Nakahara M. Geometry, Topology and Physics

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On the equator (t = s = 1), we have

= g

−1

(

+)g,where = d+d

(note that d

g = 0). Thus, ={

} deﬁnes a global connection on S

×S

Consider a Dirac operator i

∇

m+2

which couples to . The index theorem for

∇

m+2

is given by

ind i

∇

m+2

−



×S

l+1

( ) (13.97)

where

=  +

and

(

−

) is the number of + (−) chirality zero modes

of i

∇

m+2

(chirality is deﬁned in an (m + 2)-space).

Alvarez-Gaum´e and Ginsparg (1984) have shown, using an adiabatic

perturbative computation, that each winding number m

must be ±1. Moreover,

the Dirac operator i

∇

m+2

has a zero mode at p

= (t

,θ

) with (m + 2)-

dimensional chirality χ = m

=±1. Then the total winding number =



is given by the index

−

.Nowwehave

ind i

∇

m+2



×S

l+1

( ) =

2π



2π

dθ

∂w(

,θ)

∂θ

. (13.98)

We easily ﬁnd the non-Abelian anomaly from (13.98) including the

normalization. Since ch

l+1

( ) = dQ

m+1

( , ),wehave



×S

l+1

( ) =



×S

l+1

(

) +



×S

l+1

(

)



×S

m+1

(

t=1

− Q

m+1

(

s=1

(13.99)

From (11.118), we ﬁnd that

m+1

(

t=1

− Q

m+1

(

s=1

= Q

m+1

−1

g, 0) +α

= (−1)



2π



l+1

(m + 1)!

tr(g

−1

g)

m+1

+ α

. (13.100)

The index theorem is now given by

ind i

∇

m+2

= (−1)



2π



l+1

(m + 1)!



×S

tr(g

−1

g)

m+1

. (13.101)

Theorem 10.7 states that



tr(g

−1

dg)

yields the winding number of the map

g : S

→ SU(2). In the same manner, (13.101) represents the winding number of

the map g : S

m+1

→ G and is classiﬁed by π

m+1

(G) (note that S

∧S

= S

m+1

Finally, we show that the non-Abelian anomaly should be identiﬁed with

. We ﬁrst note that



×S

m+1

(

) = 0

since the integrand is independent of dθ and, thus, cannot be a volume element of

× S

.Thenwehave

ind i

∇

m+2



×S

m+1

(

g(θ)

+ ω,

g(θ)

) (13.102)

where ω = g

−1

g and

g(θ)

= d

g(θ)

+ (

g(θ)

)

= g(θ )

−1

g(θ ).

If the integrand in (13.102) is expanded in ω, only the term linear in dθ

contributes to the integral. This term Q

(ω,

g(θ)

) is proportional to

dθ ∧ (volume element in S

) and, hence, is a volume element of S

× S

.We

now have

W [ ]=



tr ω

δW [ ]

= id

w(θ, ) = 2πi



(ω,

g(θ)

). (13.103)

The explicit form of Q

is given by (13.82). For m = 4, we ﬁnd that



tr ω

δW [ ]

= 2πi



(ω,

g(θ)

)

24π



tr ω d



g(θ)

(

g(θ)

)



. (13.104)

Putting θ = 0 (g = e), we reproduce the anomalous divergence

j



24π

tr T



κλµν

∂



∂

µ ν

λ µ ν



(13.105)

which is in agreement with (13.56). The present method guarantees that the

WZ condition yields the correct result. Moreover, it reproduces the anomalous

divergence including the normalization which cannot be ﬁxed by the WZ

condition alone.

13.6 The parity anomaly in odd-dimensional spaces

So far, we have been working in even-dimensional spaces. One of the reasons

for this is that SO(2l + 1) has real or pseudo-real spinor representations but no

complex representations, hence no gauge anomaly is expected. However, we

can show that gauge theories in odd-dimensional spaces have a different kind

of anomaly called the ‘parity anomaly’, in which the parity symmetry of the

classical action is not maintained through quantization. It should be noted that

the parity anomaly in 2l + 1 dimensions is related to the Abelian anomaly in

2l + 2 dimensions as was pointed out by Alvarez-Gaum´e et al (1985).

13.6.1 The parity anomaly

Let M be a (2l + 1)-dimensional Riemannian manifold. We distinguish one

dimension from the others; that is we assume that M is of the form

× or

× ,where is a 2l-dimensional compact manifold without a boundary. We

denote the coordinate of

or S

by t while that of is denoted by x. The index

0 denotes the component in t-space while µ denotes that in x-space. For example,

the components of the γ -matrices are {γ

,γ

(1 ≤ µ ≤ 2l)}.

