Argyres–Douglas theories, S 1 reductions, and topological symmetries

Abstract

In a recent paper, we proposed closed-form expressions for the superconformal indices of the $({A}_{1},{A}_{2n-3})$ and $({A}_{1},{D}_{2n})$ Argyres–Douglas (AD) superconformal field theories (SCFTs) in the Schur limit. Following up on our results, we turn our attention to the small S 1 regime of these indices. As expected on general grounds, our study reproduces the S 3 partition functions of the resulting dimensionally reduced theories. However, we show that in all cases—with the exception of the reduction of the $({A}_{1},{D}_{4})$ SCFT—certain imaginary partners of real mass terms are turned on in the corresponding mirror theories. We interpret these deformations as R symmetry mixing with the topological symmetries of the direct S 1 reductions. Moreover, we argue that these shifts occur in any of our theories whose four-dimensional ${ \mathcal N }=2$ superconformal $U{(1)}_{R}$ symmetry does not obey an SU(2) quantization condition. We then use our R symmetry map to find the four-dimensional ancestors of certain three-dimensional operators. Somewhat surprisingly, this picture turns out to imply that the scaling dimensions of many of the chiral operators of the four-dimensional theory are encoded in accidental symmetries of the three-dimensional theory. We also comment on the implications of our work on the space of general ${ \mathcal N }=2$ SCFTs.