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arxiv: 2605.18942 · v1 · pith:5COPRLW7new · submitted 2026-05-18 · ✦ hep-th · gr-qc

Covariant unification of holographic c-functions

Niko Jokela , Jani Kastikainen , Carlos Nunez , Jos\'e Manuel Pen\'in , Helime Ruotsalainen This is my paper

Pith reviewed 2026-05-20 08:56 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords holographic c-functioncovariant formulationtop-down string backgroundsextrinsic curvatureAdS/CFTrenormalization group flowsmonotonicityKlebanov-Murugan geometry
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The pith

A covariant holographic c-function built from extrinsic curvature of codimension-two slices unifies earlier definitions without needing special coordinates or dimensional reduction.

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

The paper sets out to define a holographic c-function that works directly in general top-down string backgrounds rather than only in specially chosen coordinates. This matters for studying renormalization group flows in geometries where the holographic radial direction mixes with internal coordinates and no lower-dimensional reduction is available. The proposed definition extracts the c-function from the extrinsic curvature of codimension-two slices of the bulk geometry and reduces to known foliation-based expressions in simpler limits. When tested in conformal, confining, and flowing models including the Klebanov-Murugan geometry, the resulting function interpolates monotonically between fixed points or decreases toward zero in gapped regions as expected.

Core claim

The central claim is that the extrinsic curvature of codimension-two slices supplies a single covariant expression for the holographic c-function. This expression is independent of coordinate choice and of the existence of a consistent dimensional reduction, yet it recovers previous holographic c-functions in the appropriate limits. Evaluation in a range of top-down backgrounds shows the expected monotonic behavior: constant at conformal fixed points, decreasing to zero in confining or gapped infrared regions, and correct fixed-point values together with monotonicity evidence in the Klebanov-Murugan case where radial and internal directions mix.

What carries the argument

The extrinsic curvature of codimension-two slices of the bulk geometry, which encodes the necessary information to construct a coordinate-independent c-function.

If this is right

  • The new expression reduces exactly to prior foliation-based c-functions whenever a preferred radial coordinate or dimensional reduction is available.
  • It remains well-defined and produces the required monotonic interpolation in geometries that cannot be reduced to lower-dimensional solutions.
  • In conformal fixed-point models the function is constant, while in confining or gapped infrared regions it decreases toward zero.
  • In mixed radial-internal geometries such as Klebanov-Murugan it recovers the correct ultraviolet and infrared fixed-point values and shows evidence of monotonicity.

Where Pith is reading between the lines

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

  • The same construction may extend naturally to spacetimes where no unambiguous holographic radial direction can be identified.
  • Monotonicity might ultimately be derived from a variational principle on the bulk action rather than imposed by geometric inspection.
  • Direct comparison with entanglement-entropy definitions of the c-function could be performed in the same mixed geometries using this covariant expression.

Load-bearing premise

The construction assumes that the extrinsic curvature of codimension-two slices always yields a physically meaningful and monotonic c-function even when the radial direction mixes with internal coordinates and no consistent dimensional reduction exists.

What would settle it

A concrete counter-example would be a known top-down background in which the proposed expression either fails to reproduce the correct fixed-point values or exhibits non-monotonic behavior between ultraviolet and infrared regions.

read the original abstract

We propose a covariant holographic c-function, defined directly in a top-down background and constructed from the extrinsic curvature of codimension-two slices of the bulk geometry. The definition does not rely on a special choice of coordinates or on the existence of a consistent dimensional reduction. We show that it unifies previous foliation-based holographic c-functions into a single covariant formula, reducing to them in the appropriate limits. We evaluate the covariant expression in a range of top-down string backgrounds, including conformal models, confining geometries, flows across dimensions, and the Klebanov-Murugan geometry, in which the holographic radial direction mixes with internal coordinates and which is not the uplift of a lower-dimensional solution. In all cases, the c-function behaves as expected: it interpolates monotonically between AdS fixed points when they are present and decreases towards zero in gapped infrared regions, while in the Klebanov-Murugan case we recover the correct fixed-point values and find evidence for monotonicity. We highlight open conceptual issues, including: the lack of a universal covariant definition of the holographic radial direction in the presence of a nontrivial internal manifold; the derivation of the flow from a bulk action; and the relation to the entanglement c-function.

