On equivariant embeddings of G-bundles
Malkhaz Bakuradze, Ralf Meyer
TL;DR
This work develops equivariant stabilization results for $G$-equivariant vector bundles by linking embeddings and stable isomorphisms to multiplicities of irreducible stabiliser representations. It proves two relative theorems—an embedding criterion and a stable isomorphism criterion—under explicit rank- and orbit-type bounds, unifying ordinary, real, and quaternionic bundle cases within equivariant K-theory and extending to physics applications. The proofs rely on equivariant obstruction theory and a canonical decomposition of Hom-spaces by irreducible $H$-representations, enabling skeleta-by-skeleta extensions and a parametrised embedding argument for stable isomorphisms. Applications include a Swan-type subbundle embedding with computable bounds, and concrete implications for Bloch bundles in crystallographic systems, clarifying when stable triviality guarantees actual triviality and informing the distinction between robust and fragile topological phases. Overall, the results provide a coherent, computable framework for classifying symmetry-protected phases via equivariant bundles and offer concrete criteria for when stabilization suffices to determine topological invariants in physically relevant settings.
Abstract
For a compact group G, we give a sufficient condition for embedding one G-equivariant vector bundle into another one and for a stable isomorphism between two such bundles to imply an isomorphism. Our criteria involve multiplicities of irreducible representations of stabiliser groups. We also apply our result to ordinary nonequivariant vector bundles over the fields of quaternions, real and complex numbers and to ``real'' and ``quaternionic'' vector bundles. Our results apply to the classification of symmetry-protected topological phases of matter, providing computable bounds on the number of energy bands required to distinguish robust from fragile topological phases.