Logarithmic Graviton Module in Chiral TMG

• Logarithmic Graviton Module is an indecomposable rank-two Virasoro module emerging in three-dimensional topologically massive gravity at the chiral point, where logarithmic modes appear.
• It features a non-diagonalizable L0 with a Jordan block structure and complexified Virasoro flow that links bulk AdS3 dynamics with the logarithmic behavior in the dual conformal field theory.
• Its complete descendant tower and monodromy properties underscore deep holographic connections between chiral gravity and LCFT, emphasizing the model’s inherent non-unitarity.

The logarithmic graviton module is the indecomposable rank-two Virasoro module that appears in three-dimensional topologically massive gravity (TMG) at the chiral point $\mu \ell = 1$, where the massive graviton becomes degenerate with the left-moving graviton and a logarithmic partner mode emerges. In the contemporary AdS$_3$/LCFT$_2$ formulation, it is the module generated by the pair $(\psi_L,\psi_{\log})$, with non-diagonalizable $L_0$ action, and it admits a geometric realization in terms of a complexified Virasoro flow whose parameter is the asymptotic combination $s=\rho+i\tau$ of radial and temporal AdS$_3$ coordinates (Mvondo-She, 15 Jun 2026).

1. Chiral TMG and the origin of the logarithmic sector

Topologically Massive Gravity augments the AdS$_3$ Einstein–Hilbert action by a gravitational Chern–Simons term,

$$S_{\rm TMG} = \frac{1}{16\pi G} \int d^3x\,\sqrt{-g}\left(R+\frac{2}{\ell^2}\right) + \frac{1}{16\pi G\mu}S_{\rm CS},$$

with equations of motion

$$G_{\mu\nu}-\frac{1}{\ell^2}g_{\mu\nu}+\frac{1}{\mu}C_{\mu\nu}=0.$$

Linearized around AdS$_3$0 in transverse traceless gauge, the graviton perturbation satisfies a third-order equation that factorizes as

$_3$1

with left-moving massless, right-moving massless, and massive branches (Mvondo-She, 15 Jun 2026).

The Brown–Henneaux central charges are

$_3$2

At the chiral point

$_3$3

one has $_3$4 and $_3$5, while the linearized operators satisfy

$_3$6

Equivalently, the massive graviton branch becomes degenerate with the left-moving graviton,

$_3$7

This degeneracy is the source of the logarithmic sector: two independent solutions with distinct conformal weights away from criticality coalesce at $_3$8, and the linearized operator develops a non-diagonalizable structure (Mvondo-She, 15 Jun 2026).

In the global $_3$9 description, the left-moving graviton is a highest-weight state with

$_2$0

with $_2$1 for the usual graviton and $_2$2. The logarithmic partner appears precisely because the massless and massive weights coincide at criticality (Mvondo-She, 8 May 2026).

2. Rank-two Jordan cell and definition of the module

The logarithmic mode is obtained by differentiating the massive branch with respect to the parameter that controls the degeneracy. One standard construction is

$_2$3

while an equivalent expression is

$_2$4

For the Grumiller–Johansson solution, the mode can be written as

$_2$5

with logarithmic coefficient

$_2$6

Near the AdS$_2$7 boundary, this produces the characteristic logarithmic falloff $_2$8 (Mvondo-She, 15 Jun 2026, Mvondo-She, 8 May 2026).

The defining algebraic property is the non-diagonalizable action of $_2$9: $(\psi_L,\psi_{\log})$0 and, at level zero, also

$(\psi_L,\psi_{\log})$1

Introducing

$(\psi_L,\psi_{\log})$2

one has

$(\psi_L,\psi_{\log})$3

In the basis $(\psi_L,\psi_{\log})$4, $(\psi_L,\psi_{\log})$5 is represented by the rank-two Jordan matrix

$(\psi_L,\psi_{\log})$6

This is the standard LCFT pattern

$(\psi_L,\psi_{\log})$7

with the identification

$(\psi_L,\psi_{\log})$8

Accordingly, the logarithmic graviton module is the indecomposable Virasoro module generated by this Jordan pair and its descendants (Mvondo-She, 15 Jun 2026).

Its essential representation-theoretic feature is reducibility without decomposability. The span of $(\psi_L,\psi_{\log})$9 is an invariant submodule, but $L_0$0 cannot be split off into a separate invariant module because $L_0$1 has a $L_0$2 component. This non-diagonalizable structure is the mechanism behind logarithmic behavior in the dual field theory (Mvondo-She, 8 May 2026).

3. Complexified Virasoro flow and the geometric meaning of the module

A central development is the geometric interpretation of the logarithmic graviton module in terms of complexified Virasoro evolution. Near the AdS$L_0$3 boundary,

$L_0$4

so the logarithmic coefficient becomes

$L_0$5

Defining the complex parameter

$L_0$6

one obtains the asymptotic form

$L_0$7

Thus the same complex variable built from radial and temporal evolution appears directly in the bulk logarithmic mode (Mvondo-She, 15 Jun 2026).

