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1-Point Functions for $\mathbb{Z}_2$-orbifolds of Lattice VOAs

Maneesha Ampagouni

TL;DR

This work computes the full set of 1-point functions for the $\mathbb{Z}_2$-orbifold $V=V_L^{\theta}\oplus (V_L^T)^{\theta}$ of a lattice VOA $V_L$ with rank $k=8l$ and even, positive-definite lattice $L$. By adapting Mason–Mertens techniques to a $\mathbb{Z}_2$-twisted Zhu theory, it derives reduction and recursion formulas that express twisted and untwisted correlators in terms of modular data and lower-weight objects, and then applies these to obtain explicit closed forms for all state-types (vacuum, Heisenberg, and lattice sectors) in the twisted space. The results establish that the 1-point functions are modular forms of weight equal to the state weight (up to a character), validating modular invariance for the orbifold theory and enriching the understanding of twisted sectors in lattice VOAs. The work thus provides concrete, computable formulas for orbifold 1-point functions and demonstrates their robust modular structure, with implications for orbifold VOA theory and moonshine phenomena.

Abstract

In this paper, we compute the 1-point correlation functions of all states for the $\mathbb{Z}_2$-orbifolds of lattice vertex operator algebras.

1-Point Functions for $\mathbb{Z}_2$-orbifolds of Lattice VOAs

TL;DR

This work computes the full set of 1-point functions for the -orbifold of a lattice VOA with rank and even, positive-definite lattice . By adapting Mason–Mertens techniques to a -twisted Zhu theory, it derives reduction and recursion formulas that express twisted and untwisted correlators in terms of modular data and lower-weight objects, and then applies these to obtain explicit closed forms for all state-types (vacuum, Heisenberg, and lattice sectors) in the twisted space. The results establish that the 1-point functions are modular forms of weight equal to the state weight (up to a character), validating modular invariance for the orbifold theory and enriching the understanding of twisted sectors in lattice VOAs. The work thus provides concrete, computable formulas for orbifold 1-point functions and demonstrates their robust modular structure, with implications for orbifold VOA theory and moonshine phenomena.

Abstract

In this paper, we compute the 1-point correlation functions of all states for the -orbifolds of lattice vertex operator algebras.
Paper Structure (19 sections, 17 theorems, 245 equations)

This paper contains 19 sections, 17 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. Modular forms and elliptic-type functions
  4. Theta Functions
  5. Lattice Vertex Operator Algebra VL and the Z2-twisted module VLT
  6. Z2-orbifold of the Lattice VOA VL
  7. Z2-twisted Zhu Theory
  8. Reduction Theorems
  9. Recursion Formula
  10. Twisted Reduction Theorems
  11. Twisted Recursion Formulas
  12. Calculations in the Twisted Space
  13. In the Twisted Module
  14. Outside 2L
  15. At Heisenberg states
  16. ...and 4 more sections

Key Result

Theorem 1

For $u,v\in V_L$,

Theorems & Definitions (34)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 24 more