The Character Map in Twisted Equivariant Nonabelian Cohomology
Hisham Sati, Urs Schreiber
备注：
95 pages; v3: published version; v2: title shortened and abstract, intro & outro re-written for applied algebraic topologists, physics application split off by request from journal, now relegated to arXiv:2411.16852
发表信息：
Beijing Journal of Pure and Applied Mathematics, Vol. 2 No. 2 (2025) 515-617
The fundamental notion of non-abelian generalized cohomology gained recognition in algebraic topology as the non-abelian Poincaré-dual to "factorization homology", and in theoretical physics as providing flux-quantization for non-linear Gauss laws. However, already the archetypical example -- unstable Cohomotopy, first studied almost a century ago by Pontrjagin -- has remained underappreciated as a cohomology theory and has only recently received attention as a flux-quantizaton law ("Hypothesis H").
Here we lay out a general construction of the analogue of the Chern character map on twisted equivariant non-abelian cohomology theories (with equivariantly simply-connected classifying spaces) and illustrate the construction by spelling out a twisted equivariant form of Cohomotopy as an archetypical and intriguing running example, essentially by computing its equivariant Sullivan model.
We close with an outlook on the application of this result to the rigorous deduction of anyonic quantum states on M5-branes wrapped over Seifert 3-orbifolds.
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Topological BF Theory construction of twisted dihedral quantum double phases from spontaneous symmetry breaking
University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；Laboratoire de Physique des Solides（固体物理实验室）
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Short-range entangled topological phases of matter are closely connected to Topological Quantum Field Theory. We use this connection to classify bosonic Symmetry Protected Topological Phases in low dimensions, including the case when the symmetry involves time-reversal. To accomplish this, we generalize Turaev's description of equivariant TQFT to the unoriented case. We show that invertible unoriented equivariant TQFTs in one or less spatial dimensions are classified by twisted group cohomology, in agreement with the group cohomology proposal of Chen, Gu, Liu and Wen. We also show that inverti…
