Marathe K. Topics in Physical Mathematics

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408 Appendix D

and let p

: T

∗

M → M be the restriction of the canonical projection p :

∗

M → M to T

∗

M.Letp

∗

E (resp., p

∗

F ) be the pull-back of the bundle E

(resp., F )tothespaceT

∗

M with projection π

∗

(resp., π

∗

). Thus, we have

the diagram

∗





where σ is a morphism of vector bundles. The set of k-symbols is the set

(E,F) deﬁned by

(E,F):={σ ∈ Γ (Hom(p

∗

E,p

∗

F )) | σ(cα

,e)=c

σ(α

,e),

c>0,α

∈ T

∗

M}.

The k-symbol (or simply the symbol) of P denoted by σ

(P ) ∈ Sm

(E,F)

is deﬁned by

(P )(α

,e):=P (t)(x),

where (α

,e) ∈ p

∗

E and t ∈ Γ (E) is deﬁned as follows. We can choose

s ∈ Γ (E)andf ∈F(M ) such that s(x)=e and df (x)=α

;then

t =

(f −f(x))

It can be shown that P (t)(x) depends only on α

and e and is independent

of the various local choices made. We say that an operator P ∈ D

(E,F)

is elliptic if σ

(P ) is an isomorphism. In particular, if P is elliptic then

rank E =rankF .

The relation (D.4) suggests how to enlarge the class of diﬀerential op-

erators to a class (the class of pseudo-diﬀerential operators) large enough

to contain also the parametrix (a quasi-inverse modulo an operator of low

order) of every elliptic operator. Let k ∈ R;atotal symbol of order

k in R

is a (j × h)-matrix-valued smooth function p,withcomponents

r,s

: R

× R

→ C with compact support in the ﬁrst m-dimensional vari-

able, such that, for all multi-indices α, β, there exists a constant C

α,β

such

that

p(x, y)|≤C

α,β

(1 + |y|)

k−|β|

We denote by Sm

(j, h) the vector space of these total symbols of order k

represented by j × h matrices. Let p ∈ Sm

(j, h); we deﬁne the operator P

on S

with the relation (D.4). One can show that P is a linear continuous

operator from S

into S

and that P can be extended to a continuous

operator P

: H

) → H

s−k

), ∀s ∈ R,whereH

) denotes

vectorial Sobolev space with h components in H

). We will often write P

Appendix D 409

instead of P

. The operator P is called the pseudo-diﬀerential operator of

order k associated with p ∈ Sm

(j, h)andp is called the total symbol of P.We

observe that if l<kthen Sm

(j, h) is a subspace of Sm

(j, h). Let us denote as

(j, h)thespace∪

l<k

(j, h). If k ∈ Z then Sm

(j, h)=Sm

k−1

(j, h).

The k-symbol of P , denoted σ

(P ), is the class σ

(P )ofp in the quotient

(j, h)/ Sm

(j, h). We denote by Sm

−∞

(j, h)thespace

−∞

(j, h):=



(j, h).

We observe that p ∈ Sm

−∞

(j, h) is a matrix-valued function with com-

ponents in the Schwartz space S

, with compact support in the ﬁrst

(m-dimensional) variable. We say that P , with total symbol p,isanin-

ﬁnitely smoothing pseudo-diﬀerential operator if p ∈ Sm

−∞

(j, h). We de-

note by PD

(j, h)(resp.,PD

−∞

(j, h)) the space of pseudo-diﬀerential op-

erators of order k (resp., inﬁnitely smoothing pseudo-diﬀerential operators)

associated with total symbols in Sm

(j, h)(resp.,Sm

−∞

(j, h)). We observe

that, by the Sobolev lemma, if P ∈ PD

−∞

(j, h) then the j components

of Pf are smooth, ∀f ∈ H

), ∀s ∈ R. We say that two operators

P, Q ∈ PD

(j, h)areequivalent if P −Q ∈ PD

−∞

(j, h). One can show that,

if P ∈ PD

(h, i),Q∈ PD

(i, j)thenPQ ∈ PD

k+l

(h, j)andP

∗

∈ PD

(i, h).

