Let $\mathcal{M}_{g,n}$ be the moduli space of Riemann surfaces of genus $g$ with $n$ punctures. It is a complex orbifold (i.e. locally the quotient of a smooth complex manifold by the action of a finite group) of complex dimension $d = 3g-3+n$. This space has a natural symplectic form (in fact Kähler form with respect to the natural complex structure) $\omega$ called the Weil-Petersson symplectic form. The corresponding volume form $\omega^d / d!$ is called the Weil-Petersson volume form and the corresponding volume
\[ V_{g,n} = \int_{\mathcal{M}_{g,n}} \frac{\omega^d}{d!} \]
is what is usually called the volume of $\mathcal{M}_{g,n}$. The question is about the computation of $V_{g,n}$ and the string theory application is in particular concerned with the asymptotics of $V_{g,n}$ for $g \rightarrow + \infty$ and $n$ fixed. We can therefore assume that the Euler characteristic $\chi = 2 - 2g-n$ is negative. Wolpert has shown in 1983 that $V_{g,n} \in \pi^{6g-6+2n} \mathbb{Q}$, which is a nice general property rather than something useful for actual computations.
A basic tool to study the behavior of $V_{g,n}$ is to have a decomposition in pieces of $\mathcal{M}_{g,n}$. The idea is to replace a surface by a combinatorial object called a ribbon graph (or fat graph). A ribbon graph is an usual graph (connected and with vertices of valence at least three) which is fattened as a rubber band (equivalently, it is an usual graph with a cyclic ordering of the edges going to a given vertex). A ribbon graph has some boundary which is a disjoint union of some number $n$ of closed loops. Gluing a punctured disk along its boundary to each of these closed loops gives a closed topological surface with $n$ punctures. Let $V$ be the number of vertices and $E$ the number of edges of some ribbon graph with $n$ boundaries. The corresponding surface is of genus $g$ satisfying $2-2g=V-E+n$, we call $g$ the genus of the ribbon graph. A metrised ribbon graph is a ribbon graph endowed with some real positive number on each edge. Let $\mathcal{N}_{g,n}$ be the moduli space of metrised ribbon graphs of genus $g$ with $n$ boundaries. This space is a real orbifold and has a natural cellular decomposition given by the combinatorial type of a ribbon graph. It is easy to show that the cells of maximal dimension correspond to trivalent ribbon graphs, and that the corresponding real dimension is $6g-6+3n$.
The non-trivial fact is that for every $g$ and every $n>0$ there exists an isomorphism of orbifolds
$\mathcal{M}_{g,n} \times \mathbb{R}_+^n \simeq \mathcal{N}_{g,n}$.
In fact, there exists several such isomorphisms: one can be constructed using hyperbolic geometry, more precisely using the notion of ideal triangulations (as we assume $\chi <0$, $\mathcal{M}_{g,n}$ is also a moduli space of hyperbolic metrics), another can be constructed using the theory of Jenkins-Strebel differentials. Transferring the Weil-Petersson form to $\mathcal{N}_{g,n}$, it is possible to transfer the problem of computing $V_{g,n}$ to a problem on $\mathcal{N}_{g,n}$. The advantage is that $\mathcal{N}_{g,n}$ has a nice cellular decomposition. For the volume, only the cells of dimension maximal are relevant. Around 1990, Penner has shown in
http://projecteuclid.org/euclid.jdg/1214448257
that the number of cells of maximal dimension in the case $n=1$ is asymptotic to $\frac{(2g)!}{6g-3} (\frac{6}{e})^{2g}$ (one has to estimate the number of trivalent ribbon graphs, it is a combinatorial problem which can be solved using the theory of representations of the symmetric groups) and that the contribution of each of these cells is bounded below by $C^g$ for a positive constant $C$. Penner deduced from that a lower bound of the form $V_{g,1} \geq C^g (2g)!$. It is this result which is used by Shenker in its paper
http://ccdb5fs.kek.jp/cgi-bin/img/allpdf?200035186
on the size of non-perturbative effects in string theory.
Using similar techniques, it is possible to show that for every fixed $n$, there exists $C$ and $C'$ positive constants such that for every $g$, $(C')^{g} (2g)! \geq V_{g,n} \geq C^g (2g)!$. Here the $(2g)!$ comes from the number of cells of maximal dimensions in $\mathcal{N}_{g,n}$. Using the cellular decomposition of $\mathcal{N}_{g,n}$, it is also possible to compute the Euler characteristic of $\mathcal{M}_{g,0}$ (it is in fact easier than for the number of cells of maximal dimensions because one treats all the ribbon graphs without distinction), one finds
$\chi(\mathcal{M}_{g,0}) = \frac{B_{2g}}{2g (2g-2)}$
where $B_{2g}$ is a Bernoulli numbers which indeed grows as $(2g)!$ (an easy way to see this is to use the formula
$\zeta(2g) = \frac{(-1)^{g+1 } B_{2g} (2 \pi)^{2g}}{2 (2g)!}$
and the fact that $\zeta (2g)$ goes to $1$ for $g$ going to infinity). Remark that $B_{2g}/({2g (2g-2)})$ is the first factor appearing in the answer of suresh for the genus $g$ topological string amplitudes. It is indeed $\chi (\mathcal{M}_{g,0})$ which appears in the topological string computation and which is responsible for the $(2g)!$ behavior. In some sense, the Euler characteristic is a kind of "topological version" of the volume of a space and so the similarity between the behavior of $V_{g,n}$ and $\chi (\mathcal{M}_{g,n})$ should not be too surprising.
The conclusion of the cellular decomposition story is the estimate
$(C')^{g} (2g)! \geq V_{g,n} \geq C^g (2g)!$.
One can ask if it is possible to have a more precise control on $V_{g,n}$, $g \rightarrow +\infty$, $n$ fixed. To do better, it is necessary to have new tools and a breakthrough has been down by Mirzakhani who proved around 2007 in
http://www.ams.org/journals/jams/2007-20-01/S0894-0347-06-00526-1/S0894-0347-06-00526-1.pdf
a recursion relation between the $V_{g,n}$'s (more precisely, between more general numbers $V_{g,n}(L_1,...,L_n)$ whose $V_{g,n}$'s are the specialization at$L_1=...=L_n=0$). Using this recursion relation, she proved with Zograf in
http://arxiv.org/abs/1112.1151
that there exists a constant $C$ such that for every $n$, we have the asymptotics
$V_{g,n} =_{g \rightarrow \infty} \frac{C}{\sqrt{g}}(4 \pi^2)^{2g+n-3} (2g-3+n)! (1 + O(g))$.
A conjectural value for $C$ is $1/\sqrt{\pi}$.
Q&A (4871)
