Fermions in 2D Lorentzian Quantum Gravity

TL;DR

This paper investigates fermions in 2D Lorentzian quantum gravity by implementing Wilson fermions on triangulations, analyzing the Dirac spectrum, and probing the fractal geometry through mass gap scaling for different matter couplings.

Contribution

It introduces a numerical approach to study fermions in Lorentzian quantum gravity and measures the fractal dimension of the underlying spacetime.

Findings

Determined the Dirac-Wilson spectrum in Lorentzian backgrounds.

Measured the fractal dimension of spacetime for different matter couplings.

Found distinct fractal dimensions for $c=1/2$ and $c=4$ matter cases.

Abstract

We implement Wilson fermions on 2D Lorentzian triangulation and determine the spectrum of the Dirac-Wilson operator. We compare it to the spectrum of the corresponding operator in the Euclidean background. We use fermionic particle to probe the fractal properties of Lorentzian gravity coupled to $c=1/2$ and $c=4$ matter. We numerically determine the scaling exponent of the mass gap $M \sim N^{-1/d_H}$ to be $d_H=2.11(5)$, and $d_H=1.77(3)$ for $c=1/2$ and $c=4$, respectively