TCFT totalization versus ribbon graph complex and compatibility with Kontsevich’s map
Prove that the open TCFT functor F associated to a cyclic A∞-category with objects B induces a map F_tot: Tot^+(𝒪_B) → 𝔽^{pq}(V_B); prove that Tot^+(𝒪_B) is quasi-isomorphic to the B-coloured ribbon graph complex ℛ𝔾_B; and, under this identification, show that the generalized Kontsevich map ρ_I equals F_tot (extended γ-linearly) and is compatible with gluing of open boundaries corresponding to leaf-gluing in ribbon graphs.
References
\begin{Claim}[`Conjecture']\label{ibssm} We claim that $F$ induces a map $$F_{tot}:Tot+(\mathcal{O}_B)\rightarrow \mathcal{F}{pq}(V_B).$$ We further claim that $$ Tot+(\mathcal{O}_B)\simeq \mathcal{RG}{B}$$ and under this conjectural equivalence $\rho_I=F{tot}$ when extended $\gamma$-linearly (see diagram \ref{tot_nc}). Further this quasi-isomorphism is compatible with the operation of gluing open boundaries together and corresponds to gluing leafs together. \end{Claim} Up to signs and possible gradings, this claim should follow from Costello's work, eg Proposition 6.2.1 of and generally section 6.2 of , see also and may be straightforward to experts.\footnote{The idea is that we can reorder the tensor factors as in \ref{tensf}, quotiented by the symmetric group, by remembering their preimage under $F$. For each element of $Tot+(\mathcal{O}_B)$ we obtain a symmetric ordering of cyclic orderings of elements of $B$, provided by the coloring of free boundaries in the boundary components of a surface.}
We did not check the details on signs and gradings and thus formulated the above as a conjecture.