For titles and abstracts, please refer to the information below.

Mini-Courses

Olivia Dumitrescu (Central Michigan)

Mini-Course I: The Interplay between Higgs Bundles, Opers, and CohFT

Talk 1: Topological recursion for Hitchin fibrations of rank 2 and quantum curves

Talk 2: Interplay between CohFT and topological recursion

Talk 3: From the Hitchin component to the oper moduli

Hartmut Weiss (CAU Kiel)

Mini-Course II: The Asymptotic Geometry of Moduli Spaces of Higgs Bundles and Hyperpolygons

Talk 1: Construction of moduli spaces as hyperkähler quotients

Talk 2: Solutions of Hitchin's equations for large Higgs fields

Talk 3: Asymptotics of hyperkähler metrics

Qiongling Li (Caltech / QGM)

Mini-Course III: Harmonic Maps and Higgs Bundles

Talk 1: The relationship between Higgs bundles and harmonic maps

Talk 2: Minimal surfaces and Labourie's conjecture

Talk 3: Asymptotics of harmonic maps

In addition, Brian Collier (Maryland) will give an introductory talk on nonabelian Hodge theory to open the workshop on the Friday evening.

Participant Talks

Emily Cliff (UIUC)

On the Kirwan map for moduli stacks of Higgs bundles

Abstract: To a reductive group G and a smooth projective complex curve C, one associates the moduli stack of G-Higgs bundles on C. There is a natural map from the cohomology of the moduli stack of G-Higgs bundles to the cohomology of the moduli stack of semistable G-Higgs bundles, known as the Kirwan map. In the case of G=GL(n), Markman proved that the Kirwan map is surjective for the component of Higgs bundles of degree d coprime to n. By contrast, I explain recent work in which I show (generalizing Hitchin for SL(n)) that the Kirwan map for moduli of G-Higgs bundles fails to be surjective whenever G has disconnected centre. This is accomplished via the study of a group action on the moduli stack and the induced action on its cohomology. Time permitting, I also establish similar results for the cohomology of the moduli space (rather than stack), and for intersection cohomology. This talk is based on joint work with Thomas Nevins and Shiyu Shen.

Richard Derryberry (UT Austin)

Self-dual versions of the moduli of Higgs bundles

Abstract: I will sketch why self-dual versions of the moduli of G-Higgs bundles arise physically from the study of 4d theories of class S, and will describe an extension of the Langlands duality results of Hausel-Thaddeus (G=SL(n)) and Donagi-Pantev (arbitrary reductive G) that yields self-dual moduli spaces as a corollary.

Monica Kang (Harvard)

Flopping and slicing: understanding the effect of Mordell-Weil Torsion Z2

Abstract: I will describe how introducing a non-trivial Mordell-Weil group changes the structure of the Coulomb phases of a five-dimensional gauge theory from an M-theory compactified on an elliptically fibered Calabi-Yau threefolds with a collision of singularities of certain gauge groups. I will show how to compute topological invariants relevant for the physics, such as the Euler characteristic, Hodge numbers, and triple intersection numbers. Also, the matter representation will be determined geometrically by computing weights via intersection of curves and fibral divisors.

Christopher Mahadeo (USask)

Wrapping in 2-bands and Higgs bundles

Abstract: Topology has proven to be a crucial tool for studying conductance in materials. Topology manifests in the following way: metals and insulators are classified by the existence of an energy gap, as the latter possess a gap between valence and conduction bands while the former do not. The eigenstates and Hamiltonian of a crystal are indexed by momenta and thus give rise to a vector bundle structure over the Brillouin zone. Electronic properties of an insulator are then governed by the topological properties of the subbundle corresponding to valence bands. These bands can possess non-trivial topologies that can be realized as an obstruction to define wave functions over the entire Brillouin zone using a single phase convention. We show that in a 2-band insulator this non-trivial topology manifests as a winding number. We explore the potential application of Higgs bundles and "abelianization" to the study of topological materials.

Alessandro Malusà (QGM / USask)

Geometric quantization: some examples

Abstract: Geometric quantization is a construction motivated by and based on the techniques employed by physicists in the so-called canonical quantization. Ideally, one should look for a non-commutative deformation of the algebra of smooth functions on a symplectic manifold, the physical requirement being that the commutator of two functions should be prescribed by their Poisson bracket. Starting from the example of canonical quantization for a particle in R^{n}, I wish to motivate and introduce the basic ideas of geometric quantization on a symplectic manifold equipped with the necessary additional structure. Depending on time, I would like to mention the case of the moduli spaces of flat SU(2)- and SL(2,C)-connections over a closed Riemann surface as examples of symplectic spaces.

Ákos Nagy (Duke)

On the ends of moduli spaces of non-linear gauged sigma models

Abstract: The 2-dimensional non-linear gauged sigma models (also known as Hamiltonian Gromov-Witten theory) can be viewed as a common generalization of the vortex equations and Gromov-Witten theory. First in this talk, I will briefly introduce the theory on closed surfaces and with compact, Káhler target spaces. In his thesis, Ignasi Mundet i Riera proved the existence of a natural commodification of the moduli spaces for compact targets. This compactification however is not "metric" in the sense that it does not describe the behavior of the natural L^{2} metric near the ends of the moduli spaces. For example, Mundet's Theorem does not say whether the moduli space is complete or not (as a metric space). In a joint project with Nuno Romão, we investigated this L^{2} metric in the case when the target is projective line and the gauge group is U(1), and gave an asymptotic description of the metric. In particular we find that the "end" is at finite distance and that is the moduli space is incomplete.

Evan Sundbo (USask)

The topology of the moduli space of twisted Higgs bundles onP^{1} via quiver representations

Abstract: Higgs fields are traditionally defined as K-twisted endomorphisms of a bundle, where K is the canonical line bundle. When allowing this endomorphism to take values in an arbitrary holomorphic line bundle L instead, it is natural to consider Higgs bundles over curves of any genus g. I will sketch how the action of S^{1} on the moduli space of L-twisted Higgs bundles can give us information about the topology of this space and explain how we can use quiver representations for this same pursuit, in particular on the Riemann sphere P^{1}.

Fei Yan (UT Austin)

Nonabelian Hodge correspondence at the U(1)-fixed points

Abstract: In this talk I will describe a method to solve the nonabelian Hodge correspondence at the U(1)-fixed points in some specific Hitchin moduli spaces. I will also sketch the relation to the study of theories of class-S in four dimensions.