ERC Advanced Grant 2014 Edited Research proposal

Stability conditions, Donaldson-Thomas invariants and cluster varieties

StabilityDTCluster

Tom Bridgeland University of Sheffield 60 months

This proposal is concerned with the homological properties of Calabi-Yau threefolds, the geometric structures which play a crucial role in string theory. Rather than working directly with categories of sheaves, we focus on a closely-related class of models defined using quivers with potentials, which have themselves been the subject of intensive research over the last decade. Associated to a quiver with potential are two complex manifolds: the space of stability con- ditions and the cluster variety. Recent work by physicists Gaiotto, Moore and Neitzke suggests that there is a remarkable geometric relationship between these spaces, involving Donaldson- Thomas invariants and the Kontsevich-Soibelman wall-crossing formula. Work by the PI and others over the last couple of years has paved the way for a rigorous mathematical understanding of this relationship. Our proposal combines powerful general constructions with specific computable examples. We will work initially with a class of examples related to triangulated surfaces; here the relevant spaces can be identified with familiar objects in the topology of surfaces, including moduli spaces of quadratic differentials, projective structures and local systems. These examples already involve deep mathematics, and are closely related to quantum field theories of current interest in theoretical physics. Bridgeland StabilityDTCluster Edited grant proposal

1. Extended synopsis The two main characters in this proposal are the space of stability conditions [4, 5] and the cluster variety [16, 27]. Both have been the subject of intensive research over the last 10 years, but have rather different pedigrees. The space of stability conditions was invented to describe π- stability for D-branes in string theory, and to give a mathematical description of certain moduli spaces appearing in quantum field theory. More recently stability conditions have been applied to spectacular effect to problems in birational geometry. Cluster varieties on the other hand were invented to describe certain combinatorial structures appearing in a number of different mathematical subjects including representation theory, integrable systems and surface topology. Since then it has been appreciated that they also have close connections to homological algebra and CY3 categories. One of the main ideas of the proposal is that stability spaces and cluster varieties are linked in an extremely non-trivial way by the theory of generalised Donaldson-Thomas (DT) invariants, invented by Joyce and Kontsevich-Soibelman about five years ago [26, 29]. So far this link has only been understood in physical terms, in particular in the remarkable work of Gaiotto, Moore and Neitzke [19, 20]. A very basic and general fact is that if one considers a variation of stability conditions, the corresponding DT invariants exhibit discontinuous jumping behavior, which is described precisely by the wall-crossing formula of Kontsevich-Soibelman. The nature of this formula quickly leads to the inescapable conclusion that DT invariants should be interpreted as 1 defining Stokes data for a family of irregular connections on P parameterized by the space of stability conditions [10]. This is the main idea behind the physicists’ approach, but has not so far been satisfactorily addressed mathematically. 1.1. Background. We begin by briefly summarising some of the main mathematical objects that are important for our proposal. 1.1.1. Quivers and clusters. A quiver with potential (QWP) consists of an oriented graph Q together with a formal weighted sum W of oriented cycles in Q. Associated to a QWP is a CY3 triangulated category D = D(Q, W ): the bounded derived category of its complete Ginzburg ∼ ⊕n algebra. The Grothendieck group K0(D) = Z is the free abelian group on the vertices of Q. Define an algebraic torus ∗ ∼ ∗ n T = HomZ(K0(D), C ) = (C ) . The Euler form on K0(D) induces an invariant Poisson structure on T. When the quiver Q has no loops, the category D has a distinguished group of auto-equivalences Sph(D) generated by spherical twists in the vertex simples. Of crucial importance in the theory is the exchange graph E, whose vertices correspond to orbits of reachable t-structures in D under the action of Sph(D), and whose edges join t-structures differing by a tilt at a simple object. The cluster variety [ X = Tx x∈E is obtained by gluing overlapping copies of T, one for each vertex of the exchange graph, using certain birational Poisson maps called cluster transformations, determined by the edges of E. 1.1.2. Stability conditions. Associated to the category D is a complex manifold Stab(D) param- eterizing stability conditions on D. The points represent pairs (Z, P) consisting of a linear map Z : K0(D) → C called the central charge, and a full subcategory [ P = P(φ) ⊂ D φ∈R whose objects are said to be semistable. A basic fact is that the forgetful map

π : Stab(D) → HomZ(K0(D), C) is a local homeomorphism: deformations of the central charge lift uniquely to deformations of the stability condition. 2 Bridgeland StabilityDTCluster Edited grant proposal

The space Stab(D) carries a natural action of the group Aut(D) of auto-equivalences of the category, and we set Σ = Stab(D)/ Sph(D). There is a large open subset of Σ with a cell decomposition whose dual graph is the exchange graph E. Given a stability condition on D one can define DT invariants DT(α) ∈ Q which count semistable objects of a fixed class α ∈ K0(D) As the stability condition varies these numbers undergo discontinuous changes which are described by the Kontsevich-Soibelman (KS) wall- crossing formula discussed below. 1.1.3. Triangulated surfaces. An interesting and amenable class of examples of QWPs can be obtained [9, 31] from triangulations of marked, bordered surfaces (S, M). The associated CY3 triangulated category D = D(S, M) is independent of the chosen triangulation. For these cate- gories many of the above objects can be computed explicitly: (i) The exchange graph E coincides with the graph whose vertices are tagged ideal triangu- lations of (S, M), and whose edges are flips of such triangulations [15]. (ii) The cluster variety X is a dense open subspace of the stack of framed PGL2(C)-local systems on (S, M) [14]. (iii) The stability condition space Σ is the moduli space of meromorphic quadratic differentials on (S, M) with simple zeroes [9]. (iv) The DT invariants for a given stability condition count finite-length trajectories of the corresponding quadratic differential [9]. We emphasize that this class of examples includes some of the most basic examples in rep- resentation theory, including quivers of type A and D. They can also be viewed as Fukaya categories of certain non-compact Calabi-Yau threefolds [38]. Finally, they correspond to QFTs of current interest in theoretical physics [18, 20].

