Courant Algebroids From Categorified Symplectic Geometry

November 10, 2009

Courant Algebroids From Categorified Symplectic Geometry

Posted by John Baez

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‘Higher symplectic geometry’ is a topic that has come up recently in posts here in the $n$ -Café and in some ideas presented by Urs and others in the $n$ Lab. This is a subject I’ve been thinking a lot about lately, and I just finished a (rough!) draft of a paper that attempts to establish some connections between two different approaches to generalizing, extending, and/or categorifying symplectic geometry. So of course I happily accepted John’s invitation to write a guest post about some of this work. I hope by doing so I can contribute a little something to the ongoing discussion and learn more about the different ideas people have on the subject.

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There are many approaches in the literature that attempt to develop a more generalized notion of symplectic geometry. Some are cast in the language of ‘oidization’ (e.g. Courant algebroids, Lie $n$ -algebroids ), or in terms of supergeometry (e.g. NQ-manifolds), while others use more traditional geometric ingredients to invent interesting new structures (e.g. multisymplectic geometry, and $k$ -symplectic geometry). Some of these approaches had higher-algebraic or higher-geometric structures explicitly built in from the beginning. But many of them did not, and it is only rather recently that they have been reinterpreted in some kind of categorified way.

I’d like to focus on two approaches here. I’ll just sketch the main ideas but you can get the full story by reading the draft available for now on my web page. The first approach is what John and I call ‘categorified symplectic geometry’ or ‘ $n$ -plectic geometry’. Here one studies the categorified algebraic and geometric structures that naturally arise on manifolds equipped with a closed non-degenerate $n+1$ -form. Such manifolds show up as phase spaces in $n$ -dimensional classical field theory. The case relevant to the classical bosonic string is when $n=2$ and is called ‘2-plectic geometry’. So, just as the phase space of the classical particle is a manifold equipped with a closed, non-degenerate 2-form, the phase space of the classical string is a (finite-dimensional) manifold equipped with a closed non-degenerate 3-form.

And there is also an algebraic side to this correspondence. The symplectic form gives the space of smooth functions the structure of a Poisson algebra. Analogously, the 2-plectic form gives a Lie 2-algebra of observables associated to any 2-plectic manifold. And when the 2-plectic manifold is a compact simple Lie group $G$ , this Lie 2-algebra is closely related to the string Lie 2-algebra. (See this paper for more about Hamiltonian observables and classical strings, and this paper for more on how the string Lie 2-algebra gets involved.)

String theory, closed 3-forms and Lie 2-algebras also play important roles in the theory of Courant algebroids. Roughly, Courant algebroids are vector bundles which simultaneously combine and generalize the structures found in the tangent bundles of manifolds and quadratic Lie algebras. In particular, the underlying vector bundle of a Courant algebroid comes equipped with an antisymmetric bracket on the space of global sections. However, unlike the Lie bracket of vector fields, the bracket need not satisfy the Jacobi identity.

In a fascinating series of letters to Alan Weinstein, Pavol Ševera showed (among many other things) that so-called ‘exact Courant algebroids’ naturally arise in the context of string theory. He developed the notions of connection and curvature on exact Courant algebroids and then showed that these algebroids are classified up to isomorphism by the third de Rham cohomology of the base space. Around the same time Dmitry Roytenberg and Alan Weinstein showed that the bracket on the Courant algebroid induces an $L_{\infty}$ structure on the global sections. Roytenberg later reinterpreted this $L_{\infty}$ -algebra as a Lie 2-algebra.

So there are some very strong similarities between these two approaches! And perhaps the punchline to the story is now obvious: On any 2-plectic manifold there is an exact Courant algebroid whose Ševera class is specified by the 2-plectic form. Algebraically, this corresponds to a nice embedding of the Lie 2-algebra of observables on the 2-plectic manifold into the Lie 2-algebra induced by the bracket on the Courant algebroid’s space of global sections.

I think this is just the beginning of some deeper relationships between these two approaches, and I am optimistic that they will mesh well with the ideas that Urs presented here. Thank you for reading! Comments, questions, corrections, and suggestions are most welcome.

Posted at November 10, 2009 3:47 PM UTC

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Re: Courant Algebroids From Categorified Symplectic Geometry

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Thanks for this post!

You mention:

some ideas presented by Urs and others in the $n$ Lab.

Your link here points to $n$ -symplectic manifold. I want to emphasize that the ideas mentioned at this entry – up to just slight idiosyncracies on my part (discussed in the discussion part) – are those of Dmitry Roytenberg and Pavol Ševera, as indicated in the list of references.

A few additional ideas of mine on this topic (linked to from there) is instead at schreiber:symplectic $\infty$ -Lie algebroid.

Posted by: Urs Schreiber on November 10, 2009 4:58 PM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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I have now had a look at your notes

Christopher Rogers, Courant algebroids from categorified symplectic geometry (pdf)

This is nice stuff. On top of the new observations that you present, it also is a useful summary of the relavant aspects of a subject that is a bit notorious for having too many symbols.

Here is a “Chern-Weil theory perspective” on the issue of “connections on a Courant algebroid”, which I enjoy. I haven’t unwrapped what I say now in full gory detail to the explicit description that you consider, but I would expect that it matches.

Generally, for $\mathfrak{a}$ any $\infty$ -Lie algebroid and $X$ a manifold, a collection of flat $\mathfrak{a}$ -valued differential forms is a morphism

$A : T X \to \mathfrak{a} \,.$

In the case you are considering where $\mathfrak{a} = \mathfrak{P}(B, \Theta)$ is a Courant algebroid – a Lie 2-algebroid over a base space $B$ , and we are interested in the case that $X = B$ and that $A$ is a section of the canonical anchor morphism $\mathfrak{a} \to T X$ .

