Marathe K. Topics in Physical Mathematics

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268 8 Yang–Mills–Higgs Fields

= c(M )[|F |

+ |G|

The Lagrangian L

of the Higgs ﬁeld in interaction with the electroweak

gauge ﬁeld is given by

= c(M )[|∇

ψ|

− V (ψ)],

and the Lagrangian L

of the fermion ﬁelds in interaction with the elec-

troweak gauge ﬁeld is given by



iγ

∇



iγ

∇

In the deﬁnition of L

the ﬁrst sum extends over all left-handed doublets and

the second sum extends over all right-handed singlets and the electroweak

gauge covariant derivative ∇

is given by equation (8.55). In addition to the

above terms it is necessary to introduce Yukawa couplings to fermions, to give

masses to charged fermions. We denote by g

the Yukawa coupling constant

to fermion f. Thus, there is one Yukawa coupling constant for each family

of leptons and quarks. These are contained in the Yukawa Lagrangian L

whose detailed expression is not relevant to our discussion. Using the standard

model Lagrangian, one can construct models for the weak and electromag-

netic interactions of various types of known particles, such as leptons, and

predict the outcome of processes mediated by the heavy intermediate bosons.

A striking experimental conﬁrmation of the predictions of electroweak theory

in the late 1970s must be viewed as a big boost to the gauge theory point

of view in elementary particle physics. Several physicists had been working

on the electroweak theory. Glashow, Salam, and Weinberg working indepen-

dently arrived at essentially similar results including the mass ratios of the

electroweak bosons. The three shared the 1979 Nobel Prize for physics for

this work, paving the way for the current version of the standard model.

8.7.1 The Standard Model

The above considerations on electroweak theory can be extended to construct

the standard model of a uniﬁed theory of the strong, weak, and electromag-

netic interactions as a gauge theory with the gauge group G

deﬁned by

:= SU

(3) × SU

(2) × U

(1).

The gauge potential of the theory corresponds to spin 1 vector bosons. These

split naturally into the gauge bosons of the electroweak theory and the glu-

ons G

, 1 ≤ a ≤ 8, which correspond to the color gauge group SU

(3)

of the strong interaction with coupling constant g

. Quantized Yang–Mills

theory with the full gauge group G

is the foundation of the theory of

8.7 Electroweak Theory 269

elementary particles. However, the mechanism of mass generation for gluons,

the carrier particles of the strong interaction, is not via the Higgs mecha-

nism. It uses a quantum mechanical property called the “mass gap” in the

physics literature. This property was discovered by experimental physicists

but its mathematical foundation is unclear. Putting the quantum Yang–Mills

theory on a mathematical foundation and explaining the mass gap is one of

the seven millennium problems announced by the Clay Mathematics Insti-

tute. We refer the reader to their web page at www.claymath.org for more

details. The full Lagrangian L

of the theory can be written along the lines

of the electroweak Lagrangian given above. The Lagrangian L

contains the

following, rather large, number of free parameters:

1. three gauge coupling constants g

2. Higgs coupling constant

3. Yukawa coupling constants

4. Higgs particle mass m

5. quark-mixing matrix elements

6. number of matter families or generations (there is strong evidence that

this number is 3).

In Tables 6.1 and 6.2 we have already given the masses and electric charges

of the fermions and bosons that enter into the description of the standard

model. Initially the isospin group SU(2) of the Yang–Mills theory was ex-

pected to provide an explanation of the spectrum of strongly interacting

particles. However, it turned out that a larger group, SU(3), was needed to

explain the observed particles. The representations of SU(3) have been well

known in mathematics since the work of Weyl in the 1930s. In the 1950s, how-

ever, Weyl’s work was not included in physicists’ training. In fact, Pauli had

dubbed group theory “Gruppenpest,” and most physicists found the mathe-

matical developments quite hard to understand and felt that they would be

of little relevance to their work. The mathematicians also did not understand

physicists work and did not feel that it would contribute to a better under-

standing of group theory. When Murray Gell-Mann started to study the

representations of SU(3), he never discussed his problem with his lunchtime

companion J.-P. Serre, one of the leading experts in representation theory.

Upon returning to Caltech from Paris, where he had discussions with his

mathematician colleagues, Gell-Mann was able to show that the observed

particles did ﬁt into representations of SU(3). The simplest representation

he could use was 8-dimensional and he started to call this symmetry the

eightfold way (perhaps an allusion to Buddha’s eight steps to Nirvana). He

could ﬁt 9 of the particles into a 10-dimensional representation and was able

to predict the properties of the missing 10th particle. When this particle was

found at Brookhaven, it made the theory of group representations an essential

tool for particle theory. Gell-Mann’s work had not used the fundamental or

deﬁning representation of SU(3)asitpredictedparticles with charges equal

to a fraction of the electron charge. Such particles have never been observed.

