Supersymmetric Field Theory and Four-Manifold Invariants

I will give a short review of the physics-inspired approach to Donaldson theory based on topologically-twisted four-dimensional N=2 supersymmetric quantum field theory (SQFT) as propounded by Witten in 1988, and then describe some of my recent and ongoing work. One theme in particular is a generalization of Donaldson-Witten theory to describe invariants of smooth families of smooth, closed, oriented Riemannian 4-manifolds X using methods of SQFT and supergravity, and leading to physical derivations of relevant models of equivariant cohomology. Family Donaldson invariants are cohomology classes of the classifying space of orientation-preserving diffeomorphisms, i.e., elements of H*(BDiff^+(X)), and we propose path integral formulations of their cocycle representatives. Several interesting open questions arise concerning the observables and interpretation of the invariants, calling for a renewed math-physics dialog. I hope to (re?)ignite some interest in these issues through this talk. Time permitting, I will touch upon the second theme, a new perspective of topological twisting using the notion of ‘transfer of structure group’ associated with a continuous homomorphism between topological groups. This approach accounts for the global topology of the structure group of a quantum field theory, rather than just its simply connected cover, and leads to a proposal for twisting more general four-dimensional N=2 SQFTs. The talk will be based on collaborations with G. W. Moore, M. Roček, and R. K. Singh.