Matrix Integrals, Toda Symmetries, Virasoro Constraints and Orthogonal Polynomials

Matrix integrals, Toda symmetries, Virasoro constraints and orthogonal polynomials

M. Adler∗ P. van Moerbeke†

D´edi´eavec admiration au Professeur Paul Malliavin

Symmetries of the inﬁnite Toda lattice. The symmetries for the inﬁnite Toda lattice,

∂L 1 n (0.1) = [ (L )s,L], n = 1, 2, . . ., ∂tn 2 viewed as isospectral deformations of bi-infinite tridiagonal matrices L, are time- dependent vector fields transversal to the Toda hierarchy; bracketing a symmetry with a Toda vector field yields another vector field in the hierarchy. As is well known, the Toda hierarchy is intimately related to the Lie algebra splitting of gl(∞),

(0.2) gl(∞) = Ds ⊕ Db 3 A = As + Ab, into the algebras of skew-symmetric As and lower triangular (including the diagonal) matrices Ab (Borel matrices). We show that this splitting plays a prominent role also in the construction of the Toda symmetries and their action on τ−functions; it also plays a crucial role in obtaining the Virasoro constraints for matrix integrals and it ties up elegantly with the theory of orthogonal polynomials .

∗Department of Mathematics, Brandeis University, Waltham, Mass 02254, USA. The support of a National Science Foundation grant # DMS 95-4-51179 is gratefully acknowledged. †Department of Mathematics, Universit´ede Louvain, 1348 Louvain-la-Neuve, Belgium and Brandeis University, Waltham, Mass 02254, USA. The support of National Science Foundation # DMS 95-4-51179, Nato, FNRS and Francqui Foundation grants is gratefully acknowledged.

1 Deﬁne matrices δ and ε, with [δ, ε] = 1, acting on characters χ(z) = (χn(z))n∈Z = n (z )n∈Z as ∂ (0.3) δχ = zχ and εχ = χ . ∂z This enables us to deﬁne a wave operator S, a wave vector Ψ, P −1 1 ∞ t zi (0.4) L = SδS and Ψ = S exp 2 1 i χ(z), and an operator M, reminiscent of Orlov and Schulman’s M-operator for the KP- equation, such that ∂ (0.5) LΨ = zΨ and MΨ = Ψ, ∂z thus leading to identities of the form: ∂ (0.6) M βLαΨ = zα( )βΨ. ∂z The vector Ψ and the matrices S, L and M evolve in a way, which is compatible with the algebra splitting above,

∂Ψ 1 n ∂S 1 n (0.7) = (L )sΨ and = − (L )bS, ∂tn 2 ∂tn 2

∂L 1 n ∂M 1 n (0.8) = [ (L )s,L] and = [ (L )s,M], ∂tn 2 ∂tn 2 and the wave vector Ψ has the following representation in terms of a vector1 of τ-functions τ = (τn)n∈Z :

³ −η ´ 1 i e τ (t) Σtiz n n n (0.9) Ψ(t, z) = e 2 z q ≡ (z Ψn)n∈Z. n∈Z τn(t)τn+1(t)

The wave vector deﬁnes a t-dependent ﬂag

t t t ... ⊃ Wk−1 ⊃ Wk ⊃ Wk+1 ⊃ ...

P −i 1with η = ∞ z ∂ ; note the 1/2 appearing in Ψ 1 i ∂ti

2 of nested linear spaces, spanned by functions of z,

t k k+1 (0.10) Wk ≡ span{z Ψk, z Ψk+1,...}

Formula (0.6) motivates us to give the following deﬁnition of symmetry vector ﬁelds (symmetries), acting on the manifold of wave functions Ψ and inducing a Lax pair on the manifold of L-operators 2:

β α β α (0.11) Y α ∂ β Ψ = −(M L )bΨ and Y α ∂ β L = [−(M L )b,L]. z ( ∂z ) z ( ∂z ) It turns out that only the vector ﬁelds

n n+` (0.12) Y`,n := Y n+` ∂ n = −(M L )b, for n = 0, ` ∈ Z and n = 1, ` ≥ −1, z ( ∂z ) conserve the tridiagonal nature of the matrices L. The expressions (0.12), for n = 1, ` < −1 have no geometrical meaning, as the corresponding vector fields move you out of the space of tridiagonal matrices. This phenomenon is totally analogous to the KdV case (or pth Gel’fand-Dickey), where a certain algebra of symmetries, a representation of the sub-algebra 3 Diff(S1)+ ⊂ Diff(S1) of holomorphic vector fields ∂2 on the circle, maintains the differential nature of the 2nd order operator ∂x2 + q(x) (or pth order differential operator). According to a non-commutative Lie algebra splitting theorem, due to ([A-S- V]), stated in section 2 and adapted to the Toda lattice, we have a Lie algebra anti-homomorphism: (0.13) ∂ w ≡= {zn+`( )n, n = 0, ` ∈ Z, or n = 1, ` ≥ −1} 2 ∂z → {tangent vector fields on the Ψ-manifold} :

n+` ∂ n z ( ) 7→ Yzn+`( ∂ )n , acting on Ψ as in (0.11), ∂z ∂z to wit,

(0.14) [Y`,0, Ym,1] = `Ym+`,0 and [Y`,1, Ym,1] = (` − m)Ym+`,1.

2 sometimes YM β Lα will stand for Y α ∂ β z ( ∂z ) 3 1 + k+1 ∂ Diﬀ(S ) := span{z ∂z , k ≥ −1}

3 Transferring symmetries from the wave vector to the τ-function . An important part of the paper (section 3) is devoted to understanding how, in the general Toda context, the symmetries Y`,n acting on the manifold of wave vectors Ψ induce vector ﬁelds L`,n on the manifold of τ-vectors τ = (τj)j∈Z, for n = 0 or 1; this new result is contained in Theorem 3.2:

(0.15) ³ ´ −η 1 τ Y`,n log Ψ = (e − 1)L`,n log τ + L`,n log( ) . (Fundamental relation) 2 τδ for n = 0, ` ∈ Z and n = 1, ` ≥ −1 4 where “ Y`,n log” and “L`,n log” act as logarithmic derivatives , where L`,nf = j 5 (L`,nfj)j∈Z, and

j (2) (1) 2 (0) j (1) (0) (0.16) L`,1 = J` + (2j − ` − 1)J` + (j − j)J` , L`,0 = 2J` + 2jJ` .

Note the validity of the relation not only for inﬁnite matrices, but also for semi- inﬁnite matrices. Also note the robustness of formula (0.15): it has been shown to be valid in the KP-case (continuous) and the 2-dimensional Toda lattice (discrete) by [A-S-V]. We give here an independent proof of this relation, although it could probably have been derived from the [A-S-V]-vertex operator identity for the two- dimensional Toda lattice. The rest of the paper will be devoted to an application of the fundamental relation.

Orthogonal polynomials, skew-symmetric matrices and Virasoro constraints. Consider now in section 4 an orthonormal polynomialP basis (pnP(t, z))n≥0 ∞ i ∞ i + 2 tiz −V0+ tiz of H ≡ {1, z, z ,...} with regard to the weight ρ0(z)e 0 dz = e 0 on the interval [a, b], −∞ ≤ a < b ≤ ∞, satisfying: (0.17) 0 P∞ i ρ biz h0 0 0 P0 k k − = V0 = ∞ i =: and f0(a)ρ(a)a = f0(b)ρ(b)b = 0 (k = 0, 1,...). ρ0 0 aiz f0

The polynomials pn(t, z) are t-deformations of pn(0, z), through the exponential in the weight. Then the semi-inﬁnite vector Ψ and semi-inﬁnite matrices L and M,

n n+` ³ ´ − (M L )bΨ 4 ( )j τ L`,nτj L`,nτj+1 for instance Y`,n log Ψj := Ψ and L`,n log( τ ) ≡ τ − τ j δ j j j+1 5 (0) (1) ∂ 1 (2) (1) (1) set Jn = δn,0, Jn = + (−n)t−n, and Jn = Σi+j=n : J J : ∂tn 2 i j

4 defined by P 1 ∞ t zi ∂ (0.18) Ψ(t, z) ≡ exp 2 1 i (p (t, z)) , zΨ = LΨ and Ψ = MΨ n n≥0 ∂z are solutions of the Toda differential equations (0.7) and (0.8). Moreover, Ψ(t, z) can be represented by (0.9) with τ0 = 1 and Z P 1 ∞ i −T r V0(Z)+ ti T r Z (0.19) τn = dZ e 1 , n ≥ 1; Ωnn! Mn(a,b) here the integration is taken over a subspace Mn(a, b) of the space of n×n Hermitean matrices Z, with eigenvalues ∈ [a, b]. We prove in Theorem 4.2 that in terms of the matrices L and M, defined in 1 (0.18) and in terms of the anti-commutator {A, B} := 2 (AB+BA), the semi-infinite matrices6 (0.20) 0 m+1 m+1 m + 1 m (f0ρ0) Vm := {Q, L } = QL + L f0(L), with V−1 = Q := Mf0(L)+ (L). 2 2ρ0 are skew-symmetric for m ≥ −1 and form a representation of the Lie algebra of holomorphic vector fields Diff(S1)+, i.e. they satisfy X (0.21) [Vm,Vn] = (n − m) aiVm+n+i, m, n ≥ −1. i≥0

Thus, in terms of the splitting (0.2), we have for orthogonal polynomials the following identities:

(0.22) (Vm)b = 0, for all m ≥ −1,

leading to the vanishing of a whole algebra of symmetry vector fields YVm on the locus of wave functions Ψ, defined in (0.18); then using the fundamental relation (0.15) to transfer the vanishing statement to the τ-functions τn, we find the Virasoro- type constraints for the τn, n ≥ 0 and for m = −1, 0, 1, 2,...: X ³ ´ (2) (1) 2 (0) (1) (0) (0.23) ai(Ji+m + 2n Ji+m + n Ji+m) − bi(Ji+m+1 + n Ji+m+1) τn = 0, i≥0

6 the matrices Vm are not to be confused with the potential V0, appearing in the weight.

5 in terms of the coeﬃcients ai and bi of f0 and h0 (see (0.18)). In his fundamental paper [W], Witten had observed, as an incidental fact, that in the case of Hermite polynomials V−1 = M − L is a skew-symmetric matrix. In this paper we show that skew-symmetry plays a crucial role; in fact the Virasoro constraints (0.23) are tantamount to the skew-symmetry of the semi-inﬁnite matrices (0.20). They can also be obtained, with uninspired tears, upon substituting

k+1 (0.24) Z 7→ Z + εf0(Z)Z in the integrand of (0.19); then the linear terms in ε in the integral (0.19) must vanish and yield the same Virasoro-type constraints (0.23) for each of the integrals τn as is carried out in the appendix. At the same time, the methods above solve the “string equation”, which is : for given f0, ﬁnd two semi-inﬁnite matrices, a symmetric L and a skew-symmetric Q, satisfying

(0.25) [L, Q] = f0(L).