The above theory can be extended to vector bundles as follows. Let E

(resp., F ) be a Hermitian vector bundle of rank h (resp., j)overanm-

dimensional compact diﬀerential manifold M.LetP : Γ (E) → Γ (F )bea

linear operator from sections of E to sections of F . The operator P is called a

(linear) pseudo-diﬀerential operator of order k from E to F if locally,

in any trivialization

: U × C

→ π

−1

(U),ψ

: U × C

→ π

−1

(U)

and any chart φ : U → R

, the induced operator from sections of φ(U) ×

→ φ(U) to sections of φ(U) × C

→ φ(U) extends to an operator

that is in PD

(j, h) modulo inﬁnitely smoothing operators. We denote by

(E,F)thespace of pseudo-diﬀerential operators from E to F .A

total symbol of order k from E to F is an element p ∈ Γ (Hom(π

∗

E,π

∗

F ))

that, in any trivialization ψ

,ψ

,φ as above, induces ˜p ∈ Sm

(j, h). We

denote by Sm

(E,F)thespace of symbols of order k from E to F .In

analogy with the deﬁnition of Sm

(j, h) we may deﬁne Sm

(E,F). One

can show that for any operator P ∈ PD

(E,F) there exists a well-deﬁned

class σ

(P ) ∈ Sm

(E,F)/ Sm

(E,F), which is called the k-symbol (or

the principal symbol or simply the symbol) of P . For diﬀerential operators

from E to F this deﬁnition coincides with the previous one. We say that

P ∈ PD

(E,F) is elliptic if in the class of σ

(P ) there is an element which,

for each (x, y) ∈ T

∗

M, is an isomorphism of E

onto F

.Thus,ifP is elliptic,

rank E =rankF .WedenotebyEl

(E,F) the space of elliptic operators from

E to F .Aparametrix for P ∈ El

(E,F) is a pseudo-diﬀerential operator

410 Appendix D

Q ∈ PD

(F, E) such that

QP = id −S

and PQ = id − S

where S

and S

are inﬁnitely smoothing operators on E and F , respectively,

i.e.,

∈ PD

−∞

(E,E)andS

∈ PD

−∞

(F, F ).

We observe that a parametrix of P is an inverse of P modulo inﬁnitely

smoothing operators. One can show that every elliptic operator over a com-

pact manifold admits a parametrix which is unique up to equivalence. An

important consequence of the existence of the parametrix for an operator

P ∈ El

(E,E)withk>0 is the fact that every eigenfunction u of P is

smooth.

In order to state some fundamental properties of elliptic operators we

now recall some basic facts about the theory of continuous linear operators

between Hilbert spaces. Let H, K be Hilbert spaces and T : H → K a

linear continuous map. The adjoint T

∗

of T is the (unique) linear continuous

operator T

∗

: K → H such that

(x|Ty)=(T

∗

x|y), ∀x ∈ K, ∀y ∈ H.

If H, K are only pre-Hilbert spaces, then an operator T

∗

satisfying the above

relation does not necessarily exist, and if it does exist, it need not be unique.

The operator T is said to be compact if, for every bounded sequence {u

}

in H, the sequence {Tu

} contains a convergent subsequence. The adjoint of

a compact operator is compact and the composition of a compact operator

with a continuous operator is compact. Furthermore, if the range Im T of

T is closed, then dim Im T<∞. The operator T is said to be of trace

class if it is compact and the sequence of eigenvalues of (T

∗

T )

1/2

,counted

with their multiplicity, is summable. The operator T from a Hilbert space

H to a Hilbert space K is said to be a Fredholm operator if ker T and

ker T

∗

are ﬁnite-dimensional. It follows that, if K is inﬁnite-dimensional,

a continuous operator T cannot be compact and Fredholm. A continuous

operator P : H → K is Fredholm if and only if there exists a continuous

operator Q : K → H and compact operators S

: H → H, S

: K → K such

that

QP = id −S

and PQ = id − S

. (D.5)

Thus, Fredholm operators are the continuous operators that are invertible

modulo compact operators. In particular, from equation (D.5)itfollowsthat

the operator Q is Fredholm. If P is a Fredholm operator the index of P ,

denoted by Ind(P), is deﬁned by

Ind(P )=dimkerP − dim ker P

∗

. (D.6)

Appendix D 411

If P : H → K and Q : L → H are Fredholm operators then PQ : L → K is

a Fredholm operator and

Ind(PQ) = Ind(P ) + Ind(Q).

