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\begin{document}

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\section*{theta function}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{theta_functions}{}\paragraph*{{Theta functions}}\label{theta_functions}

[[!include theta functions - contents]]

\hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry}

[[!include complex geometry - contents]]

\hypertarget{geometric_quantization}{}\paragraph*{{Geometric quantization}}\label{geometric_quantization}

[[!include geometric quantization - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{IdeaInQuantization}{In quantization}\dotfill \pageref*{IdeaInQuantization} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Generally, a \emph{theta function} ($\theta$-function, $\Theta$-function) is a [[holomorphic section]] of a ([[polarized variety|principally polarizing]]) [[holomorphic line bundle]] over a [[complex torus]] / [[abelian variety]]. (e.g. \hyperlink{Polishchuk03}{Polishchuk 03, section 17}) and in particular over a [[Jacobian variety]] (\hyperlink{Beauville}{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]], \hyperlink{Freed87}{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 \emph{[[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]] $\mathcal{M}_\Sigma$ (\hyperlink{Hitchin90}{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 \emph{[[Jacobi theta function]]}. Notice that in their dependency not only on $\tau$ but also on $z$ these are properly called \emph{[[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 \emph{generalized theta functions} (\hyperlink{BeauvilleLaszlo93}{Beauville-Laszlo 93}).

Specifically in the context of [[number theory]]/[[arithmetic geometry]], by \emph{the} theta function one usually means the \emph{[[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 \emph{[[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 \emph{[[function field analogy]]}.

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

\hypertarget{IdeaInQuantization}{}\subsubsection*{{In quantization}}\label{IdeaInQuantization}

Theta functions are naturally thought of as being the [[space of states (in geometric quantization)|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 (\hyperlink{Tyurin02}{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|geometric]] [[quantization of Chern-Simons theory]] in the abelian case, and the [[holographic principle|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:

\begin{displaymath}
\theta(\mathbf{z},\mathbf{\tau})
  = 
  \theta\left(gauge\;field\;configuration\;on\;\Sigma\;, \; complex\;structure\;on\;\Sigma\right)
  \,.
\end{displaymath}
See (\hyperlink{AlvaresGaumeMooreVafa86}{AlvaresGaum\'e{}-Moore-Vafa 86, p. 4}, \hyperlink{Freed87}{Freed 87, section 4}, \hyperlink{FalcetoGawedzki94}{Falceto-Gawedzki 94}, \hyperlink{BunkeOlbrich94}{Bunke-Olbrich 94, around def. 4.5}, \hyperlink{GelcaUribe10a}{Gelca-Uribe 10a}, \hyperlink{GelcaUribe10b}{Gelca-Uribe 10b}, \hyperlink{GelcaHamilton12}{Gelca-Hamilton 12}, \hyperlink{GelcaHamilton14}{Gelca-Hamilton 14}, \hyperlink{Gelca14}{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'' (\hyperlink{BeauvilleLaszlo93}{Beauville-Laszlo 93}).

\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

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

Say that a \emph{system of multipliers} is a system of invertible [[holomorphic functions]]

\begin{displaymath}
e_\gamma \colon V \longrightarrow \mathbb{C}^\times \hookrightarrow \mathbb{C}
\end{displaymath}
satisfying the cocycle condition

\begin{displaymath}
e_{\gamma + \delta}(z) = e_\gamma(z + \delta) e_\delta(z)
  \,.
\end{displaymath}
Then a \emph{theta function} is a [[holomorphic function]]

\begin{displaymath}
\theta \colon V \longrightarrow \mathbb{C}
\end{displaymath}
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

\begin{displaymath}
\theta(z + \gamma) = e_\gamma(z) \theta(z)
  \,.
\end{displaymath}
e.g. (\hyperlink{Beauville}{Beauville, above prop. 2.2}), also (\hyperlink{Beauville}{Beauville, section 3.4})

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[Jacobi theta function]]


\item [[Riemann theta function]]


\item [[Ramanujan theta function]]


\item [[Dedekind eta function]]



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item the [[Verlinde formula]] computes the [[dimension]] of spaces of non-abelian theta functions


\item [[special functions]]


\item [[cubical structure on a line bundle]]


\item [[mock theta function]]


\item [[elliptic function]]



\end{itemize}
[[!include square roots of line bundles - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

