Marathe K. Topics in Physical Mathematics

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238 8 Yang–Mills–Higgs Fields

into H

(M,R). Thus, we can regard H

(M,Z) as a subset of H

(M,R).

Now H

(M,R) can be identiﬁed with the second cohomology group of the

de Rham complex H

deR

(M,R) of the base manifold M . Thus, an element

α ∈ H

(M,Z) ⊂ H

(M,R)=H

deR

(M,R)

corresponds to the class of a closed 2-form on M, which we also denote simply

by α. By applying Hodge theory we can identify H

deR

(M,R)withthespace

of harmonic 2-forms with respect to the Hodge Laplacian Δ

on 2-forms

deﬁned by Δ

= dδ + δd . Thus there exists a unique harmonic 2-form β on

M such that α =[β]. It can be shown that β is the curvature (gauge ﬁeld)

of a gauge connection on the U(1)-bundle P

over M.WenotethatΔβ =0

is equivalent to the set of two equations dβ =0,δβ=0.Thus,theharmonic

form β is a Maxwell ﬁeld, i.e., a source-free electromagnetic ﬁeld. The above

discussion proves the following theorem.

Theorem 8.1 Let P (M,U(1)) be a principal bundle over a compact, simply

connected, oriented, Riemannian manifold M. Then the Maxwell ﬁeld is the

unique harmonic 2-form representing the Euler class or the ﬁrst Chern class

(P ).

The role played by the various areas of topology and geometry in the proof

of the above theorem is as follows. Let I denote the set of isomorphism classes

of U (1)-bundles over M, H

the set of harmonic 2-forms on M , EG the set

of gauge equivalence classes of gauge ﬁelds, and CI the set of connections on

U(1)-bundles with integral curvature. Then we have the following diagram:

(M,Z) H

(M,R)

algebraic topology

[M,CP

∞

]

bundle theory

diﬀerential geometry



gauge theory

de Rham theory

deR

(M,R)

Hodge theory

theory of connections

8.2 Electromagnetic Fields 239

The fact that an integral Chern class c

(P ) is represented by the curvature

of a gauge connection in a complex line bundle associated to P also plays

an important role in geometric quantization. If M is 4-dimensional then the

Maxwell ﬁeld can be decomposed into its self-dual and anti-dual parts un-

der the Hodge star operator. The corresponding ﬁelds are related to certain

topological invariants of the manifold M such as the Seiberg–Witten invari-

ants.

Theorem 8.1 suggests where we should look for examples of source-free

electromagnetic ﬁelds. Since every U(1)-bundle with a connection is a pull-

back of a suitable U(1)-universal bundle with universal connection, it is nat-

ural to examine this bundle ﬁrst. The Stiefel bundle V

(n + k, k)overthe

Grassmann manifold G

(n + k, k)isk-classifying for SO(k). In particu-

lar, for k =2,wegetV

(n +2, 2) = SO(n +2)/SO(n)andG

(n +2, 2) =

SO(n+2)/(SO(n)×SO(2)). Similarly, V

(n+1, 1) = U (n+1)/U (n)=S

2n+1

and G

(n +1, 1) = U(n +1)/(U(n) ×U (1)) = CP

, which is the well-known

Hopf ﬁbration. Recall that the ﬁrst Chern class classiﬁes these principal

U(1)-bundles and is an integral class. When applied to the base manifold

∼

this classiﬁcation corresponds to the Dirac quantization con-

dition for a monopole. Example 6.1 corresponds to the above Hopf ﬁbration

with n = 1. The natural (or universal) connections over these bundles satisfy

source-free Maxwell’s equations. We note that the pull-back of these univer-

sal connections do not, in general, satisfy Maxwell’s equations. However, we

do get new solutions in the following situation.

Example 8.1 If M is an analytic submanifold of CP

, then the U (1)-bundle

2n+1

, pulled back by the embedding i : M→ CP

, gives a connection

on M whose curvature satisﬁes Maxwell’s equations. For example, if M =

= S

, then for each positive integer n, we have the following well-known

embedding:

: CP

→ CP

given in homogeneous coordinates z

on CP

)=(z

n−1

,...,c

n−m

,...,z

where c





1/2

. The electromagnetic ﬁeld on CP

is pulled back by f

to give a ﬁeld on CP

= S

, which corresponds to a magnetic monopole of

strength n/2. Moreover the corresponding principal U(1)-bundle is isomorphic

to the lens space L(n, 1).

