Holographic Entanglement for Chern-Simons Terms

We derive the holographic entanglement entropy contribution from pure and mixed gravitational Chern-Simons(CS) terms in AdS$_{2k+1}$. This is done through two different methods : first, by a direct evaluation of CS action in a holographic replica geometry and second by a descent of Dong's derivation applied to the corresponding anomaly polynomial. In lower dimensions $(k=1,2)$, the formula coincides with the Wald-Tachikawa formula for black hole entropy from gravitational CS terms. New extrinsic curvature corrections appear for $k\geq 3$ : we give explicit and concise expressions for the two pure gravitational CS terms in AdS$_7$ and present various consistency checks, including agreements with the black hole entropy formula when evaluated at the bifurcation surface.