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Invariant Hermitian forms on vertex algebras

Victor G. Kac, Pierluigi Möseneder Frajria, Paolo Papi

Abstract

We study invariant Hermitian forms on a conformal vertex algebra and on their (twisted) modules. We establish existence of a non-zero invariant Hermitian form on an arbitrary $W$-algebra. We show that for a minimal simple $W$-algebra $W_k(\mathfrak g,θ/2)$ this form can be unitary only when its $\tfrac{1}{2}\mathbb Z$-grading is compatible with parity, unless $W_k(\mathfrak g,θ/2)$ "collapses" to its affine subalgebra.

Invariant Hermitian forms on vertex algebras

Abstract

We study invariant Hermitian forms on a conformal vertex algebra and on their (twisted) modules. We establish existence of a non-zero invariant Hermitian form on an arbitrary -algebra. We show that for a minimal simple -algebra this form can be unitary only when its -grading is compatible with parity, unless "collapses" to its affine subalgebra.

Paper Structure

This paper contains 15 sections, 19 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. Basic definitions
  4. Twisted modules
  5. Zhu algebras
  6. The conjugate contragredient module
  7. Invariant Hermitian forms on conformal vertex algebras
  8. Examples of invariant Hermitian forms
  9. Superfermions
  10. Superbosons
  11. Affine vertex algebras
  12. Lattice vertex algebras
  13. Invariant Hermitian forms on modules
  14. Invariant Hermitian forms on $W$--algebras
  15. Appendix: $\lambda$-brackets in minimal $W$--algebras

Key Result

Lemma 2.5

The Borcherds identity Borcherds is equivalent to for all $n\in\mathbb Z$, $m\in[\gamma_a]$, $l\in[\gamma_b]$. As usual, $i_{x,y}$ means expanding in the domain $|x|>|y|$.

Theorems & Definitions (62)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Example 2.7
  • Lemma 3.1
  • proof
  • ...and 52 more