Complexified Virasoro Flow and the Logarithmic Graviton at the Chiral Point

Abstract

At the chiral point of topologically massive gravity, the massive graviton becomes degenerate with the left-moving graviton, leading to the appearance of a logarithmic mode and a corresponding rank-two Jordan structure. This logarithmic graviton plays a central role in the conjectured AdS$_3$/LCFT$_2$ correspondence, where it has been widely interpreted as the bulk counterpart of a logarithmic partner in the putative boundary logarithmic conformal field theory. In this work, we develop a geometric interpretation of this Jordan structure based on complexified Virasoro evolution. Starting from the logarithmic graviton of Grumiller and Johansson, we show that its logarithmic coefficient \[ y(\tau,\rho) = -i\tau-\ln\cosh\rho \] admits, near the AdS$_3$ boundary, the asymptotic form \[ y(\tau,\rho) = -s+\ln2+\mathcal O(e^{-2\rho}), \qquad s=\rho+i\tau. \] The same complex parameter naturally appears in the exponentiation of the Virasoro generator $L_0$ acting on a rank-two Jordan cell, \[ L_0=h\mathbf1+N, \qquad N^2=0, \] for which \[ e^{sL_0} = e^{sh}(1+sN). \] We show that the resulting Jordan evolution reproduces the characteristic logarithmic mixing of the logarithmic sector, while analytic continuation $s\rightarrow s+2\pi i$ generates the corresponding logarithmic monodromy. From this perspective, radial evolution, temporal evolution, Jordan mixing, and logarithmic monodromy may be viewed as different manifestations of a single complexified Virasoro flow. The analysis suggests a geometric interpretation of the indecomposable structures characteristic of logarithmic conformal field theory and offers a new perspective on the logarithmic graviton within the conjectural AdS$_3$/LCFT$_2$ framework.