No Shortcuts to Infinity: Weil-Petersson Distance in Calabi-Yau Complex Structure Moduli Space
Abstract
Infinite-distance limits play a central role in the Swampland program, but understanding their behaviour at intersections of boundary divisors is subtle, as it requires control over simultaneous limits of several moduli. To address this, we establish a necessary and sufficient criterion for a boundary stratum of arbitrary codimension in the complex-structure moduli space of Calabi-Yau manifolds to lie at infinite Weil-Petersson distance. In particular, intersections of infinite-distance boundary divisors necessarily remain at infinite distance. The result holds for the full Weil-Petersson metric and arbitrary trajectories, including those with unrestricted hierarchies in the saxions and axion winding. In compactifications of Type IIB string theory on Calabi-Yau threefolds, our proof is based on an inequality relating the Weil-Petersson length of a trajectory to the BPS mass associated with an auxiliary charge vector. If this mass tends to zero while the charge-to-mass ratio remains uniformly bounded, every trajectory approaching the limit has infinite length. Using asymptotic Hodge theory, we construct an auxiliary charge vector satisfying these conditions and derive the criterion directly from the limiting mixed Hodge structure associated with the degeneration. More generally, our result applies to polarized variations of Hodge structure of Calabi-Yau type. In particular, we prove Wang's conjecture on finite Weil-Petersson distance for degenerations over higher-dimensional bases.
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BibTeX
@misc{monnee2026shortcuts,
title = {{No Shortcuts to Infinity: Weil-Petersson Distance in Calabi-Yau Complex Structure Moduli Space}},
author = {Jeroen Monnee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00470},
eprint = {2610.00470},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00470},
}
RIS
TY - UNPB
TI - No Shortcuts to Infinity: Weil-Petersson Distance in Calabi-Yau Complex Structure Moduli Space
AU - Jeroen Monnee
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00470
UR - https://arxiv.org/abs/2610.00470
AB - Infinite-distance limits play a central role in the Swampland program, but understanding their behaviour at intersections of boundary divisors is subtle, as it requires control over simultaneous limits of several moduli. To address this, we establish a necessary and sufficient criterion for a boundary stratum of arbitrary codimension in the complex-structure moduli space of Calabi-Yau manifolds to lie at infinite Weil-Petersson distance. In particular, intersections of infinite-distance boundary divisors necessarily remain at infinite distance. The result holds for the full Weil-Petersson metric and arbitrary trajectories, including those with unrestricted hierarchies in the saxions and axion winding. In compactifications of Type IIB string theory on Calabi-Yau threefolds, our proof is based on an inequality relating the Weil-Petersson length of a trajectory to the BPS mass associated with an auxiliary charge vector. If this mass tends to zero while the charge-to-mass ratio remains uniformly bounded, every trajectory approaching the limit has infinite length. Using asymptotic Hodge theory, we construct an auxiliary charge vector satisfying these conditions and derive the criterion directly from the limiting mixed Hodge structure associated with the degeneration. More generally, our result applies to polarized variations of Hodge structure of Calabi-Yau type. In particular, we prove Wang's conjecture on finite Weil-Petersson distance for degenerations over higher-dimensional bases.
ER -