Convolution symmetries of integrable hierarchies, matrix models and τ-functions

Summary

Generalized convolution symmetries of integrable hierarchies of KP and 2DToda type act diagonally on the Hilbert space ℋ = L²(S¹) in the standard monomial basis. The induced transformations on the Hilbert space Grassmannian Gr_ℋ+(ℋ) may be viewed as symmetries of these hierarchies, acting upon the Sato–Segal–Wilson τ-functions, and thereby generating new solutions of the hierarchies. The corresponding transformations of the associated fermionic Fock space are also diagonal in the standard orthonormal basis, labeled by integer partitions. The Plücker coordinates of the image under the Plücker map of the element W ∈ Gr_ℋ+(ℋ) defining the initial point under the commuting KP flows are the coefficients in the single and double Schur function expansions of the associated τ-functions. These are therefore multiplied by the eigenvalues of the convolution action in the fermionic representation. Applying such transformations to standard matrix model integrals, we obtain new matrix models of externally coupled type whose partition functions are thus also seen to be KP or 2D-Toda τ-functions. More general multiple integral representations of tau functions are similarly obtained, as well as finite determinantal expressions for them.

1. Introduction: convolution symmetries of τ-functions

Solutions of integrable hierarchies of KP and 2D-Toda type are determined by their τ-functions [Sato 1981; Sato and Sato 1983; Segal and Wilson 1985]. Infinite sequences of such KP τ-functions {τ(N,t)}/N∈ℤ, depending on the infinite set of commuting flow parameters t = (t₁, t₂,... ) and an integer lattice label N, may be associated in a standard fashion (see references just cited) to elements of a “universal phase” space, viewed as an infinite Grassmann manifold or flag manifold. These satisfy the Hirota bilinear equations of the KP hierarchy and also, in certain cases (e.g., exponential flows of matrix model integrals induced by trace invariants), the equations of the Toda lattice hierarchy.