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Chern–Simons boundary variation and BF equations

Physical scope

Pure mathematics; no physical premises or empirical claim.

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ . Let $M$ be a compact oriented smooth three-manifold with boundary, $g$ a finite-dimensional real Lie algebra with invariant nondegenerate symmetric pairing, and $A \in Ω^{1} (M; g)$ a smooth connection in a fixed trivialization. For an arbitrary smooth variation $α \in Ω^{1} (M; g)$ , the Chern–Simons first variation is

$δ {CS}_{A} (α) = 2 \int_{M} ⟨ α \land F_{A} ⟩ - \int_{\partial M} ⟨ A \land α ⟩,$

where $\partial M$ has the outward-normal-first orientation. Thus the boundary term is part of the variation; the bulk Euler equation is $F_{A} = 0$ when variations are compactly supported in $int M$ (or when boundary data make the boundary variation vanish). For $B \in Ω^{1} (M; g)$ the BF action has first variation

$δ {BF}_{(B, A)} (β, α) = \int_{M} ⟨ β \land F_{A} ⟩ + \int_{M} ⟨ α \land D_{A} B ⟩ - \int_{\partial M} ⟨ B \land α ⟩,$

so interior-compact variations give the equations $F_{A} = 0$ and $D_{A} B = 0$ . No boundary condition is imposed by these bulk equations.

For $λ \in R$ fixed, the boundary potential current of $λ CS$ is $θ_{A} (α) = - λ ⟨ A \land α ⟩$ . On a compact oriented smooth surface $Σ$ (with smooth boundary allowed), put $A_{Σ} = Ω^{1} (Σ; g)$ and identify each tangent space with $Ω^{1} (Σ; g)$ . For $a \in A_{Σ}$ define $Θ_{Σ} ∣_{a} (α) = \int_{Σ} θ_{a} (α)$ , and let $Ω_{Σ} = d_{A_{Σ}} Θ_{Σ}$ . The restriction to the flat-connection solution space, wherever it is smooth, is the possibly degenerate presymplectic form. Explicitly, on tangent vectors $α_{1}, α_{2}$ it is the constant closed form

$Ω_{Σ} (α_{1}, α_{2}) = - 2 λ \int_{Σ} ⟨ α_{1} \land α_{2} ⟩ .$

At a flat connection $a$ ( $F_{a} = 0$ ), tangent variations $α$ to the flat-connection equations satisfy $D_{a} α = 0$ . For an infinitesimal gauge parameter $ϵ \in C^{\infty} (Σ; g)$ , the gauge direction is $D_{a} ϵ$ and

$Ω_{Σ} (D_{a} ϵ, α) = - 2 λ \int_{\partial Σ} ⟨ ϵ, α ⟩ .$

Consequently gauge parameters vanishing on $\partial Σ$ generate null directions on the linearized flat locus. If $ϵ$ is field-independent, the displayed boundary variation is $δ Q_{ϵ}$ for $Q_{ϵ} (a) = - 2 λ \int_{\partial Σ} ⟨ ϵ, a ⟩$ . For $λ \neq 0$ , nonzero boundary parameters need not be null and can carry such charges when the boundary conditions permit them; no charge algebra or integrability assertion for field-dependent parameters is made. If $λ = 0$ , the displayed $Θ_{Σ}$ , $Ω_{Σ}$ and $Q_{ϵ}$ vanish identically, and every gauge direction on the linearized flat locus is null.

Statement references

Chern–Simons and BF action data

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

The general Stokes theorem

Facts & Assumptions

Given: A compact oriented smooth three-manifold $M$ with boundary, a finite-dimensional real Lie algebra $g$ with the invariant nondegenerate pairing in the action definition, smooth fields $A, B$ and variations $α, β$ , and the stated ${A C}_{ω}$ assumption.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ says every countable family of nonempty sets has a choice function.

[F1]

Chern–Simons and BF action data: $F_{A} = d A + \frac{1}{2} [A, A]$ , $D_{A} X = d X + [A, X]$ , $CS (A) = \int_{M} ⟨ A \land d A + \frac{1}{3} A \land [A, A] ⟩$ , and $BF (B, A) = \int_{M} ⟨ B \land F_{A} ⟩$ .

[F2]

The general Stokes theorem: under ${A C}_{ω}$ , for an oriented smooth $n$ -manifold with boundary and a compactly supported smooth $(n - 1)$ -form $η$ , $\int_{M} d η = \int_{\partial M} j^{*} η$ with outward-normal-first boundary orientation. Here compactness of $M$ makes all smooth forms used below compactly supported.

Proof

1.1givenF1algebra

Differentiate the Chern–Simons integrand along $A + s α$ . The quadratic term contributes $⟨ α \land d A + A \land d α ⟩$ . For the cubic term, the graded bracket and invariant pairing give the cyclic identities $⟨ A \land [α, A] ⟩ = ⟨ α \land [A, A] ⟩$ and $⟨ A \land [A, α] ⟩ = ⟨ α \land [A, A] ⟩$ for degree-one $A, α$ . Thus the three differentiated terms in $\frac{1}{3} ⟨ A \land [A, A] ⟩$ are equal, and their average is $⟨ α \land [A, A] ⟩$ . The full unintegrated variation is $⟨ α \land d A + A \land d α + α \land [A, A] ⟩$ .

