Tau Functions as Widom Constants

Abstract

We define a tau function for a generic Riemann-Hilbert problem posed on a union of non-intersecting smooth closed curves with jump matrices analytic in their neighborhood. The tau function depends on parameters of the jumps and is expressed as the Fredholm determinant of an integral operator with block integrable kernel constructed in terms of elementary parametrices. Its logarithmic derivatives with respect to parameters are given by contour integrals involving these parametrices and the solution of the Riemann-Hilbert problem. In the case of one circle, the tau function coincides with Widom's determinant arising in the asymptotics of block Toeplitz matrices. Our construction gives the Jimbo-Miwa-Ueno tau function for Riemann-Hilbert problems of isomonodromic origin (Painlevé VI, V, III, Garnier system, etc) and the Sato-Segal-Wilson tau function for integrable hierarchies such as Gelfand-Dickey and Drinfeld-Sokolov. 1 cafasso@math.univ-angers.fr 2 pasha145@gmail.com 3 lisovyi@lmpt.univ-tours.fr 4 See also the work of S. Kovalevskaya [Kov] and the more recent ones on finite-gap integration; e.g. [DMN, Mat] and references therein.

$${\mathcal {N}}$$ = $$2^*$$ Gauge Theory, Free Fermions on the Torus and Painlevé VI

“…This was then shown to satisfy the Jimbo-Miwa-Ueno definition of the tau function and to reproduce the expansion of the relevant Nekrasov partition function. More recent understanding of such determinant formulas in the sphere case, together with a simplified proof, can be found in [63].…”

Liouville theory, AdS2 string, and three-point functions

“…where the factor denoted in blue was introduced in order to match it with the asymptotic form of i ± (29). The expression (48) gives two formal solutions depending on the choice of the sign in the exponent.…”

The τ-function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constant

“…The determinant of an operator A ∈ C m×m can be expanded in terms of its principal minors [11]. For a finite m × m matrix A, the minor expansion is given by This sequence obviously terminates after n = m. Now generalise the minor exansion for infinite dimensional matrix.…”

“…However the generic τ -function of the Painlevé I, II and IV equations does not seem to have a Fredholm determinant representation. The main obstacle to develop the procedure implemented in [13] and [11] is the impossibility to reduce the RHP problem of the Painlevé I, II and IV equations to a RHP on the circle. However it is expected that the generic RHP for these equations could be reduced to a RHP on the line for a jump matrix G. Then one considers the projection operator Π + to holomorphic functions on the semi-plane and define the operator T G = Π + G. For the case considered in this manuscript, this operator is trace class and therefore the Fredholm determinant (1.2) is well defined.…”

“…This paper is structured as follows. We will first setup the machinery to extend the formalism in [13], [11] to a line contour and show in 1 that the τ -function can be written as a Widom constant. Next we show in 3 that the Widom constantcoincides with the isomonodromic τ -function (1.1).…”