Schedule of the Workshop: Geometry of the Vortex Equations

João Baptista: Abelian vortices revisited
I will discuss recent results on abelian vortices, namely: (1) an interpretation of vortices as degenerate metrics on a manifold; (2) the cohomology and volume of vortex moduli spaces in gauged linear sigma-models defined over singular surfaces, simply connected Kähler manifolds, and abelian varieties.
Slides

Steve Bradlow: Vortices, principal pairs and moduli spaces in gauge theory
Starting from the abelian vortices on R^{^2} which arose in physical theories such as the Ginzburg-Landau theory of superconductivity, we will introduce several mathematically interesting generalizations and describe the contexts in which they arise. In all cases the objects can be interpreted as minimizers of gauge theoretic energy functionals, as holomorphic bundles with special features, or as zeros of symplectic moment maps. We will describe these correspondences and discuss how they lead to complementary descriptions of generalized vortex moduli spaces. Some key geometric and topological properties of the moduli spaces will be discussed.
Slides

Urs Frauenfelder: The vortex equations and symplectic Tate homology
This is joint work with Peter Albers and Kai Cieliebak. We tensor the action functional of classical mechanics with a vortex functional to get symplectic Tate homology. I will discuss some of its properties.
Notes

Óscar García-Prada: Gravitating vortices and instantons
After explaining the relation between vortices and invariant instantons - as well as more recent non-abelian generalizations -, we go on to study gravitating vortices and their relation to the coupled equations for Kähler metrics and Yang-Mills connections, considered recently in joint work with L. Álvarez-Cónsul and M. García-Fernandez.
Notes

Eduardo González: Wall-crossing and the crepant conjecture
In this talk, I will show that the graph Gromov-Witten potentials of quotients related by wall-crossings of crepant type are equivalent up to a distribution that is almost everywhere zero. This is a version of the crepant transformation conjecture of Li-Ruan, Bryan-Graber, Coates-Ruan etc, in cases where the crepant transformation is obtained by variation of GIT quotient. This is joint work with Chris Woodward.
Notes

Sergei Gukov: From vortex counting to knot homologies
I will explain a connection between (equivariant) cohomology of vortex moduli spaces and homological invariants of knots. This connection comes from a physical interpretation of Khovanov-Rozansky homology (and its colored variants) that was proposed almost ten years ago. In physics, the same system can often be looked at from a number of different angles, giving rise to relations or dualities between seemingly different mathematical objects - a famous illustration of this being the relation between Donaldson and Seiberg-Witten invariants of 4-manifolds. Similarly, the physical framework for knot homologies admits a number of equivalent descriptions which have been actively explored in recent years, including a relation to enumerative invariants of Hilbert schemes and vortex moduli spaces. This lecture is based on a series of papers with T. Dimofte, L. Hollands, A. Schwarz, M. Stošić, J. Walcher, C. Vafa, and others.
Notes

Bumsig Kim: Quasimap invariants and mirror maps
The moduli spaces of stable quasimaps unify various moduli appearing in the study of Gromov-Witten theory. We introduce big I-functions as the quasimap version of J-functions, generalizing Givental's small I-functions of smooth toric complete intersections. The J-functions are the Gromov-Witten counterparts of periods of mirror families. We discuss some advantages of I-functions, in particular an explanation of mirror maps. This is joint work with Ciocan-Fontanine.
Notes

Nick Manton: Vortices on hyperbolic surfaces, and in the dissolving limit
Vortices are usually thought of physically as particles or strings. However, in certain limits they have a rather different interpretation. Vortices on hyperbolic surfaces relate the given metric to a hyperbolic metric on another surface, with the vortex centre as a singularity. Vortices on a small surface almost dissolve, leaving almost no local information. But there is still holonomy on a surface of genus one or more, whose time-dependence yields an electric field. The moduli space metric becomes a Bergman metric in this limit.
Notes

Ignasi Mundet: Hitchin-Kobayashi correspondence on nearly singular conics
We will study the Hitchin-Kobayashi (HK) correspondence on the sequence of conics C_t = {xy=tz²} ⊂ CP², endowed with the restriction of the Fubini-Study metric, as t → 0. Fix for each t an isomorphism CP¹ ≈ C_t, and let v_t be the induced volume form on CP¹. Let P be a C*-principal bundle over CP¹. Let X be a Kähler manifold endowed with an action of C* whose restriction to S¹ is Hamiltonian, and assume that for each t there is a C*-(anti)equivariant holomorphic map φ_t: P → X, such that (P,φ_t) is stable. By the HK correspondence there is a Hermitian metric h_t on P which solves the vortex equation for the volume form v_t. Assuming that the maps φ_t are homotopic, we study the possible limits of (P,φ_t,h_t) as t → 0. The case X = CP¹ with the action of S¹ by rotations and P equal to the trivial bundle will be studied in detail.
Notes

