N=3 four dimensional field theories

We introduce a class of four dimensional field theories constructed by quotienting ordinary $\mathcal{N}=4$ $U(N)$ SYM by particular combinations of R-symmetry and $SL(2,\mathbb{Z})$ automorphisms. These theories appear naturally on the worldvolume of D3 branes probing terminal singularities in F-theory, where they can be thought of as non-perturbative generalizations of the O3 plane. We focus on cases preserving only 12 supercharges, where the quotient gives rise to theories with coupling fixed at a value of order one. These constructions possess an unconventional large $N$ limit described by a non-trivial F-theory fibration with base $Ad{S}_5\times (S^5\mathrm{/}{\mathbb{Z}}_k)$. Upon reduction on a circle the $\mathcal{N}=3$ theories flow to well-known $\mathcal{N}=6$ ABJM theories.