Volume 257, number 1,2 PHYSICS LETTERS B 21 March 1991

An action principle for self-dual Yang-Mills and Einstein equations

Stilyan Kalitzin Institute for Theoretical Physics, University of Utrecht, Princetonplein 5, NL-3584 CC Utrecht, The Netherlands

and

Emery Sokatchev l

LPTHE, Universit~ de Paris VII, F- 75251 Paris C~dex 05, France

Received 9 November 1990

We propose actions from which the self-dual Yang-Mills and Einstein equations in four dimensions follow by straightforward variation. This off-shell formulation essentially involves harmonic variables. After reformulation in harmonic space, the self- duality constraints can be derived from actions with the help of Lagrange multipliers. The latter turn out not to describe additional degrees of freedom. The off-shell theories obtained can be quantized in a standard way.

1. Introduction

The self-dual solut ions of Yang-Mil l s and Einstein equat ions have a t t racted a lot o f a t tent ion because of their possible appl ica t ion to non-per turbat ive euclid- ean field theory. A remarkable success has been the classif ication and construct ion of all f inite energy so- lutions o f the self-dual Yang-Mil l s ( S D Y M ) equa- t ions [ 1,2 ]. Usually, the self-dual equat ions are im- posed by hand, as addi t ional constraints which define a specific class of solut ions of the Yang-Mil l s or Einstein equations. It seems very interest ing to us to f ind an act ion from which the equat ions for self-dual- ity follow by straightforward variat ion. In such an approach the Yang-Mil l s (or gravi ta t ional ) field would satisfy the self-duality equat ions only on shell. Off shell they would be general unconst ra ined fields, and therefore could be quant ized by s tandard tech- niques. The problem of f inding an off-shell formula- t ion for self-dual equat ions has recently arisen in the context of string theory [ 3 ].

In general, it is bel ieved that self-dual equat ions cannot be der ived f rom actions. The reasoning is as

On leave from the Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria.

follows [4 ]. The only conceivable way o f obta ining such a constraint on the field-strength tensor is by means o f a Lagrange multiplier . However, the Lagrange mul t ip l ier usually satisfies an equat ion o f mot ion with t ime derivat ives ~. Thus the actual number o f degrees of f reedom is increased. For in- stance, take an abelian gauge field Au,(x ) in four-di- mensional euclidean space - t ime ( # = 1, 2, i = 1, 2 form a four-vector index; for the nota t ion see section 2) . The self-duality constraint is

F = P . ~ OuiA ~' + OujA ~ u = 0 . ( 1 )

A possible act ion for eq. ( 1 ) could be the following one:

S = ~ d4x PUJ)( OuiA~' + OujAi~) . (2)

Varying S (2 ) with respect to the Lagrange mul t ip l ier puJ) (x ) gives eq. ( 1 ). However, varying with respect to Au,(x) , one gets Ou~PUJ)=O. Obviously, this im- plies [~puJ) = 0, so the field puJ) ( x ) describes those degrees of freedom, which eq. ( 1 ) kills in Aui(x) .

~t In ref. [5] an action has been proposed, where the Lagrange multiplier drops out from the equations of motion. However, that action was non-linear in the constraint on the physical field. In the present paper we consider only linear actions.

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In this paper we propose a solution to the above problem in four dimensions (the generalization to 4n dimensions is straightforward). It involves a refor- mulation of the self-dual Yang-Mills and Einstein equations in harmonic space [6,7]. There one can safely employ Lagrange multipliers without increas- ing the number of degrees of freedom.

The paper is organized as follows. In section 2 we first present the basic notion of harmonic variables and their properties. Next we reformulate the self-dual Yang-Mills equations in terms of harmonic-space differential-geometry objects and obtain an action from which these equations follow. We also fix the gauges and list the propagators derived from the qua- dratic part of the action. Section 3 is devoted to the self-dual Einstein equations. They are also consid- ered in the framework of harmonic-space differential geometry. An action is presented for that case too.

