Instantons and monopoles in Yang-Mills gauge field theories

Physica ID (1980) 167-191 © North-Holland Publishing Company

I N S T A N T O N S AND M O N O P O L E S IN Y A N G - M I L L S G A U G E F I E L D T H E O R I E S

M. K. PRASAD Institute for Theoretical Physics, State University of New York, Stony Brook, Long Island, N.Y. 11794, USA

Received 14 January 1980

This is a survey article on instantons and monopoles and is intended for those who have no prior knowledge of Yang-Miils gauge field theories. With a minimal amount of physical motivation and mathematical apparatus, the basic field equations and their solutions, wherever known, are presented. Particular emphasis is put on those problems which are as yet unsolved.

Table of contents I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 2. Mathematical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 3. Yang-Mills gauge field theories: basic concepts and formulas . . . . . . . . . . . . . . . . . 170 4. Precise definition of instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5. Precise definition of monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6. Explicit instanton solutions: the complete solution . . . . . . . . . . . . . . . . . . . . . 177

6.1. 't Hooft's ~ and ~ tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 6.2. The original Belavin-Polyakov-Schwartz-Tyupkin (BPST) q = 1 instanton solution . . . . . . . 179 6.3. The Corrigan-Fairlie-'t Hooft-Wilczek ( c F r w ) ansatz and 't Hooft's solution . . . . . . . . . 179 6.4 The Atiyah-Drinfield-Hitchin-Manin (ADHM) construction . . . . . . . . . . . . . . . . • 181

7. Explicit monopole q = 1 solution: the only known solution . . . . . . . . . . . . . . . . . . 182 8. Bficldund transformations for self-dual Yaug-Mills gauge fields . . . . . . . . . . . . . . . . 183

8.1. Yang's formulation of self-dual gauge fields . . . . . . . . . . . . . . . . . . . . 184 8.2. The Atiyah--Ward (AW) Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . 185 8.3. A Manifestly gauge invariant Backlund transformation . . . . . . . . . . . . . . . . . 187

9. Non-local conservation laws for self-dual Yaug-Mills gauge fields . . . . . . . . . . . . . . . 188 10. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

1. I n t r o d u c t i o n

T h e r e is g r a d u a l l y f o r m i n g a d o g m a tha t the f o u r f u n d a m e n t a l i n t e r a c t i o n s in p h y s i c s (g rav i t a t ion ,

e l e c t r o m a g n e t i s m , w e a k , s t rong) a r e due to g a u g e f ields . T h e s c e n a r i o is a s fo l l ows :

1) Electromagnet ism-the b e s t u n d e r s t o o d o f all p h y s i c a l i n t e r a c t i o n s is a g a u g e t h e o r y tha t

d e s c r i b e s the i n t e r a c t i o n b e t w e e n the g u a g e pa r t i c l e s , ca l l ed p h o t o n s (light~), a n d e l ec t r i ca l l y c h a r g e d

pa r t i c l e s such as the e l ec t ron . W h e n i m p l e m e n t e d wi th the p r inc ip l e s o f q u a n t u m m e c h a n i c s t he t h e o r y ,

r e f e r r e d to as Q E D ( q u a n t u m e l e c t r o d y n a m i c s ) , has s p e c t a c u l a r s u c c e s s in exp la in ing e x p e r i m e n t a l

o b s e r v a t i o n . A s a c a s e in po in t , f o r the va lue o f the m a g n e t i c m o m e n t o f the e l ec t ron , t h e o r y and

e x p e r i m e n t a g r e e to n ine d e c i m a l p l aces !

2) Weak in terac t ions - the 1979 N o b e l p r i ze fo r p h y s i c s w a s a w a r d e d to G l a s h o w , Sa l am, a n d

W e i n b e r g fo r the i r w o r k on un i fy ing e l e c t r o m a g n e t i c and w e a k i n t e r a c t i o n s b y m e a n s o f a gauge

t h e o r y . T h e w e a k i n t e r a c t i o n s pa r t o f th is t h e o r y d e s c r i b e s i n t e r a c t i o n b e t w e e n the y e t u n o b s e r v e d

g a u g e pa r t i c l e s , ca l l ed i n t e r m e d i a t e b o s o n s , a n d the k n o w n pa r t i c l e s , in p a r t i c u l a r the neu t r ino .

T h o u g h no t on as s o u n d a f o o t i n g as e l e c t r o m a g n e t i s m , th is t h e o r y is hav ing r e m a r k a b l e s u c c e s s in

c o r r e l a t i n g e x p e r i m e n t a l d a t a - i n p a r t i c u l a r it p r e d i c t e d the e x i s t e n c e o f " c h a r m e d " pa r t i c l e s w h i c h

167

168 M.K. Prasad/ lnstantons and monopoles in Yang-Mills gauge field theories

was eventually discovered by Richter and Ting (and their collaborators) and for which they won the 1976 Nobel prize for physics.

3) Strong interact ions - is a gauge theory that describes the interaction between the gauge particles, called "gluons," and strongly interacting particles called hadrons (such as protons and neutrons) which are believed to be made up of (yet unobserved!) "quarks." When implemented with the principles of quantum mechanics the theory, referred to as QCD (quantum chromodynamics), explains why the proton, when probed to very short distances, behaves as if it is made up of pointlike objects, dubbed partons. The outstanding problem in QCD is to explain long distance phenomena, in particular why we do not see quarks and gluons as physical ob jec t s - the so called problem of "quark confinement." The New York Times (August 31, 1979) reported on experiments that support the quark-gluon picture of QCD.

4) G r a v i t a t i o n - the oldest known of the four interactions is universally accepted to be a gauge theory though the precise details have yet to be clarified. Ironically, gravitation has up to now resisted implementation of quantum mechanics; quantizing gravity is one of the most challenging problems in the theoretical physics of today. On a grander scale there is now great impetus in unifying by means of gauge theory, all the four fundamental interactions- the so called Super-gravity theory.

It is therefore evident that the concept of gauge fields is deeply rooted in physical phenomena. It was therefore very suprising, to physicists, that the concept of gauge fields is found to be identical to a geometrical concept called fiber bundles which has been developed by the mathematicians, entirely independent of any relationship to physical reality.

Not only is it true that gauge field is a geometrical concept, but it turns out that topological complexity is important for gauge fields. Appreciation of this point came very recently (in the last five years!) with the 't Hooft-Polyakov magnetic monopole solution and the instanton solution of Belavin, Polyakov, Schwartz and Tyupkin.

The physical implication of the magnetic monopole is quite simple: they may exist as physical objects, as Dirac speculated many years ago, and would be stable against decay, extended (not pointlike), and very massive. Magnetic monopole solution can also have indirect physical implications by their special role in theoretical models such as QCD, but such implications are yet to be clarified.

The physical implication of the instanton solution is much more subtle, in particular they are very important in QCD (some people think instantons can even solve the quark confinement problem!). Presently in the physics community there is enormous activity in this direction.

The pioneering work of Yang-Mills marks a cornerstone in theoretical physics, wherein the concept of gauge fields is established by purely physical motivation. To understand the physics of gauge fields, we urge the reader to study the original Yang-Mills paper, as no review can do justice to its clarity of thought. The purpose of this survey article is to describe to non-specialists, essential aspects of instanton and monopole solutions in Yang-Mills gauge theory. Our survey will be in a logical rather than historical order, since the latter can be quite confusing, especially in regard to the magnetic monopole solutions.

2. Mathemltical preliminaries

The purpose of this section is to establish notational conventions and present mathematical formulas that will be useful for our calculation purposes.

Consider an M dimensional space with coordinates x~ = (xl, x2 . . . . . xu) (the subscript/~ takes on the values I, 2 . . . . . M). The metric of this space is taken to be 8~, where

M.K. Prasadl Instantons and monopoles in Yang-Mills gauge field theories 169

+1 ff~=u Kronecker delta 8,, --- (2. I)

0 i f g ~ v

so that there is no distinction be tween contravar iant and covar iant indices, x~, ---x ' . In particular for any two M-vec to r s a~ and b , their scalar product is

a , b , = a ,b~8,~ = a l b l + a2b2 +" • • + aMbM, (2.2)

where we have used, as we a lways will, the Einstein summat ion convention: whenever an index appears twice in an expression, it is to be summed over its range of values.

