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.

Hycomm=BV/Δ

Posted in Mathematics by nikitamarkarian on November 22, 2009

By “$/$” I mean the homotopy quotient. This fact is implicitly contained in the classical M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa “Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes”. It follows from E.Getzler “Two-dimensional 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

Posted in Mathematics by nikitamarkarian on November 16, 2009

In this post we discussed the following situation. Let $V$ be a vector space, $f$ be a family of functions on $V$ and $\omega$ be a volume form on $V$. We describe what one can say “without analysis” about the family of integrals $\int e^f \omega$. It appears that this is possible to execute in terms of the following differential $Z/2Z$-graded BV algebra: complex is polyvector fields on $V$, differential is $\wedge df$ and $\Delta=d+ \wedge df$ (d is the de Rham differential, differential forms is identified with polyvector fields by means of the volume form $\omega$). It allows to generalize the perturbation theory for arbitrary BV algebras.

The main observation: 2-algebra 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 $\mathop{Diff}(\Omega^*, \Omega^*)$. 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. $\mathop{Diff}(\Omega^*, \Omega^*)$ is not exactly differential operators, but a thing Morita equivalent to it. Consider the differential 2-algebra $B$ that corresponds to the BV algebra produced in the previous paragraph. Its Hochschild homology as an algebra is $(\mathop{Diff}(\Omega^*, \Omega*), [\cdot, df])$ ($[,]$ 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 $O_f$ and $\omega_f$ under the Morita equivalence. As a result one has $Tor_{D(V)}^*(O_f, \omega_f)=HS_*(B)$.

 Differential operators $D(V)$ $HH(B)$ D-modules $O_f$ and $\omega_f$ $B$ as left and right $HH(B)$-module $Tor_{D(V)}^*(O_f, \omega_f)$ $HS_*(B)$
$Tor_{D(V)}^*(O_f, \omega_f)$

Spherical homology, the geometric case

Posted in Mathematics by nikitamarkarian on November 15, 2009

Let $X$ be a topological space and $\Omega^2 X$ be its double loop space. As we already discussed, $\Omega^2 X$ is acted by the operad of framed little discs. It follows that the complex of chains $C_*(\Omega^2 X)$ is acted by the chains of framed little discs, that is it is a f2-algebra. Here we consider spherical homology of this f2-algebra 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

Posted in Mathematics by nikitamarkarian on November 13, 2009

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

Posted in Mathematics by nikitamarkarian on November 10, 2009

Here I  give a definition of spherical homology, a functor from the category of BV-algebras 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 := f2-algebra.