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37
A GEOMETRIC CRITERION FOR GENERATING THE FUKAYA category
, 2010
"... Given a collection of exact Lagrangians in a Liouville manifold, we construct a map from the Hochschild homology of the Fukaya category that they generate to symplectic cohomology. Whenever the identity in symplectic cohomology lies in the image of this map, we conclude that every Lagrangian lies i ..."
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Cited by 29 (5 self)
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Given a collection of exact Lagrangians in a Liouville manifold, we construct a map from the Hochschild homology of the Fukaya category that they generate to symplectic cohomology. Whenever the identity in symplectic cohomology lies in the image of this map, we conclude that every Lagrangian lies in the idempotent closure of the chosen collection. The main new ingredients are (1) the construction of operations on the Fukaya category controlled by discs with two outputs, and (2) the Cardy relation.
SUSPENDING LEFSCHETZ FIBRATIONS, WITH AN APPLICATION TO LOCAL MIRROR SYMMETRY
, 907
"... Let Y be a smooth toric del Pezzo surface, and KY the total space of its canonical bundle. Let D b (Coh(KY)) be the bounded derived category of coherent sheaves on KY, and D b Y (Coh(KY)) the full subcategory consisting of complexes whose cohomology is supported on the zerosection Y ⊂ KY. The mirro ..."
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Cited by 13 (3 self)
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Let Y be a smooth toric del Pezzo surface, and KY the total space of its canonical bundle. Let D b (Coh(KY)) be the bounded derived category of coherent sheaves on KY, and D b Y (Coh(KY)) the full subcategory consisting of complexes whose cohomology is supported on the zerosection Y ⊂ KY. The mirror to KY (see [10,
A∞SUBALGEBRAS AND NATURAL TRANSFORMATIONS
, 2008
"... This paper explores a version of categorical localization. While the results are purely algebraic, the construction itself arose in symplectic topology, specifically in the theory of Lefschetz fibrations, and the connection with localization was arrived at a posteriori. ..."
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Cited by 12 (4 self)
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This paper explores a version of categorical localization. While the results are purely algebraic, the construction itself arose in symplectic topology, specifically in the theory of Lefschetz fibrations, and the connection with localization was arrived at a posteriori.
Hochschild homology of structured algebras
, 2011
"... Abstract. We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any PROP with A∞–multiplication—we think of such algebras as A∞–algebras “with extra structure”. As applications, we obtain an integral version of the CostelloKontsevichSo ..."
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Cited by 8 (1 self)
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Abstract. We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any PROP with A∞–multiplication—we think of such algebras as A∞–algebras “with extra structure”. As applications, we obtain an integral version of the CostelloKontsevichSoibelman moduli space action on the Hochschild complex of open TCFTs, the TradlerZeinalian action of Sullivan diagrams on the Hochschild complex of strict Frobenius algebras, and give applications to string topology in characteristic zero. Our main tool is a generalization of the Hochschild complex. The Hochschild complex of an associative algebra A admits a degree 1 selfmap, ConnesRinehart’s boundary operator B. If A is Frobenius, the (proven) cyclic Deligne conjecture says that B is the ∆–operator of a BVstructure on the Hochschild complex of A. In fact B is part of much richer structure, namely an action by the chain complex of Sullivan diagrams on the Hochschild complex [39]. A weaker version of Frobenius algebras, called here A∞–Frobenius algebras, yields instead an action by the chains on the moduli space of Riemann surfaces [8, 21]. In this paper we develop a general method
Characteristic Classes of A∞Algebras
, 2006
"... A standard combinatorial construction, due to Kontsevich, associates to any A∞algebra with an invariant inner product, an inhomogeneous class in the cohomology of the moduli spaces of Riemann surfaces with marked points. We describe an alternative version of this construction based on noncommutativ ..."
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Cited by 4 (2 self)
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A standard combinatorial construction, due to Kontsevich, associates to any A∞algebra with an invariant inner product, an inhomogeneous class in the cohomology of the moduli spaces of Riemann surfaces with marked points. We describe an alternative version of this construction based on noncommutative geometry and use it to prove that homotopy equivalent algebras give rise to the same cohomology classes. Along the way we reprove Kontsevich’s theorem relating graph homology to the homology of certain infinitedimensional