Marathe K. Topics in Physical Mathematics

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248 8 Yang–Mills–Higgs Fields

A large number of solutions of the Yang–Mills equations are known for

special manifolds. For example, the universal connections on Stiefel bundles

provide solutions of pure (sourceless) Yang–Mills equations. As we observed

in the previous section, the natural connections associated to the complex

Hopf ﬁbration satisfy source-free Maxwell’s equations. As is well known, Hopf

ﬁbrations are particular cases of Stiefel bundles. In Chapter 4, we gave a

uniﬁed treatment of real, complex, and quaternionic Stiefel bundles. Those

results can be used to arrive at solutions of pure Yang–Mills equations as

follows (for further details see [302, 377]).

Let A be the local gauge potential corresponding to the universal connec-

tion ω on the Stiefel bundle V

)(G

),U

(k)) with gauge group U

(k),

and let F

= d

A be the corresponding gauge ﬁeld. The basic computation

involves an explicit local expression for ∗F

, the Hodge dual of the gauge

ﬁeld F

. In the complex case it can be shown that ∗F

is represented by an

invariant polynomial of degree kl −1,l= n − k in F

, i.e.,

∗F = aF ∧ F ∧···∧F



 

kl−1times

where a is a constant. This expression, together with the Bianchi identities,

imply

∗ F

=0. (8.19)

In the quaternionic case, a similar expression for ∗F

leads to equation (8.19).

In the real case the expression for ∗ F

also involves the gauge potential

explicitly. These terms can be written as a wedge product of certain l-forms

ψ, which satisfy the condition d

ψ =0. As a consequence of the Maurer–

Cartan equation this, together with the Bianchi identities, implies (8.19)as

in the complex case.

Let M be a compact, oriented Riemannian manifold of dimension 4. Let

P (M, G) be a principal bundle over M with compact semisimple Lie group

G as structure group. Recall that the Hodge star operator on Λ(M)hasa

natural extension to bundle valued forms. A form α ∈ Λ

(M,ad(P )) is said

to be self-dual (resp., anti-dual)if

∗α = α (resp., ∗α = −α).

We deﬁne the self-dual part α

of a form α ∈ Λ

(M,ad(P )) by

(α + ∗α).

Similarly, the anti-dual part α

−

of a form α ∈ Λ

(M,ad(P )) is deﬁned by

−

(α −∗α).

8.3 Yang–Mills Fields 249

Over a 4-dimensional base manifold, the second Chern class and the Euler

class of P are equal and we deﬁne the instanton number k of the bundle P

k := −c

(P )[M ]=−χ(P )[M]. (8.20)

Recall that F = F

is the curvature form of the gauge connection ω,and

hence, by the theory of characteristic classes, we have

k := −c

(P )[M ]=−

8π



Tr(F ∧ F ). (8.21)

Decomposing F into its self-dual part F

and anti-dual part F

−

, we get

k =

8π



(|F

−|F

−

). (8.22)

Using F

and F

−

we can rewrite the Yang–Mills action (8.8) as follows:

(ω)=

8π



(|F

+ |F

−

). (8.23)

Comparing equations (8.22)and(8.23) above we see that the Yang–Mills

action is bounded below by the absolute value of the instanton number k,

i.e.,

(ω) ≥|k|, ∀ω ∈A(P). (8.24)

When M is 4-dimensional, we can associate to the pure Yang–Mills equations

the ﬁrst order instanton (resp., anti-instanton) equations

= ∗F

(resp., F

= −∗F

). (8.25)

The Bianchi identities imply that any solution of the instanton equations is

also a solution of the Yang–Mills equations. The ﬁelds satisfying F

= ∗F

(resp., F

= −∗F

) are also called self-dual (resp., anti-dual) Yang–Mills

ﬁelds. We note that a gauge connection satisﬁes the instanton or the anti-

instanton equation if and only if it is the absolute minimum of the Yang–Mills

action. It can be shown that the instanton equations are also invariant under

the generalized gauge transformations, which cover conformal transforma-

tions of M. This result plays a crucial role in the construction of the moduli

space of instantons. The solutions of the instanton equations in the case of

M = S

were originally called “instantons,” but this term is now used to

denote any solution of the instanton equations over a Riemannian manifold.

