On the structure of the QCD vacuum in the large-N limit

Nuclear Physics B186 (1981) 236-246 O North-Holland Publishing Company

ON THE STRUCTURE OF THE QCD V A C U U M IN THE L A R G E - N LIMIT H. FLYVBJERG and J.L. PETERSEN The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen 0, Denmark

Received 19 December 1980 We present an approximate QCD vacuum for SU(N-*oo). It is a generalization of the ferromagnetic vacuum first obtained by Savvidy for SU(2) and generalized by one of us to SU(3) and SU(4). Problems occurring for N ~>5 are handled in the large-N limit by a continuous formalism, and the vacuum obtained is characterized by N - 1 constant, commuting, color magnetic fields with an isotropic distribution of spatial directions. The energy density of this vacuum is lower than that of the perturbative vacuum by a number proportional to N 2, as expected from general large-N arguments. Like the Savvidy vacuum the large-N vacuum may decay into a variant of the domained Copenhagen vacuum. We give a lower limit on the domain size.

1. Introduction In the s e a r c h for a s o l u t i o n of t h e Q C D c o n f i n e m e n t p r o b l e m m a n y lines of a t t a c k have b e e n a d v o c a t e d . O u r w o r k is at the i n t e r s e c t i o n of two such lines: o n e line c o n s t r u c t s s i m p l e m o d e l s for the Q C D v a c u u m , the C o p e n h a g e n v a c u u m b e i n g an e x a m p l e [1-7]. A n o t h e r line tries to solve Q C D in the l a r g e - N limit, N b e i n g the n u m b e r of q u a r k colors [8]. W o r k on the C o p e n h a g e n v a c u u m has h i t h e r t o e s s e n tially b e e n b a s e d u p o n t h e g a u g e g r o u p SU(2) with ref. [7] as an e x c e p t i o n t r e a t i n g S U ( 3 ) a n d SU(4), a n d failing to g e n e r a l i z e to N t> 5. In the p r e s e n t p a p e r we t a k e the first step t o w a r d s setting up the C o p e n h a g e n v a c u u m in the l a r g e - N limit. T h e steps i n v o l v e d in setting up t h e C o p e n h a g e n v a c u u m for SU(2) are: (i) O n e calculates the effective o n e - l o o p p o t e n t i a l for a h o m o g e n e o u s m a g n e t i c b a c k g r o u n d field. T h e a s s o c i a t e d e n e r g y d e n s i t y exhibits a n o n - t r i v i a l m i n i m u m for finite b a c k g r o u n d field. A t this value the e n e r g y d e n s i t y is s m a l l e r t h a n for z e r o b a c k g r o u n d field. A s an a p p r o x i m a t e d e s c r i p t i o n of the real Q C D v a c u u m this level is u n s a t i s f a c t o r y for s e v e r a l r e a s o n s . First, r o t a t i o n i n v a r i a n c e a n d L o r e n t z invariance a r e b r o k e n b y t h e b a c k g r o u n d field. S e c o n d , the n o n - t r i v i a l m i n i m u m occurs at the " L a n d a u s i n g u l a r i t y " in the effective c o u p l i n g c o n s t a n t m e a n i n g that the o n e - l o o p p e r t u r b a t i v e c a l c u l a t i o n is unjustified. F i n a l l y the e n e r g y d e n s i t y has an i m a g i n a r y p a r t c o r r e s p o n d i n g to the e x i s t e n c e of u n s t a b l e m o d e s [1]. (ii) In the s e c o n d s t e p [ 2 - 4 ] the i m a g i n a r y p a r t of the e n e r g y d e n s i t y is r e m o v e d by including p a r t i c u l a r m o d e s in the b a c k g r o u n d field giving rise to a d o m a i n f o r m a t i o n . A l t h o u g h t h e o n e - l o o p q u a n t u m fluctuations h a v e not b e e n fully e v a l u a t e d a r o u n d 236

