# nLab metalinear structure

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# 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

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