On BV Quantization. Part I.

August 17, 2007

Posted by Urs Schreiber

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Of course, one of the questions I heard after my talk on the right structure 3-group of Chern-Simons theory here at the ESI workshop Poisson-Sigma Models, Lie Algebroids, Deformations and Higher Analogues in Vienna was

Why would we need to realize Chern-Simons theory as a 3-functor?

The true answer is

Because that’s how the dao works. It simply is the way it will turn out to be and we shouldn’t fight it.

But I am not supposed to argue like that. I am supposed to give answers that point out that it is useful to do this. That it helps to solve problems which otherwise cannot be solved.

Okay, sure. If it follows the dao, it is bound to help us solve problems. So here we go: I point out that we have a pretty good idea how to better understand 2-dimensional CFT starting from 3-functorial CFT, which crucially depends on that 3-functoriality. And in order to understand that still better, it would be helpful to understand how the Chern-Simons 3-functor comes to us from first principles.

I could go on about that point, but that’s not what my topic shall be right at the moment. Rather, there is yet another good reply:

Since (so I claim), $n$ -dimensional QFT is best understood as an $n$ -functor, it follows that if you already have a better than average understanding of $n$ -dimensional QFT, then chances are that you are already secretly using $n$ -functors yourself. Without noticing so. And in that case, it would immensely boost our overall understanding if we’d made that explicit. Since it will add to your computational prowess the glory of conceptual understanding.

That’s actually why I am here at this workshop in the first place: a priori I am not all that interested in $L_\infty$ -algebras, dG-manifolds, Courant algebroids and all the other highly involved algebraic structures that people here like to fill their blackboards with. These structures all look rather involved and somewhat messy. I wouldn’t really want to care about them – unless I knew that all this is really the shadow of very sensible structures: $L_\infty$ -algebras are just Lie $n$ -algebras, dG-manifolds are just a funny way to talk about higher Lie algebroids, Courant algebroids are just certain Lie 2-algebroids coming from something like sections of an Atiyah-2-bundle. And so on.

Only with this interpretation in mind does all this here make sense to me, and I’d dare to claim that, while the differential algebraic realization is very helpful for efficient computations, only with these interpretations do we know what the right things to do with these gadgets are.

Okay. Now there is one big topic here which is probably the most general, most powerful and most fascinating of all of these. And it is the one where I don’t yet quite know the “true” interpretation of, in the above sense:

Batalin-Vilkovisky formalism, also known as BV-Quantization.

Or, at least I didn’t. Until I heard Jae-Suk Park and Dmitry Roytenberg give talks on this.

Now I think I am beginning to see on the horizon that, possibly, BV-Quantization in fact is secretly precisely about what Isham called Quantization on a category, or rather on an $n$ -category, and what I keep referring to as the program of studying the the charged quantum $n$ -particle.

The key is once again to keep in mind the equivalence $\mathrm{qDGCA}s \stackrel{\sim}{\to} L_{\infty} \stackrel{\sim}{\to} \omega\mathrm{Lie}$ between (quasi-free) differential graded algebras, $L_{\infty}$ -algebras and Lie $n$ -algebras for arbitrary $n$ , and to keep in mind that Lie $n$ -algebras are the differential version of Lie $n$ -group – or in fact that Lie $n$ -algebroids are the differential version of Lie $n$ -groupoids – and to use that in order to translate all that differential algebra (and BV quantization is graded differential algebra taken to the extreme) back into something of manifest intrinsic meaning.

Of course I am far from fully understanding BV-quantization, let alone that translation which i would like to perform. But that won’t stop me from talking about it.

Here I start, instead of beginning to comprehensively describe BV-formalism itself (but Jim Stasheff just writes in to tell me to read

Jim Stasheff
The (secret?) homological algebra of the Batalin-Vilkovisky approach
hep-th/9712157

), by pointing out that there are indications that people in BV-theory have already, without admitting it, realized Chern-Simons theory as a 3-functor. And it seems they have even found, implicitly, just that Lie 3-algebra which I am claiming is the right Lie 3-algebra of Chern-Simons theory.

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My main source currently is the very insightful review

Dmitry Roytenberg
AKSZ-BV formalism and Courant algebroid topological field theories
hep-th/0608150

which is in based in particular on

M. Alexandrov, M. Kontsevich, A. Schwarz, O. Zaboronsky
The Geometry of the Master Equation and Topological Quantum Field Theory
hep-th/9502010 .

All this takes place in the world of “differential graded manifolds”. And we should admit that this is really nothing but defined to be the dual to the world of differential graded algebras.

(I cannot force you into thinking one way or another, but myself, I will think of dG manifolds as essentially being a way of talking about Lie $n$ -algebroids, hence about Lie $n$ -groupoids. At least “at the end of the day”, whenever that may be.)

The general story of quantization – in dG language

Now, as Dmitry Roytenberg discusses on p 5, we want to fix two duch differential graded manifolds. He calls them “source” and “target”. But I will, to emphasize the correspondence with the $n$ -particle , instead call them

$\mathrm{par}$ for “parameter space”

and $\mathrm{tar}$ for “target space”.

Moreover, Dmitry Roytenberg writes $P$ for the space (the “internal hom”, actually) of maps from the first to the second. Following my $n$ -particle conventions, I will instead call this $\mathrm{conf} := \mathrm{hom}(\mathrm{par},\mathrm{tar}) \,,$ the configuration space of maps (“fields”!) from parameter space to target space.

We imagine $\mathrm{par}$ to be a model for the $n$ -particle (a particle, a string, a membrane…), which propagates in target space $\mathrm{tar}$ . From the point of view of $\mathrm{par}$ an object in $\mathrm{conf}$ is a “field configuration” on $\mathrm{par}$ . From the point of view of $\mathrm{tar}$ an object of $\mathrm{conf}$ is a “position configuration” of $\mathrm{par}$ in $\mathrm{tar}$ .

(By the magic of the $n$ -functorial description of $n$ -bundles with connection, target space may be rather $n$ -orbifold-like, in which case a “field configuration” on $\mathrm{par}$ will be nothing but a choice of $n$ -bundle with connection on $\mathrm{par}$ . This will be important for understanding how Chern-Simons theory fits into this picture here: it is like the membrane propagating on a 3-orbifold. But I’ll get to that later.)