Introductions to the traditional notion include

\begin{itemize}%
\item D.H. Bailey et al, \emph{The Miracle of Theta Functions} (\href{http://www.cecm.sfu.ca/organics/papers/borwein/paper/html/node12.html}{web})


\item M. Bertola, \emph{Riemann surfaces and theta functions} (\href{http://www.mathstat.concordia.ca/faculty/bertola/ThetaCourse/ThetaCourse.pdf}{pdf})



\end{itemize}
Modern textbook accounts include

\begin{itemize}%
\item [[David Mumford]], \emph{Tata Lectures on Theta}, Birkh\"a{}user 1983


\item [[Alexander Polishchuk]], section 17 of \emph{Abelian varieties, Theta functions and the Fourier transform}, Cambridge University Press (2003) (\href{http://math1.unice.fr/~beauvill/pubs/poli.pdf}{review pdf})



\end{itemize}
Further discussion with an emphasis of the origin of theta functions in [[geometric quantization]] of [[Chern-Simons theory]] is in

\begin{itemize}%
\item [[Arnaud Beauville]], \emph{Theta functions, old and new}, Open Problems and Surveys of Contemporary Mathematics SMM6, pp. 99--131 (\href{http://math.unice.fr/~beauvill/pubs/thetaon.pdf}{pdf})


\item [[Andrei Tyurin]], \emph{Quantization, Classical and quantum field theory and theta functions}, AMS 2003 (\href{http://arxiv.org/abs/math/0210466}{arXiv:math/0210466v1})


\item Yuichi Nohara, \emph{Independence of polarization in geometric quantization} (\href{http://geoquant.mi.ras.ru/nohara.pdf}{pdf})


\item Gerard Lion, [[Michele Vergne]], \emph{The Weil representation, Maslov index and theta series}



\end{itemize}
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

\begin{itemize}%
\item [[Luis Alvarez-Gaumé]], [[Gregory Moore]], [[Cumrun Vafa]], \emph{Theta functions, modular invariance, and strings}, Communications in Mathematical Physics Volume 106, Number 1 (1986), 1-4 (\href{http://projecteuclid.org/euclid.cmp/1104115581}{Euclid})


\item [[Luis Alvarez-Gaumé]], [[Jean-Benoit Bost]], [[Gregory Moore]], Philip Nelson, [[Cumrun Vafa]], \emph{Bosonization on higher genus Riemann surfaces}, Communications in Mathematical Physics, Volume 112, Number 3 (1987), 503-552 (\href{http://projecteuclid.org/euclid.cmp/1104159982}{Euclid})


\item [[Daniel Freed]], around p. 30-31 of \emph{On determinant line bundles}, Math. aspects of [[string theory]], ed. S. T. Yau, World Sci. Publ. 1987, (revised \href{http://www.math.utexas.edu/~dafr/Index/determinants.pdf}{pdf}, \href{http://arxiv.org/abs/dg-ga/9505002}{dg-ga/9505002})


\item Fernando Falceto, [[Krzysztof Gawędzki]], \emph{Chern-Simons states at genus one}, Comm. Math. Phys. Volume 159, Number 3 (1994), 549-579. (\href{https://projecteuclid.org/euclid.cmp/1104254732}{Euclid})


\item Kazuhiro Hikami, \emph{Mock (False) Theta Functions as Quantum Invariants} (\href{http://arxiv.org/abs/math-ph/0506073}{arXiv:math-ph/0506073})


\item Kazuhiro Hikami, \emph{Quantum Invariants, Modular Forms, and Lattice Points II}, J. Math. Phys. 47, 102301-32pages (2006) (\href{http://arxiv.org/abs/math/0604091}{arXiv:math/0604091})


\item [[Razvan Gelca]], [[Alejandro Uribe]], \emph{From classical theta functions to topological quantum field theory} (\href{http://arxiv.org/abs/1006.3252}{arXiv:1006.3252}, \href{http://www.math.ttu.edu/~rgelca/berk.pdf}{slides pdf})


\item [[Razvan Gelca]], [[Alejandro Uribe]], \emph{Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$} (\href{http://arxiv.org/abs/1007.2010}{arXiv:1007.2010})


\item [[Razvan Gelca]], [[Alastair Hamilton]], \emph{Classical theta functions from a quantum group perspective} (\href{http://arxiv.org/abs/1209.1135}{arXiv:1209.1135})


