Loop Homotopy Algebras in Closed String Field Theory

Abstract

Barton Zwiebach constructed the `string products' on the Hilbert space of combined conformal field theory of matter and ghosts. It is well-known that the `tree level' specialization of these products forms a strongly homotopy Lie algebra. A strongly homotopy Lie algebra is given by a square zero coderivation on the cofree cocommutative connected coalgebra, on the other hand, strongly homotopy Lie algebras are algebras over the cobar construction on the commutative algebras operad. The aim of our paper is to give two similar characterizations of the structure formed by the `string products' of arbitrary genera. Our first characterization will be based on the notion of a higher order coderivation, the second characterization will be based on the machinery of modular operads. We will also discuss possible generalizations to open string field theory.Comment: LaTeX 2.09, 29 pages. Section "Loop homotopy Lie algebras - operadic approach" substantially revise

On the structure of quantum L∞ algebras

“…In the context of string field theory, this was already analyzed in [1] and further elucidated in the mathematical context in [11]. In this paper we generalize the analysis of [9] to quantum W-algebras.…”

“…In section 3, after identifying the possible origin of quantum corrections, we first define quantum L ∞ algebras. Then we will compare it to loop L ∞ algebras, the quantum corrected L ∞ algebras arising for closed string field theory (CSFT) [1,11]. In section 4 we will show in detail that the quantum W 3 algebra is organized in terms of a quantum L ∞ algebra.…”

“…We will now compare our definition for a quantum L ∞ algebra with the L ∞ algebra of closed string field theory (CSFT) [1,11]. To distinguish these two different L ∞ definitions,…”

“…we will follow Markl [11] and call the L ∞ algebra of CSFT a loop L ∞ algebra, while the definition from last section will be called quantum L ∞ algebra.…”

“…Then, the L g n products define a loop L ∞ algebra, if for any level g the following relation holds (we use the notation of [11]…”