The purpose of this lecture course is to explain the recent advances in the instanton calculus. The major event dates back to 1994 when N.Seiberg and E.Witten have presented a description of the low-energy effective action of the N=2 supersymmetric Yang-Mills theory. One of the by-products of their work was the discovery of new invariants of four-manifolds, the so-called Seiberg-Witten invariants, which are simpler to handle then Donaldson invariants, and which are at least as powerful. Despite this beautiful mathematical application the very solution of Seiberg and Witten was never given a mathematical meaning. The proposed lecture course will fulfill this task. We plan to introduce the concept of instantons in the four dimensional gauge theories and in two dimensional sigma models. We then give some general discussion of their moduli spaces, their compactifications, resolutions of singularities. We then sketch the conventional applications of these moduli spaces: Donaldson theory of four-manifold invariants, and Gromov-Witten theory of symplectic manifolds. After that we recall the notion of mirror symmetry and discuss a few examples. We then proceed with the four-dimensional analogue of mirror symmetry. For this we need a proper definition of the intersection theory on the moduli space of gauge theory instantons on R4. We give this definition and consider a few examples. Among other things we explain the relation between the resolution of singularities of this moduli space and the moduli space of instantons on the noncommutative R4 . At this point we are ready to formulate the four-dimensional mirror principle. We express the generating function of the (equivariant) intersection numbers in terms of a certain family of curves (Seiberg-Witten curves). It turns out to coincide with the tau-function of a integrable hierarchy, generalizing Toda hierarchy. In the simplest case of the rank one theory it can be identified with the Toda tau-function. The next topic of our course will be the identification of these tau-functions with the partition functions of topological strings on some simple target spaces, like CP1 or CP1 times a Landau-Ginzburg model. This identification, being an statement from experimental mathematics requires an explanation from theoretical physics, it involves notions from the modern nonperturative string theory and could a subject of another lecture course.

Zeit:       Dienstag 10-12
Ort:        HG G 43 (Hermann-Weyl-Zimmer)
Beginn:   8. April
 

M. Struwe