On the algebraic side, exponentiating the Jordan decomposition of $L_0$8 gives

$L_0$9

since $s=\rho+i\tau$0. Therefore

$s=\rho+i\tau$1

The generalized eigenstate picks up a term linear in $s=\rho+i\tau$2, which is the algebraic source of logarithmic behavior. Near the boundary, the bulk solution satisfies

$s=\rho+i\tau$3

up to the constant $s=\rho+i\tau$4 and subleading corrections. The same parameter $s=\rho+i\tau$5 therefore controls both radial-time dependence in the bulk and Jordan mixing in the boundary representation (Mvondo-She, 15 Jun 2026).

This leads to the interpretation that radial evolution, temporal evolution, and logarithmic mixing are different manifestations of a single complexified Virasoro flow. A plausible implication is that the indecomposable structure of the logarithmic module is not merely an abstract boundary feature; it is already encoded in the asymptotic bulk metric perturbation through the complex variable $s=\rho+i\tau$6 (Mvondo-She, 15 Jun 2026).

4. Descendants, monodromy, and reconstruction of the full module

The logarithmic structure persists through the full $s=\rho+i\tau$7 descendant tower. Defining

$s=\rho+i\tau$8

and using

$s=\rho+i\tau$9

one obtains

$_3$0

Hence each level $_3$1 carries the same rank-two Jordan structure,

$_3$2

The nilpotent part is unchanged across the entire descendant tower, while the eigenvalue shifts from $_3$3 to $_3$4 (Mvondo-She, 8 May 2026).

The same module can be reconstructed from monodromy. In Fefferman–Graham coordinates, near the boundary,

$_3$5

Under analytic continuation of the radial coordinate,

$_3$6

so

$_3$7

In the basis $_3$8, the monodromy operator is

$_3$9

Because $_3$0, the same unipotent monodromy acts at every descendant level,

$_3$1

This establishes the identification

$_3$2

Requiring monodromy-compatible Virasoro flow then uniquely reconstructs the full indecomposable logarithmic module (Mvondo-She, 8 May 2026).

A complementary description uses the complexified parameter $_3$3. Analytic continuation

$_3$4

gives

$_3$5

which is the full LCFT-type monodromy operator

$_3$6

The unipotent shift in $_3$7 and the Jordan monodromy of $_3$8 are therefore two descriptions of the same logarithmic graviton module (Mvondo-She, 15 Jun 2026).

5. Holographic interpretation, boundary conditions, and non-unitarity

Within the conjectured AdS$_3$9/LCFT$$S_{\rm TMG} = \frac{1}{16\pi G} \int d^3x\,\sqrt{-g}\left(R+\frac{2}{\ell^2}\right) + \frac{1}{16\pi G\mu}S_{\rm CS},$$0 correspondence, the left-moving graviton is identified with a boundary operator $$S_{\rm TMG} = \frac{1}{16\pi G} \int d^3x\,\sqrt{-g}\left(R+\frac{2}{\ell^2}\right) + \frac{1}{16\pi G\mu}S_{\rm CS},$$1 or stress tensor component, while the logarithmic graviton is identified with its logarithmic partner $$S_{\rm TMG} = \frac{1}{16\pi G} \int d^3x\,\sqrt{-g}\left(R+\frac{2}{\ell^2}\right) + \frac{1}{16\pi G\mu}S_{\rm CS},$$2. Standard LCFT two-point structure, used as context in the recent analysis, is

$$S_{\rm TMG} = \frac{1}{16\pi G} \int d^3x\,\sqrt{-g}\left(R+\frac{2}{\ell^2}\right) + \frac{1}{16\pi G\mu}S_{\rm CS},$$3

The logarithms arise precisely because $$S_{\rm TMG} = \frac{1}{16\pi G} \int d^3x\,\sqrt{-g}\left(R+\frac{2}{\ell^2}\right) + \frac{1}{16\pi G\mu}S_{\rm CS},$$4 acts as $$S_{\rm TMG} = \frac{1}{16\pi G} \int d^3x\,\sqrt{-g}\left(R+\frac{2}{\ell^2}\right) + \frac{1}{16\pi G\mu}S_{\rm CS},$$5 with $$S_{\rm TMG} = \frac{1}{16\pi G} \int d^3x\,\sqrt{-g}\left(R+\frac{2}{\ell^2}\right) + \frac{1}{16\pi G\mu}S_{\rm CS},$$6 (Mvondo-She, 15 Jun 2026).

The module has immediate consequences for the physical interpretation of chiral TMG. A rank-two Jordan block for $$S_{\rm TMG} = \frac{1}{16\pi G} \int d^3x\,\sqrt{-g}\left(R+\frac{2}{\ell^2}\right) + \frac{1}{16\pi G\mu}S_{\rm CS},$$7 is incompatible with a positive-definite inner product, so the logarithmic module is necessarily non-unitary. This matches the interpretation of the dual theory as a logarithmic conformal field theory rather than an ordinary unitary CFT (Mvondo-She, 8 May 2026).

