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Lax pairs of integrable equations in 1 ≤ d ≤ 3 dimensions as reductions of the Lax pair for the self-dual Yang-Mills equations
27 February 1995 PHYSICS LETTERS A EIS;EVIER
Physics Letters A 198 (1995) 195-200
Lax pairs of integrable equations in 1 d 3 dimensions as reductions of the Lax pair for the self-dual Yang-Mills equations M. Legar6 a, A.D. Popov b a Department of Mathematics, University of Alberta, Edmonton, Canada T6G 2G! b Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna 141980, Moscow Region, Russian Federation
Received 28 June 1994; revised manuscript received 23 September 1994; accepted for publication 11 January 1995 Communicated by A.R Fordy
Abstract
Symmetry reductions of the self-dual Yang-Mills (SDYM) equations yield various systems of differential equations in lower dimensions. We describe a general scheme of reduction of the Lax pair for the SDYM equations with respect to an arbitrary subgroup of the conformal group. The compatibility of the reduced Lax pair leads to the SDYM equations reduced under the same symmetry group. The scheme is illustrated by three examples.
The self-duality equations for the Yang-Mills fields in R 4 endowed with a diagonal metric (+1,+1,+1,+1) (R (4'°)) or ( + 1 , + 1 , - 1 , - 1 ) (R (2'2)) are examples of nonlinear integrable equations in dimension d = 4 [1,2]. The Lax pair for these equations has been given by Ward [3] as well as Belavin and Zakharov [4]. Recently it has been shown that almost all known integrable equations in 1 + 1 and 1 ÷ 2 dimensions (Korteweg-de Vries, Ernst, Kadomtsev-Petviashvili and others) may be obtained by reductions of the SDYM equations (see, e.g., Refs. [5-9] ). In most cases the reduction of the SDYM equations has been carried out with respect to translations, and the Lax pairs corresponding to the reduced equations have been derived as trivial reductions of the Lax pair for the SDYM equations. The nontrivial examples of reductions of the Lax pair for the SDYM equations to the Lax pairs for the Ernst and Painlev6 equation PvI have been considered in Refs. [9]. The symmetries involved consisted of translations, rotations and dilatations. These results Elsevier Science B.V. SSDI 0375-9601 ( 9 5 ) 00041-0
served as a starting point for our investigation. In this note, we shall describe the symmetry reduction of the Lax pair for the SDYM equations with respect to an arbitrary subgroup of their invariance group under space transformations. Accordingly, its compatibility condition shall yield the SDYM equations reduced under the same subgroup. Even if this procedure is presented only for the Euclidean space R (4'0), it can also be applied to R (2'2). Let us denote by A u the Yang-Mills ( Y M ) p o t e n tials in R (4'0) with values in the Lie algebra gl(n, C). The SDYM equations have the following form,
where /z, u . . . . . 1. . . . . 4, the gauge fields Fu~ = 3~zAv - 3 ~ A ~ + [ A u, Av ] , 0 u =- 3/ 3x ~, and eu~p,~ is the completely antisymmetric tensor in R (4'0) (e1234 = 1 ) . A corresponding Lax pair for Eqs. ( 1 ) can be written as [3,4] [D1 + iD2 - A(D3 + iD4) ]gt(x, A) = 0,
M. Legar~, A.D. Popov / Physics Letters A 198 (1995) 195-200
196
[D3 - iD4 + A(DI - i D 2 ) ] ~ ' ( x , A) = 0,
(2)
where Du = Ou + A , , and g' E C" is a vector-function (or multiplet of scalar fields) depending on the coordinates x~, of R (4'0) and a complex parameter A E CP ~. In fact, q" is a section of a vector bundle/~ "~ Z x C n over a subspace Z = R (4'°) x CP 1 of dual projective twistor space CP 3.. Eqs. (2) imply that the bundle/~ (which is the pull-back of a bundle E _~ R (4,0) x C n with self-dual connection) is holomorphic [1,3]. One can verify that the vector parts I/1 = O~ + i82 A(O3+i04), ½ = 0 3 - i 0 4 + A ( & -iO2) of the covariant derivatives in (2) define a basis of antiholomorphic vector fields with respect to the following complex structure tensor j u on R (4'°) [ 10], j~
= __,,~/.tcr~a
bundle /~ ---, Z. This is possible if the lifts of the generators (5) are infinitesimal automorphisms of J¢ [121, £ x J~ - X~J,. + J~X(,,, - ~,, ..(,,~ = O,
(6)
where E2~ is the Lie derivative along the vector field )(~ on Z. The lifted vector fields on Z, which form a realization of the Lie algebra s o ( 5 , 1), are given by 2 a = x a,
~'a=ya+2za
K, iz = K l z ' - } - ~ a # x o . Z
a,
'
Pu=Pu,
D = D,
(7)
with the following expression of the generators Z a of the S0(3) rotations on S2,
e
Za
= GcSb 0
(8)
,
o~ c
where ~7,r, -a = { eabc, o ' = b , v = c ; 8 a, o ' = 4 ; -Sa~, v = 4} is the anti-self-dual 't Hooft tensor (see Ref. [ 11 ] ), a, b . . . . . 1,2, 3, Sa parametrizes S2 ~_ C P 1, SaSa = 1, and A = (sl + is2) / ( 1 + s3 ). We obtain the definition of the self-dual 't Hooft tensor rG~a if we change the signs before 6~ and 6 a in the definition of r ~ . Using identities for the 't Hooft tensors [ 11 ], it can be shown that jucj~=_$~,
jcUVl,~=_iVlU,
J~Vf=-iV~.
