Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Genus $g$ Virasoro Correlation Functions for Vertex Operator Algebras

Michael P. Tuite, Michael Welby

TL;DR

The paper provides a genus $g$ framework for Virasoro $n$-point functions of VOAs using Schottky uniformisation and genus $g$ Zhu recursion. It constructs an explicit order-$n$ differential operator $\mathcal{D}_n(\mathbf{z})$ that acts on the normalised partition function $\Theta_V=Z_V Z_M^{-c}$ to yield the Virasoro $n$-point generating function, expressing $\mathcal{D}_n$ as a sum over Virasoro graphs with weights depending on the central charge $c$, the classical differential geometry on the surface (bidifferential, projective connection, holomorphic forms), and derivatives with respect to $3g-3$ period matrix elements. The weights are organized so that $\mathcal{D}_n$ captures Virasoro vacuum descendant correlators and obeys a precise recursive structure, with a first-principles derivation of the Ward identities and a detailed account of modular properties under homology base changes. This work links VOA correlation functions at genus $g$ to the differential geometry of moduli space, enabling modular differential equations and graph-based formulations of high-genus Virasoro correlators.

Abstract

For a simple, self-dual, strong CFT-type vertex operator algebra (VOA) of central charge $c$, we describe the Virasoro $n$-point correlation function on a genus $g$ marked Riemann surface in the Schottky uniformisation. We show that this $n$-point function determines the correlation functions for all Virasoro vacuum descendants. Using our recent work on genus $g$ Zhu recursion, we show that the Virasoro $n$-point function is determined by a differential operator $\mathcal{D}_{n}$ acting on the genus $g$ VOA partition function normalised by the Heisenberg partition function to the power of $c$. We express $\mathcal{D}_{n}$ as the sum of weights over certain Virasoro graphs where the weights explicitly depend on $c$, the classical bidifferential of the second kind, the projective connection, holomorphic 1-forms and derivatives with respect to any $3g-3$ locally independent period matrix elements. We also describe the modular properties of $\mathcal{D}_{n}$ under a homology base change.

Genus $g$ Virasoro Correlation Functions for Vertex Operator Algebras

TL;DR

The paper provides a genus framework for Virasoro -point functions of VOAs using Schottky uniformisation and genus Zhu recursion. It constructs an explicit order- differential operator that acts on the normalised partition function to yield the Virasoro -point generating function, expressing as a sum over Virasoro graphs with weights depending on the central charge , the classical differential geometry on the surface (bidifferential, projective connection, holomorphic forms), and derivatives with respect to period matrix elements. The weights are organized so that captures Virasoro vacuum descendant correlators and obeys a precise recursive structure, with a first-principles derivation of the Ward identities and a detailed account of modular properties under homology base changes. This work links VOA correlation functions at genus to the differential geometry of moduli space, enabling modular differential equations and graph-based formulations of high-genus Virasoro correlators.

Abstract

For a simple, self-dual, strong CFT-type vertex operator algebra (VOA) of central charge , we describe the Virasoro -point correlation function on a genus marked Riemann surface in the Schottky uniformisation. We show that this -point function determines the correlation functions for all Virasoro vacuum descendants. Using our recent work on genus Zhu recursion, we show that the Virasoro -point function is determined by a differential operator acting on the genus VOA partition function normalised by the Heisenberg partition function to the power of . We express as the sum of weights over certain Virasoro graphs where the weights explicitly depend on , the classical bidifferential of the second kind, the projective connection, holomorphic 1-forms and derivatives with respect to any locally independent period matrix elements. We also describe the modular properties of under a homology base change.

Paper Structure

This paper contains 17 sections, 20 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Differential Structures on Riemann Surfaces
  3. Notational conventions
  4. The Schottky uniformisation of a Riemann surface
  5. Some classical differentials on a Riemann surface
  6. Differential operators on meromorphic forms
  7. Bers Potentials and Variations of Moduli
  8. Vertex Operator Algebras and Genus $g$ Zhu Recursion
  9. Vertex operator algebras
  10. Genus $g$ correlation functions
  11. Genus $g$ Zhu recursion
  12. Genus $g$ Virasoro Correlation Functions
  13. The Virasoro Ward Identity
  14. Virasoro $n$-point generating functions
  15. Genus $g$ Virasoro graphs for Virasoro $n$-point functions
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let $F(\bm{\eta})$ be a differentiable function on ${\mathfrak{M}}$ for any local coordinates $\bm{\eta}:=\eta_{1},\ldots,\eta_{3g-3}$. Then

Theorems & Definitions (37)

  • Lemma 2.1
  • Remark 2.1
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.4
  • proof
  • ...and 27 more