2. Self-dual Yang-Mills equations

2. I. SD Y M constraints in harmonic space

We begin by recalling the formalism of SDYM the- ory in ordinary euclidean four-dimensional space E4. The Lorentz group of~ 4 is SO (4) ~ SU (2) Z SU (2), so this space can be represented as the coset

~4: {x~i} ~ SU(2) × S U ( 2 ) ' (3)

where ~ is the Poincar6 group. The indices/~= 1, 2 and i= 1, 2 are spinor indices of the two SU(2) groups, correspondingly. The coordinates x/'~ satisfy the reality condition x "~ = ~u,~ox"L In ~4 o n e consid- ers matter fields q~(x) transforming under a YM group with parameters r(x):

(0' (x) =exp[ i r (x ) ] ~o(x), (4)

as well as YM fields introduced through the covari- ant derivative

~ , = 0~ + i A ~ ( x ) ,

A~ = - i exp (it) (0~ + iAu~)exp ( - i t ) . (5)

The gauge independent object in the theory is the field-strength tensor defined by the commutator

[ ~ui, ~,J] = iFui,~J = i~u,F(ij) + i~oF~u~) • (6)

The two tensors Ft0 ) and F(u, ) in (6) correspond to the two degrees of freedom of the YM field. The the- ory is called self-dual when one of these tensors van- ishes, for instance

F~o ) = 0 . (7)

This constraint, together with the Bianchi identity for (6), implies the YM equations of motion ~U'Fui,,j= O, so one can say that eq. (7) defines an on-shell theory.

Our first aim will be to reformulate the constraint (7) in the form of an integrability condition. In this we shall follow the approach ofref. [6 ]. First we shall extend the space ~4 to the so-called harmonic space [8]:

HE4: {xUi' u±i}~ SU(2) " (8)

Unlike the case of E 4 (3), here we have divided the Poincar6 group ,~ by one of the SU (2) subgroups of the Lorentz group only (the one associated with the index ~). The remaining one is parametrized by the harmonic coordinates u ±'. The latter are defined as SU (2) matrices,

u + i u T = l , u + i = u T , (9)

carrying one SU (2) index (i) and one U ( 1 ) index ( _+ ). The functions of u ±~ are defined as harmonic expansions in terms of the irreducible products of u ±i carrying a fixed U ( 1 ) charge q:

.--'°+~J,--J. + + _ f~q)(u)= f ui, ...ui,+~uj, ...uf, (10) n=0

(here q>~0; for q < 0 the expansion is analogous). Thus, the functions (10) are effectively defined on the coset S U ( 2 ) / U ( 1 ) ~ S 2, so one can think o fH~ 4 as of the tensor product HR 4 = ~4 X S 2. It should be pointed out that the harmonic variables are in a sense similar to the twistor variables on S 2 [ 1 ]. The main difference is that u ± describe S 2 globally and the SU (2) symmetry is maintained manifest. For fur- ther comments on the relationship between the har- monic and twistor approaches see ref. [ 6 ].

The introduction of the new variables u ±~ will help us to reformulate the SDYM constraint (7) in a more suitable form. To this end we multiply (6) by u +~, u +j and use (7) to obtain

[ ~ - , ~ , + ] = 0 , (11)

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where ~ " is defined as follows:

~ + _= u+'[Ou,+iAu,(x) ] . (12)

In fact, eq. (11 ) is equivalent to the original con- straint (7). This is easily seen by using (12) and the fact that u ÷r, u +j are arbitrary commuting variables.

Remarkably, in the form (11 ) the SDYM con- straint is nothing but the integrability condition for the existence of analytic fields satisfying the covari- ant Cauchy-Riemann condition

~+~(x)=O. (13)

In the rigid case the analyticity condition

O; ~O(X, /,/)= O, O ; ~ u + i O t t i (14)

is solved by an arbitrary harmonic analytic function

~0=~0(x + ,u) , xU+---u+x ~'. (15)

Note that such analytic fields obey the Klein-Gordon equation:

[2~o= O ur Gr~o= ½0Ur(u7 u +J- u? u - 0 Gj~o

= 0u-0+~o=0. (16)

The integrability condition (11 ) has the following general solution:

A +(x, u) = - i e x p [ - i v ( x , u ) ] 0 + exp [iv(x, u ) ] ,

(17)

where v(x, u) is an arbitrary (for the time being) Lie- algebra-valued scalar field. However, the SDYM connection A ~- in ( 11 ) is a linear function of u ÷, A + =u+iAm(x) [see (12)] . This is not automati- cally so in ( 17 ), therefore we must impose a certain restriction on v(x, u). To this end we consider the derivatives with respect to the harmonic variables. One of them is

0 D + + = U + r - (18) 0u-r"

It is compatible with the SU (2) defining condition (9) (in other words, D + + is one of the two covariant derivatives on $2). It is not hard to see that A + (x, u) depends linearly on u + iff it satisfies the condition

D+ +A~- (x, u) = 0 . (19)

Indeed, by inspection of the harmonic expansion (10)

for q= + 1 one sees that only the term linear in u ÷ can survive in eq. (19).

