To obtain the number of solutions of the congruence $x^2 \equiv -1 ~ (\mod N)$ for an arbitrary integer $N$ with prime factorization $N = \prod _i p_i ^{\a_i}$ we consider the polynomial $f(x) = x^2 +1$ with $f'(x)=2x$. If $p$ is an odd prime, and $0\leq r < p$ solves $r^2+1 \equiv 0 ~ (\mod p)$ then $r \not \equiv 0 ~(\mod p)$. As a result, $f'(r) = 2 r \not \equiv 0 ~ (\mod p)$ so that there is a unique lift of $r$ to $(\mod p^2)$, and to any $(\mod p^\a)$. If $p=2$ then $r \equiv 1 ~ (\mod 2)$ so there is one solution $(\mod 2)$. Now using Theorem \ref{A.thm:3}, $f'(r) \equiv 0 ~ (\mod 2)$,  and the fact that every lift  $f(1+2q) \equiv 1 ~ (\mod 4)$ implies that there are no solutions $(\mod 4)$, and hence no solutions $(\mod 2^\a)$ for all $\a \geq 2$. Hence the number of solutions to $x^2 \equiv -1 ~ (\mod N)$ is given by,
\bea
\label{quadcon.1}
\# \Big \{ x \in \ZZ_N \,  \big | \, x^2 \equiv -1 ~ (\mod N) \Big \} = \left \{ \bma 0 && 4 | N \cr
\prod _i \Big ( 1+ (-1|p_i) \Big ) && 4 \nmid  N \cr \ema \right .
\eea
where  the Legendre symbol $(-1|p)$ evaluates to $(-1)^{{ p-1 \over 2}}$ for an odd prime $p$. For $p=2$, the number of solutions is 1, so we set $(-1|2)=0$.

\subsubsection{Quadratic residues  $x^2+x \equiv -1 ~ (\mod N)$}
\label{sec:1.5.2}

To obtain the number of solutions of the congruence $x^2+x \equiv -1 ~ (\mod N)$ for an arbitrary integer $N$ with prime factorization $N = \prod _i p_i ^{\a_i}$ we consider the polynomial $f(x) = x^2+x +1$ with $f'(x)=2x+1$.
Solving $f(x) \equiv 0 ~ (\mod N)$ is equivalent to solving,
\bea
(2x+1)^2 + 3 \equiv 0 ~ (\mod 4N)
\eea
If $9 |N$, then the congruence $y^2+3 \equiv 0 ~ (\mod 9)$ has no solutions as the quadratic residues $\mod 9$ are $0,1,4,7$, giving the residues of $y^2+3$ to be $3,4,7,1$ but not 0.  For an odd prime $p$, the congruence  $y^2 + 3 \equiv 0 ~ (\mod p)$ has $1+(-3|p)$ solutions. By quadratic reciprocity, if $p \not= 3$ then $(-3|p) = (-1|p)(3|p) =(p|3)$ which evaluates to $\pm 1$ when $p \equiv \pm 1 ~ (\mod 3)$. For $p \equiv -1 ~ (\mod 3)$ there are no solutions $y \in \ZZ$ and hence no solution of the form $y \equiv 2x+1~ (\mod p)$. For $p \equiv 1 ~ (\mod 3)$ there are two solutions $y ~ (\mod p)$. For any given $y$, there is then a unique solution $x$ to $y \equiv 2x+1~ (\mod p)$.
The number of solutions is thus given by,
\bea
\label{quadcon.2}
\# \Big \{ x \in \ZZ_N \, \big | \, x^2+x \equiv -1 ~ (\mod N) \Big \} = \left \{ \bma 0 && 9 | N \cr
\prod _i \Big ( 1+ (-3|p_i) \Big ) && 9 \nmid  N \cr \ema \right .
\eea
For $p=3$ the number of solutions is 1, so we set $(-3|3)=0$.

\sm

\subsection{Gauss sums}

For an odd prime $p$, and an arbitrary $a \in \ZZ$, the Gauss sum $g_a$ is defined by,
\bea
\label{Gauss1}
g_a = \sum _{k=0}^{p-1} (k|p) \, \ep ^{ka}
\hskip 1in
\ep = e^{2\pi i/p}
\eea
Gauss sums are directly related to the Legendre symbol,
\bea
\label{Gauss2}
g_a = (a|p) g \hskip 1in g= g_1
\eea
Indeed, for $a$ divisible by $p$ we have $g_a=0$ in view of the fact that the number of residues in $\ZZ_p^*$ equals the number of non-residues. Hence the relation holds trivially when $p | a$ as both sides vanish. For $p \nmid a$,
we use the fact that $(a|p)^2=1$, the definition of $g_a$, and the complete multiplication property of the Legendre symbol,
\bea
g_a =  (a|p) \sum _{k=0}^{p-1} (a|p) (k|p) \, \ep ^{ka} = (a|p) \sum _{k=0}^{p-1} (ka|p)  \, \ep ^{ka} = (a|p) g
\eea
The last equality follows,  for $p \nmid  a$, from the fact that the sum over $k$ is identical to the sum over $ka$. Furthermore, we have the following relation,
\bea
\label{g2}
g^2 = ( -)^{(p-1)/2} p
\eea
which follows from $g_a g_{-a} = (-1|p) g^2$ and expressing its sum over $a$ in two different ways,
\bea
\sum _{a=0}^{p-1} g_a g_{-a} = (p-1) (-1|p) g^2  =  \sum_{a,k,\ell=0}^{p-1} (k\ell |p) \ep^{(k-\ell)a}= p \sum_{k,\ell=0}^{p-1} (k\ell | p) \delta_{k,\ell} = p(p-1)
\eea
where $\delta_{k,\ell}=1$ when $k=\ell$ and zero otherwise. The result (\ref{g2}) follows upon using the first supplement $(-1|p)=(-)^{(p-1)/2}$.

\subsection{Quadratic reciprocity}

Quadratic reciprocity states that for $p,q$ distinct odd primes, we have,
\bea
(p|q)(q|p) = (-)^{(p-1)(q-1)/4}
\eea
Many proofs exist of the law of quadratic reciprocity, of which Gauss produced at least 8. Here we shall use Gauss's sums as defined in the preceding subsection. For any odd prime $q$ not equal to $p$, we begin by using (\ref{g2}) to write,
\bea
\label{pq}
g^{q-1} = (g^2)^{(q-1)/2} = (g^2| q) = (-1|q)^{(p-1)/2} (p|q)= (-)^{(p-1)(q-1)/2} (p|q)
\eea
where we used the relation $(n|p) \equiv n^{(p-1)/2} ~ (\mod p)$ of property 2. of \ref{5.prop1}  in the second equality.  Next, we use the fact that the prime power of a multiple sum evaluates as follows,
\bea
(a_1 + \cdots + a_n)^q = a_1^q + \cdots a_n^q ~ (\mod q)
\eea
This relation is valid for $a_1, \cdots, a_n \in \ZZ$, but we generalize it here to the ring of algebraic integers $\ZZ_p(\ep)$. As a result we have,
\bea
g^q = \left ( \sum_{k=0}^{p-1} (k|p) \ep^k \right )^q = \sum_{k=0}^{p-1} (k|p)^q \ep^{kq}
= \sum ^{p-1}_{k=0} (k|p) \ep^{kq} = g_q ~ (\mod q)
\eea
By (\ref{Gauss2}), we have $g_q = (q|p) g$, and upon combining with (\ref{pq}), we get,
\bea
g^{q-1} =  (-)^{(p-1)(q-1)|2} (p|q) = (q|p)
\eea
The last equality  implies the law of quadratic reciprocity upon using $(p|q)^2=1$.

\subsection{Characters}
\label{sec:charLfunc}

Consider a finite Abelian group $G$. A function $\chi: G \rightarrow \CC^*$ is said to be a \textit{character} if it is a group homomorphism, i.e. $\chi (g h) = \chi(g) \chi(h)$ for all $g,h \in G$. Characters form a group $\hat G = {\rm Hom}(G,\CC^*)$, known as the Poincar{\'e} dual of $G$.  We will denote the identity element of this group as $\chi_0$, such that $\chi_0(g) = 1$ for all $g$. The inverse elements are obtained by complex conjugation $\chi^{-1}(g) = \overline{\chi(g)}$.

\sm

As an example, consider the case of a cyclic group $G = \ZZ_n$. If we denote the order-$n$ generator by $v$, then it is clear that $\chi(v)^n = \chi(v^n) = 1$ and hence we can take $\chi(v)$ to be an $n$-th root of unity. Conversely, every $n$-th root of unity defines a character of $\ZZ_n$. Thus we conclude that $\hat G$ is in fact isomorphic to the original group, $\hat G \cong \ZZ_n$.

\sm

Characters satisfy the following relations
\bea
\label{eq:preSchurorth}
\sum_{g \in G} \chi(g) & = & \left\{ \begin{matrix} |G| & \chi = \chi_0 \\ 0 & {\rm otherwise}\end{matrix}\right.
\no \\
\sum_{\chi \in \hat G} \chi(g) & = & \left\{ \begin{matrix} |\hat G| & g = \mathds{1} \\ 0 & {\rm otherwise}\end{matrix}\right.
\eea
The first line of each of these is obvious. To prove the second line of the first relation, begin by choosing $h \in G$ such that $\chi(h) \neq 1$. Then we have
\bea
\chi(h) \sum_{g \in G} \chi(g)  = \sum_{g \in G} \chi(hg) = \sum_{g \in G} \chi(g)
\eea
where we have used the fact that $\chi$ is a homomorphism, as well as closure of the group $G$. Since $\chi(h) \neq 1$, we conclude $\sum_{g \in G} \chi(g)=0$ as per the claim. The second relation is proven exactly analogously---namely we multiply by $\chi'(g) \neq 1$ and used the fact that $\hat G$ is a group.

\sm

From these results we may obtain the following Schur orthogonality relations
\bea
\sum_{g \in G} \chi(g) \overline{\chi'(g)} & = & \left\{ \begin{matrix} |G| & \chi = \chi' \\ 0 & {\rm otherwise}\end{matrix}\right.
\no \\
\sum_{\chi \in \hat G} \chi(g) \overline{\chi(h)} & = & \left\{ \begin{matrix} |\hat G| & g = h \\ 0 & {\rm otherwise}\end{matrix}\right.
\eea
Indeed, for the former note that if $\chi, \chi' \in \hat G$ then $\chi (\chi')^{-1} \in \hat G$ as well. We may thus use the relation (\ref{eq:preSchurorth}) with $\chi$ replaced by $\chi (\chi')^{-1}$. Then noting that,
\bea
\sum_{g \in G}(\chi (\chi')^{-1})(g) = \sum_{g \in G}\chi(g) \overline{\chi'(g)}
\eea
we obtain the desired result. Likewise for the latter relation, we use the fact that,
\bea
\sum_{\chi \in \hat G} \chi (g h^{-1}) = \sum_{\chi \in \hat G} \chi(g) \overline{\chi(h)}
\eea
This completes the proof of both relations in (\ref{eq:preSchurorth}).

\newpage

\subsection{Dirichlet characters and L-functions}
\label{sec:Lseries}

A \textit{Dirichlet character} for $a \in \NN$ is a function $\chi_a: \ZZ \rightarrow \CC$ satisfying the following properties:
\begin{enumerate}
\itemsep=0in
\item Periodicity:  $\chi_a(n) = \chi_a(n+a)$ for all $ n \in \ZZ$.
\item $\chi_a(n) \neq 0$ if and only if $\gcd(n,a) = 1$.
\item Multiplicativity: $\chi_a(nm) = \chi_a(n) \chi_a(m)$ for all $n,m \in \ZZ$.
\end{enumerate}
Dirichlet characters can be thought of as characters for $ G = \ZZ_a^*$, the multiplicative group of invertible elements of $\ZZ_a$. This is an Abelian group with $|G| = \phi(a)$, where $\phi$ is Euler's totient function. In this case our orthogonality relations read as follows,
\bea
\label{eq:Dircharsortho}
\sum_{g =0}^{a-1} \chi_a(g) \overline{\chi'_a(g)} & = & \left\{ \begin{matrix} \phi(a) & \chi_a = \chi'_a \\ 0 & {\rm otherwise}\end{matrix}\right.
\no\\
\sum_{\chi_a \in \hat G} \chi_a(g) \overline{\chi_a(h)} & = & \left\{ \begin{matrix}\phi(a) & g \equiv h ~(\mod a)\\ 0 & {\rm otherwise}\end{matrix}\right.
\eea
We now introduce the Dirichlet L-function,
\bea
L_a(s, \chi) = \sum_{n= 1 }^\infty{\chi_a(n) \over n^s}
\eea
For $\chi_a$ not equal to the identity character, $L_a(s,\chi)$ converges and is analytic for ${\rm Re}\,s >1$ [23, 128]. In this region, we have the following Euler product formula,
\bea
\label{eq:DirLprod}
L_a(s, \chi) = \prod_p {1\over 1 - {\chi_a(p) \over p^s}}
\eea
the product being over prime numbers. To prove this, note that for each prime $p$, we have,
\bea
(1- \chi_a(p) p^{-s})^{-1} = \sum_{n=0}^\infty\chi_a(p)^n p^{-ns} = \sum_{n=0}^\infty \chi_a(p^n) p^{-ns}
\eea
Hence for fixed prime $q$, we have,
\bea
\prod_{p \leq q}{1\over 1 - {\chi_a(p) \over p^s}} = \sum_{n \in S_q} {\chi_a(n) \over n^s}
\eea
with $S_q$ the set of all natural numbers whose prime factors are less than or equal to $q$. Then for any $N \in \NN$, we conclude that,
\bea
\sum_{n=1}^N{\chi_a(n) \over n^s} = \prod_{p \leq q}{1 \over 1- {\chi_a(p) \over p^s}} - \sum_{\substack{n \in S_q\\ n \geq N}} {\chi_a(n) \over n^s}
\eea
with $q$ the largest prime less than or equal to $N$. Taking $N\rightarrow \infty$ gives (\ref{eq:DirLprod}).

Using this product formula, it is possible to show that $L_a(s, \chi_0)$ extends to a meromorphic function on ${\rm Res} \,s>0$, with the only pole at $s=1$. We do not reproduce the proof here; it can be found in e.g. [23, 128].

\subsubsection{Dirichlet's Theorem on Arithmetic Progressions}
We now use Dirichlet L-functions to prove Dirichlet's Theorem on arithmetic progressions,

{\theorem Let $m$ and $a$ be relatively prime natural numbers. Then there are infinitely many prime numbers $p$ such that $p \equiv m ~(\mod a)$. }
\newline

In order to prove this, we aim to prove the following stronger result,
\bea
\sum_{p \equiv m ~(\mod a)} {1\over p}  \hspace{0.2 in} \rm{is \,\,\, divergent}
\eea
which clearly implies the theorem. To do so, begin by considering the logarithm of the Dirichlet L-function, which by the product formula takes the form
\bea
\log L_a(s, \chi) = \sum_p \left(- \log\left(1- {\chi_a(p) \over p^s}\right)\right)
\eea
valid for ${\rm Re}\, s >1$. Taylor expanding the logarithm, this gives
\bea
\label{eq:LTaylorexp}
\log L_a(s, \chi) = \sum_p {\chi_a(p) \over p^s} + \sum_p \sum_{n=2}^\infty {\chi_a(p)^n \over n p^{ns}}
\eea
where, for reasons to be clear in a moment, we have split the $n=1$ term from the rest. Using the orthogonality of Dirichlet characters in (\ref{eq:Dircharsortho}), we find that
\bea
\label{eq:Dirmain}
\sum_{p \equiv m ~(\mod a)} {1\over p^s} &=& {1 \over \phi(a)} \sum_{\chi_a \in \hat G}\overline{\chi_a(m)} \sum_p {\chi_a(p) \over p^s}
\\
&=& {1 \over \phi(a)} \left[\sum_{\chi_a \in \hat G} \overline{\chi_a(m)} \log L_a(s, \chi) - \sum_{\chi_a \in \hat G} \overline{\chi_a(m)} \sum_p \sum_{n=2}^\infty {\chi_a(p)^n \over n p^{ns}} \right] \no
\eea
where $(a,m)=1$ lest $\chi_a(m) = 0$. To prove the theorem, we must now show that the right-hand side is divergent as $ s \rightarrow 1$. To do so, we begin by showing that $ \sum_p \sum_{n=2}^\infty {\chi_a(p)^n \over n p^{ns}}$ is bounded as $ s\rightarrow 1$, and hence that the second term in the difference does not diverge. To see this, note that since $|\chi_a(p)| = 0$ or 1, we have $|\chi_a(p) p^{-s}| \leq |p^{-s}| \leq \half$. Then we have
\bea
\left|\sum_{n=2}^\infty {\chi_a(p)^n \over n p^{ns}}\right| \leq \left|{\chi_a(p) \over p^s} \right|^2 \sum_{n=2}^\infty{1 \over n}\left| {\chi_a(p) \over p^s}\right|^{n-2} \leq \left|{\chi_a(p) \over p^s} \right|^2 \sum_{n=2 }^\infty \half {1 \over 2^{n-2}} = \left|{\chi_a(p) \over p^s} \right|^2 \leq {1 \over p^2}\,\,\,\,\,\,\,
\eea
Hence we conclude that
\bea
\left|  \sum_p \sum_{n=2}^\infty {\chi_a(p)^n \over n p^{ns}}\right| \leq \sum_p{1 \over p^2}< \sum_{n=1}^\infty{1 \over n^2}
\eea
the latter of which is convergent.

\sm

Having shown that the second term on the right-hand side of (\ref{eq:Dirmain}) is not divergent as $s \rightarrow 1$, we are now tasked with showing that the first term \textit{is} divergent. At the end of appendix \ref{sec:Lseries} we quoted the result that $L_a(s, \chi_0)\rightarrow \infty$ as $s \rightarrow 1$, and hence $\log L_a(s, \chi_0)$ diverges in this limit as well. To complete the proof it thus suffices to show that $L_a(1, \chi) \neq 0$ for all $\chi \neq \chi_0$.

To show this, first consider a character $\chi_a$ such that $\chi_a^2$ is not the identity character. Then we can consider the function
\bea
\lambda(s) = L_a(s, \chi_0)^3 L_a(s, \chi)^4 L_a(s, \chi^2)
\eea
By (\ref{eq:LTaylorexp}) we have
\bea
|\lambda(s)| = \left|{\rm exp}\left[\sum_{p}\sum_{n=1}^\infty{3 + 4 \chi_a(p^n)+\chi_a^2(p^n) \over n p^{n s}} \right] \right|
\eea
Letting $\theta_{n,p} = {\rm Arg}\, \chi_a(p^n)$, we then obtain
\bea
|\lambda(s)| = {\rm exp}\left| \sum_{p}\sum_{n=1}^\infty{3 + 4 \cos\theta_{n,p}+\cos 2\theta_{n,p} \over n p^{n\, {\rm Re}\,s}} \right|
\eea
But for any real $\theta$, we have
\bea
3 + 4 \cos\theta+2 \cos\theta = 2(1+\cos\theta)^2 \geq 0
\eea
Thus we conclude that
\bea
|\lambda(s) | \geq e^0 = 1
\eea
This gives the desired result. Indeed, if we had $L_a(1, \chi) = 0$, then since $L_a(s, \chi_0)$ has a simple pole at $s=1$ but $L_a(s, \chi^2)$ does \textit{not} (because $\chi_a^2 \neq \chi_0$ by assumption) the degree $\geq 4$ zero wins against the degree 3 pole, and we would have $\lambda(1) = 0$, in contradiction to the inequality just obtained. Thus we must have $L_a(1, \chi) \neq 0$.

\sm

For the case of $\chi_a^2 = \chi_0$, a similar argument can be run using $\lambda(s) = L_a(s, \chi)L_a(s, \chi_0)$. In this case one aims to show that $\log \lambda(s)  \geq \log \zeta(2 s)$. In total, one concludes that $L_a(1, \chi) \neq 0$ for all $\chi \neq \chi_0$, completing the proof of the theorem.

