Jet schemes, quantum dilogarithm and Feigin-Stoyanovsky's principal subspaces

Abstract

We analyze the structure of Feigin-Stoyanovsky's principal subspaces of affine Lie algebra from the jet algebra viewpoint. For type A level one principal subspaces, we show that their shifted multi-graded Hilbert series can be expressed either using the quantum dilogarithm or as certain generating functions “counting” finite-dimensional representations of A-type quivers. This notably results in novel fermionic character formulas for these principal subspaces. Moreover, our result implies that all level one principal subspaces of type A are “classically free” as vertex algebras.We also analyze infinite jet algebras associated to principal subspaces of affine vertex algebras L1(so5), L1(so8) and L1(g2). We derive a new character formula for the principal subspace of L1(so5), proving that it is classically free, and present evidence that the principal subspaces of L1(so8) and of L1(g2) are also classically free.