Marathe K. Topics in Physical Mathematics

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398 Appendix C

regarded as a Z-module. Thus, the category M

is the same as the category

of Abelian groups. As we will see in the next paragraph, the category of

P-modules provides a general setting for deﬁning a (co)chain complex and

(co)homology modules generalizing the classical theories. The notion of an

Abelian category, which is a generalization of the category of P-modules,

arose from Grothendieck’s attempt to unify diﬀerent cohomology theories

such as sheaf and group cohomologies by identifying the basic properties

needed in their deﬁnitions. His attempt culminated in K-theory, which has

found applications in algebraic and analytic geometry and more recently in

theoretical physics.

A chain complex over P is a pair (C

∗

,δ), where C

∗

= {C

| q ∈ Z} is a

family of P-modules and δ = {δ

: C

→ C

q−1

| q ∈ Z} is a family of P-linear

maps such that

q−1

◦ δ

=0, ∀q ∈ Z. (C.1)

An element of C

is called a q-chain.TheP-linear map δ

is called the qth

boundary operator. One usually omits the subscript for the δ

’s and also

writes δ

:= δ ◦ δ = 0 to indicate that equation (C.1)istrue.Thechain

complex (C

∗

,δ) is also represented by the following diagram:

···

∂

q−1

←− C

q−1

∂

←− C

∂

q+1

←− C

q+1

∂

q+2

←− ···

Let (C

∗

,δ) be a chain complex over P.TheP-module

∗

,δ):=Kerδ

is called the P-module of q-cycles and the P-module

∗

,δ):=Imδ

q+1

is called the P-module of q-boundaries.TheP-module

∗

,δ)=Z

∗

,δ)/B

∗

,δ)

is called the qth homology P-module of the chain complex. We will simply

write Z

, B

,andH

instead of Z

∗

,δ), B

∗

,δ), and H

∗

,δ), respec-

tively, when the complex (C

∗

,δ) is understood. The q-cycles z, z



∈ Z

are

said to be homologous if z −z



is a q-boundary, i.e., z −z



∈ B

. The family

∗

:= {H

| q ∈ Z} is called the Z-graded homology module or simply

the homology of the chain complex (C

∗

,δ). A chain complex (C

∗

,δ), is said

to be exact at C

,if

Ker δ

=Imδ

q+1

The chain complex is said to be exact if it is exact at C

, ∀q ∈ Z. Thus

the homology of a chain complex is a measure of the lack of exactness of the

chain complex.

Appendix C 399

Giventhechaincomplexes(C

∗

,δ

), (C

∗

,δ

)overP, a chain morphism

of (C

∗

,δ

)into(C

∗

,δ

) is a family f

∗

= {f

: C

→ C

| q ∈ Z} of P-linear

maps such that, ∀q ∈ Z, the following diagram commutes:

q−1

That is,

◦ f

= f

q−1

◦δ

, ∀q ∈ Z. (C.2)

It is customary to indicate the above chain morphism f

∗

by the diagram

∗

−→ C

∗

Using equation (C.2) we can show that a chain morphism f

∗

= {f

: C

→

| q ∈ Z} induces a family of P-linear maps

∗

)={H

∗

):H

∗

,δ

) → H

∗

,δ

) | q ∈ Z}

among the homology modules. We observe that chain complexes and chain

morphisms form a category and H

∗

is a covariant functor from this category

to the category of graded P-modules and P-linear maps.

Let (C

∗

,δ) be a chain complex and let D

∗

= {D

⊂ C

| q ∈ Z} be a

family of submodules such that δ

) ⊂ D

q−1

, ∀q ∈ Z. Then (D

∗

,δ

∗

)is

called a chain subcomplex of (C

∗

,δ). Let

∗

= {C

| q ∈ Z},

δ = {

| q ∈ Z},

where

: C

→ C

q−1

is the map induced by passage to the

quotient, i.e.,

(α + D

)=δ

(α)+D

q−1

. Then (C

∗

δ) is called the

quotient chain complex of (C

∗

,δ)byD

∗

Recall that a short exact sequence of P-modules is a sequence of P-

modules and P-linear maps of the type

0 −→ C

−→ D

−→ E −→ 0

such that f is injective, g is surjective and Im f =Kerg.Ashort exact

sequence of chain complexes is a sequence of chain complexes and chain

morphisms

0 −→ C

∗

−→ C

∗

−→ C

∗

−→ 0

such that, ∀q ∈ Z, the following sequence of modules is exact:

400 Appendix C

0 −→ C

−→ C

−→ 0.

