Anyonic topological order in twisted equivariant differential (TED) K-Theory

While the classification of noninteracting crystalline topological insulator phases by equivariant K-Theory has become widely accepted, its generalization to anyonic interacting phases-hence to phases with topologically ordered ground states supporting topological braid quantum gates-has remained wide open. On the contrary, the success of K-Theory with classifying noninteracting phases seems to have tacitly been perceived as precluding a K-Theoretic classification of interacting topological order; and instead a mix of other proposals has been explored. However, only K-Theory connects closely to the actual physics of valence electrons; and self-consistency demands that any other proposal must connect to K-Theory. Here, we provide a detailed argument for the classification of symmetry protected/enhanced 2-Anyonic topological order, specifically in interacting 2d semi-metals, by the twisted equivariant differential (TED) K-Theory of configuration spaces of points in the complement of nodal points inside the crystal's Brillouin torus orbi-orientifold. We argue, in particular, that: (1) topological 2d semi-metal phases modulo global mass terms are classified by the flat differential twisted equivariant K-Theory of the complement of the nodal points; (2) n-electron interacting phases are classified by the K-Theory of configuration spaces of n points in the Brillouin torus; (3) the somewhat neglected twisting of equivariant K-Theory by "inner local systems"reflects the effective "fictitious"gauge interaction of Chen, Wilczeck, Witten and Halperin (1989), which turns fermions into anyonic quanta; (4) the induced 2-Anyonic topological order is reflected in the twisted Chern classes of the interacting valence bundle over configuration space, constituting the hypergeometric integral construction of monodromy braid representations. A tight dictionary relates these arguments to those for classifying defect brane charges in string theory [H. Sati and U. Schreiber, Anyonic defect branes in TED-K-Theory, arXiv:2203.11838], which we expect to be the images of momentum-space 2-Anyons under a nonperturbative version of the AdS/CMT correspondence.