### Frobenius Algebras and the BV Formalism

#### Posted by Urs Schreiber

*guest post by Bruce Bartlett*

A nice paper appeared on the arXiv today:

Alberto Cattaneo and Pavel Mnëv, Remarks on Chern-Simons invariants.

Does the algebra $\Omega(M)$ of differentiable forms on a smooth manifold $M$ form a Frobenius algebra? Help me understand and straighten out the definitions!

Lately I have been trying to learn the BV formalism for dealing with gauge symmetry in quantum field theory. I am interested in it because it represents a body of knowledge — which the physicists have been using for ages — which is secretly all about a clever language for dealing with higher groupoids. Thus by definition all higher category enthusiasts are interested in the BV formalism!

I am using Urs as my guru (I, II, III, IV, V, VI, VII, VIII, IX, X, XI). It’s going to take a few years!

Anyhow, Cattaneo and Mnev released an interesting paper today, which seems to work out this stuff in a semi-understandable way for Chern-Simons theory.

The good news is that their main mathematical structure requires just the following geometric ingredients (see Example on page 7):

- The algebra of differentiable forms $\Omega(M)$ on a smooth manifold $M$, and

- A Lie algebra $\mathfrak{g}$ equipped with a nondegenerate ad-invariant inner product.

So here’s the good news: apparently if you understand what those two things are, and I take it we all do, then you are ready to start playing the BV game!

But it’s going to be tough.

The first question I have is the following. In the Example on page 7, they say that:

The algebra of differentiable forms $\Omega(M)$ on a closed orientable manifold forms a dg-Frobenius algebra.

A “dg-Frobenius algebra” is a differential-graded version of a Frobenius algebra. To see the precise definition, look at page 7 of the paper.

The thing which confuses me is this: I was always taught that the Fundamental Principle of TQFT is the following:

You can’t be Frobenius without being finite dimensional.

How then is the algebra of differentiable forms $\Omega(M)$ a Frobenius algebra? Is it due to their definition of nondegeneracy? People should state this explicitly, because often in the literature it will be stated that:

A pairing $(, )$ on a vector space $V$ is

nondegenerateif there exists acopairing$\gamma : \mathbb{C} \rightarrow V \otimes V$ satisfying the snake diagrams.

But I think Cattaneo and Mnev just use the following definition of nondegeneracy (it’s not clear because they don’t spell it out precisely!):

A pairing $(, )$ is nondegenerate if for every $v \in V$ we have $(v, w) = 0$ for all $w$ $\Rightarrow$ $v=0$.

This is a perfectly acceptable defintion… but it means we can’t represent the Frobenius algebra via string diagrams anymore.

Can anyone shed light on this? Is this what is going on (just a different notion of nondegeneracy), or does the word “dg” somehow radically change the idea of nondegeneracy?

## Re: Frobenius algebras and the BV formalism

Bruce, I’m not sure I could ever read that paper without some assistance, but is it possible they mean as an algebra over the ring of smooth functions?