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Generalized dilaton gravity in 2d

by Daniel Grumiller, Romain Ruzziconi, Céline Zwikel

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Submission summary

Authors (as registered SciPost users): Daniel Grumiller
Submission information
Preprint Link:  (pdf)
Date submitted: 2021-09-14 07:57
Submitted by: Grumiller, Daniel
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


Generalized dilaton gravity in 2d is the most general consistent deformation of the Jackiw-Teitelboim model that maintains local Lorentz invariance. The action is generically not power-counting renormalizable, thus going beyond the class of models typically studied. Nevertheless, all these models are exactly soluble. We focus on a subclass of dilaton scale invariant models. Within this subclass, we identify a 3-parameter family of models that describe black holes asymptoting to AdS2 in the UV and to dS2 in the IR. Since these models could be interesting for holography, we address thermodynamics and boundary issues, including boundary charges, asymptotic symmetries and holographic renormalization.

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Reports on this Submission

Anonymous Report 2 on 2021-10-30 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2109.03266v2, delivered 2021-10-30, doi: 10.21468/SciPost.Report.3762


1. A large new class of models of dilaton gravity is derived, which might be interesting.
2. The work is structurally tentalizing and raises immediate follow-up questions.


1. The calculations in section 2.3, around which much of the article centers, are densely written.
2. It is not yet clear if the new models are actually physically interesting, or whether this will end up being more a classification exercice.


This work explores the space of physically sensible models of dilaton gravity. The idea is to start with the known gauge theory (or first order) formulation of JT gravity, then generalize the underlying gauge theoretic structure from a BF theory to something known as a Poisson sigma model. They then impose that the Poisson sigma is actually describing gravity by demanding local Lorenz invariance. This makes sense, and by rewriting things in the second order fomulation (metric and dilaton) they find a general class of dilaton gravity actions corresponding with those sigma models - more general than the theories considered earlier.

They then solve the classical equations of motions, but without using any boudnary conditions, which I find peculiar, but one could choose to do this. Section 2.3 where they solve the EOM is in my opinion too densely written.

They then focus on a certain subset of new models, one could say the "most obvious" generalizations beyond the class of theories studied before, which can essentially be captured by one deformation parameter in (48). The classical solutions can have horizons, and can have a dS region behind the horizon whilst being asymptotically AdS. This last observation is surprising, and the main argument in favor of considering these deformations as interesting.

It was not clear to me what to make of this observation, but it sounds potentially interesting and could potentially turn out to be useful.

The work raises interesting follow up questions. From a structural point of view, two obvious questions are (1) whether one can go beyond classical solutions and use the Poisson sigma model formulation to compute path integrals exactly (as you can do for JT and many other dilaton gravity models), and (2) if with the appropriate boundary conditions the model as a holographic dual as quantum mechanics on (a constrained version of) the target space of the sigma model. From a physical perspective the obvious question is what to make of this dS region.
Because the work raises these interesting follow-up questions, I recommend it for publication.

I do have some recomendations / minor complaints that I think would increase the quality of the paper
1. The meaning of the notation $d\pm\omega$ in (19b) and (19d) is unclear to me.
2. Section 2.3 is quite densely written, perhaps a small appendix would be usefull.
3. There are a lot of symbols for the solutions being used, some of which are redundant (for example on could eliminate $y$ in function of $\ell$), this makes it a bit thougher for the reader to understand what is physically happening.
4. It is unclear why the authors focus on $z=1$ in section 6 as "preferred" value. To me it lookes as if the regime $z=\infty$ is most insteresting, given formula (82).
5. It is mentionned at some point that C is related to the mass, it would be useful to refer to the relevant equation there, it took me a while to find it.

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Anonymous Report 1 on 2021-10-22 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2109.03266v2, delivered 2021-10-22, doi: 10.21468/SciPost.Report.3724


1. Very clearly written
2. Finds an interesting class of new models with intriguing properties
3. Raises many interesting directions to follow up on


1. Could benefit from more details on some aspects of the models/solutions, see report


This paper studies two dimensional dilaton-gravity theories. A large new class of such models is introduced in the paper. These models are obtained from Poisson sigma models, a class of generalized gauge theories, by imposing that local Lorentz invariance is part of the gauge group. This presents the gravitational model in a first order formulation and it is shown that in the second order formulation these models correspond to having an arbitrary dilaton potential that can depend on the magnitude of the gradient of the dilaton besides the dilaton itself. Allowing such potentials results in models that are not power-counting renormalizable, but it is argued that this is not a problem since the theories possess no local degrees of freedom. The authors argue that this is the most general class of theories with this field and gauge content, based on a similar result about Poisson sigma models by Izawa. It is shown that these theories always admit linear dilaton solutions, and the phase space and symplectic form are constructed explicitly.