1.2F1algebra

Differentiate $F_{A + s α}$ at $s = 0$ ; bilinearity of the bracket gives $δ F_{A} = D_{A} α$ . Thus the BF variation before integration by parts is $\int_{M} ⟨ β \land F_{A} + B \land D_{A} α ⟩$ .

2.1A1F1F2step 1.1algebra

The graded Leibniz rule gives $d ⟨ A \land α ⟩ = ⟨ d A \land α ⟩ - ⟨ A \land d α ⟩$ . Since a two-form commutes with a one-form under wedge, $⟨ A \land d α ⟩ = ⟨ α \land d A ⟩ - d ⟨ A \land α ⟩$ . Substitute this into step 1.1, combine $2 d A + [A, A] = 2 F_{A}$ , and apply [F2] to obtain the stated Chern–Simons formula, including its boundary sign.

3.1A1F1step 2.1construct

For variations compactly supported in $int M$ , the boundary term in step 2.1 is zero. If stationarity holds for every such $α$ , then $\int_{M} ⟨ α \land F_{A} ⟩ = 0$ for every local test variation. If $F_{A}$ were nonzero at some point, nondegeneracy of the pairing would select an algebra element pairing nontrivially with one component; continuity preserves its sign on a smaller chart, and a nonnegative bump supported there would make the integral nonzero. Hence $F_{A} = 0$ on $int M$ .

3.2step 2.1algebra

For constant tangent vectors $α_{1}, α_{2}$ on the affine space $A_{Σ}$ , differentiating the potential current $θ_{A} (α) = - λ ⟨ A \land α ⟩$ from step 2.1 gives $D_{α_{1}} Θ (α_{2}) = - λ \int_{Σ} ⟨ α_{1} \land α_{2} ⟩$ and $D_{α_{2}} Θ (α_{1}) = - λ \int_{Σ} ⟨ α_{2} \land α_{1} ⟩$ . The field-space exterior derivative is their difference; antisymmetry of the wedge gives $Ω_{Σ} = - 2 λ \int_{Σ} ⟨ α_{1} \land α_{2} ⟩$ . Its coefficients are constant, so its field-space exterior derivative is zero.

4.1A1F1F2step 1.2step 2.1step 3.1algebra

Invariance of the pairing makes $D_{A}$ obey the graded Leibniz rule on paired forms. Therefore $d ⟨ B \land α ⟩ = ⟨ D_{A} B \land α ⟩ - ⟨ B \land D_{A} α ⟩$ . As in step 2.1, the two-form $D_{A} B$ commutes under wedge with $α$ . Apply [F2] to obtain the stated BF formula. For variations supported in the interior the boundary integral vanishes. Independent compactly supported $β$ and $α$ then test $F_{A}$ and $D_{A} B$ separately; the local nondegenerate-pairing bump argument in step 3.1 shows each must vanish.

4.2A1F1F2step 3.2algebra

At a flat $a$ , $D_{a}^{2} ϵ = [F_{a}, ϵ] = 0$ , so $D_{a} ϵ$ is tangent to the flat locus. For a tangent $α$ , $D_{a} α = 0$ . Invariance of the pairing yields $d ⟨ ϵ, α ⟩ = ⟨ D_{a} ϵ \land α ⟩ + ⟨ ϵ, D_{a} α ⟩$ . Integrating and applying [F2] on $Σ$ gives $\int_{Σ} ⟨ D_{a} ϵ \land α ⟩ = \int_{\partial Σ} ⟨ ϵ, α ⟩$ , which proves the gauge-contraction formula. If $ϵ ∣_{\partial Σ} = 0$ , this contraction vanishes for every tangent $α$ . If $ϵ$ is fixed under variation, direct differentiation of $Q_{ϵ} (a) = - 2 λ \int_{\partial Σ} ⟨ ϵ, a ⟩$ gives exactly the same contraction.

5.1step 4.2constructalgebra∎

For $λ \neq 0$ , nonzero boundary parameters can in fact give nonzero contractions. Take $g = R$ with pairing multiplication, the unit disk $Σ \subset R^{2}$ with its standard orientation, $a = 0$ , $ϵ (x, y) = x$ , and $α = d y$ . Then $F_{a} = 0$ , $D_{a} α = d (d y) = 0$ , and $D_{a} ϵ = d x$ . Polar coordinates give $\int_{Σ} d x \land d y = \int_{0}^{2 π} \int_{0}^{1} r d r d θ = π$ , so $Ω_{Σ} (D_{a} ϵ, α) = - 2 λ π \neq 0$ when $λ \neq 0$ . This establishes the stated possibility without asserting that every nonzero boundary parameter has a nonzero charge. If $λ = 0$ , the formulas for $Θ_{Σ}$ , $Ω_{Σ}$ and $Q_{ϵ}$ all have the common factor $λ$ and vanish; hence every gauge direction is null in this case.

Chern–Simons boundary variation and BF equations

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources