Quantum Gravity and Matter: Counting Graphs on Causal Dynamical Triangulations

An outstanding challenge for models of non-perturbative quantum gravity is the consistent formulation and quantitative evaluation of physical phenomena in a regime where geometry and matter are strongly coupled. After developing appropriate technical tools, one is interested in measuring and classifying how the quantum fluctuations of geometry alter the behaviour of matter, compared with that on a fixed background geometry. In the simplified context of two dimensions, we show how a method invented to analyze the critical behaviour of spin systems on flat lattices can be adapted to the fluctuating ensemble of curved spacetimes underlying the Causal Dynamical Triangulations (CDT) approach to quantum gravity. We develop a systematic counting of embedded graphs to evaluate the thermodynamic functions of the gravity-matter models in a high- and low-temperature expansion. For the case of the Ising model, we compute the series expansions for the magnetic susceptibility on CDT lattices and their duals up to orders 6 and 12, and analyze them by ratio method, Dlog Pad\'e and differential approximants. Apart from providing evidence for a simplification of the model's analytic structure due to the dynamical nature of the geometry, the technique introduced can shed further light on criteria \`a la Harris and Luck for the influence of random geometry on the critical properties of matter systems