Seiberg-Witten geometry, modular rational elliptic surfaces and BPS quivers

We study the Coulomb branches of four-dimensional supersymmetric quantum field theories with N = 2 supersymmetry, including the KK theories obtained from the circle compactification of the 5d N = 1 E-n Seiberg theories, with particular focus on the relation between their Seiberg-Witten geometries and rational elliptic surfaces. More attention is given to the modular surfaces, which we completely classify using the classification of subgroups of the modular group PSL(2, Z), deriving closed-form expressions for the modular functions for all congruence and some of the non-congruence subgroups. Moreover, in such cases, we give a prescription for determining the BPS quivers from the fundamental domains of the monodromy groups and study how changes of these domains can be interpreted as quiver mutations. This prescription can be also generalized to theories whose Coulomb branches contain 'undeformable' singularities, leading to known quivers of such theories.

Automated summary

We study the Coulomb branches of four-dimensional supersymmetric quantum field theories with N = 2 supersymmetry. In particular, we focus on the relation between their Seiberg-Witten geometries and rational elliptic surfaces. We classify the modular surfaces using the subgroups of the modular group PSL(2, Z) and derive closed-form expressions for the modular functions. We also provide a prescription for determining the BPS quivers using the fundamental domains of the monodromy groups and study how changes of these domains can be interpreted as quiver mutations. This prescription can be generalized to theories with 'undeformable' singularities.