LAUSR

solutions to the Seiberg–Witten equations is much simpler than instanton moduli space (it is smooth and compact), it is easier to work with the Seiberg–Witten invariants. Happily, the Seiberg–Witten invariants turn out to be just as powerful, if not more powerful, than the Donaldson invariants. In 1988, following some important questions and suggestions by Michael Atiyah, Edward Witten gave an interpretation of the Donaldson invariants as correlation functions of certain special operators in a certain quantum field theory known as “topologically twisted N = 2 supersymmetric Yang–Mills theory.” However, the path integral formulation of the invariants could not be used as a practical means of evaluating the Donaldson invariants until aspects of the low-energy dynamics of that quantum field theory were understood better. This deeper understanding was achieved in 1994 when Nathan Seiberg, together with Witten, understood the nature of the groundstates of the theory in detail. After that breakthrough there was rapid progress, culminating in the formulation of the Seiberg– Witten invariants. In fact, the Donaldson invariants can be expressed in terms of the Seiberg–Witten invariants. The relation was conjectured by Witten, at least for the case when X has b2 + > 1. The relation was derived using physical methods (path integrals and effective field theory) by G. Moore and Witten in 1997. A crucial step in that derivation involves the so-called u-plane integral. This is a very subtle, finite-dimensional integral over the complex plane, closely related to the theta-lifts used in the theory of automorphic forms. The main part of the talk will focus on some progress achieved in the past few years continuing the approach to four-manifold invariants using quantum field theory. Time permitting, at least three results will be explained. First, the topological twisting procedure of Witten can be extended to arbitrary quantum field theories with N = 2 supersymmetry. Around 2008 many new supersymmetric N = 2 field theories were discovered. Many of the new theories have the intriguing property that there is no known Lagrangian description (and probably no Lagrangian description exists). It has been a long-standing and natural question to ask if these new theories lead to new four-manifold invariants. Work with I. Nidaiev shows that, at least in the simplest of these non-Lagrangian theories, the answer is negative. Nevertheless, in the process of answering the question we still learn nontrivial facts about the Seiberg–Witten invariants. (An example is the superconformal simple type property of the Seiberg–Witten invariants.)