For the classical orthogonal polynomials, as explained in section 6, the ma- q √ f0 ∂ trices L and Q, matrix realizations of the operators z and ρ0f0 respectively, ρ0 ∂z acting on the space of polynomials, are both tridiagonal, with L symmetric and Q skew-symmetric. In addition, the matrices L and Q stabilize the ﬂag, deﬁned in (0.10), in the following sense:

(0.28) zWk ⊂ Wk−1 and T−1Wk ⊂ Wk−1.

This result is related to a classical Theorem of Bochner; see [C]. The results in this paper have been lectured on at CIMPA (1991), Como, Utrecht (1992) and Cortona (1993); see the lecture notes [vM]. Grinevich, Orlov and Schul- man made a laconic remark in a 1993 paper [GOS, p. 298] about defining symmetries for the Toda lattice. We thank A. Grünbaum, L. Haine, V. Kac, A. Magnus, A. Morozov, T. Shiota, Cr. Tracy and E. Witten for conversations, regarding various aspects of this work. We also thank S. D’Addato-Muësfor unscrambling an often messy manuscript.

Table of contents:

6 1. The Toda lattice revisited 2. Symmetries of the Toda lattice and the w2-algebra 3. The action of the symmetries on the τ-function 4. Orthogonal polynomials, matrix integrals, skew-symmetric matrices and Virasoro constraints 5. Classical orthogonal polynomials 6. Appendix: Virasoro constraints via the integrals

1 The Toda lattice revisited

On Z we deﬁne the function χ (character)

n χ : Z × C → C :(n, z) 7→ χn(z) = z and the matrix-operators δ and ε (1.1)  .   .  .. ..          0 1   -2 0       0 1   -1 0          δ =  0 1  and ε =  0 0       0 1   1 0           0   2 0  ...... acting on χ as ∂ (1.2) δχ = zχ and εχ = χ; ∂z they satisfy [δ, ε] = 1. Proposition 1.1. The inﬁnite matrix7 X ˜ j (1.3) L = ajδ ∈ D−∞,1, aj(t) = diag(aj(n, t))n∈Z, a1 = 1, j≤1

7 Dk,` (k < ` ∈ Z) denotes the set of band matrices with zeros outside the strip (k, `). The symbols ( )+, ()−, ()0 denote the projection of a matrix onto D0,∞, D−∞,−1 and D00 respectively.

7 subjected to the deformation equations ˜ ∂L ˜n ˜ ˜n ˜ (1.4) = [(L )+, L] = [−(L )−, L], n = 1, 2,.... ∂tn has, for generic initial conditions, a representation in terms of τ-functions τn ³ ´ ˜ ˜ ˜−1 γn 2 ∂ 2 (1.5) L = SδS = ..., ( ) , log γn, 1, 0,... , γn−1 ∂t1 n∈Z where8 s P −1 ∞ ˜ −n τn+1 ˜ τ(t − [δ ]) n=0 pn(−∂)τ(t)δ (1.6) γn = , S = = τn τ(t) τ(t)

The wave operator S˜ and the wave vector9

P ³ −η ´ ∞ i e τn ˜ tiz ˜ n Σ n ˜ (1.7) Ψ := e 1 Sχ(z) = z e =: (z Ψn)n∈Z τn n∈Z satisfy ˜ ˜ ˜ ˜ ˜ ∂Ψ ˜n ˜ ∂S ˜n (1.8) LΨ = zΨ, = (L )+Ψ, = −(L )−S. ∂tn ∂tn Proof: The proof of this statement can be deduced from the work of Ueno-Takasaki [U-T]; we consider only a few points: from equation (1.6), it follows that:

(1.9) S˜ = I −Aδ−1 −Bδ−2 −..., and S˜−1 = I +Aδ−1 +Bδ−2 +Aδ−1Aδ−1 +.... with ∂ p (−∂˜)τ(t) A = log τ, and B = − 2 . ∂t1 τ(t) ˜ Then, calling pkτ := pk(−∂)τ, we have P ∞ i P 8 tiz ∞ k ˜ The pk’s are the elementary Schur polynomials e 1 = 0 pk(t)z and pk(−∂) := ∂ 1 ∂ 1 ∂ α2 α3 pk(− , − , − ,...). Also [α] := (α, , , ...). ∂t1 2 ∂t2 3 ∂t3 2 3 9 P∞ z−i ∂ P∞ j set η = and Σ = tjz 1 i ∂ti 1

8 (1.10)

˜ ˜ ˜−1 0 2 −1 L = SδS = δ + (An+1 − An)n∈Zδ + (Bn+1 − Bn + AnAn+1 − An)n∈Zδ + ... ³ ∂ τ ´ = δ + log n+1 δ0 ∂t1 τn n∈Z ³ p τ p τ p τ p τ p τ ´ + − 2 n+1 + 2 n + 1 n 1 n+1 − ( 1 n )2 δ−1 + ..., τn+1 τn τn τn+1 τn n∈Z yielding the representation (1.5), except for the δ−1-term, which we discuss next. ˜ ˜ To the Toda problem is associated a ﬂag of nested planes Wn+1 ⊂ Wn ∈ Grn, ˜ n ˜ n+1 ˜ Wn ≡ span{z Ψn, z Ψn+1,...} n ˜ ∂ ˜ ∂ 2 ˜ = span z {Ψn, Ψn, ( ) Ψn,...}. ∂t1 ∂t1 ˜ ˜ ˜ −1 The inclusion Wn+1 ⊂ Wn implies, by noting Ψk = 1 + O(z ), that

˜ ∂ ˜ ˜ (1.11) zΨn+1 = Ψn − αΨn for some α = α(t). ∂t1

∂ 10 Then α(t) = log τn+1/τn and putting this expression in (1.11) yields ∂t1 ³ ´ −1 −1 −1 {τn(t − [z ]), τn+1(t)} + z τn(t − [z ])τn+1(t) − τn+1(t − [z ])τn(t) = 0.

Expanding this expression in powers of z−1 and dividing the coeﬃcient of z−1 by τnτn+1 yield 2 ∂ τn p τ p τ p τ p τ ∂t2 − 2 n+1 + 2 n + 1 n 1 n+1 − 1 = 0. τn+1 τn τn τn+1 τn Combining this relation with the customary Hirota bilinear relations, the simplest one being:

2 2 1 ∂ ∂ τn−1τn+1 − 2 τn ◦ τn + τn−1τn+1 = 0, i.e. 2 log τn = 2 , 2 ∂t1 ∂t1 τn we ﬁnd 2 p2τn+1 p2τn p1τn p1τn+1 p1τn 2 ∂ τn−1τn+1 − + + − ( ) = 2 log τn = 2 , τn+1 τn τn τn+1 τn ∂t1 τn 10{f, g} = ∂f g − f ∂g ∂t1 ∂t1

9 and thus the representation (1.5) of L˜, ending the proof of Proposition 1.1.

Henceforth, we assume L˜ as in proposition 1.1, but in addition tridiagonal: L˜ = P j ajδ ; this submanifold is invariant under the vector field (1.4); indeed, more −1≤j≤1 P ˜ j generally if L = j≤1 ajδ is a N + 1 band matrix, i.e. aj = 0 for j ≤ −N ≤ 0, then L˜ remains a N + 1 band matrix under the Toda vector fields. Moreover consider the Lie algebra decomposition, alluded to in (0.2), of gl(∞) = Ds ⊕Db 3 A = (A)s +(A)b in skew-symmetric plus lower Borel part (lower triangular, including the diagonal). Theorem 1.2. Considering the submanifold of tridiagonal matrices L˜ of propo- sitionq 1.1 and remembering the form of the diagonal matrix γ = (γn)n∈Z, with γ = τn+1 , we define a new wave operator S and wave vector Ψ : n τn

(1.12) S := γ−1S˜ and Ψ := Sχ(z)eΣ/2; also deﬁne

X∞ −1 1 k−1 −1 (1.13) L := SδS and M := S(ε + ktkδ )S . 2 1 Then the tridiagonal matrix ³ ´ γn ∂ 2 γn+1 (1.14) L = ..., 0, , log γn, , 0,... γn−1 ∂t1 γn n∈Z is symmetric and

−η −η −1 ˜ Σ/2 −1 e τ n Σ/2 e τn (1.15) Ψ = γ Ψ = e γ χ =: (z Ψn)n∈Z with Ψn := e √ . τ τnτn+1

The new quantities satisfy:

∂ log γ 1 n ∂S 1 n ∂Ψ 1 n (1.16) = (L )0 , = − (L )bS and = (L )sΨ, ∂tn 2 ∂tn 2 ∂tn 2

∂ (1.17) LΨ = zΨ and MΨ = Ψ, with [L, M] = 1, ∂z

10 and

∂L 1 n ∂M 1 n (1.18) = [ (L )s,L], = [ (L )s,M]. ∂tn 2 ∂tn 2 Proof of Theorem 1.2: For a given initial condition γ0(0), the system of partial diﬀerential equations in γ0

∂ 0 1 ˜k (1.19) log γ = (L )0; ∂tk 2 has, by Frobenius theorem, a unique solution, since

Ã ! ∂ ñ ∂ ˜k ˜k ñ ˜k ñ (1.20) L − L = −[(L )−, L ]0 + [L , (L )+]0 ∂tk ∂tn 0 ˜k ñ ˜k ñ = −[(L )−, (L )+]0 + [(L )−, (L )+]0 = 0, using [A+,B+]0 = 0 and [A−,B−]0 = 0 for arbitrary matrices A and B. Given this solution γ0(t), define S0 := γ0−1S˜ and L0 := S0δS0−1 = γ0−1Lγ˜ 0, and using ˜ ñ ˜ ∂S/∂tn = −(L )−S, compute

∂S0 ∂(γ0−1S˜) ∂γ0−1 ∂S˜ = = S˜ + γ0−1 ∂tk ∂tk ∂tk ∂tk ∂γ0 ∂S˜ = −γ0−1( )γ0−1S˜ + γ0−1 ∂tk ∂tk 1 = −γ0−1 (L˜k) S˜ − γ0−1(L˜k) S˜ 2 0 − 1 0k 0 0k 0 0 0−1 ˜ 0 0 0−1 ˜ = − (L )0S − (L )−S since L = γ Lγ and S := γ S 2µ ¶ 0k 1 0k 0 1 0k 0 = − (L ) + (L ) S = − (L ) 0 S , − 2 0 2 b where Ab0 := 2A− + A0 and As0 := A − Ab0 . It follows at once that

