The gauge-fixed vortex operator at the vacuum #
The vortex equations on the plane, for an imaginary connection a on the trivial line bundle and
a section α, are ∂bar_a α = 0 and *iF_a = (1 - |α|²) / 8; the gauge group acts by
(a, α) ↦ (a - u⁻¹ du, u α). Identify the plane with ℂ, with coordinates z = s + i t, and write
∂₁ = ∂_{1} and ∂₂ = ∂_{I} for the coordinate derivatives. A first-order deformation of the
vacuum (0, 1) is written a = i (b₁ ds + b₂ dt) and α = 1 + 2 p + 2 i q, so that the
coefficient metric |b|² + |δα|² / 4 becomes the Euclidean metric on x = (b₁, b₂, p, q) ∈ ℝ⁴. In
these coordinates the gauge-fixed linearization of the vortex equations at the vacuum is the
first-order operator L₀ x = (G, C, H₁, H₂) with
G = ∂₁ b₁ + ∂₂ b₂ + q / 2(the formal adjoint of the infinitesimal gauge action),C = -∂₁ b₂ + ∂₂ b₁ + p / 2(the linearized curvature equation),H₁ = ∂₁ p - ∂₂ q - b₂ / 2andH₂ = ∂₁ q + ∂₂ p + b₁ / 2(the real and imaginary parts of the linearized∂bar_a α).
This file defines L₀ and its formal L² adjoint L₀* as continuous linear operators on
ℝ⁴-valued Schwartz fields on ℂ. It proves the Weitzenböck identities
L₀* L₀ = L₀ L₀* = -Δ + 1 / 4 and the energy identities
‖L₀ x‖₂² = ‖L₀* x‖₂² = ‖∇x‖₂² + ‖x‖₂² / 4, for any additive Haar measure on ℂ, together with
their consequences ‖x‖₂ ≤ 2 ‖L₀ x‖₂, ‖x‖₂ ≤ 2 ‖L₀* x‖₂ and the injectivity of L₀ and L₀* on
Schwartz fields. Combined with the inverse (-Δ + 1 / 4)⁻¹ of the shifted Laplacian
(SchwartzMap.laplacianResolvent), the Weitzenböck identities show that L₀ and L₀* are
continuous linear automorphisms of the space of Schwartz fields, with inverses
L₀⁻¹ = L₀* (-Δ + 1 / 4)⁻¹ and (L₀*)⁻¹ = L₀ (-Δ + 1 / 4)⁻¹. Since (-Δ + 1 / 4)⁻¹ multiplies
the Fourier transform by (4π² |ξ|² + 1 / 4)⁻¹, the factor 4π² coming from Mathlib's
normalization of the Fourier transform, L₀⁻¹ inverts the symbol of L₀ as
L₀(ξ)* / (4π² |ξ|² + 1 / 4). Both inverses are bounded by 2 in L², the gap
1 / 2 of the vacuum. These are the vacuum inputs for the linear analysis of vortices, whose
exterior regions are modelled on the vacuum.
Main declarations #
TauCeti.Vortex.vacuumOperator: the vacuum gauge-fixed operatorL₀, characterized by the component formulasTauCeti.Vortex.vacuumOperator_apply_zerothroughTauCeti.Vortex.vacuumOperator_apply_three.TauCeti.Vortex.vacuumAdjoint: its formal adjointL₀*, characterized byTauCeti.Vortex.vacuumAdjoint_apply_zerothroughTauCeti.Vortex.vacuumAdjoint_apply_three.TauCeti.Vortex.integral_inner_vacuumOperatorandTauCeti.Vortex.integral_inner_vacuumAdjoint:∫ ⟪L₀ x, y⟫ = ∫ ⟪x, L₀* y⟫and∫ ⟪L₀* x, y⟫ = ∫ ⟪x, L₀ y⟫.TauCeti.Vortex.vacuumAdjoint_vacuumOperatorandTauCeti.Vortex.vacuumOperator_vacuumAdjoint:L₀* L₀ = -Δ + 1 / 4andL₀ L₀* = -Δ + 1 / 4.TauCeti.Vortex.integral_norm_sq_vacuumOperator:‖L₀ x‖₂² = ‖∂₁ x‖₂² + ‖∂₂ x‖₂² + ‖x‖₂² / 4, andTauCeti.Vortex.integral_norm_sq_vacuumAdjoint: the same identity forL₀*.TauCeti.Vortex.integral_norm_sq_le_four_mul_integral_norm_sq_vacuumOperatorandTauCeti.Vortex.vacuumOperator_injective: the resulting lower bound, and triviality of the kernel on Schwartz fields. Their analogues forL₀*areTauCeti.Vortex.integral_norm_sq_le_four_mul_integral_norm_sq_vacuumAdjointandTauCeti.Vortex.vacuumAdjoint_injective.TauCeti.Vortex.vacuumOperatorEquivandTauCeti.Vortex.vacuumAdjointEquiv:L₀andL₀*as continuous linear automorphisms of the Schwartz fields, with inverses given byTauCeti.Vortex.vacuumOperatorEquiv_symm_applyandTauCeti.Vortex.vacuumAdjointEquiv_symm_apply.TauCeti.Vortex.integral_norm_sq_vacuumOperatorEquiv_symm_leandTauCeti.Vortex.integral_norm_sq_vacuumAdjointEquiv_symm_le:‖L₀⁻¹ y‖₂ ≤ 2 ‖y‖₂and‖(L₀*)⁻¹ y‖₂ ≤ 2 ‖y‖₂.
References #
- A. Jaffe and C. H. Taubes, Vortices and Monopoles, Progress in Physics 2, Birkhäuser, 1980.
- C. H. Taubes, SW ⇒ Gr: from the Seiberg–Witten equations to pseudo-holomorphic curves, J. Amer. Math. Soc. 9 (1996), 845--918.
The gauge-fixed vortex operator L₀ at the vacuum, acting on deformations
x = (b₁, b₂, p, q) and producing (G, C, H₁, H₂), where G is the gauge-fixing condition, C
the linearized curvature equation and (H₁, H₂) the linearized ∂bar_a α; see the component
formulas vacuumOperator_apply_zero through vacuumOperator_apply_three.
Instances For
The formal L² adjoint L₀* of the vacuum operator, with respect to the Euclidean metrics on
the coordinates (b₁, b₂, p, q) and (G, C, H₁, H₂); see integral_inner_vacuumOperator and the
component formulas vacuumAdjoint_apply_zero through vacuumAdjoint_apply_three.
Instances For
The gauge component G = ∂₁ b₁ + ∂₂ b₂ + q / 2 of L₀.
The curvature component C = ∂₂ b₁ - ∂₁ b₂ + p / 2 of L₀.
The first ∂bar component H₁ = ∂₁ p - ∂₂ q - b₂ / 2 of L₀.
The second ∂bar component H₂ = ∂₂ p + ∂₁ q + b₁ / 2 of L₀.
The b₁ component -∂₁ G - ∂₂ C + H₂ / 2 of L₀* (G, C, H₁, H₂).
The b₂ component -∂₂ G + ∂₁ C - H₁ / 2 of L₀* (G, C, H₁, H₂).
The p component -∂₁ H₁ - ∂₂ H₂ + C / 2 of L₀* (G, C, H₁, H₂).
The q component ∂₂ H₁ - ∂₁ H₂ + G / 2 of L₀* (G, C, H₁, H₂).
The vacuum operator has constant coefficients, so it commutes with directional differentiation.
The formal adjoint has constant coefficients, so it commutes with directional differentiation.
The Weitzenböck identity at the vacuum: L₀* L₀ = -Δ + 1 / 4.
The reverse Weitzenböck identity at the vacuum: L₀ L₀* = -Δ + 1 / 4.
vacuumAdjoint is the formal L² adjoint of vacuumOperator:
∫ ⟪L₀ x, y⟫ = ∫ ⟪x, L₀* y⟫ for Schwartz fields x and y.
vacuumOperator is the formal L² adjoint of vacuumAdjoint:
∫ ⟪L₀* x, y⟫ = ∫ ⟪x, L₀ y⟫ for Schwartz fields x and y.
The vacuum energy identity: ‖L₀ x‖₂² = ‖∇x‖₂² + ‖x‖₂² / 4, where ‖∇x‖₂² is the sum of
the squared L² norms of the two coordinate derivatives ∂₁ x and ∂₂ x.
The energy identity for the adjoint: ‖L₀* x‖₂² = ‖∇x‖₂² + ‖x‖₂² / 4.
The vacuum lower bound: ‖x‖₂² ≤ 4 ‖L₀ x‖₂², that is, ‖x‖₂ ≤ 2 ‖L₀ x‖₂.
The lower bound for the adjoint: ‖x‖₂² ≤ 4 ‖L₀* x‖₂², that is, ‖x‖₂ ≤ 2 ‖L₀* x‖₂.
The vacuum operator has trivial kernel on Schwartz fields.
The adjoint of the vacuum operator has trivial kernel on Schwartz fields.
The inverses of the vacuum operator and its adjoint #
The vacuum operator is invertible: L₀ as a continuous linear automorphism of the Schwartz
fields. Its inverse is L₀⁻¹ = L₀* (-Δ + 1 / 4)⁻¹ (vacuumOperatorEquiv_symm_apply). Since
(-Δ + 1 / 4)⁻¹ multiplies the Fourier transform by (4π² |ξ|² + 1 / 4)⁻¹, this inverts the
symbol of L₀ as L₀(ξ)* / (4π² |ξ|² + 1 / 4).
Instances For
The adjoint of the vacuum operator is invertible: L₀* as a continuous linear automorphism
of the Schwartz fields. Its inverse is (L₀*)⁻¹ = L₀ (-Δ + 1 / 4)⁻¹
(vacuumAdjointEquiv_symm_apply).
Instances For
The inverse of the vacuum operator is L₀⁻¹ = L₀* (-Δ + 1 / 4)⁻¹.
The inverse of the adjoint of the vacuum operator is (L₀*)⁻¹ = L₀ (-Δ + 1 / 4)⁻¹.
The vacuum gap for the inverse: ‖L₀⁻¹ y‖₂² ≤ 4 ‖y‖₂², that is, ‖L₀⁻¹ y‖₂ ≤ 2 ‖y‖₂.
The vacuum gap for the inverse of the adjoint: ‖(L₀*)⁻¹ y‖₂ ≤ 2 ‖y‖₂.