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16
The sh Lie structure of Poisson brackets in field theory
, 1997
"... A general construction of an sh Lie algebra (L∞algebra) from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel’fand, Dickey and Dorfman. In th ..."
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Cited by 30 (4 self)
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A general construction of an sh Lie algebra (L∞algebra) from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel’fand, Dickey and Dorfman. In this way, higher order maps are constructed which combine to form an sh Lie algebra on the graded differential algebra of horizontal forms. The same construction applies for graded brackets in field theory such as the BatalinFradkinVilkovisky bracket of the Hamiltonian BRST theory or the BatalinVilkovisky antibracket. ( ∗ ) Research supported by grants from the Fonds National Belge de la Recherche Scientifique
BRST theory without Hamiltonian and Lagrangian
 J. High Energy Phys
"... We consider a generic gauge system, whose physical degrees of freedom are obtained by restriction on a constraint surface followed by factorization with respect to the action of gauge transformations; in so doing, no Hamiltonian structure or action principle is supposed to exist. For such a generic ..."
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Cited by 8 (2 self)
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We consider a generic gauge system, whose physical degrees of freedom are obtained by restriction on a constraint surface followed by factorization with respect to the action of gauge transformations; in so doing, no Hamiltonian structure or action principle is supposed to exist. For such a generic gauge system we construct a consistent BRST formulation, which includes the conventional BV Lagrangian and BFV Hamiltonian schemes as particular cases. If the original manifold carries a weak Poisson structure (a bivector field giving rise to a Poisson bracket on the space of physical observables) the generic gauge system is shown to admit deformation quantization by means of the Kontsevich formality theorem. A sigmamodel interpretation of this quantization algorithm is briefly discussed. 1
Characteristic classes of gauge systems
 Nucl. Phys. B703 (2004) 419453. CLASSES OF QMANIFOLDS 57
"... Abstract. We define and study invariants which can be uniformly constructed for any gauge system. By a gauge system we understand an (anti)Poisson supermanifold endowed with an odd Hamiltonian selfcommuting vector field called a homological vector field. This definition encompasses all the cases u ..."
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Abstract. We define and study invariants which can be uniformly constructed for any gauge system. By a gauge system we understand an (anti)Poisson supermanifold endowed with an odd Hamiltonian selfcommuting vector field called a homological vector field. This definition encompasses all the cases usually included into the notion of a gauge theory in physics as well as some other similar (but different) structures like Lie or Courant algebroids. For Lagrangian gauge theories or Hamiltonian first class constrained systems, the homological vector field is identified with the classical BRST transformation operator. We define characteristic classes of a gauge system as universal cohomology classes of the homological vector field, which are uniformly constructed in terms of this vector field itself. Not striving to exhaustively classify all the characteristic classes in this work, we compute those invariants which are built up in terms of the first derivatives of the homological vector field. We also consider the cohomological operations in the space of all the characteristic classes. In particular, we show that the (anti)Poisson bracket becomes trivial when applied to the space of all the characteristic classes, instead the latter space can be endowed with another Lie bracket operation. Making use of this Lie bracket one can generate new characteristic classes involving higher derivatives of the homological vector field. The simplest characteristic classes are illustrated by the examples relating them to anomalies in the traditional BV or BFVBRST theory and to characteristic classes of (singular) foliations. 1.
Homological (ghost) approach to constrained Hamiltonian systems,” hepth/9112002
"... I had originally intended to give a talk on homological reduction of first class constrained Hamiltonian systems, as in my joint work with Henneaux, Fisch and Teitelboim [FHST]. Since the organizers have given me the?honor? of opening the conference, I will attempt to set that work in a larger conte ..."
