Derivation of the Skyrme-Witten lagrangian from QCD

Nuclear Physics B260 (1985) 241-252 © N o r t h - H o l l a n d Publishing Company

DERIVATION OF THE S K Y R M E - W I T Y E N L A G R A N G I A N FROM QCD Alexander ZAKS*

Physics Department, Northeastern Uniuersity, Boston, MA 02115, USA Received 14 December 1985

A derivation of the Skyrme-Witten lagrangian from the Q C D action is presented. The chiral field is identified with the phases of the left- and right-handed quarks and its action is derived by integrating over the fermion modulu those phases. This integration is effected by gauging the chiral symmetry through the introduction of pure-gauge fields operating on the flavor indices. As a result, an identification of the parameters of the SW lagrangian with Q C D correlation functions is made.

The Skyrme model [1], recently revived and improved by Witten [2], has enjoyed a large following in the last year. It has become a favorite tool with elementary particle as well as nuclear physics theorists. It has been applied to calculations of hadronic spectra [3], resonances [4] and even to the evaluation of nuclear forces. In all those applications one has to choose various parameters, most frequently the Skyrme term, so that the results of calculations provide the best possible fit to the data. Witten's derivation of the model, while enjoying the beauty and elegance of relying only on general and well-established properties of QCD, is incapable of providing the actual values of those phenomenologically important parameters. It is for this reason mainly that a direct derivation of the Skyrme-Witten lagrangian from Q C D may be of not only theoretical interest but of practical value as well. To see how one may attempt such a derivation, recall the ingredients of Witten's argument. The dynamics of the Goldstone bosons are described by the fluctuations of the Q C D order parameter, X ~b = {+~et)r} ~ -b inside the ,, valley" of equivalent vacua. Those fluctuations may be described by a unitary matrix U(x) and the SU(n)~e× S U ( n ) r × U(1) chiral symmetry is represented nonlinearly as U(x) ~ V~ U(x)V r Vz, Vr ~ SU(n). To obtain from Q C D an action for the fields describing the Goldstone bosons one should, in a sense, bosonize the fermionic part of QCD. A complete bosonization in 4-dimensional spacetime is of course impossible so one may attempt some sort of partial bosonization. In particular, one may attempt to bosonize the * Supported in part by NSF grant no. NSF-PHY-811675. 241

242

A. Zaks / Skyrme- Witten lagrangian

phases of the Fermi fields by considering ~pe(x) ~ V e(x)~e(x) ~p~(x) ~ Vr(X)~r(X ) as a change of variables. This is equivalent to replacing the QCD action by

s=

f dx Eab ~)(x)[iD 8~b + A}b(x)] +~e(x) +~)a(x)[iD~ ab + ~rab(x)] IPrb(X) "{-Sg, (1)

where D is the usual color covariant derivative, Sg is the gauge action and a, b are flavor indices. Here

A e.(x ) = iVy(x) O~V¢(x), Under

Art,(x ) = iV+(x) O.Vr(X ) .

such a change of variables the order parameter X

(2)

transforms to

V~(x)X(x)V+(x) and by investigating the effective action for V~(x) and Vr(X) holding X fixed, one is probing exactly the aforementioned fluctuations of the order parameter. I wish, then, to promote V~, V~ to the status of quantum variables and investigate the partition function

z = f td ~7d ~ I[dVA [dVr ] [d gaugelexp[ S + This procedure elevates the original global SU(n)eX gauge symmetry (at the classical level):

source

terms I .

SU(n)r symmetry to a local

CAx) --, u A x ) ~ A x ) ,

Cr(x) -~ Ur(X)+r(X),

VAx)-~ VAx)u~(x),

Vr(X)-~ Vr(X)U+r(~),

A~e. ~ u~eA.u~ + iu~eO.u~ ,

(3)

