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 ${\mathbf{Th}}_1$ 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, and then into rank $3$ tensors $\dots$ This FD functor can straightforwardly act on the $d$ dimensional tensorial quantum field theory 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.

5 pages