Abstract
We study the infrared physics of 5d mathcal{N} = 1 Yang-Mills theories compactified on {mathbbm{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 . 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.
Highlights
We study the infrared physics of 5d N = 1 Yang-Mills theories compactified on S1, with a view toward 4d and 5d limits
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
One may naively think that these quiver quantum mechanics would lift to d = 1 + 1 linear sigma models in the 5d, given how the nodes of such quivers carry magnetic charges and correspond to string-like objects
Summary
We start by gathering and reviewing well-known basic facts about 5d YangMills theories, their Coulomb phases, and the Seiberg-Witten descriptions upon a circle compactification. It is straightforward to see that the mass of the W-boson/dyonic instanton is proportional to the distance between D5/NS5-branes, and the tension of the monopole string proportional to the area of the common face. If the bare coupling-squared g52 ∼ μ−0 1 is positive, the distance between two D5-branes will become smaller as φ decreases and merge at the endpoint of the moduli space. If μ0 is negative, as φ decreases, the distance between two NS5-branes will become smaller and merge at the endpoint of the moduli space 2 In both cases, the Sp(1) gauge symmetry is restored at the endpoint of the moduli space. For a positive μ0, the distance between two D5-branes becomes smaller as φ decreases and merges at the endpoint of the moduli space where the Sp(1) gauge symmetry is restored, just as in the F0 case. The endpoint theory is called the E0 theory, where no gauge symmetry restoration occurs at the boundary point [2]
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