It is largely folklore that:
(a) topological K-theory in its full twisted & equivariant & differential (TED) refinement classifies stable D-branes in string theory as well as non-interacting topological insulator phases in condensed matter theory;
(b) some non-perturbative/strongly-interacting enhancement of this classification is needed to account for M-branes in string theory as well as for interacting topological order in condensed matter theory
(a) full TED K-theory has never quite been formulated before
(b) its M-theoretic/strong-interacting enhancement has remained elusive.
This talk surveys:
(a) our construction of full TED K-theory via cohesive infinity-topos theory;
(b) our "Hypothesis H" about its M-theoretic/strongly-interacting enhancement: instead of applying TED-K to the spacetime/Brillouin-torus directly, it needs to be applied to the corresponding twisted Cohomotopy moduli space.
(c) how the resulting theory explains the nature of defect branes (such as D7-branes) in M/F-theory as well as anyonic topological order (for su(2)-anyons) in condensed matter theory.
For talk slides and further pointers see: ncatlab.org/schreiber/show/TED-K+of+Cohomotopy+and+Anyons