Nakahara M. Geometry, Topology and Physics

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We ﬁnally obtain the Lagrangian which is a scalar both under coordinate

changes and local Lorentz rotations,

≡

ψ[iγ

(∂

i



βγ

) + m]ψ (7.228)

and the scalar action

≡



√

−g

ψ[iγ

(∂

i



βγ

) + m]ψ. (7.229a)

If ψ is coupled to the gauge ﬁeld

, the action is given by



√

−g

ψ[iγ

(∂

i



βγ

) + m]ψ. (7.229b)

It is interesting to note that the spin connection term vanishes if dim M = 2.

To see this, we rewrite (7.229a) as



√

−g

ψ[iγ

←→

∂

i

{iγ

,

βγ

}+m]ψ(7.229a



)

where γ

= γ

and we have added total derivatives to the Lagrangian to

make it Hermitian. The non-vanishing components of  are 

∝[γ

,γ

]∝γ

where γ

is the two-dimensional analogue of γ

.Since{γ

,γ

}=0, the spin

connection term drops out from S

7.11 Bosonic string theory

Quantum ﬁeld theory (QFT) is occasionally called particle physics since it deals

with the dynamics of particles. As far as high-energy processes whose typical

energy is much smaller than the Planck energy (∼10

GeV) are concerned there

is no objection to this viewpoint. However, once we try to quantize gravity in

this framework, there exists an impenetrable barrier. We do not know how to

renormalize the ultraviolet divergences that are ubiquitous in the QFT of gravity.

In the early 1980s, physicists tried to construct a consistent theory of gravity

by introducing supersymmetry. In spite of a partial improvement, the resulting

supergravity could not tame the ultraviolet behaviour completely.

In the late 1960s and early 1970s, the dual resonance model was extensively

studied as a candidate for a model of hadrons. In this, particles are replaced

by one-dimensional objects called strings. Unfortunately, it turned out that

the theory contained tachyons (imaginary mass particles) and spin-2 particles

and, moreover, it is consistent only in 26-dimensional spacetime! Due to

these difﬁculties, the theory was abandoned and taken over by quantum

chromodynamics (QCD). However, a small number of people noticed that the

theory must contain the graviton and they thought it could be a candidate for the

quantum theory of gravity.

Figure 7.9. The trajectories of an open string (a) and a closed string (b). Slices of the

trajectories at ﬁxed parameter τ

are also shown.

Nowadays, supersymmetry has been built into string theory to form the

superstring theory, which is free of tachyons and consistent in ten-dimensional

spacetime. There are several candidates for consistent superstring theories. It is

sometimes suggested that complete mathematical consistency will single out a

unique theory of everything (TOE).

In this book, we study the elementary aspects of bosonic string theory in the

ﬁnal chapter. We also study some mathematical tools relevant for superstrings.

The classical review is that of Scherk (1975). We give more references in

chapter 14.

7.11.1 The string action

The trajectory of a particle in a D-dimensional Minkowski spacetime is given by

the set of D functions X

(τ ),1≤ µ ≤ D,whereτ parametrizes the trajectory.

A string is a one-dimensional object and its conﬁguration is parametrized by two

numbers (σ, τ ), σ being spacelike and τ timelike. Its position in D-dimensional

Minkowski spacetime is given by X

(σ, τ ), see ﬁgure 7.9. The parameter σ can

be normalized as σ ∈[0,π]. A string may be open or closed. We now seek an

action that governs the dynamics of strings.

We ﬁrst note that the action of a relativistic particle is the length of the world

line,

S ≡ m



ds = m



dτ(−

)

1/2

(7.230)

where

≡ dX

/dτ . For some purposes, it is convenient to take another

expression,

S =−



dτ

√

g(g

−1

− m

) (7.231)

where the auxiliary variable g ≡ g

ττ

is regarded as a metric.

Exercise 7.26. Write down the Euler–Lagrange equations derived from (7.231).

Eliminate g from (7.231) making use of the equation of motion to reproduce

(7.230).

What is the advantage of (7.231) over (7.230)? We ﬁrst note that (7.231)

makes sense even when m

= 0, while (7.230) vanishes in this case. Second,

(7.231) is quadratic in X while the X-dependence of (7.230) is rather complicated.

Nambu (1970) proposed an action describing the strings, which is

proportional to the area of the world sheet, the surface spanned by the trajectory

of a string. Clearly this is a generalization of the length of the world line of a

particle. He proposed the Nambu action,

S =−

2πα



dσ



dτ [−det(∂

∂

)]

1/2

(7.232)

where ξ

= τ, ξ

= σ and ∂

≡ ∂ X

/∂ξ

. The parameter τ

(τ

) is the

initial (ﬁnal) value of the parameter τ while α



is a parameter corresponding to

the inverse string tension (the Regge slope).

