Nikita Markarian’s mathblog

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”.