Tensor models and embedded Riemann surfaces

TL;DR

This paper explores the topological structures within tensor models for quantum gravity, revealing how embedded matrix models generate key subgraphs and how these relate to Riemann surfaces and 3D triangulations.

Contribution

It demonstrates that bubble and jacket graphs in 3D tensor models are generated by embedded matrix models and correspond to splitting surfaces, enabling a reformulation of the Boulatov model on Riemann surfaces.

Findings

Jacket graphs represent Heegaard splitting surfaces.

Embedded matrix models generate key subgraphs.

Reformulation of Boulatov model on Riemann surfaces.

Abstract

Tensor models and, more generally, group field theories are candidates for higher-dimensional quantum gravity, just as matrix models are in the 2d setting. With the recent advent of a 1/N-expansion for coloured tensor models, more focus has been given to the study of the topological aspects of their Feynman graphs. Crucial to the aforementioned analysis were certain subgraphs known as bubbles and jackets. We demonstrate in the 3d case that these graphs are generated by matrix models embedded inside the tensor theory. Moreover, we show that the jacket graphs represent (Heegaard) splitting surfaces for the triangulation dual to the Feynman graph. With this in hand, we are able to re-express the Boulatov model as a quantum field theory on these Riemann surfaces.