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Conformal Block Divisors for Discrete Series Virasoro VOA $\text{Vir}_{2k+1,2}$

Daebeom Choi

TL;DR

This work constructs and analyzes a new family of nef divisors on the moduli spaces of curves arising from discrete Virasoro VOAs Vir_{2k+1,2}, proving their positivity persists across genera. Central to the approach is the factorization property of conformal blocks and a careful analysis of F-curves to establish F-nefness and nefness/ample behavior, including a stabilization phenomenon reminiscent of critical level results. A key achievement is the Vir_{5,2}-based basis for Picard groups on \\overline{M}_{1,n}, enabling explicit contraction criteria and a description of line bundles on contractions. The paper also introduces a nonlinear, VOAs-inspired inductive framework for line bundles, yielding a family of nef (and sometimes ample) divisors that interpolate among known constructions and extend positivity techniques beyond traditional VOA settings.

Abstract

In this work, we study a family of vector bundles on the moduli space of curves constructed from representations of $\text{Vir}_{2k+1,2}$, a family of vertex operator algebras derived from the Virasoro Lie algebra. Using the relationship between rank and degree, we characterize their asymptotic behavior, demonstrating that their first Chern classes are nef on $\overline{\rm{M}}_{g,n}$ in many cases. This is the first nontrivial example of divisors arising from vertex operator algebras that are uniformly positive for all genera. Furthermore, for $g = 1$, these divisors form a $\mathbb{Q}$-basis of the Picard group of $\overline{\rm{M}}_{1,n}$, with several desirable functorial properties. Using this basis, we characterize line bundles on certain contractions of $\overline{\rm{M}}_{1,n}$. We also propose conjectures regarding the conformal blocks of Virasoro VOAs and potential generalizations. In particular, by introducing a generalization of conformal block divisors, we provide a nonlinear nef interpolation between affine and Virasoro conformal block divisors.

Conformal Block Divisors for Discrete Series Virasoro VOA $\text{Vir}_{2k+1,2}$

TL;DR

This work constructs and analyzes a new family of nef divisors on the moduli spaces of curves arising from discrete Virasoro VOAs Vir_{2k+1,2}, proving their positivity persists across genera. Central to the approach is the factorization property of conformal blocks and a careful analysis of F-curves to establish F-nefness and nefness/ample behavior, including a stabilization phenomenon reminiscent of critical level results. A key achievement is the Vir_{5,2}-based basis for Picard groups on \\overline{M}_{1,n}, enabling explicit contraction criteria and a description of line bundles on contractions. The paper also introduces a nonlinear, VOAs-inspired inductive framework for line bundles, yielding a family of nef (and sometimes ample) divisors that interpolate among known constructions and extend positivity techniques beyond traditional VOA settings.

Abstract

In this work, we study a family of vector bundles on the moduli space of curves constructed from representations of , a family of vertex operator algebras derived from the Virasoro Lie algebra. Using the relationship between rank and degree, we characterize their asymptotic behavior, demonstrating that their first Chern classes are nef on in many cases. This is the first nontrivial example of divisors arising from vertex operator algebras that are uniformly positive for all genera. Furthermore, for , these divisors form a -basis of the Picard group of , with several desirable functorial properties. Using this basis, we characterize line bundles on certain contractions of . We also propose conjectures regarding the conformal blocks of Virasoro VOAs and potential generalizations. In particular, by introducing a generalization of conformal block divisors, we provide a nonlinear nef interpolation between affine and Virasoro conformal block divisors.

Paper Structure

This paper contains 19 sections, 50 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminary
  4. Notations and conventions
  5. F-curves and the F-conjecture
  6. Vertex Operator Algebras and Virasoro VOAs
  7. Conformal Block Divisors
  8. Positivity of Conformal Block Divisors
  9. F-nefness of Conformal Block Divisors
  10. Nefness and Ampleness of Conformal Block Divisors
  11. Digression on
  12. Critical level of Virasoro conformal block divisors
  13. Differences of Virasoro conformal block divisors
  14. Basis of TEXT and applications
  15. TEXT-basis of TEXT
  16. ...and 4 more sections

Key Result

Theorem 1.1

For any $g, n$, and $n$ simple modules $W^i$, $-\mathbb{D}_{g,n}(\text{Vir}_{2k+1,2}, W^\bullet)$ is F-nef. Furthermore, if $W^i$ are nontrivial, then $-\mathbb{D}_{g,n}(\text{Vir}_{2k+1,2}, W^\bullet)$ is either zero or F-ample. For $k < 9$, $-\mathbb{D}_{g,n}(\text{Vir}_{2k+1,2}, W^\bullet)$ is ei

Theorems & Definitions (110)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Conjecture 2.1
  • Theorem 2.2
  • Theorem 2.3
  • ...and 100 more