General Mirror Pairs for Gauged Linear Sigma Models

TL;DR

This paper investigates conditions under which abelian gauged linear sigma-models exhibit nontrivial superconformal IR behavior, proposing combinatorial criteria for nonsingularity and exploring their implications for mirror symmetry and extremal transitions.

Contribution

It introduces combinatorial conditions for nonsingularity in GLSMs, analyzes their mirror symmetry properties, and connects these to known dualities like Berglund-Hubsch and Vafa-Witten.

Findings

Models with reflexive combinatorial data are generally nonsingular.

Mirror of a nonsingular GLSM is also determined by reflexive data, but exceptions exist.

A weaker, conjectured condition may suffice for nonsingularity and mirror symmetry.

Abstract

We carefully analyze the conditions for an abelian gauged linear sigma-model to exhibit nontrivial IR behavior described by a nonsingular superconformal field theory determining a superstring vacuum. This is done without reference to a geometric phase, by associating singular behavior to a noncompact space of (semi-)classical vacua. We find that models determined by reflexive combinatorial data are nonsingular for generic values of their parameters. This condition has the pleasant feature that the mirror of a nonsingular gauged linear sigma-model is another such model, but it is clearly too strong and we provide an example of a non-reflexive mirror pair. We discuss a weaker condition inspired by considering extremal transitions, which is also mirror symmetric and which we conjecture to be sufficient. We apply these ideas to extremal transitions and to understanding the way in which both Berglund-Hubsch mirror symmetry and the Vafa-Witten mirror orbifold with discrete torsion can be seen as special cases of the general combinatorial duality of GLSMs. In the former case we encounter an example showing that our weaker condition is still not necessary.