From monodromy to $SL(2,\mathbb{R})$: reconstructing the logarithmic sector of chiral TMG from virasoro flow

Abstract: We construct and analyze the logarithmic sector of chiral Topologically Massive Gravity (TMG) at the critical point $μ\ell = 1$ from the perspective of Virasoro evolution and radial monodromy in $\mathrm{AdS}3$. We show that the logarithmic graviton arises naturally as a generalized eigenstate of $L_0$, with its Jordan structure persisting uniformly across the full $SL(2,\mathbb{R})_L$ descendant tower generated by $L{-1}$. A central result is that the logarithmic mixing of primary and descendant states can be equivalently interpreted as unipotent monodromy under analytic continuation of the radial coordinate $r \to e^{2πi} r$. This establishes a direct identification between the LCFT Jordan cell structure and a geometric monodromy operator acting in the bulk. We demonstrate that requiring monodromy-compatible Virasoro flow uniquely reconstructs the full indecomposable logarithmic module, including all descendant levels, and show explicit equivalence with the logarithmic graviton module previously obtained in the linearized analysis of chiral TMG. This provides a unified representation-theoretic and geometric characterization of logarithmic gravity in $\mathrm{AdS}_3$.

• The paper demonstrates that the logarithmic sector in chiral TMG arises from the interplay of Virasoro flow and radial monodromy at the critical coupling.
• It establishes a unified framework where geometric monodromy in AdS3 is directly linked to the indecomposable rank-two Jordan cell structure in LCFT representations.
• By showing an isomorphism to the canonical logarithmic graviton module, the study paves the way for extensions to higher-spin theories and non-chiral deformations.

Reconstruction of the Logarithmic Sector in Chiral TMG via Virasoro Flow and Radial Monodromy

Introduction

This paper provides a rigorous construction and analysis of the logarithmic sector in chiral Topologically Massive Gravity (TMG) at the critical coupling $\mu \ell = 1$, utilizing the interplay between Virasoro representation theory and bulk $\mathrm{AdS}_3$ geometry. By aligning the concepts of Virasoro flow, indecomposable module structure, and geometric monodromy in the radial direction, the work achieves a unified characterization of logarithmic modes and their descendant towers in the gravitational context. The proposed framework elevates the geometric interpretation, connecting analytic continuation in the bulk to algebraic features in the boundary logarithmic conformal field theory (LCFT).

Logarithmic Modes in Chiral TMG and Jordan Structure

At the chiral point $\mu \ell = 1$, cosmological TMG exhibits a degeneracy in the spectrum of linearized perturbations due to the vanishing of the left central charge, $c_L = 0$. This degeneracy gives rise to logarithmic modes: the usual left-moving primary (boundary graviton) $\psi^L$, and its logarithmic partner $\psi^{\log}$, which emerges as a generalized Jordan eigenstate of $L_0$. The essential relation is: $L_0 \psi^{\log} = h \psi^{\log} + \psi^L \, ,$ demonstrating non-diagonalizability of $L_0$ on the subspace spanned by $\{\psi^L, \psi^{\log}\}$ and realizing a rank-two Jordan cell structure. In this representation, the nilpotent operator $\mathrm{AdS}_3$0 satisfies $\mathrm{AdS}_3$1 and encodes logarithmic mixing.

Exponentiation of $\mathrm{AdS}_3$2 yields Virasoro flow operators whose action encapsulates both scaling and logarithmic growth: $\mathrm{AdS}_3$3 Here, $\mathrm{AdS}_3$4 is the logarithmic scaling parameter, realized geometrically as the radial coordinate in $\mathrm{AdS}_3$5.

Geometric Interpretation: Radial Monodromy and Unipotency

The bulk realization of the logarithmic sector leverages Fefferman–Graham coordinates, with the radial parameter $\mathrm{AdS}_3$6. In this setup, the logarithmic partner exhibits asymptotic behavior

$\mathrm{AdS}_3$7

making explicit the $\mathrm{AdS}_3$8 dependence. Analytic continuation $\mathrm{AdS}_3$9 yields: $\mu \ell = 1$0 revealing a nontrivial unipotent monodromy—i.e., the logarithmic state does not return to itself but acquires a shift proportional to the primary. This geometric monodromy is identified at the algebraic level with the nilpotent part of $\mu \ell = 1$1, rendering the LCFT Jordan structure a consequence of bulk monodromy rather than an abstract representation-theoretic artifact.

Reconstruction of the Full Logarithmic $\mu \ell = 1$2 Module

Enforcing compatibility between monodromy and Virasoro evolution uniquely determines the module structure for all descendant levels. The $\mu \ell = 1$3 descendant tower, constructed via repeated action of $\mu \ell = 1$4, preserves the indecomposable Jordan structure: $\mu \ell = 1$5 where $\mu \ell = 1$6, and the mixing term propagates identically through all levels. The monodromy action remains uniform: $\mu \ell = 1$7 and each level realizes a fixed rank-two Jordan cell. The subspace of primary descendants $\mu \ell = 1$8 forms an invariant submodule, but the logarithmic partners cannot be decoupled, attesting to the indecomposability characteristic of LCFT representations.

Equivalence to the Logarithmic Graviton Module

The paper establishes a detailed, level-by-level isomorphism between the monodromy-derived module and the canonical logarithmic graviton module of chiral TMG, as described by Grumiller, Johansson, and collaborators. Both structures feature:

• Non-diagonalizable $\mu \ell = 1$9 action.
• Identical asymptotic and monodromy properties.
• Invariant, but indecomposable, module built from primary and logarithmic descendants.

This identification substantiates the proposal that the LCFT logarithmic sector and its bulk gravity dual are two facets of a monodromy-driven indecomposable structure.

Implications and Extensions

The geometric realization of logarithmic structure in terms of radial monodromy provides a transparent mechanism for the emergence of LCFT features in $c_L = 0$0/CFT$c_L = 0$1 holography at the chiral point of TMG. Practically, this clarifies the status of log-gravitons in the presence of boundary conditions and opens the path for systematic construction of higher-rank logarithmic modules, both in TMG and potential non-chiral deformations.

Theoretically, this approach supports a more robust dictionary between bulk analytic structures (monodromy, nilpotent flows) and boundary algebraic features (Jordan blocks, logarithmic scaling) in the context of quantum gravity and holography. It also indicates routes towards generalizations—e.g., to higher-spin theories, alternative holographic LCFT duals, or non-unitary RG flows, where monodromy phenomena are expected to play a similar role.

Conclusion

This work provides a definitive representation-theoretic and geometric framework that explains the logarithmic sector of chiral TMG as a manifestation of unipotent monodromy in the radial direction of $c_L = 0$2. Monodromy-compatible Virasoro flow reconstructs the full logarithmic $c_L = 0$3 module, identifying the LCFT indecomposable structure with concrete bulk analytic properties. This correspondence not only cements the analytic origin of log-gravitons, but also opens several lines of inquiry into the landscape of logarithmic sectors, module theory, and their holographic imprints in three-dimensional massive gravities and quantum field theories.

**** (2605.07917)