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255
The spectrum of BPS branes on a noncompact CalabiYau
, 2000
"... We begin the study of the spectrum of BPS branes and its variation on lines of marginal stability on OIP2(−3), a CalabiYau ALE space asymptotic to C 3 /Z3. We show how to get the complete spectrum near the large volume limit and near the orbifold point, and find a striking similarity between the de ..."
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Cited by 136 (13 self)
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We begin the study of the spectrum of BPS branes and its variation on lines of marginal stability on OIP2(−3), a CalabiYau ALE space asymptotic to C 3 /Z3. We show how to get the complete spectrum near the large volume limit and near the orbifold point, and find a striking similarity between the descriptions of holomorphic bundles and BPS branes in these two limits. We use these results to develop a general picture of the spectrum. We also suggest a generalization of some of the ideas to the quintic CalabiYau.
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 129 (6 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.
Introduction to Ainfinity algebras and modules
, 1999
"... These are slightly expanded notes of four introductory talks on ..."
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Cited by 123 (5 self)
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These are slightly expanded notes of four introductory talks on
Derived equivalences from mutations of quivers with potential
 ADVANCES IN MATHEMATICS 226 (2011) 2118–2168
, 2011
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Dirichlet branes, homological mirror symmetry, and stability
 Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), 395–408, Higher Ed
, 2002
"... We discuss some mathematical conjectures which have come out of the study of Dirichlet branes in superstring theory, focusing on the case of supersymmetric branes in CalabiYau compactification. This has led to the formulation of a notion of stability for objects in a derived category, contact with ..."
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Cited by 78 (1 self)
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We discuss some mathematical conjectures which have come out of the study of Dirichlet branes in superstring theory, focusing on the case of supersymmetric branes in CalabiYau compactification. This has led to the formulation of a notion of stability for objects in a derived category, contact with Kontsevich’s homological mirror symmetry conjecture, and “physics proofs” for many of the subsequent conjectures based on it, such as the representation of CalabiYau monodromy by autoequivalences of the derived category.
Hypergeometric functions and mirror symmetry in toric varieties
, 1999
"... We study aspects related to Kontsevich’s homological mirror symmetry conjecture [42] in the case of Calabi–Yau complete intersections in toric varieties. In a 1996 lecture, Kontsevich [43] indicated how his proposal implies that the groups of automorphisms of the two types of categories involved in ..."
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Cited by 72 (4 self)
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We study aspects related to Kontsevich’s homological mirror symmetry conjecture [42] in the case of Calabi–Yau complete intersections in toric varieties. In a 1996 lecture, Kontsevich [43] indicated how his proposal implies that the groups of automorphisms of the two types of categories involved in the homological mirror symmetry conjecture should also be identified. Our results provide an explicit geometric construction of the correspondence between the automorphisms of the two types of categories. We compare the monodromy calculations for the Picard–Fuchs system associated with the periods of a Calabi–Yau manifold M with the algebrogeometric computations of the cohomology action of Fourier– Mukai functors on the bounded derived category of coherent sheaves on the mirror Calabi–Yau manifold W. We obtain the complete dictionary between the two sides for the one complex parameter case of Calabi–Yau complete intersections in weighted projective spaces, as well as for some two parameter cases. We also find the complex of sheaves on W × W that corresponds to a loop in the moduli space of complex structures on M induced by a phase transition of W.
tstructures on some local CalabiYau varieties
 J. Algebra
"... Abstract. Let Z be a Fano varity satisfying the condition that the rank of the Grothendieck group of Z is one more than the dimension of Z. Let ωZ denote the total space of the canonical line bundle of Z, considered as a noncompact CalabiYau variety. We use the theory of exceptional collections to ..."
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Cited by 59 (3 self)
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Abstract. Let Z be a Fano varity satisfying the condition that the rank of the Grothendieck group of Z is one more than the dimension of Z. Let ωZ denote the total space of the canonical line bundle of Z, considered as a noncompact CalabiYau variety. We use the theory of exceptional collections to describe tstructures on the derived category of coherent sheaves on ωZ. The combinatorics of these tstructures is determined by a natural action of an affine braid group, closely related to the wellknown action of the Artin braid group on the set of exceptional collections on Z. 1.