nLab
Theta characteristic

Idea

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.

Examples

Over Riemann surfaces

For $X$ a Riemann surface, the choices of square roots of the canonical bundle correspond to the choice of spin structures (Atiyah, prop. 3.2). For $X$ of genus $g$, there are $2^{2g}$ many choices of square roots of the canonical bundle.

The function that sends a square root line bundle to the dimension of its space of holomorphic sections $\mathrm{mod}2$ is a quadratic refinement of the intersection pairing on $H^1(X,{\mathbb{Z}}_2)$ (Atiyah, theorem 2).

Related concepts

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: 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

The spaces of choice of $\Theta$-characteristics over Riemannian manifolds were discussed in

• Michael Atiyah, Riemann surfaces and spin structures, Annales Scientifiques de l’École Normale Supérieure, (1971), Quatrième Série 4: 47–62, ISSN 0012-9593, MR0286136

See also

• Gavril Farkas, Theta characteristics and their moduli (2012) (arXiv:1201.2557)