\subsubsection{Perfect squares and quadratic residues}
We close  by using the various tools of this section, including Dirichlet's theorem of arithmetic progressions, to prove the following statement:

{\theorem A number $N \in \NN$ is a perfect square if and only if it is a quadratic residue for all primes $p$ coprime to $N$.}
\newline\newline
One direction of this statement is obvious: if $N$ can be written as a perfect square, it is clear that it is also a perfect square modulo any prime. We now focus on the other direction. Assume that $N$ is a quadratic residue modulo all coprime $p$. Say that $N$ is \textit{not} a perfect square. We may then write $N$ in the form $N= j^2 p_1 \dots p_r$ for distinct primes $p_1, \dots p_r$. Since any odd power of 2 is a quadratic non-residue mod 3, we may assume that there is at least one odd factor.
\iffalse
For any prime $p$, we thus have
\bea
(N|p) = (p_1|p)\dots (p_r|p)
\eea
\fi
Take $p_1$ to be odd and consider non-zero integers $m$ and $n_i$ such that $(m|p_1) = -1$ and $(n_i|p_i) = 1$ for $i = 2, \dots, r$. By the Chinese remainder theorem, there exists a solution to the following set of congruences
\bea
\label{eq:perfectsqcong}
x \equiv 1 ~(\mod 4)\hspace{0.5 in}x \equiv m ~(\mod p_1)\hspace{0.5 in}x \equiv n_i ~(\mod p_i) \hspace{0.5 in} p_i \neq 2
\eea
If one of the prime factors $p_i = 2$, then we replace the final congruence with $x \equiv 1~(\mod 8)$. Given a particular solution $x_0$ to these congruences, the general solution takes the form $x = x_0 + 4 k p_1 \dots p_r$ for integer $k$. By Dirichlet's theorem of arithmetic progressions this takes an infinite number of prime values, and we choose one, say $x = p$, such that $p \nmid j$. By definition, we have
\bea
(p| p_1) = (m| p_1) = -1 \hspace{0.8 in} (p| p_i) = (n_i | p_i) = 1 \hspace{0.5 in} i = 2, \dots, r
\eea
Then using quadratic reciprocity, together with the first congruence of (\ref{eq:perfectsqcong}), we conclude that
\bea
(p_1| p) = -1\hspace{0.8 in} {\rm and}\hspace{0.8 in}(p_i | p) = 1 \hspace{0.3 in} i = 2, \dots, r
\eea
Thus $(N|p) = (p_1|p)\dots(p_r| p) =  -1$, and we have identified a prime $p$ coprime to $N$ for which $N$ is not a quadratic residue, contrary to hypothesis. We conclude that $N$ must be a perfect square.

\subsection*{$\bullet$ Bibliographical notes}

An excellent overview of some classic problems in arithmetic is given in the book by Cox~[144], where a historical perspective is also provided. Two other classics are the books by Serre~[128] and Hecke~[337]. Useful references for results in analytic number theory, including $L$-functions, may be found in the book by Iwaniec [23], Rademacher [46], and Siegel [53].

\newpage

\section{Riemann surfaces}
\setcounter{equation}{0}
\label{sec:RS}

A Riemann surface is a connected complex manifold of two real dimensions or equivalently a connected complex manifold of one complex dimension, also referred to as a complex curve.  In this appendix, we shall review the topology of Riemann surfaces, their homology groups,  homotopy groups, uniformization, construction in terms of Fuchsian groups, as well as their emergence from two-dimensional orientable Riemannian manifolds.

\subsection{Topology}

The topology of a Riemann surface $\Sigma$ is the topology of the underlying orientable two-dimensional manifold $\Sigma$. Throughout, we shall restrict to Riemann surfaces whose boundary $\p \Sigma$ is the union of a finite number $b$ of one-dimensional components, and a finite number $n$ of points referred to as punctures of $\Sigma$.   Given a triangulation of $\Sigma$ with $F_\Sigma$ faces, $E_\Sigma$ edges, and $V_\Sigma$ vertices, the Euler characteristic $\chi(\Sigma)$ is defined by,
\bea
\label{B.Euler}
\chi (\Sigma) = F_\Sigma - E_\Sigma + V_\Sigma
\eea
The Euler characteristic is a topological invariant in the sense that its value is independent of the triangulation chosen for a given $\Sigma$.  In suitably chosen triangulations, the operation of adding a puncture to $\Sigma$ (namely removing a point from $\Sigma$) leaves $F_\Sigma$ and $ E_\Sigma$ unchanged but diminishes $V_\Sigma$ by one while adding a one-dimensional boundary component leaves $E_\Sigma $ and $ V_\Sigma$ unchanged but diminishes $F_\Sigma$ by one. The remaining contribution to $\chi(\Sigma)$ derived from the genus $g$,  which is the number of handles on $\Sigma$, is,
\bea
\chi(\Sigma) = 2-2g -n-b
\eea
A Riemann surface with $b \not=0$ may be studied by considering its double, which is obtained by gluing $\Sigma$ and its mirror image $\Sigma '$ into a Riemann surface $\hat \Sigma$ without one-dimensional boundary components. The surface $\hat \Sigma$ is endowed with an involution $\iota$ which swaps $\Sigma$ and $\Sigma '$. Henceforth, we shall restrict attention to surfaces with $b=0$.

\sm

Compact Riemann surfaces play a central role in both Mathematics and Physics. It will often be useful to consider the  generalization to a \textit{compact Riemann surface with $n$ punctures} defined to be a compact Riemann surface with a finite number $n$ of points removed, namely with $n$ punctures. Their study may be initiated by first studying the underlying compact surface and then removing $n$ points.

\subsubsection{The Riemann-Hurwitz formula}

Holomorphic maps between Riemann surfaces allow us to relate different Riemann surfaces.
We begin by considering a non-constant holomorphic map  $f:X \to Y$ between two compact Riemann surfaces $X$ and $Y$. The degree of $f$ is the multiplicity of the function $f^{-1}$ given by the cardinality $d=|f^{-1}(y)|$ for all but finitely many points $y \in Y$. A formula valid for all points in $Y$ is obtained in terms of the \textit{degree of ramification} $e_x$ of $f$ at a point $x\in X$ which,  in local coordinates, is given by $f(z)-f(x) \sim (z-x)^{e_x}$.  The degree  of $f$ may then be defined for all $y \in Y$ by the following formula,
\bea
\label{6.degree}
d= \sum_{x \in f^{-1}(y)} e_x
\eea
and $d$ is independent of $y$. The Riemann-Hurwitz formula relates the genera $g_X$ and $g_Y$  of the surfaces $X$ and $Y$ to the degree of $f$ and the deviations from 1 of the degree of ramification,
\bea
\label{6.RHa}
2g_X -2 = d(2g_Y-2) + \sum _{x \in X} (e_x-1)
\eea
A hyper-elliptic Riemann surface $X$, viewed as a double cover of  the Riemann sphere $Y$, provides an illuminating example of the Riemann-Hurwitz formula for $d=2$ since the presence of $2n$ branch points at which $e_x=2$ readily gives $g_X= n-1$. The formula (\ref{6.RHa}) may be proven by using the expression for the Euler characteristic $\chi(X)$  in (\ref{B.Euler}) of $X$ in terms of a triangulation by $F_X$ faces, $E_X$ edges, and $V_X$ vertices (including all  the ramification points),  and similarly for $\chi(Y)$. Given that the degree of the map is $d$, we have $F_X = d F_Y$ and $E_X=d E_Y$ but $V_X= d V_Y - \sum _{x \in X} (e_x-1)$ since a vertex with ramification degree $e_x$ counts for only one vertex of $X$. Expressing the result in terms of the Euler numbers we have $\chi(X) = d \chi(Y) - \sum _{x \in X} (e_x-1)$. For a compact Riemann surface without boundary or punctures, the Euler numbers are related to the genus by $\chi(X) = 2 - 2 g_X$ and $\chi(Y) = 2 - 2g_Y$ from which (\ref{6.RHa}) follows.\footnote{For the special case of an un-ramified covering the formula $\chi(X) = d \chi(Y)$  for $\chi<0$ the formula readily follows from the fact that the Euler number is proportional to the area for unit negative curvature metrics.}

\subsubsection{Homotopy}

The homotopy groups $\pi_0(\Sigma)$ and $\pi_2(\Sigma)$ for a compact Riemann surface are trivial, and the interesting homotopy resides entirely in the first homotopy group $\pi_1(\Sigma)$, which we now define.

\sm

A closed curve $\mC: [0,1] \to \Sigma$ is a continuous map such that $\mC(1)=\mC(0)$.  Two closed curves $\mC_0,\mC_1 $ are said to be homotopic (to one another) iff there exists a family of curves $\mF(s,t)$ parametrized by $t \in [0,1]$, such that $\mF(s,t) \subset \Sigma$ and is continuous in $s,t$  for all $s,t \in [0,1]$ and  interpolates as follows: $\mF(s,0)=\cC_0(s)$ and $\mF(s,1) = \mC_1(s)$. The homotopy relation amongst closed curves is clearly an equivalence relation, whose classes are the first homotopy classes.  To make the set of classes into a group under composition of maps, we choose a base-point $P \in \Sigma$ and restrict to curves in each class that pass through $P$. Thus, we may choose the parametrization of each closed curve $\mC$ that passes through $P$  so that $\mC(0)=\mC(1)=P$. The composition $\mC_1 \circ \mC_0$ of two curves $\mC_0,\mC_1$ is given by,
\bea
\mC_1 \circ \mC_0 (s) = \left \{ \bma \mC_0(2s)  & \hbox{ for } & s \in [0,\half] \cr
\mC_1(2s-1) & \hbox{ for } & s \in [\half, 1] \ema \right .
\eea
The composition of closed curves induces a composition of first homotopy classes of curves, where
the identity element $e$ is the class of contractile curves, and the inverse is the class of curves with opposite orientation. The resulting group of classes is independent of the base-point $P$ \footnote{Independence of $P$ requires $\Sigma$ to be path-connected as we shall assume throughout.}  and is referred to as the first homotopy group, or fundamental group $\pi_1(\Sigma)$.

\sm

{\prop The Riemann sphere $\hat \CC$, the complex plane $\CC$, the Poincar\'e upper half-plane  $\cH$, and the unit disc $\cD$ are simply connected Riemann surfaces, as is any compact Riemann surface $\Sigma _0$ of genus 0,  so that their first homotopy groups are given by,
\bea
\pi_1 (\hat \CC) = \pi_1(\CC)= \pi_1(\cH) = \pi_1(\cD) = \pi_1(\Sigma _0)= \{ e \}
\eea
For a compact surface  $\Sigma$ of genus 1, namely a torus,  we have $\pi_1(\Sigma) = \ZZ^2$.}

\sm

For surfaces of genus $g \geq 2$, the fundamental group is non-Abelian, and will be discussed concretely in appendix \ref{sec:B4} in terms of representations of $SL(2,\RR)$.

\begin{figure}[htb]
\begin{center}
\tikzpicture[scale=1.05]
\scope[xshift=0cm,yshift=0cm]

% handles
\draw[ultra thick] (1.5,1.2) .. controls (2.5, 1.2) .. (3.5,2);
\draw[ultra thick] (1.8,1.2) .. controls (2.5, 1.8) .. (3.2,1.75);
\draw[ultra thick] (6.5,2) .. controls (7.5, 1.6) .. (8.5,2);
\draw[ultra thick] (6.8,1.9) .. controls (7.5, 2.2) .. (8.2,1.87);
% cycles
\draw[very thick, color=blue] plot [smooth] coordinates {(5,2) (4,2.2) (3, 2.3) (2, 2.15) (1.3,1.5) (1.5,1) ( 2, 0.85) (3,1.1) (5,2) };
\draw[very thick, color=blue] plot [smooth] coordinates {(5,2) (6,2.3) (7, 2.6) (8, 2.6) (8.8,2.2) (8.5,1.5)  (7,1.3) (5,2) };
\draw[very thick, color=red] plot [smooth] coordinates { (5,2) (4,1.8) (3,1.45) (2.7,1.4)};
\draw[very thick, color=red, dashed] plot [smooth] coordinates { (2.7,1.4) (2.5,1.2) (2.3,0.6) (2.5, 0.275) (2.9, 0.265)};
\draw[very thick, color=red] plot [smooth] coordinates { (2.9,0.265) (4,1) (5,2)};
\draw[very thick, color=red] plot [smooth] coordinates { (5,2) (6,1.8) (7,1.6) (7.3,1.7)};
\draw[very thick, color=red, dashed] plot [smooth] coordinates { (7.3,1.7) (7.9,1) (8,0.4) (7.7, 0.28)};
\draw[very thick, color=red] plot [smooth] coordinates { (7.7, 0.28) (6,1.05) (5,2)};
% contour
\draw[ultra thick] plot [smooth] coordinates {(0.7,0.6) (0.45,1.6) (1,2.8) (2.5, 3.3)  (5,2.8) (7.5, 3.3) (9,2.8) (9.5,1.6) (9,0.7) (8,0.3) (7, 0.3) (5,0.8) (3, 0.3) (2,0.1) (1,0.35) (0.7,0.6) };
% labels
\draw [color=red] (3.1,0.75) node{\small $\mA_1$};
\draw [color=red] (7.25,0.8) node{\small $\mA_2$};
\draw [color=blue] (1.3,2.2) node{\small $\mB_1$};
\draw [color=blue]  (8.8,1.3) node{\small $\mB_2$};
\draw (9.4,0.6) node{{$\Sigma$}};
% base point
\draw (5,2) node{$\bullet$};
\draw (5,2.3) node{\tiny $P$};
\endscope
\endtikzpicture
\caption{ \textit{A choice of homotopy generators $\mA_I$ and $\mB_I$ with common base point $P$ is illustrated for a compact genus-two Riemann surface $\Sigma$. }
\label{fig:B.1}}
\end{center}
\end{figure}

\subsubsection{Homology}

The top and bottom homology groups of a Riemann surface are trivial.  All the interesting homology resides in the first homology group which, for a compact Riemann surface of genus $g$ with $n$ punctures, is given by $H_1(\Sigma, \ZZ) = \ZZ^{2g+n}$. The primitive homology cycle associated with each puncture is homologous to a small circle centered at the puncture. Having chosen an orientation for the Riemann surface $\Sigma$, we introduce a binary intersection pairing $\mJ(\mC_1, \mC_2)$ on oriented 1-cycles $\mC_1$ and $\mC_2$,  which counts the number of intersections of the 1-cycles $\mC_1$ with $\mC_2$  weighed by $\pm 1$ factors for the relative orientation of the cycles at their intersection points. The pairing  $\mJ(\mC_1, \mC_2)$ is odd under swapping $\mC_1, \mC_2$.

\sm

Specializing to the case of a compact Riemann surface $\Sigma$ without punctures, the intersection form $\mJ$ is non-degenerate, and we may choose a basis for $H_1(\Sigma, \ZZ) = \ZZ^{2g}$ consisting of homology cycles $\mA_I, \mB_J$ for $I,J=1,\cdots, g$ such that,
\begin{align}
\label{B.2.1}
\mJ (\mA_I, \mA_J) & = \, \mJ (\mB_I, \mB_J)  \, = 0
\no \\
\mJ (\mA_I, \mB_J) & =  - \mJ (\mB_I, \mA_J)  =  \delta _{IJ}
\end{align}
Two different canonical bases $(\mA, \mB)$ and $(\tilde \mA, \tilde \mB)$  are related by a linear transformation that may be represented by a matrix $M$  with integer entries,
\bea
\label{B.2.2}
\left ( \begin{matrix} \tilde \mB \cr \tilde \mA \cr \end{matrix} \right )
= M \left ( \begin{matrix} \mB \cr \mA \cr \end{matrix} \right )
\eea
Here, $\mA$ and $\mB$ are column matrices with entries $\mA_I$ and $ \mB_I$, respectively.  The matrix $M$ is an element of  the rank $g$ modular group $Sp(2g,\ZZ)$ which preserves the intersection matrix~$\mJ$,
\bea
\label{B.2.3}
M^t \mJ M = \mJ
\hskip 0.7in
\mJ = \left ( \begin{matrix}0 & -I_g \cr I_g & 0 \cr \end{matrix} \right )
\hskip 0.7in
M = \left ( \begin{matrix} A & B \cr C & D \cr \end{matrix} \right )
\eea
where $A,B,C,D$ are $g \times g$ matrices with integer entries. An important subgroup of $Sp(2g,\ZZ)$ is the group $GL(g,\ZZ)$ which  consists of those $M$ that transform $\mA$-cycles into linear combinations of $\mA$-cycles and $\mB$-cycles into linear combinations of $\mB$-cycles. It is obtained by setting $B=C=0$ and $D^{-1}= A^t$.

\subsubsection{Cohomology}
\label{sec:B.1}

By the de Rham theorem, the first de Rham cohomology group is given by $H^1(\Sigma, \RR) = \RR^{2g}$ and is generated by $2g$ real-valued harmonic 1-forms on $\Sigma$. Since $\Sigma$ is a complex manifold,  harmonic one-forms may be decomposed into holomorphic and anti-holomorphic one-forms, and the cohomology may be decomposed into the Dolbeault cohomology groups,
\bea
H^1(\Sigma, \RR) =  H^{(1,0)} (\Sigma ) \oplus H^{(0,1)} (\Sigma)
\eea
generated by holomorphic $(1,0)$ forms and anti-holomorphic $(0,1)$ forms respectively. A canonical basis of the Dolbeault cohomology group  $H^{(1,0)} (\Sigma, \ZZ)= \ZZ^g$ consists of holomorphic $(1,0)$-forms  $\om_I$ with $I=1,\cdots, g$ whose periods are normalized on the cycles $\mA_I$ of the canonical homology basis $(\mA, \mB)$, as follows,
\bea
\label{B.3.1}
\oint _{\mA_I} \om_J = \delta _{IJ}
\eea
The periods on the remaining 1-cycles are then completely determined and give the matrix elements of the period matrix $\Omega$ of $\Sigma$,
\bea
\label{B.3.2}
\oint _{\mB_I} \om_J = \Omega _{IJ}
\eea
\sm
Under modular transformations $M \in Sp(2g,\ZZ)$, whose  parametrization in terms of $g \times g$ matrices $A,B,C,D$ is given in (\ref{B.2.3}), the matrix of holomorphic Abelian differentials $\om$, the period matrix $\Omega$, and its imaginary part $Y$,  transform as follows,
\bea
\label{B.3.4}
\tilde \om & = & \om (C \Omega+D)^{-1}
\no \\
\tilde \Omega & = & (A \Omega +B) (C \Omega +D)^{-1}
\no \\
\tilde Y & = & (\Omega C^t+D^t)^{-1} Y (C\Omega^* +D)
\eea
Clearly, the transformation formula for $\Omega$ reduces to the standard M\"obius transformation formula  for the special case of $g=1$.

\subsubsection{Riemann bilinear relations}
\label{sec:Riemannrelations}

A useful formula relates the integral over $\Sigma$ of a wedge product of closed forms $\varpi_1$ and $\varpi_2$ to their line integrals over canonical homology cycles,
\bea
\label{B.Riem}
\int _\Sigma \varpi_1 \wedge \varpi _2 = \sum _{K=1}^g \left ( \oint _{\mA_K} \varpi_1  \oint _{\mB_K} \varpi_2
- \oint _{\mB_K} \varpi_1  \oint _{\mA_K} \varpi_2 \right )
\eea
The formula may be established by cutting the surface $\Sigma$ along homology cycles $\mA_K$ and $\mB_K$ for $K=1,\cdots, g$, chosen so as to have a point in common, into a simply-connected domain $\Sigma_{{\rm cut}}$  in the plane, as illustrated in Figure \ref{fig:B.5}. Expressing one of the closed differentials as an exact differential of a local function  in $\Sigma_{{\rm cut}}$, for example, $\varpi _2 = d f_2$, the formula may be derived using Green's theorem for an arbitrary 1-form $\varpi$,
\bea
\int _{\Sigma_{{\rm cut}}} d \varpi = \oint _{\p \Sigma _{{\rm cut}}} \varpi
\eea
where the boundary consists of the union of cycles,
\bea
\p \Sigma _{{\rm cut}}  = \bigcup _I \Big ( \mA_I \cup \mB_I \cup \mA_I^{-1} \cup \mB_I^{-1} \Big )
\eea
as illustrated in Figure \ref{fig:B.5} for genus $g=2$.

