Classical freeness of ${\widehat{\mathfrak{sl}}}_n$ at level $1$ via combinatorics

Abstract

We use a family of Rogers--Ramanujan-type combinatorial identities of Dousse--Konan involving coloured partitions to prove classical freeness of the simple vertex operator algebras based on ${\widehat{\mathfrak{sl}}}_n$ at level $1$. These identities are used to produce Gröbner bases for the relevant arc algebras.

38 pages and many figures. Comments welcome!