Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Virasoro constraints on moduli of sheaves and vertex algebras

Arkadij Bojko, Woonam Lim, Miguel Moreira

Abstract

In enumerative geometry, Virasoro constraints were first conjectured in Gromov-Witten theory with many new recent developments in the sheaf theoretic context. In this paper, we rephrase the sheaf-theoretic Virasoro constraints in terms of primary states coming from a natural conformal vector in Joyce's vertex algebra. This shows that Virasoro constraints are preserved under wall-crossing. As an application, we prove the conjectural Virasoro constraints for moduli spaces of torsion-free sheaves on any curve and on surfaces with only $(p,p)$ cohomology classes by reducing the statements to the rank 1 case.

Virasoro constraints on moduli of sheaves and vertex algebras

Abstract

In enumerative geometry, Virasoro constraints were first conjectured in Gromov-Witten theory with many new recent developments in the sheaf theoretic context. In this paper, we rephrase the sheaf-theoretic Virasoro constraints in terms of primary states coming from a natural conformal vector in Joyce's vertex algebra. This shows that Virasoro constraints are preserved under wall-crossing. As an application, we prove the conjectural Virasoro constraints for moduli spaces of torsion-free sheaves on any curve and on surfaces with only cohomology classes by reducing the statements to the rank 1 case.
Paper Structure (43 sections, 40 theorems, 400 equations)

This paper contains 43 sections, 40 theorems, 400 equations.

Table of Contents

  1. Introduction
  2. History
  3. Moduli of sheaves and pairs
  4. Universal sheaves and descendents
  5. Joyce's vertex algebra
  6. Conformal element and Virasoro constraints
  7. Proof of Theorem \ref{['thm: main']} and other results
  8. Notation and conventions
  9. Future directions
  10. Acknowledgement
  11. Virasoro constraints
  12. Supercommutative algebras
  13. Descendent algebra
  14. Virasoro operators
  15. Weight zero descendents
  16. ...and 28 more sections

Key Result

Theorem A

The Virasoro constraints (Conjecture conj: virasorosheaves) hold for the following moduli spaces when there are no strictly semistable sheaves:

Theorems & Definitions (120)

  • Theorem A: =Theorem \ref{['thm: main result of the paper']}
  • Theorem B: =Theorem \ref{['thm: duality']} and Theorem \ref{['thm: virasorophysical']}
  • Theorem C: =Propositions \ref{['prop: wallcrossingcompatibility1']} and \ref{['prop: wallcrossingcompatibility2']}
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Conjecture 1.5: Virasoro for sheaves
  • Conjecture 1.6: Virasoro for pairs
  • Remark 1.7
  • ...and 110 more