### 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 $G$.

## Re: Eli Hawkins on Geometric Quantization, II

Thanks for blogging about this interesting topic. I have a quick question about the following paragraph:

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

objectsof the groupoid, i.e., $X$ itself.Could you provide some clarification on this? Thanks so much!

Cheers, Ari