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 [Formula: see text]-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) [Formula: see text]-electron interacting phases are classified by the K-theory of configuration spaces of [Formula: see text] 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 [Formula: see text]-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 [Formula: see text]-anyons under a nonperturbative version of the AdS/CMT correspondence.