Nakahara M. Geometry, Topology and Physics
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Proof. Suppose there exists a spin bundle over M. Thenwedefineasetof
transition functions
˜
t
ij
such that
˜
t
ij
˜
t
jk
˜
t
ki
= I over any overlapping charts U
i
, U
j
and U
k
, hence w
2
(M) is trivial. Conversely, suppose w
2
(M) is trivial, namely
f (i, j, k) = δ f
1
(i, j, k) = f
1
( j, k) f
1
(i, k) f
1
(k, i )
f
1
being a 1-cochain. We consider the 1-cochain f
1
(i, j ) defined in exercise 11.5.
If we choose new transition functions
˜
t
ij
≡
˜
t
ij
f
1
(i, j ),wehave
˜
t
ij
˜
t
jk
˜
t
ki
=[δ f
1
(i, j, k)]
2
= I
and, hence, {
˜
t
ij
} defines a spin bundle over M.
We outline some useful results:
(a)
w
1
( P
m
) = 1 w
2
( P
m
) =
1 m odd
xmeven
(11.140)
x being the generator of H
2
( P
m
;
2
).
(b)
w
1
(S
m
) = w
2
(S
m
) = 1 (11.141)
(c)
w
1
(
g
) = w
2
(
g
) = 1 (11.142)
g
being the Riemann surface of genus g.
12
INDEX THEOREMS
In physics, we often consider a differential operator defined on a manifold M.
Typical examples will be the Laplacian, the d’Alembertian and the Dirac operator.
From the mathematical point of view, these operators are regarded as maps of
sections
D : (M, E) → (M, F)
where E and F are vector bundles over M. For example, the Dirac operator is a
map F(M, E) → F(M, E), E being a spin bundle over M. If inner products are
defined on E and F, it is possible to define the adjoint of D,
D
†
: (M, F) → (M, E).
Since it is a differential operator, D carries analytic information on the
spectrum and its degeneracy. In what follows, we are interested in the zero
eigenvectors of D and D
†
,
ker D ≡{s ∈ (M, E)|Ds = 0}
ker D
†
≡{s ∈ (M, F)|D
†
s = 0}.
The analytical index is defined by
ind D = dim ker D − dim ker D
†
.
Surprisingly, this analytic quantity is a topological invariant expressed in terms of
an integral of an appropriate characteristic class over M, which provides purely
topological information on M. This interplay between analysis and topology is
the main ingredient of the index theorem.
Our exposition follows Eguchi et al (1980), Gilkey (1984), Shanahan (1978),
Kulkarni (1975) and Booss and Bleecker (1985). The reader should consult these
references for details. Alvarez (1985) contains a brief summary of this subject
along with applications to anomalies and strings.
12.1 Elliptic operators and Fredholm operators
In the following, we will be concerned with differential operators defined on
vector bundles over a compact manifold M without a boundary. We exclusively
deal with a nice class of differential operators called the Fredholm operators.
12.1.1 Elliptic operators
Let E and F be complex vector bundles over a manifold M. A differential
operator D is a linear map
D : (M, E) → (M, F). (12.1)
TakeachartU of M over which E and F are trivial. We denote the local
coordinates of U as x
µ
. We introduce the following multi-index notation,
M ≡ (µ
1
,µ
2
,...,µ
m
)µ
j
∈ ,µ
j
≥ 0
|M|≡µ
1
+ µ
2
+···+µ
m
D
M
=
∂
|M |
∂x
M
≡
∂
µ
1
+···+µ
m
∂(x
1
)
µ
1
...∂(x
m
)
µ
m
.
If dim E = k and dim F = k
, the most general form of D is
[Ds(x)]
α
=
|M |≤N
1≤a≤k
A
Mα
a
(x )D
M
s
a
(x ) 1 ≤ α ≤ k
(12.2)
where s(x) is a section of E . Note that x denotes a point whose coordinates are
x
µ
. This slight abuse simplifies the notation. A
M
≡ (A
M
)
α
a
is a k × k
matrix
which may depend on the position x . The positive integer N in (12.2) is called
the order of D. We are interested in the case in which N = 1 (the Dirac operator)
and N = 2 (the Laplacian). For example, if F is a spin bundle over M,theDirac
operator D ≡ iγ
µ
∂
µ
+m : (M, E) → (M, E) acts on a section ψ(x) of E as
[Dψ(x )]
α
= i(γ
µ
)
α
β
∂
µ
ψ
β
(x ) + mψ
α
(x ).
