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

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

2009-01-04

arxiv.org/abs/0901.0323v3

Physics

Mathematical Physics

28 pages. The effect of convolution symmetries on the Baker-Akhiezer function is clarified in v3

Scientific paper

Generalized convolution symmetries of integrable hierarchies of KP-Toda and 2KP-Toda type have the effect of multiplying the Fourier coefficients of the Baker-Akhiezer function by a specified sequence of constants. The induced action on the associated fermionic Fock space is diagonal in the standard orthonormal base determined by occupation sites and labeled by partitions. The coefficients in the single and double Schur function expansions of the associated $\tau$-functions, which are the Pl\"ucker coordinates of a decomposable element, are multiplied by the corresponding diagonal factors. Applying such transformations to matrix integrals, we obtain new matrix models of externally coupled type which are also KP-Toda or 2KP-Toda $\tau$-functions. More general multiple integral representations of tau functions are similarly obtained, as well as finite determinantal expressions for them.

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