The Satake sextic in elliptic fibrations on K3

TL;DR

This paper explores explicit formulas connecting elliptic fibrations on K3 surfaces with genus-two curves via Satake coordinates, relevant for string duality models in theoretical physics.

Contribution

It provides explicit rational maps and formulas linking Satake sextics to genus-two curves within the context of F-theory and heterotic string duality.

Findings

Derived explicit formulas for Satake sextics in elliptic fibrations.

Established the algebraic correspondence between sextic curves and Satake sextics.

Connected geometric structures to string theory dualities.

Abstract

We describe explicit formulas relevant to the F-theory/heterotic string duality that reconstruct from a specific Jacobian elliptic fibration on the Shioda-Inose surface covering a generic Kummer surface the corresponding genus-two curve using the level-two Satake coordinate functions. We derive explicitly the rational map on the moduli space of genus-two curves realizing the algebraic correspondence between a sextic curve and its Satake sextic. We will prove that it is not the original sextic defining the genus-two curve, but its corresponding Satake sextic which is manifest in the F-theory model, dual to the $\mathfrak{so}(32)$ heterotic string with an unbroken $\mathfrak{so}(28)\oplus \mathfrak{su}(2)$ gauge algebra.