Completely solvable gauge-field equations in dimension greater than four

Nuclear Physics B236 (1984) 381-396 © North-Holland Pubhshlng Company

COMPLETELY SOLVABLE GAUGE-FIELD EQUATIONS IN DIMENSION GREATER THAN FOUR R S WARD

Department of Mathematlcal Sctemes, Durham Unwersltv, Durham, England Received 29 July 1983 (Revised 24 November 1983)

This paper deals with analogues, m higher damensaons, of the self-dual Yang-Malls equations m dim4 Requanng that they be the lntegrablhty condmons for a linear system leads to a procedure for constructing thexr solutions A partial classlfieaaon of such equations ~s gaven, and three examples m dimension eight are studied in detail

1. Introduction

The self-dual Yang-Mills equations 2'~B--l~ ~ccB = F~ in four dimensions have many remarkable properties, not least of which being that they are, in some sense, "completely solvable". One possible definition of this property will be described in sect. 2; for the moment, let us note the following pieces of evidence. (i) Many explicit solutions, for example of instanton or monopole type, are known; there are constructions winch ymld all solutions of these types [1]. (n) There is a sequence of ans~itze which generate solutions of the (non-linear) self-duality equations in terms of arbitrary solutions of linear equations [2]. (iii) There exist non-local symmetry transformations which generate loop algebrae [31. The self-duality equations imply the Yang-Mllls equations, which, it is widely believed, are n o t completely solvable [4]. The purpose of this paper is to describe a way of constructing higher-dimensional completely-solvable analogues of the self-duality equations. Each of these takes the form of a linear algebraic condition on F,,, and therefore amounts to a system of first-order semi-linear equations for the gauge potential A,. In recent years, there has been interest in gauge theories in dimensions greater than four, particularly because of the way in which Higgs fields and supersymmetry can be understood through dimensional reduction from d > 4 dimensions down to d = 4 [5]. Another area in which higher dimensions are relevant is supergravlty and spontaneous compactlfication [6]. The ideas of this paper can be extended to higher-dimensional curved space-times; in particular, there are "completely solvable" 381

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analogues of the self-dual Einstein equations [7] m arbitrarily high dimension. This latter subject will not be discussed here, but will appear separately. A recent paper [8] has studied a similar problem to that of this paper, namely the finding of linear algebraic con&tions on F,, such that the Yang-Mills equations follow automatically (as is the case for the self-duahty equations m d = 4). The authors gave a partial classification of the cases that occur in dimensions d ~ 8, but did not ~dentify any completely solvable equations among the higher-dimensional examples that they discovered. The plan of this paper is as follows. In sect. 2 we shall study overdetermlned systems of linear equations, involving one or more complex parameters. The integrabihty condition for such a system is a hnear algebraic condmon on the gauge field (curvature) F,~, of the type we are interested m. This approach leads to a construction for solutions of the non-linear equations, generahzmg the "twister theory" construction for solutions of the self-duality equations [9,1, 2]. Some of the details of this are relegated to an appendix. Several classes of completely solvable equations in higher dlmensmns are listed. Sects. 3 and 4 examine in detail three particular examples occurring in this hst, all in d~mensIon d = 8. Finally, some concluding remarks are presented in sect. 5.

2. Linear systems and complete solvability To begin with, let us work in the complexified picture, and consider G L ( n , C ) gauge fields on a d-dimensional complex space C J Later we shall impose reality conditions, restricting to (say) SU(n) gauge fields on NJ. The analysis does not depend on what the gauge group is, and works for any group. Let x ~ ( # = 1,2 .... , d ) be complex coordinates on C ~, and let A u be a gauge potential on C J (represented as an n × n matrix). We introduce m complex parameters, thinking of an m-tuple of parameters as a point of complex projective m-space C P m. Let 7rP ( P = 0 , 1 . . . . . m) be homogeneous coordinates on C P m, so the m parameters are the ratios between the ~rv We want to set up a system of hnear equations revolving the gauge potential and the parameters. To this end, let V2(~r P) (a = 1 . . . . . k) be a set of polynomials in ¢rP, each homogeneous of degree q. The hnear system is V : ( ~r ) D J / ( x , Tr ) = O ,