1.1.4. Wall-crossing formula. Given a stability condition (Z, P) on our CY3 triangulated cate- gory D, there is a countable collection of rays in C of the form R>0 ·Z(E) for E ∈ D a semistable object. To each such active ray ` one can associate the formal function on the torus T X α DT` = DT (α) · x . Z(α)∈` In good cases (for example those coming from triangulated surfaces), the Hamiltonian flow of DT` gives a well-defined element S` of the group G = Bir{−,−}(T) of Poisson birational automorphisms of T. As the stability condition σ varies, the active rays ` move, and the elements S` undergo discontinuous changes. This behaviour is completely described by the KS wall-crossing formula [29], which states that for any convex sector ∆ ⊂ C, the clockwise-ordered product Y S∆ = S` ∈ G `∈∆ remains constant, providing that no active ray ` crosses the boundary of ∆. We remark that this formula makes formal sense and in fact determines the wall-crossing behaviour of DT invariants without any assumptions on the Hamiltonian flow of DT`. 1.2. DT invariants as Stokes data. The crucial observation underlying our proposal is that the wall-crossing formula can be interpreted as the statement that the elements S` ∈ G play 1 the role of Stokes data for an isomonodromic family of meromorphic connections on P taking values in the group G. This is also one of the crucial ingredients in the work of Gaiotto, Moore and Neitzke [19]. 1 To explicitly construct the connection on P corresponding to a given stability condition we must invert the Stokes map that sends a meromorphic connection to its Stokes data. This 3 Bridgeland StabilityDTCluster Edited grant proposal

∗ involves solving the following Riemann-Hilbert (RH) problem: find functions X : C → G satis- fying ∗ (RH1) As t ∈ C crosses an active ray ` in the clockwise direction, the function X has a discontinuity described by X(t) 7→ X(t) ◦ S`. (RH2) The composite X(t) ◦ exp(Z/t) → 1 as t → 0 along any ray. (RH3) The function X(t) has moderate growth as t → ∞ along any ray. Of course there are technical issues with considering connections taking values in the infinite- dimensional group G, and one of the main aims of the project is to make rigorous sense of this. Problem 1: Construct the isomonodromic family of connections whose Stokes data is given by the Donaldson-Thomas invariants. In particular, this will involve studying existence and uniqueness for solutions to the above RH problem. We expect this to be difficult. The physics paper [19] claims existence and uniqueness for a related RH problem (at least for |Z|  0), and their approach will be our starting point. The finite-dimensional situation, for example when G = GLn(C), is well understood (see Part B2), and will give a useful guide. Moreover, in the triangulated surface case we should be able understand solutions to the RH problem geometrically (see Problem 2). In the simplest case, when a ray ` contains a single stable object, the automorphism S` is a cluster transformation, and we expect the solution to Problem 1 to involve cluster theory in an essential way. In particular, the cluster variety has a degeneration to the torus T, and one can try to solve the RH problem by taking as first approximation X(t) = exp(Z/t) in the central fibre, and then solving order-by-order in the degeneration parameter. We emphasise that failure on this problem will not derail the project: the situation is rather that much of the rest of the project will give insight into how to solve this problem.

1.3. Stability conditions and cluster varieties. The stability condition space Σ and the cluster variety X are complex manifolds of the same dimension built up out of simple pieces using the combinatorics of the exchange graph. But Σ has a cell decomposition, whereas X is built by gluing tori via birational maps. One of our main aims is to unravel the highly non-trivial relationship between these two spaces. Given a solution Xσ to the RH problem at each point σ ∈ Stab(D) we can set t = 1 and apply the resulting birational automorphism to the identity id ∈ T to obtain a partially-defined holomorphic map Stab(D) 99K T.

Conjecture 1.1. The map σ 7→ Xσ(1)(id) extends to a local isomorphism F : Stab(D) → X . The main evidence for this comes from the case of triangulated surfaces. There, a stability condition [9] corresponds to a pair (S, φ) consisting of a Riemann surface S and a meromorphic quadratic differential φ. The set of such pairs can be identified with the set of projective structures on the surface by considering ratios of solutions to the equation d2y(x) − φ(x)y(x) = 0. dx2 Any projective structure has an associated framed local system, so we get a map from quadratic differentials to framed local systems. In the case when the surface has no marked points the resulting map is known to be a local isomorphism [23]. The general case has been studied by Iwaki and Nakanishi [24], who prove the required asymptotic property (RH2). A full description in the case relevant to the A2 quiver was given by Masoero [33]. Problem 2: Study the Riemann-Hilbert problem geometrically in the triangulated surface examples. 4 Bridgeland StabilityDTCluster Edited grant proposal

The first step is to construct the map of Conjecture 1.1 by linking quadratic differentials to local systems via projective structures: here we will build on the careful analysis of [24]. This first step provides a candidate solution for the RH problem specialised at id ∈ T. More ambitiously we can try to give a complete geometric description to the RH problem. The physics paper [20] uses the highly non-trivial identification between Higgs bundles and local systems to solve a closely related RH problem. Since our problem arises as the so-called conformal limit [18] of theirs, it seems reasonable to expect a geometric solution in this case also. Lurking behind the scenes in the story described above is the Hall algebra of the category D. For example, the KS wall-crossing formula is first established at this level. However Hall algebras of triangulated categories are currently poorly-understood. The triangulated surface examples can provide interesting test cases in this context. It is known [37] that spherical objects in D(S, M) up to shift are in bijection with simple closed curves on a double cover of S. This and other considerations suggest that the relevant construction from surface topology is the famous Goldman Lie algebra [21]. Problem 3: Formulate a precise relationship between the Hall algebra of the cate- gory D(S, M) and the Goldman Lie algebra. Goldman originally introduced his Lie algebra to study the Poisson geometry of the character variety, and we expect analogues of his results to apply in the more general context of Hall algebras and cluster varieties.

1.4. Geometric structures on stability spaces. It is natural to ask whether DT invariants control some kind of geometric structure on the space of stability conditions. A motivating analogy is the Frobenius structure encoded by the genus zero Gromov-Witten (GW) invariants. Note that in the geometric setting, when D = Db Coh(X) is the derived category of coherent sheaves on a Calabi-Yau threefold, the GW/DT correspondence [34, 35] shows that DT invariants are indeed closely related to GW invariants. Joyce [25] constructed holomorphic generating functions of DT invariants and used them to define a formal flat metric on Stab(D) and its associated Levi-Civita connection. Unfortunately his construction involves infinite sums which look unlikely to converge. A more conceptual approach was explained in [10] where it was shown that Joyce’s connection is related to the RH problem described above. More precisely, given a solution to this problem, the function σ 7→ Xσ(t)(id) ∈ T become constant in the limit t → ∞, and the derivative of this map then defines the Joyce connection. This gives hope that although Joyce’s sums could diverge, his metric and connection may nonetheless exist. Conjecture 1.2. There is an almost Frobenius structure [13] on Stab(D) whose metric is the Joyce metric, whose Euler vector field is given by the central charge Z, and in which the functions Z(α) for α ∈ K0(D) are the twisted periods. Note that many computed examples of spaces of stability conditions do have natural Frobenius structures [5, 6, 8], and the above conjecture is the result of detailed study of such examples. Problem 4: Provide concrete evidence for Conjecture 1.2. If we can solve Problem 2 we should be able to compute the Joyce metric and connection in the triangulated surface examples and compare with existing Frobenius structures. In the geometric case D = Db Coh(X), mirror symmetry leads one to view Stab(D) as an extension of big quantum cohomology, and one can try to get a handle on the Joyce connection using the DT/GW correspondence: this leads to a heuristic picture which does indeed identify the Joyce connection near large volume with the Levi-Civita connection on big quantum cohomology. As a final problem we would like to extend the Bridgeland-Smith result [9] to give new examples of spaces of stability conditions on CY3 categories.