If S is a compact operator then id − S is a Fredholm operator and a simple

calculation shows that Ind(id − S) = 0. Thus, for the Fredholm operator Q

in (D.5)wehave

Ind(PQ)=0

and hence

Ind(Q)=−Ind(P ).

We observe that, by deﬁnition, if P is Fredholm then P

∗

is also Fredholm

and

Ind(P

∗

)=−Ind(P).

If P ∈ PD

(E,F), then P has a well-deﬁned extension to a continuous

operator P

: H

(E) → H

s−k

(F ), ∀s. It is customary to denote P

simply

by P when the extension is clear from the context. The following theorem

contains some important properties of pseudo-diﬀerential operators.

Theorem D.2 Let E (resp., F ) be a Hermitian vector bundle over a com-

pact Riemannian manifold M and let P ∈ PD

(E,F). Then we have the

following:

1. There exists a unique P

∗

∈ PD

(F, E) such that



v, Pu



P

∗

v, u

,u∈ Γ (E),v∈ Γ (F ),

where  , 

(resp.  , 

) denotes the inner product in the ﬁbers of E

(resp., F ). P

∗

is called the formal adjoint of P . Moreover, σ

∗

(σ

(P ))

∗

2. The operator P is elliptic if and only if its formal adjoint P

∗

is elliptic.

In this case, for all s ∈ R,P

is a Fredholm operator, ker P and ker P

∗

are ﬁnite-dimensional and if we deﬁne Ind(P ):=dimkerP − dim ker P

∗

we have

Ind(P

) = Ind(P ).

3. We say that two k-symbols σ

,σ

are regularly homotopic if there exists

a homotopy σ

, 0 ≤ t ≤ 1, such that σ

(α

) is an isomorphism for all

∈ T

∗

M.Then,ifP ∈ El

(E,F), its index depends only on the regular

homotopy class of its symbol.

4. Let F = E,letE

:= ker(P −λid) denote the eigenspace associated to the

eigenvalue λ ∈ C and let V be the set of eigenvalues of P .IfP ∈ El

(E,F)

is self-adjoint and k>0, then each eigenspace of P is ﬁnite-dimensional

and each eigenvector of P is a smooth section of E.Furthermore,V is not

bounded and the Hilbert space L

(E) is a direct sum of the eigenspaces of

412 Appendix D

P , i.e.,

(E)=



λ∈V

The above theory for a single operator can be generalized to apply to diﬀer-

ential complexes that we now deﬁne. Let E

,...,E

be a set of Riemannian

vector bundles over a compact connected Riemannian manifold (M,g)and

let

: Γ (E

) → Γ (E

q+1

),q=0,...,n−1

be a set of pseudo-diﬀerential operators of some ﬁxed order k, such that

q+1

◦ L

=0, 0 ≤ q ≤ n − 2. (D.7)

The ﬁnite cochain complex (see Appendix C)

0 −→ Γ (E

)

−→ Γ (E

)

−→···

n−1

−→ Γ (E

) −→ 0

is called a diﬀerential complex of order k (or simply a diﬀerential com-

plex) and is denoted by (Γ (E),L)orsimplyby(E,L). We observe that the

generalized de Rham sequence (see Chapter 4) is an example of a sequence

which fails to be a complex, the obstruction being given by the curvature.

If (E,L) is a diﬀerential complex, then a q-cocycle (resp., q-coboundary) of

(E,L)isanelementofZ

(E,L)=kerL

(resp., B

(E,L)=ImL

q−1

). A

qth cohomology class is an element of the q-th cohomology space H

(E,L)=

(E,L)/B

(E,L)ofthecomplex(E,L). The symbol sequence associ-

ated to the diﬀerential complex (E,L)oforderk is the following sequence of

vector bundles and homomorphisms of vector bundles

0 −→ p

∗

)

−→ p

∗

)

−→ ···

n−1

)

−→ p

∗

) −→ 0.

