nLab
Theta characteristic

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

For XX a space equipped with a notion of dimension dimXdim X \in \mathbb{N} and a notion of Kähler differential forms, a Θ\Theta-characteristic of XX is a choice of square root of the canonical characteristic class of XX. See there for more details.

Examples

Over Riemann surfaces

For XX a Riemann surface, the choices of square roots of the canonical bundle correspond to the choice of spin structures (Atiyah, prop. 3.2). For XX of genus gg, there are 2 2g2^{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 mod2mod 2 is a quadratic refinement of the intersection pairing on H 1(X, 2)H^1(X, \mathbb{Z}_2) (Atiyah, theorem 2).

The following table lists classes of examples of square roots of line bundles

line bundlesquare rootchoice corresponds to
canonical bundleTheta characteristicover Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure
density bundlehalf-density bundle
canonical bundle of Lagrangian submanifoldmetalinear structuremetaplectic correction
determinant line bundlePfaffian line bundle
quadratic secondary intersection pairingpartition function of self-dual higher gauge theoryintegral Wu structure

References

The spaces of choice of Θ\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)

  • M. Bertola, Riemann surfaces and Theta Functions, August 2010 (pdf)

Revised on July 17, 2013 17:10:08 by Urs Schreiber (82.169.65.155)