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String-orthogonal polynomials, String Equations and Two-Toda Symmetries∗

M. Adler† P. van Moerbeke‡

Table of contents: 1. The 2-Toda lattice and its generic symmetries 2. A larger class of symmetries for special initial conditions 3. Borel decomposition of moment matrices, τ -functions and string-orthogonal poly- nomials 4. From string-orthogonal polynomials to the two-Toda lattice and the string equa- tion 5. Virasoro constraints on two-matrix integrals.

Consider a weight ρ(y, z)dydz = eV (y,z)dydz on IR2, with

(0.1) V (y, z) := V1(y) + V12(y, z) + V2(z) := ∞∑

1

tiy i +

∑

i,j≥1 cijy

izj − ∞∑

1

siz i,

the corresponding inner product

(0.2) 〈f, g〉 = ∫

IR2 dydzρ(y, z)f(y)g(z),

∗appeared in: Comm. Pure and Appl. Math., 50 241–290 (1997). †Department of Mathematics, Brandeis University, Waltham, Mass 02254, USA

(adler@math.brandeis.edu). The support of a National Science Foundation grant # DMS 95- 4-51179 is gratefully acknowledged.

‡Department of Mathematics, Université de Louvain, 1348 Louvain-la-Neuve, Belgium (van- moerbeke@geom.ucl.ac.be) and Brandeis University, Waltham, Mass 02254, USA (vanmoer- beke@math.brandeis.edu). The support of National Science Foundation # DMS 95-4-51179, Nato, FNRS and Francqui Foundation grants is gratefully acknowledged.

1

and the moment matrix,

(0.3) mn(t, s, c) =: (µij)0≤ij≤n−1 = (〈yi, zj〉)0≤i,j≤n−1. As a function of t, s and c, the moment matrix m∞(t, s, c) satisfies the differential equations

(0.3′) ∂m∞ ∂tn

= Λnm∞, ∂m∞ ∂sn

= −m∞Λ>n , ∂m∞ ∂cij

= Λim∞Λ> j

,

where Λ is the semi-infinite shift matrix defined below. Consider the Borel decomposition of the semi-infinite matrix1

m∞ = S−11 S2 with S1 ∈ D−∞,0, S2 ∈ D0,∞ with S1 having 1’s on the diagonal, and S2 having hi’s on the diagonal, with (see section 3)

hi = det mi+1 det mi

.

The Borel decomposition m∞ = S−11 S2 above leads to two strings (p (1)(y), p(2)(z))

of monic polynomials in one variable, constructed, in terms of the character χ̄(z) = (zn)n∈Z,n≥0, as follows:

(0.4) p(1)(y) =: S1χ̄(y) p (2)(z) =: h(S−12 )

>χ̄(z).

We call these two sequences string-orthogonal polynomials; indeed the Borel decom- position of m∞ = S−11 S2 above is equivalent to the orthogonality relations:

〈p(1)n , p(2)m 〉 = δn,mhn. We show in section 4 the string-orthogonal polynomials have the following expres- sions in terms of Schur differential polynomials2

p(1)n (y) = ∑

0≤k≤n

pn−k(−∂̃t) det mn(t, s, c) det mn(t, s, c)

yk , p(2)n (z) = ∑

0≤k≤n

pn−k(∂̃s) det mn(t, s, c) det mn(t, s, c)

zk

1Dk,` (k < ` ∈ Z) denotes the set of band matrices with zeros outside the strip (k, `). 2The Schur polynomials pk, defined by e

∑∞ 1

tiz i

= ∑∞

0 pk(t)z k and pk(t) = 0 for k < 0, ,

and not to be confused with the string-orthogonal polynomials p(k)i , k = 1, 2, lead to differential polynomials

pk(±∂̃t) = pk ( ± ∂

∂t1 ,±1

2 ∂

∂t2 ,±1

3 ∂

∂t3 , ...

)

2

To the best of our knowledge, string-orthogonal polynomials were considered for the first time, in the context of symmetric weights ρ(y, z)dydz, by Mehta (see [M] and [CMM]).

The main message of this work is to show (i) that the expressions det mN satisfy the KP-hierarchy in t and s for each N = 1, 2, ..., (section 3) (ii) that the det mN hang together in a very specific way (they form the building blocks of the 2-Toda lattice) (section 4), (iii) that det mN satisfies an additional Virasoro-like algebra of partial differential equations (section 5), as a result of so-called string equations (section 4). In [AvM2], we have obtained similar results for moment matrices associated with general weights on R (thus connecting with the standard theory of orthogonal polynomials), rather than on R2. It would not be difficult to generalize the results of this paper to more general weights ρ(y, z)dy dz on R2, besides those of (0.1).

In terms of the matrix operators

Λ :=

0 1 0 0 1

0 0 . . .

and ε :=

0 0 1 0 0

2 0 . . .

acting on χ̄ as

Λχ̄(z) = zχ̄(z), εχ̄(z) = ∂

∂z χ̄(z),

the matrices

L1 := S1ΛS −1 1 , L2 := S2Λ

>S−12 , Q1 := S1εS −1 1 , Q2 := S2ε

>S−12

interact with the vector of string-orthogonal polynomials, as follows:

(0.5) L1p (1)(y) = yp(1)(y) Q1p

(1)(y) = d

dy p(1)(y)

hL>2 h −1p(2)(z) = zp(2)(z) hQ>2 h

−1p(2)(z) = d

dz p(2)(z).

