Toward Holography on Biregular Trees

We study scalar field theory on biregular trees, as a new model for discrete holography. Biregular trees are discrete symmetric spaces associated with the bulk isometry group SU(3) over the unramified quadratic extension of a nonarchimedean field. The bulk-to-bulk and bulk-to-boundary propagators exhibit distinct features absent on the regular tree or continuum AdS spaces, arising from the semihomogeneous nature of the bulk space. We compute the two- and three-point correlators of the putative boundary dual. The three-point correlator exhibits a nontrivial "tensor structure" via dependence on the homogeneity degree of a unique bulk point specified in terms of boundary insertion points. The computed OPE coefficients show dependence on zeta functions associated with the unramified quadratic extension of a nonarchimedean field. This work initiates the formulation of holography on a family of discrete holographic spaces that exhibit features of both flat space and negatively curved space.