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

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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∞-algebras) by closed strings (L∞-algebras).

Berikashvili’s functor D defined in terms of twisting cochains is related to deformation theory, gauge theory, Chen’s formal power series connections, and the master equation in physics. The idea is advertised that some unification and understanding of the links between these topics is provided by the notion of twisting cochain and the idea of classifying twisting cochains.

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 sh-structures (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 sh-map 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 sh-morphisms. 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

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Higher homotopy generalizations of Lie-Rinehart algebras, Gerstenhaber, and Batalin-Vilkovisky algebras are explored. These are defined in terms of various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as in terms of operations involving more than two variables of the Lie triple systems kind. A basic tool is the Maurer-Cartan algebra—the algebra of alternating forms on a vector space so that Lie brackets correspond to square zero derivations of this algebra—and multialgebra generalizations thereof. The higher homotopies are phrased in terms of these multialgebras. Applications to foliations are discussed: objects which serve as replacements for the Lie algebra of vector fields on the “space of leaves ” and for the algebra of multivector fields are developed, and the spectral sequence of a foliation is shown to arise as a special case of a more general spectral sequence including as well the Hodge-de Rham spectral sequence.

Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. The purpose of the talk is to recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least as far back as the 1940’s. Prompted by the failure of the Alexander-Whitney multiplication of cocycles to be commutative, Steenrod developed certain operations which measure this failure in a coherent manner. Dold and Lashof extended Milnor’s classifying space construction to associative H-spaces, and a careful examination of this extension led Stasheff to the discovery of An-spaces and A∞-spaces as notions which control the failure of associativity in a coherent way so that the classifying space construction can still be pushed through. Algebraic versions of higher homotopies have, as we all know, led Kontsevich eventually to the proof of the formality conjecture. Homological perturbation theory (HPT), in a simple form first isolated by Eilenberg and Mac Lane in the early 1950’s, has nowadays become a standard tool to handle algebraic incarnations of higher homotopies. A basic

Abstract. We extend homological perturbation theory to encompass algebraic structures governed by operads and cooperads. Specifically, for an operad O, we define the notion of an ‘O-algebra contraction ’ and we prove that the formulas of the Basic Perturbation Lemma preserve O-algebra contractions. Over a ground ring containing the rational numbers, we give explicit formulas for constructing an O-algebra contraction from any given contraction, generalizing the so called ‘Tensor Trick’. As an illustration of our results we use them to give short proofs of the transfer and minimality theorems for O∞-algebras, where O is any Koszul operad. This subsumes, but is not restricted to, the cases of A∞, C ∞ and L∞-algebras.

Dedicated to Alan Weinstein on his 60’th birthday Abstract. Higher homotopies generalizations of Lie-Rinehart algebras, Gerstenhaber-, and Batalin-Vilkovisky algebras are explored. These are defined in terms of various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as in terms of operations involving more than two variables of the Lie triple systems kind. A basic tool is the Maurer-Cartan algebra—the algebra of alternating forms on a vector space so that Lie brackets correspond to square zero derivations of this algebra—and multi algebra generalizations thereof, and the higher homotopies are phrased in terms of these multi algebras. Applications to foliations are discussed, including objects which serve as a replacement for the Lie algebra of vector fields on the space of leaves and for the algebra of multi vector fields, as well as a construction of the spectral sequence of a foliation. 2000 Mathematics Subject Classification. Primary 17B65 17B66 53C12 57R30; secondary 17B55 17B56 17B81 18G40 53C15 55R20 70H45. Key words and phrases. Quasi-Lie-Rinehart algebra, quasi-Gerstenhaber algebra, quasi-Batalin-Vilkovisky algebra, Lie-Rinehart triple, Maurer-Cartan algebra, reductive homogeneous space, foliation, higher homotopies, Jacobi identity up to higher homotopies, spectral sequence of a foliation. 2 JOHANNES HUEBSCHMANN