Nikita Markarian’s mathblog

A kind of TQFT via BV

Posted in Mathematics by nikitamarkarian on November 30, 2009

In the definition of the spherical homology one may replace 2-sphere with any oriented 2-manifold, not only compact. It produces the following structure, that is not exactly TQFT as it is introduced e.g. in the Atiyah book and many other places, but a close thing.

So given a DG BV-algebra (better, a f2-algebra) B, for any oriented 2-manifold one has a complex and glueing of two 2-manifolds gives a map from the tensor product of complexes corresponding to pieces to the one of the result. The complex corresponding to annulus is an algebra, because composition of two annuli is an annulus again. In fact, this is the Hochschild homology of the given BV-algebra. A complex corresponding to a manifold with n border circles is a module over HH(B)^{\otimes n} and glueing is the torsion product over this algebra. For example, torus gives HH_*(HH(B)).

If the BV-algebra is the chain complex of a double loop space \Omega^2 X, the complex corresponding to a 2 -manifold M is the chain complex of the space of maps (M,X).

It seems that to get the TQFT responsible for the non-perturbative Chern-Simons theory (see e.g. the Witten’s paper) one needs a generalization of the construction described. Perhaps, the BV-algebra must be replaced with  a category (of representations of small quantum group?). And the Hochschild homology – with K-theory?

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