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237

the new background field, estimates show that the new (deeper) minimum of the energy density occurs for a finite a n d s m a l l value of the running coupling constant [4]. (iii) Finally, it has been speculated that quantum fluctuations not included in step (ii) would suffice to introduce enough tunneling between domains that rotation invariance (and eventually Lorentz invariance) could be restored [3, 4]. T h r o u g h o u t refs. [1-6] a few attempts were made to consider the gauge groups S U ( N ) for N > 2. In all cases, however, it was tacitly assumed that the relevant background field should correspond to one particular direction in color space. This question was taken up by one of us in ref. [7]. A general formalism was set up and it was shown that for gauge groups SU(3) and SU(4), the energetically preferred homogeneous background field corresponded to all magnetic fields in the Cartan algebra having the same magnitude and prescribed angles between themselves in real space. Technical difficulties prevented a solution for general N from being obtainable. In this paper we shall be mainly concerned with the first of the above-mentioned steps in constructing the large-N Copenhagen vacuum. In sect. 2 we set up some necessary formalism and notation. In sect. 3 we show that the energetically preferred background field configuration has all magnetic fields in the Cartan algebra having the same length and directions isotropically distributed in physical space*. The energy density in this case is negative (relative to the zero background field situation) and diverges like N 2 in agreement with general belief [8]. In contrast, if the background field is non-vanishing for a single color direction only, the energy density diverges as N. Finally, in sect. 4 we give conclusions and make a few preliminary remarks on the expected domain formation. The problem appears quite complicated. However, we do establish the existence of domains of arbitrary size larger than a definite minimal size which is independent of N. This last property was anticipated and used in ref. [9].

2. Formalism and notation

We consider a pure Yang-Mills theory based on the gauge group S U ( N ) for which the lagrangian is 1 ~vM = - ~ Trace ( F , , y " v ) , zg

(1)

F,,,, = i [ D , , , D~,] = O , A , , - ~ , . A , , - i [ A ~ , , A , , ] ,

(2)

where

D,

= ~,

- iA,

.

(3)

* Strictly speaking, this is true only for some auxiliary fieldscloselyconnected to the standard magnetic ones for N ~ co [cf. eqs. (10), (11)].

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H e r e A,., Ft,., D~, are N x N color matrices in the f u n d a m e n t a l representation:

A, = gAiT ~,

with trace (TaT b) = ~ 8ab.

(4)

Inclusion of the coupling constant g makes these matrices renormalization g r o u p invariant in the b a c k g r o u n d gauge calculations we shall employ. As in ref. [7], we consider a general magnetic b a c k g r o u n d field configuration of covariantly constant magnetic fields. This means that we m a y choose a gauge and a reference frame where we have N - 1 magnetic fields ~i,

i = 1. . . . .

N-

1,

(5)

constant in space and time. Here i labels the color directions in the Cartan algebra of c o m m u t i n g generators Hi, i = 1 . . . . . N - 1. (6) As in ref. [7] we would like to introduce a Cartan basis of the algebra. H o w e v e r , in the case of S U ( N ) this essentially just means that we consider individual matrix elements in the f u n d a m e n t a l representation. Let us introduce the N × N matrix of 3-vectors N-I

Bi/= ~. ~k(Hk)#=--B(~&/,

(nosummationoveri).

(7)

k=l

H e r e we have chosen a diagonal representation like

(H,,)# = (Hn)(i)sii, for i<~n,

1,

(/4.).) =

1

x

-n,

(8)

fori=n+l,

x/2n(n + 1) 0,

fori>n+l

,

for the Cartan algebra generators. Notice that we have N - 1 magnetic fields ~ ' i but N diagonal element fields B ~i~. The latter satisfy m

Y. B"' = 0 .

(9)

i~l

Also from eqs. (4), (7) and (8) we get ~ , , = 2 Trace (H,B) ~/ =

2 ~1)) n(n+l)(i~l ~ B(i)-nB(" "

(10)

For the solution for { ~ ,} which we present in the next section, we shall see that to leading o r d e r in 1 / N we have

.~n ~--~/2 B (~÷1),

for n / N fixed.

(11)

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239

T h e o n e - l o o p c a l c u l a t i o n of the effective p o t e n t i a l is b a s e d on the p a r t of the action b i l i n e a r in t h e q u a n t u m fluctuations W~ a r o u n d the b a c k g r o u n d field A~. A s t a n d a r d c a l c u l a t i o n gives SVM (bilin) = - -

12f daxY W~," "

g

3

I - - l.,.,-.~,k, it'u,, + U- - , t,k , , ~U,i k i ) g u , . I~W k , .,

ik

(12)

In this e x p r e s s i o n the field s t r e n g t h t e n s o r F u , a n d the g a u g e c o v a r i a n t d e r i v a t i v e D o involve the b a c k g r o u n d fields only. A l s o , we have i n t r o d u c e d the n o t a t i o n F ( .~ k i ) = -iz,~k~ - . . - -lZ,- .li)~ -= ( F . . ) k k -- ( F . . ) . ,

