Stone M., Goldbart P. Mathematics for Physics: A Guided Tour for Graduate Students

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12.4 Applications 425

x + X



( x)

( x + X )



Figure 12.7 Pushing forward a vector X from TM

to TN

φ(x)

be one-to-one, so, contrary to the requirement imposed on a coordinate transformation,

it may not be possible to invert the functions ξ

(x) and write the x

’s as functions of

the ξ

’s.

While we can push forward individual vectors, we cannot always push forward a

vector field X from TM to TN . If two distinct points, x

and x

, should happen to

map to the same point ξ ∈ N , and X (x

) = X (x

), we would not know whether to

choose φ

∗

[X (x

)]or φ

∗

[X (x

)]as [φ

∗

X ](ξ). This problem does not occur for differential

forms. A map φ : M → N induces a natural, and always well-defined, pull-back map

∗

(

∗

)

→

(

∗

)

which works as follows: given a form ω ∈

(

∗

)

we define φ

∗

ω as a form on M by specifying what we get when we plug the vectors

, X

, ..., X

∈ TM into it. We evaluate the form at x ∈ M by pushing the vectors X

(x)

forward from TM

to TN

φ(x)

, plugging them into ω at φ(x) and declaring the result to

be the evaluation of φ

∗

ω on the X

at x. Symbolically

[φ

∗

ω](X

, X

, ..., X

) = ω(φ

∗

, φ

∗

, ..., φ

∗

). (12.30)

This may seem rather abstract, but the idea is in practice quite simple: if the map takes

x ∈ M → ξ(x) ∈ N , and if

ω =

...i

(ξ)dξ

...dξ

, (12.31)

then

∗

ω =

...i

[ξ(x)]dξ

(x)dξ

(x) ···dξ

(x)

...i

[ξ(x)]

∂ξ

∂x

∂ξ

∂x

···

∂ξ

∂x

···dx

. (12.32)

Computationally, the process of pulling back a form is so transparent that it is easy to

confuse it with a simple change of variable. That it is not the same operation will become

clear in the next few sections where we consider maps that are many-to-one.

426 12 Integration on manifolds

Exercise 12.3: Show that the operation of taking an exterior derivative commutes with

a pull-back:



∗



= φ

∗

(dω).

Exercise 12.4: If the map φ : M → N is invertible then we may push forward a vector

field X on M to get a vector field φ

∗

X on N . Show that

[φ

∗

ω]=φ

∗



∗



Exercise 12.5: Again assume that φ : M → N is invertible. By using the coordinate

expressions for the Lie bracket along with the effect of a push-forward, show that if X ,

Y are vector fields on TM then

∗

([X , Y ]) =[φ

∗

X , φ

∗

Y ],

as vector fields on TN.

12.4.2 Spin textures

As an application of pull-backs we consider some of the topological aspects of spin

textures which are fields of unit vectors n, or “spins”, in two or three dimensions.

Consider a smooth map ϕ : R

→ S

that assigns x (→ n(x), where n is a three-

dimensional unit vector whose tip defines a point on the 2-sphere S

. A physical example

of such an n(x) would be the local direction of the spin polarization in a ferromagnetically

coupled two-dimensional electron gas.

In terms of n, the area 2-form on the 2-sphere becomes

 =

n · (dn × dn) ≡



ijk

. (12.33)

The ϕ map pulls this area-form back to

F ≡ ϕ

∗

 =

(

ijk

∂

)dx

= (

ijk

∂

) dx

(12.34)

which is a differential form in R

. We will call it the topological charge density.It

measures the area on the 2-sphere swept out by the n vectors as we explore a square in

of side dx

by dx

Suppose now that the n tends to the same unit vector n(∞) at large distance in all

directions. This allows us to think of “infinity” as a single point, and the assignment

ϕ : x (→ n(x) as a map from S

to S

. Such maps are characterized topologically by

their “topological charge”, or winding number N which counts the number of times

the image of the originating x-sphere wraps round the target n-sphere. A mathematician

would call this number the Brouwer degree of the map ϕ. It is intuitively plausible that

12.4 Applications 427

a continuous map from a sphere to itself will wrap a whole number of times, and so we

expect

N =

4π







ijk

∂



, (12.35)

to be an integer. We will soon show that this is indeed so, but first we will demonstrate

that N is a topological invariant.

In two dimensions the form F = ϕ

∗

 is automatically closed because the exterior

derivative of any 2-form is zero – there being no 3-forms in two dimensions. Even if

we consider a field n(x

, ..., x

) in m > 2 dimensions, however, we still have dF = 0.

