Luis Álvarez-Cónsul: Obstructions to the existence of gravitating vortices
In this talk, I will start by explaining some background about the self-dual Einstein–Maxwell–Higgs equations on a compact surface, and then analyze obstructions to the existence of solutions, expressed in terms of the multiplicities of the zeroes of the Higgs field and the vortex number. Joint work with Mario García-Fernandez, Oscar García-Prada, Vamsi Pritham Pingali and Chengjian Yao.
Mathew Bullimore: Vortices and generalised symmetry
There is much recent interest in generalised or categorical symmetries that go beyond the paradigm of groups and into the realm of higher groups and fusion categories. I will discuss aspects of such symmetries in the context of three-dimensional abelian GLSMs and moduli spaces of vortices. I will then speculate on generalised notions of equivariance in quasimap K-theory of toric stacks.
Mykola Dedushenko: (i) Elliptic stable envelopes as interfaces in a 3d QFT
I will review the physical realization of elliptic stable envelopes within 3d N = 4 gauge theories. This consists of two steps: (1) explaining why the (Higgs phase) SUSY vacua on a two-torus are captured by the elliptic cohomology of the Higgs branch; (2) observing that a natural SUSY interface exists that interpolates between the massive and massless regimes of the theory.
Mykola Dedushenko: (ii) Symplectic duality and the vortex partition function
Elliptic stable envelopes are known to appear in the 3d mirror symmetry of the “vertex function” (which encapsulates the K-theoretic count of vortices). Looking instead at the Higgs–Coulomb phase transition in a given 3d N = 4 theory (also known as symplectic duality), we explain this via the properties of the cigar partition function (or half-index) and the Janus interfaces from talk (i).
Aleksander Doan: Holomorphic curves and the ADHM vortex equations
I will discuss the problem of counting embedded holomorphic curves in Calabi–Yau manifolds of complex dimension three or, more generally, in symplectic six-manifolds. While the naïve count does not typically lead to an interesting geometric invariant, I will outline an ongoing project with Thomas Walpuski whose goal is to define an invariant of symplectic six-manifolds by counting embedded holomorphic curves with weight given by the number of solutions to certain gauge-theoretic equations called the ADHM vortex equations.
Andrea Ferrari: Vortices, other saddles, and wall-crossing
The enumerative geometry of vortex moduli spaces plays a key role in the study of supersymmetric gauge theories in three dimensions. For instance, in the presence of eight supersymmetry, field configurations contributing to the path integral can often be localised to moduli spaces of solutions of generalised vortex equations. In the presence of less supersymmetry, however, the path integral can receive additional contributions, which interplay in a remarkable way with those originating from vortex moduli spaces. In this talk, I will introduce some of these phenomena in simple, abelian examples, and comment on the expected relation to and potential implications for topics in mathematics such as wall-crossing.
(Joint work with M. Bullimore, H. Kim and G. Xu.)
Mario García-Fernandez: Gravitating vortices with positive curvature
In this talk I will give an overview of joint work with V. Pingali and C. Yao in arXiv:1911.09616, where we give a complete solution to the existence problem for gravitating vortices on acompact Riemann surface with non-negative topological constant c > 0.
Dan Halpern-Leistner: Infinite-dimensional geometric invariant theory and gauged Gromov–Witten theory
Harder–Narasimhan (HN) theory gives a structure theorem for vector bundles on a smooth projective curve. A bundle is either semistable, or it admits a canonical filtration whose associated graded bundle is semistable in a graded sense. After reviewing recent advances in extending HN theory to arbitrary algebraic stacks, I will discuss work in progress with Andres Fernandez Herrero to apply this general machinery to the stack of ”gauged” maps from a curve C to a G-scheme X, where G is a reductive group and X is projective over an affine scheme. Our main application is to use HN theory for gauged maps to compute generating functions for K-theoretic enumerative invariants known as gauged Gromov–Witten invariants. This problem is interesting more broadly because it can be formulated as an example of an infinite dimensional analog of the usual setup of geometric invariant theory, which has applications to other moduli problems.
