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.

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S-homology

Posted in Mathematics by nikitamarkarian on March 18, 2010

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: 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 S-homology.

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