## November 14, 2008

### 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 nondegenerate if there exists a copairing $\gamma : \mathbb{C} \rightarrow V \otimes V$ 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?

Posted at November 14, 2008 1:19 PM UTC

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### 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?

Posted by: Todd Trimble on November 14, 2008 1:45 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hi Todd, thanks yes you make a good point. I’m a bit confused though; can one make sense of the concept of a “Frobenius algebra over $C^\infty(M)$”? Can we draw string diagrams for that in a monoidal category?

Posted by: Bruce Bartlett on November 14, 2008 2:09 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

I don’t see why one can’t make sense of that. Just working in the symmetric monoidal category of say dg-modules over $\Omega^0(M)$ where the component in each degree is (I think) finitely generated and projective. Any string diagram or surface diagram calculations for the walking Frobenius object would port over to that sm category.

On the other hand, you may well be asking something else altogether, and I’m just repeating things you know very well.

Posted by: Todd Trimble on November 14, 2008 4:10 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Mmm. But does that allow us to interpret the trace map then? Currently it is integration, i.e. it goes

(1)$\int : \Omega(M) \rightarrow \mathbb{R}$
(2)$\omega \mapsto \int_M \omega$

where the integration is defined to be zero unless $\omega$ is a top-form.

If we are to think of this as a morphism in the monoidal category of dg modules over $\Omega^0(M) = C^\infty(M)$, then we need to think of $\mathbb{R}$ as a module for the algebra of smooth functions $C^\infty(M)$. Since I can’t think of a natural way to multiply a real number by a smooth function, I guess it would have to be the trivial module. But then the integration morphism wouldn’t be a map which respects the module structures (since if you multiply a form by a smooth function, you change the integral). So right now I can’t think of a way to make sense of $\Omega(M)$ as a Frobenius algebra in this monoidal category sense. I think they meant it in the sense of the second version I gave above.

Posted by: Bruce Bartlett on November 14, 2008 5:44 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Whoops, sorry, you’re right. I see now what the question was.

Posted by: Todd Trimble on November 14, 2008 5:59 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Todd Trimble wrote:

I’m not sure I could ever read that paper without some assistance

This is, by the way, one of my big hopes: that we can eventually bridge a gap:

there is so much stuff in modern (“formal”) theoretical physics which is secretly governed by higher categorical abstract nonsense. Most people working on it have no idea of the relevant theory. Most people working on higher abstract stuff have no idea about the plethora of interesting applications that already exist in disguise. And the situation does not change much because neither group is able to decipher the other group’s work! (I am overstating this a bit, but not much.)

I am hoping by uniting forces we can eventually break the dam. This will require tolerance on both sides, at least at the beginning: each side will need to tolerate the sheer ignorance of the respective other side – and try to not look contemptuously on first feeble attempts of those trying to cross the bridge from their side.

I can’t overestimate how much Todd has already helped me working on that bridge by patiently providing help with what for him were elementary facts (I just mention sheaf theory, Stone duality, Day convolution, Crans-Gray tensor product). I am hoping some day I can pay him back for that. Somehow. :-)

So let’s see how much progress we can make here, following Bruce’s question above. I’ll join you all in a moment as soon as I have finished another task…

Posted by: Urs Schreiber on November 14, 2008 2:29 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Urs, I would love to be able to sit at your feet for a few months (or years) and absorb what you’ve been trying to tell us. Any assistance I’ve provided has been but a small fraction of the range of things you think about. Someday, maybe.

Posted by: Todd Trimble on November 14, 2008 3:59 PM | Permalink | Reply to this
Read the post Local Nets and Co-Sheaves
Weblog: The n-Category Café
Excerpt: Co-sheaf condition (codescent) for Haag-Kastler nets of local quantum observables?
Tracked: November 14, 2008 3:13 PM

### Re: Frobenius algebras and the BV formalism

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.

Posted by: Ben Webster on November 14, 2008 3:49 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Okay thanks. One obstacle cleared, one hundred to go.

Posted by: Bruce Bartlett on November 14, 2008 5:45 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

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.

Posted by: Urs Schreiber on November 14, 2008 3:54 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

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.

Posted by: Aleksandar Mikovic on November 14, 2008 4:13 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hi Aleksandar,

Well that sounds reasonable to me since I am roughly familiar with the “triangulations” method for constructing TQFT’s. But I notice they didn’t say so explicitly; they don’t mention the word “simplicial” or “holonomy” in their article. Are you pointing out a fact which is well-known to the experts — that there is a simplicial toy model version of the BV formalism for Chern-Simons theory in a finite-dimensional setting — or are you mentioning a personal insight of yours here which you find helps explain what they do?

Posted by: Bruce Bartlett on November 14, 2008 5:53 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

It would be great to have a simplicial model for Chern-Simons, but this is not done yet (it is hard to construct dg Frobenius structure (even in “up to homotopy” sense) on cochains of triangulation). However, simplicial abelian CS was done in hep-th/9612009 (one needs only pairing and differential here, not the product), and simplicial non-abelian BF was done in hep-th/0610326, hep-th/0809:1160 (here one needs differential and product, but not the pairing). Those are not toy models: though they are finite-dimensional, they are exact (“homotopic” to the original TQFT) in a sense.

Posted by: Pavel Mnev on November 14, 2008 11:07 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hi Pavel,

Thanks a lot for these references! I appreciate them because they all seem to be quite readable. The paper on Discrete BF theory is especially friendly in this regard, because it includes for instance the definitions of “Gerstenhaber algebras”, “Batalin-Vilkovisky algebras”, “Q-manifolds” etc. In my short experience of the literature so far, it has been a rare occasion when the authors would stoop to giving these definitions, or even giving a reference for them.

Most of what I know about discrete TQFT I learnt from John’s quantum gravity seminar. John, perhaps you can include these papers in the references there!

Derek Wise would also be interested in this stuff I’m sure.

And also Erik Forgy, who is interested in discrete differential geometry. Erik, have you seen that paper R-torsion and linking numbers from simplicial abelian gauge theories by D. Adams? It seems to have a nice section about the discrete Hodge star operator in terms of triangulations and their duals.

Posted by: Bruce Bartlett on November 15, 2008 1:56 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hi Bruce!

Thanks for pointing out our paper. I WISH I had some time to spare. The Discrete BF Theory abstract looks very tantalizing.

I most likely saw Adams’ paper at one point (I probably have a copy of every single paper with the word “discrete” in it dating prior to 2002), but likely did not spend too much time on it. Not due to lack of interest, but due to lack of brain cells and also not being convinced that the direction was the right one for me to take. 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.

What is hard and what many people much smarter than me have worked on for years is discrete “geometrical” theories. How do you introduce a metric without breaking all the beautiful machinery of the discrete topological theory? I don’t want to give up the algebra to do geometry. The two should be interlinked in my humble and naive opinion.

That is why I was so happy when Urs joined the chase with me. As you can imagine, he quickly absorbed the relevant issues and largely solved mysteries that have been plaguing engineers and applied mathematicians since Poincare. As with most things Urs does, it will probably take years before people realize what was accomplished.

The first thing Urs discovered was that simplices are somehow “sick” (which makes it hard for me to understand why I still see him talking about simplices instead of dimaonds, but that is another story!). Simplices are fine, natural, and even beautiful when it comes to finitary topological theories, including algebraic and combinatorial properties. But when you try to introduce a metric, then simplices are no longer beautiful or natural. You must make arbitrary choices. During grad school I would refer to it as the search for the discrete Hodge star.

However, everything just snaps into place when you abandon simplices and consider “diamonds”. A diamond is basically a cubic cell complex with directed edges flowing along a preferred direction (which will turn out to be time).

I am no expert on BF theory, but from what little I do know, I think it lies kind of halfway between topology and geometry. In that case, I would not be surprised if BF theory had a nice and beautiful formulation on a simplicial cell complex. But I would also not be too surprised if BF theory on simplices forces you to make some less than natural choices.

BF theory on cell complexes, on the hand, should admit a completely natural discrete formulation. I see that (just skimming), in Pavel’s awesome looking paper (Hi Pavel!) that he looks at both simplicial and cubic cases.

I think it would be very neat to formulate BF theory within the context of discrete geometry on diamond complexes and compare the results.

Alas, I have other pressing things right now (e.g. saving the financial system from armageddon), but I am watching this conversation with one eye at least.

Best regards

Posted by: Eric on November 15, 2008 3:46 PM | Permalink | Reply to this

### locally Lorentzian latticization

The following comment is a going a bit off on a tangent to the main topic of this thread, but it is in reply to the issues about latticization mentioned above.

It is a fact that a particularly nice discrete differential calculus exists on cubical lattices. More: a particularly nice discrete Lorentzian differential geometry exists on cubical lattices. One may attribute this to the fact that simple situations tend to have easy, well-behaved tools describing them, and that these tools will fail to capture the more interesting situations and are hence uninteresting.

But in this case it might be that one shouldn’t dismiss the observation on such grounds, since there might be indications that something deeper is going on. I have been mulling over this for a long time (way too long, I am sure, but I can’t help it). It all seemed inconclusive – until I started getting some insights into the “functorial semantics” (to abuse a term) of quantum field theory in the AQFT framework #.

Blog readers may (or probably may not) recall another observation that I have been mulling over for too long now: that one can understand the topological part of 2-dimensional (and rational) conformal quantum field theory as a nonabelian 2-cocycle (with coefficients in $(\mathbf{B}Bimod(C))^I$ for $C$ the representation category of a vertex operator algebra) #.

This is like saying that a bundle with connection is – after forgetting the connection which assigns differential data to local patches – just a bunch of transition functions. So it motivates an obvious conjecture: that the entire CFT (not projected onto its topological bones) is a differential 2-cocycle #, i.e. something which is locally a smooth 2-functor such that the topological cocycle data gives the rules for how to glue these local 2-functors on intersections.

If one accepts this, then the obvious next question is: what would that local 2-functor be? And the obvious answer is: since we know from AQFT that locally, i.e. on something that looks just like the complex plane $\mathbb{C}$, 2d CFT is given by a local conformal net, that local 2-functor somehow needs to be equivalent to a local conformal net.

(Am I making myself understandable? This is supposed to be a blatantly obvious plausibility argument, but maybe I am expressing myself in un-understandable terms.)

That’s why I started trying to see the relation between 2-functors and local nets. And, lo and behold, it turns out that under some mild conditions essentially every 2-functor on piecewise lightlike 2-paths in the Minkowski plane comes with its “endomorphism co-presheaf” which is a local net #. In particular, one gets examples by approximating the 2-paths in the Minkwoski plane by the free 2-category on a 2-dimensional cubical lattice in which all edges are lightlike. (And one expects # that one can dicuss the continuum limit of the resulting lattice nets.)

And one notices that 2-functors on this Minkowski lattice relate to the discrete differential algebra of this lattice as 2-functors on smooth 2-paths relate to the qDGCA of ordinary differential forms #.

(Am I making myself understandable?)

Now I come to the main point of this message here, which is supposed to play the role of a reply to the issues of latticization mentioned above:

There is one obvious way to put the above puzzle pieces together: the above argument suggests that a 2-dimensional QFT on a topologically possibly nontrivial surface $\Sigma$ is given by a 2-cocycle for a locally Minkowskian 2-functor.

