Fano 3-Folds, Reflexive Polytopes and Brane Brick Models

TL;DR

This paper establishes a novel correspondence between all 18 regular reflexive polytopes related to smooth Fano 3-folds and 2d (0,2) gauge theories via brane brick models, linking geometry with gauge theory.

Contribution

It demonstrates, for the first time, that each of these reflexive polytopes corresponds to a 2d gauge theory realized by brane brick models, revealing a duality with the polytopes' toric diagrams.

Findings

All 18 polytopes have associated 2d gauge theories.

The mesonic moduli space generators form a lattice dual to the polytope.

The duality links geometric and gauge theoretic structures.

Abstract

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.