Topological Orders, Braiding Statistics, and Mixture of Two Types of Twisted $BF$ Theories in Five Dimensions

TL;DR

This paper develops a topological quantum field theory framework in five dimensions to classify and analyze complex topological orders involving particles, loops, and membranes, revealing exotic braiding statistics and link invariants.

Contribution

It introduces new 5D TQFT models with mixed $BF$ terms and higher-form gauge fields, extending topological order classification beyond Dijkgraaf-Witten cohomology.

Findings

Constructed all braiding processes among particles, loops, and membranes.

Derived gauge-invariant Wilson operators and braiding phases.

Identified link invariants with geometric interpretations.

Abstract

Topological orders are a prominent paradigm for describing quantum many-body systems without symmetry-breaking orders. We present a topological quantum field theoretical (TQFT) study on topological orders in five-dimensional spacetime ($5$D) in which \textit{topological excitations} include not only point-like \textit{particles}, but also two types of spatially extended objects: closed string-like \textit{loops} and two-dimensional closed \textit{membranes}. Especially, membranes have been rarely explored in the literature of topological orders. By introducing higher-form gauge fields, we construct exotic TQFT actions that include mixture of two distinct types of $BF$ topological terms and many twisted topological terms. The gauge transformations are properly defined and utilized to compute level quantization and classification of TQFTs. Among all TQFTs, some are not in Dijkgraaf-Witten cohomological classification. To characterize topological orders, we concretely construct all braiding processes among topological excitations, which leads to very exotic links formed by closed spacetime trajectories of particles, loops, and membranes. For each braiding process, we construct gauge-invariant Wilson operators and calculate the associated braiding statistical phases. As a result, we obtain expressions of link invariants all of which have manifest geometric interpretation. Following Wen's definition, the boundary theory of a topological order exhibits gravitational anomaly. We expect that the characterization and classification of 5D topological orders in this paper encode information of 4D gravitational anomaly. Further consideration, e.g., putting TQFTs on 5D manifolds with boundaries, is left to future work.