5d $E_n$ Seiberg-Witten curve via toric-like diagram

TL;DR

We introduce a systematic method to extract Seiberg-Witten curves for 5d $Sp(1)$ gauge theories with $N_f\,%28= frac{N_f}{?}$ flavors and $E_{N_f+1}$ global symmetry from toric-like $(p,q)$-web diagrams with 7-branes. By formulating the SW curve as $\, extstyle abla ext{ olinebreak} ight)$sum c_{ij} t^i w^j=0$ and enforcing boundary data from external branes, we derive explicit $N_f=6$ ($E_7$) and $N_f=7$ ($E_8$) curves that coincide with known invariant forms, and show equivalence of multiple diagrammatic presentations via Hanany-Witten coordinate transformations. In addition, we demonstrate mass decoupling and 4d limits to connect to the familiar 4d $E_7$ and $E_8$ CFT SW curves, and we establish a rank-$N$ factorization law $SW_N(U_1, frac{}{ }...)=\prod_{i=1}^N SW_1(U_i)$, suggesting a modular, pants-like structure for higher-rank theories. The results provide a versatile framework for exploring 5d uplifts of class S theories and degenerate punctures, with potential links to topological strings and dualities. Overall, the work substantiates the toric-like diagram approach as a practical tool for accessing non-toric UV fixed points in five dimensions.

Abstract

We consider 5d Sp(1) gauge theory with $E_{N_f+1}$ global symmetries based on toric(-like) diagram constructed from (p,q)-web with 7-branes. We propose a systematic procedure to compute the Seiberg-Witten curve for generic toric-like diagram. For $N_f=6,7$ flavors, we explicitly compute the Seiberg-Witten curves for 5d Sp(1) gauge theory, and show that these Seiberg-Witten curves agree with already known $E_{7,8}$ results. We also discuss a generalization of the Seiberg-Witten curve to rank-N cases.

Figures (36)

• Figure 1: A toric diagram for $E_6$
• Figure 2: A brane configuration with five flavors leading to $T_3$-diagram.
• Figure 3: A brane configuration with five flavors leading to a toric-like diagram.
• Figure 4: The configuration for the SW curve of $T_3$ toric diagram
• Figure 5: The boundary conditions toric-like diagram for $E_6$.