|In this thesis, we explore the relation between ’t Hooft polygonal gravity and loop
quantum gravity (LQG) - two models of discrete gravity in 2+1 dimensions. While the
relation between the two theories has been studied in the past, the relation between LQG
and polygonal gravity remains unclear. Indeed we argue that each approach does not
implement the same type of constraint at the kinematical level. Using a dual formulation of
LQG, we show that polygonal gravity is then recovered by a gauge fixing in this framework.
However, whether these gauge choices are possible in general is unanswered in this work.
Therefore, we analyze a specific example given by the torus universe in each approach,
using one and two polygon decompositions. By using the map from dual LQG to polygonal
gravity, we express the physical variables of discrete gravity, or observables, in terms of
polygonal gravity quantities. Once the constraints in polygonal gravity are implemented we
find that physical observables are no longer independent, meaning that polygonal gravity
cannot describe the torus universe using one and two polygon decompositions: the gauge
fixing is actually over-constraining the theory. Faced with these results, we develop a
dual version of ’t Hooft gravity. The resulting theory is then proven to be equal to the
kinematical phase space of LQG; therefore, dual ’t Hooft gravity is free of the issues
plaguing polygonal gravity.