Whenever we have a situation as just described, we have in fact also this situation: $\array{ &&&& \mathrm{conf} \\ &&& \nearrow \\ \mathrm{tar} & \leftarrow & \mathrm{conf} \times \mathrm{par} & \\ &&& \searrow \\ &&&& \mathrm{par} } \,.$

On top of this kinematical setup, there is now a bit of dynamics. That’s some structure on $\mathrm{tar}$ , possibly also some structure on $\mathrm{par}$ (but the inclined reader might remember that there are supposed to be arguments that when done really right, the latter isn’t actually extra structure).

In the context of the charged $n$ -particle, that extra piece of dynamical data is an $n$ -functor $\mathrm{tra} : \mathrm{tar} \to D$ encoding the “background field” (that’s a different kind of “field”, different from the objects of $\mathrm{conf}$ . The space $\mathrm{conf}$ conbtains the fields of the theory, while $\mathrm{tra}$ will become a field of the second quantized theory. But that’s another story…)

Here, in the BV context, this extra dynamical structure is – not too surprisingly for the cognoscenti – a symplectiv 2-form (but in the dG context!) $\Theta \,.$

(In section 4 of his article Dmitry Roytenberg spells out how this symplectic 2-form does indeed encode background $n$ -bundles with connection.)

Whichever way we think of the extra structure present on $\mathrm{tra}$ , the star-shaped diagram above makes us want to pull-push that structure through to either $\mathrm{par}$ or to $\mathrm{conf}$ . The story of that journey is the drama of the charged $n$ -particle.

Dmitry Roytenberg discusses this on p. 9 of his article. He wants to get an action functional on the space of all field configurations, so he pulls back $\Theta$ to $\mathrm{conf} \times \mathrm{par}$ and then pushes forward to $\mathrm{conf}$ by integrating over $\mathrm{par}$ .

There is a measure on $\mathrm{par}$ needed to do that: the push-forward is then simply the integration, with respect to that measure, over the fibers of $\mathrm{conf} \times \mathrm{par} \to \mathrm{par}$ .

(And I cannot refrain from saing it once again: there are indications that when everything is done following the dao, there is actually a measure arising here canonically all by itself.)

Focusing on a special case: Chern-Simons

The joy of this formalism (I haven’t even mentioned most of the further ingredients that go into this. See Dmitry’s review!!) now is its generality. Choosing different dG manifolds $\mathrm{par}$ and $\mathrm{tar}$ and different dG-symplectic 2-forms $\Theta$ , feeding these into the machinery and turning the crank, out drop lots of classical BV-action functionals which are held in high esteem.

Among them, the Chern-Simons functional.

And that’s an interesting example to have a closer look at.

Recall, Jim Stasheff and I are claiming that for any Chern-Simons theory in $(2n+1)$ -dimensions, coming from a Lie algebra $g$ with transgressive invariant polynomial $k$ on it, there is a Lie $(2n+1)$ -algebra $\mathrm{cs}_k(g)$ such that the Chern-Simons functional (at least for trivial $G$ -bundles) is a $(2n+1)$ -connection with values in this Lie $(2n+1)$ -algebra.

Moreover, we are claiming that this Lie $(2n+1)$ -algebra is actually isomorphic (not just equivalent, really ismorphic, that’s not unimportant here) to the inner derivation Lie $(2n+1)$ -algebra of the Baez-Crans-type Lie $2n-algebra$ which comes from the corresponding $(2n+1)$ -cocycle: $\mathrm{cs}_k(g) \simeq \mathrm{inn}( g_{\mu_k}) \,.$

To drive home this point: this means that in the context of $n$ -functorial QFT, on the differential level, Chern-Simons theory should be that QFT obtained by setting $\mathrm{tar} = \mathrm{cs}_k(g) \simeq \mathrm{inn}(g_{\mu_k}) \,.$ (I am slightly oversimplifying now, to get the message across. Really we’d want to pass to a Lie $(2n+1)$ -group integrating that Lie $(2n+1)$ -algebra and we might want to account for the presence of nontrivial $G$ -bundles and things like that. But at the moment this is just distraction and will be ignored.)

Okay, so let’s see what Dmitry Roytenberg, in his review of the work by M. Alexandrov, M. Kontsevich, A. Schwarz and O. Zaboronsky, says we should use for $\mathrm{tar}$ :

look at his “example 4.1” and remember to interpret that using the prescription on the bottom of p. 10.

Then notice that the Courant algebroid over a point mentined in example 4.1 is precisely the Baez-Crans Lie 2-algebra $g_{\mu = \langle \cdot, [\cdot, \cdot]\rangle}$ which we like to call the skeletal version of the String Lie 2-algebra. But think of it as the corresponding Koszul dual differential graded algebra, hence as a dG-manifold (whose even part is just a point).

Now go back to the bottom of p. 10, to see what the target space is supposed to be: it’s supposed to be essentially the shifted cotangent bundle $\Pi T^* g_\mu$

of this dG-manifold.

Maybe open the AKSZ paper at p. 7 to learn more about this.

Then notice the following:

Claim. As a free graded commutative algebra, $\Pi T^* g_\mu$ is precisely that free garded commutative algebra underlying $\mathrm{inn}(g_\mu)$ .

Proof: Just unwrap the definitions.

AKSZ are a little coy about defining the differential structure on $\Pi T^* g_\mu \,.$ They mention (p. 7 still) that there is one. And seem to assume that it is obvious. In fact, I think, too, that it is obvious. Hopefully we are agreeing with our feelings here, because I am thinking that the only obvious differential structure on $\Pi T^* g_\mu$ is precisely that on $\mathrm{inn}(g_\mu)$ .

So it seems I am saying:

Claim. When translated to the world of Lie $n$ -algebras, the claim of AKSZ-Roytenberg is that the right target for Chern-Simons theory is the Lie 3-algebra $\mathrm{inn}(g_\mu) \,.$

Which coincides, up to isomorphism, with our statement about the 3-functorial realization of Chern-Simons.