Problem 5: Calculate further examples of spaces of stability conditions for CY3 categories arising from QWPs. 5 Bridgeland StabilityDTCluster Edited grant proposal

In particular, we plan to consider quadratic differentials with non-simple zeroes, and under- stand how the different strata in the space of differentials, corresponding to different orders of zeroes, and hence different CY3 categories, fit together. We hope that this will lead to an under- standing of partial compactifications of spaces of stability conditions, which would be important in a variety of settings. Another direction is to try to generalise the Bridgeland-Smith result to cover all dimer models. This is a large class of QWPs whose associated CY3 categories include the derived categories of interesting non-projective Calabi-Yau threefolds (for example, total spaces of canonical bundles of toric del Pezzo surfaces). We expect to be able to understand such examples in terms of quadratic differentials on orbifold curves.

6 Bridgeland StabilityDTCluster Edited grant proposal

ERC Advanced Grant 2014 Research proposal [Part B2]

Stability conditions, Donaldson-Thomas invariants and cluster varieties

StabilityDTCluster

Tom Bridgeland University of Sheffield

Scientific proposal

2. State-of-the-art and objectives This proposal concerns the homological properties of Calabi-Yau threefolds. Rather than working directly with derived categories of coherent sheaves on such varieties we work with a simpler class of CY3 triangulated categories defined algebraically using a quiver with poten- tial. Associated to such a category are various objects - spaces of stability conditions, cluster varieties, Hall algebras, Donaldson-Thomas invariants - that are related in intriguing and com- plicated ways. We are convinced that unravelling these relationships will lead to a new level of understanding of the homological algebra of the CY3 condition, and more generally, of the mathematics of quantum field theory. 2.1. General context. Stability conditions were first introduced by the PI in an effort to understand work of physicist M. Douglas on Π-stability for D-branes [4, 6, 12]. The idea is that the derived category of coherent sheaves D = Db Coh(X) on a compact Calabi-Yau variety X arises in string theory as the category of topological B-branes in the non-linear sigma model associated to X. At each point on an auxilliary parameter space known as the stringy K¨ahler moduli space, there is a distinguished subcategory P ⊂ D whose objects are called BPS branes. The PI axiomatised the properties of this subcategory to give the notion of a stability condition on a triangulated category. One can then consider the space of all such stability conditions on the category D as a first mathematical approximation to the stringy K¨ahlermoduli space. This is interesting because mirror symmetry identifies this space with the complex moduli space of the mirror variety. In the last decade the theory of stability conditions has had a major impact on homological algebra. The use of geometric techniques to understand the internal structure of triangulated categories has opened up new approaches to old problems [1], and the increased flexibility allowed by considering moduli of complexes has led to spectacular progress in our understanding of the birational geometry of classical moduli spaces [2]. The real power of the theory however, is as an indespensable tool for understanding string theory and other quantum field theories. Classical mathematical approaches to QFT tend only to be valid near some limiting point: the use of homological techniques and spaces of stability conditions seems to be essential to obtain a global understanding of the physically relevant moduli spaces. The combinatorics of clusters was first identified by S. Fomin and A. Zelevinsky [16] in the context of representation theory. Soon afterwards V. Fock and A. Goncharov [14] used the same combinatorics to give rational parameterizations of moduli spaces of local systems on smooth surfaces. Before long instances of the cluster phenomenon had been identified in a wide range of mathematical fields [17]. It was Kontsevich and Soibelman [29] who first realised the link with 7 Bridgeland StabilityDTCluster Edited grant proposal

Calabi-Yau 3-categories and the emerging theory of generalized Donaldson-Thomas invariants. Their ideas were elaborated on by Keller [27] and Nagao [36] and it is this point-of-view on clusters which will be most important for us. A source of great inspiration for this proposal is the remarkable work of physicists Gaiotto, Moore and Neitzke (GMN) [19, 20]. This describes a general construction relating hyperk¨ahler geometry to DT invariants and involving cluster theory in an essential way. The material we dis- cuss below is the so-called conformal limit [18] of the original GMN story, and should be easier to understand mathematically since it remains entirely within the holomorphic world. Nonetheless, it is one of the long-term goals of our project to improve our understanding of the relationship between stability conditions and cluster varieties to the point where a mathematically rigorous understanding of the GMN construction becomes a real possibility. It is an essential feature of our proposal that the rather abstract theory concerning stability conditions and DT invariants is complemented by the study of a class of examples in which explicit computations can be carried out, and in which the general theory can be related to a rich existing body of mathematical knowledge. The class of examples that we have in mind are the quivers with potential arising from triangulations of surfaces [9, 31]. These have already been the subject of several remarkable mathematical results, some of which are discussed below. Nonetheless, we are convinced that there is much more to understand. In particular, the physics paper [20], which can be viewed as an extended example of the general theory described in [19], is concerned with precisely this class of quivers. Although we shall mostly focus on CY3 categories defined by quivers with potential, the problems we consider in this proposal are also relevant in the geometric context of the derived category of coherent sheaves on a compact Calabi-Yau threefold. For now these more complicated categories are mostly out of reach, but it is reasonable to expect that the methods we use here are sufficiently general that they will also apply in some form to this setting. In particular, the questions about geometric structures on stability conditions we raise below are particularly important in this context. It is by now well-understood that the space of stability conditions is an extended version of the stringy K¨ahlermoduli space, akin to big quantum cohomology rather than small quantum cohomology [6]. To obtain a global mathematical description of the stringy K¨ahlermoduli space, one needs some kind of Frobenius-like structure on the space of stability conditions which would enable one to pick out the required submanifold. Near large volume this is given by the usual quantum cohomology construction, but what is required is a more global construction. This then is our second long-term goal: to construct Frobenius-like structures on spaces of stability conditions which in the geometric setting will give a mathematical description of the stringy K¨ahlermoduli space. If we can make even some progress towards this goal we will have greatly enhanced our understanding of the inner-workings of mirror symmetry.

