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

N. Amburg,Andrei Mironov,Alexei Morozov,R. Yoshioka,D. Vasiliev,H. Itoyama

doi:10.1140/epjc/s10052-020-8013-8

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 mathbf{Th}_1 with singlet operators in another one mathbf{Th}_2 having an additional U(mathcal{N}) symmetry and is illustrated by the case where mathbf{Th}_1 and mathbf{Th}_2 are respectively the rank r-1 and the rank r complex tensor model. The values of FD in mathbf{Th}_1 agree with the large mathcal{N} limit of the Gaussian average of those operators in mathbf{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.

Highlights

Over the years, we have, occasionally seen cases in the study of low energy effective actions where we may even enumerate the whole set of such singlet operators and lift our considerations to all possible vacua and perturbation theory thereon
To emphasize the universal meaning of this functor, wherever possible we present a discussion in generic terms of quantum field theory, and restrict consideration to the simplest tensor models only at the points where sample calculations are needed to illustrate a power of the method
We point out a novel functorial relationship that lies in Feynman diagrams (FD) having no external line in one theory Th1 and singlet operators in another one Th2 having an additional U (N ) symmetry

Summary

The corresponding

For any rank r operator, by interpreting the contraction of any one of r indices as a symbol specifying the. We can obtain the corresponding rank r − 1 Feynman diagram. The following correspondence is established, FDσ Kσ (r−1) ↔ K(σ (r−1),σ ). This correspondence is one to one and we obtain the relation (1). Let Op(r) be the set of all operators in the rank r tensor model and FD(r−1) be the set of all Feynman diagrams in the rank r − 1 tensor model. Page 3 of 5 471 the subscript i means that the contraction by the i-th index ai is interpreted as the Wick contraction symbol. The leading term in Ni is order n and comes from the case where all of the Wick contractions of M and Mare accompanied with the ai index loops:

Nin ops

The numbers of the connected operators at each level are