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Generalized Multiple q-Zeta Values and Characters of Vertex Algebras

Antun Milas

Abstract

We analyze certain characters of vertex algebras that can be expressed using (generalized) q-MZVs. We consider: (i) characters of vertex algebras associated to arc spaces, (ii) characters (or indices) of $\mathcal{S}$-class vertex operator algebras in 4d $\mathcal{N}=2$ SCFT, (iii) supercharacters of the $\mathcal{U}$ family of vertex algebras. Along the way we also introduce a family of multiple $q$-zeta values associated to simple Lie algebras and present their conjectural properties.

Generalized Multiple q-Zeta Values and Characters of Vertex Algebras

Abstract

We analyze certain characters of vertex algebras that can be expressed using (generalized) q-MZVs. We consider: (i) characters of vertex algebras associated to arc spaces, (ii) characters (or indices) of -class vertex operator algebras in 4d SCFT, (iii) supercharacters of the family of vertex algebras. Along the way we also introduce a family of multiple -zeta values associated to simple Lie algebras and present their conjectural properties.
Paper Structure (16 sections, 23 theorems, 134 equations, 5 figures)

This paper contains 16 sections, 23 theorems, 134 equations, 5 figures.

Table of Contents

  1. Introduction and summary of results
  2. Graph series and multiple $q$-zeta values
  3. Graph series of $\Gamma_{3k+2}$, $k \geq 1$
  4. Graph series of star graphs $T_{2k+2}$
  5. Graph series and multiple $q$-zeta series
  6. Algebra of graphs series
  7. Quasi-modularity and graph series
  8. Multiple $q$-zeta values, quasi-modular forms, and characters
  9. Elliptic functions, theta functions, and quasi-modular forms
  10. Quasi-Jacobi forms
  11. Characters of $\mathcal{S}$-class vertex algebras ($g=0$)
  12. Bi-brackets of multiple $q$-zeta values of a Lie algebra $\frak{g}$
  13. Quasi-modularity for $\frak{sl}(3)$
  14. Characters and supercharacters of the $\mathcal{U}$ family of vertex algebras
  15. Supercharacters of $\mathcal{U}$ VOAs via elliptic functions
  16. ...and 1 more sections

Key Result

Theorem 1.1

We have:

Figures (5)

  • Figure 1: $C_5$ and $E_6$ graphs
  • Figure 2: Graph $\Gamma_{14}$
  • Figure 3: Graph $\Gamma_{17}$
  • Figure 4: $T_{10}$ graph
  • Figure 5: Graph $Z_{1,1,1,2,2}$

Theorems & Definitions (52)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1.4
  • Conjecture 1.5
  • Conjecture 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Lemma 2.1
  • Theorem 2.2
  • ...and 42 more