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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
A space of cyclohedra
 Discrete Comput. Geom
"... Abstract. The real points of the DeligneMumfordKnudsen moduli space Mn 0 of marked points on the sphere has a natural tiling by associahedra. We extend this idea to create a moduli space tiled by cyclohedra. We explore the structure of this space, coming from blowups of hyperplane arrangements, a ..."
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Cited by 5 (0 self)
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Abstract. The real points of the DeligneMumfordKnudsen moduli space Mn 0 of marked points on the sphere has a natural tiling by associahedra. We extend this idea to create a moduli space tiled by cyclohedra. We explore the structure of this space, coming from blowups of hyperplane arrangements, as well as discuss possibilities of its role in knot theory and mathematical physics. Acknowledgments. I am grateful to Mike Davis for pointing out the affine example given in Brown [6] and to Vic Reiner for conversations about noncrossing partitions. Thanks also go to Jack Morava and Jim Stasheff for their encouragement. Finally, I am indebted to the late Rodica Simion who provided motivation, discussions, and enthusiasm. This paper is dedicated in her memory. 1.
Associativity as Commutativity
, 2005
"... It is shown that coherence conditions for monoidal categories concerning associativity are analogous to coherence conditions for strict symmetric monoidal categories, where associativity arrows are identities. Mac Lane’s pentagonal coherence condition for associativity is decomposed into conditions ..."
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Cited by 3 (1 self)
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It is shown that coherence conditions for monoidal categories concerning associativity are analogous to coherence conditions for strict symmetric monoidal categories, where associativity arrows are identities. Mac Lane’s pentagonal coherence condition for associativity is decomposed into conditions concerning commutativity, among which we have a condition analogous to naturality and a degenerate case of Mac Lane’s hexagonal condition for commutativity. This decomposition is analogous to the derivation of the YangBaxter equation from Mac Lane’s hexagon and the naturality of commutativity. The pentagon is reduced to an inductive definition of a kind of commutativity.
Contents
, 2005
"... 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 ..."
Abstract
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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.