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

Hao Li, Antun Milas

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.

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 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 -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 are ``classically free" as vertex algebras. We also analyze infinite jet algebras associated to principal subspaces of affine vertex algebras , and . We derive a new character formula for the principal subspace of , proving that it is classically free, and present evidence that the principal subspaces of and of are also classically free.

Paper Structure

This paper contains 20 sections, 17 theorems, 159 equations, 1 table.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Affine jet and arc algebras
  4. Principal subspaces
  5. Principal subspace $W(\Lambda_0)$ of $\mathfrak{sl}_n$
  6. Jet algebra $J_\infty(A)$
  7. Jet algebra $J_\infty(B)$
  8. Jet algebra $J_\infty(H)$
  9. Quantum dilogarithm
  10. Warm up: Proof of (\ref{['sl3']}) using quantum dilogarithm
  11. General case
  12. $q$-identities from quiver representations
  13. Level one principal subspace of $B_2$
  14. Spinor representation of $B_\ell^{(1)}$
  15. Principal subspace of $L_1(\mathfrak{so}_{5})$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $A$ be the Cartan matrix $(\langle \alpha_{i},\alpha_{j} \rangle)_{1\leq i,j\leq n-1}$ of type $A_{n-1}$, $n \geq 2$, that is and

Theorems & Definitions (39)

  • Theorem 1.1
  • Example 1.2
  • Theorem 1.3
  • Definition 2.1
  • Example 2.2
  • Conjecture 2.3
  • Remark 3.1
  • Claim 3.2
  • proof
  • Example 3.3
  • ...and 29 more