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

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• The 1/N Expansion of Tensor Models Beyond Perturbation Theory
• Phase Transition in Dually Weighted Colored Tensor Models
• The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders
• The Double Scaling Limit in Arbitrary Dimensions: A Toy Model
• Canonical tensor models with local time
• Uniqueness of canonical tensor model with local time
• Super tensor models, super fuzzy spaces and super n-ary transformations
• Canonical tensor model through data analysis -- Dimensions, topologies, and geometries --
• Tensor models and 3-ary algebras