- Email: [email protected]
The Lagrangian of a self-dual gravitational field as a limit of the SDYM Lagrangian
__ l!!il
1 I March 1996
2s
‘5%
PHYSICS
ELSEWER
LETTERS
A
Physics Letters A 212 (1996) 22-28
The Lagrangian of a self-dual gravitational field as a limit of the SDYM Lagrangian Jerzy F. Plebahki Depurmtenrof Physics,
Centro de Invesrigacih
Received 2 October
I, Maciej Przanowski 2
y de Estudios Avanzados
del IPN.
AportadoPostal14-740. Mexico
1995; revised manuscript received 22 December 1995; accepted Communicated by P.R. Holland
for publication
07000. D.E, Me.rico
3 January
1996
Abstract
The action for the W(N) SDYM equations is shown to give in the limit N -+ cc the action for the six-dimensional version of the second heavenly equation. The symmetry reductinns of this latter equation to the well known equations of self-dual gravity are given. The Moyal deformation of the heavenly equations are also considered. PACT: 04.2O.C~;
11.15.-q
Recently, a great deal of interest has been shown in symmetry reductions of the SDYM equations to integrable equations of mathematical physics [ 1-16 1. In particular many papers concern the symmetry reductions of the SDYM equations to the self-dual gravity equations [6-161. In our previous works we have found the general form of the su( N) SDYM equations which in the limit N ---f co gives the sdiff (X2) g su(co) SDYM equation in the form of the second heavenly equation in six dimensions [ 17,18]. We have also shown that the symmetry reductions of that equation lead to the well known heavenly equations of self-dual gravity. We were also able to obtain the natural Moyal deformation of the heavenly equations. This deformation has been previously found by Strachan [ 12,191 and Takasaki [ 201.
t E-mail: [email protected]. z Permanent address: Institute of Physics. Technical of L6di. W6lczariska 219,93-005 Mdi, Poland. Elsevier Science B.V. PN SO375-9601(96)00025-4
University
In the present work we are going to show that the limiting process N + co leading from the su( N) SDYM equations to the heavenly equations can be also well defined on the action level. Similarconsiderations enable one to find the action for the Moyal deformation of the six-dimensional version of the second heavenly equation.
We deal with the su( N) SDYM equations in the flat four-dimensional real, simply connected flat manifold V C R4 of the metric ds2 = 2(dx C%dZ + dy 8s di;), where x, y, .i!, ji are null coordinates on V notes the symmetrized tensor product, i.e., i (dx 8 d? + d.Z @ dx). The coordinates are chosen in such a manner that the su( equations read [ 2 1]
(1) and C& dedx C&df = (x, v, 2, .?) N) SDYM
Fxy = 0,
(2a)
Fz\;= 0,
(2b)
J.F. Plebdnski. M. Przanowski/Physics
(2c)
F,i + F,., = 0,
Lerrers A 212 (1996) 22-28
sS=O,
du=
Cdu,
S=
23
dxdydzdy,
J where, as usually, F,, E SU(N) c3ccoo(v),y,v E {x, y, I, J}, stands for the Yang-MiIIs field tensor. Then, in terms of the Yang-Mills potentials A, E su( N) 8 Cm(V) which define Fpy according to the well known formula,
+A,].
F,, = [a,+&&
one rewrites Eqs. (2a)-( &A,. - &A, + &A,
+
2c) as follows,
[A.v,A?.l= 0,
- +A1 + [At.Ay]
?,Ai - &A,
(3)
= 0,
(W
= 0.
(4c)
Eq. (4a) implies that the potentials A, and AT are of the pure gauge form, i.e., there exists an SU( N)valued function g such that A,r = g-‘&g,
AT = g-‘&g.
(5)
A,.=O.
(10)
Thus the Lagrangian C defined by ( 10) can be considered to be the Lagrangian for the SDYM field. Now we let N tend to infinity. Thus we arrive at the su( co) algebra. It is well known that [ 22-261 su(oo)
2 sdiff(s*)
g the Poisson algebra on C’, (11)
where X2 is a two-dimensional real manifold. Employing the results of Refs. [ 23,241, where the case of C* being the two-torus has been considered, one can quickly find that in order to obtain the N + 0;) limit of the action ( 10) we can proceed as follows: We consider 8 to be a function on V x C* i.e., 8 = 0(x, y, X, p,p, 4) where (p, 9) are the coordinates on X2. Moreover, we make the following substitutions,
[.,
Therefore one can choose the gauge such that A, =O,
~Wx@W) + (~,.fv(J,f91},
-
(4b)
+ dyAy - $A!
