Surface Operators and Knot Homologies

TL;DR

This paper explores how four-dimensional topological gauge theories with surface operators can be used to construct and understand homological knot invariants, including categorifications of classical knot polynomials and invariants.

Contribution

It introduces a framework linking surface operators in gauge theories to the braid group actions on branes, leading to new categorifications of knot invariants.

Findings

Categorification of the Alexander polynomial

Construction of knot signature invariants

Analogues of the Casson invariant

Abstract

Topological gauge theories in four dimensions which admit surface operators provide a natural framework for realizing homological knot invariants. Every such theory leads to an action of the braid group on branes on the corresponding moduli space. This action plays a key role in the construction of homological knot invariants. We illustrate the general construction with examples based on surface operators in N=2 and N=4 twisted gauge theories which lead to a categorification of the Alexander polynomial, the equivariant knot signature, and certain analogs of the Casson invariant. This paper is based on a lecture delivered at the International Congress on Mathematical Physics 2006, Rio de Janeiro, and at the RTN Workshop 2006, Napoli.