But generally, given any such morphism of flat $\mathfrak{a}$ -valued forms, we obtain on $X$ all the secondary characteristic forms of this connection, namely all the pullbacks of all the $\infty$ -Lie algebroid cocycles on $\mathfrak{a}$ .

This is an obvious statement once we pass from the direct $\infty$ -Lie algebroid morphism to its induced morphism of Chevalley-Eilenberg algebras

$CE(T X) = \Omega^\bullet(X) \leftarrow CE(\mathfrak{a}) : A$

which I’ll denote by the same symbol. The CE-algebra of the tangent Lie algebroid is just the deRham algebra on $X$ . The CE-algebra of $\mathfrak{a}$ is what is sometimes called the “deRham complex of the Courant algebroid” or the standard complex of $\mathfrak{a}$ . If we think of $\mathfrak{a}$ as an NQ-supermanifold then this is just the dg-algebra of functions on $\mathfrak{a}$ with the specified odd vector field as the differential.

Keep in mind that for the simple case that $\mathfrak{a} = \mathfrak{g}$ is an ordinary Lie algebra, $CE(\mathfrak{a}) = CE(\mathfrak{g})$ is the ordinary Chevalley-Eilenberg algebra of $\mathfrak{g}$ . The closed elements in here are precisely the Lie algebra cocycles.

So, analogously, we are entitled to think of the closed elements $\mu$ in $CE(\mathfrak{a})$ as the $\infty$ -Lie algebroid cocycles.

Clearly, every such element $\mu \in CE(\mathfrak{a})$ of degree $k$ maps under $A$ to a closed $k$ -form $\mu(A)$ on $X$ , simply because $A$ is a morphism of dg-algebras.

For instance if $\mathfrak{a} = \mathfrak{g}$ is an ordinary semisimple Lie algebra and $\mu = \langle \cdot, [\cdot, \cdot]\rangle$ its canonical (up to normalization) Lie algebra 3-cocycle, a morphism

$A : T X \to \mathfrak{g}$

is just a flat Lie algebra valued 1-form $A$ on $X$ and

$\mu(A) = \langle A \wedge [A \wedge A]\rangle$

is the closed secondary characteristic 3-form, the “Chern-Weil image of $\mu$ ” under $A$ .

Big words for a simple concept.

Now, this story goes through entirely analogously for arbitrary $\infty$ -Lie algebroids $\mathfrak{a}$ .

So, what might be an interestring $\infty$ -Lie algebroid cocycle on a Couran Lie 2-algebroid? Well, by Roytenberg/Ševera we have that Courant algebroids $\mathfrak{a} = \mathfrak{P}(X,\Theta)$ are characterized by the fact that their Chevalley-Eilenberg algebra $CE(\mathfrak{a})$ is equipped with a bracket $\{-,-\}$ and a degree 3 element $\Theta \in CE(\mathfrak{a})$ such that $\{\Theta,\Theta\} = 0$ and such that the differential of $CE(\mathfrak{a})$ is

$d_{CE(\mathfrak{a})} = \{\Theta, -\} \,.$

But whatever that means in detail, it implies immediately that $\Theta$ is in fact an $\infty$ -Lie algebroid cocycle, since

$d_{CE(\mathfrak{a})} \Theta = \{\Theta,\Theta\} = 0 \,.$

So this means that every flat $\mathfrak{a} = \mathfrak{P}(X,\Theta)$ -valued differential form datum

$A : T X \to \mathfrak{a}$

comes canonically with a secondary curvature characteristic closed 3-form $\Theta(A)$ on $X$ , wich is the Chern-Weil image of $\Theta$ under $A$

$\array{ \Omega^\bullet(X) &\leftarrow& \mathfrak{P}(X)_{\Theta} & : A \\ \Theta(A) &\leftarrow|& \Theta } \,.$

I think this secondary curvature characteristic form which is the Chern-Weil image of the canonical $\infty$ -Lie algebroid cocycle $\Theta$ on the Courant (Lie 2-)algebroid $\mathfrak{P}(X,\Theta)$ is the closed 3-form that you discuss, whose deRham class is the “Ševera class”.

This is just meant to be a reformulation of the situation in supposedly suggestive but equivalent language. I do think, though, that this language helps to see what is going on. But possibly that’s just me.

Posted by: Urs Schreiber on November 10, 2009 6:09 PM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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Thanks for these great comments, Urs. As I read these I’m also playing catch-up on your $\infty$ -Lie theory program.

I’m beginning to like this “Chevalley-Eilenberg way” of describing $\infty$ -Lie algebroids. In particular, anytime you make an analogy with the “ordinary” CE algebra from Lie theory is very helpful. For example, I now have a much nicer picture of what you’re saying when you talk about $\infty$ -Lie algebroid cocycles. (And I like this Chern-Weil theory viewpoint as well. I agree that your secondary curvature characteristic form must be the curvature 3-form I was talking about in the draft.)

I still have some basic confusions. Roughly, I think of $CE(\mathfrak{a})$ as the algebra of functions “on $\mathfrak{a}$ ”. But I have no good notion yet of what $\mathfrak{a}$ (in general) is as a geometric object. This is partially because I’m not comfortable yet with the “ $\infty$ -geometry” setting you establish here. Perhaps, just to start somewhere, you can tell me how you encode the data in your notation for the Courant algebroid: $\mathfrak{a}=\mathfrak{B}(B,\Theta)$ . I know what $B$ is and what $\Theta$ is in this context, but I don’t understand how I get a “infinitesimal $\infty$ -Lie groupoid” from this. (Sorry, maybe this question is too ill-posed.)

