Nakahara M. Geometry, Topology and Physics

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Example 12.3. Let M be a two-dimensional orientable manifold without

boundary. Equation (12.41) reads

χ(M) =

4π





αβ

2π



(12.42a)

which is the celebrated Gauss–Bonnet theorem. For dim M = 4, it reads as

χ(M) =

32π





αβγ δ

αβ

∧

γδ

.(12.42b)

12.4 The Dolbeault complex

We recall some elementary facts about complex manifolds (see chapter 8 for

details). Let M be a compact complex manifold of complex dimension m without

a boundary. Let z

= x

+ iy

be the local coordinates and ¯z

= x

− iy

their complex conjugates. TM

denotes the tangent bundle spanned by {∂/∂z

}

and TM

−

= TM

the complex conjugate bundle spanned by {∂/∂¯z

}.The

dual of TM

is denoted by T

∗

and spanned by {dz

} while that of TM

−

∗

−

= T

∗

spanned by {d¯z

}. The space 

(M) of complexiﬁed r -forms

is decomposed as



(M) =⊕

p+q=r



p,q

(M)

where 

p,q

(M) is the space of the ( p, q)-forms, which is spanned by a basis of

the form

∧ ...∧ dz

∧ d¯z

∧ ...∧ d¯z

The exterior derivative is decomposed as d ≡ ∂ +

∂ where

∂ = dz

∧ ∂/∂z

∂ = d¯z

∧ ∂/∂¯z

They satisfy ∂

∂ +

∂∂ = ∂

∂

= 0. We have the sequences

···

∂

→ 

p,q

(M)

∂

→ 

p,q+1

(M)

∂

→··· (12.43a)

···

∂

→ 

p,q

(M)

∂

→ 

p+1,q

(M)

∂

→···. (12.43b)

We are interested in the ﬁrst sequence with p = 0,

···

∂

→ 

0,q

(M)

∂

→ 

0,q+1

(M)

∂

→···. (12.44)

This sequence is called the Dolbeault complex.

To show that (12.44) is an elliptic complex, we compute the symbol for

∂.

Let ξ = ξ

0,1

+ ξ

1,0

be a real one-form at p ∈ M,whereξ

0,1

∈ 

0,1

(M) and

1,0

= ξ

0,1

∈ 

1,0

(M).

Take an anti-holomorphic r -form ω ∈ 

0,r

(M).Weﬁnd

σ(

∂,ξ)ω =

∂( f ˜s) =

∂ f ∧˜s + f

∂ ˜s|

= ξ

0,1

∧ ω

where f (p) = 0,

∂ f (p) = ξ

0,1

, ˜s ∈ 

0,r

(M) and ˜s( p) = ω.Wehave

σ(

∂,ξ) = ξ

0,1

∧ . (12.45)

From a similar argument to that given in the previous section, it follows that the

symbol (12.45) is elliptic. Thus, the Dolbeault complex (12.44) is an elliptic

complex.

The AS index theorem takes the form

ind

∂ =







(−1)

∧

∗

−



Td(TM

)

e(TM)



vol

. (12.46)

The LHS is computed as follows. We ﬁrst note that

ker

∂

/im

∂

r−1

= H

0,r

(M)

where H

0,r

(M) is the

∂-cohomology group. Then the LHS is

ind

∂ =



r=0

(−1)

0,r

(12.47)

where b

0,r

≡ dim H

0,r

(M) is the Hodge number. This index is called the

arithmetic genus of M.

