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 $Q\hookrightarrow X$. This allows to form a square root line bundle $\sqrt{{\Lambda }^n{T}^{*}Q}$ of the canonical bundle of $Q$ and hence induces an inner product on sections of the tensor product $E{\mid }_Q\otimes \sqrt{{\Lambda }^n{T}^{*}Q}$ with the restriction of any line bundle $E$ on $X$ (a prequantum line bundle, notably).

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: 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)

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)