arXiv subjects

Tommy Shu

Publications and source records attributed to Tommy Shu.

3 recordsLinked to original sources

arXiv2026

3-Crossed Module Structure in the Five-Dimensional Topological Axion Electrodynamics

In this paper, we investigate the higher-group symmetry structure of a five-dimensional topological theory, which is described by a 3-crossed module. The model is obtained by a five-dimensional extension of topological axion electrodynamics in four dimensions. To study the symmetry structure, we couple background gauge fields to the symmetry currents via Stueckelberg couplings. We show that background gauge invariance requires modified gauge transformation laws, indicating the existence of a higher-group structure. Furthermore, we identify the underlying mathematical structure as a 3-crossed module by regarding the modified Stueckelberg couplings as curvatures of a higher-group gauge theory. We demonstrate that the gauge transformation laws derived from this algebraic structure are consistent with the analysis based on the gauge invariance. While our previous work introduced the concept of a 3-crossed module motivated by higher-group symmetries, this work provides concrete verification that this framework correctly captures the symmetry structure of physical theories.

hep-th↗

arXiv2025

3-Crossed modules, Quasi-categories, and the Moore complex

The established equivalence between 2-crossed modules and Gray 3-groups [M. Sarikaya and E. Ulualan, 2024] serves as a benchmark for higher-dimensional algebraic models. However, to the best of our knowledge, the established definitions of 3-crossed modules [Z. Arvasi, T. S. Kuzpinari, and E. \"O. Uslu, 2009] are not clearly suited for extending this equivalence. In this paper, we propose an alternative formulation of a 3-crossed module, equipped with a new type of lifting, which is specifically designed to serve as a foundation for this higher-order categorical correspondence. As the primary results of this paper, we validate this new structure. We prove that the simplicial set induced by our 3-crossed module forms a quasi-category. Furthermore, we show that the Moore complex of length 3 associated with a simplicial group naturally admits the structure of our 3-crossed module. This work establishes our definition as a robust candidate for modeling the next level in this algebraic-categorical program.

math.CT↗

arXiv2024

Topological invariants of 3-dimensional manifold with boundary by using crossed module

J.H.C. Whitehead introduced the concept of crossed modules in the early 20th century. These crossed modules are crucial for algebraic models of 2-type homotopy, which involve connected spaces with no higher than second-degree homotopy groups. They consist of two groups and certain relations between them, with known connections to 2-groups. By employing crossed modules, we can develop invariants for closed 3-dimensional and 4-dimensional manifolds. The validity of these invariants was established in a paper authored by F.Girelli, H.Pfeiffer, and E.M.Popescu([4]). Essentially, these invariants involve counting correct colors over the triangulation of a closed manifold. Interestingly, I've discovered that these invariants can also be applied to compact 3-dimensional manifolds with boundaries. Therefore, in this paper, I intend to demonstrate how these invariants can be utilized for compact 3-dimensional manifolds with boundaries, including the complements of knots.

math.GT↗