I am trying to understand the Blattner-Kostant-Sternberg pairing as it applies to geometric quantization in real polarizations whose integral manifolds are, for simplicity, compact. I have been trying to follow Sniatycki's account (Geometric Quantization and Quantum Mechanics, pp 73-75). Thus, we have a symplectic manifold $(X,\omega)$, of dim $2n$ a Hermitian bundle $L$ with connection $\nabla$ whose curvature is $\omega$.
We choose a polarization D by Lagrangian submanifolds, and a bundle of half-forms. We obtain the Bohr-Sommerfeld variety S consisting of points lying in integral submanifolds of D with trivial holonomy. This will be a finite collection of tori $S_i$ if $X$ is compact and $D$ is sufficiently nice. Now we wish to define a pairing on them.
In Sniatycki's account we project each $S_i$ to a $Q_i$ in $X/D$. We wish to obtain a density on each $Q_i$. He does this by lifting a basis for $T_x(Q_i)$ and using $\omega$ to obtain a basis for $S_i$ from this, on which the half-form has a well-defined value.
This is fine so long as $Q_i$ is of dimension $n$. But it seems to me that in many cases, e.g. a torus, the dimension will actually be 0, in which case the prescription breaks down. Obviously I am missing something. Is there a source that discusses this issue in more depth?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
I don't think you're missing anything. Yes, in the case when the Lagrangian foliation consists of tori, the $Q_i$ will just be points. The `density' at one of these points will just be a number (see for example eqn (4.51) - now $k=n$, and so $\langle\sigma_1,\sigma_2\rangle_{Q_i}$ has no arguments), and the inner product will be the sum of these numbers over all the points $Q_i$ above which the sections have support.
The main other source of this material is Śniatycki's papers from the 70s, for example "Wave Functions Relative to a Real Polarization".
