Toward Holography on Biregular Trees

TL;DR

The paper develops a discrete holographic framework on biregular trees, a semihomogeneous Bruhat--Tits structure, and derives explicit bulk-to-bulk and bulk-to-boundary propagators, a mass--dimension relation, and a BF bound for a scalar field. It computes boundary correlators, revealing a novel three-point tensor structure that depends on the degree of the unique bulk vertex where three boundary geodesics meet and expresses the associated OPE data via local zeta functions. It further establishes bulk propagator identities that reduce higher-point diagrams to unintegrated products, and discusses how these results interpolate between regular-tree (p-adic) holography and continuum AdS/CFT, while suggesting extensions to higher ${ m PGL}(n)$ buildings and ultrametric boundary theories with potential connections to nonarchimedean string theory and fracton-like models.

Abstract

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.

Figures (7)

• Figure 1: A finite subset of $(2+1,3+1)$-biregular tree $T_{2,3}$, marking vertices of alternating degrees with different colours. The boundary at infinity of the tree is schematically represented as a circle for illustration; for more details, see Sec. \ref{['sec:BDY']}.
• Figure 2: Geodesics on the biregular tree connecting boundary points $x_1, x_2, x_3$, intersecting at bulk point $o={\rm join}(x_1,x_2,x_3)$. The geodesic joining base vertex $v_0$ to the boundary points first intersects the other geodesics at $w$ as shown.
• Figure 3: The horospherical index of $z$ with respect to $v_0$ and $x$, $\langle z,x \rangle_{v_0} = d(v_0,{\rm join}(z,x,v_0)) - d(z,{\rm join}(z,x,v_0))$.
• Figure 4: (a): A configuration of four boundary points and their bulk interpretation. (b): A configuration of three boundary points.
• Figure 5: Configuration on the biregular tree for $v \in B_w$ with $x \notin \partial B_v$.