High Energy Theory Seminar

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. As shown by Gurau, Rivasseau and others, such theories possess a special solvable large N limit dominated by the “melon” diagrams. Following Witten’s work 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.