**4. Conclusions**

Chern–Simons gravity (CSG) models in *d* = 2*n* + 1 dimensions were extensively studied in the literature, starting with Witten's work for *d* = 3 [1], where the gravitational model is described by the Einstein–Hilbert Lagrangian with a cosmological constant. In the *d* > 3 case, such systems consist of specific superpositions of gravitational Lagrangians featuring all possible powers of the Riemann curvature of the given dimension, each appearing with a precise numerical coefficient. The main purpose of this paper was to propose a generalization of the CSG model, with a Lagrangian, which, in addition to the (standard) CSG Lagrangian, features new terms that are described by a frame-vector field *φa* and a scalar field *ψ*. Like the CSG, which result from the non-Abelian (nA) Chern–Simons (CS) densities, these new Lagrangians result from a new class of CS densities, which, in addition to the nA gauge field, feature an algebra-valued Higgs scalar. Like the usual nA CS densities, which result from the usual Chern–Pontryagin (CP) densities, these new CS densities are constructed in the same way, but now from the dimensional descendents of the CP densities that feature the Higgs scalar. The latter are referred to as Higgs–Chern–Pontryagin (HCS) [7–9] densities, and they are the building blocks for the generalised CSG's, namely the HCSG's [4,5] studied here.

It should be noted at this stage that the construction of HCSG's is not only confined to odd dimensions, since the HCS from which they are constructed are defined is both odd *and* even dimensions. The main reason that we have restricted our attention to odd dimensions in these preliminary investigations is that only in odd dimensions there exist CSG's, which can provide a background for the new gravitational field configurations. In even dimensional spacetimes, the HCSG models, as typified by the 3 + 1 dimensional examples in Refs. [4,5], also consist of frame-vector and scalar fields (*φ<sup>a</sup>*, *ψ*) that interact with the gravitational *Vielbein eaμ* (or the metric). These Lagrangians are invariant under gravitational gauge transformations; however, different from the odd dimensional case in this work, they do not feature (gauge-variant, pure gravity) CSG terms, with their action mixing the contribution of (*φ<sup>a</sup>*, *ψ*), *eaμ* fields.

The new fields (*φ<sup>a</sup>*, *ψ*) display non-standard dynamics, in that they feature linear 'velocity coordinates', rather than the standard 'velocity squared' kinetic terms. It may be relevant to stress that (*φ<sup>a</sup>*, *ψ*) can be seen as 'gravitational coordinates', rather than usual matter fields, since, on the level of the HCS densities from which the HCSG result, the Higgs scalar is on the same footing as the non Abelian gauge connection.

The present work, which is a continuation of that done in Ref. [6] for the the lowest dimension *d* = 3, provides the explicit expression of the HCS Lagrangians up to *d* = 7, together with an investigation of the simplest solutions for *d* = 3, 5. These solutions have the property that they do not backreact on the spacetime geometry in common, i.e., their effective energy-momentum tensor vanishes. However, while, for *d* = 3, this includes the case of BTZ BH, for *d* = 5 only a maximally symmetric AdS background is allowed. We attribute this feature to the fact that the BTZ BH possesses the same amount of symmetries as pure AdS3, being a global identification of it [14,26]. On the other hand, the case of *d* > 3 BHs in CSG are different; although their line-element is still BTZ-like [22], they are less symmetric than the AdS*d* background.

Finally, for *d* = 3, the Ref. [6] has provided (numerical) evidence for the existence of BTZ-like BH also with standard asymptotics for the fields (*φ<sup>a</sup>*, *ψ*), provided that the action is supplemented with a Maxwell field. We conjecture that a similar property holds in the higher dimensional case. In this respect, it may be interesting to consider the HCSG systems in the presence of non-Abelian matter (in *d* > 3), or Skyrme scalars, in order to search for regular solutions.

**Author Contributions:** The two authors contributed equally to the conceptualization, methodology, software, validation, formal analysis, investigation, resources, data curation, writing—original draft preparation, writing—review and editing, visualization, supervision, project administration, funding acquisition. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work of E.R. is supported by the Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT-Fundacao para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020, and by national funds (OE), through FCT, I.P., in the scope of the framework contract foreseen in the numbers 4, 5 and 6 of the article 23,of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19. We acknowledge support from the projects PTDC/FIS-OUT/28407/2017 and CERN/FIS-PAR/0027/2019. This work has further been supported by the European Union's Horizon 2020 research and innovation (RISE) programme H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740. The authors would like to acknowledge networking support by the COST Action CA16104.

**Acknowledgments:** We are grateful to Ruben Manvelyan for useful discussions.

**Conflicts of Interest:** The authors declare no conflict of interest.
