Topological quantum field theory and four-manifolds

TL;DR

This paper reviews recent advances in four-manifold invariants derived from topological quantum field theory, covering correlation functions, lower-dimensional relations, and theories with critical behavior that constrain Seiberg-Witten invariants.

Contribution

It presents new explicit calculations of Donaldson invariants for complex manifolds, explores dimensional reductions linking to three-manifold topology, and introduces theories with critical points that impose constraints on invariants.

Findings

Explicit Donaldson invariants for non-simply connected manifolds

Relations between four-dimensional theories and three-manifold topology

Constraints on Seiberg-Witten invariants from critical theories

Abstract

I review some recent results on four-manifold invariants which have been obtained in the context of topological quantum field theory. I focus on three different aspects: (a) the computation of correlation functions, which give explicit results for the Donaldson invariants of non-simply connected manifolds, and for generalizations of these invariants to the gauge group SU(N); (b) compactifications to lower dimensions, and relations with three-manifold topology and with intersection theory on the moduli space of flat connections on Riemann surfaces; (c) four-dimensional theories with critical behavior, which give some remarkable constraints on Seiberg-Witten invariants and new results on the geography of four-manifolds.