Nikita Markarian’s mathblog

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”.

Let us state a more general problem. One has a differential $Z/2Z$-graded Lie algebra $(\mathbf{g},d_{Lie})$  (in the Chern-Simons case the de Rham complex on $M$ taking values in the Lie algebra of $G$ ) with an odd non-degenerate pairing (CS: $\int \mathop{Tr} (\alpha\wedge \beta)$) such that the differential is exact (yes, for CS we have 0th and 3rd cohomology [$M$ is a homology sphere], but let us ignore it at the moment). Problem: associate with it a non-trivial object of homological algebra, or more specific an algebra over a operad.

Solution: consider the following differential $Z/2Z$-graded algebra over the BV operad (about BV operad see e.g. the Getzler’s paper). The underlying complex is the Chevalley complex $(S^* \mathbf{g}^\vee[1], d=d_{Lie}+d_{Ch})$, where $d_{Ch}=c_i^{jk}x_j\wedge x_k \partial^i$ is the Chevalley vector field. Let $D=s_{ij}\partial^i \partial^j$ be the differential operator of the second order, where $s_{ij}$ is the tensor dual to the pairing on the  Lie algebra. Define BV structure on $(S^* \mathbf{g}^\wedge[1], d)$ as follows: multiplication is the usual multiplication on the symmetric algebra and $\Delta=d+D$. Even better, we have a family of BV structures: $\Delta_\lambda= d+\lambda D$.

The only non-trivial class in cohomology of this algebra is presented by the unit and $\Delta 1=0$. The remarkable property of homotopy BV algebras is that this simplest algebra is not rigid, it could have non-trivial Massey products!  Indeed, $\Delta 1$ is a coboundary, that is $\Delta 1=d \alpha$. Then, $\Delta^2 1$ is coboundary by the relation in the operad, that is $\Delta^2 1= d\beta$ (if one have a free resolution of BV operad and in it $\Delta^2=d\Gamma$, then $b=\Gamma 1$). Now cochain $\Delta \alpha - \beta=\gamma$ is closed, this is the first Massey product (we are in $Z/2Z$-graded situation!).  Then, in our algebra action of $\Delta$ is cohomologicaly trivial, so $\Delta \gamma= d\delta$ and so on… We get Massey products numerated by natural numbers. And all Massey products are just numbers.

For more details about Massey products of BV algebra see the mentioned paper of Imma Galvez-Carrillo, Andy Tonks, Bruno Vallette.

It is natural to assume that the stationary phase approximation of the functional integral is an invariant of (a family of ) the homotopy BV algebra we produced. I hope to continue this speculation in the forthcoming posts (UPD: here).

4 Responses

1. Andrey Lazarev said, on January 4, 2010 at 8:53 am

An interesting take on Kontsevich’s ‘dual construction’. I have a couple of comments: first of all, it seems that what you (and Vallette et al.) call ‘Massey products’ is just a minimal model of an algebra over a cofibrant operad. Here the cofibrant operad is BV-infinity.

Next, there is another, more general way to associate a homological object to an acyclic algebra over any (modular) operad O. It is an algebra over the so-called dual Feynman transform F*O, see http://arxiv.org/abs/0704.2561. The operad F*O is acyclic in arity i>0 (i.e.when there is at least one input or one output) but F*O((0)) is non-trivial: it is linearly dual to FO, the usual Feynman transform of O.

I wonder whether these two points of view agree.

-Andrey

• nikitamarkarian said, on January 6, 2010 at 9:27 am

Dear Andrey, thank you for your comment!

I have a couple of comments: first of all, it seems that what you (and Vallette et al.) call ‘Massey products’ is just a minimal model of an algebra over a cofibrant operad. Here the cofibrant operad is BV-infinity.

Yes, you are right. Vallette et al. write in appendix B of their paper: “In general, we call the P∞ -operations on the homology of a P∞ -algebra the Massey products.”

Next, there is another, more general way to associate a homological object to an acyclic algebra over any (modular) operad O. It is an algebra over the so-called dual Feynman transform F*O, see http://arxiv.org/abs/0704.2561. The operad F*O is acyclic in arity i>0 (i.e.when there is at least one input or one output) but F*O((0)) is non-trivial: it is linearly dual to FO, the usual Feynman transform of O.

Thank you for pointing out the reference. Let me note that the algebra involved (with cohomology isomorphic to the base field) is not acyclic being considered as a homotopy BV algebra, simply because it is not free as a module over the Grassmann algebra $k[1, \Delta]$.

An interesting take on Kontsevich’s ‘dual construction’.

I am not sure that my post is exactly about the Kontsevich’s “dual construction”. I would say that my post is about “what we are calculating” and the Kontsevich’s “dual construction” is about “how to calculate it”. In fact, one of my aims when writing this post was to unveil the “homotopical sense” of the “dual construction”. I think this problem is connected with the difficulty you describe in the last paragraph of mentioned your and Joseph Chuang’s paper. I am not expert in the “dual construction” and the Feynman transform, I just describe my vision of it. If I am wrong I would be happy to learn more.

I wonder whether these two points of view agree.

Yes, I believe they do agree, that is answers coinside.

• Andrey Lazarev said, on January 7, 2010 at 4:57 am

Yes, I believe they do agree, that is answers coinside.’

I would be interested in elaboration of this. Maybe I should reformulate my question for clarity. Suppose that we have an acyclic (in all degrees) dg Lie algebra g (or, perhaps, some other operadic algebra). We could associate to it via the dual Feynman transform a class in the corresponding graph complex; the commutative graph complex in the Lie algebra case.

On the other hand you describe how to associate a dg BV algebra to g; it’s not acyclic and you take its minimal model. You can then associate to it (via the direct construction) a cohomology class in the suitable version of the graph complex or do something else to it.

The question is what is the relationship between these two approaches. On the face of it they describe two completely different processes and so even the statement that they agree’ needs explanation.

• nikitamarkarian said, on January 8, 2010 at 10:01 pm

I would be interested in elaboration of this.

So do I!

The question is what is the relationship between these two approaches. On the face of it they describe two completely different processes and so even the statement that they `agree’ needs explanation.

These two approaches may be used to calculate the value of the Chern-Simons functional integral as a Taylor series of $1/k$. When I say that two approaches agree I mean that they give coincident values of the integral.

The idea of my approach may be formulated (and generalized) as follows. Suppose we have an operadic algebra with pairing (contractible, but not necessary). Associate with it a “generalised BV structure” on the Andre-Quillen cohomological complex of the algebra. The question is: is it possible to express some invariants given by the dual Feynman transform in terms of homotopy structure (=Massey products) of this “generalised BV structure”? If we start with a dg Lie algebra, then this “generalized BV structure” is the usual BV structure on the Chevalley complex as I write in the post; I hope in this case the answer on the posed question is positive.