλ to V (B(S)). Also α extends to the bigger ∗-algebra generated by A+λ V (B(S)) and Theorem 4.1 remains true. We will see later that now 1 is at least toplogically cyclic. Notice that ϕλ is right linear automatically. We turn V (B(S)) into a semi-Hilbert B(S)-module, by defining the semi-inner product hf, giλ = hϕλ (f ), ϕλ (g)i . By defining the left multiplication [b.f ](t, k) = γλ−1 (t, k)bγλ (t, k)f (t, k)

(4.4)

V (B(S)) becomes a semi-Hilbert B(S)-B(S)-module and ϕλ a (B(S)-B(S)-linear) isometry. Notice that this left multiplication makes f 7→ A+λ (f ) a B(S)-B(S)-linear mapping, where La (H) is equipped with its natural B(S)-B(S)-module structure. (See the proof of Theorem (4.1) for the definition of H.) Proposition 4.2. ϕλ extends to an isomorphism between the Hilbert B(S)-B(S)-modules s s V (B(S)) and L2 (Rd , B(S))s . A fortiori, all V (B(S)) for different λ > 0 are isomorphic. d Proof. Observe that γλ−1 V (B(S)) ⊂ V (B(S)). Let b d∈ B(S) and f ∈ Cc (R ). We have −1 ϕλ (γλ χ[0,1] f b) = f b, so that ϕλ V (B(S)) ⊃ Cc (R ) ⊗ B(S). From Example 3.12 we s conclude that ϕλ V (B(S)) = L2 (Rd , B(S))s . s ϕλ is an isometry and extends as a surjective isometry from V (B(S)) to L2 (Rd , B(S))s . Clearly, this extension is an isomorphism. Remark 4.3. The preceding proof shows that the operators A+λ V (B(S)) applied successively to 1 generate a strongly dense subspace of 0B(S) Cc (Rd , B(S))s . Therefore, 1 is topologically cyclic. Notice that all results obtained so far remain valid, if we choose for γλ an arbitrary invertible element of Cb (R × Rd , B(S))s (the bounded strongly continuous functions). s

Remark 4.4. The two pictures L2 (Rd , B(S))s and V (B(S)) of the same Hilbert module are useful for two different purposes. L2 (Rd , B(S))s shows more explicitly the algebraic structure which appears simply as the pointwise operations on a two-sided module of functions with values in an algebra. The property that the module is generated by its centered elements can be seen clearly only in this picture. In Sect. 6 this yields to a systematic construction of the symmetric Fock module out of its one-particle sector L2 (Rd , B(S))s . For the limit λ → 0, however, we concentrate on the elements of the s generating subset V ⊂ V (B(S)) . (The image of f ∈ V in L2 (Rd , B(S))s under ϕλ does not converge to anything.)

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5. The Limit of the One-Particle Sector This section is the analytical heart of these notes. We compute the limit of the module V (B(S)). In Sect. 7 we point out how the results of this section can be generalized, using the algebraic results of Sect. 6, to the full system. Motivated by Remark 4.4 we give the definition of what we understand by a limit of Hilbert modules. Definition 5.1. Let V denote a vector space. A family of semi-Hilbert B-B-modules Eλ λ∈3 with linear embeddings iλ : V → Eλ is called V -related, if the B-Bsubmodule generated by iλ (V ) is Eλ . In this case iλ extends to a B-B-linear mapping from VBf onto Eλ . We turn VBf into a semi-Hilbert B-B-module Vλ , by defining the semi-inner product hf, giλ = hiλ (f ), iλ (g)i for f, g ∈ VBf . Let T1 and T2 be locally convex Hausdorff topologies on B. A semi-Hilbert B-Bmodule E is called sequentially T1 -T2 -continuous, if for all f, g ∈ E any of the four functions b 7→ hf, gbi, b 7→ hf, bgi, b 7→ hf b, gi and b 7→ hbf, gi on B is sequentially T1 -T2 -continuous. Let 3 be a net converging to λ0 ∈ 3 and B0 a ∗-subalgebra of B which is sequentially T1 -dense. We saya V -related family of sequentially T1 -T2 -continuous semiHilbert B-B-modules Eλ λ∈3 converges to Eλ0 , if limhf, giλ = hf, giλ0

(5.1)

λ

in the topology T2 for all f, g ∈ VBf 0 . We write lim Eλ = Eλ0 . λ

Remark 5.2. Some comments on this definition are in place. We are interested in the limits of the semi-inner products of elements of VBf . However, it turns out that the limit may be calculated only on the submodule VBf 0 , where B0 is a sufficiently small subalgebra of B, and in a sufficiently weak topology T2 . (If this limit took values also in B0 , we could stay with VBf 0 and forget about B. Unfortunately, this will not be the case.) By the requirement that B0 is a sequentially dense subalgebra of B in a sufficiently strong topology T1 and by the T1 -T2 -continuity conditions we assure that the semi-inner product on VBf 0 (with values in B) already determines the semi-inner product on VBf . Suppose that Eλ λ∈3\{λ0 } is V -related and sequentially T1 -T2 -continuous and that Eq. (5.1) holds. Furthermore, suppose that also the limit semi-inner product fulfills the continuity conditions and its extension to elements of VBf still takes values in B. Then VBf with extension of the semi-inner product (5.1) by T1 -T2 -continuity is a sequentially T1 -T2 -continuous semi-Hilbert B-B-module. Letting Eλ0 = VBf , the family Eλ λ∈3 is V -related, sequentially T1 -T2 -continuous and we have lim Eλ = Eλ0 . λ

Obviously, after dividing out all null-spaces, Definition 5.1 may be restricted to the case of pre-Hilbert modules. If B is a pre-C ∗ -algebra and left multiplication is norm continuous on all Eλ , we may perform a completion. Convergence of a family of Hilbert modules means that there is a familiy of dense submodules for which Definition 5.1 applies.


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Remark 5.3. If the ∗ is continuous in both topologies, then it is sufficient to check the T1 T2 -continuity conditions only for either the left or the right argument of the semi-inner product. Furthermore, if the multiplication in B0 is separately T2 -continuous, then it is sufficient to compute (5.1) on elements of the left generate of V in VBf 0 . However, there is no way out of the necessity to compute the limit on any single element in the left generate. This had been avoided in [1], so that the convergence used therein is at most a convergence of right Hilbert modules. However, notice that, in particular, the left multiplication will cause later on a big growth of the limit module. The algebraic structures in Sects. 4 and 6 cannot even be formulated without the left multiplication. Now we start choosing the ingredients of Definition 5.1 for our problem. For B0 we choose the ∗-algebra W0 = span{eiκ·p eiρ·q : κ, ρ ∈ Rd } of Weyl operators. In order to proceed, we have to recall some basic facts about the Weyl algebra. For reference see e.g. the book [13] of Petz. The Weyl algebra W is the C ∗ -algebra generated by unitary groups of elements of a C ∗ -algebra subject to Relations (2.1). By Slawny’s theorem this C ∗ -algebra is unique, so that the definition makes sense. A representation of W on a Hilbert space induces a weak topology on W. However, this topology depends highly on the representation under consideration. For instance, we identify elements b ∈ W always as operators in B(S). In this representation the operators depend strongly continuously on the parameters κ and ρ. (Such a representation is called regular. An irreducible regular representation of W is determined up to unitary equivalence.) Denote by Wp and Wq the ∗-subalgebras of W0 spanned by all eiκ·p and spanned by all eiρ·q , respectively. The Weyl operators are linearly independent, i.e. as a vector space we may identify W0 with Wp ⊗ Wq via eiκ·p eiρ·q ≡ eiκ·p ⊗ eiρ·q . Since {eiρ·q }ρ∈Rd L is a basis for Wq , we may identify B0 with Wp . We identify Wp as a subalgebra ρ∈Rd

d of L∞ (Rd ) ⊂ B(S). By the momentum algebra P we mean the ∗-subalgebra L Cb (R ) of L∞ (Rd ). Notice that the C ∗ -algebra P contains Wp . For B we choose P. We ρ∈Rd L ∞ d L (R ) ⊂ B(S). have B ⊂ ρ∈Rd

In order to define the topology T1 , we need the weak topology arising from a different L 2 d L (R ) (consisting of families representation. We define a representation π of W on ρ∈Rd fρ ρ∈Rd , where fρ ∈ L2 (Rd )) by setting 0

π(eiκ·p eiρ ·q ) fρ

0

ρ∈Rd

= eiκ·p eiρ ·q fρ−ρ0

This representation extends to elements b ∈

L ρ∈Rd

ρ∈Rd

.

L∞ (Rd ). It is, roughly speaking,

regular with respect to κ, however, ‘discrete’ with respect ρ0 . L to L∞ (Rd ) with the restriction Let I denote a finite subset of Rd . We equip ρ∈I L ∞ d of the weak topology on L (R ) induced by the representation π. We equip ρ∈Rd L ∞ d L (R ) with a different topology by considering it as the strict inductive limit ρ∈Rd L L∞ (Rd ) ; see e.g. Yosida [23, Definition I.1.6]. Clearly, a sequence of d ρ∈I

I⊂R

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M. Skeide

eiρ·q hnρ

n∈N

=

hnρ

in

ρ∈Rd n∈N

L ρ∈Rd

L∞ (Rd ), where hnρ ∈ L∞ (Rd ), con-

verges, if and only if the hnρ are different from zero only for a finite number of ρ ∈ Rd and if any of the sequences hnρ n∈N (ρ ∈ Rd ) converges in the weak topology of L ∞ d L (R ) is sequentially complete and that B0 is sequentially L∞ (Rd ). Notice that ρ∈Rd

dense in this topology. By restriction to B, we obtain the topology T1 . Notice that convergence of a sequence in the topology T1 also implies convergence in the weak topology of B(S). The topology T2 is the topology induced by matrix elements with respect to the Schwartz functions S(Rd ). Thus, hf, giλ converges to b ∈ B(S), if and only if hξ, hf, giλ ζi converges to hξ, bζi for all ξ, ζ ∈ S(Rd ). Since an element in B0 leaves invariant the domain of Schwartz functions, the multiplication with elements of B0 is a T2 -continuous operation. Also the ∗ is continuous in both topologies, i.e. Remark 5.3 applies. Of course, we choose 3 = [0, ∞), ordered decreasingly, and λ0 = 0. We return to V = A0 (R) ⊗ Cc (Rd ). Fix λ > 0 and consider V (B(S)) equipped with its semi-inner product h•, •iλ , the left multiplication (4.4) and the embeding iλ being the extension of the canonical embedding i : V → V (B(S)). Then our Eλ are iλ (VBf ). Proposition 5.4. The Eλ λ>0 form a V -related, sequentially T1 -T2 -continuous family of semi-Hilbert B-B-modules. Proof. First, we show T1 -T2 -continuity. Notice that for sequences convergence in T1 implies convergence in the weak topology and that convergence in the weak topology implies convergence in T2 . Therefore, it suffices to show that for all f, g ∈ VBf the mappings b 7→ hf, gbiλ and b 7→ hf, b.giλ are sequentially weakly continuous. However, by right B-linearity, continuity of the first mapping is a triviality. The second mapping, actually, is an inner product of element of Cc (Rd , B(S))s . By the concluding remarks in Example 3.12 we know that the mapping depends weakly continuous on b on bounded subsets. In particular, it is sequentially weakly continuous. It remains to show that the inner product maps into B. For f = χ[t,T ] fe, g = χ[s,S] ge ∈ V we have Z Z T Z S hf, giλ = dk dτ dσ fe(k)e g (k)γλ∗ (τ, k)γλ (σ, k) t

s

t

s−τ λ2

Z Z T Z S σ−τ 1 dk dτ dσ fe(k)e g (k)ei λ2 (p·k+ω(|k|)) = 2 λ t s Z Z T Z S−τ λ2 = dk dτ du fe(k)e g (k)eiu(p·k+ω(|k|)) .

(5.2)

This is the weak limit of elements in Wp and, therefore, an element of P ⊂ B. Automatically, we have hf, gbiλ ∈ B for b ∈ B. Now consider b = h(p)eiρ·q ∈ B (h ∈ P). By Eq. (2.1) and manipulations similar to (5.2) we find Z Z T Z S−τ τ λ2 dτ du fe(k)e g (k)e−i λ2 ρ·k eiu((p−ρ)·k+ω(|k|)) h(p + k)eiρ·q . hf, b.giλ = dk s−τ t (5.3) λ2


585

The integral without the factor eiρ·q is a continuous bounded function of p, i.e. an element of the momentum algebra P ⊂ B. It follows that also hf, b.giλ ∈ B. Next we evaluate the limit in (5.1). The following proposition is just a repetition of a result in [1]. However, notice that the integrations have to be performed precisely in the order indicated (i.e. the p-integration first). Proposition 5.5 ([1]). Let f, g ∈ V be given as for Eq. (5.2) and ξ, ζ ∈ S(Rd ). Then Z Z Z lim hξ, hf, giλ ζi = hχ[t,T ] , χ[s,S] i dk fe(k)e g (k) du dp ξ(p)ζ(p)eiu(p·k+ω(|k|)) . λ→0 (5.4) The factor hχ[t,T ] , χ[s,S] i is the inner product of elements of L2 (R). Proof. The matrix element of Eq. (5.2) is Z Z Z T dτ hξ, hf, giλ ζi = dk fe(k)e g (k) Z =

t

dk fe(k)e g (k)

Z

S−τ λ2 s−τ λ2

T

dτ t

Z du eiuω(|k|)

Z

S−τ λ2

dp ξ(p)ζ(p)eiup·k

du eiuω(|k|) d ξζ (uk).

s−τ λ2

For λ > 0 the order of integrations does not matter, so we may, indeed, decide to perform ξζ , the Fourier transform of ξζ, is a rapidly decreasing function. the p-integration first. d Therefore, the λ-limit in the bounds of the u-integral may by performed for almost all k (namely k 6= 0) and all τ . Depending on the sign of s − τ and S − τ , respectively, the bounds converge to ±∞. A careful analysis, involving the theorem of dominated convergence, yields the scalar product of the indicator functions in front of (5.4). The resulting function of k is bounded by a positive multiple of the function | f k(k) | which is integrable for d ≥ 2. By another application of the theorem of dominated convergence and a resubstitution of d ξζ the formula follows. Now we will show as one of our main results that the sesquilinear form on S(Rd ) given by (5.4) indeed defines an element of B. In [1] it was not clear, if (5.4) defines R any operator on S. Denote by ek the unit vector in the direction of k 6= 0 and by dek the angular part of an integration over k in polar coordinates. Lemma 5.6. Let f be an element of Cc (Rd ), ξ be an element of S(Rd ) and d ≥ 3. Furthermore, let ω be a C1 -function R+ → R+ of the form ω(r) = rω0 (r), where 0 ≤ ω0 (0) < ∞ and ω00 bounded below by a constant c > 0. Denote by ω0−1 the inverse function of ω0 extended by zero to arguments less than ω0 (0). Then Z Z Z dk f (k) du dp ξ(p)eiu(p·k+ω(|k|)) Z Z ω −1 (−p · ek )d−2 = 2π dp ξ(p) dek 00 −1 f (ω0−1 (−p · ek )ek ). ω0 (ω0 (−p · ek )) Moreover,

Z dek

ω0−1 (−p · ek )d−2 f (ω0−1 (−p · ek )ek ) ω00 (ω0−1 (−p · ek ))

as a function of p is an element of Cb (Rd ).

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Remark 5.7. Formally, we can perform the u-integration and obtain Z Z 2π dk f (k) dp ξ(p)δ p · k + ω(|k|) , where the δ-distribution is one-dimensional (not d-dimensional). The statement of the lemma arises by performing the integration over |k| first and use of the formal rules for δ-functions. However, f is in general not a test function and the domain of the |k|integration is R+ , not R. Therefore, some attention has to be paid. We will use this formal δ-notation whenever it is justified by Lemma 5.6. Proof of Lemma [5.6]. Let us write k in polar coordinates, i.e. k = rek . For fixed k 6= 0 we write the p-integral in cartesian coordinates with the first coordinate p0 being the component of p along ek . Then p has the form p = p0 ek + p⊥ with p⊥ the unique component of p perpendicular to ek . In this representation the exponent has the form iur(p0 + ω0 (r)) and we may apply the inversion formula of the theory of Fourier transThe result may be described forms to the p0 -integration followed by the u-integration. formally by the δ-function 2πδ r(p0 + ω0 (r)) for the p0 -integration. We obtain Z Z Z du dp ξ(p)eiu(p·k+ω(|k|)) = 2π dp ξ(p)δ r(p0 + ω0 (r)) Z 2π dp ξ(p)χ[0,] p · k + ω0 (|k|) . = lim →0 It is routine to check that the right-hand side is bounded uniformly in ∈ (0, 1] by a 1 . Therefore, again by the theorem of dominated convergence we positive multiple of |k| may postpone the -limit also for the k-integration and obtain Z Z Z dk f (k) du dp ξ(p)eiu(p·k+ω(|k|)) Z Z 2π dk f (k) dp ξ(p)χ[0,] p · k + ω0 (|k|) . = lim →0 Now the order of integrations no longer matters. We choose polar coordinates for the k-integration and perform first the integral over r = |k|. The above formula for finite becomes Z Z Z 2π dr rd−1 f (rek )χ[0,] r(p · ek + ω0 (r)) . dp ξ(p) dek Consider the function F (r) = r(p·ek +ω0 (r)). From the properties of ω0 it follows that ω0 (r) ≥ ω0 (0)+cr. Consequently, F (r) ≥ r(p·eRk +ω0 (0))+cr2 . If p·ek +ω0 (0) ≥ 0, then F (r) ≥ cr2 and, because d ≥ 3, the integral 1 dr rd−1 f (rek )χ[0,] r(p · ek + ω0 (r)) converges to 0 for → 0 uniformly in p · ek ≥ −ω0 (0). On the other hand, if p · ek + ω0 (0) < 0, then F (r) starts with 0 at r = 0, is negative until the second zero r0 = ω0−1 (−p · ek )) and increases monotonically faster than cr2 . We make the substitution µ = F (r) and obtain Z 1 dr rd−1 f (rek )χ[0,] r(p · ek + ω0 (r)) Z 1 r(µ)d−1 f (r(µ)ek ). = dµ 0 p · ek + ω0 (r(µ)) + r(µ)ω00 (r(µ))


The integrand is bounded by

r(µ)d−2 ω00 (r(µ))

587

sup |f (k)|. Therefore, the integral converges unik∈Rd

formly in p · ek < −ω0 (0) to the limit

r0d−2 f (r0 ek ). ω00 (r0 ) Substituting the concrete form of r0 and extending ω0−1 by ω0−1 (F ) = 0 for F ≤ ω0 (0), we obtain the claimed formula. The last statement of the lemma follows from the observation that ω0−1 is a continuous function and that if ω0−1 (−p · ek ) is big, then f (ω0−1 (−p · ek )) = 0. Corollary 5.8. The sesquilinear form on S(Rd ) given by (5.4) defines an element of B. Formally, we denote this element by hf, gi0 = 2πhχ[t,T ] , χ[s,S] i

Z

dk fe(k)e g (k)δ p · k + ω(|k|) .

Notice also the commutation relations Z g (k)δ (p − ρ) · k + ω(|k|) 2π dk fe(k)e Z g (k)δ p · k + ω(|k|) e−iρ·q . = eiρ·q 2π dk fe(k)e Again it is clear that the limit extends to the right B0 -generate of V and that the function b 7→ hf, gbi0 extends weakly continuous, i.e. a fortiori T1 -T2 -continuous, from B0 to B. It remains to show this also for the left B0 -generate. Proposition 5.9. Let again f, g ∈ V be given as for Equation (5.2) and ξ, ζ ∈ S(Rd ). Furthermore, let b = eiκ·p eiρ·q ∈ B0 . Then Z λ lim hξ, hf, b.gi ζi = δρ0 hχ[t,T ] , χ[s,S] i dk fe(k)e g (k)eiκ·k λ→0 Z Z × du dp ξ(p)eiu((p−ρ)·k+ω(|k|)) eiκ·p eiρ·q ζ (p). (5.5) Remark 5.10. δρ0 is, indeed, the Kronecker δ. So the left multiplication in the limit is no longer weakly continuous. This is the reason for our rather complicated choice of the topology T1 . Proof of Proposition 5.9. Our starting point is equation (5.3). If ρ = 0 the statement follows precisely as in the proof of Proposition 5.5. If ρ 6= 0, the expression is similar to the case ρ = 0 (where ζ is replaced by eiρ·q ζ). τ The only difference is the oscillating factor e−i λ2 ρ·k . Similarly, one argues that the λ-limit in the bounds of the u-integral may be performed first. By an application of the Riemann–Lebesgue lemma the resulting integral over the oscillating factor converges to 0. Remark 5.11. The proposition shows in a particularly simple example how the RiemannLebesgue lemma makes a lot of matrix elements disappear in the limit. This fundamental idea is due to [1]. However, in [1] the idea was not applied to the left multiplication.

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M. Skeide

Corollary 5.12. The sesquilinear form on S(Rd ), defined by (5.5), determines the element hf, b.gi0 = δρ0 eiρ·q hf, (eiκ·k g)i0 e−iρ·q eiκ·p eiρ·q in Peiρ·q ⊂ B. Moreover, the mapping b 7→ hf, b.gi0 extends sequentially T1 -T2 continuously from B0 to B. Proof. Like for ρ = 0, it follows that (5.5), indeed, defines an element of Peiρ·q . Now we observe that a matrix element hf, h.gi0 , written in the form according to Lemma 5.6, may be extended from elements in Wp to all elements h ∈ P. It suffices to show that the mapping h 7→ hf, h.gi0 is sequentially weakly continuous on P. To see this we perform first the p-integral and obtain a bounded function on ek . Inserting a sequence hn n∈N , the resulting sequence of functions on ek is uniformly bounded. By the theorem of dominated convergence we may exchange limit and ek -integration. The following theorem is proved just by collecting all the results. Theorem 5.13. The Eλ λ≥0 form a V -related, sequentially T1 -T2 -continuous family of semi-Hilbert B-B-modules and lim Eλ = E0 .

λ→0

Now we Lare going to understand the structure of E0 better. Consider E0 = B ⊗ P ⊗ V ⊗ B. Any of the summands P ⊗ V ⊗ B inherits a semi-Hilbert V ⊗B= ρ∈Rd

P-B-module structure just by restriction of the operations of E0 . Notice that the inner products differ for different indices ρ. However, the left multiplications by elements 0 h ∈ P coincide. Of course, multiplication of an element in the ρth summand by eiρ ·q from the left, is not only pointwise multiplication, but shifts this element into the (ρ+ρ0 )th summand. Next we recall that V = A0 (R) ⊗ Cc (R). The factor hχ[t,T ] , χ[s,S] i tells us that E0 is the exterior tensor product of the pre-Hilbert C-C-module A0 (R) and B ⊗ Cc (R) ⊗ B with a suitable semi-Hilbert B-B-module structure. In order to combine both observations we make the following definition. Fix ρ ∈ Rd . We turn Cc (Rd , B)s into a P-B-module by pointwise multiplication by elements of B from the right and the left multiplication defined by setting [h.f ](k) = h(p + k)f (k). Denote by Vρr the P-B–submodule of Cc (Rd , B)s generated by Cc (Rd ). We turn Vρr into a semi-Hilbert P-B-module by setting Z (ρ) hf, gi = 2π dk f (k)∗ δ (p − ρ) · k + ω(|k|) g(k). 0 Vρr . For an element fρ ρ∈Rd ∈ E we define the left action of eiρ ·q by ρ∈Rd 0 0 eiρ ·q . fρ ρ∈Rd = eiρ ·q fρ−ρ0 ρ∈Rd . The following theorem may be checked simply by inspection.

Set E =

L

Theorem 5.14. The mapping X (hρ eiρ·q ) ⊗ χ[t,T ] fρ ⊗ bρ 7−→ χ[t,T ] ⊗ hρ .(eiρ·q fρ )bρ ρ∈Rd , ρ∈Rd


589

where χ[t,T ] ∈ A0 (R), fρ ∈ Cc (Rd ) ⊂ Vρr , hρ ∈ P, and b ∈ B, (all different from 0 only for finitely many ρ ∈ Rd ) defines a surjective B-B-linear isometry E0 = B ⊗ V ⊗ B −→ A0 (R) ⊗ E. The ⊗-sign on the right-hand side is that of the exterior tensor product. Remark 5.15. Cc (Rd , B)s may be considered as a completion of Cc (Rd ) ⊗ B. The left multiplication by elements of P leaves invariant Cc (Rd , B)s and the inner product of s E0 , first restricted to P ⊗ V ⊗ B = P ⊗ VBr and then extended to P ⊗ VBr , does not distinguish between elements h ⊗ f and 1 ⊗ (h.f ). Therefore, already the comparably small spaces Vρr are sufficient to obtain an isometry. We find the commutation relations i h 0 0 (eiκ·p eiρ ·q ). fρ ρ∈Rd (t, k) = eiκ·k fρ−ρ0 (t, k) ρ∈Rd (eiκ·p eiρ ·q )

(5.6)

for elements fρ ∈ V . This means for an arbitrary element in f ∈ A0 (R) ⊗ E and 0 0 ρ, κ ∈ Rd there exists f 0 ∈ A0 (R) ⊗ E such that (eiκ·p eiρ ·q ).f = f 0 eiκ·p eiρ ·q and conversely. (The same is true already for E.) Such a possibility is not unexpected, because it already occurs for finite λ. Remark 5.16. Of course, it is true that E is non-separable. This is not remarkable due to the non-separability of W0 . However, the separabilty condition usually imposed on Hilbert modules is that of being countably generated. Clearly, E fulfills this condition, because V is separable. A much more remarkable feature is that the left multiplication is no longer weakly continuous. However, also this behaviour is not completely unexpected. It often happens in certain limits of representations of algebras that certain elements in the representation space, fixed for the limit, become orthogonal. Consider, for instance, the limit } → 0 for the canonical commutation relations or the limits q → ±1 for the quantum group SUq (2), see [17]. In both examples the limits of suitably normalized coherent vectors become orthogonal in the limit. Remark 5.17. Since B ⊂ B(S) is a pre-C ∗ -algebra, E has a semi-norm and right multiplication fulfills kf bk ≤ kf k kbk. We show that left multiplication by an element of B 2 sup hξ, hf, f iξi. Any eleacts at least boundedly on E. Indeed, we have kf k = ξ∈S(Rd ),kξk=1 ment in b = hρ ρ∈Rd ∈ B may be T1 -approximated by a sequence bn n∈N of elements in B0 , where bn = hnρ ρ∈Rd . By the Kaplansky density theorem and weak separabilty

of the unit-ball of L∞ (Rd ), we may assume that hnρ = khρ k (ρ ∈ Rd , n ∈ N). We have 2

hξ, hbn ϕλ (f ), bn ϕλ (f )iξi ≤ kbn k hξ, hϕλ (f ), ϕλ (f )iξi X 2 ≤ hξ, hϕλ (f ), ϕλ (f )iξi khρ k . ρ∈Rd

The number of ρ’s for which hnρ 6= 0 for at least one n ∈ N is finite. Our claim follows, performing the limits first λ → 0 and then n → ∞. Therefore, if necessary, we may change to the Hilbert B-B-module E where, however, B is only a pre-C ∗ -algebra. This is all that is needed for the construction of the interior tensor products in Sect. 6.

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M. Skeide

Remark 5.18. It is not difficult to see that the left and right multiplication, actually, are sequentially T1 -continuous. Therefore, all our results in this section and in Sect. 7 may be extended to the sequentially T1 -complete algebra L∞ (Rd ) ⊗ Wq .

6. Full and Symmetric Fock Modules This section together with the appendix is the algebraic heart of these notes. We recall the construction of the full Fock module which is the Hilbert module carrying Pimsner’s generalized Cuntz-Krieger algebras, see [14]. Then we show that for a suitable subcategory of the two-sided Hilbert modules, the so-called centered Hilbert modules, a construction paralleling the construction of the symmetric Fock space can be performed. The module from Sect. 4 which describes the physical system will be identified as the symmetric Fock module over Eλ . In Sect. 7 we will see that the full Fock module over A0 (R)⊗E will be the representation space on which the limits of the collective operators may be represented as elements of the corresponding generalized Cuntz-Krieger algebra. We explain the connection with B-algebras. A B-algebra with a conditional expectation is for operator-valued free probability what an algebra with a state is for usual quantum probability; see Voiculescu [22] and also [18]. The subcategory of centered B-algebras is the basis for operator-valued Bose probability; see [18]. Definition 6.1. Let E denote a semi-Hilbert B-B-module. By the full Fock module

zB (E) over E we mean the semi-Hilbert B-B-module zB (E) =

M

E n

n∈N0

(E 0 = B). By our remark in Definition 3.3 zB (E) is a pre-Hilbert module, if E is a Hilbert module and B is a C ∗ -algebra. On zB (E) we define the creators `+ (x) (x ∈ E) by setting `+ (x)xn . . . x1 = x xn . . . x1

`+ (x)1 = x

and the annihilators `(x) (x ∈ E) by setting `(x)xn . . . x1 = hx, xn ixn−1 . . . x1

`(x)1 = 0.

The creator and annihilator to the same x ∈ E are adjoint elements of La (zB (E)). Moreover, we have the relations `(x)`+ (y) = hx, yi,

(6.1)

where the algebra element hx, yi again acts as multiplication from the left. Notice that, unlike Relations (4.2), the above relations hold for all x, y ∈ E. The above definition has been introduced by Pimsner [14]. However, notice that Pimsner only considers complete modules. The first use in quantum probability occurred in Speicher [20].


591

Definition 6.2. A B-algebra is a B-B-module A with a B-B-linear multiplication M : A A → A and a B-B-linear unit mapping m : B → A, such that the associativity condition M ◦ (M id) = M ◦ (id M ) and the unit property M ◦ (m id) = id = M ◦ (id m) are fulfilled. We use the notation M (a b) = ab and m(1) = 1 (i.e. 1B = 1A = 1). A ∗-B-algebra is a B-algebra with a B-B-anti-linear involution fulfilling the usual properties. Remark 6.3. For B = C we recover the usual abstract definition of a unital complex algebra. In [22] a B-algebra is introduced as an algebra A which contains B as a subalgebra and 1B = 1A . One easily checks that this case is included in our definition as the case when m is injective, because m(B) always is a subalgebra of the algebra A. Example 6.4. The full Fock module zB (E) is turned into a B-algebra by setting (xn · · · x1 )(ym · · · y1 ) = xn · · · x1 ym · · · y1 and m(b) = b ∈ E 0 . Forgetting about the Hilbert module structure, we denote this B-algebra by TB (E) and call it the tensor B-algebra over E. For B = C we recover the usual tensor algebra over a vector space. As a B-algebra TB (E) with the canonical embedding i : E → E 1 is determined up to B-algebra isomorphism by the following universal property: An arbitrary B-B-linear mapping j : E → A into an arbitrary B-algebra A extends to a unique B-algebra homomorphism e j ◦ i. j : TB (E) → A, such that j = e Example 6.5. The ∗-algebra A` generated by all creators and annihilators on zB (E) is not necessarily unital. However, the elements of B act as double centralizers on A` . Denote by m the embedding of B into the multiplier algebra M (A` ). The ∗-subalgebra of M (A` ) generated by A` and m(B) is an example for a ∗-B-algebra where m is not necessarily injective. The generalized Cuntz-Krieger algebra is the ∗-B-subalgebra of La (zB (E)) generated by A` and B. In [14] it is shown that this algebra may be considered as the ∗-algebra generated by the module E, the ∗-algebra B, Relations (6.1) and `+ (axa0 + byb0 ) = a`+ (x)a0 + b`+ (y)b0 for all x, y ∈ E; a, a0 , b, b0 ∈ B. Example 6.6. Let V be a vector space. Then zB (VB ) = (z(V ))B . If, for instance, V = Cc (Rd ), then VB is the subspace of those functions in Cc (Rd , B) which take values only in a finite-dimensional subspace of B. Notice that, in general, Cc (Rd , B) Cc (Rd , B) 6⊂ Cc (R2d , B). However, as semi-Hilbert modules Cc (Rd , B) Cc (Rd , B) and the submodule of Cc (R2d , B) generated by elements f (k2 )g(k1 ) are isometric. The (unital) free product of (unital) algebras A, B is defined as A ? B = T (A ⊕ B)/(a ⊗ a0 − aa0 , b ⊗ b0 − bb0 , 1T − 1A , 1T − 1B ), see [20, 21]. One easily checks that TB (VBf )/(1T − 1B ) = T (V ) ? B. This shows a close connection between free two-sided modules and free products of algebras. By the

592

M. Skeide

universal property one checks the functorial property TB (E ⊕ F ) = TB (E) ?B TB (F ), where ?B denotes the B-free product of B-algebras; see [20]. (Roughly speaking, in the definition of ? one has to replace T by TB and ⊗ by . The B-free product is the coproduct in the category of B-algebras.) Thinking again about Hilbert modules, the functorial property remains true, if the tensor products and direct sums are understood as those of Hilbert modules. Now we proceed from the full to the symmetric Fock module. The symmetric Fock space may be obtained from the full Fock space via symmetrization of any n-particle sector. We want to generalize this to modules. However, in general the tensor product of modules depends on the order of the factors. Example 6.7. Denote by 4 a vector space with a basis en n∈N0 . By 4+ and 4− we denote the modules which are obtained by letting act the algebra of polynomials Chxi in one indeterminate x as creators (i.e. xen = en x = en+1 ) and as annihilators (i.e. xen = en x = en−1 , e−1 := 0), respectively. Obviously, 4+ may be identified with Chxi via en = xn . Hence, 4+ Chxi = 4+ . On the other hand, 4− Chxi = {0}. We obtain (4− ⊗ 4+ ) (4+ ⊗ 4− ) = 4− ⊗ 4+ ⊗ 4− , but (4+ ⊗ 4− ) (4− ⊗ 4+ ) = {0}. This means that, for instance, the B-tensor product of two B-algebras does not necessarily have a canonical B-algebra structure. In order to overcome this obstacle we introduce a new subcategory of the category of B-B-modules. Definition 6.8. The B-center of a B-B-module E is the set CB (E) = {x ∈ E|xb = bx (b ∈ B)}. A B-B-module E is called centered, if it is generated by its B-center. This means that for any x ∈ E there exist n ∈ N, xk ∈ CB (E), bk ∈ B (k = 1, . . . , n), such that x=

n X

x k bk =

k=1

n X

bk x k .

k=1

The following properties are checked immediately. Proposition 6.9. 1. A B-B-linear mapping maps the B-center into the B-center. 2. Any element of a centered B-B-module commutes with any element of the center of B. 3. Consequently, a B-B-module over a commutative algebra B is centered, if and only if the left and right action coincide. 4. For two centered B-B-modules E, F we have CB (E) CB (F ) ⊂ CB (E F ). Therefore, also E F is a centered B-B-module. Theorem 6.10. Let E, F be two centered B-B-modules. There is a unique B-Bmodule isomorphism F : E F → F E, called flip isomorphism, fulfilling F (x y)

for all x ∈ CB (E) and y ∈ CB (F ).

=y x

(6.2)


593

Proof. Let (x, y); x ∈ E, y ∈P F denote an arbitrary element of E × F . Since E and P ai xi and y = yj bj for suitable xi ∈ CB (E); yj ∈ F are centered, we have x = i

j

CB (F ); ai , bj ∈ B. Let x0i ∈ CB (E); yj0 ∈ CB (F ); a0i , b0j ∈ B denote another suitable choice. We find X X X a i yj x i bj = y j a i x i bj = yj a0i x0i bj ij

ij

=

X

ij

a0i yj

x0i bj

a0i yj0 b0j

x0i

=

ij

=

X

X ij

=

X

ij

Therefore, F

×

a0i yj bj x0i a0i yj0 x0i b0j .

ij

: (x, y) 7−→

X

a i yj x i bj

ij

is a well-defined mapping E × F → F E. Obviously, F × is B-B-bilinear. We show that it is balanced. Indeed, for an arbitrary a ∈ B we find X X × F (xa, y) = ai ayj xi bj = ai yj xi abj = F × (x, ay). ij

ij

Thus, by the universal property of the B-tensor product there exists a unique B-B-linear mapping F : E F → F E fulfilling F (x y)

= F × (x, y).

Of course, F fulfills (6.2). By applying F a second time (now to F E), we find F ◦ F = id. Combining this with surjectivity, we conclude that F is an isomorphism. Our proceeding in the foregoing proof is typical for all statements concerning centered B-B-modules and B-B-module homomorphisms between them. It shows that for many statements it is sufficient to check them only on elements of the center. The Btensor products of elements of the center behave formally like tensor products of vectors in usual tensor products. This means that all statements concerning centered B-algebras (i.e. B-algebras, whose module structure is centered) are clear, if the corresponding statements for algebras are understood. In particular, centered B-algebras have a natural B-tensor product. The B-tensor product of centered B-algebras is for operator-valued Bose probability what the B-free product is for operator-valued free probability, see [18]. Proposition 6.11. If E is a centered semi-Hilbert B-B-module, then hCB (E), CB (E)i is contained in the center of B. Consequently, the flip of two centered semi-Hilbert B-B-modules E and F is an isometry. Moreover, it extends as an isometry to E F . Thus, we have E F ∼ = F E. If B = B(S), the same is true for strong completions. A topological B-B-module is called topologically centered, if it contains a dense centered B-B-submodule.

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Example 6.12. VB is a centered B-B-module. It is easy to see that a B-B-module is of the form VB , if and only if it admits a module basis consisting of elements of its center. If the center of B is trivial, then the identification is canonical. If V is a pre-Hilbert space, then the exterior tensor product VB = V ⊗ B is a centered pre-Hilbert B-B-module. Assume that V is infinite-dimensional and separable. In view of Kasparov’s absorption theorem any countably generated Hilbert B-module may be considered as a complemented B-B-submodule of VB , see [8]. With the help of the flip isomorphism it is possible to define permutations on the n-fold tensor product E n of a centered B-B-module E in an obvious way. Any permutation is an element of B a (E n ) with the inverse permutation being the adjoint. If E is only topologically centered, then some attention has to be paid: We restrict to tensor products of pre-Hilbert modules in order to have a Hausdorff topology. Since the flip is an isometry, it may be extended (strongly) continuously from (CB (E))B (CB (E))B to E E. However, in general we have E E 6= F (E E). Example 6.13. Consider E = (Cc (Rd ))B with its strong Hilbert module completion L2 (Rd , B)s . On elements f, g ∈ E we have [F (f g)](k2 , k1 ) = f (k1 )g(k2 ). However, s for f, g ∈ Cc (Rd , B) ⊂ E there is in general no possibility to write f (k1 )g(k2 ) in the s form f 0 (k2 )g 0 (k1 ) for suitable f 0 , g 0 ∈ E . Definition 6.14. Let E be a centered semi-Hilbert B-B-module. We define the number operator N on zB (E) by setting N (E n ) = n. The symmetrization operator P is defined by P (E n ) being the mean over all permutations on E n . We define the symmetric Fock module 0B (E) √over+ E by setting 0B (E) = P zB (E). + (E) we define the creators a (x) = P N ` (x) and the annihilators a(x) = On 0 B √ `(x) N P for all x ∈ E. Sometimes we call `+ and ` the free creators and annihilators and a+ and a the symmetric creators and annihilators, respectively. Remark 6.15. P is a bounded self-adjoint projection. N is self-adjoint. Therefore, a+ (x) and a(x) are adjoints. One easily checks P N = N P , a+ (x)P = a+ (x) and N `+ (x) = `+ (x)(N + 1). By these relations the creators and annihilators fulfill the analogue of Relation (4.2) a(f )a+ (g) − a+ (g)a(f ) = hf, gi, if at least one of the arguments is in the B-center of E. Remark 6.16. The modules we are interested in are only topologically centered. Therefore, we have to generalize the preceding definition slightly. Suppose that E is a topologically centered pre-Hilbert B-B-module. Then the definitions of N , P and a+ (x), a(x) e of L E n (s) which is closed under P and big enough to extend to any submodule H n∈N0

contain the image of E n (n ∈ N0 ) under the canonical embedding. Also in this case e The concrete form of H e has we will speak of a symmetric Fock module 0B (E) = P H. to be clear from the context. With this agreement the module 0B(S) Cc (Rd , B(S))s from Sect. 4 is, indeed, the symmetric Fock module over the topologically centered pre-Hilbert B(S)-B(S)-module Cc (Rd , B(S))s . However, notice that the Eλ themselves are not centered, because the s s B(S)-center of Eλ is not in the image of Eλ in Eλ .


