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

Abstract

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 $$\mathbf{Th}_1$$ Th1 with singlet operators in another one $$\mathbf{Th}_2$$ Th2 having an additional $$U(\mathcal{N})$$ U(N) symmetry and is illustrated by the case where $$\mathbf{Th}_1$$ Th1 and $$\mathbf{Th}_2$$ Th2 are respectively the rank $$r-1$$ r-1 and the rank r complex tensor model. The values of FD in $$\mathbf{Th}_1$$ Th1 agree with the large $$\mathcal{N}$$ N limit of the Gaussian average of those operators in $$\mathbf{Th}_2$$ Th2 . 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.