We observe that if D

∗

is a subcomplex of C

∗

, then we have the following

short exact sequence of complexes:

0 −→ D

∗

−→ C

∗

−→ C

∗

−→ 0

where i

∗

is the inclusion morphism and π

∗

is the canonical projection mor-

phism. Let

0 −→ C

∗

−→ C

∗

−→ C

∗

−→ 0

be a short exact sequence of chain complexes and let us denote by H

the

homology modules H

∗

,δ

),i=1, 2, 3. It can be shown that there exists

a family

∗

= {h

: H

→ H

q−1

| q ∈ Z}

of linear maps such that the following homology sequence is exact:

···−→H

−→ H

q−1

−→···

The family h

∗

is called the family of connecting morphisms associated

to the short exact sequence of complexes. The corresponding homology se-

quence indicated in the above diagram is called the long exact homology

sequence.

If we consider the construction of chain complexes, but with arrows re-

versed, we have cochain complexes, which we now deﬁne. A cochain com-

plex over P is a pair (C

∗

,d), where C

∗

= {C

| q ∈ Z} is a family of

P-modules and d = {d

: C

→ C

q+1

| q ∈ Z} is a family of P-linear maps

such that

q+1

◦ d

=0, ∀q ∈ Z. (C.3)

The P-linear map d

is called the qth coboundary operator. As with chain

complexes, one usually omits the superscript for the d

and also writes d

d ◦d = 0 to indicate that equation (C.3) is true. The cochain complex (C

∗

,d)

is also represented by the diagram

···

q−2

−→ C

q−1

−→ C

q+1

−→ ···

Let (C

∗

,d) be a cochain complex over P.TheP-module

∗

,d):=Kerd

is called the P-module of q-cocycles and the P-module

∗

,d):=Imd

q−1

is called the P-module of q-coboundaries.TheP-module

Appendix C 401

∗

,d)=Z

∗

,d)/B

∗

,d)

is called the qth cohomology P-module of the cochain complex. Cochain

morphisms are deﬁned by an obvious modiﬁcation of the deﬁnition of chain

morphisms. We observe that cochain complexes and cochain morphisms form

a category and H

∗

is a contravariant functor from this category to the cate-

gory of graded P-modules and P-linear maps.

Let (C

∗

,δ) be a chain complex. Let C



be the algebraic dual of C

and

the transpose of δ.Then(C

∗

,d) deﬁned by

∗

= {C



| q ∈ Z} ,d= {

| q ∈ Z},

is a cochain complex. This cochain complex is called the dual cochain com-

plex of the the chain complex (C

∗

,δ).

As we indicated earlier, when P = Z,theP-modules are Abelian groups.

In this case one speaks of homology and cohomology groups instead of Z-

modules.

In many mathematical and physical applications the notions of chain and

cochain complexes that we have introduced are not suﬃcient and one must

consider multiple complexes. We shall only consider the double com-

plexes, also called bicomplexes, which may combine the chain and cochain

complex in one double complex. We consider this situation in detail. A mixed

double complex is a triple (C

∗

,d,δ)whereC

∗

= {C

| p, q ∈ Z} is a family

of P-modules and δ = {δ

: C

→ C

q−1

| p, q ∈ Z}, d = {d

: C

→ C

p+1

p, q ∈ Z} are families of P-linear maps such that

q−1

◦δ

=0,d

p+1

◦ d

=0,δ

p+1

◦ d

= d

q−1

◦ δ

, ∀p, q ∈ Z.

The double chain complex and double cochain complex can be deﬁned sim-

ilarly (see Bott and Tu [55] for further details). It is customary to associate

to this double complex a cochain complex (T

∗

,D) called the total cochain

complex,where



| p − q = k},

and

= d

+(−1)

where



| p −q = k},δ



{δ

| p − q = k}.

Since d

(⊕ω

)=⊕d

∈ T

k+1

and δ

(⊕ω

)=⊕δ

∈ T

k+1

,wehave

that D

maps T

to T

k+1

. It is easy to verify that D

= 0, i.e., D

k+1

◦D

0, ∀k. D is called the total diﬀerential. A slight modiﬁcation of the above

construction is used in [236] to give an interpretation of the BRST operator

in quantum ﬁeld theory as a total diﬀerential D of a suitable double complex.

