Abstract
It has recently been proven that in rank three tensor models, the antisymmetric and symmetric traceless sectors both support a large N expansion dominated by melon diagrams [1]. We show how to extend these results to the last irreducible O(N) tensor representation available in this context, which carries a two-dimensional representation of the symmetric group S3. Along the way, we emphasize the role of the irreducibility condition: it prevents the generation of vector modes which are not compatible with the large N scaling of the tensor interaction. This example supports the conjecture that a melonic large N limit should exist more generally for higher rank tensor models, provided that they are appropriately restricted to an irreducible subspace.
Highlights
Some properties of the celebrated SYK models [27,28,29,30] in the familiar context of large N quantum mechanics
It has recently been proven that in rank three tensor models, the antisymmetric and symmetric traceless sectors both support a large N expansion dominated by melon diagrams [1]
We emphasize the role of the irreducibility condition: it prevents the generation of vector modes which are not compatible with the large N scaling of the tensor interaction. This example supports the conjecture that a melonic large N limit should exist more generally for higher rank tensor models, provided that they are appropriately restricted to an irreducible subspace
Summary
We consider a bosonic theory in zero dimension with O(N ) global symmetry. The degrees of freedom are organized into a real rank-3 tensor Ta1a2a3 transforming as a product of three fundamental representations:. As standard in the literature, we will represent the interaction kernel as an ordinary fourvalent vertex, or as a stranded diagram — as shown in figure 1. Is the complete graph on four vertices, we propose to call such an interaction a complete interaction.5 By this definition, there exists other complete interactions (for instance the O(N )3-invariant interaction of [9]) that one might want to include in the action. There exists other complete interactions (for instance the O(N )3-invariant interaction of [9]) that one might want to include in the action They all become equivalent upon reduction to a tensor in a fixed irreducible representation of S3, so it is sufficient for our purpose to consider only one such interaction
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