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An Introduction to the Dirac Operator in Riemannian Geometry S. Montiel

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An Introduction to the Dirac Operator in Riemannian Geometry S. Montiel An Introduction to the Dirac Operator in Riemannian Geometry S. Montiel Universidad de Granada Varna, June 2007 1 Genealogy of the Dirac Operator • 1913 E.´ Cartan Orthogonal Lie algebras • 1927 W. Pauli Inner angular momentum (spin) of electrons • 1928 P.A.M. Dirac Dirac operator and quantum-relativistic description of electrons • 1930 H. Weyl Wave functions of neutrinos • 1937 E.´ Cartan Insurmountables difficulties to talk about spinors on manifolds • 1963 M. Atiyah and I. Singer Dirac operator on a spin Riemannian manifold 2 Genealogy of the Dirac Operator • 1963 A. Lichnerowicz (maybe I. Singer in the last 50’s) Topological obstruction for positive scalar curvature on compact spin manifolds • 1974 N. Hitchin The dimension of the space of harmonic spinors is a conformal invariant and existence of parallel spinors implies special holonomy • 1980 M. Gromov and B. Lawson More topological obstruc- tions for complete metrics with non-negative scalar cur- vature • 1981 E. Witten An elemental spinorial proof of the Schoen and Yau positive mass theorem • 1995 E. Witten Seiberg–Witten ⇒ Donaldson 3 The Wave Equation (1850-1905) • Wave equation of Maxwell and Special Relativity theories 1 ∂2u O ⊂ R3 u : O × R −→ R u = + ∆u = 0 c2 ∂t2 u(p, t) = X f(t)φ(p) f 00 + λf = 0 ∆φ − λφ = 0 where ∆ = − ∂2 − ∂2 − ∂2 and is the ratio between electro- ∂x2 ∂y2 ∂z2 c static and electrodynamic units of charge • Second order in time and space coordinates • Invariant under Lorentz transformations O(1, 3) = {A ∈ GL(4, R) | AGAt = G}, G = diag (−1, 1, 1, 1) 4 The Schr¨odinger Equation (1926) • Schr¨odinger equation of the non-relativistic Quantum Me- chanics ∂ψ O ⊂ R3 ψ : O × R −→ C −i + ∆ψ = 0 ∂t ψ(p, t) = X f(t)φ(p) f 0 + iλf = 0 ∆φ − λφ = 0 • Invariant under Galileo transformations 1 0 R3 · O(3) = {A ∈ GL(4, R) | A = , v ∈ R3, A ∈ O(3)} v A • First order in time and second order in space coordinates • Complex values 5 The Classical Dirac Operator • 1928 P.A.M. Dirac, 1930 H. Weyl Look for an equation of first order in all the variables, like this 3 i ∂ψ ∂ψ + Dψ = 0 Dψ = X γi c ∂t ∂x i=1 i whose iteration on solutions gives the wave equation. This holds iff 2 D = ∆ ⇐⇒ γiγj + γjγi = −2δij for i, j = 1, 2, 3. For example, these Pauli matrices i 0 0 1 0 i γ = γ = γ = 1 0 −i 2 −1 0 3 i 0 6 The Classical Dirac Operator and Spinor Fields 0 −1 0 • Other possible Pauli matrices γi = P γiP or γi = −γi. • Two essentially different (chirality) Dirac-Weyl equations i ∂ψ i 0 ∂ψ 0 1 ∂ψ 0 i ∂ψ ± + + + = 0 c ∂t 0 −i ∂x −1 0 ∂y i 0 ∂z • Spinor fields ψ : O × R −→ C2 expand into series ψ(p, t) = X f(t)φ(p) f 0 + iλf = 0 i 0 ∂φ 0 1 ∂φ 0 i ∂φ Dφ = + + = −λφ 0 −i ∂x −1 0 ∂y i 0 ∂z 7 The Classical Dirac Operator • To define the Dirac operator in terms of any other orthonor- mal basis {e1, e2, e3} as D = γ(e1)∇e1 + γ(e2)∇e2 + γ(e3)∇e3 we need Pauli matrices for all directions v ∈ R3. Put iv1 v2 + iv3 γ(v) = γ(v1, v2, v3) = v1γ1 + v2γ2 + v3γ3 = −v2 + iv3 −iv1 • A Lie algebra isomorphism 1 γ : (R3 = o(3), ∧) → (su(2), [ , ]) 2 • Clifford relations 3 γ(u)γ(v) + γ(v)γ(u) = −2hu, viI2, ∀u, v ∈ R 8 What Are These Spinor Fields? • For each A ∈ SU(2) there is a unique real matrix ρ(A) ∈ MR(3) such that γ(ρ(A)v) = Aγ(v)A¯t ∀v ∈ R3 • See that ρ(A) ∈ SO(3) and the map ρ : SU(2) → SO(3) is a two-sheeted (universal) covering group homomorphism • Surjective: given R ∈ SO(3), put R = s1 ◦ s2 and prove that A = γ(v1)γ(v2) ∈ SU(2) and that ρ(A) = R • Kernel: if A ∈ ker ρ then A commutes with all Pauli matrices and so A = ±I2 9 What Are These Spinor Fields? • Let φ : O → C2 be a spinor and A ∈ SU(2). Consider the open set O0 = ρ(A)t(O) and define ψ : O0 → C2, ψ(p) = A¯tφ(ρ(A)p), ∀p ∈ O0 (Dψ)(p) = 3 γ(e )(∇ ψ)(p) = 3 γ(e )A¯t(∇ φ)(ρ(A)p) Pi=1 i ei Pi=1 i ρ(A)ei = 3 A¯tγ(ρ(A)e )(∇ φ)(ρ(A)p) = A¯t(Dφ)(ρ(A)p) Pi=1 i ρ(A)ei and so Dφ = λφ ⇔ Dψ = λψ • If spatial coordinates