[Paper Review] Gepner type stability conditions on graded matrix factorizations

Yukinobu Toda|arXiv (Cornell University)|Feb 26, 2013

Algebraic structures and combinatorial models3 citations

TL;DR

This paper introduces Gepner type Bridgeland stability conditions on triangulated categories of graded matrix factorizations, conjecturing their existence for weighted homogeneous polynomials and proving it for complete intersections in Calabi-Yau manifolds of dimension ≤2. The construction relies on an autoequivalence and complex parameter pair, generalizing the Gepner point in mirror symmetry, and establishes stability of key objects like $Σ(1)$ and $Σ(2)$ via $μ$-stability and spectral analysis.

ABSTRACT

We introduce the notion of Gepner type Bridgeland stability conditions on triangulated categories, which depends on a choice of an autoequivalence and a complex number. We conjecture the existence of Gepner type stability conditions on the triangulated categories of graded matrix factorizations of weighted homogeneous polynomials. Such a stability condition may give a natural stability condition for Landau-Ginzburg B-branes, and correspond to the Gepner point of the stringy Kahler moduli space of a quintic 3-fold. The main result is to show our conjecture when the variety defined by the weighted homogeneous polynomial is a complete intersection of hyperplanes in a Calabi-Yau manifold with dimension less than or equal to two.

Motivation & Objective

• To define and study Gepner type Bridgeland stability conditions on triangulated categories of graded matrix factorizations, motivated by mirror symmetry and DT invariants.
• To conjecture the existence of such stability conditions for weighted homogeneous polynomials, particularly in the context of the Gepner point of the quintic 3-fold's Kähler moduli space.
• To provide a framework for constructing natural stability conditions on Landau-Ginzburg B-brane categories that reflect symmetries in the derived category.
• To establish the existence of these stability conditions in the case where the hypersurface defined by the polynomial is a complete intersection in a Calabi-Yau manifold of dimension ≤2.
• To verify the stability of specific objects like $Σ(1)$, $Σ(2)$, and $Σ(Ó)$ under the constructed stability condition using $μ$-stability and spectral analysis.

Proposed method

• Introduces the notion of Gepner type stability conditions via a pair $(\Phi, \lambda)$, where $\Phi$ is an autoequivalence and $\lambda$ is a complex number, satisfying $\Phi_*\sigma = \sigma \cdot \lambda$.
• Constructs a t-structure on the category of graded matrix factorizations using a grading shift and the derived category of coherent sheaves on the associated variety.
• Applies a spectral analysis of the central charge $Z_G^\dagger$ to determine stability, focusing on real and imaginary parts of the central charge on the heart of the t-structure.
• Uses $\mu$-stability on the underlying category $\mathcal{A}_W$ to analyze subobjects and quotients, ensuring vanishing of Hom groups for stability verification.
• Applies Lemma 4.14 to reduce stability checks to vanishing of Hom groups between objects with zero real central charge and the candidate stable object.
• Employs quiver representations for graded matrix factorizations in low-dimensional cases (d=4, d=6) to explicitly describe objects and verify $\mu$-stability.

Experimental results

Research questions

• RQ1 Does a Gepner type stability condition exist on the triangulated category of graded matrix factorizations for a weighted homogeneous polynomial?
• RQ2 Can such a stability condition be constructed when the associated variety is a complete intersection in a Calabi-Yau manifold of dimension ≤2?
• RQ3 How does the Gepner type condition relate to the Gepner point in the stringy Kähler moduli space of a quintic 3-fold?
• RQ4 What is the role of the autoequivalence $\Phi = \mathrm{ST}_{\mathcal{O}_X} \circ \otimes \mathcal{O}_X(1)$ and the complex parameter $\lambda = 2/5$ in defining the stability condition?
• RQ5 Which objects in the category of graded matrix factorizations are stable under the constructed Gepner type stability condition?

Key findings

• The Gepner type stability condition $\sigma_G^\dagger$ exists on the category of graded matrix factorizations for weighted homogeneous polynomials when the associated variety is a complete intersection in a Calabi-Yau manifold of dimension ≤2.
• The object $\mathbb{C}(1)$ is $\sigma_G^\dagger$-stable, verified via vanishing of Hom groups from objects with zero real central charge and $\mu$-semistability arguments.
• The object $\mathbb{C}(2)$ is $\sigma_G^\dagger$-stable, with proof relying on $\mu$-stability of quiver representations in the $d=4$ and $d=6$ cases.
• For $d=4$, the object $\mathbb{C}(2)[-1]$ is $\mu$-stable with positive slope $1/4$, and its shift $\mathbb{C}(2)[-2]$ lies in the heart $\mathcal{A}_G$, ensuring stability under $\sigma_G^\dagger$.
• For $d=6$, the object $\mathbb{C}(2)[-1]$ is $\mu$-stable with negative slope $-3/4$, and lies in $\mathcal{A}_G$, with stability under $\sigma_G^\dagger$ confirmed via Hom vanishing.
• The central charge $Z_G^\dagger$ has image in $\mathbb{Z} + \mathbb{Z}\sqrt{-1}$ for $d=4$ and in $\mathbb{Z}_{\leq 0} \times \frac{1}{2}$ for $d=6$, which is crucial for applying Lemma 4.14 to verify stability.