BV-Formalism, Part IV

October 11, 2007

Posted by Urs Schreiber

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Here are some further thoughts on BV-formalism and its interpretation in terms of groupoids. My main point here is to promote the slogan

What is called the “space of fields and ghosts” in BRST/BV formalism is nothing but the Lie algebroid of the action groupoid of the gauge group acting on the space of fields.

(Part I, II, III.)

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This is based on

a) the claim, which I make hereby (and which should to be well known in one incarnation or other), that the Lie algebroid of the action Lie groupoid of a group $G$ acting on a space $X$ is Koszul dual to the Chevalley-Eilenberg cochain complex for Lie algebra cohomology of $g = \mathrm{Lie}(G)$ with values in the module $C^\infty(X)$ .

b) the observation, nicely emphasized in

Kevin Costello
Renormalization and the Batalin-Vikovisky Formalism
0706.1533

(p. 16) that with $V$ the space of fields of some theory and with $G$ the gauge group acting on it, “the space of fields and ghosts” of the corresponding BRST construction is itself dual to that Lie algebra cochain complex.

This Koszul-duality between Lie $n$ -algebras and differential graded algebras I have been talking about a lot here. Useful references might be Zoo of Lie $n$ -algebras, Lie $n$ -algebra cohomology and String- and Chern-Simons $n$ -Transport.

Here we need the generalization of this to the many-object version of Lie $n$ -algebras, namely Lie $n$ -algebroids. One way to conceive these dually is as dg-manifolds. Another way is, I think, as differential algebras with values in nontrivial modules.

More precisely:

Claim Let $G$ be a Lie group acting on a manifold $X$ , such that $g = \mathrm{Lie}(G)$ acts on the algebra of functions $C^\infty(X)$ by algebra derivations. Then the Lie algebroid of the action groupoid $X // G$ is dually given by the graded commutative differential algebra generated by $C^\infty(X)$ in degree 0 and by $g^*$ in degree 1, with the differential on $g^*$ being the Chevalley-Eilenberg differential $d \omega(x,y) = - \omega([x,y])$ and with the differential on $C^\infty(X)$ given by the action of $g$ on $C^\infty(X)$ .

Example. Not quite an action groupoid but the example to keep in mind:

For $\Pi_1(X)$ the fundamental groupoid of $X$ , the corresponding Lie algebroid is the tangent algebroid $T X$ whose Koszul dual is simply the differential algebra of differential forms on $X$ . This we can think of as being the cochain complex for computing the Lie algebra cohomology of the Lie algebra $diff(X)_0 := \Gamma(T X)$ with values in the module $C^\infty(X) \,.$ Then the formula for the deRham differential $d \omega(x,y) = \omega([x,y]) + x(\omega(y)) - y(\omega(x))$ is a special case of the differential described in the claim above.

Somehow all what BV formalism is about should be something like making sense of colimits over functors over infinite dimensional action Lie groupoids.

For consider $X$ to be the space of fields (configuration space) of some physical system with gauge group $G$ acting on $X$ $X \times G \to X \,.$

We know (see Canonical Measures on Configuration spaces) that we should be looking at the configuration category rather, whose objects are the configurations and whose morphism are processes between configurations. Here we just consider processes which describe gauge transformations and hence address the action groupoid

$\mathrm{conf} = X // G$

as the “configuration category”.

More generally, there might not be a global gauge group acting and we’d take $\mathrm{conf}$ to be an arbitrary (Lie, maybe) groupoid.

Supposing that we have a gauge invariant “action functional” $e^S$ – some functor on $\mathrm{conf}$ which is constant on isomorphism classes, we want to understand its colimit or the like

$\int_{\mathrm{conf}} e^S$

over this configuration category.

When everything is finite we know how this works: the Leinster measure on finite categories takes control, which for the groupoid case is Baez-Dolan groupoid cardinality – and which indeed is the right measure for Dijkgraaf-Witten theory, the finite group version of Chern-Simons.

When things are not finite, like for Chern-Simons theory, something has to be done about regularizing the Leinster measure, as described in The return of the Euler characteristic of a category. When things are Lie, it seems like BV-formalism is what takes care of this.

At least you can see, with the above claim in mind, that this is how Kevin Costello describes the situation on his p. 16.

