Classical and quantum curves of 5d Seiberg's theories and their 4d limit

TL;DR

This paper explores the classical and quantum Seiberg-Witten curves of 5d N=1 SCFTs, their 4d limits, and their relation to integrable systems, providing new insights into their geometric and quantum structures.

Contribution

It constructs and compares classical and quantum curves of 5d Seiberg theories and their 4d limits using brane web techniques and q-analogues of Frobenius methods.

Findings

Classical curves constructed via five-brane web methods.

Quantum curves derived using q-analogue Frobenius approach.

Reduction to 4d curves matches known Seiberg-Witten and integrable system results.

Abstract

In this work, we examine the classical and quantum Seiberg-Witten curves of 5d N = 1 SCFTs and their 4d limits. The 5d theories we consider are Seiberg's theories of type $E_{6,7,8}$, which serve as the UV completions of 5d SU(2) gauge theories with 5, 6, or 7 flavors. Their classical curves can be constructed using the five-brane web construction [1]. We also use it to re-derive their quantum curves [2], by employing a q-analogue of the Frobenius method in the style of [3]. This allows us to compare the reduction of these 5d curves with the 4d curves, i.e. Seiberg-Witten curves of the Minahan-Nemeschansky theories and their quantization, which have been identified in [4] with the spectral curves of rank-1 complex crystallographic elliptic Calogero-Moser systems.