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

TL;DR

This paper develops a systematic method to compute 5d Seiberg-Witten curves for Sp(1) gauge theories with $E_{N_f+1}$ symmetries using toric-like diagrams, confirming known results for specific flavor numbers and extending to rank-N cases.

Contribution

It introduces a new systematic procedure for deriving Seiberg-Witten curves from toric-like diagrams, applicable to generic cases and generalizing to higher rank theories.

Findings

Computed explicit Seiberg-Witten curves for $N_f=6,7$ flavors.

Confirmed agreement with known $E_7, E_8$ results.

Proposed a generalization to rank-N theories.

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.