Abstract
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.
Highlights
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 sl2(R) generators, which in turn, generate, by exponentiation, the two-dimensional conformal transformations
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
Summary
After shortly reviewing the uniformization theorem for Riemann surfaces, we express the PSL2(R) transformations, acting on the upper half-plane, in terms of the composition of three exponentiations of the generators of sl2(R), represented by the differential operators k = zk+1∂z, k = −1, 0, 1. This will lead to consider two questions. It is easy to see that the number of parameters fixing the generators of a Fuchsian group coincides with the dimension of moduli space of complex structures of Riemann surfaces. Where the λk’s are the ones corresponding to the group element μ−1
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