Dirac Operators and Spectral Geometry Joseph C. V´arilly Notes taken by Pawe lWitkowski January 2006 Contents Introduction and Overview 4 1 Clifford algebras and spinor representations 5 1.1 Clifford algebras . 5 1.2 The universality property . 6 1.3 The trace . 7 1.4 Periodicity . 8 1.5 Chirality . 10 1.6 Spinc and Spin groups . 10 1.7 The Lie algebra of Spin(V ) ............................ 12 1.8 Orthogonal complex structures . 14 1.9 Irreducible representations of Cl(V ) ....................... 15 1.10 Representations of Spinc(V )............................ 17 2 Spinor modules over compact Riemannian manifolds 18 2.1 Remarks on Riemannian geometry . 18 2.2 Clifford algebra bundles . 19 2.3 The existence of Spinc structures . 20 2.4 Morita equivalence for (commutative) unital algebras . 22 2.5 Classification of spinor modules . 23 2.6 The spin connection . 26 2.7 Epilogue: counting the spin structures . 31 3 Dirac operators 32 3.1 The metric distance property . 32 3.2 Symmetry of the Dirac operator . 34 3.3 Selfadjointness of the Dirac operator . 35 3.4 The Schr¨odinger–Lichnerowicz formula . 36 3.5 The spectral growth of the Dirac operator . 38 4 Spectral Growth and Dixmier Traces 41 4.1 Definition of spectral triples . 41 4.2 Logarithmic divergence of spectra . 42 4.3 Some eigenvalue inequalities . 43 4.4 Dixmier traces . 46 2 5 Symbols and Traces 49 5.1 Classical pseudodifferential operators . 49 5.2 Homogeneity of distributions . 52 5.3 The Wodzicki residue . 55 5.4 Dixmier trace and Wodzicki residue . 60 6 Spectral Triples: General Theory 62 6.1 The Dixmier trace revisited . 62 6.2 Regularity of spectral triples . 66 6.3 Pre-C*-algebras . 69 6.4 Real spectral triples . 75 6.5 Summability of spectral triples . 77 7 Spectral Triples: Examples 79 7.1 Geometric conditions on spectral triples . 79 7.2 Isospectral deformations of commutative spectral triples . 82 7.3 The Moyal plane as a nonunital spectral triple . 87 7.4 A geometric spectral triple over SUq(2) . 93 A Exercises 94 A.1 Examples of Dirac operators . 94 A.1.1 The circle . 94 A.1.2 The (flat) torus . 95 2 A.1.3 The Hodge–Dirac operator on S ..................... 96 2 A.2 The Dirac operator on the sphere S ....................... 98 2 A.2.1 The spinor bundle S on S ........................ 98 S 2 A.2.2 The spin connection ∇ over S ..................... 99 A.2.3 Spinor harmonics and the Dirac operator spectrum . 101 A.3 Spinc Dirac operators on the 2-sphere . 102 A.4 A spectral triple on the noncommutative torus . 104 References 108 3 Introduction and Overview Noncommutative geometry asks: “What is the geometry of the Quantum World?” Quantum field theory considers aggregates of “particles”, which are of two general species, “bosons” and “fermions”. These are described by solutions of (relativistic) wave equations: 2 • Bosons: Klein–Gordon equation, ( + m )φ(x) = ρb(x) – “source term”; • Fermions: Dirac equation, (i∂/ − m)ψ(x) = ρf (x) – “source term”; 0 1 2 3 2 2 2 2 P3 µ µ where x = (t, ~x) = (x , x , x , x ); = −∂ /∂t + ∂ /∂~x ; and ∂/ = µ=0 γ ∂/∂x . In 0 2 j 2 order that ∂/ be a “square root of ”, we need (γ ) = −1, (γ ) = +1 for j = 1, 2, 3 and γµγν = −γνγµ for µ 6= ν. Thus, the γµ must be matrices; in fact there are four (4 × 4) matrices satisfying these relations. Point-like measurements are often ruled out by quantum mechanics; thus we replace points x ∈ M by coordinates f ∈ C(M). The metric distance on a Riemannian manifold (M, g) can be computed in two ways: dg(p, q) := inf{ length(γ : [0, 1] → M): γ(0) = p, γ(1) = q } = sup{ |f(p) − f(q)| : f ∈ C(M), kD,/ fk ≤ 1 }, where D/ is a Dirac operator with positive-definite signature (all (γµ)2 = +1) if it exists, so the Dirac operator specifies the metric. D/ is an (unbounded) operator on a Hilbert space H = L2(M, S) of “square-integrable spinors” and C∞(M) also acts on H by multiplication operators with k[D,/ f]k = k grad fk∞. Noncommutative geometry generalizes (C∞(M),L2(M, S), D/) to a spectral triple of the form (A, H,D), where A is a “smooth” algebra acting on a Hilbert space H, D is an (unbounded) selfadjoint operator on H, subject to certain conditions: in particular that [D, a] be a bounded operator for each a ∈ A. The tasks of the geometer are then: 1. To describe (metric) differential geometry in an operator language. 