[At,Ail + [A.&l
V
(6)
Henceforth we assume that this last condition holds. Consequently, Eqs. (4a) -( 4c), under condition (6)) read
1 --+{ . > . }P,
(W4
.I
TN.)+-
(12)
(.)dpdq,
22
where { . , . }p stands for the Poisson bracket (13)
&A,; - &A: + [A:,A?_.] = 0,
(74
d.rA.i - +,A, = 0.
(7b)
From (7b) it follows that there exists an su( N) -valued function 0 such that Ai = -d,.B, Substituting
A, = d,B.
(8)
forwfl =_f~(x,y,f,Y,p,q) andf2 = .h(x,y,l,7, P*4). Thus the action S defined by ( IO) is brought into the following form,
s+s,=
.I{-fe{a,e,q),
(8) into (7a) one gets +~[(a,e)(a,e)+(a?.e)(a,e)])dudpdq.
i),ar0 + a?,i?re + [ d,B, dyf?] = 0, 0 E su(N)
@C”(V).
(9)
Eq. (9) is equivalent to the su( N) SDYM equations (2a)-(2c). Now, straightforwardcalculations show that Eq. (9) can be derived from the following least action principle,
(14)
This shows that we now deal with the Lagrangian a six-dimensional space V x X2, L,
=
-fe{a,e,
+ ;w,e)me)
in
dye}p + (a,.em,m.
The similar Lagrangian in four dimensions considered in Ref. [ 271.
(15) was first
24
J.F. Pleba’nski. hf. Przanowski/Physics
It is an easy matter to prove that the Euler-Lagrange equation for C, reads
Letters A 212 (1996122-28
(c) Grant’s equation. Let now 8, = 82,
@.,z+ 0y.i;+ {&
e,,)P
= 0,
(16)
where we use the obvious notation 8, := r7,@,8,: := d,&9,. . . , etc. Ey. (16) resembIes very much the well known second heavenly equation [28] and we call it the sirdimensional version of the second heavenly equation. This equation has been found in our previous works [ 17,181 as the result of the N -+ 00 limit of Eqs. (7a), (7h). Here we show that the six-dimensional version of the second heavenly equation can he considered to be the N --f ixj limit of the su( N) SDYM equations also on the action level. Now we intend to present how the appropriate symmetry reductions af EZq. (16) lead to the heavenly equations (cf. Refs. f 17,183 ). (a) 7%efirst ~~eave~fyequation. Let @=8’ff;, = 6;,
+2+yjq, (13
Then, the function 8’ is of the form
Hence B has the form @(.Gy,~,j,p*q)
h,, + hs&!.,J - h&t,., = 0.
=.fxx+j,K-y,p,q)
(18)
Bx=-
62,
It is evident that the first heavenly equation can be
also obtained when other symmetries are assumed, for example tl; =o.
(20)
(b) The second heavenly equat~un. Here we assume the following symme~y, e!,=e,,.
(21)
Consequently, 8 takes the form ~(~,~~,~,~,~,9)
8, = -8,.
fv&Y3~,y”,p,q) = ff(x - KY - q,j,p)
=%x-l-&Y-t-p,%j)
(28)
(29)
(e) Husain 3 eq~af~~~. This equation has been found by the reduction of the Ashtekar-JacobsonSmolin equations to the sdiff(X2) chiral field equations in two dimensions [ 131. In our approach we assume the following symmetry, 8,. = 3,.
B(x,Y,:,.P,P,q)
(30)
=n(x-+a,Y+.?,JLq)
(31)
and Eq. ( 16) takes the form of Husain’s equation JLX + A,?. + &&J
- JL,,f$&l= 0.
(32)
To have a contact with some previous works [27,30,32] we rewrite Eq. (16) in terms of twospinors and then in the differential forms langauge. To this end put (-GY>P,j.P,9)
(22)
(27)
and, consequently, Eq. ( 16) is reduced to the evolution form of the second heavenly equation [30,31]
Therefore
(19)
(26)
Thus
~jnl-.,$2~,’ 1_ &,,&
= I.
(251
This is Grant’s equation [ 291. (d) The evoluti~~zfornz of the second heaver+ equation. Here we assume the symmetry
&. = 8x,
@,=B,,
= h(x +ky,p,q)
and Eq. ( 16) reads
and Eq. ( 16) is braught into the form of the first heavenly equation 1281
0; = 0,
(24)
H,, - f-I,, f N,.,.N,, - H.sI,IJ!,. = 0,
81,= -4;.
@(.&Y.l*j,P,9)
By = 0.
= (-Pi,-P”,9r,q2,~2,~,).
(33)
Hence, E!q. ( 16) reads now
and now Eq. ( 16) is brought into the form of the second heavenly equation [ 281 @.vx+ o?.,++ OX&~ - o;,, = 0.
(23)
A,B = 1,2,
(34)
J.F. Plebrinski, M. Przanowski/
where the spinorial cording to the rule
indices are to be manipulated
ac-
Physics Letters A 212 (1996) 22-28
i.e., 0 is of the form a=
(6~~1
:=
A,B,=
0 -1
1 0
,f,
= : (P),
1,2.
= -g--
a28
(35) C,
defined by (15) takes the
a28 _ --I ae
de
(36)
2 apA aijA ’
(~~,~~,r’,t-*,f*,J~). (37)
Consequently, by straightforward calculations one can show that in terms of differential forms the sixdimensional version of the heavenly equation (34) can be written as follows, dsA A dpA + drA A dqA + dJA A dqA = 0, (d~AAd~A+dsAAdqA)AdsBAdqr,AdpCAdpc
(38a) =O, (38b)
dpA A dpA A dqB Adqa A d$
Ad&
# 0.
(38~)
Thus, to obtain from (38a)-( 38c) the second heavenly equation one assumes (see (21) ) .r/, = 7”.
(39)
This leads to the equation
I a% a20 a20 -______ = 0, +--2apAapBapAapB aPAaqA @=@t-pA+@A,qB),
(40)
which is exactly the second heavenly equation as written in terms of two-spinors [ 27,30,32]. To get the first heavenly equation we put (41) Then from (38a) one infers the existence of a function 0 such that
an
an
z=aqA=
rA _I 2pA=SA+;qA,
from (41)
(44) and (42)
we obtain the relation
(cf. (17))
We now denote =
(43)
-___ -+1=0. 2 aqAaqBaqAaqB Finally,
apA&jB apAdgB
(8,,8!.,ei,8~,8,,,B,)
f%qA+pA.qB).
Consequently, (38b) and (42) give the first heavenly equation [ 27,30,32]
( )
Then the Lagrangian form
2.5
(42)
8 = fl+
$pAq,&
(45)
Analogously one can find other heavenly equations in the spinorial form. Now it is evident that the six-dimensional version of the second heavenly equation (34) implies the following conservation law, -~--_
ae aqA
(46)
According to our philosophy this conservation law overlaps the hierarchy of conservation laws for the heavenly equations. For example, in the case of the second heavenly equation, by (39), we have -ae/apA = ae/aijA. Therefore, in this case (46) gives
-~---
a@ aqA
(47)
where we substituted 0 in place of 0 (see 40) ). On the other hand, in the case of the first heavenly equation, employing (42) and (45), one quickly finds that the conservation law (46) reads
a a20 aa p-+qA aqA aqAaqBaqB -3
(48)
It is quite natural to expect that the conservation law (46) in six dimensions generates an infinite hierarchy of conservation laws in four dimensions when the heavenly equations are assumed to hold. In order to prove this statement and also to find the relation of our approach with the previous works by Boyer and one of us (J.F.P) [ 30,321 and by Strachan [ 331 we should first find the general theory of symmmetry reduction of the six-dimensional version of the second
26
J.F. Plebcinski, M. Przanowski/Physics
heavenly equation to the heavenly equations. Work on this theory is in progress. Finally, we are going to consider the Moyal deformation of Eq. ( 16). To this end we consier the SDYM equations (7a), (7b) assuming that the potentials are now the self-adjoint operator-valued functions on V c RJ acting in a Hilbert space IFI = L*(R’). Thus we now deal with the equations
Letters A 212 (1996) 22-28
+ ~wxe) * (c&O) + (aye) * (d,S)l} Pj(Pt 9) dp dq,
X
(54)
where 8 = &x,y,Z,jj,p,q)
&A_? - a&
+ &,a,,
a,&
= 0,
a;
+ dJ!;
a,1 = 0,
(49b)
A; = A.?.
= a,,
(49a)
(49c)
Then from (49b) we get ai = -a,.l?, e = &x.&?,g) Inserting
(the Weylcorrespondence), and pj = p,, (p, q) denotes the Wignerfunction for I@j) 3 i.e.,
A, = 3x8, = 8’.
(50)
(50) into (49a) one obtains
&a,8 + a+@
+ $ [a,&
a,.& = 0.
(51)
Straightforward calculations show that Eq. (5 1) can be derived from the following variational principle,
Moreover, the Moyal t-product is defined by ifi
* := exp and { .,
2
f2)M
(fl1
stands for the Moyal bracket
. }M
:=
7
= ift
sin
5”
.f23
(
>
fl = f1(x,y,f,j,p,q), f2
(58)
= fi(x,y,I,y,p,q).
