Aspects of Non-Abelian Gauge Dynamics in Two-Dimensional N=(2,2) Theories

TL;DR

This paper investigates non-Abelian gauge theories with N=(2,2) supersymmetry in two dimensions, computing indices, analyzing IR fixed points, and exploring phase structures related to Calabi-Yau manifolds and mirror symmetry.

Contribution

It provides new combinatoric formulas for the Witten index, criteria for non-singular IR fixed points, and insights into phase transitions and mirror symmetry in non-Abelian gauge theories.

Findings

Witten index computed via combinatorics for SU(k) SQCD.

Identified multiple singularities in Kahler moduli space.

Proved a mathematical conjecture relating to Calabi-Yau phase transitions.

Abstract

We study various aspects of N=(2,2) supersymmetric non-Abelian gauge theories in two dimensions, with applications to string vacua. We compute the Witten index of SU(k) SQCD with N>0 flavors with twisted masses; the result is presented as the solution to a simple combinatoric problem. We further claim that the infra-red fixed point of SU(k) gauge theory with N massless flavors is non-singular if (k,N) passes a related combinatoric criterion. These results are applied to the study of a class of U(k) linear sigma models which, in one phase, reduce to sigma models on Calabi-Yau manifolds in Grassmannians. We show that there are multiple singularities in the middle of the one-dimensional Kahler moduli space, in contrast to the Abelian models. This result precisely matches the complex structure singularities of the proposed mirrors. In one specific example, we study the physics in the other phase of the Kahler moduli space and find that it reduces to a sigma model for a second Calabi-Yau manifold which is not birationally equivalent to the first. This proves a mathematical conjecture of Rodland.