(4) Both the SDYM equations and their Lax pair in Euclidean space remain invariant under the group of conformal transformations S 0 ( 5 , 1). We shall now formulate a general algorithm of reduction of the Lax pair for SDYM equations with respect to a subgroup G of S 0 ( 5 , 1). First, let us write a realization of the Lie algebra of the group S 0 ( 5 , 1) in terms of vector fields on R (4,°), x a = 77u~x~a~, a 1
ya = ~Tu~XuG, -a
K u = 7xc~x~O F, - x u D ,
Pu = au,
D = x~8~,
(5)
where { X a, a = 1,2, 3} and {ya, a = 1,2, 3} generate two commuting SO(3)-subgroups in SO(4), P# are the generators of translations, K u are the generators of the special conformal transformations and D is the generator of dilatations. As a second step, we have to define the action of S 0 ( 5 , 1) on Z preserving the holomorphicity of the
where the coordinates sa can be expressed in terms of coordinates Yi ( i , j . . . . . 1,2) on R 2 C S2 with the relation A = Yl + iy2. Let us mention that the complex structure tensor e~ on S 2 is also invariant with respect to the transformations (7), i.e. £ 2 e ~ = O, V~ E so(5, 1). In order to reduce the linear system (2) under a subgroup G of the conformal group, we impose the conditions of G-invariance up to a gauge transformation on the potentials A s , ( x ) and the vector-function ~ ( x , A) [13], Ex¢A u = X ( A ~ + AcX~, u
=0uW~+[A~,WE],
VscCG,
(9a)
/22q~ = )?~:qt = _w~,q~,
V~ C ~,
(9b)
where the W¢'s are some functions with values in the gauge Lie algebra gl(n, C) and ~ is the Lie algebra of the group G c S 0 ( 5 , 1). When W E = O, one recovers the conditions of strict G-invariance [ 12,14], used in Examples 1 and 3 below. Thirdly, we choose, according to the method of symmetry reduction (see Ref. [ 14] and references therein), "new" coordinates on Z: the group parameter(s) which describe(s) each orbit, and the invariant coordinates OA and sr which parametrize the space of orbits and satisfy 0 ~-~( = 0 ,
£ £ OA =-- 2eOA =O,
197
M. Legard, A.D. Popov / Physics Letters A 198 (1995) 195-200
£&( - f~( = 0,
v~ c ~.
(10)
Here, the invariant coordinate sr represents the new "spectral parameter". The most general G-invariant Yang-Mills fields As`, solutions of Eq. (9a), can be written in terms of gl(n, C)-valued functions of the invariant coordinates OA and functions of the orbits coordinate(s). As for Eq. (9b), their solution is expressed in terms of an arbitrary function ~'(Oa, () of the invariant coordinates up to a GL(n, C)-valued multiplier depending on the orbit coordinate(s). Finally, we have to insert the functions A s, and ~p into the Lax pair (2). With the holomorphicity condition: 0 ~ = 0, we then find a reduced Lax pair as a set of linear differential equations in terms of functions of invariant coordinates. The compatibility of the residual Lax pair leads to the SDYM equations reduced under the subgroup G, except for one-dimensional reduced systems with trivially lifted group action. For the latter cases, the reduced SDYM equations still correspond to the compatibility conditions of the reduced linear equations if an adequate limit is performed, and Lax pairs can be obtained by multiplication of each equation with an appropriate factor. We intend to present the nontrivial reductions under all conjugated subgroups of the Euclidean group in a long paper. In the following, we illustrate the described scheme with only three examples. Example 1. Let us consider the one-dimensional group SO(2) which is generated by the vector field y3. We introduce on a stratum of R (4"0) × CP 1 the orbit variable ~p and the invariant variables {r, R, X, a, r / o r ~r = a e-it/}, where r, R, a > 0 and 0 ~< ~p, X, r / < 27r, which are related to the coordinates {xs`, h = a e -i¢} on R (4'°) × CP l (0 ~< ( < 2 ~ ) by the formulae x, = rcos(x
- ~p -
¼r/),
x3 = R c o s ( x + ~ o + ¼"r/), ~: = ½"r/- 2q~ =:~ A =
x2 = -rsin(
X - q~ - ¼ r / ) ,
x4 = - R s i n ( x + ~ o + ¼r/),
~ " e i(n/2+2~°) .
(11)
~,3 = X102 _ X2o~1 _ X3o~4 q._ X4~ 3 + 2i(,~aa - ~aa) = ~ .