(13) •/

,I ( k ~

D Iki~ = Op - tt.'%

lit

--

- A p ) = 3o - i ( ( A p ) k k -- ( A o ) , )

T h e s e are just the S U ( N ) v e r s i o n s of the o b j e c t s F~f.~~ etc. d e f i n e d in ref. [7]: If (u~), i = 1 . . . . . N - 1, is a r o o t a n d ] = 1 . . . . . N - 1 is a c o l o r index in the C a r t a n a l g e b r a , N

b" l a ) ~

1

~.

~av

i

J F.~c~ i

(14)

1

F o r a fixed b a c k g r o u n d field c o n f i g u r a t i o n , /-7- ."k~ ~ and A --oC~k~a r e given functions of the 3 x ( N - 1 ) " i n d e p e n d e n t v a r i a b l e s " M ~, i = 1 . . . . . N - 1. Eq. (12) shows that the ( i k ) v a r i a b l e s d i a g o n a l i z e the p r o b l e m : k n o w i n g the 1 - l o o p effective p o t e n t i a l for the g a u g e g r o u p S U ( 2 ) c o r r e s p o n d i n g to a single b a c k g r o u n d field B, we m a y write d o w n the l - l o o p e n e r g y d e n s i t y for o u r S U ( N ) case as ~' = R e 3t°({ M i}) = )2 V ( I B ") - B ( k ' 1 2 ) , i,k

i,k

where

11 B2 ( l o g ~B- ~ 1)

V ( B 2) = 9 - ~ 2

,

a n d A is the Q C D scale c o r r e s p o n d i n g to the r e n o r m a l i z a t i o n s c h e m e e m p l o y e d . T h e result eqs. (15) w e r e d e r i v e d in ref. [7] using a c e r t a i n m i n i m a l r e n o r m a l i z a t i o n . T h e i m a g i n a r y p a r t of the e n e r g y d e n s i t y m e a n s that the b a c k g r o u n d field is u n s t a b l e . W e i g n o r e this effect in this a n d the following section.

3. The minimal energy configuration In this section we s e e k a set of N - 1 m a g n e t i c fields {Mi, i = 1 . . . . . N - 1} o r e q u i v a l e n t l y a set of N fields B ti~ s u b j e c t to t h e c o n d i t i o n (9), such that the (real p a r t of) the e n e r g y d e n s i t y given by eq. (15) is as n e g a t i v e as possible.

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For large N it is convenient to introduce the positive density function

o(n)=-#1 f,-~

(16)

satisfying I d3Bp(B)= 1 . Eq. (9) may then be written

I

d3Bp(B)B = 0 ,

(17)

and the expression for the energy density eq. (15) ~'({B"~}) = N2

I d~B d3 B'P(B)P(B') V(IB-B'[2)"

(18)

Before we construct the distribution pro(B) minimizing this expression with the given form of V, eq. (15), let us consider a few simple examples which will establish trivial upper and lower bounds for the minimal energy density ~mi,. The potential V(B 2) in eq. (15) has a minimum Vm(B~)=

11 A4 198~.2

atB=B,

=A2 (19)

and a zero at B = B0 = ~/e A 2. If it were possible to choose the set {B "~} such that all differences ]B"i~[ [cf. eq. (13)] had the same length Bin, then, trivially, we would have a minimum energy density of (N 2 - N ) x V,,. Hence this is a lower bound:

~,. >~N(N-

1)Vm.

(20)

As was pointed out in ref. [7], this bound may be achieved for N = 2, 3, 4 but not for higher N. To get an upper bound, we assume that N is divisible by 4 (this is no loss of generality for N --, co; for small N similar constructions can be done if N is divisible by 2 or 3). We then consider a configuration of the B (i~'s where they are collected in 4 groups of identical vectors, each vector~pointing from the origin to a corner of a 1 regular tetrahedron. Each group has ~N identical vectors in it. Clearly the constraints (9) or (17) are fulfilled. If we choose the length of the sides of the tetrahedron to be Bm we get the upper bound 3 2 F~,,,<~4N V,,.

(21)

This bound is saturated for N = 4. Eqs. (20) and (21) ale enough to prove that the vacuum energy diverges as N 2.

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241

If instead we restrict our consideration to a b a c k g r o u n d field with one color degree of f r e e d o m ~ , as was d o n e in refs. [4], one finds the c o r r e s p o n d i n g m i n i m u m e n e r g y to diverge as N only. To see this we consider eqs. (7) and (8) and realize that we have to take B'n+l) = - x/r.~,,

B") = 0,

orO(1/N) fori~n+l

.

The minimal energy is o b t a i n e d for ]B("+t)] = B,,. For completeness we mention that if instead we take B ")= -~B,,e,

for i < ~ N ,

B "~ = +~Bme,

for i > ~ N ,

1

e being an arbitrary spatial unit vector, the e n e r g y density does diverge as N 2 but with a smaller factor than in the case where the B " ) ' s have different spatial directions. N o w let us construct the distribution function p,, which we claim will give the minimal energy. W e take an isotropic distribution of B ~ ' s all having the same length R: 1

~,, (B) = ~ - ~

8(IBI- R ) .

(22)

F r o m eq. (18) one gets $,[tSR] = N2

1

[(2R)2 ~(, dB 2V(B2).

(23)

It is clear from the general f o r m of V ( B 2) that this expression has a minimal value for s o m e R --/~', namely t h e / ~ which solves the e q u a t i o n V(4/~ 2) = ~ - 1~ Io4~2 d B 2 V ( B 2 ) ~ V .

(24)

For the potential eq. (15) 4/~2/A 4 = x/e,

~ ( t S a ) / ( N 2 V,,) = ~x/e = 0.824 . . . .

(25)

so that we are indeed between the two b o u n d s (20) and (21). N o w let us prove that the distribution

~(B) ~ ~a(B) = ~

I

8(IB[- g')

corresponds to a local m i n i m u m of the energy density. T h u s we consider a density of the form p ( B ) = tSa (B) + 8 • or(B)

(26a)

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with I d3 B°'(B) = 0 '

f d3BBcr(B)=O'

8>0.

(26b)

Our statement will be that when or(B) has support only in a small neighborhood of width ~7 around the sphere IB[ =/~, then the perturbations eqs. (26) will increase the energy density relative to its value for p = ilk, at least for 8 small. Let us write or(B) = ~6,~(B)o'l(/~) + o'2(B),

crx > - 1 ,

(27)

so that o-~(/~) describes a variation of the surface density on the sphere [B[ = K' and cr2(B) is non-negative. Now let us evaluate the change 8ff in the vacuum energy density due to the perturbation eqs. (26)

8~ = 8~ ~x~+ 8~ ~2~, where

8~(11=2N28 I

1 d3B°'(B)4-~

I wl=.~ d~'V(IB-B'[2)'

t ~ '¢2}= N2~ 2 f d3B d 3 B ' a r ( B ) a r ( B ') V ( ] B -

a'f).

(28)

First we show that p = ~5~ is a stationary point for the energy density. To this end we need not consider 88"~2) which is second order in 8. Hence we consider 8F~~l) first. For a fixed B for which o'2(B) ~ 0, i.e. IB] ~/~, we write

R(B)=--IBI=R(I + ~7(B)),

IB-B'I2~--2(I +rI(B))R2(1-cosO),

where 0 is the angle between B and B'. Then the integral over d ~ ' in 8~ ") takes the form 1 f<2RiS))~ d y V ( y ) = V + cr/(a)2 + O(7/3) (2R (B)) 2 ~o

(29)

The fact that no linear term in 7/(B) appears is the crucial point. It follows from the definition of /~ in eq. (24), and furthermore the constant c is positive and nonvanishing (and is easily found from the potential). U p o n insertion into eq. (28) and integration over d3B, the constant I7"gives zero because of the constraint in eqs. (26). So we get

8~x~ = 2NZSc I d3Bt72(B)[Tl(B)2 + O(T/3)]

'

(30)

where only the non-negative part tr2 of ~ contributes. Eq. (30) demonstrates that our distribution ~ is a stationary point: for small 8 and 8~'t21 is 2nd order in 8 and 8~ ~1~ is 2nd order in 7.

