Trees and spatial topology change in CDT

TL;DR

This paper solves a model of two-dimensional quantum gravity allowing limited spatial topology changes using bijections between quadrangulations and trees, deriving explicit formulas and connecting to planar maps.

Contribution

It introduces a combinatorial approach to generalized CDT, providing explicit solutions and linking the model to planar maps and their distance functions.

Findings

Explicit formulas for loop propagators and two-point functions.

Generalized CDT corresponds to the scaling limit of planar maps with finite faces.

Clarifies the relation between continuum amplitudes and planar map combinatorics.

Abstract

Generalized causal dynamical triangulations (generalized CDT) is a model of two-dimensional quantum gravity in which a limited number of spatial topology changes is allowed to occur. We solve the model at the discretized level using bijections between quadrangulations and trees. In the continuum limit (scaling limit) the amplitudes are shown to agree with known formulas and explicit expressions are obtained for loop propagators and two-point functions. It is shown that from a combinatorial point of view generalized CDT can be viewed as the scaling limit of planar maps with a finite number of faces and we determine the distance function on this ensemble of planar maps. Finally, the relation with planar maps is used to illuminate a mysterious identity of certain continuum cylinder amplitudes.