Another basic confusion I have is in the definition you give of $\mathfrak{a}$ here. In particular, I don’t know what you mean by $CE(\mathfrak{a})$ being graded commutative. Well…I should say that I know what you mean, but I don’t have nice examples and non-examples, since in the Lie algebra case $CE(\mathfrak{g})$ is always graded commutative. (Despite the remark by Jim Stasheff warning about this abuse of language, it’s still unclear to me.) Can you give me some examples here that I can keep in the back of my mind while I read on?

Posted by: Chris on November 11, 2009 10:30 PM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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I’m beginning to like this “Chevalley-Eilenberg way” of describing ∞-Lie algebroids. In particular, anytime you make an analogy with the “ordinary” CE algebra from Lie theory is very helpful.

Good. It is one of these things that after a little language barrier (but we can only communicate via language, so it’s inevitable!) become almost kindergarten stuff. It’s really all an easy fun game once one sees how the simple rules map into reality.

This is also the reason why so much nice $\infty$ -category theory is implicitly hidden in so much of physics. Physicists learned early on to value differential graded structures. They are easy to handle. But secretly all of them are $\infty$ -categorial structures.

But I have no good notion yet of what $\mathfrak{a}$ (in general) is as a geometric object. This is partially because I’m not comfortable yet with the “∞-geometry” setting you establish here.

I need to eventually find time to write a better exposition at that entry (unless somebody beats me and puts it in there for me) but here is the basic simple idea:

you probably have intuition for what a smooth groupoid is. What a smooth 2-groupoid is. So you have intuition for what a smooth $n$ -groupoid is for all $n$ .

Now think of one all whose $k$ -morphisms have infinitesimal extension. I mean, disregarding how to make this precise, this is a concept that appeals directly to intuition, right? So the statement is that this can be made precise and that this is what a Lie $n$ -algebroid is: a smooth $n$ -groupoid all whose $k$ -morphisms have infinitesimal extension.

just to start somewhere, you can tell me how you encode the data in your notation for the Courant algebroid $\mathfrak{a} = \mathcal{P}(X,\Theta)$

Ah, now you want to go the other way round and construct from a Chevalley-Eilenberg algebra the $\infty$ -Lie algebroid that it came from.

This does have an abstract answer that tells you what to do, but I don’t really have a nice way to break this down into soemthing really useful.

So if we really need we can do this, but if we don’t need to we shouldn’t waste energy on doing it. And here we don’t really need to.

For the theory-building on my Lab page it goes the other way round: we start describing some abstract nonsense that lives in the world of oo-groupoids. After playing this and that trick with these beasts, we end up with a diagram of them which we know encodes in it the answer in terms of concrete differential forms and differential geometry that we are after. So then we tun the crank on these beasts, find the infinitesimal version of these oo-Lie groupoids, take the CE-algebras of these and out comes our differential geometric data.

But if you have this dg-data in the first place, there is really no need to push it back through this machine in general. Here we don’t need to. We should work entirely with the CE dg-algebras. What the abstract nonsense in the backround only accomplishes for us now is that it is our GPS system which tells us where in the big picture all the dg-thingys we are playing with fit.

I don’t know what you mean by $CE(\mathfrak{a})$ being graded commutative.

Oh, I think you do know. Maybe I expressed myself badly. A dg-algebra is a graded algebra $A$ with a derivation $d$ of degree +1 on it. It is graded-commutative if for $a, b \in A$ two elements of the algebra of degree $n$ and $m$ , respectively, we have

$a \cdot b = (-1)^{n m} b \cdot a \,.$

For instance a Grassmann algebra. But not in general a Clifford algebra.

And here is the archetypical example to keep in mind. (In some sense this is the only example that there is, so this is a good example to keep in mind).

Take $X$ a manifold. Let $\Pi(X) = X^{\Delta^\bullet_{\mathbb{R}}}$ be the simplicial space which in degree $k$ is the space of smooth maps $\Delta^k_{\mathbb{R}} \to X$ , i.e. of smooth $k$ -simplices in $X$ .

Regarded as an $\infty$ -Lie groupoid, this is the path $\infty$ -groupoid of $X$ .

What is its CE-algebra: in degree $k$ this has the algebra of functions on the space of smooth $k$ -simplices in $X$ . So this is the space of smooth singular cochains of $X$ ! Nothing else.

The crucial point is that the product on this CE-algebra is the cup product. If you are not familiar with this, have a look at this entry where the cup product on these singular cochains is explicitly described.

From that description it is evident that here is no way that the cup product of two functions on spaces of simplices in $X$ is graded-commutative in general. Because:

the cup product builds a new function on $(n+m)$ -simplices by taking the first function and evaluating it on some sub- $m$ -simplex of any $(n+m)$ -simplex and evaluating the other one on a sub- $n$ -simplex and then taking the ordinary product of functions.

Clearly, when you swap these sub-simplices around in the big simplex, there is no need that these functions have to still take the same value, up to a sign.

So we learn from this that the $\infty$ -Lie groupoid $\Pi(X)$ is not infinitesimal.

Instead, let’s pass to its infinitesimal version: conider $\Pi^{inf}(X) \hookrightarrow \Pi(X)$ to be the simplicial smooth space that in each degree $k$ has only the subspace of the space of all $k$ -simplices in $X$ on those $k$ -simplices that have infinitesimal extension.

Again, it is easy to imagine this and you may trust that it can be given a precise meaning that matches this intuition.

But now something nice happens: when you have two adjacent infinitesimal simplices, swapping them is much less drastict than swapping two finite simplices. After all, it just means to tranlate each of them by an infinitesimal distance. That’s not much! :-)

By a cool insight emphasized by Anders Kock, one indeed finds that the cup product on functions on spaces of infinitesimal simplices does happen to be graded-commutative.