0 ∂L 1 0n 0 1 0n 0 (1.21) = [− (L )b0 ,L ] = [ (L )s0 ,L ] ∂tn 2 2

11 Observe now that the manifold of symmetric tridiagonal matrices A is invariant under the vector ﬁelds

∂A 1 n 1 n (1.22) = [− (A )b0 ,A] = [ (A )s0 ,A], ∂tn 2 2

since for A symmetric, the operations ()b0 and ()s0 coincide with the decomposition (0.2): Ab0 = Ab and As0 = As. r Now according to formula (1.5), picking γ0(0) := γ(0) = τn+1(0) as initial condi- τn(0) tion for the system of pde’s (1.19), makes L0(0) = γ0(0)−1L˜(0)γ(0)0 symmetric. Since L0 was shown to evolve according to (1.21) or (1.22), and since its initial condition is symmetric, the matrix L0 remains symmetric in t. Since L0(t) := γ0−1(t)L˜(t)γ0(t) is symmetric and since, by definition, L(t) := γ−1(t)L˜(t)γ(t) is also symmetric, we have L0(t) = L(t), and thus γ0(t) = cγ(t) for some constant c; but c must be 1, since γ0(0) = γ(0). This proves (1.14), (1.16) and the first halfs of (1.17) and (1.18). Besides multiplication of Ψ by z, which is represented by the matrix L, we also consider differentiation ∂/∂z of Ψ, which we represent by a matrix M: ∂Ψ ∂ 1 X∞ (1.23) = eΣ/2S χ + ( kt zk−1)eΣ/2Sχ ∂z ∂z 2 k Ã 1 ! X∞ 1 k−1 =: P + ktkL Ψ =: MΨ 2 1 with −1 (1.30) [L, M] = 1 and P := SεS ∈ D−∞,−1. Finally compute

∂M ∂S −1 1 k−1 −1 = (S S)(ε + Σktkδ )S ∂tn ∂tn 2 1 n−1 1 k−1 −1 ∂S −1 + nL − S(ε + Σktkδ )S S 2 2 ∂tn 1 1 1 = − (Ln) M + [ Ln,M] + M(Ln) 2 b 2 2 b 1 1 = [− (Ln) + Ln,M] 2 b 2 1 = [ (Ln) ,M], 2 s

12 ending the proof of theorem 1.2. Remark 1.2.1: Theorem 1.2 remains valid for semi-inﬁnite matrices L; the proof would only require minor modiﬁcations.

2 Symmetries of the Toda lattice and the w2-algebra Symmetries are t-dependent vector fields on the manifold of wave functions Ψ, which commute with and are transversal to the Toda vector fields, without affecting the t-variables. We shall need the following Lie algebra splitting lemmas, dealing with operators and their eigenfunctions, due to [A-S-V].

Lemma 2.1. Let D be a Lie algebra with a vector space decomposition D = D+ ⊕ D− into two Lie subalgebras D+ and D−; let V be a representation space of D, and let M ⊂ V be a submanifold preserved under the vector ﬁelds deﬁned by the action of D−, i.e., D− · x ⊂ TxM, ∀x ∈ M.

For any function p: M → D, let Yp be the vector ﬁeld on M deﬁned by

Yp(x) := −p(x)− · x, x ∈ M. (a) Consider a set A of functions p: M → D such that

Yqp = [−q−, p], ∀p, q ∈ A, holds. Then Y: p 7→ Yp gives a Lie algebra homomorphism of the Lie algebra generated by A to the Lie algebra X (M) of vector ﬁelds on M:

[Yp1 , Yp2 ] = Y[p1,p2], ∀p1, p2 ∈ A, and hence we can assume without loss of generality that A itself is a Lie algebra. (b) Suppose for a subset B ⊂ A of functions

Zq(x) := q(x)+ · x ∈ TxM, ∀x ∈ M, q ∈ B, and hence deﬁnes another vector ﬁeld Zq ∈ X (M) when q ∈ B, and such that

Zqp = [q+, p], ∀p ∈ A, q ∈ B, holds. Then [Yp, Zq] = 0, ∀p ∈ A, q ∈ B. Remark 2.1.1: A special case of this which applies to many integrable systems is: V = D0, a Lie algebra containing D, and D acts on D0 by Lie bracket, i.e., Yp(x) = [−p(x)−, x], etc.

13 Proof: To sketch the proof, let p1, p2, p ∈ A and q ∈ B; then the commutators have the following form:

[Yp1 , Yp2 ](x) = Z1(x) and [Zq, Yp](x) = Z2x, where, using D− and D+ are Lie subalgebras,

Z1 := (Yp1 (p2))− − (Yp2 (p1))− + [p1−, p2−] = [−p1−, p2]− − [−p2−, p1]− + [p1−, p2−]

= (−[p1−, p2] − [p1, p2−] + [p1−, p2−] − [p1+, p2+])− = −[p1, p2]−. and

Z2 := (Zq(p))− + (Yp(q))+ − [q+, p−] = [q+, p]− + [−p−, q]+ − [q+, p−]

= [q+, p−]− + [−p−, q+]+ − [q+, p−] = [q+, p−] − [q+, p−] = 0, ending the proof of the lemma. In the setup of the lemma, if we are given a Lie algebra (anti)homomorphism φ : g → A, we denote Yφ(x) by Yx and Zφ(x) by Zx if there is no fear of confusion. Theorem 2.2. Let L represent an infinite symmetric tridiagonal matrix, flowing according to the Toda vector fields. There is a Lie algebra anti-homomorphism ( ) ∂ vector fields on the w+ = {zn+`( )n, n = 0, ` ∈ Z or n = 1, ` ≥ −1} → 2 ∂z manifold of wave functions ∂ zn+`( )n −→ Y Ψ = −(M nLn+`) Ψ ∂z `,n b satisfying ` [Y , Y ] = δ , [Y , Y ] = `Y and [Y , Y ] = (` − m)Y . `,0 m,0 2 `+m `,0 m,1 m+`,0 `,1 m,1 m+`,1 They commute with the Toda vector fields: ∂ [Y`,n, ] = 0. ∂tk

14 −1 Note the vector ﬁelds Y`,n induce vector ﬁelds on S and L = SδS

n n+` n n+` Y`,n(S) = −(M L )bS and Y`,n(L) = [−(M L )b,L].

Proof of Theorem 2.2: Taking into account the notation of 1.13 and in view of Lemma 2.1 and the remark 2.1.1, set

0 D := gl(∞), D+ := Ds, D− := Db and D := D × D on which D acts via diagonal embedding D ,→ D0 : p 7→ (p, p).

V := D M := respectively, the space of wave operatorsS, of wave functions Ψ X∞ −1 1 k−1 −1 or of pairs (L, M) = (SδS ,S(ε + ktkδ )S ), with an 2 1 inﬁnite symmetric tridiagonal matrix L ( M nLn+`, n = 0, ` ∈ Z A := span or n = 1, ` ≥ −1 B := span {Lα, α ∈ Z} and + g := w2 with the antihomomorphism φ : g → A given by

α β β α φ(z ∂z ) := M L .

Then the vector ﬁelds take the form:

YpΨ = −pbΨ, YpS = −pbS, p ∈ A

Yp(L, M) = ([−pb,L], [−pb,M]), p ∈ A 1 n 1 n ZLn/2Ψ = 2 (L )sΨ, ZLn/2S = − 2 (L )bS = ∂ Ψ = ∂ S ∂tn ∂tn by Theorem 1.2,

1 n 1 n ∂ ZLn/2(L, M) = ([ (L )s,L], [ (L )s,M]) = (L, M) by (1.18). 2 2 ∂tn

15 Note that the vector ﬁelds

(2.1) Ym,0 ≡ YLm all m ∈ Z and Y`,1 = YML`+1 , all ` ∈ Z, ≥ −1 are tangent to M. Indeed for m < 0, the vector ﬁeld reads

∂ m m m (2.2) Ψ = Ym,0Ψ = −(L )bΨ = −L Ψ = −z Ψ for m < 0. ∂sm

The solution to this equation with initial condition Ψ(0) is given by

m Ψ = e−smz Ψ(0)(t, z) i.e., every component of the vector Ψ(0) is multiplied by the same exponential factor, and so is each τ-function: 0 −msmt−m τk(t) = τk (t)e . Since the entries of the tridiagonal matrix only depend on ratios of τ-functions, this exponential factor is irrelevant for L. In the same way Ym,0 (m ≥ 0) is tangent to the space of symmetric tridiagonal matrices, because the solution to

∂Ψ m m m m ∂Ψ (2.3) = Ym,0Ψ = −(L )bΨ = (−L + (L )s)Ψ = −z Ψ + 2 , ∂sm ∂tm is given by 1 Σt zi τ(t + 2s − [z]) Ψ = e 2 i . τ(t)

Not only the vector ﬁelds Y`,0, but also the Y`,1’s (` ≥ −1) are tangent to the space of symmetric tridiagonal matrices, because

X∞ `+1 −i (2.4) Y`,1(L) = [−(ML )b,L] , having the form a1δ + a−iδ 0 `+1 `+1 = [−ML ,L] + [(ML )s,L] `+1 `+1 = L + [(ML )s,L] = symmetric matrix for ` ≥ −1.

With these data in mind, Lemma 2.1 implies Theorem 2.2.

16 3 The action of the symmetries on τ-functions

The main purpose of this section is to show that the symmetry vector fields Y defined on the manifold of wave function Ψ induce certain precise vector fields on τ, given by the coefficients of the vertex operator expansion. The precise statement is contained in theorem 3.2. Before entering these details, we need a general statement: Lemma 3.1. Any vector field Y defined on the manifold of wave functions Ψ and commuting with the Toda vector fields induce vector fields Yˆ on the manifold of τ-functions; they are related as follows, taking into account the fact that Y log acts as a logarithmic derivative:

−η ˆ 1 ˆ τn (3.1) Y log Ψn = (e − 1) Y log τn + Y log 2 τn+1 X∞ (n) −i (n) ≡ ai z + a0 1 where Yˆ is a vector ﬁeld acting on τ-functions; the part of (3.1) containing e−η −1 is a power series in z−1 vanishing at z = ∞, whereas the other part is z-independent. For any two vector ﬁelds

0 −η ˆ ˆ 0 1 ˆ ˆ 0 τn (3.2) [Y, Y ] log Ψn = (e − 1) [Y, Y ] log τn + [Y, Y ] log , 2 τn+1 showing that the map above from the algebra of vector fields on wave functions to the algebra of vector fields on τ-functions is homomorphism. q Proof: In the computation below we use γ = τn+1 and the fact that Y := . n τn commutes with the Toda flows ∂/∂tn and thus with η: −η . e τn . .3) (log Ψn) = (log − log γn) see (1.21)(3 τn −η (e τ˙n) τ˙n 1 τn . = −η − + (log ) e τn τn 2 τn+1 −η . 1 τn . = (e − 1)(log τn) + (log ) , using [η, Y] = 0. 2 τn+1 Applying the second vector field Y0 to relation (3.3) yields

0 −η 0 ˆ 1 0 ˆ τn Y Y log Ψn = (e − 1)Y (Y log τn) + Y (Y log ) 2 τn+1

17 with

Yˆ f Y0(Yˆ log f) = Y0( ) f Yˆ 0(Yˆ f) (Yˆ 0f)(Yˆ f) = − f f 2 Yˆ 0(Yˆ f) (Yˆ 0f)(Yˆ f) = − . f f 2

Using the relation above twice leads at once to (3.2), ending the proof of Lemma 3.1. The vector fields Y`,n on Ψ induce certain precise vector fields on τ, constructed (n+1) from expressions W` appearing in the vertex operator expansion; notice the (n+1) expressions W` differ slightly from the customary ones, because of the 1/2 mul- tiplying t but not ∂/∂t:

P P∞ 1 ∂ 1 ∞ t (µi−λi) (λ−i−µ−i) (3.4) X(t, λ, µ)τ = e 2 1 i e 1 i ∂ti τ 1 Σt (µi−λi) −1 −1 = e 2 i τ(t + [λ ] − [µ ]) X∞ k X∞ (µ − λ) −`−k (k) = λ W` (τ). k=0 k! `=−∞ For instance

(0) (0) (1) (1) (2) (2) (1) (3.5) Wn = Jn = δn,0,Wn = Jn ,Wn = Jn − (n + 1)Jn with   P ∂2 P ∂ ∂  + iti for n ≥ 0  for n > 0  ∂ti∂tj ∂tj  ∂tn  i+j=n −i+j=n J (1) = 0 for n = 0 J (2) = i,j≥P1 i,j≥1 P n  n  1 ∂  1  4 (iti)(jtj) + iti ∂t for n ≤ 0. (−n)t−n for n < 0  i+j=−n i−j=−n j 2 i,j≥1

We shall also need

j (2) (1) 2 (0) (2) (1) 2 (0) (3.6) L`,1 := W` + 2jW` + (j − j)W` = J` + (2j − ` − 1)J` + (j − j)J`

j (1) (0) (1) (0) L`,0 := 2W` + 2jW` = 2J` + 2jJ` .

