$R^3$ index for four-dimensional $N=2$ field theories

Abstract

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 $\hat{I}\text{c◯}({\hat{I}}^3,u)$. We consider a supersymmetric index $I$ which receives contributions from 1/2-BPS states, generalizing the familiar Witten index $Tr(-1{)}^F{e}^{-{\hat{I}}^{2}H}$. 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 $\hat{I}\text{c◯}({\hat{I}}^3,u)$, which is continuous across the walls and exhibits the expected contributions from single particle states at large ${\hat{I}}^2$. 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 hyperkähler metric and hyperholomorphic connection on this space.

7 pages; v2: introduction expanded, minor corrections, differs from published version in PRL in that supplemental material is included as an Appendix