Higher-Spin Flat Space Cosmologies with Soft Hair

TL;DR

The paper develops near-horizon boundary conditions for flat-space 3D higher-spin gravity, revealing a fourfold $\hat{\mathfrak{u}}(1)$ current structure and a simple soft-hair entropy $S_{\mathrm{Th}}=2\pi(J_0^{+}+J_0^{-})$ that is robust against inclusion of higher spins. It constructs a twisted Sugawara map connecting near-horizon currents to asymptotic flat-space cosmology data, and derives a FW_3 algebra from Heisenberg currents, unifying soft hair with the asymptotic charges. The entropy result is shown to match more involved expressions when expressed in terms of higher-spin zero modes, and the framework naturally extends to spins $s\ge 2$ with a consistent generalization of the symmetry algebra. The work highlights the universality of soft hair and provides a concrete path to holographic interpretations of flat-space higher-spin cosmologies, with potential implications for microstate counting and holography beyond AdS.

Abstract

We present and discuss near horizon boundary conditions for flat space higher-spin gravity in three dimensions. As in related work our boundary conditions ensure regularity of the solutions independently of the charges. The asymptotic symmetry algebra is given by a set of $\hat{\mathfrak{u}}(1)$ current algebras. The associated charges generate higher-spin soft hair. We derive the entropy for solutions that are continuously connected to flat space cosmologies and find the same result as in the spin-2 case: the entropy is linear in the spin-2 zero-mode charges and independent from the spin-3 charges. Using twisted Sugawara-like constructions of higher-spin currents we show that our simple result for entropy of higher-spin flat space cosmologies coincides precisely with the complicated earlier results expressed in terms of higher-spin zero mode charges.