The comple te ly ant isymmetr ic constant tensor e,..a2...,M in M dimensional space is defined by

t + l i f / ~ , / t 2 . . . . . /zM is an even permuta t ion of 123. . .M

e,~.~,2 ..... ,u -- - 1 if/~1, ~2 . . . . . /~u is an odd permutat ion of 123. . .M

0 otherwise. (2.3)

T h e e tensor satisfies the following identity:

[ 8~i.~8,2.~," • .8~M., ,

"8~M "2 (2.4)

8~, ,uS~ ~u " "8,u,u

where I" " "I designates determinant. The determinant of a M x M matrix A,v is therefore

l det A,, - ~ e,l.,2...,uev,.~...,MA~,I.,,IA,2.~...A,,u.,, u. (2.5)

The trace of a matrix A,, is

Tr A~, - A,,,. (2.6)

A matrix is traceless if its trace is zero. For any two matrices A and B, one has:

Tr AB = Tr BA and det(AB) = (det A)(det B). (2.7)

The Hermitian conjugate and inverse of a matrix A will be designated by A + and A -~, respectively. A matrix L is Hermitian if L + = L; it is anti-Hermitian if L + = -L and it is unitary if L + = L -~.

Differentiation with respect to x~ will be denoted by 8, so that

0) 0, - 0x, ' ~x2 . . . . . ~XM " (2.8)

For matr ices which depend on x , one has the identity

ffi 1 Tr(A-~8,A) - ~ 0~, det A - ~ In det A.

For any two matr ices A and B we define their commuta to r to be

[A, B] -- A B - B A .

A simple commuta to r identity is

(2.9)

(2.10)

170 M.K. Prasad/ Instantons and monopoles in Yang-Milis gauge field theories

[I0~, A] -(0~A), (2.11)

where I is the identity matrix 8~,. To prove (2.11) we note [I0~,, A]B = O, , (AB)- AO,,B = (O~,A)B. Gauss' (or Divergence) theorem for integrals in M dimensional space will play a crucial role in our

understanding of the topology of gauge fields. Let R n be a region of M dimensional space and Rn-i the ( M - 1) dimensional subspace that forms the boundary of Rn. One can label the points of M dimensional space by M parameters Eo, a = 1, 2 . . . . . M. The coordinates in this space are given parametrically by the equations

x~, = x~,(EI, E2 . . . . . EM), t~ = 1,2 . . . . . M. (2.12)

Let us form the element of "area"

~x~, Ox~ Ox~M d'6,,.~,~ ..... J,s¢ = e,,,.,~ ...... M OE,,~ OE~"" ÜE,,M (dME)' (2.13)

where (dUE) - dE1dE2 • • • den.

Gauss' theorem can now be stated as

f 0,J~,(d'x)-- f J~,(dM-'or,), (2.14) Ru Ru-l

where

(dMx) --~ dxl dx2. . , dxu,

(dSC-lcr~) -- (M 1 - I)! e~,,~ ..... ~u d%.,,s ..... ~" (2.15)

it will be useful to remember that the volume elements in M = 3 and M = 4 spaces are

(d3x) -- dxl dx2 dx3 - r 2 dr(d~n2), where r - (x~ + x] + x]) I/~ I> O,

(d4x) -- dxl dx2 dxs dx4 - R 3 dR(dS~23), where R ~ (x~ + x] + x~ + x~) u2 >~ O. (2.16)

and the solid angles in M = 3 and M = 4 spaces are

n2-f(eno=4 - and n3.f(es23)=2 (2.17)

3. Yang-Mills gauge field theories: basic concepts and formulas

Lie algebras: A representation of Lie algebra is a set of N anti-Hermitian traceless matrices, T °, a = 1, 2 . . . . . N, obeying the equations

[T °, T b] = f~bCTC, (3.1)

where the f ' s are the (real) structure constants of some compact Lie group G. It is always possible to choose the T's so that Tr(T°T b) is proportional to 8 ~, although the constant of proportionality may depend on the representation. The Cartan inner product is defined by

(T °, T s) =" 8 "b (3.2)

and thus is proportional to the trace of the product of the matrices T ° and T s.

M.K. Prasadl lnstantons and monopoles in Yang-Mills gauge yield theories 171

Gauge fields: The basic objects in gauge theory are the Yang-Mills gauge potentials. The gauge potentials are a set of vector fields AT,(x) (where a = 1, 2 . . . . . N and tt -- 1,2, 3, 4). It is convenient to define a matrix valued vector field A~(x) by

A~, =-- gT°A~,, (3.3)

where g is a constant called the gauge coupling constant. From the matrix valued gauge potentials A~ one constructs the matrix valued gauge field strength, F,~(x), by

F ~ -- 0~A~ - Ü~A~ + [A~, A~] = [I0,~ + A~, I~ + A~]. (3.4)

Both A~ and F ~ are antihermitian traceless matrices. In explicit component form, F~ =-gT°F~ where

= - g f A~ , A , , . (3.5)

Both the matrix valued and the explicit component forms for the gauge potential and field strength will be useful for us.

We will use the expression "static guage fields" to refer to gauge potentials that are independent of

X4.

STATIC GAUGE FIELDS: Ü,A~(x) - 0 (/~ = 1, 2, 3, 4). (3.6)

The word "s ta t ic" is used because one can think of x4 as a " t ime" coordinate and xl, x2, xa as the "space" coordinates of the full four-dimensional space (x~, x2, x3, x4).

Action and energy functionals: x~ = (xl, x2, x3, x4) we define the action functional by

S-4~ f (d'x)(F~,F~.)=~, f (d'x) F~F*~>~O. For static guage fields that are independent of x4 we define the energy functional by

E ----'~g f (d'x)(F~,., F~,,)= 4!f (d'x)FT~.F:~ >t O.

For gauge potentials that depend on all four Euclidean coordinates

(3.7)

(3.8)

Gauge transformation: Both the action S and energy E functionals are invariant under the following transformations, to be referred to as gauge transformations

u u A~, • } U- IA~ ,U + U-lcg~,U, F ~ ~ U-IF~,U, (3 .9)

where U = e A'~x)r" with A'(x) being N arbitrary real functions. If F ~ vanishes, then A~ is a guage transform of zero - "pure gauge", that is to say

PURE GAUGE F,,~ = 0=~A~ = U-~O~,U (3.10)

for some U(x). Under the gauge transformation (3.9) the object OAF~ does not transform simply whereas the object

OAF~ + [AA, F~,]--, U-~{O~F~ + [AA, F~,]}U does. One therefore defines a "covariant" derivative

172 M.K. Prasadl lnstantons and monopoles in Yang-Miils gauge field theories

D~- a. + A.

so that [D~, F~,] behaves simply under gauge transformations.

(3.11)

Yang-Mills equations of motion: The physics of gauge fields is dictated by the action S and energy E functionals. When the principles of quantum mechanics are implemented in gauge theory the resulting problem seems impossible to solve exactly, and one therefore resorts to various approximation schemes. In a very important approximation scheme, called the semiclassical approximation, one finds that the important physical configurations are determined by local minima of the action S and energy E. Thus, in the context of the semiclassical approximation, one is led to study gauge field configurations that locally minimize S and E.

The extrema (not necessarily minima!) of the action S and energy E are found by standard calculus of variation techniques leading to the following Euler-Lagrange equations

O~,F~,,, + [A~,, F~,,,] = [D~,, F~,,,] = 0 (3.12)

or in explicit component form

~F~, + af abcA b l~'c = 0. (3.13)

In gauge theory the equations (3.12) and (3.13) are referred to as the Yang-MilIs equations of motion. They are coupled nonlinear partial differential equations for the gauge potential A~, and it seems unlikely that one can solve them exactly. Even if one could find solutions to (3.12), one would have to verify that they were indeed local minima of S or E and not local maxima in the functional space - a problem which in itself is quite difficult.

lnstantons and monopoles: In the last five years there has emerged a remarkable discovery which'is:

For given boundary conditions, specified by a topological charge that takes on integer values, instantons and momopoles are absolute minima of the action S and energy E functional, respectively.

The above statement implicitly assumes that S and E are finite. The topological charge q(=O, 1, 2 . . . . ) groups the gauge potential into classes by their behaviour at infinity (in Euclidean space) in such a manner that no continuous deformation of the gauge potential can change q; in mathematics such classes are known as homotopy classes.

