The n-Category Café

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?

Bruce,

Your second definition is what most people mean by non-degenerate. Obviously, it’s equivalent to the first for finite-dimensional things, but for infinite stuff you have to be more careful.

The other thing which should make you feel better is this: a dg-Frobenius algebra should mean something whose cohomology is Frobenius. Lo and behold, for the de Rham algebra, it is.

Your second definition is what most people mean by non-degenerate.

Yes, that’s what Cattaneo and Mnëv mean, I am sure. For the audience they assume the other definition of non-degeneracy is the unfamiliar one.

Dear Bruce,
The reason they have a Frobenius algebra is that they consider a toy model of Chern-Simons. This model can be thought of as a discretized version of CS, where the manifold is replaced by a symplical complex and the connection is given by the holonomy values for the edges.

Okay, so I’ll start going through the article and making – hopefully helpful – remarks on “what’s really going on”, where “really” refers to the kind of discussion we been having here.

section 2 reviews some basic BV stuff, so let me use that to quickly remind us of what we said about that recently:

in a moment we’ll be concerned with a problem of the following kind:

we’ll be given a smooth $\infty$-groupoid $conf$ equipped with a real-valued function (the “action”) on its space of objects (the “space of fields”) which is invariant under the action of the $\infty$-groupoid on its own space of objects (is “gauge invariant”). We’ll be presented with the task of integrating the action over this smooth $\infty$-groupoid.

To handle that, we take tangents and pass to the $\infty$-Lie algebroid $Lie(conf)$ corresponding to the smooth $\infty$-groupoid $conf$. This does not change the space of objects, which is still the same as before, but replaces $k$-morphisms with vectors in degree $k$ in a graded vector bundle over this space of objects: these are the tangents to the source-fibers of the spaces of $k$-morphisms of the original $\infty$-groupoid.

Essentially we’ll be saying that the things that we can integrate over $Lie(conf)$ are the invariant polynomials of $Lie(conf)$. Essentially, the integral of an invariant polynomial over the $\infty$-Lie algebroid will be obatined by: forming the $\infty$-Lie algebroid of maps from $Lie(conf)$ to itself, transgress the invariant polynomial to that space, check if it becomes a degree 0-invariant polynomial, evaluate that on the identity map.

This is the $\infty$-Lie algebraic perspective using “integration without integration” as I tried to describe here. I just mention it to provide that perspective, in case it is helpful. But in practice people usually proceed with a different imagery:

namely, one notices that the $\infty$-Lie algebroid $Lie(conf)$ can be thought of as a special kind of supermanifold, namely a “NQ”-supermanifold, which is precisely a supermanifold which is a module for the action of the “odd-line”, i.e. for $End(\mathbb{R}^{0|1})$: namely one where the $\mathbb{Z}_2$-grading is refined to a $\mathbb{N}$-grading along the canonical projection $even/odd: \mathbb{N} \to \mathbb{Z}_2$, and which is equipped with a degree +1 vector field whose graded Lie bracket with itself vanishes.

Then, the space of smooth functions on this supermanifold is a qDGCA (a quasi-free (namely free as a graded commutative algebra) differential graded-commutative algebra) which is in fact nothing but the Chevalley-Eilenberg complex of the $\infty$-Lie algebroids:

$$C^\infty(Lie(conf)) :=: CE(Lie(conf)) \,.$$

Just two perspectives on the same structure.

The supermanifold perspective has the advantage that we can make use of the theory of integration over supermanifolds to figure out what integration over our $\infty$-Lie algebroid might mean.

As reviewed here integration over supermanifolds works as follows:

1) you cook up something which behaves like a volume form (section 2.3).

2) You characterize all forms which are supposed to be integrable over submanifolds by pairs, consisting of a multivector field and such a volume form. This pair is to be thought of as being the form which one would obtain if we would/could contract the multivector field into that volume form. These pairs are called “integral forms”, section 2.5.

3) Sort of by construction, one can prove the usual standard theorems for these, such as Stokes’ theorem (section 2.6).

But the picture we arrive at this way is: we don’t want to work with differential forms on $Lie(conf)$ directly, but with multivectorfields on $Lie(conf)$.

For ordinary manifolds $X$ this just means that we use any volume form $X \in \Omega^n(X)$ to establish an isomorphism $$\Omega^\bullet(X) \simeq Multivectors_{-\bullet}(X) \,.$$ These multivectors we will call in a moment: antifields.

This isomorphism sends:

the deRham differential $d$ to an operator $\Delta$ called the BV-Laplacian. The multivectorfields come canonically equipped with the Schouten bracket (the unique extension of the Lie bracket on vector fields to a bracket on multivector fields) and this is now called the antibracket $\{\cdot,\cdot\}$.

