  The covariantization procedure is usually referred to the translation
operator, that is the derivative. Here we introduce a general method to
covariantize arbitrary differential operators, such as the ones defining the
fundamental group of a given manifold. We focus on the differential operators
representing the sl(2,R) generators, which in turn, generate, by
exponentiation, the two-dimensional conformal transformations. A key point of
our construction is the recent result on the closed forms of the
Baker-Campbell-Hausdorff formula. In particular, our covariantization receipt
is quite general. This has a deep consequence since it means that the
covariantization of the conformal group is {\it always definite}. Our
covariantization receipt is quite general and apply in general situations,
including AdS/CFT. Here we focus on the projective unitary representations of
the fundamental group of a Riemann surface, which may include elliptic points
and punctures, introduced in the framework of noncommutative Riemann surfaces.
It turns out that the covariantized conformal operators are built in terms of
Wilson loops around Poincar\'e geodesics, implying a deep relationship between
gauge theories on Riemann surfaces and Liouville theory.
