# Discrete Symmetries in Dimer Diagrams

## TL;DR

This paper uses dimer diagram techniques to identify and analyze discrete global symmetries, specifically Heisenberg groups, in quiver gauge theories on D3-branes at toric Calabi-Yau singularities, generalizing previous results.

## Contribution

It introduces a universal method to determine discrete symmetries in orbifold theories using dimer diagrams, applicable to a wide class of toric Calabi-Yau singularities.

## Key findings

- Discrete symmetries form Heisenberg groups with specific generators.
- Explicit construction of symmetries for orbifolds of C^3, conifold, and del Pezzo surfaces.
- Connection between gauge theory symmetries and torsion cycles in holographic duals.

## Abstract

We apply dimer diagram techniques to uncover discrete global symmetries in the fields theories on D3-branes at singularities given by general orbifolds of general toric Calabi-Yau threefold singularities. The discrete symmetries are discrete Heisenberg groups, with two $\mathbf{Z}_N$ generators $A$, $B$ with commutation $AB=CBA$, with $C$ a central element. This fully generalizes earlier observations in particular orbifolds of $\mathbf{C}^3$, the conifold and $Y_{p,q}$. The solution for any orbifold of a given parent theory follows from a universal structure in the infinite dimer in $\mathbf{R}^2$ giving the covering space of the unit cell of the parent theory before orbifolding. The generator $A$ is realized as a shift in the dimer diagram, associated to the orbifold quantum symmetry; the action of $B$ is determined by equations describing a 1-form in the dimer graph in the unit cell of the parent theory with twisted boundary conditions; finally, $C$ is an element of the (mesonic and baryonic) non-anomalous U$(1)$ symmetries, determined by geometric identities involving the elements of the dimer graph of the parent theory. These discrete global symmetries of the quiver gauge theories are holographically dual to discrete gauge symmetries from torsion cycles in the horizon, as we also briefly discuss. Our findings allow to easily construct the discrete symmetries for infinite classes of orbifolds. We provide explicit examples by constructing the discrete symmetries for the infinite classes of general orbifolds of $\mathbf{C}^3$, conifold, and complex cones over the toric del Pezzo surfaces, $dP_1$, $dP_2$ and $dP_3$.