LAUSR

We present the first example of a grand unified theory (GUT) with a modular symmetry interpreted as a family symmetry. The theory is based on supersymmetric $SU(5)$ in 6d, where the two extra dimensions are compactified on a ${T}_{2}/{\mathbb{Z}}_{2}$ orbifold. We have shown that, if there is a finite modular symmetry, then it can only be ${A}_{4}$ with an (infinite) discrete choice of moduli, where we focus on $\ensuremath{\tau}=\ensuremath{\omega}={e}^{i2\ensuremath{\pi}/3}$, the unique solution with $|\ensuremath{\tau}|=1$. The fields on the branes respect a generalized $CP$ and flavor symmetry ${A}_{4}\ensuremath{\ltimes}{\mathbb{Z}}_{2}$ which is isomorphic to ${S}_{4}$ which leads to an effective $\ensuremath{\mu}\ensuremath{-}\ensuremath{\tau}$ reflection symmetry at low energies, implying maximal atmospheric mixing and maximal leptonic $CP$ violation. We construct an explicit model along these lines with two triplet flavons in the bulk, whose vacuum alignments are determined by orbifold boundary conditions, analogous to those used for $SU(5)$ breaking with doublet-triplet splitting. There are two right-handed neutrinos on the branes whose Yukawa couplings are determined by modular weights. The charged lepton and down-type quarks have diagonal and hierarchical Yukawa matrices, with quark mixing due to a hierarchical up-quark Yukawa matrix.