Eq. (19) can be rewritten as a commutation relation:

[D ++, ~ + ] = 0 . (20)

Here the harmonic derivative D ÷ ÷ needs no connec- tion, since the gauge group (4) has u-independent parameters z (x). We can say that the original SDYM equation (7) has been equivalently replaced by (20), since the integrability condition (11 ) has already been solved in the form (17). Inserting (17) into (19), we find a restriction on the field v(x, u):

D ++ [ e x p ( - i v ) 0~ exp(iv) ] = 0 ,

which is the same as

0 + [exp(iv) D+ + e x p ( - i v ) ] = 0 . (21)

The conclusion is that any solution of the SDYM equation (7) can be obtained by taking a suitable field v(x, u ) satisfying (21 ), and constructing the connec- tion A ~ (x, u) (17). The latter is guaranteed to de- pend linearly on u +, so one can extract the connec- tion A ui (x) [ see ( 12 ) ], which gives a solution of the SDYM equation. The u-independent connection Aui(x ) can be constructed explicitly from v(x, u) as follows:

A~i(x) = f du e x p ( - i v ) 0 + exp(iv) u7 • (22)

The harmonic integral in (22) is defined by the fol- lowing rules:

I d u l = l , f d u + + - - (23) U (il ""Uim Ujl ""Ujn) ~ 0 •

The field v(x, u) is a gauge object with the follow- ing transformation laws. On the one hand, under the gauge group with parameter z(x) it transforms as

{exp[iv(x, u) ]} ' =exp[ iv(x , u)] exp[ - i t ( x ) ] , (24)

so that A + (17) transforms in agreement with (5). On the other hand, the solution (17) to the con- straint ( 11 ) has its own gauge freedom with an ana- lytic parameter 2(x, / t) :

[exp(iv) ] ' =exp( i2) exp( iv) ,

0~-2(x, u) = 0 . (25)

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So, exp(iv) can be called a "bridge" between two gauge frames, one with u-independent parameters r(x) and another with analytic parameters 2(x, u).

Concluding this introductory subsection we can say that the main object in our treatment of the SDYM problem will be the gauge bridge v(x, u). I f it obeys eq. (21 ), it defines a solution (in fact any solution) of the SDYM equation. Our aim in the next subsec- tion will be to write down an action for v(x, u), from which eq. (21 ) will follow as a variational equation.

2.2. The action principle

From now on we shall assume that the bridge v(x, u) is an arbitrary field, which is not subject to the constraint (21 ). Finding an action from which the on-shell equation (21 ) will follow, is in fact very easy. It is sufficient to introduce the constraint (21 ) via a Lagrange multiplier:

S = T r f d4x du P(-3)u

Xi0 + [exp(iv) D + + e x p ( - i v ) ] . (26)

Note that the Lagrange multiplier p(-3)u has U( 1 ) charge - 3 to cancel the charge of the rest of the in- tegrand (otherwise the harmonic integral would van- ish). Varying (26) with respect to p(-3)u immedi- ately produces eq. (21 ). The crucial equation is the one for P(-3)u following from the variation of S with respect to v:

0=6vS

= - T r j d4x dR fi[exp(iv) D + + e x p ( - i v ) ]

x i0+p( -3)u

= Tr J" d4x du

Xexp(iv) {D ++ [ e x p ( - i v ) fi exp(iv) ]} e x p ( - i v )

x i 0 f f p ( -3)u

~ D ++ [ e x p ( - i v ) 0+ P(-3)Uexp (iv) ] = 0 . (27)

The expression in square brackets in (27) has U( 1 ) charge - 2. Inspecting the harmonic expansion (10) for q= - 2, one easily sees that it can satisfy (27) only if it vanishes. This means that

0ff P ( - 3 ) u = 0 . (28)

