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 1990, 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:

• David Mumford: Tata Lectures on Theta I, Modern Birkhäuser Classics, Birkhäuser (1983) Springer (2007) [doi:10.1007/978-0-8176-4577-9]

• David Mumford: Tata Lectures on Theta II — Jacobian theta functions and differential equations, Modern Birkhäuser Classics, Birkhäuser (1983), Springer (2007) [doi:10.1007/978-0-8176-4578-6]

Review:

• Ching-Li Chai: Riemann’s theta function [pdf]

• 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 106 1 (1986) 1-4 [doi:10.1007/BF01210925, euclid:cmp/1104115581]

Discussion as covariantly constant sections of the Hitchin connection:

• Nigel Hitchin: Flat connections and geometric quantization, Comm. Math. Phys. 131 2 (1990) 347-380 [euclid:cmp/1104200841]

As the quantum states of abelian Chern-Simons theory 3D TQFT:

• Răzvan Gelca, Alejandro Uribe: From classical theta functions to topological quantum field theory, in: The Influence of Solomon Lefschetz in Geometry and Topology: 50 Years of Mathematics at CINVESTAV, Contemporary Mathematics 621, AMS (2014) 35-68 [arXiv:1006.3252, doi;10.1090/conm/621, ams:conm-621, slides pdf, pdf]

• Răzvan Gelca, Alastair Hamilton: Classical theta functions from a quantum group perspective, New York J. Math. 21 (2015) 93–127 [arXiv:1209.1135, nyjm:j/2015/21-4]

• Răzvan Gelca, Alastair Hamilton: The topological quantum field theory of Riemann’s theta functions, Journal of Geometry and Physics 98 (2015) 242-261 [doi:10.1016/j.geomphys.2015.08.008, arXiv:1406.4269]

Further discussion:

• Bernard Deconinck, Matthias Heil, Alexander Bobenko, Mark van Hoeij, Markus Schmies: Computing Riemann Theta Functions [arXiv:nlin/0206009]