2d Sigma Models on Non-compact Calabi-Yau and ${\mathcal N}=2$ Liouville Theory

TL;DR

This paper explores the relationship between certain 2D supersymmetric gauge theories with non-compact Calabi-Yau target spaces and ${ m N}=2$ Liouville theory, establishing an exact correspondence through large-$N$ analysis and moduli matching.

Contribution

It demonstrates an exact equivalence between Coulomb branches of specific gauge linear sigma models and ${ m N}=2$ Liouville theory, extending previous large-$N$ approximations.

Findings

Coulomb branch described by ${ m N}=2$ Liouville theory at strong coupling

Matching of complex moduli with marginal deformations in Liouville theory

Support for the proposed exact equivalence between gauge models and Liouville theory

Abstract

We consider a class of two dimensional conformal ${\mathcal N}=2$ supersymmetric $U(1)$ gauge linear sigma models with $N$ fields of charges $+1$ and $N$ fields of charges $-1$, whose Higgs branches are non-compact toric Calabi-Yau manifolds of complex dimension $2N-1$. We show, starting from large-$N$ approximation, that the Coulomb branch of these models, which opens up at strong coupling, is described by ${\mathcal N}=2$ Liouville theory and then extrapolate it to exact equivalence demanding the central charge of the Liouville theory to be $\hat{c}=2N-1$. Next we concentrate on mostly physically attractive $N=2$ and $N \geq 3$ cases and find there a perfect agreement of the set of complex moduli on the Calabi-Yau side with the marginal deformations in ${\mathcal N}=2$ Liouville theory, supporting proposed exact equivalence.