This means: given a cover $\pi : (Y = \sqcup_i O_i) \to \Sigma$ of $\Sigma$ by something like Minkowski double wedges $O_i$ (possibly with $\pi$ required to be isometric or conformal) and with $P_2(-) : MinkowskiDoubleWedges \to 2Cat$ the co-2-stack # which sends each space to the 2-catgeory of Minkowski 2-paths on it, the 2-functor in question would be the horizontal moprhism $tr$ in a span $\array{ Codesc(Y,P_2) &\stackrel{tr}{\to}& T \\ \downarrow \\ \mathcal{P}_2(\Sigma) }$ where $Codesc(Y,P_2) := \int^{[n] \in \Delta} O(\Delta^n)\otimes P_2(Y^{[n+1]})$ is the codescent 2-category # whose 2-cells are generated from

- local Minkowski 2-paths

- jumps between the patches $O_i$

- jumps between jumps on trible intersections.

Notice that in a lattice version of this the “local Minkowski 2-paths” would be cubical cells with lightlike edges, while the gluing of these on intersections would follow simplicial rules. So in the case that $P_2(-)$ is taken to be lattice 2-paths, the total 2-category $Codesc(Y, P_2)$ would encode a latticization of $\Sigma$ which is a certain mix of cubical and simplicial cells: patches of cubical cells are glued themselves in a simplicial manner.

I don’t know yet if such spans really can encode full 2-dimensional CFT. I am still thinking about this. But this is the kind of picture that I have been arriving at. To me it has a certain charme. But more of the resulting details need to be worked out. As I said, I have been mulling over this for a while now, and I feel bad towards Eric Forgy and to Jens Fjelstad for being that immensely slow on our joint projects. But I can’t help it.

Posted by: Urs Schreiber on November 19, 2008 8:37 AM | Permalink | Reply to this

### Re: locally Lorentzian latticization

Wow! That last sentence almost sounds like an apology. No apologies! You don’t know how much I enjoy watching your progress. I am so happy we have the Cafe to discuss ideas in the open like this (or watch from the side line in my case).

I feel like a piece of the puzzle just snapped into place.

“Everything is proceeding as I have foreseen.” - The Emperor

I’m also super happy to see other experts chiming in here.

One puzzle, and this is posed to David in particular…

You have convinced yourself of the importance of both the lattice and its dual. I am also convinced of the significance of the duality. However…

What if the lattice and its dual were actually just different projections of a single unified lattice?

Imagine a lattice representing both space and time. Nodes (events) would seem to appear and disappear out of existence as time elapsed. If you took a long exposure photograph of the lattice you would capture the location of each node as it flashed by. Now imagine that all the nodes can be partitioned into one of two sets. The one set can be connected to form a simplicial lattice. The other set can be connected to form a dual lattice.

BUT… what if the ability to form the lattice and its dual is an artifact of taking the long exposure photograph?

What if each edge where light like? In other words, what if a light-like edge actually connects a node of the primary lattice to a node of the dual lattice? What if the decomposition into primary and dual lattices is an artifact of “projecting down through time”?

Or put another way, what if we could take a lattice and its dual and “unpack” it into a single space-time lattice somehow?

IF that can be done, that is the procedure I have be referring to a “diamonating”.

THAT would answer any questions as to the general applicability of the work Urs and I did to arbitrary smooth manifolds.

I’ll state the conjecture again in light of the “long exposure photograph” example:

If a manifold $M$ can be triangulated, the manifold $M\times R$ can be diamonated.

Or maybe put a different way,

Any Lorentzian manifold admits a diamonation.

PS: I am also a friend of Kotiuga (as is John Baez, I believe they were dorm mates!). Small world!

Posted by: Eric on November 19, 2008 2:17 PM | Permalink | Reply to this

### Re: locally Lorentzian latticization

Eric recalled his conjecture:

If a manifold $M$ can be triangulated, the manifold $M \times \mathbb{R}$ can be diamonated.

I think this is an interesting idea. I believe the picture to have in mind is that of a discrete time evolution on a manifolds $\Sigma$ with triangulation $T$ and dual triangulation $\bar T$, such that a particle evolving in time on $\Sigma$ will see a spatial triangulation $T$ every second time step and the spatial triangulation $\bar T$ every other.

Eric, didn’t you have a pdf figure illustrating this idea? I suppose if this works (it certainly works in simple special cases, so I could say: “when this works”) it would seem to give a way to understand what I said above # with “Minkowski space” replaced everywhere more generally by “globally hyperbolic Lorentzian space”. That would be nice.

On the other hand, coming back from our tangent, one should keep in mind that for the cases that Pavel Mnëv and David Adams are interested in the manifold in question, an 3-sphere for instance, does not in general admit a globally hyperbolic structure.

To accomodate such cases, where even Eric’s $\cdots \to T \to \bar T \to T \to \bar T \to \cdots$ latticization does not work, I’m suggesting one could still profitably employ the sheafy construction which I sketched: something that locally looks for instance like $\cdots \to T \to \bar T \to T \to \bar T \to \cdots$, but where nevertheless gluing takes place globally.

Posted by: Urs Schreiber on November 19, 2008 2:35 PM | Permalink | Reply to this

### Re: locally Lorentzian latticization

Eric, didn’t you have a pdf figure illustrating this idea?

Found it!

This is not the answer, but it is maybe a start?

Making 3-diamonds from 2-simplices and their duals.

Never mind the fact I can’t count. The first slide should say “six sides” :)

Posted by: Eric on November 19, 2008 3:16 PM | Permalink | Reply to this

### Re: locally Lorentzian latticization

In my pdf, the light-like edges do not necessarily connect $T$ to $\tilde T$. Maybe that is where I went wrong?

If you have an equilateral triangulation of $R^2$ and then follow the presciption in the pdf, you might see my motivation. Maybe it was a step in the wrong direction though.

Posted by: Eric on November 19, 2008 3:23 PM | Permalink | Reply to this

### Re: locally Lorentzian latticization

Just a note on the simplest case…

Begin with a 3-cube.

Look down the major diagonal. It requires 3 time steps to traverse the edges from the beginning node (back corner) to the end node (front corner).

You will see an equilateral triangle and a regular hexagon (its dual).

By the way, for an $n$-cube, the number of nodes at each time step is the Pascal triangle.

$n = 2, 1\to 2\to 1$ (requires 2 steps to traverse 2-cube)

$n = 3, 1\to 3\to 3\to 1$ (requires 3 steps to traverse 3-cube)

etc.

Posted by: Eric on November 19, 2008 5:02 PM | Permalink | Reply to this

### Re: locally Lorentzian latticization

I’ve created a (personal) nLab entry for this concept:

Diamonation

Posted by: Eric Forgy on October 9, 2009 12:22 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

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:

$W(Lie(conf)) :=: C^\infty(TLie(conf)) =: \Omega^\bullet(Lie(conf)) \,.$

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.

Posted by: Urs Schreiber on November 14, 2008 4:58 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

I think you have confused ghosts or anti-ghosts with anti-fields. Anti-fields are generators of the Koszul complex for the surface determined by the Lagrangian.

Most of what I’ve found here so far is more to do with Frobenius algebras and/or one small aspect of what BV hath wrought.

Maybe some of the links will tell me which aspect concerns you. For me of course it’s the homological and higher homotopy aspect.

Posted by: jim stasheff on November 1, 2009 10:18 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

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…

Posted by: Urs Schreiber on November 14, 2008 5:26 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hello!

I am not sure I understood your question about higher ghosts correctly, but in standard BV language the components of the (BV-)field of Chern-Simons theory A=A0+A1+A2+A3 (0-form … 3-form pieces of the “superconnection” A) are named “ghost” A0 (corresponding to infinitesimal fiberwise rotations), “classical field” A1 (the connection), “anti-field for the classical field” A2, “anti-field for the ghost” A3. Ghost numbers are 1,0,-1,-2 respectively. No higher ghosts appear here, since the action of the “gauge” Lie algebra of fiberwise rotations on classical fields (connections) is irreducible. Tower of ghosts may appear in higher-dimensional AKSZ models, e.g. in BF theory in dimension >3. Here BV-fields are Maps(TM, g+g*[D-2]) and the tower of ghosts has height max(1,D-2).

Posted by: Pavel Mnev on November 14, 2008 9:53 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hi Pavel! Good to see you here. Thanks for taking the time to help us out.

No higher ghosts appear here

Yes, thanks for correcting that. I got myself mixed up by thinking of CS theory on the background as a 3-form gauge theory. That would indeed have higher ghosts. But the sigma model with fixed CS-backgrounbd which we are looking at of course has not.

Posted by: Urs Schreiber on November 17, 2008 12:55 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Urs, I appreciate your section-by-section explanations here. I am going through them (notwithstanding the fact that I haven’t yet gone through Tom Leinster’s gimungous post as well, which I want to do!)

I can sort of see your bigger picture. At the moment I get stuck on little things, basically just having to do with graded linear algebra. I’m not yet comfortable with these kinds of statements (as in the 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 $u,v \in F$.

What I need is a basic “graded linear algebra” worksheet. Don’t worry, I’m getting there. But I’m trying to go through what you said.

Posted by: Bruce Bartlett on November 15, 2008 2:16 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

What I need is a basic “graded linear algebra” worksheet.

In case anyone is interested, I found some relatively detailed notes on graded linear algebra via a link from Alberto Cattaneo’s webpage. Available are:

Also available are:

bla (have to enter stupid text here because the parser never accepts my ul li boxes grr x 10)

bla

Posted by: Bruce Bartlett on November 15, 2008 3:23 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

I would like to add the following gem of a introductory reference:

In this reference you can find definitions and introductory material on much of the above. In particular, Urs will be happy to know he stresses the differential forms/multivector fields picture of the BV formalism (see Section 3.1, titled de Rham theory revisited).

If anyone knows of other introductory material, please let me know.

Posted by: Bruce Bartlett on November 15, 2008 9:22 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hi Bruce,

I was offline over the weekend, needed to recover a bit. (I slept almost 24 hours on Saturday). But now I am getting back to you here:

In this reference you can find definitions and introductory material on much of the above. In particular, Urs will be happy to know he stresses the differential forms/multivector fields picture of the BV formalism (see Section 3.1, titled de Rham theory revisited).

Yes, that makes me happy! What makes me even more happy is that you are now helping me dig out useful material on this. I had missed this good stuff by Alberto Cattaneo.

(By the way, let’s also keep thinking about how this integration theory over graded homological manifolds (i.e. over $L_\infty$-algebrboid) relates to Weinstein-Hepworth integration over a stack. Both things should be two aspects of the same principle.)

Posted by: Urs Schreiber on November 17, 2008 1:01 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

I am happy to say I found another great introductory reference on the BV formalism. Can’t believe I never found it before, since it’s titled “An introduction to the Batalin-Vilkovisky formalism” :-)

For future reference, let me then make the following collection:

Bruce’s Top Three References for an Introduction to the BV Formalism and Supermathematics in General for Dummies like You and Me:

• D. Fiorenza, An introduction to the Batalin-Vilkovisky formalism, Lecture given at the Recontres Mathématiques de Glanon, July 2003. Has a fantastic motivating example: it says the BV formalism is akin to calculating $\int_{-\infty}^{+\infty} \frac{1}{1 + x^2} dx$ by extending it to the complex plane and then using the residues formula.

I would only be most pleased to update the above list. It reflects my ignorance of the literature rather than my knowledge of it.

Posted by: Bruce Bartlett on November 17, 2008 2:30 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

I guess I must toot my own horn:

[hep-th/9712157] The (secret?) homological algebra of the Batalin …
Title: The (secret?) homological algebra of the Batalin-Vilkovisky approach. Authors: Jim Stasheff. (Submitted on 16 Dec 1997) …
arxiv.org/abs/hep-th/9712157

Posted by: jim stasheff on November 17, 2008 9:57 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

You don’t need to toot your own horn. Let browsers worldwide do it for you! My impression was that this reference is not quite for dummies; one needs to have a nodding acquaintance with the theory to appreciate it. It will be in the list of Top Three References for the Discerning Practitioner of the BV formalism.

