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Linear systems and conservation laws in d = 10, N = 1 supergravity
V01ume 216. num6er 3,4
PHY51C5 LE77ER5 8
12 January 1989
L1NEAR 5 Y 5 7 E M 5 AND C 0 N 5 E R V A 7 1 0 N LAW5 1N d = 10,
1 5UPER6RAV17Y
N=
L1n9-L1e C H A U Department 4f Phy51c5, Un1ver51ty4f Ca1~f0rn1a, Dav15, CA 95616, U5A and 8. M1LEW5K1 M1cr050ft C0rp0rat10n, P05t Qff1ce80x 97017• Redm0nd, WA 98073, U5A Rece1ved 11 March 1988; rev15edmanu5cr1pt rece1ved 12 0ct06er 1988
5tart1n9 w1th the 5et 0f 119ht-11ke1n1e9ra6111tyc0nd1t10n5 1n curved d= 10, N= 1 5uper5pace we der1ve the c0rre5p0nd1n9 11near 5y5tem. 7he v1e16e1n5and c0nnect10n5 that 5at15fythe c0n5tra1nt5 are parame1r12ed u51n9 a 11near5p1n0r 5u65pace def1ned a5 the kerne1 0f the 119ht-c0nepr0ject10n 0perat0r. 7hey fu1f111an 1nf1n1te5et 0f c0n5ervat10n 1aw5.
A5 d15cu55ed 1n ref. [ 1 ] 0ne can 9enera112e the n0t10n 0f 1nte9ra6111ty a10n9 119ht-11ke 11ne5 1n 5uper5pace t0 06ta1n d = 10, N = 1 5uPer9rav1ty c0n5tra1nt5. A1th0u9h they are t00 weak t0 put the the0ry 0n 5he11 1n the 5en5e 0f E1n5te1n e4uat10n5 0f m0t10n, we cann0t exc1ude the p05516111ty that they 1ead t0 50me h19her der1vat1ve c0nf0rma1 e4uat10n5 0f m0t10n (a5 1t 15 the ca5e 0f the c0rre5p0nd1n9 f0ur-d1men510na1 the0r1e5 [2] ). We had a150 1dent1f1ed the add1t10na1 c0n5tra1nt 7,~5~)- ~6J~70,5=0 ,
(1)
7he1r 1nterpretat10n 15 the f0110w1n9:a10n9 certa1n d1rect10n5 1n 5uper5pace the c0var1ant der1vat1ve5 06ey the f1at a19e6ra. 7heref0re, a10n9 the5e d1rect10n5 they can 6e 06ta1ned 6y a 5u1ta61e tran5f0rmat10n 0f the f1at (carte51an) v1e16e1n5 and c0nnect10n5, 1,e. E,.1 =~ua~tDa1,
00a8c=~u.,~1(D~1~u8 ~) 9t~-c" ,
(3)
where D M = (~,,,, De,) are f1at c0var1ant der1vat1ve5 fu1f1111n9 the 5tandard a19e6ra {De,, D . )"= F ,,.:~1~ %, .
(4)
wh1ch 1ead5 t0 the E1n5te1n type e4uat10n5 0fm0t10n (wh1ch are a150 Wey11nvar1ant due t0 the pre5ence 0 f the d11at0n f1e1d). 1n th15 p a p e r we 5ha11 d15cu55 the C0n5e4uenCe5 0f the 119ht-11ke 1nte9ra6111ty c0n5tra1nt5 1n the C0ntext 0f the the0ry 0f 11near 5y5tem5. 1n ref. [ 3 ] 11near 5y5tem5 were c0n5tructed 1n d = 4 5uPer9rav1ty the0r1e5. 1n re1". [4] the meth0d5 0f c0n5truct1n9 and e x p a n d 1n9 the 11near 5y5tem5 1n ten d1men510n5 were deve10ped f0r the ca5e 0f the ten-d1men510na15uper-Yan9M1115 the0ry (5ee a150 ref5. [ 5 , 6 ] ) . We 5ha11 app1y t h e m t0 d = 10, N = 1 5uper9rav1ty. 7 h e 119ht-11ke 1nte9ra6111ty c0n5tra1nt5 are
where the under11n1n9 0 f t h e 1nd1ce5 den0te5 the 119ht11ke pr0ject10n
v~F~:~vt~F~,~1~({V~, V/j} - F ~ V c ) = 0 .