Ar~ --~ UrAr.U + + iu r O~U+

(4)

and reduces the phases of the fermions to gauge artifacts (so that the integration over fermions may be regarded as integration modulo the phase). The global SU(n)~× SU(n)r symmetry is represented as g.g(x)-'+ g g g £ ( x ) ,

gr(x ) ---+g r g r ( x ) , ~Jl?(x), •r(X) unchanged!,

(5)

with /_,re and Ur constant SU(n) matrices. The obvious candidate for the Skyrme-

243

A. Zaks / Skyrme-Witten lagrangian

Witten chiral field is U(x)= V£(x)V+(x) which indeed transforms under (5) as

U y ( x ) t y +. Before going any further let me put the (appropriate) objection of the alert reader at rest. I do not claim that the classical gauge symmetry (eq. (4)) of S survives quantization, neither that the action in (1) is equivalent to the QCD action. Indeed, those two are connected. For if the SU(n)~ × SU(n)r gauge symmetry would survive quantization, one could simply use it to "gauge away" A£ and A r in S to obtain the Q C D action. The gauge symmetry, however, fails to survive quantization due to the non-abelian anomaly [5].

o

=

U~

h =£,r,

24.'""'Tr{X°0'[A"0"AJ

(6)

which does not vanish even for A~ which are pure gauge. Later on, I will use this fact to derive the correct chiral action. The question that may be asked now is, if the action in (1) is not equivalent to the Q C D action what is the justification of using it? The answer to that is quite simple; the Q C D average (v.e.v.) of any function of q~£qJr, --a b and +r+£ --a b is equal to the average of the same function of (~eV~)~(V~+r) b and (~rVr+)a(Ve+e) b with S as an action! To prove this statement one has to show that the S averages of V~+~£V~ and V e + ~ V + are independent of Ve and Vr. To do that calculate the free energy W:

z= exp[ W( A.., A..,v~+sv; , v£J + vr)] = f[d~ d~ ][dgauge]exp[S + f~.V+JV£~£+ ~.VJJ

+

Vr~r]

(7)

and I have to show that

~60£,r

where

8~d£, r =

~J1 "'" O'-~n + W = 0,

(S)

V~r~V£,r. But (8) can be written as

[88a.1 . . . ~8. . . ] 8___L_w = 0, 8,O~,r

(9)

however, J and J + enter eq. (6) in a gauge-invariant way and therefore 8W/8~ is independent of J and J + [6] which proves the argument.

A. Zaks / Skyrme-Wittenlagrangian

244

So the objects of interest are functional derivatives of Z (or W) w.r.t. J and J+ integrated over V e and V~. These objects have the following integral representation:

f [dV,dV]P [ 3 j3+ ,=,=o, + 33] J z = f [ d V , dV~][dJdJ+]IdXdX +]

= ftdV, d<]tdX+d X l P ( , x , iX+)exp[Aeff(A~£, A,r, V~XVr, V+X + V~e)] = f[dV, dG][d X + d X] P(iVeXVr +, iVr X+ V~ )

Xexp[Aeff(A~,,Atzr, X+,X)], where I have used the change of variables J+----~VrJ+V}, X----~V£XV+. defined by

(10)

Aef f

iS

exp[Aeg(A~,2, A~r,X,X+)] = f[dJdJ

+ ]exp[ W(A,,.e, A~,~, J, J + ) - i t r f d 4 x ( J + X + X + J ) ] .

(11)

I will show now that W e a n be written as W I + ~ where WI(A,, At, J, J+) is gauge invariant and ~(V e, V~) is independent of J and J+. To simplify the analysis I will perform a vector gauge transformation on (7), i.e., V , ~ V,V +, Vr ~ V y + with V = Vr. Since the vector currents are not anomalous the symmetry is good at the quantum level, W(A,z, A,r, Vr+JV~, V}J+ V0 = W(A,~e,O, JU, U+J +) where U = VeV+ and A , , = iU + O,U. Now let 3o~ = iU 13U and consider

3W 3~o a

= t D~,°J~.e) = anomaly,

(12)

where anomaly is the r.h.s, of eq. (6) with A , , = iU + O~U. The solution to (12) is W(A~a ) = W I ( A , , ) + f(anomaly) 30~ where 3Wi/3o~ = 0 (i.e., W I is gauge invariant). To evaluate the integral [7] we pick a one-parameter family of U's, say U(x, "r),

A. Zaks

245

/ Skyrme- Witten lagrangim

0 G 7 G 1, so that

60 = iU+ S,U, and using this parametrization

/

(anomaly)

we have

iN. SW = - ;E 2drn2 x

[(u+

(13)

PYd

/

‘d7 0

d4X Tr /

uapa,

a,u>a,( u+a,u) + t(u+ a,u>(u+a,u>(u+ a,u)] , (14)

where NC is the number of colors. Using the antisymmetry of the product unitarity of U (i.e., &!Ji U= - Ui au), eq. (14) can be written as d4x Tr

d5y Tr

and the

[(u+ a,u)( u+a,u)( u+ a,u>(u+a,u>(u+ a,u)] [

( u+ a,u)( u+a,u)( u+a,u>(u+a,u>(u+ a,u)]