270 8 Yang–Mills–Higgs Fields

Gell-Mann was nevertheless convinced of the existence of these unobserved or

conﬁned particles as the ultimate building blocks of matter. He called them

quarks. Gell-Mann expects the quarks to be unobservable individually. They

would be conﬁned in groups making the fundamental particles.

The single mixing angle of the electroweak theory must be replaced by a

3 ×3 matrix in the study of strong interactions. The Cabibbo–Kobayashi–

Maskawa (CKM) matrix,alsoknownasthequark mixing matrix,is

a unitary matrix, which contains information on the ﬂavor changing of

quarks under weak interaction. Kobayashi and Maskawa shared the 2008

Nobel Prize in physics with Nambu for their work on the origin of broken

symmetry, which predicts the existence of at least three families of quarks.

Their work was based on the earlier work of Cabibbo, who was not honored

with the Prize. An up type quark, labeled (u, c, t), can decay into a down

type quark, labeled (d, s, b). The entries of the CKM matrix are denoted

by V

αβ

,whereα is an up type quark and β is a down type quark. The

transition probability of going from quark α to quark β is proportional to

αβ

. Using this notation, the CKM matrix V can be written as follows:

V :=

⎛

⎝

⎞

⎠

. (8.70)

The CKM matrix is a generalization of the Cabibbo angle,whichwas

the only parameter needed when only two generations of quarks were known.

The Cabibbo angle is a mixing angle between two generations of quarks. The

current model with three generations has three mixing angles. Unitarity of

the CKM matrix implies three relations involving only the absolute values of

the entries of the matrix V . These relations are

αd

+ |V

αs

+ |V

αb

=1,α= u, c, t. (8.71)

The relations of (8.71) imply that the sum of all couplings of any of the

up type quarks to all the down type quarks is the same for all generations.

This was ﬁrst pointed out by Cabibbo. These relations are called the weak

universality. Unitarity of the CKM matrix also implies the following three

phase-dependent relations.

αd

βd

+ V

αs

βs

+ V

αb

βb

=0, (8.72)

where α, β are any two distinct up type quarks.

These phase-dependent relations are crucial for an understanding of the

origin of CP violation.Forﬁxedα, β each of the relations says that the

sum of the three terms is zero. This implies that they form a triangle in the

complex plane. Each of these triangles is called a unitary triangle,andtheir

areas are the same. The area is zero for the speciﬁc choice of parameters in the

standard model for which there is no CP violation. Non-zero area indicates

8.8 Invariant Connections 271

CP violation. The CPM matrix elements are not determined by theory at

this time; various experiments for their determination are being carried out

at laboratories around the world. It was the observation of CP violation in

weak interactions that led to the existence of a third generation of quarks.

Experimental evidence for the existence of a third generation b-quark came

in 1977 through the work of the Fermi Lab, Chicago group led by Leon

Lederman. For this work he was awarded the Noble Prize in 1988.

Predictions of the standard model have been veriﬁed with great accuracy

and present there are no experimental results that contradict this model.

However, there are a number of unsatisfactory features of this theory, e.g.,

the problem of the origin of mass generation, prediction of the Higgs particle

mass, and so on.

We conclude this section with a brief look at some consistency checks

of grand uniﬁed theories from recent experimental results. Since 1989, the

Z-factory at LEP (large electron–positron collider at CERN, Geneva) has

produced a large number of electron–positron collisions for the resonance

production of Z

gauge bosons to perform a number of high-precision ex-

periments for determinatinig of the mass of asthenons (weak intermediate

vector bosons W

and Z

) and to ﬁnd lower bounds on the mass of hig-

gsons

(Higgs bosons) and the top quark. These same LEP data have been

used to perform consistency checks of grand uniﬁed theories and to study

the problem of uniﬁcation of the three coupling constants of the standard

model (see, for example, [10]). It has been shown that the standard model is

in excellent agreement with all LEP data. However, an extrapolation of the

three independent running coupling constants to high energies indicates that

their uniﬁcation within the standard model is highly unlikely. On the other

hand, the minimal supersymmetric extension of the standard model leads to

a uniﬁcation of couplings at a scale compatible with the present data. As

Richard Feynman once observed “In physics, the best one can hope for is to

be temporarily not wrong.”