595

We conclude this section with some algebraic remarks. A B-algebra A is called B-commutative, if the B-center is a commutative subalgebra of A. Notice that such an algebra is, in general, far from being commutative. √ √ √ Clearly, 0B (E) with the multiplication (P N F )(P N G) = P N (F G) (F, G ∈ zB (E)) is a centered B-commutative B-algebra. Forgetting about the Hilbert module structure, we denote this B-algebra by SB (E) and call it the symmetric tensor B-algebra over E. The symmetric tensor B-algebra is characterized up to B-algebra isomorphism by the following universal property: An arbitrary B-B-linear mapping j : E → A into an arbitrary centered B-commutative B-algebra A extends to a unique B-algebra j ◦ i. homomorphism e j : SB (E) → A, such that j = e One easily checks the functorial property SB (E ⊕ F ) = SB (E) SB (F ) by looking at elements of the center. Thinking again about centered Hilbert modules, the functorial property remains true, if the tensor product is understood as that of Hilbert modules. The functorial property of the usual Fock space is crucial for the quantum stochastic calculus of Hudson and Parthasarathy [5]. We remark that it is very well possible to introduce also the symmetric Fock module via exponential vectors. However, with the domain of exponential vectors we, definitely, leave the algebraic framework. Finally, we mention that the B-tensor product is the coproduct in the category of centered B-commutative B-algebras. In [18] we investigate a new type of Bose independence of centered B-valued quantum random variables based on centered B-algebras. We obtain the Bose analogue of Voiculescu’s operator-valued free independence, see [22]. Example 6.17. We show an example of our notion of centered B-algebras to the paper [6] by Kümmerer and Maassen. The authors study the so-called “essentially commutative Markov dilations”, which live on the algebra WB = W ⊗ B where W is a commutative von Neumann algebra and B = Mn . Clearly, WB is a centered B-commutative Balgebra. W as a commutative algebra of ‘classical white noises’ may be represented on a symmetric Fock space. Thus, WB is a centered B-commutative B-subalgebra of the centered B-algebra of adjointable operators on a suitable symmetric Fock module over B. It should be possible to generalize the results of [6] to the full B-algebra of adjointable operators on this Fock module by allowing non-commutative “white noises”. 7. The Central Limit Theorem In this section we prove in a central limit theorem that the moments of the collective creators and annihilators in the vacuum conditional expectation, represented in Sect. 4 by symmetric creators and annihilators on the symmetric Fock module 0B(S) (Cc (Rd , B(S)s ), converge to the moments of the corresponding free creators and annihilators on the full Fock module zB (A0 (R) ⊗ E) over the limit of the one-particle sector computed in Sect. 5. In a first step we show that in a pyramidally ordered product (i.e., so to speak, an anti-normally ordered product) the moments of the free operators for finite λ converge to the moments of the free operators for λ = 0. In the next step we show that nothing

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changes, if we replace for finite λ the free operators by symmetric operators. For this step an explicit knowledge of the embedding of the symmetric Fock module into the full Fock module as described in Sect. 6 is indispensable. The final step consists in showing that the limits for arbitrary monomials respect the free commutation relations given by (6.1). In the course of this section we compute a couple of T2 -limits of elements of B(S). For the sesquilinear forms on S(Rd ), defined by these algebra elements, all the limits already have been calculated by Accardi and Lu in [1]. Since the combinatorical problems of, for instance, how to write down an arbitrary monomial in creators and annihilators and so on, have been treated in [1] very carefully, we keep the proofs brief. Sometimes, we give only the main idea of a proof in a typical example. New is that the limit sequilinear forms define operators. This means that the limit conditional expectation, indeed, takes values in B(S). Also new is the interpretation of the limit of the moments of the collective operators as moments of free operators on a full Fock module in the vacuum expectation. The idea to see this, roughly speaking, by checking Relations (6.1) (see the proof of Theorem 7.3), has its drawback also in the computation of the limit of the sesquilinear forms. The structure of the proof is simplified considerably. Theorem 7.1. Let fi = χ[ti ,Ti ] fei , gi = χ[si ,Si ] gei be in V (i = 1, . . . , n; n ∈ N). Then lim h1, `(f1 ) · · · `(fn )`+ (gn ) · · · `+ (g1 )1iλ = h1, `(f1 ) · · · `(fn )`+ (gn ) · · · `+ (g1 )1i0 .

λ→0

Proof. First, we show that

Z Z gn (kn ) · · · fe1 (k1 )e g1 (k1 ) hfn . . . f1 , gn . . . g1 iλ = dkn . . . dk1 fen (kn )e Z Sn Z T 1 Z S1 Z Tn dτn dσn . . . dτ1 dσ1 γλ∗ (τ1 , k1 ) · · · γλ∗ (τn , kn )γλ (σn , kn ) · · · γλ (σ1 , k1 ) tn sn t1 s1 (7.1)

converges to the inner product on z0 in T2 . We proceed precisely as in the proof of Proposition 5.5. Here we are not very explicit, because we have been explicit there. We consider matrix elements of (7.1) with Schwartz functions ξ, ζ. The q’s in the γλ ’s disappear by extensive use of Relations (2.1), however, cause some shifts to the p’s. i and after performing the p-integral we obtain the We make the substitutions ui = σiλ−τ 2 d function ξζ (u k +. . .+u k ). Its modulo is for almost all k , . . . , k and all τ , . . . , τ n n

1 1

n

1

n

1

a rapidly decreasing upper bound for the ui -integrations. Similarly, as in the proof of Proposition 5.5 one checks that the λ-limits for the ui -integrals may be performed first. We obtain the result lim hξ, hfn . . . f1 , gn . . . g1 iλ ζi = hχ[tn ,Tn ] , χ[sn ,Sn ] i · · · hχ[t1 ,T1 ] , χ[s1 ,S1 ] i Z Z Z Z e e dk1 f1 (k1 )e g1 (k1 ) du1 · · · dkn fn (kn )e gn (kn ) dun Z dp ξ(p)ζ(p)eiun ((p+kn−1 +...+k1 )·kn +ω(|kn |)) · · · eiu1 (p·k1 +ω(|k1 |))

λ→0

of [1]. Now we proceed as in Lemma 5.6 and bring the p-integration step by step to the outer position. (Take into account that after performing the integrals over p and over ui , ki


597

(i = m + 1, . . . , n) the result is still a rapidly decreasing function on ui (i = 1, . . . , m) for almost all ki (i = 1, . . . , m). Therefore, Fubini’s theorem applies.) We obtain (by the same notational use of the δ-functions) that (7.1) converges to hfn . . . f1 , gn . . . g1 i0 = (2π)n hχ[tn ,Tn ] , χ[sn ,Sn ] i · · · hχ[t1 ,T1 ] , χ[s1 ,S1 ] i Z Z gn (kn ) · · · fe1 (k1 )e g1 (k1 ) dkn . . . dk1 fen (kn )e δ (p + kn−1 + . . . + k1 ) · kn + ω(|kn |) · · · δ p · k1 + ω(|k1 |) . Theorem 7.2. Theorem 7.1 remains true, if we replace on the left-hand side the free creators and annihilators by the symmetric creators and annihilators, i.e. lim (Aλ (f1 )· · ·Aλ (fn )A+λ (gn ) · · · A+λ (g1 ))

λ→0

= lim h1, a(ϕλ (f1 )) · · · a(ϕλ (fn ))a+ (ϕλ (gn )) · · · a+ (ϕλ (g1 ))1i λ→0

= h1, `(f1 ) · · · `(fn )`+ (gn ) · · · `+ (g1 )1i0 . Proof. Notice that a+ (ϕλ (gn )) · · · a+ (ϕλ (g1 ))1 =

√ n!P `+ (ϕλ (gn )) · · · `+ (ϕλ (g1 ))1.

Therefore, we are ready, if we show that in the sum over the permutations only the identity permutation contributes to the limit of the inner product. Applying the flip to two neighbouring elements ϕλ (gi+1 ) ϕλ (gi ) means exchanging the arguments ki+1 ↔ ki . (The σi are dummies and may be labeled arbitrarily.) We find

F (ϕλ (gi+1 ) ϕλ (gi ))

Z

Z

Si

Si+1

dσi+1 si

(ki+1 , ki ) = dσi ei

σi −σi+1 λ2

ki+1 ·ki

γλ (σi+1 , ki+1 )γλ (σi , ki )e gi (ki+1 )e gi+1 (ki ).

si+1

This differs only by the oscillating factor ei

σi −σi+1 λ2

ki+1 ·ki

from the expression

ϕλ (χ[si+1 ,Si+1 ] gei ) ϕλ (χ[si ,Si ] gei+1 ) whose inner products are known to have finite limits. This oscillating factor cannot be neutralized by any other flip operation on a different pair of neighbours. Assume, for instance, for a certain permutation π that i is the first position, counting from the right, which is changed by π. Then π may be written in the form π 0 F (i,i+1) π 00 , where F (i,i+1) is the flip of positions i and i + 1 and π 0 , π 00 are permutations involving only the positions i + 1, . . . , n. A look at the concrete form of the exponents in the oscillating factors tells us that the oscillating factor arising from F (i,i+1) will be neutralized at most on a null-set for the kj -σj -integrations (j = i + 1, . . . , n). Therefore, any non-identical permutation does not contribute to the sum over all permutations. (Notice that also here for a proper argument the theorem of dominated convergence is involved.)

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Central Limit Theorem 7.3. Theorem 7.2 remains true, if we replace on the left-hand side Aλ (f1 ) · · · Aλ (fn )A+λ (gn ) · · · A+λ (g1 ) by an arbitrary monomial in collective creators and annihilators and on the right-hand side `(f1 ) · · · `(fn )`+ (gn ) · · · `+ (g1 ) by the corresponding monomial in the free creators and annihilators. In other words, we expressed the limit of arbitrary moments of collective operators in the vacuum conditional expectation as the moments of the corresponding free operators in the vacuum expectation on the limit full Fock module. Proof. We will show that, in a certain sense, Aλ (f )A+λ (g) → hf, gi0 for λ → 0; cf. Relations (6.1). Indeed, one easily checks that [a(f )a+ (g)F ](kn , . . . , k1 ) = hf, giF (kn , . . . , k1 ) Z n X f ∗ (k)g(ki )F (k, kn , . . . , kbi , . . . , k1 ) + dk i=1

for f, g ∈ Cc (R , B) . Replacing f, g by ϕλ (f ), ϕλ (g) (f, g ∈ V ), the first summand converges precisely to what we want, namely, hf, gi0 F . In the remaing sum we may, like in the proof of Theorem 7.2, exchange the position of f ∗ and g. This produces an oscillating factor which makes the k-integral disappear in the limit. It has to be shown that in a concrete expression, e.g. like Aλ (f1 ) · · · Aλ (fn )A+λ (gn ) · · · A+λ (gm+1 )Aλ (f )A+λ (g)A+λ (gm ) · · · A+λ (g1 ) = d

s

h1, a(ϕλ (f1 )) · · · a(ϕλ (fn ))a+ (ϕλ (gn )) · · · a+ (ϕλ (gm+1 )) a(ϕλ (f ))a+ (ϕλ (g))a+ (ϕλ (gm )) · · · a+ (ϕλ (g1 ))1i, the limit a(ϕλ (f ))a+ (ϕλ (g)) → hf, gi0 for the inner pairing may be computed first. But this follows in the usual way using arguments involving the theorem of dominated convergence and the Riemann-Lebesgue lemma. Remark 7.4. It is possible to extend the preceding results in an obvious manner to elements f in the B0 -generate of V . This means that the moments of both A+ (f ) and `+ (f ) for finite λ converge to the moments of `+ (f ) on zB(S) (A0 (R)⊗E). By a slight weakening of Definition 5.1 in the sense that the generating set needs only to be topologically generating, one can show that lim zB (Eλ ) = zB (A0 (R) ⊗ E) and more or less also λ→0

lim 0B (Cc (Rd , B)s ) = zB (A0 (R) ⊗ E). However, since the notational effort and a

λ→0

precise reasoning would take a lot of time, we content ourselves with the central limit theorem. Since the moments of all creators and, henceforth, the inner products on the full Fock module are already determined by Relations (6.1) (see [14, 16]), we do not really lose information on the limit module. 8. Résumé Why do we consider the preceding theorem as a type of central limit theorem? By Eq. (2.4) the collective operators are related to a time integral over the interaction Hamiltonian. They collect, so to speak, the interaction over a certain time interval. In this sense they also may be considered as a “continuous sum” over certain random variables. By the rescaling t ,→ λt2 for the stochastic limit the “number of elements” in this “sum”


599

increases like λ12 for λ → 0. Therefore, λ12 plays the role of the number n in a usual central limit theorem. This interpretation is reconfirmed by the fact that HI (t) contains λ as a factor which, therefore, plays the role of the usual normalization √1n .

Let us recall the functorial property 0B (E ⊕ F ) = 0B (E) 0B (F ) of the symmetric Fock module. It is easy to check that creators and annihilators to orthogonal elements e ∈ E and f ∈ F factorize i.e. according to this decomposition, a# (e)a# (f ) = a# (e) a# (f ) := a# (e) id F ◦ (a# (f ) id) ◦ F , where # means ∗ or not ∗. (Actually, this is true for arbitrary elements of La (0B (E)) and La (0B (F )). However, notice that in general a+ (e)a+ (f ) 6= a+ (f )a+ (e). This follows, for instance, from a+ (be)a+ (b0 f ) = bb0 a+ (e)a+ (f ), but, a+ (b0 f )a+ (be) = b0 ba+ (f )a+ (e) = b0 ba+ (e)a+ (f ) for centered elements e and f .) It seems to be a natural idea to consider the creators and annihilators to orthogonal elements as (Bose) B-independent random variables, see [18]. However, the collective operators to different time intervalls are not independent random variables, because elements of the one-particle sector to different time intervals are not necessarilly orthogonal. Thus, the analogy to the usual central limit theorems is not complete. Nevertheless, the limits `# (χ[0,t] fe) of the collective operators A#λ (χ[0,t] fe) form stochastic processes with freely B-independent (additive) increments in the sense of Speicher [20]. For the symmetric and the full Fock space there exist the quantum stochastic calculi of Hudson-Parthasarathy [5] and Kümmerer-Speicher [7], respectively. In both cases it is allowed to tensorize the Fock space R with an initial space S. We made the initial space disappear, or better, the information about the structure of the initial space had been changed into the algebra of operators on S. Of course, our Fock modules allow for a faithful representation of all processes which are represented as quantum stochastic integrals on R⊗S. (The methods in the appendix indicate how also algebras of unbounded operators can be included in our description. The only serious obstacle consists in the fact that we are restricted to adjointable mappings.) However, in the language of Fock modules a much bigger variety of operators may be identified as creator or annihilator. For instance, we have seen that the collective operators on R ⊗ S appear naturally as creators and annihilators on the symmetric Fock module. Thus, we did not only make the initial space superfluous, but also increased the number of quantum stochastic processes which may be thought of as white noises considerably. It is noteworthy that the sets of adapted processes coincide (up to algebra isomorphism). It is natural to ask, if the more general classes of creators and annihilators (and also gauge operators) on a symmetric or full Fock module allow for a quantum stochastic calculus (being more general than the calculi in [5] and [7]). We will follow this question in the future and expect from our first rudimentary investigations that it should be possible to find rich quantum stochastic calculi. Notice that our symmetric Fock module over an arbitrary centered pre-Hilbert B-B-module is different from the symmetric Fock module introduced by Lu on which he develops in [9, 10] his quantum stochastic calculus. Lu’s symmetric Fock module is only over centered modules over a commutative algebra. In this case, we can define an isometric (but not injective) embedding from his module to ours. Our module, however, is nothing but a direct integral of a family of symmetric Fock spaces over the spectrum of the commutative algebra. We will have to check, if his apparently more general notion of “essential adaptedness”, actually allows for more quantum stochastic processes. A second question is based on our notion of centered modules. We pointed out that in order to define a tensor product of B-algebras it is necessary for them to be centered. For centered modules, however, we are able to define a natural generalization

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of bialgebras to centered B-bialgebras. We consider it as an interesting question to ask, if Schürmann’s theory of “White Noise on Bialgebras” [15] can be generalized to centered B-bialgebras, using a potential calculus on the symmetric Fock module over a centered Hilbert module. We remark that presently in [3] an attempt is made to describe the limit module again in terms of creator and annihilator densities. The algebraic relations, e.g. like (5.6), are transferred into the densities and then the “algebra” generated by the densities, is described as an “algebra” generated by relations. We insist, however, to call our limit module just a “full Fock module” and not an “interacting Fock module” as is done in [3, 1, 2]. In our opinion, the name “interacting” is given to Fock-like objects, if the inner product on the n-particle sector cannot be understood as the canonical inner product of the n-fold tensor product of one and the same (including all structures) one-particle sector by itself. One of our main results says, however, that precisely this is possible. Last but not least, we make a remark on the algebra over which our modules are. We chose a suitably dense subalgebra B of the algebra B(S) of all bounded operators on the initial space. We observed that for any λ > 0 the algebra Aλ may be represented on the same symmetric Fock module over the one-particle sector L2 (Rd , B(S))s which is a centered Hilbert module. Also in view of the appendix this module appears as the equivalent description of the physical system in terms of modules. The one-particle sector is generated by the subspace V . The inner product restricted to this subspace takes values only in the momentum algebra P. Therefore, as indicated by GNS-construction or by Remark 3.9 it is possible to restrict at any time to the P-Psubmodules generated by V . However, if we do so we lose a lot. Firstly, the one-particle sector is no longer centered and depends on λ. Thus, we do not have the possibility to interpret the GNS-module as a symmetric Fock module. Moreover, the GNS-module will no longer appear as an adequate description of the physical system. Secondly, the left multiplication appears no longer as the pointwise multiplicaton in a module of functions. We have to introduce it as an a priori operation. Finally, the apparent technical advantage of a module over a commutative algebra, actually, does not exist. The limit one-particle sector is not centered. The non-commutativity can be seen easily from the commutation relations which are fulfilled by algebra elements and elements of V . There are two possible advantages of the restriction to P. Firstly, the over-countable direct sum disappears and the limit module is now separable. Secondly, P is a C ∗ -algebra. Presently, we are not in a position to see whether our results extend to the C ∗ -completion of B or not. Thus, if one insists on considering two-sided modules over C ∗ -algebras, restriction to P is a possible way out. In this case T1 is just the weak topology of L∞ (Rd ). Appendix: Connections Between Vector Spaces and Modules In this appendix we work out the algebraic basis for our purpose to represent a quite general class of operators between Hilbert spaces on Hilbert modules. In particular, we will see that this possibility does not depend essentially on the existence of an inner product. Lemma A.1. Let G, H denote vector spaces, B a subalgebra of L(G). Furthermore, assume that g0 ∈ G is cyclic for B and that B contains a rank-one projection P0 such that P0 g0 = g0 . Then the mapping L g 7−→ Lg


601

establishes an isomorphism L(G, H) G −→ H, where denotes the tensor product over B of the L(H)-B-module L(G, H) and the B-C-module G. Proof. The mapping L g 7→ Lg is surjective (and, of course, well-defined). We show P that it is also injective. Indeed, denote by Li gi an arbitrary element of L(G, H) G. i P Li γi . Then Then there exist γi ∈ B such that gi = γi g0 . Denote L = i

X i

Li g i =

X

Li γi g 0 =

X

i

Li γ i g 0 = L g0 .

i

Suppose that L g0 6= 0. Then L P0 g0 = LP0 g0 6= 0, i.e. LP0 6= 0, i.e. Lg0 6= 0. Therefore, the mapping is injective. Theorem A.2. Let H1 , H2 be vector spaces and G and B as before. For a ∈ L(H1 , H2 ) define the mapping e a ∈ Lr (L(G, H1 ), L(G, H2 )), by setting e aL = aL. Then the mapping a 7→ e a establishes an isomorphism L(H1 , H2 ) −→ Lr (L(G, H1 ), L(G, H2 )). Moreover, if H1 = H2 = H, then the mapping is also an algebra isomorphism L(H) → Lr (L(G, H)). Proof. By the preceding lemma we may identify L(G, Hi ) G with Hi . Therefore, the mapping e a id : L(G, H2 ) G → L(G, H2 ) G may be identified with a unique element in L(H1 , H2 ). Obviously, this element is a. On the other hand, the mapping A 7→ (A id) is injective on the whole of Lr (L(G, H1 ), L(G, H2 )). (Suppose A 6= 0. Then there exists L ∈ L(G, H1 ) such that AL 6= 0, i.e. ALg = AL g = (A id)(L g) 6= 0 for a suitable g ∈ G. Henceforth, A id 6= 0.) It follows that a 7→ e a is an isomorphism and A 7→ (A id) its inverse. The last statement of the theorem is obvious. Now we consider H ⊗ G rather than H itself. Notice that the module L(G, H ⊗ G) contains H ⊗ B as a submodule, if we identify an element h ⊗ b in the latter with the mapping g 7→ h ⊗ bg in the former. Theorem A.3. Any mapping A ∈ Lr (H1 ⊗B, L(G, H2 )) extends uniquely to an element of Lr (L(G, H1 ⊗ G), L(G, H2 )). Proof. Clearly, also (H1 ⊗ B) G = H1 ⊗ G. Therefore, (A id) determines a unique a is an extension of A to L(G, H1 ⊗ G). Obviously, element a of L(H1 ⊗ G, H2 ). Then e this extension is unique. Remark A.4. The result can be generalized easily to the case when G is a right C-module. In this case B is a subalgebra of Lr (G).

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In the sequel, we assume G, H, H1 , H2 to be pre-Hilbert spaces and B = La (G). We may equip La (G, H) with an La (G)-valued inner product by setting hL, M i = L∗ M . If a ∈ La (H1 , H2 ), then e a maps La (G, H1 ) into La (G, H2 ). Moreover, ae∗ is the unique adjoint mapping in Lr (La (G, H2 ), La (G, H1 )), fulfilling hL, e aM i = hae∗ L, M i. On the other hand, if we denote by La (La (G, H1 ), La (G, H2 )) the set of all elements A ∈ Lr (La (G, H1 ), La (G, H2 )), which fulfill this adjointability condition, then (A id) is an element of La (H1 , H2 ) for all A ∈ La (La (G, H1 ), La (G, H2 )). We easily find the specializations of our foregoing theorems to adjointable mappings. Theorem A.5. The mapping a 7→ e a La (G, H2 ) is an isomorphism La (H1 , H2 ) → La (La (G, H1 ), La (G, H2 )). Moreover, for H1 = H2 it is a ∗-algebra isomorphism. An element A ∈ La (H1 ⊗ La (G), La (G, H2 )) extends uniquely to an element of La (La (G, H1 ⊗ G), La (G, H2 )). e 1 is a pre-Hilbert space which contains H1 ⊗ G as a dense Corollary A.6. Suppose H e 1 ), subspace. If an element A ∈ La (H1 ⊗ La (G), La (G, H2 )) extends to La (La (G, H La (G, H2 )), then this extension is unique. Proof. If A0 is such an extension, then (A0 id) is an extension of (A id) to an element e 1 , H2 ). Since (A id) is closable, this extension is unique. of La (H e ∈ H. We observe that η = |i e ⊗ id is an element of La (G, H ⊗ G) with Let e ⊗ id. Define the conditional expectation : La (H ⊗ G) → La (G) by adjoint η ∗ = h| setting (a) = η ∗ aη. Proposition A.7. The conditional expectation may be represented as the inner product (a) = hη, e aηi.

(A.1)

In the sense of Remark 3.4 La (G, H ⊗ G) is a pre-Hilbert module over the ∗-algebra L (G) with the separating set S0 consisting of all states, which may be written as matrix elements. Then La (G, H), η is the GNS-representation of the conditional expectation defined by Eq. (A.1). We mention an interesting application of the above methods to self-dual Hilbert modules. A Hilbert B-module E is self-dual, if the analogue of the Riesz-Fréchet theorem holds, i.e. if any element in f ∈ B r (E, B) may be represented (uniquely) as an inner product hx, •i (x ∈ E). An application of Theorem A.5 in the case H1 = H and H2 = G yields La (La (G, H), a L (G)) ∼ = La (H, G). In other words, any element in La (La (G, H), La (G)) is of the form hL, •i for a unique element L ∈ La (G, H). Let us restrict Theorems A.2 and A.5 further to bounded mappings between Hilbert spaces. (The isomorphism is, of course, an isometry.) Recognizing that such mappings are adjointable, we obtain a

B r (B(G, H1 ), B(G, H2 )) ∼ = B(H1 , H2 ) ∼ = B a (B(G, H1 ), B(G, H2 )). It is easy to check that the pre-Hilbert module norm and the operator norm on E = B(G, H) coincide. Therefore E is a self-dual Hilbert module. We recover for our special


603

case Paschke’s result [12] that bounded right linear mappings between self-dual Hilbert modules are adjointable. Moreover, a right linear mapping A defined on a submodule F of E1 into E2 allows an extension to E1 , if and only if the sesquilinear form h•, A•i on E2 × F is bounded. Now we replace H by H⊗G. Then E contains HB(G) = H ⊗B(G) and its completion HB(G) . Clearly, E is the smallest self-dual Hilbert module which contains HB(G) . Now assume that H is infinite-dimensional and separable. By the absorption theorem (see e.g. [8]) any countably generated Hilbert B(G)-module F is contained in HB(G) as a complemented submodule. The projection onto this submodule is bounded, hence, extends to a projection on E. The image of E under this projection is the the smallest self-dual Hilbert module which contains F . Since B r (E) ∼ = B(H⊗G), we are in a position to translate a couple of results from theory of operators on Hilbert spaces, e.g. the spectral theorem, to bounded operators on countably generated Hilbert B(G)modules. We will come back to this possibility in more detail in the note [19] to be published later. Note added: In the meantime (see [19]) we know that an arbitrary centered Hilbert Bs B-module E is contained in H ⊗ B for a suitable Hilbert space H. If E is strongly s complete, then it is complemented in H ⊗ B . More generally, if a Hilbert B-module E is generated by elements x for which hx, xi is an element of the center of B, then E has a left multiplication which turns it into a centered Hilbert B-B-module. This left multiplication needs not to be unique. Also an arbitrary Hilbert B-module E may be s s embedded into H ⊗ B . However, the left multiplication induced from H ⊗ B needs not to leave invariant E. Finally, there are two-sided Hilbert modules which cannot be s embedded into H ⊗ B preserving also the left multiplication. We call such modules essentially non-centered. Acknowledgement. The author wants to express his gratitude to L. Accardi for the possibility to come to the “Centro Vito Volterra”. Also the financial support from the “Deutsche Forschungsgemeinschaft” is acknowledged gratefully. It would not have been possible to enter the deep paper [1] without learning its spirit in uncountable discussions directly from the authors. Several mistakes from the first version have been corrected. For a couple of remarks added to the revised version the author owes thanks to M. Schürmann, R. Speicher and, in particular, to V. Liebscher. Many valuable remarks by two unknown referees were included in the second revision.

References 1. Accardi, L., Lu, Y.G.: The Wigner semi-circle law in quantum electro dynamics. Commun. Math. Phys. 180, 605–632 (1996) 2. Accardi, L., Lu, Y.G.: Wiener noise versus Wigner noise in quantum electrodynamics. Accardi, L. (ed.), Quantum Probability & Related Topics VIII, Singapore, New Jersey, London, Hong Kong, World Scientific, 1993 3. Accardi, L., Lu, Y.G., Volovich, I.V.: The QED interacting free Fock module. Preprint, Rome, in preparation 4. Gough, J.: On the emergence of a free noise limit from quantum field theory. Preprint, Rome, 1996 5. Hudson, R.L., Parthasarathy, K.R.: Quantum Ito’s formula and stochastic evolutions. Commun. Math. Phys. 93, 301–323 (1984) 6. Kümmerer, B., Maassen, H.: The essentially commutative dilations of dynamical semigroups on Mn . Commun. Math. Phys. 109, 1–22 (1987) 7. Kümmerer, B., Speicher, R.: Stochastic integration on the Cuntz algebra O∞ . J. Funct. Anal. 103, 372–408 (1992) 8. Lance, E.C.: Hilbert C ∗ -modules. Cambridge: Cambridge University Press, 1995

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9. 10. 11. 12. 13.

Lu, Y.G.: Quantum stochastic calculus on Hilbert modules. Submitted to Math. Z. Lu, Y.G.: Quantum Poisson processes on Hilbert modules. Submitted to Ann. I.H.P. Prob. Stat. Murphy, G.J.: C ∗ -Algebras and operator theory. Academic Press, 1990 Paschke, W.L.: Inner product modules over B ∗ -algebras. Trans. Amer. Math. Soc. 182, 443–468 (1973) Petz, D.: An invitation to the algebra of canonical commutation relations. Leuven: Leuven University Press, 1990 Pimsner, M.V.: A class of C ∗ -algebras generalizing both Cuntz-Krieger algebras and crossed products by Z. Preprint, Pennsylvania 1993, to appear in: Fields Institute Communications (D.V. Voiculescu, ed.), Memoires of the American Mathematical Society Schürmann, M.: White noise on bialgebras. (Lect. Notes Math., vol. 1544), Berlin–Heidelberg–New York: Springer, 1993 Shlyakhtenko, D.: Random gaussian band matrices and freenes with amalgamation. International Mathematics Research Notices 20, 1013–1025 (1996) Skeide, M.: Infinitesimal generators on SUq (2) in the classical and anti-classical limit. Preprint, Cottbus, 1997 Skeide, M.: A note on Bose Z-independent random variables fulfilling q-commutation relations. Preprint, Rome, 1996 Skeide, M.: Generalized matrix C ∗ -algebras and representations of Hilbert modules. Preprint, Cottbus, 1997 Speicher, R.: Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Habilitation, Heidelberg, 1994, to appear in Memoirs of the American Mathematical Society Voiculescu, D.: Dual algebraic structures on operator algebras related to free products. J. Operator Theory 17, 85–98 (1987) Voiculescu, D.: Operations on certain non-commutative operator-valued random variables. Preprint, Berkely 1992 Yosida, K.: Functional analysis. Springer, Berlin Heidelberg New York 1980

14.

15. 16. 17. 18. 19. 20.

21. 22. 23.

Communicated by H. Araki

Commun. Math. Phys. 192, 605 – 629 (1998)

Communications in


Drinfeld–Sokolov Reduction for Difference Operators and Deformations of W -Algebras I. The Case of Virasoro Algebra E. Frenkel1,? , N. Reshetikhin2 , M.A. Semenov-Tian-Shansky3,4 1 2 3 4

Department of Mathematics, Harvard University, Cambridge, MA 02138, USA Department of Mathematics, University of California, Berkeley, CA 94720, USA Université de Bourgogne, Dijon, France Steklov Mathematical Institute, St. Petersburg, Russia

Received: 20 April 1997 / Accepted: 22 July 1997

Abstract: We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-deformations of the classical W-algebras by reduction from PoissonLie loop groups. We consider in detail the case of SL2 . The nontrivial consistency conditions fix the choice of the classical r-matrix defining the Poisson-Lie structure on the loop group LSL2 , and this leads to a new elliptic classical r-matrix. The reduced Poisson algebra coincides with the deformation of the classical Virasoro algebra previously defined in [19]. We also consider a discrete analogue of this Poisson algebra. In the second part [31] the construction is generalized to the case of an arbitrary semisimple Lie algebra.

1. Introduction It is well-known that the space of ordinary differential operators of the form ∂ n +u1 ∂ n−2 + . . . + un−1 has a remarkable Poisson structure, often called the (second) Adler-GelfandDickey bracket [1, 12]. Drinfeld–Sokolov reduction [11] gives a natural realization of this Poisson structure via the hamiltonian reduction of the dual space to the affine Kacb n . Drinfeld and Sokolov [11] have applied an analogous reduction Moody algebra sl procedure to the dual space of the affinization b g of an arbitrary semisimple Lie algebra g. The Poisson algebra W(g) of functionals on the corresponding reduced space is called the classical W-algebra. Thus, one can associate a classical W-algebra to an arbitrary semisimple Lie algebra g. In particular, the classical W-algebra associated to sl2 is nothing but the classical Virasoro algebra, i.e., the Poisson algebra of functionals on the dual space to the Virasoro algebra (see, e.g., [19]). It is interesting that W(g) admits another description as the center of the universal enveloping algebra of an affine algebra. More precisely, let Z(b g)−h∨ be the center of a ?

Packard Fellow

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E. Frenkel, N. Reshetikhin, M.A. Semenov-Tian-Shansky

completion of the universal enveloping algebra U (b g)−h∨ at the critical level k = −h∨ (minus the dual Coxeter number). This center has a canonical Poisson structure. It was conjectured by V. Drinfeld and proved by B. Feigin and E. Frenkel [14, 18] that as the Poisson algebra Z(b g)−h∨ is isomorphic to the classical W-algebra W(Lg) associated with the Langlands dual Lie algebra Lg of g. In [19] two of the authors used this second realization of W-algebras to obtain their q-deformations. For instance, the q-deformation Wq (sln ) of W(sln ) was defined b n ) of a completion of the quantized universal enveloping algebra as the center Zq (sl b n )−h∨ . The Poisson structure on Zq (sl b n ) was explicitly described in [19] using Uq (sl b n ) = Wq (sln ) results of [26]. It was shown that the underlying Poisson manifold of Zq (sl n n−1 is the space of q-difference operators of the form Dq + t1 Dq + . . . + tn−1 Dq + 1. Furthermore, in [19] a q-deformation of the Miura transformation, i.e., a homomorphism from Wq (sln ) to a Heisenberg-Poisson algebra, was defined. The construction [19] of Wq (sln ) was followed by further developments: it was quantized [32, 15, 4] and the quantum algebra was used in the study of lattice models [25, 3]; the Yangian analogue of Wq (sl2 ) was considered in [8]; q-deformations of the generalized KdV hierarchies were introduced [17]. In this paper we first formulate the results of [19] in terms of first order q-difference operators and q-gauge action. This naturally leads us to a generalization of the DrinfeldSokolov scheme to the setting of q-difference operators. The initial Poisson manifold is the loop group LSLn of SLn , or more generally, the loop group of a simply-connected simple Lie group G. Much of the needed Poisson formalism has already been developed by one of the authors in [29, 30]. Results of these works allow us to define a Poisson structure on the loop group, with respect to which the q-gauge action is Poisson. We then have to perform a reduction of this Poisson manifold with respect to the q-gauge action of the loop group LN of the unipotent subgroup N of G. At this point we encounter a new kind of anomaly in the Poisson bracket relations, unfamiliar from the linear, (i.e., undeformed), situation. To describe it in physical terms, recall that the reduction procedure consists of two steps: (1) imposing the constraints and (2) passing to the quotient by the gauge group. An important point in the ordinary Drinfeld–Sokolov reduction is that these constraints are of first class, according to Dirac, i.e., their Poisson bracket vanishes on the constraint surface. In the q-difference case we have to choose carefully the classical r-matrix defining the initial Poisson structure on the loop group so as to make all constraints first class. If we use the standard r-matrix, some of the constraints are of second class, and so we have to modify the r-matrix. In this paper we do that in the case of SL2 . We show that there is essentially a unique classical r-matrix compatible with the q-difference Drinfeld–Sokolov scheme. To the best of our knowledge, this classical r-matrix is new; it yields an elliptic deformation of the Lie bialgebra structure on the loop algebra of sl2 associated with the Drinfeld “new” realization of quantized affine algebras [10, 22]. The result of the correspindingDrinfeld– Sokolov reduction is the q-deformation of the classical Virasoro algebra defined in [19]. We also construct a finite difference version of the Drinfeld–Sokolov reduction in the case of SL2 . This construction gives us a discrete version of the (classical) Virasoro algebra. We explain in detail the connection between our discrete Virasoro algebra and the lattice Virasoro algebra of Faddeev–Takhtajan–Volkov [34–36, 13]. We hope that our results will help to clarify further the meaning of the discrete Virasoro algebra and its relation to various integrable models.

Drinfeld–Sokolov Reduction for Difference Operators I.

607

The construction presented here can be generalized to the case of an arbitrary simplyconnected simple Lie group. This is done in the second part of the paper [31] written by A. Sevostyanov and one of us. The paper is arranged as follows. In Sect. 2 we recall the relevant facts of [11] and [19]. In Sect. 3 we interpret the results of [19] from the point of view of q-gauge transformations. Section 4 reviews some background material on Poisson structures on Lie groups following [29, 30]. In Sect. 5 we apply the results of Sect. 4 to the q-deformation of the Drinfeld–Sokolov reduction in the case of SL2 . In Sect. 6 we discuss the finite difference analogue of this reduction and compare its results with the Faddeev–Takhtajan–Volkov algebra. 2. Preliminaries 2.1. The differential Drinfeld–Sokolov reduction in the case of sln . Let Mn be the manifold of differential operators of the form L = ∂ n + u1 (s)∂ n−2 + . . . + un−2 (s)∂ + un−1 (s),

(2.1)

where ui (s) ∈ C((s)). Adler [1] and Gelfand-Dickey [12] have defined a remarkable two-parameter family of Poisson structures on Mn , with respect to which the corresponding KdV hierarchy is hamiltonian. In this paper we will only consider one of them, the so-called second bracket. There is a simple realization of this structure in terms of the Drinfeld–Sokolov reduction [11], Sect. 6.5. Let us briefly recall this realization. b n associated to sln ; this is the central Consider the affine Kac-Moody algebra sl extension b n → Lsln → 0, 0 → CK → sl b n , which consists of linear see [20]. Let Mn be the hyperplane in the dual space to sl functionals taking value 1 on K. Using the differential dt and the bilinear form tr AB on sln , we identify Mn with the manifold of first order differential operators ∂s + A(s),

A(s) ∈ Lsln .

b ∗ factors through the loop group LSLn c n on sl The coadjoint action of the Lie group SL n and preserves the hyperplane Mn . The corresponding action of g(s) ∈ LSLn on Mn is given by g(s) · (∂s + A(s)) = g(s)(∂s + A(s))g(s)−1 , or

(2.2)

A(s) 7→ g(s)A(s)g(s)−1 − ∂s g(s) · g(s)−1 .

Consider now the submanifold MnJ of Mn which consists of operators ∂s + A(s), where A(s) is a traceless matrix of the form   ∗ ∗ ∗ ... ∗ ∗ −1 ∗ ∗ . . . ∗ ∗    0 −1 ∗ . . . ∗ ∗ (2.3) . . . . . . . . . . . . . . . . . .    0 0 0 . . . ∗ ∗ 0 0 0 . . . −1 ∗

608


To each element L of MnJ one can naturally attach an nth order scalar differential operator as follows. Consider the equation L · 9 = 0, where   9n  9 9 =  n−1  . ... 91 Due to the special form (2.3) of L, this equation is equivalent to an nth order differential equation L·91 = 0, where L is of the form (2.1). Thus, we obtain a map π : MnJ → Mn sending L to L. Let N be the subgroup of SLn consisting of the upper triangular matrices, and LN be its loop group. If g ∈ LN and 9 is a solution of L · 9 = 0, then 90 = g9 is a solution of L0 · 90 = 0, where L0 = gLg −1 . But 91 does not change under the action of LN . Therefore π(L0 ) = π(L), and we see that π factors through the quotient of MnJ by the action of LN . The following proposition describes this quotient. Proposition 1 ([11], Proposition 3.1). The action of LN on MnJ is free, and each orbit contains a unique operator of the form   0 u1 u2 . . . un−2 un−1 −1 0 0 . . . 0 0    0   0 −1 0 . . . 0 (2.4) ∂s +  . . . . . . . . . . . . . . . . . . . . . . . .    0 0 0 ... 0 0 0 0 0 . . . −1 0 But for L of the form (2.4), π(L) is equal to the operator L given by formula (2.1). Thus, we have identified the map π with the quotient of MnJ by LN and identified Mn with MnJ /LN . The quotient MnJ /LN can actually be interpreted as the result of hamiltonian reduction. Denote by n+ (resp., n− ) the upper (resp., lower) nilpotent subalgebra of sln ; thus, n+ is the Lie algebra of N . The manifold Mn has a canonical Poisson structure, which is the restriction of the b ∗ (such a structure exists on the dual space to any Lie algebra). Lie-Poisson structure on sl n The coadjoint action of LN on Mn is hamiltonian with respect to this structure. The corresponding moment map µ : Mn → Ln− ' Ln∗+ sends ∂s + A(s) to the lowertriangular part of A(s). Consider the one-point orbit of LN ,   0 0 0 ... 0 0 −1 0 0 . . . 0 0    0 −1 0 . . . 0 0 J = . . . . . . . . . . . . . . . . . .   0 0 0 . . . 0 0 0 0 0 . . . −1 0 Then MnJ = µ−1 (J). Hence Mn is the result of hamiltonian reduction of Mn by LN with respect to the one-point orbit J. The Lie-Poisson structure on Mn gives rise to a canonical Poisson structure on Mn , which coincides with the second Adler-Gelfand-Dickey bracket, see [11], Sect. 6.5. The Poisson algebra of local functionals on Mn is called the classical W-algebra associated to sln , and is denoted by W(sln ).