\sm

The period matrix $\Omega $ is symmetric by the Riemann bilinear relations obtained by setting $\varpi_1 = \om_I$ and $\varpi _2=\om_J$ in (\ref{B.Riem}). Furthermore, the imaginary part of the period matrix,
\bea
\label{B.3.3}
Y= \Im \, \Omega
\eea
is positive definite by the Riemann relations obtained by setting $\varpi_1 = \om_I$ and $\varpi _2= \bar \om_J$ in (\ref{B.Riem}), and using the fact that $\tfrac{i}{2} \int _\Sigma \varpi \wedge  \varpi^*$ is positive definite for holomorphic 1-forms $\varpi$.

\begin{figure}[htb]
\begin{center}
\tikzpicture[scale=1.05]
\scope[xshift=0cm,yshift=0cm, scale=0.85]

% handles
\draw[ultra thick] (1.5,1.2) .. controls (2.5, 1.2) .. (3.5,2);
\draw[ultra thick] (1.8,1.2) .. controls (2.5, 1.8) .. (3.2,1.75);
\draw[ultra thick] (6.5,2) .. controls (7.5, 1.6) .. (8.5,2);
\draw[ultra thick] (6.8,1.9) .. controls (7.5, 2.2) .. (8.2,1.87);
% cycles
\draw[very thick, color=blue] plot [smooth] coordinates {(5,2) (4,2.2) (3, 2.3) (2, 2.15) (1.3,1.5) (1.5,1) ( 2, 0.85) (3,1.1) (5,2) };
\draw[very thick, color=blue] plot [smooth] coordinates {(5,2) (6,2.3) (7, 2.6) (8, 2.6) (8.8,2.2) (8.5,1.5)  (7,1.3) (5,2) };
\draw[very thick, color=red] plot [smooth] coordinates { (5,2) (4,1.8) (3,1.45) (2.7,1.4)};
\draw[very thick, color=red, dashed] plot [smooth] coordinates { (2.7,1.4) (2.5,1.2) (2.3,0.6) (2.5, 0.275) (2.9, 0.265)};
\draw[very thick, color=red] plot [smooth] coordinates { (2.9,0.265) (4,1) (5,2)};
\draw[very thick, color=red] plot [smooth] coordinates { (5,2) (6,1.8) (7,1.6) (7.3,1.7)};
\draw[very thick, color=red, dashed] plot [smooth] coordinates { (7.3,1.7) (7.9,1) (8,0.4) (7.7, 0.28)};
\draw[very thick, color=red] plot [smooth] coordinates { (7.7, 0.28) (6,1.05) (5,2)};
% contour
\draw[ultra thick] plot [smooth] coordinates {(0.7,0.6) (0.45,1.6) (1,2.8) (2.5, 3.3)  (5,2.8) (7.5, 3.3) (9,2.8) (9.5,1.6) (9,0.7) (8,0.3) (7, 0.3) (5,0.8) (3, 0.3) (2,0.1) (1,0.35) (0.7,0.6) };
% labels
\draw [color=red] (3.1,0.75) node{\small $\mA_1$};
\draw [color=red] (7.25,0.8) node{\small $\mA_2$};
\draw [color=blue] (1.3,2.2) node{\small $\mB_1$};
\draw [color=blue]  (8.8,1.3) node{\small $\mB_2$};
\draw (9.4,0.6) node{{$\Sigma$}};
% base point
\draw (5,2) node{$\bullet$};
\draw (5,2.3) node{\tiny $P$};
\endscope
\scope[xshift=13cm,yshift=1.5cm, scale=3.2]
% contour
%\draw  [thick, domain=0:90] plot ({cos(\x)},{\sin(\x)});
%\draw  [ thick, domain=0:180] plot ({cos(\x)},{sin(\x)});
%\draw  [ thick, domain=180:360] plot ({cos(\x)},{sin(\x)});
% cycles
\draw  [very  thick, color=red, domain=166:198] plot ({cosh(0.6)*cos(0)+sinh(0.6)*cos(\x)}, {cosh(0.6)*sin(0)+sinh(0.6)*sin(\x)});
\draw  [very  thick, color=red, domain=241:282] plot ({cosh(0.65)*cos(85)+sinh(0.65)*cos(\x)}, {cosh(0.65)*sin(85)+sinh(0.65)*sin(\x)});
\draw  [very  thick, color=red, domain=330:396] plot ({cosh(0.5)*cos(175)+sinh(0.5)*cos(\x)}, {cosh(0.5)*sin(175)+sinh(0.5)*sin(\x)});
\draw  [very  thick, color=red, domain=54:100] plot ({cosh(0.4)*cos(265)+sinh(0.4)*cos(\x)}, {cosh(0.4)*sin(265)+sinh(0.4)*sin(\x)});
\draw  [very  thick, color=blue, domain=201:239] plot ({cosh(0.7)*cos(40)+sinh(0.7)*cos(\x)}, {cosh(0.7)*sin(40)+sinh(0.7)*sin(\x)});
\draw  [very  thick, color=blue, domain=264:323] plot ({cosh(0.5)*cos(125)+sinh(0.5)*cos(\x)}, {cosh(0.5)*sin(125)+sinh(0.5)*sin(\x)});
\draw  [very  thick, color=blue, domain=17:73] plot ({cosh(0.7)*cos(225)+sinh(0.7)*cos(\x)}, {cosh(0.7)*sin(225)+sinh(0.7)*sin(\x)});
\draw  [very  thick, color=blue, domain=114:169] plot ({cosh(0.7)*cos(315)+sinh(0.7)*cos(\x)}, {cosh(0.7)*sin(315)+sinh(0.7)*sin(\x)});
% identifications
\draw  [thick, color=black, dashed, ->, domain=238:220] plot ({cosh(0.9)*cos(40)+sinh(0.9)*cos(\x)}, {cosh(0.9)*sin(40)+sinh(0.9)*sin(\x)});
\draw  [thick, color=black, dashed,  domain=220:203] plot ({cosh(0.9)*cos(40)+sinh(0.9)*cos(\x)}, {cosh(0.9)*sin(40)+sinh(0.9)*sin(\x)});
\draw (0.148,-0.74) [fill=black] circle(0.02cm) ;
\draw  [thick, color=black, dashed, ->, domain=68:42] plot ({cosh(0.9)*cos(225)+sinh(0.9)*cos(\x)}, {cosh(0.9)*sin(225)+sinh(0.9)*sin(\x)});
\draw  [thick, color=black, dashed,  domain=42:19] plot ({cosh(0.9)*cos(225)+sinh(0.9)*cos(\x)}, {cosh(0.9)*sin(225)+sinh(0.9)*sin(\x)});
\draw  [thick, color=black, dashed, ->, domain=281:261] plot ({cosh(0.85)*cos(85)+sinh(0.85)*cos(\x)}, {cosh(0.85)*sin(85)+sinh(0.85)*sin(\x)});
\draw  [thick, color=black, dashed, domain=261:243] plot ({cosh(0.85)*cos(85)+sinh(0.85)*cos(\x)}, {cosh(0.85)*sin(85)+sinh(0.85)*sin(\x)});
\draw  [thick, color=black, dashed, ->, domain=99:77] plot ({cosh(0.6)*cos(265)+sinh(0.6)*cos(\x)}, {cosh(0.6)*sin(265)+sinh(0.6)*sin(\x)});
\draw  [thick, color=black, dashed, domain=77:63] plot ({cosh(0.6)*cos(265)+sinh(0.6)*cos(\x)}, {cosh(0.6)*sin(265)+sinh(0.6)*sin(\x)});
\draw (0.57,0.16) [fill=black] circle(0.02cm) ;
\draw (0.251,0.53) [fill=black] circle(0.02cm) ;
\draw (-0.7,0.4) [fill=black] circle(0.02cm) ;
\draw (-0.235,0.61) [fill=black] circle(0.02cm) ;
\draw (-0.675,-0.165) [fill=black] circle(0.02cm) ;
\draw (-0.165,-0.67) [fill=black] circle(0.02cm) ;
\draw (0.57,-0.19) [fill=black] circle(0.02cm) ;

\draw (0.15, -0.82) node{\tiny $P$};
\draw (0.65, 0.2) node{\tiny $P$};
\draw (0.3, 0.63) node{\tiny $P$};
\draw (-0.78, 0.45) node{\tiny $P$};
\draw (-0.24, 0.7) node{\tiny $P$};
\draw (-0.75, -0.2) node{\tiny $P$};
\draw (-0.2, -0.75) node{\tiny $P$};
\draw (0.65, -0.2) node{\tiny $P$};
\draw [color=red] (0.73,0) node{\small $\mA_1$};
\draw [color=blue] (0.53,0.4) node{\small $\mB_1$};
\draw [color=red] (0.05,0.71) node{\small $\mA_1^{-1}$};
\draw [color=blue] (-0.5,0.6) node{\small $\mB_1^{-1}$};
\draw [color=red] (-0.75,0.1) node{\small $\mA_2$};
\draw [color=blue]  (-0.5,-0.44) node{\small $\mB_2$};
\draw [color=red] (-0.05,-0.85) node{\small $\mA_2^{-1}$};
\draw [color=blue]  (0.45,-0.5) node{\small $\mB_2^{-1}$};
% identifications
%\draw [color=red] (-0.1,0.35) node{\small $\rho(\mA_1^{-1})$};
%\draw [color=red] (0.05,-0.48) node{\small $\rho(\mA_2^{-1})$};
%\draw [color=blue] (0.2,0.2) node{\small $\rho(\mB_1)$};
%\draw [color=blue]  (-0.18,-0.2) node{\small $\rho(\mB_2)$};
\draw (0,0) node{{$\Sigma_{{\rm cut}}$}};
\endscope

\endtikzpicture
\caption{ \textit{Representation of the Riemann surface $\Sigma$ in terms of a simply connected domain $\Sigma_{{\rm cut}}$ in $\CC$ obtained by cutting $\Sigma$ along the cycles $\mA_I$ and $\mB_I$ with common base point $P$ in common, with inverse cycles pairwise identified under the dashed arrows,  illustrated here for a compact genus-two Riemann surface $\Sigma$. }
\label{fig:B.5}}
\end{center}
\end{figure}

\subsection{Metrics and complex structures}

A different point of view on Riemann surfaces, with or without punctures,  is obtained by considering connected two-dimensional orientable manifolds endowed with a Riemannian metric. This point of view is important in Physics and is widely regarded as the starting point for string perturbation theory in the Polyakov formulation.  Consider a two-dimensional oriented manifold $\Sigma$ endowed with a Riemannian metric  $\mg$. In a system of local real coordinates $\xi ^m = (\xi^1, \xi^2)$ the metric takes the form,\footnote{Throughout, the Einstein convention is used to sum over a pair of repeated upper and lower indices. }
\bea
\mg = \mg_{mn} (\xi) d\xi ^m d \xi ^n
\eea
We review the following notions:
\begin{itemize}
\item A \textit{diffeomorphism} is a smooth map $\xi \to \xi'(\xi)$  under which $\mg$ is invariant, and the components $\mg_{mn}$ transform as follows,
\bea
\mg'_{mn} (\xi') { \p \xi^{\prime \, m} \over  \p \xi^p} { \p \xi^{\prime \, n} \over  \p \xi^q} = \mg_{pq}(\xi)
\eea
Diffeomorphisms form an infinite-dimensional group ${\rm Diff}(\Sigma)$.
\sm
\item A \textit{Weyl transformation} rescales the metric $\mg$ by an arbitrary positive function $\lambda$ on $\Sigma$, namely $\mg_{mn}(\xi)  \to \mg'_{mn}(\xi) = \lambda(\xi)  \mg_{mn}(\xi)$. Weyl transformations form an infinite-dimensional group ${\rm Weyl}(\Sigma)$.
\sm
\item Generally, a \textit{conformal transformation} preserves all angles. All Weyl transformations are conformal transformations in this sense. A \textit{conformal coordinate transformation}, or  \textit{conformal diffeomorphism}  is  a diffeomorphism which is  conformal, which requires that $\mg_{mn}' (\xi') = \mu (\xi) \mg_{mn}(\xi)$ for some positive real function $\mu$.
\sm
\item An \textit{isometry} of $\mg$ preserves all distances. A diffeomorphism is an isometry of $\mg$ provided $\mg'_{mn}(\xi') = \mg_{mn}(\xi)$. Every conformal diffeomorphism is an isometry, but there exist conformal transformations, such as scaling by a constant, that are not isometries.
\end{itemize}

We have the following theorem,

{\thm An orientable connected two-dimensional surface $\Sigma$ endowed with a Riemannian metric is automatically a Riemann surface.}

\sm

To establish this result, we show that the combination of orientability and the presence of  a Riemannian metric induces a unique complex structure. The orientation may be specified by choosing an oriented volume form,
\bea
d \mu _\mg = \sqrt{\det \mg} \, \ep _{m n} \, d\xi ^m \wedge d\xi^n \hskip 1in \ep_{mn}=-\ep_{nm} , ~ \ep _{12}=+1
\eea
A reversal of orientation corresponds to reversing the sign of $\ep$. We denote by $T_p(\Sigma)$ the tangent space to $\Sigma$ at the point $p\in \Sigma$, and by $T^*_p(\Sigma)$ the cotangent space.  A convenient basis  is given by the partial derivatives $\p_m= \p/ \p \xi^m$ for $T_p (\Sigma)$ and by the differentials $d\xi^m$ for $T^* _p (\Sigma)$. The tangent space $T_p(\Sigma)$ may be viewed as the space of  vectors $v^m \p_m$ at $p$, while the cotangent space may be viewed as the space of forms $\om_m d\xi^m$ at $p$.  An \textit{almost complex structure} $\cJ$ is a map from $T_p (\Sigma)$  to itself, or equivalently from $T^*_p(\Sigma)$ to itself, whose square is minus the identity map,
\bea
\cJ : T_p (\Sigma) \to T_p (\Sigma) & \hskip 0.6in & v^m \to \cJ^m {}_n \, v^n \hskip 0.8in \cJ^2 = - I
\no \\
\cJ : T_p^* (\Sigma) \to T_p^* (\Sigma) & \hskip 0.6in & \om_m \to \cJ_m{}^n \, \om_n
\eea
Having a metric and an orientation, it is straightforward to exhibit this map,
\bea
\cJ^m {}_n & = & \sqrt{\det \mg} \, \ep _{rn} \, \mg^{rm}
\no \\
\cJ_m {}^n & = & \sqrt{\det \mg} \, \ep _{mr} \, \mg^{rn}
\eea
The metric is a covariantly constant tensor with respect to the affine connection and its associated covariant derivative $\nabla _s$, so that we have $\nabla _s \mg_{mn}=0$. As a result, $\cJ$ obeys $\nabla _s \, \cJ_m {}^n=0$ and is therefore also a covariantly constant tensor acting on $T_p(\Sigma)$ and $T^*_p(\Sigma)$. Furthermore, $\cJ$ is invariant under arbitrary Weyl rescaling and therefore depends only on the conformal class of $\mg_{mn}$.  Finally, $\cJ$ is integrable in the sense that the system of equations,
\bea
\label{B.4.holo}
\cJ_m {}^n \, \p_n f = i \, \p_m f
\eea
is integrable and thereby defines locally holomorphic functions $f$ on $\Sigma$ which endow $\Sigma$ with a \textit{complex structure}.\footnote{In dimensions higher than two, the system (\ref{B.4.holo}) has non-trivial integrability conditions, but in two dimensions there are none since the operator $\cJ-iI$ has a one-dimensional null space.}
We may use the solutions to (\ref{B.4.holo}) to define a system of local complex coordinates $z,\bar z$ in which the metric $\mg$ is locally conformally flat,
\bea
\mg = \mg_{z \bar z} |dz|^2
\eea
for a real positive function $\mg_{z \bar z}$. The conformally flat form of the metric is invariant under locally holomorphic diffeomorphisms $z \to z' (z)$. The complex manifold $\Sigma$ may be constructed by covering $\Sigma$ with coordinate charts $\cU_\alpha$ in each of which we have  local complex coordinates $(z_\alpha, \bar z_\alpha)$. In the intersection $\cU_\a \cap \cU_\b$ of the two coordinate charts, the local coordinates are related by transition functions $\f_{\a \b}(z)$ which are holomorphic and nowhere vanishing via the relations $z_\alpha(z) = \f_{\a \b} (z) z_\beta (z)$. Thus $\Sigma$ is  a Riemann surface.

\subsubsection{The Euler characteristic in terms of a metric}

The Riemann tensor on a connected two-dimensional Riemannian manifold has only one independent component, which may be expressed in terms of the Gaussian curvature $R_\mg$. The Euler characteristic of a compact Riemann surface $\Sigma$ with metric $\mg$ may be expressed in terms of $R_\mg$ by the following classic formula,
\bea
\label{B.4.Euler}
\chi (\Sigma) = { 1 \over 2 \pi} \int _\Sigma d \mu _\mg \, R_\mg
\eea
The normalization of the Gaussian curvature $R_\mg$ is such that the round sphere $S^2$ of radius 1 has area $4 \pi$ and $R_\mg=1$, so that (\ref{B.4.Euler}) indeed corresponds to genus zero. The Euler characteristic is a topological invariant. Indeed, the integral in (\ref{B.4.Euler}) is independent of the metric $\mg$ because $d \mu_\mg \, R_\mg$ is locally an exact $(1,1)$-form.

\subsection{Uniformization}

In this subsection, we present the uniformization theorem for simply connected Riemann surfaces, and then use its results to produce concrete constructions for arbitrary compact Riemann surfaces with punctures.

\sm

{\thm[The uniformization theorem]

A simply connected Riemann surface is conformally isomorphic to either one of the following Riemann surfaces,
\begin{enumerate}
\itemsep=-0.05in
\item The Riemann sphere $\hat \CC$ which is the one-point compactification of $\CC$  (also denoted $S^2$);
\item The complex plane $\CC$;
\item The complex upper half-plane $\cH$ which is isomorphic to the unit disc $\cD$.
\end{enumerate}
For an arbitrary Riemannian metric $\mg$ on $\Sigma$ in each case, there exists a Weyl transformation to a metric $\hat \mg$ such that $R_{\hat \mg}=1$ for $\hat \CC$; $R_{\hat \mg}=0$ for $\CC$; and $R_{\hat \mg}=-1$ for $\cH$ and $ \cD$. The corresponding metrics are given as follows,
\bea
\hat \mg _{S^2} = { 4 |dz|^2 \over (1+|z|^2)^2}
\hskip 0.5in
\hat \mg _\CC = |dz|^2
\hskip 0.5in
\hat \mg _{\cH} = { |dz|^2 \over (\Im z)^2}
\hskip 0.5in
\hat \mg _{\cD} = { 4 |dz|^2 \over (1-|z|^2)^2}
\eea
where $z \in \hat \CC$, $z \in \CC$, $\Im(z)>0$, and $|z|<1$ respectively.}

\sm

Each one of these spaces is maximally symmetric and may be expressed as the quotient of its isometry group by the isotropy group,
\begin{align}
\hat \CC & = SU(2)/U(1) &
\hskip 0.4 in
\CC = ISO(2)/SO(2)
\no \\
\cH & = SL(2,\RR)/SO(2) &
\hskip 0.4 in
\cD  = SU(1,1) / U(1)
\end{align}
The isometry groups $PSL(2,\RR)= SL(2,\RR)/ \ZZ_2$ and $PSU(1,1)=SU(1,1)/\ZZ_2$ are isomorphic to one another. The action on the coordinate $z$ in either case is given by,
\bea
z \to { az + b \over cz + d}
\eea
where $a,b,c,d \in \RR$ with $ad-bc=1$ for $\cH$ and $c=b^*, d=a^*$ and $|a|^2-|b|^2=1$ for $\cD$. In both cases  the elements $I$ and $-I$ leave $z$ invariant and  are identified under the center $\ZZ_2$.

