Holonomy Saddles and 5d BPS Quivers

TL;DR

This paper explores how 5d pure Yang-Mills theories compactified on a circle exhibit multiple 4d limits, leading to a rich structure of BPS quivers and spectral curves influenced by holonomy saddles.

Contribution

It introduces the concept of holonomy saddles in 5d theories, connecting them to 4d Seiberg-Witten theories and BPS quivers, with detailed examples for SU(2) and SU(3).

Findings

Multiple 4d limits arise from a single 5d theory due to holonomy saddles.

The 5d BPS quivers match the D0 probe dynamics from local Calabi-Yau geometries.

Spectral curves exhibit a N feature affecting the BPS quivers.

Abstract

We study the Seberg-Witten geometry of 5d ${\cal N}=1$ pure Yang-Mills theories compactified on a circle. The concept of the holonomy saddle implies that there are multiple 4d limits of interacting Seiberg-Witten theories from a single 5d theory, and we explore this in the simplest case of pure $SU(N)$ theories. The compactification leads to $N$ copies of locally indistinguishable 4d pure $SU(N)$ Seiberg-Witten theories in the infrared, glued together in a manner dictated by the Chern-Simons level. We show how this picture naturally builds the 5d BPS quivers which agree with the D0 probe dynamics previously proposed via the geometrically engineered local Calabi-Yau. We work out various $SU(2)$ and $SU(3)$ examples through a detailed look at the respective spectral curves. We also note a special $\mathbb{Z}_{2N}$ feature of $SU(N)_N$ spectral curves and the resulting BPS quivers, with emphasis on how the 4d holonomy saddles are affected.