Diagrammatic Representations of Higher-Dimensional Topological Orders

In 3D spacetime, topologically ordered phases of matter feature emergent particles called anyons, whose properties like fusion rules and braiding statistics are schematically represented by diagrams. Important consistency conditions, such as pentagon and hexagon relations, are encoded in diagrams. In 4D and higher spacetimes, topological orders support not only point-like particles but also spatially extended excitations, such as loops and membranes, allowing for diverse topological data about braiding, fusion, and shrinking processes. Recently, these topological data have been explored through the path-integral formalism of topological quantum field theory. Analogous to the counterpart in 3D, in this work, we construct diagrammatic representations of higher-dimensional (4D and 5D) topological orders. We introduce basic fusion and shrinking diagrams and treat them as vectors in corresponding spaces, then build complex diagrams by stacking these basic diagrams. Within the same vector spaces, we use F-, Δ-, and Δ²-symbols to transform between different bases. From these transformations, we derive consistency conditions like pentagon equations and (hierarchical) shrinking-fusion hexagon equations, which describe the consistent coexistence of fusion and shrinking data. We conjecture that all anomaly-free higher-dimensional topological orders must satisfy these conditions, with violations indicating a quantum anomaly. This work opens several promising avenues for future research such as the exploration of diagrammatic representations of braiding processes in higher dimensions as well as implications for noninvertible symmetries and Symmetry Topological Field Theory (SymTFT).