complex geometry

# Contents

## Idea

### General

Generally, a theta function ($\theta$-function, $\Theta$-function) is a holomorphic section of a (principally polarizing) holomorphic line bundle over a complex torus / abelian variety. (e.g. Polishchuk 03, section 17) and in particular over a Jacobian variety (Beauville) such as prequantum line bundles for (abelian) gauge theory. The line bundle being principally polarizing means that its space of holomorphic sections is 1-dimensional, hence that it determines the $\theta$-function up to a global complex scale factor. Typically these line bundles themselves are Theta characteristics. Expressed in coordinates $\mathbf{z}$ on the covering $\mathbb{C}^g$ of the complex torus $\mathbb{C}^g/\mathbb{Z}^g$, a $\theta$-function appears as an actual function $\mathbf{z} \mapsto \theta(\mathbf{z})$ satisfying certain transformation properties, and this is how theta functions are considered.

Those theta functions encoding sections of line bundles on a Jacobian variety $J(\Sigma)$ of a Riemann surface $\Sigma$ (determinant line bundles, Freed 87, pages 30-31) typically vary in a controlled way with the complex structure modulus $\mathbf{\tau}$ of $\Sigma$ and are hence really functions also of this variable $(\mathbf{z},\mathbf{\tau}) \mapsto \theta(\mathbf{z}, \mathbf{\tau})$ with certain transformation properties. These are the Riemann theta functions. They are the expressions in local coordinates of the covariantly constant sections of the Hitchin connection on the moduli space of Riemann surfaces $\mathcal{M}_\Sigma$ (Hitchin 90, remark 4.12). In the special case that $\Sigma$ is complex 1-dimensional of genus $g = 1$ (hence a complex elliptic curve) then such a function $(z,\tau) \mapsto \theta(z,\tau)$ of two variables with the pertinent transformation properties is a Jacobi theta function. Notice that in their dependency not only on $\tau$ but also on $z$ these are properly called Jacobi forms. Finally notice that these line bundles on Jacobian varieties have non-abelian generalizations to line bundles on moduli stacks of vector bundles of rank higher than one, whose sections may then be thought of as generalized theta functions (Beauville-Laszlo 93).

Specifically in the context of number theory/arithmetic geometry, by the theta function one usually means the Jacobi theta function (see there for more) for $z = 0$. While this is the historically first and archetypical function from which all modern generalizations derive their name, notice that at fixed $z$ as a function in $\tau$ the “theta function” is not actually a section of a line bundle anymore. The generalization in number theory of the Jacobi theta function that does again have a dependence on a twisting is the Dirichlet theta function depending on a Dirichlet character (which by Artin reciprocity corresponds to a Galois representation).

Certain integrals of theta functions yield zeta functions, see also at function field analogy.

context/function field analogytheta function $\theta$zeta function $\zeta$ (= Mellin transform of $\theta(0,-)$)L-function $L_{\mathbf{z}}$ (= Mellin transform of $\theta(\mathbf{z},-)$)eta function $\eta$special values of L-functions
physics/2d CFTpartition function $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of complex structure $\mathbf{\tau}$ of worldsheet $\Sigma$ (hence polarization of phase space) and background gauge field/source $\mathbf{z}$analytically continued trace of Feynman propagator $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$analytically continued trace of Feynman propagator in background gauge field $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$analytically continued trace of Dirac propagator in background gauge field $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s$regularized 1-loop vacuum amplitude $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / regularized fermionic 1-loop vacuum amplitude $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / vacuum energy $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$
Riemannian geometry (analysis)zeta function of an elliptic differential operatorzeta function of an elliptic differential operatoreta function of a self-adjoint operatorfunctional determinant, analytic torsion
complex analytic geometrysection $\theta(\mathbf{z},\mathbf{\tau})$ of line bundle over Jacobian variety $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$zeta function of a Riemann surfaceSelberg zeta functionDedekind eta function
arithmetic geometry for a function fieldGoss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number fieldHecke theta function, automorphic formDedekind zeta function (being the Artin L-function $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the trivial Galois representation)Artin L-function $L_{\mathbf{z}}$ of a Galois representation $\mathbf{z}$, expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps)class number $\cdot$ regulator
arithmetic geometry for $\mathbb{Q}$Jacobi theta function ($\mathbf{z} = 0$)/ Dirichlet theta function ($\mathbf{z} = \chi$ a Dirichlet character)Riemann zeta function (being the Dirichlet L-function $L_{\mathbf{z}}$ for Dirichlet character $\mathbf{z} = 0$)Artin L-function of a Galois representation $\mathbf{z}$ , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function

### In quantization

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 instance (Tyurin 02). In this context they play a proming role specifically in the quantization of higher dimensional Chern-Simons theory and of self-dual higher gauge theory. See there for more.

