Aspects of Geometry and Topology Related to High Energy Theoretical Physics

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DMS-0103877John W. MorganThe work proposed to be carried out under this grant concerns mathematical formulation and proof of some of the mathematical dualities arising from high energy physics. The first duality to be studied concerns the F-theory/heterotic string duality. The mathematical formulation of the duality is that for a certain range of parameters the moduli spaces of classical vacua for various compactifications of F-theory and of heterotic string theory should be isomorphic. We conjecture that the appropriate mathematical statement is that the moduli spaces of heterotic vacua are non-normal infinite-sheeted coverings of the F-theory moduli space associated with maximal parabolic subgroups of the fundamental group of the F-theory moduli space. These maximal parabolic subgroups are the fundamental groups of neighborhoods of certain divisors at infinity. This study involves a detailed analysis of the moduli space of semi-stable principal bundles over elliptic curves as well as the notions of differential cohomology. The second duality is the famous quantum field theory duality between SU(2) Yang-Mills theory and the abelian Seiberg-Witten theory. These dual descriptions are simply the high and low energy limits of a twisted version of supersymmetric Yang-Mills theory. One part of a mathematical understanding of this duality involves giving precise mathematical formulations of the high and low energy limits of the twisted supersymmetric theory and showing how certain correlation functions in this twisted theory converge in the sense of perturbation theory to the usual Donaldson polynomial invariants in the high energy limit and to the Seiberg-Witten invariants in the low energy limit. This part of the story is surely amenable to rigorous mathematical formulation. The more ambitious goal is to find some mathematical formalism which allows for a mathematically rigorous interpelation between these two limits -- a mathematical substitute for the mathematically undefined quantum field theory -- which would allow one to compare the high and low energy limits. The connection between mathematics and high energy theoretical physics has always been a close one. In the last few years it has taken on new aspects. With the study of global issues in quantum field theory and string theory has come a new level of interaction between geometry and topology and these areas of theoretical physics. Belief in the existence of these (to date non-rigorous) physics theories leads physicists to make predictions, i.e., conjectures in mathematics. In a strange way these conjectures play the role that physical predictions used to play. Now the physicists take confirmation that they are on the right track when these mathematical predictions can be verified by rigorous mathematics, which, since it is rigorous, can not make use of the quantum field theory paradigms that we used to make the prediction in the first place. The interest in mathematics of this interplay is that the sorts of mathematical statements that are predicted in this way have turned out to be quite novel producing conjectures unlike anything that had been seen before. Much of the progress the last ten or fifteen years in large areas of geometry and topology traces back to this source. One of the most fruitful sources of mathematical conjectures has been various dualities in physics. These occur when there is more than one classical mathematical limit of a quantum field theory or string theory. The mathematical objects that arise in the description of the classical limits then are related because they are limits of a single quantum field theory or string theory. The mathematical problem is always the same -- define rigorously what the relationship is and then establish it mathematically. The work proposed here is exactly along these lines for two such dualities.