~1.10e1~=~0(~0e 1~ ,
330
(2 )
E4. ( 2 ) 15 e4u1va1ent t0 the van15h1n9 0f certa1n ( p r 0 j e c t e d ) t0r510n5 [ 1 ] wh1ch 1n turn 1mp1y the f0110w1n9 c0nd1t10n5 •
w
v1~,.
v. . . . .
~u~ ~ , F ~ , .
~,~ = v " F . J 4 / ~1~.
~,,,-0
(5)
(6 )
7 0 fu1f111 e45. ( 5 ) the matr1ce5 9/~~ mu5t 6e pr0p0rt10na1 t0 the tran5f0rmat10n matr1ce5 0f the L0rent2 9r0up ~10m-~-~020a
m ,
(7)
V01ume216, num6er 3,4
PHY51C5 LE77ER5 8
1 0~ 1t =exp(~crj, F a h ) . ,1~
~5~=exp(0~)~
c~.:, = - c~1,,,,
12 January 1989
E~ =P(v, w)~f¢/5/~(w) D . ( 7 c0nt•d )
7he p0wer5 0f (p cance1 1n e4. (5) ref1ect1n9 the d11atat10na1 d1men510n5 0f the f1e1d5 0.~, 0.•••. 0 t h e r ten50r5 (e.9., curvature5) depend 0n der1vat1ve5 0f~0 6ut 0n1y thr0u9h the c0m61nat10n
+P(w, v)J~1/~(v) D. ,
(15a)
~.n~= P( v, w) 0f ( 75
(w)
+P(w, v),f(V/~D,0a"0,;~) (v) .
(156)
70 repr0duce the a19e6ra (2) we theref0re need t0 1mp05e the c0nd1t10n5
A5 1n ref. [4] we can 1ntr0duce 5p1n0r1a1 parametr12at10n 0 f t h e ten-d1men510na1 119ht c0ne. F0r 50me f1xed 119ht-11ke vect0r w a we c0n51der a 11near 5p1n0r 5pace def1ned a5 the kerne1 0fthe 0perat0r w~F., 1.e., a11 the 0. 5uch that
v<~F."1~E1j~=0, v"E<,~0=0.
w 4 2 ./~05j =0 .
~,5 ~5D:dp=E5(~.
(8)
(9a, 6)
0 n the 0ther hand, e4. (3) can 6e rewr1tten a5 Vj tu. w= 0 ,
(10)
(16)
7he 119ht c0ne then can 6e de5cr16ed a5 a c011ect10n 0f vect0r5
v<,= (0+5)-F<
0r m0re exp11c1t1y,
(17)
E45. (9) and ( 1 1 ) f0rm a 11near 5y5tem 1n the 5en5e that the1r 1nte9ra6111ty c0nd1t10n5 repr0duce the c0n5tra1nt5.1ndeed, fr0m
f0r 50me f1xed e" n0t 6e10n91n9 t0 the kerne1 0f 0J"F. (actua11y, at 1ea5t tw0 5uch patche5 are nece55ary t0 c0ver the wh01e c0ne, due t0 the n0ntr1v1a1 t0p0109y, c.f. re1". [4] ). W1th th15 parametr12at10n the 11near 5y5tem 6ec0me5
v0F, r.v"F/m~( {V., V/j}-F~,/[V,. ) ~=0 ,
( 0F~0) 6 9 ( 0 ) + 2 (0F"e) E.~0(0)
v.F~15V<~0:~,,=0,
v"V.0~=0.
(1 1a, 6)
(12)
we can extract the van15h1n9 0f the appr0pr1ate pr0jected t0r510n5 (the curvature5 d0 n0t enter th15 e4uat10n 51nce ~a 15 a 5ca1ar). C0n5e4uent1y, the 0ther e4uat10n
v . F :~-v~,F/'~2~({V., V : , } - F . 1 [ V < . ) 0 j ~ = 0 ,
(13)
5et5 the pr0ject10n5 0f curvature5 e4ua1 t0 2er0. 7he pre5ence 0f the add1t10na1 5ca1ar f1e1d ~ 1n the 11near 5y5tem 15 a new feature. 1t ref1ect5 the fact that the N = 1, d = 10 5uPer9rav1ty, and 0ur 5y5tem 0f c0n5tra1nt5 1n part1cu1ar, ha5 a 1ar9er 1nvar1ance 9r0up, name1y, the 5uper-Wey1 9r0up. N0t1ce that e45. (9a) and ( 1 1a) can 6e v1ewed a5 a 9enera112at10n 0f ch1ra11ty c0nd1t10n5.0n1y, 1n th15 ca5e, th1515 n0t a L0rent2 c0var1ant Y5pr0ject10n, 6ut a m0re re5tr1ct1ve 0 ( 8 ) 1nvar1ant (v-dependent) pr0ject10n. M0re prec15e1y, f0110w1n9 ref5. [4,5 ] 0ne can 1ntr0duce pr0ject10n 0perat0r5.
