# 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.

## Homology of (S^n, X)

Posted in Mathematics by nikitamarkarian on October 30, 2009

GoodwillieBurghelea-Fiedorowicz and others proved that homology of free loop space of a topological space $X$ is isomorphic to  Hochschild homology of chains $C_*(\Omega X)$ 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 Chern-Simons theory and homotopy BV algebras

Posted in Mathematics by nikitamarkarian on October 28, 2009

By the Chern-Simons theory I mean the following problem I learned long time ago from the Atiyah book . Let $M$ be a smooth real 3-manifold. Let $G$ be a compact Lie group, say $U(n)$. For any connection on the trivial $G$-bundle given by a 1-form $A$ define the Chern-Simons functional by $L(A)=\int_M 1/{4\pi} \mathop{Tr} (A\wedge dA +2/3A\wedge A\wedge A)$. (It does not depend on trivialization only up to an integer times $2\pi$). The partition function is then defined by $Z(k)=\int \exp(ikL(A))$, 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 ground-breaking Witten’s paper. The second one is the perturbative theory, it is developed in papers of Axelrod and Singer.

The perturbative Chern-Simons theory works with the stationary phase approximation $k\to \infty$. 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 $L(A)$ is quadratic-cubic. After changing variables one may consider cubic term as a perturbation of the quadratic one.  Suppose that $M$ is a homology sphere, it guarantees that the quadratic term is non-degenerate. Even if we can not calculate the leading term, which is a infinite-dimensional Gaussian integral, the perturbation theory gives us corrections to it as a series of $1/k$. 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 $M$.

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 $M$ taking values in the Lie algebra of $G$ 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 Galvez-Carrillo, Andy Tonks, Bruno Vallette “Homotopy Batalin-Vilkovisky algebras”.

## Homological algebra of perturbation theory

Posted in Mathematics by nikitamarkarian on August 19, 2009

Below there are some obvious and elementary observations about D-modules and oscillating integrals. I suspect that they are widely known, but I could not find them anywhere in literature.

In the previous post everything is based on the following observation. Let $A$ be an algebra over chains of small discs operad. Then $A$ is in particular a homotopy algebra. I propose that on Hochschild homology of $A$ there is a natural action of an operad equivalent to the
associative operad $Ass$ (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 $Ass$ on the category of twisted (semifree) complexes of $A$ and derive the previous action from this one.