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 $\mu \ell = 1$ 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 $r \to e^{2\pi i} r$. 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 $\mathrm{AdS}_3$.