Elliptic threefolds with high Mordell-Weil rank

TL;DR

This paper constructs the first known smooth elliptic Calabi-Yau threefolds with a Mordell-Weil rank of 10, analyzes their geometric and physical properties, and verifies anomaly cancellation in the associated F-theory models.

Contribution

It provides explicit examples of high-rank elliptic Calabi-Yau threefolds, computes their Mordell-Weil groups, and explores implications for F-theory compactifications.

Findings

First examples of elliptic Calabi-Yau threefolds with Mordell-Weil rank 10.

Explicit computation of Shioda homomorphism and height pairing.

Verification of anomaly cancellation conditions in the associated physical theories.

Abstract

We present the first examples of smooth elliptic Calabi-Yau threefolds with Mordell-Weil rank 10, the highest currently known value. They are given by the Schoen threefolds introduced by Namikawa; there are six isolated fibers of Kodaira Type IV. We explicitly compute the Shioda homomorphism for the generators of the Mordell-Weil group and their induced height pairing. Compactification of F-theory on these threefolds gives an effective theory in six dimensions which contains ten abelian gauge group factors. We compute the massless matter spectrum. In particular, we show that the charged singlet matter need not reside at enhancement loci of Type $I_2$, as previously believed. We relate the multiplicities of the massless spectrum to genus-zero Gopakumar-Vafa invariants and other geometric quantities of the Calabi-Yau. We show that the gravitational and abelian anomaly cancellation conditions are satisfied. We prove a Geometric Anomaly Cancellation equation and we deduce birational equivalence for the quantities in the spectrum. We explicitly describe a Weierstrass model over $\mathbb P^2$ of the Calabi-Yau threefolds as a log canonical model and compare it to a construction by Elkies and classical results of Burkhardt.