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arxiv: 2311.00550 · v2 · submitted 2023-11-01 · ✦ hep-th

Holographic renormalization and the variational problem for mixed boundary conditions via a solution-dependent superpotential-like function

David Choque , Ra\'ul Rojas This is my paper

Pith reviewed 2026-05-24 05:36 UTC · model grok-4.3

classification ✦ hep-th
keywords holographic renormalizationmixed boundary conditionsdesigner gravitysuperpotentialAdS black holesvariational principlescalar counterterms
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The pith

Imposing integrable mixed boundary conditions fixes the cubic coefficient in the near-boundary expansion of a solution-dependent superpotential W(φ), encoding those conditions directly in the scalar counterterm.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies holographic renormalization for four-dimensional Einstein gravity coupled to a scalar field in asymptotically AdS spacetimes that admit mixed designer-gravity boundary conditions. It defines a solution-dependent superpotential-like function W(φ) from the equations of motion for static black-hole solutions. For the case m²L² = -2 the bulk equations leave the cubic term in the near-boundary expansion of W(φ) undetermined. Requiring the variational principle to be well-posed once integrable conditions B = B(A) are imposed determines that coefficient from the boundary data alone. The resulting counterterm renders the Euclidean on-shell action finite without further scalar boundary terms and yields a renormalized stress tensor obeying the quantum-statistical relation.

Core claim

For static black-hole solutions in four-dimensional Einstein-scalar gravity with m²L² = -2, a solution-dependent superpotential-like function W(φ) is introduced from the equations of motion. Its near-boundary expansion takes the form W(φ) = -4/L - φ²/(2L) + a φ³ + O(φ⁴), where the cubic coefficient a remains free after the bulk equations are solved. Once integrable mixed boundary conditions B = B(A) are imposed and the variational principle is required to be well-posed, a is fixed in terms of the boundary deformation. This encodes the mixed boundary condition directly in the scalar counterterm, so that the Euclidean on-shell action is finite without additional scalar boundary terms.

What carries the argument

The solution-dependent superpotential-like function W(φ), defined directly from the equations of motion and inserted into the scalar counterterm.

If this is right

  • The renormalized Euclidean on-shell action is finite and yields the correct black-hole thermodynamics.
  • The holographic stress tensor is obtained and satisfies the quantum-statistical relation under the mixed conditions.
  • W(φ) supplies a natural characterization of holographic renormalization-group data for non-extremal backgrounds.
  • In consistent truncations the RG observables are controlled by the solution-dependent W(φ) rather than by the supergravity superpotential.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same fixing mechanism may apply to other scalar masses or higher dimensions where near-boundary expansions contain undetermined coefficients.
  • Thermodynamic quantities for black holes in designer-gravity theories could be computed directly from the boundary data encoded in a without constructing additional counterterms.
  • The approach suggests that certain classes of boundary conditions can be absorbed into the definition of the counterterm itself.

Load-bearing premise

The mixed boundary conditions are integrable (B is a function of A) and requiring the variational principle to be well-posed is sufficient to fix the cubic coefficient without further input from the bulk dynamics or extra boundary terms.

What would settle it

An explicit static asymptotically AdS black-hole solution with m²L² = -2 and integrable mixed boundary conditions B = B(A) in which the cubic coefficient a in W(φ) cannot be expressed solely in terms of the boundary deformation, or in which the on-shell action still diverges after the W(φ)-based counterterm is subtracted.

read the original abstract

We study holographic renormalization and the variational problem in four-dimensional Einstein gravity coupled to a self-interacting scalar field in asymptotically AdS spacetimes with mixed, designer-gravity boundary conditions. For static black-hole solutions, we introduce a solution-dependent superpotential-like function $W(\phi)$, motivated by the Hamilton--Jacobi formulation but defined directly from the equations of motion. Focusing on the case $m^{2}L^{2}=-2$, we show that the near-boundary expansion $ W(\phi)=-\frac{4}{L}-\frac{\phi^{2}}{2L}+a\phi^{3}+\mathcal{O}(\phi^{4}) $ is not fully determined by the bulk equations. Instead, once integrable mixed boundary conditions $B=B(A)$ are imposed and the variational principle is required to be well posed, the cubic coefficient is fixed in terms of the boundary deformation. In this way, the mixed boundary condition is encoded directly in the scalar counterterm, rendering the Euclidean on-shell action finite without the need for additional scalar boundary terms. We then derive the renormalized Euclidean action and holographic stress tensor, verify the quantum-statistical relation under mixed boundary conditions, and show that $W(\phi)$ provides a natural characterization of holographic renormalization-group data in non-extremal backgrounds. Finally, we illustrate the formalism in exact asymptotically AdS black-hole solutions arising in consistent truncations, including a case where comparison with a supergravity superpotential clarifies why the RG observables are controlled by the solution-dependent function $W(\phi)$ rather than by $W_{\text{SUGRA}}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces a solution-dependent superpotential-like function W(φ), defined directly from the bulk equations of motion for static black-hole solutions in four-dimensional Einstein-scalar gravity with m²L² = -2. It claims that, once integrable mixed boundary conditions B = B(A) are imposed and the variational principle is required to be well-posed, the cubic coefficient a in the near-boundary expansion W(φ) = -4/L - φ²/(2L) + a φ³ + O(φ⁴) is fixed in terms of the boundary deformation; this encodes the mixed conditions directly into the scalar counterterm, rendering the Euclidean on-shell action finite without additional boundary terms. The paper derives the renormalized action and holographic stress tensor, verifies the quantum-statistical relation, and illustrates the formalism with exact asymptotically AdS solutions from consistent truncations, including a comparison with the supergravity superpotential.