Boundary conditions are therefore decisive. Under strict Brown–Henneaux boundary conditions one often truncates away the logarithmic modes to recover “chiral gravity”. Allowing softened logarithmic boundary conditions admits the full logarithmic graviton module, leading to “log gravity” and a dual LCFT description (Mvondo-She, 8 May 2026). A longstanding misconception is that the chiral point automatically implies a purely chiral, holomorphically factorized theory; the one-loop partition-function analysis of Euclidean TMG instead showed that the partition function does not factorize holomorphically and, at the chiral point, has the structure expected from an LCFT, providing evidence that the logarithmic sector is not an incidental artifact of linearized analysis (Gaberdiel et al., 2010).

In that partition-function framework, the vacuum Virasoro module is accompanied by a character associated with the logarithmic partner of the stress tensor. This supports the view that the logarithmic graviton module is part of the quantum spectrum, not merely a classical degeneracy (Gaberdiel et al., 2010).

6. Related realizations and generalizations

The AdS$$S_{\rm TMG} = \frac{1}{16\pi G} \int d^3x\,\sqrt{-g}\left(R+\frac{2}{\ell^2}\right) + \frac{1}{16\pi G\mu}S_{\rm CS},$$8 logarithmic graviton module is part of a broader pattern in critical gravity. In $$S_{\rm TMG} = \frac{1}{16\pi G} \int d^3x\,\sqrt{-g}\left(R+\frac{2}{\ell^2}\right) + \frac{1}{16\pi G\mu}S_{\rm CS},$$9-dimensional critical gravity linearized about AdS, the massless and massive spin-2 modes degenerate and are replaced by logarithmic modes. The resulting pairs $$G_{\mu\nu}-\frac{1}{\ell^2}g_{\mu\nu}+\frac{1}{\mu}C_{\mu\nu}=0.$$0 satisfy a Jordan-cell relation for the AdS isometry generator, and the bulk solutions exhibit logarithmic radial dependence. These works explicitly interpret the resulting non-diagonalizable representation as the bulk counterpart of a logarithmic CFT in higher dimensions [(Bergshoeff et al., 2011); (Alishahiha et al., 2011)].

Three-dimensional variants provide further realizations. Zwei-Dreibein Gravity exhibits critical points where massive gravitons coincide with pure gauge modes and logarithmic modes appear, both in the linearized theory and in exact AdS wave solutions; at those points the theory is conjectured to be dual to an LCFT with zero central charges and nonzero new anomalies (Bergshoeff et al., 2014). In flat-space generalized massive gravity, a critical degeneracy of two massive modes produces a massive logarithmic graviton, with the pair $$G_{\mu\nu}-\frac{1}{\ell^2}g_{\mu\nu}+\frac{1}{\mu}C_{\mu\nu}=0.$$1 forming a rank-two Jordan cell for the relevant generator in the Galilean conformal algebra description (Kim et al., 2013).

A related but distinct use of the same language appears in near-extremal black holes in New Massive Gravity. There the relevant logarithmic sector is a boundary graviton sector on a near-horizon AdS$$G_{\mu\nu}-\frac{1}{\ell^2}g_{\mu\nu}+\frac{1}{\mu}C_{\mu\nu}=0.$$2 geometry: exact zero modes at extremality acquire eigenvalues linear in temperature near extremality, and their one-loop determinant yields a logarithmic correction to the entropy,

$$G_{\mu\nu}-\frac{1}{\ell^2}g_{\mu\nu}+\frac{1}{\mu}C_{\mu\nu}=0.$$3

This suggests that “logarithmic graviton module” can also denote a distinguished boundary graviton sector whose quantum fluctuations produce logarithmic thermodynamic corrections, rather than the AdS$$G_{\mu\nu}-\frac{1}{\ell^2}g_{\mu\nu}+\frac{1}{\mu}C_{\mu\nu}=0.$$4/LCFT$$G_{\mu\nu}-\frac{1}{\ell^2}g_{\mu\nu}+\frac{1}{\mu}C_{\mu\nu}=0.$$5 Jordan cell of chiral TMG (Acito et al., 11 Jun 2026).

Across these examples, the recurring structure is a degeneracy of linearized operators, the appearance of generalized eigenvectors, and a non-diagonalizable action of the relevant symmetry generator. In the specific AdS$$G_{\mu\nu}-\frac{1}{\ell^2}g_{\mu\nu}+\frac{1}{\mu}C_{\mu\nu}=0.$$6 chiral-TMG sense, however, the logarithmic graviton module is most precisely the rank-two indecomposable Virasoro module generated by $$G_{\mu\nu}-\frac{1}{\ell^2}g_{\mu\nu}+\frac{1}{\mu}C_{\mu\nu}=0.$$7 and $$G_{\mu\nu}-\frac{1}{\ell^2}g_{\mu\nu}+\frac{1}{\mu}C_{\mu\nu}=0.$$8, with descendant tower, unipotent monodromy, and complexified Virasoro flow all encoding the same Jordan structure (Mvondo-She, 15 Jun 2026, Mvondo-She, 8 May 2026).