Simpliﬁcation of the topological index can be carried out as in the case of

the de Rham complex. We refer the reader to Shanahan (1978) for the technical

details. We have



r=1

(−1)

0,r



Td(TM

) (12.48)

where Td(TM

) is the Todd class of TM

12.4.1 The twisted Dolbeault complex and the Hirzebruch–Riemann–Roch

theorem

In the Dolbeault complex, we may replace 

0,r

(M) by the tensor product bundles



0,r

(M) ⊗ V ,whereV is a holomorphic vector bundle over M,

···

∂

→ 

0,r−1

(M) ⊗ V

∂

→ 

0,r

(M) ⊗ V

∂

→···. (12.49)

The AS index theorem of this complex reduces to the Hirzebruch–Riemann–

Roch theorem,

ind

∂



Td(TM

)ch(V ). (12.50)

For example, if m = dim M = 1, we have

ind

∂

dim V



(TM

) +



(V )

= (2 − g) dim V +



2π

(12.51)

since it can be shown that



(TM

) =



e(TM) = 2 − g

g being the genus of M.

12.5 The signature complex

12.5.1 The Hirzebruch signature

Let M be a compact orientable manifold of even dimension, m = 2l.Let[ω] and

[η] be the elements of the ‘middle’ cohomology group H

(M; ). We consider a

bilinear form H

(M; ) × H

(M; ) → deﬁned by

σ([ω], [η]) ≡



ω ∧ η (12.52)

cf example 11.8. This deﬁnition is independent of the representatives of [ω] and

[η]. The form σ is symmetric if l is even (m ≡ 0 mod 4) and anti-symmetric if

l is odd (m ≡ 2 mod 4). Poincar´e duality shows that the bilinear form σ has the

maximal rank b

= dim H

(M; ) and is, hence, non-degenerate. If l ≡ 2k is

even, the symmetric form σ has real eigenvalues, b

of which are positive and b

−

of which are negative (b

+ b

−

= b

). The Hirzebruch signature is deﬁned by

τ(M) ≡ b

− b

−

. (12.53)

If l is odd, τ(M) is deﬁned to vanish (an anti-symmetric form has pure imaginary

eigenvalues). In the following, we set l = 2k.

The Hodge ∗ satisﬁes ∗

= 1whenactingona2k-form in a 4k-dimensional

manifold M and hence ∗ has eigenvalues ±1. Let Harm

(M) be the set of

harmonic 2k-forms on M. We note that Harm

(M)

∼

(M; ) and each

element of H

(M; ) has a unique harmonic representative. Harm

(M) is

separated into disjoint subspaces,

Harm

(M) = Harm

(M) ⊕ Harm

−

(M) (12.54)

according to the eigenvalue of ∗. This separation block diagonalizes the bilinear

form σ . In fact, for ω

∈ Harm

(M),

σ(ω

,ω

) =



∧ ω



∧∗ω

= (ω

,ω

)>0

where (ω

,ω

) is the standard positive-deﬁnite inner product deﬁned by (7.181).

We also ﬁnd

σ(ω

−

,ω

−

) =−



−

∧∗ω

−

=−(ω

−

,ω

−

)<0

σ(ω

,ω

−

) =−



∧∗ω

−

=−



−

∧∗ω

=−σ(ω

,ω

−

) = 0

where we have noted that α ∧∗β = β ∧∗α for any forms α and β. Hence,

σ is block diagonal with respect to Harm

(M) ⊕ Harm

−

(M) and, moreover,

= dim Harm

(M).Nowτ(M) is expressed as

τ(M) = dim Harm

(M) − dim Harm

−

(M). (12.55)

Exercise 12.2. Let dim M = 4k. Show that

τ(M) = χ(M) mod 2. (12.56)

[Hint: Use the Poincar´e duality to show that χ(M) = b

mod 2.]