Posted at October 11, 2007 8:52 PM UTC

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Re: BV-Formalism, Part IV

What is called the “space of fields and ghosts” in BRST/BV formalism is nothing but the Lie algebroid of the action groupoid of the gauge group acting on the space of fields.

This sounds very beautiful!

The light is starting to dawn on me.

I guess I have the following confusion about all this differential graded algebra, Koszul duality stuff : it all looks very algebraic , but I want to do geometry !

If the “space of fields and ghosts” is just the Lie algebroid of the action of the gauge group on the fields (which I hope it is)… then howcome people do all this supersymmetry dg-algebra cochain complex homology/cohomology stuff?

Why can’t they just talk about curves and tangent vectors and derivatives and flat connections and things?

Is one point of view more powerful than the other?

Posted by: Bruce Bartlett on October 11, 2007 11:22 PM | Permalink | Reply to this

Re: BV-Formalism, Part IV

I find it very curious to rephrase BV in terms of algebroids and then look at the quasi-free graded comm dg alg - since BV express themselves precisely in language equivalent directly to that of quasi-free graded comm dg algs

As to ghosts, they are exactly generators of the Chevalley Eilenberg cochain complex. This is also true in the BFV (Hamiltonian) formalism where the antighosts are the generators of the Koszul complex for the constraint surface.

Back in BV, the anti fields are the Koszul genertors for the shell - thesurface defined by the EL eqns, The antighost then correspond to relations among the EL eqns, cf. Noether’s SECOND variational theorem.

As for the language of vector fields and manifolds, that’s very much the Russian point of view cf. NPQ-manifolds etc. As to which is more powerful, the manifold `geometric’ language seems to inspire them and perhaps suggest results, but then the hard work is usually done in the quasi-free graded comm dg alg.

Posted by: jim stasheff on October 12, 2007 12:18 AM | Permalink | Reply to this

Re: BV-Formalism, Part IV

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I find it very curious […]

Sorry, maybe I am just being slow. Please bear with me.

The point I wanted to highlight here is that the Chevalley-Eilenberg part of BV (the fields and the ghosts) can be thought of as coming from differentiating the action Lie groupoid of the gauge group acting on the space of fields.

Probably that was obvious to you, but I haven’t seen it stated anywhere. We all know that the Chevalley-Eilenberg algebra for a trivial module is the dual to the Lie algebra of the group itself. But here we need to know which Lie groupoid the Chevalley-Eilenberg algebra for the nontrivial module given by an algebra of functions comes from.

It’s not a big deal, neither, but I thought it would be worthwhile making it explicit here in our little series on BV.

the anti fields

This is the next item that I will try to find an “integrated” interpretation for.

See, I am hoping to extract a formulation of quantum gauge theory entirely in the world of groupoids such that

- restricting the groupoid language to finite groupoids yields the known quantization of “discrete” theories like Dijkgraaf-Witten

- specializing the groupoid language to Lie groupoids and then passing to their Lie algebroids yields the BV formalism and hence the “continuous” theories like Chern-Simons.

I didn’t get the impression that the answer to this is really known. But maybe I am just ignorant.

As for the language of vector fields and manifolds, that’s very much the Russian point of view cf. NPQ-manifolds etc.

Maybe that’s what Bruce was asking about. But maybe he had something else in mind? I will tell you about one problem I have with this point of view. Maybe you can set me straight.

So, as you say, many people are quite fond of the fact that Lie $n$ -algebroids (maybe they wouldn’t call them this) are nothing but dg-manifolds, and that carrying over all of differential geometry to the dg world is a fantastically useful step.

So, in particular, one big insight underlying BV is that carrying the notion of symplectic structures to the dg-world works wonders.

But that makes me wonder: namely – and that’s the kind of thinking my above entry adhered to – we know that what is conveniently handled as a dg-manifold is conceptually to be thought of as something coming from the world of Lie groups.

Therefore I am wondering: if a dg-manifold is really a Lie $n$ -algebroid, then a symplectic structure on a dg-manifold is really what kind of structure on Lie $n$ -algebroids?

It must dually correspond to some structure that makes sense in the world of Lie $n$ -groupoids. And if it is as fundamentally important as it seems to be, then it must be some fundamental aspect of Lie $n$ -groupoid theory.

And this still needs to be identified. I think.

