# Nikita Markarian’s mathblog

## Perturbative Chern-Simons theory and spherical homology

Posted in Mathematics by nikitamarkarian on March 22, 2010

I already made some unclear hints about connections between the perturbative Chern-Simons theory and spherical homology. Below I would like to clarify it. It seems that nearly everything I am going to say is known as “BV formalism”. But until I learn this language let me continue with my own.

Given a 3-manifold and a (say) $SU(n)$ connection on it with zero cohomology, the perturbative Chern-Simons theory gives an invariant of it, that is a power series of the quantization parameter. It would be plausible to make this invariant two-step: on get from the initial data some structure  (like a homotopy algebra  over a operad) and then take an invariant of it.

The structure involved is a 1-parametric family of homotopy BV algebras that is curvy trivial and trivialized . The spherical homology gives  invariants of this structure.

Let me explain and comment last two phrases.

$S$-homology is an obvious generalization of the spherical homology. I already mentioned it here. In the present post I would like to summarise their properties.
UPD: Following M. Kapranov, I will use term manifoldic homology for the higher dimensional generalization of $S$-homology.