The diﬀerential complex is said to be elliptic if its associated symbol se-

quence is exact, i.e.,

Im σ

q−1

)=kerσ

),q=0, 1,...,n.

A single elliptic operator may be considered an elliptic complex. Given the

elliptic complex (E,L) we deﬁne the jth Laplacian, or Laplace operator

,ofthecomplex(E,L)by

:= L

∗

+ L

j−1

∗

j−1

, 0 ≤ j ≤ n. (D.8)

The operator Δ

is an elliptic operator for all j. It is self-adjoint (i.e., Δ

∗

) with respect to the inner product deﬁned by

σ

,σ

 :=



σ

(x),σ

(x)

,σ

∈ Γ (E

Appendix D 413

where  , 

is the metric on E

. Some properties of Laplace operators that

are relevant to physical applications are given in [315, 323]. The solutions

of Δ

(s) = 0 are called harmonic sections as in the particular case of the

de Rham complex. One can show that the Hodge decomposition theorem

is valid also for general elliptic complexes (see, for example, Gilkey [154]or

Wells [398]). Thus, the space of all harmonic sections is a ﬁnite-dimensional

subspace of Γ (E

) isomorphic to the jth cohomology space H

(E,L). The

index of the elliptic complex (E,L) is deﬁned by

Ind

(E,L)=



j=0

(−1)

dim H

(E,L). (D.9)

In the case of an elliptic complex consisting of a single elliptic operator P

this deﬁnition reduces to the deﬁnition of the index of P given earlier. Deﬁ-

nition (D.9) of the index is formulated using the spaces of solutions of diﬀer-

ential equations and is therefore called the analytic index. The subscript

“a”inInd

(E,L) refers to this analytic aspect of the deﬁnition. One can also

deﬁne the topological index Ind

(E,L), which can be expressed in terms

of topological invariants (characteristic classes) associated to the complex

(E,L). It turns out that the analytic index and the topological index of the

elliptic complex (E,L) coincide, i.e.,

Ind

(E,L)=Ind

(E,L). (D.10)

This is the content of the classical Atiyah–Singer index theorem and its ex-

tensions. In the following example we apply this discussion to the de Rham

complex.

Example D.1 Recall that the de Rham complex of an m-dimensional man-

ifold M is the diﬀerential complex

0 −→ Λ

(M)

−→ Λ

(M)

−→···

−→ Λ

(M) −→ 0.

The operator d is a diﬀerential operator of order 1 and its symbol is given by

(d)(α

,β

)=iα

∧ β

,β

∈ Λ

(M)

, ∀j.

It is easy to verify that the corresponding symbol sequence is exact. Thus,

the de Rham complex (Λ(M ),d) is an elliptic complex. The topological index

Ind

(Λ, d) of the de Rham complex is deﬁned by

Ind

(Λ, d)=



k=0

(−1)

(M)=χ(M),

where b

(M) is the kth Betti number of M and χ(M) is the Euler character-

istic of M.Fromequation(D.9), we have

414 Appendix D

Ind

(Λ, d)=



j=0

(−1)

dim H

(Λ, d).

By Hodge theory, H

(Λ, d) can be identiﬁed with the space of harmonic j-

forms and hence with the corresponding de Rham cohomology spaces. Now

recall from Chapter 2 that, by the classical de Rham theorem, the de Rham

cohomology is in fact topological, i.e., does not depend on the diﬀerential

structure and is isomorphic to the singular cohomology with real coeﬃcients,

i.e., H

deR

(M)

∼

(M; R), ∀q. Thus, we have a topological characterization

of the analytic index Ind

(Λ, d). This may be regarded as a very special case

of the index theorem.

One of the most extensively studied diﬀerential operators in mathematical

physics is the Dirac operator. We devote the next section to a study of this

operator by using the bundle of spinors on a Riemannian manifold.

