# Vortex Seminar

**Date:** every Tuesday, 11:00-12:00 (unless stated otherwise)

**Venue:** HIM lecture hall, Poppelsdorfer Allee 45

**Organizer: **Nuno Romão

Tuesday, September 4, 11:00-12:00

Nuno Romao (MPIM Bonn): The vortex equations

Abstract: In this talk, I shall give a preview of the upcoming activities of our group "Geometry of Gauged Vortices" at the JHTP, as well as an introduction to the vortex equations in fibre bundles over a Riemann surface.

Tuesday, September 11, 11:00-12:00

Nick Manton (University of Cambridge): Metrics from vortices

Abstract: Abelian vortices on a Riemann surface Σ (with metric) give rise to smooth Kähler metrics on the symmetric powers of Σ. This is because the N-vortex moduli space is the N-th symmetric power of Σ. The relation between these new metrics and the underlying metric on Σ is interesting and non-trivial, because vortices are not ideal points, and are sensitive to the local geometry and the topology of Σ.

Tuesday, September 18, 11:00-12:00

Tudor Dimofte (IAS Princeton): Supersymmetric vortices, 3d BPS states, and 3-manifolds

Abstract: I will review the relation between vortex partition functions for supersymmetric 2d theories and BPS indices in three dimensions, and overview some of the more interesting mathematical and physical properties enjoyed by the latter. In particular, BPS indices provide a building block for quantum invariants of (auxiliary) knots and three-manifolds, and suggest a new approach to categorification of these objects.

Tuesday, September 25: no seminar

Tuesday, October 2, 12:15-13:15

Martin Speight (University of Leeds): Vortices and the L^{^2} volume of spaces of holomorphic maps

Abstract: I will review how Baptista uses his recent work on the L^{^2} geometry of vortex moduli spaces in gauged linear sigma-models to extract a wide-ranging conjecture on the L^{^2} volume of various spaces of holomorphic maps. The latter spaces are of independent interest as soliton moduli spaces in ungauged nonlinear sigma-models, and their volumes have been computed independently in some exceptionally symmetric special cases, providing a nontrivial check on Baptista's conjecture.

Tuesday, October 9, 12:15-13:15

Amihay Hanany (Imperial College): Brane configurations and the moduli space of vortices

Abstract: This is a pedagogical lecture which explains how, using brane configurations in string theory, one can construct a quiver gauge theory that characterizes moduli spaces of vortices.

Tuesday, October 16, 11:00-12:00

Markus Szymik (University of Copenhagen): The vortex equations and stable homotopy theory

Abstract: The purpose of this talk is to explain how the vortex equations on Riemann surfaces give rise to stable homotopy classes, and to discuss what is and what should be known about these vortex classes.

Tuesday, October 23, 11:00-12:00

Joao Baptista (IST Lisbon): Abelian vortices on Kähler manifolds

Abstract: I will consider abelian vortex equations defined on compact Kähler manifolds of arbitrary dimension. When the manifold is simply connected or an abelian variety, I shall give an explicit description of the moduli spaces and explain how to compute the Kähler class and volume of the natural L^{^2}-metric. I will also recall how vortex moduli spaces relate to spaces of holomorphic maps from Kähler manifolds to toric targets.

Tuesday, October 30, 11:00-12:00

Andras Szenes (University of Geneva): Cohomology of Higgs moduli

Abstract: The topology of the moduli spaces of Higgs bundles on a Riemann surface has been in the focus of a lot of recent research. In this talk, I will give a conjectural description of the ring structure using equivariant integration. This is joint work with Tamas Hausel.

Tuesday, November 6, 11:00-12:00

Tim Perutz (UT Austin): Vortices, Seiberg-Witten monopoles and Heegaard Floer theory

Abstract: I will give an overview of the roles, known and expected, of Lagrangian submanifolds in vortex moduli spaces of Riemannian surfaces in the construction of extended TQFT-type structures in Seiberg-Witten theory and its cousin, Heegaard Floer theory. Part of the talk will discuss a theory, under construction by the speaker with Yanki Lekili, of an extension of the full Heegaard Floer 3-manifold invariants from closed 3-manifolds to 3-manifolds with boundary, via Fukaya categories of symmetric products of Riemann surfaces.

Tuesday, November 13, 17:00-18:00

Eduardo Gonzalez (UMass Boston): Algebraic approach to vortex moduli

Abstract: After a brief introduction to algebraic stacks, I will give an overview of the algebraic version of the moduli space of vortices: a stack of gauged maps with target X over C, where X is a projective variety where a reductive group acts, and C a projective curve where the vortices live. I will then introduce the notion of Mundet semi-stability for gauged maps, and define gauged Gromov-Witten invariants using this construction.

Tuesday, November 20, 11:00-12:00

Bumsig Kim (KIAS Seoul): Quasimap theory

Abstract: The moduli spaces of stable quasimaps unify various moduli appearing in the study of Gromov-Witten theory. This talk will gently introduce the notion of stable quasimaps, examples, and its applications. It is based on joint work with Ionut Ciocan-Fontanine, Hwayoung Lee and Davesh Maulik.

Tuesday, November 27: Workshop "Geometry of the Vortex Equations"

Tuesday, December 4, 11:00-12:00

Sushmita Venugopalan (TIFR Mumbai): Yang-Mills heat flow on gauged holomorphic maps

Abstract: I consider the space of gauged holomorphic maps from a Riemann

surface to a compact Kähler manifold, and study the gradient flow of a

Yang-Mills type functional, whose zeros are vortices. The flow exists for

all time. When the Riemann surface has boundary, the flow lines converge

to a vortex leading to a Hitchin-Kobayashi correspondence.

Tuesday, December 11, 10:00-11:00 + 11:15-12:00

Wednesday, December 12, 14:00-15:00 + 15:15-16:00

Fabian Ziltener (KIAS Seoul): Minicourse "Vortices and quantum Kirwan maps"

Abstract: 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. By counting symplectic vortices over the plane, one obtains a quantum deformation of this homomorphism. The map relates the equivariant Gromov-Witten theory of the symplectic manifold with the Gromov-Witten theory of the symplectic quotient. On Tuesday, I will give some geometric background on the Kirwan map, vortices and Gromov-Witten theory, and motivate the construction of a quantization of the Kirwan map. On Wednesday, I will discuss a relevant bubbling result, Fredholm theory, and decay at infinity for vortices over the plane.