P(v, w)J= ~3L~ v~,6<,w/,F:,,~ 2(v•w)
(14)
wh1ch can 6e u5ed t0 extract the v1e16e1n5 and c0nnect10n5 fr0m e45. (9) and ( 11 )
+ (eF°e)E.(0(0)=0, (0F~0)V,0<,~(0) + 2 (0F%) V.0/••(0) + (eF"e)V~0d~(0)=0.
(18)
1t can 6e expanded 1n p0wer 5er1e5 0f the parameter 01ead1n9 t0 an 1nf1n1te 5et 0f c0n5ervat10n 1aw5. E45. (18) can 6e further 51mp11f1ed 1f we make the 5pec1a1 9au9e ch01ce
p(t,,w) 15(E55--D~)=0,
p(v,w)Ja)15;,a=0,
(19)
w1th w~= (eF~e) a f1xed 119ht-11ke vect0r. We can n0w pr0ceed w1th the expan510n
~0(0)= }2 0"~>~°<->, ~:>~(0)= )2 0"~>0#%>, (20) where 0 (• 1den0te5 the pr0duct 0,~,~,2 ... ~,,. 5u65t1tut1n9 (20) 1nt0 (18) and u51n9 (19) the f1r5t few term5 1n the expan510n read
w . F -#D15~0~°)=0 ,
(21)
w ~F."nDr~01,~ ~0~= 0 .
( 22a )
2Fra"e<7c~15E1~0)(°)+w"F.ca3D~0~,1)=0,
(226) 331
v01ume 216, num6er 3,4
PHY51c5 LE77ER5 8
12 January 1989
7h15 w0rk 15 part1a11y 5upp0rted 6y D 0 E fund1n9. etc . . . . . 7 h e 1mp0rtant advanta9e 0 f t h e 5p1n0r1a1 parametr12at10n, e4. ( 17 ), 15 that the expan510n5 can 6e exp11c1t1y carr1ed 0ut t0 any 0rder. 7 h e parameter5 0 f0rm a 11near 5pace and they enter va at m05t 4uadrat1ca11y. 70 5ummar12e, the pr061em 0f 501v1n9 a h19h1y n0n11near 5et 0f t0r510n and curvature c0n5tra1nt5 f0110w1n9 fr0m (2) wa5 reduced t0 a pr061em 0f 501v1n9 a 11near 5y5tem. 1t can 6e f0rmu1ated a5 f0110w5: f1nd a 5et 0f f1e1d5 0(x, 0), 0am(x, 0) 5uch that the v1e16e1n5 and c0nnect10n5 06ta1ned fr0m e45. (15a), (156) d0 n0t depend 0n the parameter5 0. Var10u5 meth0d5 were deve10ped f0r 501v1n9 th15 type 0f pr061em5,1, they were e5pec1a11y 5ucce55fu1 1n tw0d1men510na1 the0r1e5 [8] where they 1ed t0 c0mp1ete1y 501va61e m0de15, a5 we11 a51n f0ur d1men510n5 - 1ead1n9 t0 the 501ut10n5 0f5e1f-dua1 Yan9-M1115 the0r1e5 [ 9 ]. 7 h e deve10pment 0f 51m11ar meth0d51n 5uper5ymmetr1c the0r1e5 15 an exc1t1n9 p05516111ty. ~ 5ee ref. [7] f0r a rev1ew.
