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