# Contents

## 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

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

## References

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

or

Relation to spin^c quantization is discussed in

Discussion with an eye towards Theta characteristics is in

Further references include

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

Revised on May 31, 2014 00:52:43 by Urs Schreiber (82.113.99.137)