nLab metaplectic correction (in geometric quantization)

Idea

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.

Properties

Induced inner product / Hilbert space structure

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).

Relation to $Spin$-structure and $Spin^c$-structure

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).

Relation to geometric quantization

See at

• geometric quantization – Quantum space of states.

Related concepts

• metaplectic group

The following table lists classes of examples of square roots of line bundles

line bundle	square root	choice corresponds to
canonical bundle	Theta characteristic	over Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure
density bundle	half-density bundle
canonical bundle of Lagrangian submanifold	metalinear structure	metaplectic correction
determinant line bundle	Pfaffian line bundle
quadratic secondary intersection pairing	partition function of self-dual higher gauge theory	integral Wu structure

References

For general discussion see the references listed at geometric quantization, for instance the introduction in section 7.2 of

• Sean Bates, Alan Weinstein, Lectures on geometric quantization (pdf)

or

• Martin Schottenloher, Metaplectic reduction (pdf)

• Alexander Cardona, Geometric and metaplectic quantization (pdf)

Relation to spin^c quantization is discussed in

• Paul-Emile Paradan, Spin-quantization commutes with reduction, J. Symplectic Geom. Volume 10, Number 3 (2012), 389-422. (arXiv:0911.1067, Euclid)

Discussion with an eye towards Theta characteristics is in

• Andrei Tyurin, Quantization, classical and quantum field theory and Theta-functions (arXiv:math/0210466v1)

Further references include

• L. Charles, Semi-classical properties of geometric quantization with metaplectic correction (arXiv:math.SG/0602168)