Bianchi identity: Consider the Jacobi identity for the covariant derivative D~ = 0~ + A~

[D~, [D~,, D~]] + [D~, [D~, D~]] + [D~, [DA, D~,]] - 0 . (3.14)

Now the gauge field strength F~,~ can be written as

F~, = 0~A, - 0,A, + [A~, A,] = [D~, D,]. (3.15)

Multiplying the Jacobi identity (3.14) by the completely antisymmetric tensor e~,,p gives

[D~, *F~,] - 0, where * _ i F~,, -- 2~,,A,F~,. (3.16)

The tensor *F~,, is the dual of the gauge field strength tensor F~. In gauge theory the identity (3.16) is known as the Bianchi identity.

M.K. Prasad/ lnstantons and monopoles in Yang-Miils gauge field theories 173

Self-duality: Comparing (3.16) to the Yang-Mills equation of motion (3.12) we see that any gauge field strength which is self dual

F~,~ --- *F~,~ (3.17)

automatically satisfies eq. (3.12). Eq. (3.17) is now a first order coupled non-linear partial differential equation for the gauge potential A~,, and therefore much simpler in comparison to eq. (3.12), which is second order. We will see that both instantons and monopoles are solutions to eq. (3.17).

Specification to SU(2) gauge group: Until now, we have not (for the sake of generality) specified what the compact Lie group G is. However, in order to avoid unnecessary complications, from now on we will consider specifically the simplest Lie group, namely SU(2). The essential features of instantons and monopoles are best exemplified in the SU(2) gauge theory. Of course, one can systematically generalize the SU(2) gauge theory results to any compact Lie group G gauge theory.

For SU(2) gauge group the structure constants f°~ are the three dimensional completely antisymmetric constant tensor eabc

SU(2) group: f~b~ = e ~ (a, b, c = 1, 2, 3). (3.18)

We will use a 2 x 2 matrix representation for the anti-Hermitian traceless matrices T °

O -a T~= 2-]" a = l ' 2 ' 3 ' i - V ' L - l " (3.19)

where tr°are the Pauli matrices

The Pauli matrices satisfy the following equation

o'ao'b = ~ab + i e ~ o "c.

The Cartan inner product for the representation (3.19), is

( T a, T b) = ~ a b = - 2 T r TaT b.

(3.20)

(3.21)

(3.22)

4. Precise definit ion of ins tantons

We begin by noting the trivial inequality

1 f (d4x) ½{-Tr(F~,~ - * F ~ ) ( F ~ - *F,,~)} I> 0

which implies

1 4 1 f (d x){-Tr F~..} ~> ~-~ f (d'x){-Tr F~,.*F~,.}.

(4.1)

(4.2)

since *F,,~*F~ = F ~ , , . The inequality signs in eqs. (4.1) and (4.2) become equality signs only for self-dual gauge fields F ~ = *F~,~.


Using the ant isymmetry of e ~ o and the trace property T r A B = TrBA one can prove the following identity

- T r F~*F~v =- OJ~, where J~ - - e ~ B Tr{A,F~ - ]A,A~Ao}. (4.3)

The identity (4.3) allows us to convert the integral on the right hand side of inequality (4.2) into a surface integral by means of Gauss ' theorem

"~1 f(d4x){-TrF.~*F.v}= 2-~1 R~lim s~ f (d3tT~)J~' (4.4)

where the integral on the right hand side of eq. (4.4) runs over the sphere

$~: x~,x~, = x 2 + x 2 + xl + x 2 = R 2. (4.5)

In using Gauss ' theorem, we have implicitly assumed that S is finite, which in turn implies

as R ~oo: F ~ = 0 " P U R E G A U G E " ~ Ag = U-tO~,U, (4.6)

where we have used eq. (3.10) and U is an arbitrary 2 x 2 unitary matrix with determinant one, so that

U = lV4(x, ,)- icr°V°(x, ,) with l, q4+ V ° V ~ = + 1 . (4.7)

Using eqs. (4.6) and (4.7) we therefore have

as R ~ : J~ = + ~ Tr{(U-~O,U)(U-~O~U)(U-~O~U)}, (4.8)

= + ~ . , , . . v,(o~v,)(o~ v~)(oavo. (4 .9)

The sphere {83: x~,x~, = x~+ x~+ x 2 + x~ = R 2} can be parameterized in terms of three parameters ~o(a = 1, 2, 3); x~, = x~,(~). Using eqs. (2.13) and (2.15) for M = 4, the surface element (d3~r~) becomes

(d3o.) _ 1 0x~ ea,~(d3~:)" (4.10) - 6~ ,~a 0 ~

Inserting eqs. (4.9) and (4.10) into eq. (4.4) gives

"~'~"~v'~-b-~'] \ 0 , ] (Oe ~ ) ( d O. (4.11)

It is easy to verify that the integrand of the integral on the right hand side of eqn. (4.11) satisfies the following identity

r /a_ v 12: roy° av°l Le'~'~e*~¢V'\a6 ° ] \ 06 / \06 / J 36 det L-~Z-T~r]

= 36 [ d e t e r m i n a n t [g.a] of the metric tensor g*a of the unit sphere

V*V ° = V~+ W V t + V~V~ + V3V ~= 1]. (4.12)

W e therefore have

4 87'i -2 ~ g f ( d ~ x ) { - T r F ~ , , * E , , } = ~ l i m ~ ( d ~ O V ~ =--~- q,

M.K. Prasad/ lnstantons and monopoles in Yang-MiUs gauge field theories 175

q - topological charge = 0, 1, 2 . . . . (4.13)

because while the point (6~, 62, 63) covers the spheres S~ once, the vector V ~ can cover the sphere V ~ V ~ = V~ + V ~ V ~ + V 2 V 2 + V 3 V 3 = 1 any q number of times, each time contributing a 4-dimensional solid angle f(d36) = 2¢r 2.

Thus, using eqs. (4.12) and (4.13), we establish the following inequality for action S

S ~ 83'/'2 ge q wi thq =0 , 1,2 . . . . . (4.14)

The lower bound equality sign is reached only by self-dual gauge fields F~v = *F~,~. The topological charge q in the context of instantons, is referred to as the instanton number. Eq. (4.14) shows that for any given q, self-dual gauge fields are absolute minima of the action $.

Whether the lower bound S = 8~r2/g2q can be actually realized is a dynamical, not a topological, problem and requires the construction of explicit solutions of the self-duality equations F~,~ = *F~,~ It has taken tremendous efforts from both physicists and mathematicians in order to construct explicit solutions of the self-duality equations F~,~ = *F~. The efforts began by pioneering work of Belavin, Polyakov, Schwartz and Tyupkin (BPST) for q = 1 and culminated in the work of Atiyah, Drinfield, Hitchin and Manin (ADHM) for arbitrary q. The ADHM construction was derived by use of some "state of the ar t" mathematics from differential and algebraic geome t ry - mathematics which is incomprehensible to most physicists. It is therefore ironic that the ADHM construction, while extremely difficult to derive, is extremely simple to p r e s e n t - a s we will in section 6.

5. Precise definition of monopoles

For monopoles, the gauge fields are static as defined by eq. (3.6). It is useful, in the context of monopoles, to work in explicit component forms for the gauge potential and field strength. If we introduce two vector fields B~ and E~ defined by (a, k = I, 2, 3)

abcAbAc E~ =" F~4 '= OkA~ + ~e ~'Ik./"14, (5. I)

B~ - l a ½e~m[OtA~, OmA~ obc b c - - 2 ~ k l m F l m = - - +ge AtAm], (5.2)

then the energy functional E becomes

f (d'x)F:vF:,,=½ f (d'x)[B~B~ + E[E~] (5.3) E = ~

= ½ f - - + f (d3x)B[E~, (5.4)

>>- f (d3x)B~E~" (5.5)

The inequality sign in eq. (5.5) becomes an equality sign only if B~ = E~ which is precisely the self-duality equation F~v = *F~v for static gauge fields.

The integrand in inequality eq. (5.5) can be reduced as follows

B~Ek a =- Ok(B~A~)- A~[O~B~ + ge~bcAbkB~] -- Ok(B~A~), (5.6)


where the second identity in eq. (5.6) follows from the static version of the Bianchi identity [eq. (3.16)].

