The thesis of Christopher Beasley, entitled Three Instanton Computations in Gauge Theory and String Theory, has been placed on deposit.

Any member of the University wishing to read the thesis may do so. Any objections should be submitted to me in writing. The principal advisor for this work was Professor Edward Witten.

ABSTRACT

We employ a variety of ideas from geometry and topology to perform three new instanton computations in gauge theory and string theory.

First, we consider supersymmetric QCD with gauge group SU(N_c) and with N_f flavors. In this theory, it is well known that instantons generate a superpotential if N_f = N_c - 1 and deform the moduli space of supersymmetric vacua if N_f = N_c. We extend these results to supersymmetric QCD with N_f > N_c flavors, for which we show that instantons generate a hierarchy of new, multi-fermion F-terms in the effective action.

Second, we revisit the question of which Calabi-Yau compactifications of the heterotic string are stable under worldsheet instanton corrections to the effective space-time superpotential. For instance, compactifications described by (0,2) linear sigma models are believed to be stable, suggesting a remarkable cancellation among the instanton effects in these theories. We show that this cancellation follows directly from a residue theorem, whose proof relies only upon the right-moving worldsheet supersymmetries and suitable compactness properties of the (0,2) linear sigma model. We also extend this residue theorem to a new class of ``half-linear'' sigma models. Using these half-linear models, we show that heterotic compactifications on the quintic hypersurface in CP4 for which the gauge bundle pulls back from a bundle on CP4 are stable.

Third, we study Chern-Simons gauge theory on a Seifert manifold M (the total space of a nontrivial circle bundle over a Riemann surface). When M is a Seifert manifold, Lawrence and Rozansky have shown from the exact solution of Chern-Simons theory that the partition function has a remarkably simple structure and can be rewritten entirely as a sum of local ``instanton'' contributions from the flat connections on M. We explain how this empirical fact follows from the technique of non-abelian localization as applied to the Chern-Simons path integral. In the process, we show that the partition function of Chern-Simons theory on M admits a topological interpretation in terms of the equivariant cohomology of the moduli space of flat connections on M.

Daniel Marlow
Chair, Dept. of Physics