2.2. Background. In this section we introduce some necessary background, and review some of the existing results that we wish to build on.

2.2.1. Stability conditions and clusters. Let D be a triangulated category whose Grothendieck ∼ ⊕n group K0(D) = Z is free of finite rank. Associated to D is a complex manifold Stab(D) parameterizing stability conditions on D. The points represent pairs (Z, P) consisting of a linear map Z : K0(D) → C called the central charge, and a full subcategory [ P = P(φ) ⊂ D φ∈R whose objects are said to be semistable. These structures are required to satisfy a simple set of axioms together with an extra property called the support condition. A basic fact is that the forgetful map

π : Stab(D) → HomZ(K0(D), C) 8 Bridgeland StabilityDTCluster Edited grant proposal is a local homeomorphism: deformations of the central charge lift uniquely to deformations of the stability condition. The space Stab(D) carries a natural action of the group Aut(D) of triangulated auto-equivalences of the category D. A quiver with potential (QWP) is a pair consisting of an oriented graph Q together with a formal weighted sum W of oriented cycles. We shall always assume that the quiver Q has no loops and that the potential W is a formal sum of cycles of length at least three. Associated to this data is a CY3 triangulated category D = D(Q, W ) defined to be the bounded derived category of the complete Ginzburg algebra. The category D(Q, W ) comes equipped with a canonical t-structure, whose heart A(Q, W ) ⊂ D(Q, W ) is equivalent to the category of finite-dimensional modules for the complete Jacobi algebra of (Q, W ), and is therefore a finite-length abelian category with n simple objects Si indexed by ∼ ⊕n the vertices of Q. It follows that K0(D) = Z . The objects Si are spherical and hence define spherical twist functors

TwSi ∈ Aut(D). The subgroup of Aut(D) generated by these auto-equivalences is denoted Sph(D). Of crucial importance in the theory is the exchange graph E. This has a combinatorial definition involving cluster transformations, but it was proved by Keller and others [11, 27] that it also has a clear categorical meaning: its vertices correspond to orbits of hearts in D under the action of Sph(D), while the edges join hearts that are related by the operation of tilting at a simple object. It follows immediately from this description that a large open subset of the quotient space Σ = Stab(D)/ Sph(D) has a chamber decomposition with dual graph E. Define an algebraic torus ∗ ∼ ∗ n T = HomZ(K0(D), C ) = (C ) . The Euler form on K0(D) induces a translation-invariant Poisson structure on T. The cluster variety X is obtained by gluing overlapping copies of T, one for each vertex of the exchange graph, using birational Poisson maps determined by the edges of E. 2.2.2. Quivers with potential from triangulated surfaces. The quivers with potential defined by triangulated surfaces provide a wonderful testing ground for the general theory. All the major objects one wants to consider generally are explicitly computable in these cases in terms of known objects with deep geometric content. Not only does this allow one to perform tests of general theories, it also suggests new and unexpected relationships between the abstractly- defined constructions. It is worth emphasizing that this class of examples includes some of the most basic examples in representation theory, including quivers of type A and D. The resulting CY3 categories can also be identified with Fukaya categories of non-compact Calabi-Yau threefolds [38]. Finally, these examples are intimately related to QFTs of current interest in theoretical physics [20]. Given all this, it is perhaps not surprising that this area provides such a happy hunting ground for interactions between algebra, geometry and theoretical physics. A marked bordered surface is a pair (S, M) consisting of a compact oriented surface S, possibly with boundary, and a non-empty set of marked points M ⊂ S. An ideal triangulation of (S, M) is a triangulation of S whose vertices are precisely the points M. If the triangulation is non- degenerate there is a naturally associated QWP whose quiver is illustrated in Figure 1. If two ideal triangulations are related by a flip in an edge, the associated QWPs are related by a mutation at the corresponding vertex. General results of Keller and Yang [28] then imply that the triangulated categories associated to the two QWPs are equivalent. Since any two 9 Bridgeland StabilityDTCluster Edited grant proposal ideal triangulations are related by a chain of flips it follows that there is a well-defined CY3 triangulated category D(S, M) associated to the surface (S, M). Labardini-Fragoso [31] showed how to deal with degenerate triangulations, and also slightly more general combinatorial objects called tagged triangulations.

Figure 1. Quiver associated to a non-degenerate triangulation.

In these examples we can relate the objects we are interested in to existing objects in the topology of surfaces: (1) Let Q be the quiver associated with a triangulation of (S, M). By the above, the mutation class of Q is uniquely determined by (S, M). We can form a graph E./(S, M) whose vertices are tagged triangulations and whose edges are flips.

Theorem 2.1 (Fomin, Shapiro and Thurston [15]). The exchange graph E(Q) is iso- morphic to E./(S, M).

(2) Consider a PGL2(C)-local system ∇ on the surface S \ M.A framing of ∇ is defined 1 to be a choice of flat section of the associated P -bundle over the punctured boundary ∂S \ M. One can then define a moduli space L(S, M) of framed local systems. Theorem 2.2 (Fock and Goncharov [14]). There is a dense Zariski-open subset of L(S, M) which is isomorphic to the cluster variety X (Q). (3) Let MCG(S, M) denote the mapping class group of the surface S relative to the marked points M. Recall the subgroup Sph(D) ⊂ Aut(D) generated by spherical twists. Theorem 2.3 (Bridgeland and Smith [9]). There is an isomorphism ∼ Aut(D)/ Sph(D) = MCG(S, M). (4) The pair (S, M) is determined up to isomorphism by the genus g = g(S) and a sequence of k unordered multiplicities mi ≥ 0 determining the number of marked points on each boundary component. Define a space Q(S, M) of equivalence-classes of pairs (S, φ), where S is a Riemann surface of genus g and φ is a meromorphic quadratic differential having k poles with multiplicities mi + 2. Let Quad(S, M) ⊂ Q(S, M) be the open subset consisting of differentials with simple zeroes. Theorem 2.4 (Bridgeland and Smith [9]). There is an isomorphism of complex orbifolds ∼ Stab(D)/ Aut(D) = Quad(S, M). (5) Fix a stability condition σ corresponding to a quadratic differential (S, φ) in the space Quad(S, M). There is a canonical identification between K0(D) and the so-called hat- homology group of (S, φ), and any finite-length horizontal trajectory of the differential 10 Bridgeland StabilityDTCluster Edited grant proposal

