Atiyah-Singer Revisited

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1 Atiyah-Singer Revisited Paul Baum Penn State Texas A&M Universty College Station, Texas, USA April 1, 2014

2 From E 1, E 2,..., E n obtain : 1) The Dirac operator of R n D = n j=1 E j x j 2) The Bott generator vector bundle on S n (n even) 3) The spin representation of Spin c (n)

3 W finite dimensional C vector space dim C (W ) < T : W W T Hom C (W, W ) T 2 = I = The eigen-values of T are ± i i = 1 W = W i W i W i = {v W T v = iv} W i = {v W T v = iv}

4 Bott generator vector bundle n even n = 2r S n R n+1 S n = {(a 1, a 2,..., a n+1 ) R n+1 a a a2 n+1 = 1} S n M(2 r, C) (a 1, a 2,..., a n+1 ) a 1 E 1 + a 2 E a n+1 E n+1 (a 1 E 1 + a 2 E a n+1 E n+1 ) 2 = ( a 2 1 a a 2 n+1) I = I = The eigenvalues of a 1 E 1 + a 2 E a n+1 E n+1 are ± i.

5 Bott generator vector bundle β on S n n even n = 2r S n M(2 r, C) (a 1, a 2,..., a n+1 ) a 1 E 1 + a 2 E a n+1 E n+1 β (a1,a 2,...,a n+1 ) = (i eigenspace of a 1 E 1 + a 2 E a n+1 E n+1 )

6 Bott generator vector bundle β on S n n even n = 2r β is determined by: 1 p S n, dim C (β p ) = 2 r 1 2 ch(β)[s n ] = 1

7 Since SO(n) GL(n, R) the usual map Spin c (n) SO(n) can be viewed as mapping Spin c (n) to GL(n, R). Spin c (n) GL(n, R) Let E be an R vector bundle on a paracompact Hausdorff space X. DEFINITION A Spin c datum for E is a homomorphism of principal bundles η : P F(E) whose underlying homomorphism of topological groups is Spin c (n) GL(n, R)

8 A Spin c structure for E is a homotopy class of Spin c data for E. {Spin c structures for E} := {Spin c data for E}/ where = homotopy. A Spin c vector bundle on X is an R vector bundle E on X with a given Spin c structure. A Spin c manifold M is a C manifold (with or without boundary) whose tangent bundle T M is a Spin c vector bundle.

9 Minicourse of five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. Beyond ellipticity 5. The Riemann-Roch theorem The minicourse is based on joint work with Ron Douglas and is dedicated to him.

10 ATIYAH-SINGER REVISITED This is an expository talk about the Atiyah-Singer index theorem. 1 Dirac operator of R n will be defined. 2 Some low dimensional examples of the theorem will be considered. 3 A special case of the theorem will be proved, with the proof based on Bott periodicity. 4 The proof will be outlined that the special case implies the full theorem.

11 Atiyah-Singer Index theorem M compact C manifold without boundary D an elliptic differential (or pseudo-differential) operator on M E 0, E 1, C C vector bundles on M C (M, E j ) denotes the C vector space of all C sections of E j. D : C (M, E 0 ) C (M, E 1 ) D is a linear transformation of C vector spaces.

12 Atiyah-Singer Index theorem M compact C manifold without boundary D an elliptic differential (or pseudo-differential) operator on M Index(D) := dim C (Kernel D) dim C (Cokernel D) Theorem (M.Atiyah and I.Singer) Index (D) = (a topological formula)

13 Example M = S 1 = {(t 1, t 2 ) R 2 t t2 2 = 1} D f : L 2 (S 1 ) L 2 (S 1 ) is T f 0 0 I where L 2 (S 1 ) = L 2 +(S 1 ) L 2 (S 1 ). L 2 +(S 1 ) has as orthonormal basis e inθ with n = 0, 1, 2,... L 2 (S 1 ) has as orthonormal basis e inθ with n = 1, 2, 3,....

14 Example f : S 1 R 2 {0} is a C map. S 1 R 2 {(0, 0)} f T f : L 2 +(S 1 ) L 2 +(S 1 ) is the composition L 2 +(S 1 ) M f L 2 (S 1 ) L 2 +(S 1 ) T f : L 2 +(S 1 ) L 2 +(S 1 ) is the Toeplitz operator associated to f

15 Example Thus T f is composition T f : L 2 +(S 1 ) M f L 2 (S 1 P ) L 2 +(S 1 ) where L 2 +(S 1 ) M f L 2 (S 1 ) is v fv fv(t 1, t 2 ) := f(t 1, t 2 )v(t 1, t 2 ) (t 1, t 2 ) S 1 R 2 = C and L 2 (S 1 ) P L 2 +(S 1 ) is the Hilbert space projection. D f (v + w) := T f (v) + w v L 2 +(S 1 ), w L 2 (S 1 ) Index(D f ) = -winding number (f).

