The n-Category Café

A word on geometric quantization.

Let me quickly recall what geometric quantization is about:

You all know that the Hilbert space of states of a particle charged under a hermitean line bundle with connection $(E \to X, \nabla)$ on a Riemannian manifold $X$ is that of square-integrable sections of $E$: $$Z(\bullet) = \Gamma^2(E) \,.$$

Sometimes it is useful to consider the situation not on $X$ but on the cotangent bundle $$\pi : T^* X \to X$$ – the phase space – and one wishes to concentrate on the pulled back bundle $$(\pi^* E \to T^ X, \pi^* \nabla) \,.$$ But since this now has more sections than the original bundle had we need to keep some information about $X$ around to be able to cut down the space of sections of $\pi^* E$ back to something isomorphic to $\Gamma^2(E)$. A slick way to do this is to notice that we can add to the pulled back connection $\pi^* \nabla$ the canonical 1-form $\alpha \in \Omega^1(T^* X)$, defined by $$\alpha : (v \in T_\omega T^* X) \mapsto \omega(\pi_*(v))$$.

(See for instance the beginning of That shift in dimension for a related discussion.)

This way, the line bundle with the modified connection $$(\pi^* E, \pi^* \nabla + \alpha)$$ is guaranteed to have a curvature 2-form $$F_{\pi^* \nabla + \alpha} = \pi^* F_\nabla + d\alpha$$ which has the special property that it is not only closed, as all curvature 2-forms, but also nondegenerate in that $$\omega(-,\cdot) : T T^*X \to \Lambda^2 T^* X$$ is a fiberwise isomorphism.

Over a contractible patch $U \subset X$ with coordinates $\{q^i : X \to \mathbb{R}\}$ and induced coordinates $\{q^i, p_i : T^* U \simeq U \times \mathbb{R}^n \to \mathbb{R}\}$ of the cotangent bundle $\alpha$ has the simple form $$\alpha = \sum_i p_i d q^i$$ (see the recent discussion here for more and generalizations on this) and the symplectic curvature 2-form is $$\omega := d \alpha = \sum_i d p_i \wedge d q^i + (F_\nabla)_{ij} d q^i \wedge d q^j$$ so that the fiberwise isomorphism induced by $\omega$ is essentially just that which comes from exchanging the two factors in $T^* \mathbb{R}^n \simeq \mathbb{R}^n \times \mathbb{R}^n$.

This makes it clear that the original sections of $E$ correspond to those sections of $\pi^* E$ which are constant along the fiber directions – independent, locally, of the $p_i$ – and that we can express this as follows:

The sections of $E$ correspond to those sections of $\pi^* E$ which are maximally covariantly constant without being trivial.

Namely choose any integrable distribution $F \subset T T^* X$, i.e. a subspace of vectors in each fiber such that these subspaces glue to a smooth subbundle whose sections are closed under the bracket of vector fields, with the property that

- this distribution is isotropic with respect to $\omega$, i.e. $\omega|_F = 0$

- this distribution is maximal with this property: no vector can be added to $F$ without destroying this property

then this distribution is called a polarization and

The sections of $E$ correspond to those sections of $\pi^* E$ which are covariantly constant along all $v \in F$: $$\psi \in \pi^* E, \, (\pi^*\nabla + \alpha)_{v \in F} \psi = 0$$

(To quickly see this notice that in local coordinates we can choose for instance in particular $F = \langle \frac{\partial}{\partial p_i}\rangle_i$.)

So we found the following way of talking about the space of states of the charged particle on $X$:

1) pull the situation back to phase space $T^* X$;

2) modify the connection of the pulled back bundle such that the curvature 2-form becomes non-degenerate and hence symplectic.

3) The original space of states is that of maximally covariantly constant sections of the pulled back bundle with the modified connection.

(Here the last statement is true only modulo some issues arising with dealing with the integration measure. See the discussion on pages 10-11 on Eli Hawkins’s article for more on that.)

So, finally, geometric quantization is the process of making sense of this situation after forgetting step 1). More generally, the process of making sense of this when we are just handed any space carrying a symplectic integral 2-form, which need not be the cotangent bundle of some underlying space.

Such a symplectic manifold is a classical phase space and geometric quantization says that a quantum space of states obtained from this is

- choosing a line bundle with connection realizing the integral (and symplectic) 2-form;

- choosing a polarization of the symplectic (and integral) 2-form;

- forming the space of sections covariantly constant along the polarization.

and dealing with finding an inner product on that space and turning it into a Hilbert space.

Given that Hilbert space, the algebra of observables is the $C^*$-algebra of bounded operators on it.

Eli Hawkins’ proposal

Eli Hawkins points out that there is a shortcut to this procedure if one is just interested in the algebra of observables:

the symplectic form $\omega$ gives the cotangent bundle of any space the structure of a Lie algebroid. This Lie algebroid integrates to some simply connected Lie groupoid or various quotients of that. This is naturally a symplectic Lie groupoid. The presence of the line bundle gives rise to a central extension of that Lie groupoid. And finally, that’s, as far as I understand, Eli’s main contribution here, the polarization gives rise to a notion of groupoid polarization.

And with all that in hand, the algebra of observables is nothing but the polarized convolution algebra/category algebra of the centrally extended symplectic groupoid.

A symplectic groupoid should be a groupoid internal to symplectic manifolds, though some slight variants are used here. The important point to notice to quickly see how the above can make sense is this:

the simply connected groupoid integrating the tangent Lie algebroid of a manifold $X$ is $\Pi_1(X)$. A quotient of this which still integrates the tangent Lie algebroid is the pair groupoid $X \times X$. If $X$ is a symplectic manifold, then $X \times X$ is naturally a symplectic groupoid, where the symplectic structure on the space of morphisms is the pullback of the one downstairs by the source map minus the pullback by the target map.

Now, the convolution algebra/category algebra of the pair groupoid of a finite set is just the matrix algebra on the vector spaces spanned by that finite set, which is the algebra of endomorphisms of that vector space. Eli Hawkins’ proposal is a generalization of this fact to the case that

- finite sets are replaced by manifolds

- the twist induced by the line bundle classified by the symplectic form is taken into account

- the polarization induced by that symplectic form is taken into account.

I have to run now. Maybe more later.