[Paper Review] Spaces of stability conditions

Tom Bridgeland|ArXiv.org|Nov 16, 2006

Black Holes and Theoretical Physics42 references65 citations

TL;DR

This paper surveys the mathematical theory of stability conditions on triangulated categories, linking them to string theory via $Π$-stability of D-branes and moduli spaces of superconformal field theories (SCFTs). It proposes that spaces of stability conditions may carry rich geometric structures—such as almost Frobenius manifolds—derived from central charges and central charge logarithms, with key results including the emergence of WDVV-type equations and connections to quantum cohomology and mirror symmetry.

ABSTRACT

Stability conditions are a mathematical way to understand $Π$-stability for D-branes in string theory. Spaces of stability conditions seem to be related to moduli spaces of conformal field theories. This is a survey article describing what is currently known about spaces of stability conditions, and giving some pointers for future research.

Motivation & Objective

• To survey known examples of spaces of stability conditions on smooth projective varieties and local Calabi-Yau threefolds.
• To explore the conjectural geometric structures—such as Frobenius or almost Frobenius manifolds—that may naturally arise on spaces of stability conditions.
• To investigate the deep connection between stability conditions and moduli spaces of superconformal field theories (SCFTs), particularly via mirror symmetry.
• To propose that stability conditions may be subsumed into a broader, more natural framework in the future, following current evidence from examples.
• To highlight open problems and suggest directions for future research, especially in defining global geometric structures on these spaces.

Proposed method

• Uses the definition of stability conditions on triangulated categories, involving a central charge $ Z: K(π) \to \mathbb{C} $ and slicing $ \mathcal{P} $, to define the space $ \mathrm{Stab}(\mathcal{D}) $.
• Applies tilting and mutation techniques to construct stability conditions on derived categories of coherent sheaves, especially for varieties with full exceptional collections.
• Constructs a function $ F(Z) = \sum_{\alpha \in \Lambda_+} Z(\alpha)^2 \log Z(\alpha) $ on the space of stability conditions, which satisfies the WDVV equation.
• Defines a multiplication on tangent vectors via the triple product $ \langle \theta_1, \theta_2, \theta_3 \rangle = \sum_{\alpha \in \Lambda_+} \frac{\theta_1(\alpha)\theta_2(\alpha)\theta_3(\alpha)}{Z(\alpha)} $, yielding an almost Frobenius manifold structure.
• Draws analogies between the geometry of stability conditions and quantum cohomology, particularly via Dubrovin’s conjecture linking semisimple quantum cohomology to exceptional collections.
• Proposes that Joyce’s flat connection on stability conditions for Calabi-Yau categories may provide a pathway to defining global geometric structures on $ \mathrm{Stab}(\mathcal{D}) $.

Experimental results

Research questions

• RQ1 What geometric structures, if any, can be naturally defined on the space of stability conditions?
• RQ2 How do spaces of stability conditions relate to moduli spaces of superconformal field theories (SCFTs) and mirror symmetry?
• RQ3 Can the function $ F(Z) = \sum Z(\alpha)^2 \log Z(\alpha) $ be generalized to non-exceptional or non-abelian examples, and does it always satisfy the WDVV equation?
• RQ4 Is there a deeper categorical or geometric framework that subsumes the current notion of stability conditions?
• RQ5 To what extent do the central charge and central charge logarithm functions encode the full geometry of the stability space, especially in relation to Frobenius manifolds?

Key findings

• The function $ F(Z) = \sum_{\alpha \in \Lambda_+} Z(\alpha)^2 \log Z(\alpha) $ on the space of stability conditions satisfies the WDVV equation, indicating integrable structure.
• The triple product $ \langle \theta_1, \theta_2, \theta_3 \rangle = \sum_{\alpha \in \Lambda_+} \frac{\theta_1(\alpha)\theta_2(\alpha)\theta_3(\alpha)}{Z(\alpha)} $ defines an associative multiplication on tangent vectors, yielding an almost Frobenius manifold structure.
• This almost Frobenius manifold is the almost-dual of a Frobenius manifold of Saito type associated to the surface singularity $ X = \mathbb{C}^2 / G $, linking stability conditions to singularity theory.
• The Stokes matrix of the quantum cohomology of a variety $ Z $ is conjecturally equal to the Gram matrix $ \chi(E_i, E_j) $ of a full exceptional collection, supporting Dubrovin’s conjecture.
• There is a suggestive duality: $ \mathrm{Def}(\mathcal{D}^b\mathrm{Coh}(X_1)) \cong \mathrm{Stab}(\mathcal{D}^b\mathrm{Coh}(X_2)) $ for mirror Calabi-Yau threefolds $ X_1, X_2 $, suggesting a purely algebraic-geometric mirror symmetry.
• Joyce’s recent construction of a flat connection on stability conditions for Calabi-Yau categories offers a promising route to defining global geometric structures on $ \mathrm{Stab}(\mathcal{D}) $, though extending it to derived categories remains challenging.