For a space equipped with a notion of dimension and a notion of Kähler differential forms, a -characteristic of is a choice of square root of the canonical characteristic class of . See there for more details.
For a Riemann surface, the choices of square roots of the canonical bundle correspond to the choice of spin structures (Atiyah, prop. 3.2). For of genus , there are many choices of square roots of the canonical bundle.
|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|
The spaces of choice of -characteristics over Riemannian manifolds were discussed in
Gavril Farkas, Theta characteristics and their moduli (2012) (arXiv:1201.2557)
M. Bertola, Riemann surfaces and Theta Functions, August 2010 (pdf)