8.8 Invariant Connections

In many physically interesting situations, in addition to the local internal

symmetry provided by the gauge group, an external global action of another

Lie group exists on the space of phase factors. Mathematically, we can de-

scribe the situation as follows. We consider, as before, a principal bundle

P (M, G) with gauge group G, which represents the space of phase factors.

For the external symmetry group we take a Lie group K acting on P by

bundle automorphisms. Then every element of K induces a diﬀeomorphism

of M.Thus,K may be regarded as a group of generalized gauge transfor-

I thank Prof. Ugo Amaldi of the University Milan for suggesting these particle names

and for useful discussions on the analysis of LEP data.

272 8 Yang–Mills–Higgs Fields

mations as deﬁned in Chapter 6. We ﬁx a point u

∈ P as a reference point

and let π(u

)=x

∈ M. Let J be the set of all elements f ∈ K such that

f ·u

∈ π

−1

). Thus the induced map f

∈ Diﬀ(M)ﬁxesthepointx

.We

note that the set J is a closed subgroup of K.WecallJ the isotropy sub-

group of K at x

. Deﬁne a map ψ : J → G as follows. For each f ∈ J, ψ(f)

is the unique element a ∈ G such that fu

= u

a. It is easy to see that ψ is

a Lie group homomorphism and hence induces a Lie algebra homomorphism

ψ of the Lie algebra j of J to the Lie algebra g of G.

Deﬁnition 8.1 Using the notation of the above paragraph, let Γ be a con-

nection in P with connection 1-form ω. We say that Γ is a K-invariant

connection if each f ∈ K preserves the horizontal distribution of Γ ,or,

equivalently, leaves the form ω invariant, i.e., f

∗

ω = ω.

In physical literature the group K is referred to as the external symme-

try group of the gauge potential ω.

Theorem 8.11 Let K be a group of bundle automorphisms of P (M,G) and

Γ a K-invariant connection in P with connection 1-form ω.LetΨ be the

linear map from the Lie algebra k of K to g deﬁned by

Ψ(A)=ω

(

A),A∈ k, (8.73)

where

A is the vector ﬁeld on P induced by A. Then we have

1. Ψ(A)=

ψ(A), A ∈ j;

2. if ad

is the restriction of the adjoint representation of K in k to the

subgroup J and ad is the adjoint representation of G in g (see Section

1.6), then

Ψ(ad

f(A)) = ad(ψ(f))(Ψ (A)),f∈ J, A ∈ k;

3. 2Ω

(

B)=[Ψ(A),Ψ(B)] − Ψ([A, B]), A, B ∈ k.

We say that K acts ﬁber-transitively on P if, for any two ﬁbers of P ,

there is an element of K that maps one ﬁber into the other or, equivalently, if

the induced action on the base M is transitive. In this case M is isomorphic

to the homogeneous space K/J,whereJ is the isotropy subgroup of K at

Theorem 8.12 Let K be a ﬁber-transitive group of bundle automorphisms

of P . Then there is a one-to-one correspondence between the set of K-

invariant connections in P and the set of linear maps Ψ : k → g satisfying

the ﬁrst two conditions of Theorem 8.11, the correspondence being given by

equation (8.73).

If K is ﬁber-transitive on P , the curvature form Ω of the invariant connec-

tion ω is a tensorial form of type (ad,G)thatisK-invariant. It is completely

determined by the values Ω

(

B), where A, B ∈ k. The following corollary

is an immediate consequence of condition 3 of Theorem 8.11.

8.8 Invariant Connections 273

Corollary 8.13 The K-invariant connection in P deﬁned by Ψ is ﬂat if and

only if Ψ : k → g is a Lie algebra homomorphism.

The following particular situation is often useful for the construction of

invariant connections. Suppose that k admits an ad

-invariant subspace m

that is complementary to j, i.e., k = j ⊕ m. Then we have the following.

Theorem 8.14 There is a one-to-one correspondence between the set of K-

invariant connections in P and the set of linear maps Ψ

: m → g such

that

(ad

f(A)) = ad(ψ(f))(Ψ

(A)),f∈ J, A ∈ m.

Furthermore, the curvature form Ω of this K-invariant connection satisﬁes,

∀X, Y ∈ m, the condition

2Ω

(

B)=[Ψ

(A),Ψ

(B)] − Ψ

([A, B]

) −Ψ([A, B]

where [A, B]

(resp., [A, B]

)isthem-component (resp., j-component) of

[A, B] in k.