16 RIEMANN - ROCH M compact connected Riemann surface genus of M = # of holes = 1 2 [rankh 1(M; Z)]

17 D a divisor of M D consists of a finite set of points of M p 1, p 2,..., p l and an integer assigned to each point n 1, n 2,..., n l Equivalently D is a function D : M Z with finite support Support(D) = {p M D(p) 0} Support(D) is a finite subset of M

18 D a divisor on M deg(d) := p M D(p) Remark D 1, D 2 two divisors D 1 D 2 iff p M, D 1 (p) D 2 (p) Remark D a divisor, D is ( D)(p) = D(p)

19 Example Let f : M C { } be a meromorphic function. Define a divisor δ(f) by: 0 if p is neither a zero nor a pole of f δ(f)(p) = order of the zero if f(p) = 0 (order of the pole) if p is a pole of f

20 Example Let w be a meromorphic 1-form on M. Locally w is f(z)dz where f is a (locally defined) meromorphic function. Define a divisor δ(w) by: 0 if p is neither a zero nor a pole of w δ(w)(p) = order of the zero if w(p) = 0 (order of the pole) if p is a pole of w

21 D a divisor on M { } meromorphic functions H 0 (M, D) := δ(f) D f : M C { } { } meromorphic 1-forms H 1 (M, D) := w on M δ(w) D Lemma H 0 (M, D) and H 1 (M, D) are finite dimensional C vector spaces dim C H 0 (M, D) < dim C H 1 (M, D) <

22 Theorem (R. R.) Let M be a compact connected Riemann surface and let D be a divisor on M. Then: dim C H 0 (M, D) dim C H 1 (M, D) = d g + 1 d = degree (D) g = genus (M)

23 HIRZEBRUCH-RIEMANN-ROCH M non-singular projective algebraic variety / C E an algebraic vector bundle on M E = sheaf of germs of algebraic sections of E H j (M, E) := j-th cohomology of M using E, j = 0, 1, 2, 3,...

24 LEMMA For all j = 0, 1, 2,... dim C H j (M, E) <. For all j > dim C (M), H j (M, E) = 0. χ(m, E) := n = dim C (M) n ( 1) j dim C H j (M, E) j=0 THEOREM[HRR] Let M be a non-singular projective algebraic variety / C and let E be an algebraic vector bundle on M. Then χ(m, E) = (ch(e) T d(m))[m]

25 Hirzebruch-Riemann-Roch Theorem (HRR) Let M be a non-singular projective algebraic variety / C and let E be an algebraic vector bundle on M. Then χ(m, E) = (ch(e) T d(m))[m]

26 Various well-known structures on a C manifold M make M into a Spin c manifold (complex-analytic) (symplectic) (almost complex) (contact) (stably almost complex) Spin Spin c (oriented) A Spin c manifold can be thought of as an oriented manifold with a slight extra bit of structure. Most of the oriented manifolds which occur in practice are Spin c manifolds.

27 Two Out Of Three Lemma Lemma Let 0 E E E 0 be a short exact sequence of R-vector bundles on X. If two out of three are Spin c vector bundles, then so is the third.

28 Definition Let M be a C manifold (with or without boundary). M is a Spin c manifold iff the tangent bundle T M of M is a Spin c vector bundle on M. The Two Out Of Three Lemma implies that the boundary M of a Spin c manifold M with boundary is again a Spin c manifold.

29 A Spin c manifold comes equipped with a first-order elliptic differential operator known as its Dirac operator. If M is a Spin c manifold, then T d(m) is T d(m) := exp c 1(M)/2 Â(M) T d(m) H (M; Q) If M is a complex-analyic manifold, then M has Chern classes c 1, c 2,..., c n and exp c 1(M)/2 Â(M) = P T odd (c 1, c 2,..., c n )

30 EXAMPLE. Let M be a compact complex-analytic manifold. Set Ω p,q = C (M, Λ p,q T M) Ω p,q is the C vector space of all C differential forms of type (p, q) Dolbeault complex 0 Ω 0,0 Ω 0,1 Ω 0,2 Ω 0,n 0 The Dirac operator (of the underlying Spin c manifold) is the assembled Dolbeault complex + : j Ω 0, 2j j 0, 2j+1 Ω The index of this operator is the arithmetic genus of M i.e. is the Euler number of the Dolbeault complex.

31 TWO POINTS OF VIEW ON SPIN c MANIFOLDS 1. Spin c is a slight strengthening of oriented. The oriented manifolds that occur in practice are Spin c. 2. Spin c is much weaker than complex-analytic. BUT the assempled Dolbeault complex survives (as the Dirac operator). AND the Todd class survives. M Spin c = T d(m) H (M; Q)

32 SPECIAL CASE OF ATIYAH-SINGER Let M be a compact even-dimensional Spin c manifold without boundary. Let E be a C vector bundle on M. D E denotes the Dirac operator of M tensored with E. D E : C (M, S + E) C (M, S E) S +, S are the positive (negative) spinor bundles on M. THEOREM Index(D E ) = (ch(e) T d(m))[m].

33 K 0 ( ) Definition Define an abelian group denoted K 0 ( ) by considering pairs (M, E) such that: 1 M is a compact even-dimensional Spin c manifold without boundary. 2 E is a C vector bundle on M.