609

b n , which Remark 1. For α ∈ C, let Mα,n be the hyperplane in the dual space to sl b consists of linear functionals on sln taking the value α on K. In the same way as above (for α = 1) we identify Mα,n with the space of first order differential operators α∂s + A(s),

A(s) ∈ Lsln .

The coadjoint action is given by the formula A(s) 7→ g(s)A(s)g(s)−1 − α∂s g(s) · g(s)−1 . The straightforward generalization of Proposition 1 is true for any α ∈ C. In particular, for α = 0 we obtain a description of the orbits in MnJ under the adjoint action of LN . This result is due to B. Kostant [23]. Drinfeld and Sokolov [11] gave a generalization of Proposition 1 when sln is replaced by an arbitrary semisimple Lie algebra g. The special case of their result, corresponding to α = 0, is also due to Kostant [23]. The Drinfeld–Sokolov reduction can be summarized by the following diagram: MnJ

? Mn =

MnJ /LN

-

? - Mn /LN

Mn

There are three essential properties of the Lie-Poisson structure on Mn that make the reduction work: (i) The coadjoint action of LSLn on Mn is hamiltonian with respect to this structure. (ii) The subgroup LN of LSLn is admissible in the sense that the space S of LN invariant functionals on Mn is a Poisson subalgebra of the space of all functionals on Mn . (iii) Denote by µij the function on Mn , whose value at ∂ + A ∈ Mn equals the (i, j) entry of A. The ideal in S generated by µij + δi−1,j , i > j, is a Poisson ideal. We will generalize this picture to the q-difference case. 2.2. The Miura transformation. Let Fn be the manifold of differential operators of the form   v1 0 0 . . . 0 0 −1 v2 0 . . . 0 0     0 −1 v3 . . . 0 0  (2.5) ∂s +  , . . . . . . . . . . . . . . . . . . . .   0 0 0 ... v  n−1 0 0 0 0 . . . −1 vn

610


Pn where i=1 vi = 0. We have a map m : Fn → Mn , which is the composition of the embedding Fn → MnJ and the projection π : MnJ → Mn . Using the definition of π above, m can be described explicitly as follows: the image of the operator (2.5) under m is the nth order differential operator ∂sn + u1 (s)∂sn−2 + . . . + un−1 (s) = (∂s + v1 (s)) . . . (∂s + vn (s)). The map m is called the Miura transformation. We want to describe the Poisson structure on Fn with respect to which the Miura transformation is Poisson. To this end, let us consider the restriction of the gauge action (2.2) to the opposite triangular subgroup LN− ; let µ : Mn → Ln+ ' Ln∗− be the corresponding moment map. The manifold Fn is the intersection of two level surfaces, Fn = µ−1 (J) ∩ µ−1 (0). It is easy to see that it gives a local cross-section for both actions (in other words, the orbits of LN and LN− are transversal to Fn ). Hence Fn simultaneously provides a local model for the reduced spaces Mn = µ−1 (J)/LN and µ−1 (0)/LN− . The Poisson bracket on Fn that we need to define is the so-called Dirac bracket (see, e.g., [16]), where we may regard the matrix coefficients of µ as subsidiary conditions, which fix the local gauge. The computation of the Dirac bracket for the diagonal matrix coefficients vi is very simple, since their Poisson brackets with the matrix Pn coefficients of µ all vanish on Fn . The only correction arises due to the constraint i=1 vi = 0. Denote by vi,m the linear functional on Fn , whose value on the operator (2.5) is the mth Fourier coefficient of vi (s). We obtain the following formula for the Dirac bracket on Fn : n−1 mδm,−k , n 1 {vi,m , vj,k } = − mδm,−k , i < j. n {vi,m , vi,k } =

Since Fn and Mn both are models of the same reduced space, we immediately obtain: Proposition 2 ([11], Proposition 3.26). With respect to this Poisson structure the map m : Fn → Mn is Poisson. 2.3. The q-deformations of W(sln ) and Miura transformation. In this section we summarize relevant results of [19]. Let q be a non-zero complex number, such that |q| < 1. Consider the space Mn,q of q-difference operators of the form L = Dn + t1 (s)Dn−1 + . . . + tn−1 (s)D + 1,

(2.6)

where ti (s) ∈ C((s)) for each i = 1, . . . , n, and [D · f ](s) = f (sq). Denote by ti,m the functional on Mn,q , whose value at L is the mth Fourier coefficient of ti (s). Let Rn,q be the completion of the ring of polynomials in ti,m , i = 1, . . . , N − 1; m ∈ Z, which consists of finite linear combinations of expressions of the form X c(m1 , . . . , mk ) · ti1 ,m1 . . . tik ,mk , (2.7) m1 +...+mk =M


611

where c(m1 , . . . , mk ) ∈ C. Given an operator of the form (2.6), we can substitute the coefficients ti,m into an expression like (2.7) and get a number. Therefore elements of Rn,q define functionals on the space Mn,q . In order to define the Poisson structure on Mn,q , it suffices to specify the Poisson brackets between the generators ti,m . Let Ti (z) be the generating series of the functionals ti,m : X Ti (z) = ti,m z −m . m∈Z

We define the Poisson brackets between ti,m ’s by the formulas [19] {Ti (z), Tj (w)} =

X w m (1 − q im )(1 − q m(N −j) ) Ti (z)Tj (w) z 1 − q mN

m∈Z

wq r Ti−r (w)Tj+r (z) z r=1 min(i,N X−j) w Ti−r (z)Tj+r (w), i ≤ j. δ − zq j−i+r r=1 P In these formulas δ(x) = m∈Z xm , and we use the convention that t0 (z) ≡ 1. +

min(i,N X−j)

δ

(2.8)

Remark 2. Note the difference between ti (s) and Ti (z). The former is a Laurent power series, whose coefficients are numbers. The latter is a power series infinite in both directions, whose coefficients are functionals on Mn,q . Thus, Ti (z) is just the generating function for the functionals ti,m . We use these generating functions merely to simplify our formulas for the Poisson brackets (so that we do not have to write a Poisson bracket between each individual pair ti,m and tj,k ). Remark 3. The parameter q in formula (2.8) corresponds to q −2 in the notation of [19]. Now consider the space Fn,q of n-tuples of q-difference operators (D + λ1 (s), . . . , D + λn (s)).

(2.9)

Denote by λi,m the functional on Mn,q , whose value is the mth Fourier coefficient of λi (s). We will denote by 3i (z) the generating series of the functionals λi,m : X 3i (z) = λi,m z −m . m∈Z

We define a Poisson structure on Fn,q by the formulas [19]: X w m (1 − q m )(1 − q m(N −1) ) 3i (z)3i (w), z 1 − q mN (2.10) m∈Z X wq N −1 m (1 − q m )2 3i (z)3j (w), i < j. {3i (z), 3j (w)} = − z 1 − q mN (2.11) m∈Z {3i (z), 3i (w)} =

Now we define the q-deformation of the Miura transformation as the map mq : Fn,q → Mn,q , which sends the n-tuple (2.9) to

612


L = (D + λ1 (s))(D + λ2 (sq −1 )) . . . (D + λn (sq −n+1 )),

(2.12)

i.e. X

ti (s) =

λj1 (s)λj2 (sq −1 ) . . . λji (sq −i+1 ).

(2.13)

j1 1, there exists g(s) ∈ LN , such that −1. We will prove that given A(s) ∈ Mn,q −1 α−1 n J = Mn,q , g(sq)A(s)g(s) ∈ Mn,q . Since the condition is vacuous for α = n, i.e. Mn,q J this will imply that each LN -orbit in Mn,q contains an element of the form (3.2). To prove the statement for a given α, we will recursively eliminate all entries of the αth row of A(s) (except the (α, α − 1) entry), from right to left using elementary unipotent matrices. Denote by Ei,j (x) the upper unipotent matrix whose only non-zero entry above the diagonal is the (i, j) entry equal to x. At the first step, we eliminate the (α, n) entry Aα,n of A(s) by applying the q-gauge transformation (3.1) with g(s) = α , Eα−1,n (−Aα,n (s)). Then we obtain a new matrix A0 (s), which still belongs to Mn,q but whose (α, n) entry is equal to 0. Next, we apply the q-gauge transformation by Eα−1,n−1 (−A0α,n−1 (s)) to eliminate the (α, n − 1) entry of A0 (s), etc. It is clear that at each step we do not spoil the entries that have already been set to 0. The product of the elementary unipotent matrices constructed at each step gives us an element g(s) ∈ LN α−1 . with the desired property that g(sq)A(s)g(s)−1 ∈ Mn,q To complete the proof, it suffices to remark that if A(s) and A0 (s) are of the form (3.2), and g(sq)A(s)g(s)−1 = A0 (s) for some g(s) ∈ LN , then A(s) = A0 (s) and g(s) = 1. For L of the form (2.4), πq (L) equals the operator L given by formula (2.6). Thus, we J J have identified the map πq with the quotient of Mn,q by LN and Mn,q with Mn,q /LN . Remark 4. In the same way as above we can prove the following more general statement. Let R be a ring with an automorphism τ . It rives rise to an automorphism of SLn (R) J as the set of elements of SLn (R) of the denoted by the same character. Define Mτ,n J (R) by the formula g · A = (τ · g)Ag −1 . form (2.3). Let the group N (R) act on Mτ,n Then this action of N (R) is free, and the quotient is isomorphic to the set MJτ,n (R) of elements of SLn (R) of the form (3.2) (i.e. each orbit contains a unique element of the form (3.2)). Note that the proof is not sensible to whether τ = Id or not. When τ = Id, this result is well-known. It gives the classical normal form of a linear operator. Moreover, in that case R. Steinberg has proved that the subset MJId,n (K) of SLn (K), where K is an algebraically closed field, is a cross-section of the collection of regular conjugacy classes in SLn (K) [33], Theorem 1.4. Steinberg defined an analogous cross-section for any simply-connected semisimple algebraic group [33]. His results can be viewed as group analogues of Kostant’s results on semisimple Lie algebras [23] (cf. Remark 1). Steinberg’s cross-section is used in the definition of the discrete Drinfeld– Sokolov reduction in the general semisimple case (see [31]).1 1

We are indebted to B. Kostant for drawing our attention to [33]

614


3.2. Deformed Miura transformation via q-gauge action. Let us attach to each element of Fn,q the q-difference operator 

 λ1 (s) 0 ... 0 0 −1 0 0  −1 λ2 (sq ) . . .    3 = D + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,  0  0 0 . . . λn−1 (sq −n+2 ) −n+1 ) 0 0 ... −1 λn (sq

(3.3)

Qn where i=1 λi (sq −i+1 ) = 1. J e q : Fn,q → Mn,q be the composition of the embedding Fn,q → Mn,q and Let m J J πq : Mn,q → Mn,q /LN ' Mn,q . Using the definition of πq above, one easily finds e q (3) is the operator (3.2), where ti (s) is given by formula that for 3 given by (3.3), m (2.13). Therefore we obtain e q coincides with the q-deformed Miura transformation mq . Lemma 2. The map m Remark 5. Let G be a simply-connected semisimple algebraic group over C. Let Vi be the ith fundamental representation of G (in the case G = SLn , Vi = 3i Cn ), and χi : G → C be the corresponding character, χi (g) = Tr(g, Vi ). Define a map p : G → Cn by the formula p(g) = (χ1 (g), . . . , χn (g)). By construction, p is constant on conjugacy classes. In the case G = SLn the map p has a cross-section r : Cn → SLn (C): 

 a1 a2 a3 . . . an−1 1 −1 0 0 . . . 0 0    0 −1 0 . . . 0 0 (a1 , . . . , an ) 7→  . . . . . . . . . . . . . . . . . . . .  0 0 0 . . . 0 0 0 0 0 . . . −1 0 J e 1 can coincides with the map π1 . Moreover, m The composition r ◦ p, restricted to Mn,1 be interpreted as the restriction of p to the subset of SLn consisting of matrices of the e 1 sends (λ1 , . . . , λn ) to the elementary symmetric polynomials form (3.3). Hence m

ti =

X

λj 1 λj 2 . . . λ j i ,

j1 j, is a Poisson ideal. Geometrically, the last condition means that Mn,q is a Poisson submanifold of the quotient Mn,q /LN . For the sake of completeness, we recall the notions mentioned above. Let M be a Poisson manifold, and H be a Lie group, which is itself a Poisson manifold. An action of H on M is called Poisson if H × M → M is a Poisson map (here we equip H × M with the product Poisson structure). In particular, if the multiplication map H × H → H is Poisson, then H is called a Poisson-Lie group. In this section we describe the general formalism concerning problems (i)–(iii) above. Then in the next section we specialize to M2,q and give an explicit solution of these problems. 4.2. Lie bialgebras. Let g be a Lie algebra. Recall [9] that g is called a Lie bialgebra, if g∗ also has a Lie algebra structure, such that the dual map δ : g → 32 g is a one-cocycle. We will consider factorizable Lie bialgebras (g, δ) satisfying the following conditions:

616


(1) There exists a linear map r+ : g∗ → g, such that both r+ and r− = −r+∗ are Lie algebra homomorphisms. (2) The endomorphism t = r+ − r− is g-equivariant and induces a linear isomorphism g∗ → g. Instead of the linear operator r+ ∈ Hom(g∗ , g) one often considers the corresponding element r of g⊗2 (or a completion of g⊗2 if g is infinite-dimensional). The element r (or its image in the tensor square of a particular representation of g) is called a classical r-matrix. It satisfies the classical Yang-Baxter equation: [r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ] = 0.

(4.1)

In terms or r, δ(x) = [r, x], ∀x ∈ g (here [a ⊗ b, x] = [a, x] ⊗ b + a ⊗ [b, x]). The maps r± : g∗ → g are given by the formulas: r+ (y) = (y ⊗ id)(r), r− (y) = −(id ⊗y)(r). Property (2) above means that r + σ(r), where σ(a ⊗ b) = b ⊗ a is a non-degenerate g-invariant symmetric bilinear form on g∗ . Set g± = Im(r± ). Property (1) above implies that g± ⊂ g is a Lie subalgebra. The following statement is essentially contained in [5] (cf. also [28]). Lemma 3. Let (g, g∗ ) be a factorizable Lie bialgebra. Then (1) The subspace n± = r± (Ker r∓ ) is a Lie ideal in g± . (2) The map θ : g+ /n+ → g− /n− which sends the residue class of r+ (X), X ∈ g∗ , modulo n+ to that of r− (X) modulo n− is a well-defined isomorphism of Lie algebras. (3) Let d = g ⊕ g be the direct sum of two copies of g. The map i : g∗ → d,

X 7→ (r+ (X), r− (X))

is a Lie algebra embedding; its image g∗ ⊂ d is g∗ = {(X+ , X− ) ∈ g+ ⊕ g− ⊂ d|X − = θ(X + )}, where Y ± = Y mod n± . Remark 6. The connection between our notation and that of [29] is as follows: the operator r ∈ End g of [29] coincides with the composition of r+ + r− up to the isomorphism t = r+ − r− : g∗ → g; the bilinear form used in [29] is induced by t. 4.3. Poisson-Lie groups and gauge transformations. Let (G, η) (resp., (G∗ , η ∗ )) be a Poisson-Lie group with factorizable tangent Lie bialgebra (g, δ) (resp., (g∗ , δ ∗ )). Let G± and N± be the Lie subgroups of G corresponding to the Lie subalgebras g± and n± . We denote by the same symbol θ the isomorphism G+ /N+ → G− /N− induced by θ : g+ /n+ → g− /n− . Then the group G∗ is isomorphic to {(g+ , g− ) ∈ G+ × G− |θ(g + ) = g − }, and we have a map i : G∗ → G given by i((g+ , g− )) = g+ (g− )−1 . Explicitly, the Poisson bracket on (G, η) can be written as follows: {ϕ, ψ} = hr, ∇ϕ ∧ ∇ψ − ∇0 ϕ ∧ ∇0 ψi, where for x ∈ G, ∇ϕ(x), ∇0 ϕ(x) ∈ g∗ are defined by the formulas:

(4.2)


617

d ϕ etξ x |t=0 , dt d 0 h∇ ϕ(x), ξi = ϕ xetξ |t=0 , dt h∇ϕ(x), ξi =

(4.3) (4.4)

for all ξ ∈ g. An analogous formula can be written for the Poisson bracket on (G∗ , η ∗ ). In formula (4.2) we use the standard notation a ∧ b = (a ⊗ b − b ⊗ a)/2. By definition, the action of G on itself by left translations is a Poisson group action. There is another Poisson structure η∗ on G which is covariant with respect to the adjoint action of G on itself and such that the map i : (G∗ , η ∗ ) → (G, η∗ ) is Poisson. It is given by the formula {ϕ, ψ} = hr, ∇ϕ ∧ ∇ψ + ∇0 ϕ ∧ ∇0 ψi − hr, ∇0 ϕ ⊗ ∇ψ − ∇0 ψ ⊗ ∇ϕi.

(4.5)

Proposition 4. (1) The map i : G∗ → G is a Poisson map between the Poisson manifolds (G∗ , η ∗ ) and (G, η∗ ); (2) The Poisson structure η∗ on G is covariant with respect to the adjoint action, i.e. the map (G, η) × (G, η∗ ) → (G, η∗ ) : (g, h) 7→ ghg −1 is a Poisson map. These results are proved in [29], § 3 (see also [30], § 2), using the notion of the Heisenberg double of G. Formula (4.5) can also be obtained directly from the explicit formulas for the Poisson structure η ∗ and for the embedding i. More generally, let τ be an automorphism of G, such that the corresponding automorphism of g satisfies (τ ⊗ τ )(r) = r. Define a twisted Poisson structure η∗τ on G by the formula {ϕ, ψ} = hr, ∇ϕ ∧ ∇ψ + ∇0 ϕ ∧ ∇0 ψi 0

(4.6) 0

− h(τ ⊗ id)(r), ∇ ϕ ⊗ ∇ψ − ∇ ψ ⊗ ∇ϕi, and the twisted adjoint action of G on itself by the formula g · h = τ (g)hg −1 . Theorem 1. The Poisson structure η∗τ on G is covariant with respect to the twisted adjoint action, i.e. the map (G, η) × (G, η∗τ ) → (G, η∗τ ) : (g, h) 7→ τ (g)hg −1 is a Poisson map. This result was proved in [29], § 3 (see also [30], § 2), using the notion of the twisted Heisenberg double of G. We will use Theorem 1 in two cases. In the first, G is the loop group of a finite-dimensional simple Lie group G, and τ is the automorphism g(s) → Z/N Z g(sq), q ∈ C× . In the second, G = G , and τ is the automorphism (τ (g))i → gi+1 . In the first case twisted conjugations coincide with q-gauge transformations, and in the second case they coincide with lattice gauge transformations. 4.4. Admissibility and constraints. Let M be a Poisson manifold, G a Poisson Lie group and G × M → M be a Poisson action. A subgroup H ⊂ G is called admissible if the space C ∞ (M )H of H-invariant functions on M is a Poisson subalgebra in the space C ∞ (M ) of all functions on M .

618


Proposition 5 ([29], Theorem 6). Let (g, g∗ ) be the tangent Lie bialgebra of G. A connected Lie subgroup H ⊂ G with Lie algebra h ⊂ g is admissible if h⊥ ⊂ g∗ is a Lie subalgebra. In particular, G itself is admissible. Note that H ⊂ G is a Poisson subgroup if and only if h⊥ ⊂ g∗ is an ideal; in that case the tangent Lie bialgebra of H is h, g∗ /h⊥ . Let H ⊂ G be an admissible subgroup, and I be a Poisson ideal in C ∞ (M )H , i.e. I is an ideal in the ring C ∞ (M )H , and {f, g} ∈ C ∞ (M )H for all f ∈ I, g ∈ C ∞ (M )H . Then C ∞ (M )H /I is a Poisson algebra. More geometrically, the Poisson structure on C ∞ (M )H /I can be described as follows. Assume that the quotient M/H exists as a smooth manifold. Then there exists a Poisson structure on M/H such that the canonical projection π : M → M/H is a Poisson map. Hamiltonian vector fields ξϕ , ϕ ∈ π ∗ C ∞ (M/H), generate an integrable distribution Hπ in T M . The following result is straightforward. Lemma 4. Let V ⊂ M be a submanifold preserved by H. Then V /H is a Poisson submanifold of M/H if and only if V is an integral manifold of Hπ . The integrality condition means precisely that the ideal I of all H-invariant functions on M vanishing on V is a Poisson ideal in C ∞ (M )H , and that C ∞ (V /H) = C ∞ (V )H = C ∞ (M )H /I. If this property holds, we will say that the Poisson structure on M/H can be restricted to V /H. V

-

? V /H

? - M/H

M

The Poisson structure on V /H can be described as follows. Let NV ⊂ T ∗ M |V be the conormal bundle of V . Clearly, T ∗ V ' T ∗ M |V /NV . Let ϕ, ψ ∈ C(V )H and V2 dϕ, dψ ∈ T ∗ M |V be any representatives of dϕ, dψ ∈ T ∗ V. Let PM ∈ T M be the Poisson tensor on M . Lemma 5. We have

{ϕ, ψ} = PM , dϕ ⊗ dψ ;

(4.7)

in particular, the right hand side does not depend on the choice of dϕ, dψ. Remark 7. In the case of Hamiltonian action (i.e. when the Poisson structure on H is trivial), one can construct submanifolds V satisfying the condition of Lemma 4 using the moment map. Although a similar notion of the nonabelian moment map in the context of Poisson group theory is also available [24], it is less convenient. The reason is that the nonabelian moment map is “less functorial” than the ordinary moment map. Namely, if G × M → M is a Hamiltonian action with moment map µG : M → g∗ , its restriction to a subgroup H ⊂ G is also Hamiltonian with moment µH = p ◦ µG (here p : g∗ → h∗ is the canonical projection). If G is a Poisson-Lie group, G∗ its dual, G × M → M a Poisson group action with moment µG : M → G∗ , and H ⊂ G a Poisson subgroup, the action of H still admits a moment map. But if H ⊂ G is only admissible, then the restricted action does not usually have a moment map. This is precisely the case which is encountered in the study of the q-deformed Drinfeld–Sokolov reduction.


619

5. The q-Deformed Drinfeld–Sokolov Reduction in the Case of SL2 In this section we apply the general results of the previous section to formulate a qanalogue of the Drinfeld–Sokolov reduction when G = SL2 . 5.1. Choice of r-matrix. Let g = Lsl2 . We would like to define a factorizable Lie bialgebra structure on g in such a way that the resulting Poisson-Lie structure η on LSL2 and the Poisson structure η∗q on M2,q satisfy the conditions (ii)–(iii) of Sect. 4. Let {E, H, F } be the standard basis in sl2 and {En , Hn , Fn } be the corresponding (topological) basis of Lsl2 = sl2 ⊗ C((s)) (here for each A ∈ sl2 we set An = A ⊗ sn ∈ Lsl2 ). Let τ be the automorphism of Lsl2 defined by the formula τ (A(s)) = A(sq) (we assume that q is generic). We have: τ · An = q n An . To be able to use Theorem 1, the rmatrix r ∈ Lsl⊗2 2 defining the Lie bialgebra structure on Lsl2 has to satisfy the condition (τ ⊗ τ )(r) = r. Hence the invariant bilinear form on Lsl2 defined by the symmetric part of r should also be τ -invariant. The Lie algebra Lsl2 has a unique (up to a non-zero constant multiple) invariant non-degenerate bilinear form, which is invariant under τ . It is defined by the formulas (En , Fm ) = δn,−m ,

(Hn , Hm ) = 2δn,−m ,

with all other pairings between the basis elements are 0. This fixes the symmetric part of the element r. Another condition on r is that the subgroup LN is admissible. According ∗ to Proposition 5, this means that Ln⊥ + should be a Lie subalgebra of Lsl2 . A natural example of r satisfying these two conditions is given by the formula: X 1 1X En ⊗ F−n + H0 ⊗ H0 + Hn ⊗ H−n . (5.1) r0 = 4 2 n>0

n∈Z

It is easy to verify that this element defines a factorizable Lie bialgebra structure on g. We remark that this Lie bialgebra structure gives rise to Drinfeld’s “new” realization of the quantized enveloping algebra associated to Lsl2 [10, 22, 21]. As we will see in the next subsection, r0 can not be used for the q-deformed Drinfeld–Sokolov reduction. However, the following crucial fact will enable us to perform the reduction. Let Lh be the loop algebra of the Cartan subalgebra h of sl2 . Lemma 6. For any ρ ∈ ∧2 Lh, r0 + ρ defines a factorizable Lie bialgebra structure on ∗ Lsl2 , such that Ln⊥ + is a Lie subalgebra of Lsl2 . The fact that r0 + ρ still satisfies the classical Yang-Baxter equation is a general property of factorizable r-matrices discovered in [5]. Lemma 6 allows us to consider the class of elements r given by the formula X 1 X En ⊗ F−n + φn,m · Hn ⊗ Hm , (5.2) r= 2 n∈Z

m,n∈Z

where φn,m + φm,n = δn,−m . The condition (τ ⊗ τ )(r) = r imposes the restriction φn,m = φn δn,−m , so that (5.2) takes the form r=

X n∈Z

where φn + φ−n = 1.

En ⊗ F−n +

1X φn · Hn ⊗ H−n , 2 n∈Z

(5.3)

620


5.2. The reduction. Recall that M2,q = LSL2 = SL2 ((s)) consists of the 2 × 2 matrices a(s) b(s) M (s) = , ad − bc = 1. (5.4) c(s) d(s) J and We want to impose the constraint c(s) = −1, i.e. consider the submanifold M2,q take its quotient by the (free) action of the group 1 x(s) LN = . 0 1

Let η be the Poisson-Lie structure on LSL2 induced by r given by formula (5.3). Let η∗q be the Poisson structure on M2,q defined by formula (4.6), corresponding to the automorphism τ : g(s) → g(sq). The following is an immediate corollary of Theorem 1, Proposition 5 and Lemma 6. Proposition 6. (1) The q-gauge action of (LSL2 , η) on (M2,q , η∗q ) given by formula g(s) · M (s) = g(sq)M (s)g(s)−1 is Poisson; (2) The subgroup LN ⊂ LSL2 is admissible. Thus, we have satisfied properties (i) and (ii) of Sect. 4. Now we have to choose the remaining free parameters φn so as to satisfy property (iii). The Fourier coefficients of the matrix elements of the matrix M (s) given by (5.4) define functions on M2,q . We will use the notation am for the mth Fourier coefficient of a(s). Let R2,q be the completion of the ring of polynomials in am , bm , cm , dm , m ∈ Z, defined in the same way as the ring Rn,q of Sect. 2.3. Let S2,q ⊂ R2,q be the subalgebra of LN -invariant functions. Denote by I be the ideal of S2,q generated by {cn +δn,0 , n ∈ Z} J (the defining ideal of M2,q ). Property (iii) means that I is a Poisson ideal of S2,q , which is equivalent to the J condition that {cn , cm } ∈ I, i.e. that if {cn , cm } vanishes on M2,q . This condition means that the Poisson bracket of the constraint functions vanishes on the constraint surface, i.e. the constraints are of first class according to Dirac. Let us compute the Poisson bracket between cn ’s. First, we list the left and right gradients for the functions an , bn , cn , dn (for this computation we only need the gradients of cn ’s, but we will soon need other gradients as well). It will be convenient for us to identify Lsl2 with its dual using the bilinear form introduced in the previous section. Note that with respect to this bilinear form the dual basis elements to En , Hn , and Fn are F−n , H−n /2, and E−n , respectively. Explicit computation gives (for shorthand, we write a for a(s), etc.): 1 1 a 0 b 0 −m −m 2 2 , ∇bm = s , ∇am = s c − 21 a d − 21 b 1 1 −m − 2 c a −m − 2 d b , ∇dm = s , ∇cm = s 0 21 c 0 21 d 1 1 a b 0 −m 0 −m − 2 b 0 2 , ∇ bm = s , ∇ am = s 0 − 21 a a 21 b 1 1 c d 0 −m − 2 d 0 ∇0 cm = s−m 2 d = s , ∇ . m 0 − 21 c c 21 d


621

Now we can compute the Poisson bracket between cn ’s using formula (4.6): {cm , ck } =

1X φn − φ−n + φn q n − φ−n q −n c−n+m cn+k . 2

(5.5)

n∈Z

J Restricting to M2,q , i.e. setting cn = −δn,0 , we obtain: J = {cm , ck }|M2,q

1X φm − φ−m + φm q m − φ−m q −m δm,−k . 2 n∈Z

This gives us the following equation on φm ’s: φm − φ−m + φm q m − φ−m q −m = 0. Together with the previous condition φm + φ−m = 1, this determines φm ’s uniquely: Theorem 2. The Poisson structure η∗q satisfies property (iii) of the q-deformed Drinfeld-Sokolov reduction if and only if φn =

1 . 1 + qn

Consider the r-matrix (5.2) with φn = (1 + q n )−1 . For this r-matrix, the Lie algebras defined in section Sect. 4 are as follows: g± = Lb∓ , n± = Ln∓ , where n+ = CE, n− = CF, b± = h ⊕ n± . We have: g± /n± ' Lh. The transformation θ on Lh induced by this r-matrix is equal to −τ . Explicitly, on the tensor product of the two 2-dimensional representations of sl2 ((t)), the r-matrix looks as follows:   t 0 0 0 φ s  0 −φ t δ t 0  s s ,  (5.6) t  0 0  0 −φ s 0 0 0 φ st where φ(x) =

1X 1 xn . 2 1 + qn n∈Z

Note that 2πφ(xq 1/2 ) coincides with the power series expansion of the Jacobi elliptic function dn (delta of amplitude). Now we have satisfied all the necessary properties on the Poisson structures and hence can perform the q-Drinfeld–Sokolov reduction of Sect. 3 at the level of Poisson algebras. In the next subsection we check that it indeed gives us the Poisson bracket J /LN . (2.14) on the reduced space M2,q = M2,q Remark 8. It is straightforward to identify the q → 1 limit of the reduced Poisson algebra with the classical Virasoro algebra.

622


5.3. Explicit computation of the Poisson brackets. Introduce the generating series X A(z) = an z −n , n∈Z

and the same for other matrix elements of M (s) given by formula (5.4). We fix the element r by setting φn = (1 + q n )−1 in formula (5.3) in accordance with Theorem 2. Denote X 1 − qn X (φn − φ−n )z n = zn. (5.7) ϕ(z) = 1 + qn n∈Z

n∈Z

Using the formulas for the gradients of the matrix elements given in the previous section and formula (4.6) for the Poisson bracket, we obtain the following explicit formulas for the Poisson brackets: {A(z), A(w)} = ϕ

w

A(z)A(w), zw {A(z), B(w)} = −δ A(z)B(w), z w {A(z), C(w)} = δ A(z)C(w), z w {A(z), D(w)} = −ϕ A(z)D(w), z {B(z), B(w)} = 0, wq w A(z)D(w) − δ A(z)A(w), {B(z), C(w)} = δ z z wq {B(z), D(w)} = −δ A(z)B(w), z {C(z), C(w)} = 0, w A(z)C(w), {C(z), D(w)} = δ zq w wq w {D(z), D(w)} = ϕ D(z)D(w) − δ C(z)B(w) + δ B(z)C(w). z z zq Remark 9. The relations above can be presented in matrix form as follows. Let A(z) B(z) L(z) = , C(z) D(z) and consider the operators L1 = L ⊗ id, L2 = id ⊗L acting on C2 ⊗ C2 . The r-matrix (5.6) also acts on C2 ⊗ C2 . Formula (4.6) can be written as follows: w 1 w 1 L1 (z)L2 (w) + L1 (z)L2 (w)r− {L1 (z), L2 (w)} = r− 2 z 2 z zq wq − L1 (z)r L2 (w) + L2 (w)σ(r) L1 (z), z w where


1 r−

w z

=r

w z

− σ(r)

z w

 = 

2ϕ

0 0 0

623 w z

 0 0 0 − 21 ϕ wz δ wz 0  . −δ wz − 21 ϕ wz 0  1 w 0 0 2ϕ z

J 5.4. Reduced Poisson structure. We know that M2,q = M2,q /LN is isomorphic to

t(s) 1 −1 0

(see Sect. 3). The ring R2,q of functionals on M2,q is generated by the Fourier coefficients of t(s). In order to compute the reduced Poisson bracket between them, we have to extend them to LN -invariant functions on the whole M2,q . Set e t(s) = a(s)c(sq) + d(sq)c(s).

(5.8)

t(s) are LN -invariant, and their It is easy to check that the Fourier coefficients e tm of e J restrictions to M2,q coincide with the corresponding Fourier coefficients of t(s). Let us compute the Poisson bracket between e tm ’s. Set X e tm z −m . Te(z) = m∈Z

Using the explicit formulas above, we find w {Te(z), Te(w)} = ϕ Te(z)Te(w) z wq w 2 +δ 1(z)c(w)c(wq ) − δ 1(w)c(z)c(zq 2 ), z zq

(5.9)

J where 1(z) = A(z)D(z)−B(z)C(z) = 1. Hence, restricting to M2,q (i.e. setting c(z) = 1 in formula (5.9)), we obtain: wq w w T (z)T (w) + δ −δ . {T (z), T (w)} = ϕ z z zq

This indeed coincides with formula (2.14). Remark 10. Consider the subring Se2,q of the ring R2,q , generated by cm , e tm , m ∈ Z. The ring Se2,q consists of LN -invariant functionals on M2,q , and hence it can serve as a substitute for the ring of functions on M2,q /LN . Let us compute the Poisson brackets in Se2,q . The Poisson brackets of e tm ’s are given by formula (5.9), and by construction {cm , ck } = 0. It is also easy to find that {cm , e tk } = 0. Hence Se2,q is a Poisson subalgebra of R2,q . Thus, the q-deformed Drinfeld–Sokolov reduction can be interpreted as follows. The initial Poisson algebra is R2,q . We consider its Poisson subalgebra Se2,q generated by cm ’s and e tm ’s. The ideal I of Se2,q generated by {cm + δm,0 , m ∈ Z} is a Poisson ideal. The quotient Se2,q /I is isomorphic to the q-Virasoro algebra R2,q defined in Sect. 3.

624


5.5. q-deformation of Miura transformation. As was explained in Sect. 3.2, the q-Miura transformation of [19] is the map between two (local) cross-sections of the projection J J → Mn,q /LN . In the case of LSL2 , the first cross-section πq : Mn,q λ(s) 0 −1 λ(s)−1 is defined by the subsidiary constraint b(s) = 0, and the second t(s) 1 −1 0 is defined by the subsidiary constraint d(s) = 0. The map between them is given by the formula mq : λ(s) 7→ t(s) = λ(s) + λ(sq)−1 . Now we want to recover formula (2.16) for the Poisson brackets between the Fourier coefficients λn of λ(s), which makes the map mq Poisson. We have already computed the Poisson bracket on the second (canonical) crosssection from the point of view of Poisson reduction. Now we need to compute the Poisson bracket between the functions an ’s on the first cross-section, with respect to which the map mq is Poisson. This computation is essentially similar to the one outlined in Sect. 3.2. The Poisson structure on the local cross-section is given by the Dirac bracket, which is determined by the choice of the subsidiary conditions, which fix the cross-section. The Dirac bracket has the following property (see [16]). Suppose we are given constraints ξn , n ∈ I, and subsidiary conditions ηn , n ∈ I, on a Poisson manifold M , such that {ξk , ξl } = {ηk , ηl } = 0, ∀k, l ∈ I. Let f, g be two functions on M , such that {f, ξk } and {g, ξk } vanish on the common level surface of all ξk , ηk . Then the Dirac bracket of f and g coincides with their ordinary Poisson bracket. In our case, the constraint functions are cm + δm,0 , m ∈ Z, and the subsidiary conditions are bm , m ∈ Z, which fix the local model of the reduced space. We have: {bm , bk } = 0, {cm , ck } = 0, and {am , bk } = 0, if we set bm = 0, ∀m ∈ Z. Therefore we are in the situation described above, and the Dirac bracket between am and P ak coincides with their ordinary bracket. In terms of the generating function A(z) = m∈Z am z −m it is given by the formula w A(z)A(w), {A(z), A(w)} = ϕ z which coincides with formula (2.16). Thus, we have proved the Poisson property of the q-deformation of the Miura transformation from the point of view of the deformed Drinfeld–Sokolov reduction.

6. Lattice Virasoro Algebra In this section we consider the lattice counterpart of the Drinfeld–Sokolov reduction. Our group is thus G = (SL2 )Z/N Z , where N is an integer, and τ is the automorphism of G, which maps (gi ) to (gi+1 ). Poisson structures on G which are covariant with respect to lattice gauge transformations xn 7→ gn+1 xn gn−1 have been studied already in [29] (cf.


625

also [2]). In order to make the reduction over the nilpotent subgroup N ⊂ G feasible, we have to be careful in our choice of the r-matrix. 6.1. Discrete Drinfeld–Sokolov reduction. By analogy with the continuous case, we ⊕Z/N Z as follows: choose the element r defining the Lie bialgebra structure on g = sl2 r=

X

En ⊗ Fn +

n∈Z/N Z

1 4

X

φn,m Hn ⊗ Hm ,

m,n∈Z/N Z

where φn,m + φm,n = 2δm,n . It is easy to see that r defines a factorizable Lie bialgebra structure on g. For Theorem 1 to be applicable, r has to satisfy the condition (τ ⊗τ )(r) = r, which implies that φn,m = φn−m . An element of G is an N -tuple (gi ) of elements of SL2 : a k bk gk = . c k dk We consider ak , bk , ck , dk , k ∈ Z/N Z, as the generators of the ring of functions on G. The discrete analogue of the Drinfeld–Sokolov reduction consists of taking the quotient M = GJ /N, where GJ = (GJ )Z/N Z , a b J G = , −1 d and N = N Z/N Z , acting on GJ by the formula (hi ) · (gi ) = (hi+1 gi h−1 i ). It is easy to see that

( M'

ti 1 −1 0

(6.1)

)

. i∈Z/N Z

The element r with φn,m = φn−m , φk +φ−k = 2δk,0 , defines a Lie bialgebra structure on g and Poisson structures η, η∗τ on G. According to Theorem 1, the action of (G, η) on (G, η∗τ ) given by formula (6.1) is Poisson. As in the continuous case, for the Poisson structure η∗τ to be compatible with the discrete Drinfeld–Sokolov reduction, we must have: {cn , cm }|GJ = 0.

(6.2)

Explicit calculation analogous to the one made in the previous subsection shows that (6.2) holds if and only if φn−1 + 2φn + φn+1 = 2δn,0 + 2δn+1,0 . give us a unique solution: for The initial condition φ0 = 1 and periodicity condition 2k . In what follows we restrict odd N , φk = (−1)k ; for even N , φk = (−1)k 1 − N ourselves to the case of odd N (note that in this case the linear operator id +τ is invertible). Continuing as in the previous subsection, we define e tn = an cn+1 + dn+1 cn ,

n ∈ Z/N Z.

626


These are N-invariant functions on G. We find in the same way as in the continuous case: tm } = ϕn−m e tn e tm + δn,m+1 cm cm+2 − δn+1,m cn cn+2 , {e tn , e {e tn , cm } = 0,

(6.3)

{cn , cm } = 0,

where 1 ϕk = (φk − φ−k ) = 2

0, k = 0, 6 0. (−1)k , k =

The discrete Virasoro algebra C[ti ]i∈Z/N Z is by definition the quotient of the Poisson algebra C[e ti , ci ]i∈Z/N Z by its Poisson ideal generated by ci +1, i ∈ Z/N Z. From formula (6.3) we obtain the following Poisson bracket between the generators ti : {tn , tm } = ϕn−m tn tm + δn,m+1 − δn+1,m .

(6.4)

The discrete Miura transformation is the map from the local cross-section λn 0 −1 λ−1 n to M, λn 7→ tn = λn + λ−1 n+1 .

(6.5)

It defines a Poisson map C[λ± i ]i∈Z/N Z → C[ti ]i∈Z/N Z , where the Poisson structure on the latter is given by the formula {λn , λm } = ϕn−m λn λm .

(6.6)

Remark 11. The Poisson algebra C[ti ]i∈Z/N Z can be considered as a regularized version of the q-deformed Virasoro algebra when q = , where is a primitive N th root of unity. Indeed, we can then consider t(i ), i ∈ Z/N Z, as generators and truncate in all power series appearing in the relations, summations over Z to summations over Z/N Z divided by N . This means that we replace ϕ(n ) given by formula (5.7) by ϕ( e n) =

1 N

X 1 − i ni , 1 + i

i∈Z/N Z

and δ(n ) by δn,0 . The formula for the Poisson bracket then becomes: e m−n )t(n )t(m ) + δn,m+1 − δn+1,m . {t(n ), t(m )} = φ( If we set t(i ) = ti , we recover the Poisson bracket (6.4), since it is easy to check that ϕ( e m−n ) = ϕn−m . One can apply the same procedure to the q-deformed W-algebras associated to sln and obtain lattice Poisson algebras. It would be interesting to see whether they are related to the lattice W-algebras studied in the literature, e.g., in [6, 7]. In the case of sl2 , this connection is described in the next subsection.


627

6.2. Connection with Faddeev–Takhtajan–Volkov algebra. The Poisson structures (6.4) and (6.6) are nonlocal, i.e. the Poisson brackets between distant neighbors on the lattice are nonzero. However, one can define closely connected Poisson algebras possessing local Poisson brackets; these Poisson algebras can actually be identified with those studied by L. Faddeev, L. Takhtajan, and A. Volkov. Let us first recall some results of [19] concerning the continuous case. As was explained in [19], one can associate a generating series of elements of the q-Virasoro algebra to an arbitrary finite-dimensional representation of sl2 . The series T (z) considered in this paper corresponds to the two-dimensional representation. Let T (2) (z) be the series corresponding to the three-dimensional irreducible representation of sl2 . We have the following identity [19]: T (z)T (zq) = T (2) (z) + 1, which can be taken as the definition of T (2) (z). From formula (2.15) we obtain: T (2) (z) = 3(z)3(zq) + 3(z)3(zq 2 )−1 + 3(zq)−1 3(zq 2 )−1 = A(z) + A(z)A(zq)−1 + A(zq)−1 , where A(z) = 3(z)3(zq)

(6.7)

(note that the series A(z) was introduced in Sect. 7 of [19]). From formula (2.16) we find: wq w −δ A(z)A(w). {A(z), A(w)} = δ zq z It is also easy to find wq w −δ T (2) (z)T (2) (w) − 1 δ zq z 2 wq w +δ T (w)T (wq 3 ) − δ T (z)T (zq 3 ). z zq 2

{T (2) (z), T (2) (w)} =

We can use the same idea in the lattice case. Let νn = λn λn+1 ; this is the analogue of A(z). We have: {νn , νm } = (δn+1,m − δn,m+1 )νn νm ,

(6.8)

and hence C[νi± ] is a Poisson subalgebra of C[λ± n ] with local Poisson brackets. We can = t t − 1. The Poisson bracket of t(2) also define t(2) n n+1 n n ’s is local: (2) (2) {t(2) (6.9) t(2) n , tm } = δn+1,m − δn,m+1 n tm − 1 + δn,m+2 tm tm+3 − δn+2,m tn tn+3 . (2) Unfortunately, it does not close on t(2) n ’s, so that C[ti ] is not a Poisson subalgebra of C[ti ]. But let us define formally

sn =

1 1 + t(2) n

−1 = t−1 n tn+1 =

1 . −1 (1 + νn )(1 + νn+1 )

(6.10)

628


Then from formulas (6.10) and (6.8) we find: {sn , sm } =sn sm (δn+1,m − δn,m+1 )(1 − sn − sm ) − − sn+1 δn+2,m + sm+1 δn,m+2 .

(6.11)

Thus, the Poisson bracket closes among sn ’s and defines a Poisson structure on C[si ]i∈Z/N Z . The Poisson algebra C[si ]i∈Z/N Z with Poisson bracket (6.11) was first introduced by Faddeev and Takhtajan in [34] (see formula (54)). We see that it is connected with our version of the discrete Virasoro algebra, C[ti ], by a change of variables (6.10). The Poisson algebra C[νi± ] and the Poisson map C[νi± ] → C[sn ] given by formula (6.10) were introduced by Volkov in [35] (see formulas (2) and (23)) following [34]; see also related papers [36, 13]. This map is connected with our version (6.5) of the discrete Miura transformation by a change of variables. Acknowledgement. E. Frenkel thanks P. Schapira for his hospitality at Université Paris VI in June of 1996, when this collaboration began. Some of the results of this paper have been reported in E. Frenkel’s lecture course on Soliton Theory given at Harvard University in the Spring of 1996. The research of E. Frenkel was supported by grants from the Packard and Sloan Foundations, and by the NSF grants DMS 9501414 and DMS 9304580. The research of N. Reshetikhin was supported by the NSF grant DMS 9296120.

References 1. Adler, M.: On a trace functional for formal pseudodifferential operators and the symplectic structure of the Korteweg–de Vries type equations. Invent. Math. 50, 219–248 (1979) 2. Alekseev, A., Faddeev, L., Semenov-Tian-Shansky, M.: Hidden quantum group inside Kac–Moody algebras. Commun. Math. Phys. 149, 335–345 (1992) 3. Asai, Y., Jimbo, M., Miwa, T., Pugai, Ya.: Bosonization of vertex operators for the A(1) n−1 vertex models. J. Phys. A29, 6595–6616 (1996) 4. Awata, H., Kubo, H., Odake, S., Shiraishi, J.: Quantum Wn algebras and Macdonald polynomials Comm. Math. Phys. 179, 401–416 (1996) 5. Belavin, A.A., Drinfeld V.G.: Solutions of the classical Yang-Baxter equation for simple Lie algebras. Funct. Anal. Appl. 16, 159–80 (1981) 6. Belov, A.D., Chaltikian, K.D.: Lattice analogues of W -algebras and classical integrable equations Phys. Lett., 268–274 (1993) 7. Bonora, L., Colatto, L.P., Constantinidis, C.P.: Toda lattice field theories, discrete W-algebras, Toda lattice hierarchies and quantum groups. Phys. Lett. B387, 759 (1996) 8. Ding, X.-M., Hou, B.-Y., Zhao, L.: ~-(Yangian) deformation of Miura map and Virasoro algebra. Preprint q-alg/9701014 9. Drinfeld, V.G.: Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang–Baxter equation. Sov. Math. Dokl. 27, 68–71 (1983) 10. Drinfeld, V.G.: A new realization of Yangians and quantized affine algebras. Sov. Math. Dokl. 36, 212–216 (1988) 11. Drinfeld, V.G., Sokolov, V.V.: Lie algebras and equations of Korteweg-de Vries type. Sov. Math. Dokl. 23, 457–462 (1981); J. Sov. Math. 30, 1975–2035 (1985) 12. Gelfand, I.M., Dickey, L.A.: Family of Hamiltonian structures connected with integrable nonlinear equations. Collected papers of I.M. Gelfand, vol. 1, Berlin–Heidelberg–New York: Springer-Verlag, 1987, pp. 625–646 13. Faddeev, L.D., Volkov, A.Yu.: Abelian current algebras and the Virasoro algebra on the lattice. Phys. Lett. B315, 311–8 (1993) 14. Feigin, B., Frenkel, E.: Affine Lie algebras at the critical level and Gelfand-Dikii algebras. Int. J. Math. Phys. A7, suppl. A1, 197–215 (1992)


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15. Feigin, B., Frenkel, E.: Quantum W-algebras and elliptic algebras. Comm. Math. Phys. 178, 653–678 (1996); q-alg/9508009 16. Flato, M., Lichnerowicz, A., Sternheimer, D.: Deformation of Poisson brackets, Dirac brackets and applications. J. Math. Phys. 17, 1754 (1976) 17. Frenkel, E.: Deformations of the KdV hierarchy and related soliton equations, Int. Math. Res. Notices 2, 55–76 (1996); q-alg/9511003 18. Frenkel, E.: Affine Kac-Moody algebras at the critical level and quantum Drinfeld–Sokolov reduction. PhD Thesis, Harvard University, 1991 19. Frenkel, E., Reshetikhin, N.: Quantum affine algebras and deformations of the Virasoro algebra and W-algebras Comm. Math. Phys. 178, 237–264 (1996); q-alg/9505025 20. Kac, V.: Infinite-dimensional Lie algebras. Third edition, Cambridge: Cambridge University Press, 1990 21. Kedem, R.: Singular R-matrices and Drinfeld’s comultiplication. Preprint q-alg/9611001 22. Khoroshkin, S.M., Tolstoy, V.N.: On Drinfeld’s realization of quantum affine algebras. J. Geom. Phys. 11, 445–452 (1993) 23. Kostant, B.: On Whittaker vectors and representation theory. Invent. Math. 48, 101–184 (1978) 24. Lu, J.H.: Momentum mapping and reduction of Poisson actions. In: Symplectic geometry, groupoids and integrable systems, Berkeley, 1989. P.Dazord and A.Weinstein (eds.), Berlin–Heidelberg–New York: Springer-Verlag, pp. 209–226 25. Lukyanov, S., Pugai, Ya.: Multi-point local height probabilities in the integrable RSOS models. Nucl. Phys. B473, 631 (1996) 26. Reshetikhin, N.Yu., Semenov-Tian-Shansky, M.A.: Central extensions of quantum current groups. Lett. Math. Phys. 19, 133–142 (1990) 27. Reshetikhin, N.Yu., Semenov-Tian-Shansky, M.A.: Quantum R-matrices and factorization problems. In: Geometry and Physics, essays in honor of I.M.Gelfand. S.Gindikin and I.M.Singer (eds.), pp. 533-550. Amsterdam–London–New York: North Holland, 1991 28. Semenov-Tian-Shansky, M.A.: What is a classical r-matrix. Funct. Anal. Appl. 17, 17–33 (1983) 29. Semenov-Tian-Shansky M.A.: Dressing action transformations and Poisson–Lie group actions. Publ. RIMS. 21, 1237–1260 (1985) 30. Semenov-Tian-Shansky, M.A.: Poisson Lie groups, quantum duality principle and the quantum double. Contemporary Math. 175, 219–248 31. Semenov-Tian-Shansky M.A., Sevostyanov A.V.: Drinfeld–Sokolov reduction for difference operators and deformations of W-algebras II. General semisimple case. Preprint q-alg/9702016 32. Shiraishi, J., Kubo, H., Awata, H., Odake, S.: A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions Lett. Math. Phys. 38, 33–51 (1996) 33. Steinberg, R.: Regular elements of semisimple algebraic Lie groups, Publ. Math. I.H.E.S., 25, 49–80 (1965) 34. Takhtajan, L.A., Faddeev, L.D.. Liouville model on the lattice, Lect. Notes in Phys. 246, 166–179 (1986) 35. Volkov, A.Yu.: Miura transformation on a lattice. Theor. Math. Phys. 74, 96–99 (1988) 36. Volkov, A.Yu.: Quantum Volterra model. Preprint HU-TFT-92-6 (1992) 37. Weinstein, A.: Local structure of Poisson manifolds, J. Different. Geom. 18, 523–558 (1983) Communicated by T. Miwa

Commun. Math. Phys. 192, 631 – 647 (1998)

Communications in


Drinfeld–Sokolov Reduction for Difference Operators and Deformations of W -Algebras II. The General Semisimple Case M. A. Semenov-Tian-Shansky1,3 , A. V. Sevostyanov2,3 1 2 3

Université de Bourgogne, Dijon, France Institut of Theoretical Physics, Uppsala University, Uppsala, Sweden Steklov Mathematical Institute, St.Petersburg, Russia

Received: 27 April 1997 / Accepted: 22 August 1997

Abstract: The paper is the sequel to [9]. We extend the Drinfeld–Sokolov reduction procedure to q-difference operators associated with arbitrary semisimple Lie algebras. This leads to a new elliptic deformation of the Lie bialgebra structure on the associated loop algebra. The related classical r-matrix is explicitly described in terms of the Coxeter transformation. We also present a cross-section theorem for q-gauge transformations which generalizes a theorem due to R. Steinberg. Introduction The present paper is the sequel to [9]; we refer the reader to this paper for a general introduction. Our goal is to extend the results of [9] to arbitrary semisimple Lie algebras. As an intermediate step we develop a group-theoretic framework for abstract difference equations associated with arbitrary semisimple Lie groups. A similar problem for differential equations which was solved by Drinfeld and Sokolov is linear, since it involves only the structure of the corresponding semisimple Lie algebra. Difference equations lead to the study of specific submanifolds in Lie groups which are closely related to some of its Bruhat cells. Our main technical result is a cross-section theorem for the q-gauge transformations which generalizes a theorem due to R. Steinberg. The reduction scheme outlined in [9] extends to the abstract case as well and the general conclusion remains the same: the consistency condition for the reduction imposes very rigid conditions on the underlying classical r-matrix which fix it completely. The resulting classical r-matrix is new; it yields an elliptic deformation of the Lie bialgebra structure on the loop algebras associated with so-called Drinfeld’s new realizations of quantized affine algebras [3, 10]. The explicit characterization of the set of abstract q-difference operators leads to a very simple formula for this r-matrix in terms of the Coxeter element of the corresponding Weyl group. (The role of Coxeter transformations in the theory of q-difference operators should be compared with the role of the dual Coxeter transformations in the representation theory of affine Lie algebras at the critical level which is implicit in [7, 8]). The

632

M. A. Semenov-Tian-Shansky, A. V. Sevostyanov

Coxeter element also plays a key role in the definition and the study of the generalized Miura transformation. The structure of the paper is as follows. Section 1 gives a description of the set of abstract q-difference operators associated to an arbitrary complex semisimple Lie group G. This description is based on the cross-section theorem referred to above; its proof is postponed until Sect. 3. In Sect. 2 we define a class of Poisson covariant Poisson structures on the set of q-difference G-valued connections; this definition is preceded by the description of a class of Lie bialgebra structures on loop algebras. As compared to [9], we need more details on the Poisson theory of q-gauge transformations, including the theory of the double and the twisted factorization. We then formulate our main theorem which gives an explicit description of the (unique) classical r-matrix which is compatible with the Drinfeld–Sokolov reduction for q-difference operators associated with G. At the end of Sect. 2 we also briefly discuss the reduction for finite difference operators on the lattice which yields a definition of the classical lattice W (g)-algebra extending the definition of the lattice Virasoro algebra discussed in [9]. Section 4 contains a description of the generalized Miura transformations. Finally, in Sect. 5 we compare our formulae with the formulae of Frenkel and Reshetikhin [8] for the sl(n) case. 1. Abstract q-Difference Operators The following notation will be used throughout the paper. Let G be a connected simply connected finite-dimensional complex semisimple Lie group, g its Lie algebra. Fix a Cartan subalgebra h ⊂ g and let 1 be the set of roots of (g, h). Choose an ordering in the root system; let 1+ be the system of positive roots and {α1 , ..., αl }, l = rank g, the Pl corresponding set of simple roots. For α ∈ 1, α = i=1 ni αi , we define its height by height α =

l X

ni .

i=1

Let eα ∈ g be a root vector which corresponds to α ∈ 1. Let X Ceα b=h+ α∈1+

be the corresponding Borel subalgebra and X b=h+ Ce−α α∈1+

the opposite Borel subalgebra; let n = [b, b] and n = [b, b] be their respective nil-radicals. We shall fix a nondegenerate invariant bilinear form (, ) on g. Let H = exp h, N = exp n, N = exp n, B = HN, B = HN be the Cartan subgroup, the maximal unipotent subgroups and the Borel subgroups of G which correspond to the Lie subalgebras h, n, n, b and b, respectively. Let W be the Weyl group of (g, h) ; we shall denote a representative of w ∈ W in G by the same letter. Let s1 , ..., sl be the reflections which correspond to simple roots; let s = s1 s2 · · · sl be a Coxeter element. Put N 0 = {v ∈ N ; svs−1 ∈ N }; it is easy to see that N 0 ⊂ N is an abelian subgroup, dim N s = l. Put M s = N s−1 N ; it is clear that N s−1 N = N 0 s−1 N and that N 0 ×N → M s : n0 , n 7−→ n0 s−1 n is a diffeomorphism.

Drinfeld–Sokolov Reduction for Difference Operators. II

633

Let G = LG be the loop group of G; 1 the group G will be identified with the subgroup of constant loops in G. Fix q ∈ C , |q| < 1, and let τ be the automorphism of G defined by g τ (z) = g(qz). We shall denote the corresponding automorphism of the loop algebra Lg by the same letter. Let C be another copy of G equipped with the q-gauge action of G, G × C → C; (v, L) 7−→ v τ Lv −1 .

(1.1)

The space C will be sometimes referred to as the space of q-difference connections (with values in G). Let Ms be the cell in C consisting of loops with values in M s , and S = N 0 s−1 . Theorem 1.1. The restriction of the gauge action (1.1) to N ⊂ G leaves the cell Ms ⊂ C invariant. The action of N on Ms is free and S is a cross-section of this action. The proof will be given in Sect. 3. (Its special case which corresponds to G = SL(n) is presented in [9].) Remark. A closely related theorem is due to Steinberg [17] who discussed the action of an algebraic semisimple Lie group on itself by conjugations. (In other words, the situation considered in [17] corresponds to the trivial automorphism τ = id.) Steinberg’s theorem asserts that if G is defined over an algebraically closed field, N 0 s−1 ⊂ G is a cross-section of the set of regular conjugacy classes in G. In Theorem 1.1 we replace the action of the entire group on itself by the action of its unipotent subgroup on its affine subvariety Ms . In the context of Lie algebras a similar problem was studied by B.Kostant [12], again in the case of trivial τ To motivate the above definitions let us discuss the case G = SL(n). Let us choose an order in the root system of sl(n) in such a way that positive root vectors correspond to lower triangular matrices. We may order the simple roots and choose the matrices sk , k = 1, ..., n − 1, representing the corresponding reflections in such a way that the Coxeter element s = s1 s2 ...sn−1 is represented by the matrix   0 0 ··· 0 1  −1 0 · · · 0 0     0 −1 · · · 0 0  s= . · · · · · · · · · · · · · · ·  0 0 ··· 0 0  0 0 · · · −1 0 Then the group N 0 consists of lower triangular unipotent matrices of the form   1 0 ··· 0 0  0 1 ··· 0 0    u = · · · · · · · · · · · · · · ·,  0 0 ··· 1 0  ∗ ∗ ··· ∗ 1 the set M s consists of all unimodular matrices of the form 1 We may define G, e.g., as the group of points of G over the local field C((z)), or over the ring of holomorphic functions regular in C\{0}. The concrete choice is in fact irrelevant for the validity of Theorem 1.1.

634




∗ −1  ∗ ∗  A = · · · · · ·  ∗ ∗ ∗ ∗

0 ··· −1 · · · ··· ··· ∗ ··· ∗ ···

 0 0   · · ·, −1  ∗

and the set N 0 s−1 is the set of all companion matrices of the form 

0  0   0 L= · · ·  0 1

 −1 ··· 0 0 0 −1 · · · 0 0   0 0 ··· 0 0  . ··· ··· ··· ··· ···    0 0 ··· 0 −1 u1 u2 · · · un−2 un−1

Let us associate to L a first order difference equation τ · ψ + Lψ = 0, where ψ = (ψ1 , ψ2 , ..., ψn )t is a column vector, ψk ∈ C ((z)). It is easy to see that its first component φ := ψ1 satisfies an nth order difference equation, τ n φ + un−1 τ n−1 · φ + un−2 τ n−2 · φ + ... + u1 τ · φ + φ = 0, and, moreover, ψk = τ k−1 φ, k = 1, 2, ..., n. Hence the set N 0 s−1 may be identified with the set Mn,q of all nth order q-difference operators. 2 In the general case we set, by definition, Mq (G) = Ms /LN ; as we shall see, the manifold Mq (G) carries a natural Poisson structure and may be regarded as the spectrum of a classical q-W-algebra. Remark. In [4] Drinfeld and Sokolov use a similar approach to define the set of abstract higher order differential operators associated to a given semisimple Lie algebra; the key observation that motivates their definition is that for g = sl(n) the matrix 

0 −1 0 · · ·  0 0 −1 · · ·  f = · · · · · · · · · · · ·  0 0 0 ··· 0 0 0 ···

 0 0 0 0   · · · · · · ∈ g 0 −1  0 0

is a principal nilpotent element. Accordingly, in the context of Lie algebras the set Ms ⊂ G is replaced by the affine manifold Mf = f +Lb ⊂ Lg; the main cross-section theorem then asserts that the gauge action of the subgroup LN ⊂ LG leaves Mf invariant; its restriction to Mf is free and admits a cross-section which is an affine submanifold in Lg. This cross-section provides a model for the space of higher order differential operators. Notice that there exists a well known link between principal nilpotent elements in a semisimple Lie algebra and Coxeter elements of the corresponding Weyl group provided by a theorem due to Kostant [11]. 2 Note that in this paper we use slightly different conventions, as compared to [9]; in particular, we consider the q-gauge action by lower triangular matrices as opposed to upper triangular matrices.


635

2. Poisson Structures for q-Difference Equations In this section we shall construct a class of Poisson structures on G = LG and on the space C of q-difference connections. Our main theorem asserts that there is a unique Poisson structure in this class which is compatible with the Poisson reduction over LN . We start with the description of a family of Lie bialgebra structures on loop algebras. 2.1. Factorizable Lie bialgebras associated with Lg. Let g = Lg be the loop algebra; we equip it with the standard invariant bilinear form, hX, Y i = Resz=0 (X (z) , Y (z)) dz/z. Notice that the automorphism τ satisfies hτ X, τ Y i = hX, Y i. Let d = g ⊕ g be the direct sum of two copies of g with the bilinear form hh(X1 , X2 ) , (Y1 , Y2 )ii = hX1 , Y1 i − hX2 , Y2 i .

(2.1)

Put b = Lb, b = Lb, n = Ln, n = Ln, h = Lh; let π : b → b/n, π : b → b/n be the canonical homomorphisms; the quotient algebras b/n, b/n may be canonically identified with h. Fix a linear operator θ ∈ End h which satisfies the following conditions: 1. hθX, θY i = hX, Y i for any X, Y ∈ h. 2. I − θ is invertible. We shall assume, moreover, that θ extends to an automorphism of LH which we denote by the same letter. Put o n (2.2) g∗θ = X+ , X− ∈ b ⊕ b ⊂ d; π X− = θ ◦ π (X+ ) ; let δ g ⊂ d be the diagonal subalgebra. The following assertion is well known. Proposition 2.1. d,δ g, g∗θ is a Manin triple. Thus we get a family of Lie bialgebras g, g∗θ with common double d = g ⊕ g parametrized by θ ∈ End h ; all these bialgebras are factorizable. Let ˙ +n ˙ g = n+h be the ‘pointwise Bruhat decomposition’ of the loop algebra. Let P+ , P0 , P− be the corresponding projection operators which map g onto n, h, n, respectively. The classical r-matrix associated with g, g∗θ is the kernel of the linear operator θ r+ ∈ Hom g∗θ , g , θ

r+ = P+ + (I − θ)−1 .

(2.3)

Let us also set θ r− := −θ r+∗ ; the classical Yang-Baxter identity implies that both θ r+ and θ r− are Lie algebra homomorphisms from g∗θ into g. In the definition of the Poisson structures it is sometimes convenient to use their skew-symmetric combination, 1 1 θ 1+θ θ P+ − P− + P0 ; r= r+ +θ r− = (2.4) 2 2 1−θ the “perturbation term” in (2.4) is the Cayley transform of θ,

636


θ

r0 =

1+θ P0 . 1−θ

(2.5)

The mappings θ r± give rise to group homomorphisms θ r± : (LG)∗ → LG (which we denote by the same letters). The double d = g ⊕ g has a naturalstructure of a factorizable Lie bialgebra associated with the Manin triple d,δ g, g∗θ . Hence D = G × G is a Poisson Lie group which contains both G and its dual group as Poisson subgroups. More precisely, let π : LB → LB/LN, π : LB → LB/LN be the canonical projections; the quotient groups LB/LN, LB/LN may be canonically identified with LH . Let δ G ⊂ G × G be the diagonal subgroup and G ∗ ⊂ D the subgroup defined by G∗ =

x+ , x− ∈ LB × LB; θ ◦ π(x+ ) = π(x− ) .

(2.6)

δ ∗ Proposition 2.2. (i) G, G ⊂ D are Poisson Lie subgroups with tangent Lie bialgebras ∗ ∗ g, gθ and gθ , g , respectively. (ii) Almost all elements x ∈ G admit a factorization ∗ x = x+ x−1 − , where x+ , x− ∈ G .

We shall also need the related notion of twisted factorization. Proposition 2.3. Suppose that the automorphism τ commutes with θ. Then almost all elements x ∈ G admit a twisted factorization ∗ x = xτ+ x−1 − , where x+ , x− ∈ G .

(2.7)

Factorizations (2.6) and (2.7) are unique if we assume that both x and x+ , x− are sufficiently close to the unit element. 2.2. Gauge covariant Poisson structures and reduction. Assume that θ ∈ End h commutes with τ . In that case the space C of q-difference connections admits a natural Poisson structure which is covariant with respect to the q-gauge action G × C → C. We refer the reader to [15, 16] for its construction which is based on the notion of the twisted Heisenberg double. Let A be the affine ring of G = LG; by definition, A is generated by the coefficients of the (formal) Laurent expansion of the matrix coefficients of L ∈ G in some faithful finite dimensional representation of G. Speaking informally, we may regard the elements of A as smooth functions on G or on C; the difference between the two options is in the choice of the Poisson structure. We shall write C ∞ (C) instead of A to stress the choice of this underlying Poisson structure on A. For any ϕ ∈ C ∞ (C) we denote by ∇ϕ, ∇0 ϕ its left and right gradients. Theorem 2.4. (i) For any θ ∈ End h satisfying conditions (1), (2) the bracket

, ∇ψ + θ r ∇0 ϕ , ∇0 ψ {ϕ, ψ}τ = θ r (∇ϕ) − τ ◦θ r+ ∇0 ϕ , , ∇ψ − θ r− ◦ τ −1 (∇ϕ) , ∇0 ψ ,

(2.8)

satisfies the Jacobi identity. (ii) Equip the space C of q-difference connections with the Poisson structure (2.8); then the q-gauge action defines a Poisson mapping G × C → C. (iii) The subgroup N = LN ⊂ G is admissible and hence N -invariant functions form a Poisson subalgebra in C ∞ (C).


637

(We refer the reader to [9] for a general definition of admissible subgroups.) Below we shall need another formula for the Poisson bracket {, }τ which is related to the twisted factorization in G. Let ϕ ∈ C ∞ (C); we define Zϕ ∈ g by the following relation: r+ Zϕ − τ −1 · r− Zϕ = ∇0 ϕ. Let h ∈ C be an element admitting twisted factorization, h = hτ+ h−1 − ; then

{ϕ, ψ}τ (h) = Adh+ · τ −1 · Zϕ − Adh− Zϕ , ∇ψ − ∇ϕ, Adh+ · τ −1 · Zψ − Adh− Zψ .

(2.9)

We can now state our second main theorem. For w ∈ W let Rw ∈ End Lh be the linear operator acting in the loop algebra, (Rw H) (z) = Ad w · (H (z)) . Theorem 2.5. The quotient space Ms /N =S is a Poisson submanifold in C/N if and only if the endomorphism θ is given by θ = Rs · τ , where s ∈ W is the Coxeter element. Notice that the Coxeter transformation satisfies conditions (1), (2) imposed above; since it preserves the root and weight lattices in h, it gives rise to an automorphism of LH. In the case when g = sl(2), Theorem 2.5 was proved in [9] (Theorem 2); in this case Rs = −Id. For future reference let us write down explicitly the twisted factorization problem associated with the r-matrix θ r. Proposition 2.6. Assume that θ = Rs · τ . The twisted factorization problem in the loop group G associated with the r-matrix θ r amounts to the relation −1 (2.10) , y+ ∈ LB, , y− ∈ LB, π y− = s (π (y+ )) . x = y+ y− We shall denote by G 0 ⊂ G the open subset consisting of elements admitting twisted factorization described in (2.10). We shall now explicitly describe the kernel of the corresponding classical r-matrix. The relevant part of this kernel is the ‘perturbation term’ r0 which was defined in (2.5). In the present situation we have r0 =

I + Rs · τ P0 . I − Rs · τ

(2.11)

Let Hp ∈ h, p = 1, ..., l, be the eigenvectors of the Coxeter element, Ad s(Hp ) = 2πikp

e h Hp (here h is the Coxeter number and k1 , ... kl are the exponents of g. There exists a permutation σ of the set {1, ...l} such that the basis{Hσp } is biorthogonal to {Hp } ; we may assume that hHp , Hσp i = 1.Then (r0 X) (z) =

r ∞ X X n=−∞ p=1

zn

2πikp h 2πik q n exp h p

1 + q n exp 1−

hXn , Hσp i Hp , q ∈ C, |q| < 1. (2.12)

Note that the Lie bialgebra studied by Drinfeld in [3] (see also [10]) corresponds to the “crystalline” limit q → 0 in (2.12); in this case r0 amounts to the Hilbert transform in h. It is convenient to write

638


r0 (q, z) =

r X

ψp (q, z)Hp ⊗ Hσp ,

(2.13)

p=1

where ψp (q, z) =

∞ 2πik X 1 + q n exp h p n n=−∞ 1 − q exp

2πikp h

zn.

(2.14)

Functions ψp satisfy the q-difference equations, ψp (q, z) + exp

2πikp 2πikp · ψp (q, qz) = δ (z) − exp · δ (qz) , h h

(2.15)

where δ (z) =

∞ X

zn.

(2.16)

n=−∞

2.3. Proof of Theorem 2.5. We briefly recall the geometric criterion that allows to check that a submanifold of a quotient Poisson manifold is itself a Poisson manifold. Let M be a Poisson manifold, π : M → B a Poisson submersion. Hamiltonian vector fields ξϕ , ϕ ∈ π ∗ C ∞ (B), generate an integrable distribution Hπ in T M . Proposition 2.7. Let V ⊂ M be a submanifold; W = π(V ) ⊂ B is a Poisson submanifold if and only if V is an integral manifold of Hπ . Assume that this condition holds true; let NV ⊂ T ∗ M |V be the conormal bundle of V ; clearly, T ∗ V ' T ∗ M |V /NV . Let ϕ, ψ ∈ C ∞ (W ); put ϕ∗ = π ∗ ϕ |V , ψ ∗ = π ∗ ψ |V . Let dϕ, dψ ∈ T ∗ M |V be any representatives of dϕ∗ , dψ ∗ ∈ T ∗ V . Let V2 PM ∈ V ect M be the Poisson tensor. Proposition 2.8. We have

π ∗ {ϕ, ψ} |V = PM , dϕ ∧ dψ ;

(2.17)

in particular, the r.h.s. does not depend on the choice of dϕ, dψ. Let us now apply Proposition 2.7 in the setting of Theorem 2.5. It is sufficient to check that the Hamiltonian vector fields generated by N -invariant functions on C are tangent to Ms if and only if r0 is given by (2.11). Let ϕ ∈ C ∞ (C)N ; then ϕ (v τ L) = ϕ (Lv) for all v ∈ N , and hence Z := ∇ϕ − τ · ∇0 ϕ ∈ b. Since ∇0 ϕ(L) = Ad L−1 · ∇ϕ(L), we rewrite the Poisson bracket on C in the following form:

{ϕ, ψ}τ (L) = rZ + Z − Ad L · r · τ −1 · Z + Ad L · τ −1 · Z, ∇ψ(L) ; thus in the left trivialization of T C the Hamiltonian field generated by ϕ has the following form: ξϕ (L) = rZ + Z − Ad L · r · τ −1 · Z + Ad L · τ −1 · Z. Assume that L ∈ Ms , L = vs−1 u, v ∈ N 0 , u ∈ N . Put Z = Z0 + Z+ , Z0 ∈ h,Z+ ∈ n. Then ξϕ (L) = r0 Z0 + Z0 + s−1 τ −1 · Z0 − s−1 τ −1 · r0 Z0 + X + Ad v · s−1 · Y,


639

where X ∈ n0 , Y ∈ n. On the other hand, in the left trivialization of T C the tangent space TL Mf is identified with n0 + Ad v · s−1 · n . Hence ξϕ is tangent to Mf if and only its h-component vanishes, i.e., if r0 + Id + Ad s−1 τ −1 − Ad s−1 τ −1 r0 = 0, which is equivalent to (2.11). 2.4. Lattice W-algebras. Let 0 = Z/N Z be a finite periodic lattice. Set G = G0 , g = g0 . Let τ be the automorphism of G induced by the cyclic permutation on 0, xτi = xi+1modN . We define lattice gauge transformations by g · x = g τ xg −1 . The definition of gauge covariant Poisson brackets on the space of q-difference connections has its obvious lattice counterpart. In an equally obvious way one may construct a class of Lie bialgebra structures on g = g0 which is compatible with reduction over the unipotent subgroup N = N 0 . Namely, let us consider the “pointwise Bruhat decomposition” ˙ +n, ˙ n = n 0 , h = h0 , n = n 0 g = n+h and set θ

r = P+ − P− +

I +θ P0 , θ ∈ End h. I −θ

We shall omit the details and formulate only the lattice counterpart of the main theorems. Let as usual s ∈ G be a Coxeter element. Theorem 2.9. (i) The restriction of the gauge action to N = N 0 leaves the subset Ms = Ns−1 N invariant. (ii) The restricted action is free and S = N0 s−1 , N0 = N 0 0 , is its cross-section. We shall assume for simplicity that the lattice length N is relatively prime with the Coxeter number. Theorem 2.10. The quotient Ms /N is a Poisson submanifold in the reduced space if and only if θ =Rs · τ . Remark. The condition on 0 assures that I − θ is invertible; it is likely that reduction is possible even without this assumption, but this question needs further study. For G = SL(2) this reduction was studied in detail in [9], Sect. 6. It gives a discrete version of the Virasoro algebra, which is closely connected to the lattice Virasoro algebra of [5].

3. The Cross-Section Theorem We shall prove the following assertion. Theorem 3.1. For each L ∈ N 0 s−1 N there exists a unique element n ∈ N such that nτ · L · s−1 ∈ N 0 s−1 .

640


Let Ch ⊂ W be the cyclic subgroup generated by the Coxeter element. Ch has exactly l = rank g different orbits in the root system 1(g, h). The proof depends on the structure of these orbits; for this reason we have to distinguish several cases.3 Proposition 3.2. The theorem is true for g of type Al . An elementary proof which is based on the matrix algebra is given in [9]; below we present a different proof which uses only the properties of the corresponding root system. Lemma 3.3. (i) Each orbit of Ch in 1 (g, h) consists of exactly h elements; one can order these orbits in such a way that the k th orbit contains all positive roots of height k and all negative roots of height h − k. Put nk =

M

nα , Nk = exp nk , Nk = LNk .

{α∈1+ , ht α=k}

For each k we can choose αk ∈ 1+ in such a way that nk =

h−k−1 M

nsp (αk ) .

p=0

Put npk = nsp (αk ) , Nkp = exp npk , Nkp = LNkp . Let L = v · s−1 · u, v ∈ N 0 , u ∈ N ; we must find n ∈ N such that nτ · v · s−1 · u = v0 · s−1 · n.

(3.1)

For any n ∈ N there exists a factorization n = n1 n2 · · · nl , where nk ∈ Nk ; moreover, each nk may be factorized as , npk ∈ Nkp . nk = n0k n1k · · · nh−k−1 k For any n ∈ N the element nτ · v · s−1 · u admits a representation nτ · v · s−1 · u = vs ˜ −1 u, ˜ v˜ ∈ N 0 , u˜ ∈ N ; let

−→−−−→ l h−k−1 Y Y p p u˜ = u˜ k , u˜ k ∈ Nkp , k=1

p=0

be the corresponding factorization of u. ˜ For x ∈ G we write s (x) := s · x · s−1 (this notation will be frequently used in the sequel). Lemma 3.4. We have u˜ pk = τ · s np−1 Vkp , where the factors Vkp ∈ Nkp depend only k on u, v and on nqj with j < k. 3

The proofs given below do not apply when g is a simple Lie algebra of type E6 .


641

Assume now that n satisfies (3.1); then we have v˜ = v0 , u˜ = n. This leads to the following relations: Vkp = npk , (3.2) τ · s np−1 k where we set formally n−1 k = 1. Proposition 3.5. The system (3.2) may be solved recursively starting with k = 1, p = 0. Clearly, the solution is unique. This concludes the proof for g of type Al . Let now g be a simple Lie algebra of type other than Al and E6 , l its rank. Lemma 3.6. (i) The Coxeter number h (g) is even. (ii) Each orbit of Ch in 1 (g, h) consists of exactly h elements and contains an equal number of positive and negative roots. (iii) Put M / 1+ }, np = C · eα ; 1p+ = {α ∈ 1+ ; s−p · α ∈ α∈1p +

then np ⊂ n is an abelian subalgebra, dim np = l. When g is not of type D2k+1 this assertion follows from [2] (Chapter 6, No 1.11, Prop. 33 and Chapter 3, No 6.2, Corr. 3). For g of type D2k+1 it may be checked directly. Put N p = exp np ; let N p be the corresponding subgroup of the loop group G. Let L = v · s−1 · u, v ∈ N 0 , u ∈ N ; we must find n ∈ N such that v · s−1 · u = nτ · v0 · s−1 · n−1 . Put n = n 1 n2 · · · n h , n p ∈ N p ; 2

(3.3)

the elements np will be determined recursively. Put s−1 (w) = s−1 · w · s, w ∈ N . We have Y Y ←− −→ v · s−1 (u) = τ · n−1 . (3.4) np · v0 · s−1 p We shall say that an element x ∈ G is in the big cell in G if, for all values of the argument z, the value x (z) is in the big Bruhat cell B N¯ ⊂ G. Lemma 3.7. v · s−1 (u) is in the big cell in G and admits a factorization v · s−1 (u) = x1+ · x1− , x1+ ∈ N , x1− ∈ N . Indeed, let u = uh/2 uh/2−1 · · · u1 , up ∈ N p , be a similar decomposition of u; then we have simply x− = s−1 (u1 ) . (It is clear that x1+ ∈ B actually does not have an H-component and so belongs to N .) A comparison of the r.h.s in (3.4) with the Bruhat decomposition of the l.h.s. imme−1 diately yields that the first factor in (3.3) is given by n1 = s x− . Assume that n1 , n2 , ..., nk−1 are already computed. Put mk = n1 n2 · · · nk−1 and consider the element

· v · s−1 (u) · s−1 (mk ) . Lk := s−k+1 τ m−1 k

(3.5)

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Lemma 3.8. Lk is in the big cell in G and admits a factorization Lk = xk+ xk− , xk+ ∈ N , xk− ∈ N .

(3.6)

The elements xk± are computed recursively from the known quantities. By applying a similar transform to the r.h.s. of (3.4) we get ! ! ! ← − −→ Y Y −1 k −k+1 −1 −1 −1 τ mk · τ · np · v0 · s · s (mk ) = (3.7) np L =s p

p



    − → ← −−− Y Y  · s−k n−1 . s−k+1 τ · np  v0  · s−k  n−1 p k p≥k

p≥k+1

; hence nk = sk xk− , which A comparison of (3.7) and (3.6) yields xk− = s−k n−1 k concludes the induction. 4. Generalized Miura Transform Our construction of the Miura transform for q-difference operators may be regarded as a nonlinear version of the corresponding construction for differential operators, due to Drinfeld and Sokolov [4]. Recall that the space of abstract differential operators associated with a given semisimple Lie algebra g is realized as the quotient space of the affine manifold Mf = f + Lb ⊂ Lg (the translate of Lb by a fixed principal nilpotent element f ∈ n) over the gauge action of LN . The cross-section theorem of Drinfeld and Sokolov provides a global model S for this quotient space. It is easy to see that the affine submanifold f + Lh ⊂ f + Lb is a local cross-section of the gauge action LN × Mf → Mf (i.e., the orbits of LN are transversal to f + Lh) and hence f + Lh provides a local model of the same quotient space. Thus we get a Poisson structure on f + Lh and a Poisson mapping f + Lh → S. The computation of the induced Poisson structure on f + Lh follows the general prescription of Dirac (see, e.g., [6]), but is in fact greatly simplified, since all correction terms in the Dirac formula identically vanish. One may notice that the affine manifold f + Lh ⊂ Mf is the intersection of the level surfaces of two moment maps associated with the gauge actions of the opposite triangular subgroups LN and LN ; it is this symmetry between LN and LN that accounts for cancellations in the Dirac formula. The situation in the nonlinear case is exactly similar. We pass to the formal description of our construction. Let B ⊂ G be the opposite Borel subgroup, N ⊂ B its nilradical, B = LB, N = LN . Let us consider the Poisson reduction of the space C of q-difference connections over the action of the opposite gauge group N . We equip C with the Poisson structure (2.8), where the choice of θ may be arbitrary. Proposition 4.1. (i) The q-gauge action N × C → C leaves B ⊂ C invariant. (ii) Hamiltonian vector fields on C generated by gauge invariant functions ϕ ∈ C ∞ (C)N are tangent to B ⊂ C. Corollary 4.2. B/N ⊂ C/N is a Poisson submanifold. Remark. Heuristically, the submanifold B ⊂ C corresponds to reduction at the “zero level” of the moment, hence the constraints are automatically of the first class.


643

We shall now define an embedding i : H → Ms ∩ B into the intersection of two “level surfaces”. Let w0 ∈ W be the longest element; let π ∈ Aut 1+ be the automorphism defined by π (α) = −w0 · α, α ∈ 1+ . Let Ni ⊂ N be the 1-parameter subgroup generated by the root vector eπ(αi ) , αi ∈ P . Choose an element ui ∈ Ni , ui 6= 1. Lemma 4.3. [17] w0 ui w0−1 ∈ Bsi B. We may choose ui in such a way that w0 ui w0−1 ∈ N si N . Put x = ul ul−1 ...u1 ; then f := w0 xw0−1 ∈ N s−1 N ∩ N¯ . Remark. The choice of ui is not unique, however this non-uniqueness does not affect the arguments below. Define the immersion i : H → G : x 7−→ x · f · s(x−1 ). ¯ Proposition 4.4. i(H) ⊂ N s−1 N ∩ B. ¯ For Remark. By dimension count it is easy to see that i(H) is open in N s−1 N ∩ B. ¯ it seems plausible that this is true in G = SL(n) we have simply i(H) = N s−1 N ∩ B; the general case as well. We define the corresponding embedding i : H → G for loop groups associated with H, G by the same formula. Clearly, i(H) ⊂ Ms ∩ B. Proposition 4.5. i(H) is a local cross-section of the gauge actions N × Ms → Ms , N × B → B. In other words, gauge orbits of N , N are transversal to i(H) ⊂ Ms ∩ B. Let us now assume that the Poisson structure on the space of q-difference connections is the one described in Theorem 2.5. We may consider i(H) as a (local) model of the reduced space Ms /N obtained by “fixing the gauge” by means of the “subsidiary condition” L ∈ B, or, alternatively, as a model of B/N obtained by choosing the subsidiary condition L ∈ Ms . The choice of r0 assures that both the “constraints” and the “subsidiary conditions” are of the first class. The reduced Poisson structure on i(H) may be expressed in terms of the Dirac bracket. As it appears, it is possible to avoid the actual computation of the “correction terms”. We shall prove the following assertion. Proposition 4.6. The quotient Poisson structure on i(H) is given by

(Id − τ ) (Id − Rs ) . {ϕ, ψ}i(H) = P˜H ∇ϕ, ∇ψ , P˜H = Id − Rs · τ

(4.1)

It will be convenient to introduce another parametrization of the Cartan subgroup which is related to the twisted factorization problem (2.10) in G. Lemma 4.7. B ⊂ G 0 . Proof. The twisted factorization problem in B (cf. (2.10) amounts to the relation −1 b¯ = x+ · x−1 − n− , where x+ ∈ H, x− ∈ H, n− ∈ N and x− = s (x+ ) ,

or, equivalently, b¯ = x · s(x)−1 n−1 − .

(4.2)

644


The same assertion of course holds true for H ⊂ B; in that case we have n− = 1. Let π : B → H be the projection map which assigns to h ∈ B the element x ∈ H satisfying (4.2). Lemma 4.8. Let H ⊂ G be the Cartan subgroup. The mapping p : H → H : x 7−→ x · s(x)−1 is an immersion. Put PH =

Rs · (τ − Id) (Id − Rs ) (Id − Rs · τ )

(4.3)

and define the Poisson bracket on H by {ϕ, ψ}H = hPH Dϕ, Dψi .

(4.4)

Lemma 4.9. p : H, {, }PH → H, {, }i(H) is a Poisson mapping. Hence to prove Proposition 4.6 we may use the Poisson bracket (4.4) instead of (4.1). Let ϕ, ψ ∈ C ∞ (H). Let ϕ∗ = ϕ ◦ π, , ψ ∗ = ψ ◦ π ∈ C ∞ (B) be their lifts to B defined via the twisted factorization map. (In other words, ϕ∗ b¯ = ϕ (x) , where b¯ = x · s(x)−1 n−1 − , x ∈ H.) In the right trivialization of the cotangent bundle of B the differential dϕ∗ (h) ∈ b∗ of ϕ∗ is given by dϕ∗ (h) = −τ −1 r− ∇ϕ,

(4.5)

where ∇ϕ ∈ h is the right invariant differential of ϕ evaluated at x = π (h) , and similarly for ψ ∗ . The standard embedding b∗ ⊂ g allows to regard dϕ∗ (h) as an element of g. To compute the Poisson bracket {ϕ, ψ} we may apply Proposition 2.8. We have {ϕ, ψ} (π (h)) = hPC , dϕ∗ (h) ∧ dψ ∗ (h)i .

(4.6)

Using formula (2.9) and inserting the expression (4.5) for the differentials we get (4.4). Let c : Ms → S be the canonical mapping which assigns to each L ∈ Ms the unique element L0 ∈ S lying in the same N -orbit. The generalized Miura transform m is defined by m = c ◦ i : H → S. Theorem 4.10. The generalized Miura transform is a Poisson mapping. The Poisson structure in H is given by (4.4); the Poisson structure in the target space is the reduced Poisson structure described in Theorem 2.5. The proof immediately follows from the fact that i (H) and S are different models of the quotient space Ms /N . Note that for G = SL(2) our construction of the Miura transform coincides with the one described in [9].


645

5. The SL(n) Case Our aim in this section is to compare the Poisson structures arising via the q-Drinfeld– Sokolov reduction with the results in [8]. (The case of n = 2 has been discussed in detail in [9]. Our analysis for general n is parallel to that of [9], Sect. 3, though our conventions are slightly different.) To begin with, let us list the standard facts concerning the structure of SL(n). We keep to the choice of order in the root system of sl(n) made in Sect. 1, that is, positive root vectors correspond to lower triangular matrices. We order simple roots in such a way that the Coxeter element s = s1 s2 · · · sn−1 is acting on the Cartan subalgebra h as a cyclic permutation, s−1 · diag (H1 , H2 , ..., Hn ) = diag Hn , H1 , ..., Hn−1 ; its representative in G = SL(n) is given by   0 −1 0 · · · 0  0 0 −1 · · · 0    s−1 =  · · · · · · · · · · · · · · ·  .  0 0 0 · · · −1  1 0 0 ··· 0 The automorphism π = −w0 of h is given by π · diag (H1 , H2 , ..., Hn ) = diag −Hn , −Hn−1 , ..., −H1 . We may choose the unipotent elements ui , i = 1, 2, ..., n − 1, in such a way that the principal nilpotent element f constructed in Lemma 4.3 is given by   1 −1 0 · · · 0  0 1 −1 · · · 0    f = · · · · · · · · · · · · · · ·;  0 0 0 · · · −1  0 0 0 ··· 1 the manifold Ms consists of matrices of the form  ∗ −1 0 · · ·  ∗ ∗ −1 · · ·  L = · · · · · · · · · · · ·  ∗ ∗ ∗ ··· ∗ ∗ ∗ ···

 0 0   · · ·. −1  ∗

Let x = diag(x1 , x2 , ..., xn ); the embedding i : H → Ms defined in Proposition 4.3 is given by   x1 x−1 −1 0 · · · 0 n  0 x2 x−1 −1 · · ·  0 1   . · · · · · · · · · · · · · · · i(x) := 3 =     0 0 0 · · · −1  0 0 0 · · · xn x−1 n−1 It is convenient to introduce affine coordinates on i (H) in the following way:

646




31 (z) −1  0 32 (qz)  ··· 3(z) =  · · ·  0 0 0 0

0 ··· 0 −1 · · · 0 ··· ··· ··· 0 ··· −1 0 · · · 3n q n−1 z

   . 

Let Lcan = m(3) ∈ S be the canonical form of 3,   0 −1 0 ··· 0  0  0 −1 · · · 0   Lcan =  · · · · · · · · · · · · · · ·  ;  0 0 0 · · · −1  1 u1 (z) u2 (z) · · · un−1 (z) we put up (z) := (−1)n−p−1 sn−p q n−p z . Proposition 5.1. ([9], Lemma 2) We have X 3j1 (z) 3j2 (qz) · · · 3jp q p−1 z . sp (z) =

(5.1)

1≤j1 <j2 p. 1 − q nm w n=−∞ Proof. Let ω = exp 2πi n be the primitive root of unity. The eigenvectors of s in h are ek = diag(1, ω −k , ..., ω −(n−1)k ), s · ek = ω k ek , k = 1, ..., n − 1. )(Id−Rs ) is given by the formal Laurent The kernel of the Poisson operator P˜H = (Id−τ Id−τ ·Rs series ∞ n−1 X X 1 (1 − q m ) 1 − ω k z m ek ⊗ en−k . n 1 − qn ωk w m=−∞ k=1

We have {3p (z) , 3s (w)}

(5.3) 1 (1 − q m ) 1 − ω k z m m(s−p) (ek · 3 (z))pp · en−k · 3 (w) ss q = m k n 1−q ω w m=−∞ k=1 ∞ n−1 X X 1 (1 − q m ) 1 − ω k z m = q m(s−p) ω k(s−p) 3p (z) 3s (w) . m ωk n 1 − q w m=−∞ ∞ X

n−1 X

k=1

Observe that


( 1−q n(N −1) 1 X 1 − ω k k(s−p) 1−q mn , if s = p, ω = −m −m(s−p) 1−q n 1 − qm ωk q 1−q mn , if s > p.

647

(5.4)

k

Substituting (5.4) into (5.3), we get (5.2). Formula (5.1) coincides with the q–deformed Miura transformation defined in [8]. Formula (5.2) coincides with the Poisson bracket on 3i (z)’s derived in [8]. Therefore in the case of sl(n) the Poisson algebra obtained by the difference Drinfeld–Sokolov reduction coincides with the q–deformed W–algebra introduced in [8]. Acknowledgement. The present paper is part of a joint research project which was started by E. Frenkel and N. Reshetikhin together with the first author. We are indebted to B. Kostant who has pointed out to the paper [17]; one of the authors (M.A.S.T.S.) would like to thank G. Arutyunov for useful discussion. The work of the second author was partially supported by the Royal Swedish Academy of Sciences grant no.1240.

References 1. Belavin, A.A., Drinfeld, V.G.: Solutions of the classical Yang-Baxter equation for simple Lie algebras Funct. Anal. Appl., 16, 159–80 (1981) 2. Bourbaki, N.: Groupes et algèbres de Lie ch. 4 a` 6. Paris: Hermann, 1968 3. Drinfeld, V.G.: A new realization of Yangians and quantized affine algebras. Sov. Math. Dokl. 36 (1988) 4. Drinfeld, V.G., Sokolov, V.V.: Lie algebras and equations of Korteweg–de Vries type. Sov. Math. Dokl. 23, 457–62 (1981); J. Sov. Math. 30, 1975–2035 (1985) 5. Faddeev, L.D., Volkov, A.Yu.: Abelian current algebras and the Virasoro algebra on the lattice. Phys. Lett. B315, 311–8 (1993) 6. Flato, M., Lichnerowicz, A., Sternheimer, D.: Deformation of Poisson brackets, Dirac brackets and applications. J. Math. Phys. 17, 1754 (1976) 7. Frenkel, E.: Affine Kac-Moody algebras at the critical level and quantum Drinfeld–Sokolov reduction. PhD Thesis, Harvard University, 1991 8. Frenkel, E., Reshetikhin, N.: Quantum affine algebras and deformations of the Virasoro algebra and W -algebras, Commun. Math. Phys. 178, 237–264 (1996); q-alg/9505025 9. Frenkel, E., Reshetikhin, N., Semenov-Tian-Shansky M..A.: Drinfeld–Sokolov reduction for difference operators and deformations of W -algebras I. The case of Virasoro algebra. Commun. Math. Phys. 192, 605–629 (1998) 10. Khoroshkin, S.M., Tolstoy, V.N.: On Drinfeld’s realization of quantum affine algebras. J. Geom. Phys. 11, 445–452 (1993) 11. Kostant, B.: The principal three-dimensional subgroups and the Betti numbers of a complex simple Lie group. Am. J. Math. 81, 973–1032 (1959) 12. Kostant, B.: On Whittaker vectors and representation theory. Invent. Math. 48, 101–184 (1978) 13. Kostant, B.: The solution to a generalized Toda lattice and representation theory. Adv. Math. 34, 13–53 (1980) 14. Semenov-Tian-Shansky, M.A.: What is a classical r-matrix? Funct. Anal. Appl., 17, 17–33 (1983) 15. Semenov-Tian-Shansky, M.A.: Dressing action transformations and Poisson–Lie group actions. Publ. Math. RIMS, 21, 1237–1260 (1985) 16. Semenov-Tian-Shansky, M.A. Poisson Lie groups, quantum duality principle and the quantum double. Contemp. Math., 175, 219–248 17. Steinberg, R.: Regular elements of semisimple algebraic Lie groups. Publ. Math. I.H.E.S. 25, 49–80 (1965) Communicated by G. Felder

Commun. Math. Phys. 192, 649 – 669 (1998)

Communications in


On the Aubry–Mather Theory in Statistical Mechanics A. Candel1 , R. de la Llave2,? 1 2

Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA

Received: 15 May 1996 / Accepted: 22 July 1997

Abstract: We generalize Aubry-Mather theory for configurations on the line to general sets with a group action. Cocycles on the group play the role of rotation numbers. The notion of Birkhoff configuration can be generalized to this setting. Under mild conditions on the group, we show how to find Birkhoff ground states for many-body interactions which are ferromagnetic, invariant under the group action and having periodic phase space.

1. Introduction In the motivation from Solid State Physics, Aubry-Mather theory describes the structure of solutions to the following problem. Frenkel and Kontorova proposed a very simple model for a one-dimensional crystal to describe the structure of dislocations. They considered a one-dimensional chain of identical atoms, connected by springs, placed in a periodic substrate potential V with period p. The potential energy of such a system or configuration u (indexed by the integers Z) describes the interaction of a particle u(j) with its neighbors, and is given by the formal expression S(u) =

X1 j∈Z

2

(u(j) − u(j + 1) − a)2 + V (u(j)).

The parameter a is the length of the connecting springs. The problem is then to find configurations u which minimize this potential, and to describe their structure. The first part of the sum above is the energy of the internal interaction between particles, the second is the external energy. In the absence of substrate potential, that is, when V = 0, the configurations of minimal energy are given by u(j) = aj + k. The mean ?

R. Ll. partially supported by research grants from the NSF

650

A. Candel, R. de la Llave

spacing is the rest distance of the spring. (This situation corresponds to the integrable case in the dynamical systems interpretation.) When V is large, it is reasonable to expect – and indeed it can be proved – that the particles settle around the minima of the potential V , hence that the configurations that minimize the energy are spaced with period p. Thus ground states appear as a compromise between two different periods, and one is then interested in describing their structure. The various possible configurations of the atoms in the chain are characterized by trajectories in phase space, which turns out to be orbits of a twist map of an annulus. This is obtained by looking at the variational equations that define the map. In general one finds periodic orbits, to which one can assign a rational rotation number. Other orbits have irrational rotation number, corresponding to incommensurate structures. Several authors (see [2, 6, 9]) have considered extensions of the Aubry-Mather theory to higher dimensions, that is, configurations on lattices Zn . In [2], a study is made of minimal configurations of the variational problem with nearest neighbor potentials; while in [6] the authors study variational solutions which are well-ordered (Birkhoff configurations). In fact, it is shown in [2] that in higher dimensions, there may be minimal configurations that are not well ordered. In [9] the Aubry-Mather theory is discussed for nearest neighbor potential for lattices in the plane. One can find there some interesting physical applications of this theory. For a more detailed survey of the physical significance of the theory of incommensurate crystals in several dimensions, the reader may consult [8]. The main goal of this paper is to generalize the setting of Aubry-Mather theory to configurations on sets more general than Zd and for general many-body interactions. We consider a set 3, together with potentials HB associated to finite subsets B of 3. We will need to assume that 3 admits a group action by a group G satisfying some mild assumptions. The potentials HB describe the interaction of a particle lying in 3 with its neighbors. The potential energy of such a system can be modeled by a formal sum S(u) =

X

HB (u),

B⊂3

where u : 3 → R is a configuration on 3. We will require that HB satisfy a ferromagnetic or twist condition, that they are invariant under the action of G and also satisfy some other periodicity assumption. Physically, we could think of the points of 3 as atoms, whose state is characterized by one number. The configurations describe what is the situation of all the atoms. We look for configurations u on 3 which are stationary points or minimizers for S and which have a prescribed rotation number. In our situation, the rotation number is a cocycle for the group G. Solutions to the variational equations with given rotation number always exist. But in order to find ground state configurations we have to require that the group G satisfies a (very natural) property. Essentially, this property says that we can exhaust 3 by fundamental domains for subgroups of G of finite index. We will discuss the well-order properties of the solutions that we find. These properties are the exact analog of the non-crossing properties of codimension one minimal surfaces of Riemannian manifolds, which obviously is the natural generalization of the relation between solutions to the one-dimensional Frenkel–Kontorova model and geodesics on a torus.

Aubry–Mather Theory in Statistical Mechanics

651

We think that the generality emphasizes the relevant features of the theory and that it would be of interest even if we were interested only in the one dimensional case. Nevertheless, we had several concrete examples that we wanted to consider. One of them was the following. Consider a completely homogeneous discrete subset of the hyperbolic plane, something we may call a hyperbolic crystal. This set, say 3, is assumed to be left invariant by a discrete cocompact group G0 of hyperbolic isometries, that is G0 ⊂ PSl(2, R) is its isotropy group. There is a natural potential in this situation, namely the one whose solutions are discrete harmonic functions. Our theory applies to this case, and to perturbations of it. Although hyperbolic crystals may not seem very realistic, let us mention the work of Kleiman and Saduc [5]. They suggest that amorphous structures could be regarded as projections of ordered patterns in hyperbolic space. They could be seen as models for disclinations in ordered Euclidean structures. Imagine we add wedges to a crystal. This operation decreases the curvature, thus the new crystal resembles a periodic pattern in hyperbolic space. Finally let us mention that the theory presented here applies to the Bethe lattice. Besides its applications in statistical mechanics, this seems a reasonable model for hierarchical networks, for example computer networks. It is not hard to imagine situations in which there is a penalty for computers to be out of phase and also for not being in phase with the environment that they are occupying. In the next section, we will describe the properties of the geometry of the lattice that we will need; in the following one, we will describe the properties of interactions. Then, we will state and prove the main theorem that states that there are solutions for the variational equations satisfying extra properties. Finally, we will briefly discuss some other problems. 2. Generalities on Groups and Cocycles The situation we will consider is as follows. There is a countable space 3 on which a finitely generated group G acts. We impose some conditions on the action of G on 3. First, it is effective, that is only the identity element of G acts trivially (this is no restriction because we can always pass to a quotient of G satisfying this property). Second, there is a finite fundamental domain for the action, that is, a finite subset F of 3 which intersects each orbit of G in exactly one point. For instance, 3 could be the set of vertices of some graph (we would like to keep in mind the edges as well, as we may use them for the path metric). Perhaps to avoid unnecessary complications we should assume that the graph is locally finite or uniformly locally finite (it does not become more and more populated as one goes to infinity). Now this more general situation includes the Cayley graph of a finitely generated group, graphs which are lifts of the 1-skeleton of a compact manifold to its universal covering, graphs which are quasi-isometric to manifolds of bounded geometry, etc. For example, it includes the Bethe lattice as the universal cover of the figure 8. Consult [3] for the thermodynamic properties of the Bethe lattice. In this non-commutative situation that we will consider, the role of rotation numbers in Aubry-Mather theory would be played by cocycles on the group G, that is maps σ:G→R 0

0

such that σ(γγ ) = σ(γ) + σ(γ ). The space of cocycles on G forms a real vector space, which will be denoted by H 1 (G; R).

652


We will assume that G acts on 3 with small stabilizers, meaning that if an element γ of G fixes a point of 3, then σ(γ) = 0 for any cocycle σ on G. For this to hold it is sufficient that the stabilizer of each point of 3 is a torsion group, or, more generally, that it is a group with trivial first cohomology). One technical result that we will need is the possibility of approximating arbitrary cocycles by simpler ones, namely Proposition 1. Let σ : G → R be a cocycle. Then there is a sequence of cocycles σn which converges point-wise to σ and such that σn has integral values when restricted to some subgroup Gn of finite index in G. Proof. Since G is finitely generated, H 1 (G, R) is a finite dimensional real vector space. Take a basis for it formed by cocycles with integral values, say {τ1 , . . . , τe }. Then we can write σ = a1 τ1 + . . . + an τn , some real numbers aj . For each k = 1, 2, . . . , choose rational numbers bi,k , i = 1, . . . , e, such that |ai − bi,k | < 1/k 2 . Then: |σ(γ) − b1,k τ1 (γ) − . . . − be,k τe (γ)| < |γ|1 /k 2 , where |γ|1 is the l1P -norm of γ in H 1 (G; R) with respect to the basis {τi }. Furthermore, this cocycle σk = i bi,k τi takes on rational values only. Since G is finitely generated, its image by η, say Pk , is a finitely generated subgroup of Q. Thus there is an integer m such that mPk = Z, and Pk /mPk is a finite group. In conclusion, there is a finite index subgroup of G such that σk has integer values when restricted to it. Finally, if γ ∈ G has |γ|1 < k, then for n > k we get |σ(γ) − σn (γ)| < |γ|1 /k 2 < 1/k, so that σn converges point-wise to σ as required. Also, kσ − ηk = sup |σ(γ) − σn (γ)|/|γ| ≤ sup |σ(γ) − σn (γ)|/|γ|1 < 1/k 2 , γ∈G

γ∈G

because |γ|1 ≤ |γ| as one easily verifies.

There is one property of the group that we will need to consider when proving existence of ground states, and it is that of being residually finite. This means that for each element of G, other than the identity, there is a finite index subgroup of G which does not contain it. Most familiar groups are residually finite, for instance, if R is a field and n ≥ 1, then a finitely generated subgroup of GL(n, R) is residually finite (see [12]). On the other hand, the group with presentation hx, y ; xy 2 x−1 = y 3 i is not residually finite (Baumslag–Solitar). The reason for requiring this property is that it would allow us to do a kind of renormalization needed later in our discussion. More specifically we need the following: Proposition 2. Let G, 3 be as above with G residually finite. Then there is a sequence G1 ⊃ G2 ⊃ . . . of finite index subgroups of G whose fundamental domains exhaust 3. Proof. If F is a fundamental domain for G, we can write 3 = F ∪ g1 F ∪ . . . , where gi are the nontrivial elements of G. By induction, choose a subgroup Gn of finite index in Gn−1 which does not contain gn . (Recall that residually finite is a property inherited by finite index subgroups.) Since a fundamental domain for Gn may be constructed by adding the translates of a fundamental domain of Gn−1 by representatives of elements of Gn−1 /Gn , the proof is complete.


653

3. Configurations, the Birkhoff Property and Rotation Cocycles A configuration on 3 is a map u : 3 → R. The set of all configurations on 3 has the structure of a real vector space with the obviously defined operations, and we denote it by C(3). The space of configurations is partially ordered in the following sense. We say that two configurations u, v are such that u ≤ v if u(p) ≤ v(p) for all p in 3. This makes C(3) into a partially ordered vector space. One metric we want to consider on the space of configurations is the one induced by the family of semi-norms |u(p)|, p ∈ 3, which gives rise to the pointwise convergence topology. An explicit form for this metric is the following: if we enumerate the points of 3 as p1 , p2 , · · · , pn , · · · , an explicit form is d(u, v) =

X 1 |u(pn ) − v(pn )| . 2n 1 + |u(pn ) − v(pn )| p ∈3 n

We denote kuk = d(0, u) and note that ktuk ≤ (1 + |t|)kuk for any real number t. The action of G on 3 extends to one on C(3), defined in the following manner: if γ ∈ G and u ∈ C(3), then Tγ u(p) = u(γ(p)) for any element p of 3. It is evident that this action of G on C(3) preserves the partial order on configurations. We will denote by χA the characteristic configuration associated with a subset A of 3, and which is defined as χA (p) = 1 if p ∈ A, χA (p) = 0 otherwise. We define operators RA : C(3) −→ C(3) by RA (u) = u − χA . It is clear that they are invertible and furthermore that if A, B are subsets of 3, then the operators RA and RB commute. We denote by 3α , α = 1, . . . , l the equivalence classes of 3 modulo G, and denote by Rα the operators corresponding to the sets 3α . Given any s in Zl , s = (s1 , . . . , sl ), we denote by Rs the operator Rs11 . . . Rsl l . A cocycle σ defines a configuration uσ on 3 as follows. Let F be a fundamental domain for the action of G on 3. Any p of 3 can be written as p = γq for a unique q in F . Then define uσ (p) = σ(γ). It is elementary that uσ is well-defined, for if also p = γ1 q, then γ −1 γ1 stabilizes q, so σ(γ) = σ(γ1 ) by our hypothesis. Note that uσ (q) = 0 for all the elements of F . Then, with these conventions, we say that a configuration u is of type σ if sup |u(p) − uσ (p)| < ∞.

p∈3

The set of configurations of type σ will be denoted by Oσ . Put in another way, once a fundamental domain F ⊂ 3 is fixed, any cocycle σ on G defines a configuration and Oσ is the subspace of C(3) formed by those configurations u at bounded distance from uσ . Thus Oσ is an affine space modeled on `∞ (3, R). The induced metric in Oσ is |u(γ) − v(γ)|∞ = sup |u(γ) − v(γ)|, γ∈G

which makes it into a Banach space.

654


Proposition 3. The space Oσ is independent of the fundamental domain. This is an elementary consequence of the cocycle property. Also, the same property implies: Proposition 4. The operators Tγ , γ ∈ G, and R leave Oσ invariant. Next we introduce the notion of Birkhoff configuration. A configuration u of type σ is called a Birkhoff configuration for the group G if σ(γ) ≥ (≤)sα for γ ∈ G, s ∈ Nl only when Tγ (u) ≥ (≤)Rs (u). The set of Birkhoff configurations in Oσ is denoted by Bσ . Since Rs (u)(p) = u(p) − sα if p belongs to the equivalence class 3α of 3/G, we see that u is a Birkhoff configuration if either Tγ u + σ(γ) ≥ u or Tγ u + σ(γ) ≤ u for the partial order of configurations. It also follows immediately that: Proposition 5. For every γ in G there exists s+ , s− such that Rs− (u) ≤ Tγ (u) ≤ Rs+ (u) for any u in Bσ . Although we have defined the Birkhoff property of a configuration by referring to a given cocycle, we could have avoided it. We show next that the non-intersecting property of a configuration implies that there is a cocycle for which it is Birkhoff. That is, suppose that u : 3 → R is a configuration which satisfies the following property: for each γ ∈ G and any integer k ∈ Z, we have u(γp) + k ≤ u(p) for all p ∈ 3, or

u(γp) + k ≥ u(p)

for all p ∈ 3. We are going to define a cocycle σ : G → R such that u ∈ Oσ . First, let τ + (γ) = sup(u(γp) − u(p)) p

and

τ − (γ) = inf (u(γp) − u(p)). p

Note that they are finite numbers. Furthermore, τ + is a sub-cocycle, that is, τ + (γ1 γ2 ) ≤ τ + (γ1 ) + τ + (γ2 ), and τ − is a super-cocycle. This implies that we can define τ + (γ n ) τ − (γ n ) = lim , n→∞ n→∞ n n

σ(γ) = lim

because both limits exist and are equal. We also see that τ + ((γ1 γ2 )n ) τ + (γ1 (γ2 γ1 )n−1 γ2 ) = lim ≤ σ(γ2 γ1 ) n→∞ n→∞ n n

σ(γ1 γ2 ) = lim

by elementary use of the sub-cocycle and super-cocycle properties and of the definition of σ. Therefore σ(γ1 γ2 ) = σ(γ2 γ1 ).


655

It is also elementary to check that σ(γ −1 ) = −σ(γ), and that, if γ1 and γ2 commute, then σ(γ1 γ2 ) = σ(γ1 ) + σ(γ2 ). From all these properties it follows that σ is a cocycle. Finally we show that the configuration u has σ as rotation cocycle, that is, that sup |u(p) − σ(τp )| < ∞,

p∈3

which is the same as to show that sup

|u(γq) − σ(γ)| < ∞.

γ∈G,q∈F

Here F ⊂ 3 denotes a finite fundamental domain for the action of G. Fix q ∈ F and γ ∈ G, and consider the configuration n ∈ Z 7→ u(γ n q). Since τ − (γ n ) ≤ u(γ n q) − u(q) ≤ τ + (γ n ), it follows from the definitions that it has rotation number σ(γ). As is well-known, this implies that u(q) + nσ(γ) ≤ u(γ n q) < u(q) + nσ(γ) + 1 from which it follows that |u(γq) − σ(γ)| ≤ 1 + |u(q)|. Since the fundamental domain F is finite, this implies that the rotation cocycle of u is σ, as we had claimed. Proposition 6. If u is a configuration satisfying the well-ordering property above, then there is a cocycle σ on G for which u ∈ Bσ . In particular, Corollary 1. If the group has no nontrivial cocycles, then every Birkhoff configuration is bounded. A simple example of a group without cocycles is G = ha, b; ap = bq = (a−1 b)r = 1i with 1/p + 1/q + 1/r < 1. This group can be realized faithfully as a discrete group of isometries of the hyperbolic plane. However, it has finite index subgroups which do have nontrivial cocycles. We also mention the following elementary property of Birkhoff configurations. Proposition 7. The space of Birkhoff configurations is closed for the product topology on R3 . Furthermore, for any cocycle σ, the set Bσ is non-empty and convex. Proof. For the first part, simply note that by the previous proposition the space of Birkhoff configurations is characterized by the inequalities Tγ u + n ≥ u or Tγ u + n ≥ u, so it can be written as an intersection of closed sets. The configuration uσ associated to the cocycle is Birkhoff, so Bσ is non-empty. The convexity property is obvious.

656


Similarly, we also have Proposition 8. Let σ be a cocycle in G, and consider Oσ with the Banach space topology given by uniform convergence. Then the space Bσ of Birkhoff configurations in Oσ is closed. In analogy with rational rotation numbers, we say that a configuration u has rational rotation cocycle if there is a normal finite index subgroup G0 of G and an integer valued cocycle σ : G0 → Z such that u(γ(p)) = u(p) + σ(γ) for all p ∈ 3. This definition could have been formulated in a slightly different way by using the following two propositions. Proposition 9. Let σ : G0 → Z, where G0 is a finite index normal subgroup of G with G/G0 commutative. Then σ extends to a rational cocycle on G. Proof. For γ in G, let n be any nonzero integer such that γ n ∈ G0 . Then define σ(γ) = σ(γ n )/n. It is immediate to check that σ : G → Q is well defined and, because G/G0 is commutative, it is a homomorphism. Proposition 10. Let u ∈ C(3) be a configuration such that u(γ(p)) = u(p) + σ(γ) for all p in 3 and all elements γ of a finite index normal subgroup G0 of G, and for σ : G0 → Z. Suppose that σ extends to a cocycle on G. Then u ∈ Oσ , σ the extension of the cocycle to G. Proof. Let p0 , . . . , pr be a finite set such that the translates {γ(pi ); γ ∈ G0 } fill up 3. Choose an upper bound M for the finite sets of numbers {|u(p0 )|, . . . , |u(pr )|} and {|σ(τp0 )|, . . . , |σ(τpr )|}. Then, for any p ∈ 3 we have: |u(p) − σ(τp )| = |u(γ(pk )) − σ(τp )| = |u(pk ) − σ(τpk )| ≤ 2M. We will use another property of Birkhoff configurations that we now explain. Proposition 11. Suppose that u is a Birkhoff configuration that has integer valued rotation cocycle σ : G → Z. Suppose that u is periodic with respect to some finite index subgroup G0 of G, that is, u(γ(p)) = u(p) + σ(γ) for all p in 3 and all γ in G0 . Then u is also G-periodic. The proof of this fact goes as follows. Let γ be an element of G. Then γ k ∈ G0 for some integer k because G/G0 is finite. If u is not γ-periodic, then let p be such that u(γ(p)) > u(p) + σ(γ). Using the Birkhoff property of u this implies that u(γ k (p)) > u(p) + kσ(γ), which contradicts the G0 -periodicity of u.


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4. Interaction Potentials and the Variational Problem Let S denote the collection of finite nonempty subsets of 3. An interaction potential is a collection of maps H = {HB ; B ∈ S}, where each HB : C(3) → R, such that HB (u) = HB (v) whenever u and v agree on B (so HB is interpreted as a function HB : RB → R), and for any finite subset X of 3, the series X HB (u) H(u, X) = B∩X6=∅

converges. We call H(u, X) the (total) energy of u in X. A potential is absolutely summable if the series X |HB |∞ B,p∈B

converges for all p ∈ 3. Denoting by |H|p be the value of the series above, we have a family of semi-norms in the space P of absolutely summable potentials, making it into a Frechet space. A potential H is said to be of finite range if for each p ∈ 3 there is some finite set Bp ⊂ 3 such that HB = 0 unless B ⊂ Bp for all p ∈ B. We say that H is of bounded range if there is a number M such that HB = 0 if diam(B) > M . Clearly, if H is finite range and all HB are bounded, then H ∈ P. Moreover, absolutely summable finite range potentials are dense in P. A configuration u is a ground state for the interaction potential H = {HB } if H(u, X) ≤ H(v, X) for any finite set X and any configuration v such that u = v on 3 \ X. Thus u is a ground state if the energy of any finite perturbation of u exceeds that of u. Our goal is to seek minimal configurations which belong to the spaces Oσ . Note that if u is a minimal configuration for the energy problem in Oσ , it is also a minimal configuration for the global problem in C(3), simply because if u ∈ Oσ and u = v outside some finite set X, then also v ∈ Oσ . On the other hand, note that a configuration that minimizes H(·, X) in Bσ may not necessarily be a ground state configuration. Some other properties of ground state configurations that we will use later are collected in the following proposition. Proposition 12. The set of all ground state configurations for a continuous potential H is closed in R3 with the product topology. If O denotes the space of configurations at bounded distance from a fixed one, the space of ground states in O is closed in the Banach space topology. Proof. Suppose that a sequence un of ground state configurations converges point-wise to the configuration u. If u is not a ground state configuration, then there is a finite set X and a configuration v which equals 0 outside X and such that H(u, X) − H(u + v, X) ≥ a > 0. On the finite set X, the convergence un → u is uniform, and the continuity of the potential implies that, for large n we have H(un , X) − H(un + v, X) > 0,

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contradicting the minimality of the un ’s. The same argument also proves the second part, because if un ∈ O converges to u in the Banach space topology then it also converges uniformly on each finite subset X of 3. We introduce some more definitions in order to restrict the type of potentials to be considered. An interaction potential H = {HB } is G-invariant if HγB (u) = HB (Tγ u) for all γ ∈ G, all B and all configurations u on 3. If the interaction potential H satisfies HB (Rs u) = HB (u) for all configuration u, all B and all s ∈ Zl , then we say that H has periodic phase. This periodic phase property allows us to consider the HB ’s as maps C(3)/R → R. The Banach space structure of Oσ permits us to write down the variational equations if the potential is differentiable. Thus, with the assumption of differentiability of potentials, a necessary condition for u to be a minimal configuration is that it satisfies the variational equations X ∂ HB (u) = 0. ∇H(u, X) = ∂p X∩B6=∅,p∈B

This sum converges for finite range interaction. In general, we have to make some further restrictions. Let O be a convex subset of the set of configurations. We say that an interaction H is C r -bounded on O if X kHkr,O = sup |Dj HB (u)| < ∞. u∈O

|j|≤r

Note that the space O in the definition above may be a proper subset of one of the Oσ ’s. Examples show that interaction may not be bounded in Oσ but they are in subspaces of the form O = {u; |u − uσ |∞ ≤ K}. If H is a C 1 -bounded potential on O, we define a map u 7→ V (u) on configurations u ∈ O with values in `∞ (3) by X ∂ HB (u), V (u)(p) = − ∂p B3p

and therefore the variational equations can be written as V (u) = 0. In the cases that we will consider O will be an affine space over `∞ (3), in particular one of the Oσ , and the hypothesis of C r boundedness of H will ensure that the r − 1 derivative of V exists and is uniformly bounded in the sense of derivatives of Banach spaces. Since Oσ is a Banach space isomorphic to `∞ (3) the natural interpretation of V is as a vector field on Oσ . In any case, the above definitions show that the function V associated to a C r bounded potential H on O is globally Lipschitz, that is:


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Corollary 2. If V is defined as above, then |V (u) − V (v)|∞ ≤ kHk2 |u − v|∞ . Before we state the main theorem on the existence of minimal configurations we still need a new definition concerning the potentials to be considered. This is the twist condition of Hamiltonian mechanics or the ferromagnetic property in statistical mechanics. More precisely, we say that an interaction potential H which is C 2 -bounded on O satisfies the twist condition if X ∂2 HB (u) ≤ 0 ∂p∂q

B3q

for all configurations u in O and all p, q in 3 with p 6= q. This property is obviously implied by the strong twist condition, that is ∂2 HB (u) ≤ 0 ∂p∂q for all u and all p 6= q.

5. Existence of Solutions to Variational Equations In this section we prove the existence of solution to the variational equation of a twist potential. The idea is inspired by [4] and [6]. There the strategy of the proof is to consider the flow φt : Oσ → Oσ defined by the differential equation d φt (u) = −V (φt (u)). dt Since the right-hand side is globally Lipschitz, solutions exist for all times. Then we will argue that this flow preserves the Birkhoff condition on configurations, so it is actually a flow on Bσ . Furthermore, it commutes with the operators R so that the end result is a flow on Bσ /R. Once here we will be able to use compactness and therefore fixed points for the flow. Unfortunately, for this outline to work we would need to add a further condition on the space 3, namely that it has polynomial growth function with respect to the action of G. In general this case is the exception rather than the rule. To overcome this difficulty we consider a sequence of gradient flows that converge to the one defined by the variational equations. Theorem 1. Let G be a finitely generated group acting on 3 with finitely many orbits, and with small stabilizers. Let σ be a cocycle on G and let H be an interaction potential which is C 2 -bounded on Oσ and which is G-invariant and has periodic phase. Assume that either H satisfies the strong twist condition or that H satisfies the twist condition and is of finite range. Then there is a solution of the variational equations which lies in Bσ .

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Suppose that H is of finite range. For any finite subset F in 3 consider the function HF on configuration space defined by X HB (u). HF (u) = B∩F 6=∅

Let −∇HF be the gradient vector field defined by this function and let φF be the flow defined by the differential equation d φF (u, t) = −∇HF (φF (u, t)). dt Proposition 13. The flow φF is defined on Oσ for all times. Indeed, the gradient ∇HF is a Lipschitz vector field in Oσ with the Banach space topology. Proposition 14. The flow φF (t, u) defined by the differential equation d φF (t, u) = −∇HF (φF (t, u)) dt preserves the order structure on the space of configurations. The proof of this proposition is along the lines of the one in [6]. Briefly, the idea is that the twist condition implies that the corresponding linearized equation is given by a matrix with positive entries off the diagonal. This is shown to imply the monotonicity of the gradient flow. A consequence of this and the fact that the interaction potential has periodic phase is that φt induces a flow on Bσ /R. Proposition 15. If the interaction potential H is ferromagnetic, then HF is nonincreasing along the flow, that is d HF (u) ≤ 0. dt We compute the derivative along the flow φ : O × R → O defined by the differential equation. By the chain rule we have that: d d HF (φF (u, t)) = −∇HF (u) φF (u, t) dt dt ! ∂ X =− HB (u) · ∂p B p∈BF 2 X ∂ = HB (u) ≤ 0. ∂p

!

∂ X HB (u) ∂p B

p∈BF

p∈BF

By the order preserving property, we can consider the flow on Bσ /R. The last detail we need is: Proposition 16. The space Bσ /R is compact (for the pointwise convergence metric).


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Proof. Let F be a fundamental domain for the action of G on 3. If u is a configuration, we can apply one of the operators R so that u(p) ∈ [0, 1] for p ∈ F . If furthermore u is a Birkhoff configuration, then σ(γ) ≤ u(γp) − u(p) ≤ σ(γ) + 1. Hence we can view Bσ /R as a subspace of an infinite product of circles. It is also closed, hence compact. Note that the potential HF induces a map on this quotient space because of the invariance of the interactions HB under the transformations R. Since HF is bounded below in this compact space, and it is non-increasing along the orbits of the flows, there must be a sequence tn → ∞ for which d φF (u, tn ) → 0. dt Therefore, if we start with a Birkhoff configuration u, we obtain a sequence of Birkhoff configurations φ(u, tn ) which is compact in the space Bσ /R. Thus there is a convergent subsequence whose limit is a critical point uF . Consider now an increasing sequence F0 ⊂ F1 ⊂ · · · of finite sets exhausting 3. For each of them we have a critical point un in Bσ for the gradient flow ∇Hn . After modification by the operators R, which preserves the property of being a critical point, we see that these sequence of points has a subsequence which converges point-wise to u, a configuration in Bσ . To finish the proof of the theorem we only need the following: Proposition 17. This configuration u satisfies the variational equations V (u) = 0. Proof. We have to check that V (u)(p) = 0 for any p in 3. Fix p and choose n large enough so that if B contains p then B ⊂ Fn . The convergence un → u is uniform on Fn , and ∇Hn (w)(p) = V (w)(p). It follows that u satisfies the variational equations. 6. Properties of Ground State Configurations For a configuration u and a point p in 3 define the energy of u due to p as Ep (u) =

X 1 HB (u). |B|

B3p

For X a finite subset of 3 let hX (u) =

1 X Ep (u). |X| p∈X

We would like to take the limit as X approaches 3, but there is no canonical way to do so. We will regard it as a function on finite parts of 3, X ∈ F(3) 7→ hX (u). One possibility would be to take a fundamental domain X for G, list the elements of G as {e, γ1 , γ2 , . . . } and set Xn = X ∪ . . . ∪ γn X. Then, if u is a configuration which has integer periods with respect to a finite index subgroup G0 of G, we see that the limit lim hXn (u)

n→∞

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exists and is equal to the specific energy of u in a fundamental domain for G0 . We want to mention one property about the specific energy of configurations. In [2] this property is proved under the assumption that the configuration is minimal and satisfies the Birkhoff condition. It appears that considerably less is needed. We include a proof for completeness. Proposition 18. Let the interaction potential H be of finite range, G-invariant and satisfy the strong ferromagnetic condition. Let σ be a cocycle on G. If u is a configuration which belongs to Oσ then there is a constant K(u, σ) such that H(u, X) ≤ K(σ, u) Card(X), for any finite subset X of 3. Proof. Let u be a configuration in Oσ . Let uσ be a configuration defined by the cocycle σ. Applying the ferromagnetic hypothesis in the development provided by Taylors theorem we have that for any B with HB 6= 0 there are positive constants α(B), β(B) such that X HB (u) ≤ α(B) (u(p) − u(q))2 + β(B). p,q∈B

This follows from Taylor’s theorem. The constants α and β depend on σ, on the distance between u and uσ , and on B. There are only finitely many B’s up to translation, so the invariance of H allows us to choose α and β independently of B. Next note that for any p, q and for u in Oσ we have previously shown that |u(p) − u(q)| ≤ C + |σ(γpq )|, where C is a constant and γ is any transformation that takes p to q. Therefore, if the interaction potential is of bounded range, there is a constant C such that |u(p) − u(q)| ≤ C for any p, q belonging to the same set B with HB 6= 0. We now put all this information together. Let X be a finite set and u ∈ Oσ . We have H(u, X) =

X

HB (u)

B∩X6=∅

≤

X X

α(u(p) − u(q))2 + β

B p,q∈B

≤

X

n2B (C 2 + β)

B

≤ K(u, σ)Card(X). Aubry’s fundamental lemma for the Frenkel–Kontorova model shows the nonintersecting property of ground state configurations. The analysis of Aubry and Mather of configurations on the line which are minimal for the Lagrangian shows that they satisfy a non-intersecting property, namely, if u : Z → R is a ground state configuration, then it is Birkhoff. In higher dimensions this need not be true, and Blank [2] provides the following example:


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H(u1 , u2 , u3 ) = (u1 − u2 + b1 )2 + (u1 − u3 + b2 )2 , which admits f (x1 , x2 ) = x1 x2 as a solution to the variational equations. Blank showed that a similar property holds for minimum energy configurations for the model he studies in [2]. The following proposition exhibits a similar property for the models we study. From a geometric point of view, it is essentially the maximum principle for minimal surfaces. The space 3 has no structure, but given an interaction potential H = {HB }, those sets B for which HB 6= 0 define a sort of topology on 3. So if X is a finite subset of 3, we will denote by N (X) the union of those sets B which intersect X and with HB 6= 0, and by X 0 the subset of those elements p of X for which there is some B which meets X in some other points different from p. Although these definitions may look strange, they are simply verified for models whose supporting set 3 corresponds to the vertices of a graph and the interaction potential is defined in terms of the connecting edges. First we need some new terminology. Say that a subset X of 3 is connected if for any pair of elements p, q of it, there is a sequence p0 , . . . , pk in X with p0 = p, pk = q, and sets Bi , each containing a pair pi , pi+1 , and for which HBi 6= 0. The meaning of this condition is obvious. Each connected component of 3 behaves in its own independent way, and there is no restriction in assuming our models have connected state space. Proposition 19. Let u and v be ground state configurations for the ferromagnetic potential H on the configuration space of 3. Suppose that there is a finite connected set X such that u ≥ v on N (X). Then, either u = v or u > v in all of X. Proof. We use the standard technique of the Hilbert integral in calculus of variations. Any ground state configuration must satisfy the variational equations. Thus, if p ∈ X we get X ∂ [HB (u) − HB (v)] = 0, ∂p B3p

and replacing the terms in the sum by their integrals, XZ 1 d ∂ HB (tu + (1 − t)v)dt = 0. 0 dt ∂p B3p

This expression may be written as: XX B3p q∈B

Z

1

[u(q) − v(q)] 0

∂2 HB (tu + (1 − t)v)dt = 0. ∂q∂p

Let p ∈ X 0 be such that u(p) = v(p). For q 6= p in X 0 , the term u(q) − v(q) in the sum above is ≥ 0 by hypothesis, while the integral term is strictly negative due to the ferromagnetic condition. This forces u = v through all of N (p). Now the fact that X is connected allows us to extend the equality u = v to all X. Corollary 3. Let u, v be two critical configurations on 3, which is assumed to be connected. If u ≤ v, then either u = v or u < v. Corollary 4. Let X be a connected finite subset of 3 and let u,v be two configurations which agree outside X and such that they both minimize H(·, X) with boundary conditions u3−X . Then u = v on X.

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7. Existence of Ground State Configurations The result we just proved guarantees the existence of Birkhoff critical configurations with arbitrary rotation number. In this section we show that Birkhoff ground states exist with arbitrary rotation cocycle. Here we have to assume that the group is residually finite, and that the space 3 is connected with respect to the interaction potential to be considered. Theorem 2. Let G be a finitely generated residually finite group acting on the set 3 with finitely many orbits and with small stabilizers. Let H be a finite range interaction potential on 3, making 3 connected, which is G-invariant, has periodic phase, and satisfies the strong ferromagnetic condition. If σ is a cocycle on G, then there are Birkhoff configurations on Bσ which are ground states. We start with a rational cocycle σ : G → Q, integer valued on the finite index subgroup G0 . We let F be a connected fundamental domain for G0 . Consider the Hamiltonian X HB . HF = B∩F 6=∅

We also consider an exhaustion F = F0 ⊂ F1 · · · of 3 by fundamental domains with the property that the sets B which intersect Fn are contained in Fn+1 , and they correspond to a tower of finite index subgroups of G0 . We denote by Hn the Hamiltonian corresponding to Fn . The problem is to minimize Hn with periodic boundary conditions. These boundary conditions give us the constraints for a Lagrange multiplier problem. A configuration u which is Gn -periodic is completely determined by its values on Fn . If q is a point outside Fn then there is a unique p in F such that u(q) = u(p)+σ(γ) for some γ in Gn . (There could be several γ’s but the result is independent of that because of the small stabilizers condition.) Let N denote the union of all B’s with HB 6= 0 which intersect F . The problem is then to minimize H for configurations u on N subject to the constraints determined by the periodicity condition. We obtain one linear equation of the form gq (u) = u(q) − u(p) − σ(γ) for each point q in N \ F . For p in F denote by η(p) the number of points equivalent to it in N . We start by discussing the problem of minimizing Hn over Gn -periodic configurations. First, we consider Hn as a function on RFn . This can be done because any element in this space extends in a unique way to a configuration on 3, and then we apply Hn . Now the twist condition implies that this function attains its minimum over a compact set. It follows that we obtain a periodic configuration un which minimizes Hn among periodic configurations. Proposition 20. The function Hn has a minimum on the space of Gn periodic configurations. We want to show that critical configurations of this system are ordered. The method of Lagrange multipliers provides a system of equations which we now describe. First there are real numbers λq , one for each q in N \ F , and λ0 which is either 0 or 1 (and


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so that not all λ’s are zero). The equations are as follows: if p ∈ F has no other point of its orbit in N ,then X ∂ ∂ H(u, N ) = HB (u) = 0, ∂p ∂p B3p

if the orbit of p ∈ F meets N in the points q1 , · · · , qk (other than p), then λ0

X ∂ HB (u) = λ1 + · · · + λk ∂p

B3p

for the scalars λ corresponding to the points in N \ F equivalent to p, and if q ∈ N \ F is in the orbit of p ∈ F we get λ0

X ∂ HA (u) = −λ∗ , ∂q

A3q

where the sum is over the sets A that meet F . We then see that we can take λ0 = 1 and the last two equations can be combined into one which is independent of the scalars λ and holds for any critical configuration: X ∂ HB (u) + ∂p

B3p

X

X ∂ HA (u) = 0 ∂q

q∈Op ∩N A3q

(here Op denotes the orbit of p in 3). This last equation can be rewritten in the form X B3p

nB,p

∂ HB (u) = 0 ∂p

because of the periodicity of the configuration and of the interaction potential. The index nB,p is a positive integer which comes out of the rearrangement of the sum over the A’s. Now if the interaction potential satisfies the strong ferromagnetic condition, the same technique of the Hilbert integral allows us to obtain a sort of Aubry’s fundamental lemma. Proposition 21. If the interaction potential satisfies the strong twist condition, then two Gn -periodic configurations u and v that minimize Hn and such that u ≥ v are either equal or u > v. Proof. The proof is essentially the same as the one in the previous section. First start with a point in Fn which is equivalent to no point of N (Fn ). If u = v at this point, the proof in the previous section shows the u = v in the whole neighborhood of the point. For points of Fn that have some equivalent in N (Fn ), the last equation we have written allows us to use again the Hilbert integral technique. In conclusion, if u(p) = v(p) at some point of Fn , we obtain that u = v through all of Fn , as we are assuming the fundamental domains to be connected.

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Hence, if u, v are periodic and minimize Hn we can consider the configurations u ∨ v = max{u, v} and u ∧ v = min{u, v} which are also periodic, and the twist condition implies Hn (u) + Hn (v) ≥ Hn (u ∨ v) + Hn (u ∧ v). Hence these configurations also minimize Hn . By applying the argument of the previous paragraph to the pairs u, u ∨ v, etcetera, we conclude that either u > v or v > u or they are equal. Before proceeding we note the following fact. Lemma 1. If u is a Gn -periodic configuration, so is Tγ u for any γ in G. Proof. If γ1 ∈ Gn , then by normality of Gn in G, there is γ2 in Gn such that γγ1 = γ2 γ. Hence σ(γ1 ) = σ(γ2 ) and Tγ u(γ1 p) = u(γγ1 p) = u(γ2 γp) = u(γp) + σ(γ2 ). From this the Birkhoff property follows as we now show. Proposition 22. Suppose that u is Gn -periodic and minimizes Hn . Then it is a Birkhoff configuration. Proof. We show that for any γ in G we have |Tγ u − u − σ(γ)| ≤ 1. If [σ(γ)] denotes the integer part of σ(γ), it follows from the previous lemma that the configurations u = Tγ u − [σ(γ)] and u = u − 1 are also Gn -periodic and minimize Hn . Hence the differences u − u and u − u have constant sign. Clearly these differences are invariant by Gn , and so we can determine their sign by adding over a fundamental domain for Gn . To do this, observe that for a Gn -invariant configuration v we have X Tγ v − v = σ(γ)|Fn | Fn

and in fact the sum can be taken over any fundamental domain for Gn . Hence X u − u = (σ(γ) − [σ(γ)])|Fn | ≥ 0, Fn

and similarly

X

u − u = (σ(γ) − [σ(γ)] − 1)|Fn | ≤ 0.

Fn

It follows that 0 ≤ Tγ u − u − [σ(γ)] ≤ 1, which is equivalent to the Birkhoff property. This discussion has the following consequence. We start with a tower of connected fundamental domains F = F0 ⊂ · · · ⊂ Fn ⊂ · · · corresponding to normal finite index subgroups of G0 ⊃ G1 ⊃ . . . of G, then we obtain a sequence of configurations u0 , u1 , . . . , where un is periodic with respect to Gn and minimizes Hn . Furthermore, since these configurations satisfy the Birkhoff property and the cocycle is integer valued on G0 , they are all periodic with respect to G0 . By invoking the operators R, we may assume that this sequence {un } has a subsequence which converges pointwise to a configuration u. Hence u is a Birkhoff configuration. Furthermore,


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Proposition 23. The configuration u is a ground state for H. Indeed, let v be a configuration that agrees with u outside a finite subset X of 3. Then X ⊂ Fn for some large n. We can find a larger m such that every set B which meets Fn is inside Fm . Now restrict v to Fm and extend to 3 by periodicity with respect to Gm . This does not change the value of H(v, Fn ). Then, for k ≥ n, H(v, Fn ) ≥ H(uk , Fn ). Since un converges uniformly to u on Fn , we take the limit with respect to k and the ground state property of u is verified. The case in which the cocycle σ is arbitrary is now immediate. We approximate σ by rational cocycles. For each one the argument just described produces a Birkhoff ground state. By using the operators R, we may assume this sequence converges. The limit is then a Birkhoff ground state for σ. 8. Examples and Problems The typical example is the following. Let G be a group with a finite generating set S = S −1 . With it we construct a graph whose vertices are the elements of G and whose edges are labeled by the elements of S, that is, there is an edge from g to h if h = gs, and we denote it by (g, s, gs). This graph is made into a metric space by taking each edge to be isometric to the unit interval in the real line. Furthermore, there is a natural left action of G on 0, namely, g(v, s, vs) = (gv, s, gvs), which is effective and transitive. There is a natural potential H defined by HB (u) = (u(p) − u(q))2 if p and q are connected by an edge, and trivial otherwise. The ground state configurations of this model are harmonic functions on the group G, where the discrete Laplacian is defined by 1 X u(q) − u(p), 1u(p) = n(p) q∼p where n(p) is the number of edges emanating from p, and q ∼ p means that q is connected to p by an edge. It is then elementary that any cocycle on G defines a harmonic function. This corresponds to the integrable case. The Frenkel–Kontorova model in the group G involves also the site potentials Hp (u) = V (u(p)), where V is a periodic function satisfying V 00 < 0. The variational equations are X u(gs) + V 0 (u(g)) = 0. n(g)u(g) − s∈S

Most of what was done for Z carries over to this general situation. In physics there are two types of lattices which are frequently used. One is the Euclidean lattice Zd , which is amenable, and the other goes by the name of Bethe

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lattice, non-amenable. This one is typically used to exemplify strange phenomena in comparison with the Euclidean lattices (and non-amenability is responsible for this), and also because computations for models on it are simplified by the fact that Bethe lattices are trees. From our point of view, these lattices are the graphs of free products of groups of the form G1 ∗ · · · ∗ Gn , where each Gi is a copy of the infinite cyclic group Z or of the two element group Z/2Z. They are finitely generated groups and it is elementary how to compute their first cohomology group. Therefore our theory applies to them. To conclude, we mention some problems that appear to have some interest. Of course, the theorem of existence of quasi-periodic solutions is only the first step in the very rich Aubry-Mather theory and it would be interesting to see how much generalizes to the present setting. On the other hand, there are problems that seem to be suggested by the present formulation. The first concerns Gibbs states for the model 3, H, G. It would be interesting to know what type of ground states are contained in the support of a Gibbs state. For instance, under what conditions is it true that the Dirac mass supported on a Birkhoff ground state is a Gibbs state? Examples related to this question are the Shlosman staircases (see [3]). In a more geometrical vein, it would be interesting to develop a theory of minimal surfaces in Riemannian manifolds corresponding to the relation between the Frenkel– Kontorova model and geodesics on a torus. For manifolds of dimension three the theory of minimal surfaces admits a simplicial version (Jaco and Rubinstein). Given a compact three manifold M with a triangulation, a simplicial minimal surface is described by how it should intersect the simplices of the triangulation. Passing to the universal cover of M we obtain a space 3, namely, the set of simplices, as well as an action of the fundamental group of M on 3. An interaction potential can be written down in terms of the gluing data, so that solutions to the corresponding variational equations are simplicial minimal surfaces. We also note that, given the interpretation of S as energy and of the rotation numbers as density, it is interesting physically to study the function that associates the energy per particle of the minimal configuration to a rotation number. This has been studied when 3 = Zd numerically in [9] and rigorously in [11]. We also call attention to the models of quasi-crystals based on aperiodic tilings [10]. These have actions of semi-groups that correspond to dilations. It would be interesting to know whether some version of Aubry-Mather theory could be developed for them. References 1. Aubry, S. and Le Daeron, P.Y.: The discrete Frenkel–Kontorova model and its extensions; I. Exact results for the ground states. Physica D 8, 381–422 (1983) 2. Blank, M.L.: Metric properties of minimal solutions of discrete periodic variational problems. Nonlinearity 2, 1–22 (1989) 3. Georgii, H.-O.: Gibbs measures and phase transitions. de Gruyter Studies in Mathematics 9, Berlin–New York: Walter de Gruyter, 1988 4. Golé, C.: A new proof of the Aubry-Mather’s theorem. Math. Z. 210, 441–448 (1992) 5. Kléman, M. and Sadoc, J.F.: A tentative description of the crystallography of amorphous solids. J. Physique Lett. 40, 569–574 (1979) 6. Koch, H., de la Llave, R. and Radin, C.: Aubry-Mather theory for configurations on lattices. Preprint 7. Mather, J.: Existence of quasiperiodic orbits for twists homeomorphisms of the annulus. Topology 21, 457–467 (1982) 8. Pokrovsky, V. and Talapov, A.: Theory of incommensurate crystals. Sov. Sci. Reviews, Supplement series 1 (1984)


669

9. Vallet, F.: Thermodynamique unidimensionelle, et structures bidimensionelles de quelques modeles pour des systemes incommensurables. Thesis, Univ. Paris VI (1986) 10. Senechal, M.: Quasicrystals and Geometry Cambrigde: Cambrigde University Press, 1995 11. Sen, W.M.: Phase Locking in Multi-dimensional Frenkel–Kontorova models. Preprint (1994) 12. Wehrfritz, W.M.: Infinite linear groups. Ergebnisse der Mathematik, Band 76, New York: SpringerVerlag, 1973 Communicated by Ya. G. Sinai

Commun. Math. Phys. 192, 671 – 685 (1998)

Communications in


On Eigenfunction Decay for Two Dimensional Magnetic Schrödinger Operators H. D. Cornean1 , G. Nenciu1,2 1 Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 70700 Bucharest, Romania. E-mail: [email protected] 2 Dept. of Theor. Phys., Univ. of Bucharest, P.O. Box MG 11, 76900 Bucharest, Romania. e-mail [email protected]

Received: 1 January 1997 / Accepted: 25 July 1997

Abstract: For two dimensional Schrödinger operators with a nonzero constant magnetic field perturbed by a magnetic field and a scalar potential, both vanishing arbitrarily slow at infinity, it is proved that eigenfunctions corresponding to the discrete spectrum decay faster than any exponential. Under more restrictive conditions on the perturbations, even quicker decay is obtained.

1. Introduction The aim of this paper is to study the decay properties of the eigenfunctions for the two dimensional magnetic Schrödinger operators H = (p − a(x))2 + V (x),

x ∈ R2 .

(1.1)

The two dimensional case is interesting both from physical (e.g. the theory of quantum Hall effect) and mathematical point of view; in particular, since for V = 0 and constant magnetic field B0 the spectrum of H is the well known “Landau spectrum” σL (B0 ) = {(2n + 1)B0 | n = 0, 1, . . .}

(1.2)

by adding a perturbation one obtains a very rich spectral structure: eigenvalues in the gaps of the essential spectrum, “bands” created by an infinite number of magnetic and / or scalar “wells”, etc. In this paper we shall consider a particle in a constant magnetic field perturbed by a magnetic field and a scalar potential vanishing at infinity, i.e. the “one well” problem (we stress the fact that we impose conditions on the magnetic field and not on the corresponding vector potential); in this case it is known [I, H] that the essential spectrum of H coincides with the Landau spectrum. More precisely, we shall obtain a strong decay at infinity of the eigenfunctions ψE (x), corresponding to discrete eigenvalues, E, irrespective of the fact that they lie below the essential spectrum or in between the

672

H. D. Cornean, G. Nenciu

Landau levels. In this context, a fairly straightforward application of Combes-ThomasAgmon theory gives an exponential decay, |ψE (x)| ≤ const. exp(−µ(E)|x|), where µ(E) depends upon the distance between E and the essential spectrum of H and moreover µ(E) → 0 as E approaches the essential spectrum (see however the remark at the end of the proof of Theorem 4.1). For a thorough review in this direction, including the many-body problem, see [C-F-K-S, A-H-S 1, A-H-S 2, H, Hu] and references therein. On the other hand, if the scalar potential and the perturbing magnetic field are compactly supported or have rotational symmetry, it is not hard to see that the eigenfunctions have a gaussian decay irrespective of the position of E. In this context, T. Hoffmann-Ostenhof posed the problem of the decay of eigenfunctions under less stringent conditions on the perturbation. Results concerning this question were obtained by Erdös [E] for energies below the essential spectrum. He proved that if V decays at infinity and is analytic in the angle variable then the gaussian decay of the eigenfunctions is preserved and more important, he gave an example of a potential decaying at infinity for which the decay of the ground state is slower than a gaussian. Our main result (see Theorem 4.1 for precise formulation) is that if the perturbations B 0 (x), V (x) → 0

when

|x| → ∞

(1.3)

then the eigenfunctions corresponding to E 6∈ σL (B0 ) decay faster than any exponential. We believe our result to be optimal; additional conditions have to be added in order to ensure a quicker decay of eigenfunctions and we give some results in this direction (see Theorem 4.2): if the perturbing magnetic field is compactly supported and the scalar potential has a decay of the form exp(−δ|x|β ) with 0 < β ≤ 2 then the eigenfunctions decay like exp(−µ|x|1+β/2 ). These results can be applied to the “magnetic multiple wells” problem (see [H-S, H-H, Na 1, Na 2]) in order to have a better control on the width of the mini-bands which appear in the tight-binding limit (see [B-C-D]); this will be done in a companion paper. The content of the paper is as follows. Section 2 has a preparatory character; we write down some simple properties of the family of “local” transversal gauges we shall use as well as the closed formula for the integral kernel K0 (x, x0 ; z) of the resolvent of H in the case of constant magnetic field B0 and z 6∈ σL (B0 ). Section 3 contains one of the main results of the paper: in the case when V = 0 and the magnetic field is of the form (b > 0): B(x) = B0 + b B 0 (x)

(1.4)

with B0 > 0 and B 0 uniformly bounded and smooth, we prove that for z 6∈ σL (B0 ) fixed and b → 0 then z ∈ ρ(H) and we obtain an upper bound on the integral kernel K(x, x0 ; z) of (H − z)−1 (see Theorem 4.1 for the precise statement), which essentially says that as |x − x0 | → ∞, we have |K(x, x0 ; z)| ≈ exp [−µ(b, z)|x − x0 |],

lim µ(b, z) = ∞.

b→0

(1.5)

Our method, which is interesting in itself, used for proving the above result is a “regularised” magnetic perturbation theory: the main idea is the fact that most of the “singularity” of the perturbation is given by a (nonvanishing at infinity) “gauge phase”; by an appropriate “factorisation” of this phase one obtains a much more regular perturbation theory (see [N] for another use of the same idea).

On Eigenfunction Decay for 2D Magnetic Schrödinger Operators

673

Section 4 contains the main results of the paper concerning the decay of eigenfunctions. Theorem 4.1 follows (with the technical complications due to the magnetic field) from Theorem 3.1 by substantiating the general idea that the decay at infinity of the eigenfunctions is governed by the “asymptotic” hamiltonian and its resolvent’s integral kernel behaviour. The proof of Theorem 4.2 is based on the fact that while the gaussian decay of the “free kernel” K0 may not “propagate” itself in the Neumann series when adding a perturbing scalar potential, one can use the decay properties of the potential in order to obtain a decay which is stronger than that of the potential. 2. Preliminaries In this paper we shall consider only the bidimensional case (i.e. the particle is confined in the plane x3 = 0 and the magnetic field is orthogonal to that plane). For b ∈ R and d = (d1 , d2 ) ∈ R2 , define b ∧ d ≡ (−b d2 , b d1 ) ∈ R2 .

(2.1)

Let B(x) ∈ C 1 (R2 ). We shall use the following family of vector potentials corresponding to B(x): Z 1 0 ds s B(x0 + s(x − x0 )) ∧ (x − x0 ). (2.2) a(x, x ) = 0 0

For x = 0, this is nothing but the usual transversal gauge (see e.g. [T]): Z 1 ds s B(s x) ∧ x. a(x, 0) ≡ a(x) =

(2.3)

0

If we define

f(x, x0 ) = a(x) − a(x, x0 ),

(2.4)

0

then there exists ϕ(x, x ) such that

The additional requirement

∇x ϕ(x, x0 ) = f(x, x0 ).

(2.5)

ϕ(x0 , x0 ) = 0

(2.6)

gives ϕ(x, x0 ) =

Z

x1 x01

dt f1 (t, x2 ; x0 ) +

Z

x2 x02

dt f2 (x01 , t; x0 ),

(2.7)

where xi , x0i , fi are the cartesian components of x, x0 andf respectively. From the explicit formulae (2.2), (2.4) and (2.7), it follows at once that Lemma 2.1. If |B(x)| ≤ M < ∞, then i) |ϕ(x, x0 )| ≤ 2 M |x − x0 |(|x| + |x − x0 |). ii)

a(x, x0 ) · (x − x0 ) = 0.

(2.8) (2.9)

674


iii) 0

Z

1

ds s × ∇x a(x, x ) = 0 ∂ ∂ 0 0 0 0 × B(x + s(x − x )), − B(x + s(x − x )) · (x − x0 ). ∂x2 ∂x1

(2.10)

iv) If B(x) = B0 is constant, then 1 ϕ0 (x, x0 ) = − B0 (x1 x02 − x01 x2 ), 2 1 a0 (x, x0 ) = B0 ∧ (x − x0 ). 2

(2.11)

The hamiltonian of a particle in the presence of the magnetic field and a scalar potential V is (in the transversal gauge): H = (p − a(x))2 + V (x), ∂ ∂ p = −i , , −i ∂x1 ∂x2 Z 1 Z a(x) = −x2 ds s B(s x), x1 0

(2.12) !

1

ds s B(s x) . 0

In the case of constant magnetic field, one has the hamiltonian H0 = (p − a0 (x))2 , where p = −i∇x

and a0 (x) =

1 1 − B0 x2 , B0 x1 , 2 2

(2.13) (2.14)

which is essentially self-adjoint on C0∞ R2 and its spectrum is the well known Landau spectrum (2.15) σ(H0 ) = σess (H0 ) = {(2n + 1)B0 | n = 0, 1, 2, . . .} . 2 2 For z 6∈ σ(H0 ) and g ∈ L R , we write Z (H0 − z)−1 g (x) = dx0 K0 (x, x0 ; z)g(x0 ), (H0 − z)K0 (x, x0 ; z) = δ(x − x0 ). Then takes place (see e.g. [J-P]): Lemma 2.2. Let ϕ0 (x, x0 ) = −

B0 x1 x02 − x2 x01 , 2

B0 |x − x0 |2 , 4 1 z α=− − 1 6= −1, −2, . . . . 2 B0

ψ(x, x0 ) =

Then

(2.16)


675

0

K0 (x, x0 ; z) = ei ϕ0 (x,x ) G0 (x, x0 ; z) = 0(α) i ϕ0 (x,x0 ) −ψ(x,x0 ) = e e U (α, 1; 2 ψ(x, x0 )), 4π

(2.17)

where 0 is the Euler function and U (α, γ; ξ) is the confluent hypergeometric function [A-S]. For a proof using the eigenfunctions of H0 , see [J-P]. Alternatively, for 0, [c, d] ⊂ ρ(H0 ) and z ∈ [c, d], then: B0 t)| ≤ M (β, m, c, d) < ∞. (2.18) sup tβ e−m t |U (α, 1, 2 t∈R+ From Lemma 2.2 it follows that K0 (x, x0 ; z) has a gaussian decay as |x − x0 | → ∞. We shall use this in the following form: Corollary 2.1. Let µ > 0 and T (µ, x0 ) be the operator of multiplication with eµ|x−x0 | , x0 ∈ R2 . Then for all 0 < δ < B80 and z ∈ ρ(H0 ), one has that the operators T (δ, x0 ) (H0 − z)−1 T (−2δ, x0 ) : L2 R2 → L2 R2 L∞ R2 (2.19) 2

are bounded. Proof. Since for all x, x0 ∈ R2 , δ|x − x0 |2 ≤ 2δ|x − x0 |2 + 2δ|x0 − x0 |2 one has 0

|eδ|x−x0 | K0 (x, x0 ; z)e−2δ|x −x0 | f (x0 )| ≤ B |0(α)| −2 80 −δ |x−x0 |2 |f (x0 )|. |U (α, 1; 2ψ(x, x0 ))| ≤e 4π 2

2

Since (2.18) takes place, one has that: B B0 y 2 −2 80 −δ y 2 ) ∈ L1 R2 ∩ L2 R2 U (α, 1; e 2 and the use of Young inequality ([R-S 2]) finishes the proof.

(2.20)

(2.21)

Remark. Since under a gauge transformation Uχ f (x) = ei χ(x) f (x) and Z dx0 Kχ (x, x0 ; z) = Uχ∗ (H0 − z)−1 Uχ f (x) = 2 ZR 0 dx0 e−i χ(x) K0 (x, x0 ; z)ei χ(x ) f (x0 ) =

(2.22)

R2

one has

|Kχ (x, x0 ; z)| = |K0 (x, x0 ; z)|,

i.e. the gaussian decay is valid for an arbitrary gauge.

(2.23)

676


3. Schrödinger Operators with Nonconstant Magnetic Field Suppose that we are given a magnetic field B(x) = (0, 0, B0 + B 0 (x)) , B0 > 0, ||B ||C 1 (R2 ) ≡ max {||Dα B 0 ||∞ } ≤ 1. 0

|α|≤1

(3.1)

A vector potential corresponding to B 0 is (see Sect. 2 ): a0 (x, x0 ) =

−(x2 − x02 )

Z

1

ds s B 0 (x0 + s(x − x0 )),

0

(x1 −

x01 )

Z

1

! 0

0

0

ds s B (x + s(x − x ))

(3.2)

0

Then a(x, x0 ) = a0 (x, x0 ) + a0 (x, x0 ) and |a0 (x, x0 )| < |x − x0 | , |∇x a0 (x, x0 )| < |x − x0 |.

(3.3)

We showed that there exist ϕ(x, x0 ) and ϕ0 (x, x0 ) such that: a(x) = a(x, x0 ) + ∇x ϕ(x, x0 ) = = a0 (x) + a0 (x, x0 ) + ∇x ϕ0 (x, x0 ),

(3.4)

ϕ0 (x, x0 ) = ϕ(x, x0 ) − ϕ0 (x, x0 ).

(3.5)

where We consider hamiltonians of the form: (3.6) Hb = (p − a0 − ba0 )2 , b > 0, which are essentially self-adjoint on C0∞ R2 (see e.g. [R-S] ). The main result of this section is the following perturbative expansion of the integral kernel Kb (x, x0 ; z) of the resolvent of Hb : Theorem 3.1. Let [c, d] ⊂ ρ(H0 ). Then there exists b0 > 0, depending on [c, d] such that for any 0 ≤ b ≤ b0 we have that [c, d] ⊂ ρ(Hb ) and for any z ∈ [c, d], (Hb − z)−1 has an integral kernel given by the following perturbative expansion: X 0 0 Fn (x, x0 ; z), (3.7) Kb (x, x0 ; z) = ei ϕb (x,x ) K0 (x, x0 ; z) + n≥0

where Fn (x, x0 ; z) are integral kernels corresponding to the operators Fn ∈ B L2 R2 , L2 R2 with ||Fn ||B(L2 (R2 ),L2 (R2 )) ≤ [const ([c, d]) b]n+1 .

(3.8)


677

Proof. Denote with Hb,x0 = (p − a0 (x) − ba0 (x, x0 ))2 . Then by gauge covariance: 0

0

0

0

Hb ei ϕb (x,x ) = ei ϕb (x,x ) Hb,x0 .

(3.9)

The equation satisfied by Kb takes the form: δ(x − x0 ) = (Hb − z)Kb (x, x0 ; z) = 0

0

0

0

= ei ϕb (x,x ) (Hb,x0 − z)e−i ϕb (x,x ) Kb (x, x0 ; z).

(3.10)

The main idea of the proof is to factorise the “gauge phases”. This is achieved by making the following ansatz: 0

0

Kb (x, x0 ; z) = ei ϕb (x,x ) K0 (x, x0 ; z) + F(x, x0 ; z)

(3.11)

with F to be found. One can rewrite (3.10) as: h 0 0 i δ(x − x0 ) = (Hb − z) ei ϕb (x,x ) K0 (x, x0 ; z) + F(x, x0 ; z) = 0

0

= (Hb − z)F(x, x0 ; z) + ei ϕb (x,x ) (Hb,x0 − z)K0 (x, x0 ; z).

(3.12)

But Hb,x0 = H0 − Rb (x, x0 ), where Rb (x, x0 ) = 2ba0 (x, x0 ) · (p − a0 (x)) − ib∇x a0 (x, x0 ) − b2 a0 (x, x0 ). 2

(3.13)

Since ϕ0b (x0 , x0 ) = 0, (3.12) becomes (using (2.16) ): 0

0

(Hb − z)F(x, x0 ; z) = ei ϕb (x,x ) Rb (x, x0 )K0 (x, x0 ; z).

(3.14)

Using the explicit form of K0 (x, x0 ; z) given in Lemma 2.2, the equality in (3.3) and the equality (2.9), one can derive that a0 (x, x0 ) · (p − a0 (x))K0 (x, x0 ; z) = −a0 (x, x0 ) · a0 (x, x0 )K0 (x, x0 ; z).

(3.15)

Moreover, using (2.18) with m = B0 /8, one can obtain the following estimate: B0 |x − x0 |2 0 0 (3.16) |Rb (x, x )K0 (x, x ; z)| ≤ b C(c, d) exp − 8 0

0

Write as Sb (z) the operator corresponding to ei ϕb (x,x ) K0 (x, x0 ; z) and as Tb (z) the 0 0 operator linked to ei ϕb (x,x ) Rb (x, x0 )K0 (x, x0 ; z). Using Young inequalities as in Corollary 2.1, one can see that (3.17) Tb (z), Sb (z) ∈ B L2 R2 , L2 R2 ∩ B L2 R2 , L∞ R2 and ||Sb (z)||B(L2 (R2 ),L2 (R2 )) ≤ ||K0 (y; z)||1 ≤ C1 (c, d) < ∞, ||Tb (z)||B(L2 (R2 ),L2 (R2 )) ≤ C2 (c, d) b. Define

Fn (z) = Sb (z) [Tb (z)]n+1

From (3.14) and (3.11) one has:

(3.18) ,

n ≥ 0.

(3.19)

678


(Hb − z)−1 (1 − Tb (z)) = Sb (z) and if b
0 lim bn = 0. bn = ||Bn0 ||C 1 (R2 ) , n→∞

(3.23)

Let vˆ be a unit vector in R2 and fix y ∈ R2 . Denote with T(u,y) and D(u,y) the multiplication operators with eu|ˆv·(x−y)| and eu|x−y| . Fix u > 0. Define (3.24) Hn = (p − a0 (x) − an0 (x))2 . Then there exists N (u) ≥ 1 s.t. for any n ≥ N (u) we have K ∩ σ(Hn ) = ∅ and i) T (u, y)(Hn − z)−1 T (−u, y) ∈ B L2 R2 , L2 R2 , (3.25) ∞ 2 −1 2 2 , (3.26) ii) T (u, y)(Hn − z) T (−u, y) ∈ B L R , L R 2 2 −1 2 2 . (3.27) iii) D(u, y)(Hn − z) D(−u, y) ∈ B L R , L R Moreover, the norms are uniformly bounded in y, vˆ , in n ≥ N (u) and z ∈ K. Proof. We treat only the case i), the other ones being analogous. We can apply then Theorem 3.1 if n is sufficiently large. One obtains: X Tnm (z). (3.28) (Hn − z)−1 = Sn (z) m≥0

Because both Sn (z) and Tn (z) are integral operators, using (3.16) and the explicit form of K0 (x, x0 ; z), one can prove (as in Corollary 2.1) that for any ψ ∈ L2 R2 and m ≥ 0, ||T (u, y)Sn (z)Tnm (z)T (−u, y) ψ||2 (∞) ≤ B0 x2 |0(α)| B0 t2 ≤ sup exp u t − ||U (α, 1, (B0 x2 )/2)e− 8 ||1 (2) × 4 π t∈R+ 8 m B0 t2 ||ψ||2 . (3.29) × C2 (K) bn sup exp u t − 16 t∈R+ Take now n ≥ N (u) sufficiently large such that   2  e− 4Bu0  , bn <  C2 (K)  and the proof follows.

(3.30)


679

4. Decay of Eigenfunctions In this section we consider the decay of the eigenfunctions of H = (p − a)2 + V,

(4.1)

where the scalar potential V and the magnetic field which corresponds to a, ∇ ∧ a(x) = B0 + B0 (x) ,

B0 > 0

(4.2)

satisfy the following conditions:

B 0 ∈ C 1 R2 ; lim ||B 0 ||C 1 (R2 \{|x|≤n}) = 0, n→∞ V = V1 + V2 ; V1 ∈ L2 R2 , V2 ∈ L∞ R2 , lim sup |V2 (x)| = 0. n→∞ |x|≥n

(4.3)

Under these conditions, H is essentially self-adjoint on C0∞ R2 (see e.g. [C-FK-S] ). Moreover, V is relatively compact with respect to (p − a)2 [C-F-K-S] which together with the results in [I, H] it implies that σess (H) = σ(H0 ) = {(2n + 1)B0 | n = 0, 1, 2, . . .}.