(2.1)

where q, = ~k(x", ~re) IS a column n-vector, and D~ = 0, + A, is the gauge-covanant derivative. If k > 1, eqs. (2.1) for qJ are overdeternuned, since there are k times as many equations as there are unknown functions. In order for (2.1) to have n hnearly independent solutions, it is necessary and sufficient that the commutator [V,~D,, VIDe] should vanish, which clearly gives VCaV;IF~,~=O,

(2.2)

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383

where F~ = 2 O[~A~I + [Au, A,] is the gauge field. (Square brackets enclosing mdaces denotes skew-symmetrization.) Recall that the V~ are polynommls m ~re, whereas F,~ does not depend on ~re, so eq. (2.2) is a polynomial m ~rP, and equating each coefficient of this polynomml to zero gives a set of linear equations on F~,, as desired. The number of equations is clearly

½k( k - 1 ) ( 2q + m ) The appendix describes a construcuon which demonstrates that these equations, considered as a system of first-order equations for Au, are "completely solvable", provided only that the following techmcal requirement on Vfl holds. The polynomials V~"can be written as V~e i~r p .. 7rR, where the V~p R are complex constants, and are totally symmetric in the q ind~ces P . . . R. We require that Vflp R, considered as a map from the (aP...R) space to the /~-space, be non-singular. In particular, this gwes the constraint that these two spaces have the same d~mens~on, namely

For example, if d = 4, k = 2, m = 1 and q = 1, then the reqmrement on V~p is

[

V,o

det V~I

v?,

V?o Vl"0] V1]

V1]

1/141] 4:0.

v;lj

Thas (simplest) case leads to the self-duality equations in d = 4, as we shall see in sect. 3. We can use the constraint (2.3) to obtain a classification scheme. For d ~< 11 there are a total of 11 cases, and these are set out in table 1. (The table lists 13 cases, but the three cases A2, B I and C 1 coincide, and correspond to the case of self-dual Yang-Mills m d = 4. The purpose of this triplication is to illustrate the fact that each of the classes A, B and C generahzes the self-duality equations.) In each case, the appropriate values of k, m, q and d are given, together with a quantity which measures the overdeterminacy of the equauons, and is defined as follows. The equations on F~ form a gauge-invarlant set of equations for A~. Therefore there are, in effect, ( d - 1)n 2 unknown functions, since A~ has dn 2 components, and a gauge condition such as (say) A I = 0 reduces this by n 2. Thus the number of equations minus the number of unknowns is q u a n t i t y - ½k( k - 1)( 2qm+ m ) _ ( d - 1)

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384

TABLE 1 k

m

q

d 2k 2 ( q + 1) 2(m + 1) 9

A~

k

1

1

Bq Cm

2

1

q

2 3

m 2

1 1

D

Quantity l(k

2)(3k-1) 0 12m(m- I) 10

Range k=2,3,4,5 q= 1,2,3,4 m = 1,2,3,4

times n 2. We see from table 1 that only in the c a s e s Bq (and A 2 : C 1 : B2) are there as many equations as unknowns; in the other cases we get an overdetermined set of equations for A~. Do not confuse the overdeterminacy here with that of the linear system (2.1). We have seen m this section how a system of hnear equatmns, depending on a "spectral parameter" in C P % has IntegrabIlity conditions which may be regarded as completely solvable (in the sense of the geometric construction described in the appendix). The d ~< 11 hst in table 1 can clearly be extended to arbitrarily high dimensions. But tbas is not the most general situation. For one thing, we could allow the spectral parameter to range over some more general complex manifold than C P " (for example, over an algebraic variety). For another, we could allow V~ to depend on x" and well as 7rP; in effect, this takes us from a fiat to a curved space-time background. Each of these possibilities leads to more general completely-solvable equations than those studied in this paper. In addition, there are examples which involve not just pure gauge fields, but also (say) Higgs fields. However, it may be the case that all of these arise from pure gauge field equations by dimensional reduction, as happens in the case of the Bogomolny equations for monopoles [1]. It seems reasonable to propose the following definition: that gauge field equations are completely solvable if and only if they are the integrabllity condiuons for a linear system of the most general type. It remains to make this definition more precise, and to classify all the cases that can occur. The basic idea of expressing a non-linear equation as the compatibility condition for a linear system, especially with regard to the "inverse scattering" technique, has been around for some time [10]. But until now, it does not seem to have been applied In a systematic way to gauge and other theories in arbitrary dimension.