,

where Q is a disc in Sdimensional space whose boundary is spacetime, yi = x,, i = 1. _. 4, y, = r and the last step [7] involves a factor of 5 in going from the 4-dimensional E-symbol to the 5-dimensional one. Evidently (15) is the Wess-Zumino term [8] and we have W(Apl, JU, U’J’) = W,(A,,, JU, U+J+)iNc4,, and what remains is to determine WI. By construction, WI is gauge invariant, also, since 4wz is independent of J and J+ all variations of WI w.r.t. J and Jt coincide with those of W, which at ApP= 0 will information about the variations it is clear that S”W,/SA$ at anomalous parts of the diagrams and higher-order derivatives of X, X’) can be action, A eff( APP, with

coincide with those of Wq,,. I will also need of WI w.r.t. A,,. Again, from the definition of WI A, = 0 coincide with S”WQ,,/GAI provided all are excluded*, but since those occur only in thirdW they will not enter the calculation. The effective written now as AeEf= gcfP(ApP,X, X’) - iN,Lf,,(U)

exp(ge,,(API,X,X1))=/[dJdJ+]exp

WI-i/d4xtr(X+J+J+X)

1. (16)

* Such a subtraction can be carried out unambiguously since all coefficients of anomalies are finite.

246

A. Zaks / Skyrme- Witten lagrangian

Now, as W 1 is invariant under U(x)--. U(x)V(x) U+(x)~ V+(x)U+(x) J(x) J(x)V(x), etc., ~ff will be invariant under U(x) ~ U(x)V(x)... X(x) -~ X(x)V(x)... etc. As a consequence of its gauge invariance 5~ff can be expanded in terms of gauge-invariant operators. Of those some will vanish, e.g., any one proportional to F = OtA - O~At + [At, A~] and some such as t r X + X tr(X+X) 2, etc., will be irrelevant to the dynamics of the U's. The interesting operators will be those containing covariant derivatives of X ÷ and X, e.g., tr( O,X++ U+ OyX+)( O , X - X U + O,V),

(17)

which will contribute to the kinetic term tr(OtUOtU+). The effective action ~ff actually describes a complex Higgs field X coupled in a gauge-invariant way to a pure gauge field, Ate. The global symmetry of the model is S U ( n ) e × SU(n)r with

Vy(x)V/,

v;x(x)vr

(18)

and since QCD breaks the chiral symmetry down to the diagonal SU(n) the effective potential must produce an expectation value ( X ) which will be gauge equivalent to ( X ) - I, and choosing ( X ) - I will be equivalent to fixing the unitary gauge. Since I am interested only in the low-lying excitations described by 5eff I will simply freeze X and X ÷ to be proportional to the unit matrix and expand in powers of Ate so (10) reduces to

f[dvl

V(xi)exp[ ~eff(Ate, Xo, X~)-iNc~wz(U)]

(19)

and X 0' X~- - I while the proportionality constant is to be determined. The next step is to calculate ~eff. Since the exact integration of eq. (16) is not practical one has to resort to approximations. The appropriate one in the present case, which, as I will argue later, becomes exact in the N~ 1' m limit, is saddle-point integration. In the same limit the "freezing" of X, X + to their vacuum value (up to gauge rotation) is exact as well. The saddle-point integration of (16) amounts to solving

8J = iX+'

8wi

8j + = iX,

(20)

for J and J + in terms of X and X + and saturating the integral by the "saddle point" ~eSfp(A~£,X+,X) =

WI(A,e, Jo, Jff-)-itr f d x 4 ( j ~ X + X+Jo),

(21)

where J0, Jo+ are the solutions of (20) in terms of X +, X and Ate and sp stands for saddle-point approximation. Now the X-integral is performed in the sp approxima-

A. Zaks / Skyrme- Witten lagrangian

247

tion to obtain ~ff(A~e, Xo, Xff) where X o and X 6- are the solutions to [9] 8D Xo,x~,A~=o _

8~+ 8X

Xo,X<~,A~=o= 0

(22)

which sets

Jo = J~

iX~)b=

= O,

l A O'..+ab= (lp£~r)-a b

b

= ~r~£)--a

- 8 "b.

