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# Gravity coupled to a scalar field from a Chern-Simons action: describing rotating hairy black holes and solitons with gauge fields

Dec 2022

Einstein gravity minimally coupled to a scalar field with a two-parameterHiggs-like self-interaction in three spacetime dimensions is recast in terms ofa Chern-Simons form for the algebra $g^{+}\oplus g^{-}$ where, depending on thesign of the self-interaction couplings, $g^{\pm}$ can be $so(2,2)$, $so(3,1)$or $iso(2,1)$. The field equations can then be expressed through the fieldstrength of non-flat composite gauge fields, and conserved charges are readilyobtained from boundary terms in the action that agree with those of standardChern-Simons theory for pure gravity, but with non-flat connections. Regularityof the fields then amounts to requiring the holonomy of the connections alongcontractible cycles to be trivial. These conditions are automatically fulfilledfor the scalar soliton and allow to recover the Hawking temperature andchemical potential in the case of the rotating hairy black holes presentedhere, whose entropy can also be obtained by the same formula that holds in thecase of a pure Chern-Simons theory. In the conformal (Jordan) frame the theoryis described by General Relativity with cosmological constant conformallycoupled to a self-interacting scalar field, and its formulation in terms of aChern-Simons form for suitably composite gauge fields is also brieflyaddressed.

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