Disclaimer

I am not claiming to fully understand all the BV details.While all what I said above looks quite right to me, I might be wrong somewhere. I will need to talk to a couple of people about this.

Posted at August 17, 2007 8:03 PM UTC

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Read the post On BV Quantization, Part II
Weblog: The n-Category Café
Excerpt: A review of elements of Batalin-Vilkovisky-formalism with an eye towards my claim that this describes configuration spaces which are Lie n-algebroids.
Tracked: August 18, 2007 4:33 PM

Re: On BV Quantization. Part I.

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I’ll need to refine my claim above, on how the AKSZ target space for Chern-Simons is $\mathrm{inn}(g_{\mu})^*$ . Now that I had a more careful look at what Roytenberg writes, I realize that it is a little more indirect than that: the information encoded in $g_{\mu}$ is in this formalism in some way split into a “kinetic” and an “interaction” part. More on that later.

Posted by: Urs Schreiber on August 19, 2007 12:31 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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I’ll need to refine my claim above, on how the AKSZ target space for Chern-Simons is $\mathrm{inn}(g_\mu)^*$ . Now that I had a more careful look at what Roytenberg writes, I realize that it is a little more indirect than that: the information encoded in $g_\mu$ is in this formalism in some way split into a “kinetic” and an “interaction” part. More on that later.

Actually, it is better than I originally thought.

Notice that a connection with values in just $\mathrm{inn}(g_\mu)$ comes from a 1-form $A$ and a 3-form constrained to be $\langle A \wedge A \wedge A\rangle$ up to an exact term.

Now, at the level of Lie 3-algebras this $\mathrm{inn}(g_\mu)$ happens to be isomorphic to the Chern-Simons Lie 3-algebra $\mathrm{cs}(g)$ , a connection with values in which would come from the 3-form $\langle A \wedge d A\rangle + c \langle A \wedge A \wedge A\rangle \,.$ So I was hoping that in the end the 3-group which integrates $\mathrm{inn}(g_\mu)$ would be the right structure 3-group of CS-theory, with that isomorphism somehow taking care of the fact that a 3-connection with values in just $\mathrm{inn}(g_\mu)$ is not quite the full CS action.

But, remarkably, we learn from the AKSZ-formulation of Chern-Simons theory that, indeed, the remaining part arises as the kinetic part of the action. This is that part which should in fact not come from the coupling to the 3-group gauge background.

There is, apparently, a nice story waiting to be fully unraveled here.

Posted by: Urs Schreiber on August 21, 2007 1:44 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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I wrote:

Actually, it is better than I originally thought.

But then, one should pay attention to the fact that there seem to be two different ways to perceive Chern-Simons theory, and they differ precisly in the interpretation of this term $\langle A \wedge d A\rangle$ :

A) if we consider Chern-Simons theory as a gauge theory on 3-manifolds by itself, it does make sense to split the Chern-Simons form into its “kinetic part” $\langle A \wedge d A \rangle$ and its “interaction part $\langle A \wedge A \wedge A \rangle \,.$

B) On the other hand, we can also think of Chern-Simons as a theory of maps from 3-manifolds inot a fixed target with a given $G$ -bundle on it. The action then is the pullback of the Chern-Simons form along this map. From this point of view the above splitting is not natural.

In my talk I had emphasized motivation by the second point of view, since that nicely fits into the the sequence

$n=1$ : electromagnetic 1-form coupling

$n=2$ Kalb-Ramond 2-form coupling

$n=3$ supergravity 3-form coupling

But that’s actually different from the point of view from which one should look at AKSZs stuff.

Hm…

Posted by: Urs Schreiber on August 21, 2007 5:19 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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n-dimensional QFT is best understood as an n-functor

Can you give us a sense of how much ‘freedom’ there is to devise a QFT from this viewpoint? Like, how special is Chern-Simons for $n = 3$ ? Are $n = 3$ and $n = 4$ especially interesting dimensions?

Posted by: David Corfield on August 19, 2007 12:42 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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how special is Chern-Simons for $n=3$

I am imagining that it is not entirely a coincidence that the numbers of dimensions which are both tractable and interesting coincide for QFT and $n$ -category theory:

We are all happy with 1-categories and with quantum mechanics.

Things become quite interesting as we move to 2-categories and 2D QFT. People are able to work out plenty of details here and find lots of interesting structure.

Since 2-categories form a (weak) 3-category, and since 2D QFT arises on the boundary of 3D QFT, this succes already builds a bridge into the 3-dimensional world.

But in fact, this “holographic” description of three dimensional phenomena by two-dimensional structures already pretty much exhausts the easily tractable aspects of the 3d setup: on the category side of life we tend to fall back to Gray categories, on the QFT side of life we have just recently seen Witten argue that the best bet is to define the 3D QFT by means of its boundary 2D QFT.

(But of course on both sides we can break the dimensional barrier and go all the way up to infinity by making suitably simplifying assumptions.)

Now, compare this with my argument that the 3-categorical target for Chern-Simons theory is locally $\Sigma \mathrm{INN}(\mathrm{String}_k(G)) \,.$ While this may look involved, due to the stupid notation I am using, it essentially just amounts to saying that we use the generic tractable 3-target which we can think of:

in as far as Lie 2-algebras are classified by a Lie algebra and a 3-cocycle for that, $\mathrm{String}_k(G)$ , integrating that, is more or less, in a way, the most general Lie 2-group which we can come up with. And $\Sigma \mathrm{INN}(\cdot)$ of that is just a Gray groupoid of morphisms between that.

So I’d think that 3-dimensional Chern-Simons theory is special in that it sits right at some kind of “complexity barrier”.

Posted by: Urs Schreiber on August 19, 2007 1:12 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

So this term ‘tractable’ you’re using is to be taken as an intrinsic property of the theory, not just our current ability to work with the theory?

Posted by: David Corfield on August 19, 2007 1:37 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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So this term ‘tractable’ you’re using is to be taken as an intrinsic property of the theory, not just our current ability to work with the theory?

