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

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 $μ\ell=1$. 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.