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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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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
Deformations of BatalinVilkovisky algebras
 of Banach Center Publ
, 1998
"... 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 t ..."
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Cited by 13 (0 self)
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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 upto 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
An abstract interface to higher spin gauge field theory
 J. Math. Phys
"... 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 th ..."
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Cited by 10 (4 self)
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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 physicslike, 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.
Higher order BRST and antiBRST operators and cohomology for compact Lie algebras : hepth/9810212
"... After defining cohomologically higher order BRST and antiBRST 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 tra ..."
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After defining cohomologically higher order BRST and antiBRST 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 CasimirRacah 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.
Origins and breadth of the theory of higher homotopies
, 2007
"... 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 ..."
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Cited by 7 (3 self)
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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 AlexanderWhitney 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 Hspaces, and a careful examination of this extension led Stasheff to the discovery of Anspaces 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
Structure of higher spin gauge interactions
 J. Math. Phys
, 2007
"... 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 constru ..."
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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 BRSTBV formulation of the theory will be elaborated. The relation between the vertex order expansion and homological perturbation theory will be clarified. A formal nonobstruction argument is reviewed. The syntactically derived shLie 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
On the generalized FreedmanTownsend model
 J. High Energy Phys
"... Consistent interactions that can be added to a free, Abelian gauge theory comprising a finite collection of BF models and a finite set of twoform gauge fields (with the Lagrangian action written in firstorder form as a sum of Abelian FreedmanTownsend models) are constructed from the deformation of ..."
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Consistent interactions that can be added to a free, Abelian gauge theory comprising a finite collection of BF models and a finite set of twoform gauge fields (with the Lagrangian action written in firstorder form as a sum of Abelian FreedmanTownsend models) are constructed from the deformation of the solution to the master equation based on specific cohomological techniques. Under the hypotheses of smoothness in the coupling constant, locality, Lorentz covariance, and Poincaré invariance of the interactions, supplemented with the requirement on the preservation of the number of derivatives on each field with respect to the free theory, we obtain that the deformation procedure modifies the Lagrangian action, the gauge transformations as well as the accompanying algebra. The interacting Lagrangian action contains a generalized version of nonAbelian FreedmanTownsend model. The consistency of interactions to all orders in the coupling constant unfolds certain equations, which are shown to have solutions. PACS number: 11.10.Ef 1
Towards Unifying Structures in Higher Spin Gauge Symmetry
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2008
"... This article is expository in nature, outlining some of the many still incompletely understood features of higher spin field theory. We are mainly considering higher spin gauge fields in their own right as freestanding theoretical constructs and not circumstances where they occur as part of another ..."
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Cited by 3 (0 self)
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This article is expository in nature, outlining some of the many still incompletely understood features of higher spin field theory. We are mainly considering higher spin gauge fields in their own right as freestanding theoretical constructs and not circumstances where they occur as part of another system. Considering the problem of introducing interactions among higher spin gauge fields, there has historically been two broad avenues of approach. One approach entails gauging a nonAbelian global symmetry algebra, in the process making it local. The other approach entails deforming an already local but Abelian gauge algebra, in the process making it nonAbelian. In cases where both avenues have been explored, such as for spin 1 and 2 gauge fields, the results agree (barring conceptual and technical issues) with Yang–Mills theory and Einstein gravity. In the case of an infinite tower of higher spin gauge fields, the first approach has been thoroughly developed and explored by M. Vasiliev, whereas the second approach, after having lain dormant for a long time, has received new attention by several authors lately. In the present paper we briefly review some aspects of the history of higher spin gauge fields as a backdrop to an attempt at comparing the gauging vs. deforming approaches. A common unifying structure of strongly homotopy Lie algebras
BV YangMills as a homotopy ChernSimons
"... We show that BV YangMills action can be reformulated in the homotopy ChernSimons form. The corresponding formalism is based on the constructions introduced in [4], where the YangMills equations were rewritten as the generalized MaurerCartan equations for some Homotopy Lie algebra. 1 Introduction ..."
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We show that BV YangMills action can be reformulated in the homotopy ChernSimons form. The corresponding formalism is based on the constructions introduced in [4], where the YangMills equations were rewritten as the generalized MaurerCartan equations for some Homotopy Lie algebra. 1 Introduction: ChernSimons vs YangMills ChernSimonslike theories have played the important role in both Quantum Field Theory and String Theory for a long time. The interest to such theories