Justin Hilburn: 2-categorical aspects of 3d mirror symmetry
By now it is known that many interesting phenomena in geometry and representation theory can be understood as aspects of mirrorsymmetry of 3d N = 4 SUSY QFTs. Such a QFT is associated to a hyperkähler manifold X equipped with a hyperhamiltonian action of a compact Lie group G and admits two topological twists. The first twist, which is known as the 3d B-model or Rozansky–Witten theory, is a TQFT of algebro-geometric flavor and has been studied extensively by Kapustin, Rozansky and Saulina. The second twist, which is known as the 3d A-model or 3d Seiberg–Witten theory, is a more mysterious TQFT of symplecto-topological flavor. In this talk I will discuss what is known about the 2-categories of boundary conditions for these two TQFTs. They are expected to provide two distinct categorifications of category O for the hyperkähler quotient X///G and 3d mirror symmetry is expected to induce a categorification of the Koszul duality between categories O for mirror symplectic resolutions. For abelian gauge theories this picture is work in progress with Ben Gammage and Aaron Mazel-Gee. This generalizes works of Kapustin–Vyas–Setter and Teleman on pure gauge theory.
Chiu-Chu Melissa Liu: Higgs–Coulomb correspondence and wall-crossing in abelian GLSMs
We define and compute I-functions and central charges for abelian GLSMs using virtual factorizations of Favero and Kim. In the Calabi–Yau case we provide analytic continuation for central charges by explicit integral formulas. The integrals in question are called hemisphere partition functions and we call the integral representation Higgs–Coulomb correspondence. We then use it to prove GIT stability wall-crossing for central charges.
Nick Manton: Statistical mechanics of vortices
Classical or quantized statistical mechanics of critically-coupled Abelian Higgs vortices can be modelled by free dynamics on the N-vortex moduli space, with N large. Vortex interactions are captured by the non-trivial moduli space geometry. To avoid boundary effects and satisfy Bradlow’s constraint, the vortices are defined on a compact surface of large area A, with A/N > 4π. The classical partition function depends only on the moduli space volume, and the first quantum correction at high temperature T depends on the integrated scalar curvature. Using these known geometrical quantities, we deduce the high-T equation of state of the vortex gas. When A/N is only slightly larger than 4π, the moduli space simplifies to complex projective space with its Fubini–Study geometry. Here the quantum partition function and equation of state can be calculated for any temperature. (NSM thanks S. Nasir, J. Baptista, J.M. Speight and S. Wang for their collaboration and contributions.)
Michael McBreen: Uniformizing the elliptic stable envelopes of a hypertoric variety
Elliptic stable envelopes were introduced by Aganagic and Okounkov as a key tool in the study of K-theoretic quasimap invariants for Nakajima quiver varieties and other symplectic resolutions. They bear an interesting relation to symplectic duality: the stable envelopes of a dual pair of resolutions M and N should arise from a “duality interface” living on the product M × N. I will give a gentle introduction to some of these ideas. I will then describe joint work with Artan Sheshmani and Shing-Tung Yau which uniformizes the duality interface of a hypertoric dual pair, yielding a distinguished K-theory class on an affine analogue of M × N.
Vamsi Pingali: The vortex ansatz as a fertile testing ground for certain systems of PDEs
I shall review the vortex bundle construction of García-Prada. Then I shall proceed to discuss the vortex ansatz as a way to dimensionally reduce several interesting systems of PDEs like the Kähler–Yang–Mills equations (and its offshoots) and the vector bundle Monge–Ampère equation. I shall discuss the latter in detail and end with an advertisement for studying fully nonlinear systems of PDEs arising from vector bundles (and the fact that the vortex ansatz can prove to be a powerfultesting ground for them).
Sushmita Venugopalan: Vortices on non-compact Riemann surfaces
Symplectic vortices on punctured Riemann surfaces are related by a Hitchin–Kobayashi correspondence togauged maps satisfying a semistability condition. These objects show up in various settings in gauged Gromov–Witten theory, such as in the definition of the quantum Kirwan map by Ziltener and Woodward using affine vortices; and in quasimaps to GIT quotients defined by Kim, Ciocan-Fontanine and Maulik.
Richard Wentworth: Conformal limits of parabolic Higgs bundles
Gaiotto introduced the notion of a conformal limit of a Higgs bundle and conjectured that these should identify the Hitchin component with the oper stratum in the de Rham moduli space. In the case of closed Riemann surfaces this result was proven by Dumitrescu et al., and the limits were shown to exist much more generally by Collier and the speaker. In this talk I will report on progress in the case of parabolic Higgs bundles, which were the context of Gaiotto’s original conjecture. (This is joint work with B. Collier and L. Fredrickson.)