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Vertex algebras related to regular representations of $SL_2$

Drazen Adamovic, Antun Milas

TL;DR

The paper constructs a family of vertex algebras $\mathcal{C}_p$ ($p\ge2$) that deform the regular representation of $\mathfrak{sl}_2$ and connect to chiral differential operators on $SL_2$, showing they admit a half-integer grading and are potentially quasi-lisse. It establishes explicit identifications for small $p$: $\mathcal{C}_3 \cong L_{-5/3}(\mathfrak{g}_2)$, $\mathcal{C}_4 \cong \mathcal{W}_{-23/4}(F_4,A_1+\tilde{A}_1)$, and $\mathcal{C}_5 \cong \mathcal{W}_{-30+31/5}(E_8,A_4+A_2)$, via inverse quantum Hamiltonian reduction and conformal embeddings. The work also develops modularity results for their characters, proving MLDEs of minimal orders for $p=2,3,4,5$ and expressing characters through Appell–Lerch theta-type functions, supporting quasi-lisse behavior. Overall, the paper advances the understanding of decompositions of affine $\mathfrak{g}$ embeddings into affine $W$-algebras and suggests a pathway to higher-rank generalizations and broader modular phenomena.

Abstract

We construct a family of potentially quasi-lisse (non-rational) vertex algebras, denoted by $\mathcal{C}_p$, $p \geq 2$, which are closely related to the vertex algebra of chiral differential operators on $SL(2)$ at level $-2+\frac{1}{p}$. We prove that for $p = 3$, there is an isomorphism between $\mathcal{C}_3$ and the affine vertex algebra $L_{-5/3}(\mathfrak{g}_2)$ from Deligne's series. Moreover, we also establish isomorphisms between $\mathcal{C}_4$ and $\mathcal{C}_5$ and certain affine ${W}$-algebras of types $F_4$ and $E_8$, respectively. In this way, we resolve the problem of decomposing certain conformal embeddings of affine vertex algebras into affine ${W}$-algebras. An important feature is that $\mathcal{C}_p$ is $\frac{1}{2} \mathbb{Z}_{\geq 0}$-graded with finite-dimensional graded subspaces and convergent characters. Therefore, for all $p \geq 2$, we show that the characters of $\mathcal{C}_p$ exhibit modularity, supporting the conjectural quasi-lisse property.

Vertex algebras related to regular representations of $SL_2$

TL;DR

The paper constructs a family of vertex algebras () that deform the regular representation of and connect to chiral differential operators on , showing they admit a half-integer grading and are potentially quasi-lisse. It establishes explicit identifications for small : , , and , via inverse quantum Hamiltonian reduction and conformal embeddings. The work also develops modularity results for their characters, proving MLDEs of minimal orders for and expressing characters through Appell–Lerch theta-type functions, supporting quasi-lisse behavior. Overall, the paper advances the understanding of decompositions of affine embeddings into affine -algebras and suggests a pathway to higher-rank generalizations and broader modular phenomena.

Abstract

We construct a family of potentially quasi-lisse (non-rational) vertex algebras, denoted by , , which are closely related to the vertex algebra of chiral differential operators on at level . We prove that for , there is an isomorphism between and the affine vertex algebra from Deligne's series. Moreover, we also establish isomorphisms between and and certain affine -algebras of types and , respectively. In this way, we resolve the problem of decomposing certain conformal embeddings of affine vertex algebras into affine -algebras. An important feature is that is -graded with finite-dimensional graded subspaces and convergent characters. Therefore, for all , we show that the characters of exhibit modularity, supporting the conjectural quasi-lisse property.

Paper Structure

This paper contains 22 sections, 28 theorems, 153 equations, 1 table.

Table of Contents

  1. Introduction
  2. Regular representations of vertex algebras
  3. Minimal and Weil representations
  4. Our paper and main results
  5. Notation
  6. Inverse Quantum Hamiltonian Reduction
  7. Construction of the vertex algebra $\mathcal{C}_{p}$
  8. The vertex algebra $L_{-\frac{5}{3}} (G_2)$
  9. Decomposition of $M_{(2)}$
  10. Isomorphism $\mathcal{C}_3 \cong L_{-5/3} (G_2)$
  11. The vertex algebra $\mathcal{C}_4$ as inverse QHR of $L_{-3/2} (\frak{sl}_3)$
  12. The decomposition of $L_{-3/2} (\mathfrak{sl}_3)$ as $L_{-6} (\mathfrak{sl}_2) \otimes L^{Vir}_{c_{1,4}}$--module
  13. Realization of $\mathcal{C}_p$ as an affine $W$--algebra: the cases $p=4,5$
  14. Affine $W$--algebras
  15. Affine $W$-algebra $\mathcal{W}_k(F_4,A_1+\tilde{A}_1)$
  16. ...and 7 more sections

Key Result

Theorem 1.1

For every $p \geq 1$, there is a simple vertex algebra $\mathcal{C}_p$ such that $L_{-2+\frac{1}{p}}( \mathfrak{sl}_2) \otimes L_{-2-p}( \mathfrak{sl}_2)$ is conformally embedded into $\mathcal{C}_p$, and we have the following decomposition:

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Corollary 2.2
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Corollary 3.3
  • Theorem 3.4
  • proof
  • ...and 43 more