18 ˜ (2) (2) We also introduce Wn , which diﬀers from Wn above by a factor 1/2, n + 1 (3.7) W˜ (2) := J (2) − J (1), with W˜ (2) = W (2) = J (2) for n = −1, 0, n n 2 n n n n and an operator Bm in z and t

X m+1 ∂ ∂ (3.8) Bm := −z + ntn , m ∈ Z, ∂z n>max(−m,0) ∂tn+m

∞ which restricted to functions f(t1, t2,...) of t ∈ C only yields

(2) (3.9) Bmf = Jm f for m = −1, 0, 1.

(2) The expressions Jn form a Virasoro algebra with central extension ` (3.10) [J (1),J (1)] = δ [J (1),J (2)] = `J (1) ` m 2 `+m ` m m+` `3 − ` [J (2),J (2)] = (` − m)J (2) + δ ` m `+m 12 `+m and consequently the following Virasoro commutation relations hold, upon setting

`3 − ` `(b2 − a2`2) (3.11) V = J (2) + (a` + b)J (1) , c = + , ` ` ` ` 12 2

[V`,Vm] = (l − m)V`+m + c`δ`+m,0. In particular, observe

`2 − ` (`3 − `) [W (1),W (2)] = `W (1) + δ , [W (2),W (2)] = (` − m)W (2) − 5 δ ` m `+m 2 `+m ` m `+m 12 `+m and11

(j) (j) (j) (j) (j) (j) 0 (3.12) [L`,0, Lm,1] = `L`+m,0 + c`,jδ`+m, [L`,1, Lm,1] = (` − m)L`+m,1 + c`,jδ`+m.

The purpose of this section is to prove the following relationship between the action of the symmetries on Ψ and τ.

11 0 ` 2 2 with c`,j = `(` + 2j − 1) c`,j = − 12 (5` + 24j + 7)

19 Fundamental Theorem 3.2. The following relationship holds 1³ ´ (3.13) Y log Ψ = (e−η − 1)L log τ + L log τ − (L log τ) `,n `,n 2 `,n `,n δ for n = 0, all ` ∈ Z n = 1, all ` ≥ −1 where Y`,n log and L`,n log act as logarithmic derivatives and where ³ ´ j j+1 (τδ)j = τj+1 (L`,nτ)j = L τj (L`,n)δ = L . `,n j `,n Corollary 3.2.1. The following holds for m ≥ −1 :

³ ´ ³ (2) (1) 2 (0) m+1 m + 1 m j −η (Jm + 2jJm + j Jm )τj − ML + L Ψ = z Ψj((e − 1) 2 b τj

1³(J (2) + 2jJ (1) + j2J (0))τ (J (2) + 2(j + 1)J (1) + (j + 1)2J (0))τ ´´ + m m m j − m m m j+1 ) , 2 τj τj+1 j∈Z (i) where J` is defined in (3.5). Remark that the statements of the theorem and the corollary are equally valid for semi-infinite matrices. Before giving the proof we need three lemmas. P∞ z−i ∂ Lemma 3.3. The operator Bm defined in (3.8) interacts with η = and 1 i ∂ti P∞ i 12 Σ = 1 tiz as follows

Xm X−m m−k ∂ m+k (3.14) [Bm, η] = z and BmΣ = − z ktk k=1 ∂tk k=1

−η −η [Bm, e ] = −e [Bm, η]; thus [Bm, η] = 0 when m ≤ 0 and BmΣ = 0 when m ≥ 0.

12with the understanding that Xα = 0 if α < 1. j=1

20 ∞ Moreover given an arbitrary function f(t1, t2,...) of t ∈ C , deﬁne e−ηf (3.15) Φ = eΣ/2 . f Then we have for m ≥ 0

(1) 1 −m −η W−m(f) (i) − 2 z Φ = Φ(e − 1) f , m > 0 W˜ (2) (f) (3.16) (ii) (B + mtm )Φ = Φ(e−η − 1) −m , −m ³2 f ´ −η (Bm−[Bm,η])f [Bm,η]f (iii) BmΦ = Φ (e − 1) f − f .

Proof: The ﬁrst commutation relation (3.14) follows from a straightforward computation:

h i h X i m+1 ∂ ∂ [Bm, η] = −z , η + ktk , η ∂z k>max(−m,0) ∂tk+m X∞ ∂ X∞ ∂ Xm ∂ = zm−i − z−j = zm−i . 1 ∂ti 1 ∂tj+m 1 ∂ti The third commutation relation (3.14) follows at once from the fact that the bracket [Bm,.] is a derivation and that [[Bm, η], η]. Also

X∞ X max(X−m,0) k+m k+m k+m BmΣ = − ktkz + ktk z = − ktk z . 1 k>max(−m,0) 1

Next, given Φ deﬁned as (3.15), we have that (using Bm is a derivation and (3.14)), (3.17)

e−ηf B Φ = B eΣ/2 m m f e−ηf e−ηf = B eΣ/2 + eΣ/2B f m m f ³ B f 1 [B , η]f ´ = Φ (e−η − 1) m + B Σ − e−η m f 2 m f ³ (B − [B , η])f 1 [B , η]f ´ = Φ (e−η − 1) m m + B Σ − m f 2 m f

21 which yields (3.16) (iii) for m ≥ 0, taking into account the fact that BmΣ = 0 for (1) 1 m ≥ 0. Equation (3.16) (i) is straightforward, using W−m = 2 mtm. Proving (3.16) (ii) is a bit more involved; indeed ﬁrst observe that for m ≥ 0

³ X mX−1 ´ ˜ (2) ∂ 1 1 W−mf = ntn + ntn(m − n)tm−n + (m − 1)mtm f n>m ∂tn−m 4 n=1 4 1 mX−1 1 = B−mf + ntn(m − n)tm−nf + (m − 1)mtmf; 4 n=1 4 therefore B (f) (3.19) (e−η − 1) −m f

³ ˜ (2) mX−1 ´ −η W−mf 1 1 = (e − 1) − ntn(m − n)tm−n − m(m − 1)tm f 4 n=1 4 W˜ (2) (f) = (e−η − 1) −m f mX−1 1 1 −n 1 −m+n − n(m − n)((tn − z )(tm−n − z ) − tntm−n) 4 n=1 n m − n 1 1 − m(m − 1)((t − z−m) − t ) 4 m m m ˜ (2) mX−1 −η W−m(f) 1 n−m = (e − 1) + ntnz . f 2 n=1

Using expression (3.17) for BmΦ, −m ≤ 0, and [Bm, η] = 0 when m ≤ 0, we ﬁnd mt ³ B f 1 mt ´ (B + m )Φ = Φ (e−η − 1) −m + B Σ + m −m 2 f 2 −m 2 ³ mX−1 ´ −η B−mf 1 −m+k = Φ (e − 1) − z ktk , using (3.14) f 2 k=1 W˜ (2) f = Φ(e−η − 1) −m , using (3.19), f ending the proof of (3.14) (ii) and thus of Lemma 3.3.

22 Lemma 3.4. The following holds (ν ≡ ε∂ = diag (··· , νi = i, ···))

2 (i) (PL )s = νLs + L−

(PL2) Ψ ∂ ³ ∂ τ ´ (ii) s = (e−η − 1)(2ν − I) log τ + ν z − log s . Ψ ∂t1 ∂t1 τ Proof: First consider (3.20) PLm+1 = Sεδm+1S−1 = SνδmS−1 = [S, ν]δmS−1 + νSδmS−1 = [S, ν]δmS−1 + νLm.

Since ν = εδ is diagonal, since S ∈ D−∞,0 and so [S, ν] ∈ D−∞,−1, we have

−1 −1 [S, ν]δS ∈ D−∞,0 and thus ([S, ν]δS )s = 0. This combined with the above observation leads to13

2 > (3.21) (PL )s = (νL)s = νL++ − (νL++)

= νL++ − L−ν

= ν(L++ − L−) + νL− − L−ν

= νLs + [ν, L−]

= νLs + L−, since [ν, L−] = L−,

2 since L− has one subdiagonal. In order to evaluate (PL )sΨ, we need to know LsΨ and L−Ψ. Anticipating (3.25), we have by (2.3) and (3.3)

−η ∂ ∂ τδ Y1,0Ψ = −LbΨ = 2Ψ(e − 1) log τ − Ψ log ∂t1 ∂t1 τ −η ∂ = 2Ψ(e − 1) log τ − L0Ψ, by (1.16), ∂t1 and, since Lb = 2L− + L0,

1 −η ∂ (3.22) L−Ψ = (Lb − L0)Ψ = −Ψ(e − 1) log τ, 2 ∂t1 whereas, using ³ −1 ´ 1 i τj(t − [z ]) 2 Σtiz j −1 Ψ = e z γj , τj(t)

13 L++ denotes the strictly uppertriangular part of L, i.e. the projection of L onto D1,∞

23 we have (using the logarithmic derivative)

∂Ψ −η ∂ ∂ τδ (3.23) LsΨ = 2 = 2Ψ(e − 1) log τ + zΨ − Ψ log . ∂t1 ∂t1 ∂t1 τ Using these two formulas (3.22) and (3.23), in (3.21) we have

2 (PL )sΨ = νLsΨ + L−Ψ ∂ ∂ τ = Ψ(e−η − 1)(2ν − I) log τ + zνΨ − νΨ log δ , ∂t1 ∂t1 τ ending the proof of Lemma 3.4. This lemma proves the main statement about symmetries for the s`(2, C) part of the Virasoro symmetry algebra; this is the heart of the matter.