Posted by: Bruce Bartlett on November 17, 2008 11:36 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Maybe. I tried to write it for those with the homological background who would take it on faith that physicists could be interested. perhaps I did a better job with the BFV (Hamiltonian) formalism.

Homological reduction of constrained Poisson algebras. J. Differential Geom. 45 (1997), no. 1, 221–240.

available from project Euclid

Posted by: jim stasheff on November 18, 2008 12:40 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

(By the way, let’s also keep thinking about how this integration theory over graded homological manifolds (i.e. over L ∞-algebrboid) relates to Weinstein-Hepworth integration over a stack. Both things should be two aspects of the same principle.)

Indeed. I am hoping that the Weinstein formula (given a tangent bundle interpretation by Richard Hepworth) will somehow be a special case of the BV formalism for integrating things over groupoids, as you once explained.

That is the way to make contact with the “skeptics”, and to more conventional n-category cafe fare. So it is one of my main motivations.

Posted by: Bruce Bartlett on November 17, 2008 3:30 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

I am hoping that the Weinstein formula (given a tangent bundle interpretation by Richard Hepworth) will somehow be a special case of the BV formalism for integrating things over groupoids, as you once explained.

There is a simple example one should concentrate on first. I started thinking about htis a bit in Lausanne, but somehow didn’t see the light:

namely: take your graded manifold to be $Lie(\bullet/G)$ for $G$ a Lie group and see if there is $exp(S)$ satisfying the master equation such that the BV-integral over it yields $1/|G|$, for $|G|$ the Haar-mesaure on the group $G$.

I take it that this is what Weinstein-Hepworth get (this was the motivation in the first place). So how is that reproduced using BV?

Posted by: Urs Schreiber on November 17, 2008 6:50 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Agreed. This is the example which Richard originally presented as the first challenge for the formalism.

Posted by: Bruce Bartlett on November 17, 2008 9:09 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

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.

Posted by: Pavel Mnev on November 14, 2008 10:47 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hi Pavel, thanks for these remarks.

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.

You are right in that I was thinking about Frobenius algebras in the sense of Segal’s axiomatic approach to TQFT. I didn’t recognize that they are entering in a different way here. I guess I wasn’t distinguishing between the Lagrangian and Hamiltonian formalisms — though I hasten to add I don’t really know what I am saying here. Is this somehow the fact that Frobenius algebras are entering at the level of the classical theory here, not to be confused with the Frobenius algebras which enter at the level of the quantum theory? I am using these terms in the sense of Freed as in his recent write-up Remarks on Chern-Simons theory.

Posted by: Bruce Bartlett on November 15, 2008 2:06 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Apologies for not keeping up with this thread.
As a result, the following is a response to many separate
entries within Frobenius algebras and the BV formalism
I’m well familiar with the paper of Cattaneo and Mn"ev.

“all higher category enthusiasts are interested in the BV formalism”
keeping it in mind as an infinty versionand enhancement of dg Poisson
or rather infty-Gerstenhaber algebra in a certain sense.

btw, a BV algebra is NOT = dg Poisson alg

an algebra over the ring of smooth functions? sure or still more generally
over a commutative ring in char 0 or even…

the AKSZ point of view on BV seems to have the upper hand,
but just as we can ignore the analytic issues for the time being
(is this the only reason for doing fin dim models?)
we can also ignore the (super)manifold point of view for now

in particular, this allows us to proceed in the reducible case,
hence having ghosts-of-ghosts etc to handle singularities
which are not in AKSZ but maybe in Dmitry’s review

Posted by: jim stasheff on November 15, 2008 7:59 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

“Is this somehow the fact that Frobenius algebras are entering at the level of the classical theory here, not to be confused with the Frobenius algebras which enter at the level of the quantum theory?”

Absolutely right. Frobenius algebra in Segal’s story encodes the non-perturbative quantum result. On the other hand there is another (infinite-dimensional, in “physical” setting) Frobenius algebra sitting inside the classical (BV-) fields. Perturbative path integral quantization (e.g. passing to the effective action on zero-modes) yields some algebraic structure [cyclic L_\infty enhanced with higher loops] on the space of zero-modes.

Posted by: Pavel Mnev on November 18, 2008 1:44 PM | Permalink | Reply to this

### Help

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?

Posted by: Bruce Bartlett on November 15, 2008 4:12 PM | Permalink | Reply to this

### Re: Help

Question 1. What does graded mean? Does it mean Z/2-graded, Z-graded, or N-graded? This always confuses me, because all the references use the word graded’ in different ways.

You are so right! One advantage of super’ is that it always means Z/2-graded.authors (except maybe in well established sub areas) should specify.

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) = ….. 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?

That’s often the case and always in BV or the Hamiltonian version which is BFV.

Posted by: jim stasheff on November 15, 2008 7:58 PM | Permalink | Reply to this

### technical questions

I’m sorry about these detailed technical questions, and I know the $n$-category cafe isn’t the appropriate forum for it

No, it’s okay. Give the comment a special title, as I did now (“technical questions”) and everybody can easily ignore it if not interested. Discussing it here may be much better than discussing it in some secret corner where few people see us. Imagine we’re on a conference over coffe break. We don’t shout across the whole room, but we’ll discuss something on the board. Everybody can see what we are talking about and is free to decide to ignore us and chat about something else with somebody else – or join us with good advice.

For them it means $\mathbb{Z}$-graded. They don’t say so explicitly, it seems, but that’s what they mean.

Remember that all this BV business can be thought of as applied to ordinary supermanifolds ($\mathbb{Z}_2$-graded) but is secretly all about “NQ-supermanifolds” ($\simeq$ $L_\infty$-algebroids): those supermanifolds where the $\mathbb{Z}_2$-grading is refined to an $\mathbb{N}$-grading and where we fixed a degree +1 vector field whose super Lie bracket with itself vanishes.

The negative degrees then enter once we talk about “integral forms” in terms of volume forms + multivector fields: the multivector field dual to a degree $k$ form is taken to be in degree $-k$.

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

Take $\mathbb{R} \oplus \mathbb{R}$ regarded as the span of a generator $t$ in degree 0 and a generator $s$ in degree 1. Set $\sigma(t,s) = 1$ $\sigma(s,t) = -1$ $\sigma(s,s) = 0$ $\sigma(t,t) = 0$

I’m not certain I understand the phrase “which means […]”

The symplectic form eats a bunch of graded things and spits out a number, regarded as something in degree 0. So it “annihilates” the total degree of its arguments. We say it is of homogeneous degree $-k$ if it annihilates precisely total degree of $k$, i.e. if it eats arguments of total degree $k$ and spits out a number (it spits out $0$ if the total degree does not match $k$).

Posted by: Urs Schreiber on November 17, 2008 3:50 PM | Permalink | Reply to this

### Re: technical questions

Ok thanks, got it.

Posted by: Bruce Bartlett on November 17, 2008 9:07 PM | Permalink | Reply to this

### Re: technical questions

ghosts and anti-ghosts are of opposite parity

Posted by: jim stasheff on November 17, 2008 9:59 PM | Permalink | Reply to this

### Re: technical questions

Yes, that’s what I said #, but I broke it down into two steps, which I find useful for understanding what’s going on:

From the perspective of integrating over an $L_\infty$-algebroid there is, at first, only an $\mathbb{N}$-grading, where a generator in non-negative degree $k$ is a cotangent to a source-fiber of a space of $k$-morphism in some $\infty$-groupoid (therefore $k \geq 0$.)

This gives fields (deg = 0) and ghosts (deg $\geq 1$).

But then when we want to integrate we have to pull the trick of “integrable forms” which are charatcerized by pairs consisting of a volume form and a multivector field. These multivector fields now carry the degree negative of that element which they are dual to. These are the antifields and antighosts. And this is where the $\mathbb{Z}$-grading comes from. At least in Lagrangian BV, this is the picture.

Posted by: Urs Schreiber on November 17, 2008 10:07 PM | Permalink | Reply to this

### Re: technical questions

Indeed, there are separate N-gradings to begin with - corresponding via Noether to
symmetries (ghosts) and Noether identities (antighosts) after first introducing anti-fields corresponding to the E-L equations with respect to fields. The anti side provides a resolution in the homological sense of the equations and the ghosts start as generators of the generalized Chevalley-Eilenberg complex
This provides respective differrentials which do NOT commute; that’s where life gets interesting thanks to BV.

Posted by: jim stasheff on November 18, 2008 2:10 PM | Permalink | Reply to this

### Re: technical questions

I won’t be able to spend more time thinking about this stuff this week, but one more quick comment on the problem of relating the BV integral to the volume integral over a stack:

we should keep in mind that the fact that in groupoid cardinaliy we devide by $\frac{1}{|x \to \dot|}$ or $\frac{1}{|Aut(x)|}$ (depending on wether we integrate over all objects or over isomorphism classes) “in order to” devide out overcounting obtained by counting different but gauge-equivalent points as different points.

So, one way of understanding the factor one-over-gauge-group-volume is in terms of an integral which goes only over gauge orbits, not over all points.

In BRST-BV the idea is that the full naive integral $\int_X exp(S/\hbar)$ is over a groupoid, let’s say a free action groupoid for simplicity, i.e. a global weak quotient $X//G$ whose strict quotient $X/G$ exists (as a manifold), and that we are to integrate just over the quotient $\int_X exp(S/\hbar) \delta(gauge slice) = \int_{X/G} exp(S/\hbar) = \frac{1}{|G|} \int_X exp(S/\hbar) \,.$

So here the factor $1/|G|$ comes in indirectly from the gauge fixing in the integral. This of course is accomplished in BV by means of “gauge fixing fermions”

Posted by: Urs Schreiber on November 18, 2008 9:40 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

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. 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. 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?

Posted by: David A. on November 17, 2008 9:26 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hi David,

Wow. It is great to see you here.

I left off a few adjectives in my last comment. When I said,

“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.”

I should have said, “When you are working with discrete geometrical theories…”

Of course, topology is beautiful when done discretely all by itself. Poincare is one of my heroes :)

I came at this in grad school from the perspective of applied computational physics/engineering. That many of the topological properties hold exactly when you discretize was one of the motivations to try to learn some maths (which never came easy for me) in the first place. For example, the fact that you have a coboundary map that satisfies d^2 = 0 exactly on a cell complex means that you can develop a numerical model of electromagnetic theory on that cell complex that conserves charge exactly. The missing geometrical ingredient is the discrete Hodge star. You can fairly easily come up with approximate operators, but they tend destroy the beautiful algebraic properties of the topological theory.

That is all I meant.

Cheers!

Posted by: Eric on November 17, 2008 2:49 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hi Eric,

No worries. I just wanted to make it clear what my discretization approach was and wasn’t intended for. Actually, preserving properties of the Hodge star operator in the discrete setting was an essential part of my approach. One way to see why this should be important is as follows: The Guass linking number is a topological invariant, but the usual formula for it can be written geometrically as an integral with integrand involving Hodge star and exterior derivative d. So producing an appropriate discrete Hodge star so that the discrete formula still gives the linking number for lattice links was something my discretization scheme needed to do. It turned out that to make this work out the discrete Hodge star had to be simply the duality operator mapping (co)chains of the lattice to (co)chains of the dual lattice. Hence the discretization of the abelian BF theory had to involve both the lattice and its dual.

