nLab
metaplectic correction (in geometric quantization)

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,ω)(X, \omega) induces a metalinear structure on each Lagrangian submanifold QXQ \hookrightarrow X of a given foliation by Lagrangian submanifolds (polarization). This allows to form a square root line bundle Λ nT *Q\sqrt{\Lambda^n T^* Q} of the canonical bundle of QQ and hence induces an inner product on sections of the tensor product E QΛ nT *QE|_Q \otimes \sqrt{\Lambda^n T^* Q} with the restriction of any line bundle EE on XX (a prequantum line bundle, notably).

Relation to SpinSpin-structure and Spin cSpin^c-structure

Let (X,ω)(X,\omega) be a compact symplectic manifold equipped with a Kähler polarization 𝒫\mathcal{P} hence a Kähler manifold structure JJ. A metaplectic structure is now a choice of square root Ω n,0\sqrt{\Omega^{n,0}} of the canonical line bundle Ω n,0\Omega^{n,0}. This is equivalently a spin structure on XX (see the discussion at spin structure – over Kähler manifolds).

Now given a prequantum line bundle L ωL_\omega, in this case the Dolbault quantization of L ωL_\omega coincides with the spin^c quantization of the spin^c structure induced by JJ and L ωΩ n,0L_\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

line bundlesquare rootchoice corresponds to
canonical bundleTheta characteristicover Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure
density bundlehalf-density bundle
canonical bundle of Lagrangian submanifoldmetalinear structuremetaplectic correction
determinant line bundlePfaffian line bundle
quadratic secondary intersection pairingpartition function of self-dual higher gauge theoryintegral Wu structure

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 July 14, 2013 20:35:02 by Urs Schreiber (82.113.106.95)