Symmetries of Calabi-Yau Prepotentials with Isomorphic Flops

TL;DR

This paper explores how certain symmetries in Calabi-Yau threefolds with infinite flops relate to Coxeter groups, leading to invariant prepotentials and connections to elliptic fibrations.

Contribution

It demonstrates that isomorphic flops induce Coxeter group symmetries in the prepotential, revealing new invariance properties in Calabi-Yau mirror symmetry.

Findings

Coxeter groups describe symmetries of the Kahler cone structure.

Gopakumar-Vafa invariants are preserved under these symmetries.

Invariant functions can be expressed using theta functions linked to elliptic fibrations.

Abstract

Calabi-Yau threefolds with infinitely many flops to isomorphic manifolds have an extended Kahler cone made up from an infinite number of individual Kahler cones. These cones are related by reflection symmetries across flop walls. We study the implications of this cone structure for mirror symmetry, by considering the instanton part of the prepotential in Calabi-Yau threefolds. We show that such isomorphic flops across facets of the Kahler cone boundary give rise to symmetry groups isomorphic to Coxeter groups. In the dual Mori cone, non-flopping curve classes that are identified under these groups have the same Gopakumar-Vafa invariants. This leads to instanton prepotentials invariant under Coxeter groups, which we make manifest by introducing appropriate invariant functions. For some cases, these functions can be expressed in terms of theta functions whose appearance can be linked to an elliptic fibration structure of the Calabi-Yau manifold.