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We study BPS states of 5d $\mathcal{N}=1$ $SU(2)$ Yang-Mills theory on$S^1\times \mathbb{R}^4$. Geometric engineering relates these to enumerativeinvariants for the local Hirzebruch surface $\mathbb{F}_0$. We illustratecomputations of Vafa-Witten invariants via exponential networks, verifyingfiber-base symmetry of the spectrum at certain points in moduli space, andmatching with mirror descriptions based on quivers and exceptional collections.Albeit infinite, parts of the spectrum organize in families described by simplealgebraic equations. Varying the radius of the M-theory circle interpolatessmoothly with the spectrum of 4d $\mathcal{N}=2$ Seiberg-Witten theory,recovering spectral networks in the limit.

Exponential BPS graphs and D-brane counting on toric Calabi-Yau threefolds: Part II