Chirality and self-duality in the Clifford algebra of an oriented four-space #
Let V be an oriented real inner product space of dimension four and Q a quadratic form on V
with Q v = -‖v‖², so that in CliffordAlgebra Q the generators satisfy the Clifford relation
c(v) c(w) + c(w) c(v) = -2 ⟪v, w⟫ of the spinor representations of the Seiberg–Witten equations.
The chirality element is Γ = -c(e₁) c(e₂) c(e₃) c(e₄) for a positively oriented orthonormal
basis e. It is defined here without a frame, as minus the image of the unit volume element of
⋀⁴ V under the quantization map CliffordAlgebra.equivExterior, and chirality_eq recovers
the displayed formula for every positively oriented orthonormal basis; this is the frame
independence of Γ. Moreover Γ² = 1, Γ anticommutes with every generator, and Γ is fixed
by reversal and by the grade involution. The first fact splits every Clifford module over ℝ into
the ±1 eigenspaces W⁺ and W⁻ of Γ, the second makes the generators exchange them, and the
last is the algebraic input for Γ to act self-adjointly on a Hermitian Clifford module; that
Hermitian statement is not made here.
Two-forms act through the Clifford bivector map
c₂ = CliffordAlgebra.bivectorExterior Q : ⋀² V → CliffordAlgebra Q, which sends eᵢ ∧ eⱼ to
c(eᵢ) c(eⱼ) for i ≠ j. The central identity of this file is the self-duality relation
c₂(⋆α) = Γ c₂(α),
where ⋆ is the Hodge star of the oriented space (Orientation.hodgeStar). As c₂(α) commutes
with Γ, anti-self-dual two-forms (⋆α = -α) act by zero on W⁺ and self-dual ones act by zero
on W⁻: on W⁺ only Λ⁺ survives, which is what lets the curvature equation of the
Seiberg–Witten equations be written as an equation between self-dual two-forms and trace-free
endomorphisms of W⁺.
The quantitative form of this identification is the anticommutator formula
c₂(α) c₂(β) + c₂(β) c₂(α) = -2 ⟪α, β⟫ - 2 ⟪⋆α, β⟫ Γ,
for the inner product of ⋀² V induced by that of V. For self-dual α the right-hand side is
-2 ⟪α, β⟫ (1 + Γ), so on W⁺ the two-forms anticommute up to -4 ⟪α, β⟫ and a self-dual
α squares to -2 ‖α‖². In particular self-dual two-forms act faithfully on every nonzero
positive spinor. Taking traces over a positive half-spinor space of complex dimension two, this is
the normalization ⟪C(iα), C(iβ)⟫ = 4 ⟪α, β⟫ of the Clifford map C from imaginary self-dual
two-forms to trace-free endomorphisms of W⁺; that trace statement is not made here.
Finally, Γ is natural: an isometry of the quadratic forms that carries o to o' carries the
chirality element of o to that of o'. Hence a map of Clifford modules that intertwines the
generators along an orientation-preserving isometry intertwines the chirality elements, and an
equivalence doing so restricts to the positive half-spinors. These are the frame changes and
Spinᶜ transformations under which the Seiberg–Witten equations are invariant. The same holds for
a semilinear map commuting with the generators whose twist fixes the real scalars, such as a
conjugate-linear map of complex Clifford modules; a conjugate-linear isometric equivalence of
Hermitian Clifford modules thus restricts to the positive half-spinors, as charge conjugation of
spinors does.
Main definitions #
CliffordAlgebra.chirality Q o: the chirality elementΓofCliffordAlgebra Q.CliffordAlgebra.positiveHalfSpinors o ρ: the positive half-spinorsW⁺, the+1eigenspace ofΓacting on a Clifford moduleρ : CliffordAlgebra Q → End_K W.CliffordAlgebra.positiveHalfSpinorsEquiv o ρ o' ρ' U hU: the restrictionW⁺ ≃ W'⁺of an equivalence of Clifford modulesUthat intertwines the chirality elements.CliffordAlgebra.positiveHalfSpinorsConjEquiv o ρ ρ' j hj: the restrictionW⁺ ≃ W'⁺of a conjugate-linear isometric equivalencejof Hermitian Clifford modules commuting with the generators.
Main results #
CliffordAlgebra.chirality_eq_neg_volumeElement:Γis minus the volume elementCliffordAlgebra.volumeElement Q oof the oriented four-space, andCliffordAlgebra.ofBaseChangeAux_chirality: its image in the complexified Clifford algebra is the normalized complex volume elementCliffordAlgebra.complexVolume Q o.CliffordAlgebra.chirality_eq:Γ = -c(e₁) c(e₂) c(e₃) c(e₄)for every positively oriented orthonormal basise.CliffordAlgebra.chirality_mul_self:Γ² = 1.CliffordAlgebra.chirality_mul_ι:Γ c(v) = -c(v) Γ, andCliffordAlgebra.ofBaseChangeAux_chirality_mul_ι: the same for the image ofΓin the Clifford algebra of a base change, against every vector ofA ⊗[ℝ] V.CliffordAlgebra.reverse_chirality,CliffordAlgebra.involute_chirality:Γis fixed by reversal and by the grade involution.CliffordAlgebra.chirality_mul:Γ x = involute(x) Γ, andCliffordAlgebra.commute_chirality_of_mem_even:Γcommutes with every even element.CliffordAlgebra.commute_chirality_bivectorExterior: two-forms commute withΓ, andCliffordAlgebra.mapsTo_positiveHalfSpinors: such elements preserveW⁺.CliffordAlgebra.bivectorExterior_hodgeStar: the self-duality relationc₂(⋆α) = Γ c₂(α).CliffordAlgebra.bivectorExterior_smul_eq_zero_of_hodgeStar_eq_negandCliffordAlgebra.bivectorExterior_smul_eq_zero_of_hodgeStar_eq_self: anti-self-dual two-forms annihilateW⁺, and self-dual two-forms annihilateW⁻.CliffordAlgebra.bivectorExterior_mul_bivectorExterior_add_swap: the anticommutator formulac₂(α) c₂(β) + c₂(β) c₂(α) = -2 ⟪α, β⟫ - 2 ⟪⋆α, β⟫ Γ, andCliffordAlgebra.bivectorExterior_mul_self:c₂(α)² = -‖α‖² - ⟪⋆α, α⟫ Γ.CliffordAlgebra.bivectorExterior_mul_bivectorExterior_add_swap_of_hodgeStar_eq_selfandCliffordAlgebra.bivectorExterior_mul_bivectorExterior_add_swap_of_hodgeStar_eq_neg: the anticommutator of a self-dual (anti-self-dual) two-form is-2 ⟪α, β⟫ (1 ± Γ).CliffordAlgebra.bivectorExterior_smul_bivectorExterior_smul_add_swap_of_hodgeStar_eq_selfandCliffordAlgebra.bivectorExterior_smul_bivectorExterior_smul_self_of_hodgeStar_eq_self: on positive spinors a self-dualαsatisfiesc₂(α) c₂(β) + c₂(β) c₂(α) = -4 ⟪α, β⟫andc₂(α)² = -2 ‖α‖²; the_of_hodgeStar_eq_negversions are the same on negative spinors for anti-self-dualα.CliffordAlgebra.bivectorExterior_smul_eq_zero_iff_of_hodgeStar_eq_selfandCliffordAlgebra.bivectorExterior_smul_eq_zero_iff_of_hodgeStar_eq_neg: self-dual two-forms act faithfully on nonzero positive spinors, and anti-self-dual ones on nonzero negative spinors.CliffordAlgebra.map_chirality: the algebra map induced by an isometryeof the quadratic forms sends the chirality element ofoto that of the transported orientation.CliffordAlgebra.comp_chirality_eq_chirality_comp: a map of Clifford modules intertwining the generators along an orientation-preserving linear isometry intertwines the chirality elements.CliffordAlgebra.map_mem_positiveHalfSpinors: a semilinear map of Clifford modules commuting with the generators, whose twist fixes the real scalars, preserves the positive half-spinors.
References #
- H. B. Lawson and M.-L. Michelsohn, Spin Geometry (1989), Chapter I, §5 and Chapter IV, §1.
- J. W. Morgan, The Seiberg–Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes 44, Princeton University Press, 1996, §3.1–§4.1.
The dimension Fact makes the oriented four-space finite-dimensional.
The chirality element Γ of the Clifford algebra of an oriented four-dimensional real inner
product space: minus the quantization of the unit volume element of ⋀⁴ V. When
Q v = -‖v‖², it equals -c(e₁) c(e₂) c(e₃) c(e₄) for every positively oriented orthonormal
basis e (CliffordAlgebra.chirality_eq).
Instances For
Unfolding lemma for CliffordAlgebra.chirality.
The chirality element is minus the volume element CliffordAlgebra.volumeElement Q o of the
oriented four-space.
The chirality element is the normalized complex volume element: the image of Γ in the
complexified Clifford algebra is CliffordAlgebra.complexVolume Q o = i ^ 2 c(e₁) ⋯ c(e₄).
Frame independence of the chirality element: Γ = -c(e₁) c(e₂) c(e₃) c(e₄) for every
positively oriented orthonormal basis e.
The chirality element is an involution: Γ² = 1.
The chirality element anticommutes with every generator: Γ c(v) = -c(v) Γ.
The base-changed chirality element anticommutes with every vector: for a commutative
ℝ-algebra A, the image of Γ in CliffordAlgebra (Q.baseChange A) anticommutes with ι z
for every z : A ⊗[ℝ] V.
The chirality element is fixed by reversal. Together with involute_chirality, it is fixed by
the Clifford conjugation involute ∘ reverse.
The chirality element is even: it is fixed by the grade involution.
Γ x = involute(x) Γ for every Clifford element x: the chirality element commutes with
even elements and anticommutes with odd ones.
Even Clifford elements commute with the chirality element.
Two-forms commute with the chirality element, since Γ anticommutes with each of the two
generators in a bivector.
The self-duality relation: the Clifford action of the Hodge dual of a two-form is the
chirality element times the action of the two-form, c₂(⋆α) = Γ c₂(α).
An anti-self-dual two-form α, ⋆α = -α, satisfies c₂(α) Γ = -c₂(α).
A self-dual two-form α, ⋆α = α, satisfies c₂(α) Γ = c₂(α).
The anticommutator of two two-forms #
The anticommutator of two two-forms in the Clifford algebra of an oriented four-space:
c₂(α) c₂(β) + c₂(β) c₂(α) = -2 ⟪α, β⟫ - 2 ⟪⋆α, β⟫ Γ. Since α ∧ β = ⟪⋆α, β⟫ vol, the
coefficient of Γ is minus twice the volume of α ∧ β.
The square of a two-form: c₂(α)² = -‖α‖² - ⟪⋆α, α⟫ Γ.
For a self-dual two-form α, ⋆α = α, the anticommutator is
c₂(α) c₂(β) + c₂(β) c₂(α) = -2 ⟪α, β⟫ (1 + Γ).
For an anti-self-dual two-form α, ⋆α = -α, the anticommutator is
c₂(α) c₂(β) + c₂(β) c₂(α) = -2 ⟪α, β⟫ (1 - Γ).
Anti-self-dual two-forms annihilate positive spinors: if Γ ψ = ψ and ⋆α = -α, then
c₂(α) ψ = 0.
Self-dual two-forms annihilate negative spinors: if Γ ψ = -ψ and ⋆α = α, then
c₂(α) ψ = 0.
Self-dual two-forms anticommute on positive spinors up to their inner product: if
Γ ψ = ψ and ⋆α = α, then c₂(α) c₂(β) ψ + c₂(β) c₂(α) ψ = -4 ⟪α, β⟫ ψ.
Anti-self-dual two-forms anticommute on negative spinors up to their inner product: if
Γ ψ = -ψ and ⋆α = -α, then c₂(α) c₂(β) ψ + c₂(β) c₂(α) ψ = -4 ⟪α, β⟫ ψ.
A self-dual two-form squares to -2 ‖α‖² on positive spinors: if Γ ψ = ψ and ⋆α = α,
then c₂(α) c₂(α) ψ = -2 ‖α‖² ψ.
An anti-self-dual two-form squares to -2 ‖α‖² on negative spinors: if Γ ψ = -ψ and
⋆α = -α, then c₂(α) c₂(α) ψ = -2 ‖α‖² ψ.
Self-dual two-forms act faithfully on every nonzero positive spinor: if Γ ψ = ψ,
ψ ≠ 0 and ⋆α = α, then c₂(α) ψ = 0 only for α = 0.
Anti-self-dual two-forms act faithfully on every nonzero negative spinor: if
Γ ψ = -ψ, ψ ≠ 0 and ⋆α = -α, then c₂(α) ψ = 0 only for α = 0.
The positive half-spinors W⁺ = {ψ | Γ ψ = ψ} of a Clifford module W.
Instances For
A vector is a positive half-spinor exactly when Γ ψ = ψ.
Clifford elements commuting with Γ, such as two-forms, preserve W⁺.
Naturality under oriented isometries #
Naturality of the chirality element: when Q v = -‖v‖² and Q' v = -‖v‖², the algebra
map induced by an isometry e of the quadratic forms sends the chirality element of o to that
of the transported orientation.
A map U of Clifford modules that intertwines the generators along an orientation-preserving
linear isometry f, U c(v) = c(f v) U, intertwines the chirality elements: U Γ = Γ' U.
An equivalence U of Clifford modules that intertwines the chirality elements, U Γ = Γ' U,
restricts to an equivalence of the positive half-spinors W⁺ ≃ W'⁺. By
CliffordAlgebra.comp_chirality_eq_chirality_comp, this applies to every equivalence that
intertwines the generators along an orientation-preserving isometry.
Instances For
The restriction of U to the positive half-spinors is U.
The inverse of the restriction of U to the positive half-spinors is U⁻¹.
Semilinear intertwiners #
A semilinear map U of Clifford modules over the same quadratic space, whose twist σ fixes
the real scalars and which commutes with the generators, U (c(v) w) = c(v) (U w), maps positive
half-spinors to positive half-spinors. For complex modules U can be conjugate-linear, as for
charge conjugation of spinors.
Charge conjugation of positive half-spinors: a conjugate-linear isometric equivalence j
of Hermitian Clifford modules commuting with Clifford multiplication restricts to a
conjugate-linear isometric equivalence W⁺ ≃ W'⁺ of the positive half-spinors.
Instances For
The restriction of j to the positive half-spinors is j.
The inverse of the restriction of j to the positive half-spinors is j⁻¹.