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Higher-genus quasimap wall-crossing via localization

Emily Clader, Felix Janda, Yongbin Ruan

Abstract

We give a new proof of Ciocan-Fontanine and Kim's wall-crossing formula relating the virtual classes of the moduli spaces of $ε$-stable quasimaps for different $ε$ in any genus, whenever the target is a complete intersection in projective space and there is at least one marked point. Our techniques involve a twisted graph space, which we expect to generalize to yield wall-crossing formulas for general gauged linear sigma models.

Higher-genus quasimap wall-crossing via localization

Abstract

We give a new proof of Ciocan-Fontanine and Kim's wall-crossing formula relating the virtual classes of the moduli spaces of -stable quasimaps for different in any genus, whenever the target is a complete intersection in projective space and there is at least one marked point. Our techniques involve a twisted graph space, which we expect to generalize to yield wall-crossing formulas for general gauged linear sigma models.

Paper Structure

This paper contains 21 sections, 7 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Plan of the paper
  4. Acknowledgments
  5. Definitions and setup
  6. Quasimaps and their moduli
  7. The $J$-function
  8. The wall-crossing conjecture
  9. Twisted theory
  10. Twisted graph space
  11. Definition of the twisted graph space
  12. $\mathbb{C}^*$-action and fixed loci
  13. Localization contributions
  14. Vertex contributions
  15. Edge contributions
  16. ...and 6 more sections

Key Result

Theorem 1.2

(See Theorem thm:main) Conjecture MainConj holds whenever $Z$ is a complete intersection in projective space and $n \geq 1$.

Theorems & Definitions (19)

  • Conjecture 1.1: See CFKMirror
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: Ciocan-Fontanine--Kim--Maulik CFKM
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • ...and 9 more