# Nikita Markarian’s mathblog

## Hycomm=BV/Δ

Posted in Mathematics by nikitamarkarian on November 22, 2009

By “$/$” I mean the homotopy quotient. This fact is implicitly contained in the classical M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa “Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes”. It follows from E.Getzler “Two-dimensional topological gravity and equivariant cohomology” or from part 3.4 of V.Ginzburg, M.Kapranov “Koszul duality for Operads”.

UPD: In V. Dotsenko, A. Khoroshkin, “Free resolutions via Gröbner bases” a minimal resolution of the BV operad is written down. The result confirms the title of the present post.

Let $k[\Delta]$ be the operad with one unary operation $\Delta, \deg \Delta=-1$ and relation $\Delta^2=0$. There is an obvious embedding $k[\Delta]\to BV$. Let $R$ be the operad with supercommuting unary generators $\Delta, \xi_i, \deg \Delta=-1, \deg \xi_i=-2i$ and relations $d\xi_1=\Delta$, $d\xi_2=\Delta\xi_1+\xi_1\Delta$ and so on. It has trivial cohomology. Then the homotopy quotient $BV/\Delta$ is presented by $BV\otimes_{k[\Delta]} R$.

Consider the operad in the category of topological spaces of framed little discs with marked loops as it is presented on the picture

where bold loops are inserted in all possible positions; alternatively one may think about the singular curve with bold loops collapsed. With every bold loop a circle action is associated as follows: cut the picture along the loop, then rotate the inner part and glue again. Erasing of loops give maps between spaces with different number of loops. Consider the operad that is the limit under these maps.

Equivariant homology of this operad under the circles action equals to $BV/\Delta$. On the other hand, it may be retracted on the standard resolution of $Hycomm$ via the Koszul dual operad, if one present the gravity operad (Koszul dual to $Hycom$) in terms of the equivariant homology like in the mentioned paper of Getzler; at this point one needs something like autoduality of the Gerstenhaber operad.

Question: what about the quotient of the Gerstenhaber operad by the bracket? Is it again $Hycomm$?

TWO-DIMENSIONAL TOPOLOGICAL GRAVITY AND
EQUIVARIANT COHOMOLOGY