Modeling General Asymptotic Calabi-Yau Periods

TL;DR

This paper develops a general framework using asymptotic Hodge theory to analyze the behavior of Calabi-Yau periods near boundaries in complex structure moduli space, revealing the necessity of exponential corrections except at large complex structure points.

Contribution

It introduces a systematic method to construct asymptotic period models for Calabi-Yau manifolds near various boundaries using sl(2)-data and nilpotent orbits, extending previous specific cases.

Findings

Explicit models for one- and two-moduli boundaries in Calabi-Yau threefolds.

Identification of exponential corrections as a general feature near most boundaries.

Clarification of the special role of large complex structure points.

Abstract

In the quests to uncovering the fundamental structures that underlie some of the asymptotic Swampland conjectures we initiate the general study of asymptotic period vectors of Calabi- Yau manifolds. Our strategy is to exploit the constraints imposed by completeness, symmetry, and positivity, which are formalized in asymptotic Hodge theory. We use these general principles 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, our procedure makes it possible to construct general models for these asymptotic periods. The starting point for this construction is the sl(2)-data classifying the boundary, which we use to construct the asymptotic Hodge decomposition known as the nilpotent orbit. We then use the latter to determine the asymptotic period vector. We explicitly carry out this program for all possible one- and two-moduli boundaries in Calabi-Yau threefolds and write down general models for their asymptotic periods.