Antibracket and BV-Laplacian satisfy a couple of nice mutual relations which makes these multivectorfields a BV-algebra (= differential graded Poisson algebra).

A, possibly inhomogeneous, closed differenmtial form maps under this identification of forms with multivectors to a multivector field which is annihilated by the BV Laplacian $$\Delta f = 0 \,.$$ We will be running into the situation where we want to assume that $f$ is given as an exponential of a multivectorfield called $S/\hbar$. Then the condituion that this defines a closed form is $$\Delta \exp(S/\hbar) = 0 \,.$$ This is called the quantum master equation. In terms of $S$ itself it says that $$\Delta S + \frac{1}{\hbar}\{S,S\} = 0 \,.$$ Just a weird way of looking at a closed fifferential form!

So all this terminology is just there to describe a picture dual to differential forms which arises when we want to work relative to a fixed volume form.

And everything you can do for ordinary manifolds you can do for supermanifolds. So that’s now what we do:

As far as the Chevalley-Eilenberg algebra $CE(Lie(conf))$ is the algebra of functions on our $L_\infty$-algebroid, the corresponding Weil algebra $W(Lie(conf))$ is the algebra of functions on the shifted tangent bundle:

$$W(Lie(conf)) :=: C^\infty(T[1]Lie(conf)) =: \Omega^\bullet(Lie(conf)) \,.$$

By what we just said, we imagine having picked a reference volume form on our $\infty$-Lie algebroid and use that to dualize forms to multivectors: $$Multivectors_{-\bullet}(Lie(conf)) = C^\infty(T^*[-1] Lie(conf)) \,.$$ By the standard grading gymnastics of supermanifolds we can think of multivectors (which live in negative degree) as functions on the down-shifted cotangent bundle of $Lie(conf)$.

(Notice that everything is Lagrangian here, we are not talking about phase spaces of our system. Still, this and the master action remind us of structures which we are used to from geometric quantization of phase spaces. So some noteworthy coincidence of concepts appears. There are indications that what is going on here is what is called holography: the space of histories of one system is the phase space for a system of one-dimension higher. I gave links to that somewhere, but this shall not concern us here…)

In any case, we can now do the multivector point of view gymnastics of talking about volume forms for our supermanifold $Lie(conf)$. Essentially by following our nose and keeping track of all gradings in the game the way they taught us in school, we get an integration theory.

So: recall we wanted to integrate an “action” function over an $\infty$-Lie groupoid $conf$. We see now that this means that we need to be talking about a multivector $S$ on $Lie(conf)$ satisfying the quantum master eqution $\Delta \exp(S/\hbar) = 0$.

If we have that, we can integrate over it. As described for instance in Henneaux’s Lectures on the antifield formalism (recall: “antifield” = vector field regarded as element in $Multivectorfield_{-\bullet}(X)$).

Or we can do slightly more fancy things like integrating just over submanifolds. That’s what the theory of “integrable forms” in supergeometry was set up for.

So that what Cattaneo and Mnëvrecall on their page 5, how to do integrals over submanifolds from this dual-anti-BV-perspective.

I’ll talk about section 3 in a separate comment.

I am running a bit out of time, since my beloved is waiting with pumpkin pie (or something else – I hope :-) and I need to catch a train to get to her, but just quickly some hints on:

section 3

For seeing what happens in section 3 it is good to have the $\sigma$-model perspective for CS theory in in $\infty$-version at least in the back of your mind, aspects of which are explained in Dmitry Roytenberg’s explanation of Kontsevich’s explanation of what’s going on:

Chern-Simons theory is in one way or another a theory whose target is a classifying space for smooth $G$-bundles with smooth connection over which lives some Chern-Simons 2-gerbe with connection whose transgression to the space of maps from parameter space $\Sigma$ is the action functional in question

(the Lie version of the finite group Dijkgraaf-Witten theory: there target space is the groupoid $\mathbf{B}G$, the background field is a 3-cocycle $\mathbf{B}G \to \mathbf{B}^3 U(1)$ which we transgress to $conf = Hom(\Pi_1(\Sigma), \mathbf{B}G)$. If you just internalize this appropriately in smooth $\omega$-groupoids and take care to rememeber that the background field $\mathbf{B}G \to \mathbf{B}^3 U(1)$ is an “$\infty$-anafunctor” in general, hence in the smooth case a differential cocycle, you get Chern-Simons as a $\sigma$-model).

