\item The subspace $\cS_k$ of weight $k$ cusp forms is empty for $k < 12$, while $\cS_{12}= \CC \Delta$;
\item For $k \geq 16$, we have $\cS_k = \Delta \cM_{k-12}$ and $\cM_k = \CC G_k \oplus \cS_k$.
\end{enumerate}}
For low weight these results are consistent with the results of (\ref{Glow}). Moreover, the theorem implies that $\cM$ is precisely the polynomial ring freely generated by $G_4$ and $G_6$.

\sm

To prove Theorem \ref{theorem2}, we  use the valence formula (\ref{thm1a}) of Theorem \ref{thm:3.1a}. Considering the relation (\ref{thm1a}) multiplied by 12, and then mod 2, we see that there can  be no solutions for $k$ odd.
Since modular forms have $\ord_f(p)  \geq 0$, including at the points $p=i, \rho,$ and $i \infty$, the left side of (\ref{thm1a}) is non-negative so that there can be no solutions to (\ref{thm1a}) for $k<0$. Clearly, there is also no solution when $k=2$, which concludes the proof of point 1. For $k$ even and positive, there can only be a finite number of solutions since the sum on the left of (\ref{thm1a}) is over positive or zero numbers only, proving point 2. To proceed further, it is helpful first to prove two auxiliary statements,
\bea
\label{3.G4G6}
G_4(\rho) = G_6 (i)=0
\eea
They follows from inspection of the sums,
\bea
G_4 (\rho) & = &
\sum _{(m,n)}'  { 1 \over ( m \, \rho^3 + n \, \rho )^4} = { 1 \over \rho^4} \sum _{(m,n)}'  { 1 \over ( m\, \rho^2 +  n)^4}
= \rho ^2 G_4(\rho)
\no \\
G_6 (i) & = &
\sum _{(m,n)}'  { 1 \over ( m \, i^4 + n \, i )^6} = { 1 \over i^6 } \sum _{(m,n)}'  { 1 \over ( m \, i^3 +  n)^6}
= - G_6(i)
\eea
To recast the second sum on the first line, we use $m\rho^2+n = m(-1-\rho)+n = n-m - m \rho $ so that the sum over $(m,n)$ gives $G_4(\rho)$. To recast the second sum on the second line,  we use $m \, i^3 +n = -i m +n$, so that the sum over $(m,n)$ gives $G_6(i)$.

\sm

By inspection of (\ref{thm1a}), we must have $\ord_f (i \infty)=0$ for $k=4,6, 8, 10$ since the right side of (\ref{thm1a})  is less than 1, as well as for $k=14$ since the remaining $1/6$ cannot be accounted for by non-vanishing $\ord_f(i)$ or $\ord_f(\rho)$. Therefore, the  solutions to (\ref{thm1a}) for these values of $k$  are unique and respectively given by $(\ord_f(i), \ord_f(\rho))= (0,1), \, (1,0), \, (0,2), \, (1,1), \, (1,2)$. They are generated by $G_4$, $G_6$, $G_8\sim G_4^2$, $G_{10} \sim G_4 G_6$, $G_{14} \sim G_4^2G_6$, which proves point 3. To prove point 4, we note that a cusp form has $\ord _f(i \infty) \geq 1$, which requires $k \geq 12$, and uniqueness for $k=12$. Finally for $k\geq 16$, the solutions manifestly satisfy the recursion relation $\cS_k = \Delta \cM_{k-12}$ and $\cM_k = \CC G_k \oplus \cS_k$ by inspecting (\ref{thm1a}), proving point  5.

\subsection{Modular functions and the $j$-function}
\label{sec:jfunction}

Having classified all modular forms under $SL(2,\ZZ)$ of weight $k$, we now classify all meromorphic modular functions of weight $k$. Given an arbitrary meromorphic modular function of weight $k$, we may always divide by a modular form of weight $k$, such as $G_k$, to obtain a meromorphic modular function of weight 0. Thus, it will suffice to understand the space of modular functions of weight~0. Sums, products, and ratios of modular functions of weight~0 produce again modular functions of weight 0, so that  the corresponding space forms the field of modular functions of weight 0. We seek to obtain a complete description of this space.

\sm

Clearly, the only holomorphic modular function of weight 0  is constant in $\tau$.
We begin by constructing a modular function with a single pole at the cusp and no other poles. Normalizing the residue of the pole to unity, we define the famous $j$-function by,
\bea
\label{eq:jfuncdef}
j(\tau) =  {12^3 g_2^3 \over \Delta} = { 12 ^3 \, \HE_4^3 \over \HE_4^3-\HE_6^2} = { 1 \over q} + \cO(q^0)
\eea
Inspecting (\ref{thm1a}), we have $k=0$ since we are dealing with a modular function of weight zero; $\ord _\infty (j)=-1$ since $j$ has a simple  pole at $\infty$; $\ord _i(j)=0$ since $\HE_4(i)\not=0$; and $\ord _\rho(j)=3$ since $j$ has a triple zero at $\rho$ in view of the identity $G_4(\rho)=\HE_4(\rho)=0$ established in (\ref{3.G4G6}). These values indeed satisfy  (\ref{thm1a}), and we have,\footnote{The first 19 orders in the $q$-expansion of $j$ are as follows: {\small $j(\tau) = q^{-1} +744+196884\, q+21493760 \, q^2+864299970 \, q^3+20245856256 \, q^4+333202640600 \, q^5+4252023300096 \, q^6+44656994071935\, q^7+401490886656000\, q^8+3176440229784420\, q^9+22567393309593600 \, q^{10}+146211911499519294 \, q^{11}+874313719685775360 \, q^{12}+4872010111798142520 \, q^{13}+25497827389410525184 \, q^{14}+126142916465781843075 \, q^{15}+593121772421445058560 \, q^{16}+2662842413150775245160 \, q^{17} + \cO(q^{18})$.}}
\bea
j(i \infty) = \infty \hskip 1in j(i)= 1 \hskip 1in j(\rho)=0
\eea
The classification of modular functions is given as follows.

\sm

{\thm ~ Every rational function of $j(\tau)$ is a modular function of $SL(2,\ZZ)$. Conversely, every modular function $f(\tau)$ of $SL(2,\ZZ)$ of weight 0  is a rational function of~$j$. In terms of the zeros $z_n$ and poles $p_n$ (both of which are to be counted with multiplicities, including at $i$ and $\rho$), we have the explicit formula,
\bea
f(\tau) = \prod _{n=1}^N { j(\tau) - j(z_n) \over j(\tau) - j(p_n)}
\eea}
This result completes the construction of the field of modular functions of weight 0 and, combining it with our earlier general construction of modular forms of arbitrary weight, also gives the construction for all meromorphic $SL(2,\ZZ)$-covariant functions of arbitrary weight.

\subsubsection{Modular forms with prescribed zeros}

A final comment is in order about modular forms with prescribed zeros. From (\ref{thm1a}) we see that a zero at a regular point in $F$ requires $k \geq 12$, and the addition of one regular zero requires shifting $k \to k+12$. To construct, for example, the modular forms of weight 12 with one zero at an arbitrary point $p \in F\setminus \{ i \infty, i, \rho\}$, we take a linear combination of the two modular forms available, namely $\HE_4^3$ and $\HE_6^2$, and arrange the coefficients so that a zero emerges at $p$,
\bea
f_p(\tau) = f_0 \Big ( \HE_4 (\tau)^3 \HE_6(p)^2 - \HE_4(p)^3 \HE_6(\tau)^2  \Big )
\eea
The normalization factor $f_0$ may be fixed by evaluation at any other point, including $i \infty$. One proceeds analogously for modular forms with more zeros at prescribed points.

\subsection{Modular transformations of Jacobi $\tet$-functions}

The Jacobi $\tet$-functions are subject to more delicate modular transformation laws, which we shall now derive. Recall  the definition of the basic $\tet$-function,
\bea
\tet (z|\tau) = \sum _{ n \in \ZZ} e^{ i \pi \tau n^2 + 2 \pi i n z}
\eea
Setting $z=0$ and $\tau = i t$ for $t $ real and positive, we may apply the Poisson summation formula (\ref{poisson}) to obtain the transformation rule for $S \in SL(2,\ZZ)$,
\bea
\tet (0|-\tfrac{1}{\tau}) = (-i \tau)^{  \half}\, \tet  ( 0 | \tau   )
\eea
Although originally derived for $t$ real, the formula  remains valid  for $\tau \in \cH$ provided we choose the proper branch cut for the square root. Under the modular transformation $T$, however, the $\tet$-function does not transform into itself, but instead we have,
\bea
\label{tetplusone}
\tet (z|\tau+1 )
= \tet _4(z|\tau)
\eea
where $\tet_4$ is a $\tet$-function with half-integer characteristics defined in (\ref{tet1234}). The set of four $\tet$-functions with half characteristics transforms into itself, and we have,
\begin{align}
\label{tet-trans}
\tet _1(z|\tau+1) & = \ep \, \tet _1(z|\tau)
\tet_1(\tfrac{z}{\tau}|-\tfrac{1}{\tau}) & = - i (-i \tau)^\half e^{i \pi z^2/\tau} \tet_1 (z|\tau)
\no \\
\tet _2(z|\tau+1) & = \ep \, \tet _2(z|\tau)
\tet_2(\tfrac{z}{\tau}|-\tfrac{1}{\tau}) & =  (-i \tau)^\half e^{i \pi z^2/\tau} \tet_4 (z|\tau)
\no \\
\tet _3(z|\tau+1) & =  \tet _4(z|\tau)
\tet_3(\tfrac{z}{\tau}|-\tfrac{1}{\tau}) & = (-i \tau)^\half e^{i \pi z^2/\tau} \tet_3 (z|\tau)
\no \\
\tet _4(z|\tau+1) & =  \tet _3(z|\tau)
\tet_4(\tfrac{z}{\tau}|-\tfrac{1}{\tau}) & = (-i \tau)^\half e^{i \pi z^2/\tau} \tet_2 (z|\tau)
\end{align}
where $\ep=e^{2 \pi i /8}$. One may collect these transformation laws in a single expression for each transformation, valid for all half-characteristics,
\bea
\label{4.mod2}
\tet \left [ \begin{matrix} \alpha \cr \beta \cr \end{matrix} \right ] (z |\tau+1)
& = & e^{- i \pi \alpha (1+\alpha)} \, \tet \left [ \begin{matrix} \alpha \cr \beta+ \alpha + \half  \cr \end{matrix} \right ] (z |\tau)
\no \\
\tet \left [ \begin{matrix} \alpha \cr \beta \cr \end{matrix} \right ] \left ( \tfrac{ z}{\tau}  \Big | - \tfrac{ 1}{\tau} \right )
& = & (-i \tau)^\half \, e^{2 \pi i \alpha \beta + i \pi z^2/\tau} \,
\tet \left [ \begin{matrix} \beta \cr -\alpha \cr \end{matrix} \right ] (z |\tau)
\eea
Combining the two, one finds that under a generic element $\gamma  \in SL(2, \ZZ)$,
the $\tet_1$ function transforms as follows,
\bea
\label{3.tetmod}
\tet _1 (z' |  \tau')
= \ep_\tet(\gamma)  (c\tau+d)^\half \,  e^{\pi i c z^2/(c \tau+d)} \, \tet _1 (z|\tau)
\eea
from which the transformation of the $\tet$-function with arbitrary characteristic may be deduced using (\ref{2.tetchar}), and we find,
\bea
\label{eq:Jactetmodtransf}
\tet \left [ \begin{matrix} \alpha + \half \cr \beta + \half \cr \end{matrix} \right ] \left (  z'  | \gamma \tau \right )
\ep(\a,\b\,\a',\b',\g)  (c \tau + d)^\half
e^{\pi i c z^2/(c \tau + d)} \,\,\tet \left [ \begin{matrix}  \a' + \half \cr  \b' + \half \cr \end{matrix} \right ] (z | \tau)
\eea
\bea
\ep(\a,\b\,\a',\b',\g) = \ep_\tet (\gamma) \, e^{\pi i \left \{ \a(\b+1)-\a'(\b'+1)  \right \} }
\eea
and  $z$ and the characteristics transform as follows,
\bea
z' = { z \over c \tau +d}
\hskip 0.7in
\left [ \bma \a' \cr \b' \ema \right ]  =\left [ \bma  a \a + c \b \cr b \a + d \b \ema \right ]
\hskip 0.7in
\gamma = \left ( \bma a & b \cr c & d \ema \right )
\eea
Finally, $\ep_\tet(\gamma)$ and $\ep(\a,\b\,\a',\b',\g)$ are both eight roots of unity and $\ep_\tet(\gamma)$ is independent of the characteristics $\a, \b$. Its evaluation is in terms of the generalized Legendre symbol and will be given in subsection \ref{sec:Dedekindeta} on the Dedekind $\eta$ function.

\sm

Setting $z=0$ in the theta functions gives so-called \textit{theta-constants}, whose modular transformation laws are special cases of the general transformation laws presented above. None of these are modular forms under $SL(2,\ZZ)$, though we shall see later that they are modular forms under certain congruence subgroups of $SL(2,\ZZ)$. Furthermore, the derivative $\theta'_1(0|\tau)$ is closely related to the discriminant $\Delta$ and various sums of eight powers of $\tet_2,\tet_3$, and $\tet_4$ are modular forms. We now discuss both of these constructions in turn.

\subsubsection{The discriminant $\Delta$: $\tet$-constants and product formula}
\label{sec:Discriminant}

The modular transformation law for $\tet_1'(0|\tau)$ is readily obtained from (\ref{tet-trans}) and is given below along with its eighth power,
\bea
\tet _1' (0|\tau+1)  = \ep \, \tet _1' (0|\tau)
& \hskip .8in &
\tet_1'(0|-\tfrac{1}{\tau})  =  (-i \tau)^{{ 3 \over 2}} \, \tet_1' (0|\tau)
\no \\
\tet _1' (0|\tau+1)^8  = \tet _1' (0|\tau)^8
& &
\tet_1'(0|-\tfrac{1}{\tau})^8  =  (-i \tau)^{12} \, \tet_1' (0|\tau)^8
\eea
While $\tet'_1$ itself is not a modular form, its eight power is a modular form of weight 12, and therefore must be a linear combination of $\HE_4^3$ and $\HE_6^2$. To obtain a precise relation, we appeal to the infinite product formula for $\tet_1(z|\tau)$ given in (\ref{tetprod}), and find the following expression,
\bea
\label{3.tet1}
\tet'_1(0|\tau) = \pi \tet_2(0|\tau) \tet_3(0|\tau) \tet_4(0|\tau)
2 \pi q^{1 \over 8} \prod _{n=1}^\infty (1-q^n)^3
\eea
This expression  indeed fails to be a modular form in view of the $q^{1 \over 8}$ prefactor. However,  its eight power produces a factor of $q$ which is not only consistent with being a genuine modular form, but also shows that it is a cusp form proportional to the discriminant $\Delta$. The constant of proportionality is easily fixed by matching the term of order $q$ in the Fourier expansion of both. As a result, we find not only a relation between $\theta_1'(0|\tau)^8$ and the discriminant $\Delta$ but also a product formula for $\Delta$,
\bea
\Delta (\tau) = (2 \pi)^4 \tet_1'(0|\tau)^8 = (2 \pi)^{12} q \prod _{n=1}^\infty (1-q^n)^{24}
\eea
The product formula for the discriminant shows that $\Delta(\tau)$ is nowhere vanishing for $\tau \in \cH$, and has a simple zero in $q$ at the cusp.

\sm

The Fourier coefficients $\tau(n)$ of $\Delta$, defined by,
\bea
\label{3.tau}
\Delta (\tau) = (2 \pi)^{12} \sum _{n=1}^\infty \tau (n) q^n
\eea
are referred to as the Ramanujan $\tau$-function, and have special arithmetic significance.
The lowest orders are given as follows,
\begin{align}
\tau(1) & =1 & \tau(4) & = -1472 & \tau(7) & = -16744 & \tau(10) & = -115920
\no \\
\tau(2) & = -24 & \tau(5) & = 4830 & \tau(8) & = 84480 & \tau(11) & = 534612
\no \\
\tau(3) & = 252 & \tau(6) & = -6048 & \tau(9) & = -113643 & \tau(12) & = -370944
\end{align}
The relation $1728 \Delta= (2 \pi)^{12}(E_4^3-E_6^2)$ along with $E_4^2=E_8$ allows us to express the $\tau$-function in terms of sums of divisor functions as follows,
\bea
\tau(n) & = & \sum_{m=1}^{n-1} \left ( \tfrac{200}{3} \, \sigma_3(m) \sigma _7(n-m) - 147 \, \sigma _5(m) \sigma _5(N-m) \right )
\no \\ && \quad
+  \tfrac{5}{36} \, \sigma _3(n) + \tfrac{ 7}{12} \, \sigma _5(n) + \tfrac{5}{18} \, \sigma_ 7(n)
\eea
Eliminating $\sigma_7$ in terms of $\sigma_3$ using the first formula in (\ref{3.sigma79}) confirms that $\tau(n)$ may be expressed solely in terms of $\sigma_3$ and $\sigma_5$.  Ramanujan conjectured that $\tau(n)$ is a \textit{multiplicative arithmetic function}, which satisfies,
\bea
\label{taumult}
\tau (mn) = \tau(m) \tau(n) \hskip 1in \gcd(m,n)=1
\eea
where $\gcd(m,n)$ stands for the greatest common divisor of $m$ and $n$.
Ramanujan's function is, however, not \textit{completely multiplicative} (which would mean that the first equation in (\ref{taumult}) would hold for arbitrary $m,n$, not necessarily relatively prime)  and instead satisfies,
\bea
\label{taumult2}
\tau(m) \tau(n) = \sum _{d \,|\, {\gcd} (m,n)} d^{11} \tau \left ( { mn \over d^2} \right )
\eea
Both conjectures on $\tau(n)$ were proven by Mordell in 1917 using the theory of Hecke operators applied to the cusp form $\Delta (\tau)$, and we shall provide this proof in section \ref{sec:Hecke}.

\subsubsection{Sums of eight powers}

Another way to construct a modular form using the $\tet$-constants is by sums of eighth powers. The modular transformation properties show right away that the following sums,
\bea
T_{4k} (\tau) = \tet_2(0|\tau)^{8k} + \tet_2(0|\tau)^{8k} + \tet_2(0|\tau)^{8k}
\eea
are modular forms of weight $4k$. For $k=1,2$, the corresponding spaces of modular forms are one-dimensional, so that we must have $T_4 = 2 \HE_4$ and $T_8 = 2  \HE_8 = 2 \HE_4^2$ by matching asymptotics at the cusp. In particular, we may express the modular $j$-function in terms of eighth powers of $\tet$-constants,
\bea
j(\tau) = 1728 { g_2(\tau)^3 \over \Delta(\tau)}
= 32 \, { \big (\tet _2(0|\tau)^8 + \tet _3(0|\tau)^8 + \tet _4(0|\tau)^8 \big )^3 \over
\tet _2(0|\tau)^8 \, \tet _3(0|\tau)^8 \, \tet _4(0|\tau)^8 }
\eea
It may be verified directly from the asymptotics of the $\tet$-constants at the cusp that $j(q)$ has a simple pole at the cusp with unit residue $j(q) \sim q^{-1}$.

