Nakahara M. Geometry, Topology and Physics

Подождите немного. Документ загружается.

In (14.18b), 2n∂

σ acts like a gauge potential . We also deﬁne covariant

derivatives with respect to ¯z,

∇

¯z

(n)

= g

¯zz

∇

(n)

, ∇

(n)

¯z

= g

¯zz

∇

(n)

. (14.19)

The curvature two-form of K and the scalar curvature associated with the

Christoffel symbols are

= R

zz¯z

dz ∧ d¯z =−∂

¯z

(2∂

σ)dz ∧ d¯z

=−2∂

∂

¯z

σ dz ∧ d¯z (14.20a)

= g

¯zz

Ric

¯zz

+ g

z ¯z

Ric

z ¯z

=−8e

−2σ

∂

¯z

σ. (14.20b)

Exercise 14.1. Verify that

∇

(n)

= 2e

−2σ

∂

¯z

∇

(n)

= e

−2nσ

∂

2nσ

(14.21a)

∇

¯z

(n)

= 2e

−2(n+1)σ

∂

2nσ

∇

(n)

¯z

= ∂

¯z

. (14.21b)

∇

(n)

and ∇

(n)

are mutual adjoints with respect to a properly deﬁned inner

product. Let T , U ∈

. We require that the inner product be invariant under a

holomorphic change of the coordinate z → w.Since

z ¯z

→|dw/dz|

−2

z ¯z

√

g → d

√

T → (

dw/dz)

TU→ (dw/dz)

We ﬁnd the combination

(T , U) ≡



√

g(g

z ¯z

)

TU (14.22)

is invariant under holomorphic coordinate transformations. Take T ∈

and

U ∈

n+1

.Weﬁndthat

(U, ∇

(n)

T ) =



z e

2σ

−n−1

2(n+1)σ

U 2e

−2σ

∂

¯z

=−2

−n



zT∂

¯z

(2n+1)σ

U] (partial integration)

=−2

−n



zTe

(2n+1)σ

[∂

U + (2n + 1)(∂

σ)U]

=−



√

g(g

z ¯z

)

[∇

(n+1)

T = (−∇

(n+1)

U, T ).

This shows that

(∇

(n)

)

†

=−∇

(n+1)

. (14.23a)

Exercise 14.2. Show that

(∇

(n)

)

†

=−∇

(n−1)

.(14.23b)

We deﬁne two kinds of Laplacian 

(n)

→

n±1

→



(n)

≡−∇

(n+1)

∇

(n)

=−2e

−2σ

[∂

∂

¯z

+ 2n(∂

σ)∂

¯z

] (14.24a)



−

(n)

≡−∇

(n−1)

∇

(n)

=−2e

−2σ

[∂

∂

¯z

+ 2n(∂

σ)∂

¯z

+ 2n(∂

∂

¯z

σ)]. (14.24b)

Then it follows that



(n)

− 

−

(n)

= 4ne

−2σ

(∂

∂

¯z

σ) =−

n . (14.25)

This shows, in particular, that



(0)

= 

−

(0)

(≡ 

(0)

). (14.26)

14.1.4 The Riemann–Roch theorem

Here we derive a version of the Riemann–Roch theorem from the Atiyah–Singer

index theorem following D’Hoker and Phong (1988).

Theorem 14.1. (Riemann–Roch theorem)Let

be a Riemann surface of genus

g. Then the index of the operator ∇

(n)

dim

ker ∇

(n)

− dim ker ∇

(n−1)

= (2n − 1)(g − 1). (14.27)

Proof. We use the heat kernel to evaluate the index. We ﬁrst note that ker ∇

(n)

ker 

−

(n)

and ker ∇

(n−1)

= ker 

(n−1)

(see (14.24)). The heat kernel

of 

(n)

satisﬁes



∂

∂t

+ 

(n)



(z,w;t) =



∂

∂t

+  − V



(z,w;t) = 0

where  ≡−2∂

∂

¯z

is the ﬂat-space Laplacian and

≡  −

(n)

= (1 −e

−2σ

) + 4ne

−2σ

∂

σ∂

¯z

The Laplacian  also deﬁnes a heat kernel by



∂

∂t

+ 



K (z,w;t) = 0

which is easily solved to yield

K (z,w;t) =

4πt

−|z−w|

/2t

The perturbative computation and iteration yield

(z, z



;t) = K (z, z



;t)



dw K (z,w;t − s)V

(w)

(w, z



;s)

= K (z, z



;t) +



dw K (z,w;t − s)V

(w)K (w, z



;s)



dw K (z,v;t − s)V

(v)

× K (v, w;s − s



(w)K (w, z



)

+···.