has a naturally associated class in this group. Recall that the DT invariants DT(α) for the stability condition σ can be equivalently repackaged in terms of BPS invariants Ω(α) by the multi-cover formula X 1 DT(α) = · Ω(β). k2 α=kβ These BPS invariants often have simpler expressions than the DT invariants and are conjecturally always integers. Theorem 2.5 (Bridgeland and Smith [9]). For a generic stability condition σ, the BPS invariant Ω(α) counts the number of finite-length horizontal trajectories of the corre- sponding quadratic differential φ which have hat-homology class α. Here, a saddle con- nection counts for +1, and a ring domain for −2. We hope to add to this list of results: in particular the material of Section 2.4.3 below would give another major link between homological algebra and surface topology. 2.3. Donaldson-Thomas invariants as Stokes factors. At the heart of our proposal is the idea that over the space of stability conditions on our CY3 triangulated category D is an 1 isomonodromic family of irregular connection on P taking values in the group G of birational Poisson automorphisms of the algebraic torus ∗ T = HomZ(K0(D), C ). It is this family of connections which enable us to relate the space of stability conditions to the cluster variety, and which generate the important geometric structures that we wish to study. What makes things very difficult is that the connections are given to us via their Stokes data, which is encoded by the DT invariants of the category: to obtain the connections themselves we must invert a Riemann-Hilbert map, a problem made highly non-trivial by the fact that the group G is infinite-dimensional. 2.3.1. Stokes factors in the finite-dimensional case. We shall first discuss Stokes phenomena in the case of a finite-dimensional group. The story in this case goes back at least to the work of Jimbo, Miwa and Ueno in the early 1980s. We refer to [10, Sections 2–3] and the references given there for more details. Set G = GLn(C), g = gln(C) and consider the root decomposition od od M g = h ⊕ g , g = gα, α∈Φ where h ⊂ g is the Cartan subalgebra of diagonal matrices, god ⊂ g is the subspace of matrices ∗ ∗ ∗ with zeroes on the diagonal, and Φ = {ei − ej } ⊂ h is the root system of G. We also write hreg ⊂ h for the subset consisting of diagonal matrices with distinct eigenvalues. reg od Fix matrices U = diag(u1, ··· , un) ∈ h and V ∈ g , and consider the meromorphic 1 connection on the trivial G-bundle over P U V  (1) ∇ = d − + dt. t2 t It has an irregular singularity at t = 0 and a regular singularity at t = ∞. The gauge equivalence class of ∇ at t = 0 is determined by the associated Stokes data whose definition we now recall. ∗ By a ray we mean a subset of C of the form R>0 exp(iπφ). We denote by Hr the corresponding half-plane ∗ Hr = {z = uv : u ∈ r, Re(v) > 0} ⊂ C . The Stokes rays of the connection (1) are the rays

R>0(ui − uj) = R>0 · U(α), ∗ ∗ ∗ where α = ei − ej ∈ Φ is a root, and U(α) denotes the pairing between U ∈ h and α ∈ Φ ⊂ h . 11 Bridgeland StabilityDTCluster Edited grant proposal

Given a non-Stokes ray r, a general existence result states that there is a canonical flat section Xr : Hr → G of the connection ∇, uniquely defined by the condition U/t Xr(t) · e → 1 as t → 0 in Hr.

As we vary the ray r, this canonical flat section Xr remains unchanged until r crosses a Stokes ray, where it jumps. More precisely, if ` is a Stokes ray, and r± are small perturbations of `, then the Stokes factor S` corresponding to ` is the element of G defined by

Xr+ (t) = Xr− (t) · S` for t ∈ Hr+ ∩ Hr− .

It is easy to check that S` lies in the unipotent subgroup N` ⊂ G corresponding to the roots α for which U(α) ∈ `.

We can piece the canonical sections Xr(t) together to obtain a discontinuous flat section X(t) ∗ defined for all t ∈ C : set X(t) to be equal to Xr(t) when t ∈ r. Then X(t) satisfies the Riemann-Hilbert problem (RH1) As t crosses an active ray ` in the clockwise direction, the function X has a discontinuity described by

X(t) 7→ X(t) ◦ S`. (RH2) The composite X(t) ◦ exp(U/t) → 1 as t → 0 along any ray. (RH3) The function X(t) has moderate growth as t → ∞. Consider again the connection (1). For each Stokes ray ` we can write the corresponding Stokes factor S` uniquely in the form X od S` = exp(`) where ` = α ∈ g . Z(α)∈` We can then consider the Stokes map : god → god which sends V ∈ god to the element  = P α∈Φ α encoding the Stokes data of the connection (1). The Stokes map is known to be holomorphic and a local isomorphism. Of course it depends on the chosen element U ∈ hreg. Let us now vary the matrices U, V in such a way that the Stokes data of the connection (1) remains constant. More precisely, the Stokes rays may collide and separate as U varies, but we ∗ insist that for any convex sector Σ ⊂ C the clockwise product Y SΣ = S` `∈Σ remains constant unless a Stokes ray crosses the boundary of Σ. Such variations are called isomonodromic. It can be shown that a variation is isomonodromic precisely if the residue V varies with U according to the following differential equation X (2) dVα = [Vβ,Vγ] d log γ. β+γ=α P ∗ Here we decompose V = α∈Φ Vα with Vα ∈ gα and consider an element γ ∈ Φ ⊂ h as defining a linear function on hreg. We can go further and ask how the canonical sections Xr(t) vary with U in an isomonodromic 1 reg variation. This results in a flat connection on the trivial G-bundle over P × h given by the formula U dt X dα dU  (3) ∇ = d − + V + V + , t t α α t α∈Φ 1 which restricts to the connection (1) on each fibre {U} × P . In particular, this defines a 1- parameter family of flat G-connections over hreg. 12 Bridgeland StabilityDTCluster Edited grant proposal

2.3.2. The infinite-dimensional case. We now describe a precise analogy between the finite- dimensional Stokes phenomena considered in the last section, and the theory of stability condi- tions and Donaldson-Thomas invariants. Consider the Lie algebra g of Poisson vector fields on od the algebraic torus T. It has a decomposition g = h ⊕ g , where the Cartan subalgebra

h = HomZ(K0(D), C), consists of translation-invariant vector fields on T, and the second factor od M M α g = gα = C · x . ∗ ∗ α∈K0(D) α∈K0(D) consists of Hamiltonian flows, and is the Poisson algebra of non-constant algebraic functions on T. The role of the root system Φ is played by ∗ Φ = K0(D) = K0(D) \{0}. The Lie bracket in g is given explicitly by