Significance. If the construction holds, the work supplies a concrete mechanism for handling mixed (designer-gravity) boundary conditions within holographic renormalization, yielding a natural characterization of RG data in non-extremal backgrounds via the solution-dependent W(φ). Credit is due for the explicit verification of the quantum-statistical relation and the illustrative comparison with SUGRA superpotentials in consistent truncations, which clarifies why observables are controlled by W(φ) rather than W_SUGRA.

major comments (1)
  1. [Abstract] Abstract / paragraph on the expansion of W(φ): the assertion that the bulk equations leave the cubic coefficient a undetermined while the integrable condition B = B(A) together with variational well-posedness fixes it uniquely requires an explicit step-by-step derivation (including the precise functional relation between a and B(A)) to confirm that no further constraints arise from the static equations of motion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment. We address the point below and will revise the manuscript to make the requested derivation fully explicit.

read point-by-point responses

  1. Referee: [Abstract] Abstract / paragraph on the expansion of W(φ): the assertion that the bulk equations leave the cubic coefficient a undetermined while the integrable condition B = B(A) together with variational well-posedness fixes it uniquely requires an explicit step-by-step derivation (including the precise functional relation between a and B(A)) to confirm that no further constraints arise from the static equations of motion.

    Authors: We agree that the presentation would benefit from a more explicit, self-contained derivation of the fixing of the cubic coefficient. In the revised manuscript we will expand the relevant paragraph (and the corresponding discussion in Section 3) with a step-by-step derivation: (i) recall the definition of the solution-dependent W(φ) directly from the static bulk equations of motion, which leaves the coefficient a free at this stage; (ii) impose the integrability condition that the mixed boundary data satisfy B = B(A) for some function B; (iii) require that the on-shell variation of the renormalized action vanishes for all admissible variations consistent with the boundary conditions, which yields the precise algebraic relation a = (3/2) B'(A) (or its equivalent functional form) between a and the boundary deformation; (iv) verify that the static Einstein-scalar equations impose no additional independent constraint on a once W(φ) has been defined from the bulk EOM. This relation is already implicit in the current text but will now be written out explicitly, together with a short appendix deriving the functional relation from the boundary term in the variation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines the solution-dependent W(φ) directly from the bulk equations of motion for static solutions. It explicitly states that the cubic coefficient a in the near-boundary expansion for m²L²=-2 is not fixed by those bulk equations. The authors then use the additional input of integrable mixed boundary conditions B=B(A) together with the requirement of a well-posed variational principle to determine a in terms of the boundary data. This fixes the scalar counterterm without reducing the result to a tautology by the paper's own equations, without self-citations as load-bearing premises, and without smuggling ansatze or renaming known results. The central claim therefore rests on independent boundary data rather than on any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard assumptions of asymptotically AdS gravity, the specific scalar mass, and the integrability of the mixed boundary conditions. No free parameters are explicitly fitted beyond the boundary deformation function itself. The invented entity is the solution-dependent W(φ).

axioms (2)
  • domain assumption The spacetime is asymptotically AdS with the standard Fefferman-Graham expansion.
    Invoked to define the near-boundary expansion of W(φ).
  • domain assumption The mixed boundary conditions are integrable, i.e., there exists a function B(A) such that the variational principle can be made well-posed.
    This is the condition used to fix the cubic coefficient.
invented entities (1)
  • solution-dependent superpotential-like function W(φ) no independent evidence
    purpose: To encode the mixed boundary condition in the scalar counterterm and characterize RG data.
    Defined directly from the equations of motion rather than from a superpotential equation; independent evidence would require explicit comparison with known solutions.

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