The identity [eq. (5.6)] allows us to convert the integral in eq. (5.5) into a surface integral by means of Gauss ' theorem

f (d3x)B~E~ =l:m ~ (d2O'k)B~AL (5.7)

where the integral on the right hand side of eq. (5.7) now runs over the sphere

$2,: x~xk = x~ + x~ + x~ = r 2. (5.8)

In using Gauss ' theorem we have implicitly assumed E is finite, which in turn implies

as r ~ : F~a = 0 " P U R E G A U G E " ~ Ak = U-~OkU, (5.9) as r ~ oo: okm~ ~- ~ abcAbAc T ~ a"Ikrl. 4 : O. (5.10)

where we have used eq. (3.10) and U is an arbitrary 2 x 2 unitary matrix with determinant one. Multiplying eq. (5.10) by A~ and using antisymmetry of ~abc we find Ok(A~A~) = 0, so that

as r--,o0: A~ = -g 4~ , where = I, c = constant ~>0. (5.11)

The particular combination of constants (-c/g) in eq. (5.1 I), where g is the gauge coupling constant, is chosen for later convenience. Using eq. (5. II), eq. (5.10) can be solved for A~ to give

as r ~oo: A~, = _1 ,~%~b(Ùk~c) + C~,(A~c~b). (5.12) g

Substituting eq. (5.12) into (5.2) gives B~ as r ~ o o and using eq. (5.11) we find

as r--,oo: B~A~= ++e"b~ek,m~a(otdb)(o.d~) - c b ^~ ektmOt(A,,,ck ). (5.13)

The sphere {$2,: x~x~ = x] + x~ + x] = r 2} can be parameterized in terms of two parameters ~ ( a = 1,2): Xk = Xk(~,). Using eq. (2.13) and eq. (2.15) for M = 3, the surface element d2o-k becomes

(dZm) = ~E~, Ox" Ox" 0~" 0~ # ~'#(d2~)' (5.14)

so that eq. (5.7) becomes

f f ^ 0~b0~:o 0~ (d2~:).0~e# (5.15) (d'x)BlE~ = + + lira ¢'b'¢oO~b" 82r

In deriving eq. (5.15) we have used Gauss ' theorem in reverse

lim f (dZcrk),n,,0t(A~d ~) = f (d'x),~tmO, dt(Ab#~)- 0. (5.16)

$r

It is easy to verify that the integrand of the integral on the right hand side of eq. (5.15) satisfies the following identity

M.K. Prasad/ lnstantons and monopoles in Yang-Mills gauge ]ield theories 177

O~a O~ ~j : 4 d e t L o ~ O~ ~j

=4 (determinant ]gabl of the metric tensor gob of the unit sphere

: + + 1) .

We therefore have

(5.17)

f li_mf 4~rc ¢ (d26) " k / ] ~ = - 7 q, (d3x)B~Ef = ~ s 2,

q = topological charge = 0, 1, 2 . . . . (5.18)

because while the point (6t, ~2) covers the sphere S 2 once, the vector ~a can cover the unit sphere ~o~a= ~1~1+ ~2~2+ ~3~3= 1 any q number of times, each time contributing a 3-dimensional solid angle f (d2~) = 4~.

Thus, using eq. (5.5) and eq. (5.18), we establish the following inequality for the energy E

E~> 4~rc c 2 7 q with q = 0, 1, 2 . . . . . where li_m A:A~ = ~ . (5.19)

The lower bound equality sign is reached only by static self-dual fields Bf = Eg. The topological charge q, in the context of monopoles, is referred to as the magnetic charge. Eq. (5.19) shows that for any given q, static self-dual gauge fields are absolute minima of the energy E.

Whether the lower bound E = 4~rc/g2q can be actually realized is a dynamical, not a topological, problem and requires the construction of explicit solutions of the static self-duality equations Bf = Ef and for nontrivial solutions c ~ 0 # q. The solution for magnetic charge q = 1 was given by Prasad and Sommerfield. As of now, no one has succeeded in constructing explicitly monopole solutions with magnetic charge q > 1; indeed, it is not know whether such solutions exist or not. The powerful differential, algebraic geometric techniques so useful for instantons seem to be inapplicable to monopoles. In section 7 we present the q = 1 monopole solution and in section 8, discuss attempts toward constructing multi-monopole solutions with q > 1.

6. Explicit instanton solutions

This section will be divided into four subsections. In the first subsection we present 't Hooft's ~ and tensors along with various identities that they satisfy. The a7 and ~ tensors are extremely useful for

the purpose of constructing explicit instanton solutions. We then present the original Belavin- Polyakov-Schwartz-Tyupkin (BPST) q -- 1 instanton solution and discuss its various features. In the third subsection we present the Corrigan-Fairlie-'t Hooft-Wilczek (CFFW) Ansatz which leads to instanton solutions for which q is an arbitrary integer. Here we describe the various properties of 't Hooft's solution, in particular why it is not the complete solution. The final subsection is devoted to the Atiyah-Drirdield-Hitchin-Manin (ADHM) construction which gives the complete instanton solutions for arbitrary q.

6.1. 't Hoo f t ' s ~O and ~ tensors

Define the following four vectors

178 M.K. Prasadl lnstantons and monopoles in Yang-Mills gauge field theories

I if /~ = 4 { I if /~ = 4 O ' ~ - - - + ~

icr = i f / z = a = 1, 2, 3 cr~ - i t r ° if/~ = a = 1, 2, 3" (6.1.1)

whe re I is the 2 x 2 ident i ty mat r ix and o -a a re the Paul i ma t r i ces (3.20). Us ing eq. (3.21) we see

(r;cr, + o'+~o',, = ~r,,cr+~ + o'~o'+~ = 218,...

T h e ~ and ~ t enso r s of ' t H o o f t a re def ined as fo l lows

.!. + + W,. = - ~(tr,tr. - tr .cr,). - I +

(6.1.2)

(6.1.3)

(6.1.4)

T h e essent ia l p r o p e r t y of the , / a n d ~ t ensors , us ing eq. (3.21), is tha t they are se l f -and ant i -se l f -dual , r e spec t ive ly .

_ 1 *h,v - 2e,,v~oThp = *The, (6,1.5) '~p.u l - = = - 2~A,19~, - * ~ . (6.1.6)

T h e t enso r s ~ v and ~,,v a re an t ihe rmi t i an t race less 2 x 2 ma t r i ces and t h e r e f o r e can be wr i t t en as a l inear c o m b i n a t i o n of the Paul i ma t r i ces cr ° as

_ a (Ira -=-a O'a ~"~ =- 7h'~ 2--~' ~ -- 7 / , ~ T . (6.1.7)

w h e r e the t enso r s ~ and ~ h a v e on ly real c o m p o n e n t s . Us ing eq. (3.21) ' t H o o f t has de r ived a n u m b e r of v e r y usefu l ident i t ies tha t 7) and ~ o b e y and we list s o m e of t h e m

2e,,.a'r/= #, " ~ , , = - (6.1.8) = ? E ~ m # ' r / a f l ,

° - , 7 , ~ . , ~ , = -° (6 .1 .9 ) a b - a - b vl ~,~'rl ~,~ = "O ~,~vl ~,, = 48ob, (6.1.10) a a ,7,m,= = , ~ , ~ , = 3 8 . , (6.1.11) a a )1,.¢/,.. = - ~ . ~ = . = 12, ( 6 . 1 . 1 2 )

n~,,~aa = 8~8~s - 8 ~ s 8 ~ + ~,~a, (6.1.13)

~. ~ ~# = 8 ~ 8 ~ - 8,,#8. - ~,,,a, (6.1.14)

° ~ ~ ( 6 . 1 . 1 5 ) "rh.. ~ ~ 8=b8~. + = eabcT~ sot,

- ° -~ (6.1.16)

~,~¢*t ~,*/~a = 8 , ~ - 8 ~ s ~ - 8.~)/~s + 8.¢~ ~.. (6.1.17) (~ --b" --- c

, - ~ _ o -b = - ~ ( 6 . 1 . 1 9 )

A n y a n t i s y m m e t r i c tensor T,~ == - T . , can be w r i t t e n as a sum o f se l f -dua l and an t i - se l f -dua l par ts

as f o l l o w s

r , , ~- - r , , , --½(T,,, + * T , , , )+ ~T, , , - * T,,,), (6.1.20)

w h e r e

* T ~ , =-½~,~aT=~. (6.1.21)

