## June 27, 2008

### Eli Hawkins on Geometric Quantization, II

#### Posted by Urs Schreiber

Today Eli gave the second of his two talks on $C^*$-algebraic geometric quantization at HIM, based on his Groupoid approach to quantization.

I had reported on the first talk here and summarized some related results by Landsman and Ramazan here. One nice thing Eli explained today was how his approach encompasses the one by Landsman and Ramazan.

Recall that the basic idea here is this:

given a Poisson manifold (possibly but not necessarily symplectic), thought of as the phase space of a physical system, one wants to construct the $C^*$-algebra quantizing (deforming) the Poisson Lie algebra of functions on phase space.

In more standard geometric quantization one would assume the Poisson structure to be actually symplectic, then try to build a Hilbert space from this data, cut it down a bit using a “polarization”, and finally find the above quantum deformed algebra as a subalgebra of the bounded operators of that Hilbert spaces.

In contrast, here the Hilbert space plays a secondary role or does not even appear explicitly. This is motivated by observations such as those by Landsman and Ramazan that in large classes of examples the quantum algebras turn out to be groupoid algebras of certain groupoids naturally associated with the original Poisson manifold.

Eli Hawkins’ approach aims to completely clarify this situation in that it explains in general which groupoid algebra is the right one. In brief words, the situation is simple and nice:

Every Poisson manifold $(X,\pi)$ naturally carries the corresponding Poisson Lie algebroid $(T^* X, \pi)$. If this integrates, then the integrating Lie groupoid (the source-simply connected cover or one of its quotients) is necessarily a symplectic groupoid (a groupoid with multiplicative symplectic structure on its space of morphisms) with $(X,\pi)$ the space of objects.

Using ordinary prequantization we may happen to get a line bundle on the space of morphisms (a line bundle with connection whose curvature is the given symplectic form) and furthermore – that’s Eli Hawkins’ big contribution here – there is a natural notion of polarization on the groupoid here, such that, finally, the quantum algebra in question is the groupoid convolution algebra of polarized sections of this line bundle.

The resulting groupoid $C^*$-algebra can be regarded as a $C^*$-algebraic deformation quantization of the original Poisson algebra. Notice that this is different from and really “stronger” than formal deformation quantization in terms of formal power series. See maybe my discussion here or, better, the nice introduction in the Landsman-Ramazan article.

In particular, the Landsman-Ramazan situation is recovered as follows:

recall that they observe that the $C^*$-algebraic deformation quantization of any Poisson manifold $A^*$ arising as the fiberwise dual of a Lie algebroid $A$ is the groupoid algebra of the groupoid integrating $A$.

Now, in Eli Hawkins setup we are to form the Poisson Lie algebroid over $A^*$, integrate that, cut down functions on that to polarized ones and then form the convolution algebra of those. And, lo and behold, this does reproduce the direct prescription. The reason for that is the following nice

Fact. The Lie groupoid integrating the Poisson Lie algebroid over the dual $A^*$ of a Lie algebroid $A$ is the cotangent Lie groupoid $T^* G(A) \stackrel{\to}{\to} A^*$ of the Lie groupoid $G(A)$ integrating $A$.

Then it is clear that there is a choice of polarization which divides out the cotangent fibers and hence the polarized sections on $T^* G(A)$ are just the ordinary sections on $G(A)$.

(I recall more details below. This beautiful result is discussed on p. 32.)

Eli Hawkins has a wealth of concrete examples beyond this large class of examples. Today he only found time to say a bit about the Moyal space and the noncommutative torus. But look at his article for more.

For completeness, I recall some standard definitions and then Eli’s main new definition: polarization of a Lie groupoid.

Here are the more or less standard ones:

The Poisson Lie algebroid of a manifold $X$ with Poisson tensor $\pi \in \Gamma(\Lambda^2 T X)$ has as underlying vector bundle the cotangent bundle $T^* X$, the anchor map $\array{ T^* X &\stackrel{\rho = \pi \cdot}{\to}& T X \\ & \searrow \swarrow \\ & X }$ is contraction of 1-forms with the Poisson tensor and the Lie bracket $[\cdot, \cdot]_\pi$ on sections of $T^* X$, i.e. on 1-forms in the unique one which on exact 1-forms has the property that $[d f , d g]_\pi = d\{f,g\} \,,$ with $f,g$ smooth functions on $X$ and $\{\cdot, \cdot\}$ the Poisson bracket.

A symplectic Lie groupoid is a Lie groupoid $C$ whose space of objects is a Poisson manifold and whose space of morphisms carries a symplectic structure whose symplectic form $\omega \in \Omega^2_{close}(Mor(C))$ is multiplicative in that it’s simplicial derivative vanishes $0 = \delta \omega = pr_1^* \omega - compose^* \omega + pr_2^* \omega$ on the space of composable morphisms, and such that source and target maps are homomorphisms of Poisson manifolds.

Every Lie groupoid integrating a Poisson Lie algebroid is symplectic. Picking always the source-simply connected integrating Lie groupoid this gives indeed a functor $\Sigma : Poisson manifolds \to symplectic groupoids$ (where defining the morphisms on the right is apparently slightly involved).

