Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary

TL;DR

This paper derives an exact formula for the hemisphere partition function in 2D (2,2) supersymmetric gauge theories, revealing insights into D-branes, phase boundaries, and mirror symmetry.

Contribution

It provides a novel exact Mellin-Barnes integral formula for the boundary central charge in 2D supersymmetric theories, connecting phase boundary phenomena and mirror symmetry.

Findings

Derived a general exact formula for D-brane central charge on the hemisphere.

Identified the grade restriction rule related to phase boundary convergence.

Expressed the partition function in various phases, including the large volume limit.

Abstract

We compute the partition function on the hemisphere of a class of two-dimensional (2,2) supersymmetric field theories including gauged linear sigma models. The result provides a general exact formula for the central charge of the D-brane placed at the boundary. It takes the form of Mellin-Barnes integral and the question of its convergence leads to the grade restriction rule concerning branes near the phase boundaries. We find expressions in various phases including the large volume formula in which a characteristic class called the Gamma class shows up. The two sphere partition function factorizes into two hemispheres glued by inverse to the annulus. The result can also be written in a form familiar in mirror symmetry, and suggests a way to find explicit mirror correspondence between branes.