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

TL;DR

This paper introduces a functorial relationship linking Feynman diagrams and operators across different tensor models, revealing a recursive rank-shifting process and potential bulk-boundary dualities in tensorial quantum field theories.

Contribution

It establishes a novel functorial correspondence between Feynman diagrams and operators in tensor models, connecting different ranks and geometric duals, expanding the understanding of tensor QFTs.

Findings

Feynman diagrams in one tensor model correspond to operators in another with larger symmetry.

Recursive rank shifts transform numerical data into higher-dimensional tensors.

The framework suggests a bulk-boundary duality involving triangulations and dessins d'enfant.

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 ${\bf Th}_1$ with singlet operators in another one ${\bf Th}_2$ having an additional $U({\cal N})$ symmetry and is illustrated by the case where ${\bf Th}_1$ and ${\bf Th}_2$ are respectively the rank $r-1$ and the rank $r$ complex tensor model. The values of FD in ${\bf Th}_1$ agree with the large ${\cal N}$ limit of the Gaussian average of those operators in ${\bf Th}_2$. The recursive shift in rank by this FD functor converts numbers into vectors, then into matrices, and then into rank $3$ tensors ${\ldots}$ 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.