We study these solutions and consider their topological and geometrical ap-

plications in Chapter 9.

We now give two examples of Yang–Mills ﬁelds on S

obtained by gluing

two potentials along the equator.

250 8 Yang–Mills–Higgs Fields

Example 8.2 We consider the two standard charts U

on S

(see Chap-

ter 1) and deﬁne the SU(2) principal bundle P

,SU(2)) by the transition

function

g : U

∩ U

→ SU(2),g(x)=x/|x|,

where x ∈ R

is identiﬁed with a quaternion. Essentially, the transition func-

tion g deﬁnes a map of the equator S

into SU(2), i.e., an element of the

homotopy group [g] ∈ π

(SU(2))

∼

Z.Infact,[g] is non-trivial, correspond-

ing to the non-triviality of the bundle P

. A connection A ∈A(P

,SU(2))

is speciﬁed by data consisting of a pair of su(2)-valued 1-forms A

,i=1, 2, which are restricted to U

∩ U

to obey the cocycle condition

(x)=g(x)A

−1

(x)+g(x)dg

−1

(x).

For each λ ∈ (0, 1) deﬁne the connection A

=(A

) by

=Im



xd¯x

+ |x|



=Im



¯xdx

|x|

(λ

+ |x|

)



The connection A

is self-dual with instanton number 1. The curvature of

this connection A

is F

λ+

=(F

λ+

) given by

λ+

=Im



dx ∧ d¯x

(λ

+ |x|

)



λ+

=Im



¯xdx ∧ d¯xx

|x|(λ

+ |x|

)

|x|



The basic anti-instanton over R

is described as follows. The principal bundle

−1

,SU(2)) is deﬁned by the transition function

¯g : U

∩ U

→ SU(2), ¯g(x)=¯x/|x|.

For each λ ∈ (0, 1) deﬁne the connection B

=(B

) by

=Im



¯xdx

+ |x|



=Im



xd¯x

|x|

(λ

+ |x|

)



The connection B

is anti-dual with instanton number −1. We note that

[¯g]=−[g] ∈ π

(SU(2))

∼

Z and thus [¯g] is also non-trivial. The curvature

of this connection A

is F

λ−

=(F

λ−

) given by

λ−

=Im



d¯x ∧ dx

(λ

+ |x|

)



λ−

=Im



xd¯x ∧dx¯x

|x|(λ

+ |x|

)

|x|



In the above example we have given an explicit construction of two

basic Yang–Mills ﬁelds. In fact, these ﬁelds can be obtained by applying

Uhlenbeck’s removable singularities theorem to a suitable connection on

\{(0, 0, 0, 0, 1}⊂R

to extend it to all of S

. We use this method in

our construction of the BPST instanton in Chapter 9.

8.3 Yang–Mills Fields 251

An explicit construction of the full (8k −3)-parameter family of solutions

was given by Atiyah, Drinfeld, Hitchin, and Manin ([19]). An alternative con-

struction was given by Atiyah and Ward using the Penrose correspondence.

Several solutions to the Yang–Mills equations on special manifolds of various

dimensions have been obtained in [26, 175, 186, 243, 302, 377]. Although the

solutions’ physical signiﬁcance is not clear, they may prove to be useful in

understanding the mathematical aspects of the Yang–Mills equations.

