APPLICATIONS OF ELLIPTIC OPERATORS AND THE ATIYAH SINGER INDEX THEOREM MARTIN BENDERSKY Contents 1. Review of Di®erential Geometry 2 2. De¯nition of an Elliptic Operator 5 3. Properties of Elliptic Operators 7 4. Example of an Elliptic Operator 9 5. Example: The Euler Characteristic 12 6. Example: The Signature Invariant 14 7. A Theorem of Atiyah, Frank and Mayer 18 8. Cli®ord Algebras 20 9. A Diversion: Constructing Vector Fields on Spheres using Cli®ord Algebras 23 10. Topological Invariants of the Index. The Atiyah Singer Index Theorem 26 11. Borel Hirzebruch Theory and Characteristic Classes 30 12. Veri¯cation of the Index Theorem for D 33 13. The Index of DS is the Hirzebruch L-genus 36 14. The Hirzebruch Riemann Roch Theorem 39 References 43 Date: May 3, 2006. 1 2 MARTIN BENDERSKY 1. Review of Differential Geometry References for this material are [ST], [S] or any reasonable di®erential geometry text. We ¯rst recall the de¯nition of a locally trivial vector bundle, over a pointed space X, » : F ! E ! X. E is called the total space and F = ¼¡1(x) ¼ F n,F = R or C. S We assume X = Ui with »jUi is trivial. What this means is that there is a diagram Ej ¡!Ti F n £ U Ui ? i ? # y¼ Ui = Ui On the intersections there are the maps ¡1 n T n T ¸ij = Ti ± Tj : F £ Ui Uj ! F £ Ui Uj which induce maps (also T ¸i;j denoted ¸ij) Ui Uj ¡! F (n)(F (n) is O(n) if F = R, U(n)if F = C). The f¸ijg satisfy the following composition laws: (1.1) T T ² (1) ¸ik = ¸ij ± ¸jk on Ui Uj Uk ² (2)¸ii = identity ¡1 ² (3)¸ij = ¸ ji T ² (4) ¸i;j is a continuous map from Ui Uj to F (n) Notation 1.2. ² X will denote an n dimensional, C1, compact, closed, oriented man- ifold. n @ ² rj : R ! R denotes the j-th coordinate function. In particular is @rj the usual partial with respect to the j-th coordinate. n ² xj denotes rj ± ' where ' :U ! R is a local chart, U ½ X: ² C(X; x; R) denotes the C1 functions from some neighborhood of x 2 X to R. Notice that xj may be viewed as an element of C(X; x; R). ² C(X; R) denotes the C1 global functions f : X ! R. More generally C(X; Y) denotes C1 functions from X to Y. De¯nition 1.3. A tangent vector υ, at x 2 X is a map υ : C(X; x; R) ! R such that APPLICATIONS OF ELLIPTIC OPERATORS AND THE ATIYAH SINGER INDEX THEOREM3 (1) υ(f + g) = υ(f) + υ(g) (2) υ(¸f) = λυ(f) (3) υ(f ¢ g) = υ(f)g(x) + f(x)υ(g) for f; g 2 C(X; x; R); ¸ 2 R @ ¡1 Locally υ(f) = §ai (f ± ' )j ¡1 @ri ' (x) @ ¡1 @ @ We denote (f ± Á ) by and write υ = §ai (locally). @ri @xi @xi Let T (X; x) denote the vector space of all tangent vectors at x. Then for à 2 C(X; Y) there is a linear map dà : T (X; x) ! T (Y;Ã(x)) de¯ned as follows. For υ 2 T (X; x); g 2 C(Y;Ã(x); R) dÃ(υ)(g) = υ(g ± Ã) With [ T (X) = T (X; x) x we have a commutative diagram dà T (X) ¡! T (Y) ## à X ¡! Y @ Locally if υ = §ai then @xi @ dÃ(υ) = §υ(yi ± Ã) @yi fyig local coordinates functions for a neighborhood of Ã(x) in Y. De¯nition 1.4. T ¤(X) is the dual of T (X). Locally T ¤(X) can be identi¯ed with fdfjf 2 C(X; x; R)g: From the above we have df : T (X; x) ! R on a vector υ is de¯ned by ¤ ¤ df(υ) = υ(f) 2 R. fdxi 2 T (X; x)ji = 1; ¢ ¢ ¢ ng span T (X; x). In fact @ @ def @ dxi is dual to because dxi( ) = (xj) = ±ij: @xi @xj @xi De¯nition 1.5. A smooth section, !, to the bundle T ¤(X) ¡!