Magnetic Equivariant Graded Brauer Group
Higinio Serrano, Bernardo Uribe
TL;DR
The paper introduces magnetic group symmetry into the framework of graded Brauer groups, defining the magnetic equivariant graded Brauer group GrBr_{(G,φ)}(ℂ) via non-abelian cohomology and magnetic representations. It provides a detailed, explicit decomposition of GrBr_{(G,φ)}(ℂ) into cohomological components (H^2(G,ℂ^*_{φ})), a Hom(G,ℤ/2) part, and an additional ℤ/2 factor, including explicit 2-cocycles that govern the twisted product structure. Building on Karoubi’s Banach-category approach to K-theory, the authors show that GrBr_{(G,φ)}(ℂ) parametrizes twistings of magnetic equivariant K-theory and derive degree-shift isomorphisms and 4-periodicity for certain groups. The work connects to Dyson’s ten-fold and Altland–Zirnbauer classifications through the interpretation of End(V) for irreducible graded magnetic representations, and showcases concrete computations for cyclic magnetic groups. Overall, the results give a comprehensive algebraic classification of twists in magnetic equivariant K-theory and illuminate their geometric and physical implications for symmetry-protected topological phases.
Abstract
Given a magnetic finite group, we consider the similarity classes of magnetic equivariant central simple graded algebras over the complex numbers. We call this set the magnetic equivariant graded Brauer group and its structure as an abelian group is explicitly determined. Following Karoubi, we argue that the elements of this graded Brauer group parametrize the twistings of the magnetic equivariant K-theory of a point.