  We argue that algebraic and combinatorial polytope mutations of Fano 3-folds
can be identified with mass deformations of associated 2d (0,2) supersymmetric
gauge theories realized by brane brick models. These are Type IIA brane
configurations that realize a large family of 2d worldvolume theories on probe
D1-branes at toric Calabi-Yau 4-folds. We show that brane brick models that are
related by mass deformations associated to algebraic and combinatorial polytope
mutations of Fano 3-folds have mesonic moduli spaces with the same number of
generators. We show that mesonic flavor charges of these generators form convex
reflexive lattice polytopes that are dual to the toric diagrams of the Fano
3-folds. The generating function of mesonic gauge invariant operators, also
known as the Hilbert series of the mesonic moduli space, appears to be
identical for such brane brick models under a particular refinement originating
from the U(1)_R charges in the brane brick model following the mass
deformation.
