Notes on the Atiyah-Singer index theorem

177 Pages·2013·0.971 MB·English

by  Nicolaescu L.I.

Notes on the Atiyah-Singer Index Theorem
Liviu I. Nicolaescu
Notesforatopicsintopologycourse,UniversityofNotreDame,Spring2004,Spring2013.
Lastrevision:November15,2013

i
TheAtiyah-SingerIndexTheorem
This is arguably one of the deepest and most beautiful results in modern geometry, and in my
viewisamustknowforanygeometer/topologist. Ithastodowithellipticpartialdifferentialopera-
torsonacompactmanifold,namelythoseoperatorsP withthepropertythatdimkerP,dimcokerP <
∞. In general these integers are very difficult to compute without some very precise information
about P. Remarkably, their difference, called the index of P, is a “soft” quantity in the sense that
itsdeterminationcanbecarriedoutrelyingonlyontopologicaltools. Youshouldcomparethiswith
thefollowingelementarysituation.
SupposewearegivenalinearoperatorA : Cm → Cn. Fromthisinformationalonewecannot
compute the dimension of its kernel or of its cokernel. We can however compute their difference
which,accordingtotherank-nullitytheoremforn×mmatricesmustbedimkerA−dimcokerA =
m−n.
MichaelAtiyahandIsadoreSingerhaveshowninthe1960sthattheindexofanellipticoperator
is determined by certain cohomology classes on the background manifold. These cohomology
classes are in turn topological invariants of the vector bundles on which the differential operator
acts and the homotopy class of the principal symbol of the operator. Moreover, they proved that
in order to understand the index problem for an arbitrary elliptic operator it suffices to understand
the index problem for a very special class of first order elliptic operators, namely the Dirac type
elliptic operators. Amazingly, most elliptic operators which are relevant in geometry are of Dirac
type. The index theorem for these operators contains as special cases a few celebrated results: the
Gauss-Bonnettheorem,theHirzebruchsignaturetheorem,theRiemann-Roch-Hirzebruchtheorem.
In this course we will be concerned only with the index problem for the Dirac type elliptic
operators. Wewilladoptananalyticapproachtotheindexproblembasedontheheatequationona
manifoldandEzraGetzler’srescalingtrick.
(cid:43) Prerequisites: Working knowledge of smooth manifolds, and algebraic topology (especially
cohomology). Somefamiliarity with basic notions of functionalanalysis: Hilbert spaces, bounded
linearoperators,L2-spaces.
(cid:43) Syllabus: Part I. Foundations: connections on vector bundles and the Chern-Weil construction,
calculusonRiemannmanifolds,partialdifferentialoperatorsonmanifolds,Diracoperators,[21].
PartII.Thestatementandsomebasicapplicationsoftheindextheorem,[27].
PartIII.Theproofoftheindextheorem,[27].
(cid:43) About the class There will be a few homeworks containing routine exercises which involve
the basic notions introduced during the course. We will introduce a fairly large number of new
objectsandideasandsolvingtheseexercisesistheonlywaytogainsomethingformthisclassand
appreciatetherichflavorhiddeninsidethistheorem.
ii
Notationsandconventions
•K = R,C.
•ForeveryfinitedimensionalK-vectorspaceV wedenotebyAutK(V)theLiegroupofK-linear
automorphismsofV.
•Wewillorientthemanifoldswithboundaryusingtheouternormalfirstconvention.
• We will denote by gl (K) o(n), so(n), u(n) the Lie algebras of the Lie groups GL (K) and
r
r
respectivelyU(n),O(n),SO(n).