We are particularly interested in

(z, z;t), t being small,

(z, z;t) =

4πt



dw K (z,w;t − s)V

(w)K (w, z;s) + (t).

(14.28)

If we take a coordinate system in which σ = 0atz,wehave

σ(w)  0 + ∂

σ(w − z) + ∂

¯z

σ( ¯w −¯z)

[∂

σ(w − z)

+ ∂

¯z

σ( ¯w −¯z)

+ 2∂

∂

¯z

σ |w − z|

]+···.

Due to rotational symmetry in two-dimensional space, only those terms with

one z-derivative and one ¯z-derivative survive in the integral in (14.28). Terms

proportional to ∂

σ∂

¯z

σ cancel between the second and third terms in the

expansion and we are left with terms proportional to ∂

∂

¯z

σ .Nowwehaveto

evaluate



w K (z,w;t − s)

×[2∂

∂

¯z

σ |¯w −¯z|



+ 4n( ¯w −¯z)∂

∂

¯z

σ∂

¯w

]K (w, z;s).

From the identities



w K (z,w;t − s)|w − z|



K (w, z;s)

16π

(t − s)



w|w|

exp



−

2s(t − s)

|w|



−

32π

(t − s)



w|w|

exp



−

2s(t − s)

|w|



(t − s)(2s − t)

2πt

and



w K (z,w;t − s)(¯z −¯w)∂

¯w

K (w, z;s)

32π

(t − s)



w exp



−

2s(t − s)

|w|



t − s

4πt

we ﬁnd that

(z, z;t) =

4πt

1 +3n

12π

σ +

(t). (14.29a)

We also have the diagonal part of the heat kernel

−

for 

−

(n)

−

(z, z;t) =

4πt

1 −3n

12π

σ +

(t). (14.29b)

From (14.29) and (14.20b), we obtain

ind ∇

(n)





1 −3n

12π

−

1 +3(n − 1)

12π



σ =

1 −2n

8π



=−

2n − 1

χ(

) = (2n − 1)(g − 1)

where

χ =

4π



x = 2 − 2g

is the Euler characteristic of 

14.2 Quantum theory of bosonic strings

Now we are ready to introduce Polyakov’s formulation of bosonic strings, which

is based on the path integral over geometries. Since the string action contains an

enormous symmetry, we have to pay special attention to counting independent

geometries once and only once. This is achieved by the Faddeev–Popov trick.

Our argument will be restricted to the simplest case, namely closed orientable

bosonic strings; the theory is deﬁned on Riemann surfaces.

14.2.1 Vacuum amplitude of Polyakov strings

According to the general prescription of the path integral formalism, the partition

function (vacuum-to-vacuum amplitude) of the string theory is given by

Z =

∞



g=0

∞



g=0



X γ e

−S[X,γ ]

(14.30)

Figure 14.1. The total vacuum amplitude is given by summing over g-loop amplitudes.

see ﬁgure 14.1. To avoid confusion, we denote the genus by g and the metric by

γ . The sum over genera amounts to the sum over the topologies. Z

is the g-loop

amplitude and is obtained by integrating over all metrics γ and all embeddings X.

As we shall see later, the measure

X γ is not well deﬁned and we need some

modiﬁcations. The string action S[X,γ] is taken to be

S[X,γ]≡



√

γγ

αβ

∂

4π



√

γ . (14.31)

The ﬁrst term is the Polyakov action. The second term is proportional to the Euler

characteristic

χ =

4π



√

γ = 2 − 2g

and serves as the string coupling constant; the amplitude of a loop with genus g is

suppressed by the factor e

−2λg

. Since this term is a topological invariant, it does

not affect the dynamics of the string. We are interested in Riemann surfaces of a

ﬁxed genus g and drop this term. The ﬁrst term of the action has the following

symmetries (section 7.11):

(A) Diff(

), the group of diffeomorphisms f : 

→ 

.Letξ

→ ξ



(ξ) be

the coordinate expression for f . The new metric is the pullback of the old

one whose coordinate component expression is

αβ

→ f

∗

αβ

∂ξ



∂ξ



γδ

. (14.32)

The embedding also gets transformed as

→ f

∗

= X

f. (14.33)

The invariance of the classical action takes the form

S[X,γ]=S[ f

∗

X, f

∗

γ ]. (14.34)

(B) Weyl(

), the group of two-dimensional Weyl rescalings

αβ

→ˆγ

αβ

≡ e

αβ

(14.35)

Figure 14.2. An element of ×

is obtained by the action of Diff(

) ∗Weyl(

) on

an element (X,γ) in the gauge slice.

where φ ∈ (

). The conformal invariance of S takes the form

S[X,γ]=S[X, ˆγ ]. (14.36)

The symmetries (A) and (B) must be preserved under quantization, otherwise

the theory has anomalies.

