metalinear structure

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.

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.

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

Discussion with an eye towards Theta characteristics is in

- Andrei Tyurin,
*Quantization, classical and quantum field theory and Theta-functions*(arXiv:math/0210466v1)

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