T h e combina t i ons ( T ~ - " I " , , ) c an a lso be wr i t t en as

M.K. Prasad/ lnstantons and monopoles in Yang-MUls gauge field theories 179

T~, + *T,, = ½(8~8~ - 8~8,~ _+ e~,~) T,~ (6.1.22)

D(±) 7" - ~ ~ . ~ • ~ . ( 6 . 1 . 2 3 )

In eq. (6.1.23) we have defined the projector pc±) on self-dual (+) and anti-self-dual ( - ) parts. From eqs. (6.1.13) and (6.1.14) it follows that

pc+) I a a (-) - - - I - a - a ~,~ = 271~,,~, P~,.~ - 2vl~,~l~. (6.1.24)

If the tensor T is self-dual, then

T~, = *T~,C~ ~ a T , a =0. (6.1.25)

Multiplying eq. (6.1.25) by ~ , and using eq. (6.1.10) we find

T~, = *T~,¢~ ~ a T~a = 0. (6A.26)

6.2. The original Belavin-Polyakov-Schwartz-Tyupkin q = 1 instanton solution

The BPST q = 1 instanton solution is given by

A~, = ~2 (x ~ ~,,(x_ Xo) 2- +x°)"A 2' (x - x0) 2 ---- (x - x0)~ (x - x0)~, (6.2.1)

where (x0)~ and A are five free parameters associated with the position and scale of the instanton, respectively. Using eq. (6.1.17) the gauge field strength for (6.2.1) is

F~,, o 4 A 2 = --~V-v g [(X - - X0) 2 + )[2]2 (6.2.2)

which by virtue of eq. (6.1.5) is manifestly self-dual. The action S for the field strength (6.2.2) can be easily computed to be

S = 8" / r2 / f 2. (6.2.3)

ThUS we have an explicit realization of the q = 1 instanton solution depending on 5 parameters. In fact, one can show by means of deformation theory that if there exists an SU(2) instanton solution with topological charge q, the action density (i.e., FT,~FT,,) must depend on at least ( 8 q - 3 ) parameters: 5q are the positions and scales of each single q instanton, 3q are the 3 SU(2) gauge group orientations for each instanton, and an overall SU(2) gauge group orientation is irrelevant so one subtracts 3. The ( 8 q - 3 ) parameters are essentially the "degrees of freedom" of the instanton solutions.

6.3. The Corrigan-Faidie-'t Hooft" Wilczek ( CFTW) ansatz and 't Hooft 's solution

The CFTW ansatz is given by

A~, = _ l ~ , 0 ~ In ~, (6.3.1)

where # is at the moment an arbitrary function of x~. Using eq. (6.1.17) the gauge field strength for eq. (6.3.1) is


a = 1 r ( a , a . ¢ ) .(a.¢)(a.¢)] -a 1 [(0/~o~)__2 (O~,tb)(Oo~b)] + ~""g-a l (6.3.2)

At this point, it is far from obvious how eq. (6.3.2) can be self-dual in the indices ~ and v. The requirement that (6.3.2) be self-dual is, by virtue of (6.1.26), tantamount to requiring

-~ ,F , , = O. (6.3.3)

Using eq. (6.1.16) the requirement in eq. (6.3.3) reduces drastically to a simple equation for ~k:

Vldp = 0 (I-I =- a~,a~). (6.3.4)

Thus if ~b satisfies eq. (6.3.4) the gauge field strength (6.3.2) is guaranteed to be self-dual. 't Hooft has

taken the following solution of eq. (6.3.4)

~ = 1 + "'1 (6.3.5) ° ( x - xj) ~"

The solution (6.3.5) is valid only for x # x ~ ( j = l , 2 . . . . . q). As x--*xj the function d~ becomes singular

) A 2 ~k x~xj ~ " (6.3.6)

The singularity (6.3.6), however, is a pure gauge artifact! To see this, one computes the gauge potential (6.3.1), in matrix form, near the singularity (6.3.6)

2(x - xj)~, (6.3.7) A , ~ ( x - x j ) 2"

= UTla~Uj (6.3.8)

where

Uj - ~ try. (6.3.9)

Comparing (6.3.8) and (3.10) we see that F~,, vanishes at the singularity (6.3.6) and therefore this singularity is a pure gauge artifact of no physical consequence. Using eq. (6.3.4) the action density

reduces to

+lL, o ~'o = _ ~ g VIVIIn ~k. (6.3.10) 4 x" ~.,x /~u

Inserting eq. (6.3.5) into (6.3.10), integrating over 4-dimensional Euclidean space and excluding the singularities in eq. (6.3.6) from the region of integration so that one can use Gauss' theorems, gives the

action

8w (6.3.11) S=-~q. Thus we have an explicit realization of multi-instantons with arbitrary q. From eq. (6.3.10) and

(6.3.5) it is evident that the action density depends only on 5q parameters, which one can interpret as the positions and scales of each single q instanton. Since the number of parameters is less than

M.K. Prasadl lnstantons and monopoles in Yang-MUls gauge field theories 181

(8q - 3) 't Hooft 's solution cannot be the complete instanton solution. The complete instanton solution is realized in the Atiyah-Drinfield-Hitchin-Manin (ADHM) construction which we now describe.

6.4. The A t i y a h - D r i n f i e l d - H i t c h i n - M a n i n ( A D H M ) construct ion

The ADHM construction begins with a rectangular (q + 1) x q matrix M ( x ) made up of quaternions. That is, an element Mjk of the matrix M can be written as a 2 x 2 matrix

Mjk = M ~ , (6.4.1)

where M~ are real numbers. The matrix M is chosen to be linear in x

M = B - Cx, (6.4.2)

B, C are x,-independent (q x 1) x q quaternionic matrices (6.4.3)

x -= x , ~ . (6.4.4)

Finally, M ( x ) is assumed to satisfy the nonlinear requirement

M + ( x ) M ( x ) -- R (6.4.5)

= real invertible q x q matrix. (6.4.6)

In order to construct self-dual gauge fields, it is necessary to find a (q + 1)-dimensional column vector N ( x ) obeying

N + ( x ) M ( x ) = O, (6.4.7)

N + ( x ) N ( x ) = I. (6.4.8)

The linear equation (6.4.7) can be viewed as q quaternionic conditions on the (q + 1) elements of N. Thus a solution N ( x ) of eq. (6.4.7) can always be found and the requirement of eq. (6.4.8) simply fixes its normalization.

The gauge potential in the ADHM construction is defined by

A , ( x ) = N+(x)O~,N(x). (6.4.9)

Differentiation of eq. (6.4.8)

N+(x)O~,N(x) = - (O~N+(x) )N(x) (6.4.10)

shows that A, in eq. (6.4.9) is an antihermitian traceless 2 x 2 matrix. We will now show that the gauge field strength F,~ constructed from the gauge potential in eq.

( 6 . 4 . 9 ) is manifestly self-dual.

F ~ = ~ A ~ + A~Av - (l~ ~-* v).

= (O~N+)(O~N)+ (N+O~N)(N+O~N)- (~ ~ v). = (O~N+){I - NN+}(O;N) - (iz *-~ ~,).

(6.4.11) (6.4.12) (6.4.13)

The I in eq. (6.4.13) is an ( q + 1 ) x ( q + 1) identity matrix. The quantity in the curly bracket in eq. ( 6 . 4 . 1 3 ) is simply the projection operator onto the q-dimensional quaternionic subspace orthogonal to N. Using eq. (6.4.7) and eq. (6.4.5) this can be written as a (q + 1) x (q + 1) matrix


I - N N + = M ( x ) R - 1 ( x ) M + ( x ) , (6.4.14)

where R -n is the q x q matrix inverse of the real matrix R. Differentiation of eq. (6.4.7) gives

(O~N+(x) )M(x) = - N+(O~,M(x)). (6.4.15)

so that F~, can be written as

F~v = N+(d~M)R- I (O~M+)N - (IX ~-> v). (6.4.16)

= N+C{(r~R- lc rv_ + -n + o, vR o,,}C N. (6.4.17)

= - 4N + Ct / . .R -n C + N. (6.4.18)

which, by virtue of eq. (6.1.5) is manifestly self-dual. In going from eq. (6.4.17) to eq. (6.4.18) we have used the fact that R is a real matrix so that it commutes with or,: [R, cr~ ] = 0.

That the ADHM construction give the complete ( 8 q - 3) parameter instanton solution with action S = (8¢r2lg2)q is somewhat more complicated to prove and we refer the reader to the references for detailed exposition.

