On the BPS spectrum of 5d SU(2) super-Yang-Mills

We provide a closed-form expression for the motivic Kontsevich-Soibelman invariant for M-theory in the background of the toric Calabi-Yau threefold $K_{{\mathbb{F}}_0}$. This encodes the refined BPS spectrum of $SU(2)$ 5d ${\cal N}=1$Yang-Millstheoryon$S^1\times $\mathbb{R}$^4$, corresponding to rank-zero Donaldson-Thomas invariants for $K_{{\mathbb{F}}_0}$, anywhere on the Coulomb branch.