PU(2) monopoles and gauge theoretic invariants

Florida International University, Miami FL

Investigators

Abstract

Thomas G. Leness The goals of this proposal, to be done in collaboration with P. Feehan, are to prove Witten's conjecture relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds, to understand the relation between the conditions of Seiberg-Witten simple type and Kronheimer-Mrowka simple type, and to search for possible topological constraints on these invariants. This work will be carried out by exploring the moduli space of PU(2) monopoles which contains a moduli space of anti-self-dual connections and the moduli spaces of Seiberg-Witten monopoles for certain Spin C structures. This implies that the Donaldson invariant can be expressed as a sum, over these spinc structures, of an expression given by pairing certain cohomology classes with the link of the moduli space of Seiberg-Witten monopoles in the Uhlenbeck compactification of the moduli space of PU(2) monopoles. The first phase of this work is to complete the proof that the pairing of these cohomology classes with the link of the moduli space of Seiberg-Witten monopoles can be expressed in a universal form depending only on the Seiberg-Witten invariant and the homotopy type of the manifold. This work will also yield a proof of the Kotschick-Morgan conjecture on wall-crossing formulas for Donaldson invariants. The second phase of this work is to calculate this universal form in sufficient detail to allow the computation of the explicit relation between the Donaldson and Seiberg-Witten invariants. We intend to do this calculation by using known surgery formulas for both invariants (e.g. blow-up formulas), examples where both invariants are known, and some internal symmetries of the sum mentioned above. It is possible that this relation between the Donaldson and Seiberg-Witten invariants is over-determined and thus will reveal constraints on these invariants given by the topological type of the four-manifold, as was done in earlier work with Kronheimer and Mrowka. An n-dimensional manifold is a topological space that locally looks like n-dimensional Euclidean space. Manifolds are important objects to study because they are ubiquitous: the solution set of k equations in n variables will usually be an (n-k)-dimensional manifold. The main tools for distinguishing between four-dimensional manifolds are the Seiberg-Witten and Donaldson invariants. Thus, understanding the relation between these invariants is crucial to an understanding of four-dimensional topology. In addition, the conjectures relating these invariants arise from Witten's work using quantum field theory. These methods of quantum field theory are not mathematically rigorous, so our mathematically rigorous proof of Witten's conjecture can be viewed as an extremely inexpensive form of experimental physics.