Specifically the fact that in geometric quantization of Chern-Simons theory in the abelian case, and the holographically dual partition functions of the WZW model the choice of polarization is induced from the choice of complex structure $\mathbf{\tau}$ on a given Riemann surface $\Sigma$ and for each such choice there is then a section/partition function depending on a coordinte $\mathbf{z}$ in the Jacobian $J(\Sigma)$ is reflected in the double coordinate dependence of the theta function:

$\theta(\mathbf{z},\mathbf{\tau}) = \theta\left(gauge\;field\;configuration\;on\;\Sigma\;, \; complex\;structure\;on\;\Sigma\right) \,.$

Since from the point of view of Chern-Simons theory this is a wavefunction, one might rather want to write $\Psi(\mathbf{z},\mathbf{\tau})$.

For nonabelian CS/WZW theory the same story goes through and one may the elements of the corresponding conformal blocks “generalized theta functions” (Beauville-Laszlo 93).

## Definition

Consider a complex torus $T \simeq V/\Gamma$ for given finite group $\Gamma$.

Say that a system of multipliers is a system of invertible holomorphic functions

$e_\gamma \colon V \longrightarrow \mathbb{C}^\times \hookrightarrow \mathbb{C}$

satisfying the cocycle condition

$e_{\gamma + \delta}(z) = e_\gamma(z + \delta) e_\delta(z) \,.$

Then a theta function is a holomorphic function

$\theta \colon V \longrightarrow \mathbb{C}$

for which there is a system of multipliers $\{e_\gamma\}$ satisfying the functional equation which says that for each $z \in V$ and $\gamma \in \Gamma \hookrightarrow V$ we have

$\theta(z + \gamma) = e_\gamma(z) \theta(z) \,.$

e.g. (Beauville, above prop. 2.2), also (Beauville, section 3.4)

## Examples

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

## 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)

Modern textbook accounts include

Further discussion with an emphasis of the origin of theta functions in geometric quantization of Chern-Simons theory is in

• Arnaud Beauville, Theta functions, old and new, Open Problems and Surveys of Contemporary Mathematics SMM6, pp. 99–131 (pdf)

• Andrei Tyurin, Quantization, Classical and quantum field theory and theta functions, AMS 2003 (arXiv:math/0210466v1)

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

• Gerard Lion, Michele Vergne, The Weil representation, Maslov index and theta series

Specifically the theta functions appearing in 2d CFT as conformal blocks and as spaces of sections of prequantum line bundles in quantization of Chern-Simons theory are discussed for instance in

and more generally the partition functions of connection-twisted Dirac operators on even-dimensional locally symmetric spaces is discussed in

Generalization of this from abelian to non-abelian conformal blocks to “generalized theta functions” appears in

brief review is in

• Krzysztof Gawedzki, section 5 of Conformal field theory: a case study in Y. Nutku, C. Saclioglu, T. Turgut (eds.) Frontier in Physics 102, Perseus Publishing (2000) (hep-th/9904145)

That the Riemann zeta functions are the local coordinate expressions of the covariantly constant sections of the Hitchin connection is due to

• Nigel Hitchin, remark 4.12 in Flat connections and geometric quantization, : Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. (Euclid)

Relation to elliptic genera (see also at Jacobi form)

• Kefeng Liu, section 2.4 of On modular invariance and rigidity theorems, J. Differential Geom. Volume 41, Number 2 (1995), 247-514 (EUCLID, pdf)

Theta functions for higher dimensional varieties and their relation to automorphic forms is due to

• André Weil, Sur certaines groups d’operateur unitaires, Acta. Math. 111 (1964), 143-211

see Gelbhart 84, page 35 (211) for review.

Further developments here include

• Stephen Kudla, Relations between automorphic forms produced by theta-functions, in Modular Functions of One Variable VI, Lecture Notes in Math. 627, Springer, 1977, 277–285.

• Stephen Kudla, Theta functions and Hilbert modular forms,Nagoya Math. J. 69 (1978) 97-106

• Jeffrey Stopple, Theta and $L$-function splittings, Acta Arithmetica LXXII.2 (1995) (pdf)

• Yum-Tong Siu, Theta functions in higher dimensions (pdf)

Revised on July 19, 2015 13:11:54 by Urs Schreiber (94.118.157.94)