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.
Homology of (S^n, X)
Goodwillie, BurgheleaFiedorowicz and others proved that homology of free loop space of a topological space is isomorphic to Hochschild homology of chains considered as an algerbra under the Pontryagin product. I already discussed it here, because this fact “explains” multiplication on the Hochschild homology of an algebra over the operad of chains of little discs.
Below I cite a generalization of this statement that was announced by A. Voronov in section I.5.4 of “String topology and cyclic homology.
UPD: As I learned from V. Turchin, in P. Salvatore, “Configuration spaces with summable labels” this statement is proved for mapping space from any manifolds, not only spheres.
Perturbative ChernSimons theory and homotopy BV algebras
By the ChernSimons theory I mean the following problem I learned long time ago from the Atiyah book . Let be a smooth real 3manifold. Let be a compact Lie group, say . For any connection on the trivial bundle given by a 1form define the ChernSimons functional by . (It does not depend on trivialization only up to an integer times ). The partition function is then defined by , where the integral is over all connections.
There are two ways to make sense of the integral. The first one is through quantum groups and is described in the groundbreaking Witten’s paper. The second one is the perturbative theory, it is developed in papers of Axelrod and Singer.
The perturbative ChernSimons theory works with the stationary phase approximation . Stationary points of the exponent are flat connections. The integral is the sum of contributions of all flat connections. Let us consider it locally in the neighbourhood of the trivial connection. Function is quadraticcubic. After changing variables one may consider cubic term as a perturbation of the quadratic one. Suppose that is a homology sphere, it guarantees that the quadratic term is nondegenerate. Even if we can not calculate the leading term, which is a infinitedimensional Gaussian integral, the perturbation theory gives us corrections to it as a series of . These corrections are written down by Axelrod and Singer in terms of some integrals of tensors built up from the Green function on configuration spaces of .
Obviously, these terms are a kind of some Massey products. But of what structure? An answer on this question would give us, for example, a recipe to calculate them in terms of a simplicial presentation of the manifold, without referring to any differential geometry. A (psychological) obstruction that obscures the answer is the following. The only object of homological algebra we have is the DG Lie algebra of differential forms on taking values in the Lie algebra of with the de Rham differential. But to execute the construction it is necessary that the differential would be exact. It seems that there no room for any Massey products.
The solution was found due to the recent paper Imma GalvezCarrillo, Andy Tonks, Bruno Vallette “Homotopy BatalinVilkovisky algebras”.
Homological algebra of perturbation theory
Below there are some obvious and elementary observations about Dmodules and oscillating integrals. I suspect that they are widely known, but I could not find them anywhere in literature.
Classifying operad
In the previous post everything is based on the following observation. Let be an algebra over chains of small discs operad. Then is in particular a homotopy algebra. I propose that on Hochschild homology of there is a natural action of an operad equivalent to the
associative operad (this is essentially proved in M. Brun, Z. Fiedorowicz, R. Vogt “On the multiplicative structure of topological Hochschild homology”). More generally, one may try introduce of action (whatever it means) of an operad equivalent to on the category of twisted (semifree) complexes of and derive the previous action from this one.
Thinking about a proof of this statement I formulated a more general question.
(more…)
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