Stability conditions in the mathematical Gauged Linear Sigma Model

Technical Summary: Stability Conditions in the Mathematical Gauged Linear Sigma Model

Problem Statement
The paper addresses the challenge of defining a robust mathematical framework for the Gauged Linear Sigma Model (GLSM) that generalizes existing theories, specifically the Mixed-Spin-P (MSP) fields theory for quintic threefolds and the Fan-Jarvis-Ruan (FJR) theory of LG-quasimaps.

While MSP fields successfully bridge Gromov-Witten theory and quantum singularity theory for the quintic, and FJR theory provides a counting theory for GIT quotients with super-potentials, neither framework fully generalizes to arbitrary variations of GIT (vGIT) master spaces required for studying higher-genus Gromov-Witten invariants of Calabi-Yau complete intersections in toric varieties.

A primary technical obstacle is the lack of a stability condition that guarantees the separatedness and properness of the moduli space, particularly the cosection degeneracy locus. Previous attempts, such as the ad hoc finite-automorphism condition in MSP or the "good lift" requirement in FJR theory, fail for general vGIT master spaces (e.g., where unstable loci intersect non-trivially or where good lifts do not exist). Without these geometric properties, one cannot define invariants via intersection theory or utilize virtual localization techniques effectively.

Methodology
The authors introduce a new stability condition termed $\Omega$-stability for LG-quasimaps. The methodology proceeds as follows:

1. Framework Setup: The theory is built upon an $(R$-charged) package $(V, G \leq \Gamma, \varpi, \vartheta)$, where $V$ is an affine scheme, $G$ is a reductive group, and $\Gamma$ is a reductive group containing $G$ with a character $\varpi$ inducing an isomorphism $\Gamma/G \cong \mathbb{C}^*$. The character $\vartheta$ restricts to a stability parameter $\theta$ on $G$.
2. Definition of $\Omega$-Stability: A prestable LG-quasimap $\xi = (C, \Sigma_C, u, \kappa)$ is defined as $\Omega$-stable if it satisfies three conditions involving a triple $\Omega = (S, A, \vartheta)$:
   • $\Omega$-1: The base locus is discrete and disjoint from special points.
   • $\Omega$-2: A slope condition at non-special points involving a finite set $S$ of homogeneous elements in the ring of invariants. For $f \in S$, the condition is $\min_{f \in S} \{ \frac{1}{\text{wt}_\theta(f)}\text{ord}_x(u^*f) + \text{slope}(f) \} \leq A$. Here, the slope incorporates the "R-charge" (the weight under the $\mathbb{C}^*$ action on the fibers).
   • $\Omega$-3: The $\mathbb{Q}$-line bundle $u^*L_\vartheta \otimes (\omega_C^{\log})^{\otimes A}$ is positive.
3. Analysis of Stability: The authors analyze the dependence of stability on the parameters $S$, $A$, and $\vartheta$. They demonstrate that for a fixed target, varying $A$ creates a wall-and-chamber structure similar to $\epsilon$-stable quasimaps. They show that as $A \to \infty$, the condition recovers the $\epsilon = 0^+$ stability of FJR theory.
4. Reduction to Projective Space: To prove the main theorems, the authors reduce the general case to the case of projective spaces with various R-charges. They construct a morphism from the moduli of $\Omega$-stable LG-quasimaps to a general target $X$ to the moduli of $\Omega$-stable LG-quasimaps to a projective space $\mathbb{P}^{N-1}$, preserving the stability conditions.
5. Valuative Criterion: Using the reduction, they apply the valuative criterion to prove separatedness and properness. This involves constructing extensions of families over discrete valuation rings, handling non-reduced special fibers via blow-ups and base changes, and performing a stabilization process to contract rational tails and bridges that violate the positivity condition.

Key Contributions and Results
The paper establishes the following main theorems regarding the moduli stack $\text{LGQ}^\Omega_{g,k}(X, d)$ of $\Omega$-stable LG-quasimaps:

• Theorem 2.4 (Separatedness): The stack $\text{LGQ}^\Omega_{g,k}(X, d)$ is a separated Deligne-Mumford stack of finite type. This resolves the issue of non-separated moduli spaces found in previous ad hoc constructions.
• Theorem 2.5 (Properness): If the GIT quotient $V//_\theta G$ is projective and the set $S$ is "full" (i.e., its common vanishing locus is the unstable locus), then the stack $\text{LGQ}^\Omega_{g,k}(X, d)$ is proper.
• Generalization of MSP: The new stability condition generalizes the MSP fields construction for the quintic threefold. By applying the theory to the master space $Y$ associated with the quintic, the authors recover the properness of the cosection degeneracy locus, which was previously shown via ad hoc methods.
• Extension to General Targets: The framework applies to Calabi-Yau global complete intersections in toric varieties and other vGIT master spaces where previous methods (requiring "good lifts") failed.
• Boundedness: The paper provides a proof of the boundedness of the moduli space, addressing a gap in previous literature regarding the boundedness of stable quasimaps in the non-abelian case (referencing a companion paper for the full non-abelian proof).

Significance and Claims
The authors claim that this work provides a common generalization of the MSP fields theory and the FJR GLSM theory. By introducing $\Omega$-stability, they provide a geometric platform that:

1. Guarantees the necessary separatedness and properness of the moduli spaces and their cosection degeneracy loci.
2. Allows for the definition of counting theories for a broader class of target spaces, including Calabi-Yau complete intersections in toric varieties.
3. Offers a pathway to effectively compute higher-genus Gromov-Witten invariants for these targets, potentially extending results like the BCOV Feynman rule and the Holomorphic Anomaly Equation beyond the quintic threefold.

The paper notes that while the construction is general, the specific choice of the R-charged package and the parameter $A$ may lead to different enumerative theories. It highlights that for certain cases (e.g., Fermat singularities or Calabi-Yau complete intersections in products of projective spaces), this framework has already been utilized to prove conjectures regarding the BCOV Feynman rule and the analyticity of partition functions. The work does not claim to solve all higher-genus problems immediately but establishes the foundational stability conditions required to pursue them.