(12) The solution of Eq. (9a) with W¢ = 0 has the form a2
A3 = a3 c o s ( x + q9 + ¼.q) + a4 s i n ( x + ~, + ¼r/), A4 = - a 3 s i n ( x + ~p + ¼r/) + a 4 c o s ( x + ~o + ¼r/), (13) where as` = as`(r, R, X). After imposing the holomorphicity condition on ~ , the strict invariant solution of Eq. (9b) is ~ = ~p( r, R, X, (). Substituting (13) and ~/, in the Lax pair (2), we obtain the following reduced Lax equations,
= 0,
(14a)
1
- ( ~ - ~ ) ( O ¢ + a 3 -ia4 + ((al -ia2)]~p =O, (14b) where the vector parts in (14a) and (14b) are respectively denoted by X and Y. Correspondingly, the SDYM equations ( 1 ) are reduced to [ 8 ]
!
l
1
1
Ora3 -- ORal -- Oxa4 + -~Oxa2 + [a2, a4] + [ a l , a 3 ] =0,
Ora4-1-8Ra2 + rOXa3 + -~8xal + [al,
s i n ( x - ~o - ¼r/),
a 4 ] ~- [a3, a2]
= 0,
1 1 1 1 CgRa4 -- O~raz + ~ a 4 -- r a 2 -k- ~Oxa3 -- r O x a l
- [ a l , a 2 ] + [a3,a4] = 0 ,
(15)
which agree with the compatibility conditions of the Lax pair (14) [Vx, Vy] - Vix, v] = 0.
For the lifted vector field ~-3 we get
Al = al c o s ( x - ~ - ¼r/) +
a2 = - a l s i n ( x - ~o - Jrl) + a2 c o s ( y - q~ - Jr/),
(16)
Example 2. Now we consider the four-dimensional subgroup SO(2) × ( S 0 ( 2 ) ~ T 2) of S0(5, l) generated by the vector fields X3 - Y3, X3 + Y3, P.~,/'4, with isotropy subalgebra X3-Y3 at (r, 0 , 0 , 0 ) , r > 0. The orbits are parametrized by coordinates ~9, x3 and x4, corresponding to the generators X3 + Y3, P.~ and
M. Legard, A.D. Popov / Physics Letters A 198 (1995) 195-200
198
P4 (0 ~< 0 < 27r). The invariant coordinates, satisfying Eqs. (10), are then r = (x~ 4- x~)1/2, ( = ,~ ei0. We restrict ourselves in this example to the gauge group SO(3) with generators {La,a = 1,2, 3}. For a nontrivial homomorphism e "¢x3-r~) E SO(2) t , e '~L3 E SO(3), we have Wxs-r, = -L3 [8]. Then the YM fields invariant up to a gauge have the form
A4 = - a 3 ( r ) sin[ ( a - / 3 ) (X + r/) ] + aa(r) cos[ (o~ - / 3 ) (X + r/) ].
((f
Solution of Eq. (9b) with W( = 0 is ~" = ~ ( r , ( ) . After substitution of (19) and tp, the Lax pair (2) is transformed to (cgr+~(O~+al+ia2-((a3+ia4))~=O,
ia3)L3
- ( r [ ( a l 4-ia2)L! 4- ( a 2 - ial)L2]}~ =0,
(Or_ ( 2rO~ +
(19)
(17)
where [ L~, L b ] = e a b c L c and the holomorphic solution ofEq. (9b) is ~ = ~b(r, ( ) . Inserting (17) and if, the Lax pair (2) becomes -
A2 = - a i (r) sin[ ( a +/3) (X + r/) ]
4- aa(r) sin[ (c~ - / 3 ) (X + r/) ],
cos 0"] L3, sin0 - -a3(r) F ,/
(rf
+ a2(r) sin[ ( a +/3) (X + 7-/)],
A3 = a3(r) cos[ ( a - / 3 ) (X + r/) ]
A3 = a l ( r ) L i + a2(r)L2, A4 = a2(r)L1 - a l ( r ) L 2 ,
{rOr 4- (c9( 4-
AI = al(r) cos[ ( a + f l ) ( X + ~7) ]
+ a2(r) cos[ (ce +/3) (X + r/) ],
/ sinO) L3, AI = ~ f ( r ) cos0 - -a3(r)r A2 = ( - f ( r )
( = z/exp[i2/3X - i ( 2 / 3 + 1)~] = a e -i~. The strict invariant YM fields are given by
((Or-Y-(20~+a3-ia4+((al--ia2))
~
(20)
+ i - ~ ) L3 where 7 = 2 / 3 / ( a + / 3 ) . It can be verified that the compatibility condition of the Lax equations (20) corresponds to the SDYM equations reduced under the same subgroup,
+ (al - ia2)L1 + (a2 + ial)Lzl ff = 0. For the gauge choice f = 0, the compatibility condition (16) gives the reduced SDYM equations
1
it2 + -a2 + [al,a2] - [a3,a4] = 0, r
r&l - ala3 = O, r&2 - a2a3 = O, it3 - r( a21 + a~) = O.
(/3 4-
Example 3. Let us choose the three-dimensional subgroups of S 0 ( 5 , 1) which are composed of the generators X = aX 3 +/3y3, P3, P4, where a and/3 are arbitrary real numbers such that o~ 4: -/3. We have /531// ~ 031/' ----0,
/341/t ~ 04 ~/t ----0
::::k
a4 4-
1-y r
1-3/ F
a3 4- [al, a3] 4- [a2, a4] = 0, a4 + [al,a4] + [a3,a2] = 0.