\subsection{The Dedekind $\eta$-function}
\label{sec:Dedekindeta}

The Dedekind $\eta$-function is defined as follows,
\bea
\eta (\tau) = q^{1 \over 24} \prod _{n=1}^\infty (1-q^n)
\eea
It is not a modular form of $SL(2,\ZZ)$, but its 24th power is a modular form of weight 12 and related to the discriminant  as follows,
\bea
\label{eq:deletarel}
\Delta (\tau) = (2 \pi )^{12} \eta (\tau)^{24}
\eea
Under the generators $T$ and $S$ of the modular group, $\eta$ transforms as follows,
\bea
\label{3.eta1}
\eta (\tau+1) & = & e^{2\pi i /24} \, \eta (\tau)
\no \\
\eta (-\tfrac{1}{\tau}) & = & (-i \tau)^\half  \, \eta (\tau)
\eea
The first relation may be read off from the definition of $\eta$, while the second relation may be derived first for $\tau \in i \RR^+$ using Poisson resummation, and then be extended to $\tau \in \cH$.\footnote{The square root is defined via the argument function $\arg(\tau)$ for $\tau \in \CC^* $, in terms of which  $\tau = |\tau| \exp ( i \arg(\tau))$ and $-\pi \leq \arg(\tau) < \pi$ by $\tau^\half = |\tau|^\half \exp ( i \arg(\tau)/2)$ (see for example [7]). This definition is compatible with the fact that for $\tau = i t$ with $t \in \RR^+$ we have $\eta(it), \eta(i/t)>0$. }

\sm

The transformation of $\eta(\tau)$ under an arbitrary $\gamma \in SL(2,\ZZ)$ may be expressed as follows,
\bea
\label{3.etamod}
\eta (\gamma \tau) = \ep_\eta (\gamma)  \, (c \tau +d) ^\half \eta (\tau)
\hskip 1in
\gamma = \left ( \bma a & b \cr c & d \ema \right )
\eea
with $\ep_\eta (\gamma)^{24}=1$, or $\ep : SL(2,\ZZ) \to \ZZ_{24}$. The function  $\ep_\eta (\gamma)$ depends on $\gamma$, but not on $\tau$. Its values may be determined from the transformation rules given in (\ref{3.eta1}). Note that the actions of the transformations $\gamma$ and $-\gamma$ coincide on $\tau$ but differ on the square root and on $\ep_\eta $ giving, in particular, the value $\ep_\eta (-I) = (-1)^\half$. In view of this property, $\ep_\eta (\gamma)$ is not a proper homomorphism of $SL(2,\ZZ) \to \ZZ_{24}$ but instead is referred to as a \textit{multiplier system}.  Special values may be deduced from the basic transformation laws (\ref{3.eta1}),
\bea
\ep_\eta (T^n) = e^{ 2 \pi i n /24}
\hskip 0.6in
\ep_\eta (S) = e^{-2 \pi i /8}
\hskip 0.6in
\ep_\eta (-I) =  e^{2 \pi i /4} =i
\eea
In particular, if $\gamma$ is such that either $c >0$, or $c=0$ and $d>0$, then $\ep (-\gamma) = i \ep (\gamma)$.
The Petersson formula for $\ep_\eta (\gamma)$ for a $\gamma$ with either $c> 0$, or $c=0$ and $d>0$ is given as follows, (see [7])
\bea
c \hbox{ odd: } ~ & \hskip 0.2in & \ep_\eta (\gamma)
= (d|c) \exp \left \{ \tfrac{2 \pi i }{24} \big (  (a+d)c+bd(1-c^2) -3 c \big ) \right \}
\\
c \hbox{ even: } & \hskip 0.2in &  \ep_\eta (\gamma)
= {\rm sign}(d) (c|d)  \exp \left \{ \tfrac{2 \pi i }{24} \big ( (a+d)c+bd(1-c^2) +3 d +3 -3cd \big ) \right \}
\no \eea
We recall from appendix \ref{sec:modN}, and in particular (\ref{A.jacobi}), that the Jacobi symbol $(n|N)$ for odd denominator $N$ (which is always the case here) with decomposition into distinct primes $p_i$ given by $N=\prod _i p_i ^{\a_i}$ is defined by $(n|N) = \prod_i (n|p_i)$ where $(n|p_i)$ is the Legendre symbol for an odd prime $p_i$. The definition is extended to include $n=0$ by $(0|1)=(0|-1)=1$ and negative $N$ by $(n|N)=(n| -N)$.

\sm

Knowledge of $\ep_\eta(\gamma)$ allows us to produce a formula for the multiplier $\ep _\tet(\gamma)$  encountered in the modular transformation law of the $\tet$-functions in (\ref{eq:Jactetmodtransf}) by using the relation,
\bea
\label{eq:tetpetarel}
\tet_1'(0|\tau) = 2 \pi \eta (\tau)^3
\eea
which is obtained by combining (\ref{3.tet1}) with the definition of the $\eta$-functions. The modular transformation of the left side may be derived from (\ref{3.tetmod}) while the modular transformation of the right side is given by (\ref{3.etamod}), and we obtain,
\bea
\ep _\tet (\gamma) = \ep_\eta (\gamma)^3
\eea
consistent with the fact that $\ep_\tet$ is an 8-th root of unity while $\ep_\eta$ is a 24-th root of unity.

\sm

Finally, we note that the fourth power  $\ep_\eta^4(\gamma)$ no longer suffers from the subtleties of sign reversal of $\gamma$ and the square root, and is a genuine homomorphism $\ep_\eta^4 : SL(2,\ZZ) \to \ZZ_6$ with $\ep_\eta^4(-I)=1$ and $\ep_\eta^4(\gamma_1 \gamma _2) = \ep_\eta^4(\gamma_1) \ep_\eta^4(\gamma _2)$.

\newpage

\subsection*{$\bullet$ Bibliographical notes}

The books by Apostol [8] and Lang [9] and the lecture notes by Zagier [10] offer classic introductions to the subject of modular forms.   The books by Shimura [11] and Koblitz [12] provide advanced and more abstract presentations aimed at arithmetic.  A marvelously clear and detailed presentation, to which we shall refer often in these notes, is given in the fairly recent treatise by Diamond and Shurman [13]. A general perspective from the point of view of automorphic functions is given in Bateman [14] and Iwaniec [15]. A detailed exposition of identities involving the Dedekind $\eta$-function and the role they play in the theory of modular forms may be found in the book by K\"ohler [7]. Presentations with an emphasis on $L$-functions and $q$-series expansions may be found in the books by Bump [16] and Ono [17], respectively.

\newpage

\section{Variants of modular forms}
\setcounter{equation}{0}
\label{sec:VMF}

In this section, we discuss a variety of objects that are closely related to modular forms, including quasi-modular forms, almost-holomorphic modular forms, non-holomorphic Eisenstein series, Maass forms, mock modular forms, and quantum modular forms. Other generalizations, including modular forms for congruence subgroups, vector-valued modular forms, and modular graph forms, will be introduced in sections \ref{sec:Forms}, \ref{sec:MDEsvvmfs}, and \ref{sec:MGF} respectively.

\subsection{Quasi-modular and almost-holomorphic modular forms}
\label{E2section}

Quasi-modular forms and almost-holomorphic modular forms are generalizations of the modular forms introduced earlier, which are obtained by relaxing certain parts of the definition of modular forms. We begin by reviewing the most famous of these objects, referred to as $\HE_2(\tau)$ and $\HE^*_2(\tau)$ respectively, before proceeding  to the definition and presentation of the general properties of quasi-modular forms and almost-holomorphic modular forms.

\subsubsection{$\HE_2$ and $\HE_2^*$}

We introduced the holomorphic Eisenstein series $G_k$ in (\ref{EisenG}) and (\ref{3c1}), or  equivalently $\HE_k$ in (\ref{Ektau}), which are given for even $k \geq 4$ in terms of absolutely convergent sums over a lattice $\Lambda = \ZZ \oplus \tau \ZZ$. For $k=2$, the corresponding  sum,
\bea
\sum _{\om \in \Lambda '} { 1 \over \om^2} = \sum _{{m,n \in \ZZ \atop (m,n) \not= (0,0)}} { 1 \over (m + n \tau)^2}
\eea
fails to be absolutely convergent. Instead, it is conditionally convergent and its value will depend on how the infinite sums are arranged. Physicists would  say that the sums need to be  ``regularized". There may be one or several choices that are deemed ``natural" because they preserve one property or another of modular forms.

\sm

One choice of regularization is to adopt the \textit{Eisenstein summation}  which consists in cutting off the infinite sums in a symmetrical way, as was already used in (\ref{2c2}),
\bea
\label{G2Eis}
\HE_2 (\tau) = \lim_{N\to \infty} \, \lim _{M \to \infty} \, { 1 \over 2 \zeta(2)} \sum _{n = -N  } ^N \sum_{{m=-M \atop (m,n) \not= (0,0)}} ^M  { 1 \over (m + n \tau)^2}
\eea
For finite $M$ and $N$ the sum is holomorphic in $\tau$ and the limit $M,N \to \infty$ produces a finite holomorphic  function $\HE_2(\tau)$. However, $\HE_2(\tau)$ fails to transform as a modular form under  $SL(2,\ZZ)$ transformations. This is not surprising as Theorem \ref{theorem2} tells us there are no modular forms of weight 2. Instead the transformation law of $\HE_2(\tau)$  is as follows,
\bea
\label{E2transfs}
\HE_2  ( \gamma \tau) = (c \tau +d)^2 \HE_2(\tau) + { 12 \over 2 \pi i} \, c (c \tau +d)
\eea
To show this, we decompose the sum of (\ref{G2Eis}) into contributions from $n=0$ and $n \not=0$,
\bea
\label{G2Eist}
\HE_2 (\tau) = 1 + \lim _{N \to \infty} \lim _{M \to \infty} { 1 \over  \zeta(2)} \sum _{n =1  } ^N \sum_{m=-M } ^M  { 1 \over (m + n \tau)^2}
\eea
and perform the sum over $m$ using (\ref{lif}) in the convergent limit $M \to \infty$. The remaining sum over $n$ admits a finite limit as $N \to \infty$ and evaluates as follows,
\bea
\label{3.E2sig}
\HE_2 (\tau) =  1 - 24 \sum _{n=1}^\infty \sum _{\ell=1}^ \infty \ell \, e^{2 \pi i \ell n \tau }
=1 - 24 \sum _{\ell=1}^\infty \sigma _1(\ell) q^ \ell
\eea
Alternatively, the sum over $n$ in (\ref{3.E2sig}) may be performed first and the remaining sum over $\ell$ may be  rearranged in terms of the discriminant $\Delta (\tau)$,
\bea
\label{3.E2Del}
\HE_2(\tau)  = { 1 \over 2 \pi i} \, \p_\tau \ln \Delta(\tau)
\eea
Using the modular transformation law of the weight 12 modular form $\Delta$ under $SL(2,\ZZ)$,
we deduce the transformation law of $\HE_2(\tau)$ by differentiating $\ln (\Delta)$ to obtain (\ref{E2transfs}).
Remarkable relations to the Weierstrass $\zeta$ and $\wp$ functions include the following,
\bea
\label{4.WPE}
\zeta(\thalf | \tau) = { \pi^2 \over 6} \, \HE_2(\tau)
\hskip 1in
\wp(z|\tau) = - \p_z^2 \ln \tet _1 (z|\tau) -{\pi^2 \over 3 } \, \HE_2(\tau)
\eea
The latter demonstrates how $\HE_2(\tau)$ acts as a modular connection to guarantee the proper modular transformation of $\wp$ given that of the $\tet_1$-function.

\sm

Another choice of regularization was introduced by Siegel by defining the following sum,
\bea
\label{3.Estar}
\HE_2^* (\tau) = \lim _{\ep \to 0^+} { 1 \over 2 \zeta (2)} \sum _{\om \in \Lambda '} { 1 \over \om^2 \, |\om|^\ep}
\eea
Finite $\ep>0$ renders the sum over $\Lambda'$ absolutely convergent and the limit $\ep \to 0$ turns out to be finite.
The result is a weight 2 modular form under $SL(2,\ZZ)$,
\bea
\HE_2^*(\gamma \tau)  = (c \tau +d)^2 \HE^*_2(\tau)
\eea
However, $\HE_2^*(\tau)$ fails to be holomorphic in $\tau$. The regularization factor $|\om|^\ep$ introduces non-holomorphic behavior for $\ep >0$ which leaves a non-holomorphic remnant as $\ep \to 0$,\footnote{Throughout, we shall use the notation $\tau = \tau_1 + i \tau_2$ with $\tau_1, \tau_2 \in \RR$.}
\bea
\label{eq:2E2rel}
\HE_2^*(\tau) = \HE_2 (\tau) - { 3 \over \pi \tau_2}
\eea
where the holomorphic $\HE_2(\tau)$ was given in (\ref{3.E2Del}). One verifies that the modular transformation laws of $\HE_2$ and $\HE_2^*$ map into one another under this relation.

\sm

In summary, while the Eisenstein series $\HE_k$ for $k \geq 4$ are holomorphic modular forms of weight $k$, the regularization of the infinite sum for $k=2$ allows for two ``natural" choices: one  is holomorphic but not modular, while the other  is modular but not holomorphic. One cannot enforce  both modularity and holomorphicity simultaneously.  The incompatibility between holomorphicity and modular invariance may be viewed as perhaps the first example  of what physicists refer to as an \textit{anomaly}. Eisenstein's result dates back to before  1850 !

\subsubsection{The rings of almost-holomorphic modular  and quasi-modular forms}
\label{sec:quasialmostmod}
A function $f: \cH \to \CC$ is an \textit{almost-holomorphic form of weight $k$} with respect to the full modular group $SL(2,\ZZ)$  if it transforms as follows under $\gamma \in SL(2,\ZZ)$,
\bea
f(\gamma \tau) = (c \tau +d)^k f(\tau)
\eea
and is a polynomial in $\tau_2^{-1} $ whose coefficients $f_n(q)$ are holomorphic functions of $q=e^{ 2 \pi i \tau}$,
\bea
f(\tau) = \sum _{n=0}^N f_n(q) \tau_2^{-n}
\eea
By this definition, every modular form of weight $k$ is an almost-holomorphic modular form of weight $k$, and $\HE_2^*(\tau)$ is an almost-holomorphic modular form of weight  2. Clearly, any linear combination of almost-holomorphic modular forms of weight $k$ is an almost-holomorphic modular form of weight $k$, and the product of two almost-holomorphic modular forms of weights $k$ and $\ell$ is an almost-holomorphic modular form of weight $k+\ell$, so that the space of almost-holomorphic forms a ring graded by the weight.

\sm

A function $g: \cH \to \CC$ is a \textit{quasi-modular form of weight $k$} if it is the holomorphic part $f_0(q)$ of some almost-holomorphic form $f$ of weight $k$, as defined above. By this definition, every modular form of weight $k$ is a quasi-modular form of weight $k$, and  $\HE_2(\tau)$ is a quasi-modular form of weight  2. The space of quasi-modular forms is a ring graded by the weight. More precisely, we have the following  theorem, presented here without proof.

\label{thm:almost}
The space of almost-holomorphic modular forms is a polynomial ring, graded by the weight $k$, and generated by $\HE_2^*, \HE_4$, and $\HE_6$.  The space of quasi-modular forms is a polynomial ring, graded by the weight $k$, and generated by $\HE_2, \HE_4$, and $\HE_6$. }

\subsubsection{Differential equations for quasi-modular forms}
\label{sec:introtodiffeq}

The inhomogeneous nature of the transformation law of $\HE_2(\tau)$ under $SL(2,\ZZ)$ suggests that $\HE_2(\tau)$  may be viewed geometrically as a connection in the space of modular forms $\cM$. The connection $E_2$ produces a covariant derivative $D_k$ acting on $\cM_k$,
\bea
D_k : \cM_k \to \cM_{k+2}
\hskip 1in
D_k = { 1 \over 2 \pi i} { d \over d\tau} - { k \over 12} \HE_2(\tau)
\eea
To show this, we use the fact that a modular form $f$ of weight $k$, namely $f \in \cM_k$, and its $\tau$-derivative $f'$ satisfy the following transformation rules,
\bea
f(\gamma \tau) & = & (c\tau+d)^k f(\tau)
\no \\
f'(\gamma \tau) & = & (c \tau+d)^{k+2} f'(\tau) + k c (c\tau+d)^{k+1} f(\tau)
\eea
Combining (\ref{E2transfs}) with the above relation, and the definition of $D_k$, we obtain,
\bea
D_kf(\gamma \tau) = (c \tau +d)^{k+2} D_k f(\tau)
\eea
so that $D_k f$ is a modular form of weight $k+2$, as announced. The Leibnitz formula for the product of two modular forms $f_1$ and $f_2$  of respective weights $k_1$ and $k_2$ is given as follows,
\bea
D_{k_1+k_2} \left ( f_1 f_2 \right ) = \left ( D_{k_1} f_1 \right ) f_2 + f_1 \left ( D_{k_2} f_2 \right )
\eea
When the space $\cM_{k+2}$ is one-dimensional, one easily derives a differential equation for modular forms of weight $k$.  The precise normalizations may be deduced from the asymptotic behavior at the cusp. For example, we have,
\begin{align}
D_4 \, \HE_4 & = - \tfrac{1}{3} \HE_6 & D_8 \, \HE_8 & = - \tfrac{2}{3} \HE_{10} & D_{12} \, \Delta & =0
\no \\
D_6 \, \HE_6 & = - \tfrac{1}{2} \HE_8 && & D_{12} \, \HE_{12}& = - \HE_{14}
\end{align}
Equivalent expressions may be obtained by using $\HE_8=\HE_4^2$, $\HE_{10}=\HE_4 \HE_6$, and $\HE_{14}=\HE_4^2 \HE_6$ together with Leibnitz's rule. The derivative of $\HE_{10}$ is, however, more involved since the space $\cM_{12}$ is two-dimensional.

