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

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16.3 Working in the total space 595

16.3.3 Monopole harmonics

The total-space operations and definitions in these sections may seem rather abstract.

We therefore demonstrate their power by solving the Schrödinger problem for a charged

particle confined to a unit sphere surrounding a magnetic monopole. The conventional

approach to this problem involves first selecting a gauge for the vector potential A, which,

because of the monopole, is necessarily singular at a Dirac string located somewhere on

the sphere, and then delving into properties of Gegenbauer polynomials. Eventually we

find the gauge-dependent wavefunction. By working with the total space, however, we

can solve the problem in all gauges at once, and the problem becomes a simple exercise

in Lie-group geometry.

Recall that the SU(2) representation matrices D

(θ, φ, ψ)form a complete orthonor-

mal set of functions on the group manifold S

. There will be a similar complete

orthonormal set of representation matrices on the manifold of any compact Lie group

G. Given a subgroup H ⊂ G, we will use these matrices to construct bundles associated

to a principal H -bundle that has G as its total space and the coset space G/H as its

base-space. The fibres will be copies of H , and the projection π the usual projection

G → G/H .

The functions D

(g) are not in general functions on the coset space G/H as they

depend on the choice of representative. Instead, because of the representation property,

they vary with the choice of representative in a well-defined way:

(gh) = D



(g)D



(h). (16.93)

Since we are dealing with compact groups, the representations can be taken to be unitary

and therefore

(gh)]

∗

=[D



(g)]

∗



(h)]

∗

(16.94)

= D



−1

)[D



(g)]

∗

. (16.95)

This is the correct variation under the right action of the group H for the set of functions

(gh)]

∗

to be sections of a bundle associated with the principal fibre bundle G →

G/H . The representation h (→ D(h) of H is not necessarily that defined by the label

J because irreducible representations of G may be reducible under H ; D depends on

what representation of H the index n belongs to. If D is the identity representation, then

the functions are functions on G/H in the ordinary sense. For G = SU(2) and H being

the U (1) subgroup generated by J

, the quotient space is just S

, and the projection is

the Hopf map: S

→ S

. The resulting bundle can be called the Hopf bundle. It is not

really a new object, however, because it is a generalization of the monopole bundle of

the preceding section. Parametrizing SU(2) with Euler angles, so that

(θ, φ, ψ) =J , m|e

−iφJ

−iθ J

−iψ J

|J , n, (16.96)

596 16 The geometry of fibre bundles

shows that the Hopf map consists of simply forgetting about ψ,so

Hopf : [(θ, φ, ψ) ∈ S

] (→[(θ , φ) ∈ S

]. (16.97)

The bundle is twisted because S

is not a product S

×S

. Taking n = 0 gives us functions

independent of ψ , and we obtain the well-known identification of the spherical harmonics

with representation matrices

(θ, φ) =

2L + 1

4π

(L)

(θ, φ,0)]

∗

. (16.98)

For n =  = 0 we get sections of a bundle whose Chern number is 2. These sections

are the monopole harmonics:

m;

(θ, φ, ψ) =

2J + 1

4π

m

(θ, φ, ψ)]

∗

(16.99)

for a monopole of flux



eB d(Area) = 4π. The integrality of the Chern number tells

us that the flux 4π must be an integer multiple of 2π . This gives us a geometric reason

for why the eigenvalues m of J

can only be an integer or half-integer.

The monopole harmonics have a non-trivial dependence, ∝ e

iψ

, on the choice we

make for ψ at each point on S

, and we cannot make a globally smooth choice; we

always encounter at least one point where there is a singularity. Considered as functions

on the base-space, the sections of the twisted bundle have to be constructed in patches

and glued together using transition functions. As functions on the total space of the

principal bundle, however, they are globally smooth.

We now show that the monopole harmonics are eigenfunctions of the Schrödinger

operator −∇

containing the gauge field connection, just as the spherical harmonics

are eigenfunctions of the Laplacian on the sphere. This is a simple geometrical exercise.

