# Nikita Markarian’s mathblog

## Spherical homology of a Hycomm-algebra

Posted in Mathematics by nikitamarkarian on December 22, 2010

Spherical homology of a homotopy BV algebra was a subject of some previous posts.  By Hycomm=BV/Δ, a Hycomm- algebra is a special case af a homotopy BV algebra. Below I discuss how spherical homology of such an algebra looks like.

I will follow the point of view of this post. Denote by $A_0$ a homotopy BV algebra and by $A$ the $f2$-algebra corresponding to it under the formality quasiisomorphism between BV and f2 operads. Recall that Hochschild homology $HH(A)$ of $A$ being considered as a (homotopy) algebra is equipped with a multiplication and $HH(A_0)$ is equipped with a Poisson structure. The former may be considered as a quantization of the latter. Algebra $A$ is a homotopy module over $HH(A)$, this module structure may be considered as a quantization of the trivial action  of $HH(A_0)$ on $A_0$ that sends image of Reinhart-Connes-Feigin-Tsygan differential to zero.

Now suppose that $A_0$ comes from a Hycomm algebra. In other words, $\Delta$ is homotopically (but not coherently) trivial.  This follows that the bracket of the Poisson structure on $HH(A_0)$ is trivial.  Thus algebra $HH(A)$ is commutative as it is a quantization of $HH(A_0)$. And moreover it is isomorphic to $HH(A_0)$ equipped with the standard commutative multiplication, at least formally by the quantization parameter. So the algebra structure on $HH(A)$ does not change under the quantization, but the structure of module on $A$ does. The element $m\in Ext^1_{HH(A_0)}(A_0, A_0)$ corresponding to this deformation is given by $m(a, Bb, \dots , Bc)(x)= \{ a ,b,\dots, c, x\}$, where $B$ is Reinhart-Connes-Feigin-Tsygan differential and $\{\dots\}$ is the standart generator of the  $Hycomm$ operad. One may see that the Maurer-Cartan equation follows from relations in $Hycomm$.

It seems that in a  sense an inverse statement is true: for a commutative Frobenius algebra (that is a finite-dimensional algebra with a non-degenerate pairing) a deformation of it as a module over its Hochschild homology that respects the pairing is equivalent to a deformation of the algebra  as a $Hycomm$-algebra. Or better one may consider a module with filtration over the Hochschild homology such that the associated graded module is isomorphic to the algebra itself as a module over Hochscild homology.  Anyway, given a $Hycomm$-algebra $A$ it is worth looking at the pair $(HH(A), A)$.

This approach well fits the theory of quantum cohomology as it was discussed in the previous post. There for a compact complex manifold $M$ twisted K-theory  of the loop space $K^\zeta(LM)$ was introduced. Besides there is the usual K-theory of the loop space $K(LM)$. $K(LM)$ is equipped by a product à la Chas-Sullivan. Conjecturally there is an action of $K(LM)$ on $K^\zeta(LM)$. Algebra $K(LM)$ equals to the cohomology ring $H^*(LM)$ and equals to the Hochschild homology of $H^*(M)$.  In some cases $K^\zeta(LM)$ is isomorphic to $H^*(M)$ as a vector space, for example for $\mathbb{CP}^n$. I do not know how to define a filtration on $K^\zeta(LM)$ such that the associated graded module is $H^*(M)$, it seems that one have to take into account the circle action. Anyway, as soon as this is done, we have the pair $(K(LM), K^\zeta(LM))$ that gives the desired $Hycomm$-algebra.