Topology · workbench

Symmetry breaking & topology

Explore the homotopy sequence for two spherical vacuum orbits.

Supported: SU(2)/U(1) and SO(3)/SO(2)

Homotopy exact sequenceBoth vacuum orbits are S²

\mathrm{SU}(2)/\mathrm U(1)\cong S^2

0\longrightarrow\pi_2(S^2)\cong\mathbb Z\xrightarrow{\partial=\times1}\pi_1(\mathrm{U}(1))\cong\mathbb Z\longrightarrow0

π₂ of vacuum orbit: $\mathbb Z$
Kernel of π₁(H) → π₁(G): $\mathbb Z$
Connecting multiplier: 1

A generator of π₂(S²) maps to one U(1) winding. Because SU(2) is simply connected, every stabilizer winding lies in the kernel. Both spaces have integer-valued π₂ classes; the connecting maps differ.

Embedding and generator conventions

For SU(2), U(1) is embedded as $\operatorname{diag}(e^{i\theta},e^{-i\theta})$ , with $\theta\sim\theta+2\pi$ . For SO(3), SO(2) is the subgroup of rotations about one fixed axis, with the same angular period. Choose compatible positive generators. Physical magnetic-charge units require an additional gauge-field normalization.

The exact sequence uses $\pi_2(G)=0$ and $\pi_1(S^2)=0$ for these two cases.

Method, conventions & sources

Homotopy exact sequence of H -> G -> S2, with the standard positive generators of pi2(S2) and pi1(H).

Connected groups and the indicated closed subgroups; winding generators have their standard positive orientation.

Calculation version: groups-design-1.0.0

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