Gauge ﬁelds and their associated ﬁelds arise naturally in the study of

physical ﬁelds and their interactions. The solutions of these equations are

often obtained locally. The question of whether ﬁnite energy solutions of

the coupled ﬁeld equations can be obtained globally is of great signiﬁcance

for both the physical and mathematical considerations. The early solutions of

SU(2) Yang–Mills ﬁeld equations in the Euclidean setting had a ﬁnite number

of point singularities when expressed as solutions on the base manifold R

The fundamental work of Uhlenbeck ([385,386,387]) showed that these point

singularities in gauge ﬁelds are removable by suitable gauge transformations

and that these solutions can be extended from R

to its compactiﬁcation

as singularity-free solutions of ﬁnite energy. The original proof of the

removable singularities theorem is greatly simpliﬁed by using the blown-up

manifold technique. This proof also applies to arbitrary Yang–Mills ﬁelds and,

in particular, to non-dual solutions. Note that we call a connection non-dual

if it is neither self-dual (∗F = F ) nor anti-dual (∗F = −F ), i.e., if it is not

an absolute minimum of the Yang–Mills action. Speciﬁcally, one obtains in

this way a local solution of the source-free Yang–Mills equations on the open

ball B

by removing the singularity at the origin.

Theorem 8.3 Let B

be the open ball B

⊂ R

with some metric g (not

necessarily the standard metric induced from R

). Let ω be a solution of the

Yang–Mills equations in B

\{0} with ﬁnite action, i.e.,



< ∞.

Assume that the local potential A ∈ H

\{0}) of ω has the property

that, for every smooth, compactly supported function φ ∈ C

∞

\{0}),

φA ∈ H

\{0}).Thenω is gauge-equivalent to a connection ˜ω,which

extends smoothly across the singularity to a smooth connection on B

By a grafting procedure the result of this theorem can be extended to

manifolds with a ﬁnite number of singularities as follows.

Theorem 8.4 Let M be a compact, oriented, Riemannian 4-manifold. Let

,...,p

} be any ﬁnite set of points in M.LetP

be an SU(2)-bundle

deﬁned over M \{p

,...,p

} and let ω

be a Yang–Mills connection on

P . Then the bundle P

extends to an SU(2)-bundle deﬁned over M and the

connection ω

extends to a Yang–Mills connection ω on P .

252 8 Yang–Mills–Higgs Fields

These theorems can be extended to ﬁnite energy solutions of coupled

ﬁeld equations on Riemannian base manifolds. It was shown in [386]that

the removable singularities theorems for pure gauge ﬁelds fail in dimensions

greater than 4. In dimension 3 the removable nature of isolated singularities

is discussed for the Yang–Mills and Yang–Mills–Higgs equations in Jaﬀe and

Taubes [207].

8.4 Non-dual Solutions

In general, the standard steepest descent method for ﬁnding the global min-

ima of the Yang–Mills action is not successful in 4 dimensions, even though

this technique does work in dimensions 2 and 3. In fact, the problem of

ﬁnding critical points of the Yang–Mills action seems to be similar to the 2-

dimensional harmonic map problem as well as to conformally invariant varia-

tional problems. These similarities had led some researchers to believe in the

non-existence of non-dual solutions. It can be shown that local Sobolev con-

nections, on SU(2)-bundles over R

\M (M a smoothly embedded compact

2-manifold in R

) with ﬁnite Yang–Mills action, satisfy a certain holonomy

condition. If the singular set has codimension greater than 2, then the tech-

niques used for removing the point singularities can be applied to remove

these singularities. Thus, a codimension 2 singular set, such as an S

embed-

ded in R

, provides an appropriate setting for new techniques and results. For

example, the holonomy condition implies that there exist ﬂat connections in

a principal bundle over R

that cannot be extended to a neighborhood of

the singular set S

even though the bundle itself may be topologically trivial.

It was also known that connections satisfying certain stability or symmetry

properties are either self-dual or anti-dual. In view of these results, it was

widely believed that no non-dual solutions with gauge group SU(2) existed

on the standard four sphere S

. Analogies with harmonic maps of S

to S

also appeared to support the non-existence of such solutions. In fact, it turns

out that a family of such connections may be used to obtain a non-trivial con-

nection that can be extended to a neighborhood of the singular set. These

ideas, together with the work on monopoles in hyperbolic 3-space, have been

used in [350] to prove the existence of non-dual (and hence non-minimal)

solutions of Yang–Mills equations. The existence of other non-dual solutions

has been discussed in [332, 388,389].