Because they are irreducible representations, the D

(g) are automatically eigenfunctions

of the quadratic Casimir operator

+ J

(g) = J (J + 1)D

(g). (16.100)

The J

can be either right- or left-invariant vector fields on G; the quadratic Casimir is

the same second-order differential operator in either case, and it is a good guess that it is

proportional to the Laplacian on the group manifold. Taking a locally geodesic coordinate

system (in which the connection vanishes) confirms this: J

=−∇

on the three-sphere.

The operator in (16.100) is not the Laplacian we want, however. What we need is the

∇

on the 2-sphere S

= G/H , including the connection. This ∇

operator differs

from the one on the total space since it must contain only differential operators lying

in the horizontal subspaces. There is a natural notion of orthogonality in the Lie group,

deriving from the Killing form, and it is natural to choose the horizontal subspaces to

be orthogonal to the fibres of G/H . Because multiplication on the right by the subgroup

16.3 Working in the total space 597

generated by J

moves one up and down the fibres, the orthogonal displacements are

obtained by multiplication on the right by infinitesimal elements made by exponentiating

and J

. The desired ∇

is thus made out of the left-invariant vector fields (which act

by multiplication on the right), J

and J

only. The wave operator must therefore be

−∇

= J

+ J

= J

− J

. (16.101)

Applying this to the Y

m;

, we see that they are eigenfunctions of −∇

on S

with

eigenvalues J (J + 1) − 

. The Laplace eigenvalues for our flux = 4π monopole

problem are therefore

J ,m

= (J (J +1) − 

), J ≥||, −J ≤ m ≤ J . (16.102)

The utility of the monopole harmonics is not restricted to exotic monopole physics. They

occur in molecular and nuclear physics as the wavefunctions for the rotational degrees

of freedom of diatomic molecules and uniaxially deformed nuclei that possess angular

momentum  about their axis of symmetry.

Exercise 16.3: Compare these energy levels for a particle on a sphere with those of the

Landau level problem on the plane. Show that for any fixed flux the low-lying energies

remain close to E = (eB/m

particle

)(n + 1/2), with n = 0, 1, ..., but their degeneracy is

equal to the number of flux units penetrating the sphere plus one.

16.3.4 Bundle connection and curvature forms

Recall that in Section 16.3.2 we introduced the Lie-algebra-valued functions A

(x).We

now use these functions to introduce the bundle connection form A that lives in T

∗

We set

A = A

(16.103)

and

def

= g

−1



A + δgg

−1

g. (16.104)

In these definitions, x and g are the local coordinates in which points in the total space

are labelled as (x, g), and d acts on functions of x, and the “δ” is used to denote the

exterior derivative acting on the fibre.

We have, then, that δx

= 0 and dg = 0. The

combinations δgg

−1

and g

−1

δg are respectively the right- and left-invariant Maurer–

Cartan forms on the group.

This is explained, with chararacteristic terseness, in a footnote on page 317 of L. D. Landau and E. M.

Lifshitz, Quantum Mechanics (third edition).

It is not therefore to be confused with the Hodge δ = d

†

operator.

598 16 The geometry of fibre bundles

The complete exterior derivative in the total space requires us to differentiate both

with respect to g and with respect to x, and is given by d

tot

= d +δ. Because d

, δ

and

(d + δ)

= d

+ δ

+ dδ + δd are all zero, we must have

δd + dδ = 0. (16.105)

We now define the bundle curvature form in terms of A to be

def

= d

tot

A + A

. (16.106)

To compute F in terms of A(x) and g we need the ingredients

dA = g

−1

(dA)g, (16.107)

and

δA =−(g

−1

δg)A − A(g

−1

δg) − (g

−1

δg)

. (16.108)