332
Reference5
[ 1] L.-L. Chau and 8. M11ew5k1,UCD prepr1nt 87-05 ( 1987). [2] L.-L. Chau and C.5. L1m,Phy5. Rev. Lett. 56 (1986) 294. [3] L.-L. Chau, Phy5. Lett. 8 202 (1988) 238. [4] L.-L. Chau and 8. M11ew5k1,Phy5. Lett. 8 198 (1987) 356. [5] E. A6da11a, M. F0r9er and M. Jac4ue5, CERN prepr1nt CERN-7H 4431/ 86. [6] E. W11ten,Nuc1. Phy5. 8 266 (1986) 245. [7] L.-L. Chau, 6e0metr1ca1 1nte9ra6111ty and e4uat10n5 0f m0t10n 1n phy51c5: a un1fy1n9v1ew, ta1k, 1n: Pr0c. Nanka1 W0rk5h0p (Nanka1 1n5t1tute 0f Mathemat1c5, Nanka1 Un1ver51ty,71anj1n,Ch1na, Au9u5t 1987), t0 6e pu6115hed. [8] V.E. 2akhar0v and A.V. M1kha110v, 50v. Phy5. JE7P 47 (1978) 1017. [9] M.F. At1yah and R.5. Ward, C0mmun. Math. Phy5. 55 (1977) 117; M.F. At1yah, V.6. Dr1nfe1d,N.J. H1tch1nand Yu.1. Man1n, Phy5. Lett. A 65 (1978) 185; E.F. C0rr19an, D.8. Fa1r11e, R.6. Yate5 and P. 60ddard, C0mmun. Math. Phy5. 58 (1978) 223.
PHY51C5 LE77ER5 8
12 January 1989
L1NEAR 5 Y 5 7 E M 5 AND C 0 N 5 E R V A 7 1 0 N LAW5 1N d = 10,
1 5UPER6RAV17Y
N=
L1n9-L1e C H A U Department 4f Phy51c5, Un1ver51ty4f Ca1~f0rn1a, Dav15, CA 95616, U5A and 8. M1LEW5K1 M1cr050ft C0rp0rat10n, P05t Qff1ce80x 97017• Redm0nd, WA 98073, U5A Rece1ved 11 March 1988; rev15edmanu5cr1pt rece1ved 12 0ct06er 1988
5tart1n9 w1th the 5et 0f 119ht-11ke1n1e9ra6111tyc0nd1t10n5 1n curved d= 10, N= 1 5uper5pace we der1ve the c0rre5p0nd1n9 11near 5y5tem. 7he v1e16e1n5and c0nnect10n5 that 5at15fythe c0n5tra1nt5 are parame1r12ed u51n9 a 11near5p1n0r 5u65pace def1ned a5 the kerne1 0f the 119ht-c0nepr0ject10n 0perat0r. 7hey fu1f111an 1nf1n1te5et 0f c0n5ervat10n 1aw5.
A5 d15cu55ed 1n ref. [ 1 ] 0ne can 9enera112e the n0t10n 0f 1nte9ra6111ty a10n9 119ht-11ke 11ne5 1n 5uper5pace t0 06ta1n d = 10, N = 1 5uPer9rav1ty c0n5tra1nt5. A1th0u9h they are t00 weak t0 put the the0ry 0n 5he11 1n the 5en5e 0f E1n5te1n e4uat10n5 0f m0t10n, we cann0t exc1ude the p05516111ty that they 1ead t0 50me h19her der1vat1ve c0nf0rma1 e4uat10n5 0f m0t10n (a5 1t 15 the ca5e 0f the c0rre5p0nd1n9 f0ur-d1men510na1 the0r1e5 [2] ). We had a150 1dent1f1ed the add1t10na1 c0n5tra1nt 7,~5~)- ~6J~70,5=0 ,
(1)
7he1r 1nterpretat10n 15 the f0110w1n9:a10n9 certa1n d1rect10n5 1n 5uper5pace the c0var1ant der1vat1ve5 06ey the f1at a19e6ra. 7heref0re, a10n9 the5e d1rect10n5 they can 6e 06ta1ned 6y a 5u1ta61e tran5f0rmat10n 0f the f1at (carte51an) v1e16e1n5 and c0nnect10n5, 1,e. E,.1 =~ua~tDa1,
00a8c=~u.,~1(D~1~u8 ~) 9t~-c" ,
(3)
where D M = (~,,,, De,) are f1at c0var1ant der1vat1ve5 fu1f1111n9 the 5tandard a19e6ra {De,, D . )"= F ,,.:~1~ %, .
(4)
wh1ch 1ead5 t0 the E1n5te1n type e4uat10n5 0fm0t10n (wh1ch are a150 Wey11nvar1ant due t0 the pre5ence 0 f the d11at0n f1e1d). 1n th15 p a p e r we 5ha11 d15cu55 the C0n5e4uenCe5 0f the 119ht-11ke 1nte9ra6111ty c0n5tra1nt5 1n the C0ntext 0f the the0ry 0f 11near 5y5tem5. 1n ref. [ 3 ] 11near 5y5tem5 were c0n5tructed 1n d = 4 5uPer9rav1ty the0r1e5. 1n re1". [4] the meth0d5 0f c0n5truct1n9 and e x p a n d 1n9 the 11near 5y5tem5 1n ten d1men510n5 were deve10ped f0r the ca5e 0f the ten-d1men510na15uper-Yan9M1115 the0ry (5ee a150 ref5. [ 5 , 6 ] ) . We 5ha11 app1y t h e m t0 d = 10, N = 1 5uper9rav1ty. 7 h e 119ht-11ke 1nte9ra6111ty c0n5tra1nt5 are
where the under11n1n9 0 f t h e 1nd1ce5 den0te5 the 119ht11ke pr0ject10n
v~F~:~vt~F~,~1~({V~, V/j} - F ~ V c ) = 0 .
