We present a rigorous and fully consistent K-theoretic framework for studying gapped phases of free fermions. It utilizes and profits from powerful techniques in operator K-theory, which from the point of view of symmetries such as time reversal, charge conjugation, and magnetic translations, is more general and natural than the topological version. In our model-independent approach, the dynamics are only constrained by the physical symmetries, which can be completely encoded using a suitable C * -superalgebra. Contrary to existing literature, we do not use K-theory groups to classify phases in an absolute sense, but to classify topological obstructions between phases. The Periodic Table of Kitaev is exhibited as a special case within our framework, and we prove that the phenomena of periodicity and dimension shifts are robust against disorder and magnetic fields.

Abstract. The subtle interplay between local and global charges for topological semimetals exactly parallels that for singular vector fields. Part of this story is the relationship between cohomological semimetal invariants, Euler structures, and ambiguities in the connections between Weyl points. Dually, a topological semimetal can be represented by Euler chains from which its surface Fermi arc connectivity can be deduced. These dual pictures, and the link to topological invariants of insulators, are organised using geometric exact sequences. We go beyond Dirac-type Hamiltonians and introduce new classes of semimetals whose local charges are subtle Atiyah-Dupont-Thomas invariants globally constrained by the Kervaire semicharacteristic, leading to the prediction of torsion Fermi arcs.

Abstract. Recently we introduced T-duality in the study of topological insulators. In this paper, we study the bulk-boundary correspondence for three phenomena in condensed matter physics, namely, the quantum Hall effect, the Chern insulator, and time reversal invariant topological insulators. In all of these cases, we show that T-duality trivializes the bulk-boundary correspondence.

We provide a manifestly topological classification scheme for generalised Weyl semimetals, in any spatial dimension and with arbitrary Weyl surfaces which may be non-trivially linked. The classification naturally incorporates that of Chern insulators. Our analysis refines, in a mathematically precise sense, some well-known 3D constructions to account for subtle but important global aspects of the topology of semimetals. Using a fundamental locality principle, we derive a generalized charge cancellation condition for the Weyl surface components. We analyse the bulk-boundary correspondence under a duality transformation, which reveals explicitly the topological nature of the resulting surface Fermi arcs. We also analyse the effect of moving Weyl points on the bulk and boundary topological semimetal invariants. * Electronic address: mathai.varghese@adelaide.edu.au † Electronic address: guochuan.thiang@adelaide.edu.au ‡