Abstract
We show that the counting of observables and correlators for a 3-index tensor model are organized by the structure of a family of permutation centralizer algebras. These algebras are shown to be semi-simple and their Wedderburn-Artin decompositions into matrix blocks are given in terms of Clebsch-Gordan coefficients of symmetric groups. The matrix basis for the algebras also gives an orthogonal basis for the tensor observables which diagonalizes the Gaussian two-point functions. The centres of the algebras are associated with correlators which are expressible in terms of Kronecker coefficients (Clebsch-Gordan multiplicities of symmetric groups). The color-exchange symmetry present in the Gaussian model, as well as a large class of interacting models, is used to refine the description of the permutation centralizer algebras. This discussion is extended to a general number of colors d: it is used to prove the integrality of an infinite family of number sequences related to color-symmetrizations of colored graphs, and expressible in terms of symmetric group representation theory data. Generalizing a connection between matrix models and Belyi maps, correlators in Gaussian tensor models are interpreted in terms of covers of singular 2-complexes. There is an intriguing difference, between matrix and higher rank tensor models, in the computational complexity of superficially comparable correlators of observables parametrized by Young diagrams.
Highlights
Introduced as generalizations of matrix models [1, 2] to study the discrete-to-continuum transition for discretized path integrals in quantum gravity, tensor models [3,4,5] and their further generalizations [6] were found to be tremendously more difficult to handle than the theory of matrices
We show that the counting of observables and correlators for a 3-index tensor model are organized by the structure of a family of permutation centralizer algebras
The SYK model is an active topic of research, its connections being explored with black hole physics, AdS/CFT correspondence, quantum gravity and condensed matter physics
Summary
Introduced as generalizations of matrix models [1, 2] to study the discrete-to-continuum transition for discretized path integrals in quantum gravity, tensor models [3,4,5] and their further generalizations [6] were found to be tremendously more difficult to handle than the theory of matrices. The terms in the sum are labelled by triples of Young diagrams with n boxes, with non-vanishing Kronecker coefficient These are triples R1, R2, R3 of representations of Sn such that the tensor product R1 ⊗ R2 ⊗ R3 contains the trivial under the action of the diagonal Sn. The algebra elements belonging to the matrix block labelled by the ordered triple [R1, R2, R3], along with more refined data associated with the Kronecker multiplicities, are constructed using Clebsch-Gordan coefficients of the symmetric group. Among those directions, we mention a new type of statistical models based on Young diagrams, the quest for holographic duals of tensor models and an intriguing connection between Computational Complexity Theory and correlators in matrix and tensor models. Representation theory and Young diagram combinatorics have been employed in an SYK context in [60]
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