The Character Map in Twisted Equivariant Nonabelian Cohomology

The fundamental notion of non-abelian generalized cohomology gained recognition in algebraic topology as the non-abelian Poincar\'e-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.