3. T h e c l a s s A k

In the class A k the polynomials V~ have the form V~(~r) = V~eTrP, where P ranges over 0,1. We assume that V~v is non-singular as a map from the ~-space to the aP space. So we might just as well label the coordinates on C d as x ap, with x ~ being

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385

defined in terms of these by x ~ V~pXaP. The linear system then takes the simple form ~eDap ~ = O. At this stage, the symmetry group acting on our equations Is the product of G L ( k , C) (acting on the index a) and GL(2, C) (acting on the index P). Suppose we want to reduce this symmetry group in order to preserve a euchdean metric. If k is even, the most economical way to do this is to reduce G L ( k , C ) to Sp(½k), and GL(2,C) to Sp(1). (The symplectic group Sp(r) is the group of r × r quatermonlc matrices M satisfying M*M = 1.) This works as follows. Define symplectic forms Gb and e'pQby =

El2 -~- --821 = E 3 4 =

--E43 . . . . . .

Ek( k 1) = 1 ,

with the other components equal to zero. Impose on x ~e the reality condihon X aP = EabEtpQX bQ.

Then the x ~e are, in effect, coordinates on R d = R ek, and the line-element ds 2 = eabe'pQdxaedxbQ is the euclidean metric o n R 2 k The orthogonal group SO(2k) is reduced to the subgroup [Sp(1) × Sp(½ k ) ] / Z 2. We shall now examine in detail the two cases k = 2 and k = 4, 1.e. dimensions d = 4 and d = 8. But it is worth emphasizing that each one of the A~ (k >/2) gwes a completely solvable set of equations. If k is odd, then finding a "euclidean slice" for the equations to hve on, is not quite as natural as it is for k even. In the case d = 4, the orthogonal group SO(4) is not reduced at all, since SO(4) is isomorphic to [Sp(1)× Sp(1)]/7/2- The gauge field (2-form) F,p decomposes into two irreducible pieces:

FapbQ= eabGrQ+ e'pQHab, 6 = 3 + 3,

(3.1)

where GpQ ~---~ l_ablz" FaPbQ and Haa = ½e'PQFaehQ. The integraNhty conditions for "B'POap~ = 0 are GpQ = O, since [ ~ P D a p , qrQDI, Q] = ¢rPTrQFaPbQ = ~abqrPq'rQGpQ.

If we write X 1° = y

= X 1 -[-

X 21 = ~ X

1 --

~ Z ~ - - X 3-~- lX 4,

lX 2 '

X 20

lX 2,

X 11 =

--~=X

3 q-

lX 4,

(3.2)

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then the equations GpQ = 0 become

Fv.=Fy:=O,

F~,v + F:. = 0,

(3.1)'

or, equivalently, El2 = F34,

El4 = F23,

6 3 = --F24"

(3.1)"

These are, of course, just the self-dual Yang-Mdls equations In d = 4. The G term and the H term in (3.1) are, respectively, the anti-self-dual and the self-dual parts of F~,,. The following feature of the self-duality equations Is worth recalling here, since we shall see presently that the other members of the class A k, for k even, also possess it to some extent. Tins is that the action density can be written as tr F,~F ~" = ½tr e"""~F,~F,a + 4 tr GpQG eQ .

