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Dirac operators in Riemannian geometry

Thomas Friedrich (Humboldt University Berlin)

Torino, February 2014

2

General References

Th. Friedrich, Dirac Operators in Riemannian Geometry, GraduateStudies in Mathematics No. 25, AMS 2000.

This book contains 275 references up to the year 2000

N. Ginoux, The Dirac Spectrum, Lecture Notes No. 1976, Springer 2009.

This book contains 240 references on eigenvalues of the Dirac operator up to the year

2009

H. Baum, Th. Friedrich, R. Grunewald, I. Kath, Twistor and Killingspinors on Riemannian manifolds, Teubner-Verlag Leipzig/Stuttgart1991.

This book contains 107 references on Twistor and Killing spinors up to the year 1991

3

Content

1. Motivation

2. Clifford algebras

3. Spin structures

4. The Dirac operator

5. Conformal changes of the metric

6. The index of the Dirac operator

7. Eigenvalue estimates for the Dirac operator

8. Riemannian manifolds with Killing spinors

9. The twistor operator

10. Intrinsic upper bounds for metrics on S2 and T 2

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11. Surfaces, mean curvature and the Dirac operator

12. Dirac operators depending on connections with torsion

13. The Casimir operator

14. Eigenvalue estimates for (/D)2 via deformations

15. Eigenvalue estimates for (/D)2 via twistor operator

16. Killing and twistor spinors with torsion

17. Remarks on the second Dirac eigenvalue

5

Motivation

a) From complex analysis: Consider the Cauchy-Riemann operators

=1

2(x iy), =

1

2(x + iy)

Define a differential operator P : C(R2; C2) C(R2; C2) by

P

[fg

]

= 2i

[gf

]

=

[0 ii 0

]

x

x

[fg

]

+

[0 11 0

]

y

y

[fg

]

Then x, y satisfy the Clifford relations

2x = 2y = Id, x y + y x = 0

and 4 = 4 = (Laplacian).

6

More generally: (M2n, g, J) Kahler manifold, 1 = 1,0 0,1 with

1,0 = { : (JX) = i(X)}, 0,1 = { : (JX) = i(X)},

and df = pr1,0(df) + pr0,1(df) =: f + f

Then: 2( + ) = .

Question: Does there exist a generalization of the Cauchy-Riemannoperator on a more general class of manifolds?

b) From theoretical physics: Consider a free classical particle with

m : mass, p = vm1v2/c2

: momentum, E : Energy.

Then special relativity predicts the relation

E =

c2p2 +m2c4.

7

According to the quantization rules of quantum mechanics:

E i~t, p i~, both acting on some state function

i~t =c2~2 +m2c4 Dirac equation

Question: What is the meaning of the square root?

c) From topology:

Theorem (Freedman 1982):Any unimodular quadratic form L over Z can be realized as theintersection form L = H2(X4; Z) of a 4-dimensional, compact andsimply connected topological manifold X4.

Theorem (Rochlin 1950): If M4 is smooth, closed manifolds. t. 2(M

4) = 0 then (M4) = 0 mod 16.

Theorem (Hirzebruch)1

8(M4) =

1

24

M4p1.

8

Example:

E8 =

2 1 0 0 0 0 0 01 2 1 0 0 0 0 0

0 1 2 1 0 0 0 00 0 1 2 1 0 0 00 0 0 1 2 1 0 10 0 0 0 1 2 1 00 0 0 0 0 1 2 00 0 0 0 1 0 0 2

E8 0 is of typ II and (E8) = dimE8 = 8.

Question: Does there exist a vector bundle S M4 and an ellipticdifferential operator D : (S) (S) s. t.

t-index(D) = 18(M4),

a-index(D) = dimkerD dimcokerD = 0 mod 2 ?

9

Clifford algebras

(Rn, g), e1, . . . , en an orthonormal basis. Then the (finite dimensional!)associative algebra

Cl(Rn) :=

Rn/{ei ej + ej ei = 0, e21 = 1}

is called the Clifford algebra of Rn. ClC(Rn) denotes its complexification.

Example. n = 2, g : standard euclidean scalar product.

Then e1 7 x, e2 7 y shows: ClC(R2) = MC(2)

ClC(R2) acts on C2 by endomorphisms. More generally:Thm. There exists a unique representation of smallest dimension of thealgebra ClC(Rn) on a complex vector space n:

ClC(Rn) End(n), dimn = 2[n/2].

n : space of (Dirac) spinors

10

The Spin(n) group is a two-fold covering of SO(n) and can be realizedin Cl(Rn),

Spin(n) ={x1 .... x2l, xi Rn and |xi| = 1

}.

Every vector x Rn acts on n by an endomorphism:

Rn n (x, ) 7 x n : Clifford multiplication

: Rn n n . The Spin(n)-representation Rn n splits into

Rn n = n ker() .

11

There is a universal projection of Rn n onto ker(),

p(x ) = x + 1n

n

i=1

ei ei x .

This splitting yields two differential operators of first order, the Diracoperator and the twistor operator .