Lemma 3.5. The vector ﬁelds {Y−1,1, Y0,1, Y1,1} form a representation of s`(2, C) and induce vector ﬁelds on τ as follows (in the notation (3.6)): −(ML`+1) Ψ L τ 1 µL τ L τ ¶ (3.24) Y log Ψ = b = (e−η − 1) `,1 + `,1 − ( `,1 ) `,1 Ψ τ 2 τ τ δ for ` = −1, 0, 1. Also −(L`) Ψ L τ 1 µL τ L τ ¶ (3.25) Y log Ψ = b = (e−η − 1) `,0 + `,0 − ( `,0 ) `,0 Ψ τ 2 τ τ δ for ` ∈ Z.

Proof of Lemma 3.5: Relation (3.25) will ﬁrst be established for ` = −m < 0 :

−m −m −m Y−m,0Ψ = −(L )bΨ = −L Ψ, since L ∈ Db for m ≥ 0 = −z−mΨ Ã ! 1 e−ητ = 2γ−1χ − z−m(eΣ/2 n ) 2 τn n∈Z W (1) τ = 2Ψ(e−η − 1) −m , applying (3.16)(i) componentwise to τ e−ητ Φ = eΣ/2 n τn ³ 2W (1) τ 1 2W (1) τ 2W (1) τ ´ = Ψ (e−η − 1) −m + ( −m − ( −m ) ) , τ 2 τ τ δ

24 the diﬀerence in brackets vanishing identically. The same will now be established for ` = m > 0; indeed m Ym,0Ψ ≡ −(L )bΨ m m = (−L + (L )s)Ψ ∂ = −zmΨ + 2 Ψ , using (1.16) ∂tm ∂ e−ητ = −zmΨ + zmΨ + 2eΣ/2χ ( γ−1), ∂tm τ e−ητ using Ψ = eΣ/2 γ−1χ τ e−ητ ³ ∂ ∂ ´ = 2eΣ/2χγ−1 (e−η − 1) log τ + γ γ−1 , using (3.3) τ ∂tm ∂tm ³ (1) (1) (1) ´ −η 2Wm τ 1 2Wm τ 2Wm τ = Ψ (e − 1) + ( − ( )δ) s τ 2 τ τ τj+1 ∂ −1 ∂ since γj = , and γ γ = − log γ. τj ∂tm ∂tm Relation (3.25) for ` = 0 is obvious, since Y Ψ −(L0) Ψ 1 1 µ2ντ 2ντ ¶ 1 µL τ L τ ¶ 0,0 = b = −I = (2ν−2(ν) ) = − ( ) = 0,0 − ( 0,0 ) Ψ Ψ 2 δ 2 τ τ δ 2 τ τ δ To prove (3.24), consider now m+1 (3.26) − (ML )bΨ m+1 m+1 = (−ML + (ML )s)Ψ ³ X∞ ´ m+1 ∂ 1 k+m m+1 = −z + ktk(L )s + (PL )s Ψ ∂z 2 1 ³ X ´ m+1 ∂ 1 k+m m+1 = −z + ktk(L )s + (PL )s Ψ, using (1.28) ∂z 2 k>max(−m,0) α since (L )s = 0 for α ≤ 0 ³ X ´ m+1 ∂ ∂ m+1 = −z + ktk Ψ + (PL )sΨ ∂z k>max(−m,0) ∂tk+m m+1 = BmΨ + (PL )sΨ using the deﬁnition (3.8) of Bm, −η j −1 Σ/2 e τj m m+1 = (z γj Bm(e ))j∈Z − z νΨ − ΨBm log γ + (PL )sΨ, τj

25 remembering the deﬁnition (1.21) of Ψ, using the deﬁnition (3.8) of Bm, and using the fact that Bm is a derivation.

1. For m = −1, 0, we have

m+1 m+1 PL ∈ D−∞,0 and thus (PL )s = 0.

We set m 7→ −m with m ≥ 0. Componentwise, the above expression reads, by adding and subtracting mtm/2 (m ≥ 0) and using deﬁnition (3.8) of B−m and ν = diag(··· , i, ···): −m+1 (−(ML )bΨ)j

µ ¶ −η µ ¶ j −1 mtm −m Σ/2 e τj j mtm = z γj B−m + − jz e − z Ψj B−m(log γj) + 2 τj 2 Ã ! −m −η j −1 mtm z Σ/2 e τj = z γj B−m + + 2j(1 − δ0,m)(− ) (e ) 2 2 τj µ mt ¶ 1 −zjΨ B log γ + m − zjΨ 2jδ j −m j 2 2 j 0,m ˜ (2) (1) j −η W−m(τj) W−m(τj) = z Ψj(e − 1)( + 2j ) (using (3.16)(i) and (ii)) τj τj ³ ˜ (2) ˜ (2) ´ s 1 j W−m(τj+1) W−m(τj) τj+1 + z Ψj − + (using γj = , (3.9) and (3.7)) 2 τj+1 τj τj   1 W (1) (τ ) W (1) (τ ) mt j  −m j+1 −m j  m (1) + z Ψj −2(j + 1) + 2j (using = W−m) 2 τj+1 τj 2 1 + zjΨ (−2jδ ) 2 j 0,m ³ ˜ (2) (1) 2 (0) j −η (W−m + 2jW−m + (j − j)W−m)τj = z Ψj (e − 1) τj 1³(W˜ (2) + 2jW (1) + (j2 − j)W (0) )τ + −m −m −m j 2 τj (W˜ (2) + 2(j + 1)W (1) + ((j + 1)2 − (j + 1))W (0) )τ ´´ − −m −m −m j+1 , τj+1

2 2 (0) since, in the last line, −2j = (j − j) − ((j + 1) − (j + 1)) and δ0,m = W−m. Thus

26 relation (3.24) for m = −1 and 0 follows from the simple observation (3.7) that ˜ (2) (2) W−m = W−m for m = 0, 1. 2. For m = 1 in (3.24), we use the identities in Lemma 3.3(iii) and Lemma 3.4(ii) in (3.26): 2 −(ML )bΨ (B − [B , η])τ [B , η]τ 1 τ = Ψ(e−η − 1) 1 1 − Ψ 1 − zνΨ − ΨB log δ + (PL2) Ψ τ τ 2 1 τ s using (3.16)(iii)

−η (2) ∂ ∂ 1 (2) τδ = Ψ(e − 1)(J1 − ) log τ − Ψ log τ − zνΨ − ΨJ1 log , ∂t1 ∂t1 2 τ ∂ ∂ τ +Ψ(e−η − 1)(2ν − I) log τ + zνΨ − νΨ log δ ∂t1 ∂t1 τ ∂ using (3.9), [B1, η] = (see (3.14)), and Lemma 3.4(ii), ∂t1 ³ −η (2) ∂ 1 (2) ∂ = Ψ (e − 1)(J1 + (2ν − 2) ) log τ + (J1 + (2ν − 2) ) log τ ∂t1 2 ∂t1 ´ 1 (2) ∂ − (J1 + 2ν ) log τδ , 2 ∂t1 µ L τ 1 µL τ L τ ¶¶ = Ψ (e−η − 1) 1,1 + 1,1 − ( 1,1 ) , τ 2 τ τ δ

j+1 using in the last line, the deﬁnition (3.6) of L1,1 and L1,1 τj+1 = ((L1,1τ)δ)j, establishing (3.24) for m = 1, thus ending the proof of Lemma 3.5. Proof of Theorem 3.2: The only remaining point is to establish (3.13) for n = 1 and all ` ≥ 2. To do this we use the underlying Lie algebra structure. First we have the following identity, using (3.2) and (3.1), h i h i −η ˆ ˆ 1 ˆ ˆ τn (e − 1) Yzα( ∂ )β , Yzα0 ( ∂ )β0 log τn + Yzα( ∂ )β , Yzα0 ( ∂ )β0 log ∂z ∂z 2 ∂z ∂z τn+1 h i = Y α ∂ β , Y α0 ∂ β0 log Ψn z ( ∂z ) z ( ∂z ) h i = Y log Ψn α0 ∂ β0 α ∂ β z ( ∂z ) ,z ( ∂z ) −η ˆ h i 1 ˆ h i τn = (e − 1)Y log τn + Y log . α0 ∂ β0 α ∂ β α0 ∂ β0 α ∂ β z ( ∂z ) ,z ( ∂z ) 2 z ( ∂z ) ,z ( ∂z ) τn+1

27 The terms containing (e−η − 1) in the ﬁrst and last expressions are power series in z−1, with no constant term; the second terms are independent of z. Therefore, equating constant terms yield: ³h i ´ h i τn (i) Yˆ α ∂ β , Yˆ α0 ∂ β0 − Yˆ log = 0 z ( ∂z ) z ( ∂z ) α0 ∂ β0 α ∂ β z ( ∂z ) ,z ( ∂z ) τn+1 and thus also ³h i ´ −η ˆ ˆ ˆ h i (e − 1) Y α ∂ β , Y α0 ∂ β0 − Y log τn = 0. z ( ∂z ) z ( ∂z ) α0 ∂ β0 α ∂ β z ( ∂z ) ,z ( ∂z )

Since (e−η − 1)f = 0 implies f = constant, there exists a constant c depending on α, β, α0, β0 and n such that ³h i ´ ˆ ˆ ˆ h i (ii) Y α ∂ β , Y α0 ∂ β0 − Y log τn = cαβα0β0,n; z ( ∂z ) z ( ∂z ) α0 ∂ β0 α ∂ β z ( ∂z ) ,z ( ∂z ) relation (i) says cαβα0β0 is independent of n; hence (ii) reads The two relations (i) and (ii) combined imply ³h i ´ ˆ ˆ ˆ h i (3.27) Y α ∂ β , Y α0 ∂ β0 − Y − cαβα0β0 τn = 0 z ( ∂z ) z ( ∂z ) α0 ∂ β0 α ∂ β z ( ∂z ) ,z ( ∂z ) with cαβα0β0 independent of n. Applying (3.27) to

h ∂ i h ∂ ∂ i ∂ zm+1 , z` = `zm+` and z`+1 , zm+1 = (m − `)zm+`+1 , ∂z ∂z ∂z ∂z leads to ³h i ´ ˆ ˆ ˆ (3.28) Y`,0, Ym,1 − `Y`+m,0 − c`,m τn = 0 and ³h i ´ ˆ ˆ ˆ 0 Ym,1, Y`,1 − (m − `)Ym+`,1 − cm,` τn = 0. By virtue of (3.12), we have

(3.29) [L`,0, Lm,1] − `Lm+`,0 − constant = 0.

