  Reflexive polytopes in n dimensions have attracted much attention both in
mathematics and theoretical physics due to their connection to Fano n-folds and
mirror symmetry. This work focuses on the 18 regular reflexive polytopes
corresponding to smooth Fano 3-folds. For the first time, we show that all 18
regular reflexive polytopes have corresponding 2d (0,2) gauge theories realized
by brane brick models. These 2d gauge theories can be considered as the
worldvolume theories of D1-branes probing the toric Calabi-Yau 4-singularities
whose toric diagrams are given by the associated regular reflexive polytopes.
The generators of the mesonic moduli space of the brane brick models are shown
to form a lattice of generators due to the charges under the rank 3 mesonic
flavor symmetry. It is shown that the lattice of generators is the exact polar
dual reflexive polytope to the corresponding toric diagram of the brane brick
model. This duality not only highlights the close relationship between the
geometry and 2d gauge theory, but also opens up pathways towards new
discoveries in relation to reflexive polytopes and brane brick models.