I might be completely wrong. Time will tell. But currently my impression is that there is no big principle which somehow singles out Chern-Simons among all other QFTs. It is rather a little like the harmonic oscillator: it happens to be simple enough that we can handle it (barely at the moment, one must say) and still already interesting enough to justify studying it a lot.

But I might be wrong. Maybe it turns out that CS theory is some “fixed point” of sorts in the space of all QFTs or so. But I don’t know.

Posted by: Urs Schreiber on August 20, 2007 11:55 AM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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Urs wrote:

So I’d think that 3-dimensional Chern-Simons theory is special in that it sits right at some kind of “complexity barrier”.

However, as we get better with $n$ -categories, we should be able to extend the algebraic approach to building quantum field theories to higher dimensions — at the very least for topological quantum field theories, but ideally not just those.

Indeed, this is why I got interested in $n$ -categories in the first place!

There are certainly lots of candidate TQFTs in higher dimensions. Above dimension 4, there should be enough to detect different smooth structures.

The big challenge lies in dimension 4. Win a Fields medal: create something like Donaldson theory or Seiberg–Witten theory, starting from suitable $n$ -categorical data!

(The physics jargon ‘topological’ is very misleading: TQFTs should really be called ‘diffeomorphism-invariant quantum field theories’, or something like that.)

Posted by: John Baez on August 19, 2007 1:42 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

Just in case the youngsters and/or physicists among us are not familiar with the original Chern-Simons, their approach works for any odd dim, not just n=3.

Two points:

1) the Chern-Simons form mediates the transgression between the invariant polynomial and the corresponding cocyle on the structure group, but the 2n+1-form is closed on a 2n+2-dim manifold/base space.

2) They introduced it because it captures some geometric data

Posted by: jim stasheff on August 19, 2007 2:58 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

A stupid question: what is a DG-manifold? I guess DG stands for “differential graded” and I also guess it is supposed to be some kind of an n-stack. Is just a name for an n-stack implying the fact various objects associated with ordinary manifolds (e.g. tangent bundle) become complexes? If so, why does it happen? I guess it is related to the n-categorical interpretation of the notion of a complex (I vaguely remember such an interpretation exists).

Posted by: Squark on August 24, 2007 12:13 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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what is a DG-manifold?

It’s really just the same as with supermanifolds. I think it’s used pretty much interchangeably.

So, locally it’s nothing but some differential graded-commutative algebra.

In some of the literature (for instance in AKSZ) in this context people talk about

N-manifolds

P-manifolds

Q-manifolds

and any combination of this, like NPQ.

Here “N” stands for “nom-negatively graded”. “P” standsd for Poisson, i.e. equipped with a suitably generalized Poisson structure. “Q” then denotes the existence of a nilpotent odd vector field.

So a dG manifold in this language is just a Q-manifold. If it is an NQ-manifold, trhis qualifies as a Lie $n$ -algebroid, since those just have non-negative degree. If on top of that it is P, we can start playing the BV game.

Posted by: Urs Schreiber on August 24, 2007 2:50 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

It is indeed `just’ dg-algebra, though often
they mean a dg-algebra model for an infinitesimal neighborhood of a point in a manifold, expressed in terms of what would be local coordiantes. They claim to get some geometric? intuition from this vocabulary; I guess it depends on how one has been potty trained.

Posted by: jim stasheff on August 24, 2007 3:19 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

A differential graded supercommutative algebra A should be roughly the same as a supermanifold X = Spec A acted upon by a certain supergroup G with some conditions on the action.

Let me describe G. The bosonic part of G is U(1). There is one fermionic generator theta with unity charge w.r.t. the U(1) and

{theta, theta} = 0

One can think of G as the group of “inhomogeneous symmetries” of a 1-dimensional unit charge fermionic representation of U(1).

The differentiation on A is the action of theta and the grading on A is the U(1) charge.

We need to impose an additional condition in order to ensure that functions with even U(1) charge are bosonic and those with odd fermionic. The condition is as follows. Any supermanifold has an automorphism sigma that acts as 1 on bosonic functions and as -1 on fermionic functions. We require that the element -1 of U(1) acts on X by sigma.

In principle, we need another condition to ensure the action of U(1) on the algebra is semisimple. I don’t know how to interpret it geometrically.

I have no idea what “N” means geometrically in this picture.

Also, how do Lie n-algebroids come into the picture?

Is there no relation to n-stacks, as I stipulated before? I remember hearing a lecture in which it was explained that the tangent bundle of a stack is a complex of length 2 and that DG-manifolds should yield longer complexes (it wasn’t explained what DG-manifolds are).

Posted by: Squark on August 24, 2007 3:57 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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Also, how do Lie $n$ -algebroids come into the picture?

A Lie $n$ -algebroid over a space $X$ is supposed to be given by a non-negatively graded dg manifold whose degree 0 part is $X$ and whose local algebras of functions have generators of degree $0 \leq d \leq n$ .

Setting $X = \{\mathrm{pt}\}$ this definition yields a free graded commutative algebra generated in degrees $1 \leq d \leq n$ with a nilpotent graded differential of degree 1 over it.

But this is equivalent to an $n$ -term $L_{\infty}$ -algebra. This, finally, is nothing but a Lie $n$ -algebra.

By this reasoning, an “NQ”-manifold over a space $X$ is a many object Lie $n$ -algebr, hence a Lie $n$ -algebroid.

But as I mentioned somewhere here, while a couple of people are comfortable with thinking of NQ manifolds as Lie $n$ -algebroids this way, but I am not aware of literature which expounds on this in the detail it deserves.

Posted by: Urs Schreiber on August 24, 2007 4:10 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

I think I realized what the “N” part does. If we work in the holomorphic or complex algebraic context, we should use C* instead of U(1). Then “N” amounts to replacing the group C* by the multiplicative _monoid_ of C.

“A Lie n-algebroid over a space X…”

I guess that a “Lie n-algebroid over a space X” means a Lie n-algebroid with object (0-morphism) space X.

“Setting X={pt} this definition yields a free graded commutative algebra…”

X = {pt} means that the 0 degree part is just the ground field, say C. Why does it mean the algebra has to be free?

“… This, finally, is nothing but a Lie n-algebra.”

So, for n = 1 we should get an ordinary Lie algebra. I don’t see how that works.

