In Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Mirzakhani gave a recursive formula for WP volumes of moduli spaces $\mathcal{M}_{g,n}(L)$ of bordered Riemann surfaces, and the constant term of each $\mathcal{M}_{g,n}(L)$ is a rational multiple of $\pi^{6g−6+2n}$, say $C_{g,n}$. So is there any explicit table or formula for these rationals $C_{g,n}$? As I'm studying number theory, so I'm interested in these numbers. However the recursive formula in this paper is not very friendly to me.
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2 Answers
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Norman Do calculates somes $V_{g,n}(L)$ polynomials for small $g$ and $n$ in the appendix A of his thesis:
- Norman Nam Van Do, Intersection theory on moduli spaces of curves via hyperbolic geometry (2008)
I'll copy constant terms here for future reference.
\begin{array}{|c|c|c|c|} \hline \mathrm{g}& \mathrm{n} & C_{g,n} \\ \hline 0 & 3 &1\\ \hline 0 & 4 &2\\ \hline 0 & 5 &10\\ \hline 0 & 6 &244/3\\ \hline 0 & 7 &2758/3\\ \hline \end{array} \begin{array}{|c|c|c|c|} \hline \mathrm{g}& \mathrm{n} & C_{g,n} \\ \hline 1 & 1 & 1/12\\ \hline 1 & 2 & 1/4\\ \hline 1 & 3 & 14/9\\ \hline 1 & 4 & 529/36\\ \hline 1 & 5 & 16751/90\\ \hline \end{array}
\begin{array}{|c|c|c|c|} \hline \mathrm{g}& \mathrm{n} & C_{g,n} \\ \hline 2 & 0 & 43/2160\\ \hline 2 & 1 & 29/192\\ \hline 2 & 2 & 787/480\\ \hline 2 & 3 & 1498069/64800\\ \hline \end{array}
\begin{array}{|c|c|c|c|} \hline \mathrm{g}& \mathrm{n} & C_{g,n} \\ \hline 3 & 0 & 176557/1209600\\ \hline 3 & 1 & 9292841/4082400\\ \hline 3 & 2 & 2800144027/65318400\\ \hline \end{array}
\begin{array}{|c|c|c|c|} \hline \mathrm{g}& \mathrm{n} & C_{g,n} \\ \hline 4 & 0 & 1959225867017/493807104000\\ \hline 4 & 1 & 92480712720869/987614208000\\ \hline \end{array}
\begin{array}{|c|c|c|c|} \hline \mathrm{g}& \mathrm{n} & C_{g,n} \\ \hline 5 & 0 & 84374265930915479/355541114880000\\ \hline 5 & 1 & 21185241498983729441/2824576634880000\\ \hline \end{array}
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$\begingroup$ Is there any particular reason $C_{1,4}$ is a perfect square? $\endgroup$ Oct 30, 2015 at 23:46
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The coefficients of these polynomials are top intersection numbers of psi-classes and the class $\kappa_1$ on $\overline M_{g,n}$. That's how Mirzakhani obtained a new proof of the Witten conjecture, by studying the leading coefficients. The constant term is the top intersection number of $\kappa_1$.
Carel Faber has written MAPLE program for computing top intersection numbers in the tautological ring, and in particular all these coefficients. http://math.stanford.edu/~vakil/programs/index.html
answered Oct 30, 2015 at 6:23