Modeling General Asymptotic Calabi–Yau Periods

In the quest to uncovering the fundamental structures that underlie some of the asymptotic Swampland conjectures the authors initiate the general study of asymptotic period vectors of Calabi–Yau manifolds. The strategy is to exploit the constraints imposed by completeness, symmetry, and positivity, which are formalized in asymptotic Hodge theory. The general principles are used to study the periods near any boundary in complex structure moduli space and explain that near most boundaries, leading exponentially suppressed corrections must be present for consistency. The only exception are period vectors near the well-studied large complex structure point. Together with the classification of possible boundaries, the procedure makes it possible to construct general models for these asymptotic periods. The starting point for this construction is the $sl\left(2\right)$-data classifying the boundary, which is used to construct the asymptotic Hodge decomposition known as the nilpotent orbit. The authors then use the latter to determine the asymptotic period vector. This program has been explicitly carried out for all possible one- and two-moduli boundaries in Calabi–Yau threefolds, and general models for their asymptotic periods have been written down.