Direct computation of period polynomials and classification of K3-fibred Calabi--Yau threefolds

TL;DR

This paper develops a numerical method to analyze invariants of certain string vacua, revealing constraints on their classification and identifying manifolds that cannot be realized as K3-fibred Calabi-Yau threefolds.

Contribution

It introduces a computational approach to study period polynomials and monodromy matrices, providing new insights into the classification of string vacua and Calabi-Yau manifolds.

Findings

Many studied vacua satisfy conditions for dual Type IIA models

Constraints on invariants derived from monodromy matrices

Identification of manifolds not realizable as K3-fibred Calabi-Yau threefolds

Abstract

One can assign to four-dimensional N=2 supersymmetric Heterotic string vacua a set of classification invariants including a lattice $\Lambda_S$ and vector-valued modular forms. Some of the classification invariants are constrained by the condition that the Coulomb branch monodromy matrices should be integer-valued. We computed numerically the period polynomials of meromorphic cusp forms for some rank-1 $\Lambda_S$; we then computed the monodromy matrices and extracted general patterns of the constraints on the invariants. The constraints we got imply that a large fraction of the Heterotic string vacua we studied satisfy the necessary conditions for a non-linear sigma model interpretation in the dual Type IIA description. Our computation can also be used to identify diffeomorphism classes of real six-dimensional manifolds that cannot be realized by K3-fibred Calabi--Yau threefolds.