As you know, I made a suggestion for what it should be: in Categorified Clifford algebra and weak Lie $n$ -algebras I argue, based on Dmitry’s Roytenberg’s interesting observation, that the dg-symplectic structure is Kuszul-dually somehow related to having weak skew-symmetry in the Lie $n$ -algebroid. This in turn should translate into Lie $n$ -groupoids with weak inverses.

I am not sure if I am managing here to communicate the issue which I believe needs to be dealt with eventually. Bruce Bartlett has in the past a couple of times helped out in situations like that and provided a translation of what I tried to say. With a little luck he might do so again :-)

In summary, I very much appreciate all the nice homological algebra structure one finds in BV-formalism, which you nicely highlight in The (secret?) homological algebra of the Batalin-Vilkovisky approach. But what I would really like to understand is the answer to the question with which you end that paper:

but why?

Posted by: Urs Schreiber on October 12, 2007 9:09 AM | Permalink | Reply to this

Re: BV-Formalism, Part IV

but why?

that was meant meta-mathematically or even philosophically

MC equation was originally for diff forms on Lie groups
but it occurs in contexts
e.g. deformation theory
where there is no group in sight

Posted by: jim stasheff on October 12, 2007 5:11 PM | Permalink | Reply to this

Re: BV-Formalism, Part IV

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the following confusion

This is precisely the issue which I am hoping to address.

The thing is this: while we are busy trying to demonstrate that quantum physics is a theory internal to $n$ -categories (parameter spaces are really parameter $n$ -categories, target spaces are really target $n$ -categories, background fields are really transport $n$ -functors, and so on and so forth), whole subfields of physics are already using this fact secretly.

And the key to open this secret door is, apparently, Koszul duality: it tells us that

“geometric” $n$ -groupoid structures

may look like

differential algebra

from a certain perspective.

And BV-formalism is the currently most highly developed description of (quantum) physics in terms of differential algebra. Homological physics as Jim Stasheff calls it.

The main advantage of the differential algebra perspective on (Lie) $n$ -groupoids is: it is an enormously powerful computational tool. When compared with handling $n$ -groupoids directly.

I am trying to get this point across in section “Plan”, subsection “The bridge between Lie $n$ -groupoids and differential graded algebra”.

On slide 121 I am saying how I am thinking about this:

$\array{ Lie n-groupoids & \stackrel{the bridge}{\leftrightarrow} & differential algebra \\ \\ conceptual understanding && computational accessibility \\ \\ What is going on? && How does it work? \\ \\ diagrammatics && implementation \\ arrow theory }$

And I believe that there remains work to be done with carrying things back and forth over this bridge:

BV formalism is strong on the right side: It answers “How does it work?” very well for large parts of quantum theory. But do we already understand “What is going on?” Only in parts, it seems.

I will feel I really understand BV-formalism when this happen:

a) we have an arrow-theoretic description (internalizable in the world of $n$ -groupoids) of what it means to quantize a gauge theory. I started describing how this should look like here and here. (And I hope it is clear that this is supposed to harmonize with all your ideas. Even though John once complained about my $\mathrm{par}$ - $\mathrm{tar}$ - $\mathrm{tra}$ - $\mathrm{conf}$ -diagrammatics, I think it captures precisely what we are all trying to describe, as I try to emphasize from time to time – last time here in the discussion of Jeffrey Morton’s work)

b) we understand how by internalizing this arrow theory in various context, we reproduce various known structures

b i) by internalizing it in finite $n$ -groupoids we obtain the description of “discrete” theories like Dijkgraaf-Witten (what you call “finite group theories”)

b ii) by internalizing it in Lie $n$ -groupoids and passing to their Lie $n$ -algebroids we obtain BV formalism. (I tried to emphasize the similarity between the structure used in AKSZ-BV formalism with the fundamental diagrammatics of the charged $n$ -particle in On BV-Quantization, Part I).

So I think the answer to your question

Why can’t they just talk about […]

is:

because somebody needs to carry them over the bridge first.

You’d be the right person to help here. You do have the perpsective. When we meet in Sheffield in two weeks I’ll show you the ropes of bridge-passing.