If n = 2k is even, then the Spin(n) representation splits into twoirreducible pieces,

2k = +2k 2k , x : 2k 2k .

12

Additional Spin(n)-invariant structures in n:

n real structures quaternionic structures

commutes with n 6, 7 mod 8 n 2, 3 mod 8Clifford multiplication

anti-commutes with n 0, 1 mod 8 n 4, 5 mod 8Clifford multiplication

Proposition:The representation 8k admits a Spin(8k)-invariant real structure.

The representation 8k+4 admits a Spin(8k+ 4)-invariant quaternionicstructure.

13

Spin Structures.

Idea: Attach a copy of n to every point x of a Riemannian manifold(Mn, g):

Tangent bundle:T (Mn) =

xMnTxM

n

Spinor bundle:S(Mn) =

xMnn(x)

However:

x

TxMn

n(x)

Mn

Denote by F(Mn, g) the oriented frame bundle. Mn admits a spinstructure iff the SO(n)-principal bundle F admits a reduction P F tothe group Spin(n) SO(n).

notion of a Riemannian spin manifold

14

Different spin structures:

Suppose that a discrete group acts properly discontinuous on a manifoldMn and denote by : Mn Mn := /Mn the projection onto theorbit space. Moreover, suppose that Mn admits a spin structure P. Theinduced bundle

(P) ={(m, p) Mn P : (m) = 1(p)

}

is a Spin(n)-principal bundle with the action of the spin group

(m , p) g = (m , p g) , g Spin(n) .

Moreover, acts on (P) via

(m , p) = ( m , p)

and the fixed spin bundle can be reconstructed,

P = /(P) .

15

Consider a homomorphism : {1,1} Spin(n) and introduce anew -action via the formula

(m , p) = ( m , p ()) .

The spaceP := /(P)

is still a Spin(n)-principal fiber bundle over Mn, a new spin structure ofthe manifold.

If Mn is the universal covering of Mn , then the group is isomorphicto the fundamental group 1(M

n) of Mn. In particular we proved

Theorem: If Mn admits at least one spin structure, then all spinstructures correspond to the set

Hom(1(Mn) , Z2) = H

1(Mn ; Z2) .

16

Existence of a spin structure

Consider the classifying map f : Mn BSO(n) of the tangent bundle.Mn admits a spin structure iff f lifts into the classifying space BSpin(n).Since

2(BSpin(n)) = 1(Spin(n)) = 0 and 1(BSpin(n)) = 0

we haveH2(BSpin(n); Z2) = H1(BSpin(n); Z2) = 0. Consequently, the

image of the second Stiefel-Whitney class 2 H2(BSO(n); Z2) underthe map H2(BSO(n); Z2) H2(BSpin(n); Z2) is zero. This argumentyields a necessary condition for the existence of a spin structure, namely2(M

n) = 0. Indeed , the condition is sufficient, too.

Theorem: An oriented manifold admits a spin structure iff its secondStiefel-Whitney class vanishes, 2(M

n) = 0.

17

Examples:

Sn, C(P )2n+1, . . . are spin manifolds with a unique spin structure.

Tn admits 2n different spin structures.

C(P )2n, SU(3)/SO(3), . . . are not spin manifolds.

Let (Mn, g,P) be a Riemannian spin manifold with a fixed spin structure.

The associated bundle

S := P Spin(n) n .

is the spinor bundle S .

The Levi-Civita connection can be lifted from the tangent bundle tothe spinor bundle S in a unique way.

18

Dirac operator

In an orthonormal frame e1, . . . , en

D : (S) (S), D = , D =n

i=1

ei ei.

Properties of D:

D is an elliptic differential operator of first order

D2 = S + 14Scal (Schrodinger 1932, Lichnerowicz 1962)

For the Laplacian q on differential forms in q(Mn), Hodge - de Rham

theory implies that

dimker(q) =: bq(Mn) is a topological invariant.

For the Dirac operator, dimker(D) is not a topological invariant.

19

Basic Example: (see Hitchin 1974)

Consider the Lie group Spin(3) = S3 and the basis e1, e2, e3 of its Liealgebra with the commutator relations

[e1 , e2] = 2 e3 , [e2 , e3] = 2 e1 , [e3 , e1] = 2 e2 .

We introduce a left invariant metric defined by the conditions

|e1| = |e2| = 1 , |e3| = , ei , ej = 0 if i 6= j .

The eigenvalues of the Dirac operator are given by the formulas

p() =p

+

2, p = 1, 2 . . . with multiplicity 2p

p,q() =

2 1

4pq2 + (p q)2 , p = 1, 2, . . . , q = 0, 1, . . .

with multiplicity (p+ q) .

20

The kernel of the Dirac operator corresponds to p,q = 0, i.e.

4 = 4(4pq2 + (p q)2

).

If the parameter is a transcendent number, then the kernel of theDirac operator is trivial .

If = 4p is an integer divisible by 4, then p = q is a solution. In thiscase the dimension of the kernel of the Dirac operator is at least 2p.

Remark that

lim0

p() = , lim0

p,p() = 2p , lim0

p,q() = .

21

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