Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability conditions in the mathematical Gauged Linear Sigma Model

Huai-Liang Chang, Shuai Guo, Jun Li, Wei-Ping Li, Yang Zhou

TL;DR

This work introduces Ω-stability, a unifying stability condition for LG-quasimaps in GLSM that extends Mixed-Spin-P fields to a broad class of GIT targets, including Calabi–Yau complete intersections in toric varieties. It proves that the moduli stack LGQ^{Ω}_{g,k}(X,d) is a separated Deligne–Mumford stack of finite type, and is proper when the target quotient V/\!/_{θ}G is projective and the stability set S is full. The authors develop a projective-space case as a prototype, establish a stabilization process, and then extend to the general case by constructing a comparison map to ℙ^{N-1} and checking a local valuative criterion, with boundedness proved in the abelian setting. The framework provides a geometric platform to compute higher-genus Gromov-Witten invariants for wider GLSM targets, connecting MSP theories with broader GLSM formalisms and opening paths toward BCOV-type results in new geometries.

Abstract

The theory of Mixed-Spin-P (MSP) fields was introduced by Chang-Li-Li-Liu for the quintic threefold, aiming at studying its higher-genus Gromov-Witten invariants. Chang-Guo-Li has successfully applied it to prove conjectures including the BCOV Feynman rule, Yamaguchi-Yau's polynomiality conjecture and the Holomorphic Anomaly Equation. Meanwhile, Fan-Jarvis-Ruan introduced a mathematical theory of Gauged Linear Sigma Model (GLSM), associating a counting theory to a GIT quotient with a super-potential, under suitable assumptions. This paper provides a common generalization of both works, by introducing new stability conditions in the mathematical GLSM. We show that our stability condition guarantees the separatedness and properness of the cosection degeneracy locus in the moduli. It generalizes the MSP fields construction to more general GIT quotients, including Calabi-Yau global complete intersections in toric varieties. This hopefully provides a geometric platform to effectively compute their higher-genus Gromov-Witten invariants.

Stability conditions in the mathematical Gauged Linear Sigma Model

TL;DR

This work introduces Ω-stability, a unifying stability condition for LG-quasimaps in GLSM that extends Mixed-Spin-P fields to a broad class of GIT targets, including Calabi–Yau complete intersections in toric varieties. It proves that the moduli stack LGQ^{Ω}_{g,k}(X,d) is a separated Deligne–Mumford stack of finite type, and is proper when the target quotient V/\!/_{θ}G is projective and the stability set S is full. The authors develop a projective-space case as a prototype, establish a stabilization process, and then extend to the general case by constructing a comparison map to ℙ^{N-1} and checking a local valuative criterion, with boundedness proved in the abelian setting. The framework provides a geometric platform to compute higher-genus Gromov-Witten invariants for wider GLSM targets, connecting MSP theories with broader GLSM formalisms and opening paths toward BCOV-type results in new geometries.

Abstract

The theory of Mixed-Spin-P (MSP) fields was introduced by Chang-Li-Li-Liu for the quintic threefold, aiming at studying its higher-genus Gromov-Witten invariants. Chang-Guo-Li has successfully applied it to prove conjectures including the BCOV Feynman rule, Yamaguchi-Yau's polynomiality conjecture and the Holomorphic Anomaly Equation. Meanwhile, Fan-Jarvis-Ruan introduced a mathematical theory of Gauged Linear Sigma Model (GLSM), associating a counting theory to a GIT quotient with a super-potential, under suitable assumptions. This paper provides a common generalization of both works, by introducing new stability conditions in the mathematical GLSM. We show that our stability condition guarantees the separatedness and properness of the cosection degeneracy locus in the moduli. It generalizes the MSP fields construction to more general GIT quotients, including Calabi-Yau global complete intersections in toric varieties. This hopefully provides a geometric platform to effectively compute their higher-genus Gromov-Witten invariants.
Paper Structure (28 sections, 42 theorems, 140 equations)

This paper contains 28 sections, 42 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. A brief outline of the theory
  3. The $\Omega$-stability condition
  4. The $\Omega$-stability is open
  5. The dependence on $\Omega$
  6. Dependence on $\Omega$ in the projective case
  7. Examples
  8. Projective space with various $R$-charges
  9. Complete intersections in projective spaces
  10. Comparison with fan2017mathematical
  11. Stable quasimaps
  12. Proof of main theorems: the projective space case
  13. It is a Deligne-Mumford stack
  14. Introducing curves with nonreduced components
  15. Extending the fields to a given family of curves
  16. ...and 13 more sections

Key Result

Theorem 2.4

The stack $\mathrm{LGQ}^{\Omega}_{g,k}(X,d)$ is a separated Deligne-Mumford stack of finite type.

Theorems & Definitions (87)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • ...and 77 more