(3.3)

The first term on the right-hand side of (3.3) is a "topological charge density"; its integral o v e r R 4 classifies the field topologically. If we look at finite-action fields belonging to a particular topological class, then their action is bounded below by the topological charge, and this lower bound is attained precisely when GpQ vamshes. Thus, in particular, the self-duahty equations Geo = 0 imply the Yang-Mdls equations D"F,~ = 0, since a rmnimum of the action is necessarily also a critical point. We move on now to consider the case k = 4, d = 8. Here SO(8) is reduced to [Sp(1) × Sp(2)]/Z 2, a maximal subgroup. With respect to this reduced symmetry, a 2-form F~ decomposes as follows:

fapbQ = EabapQ + Kabe Q + e'pQH, h, 28=3+

1 5 + 10,

(3.4)

where

apQ = ~1_~b=r~ebp,

1 KabPQ = 7FaPba + 7FaQb~,l 11, b = ±oP , Qz.,2.~ " ~PbQ,

(symmetric in PQ )

-- e,bGeQ

(symmetric in PQ, skew tracefree m ab)

(symmetricin ab )

The integrabdlty condition for ~reD~pq, = 0 Is ~rP~rQF~PhQ = 0, which xs equivalent to GpQ = O,

KabPQ = 0,

(3.5)

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387

a total of 18 equations. Since there are only 7 unknowns (the Au with a gauge con&tion), the system (3.5) is somewhat overdetern~ned, in contrast to the self-duality equations in d = 4. Despite this, (3.5) has many solutions, as we shall see below. If to (3.2) we add X30=t=X5+lX

6'

X40=W

X41~i~_X5_lX

6'

X31=

= --X7+IX

--~=X7+lX

8' 8'

then eqs. (3.5) become

Fv~=F,,t=F,v=Fwx=Ft~=Fw~=O, G

= Vw~ = G

= F ~ = G = F~: = 0 ,

G-C~=C,+C:=o, F,:- F~w= C,+ Gz= 0, C~+C:=C++Ci=o;

(3.5)'

or, eqmvalently,

FI~ = C.,

F13 .= _ F24 ,

E l 4 = F23 ,

Ce=C,,

C~= -g~,

&~= ~ ,

C5 = C 6 = C 7 = & ~ , ~6 = --~5=&8 = --57'

C7 = - & 8 = ~8=&7

-C5=

&~,

= - - & 6 = --55"

(3.5)"

In dimension d larger than four, there is no SO(d)-invariant 4-form e~/~ that one can use to define "self-duality" as in dimension four. However, if one reduces from SO(d) to a subgroup, then there may be a 4-form lnvanant under that subgroup. This problem was studied in [8], and the maximal subgroups of SO(d), for d~< 8, were analysed in this connection. In particular, there is a 4-form O ~ t ~= O i ~ l in dimension 8, lnvarlant under [Sp(1) × Sp(2)]/Z 2 (unfortunately, this case was missed in [8]). It can be defined as follows: given four vectors W~, X ~, Y" and ZC let O~,v~I~W~'X~Y'~Z~ be the scalar quantity 6W,,pX~Y~Z be skew-symmetrlzed over W, X, Y and Z. (In this expression, indices are lowered with the symplectlc forms e,h and e~,O: for example, X~' = X a % ~ and Y~ = Y~%'eQ.) Now O ~ t ~ defines a hnear map on the space of gauge field configurations, namely F~,, ~ !2,--,v~B-ta F ~ . , so we can look for its elgenvalues a:

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In fact, the eigenspaces of this map are exactly the three irreducible subspaces described in eq. (3.4): the G part, K part and /4 part of F,, correspond to the eigenvalues ~ = - 5 , - 1 and 3, respectively. (To check this is straightforward algebra.) Our set of 18 equaUons (3.5) is therefore eqmvalent to (3.6) with ~ = 3. Hence (3.5) xmplies the Yang-Mills equations D"F~,= 0, as a consequence of the Bianchi identities DI~F~~= 0 (cf. [8]). The analogue of (3.3) is the identity tr F,~F "~=

~tr O~,~F~"F~# + 4tr K~hpoKa~eQ+ ~-trGpQGeO.

