  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 simply-laced 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 simply-laced 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.
