Jet Schemes, Quantum Dilogarithm and Feigin-Stoyanovsky's Principal Subspaces

TL;DR

This paper explores the structure of principal subspaces in affine Lie algebras using jet algebra methods, deriving new character formulas and demonstrating classical freeness for certain types.

Contribution

It introduces novel fermionic character formulas for type A principal subspaces and extends the analysis to other affine Lie algebra types, establishing classical freeness results.

Findings

Fermionic character formulas for type A principal subspaces.

Principal subspaces are classically free as vertex algebras.

New character formulas for $ ext{so}_5$ principal subspace.

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 $L_{1}(\mathfrak{so}_5)$, $L_{1}(\mathfrak{so}_8)$ and $L_1(\frak{g}_2)$. We derive a new character formula for the principal subspace of $L_1(\mathfrak{so}_5)$, proving that it is classically free, and present evidence that the principal subspaces of $L_1(\mathfrak{so}_8)$ and of $L_1(\frak{g}_2)$ are also classically free.