Posted by: Squark on August 24, 2007 8:30 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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I guess that a “Lie $n$ -algebroid over a space $X$ ” means a Lie $n$ -algebroid with object (0-morphism) space $X$ .

Yes!

$X = \{\mathrm{pt}\}$ means that the 0 degree part is just the ground field, say $C$ . Why does it mean the algebra has to be free?

That’s an additional condition. It is certainly there and well known for $n$ -algebroids over points, i.e. Lie $n$ -algebras. I think for the kind of dg-manifolds which we need to describe Lie $n$ -algebroids we need to require that their algebras in degree $\geq 0$ have to be free as graded-commutative algebras, locally, i.e. over sufficiently small patches. But until now I haven’t really found an article that gets to the point of saying entirely explicitly what dg structure exactly and in full detail we are to regard as a Lie $n$ -algebroid.

So, for $n = 1$ we should get an ordinary Lie algebra. I don’t see how that works.

Then do the following: pick any finite dimensional vector space $V$ , form $\wedge^\bullet ( s V^*) \,,$ where $s V$ denotes $V$ regarded as being in degree 1. Then try to define any degree 1 differential $d : \wedge^\bullet ( s V^*) \to \wedge^\bullet ( s V^*)$ on that, such that $d^2 = 0$ .

You’ll find that defining the differential gives you the (dual of) a Lie bracket, while the nilpotency condition then gives the Jacobi identity.

For lots of details about how this works in general see section 3 of

Structure of Lie $n$ -algebras

For a bunch of examples of Lie $n$ -algebras for low $n$ , see

Zoo of Lie $n$ -algebras.

Posted by: Urs Schreiber on August 24, 2007 11:22 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

That’s cool. If so, what is the interpretation of Spec A from the n-algebroid viewpoint? Here A is the whole superalgebra, not only the 0-degree part. Over a point, for n = 1 we get the Lie algebra itself (regarded as an affine space).

Posted by: Squark on August 25, 2007 10:49 AM | Permalink | Reply to this

Re: On BV Quantization. Part I.

“Over a point, for n = 1 we get the Lie algebra itself (regarded as an affine space).”

Sorry, we get the _parity reversed_ Lie algebra, a fermionic affine space.

Posted by: Squark on August 25, 2007 11:22 AM | Permalink | Reply to this

Re: On BV Quantization. Part I.

A Lie 1-algebroid over a point is a Lie algebra, or wasn’t that your question?
My solw `oidification’ was precisely the
transition from being anchored to a (honest to goodness ) manifold and it’s tangent bundle to the abstract cat or alg versions thereof.

Posted by: jim stasheff on August 25, 2007 12:45 AM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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$X = \{\mathrm{pt}\}$ means that the 0 degree part is just the ground field, say $\mathbb{C}$

This is a simple but important point, which deserves attention. It’s one of these things that first look too trivial to bother with, but then turn out to be essential for getting the big picture right.

First, a simple example:

we have seen that a Lie algebra is, dually, the same as a quasi-free differential graded-commutative algebra (qDGCA) whose underlying free graded commutative algebra $\wedge^\bullet (s g^*)$ is that generated from the dual of the underlying vector space itself, regarded as sitting in degree 1, while the degree +1 differential $d : \wedge^\bullet (s g^*) \to \wedge^\bullet (s g^*)$ encodes the (dual of) the Lie bracket.

Now, this means we have taken the degree 0 part of the generators to be the 0-vector space instead of being the ground field.

One might think this makes no essential difference. Namely, if we instead take the underlying algebra to be $\wedge^\bullet ( \mathbb{R} \oplus s g^*)$ with $\mathbb{R}$ in degree 0, the graded-Leibnitz property of the differential already fixes its action on this $\mathbb{R}$ -factor to be trivial.

Moreover, for any morphism of qDGCAs of this kind, it similarly makes no real difference whether we include the “algebra $\mathbb{R}$ of functions over the point” or not: in both cases morphisms of qDGCAs correspond precisely to morphisms of the Lie algebras they come from.

Hence it seems that taking the $\mathbb{R}$ into account or not doesn’t change anything. Often it is not taken into account.

But it does make a difference! While everything is the same for 0-morphisms (the qDGCAs themselves) and for 1-morphisms, there are no 2-morphisms in the first case, where we take the space of degree 0 generators to be 0-dimensional: the 2-morphism would come from a degree -1 derivation on the qDGCA, and all of those are trivial if there is nothing in lower degree.

So, if we are thinking of qDGCAs generated in degree 1 as Lie algebras (which they are) then this looks reasonable: we don’t expect 2-morphisms between morphisms of Lie algebras.

Or do we?

That’s where it gets interesting.

Compare with the corresponding Lie groups: these have just morphisms between them, no 2-morphisms.

Ordinarily.

But then, we may regard any group as a 1-object groupoid. On the level of 0- and 1-morphisms this does not change anything:

one-object groupoids and their morphisms (functors) are precisely in bijection with groups and group homomorphisms.

But functors between groupoids have transformations going between them, which morphisms of groups do not.

These transformations, if you think about it, come from component maps which are maps from the single object to the morphisms of the target.

So, a group $G$ lives in a 1-category, but the corresponding one-object groupoid $\Sigma G$ lives in a 2-category (which, by the way, is equivalent to the 1-category of groups and “outer morphisms”).

It’s only the presence of the single object which makes the higher morphisms come into existence.

Same here: if we regard our Lie algebra as a one-object Lie algebroid, by taking in its dual qDGCA the “space of functions over the single point” into account, then we suddenly find the existence of 2-morphisms between these gadgets.

Posted by: Urs Schreiber on August 27, 2007 12:27 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

Neither am I, being barely algebroidicized myself, but such dg manifolds versus Lie n-algebras is a well traveled street for several of us.

Posted by: jim stasheff on August 25, 2007 12:42 AM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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I have no idea what N means geometrically in this picture.

By the comparison with the case where the underlying space is the point, we see that degree $k$ for the dg manifold corresponds to $k$ -morphisms for the corresponding Lie $n$ -groupoid that would integrate the corresponding Lie $n$ -algebra.

