| 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( Nc) and with Nf flavors. In this theory, it is well known that instantons generate a superpotential if Nf = Nc - 1 and deform the moduli space of supersymmetric vacua if Nf = Nc. We extend these results to supersymmetric QCD with Nf > Nc 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. |