According to the standard Faddeev–Popov formalism, the degrees of

freedom corresponding to these symmetries have to be omitted when we deﬁne

. For example, the string geometry speciﬁed by the pairs (X

,γ

) and

,γ

) should not be counted independently if they are related by an element of

Diff(

). Similarly, (X,γ) and (X, e

γ) should not be counted as independent

conﬁgurations. Unless special attention is paid, we would count the same

conﬁgurations inﬁnitely many times, which leads to disastrous divergences. It

turns out that the space of all the geometries (X,γ) can be separated into

equivalence classes (the gauge slice), any two points of which cannot be

connected by these symmetries, see ﬁgure 14.2.

To be more mathematical, let

be the space of all the embeddings X :



→

and let

be the space of all the metrics deﬁned on 

.Naively,

the path integral is deﬁned over

. Because of the symmetries (A) and

(B), however, the integral should be restricted to the quotient space (

)/G

where G = Diff(

) ∗ Weyl(

) is the gauge group.

The action of ( f, e

) on

(X,γ) ∈

( f, e

)(X,γ)= ( f

∗

X, e

∗

γ). (14.37)

The quotient

/G is called the moduli space of 

and is denoted by

Mod(

). We are also interested in the subgroup Diff

(

) of Diff(

),which

Here ∗ denotes the semi-direct product. Note that Diff(

) ∩ Weyl(

) =∅. We shall come back

to this point later.

Figure 14.3. The mapping class group (MCG) is generated by Dehn twists around a

, b

and c

(1 ≤ i ≤ g).

is a connected component of the identity map. The quotient space Teich(

) ≡

/Diff

(

) ∗ Wey l (

) is called the Te i ch m ¨uller space of 

. The general

theory of Riemann surfaces shows that Teich(

) is a ﬁnite-dimensional universal

covering space of Mod(

). Explicitly, we have

dim

Teich(

) =











0 g = 0

2 g = 1

6g − 6 g ≥ 2.

(14.38)

The group Diff(

)/Diff

(

) is known as the modular group (MG) or the

mapping class group (MCG). The MCG is generated by the Dehn twists deﬁned

in example 8.2. For the torus with genus g, the MCG is generated by 3g −1Dehn

twists around a

, b

and c

in ﬁgure 14.3. Unfortunately, these 3g −1Dehntwists

are not the minimal set of the generators. The general form of MCG for g ≥ 2is

not well understood.

From these arguments, the meaningful partition function turns out to be

≡



X γ

V (Diff ∗ Wey l)

−S[X,γ ]

(14.39)

where V (Diff ∗ Weyl) is the (inﬁnite) volume of the space of Diff(

)∗Weyl(

)

and takes care of the inﬁnite overcounting of the same geometry. The order (the

number of elements) of MCG is denoted by |MCG|. Clearly,

V (Diff ∗ Wey l) =|MCG|V (Diff

∗ Weyl). (14.40)

14.2.2 Measures of integration

We have to deﬁne a sensible measure to carry out the integration (14.39) so that

the physical degrees of freedom and the gauge degrees of freedom are separated.

This separation of degrees of freedom requires the Jacobian,

γ X → J ( physical)( gauge). (14.41)

To ﬁnd this Jacobian, we note that the Jacobian on a manifold M agrees with that

on TM. To see this, let x

) be a coordinate of a chart U (V ) of M such that

U ∩ V =∅. The Jacobian of the coordinate change is J = det(∂y

/∂x

).Take

V ∈ T

M. In components, we have V = u

∂/∂x

= v

∂/∂y

,where

= u

(∂y

/∂x

). (14.42)

} and {v

} are ﬁbre coordinates of T

M. The Jacobian

J associated with this

coordinate change is

J = det(∂v

/∂u

) = det(∂y

/∂x

) = J. (14.43)

This shows that the Jacobian at p ∈ M is the same as that on T

M. The Jacobian

J depends on p but not on the vector itself, since J depends only on p.

Example 14.2. Let (x, y) and (r,θ)be coordinates of

,wherex = r cos θ and

y = r sin θ . The Jacobian of the coordinate change is

J = det

∂(x, y)

∂(r,θ)

= r.

Let us take

V = v

∂/∂x + v

∂/∂y = v

∂/∂r + v

∂/∂θ ∈ T

) and (v

) serve as ﬁbre coordinates of T

.Since

= v

∂x /∂r + v

∂x /∂θ v

= v

∂y/∂r + v

∂y/∂θ

the associated Jacobian

J is easily calculated to be

J = det[∂(v

)/∂(v

)]=



∂x /∂r ∂ x/∂θ

∂y/∂r ∂y/∂θ



= J.

