Instantons in Non-Abeiian Gauge Theories

Non-Abelian gauge theories with any symmetry group G possess topologically nontrivial fields. This can be seen from the following consideration. In order that the Yang-Mills action be finite one must require that:

P ,,( x ) - 0(1/x^) (6.28)

93

From this fact we deduce:

-►g + o{\jx) (6.29)

where g{x) e G.

If we bound our ^ = 4 Euclidean space by a large three-dimensional sphere we obtain, according to (6.29), a map g{x): -► G. It is easy to see that all such maps are classified by the integers for any G. Let us prove this for G = SU{2). Any matrix g in this case can be written as:

^ = « 4 -h i/I • T

= g^g = I

(6.30) (6.31)

(here t are the Pauli matrices). Therefore, elements of the 5(7(2) group are in one to one correspondence with points of the sphere 5^ defined by the equation (6.31). The map of the 5^ which bounds the jc-space onto 517(2) is therefore just the map 5^ -► 5^. In this latter case all the arguments we had for 5^ -► 5^ in Section 6.1 are applicable. We have an integer q which is equal to the number of coverings given by the integral of the Jacobian. The analogue of the formula (6.4) in the present case

It is easily checked that the combination of n in the first equality is just the surface element of 5^ and hence the first term (6.32) is the Jacobian for the transformation from the jc-space (forming 5^) to the «-space.

The second equality can be checked by explicit computation or by realizing that the integrand in this formula is the only possible expression, having dimension 3 and invariant under G ® G:

g{x) -> ug{x)v (6.33)

Therefore it must be proportional to an element of the group volume.

We have obtained the following classification of Take the asymptotic form of a given A^-field at x -► oo, and determine g(x) from (6.29). After that, compute q from (6.32). Gauge fields with different q cannot be continuously deformed to one another.

There exists a more convenient expression for q than (6.32). It is given by the formula:

1

1

T rF A F (6.34)

Here in the second equality we have used a convenient notation adopted in mathematics. For each skew-symmetric p-rank tensor

7/^1 Mp define the p-form:

where the “wedge” product ^a is a skew-symmetric bilinear operation.

Its main property is:

Generally,

d x , A dx^ = — d X j A d x , (i.e. d x , a dXj = 0 )

T , A T = ( - \ y ^ T A T

The volume element of a space of dimension n can also be represented as an n-form:

, 1

dK = dx^ A • • • A dx" = — u dx'^^ a • • • a dx^”

Below we will use the operator one-form

d = dx'"

dx^^

that transforms p-forms into (p H- l)-forms:

1

dT, = - (d,T^^...Jdx^ A dx'^^ A ... A dx^^pi

From this definition the important property d^Tp = 0 follows for arbitrary Tp. The main convenience of this notation is that we avoid writing tensor indices, thus saving a lot of ink.

Let us prove now that (6.34) is equivalent to (6.32). For this let us show first that the integrand in (6.34) is a total divergence. The easiest way of doing this is to consider a variation of the integrand under

variation of the field. We have:

p{x) = Tr(F ^A F) Sp{x) = 2 Tr(F A SF)

1

F = d A + A A A = d x ^ a d x "

(6.35)

(6.36) SF dSA + A A SA F SA A A = V SA

(where F is a 2-form; d is called the exterior derivative). For pedagogi­

cal reasons let us repeat (6.35) in the ordinary notation:

P(^) = iW p Tr(F^vF^p) d^x

= Tr(F^,^F^^) d^x

Ffiv = F ^v]

SF^, = d^SA, + [A^, (5/1J ~ { p ^ v ) = V^(5/l, - V,SA^

From these equalities we obtain:

Sp{x) = 2 Tr F A {dSA F A ^a SA SA ^a A)

= 2 Tr(F A dSA) + 2 Tr(F ^a A - A ^a F) ^a SA

= 2 Tr F A d^/lH- 2 Tr dF A (5/1 = 2d(Tr F a SA)

= Tr{F,,SA^)d^x where we have used the Bianchi identity:

[V, F] = dF + /I A F - F A /I

= dx'’ A dx“^ A dx^ = 0

(6.37)

(6.38)

We have:

Sp{x) = (d^Sj i ^i x)) d^x

SJf^{x) = e^^^^ Tr(F,,SA^) ^(6.39)

Now we have to obtain the current itself. To do this let us introduce a parameter t:0 < t< 1 and consider a family of gauge fields A^{x, t) = xAJ^x). According to (6.39) we have:

dJi\x, t) , / dA^

(6.40)

> Tr{(r(a,/1, - d,AJ + AJ) A^

Integrating (6.40) on t we obtain:

jr^(x) = 1)

= - d,A, -f f[4 „ 4 J ) .4 ,)

^Tr(F*F) = d^jr%x) (*^v =

Of course, after the answer (6.41) and (6.42) is known, it can be checked by direct computation.

Substitution of (6.42) into (6.34) gives:

(6.41)

(6.42)

1 d)-;r,dV (6.43)

5*3

where we integrate over a large S^. At these distances = 0, and (6.41) can be replaced by:

^^(x) = Tr(d,A, - d,A, F lA,A,)A^

- Tr(A,A,A^)

- ^6^^^^Tr(L,L,LJ (6.44)

This proves the equivalence of (6.32) and (6.34).