So we find that the CE-algebra of functions equipped with the cupproduct on the infinitesimal simplices in $X$ is graded commutative, reflecting the fact that swicthing the position of two such functions in the product corresponds to just evaluating them on simplices an infinitesimal distance from the original simplices.

So we learn that $\Pi^{inf}(X)$ is indeed an infinitesimal $\infty$ -Lie groupoid. So it is an $\infty$ -Lie algebroid, by definition. We identify it by checking what its CE-algebra is. By Anders Kock’s results it is the deRham complex. Which we know to be the CE-algebra of the tangent Lie algebroid. So we learn that $\Pi^{inf}(X)$ is the $\infty$ -Lie groupoid incarnation of the tangent Lie algebroid.

Posted by: Urs Schreiber on November 12, 2009 1:41 AM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

But if you have this dg-data in the first place, there is really no need to push it back through this machine in general.

That’s good to hear. I suspected this was the case, since in your examples when you passed to the CE algebra point of view, everything became much more intuitive.

Oh, I think you do know. Maybe I expressed myself badly

No, no everything is fine. My question was poorly worded. Just like you suspect, I do know what graded commutative means (as well as cup product). The fact that there was this “abuse of language” remark after the definition, in addition to not being able to think of a relevant example where commutativity failing implies “non-infinitesimal” just made me feel like I had no idea what you were talking about.

But your example above is very nice, and helps clear this up. (I’m perfectly happy to not worry about the precise details of things like “infinitesimal extension” for now, and just follow my intuition.)

Thanks for the clarifications. I’m going press on and now try to understand these Poisson tensors in more detail.

Posted by: Chris on November 12, 2009 5:15 AM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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Okay, thanks for this message.

I’m going press on and now try to understand these Poisson tensors in more detail.

All right. I spent a good part of the morning expanding this entry for you.

Now there is a section Transgression cocycles that describes what it means for an invariant polynomial and a cocycle on an $\infty$ -Lie algebroid to be in transgression by a Chern-Simons element.

This is very simple. Have a look and please let me know if the text is understandable. Feel free to drop query boxes, by the way.

Then the other thing I added is a detailed descriptin of the case of ordinary semisimple Lie algebras. I spell out the Chevalley-Eilenberg algebra, the Weil algebra, the differentials, the invariant polynomials, Chern-Simons elements and cocycles in great detail. Also everything is expressed in terms of basis components.

Once you are familiar with this case, and there is nothing deep or mysterious about it, everything else is a straightforward and obvious variation on this theme.

So please have a look and let me know of whichever comments you may have.

Posted by: Urs Schreiber on November 12, 2009 10:12 AM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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Hi Urs. These notes were very helpful. I couldn’t get the drop-comment boxes to work on the page (maybe this is because it is published on your personal web?) So I’ll make some comments here.

One thing I was thinking about while reading this was how all the aspects of the $NQ$ -supermanifold approach are built in here. In the $NQ$ approach you start with a $N$ -manifold. Then you add (if you can) a symplectic structure. This gives the graded Poisson bracket, and then you look for a Hamiltonian homological vector field (the “algebroid” structure e.g. $\pi$ for $n=1$ or $\Theta$ for $n=2$ ). So in my mind there are several layers of compatibility. In the $\infty$ -Lie approach, it looks like the graded bracket is already encoded in the differential for $CE(\mathfrak{a})$ , and the existence of a cocycle in transgression with the invariant polynomial I’d guess would be equivalent to saying there exists a Hamiltonian homological vector field. Does this seem like the right kind of interpretation?

I wanted to also ask/comment on some things related to getting a hold of the corresponding $L_{\infty}$ algebra of “sections” via this co-killing process you mentioned in another comment. This is the first time I’ve seen this construction written this way (I imagine it has to be some sort of generalization of the Baez-Crans construction of Lie $n$ -algebras.) In the general case, after you do this co-killing it should give you an $L_{\infty}$ structure on this new complex $CE(\mathfrak{a}_{\mu})$ , correct? Is there some sort of nice way to see that this does indeed give a $L_{\infty}$ structure? (for example is it equivalent to saying the new differential $d_{CE(a_{\mu})}$ squares to zero?)

Now that I understand some of this more, I can appreciate this idea you mentioned:

“One needs to think a bit here about what the L ∞-algebra “of sections” of an ∞-Lie algebroid is. Roughly, the rule is that we discard the degree 0 part of the ∞-Lie algebroid and shrink it to a single point and then also discard the anchor map.”

This idea of “approximation” is really interesting. It does seem like the right description of the process that is occurring. But from one point of view it seems fairly ad-hoc. I would like understand this better, in particular, why is it a natural thing to do (maybe it isn’t?) and, possibly in some physical context, describe what we are approximating and how much information we are loosing.

Posted by: Chris on November 18, 2009 8:28 PM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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and the existence of a cocycle in transgression with the invariant polynomial I’d guess would be equivalent to saying there exists a Hamiltonian homological vector field

Well, the homological vector field is nothing but the differential on the algebra of functions. So that what makes this algebra a differential algebra.

But in the specific cases that we are looking at, it is also true that the differential is always of the form $\{\Theta, - \}$ for $\{-,-\}$ some graded bracket operation and $\Theta$ a cocycle.

In the general case, after you do this co-killing it should give you an $L_\infty$ structure on the new complex $CE(\mathfrak{a}_\mu)$ , correct?

Wait. This seems to go in the right direction, but let’s straighten this out:

there is not really an $L_\infty$ -structure on the dg-algebras $CE(\mathfrak{a}_\mu)$ . Rather, the semi-free dg-algebra itself encodes dually and equivalently an $L_\infty$ -algebra.

Semifree dg-algebras generated in positive degree are precisely the same, dually, as $L_\infty$ -algebras.

