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

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