From (53) one quickly finds that + ; [
ma@A + (a& ($8)1> w,;),
where { I$,i)}.iE~ constitutes
an orthonormal
(52)
Inserting form C
IG.i)(+jI = i.
I
(59)
basis in
X, (@.ilSl) = %k*
PJ=%. C’ .i
(59) into (54) we get the action S’“’ of the
(53) $4)
=
J Employing the Weyl-Wigner-Moyal formalism [ 3440] one can bring C(q) into the following form,
i-i@
*
{he, a&M + ; [ (&e) *
(&e)
VxR2 +
(a,@
* (+O)]}
dvdp dq.
(60)
J.F.
Pleblnski,
M. Pr:anowski/
This action leads to the Moyal deformation of the sixdimensional version of the second heavenly equation
Physics Letters A 212 Ci996) 22-28
1S] S. Chakravarty,
M.J. Ablowitz
Let!. 65 (1989) 16 1 M.J. Ablowitz
&al0 + &d$
+ {a,e,a,e},
= 0.
AS lim fr * f~ =
h-0
references
therein.
one quickly finds that the six-dimensional version of the second heavenly equation ( 16) is the fi + 0 limit of Eq. (61) and
and E.T. Newman,
(63) that the following
L.J. Mason,
Twistor
IO] S. Chakravarty, 32 (1991)
N-KC
U SDYM equations ‘2 heavenly equations su( N) SDYM equations (where c3 denotes the set of the self-adjoint operators in 7-f). Rem&. The Lagrangian ( IO) has been considered by Leznov [41] and then by Parkes [42]. Another SDYM Lagrangian has been proposed in Refs. [ 4.3.441. It has been written within the J formalism and we have not been able to find the analogy of the matrix J in self-dual gravity. We suppose that this analogy can be found when the Weyl-Wigner-Moyal formalism is used and then when the fi + 0 limit is considered. Work on this problem is in progress. We are grateful to the referee for pointing out Refs. [ 41441 and for stating the question of the uniqueness of the N + rx3 limit for the SDYM Lagrangians.
Newslett.
We are indebted to H. Garcia-CompeBn for useful discussions. One of us (M.P.) is grateful to the staff of the Department of Physics at CINVESTAV for the warm hospitality. This work is supported by CONACyT and CINVESTAV, Mexico, D.F., Mexico.
Plebaliski,
gravity, preprint,
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rheory
quantum
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( 1994)
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1079.
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and M. Przanowski.
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from 4D self-dual
hep-th19508012.
1171 J. E Plebaliski M. Przanowski. Phys. Pol. B 26 ( 1995) 889. 1. E Plebariski
681,
and H. Garcia-Compein,
field theories derived
KHTP-99-08,
1415. 63.
I I ( 1994) 927.
The universal
via Weyl-Wigner-Moyal
covering formalism.
to appear. I19
I.A.B.
Strachan.
integrable
120 121 122
The
geometry
systems, preprint
K. Takasaki,
J. Geom. Phys.
of
multidimensional
( 199.5).
I1 I.
I4 (1994)
C.N. Yang. Phys. Rev. Let!. 38 (1977)
332.
1377.
I. Hoppe, Phys. Let!. B 215 ( 1988) 706.
123
D.B. Fairlie and C.K. Zachos. Phys. Let!. B 224 ( 1989)
i 24
D.B. Fairlie.
125
E.G. Floratos,
J. Iliopoulos
B 217 (1989)
285.
(1990)
and C.K. Zachos. J. Math.
101.
Phys. 31
1088.
1261 P. Fletcher, 1271 C.P. Boyer. relativity
P. Fletcher
and G. Tiktopoulos.
Phys. Let!. B 248 ( 1990) J.D. Finley
III
and gravitation,
ed. A. Held (Plenum,
Einstein
New York.
Phys. Len.
323.
and J.F. Plebariski, memorial 1980)
in: General
volume,
pp. 241-28
1281 J.F. Plebariski. J. Math. Phys. I6 (1975) 2395. 1291 J.D.E. Grant, Phys. Rev. D 48 (1993) 2606. I301 C.P. Boyer and J. F. Plebariski, J. Math. Phys.
Vol. 2. I.
I8 ( 1977)
1022. Finley
III, J.F. Plebariski.
M. Przanowski
and H. Garcia-
( 1993) 435.
