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44
Formal solution of the master equation via HPT and deformation theory
, 1999
"... Abstract. We construct a solution of the master equation by means of standard tools from homological perturbation theory under just the hypothesis that the ground field be of characteristic zero, thereby avoiding the formality assumption of the relevant dg Lie algebra. To this end, we endow the homo ..."
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Cited by 21 (11 self)
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Abstract. We construct a solution of the master equation by means of standard tools from homological perturbation theory under just the hypothesis that the ground field be of characteristic zero, thereby avoiding the formality assumption of the relevant dg Lie algebra. To this end, we endow the homology H(g) of any differential graded Lie algebra g over a field of characteristic zero with an shLie structure such that g and H(g) are shequivalent. We discuss our solution of the master equation in the context of deformation theory. Given the extra structure appropriate to the extended moduli space of complex structures on a CalabiYau manifold, the known solutions result as a special case.
Homological reduction of constrained Poisson algebras
 J. Diff. Geom
, 1997
"... Reduction of a Hamiltonian system with symmetry and/or constraints has a long history. There are several reduction procedures, all of which agree in “nice ” cases [AGJ]. Some have a geometric emphasis reducing a (symplectic) space of states [MW], while others are algebraic reducing a (Poisson) alg ..."
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Cited by 18 (3 self)
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Reduction of a Hamiltonian system with symmetry and/or constraints has a long history. There are several reduction procedures, all of which agree in “nice ” cases [AGJ]. Some have a geometric emphasis reducing a (symplectic) space of states [MW], while others are algebraic reducing a (Poisson) algebra of observables [SW]. Some start with a momentum map whose components are constraint functions [GIMMSY]; some start with a gauge (symmetry) algebra whose generators, regarded as vector fields, correspond via the symplectic structure to constraints [D]. The relation between symmetry and constraints is particularly tight in the case Dirac calls “first class”. The present paper is concerned entirely with this first class case and deals with the reduction of a Poisson algebra via homological methods, although there is considerable motivation from topology, particularly via the models central to rational homotopy theory. Homological methods have become increasingly important in mathematical physics, especially field theory, over the last decade. In regard to constrained Hamiltonians, they came into focus with Henneaux’s Report [H] on the work of Batalin, Fradkin and Vilkovisky [BF,BV 13], emphasizing the acyclicity of a certain complex, later identified by Browning and McMullan as the Koszul complex of a regular ideal of constraints. I was able to put the
Noncommutative homotopy algebras associated with open strings
 Rev. Math. Phys
"... 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 16 (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. Contents 1 Introduction and Summary 2 1.1 A∞space and A∞algebras.............................. 2 1.2 A∞structure and classical open string field theory................. 6 1.3 Dual description; formal noncommutative supermanifold.............. 13
Homological Perturbation Theory and Associativity
, 2000
"... In this paper, we prove various results concerning DGAalgebras in the context of the Homological Perturbation Theory. We distinguish two class of contractions for algebras: full algebra contractions and semifull algebra contractions. A full algebra contraction is, in particular, a semifull algebr ..."
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Cited by 16 (10 self)
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In this paper, we prove various results concerning DGAalgebras in the context of the Homological Perturbation Theory. We distinguish two class of contractions for algebras: full algebra contractions and semifull algebra contractions. A full algebra contraction is, in particular, a semifull algebra contraction. Taking a full algebra contraction and an "algebra perturbation" as data of the Basic Perturbation Lemma, the Algebra Perturbation Lemma (or simply, FAPL) of [20] and [29] appears in a natural way. We establish here a perturbation machinery, the SemiFull Algebra Perturbation Lemma (or, simply, SFAPL) that is a generalization of the previous one in the sense that the application range of SFAPL is wider than that of FAPL. We show four important applications in which this result is essential for the construction of algebra or coalgebra structures in various chain complexes.
Transferring Algebra Structures Up to Homology Equivalence
 Math. Scand
, 1998
"... Given a strong deformation retract M of an algebra A, there are several apparently distinct ways ([9],[19], [13], [24],[15], [18], [17]) of constructing a coderivation on the tensor coalgebra of M in such a way that the resulting complex is quasi isomorphic to the classical (differential tor) [7] ba ..."