Contents
Introduction i
Notationsandconventions ii
Chapter1. GeometricPreliminaries 1
§1.1. Vectorbundlesandconnections 1
1.1.1. Smoothvectorbundles 1
1.1.2. Principalbundles 10
1.1.3. Connectionsonvectorbundles 12
§1.2. Chern-Weiltheory 21
1.2.1. ConnectionsonprincipalG-bundles 21
1.2.2. TheChern-Weilconstruction 22
1.2.3. Chernclasses 26
1.2.4. Pontryaginclasses 29
1.2.5. TheEulerclass 34
§1.3. CalculusonRiemannmanifolds 36
§1.4. ExercisesforChapter1 42
Chapter2. Ellipticpartialdifferentialoperators 45
§2.1. Definitionandbasicconstructions 45
2.1.1. Partialdifferentialoperators 45
2.1.2. Analyticpropertiesofellipticoperators 55
2.1.3. Fredholmindex 58
2.1.4. Hodgetheory 63
§2.2. Diracoperators 67
2.2.1. Cliffordalgebrasandtheirrepresentations 67
2.2.2. SpinandSpinc 76
2.2.3. GeometricDiracoperators 84
§2.3. ExercisesforChapter2 90
Chapter3. TheAtiyah-SingerIndexTheorem: StatementandExamples 93
iii
iv Contents
§3.1. Thestatementoftheindextheorem 93
§3.2. Fundamentalexamples 94
3.2.1. TheGauss-Bonnettheorem 94
3.2.2. Thesignaturetheorem 99
3.2.3. TheHodge-DolbeaultoperatorsandtheRiemann-Roch-Hirzebruchformula 103
3.2.4. ThespinDiracoperators 119
3.2.5. Thespinc Diracoperators 130
§3.3. ExercisesforChapter3 136
Chapter4. Theheatkernelproofoftheindextheorem 139
§4.1. Aroughoutlineofthestrategy 139
4.1.1. Theheatequationapproach: ababymodel 139
4.1.2. Whatreallygoesintotheproof 141
§4.2. Theheatkernel 142
4.2.1. Spectraltheoryofsymmetricellipticoperators 142
4.2.2. Theheatkernel 146
4.2.3. TheMcKean-Singerformula 153
§4.3. TheproofoftheIndexTheorem 154
4.3.1. Approximatingtheheatkernel 154
4.3.2. TheGetzlerapproximationprocess 158
4.3.3. Mehlerformula 165
4.3.4. Puttingallthemovingpartstogether 166
Bibliography 167
Index 169
Chapter 1
Geometric Preliminaries
1.1. Vectorbundlesandconnections
1.1.1. Smooth vector bundles. The notion of smooth K-vector bundle of rank r formalizes the
intuitiveideaofasmoothfamilyofr-dimensionalK-vectorspaces.
Definition 1.1.1. A smooth K-vector bundle of rank r over a smooth manifold B is a quadruple
(E,B,π,V)withthefollowingproperties.
(a)E,B aresmoothmanifoldsandV isar-dimensionalK-vectorspace.
(b)π : E → B isasurjectivesubmersion. WesetE := π−1(b)andwewillcallitthefiber(ofthe
b
bundle)overb.
(c)Thereexistsatrivializingcover,i.e.,anopencoverU = (U ) ofB anddiffeomorphisms
α α∈A
Ψ : E| = π−1(U ) → V ×U
α Uα α α
withthefollowingproperties.
(c1)Foreveryα ∈ Athediagrambelowiscommutative.
E| Ψα w V ×U
Uα α
[
’π’’) [^[proj .
U
α
(c2)Foreveryα,β ∈ Athereexistsasmoothmap
g : U := U ∩U → Aut(V), u (cid:55)→ g (u)
βα βα α β βα
suchthatforeveryu ∈ U wehavethecommutativediagram
αβ
V ×{u}
Ψα|Eu[[]
[
Eu gβα(u.)
’
Ψβ|Eu’’) u
V ×{u}
1
2 LiviuI.Nicolaescu
B is called the base, E is called total space, V is called the model (standard) fiber and π is called
thecanonical(ornatural)projection. AK-linebundleisarank1K-vectorbundle.
Remark 1.1.2. The condition (c) in the above definition implies that each fiber E has a natural
b
structureofK-vectorspace. Moreover,eachmapΨ inducesanisomorphismofvectorspaces
α
Ψ | → V ×{b}.