8.2.1 Motion in an Electromagnetic Field

We discussed above the geometric setting that characterizes source-free elec-

tromagnetic ﬁelds. On the other hand, the existence of an electromagnetic

240 8 Yang–Mills–Higgs Fields

ﬁeld has consequences for the geometry of the base space and this in turn

aﬀects the motion of test particles. Indeed, these eﬀects are well-known in

classical physics only for the electromagnetic ﬁeld. At this time there are no

known physical eﬀects associated with classical non-Abelian gauge ﬁelds such

as the Yang–Mills ﬁeld. We now give a brief discussion of these aspects of the

electromagnetic ﬁeld.

First recall that an electrostatic ﬁeld E determines the diﬀerence in po-

tential between two points by integration along a path joining them. It is

therefore reasonable to think of E as a 1-form on R

. Maxwell’s fundamental

idea was to introduce a quantity, called electric displacement D,which

has the property that its integral over a closed surface measures the charge

enclosed by the surface. One should thus think of D as a 2-form. In a uni-

form medium characterized by dielectric constant , one usually writes the

constitutive equations relating D and E as

D = E.

However, our discussion indicates that this equation only makes sense if E is

replaced by a 2-form. In fact, if we are given a metric on R

, there is a natural

way to associate with E a 2-form, namely ∗E,where∗ is the Hodge operator.

Then the above equation can be written in a mathematically precise form

D = (∗E).

Conversely, requiring such a relation between D and E, i.e., specifying the

operator ∗ : Λ

) → Λ

), is equivalent to a Euclidean metric on R

Similarly, one can interpret the magnetic tension or induction as a 2-form

B and Faraday’s law then implies that B is closed, i.e., dB =0.Thelawof

motion of a charged test particle of charge e, moving in the presence of B,is

obtained by a modiﬁcation of the canonical symplectic form ω on T

∗

the pull-back of B by the canonical projection π : T

∗

→ R

, i.e., by use

of the form

e,B

= ω + eπ

∗

The distinct theories of electricity and magnetism were uniﬁed by Maxwell

in his electromagnetic theory. To describe this theory for arbitrary elec-

tromagnetic ﬁelds, it is convenient to consider their formulation on the 4-

dimensional Minkowski space-time M

. We deﬁne the two 2-forms F and G

F := B + E ∧ dt, G := D − H ∧ dt,

where B and H are the magnetic tension and magnetic ﬁeld, respectively and

E and D are the electric ﬁeld and displacement, respectively. If J denotes

the current 3-form, then Maxwell’s equations with source J are given by

dF =0

dG =4πJ.

8.2 Electromagnetic Fields 241

The two ﬁelds F and G are related by the constitutive equationsm which

depend on the medium in which they are deﬁned. In a uniform medium (in

particular in a vacuum) we can write the constitutive equations as

G =(/μ)

1/2

∗ F.

We note that, specifying the operator ∗ : Λ

) → Λ

) determines

only the conformal class of the metric. If we choose the standard Minkowski

metric, then we can write the source-free Maxwell’s equations in a uniform

medium as

dF =0,d∗F =0.

The second equation is equivalent to δF = 0; we thus obtain the gauge ﬁeld

equations discussed earlier.

It is well known that Hamilton’s equations of motion of a particle in

classical mechanics can be given a geometrical formulation by using the phase

space P of the particle. The phase space P is, at least locally, the cotangent

space T

∗

Q of the conﬁguration space Q of the particle. For a geometrical

formulation of classical mechanics see Abraham and Marsden [1]. We now

show that this formalism can be extended to the motion of a charged particle

in an electromagnetic ﬁeld and leads to the usual equations of motion with

the Lorentz force. We choose the conﬁguration space Q of the particle as the

usual Minkowski space. The law of motion of a charged test particle with

charge e is obtained by considering the symplectic form

e,F

= ω + eπ

∗

where ω is the canonical symplectic form of T

∗

Q and π : T

∗

Q → Q is the

canonical projection. The Hamiltonian function is given by

H(q, p)=

where g

are the components of the Lorentz metric and p

are the components

of 4-momentum. In the usual system of coordinates we can write the metric

and the Hamiltonian as

= dq

− dq

H(q, p)=

− p

The matrix of the symplectic form ω

e,F

, in local coordinates x

=(q

)on

∗

Q, can be written in block matrix form as

e,F



eF −I

I 0



242 8 Yang–Mills–Higgs Fields

The Hamiltonian vector ﬁeld is given by

=(dH)