\sm

The geodesics of $\hat \CC$ are the great circles;  those of $\CC$ are straight lines; and those of $\cH$ are half-circles centered on the real axis. The geodesics of $\cD$ are the circle segments in $\cD$, centered at any $\a \in \CC$ with  $|\a|>1$, satisfying  $|z-\a|^2 = |\a|^2-1$. They intersect the unit circle $|z|=1$ orthogonally to it. The system of geodesics of $\cH$ and $\cD$ makes these spaces  into standard models for two-dimensional hyperbolic (or no-Euclidean) geometry.

\begin{figure}[htb]
\begin{center}
\tikzpicture[scale=1]
\scope[xshift=0cm,yshift=0cm]
\draw [ultra thick] (-3,0) -- (3,0);
\draw (0.2,1.3) node{$i$};
\draw (0,1) node{$\bullet$};
\draw (3,0.3) node{$\infty$};
\draw (0,0) node{$\bullet$};
\draw (0,-0.35) node{$0$};
\draw (-1,-0.35) node{$-1$};
\draw (1,-0.35) node{$1$};
\draw  [thick, color=red, domain=0:180] plot ({cos(\x)},{sin(\x)});
\draw  [thick, color=blue, domain=0:180] plot ({-0.5+2*cos(\x)},{2*sin(\x)});
\draw  [thick, color=green, domain=0:180] plot ({0.8+0.5*cos(\x)},{0.5*sin(\x)});
\draw [thick] (0,0) -- (0,3);

\draw (2.5,2.7) node{$z \in \cH$};
\endscope
\scope[xshift=9cm,yshift=-0.5cm, scale=1.2]
\draw  [ultra thick, domain=0:180] plot ({1.5*cos(\x)},{1.5+1.5*sin(\x)});
\draw  [ultra thick, domain=180:360] plot ({1.5*cos(\x)},{1.5+1.5*sin(\x)});
\draw (0.3,1.8) node{$0$};
\draw (0,1.5) node{$\bullet$};
\draw (1.5,1.5) node{$\bullet$};
\draw (-1.9,1.5) node{$-1$};
\draw (-1.5,1.5) node{$\bullet$};
\draw (1.8,1.5) node{$1$};
\draw [thick] (-1.5,1.5) -- (1.5,1.5);
\draw [thick, color=red] (0,0) -- (0,3);
\draw (0.2,3.3) node{$i$};
\draw (0.2,-0.3) node{$i$};
\draw  [thick, color=blue, domain=0:180] plot ({1.5*(-13+8*cos(\x))/(21+16*sin(\x)-8*cos(\x))},{1.5+1.5*(-4+16*cos(\x))/(21+16*sin(\x)-8*cos(\x))});
\draw  [thick, color=green, domain=0:180] plot ({1.5*(0.55-4*cos(\x))/(9.45+5*sin(\x)+4*cos(\x))},{1.5+1.5*(1.6+cos(\x))/(1.89+sin(\x)+0.8*cos(\x))});
\draw (2,3) node{$w \in \cD$};
\endscope

\endtikzpicture
\caption{\textit{Geodesics in the upper half-plane $\cH$ are semi-circles centered on $\RR$ depicted in the left figure,  with their corresponding images under the conformal map $w=(1+iz)/(1-iz)$ from $\cH$ into the unit disc $\cD$ depicted in the right figure.}
\label{fig:B.2}}
\end{center}
\end{figure}

The uniformization theorem provides us with a powerful tool to classify and construct Riemann surfaces even when they are not simply connected.  The key to the construction is the first homotopy group $\pi_1(\Sigma)$, which is trivial for simply connected $\Sigma$ but non-trivial when $\Sigma$ is not simply-connected.

\subsubsection{Genus one}

Any compact Riemann surface $\Sigma$ of genus 1 has $\pi_1(\Sigma) = \ZZ^2$ whose generators may be chosen to be the $\mA$ and $\mB$ cycles of the first homology group. The universal simply connected covering of a compact genus one surface $\Sigma$  is the complex plane $\CC$. The fundamental group is Abelian and acts on $\CC$ by translations  via a representation $\rho : \ZZ^2 \mapsto \CC$ which defines a  lattice $\Lambda$.  To specify the representation, and the lattice $\Lambda$, it suffices to assign the periods $\rho(1,0) = \om_1$ and $\rho(0,1) = \om_2$ so that the lattice is given by $\Lambda = \om_1 \ZZ \oplus \om _2 \ZZ $. The lattice is two-dimensional (over $\RR$) provided $\om_2/\om_1= \tau \in \cH$. The action of the fundamental group, via the representation $\rho$ on $\CC$, is transitive and the  quotient   $\CC/ \Lambda$ is the original genus one compact Riemann surface $\Sigma$.  Since translations in $\CC$ are isometries of the flat Euclidean metric $|dz|^2$ on $\CC$, the genus-one surface $\CC/ \Lambda$ inherits the flat metric $|dz|^2$ subject to the identifications $z \approx z+\om_1$ and $z \approx z+ \om_2$.

\subsubsection{Higher genus $g \geq 2$}
\label{sec:B4}

{\thm Every compact Riemann surface $\Sigma$ of genus $g \geq 2$ admits a Riemannian metric $\hat \mg$ of constant negative curvature $R_{\hat \mg}=-1$.}

\sm

The curvature $R_\mg$ of an arbitrary metric $\mg$ on $\Sigma$ is related to the curvature $R_{\hat \mg}$ of a metric $\hat \mg$  related to $\mg$ by a Weyl transformation $\mg = e^{2 \sigma} \hat \mg$ by the Liouville equation,
\bea
R_\mg  = e^{-2 \sigma} R_{\hat \mg} +  \Delta_\mg \sigma
\hskip 1in
\Delta_\mg \sigma = - { 1 \over \sqrt{\det \mg }}  \p_m ( \sqrt{\det \mg } \, \mg^{mn} \p_n \sigma )
\eea
where $\Delta _{\mg}$ is the Laplace-Beltrami operator on scalar functions for the metric $\mg$.
To prove the theorem one shows that, for an arbitrary metric $\mg$, the Liouville equation has a solution for $R_{\hat \mg}=-1$. The Liouville equation first arose precisely  in this context.  Before presenting the construction of higher genus surfaces we introduce the concept of a Fuchsian group.

\subsection{Fuchsian groups}
\label{sec:Fuchs}

A systematic approach to the classification and explicit construction of Riemann surfaces is through the use of Fuchsian groups. A Fuchsian group is defined to be isomorphic to a discrete subgroup of $SL(2,\RR) $. Fuchsian groups are special cases of Kleinian groups which are discrete subgroups of $SL(2,\CC)$. The elements $\gamma \not = \pm I$ of a Fuchsian group $\Gamma \subset SL(2,\RR)$ belong to one of the following three types,
\bea
\label{B.4.a}
\gamma \hbox{ \textit{elliptic} } \hskip 0.25in & & | \tr (\gamma)| < 2
\no \\
\gamma \hbox{ \textit{parabolic} } \hskip 0.08in &&  \tr (\gamma) = \pm 2
\no \\
\gamma \hbox{ \textit{hyperbolic} } &&  |\tr(\gamma)| >2
\eea
Since $\det (\gamma)=1$, $\gamma$ must be conjugate, under $SL(2,\RR)$, to one of the following matrices,
\bea
\label{B.diag}
\left ( \bma \lambda & 0 \cr 0 & \lambda ^{-1} \cr \ema \right )
\hskip 1in
\left ( \bma \pm 1  & 1 \cr 0 & \pm 1 \cr \ema \right )
\eea
The left matrix is for elliptic type with $|\lambda|=1$  and hyperbolic type with $\lambda \in \RR$, both with $\lambda \not = \pm 1$. The right matrix is  for parabolic type.

\subsubsection{Action of $\Gamma$ on $\cH$ and $\RR \cup \{ \infty \}$}

Elements $\gamma \in SL(2,\RR)$ acts on $\cH$ by M\"obius transformation,
\bea
\gamma z = \gamma (z) = { az+b \over cz + d}
\hskip 1in
\gamma = \left ( \bma a & b \cr c & d \cr \ema \right )
\eea
Since the element $\gamma =-I$ leaves every point of $\cH$ invariant, the action is effectively by the normal subgroup $PSL(2,\RR) = SL(2,\RR)/\{\pm I \}$.  The action of $SL(2,\RR)$ on $\cH$ is transitive since an arbitrary point $z=x+iy$ is the image of the point $ i \in \cH$, namely $z=\gamma_z(i) $,  for a group element $\gamma_z$ whose entries satisfy $c=0, d=a^{-1}$ and $x=ab, y=a^2$. The isotropy group of the point $i$ is the $SO(2)$ subgroup of $SL(2,\RR)$ obtained by setting $c=-b$ and $d=a$.  The isotropy subgroup of an arbitrary point $z$ is the conjugate of $SO(2)$ by the element $\gamma_z$ given by $  \gamma _z \, SO(2) \, \gamma _z^{-1}$. As a result, $\cH$ may be represented by the coset space,
\bea
\cH = SL(2,\RR) / SO(2)
\eea
The action of $SL(2,\RR)$ on $\cH$ extends to a transitive action on $\RR \cup \{ \infty \} $  so that every point $x \in \RR \cup \{ \infty \}$ may be expressed as $x=\gamma _x(\infty)$ for some $\gamma _x \in SL(2,\RR)$.

\sm

An important role will be played by the fixed points of various elements $\gamma \in SL(2,\RR)$. The classification of the elements of $SL(2,\RR)$ into elliptic, parabolic, and hyperbolic given in (\ref{B.4.a}) is equivalent to the following classification in terms of their fixed points,
\bea
\label{B.4.b}
\hbox{ \textit{elliptic element} } \hskip 0.25in & & \hbox{ one fixed point } z \in \cH, \hbox{ the other at } \bar z
\no \\
\hbox{ \textit{parabolic element} } \hskip 0.08in &&  \hbox{ one fixed point, which is in } \RR \cup \{ \infty \}
\no \\
\hbox{ \textit{hyperbolic element} } &&  \hbox{ two fixed points, which are in } \RR \cup \{ \infty \}
\eea
In particular, the classifications of (\ref{B.4.a}) and (\ref{B.4.b})  apply to the elements of an arbitrary discrete subgroup $\Gamma \subset SL(2,\RR)$. One defines,
\bea
\hbox{ \textit{elliptic point of}  } \Gamma &&
\hbox{is a point } z \in \cH \hbox{ such that there exists}
\no \\  && \hbox{ an elliptic }  \gamma \in \Gamma  \hbox{ with } \gamma (z) = z
\no \\
\hbox{ \textit{cusp of} }  \Gamma \qquad &  &
\hbox{is a point } x \in \RR \cup \{ \infty \}  \hbox{ such that there exists}
\no \\ && \hbox{ a parabolic } \gamma \in \Gamma \hbox{ with } \gamma (x) = x
\eea
If $z$ is an elliptic point of $\Gamma$, then $\gamma (z)$ is an elliptic point of $\Gamma$ for all $\gamma \in \Gamma$. If $x$ is a cusp of $\Gamma$, then $\gamma (x)$ is a cusp of $\Gamma$ for every $\gamma \in \Gamma$.

(a) If $z \in \cH$ is an elliptic point of $\Gamma$ then the set $\Gamma _z=\{ \gamma \in \Gamma \,  | \, \gamma(z)=z \}$ is a finite cyclic group. (b)  The set of elements of finite order in $\Gamma$ consist of the elliptic elements of $\Gamma$ together with $\pm I$. }

\sm

To prove (a) we note that $\Gamma _z$ is a subgroup of $\Gamma$. Representing $z=\gamma _z (i)$ we see that  $\Gamma_z$ is the set of $\gamma \in \Gamma$ for which $\gamma _z^{-1} \gamma \gamma_z (i) = i$. The isotropy group of the point $i$ was earlier  identified as $SO(2)$ so that $\Gamma _z \subset SO(2)$. Since $\Gamma_z$ is discrete and a subgroup of the compact group $SO(2)$,   $\Gamma _z$ must be finite. Since all finite subgroups of $SO(2)$ are cyclic, $\Gamma_z$ must also be cyclic.
To prove (b), we use the fact that an element $\gamma \in \Gamma$  is conjugate under $SL(2,\CC)$  to a diagonal matrix with diagonal entries $\lambda, \lambda ^{-1} \in \CC$ as in (\ref{B.diag}). Finite order implies that there exists a positive integer $n$ such that $\lambda ^n=1$. When $n=1,2$ we have $\gamma = \pm I$ while for $n >2$, $\gamma$ must be elliptic.
Conversely, every elliptic element is of finite order by (a).

\subsection{Construction of Riemann surfaces via Fuchsian groups}

The fundamental group $\pi_1(\Sigma)$ for an arbitrary compact Riemann surface $\Sigma$ of genus $g \geq 2$  is discrete and non-Abelian. The universal (i.e. simply connected) covering space $\hat \Sigma$ of $\Sigma$ may be realized as a fiber bundle over $\Sigma$ with structure group isomorphic to $\pi_1(\Sigma)$. Since $\Sigma$ admits a metric $\hat \mg$ of constant negative curvature $R_{\hat \mg}=-1$ by the above theorem, the universal covering surface admits a metric of constant negative curvature as well. Therefore, by the uniformization theorem, we conclude that $\hat \Sigma = \cH$ or the conformaly isomorphic $ \cD$. Conversely, the fundamental group $\pi_1(\Sigma)$ must act on $\hat \Sigma$ by an isometry of the constant curvature metric. Since we have $\hat \Sigma= \cH$, whose isometry group is $PSL(2,\RR)$, the action of $\pi_1(\Sigma)$ on $\cH$ must be by a representation $\rho$,
\bea
\rho : \pi_1(\Sigma) \mapsto PSL(2,\RR)
\eea
which assigns an element of $PSL(2,\RR)$ to each equivalence class of closed curves on $\Sigma$ with a specified, though arbitrary, base point $P$. To realize these assignments concretely, we choose a base point $P \in \Sigma$, as well as a set of homology generators $\mA_I$ and $\mB_J$ for $I,J=1,\cdots, g$ passing through $P$ and intersecting one another no-where else, as shown in Figure \ref{fig:B.1}.

\sm

We now cut the surface $\Sigma$ along each one of the closed curves $\mA_I$ and $\mB_J$ to unfold $\Sigma$ into a polygonal simply connected domain in $\CC$, as shown schematically for genus 2 in Figure \ref{fig:B.3}. To reconstruct the surface from the fundamental domain represented in Figure \ref{fig:B.3}, we pairwise identify the curves corresponding to $\mA_I$ and $\mA_I^{-1}$ by the transformation $\rho(\mB_I)$ thereby closing the curves $\mB_I$. Similarly, we identify the curves $\mB_I$ and $\mB_I^{-1}$ by the transformation $\rho(\mA_I^{-1})$. Clearly, the composition of all commutators must give the identity, so we must have,
\bea
\rho(\mC_1) \rho(\mC_2) \cdots \rho(\mC_g) = I \hskip 1in \rho(\mC_I) = \rho(\mA_I) \rho( \mB_I) \rho(\mA_I)^{-1} \rho(\mB_I)^{-1}
\eea
Each homotopy class of $\pi_1(\Sigma)$ contains a unique geodesic, or closed curve of minimal length in the hyperbolic metric. Choosing  the closed curves $\mA_I$ and $\mB_J$ to be geodesics, we obtain a representation of $\Sigma$ in $\cD$ by a domain $\Sigma _\cD$ whose boundary is the union of circular arcs each of which is a geodesic of $\cD$, as represented in Figure \ref{fig:B.3}. The hyperbolic  length of the closed curve $\mA_I$ is identical to the length of $\mA_I^{-1}$ and similarly the lengths of the segments $\mB_I$ and $\mB_I^{-1}$ are equal to one another.

\begin{figure}
\begin{center}
\tikzpicture[scale=3.95]

\scope[xshift=0cm,yshift=0cm]
% contour
%\draw  [thick, domain=0:90] plot ({cos(\x)},{\sin(\x)});
\draw  [ thick, domain=0:180] plot ({cos(\x)},{sin(\x)});
\draw  [ thick, domain=180:360] plot ({cos(\x)},{sin(\x)});
% cycles
\draw  [very  thick, color=red, domain=166:198] plot ({cosh(0.6)*cos(0)+sinh(0.6)*cos(\x)}, {cosh(0.6)*sin(0)+sinh(0.6)*sin(\x)});
\draw  [very  thick, color=red, domain=241:282] plot ({cosh(0.65)*cos(85)+sinh(0.65)*cos(\x)}, {cosh(0.65)*sin(85)+sinh(0.65)*sin(\x)});
\draw  [very  thick, color=red, domain=330:396] plot ({cosh(0.5)*cos(175)+sinh(0.5)*cos(\x)}, {cosh(0.5)*sin(175)+sinh(0.5)*sin(\x)});
\draw  [very  thick, color=red, domain=54:100] plot ({cosh(0.4)*cos(265)+sinh(0.4)*cos(\x)}, {cosh(0.4)*sin(265)+sinh(0.4)*sin(\x)});
\draw  [very  thick, color=blue, domain=201:239] plot ({cosh(0.7)*cos(40)+sinh(0.7)*cos(\x)}, {cosh(0.7)*sin(40)+sinh(0.7)*sin(\x)});
\draw  [very  thick, color=blue, domain=264:323] plot ({cosh(0.5)*cos(125)+sinh(0.5)*cos(\x)}, {cosh(0.5)*sin(125)+sinh(0.5)*sin(\x)});
\draw  [very  thick, color=blue, domain=17:73] plot ({cosh(0.7)*cos(225)+sinh(0.7)*cos(\x)}, {cosh(0.7)*sin(225)+sinh(0.7)*sin(\x)});
\draw  [very  thick, color=blue, domain=114:169] plot ({cosh(0.7)*cos(315)+sinh(0.7)*cos(\x)}, {cosh(0.7)*sin(315)+sinh(0.7)*sin(\x)});
% identifications
\draw  [thick, color=black, dashed, ->, domain=238:220] plot ({cosh(0.9)*cos(40)+sinh(0.9)*cos(\x)}, {cosh(0.9)*sin(40)+sinh(0.9)*sin(\x)});
\draw  [thick, color=black, dashed,  domain=220:203] plot ({cosh(0.9)*cos(40)+sinh(0.9)*cos(\x)}, {cosh(0.9)*sin(40)+sinh(0.9)*sin(\x)});
\draw (0.148,-0.74) [fill=black] circle(0.02cm) ;
\draw  [thick, color=black, dashed, ->, domain=68:42] plot ({cosh(0.9)*cos(225)+sinh(0.9)*cos(\x)}, {cosh(0.9)*sin(225)+sinh(0.9)*sin(\x)});
\draw  [thick, color=black, dashed,  domain=42:19] plot ({cosh(0.9)*cos(225)+sinh(0.9)*cos(\x)}, {cosh(0.9)*sin(225)+sinh(0.9)*sin(\x)});
\draw  [thick, color=black, dashed, ->, domain=281:261] plot ({cosh(0.85)*cos(85)+sinh(0.85)*cos(\x)}, {cosh(0.85)*sin(85)+sinh(0.85)*sin(\x)});
\draw  [thick, color=black, dashed, domain=261:243] plot ({cosh(0.85)*cos(85)+sinh(0.85)*cos(\x)}, {cosh(0.85)*sin(85)+sinh(0.85)*sin(\x)});
\draw  [thick, color=black, dashed, ->, domain=99:77] plot ({cosh(0.6)*cos(265)+sinh(0.6)*cos(\x)}, {cosh(0.6)*sin(265)+sinh(0.6)*sin(\x)});
\draw  [thick, color=black, dashed, domain=77:63] plot ({cosh(0.6)*cos(265)+sinh(0.6)*cos(\x)}, {cosh(0.6)*sin(265)+sinh(0.6)*sin(\x)});
\draw (0.57,0.16) [fill=black] circle(0.02cm) ;
\draw (0.251,0.53) [fill=black] circle(0.02cm) ;
\draw (-0.7,0.4) [fill=black] circle(0.02cm) ;
\draw (-0.235,0.61) [fill=black] circle(0.02cm) ;
\draw (-0.675,-0.165) [fill=black] circle(0.02cm) ;
\draw (-0.165,-0.67) [fill=black] circle(0.02cm) ;
\draw (0.57,-0.19) [fill=black] circle(0.02cm) ;

