Renormalization in perturbative quantum field theory is based on a Hopf algebra of Feynman diagrams. A precondition for this is locality. Therefore one might suspect that non-local field theories such as matrix or tensor field theories cannot benefit from a similar algebraic understanding. Here I show that, on the contrary, perturbative renormalization of a broad class of such field theories is based in the same way on a Hopf algebra. Their interaction vertices have the structure of graphs. This gives the necessary concept of locality and leads to Feynman diagrams defined as “2-graphs” which generate the Hopf algebra. These results set the stage for a systematic study of perturbative renormalization as well as non-perturbative aspects, e.g. Dyson-Schwinger equations, for a number of combinatorially non-local field theories with possible applications to random geometry and quantum gravity.

微扰量子场论中的重整化基于费曼图的 Hopf 代数。实现这一点的前提是地域性。因此，人们可能会怀疑诸如矩阵或张量场理论之类的非局部场理论不能从类似的代数理解中受益。在这里，我表明，相反，一大类此类场论的微扰重整化以相同的方式基于 Hopf 代数。它们的交互顶点具有图的结构。这给出了必要的局部性概念，并导致费曼图被定义为“2-图”，它生成了 Hopf 代数。这些结果为系统研究微扰重整化以及非微扰方面奠定了基础，例如戴森-施温格方程，