Virasoro Zero-Mode Flow in Critical Topologically Massive Gravity: Logarithmic Correlators and Partition Function Grading

Abstract

We study the rank-two logarithmic sector of critical Topologically Massive Gravity through the Virasoro zero-mode action \[ U(s,\bar s)=e^{-sL_0-\bar s\bar L_0}. \] Writing $L_0=h\mathbf 1+\frac{1}{2}N$ and $\bar L_0=\bar h\mathbf 1+\frac{1}{2}N$, with $N^2=0$, separates the semisimple conformal grading from the nilpotent mixing of the logarithmic pair. We apply the Virasoro zero-mode action to logarithmic two-point functions and to the conformal grading of the logarithmic contribution to the one-loop partition function. For the correlators, the semisimple part generates the ordinary conformal power law, while the nilpotent mixing governed by the combination $s+\bar s$ produces the characteristic logarithmic dependence. For the critical-TMG weights $(h,\bar h)=(2,0)$, this gives a holomorphic power law together with logarithmic dependence on the full Euclidean separation. For the partition function, the relations $q=e^{-s}$ and $\bar q=e^{-\bar s}$ express its conventional multiplicative grading in terms of the additive parameters of the same Virasoro zero-mode action, providing an exact reparameterization of the established one-loop result. The two applications capture complementary information: logarithmic correlators resolve the off-diagonal Jordan mixing, whereas the ordinary graded trace organizes conformal weights, multiplicities, and descendant content without separately resolving the nilpotent matrix element. The Virasoro zero-mode action therefore provides a unified representation-theoretic description of these boundary structures in critical TMG.