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Cited by 15 (3 self)
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Given a strong deformation retract M of an algebra A, there are several apparently distinct ways ([9],[19], [13], [24],[15], [18], [17]) of constructing a coderivation on the tensor coalgebra of M in such a way that the resulting complex is quasi isomorphic to the classical (differential tor) [7] bar construction of A. We show that these methods are equivalent and are determined combinatorially by an inductive formula first given in a very special setting in [16]. Applications to de Rham theory and Massey products are given. 1 Preliminaries and Notation Throughout this paper, R will denote a commutative ring with unit. The term (co)module is used to mean a differential graded (co)module over R and maps between modules are graded maps. When we write\Omega we mean\Omega R . The usual (Koszul) sign conventions are assumed. The degree of a homogeneous element m of some module is denoted by jmj. Algebras are assumed to be connected and coalgebras simply connected. (Co)algebras are assumed to have (co)units.(Co)algebras are, unless otherwise stated, assumed to be (co)augmented. The differential in an (co)algebra is a graded (co)derivation. The Rmodule of maps from M to N (for Rmodules M and N) is denoted by hom(M;N) (if the context requires it, we will use a subscript to denote the ground ring). The differential in this module is given by D(f) = df \Gamma (\Gamma1) jf j fd. Note that D is a derivation with respect to the composition operation whenever it is defined. In particular, End(M) = hom(M;M) is an algebra. If A is an algebra and C is a coalgebra, the module hom(C; A) is an algebra with 1 respect to the operation defined by the following diagram C f [ g  A C\Omega C \Delta ? f\Omega g  A\Omega A 6 m (1) This product is called the cup or convolution...
Homological perturbations, equivariant cohomology, and Koszul duality
"... Dedicated to the memory of V.K.A.M. Gugenheim Abstract. Our main objective is to demonstrate how homological perturbation theory (HPT) results over the last 40 years immediately or with little extra work give some of the Koszul duality results that have appeared in the last decade. Higher homotopies ..."
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Cited by 10 (6 self)
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Dedicated to the memory of V.K.A.M. Gugenheim Abstract. Our main objective is to demonstrate how homological perturbation theory (HPT) results over the last 40 years immediately or with little extra work give some of the Koszul duality results that have appeared in the last decade. Higher homotopies typically arise when a huge object, e. g. a chain complex defining various invariants of a certain geometric situation, is cut to a small model, and the higher homotopies can then be dealt with concisely in the language of shstructures (strong homotopy structures). This amounts to precise ways of handling the requisite additional structure encapsulating the various coherence conditions. Given e. g. two augmented differential graded algebras A1 and A2, an shmap from A1 to A2 is a twisting cochain from the reduced bar construction BA1 of A1 to A2 and, in this manner, the class of morphisms of augmented differential graded algebras is extended to that of shmorphisms. In the present paper, we explore small models for equivariant (co)homology via differential homological algebra techniques including homological perturbation theory which, in turn, is a standard tool to handle
Obstruction theory for objects in abelian and derived categories
"... Abstract. In this paper we develop the obstruction theory for lifting complexes, up to quasiisomorphism, to derived categories of flat nilpotent deformations of abelian categories. As a particular case we also obtain the corresponding obstruction theory for lifting of objects in terms of Yoneda Ext ..."
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Cited by 9 (1 self)
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Abstract. In this paper we develop the obstruction theory for lifting complexes, up to quasiisomorphism, to derived categories of flat nilpotent deformations of abelian categories. As a particular case we also obtain the corresponding obstruction theory for lifting of objects in terms of Yoneda Extgroups. In appendix we prove the existence of miniversal derived deformations of complexes. 1.
Openclosed homotopy algebra in mathematical physics
 J. Math. Phys
"... In this paper we discuss various aspects of openclosed homotopy algebras (OCHAs) presented in our previous paper, inspired by Zwiebach’s openclosed string field theory, but that first paper concentrated on the mathematical aspects. Here we show how an OCHA is obtained by extracting the tree part o ..."
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Cited by 9 (2 self)
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In this paper we discuss various aspects of openclosed homotopy algebras (OCHAs) presented in our previous paper, inspired by Zwiebach’s openclosed string field theory, but that first paper concentrated on the mathematical aspects. Here we show how an OCHA is obtained by extracting the tree part of Zwiebach’s quantum openclosed string field theory. We clarify the explicit relation of an OCHA with Kontsevich’s deformation quantization and with the Bmodels of homological mirror symmetry. An explicit form of the minimal model for an OCHA is given as well as its relation to the perturbative expansion of openclosed string field theory. We show that our openclosed homotopy algebra gives us a general scheme for deformation of open string structures (A∞algebras) by closed strings (L∞algebras).