Deﬁne the ‘parity’ operation P by

(t, x) →

(t, x) =−

(−t, x )

(t, x) →

(t, x) =

(−t, x )

ψ(t, x ) → ψ

(t, x) = iγ

ψ(−t, x )

ψ(t, x) →

(t, x) = i

ψ(−t, x)γ

The classical action is invariant under the parity operation,



dt dx

ψi

∇ψ →−



dt dx

ψ(−t, x)γ

i[γ

(∂

−

(−t, x ))

+ γ

(∂

(−t, x ))]γ

ψ(−t, x )



dt dx

ψ(t, x)i[γ

(∂

(t, x))

+ γ

(∂

(t, x))]ψ(t, x)

where we put t →−t in the ﬁnal line. Let us see whether this invariance is

observed by the effective action. The effective action is given by the regularized

product of the eigenvalues of i

∇.WeemploythePauli–Villars regularization to

regulate the product, that is

reg

≡¯χi

∇χ + iM ¯χχ (13.106)

is added to the original Lagrangian. The Pauli–Villars regulator χ is a spinor

which obeys bosonic statistics and the limit M →∞is understood. The

regularized determinant is

−W [ ]

det i

∇

det(i

∇+iM)



+ iM

(13.107)

where we noted that χ is bosonic. Here λ

is the i th eigenvalue of i

∇; i

∇ψ

. Under the parity operation, eigenvalues change sign,

i[γ

(∂

−

(−t, x )) + γ

(∂

(−t, x ))]iγ

(−t, x )

= iγ

[γ

(−∂

−

(τ, x )) − γ

(∂

(τ, x ))]iψ(τ, x)

=−λ

iγ

(τ, x )

where τ =−t. This shows that the effective action W [

] transforms under the

parity operation P as

W [

]→W [

]=−ln



−λ

+ iM

W [ ] (13.108)

where the bar denotes complex conjugation. (13.108) shows that the imaginary

part of W is identiﬁed with the parity-violating part

W [

]−W [

]=2ImW [ ]. (13.109)

Im W [

] is given by the η-invariant deﬁned in section 12.8. In fact,

Im W [

]= lim

M→∞



−



+ iM



= lim

M→∞



tan

−1

(M/λ

)





λ>0

1 −



λ<0



η. (13.110)

Thus, the Pauli–Villars regulator gives a regularized form for the η-invariant. We

ﬁnally have

Im W [

η =

lim

s→0





sgnλ

|λ

−2s

(13.111)

where the prime indicates the omission of zero modes.

13.6.2 The dimensional ladder: 4–3–2

It is remarkable that the parity anomaly (13.110) is closely related to the chiral

anomaly in a (2l +2)-dimensional space (Alvarez-Gaum´e et al 1985). Following

Forte (1987), we look at the dimensional ladder,

four-dimensional Abelian anomaly

↓

three-dimensional parity anomaly (13.112)

↓

two-dimensional non-Abelian anomaly.

We take M

= S

× S

as a four-dimensional space. The Abelian anomaly

is given by the index

ind i

∇

−



×S

∂



×S

( ). (13.113)

As before,

(

−

) is the number of positive (negative) chirality zero modes.

Let Q

be the Chern–Simons form of ch

( );ch

( ) = dQ

( , ).Then

≡

−

is given by



×S

( ) =



×S

(

) +



×S

(

)



×S

(

) − dQ

(

)]

24π



×S

tr(g

−1

dg)

(13.114)

where g is the gauge transformation connecting

and

;

= g

−1

(

d + d

)g. In the previous section, we have shown that also represents the

non-Abelian anomaly

2π



2π

dθ

∂w(

,θ)

∂θ

(13.115a)

where w is deﬁned by

det i

g(θ)

) = e

iw( ,θ)

det i

D( ). (13.115b)

Here

is the reference gauge potential and

g(θ)

= g

−1

(x ,θ)( + d)g(x,θ)i

D =

∂+ 

Next, we show that

is also related to the parity anomaly in three-

dimensional space. Let i

∇

be a three-dimensional Dirac operator and deﬁne

a four-dimensional Dirac operator by

[ ]≡iσ

⊗ I

∂

∂t

+ σ

⊗ i

∇

[

] (13.116)

where

is a one-parameter family of gauge potentials interpolating

t=0

and

t=1

. The Atiyah–Patodi–Singer index theorem (section 12.8) is

ind i

=−



×S

×I

( ) +

[η(t = 1) − η(t = 0)] (13.117)

where we have noted that the Dirac genus

A is trivial on S

× S

× I . Suppose

and

are related by a gauge transformation,

= g

−1

(

+ d)g (13.118a)

and consider an interpolating potential

≡ t

+ (1 − t)

.(13.118b)

Since the spectrum of i

∇

is gauge invariant, in particular Spec i

∇

(

) =

Spec i

∇

(

),theη-invariant is also gauge invariant.