2. To reconstruct (ordinary) geometry in the operator framework. 3. To develop new geometries with noncommutative coordinate algebras. The long-term goal is to geometrize quantum physics at very high energy scales, but we are still a long way from there. The general program of these lectures is as follows. (A) The classical theory of spinors and Dirac operators in the Riemannian case. (B) The operational toolkit for noncommutative generalization. (C) Reconstruction: how to recover differential geometry from the operator framework. (D) Examples of spectral triples with noncommutative coordinate algebras. 4 Chapter 1 Clifford algebras and spinor representations Here are a few general references on Clifford algebras, in reverse chronological order: [GVF, 2001], [Fri, 1997/2000], [BGV, 1992], [LM, 1989] and [ABS, 1964]. (See the bibliography for details.) 1.1 Clifford algebras n We start with (V, g), where V ' R and g is a nondegenerate symmetric bilinear form. If q(v) = g(v, v), then 2g(u, v) = q(u + v) − q(u) − q(v). Thus g is determined by the corresponding “quadratic form” q. Definition 1.1. The Clifford algebra Cl(V, g) is an algebra (over R) generated by the vectors v ∈ V subject to the relations uv + vu = 2g(u, v)1 for u, v ∈ V . The existence of this algebra can be seen in two ways. First of all, let T (V ) be the tensor L∞ ⊗n algebra on V , that is, T (V ) := k=0 V . Then Cl(V, g) := T (V )/ Idealhu ⊗ v + v ⊗ u − 2g(u, v) 1 : u, v ∈ V i. (1.1) Since the relations are not homogeneous, the Z-grading of T (V ) is lost, only a Z2-grading remains: Cl(V, g) = Cl0(V, g) ⊕ Cl1(V, g). • The second option is to define Cl(V, g) as a subalgebra of EndR(Λ V ) generated by all expressions c(v) = ε(v) + ι(v) for v ∈ V , where ε(v): u1 ∧ · · · ∧ uk 7→ v ∧ u1 ∧ · · · ∧ uk k X j−1 ι(v): u1 ∧ · · · ∧ uk 7→ (−1) g(v, uj)u1 ∧ · · · ∧ ubj ∧ · · · ∧ uk. j=1 Note that ε(v)2 = 0, ι(v)2 = 0, and ε(v)ι(u) + ι(u)ε(v) = g(v, u) 1. Thus c(v)2 = g(v, v) 1 for all v ∈ V, c(u)c(v) + c(v)c(u) = 2g(u, v) 1 for all u, v ∈ V. Thus these operators on Λ•V do provide a representation of the algebra (1.1). 5 Dimension count: suppose {e1, . , en} is an orthonormal basis for (V, g), i.e., g(ek, ek) = ±1 and g(ej, ek) = 0 for j 6= k. Then the c(ej) anticommute and thus a basis for Cl(V, g) is {c(ek1 ) . c(ekr ) : 1 ≤ k1 < ··· < kr ≤ n}, labelled by K = {k1, . , kr} ⊆ {1, . , n}. Indeed, • c(ek1 ) . c(ekr ): 1 7→ ek1 ∧ · · · ∧ ekr ≡ eK ∈ Λ V • and these are linearly independent. Thus the dimension of the subalgebra of EndR(Λ V ) generated by all c(v) is just dim Λ•V = 2n. Now, a moment’s thought shows that in the abstract presentation (1.1), the algebra Cl(V, g) is generated as a vector space by the 2n products ek1 ek2 . ekr , and these are linearly independent since the operators c(ek1 ) . c(ekr ) • are linearly independent in EndR(Λ V ). Therefore, this representation of Cl(V, g) is faithful, and dim Cl(V, g) = 2n. The so-called “symbol map”: σ : a 7→ a(1) : Cl(V, g) → Λ•V is inverted by a “quantization map”: 1 X Q : u ∧ u ∧ · · · ∧ u 7−→ (−1)τ c(u )c(u ) . c(u ). (1.2) 1 2 r r! τ(1) τ(2) τ(r) τ∈Sr To see that it is an inverse to σ, one only needs to check it on the products of elements of an orthonormal basis of (V, g). From now, we write uv instead of c(u)c(v), etc., in Cl(V, g). 1.2 The universality property Chevalley [Che] has pointed out the usefulness of the following property of Clifford algebras, which is an immediate consequence of their definition. Lemma 1.2. Any R-linear map f : V → A (an R-algebra) that satisfies 2 f(v) = g(v, v) 1A for all v ∈ V extends to an unique unital R-algebra homomorphism f˜: Cl(V, g) → A. Proof. There is really nothing to prove: f˜(v1v2 . vr) := f(v1)f(v2) . f(vr) gives the unique- ness, provided only that this recipe is well-defined. But observe that f˜(uv + vu − 2g(u, v)1) = f˜((u + v)2 − u2 − v2) − 2g(u, v) f˜(1) = [q(u + v) − q(u) − q(v) − 2g(u, v)] 1A = 0. Here are a few applications of universality that yield several useful operations on the Clifford algebra. 1. Grading: take A = Cl(V, g) itself; the linear map v 7→ −v on V extends to an automor- 2 phism χ ∈ Aut(Cl(V, g)) satisfying χ = idA, given by r χ(v1 .