Let $(M,\omega)$ be a compact symplectic manifold and $(L,\nabla)$ a prequantum line bundle. There are two schemes to quantize this data:

  1. Choose a polarization $P$ of $M$ and define the quantum Hilbert space to be sections of $L$ that are parallel along $P$. This space admits an action of the Poisson algebra of smooth functions on $M$.
  2. Choose a compatible almost complex structure. This gives a $Spin^c$ structure and define the quantum space to be the index of the corresponding Dirac operator twisted by $L$. Here there is no action of $C^\infty(M)$. When a Lie group $G$ acts on everything, the index is an element of the representation ring of $G$.

In the case that $M$ is Kähler, $L$ a holomorphic line bundle, and $P = T^{0,1}M$, the first method gives the space of holomorphic sections of $L$ and the second method gives the index of $\bar\partial_L + \bar\partial_L^*$.

The first method is more tied to physics and may seem a little ad-hoc from a mathematical point of view. The second seems more natural mathematically (since it fits in well with symplectic reduction and the index of an elliptic operator is more well-behaved mathematically than the kernel of $\nabla$ along $P$). But it seems to be a very weak notion of quantization since there is no action of $C^\infty(M)$. .

How can these two viewpoints be reconciled? Should the second version be viewed as just a more natural mathematical construction? Is there a nice way to tie it back to the physics?

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    $\begingroup$ This is a good question. I think has a good answer if one thinks of the Spin^c-quantization as taking values in KK-cocycles, as promoted particularly in articles by Landsman. But before we get to that, a remark: it is not true that all functions on phase space act on the geometric quantum states, but only those whose flow fixes the polarization leafs do. This is in general a rather small subspace of functions/observables. But still, it is a good question to ask how push-forward quantization can remember this. $\endgroup$ – Urs Schreiber Apr 11 '13 at 19:51
  • $\begingroup$ @Urs Schreiber: Thanks for your comment. I've skimmed through some of Landsman's work and it does seem to incorporate both notions of quantization. Indeed, it seems that this question is one of his motivating factors. $\endgroup$ – Eric O. Korman Apr 12 '13 at 2:27

Klaas Landsman has been proposing that geometric quantization is best understood as taking values in KK-cocycles. (A quick review of this idea is for instance around p. 134 of his student's thesis available here.) If one remembers the KK-cocycle before passing it through the Baum-Connes assembly map to compute its index (which is supposedly an iso anyway...), then this means remembering the module structure that you are asking for.

From this and various other perspectives it looks like regarding geometric quantization as a map into KK-theory is a rather attractive idea.

A collection of relevant references by Landsman, his students, and related articles is here: http://ncatlab.org/nlab/show/geometric+quantization+by+push-forward#References


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