Covariant and contravariant spinors
Let V be a real or complex vector space with a non-degenerate symmetric bilinear form B, and let q: ∧(V)→Cl(V) be the quantization map. The space q(∧2(V)) is a Lie subalgebra under commutation in the Clifford algebra. This subspace is canonically isomorph
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Covariant and contravariant spinors
Suppose V is a quadratic vector space. In Section 2.2.10 we defined a map λ : o(V ) → ∧2 (V ), which is a Lie algebra homomorphism relative to the Poisson bracket on ∧(V ). We also considered its quantization γ = q ◦ λ : o(V ) → Cl(V ), which is a Lie algebra homomorphism relative to the Clifford commutator. One of the problems addressed in this chapter is to give explicit formulas for the Clifford exponential exp(γ (A)) ∈ Cl(V ). We will compute its image under the symbol map, and express its relation to the exterior algebra exponential exp(λ(A)). These questions will be studied using the spin representation for W = V ∗ ⊕ V , with bilinear form given by the pairing. (In particular, the bilinear form on V itself need not be split, and may even be degenerate).
4.1 Pull-backs and push-forwards of spinors Let V be any vector space, and let W = V ∗ ⊕ V , equipped with the bilinear form 1 BW ((μ1 , v1 ), (μ2 , v2 )) = (μ1 , v2 + μ2 , v1 ). 2
(4.1)
We will occasionally use a basis e1 , . . . , em of V , with dual basis f 1 , . . . , f m of V ∗ . j Thus BW (ei , f j ) = 12 δi , and the Clifford relations in Cl(W ) read, in terms of super commutators, as j
[f i , f j ] = 0, [ei , f j ] = δi , [ei , ej ] = 0. We define the standard or contravariant spinor module to be ∧(V ∗ ), with generators μ ∈ V ∗ acting by exterior multiplication and v ∈ V acting by contraction. We will also consider the dual or covariant spinor module ∧(V ), with generators v ∈ V acting by exterior multiplication and μ ∈ V ∗ acting by contraction. Recall (cf. Section 3.4) that there is a canonical isomorphism of Clifford modules, ∧(V ∗ ) ∼ = ∧(V ) ⊗ det(V ∗ ), E. Meinrenken, Clifford Algebras and Lie Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 58, DOI 10.1007/978-3-642-36216-3_4, © Springer-Verlag Berlin Heidelberg 2013
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Covariant and contravariant spinors
defined by contraction. The choice of a generator Γ∧ ∈ det(V ∗ ) gives an isomorphism, called the “star operator” for the volume form Γ∧ ∗Γ∧ : ∧(V ) → ∧(V ∗ ), χ → ι(χ)Γ∧ . Thus ∗Γ∧ ◦ ε(v) = ι(v) ◦ ∗Γ∧ , ∗Γ∧ ◦ ι(μ) = ε(μ) ◦ ∗Γ∧ . In Proposition 3.11, we saw that the most general contravariant pure spinor is of the form φ = exp(−ω)κ,
(4.2)
where ω ∈ ∧2 V ∗ is a 2-form, and κ ∈ det(ann(N ))× is a volume form on V /N , for some subspace N ⊆ V . Given κ, the form φ only depends on the restriction ωN ∈ ∧2 (N ∗ ) of ω to N . By reversing the roles of V and V ∗ , the most general covariant spinor is of the form χ = exp(−π)ν,
(4.3)
where π ∈ ∧2 (V ) and ν ∈ det(S)× , for some subspace S ⊆ V . Given ν, the spinor χ only depends on the image πS ∈ ∧2 (V /S) of π . The corresponding Lagrangian subspace is F (e−π ν) = {(μ, v) ∈ V ∗ ⊕ V | μ ∈ ann(S), π(μ, ·) − v ∈ S}.
(4.4)
Note that S = F (e−π ν) ∩ V , while ann(S) is characterized as projection of F (e−π ν) to V ∗ . For a linear map Φ : V1 → V2 we denote by Φ∗ = ∧(Φ) : ∧V1 → ∧V2 the “push-forward” map, and by Φ ∗ = ∧(Φ ∗ ) : ∧V2∗ → ∧V1
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