In [332] it is shown that non-dual Yang–Mills connections exist on all

SU(2)-bundles over S

with second Chern class diﬀerent from ±1. The re-

sults are obtained by studying connections that are equivariant with respect

to the symmetry group G = SU(2) that acts on S

⊂ R

via the unique

irreducible representation. The principal orbits are 3-dimensional, reducing

the Yang–Mills equations and the instanton equations to a system of ordinary

diﬀerential equations following [389]. The inequivalent lifts of this G-action to

8.4 Non-dual Solutions 253

SU(2)-bundles over S

are called quadrupole bundles. These bundles are

naturally classiﬁed by a pair of odd positive integers (n

−

). The principal

bundle corresponding to the pair (n

−

) is denoted by P

−

)

.Thenwe

have

−

)

)=(n

− n

−

)/8.

Thus the second Chern class of the quadrupole bundle P

−

)

assumes all

integral values. The main result is:

Theorem 8.5 On every quadrupole bundle P

−

)

with n

=1and n

−

=

1 there exists a non-dual SU(2) Yang–Mills connection.

It is not known whether any non-dual connections exist for c

= ±1. The

non-dual Yang–Mills connections obtained in this work on the trivial bundle

over S

(i.e., with c

= 0) are diﬀerent from those of [350]. The proof and

further discussion may be found in [332].

Recently, non-dual solutions of the Yang–Mills equations over S

×S

and

×S

with gauge group SU(2) have been obtained in [395]. We give a brief

discussion of these results below.

Theorem 8.6 Let (m, n) be a pair of integers with |m| = |n|,whichsatisfy

the following conditions:

1. If |m| > |n|,then|m| = |n|(2l +1)+l(l +1), for l =0, 1, 2,....

2. If |n| > |m|,then|n| = |m|(2l +1)+l(l +1), for l =0, 1, 2,....

Then there exists a positive integer K

such that, for any positive odd num-

ber k>K

, there exists an irreducible SU(2)-connection A(m, n, k) over

×S

with instanton number 2mn, which is a non-minimal solution of the

Yang–Mills equations. In fact, the Yang–Mills action deﬁned by equation (8.9)

satisﬁes the inequality

2(m

+ n

+2k) −<A

(A(m, n, k)) < 2(m

+ n

+2k)+

for some  ≤ 1.

Proof :ItiswellknownthatS

is diﬀeomorphic to the complex projective

space CP

viewed as the set of 1-dimensional linear subspaces in C

.There

exists a tautological line bundle L over S

whose ﬁrst Chern number is



(L)=−1,

where c

(L) is the ﬁrst Chern class of L. Consider the standard metric on

∼

; then the ﬁrst Chern class is written as

(L)=−

4π

ω,

where ω isthevolumeformonCP

. Suppose that A

is the canonical con-

nection on L; then the curvature of A

254 8 Yang–Mills–Higgs Fields

= iω/2.

Set L(m, n)=π

∗

⊗π

∗

→ CP

×CP

, which is a complex line bundle

over the product manifold CP

×CP

,wherem, n ∈ Z,andπ

,π

are the

natural projections from CP

×CP

to the ﬁrst and second factor, respec-

tively. The ﬁrst Chern class of L(m, n)is

(L(m, n)) = −

(mω

+ nω

where ω

= π

∗

ω, i =1, 2. Let A(m, n)=π

∗

(⊗

) ⊗ π

∗

(⊗

). Then the

corresponding curvature is

F (m, n)=F

A(m,n)

(mω

+ nω

Hence L(m, n) ⊕L(m, n)

−1

→ CP

×CP

is a reducible SU(2)-bundle over

×CP

, and there exists a reducible SU(2)-connection whose curvature

F (m, n)=

(mω

+ nω

)



i 0

0 −i



We note that c

(L(m, n) ⊕ L(m, n)

