Quantum geometry of 2d gravity coupled to unitary matter

TL;DR

The paper investigates the fractal geometry of 2d gravity coupled to unitary matter by employing reparametrization-invariant geodesic-distance observables. It uses dynamical triangulations to numerically study Ising and three-state Potts models, extracting the Hausdorff dimension dh and KPZ exponents from n_1, n_phi, G_phi, and loop-length distributions. The results consistently favor dh ≈ 4 across c in [0,1], with scaling relations aligning with KPZ predictions and loop-structure matching pure gravity, suggesting a universal fractal geometry that is robust to matter backreaction. The work discusses competing theoretical predictions for dh(c), finite-size effects, and the need for larger simulations to conclusively resolve remaining tensions.

Abstract

We show that there exists a divergent correlation length in 2d quantum gravity for the matter fields close to the critical point provided one uses the invariant geodesic distance as the measure of distance. The corresponding reparameterization invariant two-point functions satisfy all scaling relations known from the ordinary theory of critical phenomena and the KPZ exponents are determined by the power-like fall off of these two-point functions. The only difference compared to flat space is the appearance of a dynamically generated fractal dimension d_h in the scaling relations. We analyze numerically the fractal properties of space-time for Ising and three-states Potts model coupled to 2d dimensional quantum gravity using finite size scaling as well as small distance scaling of invariant correlation functions. Our data are consistent with d_h=4, but we cannot rule out completely the conjecture d_H = -2α_1/α_{-1}, where α_{-n} is the gravitational dressing exponent of a spin-less primary field of conformal weight (n+1,n+1). We compute the moments <L^n> and the loop-length distribution function and show that the fractal properties associated with these observables are identical, with good accuracy, to the pure gravity case.

Figures (13)

• Figure 1: ( a) $d_h(a)$ from collapsing $n_1(r;N)$ for $N_T= 8000$--$64000$ for pure gravity. Data is collapsed in groups of three lattice sizes and in the graph we indicate the largest of each group. We show the errors computed from $\chi^2$ only for the largest lattice in order to simplify the graph. The errors for the smaller lattices are quite similar. ( b) Same as in ( a) for the Ising model for $N_T= 8000$--$128000$. ( c) Same as in ( b) for the three--states Potts model.
• Figure 2: ( a) $d_h(a)$ from collapsing $n_\varphi(r;N)$ for $N_T= 8000$--$128000$ for the Ising model the same way as described in Fig. \ref{['f:1']}. ( b)Same as in ( a) for the three--states Potts model.
• Figure 3: ( a) The small $x$ behaviour of the logarithmic derivative of $n_1(r;N)$. We use $N_T=32000$--$128000$, $d_h=4.02$ and $a=0,0.54$. ( b) Same as in ( a) for the Ising model coupled to gravity. We plot for $N_T=64000$--$256000$, $a=0,0.51$ and $d_h=4.08$. ( c) Same as in ( b) for the three--states Potts model coupled to gravity where $a=0,0.48$ and $d_h=4.10$.
• Figure 4: ( a) The small $x$ behaviour of the logarithmic derivative of $n_\varphi(r;N)$ for the Ising model coupled to gravity. We use $N_T=16000$--$256000$ and $x$ is obtained by using $d_h=4.13$, $a=0.45$. ( b) Same for the three--states Potts model coupled to gravity. We use now $d_h=4.27$ and $a=0.35$.
• Figure 5: ( a) The rescaled according to Eq. (\ref{['iii7']}) normalized spin--spin correlation function $g_\varphi(r;N)$ for the Ising model coupled to gravity. We use $N_T=2000$--$256000$, $d_h=4.0$, $a=0.51$. ( b) Same for the three--states Potts model coupled to gravity. We use $a=0.55$.