Distribution of Flux Vacua around Singular Points in Calabi-Yau Moduli Space
Tohru Eguchi, Yuji Tachikawa
TL;DR
The paper investigates how type IIB flux vacua are distributed in Calabi–Yau complex-structure moduli spaces near singular loci, using the Ashok–Douglas density $\tilde{\rho}(z,\bar z) \propto \det(-\frac{R}{2\pi}-\frac{\omega}{2\pi})$ and showing the density is integrable around conifolds, ADE fibrations, and Argyres–Douglas points. By applying Schmid’s nilpotent orbit theorem and monodromy analysis, it classifies near-singularity behavior into cases with and without logarithmic terms in the period expansions, deriving explicit growth bounds for $g_{i\bar{j}}$, $F_{ijk}$ and the resulting density in each scenario. A detailed, explicit Calabi–Yau example embedding an Argyres–Douglas point demonstrates how geometric-engineering limits and BD-type monodromies yield finite vacuum densities even in intricate degenerations; three exceptional divisors arise from blow-ups, and the AD density remains normalizable despite logarithmic structures. The results strongly suggest a finite number of vacua concentrated around each singular locus, supporting the finiteness of the flux-landscape in these regions and motivating a unified, normal-crossing approach for general proofs.
Abstract
We study the distribution of type IIB flux vacua in the moduli space near various singular loci, e.g. conifolds, ADE singularities on P1, Argyres-Douglas point etc, using the Ashok- Douglas density det(R + omega). We find that the vacuum density is integrable around each of them, irrespective of the type of the singularities. We study in detail an explicit example of an Argyres-Douglas point embedded in a compact Calabi-Yau manifold.