So simplifying a bit to get started and passing from $\infty$-groupoids to $\infty$-Lie algebroids we see that the target space Lie algebroid begins to look like the Lie 2-algebra $inn(g) = (g \to g)$ and paranmeter space is a tangent Lie algebroid $T \Sigma$, so that a $g$-valued 1-form on $\Sigma$ is a morphism $$T \Sigma \to inn(g)$$ from parameter “space” to target “space”. Now, the usual gymnastics, e.g. section 2.2 here tell us that elements of $$Hom_{L_\infty}(T\Sigma, inn(g))$$ are elements in $\Omega^\bullet(\Sigma)\otimes g$: clear once you remember that $\Omega^\bullet(\Sigma)$ is the “dual” to $T \Sigma$ and behave as in compact categroies.

So this is where this configuration space of fields (or rather history space) in 3.1 comes from.

Cattaneo and Mnëv closely follow Costello’s Renormalization and the BV-formalism here and already that made me wonder:

they don’t have higher ghosts (recall: functions on the space of tangents to the higher morphisms of $conf$), which one would however expect (we expect to see ghosts-of-ghosts-of-ghosts in a 3d sigma-model QFT such as Chern-Simons) – and which >are present in AKSZ (look at Dmitry’s review # again.) This is something I meant to sort out for myself but haven’t really yet. So I can’t help here. But maybe some reader of this comment can.

And now I need to collect my stuff and catch my train…

Hello, Bruce!

You are absolutely right to ask analytical questions about the space of fields of “physical” CS, but they are hard to answer, which is the reason why we wanted to play with the finite-dimensional model of CS, where BV formalism makes perfect sense. Passing to the true Chern-Simons is done with some cheating: we do not want to think, what differential forms precisely we take (smooth or whatever); BV Laplacian of local functionals becomes singular, in particular for the quantum part of QME \Delta S=0 the left-hand side is ill-defined. Instead one defines functional integral as a sum of Feynman graphs and the latter are defined as integrals over some cleverly compactified configuration spaces. Tricks with the BV-integral (like extracting the Laplacian) are switched to using the Stokes theorem for individual diagrams.

We said that \Omega(M) is dg Frobenius somewhat naively (well, it is indeed in your second definition). It is thought of as an ill-defined limiting case of nice finite-dimensional dg Frobenius algebras. Actually, I think, your first definition is closer to what one wants in BV formalism, as the BV Laplacian is constructed from the inverse of the pairing.

“Fundamental Principle of TQFT is the following: You can’t be Frobenius without being finite dimensional.”
- I think, you are referring to the Frobenius algebras, appearing in Segal’s axioms’ (Hamiltonian) approach to TQFT, while we are doing the Lagrangian formalism here, and the Frobenius algebra we talk about sits inside the space of fields of Lagrangian formalism.

I would dearly like to understand the basic definitions of graded mathematics. I’m going to start with the first sentence of Section 2 of Cattaneo and Mnëv. Every small step makes me stronger. Perhaps Pavel or Urs could help me?

I’m sorry about these detailed technical questions, and I know the n-category cafe isn’t the appropriate forum for it — the nLab would be a better place, but it is not operational.

First sentence of Section 2:

Let $(F, \sigma)$ be a finite-dimensional graded vector space endowed with an odd symplectic form $\sigma in \Lambda^2 F^*$ of degree $-1$, which means $\sigma(u,v) \neq 0 \Rightarrow |u| + |v| = 1$ for $u,v \in F$.

Question 1. What does graded mean? Does it mean $\mathbb{Z}/2$-graded, $\mathbb{Z}$-graded, or $\mathbb{N}$-graded? This always confuses me, because all the references use the word “graded” in different ways.

Question 2. Can you give me a simple example of a finite-dimensional graded vector space with an odd symplectic form?

Question 3. I’m ashamed to say I’m not certain I understand the phrase “which means $\sigma(u,v) \neq 0 \Rightarrow |u| + |v| = 1$ for $u,v \in F$”. Are we allowing negative gradings here? So the symplectic form takes say a vector of grade -8 and a vector of grade +9 and outputs a real number?

Hi all,

I just found my way here from the trackback to the Cattaneo-Mnev paper and saw there was some mention of one of my papers above. I don’t want to derail the discussion, but would like to clarify that the stuff I did in the abelian CS case didn’t use at all the BV technology being discussed here (which I am blissfully ignorant of). I just discretized the theory (actually it was really the abelian BF theory) in such a way that the topological quantities in the discrete theory (the partition function and expectation values of linked Wilson loops) reproduced the continuum continuum results exactly, without having to take a continuum limit. In particular this led to a simple formula for the Gauss linking number of linked lattice loops which was new as far as I know. (For a consise description see this paper rather than my longer more technical paper mentioned earlier.)

In fact I think the case of abelian CS or BF theory would make a very good testing ground, or learning ground, for understanding how this BV approach to CS or BF theory works. In the abelian case the results for the quantities of interest in the theory are well-known and quite simple: the partition function and expectation values of linked Wilson loops can be expressed in terms of Reidemeister torsion (= analytic Ray-Singer torsion) and Gauss linking numbers, respectively, as was shown by Witten in his CS paper in the continuum setting and later reproduced by me in a discrete version of the theory. I would like to see a derivation of this within the BV approach to discrete BF theory. It would be instructional for seeing how this approach works, and, to be frank, without it I will remain a bit skeptical.