The K-invariant connection in P deﬁned by Ψ

= 0 is called the canon-

ical connection in P with respect to the decomposition k = j ⊕ m.An

important special case that arises in many applications is the following. Let

H be a closed subgroup of a connected Lie group K and let M = K/H and

P = K (i.e., we consider the principal bundle K(K/H, H)). If there exists an

ad H-invariant subspace m complementary to h in k, then the h-component

ω of the canonical 1-form θ of K with respect to the decomposition k = h⊕m

deﬁnes a K-invariant connection in the bundle K (the action of K being that

by left multiplication). Conversely, any such K-invariant connection (if it ex-

ists) determines a decomposition k = h ⊕ m. Furthermore, the curvature

form Ω of the invariant connection deﬁned by ω is given by

Ω(A, B)=−

[A, B]

([A, B]

is the h-component of [A, B]ink), where A, B are left-invariant

vector ﬁelds on K belonging to m. The invariant connection ω deﬁned above

coincides with the canonical connection deﬁned at the beginning of this para-

graph.

At least locally, the Yang–Mills–Higgs equations (8.27), (8.28)maybe

regarded as obtained by dimensional reduction from the pure Yang–Mills

equations. To see this, consider a Yang–Mills connection α on the trivial

principal bundle R

m+1

× G over R

m+1

and let

+ A

m+1

, 1 ≤ i ≤ m,

be the gauge potential on R

m+1

with values in g. Suppose that this potential

does not depend on x

m+1

. Deﬁne φ := A

m+1

;thenφ can be regarded as a

Higgs potential on R

with values in g and

274 8 Yang–Mills–Higgs Fields

A = A

, 1 ≤ i ≤ m,

can be regarded as the gauge potential on R

with values in g.Letω denote

the gauge connection on R

× G corresponding to A.Thenwehave

)

=(F

)

, (F

)

im+1

=(d

φ)

1 ≤ i, j ≤ m

and the pure Yang–Mills action of F

on R

m+1

reduces to the Yang–Mills–

Higgs action (8.33). It is easy to see that this reduction is a consequence

of the translation invariance of the pure Yang–Mills system in the x

m+1

direction. In general, suppose that T is a Lie group that acts on P (M,G)

as a subgroup of Diﬀ

(P ) with induced action on M.WesaythatT is a

local symmetry group of the connection ω on P or that ω is a locally

T -invariant connection on P if the condition

ω =0, ∀A ∈ t,

is satisﬁed, where

A is the fundamental vector ﬁeld corresponding to the

element A in the Lie algebra t of T . Under certain conditions P/T is a prin-

cipal bundle over M/T and a system of equations on the original bundle can

be reduced to a coupled system on the reduced base. Invariant connections

and their applications to gauge theories over Riemannian manifolds are dis-

cussedin[129, 388, 389]. For a particular class of self-interaction potentials,

such reduction and the consequent symmetry breaking are responsible for the

Higgs mechanism [192]. For a geometrical description of dimensional reduc-

tion and its relation to the Kaluza–Klein theories, see [206]andthebookby

Coquereaux and Jadczyk [85].

Chapter 9

4-Manifold Invariants

9.1 Introduction

The concept of moduli space was introduced by Riemann in his study of the

conformal (or equivalently, complex) structures on a Riemann surface. Let us

consider the simplest non-trivial case, namely, that of a Riemann surface of

genus 1 or the torus T

. The set of all complex structures C(T

)onthetorus

is an inﬁnite-dimensional space acted on by the inﬁnite-dimensional group

Diﬀ(T

). The quotient space

M(T

):=C(T

)/ Diﬀ(T

)

is the moduli space of complex structures on T

.SinceT

with a given com-

plex structure deﬁnes an elliptic curve, M(T

) is, in fact, the moduli space of

elliptic curves. It is well known that a point ω = ω

+ iω

in the upper half-

plane H (ω

> 0) determines a complex structure and is called the modulus

of the corresponding elliptic curve. The modular group SL(2, Z)actsonH by

modular transformations and we can identify M(T

)withH/ SL(2, Z). This

is the reason for calling M (T

) the space of moduli of elliptic curves or simply

the moduli space. The topology and geometry of the moduli space has rich

structure. The natural boundary ω

= 0 of the upper half plane corresponds

to singular structures. Several important aspects of this classical example

are also found in the moduli spaces of other geometric structures. Typically,

there is an inﬁnite-dimensional group acting on an inﬁnite-dimensional space

of geometric structures with quotient a “nice space” (for example, a ﬁnite-

dimensional manifold with singularities). For a general discussion of moduli

spaces arising in various applications see, for example, [193].

