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109
A homotopy theoretic realization of string topology
, 2002
"... Abstract. Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M. In [2] Chas and Sullivan defined a product on the homology H∗(LM) of degree −d. They then investigated other structure that this product induces, including a Lie algebra structure on H∗(LM), and ..."
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Cited by 114 (12 self)
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Abstract. Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M. In [2] Chas and Sullivan defined a product on the homology H∗(LM) of degree −d. They then investigated other structure that this product induces, including a Lie algebra structure on H∗(LM), and an induced product on the S 1 equivariant homology, H S1 ∗ (LM). These algebraic structures, as well as others, came under the general heading of the “String topology ” of M. In this paper we will describe a realization of the Chas Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. We show that this ring spectrum structure extends to an operad action of the the “cactus operad”, originally defined by Voronov, which is equivalent to operad of framed disks in R 2. We then describe a cosimplicial model of this spectrum and, by applying the singular cochain functor to this cosimplicial spectrum we show that this ring structure can be interpreted as the cup product in the Hochschild cohomology, HH ∗ (C ∗ (M); C ∗ (M)).
Modular Operads
 COMPOSITIO MATH
, 1994
"... We develop a "higher genus" analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the Feynman transform, which generalizes Kontsevich's graph complexes and also the bar construction for operads. ..."
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Cited by 70 (5 self)
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We develop a "higher genus" analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the Feynman transform, which generalizes Kontsevich's graph complexes and also the bar construction for operads. We calculate the Euler characteristic of the Feynman transform, using the theory of symmetric functions: our formula is modelled on Wick's theorem. We give applications to the theory of moduli spaces of pointed algebraic curves.
Brane Topology
 In preparation
"... Consider two families of closed oriented curves in a manifold M d. At each point of intersection of a curve of one family with a curve of the other family, form a new closed curve by going around the first curve and then going around the second. Typically, an idimensional family and a jdimensional ..."
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Cited by 61 (1 self)
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Consider two families of closed oriented curves in a manifold M d. At each point of intersection of a curve of one family with a curve of the other family, form a new closed curve by going around the first curve and then going around the second. Typically, an idimensional family and a jdimensional family will produce an i + j − d + 2dimensional family. Our purpose is to describe a mathematical structure behind such interactions. 1
On Operad Structures of Moduli Spaces and String Theory
, 1994
"... We construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy Lie algebras, gravity algebras and BatalinVilkovisky algebras. These algebras appeared recently in the context of string theory, and we give a ..."
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Cited by 49 (13 self)
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We construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy Lie algebras, gravity algebras and BatalinVilkovisky algebras. These algebras appeared recently in the context of string theory, and we give a simple deduction of these algebraic structures from the formal axioms of conformal field theory and string theory.
Gerstenhaber algebras and BV algebras in Poisson geometry
"... The purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some wellknown algebraic structures such as Gerstenhaber algebras and BValgebras. Some applications are discussed. In particular, we found an explicit connection betwee ..."
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Cited by 44 (4 self)
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The purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some wellknown algebraic structures such as Gerstenhaber algebras and BValgebras. Some applications are discussed. In particular, we found an explicit connection between the KoszulBrylinski operator of a Poisson manifold and its modular class. As a consequence, we prove that Poisson homology is isomorphic to Poisson cohomology for unimodular Poisson structures. 1
Topological Open pBranes
, 2000
"... By exploiting the BV quantization of topological bosonic open membrane, we argue that flat 3form Cfield leads to deformations of the algebras of multivectors on the Dirichletbrane worldvolume as 2algebras. This would shed some new light on geometry of Mtheory 5brane and associated decoupled ..."
