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Character Identities Between Affine and Virasoro Vertex Operator Algebra Modules

Dražen Adamović, Sven Möller

Abstract

The affine vertex operator algebras for $\mathfrak{sl}_2$ and the Virasoro minimal models are related by Drinfeld-Sokolov reduction and by the Goddard-Kent-Olive coset construction. In this work, we propose another connection based on certain character identities between these vertex operator algebras and their modules. This relates the simple affine vertex operator algebras $L_k(\mathfrak{sl}_2)$ at admissible levels $k=-2+q/p$ to the rational $(q,3p)$-minimal models $L_\mathrm{Vir}(c_{q,3p},0)$, and also extends to the nonadmissible levels with $q=1$. Several special cases are particularly interesting. In the nonadmissible case $q=1$, the character identities extend to certain abelian intertwining algebras, specifically $\mathcal{V}^{(p)}$ and the doublet $\mathcal{A}^{(3p)}$. Specialising further to $p=2$, where $\mathcal{V}^{(2)}$ is the simple small $\mathcal{N}=4$ superconformal algebra of central charge $c=-9$, this recovers, via the 4d/2d-correspondence, a known identity between the Schur indices of the 4d $\mathcal{N}=4$ supersymmetric Yang-Mills theory for $\mathrm{SU}(2)$ and the 4d $\mathcal{N}=2$ $(3,2)$ Argyres-Douglas theory. In the boundary admissible case $q=2$, in a similar vein, we obtain an identity between the Schur indices of 4d $\mathcal{N}=2$ Argyres-Douglas theories of types $(A_1,D_{2n+1})$ and $(A_1,A_{6n})$. On the other hand, for integral levels, $p=1$, where both involved vertex operator algebras are strongly rational, our character identity induces a Galois conjugation between the representation categories $\mathrm{Rep}(L_{-2+q}(\mathfrak{sl}_2))$ and $\mathrm{Rep}(L_\mathrm{Vir}(c_{q,3},0))$. We conjecture that the characters are related by the action of certain Hecke operators. Finally, we also sketch how to extend the results of this paper to relaxed highest-weight and Whittaker modules.

Character Identities Between Affine and Virasoro Vertex Operator Algebra Modules

Abstract

The affine vertex operator algebras for and the Virasoro minimal models are related by Drinfeld-Sokolov reduction and by the Goddard-Kent-Olive coset construction. In this work, we propose another connection based on certain character identities between these vertex operator algebras and their modules. This relates the simple affine vertex operator algebras at admissible levels to the rational -minimal models , and also extends to the nonadmissible levels with . Several special cases are particularly interesting. In the nonadmissible case , the character identities extend to certain abelian intertwining algebras, specifically and the doublet . Specialising further to , where is the simple small superconformal algebra of central charge , this recovers, via the 4d/2d-correspondence, a known identity between the Schur indices of the 4d supersymmetric Yang-Mills theory for and the 4d Argyres-Douglas theory. In the boundary admissible case , in a similar vein, we obtain an identity between the Schur indices of 4d Argyres-Douglas theories of types and . On the other hand, for integral levels, , where both involved vertex operator algebras are strongly rational, our character identity induces a Galois conjugation between the representation categories and . We conjecture that the characters are related by the action of certain Hecke operators. Finally, we also sketch how to extend the results of this paper to relaxed highest-weight and Whittaker modules.

Paper Structure

This paper contains 24 sections, 32 theorems, 192 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Vertex Operator Algebras and Their Representations
  3. Affine Vertex Operator Algebras for sl_2
  4. Admissible Case
  5. Near-Admissible Case q=1
  6. Virasoro Vertex Operator Algebras
  7. Rational Case
  8. Logarithmic Case q=1
  9. Character Identities
  10. Verma Modules and Singular Vectors
  11. Irreducible Modules: Admissible Case
  12. Irreducible Modules: Near-Admissible Case q=1
  13. Universal Vertex Operator Algebras
  14. Integral Case p=1
  15. Galois Conjugation and Hecke Operators
  16. ...and 9 more sections

Key Result

Proposition 1

Let $q,p\in\mathbb{Z}$ with $q\geq2$, $p\geq1$ and $(p,q)=1$. Moreover, assume that $3\nmid q$. Let $r,s\in\mathbb{Z}$ with $0\leq r\leq q-2$ and $0\leq s\leq p-1$. Then there is a vector-space isomorphism between Verma modules for $L_{-2+q/p}(\sl_2)$ and $L_\mathrm{Vir}(c_{q,3p},0)$ that maps a vector of $(h_0,L_0)$-weight $(f,n)$ to a vector of $L_0$-weight $n'=-f/2+3n$. The singular vectors $\

Figures (7)

  • Figure 1: Representations of admissible level $L_{-2+q/p}(\sl_2)$.
  • Figure 2: Representations of near-admissible level $L_{-2+1/p}(\sl_2)$.
  • Figure 3: Representations of rational $L_\mathrm{Vir}(c_{q,p},0)$.
  • Figure 4: Representations of logarithmic $L_\mathrm{Vir}(c_{1,p})$.
  • Figure 5: Graded vector-space isomorphism between the two vertex operator superalgebras $\mathcal{V}^{(2)}$ and $\mathcal{A}^{(6)}$ in the image of the 4d/2d-correspondence.
  • ...and 2 more figures

Theorems & Definitions (42)

  • Proposition 1: \ref{['prop:vermasingadm']}
  • Proposition 2: \ref{['cor:charid']} (and \ref{['rem:inducedGVSI']})
  • Proposition 3: \ref{['prop:vermasingnear']}
  • Proposition 4: \ref{['cor:charid2']} (and \ref{['prop:gvsi']})
  • Proposition 5: \ref{['prop:relaxed']}
  • Proposition 6: \ref{['prop:isomorphism-whittaker']}
  • Proposition 7: \ref{['prop:fusionrings']}
  • Conjecture 1: \ref{['conj:galois']} and Conjecture \ref{['conj:Hecke']}
  • Proposition 8: \ref{['prop:FFcorrespondence']}
  • Proposition 9: \ref{['prop:charid2special']}
  • ...and 32 more