The symbol of D is a k × k
matrix
σ(D,ξ) ≡
|M |=N
A
Mα
a
(x )ξ
M
(12.3)
where ξ is a real m-tuple ξ = (ξ
1
,...,ξ
m
). The symbol is also defined
independently of the coordinates as follows. Let E
π
−→ M be a vector bundle
and let p ∈ M,ξ ∈ T
∗
p
M and s ∈ π
−1
E
( p). Take a section ˜s ∈ (M, E) such that
˜s( p) = s and a function f ∈
(M) such that f ( p) = 0anddf ( p) = ξ ∈ T
∗
p
M.
Then the symbol may be defined by
σ(D,ξ)s =
1
N!
D( f
N
˜s)|
p
. (12.4)
The factor f
N
automatically picks up the Nth-order term due to the condition
f ( p) = 0. Equation (12.4) yields the same symbol as (12.3).
If the matrix σ(D,ξ) is invertible for each x ∈ M and each ξ ∈
m
−{0},
the operator D is said to be elliptic. Clearly this definition makes sense only when
k = k
. It should be noted that the symbol for a composite operator D = D
1
◦ D
2
is a composite of the symbols, namely σ(D,ξ) = σ(D
1
,ξ)σ(D
2
,ξ). This shows
that composites of elliptic operators are also elliptic. In general, powers and roots
of elliptic operators are elliptic.
Example 12.1. Let x
µ
be the natural coordinates in
m
.IfE and F are real line
bundles over
m
, the Laplacian : (
m
, E) → (
m
, F) is defined by
≡
∂
2
∂(x
1
)
2
+···+
∂
2
∂(x
m
)
2
. (12.5)
According to (12.3), the symbol is
σ(,ξ) =
µ
(ξ
µ
)
2
.
This is in agreement with the result obtained from (12.4),
σ(,ξ)s =
1
2
( f
2
˜s)|
p
=
1
2
∂
2
∂(x
µ
)
2
( f
2
˜s)|
p
=
1
2
f
2
˜s + 2 f f ˜s + 2 f
∂ f
∂x
µ
∂ ˜s
∂x
µ
+ 2
∂ f
∂x
µ
∂ f
∂x
µ
˜s
p
=
(ξ
µ
)
2
s.
This symbol is clearly invertible for ξ = 0, and hence is elliptic.
However, the d’Alembertian
≡
∂
2
∂(x
1
)
2
+···+
∂
2
∂(x
m−1
)
2
−
∂
2
∂(x
m
)
2
(12.6)
is not elliptic since the symbol
σ(
,ξ) = (ξ
1
)
2
+···+(ξ
m−1
)
2
− (ξ
m
)
2
vanishes everywhere on the light cone,
(ξ
m
)
2
= (ξ
1
)
2
+···+(ξ
m−1
)
2
.
Exercise 12.1. Let M =
2
and consider a differential operator D of order two.
The symbol of D is of the form
σ(D,ξ) = A
11
ξ
1
ξ
1
+ 2 A
12
ξ
1
ξ
2
+ A
22
ξ
2
ξ
2
.
Show that D is elliptic if and only if σ(D,ξ) = 1 is an ellipse in ξ-space.
12.1.2 Fredholm operators
Let D : (M, E) → (M, F) be an elliptic operator. The kernel of D is the set
of null eigenvectors
ker D ≡{s ∈ (M, E)|Ds = 0}. (12.7)
Suppose E and F are endowed with fibre metrics, which will be denoted ,
E
and ,
F
, respectively. The adjoint D
†
: (M, F) → (M, E) of D is defined
by
s
, Ds
F
≡D
†
s
, s
E
(12.8)
where s ∈ (M, E) and s
∈ (M, F).Wedefinethecokernel of D by
coker D ≡ (M, F)/imD. (12.9)
Among elliptic operators we are interested in a class of operators whose
kernels and cokernels are finite dimensional. An elliptic operator D which
satisfies this condition is called a Fredholm operator.Theanalytical index
ind D ≡ dim ker D − dim coker D (12.10)
is well defined for a Fredholm operator. Henceforth, we will be concerned only
with Fredholm operators. It is known from the general theory of operators that
elliptic operators on a compact manifold are Fredholm operators. Theorem 12.1
shows that ind D is also expressed as
ind D = dim ker D − dim ker D
†
. (12.11)
Theorem 12.1. Let D : (M, E) → (M, F) be a Fredholm operator. Then
coker D
∼
=
ker D
†
≡{s ∈ (M, F)|D
†
s = 0}. (12.12)
Proof.Let[s]∈coker D be given by
[s]={s
∈ (M, F)|s
= s + Du, u ∈ (M, E)}.
