On the Landau-Ginzburg description of $(A_1^{(1)})^{\oplus N}$ invariants

TL;DR

This paper explores Landau-Ginzburg models as descriptions of non-diagonal modular invariants in tensor products of minimal superconformal models, identifying cases where such descriptions succeed or fail.

Contribution

It provides a Landau-Ginzburg interpretation for automorphism invariants and demonstrates limitations for certain exceptional cases.

Findings

Automorphism invariants are described as Landau-Ginzburg orbifolds.

The Landau-Ginzburg approach reproduces chiral rings and spectra of specific models.

Some exceptional invariants cannot be realized as orbifolds of Landau-Ginzburg models.

Abstract

We search for a \Lg\ interpretation of non-diagonal modular invariants of tensor products of minimal $n=2$ superconformal models, looking in particular at automorphism invariants and at some exceptional cases. For the former we find a simple description as \lgo s, which reproduce the correct chiral rings as well as the spectra of various Gepner--type models and orbifolds thereof. On the other hand, we are able to prove for one of the exceptional cases that this conformal field theory cannot be described by an orbifold of a \Lg\ model with respect to a manifest linear symmetry of its potential.