Asymptotic structure of $\mathcal{N}=2$ supergravity in 3D: extended super-BMS$_3$ and nonlinear energy bounds

Abstract

The asymptotically flat structure of $\mathcal{N}=(2,0)$ supergravity in three spacetime dimensions is explored. The asymptotic symmetries are spanned by an extension of the super-BMS$_3$ algebra, with two independent $\hat{u}(1)$ currents of electric and magnetic type. These currents are associated to $U(1)$ fields being even and odd under parity, respectively. Remarkably, although the $U(1)$ fields do not generate a backreaction on the metric, they provide nontrivial Sugawara-like contributions to the BMS$_3$ generators, and hence to the energy and the angular momentum. The entropy of flat cosmological spacetimes with $U(1)$ fields then acquires a nontrivial dependence on the $\hat{u}(1)$ charges. If the spin structure is odd, the ground state corresponds to Minkowski spacetime, and although the anticommutator of the canonical supercharges is linear in the energy and in the electric-like $\hat{u}(1)$ charge, the energy becomes bounded from below by the energy of the ground state shifted by the square of the electric-like $\hat{u}(1)$ charge. If the spin structure is even, the same bound for the energy generically holds, unless the absolute value of the electric-like charge is less than minus the mass of Minkowski spacetime in vacuum, so that the energy has to be nonnegative. The explicit form of the Killing spinors is found for a wide class of configurations that fulfills our boundary conditions, and they exist precisely when the corresponding bounds are saturated. It is also shown that the spectra with periodic or antiperiodic boundary conditions for the fermionic fields are related by spectral flow, in a similar way as it occurs for the $\mathcal{N}=2$ super-Virasoro algebra. Indeed, our super-BMS$_3$ algebra can be recovered from the flat limit of the superconformal algebra with $\mathcal{N}=(2,2)$, truncating the fermionic generators of the right copy.