(3.7)

The first term on the fight-hand side of (3.7) is a total divergence and therefore a candidate for a "topological charge density". But the formula (3,7) is not as easily related to a topological classification and minimum action solutions as (3.3) is, because the dimensions are wrong. Indeed, there are no finite-action solutions of the Yang-Mllls equations on R 8 (except F ~ = 0 ) [11]. In order to get finite-action solutions, one would have to "compactify" some of the dimensions. We end this section by giving an analogue of the Corrigan-Falrlie-'t HooftWllczek ansatz, A, = ~,~ 0~log ¢,

(3.8)

winch "converts" an arbitrary solution q, of the Laplace equation on R 4, into a self-dual SU(2) gauge field A,. Tins is the first of a sequence of ans~ttze which generate solutions of the self-duality equations from solutions of linear equations [2]. These ansatze generalize to our equations (3.5) in dimension eight, and the first ansatz again has the form (3.8). It is obtained by letting the "patching matrix" F discussed in the appendix have the form

F =

1

rro/~.1 ,

where/" = F(~oa, ~re) is an arbitrary scalar function of six complex variables, homogeneous of degree zero. So F is, in effect, a free function of five variables. The construction described in the appendix, for obtaining the gauge potential A,, is easily carried out (see [2] for details of the analogous calculation m d = 4), and what one ends up with is the following. Starting with the arbitrary function F, define a scalar field ~ = ~(x) by the Cauchy integral

f

,e,,pd,e

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Completely solvable gauge-field equattons

389

the contour being I % / % t = 1. Put

Aae = ~1 pQ OaQlOgq~, where T° = -7/~ = o 3, ~/1 = o l _jr_/02, and ~/o = o' - ,02, or' being the Pauli matrices. Tlus gauge potential A~ is then a solution of (3.5), depending, in effect, on a free function of five variables. If 0 is real-valued (and it is possible to choose F so that it is), then Az will be trace-free and antl-herrmtian, 1.e. will be an SU(2) gauge potential. For example, if /~ = I "1- q'f07/'l(0) 1 -- 093)-1(0) 2 -- 0)4) - 1 ,

then ~ = 1 + ( X - y ) - 2 , where X ~ = ( x 1, x 2, x 3, x 4) and Ya = (x 5, X 6, X 7, X 8) The corresponding A, may be thought of as the 1-1nstanton solution in the X-space, with the position of the instanton given by the vector Y~; or vice-versa.

4. The classes

Bqand C,.

In the case of class Bq we may express our coordinates on C d as x ~p R, where a= 1,2 and P , Q , R .... = 0 , 1 ; x ~p R has q capital Roman indices and is totally symmetric in these, so d = 2(q + 1). The linear system (2.1) becomes

rrL..rrROap R~ = O.

(4.1)

These equations have as symmetry group the product of two copies of GL(2, C), acting on the lower-case and capital indices respectively. Suppose, as m sect. 3, that we want to reduce this to a subgroup of SO(d). If q is odd, there is a natural way of doing so, which reduces the symmetry to [Sp(1) × Sp(1)]/7/2 ~ SO(4) _c S O ( d ) . This is achieved by requiring that the two symplectic forms eab and e~,Q (defined as m sect. 3) be preserved, and imposing the reality condmon xaP

R __ __ E a b EtP K . . . E t R M x b K

M

The d-dimensional euclidean metric is ds2=dxaP

RdXap R,

(4.2)

where radices are lowered by means of e and e'. Only if q is odd is (4.2) a metric line-element: if q is even, then (4.2) is skew-symmetric rather than symmetric.

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T h e case q = 1 is j u s t the self-dual Y a n g - M i l l s in d = 4 again: clearly it coincides with the case A 2. So let us exarmne the case q = 3, d = 8. H e r e a 2 - f o r m F~, F ~ e o r b s r u , b e l o n g i n g to a 28 of SO(8), d e c o m p o s e s as 28 = 3 + 15 + 7 + 3. T h e i n t e g r a b d i t y c o n d i t i o n for (4.1) is the vanishing of the 7 p a r t of F,~, i.e. a

b

e~hF~*'OR s r u ) -

_

(4.3)

O,

the parentheses denoting symmetnzation. If we define xlOOO = y = x l

X 1001 ~ W =

X1011 ~ t = x 5 + lX 6 ,

q- l X 2 '

- - X 7 q- l X 8 ~

X 1111 -~ Z ~

- - X 3-~- l X 4 ,

then eqs. (4.3) b e c o m e

F y , = 0 = F~., ~ , , , - Fw. = 0 = F y , - Fw.. Fv~ + F , : - 3&~ = 0 = F~w + F,: - 3F.,, F~-Fz:

(4.3)'

+9Ft,-9Fw~=O;

or, equivalently, El2 + F34 + 9F56 + 9Fv8 = 0, El3 -- F24 = 0,

F~4 + F23 = 0, F15 + F26 + F37 + F48 = 0, F16 -- F25 + F38 - F47 = 0, 11717- F28 - F35 -f- F46 q-- 3F57 - 3/768 = 0, F18 + F2v - F36 - F45 - 3F58 - 3F67 = 0.