(21)
These equations have been considered in Ref. [8]. With the change of variables r = lnr, a2 = r - i N 3 , a3 = r r - i N 1 , a4 = r ~-1N2, and the gauge condition al = 0, we obtain the modified Nahm equations [ 8]
.~7~ = [ ( a + / 3 ) (xlc~2 - x2al) d
+ ( a - / 3 ) (xs& - x4a3) + 2i/3(~Oa - ~hOT,)]qt = _Ox~,
-d-~TN1 = [N2, N31,
(18)
where xl = r cos [ ( a + / 3 ) ( X + r l ) ], x2 = - r sin [ (ce+ f l ) ( X 4 - r / ) ] , A = a e i2/~(n-x), r,a > 0, 0 <~ X , r / < 27r. We have selected as orbit coordinates X, x3, x4, and as invariant coordinates: r = (x{ + x2z) ½, a, r / o r
d N2 = [ N 3 , N I ] ,
d --~TN3 = exp(2yr) IN1, N2].
(22)
If we allow 3/ = 0, we retrieve the Nahm equations [ 15], and their Lax pair if each equation in (20) is multiplied by r and then transformed to the above
M. Legard, A.D. Popov / Physics Letters A 198 (1995) 195-200
variables with al = 0. The same results can be obtained if we take the limit of y to 0 in Eqs. (20) and (21). The equations of the Toda lattice with damping are derived via the algebraic reduction of the equations (22) with the matrices N1, N2, N3 C gl(n, C), defined as
( ' 0 bl 0 ... bl 0 b2 Ob2
0
bn 0
0
0 -bl
bl 0
0 bn-2 0 bn-2 0 bn- 1 0 bn-j 0
0
-b2
0 ... b2
0
d2 1 d dt 2 q'~ + t ~ q ' ~ = 2Y-2{exp[2(q'~-I - q'~ ) ] - exp[2(q,~ - qa+l ) ] }.
(25)
The corresponding Lax pair is obtained by inserting (23) in Eq. (20) and by carrying out the abovementioned changes of variables• One of the authors (A.D.R) thanks for their kind hospitality the Department of Mathematics of the University of Alberta (Edmonton) and the Max-PlanckInstitut ftir Physik (Munich), where part of this work was done. The authors thank for its kind hospitality the Centre de Recherches Math6matiques, Universit6 de Montrdal, where this work was completed• This work was supported by grant from the NSERC of Canada, by grant 93-011-140 from the Russian Foundation for Fundamental Research and by grant MAL000 from the International Science Foundation.
N1=
0 bn 0 ...
199
-bn
0
0
N2 =i References 0
b~
0
0 b.-2 0 -b.-2 0 bn-l 0 -bn-1 0
N3i (C!0c2cn0i)n ~
ca = O,
(23)
o:=1
where b~ = exp(q,~ - q~+~ ), ca = p~. The modified Nahm equations are then reduced to d ~ q , ~ = p,~, d -~rp~, = 2 exp(2y~-) {exp[2(q,_l - q,~) ] - e x p [ 2 ( q , - q~+l) ] }.
(24)
When y = 0, the latter equations coincide with the standard periodic Toda lattice equations• However, if y 4= 0, we find the equations of the Toda lattice with damping using the variable t = e x p ( y r ) ,
[1] M.E Atiyah, Classical geometry of Yang-Mills fields (Scuola Normale Superiore, Pisa, 1979); R.S. Ward and R.O. Wells Jr., Twistor geometry and field theory (Cambridge Univ. Press, Cambridge, 1990). [2] M.J. Ablowitz and P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering (Cambridge Univ. Press, Cambridge, 1991). [31 R.S. Ward, Phys. Lett. A 61 (1977) 81. [4] A.A. Belavin and V.E. Zakharov, Phys. Lett. B 73 (1978) 53. [5] R.S. Ward, Philos. Trans. R. Soc. A 315 (1985) 451; Lect. Notes Phys. 280 (1987) 106; Lond. Math. Soc. Lect. Notes Ser. 156 (1990) 246; L.J. Mason and G.A.J. Sparling, Phys. Lett. A 137 (1989) 29; J. Geom, Phys. 8 (1992) 243. [6] S. Chakravarty, M.J. Ablowitz and P.A. Clarkson, Phys. Rev. Lett. 65 (1990) 1085; I. Bakas and D.A. Depireux, Mod. Phys. Lett. A 6 (1991) 399; S. Chakravarty and M.J. Ablowitz, in: Painlev6 transcendents, eds. D, Levi and P. Winternitz (Plenum Press, New York, 1992) p. 331; S. Chakravarty, S.Kent and E.T. Newman, J. Math. Phys. 31 (1991) 2253; 33 (1992) 382. [7] T.A. lvanova and A.D. Popov, Phys. Lett. A 170 (1992) 293; I.A.B. Strachan, J. Math. Phys. 33 (1992) 2477; 34 (1993) 243; J. Tafel, J. Math. Phys. 34 (1993) 1892; M.J. Ablowitz, S. Chakravarty and L.A. Takhtajan, Commun. Math. Phys. 158 (1993) 1289.
