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Batalin–Vilkovisky algebras and twodimensional topological field theories
 265–285. AND ALGEBRAS 231
, 1994
"... Abstract: By a BatalinVilkovisky algebra, we mean a graded commutative algebra A, together with an operator A: A.+ A. such that A +1 2 = 0, and \_A,d \ — Aa is a graded derivation of A for all a e A. In this article, we show that there is a natural structure of a BatalinVilkovisky algebra on the ..."
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Cited by 123 (4 self)
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Abstract: By a BatalinVilkovisky algebra, we mean a graded commutative algebra A, together with an operator A: A.+ A. such that A +1 2 = 0, and \_A,d \ — Aa is a graded derivation of A for all a e A. In this article, we show that there is a natural structure of a BatalinVilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions. We make use of a technique from algebraic topology: the theory of operads. BatalinVilkovisky algebras are a new type of algebraic structure on graded vector spaces, which first arose in the work of Batalin and Vilkovisky on gauge fixing in quantum field theory: a BatalinVilkovisky algebra is a differential graded commutative algebra together with an operator A: A.+A such that A m+ί 2 = 0, and Δ{abc) = A(ab)c + ( V)^aA{bc) + ( l) (α ίm
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
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 36 (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
On the structure of cofree Hopf algebras
 J. reine angew. Math
"... Abstract. We prove an analogue of the PoincaréBirkhoffWitt theorem and of the CartierMilnorMoore theorem for noncocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a nondifferential B∞algebra. We construct a universal enveloping functor U2 from nondifferential B∞algebr ..."
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Cited by 36 (4 self)
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Abstract. We prove an analogue of the PoincaréBirkhoffWitt theorem and of the CartierMilnorMoore theorem for noncocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a nondifferential B∞algebra. We construct a universal enveloping functor U2 from nondifferential B∞algebras to 2associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra H is of the form U2(Prim H). We take advantage of the description of the free 2asalgebra in terms of planar trees to unravel the operad associated to nondifferential B∞algebras.
Relative formality theorem and quantisation of coisotropic submanifolds” math.QA/0501540
"... Abstract. We prove a relative version of Kontsevich’s formality theorem. This theorem involves a manifold M and a submanifold C and reduces to Kontsevich’s theorem if C = M. It states that the DGLA of multivector fields on an infinitesimal neighbourhood of C is L∞quasiisomorphic to the DGLA of mult ..."
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Cited by 30 (7 self)
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Abstract. We prove a relative version of Kontsevich’s formality theorem. This theorem involves a manifold M and a submanifold C and reduces to Kontsevich’s theorem if C = M. It states that the DGLA of multivector fields on an infinitesimal neighbourhood of C is L∞quasiisomorphic to the DGLA of multidifferential operators acting on sections of the exterior algebra of the conormal bundle. Applications to the deformation quantisation of coisotropic submanifolds are given. The proof uses a duality transformation to reduce the theorem to a version of Kontsevich’s theorem for supermanifolds, which we also discuss. In physical language, the result states that there is a duality between the Poisson sigma model on a manifold with a Dbrane and the Poisson sigma model on a supermanifold without branes (or, more properly, with a brane which extends over the whole supermanifold). 1.
Modules and Morita theorem for operads
 Am. J. of Math
"... (0.1) Morita theory. Let A, B be two commutative rings. If their respective categories of modules are equivalent, then A and B are isomorphic. This is not anymore true if A and/or B are not assumed to be commutative. Morita theory ..."
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Cited by 24 (0 self)
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(0.1) Morita theory. Let A, B be two commutative rings. If their respective categories of modules are equivalent, then A and B are isomorphic. This is not anymore true if A and/or B are not assumed to be commutative. Morita theory
On field theoretic generalizations of a Poisson algebra
 Rep. Math. Phys
, 1997
"... Rep. Math. Phys. vol. 40 (1997) p.225 ..."
Lectures on graded differential algebras and noncommutative geometry
, 1999
"... These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments. ..."
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Cited by 22 (3 self)
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These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.
shLie algebras induced by gauge transformations
 Comm. Math. Phys
"... Abstract. The physics of “particles of spin ≤ 2 ” leads to representations of a Lie algebra Ξ of gauge parameters on a vector space Φ of fields. Attempts to develop an analogous theory for spin> 2 have failed; in fact, there are claims that such a theory is impossible (though we have been unable to ..."
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Cited by 20 (4 self)
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Abstract. The physics of “particles of spin ≤ 2 ” leads to representations of a Lie algebra Ξ of gauge parameters on a vector space Φ of fields. Attempts to develop an analogous theory for spin> 2 have failed; in fact, there are claims that such a theory is impossible (though we have been unable to determine the hypotheses for such a ‘nogo ’ theorem). This led Berends, Burgers and van Dam [Bur85, BBvD84, BBvD85] to generalize to ‘field dependent parameters ’ in a setting where some analysis in terms of smooth functions is possible. Having recognized the resulting structure as that of an shlie algebra (L∞algebra), we have now reproduced their structure entirely algebraically, hopefully shedding some light on what is going on. 1.