Colored discrete spaces: higher dimensional combinatorial maps and quantum gravity

TL;DR

This paper develops combinatorial tools to systematically study higher-dimensional colored discrete spaces, their topology, and curvature, with applications to quantum gravity, tensor models, and SYK model correlators.

Contribution

It introduces a bijection with stacked 2D surfaces to classify and analyze complex discrete spaces in higher dimensions, extending previous triangulation approaches.

Findings

Classified discrete spaces by mean curvature and topology.

Analyzed new bi-pyramids and higher-dimensional generalizations.

Connected combinatorial structures to tensor models and SYK model graphs.

Abstract

In any dimension $D$, 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 the spaces which maximize the mean curvature survive. In two dimensions, this results in a theory of random discrete spheres, which converge in the continuum limit towards the Brownian sphere, a random fractal space interpreted as a quantum random space-time. In this limit, the continuous Liouville theory of $D=2$ quantum gravity is recovered. Previous results in higher dimension regarded triangulations - gluings of tetrahedra or $D$-dimensional generalizations, leading to the continuum random tree, or gluings of simple colored building blocks of small sizes, for which multi-trace matrix model results are recovered. This work aims at providing combinatorial tools which would allow a systematic study of richer building blocks and of the spaces they generate in the continuum. We develop a bijection with stacked two-dimensional discrete surfaces, and detail how it can be used to classify discrete spaces according to their mean curvature and topology. A number of blocks are analyzed, including the new infinite family of bi-pyramids, as well as toroidal and $D$-dimensional generalizations. The relation to random tensor models is detailed. A central concern is the lowest bound on the number of ($D-2$)-cells for any given blocks, or equivalently the right scaling for the associated tensor model to have a well-behaved $1/N$ expansion. We also apply our bijection to the identification of the graphs contributing at any order to the $2n$-point functions of the colored SYK model, and to the enumeration of generalized unicellular maps - spaces obtained from a single building block - according to their mean curvature.