AAG Seminar: Knots and holomorphic vector bundles

Abstract: For any N, Yang-Mills gauge theory with the gauge group SU(N) can be used to define topological invariants of 3- and 4-manifolds, which essentially determine a TQFT. In particular, the invariant associated to a closed 3-manifold has the form of a vector space, and associated to closed 4-manifolds one obtains numerical invariants. Algebraic geometry can be used to describe these invariants in many important special cases. For instance, for the 3-manifold Y given by the product of a circle and an algebraic curve C, the corresponding invariant is the (quantum) cohomology of a Fano variety, which is defined in terms of holomorphic vector bundles on C. I will discuss some structural results/conjectures about the quantum cohomology of such varieties and their implications in knot theory. This talk is based on a joint work with Nobuo Iida and Chris Scaduto.