Cyclic Cohomology for Graded 
                
                  
                
                $$C^{*,r}$$
                
                  
                    
                      C
                      
                        
                        ∗
                        ,
                        r
                      
                    
                  
                
              -algebras and Its Pairings with van Daele K-theory

Guided by the many-particle quantum theory of interacting systems, we develop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C$$^*$$ ∗ -algebras harbouring the bulk and boundary data of disordered free-fermion ground states. These we connect by a natural bulk-to-boundary short exact sequence, realising the bulk system as a quotient of the half-space theory modulo boundary contributions. To every ground state, we attach two classes in different pictures of real operator $$K$$ K -theory (or $$KR$$ KR -theory): a bulk class, using Van Daele’s picture, along with a boundary class, using Kasparov’s Fredholm picture. We then show that the connecting map for the bulk-to-boundary sequence maps these $$KR$$ KR -theory classes to each other. This implies bulk-boundary correspondence, in the presence of disorder, for both the “strong” and the “weak” invariants.

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“…Any Cor Z-valued pairing will therefore be degenerate. Kellendonk addresses this problem in recent work [29].…”

Section: Discussionmentioning

confidence: 99%