Logarithmic Graviton in Critical Gravity

• Logarithmic gravitons are generalized spin‑2 modes exhibiting logarithmic radial behavior arising at criticality in AdS3 gravity.
• They emerge when massive and massless graviton modes degenerate, forming a rank‑2 Jordan cell that underpins LCFT structures.
• Similar logarithmic features appear in four-dimensional soft graviton amplitudes and inflationary models, expanding their role in critical gravity.

Searching arXiv for relevant papers on logarithmic gravitons in chiral TMG, LCFT structure, and related critical gravity context. The term logarithmic graviton most commonly denotes a special linearized spin‑2 mode that appears when the graviton spectrum of a critical gravity theory becomes non‑diagonalisable, so that a new solution acquires logarithmic dependence in the radial coordinate and is paired with an ordinary graviton in a rank‑2 Jordan cell. In three-dimensional anti‑de Sitter gravity, this usage is standard at the chiral point of Topologically Massive Gravity (TMG), where the logarithmic graviton is the bulk counterpart of a logarithmic partner in a conjectured boundary logarithmic conformal field theory (LCFT) (Mvondo-She, 8 May 2026). In distinct four-dimensional literatures, the same phrase is also used for soft-graviton logarithms in scattering and for graviton-loop–induced secular logarithms during inflation; these uses concern logarithmic terms in amplitudes or effective fields rather than a new bulk spin‑2 mode (Laddha et al., 2018).

1. Definition and terminological scope

In the AdS$_3$ and critical-gravity literature, the logarithmic graviton is a gravitational fluctuation whose falloff near the AdS boundary contains logarithms, and which, in the dual CFT, corresponds to a logarithmic partner of the stress tensor, forming a Jordan block for $L_0$ (Gaberdiel et al., 2010). At a critical point, the usual massive and massless graviton branches degenerate, the linearized operator ceases to be diagonalisable, and a generalized eigenmode appears. In this sense the logarithmic graviton is not an independent particle species at generic couplings, but a critical mode tied to spectral degeneracy.

The term is not fully uniform across subfields. In one four-dimensional usage, “logarithmic graviton” refers to logarithmic terms in the soft-graviton expansion that produce a late-time tail in the gravitational waveform. In another, it refers to secular logarithms generated by graviton loops on de Sitter or inflationary backgrounds. These usages share the presence of logarithms but not the representation-theoretic meaning of the AdS$_3$ mode (Agrawal et al., 2023).

2. Emergence in chiral Topologically Massive Gravity

Cosmological TMG in three dimensions supplements the Einstein–Hilbert action with a gravitational Chern–Simons term. Linearized excitations around $\mathrm{AdS}_3$ organize into

$SL(2,\mathbb{R})_L \times SL(2,\mathbb{R})_R.$

At the chiral point

$\mu \ell = 1,$

the left-moving central charge vanishes,

$c_L = 0,\qquad c_R = \frac{3\ell}{G},$

and the left-moving massive graviton degenerates with a left-moving boundary graviton. The logarithmic graviton then appears as the generalized eigenmode produced by this degeneracy (Mvondo-She, 8 May 2026).

For metric perturbations

$g_{\mu\nu}=\bar g_{\mu\nu}+h_{\mu\nu},$

the logarithmic graviton can be constructed as the derivative of the massive branch at the point of degeneracy,

$$\psi^{\log}=\left.\frac{\partial \psi^M(\mu\ell)}{\partial(\mu\ell)}\right|_{\mu\ell=1}.$$

In Fefferman–Graham coordinates with $r=e^\rho$, the ordinary left-moving graviton behaves as

$L_0$0

whereas the logarithmic graviton behaves as

$L_0$1

This logarithmic falloff is the defining bulk signature of the mode.

A closely related description uses the logarithmic graviton of Grumiller and Johansson at the chiral point in global AdS$L_0$2. There the mode takes the form

$L_0$3

and near the AdS$L_0$4 boundary,

$L_0$5

This makes explicit that the logarithmic coefficient contains both radial and temporal linear pieces (Mvondo-She, 15 Jun 2026).

3. Jordan cells, Virasoro flow, and radial monodromy

On the boundary, the left-moving graviton $L_0$6 is a highest-weight state satisfying

$L_0$7

Its logarithmic partner is not an eigenstate of $L_0$8; instead,

$L_0$9

Equivalently,

$_3$0

On the span of $_3$1, $_3$2 takes Jordan form,

$_3$3

This is the standard LCFT structure: $_3$4 is the logarithmic partner of $_3$5 (Mvondo-She, 8 May 2026).

Exponentiating $_3$6 gives a one-parameter dilatation flow

$_3$7

so that

$_3$8

The extra linear term in $_3$9 is the algebraic origin of logarithmic growth, with $\mathrm{AdS}_3$0 in the bulk radial picture.

A central geometric identification is that the LCFT Jordan structure is equivalent to unipotent radial monodromy. Because

$\mathrm{AdS}_3$1

analytic continuation

$\mathrm{AdS}_3$2

induces

$\mathrm{AdS}_3$3

while $\mathrm{AdS}_3$4 remains single-valued. In the basis $\mathrm{AdS}_3$5, the monodromy is

$\mathrm{AdS}_3$6

This identifies the nilpotent part of $\mathrm{AdS}_3$7 with a bulk monodromy operator (Mvondo-She, 8 May 2026).

The same structure can be formulated as a complexified Virasoro flow. With

$\mathrm{AdS}_3$8

one has

$\mathrm{AdS}_3$9

and therefore

$SL(2,\mathbb{R})_L \times SL(2,\mathbb{R})_R.$0

Under

$SL(2,\mathbb{R})_L \times SL(2,\mathbb{R})_R.$1

the logarithmic partner transforms as

$SL(2,\mathbb{R})_L \times SL(2,\mathbb{R})_R.$2

This presents radial evolution, temporal evolution, Jordan mixing, and logarithmic monodromy as aspects of a single complex flow parameter $SL(2,\mathbb{R})_L \times SL(2,\mathbb{R})_R.$3 (Mvondo-She, 15 Jun 2026).

4. Descendants, modules, and LCFT evidence

The logarithmic structure is not confined to the primary level. For the global left-moving conformal algebra generated by $SL(2,\mathbb{R})_L \times SL(2,\mathbb{R})_R.$4, descendants are defined by

$SL(2,\mathbb{R})_L \times SL(2,\mathbb{R})_R.$5

Using

$SL(2,\mathbb{R})_L \times SL(2,\mathbb{R})_R.$6

one finds

$SL(2,\mathbb{R})_L \times SL(2,\mathbb{R})_R.$7

and

$SL(2,\mathbb{R})_L \times SL(2,\mathbb{R})_R.$8

Thus each level $SL(2,\mathbb{R})_L \times SL(2,\mathbb{R})_R.$9 furnishes a rank‑two Jordan cell, and the whole $\mu \ell = 1,$0 descendant tower forms a reducible but indecomposable logarithmic module. The same unipotent monodromy acts uniformly at every level (Mvondo-She, 8 May 2026).

This module structure is stated to coincide with the logarithmic graviton module previously obtained in linearized analyses of chiral TMG by Grumiller & Johansson and by Skenderis et al. The identification is at the level of primary relations, descendants, asymptotics, and indecomposable module structure. A principal claim is that monodromy-compatible Virasoro flow uniquely reconstructs the full indecomposable logarithmic module, including all descendant levels (Mvondo-She, 8 May 2026).

Independent evidence for the LCFT interpretation comes from the one-loop partition function of Euclidean TMG. At the chiral point $\mu \ell = 1,$1, the graviton one-loop partition function does not factorize holomorphically and takes the form

$\mu \ell = 1,$2

This has the structure expected from a logarithmic CFT, with the massive determinant supplying the sector whose single-particle content matches a field of weights $\mu \ell = 1,$3 and its descendants. In this interpretation, the ordinary left-moving stress tensor $\mu \ell = 1,$4 and its logarithmic partner $\mu \ell = 1,$5 satisfy

$\mu \ell = 1,$6

which is precisely the boundary Jordan structure associated with the bulk logarithmic graviton (Gaberdiel et al., 2010).

5. Generalizations in critical gravities

The appearance of logarithmic gravitons is not confined to chiral TMG. In higher-dimensional curvature-squared gravities at critical points, the massive and massless spin‑2 roots can also coincide, and new logarithmic solutions arise. In four dimensions, for AdS wave profiles $\mu \ell = 1,$7, the critical theory develops multiple roots and the general solution acquires logarithmic terms,

$\mu \ell = 1,$8

In general $\mu \ell = 1,$9,

$c_L = 0,\qquad c_R = \frac{3\ell}{G},$0

These modes suggest a gravity description of logarithmic CFTs in $c_L = 0,\qquad c_R = \frac{3\ell}{G},$1 dimensions (Alishahiha et al., 2011).

A detailed mode analysis in critical gravity linearized about AdS classifies logarithmic solutions into spin 2 and Proca sectors. Spin‑2 log modes are those for which the linearized Einstein tensor is a non-gauge solution of the linearized Einstein equations, whereas for Proca modes the linearized Einstein tensor takes the form of a linearized general coordinate transformation. In AdS$c_L = 0,\qquad c_R = \frac{3\ell}{G},$2, a representative logarithmic factor is

$c_L = 0,\qquad c_R = \frac{3\ell}{G},$3

The corresponding states form Jordan cells for the AdS energy generator, which suggests a holographically dual logarithmic conformal field theory (Bergshoeff et al., 2011).

In three-dimensional flat spacetime, generalized massive gravity has a critical point where two massive graviton modes degenerate. The resulting massive logarithmic graviton is

$c_L = 0,\qquad c_R = \frac{3\ell}{G},$4

and the pair $c_L = 0,\qquad c_R = \frac{3\ell}{G},$5 forms a Jordan block under the Galilean conformal generator $c_L = 0,\qquad c_R = \frac{3\ell}{G},$6. This is the flat-space analogue of the AdS$c_L = 0,\qquad c_R = \frac{3\ell}{G},$7 logarithmic graviton (Kim et al., 2013).

Zwei-Dreibein Gravity (ZDG) exhibits critical points in AdS$c_L = 0,\qquad c_R = \frac{3\ell}{G},$8 where massive graviton modes coincide with pure gauge modes and new logarithmic modes appear, both in linearized analysis and in exact AdS wave solutions of the full nonlinear theory. For the AdS wave profile $c_L = 0,\qquad c_R = \frac{3\ell}{G},$9, the critical point $g_{\mu\nu}=\bar g_{\mu\nu}+h_{\mu\nu},$0 yields

$g_{\mu\nu}=\bar g_{\mu\nu}+h_{\mu\nu},$1

and the conjectured dual LCFT has vanishing central charges but nonzero new anomalies

$g_{\mu\nu}=\bar g_{\mu\nu}+h_{\mu\nu},$2

This generalizes the New Massive Gravity critical point and embeds logarithmic AdS waves into a bigravity framework (Bergshoeff et al., 2014).

6. Distinct four-dimensional usages

In a separate line of research on soft theorems, “logarithmic graviton” refers to soft gravitons whose amplitude acquires logarithmic dependence on frequency in four dimensions. The logarithmic term in the soft expansion produces a late-time waveform tail,

$g_{\mu\nu}=\bar g_{\mu\nu}+h_{\mu\nu},$3

so that logarithms in frequency correspond to inverse-time tails in the classical waveform. This tail supplements but is distinct from linear memory. For purely massless final states, the coefficient $g_{\mu\nu}=\bar g_{\mu\nu}+h_{\mu\nu},$4 vanishes at this order (Laddha et al., 2018). A symmetry-based formulation derives the same $g_{\mu\nu}=\bar g_{\mu\nu}+h_{\mu\nu},$5 soft corrections from superrotation Ward identities, using an enlarged radiative phase space that includes gravitational tails (Agrawal et al., 2023).

In inflationary and de Sitter settings, “logarithmic graviton effects” denotes secular logarithms generated by graviton loops. Examples include the photon field strength correction

$g_{\mu\nu}=\bar g_{\mu\nu}+h_{\mu\nu},$6

and the Coulomb potential

$g_{\mu\nu}=\bar g_{\mu\nu}+h_{\mu\nu},$7

In this usage, the logarithms are attributed to inflationary gravitons and are discussed in terms of curvature-dependent field-strength renormalization, renormalization-group flow, or stochastic resummation (Glavan et al., 2023). Related work studies large temporal and spatial logarithms for massless, minimally coupled scalars, and later argues that the large logarithm in the exchange potential is tied to a gauge-independent coefficient once source and observer correlations are included (Glavan et al., 2021, Glavan et al., 2024).

A further contrast is provided by a massless, conformally coupled scalar on de Sitter. There, with the “obvious” choice for the finite part of the $g_{\mu\nu}=\bar g_{\mu\nu}+h_{\mu\nu},$8 counterterm, one-loop graviton corrections yield neither large infrared logarithms in the mode functions nor large-distance logarithmic growth in the point-source response; instead one finds a decaying logarithmic correction to the mode function and short-distance logarithmic running of the potential (Glavan et al., 2020).

In this broader usage, therefore, “logarithmic graviton” no longer denotes a Jordan-cell spin‑2 mode. It denotes logarithmic soft terms, logarithmic tails, or graviton-loop–induced secular logarithms. The AdS$g_{\mu\nu}=\bar g_{\mu\nu}+h_{\mu\nu},$9 usage remains the representation-theoretic one: a generalized eigenmode of the graviton sector at a critical point, with logarithmic radial behavior and an indecomposable LCFT interpretation (Mvondo-She, 8 May 2026).