(4.4)

Let E ∈ σdisc (H) (the discrete spectrum of H) and let ψ be a normalised eigenfunction corresponding to E. The main result of this section is that ψ decays faster than any exponential: 0 Theorem 4.1. Suppose B0 > 0 and B , V satisfy (4.3). Let K ⊂ R\σ(H0 ), K compact, E ∈ K, (H − E)ψ = 0, ψ ∈ L2 R2 . Then for any 0 ≤ u < ∞ there exists a constant C(u, K) < ∞ such that:

|ψ(x)| ≤ C(u, K)e−u|x| .

(4.5)

Remark. As proved by Erdös [E] (see also Theorem 4.2 below), under more restrictive conditions one can prove the existence of µ > 0 such as |ψ(x)| ≤ const e−µ|x|

r

,

1 < r ≤ 2.

(4.6)

The problem whether under the conditions (4.3) the decay given by Theorem 4.1 is optimal, remains open. In what follows, (4.7) g ∈ C ∞ R2 ; g = g ; ||g||C 2 (R2 ) = M < ∞. Under the conditions (4.3), one has D(H) = D((p − a)2 ) and (pj − aj )(H + i)−1 is bounded, j ∈ {1, 2}. Moreover, because [H, g] = −i{(p − a) · ∇g + ∇g · (p − a)}, it follows that ||[H, g](H + i)−1 || ≤ const(M ). Denote with 8 = [H, g]ψ = (E + i)[H, g](H + i)−1 ψ. Then ||8||2 ≤ const([c, d], M ). In what follows, we shall need a L∞ estimate on ψ.

(4.8)

680


Lemma 4.1. |ψ(x)| ≤ C(E),

(4.9)

where C(E) is bounded on compacts. Proof. As is known, (4.9) follows from the general results (Harnack type inequalities); actually in our case, for sufficiently large λ, one has (H + λ)−1 ∈ B L2 , L∞ and then the result follows from ψ = (E + λ)(H + λ)−1 ψ. Proof of Theorem 4.1. Let now vˆ be a unit vector in R2 and 0 if x ≤ 21 g ∈ C ∞ (R) , 0 ≤ g(x) ≤ 1 , g(x) = , 1 if x ≥ 1 vˆ · x gn,v ≡ g ; n ≡ x ∈ R2 | vˆ · x ≥ n . n

(4.10)

From now on, we shall denote gn,v with gn . Let Bn (x) = B0 + gn B 0 (x) , Vn = gn V (x), Z 1 an (x) = ds s Bn (s x) ∧ x,

(4.11)

0

and Hn = (p − an (x))2 + Vn (x).

(4.12)

For x(n) ∈ n , we have for all x ∈ n (remember that gn (x) = 1 on n ): an (x) = an (x, x(n) ) + ∇x ϕn (x, x(n) ) = = a(x) + ∇x δϕn (x, x(n) ),

(4.13)

δϕn (x, x(n) ) = ϕn (x, x(n) ) − ϕ(x, x(n) ).

(4.14)

where As a consequence (see (4.12) and (4.13) ): Hg2n = (p − a)2 + V g2n = = ei δϕn (.,x

(n)

)

Hn e−i δϕn (.,x

(n)

)

g2n .

(4.15)

From (4.3) and (4.10) it follows that, uniformly in vˆ , lim ||Bn 0 ||C 1 (R2 ) = 0

n→∞

lim ||gn V2 ||∞ = 0,

n→∞

;

lim ||gn V1 ||2 = 0,

n→∞

(4.16)

and then from Corollary 3.1 it follows that if E ∈ K then E ∈ ρ (p − an )2 and in addition


681

−1 ||Vn (p − an )2 − E ||B(L2 ,L2 ) ≤ −1 ≤ ||gn V1 ||2 || (p − an )2 − E ||B(L2 ,L∞ ) + −1 + ||gn V2 ||∞ || (p − an )2 − E ||B(L2 ,L2 ) → 0 , when n → ∞

(4.17)

It follows that there exists n(K) such that for n ≥ n(K), (Hn − E)−1 = −1 X n −1 ok = (p − an )2 − E , Vn (p − an )2 − E

(4.18)

k≥0

where the series in the R.H.S. of (4.18) is B L2 , L2 and B L2 , L∞ convergent. Fix now u > 0. Corollary 3.1 implies the existence of N1 (u, K) such that −1 T (−u, 0)||B(L2 ,L2 ) ≤ C1 (u, K), ||T (u, 0) (p − an )2 − E −1 T (−u, 0)||B(L2 ,L∞ ) ≤ C2 (u, K), (4.19) ||T (u, 0) (p − an )2 − E as soon as n ≥ N1 (u, K) and E ∈ K. From (4.15) it follows that: (H − E)g2n ψ = [H, g2n ]ψ ≡ 82n = = ei δϕn (Hn − E)e−i δϕn g2n ψ.

(4.20)

Together with (4.18), one has (n large enough): g2n ψ =

−1 × = ei δϕn (p − an )2 − E ok Xn −1 × Vn (p − an )2 − E e−i δϕn 82n .

(4.21)

k≥0

Because T (u, 0) commutes with V , (4.21) becomes: ||T (u, 0)g2n ψ||∞ ≤ X −1 ≤ C2 (u, K) ||Vn T (u, 0) (p − an )2 − E T (−u, 0)||kB(L2 ,L2 ) × k≥0

× ||T (u, 0)82n ||2 .

(4.22)

Since supp 82n ⊂ n \ 2n , one has: ||T (u, 0)82n ||2 ≤ e2 n u ||82n ||2 .

(4.23)

Using (4.19), there exists N (u, K) > 0 such that: −1 ||Vn T (u, 0) (p − an )2 − E T (−u, 0)||B(L2 ,L2 ) ≤ ≤ ||V1 gn ||2 C2 (u, K) + ||V2 gn ||∞ C1 (u, K) < 1

if n ≥ N (u, K).

(4.24)

Because ||g2n ||C 2 (R2 ) ≤ and using (4.24), (4.23), (4.22) and (4.8) one obtains: eu vˆ ·x |ψ(x)|g2N (u,K) ≤ C(u, K)e2uN (u,K) , (4.25) and since vˆ is arbitrary, the proof of Theorem 4.1 is completed. const n1

682


Remark. To obtain a proof of Theorem 4.1 in the Combes-Thomas scheme, in order to apply O’Connor’s Lemma, one needs to show that the essential spectrum of exp(α(1 + x2 )1/2 )H exp(−α(1 + x2 )1/2 ) ≡ H(α) is (at least) included in σL (B0 ) independently of α. One can try to prove this by showing that σess (H(α)) ⊂ σess (H0 (α)) ⊂ σL (B0 ). The second inclusion follows at once from the gaussian decay of the kernel of (H0 − z)−1 . If B 0 (x) = 0 the first inclusion (in fact equality) follows from a standard compactness argument. The hard case (at least for us) is B 0 (x) 6= 0 and actually Theorem 3.1 can be used to prove the inclusion in this case. In the rest of this section, B ∈ C∞

and B(x) = B0

if |x| ≥ 1.

(4.26)

The scalar potential V belongs to L2 + (L∞ ) and we suppose that there exist δ > 0 and 0 < β ≤ 2 such that: β

|V (x)| ≤ Ce−δ|x| ,

for |x| ≥ 1,

C > 0.

(4.27)

Then takes place Theorem 4.2. Let r = 1 + β2 , (1 < r ≤ 2) . Then there exists µ > 0 such that r

|ψ(x)| ≤ const(E) e−µ |x| .

(4.28)

One can write (see (4.11)): δ

β

|Vn (x)| ≤ Cn e− 2 |x| ,

lim Cn = 0.

(4.29)

n→∞

Take N1 sufficiently large such that if n ≥ N1 , then: Cn < dist(E, σ(H0 )),

(4.30)

and by a similar argument with that made in Theorem 4.1, one has: g2n ψ = = ei δϕn (H0 − E)−1 × X k Vn (H0 − E)−1 e−i δϕn 82n . ×

(4.31)

k≥0 r

Denote with xv = vˆ · x, with T (u, r) the multiplication operator with eu|xv | (where u > 0 and 1 ≤ r ≤ 2) and with χ = χ n . 2 Because suppVn ⊂ n2 , one has g2n ψ = = ei δϕn χ (H0 − E)−1 χ × X k × χVn (H0 − E)−1 χ e−i δϕn 82n . k≥0

(4.32)


Lemma 4.2. Let z ∈ ρ(H0 ) and 0 ≤ u
0 such that: T (u, r)χVn (H0 − E)−1 χT (−u, r) ∈ B(L2 , L2 )

(4.36)

and has a norm which goes to zero when n goes to infinity. Proof. We intend to use Young inequalities again, therefore we shall estimate for an arbitrary ψ ∈ L2 the term: |T (u, r)χVn (H0 − E)−1 χT (−u, r)ψ|(x) ≤ Z B0 |x−x0 |2 B0 |x−x0 |2 r β δ |0(α)| 8 8 dx0 e− 2 |x| e− e− × ≤ euxv Cn χ(x) 4π 0r

× |U (|x − x0 |2 )|χ(x0 )e−uxv |ψ(x0 )|.

(4.37)

We want to prove the existence of u > 0 such that for any x, x0 ∈ n2 to take place δ B0 |x − x0 |2 − uxrv + xβv + ux0r v ≥ 0. 8 2

(4.38)

First of all, |x − x0 |2 ≥ (xv − x0v )2 , so if xv ≤ x0v , the inequality is trivial. Therefore, the only interesting zone is 0 ≤ x0v ≤ xv . Define f (y) =

B0 (xv − y)2 + y r − xrv , 8u

0 ≤ y ≤ xv ,

n ≤ xv . 2

(4.39)

Then f 0 (y) = −

B0 (xv − y) + ry r−1 ; 4u

f 00 (y) =

B0 + r(r − 1)y r−2 . 4u

(4.40)

Because f 00 (y) is positive on the considered domain and f 0 (y) is changing sign in the extremities of the interval, there will be only one point z ∈ (0, xv ) in which f reaches its minimum. ∈ [0, 1]; then y = (1 − t)xv and Let t = xvx−y v B0 x2v 2 8uxr−2 v r t − 1 − (1 − t) . (4.41) f (t) = 8u B0

684


The equation f 0 (tm ) = 0 can be easily rewritten as: tm = λ(1 − tm )r−1 ; If

8u B0

λ=

4ur . B0 x2−r v

(4.42)

< 1, an approximate solution for tm is tm = λ − (r − 1)λ2 + o(λ3 ).

(4.43)

Introducing tm in f (t) one obtains: f (tm ) = −

2ur2 β 8u xv 1 + o . B0 B0

(4.44)

Choosing u sufficiently small, one has uf (y) + δ2 xβv ≥ 0. This implies: ||T (u, r)χVn (H0 − E)−1 χT (−u, r)ψ||2 ≤ B0 x2 |0(α)| Cn ||e− 8 |U (x2 )|||1 ||ψ||2 , ≤ 4π and the proof of Lemma 4.3 is complete.

(4.45)

Proof of Theorem 4.2. With the results of Lemma 4.2 and Lemma 4.3, (4.32) becomes: u ||T , r g2n ψ||∞ ≤ r−1 2 u ≤ ||T , r χ(H0 − E)−1 χT (−u, r)||B(L2 ,L∞ ) × 2r−1 X × ||T (u, r)χVn (H0 − E)−1 χT (−u, r)||kB(L2 ,L2 ) × k≥0

× ||T (u, r)82n ||2 .

(4.46)

Choose now N0 large enough, such that the above series converges. Then: u |ψ(x)| ≤ const exp − r−1 xrv , xv ≥ 2N0 , 2 where the constant is independent of vˆ .

(4.47)

Acknowledgement. We thank the referee for valuable suggestions and for pointing out that (2.18) was known.

References [A-S]

Abramovitz, M., Stegun, I.A.: Handbook of mathematical functions. National Bureau of Standards, Applied Mathematics Series 55, 1965 [A-H-S 1] Avron, J., Herbst, I., Simon, B.: Schrödinger operators with magnetic fields, I. General interactions. Duke Math. J. 45, 847–883 (1978) [A-H-S 2] Avron, J., Herbst, I., Simon, B.: Schrödinger operators with magnetic fields, II. Separation of the center mass in homogeneous magnetic fields. Ann. Phys. 114, 431–451 (1978) [B-C-D] Briet, P., Combes, J. M., Duclos, P.: Spectral stability under tunneling. Commun. Math. Phys. 126, 133–156 (1989)


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[C-F-K-S] Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators with application to quantum mechanics and global geometry. Berlin–Heidelberg–New York: Springer-Verlag, 1987 [E] Erdös, L.: Gaussian decay of the magnetic eigenfunctions. Vienna: Preprint E.S.I 184 (1994) [F-H] Feynman, R.P., Hibbs, A.: Quantum mechanics and path integrals. Hightstown, New Jersey: Mc Graw Hill, 1965 [H] Helffer, B.: On spectral theory for Schrödinger operators with magnetic potentials. Advanced Studies in Pure Mathematics 23, 113–141 (1994) [H-S] Helffer, B., Sjöstrand, J.: Multiple wells in the semi-classical limit I. Comm. in P.D.E. 9, 337–408 (1984) [H-H] Hempel, R., Herbst I.: Strong magnetic fields, Dirichlet boundaries and spectral gaps. Preprint E.S.I. 74 (1994) [Hu] Hunziker, W.: Schrödinger operators with electric or magnetic fields. Lecture Notes in Physics, Proc. Int. Conf. in Math. Phys., Lausanne 1980 [I] Iwatsuka, A.: The essential spectrum of two-dimensional Schrödinger operators with perturbed magnetic fields. J. Math. Kyoto Univ. 23, 475–480 (1983) [J-P] Joynt, R., Prange, R.: Conditions for the quantum Hall effect. Phys. Rev. B 29, 3303–3320 (1984) [K] Kato, T.: Perturbation theory for linear operators. Berlin–Heidelberg–New York: SpringerVerlag, 1976 [Na 1] Nakamura, S.: Band spectrum for Schrödinger operators with strong periodic magnetic fields. Operator Theory: Advances and Applications 78, 261–270 (1995) [Na 2] Nakamura, S., Bellisard, J.: Low energy bands do not contibute to Quantum Hall Effect. Commun. Math. Phys. 131, 283–305 (1990) [N] Nenciu, G.: Dynamics of band electrons in electric and magnetic fields: Rigorous justification of the effective hamiltonians. Rev. Mod. Phys. 63, 91–128 (1991) [R-S] Reed, M., Simon, B.: Methods of modern mathematical physics, II. New York: Academic Press, 1975 [T] Thaller, B.: The Dirac equation. Berlin–Heidelberg–New York: Springer-Verlag, 1992 Communicated by B. Simon

Commun. Math. Phys. 192, 687 – 706 (1998)

Communications in


Finite Dimensional Unitary Representations of Quantum Anti–de Sitter Groups at Roots of Unity Harold Steinacker? Theoretical Physics Group, Ernest Orlando Lawrence Berkeley National Laboratory, University of California, Berkeley, California 94720, USA and Department of Physics, University of California, Berkeley, California 94720, USA. E-mail: [email protected] Received: 27 November 1996 / Accepted: 28 July 1997

Abstract: We study irreducible unitary representations of Uq (SO(2, 1)) and Uq (SO(2, 3)) for q a root of unity, which are finite dimensional. Among others, unitary representations corresponding to all classical one-particle representations with integral weights are found for q = eiπ/M , with M being large enough. In the “massless” case with spin bigger than or equal to 1 in 4 dimensions, they are unitarizable only after factoring out a subspace of “pure gauges” as classically. A truncated associative tensor product describing unitary many-particle representations is defined for q = eiπ/M . 1. Introduction In recent years, the development of Noncommutative Geometry has sparked much interest in formulating physics and in particular quantum field theory on quantized, i.e. noncommutative spacetime. The idea is that if there are no more “points” in spacetime, such a theory should be well behaved in the UV. Quantum groups [8, 13, 7], although discovered in a different context, can be understood as generalized “symmetries” of certain quantum spaces. Thinking of elementary particles as irreducible unitary representations of the Poincaré group, it is natural to try to formulate a quantum field theory based on some quantum Poincaré group, i.e. on some quantized spacetime. There have been many attempts (e.g. [25, 20]) in this direction. One of the difficulties with many versions of a quantum Poincaré group comes from the fact that the classical Poincaré group is not semisimple. This forbids using the well developed theory of (semi)simple quantum groups, which is e.g. reviewed in [2, 23, 11]. In this paper, we consider instead the quantum Anti–de Sitter group Uq (SO(2, 3)), resp. Uq (SO(2, 1)) in 2 dimensions, thus taking advantage of much well-known mathematical machinery. In the ? Present address: Sektion Physik, Ludwig-Maximilians Universit¨ at München, Lehrstuhl Prof Wess, Theresienstr. 37, D-80333 München, Germany. E-mail: [email protected]

688

H. Steinacker

classical case, these groups (as opposed to e.g. the de Sitter group SO(4, 1)) are known to have positive-energy representations for any spin [10], and e.g. allow supersymmetric extensions [29]. Furthermore, one could argue that the usual choice of flat spacetime is a singular choice, perhaps subject to some mathematical artefacts. With this motivation, we study unitary representations of Uq (SO(2, 3)). Classically, all unitary representations are infinite-dimensional since the group is noncompact. It is well known that at roots of unity, the irreducible representations (irreps) of quantum groups are finite dimensional. In this paper, we determine if they are unitarizable, and show in particular that for q = eiπ/M , all the irreps with positive energy and integral weights are unitarizable, as long as the rest energy E0 ≥ s + 1 where s is the spin, and E0 is below some (q-dependent, large) limit. There is an intrinsic high-energy cutoff, and only finitely many such “physical” representations exist for given q. At low energies and for q close enough to 1, the structure is the same as in the classical case. Furthermore, unitary representations exist only at roots of unity (if q is a phase). For generic roots of unity, their weights are non-integral. Analogous results are found for Uq (SO(2, 1)). In general, there is a cell-like structure of unitary representations in weight space. In the “massless” case, the naive representations with spin bigger than or equal to 1 are reducible and contain a null-subspace corresponding to “pure gauge” states. It is shown that they can be consistently factored out to obtain unitary representations with only the physical degrees of freedom (“helicities”), as in the classical case [10]. The existence of finite-dimensional unitary representations of noncompact quantum groups at roots of unity has already been pointed out in [6], where several representations of Uq (SU (2, 2)) and Uq (SO(2, 3)) (with multiplicity of weights equal to one) are shown to be unitarizable. In the latter case they correspond to the Dirac singletons [5], which are recovered here as well. We also show that the class of “physical” (unitarizable) representations is closed under a new kind of associative truncated tensor product for q = eiπ/M , i.e. there exists a natural way to obtain many-particle representations. Besides being very encouraging from the point of view of quantum field theory, this shows again the markedly different properties of quantum groups at roots of unity from the case of generic q and q = 1. The results are clearly not restricted to the groups considered here and should be of interest on purely mathematical grounds as well. We develop a method to investigate the structure of representations of quantum groups at roots of unity and determine the structure of a large class of representations of Uq (SO(2, 3)). Throughout this paper, Uq (SO(2, 3)) will be equipped with a non-standard Hopf algebra star structure. The idea to find a quantum Poincaré group from Uq (SO(2, 3)) is not new: Already in [20], the so-called κ-Poincaré group was constructed by a contraction of Uq (SO(2, 3)). This contraction however essentially takes q → 1 (in a nontrivial way) and destroys the properties of the representations which we emphasize, in particular the finite dimensionality. Although it is not considered here, we want to mention that there exists a (space of functions on) quantum Anti–de Sitter space on which Uq (SO(2, 1)) resp. Uq (SO(2, 3)) operates, with an intrinsic mass parameter m2 = i(q − q −1 )/R2 where R is the “radius” of Anti–de Sitter space (and the usual Minkowski signature for q = 1) [28]. This paper is organized as follows: In Sect. 2, we investigate the unitary representations of Uq (SO(2, 1)), and define a truncated tensor product. In Sect. 3, the most important facts about quantized universal enveloping algebras of higher rank are reviewed. In Sect. 4, we consider Uq (SO(5)) and Uq (SO(2, 3)), determine the structure of the relevant irreducible representations (which are finite dimensional) and investigate

Unitary Representations of Quantum Anti–de Sitter Groups

689

which ones are unitarizable. The truncated tensor product is generalized to the case of Uq (SO(2, 3)). Finally we conclude and look at possible further developments. 2. Unitary Representations of Uq (SO(2, 1)) We first consider the simplest case of Uq (SO(2, 1)), which is a real form of U ≡ | )), the Hopf algebra defined by [8, 13], Uq (Sl(2, C [H, X ± ] = ±2X ± , [X + , X − ] = [H], 1(H) = H ⊗ 1 + 1 ⊗ H, 1(X ± ) = X ± ⊗ q H/2 + q −H/2 ⊗ X ± , S(X + ) = −qX + , S(X − ) = −q −1 X − , ε(X ± ) = ε(H) = 0,

(1)

S(H) = −H,

−n

−q | where [n] ≡ [n]q = qq−q −1 . To talk about a real form of Uq (SL(2, C)), one has to impose a reality condition, i.e. a star structure, and there may be several possibilities. Since we want the algebra to be implemented by a unitary representation on a Hilbert space, the star operation should be an antilinear antihomomorphism of the algebra. Furthermore, we will see that to get finite dimensional unitary representations, q must be a root of unity, so |q| = 1. Only at roots of unity the representation theory of quantum groups differs essentially from the classical case, and new features such as finite dimensional unitary representations of noncompact groups can appear. This suggests the following star structure corresponding to Uq (SO(2, 1)): n

H ∗ = H, which is simply

(X + )∗ = −X − ,

(2)

x∗ = e−iπH/2 θ(xc.c. )eiπH/2 ,

(3)

where θ is the usual (linear) Cartan–Weyl involution and xc.c. is the complex conjugate of x ∈ U . Since q is a phase, q c.c. = q −1 , and

provided ∗

(1(x))∗ = 1(x∗ )

(4)

(a ⊗ b)∗ = b∗ ⊗ a∗ .

(5)

∗

Then (S(x)) = S(x ), which is a non-standard Hopf algebra star structure. In particular, (5) is chosen as e.g. in [24], which is different from the standard definition. Nevertheless, this is perfectly consistent with a many-particle interpretation in Quantum Mechanics or Quantum Field Theory as discussed in [28], where it is shown e.g. how to define an invariant inner product on the tensor product with the “correct” classical limit. The irreps of U at roots of unity are well known (see e.g. [15], whose notations we largely follow), and we list some facts. Let q = e2πin/m

(6)

for positive relatively prime integers m, n and define M = m if m is odd, and M = m/2 if m is even. Then it is consistent and appropriate in our context to set (X ± )M = 0

(7)

690

H. Steinacker

(if one uses q H instead of H, then (X ± )M is central). All finite dimensional irreps are highest weight (h.w.) representations with dimension d ≤ M . There are two types of irreps: m m z, h = j, j − 2, . . . , −(d − 1) + 2n z} with dimension • Vd,z = {ejh ; j = (d − 1) + 2n j j d, for any 1 ≤ d ≤ M and z ∈ ZZ, where Heh = heh , m | \{Z • Iz1 with dimension M and h.w. (M −1)+ 2n z, for z ∈ C Z + 2n m r, 1 ≤ r ≤ M −1}.

Note that in the second type, z ∈ ZZ is allowed, in which case we will write VM,z ≡ Iz1 for convenience. We will concentrate on the Vd,z -representations from now on. Furthermore, theLfusion rules at roots of unity state that Vd,z ⊗ Vd0 ,z0 decomposes into p p ⊕d00 Vd00 ,z+z0 p Iz+z 0 , where Iz are the well-known reducible, but indecomposable representations of dimension 2M , see Fig. 1 and [15]. If q is not a root of unity, then the universal R ∈ U ⊗ U given by R = q2H ⊗H 1

∞ X l=0

q − 2 l(l+1) 1

(q − q −1 )l lH/2 + l q (X ) ⊗ q −lH/2 (X − )l [l]!

(8)

defines the quasitriangular structure of U. It satisfies e.g. σ(1(u)) = R1(u)R−1 ,

u ∈ U,

(9)

where σ(a ⊗ b) = b ⊗ a. We will only consider representations with dimension ≤ M ; then R restricted to such representations is well defined for roots of unity as well, since the sum in (8) only goes up to (M − 1). Furthermore R∗ = (R)−1 .

(10)

To see this, (3) is useful. Let us consider a hermitian invariant inner product (u, v) for u, v ∈ Vd,z . A hermitian | , (u, v)c.c. = (v, u), and inner product satisfies (u, λv) = λ(u, v) = (λc.c. u, v) for λ ∈ C it is invariant if (11) (u, x · v) = (x∗ · u, v), i.e. x∗ is the adjoint of x. If ( , ) is also positive definite, we have a unitary representation. Proposition 2.1. The representations Vd,z are unitarizable w.r.t. Uq (SO(2, 1)) if and only if (12) (−1)z+1 sin(2πnk/m) sin(2πn(d − k)/m) > 0 for all k = 1, ..., (d − 1). m m , this holds precisely if z is odd. For d − 1 ≥ 2n , it holds for isolated For d − 1 < 2n values of d only, i.e. if it holds for d, then it (generally) does not hold for d ± 1, d ± 2, . . .. m . The representations Vd,z are unitarizable w.r.t. Uq (SU (2)) if z is even and d−1 < 2n Proof. Let ejh be a basis of Vd,z with highest weight j. After a straightforward calculation, invariance implies (X − )k · ejj , (X − )k · ejj = (−1)k [k]![j][j − 1]...[j − k + 1] ejj , ejj (13)


691

for k = 1, ..., (d − 1), where [n]! = [1][2]...[n]. Therefore we can have a positive definite inner product (ejh , ejl ) = δh,l if and only if ak ≡ (−1)k [k]![j][j − 1]...[j − k + 1] is a positive number for all k = 1, ..., (d − 1), in which case ejj−2k = (ak )−1/2 (X − )k · ejj . Now ak = −[k][j − k + 1]ak−1 , and −[k][j − k + 1] = −[k][d − k +

m z] = −[k][d − k]eiπz 2n

(14)

= (−1)z+1 sin(2πnk/m) sin(2πn(d − k)/m)

1 , (15) sin(2πn/m)2

since z is an integer. Then the assertion follows. The compact case is known [15].

In particular, all of them are finite dimensional, and clearly if q is not a root of unity, none of the representations are unitarizable. We will be particularly interested in the case of (half)integer representations of type m = M Vd,z and n = 1, m even, for reasons to be discussed below. Then d − 1 < 2n always holds, and the Vd,z are unitarizable if and only if z is odd. These representations are centered around M z, with dimension less than or equal to M . Let us compare this with the classical case. For the Anti–de Sitter group SO(2, 1), H is nothing but the energy (cp. Sect. 3). At q = 1, the unitary irreps of SO(2, 1) are lowest weight representations with lowest weight j > 0, resp. highest weight representations with highest weight j < 0. For any given such lowest, resp. highest weight, we can now find a finite dimensional unitary representation with the same lowest, resp. highest weight, provided M is large enough (we only consider the (half)integer j here). These are unitary representations which for low energies look like the classical one-particle representations, but have an intrinsic high-energy cutoff if q 6= 1, which goes to infinity as q → 1. The same will be true in the 4 dimensional case. So far we only considered what could be called one-particle representations. To talk about many-particle representations, there should be a tensor product of 2 or more such irreps, which gives a unitary representation as well and agrees with the classical case for low energies. Since U is a Hopf algebra, there is a natural notion of a tensor product of two representations, given by the coproduct 1. However, it is not unitary a priori. As mentioned above, the tensor product of two irreps of type Vd,z is Vd,z ⊗ Vd0 ,z0 = ⊕d00 Vd00 ,z+z0

0 d+d −M M

p Iz+z 0,

(16)

p=r,r+2,...

where r = 1 if d + d0 − M is odd or else r = 2, and Izp is a indecomposable representation of dimenson 2M whose structure is shown in Fig. 1. The arrows indicate the rising and lowering operators. ... ...

Vp-1,z-1

...

...

W

V p-1,z+1

Fig. 1. Indecomposable representation Izp

692

H. Steinacker

In the case of Uq (SU (2)), one usually defines a truncated tensor product ⊗ by omitting all indecomposable Izp representations [24]. Then the remaining representations are unitary w.r.t. Uq (SU (2)); ⊗ is associative only from the representation theory point of view [24]. This is not the right thing to do for Uq (SO(2, 1)). Let n = 1 and m even, and consider e.g. VM −1,1 ⊗ VM −1,1 . Both factors have lowest energy H = 2, and the tensor product of the two corresponding classical representations is the sum of representations with lowest weights 4, 6, 8, . . . . In our case, these weights are in the Izp representations, while the Vd00 ,z00 have H ≥ M → ∞ and are not unitarizable. So we have to keep the Izp ’s and throw away the Vd00 ,z00 ’s in (16). A priori however, the Izp ’s are not unitarizable, either. To get a unitary tensor product, note that as a vector space,

(for p 6= 1) where

Izp = Vp−1,z−1 ⊕ W ⊕ Vp−1,z+1

(17)

W = VM −p+1,z ⊕ VM −p+1,z

(18)

as vector space. Now (X + )p−1 · vl is a lowest weight vector where vl is the vector with lowest weight of Ipz , and similarly (X − )p−1 · vh is a highest weight vector with vh being the vector of Ipz with highest weight (see Fig. 1). It is therefore consistent to consider the submodule of Ipz generated by vl , and factor out its submodule generated by (X + )p−1 ·vl ; the result is an irreducible representation equivalent to Vp−1,z−1 realized on the left summand in (17). Similarly, one could consider the submodule of Ipz generated by vh , factor out its submodule generated by (X − )p−1 · vh , and obtain an irreducible representation equivalent to Vp−1,z+1 . In short, one can just “omit” W in (17). The two V -type representations obtained this way are unitarizable provided n = 1 and m is even, and one can either keep both (notice the similarity with band structures in solidstate physics), or for simplicity keep the low-energy part only, in view of the physical application we have in mind. We therefore define a truncated tensor product as Definition 2.2. For n = 1 and even m, ˜ d0 ,z0 := Vd,z ⊗V

0 d+d −M M

Vd,z+z 0 −1 . ˜

(19)

˜ d=r,r+2,...

This can be stated as follows: Notice that any representation naturally decomposes ˜ simply means that only as a vector space into sums of Vd,z ’s, cp. (18); the definition of ⊗ the smallest value of z in this decomposition is kept, which is the submodule of irreps m (z + z 0 − 1). (Incidentally, z is the eigenvalue with lowest weights less than or equal to 2n of D3 in the classical su(2) algebra generated by {D± = ± M

(X ± )M [M ]!

, 2D3 = [D+ , D− ]},

where (X[M )]! is understood by some limes procedure.) With this in mind, it is obvious ˜ is associative: both in (V1 ⊗V ˜ 2 )⊗V ˜ 2 ⊗V ˜ 3 ), the result is simply the ˜ 3 and in V1 ⊗(V that ⊗ V ’s with minimal z, which is the same space, because the ordinary tensor product is associative and 1 is coassociative. This is in contrast with the “ordinary” truncated tensor product ⊗ [24]. Of course, one could give a similar definition for negative-energy representations. See also Definition 4.8 in the case of Uq (SO(2, 3)). ˜ d0 ,z0 is unitarizable if all the V ’s on the rhs of (19) are unitarizable. This is Vd,z ⊗V certainly true if n = 1 and m is even. In all other cases, there are no terms on the rhs of (19) if the factors on the lhs are unitarizable, since no Izp -type representations are


693

generated (they are too large). This is the reason why we concentrate on this case, and ˜ furthermore on z = z 0 = 1 which corresponds to low-energy representations. Then ⊗ defines a two-particle Hilbert space with the correct classical limit. To summarize, we have the following: ˜ d0 ,1 is unitarizable. ˜ is associative, and Vd,1 ⊗V Proposition 2.3. ⊗ How an inner product is induced from the single-particle Hilbert spaces is explained in [28]. 3. The Quantum Group Uq (SO(2, 3)) In order to generalize the above results to the 4-dimensional case, one has to use the | general machinery of quantum groups, which is briefly reviewed (cp. e.g. [2]): Let q ∈ C (αi ,αj ) and Aij = 2 (αj ,αj ) be the Cartan matrix of a classical simple Lie algebra g of rank r, where (, ) is the Killing form and {αi , i = 1, . . . , r} are the simple roots. Then the quantized universal enveloping algebra Uq (g) is the Hopf algebra generated by the elements {Xi± , Hi ; i = 1, . . . , r} and relations [8, 13, 7] Hi , Hj = 0, Hi , Xj± = ±Aji Xj± ,

q di Hi − q −di Hi = δi,j [Hi ]qi , Xi+ , Xj− = δi,j q di − q −di 1−Aji X 1 − Aji (Xi± )k Xj± (Xi± )1−Aji −k = 0, k q k=0

i 6= j,

(20)

i

q i = q di ,

where di = (αi , αi )/2,

n m

[n]qi =

= qi

qin −qi−n qi −qi−1

and

[n]qi ! . [m]qi ![n − m]qi !

(21)

The comultiplication is given by 1(Hi ) = Hi ⊗ 1 + 1 ⊗ Hi , 1(Xi± ) = Xi± ⊗ q di Hi /2 + q −di Hi /2 ⊗ Xi± .

(22)

Antipode and counit are S(Hi ) = −Hi , S(Xi+ ) = −q di Xi+ ,

ε(Hi ) = ε(Xi± ) = 0

S(Xi− ) = −q −di Xi− ,

(we use the conventions of [16], which differ slightly from e.g. [2]). | )), r = 2 and For U ≡ Uq (SO(5, C 2 −2 2 −1 Aij = , (αi , αj ) = , −1 2 −1 1

(23)

(24)

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so d1 = 1, d2 = 1/2, to have the standard physics normalization (a rescaling of ( , ) can be absorbed by a redefinition of q) The weight diagrams of the vector and P the spinor representations are given in Fig. 2 for illustration. The Weyl element is ρ = 21 α>0 α = 3 2 α1 + 2α2 .

H3

. . . . . . . . .

1

2

H2

. . . . . . . . .

Fig. 2. Vector and spinor representations of SO(2, 3)

The possible reality structures on U have been investigated in [19]. As in Sect. 2, in order to obtain finite dimensional unitary representations, q must be a root of unity. Furthermore, on physical grounds we insist upon having positive-energy representations; already in the classical case, that rules out e.g. SO(4, 1), cp. the discussion in [10]. It appears that then there is only one possibility, namely (Hi )∗ = Hi , (X1+ )∗ = −X1− , (X2+ )∗ = X2− , (a ⊗ b)∗ = b∗ ⊗ a∗ , (1(u))∗ = 1(u∗ ), (S(u))∗ = S(u∗ ),

(25) (26)

for |q| = 1, which corresponds to the Anti–de Sitter group Uq (SO(2, 3)). Again with E ≡ d1 H1 + d2 H2 , (−1)E x∗ (−1)E = θ(xc.c. ), where θ is the usual Cartan–Weyl involution corresponding to Uq (SO(5)). Although it will not be used it in the present paper, this algebra has the very important property of being quasitriangular, i.e. there exists a universal R ∈ U ⊗ U. It satisfies R∗ = (R)−1 , which can be seen e.g. from uniqueness theorems, cp. [17, 2]. In the mathematical literature, usually a rational version of the above algebra, i.e. using q di Hi instead of Hi is considered. Since we are only interested in specific representations, we prefer to work with Hi . We essentially work in the “unrestricted” specialization, i.e. (X ± )k

the divided powers (Xi± )(k) = [k]iq ! are not included if [k]qi = 0, although our results i will only concern representations which are small enough so that the distinction is not relevant. Often the following generators are more useful: p (27) hi = di Hi , e±i = [di ]Xi± , so that

hi , e±j = ±(αi , αj )e±j , ei , e−j = δi,j [hi ].

(28)

q In the present case, i.e. h1 = H1 , h2 = 21 H2 , e±1 = X1± and e±2 = [ 21 ]X2± . So far we only have the generators corresponding to the simple roots. A Cartan– Weyl basis corresponding to all roots can be obtained e.g. using the braid group action


695

introduced by Lusztig [21], (see also [2, 11]) resp. the quantum Weyl group [16, 26, 18, 2]. If ω = τi1 ...τiN is a reduced expression for the longest element of the Weyl group, where τi is the reflection along αi , then {αi1 , τi1 αi2 , ..., τi1 ...τiN −1 αiN } is an ordered set of positive roots. We will use ω = τ1 τ2 τ1 τ2 and denote them β1 = α1 , β2 = α2 , β3 = α1 + α2 , β4 = α1 + 2α2 . A Cartan–Weyl basis of root vectors of U can then be defined as {e±1 , e±3 , e±4 , e±2 } = {e±1 , T1 e±2 , T1 T2 e±1 , T1 T2 T1 e±2 } and similarly for the hi ’s, where the Ti represent the braid group on U [21]: Ti (Hj ) = Hj − Aij Hi , −Aji

Ti (Xj+ ) =

X

Ti Xi+ = −Xi− qiHi ,

(−1)r−Aji qi−r (Xi+ )(−Aji −r) Xj+ (Xi+ )(r) ,

r=0

(29) where Ti (θ(xc.c. )) = θ(Ti (x))c.c. . We find e3 = q −1 e2 e1 − e1 e2 , e−3 = qe−1 e−2 − e−2 e−1 , h3 = h1 + h2 , e4 = e2 e3 − e3 e2 , e−4 = e−3 e−2 − e−2 e−3 , h4 = h1 + 2h2 .

(30)

Similarly one defines the root vectors Xβ±l . This can be used to obtain a Poincaré– Birkhoff–Witt basis of U = U − U 0 U + where U ± is generated by the Xi± and U 0 by the Hi : for k := (k1 , . . . , kN ) where N is the number of positive roots, let Xk+ = Xβ+k1 1 . . . Xβ+kNN . Then the Xk± form a P.B.W. basis of U + , and similarly for U − [22] (assuming q 4 6= 1). Up to a trivial automorphism, (30) agrees with the basis used in [20]. The identification of the usual generators of the Poincaré group has also been given there and will not be repeated here, except for pointing out that h3 is the energy and h2 is a component of | )) subalgebras with angular momentum, see also [10]. All of the above form Uq˜ (SL(2, C appropriate q˜ (but not as coalgebras), because the Ti ’s are algebra homomorphisms. The reality structure is e∗1 = −e−1 ,

e∗2 = e−2 ,

e∗3 = −e−3 ,

e∗4 = −e−4 .

(31)

So the set {e±2 , h2 } generates a Uq˜ (SU (2)) algebra, and the other three {e±α , hα } generate noncompact Uq˜ (SO(2, 1)) algebras, as discussed in Sect. 2. 4. Unitary Representations of Uq (SO(2, 3)) and Uq (SO(5)) In this section, we consider representations of Uq (SO(2, 3)) and show that for suitable roots of unity q, the irreducible positive, resp. negative, energy representations are again unitarizable, if the highest, resp. lowest, weight lies in some “bands” in weight space. Their structure for low energies is exactly as in the classical case including the appearance of “pure gauge” subspaces for spin bigger than or equal to 1 in the “massless” case, which have to be factored out to obtain the physical, unitary representations. At high energies, there is an intrinsic cutoff. From now on q = e2πin/m . Most facts about representations of quantum groups we will use can be found e.g. in [4]. It is useful to consider the Verma modules M (λ) for a highest weight λ, which is the (unique) U - module having a highest weight vector wλ such that

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H. Steinacker

U + wλ = 0,

Hi w λ =

(λ, αi ) wλ , di

(32)

and the vectors Xk− wλ form a P.B.W. basis of M (λ). On a Verma module, one can define a unique invariant inner product ( , ), which is hermitian and satisfies (wλ , wλ ) = 1 and (u, x · v) = (θ(xc.c. ) · u, v) for x ∈ U , as in Sect. 2 [4]. θ is again the (linear) Cartan–Weyl involution corresponding to Uq (SO(5)). The irreducible highest weight representations can be obtained from the corresponding Verma module by factoring out all submodules in the Verma module. All submodules are null spaces w.r.t. the above inner product, i.e. they are orthogonal to any state in M (λ). Therefore one can consistently factor them out, and obtain a hermitian inner product on the quotient space L(λ), which is the unique irrep with highest weight λ. To see that they / U + wµ . Now for v ∈ U − wλ , it are null, let wµ ∈ M (λ) be in some submodule, so wλ ∈ + follows (wµ , v) ∈ (U wµ , wλ ) = 0. The following discussion until the paragraphPbefore Definition 4.4 is technical and may be skipped upon first reading. Let Q = ZZαi be the root lattice and Q+ = P ZZ+ αi , where ZZ+ = {0, 1, 2, . . .}. We will write λµ

if

λ − µ ∈ Q+ .

