${\mathbb{R}}^3$ Index for Four-Dimensional $\mathcal{N}=2$ Field Theories

In theories with $\mathcal{N}=2$ supersymmetry on ${\mathbb{R}}^{3,1}$, supersymmetric bound states can decay across walls of marginal stability in the space of Coulomb branch parameters, leading to discontinuities in the BPS indices $\mathrm{\Omega }(\gamma ,u)$. We consider a supersymmetric index $\mathcal{I}$ which receives contributions from $1/2$-BPS states, generalizing the familiar Witten index $\mathrm{Tr}(-1{)}^F{e}^{-\beta H}$. We expect $\mathcal{I}$ to be smooth away from loci where massless particles appear, thanks to contributions from the continuum of multiparticle states. Taking inspiration from a similar phenomenon in the hypermultiplet moduli space of $\mathcal{N}=2$ string vacua, we conjecture a formula expressing $\mathcal{I}$ in terms of the BPS indices $\mathrm{\Omega }(\gamma ,u)$, which is continuous across the walls and exhibits the expected contributions from single particle states at large $\beta$. This gives a universal prediction for the contributions of multiparticle states to the index $\mathcal{I}$. This index is naturally a function on the moduli space after reduction on a circle, closely related to the canonical hyperkähler metric and hyperholomorphic connection on this space.