Gauge Theories at Resolved and Deformed Singularities using Dimers

TL;DR

This work develops a unified framework using brane tilings (dimer diagrams) on the 2-torus to study N=1 quiver gauge theories on D3-branes at toric Calabi–Yau singularities under geometric smoothing. It provides explicit dimer-based rules for partial resolutions (including splits into multiple diagonally decoupled sectors) via Higgsing and FI terms, and for complex deformations via confinement of fractional branes, with a mirror Riemann-surface interpretation and a perfect-matchings perspective. The results enable efficient construction of the remaining gauge theories after these transitions and deepen the dictionary between toric geometry and D3-brane dynamics, offering practical tools for model-building and geometric transitions in toric settings. The methods connect zig-zag paths, perfect matchings, and mirror geometry to concrete gauge-theory dynamics, making geometric smoothing accessible in a purely combinatorial dimer language.

Abstract

The gauge theory on a set of D3-branes at a toric Calabi-Yau singularity can be encoded in a tiling of the 2-torus denoted dimer diagram (or brane tiling). We use these techniques to describe the effect on the gauge theory of geometric operations partially smoothing the singularity at which D3-branes sit, namely partial resolutions and complex deformations. More specifically, we describe the effect of arbitrary partial resolutions, including those which split the original singularity into two separated. The gauge theory correspondingly splits into two sectors (associated to branes in either singularity) decoupled at the level of massless states. We also describe the effect of complex deformations, associated to geometric transitions triggered by the presence of fractional branes with confinement in their infrared. We provide tools to easily obtain the remaining gauge theory after such partial confinement.

Figures (41)

• Figure 1: Quiver and dimer for a $\bf Z_2$ orbifold of the conifold. Faces in the dimer correspond to gauge groups, edges correspond to bifundamentals and each vertex corresponds to a superpotential term. Edges have an orientation determined by the coloring of the adjacent nodes.
• Figure 2: a) An example of a web diagram (for the theory in Figure \ref{['dconidimer_quiver']}); b) the corresponding Riemann surface $\Sigma$ in the mirror geometry.
• Figure 3: Structure of the non-trivial 3-cycles in the geometry $\mathcal{W}$. They are constructed by fibering over the segment joining $W=0$ and $W=W^*$, the $S^1$ in the $uv$ fiber (degenerating at $W=0$) times the 1-cycle in the $P(z,w)$ fiber degenerating at $W=W^*$.
• Figure 4: Dimer of the conifold with the corresponding zig-zag paths.
• Figure 5: a) Tiling of the Riemann surface (which is topologically a sphere, shown as the complex plane) for the case of D3-branes at a conifold singularity. b) The web diagram, providing a skeleton of the Riemann surface, with asymptotic legs corresponding to punctures (and hence to faces of the tiling of $\Sigma$, and zig-zag paths of the original dimer diagram).