nLab metalinear structure

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.

(Bates-Weinstein, theorem 7.16)

Related concepts

• metaplectic structure, metaplectic correction

• spin structure

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

line bundle	square root	choice corresponds to
canonical bundle	Theta characteristic	over Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure
density bundle	half-density bundle
canonical bundle of Lagrangian submanifold	metalinear structure	metaplectic correction
determinant line bundle	Pfaffian line bundle
quadratic secondary intersection pairing	partition function of self-dual higher gauge theory	integral Wu structure

References

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)