If the Poisson structure is even symplectic, then the Lie groupoid integrating the Poisson Lie algebroid is the fundamental groupoid $(X,\pi) symplectic \Rightarrow \Sigma(X,\pi) = \Pi(X) \,.$ For $X$ simply connected such that $\Pi(X) = Pair(X)$ is just the pair groupoid of $X$, then symplectic structure on $Mor(Pair(X)) = X \times X$ is $\omega \otimes (-\omega)$ for $\omega$ the symplectic form on $X$.

Now finally the definition of

A polarization on a Lie groupoid $C$ is

1) A subbundle $P \subset T_{\mathbb{C}} Mor(C)$ of the complexified tangent bundle of the space of morphisms which is involutive (i.e. integrable) in that its sections form a sub-Lie algebra of the (complexified) Lie algebra of vector fields $[P,P] \subset P \,;$

2) which is multiplicative in that for $\gamma_1, \gamma_2$ any two composable morphisms, we have $P_{\gamma_2 \circ \gamma_1} = (composition)_* (P_2)_{\gamma_1,\gamma_2} \,,$ where $P_2$ is a subbundle of the tangent bundle on the space of composable morphisms given by $P_2 := (P \times P) \cap T_{\mathbb{C}} (Mor(C) {}_t \times_s) Mor(C) \,;$

3) and which is Hermitean in that push-forward along the inversion map corresponds to complex conjugation $inv_* P = \bar P \,.$

The multiplicativity condition is the crucial one.

A prequantization on a symplectic groupoid is a line bundle $L \to Mor(C)$ with connection whose curvature is the symplectic form, $curv(L) = \omega$, and which is also multiplicative in that there is a line bundle isomorphism $\mu : pr_1^* L \otimes pr_2^* L \to (composition)^* L$ which is associative in the obvious sense (Eli referred to this as a groupoid cocycle. For $C$ a Čech groupoid this is nothing but a bundle gerbe.)

The twisted polarized convolution algebra of a symplectic, polarized and prequantized Lie groupoid is

1) of the more or less (up to some technicalities) obvious convolution algebra of sections of the prequantum line bundle $L$

2) the subalgebra of the polarized sections.

Here a section is called polarized, as usual, if its covariant derivatives along all vectors in the polarization bundle $P$ vanishes.

Finally, for the examples coming from Poisson manifolds which are (coadjoint orbits in) fiberwise linear duals $A^*$ of Lie algebroids $A$ we need

The cotangent Lie groupoid $T^* C$ of a Lie groupoid $C$ with Lie algebroid $A$ has

$Obj(T^* C) = A^*$ $Mor(T^* C) = T^* Mor(C)$ where source and target map come from left- or right invariant translating, respectively, cotangent vectors at a morphism $\gamma$ to the identity on the source of $\gamma$, ad where composition is the unique one respecting this source and target definition and covering the composition in $G$.

Posted at June 27, 2008 4:22 PM UTC

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### Re: Eli Hawkins on Geometric Quantization, II

If the Poisson structure is even symplectic, then the Lie groupoid integrating the Poisson Lie algebroid is the fundamental groupoid (X,π)symplectic⇒Σ(X,π)=Π(X). For X simply connected such that Π(X)=Pair(X) is just the pair groupoid of X, then symplectic structure on Mor(Pair(X))=X×X is ω⊗(−ω) for ω the symplectic form on X.

This confuses me, because I can think of examples where $T^*X$ has an obvious symplectic structure, but $X$ itself does not. For example, let $X$ be the configuration space of a mechanical system. (I’m particularly interested in this from the perspective of discrete Lagrangian/Hamiltonian mechanics, a la Marsden, Weinstein, et al.) The cotangent bundle has a symplectic structure, which extends to a symplectic structure on the morphisms of the pair/fundamental groupoid integrating it. However, I don’t see how this also leads to a symplectic structure on the objects of the groupoid, i.e., $X$ itself.

Could you provide some clarification on this? Thanks so much!

Cheers, Ari

Posted by: Ari Stern on June 27, 2008 10:30 PM | Permalink | Reply to this

### Re: Eli Hawkins on Geometric Quantization, II

Maybe the problem is simply that I use “$X$” where you’d use some other letter? Beware that my $X$ is the symplectic manifold. So for instance it might be $X = T^* Y$ for $Y$ any manifold. This $Y$ is what you call $X$.

(I suppose I know that I am not following the standard symbol convention. Eli used $M$ for the generic symplectic manifold. But I like $X$s :-)

Posted by: Urs Schreiber on June 27, 2008 11:27 PM | Permalink | Reply to this

### Re: Eli Hawkins on Geometric Quantization, II

Ahhh, thanks – I think that’s exactly what tripped me up. In my example, $X$ is not a symplectic manifold, but it is Poisson (at least with the trivial/zero bracket).

