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arXiv 0912.4774math.AGhep-thmath.DG
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与Seiberg-Witten曲线相关的Kummer曲面

Kummer surfaces associated with Seiberg-Witten curves

Andreas Malmendier

备注：
26 pages, LaTex
发表信息：
J. Geom. Phys. 62 (2012), no. 1, 107-123
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摘要：

通过对$\mathcal{N}=2$超对称纯$\mathrm{SU}(2)$规范理论的Seiberg-Witten曲线的基曲线$\mathbb{CP}^1$进行有理变换，得到了一族Picard秩为17的Jacobian椭圆K3曲面.将纯$\mathrm{SU}(2)$规范理论的塞伯格-威滕曲线与具有$N_f=2$无质量超多重态的$\mathrm{SU}(2)$规范理论的塞伯格-威滕曲线联系起来的等距线扩展到在族中的每个K3面上定义一个尼库林对合。我们表明，由对合的K3曲面的商的去奇异化同构于亏格2的曲线的雅可比簇的Kummer曲面。然后，我们推导出与椭圆K3表面的Yukawa耦合和纯$\mathrm{SU}(2)$规范理论的Yukawa耦合之间的关系。

英文摘要：

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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