Quadratic Differentials As Stability Conditions

QUADRATIC DIFFERENTIALS AS STABILITY CONDITIONS TOM BRIDGELAND AND IVAN SMITH Abstract. We prove that moduli spaces of meromorphic quadratic differentials with simple zeroes on compact Riemann surfaces can be identified with spaces of stability conditions on a class of CY3 triangulated categories defined using quivers with potential associated to triangulated surfaces. We relate the finite-length trajectories of such quadratic differentials to the stable objects of the corresponding stability condition. Contents 1. Introduction 1 2. Quadratic differentials 13 3. Trajectories and geodesics 22 4. Period co-ordinates 32 5. Stratification by number of separating trajectories 42 6. Colliding zeroes and poles: the spaces Quad(S; M) 56 7. Quivers and stability conditions 64 8. Surfaces and triangulations 75 9. The category associated to a surface 84 10. From differentials to stability conditions 93 11. Proofs of the main results 100 12. Examples 112 References 121 1. Introduction In this paper we prove that spaces of stability conditions on a certain class of triangulated categories can be identified with moduli spaces of meromorphic quadratic differentials. The relevant categories are Calabi-Yau of dimension three (CY3), and are described using quivers with potential associated to triangulated surfaces. The observa- tion that spaces of abelian and quadratic differentials have similar properties to spaces of stability conditions was first made by Kontsevich and Seidel several years ago. On the one hand, our results provide some of the first descriptions of spaces of stability conditions on CY3 categories, which is the case of most interest in physics. On the other, they give a precise link between the trajectory structure of flat surfaces and the theory of wall-crossing and Donaldson-Thomas invariants. T.B. is supported by All Souls College, Oxford. I.S. is partially supported by a grant from the European Research Council. 2 TOM BRIDGELAND AND IVAN SMITH Our results can also be viewed as a first step towards a mathematical understanding of the work of physicists Gaiotto, Moore and Neitzke [14, 15]. Their paper [14] de- scribes a remarkable interpretation of the Kontsevich-Soibelman wall-crossing formula for Donaldson-Thomas invariants in terms of hyperkählergeometry. In the sequel [15] an extended example is described, relating to parabolic Higgs bundles of rank two. The mathematical objects studied in the present paper are very closely related to their phys- ical counterparts in [15], and some of our basic constructions are taken directly from that paper. We hope to return to the relations with Hitchin systems and cluster vari- eties in a future publication. In another direction, the CY3 categories appearing in this paper also arise as Fukaya categories of certain quasi-projective Calabi-Yau threefolds. That relation is the subject of a sequel [6] currently in preparation. In this introductory section we shall first recall some basic facts about quadratic differentials on Riemann surfaces. We then describe the simplest examples of the categories we shall be studying, before giving a summary of our main result in that case, together with a very brief sketch of how it is proved. We then state the other version of our result involving quadratic differentials with higher-order poles. We conclude by discussing the relationship between the finite-length trajectories of a quadratic differential and the stable objects of the corresponding stability condition. As a matter of notation, the triangulated categories we consider here are most natu- rally labelled by combinatorial data consisting of a smooth surface S equipped with a collection of marked points M ⊂ S, all considered up to diffeomorphism. Initially S will be closed, but in the second form of our result S can have non-empty boundary. The quadratic differentials we consider live on Riemann surfaces S whose underlying smooth surface is obtained from S by collapsing each boundary component to a point. To avoid confusion, we shall try to preserve the notational distinction whereby S refers to a smooth surface, possibly with boundary, whereas S is always a Riemann surface, usually compact. All these surfaces will be assumed to be connected. We fix an algebraically closed field k throughout. 