If this isn’t clear, I’ll write it out in more detail now on the Lab. Just a sec.

Is there some sort of nice way to see that this does indeed give a $L_\infty$ -structure?

So, in view of what I just said, this question makes sense if you rephrase it as you did approximately in paranthesis after this:

Is there some sort of nice way to see that the differential on $CE(\mathfrak{a}_\mu)$ indeed sauqres to 0?

And there is a super-easy answer to that: the differential $d$ on $CE(\mathfrak{a}_\mu)$ is just the original differential $d_{\mathfrak{a}}$ on the original generators, and is given on the new generator $b$ by

$d : b \mapsto \mu \,,$

where $\mu$ is our cocycle. So this squares to 0 on the original generators because the original $d_{\mathfrak{a}}$ did. And it squares to 0 on the new generator because $\mu$ is a cocycle, meaning that it is a closed element for the old differential:

$d (d b) = d \mu = d_{\mathfrak{a}} \mu = 0 \,.$

This idea of “approximation”is really interesting. It does seem like the right description of the process that is occurring. But from one point of view it seems fairly ad-hoc. I would like understand this better, in particular, why is it a natural thing to do (maybe it isn’t?) and, possibly in some physical context, describe what we are approximating and how much information we are loosing.

Right, good point. I was thinking about that and have a guess now, but not worked out yet:

there is the obvious inclusion functor

$L_\infty Algebras \hookrightarrow L_\infty Algebroids \,.$

I expect that what we are looking at, the operation that computes the “ $L_\infty$ -algebra of sections” of an $L_\infty$ -algebroid is the action of an adjoint functor to this functor.

Posted by: Urs Schreiber on November 18, 2009 9:48 PM | Permalink | Reply to this

L-oo algebras dually are semifree dg-algebras

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I wrote:

If this isn’t clear, I’ll write it out in more detail now on the Lab. Just a sec.

Here we go:

I have now considerably expanded the entry $L_\infty$ -algebra.

The “Definition”-section has now three subsections:

- Original definition in terms of lots of brackets

- Reformulation in terms of semifree differential coalgebra

- Reformulation in terms of semifree differential algebra.

If the equivalence of these three formulations wasn’t clear before this might go in the direction of being helpful. But I should say that it is getting quite late around here and I just wrote this in one go, so it may be a bit rough. In particular I didn’t have the energy to put in all the right signs, yet.

Feel free to massage this.

Posted by: Urs Schreiber on November 18, 2009 11:02 PM | Permalink | Reply to this

Re: L-oo algebras dually are semifree dg-algebras

Mantra: There exists a set of signs.

Posted by: jim stasheff on November 19, 2009 2:16 PM | Permalink | Reply to this

Re: L-oo algebras dually are semifree dg-algebras

I don’t have time right now, so just a word that people should learn to live with coalgebras! Also the original masters did NOT need finite dimensionality for cohomology since they talked in terms of alternating multilinear functions.

Much as Grassmann deserves credit, Grassmann algebras seem more commonly expressed in physspeak.

Posted by: jim stasheff on November 19, 2009 2:21 PM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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I am wondering about the deeper significance of your theorem 4.4 which, as you discuss, is a variation of the old Roytenberg-Weinstein construction of an $L_\infty$ -algebra “of sections” from a Courant algebroid.

I am thinking this should be a special case of a systematic process which reads in any symplectic $n$ -Lie algebroid $\mathfrak{P}(X,\Theta)$ and spits out a Lie $n$ -algebra.

I believe the story here is the following:

as discussed above, a symplectic Lie $n$ -algebroid and hence in particular a Courant algebroid comes naturally equipped with a degree $(n+1)$ cocycle $\Theta$ .

Now, around here it has become second nature for us that whenever someone hands us an $\infty$ -Lie algebroid $\mathfrak{a}$ and a cocycle $\mu$ on it, we have a reflex to form the “String-like extension” $\mathfrak{a}_\mu$ obtained by co-killing $\mu$ .

In terms of Chevalley-Eilenberg algebras this is simple: the dg-algebra $CE(\mathfrak{a}_\mu)$ is that obtained from $CE(\mathfrak{a})$ together with one new generator $b$ in degree $n$ with differential $d_{CE(\mathfrak{a}_\mu)}$ acting as $d_{CE(\mathfrak{a})}$ on all the original generators and acting on $b$ as

$d_{CE(\mathfrak{a}_\mu)} b = \mu \,.$

By the graded Leibnitz rule satisfied by the derivation $d_{CE(\mathfrak{a}_\mu)}$ this fixes it uniquely.

So for $\mathfrak{a} = \mathfrak{g}$ an ordinary semisimple Lie algebra and for $\mu$ the canonical (up to normalization) Lie algebra 3-cocycle, this procedure yields the String Lie 2-algebra $\mathfrak{g}_\mu$ .

But the procedure extends entirely analogously to $\infty$ -Lie algebroids in the obvious fashion.

This means that faced with a Courant algebroid $\mathfrak{a} = \mathfrak{P}(X, \Theta)$ our first reaction is to construct a new Lie 2-algebroid $\mathfrak{a}_\Theta$ obtained by co-killing $\Theta$ .

Its Chevalley-Eilenberg algebra $CE(\mathfrak{a}_\Theta)$ looks essentially like the original one, but it has one new generator in degree 2 satisfying

$d_{CE(\mathfrak{a}_\Theta)} b := \Theta \,.$

This $\mathfrak{a}_\Theta = \mathfrak{P}(X,\Theta)_\Theta$ is another Lie 2-algebroid that canonically comes with any Courant algebroid.

I am thinking that the Lie 2-algebras of Roytenberg-Weinstein and now the one you present, which are induced from the structure of a Courant algebroid, should be thought of as being approximations to the Lie 2-algebra of global sections canonically associated with this “String-like extended” Lie 2-algebroid obtained by co-killing an $\infty$ -Lie algebroid cocycle.