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Cited by 37 (1 self)
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By exploiting the BV quantization of topological bosonic open membrane, we argue that flat 3form Cfield leads to deformations of the algebras of multivectors on the Dirichletbrane worldvolume as 2algebras. This would shed some new light on geometry of Mtheory 5brane and associated decoupled theories. We show that, in general, topological open pbrane has a structure of (p + 1)algebra in the bulk, while a structure of palgebra in the boundary. The bulk/boundary correspondences are exactly as of the generalized Deligne conjecture (a theorem of Kontsevich) in the algebraic world of palgebras. It also imply that the algebras of quantum observables of (p − 1)brane are “close to ” the algebras of its classical observables as palgebras. We interpret above as deformation quantization of (p − 1)brane, generalizing the p = 1 case. We argue that there is such quantization based on the direct relation between BV master equation and Ward identity of the bulk topological theory. The path integral of the theory will lead to the explicit formula. We also discuss some applications to
Twodimensional topological gravity and equivalent cohomology
 Commun. Math. Phys
, 1994
"... The analogy between topological string theory and equivariant cohomology for differentiable actions of the circle group on manifolds has been widely remarked on. One of our aims in this paper is to make this analogy precise. We show that topological string theory is the “derived functor ” of semire ..."
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Cited by 35 (2 self)
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The analogy between topological string theory and equivariant cohomology for differentiable actions of the circle group on manifolds has been widely remarked on. One of our aims in this paper is to make this analogy precise. We show that topological string theory is the “derived functor ” of semirelative cohomology, just as equivariant cohomology is the derived functor of basic cohomology. That homological algebra finds a place in the study of topological string theory should not surprise the reader, granted that topological string theory is the conformal field theorist’s algebraic topology. In [7], we have shown that the cohomology of a topological conformal field theory carries the structure of a BatalinVilkovisky algebra (actually, two commuting such structures, corresponding to the two chiral sectors of the theory). In the second part of this paper, we describe the analogous algebraic structure on the equivariant cohomology of a topological conformal field theory: we call this structure a gravity algebra. This algebraic structure is a certain generalization of a Lie algebra, and is distinguished by the fact that it has an infinite sequence of independent operations {a1,...,ak}, k ≥ 2, satisfying quadratic relations generalizing the Jacobi rule. (The operad underlying the category of gravity algebras has been studied independently by GinzburgKapranov [9].)
Topological strings in generalized complex space
"... A twodimensional topological sigmamodel on a generalized CalabiYau target space X is defined. The model is constructed in BatalinVilkovisky formalism using only a generalized complex structure J and a pure spinor ρ on X. In the present construction the algebra of Qtransformations automatically ..."
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Cited by 31 (1 self)
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A twodimensional topological sigmamodel on a generalized CalabiYau target space X is defined. The model is constructed in BatalinVilkovisky formalism using only a generalized complex structure J and a pure spinor ρ on X. In the present construction the algebra of Qtransformations automatically closes offshell, the model transparently depends only on J, the algebra of observables and correlation functions for topologically trivial maps in genus zero are easily defined. The extended moduli space appears naturally. The familiar action of the twisted N = 2 CFT can be recovered after a gauge fixing. In the open case, we consider an example of generalized deformation of complex structure by a holomorphic Poisson bivector β and recover holomorphic noncommutative Kontsevich ∗product. 1
Perturbative Topological Field Theory
 Commun. Math. Phys
"... We give a review of the application of perturbative techniques to topological quantum field theories, in particular threedimensional ChernSimonsWitten theory and its various generalizations. To this end we give an introduction to graph homology and homotopy algebras and the work of Vassiliev and K ..."
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Cited by 30 (2 self)
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We give a review of the application of perturbative techniques to topological quantum field theories, in particular threedimensional ChernSimonsWitten theory and its various generalizations. To this end we give an introduction to graph homology and homotopy algebras and the work of Vassiliev and Kontsevich on perturbative knot invariants. 1. Introduction In these lecture notes we will give a review of some recent mathematical developments in topological field theory following the work of Kontsevich, Axelrod and Singer, Vassiliev, BarNatan, Witten and others. Quite remarkable these new ideas are all related to oldfashioned perturbative techniques in field theory. Indeed, it is an interesting comment on the development of the interaction between physics and mathematics that these days pure mathematicians are calculating Feynman diagrams whereas this skill is slowly disappearing among a large fraction of theoretical physics students. Our starting point in all this will be mostly thre...