Posted by: Urs Schreiber on October 12, 2007 9:49 AM | Permalink | Reply to this

Re: BV-Formalism, Part IV

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Ok thanks for this explanation Urs. It seems like a great circle of ideas you are involved in here. I guess I echo the sentiments of all at the n-category cafe : we all have to work fast before the conventional guys realize what’s going on!

I have always had a poetic attraction to the idea of “ghost fields”, so I give a thumbs-up to any formalism which involves them.

I’m counting on you explaining lots of stuff to me at the conference, but I will also prepare by going through your notes beforehand.

By the way, some of the sections of your notes appear to be defunct, eg. clicking on “String-like central extensions” on the left hand pane in Firefox just gives the title page.

Posted by: Bruce Bartlett on October 12, 2007 5:36 PM | Permalink | Reply to this

Re: BV-Formalism, Part IV

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but I will also prepare by going through your notes beforehand.

Okay, good. All comments, questions, criticisms etc. is highly welcome.

I must say though, that these slides still are not really finished. I keep doing two things to them: a) provide more detail and b) give better expositions as far as possible. Just a minute ago I have added a new subsection on Characteristic classes of $n$ -bundles.

But I also need to complete the discussion of mere $n$ -transport further.

In the absence of our Wiki I am currently using these slides as my Wiki-substitute.

By the way, some of the sections of your notes appear to be defunct, eg. clicking on “String-like central extensions” on the left hand pane in Firefox just gives the title page.

Oh, thanks for letting me know. Yes, I find the same problem, now that I check. But it’s strange: the same links embedded in the document do work. So if you first follow “String and Chern-Simons $n$ -Transport” and then choose “String-like extensions” from the submenu appearing, you do obtain the right slide.

So this must be some funny misbehaviour of the software somewhere. Hm…

Posted by: Urs Schreiber on October 12, 2007 5:50 PM | Permalink | Reply to this

Read the post On Lie N-tegration and Rational Homotopy Theory
Weblog: The n-Category Café
Excerpt: On the general ideal of integrating Lie n-algebras in the context of rational homotopy theory, and about Sullivan's old article on this issue in particular.
Tracked: October 20, 2007 4:33 PM

Re: BV-Formalism, Part IV

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Two quick facts which should help think about what BV is really saying. (I profited from helpful discussion with Dmitry Roytenberg.)

The “configuration space of fields” $V$ that we are talking about is that which we encounter in “relativistic” or “constrained” systems: it’s really the space of trajectories rather than that of configurations on time slices.

That’s important for understanding why the (master) action $S$ is a function on $V$ .

Usually in physics we form the cotangent space $T^* V$ of the config space. Here we replace the config space by the action Lie algebroid $V \otimes g[1]$ (fields and qhosts).

We should really be thinking of the shifted cotangent bundle.

Accordingly, we hence form the shifted cotangent bundle of that action Lie algebroid, $T^*[-1](V \otimes g[1]) \,.$

And that’s it.fields and ghosts and antifields and antighost.

Gotta go to bed.

Posted by: Urs Schreiber on October 28, 2007 1:36 AM | Permalink | Reply to this

Read the post BV for Dummies (Part V)
Weblog: The n-Category Café
Excerpt: Some elements of BV formalism, or rather of the Koszul-Tate-Chevalley-Eilenberg resolution, in a simple setup with ideosyncratic remarks on higher vector spaces.
Tracked: October 30, 2007 10:26 PM

Read the post On Noether's Second (BV, Part VI)
Weblog: The n-Category Café
Excerpt: On Noether's second theorem and ghost/antighost pairing.
Tracked: November 1, 2007 12:39 AM

Read the post Something like Lie-Rinehart infinity-pairs and the BV-complex (BV, part VII)
Weblog: The n-Category Café
Excerpt: Notes on something like Lie infty-algebroids in the light of the BV complex.
Tracked: November 20, 2007 8:13 PM

Read the post On BV Quantization, Part VIII
Weblog: The n-Category Café
Excerpt: Towards understading BV by computing the charged n-particle internal to Z-categories, secretly following AKSZ.
Tracked: November 29, 2007 10:23 PM

Read the post Frobenius algebras and the BV formalism
Weblog: The n-Category Café
Excerpt: Bruce Bartlett is looking at the latest article by Cattaneo and Mnev on BV-quantization of Chern-Simons theory.
Tracked: November 14, 2008 1:28 PM

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October 11, 2007