D.4 The Dirac Operator

Spinors were ﬁrst introduced by physicists to study representations of the

universal covering group Spin(3) = SU(2) of the rotation group SO(3). The

non-relativistic theory of spin was developed by Pauli. Pauli’s spin matrices

are the generators of the spin group SU(2), which was interpreted as repre-

senting an intrinsic angular momentum of the electron. The relativistic wave

equation for the electron was discovered by Dirac [102]. The wave functions

of the electron occurring in Dirac’s equation belong to the 4-dimensional rep-

resentation of Spin(3, 1) = SL(2, C), which is the universal covering group,

of the connected component of the identity of the Lorentz group. The formal

treatment of spinors in these and other related works was replaced by a geo-

metrical deﬁnition by E. Cartan. In the introduction to his famous book on

the theory of spinors [71], Cartan writes:

... because of this geometrical origin, the matrices used by physicists in

Quantum Mechanics appear of their own accord, and we can grasp the

profound origin of the property, possessed by Cliﬀord algebras, of repre-

senting rotations in space having any number of dimensions. Finally this

geometrical origin makes it very easy to introduce spinors into Rieman-

nian geometry, and particularly to apply the idea of parallel transport

to these geometrical entities.

It is this generalized Dirac operator that plays a fundamental role in index

theory and in particular in the study of moduli spaces of gauge ﬁelds and in

the Seiberg–Witten theory.

Our discussion of the Dirac operator is based on Cliﬀord bundles over

Riemannian manifolds. A typical ﬁber of a Cliﬀord bundle is a Cliﬀord

Appendix D 415

algebra.Letn = r + s and let g

be the standard pseudo-metric on R

signature (r, s). Let Pin(r, s) be the group of the elements of C(r, s)thatare

products of a ﬁnite number of elements γ(v) ∈ γ(R

) ⊂ C(r, s), such that

(v, v)=±1. The group Pin(r, s) is called the Pin group of (R

). The

subgroup of Pin(r, s) consisting of the elements which are products of an even

number of elements in γ(R

) is called the spinor group of (R

)andis

denoted by Spin(r, s). Let α : R

→ C(r, s) be the map deﬁned by

α(v)=−v, ∀v ∈ R

This map extends to a unique algebra automorphism of C(r, s) such that

= id.Letu ∈ Pin(r, s). By identifying R

with γ(R

) we deﬁne the map



: R

→ R



(v):=α(u)vu

−1

∈ γ(R

)

∼

One can show that



is an invertible linear map of R

into itself. In fact,

it turns out that



∈ O(r, s), ∀u ∈ Pin(r, s).

The map



Ad : u →



of Pin(r, s) → O(r, s)

is called the twisted adjoint representation of Pin(r, s). It is a surjective

homomorphism whose restriction to Spin(r, s)is,forn>2, a two-sheeted

universal covering of SO(r, s). In what follows we shall restrict ourselves to

the case of signature (n, 0) and denote Spin(n, 0) by Spin(n). The general

case can be treated along similar lines.

Let E be an oriented Riemannian vector bundle of rank n>2andlet

SO(E) be the bundle of oriented orthonormal frames in E.Aspin structure

Spin(E)onE is a universal extension of the bundle SO(E) to the group

Spin(n). The bundle Spin(E) is a principal Spin(n)-bundle over M and is

called the spin frame bundle of E.

Theorem D.3 Let E be an oriented Riemannian vector bundle over a man-

ifold M. E admits a spin structure if and only if its second Stiefel–Whitney

class w

(E) is zero. Furthermore, if w

(E)=0, then the spin structures on

E are in one-to-one correspondence with the elements of H

(M; Z

If M is an oriented Riemannian manifold we deﬁne a spin structure on M

to be a spin structure on TM.Aspin manifold is an oriented Riemannian

manifold M with w

(M):=w

(TM)=0, together with a spin structure on

M. We will denote by Spin(M ) the corresponding spin structure on TM.By

abuse of language Spin(M) is called the spin frame bundle of M.

