On axially symmetric selfdual gauge field configurations in 4p dimensions

Volume 162B, number 4,5,6

PHYSICS LETTERS

14 November 1985

ON AXIALLY SYMMETRIC SELFDUAL GAUGE FIELD CONFIGURATIONS I N 4p D I M E N S I O N S A. C H A K R A B A R T I a b c d e

a, T.N. S H E R R Y b,c and D.H. T C H R A K I A N d,e

Centre de Physique Thborique, l~cole Polytechnique, F-91128 Palaiseau Cedex, France Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B9 Department of Mathematical Physics, University College, Galway, Ireland Physikalisches Institut der UniversitiJt Bonn, Nussallee 12, 5300 Bonn 1, Fed. Rep. Germany Department of Mathematical Physics, St. Patrick's College, Maynooth, Ireland

Received 5 August 1985

Axially symmetric gauge field configurations are studied using generalized selfduality conditions for 4p dimensions (p = 2, 3, ...). A generalization of the Liouville equation arising for p = 1, is found. The dimensional reduction to a monopole in seven dimensions is performed. Uses of a coordinate transformation compactifying the time domain, are indicated.

Higher order gauge field systems in d ( > 4) dimensions have been studied in previous papers [ 1 - 3 ] . In particular for d = 4p (p = integer) the selfdual Finite action solutions of the conformally invariant action density proposed [2] can be directly related to the higher order topological invariants - the C h e m Pontryagin (CP) integrals. Denoting the 2p form constructed from the curvature by F ( 2 p ) , this action density and selfduality are expressed by c3 = tr F ( 2 p ) 2,

(1)

F ( 2 p ) = *F(2p),

(2)

where * denotes the Hodge dual. The system (1), when subjected to some suitable dimensional reduction, yields [3] the Yang-Mills-Higgs (YMH) system, augmented by higher order functions of the curvature and Higgs fields which are suppressed relative to the YMH sector by the square of the (very small) radius of a sphere of compactification. As such (1) can be considered a generalisation of the Yang-Mills (YM) system, and we henceforth call it generalised YM (GYM). The selfdual solutions of (2), which for p = 2 were also found by Grossmann et al. [4], can be character340

ised [2] in the Schwarz [5] gauge by the radial function g=(r 2-X2)/(r 2+X2),

r 2=x/~ x u ,

la = 1 ... 4 p .

For p = 1 these are simply the BPST [6] solutions of YM, and hence we shall call them generalised BPST field configurations, GBPST. After the spherically symmetric GBPST, the next natural step is to explore the axially symmetric solutions of (2). This is the purpose of the present work, and as in ref. [2], we study the case p = 2 in detail. The method can be canonically generalised for p > 2. A remarkable generalisation of the LiouviUe equation which arises, in this context, f o r p = 1 is thus dis. covered - eq. (45). Following Witten [7] (p = 1), the imposition of axial symmetry on the gauge field in eight (p = 2) euclidean dimensions can be viewed as using the line element ds 2 = d x

dx u=dr 2+r2dI26+dt

= r 2 [(dr 2 + d t 2 ) / r 2 + d~26],

2 (3)

where t = Xs, r = ~ (i = 1 ..... 7), and dI26 = d x a d X a , where x a (a = 1 ... 6) are the coordinates on the six-sphere S 6 . The form (3) of ds 2 displays the 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

conformal equivalence with H 2 X S6 . We look for solutions symmetric with respect to S6 . We use, as previously [2], the fixed point formalism in the Schwarz gauge [5]. On this fixed point the metric corresponding to (3) takes the values guvlS 6 = (r28ab , 1, 1),

(4)

where we denote the indices/a = ~, a, with x~ =(x 1 ,x2) = (t,r) anda = 1 ... 6 labelling x a, the coordinates on S6 . The choice of our gauge group is some group in which 8 X 8 SO(8) spinor representations can be imbedded. The reason for this is that the GBPST field configurations [2] lie in the 2P X 2P (spinor) representation of the algebra of SO(4p). In what follows, in our axially symmetric Ansatz, we use the 8 X 8 spinor representation Luv of SO(8), in terms of the 8 X 8 Dmatrices Fa = (F 1 ..... F6) and F 7 = iF 1 ... F 6 : L

= (L b = - -'2ro r ,Lo 7 _a. - vr

,L

= 1 r7r ,

L78 = (1/2i)r7).

