An isomorphism between the fusion algebras of $V_L^+$ and type $D^{(1)}$ level 2

Abstract

The fusion algebra of the vertex operator algebra $V_L^+$ for a rank 1 even lattice $L$ is explicitly shown to be isomorphic to the fusion algebra of the Kac-Moody algebra of type $D^{(1)}$ at level 2 in almost all cases.

Shortly after posting this article, C. Schweigert pointed out to us that our result was already known in physics literature. The paper is thus mainly useful for looking up the explicit fusion algebra and S-matrix of type D level 2