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

We show that a graded commutative algebra A with any square zero odd differential operator is a natural generalization of the Batalin–Vilkovisky algebra. While such an operator of order 2 defines a Lie algebra structure on A, an operator of an order higher than 2 (Koszul–Akman definition) leads to the structure of a strongly homotopy Lie algebra (L∞–algebra) on A. This allows us to give a definition of a Batalin–Vilkovisky algebra up-to homotopy. We also make an important conjecture generalizing Kontsevich formality theorem to the Batalin–Vilkovisky algebra level. 1. Introduction. Batalin–Vilkovisky algebras are graded commutative algebras with an extra structure given by a second order differential operator of square 0. The simplest example is the algebra of polyvector fields on a vector space R n. There is a second order square zero differential operator on this algebra. This operator comes

A comprehensive approach to the theory of higher spin gauge fields is proposed. By explicitly separating out details of implementation from general principles, it becomes possible to focus on the bare minimum of requirements that such a theory must satisfy. The abstraction is based on a survey of the progress that has been achieved since relativistic wave equations for higher spin fields were first considered in the nineteen thirties. As a byproduct, a formalism is obtained that is abstract enough to describe a wide class of classical field theories. The formalism, viewed as syntax, can then be semantically mapped to a category of homotopy Lie algebras, thus showing that the theory in some sense exists, at least as an abstract mathematical structure. Still, a concrete physics-like, implementation remains to be constructed. Lacking deep physical insight into the problem, an implementation in terms of generalized vertex operators is set up within which a brute force iterative determination of the first few orders in the interaction can be attempted.

In a previous paper, higher spin gauge field theory was formulated in an abstract way, essentially only keeping enough machinery to discuss gauge invariance of an action. The approach could be thought of as providing an interface (or syntax) towards an implementation (or semantics) yet to be constructed. The structure then revealed turned out to be that of a strongly homotopy Lie algebra. In the present paper, the framework will be connected to more conventional field theoretic concepts. The Fock complex vertex operator implementation of the interactions in the BRST-BV formulation of the theory will be elaborated. The relation between the vertex order expansion and homological perturbation theory will be clarified. A formal non-obstruction argument is reviewed. The syntactically derived sh-Lie algebra structure is semantically mapped to the Fock complex implementation and it is shown that the equations governing the higher order vertices are reproduced. Global symmetries and subsidiary conditions are discussed and as a result the tracelessness constraints are discarded. Thus all equations needed to compute the vertices to any order are collected. The framework is general enough to encompass all possible interaction terms. Finally, the abstract framework itself will be strengthened by showing

After defining cohomologically higher order BRST and anti-BRST operators for a compact simple algebra G, the associated higher order Laplacians are introduced and the corresponding supersymmetry algebra Σ is analysed. These operators act on the states generated by a set of fermionic ghost fields transforming under the adjoint representation. In contrast with the standard case, for which the Laplacian is given by the quadratic Casimir, the higher order Laplacians W are not in general given completely in terms of the Casimir-Racah operators, and may involve the ghost number operator. The higher order version of the Hodge decomposition is exhibited. The example of su(3) is worked out in detail, including the expression of its higher order Laplacian W.

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

We show that BV Yang-Mills action can be reformulated in the homotopy Chern-Simons form. The corresponding formalism is based on the constructions introduced in [4], where the Yang-Mills equations were rewritten as the generalized Maurer-Cartan equations for some Homotopy Lie algebra. 1 Introduction: Chern-Simons vs Yang-Mills Chern-Simons-like theories have played the important role in both Quantum Field Theory and String Theory for a long time. The interest to such theories

Consistent interactions among a set of two-form gauge fields in four dimensions are derived along a Hamiltonian cohomological procedure. It is shown that the deformation of the BRST charge and BRST-invariant Hamiltonian for the free model leads to the Freedman-Townsend interaction vertex. The resulting interaction deforms both the gauge transformations and reducibility relations, but not the algebra of gauge transformations. PACS number: 11.10.Ef 1

. A Leibniz n-algebra is a vector space equipped with an n-ary operation which has the property of being a derivation for itself. This property is crucial in Nambu mechanics. For n = 2 this is the notion of Leibniz algebra. In this paper we prove that the free Leibniz (n+1)-algebra can be described in terms of the n-magma, that is the set of n-ary planar trees. Then it is shown that the n-tensor power functor, which makes a Leibniz (n + 1)-algebra into a Leibniz algebra, sends a free object to a free object. This result is used in the last section, together with former results of Loday and Pirashvili, to construct a small complex which computes Quillen cohomology with coe#cients for any Leibniz n-algebra . Mathematics Subject Classification (2000): 17Axx, 70H05. 1. Introduction Leibniz algebras were introduced by the second author in [4]. They play an important role in Hochschild homology theory [4], [5] as well as in Nambu mechanics ([6], see also [1]). Let us recall that a Leibni...

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