 α1 α2  α2 α1 α1+α2 (θ1, x ), (θ2, x ) = θ1(α2) · x − θ2(α1) · x + χ(α1, α2) · x . It is natural to consider completions of the Lie algebra g, but we are not in a position to be precise about this at present. Similarly, it is not clear exactly what one should take to be the group G in general, but for now we will take it be the group

G = Bir{−,−}(T) of birational Poisson automorphisms of T. Given a stability condition σ = (Z, P) we define a ray R>0 exp(iπφ) to be active if the subcategory P(φ) ⊂ D is nonzero. Associated to each such ray ` = R>0 exp(iπφ) there is a formal power series X α DT` = DT(α) · x , Z(α)∈` where DT (α) ∈ Q is the DT invariant counting σ-semistable objects of class α. In general Q this can be thought of as living in the nilpotent topological Lie algebra Z(α)∈` gα, and hence defining an element S` of the unipotent topological group obtained by formal exponentiation. However, in the examples coming from triangulated surfaces, the formula ∗  S` (x) = exp DT`, − (x) ∗ gives a well-defined automorphism S` of the field of rational functions C(T), and hence we can view S` as living in the group G. We can interpret the elements S` ∈ G as defining Stokes factors for an irregular connection of the form Z f  ∇ = d − + dt, t2 t where f lies in god, or more realistically, in some completion thereof. The Kontsevich-Soibelman wall-crossing formula is precisely the condition that this family of connections is isomonodromic. Note that if we patch together the canonical solutions to this connection as in the last section the resulting function will satisfy the Riemann-Hilbert problem discussed in Part B1. One of the main aims of the proposal is to rigorously construct the isomonodromic family of connections whose Stokes data is given by Donaldson-Thomas invariants. This is a difficult problem. Our approach will be threefold: (a) Analytic methods as in [19] should give existence of solutions for large |Z|, (b) Use the degeneration of the cluster variety to the torus T and try to construct a solution order-by-order in the degeneration parameter, (c) Understand what to expect by studying the triangulated surface examples (see below). We re-emphasise that failure on this problem will not derail the project: the situation is rather that much of the rest of the project will generate insight into how to solve this problem. 13 Bridgeland StabilityDTCluster Edited grant proposal

2.4. Stability conditions and cluster varieties. A major goal of our proposal is to under- stand the relationship between the space of stability conditions of a QWP and the corresponding cluster variety. We aim to prove that there is a natural holomorphic map between these spaces which is a local isomorphism. We expect this map to be given in terms of the canonical solutions to the isomonodromic family of connections discussed in the last section. More precisely, given a canonical solution Xσ(t) to the connection determined by the stability condition σ ∈ Stab(D), we can evaluate at t = 1 and apply to the identity id ∈ T. If we then let the stability condition σ vary we will obtain a partially-defined holomorphic map Stab(D) 99K T.

Conjecture 2.6. The map σ 7→ Xσ(1)(id) extends to a local isomorphism F : Stab(D) → X . To prove this conjecture directly would involve a complete solution to Problem 1. We will first try to understand the map F geometrically in the triangulated surface examples.

2.4.1. Projective structures and quadratic differentials. To understand the relationship between stability conditions and the cluster variety in the traingulated surface examples we need to say something about projective structures. We first do this in the simpler holomorphic case corresponding to |M| = 0, although this is not strictly relevant since the definitions and results of Section 1.2.2 definitely need M to be non-empty. 1 Recall that a projective structure on a smooth surface S is an atlas of charts φα : Uα → P whose transition functions are constant elements of PGL2(C). Such a structure induces, in particular, a complex structure on S. Let P(g) denote the space parameterizing equivalence- classes of surfaces S of genus g equipped with a projective structure, and let Q(g) be the space of equivalence-classes of Riemann surfaces C equipped with a holomorphic quadratic differential. Both spaces have obvious maps to the space M(g) of Riemann surfaces of genus g: p: Q(g) → M(g), q : P(g) → M(g). The map p is a holomorphic vector bundle, and P(g) is an affine bundle over it: for any Riemann surface C ∈ M(g), the space of projective structures inducing the given complex 0 ⊗2 structure is a torsor for the vector space H (C, ωC ). Indeed, given a projective structure on S and a quadratic differential φ we can consider the differential equation d2y(x) (4) − φ(x)y(x) = 0, dx2 where x is a co-ordinate compatible with the given projective structure. Ratios of solutions to this give a new projective structure. Conversely, given two projective structures with the same underlying Riemann surface, one can take the Schwarzian derivative of one with respect to a local co-ordinate for the other, thus defining a quadratic differential on S. Any projective structure determines a PGL2(C)-local system on the underlying smooth surface S in the obvious way. We write L(g) for the space of such local systems. Theorem 2.7. The map P(g) → L(g) sending a projective structure to its monodromy repre- sentation is a local analytic isomorphism. On the other hand, choosing a holomorphic section of p: Q(g) → M(g) (and these do exist) gives an isomorphism of complex orbifolds Q(g) =∼ P(g). Putting these results together we see that there exist naturally-defined local isomorphisms Q(g) → L(g).

2.4.2. Meromorphic projective structures. To use the ideas of the last section to relate stability spaces to cluster varieties we need to generalise them to include spaces of meromorphic objects. Meromorphic quadratic differentials were treated in full generality in [9], but meromorphic projective structures do not seem to have a systematic treatment in the literature. Recall the space Q(S, M) defined in Section 1.2.2 (4) above. 14 Bridgeland StabilityDTCluster Edited grant proposal

Conjecture 2.8. There is a space P(S, M) of projective structures with appropriate singularities which can be identified non-canonically with Q(S, M). Taking monodromy gives an analytic map H : Q(S, M) → Loc(S, M) which is moreover a local isomorphism. An important question is whether the image of the map H lies in the dense open subset of Loc(S, M) given by the cluster variety X . If so we can use the Theorems 2.2 and 2.4 to conclude that there exist naturally-defined local analytic isomorphisms H : Stab(D) → X in the triangulated surface examples.

