Fusion rules and complete reducibility of certain modules for affine Lie algebras

TL;DR

This paper introduces a new method using fusion rules for affine vertex operator algebras to analyze module reducibility and branching rules at negative levels, providing new proofs and decompositions for affine Lie algebra modules.

Contribution

It develops a novel approach leveraging fusion rules to establish complete reducibility and explicit module decompositions at negative levels for affine Lie algebras of types A, C, E, and F.

Findings

Proved closure of certain modules under fusion for affine type A.

Established complete reducibility of level -1 modules for types A and C.

Decomposed modules for types E6 and F4 at negative levels.

Abstract

We develop a new method for obtaining branching rules for affine Kac-Moody Lie algebras at negative integer levels. This method uses fusion rules for vertex operator algebras of affine type. We prove that an infinite family of ordinary modules for affine vertex algebra of type A investigated in Adamovi\'c and O. Per\v{s}e (2008) is closed under fusion. Then we apply these fusion rules on explicit bosonic realization of level -1 modules for the affine Lie algebra of type $A_{\ell-1}^{(1)}$, obtain a new proof of complete reducibility for these representations, and the corresponding decomposition for $\ell \ge 3$. We also obtain the complete reducibility of the associated level -1 modules for affine Lie algebra of type $C_{\ell}^{(1)}$. Next we notice that the category of $D_{2 \ell -1}^{(1)}$ modules at level $- 2 \ell +3 $ obtained in Per\v{s}e (2012) has the isomorphic fusion algebra. This enables us to decompose certain $E_6 ^{(1)}$ and $F_4 ^{(1)}$--modules at negative levels.