α E
b
(cid:116)(cid:117)
Here is some terminology we will use frequently. Often instead of (E,π,B,V) we will write
E →π B or simplyE. The inversesof Ψ−1 are calledlocal trivializationsof the bundle (overU ).
α α
Themapg iscalledthegluingmapfromtheα-trivializationtotheβ-trivialization. Thecollection
βα
(cid:110) (cid:111)
g : U → Aut(V); U (cid:54)= ∅
βα αβ αβ
iscalleda(Aut(V))-gluingcocycle(subordinatedtoU)sinceitsatisfiesthecocyclecondition
g (u) = g (u)·g (u), ∀u ∈ U := U ∩U ∩U , (1.1.1)
γα γβ βα αβγ α β γ
where”·”denotesthemultiplicationintheLiegroupAut(V). Notethat(1.1.1)impliesthat
g (u) ≡ 1 , g (u) = g (u)−1, ∀u ∈ U . (1.1.2)
αα V βα αβ αβ
Example1.1.3. (a)Avectorspacecanberegardedasavectorbundleoverapoint.
(b) For every smooth manifold M and every finite dimensional K-vector space we denote by V
M
thetrivialvectorbundle
V ×M → M, (v,m) (cid:55)→ m.
(c)ThetangentbundleTM ofasmoothmanifoldisasmoothvectorbundle.
π π
(d) If E → B is a smooth vector bundle and U (cid:44)→ B is an open set then E | → U is the vector
U
bundle
π−1(U) →π U.
(e) Recall that CP1 is the space of all one-dimensional subspaces of C2. Equivalently, CP1 is the
quotientofC2\{0}modulotheequivalencerelation
p ∼ p(cid:48) ⇐⇒ ∃λ ∈ C∗ : p(cid:48) = λp.
For every p = (z ,z ) ∈ C2 \{0} we denote by [p] = [z ,z ] its ∼-equivalence class which we
0 1 0 1
viewasthelinecontainingtheoriginandthepoint(z ,z ). Wehaveaniceopencover{U ,U }of
0 1 0 1
CP1 definedby
U := {[z ,z ]; z (cid:54)= 0}.
i 0 1 i
ThesetU consistsofthelinestransversaltotheverticalaxis,whileU consistsofthelinestransver-
0 1
saltothehorizontalaxis. Theslopem = z /z ofthelinethrough(z ,z )isalocalcoordinated
0 1 0 0 1
overU andtheslopem = z /z isalocalcoordinateoverU . Ontheoverlapwehave
0 1 0 1 1
m = 1/m .
1 0
Let
y z
E = {(x,y;[z ,z ]) ∈ C2×CP1; = 1, i.e. yz −xz = 0}
0 1 0 1
x z
0
ThenaturalprojectionC2×CP1 → CP1inducesasurjectionπ : E → CP1. Observethatforevery
[p] ∈ CP1 the fiber π−1(p) can be naturally identified with the line through p. We can thus regard
NotesontheAtiyah-SingerindexTheorem 3
E as a family of 1-dimensional vector spaces. We want to show that π actually defines a structure
ofasmoothcomplexlinebundleoverCP1. Set
(cid:110) (cid:111)
E := π−1(U ) = (x,y;[z ,z ]) ∈ E; z (cid:54)= 0 .
i i 0 1 i
Weconstructamap
Ψ : E → C×U , E (cid:51) (x,y;[z ,z ]) (cid:55)→ (x,[z ,z ])
0 0 0 0 0 1 0 1
and
Ψ : E → C×U , E (cid:51) (x,y;[z ,z ]y) (cid:55)→ (y,[z ,z ])
1 1 1 1 0 1 0 1
ObservethatΨ isbijectivewithinverseΨ−1 : C×U → E isgivenby
0 0 0 0
z
C×U (cid:51) (t;[z ,z ]) (cid:55)→ (t, 1t;[z ,z ]) = (t,m t;[z ,z ]).