I am vaguely aware that finding an appropriate discrete Hodge star is also important in computational electromagnetism contexts, but it could very well be that the one I used is not the best one for that context. However, for reproducing the linking number stuff in discretized abelian BF theory I am convinced that a “lattice-dual lattice ” approach is unavoidable. That’s why I am very interested to see how the linking number stuff can be reproduced in the present BV approach to discrete BF theory in the abelian case (which doesn’t use a lattice-dual lattice setup as far as I can tell). Frankly I don’t believe it is possible, but I would be glad to be proved wrong!

Posted by: David A. on November 18, 2008 2:28 AM | Permalink | Reply to this

### Discrete Hodge Star

Interesting!

You are probably aware of some nice research by a friend of mine, Alain Bossavit. He has done quite a lot to formulate a discrete Hodge using the lattice/dual lattice approach from the applied numerical methods perspective. It has been many years since I thought about this in any details. Other names that come to mind include Ralf Hiptmair. He is quite good too. I’m sure I have studied your work at some point. It sounds neat!

I WISH I had some time to sit down with your paper. If things settle down a bit with work, I will have a look again.

What you are referring to as a discrete Hodge star, I’m guessing would be something I’ve referred to as a “topological” Hodge star. It associates for each $k$-cell of an $n$-dimensional lattice, an $(n-k)$-cell of the dual lattice. I call it “topological”, because I’m guessing you do not require, for example, a 1-cell in 3-d to be “perpendicular” to its dual 2-cell. It just needs to be associated with it. Is that right?

Regarding the need for a dual lattice, I think this can be traced to your choice of a simplicial lattice maybe? You may not need a dual if you work with diamonds for instance.

Question (I’m typing fast because I can only spare a few moments!), when the Hodge appears in your work, is it combined with a coboundary? I ask because if you combine the topological Hodge with the coboundary on the dual lattice, it becomes the boundary operator on the primary lattice, i.e.

$\partial = * d *.$

Maybe it is related.

If you want to construct a discrete geometrical Hodge star on the lattice/dual lattice, you can end up with some spurious geometries with things like negative areas, etc. This does not happen with the topological Hodge.

I will leave with one wild thought…

Urs and I worked with “diamonds” which are just special cubes with certain desirable properties. We constructed a very satisfying discrete geometry on these diamond complexes. HOWEVER, we only demonstrated the special case of a globally cubic complex, when I believe that all you need is a local cubic property (which I haven’t been able to enunciate yet).

We know that a smooth manifold $M$ can be triangulated. From this triangulation, you can construct the dual lattice. I think that there should be some way to combine the lattice and dual in such a way that can be extruded into a diamond complex.

My conjecture is that any smooth manifold of the form $M\times R$ can be “diamonated”, just as $M$ can be “triangulated”. This procedure will likely involve constructing the dual lattice. Somewhere around here you can find my first attempt. I’m not sure if it works. I think there may have been some trouble with it (I don’t remember), but I think something like that will work. That would conceivably provide a bridge between what you did and what Urs and I did.

Gotta run!

Posted by: Eric on November 18, 2008 4:06 AM | Permalink | Reply to this

### Re: Discrete Hodge Star

Thanks for your interest Eric. I exchanged a few emails with Alain Bossavit back in 1997, and also met Robert Kotiuga around that time, which is how I came to hear about the issue of discrete Hodge operator in the computational electromagnetism context. Unfortunately I was too preoccupied with other things to look into it in detail at that time, although would be glad to do so at some point. (It would be nice to do something practically useful besides all the abstract stuff!)

Yes, what I used sounds like the topological Hodge star you mentioned, and yes its interplay with the coboundary and boundary operators on the lattice and its dual that you mentioned was also exploited.
The discretization of the abelian BF theory worked for any polyhedral cell decomposition of the manifold, not just triangulations, and the dual lattice was needed on general grounds. (In fact the doubling of the degrees of freedom this leads to seems to be a generic feature of discretizations of field theories with odd number of derivatives. E.g., in lattice formulation of fermions (involving first order derivatives) the problem goes under the name “lattice fermion doubling”.)

Although constructing the discrete BF theory and reproducing the topological stuff (linking numbers etc) worked for general cell decomposition, there was another part where triangulations were specifically required. This was for embedding the discrete theory into the continuum one by mapping the cochains into differential forms in such a way that the discrete and continuum Lagrangians were equal. (For some reason I thought it was desirable to do that, but then afterwards I wasn’t sure why - that was probably the reason I never submitted that paper for publication.) Anyway, roughly the way it was done was to use the fact that the barycentric subdivision of a triangulation can be embedded into both the original triangulation and its dual, so there is a way to map cochains of both the triangulation and its dual to diff forms using the Whitney map on cochains of the barycentric subdivision.

Thanks for mentioning your ideas and paper with Urs. I will bear it in mind and take a look when I have the chance.

Posted by: David A. on November 19, 2008 6:36 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Frankly I don’t believe [getting linking numbers from BV] is possible, but I would be glad to be proved wrong!

I am hoping that Pavel or some other BV expert sees this and drops a comment.

But what reason do you have to believe that it is not possible? The BV formalism is not supposed to be an extra assumption, but just a tool to for writing down a gauge-fixed path integral. If you can evaluate your gauge-fixed path integral by other means, it ought to be reproduced by the BV integra. What do you think goes wrong?

Posted by: Urs Schreiber on November 18, 2008 8:59 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

I was specifically referring to the problem of reproducing the linking numbers in the discrete BF theory. In the continuum formulation I’m sure this is no problem in the BV approach. As you mentioned, it is just a way to do the gauge fixing and I’m sure it gives the same results as before. However, in the discrete setting it matters very much how you formulate the discretization. There are various ways to discretize, but most of them will not exactly reproduce the topological stuff like linking numbers - you would typically only recover these in some continuum limit. My experience with trying to do this leads me to believe that to reproduce the linking numbers exactly without taking a continuum limit you need to discretize in a specific way that involves the lattice and its dual. Since Pavel’s approach to discrete BF theory didn’t involve this I had a hard time believing that it would exactly reproduce the topological quantities that arise from partition function and obsevables associated with linked knots in the continuum theory. However, from his comment below, and from looking at his papers again, it seems his goals were different in any case, so my comment should not be interpreted as criticism or suggestion that there was some flaw in his papers. In fact they look very nice! But I guess it is good to be clear about the various goals that one can have in pursuing discrete BF theory, and I hope my comments have helped to shed some light on that.

Posted by: David A. on November 19, 2008 4:20 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hello, David

“Frankly I don’t believe it is possible, but I would be glad to be proved wrong!”

I will try to comment on this. The quantity I was concerned with in my work on discrete non-abelian BF is the effective action on zero-modes (i.e. on [shifted] de Rham cohmology). It incorporates Massey operations on cohomology (generated by the tree part of effective action) and certain measure on zero-modes (generated by 1-loop part of effective action). The 1-loop part was considered modulo constants, and the constant term we omitted is \hbar dim(g) log “det (d)”. Here dim(g) is the dimension of gauge group and “det (d)” (when properly gauge fixed) is the torsion you were computing. So, I was considering non-abelian BF, normalized by abelian BF.

If you are interested in the partition function for non-abelian BF (I do not have a good intuition, what this quantity should be), then it presumably should be obtained by integrating this effective action on zero-modes, but some additional work has to be done. First, the effective action should be globalized, i.e. computed in the background of arbitrary flat connection (while I was concerned only with the neighborhood of zero connection). Second, one should understand how to make sense of the integral over zero-modes.

I did not consider observables, but by construction they can be transferred to the discrete setting (evaluating the BV integral over UV fields with the insertion of observable). But the “simplicial observable” will most likely be not simplicially local (and probably there will be no explicit formula for it either), and it will not be convenient to work with such an object. The easy thing is to transfer the “A-knot” (Wilson loop of connection A) to the triangulation - it is simplicially local. But it does not give you any knot or link invariants, as the A field in BF theory is not self-interacting. The only thing the VEV of this observable could detect would be the class of the knot in fundamental group of the manifold, which is not very exciting. The knot/link invariants may only be obtained from observables containing the field B, and they should transfer to the triangulation in bizarre way.

Posted by: Pavel Mnev on November 18, 2008 11:25 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hi Pavel,

Thanks for your comment. As mentioned above in my reply to Urs, it seems the goals in your study of discrete BF theory are different from the ones I had. If I understand correctly, your aim is basically to construct a “finite-dimensional” version of BF theory that has key properties of the continuum theory and in which quantities of interest such as the quantum effective action can be studied. Then in your new paper with Cattaneo you use the insights gained form this to construct new invariants in the framework of perturbative CS theory - is that right? It seems very nice.
The goal in my discrete BF work was different though - I just wanted to cook up a discretization which exactly reproduced the topological quantities of the theory in a discrete setting, without caring about anything else. It worked out in the abelian case, but not for the non-abelian case so far…

By the way, probably you are already aware of it but I would just like to mention that there is an interesting observable for knots in the non-abelian BF framework. It involves both the B and A fields (where A is the gauge field whose curvature is F) and gives the Alexander-Conway knot invariant as its expectation value – see hep-th/9407070. (In fact it seems that this can already be obtained from an observable in the abelian BF theory – see hep-th/9609205.) Also, the partition function in the non-abelian BF theory has been evaluated and found to give the integral of Ray-Singer torsion over the gauge orbit space of flat connections – I expect this is reviewed in the Physics Report article “Topological field theory” from 1991.

Posted by: David A. on November 19, 2008 4:31 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Thank you for the references.

“I just wanted to cook up a discretization which exactly reproduced the topological quantities of the theory in a discrete setting, without caring about anything else. “

Actually, the discretization I studied for BF does the same (by construction of effective theory): it exactly reproduces quantities of continuum BF theory (e.g. the effective action on zero-modes) for arbitrary triangulation. The problem though is that the action for the triangulation is given itself as a path integral and it is not easy to write closed formula for it. Observables may also be transferred to the triangulation and their VEVs in simplicial theory (given by finite-dimensional integrals) should reproduce exactly the VEVs in continuum theory.

Posted by: Pavel Mnev on November 19, 2008 6:56 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

In that case I will have to revert to my previous statement that I have a hard time believing that the topological quantities arising from the partition function and observables associated with knots can be exactly reproduced in your discrete theory, but I would be glad to be proved wrong.

Can you name one such quantity that you can demonstrate is exactly reproduced in your approach? If observables are problematic in your setting then just consider the partition function. In the abelian BF theory it is given by the Reidemeister-Ray-Singer torsion (with trivial flat connection), while in the non-abelian BF it is given by the integral of the R-R-S torsion over the orbit space of flat gauge fields. Can you demonstrate that either of these is reproduced in your approach to discretizarion of BF theory in either the abelian or non-abelian cases?
If for some reason this is problematic in your setting, can you name one explicit topological invariant (of either manifolds or knots) that you can show is exactly reproduced in your discrete theory?

My experience with trying to discretize the abelian BF theory is that it is impossible to exactly reproduce the Reidemeister torsion and linking numbers in the discrete theory unless the discretization involves the lattice and its dual in a specific way. That is why I don’t believe that your theory will be able to manage this, since it discretizes in a different way. But I would be glad to be proved wrong (and learn something in the process) if you can provide an actual demonstration of the exact reproduction of one of these topological quantities in your discrete setting.

Posted by: David A. on November 20, 2008 2:13 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

My experience with trying to discretize the abelian BF theory is that it is impossible to exactly reproduce the Reidemeister torsion and linking numbers in the discrete theory unless the discretization involves the lattice and its dual in a specific way.

I should sit down and study your and Pavel Mnëv’s articles on this in more detail, but let me ask a question anyway:

do you really mean to say that the right result is obtained in principle only for a specific choice of trinagulation (and dual triangulation)?