7. Explicit m o n o p o l e q - 1 solution

For monopoles the gauge fields are static and for q = 1 one takes the ansatz

1 X a h(r) , A [ = l e ° ~ b Xb a(r) , (7.1) AS = ~--~- g r

where h and a are for the moment arbitrary functions of r. The ansatz (7.1) can be motivated by the asymptotic (as r-~oo) forms eqs. (5.11) and (5.12) with all tensor indices constructed from only 3 Euclidean space coordinates x° (a = 1,2,3)-so that there is no special direction, i.e., a kind of "spherical symmetry".

From the ansatz eq. (7.1) we compute the vector fields B~ and E~ defined in eqs. (5.2) and (5.1) to be

1 , ~ [ , a \ . x °x k

where for any function f ( r ) we define f ' = d f (r ) ldr . The static self-duality equations B[ = E[ therefore become

(ra)' = h(ra - 1), (7.4)

h' + h ( a _ 1 ) = a2 + a ' -a ' r (7.5)

Solving eq. (7.4) for h and substituting into eq. (7.5) we obtain

1 (In Q)" = Q2, where Q - r (ra - 1). (7.6)

M.K. Prasad[ Instantons and monopoles in Yang-Mills gauge fieM theories 183

Equation (7.6) can be easily integrated, giving

1 1 4 Cr ] 1 a = - 1 h = - - C coth(Cr + r0), (7.7) r -sinh(~rr+r0) ' r

where C (which we take to be positive) and r0 are constants of integration. We now require both AS and A~ to be finite (nonsingular) for all 3-dimensional Euclidean space and

this forces us to take r0 -- 0 and the minus sign in eq. (7.7). Thus the gauge potentials

1 x a [1 - Cr coth(Cr)], (7.8) A: =~r-- f

A~ lea,b X b [1 Cr ] = g r - r sinh-(-Cr) J" (7.9)

are finite nonsingular solutions of the static self-duality equations B~ = E~. Using eqs. (5.5), (5.6) and (5.1) it is easy to verify

a a = = OkOk[2A4A4 ]. (7.10) B~ E~ ~ B,Ek ~ a a

SO that the energy associated with the gauge potentials eq. (7.8) and (7.9) is

E=~--~ f (d3X)OkO,{~[1-Cr coth(Cr)]2}. (7.11)

Since the integrand of eq. (7.11) is nonsingular, one can use Gauss' theorem to find for the energy

E = 4~'C[g 2. (7.12)

Comparing with eq. (5.19) we see that we indeed have an explicit realization of a q = 1 monopole solution.

To this day almost nothing is known about explicit monopole solutions with q > 1. The algebraic geometric techniques which led to the ADHM construction for instantons do not seem to be applicable here. In the next section we describe attempts using Bficklund transformations that give some insights- though no concrete solution to the problems involved.

8. Biicklund transformations for self-dual gauge fields

This section is devoted to B~icklund transformations for self-dual gauge fields which give some insight-though, unfortunately, no concrete solutions to the problems involved. For our purpose, a Bficklund transformation (BT) is a transformation which generates locally "new" solutions of the self-duality equations from "old" ones. Since the self-duality equations are complicated nonlinear coupled partial differential equations, the BT can be quite useful if only trivial solutions are known.

This section will be subdivided into three subsections: In the first subsection we present Yang's formulation of self-dual guage fields, which is the starting point for all Bficklund transformation discussions. We then present the Atiyah-Ward ans~itz in the context of the instanton problem. Historically, the Atiyah-Ward (AW) ansfitze were proposed as a means of generating the complete instanton solution, but because of its global singularity problems was abandoned in favor of the more useful algebraic geometric techniques that eventually led to the ADHM construction. Finally, we present a BT in a manifestly gauge invariant formulation of self-dual gauge fields that was very


recently discovered. Unfortunately, like the AW construction, this new BT also has severe singularity problems when viewed globally- a feature that seems characteristic of all such transformations.

Unless one has a better understanding (or perhaps control) of the global singularities generated by these BT, it is not at all clear what their significance is in relation to instantons and monopoles.

8. I. Yang' s formulation o f self-dual gauge fields

The essential idea of Yang is to consider an analytic continuation of the gauge potential A~ (in matrix form) into complex space where xl, x2, x3 and x4 are complex. The self-duality equations F~,~ = *F~ are then valid also in complex space, in a region containing real space where the x's are real. Now consider four new complex variables y, ~, z and ~ defined by

Vr2y -- xl +ix2, V~y - x l - i x2, (8.1.1)

X/2z -- x3 - i x4, V~e - x3 + i x4.

It is simple to check that the self-duality equations F~, = *F~ reduce to

Fyz = 0, (8.1.2)

Fg~ = 0, (8.1.3)

Fyg + Fz~ = 0. (8.1.4)

Equations (8.1.2) and (8.1.3) can be immediately integrated, since they are pure gauge, to give

Ay = D - I D , Az = D - I D , (8.1.5)

A~ = ~ - 1 ~ , Az = D-IDa, (8.1.6)

where D and/5 are arbitrary 2 × 2 complex matrix functions of y, y, z and ~ with determinant = 1 (for SU(2) gauge group) and Dy - 0rD etc.

For real gauge fields A,, - -A~ (the symbol = is used for equations valid only for real values of x:, x2, x3 and x4), we require

/5 - (D+) -1. (8.1.7)

Gauge transformations eq. (3.9) are the transformations

D ~ D U , D ~ D U , U + U - L (8.1.8)

where U is a 2 x 2 complex matrix function of y, y, z and £ with determinant = 1. Under transformation (8.1.8), eq. (8.1.7) remains unchanged.

We now define the Hermitian matrix J as

j •, D ~ -I - DD +. (8.1.9)

J has the very important property of being invariant under the gauge transformation eq. (8.1.8). The only nonvanishing field strengths in terms of J become

Fue = -D-I(J-IJ,)eJD (u, v = y, z). (8.1.10)

and the remaining self-duality equation (8.1.4) takes the form

(J-JJy)9 + (J-IJz)f = O. (8.1.11)

M.K. Prasad/ lnstantons and monopoles in Yang-MiUs gauge Yield theories 185

The action density in terms of ] is

Y(J) --- -½Tr F~F~v = - 2 Tr(F,~F~ + F,~F~,) = -2 Tr{(J-IJ,)~(J-IJ~)~ - (J-IJy),(J-IJ~)~}.

For static gauge fields, eq. (3.6) becomes

STATIC GAUGE FIELDS: OzA~,(x) = a~A,,(x) (/~ = 1, 2, 3, 4). (8.1.13)

so that for static fields Jz = J~. The gauge potential Af in matrix form is A4 = g(o'°/2i)A~ = ( i / ~ ) (Az- At )= (i/%/2) (D-~D~- [)-~D~) s ince / ) , =/3~ for static fields. Therefore, the energy density eq. (7.10) for static self-dual fields takes on the following form

~(J) = --akak Tr(AtAJ = -(0ya~ + a~a~) Tr(J~Yz). (8.1.14)

The action S and energy E are given by

s = ~ f (d'x)~(]), E f ~ f (d3x)~(]). (8.1.15)

In closing this subsection, we remark that Y is arbitrary within the transformation

(8.1.12)

J ~ l?(y, [)JV(y, z), I~ - V +, (8.1.16)

where V is an arbitrary function of y and z. Under the transformation eq. (8.1.16) the gauge potentials A,, and therefore the field strength F~,~, remain unchanged.

8.2. The Atiyah-Ward ansdtze

The Atiyah-Ward (AW) construction begins by an explicit parametrization of the matrix Y

i !

j = P ¢

and for real - - +

gauge fields A~, - - A ~ , we require

(8.2.1)

ffi real p - p * (p* ffi complex conjugate of p). (8.2.2)

The self-duality equations (8.1.11) take the form

~Z] In 4' + (P '~ ;2 p~St) = 0,

o

[] ffi a~a~ = 2( a,a~ + a~a,).