∂H

∂x

αβ

e,F

∂

where ω

αβ

e,F

are the elements of (ω

e,F

)

−1

. The corresponding Hamilton’s equa-

tions are given by

= X

= ω

αβ

e,F

∂H

∂x

i.e.,

∂H

∂p

, (8.4)

= −

∂H

∂q

+ eF

∂H

∂p

. (8.5)

Equation (8.5) implies that the 3-momentum p =(p

) of the particle

satisﬁes the equation

(E + p × B), (8.6)

where E and B are respectively the electric and the magnetic ﬁelds. The

equation (8.6) is the well-known equation of motion of a charged particle in

an electromagnetic ﬁeld subject to the Lorentz force. Now the electromag-

netic ﬁeld is a gauge ﬁeld with Abelian gauge group U(1). The orbits of

contragredient action of U(1) on the dual of its Lie algebra u(1) are trivial.

Identifying u(1) and its dual with R, we see that an orbit through e ∈ R

is the point e itself. Thus, in this case, the choice of an orbit is the same

thing as the choice of the unit of charge. This construction can be general-

ized to gauge ﬁelds with arbitrary structure group and, in particular, to the

Yang–Mills ﬁelds to obtain the equations of motion of a particle moving in

a Yang–Mills ﬁeld. A detailed discussion of this is given in Guillemin and

Sternberg [179].

8.2.2 The Bohm–Aharonov Eﬀect

We discussed above the eﬀect of the geometry of the base space on the prop-

erties of the electromagnetic ﬁelds deﬁned on it. We now discuss a property

of the electromagnetic ﬁeld that depends on the topology of the base space.

In Example 6.1 of the Dirac monopole we saw that the topology of the base

space may require several local gauge potentials to describe a single global

gauge ﬁeld. In fact, this is the general situation. In classical theory only the

electromagnetic ﬁeld was supposed to have physical signiﬁcance while the

potential was regarded as a mathematical artifact. However, in topologically

8.2 Electromagnetic Fields 243

non-trivial spaces the potential also becomes physically signiﬁcant. For ex-

ample, in non-simply connected spaces the equation dF =0,satisﬁedbya

2-form F , deﬁnes not only a local potential but a global topological prop-

erty of belonging to a given cohomology class. Bohm and Aharonov ([8, 7])

suggested that in quantum theory the non-local character of electromagnetic

potential A = A

, in a multiply connected region of space-time, should

have a further kind of signiﬁcance that it does not have in the classical theory.

They proposed detecting this topological eﬀect by computing the phase shift



around a closed curve not homotopic to the identity and computing

its eﬀect in an electron interference experiment. We now discuss the special

case of a non-relativistic charged particle moving through a vector potential

corresponding to zero magnetic ﬁeld to bring out the eﬀect of potentials in

the absence of ﬁelds.

Consider a long solenoid placed along the z-axis with its center at the

origin. Then for motion near the origin the space may be considered to be

minus the z-axis. A loop around the solenoid is then homotopically non-

trivial (i.e., not homotopic to the identity). Thus, two paths c

, c

respectively

joining two points p

, p

on opposite sides of the solenoid are not homotopic.

If we send particle beams along these paths from p

to p

, we can observe the

resulting interference pattern at p

.Letψ

, i =1, 2, denote the wave function

corresponding to the beam passing along c

when the vector potential is zero.

When the ﬁeld is switched on in the solenoid, the paths c

, c

remain in the

ﬁeld-free region, but the vector potential in this region is not zero. The wave

function ψ

is now changed to ψ

/

,where

= q



,j=1, 2.

It follows that the total wave function of the system is changed as follows:

+ ψ

→ ψ

/

+ ψ



. (8.7)

The interference eﬀect of the recombined beams will thus depend on the

phase factor

i(S

−S

)/

= e

(iq/)



where c is the closed path c

− c

. The predicted eﬀect, called the Bohm–

Aharonov eﬀect, was conﬁrmed by experimental observations. This result

ﬁrmly established the physical signiﬁcance of the gauge potential in quantum

theory. Extensions of the Bohm–Aharonov eﬀect to non-Abelian gauge ﬁelds

have been proposed, but there is no physical evidence for this eﬀect at this

time. The prototype of non-Abelian gauge theories is the theory of Yang–

Mills ﬁelds. Many of the mathematical considerations given above for the

electromagnetic ﬁelds extend to Yang–Mills ﬁelds, which we discuss in the

next section.

