## Perturbative Chern-Simons theory and spherical homology

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) 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

-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:** Things described here and in some previous posts are particular cases of the topological chiral homology introduced in J. Lurie, “Derived Algebraic Geometry VI: E_k Algebras”. The only difference is that I need framed discs. I thank Damien for this comment.

**UPD:** Following M. Kapranov, I will use term *manifoldic homology* for the higher dimensional generalization of -homology.

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