Symplectic instanton homology and Floer theory
Christopher Woodward (Rutgers)
I will describe work with K Wehrheim, S. Mau and C. Manolescu.
Instanton homology is an invariant of homology spheres Y whose
differential counts rank two instantons on Y x R. A conjecture
of Atiyah and Floer relates this with Lagrangian Floer homology
in the moduli space of flat bundles on a Heegard surface X for Y.
I will describe how to prove that the latter is a topological
invariant without reference to gauge theory using "pseudoholomorphic
quilts", and extend the definition of instanton homology to
include three-manifolds with knots and bundles of arbitrary rank.
The resulting theory is a symplectic version of Donaldson-Floer
theory, and satisfies many of the axioms of tft with corners.
Unfortunately the invariants are quite difficult to compute.