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Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules

Bin Gui

Abstract

A unitary and strongly rational vertex operator algebra (VOA) $V$ is called strongly unitary if all irreducible $V$-modules are unitarizable. A strongly unitary VOA $V$ is called completely unitary if for each unitary $V$-modules $W_1$, $W_2$ the canonical nondegenerate Hermitian form on the fusion product $W_1\boxtimes W_2$ is positive. It is known that if $V$ is completely unitary, then the modular category of unitary $V$-modules is unitary [Gui19b], and all simple VOA extensions of V are automatically unitary and moreover completely unitary [Gui22, CGGH23]. In this paper, we give a geometric characterization of the positivity of the Hermitian product on $W_1$ and $W_2$, which helps us prove that the positivity is always true when the fusion product $W_1\boxtimes W_2$ is an irreducible and unitarizable $V$-module. We give several applications: (1) We show that if $V$ is a unitary (strongly rational) holomorphic VOA with a finite cyclic unitary automorphism group $G$, and if $V^G$ is strongly unitary, then $V^G$ is completely unitary. This result applies to the cyclic permutation orbifolds of unitary holomophic VOAs. (2) We show that if $V$ is unitary and strongly rational, and if $U$ is a simple current extension which is unitarizable as a $V$-module, then $U$ is a unitary VOA.

Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules

Abstract

A unitary and strongly rational vertex operator algebra (VOA) is called strongly unitary if all irreducible -modules are unitarizable. A strongly unitary VOA is called completely unitary if for each unitary -modules , the canonical nondegenerate Hermitian form on the fusion product is positive. It is known that if is completely unitary, then the modular category of unitary -modules is unitary [Gui19b], and all simple VOA extensions of V are automatically unitary and moreover completely unitary [Gui22, CGGH23]. In this paper, we give a geometric characterization of the positivity of the Hermitian product on and , which helps us prove that the positivity is always true when the fusion product is an irreducible and unitarizable -module. We give several applications: (1) We show that if is a unitary (strongly rational) holomorphic VOA with a finite cyclic unitary automorphism group , and if is strongly unitary, then is completely unitary. This result applies to the cyclic permutation orbifolds of unitary holomophic VOAs. (2) We show that if is unitary and strongly rational, and if is a simple current extension which is unitarizable as a -module, then is a unitary VOA.
Paper Structure (38 sections, 39 theorems, 329 equations)

This paper contains 38 sections, 39 theorems, 329 equations.

Table of Contents

  1. Introduction
  2. VOA modules and change of coordinates
  3. VOAs and their modules
  4. Change of coordinates
  5. Introduction to conformal blocks
  6. VOA bundles
  7. Definition of conformal blocks
  8. Examples in genus $0$
  9. Basic properties of conformal blocks
  10. Sewing conformal blocks
  11. Unitary VOA modules and conjugate conformal blocks
  12. Unitary VOAs and unitary modules
  13. Conjugate conformal blocks
  14. Reflection positive conformal blocks
  15. Sewing $\mathfrak X$ and $\mathfrak X^*$, and reflection positive conformal blocks
  16. ...and 23 more sections

Key Result

Theorem 1.12

$\mathcal{U}$ is a representation of $\mathbb G$ on $\mathbb W$. Namely, $\mathcal{U}(\mathbf{1})=\mathbf{1}$ (here $\mathbf{1}(z)=z$), and $\mathcal{U}(\alpha\circ\beta)=\mathcal{U}(\alpha)\circ\mathcal{U}(\beta)$ if $\alpha,\beta\in\mathbb G$. In particular, $\mathcal{U}(\alpha)$ has inverse $\mat

Theorems & Definitions (157)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 147 more