12.5.2 The signature complex and the Hirzebruch signature theorem

Let M be an m-dimensional compact Riemannian manifold without a boundary

and let g be the given metric. Consider an operator

≡ d + d

†

. (12.57)

is a square root of the Laplacian:

= dd

†

+ d

†

d = . To show that

is elliptic, it sufﬁces to verify that  is elliptic since the symbol of a product of

operators is the product of symbols. Let us compute the symbol of . As for d,

we have σ(d,ξ)ω = ξ ∧ ω.Asford

†

, it can be shown that (Palais 1965, pp77–8)

σ(d

†

,ξ) =−i

. (12.58)

Here i

: 

(M) → 

r−1

(M) is an interior product deﬁned by (cf. (5.79))

(dx

∧ ...∧ dx

)

≡



j =1

(−1)

j +1

∧ ...∧ d ˆx

∧ ...∧ dx

where the one-form under ˆ is omitted and we put ξ = ξ

. Now the symbol

of the Laplacian is obtained from (12.58) as

σ(,ξ)ω = σ(dd

†

+ d

†

d,ξ)ω =−[ξ ∧ i

(ω) +i

(ξ ∧ ω)]

=−i

(ξ) ∧ ω =−ξ 

where ω is an arbitrary r-form and the norm is taken with respect to the given

Riemannian metric. Finally, we obtain

σ(,ξ) =−ξ

. (12.59)

Thus, the Laplacian  is elliptic and so is

= d +d

†

Since the Laplacian  =

is self-dual on 

∗

(M), the index of  vanishes

trivially. It is also observed that

†

on 

∗

(M) and, hence, ind = 0.

To construct a non-trivial index theorem, we have to ﬁnd a complex on which

=

†

Exercise 12.3. Consider the restriction

of to even forms,

: 

(M) →



(M) where 

(M) ≡⊕

(M) and 

(M) ≡⊕

2i+1

(M) .The

adjoint of

≡

e†

: 

(M)

→ 

(M)

. Show that

ind

= dim ker

− dim ker

= χ(M).

[Hint: Prove ker

=⊕Harm

(M) and ker

=⊕Harm

2i+1

(M).This

complex, although non-trivial, does not yield anything new.]

If dim M = m = 2l,wehave∗∗η = (−1)

η for η ∈ 

(M) .Wedeﬁne

an operator π : 

(M) → 

m−r

(M) by

π ≡ i

r(r−1)+l

∗ . (12.60)

Observe that π is a ‘square root’ of (−1)

∗∗=1. In fact, for ω ∈ 

(M) ,

ω = i

r(r−1)+l

π(∗ω) = i

r(r−1)+l+(2l−r)(2l−r−1)+l

∗∗ω

= i

∗∗ω = (−1)

∗∗ω = ω (12.61)

where we have noted that r ≡ r

mod 2. We easily verify (exercise) that

{π,

}=π + π = 0. (12.62)

Let π act on 

∗

(M) =⊕

(M) .Sinceπ

= 1, the eigenvalues of π are ±1.

Then we have a decomposition of 

∗

(M) into the ±1 eigenspaces 

(M) of π



∗

(M) = 

(M) ⊕

−

(M). (12.63)

Since

anti-commutes with π, the restriction of to 

(M) deﬁnes an elliptic

complex called the signature complex,

: 

(M) → 

−

(M) (12.64)

where

≡ |



(M)

. The index of the signature complex is

ind

= dim ker

− dim ker

−

= dim Harm(M)

− dim Harm(M)

−

(12.65)

where

−

≡

†

: 

−

(M) → 

(M) and Harm(M)

≡{ω ∈ 

(M)|

ω =

0}. On the RHS of (12.65), all the contributions except those from the harmonic

l-forms cancel out. To see this, we separate ker

and ker

−

ker

= Harm

(M)

⊕



0≤r<l

[Harm

(M)

⊕ Harm

m−r

(M)

]

where Harm

(M)

≡ Harm(M)

∩ 

(M).Ifω ∈ Harm

(M),wehave

ω±πω ∈ Harm

(M)

⊕Harm

m−r

(M)

.Thenamapω+πω → ω−πω deﬁnes

an isomorphism between Harm

(M)

⊕ Harm

m−r

(M)

and Harm

(M)

−

⊕

Harm

m−r

(M)

−

. Now the index simpliﬁes as

ind

= dim Harm

(M)

− dim Harm

(M)

−

(12.66)

where we put l = 2k as before (the index vanishes if l is odd). It is important to

note that Harm

(M)

= Harm

(M) since π =∗in Harm

(M), see (12.54).

