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

We study the Coulomb branches of the rank-one 4d $\mathcal{N} = 2$ quantum field theories, including the KK theories obtained from the circle compactification of the 5d $\mathcal{N}= 1$ $E_n$ Seiberg theories. The focus is set on the relation between their Seiberg-Witten geometries and rational elliptic surfaces, with more attention being given to the modular surfaces, which we completely classify using the classification of subgroups of the modular group ${\rm PSL}(2,\mathbb{Z})$. We derive closed-form expressions for the modular functions for all congruence and some of the non-congruence subgroups associated with these geometries. Moreover, we show how the BPS quivers of these theories can be determined directly from the fundamental domains of the monodromy groups and study how changes of these domains can be interpreted as quiver mutations. This approach can also be generalized to theories whose Coulomb branches contain `undeformable' singularities, leading to known quivers of such theories.