# Fusion Rules for the Lattice Vertex Operator Algebra $V_L$

## TL;DR

This paper determines the fusion rules for certain modules of the lattice vertex operator algebra $V_L$, extending known results to include twisted modules and their fusion products.

## Contribution

It explicitly computes the fusion products involving twisted modules of $V_L$, which were previously unknown.

## Key findings

- Fusion product of untwisted modules is well-known.
- Derived fusion rules involving twisted modules $V_L^{T_{ heta}}$.
- Extended the understanding of module interactions in lattice VOAs.

## Abstract

For a positive-definite, even, integral lattice $L$, the lattice vertex operator algebra $V_L$ is known to be rational and $C_2$-cofinite, and thus the fusion products of its modules always exist. The fusion product of two untwisted irreducible $V_L$-modules is well-known, namely $V_{L+\lambda} \boxtimes_{V_L} V_{L+\mu} = V_{L + \lambda + \mu}$. In this paper, we determine the other two fusion products: $V_{L+\lambda} \boxtimes_{V_L} V_L^{T_{\chi}}$ and $V_L^{T_{\chi_1}} \boxtimes_{V_L} V_L^{T_{\chi_2}}$.