A metalinear structure on a manifold of dimension exists precisely if the Chern class of the canonical bundle is divisible by 2. So a metalinear structure is equivalent to the existence of a square root line bundle ( Theta characteristic ).
This means that for any hermitean line bundle, sections of the tensor product have a canonical inner product (if is compact and orientable). This is the use of metalinear structure in metaplectic correction.
|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|
Lecture notes include
Discussion with an eye towards Theta characteristics is in