1.1. Quadratic differentials. A meromorphic quadratic differential φ on a Riemann ⊗2 surface S is a meromorphic section of the holomorphic line bundle !S . We empha- size that all the differentials considered in this paper will be assumed to have simple zeroes. Two quadratic differentials φ1; φ2 on Riemann surfaces S1;S2 are considered to ∗ be equivalent if there is a holomorphic isomorphism f : S1 ! S2 such that f (φ2) = φ1. Let S be a compact, closed, oriented surface, with a non-empty set of marked points M ⊂ S. We assume that if g(S) = 0 then jMj > 3. Up to diffeomorphism the pair (S; M) is determined by the genus g = g(S) and the number d = jMj > 0 of marked points. We use this combinatorial data to specify a union of strata in the space of meromorphic quadratic differentials; this will be less trivial later when we allow S to have boundary. QUADRATIC DIFFERENTIALS AS STABILITY CONDITIONS 3 By a quadratic differential on (S; M) we shall mean a pair (S; φ), where S is a compact and connected Riemann surface of genus g = g(S), and φ is a meromorphic quadratic differential with simple zeroes and exactly d = jMj poles, each one of order 6 2. Note that every equivalence class of such differentials contains pairs (S; φ) such that S is the underlying smooth surface of S, and φ has poles precisely at the points of M. A quadratic differential (S; φ) of this form determines a double cover π : S^ ! S, called the spectral cover, branched precisely at the zeroes and the simple poles of φ. This cover has the property that π∗(φ) = ⊗ for some globally-defined meromorphic 1-form . We write S^◦ ⊂ S^ for the complement of the poles of . The hat-homology group of the differential (S; φ) is defined to be ^ ^◦ − H(φ) = H1(S ; Z) where the superscript indicates the anti-invariant part for the action of the covering involution. The 1-form is holomorphic on S^◦ and anti-invariant, and hence defines a de Rham cohomology class, called the period of φ, which we choose to view as a group homomorphism Z ^ Zφ : H(φ) ! C; γ 7! : γ There is a complex orbifold Quad(S; M) of dimension n = 6g − 6 + 3d parameterizing equivalence-classes of quadratic differentials on (S; M). We call a quadratic differential complete if it has no simple poles; such differentials form a dense open subset Quad(S; M)0 ⊂ Quad(S; M). ^ The homology groups H(φ) form a local system over the orbifold Quad(S; M)0.A slightly subtle point is that this local system does not extend over Quad(S; M), but rather has monodromy of order 2 around each component of the divisor parameterizing differentials with a simple pole. It therefore defines a local system on an orbifold Quad~(S; M) which has larger automorphism groups along this divisor. There is a natural map Quad~(S; M) ! Quad(S; M); which is an isomorphism over the open subset Quad(S; M)0, and which induces an isomorphism on coarse moduli spaces. Fixing a free abelian group Γ of rank n, we can also consider an unramified cover Γ Quad (S; M) −! Quad~(S; M) of framed quadratic differentials, consisting of equivalence classes of quadratic differentials as above, equipped with a local trivalization Γ ∼= H^ (φ) of the hat-homology local system. 4 TOM BRIDGELAND AND IVAN SMITH In Section 4 we shall prove the following result, which is a variation on the usual existence of period co-ordinates in spaces of quadratic differentials. For this we need to assume that (S; M) is not a torus with a single marked point. Γ Theorem 1.1. The space of framed differentials Quad (S; M) is a complex manifold, and there is a local homeomorphism Γ (1.1) π : Quad (S; M) −! HomZ(Γ; C); obtained by composing the framing and the period. Γ In the excluded case the space Quad (S; M) is not a manifold because it has generic automorphism group Z2. 1.2. Triangulations and quivers. Suppose again that S is a compact, closed, oriented surface with a non-empty set of marked points M ⊂ S. For the purposes of the following discussion we will assume that if g(S) = 0 then jMj > 5. By a non-degenerate ideal triangulation of (S; M) we mean a triangulation of S whose vertex set is precisely M and in which every vertex has valency at least 3. To each such triangulation T there is an associated quiver Q(T ) whose vertices are the midpoints of the edges of T , and whose arrows are obtained by inscribing a small clockwise 3-cycle inside each face of T , as illustrated in Figure 1. Figure 1. Quiver associated to a triangulation. There are two obvious systems of cycles in Q(T ), namely a clockwise 3-cycle T (f) in each face f, and an anticlockwise cycle C(p) of length at least 3 encircling each point p 2 M. We define a potential W (T ) on Q(T ) by taking the sum X X W (T ) = T (f) − C(p): f p QUADRATIC DIFFERENTIALS AS STABILITY CONDITIONS 5 Consider the derived category of the complete Ginzburg algebra [16, 21] of the quiver with potential (Q(T );W (T )) over k, and let D(T ) be the full subcategory consisting of modules with finite-dimensional cohomology. It is a CY3 triangulated category of finite type over k, and comes equipped with a canonical t-structure, whose heart A(T ) ⊂ D(T ) is equivalent to the category of finite-dimensional modules for the completed Jacobi algebra of (Q(T );W (T )).

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