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Introduction to Ainfinity algebras and modules
, 1999
"... These are slightly expanded notes of four introductory talks on ..."
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Cited by 117 (4 self)
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These are slightly expanded notes of four introductory talks on
Differential invariants and curved BernsteinGelfandGelfand sequences
 JOUR. REINE ANGEW. MATH
, 2000
"... We give a simple construction of the BernsteinGelfandGelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry. This method permits us to define the additional structure of a bilinear differential “cup product ” on this sequence, satisfying a Leibniz ru ..."
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Cited by 78 (6 self)
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We give a simple construction of the BernsteinGelfandGelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry. This method permits us to define the additional structure of a bilinear differential “cup product ” on this sequence, satisfying a Leibniz rule up to curvature terms. It is not associative, but is part of an A∞algebra of multilinear differential operators, which we also obtain explicitly. We illustrate the construction in the case of conformal differential geometry, where the cup product provides a widereaching generalization of helicity raising and lowering for conformally invariant field equations.
Dbranes, Categories and N = 1 Supersymmetry
, 2000
"... We show that boundary conditions in topological open string theory on CalabiYau manifolds are objects in the derived category of coherent sheaves, as foreseen in the homological mirror symmetry proposal of Kontsevich. Together with conformal field theory considerations, this leads to a precise crit ..."
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Cited by 53 (0 self)
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We show that boundary conditions in topological open string theory on CalabiYau manifolds are objects in the derived category of coherent sheaves, as foreseen in the homological mirror symmetry proposal of Kontsevich. Together with conformal field theory considerations, this leads to a precise criterion determining the BPS branes at any point in CY moduli space, completing the proposal of Πstability.
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
Homological mirror symmetry with higher products
 in Proceedings of the Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds, 247–259. AMS and International
, 2001
"... The homological mirror symmetry conjecture formulated by M. Kontsevich in [6] claims that derived categories of Fukaya’s symplectic A∞categogy F(M) of a CalabiYau manifold M and of coherent sheaves on a mirror dual CalabiYau manifold X are equivalent. In particular, this means that one can identif ..."
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Cited by 41 (6 self)
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The homological mirror symmetry conjecture formulated by M. Kontsevich in [6] claims that derived categories of Fukaya’s symplectic A∞categogy F(M) of a CalabiYau manifold M and of coherent sheaves on a mirror dual CalabiYau manifold X are equivalent. In particular, this means that one can identify the associative product on Extgroups between coherent sheaves on X with the corresponding product in the Floer cohomology of Lagrangians submanifolds in M (defined by Fukaya in [2]). The drawback of this conjecture is that one has an A∞category on the symplectic side of the story and the usual category on the complex side, so one has to make the usual category out of F(M). In this note we fix this problem by constructing an A∞category on the complex side and formulate a more general conjecture involving A∞categories on both sides. Let X be a compact complex manifold equipped with a hermitian metric. Inspired by Merkulov’s paper [7] we define an A∞category
Noncommutative homotopy algebras associated with open strings
 REV. MATH. PHYS
, 2003
"... We discuss general properties of A∞algebras and their applications to the theory of open strings. The properties of cyclicity for A∞algebras are examined in detail. We prove the decomposition theorem, which is a stronger version of the minimal model theorem, for A∞algebras and cyclic A∞algebras a ..."
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Cited by 28 (5 self)
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We discuss general properties of A∞algebras and their applications to the theory of open strings. The properties of cyclicity for A∞algebras are examined in detail. We prove the decomposition theorem, which is a stronger version of the minimal model theorem, for A∞algebras and cyclic A∞algebras and discuss various consequences of it. In particular it is applied to classical open string field theories and it is shown that all classical open string field theories on a fixed conformal background are cyclic A∞isomorphic to each other. The same results hold for classical closed string field theories, whose algebraic structure is governed by cyclic L∞algebras.