200
M. Legar~, A.D. Popov / Physics Letters A 198 (1995) 195-200
[8] M. Kovalyov, M. Legar6 and L. Gagnon, J. Math. Phys. 34 (1993) 3425. [9] L.J. Mason and N.M.J. Woodhouse, Nonlinearity 1 (1988) 73; 6 (1993) 569; J. Fletcher and N.N.J. Woodhouse, Lond. Math. Soc. Lect. Notes Ser. 156 (1990) 260. I10] N.M.J. Woodhouse, Class. Quantum Grav. 2 (1985) 257. I 11 ] M.K. Prasad, Physica D 1 (1980) 167. [ 12] A. Lichnerowicz, Gdom6trie des groupes de transformations (Dunod, Paris, 1958); S. Kobayashi, Transformation groups in differential geometry (Springer, Berlin, 1972). [13] P. Forg~ics and N,S. Manton, Commun. Math. Phys. 72 (1980) 15; J. Harnad, S. Shnider and L. Vinet, J. Math. Phys. 21 (1980) 2719; R. Jackiw and N.S. Manton, Ann. Phys. 127 (1980) 257;
V. Hussin, J. Negro and M.A. del Olmo, lnvariant connections in a non-Abelian principal bundle, preprint CRM-1872, Montreal (1993). [14] P.J. Olver, Applications of Lie groups to differential equations (Springer, Berlin, 1986); P. Wintemitz, in: NATO ASI Series C, Vol. 310. Partially integrable evolution equations in physics, eds. R. Conte and N. Boccara (Kluwer, Dordrecht, 1990) p. 515. [151 W. Nahm, in: Monopoles in quantum field theory, eds. N.S. Craigie, P. Goddard and W. Nahm (World Scientific, Singapore, 1981 ); N.J. Hitchin, Commun. Math. Phys. 89 (1983) 145; J. Hurtubise, Commun. Math. Phys. 120 (1989) 613; T.A. Ivanova, Russ. Math. Surv. 46 (1991) 174; T.A. lvanova and A.D. Popov, Lett. Math. Phys. 23 ( 1991 ) 29.
Physics Letters A 198 (1995) 195-200
Lax pairs of integrable equations in 1 d 3 dimensions as reductions of the Lax pair for the self-dual Yang-Mills equations M. Legar6 a, A.D. Popov b a Department of Mathematics, University of Alberta, Edmonton, Canada T6G 2G! b Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna 141980, Moscow Region, Russian Federation
Received 28 June 1994; revised manuscript received 23 September 1994; accepted for publication 11 January 1995 Communicated by A.R Fordy
Abstract
Symmetry reductions of the self-dual Yang-Mills (SDYM) equations yield various systems of differential equations in lower dimensions. We describe a general scheme of reduction of the Lax pair for the SDYM equations with respect to an arbitrary subgroup of the conformal group. The compatibility of the reduced Lax pair leads to the SDYM equations reduced under the same symmetry group. The scheme is illustrated by three examples.
The self-duality equations for the Yang-Mills fields in R 4 endowed with a diagonal metric (+1,+1,+1,+1) (R (4'°)) or ( + 1 , + 1 , - 1 , - 1 ) (R (2'2)) are examples of nonlinear integrable equations in dimension d = 4 [1,2]. The Lax pair for these equations has been given by Ward [3] as well as Belavin and Zakharov [4]. Recently it has been shown that almost all known integrable equations in 1 + 1 and 1 ÷ 2 dimensions (Korteweg-de Vries, Ernst, Kadomtsev-Petviashvili and others) may be obtained by reductions of the SDYM equations (see, e.g., Refs. [5-9] ). In most cases the reduction of the SDYM equations has been carried out with respect to translations, and the Lax pairs corresponding to the reduced equations have been derived as trivial reductions of the Lax pair for the SDYM equations. The nontrivial examples of reductions of the Lax pair for the SDYM equations to the Lax pairs for the Ernst and Painlev6 equation PvI have been considered in Refs. [9]. The symmetries involved consisted of translations, rotations and dilatations. These results Elsevier Science B.V. SSDI 0375-9601 ( 9 5 ) 00041-0
served as a starting point for our investigation. In this note, we shall describe the symmetry reduction of the Lax pair for the SDYM equations with respect to an arbitrary subgroup of their invariance group under space transformations. Accordingly, its compatibility condition shall yield the SDYM equations reduced under the same subgroup. Even if this procedure is presented only for the Euclidean space R (4'0), it can also be applied to R (2'2). Let us denote by A u the Yang-Mills ( Y M ) p o t e n tials in R (4'0) with values in the Lie algebra gl(n, C). The SDYM equations have the following form,