(4.3)"

This system of e q u a t i o n s is like (3.5) in that it a d m i t s ansatze of the sort m e n t i o n e d at the end of sect. 3. But it is unlike (3.5) in not being o v e r d e t e r m i n e d (here, there are as m a n y equations, n a m e l y seven, as there are unknowns), a n d in that it does not i m p l y the Y a n g - M i l l s equations.

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391

Let us move on, finally, to consider the class C,,. This case 1S, in m a n y ways, c o m p l e m e n t a r y to class A k. As coordinates we m a y use x ~p, where a is a 2-dimensional index and P is an (m + 1)-dimensional index. If m is odd, we can define a euclidean structure exactly as was dome in sect. 3 (the definitions of the symplectic forms eah and e'pQ being adjusted in an obvious way to take account of the changed ranges of the Indices). The dimension of the space is d = 2 ( m + 1), and the orthogonal group S O ( d ) is broken down to [Sp(1)× Sp(~m + ½)]/Z 2. The case m = 1 is that of the self-duahty equations in d = 4. The next interesting case is that of m = 3, d = 8. Here, the decomposition of the field tensor F~,, is just like (3 4), but with capital and lower-case indices swapped, namely

FapbQ=epQGab+KabpQ+eahHeQ,

(4.4)

28 = 3 + 15 + 10.

The hnear system is

~reDap6=

0 and its lntegrablhty condition is

HpQ =

0,

(4.5)

a set of 10 equations. If we set

X 10 =

X 21 = y

X 11 =

--

X 12 =

X 23 =

X 13 =

--

=

X 20 =

i =

X 22 =

X 1 4-

Z =

--X

X 5 +

W =

IX 2, 3 4-

IX 4,

lX6~ - - X 7 4-

IX 8,

then eqs. (4.5) become

C~=o = F~, £,~ = o = & ~ , F , z + C ~ = o = ~z + &...

C,-Fw~=O=C,-F~:, F.,-Cz

= 0 = F,~- C ~ ;

(4.5)'

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R.S Ward / Completel; soh~able gaugefield equattons

or, equivalently,

--~3"

~2 = --~4'

~3 = ~4'

~4 =

66= -~8,

67=~8,

~8= -~,

Ft5 + F26 + F37 + F4s = 0 , Fi6 - F25 + F38 - F47 = 0, F17 - F28 - F35 + F.6 = 0 ,

F]8 + F27 - F36 - F45 = 0.

(4.5)"

Note that these equations precisely complement eqs. (3.5), and that they imply the set of seven equations (4.3). In this case, as in the case A n, there is an invariant 4-form O , ~ ; it is unique up to scale, and is defined as in sect. 3 but with capital and lower-case indices interchanged. However, eqs. (4.5) are not elgenvalue equations of the form (3.6); indeed, (4.5) does not Imply the Yang-Mills equations. But it lmphes equations which are obtained from a slightly modified lagranglan. For one has the identity tr F ~ F "~ + 16 tr G, bG"h= - ½ t r O , ~ F ~ F ~B + 8trHeQH eQ

(4.6)

Since the O term in (4.6) is a total divergence, the vanishing of HpQ, i.e. the system (4.5), Implies the Euler-Lagrange equations obtained from the lagrangian on the left-hand side of (4.6). This "modified" action density is gauge-lnvariant, positivedefinite and [Sp(1)× Sp(2)]/Z 2 invarlant, but not SO(8) invariant. To summarize: the equations in classes Bq and Cm, although completely solvable, do not imply the Yang-Mills equations (except for q = m = 1). But they may be of interest in the context of lagrangians other than the "standard" Yang-Mills one.