(5)

The change in i --> - i in (5) interchanges selfdual and antiselfdual solutions. Using the formalism developed in refs. [8], our Ansatz for the 8 X 8 SO(8) connection field can be stated:

Aa[s6=%(r,t)L78 , a=t,r, A a IS6 = ¢(r, t)La7 + x(r, t)La8.

(6)

The field strengths are [8] F~Is6 =f~L78,

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Volume 162B, number 4,5,6

(7)

(lO)

In (9) and (I0), the four-form field is explicitly given by

Y

o= {e

where the square brackets indicate cyclic summation. Substituting (4), (7) and (8) into (9) and using the identity

(Lp [v,Lpol

} = (1/4!)

epuporxKn {Lrix,Ln ] ),

(1 I)

one finds the following selfduality equations: el)t = ~dr,

.o~t = - % ,

(12a)

3r-2S 2 + Sftr - 2 ( q Y t ~ r - C~r~t) = 0.

(12b)

The corresponding action integral for selfdual field configurations, which is equal to the fourth CP integral, is proportional to

e~

ffa (s%-6 [¢013X+-~¢3aiX3 + ¢3(1¢2 _ _~)0#X

-- X3(1X 2 -- ~)0#¢ ] )(r 2 + t2) 7/2 dr dt.

(13)

In (12a, b) we have reverted from using the indices to the explicit labels t and r, as these will be used below to denote differentiations with respect to t and r respectively. The selfduality equations (12) can now be treated as in ref. [7]. We state the results briefly. Setting

at=-Or~,

FablS6=--SLab,

Faals6 = Q~ La7 + ..~aLa8,

c5 = V~-tr FuvpoFUVPa.

a r=ot~b,

¢0-ix0=f(z),

¢ = ¢ 0 e~0, X =x0 e~,

z=r+it,

~=r-it,

(14)

one obtains the integral of (12a) by requiring that f ( z ) be analytic [7], yielding

where fail = o aft-- O#a , S = 1 - - ¢ 2 - - X 2,

~

= a¢

-

%x,

sg~ =

a~,x + % ¢ .

We are now in a position to impose the condition of selfduality (2), which for p = 2, and on the space H 2 X S6 takes the form

Fpvpo = (1/4!) Vr-geuupa.rxKnF ,xKn • The corresponding action density is of course

c)Yt = i[x/f az(X/fe~° ) -V/flOF(v/~e~)],

(15a)

C);r= [X/~0z (X/f e~°) + V~-0r (x//Te~°)].

(15b)

(8)

(9)

Eq. (12b) now reduces to

Oz OFt~ = (3/4r2)(e2~ f.~ - 1) - 2 0z(e~ X/~) az--(e V/if)/(

f f - 1).

(16)

Finally choosing the arbitrary analytic function f ( z ) 341

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to satisfyff = 1, (16) reduces further to (02 + Ot2)$ = (3/r2)(e 2qj - 1) - [2e 2 . / ( e 2. - 1)1 [(or0) 2 + Ot0)21.

(17)

The i~.dex 1 solution of the corresponding equation in four dimensions [7] (a} + 02)0 = r - 2 ( e 2¢ -- 1),

The appropriate Bogomolnyi equations are now

Fi]kl = !e2i]klmnp (Fmn , Dp¢}.

(26a)

(f'/J")'= (3/r2)(f 2 -- 1) - 2f'2/(f 2 -- 1).

(265)

(20)

(21)

Setting f = e ~° we then find from (26b), the static version of(17) 4 " = (3/r2)(e 2~0 - 1) -- [2e 2¢/(e 2~° - 1)l 4 '2. (27) Unlike the BPST solutions [2] of (2) in 4/7 dimensions, the PS monopole equations in (4p - 1) dimensions cannot be integrated collectively, that is the solution (22) of the p = 1 BPS equation (21) does not satisfy the p = 2 equations (26a) or (27). Assuming that a solution to (260) [or (27)] exists we study some of its properties. For limiting values of r, (26a) is compatible with

(i)

f = 1 + elf2 + ... (r -+ 13)

(28)

(ii)

f=c2r3e -xr

(29)

(X>0,r~).