\sm

Given that $\HE_2$ may naturally be interpreted as a \textit{connection} in $\cM_k$ there should be a natural notion of \textit{curvature}. The associated curvature should transform tensorially, namely it should be a modular form, and should be obtained from the derivative and the square of the connection. Given the transformation law (\ref{E2transfs}) for $\HE_2$, we obtain the transformations for its derivative with respect to $\tau$ and its square,
\bea
\HE_2' ( \tau'  ) & = & (c \tau +d)^4 \HE_2'(\tau) + 2 c (c \tau+d)^3 \HE_2 (\tau) + \tfrac{ 12 c^2}{2 \pi i}  (c\tau+d)^2
\no \\
\HE_2  ( \tau' )^2 & = & (c \tau +d)^4 \HE_2(\tau)^2 + \tfrac{ 24 }{ 2 \pi i} \, c (c \tau +d) ^3 \HE_2 (\tau)
+ \left ( \tfrac{ 12 c}{2 \pi i}\right )^2 (c\tau+d)^2
\eea
As a result, the following combination transforms as a modular form of weight 4,
\bea
{ 1 \over 2 \pi i} \HE_2' ( \tau'  ) -{1 \over 12} \HE_2(\tau')^2 =
(c \tau +d)^4 \left (  { 1 \over 2 \pi i} \HE_2' ( \tau ) -{1 \over 12} \HE_2(\tau)^2\right )
\eea
Since this combination is both holomorphic in $\tau$ and a modular form of weight 4, it must belong to $\cM_4$. But by Theorem \ref{theorem2}, the space $\cM_4$ is dimension 1 and is generated by $\HE_4$. Matching the value at $q=0$, we obtain the relation,
\bea
{ 1 \over 2 \pi i} { d \over d \tau} \HE_2 ( \tau ) -{1 \over 12} \HE_2(\tau)^2 = - { 1 \over 12} \HE_4(\tau)
\eea
It is natural to interpret $\HE_4$ as the curvature, given indeed by a linear combination of the derivative and the square of $\HE_2$.
We note that the coefficient multiplying the square is half of what one would have expected if the derivative had been applied to a modular form of weight 2. This is as usual when obtaining the curvature from the connection.

\subsection{Non-holomorphic Eisenstein series}
\label{sec:4.2}

We next introduce non-holomorphic Eisenstein series, which may be viewed as generalizing the Riemann $\zeta$-function, modular forms, and almost-holomorphic modular forms. Given a lattice $\Lambda = \ZZ + \tau \ZZ$, a non-holomorphic Eisenstein series may be defined by the following lattice sum,\footnote{The notation $E_s(\tau)$ used for the non-holomorphic Eisenstein series should not be confused with the notation $\HE_s(\tau)$ used for the holomorphic Eisenstein series.} or so-called \textit{Kronecker-Eisenstein sum},
\bea
\label{3.nhE}
E_s(\tau) = \sum _{\om \in \Lambda '} { \tau_2^s \over \pi^s |\om|^{2s}} = \sum _{m,n \in \ZZ}' { \tau_2^s \over \pi^s |m + n \tau|^{2s}}
\eea
where $\tau_2=\Im(\tau)$ and the prime on the sum indicates that the contribution $m=n=0$ is to be omitted from the sum. The sum is absolutely convergent for $\Re(s)>1$. The lattice sum definition of $E_s(\tau)$ may be used to show that $E_s$ is modular invariant,
\bea
E_s(\gamma \tau) = E_s(\tau) \hskip 1in \gamma = \left ( \bma a & b \cr c & d \ema \right ) \in SL(2,\ZZ)
\eea
It may also be used to factor the non-trivial common divisors of $m$ and $n$ in  the double sum,
\bea
E_s(\tau) = \zeta(2s) \sum_{{c,d \in \ZZ \atop \gcd(c,d)=1}}' { \tau_2^s \over \pi^s |c \tau +d|^{2s}}
\eea
where $\zeta(2s)$ is the Riemann $\zeta$-function and the remaining sum is over pairs $(c,d) \in \ZZ^2$ that have no common divisors strictly greater than 1. This expression in turn allows us to write $E_s(\tau)$ as a Poincar\'e series over the coset $\Gamma_\infty \backslash SL(2,\ZZ)$ as follows,
\bea
\label{eq:nonholoPoincare}
E_s(\tau) = { \zeta (2s) \over \pi^s} \sum _{\gamma \in \Gamma _\infty \backslash SL(2,\ZZ)} \left ( \Im \gamma \tau \right )^s
\eea
This relation implies that $E_s(\tau)$ vanishes for all $\tau$ whenever $2s$ is a zero of the Riemann $\zeta$-function, including at its zeros on the negative real axis, as well as at its non-trivial zeros when $\Re(4s)=1$.
Finally, the non-holomorphic Eisenstein series $E_s(\tau)$ is an eigenfunction of  the Laplace equation,
\bea
\label{eq:EisLapl}
\Delta E_s(\tau) = s(s-1) E_s(\tau)
\hskip 1in
\Delta = 4  \tau_2^2 \p_\tau \p _{\bar \tau}=  \tau_2^2 ( \p_{\tau_1}^2 + \p_{\tau_2}^2)
\eea
as may be verified by applying the Laplace operator term-wise in the sum of (\ref{3.nhE}).

\subsubsection{Analytic continuation of $E_s(\tau)$}

While $E_s(\tau) $ is defined by the series (\ref{3.nhE}) for $\Re(s)>1$, it may be analytically continued in~$s$ throughout the complex $s$-plane to produce a meromorphic function, just as was possible for the Riemann $\zeta$-function. To exhibit the analytic structure of $E_s(\tau) $ in $s$, we use the following integral representation for the summands,
\bea
\label{4.d6}
\Gamma (s) E_s(\tau)  = \int _0 ^\infty { dt \over t} \, t^s \left ( \sum _{m,n\in \ZZ} e^{ - \pi t |m +n\tau|^2 /\tau_2} -1 \right )
\eea
The subtraction  of $1$ eliminates the zero mode. The non-analyticity in $s$ arises from the small $t$ regime. As we did in the case of the Riemann $\zeta$-function, we split the integral at $t=1$ and Poisson re-sum the integrand for $0 < t < 1$ in both $m$ and $n$ to obtain,
\bea
\label{4.d6a}
\Gamma (s) E_s(\tau)  & = &
\int _1 ^\infty { dt \over t} \, t^s \left ( \sum _{m,n\in \ZZ} e^{ - \pi t |m +n\tau|^2 /\tau_2} -1 \right )
\no \\ &&
+ \int _0 ^1 { dt \over t} \, t^s \left ( {1 \over t} \sum _{m,n\in \ZZ} e^{ - \pi   |m +n\tau|^2/(t\tau_2)} -1 \right )
\eea
Next, we change variable in the second integral $t \to 1/t$, collect the resulting integrals for $t \in [1,\infty [$,
and analytically continue a trivial remaining integral. The result is as follows,
\bea
\label{4.d7}
\Gamma (s)  E_s(\tau) =  - {1 \over s(1-s) }+
\int _1 ^\infty { dt \over t} \, \Big ( t^s+ t^{1-s} \Big )  \left ( \sum _{m,n\in \ZZ} e^{ - \pi t |m +n\tau |^2 /\tau_2} -1 \right )
\eea
The first term on the right side is meromorphic in $s \in \CC$, while the integral is manifestly holomorphic throughout  $\CC$. As a result, $E_s(\tau) $ admits a meromorphic analytic continuation to $s \in \CC$ with a simple pole at $s=1$ with residue 1. Moreover, the expression is manifestly invariant under $s \to 1-s$ which implies  the functional equation,
\bea
\label{4.d8}
\Gamma (s) E_s(\tau) = \Gamma (1-s) E_{1-s}(\tau)
\eea
In particular, the functional relation requires $E_s(\tau)=0$ whenever $s=-1,-2, \cdots$ in view of the fact that the right side is finite for these values and that $\Gamma (s)$ on the left side  is infinite. This observation reproduces our earlier observation that $E_s(\tau)=0$ whenever $2s$ is a zero of the Riemann $\zeta$-function.

\subsubsection{Fourier series of $E_s(\tau)$}

The Fourier series of $E_s(\tau)$ for all $\Re(s)>1$ may be derived from the integral representation in (\ref{4.d6}),
this time by Poisson re-summation  in the variable $m$ only, using the formula,
\bea
\sum _{m,n \in \ZZ}  e^{- \pi t |m+n \tau|^2/\tau_2}
= \sqrt{ { \tau _2 \over t}} \sum _{m,n \in \ZZ} e^{2 \pi i m n \tau_1} \, e^{ - \pi m^2\tau_2/t - \pi n^2 t \tau_2}
\eea
In the mathematics literature this procedure is referred to as the Chowla-Selberg method.
Substituting this result into (\ref{4.d6}) and  splitting the sum over $m,n$ into contributions for $m n \not=0 $ and $mn=0$, we have
\bea
\Gamma (s) E_s(\tau)=  A_s(\tau)+B_s(\tau)
\eea
\bea
\label{4.d9}
A_s(\tau) & = & \sqrt{\tau_2} \sum _{m \not=0} \sum _{n \not=0}  e^{2 \pi i mn \tau_1} \int _0^\infty { dt \over t} \,
t^{s-\half}  \, e^{- \pi \tau_2 (m^2/t + n^2 t)}
\no \\
B_s(\tau) & = &   \int _0^\infty { dt \over t} \,
t^s \left ( -1 + 2 \sum _{n =1}^\infty   \sqrt{{\tau_2 \over t}} \,  e^{ - \pi n^2 t \tau_2}
+  \sum _{m \in \ZZ}  \sqrt{{\tau_2 \over t}} \,  e^{ - \pi m^2  \tau_2/ t } \right )
\eea
The integral $A_s(\tau)$ is manifestly holomorphic in $s$ throughout $\CC$, and may be computed by performing the $t$-integral in terms of a $K$-Bessel function,
\bea
\label{4.d10}
A_s(\tau) = 2 \sqrt{\tau_2} \sum _{m,n \not=0} \left | { m \over n} \right |^{s-\half} e^{2 \pi i mn \tau_1} K_{s-\half} ( 2 \pi \tau_2  |mn| )
\eea
The integral $B_s(\tau)$ may be computed by  Poisson re-summing the $m$-sum in the integrand,
\bea
B_s(\tau) =  \sum _{n\not = 0} \int _0^\infty { dt \over t} \,
t^s \left (  \sqrt{{\tau_2 \over t}} \,  e^{ - \pi n^2 t \tau_2}
+  e^{ - \pi n^2  t/ \tau_2 } \right )
\eea
which is readily evaluated in terms of the Riemann $\zeta$-function or associated $\xi$-function,
\bea
\label{4.d11}
B_s(\tau) & = & 2 \Gamma (s) \pi^{-s} \tau_2^s \zeta (2s) + 2 \Gamma (s-\thalf ) \pi^{\half -s} \tau_2^{1-s} \zeta (2s-1)
\no \\
& = &
\tau_2^s \xi (2s) + \tau_2^{1-s} \xi (2s-1)
\eea
where $\xi(s)$ is the function computed in (\ref{xi}). We may now verify the functional relation (\ref{4.d8}) directly in terms of the transformation properties of $A_s(\tau)$ and $B_s(\tau)$. For $A_s(\tau)$ this is manifest upon using the symmetry under interchange of $m$ and $n$, while for $B_s(\tau)$ it is a consequence of  the relation $\xi(1-s)=\xi(s)$.   Assembling both contributions, and recasting $A_s(\tau)$ in terms of a summation over a single variable $N=mn$, we obtain the celebrated Fourier series decomposition for  $ E_s(\tau)$,
\bea
\label{4.d12}
\Gamma (s) E_s(\tau) & = &
2 \Gamma (s) \pi^{-s} \, \zeta (2s) \, \tau_2^s  + 2 \Gamma (s-\thalf ) \pi^{\half -s}  \, \zeta (2s-1) \, \tau_2^{1-s}
\no \\ &&
+ 4 \sqrt{\tau_2} \sum _{N \not=0} |N| ^{\half-s} \sigma _{2s-1}(|N|) \, e^{2 \pi i N \tau_1} K_{s-\half} ( 2 \pi \tau_2  |N| )
\eea
where $\sigma _\alpha (n)$ is defined to be the sum  $\sigma _\alpha (n) = \sum _{d |n} d^\alpha$ of positive divisors $d$ of $n$.  One verifies that each Fourier mode, including the power-behaved terms on the first line, individually satisfy the Laplace eigenvalue equation, as required by the fact that $\Delta$ commutes with the translation operator $\p_{\tau_1}$.

\sm

For integer $s=k$ the Bessel function in the Fourier series becomes a spherical Bessel function $K_{k-\half}(x)$, which is a combination of an exponential in $x$ and a polynomial in $1/x$. As a result, the  Fourier series of $E_k(\tau)$ simplifies and may be recast as follows,
\bea
\label{Eas}
E_k(\tau) &=& (- )^{k+1} { B_{2k} \over (2k)!} \, y^k
+ { 4 \, (2k-3)! \, \zeta (2k-1) \over (k-2)! \, (k-1)! \, y^{k-1}}
\no \\ && \quad
+{2\over (k-1)! }\sum_{N=1}^\infty N^{k-1}\sigma_{1-2k}(N)P_k(Ny) \left(q^N + \bar q^N \right)
\eea
where $q=e^{2 \pi i \tau}$ and $y = 4 \pi \tau_2$. Furthermore,  $B_{2k}$ are the Bernoulli numbers  and $P_k(x)$ is a polynomial in $1/x$ given by,
\bea
P_k(x) = \sum_{m=0}^{k-1}{(k+m-1)! \over m! \, (k-m-1)! \, x^m}
\eea
Non-holomorphic Eisenstein series with integer $s$ will play a crucial role in the theory of modular graph functions and forms, to be developed in section \ref{sec:MGF}.

\subsection{Maass forms}
\label{sec:4.3}

The non-holomorphic Eisenstein series discussed in the previous subsection are special examples of a more general class of non-holomorphic objects known as Maass forms. A  non-holomorphic function $f(\tau)$ is called a Maass form if it is $SL(2,\ZZ)$-invariant, satisfies a Laplace eigenvalue equation,
\bea
\Delta f(\tau) = \lambda f(\tau) \hspace{0.8 in} \Delta =4 \tau_2^2 \p_\tau \p_{\bar \tau} = \tau_2^2 (\p_{\tau_1}^2 + \p_{\tau_2}^2)
\eea
for $\lambda \in \CC$, and has at most polynomial growth at the cusps. Without loss of generality, we will write $\lambda = s(s-1)$ and denote the space of such Maass forms by $\cN(s)$, where $\cN(s) = \cN(1-s)$. From (\ref{eq:EisLapl}) we see that the non-holomorphic Eisenstein series $E_s(\tau)$ are elements of $\cN(s)$. Indeed, below we will see that for generic $s\in \CC $ the non-holomorphic Eisenstein series are the \textit{only} elements of $\cN(s)$.

\sm

The Fourier expansions of generic Maass forms admit surprisingly simple closed form expressions. In general, non-holomorphic modular forms admit Fourier expansions in separate powers of $q$ and $\bar q$, of the type,
\bea
f(\tau) = \sum_{M,N \in \ZZ} \widetilde{c}_{M,N}(\tau_2) q^M \bar q^N
\eea
with $\widetilde{c}_{M,N}(\tau_2)$ being functions of $\tau_2$ but not of $\tau_1$, since the latter would violate invariance under $T: \tau \rightarrow \tau + 1$. It is often useful to absorb all $\tau_2$ dependence, including that of $q$ and $\bar q$, into the Fourier coefficients, giving an alternative form for the Fourier expansion as,
\bea
f(\tau)= \sum_{N\in \ZZ} c_N(\tau_2) \,e^{2 \pi i N \tau_1}
\eea
We will now see that, for Maass forms, the coefficients $c_N(\tau_2)$ can be fixed uniquely, up to overall constants. Indeed, inverting the Fourier expansion gives,
\bea
c_N(\tau_2) = \int_0^1 d \tau_1 \,e^{-2 \pi i N \tau_1 } \,f(\tau)
\eea
and we may make use of the eigenvalue equation to obtain,
\bea
& = & {1\over s(s-1) } \int_0^1 d\tau_1\, e^{-2 \pi i N \tau_1 } \,\Delta f(\tau)
\no \\
& = & {\tau_2^2 \over s(s-1)} \Big [ (2 \pi i N)^2 c_N(\tau_2) + \p_{\tau_2}^2 c_N(\tau_2) \Big ]
\eea
This gives a second order differential equation for the coefficients $c_N(\tau_2)$,
\bea
\tau_2^2 \p_{\tau_2}^2 c_N(\tau_2) = \Big ( s(s-1) +(2 \pi N \tau_2)^2 \Big ) c_N(\tau_2)
\eea
For $N=0$ the general solution is of the form,
\bea
c_0(\tau_2) = a \,\tau_2^s+ b \,\tau_2^{1-s}
\eea
For $N\neq 0$ we obtain a modified Bessel equation with general solutions $I_{s-1/2}$ and $K_{s-1/2}$. Discarding the solutions $I_{s-1/2}$ since they grow exponentially at the cusp, we obtain,
\bea
c_N(\tau_2) = \alpha_N \,\sqrt{\tau_2} \,K_{s - 1/2}\left(2 \pi \tau_2|N|\right) \hspace{0.5in} N \neq 0
\eea
We conclude that a general element of $\cN(s)$ has a Fourier expansion of the form,
\bea
\label{eq:Maassfourier}
f(\tau) = a \,\tau_2^{s}+ b\, \tau_2^{1-s} + \sum_{N \neq 0} \alpha_N \,\sqrt{\tau_2} \,e^{2 \pi i N \tau_1 }K_{s-1/2}\left(2 \pi \tau_2 |N|\right)
\eea
Note that the Fourier expansions of the non-holomorphic Eisenstein series were obtained already in (\ref{4.d12}), and match with the above result upon substituting,
\bea
a = {2 \zeta(2s) \over \pi^{s}}
\hspace{0.4 in}
b= {2 \Gamma(s-\half) \pi^{\half - s} { \zeta(2s-1)} \over \Gamma(s)}
\hspace{0.4 in}
\alpha_N = {4 |N| ^{\half - s} \sigma_{2s-1}(|N|) \over \Gamma(s)}
\eea

\sm

The constant part of the Fourier expansion, i.e. the term $a \,\tau_2^s+ b\, \tau_2^{1-s}$, is known as the \textit{Laurent polynomial} of the Maass form. Though it may at first sight seem that the space of Laurent polynomials is two-dimensional and spanned by $a$ and $b$, in fact this space is only one-dimensional. This may be shown by considering Maass forms on the truncated fundamental domain,
\bea
F_L = \left\{\tau \in \cH, \,\,|\tau| \geq 1,\,\, |\mathrm{Re}(\tau)| \leq \half,\,\, \tau_2 \leq L\right\}
\eea
Indeed, consider two Maass forms $f,g \in \cN(s)$ with Fourier expansions
\bea
f(\tau) = \sum_{N \in \ZZ} c_N(\tau_2) e^{2 \pi i N \tau_1}\hspace{0.6 in}g(\tau) = \sum_{N \in \ZZ} d_N(\tau_2) e^{2 \pi i N \tau_1}
\eea
Green's theorem on the region $F_L$ states that if ${\p\over \p n}$ is the normal derivative on $\p F_L$ and $ds$ is the differential arc length, then we have
\bea
\int_{\p F_L} ds \left(f {\p g \over \p n} - d {\p f \over \p n} \right) =  \int_{F_L} {d \tau_1 d \tau_2 \over \tau_2^2} (f \Delta g - g \Delta f) =0
\eea
where the last equality follows from the eigenvalue equation. The left side may be evaluated in terms of the Fourier coefficients. Noticing that modular invariance reduces the boundary integral to an integral over $\tau_2 = L$ (since the remaining boundaries of the fundamental domain give equal and opposite contributions), we have
\bea
\int_{\p F_L} ds \left(f {\p g \over \p n} - f {\p f \over \p n} \right) & =& \int_{-\half}^{\half} d \tau_1 \sum_{N,M\in \ZZ} \left(c_N(L) d'_M(L) - c'_N(L) d_M(L) \right)e^{2 \pi i (N+M) \tau_1}
\no\\
&=& \sum_{N\in \ZZ} \left[c_N(L) d'_{-N}(L) - c'_N(L) d_{-N}(L) \right]
\eea
where the derivative is defined by,
\bea
c'_N(L) = {\p c_N(\tau_2) \over \p \tau_2} \Big |_{\tau_2 = L}
\eea
and similarly for $d'_{-N}(L)$. In particular, this implies that $c_0 d'_0 - c'_0 d_0 = 0$, and if we write,
\bea
c_0(\tau_2) = a \,\tau_2^s+ c\, \tau_2^{1-s} \hspace{0.7 in} d_0(\tau_2) = b \,\tau_2^s+ d\, \tau_2^{1-s}
\eea
then for $s \neq  {1\over 2}$ we obtain $ad-bc=0$, which indeed means that the space of Laurent polynomials has dimension one. In the case that $s = \half$ this space is also obviously of dimension one, since then the two factors of $\tau_2$ have the same power.

