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Five-Dimensional Gauge Theories from Shifted Web Diagrams

Brice Bastian, Stefan Hohenegger, Amer Iqbal, Soo-Jong Rey

TL;DR

Shifted toric web diagrams extend the extended Kähler moduli space of Calabi–Yau threefolds and enable new five-dimensional quiver gauge theories to emerge as decompactification limits of six-dimensional parents. The authors illustrate the mechanism with explicit examples, deriving 5d theories such as $SU(2)$ with $N_f=2$ and $U(3)$ with $N_f=4$, and then generalize to a two-parameter series that yields AB hexagon quivers $([U(A{+}1)]^B,2A\,\mathbf{F},(B{-}1)\,\mathbf{BF})$ in five dimensions. They also connect these 5d limits to a divisor-based pattern $[U((N{+}1)/D)]^{D-1}$, enriching the geometric engineering perspective and aligning with contemporary 5d classifications. The work opens avenues for further exploration of shifted-web limits and their gauge-theoretic implications in the broader landscape of high-dimensional quantum field theories.

Abstract

In previous works (arXiv:1610.07916, arXiv:1711.07921, arXiv:1807.00186) we studied a class of toric Calabi-Yau threefolds which engineer six-dimensional supersymmetric gauge theories with gauge group $U(N)$ and adjoint matter. The Kähler moduli space of these manifolds can be extended through flop transformations to include regions which are described by so-called shifted toric web diagrams. In this paper we analyse gauge theories that are engineered by these shifted toric web diagrams and argue that in specific limits, some of the them engineer five-dimensional quiver gauge theories with gauge group $G\subset U(N)$ and with fundamental and bi-fundamental matter. We discuss several examples in detail and describe how the matter sector is obtained from the six-dimensional parent theory.

Five-Dimensional Gauge Theories from Shifted Web Diagrams

TL;DR

Shifted toric web diagrams extend the extended Kähler moduli space of Calabi–Yau threefolds and enable new five-dimensional quiver gauge theories to emerge as decompactification limits of six-dimensional parents. The authors illustrate the mechanism with explicit examples, deriving 5d theories such as with and with , and then generalize to a two-parameter series that yields AB hexagon quivers in five dimensions. They also connect these 5d limits to a divisor-based pattern , enriching the geometric engineering perspective and aligning with contemporary 5d classifications. The work opens avenues for further exploration of shifted-web limits and their gauge-theoretic implications in the broader landscape of high-dimensional quantum field theories.

Abstract

In previous works (arXiv:1610.07916, arXiv:1711.07921, arXiv:1807.00186) we studied a class of toric Calabi-Yau threefolds which engineer six-dimensional supersymmetric gauge theories with gauge group and adjoint matter. The Kähler moduli space of these manifolds can be extended through flop transformations to include regions which are described by so-called shifted toric web diagrams. In this paper we analyse gauge theories that are engineered by these shifted toric web diagrams and argue that in specific limits, some of the them engineer five-dimensional quiver gauge theories with gauge group and with fundamental and bi-fundamental matter. We discuss several examples in detail and describe how the matter sector is obtained from the six-dimensional parent theory.

Paper Structure

This paper contains 12 sections, 33 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Specific examples
  3. Example: $(N,M)=(3,1)$
  4. Example: $(N,M)=(5,1)$
  5. Shift $\delta=2$
  6. Shift $\delta=3$
  7. A Two Parameter Series of Five-dimensional Theories
  8. Conclusions
  9. Decompactifying limit and the partition function
  10. Intersection numbers for $X_{N,M}$
  11. Infinite toric fan
  12. $X_{N,M}^{(\delta)}$ and intersection numbers

Figures (14)

  • Figure 1: Web diagram for $X_{N,1}^{(\delta)}$. The labelling of the lines at the top of the diagram is understood to be modulo $N$, such that for $\delta=0$ we get back the 'usual' web diagram $X_{N,1}$.
  • Figure 2: Web diagram of $X_{3,1}^{(1)}$.
  • Figure 3: (a) Gluing two copies of $\mathbb{F}_1$. (b) Same geometry after cutting along the red line and re-gluing along the line labelled $-E_1$.
  • Figure 4: (a) Geometry of Fig. \ref{['Fig:31StartingF1F1']} after a flop transformation of the line $-E_1$. (b) Same geometry after an $SL(2,\mathbb{Z})$ transformation.
  • Figure 5: Decompactified web diagram of $X_{3,1}^{(1)}$.
  • ...and 9 more figures