Super tensor models, super fuzzy spaces and super n-ary transformations

TL;DR

This paper extends tensor models to super fuzzy spaces, introducing super n-ary transformations and super Lie algebras, aiming to develop background independent models for supersymmetric space generation.

Contribution

It proposes a super algebraic framework for rank-three tensor models and super fuzzy spaces, including super n-ary transformations and their algebraic properties.

Findings

Super n-ary transformations form finite closed super Lie algebras.

Super fuzzy spaces satisfy a cyclicity condition on their function algebras.

Super tensor models could serve as background independent models for supersymmetric space generation.

Abstract

By extending the algebraic description of the bosonic rank-three tensor models, a general framework for super rank-three tensor models and correspondence to super fuzzy spaces is proposed. The corresponding super fuzzy spaces must satisfy a certain cyclicity condition on the algebras of functions on them. Due to the cyclicity condition, the symmetry of the super rank-three tensor models are represented by super n-ary transformations. The Leibnitz rules and the fundamental identities for the super n-ary transformations are discussed from the perspective of the symmetry of the algebra of a fuzzy space. It is shown that the super n-ary transformations of finite orders which conserve the algebra of a fuzzy space form a finite closed n-ary super Lie algebra. Super rank-three tensor models would be of physical interest as background independent models for dynamical generation of supersymmetric fuzzy spaces, in which quantum corrections are under control.