A new perspective on matter coupling in 2d quantum gravity

TL;DR

This work investigates how conformal matter couples to a non-perturbative 2d Lorentzian quantum gravity model by combining a high-temperature expansion with Monte Carlo simulations of gravity+matter on causal triangulations. The results show that the Ising model coupled to Lorentzian 2d gravity retains flat-space critical exponents (Onsager class) and a Hausdorff dimension $d_H=2$, despite large geometric fluctuations and a dynamical triangulation structure. The absence of baby universes is identified as the key factor yielding weak matter–geometry coupling, distinguishing the Lorentzian theory from Liouville gravity and potentially avoiding the $c=1$ barrier. Overall, the paper provides evidence that Lorentzian 2d gravity constitutes a genuine, smoother alternative to Euclidean approaches, with implications for non-perturbative quantum gravity and the study of matter–geometry interactions.

Abstract

We provide compelling evidence that a previously introduced model of non-perturbative 2d Lorentzian quantum gravity exhibits (two-dimensional) flat-space behaviour when coupled to Ising spins. The evidence comes from both a high-temperature expansion and from Monte Carlo simulations of the combined gravity-matter system. This weak-coupling behaviour lends further support to the conclusion that the Lorentzian model is a genuine alternative to Liouville quantum gravity in two dimensions, with a different, and much `smoother' critical behaviour.

Figures (6)

• Figure 1: A typical discrete history of pure Lorentzian gravity with volume N=1024.
• Figure 2: A two-dimensional geometry with a 'pinch' of length 1.
• Figure 3: The triangles contributing to the weight at the vertex $i$.
• Figure 4: The critical cosmological constant as a function of the Ising coupling $\beta$, as measured by Monte Carlo simulations ($t\!=\! 32$, $N\!=\! 2048$), and compared to the corresponding high-$T$ expansions $F_1(\beta)$ and $F_2(\beta)$ at order 0 and order 6.
• Figure 5: The move used in the Monte Carlo updating of the geometry.