# nLab Theta characteristic

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

cohomology

# Contents

## Idea

For $X$ a space equipped with a notion of dimension $\mathrm{dim}X\in ℕ$ 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 ${H}^{1}\left(X,{ℤ}_{2}\right)$ (Atiyah, theorem 2).

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

## 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

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

Revised on July 10, 2012 16:43:44 by Urs Schreiber (89.204.138.228)