Current algebras and categorified quantum groups

Abstract

We identify the trace, or zeroth Hochschild homology, of categorified quantum groups of ADE type with the corresponding current algebra of the same type. To prove this, we show that 2-representations defined using categories of modules over cyclotomic (or deformed cyclotomic) quotients of Khovanov-Lauda-Rouquier algebras correspond to local (or global) Weyl modules. We also investigate the implications for centers of categories in 2-representations of categorified quantum groups.