The Gaussian factor is universal in the sense that it is independent of the momenta $k_i$ and polarization tensors $\ep_i$ of the external states and actually independent of the specific string theory under consideration. Modular invariance allows one to choose the fundamental domain $\cM_1$ for $\cH / SL(2,\ZZ)$ that is adapted to the particular choice for the flow of loop momentum. For our choice of loop momentum traversing the $\mA$-cycle, the choice of fundamental domain is the standard one, $\cM_1 = \{ \tau \in \cH, ~ |\tau|\geq 1, ~ |\Re(\tau)| \leq \half \}$. The high energy behavior of the loop momentum is governed by the magnitude of $\Im (\tau)$, which is uniformly bounded away from zero by our choice of fundamental domain. The uniform Gaussian suppression at large loop momenta implies UV finiteness as all other factors are either only exponential or polynomial in $p$.
Higher genus amplitudes have a generalization of the above factor which involves the period matrix of the corresponding higher genus surface, but the structure is otherwise analogous to the one-loop case.

\subsection{Effective interactions from the four-graviton amplitude}
\label{sec:12.7}

The tree-level, one-loop, and two-loop string amplitudes listed in (\ref{12.amps})  for four gravitons have important applications to the evaluation of string theory corrections to supergravity in the low energy limit. Type~IIB supergravity and its relation to Type~IIB superstring theory will be reviewed in section \ref{sec:IIB}. In preparation for those discussions and the comparison of perturbative results with the implications from supersymmetry and $SL(2,\ZZ)$-duality of Type~IIB string theory, we shall here evaluate the behavior of the partial amplitudes in (\ref{12.bamps}) and the full amplitudes in (\ref{12.amps}) in the low energy limit.

\sm

To illustrate the emergence of low energy effective interactions from superstring amplitudes, we recall the expansion in powers of the string coupling $g_s$ of the four-graviton amplitude, given in (\ref{8a1}) and repeated here for convenience,
\bea
\label{8a1a}
\cA (\ep_i; k_i) = \kappa_{10}^2 \cR^4 \sum _{h=0}^\infty g_s^{2h-2} \cA^{(h)}   (s,t,u)
\eea
where $\cR$ stands for the on-shell linearized Weyl tensor, whose expression was given in (\ref{8a2}) and (\ref{8a3}), while the expressions for $\cA^{(0)} , \cA^{(1)} , \cA^{(2)} $ were given in (\ref{12.amps}).

\subsubsection{Low energy expansion at tree-level}

We concentrate first on the tree-level contribution $\cA^{(0)} $ and use the following series expansion for the ratio of $\Gamma$-functions,
\bea
\label{12.exp}
{ \Gamma (1-s) \Gamma (1-t) \Gamma (1-u)  \over \Gamma (1+s) \Gamma (1+t) \Gamma (1+u)}
= \exp \left \{ \sum _{n=1} ^\infty { 2 \zeta (2n+1) \over 2n+1} (s^{2n+1} + t^{2n+1} + u^{2n+1}) \right \}
\eea
The argument of the exponential must clearly be odd in $s,t,u$, since the left side is mapped to its inverse under simultaneous sign reversal of $s,t,u$. The linear term cancels by $s+t+u=0$. The amplitude $\cA^{(0)} $ is a symmetric function of $s,t,u$ and may therefore be expanded in powers of the two remaining symmetric polynomials in $s,t,u$,
\bea
\sigma _2 = s^2 + t^2 + u^2
\hskip 1in
\sigma _3 = s^3 + t^3 + u^3 = 3 stu
\eea
The first few terms in the expansion in powers of $\sigma _2$ and $\sigma _3$ are given by,
\bea
\label{8a5}
\cA^{(0)}  = { 1 \over stu} + 2 \zeta (3) + \sigma _2 \, \zeta (5)
+ {2 \over 3}  \sigma _3 \, \zeta (3)^2 +\half \sigma _2^2 \, \zeta (7) + \cO(s_{ij})^5
\eea
The first term on the right side  in the expansion is non-analytic at low energy and corresponds to the exchange of massless gravitons and other massless bosons in Type IIB supergravity.  The remaining terms on the right side  are all local and successively correspond to effective interactions which are schematically of the form $D^{2k} \cR^4$ for $k=0, 2, 3$ and $4$. The effective interaction $D^2 \cR^4$ is missing in this list because of the relation $s+t+u=0$.

\subsubsection{Transcendental weight}
\label{sec:12.7.2}

An important notion that has arisen over the past few years, starting in the study of quantum field theory amplitudes, is that of \textit{transcendental weight}. A natural starting point for its definition is with zeta-values and multiple zeta-values,
\bea
\label{12.MZV}
\zeta(k) = \sum _{n=1}^\infty { 1 \over n^k}
\hskip 1in
\zeta(k_1, \cdots, k_p) = \sum _{n_1\geq  n_2 \geq \cdots \geq n_p \geq 1} { 1 \over n_1^{k_1} \cdots n_p^{k_p}}
\eea
for integer $k, k_1 \geq 2$ and $k_2, \cdots, k_p \geq 1$. When $k$ is even, $\zeta(k) \in \pi^k \QQ$. Transcendental weight provides a grading upon multiplication and one assigns weight 0 to any algebraic number, weight 1 to $\pi$, and thus weight $k$ to $\zeta (k)$ for $k$ even. By extension, one assigns weight $k$ to $\zeta (k)$ for all $k$ and weight $k=k_1+\cdots +k_p$ to the multiple zeta value $\zeta(k_1, \cdots, k_p)$.

\sm

With these assignments of transcendental weight, we observe that the argument of the exponential on the right of (\ref{12.exp}) has transcendental weight 0 provided we assign weight $-1$ to $s,t,$ and $u$. Each term in the expansion of the tree-level amplitude in (\ref{8a5}) thus has weight 3. The amplitude is said to exhibit \textit{uniform transcendentality}.

\sm

Remarkably, string theory has non-trivial transcendentality properties already at tree-level. This is in contrast with the case of quantum field theory where tree-level amplitudes have trivial transcendentality and it is only through the expansion of the $\Gamma$-functions that arise in dimensional regularization at loop level that non-trivial transcendentality arises. Uniform transcendentality, observed here for tree-level Type~II string amplitudes, is shared in quantum field theory by the maximally supersymmetric $\cN=4$ Yang-Mills theory in four dimensions. A thorough understanding of how this property arises, and for precisely which correlators it holds, remains an open problem.

\subsection{Genus one in terms of modular graph functions}

The genus-one amplitude $\cA^{(1)}$ is obtained in (\ref{12.amps}) by integrating the partial amplitude $\cB^{(1)}(s,t,u|\tau)$, which  is a modular function in $\tau$, over the genus-one moduli space $\cM_1$.  The partial amplitude in turn is given by the  integration over four copies of the torus $\Sigma$ with modulus $\tau$ in (\ref{12.bamps}).
For fixed $\tau \in \cH$, the scalar Green function $G(z|\tau)$ on the torus  is smooth throughout the torus except for the logarithmic singularity at $z=0$ given by $G(z|\tau) \approx - \ln |z|^2$. As a result, for fixed $\tau$, the integrals which define $\cB^{(1)}(s,t,u|\tau)$ as a function of $s, t, u$  are absolutely convergent for $\Re(s), \Re (t), \Re (u) <1$, and may be expanded in a Taylor series in powers of $s,t,u$ with radius of convergence $|s|, |t|, |u| <1$. Since the function $\cB^{(1)}(s,t,u|\tau)$ is symmetric in $s,t,u$, we may organize this expansion in powers of the symmetric polynomials $\sigma _2 $ and $\sigma _3$ in $s,t,u$ as we did for the tree-level amplitude,
\bea
\label{8c4}
\cB^{(1)} (s,t,u|\tau) = \sum _{p, q =0} ^\infty \cB _{(p,q)}^{(1)} (\tau) \, {\sigma _2 ^p \, \sigma _3^q \over p ! \, q !}
\eea
Since $\cB^{(1)} (s,t,u|\tau)$ is a modular function in $\tau$ each coefficient $\cB_{(p,q)}^{(1)}(\tau)$ is a modular function of $\tau$. The coefficients may be computed by expanding the exponential in the integrand of (\ref{12.bamps}) in powers of its argument to a given order $w$, which is the overall degree in the variables $s,t,u$, given by $w=2p+3q$, so that we have,
\bea
\label{8c5}
\sum _{{p, q \geq 0 \atop 2p+3q=w}}  \cB _{(p,q)}^{(1)} (\tau) \, {\sigma _2 ^p \, \sigma _3^q \over p ! \, q !} = { 1 \over w!} \prod _{i=1}^ 4 \int _\Sigma { d^2 z_i \over \Im \tau} \,  \left (
\sum _{i < j} s_{ij} G(z_i-z_j |\tau) \right )^w
\eea
The coefficients $\cB_{(p,q)}^{(1)}(\tau)$ for different values of $p,q$ satisfying $w=2p+3q$ may then be sorted out by further expanding the $w$-th power on  the right side. The expansion may be presented  graphically, and the modular function corresponding to a given graph is precisely one of the  \textit{modular graph functions} discussed in section \ref{sec:MGF}.  Each graph has four vertices, namely the points $z_i$ for $i=1,2,3,4$, and $w$ edges. An  edge may connect any pair of distinct vertices, but is not allowed to begin and end on the same vertex.

\sm

The expansion is simplified by the following observations. Any modular graph function corresponding to
\begin{itemize}
\itemsep=0in
\item a graph containing a vertex on which only a single edge begins or ends vanishes;
\item a graph which becomes disconnected upon removing one edge vanishes;
\item a graph which becomes disconnected upon removing a single vertex factorizes into the modular graph functions of the resulting connected components.
\end{itemize}
The graphs corresponding to the modular graph functions which contribute to the modular functions $\cB_{(p,q)}(\tau)$ up to weight $w=5$ are presented in Figure \ref{fig:8}. Instead of organizing the graphs by weight $w$, they may alternatively be organized by loop order, which is indicated in Figure \ref{fig:8} by the colored rectangular boxes.

\sm

The weight $w$ of the graphs contributing to $\cB_{(p,q)} (\tau)$ for $w=2p+3q$ may now be seen to coincide with the  definition of \textit{transcendental weight} given in subsection \ref{sec:12.7.2}, provided we continue to assign weight $-1$ to $s,t,u$ and weight 0 to $\tau$.

\sm

By expanding (\ref{8c5}) and using identities such as the one in (\ref{eq:MGFidentities}), one finds,
\bea
\label{12.calB}
\cB_{(0,0)}^{(1)} &=& 1
\no \\
\cB_{(1,0)}^{(1)} &=& E_2
\no\\
\cB_{(0,1)}^{(1)} &=& \tfrac{5}{3} E_3 + \tfrac{1}{3} \zeta(3)
\no\\
\cB_{(2,0)}^{(1)} &= &  2 C_{2,1,1} + E_2^2 - E_4
\no \\
\cB_{(1,1)} ^{(1)} & = &  \tfrac{7}{3} C_{3,1,1} +\tfrac{5}{3} E_2 E_3 +\tfrac{1}{3} \zeta(3)E_3 -\tfrac{34}{15}E_5 +\tfrac{1}{5} \zeta(5)
\eea
up to weight $w=5$ included. The graphs that contribute are shown in Figure \ref{fig:8}.

\begin{figure}[htb]
\begin{center}
\tikzpicture[scale=1.4]
\begin{scope}[xshift=10cm]
\draw (0,5.1) node{$\cB_{(1,0)}^{(1)} D^4\cR^4$} ;
\draw (2,5.1) node{$\bullet$} ;
\draw (3,5.1) node{$\bullet$} ;
\draw (2,5.1) to [bend left=40] (3,5.1) ;
\draw (2,5.1) to [bend right=40] (3,5.1) ;
\draw (0,3.7) node{$\cB_{(0,1)}^{(1)} D^6 \cR^4 $} ;
\draw (2,3.4) node{$\bullet$} ;
\draw (2.5,4.1) node{$\bullet$} ;
\draw (3,3.4) node{$\bullet$} ;
\draw (2,3.4) to (2.5, 4.1) ;
\draw (3,3.4) to  (2.5, 4.1) ;
\draw (2,3.4) to  (3,3.4) ;
\draw (4,3.7) node{$\bullet$} ;
\draw (5,3.7) node{$\bullet$} ;
\draw (4,3.7) to [bend left=40] (5,3.7) ;
\draw (4,3.7) to  (5,3.7) ;
\draw (4,3.7) to [bend right=40] (5,3.7) ;
\draw (0,2.2) node{$\cB_{(2,0)}^{(1)} D^8\cR^4$} ;
\draw (2,1.7) node{$\bullet$} ;
\draw (2,2.7) node{$\bullet$} ;
\draw (3,1.7) node{$\bullet$} ;
\draw (3,2.7) node{$\bullet$} ;
\draw (2,1.7) to (3,1.7) ;
\draw (2,1.7) to (2,2.7) ;
\draw (3,1.7) to (3,2.7) ;
\draw (2,2.7) to (3,2.7) ;
\draw (4,1.9) node{$\bullet$} ;
\draw (4.5,2.6) node{$\bullet$} ;
\draw (5,1.9) node{$\bullet$} ;
\draw (4,1.9) to (4.5,2.6) ;
\draw (5,1.9) to  (4.5,2.6) ;
\draw (4,1.9) to [bend left=40] (5,1.9) ;
\draw (4,1.9) to [bend left=-40] (5,1.9) ;
\draw (6,2) node{$\bullet$} ;
\draw (7,2) node{$\bullet$} ;
\draw (6,2) to [bend left=50] (7,2) ;
\draw (6,2) to [bend left=15] (7,2) ;
\draw (6,2) to [bend right=15] (7,2) ;
\draw (6,2) to [bend right=50] (7,2) ;
\draw (0,0.3) node{$\cB_{(1,1)}^{(1)} D^{10}\cR^4$} ;
\draw (4,0) node{$\bullet$} ;
\draw (4,1) node{$\bullet$} ;
\draw (5,0) node{$\bullet$} ;
\draw (5,1) node{$\bullet$} ;
\draw (4,0) to [bend left=40]  (5,0) ;
\draw (4,0) to [bend left=-40]  (5,0) ;
\draw (4,0) to (4,1) ;
\draw (5,0) to (5,1) ;
\draw (4,1) to (5,1) ;
\draw (6,0.2) node{$\bullet$} ;
\draw (6.5,0.9) node{$\bullet$} ;
\draw (7,0.2) node{$\bullet$} ;
\draw (6,0.2) to (6.5,0.9) ;
\draw (7,0.2) to  (6.5,0.9) ;
\draw (6,0.2) to [bend left=40] (7,0.2) ;
\draw (6,0.2) to  (7,0.2) ;
\draw (6,0.2) to [bend left=-40] (7,0.2) ;
\draw (8,0.3) node{$\bullet$} ;
\draw (9,0.3) node{$\bullet$} ;
\draw (8,0.3) to [bend left=50] (9,0.3) ;
\draw (8,0.3) to [bend left=20] (9,0.3) ;
\draw (8,0.3) to (9,0.3) ;
\draw (8,0.3) to [bend right=20] (9,0.3) ;
\draw (8,0.3) to [bend right=50] (9,0.3) ;
\draw (4,-2) node{$\bullet$} ;
\draw (4,-1) node{$\bullet$} ;
\draw (5,-2) node{$\bullet$} ;
\draw (5,-1) node{$\bullet$} ;
\draw (4,-2) to (5,-2) ;
\draw (4,-2) to (5,-1) ;
\draw (4,-2) to (4,-1) ;
\draw (5,-2) to (5,-1) ;
\draw (4,-1) to (5,-1) ;
\draw (6,-1.8) node{$\bullet$} ;
\draw (6.5,-1.1) node{$\bullet$} ;
\draw (7,-1.8) node{$\bullet$} ;
\draw (6,-1.8) to [bend left=20] (6.5,-1.1) ;
\draw (6,-1.8) to [bend left=-20] (6.5,-1.1) ;
\draw (7,-1.8) to [bend left=20] (6.5,-1.1) ;
\draw (7,-1.8) to [bend left=-20] (6.5,-1.1) ;
\draw (6,-1.8) to  (7,-1.8) ;
\draw [red, very thick] (1.6,-2.5) to (3.4,-2.5);
\draw [red, very thick] (1.6, -2.5) to (1.6, 5.5);
\draw [red, very thick] (3.4,-2.5) to (3.4,5.5);
\draw [red, very thick] (1.6, 5.5) to (3.4, 5.5);
\draw [blue, very thick] (3.6,-2.5) to (5.4,-2.5);
\draw [blue, very thick] (3.6, -2.5) to (3.6, 5.5);
\draw [blue, very thick] (5.4,-2.5) to (5.4,5.5);
\draw [blue, very thick] (3.6, 5.5) to (5.4, 5.5);
\draw [green, very thick] (5.6,-2.5) to (7.4,-2.5);
\draw [green, very thick] (5.6, -2.5) to (5.6, 5.5);
\draw [green, very thick] (7.4,-2.5) to (7.4,5.5);
\draw [green, very thick] (5.6, 5.5) to (7.4, 5.5);
\draw (2.5, -2.8) [red] node{one-loop};
\draw (4.5, -2.8) [blue] node{two-loops};
\draw (6.5, -2.8) [green] node{three-loops};
\end{scope}
\endtikzpicture
\end{center}
\caption{\textit{Modular graph functions contributing up to the genus-one four-graviton amplitude, up to  weight $w=5$ included. }\label{fig:8}}
\end{figure}

\subsection{Genus two in terms of modular graph functions}

The genus-two amplitude  $\cA^{(2)}$ is obtained in (\ref{12.amps}) by integrating $\cB^{(2)}(s,t,u|\Omega)$ over the genus-two moduli space $\cM_2$.  The partial amplitude $\cB^{(2)}(s,t,u|\Omega)$ in turn is given by the  integration over four copies on the genus-two surface $\Sigma$ with moduli $\Omega$ in (\ref{12.bamps}). For fixed $\Omega \in \cH_2$, the scalar Arakelov Green function $\cG(z,w|\Omega)$ is smooth throughout $\Sigma$ except for the logarithmic singularity at $z=w$ given by $\cG(z,w|\Omega) \approx - \ln |z-w|^2$. As a result, for fixed~$\Omega$, the integrals that define $\cB^{(2)}(s,t,u|\Omega)$ as a function of $s, t, u$  are absolutely convergent for $\Re(s), \Re (t), \Re (u) <1$ and may be expanded in a Taylor series in powers of $s,t,u$ with radius of convergence $|s|, |t|, |u| <1$. This situation is completely analogous to the genus-one case. The presence of the additional  measure factor $|\cY|^2$ is inconsequential for these results to hold as $\cY$ is holomorphic on $\Sigma$.

