Fractons on curved spacetime in 2 + 1 dimensions

We have incorporated the following two new paragraphs:

- Below Eq. (3.42)

“In summary, we have brought the Chern--Simons theory based on the fracton algebra, with and without cosmological constant, from the first order to the second order formulation. In the second order formulation, the Lagrangian depends on the geometric quantities that describe an Aristotelian geometry, $(\tau_{\mu},h_{\mu\nu})$. This is analogous to the reformulation of Chern--Simons theories based on the Poincar\'e or (A)dS algebras to-three dimensional gravity in the metric formulation~\cite{Achucarro:1987vz,Witten:1988hc}, but without Lorentzian boost symmetry. Additionally, our theory naturally incorporates the coupling of a fractonic electromagnetic field, described by $(\phi,A_{\mu\nu})$ to the Aristotelian gravitational theory, such that we have a generalization of dipole conservation to curved and unrestricted Aristotelian geometry. Further physical implications of this theory are discussed in~\cite{Huang:2023zhp}.”

- In page 17 in the discussion section:

“\item[Aristotelian black holes] The circularly symmetric solution described in Section \ref{sec:solutions-charges}, with negative cosmological constant, shares many properties with the BTZ black hole in General Relativity. It is therefore natural to ask whether it is possible to define a notion of an ``Aristotelian black hole’’. Given the significant differences between the properties of Aristotelian and Riemannian geometries, one might attempt to extend the concept of an event horizon to Aristotelian geometries, as well as their thermal properties, as was done, for example, in the case of Carrollian gravitational theories \cite{Ecker:2023uwm}. One possible approach is to leverage the isomorphism between the dipole algebra with a negative cosmological constant and the three-dimensional Poincaré algebra to describe thermal solutions within the Chern-Simons formulation of the Aristotelian theory. In particular, we would like to study whether the flat-space generalization of the Cardy formula \cite{Bagchi:2012xr,Barnich:2012xq} could play a significant role in describing the thermal properties of Aristotelian black holes, as well as its connection to the BMS-type asymptotic conditions outlined in \eqref{eq:as_cond}.”