Note on the Vanishing Finite Contribution of the Skenderis Solodukhin Counterterm Lagrangian to the TMG Action Offset for BTZ Black Holes

Holographic complexity, particularly via the Complexity=Action (CA) conjecture, relates the complexity of a boundary state to the gravitational action on the bulk Wheeler-DeWitt (WDW) patch. In 3D Topologically Massive Gravity (TMG), the gravitational Chern-Simons (GCS) term modifies the standard Einstein-Hilbert action. We investigate whether the GCS term contributes a finite offset, I0(GCS), to the WDW action, potentially related to the TMG coupling parameter mu and the background mass. Following holographic renormalization principles for TMG established by Skenderis and Solodukhin (arXiv:0906.4926), a key counterterm Lagrangian, L(2,ct), is required. This contributes to the total counterterm action as I(CT, GCS) = (1/mu) * Integral[L(2,ct) d^2x]. This note details the explicit calculation of the finite part arising solely from the integral of L(CT, TMG) = L(2,ct) / mu over the t=0 asymptotic boundary segment. We regularize using Fefferman-Graham coordinates for a non-rotating BTZ black hole background. The calculation yields a vanishing result ( I0(GCS) = 0 ) in the limit where the regulator is removed. This suggests that the expected non-zero offset, which should be proportional to the TMG charge modification ( J(TMG) - J(GR) = M(GR) / mu ), likely arises from the cancellation between divergences in the on-shell bulk/boundary GCS action terms and the full set of TMG counterterms. The offset does not appear to originate solely from the finite part of the integral of L(CT, TMG) itself.