One needs to think a bit here about what the $L_\infty$ -algebra “of sections” of an $\infty$ -Lie algebroid is. Roughly, the rule is that we discard the degree 0 part of the $\infty$ -Lie algebroid and shrink it to a single point and then also discard the anchor map.

For instance when we do this to the tangent Lie algebroid $T X$ we take the deRham dg-algebra $\Omega^\bullet(X)$ , which is generated over its degree 0 bit $C^\infty(X)$ by its degree 1 bit (the 1-forms). Discarding the anchor map makes the deRham differential on 1-forms become

$(d_{dR} \omega)(v,w) = \omega([v,w]) + discarded$

and hence all that remains is, indeed, the CE-algebra of the Lie algebra of vector fields.

Apply a similar truncation procedure to the String-like extension-by-co-killing $\mathfrak{a}_\Theta = \mathfrak{P}(X,\Theta)_\Theta$ and you get the Lie 2-algebra which you describe.

This is obvious in the case where $X = pt$ . In this case the above procedure reproduces literally the story of the String Lie 2-algebra.

So I would say that the Lie 2-algebra that you describe should e regarded as a generalization of the Lie 2-algebra of section of the Lie 2-algebroid obtained from the Courant Lie 2-algebroid by co-killing the canonical cocycle $\Theta$ .

Posted by: Urs Schreiber on November 10, 2009 7:17 PM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

I am confused. You and John study n-plectic geometry, manifolds with n+1 form given. So for n=1 get symplectic geometry and in your draft you study n=2. Now over on the n-lab there is a page on n-symplectic geometry, where for n=0 get symplectic geometry and for n=2 you get courant algebroids, which you explain are related to 2-plectic geometry. But what about Poisson geometry, n=1-symplectic geometry? Is there supposed to be 1.5-plectic geometry?

Posted by: Maarten Bergvelt on November 15, 2009 1:50 PM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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Hi Maarten. Thanks for this comment. Trying to understand how the $n$ in $n$ -plectic is related to the $n$ in Severa’s work (described at the n-symplectic geometry page) was one of the motivations for writing this paper. I’m working on that now as well as trying to understand both approaches in the $\infty$ -Lie algebroid context developed by Urs.

I have some rough ideas about the relationship between the two, but they are very premature at the moment. But the quick (and unsatisfactory) answer is that $n$ -plectic geometry is not the same thing as $n$ -symplectic geometry.

Posted by: Chris on November 15, 2009 8:45 PM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

Thanks, Chris, to confirm that it does not seem so straightforward. Severa’s paper has a very interesting table but is not exactly easy to understand.

Posted by: Maarten Bergvelt on November 15, 2009 11:01 PM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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Severa’s paper has a very interesting table but is not exactly easy to understand.

What’s the first item you feel is not easy to understand? I believe I understand the table, so maybe we can sort it out here.

Posted by: Urs Schreiber on November 16, 2009 5:05 PM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

Thanks, Urs, I tried to say that the paper was hard to understand. For instance it is not clear to me how a symplectic manifold is the n=0 case of a chain of constructions. I looked at the n-lab page for symplectic manifold but found no enlightment. Maybe we should move over to the n-lab and explain it there?

Posted by: Maarten Bergvelt on November 17, 2009 1:09 AM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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For instance it is not clear to me how a symplectic manifold is the $n=0$ case of a chain of constructions.

Okay:

in NQ-supermanifold language it works like this: an $n$ -symplectic manifold is an NQ-supermanifold equipped with a 2-form which is closed, non-degenerate and “of degree $n$ ”.

Being of degree 0 means that only vector fields allong degree 0 coordinates may be sent to non-vanishing values. Hence for the form to be non-degenerate, there may not be any higher degree coordinates, hence the NQ-supermanifold must be an ordinary manifold. On that the 2-form is then a closed non-degenerate 2-form, hence a symplectic form.

More in Lie theoretic language I have also added a bit more detail on this case here, as well as on the other cases.

Posted by: Urs Schreiber on November 17, 2009 9:04 AM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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Okay, I did spend some time on the n-lab, but didn’t find answers to my doubtlessly stupid questions, and as everything is rather dispersed there I am coming back here to get enlightment. I’ll try and explain what I understand, and ask questions along the way. Please let me know if we should move this discussion to some other place.

It seems that the first things to understand are Lie algebroids and higher siblings, and the various formulations of these concepts.

Well, Lie algebroids are among other things vectorbundles $E\to X$ , together with an anchor map from section of $E$ to sections of the tangent bundle of $X$ . Now if you learned your geometry from Hartshorne you know that this corresponds locally to a module $M$ over a commutative ring $A$ , together with an $A$ -module morphism $M\to \Der(A)$ . To get the vector bundle back, you just take the symmetric algebra $S=\Sym(M)$ and $E$ is $\Spec(S)$ .

But people now tell us something weird: to understand Lie algebroids you should not take $\Sym(M)$ as your coordinate ring, but $\tilde S=\Sym(\Pi M^{*})$ , where $\Pi$ is the even-odd parity flip. Then $\Spec(\tilde S)$ is an example of a $N$ -manifold: you give elements of $\Pi M^{*}$ degree and parity 1, so the coordinate ring becomes $\mathbb{N}$ graded, with compatible $\mathbb{Z}_{2}$ grading.

That seems bizarre: why not $\Sym(\Pi M)$ or $\Sym(M^{*})$ ? For instance, if $E$ was the cotangent bundle to X, we are told that we really are interested in the tangent bundle, superized. Is there any relation with the Legendre transform from classical mechanics? There you also move from cotangent to tangent bundle, to move from Hamiltonian to Lagrangian mechanics, but you need the Lagrangian to do this. Is the $N$ -manifold trick a clever way to do a Legendre transform without having to specify a Lagrangian?