Exercise 7.27. The action S is required to have no dimension. We take σ and τ to

be dimensionless. Show that the dimension of α



is [length]

Although the action provides a nice geometrical picture, it is not quadratic

in X and it turned out that the quantization of the theory was rather difﬁcult. Let

us seek an equivalent action which is easier to quantize. We proceed analogously

to the case of point particles. A quadratic action for strings is called the Polyakov

action (Polyakov 1981) and is given by

S =−

4πα



dσ



dτ

√

−gg

αβ

∂

(7.233)

where g = det g

αβ

and g

αβ

= (g

−1

)

αβ

. If the string is open, the trajectory is

a sheet while if it is closed, it is a tube, see ﬁgure 7.9. It is shown here that the

action (7.233) agrees with (7.232) upon eliminating g. It should be noted though

that this is true only for the Lagrangian. There is no guarantee that this remains

true at the quantum level. It has been shown that the quantum theory based on the

respective Lagrangians agrees only for D = 26. The action (7.233) is invariant

under

(i) local reparametrization of the world sheet

τ → τ



(τ, σ ) σ → σ



(τ, σ ) (7.234a)

(ii) Weyl rescaling

αβ

→ g



αβ

≡ e

φ(σ,τ)

αβ

(7.234b)

(iii) global Poincar´einvariance

→ X

µ

≡ 

 ∈ SO(D −1, 1) a ∈

. (7.234c)

These symmetries will be worked out later.

Exercise 7.28. Taking advantage of symmetries (i) and (iii), it is always possible

to choose g

αβ

in the form g

αβ

= η

αβ

. Write down the equation of motion for X

to show that it obeys the equation

αβ

∂

= 0. (7.235)

7.11.2 Symmetries of the Polyakov strings

The bosonic string theory is deﬁned on a two-dimensional Lorentz manifold

(M, g). The embedding f : M →

is deﬁned by ξ

→ X

where

{ξ

}=(τ, σ ) are the local coordinates of M. We assume the physical spacetime

is Minkowskian (

,η)for simplicity. The Polyakov action

S =−



√

−gg

αβ

∂

µν

(7.236)

is left invariant under the coordinate reparametrization Diff(M) since the volume

element

√

−gd

ξ is invariant and g

αβ

∂

is a scalar.

Now we are ready to derive the equation of motion. Our variational

parameters are the embedding X

and the geometry g

αβ

. Under the variation

δ X

, we have the Euler–Lagrange equation

∂

(

√

−gg

αβ

∂

) = 0. (7.237a)

Under the variation δg

αβ

, the integrand of S changes as

δ(

√

−gg

αβ

∂

) = δ

√

−gg

αβ

∂

√

−gδg

αβ

∂

=−

√

−gg

γδ

δg

γδ

αβ

∂

√

−gδg

αβ

∂

where proposition 7.2 has been used. Since this should vanish for any variation

δg

αβ

, we should have

αβ

= ∂

∂

−

αβ

γδ

∂

) = 0. (7.237b)

This is solved for g

αβ

to yield

αβ

= ∂

∂

µν

(7.238)

showing that the induced metric (the RHS) agrees with g

αβ

. Substituting (7.238)

into (7.236) to eliminate g

αβ

, we recover the Nambu action,

S =−



−det(∂

∂

). (7.239)

By construction, the action S is invariant under local reparametrization of

M, {ξ

}→{ξ

α

(ξ)}. In addition to this, the action has extra invariances. Under

the global Poincar

e transformation in D-dimensional spacetime,

→ X

µ

≡ 

+ a

(7.240)

the action S transforms as

S →−



√

−gg

αβ

∂

(

+ a

)∂

(

+ a

)η

µν

=−



√

−gg

αβ

∂

(



µν

From 



µν

= η

κλ

,weﬁndthatS is invariant under global Poincar´e

transformations. The action S is also invariant under the Weyl rescaling,

αβ

(τ, σ ) → e

2σ(τ,σ)

α,β

(τ, σ ) keeping (τ, σ ) ﬁxed. In fact, S transforms as

S →−



−e

4σ

−2σ

αβ

∂

µν

and hence is left invariant. Note that the Weyl rescaling invariance exists only

when M is two dimensional, making strings prominent among other extended

objccts such as membranes.

Since dim M = 2, we can always parametrize the world sheet by the

isothermal coordinate (example 7.9) so that

αβ

= e

2σ(τ,σ)

αβ

. (7.241)

Then the Weyl rescaling invariance allows us to choose the standard metric η

αβ

on the world sheet. The metric g

αβ

has three independent components while the

reparametrization has two degrees of freedom and the Weyl scaling invariance

has one. Thus, so long as we are dealing with strings, we can choose the standard

metric η

αβ

We end our analysis of Polyakov strings here. Polyakov strings will be

quantized in the most elegant manner in chapter 14.

