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Flux quantization in the specified SU(2) adjoint-Higgs sector

Physical scope

Conditional classical flux quantization for the stated global SU(2) form, adjoint Higgs asymptotic reduction to U(1), and fundamental-doublet probe lattice. No charge lattice for SO(3) or other global forms, quantum state, or empirical monopole claim is asserted.

Statement

Assume ${A C}_{ω}$ . On oriented Euclidean $R^{3}$ with $ϵ_{123} = 1$ , use the zero-potential static model with global group $S U (2)$ , $T_{a} = - i σ_{a} / 2$ , $κ (X, Y) = - 2 {Tr}_{2} (X Y)$ , $F_{i j} = \partial_{i} A_{j} - \partial_{j} A_{i} + [A_{i}, A_{j}]$ , $D_{i} Φ = \partial_{i} Φ + [A_{i}, Φ]$ , and $B_{i} = \frac{1}{2} ϵ_{i j k} F_{j k}$ . Let $g > 0$ be dimensionless, $v > 0$ of dimension ${l e n g t h}^{- 1}$ , and $A_{i}, Φ$ be smooth static fields. Assume $∣ Φ ∣_{κ} \to v$ , $Φ (R \cdot) / v \to n_{\infty}$ in $C^{\infty} (S^{2}, s u (2))$ , and the boundary reduction specified below. Let $D_{N}, D_{S} \subset S^{2}$ be the closed northern and southern hemispheres, and choose smooth lifts $u_{α}$ on neighborhoods of $D_{α}$ with ${Ad}_{u_{α}} T_{3} = n_{\infty}$ and transition $u_{S} = u_{N} h$ on their common equator, where $h (φ) = \exp (θ (φ) T_{3})$ and $θ : [0, 2 π] \to R$ is a smooth lift. For $ι_{R} : S^{2} \to S_{R}^{2}$ , $ι_{R} (ω) = R ω$ , define on each $D_{α}$ the fixed-frame angular connection, curvature, and normalized Higgs field by

$A_{α, R} : = {Ad}_{u_{α}^{- 1}} (ι_{R}^{*} A) + u_{α}^{- 1} d u_{α}, F_{α, R} : = {Ad}_{u_{α}^{- 1}} (ι_{R}^{*} F), q_{α, R} : = {Ad}_{u_{α}^{- 1}} (Φ (R \cdot) / v) .$

Assume that $A_{α, R}$ converges pointwise on each closed hemisphere to $A_{α, \infty}$ , which extends as a $C^{1}$ one-form to a neighborhood of $D_{α}$ . Define its curvature by $F_{α, \infty} : = d A_{α, \infty} + \frac{1}{2} [A_{α, \infty} \land A_{α, \infty}]$ , and assume it extends as a $C^{1}$ two-form to a neighborhood of $D_{α}$ . Also assume $F_{α, R} \to F_{α, \infty}$ uniformly on each closed hemisphere. The fixed-frame right-action transition gives $A_{S, \infty} = {Ad}_{h^{- 1}} A_{N, \infty} + h^{- 1} d h$ . The smooth Higgs-field limit gives $q_{α, R} \to T_{3}$ uniformly. Define $D_{t a n} n_{\infty} ∣_{D_{α}} : = {Ad}_{u_{α}} [A_{α, \infty}, T_{3}]$ and require it to vanish. Define $B_{\infty} : S^{2} \to s u (2)$ by ${Ad}_{u_{α}} F_{α, \infty} = B_{\infty} d Ω$ on $D_{α}$ , with $d Ω$ the outward unit-sphere area form. Curvature covariance identifies the two definitions on the equator. Equivalently $B_{\infty} (ω) = \lim_{R \to \infty} R^{2} \sum_{i} ω_{i} B_{i} (R ω)$ uniformly on each hemisphere. Require $[B_{\infty} (ω), n_{\infty} (ω)] = 0$ , which is the stated stabilizer-algebra condition. The equator is oriented as the boundary of $D_{N}$ , and the admissible external electric probes include the fundamental doublet. If the surface flux limit

$m = \lim_{R \to \infty} \int_{S_{R}^{2}} κ (Φ, B_{i}) {\hat{x}}_{i} d S$

exists, define

$ν : = - \frac{θ (2 π) - θ (0)}{4 π} .$

Then $ν \in Z$ and the classical magnetic surface charge and energy-normalized surface term are

$m = 4 π v ν, T = \frac{m}{g^{2}} = \frac{4 π v}{g^{2}} ν .$

This is flux quantization for the stated global form, boundary reduction, and probe lattice; it does not assert a universal normalization for other groups or global forms.

Statement references

David Tong, §2.1.1, Dirac quantization and SU(N) magnetic charge

Jeffrey A. Harvey, §1.4, Higgs winding and charge quantization

Specified SU(2) adjoint-Higgs boundary data

Stokes theorem on a compact oriented surface region

Normalized invariant form and SU(2) commutators

Facts & Assumptions

[A1]

${A C}_{ω}$ is the Axiom of Countable Choice. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

The sector fixes $S U (2)$ , $T_{3} = - i σ_{3} / 2$ , the right-action transition law, $Φ \to v n_{\infty}$ , the adapted local frames, and the surface-flux definition. Static SU(2) adjoint-Higgs Prasad–Sommerfield monopole sector

[F2]

For the chosen basis, $[T_{a}, T_{b}] = ϵ_{a b c} T_{c}$ and $κ$ is positive definite, symmetric, and $Ad$ -invariant with $κ (T_{a}, T_{b}) = δ_{a b}$ . Compactness and normalized invariant form for SU(2)

[F3]

For every compact oriented surface region with finitely many smooth boundary arcs and every $C^{1}$ one-form, $\int_{D} d α = \int_{\partial D} α$ with outward-normal-first boundary orientation; the result assumes ${A C}_{ω}$ . Stokes for C1 one-forms on compact surface regions

Proof

Given: [A1], [F1]–[F3], the smooth limiting reduction and local frames in the Statement, and the stated transition lift.

1.1F1F2givenalgebra

In either limiting adapted frame, $n_{\infty} = T_{3}$ and $D_{t a n} n_{\infty} = 0$ , so $[A_{α, \infty}, T_{3}] = 0$ . Writing $A_{α, \infty} = a_{α}^{1} T_{1} + a_{α}^{2} T_{2} + a_{α}^{3} T_{3}$ , the commutator relations in [F2] give $[A_{α, \infty}, T_{3}] = a_{α}^{2} T_{1} - a_{α}^{1} T_{2}$ ; hence $A_{α, \infty} = a_{α} T_{3}$ for a real $C^{1}$ one-form $a_{α}$ . Its curvature is $F_{α, \infty} = d a_{α} T_{3}$ , since $[T_{3}, T_{3}] = 0$ .

2.1F1F2step 1.1algebra

For finite $R$ , the two-form $F$ pulled back by $ι_{R}$ is the magnetic flux form: $ι_{R}^{*} F = (B_{i} {\hat{x}}_{i}) d S_{R}$ under the outward orientation. By $Ad$ -invariance of $κ$ and the frame definitions, the flux at radius $R$ therefore satisfies $\frac{1}{v} \int_{S_{R}^{2}} κ (Φ, B_{i}) {\hat{x}}_{i} d S = \sum_{α = N, S} \int_{D_{α}} κ (q_{α, R}, F_{α, R}) .$ The uniform limits of $q_{α, R}$ and $F_{α, R}$ imply uniform convergence of the integrands to $κ (T_{3}, F_{α, \infty})$ . Each hemisphere has finite area, so the absolute difference of its integrals is bounded by its area times the sup norm of the integrand difference and tends to zero. Using $κ (T_{3}, T_{3}) = 1$ , we obtain $m / v = \sum_{α} \int_{D_{α}} κ (T_{3}, F_{α, \infty}) = \int_{D_{N}} d a_{N} + \int_{D_{S}} d a_{S}$ .

3.1A1F1F3step 2.1algebra

Apply [F3] separately to the two closed hemispheres. Their induced equatorial orientations are opposite, so $\int_{D_{N}} d a_{N} + \int_{D_{S}} d a_{S} = \int_{C} (a_{N} - a_{S})$ . The limiting right-action transition stated above, with $h = \exp (θ T_{3})$ , and the fact that both limiting forms lie in the stabilizer algebra give $a_{S} = a_{N} + d θ$ . Therefore the flux is $- \int_{C} d θ = - (θ (2 π) - θ (0))$ .

4.1F1step 3.1algebra∎

In the fundamental representation, $\exp (θ T_{3}) = diag (e^{- i θ / 2}, e^{i θ / 2})$ . Since $h$ is a closed SU(2) loop, equality of its endpoint matrices gives $e^{- i (θ (2 π) - θ (0)) / 2} = 1$ , hence $θ (2 π) - θ (0) = 4 π k$ for some $k \in Z$ . The definition $ν = - k$ and step 3.1 yield $m / v = 4 π ν$ , so $m = 4 π v ν$ and $T = 4 π v ν / g^{2}$ .

Remarks

The diagram records that both local lifts project to the same asymptotic Higgs direction. The lower map is $p (u) = {Ad}_{u} T_{3}$ ; the relation $u_{S} = u_{N} h$ with $h$ in the stabilizer makes the square commute. The proof above uses the transition formula and Stokes theorem directly; it does not rely on the diagram.

Flux quantization in the specified SU(2) adjoint-Higgs sector

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources