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Universal Gibbons-Hawking-York term for theories with curvature, torsion and non-metricity

by Johanna Erdmenger, Bastian Heß, Ioannis Matthaiakakis, René Meyer

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Submission summary

Authors (as registered SciPost users): Johanna Erdmenger · Ioannis Matthaiakakis · Rene Meyer
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-02-15
Date submitted: 2022-12-12 10:05
Submitted by: Matthaiakakis, Ioannis
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
  • High-Energy Physics - Theory
Approach: Theoretical


Motivated by establishing holographic renormalization for gravitational theories with non-metricity and torsion, we present a new and efficient general method for calculating Gibbons-Hawking-York (GHY) terms. Our method consists of linearizing any nonlinearity in curvature, torsion or non-metricity by introducing suitable Lagrange multipliers. Moreover, we use a split formalism for differential forms, writing them in $(n-1)+1$ dimensions. The boundary terms of the action are manifest in this formalism by means of Stokes' theorem, such that the compensating GHY term for the Dirichlet problem may be read off directly. We observe that only those terms in the Lagrangian that contain curvature contribute to the GHY term. Terms polynomial solely in torsion and non-metricity do not require any GHY term compensation for the variational problem to be well-defined. We test our method by confirming existing results for Einstein-Hilbert and four-dimensional Chern-Simons modified gravity. Moreover, we obtain new results for Lovelock-Chern-Simons and metric-affine gravity. For all four examples, our new method and results contribute to a new approach towards a systematic hydrodynamic expansion for spin and hypermomentum currents within AdS/CFT.

Published as SciPost Phys. 14, 099 (2023)

Reports on this Submission

Anonymous Report 3 on 2023-1-25 (Invited Report)


1) Potentially interesting direction both in theoretical as well in applied (i.e. condensed matter) research research.

2) Very clear presentation of the technical points.


1) One would have expected at least one simple application of their result i.e. a boundary one-point function of an exact bulk solution with torsion and nonmetricity.


This work is relevant, well written and can be published in its present form. I urge the authors to consider continuing their research in this direction and think of specific examples.

Requested changes

No changes required.

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Anonymous Report 2 on 2023-1-18 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2211.02064v2, delivered 2023-01-18, doi: 10.21468/SciPost.Report.6557


1. Has a universal, model-independent scope

2. Solves a very specific technical problem, namely how to derive Gibbons-Hawking-York-like boundary terms, for generic gravity(-like) theories

3. Uses minimal input and provides maximal generality, including applications to more exotic, higher-derivative theories and/or theories with torsion/non-metricity


1. Provides no genuinely new example not considered in the previous literature, except for results relegated to an appendix (and the addition of torsion in the Lovelock example)

2. Not sure this is a weakness per se, but the two examples provided both differ by some numerical factor from corresponding earlier results in the literature

3. Parts of the paper are repetitive since there is a fairly long summary in section 2 (4 pages), and the actual derivation comes only after the two examples for this derivation


The universality of the results presented in this paper could make this a standard reference for future purposes, especially for researchers interested in holography applied to metric affine gravity theories. While the Gibbons-Hawking-York boundary term is only one ingredient for this purpose, it is undoubtedly an important ingredient. It could have been nice to also include a generalization to supergravity, but the paper is worthwhile as it is.

In my opinion, the paper is suitable for publication in SciPost Physics, both in terms of topic and quality.

Requested changes

The paper could be published in SciPost Physics in its present form. I would have preferred a more condensed version of the presentation, basically eliminating section 2 and inserting instead section 4, and then ending with the examples of section 3. However, I realize that this is a matter of taste, so this should not be considered a requested change but merely an optional suggestion.

  • validity: top
  • significance: high
  • originality: ok
  • clarity: good
  • formatting: perfect
  • grammar: perfect

Anonymous Report 1 on 2023-1-12 (Invited Report)


1. Presents a general methods for deriving GHY terms.

2. Exemplifies/verifies the result in Einstein-Hilbert gravity, Chern-Simons modified gravity and Lovelock-Chern-Simons gravity.

3, The disposition of the presentation is very clear and pedagogical.


No significant weaknesses


The acceptance criteria are met.

Requested changes

No changes required


  • validity: high
  • significance: top
  • originality: high
  • clarity: high
  • formatting: good
  • grammar: excellent

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