0 0 1 0 1 0 0 1
z
0
Thecomposition
Ψ ◦Ψ−1 : C×U → C×U
1 0 01 01
isgivenby
C×U (cid:51) (s;[p]) (cid:55)→ (g ([p])s,[p]),
01 10
where
U (cid:51) [p] = [z ,z ] (cid:55)→ g ([p]) = z /z = m ([p]) ∈ C∗ = GL (C).
01 0 1 10 1 0 0 1
Thecomplexlinebundleconstructedaboveiscalledthetautologicallinebundle. (cid:116)(cid:117)
GivenasmoothmanifoldB,avectorspaceV,anopencoverU = (U ) ofB,andagluing
α α∈A
cocyclesubordinatedtoU
g : U → Aut(V)
βα αβ
wecanconstructasmoothvectorbundleasfollows. Considerthedisjointunion
(cid:97)
X = V .
Uα
α∈A
DenotebyE thequotientspaceofX modulotheequivalencerelation
V (cid:51) (v ,u ) ∼ (v ,u ) ∈ V ⇐⇒ u = u = u ∈ U , v = g (u)v .
Uα α α β β Uβ α β αβ β βα α
SinceweglueopensetsofsmoothmanifoldsviadiffeomorphismswededucethatE isnaturallya
smoothmanifold. Moreover,thenaturalprojectionsπ : V → U arecompatiblewiththeabove
α Uα α
equivalencerelationanddefineasmoothmap
π : E → B.
The natural maps Φ : V → E | are diffeomorphisms and their inverses Ψ = Φ−1 satisfy
α Uα Uα α α
all the conditions in Definition 1.1.1. We will denote the vector bundle obtained in this fashion by
(U,g ,V)orby(B,U,g ,V).
•• ••
Definition 1.1.4. Suppose (E,π ,B,V) and (F,π ,B,W) are smooth K-vector bundles over B
E F
ofrankspandrespectivelyq. Assume{U ,Ψ } isatrivializingcoverforπ and{V ,Φ } isa
α α α E β β β
trivializing cover for π . A vector bundle morphism from E −π→E B to F −π→F B is a smooth map
F
T : E → F satisfyingthefollowingconditions.
4 LiviuI.Nicolaescu
(a)Thediagrambellowiscommutative.
E T w F
[ (cid:26) .
πE[] (cid:26)(cid:29)πF
B
(b) The map T is linear along the fibers, i.e. for every b ∈ B and every α ∈ A, b ∈ B such that
b ∈ U ∩V thecompositionΦ T| Ψ | : V → W islinear,
α β β Fb α Eb
E w V ×{b}
b
Ψα|Eb
T|Eb linear.
Fu w W ×u {b}
b
Φβ|Fb
ThemorphismT iscalledanisomorphismifitisadiffeomorphism.
We denote by Hom(E,F) the space of bundle morphisms E → F. When E = F we set
End(E) := Hom(E,E). A gauge transformation of E is a bundle automorphism E → E. We
willdenotethespaceofgaugetransformationsofE byAut(E)orG .
E
We will denote by VBK(M) the set of isomorphism classes of smooth K-vector bundles over
M. (cid:116)(cid:117)
π
Definition1.1.5. AsubbundleofE → B isasmoothsubmanifoldF (cid:44)→ E withthepropertythat
π
F → B isavectorbundleandtheinclusionF (cid:44)→ E isabundlemorphism. (cid:116)(cid:117)
Definition1.1.6. SupposeE → M isarankr K-vectorbundleoverM. AtrivializationofE isa
bundleisomorphism
Kr → E.
M
ThebundleE iscalledtrivializableifitadmitstrivializations. Atrivialized vectorbundleisapair
(vectorbundle,trivialization). (cid:116)(cid:117)
Example1.1.7. (a)Abundlemorphismbetweentwotrivialvectorbundles
T : V → W
B B
isasmoothmap
T : B → Hom(V,W).
(b) If we are given two vector bundles over B described by gluing cocycles subordinated to the
sameopencover
(U,g ,V), (U,h ,W)
•• ••
thenabundlemorphismcanbedescribedasacollectionofsmoothmaps
T : U → Hom(V,W)
α α