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.

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

Discussion with an eye towards Theta characteristics is in

Revised on July 10, 2012 18:28:06 by Urs Schreiber (134.76.83.9)