Degree 0 then corresponds to the space of 0-morphisms, i.e. that of objects.

From this point of view it is clear why negative degree is excluded. But compare David Ben-Zvi’s remark here, also regarding your remark about stacks:

David Ben-Zvi points out that negative degree $-k$ of the dg-manifold somehow corresponds to stuff at level $k$ that has been quotiented out.

Posted by: Urs Schreiber on August 24, 2007 4:56 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

David Ben-Zvi points out that negative degree −k of the dg-manifold somehow corresponds to stuff at level k that has been quotiented out.

That’s fairly visible in the BFV (Batalin-Fradkin-Vilkovisky) construction for constrained Hamiltonian systems.
Ghosts are adjoined in positive degree 1 corresponding to a basis of the Lie algebra of symmetries and
antighosts are adjoined in degree -1
corresponding to a generating set of
of constraints.

The corresponding quasi-free dg algebra
combines the Chevalley-Eilenberg complex for the Lie alg acting on the original
functions on the phase space with the
Koszul complex for the ideal generated by the constraints.

Mathematical treatments are given in (and many ohter places):

@article{jds:bull,
author={ J.D. Stasheff},
title={Constrained {P}oisson algebras and strong homotopy representations},
journal={ Bull. Amer. Math. Soc. },
pages={287-290 },
year=1988}

@article{jim:hrcpa,
author={J.D. Stasheff},
title={Homological reduction of constrained {P}oisson algebras},
journal={ J. Diff. Geom.},
volume={45},
pages={ 221-240},
year=1997}

Posted by: jim stasheff on August 25, 2007 12:56 AM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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I am still hoping to get a better understanding of negatively graded dg-algebras from a perspective that better fits into my worldview.

Today I thought that maybe I can think of the non-positively graded parts as constituting an $n$ -module for the Lie $n$ -algebra in the positively graded part.

I can think of a Lie 1-algebroid as a Lie-Rinehart pair: a Lie algebra acting on an algebra of functions. In the dg-version the Lie algebra sits in degree +1, the functions it acts on in degree 0.

If we take this picture and think of categorification, we’d expect to see a “2-Lie-Rinehart pair”: a Lie 2-algebra acting on a two-space of functions.

That data would plausibly assemble itself into something concentrated in four different degrees: the Lie 2-algebra in degrees +1 and +2, the 2-algebra it acts on in degree 0 and -1.

In fact, a dg-algebra (with differential of degree +1 as usual) concentrated in degrees 0, -1, -2 is nothing but a cochain complex. So a 2- or 3-vector space. Combining that with a dg-thing in degree +1 and +2 coming from a Lie 2-algebra could describe the action of that Lie 2-algebra on that 2-vector space.

So, I am thinking, could it be that, as I can think of positively graded dg-algebras as being Lie versions of Lie $n$ -groups, arbitrarily graded dg-algebras can be thought of as encoding representations of Lie $n$ -groups on $n$ -vector spaces?

Posted by: Urs Schreiber on October 16, 2007 8:07 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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Squark wrote:

I remember hearing a lecture in which it was explained that the tangent bundle of a stack is a complex of length 2 and that DG-manifolds should yield longer complexes (it wasn’t explained what DG-manifolds are).

Hello! That’s nice. Alissa Crans and I took the liberty of calling a length 2 chain complex of vector spaces a ‘2-vector space’. This terminology could easily be considered pompous or deceptive, but we used it for a reason: chain complexes of length 2 are a nice way to categorify the concept of vector space, since:

$[length \; 2 \; chain complexes of vector spaces] \simeq$ $[categories internal to Vect]$

I can easily believe that the tangent spaces of a differentiable stack turn out to be 2-vector spaces of this sort, since I already know that every smooth category has tangent spaces of this sort. Alissa and I focussed on a special case: the tangent space at the identity of a ‘Lie 2-group’ is a ‘Lie 2-algebra’.

More generally,

$[length \; n \; chain complexes of vector spaces] \simeq$ $[strict n-categories internal to Vect]$

Now, what about DG-manifolds? Section 2 of this paper has a nice intro to ‘ $NQ$ -manifolds’, which I believe are almost the same as ‘differential graded manifolds’:

Dmitry Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids.

Via a clever Koszul duality trick, $NQ$ -manifolds of degree $n$ are a practical, pedestrian way of describing Lie $n$ -algebroids. If we knew what Lie $n$ -algebroids really were, these in turn would be a practical, pedestrian way of describing Lie $n$ -groupoids — which in turn would be a practical, pedestrian way of presenting differentiable ( $n$ -1)-stacks.

In case it’s not obvious: if we knew what all these different things were, they would all be different ways of categorifying the concept of manifold $n$ -1 times!

But, I say “would be” for various reasons. First of all, Lie $n$ -algebroids and Lie $n$ -groupoids seem to be well-understood only up to $n = 2$ . Second of all, there’s a strange fly in the ointment. When you try to integrate a Lie algebroid, it seems you don’t get a Lie groupoid! Instead, it seems you get a Lie 2-groupoid!

I’ve been learning a huge amount about this stuff here at the ESI. I’ve got a lot of catching up to do. It’s quite mind-boggling how far the categorification of differential geometry has progressed, mostly under various disguises — like ‘NQ-manifolds’.

I’ve especially profited from conversations with the ebullient Chenchang Zhu — a student of Alan Weinstein who has done a lot of work on integrating Lie algebroids to get Lie 2-groupoids. I think I’m beginning to see the reason we don’t get Lie groupoids, and maybe even the solution.

Posted by: John Baez on August 24, 2007 5:00 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

Hey!

“I can easily believe that the tangent spaces of a differentiable stack turn out to be 2-vector spaces of this sort, since I already know that every smooth category has tangent spaces of this sort.”

Given a certain geometrical category C (e.g. smooth manifolds, complex analytic manifolds, algebraic varieties) groupoids internal to C and “C-stacks” appear to me rather similar. Both are supposed to be some kind of a hybrid between objects of C and groupoids.

Do they yield equivalent 2-categories? Or is there some crucial difference between the two?