\draw (0.15, -0.82) node{\tiny $P$};
\draw (0.65, 0.2) node{\tiny $P$};
\draw (0.3, 0.63) node{\tiny $P$};
\draw (-0.78, 0.45) node{\tiny $P$};
\draw (-0.24, 0.7) node{\tiny $P$};
\draw (-0.75, -0.2) node{\tiny $P$};
\draw (-0.2, -0.75) node{\tiny $P$};
\draw (0.65, -0.2) node{\tiny $P$};
\draw [color=red] (0.73,0) node{\small $\mA_1$};
\draw [color=blue] (0.53,0.4) node{\small $\mB_1$};
\draw [color=red] (0.05,0.71) node{\small $\mA_1^{-1}$};
\draw [color=blue] (-0.5,0.6) node{\small $\mB_1^{-1}$};
\draw [color=red] (-0.8,0.1) node{\small $\mA_2$};
\draw [color=blue]  (-0.5,-0.44) node{\small $\mB_2$};
\draw [color=red] (-0.05,-0.85) node{\small $\mA_2^{-1}$};
\draw [color=blue]  (0.45,-0.5) node{\small $\mB_2^{-1}$};
% identifications
\draw [color=red] (-0.1,0.35) node{\small $\rho(\mA_1^{-1})$};
\draw [color=red] (0.05,-0.48) node{\small $\rho(\mA_2^{-1})$};
\draw [color=blue] (0.2,0.2) node{\small $\rho(\mB_1)$};
\draw [color=blue]  (-0.18,-0.2) node{\small $\rho(\mB_2)$};
\draw (1,-0.8) node{{$\cD$}};
\endscope
\endtikzpicture
\caption{\textit{A fundamental domain for  a compact genus-two Riemann surface $\Sigma$  in the hyperbolic disc $\cD$. The surface $\Sigma$ is recovered by  identifying the curves $\mA_I $ with $ \mA_I^{-1}$  and identifying the curves $\mB_I$ with $ \mB_I^{-1}$.} \label{fig:B.3}}
\end{center}
\end{figure}

\subsubsection{Compactification of $\Gamma \backslash \cH$}

We choose the standard topology of $\cH$ generated by open sets given by open metric discs.  Consider a Fuchsian group $\Gamma \subset SL(2,\RR)$ whose cusps form a set $\cC \subset \RR \cup \{ \infty \}$. To $\Gamma$ we associate the space,
\bea
\bar \cH = \cH \cup \cC
\eea
To specify the topology of $\bar \cH$ we add open sets that contain the cusps in $\cC$ to the topology of $\cH$. For the cusp $x= \infty$ and the cusps $x \not = \infty$,  these open sets may be chosen as follows,
\bea
x = \infty & \hskip 0.5in & \{ \infty \} \cup \big \{ z \in \cH \, \hbox{ such that } \, \Im(z) > y >0 \hbox{ for } y \in \RR^+ \big \}
\no \\
x \not= \infty & \hskip 0.5in & \{ x \} \cup \big \{ z \in \cH \, \hbox{ such that } \, |z-x-iy| < y \hbox{ for }  y \in \RR^+ \big \}
\eea
The space $\bar \cH$ equipped with this topology is clearly not compact. In fact, it is not even locally compact.\footnote{A topological space $X$ is locally compact if every point in $X$ has an open neighborhood which is contained in a compact subset of $X$.}  The group $\Gamma$ consistently acts of $\bar \cH$ which allows us  to define the quotient,
\bea
X(\Gamma) = \Gamma \backslash \bar \cH
\eea
The quotient space $\Gamma \backslash \bar \cH$ is locally compact. Finally, we state the following proposition  here without proof.

{\prop \label{prop:B6} If the space $\Gamma \backslash \bar \cH$ is compact then the number of $\Gamma$-inequivalent cusps (resp. the number of elliptic points) is finite.}

\subsection*{$\bullet$ Bibliographical notes}

Classic references on Riemann surfaces are the books by Gunning~[338] and by Farkas and Kra~[339]. Comprehensive lecture notes by Bost may be found in the collection~[340]. A detailed treatment of modular curves and their compactification may be found in Chapters 2 and 3 of [13], where a proof of Proposition \ref{prop:B6} is also given.

\newpage

\section{Line bundles on Riemann surfaces}
\setcounter{equation}{0}
\label{sec:LB}

In this appendix, we shall present a brief review of complex line bundles over Riemann surfaces. We begin by
giving a general mathematical definition of a complex line bundle over a Riemann surface $\Sigma$, the topological classification of line bundles, divisors, the Riemann-Roch theorem, and various dimension formulas including the dimension of moduli space. We then discuss the same topic of line bundles from a more physics-oriented point of view as spaces of vector fields, forms, and spinors.

\subsection{Holomorphic line bundles on a Riemann surface}

A complex line bundle $L$ on a Riemann surface $\Sigma$ provides an assignment of a one-dimensional complex vector space $L_p$ to each point $p \in \Sigma$. A local description of the line bundle $L$  may be given by using a covering of $\Sigma$ with  open sets $\cU_\a$ so that $\bigcup _\a \cU_\a = \Sigma$.  Introducing local complex coordinates $z_\a, \bar z_\a$ in each open set makes $\cU_\a$  into a coordinate chart. Locally in each coordinate chart, the bundle is a direct product $\cU_\a \times \CC$ and each section $f$ of $L$ reduces to a smooth $\CC$-valued function $f_\a$ on $\cU_\a$. In the intersection $\cU_\a \cap \, \cU_\beta$ the functions $f_\a$ and $f_\b$ are related by transition functions $\f_{\a \b}$,
\bea
\label{6.A1}
f_\a = \f_{\a \b} f_\b \hskip 0.5in \hbox{ in } \quad \cU_\a \cap \, \cU_\b
\eea
The transition functions $\f_{\a \b}$ are nowhere vanishing and, for a holomorphic line bundle, must be holomorphic functions. Considering now the intersection of three coordinate charts, we obtain a condition on products of transition functions,
\bea
\label{6A.prod}
\f_{\a \b} \, \f_{\b \g} = \f _{\a \g} \hskip 0.5in \hbox{ in } \quad \cU_\a \cap \, \cU_\b \cap \, \cU_\g
\eea
Note that, for a holomorphic line bundle $L$, the transition functions are holomorphic, but the sections $f$ and the
local functions $f_\a$ need not be holomorphic.

\subsubsection{Topological classification}

To obtain the topological classification of line bundles over $\Sigma$, we solve the non-vanishing condition on the transition functions $\f_{\a \b}$ in terms of exponentials, $\f_{\a \b} = \exp ( 2\pi i \psi_{\a \b})$, where $\psi_{ \a \b}$ is defined by $\f_{\a \b}$ up to the addition of an arbitrary integer $n_{\a \b}$. The product relation (\ref{6A.prod}) is then equivalent to the fact that the combinations,
\bea
c_{\a\b\g} = \psi_{\a \b} + \psi _{\b \g} - \psi _{\a\g}
\eea
are integers since their exponential equals 1 by (\ref{6A.prod}). The transformations $\psi_{\a\b} \to \psi _{\a\b}+ n_{\a \b}$ imply  transformations of $c_{\a\b\g}$ by an \textit{exact cocyle} $n_{\a\b} + n_{\b \g} - n_{\a\g}$,
\bea
\label{6A.gauge}
c_{\a \b \g} \to c_{\a\b\g} + n_{\a\b} + n_{\b \g} - n_{\a\g}
\eea
By construction in terms of $\psi_{\a\b}$, the integers $c_{\a\b\g}$ satisfy the following \textit{closed cocycle} relation on the intersection of four coordinate charts,
\bea
\label{6A.cocycle}
c_{\a \b \g} - c_{\b \g \delta} +c_{\g \delta \a} - c_{\delta \a \b} =0
\eea
The space of closed cocycles $c_{\a\b\g}$ satisfying (\ref{6A.cocycle}) modulo exact cocycles $n_{\a\b} + n_{\b \g} - n_{\a\g}$ is the second \v Cech cohomology group of $\Sigma$ with integer coefficients, denoted $H^2(\Sigma, \ZZ)$.

\sm

One may translate the \v Cech cohomology formulation of line bundles into the language of differential forms, de Rham cohomology, and gauge fields, which is more familiar to physicists. It will be convenient here to assume that $\Sigma$ is compact so that one may extract from the covering of $\Sigma$ by open sets $\cU_\a$ a covering with only a finite number of open sets, and avoid issues of convergence.

\sm

Each line bundle $L$ on $\Sigma$ corresponds to a cohomology class of $H^2(\Sigma, \ZZ)$, labeled by the first Chern class $c_1(L)$. In turn, the first Chern class $c_1(L)$  may be represented in terms of a de Rham cohomology class $[F]$ where $F$ is the curvature 2-form of a $U(1)$ connection 1-form. In each coordinate chart $\cU_\a$ a representative $F$ of the class $[F]$ may be expressed in terms of a local connection form $A_\a$ by  $F = d A_\a$. The transition functions for the connection are,\footnote{To be precise the de Rham theorem states an isomorphism between the de Rham cohomology group $H^2_{{\rm dR}}(\Sigma)$ and the Cech cohomology group with real coefficients $H^2(\Sigma, \RR)$.}
\bea
A_\a - A_\b = d \lambda _{\a \b}
\eea
with $c_{\a\b\g} = \lambda_{\a\b} + \lambda _{\b \g} - \lambda _{\a \g}$. The first Chern class, defined by,
\bea
\label{C.Chern1}
c_1(L) = { i \over 2 \pi} \int _\Sigma F
\eea
takes integer values, so that $H^2(\Sigma, \ZZ) =\ZZ$.  For later use, we note that the first Chern class of the tensor product of two line bundles $L_1$ and $L_2$ over $\Sigma$ is given as follows,
\bea
\label{C.Chern2}
c_1 (L_1 \otimes L_2) = c_1 (L_1) + c_1(L_2)
\eea
We conclude that the topological classification of line bundles $L$ over a compact Riemann surface $\Sigma$ is in terms of a single integer $d$, referred to as the degree of $L$,  which labels the first Chern class $c_1(L)$ of $L$.

\subsubsection{Holomorphic classification: divisors}

Holomorphic line bundles in a given topological class characterized by the degree $c_1(L)=d$ are not necessarily equivalent holomorphically. The space of all holomorphic line bundles for given degree $d$ is referred to as the Picard variety,
\bea
{\rm Pic}_d (\Sigma) = \left \{ \hbox{holomorphic line bundles } L \hbox{ on } \Sigma \hbox{ with } c_1(L)=d \right \}
\eea
The special case of $d=0$ corresponds to the \textit{Jacobian variety} $J(\Sigma) = {\rm Pic}_0(\Sigma)$.

\sm

A convenient way to characterize a holomorphic line bundle is by its {\rm divisor}, which is a complex codimension-one subset of $\Sigma$ given by a formal sum over distinct points $p_\a \in \Sigma$,
\bea
\label{6A.div}
D = \sum _{\a=1}^N n_\a p_\a
\eea
The integer  $n_\a$ denotes the order of the point $p_\a$ in the divisor $D$ and may take positive or negative values, while points for which $n_\a=0$ are usually omitted from the sum.

\sm

To construct the line bundle $L$ associated with a divisor~$D$, we introduce coordinate charts  $\cU_\a$  with  $p_\a \in \cU_\a$ and a local complex coordinate $z_\a$ in $\cU_\a$ that vanishes at  $p_\a$ for each $\a = 1,\cdots ,N$. By choosing each $\cU_\a$ small enough we can make them mutually disjoint $\cU_\a \cap \, \cU_\b= \emptyset $ for $\a \not= \b$. We denote the open complement to all the points $p_\a$ by  $\cU_\infty = \Sigma \setminus \{ p_1, \cdots, p_N\}$ so that $\cU_\infty \bigcup _\a \cU_\a = \Sigma$. The line bundle $L$ corresponding to the divisor $D$ is then constructed by taking transition functions in the intersections $\cU_\a \cap \, \cU_\infty$ to be $z_\a ^{n_\a}$, the integers $n_\a$ corresponding to the order of the point $p_\a$ in the divisor $D$ in (\ref{6A.div}). A section $f$ of $L$, described locally by functions $f_\a$ on $\cU_\a$ and $f_\infty$ in $\cU_\infty$, satisfies $f_\a = z_\a ^{n_\a} f_\infty$. Conversely, given a line bundle $L$ and a section $f$, the corresponding divisor may be recovered uniquely from the zeros and poles of $f$. Since the open sets $\cU_\a$ are mutually disjoint, all intersections with more than two $\cU$ coordinate charts are empty. As a result, the cocycle conditions (\ref{6A.prod}) and (\ref{6A.cocycle})  are automatically satisfied.

\sm

Adding the contributions from each transition function to the first Chern class, we find,
\bea
d = c_1(L) = \sum_{\a =1}^N n_\a
\eea
In other words, the degree $d$ of $L$ is the total number of zeros (counted with their orders $n_\a$ when $n_\a>0$) minus the total number of poles (counted with their orders $-n_\a$ when $n_\a <0$).
A topologically trivial bundle $L$ with $d=0$ has an equal number of zeros and poles (counted with their orders) and its meromorphic sections are the meromorphic functions on~$\Sigma$. One may introduce an equivalence relation between divisors such that $D_1$ and $D_2$ are equivalent to one another if they differ by a divisor of degree zero. When this is the case, the ratio $f_2/f_1$ of any two sections $f_1$ and $f_2$ of $D_1$ and $D_2$ respectively, has degree zero.  The equivalence class with representative divisor $D$ is denoted by $[D]$.

\subsubsection{Examples}

A familiar example in Physics is provided by the Dirac magnetic monopole, which may be thought of having a constant field strength $F$ on the Riemann sphere $\hat \CC$. Denoting the North and South poles by $p_\pm$, we  introduce coordinate charts $\cU_\pm = \hat \CC \setminus \{ p_\mp \}$ and local coordinates $z_\pm$ which vanish at  $p_\pm$. Choosing $f_-=1$ and $f_+=z_+^{n_+}$ gives $\psi _{+-} = n_+ \ln (z_+)/(2 \pi i) $ and $A_+ - A_-=  n_+ dz_+ /(2 \pi i z_+)$, so that $c_1(L)=n_+$ for a Dirac monopole of charge $n_+$.

\sm

The tangent bundle $T \Sigma$ over a Riemann surface $\Sigma$ is a bundle of rank 2 over the reals. Using the complex structure of $\Sigma$ it may be decomposed into the direct sum of its holomorphic component $T\Sigma _{(1,0)}$ and its complex conjugate $T\Sigma _{(0,1)}$. The cotangent bundle $T^*\Sigma$ may be decomposed analogously,
\bea
T \Sigma & = & T\Sigma _{(1,0)} \oplus T\Sigma _{(0,1)}
\no \\
T^* \Sigma & = & T^*\Sigma _{(1,0)} \oplus T^*\Sigma _{(0,1)}
\eea
The bundles $T\Sigma _{(1,0)}$ and $T^*\Sigma _{(1,0)}$ are holomorphic line bundles over $\Sigma$. The holomorphic cotangent bundle $T^*\Sigma _{(1,0)}$ is also referred to as the \textit{canonical bundle} over $\Sigma$ and denoted $K$, while $ T\Sigma _{(1,0)}$ is isomorphic to $K^{-1}$.

\sm

This decomposition may be rendered explicit on the sections of the bundles.  In terms of real local coordinates $\xi^m$ on $\Sigma$ a section of $T\Sigma$ is a  vector field and may be expressed as $v^m  \p_m$ while a section of $T^*\Sigma$ is a  differential one-form that takes the form $\om_m d\xi^m$.   In local complex coordinates $z, \bar z$ they may be decomposed as follows,
\bea
v^m \, \p_m & = & v^z \, \p_z + v^{\bar z} \, \p_{\bar z}
\no \\
\om_m \, d\xi^m & = & \om_z \, dz + \om_{\bar z} \, d{\bar z}
\eea
where $v^z \, \p_z $ is a section of $T\Sigma _{(1,0)}$ and $\om_z \, dz$ is a section of $T^*\Sigma _{(1,0)}$ and similarly for their complex conjugates. Their transition functions are given as follows,
\bea
(v^z)_\a (z_\a) & = & (v^z)_\b (z_\beta) \, \left (  \p z_\a / \p z_\b \right )
\no \\
(\om_z)_\a (z_\a) & = & (\om_z)_\b (z_\b) \, \left (  \p z_\a / \p z_\b \right )^{-1}
\eea
where $\p z_\a / \p z_\b$ is a holomorphic nowhere vanishing transition function in $\cU_\a \cap \, \cU_\b$.

\sm

Finally, a spin bundle $S$ on $\Sigma$ is a holomorphic line bundle whose tensor square is isomorphic to the canonical bundle $K$,
\bea
S \otimes S \approx K
\hskip 1in c_1(S) = \thalf c_1(K)
\eea
The relation between the Chern numbers follows from  (\ref{C.Chern2}). By a slight abuse of notation, a spin bundle is some times denoted by $K^\half$. For $g=0$ the spin bundle is unique. However, for $g\geq 1$, the relation to the canonical bundle  does not specify the spin bundle $S$ uniquely. The different spin bundles are referred to as \textit{spin structures}.  On a surface of genus $g$, there are $2^{2g}$ independent spin structures.

\subsection{Holomorphic sections and the Riemann-Roch theorem}

As emphasized earlier, while the transition functions of a holomorphic line bundle are holomorphic functions, the sections of a holomorphic line bundle are not generally holomorphic. The Cauchy-Riemann operator $\bar \p _L$ for a holomorphic line bundle $L$  acts on the sections of $L$ by the ordinary Cauch-Riemann operator $\bar \p = d\bar z^\a \p_{\bar z_\a} $ with respect to the local coordinate $z_\a$ in each open set $\cU_\a$. This action is covariant without the need for a connection, as may be seen by applying $\bar \p$ to local sections in the overlap of open sets $\cU_\a$ given by (\ref{6.A1}),
\bea
\bar \p f_\a = \f_{\a \b} \, \bar \p f_\b   \hskip 0.6in \hbox{in} \hskip 0.6in \cU_\a \cap \cU_\b
\eea
The subspace of all holomorphic sections of a holomorphic line bundle plays a special role and contains a lot of information on the line bundle. It may be defined as follows. To a holomorphic line bundle $L$ we associate the vector space of its holomorphic sections,
\bea
{\rm Ker} (\bar \p_L) = \{ \hbox{holomorphic sections of } L \}
\eea
These spaces are finite-dimensional and their dimensions are related by the Riemann-Roch theorem, which will be proven, in a slightly different guise, in appendix \ref{sec:C.RR}.

{\thm The Riemann-Roch theorem for a holomorphic line bundle $L$ states
\bea
\dim {\rm Ker} \left ( \bar \p_L \right )  - \dim {\rm Ker} \left ( \bar \p _{K \otimes L^{-1} } \right ) = c_1(L) + \half \chi(\Sigma)
\eea
where $K$ is the canonical bundle, $\chi(\Sigma)$ is the Euler characteristic of $\Sigma$, and $\bar \p _L$ is the Cauchy Riemann operator acting on $L$. }

\subsection{Vanishing Theorem and dimension formulas}

In this subsection, we shall state a vanishing theorem and combine its implications with the Riemann-Roch theorem to obtain general expressions for the dimensions of the spaces of homomorphic sections of line bundles. We shall specialize to the case of compact $\Sigma$ and make use of the fact that the Euler number is then given in terms of the genus $g$ of $\Sigma$ by $\chi(\Sigma) =2 - 2g$.

