Kummer surfaces associated with Seiberg-Witten curves

Details Kummer surfaces associated with Seiberg-Witten curves Kummer surfaces associated with Seiberg-Witten curves

2009-12-24

arxiv.org/abs/0912.4774v3

J. Geom. Phys. 62 (2011), no. 1, 107-123

Mathematics

Algebraic Geometry

26 pages, LaTex

Scientific paper

10.1016/j.geomphys.2011.09.010

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.

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