  We present an implementation in conformal field theory (CFT) of local finite
conformal transformations fixing a point. We give explicit constructions when
the fixed point is either the origin or the point at infinity. Both cases
involve the exponentiation of a Borel subalgebra of the Virasoro algebra. We
use this to build coherent state representations and to derive a close analog
of Wick's theorem for the Virasoro algebra. This allows to compute the
conformal partition function in non trivial geometries obtained by removal of
hulls from the upper half plane. This is then applied to stochastic Loewner
evolutions (SLE). We give a rigorous derivation of the equations, obtained
previously by the authors, that connect the stochastic Loewner equation to the
representation theory of the Virasoro algebra. We give a new proof that this
construction enumerates all polynomial SLE martingales. When one of the hulls
removed from the upper half plane is the SLE hull, we show that the partition
function is a famous local martingale known to probabilists, thereby
unravelling its CFT origin.
