Monodromy, Logarithmic Sectors, and Two-Point Functions in Critical Topologically Massive Gravity

Abstract: We investigate the structure of logarithmic modes in critical topologically massive gravity (CTMG) at the chiral point $μ\ell=1$ from the perspective of analytic continuation and monodromy. Starting from the degeneration of massive and left-moving graviton modes, we construct the logarithmic mode as a derivative in parameter space and show that it acquires a natural multivalued structure upon complexification of the radial coordinate. We demonstrate that this multivaluedness induces a nontrivial monodromy action on the space of linearized solutions, under which the left-moving and logarithmic modes form an indecomposable (Jordan block) representation. This monodromy is unipotent and provides a bulk realization of the logarithmic structure typically associated with logarithmic conformal field theories. We further show that the monodromy representation alone is sufficiently constraining to determine the characteristic logarithmic form and mixing structure of two-point functions, up to normalization, without assuming logarithmic conformal field theory data a priori. These results suggest a geometric interpretation in which logarithmic modes act as sources of branchlike behavior in the bulk, analogous to twist fields that generate monodromy. While this perspective is compatible with proposed connections to branched coverings, Hurwitz theory, and integrable hierarchies, establishing a precise correspondence is left for future work.