28 According to (3.25) we have ˆ Y`,0 = L`,0 implying by subtracting (3.29) from (3.28) ˆ [L`,0, Ym,1 − Lm,1] = constant, for `, m ∈ Z, m ≥ −1. ∂ The only operator commuting (modulo constant) with all L`,0 = 2 + (−`)t−` is ∂t1 given by linear combinations of a constant, tα and ∂/∂tα, i.e.: X∞ ˜ (m) (1) (m) .30) Ym,1 − Lm,1 = cj Jj+m, for m ≥ 2, (c−m = 0)(3 j=−∞ = 0 for m = −1, 0, 1. ˜ Putting Ym,1 from (3.30) into the second relation of (3.28) implies (modulo constants) h ∞ ∞ i X (m) (1) X (`) (1) (m+`) (1) Lm,1 + cj Jj+m, L`,1 + cj Jj+` = (m − `)(Lm+`,1 + Σcj Jj+m+`) j=−∞ j=−∞ which also equals by explicit computation, using (3.12): (`) (1) (m) (1) = (m − `)Lm+`,1 − Σcj (j + `)Jm+j+` + Σcj (j + m)Jm+j+`. Comparing the coeﬃcients of the J (1)’s in two expressions on the right hand side yields (m+`) (m) (`) (3.31) (m − `)cj = (m + j)cj − (` + j)cj provided m + j + ` 6= 0 (m) with cj = 0 for m = −1, 0, 1, all j ∈ Z. Setting ` = 0 in (3.31), yields (m) (0) (m) (0) j(cj − cj ) = 0 and thus cj = cj for j 6= 0, −m, implying (m) cj = 0 all m ≥ −1 and j 6= 0, −m. Also, setting j = 0 and ` = −1 in (3.31) yields (m) (m−1) (−1) mc0 = (m + 1)c0 − c0 , for m ≥ 2, (−1) (1) implying by induction, since c0 = c0 = 0 (m) c0 = 0 for all m ≥ −1, concluding the proof.

29 Proof of Corollary 3.2.1: According to theorem 3.2, the vector field m + 1 m + 1 Y + Y = −(MLm+1 + Lm) , m,1 2 m,0 2 b acting on Ψ, induces on τj the vector field m + 1 Lj + Lj = J (2) + (2j − m − 1)J (1) + (j2 − j)δ m,1 2 m,0 m m m,0 m + 1 + (2J (1) + 2jδ ) 2 m m,0 (2) (1) 2 (0) = Jm + 2jJm + j Jm establishing the corollary. Example : symmetries at t = 0. By (2.4), the symmetries take on the following form `+1 `+1 Y`,1L = L + [(ML )s,L], with (see (1.23)) X∞ X∞ −1 1 k−1 1 k−1 M = SεS + ktkL = P + ktkL . 2 1 2 1 Therefore, at t = 0, ¯ ¯ `+1 `+1 Y`,1L¯ = L + [(PL )s,L], t=0 with `+1 ` −1 ` (PL )s = ([S, ν]δ S )s + (νL )s, upon using (3.20). In the formula above, the wave operator S = γ−1S˜ is given by (1.9) and one finds, by a computation similar to (3.21), ` ` ` (νL )s = ν(L )s + [ν, (L )−].

If bi and ai stand for the diagonal and off-diagonal elements of L respectively, i.e. L = δ−1a + bδ0 + aδ, we have, in view of Lemma 3.4, ¯ ¯ ¯ ¯ Y−1,1L¯ = I, Y0,1L¯ = L t=0 t=0 and ( ¯ ˙ 2 2 2 ¯ bi = bi + (2i + 1)ai − (2i − 3)ai−1 Y1,1L¯ : t=0 a˙ i = ai((i + 1)bi+1 − (i − 1)bi). The subsequent symmetry vector fields can all be computed and are non-local; for P ∂ n−1 −1 −1 ˜ instance Y2,1L involves the coefficients log τn = bi of δ in γ S (see (1.9)). ∂t1 0

30 4 Orthogonal polynomials, matrix integrals, skew- symmetric matrices and Virasoro constraints

Remember from the introduction the orthogonal (orthonormal) polynomial basis of H+ = {1, z, z2,...} on the interval [a, b], − ∞ ≤ a < b ≤ ∞, (4.1) r 1 p˜r(t, z) = z + ... (monic) and pr(t, z) = q p˜r(t, z) (orthonormal), r ≥ 0, hr(t) P i with regard to the t-dependent inner product (via the exponential e tiz ):

Z P P b ∞ i i −V0(z)+ tiz tiz (4.2) hu, vit = uvρtdz, with ρt(z) = e 1 = ρ0(z)e ; a i.e., hp˜i, p˜jit = hiδij and hpi, pjit = δij.

Then the semi-inﬁnite vector (of h , i0-orthonormal functions)

1 i 1 i Σtiz Σtiz > (4.3) Ψ(t, z) := e 2 p(t, z) := e 2 (p0(t, z), p1(t, z),...) , satisﬁes the orthogonality relations

(4.4) h(Ψ(t, z))i, (Ψ(t, z))ji0 = hpi(t, z), pj(t, z)it = δij,

The weight is assumed to have the following property14: (4.5) 0 P∞ i ρ biz h0(z) 0 0 P0 k k − = V0 = ∞ i = with ρ0(a)f0(a)a = ρ0(b)f0(b)b = 0, k ≥ 0. ρ0 0 aiz f0(z) Deﬁne semi-inﬁnite matrices L and P such that ∂ (4.6) zp(t, z) = L(t)p(t, z), p(t, z) = P p(t, z) ∂z The ideas of Theorem 4.1 are due to Bessis-Itzykson-Zuber [BIZ] and Witten [W].

14 0 the choice of f0 is not unique. When V0 is rational, then picking f0 = (polynomial in the denominator) is a canonical choice.

31 Theorem 4.1. The semi-inﬁnite vector Ψ(t, z) and the semi-inﬁnite matrices L(t) 1 P∞ k−1 (symmetric), and M(t) := P (t) + 2 1 ktkL , satisfy ∂ (4.7) zΨ(t, z) = L(t)Ψ(t, z), and Ψ(t, z) = MΨ(t, z) ∂z and

∂L 1 n ∂M 1 n ∂Ψ 1 n (4.8) = [(L )s,L], = [(L )s,M], and = (L )s, Ψ; ∂tn 2 ∂tn 2 ∂tn 2

2 the wave vector Ψ(t, z) and the L -norms hn(t) admit the representation

³ −1 ´ 1 i τ (t − [z ]) τ (t) Σtiz n n n+1 (4.9) Ψ(t, z) = e 2 z q and hn(t) = n≥0 τ (t) τn(t)τn+1(t) n with Z 1 i −T r V0(Z)+ΣtiT r Z (4.10) τn(t) = e dZ; Ωnn! Mn(a,b) the integration is taken over the space Mn(a, b) of n × n Hermitean matrices with eigenvalues ∈ [a, b].

˙ ∂ Proof: Step 1. Suppose Ψ = BΨ and P (z, ∂z )Ψ = PΨ where B and P are matrices and P is a polynomial with constant coeﬃcients. Then

P˙ = [B, P].

∂ Indeed, this follows from diﬀerentiating P (z, ∂z )Ψ = PΨ, and observing that

P BΨ = P Ψ˙ = P˙ Ψ + PΨ˙ = P˙ Ψ + PBΨ.

Step 2. The matrix L ∈ D−∞,1 is symmetric because the operation of multiplication by z is symmetric with respect to h , i on H+ and is represented by L in the basis pi. Moreover, P ∈ D−∞,−1 and [L, P ] = [L, M] = 1. Also for k ≥ 0,

P ∞ P µ ¶ ∂Ψ ∂p 1 ∞ i 1 X 1 ∞ i 1 tiz i−1 tiz i−1 (4.11) = e 2 1 + itiz pe 2 1 = P + ΣitiL Ψ = MΨ. ∂z ∂z 2 1 2 establishing (4.7).

32 Step 3. We now prove the ﬁrst statement of (4.8). Since ∂pk/∂ti is again a polynomial of the same degree as pk, we have: X ∂pk (i) (i) (4.12) = Ak` p`,A ∈ Db. ∂ti 0≤`≤k The precise nature of A(i) is found as follows: for ` < k, ∂ Z 0 = pk(z)p`(z)ρt(z)dz ∂ti Z Z Z ∂p ∂p ∂ j k ` −V0+Σtj z = p`ρt(z)dz + pk ρt(z)dz + dz( e )pkp`, using hpi, pji = δij ∂ti ∂ti ∂ti Z X Z X (i) i = Akj pjp`ρt(z)dz + (L )kjpjp`ρt(z)dz, using (4.12) and (4.6) j≤k j (i) i = Ak` + (L )k` and for ` = k, Z Z X Z X ∂ 2 (i) i 0 = (pk(z)) ρt(z)dz = 2 Akmpmpkρt(z)dz + (L )kjpjpkρt(z)dz ∂ti m≤k j (i) i = 2Akk + (L )kk, implying 1 1 (4.13) A(i) = −(Li) − (Li) = − (Li) , − 2 0 2 b and thus

∂Ψ ∂ Σ/2 1 Σ/2 i 1 Σ/2 i 1 i i 1 i = e p = e z p − e (L )bp = (L − (L )b)Ψ = (L )sΨ. ∂ti ∂ti 2 2 2 2 Now using step 1, we have immediately (4.8). So Ψ satisﬁes the Toda equations (1.16) and behaves asymptotically as: ³ ´ 1 Σt zi 1 n −1 (4.14) Ψ = e 2 i √ z (1 + O(z )) hn n∈Z Therefore in view of (1.15), we must have q s τn+1 hn = γn = . τn

33 Step 4. The integration (4.10) is taken with respect to the invariant measure Y Y (4.15) dZ = dZii d(ReZij)d(ImZij). 1≤i≤n 1≤i

(4.17) VmΨ := −(Vm)bΨ = 0, for m ≥ −1; 1 + the symmetries Vm form a (non-standard) representation of Diﬀ(S ) : X (4.18) [Vm, Vn] = (m − n) aiVm+n+i, −1 ≤ n, m < ∞, i≥0

34 and are deﬁned by the semi-inﬁnite matrices15 16 m + 1 (4.19) V := {Q, Lm+1} = QLm+1 + Lmf (L), with Q := Mf (L) + g (L), m 2 0 0 0 which are skew-symmetric on the locus of Ψ(t, z) above. Moreover Q is a solution of the “string equation”

(4.20) [L, Q] = f0(L), and the τ-vector satisﬁes the Virasoro constraints (4.21) X ³ ´ (n) (2) (1) 2 (0) (1) (0) Vm τn = ai(Ji+m + 2n Ji+m + n Ji+m) − bi(Ji+m+1 + n Ji+m+1) τn = 0, i≥0

for m = −1, 0, 1, . . . , n = 0, 1, 2,..., (n) with the Vm , m ≥ −1 (for ﬁxed n ≥ 0) satisfying the same Virasoro relations as (4.18), except for an additive constant.

In preparation of the proof we give some elementary lemmas.

