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.

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.

(Bates-Weinstein, theorem 7.16)

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

Lecture notes include

- Sean Bates, Alan Weinstein,
*Lectures on the geometry of quantization*, (pdf)

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