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Abstract |
In any dimension, the Euclidean Einstein-Hilbert action, which describes gravity in the absence of matter, can be discretized over random discrete spaces obtained by gluing families of polytopes together in all possible ways. In the physical limit of small Newton constant, only discrete spaces which maximize the mean curvature survive. In two dimensions, this results in a theory of random discrete spheres, which, in the continuum limit, converge towards a fractal continuous space called the Brownian sphere, which is interpreted as a quantum space-time. In this limit, the Liouville continuous theory of two-dimensional quantum gravity is recovered.
Previous results in higher dimension regarded random triangulations (gluings of tetrahedra or higher dimensional generalizations) or gluings of simple building blocks of small size. For these polytopes, we recover at best the two-dimensional results. This work aims at providing combinatorial tools, which would allow a systematic study of more complicated building blocks and of the continuous quantum space-times they generate in the continuum limit. We develop a bijection with stacked discrete surfaces and explain how it can be used to characterize the discrete spaces that survive in the physical limit of small Newton constant. |
