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Geometric dualities in 4d field theories and their 5d interpretation

Sebastian Franco, Amihay Hanany

TL;DR

The paper develops a unified framework for studying dualities and flows among 4d N=1 gauge theories from D3-branes at toric singularities by using (p,q) webs to encode geometry and derive 4d quivers, while simultaneously linking to five-dimensional SU(2) theories with flavors. It shows how toric duals are related to Seiberg duality and how geometric transitions correspond to higgsings in four dimensions and flavor mass tunings in five dimensions, with BPS spectra and monopole tensions varying continuously across phases. This three-way correspondence provides a practical toolkit to generate toric phases (including the various dP_n and F0) and to study quiver symmetries and the dynamics of BPS states within a geometric and field-theoretic context. The results illuminate the role of marginal stability in 5d as a mirror of Seiberg duality in 4d and offer a versatile bridge between toric geometry, gauge theory dualities, and higher-dimensional BPS physics.

Abstract

We study four-dimensional N=1 gauge theories arising on D3-branes probing toric singularities. Toric dualities and flows between theories corresponding to different singularities are analyzed by encoding the geometric information into (p,q) webs. A new method for identifying global symmetries of the four-dimensional theories using the brane webs is developed. Five-dimensional theories are associated to the theories on the D3-branes by using (p,q) webs. This leads to a novel interpretation of Seiberg duality, as crossing curves of marginal stability in five dimensions.

Geometric dualities in 4d field theories and their 5d interpretation

TL;DR

The paper develops a unified framework for studying dualities and flows among 4d N=1 gauge theories from D3-branes at toric singularities by using (p,q) webs to encode geometry and derive 4d quivers, while simultaneously linking to five-dimensional SU(2) theories with flavors. It shows how toric duals are related to Seiberg duality and how geometric transitions correspond to higgsings in four dimensions and flavor mass tunings in five dimensions, with BPS spectra and monopole tensions varying continuously across phases. This three-way correspondence provides a practical toolkit to generate toric phases (including the various dP_n and F0) and to study quiver symmetries and the dynamics of BPS states within a geometric and field-theoretic context. The results illuminate the role of marginal stability in 5d as a mirror of Seiberg duality in 4d and offer a versatile bridge between toric geometry, gauge theory dualities, and higher-dimensional BPS physics.

Abstract

We study four-dimensional N=1 gauge theories arising on D3-branes probing toric singularities. Toric dualities and flows between theories corresponding to different singularities are analyzed by encoding the geometric information into (p,q) webs. A new method for identifying global symmetries of the four-dimensional theories using the brane webs is developed. Five-dimensional theories are associated to the theories on the D3-branes by using (p,q) webs. This leads to a novel interpretation of Seiberg duality, as crossing curves of marginal stability in five dimensions.

Paper Structure

This paper contains 19 sections, 22 equations, 21 figures, 1 table.

Table of Contents

  1. Introduction
  2. A review of $(p,q)$ webs and toric geometry
  3. $(p,q)$ webs and five dimensional theories
  4. Toric geometry
  5. Four dimensional quivers from $(p,q)$ webs
  6. Geometric transitions
  7. Blow-ups, unhiggsing and $(p,q)$ webs
  8. The two phases of $F_0$
  9. Higgsings as blow-downs
  10. An application, partial resolutions of $\hbox{$\,{\rm C}$}^3/({ Z Z}_3 \times { Z Z}_3)$
  11. Symmetries of the four dimensional gauge theory
  12. Geometric transitions from the perspective of five dimensional theories
  13. Five dimensional interpretation of the theories
  14. Different limits in moduli space, a first example
  15. Geometrical blow-ups as tuning bare masses
  16. ...and 4 more sections

Figures (21)

  • Figure 1: The three alternative perspectives that will be developed in this paper. The connections between them will be made using $(p,q)$ webs.
  • Figure 2: A $(p,q)$ web corresponding to an $SU(2)$ theory with one flavor.
  • Figure 3: Toric representation of $F_0={\rm I P}^1 \times {\rm I P}^1$.
  • Figure 4: Possible D3, D5 and D7-branes located at 0-cycles and wrapping compact 2 and 4 cycles, respectively.
  • Figure 5: A $(p,q)$ web for $dP_1$ and its associated quiver.
  • ...and 16 more figures