Now the crucial point comes. Eq. (28) can actually be solved. Its general solution is

e(-3) = 0+~a u t~u~ ) = O+ a ( - 4 ) (29)

where a ( - 4) (x, u ) is an arbitrary scalar field of charge - 4 . At the same time, the action(26) has the ob- vious gauge freedom

p(-3), = p ( - 3 ) + 0~- b ( -4) , (30)

where b(-a)(y, u) is also an arbitrary field. So, the solution (29) is pure gauge. The conclusion is: the equation of motion for the Lagrange multiplier e(-3) does not describe any new degrees of freedom. This is an exceptional feature of the problem at hand, unlike most other cases where Lagrange multipliers are employed. Note that the same trick could be per- formed in a space-time with 4n dimensions. There eq. (7) generalizes the notion of self-duality, and the general solution to eq. (28) is of the form of the first equation in (29).

A few words about the gauge-transformation prop- erties of the action (26). It is obviously invariant un- der the u-independent gauge transformations (24) [ since D + ÷ r (x) = 0 ]. I f we wish to make it invariant under the analytic transformations (25) as well, we have to assume the following transformation law for p( -3) (use the analyticity of X):

p(-3), =exp( i2) e ( - 3 ) e x p ( - - i 2 ) (31) ,u

2. 3. Gauge fixing and quantization

The action (26) involves unconstrained off-shell gauge fields, so one can quantize it using the standard technique. First, one must fix the two gauge invari- ances ( 2 4 ) a n d (30). The appropriate gauge-fixing condition for the z gauge group (24) is

~ du v(x, u)=O. (32)

Indeed, in the abelian case (24) means 6v(x, u ) = - r ( x ) , s o

5 f d u v = - f d u z ( x ) = - r ( x ) . (33)

Thus, condition ( 32 ) completely fixes the parameter r (x) . The condition fixing the abelian gauge invari- ance (30) is

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0~-P(-3)u=0 • (34)

Indeed,

8 (0~- P(-3)u) = 0~- 0~+b(-4) = - D b (-4). (35)

The operator [] in ( [ 5 ) is invertible, so (34) prop- erly fixes the gauge.

The gauge invariance (25), (31 ) with analytic pa- rameters/l need not be fixed. The reason is that the parameters obey the on-shell equation (16). The lat- ter has a unique solution under suitable boundary conditions, so the parameters do not carry true (off- shell) degrees of freedom.

The two gauge-fixing conditions ( 32 ) and (34) can be implemented in the action with the help of Lagrange multipliers:

Sg.f. = ~ d4x du Tr[p(x)v(x, u)

+A (+4)(x, u) 0~- e(-3)~(x, u) ] . (36)

In the non-abelian case one needs Faddeev-Popov ghosts. In that case the transformation law (24) is non-polynomial:

By= - , - i [ v , r] + .... (37)

and so must be the coupling of v to the ghosts ( v= vat a, [t ~, t b ] =if~b~t~):

Sgh = ~ d4x du c ' a ( x ) [¢~ab+fabcvc (x , U) + . . . ] c b ( x ) .

(38)

In order to find the propagators one considers the bilinear action:

So = f d4xdu (e(-3)u ~+D++v+pv

+A (+4) O~P(-3)~+c'aca). (39)

The derivation of the propagators involves some knowledge of harmonic analysis. We refer the reader to ref. [ 9 ] and here give only the results. The non- vanishing propagators are

<v (1 )p ~_ 3 ) (2 ) )_ u+u; u~ i Oui ~(x l -x2) u? u~ []

( v( 1 )p(2) ) =g(x , - x 2 ) ,

<p(~-3)(I)A(+4)(2) )

=~(--4'4)(Ul, U2) u'~i Oil i ~(Xl --X2) , []

<c'a( 1 )cb(2) > m(~ab(~(X 1 --X2). (40)

Note the absence of a propagator for the gauge field (v (1 )v (2 )> .

A detailed study of the quantum properties of this theory is beyond the scope of the present paper. We would like to mention only that new specific diver- gences may occur in the harmonic sector.

3. Self-dual Einstein equations

T h e Einstein metrics that satisfy a self-duality con- dition in four dimensions are also called hyper-K~ler metrics. The treatment of the self-dual Einstein equa- tions is analogous to that of the Yang-Mills ones. The main difference is that the gauge group now affects the coordinates of space-time. The principle of har- monic analyticity, which is once again the comer- stone of the construction, now involves an analytic basis in space-time. As was shown in ref. [ 7 ], the solutions of the hyper-K~ihler constraints are para- metrized by two arbitrary analytic fields. They serve as prepotentials generating the bridge defining the analytic basis [an analog of the gauge bridge v(x, u) in section 2]. In the off-shell theory both the prepo- tentials and the bridge become unconstrained dy- namical variables. The equations of motion follow- ing from the action will be just the analyticity conditions for the prepotentials and the relation be- tween the bridge and the prepotentials.

3.1. Hyper-Kiihler constraints in harmonic space

Our starting framework is the riemannian space ~4 (with euclidean signature), in which acts the diffeo- morphism group

x lZit = x lti "~- "~ l~i ( x ) . (41)

The matter fields can carry tangent-space indices

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~0,#...u... (x), where or, fl transform under a local SU (2) group, and i, j, ... are rigid SU(2) indices. The co- variant derivatives are defined as follows:

~ a i = e o . "UkOuk Jr o)oa(a~)A ar. ( 4 2 )

Here eo, i u k ( x ) are vierbein fields and o9,,#~)(x) are connections [A #r are generators of the local SU(2) group ].

In general, the commutator of two covariant deriv- atives (42) defines the curvature (Riemann) tensor:

[ ~a i , ~#j] = (~ijR(a#)(ra) +~aBR( i j ) ( ra) ) A~'a. (43)

Like in the Yang-Mills case, self-duality means the vanishing of one half of this tensor, for instance,

R(o)(y~) =0, (44)

o r

[ ~'.(i, ~#j)] = 0 . (45)

Note that this is a particular case of the so-called hy- per-Kiihler geometry [ 10 ]. The latter is defined in ~ 4 n by the condition that the holonomy group is a subgroup of Sp(n) [rather than the most general O (4n) ]. In four dimensions this condition is equiv- alent to self-duality.

The reformulation of this theory in harmonic space is very similar to that of the Yang-Mills one. One re- places eq. (45) by two constraints:

[ ~ + , ~ - ] = 0 ~ - R ++[~] = ~ , ~ e u + i u + : R ( u ) = O , (46)

[ ~ + + , ~ + ] = 0 ~ ~ + = u + ~ i . (47)

Note that although the space-time ~4 has been ex- tended to H R 4 ( 8 ), the diffeomorphism group is still (41 ), i.e. the harmonic coordinates u _+i do not trans- form. Once again, eq. (46) is the integrability con- dition for the existence of harmonic analytic fields defined by the Cauchy-Riemann condition

~ + q%.. .(x, u ) = 0 . (48)

The main step in solving eq. (46) is the definition of an analytic basis in harmonic space:

x:~ + = x " u ; ~ +v~'+(x, u), x~- = x ~ u 7 ,

There the analyticity condition (48) fields) becomes simply:

U +9 "

(49)

(for scalar

0 oz.~0 - ~ ~0=0 ~ ~0=~0(x3, u-+) . (50)

Eq. (50) and its solution are covariant under the spe- cial diffeomorphisms

(x~+) ' = x ~ + +Z~+ (x + , u -+ ) , (51)

with analytic parameters 2 ~+. Therefore the "bridge" v u+ (x, u) connecting the two bases (x "i, u -*i) and (XA u-+ , U +_i) has the following transformation law:

v"+'(x ', u' )

=vU+ (x, u )+2~+ (x +, u ) - r U i ( x ) u : - . (52)

The complete solution of the hyper-Kiihler con- straints (46), (47) goes through several more steps. We refer the reader to ref. [ 7 ] for the details. Here we only give the net result. Any solution of the hyper- Kiihler constraints can be constructed from two ar- bitrary analytic objects:

~ ( + 4 ) (Xhl- , U + ) , (~/j+ ( X A+ , U - - ) , +

O + G ~ ( + 4 ) _ _ + + Au-* - 0~u£:~ = 0 . (53)

They are defined up to gauge transformations,

~ ( P ( + 4 ) = 0 + +/~ + + Jr 0 + + • g+ "(~/t+ ,

a~" = ~- ~ + ~ + a - ~ + + (54) - - VA#+~ -z. v - - ...+Au+~ ,

where O+ + = u + i O /Ou - i , 02~ - O / O x ~ + and +~++= 2 + + (x~-, u) is a new analytic (pregauge) parameter. The bridge v u+ (x, u) defining the analytic basis (49) is not an independent object. It can be found as a so- lution of the harmonic differential equation

~ + + v U + = ½ H ' U V ( O 2 v ~ ( + 4 ) + O + + , ~ + ) . (55)

Here

~ + + = 0 + + + ~ + + v "+02~+(x~+-v"+)0A+~ (56)

is the harmonic covariant derivative in the analytic basis (49) [in the r basis ( x ui, u +-i) it is simply ~ + + = 0 + + ] , and

OauL# , - 0A~5¢ u . (57) H U ~ H ~ = 3 ~ , H u ~ - - + - +

Note that the analytic prepotential L# + is essentially pure gauge, since the analytic parameter 2 a+ in (54) can gauge it away.

Once the solution of eq. (55) has been found, one

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can explicitly construct the hyper- I~hler metric in terms of the bridge v +" [7] .

To summarize, the content o f the self-dual Ein- stein theory is described by the two analytic prepo- tentials ,~(4+), o~O~F. The bridge vU+(x, u) leading from the prepotentials to the self-dual metric is found from the differential equation (55). Putting this the- ory off shell would mean relaxing the analyticity con- ditions on L~ a~4+), L# + and the relation (55), i.e. treating &e(4+), L#+ and v "+ as arbitrary indepen- dent fields. The above constraints should then be ob- tained from an action.

3.2. The action

As explained at the end of the preceding subsec- tion, from now on we shall regard L :(4+), L#~ and v "+ as independent fields. The analyticity conditions on L~ (4+), L :+ and the relation (55) are imposed via Lagrange multipliers:

S= f d4xA du {p(-s)u 0+ .~(+4)+Q(-2).v 0,]+ ~a+

+ R (-3)[ @ + +vU+_ ½nu'(0X, L# < +4)+ 0+ +~/7~ ) ]}. (58)

This action has been written down in the analytic ba- sis (x~ -+ , u -+ ), although it can be rewritten in the original one (x "i, u -+ ) equally well. Note the U( 1 ) charges of the Lagrange multipliers P, Q, R, matching those o f the other terms in (58). The transformation laws of P, Q, R under the gauge groups can be found using (51), (52), ( 5 4 ) a n d the invariance o f the action.

The important point about this action is to make sure that the Lagrange multipliers do not describe new degrees o f freedom. To this end we first consider the variation with respect to v "+. From (56) we find

8(.~++v .+)

= ( ~ + + S v ' + _ 8v a+ 0.+av'+) ( 1 _ 0 - v + ) -1 ~, (59)

where ( 1 - 0 -v + ) ~ -=8~, - 0~.v +". Then from the ac- tion (58) we obtain the equation

~++R(-3)

= (0ff, v "+ - 0j, -@ ++ v "+ )R~ -3) - 0ff.v "+ R(, -3)

_ ~,r~+2) , R ( - 3 ) ( 6 0 )

It is not hard to show that for any matrix M (+2) one can find a suitable rotation o f R ~- 3):

R ( - 3 ) ~ m ~ R ~ -3) '

M ( + 2 ) ~ _ ( ~ + + a .. u - m u ) , n - l ~ (61)

after which eq. (60) becomes ~ + + R ~ -3) =0 . Re- written in the basis (x "i, u-+i), this is just 0++R~ -3) =0 . As discussed in subsection 2.2, such an equation for a negative-charged field has only a trivial solution, R~ -3) =0 . Next, the variation o f ~(+4) and ~ + in (58) directly produces the equa- tions 0+.P<-5)"=0, 0+.Q(-2)""=0. As before [see eq. (28 ) ] , these equations have general solutions, which coincide with the gauge freedom in the Lagrange multipliers P, Q in the action (58). Thus, on shell one of the Lagrange multipliers (R) van- ishes, the other two (P, Q) are pure gauges, so they do not increase the number o f degrees o f freedom.

In conclusion we would like to stress once again that the introduction o f harmonic variables has been instrumental for finding an action principle for self- dual equations in four dimensions.

A c k n o w l e d g e m e n t

E.S. would like to thank F. Delduc, G. ' t Hooft, V. Ogievetsky and C. Vafa for stimulating discussions.

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