244 8 Yang–Mills–Higgs Fields

8.3 Yang–Mills Fields

Yang–Mills ﬁelds form a special class of gauge ﬁelds that have been exten-

sively investigated. In addition to the references given for gauge ﬁelds we give

the following references which deal with Yang–Mills ﬁelds. They are Douady

and Verdier [114], [59, 60, 386], and [410,411].

Let M be a connected manifold and let P (M,G) be a principal bundle

over M with Lie group G as the gauge group. In this section we restrict

ourselves to the space A(P ) of gauge connections as our conﬁguration space.

If ω is a gauge connection on P then, in a local gauge t ∈ Γ (U, P ), we have

the corresponding local gauge potential

= t

∗

ω ∈ Λ

(U, ad(P)).

Locally, a gauge transformation g ∈Greduces to a G-valued function g

U, and its action on A

is given by

·A

=(adg

) ◦A

+ g

∗

Θ,

where Θ is the canonical 1-form on G. It is customary to indicate this gauge

transformation as follows

:= g · A = g

−1

Ag + g

−1

when the section t is understood.

The gauge ﬁeld F

is the unique 2-form on M with values in the bundle

ad(P) satisfying

= s

where Ω is the curvature of the gauge connection ω. In the local gauge t we

can write

= d

= dA

where the bracket is taken as the bracket of bundle-valued forms.

We note that it is always possible to introduce a Riemannian metric on

a vector bundle over M. The manifold M itself admits a Riemannian met-

ric. We assume that metrics are chosen on M and the bundles over M,and

the norm is deﬁned on sections of these bundles as an L

-norm if M is not

compact. However, to simplify the mathematical considerations, we assume

in the rest of this chapter that M is a compact, connected, oriented, Rie-

mannian manifold and that G is a compact, semisimple Lie group, unless

otherwise indicated. For a given M and G, the principal G-bundles over M

are classiﬁed by [M; BG], the set of homotopy classes of maps of M into the

classifying space BG for G.Forf ∈ [M; BG], let P

be a representative of

the isomorphism class of principal bundles corresponding to f.LetA(P

)

be the space of gauge potentials on P

. Then, the Yang–Mills conﬁguration

space A

is the disjoint union of the spaces A(P

), i.e.,

8.3 Yang–Mills Fields 245

= ∪{A(P

) | f ∈ [M; BG]}.

The isomorphism class [P

] is uniquely determined by the characteristic

classes of the bundle P

. For example, if dim M =4andG = SU(2),

then c

) = 0 and hence [P

] is determined by the second Chern class

)=c

), where V

is the complex vector bundle of rank 2 associated

to P

by the deﬁning representation of SU(2). Now the class c

), evalu-

ated on the fundamental cycle of M, is integral, i.e., c

)[M] ∈ Z.Thus,in

this case P

is classiﬁed by an integer n(f)andwecanwriteA(P

)asA

n(f)

or simply as A

. In the physics literature, this number n is referred to as the

instanton number of P

. From both the physical and mathematical point

of view, this is the most important case.

We now restrict ourselves to a ﬁxed member A(P

) of the disjoint union

and write simply P for P

. Then the Yang–Mills action or (func-

tional) A

is deﬁned by

(ω)=

8π



, ∀ωA(P ). (8.8)

To ﬁnd the corresponding Euler–Lagrange equations, we take into account

the fact that the space of gauge potentials is an aﬃne space and hence it is

enough to consider variations along the straight lines through ω of the form

= ω + tA, where A ∈ Λ

(M,ad P ).

