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The asymptotically flat structure of $\mathcal{N}=(2,0)$ supergravity inthree spacetime dimensions is explored. The asymptotic symmetries are spannedby 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 providenontrivial Sugawara-like contributions to the BMS$_3$ generators, and hence tothe energy and the angular momentum. The entropy of flat cosmologicalspacetimes with $U(1)$ fields then acquires a nontrivial dependence on the$\hat{u}(1)$ charges. If the spin structure is odd, the ground statecorresponds to Minkowski spacetime, and although the anticommutator of thecanonical 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 theground state shifted by the square of the electric-like $\hat{u}(1)$ charge. Ifthe 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 themass of Minkowski spacetime in vacuum, so that the energy has to benonnegative. The explicit form of the Killing spinors is found for a wide classof configurations that fulfills our boundary conditions, and they existprecisely when the corresponding bounds are saturated. It is also shown thatthe spectra with periodic or antiperiodic boundary conditions for the fermionicfields 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 canbe recovered from the flat limit of the superconformal algebra with$\mathcal{N}=(2,2)$, truncating the fermionic generators of the right copy.

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