  Following the 1984 seminal work of Belavin, Polyakov and Zamolodchikov on
two-dimensional conformal field theories, Toda conformal field theories were
introduced in the physics literature as a family of two-dimensional conformal
field theories that enjoy, in addition to conformal symmetry, an extended level
of symmetry usually referred to as W-symmetry or higher-spin symmetry. More
precisely Toda conformal field theories provide a natural way to associate to a
finite-dimensional simple and complex Lie algebra a conformal field theory for
which the algebra of symmetry contains the Virasoro algebra. In this document
we use the path integral formulation of these models to provide a rigorous
mathematical construction of Toda conformal field theories based on probability
theory. By doing so we recover expected properties of the theory such as the
Weyl anomaly formula with respect to the change of background metric by a
conformal factor and the existence of Seiberg bounds for the correlation
functions.