Ignasi Mundet: Hamiltonian Gromov-Witten invariants and nodal curves
I will review some of the geometric aspects of the definition of Hamiltonian Gromov-Witten invariants of compact symplectic manifolds endowed with Hamiltonian actions of the circle, with a domain curve allowed to move in moduli and eventually become singular. I will particularly emphasize the relevant compactness theorem, which will also play a role in my Tuesday talk (on the HK correspondence on nearly singular conics). This is joint work in progress with G. Tian.
Notes

Tim Nguyen: Seiberg-Witten theory and Lagrangian correspondences between vortex moduli spaces
We discuss how the Seiberg-Witten equations on 3-manifolds with boundary yield Lagrangian submanifolds of the configuration spaces at the boundary. This includes the case for 3-manifolds with cylindrical ends, in which we obtain immersed Lagrangians within the vortex moduli spaces at infinity. We then apply a TQFT-like construction due to Donaldson to obtain the Seiberg-Witten invariants of closed 3-manifolds via a composition of these Lagrangian correspondences.
Notes

Andreas Ott: Non-local vortices via holonomy perturbations
Gauged Gromov-Witten invariants are defined by counting solutions of the vortex equations. One of the main problems in this definition is transversality for the boundary strata of the moduli space of vortices. To overcome this problem, I will introduce a holonomy perturbation scheme for the vortex equations. The main idea is to let the almost complex structure depend on the solutions themselves, via a classifying map for the action of the group of gauge transformations on the configuration space for the vortex equations.
Notes

Richard Szabo: Quiver gauge theories and nonabelian vortices
We review equivariant dimensional reduction of vector bundles over spaces of the form M×G/H, where M is a Kähler manifold and G/H is a homogeneous space. Given a G-module, by twisting with a particular bundle over G/H we obtain a G-equivariant unitary bundle with G-equivariant connection over M×G/H. The Donaldson-Uhlenbeck-Yau equations on these spaces reduce to vortex-type equations in a particular quiver gauge theory on M. Standard vortex equations and Seiberg-Witten monopole equations are particular examples. We also explain the modifications obtained through noncommutative deformations of these spaces, and the extension to non-Kähler geometries with torsion on G/H.
Notes

Sushmita Venugopalan: Classification of affine vortices
I present a classification result for affine vortices that generalizes the 1980 result of Jaffe and Taubes. The target X is a Kähler manifold with Hamiltonian action of a compact Lie group K which extends to an action of the complexification G of K. We take X to be either a compact manifold or a complex vector space with a convex linear K-action, and assume that the action of G on the semi-stable locus of X has finite stabilizers. There is a Hitchin-Kobayashi correspondence for K-vortices, and it leads to a classification of K-vortices on the complex plane. The main tool used in the proof of our result is a corresponding result for compact Riemann surfaces with boundary; the main argument uses a gradient flow method.
Notes

Chris Woodward: Gauged Gromov-Witten theory and applications
I will describe the construction, joint with Eduardo González, of gauged Gromov-Witten invariants as integrals over moduli spaces of symplectic vortices in the case of algebraic target, based on earlier symplectic definitions by Mundet, Salamon, Gaio, and Ott. I will then describe some recent applications, such as the computation of quantum cohomology of toric orbifolds in which many of the invariants can be made explicit.
Notes

Fabian Ziltener: A quantum Kirwan map and symplectic vortices
Given a Hamiltonian Lie group action on a symplectic manifold, the Kirwan map is a natural ring homomorphism from the equivariant cohomology of the manifold to the cohomology of the symplectic quotient (i.e., the reduced space in physical terms). I will explain how to construct a quantum deformation of this map, by counting symplectic vortices over the plane. The map relates the equivariant Gromov-Witten theory of the symplectic manifold with the Gromov-Witten theory of the symplectic quotient. Its construction involves a bubbling result and Fredholm theory for vortices over the plane.
Slides