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.

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