\item [[Razvan Gelca]], [[Alastair Hamilton]], \emph{The topological quantum field theory of Riemann's theta functions} (\href{http://arxiv.org/abs/1406.4269}{arXiv:1406.4269})


\item [[Razvan Gelca]], \emph{Theta Functions and Knots}, World Scientific 2014 (\href{http://www.worldscientific.com/doi/suppl/10.1142/8872/suppl_file/8872_chap01.pdf}{prologue pdf}, \href{http://www.worldscientific.com/worldscibooks/10.1142/8872}{publisher page})


\item Johan Martens [[Jørgen Andersen]], notes by S\o{}ren J\o{}rgensen, p. 53 of \emph{Topological quantum field theories and moduli spaces}, 2011 (\href{http://maths.fuglede.dk/noter/tqftms.pdf}{pdf})



\end{itemize}
and more generally the [[partition functions]] of connection-twisted Dirac operators on even-dimensional locally symmetric spaces is discussed in

\begin{itemize}%
\item [[Ulrich Bunke]], [[Martin Olbrich]], \emph{Theta and zeta functions for locally symmetric spaces of rank one} (\href{http://arxiv.org/abs/dg-ga/9407013}{arXiv:dg-ga/9407013})

\end{itemize}
Generalization of this from abelian to non-abelian [[conformal blocks]] to ``generalized theta functions'' appears in

\begin{itemize}%
\item [[Arnaud Beauville]], [[Yves Laszlo]], \emph{Conformal blocks and generalized theta functions}, Comm. Math. Phys. \textbf{164} (1994), 385 - 419, \href{http://projecteuclid.org/euclid.cmp/1104270837}{euclid}, \href{http://arxiv.org/abs/alg-geom/9309003}{alg-geom/9309003}, \href{http://www.ams.org/mathscinet-getitem?mr=1289330}{MR1289330}

\end{itemize}
brief review is in

\begin{itemize}%
\item [[Krzysztof Gawedzki]], section 5 of \emph{Conformal field theory: a case study} in Y. Nutku, C. Saclioglu, T. Turgut (eds.) \emph{Frontier in Physics} 102, Perseus Publishing (2000) (\href{http://xxx.lanl.gov/abs/hep-th/9904145}{hep-th/9904145})

\end{itemize}
That the Riemann zeta functions are the local coordinate expressions of the covariantly constant sections of the [[Hitchin connection]] is due to

\begin{itemize}%
\item [[Nigel Hitchin]], remark 4.12 in \emph{Flat connections and geometric quantization}, : Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. (\href{http://projecteuclid.org/euclid.cmp/1104200841}{Euclid})

\end{itemize}
Relation to [[elliptic genera]] (see also at \emph{[[Jacobi form]]})

\begin{itemize}%
\item [[Kefeng Liu]], section 2.4 of \emph{On modular invariance and rigidity theorems}, J. Differential Geom. Volume 41, Number 2 (1995), 247-514 (\href{http://projecteuclid.org/euclid.jdg/1214456221}{EUCLID}, \href{http://www.math.ucla.edu/~liu/Research/loja.pdf}{pdf})

\end{itemize}
Theta functions for higher dimensional varieties and their relation to [[automorphic forms]] is due to

\begin{itemize}%
\item [[André Weil]], \emph{Sur certaines groups d'operateur unitaires}, Acta. Math. 111 (1964), 143-211

\end{itemize}
see \href{Langlands+program#Gelbhart84}{Gelbhart 84, page 35 (211)} for review.

Further developments here include

\begin{itemize}%
\item [[Stephen Kudla]], \emph{Relations between automorphic forms produced by theta-functions}, in \emph{Modular Functions of One Variable VI}, Lecture Notes in Math. 627, Springer, 1977, 277--285.


\item [[Stephen Kudla]], \emph{Theta functions and Hilbert modular forms},Nagoya Math. J. 69 (1978) 97-106


\item [[Jeffrey Stopple]], \emph{Theta and $L$-function splittings}, Acta Arithmetica LXXII.2 (1995) (\href{http://matwbn.icm.edu.pl/ksiazki/aa/aa72/aa7221.pdf}{pdf})


\item Yum-Tong Siu, \emph{Theta functions in higher dimensions} ([[SiuHigherTheta.pdf:file]])



\end{itemize}
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