Complexified Virasoro Zero-Mode Flow and the Logarithmic Graviton at the Chiral Point

Abstract

We investigate the relation between Virasoro zero-mode evolution and the bulk logarithmic graviton in topologically massive gravity at the chiral point. By introducing additive coordinates adapted to the global time and radial dependence of the logarithmic mode, we obtain the expression \begin{equation} y(s,\bar s) = -s+\ln2 -\ln\!\left(1+e^{-(s+\bar s)}\right), \qquad s=ρ+iτ, \quad \bar s=ρ-iτ. \end{equation} This relation provides a direct bulk realization of the additive coordinates previously associated with the joint Virasoro zero-mode evolution. Near the AdS$_3$ boundary, the logarithmic coefficient becomes linear in $s$ up to an additive constant and exponentially suppressed corrections, while its exact finite-radius dependence is governed by the diagonal combination $s+\bar s=2ρ$. Upon complexification of the radial coordinate, this combination also organizes the periodic branch structure underlying the previously established bulk monodromy of the logarithmic graviton. The same diagonal coordinate controls the nilpotent part of the Virasoro zero-mode representation, providing a common additive-coordinate description of the representation-theoretic Jordan structure and the radial analytic structure of the bulk solution. The corresponding operations remain distinct: finite zero-mode evolution is algebraic, whereas bulk monodromy arises from analytic continuation and winding of the multivalued gravitational solution.