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Conformal Dimensions On Causal Random Geometry

Ryan Barouki, Henry Stubbs, John Wheater

TL;DR

This work investigates Ising matter coupled to 2D causal dynamical triangulations (CDT) and demonstrates that, in the quenched CDT setting, the conformal weights remain the flat-lattice values ($h_\sigma=1/16$, $h_\psi=1/2$), indicating a KPZ-like shift is avoided. The authors develop a defect-based Ising formulation on CDT using the plaquette representation, construct Dehn twist operators, and embed these within a continuum framework via the Lamperti–Ney process to realize a CDT random measure $d\mu=L(t)dt$, thereby deriving a continuum Dehn twist corresponding to $e^{2\pi i (L_0-\bar L_0)}$. A key result is that CDT’s one-dimensional geometry yields no KPZ shift, with the scaling dimensions matching those on a fixed lattice, while still enabling a consistent connection to Hořava–Lifshitz gravity through a projectable HL-like Hamiltonian. The paper further outlines how these results may extend to the annealed ensemble and discusses the broader implications for 2D quantum gravity models that impose causal structure.

Abstract

We investigate the interaction between matter and causal dynamical triangulations (CDT) in the context of two-dimensional quantum gravity. We focus on the Ising model coupled to CDT, contrasting this with Liouville gravity and the relation to the Knizhnik-Polyakov-Zamolodchikov (KPZ) formula. We demonstrate analytically for the quenched model that the conformal dimensions of fields on CDT align with those on a fixed lattice. We do this using a combination of lattice methods and adapting the Duplantier-Sheffield framework to CDT, emphasizing the one-dimensional nature of CDT and its description via a stochastic differential equation.

Conformal Dimensions On Causal Random Geometry

TL;DR

This work investigates Ising matter coupled to 2D causal dynamical triangulations (CDT) and demonstrates that, in the quenched CDT setting, the conformal weights remain the flat-lattice values (, ), indicating a KPZ-like shift is avoided. The authors develop a defect-based Ising formulation on CDT using the plaquette representation, construct Dehn twist operators, and embed these within a continuum framework via the Lamperti–Ney process to realize a CDT random measure , thereby deriving a continuum Dehn twist corresponding to . A key result is that CDT’s one-dimensional geometry yields no KPZ shift, with the scaling dimensions matching those on a fixed lattice, while still enabling a consistent connection to Hořava–Lifshitz gravity through a projectable HL-like Hamiltonian. The paper further outlines how these results may extend to the annealed ensemble and discusses the broader implications for 2D quantum gravity models that impose causal structure.

Abstract

We investigate the interaction between matter and causal dynamical triangulations (CDT) in the context of two-dimensional quantum gravity. We focus on the Ising model coupled to CDT, contrasting this with Liouville gravity and the relation to the Knizhnik-Polyakov-Zamolodchikov (KPZ) formula. We demonstrate analytically for the quenched model that the conformal dimensions of fields on CDT align with those on a fixed lattice. We do this using a combination of lattice methods and adapting the Duplantier-Sheffield framework to CDT, emphasizing the one-dimensional nature of CDT and its description via a stochastic differential equation.

Paper Structure

This paper contains 22 sections, 10 theorems, 116 equations, 14 figures.

Table of Contents

  1. Introduction
  2. 2D quantum gravity
  3. Graphs
  4. Euclidean triangulations
  5. Causal triangulations
  6. Coupling matter
  7. Liouville gravity à la Duplantier-Sheffield
  8. The Ising model and topological defects
  9. The Ising model in the plaquette formalism
  10. The spin-flip defect
  11. The duality defect
  12. The Ising fusion category
  13. Dehn twist operators on the lattice
  14. Stochastic formulation of CDT
  15. The Lamperti-Ney Process
  16. ...and 7 more sections

Key Result

Lemma 3.1

For all $C \in \mathcal{C}_\infty \setminus D$, each defect $\phi = \psi, \sigma$ and every height $t$ there is an operator $\mathbf T_\phi$ acting on the Hilbert space $\mathcal{H}_\phi$ at height $t$ satisfying the relations where the operators $\mathds 1_\phi$ act as the identity in the presence of a vertical defect $\phi$, i.e. $\mathds{1}_\phi\ket{\{h_i\}} = \ket{\{h_i\}}$.

Figures (14)

  • Figure 1: An instance of a large planar map with 30,000 vertices generated by Jérémie Bettinelli JerBet.
  • Figure 2: A causal triangulation of height 3, and the action of the map $\beta$ to its associated tree.
  • Figure 3: A dyadic decomposition of $e^{\gamma\phi}$ with $\gamma = 2$ (left) and $\gamma=0.5$ (right)
  • Figure 4: The graph $\widetilde{G}$ (orange) formed from the original square lattice (black), and its dual (gray, dashed).
  • Figure 5: Horizontal (orange) and vertical plaquettes for a single CDT strip where the dots show the location of the weights $d_v$.
  • ...and 9 more figures

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: Euclidean and quantum balls
  • Definition 2.6: Euclidean and quantum scaling
  • Lemma 3.1: Dehn twist operators
  • proof
  • Corollary 3.2: Eigenvalues of Dehn twist operators
  • proof
  • ...and 17 more