Analytic Periods via Twisted Symmetric Squares

TL;DR

This paper develops a method to compute the periods of one-parameter K3 manifolds using symmetric square operators, enabling explicit inversion of mirror maps and aiding in Swampland Distance Conjecture checks.

Contribution

It introduces a novel approach using twisted symmetric squares of Picard-Fuchs operators to derive globally valid period expressions for K3 manifolds.

Findings

Explicit formulas for K3 periods in terms of elliptic integrals

Global validity of period expressions across moduli space

Facilitation of mirror map inversion and distance calculations

Abstract

We study the symmetric square of Picard-Fuchs operators of genus one curves and the thereby induced generalized Clausen identities. This allows the computation of analytic expressions for the periods of all one-parameter K3 manifolds in terms of elliptic integrals. The resulting expressions are globally valid throughout the moduli space and allow the explicit inversion of the mirror map and the exact computation of distances, useful for checks of the Swampland Distance Conjecture. We comment on the generalization to multi-parameter models and provide a two-parameter example.