That would seem strange, on general grounds. It would seem that if you have a discretized topological theory where any output depends on the choice of triangulation then you have a problem in your formalism, since one would demand that the result must not depend on the choice of triangulation.

So maybe what you really mean is: unless one chooses a specific clever triangulation it becomes operationally extremely hard to carry out the computations of the output of the theory? I.e. that only for a specific choice of triangulation is the computation tractable.

That I have no problem to believe. And it would actually be consistent with that Pavel said, who wrote #, I think, that the expressions for his discrete BV version of BF-theory are guaranteed to reproduce the continuum theory, but are hard to evaluate in practice.

I think this is a common phenomenon in theoretical physics. It seems to me that instead of suggesting that Pavel’s formalism misses some results, it would indicate that your and his result would have a nice synthesis:

namely, as far as I understand what you both said, it would seem that your approach would give a specific realization of Pavel’s general scheme in which that general scheme becomes effectively computable.

But, as I said, I should better sit down and read your articles in detail…

Posted by: Urs Schreiber on November 20, 2008 10:16 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

“So maybe what you really mean is: unless one chooses a specific clever triangulation it becomes operationally extremely hard to carry out the computations of the output of the theory?”

No, there is no constraint on the triangulation; any one is fine. The point is that, for arbitrary triangulation, one needs to specify a prescription for constructing the discrete theory: The discrete fields need to be specified (often these will be cochains associated with the triangulation), as well as a Lagangian, or action functional for them.

When doing this work I started out taking the discrete fields to be cochains of the triangulation and tried to cook up an action functional such that the topological stuff was exactly reproduced in the discrete QFT. But it didn’t work, and in specific simple examples such as when the manifold is the circle I could prove that it was impossible to make this work in this setting. Then I realized that if the discrete A-field is a cochain of the triangulation and the discrete B-field is a cochain of the *dual* of the triangulation then there is a natural candidate for the action functional, and, using it, the topological stuff (R-torsion and linking numbers) was exactly reproduced.

Posted by: David A. on November 21, 2008 3:40 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

“can you name one explicit topological invariant (of either manifolds or knots) that you can show is exactly reproduced in your discrete theory?”

Sure, I think I already mentioned it: the effective action on cohomology. It is not a number, but a function. But it is an interesting invariant of manifolds. It may be computed exactly from discrete theory (which is guaranteed by construction). 1-loop part of this function should be the RRS torsion, while tree part generates Massey products on cohomology. So, this object is essentially the function you want to integrate over moduli space of flat connections, to produce the partition function, but I considered it only perturbatively near zero connection. This effective action on cohomology is explicitly computed for some examples in 0809.1160 (section 7.3) and is shown to distinguish (sometimes) rationally homotopic manifolds. So it is indeed an interesting invariant of manifolds.

Posted by: Pavel Mnev on November 20, 2008 11:22 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Thanks, I understand better what you mean now and it is very interesting.
I didn’t have a chance to study your discrete BF paper in any detail yet but would like to ask a few questions:

(1) To evaluate the effective action for some specific manifold, presumably you use a specific triangulation of the manifold. (That seems to be the case in the examples in your paper.) But the answer should be independent of the choice of triangulation. So I guess you have a theorem somewhere in the paper that says this. Can you direct me to it please?

(2) I would like to understand more precisely the relation of your invariant to R-torsion. The latter depends on the flat connection, and you say that you reproduce this “perturbatively near the zero connection”. Does this mean that if I think of R-torsion as a function of flat connection then your invariant is its derivative evaluated at the zero connection in some sense?

(3) I guess it would be straightforward to evaluate the R-torsion for the specific manifolds in the examples you gave in the paper and confirm explicitly the relation between R-torsion and your invariant in these cases. If you have done, or will do, these calculations I would be interested to know the outcomes.

Posted by: David A. on November 21, 2008 4:07 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

(1) This is a special case of statement 4 on p.46 of 0809.1160 and uses the obvious fact that there is a path in the space of induction data [(forms on M) to (de Rham cohomology of M)], which connects the induction data (i) obtained by composition
[(forms on M) to (cochains of triangulation T1) to (cohomology of M)] and induction data (ii) obtained as [(forms on M) to (cochains of triangulation T2) to (cohomology of M)].

There also exists an independent purely finite-dimensional argument that shows that effective action induced from simplicial BF theory on triangulations related by Pachner moves are same (modulo canonical transformations), but this is not written in the paper (I will probably include this in an update).

Another comment here: I used specific cell decompositions (they are cubic ones, not triangulations) in my explicit examples, because, e.g. I cannot write a closed formula for discrete BF action for a square, but I can do that for a square glued into a cylinder (section 6.4.3 of 0809.1160). From this I can glue the Klein bottle with special cell decomposition (with one 2-cell, two 1-cells and one 0-cell) where I know the discrete BF action and can induce now (evaluating the integral over the 2-cell and one of 1-cells) the effective action on cohomology.

(2) Now I am somewhat confused myself about what is called “R-torsion” and “R-torsion as a function of flat connection”. What I mean is (statement 11, p.71 of 0809.1160 - this is abstract version) that there is a map U which sends Maurer-Cartan elements of de Rham cohomology H^1(M,g) into flat connections on M. In small (but finite) neighborhood of zero this map is a bijection. The invariant part of effective action on cohomologies is its restriction to the Maurer-Cartan set MC\subset H^1(M,g), and it is expressed in terms of the map U and the determinant that I would call the non-abelian torsion as
(*) S’|_{MC} (x) = \hbar Str log(1+K [U(x),\bullet])
Where super-trace is over the space of g-valued differential forms, x is 1-cohomology, satisfying Maurer-Cartan equation, K is some chain homotopy from forms to cohomology, e.g. the Hodge chain homotopy
K=d^*/(d d^* + d^* d)
Quantity (*) is a function on MC \subset H^1(M,g) and thus is a function on the neighborhood of zero on the moduli space of flat connections (normalized so that S’(0)=0, thus the contribution of determinant from the abelian BF theory is thrown away). I would say that it is \hbar\times (torsion of flat connection U(x)). But I do not know if this is the usual notion of torsion.

So the invariant I was studying is the “torsion”
Sdet(1+K [A,\bullet]) = “det(d_A)/det(d)”
as a function on finite neighborhood of zero on moduli space of flat connections.

(3) Probably, I do not know. Could you give a reference where the procedure of computing the torsion is described?

Posted by: Pavel Mnev on November 21, 2008 3:29 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Thanks for the details. The structure of your invariant looks like what one would expect for the log of the torsion ratio “det(d_A)/det(d)” where det(d_A) is the analytic Ray-Singer torsion. But I recommend that you check this. A nice exposition of the Ray-Singer torsion is given in

D. Ray and I. Singer, “Analytic torsion” Proc. Sympos. Pure Math. Vol.XXIII (1971) 167-181

Once you verify that your invariant is indeed RS torsion ratio then you know that it is a topological invariant and there is no need to worry about showing independence of choice of triangulation in the combinatorial description of it that you derived.

The analytic Ray-Singer torsion was shown (independently) by W. Muller and J. Cheeger to coincide with the combinatorial Reidemeister torsion (also called R-torsion). The homotopy stuff that you mentioned sounds like something that might very well have been used in the proofs of that result. So I suggest you take a look at the papers:

W. Muller, Adv. Math. 28 (1978) 233
J. Cheeger, Ann. Math. 109 (1979) 259

(There was another more recent paper giving a proof of this as well, but i can’t remember the reference.)

The combinatorial R-torsion is given by a “superdeterminant” of the coboundary operator (with zero-modes excluded in an suitable way) - the precise definition is in the above papers. To evaluate it in practice for a specific triangulation you just need to work out the matrix expression for the coboundary operator between the cochain spaces (with the cells of the triangulation providing a canonical basis for each (co)chain space) and then calculate the determinants. It should be straightforward to do this for for the examples in your paper using the specific cell decompositions that you have there, if you want to check explicitly that your invariant coincides with the torsion ratio in these cases.

At this point I think I can understand why it is possible to obtain the torsion in your discrete BF theory despite my expectations that this shouldn’t be possible. I guess it is because the action functional you construct is non-local, whereas I was just considering particular types of local actions.

Posted by: David A. on November 22, 2008 10:39 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Thanks again for the references.

“The combinatorial R-torsion is given by a “superdeterminant” of the coboundary operator (with zero-modes excluded in an suitable way) - the precise definition is in the above papers.”

Could you please point me more specifically to this definition?

“To evaluate it in practice for a specific triangulation you just need to work out the matrix expression for the coboundary operator between the cochain spaces (with the cells of the triangulation providing a canonical basis for each (co)chain space) and then calculate the determinants.”

I feel very uneasy about that. If I have cell decomposition (I am supposing, it is either a triangulation or a cubic cell decomposition) $\Xi$ of manifold $M$ then the coboundary operator $d^\Xi$ is of course an obvious matrix defined by adjacency of cells. From what I was doing, the formula for the invariant on the level of cell decomposition is:

(1)$\log \det(d+[A,\bullet])/\det(d)'':= Str_{\Omega^\bullet(M,g)} \log (1+K [A,\bullet])=$
(2)$=\left(Str_{C^\bullet(\Xi,g)}\log (1+K^\Xi \sum_{n=1}^\infty\frac{1}{n!} l^\Xi_{n+1}(A^\Xi,\ldots,A^\Xi,\bullet)\right) + \sum_{n=2}^\infty \frac{1}{n!} q^\Xi_n (A^\Xi,\ldots, A^\Xi)$

where $A\in \Omega^1(M,g)$ is a flat connection on $M$, $A^\Xi$ is its image under the projection from forms to cell cochains $\Omega^\bullet(M,g)\rightarrow C^\bullet (M,g)$ (given by integrals over cells, in particular $A^\Xi=\sum_{\sigma\in\Xi}e_\sigma \int_\sigma A$ where sum is taken over 1-cells of $\Xi$ and $\{e_\sigma\}$ are standard basis cochains associated to cells); $K$ is some chain homotopy, contracting forms to de Rham cohomology, $K^\Xi$ is some chain homotopy, contracting cell cochains of $\Xi$ to de Rham cohomology (for instance, one can take $K^\Xi=\frac{(d^\Xi)^T}{d^\Xi\, (d^\Xi)^T+(d^\Xi)^T d^\Xi}$; $T$ is the transposition in standard basis for cell cochains). Operations $l^\Xi_n: \wedge^n C^\bullet(\Xi,g)\rightarrow C^\bullet(\Xi,g)$, $q^\Xi_n: \wedge^n C^\bullet(\Xi,g)\rightarrow \mathbb{R}$ are the quite hard to compute (well, they may be computed order by order in perturbation theory) and are the proper discretization of the operation $[,]$ (wedge product on forms, tensor the Lie bracket in g) on $\Omega^\bullet(M,g)$ which is used in the covariant differential $d_A:=d+[A,\bullet]$. These operations $l_n$, $q_n$ are what the buzz iz about in my consideration of discrete BF theory. So this formula may be interpreted, if I understand correctly, as the statement of equality of (normalized) Ray-Singer torsion for $M$ in the background of flat connection $A$ and some combinatorial torsion for cell decomposition $\Xi$ in the background of the projected connection $A^\Xi$. This combinatorial torsion uses of course the coboundary operator $d^\Xi$, but also the operations $\{l^\Xi_n\}$, $\{q^\Xi_n\}$ which make the story non- trivial.

