Enhanced asymptotic symmetry algebra of $(2+1)$-dimensional flat space

In this paper we present a new set of asymptotic boundary conditions for Einstein gravity in $(2+1)$-dimensions with a vanishing cosmological constant that are a generalization of the Barnich-Compère boundary conditions [G. Barnich and G. Compere, Classical Quantum Gravity 24, F15 (2007)]. These new boundary conditions lead to an asymptotic symmetry algebra that is generated by a ${\mathfrak{bms}}_3$ algebra and two affine $\widehat{\mathfrak{u}}(1)$ current algebras. We then apply these boundary conditions to topologically massive gravity (TMG) and determine how the presence of the gravitational Chern-Simons term affects the central extensions of the asymptotic symmetry algebra. We furthermore determine the thermal entropy of solutions obeying our new boundary conditions for both Einstein gravity and TMG.