Aspects of 5d Seiberg-Witten theories on $$ {\mathbbm{S}}^1 $$

Abstract

Abstract We study the infrared physics of 5d $$ \mathcal{N} $$ N = 1 Yang-Mills theories compactified on $$ {\mathbbm{S}}^1 $$ 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 $$ {\mathbbm{S}}^1 $$ 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 dP2 theory. Along the way, we also offer a general field theory proof of the odd shift of electric charge on Sp(k)π instanton solitons, previously observed via geometric engineering for low-rank supersymmetric theories.