Integrable N = 2 Landau-Ginzburg Theories from Quotients of Fusion Rings

TL;DR

This paper constructs new integrable N=2 Landau-Ginzburg theories from quotients of fusion rings, revealing connections between fusion rings, superpotentials, and quantum integrability, and providing explicit correlation function calculations.

Contribution

It introduces a novel class of superpotentials based on $SO(N)_K$ fusion rings, extending the framework of integrable N=2 theories and analyzing their topological properties.

Findings

Superpotentials are natural quotients of $SO(N)_K$ fusion rings.

Correlation functions are expressed as sums of Verlinde dimensions.

Related theories $SO(2n+1)_{2k+1}$ and $SO(2k+1)_{2n+1}$ are isomorphic.

Abstract

The discovery of integrable $N=2$ supersymmetric Landau-Ginzburg theories whose chiral rings are fusion rings suggests a close connection between fusion rings, the related Landau-Ginzburg superpotentials, and $N=2$ quantum integrability. We examine this connection by finding the natural $SO(N)_K$ analogue of the construction that produced the superpotentials with $Sp(N)_K$ and $SU(N)_K$ fusion rings as chiral rings. The chiral rings of the new superpotentials are not directly the fusion rings of any conformal field theory, although they are natural quotients of the tensor subring of the $SO(N)_K$ fusion ring. The new superpotentials yield solvable (twisted $N=2$) topological field theories. We obtain the integer-valued correlation functions as sums of $SO(N)_K$ Verlinde dimensions by expressing the correlators as fusion residues. The $SO(2n+1)_{2k+1}$ and $SO(2k+1)_{2n+1}$ related topological Landau-Ginzburg theories are isomorphic, despite being defined via quite different superpotentials.