group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $X$ a space equipped with a notion of dimension $dim X \in \mathbb{N}$ and a notion of Kähler differential forms, a $\Theta$-characteristic of $X$ is a choice of square root of the canonical characteristic class of $X$. 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 $\Sigma$ a Riemann surface, the choices of square roots of the canonical bundle correspond to the choice of spin structures.
For $X$ of genus $g$, there are $2^{2g}$ many choices of square roots of the canonical bundle.
The first statement remains true in higher dimensions over Kähler manifolds, see at Spin structure – On Kähler manifolds.
The function that sends a square root line bundle to the dimension of its space of holomorphic sections $mod \;2$ is a quadratic refinement of the intersection pairing on $H^1(X, \mathbb{Z}_2)$.
This is due to (Atiyah, theorem 2). A motivational survey in broader context of quadratic refinements of the intersection pairing in higher dimensions is in (Hopkins-Singer 02, section 2.1).
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 $\Theta$-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 $2k+1$ 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).
The following table lists classes of examples of square roots of line bundles
The spaces of choices of $\Theta$-characteristics over Riemannian manifolds were originally discussed in
See also
M. Bertola, Riemann surfaces and Theta Functions, August 2010 (pdf)
Gavril Farkas, Theta characteristics and their moduli (2012) (arXiv:1201.2557)
The relation of Theta characteristics on intermediate Jacobians to self-dual higher gauge theory was first recognized in
and the argument there was made rigorous in
Related arguments revolving around characteristic elements for the intersection pairing appear in