Aspects of 5d Seiberg-Witten Theories on $\mathbb{S}^1$

TL;DR

This paper explores the intricate Coulombic moduli spaces and wall-crossing phenomena in 5d $ ext{N}=1$ Yang-Mills theories compactified on $ ext{S}^1$, revealing connections to 4d Seiberg-Witten geometries and BPS string invariants.

Contribution

It provides a detailed analysis of the global structures of Coulomb moduli spaces and the embedding of 4d geometries within 5d theories, including a proof of electric charge shifts in instanton solitons.

Findings

Coulomb phase boundaries relate to wall-crossings by 5d BPS particles.

Elliptic genera of magnetic BPS strings encode 4d wall-crossing memory.

A general field theory proof of the odd electric charge shift in Sp$(k)_ ext{π}$ instantons.

Abstract

We study the infrared physics of 5d $\cal N=1$ Yang-Mills theories compactified on $\mathbb{S}^1$, with a view toward 4d and 5d limits. Global structures of the simplest Coulombic moduli spaces are outlined, with an emphasis on how multiple planar 4d Seiberg-Witten geometries are embedded in the cigar geometry of a single 5d theory on $\mathbb{S}^1$. The Coulomb phase boundaries in the decompactification limit are given particular attention and related to how the wall-crossings by 5d BPS particles turn off. On the other hand, the elliptic genera of magnetic BPS strings do wall-cross and retain the memory of 4d wall-crossings, which we review with the example of dP$_2$ theory. Along the way, we also offer a general field theory proof of the odd shift of electric charge on Sp$(k)_\pi$ instanton solitons, previously observed via geometric engineering for low-rank supersymmetric theories.