Appendix D

Operator Theory

D.1 Introduction

It is well known that pure gauge theories cannot describe interactions that

have massive carrier particles. A resolution of this problem requires the in-

troduction of associated ﬁelds. We discussed these ﬁelds and their couplings

in Chapter 6. As with pure gauge ﬁelds, the coupled ﬁelds are solutions of

partial diﬀerential equations on a suitable manifold. The theory of diﬀeren-

tial operators necessary for studying these ﬁeld equations is a vast and very

active area in mathematics and physics. In this appendix we give a brief

introduction to the parts of operator theory that are relevant to the applica-

tions to ﬁeld theories. Special operators and corresponding index theorems

are also discussed here. We are interested in the situation where the oper-

ators are linear diﬀerential or pseudo-diﬀerential operators on modules of

smooth sections of vector bundles over a suitable base manifold. Standard

references for this material are Gilkey [154], Palais [310], and Wells [398]. In

what follows we consider complex vector bundles over a compact base mani-

fold. This simpliﬁes many considerations, although the corresponding theory

over non-compact base manifold and for real vector bundles can be developed

along similar lines. Sobolev spaces play a fundamental role in the study of

diﬀerential operators and in particular, in the study of operators arising in

gauge theories. The analysis of diﬀerential operators is greatly simpliﬁed by

the use of these spaces, essentially because they are Hilbert spaces in which

diﬀerential operators, which are not continuous in the usual L

spaces, are

continuous. For an introduction to the theory of Sobolev spaces, see for ex-

ample, Adams [5]. We now give a brief account of some important aspects of

these spaces, which are needed in this appendix.

403

404 Appendix D

D.2 Sobolev Spaces

Recall that a multi-index α in dimension n is an ordered n-tuple of non-

negative integers, i.e., α =(α

,α

,...,α

). The number |α| = α

+α

+···+

is called the length of α. We will use the notation

∂

= ∂

|α|

/∂x

···∂x

and D

:= (− i)

|α|

∂

to indicate that derivatives are taken with respect to the variable x =

,...,x

) ∈ R

, and we will write ∂

and D

when this variable is un-

derstood. Let ν be a measure on R

and let L

(ν)betheHilbertspaceof

complex-valued functions on R

, which are square integrable with respect to

ν, with the inner product deﬁned by

(f,g) → (f|g):=



fgdν.

In the above deﬁnition of inner product on L

(ν) we have followed the con-

vention commonly used in the physics literature. We alert the reader that

most mathematical works use the convention of linearity in the ﬁrst variable

and semi-linearity in the second. However, both induce the same norm on

(ν) and the use of one or the other convention is simply a matter of choice.

Let μ

or simply μ denote the Lebesgue measure on R

. Let us denote by

f or

Ff the Fourier transform of the Lebesgue integrable function f : R

→ C.

By deﬁnition,

Ff(x) ≡

f(x):=(2π)

−n/2



f(y)e

−ix·y

dμ(y),

where x·y = x

+···+x

.LetS

denote the Schwartz space of (rapidly

decreasing) smooth complex-valued functions on R

such that, ∀p ∈ N and

for every multi-index α, there exists a real positive constant C

α,p

such that

(1 + |x|

)

f(x)|≤C

α,p

, ∀x ∈ R

The space S

is a dense subspace of L

(μ). The Parseval relation (f |g)=

(Ff|Fg) implies that S

is mapped isometrically onto itself by the Fourier

transform F . Using this property, it can be shown that F has a unique ex-

tension to a unitary operator on L

(μ). If α =(α

,...,α

) is a multi-index,

let us denote by M

the operator of multiplication on S

deﬁned by

f)(x)=x

f(x), ∀x ∈ R

where x

:= x

···x

. Then one can show that, on S

◦ F =(−1)

|α|

F ◦ M

, (D.1)

Appendix D 405

F ◦ D

= M

◦ F. (D.2)

We note that relation (D.2) is used in passing from coordinate representation

to the momentum representation in quantum mechanics.

Let ν

,s∈ R, denote the measure on R

deﬁned by

dν

(x)=(1+|x|

)

dμ(x).

For f ∈S

the Sobolev s-norm is deﬁned by

|f|



|Ff|dν

The Sobolev space H

) is the completion of S

in this norm. It is

customary to refer to H

)astheSobolev s-completion, or simply the

Sobolev completion,ofS

.Weobservethatifs<tthen L

(ν

) ⊂ L

(ν

)

(in particular, L

(ν

)=L

(μ)) and, for s ≥ 0,

):={f ∈ L

(μ) | Ff ∈ L

(ν

)}.

Furthermore, if f,g ∈ H

),s≥ 0, then their inner product is deﬁned by

(f|g)



Ff · Fgdν

Thus, for s ≥ 0, H

)=F

−1

(ν

)) and F maps H

) isometrically

onto the Hilbert space L

(ν

). Due to the relation (D.2) we can consider

),s≥ 0, as the space of square summable complex-valued functions

f on R

whose distributional derivative D

f is a function in L

(μ), for all

multi-indices α such that |α|≤s. In the following theorem we list three useful

results concerning Sobolev spaces.

Theorem D.1 Let H

),s ∈ R denote the Sobolev s-completion of the

Schwartz space S

. Then we have the following:

1. (Sobolev lemma) Let B

⊂ C

) denote the Banach space of complex-

valued functions f of class C

with norm |·|

k deﬁned by

|f|

:= sup



|α|≤k

If s>n/2+k and f ∈ H

),thenf may be considered a function of

class C

, and the natural injection of H

) into the Banach space B

is continuous.

