Event

We review the double line notation for the Feynman diagram expansion of N by N matrix models. In the ‘t Hooft large N limit only the planar diagrams survive, and the dual graphs may be thought of as discretized random surfaces. We proceed to theories where the dynamical degrees of freedom are rank-3 tensors with distinguishable indices, each of which takes N values. Their Feynman diagrams may be drawn using colored triple lines (red, blue, green), while the dual graphs are made out of tetrahedra glued along their triangular faces. Such theories possess a special solvable large N limit dominated by the “melon” diagrams. We discuss quantum mechanical models of fermionic rank-3 tensors and their similarity with the Sachdev-Ye- Kitaev disordered model. We then use the large N Schwinger-Dyson equations to study the conformal dimensions of certain composite operators. Gauging the global symmetry in the quantum mechanical models removes the non-singlet states; therefore, one can search for their well-defined gravity duals. We note that the models possess a vast number of gauge-invariant operators involving higher powers of the tensor field. Finally, we discuss similar models of a commuting rank-3 tensor in dimension d. While the quartic interaction is not positive definite, we study the large N Schwinger-Dyson equations and show that their solution is consistent with conformal invariance.