Statistics, 't Hooft Anomaly, and the Else-Nayak Index: a careful comparison of concepts

Abstract: Generalized symmetries and topological excitations, as well as symmetry anomalies and the statistics of topological excitations, are widely believed to be related. There are, however, pitfalls in how this relation is established. A lattice truncation of a symmetry transformation to a finite patch gives a symmetry patch operator that creates symmetry defects at its boundary. This geometric picture resembles a hopping operator creating topological excitations at the boundary of its support, but does not provide well-defined statistics, let alone guarantee agreement with the symmetry anomaly. A more natural and robust relation is that the hopping operators of topological excitations are symmetric: they commute with symmetry transformations. Under suitable assumptions, this condition yields a one-to-one correspondence between statistics and anomalies. We further couple boundary matter to a DW gauge field in one higher dimension to explain this relation from the perspective of gauging. A hopping operator is, in essence, a gauge-invariant operator acting on the physical degrees of freedom after gauging. Once the gauge-field background is fixed, global symmetry comes from gauge transformations that preserve that background, while the symmetric condition on hopping is precisely the remaining gauge invariance. This distinction clarifies potential misconceptions in the literature and provides a more reliable framework for comparing symmetries and topological excitations.