Abstracts for Vortex Seminar

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.

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.

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.

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.