Anyway, let’s follow the $N$ -manifold way. We still have the anchor map $\rho\colon M\to \Der(A)$ , so dualizing this gives $\rho^{*}\colon \Der(A)^{*}\to M^{*}$ . But $\Der(A)^{*}$ is just $\Omega_{1}(A)$ , the universal module of differentials of $A$ . In particular, we have the deRham differential $d: A\to \Omega_{1}(A)$ which combines with $\rho^{*}$ by universality to give a map $Q$ making the diagram commute: $\begin{matrix} A& \overset{Q}{\to} & M^{*} \\ &\underset{d}{\searrow}&\, \uparrow \rho^{*}\\ & & \Omega_{1}(A) \end{matrix}$ Next we have the compatibilty conditions on a Lie algebroid; these give maps

(1)

M\wedge M\overset{[,]}{\to}M\overset{\rho}{\to} \Der(A)

and

(2)

M\wedge M\overset{\rho\wedge\rho}{\to}\Der(A)\wedge\Der(A)\overset{[,]}{\to} \Der(A)

that are supposed to be equal. Now I haven’t checked the details but it seems that you should define on $\Sym(\Pi M^{*})$ an odd derivation $Q$ : on degree zero elements as above, on degree 1 elements by the dual of the commutator, and then the equality of two maps (1) and (2) above should give $Q^{2}=0$ . Is that right? How do you change this to get in this setting for instance Courant Lie algebroids, where the bracket on $M$ is not longer a Lie algebra?

Next we should introduce symplectic forms, but let me postpone this a bit.

Posted by: Maarten Bergvelt on November 24, 2009 4:25 PM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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Hi Maarten and Urs. As I started writing this, I noticed Urs replied already, most likely with much better information. But I’ll say a few quick things regardless, that I hope will not create confusion.

I first read about this idea of passing from a Lie algebroid to $\Pi E$ in Roytenberg’s thesis (available here ). He describes how a Lie algebroid structure on the vector bundle is equivalent to a degree 1 square zero derivation $d$ on the exterior algebra $\Gamma(\Lambda E^{*})$ . This square zero condition encodes all the properties of the Lie bracket and the anchor map, in way that I think is identical to your description above. (So far isn’t this just something like Koszul duality?)

To go to the graded supermanifold context one naturally interprets $\Gamma(\Lambda E^{*})$ as the algebra of functions on $\Pi E$ and this basically gets you to the sheaf you describe above. The differential $d$ becomes the homological vector field $Q$ .

Maarten, I don’t know how Courant algebroids can come in here without using the symplectic NQ-structure. But I believe it’s true that if $Q$ is odd but not grade 1, then you get $L_{\infty}$ structures instead of strict Lie. So that’s an example of how to encode “higher” structures in this picture.

Finally Roytenberg mentions Legendre transforms in this super-context, but I think for dealing with Lie bialgebroids. Not directly related, but similar enough that maybe you’d find it interesting.

(BTW any mistakes or errors above are from my misinterpretation of Roytenberg’s work, and should be credited to me, not him :) )

Thanks to both of you for continuing to comment on things related to this post. I’ve been quiet on this thread lately because my last qualifying exam is next week. So I’ve been busy preparing for that.

Posted by: Chris on November 25, 2009 12:28 AM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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Thanks, Urs and Chris, for lots of food for thought.

Some quick questions.

1. A symplectic manifold then seems to be both a n=0 and n=1 case of the construction: for an NQ manifold with weight 0 form you just get a classical symplectic manifold and the case n=1 is supposed to be the case of a Poisson Lie algebroid, and a symplectic manifold gives also a Poisson manifold. The two NQ-supermanifolds that you get are certainly not isomorphic in the obvious way. Isn’t this a problem? For instance, according to Severa you have different quantizations depending on the value of n. So it seems to matter for the quantisation how you think about a symplectic manifold. That seems strange. Or is there some fancy notion of isomorphism that makes the two incarnations of a symplectic manifold secrtely the same?

2. This is about the n-plectic manifolds as discussed in Chris (very clear!) draft. These are just manifolds $X$ with a closed n+1 form which is non-degenerate. This gives an injective map $i\colon TX\to \Lambda^n T^{*}X$ , and the image of $i$ is the space of hamiltonian forms. Then there is an inverse map from hamiltonian forms to Hamiltonian vector fields, and the Hamiltonian vector fields are used to build various other structures. There should be a notion of n-sson manifold where you have just a map in the other direction, $i^\sharp\colon \Lambda^{n} T^{*}X\to TX$ , maybe restricted to a submanifold (the n-Poisson analog of the Hamiltonian forms), satisfying some higher Lie condition, the higher analog of the Schouten bracket vanishing. Maybe there is a foliation of n-sson manifolds by n-plectic manifolds?Does anybody knows such beasts? Are they just the NQ-manifolds with weight n form?

Chris, maybe you should not be reading this, and get ready for your exam. Good luck!

Posted by: Maarten Bergvelt on November 25, 2009 2:17 AM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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So it seems to matter for the quantisation how you think about a symplectic manifold.

I think the geometric quantization prescription for a Poisson Lie algebroid will reproduce in the case that the Poisson structure comes from a symplectic structure the ordinary geometric quantization of that symplectic structure.

This is described nicely, I think, by Eli Hawkins (arXiv, blog I, II, III, wiki)

Posted by: Urs Schreiber on November 25, 2009 9:53 AM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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Chris,

yes, that’s correct. Except this bit here:

But I believe it’s true that if $Q$ is odd but not grade 1, then you get $L_\infty$ -structures instead of strict Lie.