34 Set K 0 ( ) = {(M, E)}/ where the the equivalence relation is generated by the three elementary steps Bordism Direct sum - disjoint union Vector bundle modification Addition in K 0 ( ) is disjoint union. (M, E) + (M, E ) = (M M, E E ) In K 0 ( ) the additive inverse of (M, E) is ( M, E) where M denotes M with the Spin c structure reversed. (M, E) = ( M, E)

35 Isomorphism (M, E) is isomorphic to (M, E ) iff a diffeomorphism ψ : M M preserving the Spin c -structures on M, M and with ψ (E ) = E.

36 Bordism (M 0, E 0 ) is bordant to (M 1, E 1 ) iff (Ω, E) such that: 1 Ω is a compact odd-dimensional Spin c manifold with boundary. 2 E is a C vector bundle on Ω. 3 ( Ω, E Ω ) = (M 0, E 0 ) ( M 1, E 1, ) M 1 is M 1 with the Spin c structure reversed.

37 (M 0, E 0 ) ( M 1, E 1 )

38 Direct sum - disjoint union Let E, E be two C vector bundles on M (M, E) (M, E ) (M, E E )

39 Vector bundle modification (M, E) Let F be a Spin c vector bundle on M Assume that dim R (F p ) 0 mod 2 p M for every fiber F p of F 1 R = M R S(F 1 R ) := unit sphere bundle of F 1 R (M, E) (S(F 1 R ), β π E)

40 S(F 1 R ) M π This is a fibration with even-dimensional spheres as fibers. F 1 R is a Spin c vector bundle on M with odd-dimensional fibers. The Spin c structure for F causes there to appear on S(F 1 R ) a C-vector bundle β whose restriction to each fiber of π is the Bott generator vector bundle of that even-dimensional sphere. (M, E) (S(F 1 R ), β π E)

41 Addition in K 0 ( ) is disjoint union. (M, E) + (M, E ) = (M M, E E ) In K 0 ( ) the additive inverse of (M, E) is ( M, E) where M denotes M with the Spin c structure reversed. (M, E) = ( M, E)

42 DEFINITION. (M, E) bounds (Ω, Ẽ) with : 1 Ω is a compact odd-dimensional Spin c manifold with boundary. 2 Ẽ is a C vector bundle on Ω. 3 ( Ω, Ẽ Ω) = (M, E) REMARK. (M, E) = 0 in K 0 ( ) (M, E) (M, E ) where (M, E ) bounds.

43 Consider the homomorphism of abelian groups K 0 ( ) Z (M, E) Index(D E ) Notation D E is the Dirac operator of M tensored with E.

44 It is a corollary of Bott periodicity that this homomorphism of abelian groups is an isomorphism. Equivalently, Index(D E ) is a complete invariant for the equivalence relation generated by the three elementary steps; i.e. (M, E) (M, E ) if and only if Index(D E ) = Index(D E ).

45 BOTT PERIODICITY Z j odd π j GL(n, C) = 0 j even j = 0, 1, 2,..., 2n 1

46 Why does Bott periodicity imply that K 0 ( ) Z (M, E) Index(D E ) is an isomorphism?

47 To prove surjectivity must find an (M, E) with Index(D E ) = 1. e.g. Let M = CP n, and let E be the trivial (complex) line bundle on CP n E=1 C = CP n C Index(CP n, 1 C ) = 1 Thus Bott periodicity is not used in the proof of surjectivity.

48 Lemma used in the Proof of Injectivity Given any (M, E) there exists an even-dimensional sphere S 2n and a C-vector bundle F on S 2n with (M, E) (S 2n, F ). Bott periodicity is not used in the proof of this lemma. The lemma is proved by a direct argument using the definition of the equivalence relation on the pairs (M, E).

49 Let r be a positive integer, and let Vect C (S 2n, r) be the set of isomorphism classes of C vector bundles on S 2n of rank r, i.e. of fiber dimension r. Vect C (S 2n, r) π 2n 1 GL(r, C)

50 PROOF OF INJECTIVITY Let (M, E) have Index(M, E) = 0. By the above lemma, we may assume that (M, E) = (S 2n, F ). Using Bott periodicity plus the bijection Vect C (S 2n, r) π 2n 1 GL(r, C) we may assume that F is of the form F = θ p qβ θ p = S 2n C p and β is the Bott generator vector bundle on S 2n. Convention. If q < 0, then qβ = q β.

51 Index(S 2n, β) = 1 Index(S 2n, θ p ) = 0 Therefore Index(S 2n, F ) = 0 = q = 0 Hence (S 2n, F ) = (S 2n, θ p ). This bounds and so is zero in K 0 ( ). QED (S 2n, θ p ) = (B 2n+1, B 2n+1 C p )

52 Define a homomorphism of abelian groups K 0 ( ) Q (M, E) ( ch(e) Td(M) ) [M] where ch(e) is the Chern character of E and Td(M) is the Todd class of M. ch(e) H (M, Q) and Td(M) H (M, Q). [M] is the orientation cycle of M. [M] H (M, Z).

53 Granted that K 0 ( ) Z (M, E) Index(D E ) is an isomorphism, to prove that these two homomorphisms are equal, it suffices to check one nonzero example.

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