One can also formally write the first term in (2) (the log of combinatorial determinant) as $\log\det(d^\Xi_{A^\Xi})/\det(d^\Xi)''$ where the coboundary operator twisted by discrete flat connection is $d^\Xi_{A^\Xi}:=d^\Xi+\sum_{n=1}^\infty\frac{1}{n!}l^\Xi_{n+1}(A^\Xi,\ldots,A^\Xi,\bullet)$

I wonder what is the relation of (2) to R-torsion (I still have no understanding of what it is, in non-abelian case, in presence of a flat connection). From what you say, they should be equal, but how could these complicated operations $l_n$, $q_n$ mysteriously disappear?

I would like to ask another question here: I am looking at your paper hep-th/9709147 and it seems that you use the explicit results for R-torsion for some 3-manifolds, but for the detailed computation you refer to  which is “in preparation”. Where I could find this text? - I would very much like to look at these computations.

“I guess it is because the action functional you construct is non-local, whereas I was just considering particular types of local actions.”

Well, it is sort of local - it is given as a sum of contributions of cells. These contributions are universal, depend only on the dimension of simplex/cube and the restriction of discrete fields to the cell. But of course for a given, say, cube, this contribution may contain terms, mixing different faces of this cube.

Posted by: Pavel Mnev on November 22, 2008 11:27 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Just in case it might help, the book The Laplacian on a Riemannian Manifold by Rosenberg contains a section (Google Books link) on Reidemeister torsion. No doubt you know this stuff already. I didn’t.

Posted by: Bruce Bartlett on November 23, 2008 12:55 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Actually, I do not know this stuff, and I am precisely trying to fill this gap in my understanding. Thanks very much for a helpful reference.

Now I think I am beginning to understand that the definition of R-torsion is non-perturbative in the connection: you work with the complex of “twisted cochains” (where the twist is defined by holonomies of the connection, which are not assumed to be close to 1) = $\pi_1$-equivariant cochains on the simply-connected cover of $M$. On the other hand, I was working with usual (non-twisted) cochains, without using the simply-connected cover, and I worked perturbatively in the connection (i.e. supposing that its holonomies are close to 1). The non-trivial thing is then to express the covariant derivative $d_A$ in terms of non-twisted cochains - here you have to start the story of induced $L_\infty$-structure on (non-twisted) cochains etc.

I will try to understand the relation of approaches.

Posted by: Pavel Mnev on November 23, 2008 2:20 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Pavel, it has been a long time since I worked on this stuff so I will need to re-familiarize myself with it to properly answer your questions. However, for what it is worth, my impression is that what you have is a kind of perturbative expansion of the log of the torsion ratio. The torsion is defined from a flat connection d_A (which can be locally written as d + A ) on the space of differential forms with values in a flat vector bundle. Flatness just means that (d_A)^2 = 0, so cohomology can be defined in the usual way, but is now “twisted”, not the usual de Rham cohomology.

Because of a correspondence between flat connections and maps \pi_1(M) -> G (where G is the gauge group), this stuff can equivalently be formulated in terms of \pi_1-equivariant cochains on the simply connected cover of M. That is often taken as the starting point for definitions of torsion in the literature. But I suspect that for understanding the connection between torsion and your stuff it will be better to work in the other picture of forms or cochains with “values in the flat vectorbundle” over M. You don’t need to invoke the simply-connected cover of M.

“I would like to ask another question here: I am looking at your paper hep-th/9709147 and it seems that you use the explicit results for R-torsion for some 3-manifolds, but for the detailed computation you refer to  which is “in preparation”. Where I could find this text? - I would very much like to look at these computations.”

If I remember rightly, I put them in an appendix of the article “Introduction to Chern-Simons gauge theory on general 3-manifolds” in “Mathematical Methods in Physics (Proceedings of Londrina Conf., 1999) p.1-43, published by World Scientific (2000). (Warning: that article was written in a big hurry to meet a deadline and I screwed up some things. However, the specific computations you asked about should be OK.) I lost the latex file for the article, and don’t have a copy, so hopefully your library has it; otherwise let me know and I’ll try to get hold of it some other way.

Bruce, thanks a lot for pointing out Rosenberg’s book! It looks like a “must have”. A small comment about his discussion of R-torsion: On page 154 he says that it is only defined when the cohomology groups of the chain complex are trivial. Not true - it can be defined and is a topological invariant also in the non-trivial case – then it is a “function of the cohomology”. The situation is the same for the analytic torsion.

Posted by: David A. on November 24, 2008 1:08 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Pavel, I found the book with the article in our library and checked that it does contain the details of the calculations you asked about. If you have trouble getting hold of it I can send you a scanned copy.

Posted by: David A. on November 25, 2008 9:07 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Thanks very much! Could you then please send me the scan to pmnev@pdmi.ras.ru ? - This book is not in our library.

Posted by: Pavel Mnev on November 25, 2008 11:36 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Pavel (or anyone else, for that matter), is it possible to say in mathematical terms what is meant by the phrase ‘effective action on cohomology’? Thanks.

Posted by: Simon Willerton on November 20, 2008 1:39 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Of course. Definition, adapted for BF theory is as follows. Let V be a dg Lie algebra and
(1) V=H(V) + V”
be its splitting into cohomology plus an acyclic subcomplex. Let F=V+V^*[-2] be the space of fields of associated abstract BF theory, and let also F’=H(V) + (H(V))^*[-2] ,
F”=V” + V”^*[-2]. Thus
(2) F=F’+F”
is the splitting of fields into “infrared” and “ultraviolet” parts defined by the splitting (1). We regard (2) as a vector bundle over F’ with fiber F”. The “effective action on cohomology” S’ is a function on F’ defined by fiber BV integral
(3) exp(S’/\hbar)=\int exp(S/\hbar)
where S \in Fun(F) is the BF action asociated to the dg Lie structure on V, and the integral is taken over some Lagrangian subspace L \subset F” (the “gauge fixing” of integral (3)). Different choices of L in general lead to different effective actions S’, but they have to be related by BV canonical transformation (infinitesimally: S’~S’+{S,R}+\hbar \Delta R for some generator R). So in principle the effective action should be regarded modulo canonical transformations (or more precisely: modulo _some_ canonical transformations, that arise from changes of L).

Standard BF theory is the case V= (differential forms on manifold M)\otimes (some Lie algebra g), V’=(de Rham cohomology of M)\otimes g. The de Rham cohomology may be embedded into forms e.g. as harmonic forms w.r.t. some Riemannian metric, or in some other way.

Posted by: Pavel Mnev on November 20, 2008 2:17 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

[Writing Pavel’s last comment in Itex - DC]

Of course. Definition, adapted for BF theory is as follows. Let V be a dg Lie algebra and

(1) $V = H(V) + V''$

be its splitting into cohomology plus an acyclic subcomplex. Let $F = V+V^*[-2]$ be the space of fields of associated abstract BF theory, and let also

$F' = H(V) + (H(V))^*[-2],$

$F'' =V'' + V''^*[-2].$

Thus

(2) $F = F' + F''$

is the splitting of fields into “infrared” and “ultraviolet” parts defined by the splitting (1). We regard (2) as a vector bundle over $F'$ with fiber $F''$. The “effective action on cohomology” $S'$ is a function on $F'$ defined by fiber BV integral

(3) $exp(S'/\hbar)=\int exp(S/\hbar)$

where $S \in Fun(F)$ is the BF action asociated to the dg Lie structure on $V$, and the integral is taken over some Lagrangian subspace $L \subset F''$ (the “gauge fixing” of integral (3)).

Different choices of $L$ in general lead to different effective actions $S'$, but they have to be related by BV canonical transformation (infinitesimally: $S' ~ S' + \{S, R\} + \hbar \Delta R$ for some generator $R$). So in principle the effective action should be regarded modulo canonical transformations (or more precisely: modulo _some_ canonical transformations, that arise from changes of $L$).

Standard BF theory is the case $V = (differential forms on manifold M)\otimes (some Lie algebra g)$, $V' = (de Rham cohomology of M)\otimes g$. The de Rham cohomology may be embedded into forms e.g. as harmonic forms w.r.t. some Riemannian metric, or in some other way.

Posted by: David Corfield on November 20, 2008 2:49 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hi David, indeed it’s good to have you make some comments here. Of the three points you raised, I think I am too much of a beginner in these issues to say anything intelligent at this stage, so it’s best if I keep my mouth shut.

Posted by: Bruce Bartlett on November 17, 2008 3:22 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hi Bruce, thanks for the welcome!

Posted by: David A. on November 18, 2008 2:34 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hi David,

I just noticed something neat in your paper.

You define a “simplicial wedge product” (Equation 2.5)

$x\wedge^K y := A^K(W^K(x)\wedge W^K(y))$

This product is not an algebra morphism

$W^K(x\wedge^K y) \ne W^K(x)\wedge W^K(y)$

because

$W^K A^K \ne 1.$

HOWEVER, you can go the other way. Define a “continuum cup product”

$\alpha\smile^C \beta := W^k(A^K(\alpha) \smile A^K(\beta)).$

This IS an algebra morphism

$A^K(\alpha\cup^C\beta) = A^K(\alpha)\cup A^K(\beta)$

because

$A^K W^K = 1.$

This “continuum cup product” is not commutative at the cochain level, but is when you pass to cohomology.

The noncommutativity turns out to be a feature rather than a bug.

I haven’t read beyond Equation 2.5 in your paper yet, but couldn’t resist pointing this out. This could provide a bridge between your work and ours. Fun!

More later…

Eric

Posted by: Eric on November 23, 2008 3:11 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

I read a little more and see that you do mention cup and cap so can imagine you will object to the relation.

Instead of cup product, you should consider the product in our paper. Everything I said above applies…

I was never a big fan of the simplicial wedge product because of its failure to be an algebra morphism. The first paper where I saw the simplicial wedge product appear was in Dodziuk’s 1976 paper

Finite difference approach to the Hodge theory of harmonic forms Amer. J. Math., 98 (1976), 79-104

By the way, have you seen Kotiuga’s papers (I imagine you have)

Kotiuga, P.R. Analysis of Finite Element Matrices Arising from Discretizations of Helicity Functionals, Journal of Applied Physics, 67(9), May 1990, pp. 5815-5817.

Kotiuga, P.R., Helicity Functionals and Metric Invariance in Three Dimensions, IEEE Transactions on Magnetics, MAG-25, (4), July 1989, pp. 2813-2815.

Kotiuga, P.R., Variational Principles for Three-Dimensional Magnetostatics Based on Helicity, Journal of Applied Physics, 63(8), April 1988, pp.3360-3362.

Kotiuga, P.R., Hodge Decompositions and Computational Electromagnetics, Ph.D. Thesis, McGill University, Montreal, Canada, 1985.

Helicity functionals arising from problems in applied computational electromagnetics are essentially “discrete BF theories” in three dimensions.

Posted by: Eric on November 23, 2008 3:31 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hi again David,

I’m really enjoying reading your paper. It has all the ingredients that are close to my heart.

My poor brain is being pushed to the limits, but I’m beginning to think that I have seen your paper at one point, but was not mathematically ready to appreciate it yet at the time (I was an engineer after all!). Then Urs and I worked together in a flash and wrote our paper. Now, it is fun to look back at your paper after the stuff Urs and I did with better understanding.

Here are a few thoughts…

Thought 1

One trick I learned from Urs is that if you have a well-behaved topological operator that has the desired algebraic properties of a metric operator you are trying to discover, then you can begin with the topological operator and “deform” it.

The topological Hodge operator $*$ in your paper certainly has the desirable properties. I wonder if there is a way to deform it, i.e.

$\star = \mu^{-1} * \mu$

where $\star$ is the (metrical) Hodge operator. That would be neat. I imagine that might bring you full circle to things Bossavit and Hiptmair have done.

Thought 2

It looks like

$T^{\bar K} T^K$

is the “graph Laplacian”.

It would be neat if you could construct a Dirac operator.

