Special Joyce structures and hyperkähler metrics (Ivan TULLI, University of Sheffield)

14.11.2024 16:15

Joyce structures were introduced by T. Bridgeland in the context of the space of stability conditions Stab(X) of a 3d Calabi-Yau category X and its associated Donaldson-Thomas (DT) invariants. They are described by a certain C^*-family of non-linear flat connections on the tangent bundle T(Stab(X)) of Stab(X) and, under certain non-degeneracy conditions, encode a complex-hyperkähler structure on T(Stab(X)). On the other hand, the work of Gaiotto-Moore-Neitzke (GMN) shows how to construct from the Coulomb branch M of a 4d N=2 theory and its associated BPS indices, a real-hyperkähler (HK) geometry on the cotangent bundle T*M of M. This HK metric is the instanton corrected metric associated with the 3d theory obtained by compactifying the 4d N=2 theory on S^1. In this talk, I will try to reinterpret the GMN construction in a way that is similar to Joyce structures. The resulting structure is called a "special Joyce structure", and encodes an HK geometry on T*M. At least in the case where the BPS indices are uncoupled, the HK metric encoded by the corresponding special Joyce structure is shown to match the GMN HK metric. As a by-product of this description, we will see that the GMN HK geometry in the uncoupled case can be encoded in the geometry of the base M, together with a single function J on T^*M determined by the BPS indices. The function J seems to be related to a function previously studied by S. Alexandrov and B. Pioline in arXiv:1808.08479, which they call the "instanton generating function". This talk is based on arXiv:2403.00548.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-07, Séminaire "Physical Mathematics Seminar"

entrée libre