Rationality of Lorentzian Lattice CFTs And The Associated Modular Tensor Category

TL;DR

This paper classifies modules of rational Lorentzian lattice vertex operator algebras, constructs their associated modular tensor categories, and provides explicit examples linking to Kac-Moody algebras.

Contribution

It provides a complete classification of irreducible modules and the associated modular tensor categories for rational Lorentzian lattice VOAs, including explicit modular data and fusion rules.

Findings

Classified irreducible modules via lattice quotient classes.

Constructed explicit modular tensor categories for these VOAs.

Connected the lattice VOAs to Kac-Moody algebra MTCs.

Abstract

We classify the irreducible modules of a rational Lorentzian lattice vertex operator algebra (LLVOA) based on an even, self-dual Lorentzian lattice $\Lambda\subset\mathbb{R}^{m,n}$ of signature $(m,n)$. We show that the set of isomorphism classes of irreducible modules of the LLVOA are in one-to-one correspondence with the equivalence classes $\Lambda_0^\circ/\Lambda_0$ for a certain subset $\Lambda_0^\circ\subset\mathbb{R}^{m,n}$ and a full rank sublattice $\Lambda_0\subset\Lambda$. We also classify the intertwining operators between the modules and calculate the fusion rules. We then describe the standard construction of modular tensor category (MTC) associated to rational LLCFTs. We explicitly construct the modular data and braiding and fusing matrices for the MTC. As a concrete example, we show that the LLCFT based on a certain even, self-dual Lorentzian lattice of signature $(m, n)$, with $m$ even, realizes the $D(m \bmod 8)$ level 1 Kac-Moody MTC.