Posted by: Squark on August 25, 2007 8:27 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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groupoids internal to $C$ and “ $C$ -stacks” appear to me rather similar

I had a long discussion with Bruce Bartlett a while ago about this question, or at least about something very closely related:

for $S$ any site, what precisely is the relation between

- categories internal to sheaves over $S$

- stacks on $S$

Mike Shulman finally released us from our ignorance by offering this excellent useful comment.

Posted by: Urs Schreiber on August 25, 2007 8:35 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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Squark wrote:

Given a certain geometrical category $C$ (e.g. smooth manifolds, complex analytic manifolds, algebraic varieties) groupoids internal to $C$ and “ $C$ -stacks” appear to me rather similar. Both are supposed to be some kind of a hybrid between objects of $C$ and groupoids.

Right!

Do they yield equivalent 2-categories? Or is there some crucial difference between the two?

As far as I can tell, the morphisms between $C$ -stacks are more flexible than the morphisms between groupoids internal to $C$ . Say $C$ is the category of smooth manifolds, so we have Lie groupoids versus differentiable stacks.

A morphism $f: X \to Y$ between Lie groupoids involves a smooth map from the manifold of objects of $X$ to the manifold of objects of $Y$ . If we instead worked with smooth stacks, we could also get morphisms that looked locally of this form, but not globally. And, this extra flexibility is very useful.

I believe this difference is why people talk about the differentiable stack ‘presented by’ a Lie groupoid.

This difference is also why Toby Bartels invented ‘smooth anafunctors’.

I also think this difference has something to do with the scary term ‘Hilsum–Skandalis map’ — but don’t ask me what that term actually means. Personally I think it would be skandalous to use such an uninformative term for any concept of real importance.

Posted by: John Baez on August 27, 2007 11:06 AM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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As far as I understand, a stack presented by a Lie groupoid $G$ is precisely the stack of principal $G$ -bundles, also knowns as $G$ -torsors.

Hence the stack is like a category of modules (acted-on things) and whenever we have a morphism between categories of modules we say we have a Morita morphism between the things that act.

Here these Morita morphisms are called either Morita, or Hilsum-Skandalis, or anafunctors, or…

At least roughly. Somebody should figure out all the technical details and clean this issue up once and for all. It keeps coming up again and again. Last time here on secret blogging seminar.

Posted by: Urs Schreiber on August 27, 2007 11:55 AM | Permalink | Reply to this

Re: On BV Quantization. Part I.

“As far as I understand, a stack presented by a Lie groupoid G is precisely the stack of principal G-bundles, also knowns as G-torsors.

Hence the stack is like a category of modules (acted-on things) and whenever we have a morphism between categories of modules we say we have a Morita morphism between the things that act.”

Can you give an example of two non-equivalent Lie groupoids G1 and G2 with equivalent corresponding stacks i.e. the category of principal G1-bundles is (functorially) equivalent to the category of principal G2-bundles over any base X? Or is it that we have “more morphisms” but “the same amount” of isomorphisms?

Also, does any stack correspond to some groupoid?

Posted by: Squark on August 27, 2007 11:10 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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The main `problem’ with thinking about differentiable stacks in terms of smooth groupoids is that smooth functors that have no pseudo-inverse become equivalences on passing to stacks. Likewise, there are equivalences of stacks which do not come from a smooth functor. My personal favourite way of handling this is via a (bi)category of fractions - formally invert the smooth functors which are fully faithful and essentially surjective (these concepts can be defined internally) - see Pronk’s 1996 paper. Then morphisms in the localised category are spans

$X \stackrel{\sim}{\leftarrow} Z \to Y$

with the left functor fully faithful and essentially surjective. This morphism is an isomorphism precisely when the right functor is also fully faithful and essentially surjective, and the inverse to such a span $(f,g)$ is $(g,f)$ . This presentation of a generalised morphism is not unique - technically it is an equivalence class of spans, but we just pick a representative. Think if you like of Segal’s work on classifying spaces of topological categories.

The point is that if $G_1 \to G_2$ is fully faithful and ess. surj., the respectve categories of principal bundles over all bases $X$ are equivalent. Thus if a pair of smooth groupoids are connected by span they can have the same categories of principal bundles, with no smooth functor between them at all! (note that there will be a functor on underlying groupoids internal to $Set$ if you don’t mind applying the axiom of choice).

Also, does any stack correspond to some groupoid?

As long as `any stack’ means Artin, Deligne-Mumford, differentiable or topological stack, in approximately decreasing order of rigidity. Essentially these types of stacks are defined to be ones equipped with an appropriately fibration-like map from a representable stack, represented by a scheme, manifold or topological space, and the weak kernel-pair of that map gives rise to a representable stack and the two spaces representing these two stacks form the objects and arrows of an internal groupoid.

Consider if you will schemes: they are ringed spaces that locally look like $\mathrm{Spec}(R)$ for some ring $R$ . Or manifolds: topological spaces that look locally like $\mathbf{R}^n$ . These two examples give rise to internal groupoids, via intersections of coordinate patches. No-one expects this way of thinking about manifolds or schemes to be unique, and likewise with stacks.

Posted by: David Roberts on August 28, 2007 6:12 AM | Permalink | Reply to this

Re: On BV Quantization. Part I.

Thanks, David R., for supplying clear and precise information on stacks versus internal groupoids!!!

Posted by: John Baez on August 28, 2007 9:08 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

“The main problem with thinking about differentiable stacks in terms of smooth groupoids is that smooth functors that have no pseudo-inverse become equivalences on passing to stacks.”

Can you given an example of such a functor?

“Essentially these types of stacks are defined to be ones equipped with an appropriately fibration-like map from a representable stack, represented by a scheme, manifold or topological space, and the weak kernel-pair of that map gives rise to a representable stack and the two spaces representing these two stacks form the objects and arrows of an internal groupoid.”

What’s a “weak kernel-pair”?

Posted by: Squark on September 6, 2007 4:14 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

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If $X$ is a space on which a topological group $G$ acts freely, then there is a functor $Q$ from the action groupoid $X//G$ of $G$ on $X$ to the category defined by the space $X/G$ . If the quotient map does not admit a section (say $X$ is the total space of a nontrivial prinicpal $G$ -bundle), then there is no pseudoinverse to $Q$ . I know, (but haven’t proved it for myself) that the stacks associated to $X//G$ and $X/G$ are equivalent.

