Geodesics in the extended K\"ahler cone of Calabi-Yau threefolds

TL;DR

This paper analyzes the structure of the extended K"ahler cone in Calabi-Yau threefolds with $h^{1,1}=2$, solving geodesic equations to understand topology change and flop transitions relevant for M-theory compactifications.

Contribution

It provides a classification of intersection forms, explicit solutions to geodesic equations, and a detailed study of the effective cones for all $h^{1,1}=2$ threefolds from key datasets.

Findings

Geodesics crash into walls signaling supergravity breakdown.

All three intersection form cases are realized in examples.

Infinite flop sequences and isomorphic flops are common phenomena.

Abstract

We present a detailed study of the effective cones of Calabi-Yau threefolds with $h^{1,1}=2$, including the possible types of walls bounding the K\"ahler cone and a classification of the intersection forms arising in the geometrical phases. For all three normal forms in the classification we explicitly solve the geodesic equation and use this to study the evolution near K\"ahler cone walls and across flop transitions in the context of M-theory compactifications. In the case where the geometric regime ends at a wall beyond which the effective cone continues, the geodesics "crash" into the wall, signaling a breakdown of the M-theory supergravity approximation. For illustration, we characterise the structure of the extended K\"ahler and effective cones of all $h^{1,1}=2$ threefolds from the CICY and Kreuzer-Skarke lists, providing a rich set of examples for studying topology change in string theory. These examples show that all three cases of intersection form are realised and suggest that isomorphic flops and infinite flop sequences are common phenomena.