Then η(t = 0) = η(t = 1)

and the APS index theorem (13.117) yields

spectral ﬂow = ind i

(

)



×S

( ) =



×S

(

) − Q

(

)]



×S

−1

dg, 0) = . (13.119)

Thus, the spectral ﬂow of the three-dimensional theory is given by the index

In summary, the map g : S

×S

→ G is understood in three different ways:

(1) g is a transition function at the boundary of two patches of a G bundle over

× S

. It yields the index of the four-dimensional Abelian anomaly.

(2) Suppose

and

= g

−1

(

+ d)g are gauge potentials on S

× S

The gauge transformation function g measures the spectral ﬂow

between

Spe i

∇

(

) and Spe i

∇

(

(3) g : S

× S

→ G induces a map S

→ , the winding number of which

is identiﬁed with the non-Abelian anomaly in two-dimensional space.

Thus, we have obtained the ‘dimensional ladder’ 4–3–2. The extension to higher

dimensions is obvious.

Note that there is no gauge anomaly in odd-dimensional spaces.

BOSONIC STRING THEORY

In the present chapter, we study the one-loop amplitude of bosonic string

theory. Our example is the simplest one: closed, oriented bosonic strings in 26-

dimensional Euclidean space.

The action is the Polyakov action

S =

2π





√

γγ

αβ

∂

−

4π





√

γ (14.1)

where 

is a Riemann surface with genus g. The second term is proportional

to the Euler characteristic χ = 2 − 2g and, hence, determines the relative ratio

of multi-loop amplitudes; the g-loop amplitude is proportional to exp(−λg).We

have not written down the possible counter terms explicitly.

In the following sections, we work out the path integral formalism of bosonic

strings. We ﬁrst develop the necessary mathematical tools, namely differential

geometry on Riemann surfaces. Then the path integral expression for the vacuum

amplitude is written down. As an example, we compute the one-loop vacuum

amplitude. Our exposition is based on D’Hoker and Phong (1986), Polchinski

(1986) and Moore and Nelson (1986). There are many surveys of these topics,

for example, Alvarez-Gaum´e and Nelson (1986), Bagger (1987), D’Hoker and

Phong (1988) and Weinberg (1988).

14.1 Differential geometry on Riemann surfaces

Riemann surfaces are real two-dimensional manifolds without boundary. In our

study of topology and geometry, we referred to them in various places. Here

we summarize the basic facts on Riemann surfaces, which will make this chapter

self-contained. We also introduce several new aspects of Riemann surfaces, which

provide enough background for the study of bosonic string amplitudes.

14.1.1 Metric and complex structure

Let 

be a Riemann surface of genus g. Itwasshowninexample7.9thatwe

may introduce, in any chart U ,theisothermal coordinates (ξ

,ξ

) in which the

metric is conformally ﬂat:

g = e

2σ(ξ)

(dξ

⊗ dξ

+ dξ

⊗ dξ

). (14.2)

The reason for D = 26 will be clariﬁed in section 14.2.

Introduce the complex coordinates

z = ξ

+ iξ

¯z = ξ

− iξ

. (14.3)

Forms and vectors are spanned by

dz = dξ

+ id ξ

d¯z = dξ

− id ξ

(14.4a)

∂



∂

∂ξ

− i

∂

∂ξ



∂

¯z



∂

∂ξ

+ i

∂

∂ξ



. (14.4b)

In terms of the complex coordinates, the metric takes the form

g =

2σ(z,¯z)

[dz ⊗ d¯z + d¯z ⊗ dz]. (14.5)

The components of g are

z ¯z

= g

¯zz

2σ

= g

¯z¯z

= 0 (14.6a)

z ¯z

= g

¯zz

= 2e

−2σ

= g

¯z ¯z

= 0. (14.6b)

Let V be another chart of 

such that U ∩ V =∅.Let(w, ¯w) be the

complex coordinates in V . The metric in V is

g = e

2σ



(w, ¯w)

dw ⊗ d ¯w. (14.7)

The two expressions (14.5) and (14.7) should agree on U ∩ V ,

2σ(z,¯z)

dz ⊗ d¯z = e

2σ



(w, ¯w)

dw ⊗ d ¯w.

