Algebraic aspects of Chen's twisting cochain

Gugenheim, V. K. A. M.; Lambe, Larry A.; Stasheff, Jim

doi:10.1215/ijm/1255988274

Cited by 55 publications

(49 citation statements)

References 3 publications

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“…For A ∞ -algebras, it was proved by Kadeishvili [29]. For the construction of minimal models of A ∞ -structures, in particular on the homology of a differential graded algebra, homological perturbation theory (HPT) is developed by [19,27,20,21,22,23], for instance, and the form of a minimal model is also given explicitly and more recently in [45,37]. There are various results referred to as minimal model theorems: the weakest form asserts the existence of a quasi-isomorphism as A ∞ -algebras H(A) → A for an A ∞ -structure on H(A), by noticing that all the relevant obstructions vanish because the homology of A and H(A) agree.…”

Section: Minimal Model Theorem and Decomposition Theoremmentioning

confidence: 99%

“…For an A ∞ -or L ∞ -algebra, the minimal model theorem is now combined with various stronger results; those employing the techniques of homological perturbation theory (HPT) (for instance see [19,27,20,21,22,23]), what is called the decomposition theorem in [31,33], Lefèvre's approach [40], etc. These theorems are very powerful and make clear the homotopy invariant nature of the algebraic properties (for instance [42,28,33]).…”

Section: Introductionmentioning

confidence: 99%

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Homotopy Algebras Inspired by Classical Open-Closed String Field Theory

Kajiura

Stasheff

2006

Commun. Math. Phys.

154

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We define a homotopy algebra associated to classical open-closed strings. We call it an open-closed homotopy algebra (OCHA). It is inspired by Zwiebach's open-closed string field theory and also is related to the situation of Kontsevich's deformation quantization. We show that it is actually a homotopy invariant notion; for instance, the minimal model theorem holds. Also, we show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures (A ∞ -algebras) by closed strings (L ∞ -algebras).

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Section: Minimal Model Theorem and Decomposition Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Homotopy Algebras Inspired by Classical Open-Closed String Field Theory

Kajiura

Stasheff

2006

Commun. Math. Phys.

154

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“…The proof of the existence of Q can be handled effectively by the "step-by-step obstruction" methods of homological perturbation theory [G,GM,GSta,GLS,HK]. We adapt the details to this case, rather than appealing to the general theory.…”

Section: Differential Graded Commutative Algebrasmentioning

confidence: 99%

Homological reduction of constrained Poisson algebras

Stasheff¹

1997

J. Differential Geom.

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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 1-3], 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 FBV construction into the context of homological perturbation theory [S1] and, together with Henneaux et al [FHST], extend the construction to the case of non-regular geometric constraints of first class. Independently, using a mixture of homological and C 1 -patching techniques, Dubois-Violette extended the construction to regular but not-necessarily-firstclass constraints [D-V].I am grateful to all of the above for their input and inspiration, whether in their papers or in conversation. The present version has also profitted from conversations at the MSRI Workshop on Symplectic Topology. Finally, I would like to express my thanks to the referee who has read several versions with extreme care, suggesting extensive improvements, both factual and stylistic. While early revision was in progress, Kimura sent me a copy of [Ki] which has also had a significant influence on the present exposition, as has his continued interaction while with me at UNC as an NSF Post-Doc.

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“…The minimal model theorem for A ∞ -algebras was proved by Kadeishvili [30]. For the construction of minimal models of A ∞ -structures, homological perturbation theory was developed by [17,28,18,21,19,20], for instance, and the form of a minimal model is then given explicitly in [54,41]. Also, the existence of a stronger theorem, called the decomposition theorem in [33,36], is mentioned in [40] and proved by employing a kind of homological perturbation theory in [33,36] (see also [47]).…”

Section: Maurer-cartan Equation Minimal Model and Tree Open-closed Smentioning

confidence: 99%

Open-closed homotopy algebra in mathematical physics

Kajiura

Stasheff

2006

Journal of Mathematical Physics

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In this paper we discuss various aspects of open-closed homotopy algebras (OCHAs) presented in our previous paper, inspired by Zwiebach's open-closed 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 open-closed string field theory. We clarify the explicit relation of an OCHA with Kontsevich's deformation quantization and with the B-models 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 open-closed string field theory. We show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures ($A_\infty$-algebras) by closed strings ($L_\infty$-algebras).Comment: 38 pages, 4 figures; v2: published versio

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Algebraic aspects of Chen's twisting cochain

Cited by 55 publications

References 3 publications

Homotopy Algebras Inspired by Classical Open-Closed String Field Theory

Homotopy Algebras Inspired by Classical Open-Closed String Field Theory

Homological reduction of constrained Poisson algebras

Open-closed homotopy algebra in mathematical physics

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