Boundary Toda Conformal Field Theory from the path integral

Abstract: Toda Conformal Field Theories (CFTs hereafter) are generalizations of Liouville CFT where the underlying field is no longer scalar but takes values in a finite-dimensional vector space. The algebra of symmetry of such models is given by $W$-algebras, which contain the Virasoro algebra as a subalgebra. In contrast with Liouville CFT, and in the presence of a boundary, there may exist non-trivial automorphisms of the associated $W$-algebra, corresponding to different boundary conditions for the field of the theory. Based on this particular feature, we provide in this document a probabilistic construction of Toda CFTs on a Riemann surface with or without boundary. To be more specific we define different classes of models in relation with the different types of boundary conditions suggested by the non-triviality of the automorphism group of the $W$-algebra. To do so, we rely on a probabilistic framework based on Gaussian Free Fields and Gaussian Multiplicative Chaos, and make sense of "Cardy's doubling trick" to construct the underlying field of the theory.