J.F. Plebariski, J. Math. Phys. 26 ( 1985) 229. 1331 I.A.B. Strachan, Class. Quantum Grav. IO ( 199.7) 1417. 1341 H. Weyl, Z. Phys. 46 (1927) I. 13.51E.P. Wigner, Phys. Rev. 40 ( 1932) 749. I361 J.E. Moyal. Proc. Cambridge Philos. Sot. 45 ( 1949) 99. I371 F. Bayen. M. Flato, C. Fronsdal. A. Lichnerowicz and D. Stemheimer. Ann. Phys. (NY) I1 1 (1978) 61. I I I. 1381 J.F. Plebariski. preprint No. 69, Institute of Physics. Nicolaus 1321 C.P. Boyerand
Ward, Nucl.
121
semiclassical,
Phys. A 7 (1992)
CompeBn, Phys. Lett. A I81
1I 1 KS. 121R.S. 13j R.S.
30 (1990)
Phys. 34 (1993)
M.
Ueno, Integrable
1311 J.D. References
Phys.
(World
Phys. Lett. B 282 (1992)
Acta Phys. Pol. B 25
118
I 1. and
1458.
[ 141 C. Castro, J. Math.
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random.
L.J. Mason and E.T. Newman,
Strachan,
[ 15 I J. E
hlath.
and Z. Popowicz
131 V. Husain, Class. Quantum
Gathering all that one concludes relations hold:
199
I99 I )
Singapore.
121 I.A.B.
and inverse
systems, W-algebras and ali
classical,
I I I Q. Han Park, In!. J. Mod.
lim S’“’ = S CO’ n-o
lecture notes series, equations
Commun.
integrable fields:
eds. PA. Garbaczewski
[ 91
Phys. Rev.
659.
that, in: Non-linear
(62)
LMS
evolution
Univ. Press, Cambridge,
18 1 I. Bakas, Self-duality,
= {.fltf2)Pq
lim{fl,f?}M
h-n
nonlinear
(Cambridge
(1989)
f1f2,
and P.A. Clarkson,
scattering 171 L.J. Mason
and P.A. Clarkson.
1085.
Vol. 149. Solitons.
(61)
27
Copernicus
University.
Toruri ( 1969).
28
J.F. Plebcinski. M. Przunowski/Physics
191 M. Hillery.
R.F. O’Connell,
Phys. Rep. 106 (1984)
M.O.
Scully
and E.P. Wigner,
121.
401 W.I. Tatarskij, Usp. Fiz. Nauk 139 (1983) 587. 41 I A.N. Leznov, Theor. M. Phys. 73 (1988) 1233.
Letters A 212 (1996) 22-28
[ 421
A. Parkes, Phys. Lett. B 286 (1992)
1431 V.P. Nair and J. Schiff,
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1441 B. Hou and L. Chao. Phys. Len. B 298 (1993)
423 103.
1 I March 1996
2s
‘5%
PHYSICS
ELSEWER
LETTERS
A
Physics Letters A 212 (1996) 22-28
The Lagrangian of a self-dual gravitational field as a limit of the SDYM Lagrangian Jerzy F. Plebahki Depurmtenrof Physics,
Centro de Invesrigacih
Received 2 October
I, Maciej Przanowski 2
y de Estudios Avanzados
del IPN.
AportadoPostal14-740. Mexico
1995; revised manuscript received 22 December 1995; accepted Communicated by P.R. Holland
for publication
07000. D.E, Me.rico
3 January
1996
Abstract
The action for the W(N) SDYM equations is shown to give in the limit N -+ cc the action for the six-dimensional version of the second heavenly equation. The symmetry reductinns of this latter equation to the well known equations of self-dual gravity are given. The Moyal deformation of the heavenly equations are also considered. PACT: 04.2O.C~;
11.15.-q
Recently, a great deal of interest has been shown in symmetry reductions of the SDYM equations to integrable equations of mathematical physics [ 1-16 1. In particular many papers concern the symmetry reductions of the SDYM equations to the self-dual gravity equations [6-161. In our previous works we have found the general form of the su( N) SDYM equations which in the limit N ---f co gives the sdiff (X2) g su(co) SDYM equation in the form of the second heavenly equation in six dimensions [ 17,18]. We have also shown that the symmetry reductions of that equation lead to the well known heavenly equations of self-dual gravity. We were also able to obtain the natural Moyal deformation of the heavenly equations. This deformation has been previously found by Strachan [ 12,191 and Takasaki [ 201.
t E-mail: [email protected]. z Permanent address: Institute of Physics. Technical of L6di. W6lczariska 219,93-005 Mdi, Poland. Elsevier Science B.V. PN SO375-9601(96)00025-4
University
In the present work we are going to show that the limiting process N + co leading from the su( N) SDYM equations to the heavenly equations can be also well defined on the action level. Similarconsiderations enable one to find the action for the Moyal deformation of the six-dimensional version of the second heavenly equation.