The weak kernel pair of a functor $F:X\to Y$ is the weak pullback of $F$ along itself.

Posted by: David Roberts on September 7, 2007 3:43 AM | Permalink | Reply to this

Re: On BV Quantization. Part I.

“Also, does any stack correspond to some groupoid?”

Now I realize that the answer is “no”. The space of objects of a differentiable stack might not be a differentiable manifold. Take for example the quotient of R by x |-> -x. It is a stack but the ray [0, infinity) is not a manifold: it’s a manifold with boundary. If we allow quotients by positive-dimensional groups, things become even worse. Consider, for instance, the quotient of R^n by R*. It contains RP^n-1 and an additional “point with automorphisms” corresponding to the 0 vector in R^n. The additional point is clearly not disconnected from the rest of the space. This is not a manifold of any kind. Maybe the answer would be “yes” in the context of topological spaces?

Posted by: Squark on August 28, 2007 8:23 AM | Permalink | Reply to this

Re: On BV Quantization. Part I.

“Now I realize that the answer is no”

I changed my mind. For the quotient of X by G, the space of objects can be taken X and the space of morphisms X x G. The “source” map is the trivial projection on x, the “target” map is the action of G on X.

Posted by: Squark on August 28, 2007 8:26 AM | Permalink | Reply to this

Re: On BV Quantization. Part I.

$MathML-enabled post (click for more details).$

I just had a revelation. This is what the answer must be:

NQ manifolds with generators in degree up to $n$ are the same as Lie $n$ -algebroids with strict skew-symmetry but weak Jacobi identiy.

That’s the known part. Now, I predict

NP manifolds with generators in degree up to $n$ are the same as Lie $n$ -algebroids with strict Jacobi identity but weak skew symmetry.

Then, finally, general weak Lie $n$ -algebroids correspond to

NPQ manifolds, where the Poisson bracket $(\cdot,\cdot)$ and the differential $d$ are related by an element $S$ of the algebra as $d = (S,\cdot) \,.$

That’s what it must all mean.

So, that would imply, I guess, that general Lie $n$ -algebroids correspond to non-negativley graded classical BV algebras.

Hm…

Posted by: Urs Schreiber on August 25, 2007 7:12 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

Careful! Watch out for weak skew-symm. At least the associative analog is quite a different story. The simplicial/singualr cup product is strictly associative but homotopy commutative, cf. \cup_1 and that’s jsut the tip of the iceberg.

Is there a tex version of the Bourbaki symbol for a dangerous turning as in a road sign?

Posted by: jim stasheff on August 26, 2007 9:31 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

$MathML-enabled post (click for more details).$

A while ago I wrote:

I just had a revelation.

Yesterday a followup revelation.

general weak Lie $n$ -algebroids correspond to NPQ manifolds

Had an extensive discussion of this with Pavol Ševera. We agreed that one ought to be looking for some kind of obvious and/or natural thing in between $A_\infty$ and $L_\infty$ . You know, whatever truly weak Lie $n$ -algebras are, it must have a description in terms of something we already sort of know.

$A_\infty$ comes from tensor co-algebra with co-differential. $L_\infty$ from (graded) symmetric tensor co-algebra with codifferential.

“Clearly” the strict skew symmetry of the Lie $n$ -algebra bracket $n$ -functor translates into the graded commutativity of the coalgebra.

What we are looking for is a situation where two things graded commute only up to an additve correction (since composing higher coherence morphisms in this game is just addition).

But this can only mean one thing: we need Clifford algebra instead of exterior algebra.

(Sorry for the boldface. Couldn’t help it.)

Wednesday’s conjecture: Lie $n$ -algebras with arbitrary coherently weak skew symmetry and arbitrary coherent weak Jacobi identity correspond to graded differential Clifford algebras.

You see, take a Lie algebra and the corresponding Chevalley Eilenber algebra $\wedge^\bullet ( s g^*, d ) \,.$

Then replace the exterior algebra (of left invariant differential forms on $G$ , really), with the Clifford algebra coming from some bilinear form on $g$ , but such that the $\mathbb{Z}$ -grading is respected.

So we throw in one single degree 2-generator $b$ and demand that for $t^a$ and $t^b$ any two elements in $s g^*$ , instead of

$t^a t^b = - t^b t^a$

we have

$k_{ab} t^a t^b = b \,,$

where $k_{ab}$ are the components of the symmetric bilinear form.

For this to be compatible with the differential, we find that $k$ has to be invariant. Just as if $k_{ab}$ were the components of a symplectic 2-form on the dg-manifold.

Posted by: Urs Schreiber on September 27, 2007 6:29 PM | Permalink | Reply to this

Read the post Detecting Higher Order Necklaces
Weblog: The n-Category Café
Excerpt: Nils Baas on higher order structures,Enrico Vitale on weak cokernels and a speculation on weak Lie n-algebras triggered by discussion with Pavol Severa.
Tracked: September 28, 2007 12:44 AM

Read the post BV-Formalism, Part IV
Weblog: The n-Category Café
Excerpt: Lie algebroids of action groupoids and their relation to BRST formalism.
Tracked: October 11, 2007 9:47 PM

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

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 10:09 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 8:10 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 10:29 PM

Read the post Smooth 2-Functors and Differential Forms
Weblog: The n-Category Café
Excerpt: An article on the relation between smooth 2-functors with values in strict 2-groups, and an outline of the big picture that this sits in.
Tracked: February 6, 2008 12:06 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 1:30 PM

Re: On BV Quantization. Part I.

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Does the cafe also have a piece on the BV formalism (that should be plural) especially before quantization?

Posted by: jim on October 14, 2013 3:37 PM | Permalink | Reply to this

Re: On BV Quantization. Part I.

$MathML-enabled post (click for more details).$

The next in the BV-series is presented as

A review of some basics of classical BV formalism.

Ideally, anything useful would find its way to the nLab entry.

Posted by: David Corfield on October 15, 2013 10:09 AM | Permalink | Reply to this

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