I’ll also take the opportunity to clarify the purpose of my discretization scheme in response to what Eric wrote above:

“When you are working with discrete theories, I can understand a temptation to begin with topological theories. Discrete (or I should say “finitary”) topological theories are easy! Plus, all you have to do is add a metric later, right? Not so fast.”

In fact for me it was the other way around: When you are working with topological field theories (as I was) it is natural to try to discretize them in such a way that their topological invariants are exactly reproduced in the discrete setting. It is a general theme of topology that topological quantities have both continuous (smooth) and discrete (e.g. simplicial) descriptions. E.g.: Betti numbers of manifolds, Reidemeister-Ray-Singer torsion. So it is natural to hope that topological quantum field theories should have such discretizations. Besides its intrinsic interest, one goal is to find new and simpler ways of evaluating the invariants that arise in these theories via discrete descriptions. The “lattice-dual lattice” discretization I introduced worked fine for this in the abelian BF case. I don’t know if it has any uses beyond this to “geometrical” theories that Eric alluded to, but at any rate it has done what it was designed to do. (I had hoped to extend it to the much more interesting case of non-abelian BF theory; it seems hard but maybe one day I’ll get around to that.)

Finally I have a question for the experts about the status of the 3-manifold invariants in the CS perturbation theory after the Cattaneo-Mnev (C-M) paper. The prior situation as I undestood it was as follows: Perturbative expansion is done around a flat connection A_f. First, Axelrod and Singer showed (in ref.[1] of C-M) that the terms in the expansion are finite and metric-independent in the case when A_f is “irreducible” (i.e, the cohomology associated with the flat connection vanishes). Irreducibility was required because otherwise the propagator was ill-defined. The Axelrod-Singer result was extended to reducible flat connections by yours truly in hep-th/9704159. (This was done via a refinement of the Faddeev-Popov procedure to get a well-defined propagator in this case.) The degree zero part of the cohomology of the flat connection was allowed to be non-zero, but the degree 1 part still had to vanish (i.e., A_f had to be an isolated point in the moduli space of flat connections). Apparently no one noticed though, since this work was ignored and later Cattaneo and Bott derived essentially the same result again, along with other stuff about the structure of the invariants (ref.[2] of C-M). However, the problem remained of showing finiteness and topological invariance of the terms in the perturbative expansion in the general case of expansion around a flat connection without any restrictions on its cohomology. That always seemed to me to be a really tough problem. So my question is: has this problem now been solved in the C-M paper?

I would like exmamples on how to visualize ghost-of-ghosts-of-ghosts. I asked Urs by email:

Urs: take the differential graded algebra which comes from a single generator g1 in degree 1 and a generator g2 in degree 2 with the differential such that d g1 = g2 and d g2 = 0. Then this can be regarded as the BRST complex for 2-form gauge theory over the point (so it’s a very very very simple special case). g1 is then the ghost and g2 the ghost-of-ghost.

me: In this case, can I can visualize the geometric picture of g2 as a plane, and g1 as the the line generated by the vector product of the generating vectors of g2?

Urs: Hi Daniel, you can visualize g2 as an abstract 2-form and g1 as an abstract 1-form. Yes.

So, my question is, by going up in the form ladder, can I visualize the (ghost-of)^n as an n-surface? Is it possible to define a ghost “brane” and anti “brane” in this setting? Not necessaraly related to branes, but just sayng so that one can get a rough idea of it. That is, you have that surface, so why not trying to come with fields on that ghost or anti ghost surface? Well, it’s just an idea, I can accept that those (ghost of)^n are mostly useless and vacuous.

“But now suppose that you are really being careful: you find that there are two different paths of gauge transformations from A 0 to A 1

Suppose that, while different, these two paths are “essentially the same”, again, in that there is a path of paths P connecting them (in the space of gauge transformations):

Dual to this bivector is a 2-form. This 2-form (on the space of gauge transformations!) is called a ghost-of-ghosts.

one theory may have just ordinary ghosts, the other may have ghosts-of-ghosts and still both theories are “the same”.”

The way you do it, it seems that as long as you can span a n-configurational space with paths, you can say that it is the same physics. So, can I say that a k-fold ghost it is a k-dimensional subspace of the n-configurational subspace? So, the long sequence has 2k+1 steps, with the same physics.

For example, SU(2) is diffeomorphic to a 3-sphere. Then, we have at most a 4-ghost.

Bruce asked a question about integration over supermanifolds. A reply is developing on the $n$Lab here.