Novel moduli space in modular flavor models : a case study for modular $T^{\mathrm{\prime }}$ seesaw model with hidden $SU(2)$ gauge symmetry

We study flavor phenomenologies in a basis of a double covering of modular $A_4$ group with a hidden $SU(2)$ symmetry, in which we work on regions at nearby two fixed points and three special points. These special points of $SL(2,\mathbb{Z})$ moduli space are statistically favored in flux compactifications of Type IIB string theory. Neutrino masses are approximately obtained by using the seesaw mechanism due to our two additional symmetries. We perform chi square numerical analysis for each of the fixed/special points in the normal and inverted hierarchies and demonstrate predictions for each case. Then, we briefly discuss the possible signature of hidden gauge bosons at collider experiments from the hidden $SU(2)$ symmetry.