A kind of TQFT via BV
In the definition of the spherical homology one may replace 2sphere with any oriented 2manifold, 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.
Hycomm=BV/Δ
By “” I mean the homotopy quotient. This fact is implicitly contained in the classical M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa “KodairaSpencer Theory of Gravity and Exact Results for Quantum String Amplitudes”. It follows from E.Getzler “Twodimensional topological gravity and equivariant cohomology” or from part 3.4 of V.Ginzburg, M.Kapranov “Koszul duality for Operads”.
UPD: In V. Dotsenko, A. Khoroshkin, “Free resolutions via Gröbner bases” a minimal resolution of the BV operad is written down. The result confirms the title of the present post.
Oscillating integrals and spherical homology
In this post we discussed the following situation. Let be a vector space, be a family of functions on and be a volume form on . We describe what one can say “without analysis” about the family of integrals . It appears that this is possible to execute in terms of the following differential graded BV algebra: complex is polyvector fields on , differential is and (d is the de Rham differential, differential forms is identified with polyvector fields by means of the volume form ). It allows to generalize the perturbation theory for arbitrary BV algebras.
The main observation: 2algebra associated with the Gerstenhaber algebra of polyvector fields is Hochschild cochains, and its Hochschild homology as an algebra is the ring of differential operators on total differential forms . This fact is contained in R.Nest, B.Tsygan “On the Cohomology Ring of an Algebra” and in D.Tamarkin, B.Tsygan, “The ring of differential operators on forms in noncommutative calculus”, see also the draft note from this post. is not exactly differential operators, but a thing Morita equivalent to it. Consider the differential 2algebra that corresponds to the BV algebra produced in the previous paragraph. Its Hochschild homology as an algebra is ( is the usual commutator), that is a thing which is Morita equivalent to differential operators. Left and right modules over it provided by the statement correspond to and under the Morita equivalence. As a result one has .
Differential operators 

Dmodules and 
as left and right module 
Spherical homology, the geometric case
Let be a topological space and be its double loop space. As we already discussed, is acted by the operad of framed little discs. It follows that the complex of chains is acted by the chains of framed little discs, that is it is a f2algebra. Here we consider spherical homology of this f2algebra and show that two definitions (this and that) give the same result. It follows, by the way, that they coincide being applied to free algebras.
Spherical homology II
Here I give the second definition of the spherical homology based on a statement about the Hochschild homology of a BV(f2)algebra that has an independent interest.
Spherical homology I
Here I give a definition of spherical homology, a functor from the category of BValgebras to the category of complexes, that is analogous to the Hochschild homology of associative algebras.
Notation: algebra over the operad of chains of framed little discs := f2algebra.
leave a comment