\subsubsection{The space of Maass forms}

As was mentioned above, at generic values of $s\in \CC$ the space of Maass forms $\cN(s)$ consists only of non-holomorphic Eisenstein series $E_s(\tau)$.  More generally, at non-generic values of~$s$, the space of Maass forms can include cusp forms. In the current context, cusp forms are defined to be the subset of elements of $\cN(s)$ such that the Laurent polynomial vanishes, that is to say $a=b=0$ in (\ref{eq:Maassfourier}). We will denote the space of cusp forms by $\cS(s)$. The non-holomorphic Eisenstein series are clearly not cusp forms for any $s$. For generic values of~$s$, the set $\cS(s)$ will be empty.

\sm

The structure of $\cN(s)$ can be summarized as follows:
\label{thm:Maassspace}
The space of Maass forms $\cN(s)$ is given by
\bea
\cN(s) = \left\{ \begin{matrix} \CC E_s(\tau) & & \mathrm{Re} \,s > \half \,\,\mathrm{and}\,\, s \notin [0, 1]  \\
\CC  E_s(\tau)  \oplus \cS(s) &&  \mathrm{Re}\, s = \half\,\,\,\mathrm{or}\,\,\, s \in [0, 1 )  \\ \CC &&s = 1  \end{matrix} \right.
\eea}

\sm

The proof of this theorem follows almost immediately from the fact, proven above, that the space of Laurent polynomials has dimension 1. This means that for any $f(\tau) \in \cN(s)$, there exists a constant $c$ such that the combination $f(\tau) - c E(\tau,s)$ has trivial Laurent polynomial, i.e. is a cusp form.

\sm

To complete the proof, we need only prove that the space of cusp forms is empty for all values of $s$ which  satisfy neither $\mathrm{Re}\, s = \half$ nor $s \in [0, 1 )$. The non-existence for $s=1$ is immediate: in this case the cusp form would be harmonic, i.e. satisfy a Laplace eigenvalue equation with zero eigenvalue, and hence would have to vanish by the maximum modulus principle. More generally we may argue as follows. Above we have seen that it is possible to find a constant $c$ such that for any $f(\tau) \in \cN(s)$, the combination  $h(\tau) := f(\tau) - c E_s(\tau)$ is a cusp form. Since cusp forms are square-integrable over the fundamental domain $F = SL(2, \ZZ)\backslash\cH$ equipped with the Poincar\'e measure $d^2\tau / \tau_2^2$,
\bea
\int_{F}  {d^2 \tau \over \tau_2^2 }\,|h(\tau)|^2  < \infty
\eea
we may manipulate the following absolutely convergent integral by using the differential equation $\Delta h = s(s-1) h$  satisfied by $h$, and Green's theorem,
\bea
s(s-1) \int_{F}  {d^2 \tau \over \tau_2^2 }|h(\tau)|^2 =
\int_F  {d^2 \tau \over \tau_2^2 }\Delta h(\tau)\, \overline{h(\tau)}
= -  4\int_{F}  {d^2 \tau}  \left | \p_\tau h \right |^2
\eea
Since the right side is real and non-positive,  cusp forms must have real $s(s-1) \leq 0$, which restricts us to $s \in [0,1]$ or $\mathrm{Re}\, s = \half$.

\sm

In closing, let us note that the triviality of the space of cusp forms for $\mathrm{Re} \,s \neq \half$ and $s \not \in \RR$ implies that for these values the space $\cN(s)$ is one-dimensional and hence there must be a functional relation relating $E_s(\tau)$ and $E_{1-s}(\tau)$, since both have Laplace eigenvalue $s(s-1)$. Indeed, precisely such a relation was identified in (\ref{4.d8}).

\subsubsection{Cusp forms}

We have just shown that the space $\cS(s)$ of cusp forms is empty unless $\mathrm{Re}\, s = \half\,\,\,\mathrm{or}\,\,\, s \in [0, 1 )$. Actually, one can further show that $\cS(s)$ vanishes for $s \in [0,1)$, and hence that all non-trivial cusp forms must exist at $\mathrm{Re}\, s = \half$ and away from the real axis: we refer to [18] for the proof. Denoting $s = \half + i t$ for $ t \in \RR$, the corresponding eigenvalue is $\lambda = s(s-1) = - \left({1\over 4} + t^2\right)$.

\sm

For any $t \in \RR$, the space $\cS\left(\half + i t\right)$ is finite-dimensional. To show this, begin with $M$ elements $v_1(\tau), \dots, v_M(\tau) \in \cS\left(\half + i t\right)$. Given an additional element $\widetilde v(\tau)$, it is possible to choose coefficients $c_i \in \CC$ such that the linear combination,
\bea
v(\tau) :=\widetilde v(\tau) - \sum_{i=1}^M c_i v_i(\tau)
\eea
has vanishing Fourier coefficients up to order $M$. We will now show that for $M$ sufficiently large but finite, the vanishing of Fourier coefficients up to order $M$ implies vanishing to all orders. In other words we will have $v(\tau) =0$ exactly, and hence all elements $\widetilde v(\tau)\in \cS\left(\half + i t\right)$ can be written as a linear combination of a finite number $M$ of other elements.

\sm

To prove the desired result, begin by recalling that a cusp form $v(\tau)$ is bounded on the upper-half plane, and hence that $|v(\tau)|$ must have a maximum at some location $\tau^{(0)} = \tau_1^{(0)} + i \tau_2^{(0)}$. From the form of the Fourier expansion (\ref{eq:Maassfourier}) and orthogonality of exponentials, we have,
\bea
\a_N \,\sqrt{\tau_2^{(0)}}\, K_{it}(2 \pi \tau_2^{(0)} |N|) = \int_{-1/2}^{1/2} d \tau_1 \,v(\tau_1 + i {\tau_2^{(0)}/2})\, e^{- 2 \pi i N \tau_1}\left({\sqrt{2} K_{it} (2 \pi \tau_2^{(0)} |N|) \over K_{it} ( \pi \tau_2^{(0)} |N|) } \right)
\eea
It then follows that,
\bea
\left|v(\tau^{(0)}) \right| < 2 \sqrt{2}\, \left|v(\tau^{(0)})\right|\, \sum_{|N| > M} {K_{it}(2 \pi \tau_2^{(0)} N) \over K_{it}( \pi \tau_2^{(0)} N) }
\eea
If $M$ is sufficiently large (depending on $t$), we may estimate the Bessel functions by exponentials, giving the inequality,
\bea
\left|v(\tau^{(0)})\right| \leq \left|v(\tau^{(0)})\right| \,C e^{- \pi M  \tau_2^{(0)}}
\eea
for some proportionality constant $C$. Hence if $M\geq (\ln C) / (\pi  \tau_2^{(0)}) $ then we conclude that $|v(\tau^{(0)})|=0$, and hence that $v(\tau)$ is identically zero.

\sm

We have just shown that $\cS\left(\half + i t\right)$ is finite-dimensional for any $t$. Furthermore, though we will not reproduce the proof here, it has been shown that there exist an infinite number of values of $t$ for which the space of cusp forms is non-empty.  However, there is no exact value of $t$ for which this is known to be the case. The smallest approximate values of $t$ for which cusp forms are believed to exist are,
\bea
\label{eq:discreteseriesvalues}
t \approx\, 9.53369, \,\, 12.17300, \,\, 13.77975, \,\, 14.35850,\,\,16.13807,\,\,16.64425,\,\,17.73856, \, \dots
\eea
In particular, we see that there is only a \textit{discretum} of such values.

\subsubsection{Comments on the physical interpretation of the spectrum}

The fundamental domain for $SL(2,\ZZ)$ is non-compact as it has a puncture at the cusp. From the point of view of scattering theory, one may view the cusp as an asymptotic region from which \textit{a free wave} in the continuous spectrum  can be sent in. This wave is indeed oscillatory and of the form $\tau_2^{\half + it}$ for real $t$,  from the first constant term in the Fourier expansion. Since the fundamental domain is otherwise bounded, the wave hits a wall at $|\tau|=1$ and must be reflected out through the puncture and is asymptotically of the form $\tau_2^{\half -i t}$. This phenomenon is analogous to one-dimensional scattering on a semi-infinite line and explains in physical terms why the Eisenstein solutions have $E_{\half+it}(\tau) = E_{\half -it} (\tau)$.

\sm

The cusp forms are wave functions of bound states, as is confirmed by the fact that they decay exponentially towards the cusp. In most quantum mechanics problems, the number of bound states in the presence of a wall would be finite. But there are exceptions, such as the Coulomb problem. Here, it is the special properties of the hyperbolic metric on the  fundamental domain of $SL(2,\ZZ)$ that produce an infinite number of cusp forms, or bound states, with a spectrum that rises all the way to infinity.

\subsection{Spectral decomposition}

We have now discussed various aspects of the spectrum of the Laplacian on the fundamental domain $SL(2,\ZZ) \backslash \cH$.
It is possible to extract from this data the following orthonormal basis of eigenfunctions of the Laplacian:
\begin{itemize}
\item \textbf{Discrete series:} The constant function $v_0 := \sqrt{\mathrm{Area}(F)} = \sqrt{3/ \pi}$ (which is trivially an eigenfunction of the Laplacian) together with an orthonormal basis of cusp forms $v_n(\tau)$ with corresponding eigenvalues $s_n(s_n-1) = - \left({1\over 4} + t_n^2 \right)$ for $t_n \in \RR$. The first few values of $t_n$ are given in (\ref{eq:discreteseriesvalues}).
\item \textbf{Continuous series:} The non-holomorphic Eisenstein series $E_s(\tau)$ with $s= \half + i t$ and $t\in \RR$ taking a continuum of values.
\end{itemize}

\sm

Orthogonality here is defined relative to the Petersson inner product with the Poincar\'e measure $d^2 \tau / \tau_2^2$ on the fundamental domain, i.e.
\bea
\langle f, g \rangle = \int_{F} {d^2 \tau \over \tau_2^2} f(\tau) \overline{g(\tau)}
\eea
Let us use this inner product to verify the orthogonality claimed above. Consider the inner product between a non-holomorphic Eisenstein series $E_s(\tau) $  and a cusp form $v_n(\tau)$,
\bea
\langle E_s(\tau) , \, v_n( \tau) \rangle = \int_{F} {d^2 \tau \over \tau_2^2} E_s(\tau) \overline{v_n( \tau)}
\eea
This inner product converges since cusp forms have exponential decay at the cusps, whereas non-holomoprhic Eisenstein series have at most polynomial growth. For $\mathrm{Re}\, s >1$, we may proceed by using the Poincar{\'e} series for $E_s(\tau) $ given in (\ref{eq:nonholoPoincare}), which gives
\bea
\langle E_s(\tau) , \, v_n( \tau) \rangle &=& {\zeta(2s) \over \pi^s}\int_{F} {d^2 \tau \over \tau_2^2} \,  \sum_{\gamma \in \Gamma_\infty \backslash SL(2, \ZZ)}(\mathrm{Im}\, \gamma \tau)^s \,\overline{v_n(\tau)}
\no\\
&=&  {\zeta(2s) \over \pi^s} \int_{-\half}^\half d\tau_1 \int_0^\infty  {d \tau_2 \over \tau_2^2}\,\tau_2^s\, \overline{v_n( \tau)}
\eea
where in the second equality we have used modular invariance to unfold the integral over the fundamental domain, i.e. we have replaced the integral over $SL(2,\ZZ)$ orbits in $F$ with an integral over the strip $\Gamma_\infty \backslash \cH$,
\bea
\int_{SL(2,Z)\backslash \cH} ~~ \sum_{\Gamma_\infty \backslash SL(2,\ZZ)} \boldsymbol{\cdot} \,\,\,\,\,=\,\,\,\,\, \int_{\Gamma_\infty \backslash \cH} \boldsymbol{\cdot}
\eea
This is the analog of the unfolding trick introduced for periodic functions in section \ref{sec:unfolding}.
The integral over $\tau_1$ may now be performed and isolates the Laurent polynomial of $v_n( \tau)$, which is vanishing by definition. This gives the desired orthogonality for $\mathrm{Re}\, s >1$. The analogous result for $\mathrm{Re}\, s <1$ may be obtained by analytic continuation.

\sm

An arbitrary square integrable modular function can now be decomposed into this orthogonal basis. We state without proof the following spectral decomposition theorem,
{\thm [Roelcke-Selberg]
Let $v_0 = \sqrt{3 / \pi}$ be the normalized constant function  and $\{v_n(\tau)\}_{n \geq 1}$ be an orthonormal basis of cusps forms. Then any square integrable modular function $f(\tau)$ admits the following decomposition,
\bea
f(\tau) = \sum_{n \geq 0} \langle f(\tau), v_n(\tau)\rangle\, v_n(\tau) + {1 \over 4 \pi} \int_{-\infty}^\infty dt \, \left\langle f(\tau),  \,E_{\thalf + i t} (\tau) \right\rangle E_{\thalf + i t} (\tau)\,\,
\eea

The reader may be worried about the convergence properties of the continuous portion of this decomposition. Indeed, $E_s(\tau)$ is not square-integrable because of its Laurent polynomial. However, when $\mathrm{Re}\, s = \half$, one can show that $|E_s(\tau)| \ll \sqrt{\tau_2}$ when $\tau_2 \rightarrow \infty$, and the extra integration over $t$ in the decomposition formula saves a factor of $\log \tau_2$, which is sufficient to turn the right side into a square-integrable function.

\subsection{Maass forms of general weight}

Thus far we have focused on Maass forms of weight $(0,0)$. We now discuss Maass forms of arbitrary (integer) weight $k$, i.e. those satisfying
\bea
f\left({a \tau + b \over c \tau + d}\right) = (c \tau + d)^k f(\tau)\hspace{0.5 in} \left(\begin{matrix} a & b \\ c & d\end{matrix} \right) \in SL(2, \ZZ)
\eea
The Maass property in this case must be recast in terms of a weight-$k$ Laplacian
\bea
\Delta_k = 4 \tau_2^{2-k} \p_{\tau} \tau_2^k \p_{\bar{\tau}} = \Delta - 2 i k \tau_2 \p_{\bar \tau}
\eea
in order to get a modular covariant object.

Of particular interest are the \textit{harmonic} Maass forms of weight $k$, i.e. those satisfying an eigenvalue equation $\Delta_k f = 0$. Unlike in the case of weight 0, for non-zero weight the space of harmonic Maass forms is non-trivial.
\sm
A simple example of a weight-2 harmonic Maass form is the almost-holomorphic Eisenstein series $\HE_2^*(\tau)$ introduced in section \ref{E2section}. Indeed, from (\ref{eq:2E2rel}) and the fact that $\HE_2(\tau)$ is holomorphic we have
\bea
\Delta_2 \HE^*_2(\tau)=  - {3 \over \pi} (\Delta - 4 i \tau_2 \p_{\bar \tau}) \tau_2^{-1} = - {3 \over \pi} ( 2 \tau_2^{-1} -2 \tau_2^{-1} )  = 0
\eea

\sm

The space of weight-$k$ harmonic Maass forms $\cI_k$ admits an interesting decomposition which will feature prominently in the subsequent section. This decomposition can be captured by the following exact sequence,
\bea
\label{eq:weightkMaassspace}
0 \rightarrow \cM^!_k \rightarrow \cI_k \rightarrow \cM^!_{2-k} \rightarrow 0
\eea
with $\cM^!_k$ the space of weight-$k$ modular functions. The content of this exact sequence is two-fold. First, it says that the space $\cM^!_k$ of weight-$k$ modular functions is a subset of the space $\cI_k$ of weight-$k$ harmonic Maass forms. This follows trivially from the fact that modular functions are annihilated by $\Delta_k$.  Second, it says that the space $\cM^!_{2-k}$ of weight-$(2-k)$ modular functions is a quotient of $\cI_k$ by $\cM^!_k$. This follows from the fact that, given an element $f(\tau) \in \cI_k$, the function $g(\tau):=\tau_2^{k} \p_{ \tau} \overline{f(\tau)}$ is holomorphic and of weight $2-k$, and vanishes if and only if $f(\tau)$ is in $\cM^!_k$.

\subsection{Mock modular forms}

In section \ref{E2section} we saw that there exist two variants of the weight-two holomorphic Eisenstein series, namely the function $\HE_2(\tau)$ defined in (\ref{G2Eis}) which is holomorphic but not modular, and the function $\HE_2^*(\tau)$ defined in (\ref{3.Estar}) which is modular but not holomorphic. The former is an example of a quasi-modular form, whereas the latter is an example of an almost-holomorphic modular form. Section \ref{sec:quasialmostmod} described an isomorphism between the sets of quasi-modular forms and almost-holomorphic modular forms.