\sm

Since the function $\cB^{(2)}(s,t,u|\Omega)$ is symmetric in $s,t,u$, we may organize its Taylor expansion in powers of $s,t,u$ via the symmetric polynomials   $\sigma _2 $ and $\sigma _3$   as we did for the tree-level amplitude and the genus-one partial  amplitude,
\bea
\label{8c4a}
\cB^{(2)} (s,t,u|\Omega) = \sum _{p, q =0} ^\infty \cB _{(p,q)}^{(2)} (\Omega) \, {\sigma _2 ^p \, \sigma _3^q \over p ! \, q !}
\eea
Since $\cB^{(2)} (s,t,u|\Omega)$ is a modular function of $\Omega$, namely it is invariant under the genus-two modular group $Sp(4,\ZZ)$, each coefficient $\cB_{(p,q)}^{(2)}(\Omega)$ is itself a modular function of $\Omega$. The coefficients may be computed by expanding the exponential in the integrand of (\ref{12.bamps}) in powers of its argument to a given order $w$, which is related to the overall degree in the variables $s,t,u$, given by $w+2=2p+3q$, so that we have,
\bea
\label{8c5a}
\sum _{{p, q \geq 0 \atop 2p+3q=w+2}}  \cB _{(p,q)}^{(2)} (\Omega) \, {\sigma _2 ^p \, \sigma _3^q \over p ! \, q !} = { 1 \over w!}  \int _{\Sigma^4}  { |\cY|^2 \over (\det \Im \Omega)^2}   \left (
\sum _{i < j} s_{ij} \cG(z_i, z_j |\Omega) \right )^w
\eea
Note that the shift to $w+2$ included  in the range of summation on the left side is due to the fact that $\cY$ is linear in the variables $s,t,u$.

\subsubsection{Contributions to low weight}
\label{sec:12.9.1}

Linearity of $\cY$ in $s,t,u$ immediately implies that the genus-two contribution to the $\cR^4$ interaction without derivatives must vanish $\cB_{(0,0)}^{(2)}=0$. As was already mentioned earlier, the $D^2 \cR^4$ contribution vanishes in view of the momentum conservation identity $s+t+u=0$.

\sm

The leading low energy contribution is the $D^4 \cR^4$ interaction obtained from the zero-th order expansion term of the exponential in (\ref{8c5}). The remaining integrations required to evaluate $\cB^{(2)} _{(1,0)}$ may be carried out using the Riemann bilinear relations, reviewed in appendix~\ref{sec:RS}, and yield an $\Omega$-independent contribution.

\sm

The coefficient $\cB^{(2)}_{(0,1)}$ has non-trivial dependence on $\Omega$, denoted by $64 \, \f(\Omega)$. Assembling these results to order $D^6\cR^4$ we have,
\bea
\cB^{(2)}(s,t,u|\Omega) = 32 (s^2+t^2+u^2) + 192 \, stu \, \f (\Omega)+ \cO(s_{ij}^4 )
\eea
The genus-two modular function $\f(\Omega)$  is given in terms of the Arakelov Green function by,
\bea
\f(\Omega) = { 1 \over 8}
\int _{\Sigma ^2} \om_I(x) \om_J(y)  \Big ( \bar \omega^I(x) \bar \omega^J(y) - 2 \bar \om ^J(x) \bar \om ^I(y) \Big )
\cG(x,y|\Omega)
\eea
where $\bar \om^I= (Y^{-1})^{IJ} \, \overline{\om_J}$. This object was identified with the spectral invariant introduced by Kawazumi and Zhang. Higher order contributions to $\cB^{(2)}(s,t,u|\Omega) $ may similarly be expressed in terms of integrals of products of Arakelov Green functions, and may be viewed as higher genus generalizations of the genus-one modular graph functions investigated in section \ref{sec:MGF}. Their general structure has, however, been less well-studied and is much less well-understood than that of their genus-one counterparts.

\subsection{Integration over the genus-one moduli space}

When considering one-loop or two-loop amplitudes we have, so far,  focused attention on the structure of the partial
amplitudes $\cB^{(1)}(s,t,u|\tau)$ and $\cB^{(2)}(s,t,u|\Omega)$, for which we have obtained a systematic low energy expansion in terms of modular graph functions, both at genus one and genus two. For fixed $\tau$ and $\Omega$, this expansion is absolutely convergent for $\Re(s), \Re(t), \Re(u)  <1$, and admits an analytic continuation in $s,t,u$ whose sole singularities are simple poles at positive integers in $s,t,u$, corresponding to the exchanges of massive string states, as may be seen from the operator product expansion.

\sm

Obtaining the corresponding physical string  amplitudes $\cA^{(1)}(s_{ij})$ and $\cA^{(2)}(s_{ij})$ further requires the integrations over the moduli spaces of genus-one and genus-two Riemann surfaces shown in (\ref{12.amps}).  These integrations are absolutely convergent only when $s,t,u$ are purely imaginary, in which case $\cB^{(1)}(s,t,u|\tau)$ and $\cB^{(2)}(s,t,u|\Omega)$ may be uniformly bounded on their respective moduli spaces. For the case of genus one, the lack of convergence originates from the integration in the region near the cusp. This may be seen by expressing the genus-one Green function $G(z|\tau)$ in the co-moving coordinates $z=x+y \tau$ for $x,y \in \RR/\ZZ$ first introduced in section \ref{5.LapTor} and given by (\ref{10.green-theta}),
\bea
G(x+y \tau | \tau) = - \ln \left | { \tet_1(x+y \tau |\tau) \over \eta(\tau) } \right |^2 + 2 \pi \tau_2 y^2
\eea
For $\tau_2 \to \infty$ and fixed $x$ and $|y| < \thalf$, the Green function behaves as follows,
\bea
G(x+y \tau | \tau) \approx 2 \pi \tau_2 (y^2 -|y|)
\eea
which diverges as $\tau_2 \to \infty$.

\sm

The analytic continuation in $s,t,u$ of the integrals over moduli now produces double and additional simple poles at all positive integers, as well as branch cuts in $s,t,u$ that originate at every non-negative  integer, including zero.
These singularities are fully expected on physical grounds. The poles produce  mass renormalization and non-zero decay widths of massive string states, while the branch cuts correspond to the decay of a single string into a two-string state. The presence of the branch cuts implies that the string amplitudes $\cA$ do not admit a convergent low energy Taylor expansion. We now review the derivation of the branch cuts that arise in the low energy expansion for the case of one-loop amplitudes.% the case of two-loop amplitudes being similar but more involved.

\subsubsection{Partitioning the genus-one moduli space}

To proceed with the analysis of the branch cuts produced in the low energy limit, we partition the moduli space $\cM_1$ of genus-one Riemann surfaces into a region near the cusp $\cM_R$ and its complement $\cM_L$, as illustrated in the Figure \ref{fig:12.4}, and given as follows for $L >1$,
\bea
\cM_R & = & \cM_1 \cap \{ \tau_2 >L \}
\hskip 1in \cM_L \cap \cM_R = \emptyset
\no \\
\cM_L & = & \cM_1 \cap \{ \tau_2 \leq L \}
\hskip 1in \cM_L \cup \cM_R = \cM_1
\eea

\begin{figure}[htb]
\begin{center}
\tikzpicture[scale=0.7]
\scope[xshift=-5cm,yshift=0cm]
\draw[thick] (-5,0) -- (7,0);
\draw[thick] (0,0) -- (0,10);
\draw[ultra thick, color=red] (-3,8) -- (3,8);
\draw[thick, dotted] (-2,0) -- (-2,3.464);
\draw[thick, dotted] (2,0) -- (2,3.464);
\draw[ultra thick] (-2, 3.464) -- (-2, 10);
\draw[ultra thick] (2 , 3.464) -- (2, 10);
\draw [ultra thick] (-2,3.464) arc (120:60:4 and 4);
\draw [dashed] (-4,0) arc (180:120:4 and 4);
\draw [dashed] (4,0) arc (0:60:4 and 4);
\draw (0,-0.7) node{$0$};
\draw (-2,-0.7) node{$-\half $};
\draw (2,-0.7) node{$\half $};
\draw (-4,-0.7) node{$-1$};
\draw (4,-0.7) node{$1$};
\draw (5.5,6) node{$\cM_L=\cM_1 \cap \{ \tau_2<L\} $};
\draw (5.5,9) node{$\cM_R=\cM_1 \cap \{ \tau_2>L\} $};
\draw (7,0.5) node{$\RR$};
\draw (-5,8) node{$\cH_1$};
\draw (-0.5,4.5) node{\small $i$};
\draw (-0.5,8.5) node{\small $iL$};
\draw (0,4) node{$\bullet$};
\draw (2,3.464) node{$\bullet$};
\draw (-2,3.464) node{$\bullet$};
%\draw (3.2,3.8) node{$e^{2 \pi i /6}$};
%\draw (-3,3.8) node{$e^{2 \pi i /3}$};
\draw (0,11) node{$i \infty$};
\endscope
\endtikzpicture
\end{center}
\caption{\textit{Partition of the genus-one moduli space  $\cM_1$ into $\cM_R$ and $\cM_L$. }\label{fig:12.4}}
\end{figure}

We shall think of $L$ as a free parameter, but in practice take $L \gg 1$ so that $\cM_R$ is a small neighborhood of the cusp. The amplitude $\cA^{(1)}$ is  a sum of the corresponding integrals,
\bea
\cA^{(1)} (s_{ij} ) = \cA^{(1)}_L (s_{ij} ) + \cA^{(1)}_R (s_{ij} )
\eea
\bea
\cA^{(1)}_{L,R} (s_{ij} ) = { \pi \over 16} \int _{\cM_{L,R}}   { |d \tau|^2 \over (\Im \tau)^2} \, \cB^{(1)}  (s,t,u |\tau)
\eea
The region $\cM_L$ is compact and the integrand $\cB^{(1)}  (s,t,u |\tau) $ admits a uniform low energy expansion in the region of convergence $|s|, |t|, |u|<1$. Therefore, the resulting contribution $\cA^{(1)}_L (s_{ij} )$ will be analytic in $s,t,u$ but dependent on $L$.

\sm

The region $\cM_R$ is not compact and $\cB^{(1)}  (s,t,u |\tau) $ cannot be handled  with a uniformly convergent low energy expansion. Instead we need a treatment of this contribution that is exact up to contributions exponentially suppressed in $L$. The resulting $\cA^{(1)}_R (s_{ij} )$ will exhibit branch cuts in $s,t,u$ and its $L$-dependence will cancel the $L$-dependence of $\cA^{(1)}_L (s_{ij})$.

\subsubsection{Integrals involving Eisenstein series and modular graph functions}

To carry out the integrations over the fundamental domain of the partial amplitudes $\cB^{(1)}_{(p,q)} (\tau)$, one invariably encounters integrals of Eisenstein series, products of Eisenstein series, and other modular graph functions,  over the truncated fundamental domain $\cM_L$. These integrals are of interest in their own right and we dedicate this subsection to deriving some general results and applying these results to the simplest cases, and then close by quoting the integral of the product of three Eisenstein series from Zagier's work. We begin with the following straightforward  application of Stokes's theorem to an arbitrary modular graph function $\cC$ of weight $(0,0)$, or to an arbitrary modular graph form $\cC^+$  of weight $(0,2)$,
\bea
\label{12.ints}
\int _{\cM_L} { d^2 \tau \over \tau_2^2} \, \Delta \cC & = & \int _0^1 d\tau _1 \, \p_{\tau_2} \cC(\tau_1, \tau_2) \Big | _{\tau_2 =L}
\no \\
\int _{\cM_L} { d^2 \tau \over \tau_2^2} \, \nabla \cC^+ & = & \int _0^1 d\tau _1 \, \cC^+(\tau_1, \tau_2) \Big | _{\tau_2 =L}
\eea
where the result on the right side may be read off from the constant Fourier mode part of $\cC$ and $\cC^+$. The result may be readily used to evaluate the integral of a non-holomorphic Eisenstein series or products thereof. It turns out that the simplest expressions are obtained in terms of the normalized Riemann zeta function $\zeta^*(s)$ and the normalized non-holomorphic Eisenstein series $E^*_s(\tau)$, defined by,
\bea
\zeta ^*(s) =  { \Gamma (s/2) \zeta (s) \over \pi^{{s \over 2}}}
\hskip 1in
E_s^*(\tau) = \thalf \Gamma (s) E_s(\tau)
\eea
The normalization guarantees the functional equations $\zeta^*(1-s) = \zeta^*(s)$ and $E^*_{1-s} (\tau) = E^*_s(\tau)$, in addition to the following simple Fourier expansion,
\bea
\label{12.Eisas}
E^*_s(\tau) = \zeta ^*(2s) \, \tau_2^s + \zeta ^*(2s-1) \, \tau_2^{1-s} + \cO(e^{-2 \pi \tau_2})
\eea
In terms of these functions, we have the following result.

\label{12.Eisint}
The following integrals involving Eisenstein series are given, up to exponentially decaying terms $\cO(e^{-2 \pi \tau_2})$ at the cusp, by,
\bea
\int _{\cM_L} { d^2 \tau \over \tau_2^2} \, E_s^* & = & \sum _{x=s,1-s} \zeta ^*(2x) { L^{x-1} \over x-1}
\\
\int _{\cM_L} { d^2 \tau \over \tau_2^2} \, E_s^*E_t^* & = & \sum _{x=s,1-s} \, \sum_{y=t,1-t} \zeta ^*(2x) \zeta ^*(2y) { L^{x+y-1} \over x+y-1}
\no \\
\int _{\cM_L} { d^2 \tau \over \tau_2^2} \, E_s^*E_t^* E^*_u & = &
\zeta^*(w-1) \zeta^*(w-2s) \zeta^*(w-2t) \zeta ^*(w-2u)
\no \\ &&
+ \sum _{x=s,1-s} \, \sum_{y=t,1-t} \sum _{z=u,1-u} \zeta ^*(2x) \zeta ^*(2y) \zeta^*(2z) { L^{x+y+z-1} \over x+y+z-1}
\no
\eea
where $w=s+t+u$ in the last line. }

\sm

The first line in Lemma \ref{12.Eisint} may be proven by substituting $E_s^*= \Delta E_s^*/s(s-1)$ in the integrand  and then  using the first line on (\ref{12.ints}). The second line in Lemma \ref{12.Eisint} may be proven by expressing the combination $\Delta E_s^* \, E^*_t - E_s^* \Delta E^*_t $ in two different ways,
\bea
\Delta E_s^* \, E^*_t - E_s^* \Delta E^*_t
& = &
\Big  ( s(s-1) - t(t-1) \Big ) E^*_s E^*_t
\\ & = &
\nabla \left ( { E^*_s \bar \nabla E_t^* - E^*_t \bar \nabla E_s^* \over 2 \tau_2^2}     \right )
+ \bar \nabla \left ( { E^*_s \nabla E_t^* - E^*_t  \nabla E_s^* \over 2 \tau_2^2}     \right )
\no
\eea
and then using the expression on the right side of the first line to eliminate $E_s^* E_t^*$ from the integrand of \ref{12.Eisint} in favor of the total derivative terms in the second line. Finally, to integrate the total derivative terms, we use the second line in (\ref{12.ints}) and its complex conjugate relation along with the asymptotics (\ref{12.Eisas}) of the Eisenstein series.  The third line in Lemma \ref{12.Eisint} was proven by Zagier using different methods, as will be indicated in the bibliographical notes.

\subsubsection{The non-analytic contribution $\cA_R^{(1)}$}

To evaluate $\cB^{(1)}(s,t,u|\tau)$ for $\tau \in \cM_R$, we cannot use  (\ref{8c4}) since this expansion is not uniformly convergent throughout $\cM_R$. Instead, our starting point is the original expression in (\ref{12.amps}), which we shall evaluate exactly up to terms that are exponentially suppressed in~$L$. To do so, we use co-moving coordinates, $z_i = x_i + y_i \tau$ with $i=1,2,3,4$ and partition the integration region $\Sigma ^4$ into the six possible orderings of $y_1,y_2, y_3, y_4$ up to cyclic permutations.  Doing so allows us to decompose $\cA^{(1)}_R(s_{ij})$ into a sum of six more elementary contributions, which are pairwise equal to one another,
\bea
\cA^{(1)}_R(s_{ij}) = 2\cA_*(L; s,t) + 2\cA_*(L; t,u) + 2\cA_*(L; u,s) \hskip 0.5in s+t+u=0
\eea
The function $\cA_*(L; s,t) $ may be computed exactly, up to exponentially suppressed terms,
\bea
\cA_*(L; s,t)  =  \mC (s,t;0) \, \ln(-4 \pi Ls) + { \p \over \p \ep} \mC (s,t;\ep) \Big |_{\ep=0} + \cO(e^{ - 2 \pi L})
\eea
where $\mC$ is conveniently expressed as the sum of two integral representations,
\bea
\mC(s,t;\ep) & = & - { 8 \pi s^4 \over \Gamma (3+\ep)}
\int _0^1 dx \int _0^1 dy \, x^{5+2\ep} y^{2+\ep} (1-y)^{2+\ep} W(s, -sxy+t(1-x))
\no \\ &&
- 4 \pi s^7 \sum_{k=0}^\infty s^k t^k \int _0^1 \! dx \, { x^{k+3+\ep} (1-x)^{k+3+\ep} \over 2 \Gamma (k+4+\ep) \, k!}
\left [ { \p^k \over \p \mu^k} W(s,\mu-sx) \Big |_{\mu=0} \right ]^2
\qquad
\eea
The function $W$ is closely related to the Virasoro-Shapiro amplitude,
\bea
W(s,t) = { 1 \over stu} \left ( { \Gamma (1-s) \Gamma (1-t) \Gamma (1-u) \over \Gamma (1+s) \Gamma (1+t) \Gamma (1+u)} -1 \right )
\hskip 1in u=-s-t
\eea
Although the above expression may look daunting at first, it analytical structure is manifest. The function $W(s,t)$ is meromorphic throughout $s,t \in \CC$, holomorphic in discs of unit radius near the origin, and has only simple poles at positive integer values of $s,t,u$, and thus admits a Taylor expansion in $s,t$. Substituting this Taylor expansion into the integrals defining $\mC$, we see that the integrals preserve the holomorphicity in $s,t,u$ near the origin, so that $\mC$ itself has a convergent Taylor expansion  near the origin, which starts at order $s^4$. To this lowest order we have,
\bea
\label{ARtot}
\mC(s,t;\ep) = -  16 \pi \zeta(3) s^4 { \Gamma (3 + \ep) \over \Gamma (7+2 \ep)} + \cO(s^5, s^4t)
\eea
The coefficients of the expansion to all orders  inherit contributions linear and bilinear in odd zeta-values from the expansion of the Virasoro-Shapiro amplitude,  with rational coefficients in the coefficient $\mC(s,t;0)$  of the $\ln(-4 \pi Ls)$ term, and additional harmonic sums in the $\p_\ep \mC(s,t;\ep) \big |_{\ep=0}$ analytic contribution to $\cA_*(L;s,t)$. The discontinuity of $\ln(-4 \pi Ls)$  near the origin reproduces the square of the tree-level amplitude by unitarity.

