• CDT
• Monte Carlo methods
• topological charge
• Yang-Mills theory

Causal Dynamical Triangulation (CDT) is a discrete and non-perturbative approach to quantum gravity which acquired a remarkable success and interest in the last decades. There are at least two main reasons behind this: first, according to Weiberg’s assumption about the renormalization of an asymptotically safe theory, CDT aims to search for UV fixed points of a quantum gravity theory described by Einstein-Hilbert action. Second, thanks to its triangulated-lattice formulation, it is quite simple to perform numerical simulations based on Monte-Carlo methods.

Grounded on this theoretical framework and its computational implementation, it is natural and useful to study what happens if CDT is coupled with other models which have a well-known lattice formulation, for instance Yang-Mills theory. Indeed, studying topology of gauge fields plays a key in role in Quantum ChromoDynamics (QCD) with respect to the dependence on the topological parameter θ of CP violation, but could also help to better characterize the effective space-time topology which has a remarkable impact on phase transitions of geometries in CDT and in particular on the order parameters defining them.

Up to now, gauge fields and their topology have been successfully implemented and studied on two-dimensional triangulated manifolds, but it is far less simple to extent the same analysis in four dimensions where CDT is expected to reproduce De-Sitter universes and has a more expensive and intricate numerical setting. Nevertheless, a very first 4d implementation has been proposed in this thesis, providing some interesting results.

The starting point for this work resides in the construction of the dual graph of the triangulation where gauge variables live and thus of the plaquettes from which it is possible to retrieve the information about the strength field tensor and its flux through their surfaces. The algorithm implemented to update links, i.e SU(N) matrices, is a standard heath-bath process for N=2 and the Cabibbo-Marinari one for N>2, properly adjusted in order to fit the curved geometry.

The main observables which have been investigated in this dissertation are the Yang-Mills action and especially the topological charge Q. About the latter, a first definition based on simplicial triangulations has been proposed, taking particularly care of its pseudo-scalar nature. Indeed, to make this property satisfied, it was necessary to construct a consistent global orientation of the local charts associated with each site of the dual graph (barycenter of simplices building the triangulation), even though CDT model is diffeomorphism invariant and thus its implementation coordinate-free. By means of such definition and a standard cooling procedure aimed to remove the UV fluctuations, a non-trivial topological structure has been found out for manifolds of toroidal topology TxT³, while spherical ones TxS³ seem to be lacking based on data collected so far.

In the first scenario it was shown that the system sets on meta-stable states, (anti-)instantons configurations, which are associated with equally displaced values of Q. The action gap between two consecutive topological sectors was also measured and for quasi-flat geometries it was shown that the topological susceptibility scales with the fourth power of the lattice spacing a, supposed that a depends on the bare coupling constant as in the exact flat case. The whole analysis has been carried out initially on quasi-flat triangulations and in a second moment on fixed geometries evolved by gravity Monte-Carlo moves.

Despite the static analysis, the presence of non-trivial topology is a promising result which encourages to better understand and investigate both the definition of Q and its behavior over all the phase diagram and the possibility of formulating a dynamical Monte-Carlo algorithm capable of updating gauge links and geometry at same time.