Discrete Symmetries in Dimer Diagrams

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, $d{P}_1$, $d{P}_2$ and $d{P}_3$.

42 pages + Appendices

References in corpus (14)

• Symmetries and Strings in Field Theory and Gravity
• The Swampland: Introduction and Review
• Constraints on symmetry from holography
• Brane Tilings
• The Master Space of N=1 Gauge Theories
• Counting Chiral Operators in Quiver Gauge Theories
• Mastering the Master Space
• Exactly marginal deformations of quiver gauge theories as seen from brane tilings
• Finite Heisenberg Groups in Quiver Gauge Theories
• E3-brane instantons and baryonic operators for D3-branes on toric singularities
• Discrete Gauge Symmetries and the Weak Gravity Conjecture
• Central Extensions of Finite Heisenberg Groups in Cascading Quiver Gauge Theories
• Finite Heisenbeg Groups and Seiberg Dualities in Quiver Gauge Theories
• Finite Heisenberg Groups from Nonabelian Orbifold Quiver Gauge Theories

Cited by in corpus (2)

• Discrete Symmetries, Weak Coupling Conjecture and Scale Separation in AdS Vacua
• Implications of the weak gravity conjecture in anomalous quiver gauge theories