Our aim now is to find an instanton solution with ^ = 1. As in the case of the n-field we can avoid solving the Yang-Mills equations themselves, by considering instead the “square root” of them. Let us use an identity:

S = 1

4eo Tr d^x

=

^{¿ 1}

^J

Tr((F,„ - *FJ^) +

¿2 ^j

^{Tr(F,. d^x}

We see that if we find a solution of the “duality” equation F = *F_{fiv fiV}

(6.45)

(6.46)

then the action for a fixed q will be minimal. Actually, it is trivial to check that if the first-order equations (6.46) are satisfied, then the Yang-Mills equations

(6.47)

y^F'^y = 0

will also be satisfied (the converse is not true). To check this, let us differentiate (6.46):

^ 0 (6.48)

(where the last equality is a consequence of the Bianichi identity:

+ ^P^yoL F ^y^ap “ ^ if ^ap ~ ^a^p ~ ^p^a F ^^])

(6.49) Notice by the way that for constant this reduces to the Jacobi identity:

lAp, Ay']'] + lAp, [Ay, AJ] + lAy, Ap]] = 0 (6.50) We see that the “duality” equations (6.46) are in some sense a four dimensional analogue of the Cauchy-Riemann equations (6.9).

Their most surprising property is that they possess multi-instanton solutions. Before discussing them let us present a solution with q = i.

The ansatz for this solution can be found by the following trick.

Let us consider instead of the gauge group S'1/(2), a group St/(2) ® SC/(2) ^ 0(4). Then equations (6.46) will have the symmetry group 0(4) (X) 0(4), where the first factor is space rotations, and the second isotopic rotations. We shall be looking for a solution which breaks 0(4)® 0(4) but preserves the single 0(4) formed by simultaneous rotations in jc-space and isotopic space. After that we shall return to S0(2).

Generators of 0(4) are described by matrices skew-symmetric in (a, j8), which represent a rotation in the (a, jS)-plane. Therefore, gauge fields for this group also have these indices:

A , = ( X )

The most general 0(4)-symmetric ansatz is given by:

Afix) = (r^ =

(6.51)

(6.52) On symmetry grounds it must be compatible with the Yang-Mills and duality equations. The six fields Al^ can be split into two sets, each of three fields, and each corresponding to an SU(2). This splitting is described by:

a; = ^(Af + = kri^fAl^

(6.53)

(here rj^i^Q = etc.). The origin of this decomposition is based on a check that the commutation relations:

^jaP jyd-^ _ ^ctyjPS _|_ ^p Sja y

_ ^a ^ jpy _ ^Pyjoi6

get decomposed as:

= [ x ; , x ; ] = o for

2^2^abc^be i ^ao) We conclude from (6.52) and (6.53) that the ansatz:

(6.54)

(6.55) is bound to be compatible with the duality equation. The substitution of (6.55) into (6.46) gives after some simple calculations:

Alix) =²y?a^v(^v - ^v) (jc - ay + K M =

(6.56) { { x - a f + p y

with arbitrary scale parameter p and position parameter a^.

This Non-Abelian instanton can be viewed as a magnetic dipole of size p. If we consider now the contribution of one instanton to the partition function Z we find, just as in case of n-field, several factors.

First of all we have a factor e “^*=^ = which gets replaced, after taking account of the one loop correction by where, for SU(N)

is an effective coupling for the size p. The contribution has to be integrated over p and a. The measure must be both scale and transla- tionally invariant. The only combination with these properties is

dp p~^. We find from this consideration:

7(1) ,

-^INST ^{y g -}8^«2^/g2^(p) lN/3 (6.58) (F is the 4-volume).

As happened in the case of the /i-field, the instanton contribution has an infrared divergence. This implies that in the multi-instanton picture.

individual instantons tend to grow and to overlap. The naive dilute gas approximation is certainly inapplicable then, and we should expect something like dissociation of dipole-like instantons to their elementary constituents, as happened in the case of the /i-field. However, even one loop computations on the multi-instanton background have not yet been performed, and nothing similar to the Coulomb plasma of the previous section has been discovered. This is connected partly with the fact that multi-instanton solutions have not been explicitly parame­

trized up to now. I expect many interesting surprises await us, even on the one loop level, in this hard problem.

There exists an interesting phenomenological description of the instanton liquid, which explains some qualitative features of hadrons.

This approach! will not be discussed in this book.

So, our conclusion is that on the present level of understanding of instanton dynamics, we cannot obtain any exact dynamical statements concerning Non-Abelian gauge theory. In the case of /i-fields the situation is slightly better, since we were able to demonstrate the appearance of the mass gap on a qualitative level. Even in this case one would like to have much deeper understanding of the situation. There are reasons to believe that some considerable progress will be achieved in the near future. In the case of gauge fields we have to pray for luck.

At the same time, the existence of fields with topological charge has a deep qualitative influence on the dynamical structure of the theory. We describe some of this in the next section.

Im Dokument Gauge Fields and Strings (Seite 103-110)