This is because

dF =



ijk

∂

. (12.36)

If we plug infinitesimal vectors into the dx

to get their components δx

, we have to

evaluate the triple-product of three vectors δn

= ∂

δx

, each of which is tangent to

the 2-sphere. But the tangent space of S

is two-dimensional, so any three tangent vectors

, t

, are linearly dependent and their triple-product t

· (t

× t

) is therefore zero.

Although it is closed, F = ϕ

∗

 will not generally be the d of a globally defined

1-form. Suppose, however, that we vary the map so that n (x) → n(x) + δn(x). The

corresponding change in the topological charge density is

δF = ϕ

∗

[n · (d(δn) × dn)], (12.37)

and this variation can be written as a total derivative:

δF = d{ϕ

∗

[n · (δn × dn)]} ≡ d{

ijk

δn

∂

}. (12.38)

In these manipulations we have used δn · (dn × dn) = dn · (δn × dn) = 0, the triple-

products being zero for the linear-dependence reason adduced earlier. From Stokes’

theorem, we have

δN =



δF =



∂S



ijk

δn

∂

. (12.39)

Because the sphere has no boundary, i.e. ∂S

=∅, the last integral vanishes, so we

conclude that δN = 0 under any smooth deformation of the map n(x). This is what we

mean when we say that N is a topological invariant. Equivalently, on R

, with n constant

at infinity, we have

δN =



δF =







ijk

δn

∂

, (12.40)

where  is a curve surrounding the origin at large distance. Again δN = 0, this time

because ∂

= 0 everywhere on .

428 12 Integration on manifolds

In some physical applications, the field n winds in localized regions called skyrmions.

These knots in the spin field behave very much as elementary particles, retaining their

identity as they move through the system. The winding number counts how many

skyrmions (minus the number of anti-skyrmions, which wind with opposite orienta-

tion) there are. To construct a smooth multi-skyrmion map ϕ : R

→ S

with positive

winding number N, take a set of N + 1 complex numbers λ, a

, ..., a

and another set

of N complex numbers b

, ..., b

such that no b coincides with any a. Then put

iφ

tan

= λ

(z − a

)...(z − a

)

(z − b

)...(z − b

)

(12.41)

where z = x

+ix

, and θ and φ are spherical polar coordinates specifying the direction

n at the point (x

, x

). At the points z = a

the vector n points straight up, and at the

points z = b

it points straight down. You will show in Exercise 12.12 that this particular

n-field configuration minimizes the energy functional

E[n]=



(

∂

n · ∂

n + ∂

n · ∂

)





|∇n

+|∇n

(12.42)

for the given winding number N . In the next section we will explain the geometric origin

of the mysterious combination e

iφ

tan θ/2.

12.4.3 The Hopf map

You may recall that in Section 10.2.3 we defined complex projective space CP

to be

the set of rays in a complex (n + 1)-dimensional vector space. A ray is an equivalence

class of vectors [ζ

, ζ

, ..., ζ

n+1

], where the ζ

are not all zero, and where we do not

distinguish between [ζ

, ζ

, ..., ζ

n+1

] and [λζ

, λζ

, ..., λζ

n+1

] for non-zero complex

λ. The space of rays is a 2n-dimensional real manifold: in a region where ζ

n+1

does not

vanish, we can take as coordinates the real numbers ξ

, ..., ξ

, η

, ..., η

where

+ iη

n+1

, ξ

+ iη

n+1

, ..., ξ

+ iη

n+1

. (12.43)

Similar coordinate charts can be constructed in the regions where other ζ

are non-zero.

Every point in CP

lies in at least one of these coordinate charts, and the coordinate

transformation rules for going from one chart to another are smooth.

The simplest complex projective space, CP

, is the real 2-sphere S

in disguise.

This rather non-obvious fact is revealed by the use of a stereographic map to make the

equivalence class [ζ

, ζ

]∈CP

correspond to a point n on the sphere. When ζ

is non-

zero, the class [ζ

, ζ

] is uniquely determined by the ratio ζ

/ζ

=|ζ

/ζ

iφ

, which we

plot on the complex plane. We think of this copy of C as being the xy-plane in R

.We

then draw a straight line connecting the plotted point to the south pole of a unit sphere

12.4 Applications 429

θ/2

/ ζ

= ζ

Figure 12.8 Two views of the stereographic map between the 2-sphere and the complex plane.

The point ζ = ζ

/ζ

∈ C corresponds to the unit vector n ∈ S

circumscribed about the origin in R

. The point where this line (continued, if necessary)

intersects the sphere is the tip of the unit vector n.