\sm

Having introduced non-holomorphic modular forms in the previous subsections, it is natural to ask if their holomorphic parts lead to interesting analogs of quasi-modular forms. In particular, we will consider the holomorphic parts of harmonic  ``weak" Maass forms,\footnote{The adjective ``weak" indicates the potential for exponential growth at the cusps; in contrast, a ``Maass form" is required to have at most polynomial growth at the cusp.}
which are known as \textit{mock modular forms}.\footnote{Mock modular forms made their first appearance in a letter of Ramanujan to Hardy, in which they were referred to as \textit{mock theta functions}. In the modern definition, a mock theta function is a $q$-series $H(q) = \sum_{n \geq 0} a_n q^n$ such that for some $\lambda \in \QQ$ the series $q^\lambda H(q)$ gives a mock modular form of weight $\half$, with the shadow (defined below) being a weight-$3\over 2$ function of the form $\sum_{n \in \ZZ} \eps(n) n q^{|\kappa| n^2}$ with $\kappa \in \QQ$ and $\eps$ an odd periodic function.
} Given a harmonic Maass form $F^*(\tau)$, we may extract its holomorphic part as follows. We begin by defining,
\bea
g(\tau) = \tau_2^k {\p \over \p  \tau} \overline{F^*(\tau)}
\eea
By the condition that $F^*(\tau)$ be harmonic, it follows that $g(\tau)$ is holomorphic, since $\Delta_k = \tau_2^{2-k} \p_\tau \tau_2^k \p_{\overline \tau}$.
If one can identify a new function $g^*(\tau)$ such that its image under $\tau_2^k \p_{\overline \tau}$ is $\overline{g(\tau)}$, then it is clear that,
\bea
F(\tau) := F^*(\tau) - g^*(\tau)
\eea
will be holomorphic. Such a function $g^*(\tau)$ can be obtained by simply inverting the differential operator $\tau_2^k \p_{\overline \tau}$, giving the expression,
\bea
g^*(\tau) = \left({i\over 2} \right)^{k-1} \int_{- \overline \tau}^{i\infty}d z\,\,  {\overline{g(-\overline{z})}\over (z+\tau)^k}
\eea
known as a non-holomorphic Eichler integral. This integral is independent of the path since the integrand is holomorphic in $z$. %It converges when $g(\tau)$ is a cusp form, but can be analytically continued to the non-cuspidal case as long as $k \neq 1$ and $n \neq 0$.

\sm

The function $F(\tau)$ is holomorphic, but will in general fail to be modular, since $g^*(\tau)$ is not modular. The function $g(\tau)$ in terms of which $F(\tau)$ is defined is referred to as the \textit{shadow}, and is a weight-$(2-k)$ modular function. The pair of $F(\tau)$ and $g(\tau)$ are often together referred to as a mock modular form, though in some contexts the function $F(\tau)$ alone is given that name.

\sm

Given a harmonic Maass form $F^*(\tau)$, it is always possible to extract from it a unique mock modular form by taking the holomorphic part in the way described above. Conversely, given a mock modular form, it is always possible to complete it to a harmonic Maass form by taking the sum of $F(\tau)$ with the function $g^*(\tau)$ obtained from the shadow. This establishes an isomorphism between the two spaces, exactly analogous to the isomorphism between the spaces of almost-holomorphic modular forms and quasi-modular forms given in section \ref{E2section}. The space of mock modular forms then fits into an exact sequence of the form in (\ref{eq:weightkMaassspace}). By nature of it being an exact sequence, the map from the space of mock modular forms to the space of shadows (known as the \textit{shadow map}) is surjective.

\subsubsection{Examples of mock modular forms}

We now give some concrete examples of mock modular forms.

\sm

A first example is provided by $\HE_2(\tau)$. In the previous section we showed that $\HE_2^*(\tau)$ is a weight-2 harmonic Maass form, and hence by definition the holomorphic part of $\HE_2^*(\tau)$, namely $\HE_2(\tau)$, is a weight-2 mock modular form.  The shadow $g(\tau)$ in this case is simply a constant, namely $g(\tau) = - {12 \over \pi}$. It is easy to check that this constant gives $g^*(\tau) = - {3\over \pi \tau_2}$, reproducing the result in (\ref{eq:2E2rel}). Thus we see that $\HE_2(\tau)$ is both quasi-modular and mock modular. This is not in general the case:  quasi-modular forms are generically not mock modular. For example, $(\HE_2(\tau))^2$ is a quasi-modular form, but fails to be mock modular since its modular completion is not a harmonic Maass form.

\sm

A second example, which appears rather unexpectedly in physics via the elliptic genus of strings propagating on a K3 manifold, is the following function,
\bea
F^*(\tau) = F(\tau) + {6 \over \sqrt{ i}}\int_{- \overline{\tau}}^\infty dz \, { \overline{\eta(-\overline{z})^3}\over (z+\tau)^{1/2}}
\eea
with $F(\tau)$ defined as,
\bea
\label{eq:mmf2}
F(\tau) = {1 \over \eta(\tau)^3}\bigg ( - \HE_2(\tau)
+ 24 \sum_{\substack{r > s > 0 \\ r - s = 1 \,\, \mathrm{mod}\,\, 2}} (-1)^r \, s \, q^{rs/2} \bigg)
\eea
It is known that $F^*(\tau)$ is modular with weight $\half$. Hence its holomorphic part $F(\tau)$ must be a mock modular function. The shadow in this case is $g(\tau) = 3 \sqrt{2} \,\eta(\tau)^3$.

\sm

A curious fact is that the first few terms in the Fourier expansion of $F(\tau)$,
\bea
F(\tau) = q^{-{1\over 8}} \left( -1 + 45 q + 231 q^2 + 770 q^3 + 2277q^4 + 5796 q^5 + \dots\right)
\eea
give dimensions of irreducible representations of the Mathieu group $M_{24}$, a sporadic simple group of order 244,823,040. This correspondence goes under the name of \textit{Mathieu moonshine}, and has been the subject of much research.

\sm

As a third and final example, we mention with a classic result of Zagier, which states that the following two functions are harmonic with $k={3\over 2}$,
\bea
F_0^*(\tau) &=& 3 \sum_{n \geq 0} H(4 n) \, q^n
+ 6 \tau_2^{-1/2} \sum_{n \in \ZZ} \beta(4 \pi n^2 \tau_2) \, q^{-n^2}
\no\\
F_1^*(\tau) &=& 3 \sum_{n>0} H(4n-1) \, q^{n- {1\over 4}}
+ 6 \tau_2^{-1/2} \sum_{n \in \ZZ} \beta\left(4 \pi (n+ 1/2)^2 \tau_2\right) q^{-\left(n+\half\right)^2}
\eea
where $\beta(t)$ is an incomplete gamma function
\bea
\beta(t) = {1\over 16 \pi} \int_1^\infty du\, u^{-3/2} e^{- ut }
\eea
and $H(n)$ are the Hurwitz class number of imaginary quadratic fields. The latter are defined for $n>0$ to be the number of $PSL(2, \ZZ)$ equivalence classes of integral binary quadratic forms of discriminant $-n$, weighted by the reciprocal of the number of their automorphisms. The first few non-zero entries are given by
\bea
H(3) = {1\over 3} \hspace{0.5 in} H(4) = \half \hspace{0.5 in} H(7) = H(8) = H(11) = 1
\eea
For $n=0$ we have $H(0) = -{1\over 12}$ and for $n<0$ we have $H(n) = 0$.

\sm

The holomorphic parts $F_{0,1}(\tau)$ of $F^*_{0,1}(\tau)$ are easy to read off,
\bea
\label{eq:mmf3}
F_0(\tau) &=& 3 \sum_{n \geq 0} H(4 n) q^n
\no\\
F_1(\tau) &=&3 \sum_{n>0} H(4n-1) q^{n- {1\over 4}}
\eea
and have corresponding shadows
\bea
g_0(\tau) = - {3\over 4 \pi} \vartheta_3(0 | 2 \tau) \hspace{0.5 in} g_1(\tau) = - {3\over 4 \pi} \vartheta_2(0 | 2 \tau)
\eea
It is important to note that these functions are not mock modular forms in the strict sense discussed above, since under $\tau \rightarrow - 1/\tau$ the original harmonic functions  $F^*_{0,1}(\tau)$  are not individually invariant, but rather transform into one another as
\bea
\binom{F^*_0(-1/\tau)}{F^*_1(-1/\tau)} = - {(-i \tau)^{3/2}\over \sqrt{2}} \left(\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}  \right)  \binom{F^*_0(\tau)}{F^*_1(\tau)}
\eea
There are two ways of interpreting this fact. The first is to think of $F^*_{0,1}(\tau)$  as forming a \textit{vector-valued} harmonic Maass form. In this interpretation the functions $F_{0,1}(\tau)$ likewise form a vector-valued mock modular form. Vector-valued modular forms will be discussed in more detail in section \ref{sec:MDEsvvmfs}.

\sm

The second approach is to notice that $F^*_{0}(\tau)$ is invariant under $T$ and $F^*_{1}(\tau)$ is invariant under $T^4$. Hence we can say that $F^*_{0}(\tau)$ individually is modular invariant under the subgroup of $SL(2, \ZZ)$ generated by $T$ and $ST^4S$. This subgroup is denoted by $\Gamma_0(4)$, and is one of the so-called \textit{congruence subgroups} of $SL(2,\ZZ)$, discussed in detail in section \ref{sec:Cong}. In this interpretation one would say that $F^*_{0}(\tau)$ is a harmonic Maass form for the congruence subgroup $\Gamma_0(4)$, and that $F_{0}(\tau)$ is likewise a mock modular form for this subgroup. Modular forms for congruence subgroups will be discussed in detail in section \ref{sec:Forms}.

\subsection{Quantum modular forms}

Finally, let us briefly mention the notion of a \textit{quantum modular form}.\footnote{As far as we are aware, there is no direct connection between the term ``quantum" used here and the usual notion of ``quantum" in Physics. }  Unlike for standard modular forms for which the domain is the upper half-plane, quantum modular forms are defined on the space of cusps $\QQ \cup \{i \infty\}$ (or some subset thereof). Since this domain is a discrete space, quantum modular forms do not obey the usual analyticity properties of modular forms. Furthermore, they are not modular in the standard sense. Instead, a weight-~$k$ quantum modular form $f(\tau)$  is defined such that for all $\gamma \in SL(2, \ZZ)$, the difference
\bea
h_\gamma(\tau):=  f(\tau) - \eps_\gamma (c \tau + d)^k f\left({a\tau+b \over c \tau + d}\right)
\eea
is an analytic function on the upper half-plane. Here we have allowed for a multiplier phase $\eps_\gamma$, similar to that appearing in the transformation rules for the Dedekind $\eta$-function in section \ref{sec:Dedekindeta}.

\sm

It is simplest to illustrate this definition by proceeding directly to an example. Consider the following $q$-expansion,
\bea
F(q) = \sum_{n=0}^\infty \prod_{i=0}^{n-1}(1-q^{i+1})
\eea
known as Kontsevich's strange function. This function does not converge on any open subset of $\CC$, but is well-defined at roots of unity, i.e. when $\tau \in \QQ$. We now define the closely related object,
\bea
\phi(\tau) = q^{1\over 24 } F(q)
\eea
It follows from this definition that $\phi(\tau+1) = e^{2\pi i /24} \phi(\tau)$.

\sm

To understand the behavior under $S$, we first note that the $T$ transformation just given implies $\phi(n) = e^{2 \pi i n /24} \phi(0) =  e^{2 \pi i n /24}$ for $ n \in \ZZ$. On the other hand, Kontsevich and Zagier have shown that for $\tau = -{1\over n}$, one has,
\bea
\phi\left(-{1\over n} \right) \sim n^{3/2}e^{2 \pi i {(3+ n) \over 24}} + \sum_{j=0}^\infty c_j \left({2\pi i \over n} \right)^j
\eea
as $n \rightarrow \infty$, with the first few coefficients in the expansion being,
\bea
c_0 = 1 \hskip 0.6in c_1 = {23\over 24} \hskip 0.6in c_2 = {1681 \over 1152}
\eea
Using the $T$ transformation, the first term above can be rewritten as,
\bea
n^{3/2}e^{2 \pi i \over 8} \phi(n) = (-in)^{3/2} \phi(n)
\eea
and hence we conclude that the function
\bea
h_S(n) :=\phi(n) + (-i n)^{3/2} \phi(-1/n)
\eea
has a well-defined Taylor expansion at $\tau = 0$, and so is smooth there. Indeed, the $n$-th derivative of $h_S(n)$ at $\tau = 0$ is given by $c_n \left({2\pi i / n} \right)^n$. Though we do not give details here, the function $h_S(n)$ can furthermore be extended to a real-analytic function on $\CC$ (except for at $\tau=0$), so $\phi(\tau)$ is an example of a quantum modular form.

\subsection*{$\bullet$ Bibliographical notes}

Classic introductions to spectral analysis on symmetric spaces with applications to Maass forms and non-holomorphic Eisenstein series  may be found in the books by Kubota~[19], Terras~[20, 21], and Iwaniec~[22, 23]. {A discussion aimed at physicists, and focused on the definition over the adeles, can by found in the book by Fleig, Gustafsson, Kleinschmidt, and Persson [24].} Group-theoretic aspects of the spectral decomposition of automorphic forms are expounded on in the book by Gel'fand, Graev, and Pyatetskii-Shapiro~[25]. Faddeev's approach, which is based on operator perturbation theory, and perhaps closer to physics, is reviewed in detail and commented upon in  Lang~[18].  A construction of cusp forms which are not Maass forms was given in  [26].   The spectral decomposition theorem has recently been used in a physics context in [27, 28, 29]. Generalizations to automorphic forms under the group $GL(n,\RR)$ are developed in the book by Goldfeld [30].

\sm

A fundamental role was played by quasi-modular forms in Viazovska's proof of the sphere packing problem in eight dimensions, namely that no packing is more dense than the one corresponding to the root lattice of $E_8$. The original paper may be found in [31], while the generalization to sphere packing in dimension 24 was settled in [32].

\sm

Reviews on mock modular forms include the lecture notes by Zagier~[33] in the mathematics literature and [34, 35] in the physics literature. The second example of a mock modular form, given in (\ref{eq:mmf2}), appears in physics as the elliptic genus of a K3 manifold [36, 37]. The connection with dimensions of representations of the Monster group $M_{24}$ was observed in [38], beginning the program of Mathieu moonshine. Reviews of moonshine aimed at physicists may be found in [35, 39, 40, 41] and references therein.
The third example of mock modular forms, given in (\ref{eq:mmf3}), also arises in at least one physics context, as the partition function of $\cN=4$ super Yang-Mills theory with gauge group $SO(3)$ on $\CC \mathbb{P}^2$ [42].

\sm

Quantum modular forms were introduced in mathematics  in [43, 44, 45] but their role in physics remains unexplored to date, despite the word ``quantum" in their name.

\newpage

\section{Quantum fields on a torus}
\setcounter{equation}{0}
\label{sec:TorusQFT}

In this section, we shall present some immediate applications of the theory of elliptic and modular functions to problems of physical interest, including the construction of the Green functions and the functional determinants for the $bc$ system, the scalar,  and the spinor fields on the torus. In particular, we shall show how the singular terms in the operator product expansion of holomorphic fields for the $bc$ system  essentially determine arbitrary correlation functions on the torus in terms of elliptic functions.

\subsection{Quantum fields}

A quantum field is a map from a manifold $\Sigma$ to a linear operator acting on a vector space~$\mF$, or Fock space,  which is equipped with a Hermitian inner product. If the inner product is positive definite then $\mF$ is a Hilbert space familar from standard quantum theory, but we shall consider quantum field theories where the inner product is not necessarily definite  as well. We shall be interested in the special case where the manifold $\Sigma$ is a (connected) Riemann surface which may be compact or non-compact, with or without a boundary. Quantum fields on the complex plane $\Sigma =\CC$ describe the continuum limit and critical phenomena  of two-dimensional statistical mechanics systems, such as the Ising, Potts, XY, and non-linear sigma models. Quantum fields on a general Riemann surface $\Sigma$ provide the proper setting for string perturbation theory, where probability amplitudes are obtained via an expansion in powers of the string coupling constant. The expansion is obtained by summing contributions from Riemann surfaces of all genera and, for each genus, by integrating  over the corresponding moduli space of Riemann surfaces, as will be explained in more detail in section \ref{sec:SA}.

\subsubsection{Conformal fields}
\label{sec:conffields}

A conformal field $\phi$ on $\Sigma$ may be defined as follows. We consider two arbitrary  coordinate charts $\cU, \cU' \subset \Sigma$ with local complex coordinate systems  $(z,\bar z)$ and $(z',\bar z')$. Assuming that these coordinate charts have a non-empty intersection $\cU \cap \, \cU' \not = \emptyset$, the coordinate systems in the overlap $\cU \cap \, \cU'$ are related to one another by a holomorphic transition function $z'=f(z)$ whose derivative $f'(z)$ is non-zero throughout $\cU \, \cap \, \cU'$. Any such transformation of the form $z \to z'=f(z)$ is a locally conformal transformation, namely it preserves all angles. For more details see appendix \ref{sec:RS}. In the coordinate systems $(z,\bar z)$ and $(z',\bar z')$ the field $\phi$ is given by $\phi(z,\bar z)$ and $\phi'(z', \bar z')$, respectively. The field $\phi$ is a conformal (primary) field of conformal weight $(h, \tilde h)$,   provided $\phi$ and $\phi'$ are related as follows,
\bea
\phi'(z', \bar z') (dz')^h (d\bar z')^{\tilde h} = \phi(z,\bar z) (dz)^h (d \bar z)^{\tilde h}
\eea
or  in view of the holomorphicity of $f(z)$,
\bea
\label{5.conf}
f'(z) ^{h} \,  \overline{f'(z)} ^{\tilde h} \, \phi'(z', \bar z') =    \phi(z,\bar z)
\eea
Equivalently, a conformal field may be viewed as a section of an operator-valued  holomorphic line bundle over $\Sigma$. Clearly, $h$ and $ \tilde h$  must be the same for all charts $\cU \subset \Sigma$.

\sm

Local conformal transformations form an infinite-dimensional  Lie algebra, referred to as the \textit{Virasoro algebra}. Its generators $L_m$ for $m \in \ZZ$ obey the following structure relations,
\bea
\label{eq:Virasoro}
{} [L_m, L_n] = (m-n) L_{m+n} + { \mc \over 12} m(m^2-1) \delta_{m+n,0}
\eea
Here $\mc$ is a real-valued constant referred to as the \textit{central charge} of the Virasoro algebra. The last term is proportional to the identity operator in the representation space of the Virasoro algebra, and therefore commutes with all $L_m$. When $\mc \not=0$, the Virasoro algebra is a central extension of the \textit{Witt algebra}, which has $\mc=0$.

