nLab metaplectic correction (in geometric quantization)




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.


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 (a “Theta characteristic”, see below) 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} (a Theta characteristic for the complex analytic space XX). 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


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


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)

Last revised on January 2, 2015 at 19:33:04. See the history of this page for a list of all contributions to it.