An obvious choice might be something like

$D^K = d^K + \partial^K.$

Maybe a less obvious choice would be some kind of hybrid

$D^{(K,\widehat K)} = d^K + \partial^{\widehat K}.$

This, or something like it, would require combining the primary and dual lattices. For that, I think this idea I’ve been rambling about called “diamondation” would come in, i.e. begin with a triangulation and its dual, and “unpack” it into a single cell complex called a diamond complex, which is the subject of our paper, where we demonstrated a discrete Dirac operator.

Thought 3

In your paper, you seemed to be happy that your construction of Hodge duality does not require an ordering of nodes. The requirement of an ordering is due to the use of the cup product.

Our construction does not require an ordering of the nodes, but rather a partial ordering. This partial ordering is related to causality.

If you take a diamond complex, which comes with a partial ordering, representing a diamondation of a Lorentzian manifold $M\times R$ and “project” it down in time via the “long exposure photograph” idea, then you end up with a oriented triangulation of $M$.

Your construction does not require an ordering of nodes, but requires oriented simplices. Here is another conjecture:

Given a triangulation $\bigtriangleup_M$ of a manifold $M$ and the corresponding diamondation $\Diamond_M$ of $M\times R$, the orientation of $\bigtriangleup_M$ is inherited from the partial ordering of $\Diamond_M$.

This could pave another bridge between what you did and what we did.

On the other hand, after seeing the simplicial wedge product in your paper, I am pretty sure that the two constructions are not equivalent, but maybe “dual” to one another.

In a way, you start with a product on a simplicial complex and antisymmetrize to get a new product that has many desirable properties that you see in the continuum. But you are missing one important property.

The simplicial wedge product is not associative.

Urs and I went a different route. Borrowing from work done by Dimakis and Mueller-Hoissen, we deal with a slightly different discrete product that is not commutative, but is associative.

This discrete product maps to the continuum via an algebra morphism so that the continuum algrebra remains closed. Here is another factoid:

The continuum commutative, but nonassociative algebra of Whitney forms is not closed.

In other words, the wedge product of two Whitney forms is generally not another Whitney form.

I don’t know what to call it, but

The continuum noncommutative, but associative algebra of Dimakis-Mueller-Hoissen-Schreiber-Forgy forms is closed.

A DMHSF form is essentially the image of an analogue of the Whitney map applied to diamonds instead of simplices (with corresponding obvious de Rham map).

When you are working with discrete field theories, you have a choice to make:

1. Commutativity with nonassociativity

2. Associativity with noncommutativity

The work that Urs and I did can be thought of as representing Choice 2. Your stuff seems to represent Choice 1. I suspect that deep down the two choices are related somehow. Dennis Sullivan and his student Scott Wilson also followed Choice 1, but when I tried to talk to them it was a documented disaster.

Gotta run for now!

Eric

PS: I feel obliged to give this disclaimer. Most people who know me already know how clueless I am, but I feel the need to warn new people. I’m not a mathematician, a physicist, or an engineer, but I know a dangerously small amount of all three. Plus, I work in finance now! I often say things because I “feel” they are right. Most of the time, there is a grain of truth to what I say, but often I mix up terminology, e.g. I might say “BF theory” when “Chern Simons theory” is more appropriate. Sometimes I’m just plain wrong. Nevertheless, if you’re patient, on occasion, I am capable of helping to push good ideas along.

Posted by: Eric on November 23, 2008 4:22 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Hi Eric,

Thanks for your comments and thoughts, and sorry for my slow response. Actually I wasn’t anticipating any comments addressed to me away from our previous discussion, and just saw yours now while taking a look at what was going on in the rest of the thread. (I’m still a bit unfamiliar with these things.) No need for the disclaimer - it is clear you have thought a lot about this stuff and have interesting things to say about it. Don’t worry about going with “feelings” for what is right, and loose terminology - I do that too!

Regarding geometric discretization, it seems that it is impossible to find one that has *all* the properties of the continuum setting that one would like. There is always some sacrifice that must be made. So I guess the thing to do is taylor the discretization scheme to the specific problem one is interested in, trying to preserve the properties that are most important in that context while sacrificing the ones that happen to be less important in the considered context. That was certainly the case for the scheme I cooked up: It works as desired for reproducing the topological stuff of the abelian BF theory; if it turns out to have any other applications beyond that then great, but if not then no problem - it already did the job it was supposed to do.

It seems that, generally, people interested in geometric discretization in various contexts cook up their own scheme that works best for what they want to do. Consequently there are quite a few discretization schemes out there at this point. For example, in this paper a Caltech applied math group introduced their own new scheme that apparently works best for the problems they are interested in (but which looks like it would be no good for, e.g., discretization of abelian BF theory).

Regarding the possibility of constructing a Dirac operator in the discrete setting, that is indeed something I thought about back in those days. But to do anything interesting and physically relevant with it you need to couple it to gauge fields, and I didn’t see any natural way to do that within my scheme. So I didn’t get anywhere with that.

Thanks very much for the references to Kotiuga’s papers and for pointing out that there is a connection between the helicity functionals and BF theory in 3 dimensions. That is definitely something I would like to check out and see it my scheme can be of any use there.

I wish I had time to take a closer look at the your paper with Urs and give some thoughts on it, but it is hard to find any spare time right now. I will remember it though, and get back to this when I have the chance.

Posted by: David A. on November 25, 2008 8:58 AM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

The classical map inducing an algebra iso of de Rham and singular or simplicial or Cech cohomologies is not an algebra map, but, surprise!, it is strongly homotopy multiplictive.

MR0425956 (54 #13906) Bousfield, A. K.; Gugenheim, V. K. A. M. On deRham theory and rational homotopy type. Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94 pp.

MR0418083 (54 #6127) Gugenheim, V. K. A. M. On the multiplicative structure of the de Rham theory. J. Differential Geometry 11 (1976), no. 2, 309–314.

Posted by: jim stasheff on November 23, 2008 10:10 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

I tracked down a pdf of the second paper

On the multiplicative structure of the de Rham theory

I was happy to see the last two references to Whitney’s papers. Those are two of my all time favorites!

Reference  also looks very interesting

J. Stasheff & S. Halperin, Differential homological algebra in its own rite, Proc. Advanced Study Inst. Algebraic Topology, Aarhus, 1970.

;)

Here is an interesting quote:

Additionally, the theorem asserts that the isomorphism

$H(\rho): H(A^*)\to H(C^*)$

is a map of algebras. Of course, $\rho$ itself is not a map of algebras; $A^*$ is commutative and $C^*$ is not.

Situations of this sort - maps which are not multiplicative but become so in homology - are encountered elsewhere, and it has been observed that often this phenomenon is associated with the existence of a whole family of “higher homotopies”; cf. e.g. .

Wow. This brings in all kinds of things I wish I knew more about. Very fascinating.

There are at least three “discrete algebras” that relate to all this. The algebra $C^*$ is the one whose product is cup product. Then you have David’s simplicial wedge product, which is essentially an antisymmetrization of the cup product and finally you have the discrete product we worked with. I know that the simplicial wedge product leads to some higher algebra stuff, because this is what Sullivan was talking about. I have also always suspected a relation between our product and cup product, but think it is a little different. It shares the property of being associative and noncommutative with the cup product though.

I wonder if our product might somehow relate to higher homotopies…

Posted by: Eric on November 25, 2008 10:29 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

to homotopy associativity

what’s a reference for your product?

Posted by: jim stasheff on November 25, 2008 10:44 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

what’s a reference for your product?

Well, you have our paper:

Discrete Differential Geometry on Causal Graphs

But the basic algebra part comes from Dimakis.

There are a ton of papers that might be down your alley, but this is probably a good place to start:

Discrete Riemannian Geometry

Posted by: Eric on November 25, 2008 11:17 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

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.

Posted by: Daniel de França MTd2 on November 19, 2008 2:25 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

can I visualize the (ghost-of)^n as an n-surface

Not in the sense in which you seem to imagine it.

so that one can get a rough idea of it

The rough idea is actually simple:

Consider some physical system which you describe by some symbols.

Say here is one symbol with which you describe the state of the system $A_0$ and here is another symbol $\;\;\;\;\;\;\;\; A_1$ with which you indicate another state of the system.

Now imagine that, even though you are using different symbols, both of these states in nature correspond to the same state. Imagine there is a transformation, think literally of a a continuous path of operations $A_0 \stackrel{\phi}{\to} A_1$ which turns your entity $A_0$ into $A_1$: a gauge transformation.

Now suppose that there is a smooth way to turn $A_0$ into $A_1$ by changing it gradually:

first we apply a little bit of gauge transformation to $A_0$ to arrive at $A_{1/2}$. Then we apply a little more gauge transformation to get from $A_{1/2}$ to $A_1$.

Or break it down into even finer steps $A_0 \to A_{1/4} \to A_{1/2} \to A_{3/4} \to A_1 \,.$

I am being sketchy, but I suppose you can get the idea: in such a situation, where gauge transformations can vary smoothly, we can take the tangent to the path $A_0 \to A_{1/4} \to A_{1/2} \to A_{3/4} \to A_1$ along which we are gauge transforming. This tangent is a vector. The linear dual to it is a differential form. This differential form they call a ghost, just to confuse you.

(Well, not just to confuse you. Feynman coined the term because after perturbative quantization these 1-forms show up as spurious particles running through Feynman’s diagrams. But let that not concern us here.)

So, in some way, ghosts are about 1-dimensional entities: namely about paths in the space of possible gauge transformations.

But now suppose that you are really being careful: you find that there are two different paths of gauge transformations from $A_0$ to $A_1$: this one $\array{ & \nearrow \searrow \\ A_0 && A_1 }$

and this one:

$\array{ A_0 && A_1 \\ & \searrow \nearrow } \,.$

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):

$\array{ & \nearrow \searrow \\ A_0 &\Downarrow^P& A_1 \\ & \searrow \nearrow } \,.$

Again suppose that everything is smooth. Then we should be able to compute the tangent bivector to $P$ at A_0, some infinitesimal version of

$\array{ & \nearrow \\ A_0 \\ & \searrow } \,.$

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

So, in some way $n$-fold ghosts of ghosts really are about $n$-dimensional surfaces. But careful with where these surfaces live: in the abstract space of gauge transformations.

I am telling this story at this rough level because it seems that this is what you are looking for. What I said is not precise, but is the cartoon of a precise story underlying it.

The precise story reads like this:

The $k$-fold ghosts of ghosts are the cotangents to the source-fibers of the space of $k$-morphisms of the configuration-“space” of the physical system, which is not a space but a smooth $\infty$-groupoid.

You may read “$\infty$-stack” for “smooth $\infty$-groupoid” if you prefer that.

So when above I was talking about “paths pf paths in the space of gauge transformations” I was slightly lying: these are really 2-morphisms in some $\infty$-groupoid.

But, on the other hand, there are situations where both coincide. This here is an instructive example to keep in mind:

Categorical degree is not an intrinsic notion. Meaning: one theory may have just ordinary ghosts, the other may have ghosts-of-ghosts and still both theories are “the same”.

For instance consider the gauge group $G = SU(3)$ of quantum chromo dynamics. This is an ordinary group. But there is a 2-group weakly equivalent to it, which is much bigger, it goes by the name $(\Omega G \to P G)$. Its objects are paths in $G$, it’s morphisms are homotopy classes of surfaces in $G$!

QCD with gauge group $G$ is equivalent to 2-QCD with gauge group $\Omega G \to P G$.

With this 2-group $(\Omega G \to P G)$ used to interpret the “rough story” I told at the beginning of this comment the rough imagery about paths and paths-of-paths becomes a rather exact picture of what is really going on.

