Level-Rank Duality for Vertex Operator Algebras of types B and D

TL;DR

This paper explores a level-rank duality for vertex operator algebras of types B and D, revealing a correspondence between commutant VOAs in tensor products and fixed point subalgebras, advancing understanding of algebraic symmetries.

Contribution

It establishes a new level-rank duality for VOAs of types B and D, linking commutant VOAs with fixed point subalgebras via abelian group actions.

Findings

Commutant VOAs can be realized as fixed point subalgebras of dual VOAs.

The duality involves simple current extensions and abelian group symmetries.

Results extend the understanding of level-rank duality in Lie algebra-related VOAs.

Abstract

For the simple Lie algebra $ \frak{so}_m$, we study the commutant vertex operator algebra of $ L_{\hat{\frak{so}}_{m}}(n,0)$ in the $n$-fold tensor product $ L_{\hat{\frak{so}}_{m}}(1,0)^{\otimes n}$. It turns out that this commutant vertex operator algebra can be realized as a fixed point subalgebra of $L_{\hat{\frak{so}}_{n}}(m,0)$ (or its simple current extension) associated with a certain abelian group. This result may be viewed as a version of level-rank duality.