\sm

Actually, conformal transformations may be allowed to act independently on the holomorphic and anti-holomorphic coordinates, in which case we have two copies of the Virasoro algebra, whose generators are denoted $L_n$ acting on $z$ and $\tilde L_n$ acting on $\bar z$. These two Virasoro algebras commute so that $[L_m, \tilde L_n]=0$ for all $m,n \in \ZZ$. The conformal weights $h, \tilde h \in \RR$ characterize the representation of the Virasoro algebra under which the field $\phi$ transforms,
\bea
\label{5.Lphi}
{} [L_m, \phi(z,\bar z) ] & = & z^{m+1} \p_z \phi (z, \bar z) + h(m+1) z^m \phi(z, \bar z)
\no \\
{} [\tilde L_m, \phi(z, \bar z ) ] & = & \bar z^{m+1} \pbz \phi (z,\bar z) + \tilde h(m+1) \bar z^m \phi(z, \bar z)
\eea
The {\sl minimal models} correspond to the irreducible representations of the Virasoro algebra for $0 < \mc <1$, for which the spectrum of allowed conformal weights $h$ (of primary conformal fields) is finite. Minimal models have rational conformal weights $h, \tilde h$ and a rational central charge $\mc$. They are special cases of the more general class of ``rational conformal field theories," discussed further in section \ref{eq:CMandNLSM}. For $0 < \mc < 1$, the unitary minimal models constitute a further subclass of conformal field theories that are so-called \textit{unitary}, and in particular are associated with $\mF$ being a genuine Hilbert space. Many conformal field theories may be realized in terms of free fields.

\subsubsection{Free conformal fields}

We shall limit our examples in the sequel  to free conformal field theories. A free quantum field satisfies a linear partial differential equation on $\Sigma$, given by a differential operator $\cD$. The order of the operator $\cD$ is usually of order no larger than 2. All correlation functions of a free quantum field may be evaluated in terms of the Green function $\cD^{-1} $ and the functional determinant of $\cD$.  For free conformal fields on an arbitrary genus-one Riemann surface~$\Sigma$,  the correlators and determinants may be expressed in terms of elliptic functions and modular forms. It turns out that the quantization of free conformal fields already provides a sufficiently rich structure for all of  string perturbation theory on an arbitrary flat space-time manifold, including Minkowski space-time with or without partial toroidal compactification, as will be further detailed in sections \ref{sec:SA} and \ref{sec:Toroidal}.

\subsection{The $bc$ system}
\label{sec:5.bc}

An important example of a free conformal field theory on a Riemann surface $\Sigma$, that has an immediate connection with elliptic functions and modular forms,  is provided by a system of two quantum fields  $b$ and $c$  governed by a first order action $S[b,c]$,
\bea
S[b,c] = { 1 \over 2 \pi} \int _\Sigma d^2 z \, b(z,\bar z) \pbz c(z,\bar z)
\hskip 1in
d^2 z = { i \over 2} dz \wedge d\bar z
\eea
Invariance of $S[b,c]$ under local conformal transformations requires  both fields to have $\tilde h=0 $ and their remaining conformal weights to be conjugate: if $b$ has conformal weight $(h,0)$ then~$c$ must have conformal weight $(1-h,0)$. The field equations for $b$ and $c$ are obtained using the variational principle for $S[b,c]$, and are given by,
\bea
\pbz b = \pbz c=0
\eea
so that $b$ and $c$ are locally holomorphic functions of $z$ on $\Sigma$. The quantum field theory with these assignments is referred to as the $bc$ system of weight $h$. Below we shall consider the $bc$ system on the annulus and on the torus. The $bc$ system on a Riemann surface of arbitrary genus will be discussed in appendix \ref{app.bc}.

\subsubsection{The $bc$ system on an annulus}

An annulus is conformally isomorphic to a cylinder. Choosing the local complex coordinate $z'=x+iy$ on the cylinder with $x,y \in \RR$ and the periodic identification $x \approx x+1$, the coordinate $z$ on the annulus is given by the conformal map $z = e^{2 \pi i z'} = e^{2 \pi i (x+iy)}$.  Holomorphicity and periodicity of the fields $b$ and $c$ on the cylinder produce the following Fourier expansion in the coordinate $z'=x+iy$,\footnote{We use proportionality signs here in order to leave the precise normalization to be fixed on the annulus.}
\bea
b'(z') \sim \sum _{n \in \ZZ} b_n e^{- 2 \pi i n (x+i y)}
\hskip 1in
c'(z') \sim \sum _{n \in \ZZ} c_n e^{- 2 \pi i n (x+i y)}
\eea
If the fields $b$ and $c$ have conformal weights $(h,0)$ and $(1-h,0)$ respectively, then the fields on the annulus are obtained by the conformal transformation (\ref{5.conf}), and the Fourier expansions become  Laurent expansions in $z=e^{2 \pi i z'}$ given as follows,
\bea
b(z) = \sum_{n \in \ZZ} b_n \, z^{-n-h}
\hskip 1in
c(z) = \sum_{n \in \ZZ} c_n z^{-n-1+h}
\eea
Considering the fields $b$ and $c$ as quantum operators, the coefficients $b_n$ and $c_n$ are operators. The closest connection with holomorphic and meromorphic functions will be obtained by requiring the operators to satisfy anti-commutation relations,
\bea
\label{5.ACom}
\{ b_m, b_n \} = \{ c_m, c_n \} =0
\hskip 1in
\{ b_m, c_n\} = \delta _{m+n,0}
\eea
where $\{A,B\} = AB+BA$ is the anti-commutator and $\delta_{m+n,0}$ is the Kronecker  symbol. We choose a polarization in which $b_n$ and $c_n$ are annihilation operators  for $n >0$ while $b_n^\dagger = b_{-n}$ and $c_n^\dagger = c_{-n}$  are creation operators. The ground state $|0\>$  is annihilated by all $b_n, c_n$ for $n >0$. The anti-commutation relations for $b_0, c_0$ define a Clifford-Dirac algebra whose unique irreducible representation has dimension 2. We shall define the ground state by,
\begin{align}
b_0|0\> & = \< 0 | c_0 = 0    & b_n |0\>&=0 & c_n |0\> &=0 & n \geq 1
\end{align}
The Fock space $\mF$ is then generated by applying arbitrary polynomials in $b_n^\dagger$ and $c_n^\dagger$ for all possible values of $n >0$  to  the ground state $|0 \>$.

\subsubsection{The operator product expansion}

The fields $b(z)$ and $c(z)$ are holomorphic in $z$ on the annulus. The product $c(z) b(w)$ of operators, however, is singular as $z \to w$. This may be seen  by applying the operator to the ground state and using the fact that $|0\>$ is  annihilated by all $b_n$ with $n >0$,
\bea
\label{5.OPE1}
c(z) b(w) |0\> = \sum_{m \in \ZZ} c_m z^{-m-1+h} \sum _{n \leq 0} b_n w^{-n-h} |0 \>
\eea
The contributions from $m \leq 0$ are non-singular  as $z \to w$ while for $m>0$ we may replace the product $c_m b_n$ by the anti-commutator $\{ c_m, b_n \} = \delta_{m+n,0}$  in view of $c_m |0 \>=0$. The resulting series is geometric in $w/z$, absolutely convergent for $|z/w|<1$, and may be analytically continued in $z$ and $w$ throughout the annulus,
\bea
\label{5.OPE2}
c(z) b(w) |0\> = \sum_{m>0} z^{-m-1+h} w^{m-h} |0 \> + \hbox{ regular }
= { 1 \over z-w} \, |0\> + \hbox{ regular }
\eea
where ``regular" stands for all contributions that have a finite limit as $z \to w$.  The pole in $z-w$ arises due to the infinite nature of the series and is universal in the following sense.  If we apply the operator $c(z) b(w)$ to an arbitrary Fock space state, obtained by acting on $|0\>$ with an arbitrary polynomial in $b_n^\dagger $ and $c_n^\dagger $ with $n>0$, then only a finite number of terms in the infinite series (\ref{5.OPE2}) will be affected, which leaves the singularity unchanged. For this reason, one promotes the singularity to an operator statement,
\bea
\label{5.OPE3}
c(z) b(w) = { I_\mF \over z-w} + \hbox{ regular }
\eea
where $I_\mF$ is the identity operator in the Fock space $\mF$. The Laurent series for the operator $b(z)c(w)$ is absolutely convergent for $|w/z|<1$, produces the pole $I_\mF/(z-w)$, and may be analytically continued to arbitrary $z,w $ in the annulus. Comparing with the pole in $c(z)b(w)$, we observe that the sign reversal is consistent with the fact that $b$ and $c$ satisfy anti-commutation relations. Actually, the pole term is only the first in an infinite Laurent series expansion of the operator $c(z)b(w)$ in powers of $z-w$. The general expansion is referred to as the \textit{operator product expansion} of the operators $c(z)$ and $b(w)$,
\bea
c(z) b(w) = {I_\mF \over z-w} + \sum _{n=0}^\infty (z-w)^n j^{(n+1)}(w)
\eea
The coefficients $j^{(n+1)}(w)$ for $n >0$ are conformal primary fields of weight $(n+1,0)$. The transformation law for $j^{(1)}$ is more complicated and will be discussed later. The operators $b(z) b(w)$ and $c(z) c(w)$ similarly have operator product expansions. In view of the anti-commutation relations (\ref{5.ACom}) we have $b(z)^2=c(z)^2=0$ and,
\bea
\label{5.OPE5}
b(z) b(w) &= & (z-w) (\p_w b(w)) \, b(w) + \cO\big ( (z-w)^2\big )
\no \\
c(z) c(w) & = & (z-w) (\p_w c(w)) \, c(w) + \cO\big ( (z-w)^2\big )
\eea
so that the operator product expansion is entirely regular.

\subsection{Correlators of the $bc$ system on the torus}

We now consider the $bc$ system on an arbitrary torus $\CC/\Lambda$ with $\Lambda = \ZZ \oplus \tau \ZZ$ and $\Im(\tau)>0$. The fields $b$ and $c$ are meromorphic and doubly periodic, and are thus elliptic functions. On the torus the $bc$ systems for different integer values of $h$ are all isomorphic to one another, in view of the fact that the torus has a holomorphic differential $dz$ which is nowhere vanishing. As a result, a field $\phi(z) (dz)^h$ of conformal weight $(h,0)$  is isomorphic to a scalar field $\phi(z)$ of weight $(0,0)$.  This isomorphism allows us for a  direct evaluation of all correlators of $b,c$ fields on the torus.  A correlator is the matrix element in the vacuum state $|0\>$ of a product of $b$ and $c$ fields  evaluated at distinct points on $\Sigma$,
\bea
A= \< 0 | c(z_1) \cdots c(z_N) b(w_1) \cdots b(w_M) |0\>
\eea
The correlators are meromorphic in $z_n$ for all $n=1,\cdots, N$ and in $w_m$ for all $m=1,\cdots, M$.
We shall now show that the positions of their zeros and poles in each variable are known from the operator product expansions discussed earlier. In view of (\ref{5.OPE5}), the correlator has a simple zero when two points $z_n$ and $z_{n'}$ coincide, as well as when two points $w_m$ and $w_{m'}$ coincide. In view of (\ref{5.OPE3}), the correlator has a simple pole whenever a point $z_n$ approaches a point $w_m$, whose residue is given by a correlator with one fewer $b$ and one fewer $c$,
\bea
A = { (-)^{N-1+m-n} \over z_n - w_m}
\< 0 | c(z_1) \cdots \widehat{c(z_n)} \cdots c(z_N) b(w_1) \cdots \widehat{b(w_m)} \cdots  b(w_M) |0\>
+ \hbox{ regular }
\quad
\eea
The sign factor arises from permuting the positions of the fields, the hatted operators are to be omitted,    and ``regular" includes all terms that are non-singular as $z_n \to w_m$.

\sm

Considered as a function of $z_n$,  the correlator  has precisely $M$ simple poles at $w_1, \cdots, w_M$. In view of Theorem \ref{thm:1} it must have $M$ zeros in $z_n$. These zeros must include the $N-1$ points $z_{n'}$ with $n'\not= n$, so we must have $M\geq N-1$. Similarly, as a function of $w_m$, the correlator has precisely $N$ simple poles at $z_1, \cdots, z_N$. In view of Theorem \ref{thm:1} it must have $N$ zeros in $w_m$  which must include the $M-1$ points $w_{m'}$ with $m' \not= m$, so we must also have $N \geq M-1$. Combing both inequalities allows for either $M=N$ or $M-N=\pm 1$. The latter is excluded by symmetry under reversing the sign of both $b$ and $c$. Thus only $M=N$ gives a non-vanishing correlator. We shall now  evaluate the correlators $A$ for $M=N$ in terms of $\tet$-functions using Theorems \ref{thm:1},  \ref{thm:2.5}, and \ref{thm:2.7}.

\subsubsection{The 4-point correlator}

We begin with the correlator $\<0| c(z_1) c(z_2) b(w_1) b(w_2) |0\>$. As a function of $z_1$, it has a simple zero at $z_2$, simple poles at $w_1$ and $w_2$, and no other poles. In view of the second relation in Theorem \ref{thm:1}, it must have a second zero, whose position is determined by the third relation in Theorem \ref{thm:1} to be given by $-z_2+w_1+w_2$ (mod $\Lambda$). Proceeding analogously for $z_2$ using Theorem \ref{thm:2.5} we obtain the following expression for the correlator,
\bea
\<0| c(z_1) c(z_2) b(w_1) b(w_2) |0\> = A_0 \, { \tet_1(z_1+z_2-w_1-w_2) \tet_1(z_1-z_2) \tet_1(w_1-w_2) \over
\tet_1(z_1-w_1) \tet_1(z_1-w_2) \tet_1(z_2-w_1) \tet_1(z_2-w_2) }
\eea
Here, $A_0$ is independent of $z_1,z_2,w_1,w_2$ and all dependence on $\tau$  has been suppressed. We may use this expression to obtain the correlator $\<0 |c(z_1) b(w_2)|0\>$ by letting $z_2\to w_1$,
\bea
\label{5.res}
\<0| c(z_1) c(z_2) b(w_1) b(w_2) |0\> = { 1 \over z_2-w_1} \, { A_0 \over \tet_1'(0)} + \hbox{ regular}
\eea
The two-point correlator is constant and saturated by the zero modes of the $c$ and $b$ fields,
\bea
\<0| c(z_1) b(w_2) |0\> ={ A_0 \over \tet_1'(0)}
\eea
consistently with the fact that no function on a torus can have a single simple pole. Alternatively, we may use Theorem \ref{thm:2.7} to express the correlator in terms of the Weierstrass $\zeta$-function and its derivatives. Since the poles in $z_1, z_2$ of the correlator at $w_1, w_2$ are simple, and must have opposite residue, we have,
\bea
\<0| c(z_1) c(z_2) b(w_1) b(w_2) |0\> = A_3+ \sum _{i=1}^2 A_i \big \{ \zeta (z_i-w_1) -  \zeta(z_i-w_2) \big \}
\eea
where $A_1, A_2, A_3$ are independent of $z_1, z_2,w_1, w_2$. Matching the residues with (\ref{5.res}), we find $A_1=-A_2=-A_0/\tet_1'(0)$. This condition also shows that the correlator has zeros at $z_1=z_2$ and $w_1=w_2$ provided $A_3=0$. We leave it as an exercise to the reader to express the correlator in terms of the Weierstrass $\wp$ function and to show that these three expressions may be related to one another with explicit identities.

\subsubsection{Arbitrary correlators}

Correlators for  arbitrary $N$ may be evaluated iteratively.  As a function of $z_1$ the correlator has $N$ simple poles at $w_1, \cdots, w_N$, and $N-1$ known zeros at $z_2,\cdots, z_N$. The missing zero is given by Theorem \ref{thm:2} by the sum $-\sum_{n=2}^N z_n+ \sum_{n=1}^N w_n$. Putting all together, we have,
\bea
A= \hat A_1 \, { \tet_1 (D)  \, \prod _{n=2}^N \tet_1(z_1-z_n)
\over \prod _{n=1}^N \tet_1(z_1-w_n)}
\hskip 1in
D= \sum _{n=1} ^N (z_n-w_n)
\eea
where $\hat A_1$ is independent of $z_1$, but depends on all other $z_n$ and $w_n$. Again, all dependence on $\tau$ has been suppressed. Proceeding iteratively on the points $z_n$ we find that the factor $\tet_1(D)$ always accounts for the missing zero. Putting all together, we obtain,
\bea
A = A_0 \,  { \tet_1 (D)  \, \prod _{m<n} \tet_1(z_m-z_n) \tet_1 (w_m-w_n)
\over \prod _{m,n} \tet_1(z_m-w_n)}
\eea
where $A_0$ depends only on $\tau$. We leave it as an exercise for the reader to express the correlator also in terms of Weierstrass $\wp$- and $\zeta$-functions using Theorems \ref{thm:2} and \ref{thm:2.7} respectively.

\subsection{The scalar field }

A real-valued scalar field $\f$  is governed  by a second order differential operator on the torus, namely the Laplace-Beltrami operator. In local complex coordinates $(z,\bar z)$ on the torus $\Sigma =\CC/\Lambda$ with $\Lambda = \ZZ \oplus \tau \ZZ$, its action may be expressed as follows,
\bea
S[\f] = { 1 \over 2 \pi} \int _\Sigma d^2 z \, \p_z \f \, \pbz \f
\eea
The corresponding field equation  $\p_z \pbz \f=0$ implies that $\p_z \f$ is a locally holomorphic field while $\pbz \f$ is a locally anti-holomorphic field. On an annulus, these quantum fields admit  Laurent expansions,
\bea
i \p_z \f (z, \bar z) = \sum_{m \in \ZZ} \f_m \, z^{-m-1} \hskip 1in
i \pbz \f (z, \bar z) = \sum_{m \in \ZZ} \tilde \f_m \, \bar z^{-m-1}
\eea
where the coefficient operators $\f_m$ and $\tilde \f_m$  satisfy commutation relations,
\bea
{} [ \f_m, \f_n] = m \delta _{m+n,0} \hskip 1in {} [ \tilde \f_m, \tilde \f_n] = m \delta _{m+n,0}
\eea
We choose a polarization such that $\f_{-n} = \f_n^\dagger $ and $\tilde \f_{-n} = \tilde \f_n^\dagger$, and define the ground state $|0\>$ to be  annihilated by $\f_n|0 \>= \tilde \f_n |0\>=0$ for $n \geq0$. The Fock space is constructed by applying arbitrary polynomials in $\f_n^\dagger$ for $n >0$ to $|0\>$. Each such state has positive norm so that the Fock space for a scalar field is actually a  Hilbert space, which we denote by $\mF$.

\sm

The scalar field theory is conformal and has two commuting Virasoro algebras generated by $L_m$ and $\tilde L_m$ whose expressions in terms of $\f_m$ and $\tilde \f_m$ are given as follows,\footnote{The offset $-\tfrac{1}{24}$ in $L_0$ arises from the conformal transformation of the expression on the cylinder (which has no such offset) to the annulus. It may also be viewed as the sum of the zero-point energies of the harmonic oscillators, using the fact that the sum of all positive integers $m$ gives $\zeta(-1)=-\tfrac{1}{12}$.}
\bea
L_0 = - { 1 \over 24} + \half \f_0^2 + \sum _{n=1}^\infty \f_{-n} \f_n
\hskip 1in
L_m = \half \sum_{n\in \ZZ} \f_{m-n} \f_n \quad \hbox{ for } m \not=0
\eea
and similarly for $\tilde L_m$.  The operators $\p_z \f$ and $\pbz \f$ are conformal primary fields of weights $(1,0)$ and $(0,1)$ respectively, and satisfy the corresponding transformations given in (\ref{5.Lphi}).

