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A Matrix Rank Formula for Vector Bundles of Vertex Operator Algebra Coinvariants and Conformal Blocks

Xiangrui Luo

Abstract

We introduce FA-matrices for computing ranks of vector bundles of coinvariants and conformal blocks associated with modules over vertex operator algebras on the moduli space of stable pointed curves, unifying the notions of fusion and averaging matrices and generalizing Ueno's work. To illustrate, we compute ranks of vector bundles determined by pointed VOAs and the tensor product of certain VOAs, as well as other examples. As an application, positivity properties of their first Chern classes are analyzed.

A Matrix Rank Formula for Vector Bundles of Vertex Operator Algebra Coinvariants and Conformal Blocks

Abstract

We introduce FA-matrices for computing ranks of vector bundles of coinvariants and conformal blocks associated with modules over vertex operator algebras on the moduli space of stable pointed curves, unifying the notions of fusion and averaging matrices and generalizing Ueno's work. To illustrate, we compute ranks of vector bundles determined by pointed VOAs and the tensor product of certain VOAs, as well as other examples. As an application, positivity properties of their first Chern classes are analyzed.
Paper Structure (21 sections, 44 theorems, 149 equations)

This paper contains 21 sections, 44 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Notation and Conventions
  4. Formal Factorization Property and Formal FA-Matrix
  5. Vector Bundle of Coinvariants on $\overline{\mathcal{M}}_{g,n}$ and FA-Matrix
  6. Vector Bundle of Coinvariants on Tensor Product of VOAs
  7. Pointed vertex operator algebra
  8. Ranks of vector bundles of coinvariants associated with pointed VOAs
  9. Coinvariants and conformal block divisor associated with a pointed VOA.
  10. Positivity of coinvariant divisors associated with a pointed VOA
  11. Examples of Pointed VOAs
  12. Symmetric Coinvariant Divisors and Irreducible Module of Order two
  13. Symmetric coinvariant divisors and Virasoro VOAs
  14. Some examples of rank formulae
  15. Virasoro VOAs in the discrete series.
  16. ...and 6 more sections

Key Result

Theorem 1

(Theorem cor_2, matrix rank formula) Let $V$ be a strongly rational VOA (c.f., Definition def_strongly_rational), $\{W_{1},\cdots,W_{l}\}$ be the collection of all irreducible admissible $V$-modules, up to isomorphism. For each irreducible module $W$ and $g\ge 0$, define an $l\times l$ matrix $R_{W, Moreover, if $g\ge 1$, one also has that where $\mathrm{Tr}$ denotes the trace of a matrix.

Theorems & Definitions (97)

  • Theorem 1
  • Definition 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • Theorem 6
  • Proposition 7
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 87 more