Nikita Markarian’s mathblog

Manifoldic homology and Chern-Simons formalism

Posted in Mathematics by nikitamarkarian on July 30, 2011

My preprint arXiv:1106.5352v1 is mostly inspired by the perfect explanation of Damien Calaque of the Kevin Costello’s project. Given a $e_n$-algebra $A$ and a compact parallelized $n$-manifold without borders $M$ a morphism from homology of the Lie algebra $L(A)$ associated with $A$ to some manifoldic homology is constructed in the preprint. I guess that for a proper $A$ this morphism gives perturbative Chern-Simons invariants of $M$ and that this construction is closely connected with the Costello’s approach. Some comments are under the cut.
Let $A$ be a $e_n$-algebra. Then $A[1-n]$ is equipped with a structure of $L_\infty$-algebra. Denote this $L_\infty$-algebra by $L(A)$. In the preprint I explicitly write down the morphism from homology of $L(A)$ to the manifoldic homology of $A$ on a parallelized n-manifold without borders. This morphism is the composition $H_*(L(A))\to HM_*(U^n(A))\to HM_*(A)$, where the first arrow is the tautological morphism and the second arrow is induced by the canonical morphism.

How is it connected with the Chern-Simons perturbative invariants? The point is that for some algebras $A$ the space $HM_*(A)$ is one-dimensional for any compact $M$ without borders, thus we get an invariant of $M$ lying in the cohomology of $L(A)$. To get an example of such an algebra consider a commutative algebra $C$, then consider it as $e_{n-1}$-algebra and take its deformation complex $Def_{e_{n-1}}(C)$. It has a structure of $e_{n}$-algebra.

Conjecture. For $n>1$, a compact $n$-manifold $M$ without borders and a commutative algebra $C$ the manifoldic homology $HM_*(Def_{e_{n-1}}(C))$ is the shifted derived de Rham complex of $C$. In particular, for the polynomial algebra $C$ the manifoldic homology is one-dimensional.

Note that this conjecture is consistent with the rigidity of the manifoldic homology that was discussed here.

For $n=1$ the statement is that Hochschild homology of the ring of differential operators is one-dimensional (it sits in the degree $2n$, where $n$ is the dimension of the affine space). This fact plays a crucial role the Bressler-Tsygan-Nest proof of the Riemann-Roch theorem cited in the Costello’s paper. Note that the morphism from the homology of the Lie algebra of differential operators to Hochschild homology of differential operators is very important there.