Let us derive this Jacobian in an indirect but suggestive way. We normalize

themeasured

v as

1 =



v exp(−

v

) =



exp[−

+ v

)].

We also have v



= v

+ r

. Noting that the Jacobian is independent of v

and v

,wehave

1 = J



exp[−

)]=Jr

−1

This normalization of the measure differs by a constant factor from the conventional one.

from which we ﬁnd J = r . We use this procedure to ﬁnd the functional measure

of string theory.

This analysis enables us to write

δγ δ X = J δ(physical) δ(gauge) (14.44)

where δγ (δX) is a small variation of the metric γ (the embedding X)andis

regarded as an element of T

(

)(T

). The meaning of the RHS becomes

clear in a moment.

Consider the diffeomorphism generated by an inﬁnitesimal vector ﬁeld δv

on 

.Sinceδv is inﬁnitesimal, it belongs to Diff

(

) rather than the full group

Diff(

). The changes of the metric and the embedding under δv are (see (7.120))

αβ

= (

δv

γ)

αβ

=∇

δv

+∇

δv

X = δv

∂

X. (14.45)

The changes of γ and X under an inﬁnitesimal Weyl rescaling e

δφ

are

αβ

= δφγ

αβ

X = 0. (14.46)

These changes belong to unphysical (gauge) degrees of freedom. In general, a

small change of metric is given by

δγ

αβ

= δ

αβ

+ δ

αβ

+ (physical change)

= δφγ

αβ

+∇

δv

+∇

δv

+ δt

∂

∂t

αβ

(t) (14.47)

where the last term is called the Te ic hm ¨uller deformation of the metric, which

can neither be described by a diffeomorphism nor by a Weyl rescaling. As

mentioned before, {i } is a ﬁnite set, 1 ≤ i ≤ n = dim

Teich(

).Itis

convenient for later purposes to separate δγ into a traceless part and a part with a

non-zero trace. We write

δγ

αβ

= δ

φγ

αβ

+ (P

δv)

αβ

+ δt

iαβ

(t) (14.48)

where T

iαβ

is the traceless part of the Teichm¨uller deformation,

iαβ

≡

∂γ

αβ

∂t

−

αβ

γδ

∂γ

γδ

∂t

. (14.49)

The operator P

is deﬁned by

δv)

αβ

≡∇

δv

+∇

δv

− γ

αβ

(∇

δv

) (14.50)

and picks up the traceless part of δ

αβ

while δ

φ is deﬁned by

φ = δφ +



∇

δv

+ trace part of δt

∂γ

∂t



(14.51)

It should be kept in mind that we introduce the tangent space only to obtain the Jacobian. The

tangent space itself has no physical relevance.

where we do not need the explicit form in the parentheses.

As for the embeddings, we consider the quotient

/Diff(

). An arbitrary

embedding X is obtained by the action of Diff(

) on some

X ∈ /Diff(

Then a small change of the embedding is expressed as

δ X = δv

∂

X + δ

X (14.52)

where the ﬁrst term represents the change of X generated by δv while the second

is not associated with diffeomorphisms. Now the measure should look like

δγ δ X = J d

t δv δφ δ

X . (14.53)

To deﬁne the measure, we need to specify a metric on the tangent space, see

example 14.2. We restrict ourselves to the so called ultralocal metric which is

quadratic and depends on γ

αβ

but not on ∂γ

αβ

. Deﬁne a metric for symmetric

second-rank tensors by

δh



√

γ(G

αβγ δ

+ uγ

αβ

γδ

)δh

αβ

δh

γδ

(14.54a)

where u > 0 is an arbitrary constant and

αβγ δ

≡ γ

αγ

βδ

+ γ

αδ

βγ

− γ

αβ

γδ

. (14.55)

It is readily veriﬁed that G is the projection operator to the traceless part

(tr G

αβγ δ

δh

γδ

= γ

αβ

αβγ δ

δh

γδ

= 0) while uγ

αβ

γδ

is that to the trace part.

In a ﬁnite-dimensional manifold, a metric deﬁnes a natural volume element. In

the present case, however, the measure cannot be deﬁned explicitly and we have

to deﬁne it implicitly in terms of the Gaussian integral (see example 14.2),



δh exp(−

||δh||

) = 1. (14.56a)

Similarly, the metrics for a scalar δφ, a vector δv and a map δ X

are deﬁned by

δφ



√

γδφ

(14.54b)

δv



√

γγ

αβ

δv

(14.54c)

δX 



√

γδX

δ X

. (14.54d)

With these metrics, the measures are deﬁned by



δφ exp(−

||δφ||

) = 1 (14.56b)



δv exp(−

||δv||

) = 1 (14.56c)



δ X exp(−

||δ X ||

) = 1. (14.56d)