5. Concluding remarks We have seen that there are gauge-field equations in dimension greater than four which share some of the remarkable properties of the self-duality equations in 4d. However, the case of dimension four is rather special, for reasons arising from the choice of t r F 2 as the action functional. If higher dimensions are to be relevant, then some process of dimensional reduction, such as spontaneous compactification, has to be involved. To this end, one should extend the considerations of the present paper (which deals only with flat d-dimensional space) to curved spaces. This can be done, and will be described elsewhere.

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Completely solvable equations are few and far between, in the sense that almost every non-linear equation is, it would seem, not completely solvable. But despite being rather special, completely-solvable equations (in gauge theories, or in other cases such as CP" models) are worth studying, for what they can tell us about non-linear and non-perturbatlve phenomena in general. The author would like to thank C. Devchand and D.B. Fairhe for useful conversations.

Appendix This appendix describes a way in which the solutions of the lntegrablhty conditions arising from a linear system V,"(~re ) D , } = 0 can be constructed. It justifies our calling such integrabllity equations "completely solvable". The construction is best described in the language of vector bundles and geometry, but the discussion here will largely avoid such sophistication. Because of the assumption that V~t, s be non-singular, we may denote the coordinates on C a as x"*" s (symmetric in P . . . S ) . The linear system now reads 7rP...TrSo, p s~k = O,

the lntegrablhty conditions for which are skewed over ab ] F,p sbr w symmetrlzed over P .. W / = 0.

(A.1)

For the sake of simplicity let us take m = 1, so that the capital indices P, Q . . . . range over 0,1. A later remark will describe what happens when m > 1. Solutions of (A.1) may be constructed as follows. Let F = F(~o "p R, ~ro) be a non-singular n × n matrix of complex-analytic functions of kq + 2 variables, homogeneous of degree zero. The symbol ~0"~ R has q - 1 capital indices, and is totally symmetric in these. Thus (taking the homogeneity into account) F 1s defined on a region of C P kq+I. We reqmre this region to be such that

G ( x , ~r)= F ( x "e Rs~rs, ~rr)

(A.2)

IS analytic in a neighbourhood of [~rl/rro] = 1, for each fixed x. In other words, G is analytic on an annular region in the ~ri/rr0 Argand plane. It follows from (A.2) that

,c(x,

= 0.

(A.3)

(The Indices P, Q . . . . are raised and lowered using the symplectic form e' as in sect.

3.)

394

R S Ward / Completely solvable gauge-field equanons

Now "split" G by finding non-singular n × n matrices H(x, w) and /:/(x, w), where H is analytic for IWl/%l ~< 1 and/2/is analytic for 1~/%1 >/1 (including ~ ) , such that G =/~/H t.

(A.4)

H lwe...wSO~e sH=I21-17rP...~rSO,e sl;t.

(A.5)

From (A.3) we see that

Now the left-hand side of (A.5) is analytic for IWl/%l ~< 1, and the nght-hand side for ITh/%l >/1, whence each side is analytic for all w e, and so (by a form of Liouville's theorem) must be a polynoImal in w e of degree q:

H IwP...wsO**e sH=~re...~rSA,e s,

(A.6)

where the A,p s are functions only of x (not of we). This defines the G L ( n , C ) gauge potential A,. The field equations (A.1) are automatically satisfied, as one can see by operating on (A.6) with wr...crWObv w and skewing over ab. So the construction generates solutions out of (essentially arbitrary) matrix functions F. Furthermore, it produces all solutions of (A.1), this is not obvious, but can be proved without too much difficulty. REMARKS

(i) The splitting (A.4) is not, in general, possible, we have to choose F in such a way that it ts possible, for all x. (For example, if n = 1 and F is a 1 × 1 matrix, then the condition is that log G should be single-valued on the region of analytlcity specified for G.) Those values of x where the splitting (A.4) is impossible will be singular points of A,. (ii) The splitting is not unique, but clearly admits the freedom H ~ HA, I2I ~ I~IA, where A = A(x) is a non-singular n × n matrix of functions of x only. From eq. (A.6) we see that this reduces on A~ the transformation