\sm

The operator product expansion of two holomorphic fields is given by,
\bea
\label{5.OPEf}
\p_z \f(z) \p_w \f(w) = { 1 \over (z-w)^2} - 2 \, T(w) + \cO(z-w)
\eea
where the Laurent expansion of the stress tensor $T(w)$ for the holomorphic component of the scalar field is given in terms of the Virasoro generators $L_m$,
\bea
T(w) = \sum _{n\in \ZZ} L_n \, w^{-n-2}
\eea
and similarly for the anti-holomorphic field $\pbz \f$ and the generators $\tilde L_m$.

\sm

We may attempt to use the operator product expansion (\ref{5.OPEf}) to evaluate the correlator of holomorphic fields $\p_z\f(z)$, just as we had in the case of the $bc$ system. For simplicity, we choose matrix elements  between the ground state $|0\>=|0,0\>$ on the torus $\Sigma$,
\bea
\< 0 | \p_z \f(z_1) \cdots \p_z \f (z_N) |0\>
\eea
The correlator vanishes for odd $N$ and is meromorphic in all its points $z_n$ for even $N$. As a function of $z_n$,  the correlator has a double pole at every point $z_m$ with $m \not=n$. For example, for the 2-point correlator, the double pole shows that the correlator must be of the form,
\bea
\< 0 | \p_z \f(z_1)  \p_z \f (z_2) |0\> = \wp(z_1-z_2) + A_4
\eea
where $A_4$ is independent of $z_1, z_2$. In contrast with the $bc$ system, the zeros of the correlator are not manifest now, and the constant $A_4$ cannot be fixed by using known zeros. It may be determined, however, by using not just the pole term in the operator product expansion, but also the term proportional to the stress tensor $T$, and we find $A_4=0$. For higher point correlators, however, the corresponding constant terms will not be so easy to evaluate. Instead we shall turn to evaluating the full correlators of the field $\f$ itself.

\subsection{The scalar Green function on the torus}

The scalar Laplace-Beltrami operator on the torus has a zero mode and is not invertible on all scalar functions.
However, an inverse may be defined on its range, namely on functions orthogonal to constant functions on $\Sigma$. To construct this inverse, we shall present two different approaches, the first based on Fourier series, and the second using $\tet$-functions.

\subsubsection{The Laplacian on the torus}
\label{5.LapTor}

An arbitrary Riemannian metric on the torus $\Sigma$ may be rescaled to a flat metric using a $\Sigma$-dependent Weyl transformation, as shown in more detail in appendix \ref{sec:RS}. A further rescaling by a constant Weyl transformation allows us to obtain a flat metric with unit area. This reduced metric depends on a single complex modulus parameter $\tau \in \cH$, and we represent the torus  by the quotient $\Sigma = \CC/\Lambda$ for the lattice $\Lambda = \ZZ + \ZZ \tau$. The points on $\Sigma$ may be parametrized  by a complex coordinate $z= x + y \tau$ for $x,y \in \RR/\ZZ$, for example in the range $0 \leq x,y \leq 1$ (see Figure \ref{2.fig:2}). In these coordinates,  the flat metric $\mg$  and associated volume form $d \mu_\mg$ of unit area are given by,
\bea
\mg = { |dz|^2 \over \tau _2} = { |dx+ \tau dy|^2 \over \tau_2}
\hskip 1in
d\mu_\mg = {i \over 2} { d z \wedge d \bar z \over \tau_2} =  dx \wedge dy
\eea
We define an inner product on the space of complex-valued functions $f_1,f_2 : \Sigma \mapsto \CC$ by,
\bea
\label{5.f1f2}
\< f_1 | f_2 \>  = \int _\Sigma d\mu _\mg \, \bar f_1 \, f_2
\eea
making the space of  bounded functions on $\Sigma$ into an $L^2 (\Sigma)$ Hilbert space.  The Cauchy-Riemann operators in these coordinates take the form,
\bea
\p_z = { 1 \over 2 i \tau_2} ( \p_y - \bar \tau \p_x)
\hskip 1in
\pbz = - { 1 \over 2 i \tau_2} ( \p_y - \tau \p_x)
\eea
while the Laplace-Beltrami operator, or simply \textit{Laplacian},  is given by,
\bea
\Delta_\mg = - 4 \tau _2 \pbz \p_z = - { 1 \over \tau_2} (\p_y - \tau \p_x ) (\p_y - \bar \tau \p_x )
\eea
The Laplacian is self-adjoint with respect to the inner product (\ref{5.f1f2}) on $L^2 (\Sigma)$ and positive. It is invariant under translations in $x,y$ and thus commutes with the operators $i \p_x$ and $i \p_y$ that generate infinitesimal translations in the $x$ and $y$ directions respectively. For fixed $\tau $, the Laplacian $\Delta_\mg$  has a discrete spectrum with a single zero mode, namely the constant function on $\Sigma$, so that ${\rm Ker \Delta_\mg} = \CC$. Its range is the set of  functions orthogonal to constant functions.

\subsubsection{The scalar Green function via Fourier series}

Since the self-adjoint operators $i \p_x$ and $i \p_y$ mutually commute and commute with $\Delta _\mg$, the Laplacian is diagonal in a basis where $i \p_x$ and $i \p_y$  are simultaneously diagonal. Such a basis is given by the following orthonormal basis of doubly periodic functions on $\Sigma$,
\bea
\label{10.orthon}
f_{m,n}(x,y) & = & e^{2 \pi i (nx-my)}
\no \\
\< f_{m',n'} | f_{m,n} \> & = & \delta _{m',m} \, \delta _{n',n}
\hskip 1in m,m',n,n' \in \ZZ
\eea
They also satisfy the completeness relation,
\bea
\label{10,comp}
\sum_{m,n \in \ZZ} \overline{ f_{m,n}}(u,v) f_{m,n}(x,y) = \delta (x-u) \delta (y-v)
\eea
where the Dirac $\delta$ functions on the right are periodic in their argument with period 1.
One verifies that the Laplace operator is indeed diagonal in this basis, and we find,
\bea
\Delta_\mg \, f_{m,n}= \lambda _{m,n}(\tau)   \, f_{m,n}
\hskip 1in
\lambda _{m,n} (\tau) = 4 \pi^2 \, { |m+  n\tau |^2 \over \tau_2}
\eea
The zero eigenvalue corresponds to the constant function with $m=n=0$, while all other eigenvalues are strictly positive, as expected. The Green function is customarily defined as the inverse of $\Delta_\mg /4\pi$ on the space of functions orthogonal to constants. In view of translation invariance, we have $G(z,w|\tau)=G(z-w|\tau)$ with $z= x + y \tau$,  and $G(z|\tau)$ is given by,
\bea
\label{10.green1}
G(z|\tau) = \sum _{(m,n) \not= (0,0)} { \tau_2  \over \pi |m+n \tau|^2} \, e^{ 2\pi i (nx - my)}
\eea
Applying the Laplacian, and using the completeness relation (\ref{10.orthon}), one verifies,
\bea
\label{10.green2}
\Delta _\mg G(z-w|\tau) = 4 \pi \tau_2 \, \delta^{(2)} (z-w) - 4 \pi
\eea
where the 2-dimensional $\delta$-function, with $w=u+\tau v$, is defined by,
\bea
\tau_2 \, \delta^{(2)} (z-w) =\delta (x-u) \delta (y-v)
\eea
By construction, the right side of (\ref{10.green2})  is orthogonal to constant functions. Actually, the Green function obtained in (\ref{10.green1})  itself is orthogonal to constant functions,
\bea
\label{10.norm}
\int _\Sigma d \mu _\mg(z) \, G(z-w|\tau) =0
\eea
since the zero mode $(m,n)=(0,0)$ has been removed from the sum in (\ref{10.green1}).

\subsubsection{The scalar Green function via $\tet$-functions}

We may take equations (\ref{10.green2}) and (\ref{10.norm}) as defining $G$ and express both in terms of the complex variables $z,w$,
\bea
\label{10.green3}
-  \p_z \pbz G(z-w|\tau) =  \pi \delta^{(2)} (z-w) - { \pi \over \tau_2}
\eea
The solution for $G$ is easy to construct in terms of $\tet$-functions, as we shall now show. This method will generalize to constructing the Green function on Riemann surfaces of arbitrary genus, whereas the method of Fourier series of the preceding section does not generalize since higher genus Riemann surfaces have no continuous isometries.

\sm

We proceed as follows for fixed $\tau \in \cH$. For $|z-w|\ll 1$ and  $|z-w|\ll\tau_2$,  the global structure of the torus is negligible and the Green function must approach its expression on $\CC$  which is given by $G(z-w|\tau) \approx - \ln |z-w|^2$. The function $\tet_1(z-w|\tau)/\tet_1'(0|\tau)$ behaves as $z-w$ for small $|z-w|$ and is therefore a reasonable candidate to play the role of $z-w$ on the torus. However, $\tet_1(z-w|\tau)$ cannot be a doubly periodic function as it is holomorphic and would then have to be constant by Liouville's theorem. Still, given the transformation laws of $\tet_1(z-w|\tau)$ under $z \to z+1$ and $z \to z+\tau$ given in (\ref{2.theta-shift}), one readily finds a simple additive term to restore double periodicity, and $G(z,w|\tau)=G(z-w|\tau)$ is given by,
\bea
G(z|\tau) = - \ln \left | { \tet _1 (z|\tau) \over \tet _1 '(0|\tau)} \right |^2 + 2 \pi {(\Im z)^2 \over \tau_2 } + G_0 (\tau)
\eea
Upon applying the Laplace operator, the first term produces the $\delta$-function, while the second term produces the constant on the right side of (\ref{10.green3}).  The $z$-independent term $G_0(\tau)$ may be determined  by imposing the normalization condition (\ref{10.norm}).

\sm

To compute $G_0(\tau)$ it is convenient to choose the range of integration $-\half \leq x,y \leq \half$. The integrals of the last two terms in $G$ are readily evaluated. To compute the   integral of the first term in $G$ we use the product formula for $\tet_1$ given in (\ref{tetprod}),  expand the logarithm of the factors $(1-q^n e^{ \pm 2 \pi i z})$ in powers of $q$, observe that their integral vanishes term-by-term, and finally compute the integral of $\ln |\sin (\pi z)|^2$ by the same expansion method. Putting all together, we obtain the properly normalized scalar Green function,
\bea
\label{10.green-theta}
G(z|\tau) = - \ln \left | { \tet _1 (z|\tau) \over \eta (\tau)} \right |^2 + 2 \pi {(\Im z)^2 \over \tau_2 }
\eea
The expression (\ref{10.green1}) for $G$ may be related to (\ref{10.green-theta}) by carrying out the sum over $m$ first  in (\ref{10.green1}) using the following Lemma.
\label{Lipschitz}
The Lipschitz formula for $\Re(z)>0$, $\Re(s)>0$ and $0< \alpha \leq 1$ is given by,
\bea
\sum_{n \in \ZZ} { e^{2 \pi i n \a} \over (z+in)^s} = { (2 \pi)^s \over \Gamma(s)} \sum_{m=0} ^\infty (m+\a)^{s-1} \, e^{-2 \pi z(m+\a)}
\eea}
The proof may be found in Rademacher [46].

\subsubsection{Modular properties}

A modular transformation $ \gamma \in SL(2,\ZZ)$ leaves the metric $\mg$ and the Laplacian $\Delta _\mg$  invariant provided $\tau$ and the complex coordinate $z$ transform according to the following rules,
\bea
\tau' = \gamma \tau = { a \tau + b \over c \tau + d}
\hskip 0.7in
z' = \gamma z= { z \over c \tau +d}
\hskip 0.7in
\gamma = \left ( \bma a & b \cr c & d \ema \right )
\eea
The transformations on the real coordinates are as follows,
\bea
\left ( \bma  x' \cr  y' \ema \right ) = \left ( \bma a & -b \cr -c & d \ema \right ) \left ( \bma x \cr y \ema \right )
\eea
The spectrum of the Laplacian is invariant under $SL(2,\ZZ)$, as the eigenvalues $\lambda_{m,n}(\tau)$ of $\Delta_\mg$ transform as follows,
\bea
\lambda _{m,n} (x,y) =  \lambda _{m',n'}(x',y')
\hskip 1in
\left ( \bma  m' \cr  n' \ema \right ) = \left ( \bma a & -b \cr -c & d \ema \right ) \left ( \bma m \cr n \ema \right )
\eea
in view of the relation $n'x'-m'y'=nx-my$. It readily follows that also the normalized Green function, given in (\ref{10.green1}), is modular invariant,
\bea
\label{10.G-mod}
G\left ( {z-w \over c \tau + d} \bigg | { a \tau + b \over c \tau +d} \right ) = G(z-w|\tau)
\eea
Modular invariance may be verified directly on the expression for the Green function in terms of $\tet$-functions in (\ref{10.green-theta}) by using the transformation laws of $\tet_1$ given in (\ref{4.mod2}).

\subsection{Scalar determinant and the first Kronecker limit formula}

The functional determinant of $\Delta= \Delta_\mg/4 \pi$, considered as an operator on the subspace of functions orthogonal to constants,  is formally given by the product of the eigenvalues $\lambda_{m,n} (\tau)/4 \pi $. However, this product does not converge.  To define the determinant we introduce the following $\zeta$-function associated with $\Delta$,
\bea
\label{4d5}
\zeta _{\Delta} (\tau, s) = \sum _{(m,n ) \not = (0,0)}   \left ( {4 \pi \over \lambda_{m,n}} \right )^s= E_s(\tau)
\eea
where $E_s(\tau)$ is the non-holomorphic Eisenstein series introduced in section \ref{sec:4.2}. The series for $E_s(\tau)$ is absolutely convergent for $\Re(s) >1$. Its analytic continuation to $s \in \CC$ has a simple pole at $s=1$, is modular invariant, and satisfies the functional relation,
\bea
\label{4d8}
\Gamma (s) E_s(\tau) = \Gamma (1-s) E_{1-s}(\tau)
\eea
Taking the limit $s \to 0$ of (\ref{4.d12}) gives  $ \zeta_\Delta(\tau,0)=-1$, while the determinant is given by,
\bea
\label{4d5a}
\ln \Det ' \Delta = - \zeta_{\Delta}  '(\tau, 0)
- \p_s E_s(\tau) \Big |_{s=0}
\eea
To compute the determinant, we use the expressions for $E_s(\tau)$ in terms of the functions $A_s(\tau)$ and $B_s(\tau)$ defined in (\ref{4.d9}) and (\ref{4.d10}),
\bea
\zeta'_\Delta(\tau, 0) = A_0(\tau)  + {\pi \tau_2 \over 3} - \ln \tau_2 - \ln (4 \pi)
\eea
The last three terms on the right side arise from differentiating $B_s(\tau)/\Gamma(s)$ at $s=0$. To compute $A_0(\tau)$, we simplify the summand in (\ref{4.d10}) by using the expression for the modified Bessel function in terms of elementary functions,  $K_{-\half}(x)= \sqrt{\pi/2x} \, e^{-x}$. We discover that $A_0(\tau)$ becomes a sum over holomorphic and anti-holomorphic dependence on $\tau$,
\bea
A_0(\tau) = \sum _{m,n \not=0} { 1 \over |m| } \, \exp \Big \{ 2 \pi i mn  \tau _1 -2 \pi |mn| \tau_2  \Big \}
\eea
which may be compactly expressed as follows,
\bea
A_0(\tau) = -2 \ln \prod _{m=1}^\infty (1-q^m)(1-\bar q^m)
\eea
We may express the final combined result in terms of the Dedekind $\eta(\tau)$ function,
\bea
\zeta'_\Delta(\tau, 0) = - \ln \left ( 4 \pi \tau _2  |\eta (\tau)|^4 \right )
\eea
Combining this with $\zeta _\Delta (\tau, 0)=-1$ and (\ref{4d5a}), we obtain,
\bea
\Det ' \Delta =   4 \pi \tau _2  |\eta (\tau)|^4 = 4 \pi  \Det ' \Delta _\mg
\eea
which is manifestly modular invariant.

\subsubsection{The first Kronecker limit formula}

The first Kronecker limit formula gives the residue and the finite part of $\zeta_\Delta(\tau, s)$ at its pole $s=1$ by the following expression,
\bea
\zeta_\Delta (\tau, s) = { a_{-1} \over s-1} + a_0 + \cO (s-1)
\eea
Using the functional relation (\ref{4d8}), it is immediate that $a_{-1}=-1$. From the same formula we get also $a_0$ by,
\bea
a_0 = { d \over ds}  \Big ( (s-1) E_s(\tau) \Big ) \Big |_{s=1} = - \ln \left ( 4 \pi \tau _2  |\eta (\tau)|^4 \right )  + 2 \gamma
\eea
where $\gamma $ is Euler's constant as it arises in $\Gamma '(1)=-\gamma$.

\subsubsection{The scalar partition function}
\label{sec:partitions}

The scalar determinant is intimately related to the partition function for the free scalar field in statistical mechanics. A partition function $Z(\beta)$  for a Hamiltonian $H$ acting in a Hilbert space $\mF$, evaluated at an inverse temperature $\beta$ is given by,
\bea
Z(\beta)= \Tr _\mF \left ( e^{-\beta H} \right )
\eea
where $\Tr_\mF$ is the trace evaluated in the Hilbert space $\mF$. The scalar field has two independent sets of oscillators, namely $\f_m$ and $\tilde \f_m$, with independent Virasoro generators. In particular, the generators $L_0$ and $\tilde L_0$ are given by a sum of decoupled harmonic oscillator Hamiltonians plus the term due to the zero mode.
Instead of the real-valued inverse temperature $\beta>0$, we shall complexify $\beta$ and  let $\beta \to -2 \pi i \tau$ whose real part is still positive in view of $\Im(\tau) >0$. The partition function for $L_0$ is then given as follows,
\bea
\Tr \left ( q^{ L_0} \right ) = q^{-\tfrac{1}{24} + \tfrac{1}{2} \f_0^2} \prod _{n=1}^ \infty { 1 \over (1-q^n)}
= q^{\tfrac{1}{2} \f_0^2} \, \eta (\tau)^{-1}
\eea
where the trace is taken only over the oscillator part of the Hilbert space $\mF$. Proceeding similarly for $\tilde L_0$ and complexifying its inverse temperature $\beta \to 2 \pi i \bar \tau$, we obtain the combined partition function,
\bea
\Tr  \left ( q^{L_0} \bar q ^{\tilde L_0} \right )
= q^{\tfrac{1}{2} \f_0^2} \bar q^{\tfrac{1}{2} \tilde \f_0^2} { 1 \over |\eta (\tau)|^2}
\eea
The zero mode operators $\f_0$ and $\tilde \f_0$ are self-adjoint and have a continuous spectrum of real eigenvalues. Physically, these eigenvalues are the momenta of left- and right-movers. Periodicity in $\tau_1 \to \tau_1 +1$ requires $\f_0^2 - \tilde \f_0^2 \in 2 \ZZ$. When the field $\f$ takes values in $\RR$, we require the eigenvalues of $\f_0$ and $\tilde \f_0$ to be equal to one another, and to range over $\RR$. The full trace is then given by,
\bea
\Tr _\mH \left ( q^{L_0} \bar q ^{\tilde L_0} \right ) =
{ 1 \over |\eta (\tau)|^2} \int _\RR dp \, | q | ^{p^2} = { 1 \over \sqrt{ \tau_2} \, |\eta (\tau)|^2} = \left ( \Det ' \Delta _\mg \right )^{-{1 \over 2}}
\eea
which is precisely the modular invariant partition function evaluated using the functional integral.