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Second, eq. (30) d e m o n s t r a t e s that tS~ is a stable point with respect to variations having 0"2 # 0: for 0"2 fixed, we m a y take ~ small e n o u g h that 6~ ~1~ d o m i n a t e s over 8~ 'C2~ since 6 ~ ~1~ is 1st o r d e r in 6. But eq. (30) shows that 6 ~ ~ is positive. It only remains to investigate the stability in the case 0"2 = 0, where 6F~~ vanishes identically. In this case we can show that 6F~c2~ > 0 i n d e p e n d e n t of 8, so that our result here is stronger than when 0"2 # 0. W h e n B and B ' are of l e n g t h / ~ in eq. (28), we get (putting 6 = 1)

6~12~ = N 2 I dO d.(2'0-1(B)0-1(B')V(2/~ 2 - 2 B • B'), where B • B ' = / ~ 2 cos 0 and d O and d O ' are solid angles pertaining to B and B'. Performing a Taylor series expansion of V a r o u n d the value 2/~ 2 of the a r g u m e n t gives

8 ~ 2 ' = N 2 ~ ( - ) " V ' n ' ( 2 / ~ 2 ) ~ .2" w I d ~ d / T o - ~ ( B ) o ' ~ ( B ' ) ( B . B ' ) ~.

(31)

n =0

H e r e the double integral is a sum of terms of the form

[ff

P q r] 2

d.O0-1(B)B~ByB:

,

p+q+r=n,

each term having positive coefficients. This establishes that 6~ ~z) is positive definite if (-)"V'"'(2/~2)> 0,

for all n I--2.

(32)

T h e terms with n = 0 and n = 1 in eq. (31) vanish identically by virtue of the constraints in eq. (26). Conditions (32) are easily seen to be fulfilled by the potential eq. (15). Let us m a k e a few remarks a b o u t the result. First, the use of the Taylor expansion in eq. (31) is justified since the only singularity of V is at the origin, and we only use the expansion for 0 <~ 2/~ 2 - 2 B . B ' <~4/~ 2. Second, if we had a polynomial potential, i.e. a potential with derivatives vanishing f r o m some finite order, the configuration described by tSk would be connected to intinitely m a n y o t h e r configurations having the same energy: namely, taking a 0-~ with only very high angular m o m e n t s non-vanishing would leave 6 ~ ~2~= 0. Third, we should point out some variations r o u n d tSa which we have not considered above. We have required ,5 to be small for (26) to be a perturbation of tSa. This r e q u i r e m e n t c o r r e s p o n d s to perturbing only a few of the N magnetic fields (B~)~ = ~..... u away from the sphere of radius R in field space. If we put 6 = 1 in (26), but still require 7/to be small, what we have is a p e r t u r b a t i o n of the sphere itself, i.e. of the support of p. O n e such p e r t u r b a t i o n to a n o t h e r sphere with a radius different f r o m / ~ has been considered, but general perturbations to elliptic or m o r e complicated supports have not been considered.

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Finally we may go back to the expression eq. (10) giving the magnetic fields ,~ ~in terms of the B(~)'s. For the given distribution function t~a(B) there will be N ! different sets of B " ) ' s corresponding to the same physical background field configuration. When N is large, nearly all of these sets will make the first sum in eq. (10) vanish (for n large). In this sense eq. (11) is true for our distribution: also the magnetic fields M ~are distributed isotropically in space and all have the same length to leading order in 1/N. 4. Conclusion We have gone through the first step in setting up the " C o p e n h a g e n v a c u u m " for N ~ co. The field configuration having minimal energy turns out to be very easy to describe: all magnetic fields lying in the Cartan-algebra should have the same characteristic length. In renormalization group invariant units the most favorable strength of the magnetic field is reduced relative to its SU(2) value by the factor [eqs. (11), (19), (25)]

14 (su(co))l/l~ (su(2))l

= e 1"'/42 = 0.91 . . . .

(33)

The various color components are isotropically distributed in space. The value of the energy density grows with N as N 2, as anticipated from general N --) co result. The same is true of the imaginary part due to unstable modes. Hence we would expect domain formation to take place for any value of N much in the same way as it occurs in SU(2) (refs. [2-4]). In a first approximation domains in SU(2) may be described as forming a "lattice of spaghetti" [3, 4] or color magnetic flux tubes, each carrying an SU(2) unit of 't Hooft flux [4, 10, 11]. Similarly, in our case we might imagine many different spaghetti directions corresponding to the many different directions of magnetic fields. The generalization of the necessary analysis appears rather complicated, however, and we only give a few general remarks. In refs. [4, 11] an appealing relation was provided between the emergence of a gauge-invariant ZN flux and periodicity conditions on the spaghetti vacuum. In our case we should have different periodicity lattices for different color directions and we do not know how to formulate this in a gauge-invariant way. On the other hand, the physics of domain formation seems rather easy to generalize. In SU(2) the vacuum energy receives a contribution from all possible field modes (one-loop approximation): co

g'VAc(SU(2))=const. I_ dk3 Z OO

{(2B(n+3)+k])l/2+(2B(n-½)+k])l/2}.