~1.10e1~=~0(~0e 1~ ,
330
(2 )
E4. ( 2 ) 15 e4u1va1ent t0 the van15h1n9 0f certa1n ( p r 0 j e c t e d ) t0r510n5 [ 1 ] wh1ch 1n turn 1mp1y the f0110w1n9 c0nd1t10n5 •
w
v1~,.
v. . . . .
~u~ ~ , F ~ , .
~,~ = v " F . J 4 / ~1~.
~,,,-0
(5)
(6 )
7 0 fu1f111 e45. ( 5 ) the matr1ce5 9/~~ mu5t 6e pr0p0rt10na1 t0 the tran5f0rmat10n matr1ce5 0f the L0rent2 9r0up ~10m-~-~020a
m ,
(7)
V01ume216, num6er 3,4
PHY51C5 LE77ER5 8
1 0~ 1t =exp(~crj, F a h ) . ,1~
~5~=exp(0~)~
c~.:, = - c~1,,,,
12 January 1989
E~ =P(v, w)~f¢/5/~(w) D . ( 7 c0nt•d )
7he p0wer5 0f (p cance1 1n e4. (5) ref1ect1n9 the d11atat10na1 d1men510n5 0f the f1e1d5 0.~, 0.•••. 0 t h e r ten50r5 (e.9., curvature5) depend 0n der1vat1ve5 0f~0 6ut 0n1y thr0u9h the c0m61nat10n
+P(w, v)J~1/~(v) D. ,
(15a)
~.n~= P( v, w) 0f ( 75
(w)
+P(w, v),f(V/~D,0a"0,;~) (v) .
(156)
70 repr0duce the a19e6ra (2) we theref0re need t0 1mp05e the c0nd1t10n5
A5 1n ref. [4] we can 1ntr0duce 5p1n0r1a1 parametr12at10n 0 f t h e ten-d1men510na1 119ht c0ne. F0r 50me f1xed 119ht-11ke vect0r w a we c0n51der a 11near 5p1n0r 5pace def1ned a5 the kerne1 0fthe 0perat0r w~F., 1.e., a11 the 0. 5uch that
v<~F."1~E1j~=0, v"E<,~0=0.
w 4 2 ./~05j =0 .
~,5 ~5D:dp=E5(~.
(8)
(9a, 6)
0 n the 0ther hand, e4. (3) can 6e rewr1tten a5 Vj tu. w= 0 ,
(10)
(16)
7he 119ht c0ne then can 6e de5cr16ed a5 a c011ect10n 0f vect0r5
v<,= (0+5)-F<
0r m0re exp11c1t1y,
(17)
E45. (9) and ( 1 1 ) f0rm a 11near 5y5tem 1n the 5en5e that the1r 1nte9ra6111ty c0nd1t10n5 repr0duce the c0n5tra1nt5.1ndeed, fr0m
f0r 50me f1xed e" n0t 6e10n91n9 t0 the kerne1 0f 0J"F. (actua11y, at 1ea5t tw0 5uch patche5 are nece55ary t0 c0ver the wh01e c0ne, due t0 the n0ntr1v1a1 t0p0109y, c.f. re1". [4] ). W1th th15 parametr12at10n the 11near 5y5tem 6ec0me5
v0F, r.v"F/m~( {V., V/j}-F~,/[V,. ) ~=0 ,
( 0F~0) 6 9 ( 0 ) + 2 (0F"e) E.~0(0)
v.F~15V<~0:~,,=0,
v"V.0~=0.