Since

dw ⊗ d ¯w =[(∂w/∂z) dz + (∂w/∂ ¯z)d¯z]⊗[(∂ ¯w/∂z) dz + (∂ ¯w/∂ ¯z)d¯z]

∝ dz ⊗ d¯z

we must have ∂w/∂¯z = ∂ ¯w/∂z = 0. [Another possibility, ∂w/∂z = ∂ ¯w/∂ ¯z = 0

is ruled out if (z, ¯z) and (w, ¯w) deﬁne the same orientation.] Thus, it follows that

w = w(z) ¯w =¯w(¯z) (14.8)

which veriﬁes that 

is a complex manifold. We also have

2σ(z,¯z)

= e

2σ



(w, ¯w)

|∂w/∂z|

. (14.9)

14.1.2 Vectors, forms and tensors

Let M = 

. The components of vector ﬁelds V

∂/∂z ∈ TM

and V

¯z

∂/∂ ¯z ∈

−

transform as

= (∂w/∂z)V

¯w

= (∂ ¯w/∂ ¯z)V

¯z

. (14.10)

The components of differential forms w

dz ∈ 

1,0

(M) and w

¯z

d¯z ∈ 

0,1

(M)

transform as

= (∂w/∂ z)

−1

¯w

= (∂ ¯w/∂ ¯z)

−1

¯z

. (14.11)

These are identiﬁed with sections of the holomorphic (anti-holomorphic) line

bundles over M = 

, for which the transition functions are holomorphic (anti-

holomorphic). The metric provides a natural isomorphism between TM

and



0,1

(M) through

¯z

= g

¯zz

, V

= g

z ¯z

¯z

. (14.12)

Similarly, TM

−

is isomorphic to ω

1,0

(M):

= g

z ¯z

¯z

, V

¯z

= g

¯zz

. (14.13)

In general, given an arbitrary tensor, the metric allows us to trade all the ¯z-indices

for z-indices. It is easy to see that

./,-

z...z

./,-

¯z...¯z

z...z

,-./

¯z...¯z

,-./

→ T

./,-

z...z

,-./

= (g

z ¯z

)

z ¯z

)

./,-

z...z

./,-

¯z...¯z

z...z

,-./

¯z...¯z

,-./

. (14.14)

This correspondence is an isomorphism. For example, observe that

z ¯z

¯z

→ g

z ¯z

¯z

= T

Thus, it is only necessary to consider tensors with pure z-indices. For these

tensors, we assign the helicity. Since T has z-indices only, it transforms under

z → w as

T →



∂w

∂z



T (14.15)

where n ∈

is given by the number of upper z-indices minus the number of lower

z-indices. For example,

→ T



∂w

∂z



All that matters is the difference between the number of upper indices and the

number of lower indices. The tensor T

is left invariant under z → w and is

regarded as a scalar. The number n is called the helicity. The set of helicity-n

tensors is denoted by

≡{T

./,-

z...z

,-./

|q − p = n}. (14.16)

The helicity characterizes the irreducible representation of U(1) = SO(2).

Sofarwehaveassumedn is an integer. It can be shown that n =

corresponds to the spinor ﬁeld on 

. In fact, the existence of spinors on the

Riemann surfaces is guaranteed by the triviality of the second Stiefel–Whitney

class of 

.Theset

is identiﬁed with the holomorphic line bundle K over 

Then

1/2

is the square root of K : S

= K =

where S

is the positive-

chirality spin bundle. Similarly, we have

−1

K = S

−

where S

−

is the

negative-chirality spin bundle.

Example 14.1. In real indices, the helicity ±1 vectors are given by V

±iV

.This

follows since

∂

∂ξ

+ V

∂

∂ξ

= (V

+ iV

)∂

+ (V

− iV

)∂

¯z

We put V

= V

+ iV

and V

¯z

= V

− iV

 V

. The helicity ±2 tensors are

± iT

,whereT is a symmetric traceless tensor of rank two. In fact, we ﬁnd



∂

∂ξ

⊗

∂

∂ξ

−

∂

∂ξ

⊗

∂

∂ξ



+ T



∂

∂ξ

⊗

∂

∂ξ

∂

∂ξ

⊗

∂

∂ξ



= 2(T

+ iT

)∂

⊗ ∂

+ 2(T

− iT

)∂

¯z

⊗ ∂

¯z

Clearly T

= 2(T

+ iT

) has helicity +2andT

¯z¯z

= 2(T

− iT

) has

helicity −2 (note that g

z ¯z

¯z¯z

= T

14.1.3 Covariant derivatives

The only non-vanishing Christoffel symbols of 

are (see (8.69))



= g

z ¯z

∂

z ¯z

= 2∂

σ

¯z

¯z¯z

= g

¯zz

∂

¯z

¯zz

= 2∂

¯z

σ. (14.17)

For tensors in

, we deﬁne two kinds of covariant derivative: ∇

(n)

→

n+1

and ∇

(n)

→

n−1

.Let

./,-

z...z

,-./

∈

(q − p = n).

We deﬁne

∇

(n)

z...z

= g

z ¯z

∇

¯z

z...z

= g

z ¯z

[∂

¯z

+ (q − p)

¯zz

z...z

= g

z ¯z

∂

¯z

z...z

(14.18a)

∇

(n)

z...z

=∇

z...z

=[∂

+ (q − p)

z...z

= (∂

+ 2n∂

σ)T

z...z

. (14.18b)

We use S

, instead of 

, to denote the spin bundles. The symbol 

is reserved for Laplacians.