Dunajski M. Solitons, Instantons, and Twistors

Подождите немного. Документ загружается.

106 6: Gauge ﬁeld theory

The standard model unifying electromagnetic and nuclear interactions is

an example of gauge theory with G = SU(3) × SU(2) × U(1). The 12 gauge

bosons in this model consist of a massless photon, 8 gluons, and 3 massive

W and Z bosons which carry the weak nuclear force. The electromagnetic

interaction is long ranged as the photon is massles, but both nuclear forces

are short ranged. In the case of the weak interaction this is due to the corre-

sponding gauge bosons being massive. In the case of strong interaction there

is a still poorly understood mechanism, called conﬁnement, which implies that

the forces between quarks increase when the quarks become separated. This

effectively limits the range of strong interactions to 10

−15

meters. See [168] for

a very good presentation of QCD and other gauge theories in the context of

particle physics.

It is fair to say that the concept of gauge symmetry gave rise to the greatest

revolution in physics in the second half of the twentieth century. It has lead to

several Nobel Prizes awarded for theoretical work: In 1979 to Glashow, Salam,

and Weinberg for their work on gauge theory of electroweak interactions done

in the 1960s. In 1999 to t’Hooft and Veltman for their work on renormaliz-

ability of quantum gauge theories done in the early 1970s. In 2004 to Gross,

Politzer, and Wilczek for their work on asymptotic freedom done in the early

1970s. In 2008 to Nambu for his discovery in the 1960s of the mechanism

of spontaneously broken symmetry in particle physics and to Kobayashi and

Maskawa for their work on CP violation done in the 1970s. More prizes are

likely to follow if the LHC discovers the Higgs particle and other forms of

matter (see the footnote on page 26).

The pure mathematical studies of gauge theory initiated by the Oxford

school of Atiyah led to advances in differential geometry and eventually to

solutions of several long-standing problems in topology of lower dimensional

manifolds [39, 40]. The twistor techniques, proposed by Penrose [129] and

used by Ward [169] to solve the anti-self-dual sector of the gauge ﬁeld equa-

tions, proved to be a universal language for most lower dimensional integrable

systems describing solitons. The gauge theory lead to Fields medals which

carry the weight of Nobel Prize in mathematics: In 1986 to Donaldson for

his gauge-inspired work on topology of four manifolds. In 1990 to Witten (the

ﬁrst physicist to be awarded the medal) for his work on mathematical aspects

of quantum gauge theories. In 1998 to Kontsevich for a rigorous formulation

of the Feynman integral in topological ﬁeld theories.

6.1 Gauge potential and Higgs ﬁeld

In this section we consider gauge theory in (D + 1)-dimensional Minkowski

space M, ...with a preferred volume from. Let the gauge potential A = A

6.1 Gauge potential and Higgs ﬁeld 107

be a one form with values in the Lie algebra g of some Lie group G and let

D = d + A be the covariant derivative. Let

F =

µν

∧ dx

= dA+ A ∧ A

be the gauge ﬁeld of A. Its components are given by

µν

= ∂

− ∂

+[A

, A

]=[D

, D

Gauge transformations identify (A, A



) and (F, F



), where



= gAg

−1

− dgg

−1

, F



= gFg

−1

, and g = g(x

) ∈ G. (6.1.1)

The ﬁeld satisﬁes the Bianchi identity

DF = dF +[A, F ]=d

A + dA∧ A − A ∧ dA (6.1.2)

+ A∧ dA+ A

− dA∧ A − A

=0.

In addition to the gauge potential we also introduce a Higgs ﬁeld  : R

D+1

→ g

in the adjoint representation, that is, D = d +[A,]. Its gauge transforma-

tion is





= gg

−1

. (6.1.3)

We should note that the standard model of elementary particles uses a different

set-up for the Higgs ﬁeld. There one regards  as a multiplet with complex

scalar ﬁelds transforming in the fundamental representation of G. In particular

in the electroweak theory one takes G = U(2), and the Higgs ﬁeld is a complex

doublet. This theory does not admit solitons. We shall see that choosing the

adjoint representation (6.1.3) will allow solitons in the form of non-abelian

monopoles. Solitons also play a role in supersymmetric gauge theories which

we do not discuss. The reader should consult [41], [162].

Notation

Let ∗ : 

→ 

D+1−p

be the linear map deﬁned by

∗(dx

∧···∧dx



|det(η)|

(D +1− p)!

···µ

p+1

···µ

D+1

p+1

∧···∧dx

D+1

In this chapter we shall use the letters a, b, c,... to denote the Lie algebra

indices. If G = SU(2) we choose a basis T

, a =1, 2, 3 for the Lie algebra of

SU(2) (anti-Hermitian, traceless 2 × 2 matrices) such that

, T

]=−ε

abc

, T

iσ

, and Tr(T

)=−

Note the sign difference between the inhomogeneous terms in (6.1.1) and (3.3.13). This is

consistent with A = −Udρ − Vdτ on R

108 6: Gauge ﬁeld theory

Here σ

are the Pauli matrices, and a general group element is g = exp(α

)

with α

real. The components of D and F with respect to this basis are given

)

= ∂



− ε

abc



and F

µν

= ∂

− ∂

− ε

abc

. (6.1.4)

If D + 1 = 4, the dual of the ﬁeld tensor is given by

(∗F )

µν

=(1/2)ε

µναβ

αβ

. (6.1.5)

A two form F =(1/2)F

µν

∧ dx

is called self-dual (SD) or anti-self-dual

(ASD) if ∗F = F or ∗F = −F , respectively.

In terms of differential forms

−Tr( F ∧∗F )=−

Tr( F

µν

x =

µν

µν a

where d

x =

µναβ

∧ dx

, and we have used the identities

µναβ

x = −dx

∧ dx

and ε

αβρσ

µνρσ

= −4δ

[µ

[α

ν]

β]

6.1.1 Scaling argument

Let us now examine how Derrick’s scaling argument used in Section 5.3.2

applies to gauge ﬁelds. Consider the ﬁelds (A,) given by a potential one-form

and a scalar Higgs ﬁeld, respectively, with energy functional

E =



x[|F |

+ |D|

+ U()] = E

+ E

D

+ E

The terms like |F |

here are positive deﬁnite and correspond to energy in the

D + 1 splitting, that is,

F =

∧ dx

+ E

∧ dt and |F |

∼−Tr( B

+ E

) = −Tr( F

µν

(Note that the trace is negative deﬁnite for su(n).) We are interested in static,

ﬁnite-energy critical points of this functional. Let A(x),(x) be such a critical

point, and let



(c)

(x)=(cx), A

(c)

(x)=cA(cx), F

(c)

(x)=c

F (cx), and

(c)



(c)

= cD(cx).

This leads to

(c)

D−4

D−2

D

and dE

(c)

/dc|

c=1

= 0 yields

(D − 4)E

+(D − 2)E

D

+ DE

=0. (6.1.6)

6.1 Gauge potential and Higgs ﬁeld 109

Therefore E

(c)

can be stationary provided that 0 ≤ D ≤ 4. Derrick’s scaling

argument can only rule out some dimensions and conﬁgurations. It is not

necessarily true that topological solitons exists if (6.1.6) holds. The following

solutions can nevertheless be shown to exist:

D =1. Gauged kinks generalizing solution (5.3.18).

D =2. Vortices with E

= E

in the Ginsburg–Landau model. Consult the

monograph [114] for detailed discussion of these solitons.

D =3. Non-abelian monopoles with E

= E

D

D =4. Solutions are possible with E

D

= E

= 0. These are instantons in

pure gauge theory.

6.1.2 Principal bundles

The mathematical formalism behind gauge theory is that of a connection on

a principal bundle π : P → M with a structure group G. This section is not

meant to be an introduction to this formalism – the deﬁnitions and proofs can

be found in [43, 62, 94] – but is included for more mathematically inclined

readers who want to place gauge theory in a geometric context. Other readers

can skip it.