(23)

Before turning to the analysis of the expansion of ~ there is the question of the validity of the saddle-point approximation. If indeed it is a valid approximation it must be a leading order of an expansion in a small parameter. First observe that the integral

]

f[dJdX]r-fliX(xi)exp[W(J)-ifJX

(24)

can be performed exactly by reversing the sequence of eq. (10) and integrating over X first yielding

8----.-L-eW(J) = z(I~i o(xi) ) ~Ii 8J(xi) J=O

'

(25)

where O is the operator to which J couples. On the other hand, doing the J-integral by sp approximation (24) reduces to

f[dX] 17iiX(xi)e(X).

(26)

Now, since I wish to consider ~ as a bona fide quantum action (26) has to be integrated exactly. If, however, I ignore that for a moment and approximate (26) by the saddle point I get

8X o-

J=O,

Xo = ( 0 ) ,

f dxI-IiX(x~)e~(X)=z1-]x°'i Z = e w(°) = e ~(x°) .

(27)

And obviously eq. (27) yields the correct result (in leading order) if ( I ~ i O ( x i ) ) = + O(e). And this indeed is the case for QCD if O is (color) gauge invariant [10] (in the present case ~ k ) and e = 1/N c. Now, the approximation in (27) to the X-integral is exact if ~ - N C, which is indeed the case for the SW action [8]. A simple analysis will reveal that the subleading contributions to (25) (i.e. down by powers of 1/No) corresponding to various connected contributions to (1-lOt}

Iqi(O(x~) )

248

A. Zaks / Skyrme. Witten lagrangian

agree exactly with the corresponding subleading contributions to the integral in (27). This establishes the claim that (19) is the leading order in 1/N c expansion. At this point, all that remains to be done is the expansion of 5 in powers of Ag~. Notice, however, that there is an alternative expansion one could use. Utilizing the SU(n)~¢ gauge invariance of 5,

5( <~, xo, x~ ) = ~(o, UXo, x; u+ ),

(28)

the SU(n)~X SU(n)~ invariance of 5(X, X+),

5( X,X+)= f d4x[V( X+X)- ZI( X+X)trO.X+O.X + Z2( X+X)tr O2X+ O2X + ...]

(29)

and substituting X = UXo, X +-- X~ U + into eq. (29) one can relate the various coefficients in the SW lagrangian to the Z ' s in (29) (the Z~'s are of course independent of U). This will involve evaluating various n-point proper vertices (2PI functions) of the chiral order parameters which proves to be tedious and not very revealing. Returning to the task at hand, I wish to expand 5 as

5(v+oy, x + , x ) = E ~ 1 tl

f d4xl.. " f d4x. C~[b~.;;:a.b.(xl...x. )

X H(iU+(xi) i

r<°~),

~-'t~a...~.

<°b)°
o~,

8A,, (xl)

1

. ~n ..8

(30)

Op,i U ( x i ) ) a'b' ,

a b

A,~ "(x,)

~(A~ ,x+,x)

X +

,x,A=0

,

where, as indicated the derivatives are taken with constant X +, X and at A = 0. The first derivative, i.e., C~b(x), is given by (supressing euclidean and flavor indices)

85

aA(x)

x.x +

8W, J,J+ r 4 [ 8W, 8J(y)

~A(x)

+Jd Y[aJT;)8ACx)

~W I 8 J + ( y ) 8J+Cy) 8ACx)

( 8JCy) 6 J + C y ) ) ] - i X+Cy)3--~(-~(x)+ 6ACx----~XCy) , where 3W/6J and S W / / ~ J + a r e evaluated at constant A and 8J/6A and constant X, X +. Using the definition of X and X + (31) reduces to 65

~A;~(x)

A=0 =

6WI

eAX~(x)

(31)

6J+/SA

A =(f)(x)7,+~(x))=(J,~(x))=O, =0

at

(32)

A. Zaks / Skyrme-Wittenlagrangian

249

where the average is taken with the usual Q C D action (since at A ~ , = 0 W I is equivalent to Woci) and X = X 0 implies J = 0). The coefficient of the bilinear term calbl,azb2~ _,, ,x, y ) is given by

'32

~ x,x+ =

82WI

'3A,(x) '3A,(y)

J,J+

'3A,(x) 6 a , ( y ) '32WI '3J(z) x x + '3A,(x)'3J(z) '3A~(y) .