To conclude: you have the right intuition if $n$-fold ghosts of ghosts make you think of $n$-dimensional surfaces. But these surfaces are not surfaces in space or spacetime – but in the abstract space of parameters with which you describe a physical theory.

Posted by: Urs Schreiber on November 19, 2008 8:07 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

going from n-surfaces to infty-groupoids is a bridge too far

on the other hand, the space of parameters is quite appropriate - why call it abstract?
because outside of space-time? or of the bundle you came in with?

IF it helps (perhaps for intuition but not for computation), think of the ghosts as sections of an auxiliary ghost’ bundle
and similarly for ghosts-of-ghosts, anti-ghosts, anti-fields,… each with its own bundle

Posted by: jim stasheff on November 21, 2008 2:06 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

going from $n$-surfaces to $\infty$-groupoids is a bridge too far

Actually its quite a short step:

whatever you think of when you think $\infty$-groupoids, the archetypical example is supposed to the the fundamental $\infty$-groupoid of a space, whose $k$-cells are $k$-surfaces in that space.

And $\infty$-Lie integration theory tells us: up to discrete quotients, all smooth $\infty$-groupoids are fundamental groupoids of some (generalized) space.

So, while the language may sound overpowered, the bridge is really short.

Posted by: Urs Schreiber on November 21, 2008 2:14 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

Daniel writes:
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.

Definitely NOT vacuous and useless only if all you care about is H in degree 0. In so far as you can picture’ an n-form, you
might could do so for ghosts of ghosts of … but they are fields’ in their own right/rite. What would an n-surface picture do for you?

Posted by: jim stasheff on November 21, 2008 1:46 PM | Permalink | Reply to this

### Re: Frobenius algebras and the BV formalism

“What would an n-surface picture do for you?”

To raise my couscious awareness of whenever I see a surface, I think of ghosts. Maybe there is a way, not necessaraly linked to BV formalism, to build theories by gluing, embending intesecting, tying, linking ghosts and later putting a suitable field at the level 0, just like one manipulate topologies to get new ones, except for the field part.

It would be fun.

Posted by: Daniel de França MTd2 on November 21, 2008 2:55 PM | Permalink | Reply to this

### Re: Frobenius Algebras and the BV Formalism

“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.

Posted by: Daniel de França MTd2 on November 20, 2008 12:48 AM | Permalink | Reply to this

### Re: Frobenius Algebras and the BV Formalism

OOops, I was thinking about SU(3),no SU(2), since I was following Urs’ QCD example. Tom Leinster corrected me, but his post was erased by mistake with mine, when someone deleted a double post of mine.

Posted by: Daniel de França MTd2 on November 20, 2008 2:47 AM | Permalink | Reply to this

### Re: Frobenius Algebras and the BV Formalism

What would a SU(3) be like? I found something:

http://dukespace.lib.duke.edu/dspace/bitstream/10161/826/1/D_Xu_Feng_a_200808.pdf

On p. 6 he says SU(3) holonomy for 6 dimensions on a riemmanian manifold is Calabi Yau. Can I say that a k-fold ghost is sometimes simulated by compactification k dimensions of Type I strings?

Posted by: Daniel de França MTd2 on November 20, 2008 3:19 AM | Permalink | Reply to this

### Re: Frobenius Algebras and the BV Formalism

Can I say that a $k$-fold ghost is sometimes simulated by compactification $k$ dimensions of Type I strings?

No.

And you shouldn’t try to say things like this at the moment. My impression was that you are still trying to get a handle on BRST formalism in general and then notion of ghosts-of-ghosts in particular. It seems you need some elementary facts. For that it seems unlikely to be helpful to follow every wild speculation that comes to mind, if I may say so.

If you really want to understand ghosts-of-ghosts we should do the following:

we should start by discussing the BRST complex for abelian 1-form gauge theory, aka electromagnetism. This involves ghosts which correspond to the fact that a gauge transformation between a 1-form $A_0$ and a 1-form $A_1$ is a function $\lambda$ such that $A_1 = A_0 + d\lambda$.

If and when you are comfortable with that, we should pass to abelian 2-form gauge theory, aka Kalb-Ramond-electromagnetism. This involves ghosts-of-ghosts which correspond to the fact that

a gauge transformation of a 2-form $B_0$ to a 2-form $B_1$ is itself a 1-form $A_0$ such that $B_1 = B_0 + d A_0 \,.$

$\array{ & \nearrow \searrow^{A_0} \\ B_0 && B_1 } \,.$

But there may be another 1-form $A_1$ accomplishing this

$B_1 = B_0 + d A_1 \,.$

$\array{ B_0 && B_1 \\ & \searrow \nearrow_{A_1} } \,.$

This happens (locally, and for our purpose this is all we care about) if and only if there is a gauge-transformation between these two gauge transformations, a “gauge-of-gauge”-transformation, namely a function $\lambda$ such that

$A_1 = A_0 + d\lambda$

$\array{ & \nearrow \searrow^{A_0} \\ B_0 &\Downarrow^\lambda& B_1 \\ & \searrow \nearrow_{A_1} } \,.$

So 2-form electromagnetism has (locally, but never mind)

- fields which are given by 2-forms $B$

- gauge transformations which are given by 1-forms $A$

- gauga of gauge transformations which are given by 0-forms= functions $\lambda$.

If one thinks obout this correctly this just says that (smooth) 2-forms naturally live in a (smooth) 2-groupoid, this is the 2-stack, if one wishes, of configurations of 2-form electromagnetism (locally).

Whatever this really means in detail, it should be clear from the above pictures that:

- infinitesimal gauge transformations of 2-forms are vectors on some space. The dual 1-forms to these vectors are the ghosts of 2-form electromagnetism.

- infinitesimal gauge-of-gauge transformations of 2-forms are bivectors on some space. The dual 2-forms to these bivectors are the ghosts-of-ghosts of 2-form electromagnetism.

Posted by: Urs Schreiber on November 20, 2008 9:56 AM | Permalink | Reply to this

### Re: Frobenius Algebras and the BV Formalism

I will just try to justify why my speculation was wrong so that I can stop thinking about wrong things, and after, I will restate a question that asked you right above in a clearer way, given your explanation.

Anyway, I am sorry, I asked the thing about the SU(3) and strings because something you told made think wrongly about something mechanic:

“But these surfaces are not surfaces in space or spacetime – but in the abstract space of parameters with which you describe a physical theory.”

Then I reasoned in this way: the parameters of a theory can live in in surface, but each point of space-time has its configuration. So given that the parameters can “slide” in that surface, and each point of that space time for a theory has its one own precise set of paramters, and that surface is a kind of restriction for the parameters that are varied for each point of space time. But, if I had extra dimensions, I could build a “mechanical” model for this parameter space, to constrain it, in which that surface would be an holonomy. I was wrong because, as you said even before, these ghosts cannot have nothing to do with fields, and I was trying, again, to dress them with fields! Right?

Now, I will restate a previous question:

The way you find the k-ghosts, it seems that you are taking an homological sequence until you find an exact q-ghost, right? So, if we have a k-form field, the highest number of ghosts we can have is k-1.

Posted by: Daniel de França MTd2 on November 20, 2008 2:36 PM | Permalink | Reply to this

### Re: Frobenius Algebras and the BV Formalism

So, if we have a $k$-form field, the highest number of ghosts we can have is $k-1$.

For $k$-form field theory the highest degree of ghosts is $k$. Compare the above discussion.

Posted by: Urs Schreiber on November 20, 2008 5:36 PM | Permalink | Reply to this

### Re: Frobenius Algebras and the BV Formalism

I am sorry, my mistake was counting the number of n-forms of ghosts instead of its k folding number.

Posted by: Daniel de F. on November 20, 2008 8:20 PM | Permalink | Reply to this

### Re: Frobenius Algebras and the BV Formalism

For the following to be true:

So, if we have a k-form field, the highest number of ghosts we can have is k−1.

For k-form field theory the highest degree of ghosts is k. Compare the above discussion.

There must be some additional assumptions which I am unable to guess. Perhaps: k-form field theory means one which in the full BV formalism involves at most k-ghosts? Surely the degree of forms in the original Lagrangian does not restrict the ghost tower.

Posted by: jim stasheff on November 21, 2008 2:01 PM | Permalink | Reply to this

### Re: Frobenius Algebras and the BV Formalism

There must be some additional assumptions which I am unable to guess.

By $n$-form gauge theory I mean a theory whose fields are (locally) differential forms with values in a Lie $n$-algebra.

So: 1-form gauge theory is ordinary gauge theory. Fields are locally 1-forms. This has ghosts of degree 1.

And so on.

Posted by: Urs Schreiber on November 21, 2008 2:11 PM | Permalink | Reply to this

### Re: Frobenius Algebras and the BV Formalism

“So, if we have a k-form field, the highest number of ghosts we can have is k−1.”

Could be that the number of ghosts would be $(k)^{2}$, because given the analogy of surfaces, you would have $\frac{(k-1)!}{(k-1-p)!(p)!}$ for every p-form ghost?

“Surely the degree of forms in the original Lagrangian does not restrict the ghost tower.”

I don’t understand. Can you explain?

Posted by: Daniel de França MTd2 on November 21, 2008 3:13 PM | Permalink | Reply to this

### Re: Frobenius Algebras and the BV Formalism

Remember that the BV formalism starts with an action or equivalently a Lagrangian subthing equivalent to the solution space of Equations of Morion

then we adjoin antifields corresponding the the EL eqns

but they satisfy relations corresponding to Noether identities so we adjoint antifields of antifields’

corresponding to Noether identities we have ghosts (Noether’s 2nd Theorem)
and so on up on both sides

Posted by: jim stasheff on November 21, 2008 11:28 PM | Permalink | Reply to this

### Re: Frobenius Algebras and the BV Formalism

“but they satisfy relations corresponding to Noether identities so we adjoint antifields of antifields’”

I don’t get this part. Can you give a simple example explicitly? What are antifields of antifiels? I thought it would be something like :

…-> 2- antighost -> 1 - anti ghost -> antifield -> field -> 1-ghost -> 2-ghost

Also when you said tower of ghosts, I understood as it being the “size of the sequence”.

Wouldn’t be that adding floors to that tower be cheating, like modifying a theory by changing its dynamics?

Posted by: Daniel de França MTd2 on November 22, 2008 2:52 AM | Permalink | Reply to this

### Re: Frobenius Algebras and the BV Formalism

In response to:
antifields of antifields’

I don’t get this part. Can you give a simple example explicitly? What are antifields of antifiels? I thought it would be something like :
-> 2- antighost -> 1 - anti ghost -> antifield -> field -> 1-ghost -> 2-ghost

yes, antighost is a better name for what I called antifield of antifields

the arrows you write really go from the
tail object to a polynomial (over the ground ring = space of fields) in the objects further along in the sequence
so that the sequence is exact

e.g. antifield -> field (roughly) gives the de Rham cohomology of the Lagrangian

for full details, see the book by Henneaux and Teitelbohm or my papers
I find the Hamiltonian BFV version easy to `picture’

Posted by: jim stasheff on November 22, 2008 2:08 PM | Permalink | Reply to this

### Re: Frobenius Algebras and the BV Formalism

I’m sorry, the generic sequence I meant is:

…-> 2- antighost -> 1 - anti ghost -> antifield -> field -> 1-ghost -> 2-ghost ->…

Posted by: Daniel de F. on November 22, 2008 3:21 AM | Permalink | Reply to this

### Re: Frobenius Algebras and the BV Formalism

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

Posted by: Urs Schreiber on December 2, 2008 9:21 AM | Permalink | Reply to this

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