\subsubsection{The analytic contribution $\cA_L^{(1)}$}

The contribution $\cA^{(1)}_L(s_{ij})$ is analytic in $s,t,u$ near the origin and we may use the expansion of $\cB^{(1)}  (s,t,u |\tau)$ in terms of modular graph functions of (\ref{8c4}) to evaluate $\cA_L^{(1)}(s_{ij})$ order by order in powers of $s,t,u$,
\bea
\cA^{(1)} _L(s_{ij}) = \sum_{p,q=0}^\infty \cA_{(p,q)}^{(1)} \, { \sigma _2^p \sigma _3^q \over p! \, q!}
\eea
Since the non-analytic part $\cA_R^{(1)} (s_{ij})$ starts contributing to $\cA^{(1)}(s_{ij}) $ only at order $s^4$, all its lower orders are given entirely by $\cA^{(1)}_L(s_{ij})$.  The corresponding amplitudes are readily evaluated, and we find,
\begin{align}
\label{12.loen}
\cA_{(0,0)}^{(1)} & = { \pi \over 3} & \cA_{(0,1)}^{(1)} & = {\pi  \over 9} \zeta (3)
\cA_{(1,0)}^{(1)} & = 0 & \cA_{(1,1) }^{(1)} & = {29 \pi  \over 540}\zeta(5)
\end{align}
The remaining coefficients at weight four are given as follows,
\bea
\label{ALtot}
\cA_{(2,0)} ^{(1)}=
{ 4 \pi \zeta(3)  \over 45} \left [
\ln (2 L) + {\zeta '(4) \over \zeta (4) } - {\zeta '(3) \over \zeta (3) }  -{ 1 \over 4} \right ]
\eea

\subsubsection{Assembling analytic and non-analytic contributions}

Assembling the contributions to $\cA_L^{(1)}$ in (\ref{ALtot}) and to $\cA_R^{(1)}$ in (\ref{ARtot}) gives $\cA^{(1)}(s_{ij})$, the total amplitude,  to order $\cO(s_{ij}^4)$ included. We see explicitly that all $L$-dependence cancels as required for the consistency of the calculation. It is instructive to rearrange  the total amplitude as the sum
of ``analytic" and ``non-analytic" pieces,
\bea
\label{suman}
\cA^{(1)}(s_{ij}) = 2\pi \Big ( \cA_L ^{(1)} (L;s_{ij}) + \cA_R ^{(1)} (L;s_{ij}) \Big ) =
\cA_{{\rm an}}(s_{ij}) +\cA_{{\rm non-an}}(s_{ij})
\eea
The reason for the quotation marks on  analytic and non-analytic is that the non-analytic piece actually contains also analytic contributions, so that the nomenclature is natural and suggestive but not entirely precise.
The ``analytic'' piece $ \cA_{{\rm an}} (s_{ij})$   is given by,
\bea
\label{Aan}
\cA_{{\rm an}} (s_{ij}) = { 2 \pi^2 \over 3} \left ( 1 + {\zeta (3) \sigma _3 \over 3}  + \cO(s_{ij}^5)\right )
\eea
and the  ``non-analytic'' piece $ \cA_{{\rm non-an}} (s_{ij})$ has the form,
\bea
\label{Anonan}
\cA_{{\rm non-an}} (s_{ij}) = {2\pi^2 \over 3}  \left ( \cA_{{\rm sugra}} + \cA_4   +  \cO(s_{ij}^5) \right )
\eea
The lowest-order term $ \cA_{{\rm sugra}}$ is a regularized version of the ten-dimensional one-loop supergravity amplitude.  The contribution $\cA_4$ is given by,
\bea
\label{Ahatfour}
\cA_4  =  { 4 s^4 \over 15} \, \zeta (3)  \left ( - \ln (- 2 \pi s) +  {\zeta '(4) \over \zeta (4) } - {\zeta '(3) \over \zeta (3) } - \gamma  +{63 \over 20} \right ) +
\hbox{ cycl } (s,t,u)
\eea
where $\gamma$ is the Euler constant. The discontinuity of $ \cA_{{\rm non-an}} (s_{ij}) $, namely the  coefficient of the $\ln(-2\pi s)$ terms in (\ref{Ahatfour}), reproduce those obtained in [93].

\subsection{Integration over the genus-two moduli space}
\label{12.11}

The function $\cB^{(2)} _{(1,0)} (\Omega)$ needed to obtain the coefficient of the $D^4\cR^4$ interaction was found to be independent of moduli $\Omega$ in subsection \ref{sec:12.9.1}. Thus, its integral over moduli space $\cM_2$ is given by the volume of $\cM_2$ with the Siegel metric on the Siegel upper half space,
\bea
ds^2 = \sum_{I,J,K,L} Y^{-1}_{IJ} \, Y^{-1}_{KL} \, d\bar \Omega_{JK} \, d\Omega_{IL}
\eea
of constant negative curvature.  The volume form $d \mu_2$ and the value of the volume, computed by Siegel, are given by,
\bea
\int _{\cM_2} d \mu_2  = { 8 \pi^3 \over 270}
\hskip 1in
d \mu_2 = { |d \Omega_{11} \wedge d \Omega_{12} \wedge d \Omega _{22}  |^2 \over (\det \Im \Omega)^3}
\eea
As a result, the low energy effective interaction to this order is given by,
\bea
\label{12.gen2a}
\cA^{(2)}_{(1,0)} = { 4 \pi^4 \over 270} (s^2+t^2+u^2) \cR^4
\eea

The function $\cB^{(2)} _{(0,1)} (\Omega)$ is proportional to the Kawazumi-Zhang invariant $\f$. Its integral over $\cM_2$ is convergent and yields the contribution $\cA^{(2)}_{(0,1)} $ to the low energy effective action. Evaluation of the integral is made possible by the fact that $\f$ satisfies an inhomogeneous Laplace eigenvalue equation,
\bea
(\Delta - 5) \f = - 2 \pi \delta_{SN}
\eea
where $\Delta$ is the Laplace-Beltrami operator on the Siegel upper half space for the Siegel metric,
\bea
\Delta = \sum_{I,J,K,L} 4 \, Y_{IK} \, Y_{JL} \bar \p_{IJ} \p_{KL}
\hskip 1in
\p_{IJ} = \half (1+\delta_{IJ}) { \p \over \p \Omega_{IJ}}
\eea
and $\delta_{SN}$ is the Dirac $\delta$-function supported on the separating degeneration node.
The coefficient $\cB_{(0,1)}^{(2)}$ is then given by the integral of the Kawazumi-Zhang invariant over $\cM_2$,
\bea
\cB_{(0,1)} ^{(2)} = \pi \int _{\cM_2} d\mu_2 \,  \f = {\pi \over 5} \int _{\cM_2} d \mu_2 \Big ( \Delta \f + 2 \pi \delta_{SN} \Big ) = {2 \pi^3 \over 45}
\eea
The integral may be evaluated using Stokes's theorem and the asymptotic behavior for the Kawazumi-Zhang invariant, and we find,
\bea
\label{12.gen2b}
\cA_{(0,1)} ^{(2)} =  {2 \pi^3 \over 45} (s^3+t^3+u^3) \cR^4
\eea
For possible verifications to genus-three order, we refer to the references of this section.

\subsection*{$\bullet$ Bibliographical notes}

Classic books  by Green, Schwarz, and Witten [169, 170] and by Polchinski [171, 172] present  comprehensive perspectives on string theory. A pedagogical introduction, aimed at undergraduate and graduate students, is given in the book by Zwiebach [173]. Other accounts may be found in the books by Johnson [174], by Becker, Becker, and Schwarz [175], by Blumenhagen, L\"ust, and Theissen  [176], and by Kiritsis  [177]. {A beautiful discussion of the connection between modular/automorphic forms and string amplitudes can be found in the book by Fleig, Gustafsson, Kleinschmidt, and Persson [24].}

\sm

The early development of string theory was dominated by the study of string amplitudes. The Ramond and Neveu-Schwarz sectors of the RNS formulation were introduced in [178] and [179], respectively. The different spin structure sectors and the Gliozzi-Scherk-Olive (GSO) projection onto supersymmetric string spectra were introduced in [180].
The Polyakov formulation of string theory, along with many other key problems in modern theoretical physics, is presented in the book by Polyakov [181]. Friedan's renormalization of the non-linear $\sigma$-model and the relation between Einstein's equations and conformal invariance may be found in [182]. Generalizations with worldsheet fermions are given in [183], and including torsion in [184] and the dilaton in [185]. The $\alpha'$ expansion of the effective action including the metric, anti-symmetric tensor field, and dilaton is presented in detail in the book of Princeton Institute for Advanced Study lecture notes [186].

\sm

The standard treatment of the decoupling of negative norm states for both the bosonic and superstrings  is clearly reviewed in  [169]. The modern BRST-based approach to the covariant quantization of superstrings in the RNS formulation was developed in [51], and reviewed in [169, 170] and [171, 172]. The structure of gauge and gravitational anomalies, relevant to supergravity and string theories in various dimensions, was obtained in [187]. The cancellation of anomalies in the Type I theory with gauge group $SO(32)$ was famously discovered in [188]. Useful lecture notes on anomalies may be found in [189].

\sm

Reviews of superstring perturbation theory may be found in [190, 191] for the RNS formulation and in  [192] for the pure spinor formulation.  Lectures accessible to a more mathematically oriented readership are in [186], while an elaboration of  the mathematical underpinning of the RNS formulation is provided in [193, 194]. A broad overview of the current status of superstring perturbation theory, and open problems,  is given in [195], which contains an extensive bibliography.

\sm

The tree-level and one-loop four graviton amplitudes were derived in the original papers  in which the Type II [196]  and Heterotic strings [197, 198]  were discovered. The existence of the analytic continuation of the one-loop four graviton amplitude was shown in  [199, 200].

\sm

The fact that the low energy expansion of string theory leads to $\cN=4$ super-Yang-Mills theory and $\cN=8$ supergravity in four space-time dimensions was discovered  in [201].  The systematic analysis of the low energy effective interactions induced by the tree-level four-graviton amplitude was initiated in [202, 203],
while the effects at genus-one were obtained in  [204, 205, 93]. The low energy expansion, up to order $D^8\cR^4$, including the logarithmic branch cuts and the transcendentality properties of the amplitude  discussed in section \ref{sec:12.7.2} were obtained in [206].

\sm

The original Rankin-Selberg method for integrating cusp forms over the fundamental domain for $SL(2,\ZZ)$
was developed in [207, 208], and was extended to automorphic functions of non-rapid decay in [209]. { It was applied to string amplitudes in [210] and subsequent works [211, 212]. } The proof of the last integral identity in Lemma \ref{12.Eisint} is in [209], while methods to carry out integrals of modular graph functions may be found  in [213] and references therein.

\sm

Considerable progress has been made recently towards evaluating tree-level and genus-one string amplitudes with arbitrary numbers of external states in [214, 215, 216, 217, 218, 219] and references therein. The role of motivic and multiple zeta-values played in the form of the amplitudes was examined in [220, 221, 222].
Genus-one amplitudes have also recently been reproduced from ${\cal N}=4$ supersymmetric Yang--Mills theory by considering a flat-space limit of $AdS_5\times S^5$  [223].

\sm

For genus-two amplitudes, the measure on supermoduli space was derived in the RNS formulation in [224, 225, 226] and reproduced using algebraic-geometric methods in [227]. The full amplitude for four gravitons was obtained in [228] in the RNS formulation, and extended to include external fermion states in   [229, 230] using the pure spinor formulation. The overall normalization of the amplitude was obtained in  [231, 232].

\sm

Recently, the genus-two amplitudes with five massless states were constructed in [233, 234] using an amalgam of methods from the RNS and pure spinor formulations and. For external NS boson states and even spin structure, these results were confirmed from first principles in the RNS formulation in [235].

\sm

The generalization of  modular graph functions to genus two and higher genus was developed in [124] while the relation between the coefficient of the $D^6\cR^4$ term and the invariant of Kawazumi  [236] and Zhang [237] was identified in [238] and used to derive the corresponding differential equation in  [239]. Degenerations of genus-two modular forms naturally produce non-holomorphic versions of Jacobi forms, systematic treatment of which may be found in the book by Eichler and Zagier [240].  A $\tet$-lift representation of the genus-two Kawazumi-Zhang invariant was obtained in [241]. For the  derivation of the results in section \ref{12.11} we refer to  [239].

\sm

For amplitudes at genus three and beyond, no first principle derivations are available at the time of this writing.
A proposal for the measure on supermoduli space in the RNS formulation was advanced in [242],
building on  some earlier unsuccessful attempts in [243]. The proposal of [242] was critiqued, however, in the appendix of [244].  The leading low energy effective interaction in Type II string theory was obtained using the pure spinor formulation in [245]. The corresponding expression for the amplitude, however, diverges beyond the leading low energy limit so that no independent checks on the unitarity of the amplitude is available. A proposal for the full genus-three four graviton amplitude has been advanced in [246], but details of the derivation are not yet available. Proposals for the measure on supermoduli space in the RNS formulation at genus four may be found in [247, 248].

\newpage

\section{Toroidal compactification}
\setcounter{equation}{0}
\label{sec:Toroidal}

In this section, we discuss the modular properties of quantum field theory of scalar fields that take values in a $d$-dimensional torus with a flat metric and constant anti-symmetric tensor. The problem is of great interest in quantum field theory and string theory in view of the fact that such \textit{toroidal compactifications} admit solutions using free field theory methods on the worldsheet, preserve Poincar\'e supersymmetries, and may be used to relate different perturbative string theories via \textit{T-duality}. Toroidal compactifications produce large duality groups that generalize the modular group $SL(2,\ZZ)$ considered thus far.

\subsection{Conformal field theory on flat manifolds}

In section \ref{sec:TorusQFT} we considered the functional integral for a single free scalar field $\varphi$ that  takes values in $\RR$. The situation is readily generalized to the case of  $d$ free scalar fields taking values in a flat Euclidean $\RR^d$. As discussed in section \ref{sec:SA}, the generalization to $d$ scalar fields taking values in (the local coordinate systems of) an arbitrary space-time manifold in general leads to fully interacting quantum field theories on the worldsheet that cannot be solved exactly.

\sm

There is one class of theories, however, where an exact quantization may be obtained, namely when all the fields $G,B,\Phi$ are constant, and the manifold $M$ is flat. Although flat, $M$ may  have non-trivial topology as in the case of a $d$-dimensional torus $M=T_d$, which is the case we shall consider here.  Constant $\Phi$ gives rise to a term proportional to the Euler number of $\Sigma$ and weighs the functional integral by a power of the string coupling which is independent on the quantum field $X$. Finally, the term involving $B_{\mu \nu}$ is a topological term when $B$ is constant. It does not contribute when the topology of $M$ is trivial, but when $M$ is a torus, it does have very interesting physical effects which we shall now examine. For constant $G,B,\Phi$, the $X$-dependent part of the action thus reduces to,
\bea
I[X, g]  =
{ 1 \over 4 \pi \alpha ' } \int _\Sigma d \mu _g \, \Big  (  g^{mn} G_{\mu \nu}
- i \, \ep^{mn}  B_{\mu \nu}   \Big ) \p_m X ^\mu  \p_n X ^\nu
\eea
Henceforth, we shall restrict to the case where the worldsheet  surface is a torus of modulus~$\tau$, so that $\Sigma = \CC /\Lambda$ with $\Lambda = \ZZ + \tau \ZZ$. We shall choose a system of complex coordinates $z,\bar z$ on $\Sigma$ in which the metric $g$ takes the form,
\bea
g_{mn} \, d\xi ^m d \xi^n = { |dz|^2 \over \tau_2}
\eea
or in components we have $g_{zz}=g_{\bar z \bar z}=0$ and $g_{z \bar z} = 1/(2\tau_2)$. Furthermore,  the space-time torus $T_d$ is given by $\RR^d / L$ for some $d$-dimensional lattice $L$.

\subsection{Lattices and tori of dimension $d$}

A lattice $L$ of rank $d$, also referred to as a lattice of dimension $d$,  may be represented in terms of a basis of $d$ vectors $v_1 , \cdots, v_d $  in $\RR^d$,
\bea
L = \Big \{ \ell = \sum_{i=1}^d n_i v_i  ~~ n_i \in \ZZ \Big \}
\eea
It will be useful to  view the assignment of basis vectors as obtained by a $GL(d,\RR)$ transformation applied to a standard basis of $d$ vectors, such as for example $v_1^0=(1,0,\cdots, 0)$, $v_2^0 = (0,1,0,\cdots, 0), \cdots, v_n^0 =(0,\cdots, 0, 1)$. Decomposing $GL(d,\RR) = SL(d,\RR) \times \RR^+$, the factor $\RR^+$ corresponds to overall rescaling of the lattice.  An arbitrary  set of $d$ vectors $v_i' \in L$ for $i=1,\cdots, d$ may be decomposed in the basis $v_i$,
\bea
v'_i = \sum_{j=1}^d M_{ij} v_j \hskip 1in M_{ij} \in \ZZ
\eea
and generates a sub-lattice $L' \subset L$. Lattices may be viewed as Abelian groups so that $L'$ is a subgroup of $L$.  The index of $L'$ as a subgroup of $L$ is given as follows,
\bea
[L:L'] = |\det M|
\eea
Equivalently, the unit lattice cell of $L'$ contains $|\det M|$ copies of the unit lattice cell of $L$. In the special case where $\det M=1$, we have $L'=L$ and therefore $M$ is an automorphism of~$L$. The group of all automorphisms of $L$ is $SL(d,\ZZ)$, thereby generalizing to $d$ dimensions the result familiar from the two-dimensional torus. When $\det M=-1$ the lattice $L'$ is the mirror image of $L$, reversing the orientation of the lattice.

\sm

Introducing the equivalence relation between two lattices that are related to one another by an overall rescaling in $\RR^+$, an overall rotation in $SO(d)$, and a lattice automorphism in $SL(d,\ZZ)$, we may identify the space of all inequivalent lattices as follows,
\bea
\hbox{space of inequivalent lattices } = SL(d,\ZZ) \backslash SL(d,\RR)/SO(d,\RR)
\eea
In the case of a two-dimensional lattice $L$ with $d=2$, we may identify $SL(2,\RR)/SO(2)$ with the Poincar\'e upper half-plane $\cH$ and the entire coset $SL(2,\ZZ) \backslash SL(2,\RR)/SO(2,\RR)$ with the moduli space of  toroidal Riemann surfaces.  The dynamics of point particles propagating on the torus $T_d=\RR^d/L$ is sensitive to the lattice geometry of this quotient. We shall see below that the dynamics of strings propagating on the torus $\RR^d/L$ is sensitive to a different equivalence class of lattices, which is ``smaller" than  $SL(d,\ZZ) \backslash SL(d,\RR)$ thanks to the quintessentially string-theoretic phenomenon of $T$-duality.

\subsection{Fields taking values in a torus}

We begin by reviewing the construction of the space of $d$ scalar fields $X(z)$ that take values in the torus $\RR^d / L$ and are functions on a worldsheet $\Sigma= \CC/ \Lambda$ with the topology of a two-dimensional torus, whose complex structure is given by its modulus $\tau \in \cH$ so that $\Lambda = \ZZ \oplus \tau \ZZ$.  The space of maps,
\bea
X : \, \Sigma = \CC/\Lambda \, \to \, T_d= \RR^d /L
\eea
is disconnected and may be decomposed into the union of an infinite number of connected components which are labelled by two copies of the lattice $L$. In a sector labelled by a pair $(\ell_A, \ell_B) \in L\times L$, the field $X^\mu$ satisfies the following monodromy conditions,
\bea
X (z+1) & = & X (z) + \ell _A
\no \\
X (z+\tau) & = & X (z) + \ell  _B
\hskip 1in \ell _A, \ell_B \in L
\eea
The field $X$ may be decomposed into a purely periodic part $Y$, which satisfies,
\bea
\eea
and a special solution to the monodromy conditions. The latter  is arbitrary but it will be convenient to choose it  linear in $z$ and $\bar z$. Since the field $X$ and the vectors $\ell_A, \ell_B$  are real-valued, we have the following decomposition,
\bea
X _{\ell_A, \ell_B} (z) = Y  (z) + z \, { \ell _B - \bar \tau \ell  _A \over \tau - \bar \tau}
- \bar z \, { \ell  _B - \tau \ell  _A \over \tau - \bar \tau}
\eea
Such a configuration is sometimes referred to as a \textit{worldsheet instanton}. The lattice elements $\ell_A, \ell_B$ are related to the momentum and the winding modes of strings on $T_d$, as we shall clarify below. The integration over all $X : \Sigma \to T_d$ decomposes into a sum over sectors labelled by $L \times L$. The  functional integral takes the following form,
\bea
\int _{{\rm maps}(\Sigma \to T_d)} D X e^{-I[X,g]}
= {\rm Vol} (T_d) \sum _{\ell _A, \ell_B \in L} \int _{{\rm maps}(\Sigma \to \RR^d)} D' Y \, e^{-I[X_{\ell_A, \ell_B}, g]}
\eea
For given worldsheet metric $g$, the measures $DX$ and $D 'Y$ are evaluated using the norm of (\ref{12.measure}).
The factor ${\rm Vol } (T_d)$ arises from the normalizable zero mode of the field $Y$, and the integration $D' Y$ is over all  non-zero modes, i.e. all functions $Y$ orthogonal to constants. The contribution to the action of $Y$ decouples from the contribution of the momentum and winding modes, which may be evaluated explicitly,
\bea
I[X_{\ell_A, \ell_B}, g] = I[Y, g] + { 1 \over 4 \pi \alpha ' \tau_2} (\ell ^\mu _B - \bar \tau \ell ^\mu _A)
(\ell ^\nu _B -  \tau \ell ^\nu _A) (G_{\mu \nu} +  B_{\mu \nu} )
\eea
The integral over $Y$ factorizes, the dependence on $B_{\mu \nu}$ cancels out in this integral and reduces to the functional integrals computed earlier in section \ref{sec:TorusQFT}. The remaining factor is given by the following lattice sum,
\bea
\label{4h9}
Z (L, \tau) = {\rm Vol} (T_d)  \sum _{\ell _A, \ell_B \in L} \exp \left \{ -
{ 1 \over 4 \pi \alpha ' \tau_2} (\ell ^\mu _B - \bar \tau \ell ^\mu _A)
(\ell ^\nu _B -  \tau \ell ^\nu _A) (G_{\mu \nu} +  B_{\mu \nu} ) \right \}
\eea
The dependence on $B_{\mu \nu}$ reduces to $-i B_{\mu \nu} \, \ell ^\mu _A \, \ell ^\nu _B / (2 \pi \alpha ')$ and is independent of $\tau$, as is expected for a topological term. Before investigating the properties for a general torus $T_d$, we first look at the much simpler case of $d=1$.

\subsection{T-duality on a circle}

In this subsection, we study the simplest case of toroidal compactification when $d=1$ and exhibit the phenomenon of $T$-duality for a circle. We take $G=1$, and introduce the lattice $L$ associated with a circle of radius $R$, namely $L = 2 \pi R \, \ZZ$. We label the monodromy vectors $\ell_A$ and $\ell_B$, which are one-dimensional in this case,  as follows,
\bea
\label{4j0}
\ell _A & = & 2 \pi R \, n
\no \\
\ell_B & = & 2 \pi R \, m \hskip 1in m,n \in \ZZ
\eea
The sum over momentum and winding modes then simplifies to,
\bea
\label{4j1}
Z(R, \tau) = \sqrt{R^2 \over \alpha'} \sum _{m,n \in \ZZ}  \exp \left \{ -
{ \pi R^2 \over \alpha ' \tau_2} | m - \tau n|^2  \right \}
\eea
The prefactor in $\sqrt{R^2/\alpha'}$ arises from the zero mode of the field $Y$ which, for a circle of radius $R$, gives a finite contribution. The above sum $Z(R,\tau)$  is a rather familiar looking object. To exhibit $T$-duality, we  perform a Poisson resummation on both $m,n$. To do so we compute the Fourier transform in both variables,
\bea
\int _{\RR^2} dm \, dn  \, e^{ - 2 \pi i (m x + ny)} \, \exp \left \{ - { \pi R^2 \over \alpha ' \tau_2} | m - \tau n|^2
\right \} = { \alpha' \over R^2} \, \exp \left \{ - { \pi \alpha ' \over R^2 \tau_2} | y+\tau x|^2 \right \}
\eea
so that Poisson resummation gives us,
\bea
Z(R, \tau) = \sqrt{\alpha '  \over R^2} \sum _{m,n \in \ZZ}  \exp \left \{ -
{ \pi \alpha '  \over R^2 \tau_2} | m + \tau n|^2  \right \}
\eea
Noticing that Poisson resummation has inverted the dependence on $R^2/\alpha'$, we have,
\bea
Z(R,\tau) = Z\left ( { \alpha ' \over R}, \tau \right )
\eea
Thus, a free bosonic string on a circle of radius $R$ has exactly the same partition function as a free bosonic string on a circle of radius $\alpha ' / R$. While we have proven this above for the free string on a genus-one worldsheet only, the effect actually is valid on surfaces of arbitrary genus, and thus holds for fully interacting string theory. The effect is referred to as \textit{$T$-duality}.

\sm

We may organize the double sum in a slightly different way which makes the decomposition into momentum and winding modes transparent. Performing a Poisson resummation on (\ref{4j1}) in $m$ only, we obtain the representation,
\bea
\label{eq:compbosZ}
Z(R,\tau) = (\tau_2)^\half \sum _{m,n \in \ZZ} e^{2 \pi i \tau p_L^2} \, e^{-2 \pi i \bar \tau p_R^2}
\eea
where the left and right momenta are given in terms of the integers $m,n$ by,
\bea
p_L = \half \left ( { \sqrt{\alpha'} \over  R}  m + { R \over \sqrt{\alpha'}} n \right )
& \hskip 1in & p_L+p_R = { \sqrt{\alpha'} \over  R}  m
\no \\
p_R = \half \left ( { \sqrt{\alpha'} \over  R}  m - { R \over \sqrt{\alpha'}} n \right )
& \hskip 1in & p_L-p_R = { R \over \sqrt{\alpha'}} n
\eea
The lattice points $p_L+p_R \in \sqrt{\alpha'}/R \, \ZZ$ are clearly associated with the total momentum of the string, while the lattice points $p_L-p_R \in R/\sqrt{\alpha'} \, \ZZ$ are associated with the winding modes.

\sm

The space of all possible compactifications on the circle is labeled by the radius $R$ modulo the identification $R \equiv \alpha '/R$, a space we may denote by $\RR^+/\ZZ_2$. Clearly, the lattices $L=\sqrt{\alpha'}/R \, \ZZ$ and $L^+=R/\sqrt{\alpha'} \, \ZZ$ are dual to one another, a fact that is reflected in the relation $(p_L+p_R)(p_L-p_R) = mn \in \ZZ$. Thus, the lattice sum may be written alternatively as,
\bea
Z(R,\tau) = (\tau_2)^\half \sum _{p \in L} \sum _{ w\in L^+} e^{ \pi i \tau  (p+w)^2/2} \, e^{- \pi i \bar \tau  (p-w)^2/2}
\eea
At the radius $R = \sqrt{\alpha'}$ the conformal field theory is self-dual and enjoys an enhanced symmetry to the $SU(2)$ Kac-Moody algebra. The radius $R = \sqrt{2 \alpha'}$ is also special in a different sense: it is at this point that the $\mc=1$ Virasoro algebra representation of the conformal field theory becomes reducible to two $\mc=\half$ fermions.

\sm

$T$-duality has several important consequences for string theory, which include,
\begin{enumerate}
\item The fact that (bosonic) string theory on a circle of radius $R$ is physically equivalent to a string theory on a circle of radius $\alpha '/R$ means that we can never physically probe distances smaller than $\sqrt{\alpha '}$. Analogous arguments exist for the energy of scattering processes, namely probing a string with energy $E$ and with energy $1/(\alpha ' E)$ are physically equivalent. It means that string theory has built in its dynamics a {\sm smallest string length scale} $\sqrt{\alpha'}$.
\item Actually, the above picture is modified in the case of superstring theories. In Type II string theories, compactification of Type IIA on a circle of radius $R$ is physically equivalent to Type IIB theory on a circle of radius $\alpha '/R$. In Heterotic string theories, the $E_8 \times E_8$ theory is invariant under $R \leftrightarrow \alpha ' /R$ but the $Spin (32)/\ZZ_2$ theory is exchanged with the theory of open and closed strings Type I.
In both cases, $T$-duality provides a mechanism for a partial unification of the five 10-dimensional superstring theories.
\item $T$-duality extends to string theories on curved manifolds under certain conditions. For example, on Calabi-Yau spaces, it is promoted to \textit{mirror symmetry}.
\end{enumerate}

\subsection{T-duality on a torus $T_d$}

For a general torus $T_d= \RR^d /L$ with $d \geq 2$ new effects arise. We consider the lattice sum of (\ref{4h9}) and introduce a basis $\{ v_1 ^\mu , \cdots, v_d ^\mu \}$ with $\mu =1, \cdots,d$ for the lattice $L$. Generalizing the conventions of (\ref{4j0}) used for the circle, we have,
\bea
\ell _A ^\mu & = & 2 \pi \sum _{i=1}^d n^i  \, v_i ^\mu \hskip 1in n^i \in \ZZ
\no \\
\ell _B ^\mu & = & 2 \pi \sum _{i=1}^d m^i \, v_i ^\mu  \hskip 1in m^i \in \ZZ
\eea
What enters $Z(L,\tau)$ is the combination,
\bea
{1 \over \alpha '} (G_{\mu \nu} + B_{\mu \nu} ) v^\mu _i v^\nu _j = \cG _{ij}+ \cB_{ij}
\eea
where $\cG_{ij}$ and $\cB_{ij}$ are respectively symmetric and anti-symmetric under the interchange of the indices $i,j$.
Using matrix notation for $m,n,\cG, \cB$, we have,
\bea
\label{4k2}
Z (L, \tau) = {\rm Vol} (T^d)  \sum _{m, n \in \ZZ^d} \exp \left \{ -
{ \pi \over \tau_2} (m + \tau n)^\dagger  (\cG +\cB) (m +  \tau n) \right \}
\eea
Since $v_i$ forms a basis of $L$, the volume of the torus is given by,
\bea
{\rm Vol}(T_d) = (\det \cG)^\half
\eea
up to some factor of $2 \pi$ which will be of no interest here.

\subsubsection{A first look at $T$-duality}

The argument of the exponential may be decomposed as follows,
\bea
-  { \pi \over \tau_2} (m + \tau n)^\dagger  (\cG +\cB) (m +  \tau n)
& = &
- { \pi \over \tau_2} (m + \tau_1 n)^t \cG (m + \tau_1 n)
\no \\ &&
- \pi \tau_2 n^t \cG n - 2 \pi i m^t \cB n
\eea
As a result, each term in the exponential is invariant under a shift in $\cB$ by an arbitrary anti-symmetric matrix with integer entries, or in components,
\bea
\cB_{ij}  \to \cB_{ij}  + b_{ij} \hskip 1in b_{ij}=-b_{ji}\in \ZZ
\eea
This invariance is an immediate consequence of the fact that, for constant $B$,  the $B$-term  is topological and its values on the field configurations for the torus $T_d$ are quantized. The corresponding symmetry is the analogue of the $T$-transformation $\tau_1 \to \tau_1 + 1$  for $SL(2,\ZZ)$, and will provide some of the generators of the full $T$-duality group for $T_d$.

\sm

Next, we wish to obtain the analogue of the $S$-transformation. To do so, we Poisson resum in both $m,n$.
It is helpful to express the quadratic form combining $m,n$ into a single column matrix, so that,
\bea
- { \pi \over \tau_2} (m + \tau n)^\dagger  (\cG +\cB) (m +  \tau n)
- \pi \left ( \begin{matrix}m \cr n \cr \end{matrix} \right ) ^t \cM \left ( \begin{matrix} m \cr n \cr \end{matrix} \right )
\eea
where $\cM$ is given by,
\bea
\cM = {1 \over \tau_2}  \left (  \begin{matrix}  \cG &  & \tau_1  \cG + i \tau_2 \cB \cr && \cr
\tau_1 \cG - i \tau_2 \cB & & (\tau_1^2 + \tau_2^2) \cG \cr \end{matrix} \right )
\eea
The inverse matrix may be expressed as follows,
\bea
\cM^{-1} = {1 \over \tau_2} \left ( \begin{matrix} 0 && I \cr && \cr -I && 0 \cr \end{matrix} \right )^t
\left (  \begin{matrix}  \tilde \cG &  & \tau_1  \tilde \cG + i \tau_2 \tilde \cB \cr && \cr
\tau_1 \tilde \cG - i \tau_2 \tilde \cB & & (\tau_1^2 + \tau_2^2) \tilde \cG \cr \end{matrix} \right )
\left ( \begin{matrix} 0 && I \cr && \cr -I && 0 \cr \end{matrix} \right )
\eea
where $\tilde \cG$ and $\tilde \cB$ are given by the $S$-transformation,
\bea
\tilde \cG = \Big ( \cG + \cB^t \, \cG^{-1} \, \cB \Big )^{-1}
\hskip 1in
\tilde \cB = \cG^{-1} \cB \, \tilde \cG
\eea
The action of $S^2$ is obtained by repeating the transformation,
\bea
\hat \cG = \Big ( \tilde \cG + \tilde \cB^t \, \tilde \cG^{-1} \, \tilde \cB \Big )^{-1}
\hskip 1in
\hat \cB = \tilde \cG^{-1} \tilde \cB \, \hat \cG
\eea
and eliminating $\tilde \cB$ and $\tilde \cG$. The result is $\hat \cB=\cB$ and $\hat \cG = \cG$, so that the action of $S^2$ on $\cG, \cB$ is the identity.\footnote{Although in $SL(2,\ZZ)$ the matrix $S$ squares to $S^2=-I$, its action on $\tau$ reduces to the identity; the above statement is the direct analogue thereof.} We shall show shortly that  $S$ and the integer translations of $\cB$ generate the duality group $SO(d,d, \ZZ)$.

\subsubsection{Holomorphic block decomposition}

As in the case of the scalar field theory valued in the circle, it is instructive to perform a Poisson resummation only on the integers $m$, and expose the holomorphic structure in $\tau$. To do so, we start from the decomposition of the quadratic form given earlier, and we need the following Fourier transform,
\bea
\int _{\RR^d} d^d x \, e^{-2 \pi i x^t m} \, \exp \left \{
- { \pi \over \tau_2} (x + \tau_1 n)^t \cG (x + \tau_1 n)  - 2 \pi i x^t \cB n \right \}
\no \\ && \qquad =
{ (\tau_2)^{{ d \over 2}} \over (\det \cG)^\half } \exp \Big \{
2 \pi i \tau   p_L^2 - 2 \pi i \bar \tau  p_R^2 \Big \}
\eea
where we obtain after minimal simplifications,
\bea
4 p_L^2 & = & (m+\cB n)^t \cG^{-1} (m+\cB n) + n^t \cG n + 2 m^t n
\no \\
4 p_R^2 & = & (m+\cB n)^t \cG^{-1} (m+\cB n) + n^t \cG n - 2 m^t n
\eea
Note that while the integers $n$ were  defined with an upper index, as were the old $m$, by contrast the Poisson resummation variable $m$ have lower indices. So, it is instructive to write out the above relations in terms of indices,
\bea
4 p_L^2 & = &  \Big (m_i+ \cB_{ii'}  n^{i'} \Big  ) (\cG^{-1})^{ij} \Big (m_j+ \cB_{jj'} n^{j'} \Big ) + n^i \cG_{ij}  n^j  + 2  m_i  n^i
\no \\
4 p_R^2 & = &  \Big (m_i+ \cB_{ii'}  n^{i'} \Big  ) (\cG^{-1})^{ij} \Big (m_j+ \cB_{jj'} n^{j'} \Big ) + n^i \cG_{ij}  n^j  - 2  m_i  n^i
\no
\eea
In evaluating the partition function, the factor ${\rm Vol} (T^d)$ cancels out, and we find,
\bea
Z(L,\tau) = (\tau_2)^{{d \over 2}} \sum _{m,n \in \ZZ^d} \exp \Big \{ 2 \pi i \tau \, p_L^2 - 2 \pi i \bar \tau \, p_R^2 \Big \}
\eea
with $p_L^2, p_R^2$ functions of $m,n$ as defined above. The invariance under integer shifts of the matrix $\cB$ is now manifested by the fact that such a shift may be compensated in the sum by a shift in the integers $m$.

