Communications in Mathematical Physics

, Volume 355, Issue 2, pp 561–602 | Cite as

Differential Topology of Semimetals

  • Varghese Mathai
  • Guo Chuan Thiang


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.


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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of AdelaideAdelaideAustralia

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