Lemma 4.3. Consider operators S1 and Sf acting on a suitable space of functions √ √ 17 of z, such that [S1, z] = 1 and Sf = fS1 f; then the following holds : (i) Sf = {S1, f} and [Sf , z] = f, (ii) [{S1, h1}, {S1, h2}] = {S1, (h1, h2)} (iii){S1, h1h2} = {{S1, h1}, h2} = {{S1, h2}, h1} m+1 n+1 2 m+n+1 m+n+1 (iv) [{S1, fz }, {S1, fz }] = (n−m){S1, f z } = (n−m){{S1, fz }, f} m+1 1 (v) the operators {S1, z }, m ∈ Z form a representation of Diﬀ(S ),

m+1 n+1 m+n+1 [{S1, z }, {S1, z }] = (n − m){S1, z }

P i m+1 (vi) given f(z) = i≥0 aiz , the {S1, fz }, m ∈ Z also form a representation of Diﬀ(S0): X m+1 n+1 m+n+i+1 [{S1, fz }, {S1, fz }] = (n − m) ai{S1, fz }, i≥0

15in terms of the anticommutator (0.20) 0 0 0 16 (f0ρ0) f0−h0 f0ρ0 set g0 := = , with h0 := − 2ρ0 2 ρ0 17 0 0 (h1, h2) = h1h2 − h2h1 denotes the Wronskian

35 −1 P i the map to the standard generators, f (z) = i≥−k a¯iz , X m+1 m+1 −1 m+1 m+i+1 {S1, fz } 7→ {S1, z } = {S1, f · fz } = a¯i{S1, fz }. i≥−k n n−1 Proof: [S1, z] = 1 implies [S1, z ] = nz , since [S1,.] is a derivation, and thus 0 [S1, h] = h , which leads to q q q q 1 1 S = fS f = f[S , f] + fS = (2fS + f 0) = (2fS + [S , f]) = {S , f}. f 1 1 1 2 1 2 1 1 1 The second part of (i),(ii) and (iii) follows by direct computation; (iv) is an imme- diate consequence of (ii), (iii) and the Wronskian identity (fzm+1, fzn+1) = (n − m)f 2zm+n+1. The Virasoro relations (v) follow immediately from (iv), whereas (vi) follows from the argument:

m+1 n+1 2 m+n+1 [{S1, fz }, {S1, fz }] = (n − m){S1, f z } m+n+1 = (n − m){{S1, fz }, f}, by (iii) X m+n+1 i = (n − m) ai{{S1, fz }, z } i≥0 X m+n+i+1 = (n − m) ai{S1, fz }, by (iii) i≥0 ending the proof of lemma 4.3. Lemma 4.4. Consider the function space H = {. . . , z−1, 1, z, . . .} with a real in- R b ner product hu, viρ = a uvρdz, −∞ ≤ a < b ≤ ∞ with regard to the weight P i ρ, with ρ(a) = ρ(b) = 0; also consider an arbitrary function f = i≥0 aiz with f(a)ρ(a)am = f(b)ρ(b)bm = 0, for m ∈ Z. Then the ﬁrst-order diﬀerential operator from H to H d S = f + g, dz is skew-symmetric for h , iρ if and only if S takes on the form s f d q 1 d √ d 1 ρ0 (4.22) S = fρ = {S , f}, where S = √ ρ = + ρ dz 1 1 ρ dz dz 2 ρ

m+1 m+1 So, the operators {S1, z } and {S1, fz }, for m ∈ Z, form representations of 1 Diﬀ(S ) in the space of skew-symmetric operators so(H, <, >ρ).

36 Proof: First compute the expressions:

Z b du Z b hSu, vi = f vρ dz + guvρdz a dz a Z b Z b b d = uvfρ |a − u (vfρ)dz + guvρdz a dz a Z b h d i = ρu ρ−1(− f + g)(ρv) dz, using fρ(a) = fρ(b) = 0 a dz and Z b d hu, Svi = ρu(f + g)vdz. a dz Imposing S skew, i.e., hSu, vi = hu, ST vi = −hu, Svi, leads to the operator identity

d d ρ−1(− f + g)ρ = −(f + g), dz dz

1 −1 0 in turn, leading to g = 2 ρ (fρ) ; thus S takes on the form (4.22). The last part of the proof of Lemma 4.4 follows at once from the above and Lemma 4.3 (iv) and (v).

Lemma 4.5. Consider the above inner-product hu, vi in the space H+ = {1, z, z2,...}, for the weight ρ having a representation of the form

0 P i ρ i≥0 biz h P (4.23) − = i ≡ . ρ i≥0 aiz f

+ Let H have an orthonormal basis of functions (ϕk)k≥0; then the operators m+1 + + {S1, fz } for m ≥ −1 are maps from H to H and its representing matri- m+1 ces in that basis (i.e., (h{S1, fz }ϕk, ϕ`i)k,`≥0), for m ≥ −1 form a closed Lie algebra ⊂ so(0, ∞)18. m+1 + + Remark: The operators {S1, z } do not map H in H . Proof: The operators, n o m+1 m+1 {S1, fz } = {S1, f}, z : H → H, for m ≥ −1,

18so(0, ∞) denotes the Lie algebra of semi-inﬁnite skew-symmetric matrices.

37 which are skew-symmetric by Lemma 4.4, preserve the subspace H+, since by virtue of (4.22),

d (f · ρ)0 d f 0 − h (4.24) {S , f} = f + = f + 1 dz 2ρ dz 2

0 contains holomorphic series f and f − h, by (4.23). In a basis of functions (ϕk)k≥0, orthonormal with respect to h , iρ, the corresponding matrices will also be skew- symmetric. Proof of Theorem 4.2: According to Lemma 4.5, the operators (4.25) 1 d n 1 d o (ρ0,f0) √ m+1 √ m+1 Tm := Tm := {√ ρ0, f0z } = {√ ρ0, f0}, z , m ≥ −1 ρ0 dz ρ0 dz map H+ into H+ and form an algebra with structure constants: X (4.26) [Tm,Tn] = (n − m) aiTm+n+i, m, n ≥ −1. i≥0

Under the map φ (Theorem 2.2), the operators Tm get transformed into matrices Vm = φ(Tm), such that

(4.27) TmΨ(t, z) = VmΨ(t, z); namely (see footnote 15) (4.28) Ã ! ³n 1 d √ o´ d Q := V−1 = φ(T−1) = φ √ ρ0, f0 = φ f0 + g0 = Mf0(L) + g0(L) ρ0 dz dz and

.29) Vm : = φ(Tm)(4 m+1 = φ({T−1, z }) 1 = φ(zm+1T + [T , zm+1]) −1 2 −1 1 d = φ(zm+1T + [f , zm+1]) −1 2 0 dz m + 1 = φ(zm+1T + f zm) −1 2 0

38 m + 1 = QLm+1 + Lmf (L) 2 0 X X X i+m+1 (i + 1)ai+1 − bi i+m+1 m + 1 i+m = aiML + L + aiL i≥0 i≥0 2 2 i≥0 X X i+m+1 i + m + 1 i+m bi i+m+1 = ai(ML + L ) − L , i≥0 2 i≥0 2 where we used X 0 X i (f0ρ0) 1 i f0 = aiz and g0 = = ((i + 1)ai+1 − bi)z . i≥0 2ρ0 2 i≥0

In addition, according to Lemma 4.5, the z-operators Tm are skew-symmetric 1 + with regard to h , i0 and thus form a representation of Diff(S ) ( ) 1 + + skew-symmetric operators Diff(S ) −→ so(H , h , i0) := + , on H , h , i0 P i tiz with structure constants given by (4.18). The components e pn(t, z), n ≥ 0 + of Ψ(t, z) form an orthonormal basis of H , h , i0, with regard to which the operators Tm are represented by semi-infinite skew-symmetric matrices; i.e., the anti- homomorphism φ restricts to the following map

+ φ : so(H , h , i0) −→ so(0, ∞) (anti-homomorphism).

Hence the matrices Vm, m ≥ −1 are skew-symmetric (i.e., (Vm)b = 0) and thus, using (4.29), we have

.30)0 = VmΨ = −(Vm)bΨ(4   X X i+m+1 i + m + 1 i+m bi i+m+1 =  ai(ML + L )b − (L )b Ψ. i≥0 2 i≥0 2 In the ﬁnal step, Theorem 3.2 and Corollary 3.2.1 lead to the promised τ-constraints (4.21), modulo a constant, i.e.,

(k) (k) (4.31) Vm τk = cm τk m ≥ −1, k ≥ 0. By (3.27), this constant is independent of k, i.e.,

(k) (0) cm = cm ;

39 (0) upon evaluating (4.31) at k = 0 and upon using τ0 = 1, we conclude cm = 0, yielding (4.21), as claimed. Finally the map

Tm 7−→ Vm, m ≥ −1 is an anti-homomorphism (modulo constants) by Lemma 3.1; we also have

(4.32) φ :[T−1, z] = f0(z) 7−→ [L, Q] = f0(L), which is the “string equation”, concluding the proof of Theorem 4.2. −1 P i Remark 4. : Note that, if f0 = i≥−k a¯iz , the map X ¯ (4.33) Tm 7→ Tm = a¯iTm+i, m ≥ k − 1 i≥−k

1 sends Tm into the standard representation of Diﬀ(S ): ¯ ¯ ¯ (4.34) [Tm, Tn] = (n − m)Tm+n, m, n ≥ k − 1, according to Lemma 4.3. Example. In the next section we shall consider the classical orthogonal polynomials; we consider here, for a given polynomial q(z), in the interval [a, b] the weight

α q(z) β ρ0 = (z − a) e (z − b) , with f0 = (z − a)(z − b) and α, β ∈ Z, ≥ 1.

It implies that (see footnote 15)

f0(a)ρ0(a) = f0(b)ρ0(b) = 0 and that both f0 and

0 (f0ρ0) (z − a)(z − b) α 0 β a + b g0 = = z − ( + q + ) − , 2ρ0 2 z − a z − b 2 are polynomial. Then Q = Mf0(L) + g0(L) is skew-symmetric and [L, Q] = f0(L). The Virasoro constraints (4.21) are then given by a ﬁnite sum.

40 5 Classical orthogonal polynomials

It is interesting to revisit the classical orthogonal polynomials, from the point of view of this analysis. As a main feature, we note that, in this case, not only is L (multiplication by z) symmetric and tridiagonal, but there exists another operator, a ﬁrst-order diﬀerential operator, which yields a skew-symmetric and tridiagonal matrix. It is precisely given by the matrix Q! The classical orthogonal polynomials are characterized by Rodrigues’ formula,

1 d n n pn = ( ) (ρ0X ), Knρ0 dz

−V0 Kn constant, X(z) polynomial in z of degree ≤ 2, and ρ0 = e .Compare Rodrigues’ formula for n = 1 with the one for g0, ( see theorem 4.2, footnote 15)

K1 1 d 1 d p1 = (ρ0X) and g0 = (ρ0f0), 2 2ρ0 dz 2ρ0 dz which leads to the natural identiﬁcation K g = 1 p and f = X 0 2 1 0 and thus d d K p d X0 − XV 0 T = f + g = X + 1 1 = X + 0 . −1 0 dz 0 dz 2 dz 2 0 0 Since both (degree X) ≤ 2 and (degree (X − XV0 )) ≤ 1, as will appear from the table below, we have that T−1, acting on polynomials, raises the degree by at most 1: X T−1pk(0, z) = Qkipi(0, z), i≤k+1 while, since Q is skew-symmetric,

(6.1) T−1pk(0, z) = (Qp(0, z))k = −Qk,k−1pk−1(0, z) + Qk,k+1pk+1(0, z), together with

(6.2) zpk(0, z) = Lk−1,kpk−1(0, z) + Lk,kpk(0, z) + Lk,k+1pk+1(0, z). This implies at the level of the ﬂag

t t t ... ⊃ Wk−1 ⊃ Wk ≡ span{(Ψ)k, (Ψ)k+1,...} ⊃ Wk+1 ⊃ ...

