Surface Operators and Knot Homologies

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 oil branes oil 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.