Low-Dimensional Manifolds and Gauge Theory

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9704507 Morgan The study to be carried out in this project is two-fold. Both aspects of the study lie at the intersection of geometry and topology, on the one hand, and theoretical physics (quantum field theory and string theory) on the other. One aspect concerns gauge-theoretic invariants of four-dimensional manifolds, in particular, the Donaldson polynomial invariants and the Seiberg-Witten invariants. Here the aim is to understand the relation of these invariants to more classical geometric descriptions of a four-manifold, for example, a handlebody description. The other aspect is the study of holomorphic vector bundles over elliptically fibered manifolds and the relation of these to families of K3 surfaces that arise in F-theory in mathematical physics. Here the aim is to establish mathematical results for both sides of the putative F-theory duality and then to check that the predictions of this duality do indeed hold. There has been a resurgence in recent years of the interactions between the areas of geometry and topology in mathematics and the areas of quantum field theory and string theory in theoretical physics. This interaction has proved very fruitful for both disciplines -- the physics brings new methods to the study of low dimensional spaces, and the mathematics provides the framework in which the studies in theoretical high-energy physics are done. This project will investigate in more detail two of these interactions. One is the study of four-dimensional spaces, using tools provided by theoretical physics. These tools have yielded much insight in the last few years, and there is much more information about four-dimensional spaces that should be accessible using these tools. The other aspect of the project involves the mathematics of a new physical theory. This theory makes very precise predictions about two kinds of mathematical objects that, on the face of it, bear little relation to each other. On the other hand, these mathemat ical objects can be studied in their own right by purely mathematical tools that have been developed independently of physics. The results of the mathematical analyses of these objects can be used to check the predictions of the physical theory in much the same way as physical experiments are used. ***