We show that there is a surjection ker D
†
→ coker D, namely any [s]∈coker D
has a representative s
0
∈ ker D
†
.Defines
0
by
s
0
≡ s − D
1
D
†
D
D
†
s. (12.13)
We find s
0
∈ ker D
†
since D
†
s
0
= D
†
s − D
†
D(D
†
D)
−1
D
†
s = D
†
s − D
†
s =
0. Next, let s
0
, s
0
∈ ker D
†
and s
0
= s
0
. We show that [s
0
] =[s
0
] in
(M, F)/ im D.If[s
0
]=[s
0
], there is an element u ∈ (M, E) such that
s
0
− s
0
= Du.Then0=u, D
†
(s
0
− s
0
)
E
=u, D
†
Du
E
=Du, Du
F
≥ 0,
hence Du = 0, which contradicts our assumption s
0
= s
0
. Thus, the map
s
0
→[s] is a bijection and we have established that coker D
∼
=
ker D
†
.
12.1.3 Elliptic complexes
Consider a sequence of Fredholm operators,
···→(M, E
i−1
)
D
i−1
−→ (M, E
i
)
D
i
−→ (M, E
i+1
)
D
i+1
−→··· (12.14)
where {E
i
} is a sequence of vector bundles over a compact manifold M.The
sequence (E
i
, D
i
) is called an elliptic complex if D
i
is nilpotent (that is
D
i
◦D
i−1
= 0) for any i . The reader may refer to (M, E
i
) =
i
(M) and D
i
= d
(exterior derivative) for example. The adjoint of D
i
: (M, E
i
) → (M, E
i+1
)
is denoted by
D
†
i
: (M, E
i+1
) → (M, E
i
).
The Laplacian
i
: (M, E
i
) → (M, E
i
) is
i
≡ D
i−1
D
†
i−1
+ D
†
i
D
i
. (12.15)
The Hodge decomposition also applies to the present case,
s
i
= D
i−1
s
i−1
+ D
†
i
s
i+1
+ h
i
(12.16)
where s
i±1
∈ (M, E
i±1
) and h
i
is in the kernel of
i
,
i
h
i
= 0.
Analogously to the de Rham cohomology groups, we define
H
i
(E, D) ≡ ker D
i
/imD
i−1
. (12.17)
As in the case of the de Rham theory, it can be shown that H
i
(E, D) is isomorphic
to the kernel of
i
. Accordingly, we have
dim H
i
(E, D) = dim Harm
i
(E, D) (12.18)
where Harm
i
(E, D) is a vector space spanned by {h
i
}.Theindex of this elliptic
complex is defined by
ind D ≡
m
i=0
(−1)
i
dim H
i
(E, D) =
m
i=0
(−1)
i
dim ker
i
. (12.19)
The index thus defined generalizes the Euler characteristic, see example 12.2.
How is this related to (12.10)? Consider the complex (M, E)
D
→ (M, F).
We may formally add zero on both sides,
0
i
→ (M, E)
D
→ (M, F)
ϕ
→ 0 (12.20)
where i is the inclusion. The index according to (12.19) is
dim ker D −{dim (M, F) − dim imD}=dim ker D − dim cokerD
where we have noted that dim im i = 0, ker ϕ = (M, F) and coker D =
ker ϕ/ im D. Thus, (12.19) yields the same index as (12.10).
It is often convenient to work with a two-term elliptic complex which has the
same index as the original elliptic complex (E, D).Thisrolling up is carried out
by defining
E
+
≡⊕
r
E
2r
, E
−
≡⊕
r
E
2r+1
(12.21)
which are called the even bundle and the odd bundle, respectively.