Example 2.1. Consider the case of the A2 quiver. This arises as the QWP associated to any triangulation of the marked bordered surface (S, M) consisting of a disc with 5 marked points on its boundary. It follows from the results of either [8] or [9] that there is an isomorphism of complex manifolds ∼ 2 Σ = Stab(D)/ Sph(D) = C \ ∆, where the discriminant locus is 2 2 3 ∆ = {(a, b) ∈ C : 27a − 4b = 0}. 2 One should view (a, b) ∈ C as corresponding to the quadratic differential φ(x) = (x3 + ax + b)dx⊗2 1 2 on P with a pole of order 7 at ∞. The subset C \ ∆ then corresponds to differentials with simple zeroes. The cluster variety X is the affine del Pezzo surface of degree 5 given by the equations

xi−1xi+1 = 1 + xi, i ∈ Z/5Z. It turns out that there is a holomorphic map H :Σ → X which is a local isomorphism. This map is obtained as above by considering the Schr¨odingerequation with cubic potential d2y(x) (5) − (x3 + ax + b) · y(x) = 0. dx2 More precisely, the rank two, first order differential equation associated to (5) defines a point in the space of labelled PGL2(C)-local systems on (S, M). We thus get a partially-defined holomor- phic map 2 H : C 99K X , and Masoero [33] has shown that this map is in fact globally-defined, and is a surjective local isomorphism.

It is natural to ask why the map H of Conjecture 1.8 has anything to do with the map F of Conjecture 2.6. It is probably impossible to prove anything here since the RH problem evaluated at id ∈ T will certainly have no unqiueness properties, and the map H is also very ∼ much non-unique since it depends on a choice of identification P(S, M) = Q(S, M). Nonetheless, ∗ we can introduce the variable t into the map H by using the C action on quadratic differentials (or equivalently stability conditions), and it then becomes interesting to check the conditions of the RH problem. The jumping behaviour (RH1) is inherent in the statement that we have a map to the cluster variety: if we use the horizontal trajectories of the differential to determine the triangulation defining the cluster co-ordinates then rotating the differential will make these co-ordinates jump in the required way. Moreover, arguments from WKB analysis as in [24] show that the maps described above have the required asymptotics (RH2). The property (RH3) seems really to de- pend on the identification between quadratic differentials and projective structures one chooses: it can be checked in the A2 example above that in fact H becomes constant as t → ∞. 15 Bridgeland StabilityDTCluster Edited grant proposal

A more ambitious problem is to give a geometric description of a full solution to the RH problem in the triangulated surface case. The physics paper [20] achieves this for a closely related RH problem in terms of Hitchin systems. Since our RH problem is the so-called conformal limit [18] of theirs, it seems reasonable to expect a geometric solution in this case also. 2.4.3. Goldman bracket. Goldman constructed a Lie algebra whose elements are free homotopy classes of oriented loops in a fixed oriented smooth surface S. The bracket is given by

X hγ1,γ2ip (6) [γ1, γ2] = (−1) γ1 ∗p γ2. p∈γ1∩γ2

Here we assume that γ1, γ2 are representatives of the given homotopy classes that intersect transversely. The notation γ1 ∗p γ2 means the concatenation of the two loops obtained by joining them at a given point p in their intersection, and hγ1, γ2ip ∈ {±1} measures the relative orientation of the two loops at p. Consider now a marked, bordered surface (S, M) and assume for technical reasons that all marked points lie on the boundary. Take a point (S, φ) ∈ Quad(S, M): this consists of a Riemann surface S whose underlying smooth surface is obtained from S by contracting the boundary circles of S, and a meromorphic quadratic differential φ on S with simple zeroes and poles at the images of the boundary circles. It was proved in [37] that there is a bijection γ 7→ Sγ between homotopy classes of paths on S connecting zeroes of φ (note that these lift to closed loops on the double cover of S determined by φ), and spherical objects in the category D = D(S, M) considered up to shift. Moreover, under this correspondence, there are isomorphisms ∗ ∼ M HomD(Sγ1 ,Sγ2 ) = C · p, p∈γ1∩γ2 which preserve the Z/2-grading, given on the right-hand side by the sign hγ1, γ2ip ∈ {±1} as above. At this point we are not able to be precise about exactly what flavour of Hall algebra we should consider. Nonetheless, the basic idea behind any Hall algebra is that the product of two objects should be the sum of all possible extensions between them. If one considers the corresponding commutator bracket, then using the CY3 condition we would therefore expect an expression of the form X |x| (7) [S1,S2] = (−1) S1 ∗x S2, ∗ x∈HomD(S1,S2) where |x| ∈ Z/2 is the degree of the class x, and S1 ∗x S2 denotes the object obtained by extending S1 by S2, or vice versa, depending on the sign of |x|. Formula (7) is very schematic, but is clearly closely analogous to (6). Our goal is to make this precise. We note that Goldman was originally motivated by applications to the Poisson geometry of the character variety. In particular he constructed a homomorphism from his Lie algebra to the Poisson algebra of functions on the character variety. In our more general setting this should become a homomorphism from the Hall Lie algebra to the Poisson algebra of functions on the cluster variety. 2.5. Geometric structures on stability spaces. There are many examples in which the space Stab(D) has the natural structure of a Frobenius manifold (or rather an almost Frobenius manifold, see [13]). We list some examples and references

(a) The CYd version of the A2 quiver is dealt with in [8]. (b) The space of stability conditions on the derived category of the total space of the canon- 2 2 ical bundle of P is treated in [5] and related to the quantum cohomology of P . (c) Mirror symmetry suggests that when X is a smooth projective Calabi-Yau threefold, Stab D(X) should be an extension of the big quantum cohomology of X [6, Section 7]. 16 Bridgeland StabilityDTCluster Edited grant proposal

(d) Unfolding spaces of quasi-homogeneous surface singularities are expected to coincide with spaces of stability conditions on the corresponding Fukaya-Seidel categories [6, Section 7]. (e) The spaces of quadratic differentials appearing in [9] are also expected to have Frobenius structures [30]. In any case it is natural to ask whether the G-valued connections considered in Section 1.3.2 can be used to induce any interesting geometry on the space Stab(D).

