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Multicritical Dynamical Triangulations and Topological Recursion

Hiroyuki Fuji, Masahide Manabe, Yoshiyuki Watabiki

Abstract

We explore a continuum theory of multicritical dynamical triangulations and causal dynamical triangulations in two-dimensional quantum gravity from the perspective of the Chekhov-Eynard-Orantin topological recursion. The former model lacks a causal time direction and is governed by the two-reduced $W^{(3)}$ algebra, whereas the latter model possesses a causal time direction and is governed by the full $W^{(3)}$ algebra. We show that the topological recursion solves the Schwinger-Dyson equations for both models, and we explicitly compute several amplitudes.

Multicritical Dynamical Triangulations and Topological Recursion

Abstract

We explore a continuum theory of multicritical dynamical triangulations and causal dynamical triangulations in two-dimensional quantum gravity from the perspective of the Chekhov-Eynard-Orantin topological recursion. The former model lacks a causal time direction and is governed by the two-reduced algebra, whereas the latter model possesses a causal time direction and is governed by the full algebra. We show that the topological recursion solves the Schwinger-Dyson equations for both models, and we explicitly compute several amplitudes.

Paper Structure

This paper contains 26 sections, 3 theorems, 132 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Dynamical Triangulation (DT)
  3. Pure DT
  4. Formulation
  5. Amplitudes and Schwinger-Dyson equation
  6. Multicritical DT
  7. Formulation
  8. Amplitudes and Schwinger-Dyson equation
  9. Perturbative amplitudes and topological recursion
  10. Relation between multicritical DT and JT gravity
  11. Causal Dynamical Triangulation (CDT)
  12. Pure CDT
  13. Formulation
  14. Amplitudes and Schwinger-Dyson equation
  15. Multicritical CDT
  16. ...and 11 more sections

Key Result

Proposition 2.1

The amplitudes $\widetilde{\mathcal{F}{\space}}_N^{{\rm conn}}(\xi_1,\ldots,\xi_N)$, determined by equation sp_sd_m_crit_dt and also providing solutions to the SD equation sd_multicritical, satisfy where $\tilde{C}_{N-1}(\bm{\xi}_{I \backslash \{1\}})$ is a function of $\bm{\xi}_{I \backslash \{1\}}$.

Figures (1)

  • Figure 1: Two configurations of a 2D surface with a boundary. $\bullet$ Left panel: An example of a DT or CDT configuration that satisfies the no big-bang condition. (The configuration does not create a universe from nothing; configurations of the type shown in the right panel are therefore excluded.) $\bullet$ Right panel: An example of a CDT configuration that does not satisfy the no big-bang condition. (The red point represents the creation of a universe from nothing. The region including the red point and its surroundings is inaccessible from the lower-right universe, indicating that this configuration preserves causality.)

Theorems & Definitions (3)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 3.1