where /z, u . . . . . 1. . . . . 4, the gauge fields Fu~ = 3~zAv - 3 ~ A ~ + [ A u, Av ] , 0 u =- 3/ 3x ~, and eu~p,~ is the completely antisymmetric tensor in R (4'0) (e1234 = 1 ) . A corresponding Lax pair for Eqs. ( 1 ) can be written as [3,4] [D1 + iD2 - A(D3 + iD4) ]gt(x, A) = 0,
M. Legar~, A.D. Popov / Physics Letters A 198 (1995) 195-200
196
[D3 - iD4 + A(DI - i D 2 ) ] ~ ' ( x , A) = 0,
(2)
where Du = Ou + A , , and g' E C" is a vector-function (or multiplet of scalar fields) depending on the coordinates x~, of R (4'0) and a complex parameter A E CP ~. In fact, q" is a section of a vector bundle/~ "~ Z x C n over a subspace Z = R (4'°) x CP 1 of dual projective twistor space CP 3.. Eqs. (2) imply that the bundle/~ (which is the pull-back of a bundle E _~ R (4,0) x C n with self-dual connection) is holomorphic [1,3]. One can verify that the vector parts I/1 = O~ + i82 A(O3+i04), ½ = 0 3 - i 0 4 + A ( & -iO2) of the covariant derivatives in (2) define a basis of antiholomorphic vector fields with respect to the following complex structure tensor j u on R (4'°) [ 10], j~
= __,,~/.tcr~a
bundle /~ ---, Z. This is possible if the lifts of the generators (5) are infinitesimal automorphisms of J¢ [121, £ x J~ - X~J,. + J~X(,,, - ~,, ..(,,~ = O,
(6)
where E2~ is the Lie derivative along the vector field )(~ on Z. The lifted vector fields on Z, which form a realization of the Lie algebra s o ( 5 , 1), are given by 2 a = x a,
~'a=ya+2za
K, iz = K l z ' - } - ~ a # x o . Z
a,
'
Pu=Pu,
D = D,
(7)
with the following expression of the generators Z a of the S0(3) rotations on S2,
e
Za
= GcSb 0
(8)
,
o~ c
where ~7,r, -a = { eabc, o ' = b , v = c ; 8 a, o ' = 4 ; -Sa~, v = 4} is the anti-self-dual 't Hooft tensor (see Ref. [ 11 ] ), a, b . . . . . 1,2, 3, Sa parametrizes S2 ~_ C P 1, SaSa = 1, and A = (sl + is2) / ( 1 + s3 ). We obtain the definition of the self-dual 't Hooft tensor rG~a if we change the signs before 6~ and 6 a in the definition of r ~ . Using identities for the 't Hooft tensors [ 11 ], it can be shown that jucj~=_$~,
jcUVl,~=_iVlU,
J~Vf=-iV~.
(4) Both the SDYM equations and their Lax pair in Euclidean space remain invariant under the group of conformal transformations S 0 ( 5 , 1). We shall now formulate a general algorithm of reduction of the Lax pair for SDYM equations with respect to a subgroup G of S 0 ( 5 , 1). First, let us write a realization of the Lie algebra of the group S 0 ( 5 , 1) in terms of vector fields on R (4,°), x a = 77u~x~a~, a 1
ya = ~Tu~XuG, -a
K u = 7xc~x~O F, - x u D ,
Pu = au,
D = x~8~,
(5)
where { X a, a = 1,2, 3} and {ya, a = 1,2, 3} generate two commuting SO(3)-subgroups in SO(4), P# are the generators of translations, K u are the generators of the special conformal transformations and D is the generator of dilatations. As a second step, we have to define the action of S 0 ( 5 , 1) on Z preserving the holomorphicity of the
where the coordinates sa can be expressed in terms of coordinates Yi ( i , j . . . . . 1,2) on R 2 C S2 with the relation A = Yl + iy2. Let us mention that the complex structure tensor e~ on S 2 is also invariant with respect to the transformations (7), i.e. £ 2 e ~ = O, V~ E so(5, 1). In order to reduce the linear system (2) under a subgroup G of the conformal group, we impose the conditions of G-invariance up to a gauge transformation on the potentials A s , ( x ) and the vector-function ~ ( x , A) [13], Ex¢A u = X ( A ~ + AcX~, u
=0uW~+[A~,WE],
VscCG,
(9a)
/22q~ = )?~:qt = _w~,q~,
V~ C ~,
(9b)
where the W¢'s are some functions with values in the gauge Lie algebra gl(n, C) and ~ is the Lie algebra of the group G c S 0 ( 5 , 1). When W E = O, one recovers the conditions of strict G-invariance [ 12,14], used in Examples 1 and 3 below. Thirdly, we choose, according to the method of symmetry reduction (see Ref. [ 14] and references therein), "new" coordinates on Z: the group parameter(s) which describe(s) each orbit, and the invariant coordinates OA and sr which parametrize the space of orbits and satisfy 0 ~-~( = 0 ,
£ £ OA =-- 2eOA =O,
197
M. Legard, A.D. Popov / Physics Letters A 198 (1995) 195-200
£&( - f~( = 0,
v~ c ~.