(The parameters can eventually be constrained by fix-

gives finite energy monopoles for (22)

It is interesting to compare the analogous situation in reducing from eight to seven dimensions. Identifying A 8 with the 8 X 8 SO(8) valued Higgs scalar
~PlxT=h(r)L78, ATIx7 =0, Aalx7 =r -1 [f(r)La7 +g(r)La8], a = 1..... 6,

(24d)

h =f'/f,

Before discussing (17) further we note the relationship between the solutions of the selfduality equation (2) in four dimensions (p = 1) and the corresponding Prasad-Sommerfield (PS) solutions in three dimensions. There, the static version of (18) gives the PS monopole. With 0t0 --- 0,

e ~ = cr/sinh cr.

D
(19)

general feature already noted in the BPST formalism [2,4,10], that essentially the same one instanton solution can be implemented in 4/7 dimensions. The time translation and scale symmetries of(17) show the solution (19) still holds under

d2~k/dr 2 = r-2(e 2¢ - 1)

(24c)

(18)

is also a solution of(17). This is an expression of the

z = r + it ~ )~[r + i(t - e)].

D7cb]x~ = h~L78 ,

(25) It turns out that in (23) and (24) one can setg = 0 without any loss in the generality of the solutions of (25), which reduces to

namely [9] e, = { [t 2 + (r - 1)2] [t 2 + (1"+ 1) 2 ] }1/2 1 + t2 +r 2

14 November 1985

(23)

where all t-dependence has been suppressed. The field strengths Fi] and Die are computed [8] to be

Fa7 Ix7 = - r - l ( f ' l , a7 +g'La8),

(24a)

Fab Ix7 = - r - 2 ( 1 - f 2 _ g2)Lab '

(24b)

ing scale and normalisation.) The situation at r ~ 0 is the same as in d = 3, (p = I) and (28) gives a regular behaviour as r ~ 0 as can be seen from (24b) (with g = 0). At r -* ~ , r 3 appears in (29) instead of r for d = 3. This power corresponds to the coefficient 3 in (26b), and we will see below that indeed it is always equal to (2p - 1). This property is related to the convergence of the (seven-dimensional) "energy" integral. This integral

f~?d7x=f

tr(Fi2k I +4{Fi[i, Dk ] ¢}2) d7 x

is equal, by virtue of the Bogomolnyi equation (25), to the "monopole" charge

p=4eiiklmn p f treFiiFktFmn dSp, 342

(30)

(31)

where we have used Stokes' theorem in (31), consistently with our assumption that smooth solutions exist. It is then evident from (24) and (29) that (31) reduces just to an angular integration, giving the unit "monopole" flux. We now return to the eight-dimensional system and examine (17) in the light of the above. A class of Ymite action solutions (for d = 8) can also be studied by using static techniques using a trick exploited by one of us in a series of previous papers [11]. Let r + it = tanh(p + it).

(32)

One can start with an Ansatz in terms of(t), r) instead of (6). This is the approach used before [11]. Here, for brevity, we start at the level of (17), which is transformed, using (44), to (02 + 02)~ ' = (3/sinh2p)(e 20

-

-

1)

- [2e2~°/(e 2~° - I)1 [(apO) 2 + (0tO)2].

(33)

Under the rescaling [using primes to avoid confusion with (32)]

r-*t'/~,

p-*r'/~,

a-~

(38)

(35) reduces to (27). This is a particular case of the "monopole limit" of instanton sequences studied extensively in previous papers [ 11 ]. To exploit this technique to construct BPS monopoles one cannot start from a particular case like (36). One has to use a more general solution incorporating the scale parameter tx of (38) in an appropriate fashion [11]. This will not be attempted here. We conclude by pointing out certain remarkable aspects of (26). In four dimensions when in 't Hooft or J a c k i w - N o h i - R e b b i solutions [ 12], one puts all centres on the time axis one obtains Witten's solutions [7] in a different gauge. The latter ones are thus contained in the former ones as particular cases. Let us examine the same situation in eight dimensions. Let

Au(x ) = Luu Oj(x).

