Ersatz gravity and black-hole thermodynamics from Manin gauge theory with noncompact gauge group

We show that a three-dimensional Manin gauge theory with gauge group $\mathrm{SL}(2;\mathbb{R})$ (i.e. Yang-Mills theory, the third-way theory, or the imaginary third-way theory) minimally coupled to Einstein gravity admits a dual interpretation as Einstein gravity with an exotic coupling to a Manin gauge theory, where the roles of dreibein/spin connection and field strength/gauge potential are interchanged. The dual, or ersatz, gravitational metric ${\widehat{g}}_{\mu \nu }\sim \mathrm{tr}((\star F{)}_{\mu }(\star F{)}_{\nu })$ is a classical double copy of the gauge field strength $F_{\mu \nu }$ (as opposed to the usual double copy of the gauge potential $A_{\mu }$). If matter exclusively couples to $\widehat{g}$ (for example, in a gravitational decoupling limit), then one can formulate black-hole thermodynamics with regards to the ersatz metric. In particular, a black-hole solution for the ersatz metric ${\widehat{g}}_{\mu \nu }$ (made of Yang-Mills fields) radiates ersatz Hawking radiation and obeys the laws of black-hole thermodynamics.