Using eqs. (8.2.3)-(8.2.5) one can reduce the action density eq. (8.1.12) to the following

(8.2.3)

(8.2.4)

(8.2.5)

186 M.K. Prasad# Instantons and monopoles in Yang-Mills gauge field theories

r ,.,.(o.o.;,.,.)-,.,. (o.o.;,,.) l ~(4,, p, p)= -12[][2] In4, L+,a,O ' (4,,4,~ P,P,)_a,O, (4,'4,'.cil~ p,l~,)J" (8.2.6)

The CFTW ansatz and 't Hooft ' s solution of section 6 have a simple form in terms of 4,, p, and p C F T W Ansatz : py = 4,~,

' t Ho o f t ' s solution: [~4, = O,

5e(4,, p, p) = -½1--ll-q In 4,.

4,---1+ (8.2.7)

Atiyah and Ward call the CFTW ansatz eq. (8.2.7) ansatz ~ and from it they construct an infinite hierarchy of ans~itze ~t( l = 2, 3 . . . ) which allows an explicit construction of 4,, p, and ~ that automatically satisfy the self-duality equations (8.2.3)-(8.2.5). In order to construct ~t we will need some preliminary results.

The positive definite Hermitian matrix J = D D + can be factored into a product of upper and lower (or vice versa) triangular matrices as follows

Y = R R + = RIR I÷,

o] R = LplV,- ~ ~/-'~ " 1/Vc~--r j '

4, -: real, p = p . , 4,i = real, t~1 = pl..

(8.2.8)

(8.2.9)

It is evident from (8.2.8) that one can choose a gauge so that D = R or D = R 1 and it is easy to check that in both gauges the self-duality equations have the form of eq. (8.2.3)-(8.2.5) (in the case of D = R 1

all the 4,, p, p are replaced by 4,1, p~, pt). From eq. (8.2.8) we see that R-mR I is a unitary matrix so that we can always make a gauge transformation from the R gauge to the R I gauge. We therefore are led to the following theorem.

Theorem 1: If (4,,p,p) satisfy eqs. (8.2.3)-(8.2.5) then so do (4,1, pt, pl) defined by

4,1= 4, , pl ~+___~p~. -- p ' - - (8.2.10)

Furthermore, the gauge potentials constructed from (4, t ,p~,pl) are a gauge transform of those constructed from (4,, p, p~) so that

ff(4,t, pl, pl) = 5e(4,, p, p). (8.2.11)

since the action density is invariant under a gauge transformation. Note that I is a discrete transformation since when operated twice it gives an identity (i.e., 4,n = 4,, pn = p, pu = p).

We now state a second theorem which is very simple to prove.

Theorem 2: If (4,, p, p) satisfy eqs. (8.2.3)-(8.2.5) then so do (4,B, pS, pe) defined by

_1 (8.2.12)

M.K. Prasadl lnstantons and monopoles in Yang-Mills gauge field theories 187

Unlike eq. (8.2.10), one has to solve differential (not algebraic) equations to find p8 and ~B. It is very important to note that

~B _ _ pB.. (8.2.13)

and therefore violates the reality condition eq. (8.2.2). Furthermore, acting with the operator B twice is a trivial operation (i.e., ~b BB= ~b,p~ B= pz ,~B = ~ , etc.) in that it does not change the gauge potential. Therefore, in order to use B more than once, one must interpose the I operation of Theorem 1 between two B ' s .

Using eqs. (8.2.6), (8.2.3) and (8.2.12) the action density of ~b a, pB, ASB is

.y(~bs, ps, ~B) = bo(~b, p, jS) - ½[:][:] In ~b B. (8.2.14)

In general [7[=] In ~ba# 0, so that (~b s, O 8, j5 a) cannot be a gauge transformation of (~, p, ~). The AW ansatz Mr(! = 2, 3, 4 . . . ) can now be defined by the following chain of operations

,~l (m)> ,~2 (m)) ~3 is1)> . . ", (8.2.15)

where ~1 is the CFTW ansatz eq. (8.2.7) and ( B I ) means operate with I f i r s t and then with B as defined in Theorems 1 and 2, respectively. Using eqs. (8.2.11) and (8.2.14) one finds the action density for the At ansatz to be

i

~ ( ~ t + l ) = ~--0 {-2~:][:] In [(BI)k~b]}, (8.2.16)

where ~b is th CFTW ansatz ~b defined in eq. (8.2.7) and ( B I ) ~ means operate with ( B I ) k number of times.

Now because of eq. (8.2.13), all the even ans~itze ~2~ will not in general satisfy the reality condition eq. (8.2.2) whereas all the o d d ansatz d2t+l will.

A much more serious consequence of eq. (8.2.13) is the following. If one explicitly computes the action density b°(~t) for 1 > 1, one finds it is infested with severe singularities - singularities which are not gauge artifacts (as was the case with 't Hoof t ' s solution in subsection 3 of section 6). It is at the moment not at all clear how to cope with such singularities and therefore how the AW ansatz is related to the A D H M construction which is known to give the complete instanton solution.

8.3. M a n i f e s t l y g a u g e i n v a r i a n t B a ' c k l u n d t r a n s f o r m a t i o n

The AW construction requires an explicit parameterization of J and thereby breaks manifest gauge invariance. Recent ly a new B/icklund transformation directly in terms of the gauge invariant matrix J was found and we now describe it.

Let J and J ' be two 2 x 2 Hermitian matrices that satisfy the following equation

j - i j y _ t , - l r , _ - - y -- e ia(J-]J ' )~, (8.3.1)

j , j - i = j j , - i + f lI , (8.3.2)

where a and/3 are real constants. If we take the Hermitian conjugate of eq. (8.3.1) and use eq. (8.3.2) we obtain

j - i j z _ j , - i j , = _eio(j-ij ,)~. (8.3.3)


It therefore follows that

(j-Iy,)~ + (y-Iyz)~ = (y,-,y,y)~ + (y,-,y,)~. (8.3.4)

so that both J and J ' are solutions of the selflduality eq. (8.1.11). If J # J ' and det J = + l then det J ' = - l . To see this, define P~-J 'J -~ so that, by eq. (8.3.2),

P~ =/3P + L By the Cayley-Hamilton theorem P[- /3 +Tr P] = I[1 +det P]. If det P # -1 then P is proportional to the identity matrix and by an appropriate rescaling one has J ' = J which is a trivial solution of eq. (8.3.1) and eq. (8.3.2). Therefore, for J ' # J one must require det P = - 1 and Tr P =/3.

However, for SU(2) gauge theory J = DD + which is not only Hermitian but positive definite so that det J > 0. Since det J ' = - 1 it cannot describe SU(2) gauge fields. This problem is the analog of eq. (8.2.13) for the AW construction. As in AW construction, to get det J ' = 1, one must apply the Bticklund transformation eqs. (8.3.1) and (8.3.2) and even number of times. Unfortunately the singularity problems associated with the AW construction seem to persist for this new construction

also. We close this section by remarking that the BT so far discussed are not strong in the sense that they

do not imply the "new" and "old" solutions of self-duality equations are independently satisfied. That is, for example, eq. (8.3.4) does not imply that (y - lyy )y + (y-Iyz)~ = 0 and (y , - Iy ,y )y + (y , - Iy , z ) e = 0. The classical BT in the context of solitons (e.g., Sine-Gordon, KdV eq. etc.) are in fact strong and quite useful in generating non-trivial soliton solutions from trivial solutions. It is not known whether a strong BT exists for self--dual Yang-Mills equations.

9. Non-local conservation laws for serf-dual Yang-Mllis gauge fields

In this final section we construct, using the manifestly gauge invariant formalism of section 8, an infinite set of non-local conservation laws for self-dual gauge fields. Such conservation laws are known to be useful in the context of solitons (see references) and it is hoped that they will shed light

upon the hidden symmetries of self-dual gauge fields. By a conservation law we simply mean an equation of the form O, V~ = 0-strictly speaking, since we

are in Euclidean space, we should call such an equation a continuity equation. In the complex notation

of section 8 we see that

a,v, = a, vg + a~v, + a.v, + a,v..

so that the self-duality equation (8.1.11)

(9.1)

V(y") = O~X ("), V(7 ) = -O~X c"), n ~> 1. (9.4)

( j - I jy)~ + (j-I j~)~ = 0. (9.2)

is in fact a conservation law (for static fields J: = J~). It has been recently found that from eq. (9.2) one can in fact construct an infinite set of non-local

conservation laws. The construction is as follows. Let us assume we have constructed the nth

conservation law

( V(y'))~ + (V~'))~ = O, n ~> 1. (9.3)

Equation (9.3.) implies that there exists a 2 x 2 matrix function X(')(y, Y, z, ~) such that


The "(n + l)th" conservation law is then defined as

(V~,'+~)~ + (V~'+'~), = O,

vO ,n+l) = (•y + J-Ijy)X{n) ,

V(? +1) = (O~ + J- ' l~)X ('~ (n >10).

