geometric quantization higher geometric quantization
geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
In the context of geometric quantization a metaplectic correction is a choice of metaplectic structure on the given symplectic manifold. It allows to make the space of states into a Hilbert space.
It is called a correction mostly for historical reasons, since it was not included in all constructions from the beginning.
A metaplectic structure on a symplectic manifold $(X, \omega)$ induces a metalinear structure on each Lagrangian submanifold $Q \hookrightarrow X$ of a given foliation by Lagrangian submanifolds (polarization). This allows to form a square root line bundle $\sqrt{\Lambda^n T^* Q}$ of the canonical bundle of $Q$ (a “Theta characteristic”, see below) and hence induces an inner product on sections of the tensor product $E|_Q \otimes \sqrt{\Lambda^n T^* Q}$ with the restriction of any line bundle $E$ on $X$ (a prequantum line bundle, notably).
Let $(X,\omega)$ be a compact symplectic manifold equipped with a Kähler polarization $\mathcal{P}$ hence a Kähler manifold structure $J$. A metaplectic structure is now a choice of square root $\sqrt{\Omega^{n,0}}$ of the canonical line bundle $\Omega^{n,0}$ (a Theta characteristic for the complex analytic space $X$). This is equivalently a spin structure on $X$ (see the discussion at spin structure – over Kähler manifolds).
Now given a prequantum line bundle $L_\omega$, in this case the Dolbault quantization of $L_\omega$ coincides with the spin^c quantization of the spin^c structure induced by $J$ and $L_\omega \otimes \sqrt{\Omega^{n,0}}$.
This appears as (Paradan 09, prop. 2.2).
See at
The following table lists classes of examples of square roots of line bundles
For general discussion see the references listed at geometric quantization, for instance the introduction in section 7.2 of
or
Martin Schottenloher, Metaplectic reduction (pdf)
Alexander Cardona, Geometric and metaplectic quantization (pdf)
Relation to spin^c quantization is discussed in
Discussion with an eye towards Theta characteristics is in
Further references include