\subsection{Spinor fields}

The orthogonal Lie algebra $SO(d)$ has spinor representations in addition to its tensorial representations. A spinor representation may be viewed either as  a double-valued representation of the group $SO(d)$ or as a single-valued representation of the double cover $Spin(d)$. A manifold supports spinor fields provided it is orientable and its second Stiefel-Whitney class vanishes. These conditions are satisfied in $d=2$ for Riemann surfaces.  In terms of bundles, one decomposes the cotangent bundle $T^*\Sigma$ into a direct sum of its holomorphic and anti-holomorphic part $T^*\Sigma = K \oplus K^*$ where $K$ is referred to as the canonical bundle. A spin bundle $S$ is a holomorphic line bundle whose square is isomorphic to the canonical bundle $S^{\otimes 2} \approx K$, and a spinor field is then an operator-valued section of $S$. For more details on Riemann surfaces see appendix \ref{sec:RS}, and for the general construction of line bundles on an arbitrary Riemann surface see appendix \ref{sec:LB}.

\sm

Different realizations of spinor fields, corresponding to different spin bundles,  on a given Riemann surface $\Sigma$,  are referred to as  \textit{spin structures}. They are labeled by  the first homology class of the surface with $\ZZ_2$ coefficients, namely $H_1(\Sigma, \ZZ_2)$. For example, the homology group of a compact genus $g$ Riemann surface is isomorphic to $\ZZ^{2g}$ and therefore the different spin structures are labeled by $\ZZ_2$ on each one of the $2g$ independent homology cycle.

\sm

For the annulus, there is a single homology cycle, and thus two different spin structures, referred to as Neveu-Schwarz (NS) or Ramond (R). Denoting a holomorphic spinor field of conformal weight $(h,0)$ by $\psi$, its Laurent series on the cylinder and on the annulus, with coordinates $z'=x+iy$ and   $z=e^{2 \pi i z'}$ respectively,  are given as follows,
\begin{align}
\hbox{NS} && \psi '(z')  & = \sum _{r \in \ZZ+\half} \psi_r \, e^{-2 \pi i r(x+iy)} & \psi (z) & = \sum_{r \in \ZZ+\half}
\psi_r \, z^{-r -h}
\no \\
\hbox{R} \, && \psi '(z')  & = \sum _{n \in \ZZ} \psi_n \, e^{-2 \pi i n(x+iy)} & \psi (z) & = \sum_{n \in \ZZ}
\psi_n \, z^{-r -h}
\end{align}
A free spinor field on a Riemann surface  is governed by a first order differential operator, namely by a generalized  $bc$ system in which the fields $b$ and $c$ are allowed either NS or R identifications. An important special case is that of a spinor field of conformal weight $(\half, 0)$, in which case the fields $b$ and $c$ have the same conformal weight $(\half, 0)$ and are quantized using anti-commutation relations as in the $bc$ system we discussed earlier.\footnote{Another important case is the superghost $\beta \gamma$ system where $b=\beta$ has weight $({3 \over 2}, 0)$ and $c=\gamma$ has weight $(-\half,0)$ and the oscillators obey commutation relations.}  In this case the NS spin structure makes the field $\psi(z)$ single-valued on the annulus, while the R spin structure makes the field $\psi(z)$ double-valued on the annulus.

\sm

For the torus, a canonical choice of homology cycles $\mA$ and $\mB$ is given by the identifications $z\to z+1$ and $z \to z+\tau$ respectively. The four spin structures are distinguished according to whether the field is periodic or anti-periodic around each cycle. The signs are inherited from the cylinder,
\bea
\psi (z+1) & = & - e^{2 \pi i \a} \psi(z)
\no \\
\psi (z+\tau) & = & - e^{2 \pi i \beta} \psi(z)
\eea
for $\a, \b \in \{ 0 , \thalf \}$ (mod 1) with the values 0 and $\thalf$ corresponding to NS and R respectively. Under the modular transformations $S$ and $T$ the four spin structures transform as follows,
\begin{align}
S: &  \left [ \bma \a \\ \b \ema \right ] \rightarrow \left [ \bma \b \\ \a \ema \right ]
& T: & \left [ \bma \a \\ \b \ema \right ] \rightarrow \left [ \bma \a \\ \a+\b +\thalf  \ema \right ]
\end{align}
where the addition $\a+\b+\thalf$ is evaluated mod 1. The spin structure $\a=\b=\thalf $ is referred to as the \textit{odd spin structure}, while the three other spin structures are referred to as \textit{even spin structures}. Under modular transformations, the odd spin structure maps to itself, while the three even spin structures are permuted amongst one another. This transformation rule precisely corresponds to the transformation rule for Jacobi $\tet$-functions with half-integer characteristics given in (\ref{4.mod2}). This is no accident, as will be further clarified below. Furthermore, in view of the existence of the holomorphic and nowhere vanishing differential $dz$, spinor fields on the torus for a given spin structure but different conformal weights $(h,0)$ are all isomorphic to one another. Therefore, we may focus attention on the case $h=\half$. For two functions $f_1$ and $f_2$ with the same spin structure (or more accurately for two sections of a given spin bundle $S$), we define a Hermitian inner product,
\bea
\<f_1 |f_2\> = \int _\Sigma d\mu_\mg \bar f_1 f_2
\eea
which has a positive definite norm and produces a Hilbert space $L^2(\Sigma, S)$.

\subsection{Spinor Green functions on the torus}

In this subsection, we shall derive the Green function for the spinor field on a torus $\Sigma = \CC/\Lambda$ with $\Lambda = \ZZ \oplus \tau \ZZ$ for  each one of the four spin structures. Just as we did for the scalar field, we offer two perspectives: a calculation via Fourier analysis and a calculation directly in terms of $\tet$-functions.

\subsubsection{Odd spin structure Green function}

For the odd spin structure $\a=\b=\half $, the Green function may be obtained from the results of the scalar case, because the spinor field  is doubly periodic and has the constant spinor as a zero mode. We define the Green function $S_1(z|\tau)$ that satisfies the following equation,
\bea
\pbz S_1(z-w|\tau) = \pi \delta ^{(2)}(z-w) - {\pi \over \tau_2}
\eea
by analogy with the Laplace equation satisfied by the scalar Green function $G(z-w|\tau)$ in (\ref{10.green3}).
The solution is given in terms of the scalar Green function by,
\bea
S_1(z-w|\tau) = - \p_z G(z-w|\tau) = \p_z \ln \tet_1(z-w|\tau) + { 2 \pi i \over \tau_2} \, \Im (z-w)
\eea
up to an additive constant which amounts to adding a term proportional to the Dirac zero mode.
Since $S_1(z-w|\tau)$ has a single pole in $z$ at $w$, and is doubly periodic, it cannot possibly be meromorphic
as no elliptic functions with a single simple pole exist. The second term on the right is present to compensate.
Alternatively, a term-by-term differentiation of the Kronecker-Eisenstein series of $G(z-w|\tau)$ in (\ref{10.green1}) gives the following expression for $S_1$,
\bea
S_1(z-w|\tau) = \sum _{(m,n) \not= (0,0)} {  e^{ 2 \pi i (nx-my)} \over m + n \tau}
\eea
The sum is only conditionally convergent and requires regularization. An alternative Green function $\tilde S_1(z;w_1,w_2|\tau) $ allows for two poles in $z$ at $w_1$ and $w_2$ with opposite residue. Requiring anti-symmetry in $w_1, w_2$ produces a unique Green function,
\bea
\tilde S_1(z;w_1,w_2|\tau ) & = & S_1(z-w_1|\tau) + S_1(w_1-w_2|\tau) + S_1 (w_2-z|\tau)
\no \\
& = & \zeta (z-w_1|\tau) + \zeta (w_1-w_2|\tau) + \zeta(w_2-z|\tau)
\eea
where the expression is recast in terms of the Weierstrass $\zeta$-function on the second line above.

\subsubsection{Even spin structure Green functions via Fourier analysis}

The chiral Dirac operator $\pbz$ is not self-adjoint. However, it is a linear combination with constant coefficients of two self-adjoint operators $i \p_x$ and $i \p_y$,
\bea
\label{5.pbz}
\pbz = { 1 \over 2 \tau_2} ( i \p_y - \tau\,  i \p_x)
\eea
Since the operators $i\p_x$ and $i \p_y$ mutually commute, the operator $\pbz$ may be diagonalized in the basis in which $i\p_x$ and $i \p_y$ are diagonal. However, since the coefficients of the linear combination in (\ref{5.pbz}) are complex, the eigenvalues of $\pbz$ are complex.

\sm

For a given even spin structure $[\kappa ]= \ss$, the operators $i \p_x$ and $i \p_y$ are diagonalized in the following basis,
\bea
f [\kappa]  _{m,n} (x,y)  =  \exp \Big \{ 2 \pi i ( (n+\beta+\thalf )x - (m+\alpha +\thalf ) y) \Big \}
\eea
The eigenvalues of the operator $\pbz$ are given by,
\bea
\lambda [\kappa]  _{m,n} = { \pi \over \tau_2} \Big (m+\a + \thalf  + (n + \b + \thalf ) \tau \Big )
\eea
Since $\Im (\tau) >0$ and $\a, \b$ do not both equal $\thalf$ mod 1, none of the eigenvalues vanishes. The Green function for the operator $\tau_2 \pbz$ is then given by,
\bea
S[\kappa]  (z|\tau) = { 1 \over \tau_2} \sum_{m,n \in \ZZ} { f[\kappa]  _{m,n} (x,y) \over \lambda [\kappa] _{m,n}}
=\sum_{m,n \in \ZZ} { e^{2 \pi i \{ (n +\b +\half ) x - (m+\a+\half ) y \} } \over m+ \a+ \half  + (n +\b+\half ) \tau}
\eea
so that $S[\kappa]$ satisfies,
\bea
\pbz S[\kappa] (z-w|\tau) = \pi \delta (z-w)
\eea
where the integral of $\delta(z-w)$ over $\Sigma$ is normalized to $\tau_2$.

\subsubsection{Even spin structure Green functions via $\tet$-functions}

We seek a meromorphic function $S[\kappa](z-w|\tau)$, which has a single simple pole  in a fundamental parallelogram for $\Lambda$, is doubly periodic with periods in $2 \Lambda$, and behaves as follows under translations in $\Lambda /(2 \Lambda)$,
\bea
S[\kappa ] (z+1|\tau ) & = & - e^{2 \pi i \alpha} \, S [\kappa ] (z|\tau) \hskip 1in [\kappa] = \ss
\no \\
S[\kappa ] (z+\tau|\tau ) &  = & - e^{2 \pi i \beta} \,  S [\kappa ] (z|\tau)
\eea
For even spin structures $\kappa$, the simple pole in $\CC/\Lambda$ and the above translations are realized uniquely by the ratio $\tet [\kappa] (z|\tau)/\tet_1(z|\tau)$, and the normalization of the residue at $z=w$ gives,
\bea
S [\kappa]  (z-w|\tau) = { \tet [\kappa]  (z-w|\tau)
\, \tet _1 '(0|\tau) \over \tet [\kappa]  (0|\tau) \, \tet _1 (z-w|\tau) }
\eea
As an elliptic function in $\CC/(2 \Lambda)$, the \textit{Szeg\"o kernel} $S[\kappa](z-w|\tau)$ has four simple poles, namely at $z\equiv 0, \tau+1$ with residue 1, and at $z \equiv 1, \tau$ with residue $-1$.

\subsection{Spinor determinant and the second Kronecker limit formula}

The eigenvalues of the chiral Dirac operator $\pbz$ are complex, rendering the $\zeta$-function approach to evaluating its determinant inapplicable. Instead we shall make use of the fact that the dependence of the operator $\tau_2 \pbz$  on the modulus $\tau$ is holomorphic and we shall seek to construct an object $\Det (\pbz) $ that is holomorphic in $\tau$.

\sm

What we do know how to evaluate using $\zeta$-function methods  is the determinant of the Laplace operator $\Delta _\kappa=  -4 \tau_2 \p_z \pbz$ on a spinor with even spin structure $\kappa$. The Laplace operator is self-adjoint, has positive spectrum, and no zero modes,
\bea
\lambda [\kappa] _{m,n} = { 4 \pi^2 \over \tau_2} \big | m+\a +(n+\beta) \tau \big |^2
\hskip 1in
\kappa = \left [ \bma \half +\a \\ \half +\b \ema \right ]
\eea
In fact we may generalize the problem by taking the characteristics $\a, \b$ to be arbitrary with the sole restriction that $(\a,\b) \not \equiv (0,0) $ mod 1.  The corresponding $\zeta$-function is defined by the Kronecker-Eisenstein sum,
\bea
\zeta _{\Delta_\kappa}  (\tau, s) = \sum _{m,n \in \ZZ} { \tau_2^s \over \pi^s |m+\alpha + (n + \beta )\tau|^{2s}}
\eea
which is absolutely convergent for $\Re(s)>1$. We shall show below that $\zeta_{\Delta_\kappa}(\tau, s)$ may be analytically continued in $s$ throughout the complex plane. The determinant is obtained from this analytic continuation as usual,
\bea
\ln \Det \Delta _\kappa = - \zeta _{\Delta _\kappa} '(\tau, 0)
\eea
We recast the Kronecker-Eisenstein sum in terms of the following  heat-kernel representation,
\bea
\label{4d6}
\Gamma (s) \zeta_{\Delta _\kappa}  (\tau, s) = \int _0 ^\infty { dt \over t} \, t^s  \sum _{m,n\in \ZZ}
\exp \left \{  - { \pi t \over \tau_2}  \big |m+\alpha  +(n+\beta) \tau \big |^2  \right \}
\eea
and Poisson re-sum either in the summation variable  $m$ if $\beta \not=0$ or in $n$  if $\a\not=0$ (but not in both). We shall work out the case $\beta \not=0$,  the other case being analogous, using the formula,
\bea
\sum _{m,n \in \ZZ}  e^{- \pi t |m+\alpha +(n+\beta) \tau|^2/\tau_2}
= \sqrt{ { \tau _2 \over t}} \sum _{m,n \in \ZZ} e^{2 \pi i m (\alpha + (n+\beta)  \tau_1} \, e^{ - \pi m^2\tau_2/t - \pi (n+\beta)^2 t \tau_2}
\eea
Substituting this result into (\ref{4d6}) and  splitting the sum over $m,n$ into contributions for $m  \not=0 $ and $m=0$, we have $\Gamma (s)  \zeta_{\Delta _\kappa} (\tau,s) =  A_s(\tau)+B_s(\tau)$ with,
\bea
\label{4d9}
A_s(\tau) & = & \sqrt{\tau_2} \sum _{m \not=0} \sum _n e^{2 \pi i m(\alpha + (n+\beta)  \tau_1)}
\int _0^\infty { dt \over t} \,
t^{s-\half}  \, e^{- \pi \tau_2 (m^2/t + (n+\beta) ^2 t)}
\no \\
B_s(\tau) & = &  \sqrt{\tau_2} \int _0^\infty { dt \over t} \,
t^{s-\half}  \sum _{n}   \,  e^{ - \pi (n+\beta) ^2 t \tau_2}
\eea
The integral $A_s(\tau)$ is manifestly holomorphic in $s$ throughout $\CC$, and may be evaluated  by performing the $t$-integral in terms of a Bessel function,
\bea
\label{4d10a}
A_s(\tau) = 2 \sqrt{\tau_2} \sum _{m \not=0} \sum _n
\left | { m \over n+\beta } \right |^{s-\half} e^{2 \pi i m (\alpha + (n+\beta)  \tau_1)}
K_{s-\half} \big ( 2 \pi \tau_2  |m(n+\beta) | \big )
\eea
To evaluate the determinant, we only need $A_0(\tau)$, which may be read off from the above formula by setting $s=0$
and using the expression for the spherical Bessel $K_{-\half} (x) = \sqrt{\pi/2x} \, e^{-x}$,
\bea
A_0(\tau) = \sum _{m=1}^ \infty \sum _n { 1 \over m} \, e^{-2 \pi \tau_2 m |n+\beta|} \left (
e^{ 2 \pi i m (\alpha + (n+\beta) \tau_1) } + e^{ - 2 \pi i m (\alpha + (n+\beta) \tau_1) } \right )
\eea
Without loss of generality we may take $0< \beta < 1$ to carry out the sum and obtain,
\bea
A_0(\tau) = - \sum _{n=0}^\infty  \ln \left | 1 - q^{n+\beta} \, e^{2 \pi i \alpha} \right |^2
- \sum _{n=1}^\infty \ln \left | 1 - q^{n-\beta} \, e^{- 2 \pi i \alpha} \right |^2
\eea
Finally, we compute $B_s(\tau)$ by performing the integrals under the sum,
\bea
B_s(\tau) =  \Gamma (s-\thalf) \pi ^{\half -s} \tau_2 ^{1-s} \sum _n { 1 \over |n+\beta|^{2s-1}}
\eea
The sum is absolutely convergent for $\Re(s)>1$ and may be evaluated in terms of Hurwitz $\zeta$-functions. The analytic continuation of the Hurwitz $\zeta$-function is well-known, and the final result is given by,
\bea
\zeta _{\Delta_\kappa}  '(\tau, 0) = - \ln \left | { \tet [\alpha \, \beta ] (0|\tau) \over \eta (\tau) } \right |^2
\eea
This result is a version of the \textit{second Kronecker limit formula}.

\sm

The determinant obtained above is for the $bc$ system in which both $b$ and $c$ are conformal spinor fields of weight $(\thalf, 0)$, along with their complex conjugate fields. The natural candidate for the determinant of the chiral operator $\pbz$ is obtained by retaining only the part with holomorphic dependence on $\tau$,
\bea
\Det \, \pbz \Big |_{bc} = { \tet [\alpha \, \beta ] (0|\tau) \over \eta (\tau) }
\eea
The spinor fields that are required in string theory for the worldsheet matter fermions are further restricted to $\psi=b=c$, so that the $b$ and $c$ fields are actually identical fields. The functional chiral determinant of the operator $\pbz$  in this case is the square root of the determinant of the $bc$ system,
\bea
\Det \, \pbz \Big |_\psi = { \tet [\alpha \, \beta ] (0|\tau)^\half  \over \eta (\tau)^\half }
\eea