Determining F-theory matter via Gromov-Witten invariants

TL;DR

This paper develops a method using Gromov-Witten invariants, mirror symmetry, and toric geometry to determine matter content in F-theory compactifications on elliptically fibered Calabi-Yau manifolds over Hirzebruch surfaces.

Contribution

It introduces an algorithm to approximate Mori cones and identify matter representations via mirror symmetry and toric methods in F-theory compactifications.

Findings

The algorithm accurately approximates Mori cones for certain geometries.

Mirror symmetry helps determine if classes correspond to irreducible curves.

The method reveals matter content from geometric data of Calabi-Yau manifolds.

Abstract

We show how to use Gromov-Witten invariants to determine the matter content of F-theory compactifications on elliptically fibered Calabi-Yau manifolds $X$ over Hirzebruch surfaces. To determine the representations of these matter multiplets under the gauge algebra $\mathfrak{g}$, we use toric methods to embed the weight lattice of $\mathfrak{g}$ into the integer homology lattice of $X$. We then apply mirror symmetry to determine whether classes in this lattice which correspond to weights of given representations are represented by irreducible curves. Applying mirror symmetry efficiently to such geometries requires obtaining good approximations to their Mori cones. We propose an algorithm for obtaining such approximations. When the algorithm yields a smooth cone, we find that the latter in fact coincides with the Mori cone of $X$ and already contains information on the matter content of compactifications on $X$. Our algorithm relies on studying toric ambient spaces for the Calabi-Yau hypersurface $X$ which are merely birationally equivalent to fibrations over Hirzebruch surfaces. We study the flops relating such varieties in detail.