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A Riemann-Hilbert problem from stability conditions

Banff International Research Station for Mathematical Innovation and Discovery

Given an algebra $A$ and a set of automorphisms, one can define a Riemann-Hilbert (RH) problem, aimed to find meromorphic connections on the $\mathrm{Aut}(A)$-principl bundle over $\bC$ with prescribed generalised monodromy. It is of particular interest considering a family of `isomonodromic' RH problems on $A$, parametrised by a complex manifold $M$, as this induces interesting geometric structures on the space of parameters $M$. In the context of stability conditions, there is a Riemann-Hilbert problem naturally attached to a CY3 category endowed with a generalised Donaldson-Thomas theory counting semistable objects. It is defined on the torus algebra of character on the free abelian group generated by classes of simple stable objects and depends on the choice of a stability condition. The wall-crossing formulae from Donaldson-Thomas theory are interpreted as isomonodromy conditions. I will introduce this topic and discuss some recent developments, focusing on an example associated with some special quivers.

Mathematics; Algebraic geometry; Associative rings and algebras; Representation theory

video/mp4

Author affiliation: University of Sheffield

2019-05-01

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/70009

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/