(10)
Here, the invariant coordinate sr represents the new "spectral parameter". The most general G-invariant Yang-Mills fields As`, solutions of Eq. (9a), can be written in terms of gl(n, C)-valued functions of the invariant coordinates OA and functions of the orbits coordinate(s). As for Eq. (9b), their solution is expressed in terms of an arbitrary function ~'(Oa, () of the invariant coordinates up to a GL(n, C)-valued multiplier depending on the orbit coordinate(s). Finally, we have to insert the functions A s, and ~p into the Lax pair (2). With the holomorphicity condition: 0 ~ = 0, we then find a reduced Lax pair as a set of linear differential equations in terms of functions of invariant coordinates. The compatibility of the residual Lax pair leads to the SDYM equations reduced under the subgroup G, except for one-dimensional reduced systems with trivially lifted group action. For the latter cases, the reduced SDYM equations still correspond to the compatibility conditions of the reduced linear equations if an adequate limit is performed, and Lax pairs can be obtained by multiplication of each equation with an appropriate factor. We intend to present the nontrivial reductions under all conjugated subgroups of the Euclidean group in a long paper. In the following, we illustrate the described scheme with only three examples. Example 1. Let us consider the one-dimensional group SO(2) which is generated by the vector field y3. We introduce on a stratum of R (4"0) × CP 1 the orbit variable ~p and the invariant variables {r, R, X, a, r / o r ~r = a e-it/}, where r, R, a > 0 and 0 ~< ~p, X, r / < 27r, which are related to the coordinates {xs`, h = a e -i¢} on R (4'°) × CP l (0 ~< ( < 2 ~ ) by the formulae x, = rcos(x
- ~p -
¼r/),
x3 = R c o s ( x + ~ o + ¼"r/), ~: = ½"r/- 2q~ =:~ A =
x2 = -rsin(
X - q~ - ¼ r / ) ,
x4 = - R s i n ( x + ~ o + ¼r/),
~ " e i(n/2+2~°) .
(11)
~,3 = X102 _ X2o~1 _ X3o~4 q._ X4~ 3 + 2i(,~aa - ~aa) = ~ .
(12) The solution of Eq. (9a) with W¢ = 0 has the form a2
A3 = a3 c o s ( x + q9 + ¼.q) + a4 s i n ( x + ~, + ¼r/), A4 = - a 3 s i n ( x + ~p + ¼r/) + a 4 c o s ( x + ~o + ¼r/), (13) where as` = as`(r, R, X). After imposing the holomorphicity condition on ~ , the strict invariant solution of Eq. (9b) is ~ = ~p( r, R, X, (). Substituting (13) and ~/, in the Lax pair (2), we obtain the following reduced Lax equations,
= 0,
(14a)
1
- ( ~ - ~ ) ( O ¢ + a 3 -ia4 + ((al -ia2)]~p =O, (14b) where the vector parts in (14a) and (14b) are respectively denoted by X and Y. Correspondingly, the SDYM equations ( 1 ) are reduced to [ 8 ]
!
l
1
1
Ora3 -- ORal -- Oxa4 + -~Oxa2 + [a2, a4] + [ a l , a 3 ] =0,
Ora4-1-8Ra2 + rOXa3 + -~8xal + [al,
s i n ( x - ~o - ¼r/),
a 4 ] ~- [a3, a2]
= 0,
1 1 1 1 CgRa4 -- O~raz + ~ a 4 -- r a 2 -k- ~Oxa3 -- r O x a l
- [ a l , a 2 ] + [a3,a4] = 0 ,
(15)
which agree with the compatibility conditions of the Lax pair (14) [Vx, Vy] - Vix, v] = 0.
For the lifted vector field ~-3 we get
Al = al c o s ( x - ~ - ¼r/) +
a2 = - a l s i n ( x - ~o - Jrl) + a2 c o s ( y - q~ - Jr/),
(16)
Example 2. Now we consider the four-dimensional subgroup SO(2) × ( S 0 ( 2 ) ~ T 2) of S0(5, l) generated by the vector fields X3 - Y3, X3 + Y3, P.~,/'4, with isotropy subalgebra X3-Y3 at (r, 0 , 0 , 0 ) , r > 0. The orbits are parametrized by coordinates ~9, x3 and x4, corresponding to the generators X3 + Y3, P.~ and
M. Legard, A.D. Popov / Physics Letters A 198 (1995) 195-200
198
P4 (0 ~< 0 < 27r). The invariant coordinates, satisfying Eqs. (10), are then r = (x~ 4- x~)1/2, ( = ,~ ei0. We restrict ourselves in this example to the gauge group SO(3) with generators {La,a = 1,2, 3}. For a nontrivial homomorphism e "¢x3-r~) E SO(2) t , e '~L3 E SO(3), we have Wxs-r, = -L3 [8]. Then the YM fields invariant up to a gauge have the form
A4 = - a 3 ( r ) sin[ ( a - / 3 ) (X + r/) ] + aa(r) cos[ (o~ - / 3 ) (X + r/) ].
