nLab
metalinear structure

Contents

Definition

A metalinear structure on a smooth manifold of dimension nn is a lift of the structure group of the tangent bundle along the group extension Ml(n)GL(n)Ml(n) \to GL(n) of the general linear group by the metalinear group.

Properties

Obstruction and existence

A metalinear structure on a manifold QQ of dimension nn exists precisely if the Chern class of the canonical bundle nT *Q\wedge^n T^*Q is divisible by 2. So a metalinear structure is equivalent to the existence of a square root line bundle nT *Q\sqrt{\wedge^n T^* Q} ( Theta characteristic ).

This means that for EQE \to Q any hermitean line bundle, sections of the tensor product E nT *QE \otimes \sqrt{\wedge^n T^* Q} have a canonical inner product (if QQ is compact and orientable). This is the use of metalinear structure in metaplectic correction.

Relation to metaplectic structure

Theorem

Let (X,ω)(X,\omega) be a symplectic manifold and LTXL \subset T X a subbundle of Lagrangian subspaces of the tangent bundle. Then TXT X admits a metaplectic structure precisely if LL admits a metalinear structure.

(Bates-Weinstein, theorem 7.16)

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

Lecture notes include

Discussion with an eye towards Theta characteristics is in

Revised on January 2, 2015 19:47:59 by Urs Schreiber (127.0.0.1)