Pro-Tensor Network

TL;DR

The paper introduces pro-tensor networks, a rigorous graphical framework for many-body theories that generalizes tensor networks and applies to topological and symmetry studies.

Contribution

It develops a novel pro-tensor network framework that relaxes traditional assumptions, enabling broader exploration of many-body physics and topological phenomena.

Findings

Recover the Levin-Wen model as a pro-tensor network

Generalize Kitaev and Kong's particle characterization

Apply to symmetric tensor networks and topological holography

Abstract

We introduce the pro-tensor network, a categorification of the tensor network, as a fully rigorous yet graphically transparent framework for studying the collection of many many-body theories, which we dub many-many-body theory. We provide a comprehensive toolbox for the graphical calculations using pro-tensor networks. As applications, we recover the Levin-Wen model as a "uniform" pro-tensor network and generalize a result of Kitaev and Kong by characterizing particles as modules over promonads. One can also interpret the string-net pro-tensor network as the space of symmetric tensor networks, thus our framework also applies to the study of generalized symmetry and topological holography. Notably, our generalization dispenses with the assumptions of semisimplicity, finiteness, and rigidity, potentially facilitating the exploration of many-body physics beyond these constraints.