Let ω be a connection one-form with values in g whose vertical component

is the Maurer–Cartan one-form, and let  be its curvature. In local coordinates

ω = γ

−1

Aγ + γ

−1

dγ and  = dω + ω ∧ ω = γ

−1

F γ, (6.1.7)

where (A, F ) are the gauge potential and gauge ﬁeld on M and γ : M → G

takes values in the gauge group. The G-valued transition functions act on the

ﬁbres by left multiplication. If U and U



are two overlapping open sets in M,

and g



= g is the transition function, then the local ﬁbre coordinates γ and γ



are related by γ



= gγ . The connection and the curvature will be well deﬁned

in the overlap region if

−1

Aγ + γ

−1

dγ = γ

−1



+ γ

−1

dγ



and γ

−1

F γ = γ

−1



These relations hold if A, A



, F, F



are related by the gauge transformations

(6.1.1).

Any section γ = γ (x) can be used to pull back ω and  to the base space,

so that the pulled-back connection A = γ

∗

(ω) is the gauge potential and the

pulled-back curvature F = γ

∗

() is the gauge ﬁeld. The gauge transformations

(6.1.1) correspond to changes of the section. If the bundle is non-trivial (e.g.

the Dirac monopole), the global section does not exist and the gauge potentials

can be only deﬁned locally.

110 6: Gauge ﬁeld theory

Given ω we can deﬁne the splitting of TP into the horizontal and vertical

components:

TP = H(P) ⊕ V(P),

where the horizontal vectors belong to the kernel of ω, that is, H(P) consists

of vector ﬁelds X on P such that X

ω = 0. The basis of the distribution

V(P) is given by left-invariant vector ﬁelds, and the horizontal distribution is

spanned by

∂

∂x

− A

(x)R

, a =1,...,dim g

where R

are the right invariant vector ﬁelds on G such that

, R

]=− f

The curvature is the obstruction to the integrability of the horizontal distribu-

tion as

, D

]=−F

µν

The negative sign on the RHS is consistent with

F =

µν

∧ dx

and A = A

where [T

, T

]= f

6.2 Dirac monopole and ﬂux quantization

Consider Maxwell electrodynamics as a U(1)-gauge theory on R

3,1

with a ﬁeld

given by

F = E

∧ dt +

ijk

∧ dx

The Maxwell equations with a source one-form J = J

are

d ∗ F = ∗J and dF =0.

If there is no source the duality F →∗F corresponds to the symmetry between

the electric and magnetic ﬁelds E and B.

The lack of magnetic charges is a consequence of the contractibility of the

space and follows from the Bianchi identity. Allowing the U(1) bundle to be

deﬁned on the complement of a point will allow magnetic charges. Introduce

two coordinate patches U

and U

−

in R

−{0} covering the regions z > −ε

6.2 Dirac monopole and ﬂux quantization 111

and z <ε, respectively. Topologically R

−{0} is S

× R, and we can think of

as a two-patch covering of the sphere. Consider the gauge potentials which

are regular in the overlap region:

4πr

z ± r

(xdy − ydx)=

4π

(±1 − cos θ )dφ, g = const.

They are related by a gauge transformation

= A

−

2π

d tan

−1

(y/x)=A

−

2π

dφ,

where

x = r sin θ cos φ, y = r sin θ sin φ, and z = r cos θ, 0 ≤ θ<π,0 ≤ φ<2π.

The ﬁeld is given by F = dA

in U

,so

F =

4πr

(xdy ∧ dz + ydz ∧ dx + zdx ∧ dy)=

4π

sin θ dθ ∧ dφ,

E = 0 and B =

4πr

. (6.2.8)

The two-form F is closed but not exact.

We interpret this solution as a magnetic monopole, because it is analogous

to the electric ﬁeld of a point charge q given by

E =

4πr

and B =0

(with A = −qdt/r). In Dirac’s approach [38] the coordinate patches were not

used which led to the appearance of non-physical ‘string singularities’ along the

z-axis (at θ = π or θ = 0). A gauge transformation moves the singular string to

any other half-axis which starts at r = 0 and ends at ∞.

