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197
Derived categories of coherent sheaves and triangulated categories of singularities
, 2005
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The stable derived category of a Noetherian scheme
 COMPOS. MATH
, 2004
"... For a noetherian scheme, we introduce its unbounded stable derived category. This leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. Some applications are included, for instance an anal ..."
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Cited by 95 (12 self)
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For a noetherian scheme, we introduce its unbounded stable derived category. This leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. Some applications are included, for instance an analogue of maximal CohenMacaulay approximations, a construction of Tate cohomology, and an extension of the classical Grothendieck duality. In addition, the relevance of the stable derived category in modular representation theory is indicated.
Mirror symmetry for weighted projective planes and their noncommutative deformations
, 2004
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Topological Correlators in Landau–Ginzburg Models with Boundaries
 Adv. Theor. Math. Phys
"... We compute topological correlators in LandauGinzburg models on a Riemann surface with arbitrary number of handles and boundaries. The boundaries may correspond to arbitrary topological Dbranes of type B. We also allow arbitrary operator insertions on the boundary and in the bulk. The answer is giv ..."
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Cited by 82 (3 self)
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We compute topological correlators in LandauGinzburg models on a Riemann surface with arbitrary number of handles and boundaries. The boundaries may correspond to arbitrary topological Dbranes of type B. We also allow arbitrary operator insertions on the boundary and in the bulk. The answer is given by an explicit formula which can be regarded as an openstring generalization of C. Vafa’s formula for closedstring topological correlators. We discuss how to extend our results to the case of LandauGinzburg orbifolds.
LandauGinzburg Realization of Open String TFT
, 2003
"... We investigate Btype topological LandauGinzburg theory with one variable, with D2brane boundary conditions. The allowed brane configurations are determined in terms of the possible factorizations of the superpotential, and compute the corresponding open string chiral rings. These are characterized ..."
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Cited by 75 (20 self)
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We investigate Btype topological LandauGinzburg theory with one variable, with D2brane boundary conditions. The allowed brane configurations are determined in terms of the possible factorizations of the superpotential, and compute the corresponding open string chiral rings. These are characterized by bosonic and fermionic generators that satisfy certain relations. Moreover we show that the disk correlators, being continuous functions of deformation parameters, satisfy the topological sewing constraints, thereby proving consistency of the theory. In addition we show that the open string LG model is, in its content, equivalent to a certain triangulated category introduced by Kontsevich, and thus may be viewed as a concrete physical realization of it.
KhovanovRozansky Homology and Topological Strings
, 2005
"... We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted direct ..."
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Cited by 59 (13 self)
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We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted directly in topological string theory. We use this approach in various examples to predict the sl(N) knot homology groups for all values of N. We verify that our predictions pass some nontrivial checks. Dedicated to the memory of F.A. Berezin.
Dbranes on CalabiYau manifolds
, 2004
"... In this review we study BPS Dbranes on Calabi–Yau threefolds. Such Dbranes naturally divide into two sets called Abranes and Bbranes which are most easily understood from topological field theory. The main aim of this paper is to provide a selfcontained guide to the derived category approach to ..."
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Cited by 59 (8 self)
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In this review we study BPS Dbranes on Calabi–Yau threefolds. Such Dbranes naturally divide into two sets called Abranes and Bbranes which are most easily understood from topological field theory. The main aim of this paper is to provide a selfcontained guide to the derived category approach to Bbranes and the idea of Πstability. We argue that this mathematical machinery is hard to avoid for a proper understanding of Bbranes. Abranes and Bbranes are related in a very complicated and interesting way which ties in with the “homological mirror symmetry ” conjecture of Kontsevich. We motivate and exploit this form of mirror symmetry. The examples of the quintic 3fold, flops and orbifolds are discussed at some length. In the latter
Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves
, 2005
"... We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau ..."
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Cited by 57 (10 self)
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We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau
Compact generators in categories of matrix factorizations
 MR2824483 (2012h:18014), Zbl 1252.18026, arXiv:0904.4713
"... Abstract. We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. We exhibit the stabilized residue field as a compact generator. This implies a quasiequivalence between the category of matrix factorizations and the dg derived category of an ex ..."
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Cited by 52 (1 self)
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Abstract. We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. We exhibit the stabilized residue field as a compact generator. This implies a quasiequivalence between the category of matrix factorizations and the dg derived category of an explicitly computable dg algebra. Building on this quasiequivalence we establish a derived Morita theory which identifies the functors between matrix factorization categories as integral transforms. This enables us to calculate the Hochschild chain and cochain complexes of matrix factorization categories. Finally, we give interpretations of the results of this work in terms of noncommutative geometry modelled on dg categories. Contents
On the boundary coupling of topological LandauGinzburg models
, 2003
"... I propose a general form for the boundary coupling of Btype topological LandauGinzburg models. In particular, I show that the relevant background in the open string sector is a (generally nonAbelian) superconnection of type (0, 1) living in a complex superbundle defined on the target space, whi ..."
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Cited by 50 (3 self)
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I propose a general form for the boundary coupling of Btype topological LandauGinzburg models. In particular, I show that the relevant background in the open string sector is a (generally nonAbelian) superconnection of type (0, 1) living in a complex superbundle defined on the target space, which I allow to be a noncompact CalabiYau manifold. This extends and clarifies previous proposals. Generalizing an argument due to Witten, I show that BRST invariance of the partition function on the worldsheet amounts to the condition that the (0, ≤ 2) part of the superconnection’s curvature equals a constant endomorphism plus the LandauGinzburg potential times the identity section of the underlying superbundle. This provides the target space equations of motion for the open topological model.