The (9.5) we note the following

(V~"+'~) 9 + (V~"+'))~.

= (o3y + J-l/y)(a~X(n) ) + (0 z +/-IJz)(o~)((n) )

= -(o , + J - '4 ) V~F + (o~ + J-'y,) V~ "~

= -[lO, + J - ' 4 , lO, + J-~J,]x c'-')

--~0.

(9.5)

(9.6)

(9.7)

proof is by induction. The induction starts with X (°) = I and then V~y~)= Y-~Y, etc. To prove eq.

(9.8) (9.9)

(9.10)

(9.11)

Eqs. (9.8) and (9.9) follow from eqs. (9.2) and (9.4) respectively, while in eq. (9.10) we have used the induction hypothesis eqs. (9.6) and (9.7). Thus we have constructed an infinite set of non-local (since X c") is a non-local function of V ~"~) conservation laws for self-dual gauge fields in terms of Hermitian and gauge invariant matrix J. The implications of these conservation laws have yet to be explored.

10. Conclusions

To make a long story short, all instanton solutions are known explicitly whereas only the simplest monopole solution is known explicitly. Almost nothing is known about explicit multimonopole solutions, and therefore any concrete results in this direction would be extremely important.

We presented two distinct Biicklund transformations for self-dual gauge fields which we hoped could generate explicit multimonopole solutions from the known one, but because of global singularities which we do not know how to control, this hope has not been realized.

We also presented an infinite set of non-local conservation laws which we believe can shed light on the hidden symmetries of self-dual gauge fields.

Acknowledgement

I thank Professor Craig Tracy for encouraging me to write a survey article on instantons and monopoles. This work was partially financed by National Science Foundation grant PHY 79--06376.

Notes and references

We now present, for each of the previous sections, relevant references and remarks. We then summarize some recent developments in regard to monopoles.

It will be convenient to use the following abbreviations for some frequently quoted journals: SA - Scientific American

190 M.K. Prasadl lnstantons and monopoles in Yang-Mills gauge Yield theories

PL - Physics Letters NP - Nuclear Physics C M P - Communications in Mathematical Physics JETP - Soviet Journal for Experimental and Theoretical Physics PR - Physical Review P R L - Physical Review Letters J M P - Journal of Mathematical Physics

1. General introduction to gauge theories, quarks and gluons, and supergravity can be found in: [1] S. Weinberg, SA, July 1974, pp. 50-54. [2] S.L. Glashow, SA, October 1975, pp. 38--50. [3] D.Z. Freedman and P. van Nieuwenhuizen, SA, February, 1978, pp. 126-143.

A general introduction to solitons, instantons and monopoles can be found in [4] C. Rebbi, SA, February, 1979, pp. 92-116. 2. The mathematics of this section is expounded in several textbooks from which we note the following two for their clarity:

[5] I. Satake, Linear Algebra, Pure and Applied Mathematics (New York, 1975). [6] J.L. Anderson, Principles of Relativity Physics (Academic Press, New York 1967). 3. Yang-Mills gauge theory is a creation of:

[7] C.N. Yang and R.L. Mills, PR 96 (1954) 191. 4. Instantons are a creation of

[8] A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu. S. Tyupkin, PL B59 (1975) 85. 5. Monopoles are a creation of

[9] G. 't Hooft, NP B79 (1974) 276. [10] A.M. Polyakov, JETP Letters 20, (1974) 194 and JETP 41 (1975) 988. The monopoles that are studied in this survey are a very special limit of the 't Hooft-Polyakov monopoles as defined in: [11] M.K. Prasad and C.M. Sommerfield, PRL 35 (1975) 760. In this special limit, the fourth component of the Yang-Mills gauge potentials A~ becomes the "Higgs field" of the 't

Hooft-Polyakov monopole theory. The precise definition of monopoles in this limit is due to: [12] E.B. Bogomorny, Soviet Journal of Nuclear Physics 24 (1976) 449. [13] S. Coleman, S. Parl~e, A. Neveu and C.M. Sommerfield, PR D15 (1977) 544.

't Hooft's ,~ and ~ tensors along with various identities they satisfy are given in [14] G. 't Hooft, PR DI4 (1976) 3432. The original q = 1 instanton solution is due to: [15] A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu. S. Tyupkin PL B59 (1975) 85. The deformation theory used in arriving at the (8q - 3) parameters can be found in [16] A. Schwartz, PL 67B (1977) 172. [17] R. Jackiw and C. Rebbi, PL 67B (1977) 189. [18] L. Brown, R. Carlitz and C. Lee, PR D16 (1977) 417. Multi-instanton solutions with q > 1 were first constructed by [19] E. Witten, PRL 38 (1977) 121. [20] C.K. Peng, Scientia Sinica XX 3 (1977) 345. The C F r W ansatz is due to [21] E. Corrigan and D. Fairlie, PL 67B (1977) 69. [22] G. 't Hooft (Unpublished). [23] F. Wilczek in Quark Confinement and Field Theory, eds. D. Stump and D. Weingarten (Wiley, New York, 1977). It was 't Hooft who realized that the CFTW ansatz can give acceptable multi-instanton solutions: that is, he showed the

singularities occurring in the solution are pure gauge artifacts. ' t Hooft's solution was generalized by: [24] R. Jackiw, C. Nohl and C. Rebbi, PR D 15 (1977) 1642. The ADHM construction is due to: [25] M.F. Atiyah, N.J. Hitchin, V.G. Drinfield and Yu. L Manin, PL A65 (1978) 185. The ADHM construction is elaborated in: [26] N. Christ, EJ . Weinberg and N.K. Stanton, PR DIg (1978) 2013. [27] E. Corrigan, D. Fairlie, S. Tempelton and P. Goddard, NP BI40 (1978) 31.

7. The explicit monopole q = 1 solution is due to [28] M.K. Prasad and C.M. Sommerfield, PRL 35 (1975) 760.

g. Biicklund transformations in the context of solitons can be found in the review:


[29] A.C. Scott, F.Y.F. Chu and D.W. McLaughlin, Proceedings of IEEE (USA) 61 (1973) 1443. Yang's formulation of self-dual gauge fields can be found in: [30] C.N. Yang, PRL 38 (1977) 1377. The Atiyah Ward (AW) ansatz is due to: [31] M.F. Atiyah and R.S. Ward, CMP 55 (1977) 117. The AW ansatz is elaborated in [32] E. Corrigan, D. Fairlie, R. Yates and P. Goddard, CMP 58 (1978) 223. [33] M.K. Prasad, PR DI7 (1978) 3243 (the singularity problems of the AW ansatz are discussed here). [34] Y. Brihaye, D. Fairlie, J. Nuyts and R. Yates, JMP 19 (1978) 2528. The gauge invariant Biicklund transformation is due to: [35] M.K. Prasad, A. Sinha and Ling-Lie Chau Wang, PRL 43 (1979) 750. which is motivated by the work of [36] K. Pohlmeyer, Universit~t Freiburg Preprint 1978. Attempts to generate multimonopole solutions can be found in [37] N.S. Manton, NP B135 (1978) 319, who uses the CFTW ansatz. [38] M.A. Lohe, NP B142 (1978) 236, who uses the AW ansatz. [39] D.J. Bruce, NP B142 (1978) 253, who shows the singularity problems of the previous work.

9. The infinite set of non-local conservation laws for self-dual gauge fields is due to: [40] M.K. Prasad, A. Sinha and Ling-Lie Chau Wang, PL 87B (1979) 237. Recent developments in regard to monopoles

It has been recently shown that the explicit monopole q = 1 solution has only 3 degrees of freedom corresponding to its position in 3-dimensional Euclidean space. The rigorous proof of this can be found in:

[41] R. Akhoury, J. Jun and A.S. Goldhaber, PR D21 (1980) 454, which complements work done by: [42] S.L. Adler, PR D19 (1979) 2997.

If there exists an explicit multimonopole solution with q > 1 then it must have at least (4q - !) degrees of freedom. The proof of this result and its interpretation can be found in:

[43] E.J. Weinberg, PR D20 (1979) 936.

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Instantons and monopoles in Yang-Mills gauge field theories