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66
Stability structures, motivic DonaldsonThomas invariants and cluster transformations
, 2008
"... ..."
Curve counting via stable pairs in the derived category
, 2009
"... For a nonsingular projective 3fold X, we define integer invariants virtually enumerating pairs (C,D) where C ⊂ X is an embedded curve and D ⊂ C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting in ..."
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Cited by 114 (21 self)
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For a nonsingular projective 3fold X, we define integer invariants virtually enumerating pairs (C,D) where C ⊂ X is an embedded curve and D ⊂ C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting invariants are conjecturally equivalent, after universal transformations, to both the GromovWitten and DT theories of X. For CalabiYau 3folds, the latter equivalence should be viewed as a wallcrossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric CalabiYau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We
Configurations in abelian categories. III. Stability conditions and invariants
, 2004
"... This is the third in a series of papers on configurations in an abelian category A. Given a finite partially ordered set (poset) (I, ≼), an (I, ≼)configuration (σ, ι, π) is a finite collection of objects σ(J) and morphisms ι(J, K) or π(J,K) : σ(J) → σ(K) in A satisfying some axioms, where J, K are ..."
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Cited by 45 (16 self)
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This is the third in a series of papers on configurations in an abelian category A. Given a finite partially ordered set (poset) (I, ≼), an (I, ≼)configuration (σ, ι, π) is a finite collection of objects σ(J) and morphisms ι(J, K) or π(J,K) : σ(J) → σ(K) in A satisfying some axioms, where J, K are subsets of I. Configurations are a tool for describing how an object X in A decomposes into subobjects. The first paper defined configurations and developed their basic properties. The second studied moduli spaces of (I, ≼)configurations in A, using the theory of Artin stacks. It proved wellbehaved moduli stacks exist when A is the abelian category of coherent sheaves on a projective Kscheme P, or of representations of a quiver Q. This paper introduces stability conditions (τ, T, �) on A. We show the moduli spaces Mst, Mss(I, ≼, κ, τ)A of τstable or τsemistable configurations are constructible subsets in the moduli stack M(I, ≼, κ)A of all configurations. Using the theory of constructible functions on Artin stacks, we prove many identities relating pushforwards of the characteristic functions δst, δss(I, ≼, κ, τ) of Mst, Mss(I, ≼, κ, τ)A. We define systems of invariants Ist, Iss(I, ≼, κ, τ) of A and (τ, T, �)
Polynomial Bridgeland stability conditions and the large volume limit
 Geom. Topol
"... ABSTRACT. We introduce the notion of a polynomial stability condition, generalizing Bridgeland stability conditions on triangulated categories. We construct and study a family of polynomial stability conditions for any normal projective variety. This family includes both Simpsonstability, and large ..."
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Cited by 38 (4 self)
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ABSTRACT. We introduce the notion of a polynomial stability condition, generalizing Bridgeland stability conditions on triangulated categories. We construct and study a family of polynomial stability conditions for any normal projective variety. This family includes both Simpsonstability, and large volume limits of Bridgeland stability conditions. We show that the PT/DTcorrespondence relating stable pairs to DonaldsonThomas invariants (conjectured by Pandharipande and Thomas) can be understood as a wallcrossing in our family of polynomial stability conditions. Similarly, we show that the relation between stable pairs and invariants of onedimensional torsion sheaves (proven recently by the same authors) is a wallcrossing formula. CONTENTS
Moduli stacks and invariants of semistable objects on K3
, 2007
"... For a K3 surface X and its bounded derived category of coherent sheaves D(X), we have the notion of stability conditions on D(X) in the sense of T.Bridgeland. In this paper, we show that the moduli stack of semistable objects in D(X) with a fixed numerical class and a phase is represented by an Arti ..."
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Cited by 32 (9 self)
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For a K3 surface X and its bounded derived category of coherent sheaves D(X), we have the notion of stability conditions on D(X) in the sense of T.Bridgeland. In this paper, we show that the moduli stack of semistable objects in D(X) with a fixed numerical class and a phase is represented by an Artin stack of finite type over C. Then following D.Joyce’s work, we introduce the invariants counting semistable objects in D(X), and show that the invariants are independent of a choice of a stability condition.
Holomorphic generating functions for invariants counting coherent sheaves on CalabiYau 3folds
 Geometry and Topology
, 2007
"... sheaves on X, and Stab.T / the complex manifold of Bridgeland stability conditions ..."
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Cited by 31 (4 self)
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sheaves on X, and Stab.T / the complex manifold of Bridgeland stability conditions
Wall Crossing from Boltzmann Black Hole Halos
, 2011
"... A key question in the study of N = 2 supersymmetric string or field theories is to understand the decay of BPS bound states across walls of marginal stability in the space of parameters or vacua. By representing the potentially unstable bound states as multicentered black hole solutions in N = 2 s ..."
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Cited by 30 (8 self)
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A key question in the study of N = 2 supersymmetric string or field theories is to understand the decay of BPS bound states across walls of marginal stability in the space of parameters or vacua. By representing the potentially unstable bound states as multicentered black hole solutions in N = 2 supergravity, we provide two fully general and explicit formulæ for the change in the (refined) index across the wall. The first, “Higgs branch” formula relies on Reineke’s results for invariants of quivers without oriented loops, specialized to the Abelian case. The second, “Coulomb branch ” formula results from evaluating the symplectic volume of the classical phase space of multicentered solutions by localization. We provide extensive evidence that these new formulæ agree with each other and with the mathematical results of Kontsevich and Soibelman (KS) and Joyce and Song (JS). The main physical insight behind our results is that the BoseFermi statistics of individual black holes participating in the bound state can be traded for MaxwellBoltzmann statistics, provided the (integer) index Ω(γ) of the internal degrees of freedom carried by each black hole is replaced by an effective (rational) index Ω̄(γ) = mγ Ω(γ/m)/m 2. A similar map also exists for the refined index. This observation provides a physical rationale for the appearance of the rational DonaldsonThomas invariant Ω̄(γ) in the works of KS and JS. The simplicity of the wallcrossing formula for rational invariants allows us to generalize the “semiprimitive wallcrossing formula ” to arbitrary decays of the type γ →Mγ1 +Nγ2 with M = 2, 3.