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

Abstract

AbstractA 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.