We find that

F = (d + δ)A + A

= g

−1



dA + A

= g

−1

Fg, (16.109)

where

F =

µν

, (16.110)

and

µν

= ∂

− ∂

+[A

, A

]. (16.111)

Although we have defined the connection form A in terms of the local bundle coordi-

nates (x, g), it is, in fact, an intrinsic quantity, i.e. it has a global existence, independent

of the choice of these coordinates. A has been constructed so that

(i) A vector is annihilated by A if and only if it is horizontal. In particular, A(D

) = 0

for all covariant derivatives D

(ii) The connection form is constant on left-invariant vector fields on the fibres. In

particular, A(L

) = i

Between them, the globally defined fields D

∈ H

(P) and L

∈ V

(P) span the

tangent space TP

. Consequently the two properties listed above tell us how to evaluate

A on any vector, and so define it uniquely and globally.

16.3 Working in the total space 599

From the globally defined and gauge invariant A and its associated curvature F, and for

any local gauge-choice section σ : (U ⊂ M ) → P, we can recover the gauge-dependent

base-space forms A and F as the pull-backs

A = σ

∗

A, F = σ

∗

F, (16.112)

to U ⊂ M of the total-space forms. The resulting forms are

A =



−1

σ + σ

−1

∂

, F =



−1

µν

, (16.113)

and coincide with the equations connecting A

with A

and F

µν

with F

µν

that we

obtained in Section 16.3.2. We should take care to note that the dx

that appear in A

and F are differential forms on M , while the dx

that appear in A and F are differential

forms on P. Now the projection π is a left inverse of the gauge-choice section σ , i.e.

π ◦σ = identity. The associated pull-backs are also inverses, but with the order reversed:

∗

◦ π

∗

= identity. These maps relate the two sets of “dx

”by

= σ

∗





,ordx

= π

∗





. (16.114)

We now explain the advantage of knowing the total space connection and curvature

forms. Consider the Chern character ∝ tr F

on the base-space M . We can use the bundle

projection π to pull this form back to the total space. From

µν

= (gσ

−1

)

−1

µν

(gσ

−1

), (16.115)

we find that

∗



tr F

= tr F

. (16.116)

Now A, F and d

tot

have the same calculus properties as A , F and d. The manipulations

that give

tr F

= d tr



AdA+



also show, therefore, that

tr F

= d

tot



A d

tot

A +



. (16.117)

There is a big difference in the significance of the computation, however. The bundle

connection A is globally defined; consequently, the form

(A) ≡ tr



A d

tot

A +



(16.118)

600 16 The geometry of fibre bundles

is also globally defined. The pull-back to the total space of the Chern character is d

tot

exact! This miracle works for all characteristic classes: but on the base-space they are

exact only when the bundle is trivial; on the total space they are always exact.

We have seen this phenomenon before, for example in Exercise 15.7. The area form

d[Area]=sin θ dθ dφ is closed but not exact on S

. When pulled back to S

by the Hopf

map, the area form becomes exact:

Hopf

∗

d[Area]=sin θ dθ dφ = d(−cos θ dφ + dψ). (16.119)

16.3.5 Characteristic classes as obstructions

The generalized Gauss–Bonnet theorem states that, for a compact orientable even-

dimensional manifold M , the integral of the Euler class over M is equal to the Euler

character χ(M ). Shiing-Shen Chern used the exactness of the pull-back of the Euler

class to give an elegant intrinsic proof

of this theorem. He showed that the integral of

the Euler class over M was equal to the sum of the Poincaré–Hopf indices of any tangent

vector field on M , a sum we independently know to equal the Euler character χ(M ).We

illustrate his strategy by showing how a non-zero ch

(F) provides a similar index sum

for the singularities of any section of an SU(2)-bundle over a four-dimensional base-

space. This result provides an interpretation of characteristic classes as obstructions to

the existence of global sections.

