Title: Asymptotics of Higgs bundles and hyperpolygons
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| Date | Mon 19 Jun — Wed 21 Jun |
| Time | 09:00 — 16:00 |
| Location | 1523-318 Fys. Aud. |
Abstract
Since its discovery by Hitchin in 1987, the moduli space of Higgs bundles on a Riemann surface has become a key object at the intersection of a number of mathematical disciplines. In geometry, it is a noncompact Calabi-Yau manifold with a distinguished hyperkaehler structure arising as an infinite-dimensional quotient. In physics, it is a solution space to a version of the Yang-Mills equations and possesses a nontrivial "mirror symmetry". In dynamics, it is a completely integrable Hamiltonian system that captures many classically-known mechanical systems. In representation theory, the moduli space is closely related to natural fibrations occurring in Lie theory and its mirror symmetry is intertwined with Langlands duality.
The moduli space's hyperkähler metric has proven to be one of its most elusive features. While the metric is known to exist and is complete in most circumstances, it so far has not been written down explicitly. Recent works by Gaiotto-Moore-Neitzke, Mazzeo-Swoboda-Weiss-Witt, and Fredrickson have found new characterizations of the hyperkaehler metric, by comparing it to a so-called "semi-flat metric". In particular, the work of Mazzeo-Swoboda-Weiss-Witt studies the "ends" of the moduli space of Higgs bundles in low rank, where "limiting configurations" are found that partially compactify the moduli space. This partial compactification is essential to understanding the metric.
At the same time, the works of Godinho-Mandini, Biswas-Florentino-Godinho-Mandini, and Fisher-Rayan make a connection between parabolic Higgs bundles on the punctured sphere and hyperpolygons, whose moduli spaces arise as finite-dimensional hyperkaehler quotients. Hyperpolygon spaces can be studied asymptotically in the same way as ordinary Higgs bundles, allowing us to compare the metric on hyperpolygon space to the one on the larger parabolic Higgs space.
This lecture series will introduce Higgs bundles and their moduli spaces, from both the gauge-theoretic and algebro-geometric points of view. We then give an introduction to the asymptotic analysis of the Higgs bundle moduli space and its hyperkähler metric, with a particular focus on the low-rank situation. Finally, we discuss hyperpolygons and how the limiting programme can be carried out in that case.
Steven Rayan (University of Saskatchewan)
Hartmut Weiß (Christian-Albrechts-Universität zu Kiel (CAU))
REGISTRATION: send an e-mail to qgm@au.dk.
Programme
Monday 19 June 2017 (Fys. Aud.)
Intro to moduli space of Higgs bundles: gauge theory (Compact Riemann surfaces, Yang-Mills / Hitchin equations and Hermitian metrics, Hitchin's construction, integrable system, etc.) by Hartmut Weiss.
10:00-10:45 Lecture
11:00-11:45 Lecture
Lunch
Intro to moduli space of Higgs bundles (algebraic geometry (Higgs bundles and stability condition from algebraic point of view, Nitsure construction, spectral curves and Hitchin fibration, C*-action, etc.) by Steven Rayan.
14:00-14:45 Lecture
15:00-15:45 Lecture
Tuesday 20 June 2017 (Fys. Aud.)
Topology of Hitchin system (Morse theory of U(1)-action, fixed points as quivers, calculation of Poincare polynomials in low rank and genus, quivers as motivation for hyperpolygons) by Steven Rayan
10:00-10:45 Lecture
11:00-11:45 Lecture
Lunch
Asymptotics of Higgs bundles (Partial compactification of moduli space, limiting configurations) by Hartmut Weiss.
14:00-14:45 Lecture
15:00-15:45 Lecture
Wednesday 21 June 2017 (Fys. Aud.)
Hyperpolygons and parabolic Higgs bundles by Steven Rayan
10:00-10:45 Lecture
11:00-11:45 Lecture
Lunch
Asymptotics of Higgs bundles (continued) and asymptotics of hyperpolygons (work in progress) by Hartmut Weiss.
14:00-14:45 Lecture
15:00-15:45 Lecture
