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

TL;DR

This work aims to realize mirror symmetry for Calabi–Yau spaces realized as complete intersections in toric varieties through 2D abelian gauge theories. It develops a Gauged Linear Sigma Model (GLSM) framework and a dual, twisted GLSM approach, yielding explicit moduli mappings (the monomial-divisor mirror map) and an algebraic condition for anomaly-free, nonanomalous $R$-symmetry that fixes the IR fixed point data. The paper provides explicit constructions and maps, and discusses obstructions arising from the lack of global symmetries and nontrivial dynamics, proposing a symmetric, anomaly-aware Lagrangian as a route toward deriving the conjecture. While the proposed duality framework captures essential features and clarifies how mirror symmetry could emerge from abelian duality, it stops short of a complete derivation, highlighting sign and anomaly issues that require further refinement. Overall, the work clarifies the connection between GLSMs and mirror symmetry in this class of models and outlines concrete avenues for achieving a rigorous derivation.

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.