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

Abstract: This article is the first of a two-part series dedicated to studying the symmetries enjoyed by the probabilistic construction of the $\mathfrak{sl}_3$ boundary Toda Conformal Field Theory. Namely in the present document we show that this model enjoys higher-spin symmetry in the form of Ward identities, both local and global. 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 this enhanced level of symmetry. This answers a question raised in the physics literature as to whether Toda theory still enjoys higher-spin symmetry in the boundary case. The second part of this series will be dedicated to computing the singular vectors of the theory and showing that they give rise to higher equations of motion as well as, under additional assumptions, BPZ-type differential equations for the correlation functions.