Contents

Contents

Definition

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

Properties

Obstruction and existence

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

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

Relation to metaplectic structure

Theorem

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

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

References

Lecture notes include

Discussion with an eye towards Theta characteristics is in

Last revised on January 2, 2015 at 19:47:59. See the history of this page for a list of all contributions to it.