  We consider the structure of current and stress tensor two-point functions in
conformal field theory with a boundary. The main result of this paper is a
relation between a boundary central charge and the coefficient of a
displacement operator correlation function in the boundary limit. The boundary
central charge under consideration is the coefficient of the product of the
extrinsic curvature and the Weyl curvature in the conformal anomaly. Along the
way, we describe several auxiliary results. Three of the more notable are as
follows: (1) we give the bulk and boundary conformal blocks for the current
two-point function; (2) we show that the structure of these current and stress
tensor two-point functions is essentially universal for all free theories; (3)
we introduce a class of interacting conformal field theories with boundary
degrees of freedom, where the interactions are confined to the boundary. The
most interesting example we consider can be thought of as the infrared fixed
point of graphene. This particular interacting conformal model in four
dimensions provides a counterexample of a previously conjectured relation
between a boundary central charge and a bulk central charge. The model also
demonstrates that the boundary central charge can change in response to
marginal deformations.
