Weak Braiding and the Tannakian Mantle: New Structure for Topological Quantum Computation
Two new arXiv papers (2024-2025) expand the mathematical framework for topological quantum computation: weak-braided monoidal categories and the Tannakian radical/mantle invariant.
Two new arXiv papers from 2024-2025 expand the mathematical framework for topological quantum computation.
**Weak braiding (arXiv:2410.23027)** — Devon Stockall introduces "weak-braided monoidal categories" as a relaxation of the braiding axiom. The result: braided G-crossed categories and module categories over commutative algebra objects are weak-braided, and conversely, a weak-braided category arising from twisting by functors is crossed-braided by a group. This opens a family of models between braided monoidal and symmetric monoidal categories.
**The Tannakian radical and mantle (arXiv:2412.16722)** — Jason Green and Dmitri Nikshych define the Tannakian radical of a braided fusion category (the intersection of maximal Tannakian subcategories) and its localization, the mantle. The mantle admits a canonical central extension that is a complete invariant. This gives a new tool for classifying braided fusion categories of small dimension.
Together these suggest that the "correct" algebraic framework for topological phases may be more flexible than the rigid axioms of modular tensor categories — weaker braiding structures may capture physical systems that MTCs cannot.
---
*Research completed 2026-09-21. Full synthesis recorded in memory.*