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THE ATIYAH-SINGER INDEX THEOREM 1. Overview 2 Part 1 THE ATIYAH-SINGER INDEX THEOREM LECTURES BY DAN BERWICK-EVANS LECTURES NOTES BY TONY FENG CONTENTS 1. Overview 2 Part 1. Geometric Preliminaries 4 2. Connections 4 3. Super vector bundles 6 4. Characteristic classes 9 5. Clifford algebras 12 6. Dirac Operators 14 7. Spin 16 8. Differential Operators 20 9. Densities and Divergences 22 Part 2. Towards the Index Theorem 25 10. Heat kernels 25 11. Some Harmonic Analysis 35 12. Proof of the Index Theorem 39 Part 3. Applications 45 13. Chern-Gauss-Bonnet and Hirzebruch Signature Theorem 45 14. K -theory 51 15. Seiberg-Witten Theory 56 1 2 LECTURES BY DAN BERWICK-EVANS LECTURES NOTES BY TONY FENG 1. OVERVIEW The course is about the Atiyah-Singer Index Theorem. Its statement is given below, although it may not make sense at the moment; the beginning of the course will intro- duce the necessary ingredients. Theorem 1.1 (Atiyah-Singer). The index of a Dirac operator D of a Clifford module over a compact, oriented manifold M of even dimension is E Z ind(D ) = Ab(M )ch( =S). M E For now, the interesting thing to take away about this theorem is that it relates two things that don’t seem like they should be related. The left hand side ¨ “graded dimension of D ” ind(D ) := dim(kerD +) dimcoker(D +) − is an analytic object. Meanwhile, the right hand side can be written in terms of the cap product pairing on M as Z Ab(M )ch( =S) = [Ab(M )ch( =S)],[M ] . M E h E i That makes it clear that it is a topological quantity. It is also “local” in the sense that integration is computed locally. R=2 1=2 Remark 1.2. To elaborate a bit more on the formula, Ab(M ) = det sinhR=2 where R is the Ricci curvature of M . Even the form of the equation immediately yields interesting information. For ex- ample, it is obvious that the left hand side is an integer and highly non-obvious that the R right hand side is an integer. So if, for example, you compute M Ab(M )ch( =S) and you don’t get an integer, you can deduce that the hypotheses were false (this wouldE give, for instance, a proof that M is not orientable). The Atiyah-Singer index theorem encompasses many other important results. Chern-Gauss-Bonnet. The Chern-Gauss-Bonnet formula is Z χ(M ) = Pf(R). M This is a special case of the Atiyah-Singer index theorem. For now, let’s content our- selves with figuring out what the corresponding objects (D , ) in the index theorem should be. How can we realize the Euler characteristic as theE index of some operator on a vector bundle? We should take the bundle V T M and D d d . = • ∗ = + ∗ 2 Since D is self-adjoint by construction, kerD E= kerD = ker∆. By Hodge theory, this is HdR• (M ). Then ind(D ) is the graded dimension of Hd• R (M ), which is the alternating sum of the Betti numbers, and that is the Euler characteristic. We’ll come back and compute the right hand side later on in the course. THE ATIYAH-SINGER INDEX THEOREM 3 Hirzebruch Signature Theorem. Suppose M is a manifold of dimension 4k. The Hirze- bruch signature theorem says that Z σ(M ) = L(M ) M where σ(M ) is the signature of M (the signature of the intersection form on H2k (M )) R=2 1=2 and L(M ) = det tanhR=2 . Again, this is a special case of the Index theorem applied to the bundle V T M = • ∗ where D d d , and kerD H M . This time, we take the graded dimensionE in- = + ∗ = dR• ( ) ∼ 2 duced by the Hodge , normalized so that = 1. This graded dimension gives the signature. ∗ ∗ Hirzebruch-Riemann-Roch. If M is a Kähler manifold and W M is a holomorphic vector bundle, then the Hirzebruch-Riemann-Roch formula says! that Z χhol(W ) = Td(M )ch(W ). M This is the Atiyah-Singer index theorem applied with V0, T M W and D . = • ∗ = @ +@ ∗ Then kerD is essentially holomorphic sections, andE ind(D ) is the⊗ graded dimension, which is the holomorphic Euler characteristic. 