Tuesday, August 29 |
07:00 - 08:30 |
Breakfast (Vistas Dining Room) |
08:30 - 09:20 |
Jan Swoboda: The large-scale geometry of Higgs bundle moduli spaces ↓ In this talk I will explain recent joint work with Rafe Mazzeo, Hartmut Wei{\ss} and Frederik Witt on the asymptotics of the natural L^2-metric G_{L^2} on the moduli space \mathcal M of rank-2 Higgs bundles over a Riemann surface X as given by the set of solutions to the self-duality equations
\begin{cases}
0=\bar\partial_A\Phi\\
0=F_A+[\Phi\wedge\Phi^{\ast}]
\end{cases}
for a unitary connection A and a Higgs field \Phi on X. I will show that on the regular part of the Hitchin fibration (A,\Phi)\mapsto\det\Phi this metric is well-approximated by the semiflat metric G_{\operatorname{sf}} coming from the completely integrable system on \mathcal M. This also reveals the asymptotically conic structure of G_{L^2}, with the (generic) fibres of the above fibration being asymptotically flat tori. This result confirms some aspects of a more general conjectural picture made by Gaiotto, Moore and Neitzke. Its proof is based on a detailed understanding of the ends structure of \mathcal M. The analytic methods used here in addition yield a complete asymptotic expansion of the difference G_{L^2}-G_{\operatorname{sf}} between the two metrics, with leading order term decaying at some polynomial rate as \|\Phi\|\to\infty. (TCPL 201) |
09:30 - 10:20 |
Laura Fredrickson: The ends of the Hitchin moduli space in higher rank ↓ Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichm\"uller theory, and the geometric Langlands correspondence. In this talk, I'll describe what solutions of SL(n,\mathbb{C})-Hitchin's equations ``near the ends'' of the moduli space look like, and the resulting compactification of the Hitchin moduli space. Wild Hitchin moduli spaces are an important ingredient in this construction. This construction generalizes Mazzeo-Swoboda-Weiss-Witt's construction of SL(2,\mathbb{C})-solutions of Hitchin's equations where the Higgs field is ``simple.'' (TCPL 201) |
10:20 - 11:00 |
Coffee Break (TCPL Foyer) |
11:00 - 12:00 |
Interaction/Collaboration Period (TCPL 201) |
12:00 - 13:30 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |
13:30 - 14:20 |
Mark Stern: Monotonicity and Betti Number Bounds ↓ In this talk I will discuss the application of techniques initially developed to study singularities of Yang Mill's fields and harmonic maps to obtain Betti number bounds, especially for negatively curved manifolds. (TCPL 201) |
14:30 - 14:50 |
Group Photo ↓ Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo! (TCPL Foyer) |
14:50 - 15:30 |
Coffee Break (TCPL Foyer) |
15:30 - 16:20 |
Sergey Cherkis: Instantons, Bows, and Gauge Theory Mirror Symmetry ↓ Bow moduli spaces have various applications. They are isomorphic to moduli spaces of Yang-Mills instantons on Asymptotically Locally Flat spaces. We begin by defining the instantons topological charges and identifying the relevant balanced and cobalanced bow representations.
Bow moduli spaces can also be identified with moduli spaces of vacua of quantum supersymmetric gauge theories. We identify gauge theories for which bows deliver both their Higgs and Coulomb branches. Moreover, for these theories the gauge theory mirror symmetry acts as isometry interchanging the types of the branches.
Two independent approaches are presented: 1. the generalizations of the ADHM-Nahm transform, that we call Up and Down transforms (developed in collaboration with Mark Stern and Andres Larrain-Hubach) and 2. the monad construction approach (developed in collaboration with Jacques Hurtubise). (TCPL 201) |
16:30 - 17:20 |
Amihay Hanany: Coulomb branch, Higgs branch, and minimally unbalanced quivers ↓ Among the set of all finite dimensional HyperKahler cones, there is a special family, constructed using quivers, which we term "minimally unbalanced" quivers. This family turns out to solve a problem in super symmetric gauge theories in five dimensions. After an introduction to HyperKahler cones and Coulomb branches, I will present the family and discuss its properties, perhaps going over some physics of five dimensions. (TCPL 201) |
17:30 - 19:30 |
Dinner (Vistas Dining Room) |