Direct computation shows that the gauge ﬁeld, corresponding to the gauge

potential ω

,isgivenby

= F

+ td

A + t

(A ∧A). (8.9)

Using equations (8.8)and(8.9)weobtain

(ω

)

|t=0





+ td

A + t

(A ∧A)|



|t=0



A>dv



<δ

,A>dv

The above result is often expressed in the form of a variational equation

δA

(ω)(A)=2δ

,A. (8.10)

A gauge potential ω is called a critical point of the Yang–Mills functional

δA

(ω)(A)=0, ∀A ∈ Λ

(M,ad P ). (8.11)

246 8 Yang–Mills–Higgs Fields

The critical points of the Yang–Mills functional are solutions of the corre-

sponding Euler–Lagrange equation

=0. (8.12)

The equations (8.12) are called the pure (or sourceless) Yang–Mills equa-

tions. A gauge potential ω satisfying the Yang–Mills equations is called

the Yang–Mills connection and its gauge ﬁeld F

is called the Yang–

Mills ﬁeld. Using local expressions for d

and δ

it is easy to show that

= ±∗d

∗. From this it follows that ω satisﬁes the Yang–Mills equa-

tions (8.12) if and only if it is a solution of the equation

∗ F

=0. (8.13)

In the following theorem we have collected various characterizations of the

Yang–Mills equations.

Theorem 8.2 Let ω be a connection on the bundle P (M,G) with ﬁnite

Yang–Mills action. Then the following statements are equivalent:

1. ω is a critical point of the Yang–Mills functional;

2. ω satisﬁes the equation δ

=0;

3. ω satisﬁes the equation d

∗F

=0;

4. ω satisﬁes the equation ∇

=0, where ∇

= d

+ δ

is the Hodge

Laplacian on forms.

The equivalence of the ﬁrst three statements follows from the discussion

given above. The equivalence of the statements 2 and 4 follows from the

identity

∇

 = d



+ δ



= δ



The second equality in the above identity follows from the Bianchi identity

=0. (8.14)

This identity is a consequence of the Cartan structure equations and ex-

presses the fact that, locally, F

is derived from a potential. It is customary

to consider the pair (8.12)and(8.14)or(8.13)and(8.14) as the Yang–Mills

equations. This is consistent with the fact that they reduce to Maxwell’s

equations for the electromagnetic ﬁeld F , when the gauge group G is U(1)

and M is the Minkowski space.

In local orthonormal coordinates, the Yang–Mills equations can be written

∂F

∂x

+[A

]=0, (8.15)

where the components F

of the 2-form F

are given by

∂A

∂x

−

∂A

∂x

+[A

]. (8.16)

8.3 Yang–Mills Fields 247

Thus, we see that the Yang–Mills equations are a system of non-linear, second

order, partial diﬀerential equations for the components of the gauge potential

A. The presence of quadratic and cubic terms in the potential represents the

self-interaction of the Yang–Mills ﬁeld.

In (8.8) we deﬁned the Yang–Mills action A

on the domain A(P ). This

domain is an aﬃne space and hence is contractible. But there is a large sym-

metry group acting on this space. This is the group G of gauge transformations

of P . By deﬁnition, a gauge transformation φ ∈Gis an automorphism of the

bundle P , which covers the identity map of M and hence leaves each x ∈ M

ﬁxed. Under this gauge transformation, the gauge ﬁeld transforms as

→ F

:= φ

−1

φ.

Therefore, the pointwise norm |F

is gauge-invariant. It follows that the

Yang–Mills action A

(ω) is gauge-invariant and hence induces a functional

on the moduli space M = A/G of gauge connections. This functional is also

called the Yang–Mills functional (or action). In order to relate the topology

of the moduli space M to the critical points of the Yang–Mills functional,

it is necessary to compute the second variation or the Hessian of the

Yang–Mills action

(ω):=

(ω

)

|t=0

One can verify that the Hessian, viewed as a symmetric, bilinear form, is

given by the following expression:

(ω)(A, B)=2



(d

A, d

B + F

,A∧B + B ∧ A).

Analysis of the Yang–Mills equations then proceeds by our obtaining various

estimates using these variations.

We note that if φ ∈ Aut(P ) covers a conformal transformation of M (also

denoted by φ), then this φ ∈ Diﬀ(M) induces a conformal change of metric.

It is given by

∗

g = e

g, f ∈F(M). (8.17)

Neither the pointwise norm |F

nor the Riemannian volume form v

are

invariant under this conformal change, and the integrand in the Yang–Mills

action transforms as follows:

→ (e

−4f

)(e

). (8.18)

It follows that, in the particular case of m = 4, the Yang–Mills action is

invariant under the generalized gauge transformations that cover conformal

transformations of M.