Geometric engineering, mirror symmetry and 6d(1,0) → 4d(N=2)

We study compactification of 6 dimensional (1,0) theories on T-2. We use geometric engineering of these theories via F-theory and employ mirror symmetry technology to solve for the effective 4d N = 2 geometry for a large number of the (1,0) theories including those associated with conformal matter. Using this we show that for a given 6d theory we can obtain many inequivalent 4d N = 2 SCFTs. Some of these respect the global symmetries of the 6d theory while others exhibit SL(2, Z) duality symmetry inherited from global diffeomorphisms of the T-2. This construction also explains the 6d origin of moduli space of 4d affine ADE quiver theories as flat ADE connections on T-2. Among the resulting 4d N = 2 CFTs we find theories whose vacuum geometry is captured by an LG theory (as opposed to a curve or a local CY geometry). We obtain arbitrary genus curves of class S with punctures from toroidal compactification of (1,0) SCFTs where the curve of the class S theory emerges through mirror symmetry. We also show that toroidal compactification of the little string version of these theories can lead to class S theories with no punctures on arbitrary genus Riemann surface.