The magnetic ﬂux through a sphere around r = 0 is given by

Q =



F =



−



− A

−

2π



dφ =

2π

(increase of φ)

because the equator C has opposite orientations as the boundary in the two

regions U

Dirac has argued from the QM insight: The potential A couples to parti-

cles/ﬁelds of electric charge q via covariant derivative:

D = d − iqA,

112 6: Gauge ﬁeld theory

where  is a wave function, and 

= exp (iqgφ/2π )

−

so that

D

=(d −iqA

)



d −iqA

−

iqg

2π

dφ



exp



iqgφ

2π





−

= exp



iqgφ

2π



D

−

We require exp(igqφ/2π ) to be well deﬁned on C so that the gauge transfor-

mations make sense, and we obtain the Dirac quantization condition

gq =2π N, N ∈ Z, (6.2.9)

where g and q are any magnetic and electric charges of particles (there would

be a factor  on the RHS if we did not set it to 1). If we accept that all electric

charges are integer multiples of the electron charge e then magnetic charges g

must satisfy ge =2π N, and the minimal magnetic charge is g =2π/e. If there

existed just one magnetic monopole in the universe, we would understand

electric charge quantization. Each electric charge would be an integer multiple

of 2π/g.

The Dirac monopole is not a soliton because of its singularity at r = 0. Its

energy density decays like 1/r

, and therefore the monopole mass diverges

linearly. This divergence can be regularized, and leads to a large ﬁnite mass. So

far no magnetic monopoles have been detected experimentally – all magnets

seem to have two poles.

6.2.1 Hopf ﬁbration

From the the differential geometric perspective the monopole number clas-

siﬁes the principal U(1)-bundles over a two-sphere S

. The two-sphere is

homotopy equivalent to R

−{0} in the following sense: Let j : S

−→ R

the inclusion, and let p : R

−{0}−→S

be the projection x −→ x/|x|. Then

p ◦ j = Id and j ◦ p is homotopic to the identity map by

f (x, t)=tx −

(1 − t)x

|x|

Transition functions for U(1)-bundles over S

are continuous maps from

× R to S

× R and the bundles are classiﬁed by the degree of these maps

restricted to S

. The connection one-form (6.1.7) ω is given by

ω =



+ dψ

on U

−

+ dψ

−

on U

−

This is globally deﬁned on P which gives the transition relations e

iψ

−

= ge

iψ

The bundle will be a manifold if the transition function is of the form e

(iNφ)

for

6.2 Dirac monopole and ﬂux quantization 113

N ∈ Z. This gives a bundle with the ﬁrst Chern number −N given by

= −

2π



as in general C

= i/2π, and the curvature is purely imaginary for the U(1)

principal bundle, that is,  = iF (see Section 6.4.1 for the deﬁnition of Chern

numbers). This number (which is the negative of the monopole number) only

depends on the transition function, and not on the fact that A satisﬁes the

Maxwell equations.

The case N = 0 corresponds to a trivial bundle, and N = −1 gives the Hopf

bundle S

→ S

. We shall present a description of the Hopf bundle in terms

of complex numbers. The total space of the bundle is P = S

⊂ C

which is

explicitly given by

+ Z

=1, where (Z

, Z

) ∈ C

Any complex line A

+ A

= 0 through the origin in C

intersects the

three-sphere in a circle S

. Each circle is a one-dimensional ﬁbre over a point

, A

) in the space of complex lines in C

. To specify such a point one only

needs a ratio of complex numbers A

and A

which is allowed to be inﬁnite.

The space of these ratios is a Riemann sphere CP

(see Appendix B). As a real

manifold it is diffeomorphic to S

, with the explicit map given by (5.4.30). The

projection π : S

→ S

, Z

) −→

in a patch of S

where Z

= 0. The bundle has N = −1 and therefore is not dif-

feomorphic to S

× S

. To see that N = −1setλ = Z

= tan (θ/2) exp (iφ)

and

λ = Z

, so that the Riemannian metric on the sphere is

|dλ|

(1 + |λ|

)

= dθ

+sin

θdφ

Parameterize the three-sphere by

= cos (θ/2)e

iξ

and Z

= sin (θ/2)e

iξ

where (ξ

,ξ

,θ) ∈ R

× S

. The projection is

π(ξ

,ξ

,θ)=



λ +

1+|λ|

λ −

i(1 + |λ|

)

−1+|λ|

1+|λ|



=(sinθ cos φ,sin θ sin φ,cos θ ),

where φ = ξ

− ξ

. We shall read off the patching function from two local

sections

s(λ)=

(1,λ)



1+|λ|

and

λ)=

(

λ, 1)



1+|

λ|

114 6: Gauge ﬁeld theory

where (Z

, Z



+ |Z

is a unit vector in C

corresponding to a point

in S

. Now

λ)=

λ(1,λ)



1+|

λ|



|λ|



−1

s(λ)

so the transition function is e

−iφ

, and N = −1. In Appendix B the Chern num-

ber N has a holomorphic interpretation as the Hopf bundle is the restriction

of the tautological line bundle O(−1) over CP

to real ﬁbres.