\sm

We begin by obtaining some immediate consequences of the Riemann-Roch theorem. Since the only holomorphic functions on $\Sigma$ are the constant functions, the corresponding bundle $L$ is the trivial bundle with $c_1(L)=0$ and  we have ${\rm Ker} (\bar \p_L)= \CC$ and  $\dim {\rm Ker} (\bar \p_L) =1$. Using the Riemann-Roch theorem for the trivial bundle $L$, we find
\bea
\dim {\rm Ker} (\bar \p_K) =g
\eea
For $g=0$ there are no holomorphic 1-forms. For $g=1$ we recover the observation, made long ago, that the space of holomorphic 1-forms is one-dimensional. For $g \geq 1$, there are $g$ linearly independent holomorphic 1-forms on $\Sigma$ already identified in appendix \ref{sec:B.1}.    Setting $L=K$, and using the fact that $K \otimes K^{-1}$ is the trivial bundle,  the Riemann-Roch theorem implies $c_1(K)=2g-2$. For $g=1$ we recover the fact, observed long ago,  that the holomorphic 1-form $dz$ is nowhere vanishing, while for $g\geq 2$ we obtain the result that every holomorphic one-form has $2g-2$ zeros (counted with multiplicities).

\sm

More generally, representing the divisor class $[L]$ of an arbitrary line bundle $L$ by a tensor power of the canonical bundle  $[K^n]$ for $n \in \ZZ/2$, where it is understood that $K^\half =S$ is a spin bundle with specified spin structure, we have the following vanishing theorem:

{\thm The following holds}
\bea
\label{6A.van}
\dim {\rm Ker} (\bar \p_L) =0 \hskip 0.4in
\left \{ \bma g=0 & \hbox{ and } & n >0 \cr g\geq 2 & \hbox{ and } & n <0  \cr \ema \right .
\eea

The entry for $g=0$ states the absence of holomorphic forms or spinors of degree greater than 0 on the sphere. The entry for $g \geq 2$ states that there are no holomorphic vector fields on a higher genus Riemann surface. The proof of this theorem will be giving in appendix \ref{sec:C.VT}, in a slightly different guise.

\sm

Combining the results of the vanishing theorem with the results obtained from the Riemann-Roch theorem for arbitrary divisor class  $[L]=[K^n]$ and using the relations $c_1(L)= c_1(K^n) = n c_1(K) $ for  $n \in \ZZ/2$ we obtain for  $n \not =  \half$,
\bea
\label{6A.dims1}
\dim {\rm Ker} (\bar \p _L) = \left \{ \bma
(2n-1)(g-1) & \hbox{ for } &  g \geq 2 & \hbox{and} & n \geq { 3 \over 2} & \cr
g &  &  g \geq 0 &  & n=1 & \cr
1 &  &  g \geq 0 &  & n=0 & \cr
1 &  &  g =1 &  & n \not= \half & \cr
1-2n && g=0 && n \leq -\half \cr
\ema \right .
\eea
For $n$ a half-odd-integer, these dimensions are independent of the spin structure of~$L$.  But for $n =  \half$ the dimension of ${\rm Ker} (\bar \p _L) $ depends on the spin structure of $L$. Spin structures are partitioned into even and odd depending on whether $\dim {\rm Ker} (\bar \p _L) $ is even or odd,
\bea
\label{6A.dims2}
\dim {\rm Ker} (\bar \p _L) = \left \{ \bma
1 ~ (\mod 2) &  & g \geq 1 &  & n = \thalf &\hbox{(odd spin structures)} \cr
&&&& \cr
0 ~ (\mod 2) & & g \geq 1 &  & n = \thalf & \hbox{(even spin structures)} \cr
\ema \right .
\eea
The dimensions in (\ref{6A.dims2}) equal $1$ and $0$ respectively throughout moduli space for $g=1,2$ and at generic points in moduli space for arbitrary $g \geq 3$.  For $g \geq 3$, the dimensions may increase by a multiple of 2 at certain sub-varieties of moduli space, including at the locus of hyper-elliptic Riemann surfaces.

\subsection{Tensors and spinors on $\Sigma$}
\label{sec:C4}

In Physics, we will deal with scalar, vector, tensor, and spinor fields on a Riemann surface~$\Sigma$ viewed as a two real dimensional surface equipped with a Riemannian metric,
\bea
\mg = \mg_{mn} (\xi) \, d \xi^m d \xi^n
\eea
in a system of real coordinates $\xi^m$.  We shall now analyze complex line bundles, covariant derivatives and Laplace-Beltrami operators on line bundles in a tensorial formulation, and use it to prove the Riemann-Roch and vanishing theorems.

\sm

An arbitrary tensor field $\mt_{m_1 \dots  m_n}$ of rank $n$ may be decomposed into a direct sum of symmetric  traceless tensors and scalars. Indeed, any anti-symmetric pair of indices may be contracted with the tensor $\sqrt{\mg} \, \ep_{mn}$ to produce a scalar, while the metric $\mg_{mn}$ may be used to eliminate all traces. The resulting symmetric traceless tensor may be further decomposed into the eigenspaces of the complex structure $\cJ$.  The convention $v^z \p_z = v^z (dz)^{-1}$ may be used to identify a form of negative weight with a vector field of opposite weight. Henceforth,  we shall suppress the $z$ and $\bar z$  indices on these reduced tensors, so that $\mt= t_{z, \cdots z}$ and $\bar \mt = t_{\bar z \cdots, \bar z}$. Thus, all tensor fields decompose into a direct sum of the following spaces,
\bea
T_{(n,0)}^* = \Big \{ \mt(z) : \Sigma \to \CC ~ \hbox{such that} ~ \mt'(z') (dz')^n = \mt(z) (dz)^n \Big \}
\eea
The space  $T_{(1,0)}^*$ may be identified with the space of sections of the canonical bundle $K$, while the spaces  $T^*_{(n,0)}$ for $n >0$ may be identified with the space of sections of the $n$-th tensor power $K^n$ of the canonical bundle $K$.  Finally, we include spinors by allowing $n$ to take half-integer values which further requires specifying a spin structure.

\subsubsection{Covariant derivatives and Laplace-Beltrami operators}

On each space of sections $T_{(n,0)}^*$ we define an $L^2(\Sigma)$ inner product as follows,
\bea
\< \mt_1 |  \mt_2 \> = \int _\Sigma d\mu_\mg \, (\mg_{z \bar z})^{-n} \, \overline{ \mt_1 (z)} \, \mt_2(z)
\eea
The Cauchy-Riemann operator $\bar \p = (d\bar z) \pbz$ acts covariantly on the spaces $T_{(n,0)}^*$, without the need for a Christoffel connection,
\bea
\bar \p : T_{(n,0)}^* \to T^*_{(n,1)} ~~ \hbox{with} ~~ \mt(z) (dz)^n \to \big ( \pbz \mt(z) \big ) (dz)^n d\bar z
\eea
It is often more convenient to use a covariant derivative which maps a space into a space of the same type. This may be achieved by the Cauchy-Riemann operator acting on $T^*_{(n,0)}$,
\bea
\nabla ^z_{(n)} : T_{(n,0)}^* \to T_{(n-1,0)}^* \qquad \hbox{with} \qquad \mt(z) (dz)^n \to \nabla ^z_{(n)} \mt (z) = \big ( g^{z\bar z} \pbz \mt(z) \big )  (dz)^{n-1}
\eea
To define the adjoint operator $- \nabla _z ^{(n-1)} $ of $\nabla ^z_{(n)}$ with respect to the $L^2(\Sigma)$ inner product  does require a Christoffel connection and is given by,
\bea
\nabla _z ^{(n)} : T_{(n,0)}^* \to T_{(n+1,0)}^* ~~ \hbox{with} ~~ \mt(z) (dz)^n \to \big ( \nabla _z \mt(z) \big ) (dz)^{n+1}
\eea
The adjoint operators, and the associated Laplace-Beltrami operators,  are as follows,
\bea
\left ( \nabla ^z _{(n)} \right )^\dagger = - \nabla _z ^{(n-1)}
& \hskip 0.8in &
\Delta ^+_{(n)} = - 2\nabla ^z _{(n+1)}  \nabla _z ^{(n)}
\no \\
\left ( \nabla _z ^{(n)} \right )^\dagger = - \nabla ^z _{(n+1)}
\Delta ^-_{(n)} =  - 2\nabla _z ^{(n-1)} \nabla ^z _{(n)}
\eea
The Laplace-Beltrami  operators both map  $T_{(n,0)} \to T_{(n,0)}$. The non-zero eigenvalues and their corresponding eigenfunctions are related. To see this, we observe that,
\bea
\Delta ^+_{(n)} \psi = \lambda \psi & \quad \Longrightarrow \quad & \Delta ^-_{(n+1)} \left ( \nabla ^{(n)} _z \psi \right ) = \lambda \left ( \nabla ^{(n)} _z \psi \right )
\no \\
\Delta ^-_{(n)} \f = \mu \f & \quad \Longrightarrow \quad & \Delta ^+_{(n-1)} \left ( \nabla _{(n)} ^z \f \right )
= \mu \left ( \nabla _{(n)} ^z \f \right )
\eea
When $\lambda, \mu \not=0$, these maps are invertible.
As a result, the spectra of non-vanishing eigenvalues of $\Delta ^+_{(n)}$ and $\Delta ^-_{(n+1)}$ coincide and the one-to-one map between their eigenfunctions is provided by the above relations. No such correspondence follows for the kernels of these operators. Since the Laplace-Beltrami operators are positive, their kernels are related to the kernels of the Cauchy-Riemann operators as follows,
\bea
{\rm Ker} \, \Delta ^+_{(n)} & = & {\rm Ker} \, \nabla _z ^{(n)}
\no \\
{\rm Ker} \, \Delta ^-_{(n)} & = & {\rm Ker} \, \nabla ^z _{(n)}
\eea
In the subsequent sections, we shall relate the dimensions of these kernels to one another by the Riemann-Roch and vanishing theorems.

\subsection{Proof of the Riemann-Roch theorem}
\label{sec:C.RR}

The kernel of $\nabla _{(n)}^z$ is the space of holomorphic differentials of rank $n$ for $n \geq 1$, and holomorphic vector fields of rank $-n$ for $n\leq -1$. In this formulation, the Riemann-Roch theorem for a compact Riemann surface $\Sigma$ may be stated as follows,
\bea
\dim {\rm Ker} \, \nabla ^z _{(n+1)} - \dim {\rm Ker} \, \nabla _z ^{(n)}  = - \left ( n + \thalf \right ) \chi(\Sigma)=(2n+1)(h-1)
\eea
where $\chi (\Sigma)= 2 - 2h$ is the Euler number of $\Sigma$. The Riemann-Roch theorem may equivalently be recast in terms of Laplace-Beltrami operators,
\bea
\label{C.RR1}
\dim {\rm Ker} \, \Delta _{(n+1)} ^- - \dim {\rm Ker} \, \Delta _{(n)}^+    = (2 n + 1 ) (h-1)
\eea
To prove this result, we use the fact established earlier  that the non-zero eigenvalues of $\Delta _{(n)}^+$ and $\Delta _{(n+1}^-$ are identical to obtain the following relation,
\bea
\label{C.dims}
\dim {\rm Ker} \, \Delta _{(n+1)} ^- - \dim {\rm Ker} \, \Delta _{(n)}^+
= \Tr \left ( e^{ - s \Delta _{(n+1)}^- } \right )  - \Tr \left ( e^{ - s \Delta _{(n)}^+ } \right )
\eea
valid for all $0 < s \in \RR$. In particular, the right side may be evaluated in the limit $s \to 0$ where the short-time asymptotics of the heat-kernel may be used to derive an explicit formula. The short-time expansion, or Bott-Seeley expansion,  is given as follows,
\bea
\Tr \left ( f \, e^{-s \Delta _{(n)}^\pm } \right ) =  { 1 \over 4 \pi s} \int _\Sigma d\mu _\mg f
+{ 1 \pm 3 n \over 12 \pi} \int _\Sigma d \mu_\mg R_\mg f + \cO(s)
\eea
Setting $f=1$, substituting the expressions into (\ref{C.dims}), and using the integral expression for the Euler number, we
establish (\ref{C.RR1}), which proves the Riemann-Roch theorem.

\subsection{Proof of the vanishing theorem}
\label{sec:C.VT}

In terms of the differential operators $\nabla ^z _{(n)}$ and $\nabla ^z _{(n)}$ the vanishing theorem read as follows,
\bea
\label{B.vanish}
\dim {\rm Ker} \, \nabla ^z _{(n)} =0  & \hskip 1in & g=0 \hbox{ and } n >0
\no \\
\dim {\rm Ker} \, \nabla ^z _{(n)} =0  & \hskip 1in & g\geq 2  \hbox{ and } n <0
\eea
The proof of these theorems uses the following equations from differential geometry,
\bea
{} [ \nabla _m, \nabla _n] \, \mt_p & = & - \mt_q \, R^q {}_{pmn}
\no \\
{} [ \nabla _m, \nabla _n] \, \mt^p & = & + \mt^q \, R^p{}_{qmn}
\eea
where $R^q{}_{pmn}$ is the Riemann tensor, and $t_q$ is a rank one tensor, i.e. a one-form. Adapted to the case of two dimensions, and working in complex coordinates,  the formulas reduce to,
\bea
{} [ \nabla _z , \nabla_{\bar z} ] \, \mt_z =  - \mg_{z \bar z} R_\mg \, \mt_z
\eea
where $R_\mg$ is the Gaussian curvature (normalized to be one for the round sphere of unit radius). Formulas for  tensors and spinors $\mt_{z \cdots z}= \mt \in T_{(n,0)}^*$ of arbitrary rank $n \in \ZZ/2$ may be derived by taking the tensor product and the square root, and we find,
\bea
{} [ \nabla _z , \nabla_{\bar z} ] \, \mt  =  - n \, \mg_{z \bar z} \, R_\mg \, \mt
\eea
We are now ready to prove the vanishing theorem. We use the following rearrangement formula, obtained by integrating by parts (no boundary terms arise since $\Sigma$ has no boundary),
\bea
\label{B.sq}
\int _\Sigma d \mu_\mg \, (\mg_{z \bar z})^{n-1} |\nabla _{\bar z} \, \mt |^2
= \int _\Sigma d \mu_\mg  \Bigg ( (\mg_{z \bar z})^{n-1} \, | \nabla _{\bar z} \bar \mt |^2
+ n (\mg_{z \bar z})^n R_\mg \, |\mt |^2  \Bigg )
\eea

$\bullet$ For the case $g=0$, we may choose a metric of constant positive curvature $R_\mg=1$ on the sphere. Since we have $n >0$ the right side of (\ref{B.sq})  is positive definite in $\mt$ so that the left side cannot vanish unless $\mt=0$. As a result there are no holomorphic differential forms of any positive weight $n >0$ on the sphere which proves the result on the first line of (\ref{B.vanish}).

$\bullet$ For the case $g\geq 2$, we may choose a metric of constant negative curvature $R_\mg=-1$. Since we have  $n<0$ the right side of  (\ref{B.sq}) is again positive definite so that the left side cannot vanish unless $\mt=0$. As a result there are no holomorphic vector fields of any negative weight $n<0$ on a higher genus Riemann surface. In particular, this implies that there are no conformal Killing vectors or conformal Killing spinors on a Riemann surface of genus $g \geq 2$, and thus no continuous symmetries.

$\bullet$ For the case  $g=1$, and a flat metric $R_\mg=0$ on the torus, the relation (\ref{B.sq}) imposes no restrictions.

\subsection{The dimension of moduli space}

The moduli space $\cM_g$ of  compact Riemann surfaces of arbitrary genus $g$ is the space of inequivalent complex structures or, equivalently, the space of inequivalent conformal structures. Complex structures and conformal structures may be parametrized  with the help of the Weyl-invariant tensor $\cJ_m {}^n = \sqrt{\mg} \, \ep_{mp}\, \mg ^{pn}$.  Two tensors that are related by a diffeomorphsim of $\Sigma$ are equivalent to one another, so that the moduli space is given by,
\bea
\cM_g = \{ \cJ \} / {\rm Diff}(\Sigma)
\eea
To compute the dimension of $\cM_g$, we evaluate the dimension of its tangent space at a given complex structure with local complex coordinates $z, \bar z$ in terms of which the metric is given by $\mg =  2\mg_{z \bar z} |dz|^2$, and the complex structure tensor $\cJ$ has the following components,
\bea
\cJ_z{}^{\bar z} = - \cJ_{\bar z} {}^z = 0
\hskip 1in
\cJ_z{}^z = - \cJ_{\bar z} {}^{\bar z} = i
\eea
Next, we compute the variation of the complex structure tensor, as the metric $\mg$ is varied, \textit{leaving the system of complex coordinates $z, \bar z$ unchanged. } The variation of the metric is used to first order,
\bea
\delta \mg = 2 \delta \mg_{z\bar z} \, |dz|^2 + \delta \mg_{zz} \, dz^2 + \delta \mg_{\bar z \bar z} \, d\bar z^2
\eea
The effect of the infinitesimal Weyl transformation $\delta \mg_{z \bar z}$ cancels out of the variation of $\cJ$ since the complex structure is invariant under Weyl transformations. Thus, we obtain the following variations, to first order in $\delta \mg_{zz}$ and $\delta \mg_{\bar z \bar z}$,
\begin{align}
\delta \cJ_z{}^z & =0 &
\delta \cJ_z{}^{\bar z} & =   + i \, \mg_{z \bar z} \, \delta \mg^{\bar z \bar z}  ~~ \in ~ T^*_{(1,-1)}
\no \\
\delta  \cJ_{\bar z} {}^{\bar z} & = 0 &
\delta  \cJ_{\bar z} {}^z & =   -i \, \mg_{\bar z z} \, \delta \mg^{z z} ~~ \in ~ T^*_{(-1,1)}
\end{align}
Moduli deformations, however, are variations of complex structure modulo diffeomorphisms. The action of an infinitesimal  diffeomorphism $v^m \p_m$ on the metric is given as follows,
\bea
\delta _v \mg_{mn} = \nabla _m v_n + \nabla _n v_m
\eea
Under this variation, the complex structure transforms as follows,
\bea
\delta _v \, \cJ_z{}^{\bar z} & = & - i \, \nabla _z^{(1)}   v^{\bar z}
\no \\
\delta _v \, \cJ_{\bar z} {}^z & = & + i \, \nabla _{\bar z}^{(-1)}    v^z
\eea
Vectors in the tangent space $T_\mg \cM_g$  to moduli space $\cM_g$ at a given metric $\mg$,  may be identified with complex structure deformations that are in the orthogonal complement to the ranges of these operators, evaluated at the metric $\mg$,
\bea
T_\mg \cM_g = \Big ( {\rm Range } ~ \nabla _z ^{(1)} \Big ) ^\dagger
\oplus \Big ( {\rm Range } ~ \nabla _{\bar z} ^{(-1)} \Big ) ^\dagger
\eea
But the orthogonal complement to the range of any operator is the kernel of the adjoint of that operator.
Since we have $(\nabla _z ^{(1)} )^\dagger = - \nabla ^z _{(2)} $, we find,
\bea
T_\mg \cM_g = {\rm Ker} \nabla ^z _{(2)} \oplus {\rm Ker} \nabla ^{\bar z} _{(-2)}
\eea
First of all, we note that the tangent space to $\cM_g$ is split according to holomorphic and anti-holomorphic directions, so that $\cM_g$ is a complex manifold (actually, more precisely a complex orbifold). The dimension of moduli space may be read off from the table of dimensions given in the preceding subsection,
\bea
\label{eq:modspacedim}
\dim_\CC \cM_g = \left \{ \begin{matrix} 0 & g=0 \cr 1 & g=1 \cr 3g-3 & g \geq 2 \cr \end{matrix} \right .
\eea
The combination $\delta \cJ_{\bar z} {}^z$ and its complex conjugate are often referred to as \textit{Beltrami differentials}. The elements of the dual space ${\rm Ker} \nabla ^z _{(2)}$ are the \textit{holomorphic quadratic differentials}.