But the half-plane r E [0, oo1,

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Volume 162B, number 4,5,6

(39)

For axial symrnetry f ( x ) = f(r, t), and at the fixed

t E [_oo,+=o1

point x 7 = r,x a = O,

is now mapped on the strip O E [0,°°],

r E [-rr, +rr].

(34)

Ftr = hlL78 , Fta = h2La8 + h3La7,

Hence, setting arO = 0, dO/dp = ~p, (33) reduces to

Fab = --glLab , Era =g2La7 +g3La8 ,

Opp = (3/sinh2p)(e 2¢ - 1)

where (with frr = a2f etc.)

- [2e 2~/(e 2~° - 1)1 ~2,

(35)

the solution of which can, if appropriately constructed, have finite action in eight dimensions since r has a compact domain, 27r. Indeed, the index-1 solution (19) of (17) is now just

e ¢ = (cosh p ) - l .

(36)

(35')

A transformation equivalent to (20) can also be implemented in (33). For simplicity, we just note a linear deformation of (36): Up to first order in e I and e2 e ~° = (cosh p ) - I + (e I c o s t + e 2 sinr) tanh2p

gl =f2 +f2 +(2/r)fr '

,2--r,,+g h3 =g3 =frt - f r f t •

(41)

The selfduality conditions imply two constraints. One is exactly as in four dimensions

f r r + f , ' +f2r +f2 +(2[r)fr=O"

One can write (35) also as (e 2~0 - 2~)0o = 6(sinh p)-2(e2qJ - 1) 2.

hl =frr +ftt,

(37)

(40)

(42)

But now there is an additional one g l2 - h 2 - h2 = 0,

(43)

which is indeed satisfied for f = ln(r 2 + t2) -1

for index zero,

f = ln(1 + (r 2 + t2) - 1 )

for indexone.

(44a) (44b)

is a solution of (33). 343

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But for two centres, namely, f = In[1 + [ r 2 + ( t - e ) 2] - 1 + [r2+(t+e)2]-1]

(44c)

there is already incompatibility with (43). We will not try to inject the most general solution of (42) into (43) to classify all possibilities. But evidently there are difficulties with the Ansatz (39) f o r p > 1. But in contrast, the Witten-type Ansatz (6) has been shown to lead f'mally to a single constraint (17).

Even more remarkably (17} turns out to be the f~rst non-degenerate case in the hierarchy o f 4p dimensions (p = 1, 2, 3 .... ). Here we will only state that for arbitrary p a direct generalisation of (6) leads to two equations, one of which is simply (12a) which can be integrated directly to yield the hierarchy of equations (12b) or (17): (a 2 + a2)~ = [ ( 2 p -

1)/r 2] (e 2~

One o f us (A.Ch.) acknowledges with pleasure helpful, clarifying discussions with P. Forg~ics, Z. Horvdth and L. Palla in Budapest. D.Tch. would like to thank GaM. O'Brien for numerous very helpful discussions, and the Alexander yon Humboldt-Foundation for their support during his stay in Bonn.

References [1] [2] [3] [4] [5] [6] [7] [8]

- 1)

-:[2(p - l)e 2~/(e 2~ - 1)] [(arC,)2 + (ate)2], t=X4p , r = ( x 12+ . . . + ~v2 ( 4 p _ l ) ~I/2 j .

(45)

No additional terms arise on the right hand side for p > 2. This structure also guarantees the validity of the one instanton solution (19) for all p. We Fred it satisfactory to have discovered in eight dimensions the first indication of the full canonical structure which was obscured in four dimensions, the last term of (45) disappearing for p = 1. We hope to present elsewhere a more detailed study o f (45), at least for p = 2.

344

14 November 1985

[9] [ 10] [11] [ 12]

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