On the duality of F-theory and the CHL string in seven dimensions

TL;DR

This paper explores the duality between F-theory and the CHL string in seven dimensions, establishing algebraic correspondences between specific K3 surfaces and providing explicit parametrizations in special cases.

Contribution

It reveals algebraic correspondences between polarized K3 surfaces in F-theory and CHL string duality, and derives explicit parametrizations in special involution cases.

Findings

Algebraic correspondences between K3 surfaces in dual theories

Explicit parametrization for dual F-theory compactifications

Coincidence of moduli spaces under special involutions

Abstract

We show that the duality between F-theory and the CHL string in seven dimensions defines algebraic correspondences between K3 surfaces polarized by the rank-ten lattices $H \oplus N$ and $H\oplus E_8(-2)$. In the special case when the F-theory admits an additional anti-symplectic involution or, equivalently, the CHL string admits a symplectic one, both moduli spaces coincide. In this case, we derive an explicit parametrization for the F-theory compactifications dual to the CHL string, using an auxiliary genus-one curve, based on a construction given by Andr\'e Weil.