The $Q$ is always of degree 1. We are talking about dg-algebras and their differential is always of degree 1 (+1 for the cochain dg-algebras that we are talking about, -1 for the chain complex dg-algebras).

It’s the degree of the generators of the graded Grassmann algebra $\wedge^\bullet \mathfrak{g}^*$ that the differential is acting on which controls the categorical degree. If these generators sit in degree 1 to degree $n$ , then the $Q$ acting on them encodes a Lie $n$ -algebra. If they are indegree 0 to $n$ then a Lie $n$ -algebroid. All of this weak (i.e. with homotopy coherent Jacobi identity).

The case of an ordinary Lie algebra is that where all generators are in degree 1.

A strict $\infty$ -Lie algebra, also called a graded Lie algebra, is one such that $Q$ is at most co-binary, i.e. $Q(\mathfrak{g}^*) \subset \mathfrak{g}^* \oplus \mathfrak{g}^* \wedge \mathfrak{g}^*$ .

What makes a Courant algebroid different from, say, a Poisson Lie algebroid is that it has generators in degree 0 (the “ $x$ “s), 1 (the “ $\theta$ “s) and 2 (the “ $p$ “s).

Posted by: Urs Schreiber on November 25, 2009 9:42 AM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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Just a very quick reply:

The Q is always of degree 1. We are talking about dg-algebras and their differential is always of degree 1 (+1 for the cochain dg-algebras that we are talking about, -1 for the chain complex dg-algebras).

Whoops! Thanks, Urs. I should have realized that. What I was thinking of was the construction of $L_{\infty}$ structures from homological vector fields given in the paper by Alexandrov, Kontsevich, Schwarz, and Zaboronsky: The Geometry of the Master Equation and Topological Quantum Field Theory

I haven’t looked at that paper in a while, so I got the details scrambled.

Posted by: Chris on November 26, 2009 10:14 AM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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By the way, just in case you aren’t aware of it (probably you are):

the best place to start reaading about AKSZ theory is of course Dmitry Roytenberg’s review of the original article.

Another by the way: did you see this reply to you that I posted a while ago?

Maybe you didn’t see that. This tried to explain all this in some detail. But it’s clearly still far from any perfection. I’d be interested in hearing your comments about which points are not clear yet, and which are.

Posted by: Urs Schreiber on November 26, 2009 10:52 AM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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The best place to start reaading about AKSZ theory is of course Dmitry Roytenberg’s review of the original article.

Thanks for this reference. I actually haven’t looked at it. I should soon, now that I have a better idea of how graded manifolds and Courant algebroids are related.

I recently did see all the new stuff you added to the $L_{\infty}$ page. Thank you! But I haven’t had the time yet to really give it a proper reading. I was planning on helping out by adding all the signs and whatnot. I’ll be sure to provide some feedback too.

As you probably have realized, out of these three different formulations the semifree differential algebra approach we’ve been discussing here is the one newest to me. (Actually the first “official” definition of a $L_{\infty}$ algebra that I learned in a class was the coalgebra definition.)

Thanks again!

Posted by: Chris on November 26, 2009 11:55 AM | Permalink | Reply to this

Re: Courant Algebroids From Categorified Symplectic Geometry

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why not $Sym(\Pi M)$ …

As long as we haven’t specified a differential, the difference is pretty much inessential.

… or [why not on] $Sym(M^*)$

Because the elements of $M^*$ are cotangents to the space of 1-morphisms of the Lie groupoid integrating the Lie algebroid: the $\mathbb{N}$ -degree on the CE-algebra is the categorical degree of morphisms which makes all the difference bewteen the $\infty$ -categorical nature of Lie algebroids/NQ-supermanifolds and the plain spatial nature of bare supermanifolds.

You may apply throughout the monoidal Dold-Kan corresondence to see the “true” nature of the dg-algebras that we are talking about: they are images of what more naturally are really cosimplicial algebras, which in turn are algebras of functions on simplcial objects. These simplicial objects model $\infty$ -groupoids. The $\mathbb{N}$ -degree that we are seeing is that of the $k$ -morphisms in these beasts.

For instance, if $E$ was the cotangent bundle to $X$ , we are told that we really are interested in the tangent bundle, superized.

Of sections of the tangent bundle, yes. And conversely. I just wrote an entry on the shifted tangent bundle in reply on this point. Not sure if that will tell you anything new, but maybe have a look.

Is there any relation with the Legendre transform from classical mechanics?

Yes. I quickly collected some links here. The second one is most detailed.

Is the $N$ -manifold trick a clever way to do a Legendre transform without having to specify a Lagrangian?

I’ll teach you call the good natural theory of Lie algebroids a trick! ;-)

But, no, there is no Legendre transformation without a Lagrangian. See the above references for more details, I don’t have the energy for that now. You may pay me back for my efforts by summarizing the main points afterwards here, for all of us.

Now I haven’t checked the details but it seems that you should define on $Sym(\Pi M^*)$ an odd derivation $Q$ : on degree zero elements as above, on degree 1 elements by the dual of the commutator, and then the equality of two maps (1) and (2) above should give $Q^2 = 0$ . Is that right?

That’s right, yes. The formula that defines the differential is now here.

Posted by: Urs Schreiber on November 24, 2009 7:40 PM | Permalink | Reply to this

standard Courant algebroid

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I have begun an entry with notes on the standard Courant algebroid, that on $T X \oplus T^* X$ .

So far this contains just very simple statements, but told from the perspective of $\infty$ -Lie theory.

Eventually I am going to discuss generalized complex geometry and geometric T-duality in this fashion.

Posted by: Urs Schreiber on December 3, 2009 12:09 AM | Permalink | Reply to this

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November 10, 2009