Exercise 7.29. Let (M, g) and (N, h) be Riemannian manifolds. Take a chart U

of M in which the metric g takes the form

g = g

µν

(x ) dx

⊗ dx

Take a chart V of N on which h takes the form

h = G

αβ

(φ)dφ

⊗ dφ

Amapφ : M → N deﬁned by x → φ(x) is called a harmonic map if it satisﬁes

√

∂

[

√

µν

∂

]+

βγ

∂

µν

= 0. (7.242)

Show that this equation is obtained by the variation of the action

S ≡



√

µν

∂

αβ

(φ) (7.243)

with respect to φ. Applications of harmonic maps to physics are found in Misner

(1978) and S´anchez (1988). Mathematical aspects have been reviewed in Eells

and Lemaire (1968).

Problems

7.1 Let ∇ be a general connection for which the torsion tensor does not vanish.

Show that the ﬁrst Bianchi identity becomes

{R(X, Y )Z }= {T (X, [Y, Z])}+ {∇

[T (Y, Z)]}

where

is the symmetrizer deﬁned in theorem 7.2. Show also that the second

Bianchi identity is given by

{(∇

R)(Y, Z )}V = {R(X, T (Y, Z ))}V

where

symmetrizes X , Y and Z only.

7.2 Let (M, g) be a conformally ﬂat three-dimensional manifold. Show that the

Weyl–Schouten tensor deﬁned by

λµν

≡∇

Ric

λµ

−∇

Ric

λν

−

λµ

∂

− g

λν

∂

)

vanishes. It is known that C

λµν

= 0 is the necessary and sufﬁcient condition for

conformal ﬂatness if dim M = 3.

7.3 Consider a metric

g =−dt ⊗ dt + dr ⊗ dr + (1 −4µ

dφ ⊗ dφ + dz ⊗ dz

where 0 <µ<1/2andµ = 1/4. Introduce a new variable

(

φ ≡ (1 − 4µ)φ

and show that the metric g reduces to the Minkowski metric. Does this mean that

g describes Minkowski spacetime? Compute the Riemann curvature tensor and

show that there is a stringlike singularity at r = 0. This singularity is conical (the

spacetime is ﬂat except along the line). This metric models the spacetime of a

cosmic string.

COMPLEX MANIFOLDS

A differentiable manifold is a topological space which admits differentiable

structures. Here we introduce another structure which has relevance in physics.

In elementary complex analysis, the partial derivatives are required to satisfy the

Cauchy–Riemann relations. We talk not only of the differentiability but also of

the analyticity of a function in this case. A complex manifold admits a complex

structure in which each coordinate neighbourhood is homeomorphic to

and

the transition from one coordinate system to the other is analytic.

The reader may consult Chern (1979), Goldberg (1962) or Greene (1987)

for further details. Grifﬁths and Harris (1978), chapter 0 is a concise survey of

the present topics. For applications to physics, see Horowitz (1986) and Candelas

(1988).

8.1 Complex manifolds

To begin with, we deﬁne a holomorphic (or analytic) map on

.Acomplex-

valued function f :

→ is holomorphic if f = f

+ i f

satisﬁes the

Cauchy–Riemann relations for each z

= x

+ i y

∂ f

∂x

∂ f

∂y

∂ f

∂x

=−

∂ f

∂y

. (8.1)

Amap( f

,..., f

) :

→

is called holomorphic if each function f

(1 ≤ λ ≤ n) is holomorphic.

8.1.1 Deﬁnitions

Deﬁnition 8.1. M is a complex manifold if the following axioms hold,

(i) M is a topological space.

(ii) M is provided with a family of pairs {(U

,ϕ

)}.

(iii) {U

} is a family of open sets which covers M.Themapϕ

is a

homeomorphism from U

to an open subset U of

. [Hence, M is even

dimensional.]

(iv) Given U

and U

such that U

∩ U

=∅,themapψ

= ϕ

◦ ϕ

−1

from

∩U

) to ϕ

∩U

) is holomorphic.

The number m is called the complex dimension of M and is denoted as

dim

M = m. The real dimension 2m is denoted either by dim M or simply

by dim M.Letz

= ϕ

( p) and w

= ϕ

( p) be the (complex) coordinates of

a point p ∈ U

∩ U

in the charts (U

,ϕ

) and (U

,ϕ

), respectively. Axiom

(iv) asserts that the function w

= u

+ iv

(1 ≤ ν ≤ m) is holomorphic in

= x

+ iy

, namely

∂u

∂x

∂v

∂y

∂u

∂y

=−

∂v

∂x

1 ≤ µ, ν ≤ m.

These axioms ensure that calculus on complex manifolds can be carried out

independently of the special coordinates chosen. For example,

is the simplest

complex manifold. A single chart covers the whole space and ϕ is the identity

map.

