Edge length dynamics on graphs with applications to p-adic AdS/CFT

Abstract

We formulate a Euclidean theory of edge length dynamics based on a notion of Ricci curvature on graphs with variable edge lengths. In order to write an explicit form for the discrete analog of the Einstein-Hilbert action, we require that the graph should either be a tree or that all its cycles should be sufficiently long. The infinite regular tree with all edge lengths equal is an example of a graph with constant negative curvature, providing a connection with p-adic AdS/CFT, where such a tree takes the place of anti-de Sitter space. We compute simple correlators of the operator holographically dual to edge length fluctuations. This operator has dimension equal to the dimension of the boundary, and it has some features in common with the stress tensor.

Tensor network and (p-adic) AdS/CFT

“…Therefore a tensor network based on the Bruhat-Tits tree would have much more symmetry than its counterpart living on a regular tessellation. This is inspired by recent proposals for the p-adic AdS/CFT correspondence [26][27][28], which generalize the AdS/CFT dictionary to the situation where the boundary theory lives…”

“…28 In this subsection we will mostly focus on the case with boundary dimension d = 1. 29 The normalizations of our smearing function and the bulk-bulk kernel will be fixed later using two-point correlation functions and will turn out to be consistent with this identification.…”

“…In general when we start discussing fluctuations around a background, then it seems natural that different edges should in general have different weights. A first step has been taken in [28], where the weights are interpreted as metric fluctuations. The Laplacian also depends on these weights by…”

Holographic dual of the five-point conformal block

“…Indeed, it turns out, the problem of various higher-point blocks is relatively easily addressed over the p-adics [27], and owing to the close ties with the usual AdS/CFT setup (see, e.g., Refs. [25,26,6,28,29,30,31,32,33,34,35]) we are able to extract key new insights into the analogous calculation over reals. The p-adic results generalize the result of Ref.…”

The boundary theory of a spinor field theory on the Bruhat-Tits tree

“…D 2∆−1 is the 1-dim 2∆ − 1-th order Vladimirov derivative operator [9] up to a factor depending on ∆.…”