Complex integrable systems, Hyperkahler metrics, and Joyce structures

Ivan Tulli (University of Sheffield)

Tuesday 18th November 16:00-17:00
Maths 311B

Abstract

Joyce structures were introduced by T. Bridgeland as a conjectural geometric structure on spaces of stability conditions, encoded by Donaldson-Thomas invariants. In subsequent work by T. Bridgeland and I. Strachan it was shown that, under a certain non-degeneracy condition, a Joyce structure defines a complex - hyperkahler structure on the tangent bundle of the space of stability conditions. On the other hand, in the context of 4d N=2 SUSY gauge theories there is a construction due to Gaiotto-Moore-Neitzke, which produces a hyperkahler metric with a compatible integrable system structure, encoded by BPS indices of the theory. Both stories involve similar conjectural constructions that involve solving families of Riemann-Hilbert problems determined by DT/BPS invariants. In this talk I will try to clarify the similarities and differences of both constructions, by describing a "Joyce structure"-like description of (real) hyperkahler metrics with compatible integrable system structure. This is based on work in progress and is a sequel to arXiv:2403.00548.

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