Let {(U

,ϕ

)} and {(V

,ψ

)} be atlases of M. If the union of two atlases is

again an atlas which satisﬁes the axioms of deﬁnition 8.1, they are said to deﬁne

the same complex structure. A complex manifold may carry a number of complex

structures (see example 8.2).

8.1.2 Examples

Example 8.1. In exercise 5.1, it was shown that the stereographic coordinates of

a point P(x, y, z) ∈ S

−{North Pole} projected from the North Pole are

(X, Y ) =



1 − z



while those of a point P(x , y, z) ∈ S

−{South Pole} projected from the South

Pole are

(U, V ) =



1 + z

−y

1 + z



[Note the orientation of (U, V ) in ﬁgure 5.5.] Let us deﬁne complex coordinates

Z = X + iY,

Z = X − iY, W = U + iV, W = U − iV.

W is a holomorphic function of Z,

W =

x − iy

1 + z

1 − z

1 + z

(X − iY ) =

X − iY

+ Y

Thus, S

is a complex manifold which is identiﬁed with the Riemann sphere

∪{∞}.

Example 8.2. Take a complex plane

and deﬁne a lattice L(ω

,ω

) ≡{ω

m +

n|m, n ∈ } where ω

and ω

are two non-vanishing complex numbers such

Figure 8.1. Two complex numbers ω

and ω

deﬁne a lattice L(ω

,ω

) in the complex

plane.

/L(ω

,ω

) is homeomorphic to the torus (the shaded area).

that ω

/ω

/∈ ; see ﬁgure 8.1. Without loss of generality, we may take

Im(ω

/ω

)>0. The manifold /L(ω

,ω

) is obtained by identifying the points

, z

∈ such that z

− z

= ω

m + ω

n for some m, n ∈ . Since the

opposite sides of the shaded area of ﬁgure 8.1 are identiﬁed,

/L(ω

,ω

) is

homeomorphic to the torus T

. The complex structure of naturally induces

that of

/L(ω

,ω

). We say that the pair (ω

,ω

) deﬁnes a complex structure on

. There are many pairs (ω

,ω

) which give the same complex structure on T

When do pairs (ω

,ω

) and (ω



,ω



) (Im(ω

/ω

)>0, Im(ω



/ω



)>0)

deﬁne the same complex structure? We ﬁrst note that two lattices L(ω

,ω

) and

L(ω



,ω



) coincide if and only if there exists a matrix





∈ PSL(2,

) ≡ SL(2, )/

such that













. (8.2)

This statement is proved as follows.

Suppose













where





∈ SL(2,

The group SL(2, ) has been deﬁned in (2.4). Two matrices A and −A are identiﬁed in PSL(2, ).

Since ω



,ω



∈ L(ω

,ω

),weﬁndL(ω



,ω



) ⊂ L(ω

,ω

).From







d −b

−ca







we also ﬁnd L(ω

,ω

) ⊂ L(ω



,ω



). Thus, L(ω

,ω

) = L(ω



,ω



).Conversely,

if L(ω

,ω

) = L(ω



,ω



), ω



and ω



are lattice points of L(ω

,ω

) and can be

written as ω



= dω

+ cω

and ω



= bω

+ aω

where a, b, c, d ∈ .Also

and ω

may be expressed as ω

= d



+ c



and ω

= b



+ a



where



, b



, c



, d



∈ .Thenwehave























from which we ﬁnd













Equating the determinants of both sides, we have (a



−b



)(ad −bc) = 1. All

the entries being integers, this is possible only when ad − bc =±1. Since







= Im



bω

+ aω

dω

+ cω



ad − bc

|c(ω

/ω

) + d|







> 0

we must have ad − bc > 0, that is,





∈ SL(2,

In fact, it is clear that





∈ SL(2,

)

deﬁnes the same lattice as

−





and we have to identify those matrices of SL(2,

) whichdifferonlybytheir

overall signature. Thus, two lattices agree if they are related by PSL(2,

) ≡

SL(2,

Assume that there exists a one-to-one holomorphic map h of

/L(ω

,ω

)

onto

/L( ˜ω

, ˜ω

) where Im(ω

/ω

)>0, Im( ˜ω

/ ˜ω

)>0. Let p : →

/L(ω

,ω

) and ˜p : → /L( ˜ω

, ˜ω

) be the natural projections. For example,

p maps a point in

to an equivalent point in /L(ω

,ω

). Choose the origin 0

and deﬁne h

∗

(0) to be a point such that ˜p ◦ h

∗

(0) = h ◦ p(0) (ﬁgure 8.2),

∗

−−−−−−−−−−→



˜p

/L(ω

,ω

)

−−−−−−−−−−→ /L( ˜ω

, ˜ω

(8.3)