We deal with the su( N) SDYM equations in the flat four-dimensional real, simply connected flat manifold V C R4 of the metric ds2 = 2(dx C%dZ + dy 8s di;), where x, y, .i!, ji are null coordinates on V notes the symmetrized tensor product, i.e., i (dx 8 d? + d.Z @ dx). The coordinates are chosen in such a manner that the su( equations read [ 2 1]
(1) and C& dedx C&df = (x, v, 2, .?) N) SDYM
Fxy = 0,
(2a)
Fz\;= 0,
(2b)
J.F. Plebdnski. M. Przanowski/Physics
(2c)
F,i + F,., = 0,
Lerrers A 212 (1996) 22-28
sS=O,
du=
Cdu,
S=
23
dxdydzdy,
J where, as usually, F,, E SU(N) c3ccoo(v),y,v E {x, y, I, J}, stands for the Yang-MiIIs field tensor. Then, in terms of the Yang-Mills potentials A, E su( N) 8 Cm(V) which define Fpy according to the well known formula,
+A,].
F,, = [a,+&&
one rewrites Eqs. (2a)-( &A,. - &A, + &A,
+
2c) as follows,
[A.v,A?.l= 0,
- +A1 + [At.Ay]
?,Ai - &A,
(3)
= 0,
(W
= 0.
(4c)
Eq. (4a) implies that the potentials A, and AT are of the pure gauge form, i.e., there exists an SU( N)valued function g such that A,r = g-‘&g,
AT = g-‘&g.
(5)
A,.=O.
(10)
Thus the Lagrangian C defined by ( 10) can be considered to be the Lagrangian for the SDYM field. Now we let N tend to infinity. Thus we arrive at the su( co) algebra. It is well known that [ 22-261 su(oo)
2 sdiff(s*)
g the Poisson algebra on C’, (11)
where X2 is a two-dimensional real manifold. Employing the results of Refs. [ 23,241, where the case of C* being the two-torus has been considered, one can quickly find that in order to obtain the N + 0;) limit of the action ( 10) we can proceed as follows: We consider 8 to be a function on V x C* i.e., 8 = 0(x, y, X, p,p, 4) where (p, 9) are the coordinates on X2. Moreover, we make the following substitutions,
[.,
Therefore one can choose the gauge such that A, =O,
~Wx@W) + (~,.fv(J,f91},
-
(4b)
+ dyAy - $A!
[At,Ail + [A.&l
V
(6)
Henceforth we assume that this last condition holds. Consequently, Eqs. (4a) -( 4c), under condition (6)) read
1 --+{ . > . }P,
(W4
.I
TN.)+-
(12)
(.)dpdq,
22
where { . , . }p stands for the Poisson bracket (13)
&A,; - &A: + [A:,A?_.] = 0,
(74
d.rA.i - +,A, = 0.
(7b)
From (7b) it follows that there exists an su( N) -valued function 0 such that Ai = -d,.B, Substituting
A, = d,B.
(8)
forwfl =_f~(x,y,f,Y,p,q) andf2 = .h(x,y,l,7, P*4). Thus the action S defined by ( IO) is brought into the following form,
s+s,=
.I{-fe{a,e,q),
(8) into (7a) one gets +~[(a,e)(a,e)+(a?.e)(a,e)])dudpdq.
i),ar0 + a?,i?re + [ d,B, dyf?] = 0, 0 E su(N)
@C”(V).
(9)
Eq. (9) is equivalent to the su( N) SDYM equations (2a)-(2c). Now, straightforwardcalculations show that Eq. (9) can be derived from the following least action principle,
(14)
This shows that we now deal with the Lagrangian a six-dimensional space V x X2, L,
=
-fe{a,e,
+ ;w,e)me)
in
dye}p + (a,.em,m.
The similar Lagrangian in four dimensions considered in Ref. [ 271.
(15) was first
24
J.F. Pleba’nski. hf. Przanowski/Physics
It is an easy matter to prove that the Euler-Lagrange equation for C, reads
Letters A 212 (1996122-28
(c) Grant’s equation. Let now 8, = 82,
@.,z+ 0y.i;+ {&
e,,)P
= 0,
(16)
where we use the obvious notation 8, := r7,@,8,: := d,&9,. . . , etc. Ey. (16) resembIes very much the well known second heavenly equation [28] and we call it the sirdimensional version of the second heavenly equation. This equation has been found in our previous works [ 17,181 as the result of the N -+ 00 limit of Eqs. (7a), (7h). Here we show that the six-dimensional version of the second heavenly equation can he considered to be the N --f ixj limit of the su( N) SDYM equations also on the action level. Now we intend to present how the appropriate symmetry reductions af EZq. (16) lead to the heavenly equations (cf. Refs. f 17,183 ). (a) 7%efirst ~~eave~fyequation. Let @=8’ff;, = 6;,
+2+yjq, (13
Then, the function 8’ is of the form
Hence B has the form @(.Gy,~,j,p*q)
h,, + hs&!.,J - h&t,., = 0.
