Orientations of Orbi-K-Theory measuring Topological Phases and Brane Charges

TL;DR

This paper explores the role of unstable cohomology theories, like cohomotopy, in classifying topological phases and brane charges, and introduces a new model of twisted orbifold K-theory to measure these fragile phases.

Contribution

It provides an explicit realization of unstable complex/quaternionic orientations in equivariant K-theory using real division algebras within a novel twisted orbifold K-theory framework.

Findings

Cohomotopy captures the fine-grained classification of topological phases.

Stable cohomology measures fragile phases via complex/quaternionic orientations.

Application to classifying 2-band crystalline topological insulators and M-string charges.

Abstract

Topological Chern phases of quantum materials, as well as brane charges on M-theory orbifolds, have famously been argued to be classified by (orbi) topological K-theory, or possibly by other stable and, notably, complex-oriented cohomology theories, such as elliptic cohomology or Morava K-theory. However, closer inspection reveals that the most fine-grained "fragile" microscopic classification in both cases is in (orbi) Cohomotopy, which is the primordial "unstable" or nonabelian generalized cohomology. Coarsening takes the latter (fragile) to the former (stable) cohomology along an nonabelian cohomology operations. But what then is the role of complex orientation on the stable side? We observe here (i) that over gapped nodal lines in the 2D Brillouin torus and on probe M5-branes in 11D spacetime, the cohomotopical phases/charges lift through the complex/quaternionic Hopf fibration, and (ii) that measuring this fragile situation in stable cohomology means equivalently to ask for universal complex/quaternionic orientation on stable cohomology in four/ten dimensions! Then we give an explicit realization of such unstable four/ten-dimensional complex/quaternionic orientation in U(2)/Sp(2)-equivariant K-theory, using real division-algebraic tools within a new model of twisted orbifold K-theory in cohesive homotopy theory; and we explain this as an extraordinary character map from orbi Cohomotopy-twisted Cohomotopy to relative orbi K-theory. Finally, we discuss an application to the classification of 2-band crystalline topological insulator phases sensitive to the topology in the gapping process of their nodal line semimetal parent phase, and to the measurement of M-string charges inside M5-brane probes.