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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>SU</a:mi> </a:mrow> <a:mo stretchy="false">(</a:mo> <a:mn>3</a:mn> <a:mo stretchy="false">)</a:mo> </a:math> 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 operator product expansion 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.