What I think was confusing me even more, though, is that I find it somewhat unnatural to think of $T^*X$ as a Lie algebroid – I tend to think of $A=TX$ as the Lie algebroid (with the identity anchor), and $A^*=T^*X$ as its dual. Now, the pair/fundamental groupoid integrates $TX$, but I wasn’t immediately able to see how this groupoid inherits the symplectic structure of $T^*X$.

For me, a more natural formulation is to look at $(TX,T^*X)$ as a Lie bialgebroid, which links together the structures of both the tangent and cotangent bundles. I found a nice paper of Mackenzie and Xu (http://arxiv.org/abs/dg-ga/9712012), in which they discuss the relationship between symplectic groupoids and Lie bialgebroids over Poisson manifolds (section 5). In particular, they prove (Theorem 5.3) that $(AG,A^*G) = (TX,T^*X)$ if and only if the groupoid $G$ is symplectic. Since the pair/fundamental groupoid $G$ integrates to the tangent bundle, we have $AG=TX$, and so this groupoid is symplectic.

Thanks again!
Ari

Posted by: Ari Stern on June 28, 2008 2:28 AM | Permalink | Reply to this

### Re: Eli Hawkins on Geometric Quantization, II

Argh, I just realized that I made a crucial transposition in the statement of that theorem. It actually requires $(AG,A^*G) \cong (T^*M,TM)$, so $G$ has to integrate the cotangent bundle, not the tangent bundle. So, it’s not quite as straightforward as all that (although one can probably take care of it by using the Legendre transform). Anyway, thanks again … this really helped clear some things up for me.

Cheers, Ari

Posted by: Ari Stern on June 28, 2008 5:31 AM | Permalink | Reply to this

### Re: Eli Hawkins on Geometric Quantization, II

In my example, $X$ is not a symplectic manifold, but it is Poisson (at least with the trivial/zero bracket).

Okay, but did I make it clear that my $X$ cou be your $T^* X$?

Let me get back to the original statement (which you know well, I gather, but just for the sake of the discussion here), using different letters:

Let $P$ be a Poisson manifold. Then we can form the Poisson Lie algebroid on $P$ whose underlying vector bundle is $T^* P$. If $P$ even happens to be symplectic, for instance because $P = T^* X$ for any manifold $X$, then the Poisson Lie algebroid $T^* P$ integrates to the pair groupoid of $P$: $G(T^* P) = Pair(P)$.

Let’s assume that indeed $P = T^* X$.

Then we can pick a polarization on $Pair(P)$ such that polarized functions on $Mor(Pair(P))$ are just functions on $Pair(X)$, so that finally the groupoid $C^*$-algebra of polarized functions on $Pair(P)$ is the groupoid $C^*$-algebra of $Pair(X)$, which is the algebra of compact operators on $L^2(X)$.

Posted by: Urs Schreiber on June 29, 2008 11:37 AM | Permalink | Reply to this

### Re: Eli Hawkins on Geometric Quantization, II

find it somewhat unnatural to think of $T^* X$ as a Lie algebroid – I tend to think of $A = T X$ as the Lie algebroid (with the identity anchor), and $A^* = T^* X$ as its dual.

Right. It is kind of important in this business that there are two different crucial structures on $T^* X$:

1) $T^* X$ is the fiberwise linear dual of the vector bundle $T X$ underlying the tangent Lie algebroid. As such, $T^* X$ is naturally a Poisson manifold (which happens to even be symplectic.)

2) If $X$ itself is already Poisson, then $T^* X$ is also the vector bundle underlying the Poisson Lie algebroid $\array{ T^* X &\stackrel{\rho}{\to}& T X \\ & \searrow \swarrow \\ & X } \,,$ where $\rho$ is contraction with the Poisson bivector.

Posted by: Urs Schreiber on June 29, 2008 11:42 AM | Permalink | Reply to this

### Re: Eli Hawkins on Geometric Quantization, II

Now, the pair/fundamental groupoid integrates $T X$, but I wasn’t immediately able to see how this groupoid inherits the symplectic structure of $T^* X$.

And actually it does not. The statement was rather this:

if $P$ (which I originally denoted $X$, but let’s call it $P$ now) happens to be a symplectic manifold with symplectic form $\omega$, then the pair groupoid $Pair(P)$ is naturally a symplectic groupoid.

Posted by: Urs Schreiber on June 29, 2008 11:49 AM | Permalink | Reply to this

### Re: Eli Hawkins on Geometric Quantization, II

In the framework of Lie-Rinehart algebras, a related construction including the notion of polarization can be found in my paper

Poisson cohomology and quantization.
J. reine angew. Math.
408 (1990), 57–113.

A survey:
Lie-Rinehart algebras,
descent, and quantization, in: Galois theory, Hopf algebras,
and semiabelian categories, Fields Institute Communications
43 (2004), 295–316, Amer.Math. Society, Providence, R. I.,
{\tt math.SG/0303016}

Posted by: Johannes Huebschmann on October 27, 2008 6:59 PM | Permalink | Reply to this

### Re: Eli Hawkins on Geometric Quantization, II

I started saving some of this stuff at $n$Lab:symplectic groupoid.

Posted by: Urs Schreiber on November 2, 2009 11:28 PM | Permalink | Reply to this

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