Now the index (12.66) reduces to the Hirzebruch signature,

ind

= τ(M). (12.67)

The derivation of the topological index is rather technical and we simply

quote the result from Shanahan (1978). Let ∧

∗

M be the subspace of ∧T

∗

such that 

(M) = (M, ∧

∗

M ).Thenwehave

topological index = (−1)



ch(∧

∗

M −∧

−

∗

M )

Td(TM

)

e(TM)



vol

= 2





i=1

tanh x



vol





i=1

tanh x



vol

where the last equality is true only for the 2l-forms in the expansion and x

(TM ). Now we have obtained the Hirzebruch signature theorem

τ(M) =



L(TM)|

vol

(12.68)

where L is the Hirzebruch L-polynomial deﬁned by (11.91). Since L is even in

, τ(M) vanishes if m = 2 mod 4. For example, τ(M) = 0form = 2. If m = 4,

we have

τ(M) =



(TM) =−

24π



. (12.69)

As in the case of the Dolbeault complex, we may twist the signature complex,

see Eguchi et al (1980), for example.

12.6 Spin complexes

The ﬁnal example of classical complexes is the spin complex. This complex is

very important in physics since it describes Dirac ﬁelds interacting with gauge

ﬁelds and/or gravitational ﬁelds.

12.6.1 Dirac operator

Let us consider a spin bundle S(M) over an m-dimensional orientable manifold

M. We shall denote the set of sections of this bundle by (M) = (M, S(M)).

We assume that m = 2l is an even integer. The spin group SPIN(m) is generated

by m Dirac matrices {γ

}, which satisfy

α†

= γ

(12.70a)

{γ

,γ

}=2δ

αβ

. (12.70b)

Throughout this chapter we assume that the metric has the Euclidean signature.

The Clifford algebra is generated by

1;γ

;γ

(α

<α

);...;

...γ

(α

<... < α

);...;γ

...γ

The last generator is of particular importance and we deﬁne

m+1

≡ i

...γ

. (12.71)

Our convention is such that (γ

m+1

)

= I and (γ

m+1

)

†

= γ

m+1

. It can be shown

from the general theory of the Clifford algebra that the γ

are represented by

× 2

matrices with complex entries. It is convenient to take a representation of

{γ

} such that γ

m+1

is diagonal,

m+1



1 0

0 −1



(12.72)

where 1 here is the 2

l−1

× 2

l−1

unit matrix.

Example 12.4. For m = 2, we take

= σ

= iγ

= σ

being the Pauli matrices,







0 −i





0 −1



For m = 4, we may take



0iα

−i ¯α



= (I

, −iσ ), ¯α

= (I

, iσ )

=−γ



0 −I



A Dirac spinor ψ ∈ (M) is an irreducible representation of the Clifford

algebra but not that of SPIN(2l). Irreducible representations of SPIN(2l) are

obtained by separating (M) according to the eigenvalues of γ

m+1

.Since

(γ

m+1

)

= I , the eigenvalues of γ

m+1

, called the chirality,mustbe±1. Then

(M) is separated into two eigenspaces

(M) = 

(M) ⊕ 

−

(M) (12.73)

where γ

m+1

=±ψ

for ψ

∈ 

(M). The projection operators

onto



are given by

≡

(I + γ

m+1

) =



1 0



(12.74a)

−

≡

(I − γ

m+1

) =



0 1



. (12.74b)

Thus, we may write





∈ 

(M), ψ

−



−



∈ 

−

(M). (12.75)

The reader should verify that

−

= 1,(

)

+ −

= ψ

and

∓

= 0.