=.fxx+j,K-y,p,q)
(18)
Bx=-
62,
It is evident that the first heavenly equation can be
also obtained when other symmetries are assumed, for example tl; =o.
(20)
(b) The second heavenly equat~un. Here we assume the following symme~y, e!,=e,,.
(21)
Consequently, 8 takes the form ~(~,~~,~,~,~,9)
8, = -8,.
fv&Y3~,y”,p,q) = ff(x - KY - q,j,p)
=%x-l-&Y-t-p,%j)
(28)
(29)
(e) Husain 3 eq~af~~~. This equation has been found by the reduction of the Ashtekar-JacobsonSmolin equations to the sdiff(X2) chiral field equations in two dimensions [ 131. In our approach we assume the following symmetry, 8,. = 3,.
B(x,Y,:,.P,P,q)
(30)
=n(x-+a,Y+.?,JLq)
(31)
and Eq. ( 16) takes the form of Husain’s equation JLX + A,?. + &&J
- JL,,f$&l= 0.
(32)
To have a contact with some previous works [27,30,32] we rewrite Eq. (16) in terms of twospinors and then in the differential forms langauge. To this end put (-GY>P,j.P,9)
(22)
(27)
and, consequently, Eq. ( 16) is reduced to the evolution form of the second heavenly equation [30,31]
Therefore
(19)
(26)
Thus
~jnl-.,$2~,’ 1_ &,,&
= I.
(251
This is Grant’s equation [ 291. (d) The evoluti~~zfornz of the second heaver+ equation. Here we assume the symmetry
&. = 8x,
@,=B,,
= h(x +ky,p,q)
and Eq. ( 16) reads
and Eq. ( 16) is braught into the form of the first heavenly equation 1281
0; = 0,
(24)
H,, - f-I,, f N,.,.N,, - H.sI,IJ!,. = 0,
81,= -4;.
@(.&Y.l*j,P,9)
By = 0.
= (-Pi,-P”,9r,q2,~2,~,).
(33)
Hence, E!q. ( 16) reads now
and now Eq. ( 16) is brought into the form of the second heavenly equation [ 281 @.vx+ o?.,++ OX&~ - o;,, = 0.
(23)
A,B = 1,2,
(34)
J.F. Plebrinski, M. Przanowski/
where the spinorial cording to the rule
indices are to be manipulated
ac-
Physics Letters A 212 (1996) 22-28
i.e., 0 is of the form a=
(6~~1
:=
A,B,=
0 -1
1 0
,f,
= : (P),
1,2.
= -g--
a28
(35) C,
defined by (15) takes the
a28 _ --I ae
de
(36)
2 apA aijA ’
(~~,~~,r’,t-*,f*,J~). (37)
Consequently, by straightforward calculations one can show that in terms of differential forms the sixdimensional version of the heavenly equation (34) can be written as follows, dsA A dpA + drA A dqA + dJA A dqA = 0, (d~AAd~A+dsAAdqA)AdsBAdqr,AdpCAdpc
(38a) =O, (38b)
dpA A dpA A dqB Adqa A d$
Ad&
# 0.
(38~)
Thus, to obtain from (38a)-( 38c) the second heavenly equation one assumes (see (21) ) .r/, = 7”.
(39)
This leads to the equation
I a% a20 a20 -______ = 0, +--2apAapBapAapB aPAaqA @=@t-pA+@A,qB),
(40)
which is exactly the second heavenly equation as written in terms of two-spinors [ 27,30,32]. To get the first heavenly equation we put (41) Then from (38a) one infers the existence of a function 0 such that
an
an
z=aqA=
rA _I 2pA=SA+;qA,
from (41)
(44) and (42)
we obtain the relation
(cf. (17))
We now denote =
(43)
-___ -+1=0. 2 aqAaqBaqAaqB Finally,
apA&jB apAdgB
(8,,8!.,ei,8~,8,,,B,)
f%qA+pA.qB).