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 / 1 minor

Summary. The paper proposes a covariant holographic c-function constructed from the extrinsic curvature of codimension-two slices of the bulk geometry in top-down string backgrounds. It claims this definition unifies previous foliation-based c-functions into a single formula that reduces to them in appropriate limits, does not require special coordinates or consistent dimensional reduction, and exhibits the expected monotonic interpolation between AdS fixed points (or decrease to zero in gapped regions) when evaluated in conformal models, confining geometries, dimension-changing flows, and the Klebanov-Murugan geometry.

Significance. If the construction can be made fully covariant without auxiliary choices, the unification would be a useful technical advance for studying RG flows directly in ten-dimensional uplifts. The explicit evaluations across multiple backgrounds, including recovery of correct fixed-point values in the Klebanov-Murugan case, constitute concrete supporting evidence.

major comments (1)
  1. [Abstract] Abstract and the discussion of open issues: the central claim that the definition 'does not rely on a special choice of coordinates' is in tension with the explicit listing of 'the lack of a universal covariant definition of the holographic radial direction in the presence of a nontrivial internal manifold' as an unresolved conceptual issue. Defining the codimension-two slices requires identifying this direction, which remains case-by-case when the radial coordinate mixes with internal directions; this choice is load-bearing for the covariance and unification assertions.
minor comments (1)
  1. The manuscript would benefit from an explicit algorithmic prescription or coordinate-independent criterion for selecting the slices in the Klebanov-Murugan example, even if only as a working definition for the checks performed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting this point of presentation. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the discussion of open issues: the central claim that the definition 'does not rely on a special choice of coordinates' is in tension with the explicit listing of 'the lack of a universal covariant definition of the holographic radial direction in the presence of a nontrivial internal manifold' as an unresolved conceptual issue. Defining the codimension-two slices requires identifying this direction, which remains case-by-case when the radial coordinate mixes with internal directions; this choice is load-bearing for the covariance and unification assertions.

    Authors: We appreciate the referee identifying this tension in our wording. The proposed c-function is defined via the extrinsic curvature of codimension-two slices, a tensorial quantity that yields a coordinate-independent expression once the slices are fixed; the unification with prior foliation-based definitions follows directly from this geometric construction. We agree, however, that when the holographic radial direction mixes with internal coordinates, the identification of an appropriate foliation is not yet furnished by a universal, fully covariant prescription and must be performed on a case-by-case basis. This is exactly the open conceptual issue we already flag in the manuscript. To remove the apparent inconsistency, we will revise the abstract to state that the definition is covariant given a choice of codimension-two foliation, and we will expand the discussion of open issues to clarify the status of slice selection in mixed geometries such as the Klebanov-Murugan background. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new geometric definition is independent of inputs

full rationale

The paper introduces a covariant c-function constructed directly from the extrinsic curvature of codimension-two slices as a new proposal that unifies prior foliation-based expressions only in appropriate limits. This is presented as a consistency check rather than a definitional reduction. No fitted parameters are renamed as predictions, no load-bearing self-citations justify the central premise, and no ansatz is smuggled via prior work. The manuscript explicitly flags the lack of a universal radial direction definition as an open issue, confirming the construction is offered as an independent geometric object rather than forced by its own inputs. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proposal rests on standard holographic assumptions rather than new fitted parameters or invented entities; the key premise is that extrinsic curvature supplies a suitable c-function without coordinate specialization.

axioms (1)
  • domain assumption The holographic principle applies to the top-down string backgrounds considered, allowing geometric quantities to encode field-theory RG data.
    Implicit throughout holographic c-function literature and required for any such construction to be meaningful.

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