Abstract
We study invariant operators in general tensor models. We show that representation theory provides an efficient framework to count and classify invariants in tensor models of (gauge) symmetry Gd=U(N1)⊗⋯⊗U(Nd). As a continuation and completion of our earlier work, we present two natural ways of counting invariants, one for arbitrary Gd and another valid for large rank of Gd. We construct bases of invariant operators based on the counting, and compute correlators of their elements. The basis associated with finite rank of Gd diagonalizes the two-point function of the free theory. It is analogous to the restricted Schur basis used in matrix models. We show that the constructions get almost identical as we swap the Littlewood–Richardson numbers in multi-matrix models with Kronecker coefficients in general tensor models. We explore the parallelism between matrix model and tensor model in depth from the perspective of representation theory and comment on several ideas for future investigation.
Highlights
This paper is meant to be a comprehensive revision and completion of our earlier work [42]
We found two different bases, one valid for arbitrary values of the ranks of the symmetry group, and a second basis of invariants which applies for large ranks
We show explicitly in Eq (3.19) that the counting of elements of both bases agrees for large ranks
Summary
We start by setting our notation and reviewing some essential facts of elementary representation theory which will be relevant throughout this work. Under the action of the gauge group Gd, Vn and V n split into orbits that correspond to irreducible representations (irreps) of Gd. It is known that the irreducible representations of Gd are labeled by a collection of d Young diagrams with n boxes each In what follows, operators of colored tensor fields will be labeled by irreps of the symmetric group Snd and the gauge group Gd. Schur-Weyl duality: Schur-Weyl duality states that, as the action of Sn and the action of diagonal U(N) on (CN )⊗n commute, we have the multiplicity-free product decomposition (CN )⊗n =. Projectors in (4.8) act in the space where the n tensor fields have been symmetrized, implementing the manifest symmetry of Φ⊗n under permuting any of the tensor copies.
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