−1

)=0and

−c

(L(m, n) ⊕ L(m, n)

−1

4π

det(F (m, n)) =

8π

∧ ω

It follows that if m = n (resp., m = −n), then A(m, n) is a reducible self-dual

(resp., anti-dual) SU(2)-connection on L(m, n) ⊕ L(m, n)

−1

with instanton

number 2m

(resp. −2m

). When |m| = |n|,thenA(m, n) is a reducible non-

minimal Yang–Mills connection on L(m, n) ⊕ L(m, n)

−1

with degree 2mn

since F (m, n) is neither self-dual nor anti-dual. Grafting the standard in-

stanton and anti-instanton solutions on these non-dual solutions gives the

required family of solutions. 

For Yang–Mills ﬁelds on S

× S

we have the following result.

Theorem 8.7 There exists a positive integer K

such that, for any positive

odd number k>K

, there exists an irreducible SU(2)-connection A(k) over

×S

with instanton number 0 that is a non-minimal solution of the Yang–

Mills equations and its action satisﬁes the inequality

(A(k)) > 4k.

A special non-dual Yang–Mills ﬁeld on S

× S

wasusedin[269]asthe

source ﬁeld in the construction of generalized gravitational instantons. From

a physical point of view, one can consider the family of non-dual solutions to

be obtained by studying the interaction of instantons, anti-instantons, and

8.5 Yang–Mills–Higgs Fields 255

the ﬁeld of a ﬁxed background connection, which is an isolated non-minimal

solution of the Yang–Mills equations.

8.5 Yang–Mills–Higgs Fields

In this section we give a brief discussion of the most extensively studied cou-

pled system, namely, the system of Yang–Mills–Higgs ﬁelds, and touch upon

some related areas of active current research. From the physical point of view

the Yang–Mills–Higgs system is the starting point for the construction of real-

istic models of ﬁeld theories, which describe interactions with massive carrier

particles. From the mathematical point of view the Yang–Mills–Higgs system

provides a set of non-linear partial diﬀerential equations, whose solutions have

many interesting properties. In fact, these solutions include solitons, vortices,

and monopoles as particular cases. Establishing the existence of solutions of

the Yang–Mills–Higgs system of equations has required the introduction of

new techniques, since the standard methods in the theory of non-linear par-

tial diﬀerential equations are, in general, not applicable to this system. We

now give a brief general discussion of various couplings of Yang–Mills ﬁeld to

Higgs ﬁelds, with or without self-interaction.

Let (M, g) be a compact Riemannian manifold and P (M,G) a principal

bundle with compact semisimple gauge group G.Leth denote a ﬁxed bi-

invariant metric on G.Themetricsg and h induce inner products and norms

on various bundles associated to P and their sections. We denote all these

diﬀerent norms by the same symbol, since the particular norm used is clear

from the context. The Yang–Mills–Higgs conﬁguration space is deﬁned

:= A(P ) × Γ (ad P ),

i.e.,

⊂{(ω, φ) ∈ Λ

(P, g) × Λ

(M,ad P )},

where ω is a gauge connection and φ is a section of the Higgs bundle ad P .In

the physics literature φ is called the Higgs ﬁeld in the adjoint representation.

In the rest of this section we consider various coupled systems with such Higgs

ﬁeld.

A Yang–Mills–Higgs action A

, with self-interaction potential V :

→ R

, is deﬁned on the conﬁguration space C by

(ω,φ)=c(M)





+ c

φ|

+ c

V (|φ|

)



, (8.26)

where c(M)isanormalizing constant that depends on the dimension of

M (c(M)=

8π

for a 4-dimensional manifold) and c

are the coupling

constants. The constant c

measures the relative strengths of the gauge

256 8 Yang–Mills–Higgs Fields

ﬁeld and its interaction with the Higgs ﬁeld. When c

= 0, the constant

measures the relative strengths of the Higgs ﬁeld self-interaction and

the gauge ﬁeld–Higgs ﬁeld interaction. A Yang–Mills–Higgs system with

self-interaction potential V is deﬁned as a critical point (ω, φ) ∈Cof the

action A

. The corresponding Euler–Lagrange equations are

+ c

[φ, d

φ]=0, (8.27)

