Nikita Markarian’s mathblog

Perturbative Chern-Simons theory and spherical homology

Posted in Mathematics by nikitamarkarian on March 22, 2010

I already made some unclear hints about connections between the perturbative Chern-Simons theory and spherical homology. Below I would like to clarify it. It seems that nearly everything I am going to say is known as “BV formalism”. But until I learn this language let me continue with my own.

Given a 3-manifold and a (say) $SU(n)$ connection on it with zero cohomology, the perturbative Chern-Simons theory gives an invariant of it, that is a power series of the quantization parameter. It would be plausible to make this invariant two-step: on get from the initial data some structure  (like a homotopy algebra  over a operad) and then take an invariant of it.

The structure involved is a 1-parametric family of homotopy BV algebras that is curvy trivial and trivialized . The spherical homology gives  invariants of this structure.

Let me explain and comment last two phrases.

Begin with the “curvy trivial” (horrible term) deformation. For any algebra over a operad there exists the standard notion of the deformation complex $\mathbf{Def}$: differentiations of the free cooperad generated by the complex dual to the algebra (for a BV algebra see Imma Galvez-Carrillo, Andy Tonks, Bruno Vallette “Homotopy Batalin-Vilkovisky algebras”). Honest 1st-order deformations form its subcomplex $\mathbf{HDef}$ and the rest is responsible for “gerby” or “curvy” deformations. Let us consider a familiar example: (homotopy) associative algebra $A$. The deformation complex $\mathbf{Def}$ is the Hochschild cohomological complex $C^*(A,A)$ and honest 1-order $A_\infty$  deformations are described by the subcomplex $\mathbf{HDef}=C^{\ge 1}(A,A)$. For a non-central element $a\in A$ of degree 1 the 1-cocycle $[a, \dots ]$ is a coboundary in $\mathbf {Def}$, but it gives a non trivial 1st-order deformation of $A$. This is an example of what I call a curvy trivial deformation.

Definition. Call a 1st-order deformation of  an algebra over a operad curvy trivial if the corresponding class lies in the kernel of the embedding morphism $H^*(\mathbf{HDef})\to H^*(\mathbf {Def})$ and call it trivialized if the class is lifted to the cohomology of $cone (\mathbf{HDef}\to \mathbf{Def})$. A smooth family of algebras such that  the tangent 1st-order deformation at any point is curvy trivial (and trivialized) we will call curvy trivial (and trivialized).

An example of a curvy trivial deformation of a dg BV algebra is given by the cocycle $\{f,\dots \}$   for a $\Delta$-closed $f$ and choice of $f$ is the trivialization.

A classical example of  a curvy trivial and trivialized family is the following. Consider a family of Calabi-Yau manifolds. Comlexes of polyvector fields form a family of dg BV algebras that is curvy trivial and trivialized. Indeed, deformation of the BV algebra corresponding to a deformation of Calabi-Yau is given by cocycle $\{f,\dots \}$, where $f$ is the tangent vector to the moduli space of Calabi-Yau manifolds.

Now the spherical homology come into play. The spherical homology (and any $S$-homology for a compact oriented $S$) is obviously acted by the deformation complex of BV algebras (or f2-algebras) $\mathbf{Def}$. It follows that the spherical homology of a curvy trivial and trivialized family is equipped with a connection. A section of this local system obey some differential equation that is an invariant of the family. For example, the unit of the algebra gives such a section. For the example above about a family of Calabi-Yau manifolds this section is the integral of the volume form.

Here I associated a BV algebra with an oscillating integral. On may easily see that for a family of functions corresponding family of BV algebras is curvy trivial and trivialized. And (as it follows from the mentioned post) the perturbation theory deals with the local system of spherical homologies. From this perspective phrase “a curvy trivial and trivialized family of BV algebras gives the perturbative Chern-Simons invariant” sounds like a synonym for “the perturbative Chern-Simons invariant is given by an oscillating integral”.

Suppose the 3-manifold is triangulated. What one should place in a simplex? That is what is a perturbative analog of  6j-symbol (this question is posed as problem 7.5 in )? I guess the answer should come from the quasiclassical quantum group theory (connection between BV and quantum group was already mentioned here). Is it classical $r$-matrix that responsible for trivialization of the curvy trivial family of BV algebras?

Finally, let $M$ be a 3d manifold that is a product of a surface $S$ and a circle. The usual Chern-Simons theory is not applicable in this case, because it is not a rational homological sphere.  But the BV algebra in hand may be easily described. Consider the moduli space  of connections on $M$. It equals to the space of pairs (connection on $S$, its automorphism). At a general point this is simply moduli space of connections on $S$. It is equipped with a symplectic form $\omega$ (note that it may be produced (for a surface with holes) from the classical $r$-matrix, see V.V.Fock, A.A.Rosly, “Poisson structure on moduli of flat connections on Riemann surfaces and r-matrix”). The BV algebra is the de Rham complex on this moduli space and $\frac{1}{h} \Delta$ is the Brylinski differential associated with $\omega$ plus the de Rham differential.

In the paper of Hitchin, papers of Simpson and others a hyperkahler structure on the space of local systems on a complex curve was introduced. I wonder if this structure is connected with the family of BV algebras mentioned in the previous paragraph?