Tensor models and hierarchy of n-ary algebras

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.

13 pages, some references updated and corrected

References in corpus (10)

• Gauge Symmetry and Supersymmetry of Multiple M2-Branes
• Algebraic structures on parallel M2-branes
• Modeling Multiple M2's
• EPRL/FK Group Field Theory
• Lost in Translation: Topological Singularities in Group Field Theory
• Quantized Nambu-Poisson Manifolds and n-Lie Algebras
• Tensor model and dynamical generation of commutative nonassociative fuzzy spaces
• The fluctuation spectra around a Gaussian classical solution of a tensor model and the general relativity
• The lowest modes around Gaussian solutions of tensor models and the general relativity
• Emergent general relativity in fuzzy spaces from tensor models