Tensor models and hierarchy of n-ary algebras

TL;DR

This paper explores the hierarchy of n-ary algebras underlying tensor models, revealing their role in generating symmetries in quantum gravity models and extending previous work on 3-ary algebra symmetries.

Contribution

It introduces a hierarchy of n-ary algebras as the symmetry generators of tensor models, generalizing the known 3-ary algebra symmetry to include higher n-ary transformations.

Findings

Hierarchy of n-ary algebras underpins tensor model symmetries

Finite n-ary Lie subalgebras preserve algebra invariance

Higher n-ary transformations generate non-local symmetries in tensor models

Abstract

Tensor models are generalization of matrix models, and are studied as models of quantum gravity. It is shown that the symmetry of the rank-three tensor models is generated by a hierarchy of n-ary algebras starting from the usual commutator, and the 3-ary algebra symmetry reported in the previous paper is just a single sector of the whole structure. The condition for the Leibnitz rules of the n-ary algebras is discussed from the perspective of the invariance of the underlying algebra under the n-ary transformations. It is shown that the n-ary transformations which keep the underlying algebraic structure invariant form closed finite n-ary Lie subalgebras. It is also shown that, in physical settings, the 3-ary transformation practically generates only local infinitesimal symmetry transformations, and the other more non-local infinitesimal symmetry transformations of the tensor models are generated by higher n-ary transformations.