~1=0

The difference between the first and last term in the curly bracket is due to the energy difference of the two different polarization states of the gluons relative to the

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245

b a c k g r o u n d field. F o r n = 0 we get an i m a g i n a r y p a r t w h e n Ik3L < , / B ,

(35)

w h e r e k3 is a t r a n s l a t i o n a l m o m e n t u m of the field m o d e . T h e effect of d o m a i n f o r m a t i o n m a y be v i e w e d as a c u t t i n g off of these low m o m e n t u m m o d e s . This is in a c c o r d with the fact that d o m a i n s are c h a r a c t e r i z e d by the a r e a a = 21r/B.

(36)

T h e s p e c t r u m eq. (34) c o m e s from t h e differential o p e r a t o r in the b i l i n e a r p a r t of the action, eq. (12). In fact, again this e q u a t i o n d e c o u p l e s different values of (i, k). T h u s we might e x p e c t the (positive helicity p a r t of the) IWk~rS tO be p e r i o d i c in a p l a n e p e r p e n d i c u l a r to the field B ~ik~ = B I') - B Ik) ,

(37)

27/adk --iB,k~l •

(38)

with lattice a r e a

But f r o m o u r d i s t r i b u t i o n ~ ( B ) we k n o w the d i s t r i b u t i o n of the differences (37). T h e s e d i f f e r e n c e s r a n g e from 0 to 2/~ and c o r r e s p o n d i n g l y the d o m a i n sizes r a n g e f r o m rr//? to oo w i t h / ~ ' given by eq. (25). T h e m a i n p o i n t we w a n t to m a k e is that t h e r e c o n t i n u e s to be a finite m i n i m u m (characteristic) d o m a i n size in the N - - , oc limit. T h a t m i n i m u m size is a f a c t o r e -1/4 t i m e s the SU(2) value. W e have p r o f i t e d f r o m c o n v e r s a t i o n s with H.B. Nielsen a n d P. O i e s e n .

Note added in proof A f t e r s u b m i s s i o n of this p a p e r we r e c e i v e d a p r e p r i n t by J e ~ a b e k , W o s i e k a n d Z a l e w s k i [12] t r e a t i n g the s a m e p r o b l e m with the s a m e result.

References [1] N.K. Nielsen and P. Olesen, Nucl. Phys. B144 (1978) 376 [2] N.K. Nielsen and P. Olesen, Phys. I.ett. 79B (1978) 304; J. Ambjorn, N.K. Nielsen and P. Olesen, Nucl. Phys. B152 (1979) 75 [3] ll.B. Nielsen and P. Olesen, Nuel. Phys. B160 (1979) 380 [41 J. Ambjorn and P. Olesen, Nucl. Phys. B170 [FSI] (1980) 60; 265 [5] H.B. Nielsen and M. Ninomiya, Nucl. Phys. B156 (1979) 1 [6] S.G. Matinyan and G.K Savvidy, Nucl. Phys. B134 (1978) 539; G.K. Savvidy, Phys. Lett. 71B (1977) 133; M.J. Duff and M. Ram6n-Medrano, Phys. Rex,. D12 (1976) 3357; H. Pagels and E. Tomboulis, Nucl. Phys. B143 119781 485 [7] H. Flyvbjerg, Nucl. Phys. B176 (1980) 379

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[8] G. 't Hooft, Nucl. Phys. B72 (1974) 461; B75 (1974) 461; E. Witten, Cargese Lectures (1979); S. Coleman, Erice Lectures (1979) [9] P. Olesen and J.L. Petersen, Nucl. Phys. B181 (1981) 157 [10] G. 't Hooft, Nucl. Phys. B138 (1978) 1; B153 (1979) 141 [11] J. Ambj~Irn, B. Felsager and P. Olesen, Nucl. Phys. B175 (1980) 349 [12] M. Jezabek, J. Wosiek and K. Zalewski, Quasiclassical vacua for gauge groups SU(N--,oc), Jagellonian Univ. preprint TPJU-2/1981 (January, 1981)