\subsubsection{$T$-duality in terms of the lattice}

The description of the torus with both an arbitrary lattice $L$ and an arbitrary metric $G$ is in fact redundant. By general linear transformation, we may always make the metric equal to the identity. Geometrically, this is equivalent to expressing the metric $\cG_{ij}$  in terms of an orthonormal frame $e_\mu {}^a$ and its inverse $e_a {}^\mu$ by
\bea
\cG_{ij} = e_i {}^a e_j{}^b \delta _{ab}
\eea
where $i$ are ``Einstein indices" and $a$ are \textit{frame indices}. The lattice $L$ with identity metric is then generated by the vector $ e_a {}^i$, while the dual lattice $L^+$ is generated by the dual vector $e_i {}^a$. Defining also the $B$-field in frame indices,
\bea
\mB _{ab} = \cB_{ij} \, e_a{}^i e_b{}^j
\eea
and defining the lattice and dual lattice vectors,
\bea
\mu _a & = & m_i \, e_a{}^i
\no \\
\nu ^a & = & n^i \, e_i {}^a
\eea
we find that $p_L^2$ and $p_R^2$ may be interpreted as square of vectors, respectively given by,
\bea
p_L & = & \half \Big (   \mu + \mB \, \nu + \nu   \Big )
\hskip 1in \mu \in L^+
\no \\
p_R & = & \half \Big ( \mu + \mB \, \nu -  \nu    \Big )
\hskip 1in \nu \in L
\eea
\bea
p_L\cdot p_L - p_R \cdot p_R = \mu \cdot \nu = m^t n \in \ZZ
\eea
To parametrize the space of all lattices, we proceed as follows. Given a lattice, it may be deformed continuously provided we maintain the condition $p_L\cdot p_L - p_R \cdot p_R  \in \ZZ$. This condition is invariant under the group $SO(d,d;\RR)$. But rotations under $SO(d)$ on both $p_L$ and $p_R$ are physically indistinguishable from the original lattice. Therefore, any lattice and its dual may be parametrized by a point in the coset space,
\bea
SO(d,d;\RR) / (SO(d;\RR) \times SO(d;\RR))
\eea
The dimension of this space is,
\bea
\half (2d)(2d-1) - 2 \times \half d(d-1) = d^2
\eea
This dimension is easily understood in the original formulation with arbitrary metric $G$ and $B$-field in a square torus: together they have $d^2$ components.

\sm

Two different points in $SO(d,d;\RR) / (SO(d;\RR) \times SO(d;\RR))$ may correspond to the same lattice.
We already know that this is the case when $\cB_{ij}$ are shifted by integers, and when the $S$ transformation is applied to both $\cG$ and $\cB$. The shift is realized in terms of an anti-symmetric matrix $K$ with integer entries,
\bea
\gamma = \left ( \begin{matrix} I + K & - K \cr K & I - K \cr \end{matrix} \right )
\hskip 1in
\left ( \begin{matrix} \tilde p_L \cr \tilde p_R \cr \end{matrix} \right ) =  \gamma
\left ( \begin{matrix} p_L \cr  p_R \cr \end{matrix} \right )
\eea
whose effect is to shift $\cB \to \cB + 2 K$. The matrix $\gamma$ is easily seen to belong to $SO(d,d;\RR)$, and to be  a translation or \textit{parabolic} element, such that,
\bea
\gamma ^n = \left ( \begin{matrix} I + nK & - nK \cr nK & I - nK \cr \end{matrix} \right )
\eea
The $S$-transformation is given by,
\bea
S = \left ( \begin{matrix} 0 & I \cr -I & 0 \cr \end{matrix} \right )
\eea
The space of inequivalent tori $T_d$ from the string point of view are thus parametrized by the double coset space,
\bea
SO(d,d;\ZZ) \backslash SO(d,d; \RR) / SO(d;\RR) \times SO(d;\RR)
\eea
In the case of heterotic string, the result extends to include the left-moving 16 dimensional torus, and the duality group is $SO(d, d+16;\ZZ)$.

\subsection{Rationality and complex multiplication}
\label{eq:CMandNLSM}

In section \ref{sec:3dgrav} we introduced the torus partition function of a two-dimensional conformal field theory, e.g.  in (\ref{eq:CFTtoruspartfunct}).  A CFT is said to be \textit{rational} if the sum over characters is finite. In a sense, a rational CFT is one which is exactly solvable, and great effort has been expended towards obtaining a classification of such theories.
In this section, we ask when a $T_2$ non-linear sigma model is rational.

\subsubsection{The case $T_1$}

To begin, let us consider the simpler case of an $S^1$ non-linear sigma model, with the circle having radius $R$. In this case, the partition function may be computed to be
\bea
Z(R, \tau) = {1 \over |\eta(\tau)|^2} \sum_{m,n \in \ZZ} q^{\half \left({m \over R} + {n R \over 2}\right)^2}\bar q^{\half \left({m \over R} -{n R \over 2}\right)^2}
\eea
This comes from (\ref{eq:compbosZ}) by choosing $\alpha' = 2$ and dressing with a factor of $|\eta(\tau)|^2$ to account for Virasoro descendants. To address the question of rationality, we must ask when this can be written as a sum over a finite number of characters. It turns out that this is possible if $R^2 \in \QQ$. Indeed, say that $R^2 = {2k \over \ell}$ for $k, \ell \in \ZZ$. In this case we can write the partition function as,
\bea
Z\left(\sqrt{{2k \over \ell}},\, \tau\right) ={1 \over |\eta(\tau)|^2}\sum_{m,n\in \ZZ} q^{{1 \over 4 k \ell} (m \ell + n k)^2} \overline q^{{1 \over 4 k \ell} (m \ell - n k)^2}
\eea
It will be useful to write
\bea
m = \m + 2 k r \hspace{0.8 in} n = \n + 2 \ell s
\eea
where $\m \in \{0,\dots, 2k -1\}$, $\n \in \{0,\dots, 2\ell-1\}$, and $r, s \in \ZZ$.
We may then split the sums as
\bea
Z\left(\sqrt{{2k \over \ell}},\, \tau\right) ={1 \over |\eta(\tau)|^2}\sum_{\m =0}^{2k-1} \sum_{\n =0}^{2 \ell-1} \sum_{r, s \in \ZZ}q^{{1 \over 4 k \ell} (2 k \ell (r+s) + \m \ell + \n k)^2} \overline q^{{1 \over 4 k \ell} (2 k \ell (r-s) +\m \ell - \n k)^2}
\eea
We now again change summation variables to,
\bea
\tilde{\m} = \m \ell + \n k \hspace{0.5 in} \tilde{\n} = \m \ell - \n k \hspace{0.5in} \rho = r+s \hspace{0.5 in} \sigma = r - s
\eea
By definition we have,
\bea
\label{eq:rattorus}
\tilde{\m} + \tilde{\n} \equiv 0 \,\,\,({\rm mod}\,\,2 \ell) \hspace{0.8 in}\tilde{\m} - \tilde{\n} \equiv 0 \,\,\,({\rm mod}\,\,2k)
\eea
Hence we can rewrite the partition function as a finite sum,
\bea
Z\left(\sqrt{{2k \over \ell}},\, \tau\right)  = \sum'_{\substack{\tilde{\m}, \tilde{\n} = 0,\dots,2k \ell - 1}} \chi_{\tilde \m}(\tau) \overline \chi_{\tilde \n}(\bar \tau)
\eea
where the primed sum denotes a sum subject to restrictions (\ref{eq:rattorus}). The result has been written in terms of the characters,
\bea
\chi_{\tilde \m}(\tau) = {1 \over \eta(\tau)}\sum_{\rho \in \ZZ}q^{k \ell \left(\rho + {\tilde \m \over 2 k\ell } \right)^2}
\eea
That these are the correct characters for $T_1$ is confirmed by noting that the compact boson at radius $R^2 = {2k \over \ell}$ is equivalent to a $\ZZ_\ell$ orbifold of the $U(1)_k$ current algebra theory.

\subsubsection{The case $T_2$}

We now proceed to the case of $M=T_2$. For the moment we will consider only tori that are trivial products of circles, with radii $R_1$ and $R_2$.  Then the CFT factorizes into two $S^1$ CFTs studied above, and the total CFT is rational if and only if the two components CFTs are. In other words, we must take $R_{1,2}^2 \in \QQ$. Say concretely that,
\bea
R_1^2 = {2k_1 \over \ell_1} \hspace{0.8 in}R_2^2 = {2k_2 \over \ell_2} \hspace{0.5 in} k_i, \ell_i \in \ZZ
\eea
The complex structure of the spacetime torus, which we will denote as $\rho$ to avoid confusion with the complex structure $\tau$ of the worldsheet torus, is given by $\rho = i R_1 / R_2$ up to $SL(2, \ZZ)$ transformations. Hence  $\rho$ satisfies a quadratic equation with integer coefficients,
\bea
\ell_1 k_2 \rho^2 + \ell_2 k_1 = 0
\eea
This is precisely the condition that the spacetime $T_2$ admit complex multiplication. Though we have restricted ourselves to tori which are trivial products of circles here, one can argue by similar methods that for generic elliptic curves $\CC/L$, rationality of the CFT is tantamount to the presence of complex multiplication on $\CC/L$.

\subsection*{$\bullet$ Bibliographical notes}

Toroidal compactification of higher space-time dimensions  goes back to [249] and was generalized to the Heterotic string in [250, 251]. It  was used to map out the moduli space of $\mc=1$ unitary conformal field theories in [252].  Comprehensive overviews of toroidal compactification of the bosonic string may be found in the first volume of  Polchinski's book [171], while toroidal compactification of the superstring, T-duality, and related topics are discussed in detail in the second volume of Polchinski's book  [172] as well as in the book by Kiritsis [177]. Further useful references on correlation functions for compact scalars, representations of current and Kac-Moody algebras and related topics may be found in [253] and [254]. For the contents of section \ref{eq:CMandNLSM}, we refer again to  [159], and to section \ref{sec:CM}  on complex multiplication of these lecture notes.

\newpage

\section{S-duality of Type IIB superstrings}
\setcounter{equation}{0}
\label{sec:IIB}

In this section, we shall draw together a number of different strands of inquiry addressed earlier  in these lecture notes.  We shall study the interplay between superstring amplitudes, their low energy effective interactions, Type~IIB supergravity, and the S-duality symmetry of Type~IIB superstring theory. We begin with a brief review of Type~IIB supergravity which, in particular, provides the massless sector of Type IIB superstring theory. We then discuss how the $SL(2,\RR)$ symmetry of Type IIB supergravity is reduced to the $SL(2,\ZZ)$ symmetry of Type IIB superstring theory via an anomaly. We conclude with a discussion of how the low energy effective interactions induced by string theory on supergravity may be organized in terms of modular functions and forms under this $SL(2,\ZZ)$ symmetry, and match the predictions provided by perturbative calculations of section \ref{sec:SA}.

\subsection{Type IIB supergravity}
\label{sec:IIBSUGRA}
In section \ref{sec:SA} we discussed the expansion of bosonic and superstring amplitudes in powers of the string coupling constant~$g_s$. While this expansion produces only an asymptotic series in~$g_s$, it has the advantage of being valid for all energy ranges. A different, and complementary, expansion of string theory is in powers of $\alpha'$, also referred to as the \textit{low energy expansion}. This is the expansion that was used for the string amplitudes in subsection~\ref{sec:12.7}. To leading order in this approximation, one omits the effects of all the massive string states, whose mass-squares are set  by the string scale $1/\alpha'$. The remaining massless states precisely make up the contents of the corresponding supergravity.

\sm

To each 10-dimensional perturbative string theory, namely Type~I, Type~IIA, Type~IIB, and Heterotic $E_8 \times E_8$ and ${\rm Spin}(32)/\ZZ_2$ there is a corresponding supergravity. The Type~II theories have the maximal number of 32 supersymmetries, while the other three have 16 supersymmetries. We shall focus here on Type~IIB as its corresponding Type~IIB superstring theory has $SL(2,\ZZ)$ duality symmetry, which is the focus of these lecture notes.

\sm

The  contents of Type IIB supergravity consist of the following fields,
\bea
G \, & \hskip 1in & \hbox{ the metric, whose excitations are the graviton}
\no \\
C_2 & \hskip 1in & \hbox{ the complex 2-form potential}
\no \\
\tau \, & & \hbox{ the complex axion/dilaton field}
\no \\
C_4 && \hbox{ the real 4-form potential, whose field strength is self-dual}
\no \\
\psi \, && \hbox{ chirality $+$ Weyl spinor gravitino}
\no \\
\lambda \, && \hbox{ chirality $-$ Weyl spinor dilatino}
\eea
The fields $G, C_2, \tau$, and $C_4$ are bosonic fields, while $\psi, \lambda$ are fermions. Here we shall concentrate on the properties and field equations of the bosonic fields. The field strength of $C_2$ is $F_3=dC_2$, while the field strengths of $\tau$ and $C_4$ are given as follows,
\begin{align}
P & =  f^2 \, dB  & B & = (1+i\tau)/(1-i\tau)
\no \\
Q & =  f^2 \, \Im (B d \bar B) & f^2 & = (1-|B|^2)^{-1}
\no \\
F_5 & =  d C_4 + \tfrac{i}{16} \big ( C_2  \wedge \bar F_3 - \bar C_2 \wedge F_3 \big ) & \star F_5 & = F_5
\end{align}
The axion-dilaton field $\tau$ is valued in the hyperbolic upper half-plane $\tau_2=\Im \tau >0$.  The  field equation for $F_5$ coincides with its Bianchi identity $dF_5 =\tfrac{i}{8} F_3  \wedge \bar F_3$ in view of its self-duality relation. In terms of the components of the fields $P,Q,F_5$, and $K  = f(F_3 - B \bar F_3) $,
\begin{align}
P & =P_\mu \, dx^\mu & K &  = \tfrac{1}{3!} K_{\mu \nu \rho} \, dx^\mu \wedge dx^\nu \wedge dx^\rho
\no \\
Q& =Q_\mu \, dx^\mu & F_5 & = \tfrac{1}{5!}F_{5\mu \nu \rho \sigma \tau} \, dx^\mu \wedge dx^\nu \wedge dx^\rho \wedge dx^\sigma \wedge dx ^\tau
\end{align}
the remaining field equations are conveniently expressed as follows,
\bea
0 & = & \nabla ^\mu P_\mu - 2 i Q^\mu P_\mu + \tfrac{1}{24} K_{\mu \nu \rho} K^{\mu \nu \rho}
\no \\
0 & = & \nabla ^\rho K_{\mu \nu \rho} - i Q^\rho K_{\mu \nu \rho} - P^\rho \bar K_{\mu \nu \rho} +\tfrac{2i}{3} F_{5 \mu \nu \rho \sigma \tau} K^{\rho \sigma \tau}
\no \\
0 & = & R_{\mu \nu} -P_\mu \bar P_\nu - \bar P_\mu P_\nu - \tfrac{1}{6} (F_5^2)_{\mu \nu}
- \tfrac{1}{4} \Re( K_\mu {}^{\rho \sigma} \bar K_{\nu \rho \sigma} ) + \tfrac{1}{48} G_{\mu \nu} K^{\rho \tau \sigma} \bar K_{\rho \tau \sigma}
\eea
where $R_{\mu \nu}$ is the Ricci tensor for the metric $G_{\mu \nu}$.   The axion-dilaton field $\TAU$ may be related to the axion field $\chi$ and the dilaton field  $\Phi$,
\bea
\TAU = \chi + i e^{- \Phi}
\eea
In view of the self-duality condition on $F_5$, there does not exist a Lorentz-covariant action from which the field equations of Type IIB supergravity may be deduced using the standard variational principle. Instead one may define the  following action,\footnote{Our notation for the contraction of two rank $n$ anti-symmetric tensors is $F_n \cdot F'_n = \tfrac{1}{n!} F_{n \mu_1 \cdots \mu_n} F_n ^{\prime \mu_1 \cdots \mu_n}$.}
\bea
S_{{\rm IIB}} & = & { 1 \over 2 \kappa_{10}^2} \int d\mu_G \left (
R - { \p_\mu \TAU \p^\mu \bar \TAU \over 2 \TAU_2^2}
-   { (F_3^1 - \TAU F_3^2 )\cdot (F_3^1 - \bar \TAU F_3^2 ) \over 2 \, \TAU_2}
- {1 \over 4} F_5 \cdot F_5 \right )
\no \\ &&
- {2 \over \kappa_{10}^2} \int C_4 \wedge F_3^1 \wedge F_3^2
\eea
Here $d\mu_G$ is the invariant volume form for the metric $G$; $\kappa_{10}^2 $ is related to the 10-dimensional Newton constant; and  we have $F_3 = F_3^1 + i F_3 ^2$ and $\TAU=\TAU_1 + i \TAU_2$ for real field components $F_3^1, F_3^2, \TAU_1, \TAU_2$. Importantly, the action $S_{{\rm IIB}}$ should be considered for a field $F_5$ that  is \textit{unconstrained by the self-duality condition}. The field equations for Type IIB supergravity may be obtained from this action by varying the \textit{unconstrained field $F_5$}, along with all the other fields,  and only subsequently imposing the self-duality condition $\star F_5 = F_5$.

\subsection{From $SL(2,\RR)$ to $SL(2,\ZZ)$}

The Type IIB supergravity field equations are invariant under the group $SL(2,\RR)$,  under which the metric $G$ and the anti-symmetric fields $C_4$ and $F_5$ are invariant, and the  fields $P, Q, K, \tau$ and $F_3=F_3^1+iF_3^2$  transform  as follows,
\begin{align}
P & \to  P'= e^{2 i \theta} P  & \tau & \to   \tau' = (a \tau +b)/(c \tau+d)
\no \\
Q & \to  Q' = Q + d \theta & F_3^1 & \to (F_3^1)'= a F_3^1 + b F_3^2
\no \\
K & \to  K' = e^{i \theta } K & F_3^2 & \to (F_3^2)'= c F_3^1 + d F_3^2
\end{align}
The variable $\theta$ is given by,
\bea
e^{i \theta} = \left ( { (\tau+i)  ( a \bar \tau -i c \bar \tau +b-id  )  \over (\bar \tau-i) \big ( a \tau + ic \tau +b+id \big )}  \right )^\half
\eea
The axion-dilaton takes values in the coset $SL(2,\RR)/U(1)$ and the phase $e^{i \theta}$ performs a field-dependent $U(1)$ rotation. The gravitino field $\psi$ and the dilatino field $\lambda$ transform with $U(1)$ phase factors under $SL(2,\RR)$, given by $\psi \to \psi' = e^{ i \theta /2} \psi $ and $\lambda \to \lambda ' = e^{3i \theta /2} \lambda$.

