Kummer surfaces associated with Seiberg-Witten curves

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.

26 pages, LaTex

References in corpus (3)

• Compactifications of F-Theory on Calabi--Yau Threefolds -- II
• F-theory and Orientifolds
• Special Kahler Manifolds

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• Coulomb branch surgery: Holonomy saddles, S-folds and discrete symmetry gaugings
• Minimal Weierstrass equations for genus 2 curves
• Isogenies of certain K3 surfaces of rank 18