The Dirac operator in a curved space is given by (section 7.10)

∇ψ ≡ iγ

∇

∂/∂x

ψ = iγ

(∂

+ ω

)ψ (12.76)

where ω

iω

αβ



αβ

is the spin connection and γ

= γ

. Let us prove

that i

∇ is elliptic. Let f be a function deﬁned near p ∈ M such that f (p) = 0

and iγ

∂

f ( p) = iγ

≡ i

ξ .

Takeasection

ψ ∈ (M) such that

ψ(p) = ψ.

From (12.4), we have

σ(i

∇,ξ)ψ = i

∇( f

ψ)|

= (i

∇ f )

ψ|

= i

ξψ

which shows that

σ(i

∇,

ξ) = i

ξ. (12.77)

If we note that

ξ = ξ

= ξ

, we ﬁnd that (12.77) is invertible for

ξ = 0, hence i

∇ is an elliptic operator.

It can be shown that {γ

} is taken in the form



0iα

−i ¯α



†

=¯α

(12.78)

Note the minor abuse of the notation.

For a vector A = A

, A denotes γ

see example 12.4 for m = 2 and 4. Then (12.76) becomes

∇=



0 D

†

D 0



(12.79)

where

D ≡¯α

(∂

+ ω

) D

†

≡−α

(∂

+ ω

). (12.80)

Hence, D

†

is, indeed, the adjoint of D (note that ∂

+ ω

is anti-Hermitian). For





∈ 

(M)

we have

∇







0 D

†

D 0







Dψ



while for



−



∈ 

−

(M)

we have

∇



−





†

−



Hence, D = i

∇

: 

(M) → 

−

(M) and D

†

= i

∇

−

: 

−

(M) →



(M). Now we have a two-term complex



(M)

−→

←−

†



−

(M) (12.81)

called the spin complex. The analytical index of this complex is

ind D = dim ker D − dim ker D

†

= ν

− ν

−

(12.82)

where ν

(ν

−

) is the number of zero-energy modes of chirality +(−).

Let us apply the AS index theorem to this case. Without getting into the

details of the Clifford algebra and the spin complex, we simply write down the

result. The AS index theorem for the spin complex (12.81) is

− ν

−



ch(

(M) − 

−

(M))

Td(TM

)

e(TM)



vol



A(TM)|

vol

(12.83)

where

A is the Dirac genus deﬁned by (11.94). Since

A contains only 4 j -forms,

−ν

−

vanishes unless m = 0 mod 4. Of course, this does not necessarily imply

= ν

−

= 0. The proof of (12.83) will be given later in sections 12.9 and 12.10.

12.6.2 Twisted spin complexes

In physics, a spinor ﬁeld may belong to a representation of a group G.For

example, the quark ﬁeld in QCD belongs to the 3 of SU(3). A spinor which

belongs to a representation of G is a section of the product bundle S(M) ⊗

E,whereE is an associated vector bundle of P(M, G) in an appropriate

representation. The Dirac operator D

: 

(M) ⊗ E → (M)

−

⊗ E in this

case is

= iγ

(∂

+ ω

)

(12.84)

where

is the gauge potential on E. The AS index theorem for this twisted spin

complex is

− ν

−



A(TM)ch(E)|

vol

. (12.85)

For dim M = 2, we have

− ν

−



(E) =

2π



tr (12.86)

while for dim M = 4,

− ν

−



[ch

(E) +

(TM)ch

(E)]

−1

8π



dim E

192π



. (12.87)

Example 12.5. Let

M = T

= S

×···×S

, -. /

2l times

Then we ﬁnd

A(TM) =



⊕





A(TS

) = 1.

We also have

A(TS

) = 1. Accordingly, the index of these bundles is

− ν

−



ch(E)|

vol

. (12.88)

Example 12.6. Let us consider the monopole bundle P(S

, U(1)).If is the

local gauge potential, the ﬁeld strength is

= d . The index theorem is

− ν

−

2π



=−

2π



F (12.89)

where

= iF. As was shown in section 10.5, the RHS represents the winding

number π

(U(1)) = and analytical information (the LHS) is now expressed in

a topological way (the RHS).