If ζ

were zero, n would end up at the north pole, where the R

coordinate z takes

the value z = 1. If ζ

goes to zero with ζ

fixed, n moves smoothly to the south pole

z =−1. We therefore extend the definition of our map to the case ζ

= 0 by making the

equivalence class [0, ζ

] correspond to the south pole. We can find an explicit formula

for this map. Figure 12.8 shows that ζ

/ζ

= e

iφ

tan θ/2, and this relation suggests the

use of the “t”-substitution formulæ:

sin θ =

1 + t

, cos θ =

1 − t

1 + t

, (12.44)

where t = tan θ/2. Since the x, y, z components of n are given by

= sin θ cos φ,

= sin θ sin φ,

= cos θ, (12.45)

we find that

+ in

2(ζ

/ζ

)

1 +|ζ

/ζ

, n

1 −|ζ

/ζ

1 +|ζ

/ζ

. (12.46)

We can multiply through by |ζ

= ζ

, and so write this correspondence in a more

symmetrical manner:

+ ζ

|ζ

+|ζ



− ζ

|ζ

+|ζ



|ζ

−|ζ

|ζ

+|ζ

. (12.47)

430 12 Integration on manifolds

This last form can be conveniently expressed in terms of the Pauli sigma matrices

=σ





, =σ



0 −i

i 0



, =σ



0 −1



(12.48)

= (z

, z

)







= (z

, z

)



0 −i

i 0





= (z

, z

)



0 −1





, (12.49)

where







|ζ

+|ζ





(12.50)

is a normalized 2-vector, which we think of as a spinor.

The correspondence CP

* S

now has a quantum-mechanical interpretation: any

unit 3-vector n can be obtained as the expectation value of the ˆσ matrices in a normalized

spinor state. Conversely, any normalized spinor ψ = (z

, z

)

gives rise to a unit

vector via

= ψ

†

ˆσ

ψ. (12.51)

Now, since

1 =|z

+|z

, (12.52)

the normalized spinor can be thought of as defining a point in S

. This means that the

one-to-one correspondence [z

, z

]↔n also gives rise to a map from S

→ S

. This is

called the Hopf map:

Hopf : S

→ S

. (12.53)

The dimension reduces from three to two, so the Hopf map cannot be one-to-one. Even

after we have normalized [ζ

, ζ

], we are still left with a choice of overall phase. Both

, z

) and (z

iθ

, z

iθ

), although distinct points in S

, correspond to the same point in

, and hence in S

. The inverse image of a point in S

is a geodesic circle in S

. Later,

we will show that any two such geodesic circles are linked, and this makes the Hopf

map topologically non-trivial, in that it cannot be continuously deformed to a constant

map – i.e. to a map that takes all of S

to a single point in S

12.4 Applications 431

Exercise 12.6: We have seen that the stereographic map relates the point with spherical

polar coordinates θ, φ to the complex number

ζ = e

iφ

tan θ/2.

We can therefore set ζ = ξ +iη and take ξ, η as stereographic coordinates on the sphere.

Show that in these coordinates the sphere metric is given by

g( , ) ≡ dθ ⊗ dθ + sin

θ dφ ⊗ dφ

(1 +|ζ |

)

(dζ ⊗ dζ + dζ ⊗ dζ)

(1 + ξ

+|η|

)

(dξ ⊗ dξ + dη ⊗ dη),

and that the area 2-form becomes

 ≡ sin θ dθ ∧ dφ

(1 +|ζ |

)

dζ ∧ dζ

(1 + ξ

+ η

)

dξ ∧ dη. (12.54)

12.4.4 Homotopy and the Hopf map

We can use the Hopf map to factor the map ϕ : x (→ n(x) via the 3-sphere by specifying

the spinor ψ at each point, instead of the vector n, and so mapping indirectly

ϕ : R

→ S

Hopf

→ S

It might seem that for a given spin-field n(x) we can choose the overall phase of ψ(x) ≡

(x), z

(x))

as we like; however, if we demand that the z

’s be continuous functions of

x then there is a rather non-obvious topological restriction which has important physical

consequences. To see how this comes about, we first express the winding number in

terms of the z

. We find (after a page or two of algebra) that

F = (

ijk

∂

) dx



i=1

(

∂

− ∂

∂

)

, (12.55)

and so the topological charge N is given by

N =

2πi





i=1

(

∂

− ∂

∂

)

. (12.56)

432 12 Integration on manifolds

Now, when written in terms of the z

variables, the form F becomes a total derivative:

F =



i=1

(

∂

− ∂

∂

)

= d





i=1



∂

− (∂





. (12.57)

Furthermore, because n is fixed at large distance, we have (z

, z

) = e

iθ

, c

) near

infinity, where c

, c

are constants with |c

+|c

= 1. Thus, near infinity,



i=1



∂

− (∂



→ (|c

+|c

)dθ = dθ . (12.58)

We combine this observation with Stokes’ theorem to obtain

N =

2πi







i=1



∂

− (∂



2π





dθ . (12.59)

Here, as in the previous section,  is a curve surrounding the origin at large distance.