Correspondingly we consider the operators
A ≡⊕
r
(D
2r
+ D
†
2r−1
), A
†
≡⊕
r
(D
2r+1
+ D
†
2r
). (12.22)
We readily verify that A : (M, E
+
) → (M, E
−
) and A
†
: (M, E
−
) →
(M, E
+
).FromA and A
†
, we construct the two Laplacians
+
≡ A
†
A =⊕
r,s
(D
2r+1
+ D
†
2r
)(D
2s
+ D
†
2s−1
)
=⊕
r
(D
2r−1
D
†
2r−1
+ D
†
2r
D
2r
) =⊕
r
2r
(12.23a)
−
≡ AA
†
=⊕
r
2r+1
. (12.23b)
Then we have
ind(E
±
, A) = dim ker
+
− dim ker
−
=
(−1)
r
dim ker
r
= ind(E, D). (12.24)
Example 12.2. Let us consider the de Rham complex (M) over a compact
manifold M without a boundary,
0
i
→
0
(M)
d
→
1
(M)
d
→···
d
→
m
(M)
d
→ 0 (12.25)
where m = dim M and d stands for d
r
:
r
(M) →
r+1
(M). H
r
(E, D) defined
by (12.25) agrees with the de Rham cohomology group H
r
(M, ). The index is
identified with the Euler characteristic,
ind(
∗
(M), d) =
m
r=0
(−1)
r
dim H
r
(M; ) = χ(M). (12.26)
We found in chapter 7 that b
r
≡ dim H
r
(M. ) agrees with the number of linearly
independent harmonic r -forms: dim H
r
(M, ) = dim Harm
r
(M) = dim ker
r
,
where
r
is the Laplacian
r
= (d + d
†
)
2
= d
r−1
d
†
r−1
+ d
†
r
d
r
(12.27)
d
†
r
:
r+1
(M) →
r
(M) being the adjoint of d
r
.Nowwefindthat
χ(M) =
m
r=0
(−1)
r
dim ker
r
. (12.28)
This relation is very interesting since the LHS is a purely topological quantity
which can be computed by triangulating M, for example, while the RHS is given
by the solution of an analytic equation
r
u = 0. We noted in example 11.6 that
χ(M) is given by integrating the Euler class over M: χ(M) =
M
e(TM).Now
(12.28) reads
m
r=1
(−1)
r
dim ker
r
=
M
e(TM). (12.29)
This is a typical form of the index theorem. The RHS is an analytic index while
the LHS is a topological index given by the integral of certain characteristic
classes. In section 12.3, we derive (12.29) from the Atiyah–Singer index theorem.
The two-term complex is given by
+
(M) ≡⊕
r
2r
(M)
−
(M) ≡⊕
r
2r+1
(M). (12.30)
The corresponding operators are
A ≡⊕
r
(d
2r
+ d
†
2r−1
) A
†
≡⊕
r
(d
2r−1
+ d
†
2r
). (12.31)
It is left as an exercise to the reader to show that
ind(
±
(M), A) = dim ker A
+
− dim ker A
−
= χ(M). (12.32)
12.2 The Atiyah–Singer index theorem
12.2.1 Statement of the theorem
Theorem 12.2. (Atiyah–Singer index theorem)Let(E, D) be an elliptic
complex over an m-dimensional compact manifold M without a boundary. The
index of this complex is given by
ind(E, D) = (−1)
m(m+1)/2
M
ch
⊕
r
(−1)
r
E
r
Td(TM
)
e(TM)
vol
. (12.33)
In the integrand of the RHS, only m-forms are picked up, so that the integration
makes sense. [Remarks: The division by e(TM) can really be carried out at the
formal level. If m is an odd integer, the index vanishes identically, see below.
Original references are Atiyah and Singer (1968a, b), Atiyah and Segal (1968).]
The proof of theorem 12.2 is found in Shanahan (1978), Palais (1965) and
Gilkey (1984). The proof found there is based on either K -theory or the heat
kernel formalism. In section 13.2, we give a proof of the simplest version of the
Atiyah–Singer (AS) index theorem for a spin complex. Recently physicists have
found another proof of the theorem making use of supersymmetry. This proof is
outlined in sections 12.9 and 12.10. Interested readers should consult Alvarez-
Gaum´e (1983) and Friedan and Windey (1984, 1985) for further details.
