Virasoro flow, monodromy, and indecomposable structures in critical AdS$_3$ topologically massive gravity

TL;DR

This paper develops a representation-theoretic framework for understanding the role of monodromy and indecomposable structures in the asymptotic symmetry evolution of critical topologically massive gravity at the chiral point.

Contribution

It introduces a unified complex flow description of symmetry evolution, highlighting the emergence of logarithmic modes and indecomposable structures in the theory.

Findings

L_0 becomes non-diagonalizable with a nilpotent component at the chiral point.

Logarithmic modes are natural generalized eigenstates of L_0.

The sector decomposition aligns with invariant and generalized invariant subspaces.

Abstract

We develop a representation-theoretic framework for the relation between asymptotic symmetry evolution and monodromy in critical topologically massive gravity at the chiral point $\mu \ell=1$. We show that continuous evolution generated by the Virasoro zero mode $L_0$ and analytic continuation around branch points can be unified as different regimes of a single complex one-parameter flow. At the chiral point, $L_0$ becomes non-diagonalizable and takes the form $L_0=h \mathbf{1}+N$, with $N$ nilpotent. We demonstrate that this nilpotent component governs identical mixing structures in both real and imaginary flow parameters, producing linear mixing under continuous evolution and logarithmic mixing under monodromy. In this sense, the logarithmic sector is characterized by a single indecomposable structure in state space probed uniformly by both transformations. Logarithmic modes arise naturally as generalized eigenstates of $L_0$, and the sector decomposition admits an algebraic interpretation in terms of invariant and generalized invariant subspaces. This provides a unified description of logarithmic structures in critical topologically massive gravity and clarifies their role in the representation theory of asymptotic symmetries.