Asymptotic Weil–Petersson diameter of moduli space

Determine the asymptotic growth of the Weil–Petersson diameter of the moduli space \(\mathcal{M}_g\) of genus-\(g\) Riemann surfaces, in particular whether \(\operatorname{diam}^{(\mathrm{WP})}(\mathcal{M}_g)\) grows no faster than a constant multiple of \(\sqrt{g}\) as \(g\to\infty\), as required by the AdS Refined Distance Conjecture.

Background

For large NN, the conformal-manifold metric of the unpunctured AN−1A_{N-1} Class S\mathcal{S} theories is proportional to the Weil–Petersson metric on the moduli space Mg\mathcal{M}_g of genus-gg Riemann surfaces. The authors translate the AdS Refined Distance Conjecture into the proposed upper bound diam⁡(WP)(Mg)≲g−1\operatorname{diam}^{(\mathrm{WP})}(\mathcal{M}_g)\lesssim\sqrt{g-1}.

Known mathematical results provide a lower bound of order g\sqrt{g} and an upper bound of order glog⁡g\sqrt{g}\log g, leaving unresolved the actual asymptotic behavior. Establishing the g\sqrt{g} scaling would mean that the lower bound is asymptotically saturated and would support the AdS Refined Distance Conjecture in this setting; faster growth would disprove that conjectural bound.

References

Mathematicians have studied the diameter of this moduli space, but the asymptotic dependence of $\text{diam}{\text{(WP)}(\mathcal{M}_g)$ as a function of $g$ is not known; it is not even known whether this quantity has a well-defined limit. The most recent result we could find is that in the $g\rightarrow\infty$ limit, there are both lower and upper bounds $\sqrt{g}\lesssim\text{diam}{\text{(WP)}(\mathcal{M}_g)\lesssim\sqrt{g}\log(g)$. The AdS RDC bound rdc444 implies then that the lower bound bsat must be saturated (with a not too large numerical coefficient), and that any growth of the diameter faster than $\sqrt{g}$ is impossible. Thus, in this case, Swampland principles are making a prediction about an open mathematical problem!

— Stress-Testing Swampland Bounds with Class S Theories  (2608.28333 - Fenati et al., 28 Aug 2026) in Section “Testing the Refined Distance Conjecture,” immediately following Eq. (\ref{bsat})