\sm

The $SL(2,\RR) $ symmetry discussed above is a symmetry of the classical supergravity field equations, as well as of the action $S_{{\rm IIB}}$ and the self-duality condition $\star F_5 = F_5$. A classical symmetry cannot always be promoted to a symmetry of the corresponding quantized field theory, in which case the symmetry is said to suffer an \textit{anomaly}. Phase rotations on chiral fermions often suffer anomalies, and this is also the case here with the field-dependent $U(1)$ transformations acting on the gravitino and dilatino chiral fermions. A treatment of anomalies is beyond the scope of this paper, so we shall just state the result.

\sm

It turns out that the continuous $SL(2,\RR)$ symmetry is broken by the $U(1)$-anomaly, but the quantum theory retains a discrete symmetry, under which the axion field is shifted by an integer $ \chi \to \chi'= \chi + b$ with $b \in \ZZ$. The discrete nature of the remaining symmetry is due to the fact that the axion field couples to the topological charge density of the $D_{-1}$ instantons in Type IIB superstring theory, whose total charge is quantized. The shift in the axion field $ \chi \to \chi' = \chi + b$ with $b \in \ZZ$ that survives quantization also shifts the complex axion-dilaton field $\TAU \to \TAU' = \TAU+b$, and is supplemented by the transformation $\TAU \to -1/\TAU$. Together these transformations generate the discrete group $SL(2,\ZZ)$. This symmetry extends to a symmetry of the entire Type IIB superstring theory.

\sm

This phenomenon is actually familiar from an extension of the Standard Model in which an axion field is included. Continuous shifts in the axion field again suffer an anomaly but discrete shifts survive. In this case the discrete nature of the shift is due to the coupling of the axion field to the topological charge density of Yang-Mills instantons whose topological charge is quantized in terms of the second Chern class of the gauge group.

\subsection{Low energy effective interactions}

Supergravity, as a classical theory, is valid for all values of the string coupling. Thus, it permits  investigations into certain strong coupling phenomena such as solitonic states, NS branes and D-branes. Supergravity has been an invaluable tool in the search for semi-realistic compactifications of superstring theory, such as on tori, oribifolds, and Calabi-Yau spaces.

\sm

However, supergravity alone is  not a consistent quantum theory. In 10 space-time dimensions, it exhibits UV divergences starting at one loop. The UV convergence situation is somewhat improved by lowering the dimension of space-time. In four space-time dimensions the four-graviton amplitude is UV convergent up to five loops, but it is likely that non-renormalizable UV divergences start occurring at seven loops, rendering the quantum theory inconsistent, or more accurately, incomplete. Its UV completion is precisely superstring theory. From the vantage point of superstring theory, supergravity should be thought of as an \textit{effective low energy  field theory} that captures the dynamics of the massless states of Type IIB string theory, to leading order  in the $\alpha'$ expansion.

\sm

The validity of an effective field theory may be extended beyond its leading order contributions. Viewed as an effective field theory of Type IIB superstring theory, the corresponding supergravity may be supplemented with contributions of higher order in $\alpha'$. Such contributions are due to the effects of massive string states whose mass, we recall, is of order $1/\sqrt{\alpha'}$ and are referred to as \textit{effective interactions} In the approximation where the momenta used to probe string amplitudes are small compared to $1/\sqrt{\alpha'}$, the effective interactions are local, as is illustrated  in Figure \ref{13.fig:4} for the exchange of a massive string state. Although these string induced effective interactions are highly suppressed they provide systematic corrections to supergravity in an expansion in powers of $\alpha'$.

\begin{figure}[htb]
\begin{center}
\tikzpicture[scale=0.65]
\begin{scope}[xshift=10cm]
\draw [very thick] (-1,1) to (0,0) ;
\draw [very thick] (-1,-1) to (0,0) ;
\draw [ultra thick] (0,0.05) to (2,0.05) ;
\draw [ultra thick] (0,-0.05) to (2,-0.05) ;
\draw [very thick] (2,0) to (3,1) ;
\draw [very thick] (2,0) to (3,-1) ;
\draw [very thick,->] (4,0) to (4.5,0) ;
\draw [very thick] (6,1) to (7,0) ;
\draw [very thick] (6,-1) to (7,0) ;
\draw [very thick] (7,0) to (8,1) ;
\draw [very thick] (7,0) to (8,-1) ;
\draw (7,0) node{$\bullet$};
\end{scope}
\endtikzpicture
\caption{\textit{The exchange of a massive string state, indicated by the thick line in the left panel, induces a local effective interaction indicated by a thick dot in the right panel. }\label{13.fig:4}}
\end{center}
\end{figure}

The space-time we observe has dimension four and is approximately flat when its curvature is measured in units of the Planck scale. Therefore, if the space-time of superstring theory is to be 10 then the six extra dimensions must have  radii that are smaller than the smallest length scales accessible to experiment today. Physically realistic superstring theories are often based on space-times of the form $\RR^{10-d} \times M_d$ where $M_d$ is a compact manifold or orbifold and the physically observed dimension of space time corresponds to $d=6$. For the case where $M_d$ is a flat torus, superstring perturbation theory continues to be well-understood.  When $M_d$ is an orbifold of a torus,  the space $M_d$ is flat away from isolated point-like singularities, and string theory on toroidal orbifold spaces  still lends itself to reasonably  explicit solutions. However, when $M_d$ is an arbitrary curved manifold, the predictions of superstring theory are much more difficult to obtain. This is the case even when space-time is a curved maximally symmetric space such as $AdS_5 \times S^5$, for which the quantization of the superstring is still largely an unresolved problem. An important exception is when the curvature of $M_d$ is uniformly small compared to the Planck scale, a case that is referred to as the large radius expansion, and that we shall discuss next.

\sm

Restricting, for example, to purely gravitational effects, the supergravity Lagrangian reduces to the Einstein-Hilbert term given by the Ricci scalar for the space-time metric. In Type II superstring theory, for example, the lowest order $\alpha'$ correction to the  Einstein-Hilbert action is given by a term which we symbolically represent by $\cR^4$ where $\cR$ stands for the Riemann tensor, and the contribution to the action is obtained via a special contraction of the four factors, consistent with space-time supersymmetry, as will be explained below. This effective interaction arises at order $(\alpha')^3$. Higher order effective interactions involve more derivatives and higher powers of $\alpha'$ and are symbolically represented by $D^{2k} \cR^4$ where $D$ represents a covariant derivative, again suitably contracted.

\subsection{$SL(2, \ZZ)$ duality in Type IIB superstring theory}

Type IIB superstring theory exhibits $SL(2, \ZZ)$ duality symmetry which leaves the space-time metric $G_E$  in Einstein frame and the self-dual five form invariant, and transforms the 3-form flux field strengths linearly into one another,
\bea
\left ( \bma F_3^1 \cr F_3^2 \ema \right ) \to
\left( \begin{matrix} a & b \cr c & d \cr  \end{matrix} \right) \left ( \bma F_3^1 \cr F_3^2 \ema \right )
\hskip 1in
\left ( \begin{matrix} a & b \cr c & d \cr  \end{matrix} \right ) \in SL(2,\ZZ)
\eea
As for the axion-dilaton field obtained by combining the dilaton field $\Phi$ and the axion field $\chi$ into a complex scalar field $\TAU = \chi + i e^{-\Phi}$, transformations under $SL(2,\ZZ)$ occur as follows,
\bea
\label{8b1}
\TAU \to { a \TAU + b \over c \TAU + d}
\eea
The transformation properties of the fermionic fields are more complicated, and will not be reproduced here. In fact, when one accounts for the fermionic fields the true duality symmetry of Type IIB is \textit{not} $SL(2, \ZZ)$, but rather an extension of it known as the meta-linear group $ML^+(2, \ZZ)$; see [255] for details. This subtlety will not affect the purely bosonic analysis below.

Being a symmetry of Type IIB string theory, $SL(2,\ZZ)$ must be a symmetry of the low energy effective action of the theory. The effective interactions accessible from the four-graviton amplitude are of the form $D^{2k} \cR^4$, as explained above, but they have coefficients that depend upon the vacuum expectation value of the dilaton $g_s = e^\Phi$ and the axion $\chi$. Expressing the effective action in terms of the Einstein frame metric $G_E$, with associated Riemann tensor $\cR_E$, and the vacuum expectation value $\tau$, we find to  order $(\alpha ')^7$,
\bea
\label{4b2}
\int d\mu_G  \Big ( \cE_0 (\TAU) \cR_E^4 + \cE_4 (\TAU) D^4 \cR_E^4 +\cE_6 (\TAU) D^6 \cR_E^4 + \cE_8 (\TAU) D^8 \cR_E^4 + \cO((\alpha ')^8) \Big )
\eea
where the coefficients $\cE_{2k}(\TAU)$ are real-valued scalar functions of $\TAU$. Derivatives of $\tau$ may occur as well but will be systematically omitted here for simplicity.  Since the action must be $SL(2,\ZZ)$-invariant and, in Einstein frame, $G_E$ and $\cR_E$ are invariant as well, we see that each coefficient $\cE_{2k}(\TAU)$ must be invariant,
\bea
\cE_{2k} \left (  { a \TAU + b \over c \TAU + d}  \right ) = \cE_{2k} (\TAU)
\hskip 1in
\left ( \begin{matrix} a & b \cr c & d \cr  \end{matrix} \right ) \in SL(2,\ZZ)
\eea
a property which makes them into \textit{non-holomorphic modular functions}.

\sm

String perturbation theory calculations are carried out in the string frame metric which we shall denote by $G_{\mu \nu}$ and which is related to the Einstein frame metric by $G_{\mu \nu} = e^{\Phi /2} G_{E \mu \nu}$. Converting the expression for the effective action of (\ref{4b2}) to string frame may be done by using the following relations (for constant values of $\tau$),
\bea
\sqrt{G_E} \, \cE_{2k} (\TAU) \, D_E ^{2k} \cR_E ^4 = e^{ (k-1)\Phi/2} \, \sqrt{G} \, \cE_{2k} (\TAU) \, D ^{2k} \cR ^4
\eea
where $D$ and $\cR$ now stand respectively for the covariant derivative and Riemann tensor in the string frame. Now let us carry out the following exercise: express combinations of the non-holomorphic Eisenstein series with effective interactions $D_E^{2k} \cR_E ^4$ in the Einstein frame such that the leading perturbative contribution in string frame is tree-level,
\bea
\half \pi^{3/2} \sqrt{G_E} \, E_{{ 3 \over 2}} (\TAU)  \cR_E ^4 & = &
e^{-2 \Phi} \zeta (3) \cR^4 +{ \pi^2 \over 3} \cR^4 + \cdots
\no \\
\half \pi^{5/2} \sqrt{G_E} \, E_{{ 5 \over 2}} (\TAU) D_E^4 \cR_E ^4 & = &
e^{-2 \Phi} \zeta (5) D^4 \cR^4 +{ 2 \pi^4 \over 135} e^{2 \Phi} D^4 \cR^4 + \cdots
\eea
Comparing the terms in the above table which have the $\Phi$-dependence of tree-level perturbative contributions with the known result of (\ref{12.loen}) at genus one and (\ref{12.gen2a}) and (\ref{12.gen2b})  at genus two, we see that their dependence on odd zeta-values is precisely reproduced. This observation would suggest that the coefficients $\cE_{2k}(\TAU)$ in (\ref{4b2}) may be given by the  non-holomorphic Eisenstein series.

\sm

\subsection{Eisenstein series from supersymmetry and S-duality}

We will now show how supersymmetry and S-duality can be used to prove the identifications between $\cE_{2k}(\tau)$ and the appropriate Eisenstein series. We will begin with the case of $\cE_{0}(\tau)$, namely the coefficient of $\cR^4$.  Because the metric $G$ is in the same supermultiplet as all of the other supergravity fields reviewed in section \ref{sec:IIBSUGRA}, supersymmetry requires that  the $\cR^4$ term in the low energy effective action be accompanied by a number of analogous terms related by supersymmetry. To make this more concrete, we first introduce some notation. As we have discussed above, the complex 2-form potential $C_2$, and consequently the corresponding field strength $F_3$,  transforms as a doublet under $SL(2, \ZZ)$. It is useful to package these field strengths into an $SL(2,\ZZ)$ singlet via,
\bea
M = {e^{i \phi}\over \sqrt{- 2 i \tau_2}}(F_3^{1} - \tau F_3^{2}) %\hspace{0.5 in}M^* = {e^{-i \phi}\over \sqrt{- 2 i \tau_2}}(F_3^{(1)} - \overline{\tau} F_3^{(2)})
\eea
where the superscript on $F_3$ is a doublet index. We further define the ``super-covariantized" combination,
\bea
\widehat M_{\mu \nu \rho} = M_{\mu \nu \rho} - 3 \overline{\psi}_{[\mu} \gamma_{\nu \rho]}\lambda - 6 i \overline{\psi}^*_{[\mu}\gamma_\nu \psi_{\rho]}
\eea
obtained by dressing $M$ with appropriate combinations of the gravitino and dilatino fields $\psi$ and $\lambda$. The combination $\widehat M$ has the property that under supersymmetry transformations it does not contain derivatives of the transformation parameter.

\sm

In terms of this supercovariant field strength, the supersymmetric completion to the $\cR^4$ term can be written as [256],
\bea
\label{eq:S3def}
&\vphantom{.}&S^{(3)} = \int d \mu_G \left(\cE_{0}^{(12,-12)} \lambda^{16} + \cE_{0}^{(11,-11)} \widehat M \lambda^{14} + \dots + \cE_{0}^{(8,-8)} \widehat M^8 \right.
\no\\
&\vphantom{.}&\hspace{1.5in}\left. + \dots +   \cE_{0}^{(0,0)}\cR^4 + \dots + \cE_{0}^{(-12,12)} (\lambda^*)^{16}\right)
\eea
where $\lambda^{16}$ and $\widehat M \lambda^{14}$ (which will be the main terms of interest to us below) are short for the following contractions,
\bea
\lambda^{16} &=& {1 \over 16!} \eps_{a_1\dots a_{16}} \lambda^{a_1} \dots \lambda^{a_{16}}
\no\\
\widehat M \lambda^{14} &=& {1 \over 14!} \widehat M_{\mu \nu \rho} (\gamma^{\mu \nu \rho} \gamma^0)_{a_{15} a_{16}} \eps_{a_1\dots a_{16}} \lambda^{a_1} \dots \lambda^{a_{14}}
\eea
The $SL(2,\ZZ)$ symmetry of Type IIB demands that the coefficient functions $\cE_{0}^{(w,-w)}(\tau)$ be modular forms of holomorphic weight $w$ and anti-holomorphic weight $-w$. The coefficient function $\cE_{0}^{(0,0)}(\tau) = \cE_{0}(\tau)$ is the original modular invariant coefficient function of $\cR^4$. The coefficient functions are furthermore required by supersymmetry to satisfy
\bea
\label{eq:relbetweenEww}
\cE_{0}^{(w,-w)}(\tau) = D_{w-1}\dots D_0 \cE_{0}(\tau)
\eea
with $D_w = i \tau_2 \left( {\p \over \p \tau} - i {w \over 2 \tau_2} \right)$ the covariant derivative mapping weight $(w,w')$ modular forms to weight $(w+1, w'-1)$ modular forms. Because of these relations,  constraints on $\cE_{0}^{(w,-w)}$ for any $w$ can imply non-trivial constraints on $\cE_{0}$. We will now see how to use supersymmetry to obtain differential constraints on $\cE_{0}^{(12,-12)}$ and $\cE_{0}^{(11,-11)}$ in particular, which will suffice to prove that $\cE_{0}$ is, up to normalization, equivalent to $E_{3\over 2}(\tau)$.

\sm

Let us begin by schematically writing the low-energy effective action as
\bea
S = {1 \over (\alpha')^4} \left[ S^{(0)} + (\alpha')^3 S^{(3)} + (\alpha')^4 S^{(4)}+ (\alpha')^5 S^{(5)} + \dots  \right]
\eea
with $S^{(3)}$ in particular given in (\ref{eq:S3def}) above. Likewise, a generic SUSY transformation $\delta$ can be expanded in powers of $\alpha'$ as
\bea
\label{eq:SUSYvarexp}
\delta = \delta^{(0)} +  (\alpha')^3 \delta^{(3)} + (\alpha')^4 \delta^{(4)} + (\alpha')^5 \delta^{(5)} + \dots
\eea
Requiring that the action has supersymmetry implies that we must have
\bea
\delta^{(0)} S^{(0)} = \delta^{(0)} S^{(3)} + \delta^{(3)} S^{(0)}  = \delta^{(0)} S^{(5)} + \delta^{(5)} S^{(0)} = \dots = 0
\eea
on the equations of motion.

