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.
In complex analytic geometry and at least if the Theta characteristic is principally polarizing then its holomorphic sections are called theta functions. In particular for line bundles over the Jacobian variety of a Riemann surface they are called Riemann theta functions.
For of genus , there are many choices of square roots of the canonical bundle.
In the context of geometric quantization a metaplectic structure on a polarization is a square root of a certain line bundle. In the special case of Kähler polarization this is a square root precisely of the canonical line bundle of the underlying complex manifold and hence is a -characteristic. Also, equivalently this is a Spin structure, see at spin structure – Over a Kähler manifold. For more on this see at geometric quantization – Quantum states as index of Dolbeault-Dirac operator.
Notice that generalizing from complex analytic geometry to algebraic geometry over other bases, then the analog of a Kähler polarization is a polarized variety. Hence a choice of Theta characteristic on a polarized variety is the analog of a metaplectically corrected Kähler manifold.
A special square root of the canonical bundle on intermediate Jacobians in dimension thought of as moduli spaces of (flat) circle (2k+1)-bundles with connection has a unique section the partition function of abelian self-dual higher gauge theory (see there for details). (Witten 96, Hopkins-Singer 02).
|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 choices of -characteristics over Riemannian manifolds were originally discussed in
M. Bertola, Riemann surfaces and Theta Functions, August 2010 (pdf)
Gavril Farkas, Theta characteristics and their moduli (2012) (arXiv:1201.2557)
and the argument there was made rigorous in