We shall be concerned with the construction of the moduli spaces of in-

stantons and in studying their topological and geometric properties in this

chapter. The simplest of these moduli spaces is the space M

(M) of instan-

tons with instanton number 1 on a compact, simply connected Riemannian

manifold M with positive deﬁnite intersection form. Donaldson showed that

K. Marathe, Topics in Physical Mathematics, DOI 10.1007/978-1-84882-939-8 9, 275

 Springer-Verlag London Limited 2010

276 9 4-Manifold Invarian t s

this moduli space M

(M) is a compact, oriented, 5-dimensional manifold

with singularities and that it provides a cobordism between M and a sum

of a certain number of copies of CP

. This result has led to the existence of

new obstructions to smoothability of 4-manifolds and to some surprising and

unexpected results about exotic diﬀerential structures on the standard Eu-

clidean space R

. In Section 9.2 we give an explicit construction of the moduli

space M

of the BPST-instantons of instanton number 1 and indicate the

construction of the moduli space M

of the complete (8k − 3)-parameter

family of instanton solutions over S

with gauge group SU(2) and instan-

ton number k. The moduli spaces of instantons on an arbitrary Riemannian

4-manifold with semisimple Lie group as gauge group are also discussed here.

A brief account of Donaldson’s theorem on the topology of moduli spaces

of instantons and its implications for smoothability of 4-manifolds can be

found in Section 9.3. The investigation of the Riemannian geometry of these

moduli spaces was begun after Donaldson’s work. The results obtained for

the metric and curvature of M

are given and a geometrical interpretation

of Donaldson’s results is also indicated in this section. Donaldson’s polyno-

mial invariants are discussed in Section 9.4. The generating function of these

invariants and its relation to basic classes is also given there. In the summer

of 1994 Witten announced in his lecture at the International Congress on

Mathematical Physics in Paris that the Seiberg–Witten theory can be used

to obtain all the information contained in the Donaldson invariants. His paper

[407] gave an explicit formula relating the SW invariants and the Donaldson

invariants. We discuss this work in Section 9.5. Witten’s derivation of this

fantastic formula was based on physical reasoning. There is now a mathe-

matical proof for a large class of manifolds. We indicate this result and some

of its applications in the same section.

9.2 Moduli Spaces of Instantons

A complete set of solutions of instanton equations on S

was obtained by

the methods of algebraic geometry and the theory of complex manifolds.

These methods have also been used in the study of the Yang–Mills equa-

tions over Riemann surfaces. Since this approach is not discussed here, we

simply indicate below some references where this and related topics are de-

veloped. They are [17,19,51,67,256]. See also the book [143]byFriedmanand

Morgan.

Let M be a compact, connected, oriented Riemannian manifold of dimen-

sion 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) has a natural extension to bundle valued forms and this is used to

deﬁne the self-dual and anti-dual forms in Λ

(M,ad P ) and to decompose a

form α ∈ Λ

(M,ad P ) into its self-dual and anti-dual parts. A G-instanton

9.2 Moduli Spaces of Instantons 277

(resp., G-anti-instanton)overM is a self-dual (resp., anti-dual) Yang–Mills

ﬁeld on the principal bundle P .IfF = F

is the gauge ﬁeld corresponding

to the gauge connection ω,then

∗F = F (9.1)

is called the instanton equation and

∗F = −F (9.2)

is called the anti-instanton equation. Over a 4-dimensional base manifold

M, the second Chern class and the Euler class of P are equal and we deﬁne

the instanton number k of a G-instanton by

k := −c

(P )[M ]=−χ(P )[M]. (9.3)

Recall that using the decomposition of F = F

into its self-dual part F

and

anti-dual part F

−

(see Chapter 8) we get

k =

8π



(|F

−|F

−

) (9.4)

and

(ω)=

8π



(|F

+ |F

−

). (9.5)

Comparing equations (9.4)and(9.5) 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 ) (9.6)

and the connections satisfying the instanton or the anti-instanton equations

are the absolute minima of the action. In particular, for a self-dual (∗F = F )

Yang–Mills ﬁeld, i.e., a G-instanton, we have

8π



(F



)=|k|.

Similarly, for an anti-dual (∗ F = −F ) Yang–Mills ﬁeld, i.e., a G-anti-

instanton, we have

8π



(F

−



)=−|k|.

We now discuss the most well known family of instanton solutions, namely

the BPST instantons (Belavin–Polyakov–Schwartz–Tyupkin instan-

tons)[36]. It can be shown that the BPST instanton solution over the base

manifold R

can be extended to the conformal compactiﬁcation of R

, i.e.,

. This extension is characterized by a self-dual connection on a non-trivial