\subsection*{$\bullet$ Bibliographical notes}

A summary of the ingredients of holomorphic line bundles on compact Riemann surfaces, accessible to physicists, was given in the review paper on string perturbation theory~[190] and in the lecture notes~[340]. A  fundamental and useful reference  is the book by Fay~[341]. Helpful discussions of line bundles characterized by divisors may be found in the book by Farkas and Kra [339], while the deformation theory of complex structures for Riemann surfaces is treated quite explicitly in the book by Schiffer and Spencer [342].  More general treatments of vector bundles on complex and K\"ahler manifolds may be found in the books by Gunning [343] and Kodaira [344]. The Bott-Seeley expansion in the context of two-dimensional Riemannian manifolds is discussed and proven in [345] and [190].

\newpage

\section{Higher genus $\tet$-functions and meromorphic forms}
\setcounter{equation}{0}
\label{sec:Theta}

In this final appendix, we shall review the modular geometry of the Siegel half-space at higher rank, Riemann $\tet$-functions of higher rank, the embedding of higher-genus Riemann surfaces into the Jacobian variety via the Abel map,  and use these ingredients to construct the prime form, the Szeg\"o kernel, and other meromorphic differential forms on higher-genus Riemann surfaces.

\subsection{The Siegel half space}
\label{sec:A1}

The rank $g$ Siegel half space $\cH_g$  may be defined  as the space of complex $g \times g$ symmetric matrices with positive definite imaginary part,
\bea
\label{A1}
\cH_g = \Big \{ \Omega \in \CC^{g \times g}, ~ \Omega^t = \Omega, ~ Y= \Im (\Omega)>0 \Big \}
\eea
Its dimension is  $\dim \cH_g = \half g (g+1)$. Alternatively, $\cH_g$ is given as the coset space
\bea
\label{A2}
\cH_g = Sp(2g, \RR) / U(g)
\eea
The group $Sp(2g, \RR)$ acts on $\Omega$ by
\bea
\label{A3}
\Omega \to \Omega ' = (A\Omega +B) (C \Omega +D)^{-1}
\eea
where the $g\times g$ real matrices $A,B,C,D$ are given in terms of $M \in Sp(2g,\RR)$ by,
\bea
\label{A4}
M = \left ( \begin{matrix} A & B \cr C & D \cr \end{matrix} \right )
\hskip 0.8in
M^t \mJ M = \mJ
\hskip 0.8in
\mJ = \left ( \begin{matrix} 0 & -I \cr I & 0 \cr \end{matrix} \right )
\eea
The unitary subgroup $U(g)$  is generated by setting $C=-B$ and $ D=A$ subject to the conditions $A^tA+B^tB=I$ and $A^tB-B^tA=0$. Since the group $U(g)$ contains a $U(1)$ factor, $\cH_g$ is a K\"ahler manifold, with the following $Sp(2g,\RR)$-invariant K\"ahler metric,
\bea
\label{A5}
ds^2_g = \sum_{I,J,K,L=1}^g (Y^{-1})^{IJ} \, d \Omega _{IK} \, (Y^{-1} )^{KL} \, d \bar \Omega _{JL}
\eea
The associated Laplace-Beltrami operator  $\Delta$ on scalar functions of $\cH_g$ is given by,
\bea
\label{A6}
\Delta = \sum _{I,J,K,L=1}^g 4 \, Y_{IJ} Y_{KL} \, \p^{IK} \, \bar \p^{JL}
\eea
where the derivatives with respect to the components of $\Omega_{IJ}$ are defined as follows,
\bea
\p^{II} = { \p \over  \p\Omega _{II}}
\hskip 1in
\p^{IJ} = \half { \p \over  \p \Omega _{IJ}} \qquad \hbox{ for }  J \not= I
\eea
For genus one, $\cH_1=\cH$ coincides with the Poincar\'e upper half-plane, $ds_1^2$ with the Poincar\'e metric on $\cH_1$, and $\Delta$ with the Laplace-Beltrami  operator on scalars functions on $\cH_1$.

\subsection{The Riemann theta function}

The \textit{Riemann $\tet$-functions} are complex-valued functions  on $\cH_g \times \CC^g$ that generalize the Jacobi $\tet$-functions on $\cH \times \CC$ to arbitrary rank $g$. Just as for  Jacobi $\tet$-functions, Riemann $\tet$-functions  may be considered for arbitrary complex characteristics $\delta = [\delta' |\delta '']$ with $\delta ', \delta '' \in \CC^{2g}$. In terms of local coordinates $\Omega$ on  $\cH_g$ and $\zeta = (\zeta _1, \cdots, \zeta _g)^t $ on $ \CC^g$, the Riemann $\tet$-function for characteristic $\delta$ may be defined by the following absolutely convergent series,
\bea
\tet [\delta] (\zeta| \Omega)
\equiv
\sum _{n \in \ZZ^g }
\exp \biggl (i \pi (n + \delta ') ^t \Omega (n+ \delta ') + 2\pi i (n+\delta ') ^t  (\zeta + \delta '') \biggl ) \, .
\eea
Upon shifting by $m,  n \in \ZZ^g$ we have the periodicity relations,
\bea
\tet [\delta] (\zeta + m + \Omega n| \Omega ) & = &
e^{ -i \pi n^t \Omega n - 2 \pi i n^t (\zeta + \delta ') + 2 \pi i m^t \delta '' } \, \tet [\delta] (\zeta | \Omega)
\no \\
\tet [\delta' +n | \delta'' + m ] (\zeta | \Omega ) & = &
e^{ 2 \pi i m ^t \delta'} \,  \tet [\delta' | \delta ''] (\zeta | \Omega)
\eea
The $\tet$-function without characteristics is often denoted by $\tet (\zeta| \Omega) = \tet [0] (\zeta| \Omega)$.

\sm

We shall focus on \textit{half-integer characteristics} $\delta$ with values $\delta', \delta '' \in (\ZZ/2\ZZ) ^g$ corresponding to a spin structure. The parity $4\delta ' \cdot \delta ''$ (mod 2) of a $\tet$-function with half-integer characteristics  depends on $\delta$ and, at a point $(\zeta | \Omega)$ where $\tet [\delta ](\zeta |\Omega) \not=0$,  is defined by,
\bea
\tet [\delta ] (- \zeta | \Omega ) = (-1) ^{4 \delta ' \cdot \delta ''} \tet [\delta ](\zeta | \Omega)
\eea
The spin structure $\delta$ is referred to as an even or odd spin structure according to whether the integer $4\delta ' \cdot \delta ''$ is even or odd.

\subsubsection{Modular transformations}

The discrete (or \textit{arithmetic}) subgroup $Sp(2g,\ZZ)$ of $Sp(2g, \RR)$ is the group of modular transformations,  whose action on the Siegel upper half-space $\cH_g$ is given by (\ref{A3}), but now with $M \in Sp(2g, \ZZ)$ with the entries of the matrices $A,B,C,D$ being integers.  Under a modular transformation $M \in Sp(2g,\ZZ)$, a half-integer characteristic $\delta = [\delta ' | \, \delta '']$ transforms into another half-integer  characteristic  $\ti \delta = [\ti \delta ' | \, \ti \delta '']$ given as follows,
\bea
\left ( \begin{matrix}  \tilde \delta ' \cr \tilde \delta '' \end{matrix} \right )
\left ( \begin{matrix}D & -C \cr -B & A \cr \end{matrix} \right )
\left ( \begin{matrix} \delta ' \cr \delta '' \cr \end{matrix} \right )
+ \half \, {\rm diag} \left ( \begin{matrix} CD^t \cr AB^t \cr \end{matrix} \right )
\hskip .7 in
M= \left ( \begin{matrix}A & B \cr C & D \cr \end{matrix} \right )
\eea
Under joint modular transformations of $\Omega$ and $\delta$, the $\tet$-function transforms as follows,
\bea
\tet [\tilde \delta ] \left (\{ (C\Omega +D)^{-1} \}^t  \zeta | \tilde \Omega \right ) =
\ep (\delta, M) \, \det (C\Omega + D)^{\half} \, \tet [\delta ](\zeta | \Omega)
\eea
where $\ep(\delta, M) ^8=1$ and the precise dependence of $\ep ( \delta, M)$ on its arguments  is given in [341, 346].

\subsubsection{Riemann relations on $\tet$-functions}

For each spin structure $\lambda$, there exists a  Riemann relation which may be expressed as a quadrilinear sum over all spin structures $\kappa$,
\be
\label{D.RR}
\sum _\kappa \<\kappa | \lambda \>
\tet [\kappa ](\zeta _1 ) \tet [\kappa ](\zeta _2)
\tet [\kappa ](\zeta _3 ) \tet [\kappa ](\zeta _4)
= 4\,
\tet [\lambda ] (\zeta _1 ') \tet [\lambda ] (\zeta _2 ')
\tet [\lambda ] (\zeta _3 ') \tet [\lambda ] (\zeta _4 ')
\ee
The signature symbol for the pairing of two spin structures is defined by,
\bea
\< \kappa | \lambda \> = e^{4 \pi i ( \kappa' \cdot  \lambda'' - \kappa '' \cdot \lambda ')}
\eea
and the relation between the vectors $\zeta$ and $\zeta '$ is given in terms of a matrix $\Lambda$,
\be
\left ( \begin{matrix}
\zeta _1 ' \cr  \zeta _2 ' \cr \zeta _3 ' \cr \zeta _4 ' \cr \end{matrix}  \right )
= \Lambda
\left ( \begin{matrix}
\zeta _1  \cr  \zeta _2  \cr \zeta _3  \cr \zeta _4  \cr \end{matrix} \right )
\qquad \qquad
\Lambda =
\half \left (\begin{matrix}
1 &  1 &  1 &  1 \cr
1 &  1 & -1 & -1 \cr
1 & -1 &  1 & -1 \cr
1 & -1 & -1 &  1 \cr \end{matrix} \right )
\ee
which satisfies $\Lambda ^2 = I$. In the special case where $\zeta = \zeta '=0$, only even spin structures  $\kappa =\delta $ contribute to the sum and we recover a Riemann identity for each odd spin structure $\lambda = \nu$,
\be
\sum _\delta \<\nu |\delta \> \tet [\delta ]^4 (0,\Omega )  =0
\ee
More generally, if $\zeta_1+\zeta_2+\zeta_3+\zeta_4=0$ then $\zeta_1'=0$,  the right side of (\ref{D.RR}) vanishes for any odd spin structure $\lambda$, and the Riemann relations become,
\be
\sum _\kappa \<\kappa | \lambda \>
\tet [\kappa ](\zeta _1 ) \tet [\kappa ](\zeta _2)
\tet [\kappa ](\zeta _3 ) \tet [\kappa ](\zeta _4)
= 0
\ee
For genus one, there is only a single odd spin structure and the Riemann relation reduces to the famous Jacobi identity $\tet_2^4 - \tet _3^4 +\tet_4^4 =0$.

\subsection{The Jacobian, Abel map, and Riemann vanishing theorem}

We recall from appendix \ref{sec:RS} that a canonical basis may be chosen for the holomorphic $(1,0)$-forms $\om_I$ with  $I=1,\cdots, g$ on a compact Riemann surface of genus $g$ by normalizing the integrals of $\om_I$  on the $\mA$-cycles of a canonical basis $(\mA, \mB)$ of generators for the homology group $H_1 (\Sigma, \ZZ)$. The integrals on the $\mB$-cycles then give the period matrix $\Omega$ of the Riemann surface $\Sigma$ in the homology basis $(\mA, \mB)$,
\bea
\oint _{\mA_I} \om _J = \delta _{IJ}
\hskip 1in
\oint _{\mB_I} \om _J = \Omega _{IJ}
\eea
Under a symplectic change of canonical homology basis, the holomorphic $(1,0)$-forms $\om$ and period matrix $\Omega$ transform as given in (\ref{B.3.4}).  In view of the Riemann relations, we have $\Omega ^t = \Omega$ and $\Im (\Omega) >0$, so that the period matrix of a compact Riemann surface takes values in the Siegel half space $\cH_g$. The set of $\mA$ and $\mB$ periods defines a lattice $\Lambda = \ZZ^g + \Omega \ZZ^g$ in $\CC^g$, whose quotient is an Abelian variety referred to as the the Jacobian variety,
\bea
J(\Sigma) = \CC^g / \{ \ZZ^g + \Omega \ZZ^g\}
\eea
In other words, the Jacobian is a $g$-dimensional complex torus. For genus $g=1$, this torus gives an equivalent representation of the Riemann surface itself. For higher genus $g \geq 2$, this simple correspondence no longer holds, but is replaced by a more subtle identification produced by the Abel map.

\sm

Given a base point $z_0 \in \Sigma$, the Abel map sends a divisor $D$ of $n$ points $z_i \in \Sigma$ with weights $q_i \in \ZZ$ for $i=1,\cdots,n$, formally denoted by  $D=q_1 z_1 + \cdots  q_n z_n$, into $\CC^g$ by,
\bea
\label{abelmap}
D= q_1 z_1 + \cdots + q_n z_n
\to
\sum _{i=1} ^n q_i \int _{z_0} ^{z_i} (\omega _1, \cdots , \omega _g)
\eea
where the $g$-tuple $(\omega _1, \cdots , \omega _g)$ stands for the vector of holomorphic $(1,0)$-forms $\om_I$.
The Abel map into $\CC^g$ is multiple valued because each integral $\int _{z_0} ^{z_i} \om_I$ is multiple-valued in $\CC$, but the Abel map is single valued as a map into the Jacobian $J(\Sigma)$.

\sm

{\thm The Riemann vanishing theorem states that $\tet(\zeta|\Omega)=0$ if and only if there exist $g-1$ points $p_1, \cdots, p_{g-1} \in \Sigma$ such that,
\bea
\label{zetaDel}
\zeta = p_1+\cdots p_{g-1} - \Delta (z_0)
\hskip 1in
\zeta _I =  \sum_{i=1}^{g-1} \int _{z_0} ^{z_i} \om_I - \Delta _I(z_0)
\eea
where $\Delta _I (z_0)$ are the components of the Riemann vector for base-point $z_0$, given by,
\bea
\Delta _I (z_0) = - \half - \half \Omega_{II} + \sum_{J\not= I} \oint _{\mA_J} \om_J(z) \int ^z _{z_0} \om_I
\eea
The combination $\zeta$ is independent of the base-point. }

The proof of the Riemann vanishing theorem may be found in any book on $\tet$-functions, or in the physics literature in [190]. For the special case of genus $g=1$, the set of points $p_i$ is empty, and the Riemann vector reduces to $\Delta = - \half - \half \tau$ which merely states that $\tet ( \thalf + \thalf \tau |\tau)=0$, a fact we established long ago.

\sm

An important application of the Riemann vanishing theorem is to the existence of the following holomorphic $(1,0)$-forms, defined for any odd spin structure $\nu$ by
\bea
\label{omnu}
\om_\nu(z) = \sum _I \om _I(z) \p^I \tet [\nu] (0|\Omega )
\eea
Its $2(h-1)$ zeros are all double zeros at points $p_i$ such that $\zeta_I=\nu_I$ in (\ref{zetaDel}).  Therefore, $\om_\nu(z)$ admits a holomorphic square root $h_\nu(z)$. Since the square of $h_\nu$ is  $\om_\nu(z)$, which is a section of the canonical bundle, $h_\nu$ is a section of a spin bundle with spin structure $\nu$. For each odd spin structure $\nu$, the holomorphic $(\half, 0)$-form $h_\nu$ is unique up to a sign.

\subsection{The prime form}

To construct the  \textit{prime form} we consider an arbitrary  odd spin structure $\nu$. Its associated holomorphic section $h_\nu(z)$ of the spin bundle with spin structure $\nu$ has $g-1$ zeros. The prime form is a $(-\thalf,0)$-form in both variables $z$ and $w$, defined by,
\bea
E(z,w|\Omega) = {\tet [\nu ] (z-w| \Omega) \over h_\nu (z) h_\nu (w)}
\eea
The argument $z-w$ of the $\tet$-function stands for the Abel map of (\ref{abelmap}) with divisor  $D=z-w$. The form $E(z,w|\Omega)$ defined this way is  independent of $\nu$. It is holomorphic in $z$ and $w$ and has a unique simple zero at $z=w$. It is single valued when $z$ is moved around $\mA_I$ cycles, but has non-trivial monodromy when $z\to z+\mB_I$ is moved around $\mB_I$ cycles,
\bea
E(z+\mA_I,w|\Omega) & = &E(z,w|\Omega)
\no \\
E(z+\mB_I,w |\Omega) & = & - \exp \biggl ( -i \pi \Omega _{II} + 2 \pi i \int ^z _w \!
\omega _I \biggr ) E(z,w|\Omega)
\eea
Therefore, it is properly defined on the simply connected covering space of $\Sigma \times \Sigma$, or it may be defined in a fundamental domain for $\Sigma$ in the complex plane.

\subsection{Holomorphic differentials}

Combining the Riemann-Roch and vanishing theorems, the dimension of the space of holomorphic $(n,0)$-forms is denoted by $\Upsilon(n) = \dim {\rm Ker} \, \pbz ^{(n)}$ and is given by,
\be
\Upsilon (n)
\left \{
\begin{matrix}
0 & n < 0\, , \ {\rm and} \ n=1/2 \ {\rm even \ spin \ structure} \cr
1 & n=0\, ,   \ {\rm and} \ n=1/2 \ {\rm odd \ spin \ structure}  \cr
g & n=1 \cr  (2n-1) (g-1) & n \geq 3/2, \ {\rm and} \, g\geq 2  \end{matrix}
\right .
\ee
The dimensions listed for $n=1/2$ are for generic moduli and are valid for exceptional moduli mod 2. A basis  of holomorphic differentials is denoted by $\phi _a ^{(n)}$, $a=1, \cdots , \Upsilon (n)$. They are holomorphic sections of the line bundles $K^n$, the $n$-th power of the canonical bundle $K$, for which the number of zeros is given by,
\be
\# \ {\rm zeros}  \ \phi ^{(n)} _a = c_1(K^n) = 2n(g-1)
\ee
For a given $a$, there are $\Upsilon(n)-1$ zeros fixed to be $z_b $ with $b \not=a$, and another $g$ zeros due to the Riemann vanishing theorem, giving a total of $2n(g-1)$, in agreement with the value of the first Chern class.
For $n=0$, they are constants, for $n=1/2$ and $\nu $ odd they are denoted by $h_\nu (z)$, while for $n=1$ they are the holomorphic Abelian differentials  $\omega _I$, for $I=1,\cdots ,g$.