((f
Solution of Eq. (9b) with W( = 0 is ~" = ~ ( r , ( ) . After substitution of (19) and tp, the Lax pair (2) is transformed to (cgr+~(O~+al+ia2-((a3+ia4))~=O,
ia3)L3
- ( r [ ( a l 4-ia2)L! 4- ( a 2 - ial)L2]}~ =0,
(Or_ ( 2rO~ +
(19)
(17)
where [ L~, L b ] = e a b c L c and the holomorphic solution ofEq. (9b) is ~ = ~b(r, ( ) . Inserting (17) and if, the Lax pair (2) becomes -
A2 = - a i (r) sin[ ( a +/3) (X + r/) ]
4- aa(r) sin[ (c~ - / 3 ) (X + r/) ],
cos 0"] L3, sin0 - -a3(r) F ,/
(rf
+ a2(r) sin[ ( a +/3) (X + 7-/)],
A3 = a3(r) cos[ ( a - / 3 ) (X + r/) ]
A3 = a l ( r ) L i + a2(r)L2, A4 = a2(r)L1 - a l ( r ) L 2 ,
{rOr 4- (c9( 4-
AI = al(r) cos[ ( a + f l ) ( X + ~7) ]
+ a2(r) cos[ (ce +/3) (X + r/) ],
/ sinO) L3, AI = ~ f ( r ) cos0 - -a3(r)r A2 = ( - f ( r )
( = z/exp[i2/3X - i ( 2 / 3 + 1)~] = a e -i~. The strict invariant YM fields are given by
((Or-Y-(20~+a3-ia4+((al--ia2))
~
(20)
+ i - ~ ) L3 where 7 = 2 / 3 / ( a + / 3 ) . It can be verified that the compatibility condition of the Lax equations (20) corresponds to the SDYM equations reduced under the same subgroup,
+ (al - ia2)L1 + (a2 + ial)Lzl ff = 0. For the gauge choice f = 0, the compatibility condition (16) gives the reduced SDYM equations
1
it2 + -a2 + [al,a2] - [a3,a4] = 0, r
r&l - ala3 = O, r&2 - a2a3 = O, it3 - r( a21 + a~) = O.
(/3 4-
Example 3. Let us choose the three-dimensional subgroups of S 0 ( 5 , 1) which are composed of the generators X = aX 3 +/3y3, P3, P4, where a and/3 are arbitrary real numbers such that o~ 4: -/3. We have /531// ~ 031/' ----0,
/341/t ~ 04 ~/t ----0
::::k
a4 4-
1-y r
1-3/ F
a3 4- [al, a3] 4- [a2, a4] = 0, a4 + [al,a4] + [a3,a2] = 0.
(21)
These equations have been considered in Ref. [8]. With the change of variables r = lnr, a2 = r - i N 3 , a3 = r r - i N 1 , a4 = r ~-1N2, and the gauge condition al = 0, we obtain the modified Nahm equations [ 8]
.~7~ = [ ( a + / 3 ) (xlc~2 - x2al) d
+ ( a - / 3 ) (xs& - x4a3) + 2i/3(~Oa - ~hOT,)]qt = _Ox~,
-d-~TN1 = [N2, N31,
(18)
where xl = r cos [ ( a + / 3 ) ( X + r l ) ], x2 = - r sin [ (ce+ f l ) ( X 4 - r / ) ] , A = a e i2/~(n-x), r,a > 0, 0 <~ X , r / < 27r. We have selected as orbit coordinates X, x3, x4, and as invariant coordinates: r = (x{ + x2z) ½, a, r / o r
d N2 = [ N 3 , N I ] ,
d --~TN3 = exp(2yr) IN1, N2].
(22)
If we allow 3/ = 0, we retrieve the Nahm equations [ 15], and their Lax pair if each equation in (20) is multiplied by r and then transformed to the above
M. Legard, A.D. Popov / Physics Letters A 198 (1995) 195-200
variables with al = 0. The same results can be obtained if we take the limit of y to 0 in Eqs. (20) and (21). The equations of the Toda lattice with damping are derived via the algebraic reduction of the equations (22) with the matrices N1, N2, N3 C gl(n, C), defined as
( ' 0 bl 0 ... bl 0 b2 Ob2
0
bn 0
0
0 -bl
bl 0
0 bn-2 0 bn-2 0 bn- 1 0 bn-j 0
0
-b2
0 ... b2
0
d2 1 d dt 2 q'~ + t ~ q ' ~ = 2Y-2{exp[2(q'~-I - q'~ ) ] - exp[2(q,~ - qa+l ) ] }.
(25)
The corresponding Lax pair is obtained by inserting (23) in Eq. (20) and by carrying out the abovementioned changes of variables• One of the authors (A.D.R) thanks for their kind hospitality the Department of Mathematics of the University of Alberta (Edmonton) and the Max-PlanckInstitut ftir Physik (Munich), where part of this work was done. The authors thank for its kind hospitality the Centre de Recherches Math6matiques, Universit6 de Montrdal, where this work was completed• This work was supported by grant from the NSERC of Canada, by grant 93-011-140 from the Russian Foundation for Fundamental Research and by grant MAL000 from the International Science Foundation.
N1=
0 bn 0 ...
199
-bn
0
0
N2 =i References 0
b~
0
0 b.-2 0 -b.-2 0 bn-l 0 -bn-1 0
N3i (C!0c2cn0i)n ~
ca = O,
(23)
o:=1
where b~ = exp(q,~ - q~+~ ), ca = p~. The modified Nahm equations are then reduced to d ~ q , ~ = p,~, d -~rp~, = 2 exp(2y~-) {exp[2(q,_l - q,~) ] - e x p [ 2 ( q , - q~+l) ] }.
(24)
When y = 0, the latter equations coincide with the standard periodic Toda lattice equations• However, if y 4= 0, we find the equations of the Toda lattice with damping using the variable t = e x p ( y r ) ,
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