A~ ~ A-1A~A + A-IO, A. So the sphtting freedom corresponds to gauge freedom in Au. (iu) The reason for requlnng Vfp s to be non-singular is as follows. If we did not impose this restriction, we could still carry out the construction to obtain an object A.p s- But in order to be able to interpret this as a gauge potential A., we have to be able to invert V~p s. (iv) The geometnc picture may be summarized as follows. For each w P, the k vectors Vl~(~r). . . . . Vk~(Ir) span a complex k-plane Z in C d. The linear equations V~D,~ = 0 say that ~k is covariantly constant over each of these k-planes, the

R.S Ward / Completely solvable gauge-field equatwns

395

consistency c o n d i t i o n for which is that the c u r v a t u r e F ~ should vanish when restricted to a n y of the k-planes. This is w h a t eq. (A.1) says. T h e space of all k - p l a n e s g e n e r a t e d b y the Vf is an o p e n subset 6~ of C P kq+l, a n d the gauge field d e t e r m i n e s a vector b u n d l e E over ~ : the fibre E z over a p o i n t Z in ~, is defined to be the n - d i m e n s i o n a l vector space E z = ( ~ on ZI V [ D J / = O on Z ) . T h e basic t h e o r e m says that solutions of (A.1) in C d c o r r e s p o n d to vector b u n d l e s E over ~ , satisfying the c o n d i t i o n that E restricted to x (defined below) is trivial. This x is a C P 1 in C P kq+l which c o r r e s p o n d s to x ~ ~ ; in terms of the c o o r d i n a t e s (c0ae R, ~T) on C P kq+l, x is given b y ¢daP

R : xaP

RST.[S '

in terms of x aP s. T h i n k of the ~re, for fLxed x, as h o m o g e n e o u s c o o r d i n a t e s on x. The m a t r i x F is a " p a t c h i n g m a t r i x " winch d e t e r m i n e s the vector b u n d l e E. T h e idea ts that one cuts ~ into two pieces, c o r r e s p o n d i n g to I~q/~r0l ~< 1 a n d I~rl/~r0] >/1, a n d on the o v e r l a p one specifies the p a t c h i n g m a t r i x F. The c o n d i t i o n that E restricted to x be trivial j u s t says that F is " s p l i t t a b l e " for that p a r t i c u l a r x. W e see, therefore, that solutions of the n o n - l i n e a r e q u a t i o n s (A.1) c o r r e s p o n d to vector b u n d l e s E which are required to satisfy a regularity condition, b u t no differential equation. (v) All ttns was for m = 1. It r e a d i l y extends to m > 1, a n d one again has a t h e o r e m like that in the previous remark. But n o w one needs m o r e than two patches, in fact m + 1 in general. So one has to start with several matrices F, one for each overlap b e t w e e n pairs of patches. Things b e c o m e s o m e w h a t m o r e c o m p l i c a t e d , b u t the s p l i t t i n g - t y p e c o n s t r u c t i o n works as it d i d for m = 1.

References [1] M F Atlyah, N J I-htctun, V G Dnnfeld and Yu I Mamn, Phys Lett 65A (1978) 185, E Corrlgan and P Goddard, Commun Math Phys 80 (1981)575, N J Hltctun, Commun Math Phys 83 (1982) 579, 89 (1983) 145 [2] M F Atlyah and R S Ward, Commun Math Phys 55 (1977)117, E F Corngan, D B Fmrhe, R G Yates and P Goddard, Commun Math Phys 58 (1978) 223, R S Ward, Commun Math Phys 80 (1981)563 [3] L -L Chau, Ge M -L and Wu Y -S, Phys Rev D25 (1982) 1086, L Dolan, Phys Lett 113B (1982) 387 [4] S G Matmyan, G K Savvldy and N G Ter-Arutyunyan-Savvady. JETP (Sov Phys ) 53 (1981) 421, E S Nlkolaevskn and L N Shur, JETP Lett 36 (1982) 218 [51 P Forgb.cs and N S Manton, Commun Math Phys 72 (1980) 15, E Cremmer, in Supergrawty "81, eds S Ferrara and J G Taylor, (Cambridge Umverslty Press, 1982) pp 313-368

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[6] P G 0 Freund and MA

[7] [8] [9] IO] ill]

Completely solvable gauge-field equations

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