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On Correspondences Between Toric Singularities and (p,q)-webs

Bo Feng, Yang-Hui He, Francis Lam

TL;DR

This work resolves a long-standing puzzle about how toric singularities, toric diagrams, and (p,q)-webs encode 4D ${ m N}=1$ gauge theories by distinguishing two kinds of (p,q)-webs: toric-(p,q)-webs, which map one-to-one with toric diagrams, and quiver-(p,q)-webs, which can correspond to multiple gauge theories and miss non-chiral matter. Through detailed case studies of partial resolutions of $ abla \, oldsymbol{ m C}^3/(oldsymbol{ m Z}_3 imes oldsymbol{ m Z}_3)$—including $F_2$, $PdP2$, and the $PdP3$ family—the authors show that toric webs and quiver webs can diverge in content, while higgsing and splitting operations provide a controlled way to relate different phases and geometries. Field-theory checks of higgsings (e.g., PdP2→dP1 and PdP3b→dP2) corroborate the web-based pictures, illustrating how giving vacuum expectation values to specific bifundamentals corresponds to brane-splitting or leg-merging in the web. The work concludes that toric-(p,q)-webs offer a more fundamental, geometry-driven correspondence, while quiver-(p,q)-webs serve as a practical tool with intrinsic ambiguities when non-chiral matter or leg ordering is involved, thereby clarifying how to read off higgsings and blowups across the web-geometric framework.

Abstract

We study four-dimensional N=1 gauge theories which arise from D3-brane probes of toric Calabi-Yau threefolds. There are some standing paradoxes in the literature regarding relations among (p,q)-webs, toric diagrams and various phases of the gauge theories, we resolve them by proposing and carefully distinguishing between two kinds of (p,q)-webs: toric and quiver (p,q)-webs. The former has a one to one correspondence with the toric diagram while the latter can correspond to multiple gauge theories. The key reason for this ambiguity is that a given quiver (p,q)-web can not capture non-chiral matter fields in the gauge theory. To support our claim we analyse families of theories emerging from partial resolution of Abelian orbifolds using the Inverse Algorithm of hep-th/0003085 as well as (p,q)-web techniques. We present complex inter-relations among these theories by Higgsing, blowups and brane splittings. We also point out subtleties involved in the ordering of legs in the (p,q) diagram.

On Correspondences Between Toric Singularities and (p,q)-webs

TL;DR

This work resolves a long-standing puzzle about how toric singularities, toric diagrams, and (p,q)-webs encode 4D gauge theories by distinguishing two kinds of (p,q)-webs: toric-(p,q)-webs, which map one-to-one with toric diagrams, and quiver-(p,q)-webs, which can correspond to multiple gauge theories and miss non-chiral matter. Through detailed case studies of partial resolutions of —including , , and the family—the authors show that toric webs and quiver webs can diverge in content, while higgsing and splitting operations provide a controlled way to relate different phases and geometries. Field-theory checks of higgsings (e.g., PdP2→dP1 and PdP3b→dP2) corroborate the web-based pictures, illustrating how giving vacuum expectation values to specific bifundamentals corresponds to brane-splitting or leg-merging in the web. The work concludes that toric-(p,q)-webs offer a more fundamental, geometry-driven correspondence, while quiver-(p,q)-webs serve as a practical tool with intrinsic ambiguities when non-chiral matter or leg ordering is involved, thereby clarifying how to read off higgsings and blowups across the web-geometric framework.

Abstract

We study four-dimensional N=1 gauge theories which arise from D3-brane probes of toric Calabi-Yau threefolds. There are some standing paradoxes in the literature regarding relations among (p,q)-webs, toric diagrams and various phases of the gauge theories, we resolve them by proposing and carefully distinguishing between two kinds of (p,q)-webs: toric and quiver (p,q)-webs. The former has a one to one correspondence with the toric diagram while the latter can correspond to multiple gauge theories. The key reason for this ambiguity is that a given quiver (p,q)-web can not capture non-chiral matter fields in the gauge theory. To support our claim we analyse families of theories emerging from partial resolution of Abelian orbifolds using the Inverse Algorithm of hep-th/0003085 as well as (p,q)-web techniques. We present complex inter-relations among these theories by Higgsing, blowups and brane splittings. We also point out subtleties involved in the ordering of legs in the (p,q) diagram.

Paper Structure

This paper contains 19 sections, 15 equations, 2 figures.

Table of Contents

  1. Introduction and Summary
  2. A Brief Reminder on $(p,q)$-Webs of Branes
  3. The $(p,q)$-Web
  4. Grid Diagrams, Toric Diagrams and Four-dimensional Gauge Theory
  5. Case Studies: Partial Resolutions of $\hbox{$\,{\rm C}$}^3/{ Z Z}_3 \times { Z Z}_3$
  6. The $F_2$ Singularity
  7. The $PdP2$ Singularity
  8. The $PdP3$ Family of Singularities
  9. The $PdP3b$ Singularity
  10. The $PdP3c$ Singularity
  11. The $PdP3a$ Singularity
  12. Higgsing, Splitting and Blowing Up
  13. Higgsing: Checks in the Field Theory
  14. From $PdP2$ to $dP1$
  15. From $PdP3b$ to $dP2$
  16. ...and 4 more sections

Figures (2)

  • Figure 1: The Brane Box model of for the orbifold theory $\hbox{$\,{\rm C}$}^3/{ Z Z}_{4}$ with action $(1,1,2)$. We have also included the adjacency matrix $A$ for reference.
  • Figure 2: (a) The brane box model for the matter content of $\hbox{$\,{\rm C}$}^3/{ Z Z}_3\times { Z Z}_3$ and (b) its corresponding quiver-$(p,q)$-web. Notice that we have three equivalent groups of external legs which we have labelled as $(123)$, $(456)$ and $(789)$. For every group, the order of nodes has not been specified, the $(p,q)$ charges have been written in square brackets. For reference, we have included the quiver matrix for the theory.