Estimating volume of moduli space of genus-g Riemann surface with n marked points

Question

I wanted to know how can I estimate the volume of the moduli space of a Riemann surface of genus g and having n marked points. I am reading some old string theory papers which discuss divergences in string perturbation theory. In this respect authors such as Gross, Periwal and Shenker say that the partition function in closed bosonic string theories asymptotically grows as (2g)!, where g is the genus and they claim that this estimate can be obtained from the volume of moduli space. It would be great if anyone of you can suggest some reference where I learn about finding out the volume of moduli space of genus-g surfaces with n punctures.

This post imported from StackExchange Physics at 2014-09-05 08:30 (UCT), posted by SE-user user29126

Comments

Perhaps this question should (also?) be asked on math.SE?

This post imported from StackExchange Physics at 2014-09-05 08:30 (UCT), posted by SE-user Danu

Answer 1

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 $1/\sqrt{\pi}$.

Comments

@40227 Nice answer. I just carried out a minor correction of a typo, replaced $\mathcal{M}_{g,0}$ by $\chi(\mathcal{M}_{g,0})$ in your answer. I hope that is okay with you. I must confess on being too lazy to lookup the exact volume computation. I am glad I didn't as I got learn about the Mirzakhani work.

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@ suresh Many thanks for the correction of the typo.

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Thank you very much for your detailed and very nicely explained reply. This was really helpful.

Answer 2

A more concrete example where the analog of the Gross-Periwal-Shenker divergence is clear to see is given by the topological partition function in type IIA string theory. Section 2 of this paper by Gopakumar and Vafa  is relevant for you. Here $\mathcal{F}_g$ is the genus $g$ contribution. One has (Eq. 2.3 in the paper)

$\mathcal{F}_g \sim \frac{B_g}{2g(2g-2)} \frac{B_{g-1}}{(2g-2)!}$

which for large $g$ grows as $(2g)!$ using properties of the Bernoulli numbers $B_g$. (I think Gopakumar-Vafa use a strange notation for Bernoulli numbers -- they mean $B_{2g}$ when they write $B_g$!) The paper by Faber-Pandaripande might be useful starting point for you if you wish to know more about computations associated with $\mathcal{M}_{g,n}$.