Tensor models and hierarchy of n-ary algebras

TL;DR

The paper addresses how rank-3 tensor models, which describe dynamical fuzzy spaces, exhibit a rich symmetry structure that extends beyond the previously identified 3-ary transformations. It develops a general framework for metric-invariant n-ary transformations by generalizing the commutator to p-ary operations and enforcing Leibniz-type rules, revealing a hierarchical hierarchy of n-ary algebras whose closures depend on invariance constraints. The rank-3 tensor model’s SO(N) symmetry emerges from these n-ary transformations, with locality in physically realistic fuzzy spaces restricting 3-ary transformations to local symmetries, while higher arities encode increasingly nonlocal infinitesimal symmetries. This provides a systematic algebraic perspective on the symmetries of tensor models and fuzzy spaces, suggesting a path to capture nonlocal structures in quantum gravity 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.