[Paper Review] Stability conditions on triangulated categories

Tom Bridgeland|ArXiv.org|Dec 17, 2002

Black Holes and Theoretical PhysicsPhysics and Astronomy6 references56 citations

TL;DR

This paper introduces stability conditions on triangulated categories as a categorical generalization of Harder-Narasimhan filtrations, using a central charge map and slicing into semistable subcategories. It proves that the space of locally-finite stability conditions on the derived category of a genus-one curve is naturally a manifold diffeomorphic to the universal cover of SL(2,R), with a free and transitive action by the group GL⁺(2,R).

ABSTRACT

This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from M.R. Douglas's notion of $Π$-stability. From a mathematical point of view, the most interesting feature of the definition is that the set of stability conditions $\Stab(\T)$ on a fixed category $\T$ has a natural topology, thus defining a new invariant of triangulated categories. After setting up the necessary definitions I prove a deformation result which shows that the space $\Stab(\T)$ with its natural topology is a manifold, possibly infinite-dimensional.

Motivation & Objective

• To define and formalize stability conditions on triangulated categories as a categorical framework for generalizing Harder-Narasimhan filtrations.
• To endow the set of stability conditions with a natural topology, making it a topological invariant of the category.
• To establish that the space of locally-finite stability conditions on the derived category of a curve is a manifold.
• To determine the global structure of the stability manifold for elliptic curves using group actions and representation theory.

Proposed method

• Define a stability condition as a pair (Z, P), where Z is a central charge homomorphism to ℂ and P assigns full subcategories P(φ) to real phases φ.
• Require that objects in P(φ) have central charge Z(E) = m(E)exp(iπφ) with m(E) > 0, and that P(φ+1) = P(φ)[1].
• Impose the orthogonality condition: Hom(A₁, A₂) = 0 for Aⱼ ∈ P(φⱼ) with φ₁ > φ₂.
• Demand that every nonzero object admits a finite filtration with semistable factors of strictly decreasing phases.
• Introduce the local-finiteness condition to ensure the heart of the t-structure is of finite length and the topology is well-behaved.
• Use deformation theory and the central charge map to show that the space Stab(𝒟) is locally homeomorphic to a complex vector space, hence a manifold.

Experimental results

Research questions

• RQ1 What conditions define a stability condition on a triangulated category, and how do they generalize classical Harder-Narasimhan filtrations?
• RQ2 How can one topologize the set of stability conditions to form a new invariant of triangulated categories?
• RQ3 What is the global geometric structure of the space of stability conditions on the derived category of a smooth projective curve?
• RQ4 How does the group GL⁺(2,R) act on the stability manifold of an elliptic curve, and what is its orbit structure?

Key findings

• The space of locally-finite stability conditions on the derived category of a smooth projective curve of genus one is a manifold diffeomorphic to the universal cover of SL(2,R).
• The action of the group GL⁺(2,R) on this space is free and transitive, so Stab(X) ≅ GL⁺(2,R) as a manifold.
• The central charge map Z: K(X) → ℂ induces a local homeomorphism from Stab(X) to the complex vector space Hom(𝒩(X), ℂ), which is two-dimensional.
• The quotient Stab(X)/Aut(D(X)) is isomorphic to GL⁺(2,R)/SL(2,Z), a ℂ*-bundle over the moduli space of elliptic curves.
• The standard stability condition with Z(E) = −deg(E) + i·rank(E) is unique up to the action of GL⁺(2,R), and all stability conditions arise from it via group action.
• The proof relies on the fact that all indecomposable sheaves are semistable in any stability condition, and that the central charge cannot be real-valued, forcing it to be an orientation-preserving isomorphism.