φ + c



(|φ|

)φ =0, (8.28)

where V



(x)=dV/dx.Equations(8.27), (8.28) are called the Yang–Mills–

Higgs ﬁeld equations with self-interaction potential V . In the physics litera-

ture, it is customary to deﬁne the current J by

J = −c

[φ, d

φ]. (8.29)

Using this deﬁnition of the current we can rewrite equation (8.27) as follows:

= J. (8.30)

Note that in these equations δ

is the formal L

-adjoint of the corresponding

map d

. Thus, in (8.27) δ

is a map

: Λ

(M,ad P ) → Λ

(M,ad P )

and in (8.28) δ

is a map

: Λ

(M,ad P ) → Λ

(M,ad P )=Γ (ad P ).

The pair (ω, φ) also satisﬁes the following Bianchi identities

=0, (8.31)

·d

φ =[F

,φ]. (8.32)

Note that these identities are always satisﬁed whether or not equa-

tions (8.27), (8.28) are satisﬁed.

We note that no solution in closed form of the Yang–Mills–Higgs equations

with self-interaction potential is known for c

> 0, but existence of spherically

symmetric solutions is known. Also, existence of solutions for the system is

known in dimensions 2 and 3. It can be shown that the solutions of the

system of equations (8.27), (8.28)onR

satisfy the following relation (Jaﬀe

and Taubes [207]):

(n −4)|F

+(n − 2)c

φ|

+ nc

|V (|φ|

)| =0.

From this relation it follows that for c

≥ 0,c

≥ 0 there are no non-trivial

solutions for n>4, and for n = 4 every solution decouples (i.e., is equivalent

8.5 Yang–Mills–Higgs Fields 257

to a pure Yang–Mills solution.) For the Yang–Mills–Higgs system on R

with

=1andc

≥ 0 we have the following result [169].

Theorem 8.8 Let V (t)=(1− t)

(1 + at),a≥ 0. Then for c

suﬃciently

small, there exists a positive action solution to equations (8.27), (8.28)which

is not gauge equivalent to a spherically symmetric solution. Furthermore, for

smaller still, there exists a solution that has the above properties and is,

in addition, not a local minimum of the action.

Writing c

=1andc

=0in(8.26) we get the usual Yang–Mills–Higgs

action

(ω,φ)=c(M)



[|F

+ |d

φ|

]. (8.33)

The corresponding Yang–Mills–Higgs equations are

+[φ, d

φ]=0, (8.34)

φ =0. (8.35)

In the above discussion the Higgs ﬁeld was deﬁned to be a section of the

Lie algebra bundle ad P . We now remove this restriction by deﬁning the

generalized Higgs bundle

:= P ×

to be the vector bundle associated to P by the representation ρ of the gauge

group G on the vector space H (which may be real or complex). We as-

sume that H admits a G-invariant metric. A section ψ ∈ Γ (H

) is called

the generalized Higgs ﬁeld. In what follows we shall drop the adjective

“generalized” when the representation ρ is understood.

8.5.1 Monopoles

Finite action solutions of equations (8.27), (8.28) are called solitons. Locally,

these solutions correspond to time-independent, ﬁnite-energy solutions on

R × M. The Yang–Mills instantons discussed earlier are soliton solutions of

equations (8.27), (8.28) corresponding to φ =0,c

= 0. The soliton solutions

on a 2-dimensional base are known as vortices and those on a 3-dimensional

base are known as monopoles. Vortices and monopoles have many properties

that are qualitatively similar to those of the Yang–Mills instantons.

When M is 3-dimensional, we can associate to the Yang–Mills–Higgs equa-

tions the ﬁrst order Bogomolnyi equations [46]

= ±∗d

φ. (8.36)