2. (Rellich lemma) Let s<tand let {f

} be a sequence of functions in S

with support in a compact subset of R

and such that, for some constant

K, |f

≤ K, ∀i.Thenasubsequenceof{f

} converges in H

).Thus,

the natural injection of H

) into H

) is compact when restricted

to functions with support in a ﬁxed compact subset of R

406 Appendix D

3. For al l s ∈ R,therestrictiontoS

×S

of the L

(μ)-inner product extends

to a bilinear map of H

−s

) × H

) → C, which gives a canonical

identiﬁcation of H

−s

) with the dual of H

The above construction of Sobolev spaces can be generalized to sections of

vector bundles over a manifold as follows. Let E be a Hermitian vector bundle

over an m-dimensional compact manifold M with n-dimensional ﬁbers. Let

{(U

,φ

)} be an atlas on M with a ﬁnite family of charts such that φ

⊂ R

and let {ρ

} be a smooth partition of unity subordinate to this

atlas. Let us suppose that the atlas is chosen to give a local representation

{(U

,ψ

)} of the bundle E so that ψ

: U

× C

→ π

−1

). If ξ ∈ Γ (E),

then ψ

−1

◦ρ

ξ induces, through the chart φ

, a section of V

×C

over V

which can be considered a map of V

into C

, denoted by f

. This map has

compact support and can be extended to a map, also denoted by f

,ofR

into C

with the same support. Thus, f

is an n-tuple f

α,i

, i =1,...,n,of

elements of S

. Therefore, we may deﬁne

ξ



α,i

It is easy to see that 

is a norm on Γ (E)andthatΓ (E)withthisnormis

a pre-Hilbert space. Let H

(E)betheHilbertspaceobtained by completion

of this normed vector space. One can show that norms related to diﬀerent

partitions of unity and to other choices made in the deﬁnition are equivalent.

Thus, H

(E) is topologically a Hilbert space. The results of Theorem D.1

have obvious generalizations to this global case.

D.3 Pseudo-Diﬀerential Operators

The results on U ⊂ R

are generally referred to as local, while those on a

manifold (obtained by patching toghether these local results) are referred to

as global. We use a vector notation which easily generalizes to the global case.

Let U ×C

be the trivial complex vector bundle over an open subset U ⊂ R

and let Γ

be the space of smooth sections of this bundle. A diﬀerential

operator of order k from U × C

to U × C

is a linear map P : Γ

→ Γ

which can be written as

P =



|α|≤k

, (D.3)

where a

is a (j × h)-matrix-valued function on U.LetU = R

and let

≡S

⊕···⊕S

  

h times

Appendix D 407

denote the space of sections of R

× C

.AnelementofS

is an h-tuple

of functions in S

.LetI denote the operator on functions f : R

→ C

deﬁned by (If)(x):=f(−x). Then M

◦I=(−1)

|α|

I◦M

.Thisrelation

and equations (D.1)and(D.2)implythat

= I◦F ◦ M

◦ F.

Using this equation we obtain the following expression for the operator P on

f ∈S

(Pf)(x)=(2π)

−m/2



ix·y

p(x, y)(Ff)(y)dμ(y), (D.4)

where p(x, y) is the matrix-valued function deﬁned by

p(x, y)=



|α|≤k

(x)y

The function p is called the total symbol of P , while the function σ

(P )

deﬁned by

(P )(x, y)=



|α|=k

(x)y

is called the k-symbol or the principal symbol (or simply the symbol)

of P .LetE (resp., F ) be a Hermitian vector bundle of rank h (resp., j)

over an m-dimensional 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) diﬀerential operator of order k from E to F if locally, in a

coordinate neighborhood U , P has the expression (D.3). Alternatively we can

express this deﬁnition by saying that P factors through the k-jet extension

(E)ofE, i.e., there exists a vector bundle morphism f : J

(E) → F such

that P = f

∗

◦j

,wheref

∗

: Γ (J

(E)) → Γ (F ) is the map induced by f, i.e.,

the following diagram commutes

Γ (J

(E)) Γ (F )

∗

Γ (E)





In fact, this formulation can easily be extended to deﬁne a non-linear dif-

ferential operator of order k between sections of arbitrary ﬁber bundles. We

denote by D

(E,F)thespace of linear diﬀerential operators of order

k from E to F . We now give a direct deﬁnition of the symbol of an operator

P ∈ D

(E,F), which generalizes the deﬁnition of k-symbol given above in

the local case. Let

∗

M = T

∗

M \{the image of the zero section}