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Tau functions and the limit of block Toeplitz determinants

Abstract : A classical way to introduce tau functions for integrable hierarchies of solitonic equations is by means of the Sato-Segal-Wilson infinite-dimensional Grassmannian. Every point in the Grassmannian is naturally related to a Riemann-Hilbert problem on the unit circle, for which Bertola proposed a tau function that generalizes the Jimbo-Miwa-Ueno tau function for isomonodromic deformation problems. In this paper, we prove that the Sato-Segal-Wilson tau function and the (generalized) Jimbo-Miwa-Ueno isomonodromy tau function coincide under a very general setting, by identifying each of them to the large-size limit of a block Toeplitz determinant. As an application, we give a new definition of tau function for Drinfeld-Sokolov hierarchies (and their generalizations) by means of infinite-dimensional Grassmannians, and clarify their relation with other tau functions given in the literature.
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https://hal.archives-ouvertes.fr/hal-00989747
Contributor : Mattia Cafasso <>
Submitted on : Monday, May 12, 2014 - 1:35:43 PM
Last modification on : Monday, March 9, 2020 - 6:15:54 PM

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  • HAL Id : hal-00989747, version 1
  • ARXIV : 1404.5149

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Mattia Cafasso, Chao-Zhong Wu. Tau functions and the limit of block Toeplitz determinants. International Mathematics Research Notices, Oxford University Press (OUP), 2015. ⟨hal-00989747⟩

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