Correspondence between Feynman diagrams and operators in quantum field theory that emerges from tensor model

Abstract

A novel functorial relationship in perturbative quantum field theory is pointed out that associates Feynman diagrams (FD) having no external line in one theory Th 1 with singlet operators in another one Th 2 having an additional U (N ) symmetry and is illustrated by the case where Th 1 and Th 2 are respectively the rank r − 1 and the rank r complex tensor model. The values of FD in Th 1 agree with the large N limit of the Gaussian average of those operators in Th 2 . The recursive shift in rank by this FD functor converts numbers into vectors, then into matrices, then into rank 3 tensors and so on. This FD functor can straightforwardly act on the d dimensional tensorial quantum field theory (QFT) counterparts as well. In the case of rank 2-rank 3 correspondence, it can be combined with the geometrical pictures of the dual of the original FD, namely, equilateral triangulations (Grothendieck's dessins d'enfant) to form a triality which may be regarded as a bulk-boundary correspondence.

On the counting tensor model observables as U(N) and O(N) classical invariants

“…This is a surprising feature reminiscent of dimensional reduction/uplift. This link has been deepen very recently by Amburg et al [18] as it generalizes at any d (states in rank d corresponds to diagrams in rank d −1). This of course deserves careful study as it may reveal important properties similar of that of gauge/gravity duality for tensor models.…”

“…The properties of B R; r,ν r i; m r have been listed in [7] (see page 17,18). First, decompose the above (4.16), exploiting the branching coefficient…”

“…We can easily foresee that the quantum field theory calculations heavily rely on the diagrammatics and combinatorics of these objects. Hence, a systematic combinatorial study of U(N) and O(N) classical invariants has been launched in the recent years bringing already a wealth of core results [8,9,10,11,12,13,14,15,16,7,17,18] A preferred way of enumerating these invariants mainly rests on algebraic techniques of the symmetric groups. There are a lot of reasons why the use symmetric groups has become a natural reflex and a dominating tool in the combinatorial study of tensor invariants.…”

W-representation of Rainbow tensor model

“…For the Gaussian average of the rank r operator O (r) in the rank r complex tensor model, there is a limit relation with O (r+1) r+1 in the rank r + 1 model [10], i.e.,…”

“…As the generalizations of matrix models from matrices to tensor, tensor models become very useful in the deep study of higher dimensional quantum gravity [7]- [9]. Quite recently, the operators/Feynman diagrams correspondence in quantum field theory was provided [10]. For the number of Feynman diagrams with n propagators in the rank r − 1 complex tensor model, it is equal to the number of singlet operators with 2n vertices in the rank r complex tensor model.…”