  In this paper, we study the logarithmic terms in the partition functions of
CFTs with boundaries (BCFTs). In three dimensions, their coefficients give the
boundary central charges, which are conjectured to be monotonically decreasing
functions under the RG flows. We present a few supporting evidences from field
theory calculations. In two dimensions, we give a holographic construction
(AdS/BCFT) for an arbitrary shape of boundary and calculate its logarithmic
term as well as boundary energy momentum tensors, confirming its consistency
with the Weyl anomaly. Moreover, we give perturbative solutions of gravity
duals for the three dimensional BCFTs with any shapes of boundaries. We find
that the standard Fefferman-Graham expansion breaks down for generic choices of
BCFT boundaries.

High Energy Physics - Theory;Strongly Correlated Electrons;General Relativity and Quantum Cosmology