Let σ : M → P be a section of an SU(2) principal bundle P over a four-dimensional

compact orientable manifold M without boundary. For any SU(n) group we have

(F) ≡ 0, but



(F) =−

8π



tr (F

) = n (16.120)

can be non-zero.

The section σ will, in general, have points x

where it becomes singular. We punch

infinitesimal holes in M surrounding the singular points. The manifold M



= (M \holes)

will have as its boundary ∂M



a disjoint union of small 3-spheres. We denote by the

image of M



under the map σ : M



→ P. This will be a submanifold of P, whose

boundary will be equal in homology to a linear combination of the boundary components

of M



with integer coefficients. We show that the Chern number n is equal to the sum of

these coefficients.

We begin by using the projection π to pull back ch

(F) to the bundle, where we

know that

∗

(F) =−

8π

tot

(A). (16.121)

S.-J. Chern, Ann. Math., 47 (1946) 85. This paper is a readable classic.

16.3 Working in the total space 601

Now we can decompose ω

(A) into terms of different bi-degree, i.e. into terms that are

p-forms in d and q-forms in δ:

(A) = ω

+ ω

. (16.122)

Here the superscript counts the form-degree in δ, and the subscript the form-degree in

d. The only term we need to know explicitly is ω

. This comes from the g

−1

δg part of

A, and is

= tr



−1

δg)δ(g

−1

δg) +

−1

δg)



= tr



−(g

−1

δg)

−1

δg)



=−

−1

δg)

. (16.123)

We next use the map σ : M



→ P to pull the right-hand side of (16.121) back from P

to M



. We recall that acting on forms on M



we have σ

∗

◦ π

∗

= identity. Thus



(F) =





(F) =





∗

◦ π

∗

(F)

=−

8π





∗

tot

(A)

=−

8π



tot

(A)

=−

8π



∂

(A)

24π



∂

−1

δg)

. (16.124)

At the first step we have observed that the omitted spheres make a negligible contribution

to the integral over M , and at the last step we have used the fact that the boundary of

has significant extent only along the fibres, so all contributions to the integral over ∂

come from the purely vertical component of ω

(A), which is ω

=−

−1

dg).

We know (see Exercise 15.8) that for maps g (→ U ∈ SU(2) we have



tr (g

−1

dg)

= 24π

× winding number.

We conclude that



(F) =

24π



∂

−1

δg)



singularities x

(16.125)

602 16 The geometry of fibre bundles

where N

is the Brouwer degree of the map σ : S

→ SU(2)

∼

on the small sphere

surrounding x

It turns out that for any SU(n) the integral of tr (g

−1

δg)

is 24π

times an integer

winding number of g about homology spheres. The second Chern number of a SU(n)-

bundle is therefore also equal to the sum of the winding-number indices of the section

about its singularities. Chern’s strategy can be used to relate other characteristic classes

to obstructions to the existence of global sections of appropriate bundles.

16.3.6 Stora–Zumino descent equations

In the previous sections we met the forms

A = g

−1

Ag + g

−1

δg (16.126)

and

A = σ

−1

Aσ + σ

−1

dσ . (16.127)

The group element g labelled points on the fibres and was independent of x, while σ(x)

was the gauge-choice section of the bundle and depended on x. The two quantities A

and A look similar, but are not identical. A third superficially similar but distinct object

is met with in the BRST (Becchi–Rouet–Stora–Tyutin) approach to quantizing gauge

theories, and also in the geometric theory of anomalies. We describe it here to alert the

reader to the potential for confusion.

