Towards Mirror Symmetry as Duality for Two-Dimensional Abelian Gauge Theories

TL;DR

This paper explores mirror symmetry in two-dimensional abelian gauge theories with Calabi--Yau targets, proposing a duality framework that relates different theories and could lead to a derivation of mirror symmetry.

Contribution

It formulates mirror symmetry as a duality in abelian gauge theories for Calabi--Yau spaces, providing explicit mappings and addressing challenges in deriving the symmetry.

Findings

Duality relates inequivalent theories with isomorphic low-energy limits

Proposes a symmetric Lagrangian to overcome duality obstacles

Fails to fully derive the mirror symmetry conjecture

Abstract

Superconformal sigma models with Calabi--Yau target spaces described as complete intersection subvarieties in toric varieties can be obtained as the low-energy limit of certain abelian gauge theories in two dimensions. We formulate mirror symmetry for this class of Calabi--Yau spaces as a duality in the abelian gauge theory, giving the explicit mapping relating the two Lagrangians. The duality relates inequivalent theories which lead to isomorphic theories in the low-energy limit. This formulation suggests that mirror symmetry could be derived using abelian duality. The application of duality in this context is complicated by the presence of nontrivial dynamics and the absence of a global symmetry. We propose a way to overcome these obstacles, leading to a more symmetric Lagrangian. The argument, however, fails to produce a derivation of the conjecture.