nLab Riemann theta function

Idea

The Riemann theta functions are a special class of theta functions. They are coordinate-expressions for the (essentially unique) holomorphic sections of certain line bundles over Jacobian varieties of compact Riemann surfaces $\Sigma$.

As such these are functions of two parameters, a complex modulus $\mathbf{\tau}$ varying in the Siegel upper half-space and encoding the complex structure on $\Sigma$, as well as a parameter $\mathbf{z}$ varying in $\mathbb{C}^g$ and encoding a point in the Jacobian variety $\mathbb{C}^g/\mathbb{Z}^g$. As such the Riemann theta functions are the local coordinate expressions of the covariantly constant sections of the Hitchin connection on the moduli space of Riemann surfaces (for circle group gauge group) (Hitchin 90, remark 4.12).

The partition function of a 2d CFT/string is naturally a Riemann theta function (AlvaresGaume-Moore-Vafa 86).

Related concepts

• Theta characteristic

References

A standard account is

• David Mumford, Tata Lectures on Theta, Birkhäuser 1983

Review includes

• Wikipedia, Riemann theta function

A review with an eye towards the interpretation of the Riemann theta functions as partition functions of 2d CFT/string models is in

• Luis Alvarez-Gaumé, Gregory Moore, Cumrun Vafa, Theta functions, modular invariance, and strings, Communications in Mathematical Physics Volume 106, Number 1 (1986), 1-4 (Euclid)

Discussion as covariantly constant sections of the Hitchin connection is in

• Nigel Hitchin, Flat connections and geometric quantization, : Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. (Euclid)