An article that we are finalizing:
Hisham Sati, $\;$ Urs Schreiber:
Anyonic topological order in TED K-theory
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on extending the K-theory classification of topological phases to anyonic topological order.
Abstract. While the classification of non-interacting crycrystalline 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 non-interacting topological 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 $\mathfrak{su}(2)$-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:
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;
$n$-electron interacting phases are classified by the K-theory of configuration spaces of $n$ points in the Brillouin torus;
the somewhat neglected twisting of equivariant K-theory by “inner local systems” reflects the effective "fictitious" gauge interaction of Chen, Wilczeck, Witten & Halperin (1989), which turns fermions into anyonic quanta;
the induced $\mathfrak{su}(2)$-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 $[$SS22-Any$]$, which we expect to be the images of momentum-space $\mathfrak{su}(2)$-anyons under a non-perturbative version of the AdS/CMT correspondence.
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