41 that t t t t zWk ⊂ Wk−1 and T−1Wk ⊂ Wk−1.

Thus the recursion operators z and T−1 serve to characterize the flag and so the wave vector Ψ. It is interesting to speculate on considering “(p, q)-cases”, where, for instance, p t t t t z Wk ⊂ Wk−p and T−1Wk ⊂ Wk−q. The existence of two operators, a symmetric and a skew-symmetric one, both represented by tridiagonal matrices, probably characterize the orthogonal polynomials on the line. Related, it is interesting to point out a conjecture by Karlin and Szegöand a precise formulation by Al-Salam and Chihara, were classical orthogonal polynomials are characterized by orthogonality and the existence of a differentiation formula of the form

0 f0(z)pn(z) = (αnz + βn)pn(z) + γnpn−1(z). We now have the following table:

Hermite Laguerre Jacobi 2 e−V0(z)dz e−z dz e−zzαdz (1 − z)α(1 + z)βdz

(a, b)(−∞, ∞) (0, ∞)(−1, 1)

d d d 1 2 d T−1 = f0 dz + g0 dz − z z dz − 2 (z − α − 1) (1 − z ) dz 1 − 2 ((α + β + 2)z + (α − β))

string [L, Q] = 1 [L, Q] = L [L, Q] = 1 − L2 equation

We now give a detailed discussion for each case: (a) weight e−z2 dz. The corresponding (monic) orthogonal Hermite polynomials satisfy the classic relations n d zp˜ = p˜ +p ˜ , and p˜ = np˜ . n 2 n−1 n+1 dz n n−1 Therefore the matrices deﬁned by n d n zp˜ = p˜ +p ˜ ( − z)˜p = p˜ − p˜ n 2 n−1 n+1 dz n 2 n−1 n+1

42 can be turned simultaneously into symmetric and skew-symmetric matrices L and Q = M − L respectively, by an appropriate diagonal conjugation. The string equation reads [L, Q] = 1 and the matrix integrals τn satisfy

(n) (2) (1) (1) 2 Vm τn = (Jm + 2nJm − 2Jm+2 + n δm,0)τn = 0, m = −1, 0, 1,... upon using formula (7.12) for a0 = 1, b1 = 2 and all other ai, bi = 0; this captures the original case of Bessis-Itzykson-Zuber and Witten [ ]; Witten had pointed out in his Harvard lecture that M − L is a skew-symmetric matrix. (b) weight e−zzαdz. Again the classic relations for (monic) Laguerre polynomials,

zp˜n = n(n + α)˜pn−1 + (2n + α + 1)˜pn +p ˜n+1 d z p˜ = n(n + α)˜p + np˜ dz n n−1 n yield symmetric and skew-symmetric matrices L and Q, after conjugation of

zp˜n = n(n + α)˜pn−1 + (2n + α + 1)˜pn +p ˜n+1 ∂ (2z − (z − α − 1))˜p = n(n + α)˜p + 0.p˜ − p˜ . ∂z n n−1 n n+1

Setting a1 = 1, b0 = −α, b1 = 1 and all other ai = bi = 0, yields

(n) (2) (1) (1) (1) Vm τn = (Jm + 2nJm + αJm − Jm+1 + n(n + α)δm,0)τn = 0, m = 0, 1, 2,..., and the string equation [L, Q] = L; this case was investigated by Haine and Horozov [HH]. (c) weight (1 − z)α(1 + z)βdz. The matrices L and Q will be deﬁned by the operators acting on (monic) Jacobi polynomials

zpñ = An−1pñ−1 + Bnpñ +p ñ+1 0 1 α + β −1 2 d (f0ρ0) −( + ) ((1 − z ) + )˜pn = −An−1pñ−1 +p ñ+1 n + 1 2 dz 2ρ0 with 4n(n + α + β)(n + α)(n + β) A = n−1 (2n + α + β)2(2n + α + β + 1)(2n + α + β − 1) α2 − β2 B = − . n (2n + α + β)(2n + α + β + 2)

43 Setting a0 = 1, a1 = 0, a2 = −1, b0 = α − β, b1 = α + β and all other ai = bj = 0 leads to the constraints

(2) (2) (1) (1) (1) (1) 2 (Jm −Jm−2−2nJm−2+2nJm +(α−β)Jm−1+(α+β)Jm −n δm,2+n(α−β)δm,1)τn = 0, m = 1, 2, 3,... and the string equation [L, Q] = 1 − L2. Gegenbauer (α = β = λ − 1/2) and Legendre (α = β = 0) polynomials are limiting cases of Jacobi polynomials.

6 Appendix: Virasoro constraints via the integrals

The Virasoro constraints for the integrals can be shown in a direct way: P j Lemma A.1. Given f0(z) = j≥0 ajz , the following holds for k ≥ −1: (A.1) P ¯ ³ 2 ´ P ∞ i ∂ k+1 ∂ X X ∂ ∂ ∞ i tiZ εf0(Z)Z ¯ 2 tiZ e 1 e ∂Z dZ¯ = dZ ar + 2n + n δr+k e 1 , ∂ε ε=0 ∂t ∂t ∂t r≥0 i+j=r+k i j r+k where ∂/∂tj = 0 for j ≤ 0.

Proof: We break up the proof of this Lemma in elementary steps, involving the diagonal part of dZ, i.e., 2 dZ = dz1 . . . dzn∆(z) × angular part. At ﬁrst, we compute for k ≥ 0:

³ Pn ´ 2 ∂ aε zk+1 ∂ (A.2) e i=1 i ∂zi ∆(z) ∆(z) ∂ε

¯ ∂ k+1 k+1 ¯ = 2 log ∆(z1 + aεz1 , . . . , zn + aεzn )¯ ∂εÃ ! ε=0 X P i j k = 2a zαzβ + (n − 1) zα − an(n − 1)δk,o 1≤α<β≤n i+j=k 1≤α≤n i,j>0 and for k ≥ −1 P ¯ aε n zk+1 ∂ ¯ ∂ i=1 i ∂zi ∂ε e dz1 . . . dzn¯ (A.3) ε=0 dz1 . . . dzn

44 Yn ¯ ∂ k ¯ = a (1 + ε(k + 1)zα)¯ ε=0 ∂ε α=1 X k = a(k + 1) zα. 1≤α≤n

Note that both expressions (A.2) and (A.3) vanish for k=-1. Also, we have for k ≥ 1, P i tizα 1≤i≤∞ P 2 ∂ 1≤α≤n ∂t ∂t e i+j=k i j P X (A.4) i,j>0 P = 2 zi zj + (k − 1) zk i α β α tizα 1≤α<β≤n 1≤i≤∞ i+j=k 1≤α≤n e 1≤α≤n i,j>0 and P i tizα ∂ 1≤i≤∞ X (A.5) (2n log) e 1≤α≤n = 2n zk . ∂t α k 1≤α≤n

Summing up (A.2) and (A.3) yields for k ≥ 0:   Pn k+1 ∂ ¯ ∂ aε zi ∂z 2 ¯ e i=1 i ∆ (z)dz1 . . . dzn¯  X  ∂ε ε=0  P i j k  2 = a 2 zαzβ + (2n + k − 1) zα − n(n − 1)δk ; ∆ (z)dz1 . . . dzn  1≤α<β≤n  i+j=k 1≤α≤n i,j>0 this expression vanishes for k = −1. This expression equals the sum of (A.4) and (A.5); thus P   i tizα 1≤i≤∞ P 2 P  ∂ ∂ 2 1≤α≤n n k+1 ∂ ¯ a + 2n + δkn e aε z ¯ ∂ti∂tj ∂tk ∂ i=1 i ∂zi 2 i+j=k ∂ε e ∆ (z)dz1 . . . dzn¯ ε=0 = i,j>0 P 2 i ∆ (z)dz1 . . . dzn tizα 1≤i≤∞ e 1≤α≤n establishing Lemma A.1. −V Theorem A.2. Let the weight ρ0 = e 0 have a representation (not necessarily unique) of the form P i biz h 0 Pi≥0 0 V0 = i ≡ . i≥0 aiz f0

45 Then the matrix integral τ = In satisﬁes the KP equation, and Virasoro-like con- n Ωnn! straints: X ³ ´ (2) (1) 2 (0) (1) (0) ai(Ji+m + 2n Ji+m + n Ji+m) − bi(Ji+m+1 + n Ji+m+1) τn = 0 with m ≥ −1. i≥0 Proof: Shifting the integration variable Z by means of

m+1 Z 7→ Z + εf0(Z)Z , m ≥ −1 and using the notation i Φ(Z) = e−T r V0(Z)+ΣtiT r Z , we compute, since the integral remains unchanged, that Z ¯ ∂ εf (Z)Zm+1 ∂ ¯ 0 = e 0 ∂Z Φ(Z)dZ¯ ∂ε M ε=0 Z n P∞ ¯ ∂ −T r V (Z+εf (Z)Zm+1) t T r(Z+εf (Z)Zm+1)` εf (Z)Zm+1 ∂ ¯ = e 0 0 e `=1 ` 0 e 0 ∂Z (dZ)¯ ∂ε M ε=0 Z n P∞ ¯ ∂ ε(−T r V 0(Z)f (Z)Zm+1+ `t T r f (Z)Zm+`)+O(ε2) εf (Z)Zm+1 ∂ ¯ = e 0 0 `=1 ` 0 Φ(Z)e 0 ∂Z dZ¯ ∂ε ε=0 ÃMn Z X∞ m+1 m+` = −T r h0(Z)Z + `t`T r f0(Z)Z Mn `=1 ! X ³ X ∂2 ∂ ´ + a + 2n + n2δ Φ(Z)dZ i ∂t ∂t ∂t i+m i≥0 α+β=i+m α β i+m using Lemma A.1. Ã Z X i+m+1 = − biT r Z Mn i≥0 ! X ³X∞ X ∂2 ∂ ´ + a `t T r Zi+m+` + + 2n + n2δ Φ(Z)dZ i ` ∂t ∂t ∂t i+m i≥0 `=1 α+β=i+m α β i+m Z Ã X ∂ = − b − nb δ i ∂t 0 m+1 Mn i≥0,i>−m−1 i+m+1 ! X ³X∞ ∂ X ∂2 ∂ ´ + a `t + + 2n + n2δ Φ(Z)dZ i ` ∂t ∂t ∂t ∂t i+m i≥0 `=1 i+m+` α+β=i+m α β i+m X ³ ´ (2) (1) 2 (0) (1) (0) = ai(Ji+m + 2n Ji+m + n Ji+m) − bi(Ji+m+1 + n Ji+m+1) τn, i≥0 ending the proof of Theorem A.2.

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