Cyclic operads and homology of graph complexes

We will consider P-graph complexes, where P is a cyclic operad. P-graph complexes are natural generalizations of Kontsevich's graph complexes -- for P = the operad for associative algebras it is the complex of ribbon graphs, for P = the operad for commutative associative algebras, the complex of all graphs. We construct a `universal class' in the cohomology of the graph complex with coefficients in a theory. The Kontsevich-type invariant is then an evaluation, on a concrete cyclic algebra, of this class. We also explain some results of M. Penkava and A. Schwarz on the construction of an invariant from a cyclic deformation of a cyclic algebra. Our constructions are illustrated by a `toy model' of tree complexes.Comment: LaTeX 2.09 + article12pt,leqno style, 10 page

Loop Homotopy Algebras in Closed String Field Theory

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

Homotopy Algebras via Resolutions of Operads

The aim of this brief note is mainly to advocate our approach to homotopy algebras based on the minimal model of an operad. Our exposition is motivated by two examples which we discuss very explicitly - the example of strongly homotopy associative algebras and the example of strongly homotopy Lie algebras. We then indicate what must be proved in order to show that these homotopy algebraic structures are really `stable under a homotopy.' The paper is based on a talk given by the author on June 16, 1998, at University of Osnabrueck, Germany.Comment: LaTeX 2.09, 9 pages; `indecomposables' changed to `decomposables