2.5.1. Joyce connection. If we can naively invert the Stokes map in the infinite-diemnsional context described in Section 1.3.2, the result will be a collection of holomorphic functions fα : Stab(D) → C which together satisfy the isomonodromy differential equation X (8) dfα = hβ, γi · fβfγ · d log γ. β+γ=α

∗ Here α, β, γ are elements of K0(D) , and hβ, γi ∈ Z denotes the Euler form. The element γ ∈ K0(D) defines a holomorphic function on Stab(D) by setting γ(Z, P) = Z(γ). As in (3) we obtain a 1-parameter family of flat connections on the trivial G-bundle over Stab(D). The limit as t → ∞ takes the form

X α (9) ∇ = d − fαx · d log α. ∗ α∈K0(D)

This connection takes values in (some completion of) the subalgebra g− ⊂ g consisting of vector fields which are anti-invariant under the inverse map (−1): T → T. Explicitly, this is the α −α od ∗ subalgebra spanned by the elements x + x ∈ g for all classes α ∈ K0(D) . There is a Lie algebra homomorphism I : g− → gl(h) where

h = Tid(T) = HomZ(K0(D), C).

Geometrically this is because the flow of any vector field in g− preserves id ∈ T and hence induces an endomorphism of the tangent space. In terms of the above generators we have

α −α I(x + x ) = Pα ∈ gl(h) where Pα(θ) = 2θ(α)hα, −i. We can naively apply I to the connection (9) to give a connection on the tangent bundle of Stab(D) defined by the formula X (10) ∇ = d − fαPα · d log α. ∗ α∈K0(D) Of course convergence issues now become acute. The expression (10) was first written down by Joyce [25], who showed that, ignoring convergence issues, it is the Levi-Civita connection for the metric X (11) g(θ1, θ2) = fα · θ1(α)θ2(α). α

One can make other constructions in the same spirit. For example, the central charge Z : K0(D) → C defines a vector field E on Stab(D) whose gradient V = ∇(E) is the endomorphism X V = fαPα ∈ gl(h). α One can formally check that V is covariantly constant with respect to the Joyce connection, and skew-symmetric with respect to the metric (11). 17 Bridgeland StabilityDTCluster Edited grant proposal

2.5.2. Possible Frobenius structure. The structures appearing in the last section are strongly reminiscent of structures naturally occuring in the theory of Frobenius manifolds. Based on common features of the examples listed above we suggest the following idea. Conjecture 2.9. There is an almost Frobenius structure on Stab(D) whose metric is the Joyce metric, whose Euler vector field is given by the central charge Z, and in which the functions Z(α) for α ∈ K0(D) are twisted periods. We note that the Poisson bracket of the Frobenius manifold then becomes the obvious Poisson structure on Stab(D) induced by the Euler form on K0(D), and the Levi-Civita connection is the Joyce connection (10). We hope to provide some concrete evidence for this conjecture. This is a difficult problem, but even limited progress would be very exciting. Here are some approaches we can take: (a) Try to compute the Joyce connection in some simple cases (for example those coming from triangulated surfaces) or even just the A2 case. (b) Use the DT/GW correspondence to give a rough calculation of the Joyce connection in the geometric case D = Db Coh(X). We discuss this below. (c) Try to go the other way: use the existing structure on a semi-simple Frobenius manifold to construct a family of G-valued connections which has the same Stokes data as that arising from the DT invariants in some corresponding example. (d) Try to understand better the formal analogies between the natural structure on Stab(D) (a family of connections with values in G) and the natural structure on a Frobenius manifold (a family of connections with values in GLn(C). 2.5.3. DT/GW heuristics. As mentioned in item (b) above, one can try to understand the Joyce connection in the case when D = Db Coh(X) is the derived category of a smooth projective Calabi-Yau threefold X. In fact stability conditions are only beginning to be understood in this case [3, 32], but mirror symmetry gives a clear picture of what to expect. The following reasoning should be considered completely heuristic: our aim will be to tighten it up into a clear argument, relying perhaps on some explicit convergence conjectures. A neighbourhood of ∞ in the complexified K¨ahlercone of X is expected to embed into Stab(D), with the periods Z given in terms of the K¨ahler parameters β + iω by the mirror map. Fix R > 0 and consider the function X FR(Z) = N(α) exp(RZ(α)). Re Z(α)<0 Near large volume the dominant terms should be given by the sum over Chern classes α sup- ported in dimension ≤ 1. Moreover, the corresponding DT invariants count Gieseker/Simpson semistable sheaves supported in dimension ≤ 1. Consider now setting the K¨ahlerparameters to be (B + iω)/R and sending R → 0, thus going to large volume at the same time as sending R → 0. Combining an identity of Toda [7, Theorem 7.4] with the DT/GW correspondence [34, 35] we can deduce that X (12) lim R · FR(Z) = GW(β, 0) exp hB + iω, βi − ζ(3) · χ(X), R→0 β>0 where the term on the right comes from the MacMahon function and corresponds to zero- dimensional sheaves. The expression on the right of (12) is exactly the Gromov-Witten prepotential F. We can now argue further that as we go to large volume, the function fα on Stab(D) tends to the constant DTα. It follows from this that in period co-ordinates (z1, ··· , zn) the Joyce metric becomes  ∂  X ∂3F ∂ ∇ ∂ = kl , ∂zi ∂zj ∂zi∂zj∂zk ∂zl k,l 18 Bridgeland StabilityDTCluster Edited grant proposal where ij represents the Euler form. This is then consistent with the fact that near large volume the stringy K¨ahlermoduli space is given by  ∂F ∂F  z1, ··· , zi, , ··· , , ∂z1 ∂zn where (z1, ··· , z2n) are period co-ordinates for which ij = δi,n+i.

2.5.4. Further examples. We can try to generalise the Bridgeland-Smith theorem to include quadratic differentials with higher-order zeroes. For example, if we consider differentials on a torus with a double pole and a double zero we obtain the QWP corresponding to the resolved conifold. The space of quadratic differentials Q(S, M) on a fixed surface (S, M) considered in Section 1.2.2 (4), splits into strata depending on the numbers of zeroes of different orders. The open stratum is precisely the space Quad(S, M) appearing in [9]. This leads to a picture where each stratum of Q(S, M) corresponds to a space of stability conditions (modulo auto-equivalences) on a different CY3 category, with these categories being related by basic operations (taking the quotient by a spherical object). It would be very inter- esting to understand how these different strata fit together in the homological world: this would provide a window onto important question of partial compactifications of stability spaces, which could have important applications in other areas. In a slightly different direction we will try to generalise the Bridgeland-Smith theorem to include all dimer models. The resulting CY3 categories include the derived categories of geo- 2 metrically interesting examples, such as local P . It seems possible that this could be done using quadratic differentials on orbifold curves, but this would require further investigation. This could be very worthwhile because these examples cover various interesting examples of non-compact CY threefolds.

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