Fabrizio Del Monte (CRM)
Title:听Isomonodromic tau functions on Riemann Surfaces through free fermions and four-dimensional QFTs
Abstract:听Using arguments from two-dimensional Conformal Field Theory, Gamayun Iorgov and Lisovyy provided in 2012 an explicit expression for the tau function of the sixth
Painlev茅 equation as a Fourier transform of Virasoro conformal blocks of a free fermion CFT, which has an explicit combinatorial (convergent) expansion as the so-called
dual partition function of a corresponding four-dimensional supersymmetric Quantum Field Theory. This "Kiev formula" has been later generalized to more general isomonodromic problems on the sphere, with both regular and irregular punctures, and the combinatorial expansion has been shown to arise from the minor expansion of an associated Fredholm determinant, in terms of which the tau function can be formulated.
In this talk I will show how the above picture can be generalized to the case of Riemann Surfaces with nonzero genus and marked points, where new features arise due to the nontriviality of the moduli space of flat connections. I will consider in detail the example of the punctured torus, for which I will show that the tau function can be written as a free fermion conformal block from two-dimensional CFT, and as a Fredholm determinant of Cauchy operators, whose minor expansion reproduces the conformal block itself.听
听
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