- 01/12/21: Riemann-Hilbert problems, Hitchin systems and the conformal limit
- 06/03/20: An update on hyperkahler metrics on moduli of Higgs bundles
January 12, 2021
TITLE: Riemann-Hilbert problems, Hitchin systems and the conformal limit
Given a Riemann surface C equipped with a meromorphic quadratic differential, one can define two natural families of flat sl(2)-connections on C. One of these families consists of sl(2)-opers (Schrodinger equations); the other is determined by a solution of Hitchin’s equations on C. For either of these families, the monodromy data is expected to be the solution of a Riemann-Hilbert problem over CP1, with Stokes phenomena determined by generalized Donaldson-Thomas invariants. In the case of sl(2)-opers, this Riemann-Hilbert problem should be identified with a special case of the one described by Tom Bridgeland in his lecture series. I will describe these two Riemann-Hilbert problems and the expected relation between them.
June 3, 2020
TITLE: An update on hyperkahler metrics on moduli of Higgs bundles
ABSTRACT: In joint work with Davide Gaiotto and Greg Moore, we gave a new conjectural construction of the hyperkahler metric on moduli spaces of Higgs bundles, in which the key new ingredient was counts of BPS states (Donaldson-Thomas-type invariants).
Through the work of various authors, including me, Dumas, Fredrickson, Mazzeo, Swoboda, Weiss, Witt, there is now some evidence that this conjectural picture is correct. On the one hand, some of the asymptotic predictions which follow from the conjecture have been proven; on the other hand, there is numerical evidence that the conjecture is correct even far away from the asymptotic limit. I will review these developments.