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.

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.

Examples

Over Riemann surfaces

Proposition

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.

(Atiyah, prop. 3.2).

Remark

The first statement remains true in higher dimensions over Kähler manifolds, see at Spin structure – On Kähler manifolds.

Proposition

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).

As metaplectic and Spin structure over (Kähler-)polarized varieties

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.

Over intermediate Jacobians

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).

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 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

References

The spaces of choices of $\Theta$-characteristics over Riemannian manifolds were originally 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

• 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

• Edward Witten, Five-Brane Effective Action In M-Theory, J.Geom.Phys.22:103-133,1997 (arXiv:hep-th/9610234)

and the argument there was made rigorous in

• Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology, and M-Theory

Related arguments revolving around characteristic elements for the intersection pairing appear in

• Bjorn Poonen, Eric Rains, Self cup products and the theta characteristic torsor (arXiv:1104.2105)