¼ X is called a 1-form. 4 MARTIN BENDERSKY Locally ! = §aidxi where ai 2 C(U; R); (U ⊆ X): For example if @ @ @ f 2 C(X; x; R); df = § (f)dxi In fact ai = §ajdxj( ) = df( ) = @xi @xi @xi @ (f): @xi Notation 1.6. To generalize the above we de¯ne: S ²¤k(X) = ¤k(T ¤(X; x)) Lx2X ²¤¤(X) = ¤k(X) k ²A smooth section to the bundle ¤k(X) ! X is a k¡form: The space of k-forms is denoted ­k(X; R). ²A smooth section to the bundle ¤¤(X) ! X is a di®erential form:The space of di®erential-forms is denoted ­¤(X; R). There is a unique map (1.7) d : ­k(X; R) ! ­k+1(X; R) such that: ² d(f) = df(df as in the paragraph after 1:4) where f 2 C(X; R) is thought of as a 0-form (R = ­0). ² d(¹ ^ ¿) = d¹ ^ ¿ + (¡1)k¹ ^ d¿(¹ 2 ­k(X; R)) ² d2 = 0 Locally if ! = §aI dxI then @ d! = § (aI )dxj ^ dxI I;j @xj (I a sequence of non-negative integers i1 < i2 ¢ ¢ ¢ < ik; dxI = dx1 ^ dx2 ¢ ¢ ¢ ^ dxk) De¯nition 1.8. A volume element is a choice of basis for ¤n(T ¤(X; x)): For example dx1 ^ ¢ ¢ ¢ ^ xn. APPLICATIONS OF ELLIPTIC OPERATORS AND THE ATIYAH SINGER INDEX THEOREM5 2. Definition of an Elliptic Operator Notation 2.1. n ² Let U ½ R be an open set, t = (t1; ¢ ¢ ¢ tn) an n-tuple of non- negative integers. jtj = §ti: De¯ne 1 @jtj Dt = jtj t1 t2 tn i @x1 @x2 ¢ ¢ ¢ @xn ¤ t t1 t2 tn ² If v 2 T (X; x) with v = §vidxi then v = v1 v2 ¢ ¢ ¢ vn 2 R. Let A; B be ¯nite dimensional complex vector spaces, C(U; A);C(U; B) C1 functions, as before (with the compact open topology). De¯nition 2.2. A linear di®erential operator of order r is a linear map D : C(U; A) ! C(U; B) such that t D(f) = § gtD (f); f 2 C(U; A) jtj·r 1 where gt is a C function, gt : U ! Hom(A; B) i.e. gt(u)is a matrix. Now suppose E; F are complex vector bundles over X. De¯nition 2.3. A di®erential operator, D, is a linear map D : ¡(X;E) ! ¡(X;F ) (¡ is the space of C1 sections with the compact open topology) such that there exist open sets Ui which cover X where EjUi ;F jUi are trivial and there exist linear di®erential operators, Di on Ui such that D(f)jUi = Di Let ¼ : T ¤(X) ! X be the projection. For q 2 X letg : X ! R such ¤ that vq = dg(q) 2 T (X; q) and s 2 ¡(E) such that s(q) = e; ¸ 2 R. Now write ¡i¸g i¸g r e D(e s) = ¸ pr(s; g) + ¢ ¢ ¢ ¸1p1 + p0 2 ¡(X;F ): For ¯xed g, s ! pr(s; g) is a homomorphism of Eq ! Fq which only depends on the derivative of g, i.e. v. 6 MARTIN BENDERSKY ¤ ¤ De¯nition 2.4. The symbol σD : ¼ (E) ! ¼ (F ) of the operator, D is de¯ned by 1 σ (v; e) = p (s; g) D ijtj r for D and r-th order di®erential operator. Example 2.5. Let D = @ @ : Then @xi1 @xi2 @ @ e¡¸g (ei¸gs) = @xi1 @xi2 @ @g e¡i¸g [i¸ ei¸gs + lower terms in ¸] = @xi1 @xi2 @g @g i2¸2 s + lower terms in ¸ @xi1 @xi2 @g In local coordinates we suppose dgx = §»idxi;»i = then @xi jtj ¡i¸g @ i¸g jtj jtj t e (e s)q = i ¸ » s(q) + lower terms in ¸ t1 tn @x1 ¢ ¢ ¢ @xn Finally t σD(vq; e) = § gt(x)v e 2 F jtj=r t t i.e. σD is the leading term of D with v substituted for D . De¯nition 2.6. D is an elliptic operator if σD(v) is an isomorphism for all v 6= 0. ¤ ¤ i.e. σD : ¼ (E) ! ¼ (F ) is an isomorphism of the ¯bers away from the zero section of T ¤(X) ! X. APPLICATIONS OF ELLIPTIC OPERATORS AND THE ATIYAH SINGER INDEX THEOREM7 3. Properties of Elliptic Operators Reference for this section is [P]. Theorem 3.1. If D is an elliptic di®erential operator then D : ¡(X;E) ! ¡(X;F ) has ¯nite dimensional kernel and ¯nite dimensional cokernel. Proof : [P] page 178. De¯nition 3.2. The index of the elliptic operator D is de¯ned to be IndexD = dimC( kernel D) ¡ dimC( cokernel D). Theorem 3.3. (Stability of the Index) Index D is invariant under \deformations" through elliptic operators. Proof : [P] page 185. Corollary 3.4. The index depends only on the symbol. Proof : If D1 and D2 have the same symbol then t(D1)¡(1¡t)D2 is a deformation of D1 to D2 which is elliptic at each stage. q:e:d: We now assume E and F have a Hermitian inner product. De¯nition 3.5. A formal adjoint for D is a di®erential operator, D¤ such that for s 2 ¡(E); t 2 ¡(F ) Z Z ¤ hD(s); tiF = hs; D (t)iE X X Theorem 3.6. For a ¯xed metric, there is a unique formal adjoint. Furthermore ²σ(D¤) = (σ(D))¤ ²cokernel D = kernel D¤ ¤ ²Index D = dimCkernelD ¡ dimCkernelD ²Index D = ¡Index D¤ Proof : Locally the existence of D¤ is integration by parts.