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Curve counting via stable pairs in the derived category
"... Abstract. 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 resu ..."
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Cited by 44 (9 self)
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Abstract. 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
Stability conditions on K3 surfaces
"... Abstract. This paper contains a description of one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic K3 surface. 1. ..."
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Cited by 43 (5 self)
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Abstract. This paper contains a description of one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic K3 surface. 1.
Noncommutative DonaldsonThomas theory and the conifold
, 2008
"... Given a quiver algebra A with relations defined by a superpotential, this paper defines a set of invariants of A counting framed cyclic Amodules, analogous to rank1 Donaldson–Thomas invariants of Calabi–Yau threefolds. For the special case when A is the noncommutative crepant resolution of the th ..."
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Cited by 27 (0 self)
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Given a quiver algebra A with relations defined by a superpotential, this paper defines a set of invariants of A counting framed cyclic Amodules, analogous to rank1 Donaldson–Thomas invariants of Calabi–Yau threefolds. For the special case when A is the noncommutative crepant resolution of the threefold ordinary double point, it is proved using torus localization that the invariants count certain pyramidshaped partitionlike configurations, or equivalently infinite dimer configurations in the square dimer model with a fixed boundary condition. The resulting partition function admits an infinite product expansion, which factorizes into the rank1 Donaldson–Thomas partition functions of the commutative crepant resolution of the singularity and its flop. The different partition functions are speculatively interpreted as counting stable objects in the derived category of Amodules under different stability conditions; their relationship should then be an instance of wall crossing in the space of stability conditions on this triangulated category.
Stability conditions on a noncompact CalabiYau threefold
"... Abstract. We study the space of stability conditions on the noncompact CalabiYau threefold X which is the total space of the canonical bundle of P 2. We give a combinatorial description of an open subset of Stab(X) and state a conjecture relating Stab(X) to the Frobenius manifold obtained from the ..."
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Cited by 26 (1 self)
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Abstract. We study the space of stability conditions on the noncompact CalabiYau threefold X which is the total space of the canonical bundle of P 2. We give a combinatorial description of an open subset of Stab(X) and state a conjecture relating Stab(X) to the Frobenius manifold obtained from the quantum cohomology of P 2. We give some evidence from mirror symmetry for this conjecture. 1.
Central charges, symplectic forms, and hypergeometric series in local mirror symmetry
 hepth/0404043 46 A. Iqbal and A.K. KashaniPoor, The Vertex on a Strip. hepth/0410174
"... Abstract. We study a cohomologyvalued hypergeometric series which naturally arises in the description of (local) mirror symmetry. We identify it as a central charge formula for BPS states and study its monodromy property from the viewpoint of Kontsevich’s homological mirror symmetry. In case of loc ..."
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Cited by 22 (1 self)
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Abstract. We study a cohomologyvalued hypergeometric series which naturally arises in the description of (local) mirror symmetry. We identify it as a central charge formula for BPS states and study its monodromy property from the viewpoint of Kontsevich’s homological mirror symmetry. In case of local mirror symmetry, we will identify a symplectic form, and will conjecture an integral and symplectic monodromy property of a relevant hypergeometric series of Gel’fandKapranovZelevinski type.
Noncommutative twotori with real multiplication as noncommutative projective varieties
 J. Geom. Phys
"... Abstract. We define analogues of homogeneous coordinate algebras for noncommutative twotori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate a ..."
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Cited by 21 (4 self)
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Abstract. We define analogues of homogeneous coordinate algebras for noncommutative twotori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate algebras. We give a criterion for such an algebra to be Koszul and prove that the Koszul dual algebra also comes from some noncommutative twotorus with real multiplication. These results are based on the techniques of [14] allowing to interpret all the data in terms of autoequivalences of the derived categories of coherent sheaves on elliptic curves.
AN INTEGRAL STRUCTURE IN QUANTUM COHOMOLOGY AND MIRROR SYMMETRY FOR TORIC ORBIFOLDS
, 2009
"... We introduce an integral structure in orbifold quantum cohomology associated to the Kgroup and the b Γclass. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the LandauGinzburg model under mirror symmetry. By assuming the ..."
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Cited by 18 (1 self)
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We introduce an integral structure in orbifold quantum cohomology associated to the Kgroup and the b Γclass. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the LandauGinzburg model under mirror symmetry. By assuming the existence of an integral structure, we give a natural explanation for the specialization to a root of unity in Y. Ruan’s crepant resolution conjecture [66].
Spaces of stability conditions
"... Abstract. Stability conditions are a mathematical way to understand Πstability for Dbranes 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 conditi ..."
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Cited by 17 (3 self)
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Abstract. Stability conditions are a mathematical way to understand Πstability for Dbranes 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. 1.
Stability conditions and crepant small resolutions
, 2007
"... In this paper, we describe the spaces of stability conditions on the triangulated categories associated to three dimensional crepant small resolutions. The resulting spaces have chamber structures such that each chamber corresponds to a birational model together with a special FourierMukai transfor ..."
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Cited by 15 (6 self)
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In this paper, we describe the spaces of stability conditions on the triangulated categories associated to three dimensional crepant small resolutions. The resulting spaces have chamber structures such that each chamber corresponds to a birational model together with a special FourierMukai transform. We observe that these spaces are covering spaces over certain open subsets of finite dimensional vector spaces, and determine their deck transformations.