\sm

We now consider the first two terms in the action,
\bea
\cL^{(3)} = \cE_{0}^{(12,-12)} \lambda^{16} + \cE_{0}^{(11,-11)} \widehat M \lambda^{14}
\eea
A crucial feature of these two terms is that they are related by a subset of the SUSY transformations that do not mix with any of the other terms at this order, though we will have to keep track of variations of the lowest order action $S^{(0)}$ by $\delta^{(3)}$. It is a straightforward exercise to find the lowest-order $\delta^{(0)}$ supersymmetry variation of $\cL^{(3)}$. Denoting the supersymmetry parameter by $\eps$ and keeping only terms proportional to $\lambda^{16} \psi_\mu^*\eps$ gives
\bea
\delta^{(0)} \cL^{(3)} |_{\lambda^{16} \psi_\mu^* \eps} = - 8i \left( \cE_{0}^{(12,-12)}(\tau) + 108 D_{11} \cE_{0}^{(11,-11)}(\tau)\right)
\eea
On the other hand, the $\delta^{(3)}$ variation of $S^{(0)}$ cannot produce any such term. Hence we require $\delta^{(0)} \cL^{(3)}|_{\lambda^{16} \psi_\mu^* \eps} = 0 $ alone, which demands
\bea
\label{eq:SUSYconst1}
D_{11} \cE_{0}^{(11,-11)}(\tau) = - {1\over 108} \cE_{0}^{(12,-12)} (\tau)
\eea
Further constraints can be obtained by considering the term in the supersymmetry variation proportional to $\lambda^{16} \lambda^* \eps^*$. In this case the computation is slightly more involved since now there \textit{is} a contribution from the $\delta^{(3)}$ variation of $S^{(0)}$. The analysis will not be reproduced here, but the result is the following constraint
\bea
\label{eq:SUSYconst2}
\overline{D}_{-12}\, \cE_{0}^{(12,-12)}(\tau) + 3240\, \cE_{0}^{(11,-11)}(\tau) - 90 g = 0
\eea
where $g(\tau, \overline \tau)$ is an unknown function. A final constraint may be obtained by demanding closure of the SUSY algebra on $\lambda^*$, giving
\bea
\label{eq:SUSYconst3}
32 D_{11} g = \cE_{0}^{(12,-12)}(\tau)
\eea
We have now obtained three constraints (\ref{eq:SUSYconst1}), (\ref{eq:SUSYconst2}), and (\ref{eq:SUSYconst3}). Combining them, we may derive a Laplace eigenvalue equation for $\cE_{0}^{(12,-12)}$,
\bea
\Delta \,\cE_{0}^{(12,-12)}(\tau)= \left(-132 + {3\over 4}\right) \cE_{0}^{(12,-12)}(\tau)
\eea
where we have noted that, when acting on weight-$(12,-12)$ modular functions, the Laplacian is given by $\Delta = 4 D_{11} \overline{D}_{-12}$.

\sm

It remains only to translate this differential constraint on $\cE_{0}^{(12,-12)}(\tau)$ to a differential constraint on $\cE_{0}(\tau)$, which can be done using the relation (\ref{eq:relbetweenEww}) between the two. This  gives the following Laplace eigenvalue equation for $\cE_{0}(\tau)$,
\bea
\Delta \cE_{0}(\tau) = {3\over 4}\cE_{0}(\tau)
\eea
Together with the asymptotic expansion obtained from string perturbation theory, this equation uniquely fixes $\cE_{0}(\tau)$ to be the non-holomorphic Eisenstein series $E_{3\over 2}(\tau)$. Indeed, we saw already in (\ref{eq:EisLapl}) that the Eisenstein series satisfy precisely such eigenvalue equations.

\sm

We next briefly outline the analogous calculation for the $D^4 \cR^4$ term. As before, the first step is to identify a subset of the terms appearing in $S^{(5)}$ which mix with each other, but not with other terms, under some subset of the supersymmetry transformations. In the current case, a judicious choice is the following,
\bea
\cL^{(5)} = \lambda^{16}\widehat M^4 \cE_4^{(14,-14)} + \lambda^{15} \gamma^\mu \psi_\mu^* \widehat M^4 \cE_4^{(13,-13)} + \lambda^{16} \widehat M^2 \widehat M_{\rho_1 \rho_2 \rho_3}\widehat M^{\rho_1 \rho_2 \rho_3} \widetilde{\cE}_4^{(13,-13)}
\eea
As before, modular invariance demands that the coefficient functions $\cE_4^{(w,-w)}$ be modular forms of weight-$(w,-w)$, with $\cE_4^{(0,0)} = \cE_4$ the coefficient of $D^4 \cR^4$ itself. Note that there are two coefficient functions of weight-$(13,-13)$ above, which we have denoted by $\cE_4^{(13,-13)}$ and $\widetilde{\cE}_4^{(13,-13)}$. We now proceed by again checking invariance under supersymmetry. Considering the term proportional to $\lambda^{16} \psi_\mu^* \eps$, we find only the contribution from $\delta^{(0)} \cL^{(5)}$, and demanding that this vanishes gives the constraint
\bea
2 D_{13} \, \cE_4^{(13,-13)} - 11 \, \cE_4^{(14,-14)} = 0
\eea
Considering instead the term proportional to $\lambda^{16} \lambda^* \eps^*$, we obtain both a contribution from the $\delta^{(0)}$ variation of $\cL^{(5)}$ as well as the $\delta^{(5)}$ variation of $S^{(0)}$. Demanding the vanishing of the combination gives the constraint
\bea
2 \overline{D}_{-14} \, \cE_4^{(14,-14)}+ 15 \, \cE_4^{(13,-13)}
- {9 i \over 16} \, \widetilde{\cE}_4^{(13,-13)} - 1080 i \, g_1 - {3\over 4} i \, g_2 = 0
\eea
with $g_1(\tau, \overline{\tau})$ and $g_2(\tau, \overline{\tau})$ undetermined functions. Closure of the supersymmetry algebra on $\lambda^*$ further gives rise to the following three equations,
\bea
-192 i \, D_{13} g_1 = \cE_4^{(14,-14)} \hspace{0.3 in} -108 \, g_1
= \widetilde \cE_4^{(13,-13)} \hspace{0.3 in} i \, g_2 + 191 i g_1 = \half  \cE_4^{(13,-13)}
\eea
These constraints can then be combined to obtain a Laplace eigenvalue equation for $\cE_4^{(14,-14)}$, namely
\bea
\Delta \cE_4^{(14,-14)} = \left(-182 + {15 \over 4} \right) \, \cE_4^{(14,-14)}
\eea
Recalling that $\cE_4^{(14,-14)} = D_{13} \dots D_0 \cE_4$, we conclude that $\cE_4$ satisfies the equation
\bea
\Delta \cE_4(\tau) = {15 \over 4}\, \cE_4(\tau)
\eea
This eigenvalue equation, together with the asymptotics at the cusp, fix $\cE_4$ to be equal to $E_{5/2}(\tau)$ up to an overall constant factor.

\sm

Finally, for the coefficient of the $D^6 \cR^4$ term, analogous steps can again be carried out, though one now obtains an \textit{inhomogeneous} Laplace eigenvalue equation,
\bea
\label{eq:D6R4diffeq}
(\Delta - 12) \cE_{6}(\tau) = - 6 \pi^3 E_{3/2}(\tau)^2
\eea
the solution to which is no longer an Eisenstein series. No closed form solution is known to this differential equation, but it is possible to get a closed form expression for the Laurent polynomial $\cE_6^{(0)}$ of $\cE_6$, namely the portions of $\cE_6$ which are power-law in $\tau_2$ and independent of $\tau_1$. Fortunately, the Laurent polynomial pieces are the most interesting for the present purposes, since they are the ones which can be compared to calculations in string perturbation theory.

\sm

To obtain the Laurent polynomial $\cE_6^{(0)}(\tau)$ we begin by inserting the Fourier series of $E_{3/2}(\tau)$, given in (\ref{4.d12}), into (\ref{eq:D6R4diffeq}) and keeping only the Laurent polynomial pieces,
\bea
(\Delta - 12) \cE_{6}^{(0)}(\tau) &=& - 6\left(\left(2 \zeta(3) \tau_2^{3/2} + {2 \over 3}\pi^2 \tau_2^{-1/2}\right)^2 \right.
\no\\
&\vphantom{.}& \hspace{0.5 in} \left.+ 64 \pi^2 \tau_2 \sum_{N \neq 0} |N|^{-2} \sigma_2(|N|)^2 K_1(2 \pi \tau_2 |N|)^2 \right)
\eea
Because the Laplacian is given by $\Delta = \tau_2^2 (\p_{\tau_1}^2 + \p_{\tau_2}^2)$, and hence the derivatives $\p_{\tau_2}^2$ are dressed with factors of $\tau_2^2$, the function $\cE_6^{(0)}$ must be a polynomial in $\tau_2$ with powers being those appearing on the right side, namely $\tau_2^3$, $\tau_2$, and $\tau_2^{-1}$, together with the powers solving the homogeneous equation $(\Delta - 12) f = 0$, namely $\tau_2^{-3}$ and $\tau_2^4$. The term $\tau_2^4$ can be ruled out on physical grounds, since there should be no term more singular than the tree-level $\tau_2^3$ term. On the other hand, the $\tau_2^{-3}$ contribution corresponds to a three-loop contribution which can be (and indeed is) non-zero.

\sm

The general Ansatz for $\cE_{6}^{(0)}(\tau)$ is then,
\bea
\cE_{6}^{(0)}(\tau) = a_1 \tau_2^3 + a_2 \tau_2 + a_3 \tau_2^{-1} + a_4 \tau_2^{-3}
\eea
By plugging into (\ref{eq:D6R4diffeq}) we may uniquely fix $a_1, a_2,a_3$, giving,
\bea
\cE_{6}^{(0)}(\tau) = 4 \zeta(3)^2 \tau_2^3 + {4\over 3}\pi^2 \zeta(3) \tau_2 + {4 \pi^4 \over 15} \tau_2^{-1} + a_4 \tau_2^{-3}
\eea
These match with the expectations from string perturbation given in (\ref{8a5}), (\ref{12.loen}), and (\ref{12.gen2b}) at tree-level, one-loop, and two-loops, respectively. As for the three-loop coefficient $a_4$, this cannot be fixed by simply inserting the Ansatz into (\ref{eq:D6R4diffeq}), since this term already solves the homogenous equation. Instead, this coefficient was fixed in [257] by multiplying both sides of the Laplace equation by $E_4(\tau)$ and integrating over the fundamental domain, using the results presented in Lemma \ref{12.Eisint}. The final result is
\bea
a_4 = {8 \pi^6 \over 8505}
\eea
We note in closing that beyond $D^6 \cR_4$, the terms in the low energy effective action are no longer protected by supersymmetry, and thus we do not expect to be able to extend the above analysis to study them.

\sm

\subsubsection{Non-renormalization Theorems}
\label{sec:nonrenorm}

Perhaps the most striking implications of the modular structure of the $\cR^4$, $D^4 \cR^4$, and $D^6 \cR^4$ effective interactions is that they make predictions for the amplitudes at all orders of perturbation theory, i.e. all genera. From the Fourier series expansion of the functions $E_{3 \over 2}(\TAU) $ we see that, aside from the leading perturbative contribution, there is only one other perturbative contribution, given by the second term, and behaving as $\TAU_2 ^{-\half}$. In the string frame, this gives a contribution of order $\TAU_2^0$ which corresponds to genus one. This means that from superstring perturbation theory, the $\cR^4$ effective interaction receives contributions from genus zero and genus one only. There are no contributions to genus 2 and higher. Similarly, for the $D^4 \cR^4$ effective interaction, there is a genus zero contribution and the only other contribution behaves as $\TAU_2^{- {3 \over 2}} $ in the Einstein frame, or $\TAU_2^{-2}$ in the string frame. Hence there is no genus one correction to the $D^4 \cR^4$ effective interaction but there is a non-zero genus two contribution. There are no higher contributions.

\subsection*{$\bullet$ Bibliographical notes}

The existence of a web of dualities between different supergravities and string theories  was proposed in [258], and led to the discovery of M-theory in [259]. The action of $SL(2,\ZZ)$ on branes in Type IIB was developed in [260], while evidence of the duality between Type I and Heterotic string theory was given in [261]. Useful lecture notes may be found in [262], and an early account of the role played by automorphic functions in dualities is in [263, 264].
The textbooks by Polchinski [171, 172], Johnson [174], and by Becker, Becker and Schwarz [175] provide excellent and comprehensive accounts of string dualities.

\sm

The use of S-duality to constrain the low-energy effective action goes back to [265], where the coefficient of the $\cR^4$ term was related to the non-holomorphic Eisenstein series $E_{3/2}(\tau)$ using results from tree-level string theory and D-instantons. A derivation  using supersymmetry, in the way reviewed above, was provided in [256] and extended  in [266]. That the same techniques could also be applied to the $D^4 \cR^4$ term was proposed in [256], and used to provide a full derivation in [267]. {As we have mentioned in the text, there is no closed form expression for the $D^6 \cR^4$ term, though it is known to satisfy a Laplace eigenvalue equation; this property and others were discussed in [268, 269, 257, 270, 271]. An implicit expression for the  $D^6 \cR^4$ term as a two-loop Schwinger integral can be found in [272]. }
\sm

As mentioned in section \ref{sec:nonrenorm}, the results quoted in the previous paragraph imply a number of non-renormalization theorems, some of which were anticipated from superstring perturbation theory in [273, 192].  For the $\cR^4$ term, the S-duality and supersymmetry analysis predict the absence of perturbative corrections beyond one-loop, and indeed the vanishing at genus two was proven in [228]. Conversely, for the $D^4 \cR^4$ term there is expected to be no genus one correction, as was verified in [205], but there is a non-zero genus-two contribution which was successfully computed and matched in [231].

\sm

The systematic study of genus-one contributions to effective interactions of the form $D^{2 \ell} \cR^4$ with $\ell \geq 3$ was initiated in [93, 94]. From this grew the notion of modular graph functions [99, 104], introduced in section \ref{sec:MGF}.

\sm

One of the hallmarks of superstring theory is the infinite tower of BPS states, which includes black hole microstates. If the string background under consideration contains a torus, then the functions which count these microstates often enjoy nice modular or automorphic properties. A particularly well-studied setup is string theory compactified to 4d with $\cN=4$ supersymmetry. This can be engineered by considering Type II superstrings on $K3 \times T^2$ or Heterotic strings on $T^6$. Either way, the theory is expected to have an $SL(2, \ZZ) \times SO(22,6,\ZZ)$ duality symmetry, with the corresponding counting functions expressible as Siegel modular forms [274]. The contribution of multi-centered black holes was studied in [275], while the single-centered black holes were shown to have interesting connections with mock modular and Jacobi forms in [34]. Recent connections have been made with Hurwitz class numbers [276] and class groups [277] as well. More generally, the tools of Rademacher sums and Farey tail expansions have been applied to the study of microstate counting in [278, 279, 280]. A pedagogical introduction to the relation between blackhole microstate counting and automorphic forms can be found in [281].

\newpage

\section{Dualities in $\cN=2$ super Yang-Mills theories}
\setcounter{equation}{0}
\label{sec:SW}

In this penultimate section, we shall  discuss dualities in Yang-Mills theories with extended supersymmetry in four-dimensional Minkowski space-time. We briefly review supersymmetry multiplets of states and fields and the construction of supersymmetric Lagrangian theories with  $\cN=1,2,4$ Poincar\'e supersymmetries. We then discuss the $SL(2,\ZZ)$ Montonen-Olive duality properties of the maximally supersymmetric $\cN=4$ theory and  the low energy effective Lagrangians for $\cN=2$ theories via the Seiberg-Witten solution. We shall close this section with a discussion of dualities of $\cN=2$ superconformal gauge theories, which possess interesting spaces of marginal gauge couplings. In some cases these spaces of couplings can be identified with the moduli spaces for Riemann surfaces of various genera.

\subsection{Super Yang-Mills: states and fields}

The quantum field theories we shall consider here are invariant under the Poincar\'e algebra, whose generators of translations $P_\mu$ and Lorentz transformations $L_{\mu \nu}$ satisfy the following structure relations,\footnote{Our conventions are as follows. The non-zero components of the Minkowski metric $\eta_{\mu \nu}$ are  $-\eta_{00} =\eta_{11} =\eta_{22}=\eta_{33}=1$. The Clifford-Dirac algebra is $\{ \gamma ^\mu, \gamma ^\nu \} = -2 \eta^{\mu \nu} I_4$, where $I_d$ is the $d \times d$ identity matrix. In a basis where the chirality matrix $\gamma _5= \left ( \begin{smallmatrix} I_2 & 0 \cr 0 & -I_2 \end{smallmatrix} \right )$ is diagonal, we have  $\gamma ^\mu = \left ( \begin{smallmatrix} 0 & \sigma ^\mu \cr \bar \sigma ^\mu & 0 \end{smallmatrix} \right )$ where $\sigma ^0 = \bar \sigma ^0 =- I_2$ and $\sigma ^i= - \bar \sigma ^i$ are the standard Pauli matrices for $i=1,2,3$. Two-component spinor indices $\a, \dot \a$ are raised and lowered with the help of the anti-symmetric symbols $\epsilon^{\a \b},\epsilon ^{\dot \a \dot \b}$ and $\epsilon_{\a \b},\epsilon _{\dot \a \dot \b}$ normalized to  $\epsilon^{12}=\epsilon ^{\dot 1 \dot 2}=-\epsilon_{12}=-\epsilon _{\dot 1 \dot 2}=1$.  We use the notations $\psi \chi= \psi^\a \chi_\a$, $\bar \psi \bar \chi = \bar \psi _{\dot \a} \bar \chi ^{\dot \a}$, and $\chi \sigma ^\mu \bar \psi = \chi^\a \sigma ^\mu _{\a \dot \a} \bar \psi ^{\dot \a}$.  For more details on spinor index notation, see for example [282].}
\begin{align}
[L_{\m \n}, L_{\rho \sigma}] &= \eta_{\n \rho} L_{\mu \sigma} + \eta_{\m \sigma} L_{\n \rho} - \eta_{\m \rho} L_{\n \sigma} - \eta_{\n \sigma} L_{\m \rho}
\no\\
{[L_{\m \n}, P_\rho]} &= \eta_{\n \rho} P_\m - \eta_{\m \rho} P_\n
\no \\
{} [P_\mu, P_\nu] & = 0
\end{align}
In supersymmetric theories the Poincar\'e  algebra is extended to a super Poincar\'e algebra by supplementing $P_\mu$ and $L_{\mu \nu}$ with $\cN$ supercharges $Q_\a^I$ and their adjoints $\bar Q_{\dot \a I} = (Q^I_\a)^\dagger$ for spinor index $\a =1,2$ and $I=1,\cdots, \cN$.  The supercharges commute with $P_\mu$, are Weyl spinors under the Lorentz generators $L_{\mu \nu}$, and obey the following anti-commutation relations,
\bea
\label{15.QQ}
\{ Q_\a ^I, \bar Q_{\dot \b J} \} & = & 2 \, \sigma ^\mu _{\a \dot \b} \, P_\mu \, \delta ^I _J \hskip 0.6in I,J =1 , \cdots, \cN
\no \\
\{ Q_\a ^I, Q_\b ^J \} & = & 2\,  \epsilon _{\a \b} \, Z^{IJ}
\eea
The generators $Z^{IJ} = - Z^{JI}$ can arise only when $\cN\geq 2$. They are central generators in the Poincar\'e superalgebra since they commute with one another and with all other generators. When $Z^{IJ}=0$, the supersymmetry algebra is invariant under a $U(\cN)_R$ automorphism group which, in Physics, is referred to as the R-symmetry group.

