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

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.