Higher-spin symmetry in the $\mathfrak{sl}_3$ boundary Toda conformal field theory

This article is dedicated to studying the symmetries enjoyed by the probabilistic construction of the $\mathfrak{sl}_3$ boundary Toda Conformal Field Theory. Namely we show that this model enjoys higher-spin symmetry in the form of Ward identities, higher equations of motion and under additional assumptions BPZ-type differential equations. To do so we consider the $\mathfrak{sl}_3$ Toda theory on the upper-half plane and rigorously define the descendant fields associated to the Vertex Operators. We then show that we can express local as well as global Ward identities based on them, for both the stress-energy tensor and the higher-spin current that encodes the $W$-symmetry. On the other hand, we show that specific Vertex Operators called degenerate fields give rise to singular vectors, which become higher equations of motion at the level of correlation functions. As a corollary we obtain BPZ-type differential equations satisfied by a family of correlation functions, which were unknown in the physics literature.