The family of S

’s in the Hopf bundle consists of circles (sometimes called

the Clifford parallels) which twists around each other with linking number

equal to one. This can be visualized by a stereographic projection of S

to R

where the ﬁbres of the Hopf bundle map to circles in the Euclidean three-space.

This picture has motivated the early development of twistor theory [132].

An alternative description of the Hopf bundle is as the principal ﬁbre bundle

with a total space SU(2), and the ﬁbration

SU(2) → SU(2)/U(1) = S

where the U(1) is identiﬁed with a subgroup of SU(2) consisting of diagonal

matrices. The right action of S

on SU(2) is (Z

, Z

iα

=(Z

iα

, Z

iα

), where

+ |Z

= 1. This action ﬁxes the ratio Z

∈ CP

= S

6.3 Non-abelian monopoles

Choose the gauge group G = SU(2) and consider the Lagrangian density

L = −

µν

µνa



− U() (6.3.10)

which is gauge invariant if U() is gauge invariant. Choose

U()=

c(||

− v

)

, where ||

= 



,v∈ R

and c is a constant.

The ﬁniteness of the energy is assured by

||−→v, D

 −→ 0, and F

µν

−→ 0, as r →∞. (6.3.11)

The ﬁeld equations in components relative to the standard basis T

can be

derived using (6.1.4):

µν

)

= −ε

abc



)

and (D

)

= −c(||

− v

)

(6.3.12)

6.3 Non-abelian monopoles 115

One solution is A =0, = 

= const, with |

| = v. This ground state is

not unique if v = 0, and as a consequence the gauge group is spontaneously

broken to U(1). This is because we need 

= g

−1

to preserve the ground

state.

The reasons for introducing the Higgs ﬁeld have to do with a description

of short-range interactions in gauge theory carried by massive bosons. Pure

Yang–Mills (YM) ﬁelds describe massless particles, with a number of gluons

given by the dimension of the gauge group. Adding a mass term of the

form m

Q(A) which is quadratic in A to the Lagrangian would necessarily

break the gauge invariance. A way around this is to include an additional

ﬁeld  such that  −→ 

with 

= const at spatial inﬁnity. Strictly speak-

ing the minimum of the potential is not unique, but can be transformed

to 

by a gauge transformation. Then D

= A

, and a gauge-invariant

term |D|

induces a term quadratic in A with the mass determined by the

constant 

6.3.1 Topology of monopoles

In this section we shall consider a soliton – static ﬁnite-energy solution to

(6.3.12). We shall chose a gauge A

=0,sothatD

( f ) = 0 where f is any

Lie algebra–valued ﬁeld which does not depend on time. For notational con-

venience we shall set v = 1, which is always possible if v = 0 by a rescaling of

. Let

 =



||

The Higgs ﬁeld at inﬁnity carries all the topological information. The bound-

ary conditions imply that



∞

= lim

r→∞



deﬁnes a map from S

∞

(the two-sphere at the spatial inﬁnity) to the unit two-

sphere in the Lie algebra (as || = v asymptotically to ensure ﬁnite energy).

This ﬁeld is topologically classiﬁed by its degree N (see formula (A7)) related

to the outward magnetic ﬂux at inﬁnity.

The non-abelian monopoles are different from the Dirac monopole in

that the ﬁelds are smooth everywhere in R

, but there is a connection: the

gauge group SU(2) is broken down to U(1) at inﬁnity in R

by the ﬁeld

. A non-abelian monopole looks like a Dirac N-pole when viewed from

a distance. The only difference is that now we have two long-range ﬁelds

 and F unlike the one ﬁeld F (6.2.8) for the Dirac monopole. For short