The semi-infinite matrix L1 (respectively L2) is lower-triangular (resp. lower-triangular), with one subdiagonal above (resp. below); Q1 (resp. Q2) is strictly lower-triangular (resp. strictly upper-triangular). In Theorem 4.1 we prove the matrices Li and Qi satisfy so-called “string equations”

(0.6) Q1 + ∂V

∂y (L1, L2) = 0, Q2 +

∂V

∂z (L1, L2) = 0.

3

When

V (y, z) = `1∑

1

tiy i + cyz −

`2∑

1

siz i,

then (0.6) implies that L1 is a `2 + 1-band matrix, and L2 is a `1 + 1-band matrix. Moreover, as a function of (t, s), the couple L := (L1, L2) satisfies the “ two-

Toda lattice equations”, and as a function of cα,β, L satisfies another hierarchy of commuting vector fields; so, in terms of an appropriate Lie algebra splitting ( )+ and ( )−, to be explained in section 1, we have

(0.7) ∂L

∂tn = [(Ln1 , 0)+, L]

∂L

∂sn = [(0, Ln2 )+, L],

∂L

∂cα,β = −[(Lα1Lβ2 , 0)−, L],

and, what is equivalent, the moment matrix m∞ satisfies the differential equations (0.3’), with solution (thinking of tn = cn0 and sn = −c0n):

m∞(t, s, c) = ∑

(rαβ)α,β≥0∈Z∞ (α,β) 6=(0,0)

∏

(α,β)

c rαβ αβ

rαβ!

Λ

∑ α≥1 αrαβm∞(0, 0, 0)Λ

> ∑

β≥1 βrαβ ;

in particular m∞(t, s, 0) = e

∑∞ 1

tnΛnm∞(0, 0, 0)e− ∑∞

1 snΛ>n .

Thus the integrable system under consideration in this paper is a cαβ-deformation of the 2-Toda lattice, which itself is an isospectral deformation of a couple of matrices (L1, L2), in general bi-infinite, depending on two sequences of times t1, t2, . . . and s1, s2, . . ..

Moreover, the determinant of the moment matrix has many different expressions; in particular, in terms of the moment matrix at t = s = 0, using the matrix EN(t) :=

(the first N rows of e ∑∞

1 tnΛn) of Schur polynomials pn(t) (see section 3). It can also

be expressed as a 2-matrix integral reminiscent of 2-matrix integrals in string theory and in terms of the diagonal elements hi of the upper-triangular matrix S2:

(0.8)

N ! det mN(t, s, c) = N ! det ( EN(t)m∞(0, 0, c)EN(−s)>

)

= ∫ ∫

u,v∈IRN e ∑N

k=1 V (uk,vk)

∏

i

Finally τN(t, s, c) is a “τ -function” in the sense of Sato, separately in t and s 3.

The string equations (0.6) play an important role: they have many conse- quences! In Theorem 5.1, we show τN = det mN(t, s, c) satisfies the following set of constraints4

(0.9)

( J

(2) i + (2N + i + 1)J

(1) i + N(N + 1)J

(0) i + 2

∑

r,s≥1 rcrs

∂

∂ci+r,s

) τN = 0

( J̃

(2) i − (2N + i + 1)J̃ (1)i + N(N + 1)J̃ (0)i + 2

∑

r,s≥1 scrs

∂

∂cr,s+i

) τN = 0,

for i ≥ −1 and N ≥ 0. When V12 = cyz, the relations (0.9) reduce to an inductive system of partial differ- ential equations in t and s, for τ0, τ1, τ2, . . ., (Corollary 5.1.1)

(0.10)

( J

(2) i + (2N + i + 1)J

(1) i + N(N + 1)J

(0) i

) τN + 2c pi+N(∂̃t)pN(−∂̃s)τ1 ◦ τN−1 = 0

( J̃

(2) i − (2N + i + 1)J̃ (1)i + N(N + 1)J̃ (0)i

) τN + 2c pN(∂̃t)pi+N(−∂̃s)τ1 ◦ τN−1 = 0

3Sato’s τ -function τ(t) in t ∈ C∞ is the determinant of the projection e− ∑

tiz i

W → H+ = {1, z, z2, . . .}, where W is a fixed span of functions in z with poles at z = ∞ of order k = 0, 1, 2, . . .. Equivalently, a τ -function satisfies the bilinear relations

∮

C

τ(t− [z−1])τ(t′ + [z−1])e ∑∞

1 (ti−t′i)zidz = 0,

where C is a small contour about z = ∞ and where [α] := ( α, α

2

2 , α3

3 , ... ) ∈ C∞. The bilinear

relations imply that τ(t) satisfies the KP-hierarchy. 4in terms of the customary Virasoro generators in t1, t2, ...:

J (0)n = δn0, J (1) n =

∂

∂tn + (−n)t−n, J (1)0 = 0

J (2)n = ∑

i+j=n

: J (1)i J (1) j :=

∑

i+j=n

∂2

∂ti∂tj + 2

∑

−i+j=n iti

∂

∂tj +

∑

−i−j=n (iti)(jtj),

where “: :” denotes normal ordering, i.e., always pull differentiation to the right, and where a s

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