\sm

Given any set of $\Upsilon (n)$ points $z_1, \cdots , z_{\Upsilon (n)}$ on the surface, we may choose a basis $\hat \phi ^{(n)} _a$ for the holomorphic $n$-differentials normalized at the points $z_b$ by,
\be
\label{diffnorm}
\hat \phi _a ^{(n)} (z_b) = \delta _a ^b
\ee
The holomorphic differentials with this normalization may be exhibited explicitly in terms of the prime form $E(z,w)$, the $h/2$ differential $\sigma (z)$ and $\vartheta$-functions. For $n \geq 3/2$, we have,
\be
\label{diffexp}
\hat \phi _a ^{(n)} (z)
{\vartheta [\delta ](z-z_a + \sum z_b -(2n-1)\Delta) \over
\vartheta [\delta ] ( \sum z_b -(2n-1)\Delta)}
{\prod _{b\not= a} E(z,z_b) \over \prod _{b\not= a} E(z_a,z_b)}
\biggl ( {\sigma (z) \over \sigma (z_a)} \biggr ) ^{2n-1}
\ee
Here, $\sigma (z)$ is a holomorphic $(\tfrac{h}{2}, 0)$-form without zeros or poles defined, up to a constant, by the following ratio,
\bea
{\sigma (z) \over \sigma (w)} & = &
{\vartheta (z-\sum p_i +\Delta) \over \vartheta (w-\sum p_i +\Delta)}
\prod _{i=1} ^h {E(w,p_i) \over E(z,p_i)}
\eea
where $p_i$, $i=1,\cdots , h$ are arbitrary points on the surface. Note that $\sigma(z)$ is single valued around $\mA_I$ cycles but multivalued around $\mB_I$ cycles in the following way
\bea
\sigma (z+\mB_I) &=& \sigma (z) \exp \bigl \{ -i\pi (h-1) \Omega _{II} + 2 \pi i
\Delta _{Iz} \bigr \}
\eea
Besides the $\Upsilon(n)-1$ zeros $z_b$, $b\not=a$, the differential $\hat \phi _a ^{(n)}(z)$ has $h$ additional zeros. The differential  $\hat \phi _a ^{(n)}(z)$ is an $(n,0)$-form in $z$, a $(-n,0)$-form in $z_a$ and a $(0,0)$-form $z_b$ with $b\not=a$. For $n=1$, we have,
\be
\hat \phi ^{(1)} _a (z)
{\tet  (z-z_a  + \sum z_b -w_0 - \Delta ) \over
\tet (  \sum z_b -w_0 - \Delta ) \ E(z,w)} \
\prod _{b\not= a} {E(z,z_b) \over E(z_a,z_b)}
{ E(z_a,w_0) \over E(z, w_0)}
\ {\sigma (z) \over \sigma (z_a)} \, .
\ee

\subsection{Meromorphic differentials}

Meromorphic differentials play the role of Green functions for the Cauchy-Riemann operators acting on holomorphic line bundles. The  Green function $G_{n}(z,w)= G_n (z,w;z_1,\cdots, z_{\Upsilon(n)})$ for the operator $\p _{\bar z} ^{(n)}$ is a meromorphic $(n,0)$ form in $z$ and $(1-n,0)$-form in $w$. When $n\geq 3/2$ for general spin structure $\delta$  and $n=1/2$ for even spin structure $\delta$, the Green function $G_n$ may be defined by the following relations,
\bea
\p _{\bar z} ^{(n)} G _n (z,w)  & = & + 2 \pi \delta (z,w)
\\
\p _{\bar w} ^{(1-n)} G_n(z,w) & = & - 2 \pi \delta (z,w)
+ 2\pi \sum _{a=1} ^{\Upsilon (n)} \hat \phi _a ^{(n)} (z) \delta (w,z_a)
\eea
The properly normalized holomorphic $n$-differentials $\hat \phi _a ^{(n)}$ are defined in (\ref{diffnorm}) and (\ref{diffexp}). Setting $z=z_a$, we have $\p _{\bar w} ^{(1-n)} G_n(z_a,w)=0$, so that $G_n(z_a,w)=0$.
Explicit expressions for the Green's function are
\be
G_n (z,w)
{\vartheta [\delta ]\big (z-w + \sum z_b -(2n-1)\Delta \big ) \over
\vartheta [\delta ]\big ( \sum z_b -(2n-1)\Delta \big ) \ E(z,w)} \
{\prod _{a} E(z,z_a) \over \prod _{a} E(w,z_a)} \
\biggl ( {\sigma (z) \over \sigma (w)} \biggr ) ^{2n-1}
\ee
For $n=1/2$, this reduces to the standard form of the Szeg\" o kernel, usually denoted by
\be
S_\delta (z,w) =
{\vartheta [\delta ]\big (z-w  \big ) \over
\vartheta [\delta ]\big ( 0 \big ) \ E(z,w)}
\ee
For $n=1$, the Green's function $G_1(z,w)=G_1(z,w;z_1,\cdots ,z_h, w_0)$
is the Abelian differential of the third kind, satisfying
\bea
\p _{\bar z} ^{(1)} G _1 (z,w)  & = & + 2 \pi \delta (z,w) - 2\pi
\delta (z,w_0)
\\
\p _{\bar w} ^{(0)} G_1 (z,w) & = & - 2 \pi \delta (z,w)
+ 2\pi \sum _{a=1} ^h \hat \phi _a ^{(1)} (z) \ \delta (w,z_a)
\eea
and explicitly given by the following expression
\be
{\vartheta \big (z-w-w_0 + \sum z_b - \Delta \big ) \over
\vartheta \big ( -w_0 + \sum z_b - \Delta \big ) \ E(z,w)} \
{\prod _{a} E(z,z_a) E(w,w_0) \over \prod _{a} E(w,z_a) E(z, w_0)} \
{\sigma (z) \over \sigma (w)} \, .
\ee
The combination $\p_z \p_w \ln E(z,w)$ is a meromorphic differential (Abelian of the second kind) with a single double pole at
$z=w$. Its integrals around homology cycles are given by
\bea
\label{prf}
\oint _{\mA_I} \! dz \p_z \p_w \ln E(z,w) & = & 0
\nonumber \\
\oint _{\mB_I} \! dz \p_z \p_w \ln E(z,w) & = & 2 \pi i \omega _I(w)
\eea
and will be of use throughout.

\subsection{The  $b c$ system}
\label{app.bc}

A useful unification of the formulas for holomorphic and meromorphic forms is provided by the conformal field theory correlators of anti-commuting fields $b$ and $c$ of weights $(n,0)$ and $(1-n,0)$.  This quantum field theory was already introduced in section \ref{sec:5.bc} for a Riemann surface of genus 1, namely a torus. Here,  we shall  generalize the correlators to arbitrary genus $g \geq 2$. Since the roles of $b$ and $c$ are swapped by letting $n \to 1-n$, we shall  restrict to considering the cases $n \geq \thalf$ without loss of generality.

\sm

As in the case of the torus, the fields $b,c$ are locally holomorphic,
\bea
\pbz b = \pbz c=0
\eea
However, as  $b$ and $c$  fields approach one another poles may develop. The poles may be represented schematically by the singular contribution to the operator product expansion, given by a simple pole with unit residue,
\bea
b(z) \, c(w) \sim { 1 \over z-w}
\eea
All poles must arise from these operator coincidences.  In view of the anti-commuting nature of the fields, the operator product of two like fields produces a simple zero,
\bea
b(z_1) b(z_2) & \sim & (z_1-z_2) \, \p b(z_2)  \ b(z_2)
\no \\
c(w_1) c(w_2) & \sim & (w_1-w_2) \, \p c(w_2) \, c(w_2)
\eea
In the case of the torus we used Fourier analysis to solve for the fields and their correlators. Since higher-genus Riemann surfaces have no continuous isometries, such as translation symmetry, Fourier analysis is no longer applicable.

\sm

Now consider the correlator of an arbitrary number of the $b$ and $c$ fields of weight $(n,0)$ and $(1-n,0)$ respectively,
\bea
\< b(z_1) \cdots b(z_N) \, c(w_1) \cdots c(w_M)\>
\eea
Viewed as a function of $z_1$, the correlator has simple zeros at $z_2, \cdots, z_N$ and simple poles at $w_1, \cdots, w_M$.  But since all poles arise from operator coincidences, and no other poles can appear, the number of poles in $z_1$ is exactly $M$, and thus the $n$-form must have $M+2n(g-1)$ zeros. Now $N-1$ of these zeros are specified by the positions of the points $z_2, \cdots, z_N$, leaving an extra $g$ zeros due to the Riemann vanishing theorem, for a total of $N-1+g$ zeros. Equating the number of zeros computed in these two different ways gives,
\bea
\label{D.const}
\eea
The correlator must vanish unless the number of $b$ and $c$ fields satisfies this relation.

\subsubsection{The case $n = \thalf$ with even spin structure}

In the case $n = \half$ with even spin structure $\delta$ and generic moduli,  neither the $b$ nor the $c$ field has zero modes.
By using Wick contractions, the correlator is given by,
\bea
\< b(z_1) \cdots b(z_M) \, c(w_M) \cdots c(w_1)\> = \det  S_\delta (z_i, w_j)
\eea
where $S_\delta$ is the Szeg\"o kernel for even spin structure $\delta$, the determinant is taken of the $M \times M$ matrix $S(z_i, w_j)$, and the numbering of the points $w_j$ has been chosen  in descending order so that the sign multiplying the determinant on the right side is positive.  An alternative formula is obtained by matching poles and zeros in each variable, and we obtain,
\bea
\< b(z_1) \cdots b(z_M) \, c(w_M) \cdots c(w_1) \> =
{ \tet [\delta] (D|\Omega) \over \tet [\delta] (0|\Omega) } { \prod _{i<j} E(z_i,z_j) E(w_i,w_j) \over \prod _{i,j} E(z_i, w_j)}
\eea
where the divisor is $D=(z_1-w_1)+\cdots + (z_M - w_M)$. This formula is often referred to as the \textit{bosonized correlator}  as it may be obtained by representing the spin-$\thalf$ fermion fields $b,c$ in terms of the chiral half of a complex scalar field.

\subsubsection{The case $n = \thalf$ with odd spin structure}

In the case $n=\half$ with  odd spin structure $\nu$ and generic moduli, there is a single zero mode $h_\nu$ of the Dirac equation. Clearly, the two-point correlator $\<b (z) c(w)\>$  must be saturated by the zero modes and be proportional to $h_\nu(z)h_\nu(w)$.  For higher point functions, we need a Green function. The Green function may be defined by an equation that is orthogonal to the Dirac zero mode. Such an equation is not unique, but different definitions will produce the same correlator. Preserving meromorphicity, we choose the following definition,
\bea
\pbz S_\nu(z,w) = 2 \pi \delta(z,w) - 2 \pi \delta (z,w_0) { h_\nu (w) \over h _\nu (w_0)}
\eea
which is orthogonal to $h_\nu(z)$ by construction. The general correlator is then given by,
\bea
\< b(z_1) \cdots b(z_M) \, c(w_M) \cdots c(w_1) \>
= \sum _{i,j=1}^M (-)^{i+j} \< b(z_i) c(w_j) \> \, \det S_\nu (z_k, z_l) \Big |_{{k \not= i, \atop  l \not= j}}
\eea
The constant of proportionality in $\<b (z) c(w)\> \sim h_\nu(z)h_\nu(w)$ is given by the functional determinant of the Dirac operator on $n=\half$ sections with odd spin structure $\nu$.

\subsubsection{The case $n = 1$}

In any non-vanishing correlator for $n=1$, the $b$ field has $g$ zero modes, while the $c$ field has one zero mode, namely the constants.  We shall normalize the correlator so that the simplest non-zero amplitude is given by,
\bea
\cG_0(z_1, \cdots, z_g; w) = \< b(z_1) \cdots b(z_g) c(w) \> = \det \om_I (z_i)
\eea
where the determinant is taken of the $g \times g$ matrix with entries $\om_I(z_i)$ for $i, I=1,\cdots, g$. The correlator with an arbitrary number of $b,c$ fields subject to the condition (\ref{D.const}),
\bea
\cG_M(z_1, \cdots w_{M+1} ) = \< b(z_1) \cdots b(z_{g+M})\,  c(w_1) \cdots c(w_{M+1}) \>
\eea
with this normalization is given by
\bea
\cG_M (z_1,  \cdots, w_{M+1}) =  { \tet \left ( D \right )  \over Z^3 }  \,
{ \prod _{i<j} E(z_i,z_j) \prod _{a<b} E(w_a,w_b) \prod _i \sigma (z_i)  \over \prod _{i,a} E(z_i,w_a) \prod _a \sigma (w_a)  }
\label{defcgn}
\quad
\eea
where $i,j =1, \cdots, g+M$ and $a,b=1,\cdots, M+1$ and $D$ is defined by
\bea
D = \sum _{i=1}^{g+M}  z_i - \sum _{a=1}^{M+1} w_a - \Delta
\eea
The factor $Z$ is the chiral boson partition function  determined by setting $M=0$,
\bea
\det \om_I(p_i) = { \tet \left ( p_1 + \cdots p_g -q -\Delta \right )  \over Z^3 }  \,
{ \prod _{i<j} E(p_i,p_j) \prod _i \sigma (p_i)  \over \prod _{i} E(p_i,q) \sigma (q)  }
\eea
for arbitrary points $q$ and $p_i$ with $i=1,\cdots, g$.
With this normalization the correlator transforms under modular transformations by,
\bea
\tilde \cG_M(z_1, \cdots, w_{M+1}) = \det (C \Omega +D)^{-1} \cG_M(z_1, \cdots, w_{M+1})
\label{trflawG}
\eea

\subsubsection{The case $n \geq \tfrac{3}{2}$}

For $n \geq \tfrac{3}{2}$, the $b$ field has $\Upsilon = (2n-1)(g-1)$ zero modes, while the $c$ field has no zero modes.
We have the following bosonization formula,
\bea
\< b(z_1) \cdots b(z_{M+\Upsilon}) \, c(w_1) \cdots c(w_M) \> =
{ \tet [\delta] (\zeta |\Omega) \over Z} { \prod _{a<b} E(z_a,z_b) \prod _{i<j} E(w_i,w_j) \prod _a \sigma (z_a) \over \prod _{i,a} E(z_a, w_i) \prod _i \sigma (w_i)}
\no \\
\eea
where $a,b=1,\cdots , M+ \Upsilon$, while $i,j=1, \cdots , M$ and $\zeta$ is given by,
\bea
\zeta = \sum _{a=1}^{M + \Upsilon} z_a - \sum _{i=1}^M w_i - (2n-1) \Delta
\eea
In the special case where $M=0$, we recover a purely holomorphic correlator, which is entirely saturated by the zero modes of the field $b$,
\bea
\< b(z_1) \cdots b(z_\Upsilon)  \> =
{ \tet [\delta] (\zeta |\Omega) \over Z}  \prod _{a<b} E(z_a,z_b) \prod _a \sigma (z_a)
\eea

\subsection*{$\bullet$ Bibliographical notes}

Classic introductions to Siegel modular forms and higher rank modular geometry may be found in the books by Siegel~[347, 348], Igusa~[346], and Klingen~[349]. Useful lecture notes on higher rank modular forms are those by van der Geer~[350]. The connection between higher rank $\tet$-functions and Riemann surfaces is spelled out in the classic book by Fay~[341]. The connection between higher rank $\tet$-functions and non-linear equations may be found in the survey article by Dubrovin [351]. The solution of the $bc$ quantum system in terms of $\tet$-functions, using the work of [341], was given in [60].

\newpage

\providecommand{\href}[2]{#2}\begingroup\raggedright\begin{thebibliography}{100}

[1]
A.~Erd\`elyi, ed., \emph{Higher transcendental functions}, vol.~2 of
\emph{Bateman Manuscript Project}, Robert E. Krieger Publishing Company

[2]
A.~Weil, \emph{Elliptic functions according to Eisenstein and Kronecker},
vol.~88, Springer Science \& Business Media (1999).

[3]
E.T.~Whittaker and G.N.~Watson, \emph{A course of modern analysis}, Cambridge
University press (1920).

[4]
D.~Mumford, \emph{{Tata lectures on Theta. I (modern Birkh{\"a}user
classics)}}, Birkh{\"a}user Boston Incorporated (2007).

[5]
S.~Lang, \emph{Elliptic functions},  in \emph{Elliptic functions}, pp.~5--21,
Springer (1987).

[6]
F.~Oberhettinger and W.~Magnus, \emph{{Anwendung der elliptischen Funktionen in
Physik und Technik}}, vol.~55, Springer-Verlag (2013).

[7]
G.~K{\"o}hler, \emph{Eta products and theta series identities}, Springer
Monographs in Mathematics, Springer Science \& Business Media (2011).

[8]
T.A.~Apostol, \emph{Modular Functions and Dirichlet Series in Number Theory},
Graduate Texts in Mathematics, number 41, Springer Verlag (1976).

[9]
S.~Lang, \emph{Introduction to Modular Forms}, vol.~222 of \emph{Grundlehren
der mathematischen Wissenschaften}, Springer-Verlag (1976).

[10]
D.~Zagier, \emph{Elliptic modular forms and their applications},  in \emph{The
1-2-3 of modular forms}, pp.~1--103, Springer (2008).

[11]
G.~Shimura, \emph{Introduction to the arithmetic theory of automorphic
functions}, vol.~1, Princeton university press (1971).

[12]
N.I.~Koblitz, \emph{Introduction to elliptic curves and modular forms},
vol.~97, Springer Science \& Business Media (2012).

[13]
F.~Diamond and J.M.~Shurman, \emph{A first course in modular forms}, vol.~228
of \emph{Graduate Texts in Mathematics}, Springer (2005).

[14]
A.~Erd\`elyi, ed., \emph{Higher transcendental functions}, vol.~3 of
\emph{Bateman Manuscript Project}, Robert E. Krieger Publishing Company

[15]
H.~Iwaniec, \emph{Topics in classical automorphic forms}, vol.~17 of
\emph{Graduate Studies in Mathematics}, American Mathematical Soc. (1997).

[16]
D.~Bump, \emph{Automorphic Forms and Representations}, vol.~55 of
\emph{Cambridge Studies in Advanced Mathematics}, Cambridge University Press

[17]
K.~Ono, \emph{The Web of Modularity: Arithmetic of the Coefficients of Modular
Forms and $q$-series}, vol.~102 of \emph{CBMS}, American Mathematical Society

[18]
S.~Lang, \emph{SL(2,R)}, vol.~105, Springer Science \& Business Media (2012).

[19]
T.~Kubota, \emph{Elementary theory of Eisenstein series}, Halsted Press (1973).

[20]
A.~Terras, \emph{Harmonic analysis on symmetric spaces and applications I},
Springer Science \& Business Media (1985).

[21]
A.~Terras, \emph{Harmonic analysis on symmetric spaces and applications II},
Springer Science \& Business Media (1988).

[22]
H.~Iwaniec, \emph{Spectral methods of automorphic forms}, vol.~53 of
\emph{Graduate Studies in Mathematics}, American Mathematical Soc. (2002).

[23]
H.~Iwaniec and E.~Kowalski, \emph{Analytic number theory}, vol.~53, American
Mathematical Soc. (2021).

[24]
P.~Fleig, H.P.A.~Gustafsson, A.~Kleinschmidt and D.~Persson, \emph{{Eisenstein
series and automorphic representations}}, Cambridge University Press (6,
2018), [\href{https://arxiv.org/abs/1511.04265}{{\ttfamily 1511.04265}}].

[25]
I.~Gel'land, M.~Graev and I.~Pyatetskii-Shapiro, \emph{Representation theory
and automorphic functions}, {\emph{(Transl. from Russian) Philadelphia:
Saunders} (1969) }.

[26]
F.~Brown, \emph{A class of non-holomorphic modular forms iii: real analytic
cusp forms for $sl_2(\mathbb{Z})$}, {\emph{arXiv preprint arXiv:1710.07912}
(2017) }.

[27]
N.~Benjamin, S.~Collier, A.L.~Fitzpatrick, A.~Maloney and E.~Perlmutter,
\emph{{Harmonic analysis of 2d CFT partition functions}},
\href{https://doi.org/10.1007/JHEP09(2021)174}{\emph{JHEP} {\bfseries 09}
(2021) 174} [\href{https://arxiv.org/abs/2107.10744}{{\ttfamily

[28]
S.~Collier and E.~Perlmutter, \emph{{Harnessing S-Duality in $\mathcal{N}=4$
SYM \& Supergravity as $SL(2,\mathbb{Z})$-Averaged Strings}},
\href{https://arxiv.org/abs/2201.05093}{{\ttfamily 2201.05093}}.

[29]
H.~Paul, E.~Perlmutter and H.~Raj, \emph{{Integrated Correlators in
$\mathcal{N}=4$ SYM via $SL(2,\mathbb{Z})$ Spectral Theory}},
\href{https://arxiv.org/abs/2209.06639}{{\ttfamily 2209.06639}}.

[30]
D.~Goldfeld, \emph{Automorphic Forms and L-functions for the Group
$GL(2,\RR)$}, vol.~99 of \emph{Cambridge Studies in Advanced Mathematics},
Cambridge University Press (2006).

[31]
M.~Viazovska, \emph{{The sphere packing problem in dimension $8$}},
\href{https://doi.org/10.4007/annals.2017.185.3.7}{\emph{Annals of
Mathematics} {\bfseries 185} (2017) 991 }.

[32]
H.~Cohn, A.~Kumar, S.~Miller, D.~Radchenko and M.~Viazovska, \emph{{The sphere
packing problem in dimension $24$}},
\href{https://doi.org/10.4007/annals.2017.185.3.8}{\emph{Annals of
Mathematics} {\bfseries 185} (2017) 1017 }.

[33]
D.~Zagier, \emph{Ramanujan's mock theta functions and their applications},