(1 1a, 6)
(12)
we can extract the van15h1n9 0f the appr0pr1ate pr0jected t0r510n5 (the curvature5 d0 n0t enter th15 e4uat10n 51nce ~a 15 a 5ca1ar). C0n5e4uent1y, the 0ther e4uat10n
v . F :~-v~,F/'~2~({V., V : , } - F . 1 [ V < . ) 0 j ~ = 0 ,
(13)
5et5 the pr0ject10n5 0f curvature5 e4ua1 t0 2er0. 7he pre5ence 0f the add1t10na1 5ca1ar f1e1d ~ 1n the 11near 5y5tem 15 a new feature. 1t ref1ect5 the fact that the N = 1, d = 10 5uPer9rav1ty, and 0ur 5y5tem 0f c0n5tra1nt5 1n part1cu1ar, ha5 a 1ar9er 1nvar1ance 9r0up, name1y, the 5uper-Wey1 9r0up. N0t1ce that e45. (9a) and ( 1 1a) can 6e v1ewed a5 a 9enera112at10n 0f ch1ra11ty c0nd1t10n5.0n1y, 1n th15 ca5e, th1515 n0t a L0rent2 c0var1ant Y5pr0ject10n, 6ut a m0re re5tr1ct1ve 0 ( 8 ) 1nvar1ant (v-dependent) pr0ject10n. M0re prec15e1y, f0110w1n9 ref5. [4,5 ] 0ne can 1ntr0duce pr0ject10n 0perat0r5.
P(v, w)J= ~3L~ v~,6<,w/,F:,,~ 2(v•w)
(14)
wh1ch can 6e u5ed t0 extract the v1e16e1n5 and c0nnect10n5 fr0m e45. (9) and ( 11 )
+ (eF°e)E.(0(0)=0, (0F~0)V,0<,~(0) + 2 (0F%) V.0/••(0) + (eF"e)V~0d~(0)=0.
(18)
1t can 6e expanded 1n p0wer 5er1e5 0f the parameter 01ead1n9 t0 an 1nf1n1te 5et 0f c0n5ervat10n 1aw5. E45. (18) can 6e further 51mp11f1ed 1f we make the 5pec1a1 9au9e ch01ce
p(t,,w) 15(E55--D~)=0,
p(v,w)Ja)15;,a=0,
(19)
w1th w~= (eF~e) a f1xed 119ht-11ke vect0r. We can n0w pr0ceed w1th the expan510n
~0(0)= }2 0"~>~°<->, ~:>~(0)= )2 0"~>0#%>, (20) where 0 (• 1den0te5 the pr0duct 0,~,~,2 ... ~,,. 5u65t1tut1n9 (20) 1nt0 (18) and u51n9 (19) the f1r5t few term5 1n the expan510n read
w . F -#D15~0~°)=0 ,
(21)
w ~F."nDr~01,~ ~0~= 0 .
( 22a )
2Fra"e<7c~15E1~0)(°)+w"F.ca3D~0~,1)=0,
(226) 331
v01ume 216, num6er 3,4
PHY51c5 LE77ER5 8
12 January 1989
7h15 w0rk 15 part1a11y 5upp0rted 6y D 0 E fund1n9. etc . . . . . 7 h e 1mp0rtant advanta9e 0 f t h e 5p1n0r1a1 parametr12at10n, e4. ( 17 ), 15 that the expan510n5 can 6e exp11c1t1y carr1ed 0ut t0 any 0rder. 7 h e parameter5 0 f0rm a 11near 5pace and they enter va at m05t 4uadrat1ca11y. 70 5ummar12e, the pr061em 0f 501v1n9 a h19h1y n0n11near 5et 0f t0r510n and curvature c0n5tra1nt5 f0110w1n9 fr0m (2) wa5 reduced t0 a pr061em 0f 501v1n9 a 11near 5y5tem. 1t can 6e f0rmu1ated a5 f0110w5: f1nd a 5et 0f f1e1d5 0(x, 0), 0am(x, 0) 5uch that the v1e16e1n5 and c0nnect10n5 06ta1ned fr0m e45. (15a), (156) d0 n0t depend 0n the parameter5 0. Var10u5 meth0d5 were deve10ped f0r 501v1n9 th15 type 0f pr061em5,1, they were e5pec1a11y 5ucce55fu1 1n tw0d1men510na1 the0r1e5 [8] where they 1ed t0 c0mp1ete1y 501va61e m0de15, a5 we11 a51n f0ur d1men510n5 - 1ead1n9 t0 the 501ut10n5 0f5e1f-dua1 Yan9-M1115 the0r1e5 [ 9 ]. 7 h e deve10pment 0f 51m11ar meth0d51n 5uper5ymmetr1c the0r1e5 15 an exc1t1n9 p05516111ty. ~ 5ee ref. [7] f0r a rev1ew.
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Reference5
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