Symplectic Vortex Equation and Adiabatic Limits

Abstract

This thesis consists of four parts, on four separate topics in the study of the symplectic vortex equation and their adiabatic limits.

In the first part, we constructed a compactication of the moduli space of twisted holomorphic maps with Lagrangian boundary condition. It generalizes the compactness theorem of Mundet-Tian in the case of closed Riemann surfaces to the case of bordered Riemann surfaces, and it is the first step in developing the open-string analogue of Mundet-Tian's program.

In the second part, we studied the Morse theory of Lagrange multipliers, which is based on a joint work with Stephen Schecter. We also considered two adiabatic limits by varying a real parameter in this theory, which result in two different homology group. Via the homotopy provided by the variation of we prove that the two homology groups are isomorphic.

In the third part, we considered a U(1)-gauged linear σ-model and its low-energy adiabatic limits. Via adiabatic limits, we managed to classify all affine vortices with target the complex vector space and diagonal

U(1)-action, and we identify their moduli spaces, which generalizes Taubes' result. This also gives a precise meaning of the "point-like instantons" described by Witten. We also computed the associated quantum Kirwan map by compactifying the moduli space.

In the fourth part, we introduce a new type of equations. It is a generalization of Witten's equation for a quasi-homogeneous polynomial W, by coupling a gauge field. The purpose of this generalization is to realize the geometric Landau-Ginzburg/Calabi-Yau correspondence predicted in string theory. This part is based on a work in progress joint with Gang Tian.