Quantum aspects of non-invertible flavor symmetries in intersecting/magnetized D-brane models

We discuss selection rules of chiral matters in type IIA intersecting and IIB magnetized D-brane models on toroidal orbifolds. Since the chiral matters on toroidal orbifolds are labeled by a certain conjugacy class of the gauged orbifold group, the selection rules involve non-trivial fusion rules. We find that the representation of the chiral matters is described by a D4 flavor symmetry for an even number of magnetic fluxes or winding numbers at tree level. Furthermore, the D4 symmetry still remains even when we take into account loop effects. We also study non-perturbative effects such as D-brane instantons.