For η ∈ Q, denote (see [4]) Par(η) := {k ∈ ZZ+N ;

X

ki βi = η}.

(33)

(34)

Let M (λ)η be the weight space with weight λ − η in M (λ). Then its dimension is given by |Par(η)|. If M (λ) contains a highest weight vector with weight σ, then the multiplicity of the weight space M (λ)/M (σ) η is given by |Par(η)|−|Par(η +σ −λ)|, and so on. We will see how this allows to determine the structure, i.e. the characters of the irreducible highest weight representations. As usual, the character of a representation V (λ) with maximal weight λ is the function on weight space defined by X dim V (λ)η e−η , (35) ch(V (λ)) = eλ η∈Q+

where eλ−η (µ) := e(λ−η,µ) , and V (λ)η is the weight space of V (λ) at weight λ − η. The characters of inequivalent highest weight irreps (which are finite dimensional at roots of unity) are linearly independent. Furthermore, the characters of Verma modules are the same as in the classical case [12, 4], X |Par(η)|e−η . (36) ch(M (λ)) = eλ η∈Q+

In general, the structure of Verma modules is quite complicated, and the proper technical tool to describe it is its composition series. For a U -module M with a maximal weight, consider a sequence of submodules . . . ⊂ W2 ⊂ W1 ⊂ W0 = M such that Wk /Wk+1 is irreducible, and thus Wk /Wk+1 ∼ = L(µk ) for some µk . (If the series is finite, it is sometimes called a Jordan–Hölder series. For roots of unity it is infinite, but this is not a problem for our arguments. Wk+1 can be constructed inductively by fixing a maximal submodule of Wk ). While k may not be unique, it Pthe submodules WP ch(L(µk )). Since the is obvious that we always have ch(M ) = ch(Wk /Wk+1 ) =


697

characters of irreps are linearly independent, this decomposition of ch(M ) is unique, and so are the subquotients L(µk ). We will study the composition series of the Verma module M (λ), in order to determine the structure of the corresponding irreducible highest weight representation. Our main tool to achieve this is a remarkable formula by De Concini and Kac for det(M (λ)η ), the determinant of the inner product matrix of M (λ)η . Before stating it, we point out its use for determining irreps: Lemma 4.1. Let wλ be the highest weight vector in an irreducible highest weight representation L(λ) with invariant inner product. If (wλ , wλ ) 6= 0, then ( , ) is non-degenerate, i.e. (37) det(L(λ)η ) 6= 0 for every weight space with weight λ − η in L(λ). Proof. Assume to the contrary that there is a vector vµ which is orthogonal to all vectors of the same weight, and therefore to all vectors of any weight. Because L(λ) is irreducible, there exists an u ∈ U such that wλ = u · vµ . But then (wλ , wλ ) = (wλ , u · vµ ) = (u† · wλ , vµ ) = 0, which is a contradiction. Now we state the result of De Concini and Kac [4]: det(M (λ)η ) =

Y

Y

[mβ ]dβ

β∈R+ mβ ∈IN

q (λ+ρ−mβ β/2,β) − q −(λ+ρ−mβ β/2,β) q dβ − q −dβ

|Par(η−mβ β)|

(38) in a P.B.W. basis for arbitrary highest weight λ, where R+ denotes the positive roots (cp. Sect. 3), and dβ = (β, β)/2. To get some insight, notice first of all that due to |Par(η − mβ β)| in the exponent, the product is finite. Now for some positive root β, let kβ be the smallest integer such that (λ+ρ−kβ β/2,β)

−(λ+ρ−kβ β/2,β)

−q = 0, and consider the weight space D(λ)kβ ,β := [kβ ]dβ q q dβ −q −dβ at weight λ − kβ β, i.e. ηβ = kβ β. Then |Par(ηβ − kβ β)| = 1 and det(M (λ)ηβ ) is zero, so there is a highest weight vector wβ with weight λ − ηβ (assuming for now that there is no other null state with weight larger than (λ − ηβ )). It generates a submodule which is again a Verma module (because U does not have zero divisors [4]), with dimension |Par(η−kβ β)| at weight λ−η. This is the origin of the exponent. However the submodules generated by the ωβi are in general not independent, i.e. they may contain common highest weight vectors, and other highest weight vectors besides these wβi might exist. Nevertheless, all the highest weights µk in the composition series of M (λ) are precisely obtained in this way. This “strong linkage principle” will be proven below, adapting the arguments in [12] for the classical case. While it is not a new insight for the quantum case either [6, 1], it seems that no explicit proof has been given at least in the case of even roots of unity, which is most interesting from our point of view, as we will see. To make the structure more transparent, let INβT be the set of positive integers k with m [k]β = 0, and INβR the positive integers k such that (λ + ρ − k2 β, β) ∈ 2n ZZ. Then

D(λ)k,β = 0 ⇔ k ∈ INβT The second condition is k = 2 (λ+ρ,β) (β,β) +

m 2 Z, 2n (β,β) Z

or

k ∈ INβR .

(39)

which means that

λ − kβ = σβ,l (λ),

(40)

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H. Steinacker

m where σβ,l (λ) is the reflection of λ by a plane perpendicular to β through −ρ + 4nd lβ, β for some integer l. For general l, σβ,l (λ) ∈ / λ + Q; but k should be an integer, so it is natural to define the affine Weyl group Wλ of reflections in weight space to be generated by those reflections σβi ,li in weight space which map λ into λ + Q. For q = e2πin/m , two such allowed reflection planes perpendicular to βi will differ by multiples of 21 M(i) βi ; here M(i) = m for di = 21 , while for di = 1, M(i) = m or m/2 if m is odd or even, respectively. Thus Wλ is generated by all reflections by these planes. Alternatively, it is generated by the usual Weyl group with reflection center −ρ, and translations by M(i) βi . Now the strong linkage principle states the following:

Proposition 4.2. L(µ) is a composition factor of the Verma module M (λ) if and only if µ is strongly linked to λ, i.e. if there is a descendant sequence of weights related by the affine Weyl group as λ λi = σβi ,li (λ) . . . λkj...i = σβk ,lk (λj...i ) = µ.

(41)

Proof. The main tool to show this is the formula (38). Consider the inner product matrix Mk,k0 := (Xk− wλ , Xk−0 wλ ); it is hermitian, since q is a phase. One can define an analytic continuation of it as follows: for the same P.B.W. basis, let Bk,k0 (q, λ) := (Xk− wλ , Xk−0 wλ )b be the matrix of the invariant bilinear form defined as in [4], which is manifestly analytic in q and λ (one considers q as a formal variable and replaces | and arbitrary q → q −1 in the first argument of ( , )b ). Then (38) holds for all q ∈ C 0 0 complexified λ [4]. For |q| = 1 and real λ, Bk,k (q, λ) = Mk,k . Let λ0 = λ + hρ and 0 0 | ; then B q 0 = qeiπh for h ∈ C k,k0 (q , λ ) is analytic in h, and hermitian for h ∈ IR. Furthermore, one can identify the modules M (λ0 ) for different h via the P.B.W. basis. In this sense, the action of Xi± is analytic in h (it only depends on the commutation relations of the Xβ± ). Now it follows (see Theorem 1.10 in [14], chapter 2 on matrices which are analytic in h and normal for real h) that the eigenvalues ej of Bk,k0 (q 0 , λ0 ) are analytic in h, and there exist analytic projectors Pej on the eigenspaces Vej which span the entire vectorspace (except possibly at isolated points where some eigenvalues coincide; for h ∈ IR however, the generic eigenspaces are orthogonal and therefore remain independent even at such points). These projectors provide an analytic basis of eigenvectors of Bk,k0 (q 0 , λ0 ). Now let M Vk := V ej , (42) ej ∝hk

i.e. the sum of the eigenspaces whose eigenvalues ej have a zero of order k (precisely) at h = 0. Of course, (Vk , Vk0 )b = 0 for k 6= k 0 . The Vk span the entire space, they have an analytic basis as discussed, and have the following properties: Lemma 4.3. 1) (vk , v)b = o(hk ) for vk ∈ Vk and any (analytic) v ∈ M (λ0 ). k X X 2) Xi± vk = al vl + hl bl vk−l for vl ∈ Vl and al , bl analytic. In particular at h = 0, l≥k

l=1

M k := ⊕n≥k Vn is invariant.

(43)


699

Proof. 1) Decomposing v according to ⊕l Vl , only the (analytic) component in Vk contributes in (vk , v)b , with a factor hk by the definition of Vk (o(hk ) means at least k factors of h). P 2) Decompose Xi± vk = ej aej vej with analytic coefficients aej corresponding to the eigenvalue ej . For any vej appearing on the rhs, consider (vej , Xi± vk )b = aej (vej , vej )b = c aej ej (with c 6= 0 at h = 0, since vej might not be normalized). But the lhs is (Xi∓ vej , vk )b = o(hk ) as shown above. Therefore aej ej = o(hk ), which implies 2). In particular, M (λ)/M 1 is irreducible and nothing but L(λ). (The sequence of submodules ... ⊂ M 2 ⊂ M 1 ⊂ M (λ) is similar to the Jantzen filtration [12].) By the definition of M k , we have X k ord(det(M (λ)η )) = dim Mλ−η , (44) k≥1 k is the weight space of M k at weight λ−η, and ord(det(M (λ)η )) is the order where Mλ−η of the zero of det(M (λ)η ) as a function of h, i.e. the maximal power of h it contains. Now from (38), it follows that X X X k ch(M k ) = eλ ( dim Mλ−η )e−η η∈Q+ k≥1

k≥1

=e

λ

X

ord(det(M (λ)η ))e−η

η∈Q+

=

X

(

β∈IR+

=

X

X

+

n∈INβT

(

X

β∈IR+ n∈IN T β

X n∈INβR

+

X

)eλ

X

|Par(η − nβ)|e−η

η∈Q+

)ch(M (λ − nβ)),

(45)

n∈INβR

where we used (36). Now we can show (4.2) inductively. Both the left and the right side of (45) can be decomposed into a sum of characters of highest weight irreps, according to their composition series. These characters are linearly independent. Suppose that L(λ − η) is a composition factor of M (λ). Then the corresponding character is contained in the lhs of (45), since M (λ)/M 1 is irreducible. Therefore it is also contained in one of the ch(M (λ − nβ)) on the rhs. Therefore L(λ − η) is a composition factor of one of these M (λ − nβ), and by the induction assumption we obtain that µ ≡ λ − η is strongly linked to λ as in (41). Conversely, assume that µ satisfies (41). By the induction assumption, there exists a n ∈ INβT ∪ INβR such that L(µ) is a subquotient of M (λ − nβ). Then (45) shows that L(µ) is a subquotient of M (λ). Obviously this applies to other quantum groups as well. In particular, we recover the well-known fact that for q = e2πin/m , all (Xi− )M(i) wλ are highest weight vectors, and zero in an irrep. reNow we can study the irreps of Uq (SO(5)) and Uq (SO(2, 3)). First, there Pexist m k markable nontrivial one dimensional representations wλ0 with weights λ0 = 2n i αi for integers ki . By tensoring any representation with wλ0 , one obtains another representation with the same structure, but all weights shifted by λ0 . We will see below that

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by such a shift, representations which are unitarizable w.r.t. Uq (SO(2, 3)) are in one to one correspondence with representations which are unitarizable w.r.t. Uq (SO(5)). It is therefore enough to consider highest weights in the following domain: Definition 4.4. A weight λ = E0 β3 + sβ2 is called basic if 0 ≤ (λ, β3 ) = E0
|ν|−τ

∀ν ∈ Z` , ν 6= 0 ,

(1.4) √ for some positive constants C0 and τ (here and henceforth |ν| = ν · ν, while kνk = P` j=1 |νj |, if ν = (ν1 , . . . , ν` )); (3) f has the form f = (f1 , . . . , f` ), with of the class Cˆ (p) (T` ) introduced in P each fj iν·α [BGGM], for some p: namely fj (α) = ν∈Z fjν e , fν = f−ν , with fj0 = 0 and, for ν 6= 0, N (j) ||ν|| X c(j) n + dn (−1) fjν = , (1.5) n |ν| n≥p+`

(j) for some N ≥ p + ` and some constants c(j) n , dn ; and (4) N(ε), called a “counterterm”, has to be fixed in order to make the equations of motion for the model (1.3) linearizable. For instance we can choose fjν = aj |ν|−b , with b = p + ` and a = (a1 , . . . , a` ) ∈ R` . In the following we shall deal explicitly with such a function: the proof can be trivially extended to the class of functions (1.5).

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F. Bonetto, G. Gallavotti, G. Gentile, V. Mastropietro

Theorem 1.4. Given the Hamiltonian (1.1), with ω0 satisfying the diophantine condition (1.4), F ≡ 0 and f = (f1 , . . . , f` ), with each fj ∈ Cˆ (p) (T` ), there exist two positive constants ε0 and p0 = 2 + 3τ , and a function N(ε) analytic in ε for |ε| < ε0 , such that the equations of motion corresponding to the Hamiltonian (1.3) admit linearizable ˙ = ω) solutions in C (0) (T` ) analytic in ε (i.e. conjugable to the linear flow defined by ψ for |ε| < ε0 , provided p > p0 , i.e. solutions described by (1.7), (1.8) below. 1.5. The equations of motion for the Hamiltonian (1.3) are given by dαj = ω0j + εfj (α) + Nj (ε) , dt dAj = −εA · ∂αj f(α) . dt

(1.6)

We look for motions of the form α(t) = ω0 t + h(ω0 t) , A(t) = A0 + H(ω0 t) ,

h(ψ) =

∞ X X

H(ψ) =

hν(k) eiν·ψ εk ,

k=1 ν∈Z ∞ X X

(1.7) Hν(k) eiν·ψ

k

ε ,

k=1 ν∈Z

with h odd and H even in ψ so that the equations for h and H become (ω0 · ∂ψ ) hj (ψ) = εfj (ψ + h(ψ)) + Nj (ε) , (ω0 · ∂ψ ) Hj (ψ) = −ε[A0 + H(ψ)] · ∂αj f(ψ + h(ψ)) ,

(1.8)

where ∂α denotes derivative with respect to the argument, and N(ε) has to be so chosen that the right-hand side of the first equation in (1.8) has vanishing average (see Sect. 2.1 below). A “solution” to (1.8) can be given a meaning as soon as h, H are continuous by requiring equality of the Fourier transforms of both sides (regarded as distributions, see [BGGM]). We see from (1.8) that the equation for h is closed, so that, as long as we are interested only in the function h, i.e. in the analytic linearizability of (1.6), we can confine ourselves to studying only the first equation in (1.8). This is the equation that one has to solve to linearize the flow generated by dα/dt = ω0 + εf(α) + N(ε): hence it is not surprising that the equation for H can be easily solved once h is known: see Sect. 2.3 below. Note that since f is supposed even, then we expect that h is odd and H is even: hence while the equation for H does not seem to hit any obvious compatibility problems we see that the equation for h does, unless N(ε) is suitably chosen. In fact the function εf(ψ + h(ψ)), being even has no a priori reason to have a vanishing integral over ψ (as it should, being equal to (ω0 · ∂ ψ )h(ψ)). 2. Formal Solution and Graph Representation 2.1. We study now Eqs. (1.8) with fjν = aj |ν|−b replaced with fjν e−κ|ν| . The parameter κ is taken κ > 0, and, after computing the coefficients hν(k) in (1.7), which will depend on κ, one will perform the limit κ → 0 (“Abel’s summation”).

Quasi Linear Flows on Tori

711

The formal solubility of (1.8) with fjν replaced with fjν e−κ|ν| follows from [E2], where more general interaction potentials are considered (see also [GM2], Sect. 8.1, where the formalism is similar to the one used here). (k) One has h(k) j0 = Hj0 = 0 ∀k ≥ 1, while, when ν 6= 0, for k = 1, h(1) jν =

fjν , iω0 · ν

(1) Hjν =−

iνj A0 · fν , iω0 · ν

(2.1)

and, for k ≥ 2, h(k) jν =

1 X 1 iω0 · ν p! ν p>0

(k) Hjν

X

p>0

X

(iν0j ) ·

Hν(˜k) · fν 0 ˜

p Y

iν 0 · hν(kss )

p>0

X

A0 · fν 0

P∞

Nj(k) = −

k=1

p Y

iν 0 · hν(kss ) ,

Nj(k) εk , with Nj(k) defined by Nj(1) = −fj0 and, for k ≥ 2,

X 1 p! p>0

(2.2)

s=1

k1 +...+kp =k−1

provided Nj (ε) =

(iν0j ) ·

0 +ν 1 +...+ν p =ν

X

·

s=1

˜ 1 +...+kp =k−1 k+k

1 X 1 − iω0 · ν p! ν

iν 0 · hν(kss ) ,

0 +ν 1 +...+ν p =ν

X

·

p Y

k1 +...+kp =k−1 s=1

0 +ν 1 +...+ν p =ν

1 X 1 =− iω0 · ν p! ν+ν ˜

X

fjν 0

X ν 0 +ν 1 +...+ν p =0

fjν 0

X

p Y

iν 0 · hν(kss ) .

(2.3)

k1 +...+kp =k−1 s=1

Equality (2.3) assures the formal solubility of (1.8). The function f is even, hence h is odd and H is even. If f is analytic (κ > 0) the convergence of the series defining the functions h and H is a corollary of [E1,E2] (see also [GM2], Theorem 1.4), but the convergence radius is not uniform in κ (it shrinks to zero when κ → 0). The aim of the present paper is to show that, if f belongs to the class of functions Cˆ (p) (T` ), then there are cancellation mechanisms that imply convergence of the series and, therefore, analyticity in ε of the equations of motion solution. 2.2. We shall use a representation of (2.2) in terms of “Feynman graphs” following the rules in [BGGM], Sect. 3: the reader not familiar with [BGGM] can find in Appendix A1 below a brief but selfconsistent description of the graphs. See [GGM] for the terminology motivation. The only difference will be that that the “value” of a graph ϑ is now given by Y (iν v0 · fν ) v , (2.4) Val(ϑ) = iω · ν 0 λ v v w, has vw on the path of a real or virtual resonance V . Then the Q σw ν x) change of variables U ∈ U(B1v (P )) constructs a graph (ϑ0 , {ν λ }, w∈B1v (P ) Uvw in which the line incoming into the resonance carries some momentum −ν while the outgoing line carries a momentum ν: hence in the new graph the cluster V is no longer a resonance; or, viceversa, it can happen that a virtual resonance becomes real. To avoid this “interference between ultraviolet and infrared cancellations" we must exclude the resonances (virtual or real) from the interpolations. 4.7. Then we see that (4.22) leads to the following “path expansion” for Sk (ϑ0 ) summarizing our analysis X X X∗ Y X X |ν|s |W (ϑ0 , ν)| = |ν|s (4.27) ν

ν

nZ

0 1

Y w∈B1v (P )

{hx } P ∈P {ν λ } v∈Mh (P )

dtw

X

o Ov fν1 v (tv ) (ν v (tv ))mv Yv (tv ) RD(ϑ0 ) ,

||mv ||=pv

where (1) P is a partial pavement of the graph with non overlapping “paths” such that: (1.1) a path P (v, z) is a connected set of comparable lines (“ordered paths”) connecting the

728


node v to the node z < v; (1.2) the resonance paths are not contained in any path; (1.3) for any line λ which is not contained in any resonance path there is one path in P covering it: λ ∈ P (v, z) for some P (v, z) ∈ P ; (2) Mh (P ) is the collection of upper end nodes of the paths in P , and Me (P ) of lower end nodes; (3) Bv (P ) is the set of nodes w immediately preceding v such that (3.1) vw is not in any resonance path, and (3.2) a path P (v, z(v, w)) ∈ P starting from v passes through w, and B1v (P ) is the set of nodes w ∈ Bv (P ) which are out of order with respect to v, i.e. such that (4.28) 2hv > 25 pv |ν λw | ; (4) Yv (tv ) is defined as Q 4 ˜ vw (tv )) , w∈B¯ v (P )\B1v (P ) (2 pv x Yv (tv ) = 1,

if v ∈ Mh (P ) , otherwise ,

(4.29)

if x ˜ vw (tv ) is the vector defined via the implicit relation ˜ vw (tv ) · ν λw , |ν v (tv )| = 25 pv x

(4.30)

˜ vw (tv )) depends on ν λw but not on the individual external so that |x ˜ vw (tv ))| ≤ 1 and x momenta which add to ν λw and B˜v (P ) is the set of nodes verifying (3.2) in item (3); (5) the operator Ov is defined as Ov (ν v (tv ))mv Yv (tv ) fν v (tv ) = (4.31) ∂ |B1v (P )| 1 Yv (tv ) mv ηv f ( ν) ¯ (2 ν) ¯ , = ν ¯ ν=ν ¯ v (tv ) |ν| ¯ |Bv (P )|−|B1v (P )| ∂ ν¯ |B1v (P )| with ηv = 1 if v ∈ Me (P ), and ηv = 0 otherwise, and fν1 defined after (4.24); (6) the sum over {ν λ } has the restriction that the external momentum configuration {ν x } is compatible with the scales {hx }; (7) RD(ϑ0 ) is the same for all graphs involved in the cancellations mechanisms, as the moduli of the momenta do not change under the action of the change of variables (4.2), and the signs are taken into account by the interpolation formula (4.3) (see Remark 4.2). 4.8. We can bound Y

|ν|s

Ov fν1 v (tv ) (ν v (tv ))mv Yv (tv ) ≤ v∈ϑ0

≤

Y

D1 D2pv qv ! ppvv −qv 2hv (1−b+s+ηv ) ≤

v∈ϑ0

n

ϑ0

≤ max C3k P ∈P

D3 D4pv pv ! 2hv (1−b+s+ηv ) ,

v∈ϑ0

for suitable constants Dj , and use X X |ν|s RW (ϑ0 , ν) ≤ {ν}

Y

h Y v≤v0

Q

η

2τ ≤ v∈ϑ0 |ν v |

iX Y h

η

Q

2hv (pv +s+`+ 2 τ −b)

pv !

{hx } v≤v0

(4.32)

η

hv 2 τ , so that v∈ϑ0 2

Y P (v,z)∈P

2(hz −hv )

io

(4.33) ,


729

where the number of pavements P is estimated by 2k , see Appendix A2, in [BGGM]. Then, setting b = 2 + s + ` + η2 τ + µ, with µ > 0, and exploiting the identity X

hv pv =

v 2(` + 1) continuous derivatives (if ` is the dimension of the frequency vector of the quasi periodic potential and τ is supposed to be τ > `−1), with no other restriction on the potential regularity. However, in order to perform a meaningful comparison between the two results, one has to consider carefully the exact form of the interaction potential. 5.2. The problem studied in [DS,R,Pa] is the Schrödinger equation h

−

i d2 + εV (x) ψ(x) = Eψ(x) , 2 dx

where V (x) is a quasi periodic function of the form X V (x) = eiω·νx Vν ,

(5.1)

(5.2)

ν∈Z`−1

with ω ∈ R`−1 satisfying a diophantine condition. The problem to find eigenvalues and eigenfunctions of (5.1) can be easily seen, see for instance [G2], to be equivalent to solving the equations of motion of the classical mechanics system described by the Hamiltonian H=

i q2 h p2 +ω·B+ E − εV (β) , 2 2

(5.3)

with (p, q) ∈ R2 and (B, β) ∈ R`−1 × T`−1 . In fact the evolution equation for the coordinate q is the eigenvalue equation (5.1). Then it is possible to introduce a canonical transformation C : (p, q) → (A1 , α1 ), [G2], such that the Hamiltonian (5.3) becomes H=

√

EA1 + ω · B + εf (α1 , β) ,

A1 sin2 α1 V (β) , f (α1 , β) = − √ E

(5.4)

which can be reduced to the form (1.3), with A = (A1 , B) ∈ R` , α = (α1 , β) ∈ T` , and f(α) = (f (α), 0, . . . , 0). For the proof of such an assertion, we refer to [G2]. And the equations of motions for β give β(t) = β 0 + ωt, and the derivatives whose number can grow up indefinitely, in the expansion described in Sect. 2, are those acting on the α1 variable: however the perturbation is always analytic in α1 . Thus the assumptions on the interaction potential V can be weakened, compared to the ones following from the general result in Theorem 1.4, simply because the onedimensional Schrödinger equation can be reduced to a classical mechanics problem with Hamiltonian of the form (1.1), but the interaction term depends analytically on α1 , independently on the regularity of the quasi periodic potential. In this case the existence of the counterterm can be proved without exploiting ultraviolet cancellations, and the infrared cancellations are sufficient to give convergence of the perturbative series, provided the quasi periodic potential is so regular to guarantee


731

the summability on the Fourier components in the perturbative series: the analysis in Sects. 3, 4 gives p > ` + 3τ , see Appendix A3 for details. Then, if τ > ` − 1, one has p > 4` − 3. With respect to [Pa], the result is weaker for ` ≥ 3 but, in some respects, better for ` = 2. The result in [Pa] has been obtained by using the Moser-Nash techniques for KAM theory, and it is known that the class of differentiability of the perturbations of integrable systems can be raised in the KAM theory above Moser’s result, [P]: then one can conjecture that also for the Schrödinger equation the ideas in [P] could lead to p > 2`. Our result p > 4` − 3 can be considered, for ` = 2, a partial improvement of [Pa] in this direction. We stress that with the techniques described in the present paper, the ultraviolet cancellations do not enter into the analysis to obtain analyticity in the perturbative parameter of the eigenvalue E and of the corresponding eigenfunction ψ(x) in (5.1). It follows that the techniques of [E2] imply , in this case, our results, although the question was not relevant for that paper. 5.3. The situation is essentially identical if one consider the Schrödinger equation h

−

i d2 + U (x) + εV (x) ψ(x) = Eψ(x), dx2

(5.5)

where U (x) is a periodic potential with frequency ω2 and V (x) a quasi periodic function of the form X eiω·νx Vν , (5.6) V (x) = ν∈Z`−2

with ω ∈ R`−2 , such that ω2 and ω satisfy a diophantine condition. The Hamiltonian of the corresponding classical mechanics problem is H=

i q2 h p2 + ω2 B2 + ω · B + E − U (β2 ) − εV (β) , 2 2

(5.7)

with (p, q) ∈ R2 , (B2 , β2 ) ∈ R1 × T1 , and (B, β) ∈ R`−2 × T`−2 . If ε = 0, the Hamiltonian is integrable, [G2,C], so that (5.7) becomes H = ω01 A1 + ω02 A2 + ω · B + ε A1 f (α1 , α2 , β) , f (α1 , α2 , β) = −G(α1 , α2 ) V (β) ,

(5.8)

where G(α1 , α2 ) is a function which depends analytically on α1 , [C], Sect. V,VI, independently on the regularity of U and V in (5.5). The fact that the interaction is proportional only to A1 (i.e. independent on the other action variables) implies that the equations of motion for α2 and β can be trivially integrated and give α2 = α20 + ω02 t and βj = βj0 + ω0j t, 2 ≤ j ≤ `. Then we can reason as in Sect. 5.2, and the same conclusions hold.

Appendix A1. Graphs and Graph Rules We lay down one after the other, on a plane, k pairwise distinct unit segments oriented from one extreme to the other: respectively the “initial point” and the “endpoint” of the oriented segment. The oriented segment will also be called “arrow”, “branch” or “line”. The segments are supposed to be numbered from 1 to k.

732


The rule is that after laying down the first segment, the “root branch”, with the endpoint at the origin and otherwise arbitrarily, the others are laid down one after the other by attaching an endpoint of a new branch to an initial point of an old one and by leaving free the new branch initial point. The set of initial points of the object thus constructed will be called the set of the graph “nodes” or “vertices”. A graph of “order” k is therefore a partially ordered set of k nodes with top point the endpoint of the root branch, also called the “root” (which is not a node); in general there will be several “bottom nodes” (at most k − 1). We denote by ≤ the ordering relation, and say that two nodes v, w are “comparable” if v < w or w < v. With each graph node v we associate an “external momentum” or “mode” which is simply an integer component vector ν v 6= 0; with the root of the graph (which is not regarded as a node) we associate a label j = 1, . . . , `. For each node v, we denote by v 0 the node immediately following v and by λv ≡ v 0 v the branch connecting v to v 0 (v will be the initial point and v 0 the endpoint of λv ). If v1 is the node immediately preceding the root r (“highest node”) then we shall write v10 = r, for uniformity of notation (recall that r is not a node). We consider “comparable” two lines λv , λw , if v, w are such. If pv is the number of branches entering the node v, then each of the pv branches can be thought of as the root branch of a “subgraph” having root at v: the subgraph is uniquely determined by v and one of the pv nodes w immediately preceding v. Hence if w0 = v it will be denoted ϑvw . We shall call “equivalent” graphs which can be overlapped by (1) changing the angles between branches emerging from the same node, or (2) permuting the subgraphs entering into a node v in such a way that all the labels match. The number of (non-equivalent numbered) graphs with k branches is bounded by 4k k!, [HP].

Appendix A2. Proof of Lemma 3.6 We consider all the graphs we obtain by detaching from each resonance the subgraph with root vVb , if vVb is the node in which the resonance line λV enters, then reattaching it to all the nodes w ∈ V . We call this set of contributions a “resonance family”. If one sets ζ ≡ ω0 · ν λV = 0, no propagator changes inside the resonance, and the only effect of the above operation is that in the factor h χ(2−nλ ω0 · ν λV ) i (ν v0 · fν va ) (ν vb · fν v1 ) V V V (ω0 · ν λV )2 V

(A2.1)

appearing in (3.3) the external momentum ν vb assumes successively the values ν w , V w ∈ V . In this way, by summing over all P the trees belonging to a given resonance family, we build a quantity proportional to w∈V ν w = 0, by Definition 3.3 of real resonance. It is important to note that, by the definition of real resonance, the scale of the lines internal to V cannot change too much, certainly not enough to break the cluster V (i.e. no scale of a line internal to V can become smaller than nλV .)


733

Appendix A3. Regularity of the Potential for the Schrödinger Equation in a Quasi Periodic Potential The equations of motion of the Hamiltonian system (5.4) for the angle variables are 1 dα1 = k − ε √ (sin2 α1 )V (ω0 t) + N1 , dt E dβ = ω0 , dt

(A3.1)

√ where E = k + N1 , which can be discussed as in Sect. 2. We look for a “Bloch wave” with momentum k and energy E assuming that the vector (k, ω0 ) is diophantine (i.e. a quasi periodic solution with rotation vector (k, ω0 )). Following [G2] we regard the √1E in (A3.1) as a parameter to be fixed later: we can deduce a function of ε and in fact that the solution to (A3.1), and in particular N1 = N1 (ε, E) as√ ε √ E, is analytic in E and therefore the “dispersion relation” equation E = k +N1 (ε, E) can be easily solved (see [G2]). A formula in terms of graphs (2.6) can be still obtained, where Val(ϑ), defined in (2.4), becomes Y 1 ν v 0 s ν v Vν v −√ , (A3.2) Val(ϑ) = E kνλv + ω0 · ν λv v 3τ + ` − 1. The equations of motion for the action variables give A1 ∂ sin2 α1 dA1 = ε√ V (ω0 t) , dt E ∂α1 A1 ∂V (β) dB = ε √ sin2 α1 , dt ∂β β=ω0 t E

(A3.4)

so that we can reason as above, with the only difference that the highest node of the graph v1 has a factor ν v1 which requires, to guarantee the summability on ν v1 , V ∈ C (p) (T`−1 ), with p > 3τ + `. Then one has to require at least V ∈ C (p) (T`−1 ), p > 3τ + `, in order to have h1 ∈ C (1) (T1 ), as it has to be for the Schrödinger equation (5.1) to be meaningful, if one recalls that (1) the wave function ψ(x) solving (5.1) has to be of class C (1) for V ∈ C (0) , and (2) ψ(x) = q(x), where q is the variable related with α1 by the canonical transformation C defined before (5.4).5 5 Note that if we confine ourselves to the classes of functions introduced in Sect. 1.3, item (3), then we have to require V ∈ Cˆ (p) (T`−1 ), p > 3τ + 1, in order to have ψ ∈ C (1) .

734


Appendix A4. Comparison between Moser’s Counterterms Theorem and the Counterterms Conjecture in [G1] A 4.1. In [M1] a perturbation theory for quasi-periodic solutions of a nonlinear system of ordinary differential equations is developed. Up to a (trivial) coordinate transformation, the system can be written in the form dx = ω + εf(x, y; ε) , dt dy = y + εg(x, y; ε) , dt

(A4.1)

where x ≡ (x1 , . . . , xn ) ∈ Rn , y ≡ (y1 , . . . , ym ) ∈ Rm , ω ∈ Rn , is a constant m × m matrix with eigenvalues 1 , . . . , m , and f and g are functions with period 2π in x1 , . . . , xn and analytic in x, y and ε (in suitable domains). If the characteristic numbers ω1 , . . . , ωn , 1 , . . . , n verify the “generalized diophantine condition” n m X X −1 C0 i ν j ωj + µi i ≥ |ν|τ + 1 , j=1

(A4.2)

i=1

Pn with ν ≡ (ν1 , . . . , νm ) ∈ Zn , |ν| = j=1 νj , and (µ1 , . . . , µm ) ∈ Zm , then there exists unique analytic vector valued functions λ(ε) and m(ε) and a unique analytic matrix valued function M (ε) such that the modified system dx = ω + εf(x, y; ε) + λ(ε) , dt dy = y + εg(x, y; ε) + m(ε) + M (ε)y , dt

(A4.3)

admits a quasi periodic solution with the same characteristic number as the unperturbed one, [M1], Theorem 1. A 4.2. Let us consider the case in which m = n = `, = 0, and there exists a function H0 = ω · y + εf (x, y; ε) such that f = ∂y f and g = −∂x f . Then the system (A4.1) becomes the system studied in [GM2], Sect. 8. Under the same hypotheses, if moreover f (x, y; ε) ≡ εy · f(x) for some function f, (A4.1) and (A4.3) become the equations of motion of systems described by the Hamiltonians, respectively, (1.1) and (1.3). In fact the linearity in the action variables of the term added to the Hamiltonian H0 in (1.3) leads to a term independent of the action variables in the equations of motion, i.e. N(ε) ≡ λ(ε), while the counterterms m(ε) and M (ε) are identically vanishing as a consequence of the symplectic structure of the equations of motion (as one can argue a posteriori from Theorem 1.4 in Sect. 1). In the general case in which the function f (x, y; ε) appearing in the Hamiltonian H0 depends arbitrarily (but always analytically) on y, the systems studied in [M1] (under the same hypotheses as above) and in [GM2] are no longer equal to each other, i.e. the modified system (A4.3) is not the system with Hamiltonian considered in Eq. (1.10) of [GM2], so that Theorem 1.4 in [GM2] cannot be reduced to the results of [M1]: in fact not only there will be no more a trivial relation between the counterterms N(ε) and λ(ε), but also the equations of motion solutions will be different from each other.


735

Note however that the result following from Moser’s theorem applied to such a system (i.e. a Hamiltonian system with = 0) can be (trivially) reproduced with our techniques. Also an extension of our techniques to Hamiltonian systems (verifying the anisochrony condition) such that 6= 0 could been envisaged:6 an example in this direction is in [Ge], where has eigenvalues 1 = . . . = `−1 = 0, ` = g 2 , and the existence of a counterterm M (ε) analytic in ε is proven (while λ(ε) ≡ m(ε) ≡ 0 again for the symplectic structure of the equations of motion). Acknowledgement. Partially supported by the European Network on: “Stability and Universality in Classical Mechanics", # ERBCHRXCT940460. G.Ge. acknowledges support from the EC program TMR and IHES. We are grateful to L. H. Eliasson for kindly signaling an error in our concluding remarks in Sect. 5.

References [BGGM] Bonetto, F., Gallavotti, G., Gentile, G., Mastropietro, V.: Lindstedt series, ultraviolet divergences and Moser’s theorem. Preprint IHES/P/95/102 (1995) [C] Chierchia, L.: Absolutely continuous spectra of quasi periodic Schrödinger operators. J. Math. Phys. 28, 2891–2898 (1987) [DS] Dinaburg, E.I., Sinai, Ya.G.: The one dimensional Schrödinger equation with a quasi periodic potential. Funct. Anal. and Appl. 9, 279–289 (1975) [E1] Eliasson, L.H.: Absolutely convergent series expansions for quasi-periodic motions. Mathematical Physics Electronic Journal 2, (4): 1–33 (1996), http://mpej @math. utexas.edu. [E2] Eliasson, L.H.: Hamiltonian systems with linear normal form near an invariant torus. In: Nonlinear Dynamics, Ed. G. Turchetti, Bologna Conference, 30/5 to 3/6 1988, Singapore: World Scientific, 1989 [EV] Ecalle, J., Vallet, B.: Prenormalization, correction, and linearization of resonant vector fields or diffeomorphisms, Preprint 95-32, Université de Paris Sud - Mathematiques, Paris (1995) [G1] Gallavotti, G.: A criterion of integrability for perturbed harmonic oscillators. “Wick Ordering” of the perturbations in classical mechanics and invariance of the frequency spectrum. Commun. Math. Phys. 87, 365–383 (1982) [G2] Gallavotti, G.: Classical Mechanics and Renormalization Group. Lectures notes, 1983 Erice summer school, Ed. A. Wightman, G. Velo, N.Y.: Plenum, 1985 [G3] Gallavotti, G.: Quasi-integrable mechanical systems. In: Critical Phenomena, Random Systems, Gauge Theories. Les Houches, Session XLIII (1984), Vol. II, Ed. K. Osterwalder & R. Stora, Amsterdam: North Holland, (1986) pp. 539–624 [G4] Gallavotti, G.: Twistless KAM tori. Commun. Math. Phys. 164, 145–156 (1994) [GGM] Gallavotti, G., Gentile, G., Mastropietro, V.: Field theory and KAM tori. Mathematical Physics Electronic Journal 1, (5): 1–13 (1995), http://mpej@ math. utexas. edu [Ge] Gentile, G.: Whiskered tori with prefixed frequencies and Lyapunov spectrum. Dynam. Stability of Systems 10, 269–308, (1995) [GM1] Gentile, G., Mastropietro, V.: KAM theorem revisited. Physica D 90, 225–234 (1996) [GM2] Gentile, G., Mastropietro, V.: Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications. Rev. Math. Phys. 8, 393–444 (1996) [HP] Harary, F., Palmer, E.: Graphical enumeration. New York: Academic Press, 1973 [M1] Moser, J.: Convergent series expansions for quasi periodic motions. Math. Ann. 169, 136-176 (1967) [M2] Moser, J.: On the construction of almost periodic solutions for ordinary differential equations. Proceedings of the International Conference of functional analysis and related topics. Tokyo, 60– 67, (1969) [Pa] Parasyuk, I.S.: On instability zones of the Schrödinger equation with a quasi periodic potential. Ukr. Mat. Zh. 30, 70–78 (1985) 6

The anisochronus case with = 0 is simply the KAM theorem.

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[PF]

Pastur, L.A., Figotin, A.: Spectra of random and almost-periodic operators. Grundlehren der Mathematischen Wissenschaften 297, Berlin: Springer, 1992 ¨ Pöschel, J.: Uber invarianten Tori on differenzierbaren Hamiltonschen Systemen, Bonn. Math. Schr. 120, 1–120 (1980) Rüssmann, H.: One dimensional Schrödinger equation. Ann. New York Acad. Sci. 357, 90–107 (1980) Siegel, C.L.: Iterations of analytic functions. Ann. Math. 43, No.4, 607–612 (1943)

[P] [R] [S]

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