Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus

Abstract

We compute the monodromy dependence of the isomonodromic tau function on a torus with n Fuchsian singularities and SL(N) residue matrices by using its explicit Fredholm determinant representation. We show that the exterior logarithmic derivative of the tau function defines a closed one-form on the space of monodromies and times, and identify it with the generating function of the monodromy symplectomorphism. As an illustrative example, we discuss the simplest case of the one-punctured torus in detail. Finally, we show that previous results obtained in the genus zero case can be recovered in a straightforward manner using the techniques presented here.

Preface to resurgent asymptotics, Painlevé equations and quantum field theory focus issue

“…This role is showcased in the paper by Lisovvy and Naiduk, who provide a rigorous derivation for a formula expressing coefficients of Heun equation using classical Virasoro conformal blocks, allowing these coefficients to be computed to arbitrary order [7]. Del Monte et al explore the monodromy dependence of isomonodromic tau functions, and provide a detailed discussion of the behaviour of these functions on the torus [5].…”