The rest of the paper focuses on a three parameter sub-family of these new models, with one parameter introducing the minimal new feature, i.e. gradient of the dilaton to the fourth power. The linear dilaton solutions are explicitly written down for this sub-family. There are three distinct cases depending on the value of a discriminant formed from the three parameters of the model. A well defined asymptotic region and vacuum exists for positive discriminant and the rest of the paper focuses on this case.

The authors proceed to analyze the linear dilaton solutions and they find that, depending on the values of the parameters of the model, they can correspond to asymptotically $AdS_2$ or flat black hole solutions. Moreover, near the origin (defined where the dilaton is zero), the solution looks like de Sitter, and the geometry can be positively curved at the horizon. The behavior of the Ricci scalar in different regions of the spacetime is analyzed in detail as a function of the parameters of the model. The vacuum of these theories is Poincare $AdS_2$. For generic values of the parameters, the black hole solutions are found to be Lifshitz-like, in the sense that there is a non-trivial scaling exponent between the mass and the temperature. In the limit of large “anisotropy”, the solutions limit to a locally $AdS_2$ spacetime in the exterior, glued to a locally $dS_2$ spacetime in the interior along the horizon.

The authors then proceed to discuss the Euclidean solution and examine its thermodynamic properties, by calculating the renormalized on-shell action, the Hawking temperature, and the Wald entropy of the horizon, these fit together consistently as expected. The solutions are found to be thermodynamically stable.

Finally, the boundary conditions and asymptotic symmetries are laid out for the isotropic models, and the corresponding asymptotic charges are constructed.

Two dimensional dilaton-gravity theories are very useful arenas to tackle non-perturbative questions about gravity. JT gravity and some of its generalizations have been subject to much recent interest in this regard, providing new insights into the black hole information problem, spacetime emergence in AdS/CFT, and raising new puzzles about quantum gravity. This paper provides a large extension of this class of models, working out many of the classical aspects of the interesting solutions, paving a way to a quantum study. Moreover, extremely intriguing features are found, such as the de Sitter interiors, and the Lifschitz scaling in the absence of boundary anisotropy, all deserving further study. I therefore heartily recommend this excellent paper for publication.

I have a couple of small questions that the authors could optionally consider addressing:

-What are $a_0$ and $a_1$ in (12)? They are left undefined but clash with the parameters of the three parameter model in sec. 5.

-Is it obvious that the Izawa proof generalizes in the presence of the local Lorentz invariance constraint on $P_{IJ}$?

-Is JT gravity reachable as a limiting case of the positive discriminant class? The fact that $R>0$ at the origin always suggests that it’s not, since in JT gravity $R$ is constant negative everywhere. Similar comments are made in the paper about the zero discriminant case, but this point about JT gravity maybe worth emphasizing. Could there be other such “islands” in model space whose solutions cannot be deformed into each other?

-How does the Kruskal extension/Penrose diagram of these solutions look like? Is it a two boundary strip as in the case of JT gravity, with alternating $X\rightarrow\infty$ and $X\rightarrow-\infty$ boundaries, but now with de Sitter-like interior regions?

-The discussion of Lifschitz scaling around (71) seems maybe a bit misleading, after all, this anisotropy near the boundary maybe removed by a change of coordinates (the same change that puts the vacuum $AdS_2$ in standard Poincare coordinates). Of course, comparing (92) and (93) or looking at (96) shows clearly the Lifschitz-like physical features.

-For negative $z$ in empty $AdS_2$ (73), the origin and the asymptotic boundary are swapped, which may affect the Euclidean analysis in the BH case (among other things). Does this affect the conclusion of negative heath capacity for this branch?

-Where was the dilaton scale invariance (45) important for the paper?

-In the asymptotic symmetry group, the factor of $C_\infty(S^1)$ presumably refers to the exponentiated version of $G $not being required to be bijective. How can one see that there are no such global constraints on the finite versions of $G$-transformations?

-Before (105): yields the charges->yields the variation of the charges?

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