An introduction to tensor models: from random geometry to melonic CFTs - Lecture I

Abstract

Tensor models are particularly interesting due to their melonic large-N limit which is richer than the large-N limit of vector models but simpler than the planar limit of matrix models. Tensor models were first introduced in zero dimension in the context of random geometry and quantum gravity. They were then extended to quantum mechanical models in one dimension as an alternative to the Sachdev-Ye-Kitaev model without disorder. Finally, they were generalized in higher dimensions as toy models for strongly-coupled QFTs. In this context, they give rise in the infrared to a new kind of conformal field theories analytically accessible, called melonic CFTs.

In these lectures, after reviewing the large-N expansion of matrix models, I will introduce tensor models and derive their melonic large-N limit. In both cases, I will present some applications to random geometry and quantum gravity. The second part of the lectures will focus on melonic CFTs. In particular, I will review the bosonic long-range 0(N)3 model giving rise at large N to a unitary CFT in the infrared.