Let E be an oriented Riemannian vector bundle of rank n;theCliﬀord

bundle of E, denoted by Cl(E), is the associated bundle (C(n) ≡ C(n, 0))

Cl(E):=SO(E) ×

C(n),

416 Appendix D

where r : SO(n) → Aut(C(n)) is the canonical representation of SO(n)into

the automorphism group of C(n). It is easy to see that Cl(E)

= C(E

), for

all x in the base space of E.Thus,Cl(E) is a bundle of Cliﬀord algebras. If

the bundle E admits a spin structure with Spin(n)-bundle Spin(E), a real

(resp., complex) spinor bundle of E is an associated vector bundle of the

form

Spin(E) ×

where ρ is a representation of Spin(n) in the homomorphisms of the real

(resp., complex) vector space V . Diﬀerent spinor bundles of E may be ob-

tained with diﬀerent choices of Spin(E)andρ. We will denote by S(E, ρ)(or

simply S(E)) the spinor bundle of E deﬁned by ρ. We observe that the vector

space V of the representation ρ is a left module over C(n). Thus, S(E)

a left module over C(E

)

∼

C(n)andS(E) is a bundle of left modules. We

observe that the Cliﬀord bundle Cl(E) is a particular case of spinor bundle

because

Cl(E)=Spin(E) ×

C(n),

where Ad is the representation of Spin(n) into the automorphisms of C(n)

such that Ad

v = uvu

−1

,u∈ Spin(n) ⊂ C(n). It follows that Ad

−1

= id.

Furthermore, the canonical vector space isomorphism of the exterior algebra

Λ(R

)withC(n) (not an algebra isomorphism) gives rise to a canonical

isomorphism

Λ(E)

∼

Cl(E). (D.11)

Let M be an oriented Riemannian m-manifold with w

(M)=0andlet

S(M):=S(TM) be a real spinor bundle of TM. Let us suppose that S(M)

is Riemannian and has a Riemannian connection with covariant derivative ∇.

The Dirac operator D relative to ∇ is the ﬁrst order diﬀerential operator

D on sections of spinor bundle Γ (S(M)) deﬁned by

Dσ :=



j=1

·∇

σ, σ ∈ Γ (S(M)), (D.12)

where, {e

,...,e

} is an orthonormal basis of T

M ∀x ∈ M,and· denotes

Cliﬀord multiplication. In Minkowski space the Dirac operator has the ex-

pression

D =



∂

. (D.13)

From this it follows that D

= g

∂

, which is the Laplacian in Minkowski

space. Thus, the Dirac operator can be regarded as the square root of the

Laplacian.

The Dirac operator is an elliptic operator with symbol

(D)(u

)=iu

·v

∈ C(T

M).

Appendix D 417

We observe that a Riemannian connection always exists on SO(M). In fact,

by the fundamental theorem of Riemannian geometry, SO(M)(M is a Rie-

mannian manifold) admits a canonical torsion-free Riemannian connection

which can be lifted to Spin(M) and thus to any spinor bundle on M.Thus,

we have a canonical Riemannian connection also on Cl(M):=Cl(TM). Un-

der the canonical isomorphism (D.11)withE = TM, the Dirac operator

D relative to the canonical Riemannian connection on Cl(M) satisﬁes the

relation

∼

d + δ.

A Dirac bundle over a Riemannian manifold M is a Riemannian spinor

bundle S(M)ofTM with a Riemannian connection ω on S(M) such that:

1. for all x ∈ M and for all unit vectors e ∈ T

M we have

e ·u

,e· v

 = u

, ∀u

∈ S(M)

;

2. if ∇ is the covariant derivative of the canonical Riemannian connection on

M,then

∇

(φ ·σ)=∇φ · σ + φ ·∇

σ.

The space Γ

(S(M)) of sections of S(M) with compact support has an inner

product deﬁned by

(σ

,σ

):=



σ

,σ

. (D.14)

With respect to this inner product, the Dirac operator of a Dirac bundle is

formally self-adjoint, i.e.

(Dσ

,σ

)=(σ

, Dσ

),σ

,σ

∈ Γ

(S(M)).

The Dirac operator of a Dirac bundle S(M) over a complete Riemannian

manifold M can be extended to the space L

(S(M)) of square integrable

sections of S(M ) and one can show that this extension is essentially self-

adjoint, i.e., the closure of D is self-adjoint. We have given several examples

of index theorems in Chapter 5.