Consequently, (38b) and (42) give the first heavenly equation [ 27,30,32]
( )
Then the Lagrangian form
2.5
(42)
8 = fl+
$pAq,&
(45)
Analogously one can find other heavenly equations in the spinorial form. Now it is evident that the six-dimensional version of the second heavenly equation (34) implies the following conservation law, -~--_
ae aqA
(46)
According to our philosophy this conservation law overlaps the hierarchy of conservation laws for the heavenly equations. For example, in the case of the second heavenly equation, by (39), we have -ae/apA = ae/aijA. Therefore, in this case (46) gives
-~---
a@ aqA
(47)
where we substituted 0 in place of 0 (see 40) ). On the other hand, in the case of the first heavenly equation, employing (42) and (45), one quickly finds that the conservation law (46) reads
a a20 aa p-+qA aqA aqAaqBaqB -3
(48)
It is quite natural to expect that the conservation law (46) in six dimensions generates an infinite hierarchy of conservation laws in four dimensions when the heavenly equations are assumed to hold. In order to prove this statement and also to find the relation of our approach with the previous works by Boyer and one of us (J.F.P) [ 30,321 and by Strachan [ 331 we should first find the general theory of symmmetry reduction of the six-dimensional version of the second
26
J.F. Plebcinski, M. Przanowski/Physics
heavenly equation to the heavenly equations. Work on this theory is in progress. Finally, we are going to consider the Moyal deformation of Eq. ( 16). To this end we consier the SDYM equations (7a), (7b) assuming that the potentials are now the self-adjoint operator-valued functions on V c RJ acting in a Hilbert space IFI = L*(R’). Thus we now deal with the equations
Letters A 212 (1996) 22-28
+ ~wxe) * (c&O) + (aye) * (d,S)l} Pj(Pt 9) dp dq,
X
(54)
where 8 = &x,y,Z,jj,p,q)
&A_? - a&
+ &,a,,
a,&
= 0,
a;
+ dJ!;
a,1 = 0,
(49b)
A; = A.?.
= a,,
(49a)
(49c)
Then from (49b) we get ai = -a,.l?, e = &x.&?,g) Inserting
(the Weylcorrespondence), and pj = p,, (p, q) denotes the Wignerfunction for I@j) 3 i.e.,
A, = 3x8, = 8’.
(50)
(50) into (49a) one obtains
&a,8 + a+@
+ $ [a,&
a,.& = 0.
(51)
Straightforward calculations show that Eq. (5 1) can be derived from the following variational principle,
Moreover, the Moyal t-product is defined by ifi
* := exp and { .,
2
f2)M
(fl1
stands for the Moyal bracket
. }M
:=
7
= ift
sin
5”
.f23
(
>
fl = f1(x,y,f,j,p,q), f2
(58)
= fi(x,y,I,y,p,q).
From (53) one quickly finds that + ; [
ma@A + (a& ($8)1> w,;),
where { I$,i)}.iE~ constitutes
an orthonormal
(52)
Inserting form C
IG.i)(+jI = i.
I
(59)
basis in
X, (@.ilSl) = %k*
PJ=%. C’ .i
(59) into (54) we get the action S’“’ of the
(53) $4)
=
J Employing the Weyl-Wigner-Moyal formalism [ 3440] one can bring C(q) into the following form,
i-i@
*
{he, a&M + ; [ (&e) *
(&e)
VxR2 +
(a,@
* (+O)]}
dvdp dq.
(60)
J.F.
Pleblnski,
M. Pr:anowski/
This action leads to the Moyal deformation of the sixdimensional version of the second heavenly equation
Physics Letters A 212 Ci996) 22-28
1S] S. Chakravarty,
M.J. Ablowitz
Let!. 65 (1989) 16 1 M.J. Ablowitz
&al0 + &d$
+ {a,e,a,e},
= 0.
AS lim fr * f~ =
h-0
references
therein.
one quickly finds that the six-dimensional version of the second heavenly equation ( 16) is the fi + 0 limit of Eq. (61) and
and E.T. Newman,
(63) that the following
L.J. Mason,
Twistor
IO] S. Chakravarty, 32 (1991)
N-KC
U SDYM equations ‘2 heavenly equations su( N) SDYM equations (where c3 denotes the set of the self-adjoint operators in 7-f). Rem&. The Lagrangian ( IO) has been considered by Leznov [41] and then by Parkes [42]. Another SDYM Lagrangian has been proposed in Refs. [ 4.3.441. It has been written within the J formalism and we have not been able to find the analogy of the matrix J in self-dual gravity. We suppose that this analogy can be found when the Weyl-Wigner-Moyal formalism is used and then when the fi + 0 limit is considered. Work on this problem is in progress. We are grateful to the referee for pointing out Refs. [ 41441 and for stating the question of the uniqueness of the N + rx3 limit for the SDYM Lagrangians.
Newslett.
We are indebted to H. Garcia-CompeBn for useful discussions. One of us (M.P.) is grateful to the staff of the Department of Physics at CINVESTAV for the warm hospitality. This work is supported by CONACyT and CINVESTAV, Mexico, D.F., Mexico.
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lim{fl,f?}M
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