Ivancevic_Applied-Diff-Geom

728 Applied Differential Geometry: A Modern IntroductionLet X be a manifold that admits a spinor bundle S. Let {e i } be a localorthonormal basis of sections of the tangent bundle T X and 〈 , 〉 be theHermitian structure as in the above definition of spinors. Then, for anysection ψ ∈ Γ(X, S), the expression 〈e i e j ψ, ψ〉 is purely imaginary at eachpoint x ∈ X. As a proof, by skew–adjointness of Clifford multiplicationand the fact that the basis is orthonormal,〈e i e j ψ, ψ〉 = − 〈e j ψ, e i ψ〉 = 〈ψ, e j e i ψ〉 − 〈ψ, e i e j ψ〉 = −〈e i e j ψ, ψ〉.Given a spinor bundle S over X, the Dirac operator on S is a first orderdifferential operator on the smooth sections D : Γ(X, S + ) → Γ(X, S − ),defined as the compositionD : Γ(X, S + ) ∇ → Γ(X, S + ) ⊗ T ∗ X g → Γ(X, S + ) ⊗ T X • → Γ(X, S − ),where the first map is the covariant derivative, with the Spin–connectioninduced by the Levi–Civita connection on X (see [Roe (1988)]), the secondis the Legendre transform given by the Riemannian metric, and the thirdis Clifford multiplication.It is easy to check (for details see [Roe (1988)]) that this corresponds tothe following expression in coordinates: Ds = e k · ∇ k s.An essential tool in Spin geometry, which is very useful in SW gaugetheory (see e.g. [Jost et. al. (1995); Kronheimer and Mrowka (1994a);Taubes (1994); Witten (1994)]), is the Weitzenböck formula.Given a smooth vector bundle E over a Spin c −-manifold X and aconnection A on E, the twisted Dirac operator D A : Γ(X, S + ⊗ E) →Γ(X, S − ⊗E) is the operator acting on a section s⊗e as the Dirac operatoron s and the composite of the covariant derivative ˜∇ A and the Cliffordmultiplication on e:D A : Γ(X, S + ⊗ E) ∇⊗1+1⊗ →˜∇ AΓ(X, S + ⊗ E ⊗ T ∗ X) →gg→ Γ(X, S + ⊗ E ⊗ T X) → • Γ(X, S − ⊗ E).The twisted Dirac operator D A satisfies the Weitzenböck formula:D 2 As = (∇ ∗ A∇ A + κ 4 + −i4 F A)s,where ∇ ∗ A is the formal adjoint of the covariant derivative with respectto the Spin−-connection on the Spinor bundle, and with respect to theconnection A on E, ∇ A = ∇ ⊗ 1 + 1 ⊗ ˜∇ A ; κ is the scalar curvature on