SIGMA MODELS IN STRING AND M-THEORY

Abstract

In this thesis, we explore the topic of applying sigma models to understand aspects of string theory and M-theory.

Firstly, we describe supersymmetric A-branes and B-branes in open N = (2,2) dynamically gauged nonlinear sigma models (GNLSM), placing emphasis on toric manifold target spaces. For a subset of toric manifolds, these equivariant branes have a mirror description as branes in gauged Landau-Ginzburg models with neutral matter. This leads us to an equivariant version of the mathematical duality known as homological mirror symmetry. We then study correlation functions in the topological A-twisted version of the GNLSM, and identify their values with open Hamiltonian Gromov-Witten invariants. Supersymmetry breaking can occur in the A-twisted GNLSM due to nonperturbative open symplectic vortices, and we canonically BRST quantize the mirror theory to analyze this phenomenon.

Secondly, we study the ground states and left-excited states of the A_{k-1} N = (2,0) little string theory. Via a theorem by Atiyah, these sectors can be captured by a supersymmetric nonlinear sigma model on CP^1 with target space the based loop group of SU(k). The ground states, described by L^2-cohomology classes, form modules over an affine Lie algebra, while the left-excited states, described by chiral differential operators, form modules over a toroidal Lie algebra. This provides us with a derivation and generalization of a mathematical duality conjectured by Braverman and Finkelberg. We also apply our results to analyze the 1/2 and 1/4 BPS sectors of the M5-brane worldvolume theory.