Level-rank duality via tensor categories

TL;DR

This paper introduces a novel approach to derive branching rules for conformal embeddings and establishes a braided equivalence between certain representation categories, revealing a new layer of symmetry in affine Lie algebra representations.

Contribution

It provides a new method to derive branching rules for conformal embeddings and proves a braided equivalence between degree zero representation categories of affine Lie algebras.

Findings

Derived new branching rules for conformal embedding $( ext{sl}_n)_m imes ( ext{sl}_m)_n o ( ext{sl}_{nm})_1$

Proved braided equivalence between categories $ ext{C}( ext{sl}_n)_m^0$ and $ ext{C}( ext{sl}_m)_n^0$ with reversed braiding

Revealed a symmetry between representation categories of affine Lie algebras

Abstract

We give a new way to derive branching rules for the conformal embedding $$(\asl_n)_m\oplus(\asl_m)_n\subset(\asl_{nm})_1.$$ In addition, we show that the category $\Cc(\asl_n)_m^0$ of degree zero integrable highest weight $(\asl_n)_m$-representations is braided equivalent to $\Cc(\asl_m)_n^0$ with the reversed braiding.