Hyper-Kähler metrics from isomonodromy

Timothy Moy (University of Cambridge)

Tuesday 16th April 16:00-17:00 311B

Abstract

Joyce structures of class S[A_1] are complexified hyper-Kähler metrics with extra symmetries existing on the tangent bundles of spaces of meromorphic quadratic differentials. Bridgeland and Masoero obtained the metric in terms of elementary (rational) functions in the case of quadratic differentials with a single pole of order five on the Riemann sphere, via the calculation of the isomonodromy of a family of ODE. I will review this construction and explain how it can be generalised to quadratic differentials on the Riemann sphere with a pole of order 2n + 1. This leads to a family of hyper-Kähler metrics of dimension 4n, compatible in the sense of Bridgeland and Strachan, with the natural symplectic structure on the base space. This talk is based on joint work with Maciej Dunajski in ArXiv:2402.14352.

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