nLab
theta function

Idea

A theta function ($\vartheta$-function, $\Theta$-function) is a holomorphic section of a holomorphic line bundle over a complex torus? / abelian variety.

Expressed in local coordinates ${ℂ}^g$ it becomes an actual function, satisfying certain transformation properties.

Theta functions are naturally thought of as being the states in the geometric quantization of the given complex space, the given holomorphic line bundle being the prequantum line bundle and the condition of holomorphicity of the section being the polarization condition. See for instace (Tyurin).

Examples

• Riemann theta function

• Jacobi theta function

• Ramanujan theta function

Related concepts

• special functions

• mock theta function?

References

Introductions to the traditional notion include

• D.H. Bailey et al, The Miracle of Theta Functions (web)

• M. Bertola, Riemann surfaces and theta functions (pdf)

A modern textbook account is

• Alexander Polishchuk, Abelian varieties, Theta functions and the Fourier transform, Cambridge University Press (2003)

Further discussion with an emphasis of the origin of theta functions in geometric quantization is in

• Arnaud Beuville, Theta functions, old and new (pdf)

• Andrei Tyurin, Quantization, Classical and quantum field theory and theta functions (arXiv:math/0210466v1)

• Yuichi Nohara, Independence of polarization in geometric quantization (pdf)