Large N Tensor and SYK Models

The SYK model consists of Majorana fermions that interact randomly four at a time. A holographic dual may exist for this model, which makes it interesting in the study of quantum gravity. It has been found that the SYK model is similar to large N tensor models: in both models, only the melonic diagrams survive in the large N limit. In this paper, we explore the large N tensor model with O(N)^3 symmetry containing two flavours of Majorana fermions in the fundamental representation, the quartic Hamiltonian of which depends on a real parameter. We derive the kernels of the four point functions. With these, we calculate the scaling dimensions of several types of conformal primaries. We also find a duality relation between two Hamiltonians of different real parameters. This is not a perfect duality, because the normalization of energy scales with the transformation. Nevertheless, the ratios of the energies are the same, and the operator dimensions are preserved. Last, for real parameters > 1 or < 0 the scaling dimensions of one of the conformal primaries become complex, rendering the model unstable.