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Classical SU(2) monopole energy bound and Bogomolny equations

Physical scope

Conditional classical energy identity and lower bound for smooth finite-energy fields in the specified static SU(2) adjoint-Higgs Prasad–Sommerfield sector. It asserts neither existence of a saturating solution nor any quantum particle or supersymmetry state.

Statement

Assume ${A C}_{ω}$ . Use natural units on Minkowski $R^{1, 3}$ with $η = diag (- 1, 1, 1, 1)$ and oriented Euclidean spatial slice with $ϵ_{123} = 1$ . Fix the global group $S U (2)$ , basis $T_{a} = - i σ_{a} / 2$ , and positive $Ad$ -invariant form $κ (X, Y) = - 2 {Tr}_{2} (X Y)$ with $κ (T_{a}, T_{b}) = δ_{a b}$ . Let $g > 0$ be dimensionless and let $A_{i}, Φ$ be smooth static fields in the zero-potential adjoint-Higgs model, with $[A_{i}] = [Φ] = {l e n g t h}^{- 1}$ , $F_{i j} = \partial_{i} A_{j} - \partial_{j} A_{i} + [A_{i}, A_{j}]$ , $D_{i} Φ = \partial_{i} Φ + [A_{i}, Φ]$ , and $B_{i} = \frac{1}{2} ϵ_{i j k} F_{j k}$ . Require the model's smooth finite-energy boundary reduction: $∣ Φ ∣_{κ} \to v > 0$ for a fixed $v$ of dimension ${l e n g t h}^{- 1}$ , with $n_{\infty} (ω) = \lim_{R \to \infty} Φ (R ω) / v$ in $C^{\infty} (S^{2}, s u (2))$ . Use the adapted limiting connections $A_{α, \infty}$ of the static-sector Definition: the tangential condition is $[A_{α, \infty}, T_{3}] = 0$ in each adapted frame. The scaled radial limit $B_{\infty} (ω) : = \lim_{R \to \infty} R^{2} \sum_{i} ω_{i} B_{i} (R ω)$ exists uniformly on each closed hemisphere and satisfies $[B_{\infty} (ω), n_{\infty} (ω)] = 0$ . Suppose the energy

$E = \frac{1}{2 g^{2}} \int_{R^{3}} \sum_{i} (κ (B_{i}, B_{i}) + κ (D_{i} Φ, D_{i} Φ)) d^{3} x$

is finite and the model's asymptotic reduction and surface limit

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

exist. Put $T = m / g^{2}$ . Then both identities hold:

$E = \frac{1}{2 g^{2}} \int_{R^{3}} \sum_{i} κ (B_{i} - D_{i} Φ, B_{i} - D_{i} Φ) d^{3} x + T,$

$E = \frac{1}{2 g^{2}} \int_{R^{3}} \sum_{i} κ (B_{i} + D_{i} Φ, B_{i} + D_{i} Φ) d^{3} x - T .$

Consequently $E \geq ∣ T ∣$ . If $T > 0$ , equality holds exactly when $B_{i} = D_{i} Φ$ for every $i$ and every $x$ ; if $T < 0$ , equality holds exactly when $B_{i} = - D_{i} Φ$ everywhere. If $T = 0$ , equality holds exactly when $B_{i} = 0$ and $D_{i} Φ = 0$ everywhere. These are conditional classical deductions for the adopted energy functional; the word BPS here refers only to classical energy saturation and asserts no quantum state or supersymmetry central-charge relation.

Statement references

David Tong, §2.1.2, the monopole equations and energy bound

Jeffrey A. Harvey, §2.1, symmetric monopoles and the Bogomolny bound

Static SU(2) adjoint-Higgs model and boundary flux

Principal curvature and Bianchi identity

Positive invariant form on SU(2)

Divergence theorem on bounded C1 domains

Facts & Assumptions

[A1]

${A C}_{ω}$ is the Axiom of Countable Choice, required by the exact Bianchi and divergence suppliers. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

The sector fixes $F_{i j} = \partial_{i} A_{j} - \partial_{j} A_{i} + [A_{i}, A_{j}]$ , $D_{i} Φ = \partial_{i} Φ + [A_{i}, Φ]$ , $B_{i} = \frac{1}{2} ϵ_{i j k} F_{j k}$ , the positive coupling $g$ , the energy functional, and the finite surface limit. Static SU(2) adjoint-Higgs Prasad–Sommerfield monopole sector