+ f d4Z

'3~w~

'3J+(z) .,x+]

+ '3Au(x)'3J+(x) 8--~,(y)

(33)

"

The first term on the r.h.s, of eq. (33) is simply ( J~* ,b (x)J;,ca(Y))c again, calculated in the Q C D vacuum. The other terms can be calculated as follows:

'3

'3AtL(X)[~j~-~"~''WI]~ -(J~t~e(X)~r(X)~j£(z)),

(34)

where again A~ = 0 at X = X 0 imply that the averages reduce to the Q C D ones. Using '3~/'3X= - J + and '3~/'3X += - J

'3J(x) x, '3A~,(y) x+

'3 '3X+(x)

'3A.(y)

r

'32WI

=-fdz

'3w~ '3 '3x+(x) '3A~,(y) '32w,

'3J(z)

[ '3J(z) '3A,(y) '3X+(x)

'3J+(z) '3A,(y) 8X+(z)

(35)

]1 '3x -

'3j + '3X+-

and a similar expression for

'3x t '3x +1=

'3 [ '3~ ] [ '3X+

'3J+/SA.

~

=

'3j+(~) ],

~ T ) = w'

,

'32

] 1

'3j+'3j+ WI

,

(36)

250

A. Zaks / Skyrme-Witten lagrangian

The evaluation of eqs. (33)-(36) is considerably simplified if one defines X s = X + In these coordinates J~ couples to ~ + and Jp to ~i'/5~ and the present choice of the direction of the chiral condensate is the standard one, (~/~+b)_ 8~b (~/~75q~b)= 0. Using the new coordinates and some straightforward but still tedious algebra, the last two terms in eq. (33) combine to give 3- X, iXp = X + - X and Js = J+3- J, /Jp = J + - J -

82

~ f d4z f d4z'(J/'Q(x)Tki(z)'yS'U(z) 8X~/(z)SX~p'(Z ' )

X ~}(Xp, Xs)(Jf~a(y)~k(z')ys~/(x')),

(37)

where I have used (J.~(x)~(y)6(y)) = 0 (which follows from (~'/5~) = 0). Now, to leading order in 1/N c the current correlation function (in momentum space) can be written as [10]

(J~z(P)J;z(-P))- --½8adScb(PZS~'~--P~'P")

+ ~i p2 +m: '

where the sum is over all intermediate meson states. Here m i are the meson masses and .f2 their coupfings to J~. If we are interested only in low-momentum dynamics, i.e., p2 << mo where m 0 is the lightest of the massive mesons, the poles in the sum on the 1.h.s. of (38) will be dominated by the mass and

(j~,.e(p)j~.e(_p))_½f~r2(6~, ~ p~p~ 1/N c

while eq. (37), to leading order in propagator) [9] contributes '

2 m7

(39)

(note that ~2~/~X2 is the pion inverse

2P"P"

-- 2~adScrf~ p2

(40)

Inserting this into the expansion in eq. (30) and using the fact that

o (v + o

v)-o.(v + o v)= [v + o y , v + 0 y ] ,

(41)

the action to this order becomes

f d4x{lfZtr(OyOy+)+~tr([U + Oy, U + 0 , U ] 2 ) ) ,

(42)

A. Zaks / Skyrme-Witten lagrangian

251

where* 1 / e 2 = EifZ/m2 and the terms with more than 4 derivatives were neglected. Notice however, that there may be additional contributions to the Skyrme terms. The next two terms in the expansion have coefficients proportional to ( J;~(xi)J~ff(x2)J~fg(x3))c and ( J;~(x1)JCff(x2)J~(x3)Jl~(x~))c **. Those current correlation functions may possess "contact terms" of the form [10] (3bc3deSul-3d,fihe3f,,)3,,fl4(X~- X2) O~fi'(X2 --X3) for the 3-point function and ( 3bc3de3fe,3,~,,~da~be~fg~hc)~et~l,~4(Xl--X2)~4(X2--X3)~4(X3--X4) for the 4-point function. Those, however, will be accompanied by terms analogous to those in eq. (37) for the two-point function. In particular the three-point function contribution to the coefficient of (U + OU) 3 will be accompanied by the terms

( .

d

d4yl... d 4Y4 A~(xlYx)Aj,(x2Y2)Aa(x3,1 1 2 Y3, Y4)~2(Y1, Y3)~2(Y2, Y4), (43a)

f

d4yl...

1 1 1 d 4y3A,(xxyl)A~(xzyz)A~(x3Y3)~3(yl, Y2, Y3),

f d4yld4y2B~,(xlxzyt)A~(x3Y2),~2(YlY2),

(43b)

(43c)

where

Al(xy) = (J~eCx)~(y)ys~(y)) , A~(x, y, z) = (J~eCx)f~Cy)ys~(y)~(z)ys~(z)),

B~(x, y, z) = ( L , ( x ) L t ( y ) f f ( z ) r ~ ( z ) ) and

~ ( x y ) = a ~ / s X ~ ( x ) 8Xv(y),

~,(xy~) = 8'~/sXv(x) 8Xv(y) axv(~).

Of these terms (43c) has the exact same structure as the "contact" term for (JJJ) and the two terms will cancel. And a similar thing will happen with the four-point function as well. Before concluding let me make one remark about renormalization. Though all the formal manipulations leading to the final result I have ignored the question of UV divergences. The way one should think about this problem is as follows. Imagine defining the path integral by some nonperturbative gauge-invariant regulator (e.g., introducing higher covariant derivatives). Then, all the formal manipulations leading to the coefficients of the chiral lagrangian are well defined. Then, one should renormalize the various correlation functions as done in standard QCD calculations. * A l l these t e r m s h a v e the s a m e N, d e p e n d e n c e as f 2 . * * A s m e n t i o n e d earlier, (J1JlJl) and (JlJiJ1J1) s h o u l d be c a l c u l a t e d w i t h the a n o m a l o u s t e r m s excluded.

252

A. Zaks / Skyrme-Witten lagrangian

Using the method of partial bosonization, implemented by gauging the S U ( n ) x SU(n) chiral symmetry of a vector-like gauge theory (QCD), I have shown that the dynamics of the Goldstone bosons is described by a pure gauge field coupled to a set of scalar fields. These scalar fields are basically the expectation values of the chiral order parameters in an external left-handed vector field. As usual, in a situation like this, it is possible to investigate the low-lying spectrum by freezing the Higgs field to its vacuum value and the resulting action for the gauge fields describes the dynamics of the massless degrees of freedom. A straightforward generalization of the technique presented here can be used to analyze the coupling of the vector mesons to the chiral lagrangian. The idea is to introduce an additional source for the vector currents which after extracting the fermion phases will be of the form, f/,eUJ~,7~,U+q~e and ~rVJ~,y~,V++r along with the appropriate order parameter J,V~. This will transform the terms U ÷ O,U and V + O~,Vin eq. (1) to U + ( 0, + J,) U and V ÷ ( 0, + J,) V, gauging the vector part of the chiral symmetry defined in eq. (5) and continuing along a line of arguments similar to the one followed in the paper. This program, however, will not be pursued here. I am indebted to V.P. Nair for suggesting the idea of partial bosonization and for helpful conversations. References [1] T.H.R. Skyrme, Proc. Roy. Soc. A260 (1961) 127 [2] E. Witten, Nucl. Phys. B223 (1983) 433; A.P. Balachandran, V.P. Nair, S.G. Rajeev and A. Stem, Phys. Rev. Lett. 49 (1982) 1124; Phys. Rev. D27 (1969) 1153 [3] G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B228 (1983) 552; G.S. Adkins and C.R. Nappi, Phys. Lett. 137B (1984) 251 [4] J.D. Breit and C.R. Nappi, Phys. Rev. Lett. 53 (1984) 889; J.D. Breit, Univ. of Penn. preprint UPR 0271T [5] S.L. Adler, Phys. Rev. 177 (1969) 2426; J.S. Bell and R. Jackiw, Nuovo Cim. 60 (1969) 47 [6] W. Bardeen, Phys. Rev. 184 (1969) 1848 [7] B. Zumino, Les-Houches lectures, August 1983; A. Dhar and S.R. Wadia, Phys. Rev. Lett. 52 (1984) 959 [8] J. Wess and B. Zumino, Phys. Lett. 3713 (1971) 95; E. Witten, Nucl. Phys. B223 (1983) 422 [9] E.S. Abers and B.W. Lee, Phys. Reports 9 (1973) No. 1 [10] E. Witten, Nucl. Phys. B160 (1979) 57