\sm

Applying a supercharge to a state changes the spin  by $\pm \thalf$ so that the values of the spin in the multiplets of the Poincar\'e superalgebra must increase with $\cN$. When $\cN>4$, the multiplets necessarily involve states of spin greater than 1, which cannot be accommodated in Yang-Mills theory. For $\cN\leq 8$ they can, however, be accommodated in supergravity, but this requires states of spin 2. Thus $\cN=4$ is the maximal number of Poincar\'e supersymmetries allowed for supersymmetric Yang-Mills theories in four dimensions.

\subsubsection{States}

The spectrum of states in a supersymmetric theory exhibits a characteristic pattern. All states have positive or zero energy.   If the state of lowest energy in the spectrum---namely the ground state---has exactly zero energy then  supersymmetry is said to be \textit{unbroken} or \textit{manifest}. In this case  all positive energy states occur in boson-fermion  pairs of equal mass, momentum, and internal quantum numbers. If the ground state has strictly positive energy, however, the ground state is not supersymmetric and supersymmetry is said to be \textit{spontaneously broken}. Bosons and fermions of identical internal quantum numbers do not generally have the same mass.

\sm

In this section we shall consider only the case where supersymmetry is manifest.  For the case of $\cN=1$, this is the complete story. For $\cN\geq 2$, there is an additional interplay between  the mass of the state and its central charge.

\sm

For $\cN=2$ there is a single complex-valued central charge $Z=Z^{12}$, which we decompose into its absolute value and its phase, $Z=e^{i \f} |Z|$ for $\f \in \RR$. We consider a state with mass $M \not= 0$ and use a Poincar\'e transformation to the rest frame of this state with momentum  $P^\mu = (M,0,0,0)$. In terms of the combinations $ \cQ^\pm _\a= \thalf (Q^1 _\a \mp e^{- i \f} \sigma ^0_{\a \dot \b} Q^{2 \dot \b})$  the anti-commutation relations of (\ref{15.QQ}) are equivalent to the relations,
\bea
\big \{ \cQ^\pm _\a, (\cQ^\pm _\b)^\dagger \big \} =  \delta _{\a}^{\b} (M \pm |Z|)
\eea
while all other anti-commutators vanish identically. Since the operators $\big \{ \cQ^\pm _\a, (\cQ^\pm _\a)^\dagger \big \}$  are self-adjoint and positive in a unitary theory, we obtain the following so-called \textit{BPS bound} between the mass $M$ and the central charge $Z$ of any state,
\bea
M \geq |Z|
\eea
In the limit where the state becomes massless, its central charge must vanish. Of particular interest are the massive  states for which the BPS bound is saturated, say by $M=|Z|$. In these states, which are referred to as BPS states,   the supersymmetry algebra reduces to,
\bea
\big \{ \cQ^+ _\a, (\cQ^+ _\b)^\dagger \big \} = 2M \delta _\a ^\b
\hskip 1in
\big \{ \cQ^- _\a, (\cQ^- _\b)^\dagger \big \} = 0
\eea
while all other anti-commutators vanish. The  operators $\cQ^-_\a$ and $(\cQ^-_\a)^\dagger$ both annihilate the BPS states which means that BPS states are invariant under half of the total number of Poincar\'e supersymmetries. More specifically they are referred to as $\thalf$-BPS states, and they make up a supermultiplet that is half as long as the non-BPS  supermultiplets.

\sm

For $\cN=4$, the matrix of central charges $Z^{IJ}=-Z^{JI}$ has two complex-valued eigenvalues $Z_1$ and $Z_2$. The BPS bound is now $M \geq |Z_1|, |Z_2|$. When $|Z_1| \not= |Z_2|$ one obtains $\thalf$-BPS states by setting $M$ equal to the maximum of $|Z_1|$ and $|Z_2|$. But when $|Z_1|=|Z_2|$, a further shortening of the supersymmetry multiplet takes place  to $\tfrac{1}{4}$-BPS states. It is a general result of CPT invariance in four space-time dimensions that supersymmetric Yang-Mills theories  with $\cN=3$ automatically enjoy the full $\cN=4$ supersymmetry, so that the case $\cN=3$ need not be considered separately.

\subsubsection{Fields}
\label{sec:supermults}

A super Yang-Mills theory with gauge group~$G$ and associated Lie algebra~$\mathfrak{g}$ is constructed in terms of the customary fields of four-dimensional quantum field theory absent gravity, namely spin-1 gauge fields $A_\mu$  in the  adjoint representation of~$G$,  together with spin-$\thalf $ left Weyl fermion fields $\psi_\a, \lambda_\a$ and spin-0 scalar fields $\phi,h$  transforming under various representations of $G$. We recall that, in even dimensions,  a Dirac spinor is the direct sum of two Weyl spinors of opposite chirality and that, in four dimensions, a Majorana spinor is equivalent to a Weyl spinor.  The field multiplets for $\cN=1,2,4$ are as follows:

\begin{itemize}
\itemsep=0in
\item $\cN=1$ \textit{Gauge multiplet} $(A_\mu, \lambda_\a)$ in the adjoint representation of $G$,  where $A_\mu$ is the gauge field and $\lambda_\a$ is a left Weyl fermion referred to as the gaugino;
\item $\cN=1$ \textit{Chiral multiplet} $(\psi_\a, \phi)$ in a representation $\cR$ of $G$, where $\psi_\a$ is a left Weyl fermion and $\phi$ is a complex scalar field;\\
\item $\cN=2$ \textit{Gauge multiplet} $(A_\mu, \lambda_\a, \tilde \lambda_\a , \phi )$ in the adjoint representation of $G$, where $A_\mu$ and $\phi$ are singlets under $SU(2)_R$  while $(\lambda_\a , \tilde \lambda_\a)$ is a doublet;
\item $\cN=2$ \textit{Hypermultiplet} $(h, \tilde h, \psi_\a, \tilde \psi_\a)$ in a representation $\cR$ of $G$,
where $(h, \tilde h)$ is a doublet under $SU(2)_R$ while $\psi_\a, \tilde \psi _\a$ are singlets;\\
\item $\cN=4$ \textit{Gauge multiplet} $(A_\mu, \lambda _\a^A, \phi^I)$ in the adjoint representation of $G$ where $\lambda^A _\a$ with $A=1,\cdots,4$ are left Weyl fermions in the ${\bf 4}$ of $SU(4)_R$  and $\phi^I$ with $I=1,\cdots, 6$ are complex scalars in the ${\bf 6}$ of $SU(4)_R$ while $A_\mu$ is a singlet.
\end{itemize}

The fields of an $\cN=1$ theory may be collected in fully off-shell superfields with linear transformations under the action of the super Poincar\'e algebra, as will be made explicit in the next subsection. The construction of superfields for higher numbers of supersymmetries is, however, much more complicated and will not be addressed here. For this reason it will be useful to decompose the field content of the $\cN=2$ and $\cN=4$ theories in terms of $\cN=1$ superfields. In this spirit, the $\cN=2$ gauge multiplet is the direct sum of an $\cN=1$ gauge multiplet and an $\cN=1$ chiral multiplet, both in the adjoint representation of $G$. The $\cN=2$ hypermultiplet is a sum of two $\cN=1$ chiral multiplets in complex conjugate representations of one another $\cR \oplus \bar \cR$. Similarly, the $\cN=4$ gauge multiplet is the direct sum of an $\cN=2$ gauge multiplet and an $\cN=2$ hypermultiplet, both in the adjoint representation of $G$. Note that for $\cN=4$ theories the  $U(1)_R$ factor of the automorphism group $U(\cN)_R$ of the supersymmetry algebra   is anomalous and fails to be  a quantum symmetry. The charge assignments of the $U(1)_R$ factor in the case of $\cN=1$ and $\cN=2$ will not be important in the sequel, and we shall not spell them out here.

\sm

Before moving on, we remark that when $\cR$ is a pseudoreal representation, and hence admits an antisymmetric invariant tensor $\eps_{IJ}$, the fields $h_I$ and $\tilde h^J$ in the $\cN=2$ hypermultiplet (with indices $I,J$ in the representation $\cR$ shown explicitly) can be constrained to satisfy,
\bea
h_I = \eps_{IJ} \tilde h^J
\eea
This constraint is compatible with $\cN=2$ supersymmetry, and halves the number of degrees of freedom in the multiplet. The resulting multiplet is referred to as a \textit{half-hypermultiplet}. More generally, given a complex representation $\cR$ there is a standard pseudo-real structure on $\cR \oplus \overline \cR$, and hence we may define half-hypermultiplets in the representation $\cR \oplus \overline \cR$. These are precisely the same as full hypermultiplets in the complex representation $\cR$.

\subsection{Super Yang-Mills: Lagrangians}

Two types of supersymmetric Lagrangians will be of interest to us: those that correspond to renormalizable quantum field theories, and those that collect the low energy effective interactions of some arbitrary Lagrangian or non-Lagrangian quantum field theory. In either case, we shall restrict attention to Lagrangians in which bosonic fields have a total of at most two derivatives while fermions have a total of at most one derivative acting on any field. Furthermore, the Lagrangians for theories with $\cN=2$ and  $\cN=4$ supersymmetries will be expressed in terms of $\cN=1$ chiral and vector superfields, using the decomposition of the corresponding multiplets given in the penultimate paragraph of the preceding subsection. Throughout we shall assume that the gauge group $G$ is compact with Lie algebra $\mathfrak{g}$, and of the form of a semi-simple group times a certain number of $U(1)$ factors.

\sm

To introduce $\cN=1$ superfields, we supplement the coordinates $x^\mu$ of Minkowski space-time with Grassmann
spinor variables $\theta ^\a$ and $\bar \theta _{\dot \a}$ of odd grading.  The superfield $\Phi$ for the $\cN=1$ chiral multiplet containing the fields $(\phi, \psi _\a)$  is given by,
\bea
\Phi = \phi + 2 \psi_\a \theta^\a + F \theta_\a \theta^\a
\eea
where $F$ is a non-dynamical complex-valued scalar \textit{auxiliary field}. The chiral superfield $\Phi$ is generally matrix-valued with components $\Phi^i$ and transforms under an arbitrary representation $\cR$ of the gauge algebra $\mathfrak{g}$ with $i = 1 ,\cdots, \dim \cR$. The superfield $V$ for the $\cN=1$ gauge multiplet satisfies a reality condition $V^\dagger = V$ and transforms in the adjoint representation of the gauge algebra~$\mathfrak{g}$. The general decomposition of the superfield $V$  is more complicated than that of the chiral multiplet and we present it here in \textit{Wess-Zumino gauge} only,
\bea
V = - \theta \sigma^\m \overline\theta _\m + i \theta \theta \bar \theta \bar \lambda -  i \bar \theta \bar \theta  \theta  \lambda + \half \theta\theta \bar\theta \bar \theta D
\eea
where $D$ is a real-valued scalar auxiliary field. The vector superfield $V$ is generally matrix-valued and may be  decomposed $V= \sum_a T^a V^a$ onto the generators  $T^a$ of $\mathfrak{g}$, with $a=1, \cdots, \dim \mathfrak{g}$, in the adjoint representation of the gauge algebra $\mathfrak{g}$.

\subsubsection{$\cN=1$ Lagrangians}

In terms of the $\cN=1$ gauge superfield $V$  and chiral superfield $\Phi$, the most general Lagrangian density, under the assumptions spelled out earlier,   is of the following form,
\bea
\label{eq:4dN2Lag}
\cL = \int d^4 \theta \, K(e^V \Phi, \Phi^\dagger) + \mathrm{Re}\int d^2 \theta \Big (  U(\Phi) + \tau_{ab}(\Phi)  W^a W^b \Big )
\eea
in superspace notation. Here $K$ is the real-valued \textit{K{\"a}hler potential},  $U$ is the \textit{superpotential},
and $\tau_{ab}$ is the matrix of couplings and mixings of the field strengths $W_\a^a$ of the gauge fields $V^a$. The functions $U(\Phi)$ and $\tau_{ab}(\Phi)$ are locally holomorphic in $\Phi$.  The superfield $W_\a= \sum _a W_\a ^a T^a$ is obtained from the vector superfield $V$ by taking,
\bea
W_\alpha  = -{1\over 4} \bar D \bar D \left ( e^{-V} D_\alpha e^V \right )
\hspace{0.7 in}
D_\a = {\p \over \p \theta^\alpha} + i \sigma^\m_{\a \dot \a} \bar \theta^{\dot \a} \p_\m
\eea
The field $W_\a$ is a chiral superfield since $\bar D ^{\dot \a} W_\b=0$ by construction.

\subsubsection{Renormalizable $\cN=2$ Lagrangians}

A general four-dimensional  $\cN=2$ Lagrangian theory is specified by a gauge algebra $\mathfrak{g}$ which determines the vector multiplets, as well as a representation $\cR$ of $\mathfrak{g}$ for the hypermultiplets, or half-hypermultiplets if $\cR$ is pseudo-real.\footnote{This data is not quite complete: in general, additional global data such as the spectrum of line operators is necessary to fully specify the theory. We will discuss such additional data in the context of $\cN=4$ Yang-Mills in section \ref{sec:SL2ZN4YM}, but will otherwise largely ignore this subtlety. As long as one is concerned only with the local operator spectrum of the theory, no harm arises from this omission.}
The $\cN=2$ gauge multiplet consists of an $\cN=1$ gauge multiplet $V$ and an $\cN=1$ chiral multiplet $\Phi$ in the adjoint representation of $\mathfrak{g}$, while the $\cN=2$ hypermultiplet consists of two $\cN=1$ chiral multiplets $H, \tilde H$ in the representations $\cR$ and $\bar \cR$ of~$\mathfrak{g}$, respectively. The $\cN=2$ half-hypermultiplet consists of a single $\cN=1$ chiral multiplet $H$ in a pseudo-real representation $\cR$.

\sm

Let us consider first a renormalizable  $\cN=2$  theory without hypermultiplets. We begin by arguing that the superpotential $U(\Phi)$  must vanish. Indeed, for the theory to be renormalizable, one requires that $\tau_{ab}$ be independent of $\Phi$ and that the superpotential $U(\Phi)$ be at most cubic in the chiral superfield $\Phi$.
Upon integrating over superspace, the final two terms of the Lagrangian (\ref{eq:4dN2Lag}) give,
\bea
\mathrm{Re}\int d^2 \theta \,U(\Phi)&=& \mathrm{Re} \Big \{
F^a \p_a U(\phi) + \psi^a \psi^b \p_a \p_b U(\phi)  \Big \}
\\
\mathrm{Re} \int d^2 \theta \,\tau_{ab}\, W^a W^b
- \mathrm{Re}\left\{\tau_{ab} \left({i\over 2} F_{\m \n}^a F^{b \m \n} - \half F_{\m \n}^a \tilde F^{b \m \n} + \overline \lambda^a \overline \sigma^\m D_\m \lambda^b \right)  \right\}
\no
\eea
where $\p_a = \p / \p \phi^a$.  To obtain a Lagrangian with  $\cN=2$ supersymmetry, the combined contributions of the $\cN=1$ gauge and chiral multiplets must be invariant under the $SU(2)_R$ symmetry rotating $\psi_\a$ into $\lambda_\a$, which is possible only if $U(\Phi)$ is linear in $\Phi$. Upon integrating out the auxiliary field $F$, a linear $U(\Phi)$ would add a constant $|\p U|^2$ to the energy density and spontaneously break supersymmetry. Therefore, in a theory with supersymmetric vacua, the superpotential  $U$ is a constant that may set to zero without loss of generality.

\sm

What this means is that the only potential for the scalar fields $\phi^a$ comes from the K{\"a}hler potential. To proceed, we shall assume for simplicity that the gauge group $G$ is a simple Lie group, so that there is a single $\cN=1$ vector superfield $V$ transforming under the adjoint representation of $\mathfrak{g}$. For a renormalizable theory, the K{\"a}hler potential must be \textit{canonical} i.e. $K(e^V \Phi, \Phi^\dagger) = \mathrm{Tr} \left(\Phi^\dagger e^{V} \Phi\right)$. In that case the auxiliary field $D$ may be integrated out using its field equations and the resulting scalar potential takes the form,
\bea
\label{eq:vectmultscalarpot}
V_{{\rm gauge}}(\phi, \bar \phi) = \mathrm{Tr}\left( i [\phi, \bar \phi] \right)^2
\eea
The structure of this potential will be of central importance in disentangling the vacuum structure of the $\cN=2$ theories.

\sm

Next, we include an $\cN=2$ hypermultiplet, which  consists of $\cN=1$ chiral superfields $H=(h,\psi_\a)$ and $\tilde H = (\tilde h, \tilde \psi_\a)$ in representations $\cR$ and $\bar \cR$ of $\mathfrak{g}$, respectively.   The representation $\cR \oplus \bar \cR$ may be reducible into a direct sum of irreducible representations $\cR_I$ and $ \bar \cR_I$ of $\mathfrak{g}$,
\bea
\cR = \bigoplus _I n_I \,  \cR_I \oplus \bar \cR_I \hskip 1in
\sum_I n_I  =N_f
\eea
where $n_I$ is the number of hypermultiplets in representation $\cR_I$, and $N_f$ is the total number of hypermultiplets in all representations $\cR_I$. We decompose the hypermultiplet accordingly into hypermultiplets $H_I$ and $\tilde H_I$ in the representations $\cR_I$ and $\bar \cR_I$ respectively.  The scalar fields $h$ and $\tilde h$ in the hypermultiplets produce the following addition to the scalar potential,
\bea
V_{{\rm hyper}} (h, \tilde h, \phi) = \sum_{i,j=1}^{\dim \cR} h_i  (\phi_a T^a )_{ij}  \tilde h_j + \sum _I \sum _{s,t =1}^{n_I} M_{I,s,t} h_{I,s}  \tilde h_{I,t}
\eea
Here $s,t$ are flavor indices running from $1, \dots, n_I$ in each irreducible representation $\cR_I$, while $i,j$ are the indices in the representation $\cR$, and $\phi$ is the scalar component of the superfield~$\Phi$. The matrix $M_{I,s,t}$ contains parameters that can be interpreted as masses and mixings of the hypermultiplets in a given representation $\cR_I$ of $\mathfrak{g}$. Gauge invariance precludes mixings between hypermultiplets $\cR_I$ and $\cR_J$ with $I \not= J$. Combining this scalar potential with that in (\ref{eq:vectmultscalarpot}) allows for interesting structure for the moduli space of vacua.

\subsection{Low-energy Lagrangian on the Coulomb branch}