[F2]

The form $κ$ is a positive-definite symmetric real bilinear form and is $Ad$ -invariant, so $κ ([X, Y], Z) + κ (Y, [X, Z]) = 0$ . Compactness and normalized invariant form for SU(2)

[F3]

In every local section, $F = d A + \frac{1}{2} [A \land A]$ and $D_{A} F = 0$ ; contracting the latter with $ϵ_{i j k}$ gives $D_{i} B_{i} = 0$ . Principal curvature, Cartan structure equation, and Bianchi identity

[F4]

For a bounded $C^{1}$ domain and a $C^{1}$ vector field on its closure, the integral of the divergence equals the outward boundary flux; this theorem assumes ${A C}_{ω}$ . Divergence on a bounded C1 Euclidean domain

Proof

Given: [A1], [F1]–[F4], smooth fields, and the finite-energy and surface-limit hypotheses.

1.1A1F1F2F3algebra

Define $J_{i} = κ (Φ, B_{i})$ . The product rule gives $\partial_{i} J_{i} = κ (\partial_{i} Φ, B_{i}) + κ (Φ, \partial_{i} B_{i})$ . By [F1]–[F3], the sum is $κ (D_{i} Φ, B_{i}) + κ (Φ, D_{i} B_{i}) = κ (D_{i} Φ, B_{i})$ : the two connection terms cancel by Ad-invariance and $D_{i} B_{i} = 0$ by Bianchi.

2.1A1F1F2F4step 1.1algebra

On each closed ball $B_{R}$ , $J$ is $C^{1}$ , so [F4] and step 1.1 give $\int_{B_{R}} \sum_{i} κ (D_{i} Φ, B_{i}) d^{3} x = \int_{S_{R}^{2}} κ (Φ, B_{i}) {\hat{x}}_{i} d S$ . Positive definiteness in [F2] gives $2 ∣ κ (D_{i} Φ, B_{i}) ∣ \leq κ (D_{i} Φ, D_{i} Φ) + κ (B_{i}, B_{i})$ , so finite energy makes the cross term integrable. Taking $R \to \infty$ and using the declared surface limit yields $\int_{R^{3}} \sum_{i} κ (B_{i}, D_{i} Φ) d^{3} x = m$ .

3.1F1F2step 2.1algebra

Expand the first square: its integral is $\int \sum_{i} (κ (B_{i}, B_{i}) + κ (D_{i} Φ, D_{i} Φ) - 2 κ (B_{i}, D_{i} Φ)) d^{3} x$ . By [F1] the first two terms give $2 g^{2} E$ , and by step 2.1 the cross term integrates to $2 m$ ; multiplying by $1 / (2 g^{2})$ and adding $T = m / g^{2}$ gives the first identity. Expanding with the plus sign changes the cross term to $+ 2 m$ , so subtracting $T$ gives the second identity. All integrals converge by [F1], [F2], and step 2.1.

4.1F1F2step 3.1algebra∎

Each square integral in step 3.1 is nonnegative by [F2], giving $E \geq T$ and $E \geq - T$ , hence $E \geq ∣ T ∣$ . For $T > 0$ , equality in $E \geq T$ holds exactly when the integral of the continuous nonnegative function $\sum_{i} κ (B_{i} - D_{i} Φ, B_{i} - D_{i} Φ)$ is zero, which is equivalent to $B_{i} = D_{i} Φ$ everywhere; the converse follows from the first identity. For $T < 0$ , the second identity gives equivalently $B_{i} = - D_{i} Φ$ . For $T = 0$ , equality means $E = 0$ ; positivity and continuity of the energy density imply $B_{i} = D_{i} Φ = 0$ everywhere, and these conditions also make $E = 0$ .

Remarks

For anti-Hermitian $X, Y$ , put $X_{H} = i X$ and $Y_{H} = i Y$ . Then $κ (X, Y) = 2 {Tr}_{2} (X_{H} Y_{H})$ . Consequently the energy coefficient $κ / (2 g^{2})$ becomes ${Tr}_{2} / g^{2}$ , and the surface coefficient $κ / g^{2}$ becomes $2 {Tr}_{2} / g^{2}$ . Both match Tong's equation (2.16) at $g^{2} = e^{2}$ . The proof is a classical model deduction and establishes no existence result for any specified topological sector.

Classical SU(2) monopole energy bound and Bogomolny equations

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources