Kummer surfaces associated with Seiberg-Witten curves

TL;DR

This paper explores the geometric structures of K3 and Kummer surfaces derived from Seiberg-Witten curves, revealing new relations between gauge theory couplings and complex surface involutions.

Contribution

It introduces a novel geometric construction linking Seiberg-Witten curves to Kummer surfaces via rational transformations and Nikulin involutions.

Findings

Construction of a family of Jacobian elliptic K3 surfaces with Picard rank 17.

Identification of a Nikulin involution extending the isogeny between Seiberg-Witten curves.

Derivation of a relation between Yukawa couplings in gauge theory and elliptic K3 surfaces.

Abstract

By carrying out a rational transformation on the base curve $\mathbb{CP}^1$ of the Seiberg-Witten curve for $\mathcal{N}=2$ supersymmetric pure $\mathrm{SU}(2)$-gauge theory, we obtain a family of Jacobian elliptic K3 surfaces of Picard rank 17. The isogeny relating the Seiberg-Witten curve for pure $\mathrm{SU}(2)$-gauge theory to the one for $\mathrm{SU}(2)$-gauge theory with $N_f=2$ massless hypermultiplets extends to define a Nikulin involution on each K3 surface in the family. We show that the desingularization of the quotient of the K3 surface by the involution is isomorphic to a Kummer surface of the Jacobian variety of a curve of genus two. We then derive a relation between the Yukawa coupling associated with the elliptic K3 surface and the Yukawa coupling of pure $\mathrm{SU}(2)$-gauge theory.