nLab theta function

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 analogy	theta 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 CFT	partition 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 operator	zeta function of an elliptic differential operator	eta function of a self-adjoint operator	functional determinant, analytic torsion
complex analytic geometry	section $\theta(\mathbf{z},\mathbf{\tau})$ 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 surface	Selberg zeta function		Dedekind eta function
arithmetic geometry for a function field		Goss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number field	Hecke theta function, automorphic form	Dedekind 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 coordinate $\mathbf{z}$ in the Jacobian $J(\Sigma)$ is reflected in the double coordinate dependence of the theta function:

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

See (AlvaresGaumé-Moore-Vafa 86, p. 4, Freed 87, section 4, Falceto-Gawedzki 94, Bunke-Olbrich 94, around def. 4.5, Gelca-Uribe 10a, Gelca-Uribe 10b, Gelca-Hamilton 12, Gelca-Hamilton 14, Gelca 14).

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

• Jacobi theta function

• Riemann theta function

• Ramanujan theta function?

• Dedekind eta function

Related concepts

• the Verlinde formula computes the dimension of spaces of non-abelian theta functions

• special functions

• cubical structure on a line bundle

• mock theta function?

• elliptic function

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

line bundle	square root	choice corresponds to
canonical bundle	Theta characteristic	over Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure
density bundle	half-density bundle
canonical bundle of Lagrangian submanifold	metalinear structure	metaplectic correction
determinant line bundle	Pfaffian line bundle
quadratic secondary intersection pairing	partition function of self-dual higher gauge theory	integral Wu structure

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

• David Mumford, Tata Lectures on Theta, Birkhäuser 1983

• Alexander Polishchuk, section 17 of Abelian varieties, Theta functions and the Fourier transform, Cambridge University Press (2003) (review pdf)

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

• 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)

• Luis Alvarez-Gaumé, Jean-Benoit Bost, Gregory Moore, Philip Nelson, Cumrun Vafa, Bosonization on higher genus Riemann surfaces, Communications in Mathematical Physics, Volume 112, Number 3 (1987), 503-552 (Euclid)

• Daniel Freed, around p. 30-31 of On determinant line bundles, Math. aspects of string theory, ed. S. T. Yau, World Sci. Publ. 1987, (revised pdf, dg-ga/9505002)

• Fernando Falceto, Krzysztof Gawędzki, Chern-Simons states at genus one, Comm. Math. Phys. Volume 159, Number 3 (1994), 549-579. (Euclid)

• Kazuhiro Hikami, Mock (False) Theta Functions as Quantum Invariants (arXiv:math-ph/0506073)

• Kazuhiro Hikami, Quantum Invariants, Modular Forms, and Lattice Points II, J. Math. Phys. 47, 102301-32pages (2006) (arXiv:math/0604091)

• Razvan Gelca, Alejandro Uribe, From classical theta functions to topological quantum field theory (arXiv:1006.3252, slides pdf)

• Razvan Gelca, Alejandro Uribe, Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$ (arXiv:1007.2010)

• Razvan Gelca, Alastair Hamilton, Classical theta functions from a quantum group perspective (arXiv:1209.1135)

• Razvan Gelca, Alastair Hamilton, The topological quantum field theory of Riemann’s theta functions (arXiv:1406.4269)

• Razvan Gelca, Theta Functions and Knots, World Scientific 2014 (prologue pdf, publisher page)

• Johan Martens Jørgen Andersen, notes by Søren Jørgensen, p. 53 of Topological quantum field theories and moduli spaces, 2011 (pdf)

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

• Ulrich Bunke, Martin Olbrich, Theta and zeta functions for locally symmetric spaces of rank one (arXiv:dg-ga/9407013)

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

• Arnaud Beauville, Yves Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), 385 - 419, euclid, alg-geom/9309003, MR1289330

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)