On a theorem of Kontsevich

Abstract

In [12] and [13], M. Kontsevich introduced graph homology as a tool to compute the homology of three infinite dimensional Lie algebras, associated to the three operads "commutative," "associative" and "Lie." We generalize his theorem to all cyclic operads, in the process giving a more careful treatment of the construction than in Kontsevich's original papers. We also give a more explicit treatment of the isomorphisms of graph homologies with the homology of moduli space and Out(F r ) outlined by Kontsevich. In [4] we defined a Lie bracket and cobracket on the commutative graph complex, which was extended in [3] to the case of all cyclic operads. These operations form a Lie bi-algebra on a natural subcomplex. We show that in the associative and Lie cases the subcomplex on which the bi-algebra structure exists carries all of the homology, and we explain why the subcomplex in the commutative case does not.

Modular envelopes, OSFT and nonsymmetric (non-$\Sigma$) modular operads

“…[ April 15, 2014] with the spikes representing elements of C. Our pancakes appeared as 'spiders' in [5,32] Let Fin (resp. Cyc) denote the category of finite (resp.…”

“…where indicates that the corresponding term has been omitted, and C ′ i ∪ C ′′ j \{u, v} is cyclically ordered as in (5).…”

“…with C i ∪ C j \ {u, v} cyclically ordered as in (5). If i = j, we define (12) as the multicyclically ordered set given by the decomposition…”

“…By the associativity [32, Theorem II. 5.42] of the contractions, ̟ Γ = ̟ K , it is therefore enough to prove (34) for graphs K with one vertex. Graphically, such a K is a non-planar 'tick' with g 'bellies' as in where g = 3.…”

Dual Feynman transform for modular operads

“…Similarly we define a 1-orientation to be an ordering of the vertices together with an orientation of every edge of G, again modulo even permutations. A d-orientation on G determines a non-zero element of Det(Edge(G)) ⊗ Det d (H 1 (G)); see, e.g., [5]; reversing the orientation means negating the corresponding element.…”