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Moduli Spaces of Gauge Theories from Dimer Models: Proof of the Correspondence

Sebastian Franco, David Vegh

TL;DR

This work resolves the Mathematical Dimer Conjecture for toric quivers embedded on a two-torus by showing a one-to-one correspondence between GLSM fields and perfect matchings in the brane tiling, and by proving that the toric diagram equals the Newton polygon of the dimer's characteristic polynomial $P(x,y)$. The authors reformulate F-term equations using gauge transformations and magnetic flux coordinates, yielding a natural identification of GLSM variables with perfect matchings and deriving toric coordinates from height-function slopes. They connect the Forward Algorithm's GLSM data to the dimer framework via the KT product and a left-inverse relation $T=K_L^{-1}K$, demonstrating that height changes $(h_x,h_y)$ reproduce the toric diagram coordinates and thus encode the moduli space geometry. This result strengthens dimers as a robust, computationally efficient tool for analyzing toric quiver gauge theories and their AdS/CFT duals, with clear paths for extending to non-toric phases and higher-dimensional generalizations.

Abstract

Recently, a new way of deriving the moduli space of quiver gauge theories that arise on the world-volume of D3-branes probing singular toric Calabi-Yau cones was conjectured. According to the proposal, the gauge group, matter content and tree-level superpotential of the gauge theory is encoded in a periodic tiling, the dimer graph. The conjecture provides a simple procedure for determining the moduli space of the gauge theory in terms of perfect matchings. For gauge theories described by periodic quivers that can be embedded on a two-dimensional torus, we prove the equivalence between the determination of the toric moduli space with a gauged linear sigma model and the computation of the Newton polygon of the characteristic polynomial of the dimer model. We show that perfect matchings are in one-to-one correspondence with fields in the linear sigma model. Furthermore, we prove that the position in the toric diagram of every sigma model field is given by the slope of the height function of the corresponding perfect matching.

Moduli Spaces of Gauge Theories from Dimer Models: Proof of the Correspondence

TL;DR

This work resolves the Mathematical Dimer Conjecture for toric quivers embedded on a two-torus by showing a one-to-one correspondence between GLSM fields and perfect matchings in the brane tiling, and by proving that the toric diagram equals the Newton polygon of the dimer's characteristic polynomial . The authors reformulate F-term equations using gauge transformations and magnetic flux coordinates, yielding a natural identification of GLSM variables with perfect matchings and deriving toric coordinates from height-function slopes. They connect the Forward Algorithm's GLSM data to the dimer framework via the KT product and a left-inverse relation , demonstrating that height changes reproduce the toric diagram coordinates and thus encode the moduli space geometry. This result strengthens dimers as a robust, computationally efficient tool for analyzing toric quiver gauge theories and their AdS/CFT duals, with clear paths for extending to non-toric phases and higher-dimensional generalizations.

Abstract

Recently, a new way of deriving the moduli space of quiver gauge theories that arise on the world-volume of D3-branes probing singular toric Calabi-Yau cones was conjectured. According to the proposal, the gauge group, matter content and tree-level superpotential of the gauge theory is encoded in a periodic tiling, the dimer graph. The conjecture provides a simple procedure for determining the moduli space of the gauge theory in terms of perfect matchings. For gauge theories described by periodic quivers that can be embedded on a two-dimensional torus, we prove the equivalence between the determination of the toric moduli space with a gauged linear sigma model and the computation of the Newton polygon of the characteristic polynomial of the dimer model. We show that perfect matchings are in one-to-one correspondence with fields in the linear sigma model. Furthermore, we prove that the position in the toric diagram of every sigma model field is given by the slope of the height function of the corresponding perfect matching.

Paper Structure

This paper contains 12 sections, 41 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Toric quivers and brane tilings
  3. Geometry of the tiling embedding from conformal invariance
  4. Height function
  5. Toric geometry from gauge theory
  6. The conjecture
  7. The proof
  8. Solving F--term conditions: gauge transformations and magnetic coordinates
  9. The GLSM fields are perfect matchings
  10. Height changes as positions in a toric diagram
  11. Conclusions
  12. Appendix

Figures (10)

  • Figure 1: Quiver diagram for Model II of $dP_2$.
  • Figure 2: Periodic quiver for Model II of $dP_2$. We show several fundamental cells.
  • Figure 3: Two plaquettes are equal once the F--term equation for the common field is imposed.
  • Figure 4: Brane tiling for Model II of $dP_2$.
  • Figure 5: A bipartite graph tiling the Klein Bottle.
  • ...and 5 more figures