change through R ∈ SO(3), then com- ponents of spinors change through ρ−1(R) ∈ SU(2)(?) i θ e 2 0 • ρ −i θ = Rθ is a rotation of angle θ around the 0 e 2 x-axis 10 Speaking Bundle Language • 1963 Atiyah and Singer ? O ⊂ R3 is a chart domain of an oriented Riemannian three- manifold M ? φ : O → C2 is the local expression of a section of a complex vector bundle ΣM with fiber C2 associated to a virtual (?) principal bundle with structure group SU(2) ? Lift transition functions fij : Ui ∩ Uj → SO(3) to maps gij : Ui ∩ Uj → SU(2) and define hijk : Ui ∩ Uj ∩ Uk → Z2 = {+1, −1} according to gik = ±(gjkgij). This h is a cocycle and de- 2 fines the second Stiefel-Whitney class w2(M) ∈ H (M, Z2) 11 Speaking Bundle Language ? ΣM must have a Hermitian metric h , i and a covariant derivative ∇ which parallelizes the metric ? A bundle map γ : T M → EndC(ΣM) with γ(u)γ(v) + γ(v)γ(u) = −2hu, viI2 compatible with both h , i and ∇ called a Clifford multi- plication because it determines a complex representation of each Clifford algebra C`(TpM) on the space ΣpM ? In this frame, the Dirac operator is 3 Dψ = X γ(ei)∇eiψ i=1 where e1, e2, e3 is an orthonormal basis of TpM An Introduction to the Dirac Operator in Riemannian Geometry S. Montiel Universidad de Granada Varna, June 2007 12 Emergence Kit for Riemannian Geometry Notation • Let M be a Riemannian manifold, h , i the metric and ∇ the Levi-Civit`a connection • R will be the Riemannian curvature operator R(X, Y )Z = ∇X∇Y Z − ∇Y ∇XZ − ∇[X,Y ]Z, X, Y, Z ∈ Γ(T M) and also Riemannian curvature tensor of M, given by R(X, Y, Z, W ) = hR(X, Y )Z, W i, X, Y, Z, W ∈ Γ(T M) • The single and double contractions of this four-covariant tensor n n Ric(X, W ) = X R(X, ei, ei, W ) S = X R(ei, ej, ej, ei) i=1 i,j=1 are the Ricci tensor and the scalar curvature of M, respec- tively 13 Exterior Geometry for Riemannian Geometers ∗ n k • The exterior bundle Λ (M) = Lk=1 Λ (M) inherits the met- ric and the connection • Lemma 1 Music and products are parallel [ [ ] ] ∇X(Y ) = (∇XY ) , ∇X(α ) = (∇Xα) ∇X(ω ∧ η) = (∇Xω) ∧ η + ω ∧ (∇Xη) ∇X(Y y ω) = (∇XY )y ω + Y y (∇Xω) • Exterior product and inner product are adjoint each other hX[ ∧ ω, ηi = hω, Xy ηi • Riemannian expressions for an old friend and its adjoint n n [ ∧ ∇ − ∇ d = X ei ei δ = X eiy ei i=1 i=1 14 Hermitian Bundles • Let ΣM be a rank N complex vector bundle over M, h , i a Hermitian metric and ∇ a unitary connection with Xhψ, φi = h∇Xψ, φi + hψ, ∇Xφi ψ, φ ∈ Γ(ΣM), X ∈ Γ(T M) • The Levi-Civit`a connection allows us to perform second derivatives (∇2 )( ) = ∇ ∇ − ∇ ψ X, Y X Y ψ (∇X Y )ψ • The skew-symmetric part RΣM (X, Y )ψ = (∇2ψ)(X, Y ) − (∇2ψ)(Y, X) is tensorial in ψ. It is the curvature operator of (ΣM, ∇) • Skew-symmetry and Bianchi identity hRΣM (X, Y )ψ, φi = −hψ, RΣM (X, Y )φi ΣM ΣM ΣM (∇ZR )(X, Y ) + (∇Y R )(Z, X) + (∇XR )(Y, Z) = 0 15 Hermitian Bundles • As a consequence N ΣM ΣM α(X, Y ) = tr iR (X, Y ) = − XhR (X, Y )ψk, iψki k=1 is a closed two-form with 2πZ-periods • The first Chern class 1 2 c1(ΣM) = [ α] ∈ H (M, Z) 2π does not depend on the connection ∇ • When N = 1 (complex line bundles case) 1 1 2 c1 : (H (M, S ), ⊗) → (H (M, Z), +) is an isomorphism 16 The Rough Laplacian • Second derivatives allow to define the rough Laplacian n 2 2 ∆ : Γ(ΣM) → Γ(ΣM) ∆ψ = −tr ∇ ψ = − X(∇ ψ)(ei, ei) i=1 • It is an L2-symmetric non-negative operator, because Z h∆ψ, φi = Z h∇ψ, ∇φi M M for sections of compact support • It is an elliptic second order differential operator and so it has a real discrete non-bounded spectrum • When ΣM = M × C, ∆ is the usual Laplacian 17 Elliptic Differential Operators • Lemma 2 Let L : Γ(E) → Γ(F ) be an elliptic differential operator taking sections of a vector bundle E on sections of another vector bundle F on a compact Riemannian manifold M. Then both ker L and coker L are finite-dimensional and the index of L, defined by ind L = dim ker L − dim coker L = dim ker L − dim ker L∗, where L∗ : Γ(F ) → Γ(E) is the formal adjoint of L with respect to the L2-products, depends only on the homo- topy class of L.
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