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Cited by 3 (0 self)
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I had originally intended to give a talk on homological reduction of first class constrained Hamiltonian systems, as in my joint work with Henneaux, Fisch and Teitelboim [FHST]. Since the organizers have given me the?honor? of opening the conference, I will attempt to set that work in a larger context, namely that of ghost techniques in mathematical physics. What are ‘ghosts ’ and what are they doing in physics? The name reflects the fact that they are new, auxiliary variables that are NOT physical, but are added to the system for computational reasons. An analogy familiar to many mathematicians is that of a resolution in homological algebra the generators added to construct the resolution are the analogs of ghosts. Indeed it is more than an analogy in some cases; I first became seriously interested in the subject when I read a preprint of Browning and McMullan [BM] in which certain ‘antighosts ’ were clearly identified as generators of the Koszul resolution of an appropriate ideal. But I am getting ahead of my story, both conceptually and historically. My intention this morning is to set the stage for a set of techniques and results which
Deformation Quantization and Reduction
, 2007
"... Abstract. This note is an overview of the Poisson sigma model (PSM) and its applications in deformation quantization. Reduction of coisotropic and prePoisson submanifolds, their appearance as branes of the PSM, quantization in terms of L∞ and A∞algebras, and bimodule structures are recalled. As a ..."
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Cited by 1 (1 self)
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Abstract. This note is an overview of the Poisson sigma model (PSM) and its applications in deformation quantization. Reduction of coisotropic and prePoisson submanifolds, their appearance as branes of the PSM, quantization in terms of L∞ and A∞algebras, and bimodule structures are recalled. As an application, an “almost ” functorial quantization of Poisson maps is presented if no anomalies occur. This leads in principle to a novel approach for the quantization of Poisson–Lie groups. 1.
Université des Sciences et Technologies de Lille
, 2003
"... Abstract. A LieRinehart algebra (A, L) consists of a commutative algebra A and a Lie algebra L with additional structure which generalizes the mutual structure of interaction between the algebra of smooth functions and the Lie algebra of smooth vector fields on a smooth manifold. LieRinehart algeb ..."
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Abstract. A LieRinehart algebra (A, L) consists of a commutative algebra A and a Lie algebra L with additional structure which generalizes the mutual structure of interaction between the algebra of smooth functions and the Lie algebra of smooth vector fields on a smooth manifold. LieRinehart algebras provide the correct categorical language to solve the problem whether Kähler quantization commutes with reduction which, in turn, may be seen as a descent problem.
BFVCOMPLEX AND HIGHER HOMOTOPY STRUCTURES
, 2007
"... Abstract. We present an alternative approach to induced higher homotopy structures constructed by the ’basic perturbation lemma’. This approach is motivated by physical considerations and makes use of operads and their representations. As an application we prove that the BFVcomplex controls the for ..."
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Abstract. We present an alternative approach to induced higher homotopy structures constructed by the ’basic perturbation lemma’. This approach is motivated by physical considerations and makes use of operads and their representations. As an application we prove that the BFVcomplex controls the formal deformations of coisotropic submanifolds. This is established by identifying the P∞algebra structure on the normal bundle of a coisotropic submanifold as a homotopy structure induced from the BFVcomplex. Then we provide the connection to the geometric picture in the new framework. Contents
arXiv version: fonts, pagination and layout may vary from GTM published version Perturbative expansion of Chern–Simons theory
"... SAWON An overview of the perturbative expansion of the Chern–Simons path integral is given. The main goal is to describe how trivalent graphs appear: as they already occur in the perturbative expansion of an analogous finitedimensional integral, we discuss this case in detail. 81T18; 57M27, 58J28, ..."
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SAWON An overview of the perturbative expansion of the Chern–Simons path integral is given. The main goal is to describe how trivalent graphs appear: as they already occur in the perturbative expansion of an analogous finitedimensional integral, we discuss this case in detail. 81T18; 57M27, 58J28, 81T13 1
BVFCOMPLEX AND HIGHER HOMOTOPY STRUCTURES
, 2006
"... Abstract. We present an alternative approach to higher derived homotopy structures induced by the ’basic perturbation lemma’. This approach is motivated by physical considerations and makes use of operads and their representations. As an application we prove that the BVFcomplex controls the formal ..."
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Abstract. We present an alternative approach to higher derived homotopy structures induced by the ’basic perturbation lemma’. This approach is motivated by physical considerations and makes use of operads and their representations. As an application we prove that the BVFcomplex controls the formal deformations of coisotropic submanifolds – at least locally or under assumptions on the topology of the coisotropic submanifold. This is established by identifying the P∞algebra structure on the normal bundle of a coisotropic submanifold as a derived homotopy structure of the BVFcomplex. Then we provide the connection to the geometric picture in the new framework. Contents