Now



dθ is the total change in θ as we circle the boundary. While the phase e

iθ

has

to return to its original value after a round trip, the angle θ can increase by an integer

multiple of 2π . The winding number

dθ/2π can therefore be non-zero, but must be

an integer.

We have uncovered the rather surprising fact that the topological charge of the map

ϕ : S

→ S

is equal to the winding number of the phase angle θ at infinity. This is the

topological restriction referred to in the preceding paragraph. As a byproduct, we have

confirmed our conjecture that the topological charge N is an integer. The existence of this

integer invariant shows that the smooth maps ϕ : S

→ S

fall into distinct homotopy

classes labelled by N . Maps with different values of N cannot be continuously deformed

into one another, and, while we have not shown that it is so, two maps with the same

value of N can be deformed into each other.

Maps that can be continuously deformed one into the other are said to be homotopic.

The set of homotopy classes of the maps of the n-sphere into a manifold M is denoted

by π

(M ). In the present case M = S

. We are therefore claiming that

) = Z, (12.60)

where we are identifying the homotopy class with its winding number N ∈ Z.

12.4.5 The Hopf index

We have so far discussed maps from S

to S

. It is perhaps not too surprising that such

maps are classified by a winding number. What is rather more surprising is that maps

12.4 Applications 433

Figure 12.9 A twisted cable with N = 5.

ϕ : S

→ S

also have an associated topological number. If we continue to assume

that n tends to a constant vector at infinity so that we can think of R

∪ {∞} as being

, this number will label the homotopy classes π

) of fields of unit vectors n in

three dimensions. We will think of the third dimension as being time. In this situation an

interesting set of n fields to consider are the n(x, t) corresponding moving skyrmions.

The world-lines of these skyrmions will be tubes outside of which n is constant, and

such that on any slice through the tube, n will cover the target n-sphere once.

To motivate the formula we will find for the topological number, we begin with a

problem from magnetostatics. Suppose we are given a cable originally made up of a

bundle of many parallel wires. The cable is then twisted N times about its axis and bent

into a closed loop, the end of each individual wire being attached to its beginning to

make a continuous circuit (Figure 12.9). A current I flows in the cable in such a manner

that each individual wire carries only an infinitesimal part δI

of the total. The sense of

the current is such that as we flow with it around the cable each wire wraps N times

anticlockwise about all the others. The current produces a magnetic field B. Can we

determine the integer twisting number N knowing only this B field?

The answer is yes. We use Ampère’s law in integral form,



B · dr = (current encircled by ). (12.61)

We also observe that the current density ∇×B = J at a point is directed along the tangent

to the wire passing through that point. We therefore integrate along each individual wire

as it encircles the others, and sum over the wires to find



wires i

δI

B · dr



B · J d

x =



B · (∇×B) d

x. (12.62)

We now apply this insight to our three-dimensional field of unit vectors n(x). The quantity

playing the role of the current density J is the topological current



σµν



ijk

∂

. (12.63)

434 12 Integration on manifolds

Observe that div J = 0. This is simply another way of saying that the 2-form F = ϕ

∗



is closed.

The flux of J through a surface S is

I =



J · dS =



F (12.64)

and this is the area of the spherical surface covered by the n’s. A skyrmion, for example,

has total topological current I = 4π, the total surface area of the 2-sphere. The skyrmion

world-line will play the role of the cable, and the inverse images in R

of points on S

correspond to the individual wires.

In form language, the field corresponding to B can be any 1-form A such that

dA = F. Thus

Hopf



B · J d

x =

16π



AF (12.65)

will be an integer. This integer is the Hopf linking number,orHopf index, and counts

the number of times the skyrmion twists before it bites its tail to form a closed-loop

world-line.

There is another way of obtaining this formula, and of understanding the number

16π

. We observe that the two-form F and the one-form A are the pull-back from S

along ψ of the forms

F =



i=1

(

− dz

)

A =



i=1

(

− z

)

, (12.66)

respectively. If we substitute z

1,2

= ξ

1,2

+ iη

1,2

, we find that

AF = 8(ξ

dη

dξ

dη

− η

dξ

dη

+ ξ

dη

dξ

dη

− η

dξ

dη

). (12.67)

We know from Exercise 12.2 that this expression is eight times the volume 3-form on

the 3-sphere. Now the total volume of the unit 3-sphere is 2π

, and so, from our factored

map x (→ ψ (→ n we have that

Hopf

16π



∗

(AF) =

2π



∗

d(Volume on S

) (12.68)

is the number of times the normalized spinor ψ(x) covers S

as x covers R

. For the

Hopf map itself, this number is unity, and so the loop in S

that is the inverse image of

a point in S

will twist once around any other such inverse image loop.