The n-Category Café

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 $G$.