The following corollary is a direct consequence of theorem 12.2.
Corollary 12.1. Let (M, E)
D
→ (M, F) be a two-term elliptic complex. The
index of D is given by
ind D = dim ker D − dim ker D
†
= (−1)
m(m+1)/2
M
(chE − chF)
Td(TM
)
e(TM)
vol
. (12.34)
12.3 The de Rham complex
Let M be an m-dimensional compact orientable manifold with no boundary. By
now we are familiar with the de Rham complex,
···
d
→
r−1
(M)
d
→
r
(M)
d
→
r+1
(M)
d
→··· (12.35)
where
r
(M) = (M, ∧
r
T
∗
M ). We complexified the forms so that we may
apply the AS index theorem. The exterior derivative satisfies d
2
= 0. To show
that (12.35) is an elliptic complex, we have to show that d is elliptic. To find the
symbol for d, we note that
σ(d,ξ)ω = d( f ˜s)
|
p
= d f ∧˜s + f d˜s|
p
= ξ ∧ ω
where p ∈ M,ω ∈
r
p
(M) , f (p) = 0, d f ( p) = ξ, ˜s ∈
r
(M) and
˜s( p) = ω; see (12.4). We find
σ(d,ξ) = ξ ∧ . (12.36)
This defines a map
r
(M) →
r+1
(M) and is non-singular if ξ = 0.
Thus, we have proved that d :
r
(M) →
r+1
(M) is elliptic and, hence,
(12.35) is an elliptic complex. Note, however, that the operator d :
k
(M) →
k+1
(M) is not Fredholm since ker d is infinite dimensional. To apply the index
theorem to this complex, we have to consider the de Rham cohomology group
H
r
(M) instead. The operator d is certainly Fredholm on this space.
Let us find the index theorem for this complex. We note that
dim
H
r
(M; ) = dim H
r
(M; ). Hence, the analytical index is
ind d =
m
r=0
(−1)
r
dim H
r
(M; )
=
(−1)
r
dim H
r
(M; ) = χ(M) (12.37)
where χ(M) is the Euler characteristic of M. Suppose M is even dimensional,
m = 2l. The RHS of (12.33) gives the topological index
(−1)
l(2l+1)
M
ch
m
⊕
r=0
(−1)
r
∧
r
T
∗
M
Td(TM
)
e(TM)
vol
. (12.38)
The splitting principle yields
ch
m
⊕
r=0
(−1)
r
∧
r
T
∗
M
= 1 −ch(T
∗
M ) + ch(∧
2
T
∗
M ) +···+(−1)
m
ch(∧
m
T
∗
M )
= 1 −
m
i=1
e
−x
i
(TM ) +
i< j
e
−x
i
e
−x
j
(TM ) +···
+ (−1)
m
e
−x
1
e
−x
2
...e
−x
m
(TM )
=
m
i=1
(1 −e
−x
i
)(TM )
where we have noted that x
i
(T
∗
M ) =−x
i
(TM ).[LetL be a complex line
bundle and L
∗
be its dual bundle. L ⊗ L
∗
is a bundle whose section is a map
→ at each fibre of L. L ⊗ L
∗
has a global section which vanishes nowhere
(the identity map, for example) from which we can show L⊗L
∗
is a trivial bundle.
We have c
1
(L ⊗L
∗
) = c
1
(L)+c
1
(L
∗
) = 0, hence x(L
∗
) =−x (L). The splitting
principle yields x
i
(T
∗
M ) =−x
i
(TM ).] We also have
Td(TM
) =
m
i=1
x
i
1 − e
−x
i
(TM )
e(TM) =
l
i=1
x
i
(TM ).
Substituting these in (12.38), we have
ind d =
M
(−1)
l(2l+1)
(−1)
l
l
i=1
x
i
(TM )
=
M
e(TM). (12.39)
If m is odd, it can be shown that (Shanahan (1978), p22)
ind d = 0 (12.40)
which is in harmony with the fact that e(TM) = 0ifdimM is odd. In any case,
the index theorem for the de Rham complex is
χ(M) =
M
e(TM). (12.41)