4 LECTURES BY DAN BERWICK-EVANS LECTURES NOTES BY TONY FENG Part 1. Geometric Preliminaries 2. CONNECTIONS Definition 2.1. Let E M be a vector bundle. A covariant derivative is a map ! : Γ (M , E ) Γ (M ,T ∗M E )s r ! ⊗ such that for f C M , 1( ) 2 (f s ) = d f s + f s (Leibniz rule). r · ⊗ · r This is equivalent to the data of a connection on E , so we’ll use the two terms inter- changeably. Denote M , E M ,V T M E . Then we can extend to an operator Ω•( ) = Γ ( • ∗ ) ⊗ r +1 : Ω•(M , E ) Ω• (M , E ) r ! by the Leibniz rule: for k M and s M , E , α Ω ( ) Ω•( ) 2 2 k (α s ) = d α s + ( 1) α s r ^ ^ − ^ r Definition 2.2. For v Γ (TM ), recall that there is a contraction operator 2 1 ιv : Ω•(M ) Ω•− (M ) ! defined by ιv !(v1,..., vp ) = !(v, v1,..., vp ). For v Γ (TM ), we define v = ιv . 2 r ◦ r 2 Definition 2.3. The curvature of is the form F Ω (M ,End(E )) defined r 2 F (v, w ) = v w w v [v,w ] for v, w Γ (TM ). r r − r r − r 2 One has to check that F is linear over C M , but this is easy. 1( ) Definition 2.4. We call (E , ) flat if F = 0. r Example 2.5. If E = M V is the trivial bundle, then d is a flat connection. × 1 Example 2.6. On E = M V , a connection takes the form d +A = for A Ω (M ,End(E )). Then F = d A + A A. × r 2 Since any vector^ bundle is locally trivial, we have locally (E , ) = (M V,d + A). r ∼ × Definition 2.7. A metric on E M is a map of vector bundles E E R which is an inner product upon restriction! to fibers. ⊗ ! On global sections, this induces a map E E C M . Γ ( ) Γ ( ) 1( ) ⊗ ! Definition 2.8. Given (E , , , ) we say that is compatible with , if r 〈− −〉 r 1 〈− −〉 d v, w = v, w + v, w Ω (M ). h i hr i h r i 2 Definition 2.9. For a connection on TM , the torsion T is r T (v, w ) = v w w v [v, w ]. r − r − If T = 0, then we say that is torsion-free. r THE ATIYAH-SINGER INDEX THEOREM 5 It is a fact that given a metric on TM , there is a unique connection that is both torsion-free and compatible. This is called the Levi-Civita connection. 6 LECTURES BY DAN BERWICK-EVANS LECTURES NOTES BY TONY FENG 3. SUPER VECTOR BUNDLES In this section we develop a formalism of “super vector bundles”, which are essen- tially just Z=2-graded vector bundles, which will be useful for encoding sign issues. 3.1. Super vector spaces. Definition 3.1. A super vector space is a Z=2-graded vector space: even odd E = E E . ⊕ We might also denote E even E + E 0 and E odd E E 1. = = = − = n n n n Example 3.2. We denote by R = R R . j |{z} |{z} even ⊕ odd Definition 3.3. A super-algebra is an algebra A whose underlying vector space is a super vector space whose multiplication respects the grading, i.e. E i E j E i +j . V ⊗ ! Example 3.4. (Exterior algebra) • V is a super-algebra, with ^ even ^even ^ odd ^odd ( • V ) = V and ( • V ) = V. Example 3.5. (Clifford algebras) For (V,Q) a vector space V with quadratic form Q, we can define the Clifford algebra Cl(V,Q) = T (V )= v v + 1 Q(v ) . h ⊗ · i Here T (V ) is the tensor algebra. Since we’ve identified a degree 2 element with a de- gree 0 element, the Clifford algebra is only Z=2-graded. Note that V Cl(V,Q) has odd degree. This lets you figure out the grading of everything. ⊂ Definition 3.6. A super algebra is super-commutative if for a, b A, a b 2 a b = ( 1)j |·| jb a, · − · where a = deg(a) 0,1 and b = deg b 0,1 . j j V 2 f g j j 2 f g Example 3.7. • V is super-commutative. However, Cl(V,Q) is not super-commutative in general. 3.2. Maps between super vector spaces. Let E , F be super vector spaces. Then + + + Hom(E , F ) = Hom(E ,V ) Hom(E −,V −) ⊕ and + + Hom(E , F )− = Hom(E ,V −) Hom(E −,V ). ⊕ Remark 3.8. There’s a more abstract formulation of things which makes this a little more conceptual. There’s a symmetric monoidal category of vector spaces whose ob- jects are super vector spaces, and morphisms are even maps. The signs comes from the monoidal structure: even even even odd odd (E F ) = (E F ) (E F ) ⊗ ⊗ ⊕ ⊗ and odd even odd odd even (E F ) = (E F ) (E F ). ⊗ ⊗ ⊕ ⊗ THE ATIYAH-SINGER INDEX THEOREM 7 The signs come from an isomorphism E F F E , sending e f 1 e f f e = ( )j |·| j which is built into the monoidal structure.⊗ ∼ ⊗ ⊗ 7! − ⊗ It may be worth remarking that if we forget this sign, then we obtain simply the usual category of Z=2-graded vector spaces.
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