Diagrammatics, Pentagon Equations, and Hexagon Equations of Topological Orders with Loop- and Membrane-like Excitations

TL;DR

This paper develops diagrammatic frameworks for understanding 4D and higher-dimensional topological orders with loop- and membrane-like excitations, introducing new equations that constrain their topological data and suggest criteria for anomaly detection.

Contribution

It systematically constructs diagrammatic representations for higher-dimensional topological orders and introduces pentagon and hexagon equations that govern their topological data.

Findings

Discovered pentagon equations constraining fusion data.

Formulated hexagon equations for shrinking-fusion processes.

Proposed conditions for anomaly-free higher-dimensional topological orders.

Abstract

In spacetime dimensions of 4 (i.e., 3+1) and higher, topological orders exhibit spatially extended excitations like loops and membranes, which support diverse topological data characterizing braiding, fusion, and shrinking processes, despite the absence of anyons. Our understanding of these topological data remains less mature compared to 3D, where anyons have been extensively studied and can be fully described through diagrammatic representations. Inspired by recent advancements in field theory descriptions of higher-dimensional topological orders, this paper systematically constructs diagrammatic representations for 4D and 5D topological orders, generalizable to higher dimensions. We introduce elementary diagrams for fusion and shrinking processes, treating them as vectors in fusion and shrinking spaces, respectively, and build complex diagrams by combining these elementary diagrams. Within these vector spaces, we design unitary operations represented by \(F\)-, \(\Delta\)-, and \(\Delta^2\)-symbols to transform between different bases. We discover \textit{pentagon equations} and \textit{(hierarchical) shrinking-fusion hexagon equations} that impose constraints on the legitimate forms of these unitary operations. We conjecture that all anomaly-free higher-dimensional topological orders must satisfy these conditions and any violations indicate a quantum anomaly. This work opens promising avenues for future research, including the exploration of diagrammatic representations involving braiding and the study of non-invertible symmetries and symmetry topological field theories in higher spacetime dimensions.