Combinatorial and Algebraic Mutations of Toric Fano 3-folds and Mass Deformations of 2d (0,2) Quiver Gauge Theories

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.