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

Abstract

We study the infrared physics of 5d $\mathcal{N}=1$ 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.

arXiv comment77 pages, 28 figures

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