Rather than attempting to define this new differential form rigorously, we will first

explain how to calculate with it, and only then indicate what it is. We begin by considering

a fixed connection form A on M, and its orbit under the action of the group G of gauge

transformations. These elements of this infinite dimensional group are maps g : M → G

equipped with pointwise product g

(x) = g

(x)g

(x). This g(x) is neither the fibre

coordinate g, nor the gauge choice section σ(x). The gauge transformation g(x) acts on

A to give A

where

= g

−1

Ag + g

−1

dg. (16.128)

We now introduce an object

v(x) = g

−1

δg, (16.129)

and consider

A = A

+ v = g

−1

Ag + g

−1

dg + g

−1

δg. (16.130)

This 1-form appears to be a hybrid of the earlier quantities, but we will see that it has

to be considered as something new. The essential difference from what has gone before

16.3 Working in the total space 603

is that we want v to behave like g

−1

δg, in that δv =−v

, and yet to depend on x.In

particular we want δ to behave as an exterior derivative that implements an infinitesimal

gauge transformation that takes g → g + δg. Thus,

δ(g

−1

dg) =−(g

−1

δg)(g

−1

dg) + g

−1

δdg

=−(g

−1

δg)(g

−1

dg) − (g

−1

dg)(g

−1

δg) + (g

−1

dg)(g

−1

δg) − g

−1

dδg

=−v(g

−1

dg) − (g

−1

dg)v − dv, (16.131)

and hence

δA

=−vA

− A

v −dv . (16.132)

Previously g

−1

dg ≡ 0, and so there was no “dv ”inδ(gauge field).

We can define a curvature associated with A

def

= d

tot

A + A

, (16.133)

and compute

F = (d + δ)(A

+ v ) + (A

+ v )

= dA

+ dv +δA

+ δv + (A

)

+ A

v +vA

+ v

= dA

+ (A

)

= g

−1

Fg. (16.134)

Stora calls (16.134) the Russian formula.

Because F is yet another gauge transform of F, we have

tr F

= tr F

= (d + δ) tr



A(d + δ)A +



(16.135)

and can decompose the right-hand side into terms that are simultaneously p-foms in d

and q-forms in δ.

The left-hand side, tr F

= tr F

,of(16.135) is independent of v. The right-hand side

of (16.135) contains ω

(A) which we expand as

+ v ) = ω

) + ω

(v, A

) + ω

(v, A

) + ω

(v). (16.136)

As in the previous section, the superscript counts the form-degree in δ, and the subscript

the form-degree in d. Explicit computation shows that

) = tr



)



(v, A

) = tr (v dA

(v, A

) =−tr (A

(v) =−

. (16.137)

604 16 The geometry of fibre bundles

For example,

(v) = tr



v δv +



= tr



v(−v

) +



=−

. (16.138)

With this decomposition, (16.117) falls apart into the chain of descent equations

tr F

= dω

δω

) =−d ω

(v, A

δω

(v, A

) =−d ω

(v, A

δω

(v, A

) =−d ω

(v),

δω

(v) = 0. (16.139)

Let us verify, for example, the penultimate equation δω

(v, A

) =−d ω

(v). The

left-hand side is

−δ tr (A

) =−tr (−Av

− v A

− dvv

) = tr (dvv

), (16.140)

the terms involving A

having cancelled via the cyclic property of the trace and the fact

that A

anticommutes with v. The right-hand side is

−d



−

tr v



= tr (dvv

) (16.141)

as required.

The descent equations were introduced by Raymond Stora and Bruno Zumino as a

tool for obtaining and systematizing information about anomalies in the quantum field

theory of fermions interacting with the gauge field A

. The ω

(v, A

) are p-forms in the

, and before use they are integrated over p-cycles in M . This process is understood to

produce local functionals of A

that remain q-forms in δg. For example, in 2n space-time

dimensions, the integral

I[g

−1

δg, A



−1

δg, A

) (16.142)

has the properties required for it to be a candidate for the anomalous variation δS[A

] of

the fermion effective action due to an infinitesimal gauge transformation g → g + δg.

In particular, when ∂M =∅, we have

δI [g

−1

δg, A



δω

(v, A

) =−



dω

2n−1

(v, A

) = 0. (16.143)

This is the Wess–Zumino consistency condition that δ(δS) must obey as a consequence

of δ

= 0.