((JUllm Jl J\foth r:rno hool Jool'1W I 
Fo/ H, /\"' 2, ('1 · 51} . . ·tu91u t IJQ(J{i. 

Global Solutions of Yang-Mili Equat ion 

Qike n g Lu 1 
l11 stit.nlt' oí ~ l uthl'lllHtk:-;, A<·1Hll'll\Y uf t<.lill.il1 •1m1ti1'.S 1utcl )"'-lt•m Sril'llrt· 

hirn .. '!'<.· J\mdemy oí Sdcnccs, l3ciji11 g 1000 O. hinn 
11 1qik@public .btn .11c t. c11 

ABSTfu\ C T 
:loba! nnd explicil sol utions of Ynn g-t-.füls cquntions are gi,·en in the M in kowski 

spacc, coníormnl spncc nnd t hc dc-S ittcr spnccs of arbitrnry cosmology rons tnnt s. 
Thc m<'lhod uscd is ro ndudcd i11lo o p,cncrnl th co rcm. 

R ESUMEN 
Solu cione:. explici t as y globo les de lo e<:uacio ncs de Yang-Mills son dadns c r1 

et es pacio dc Minkowski, eu cs pnclos co nfo rm es y en los cspaci06 de De- iUc r co n 
co ns tnnl CB C'0'-1nológkns 11rbi 1 rnrlns. l~ l mótodo usado oo ndu)"'l' en un tcorc rnu 
gcncrnl . 

Kcy word~ um l phro 11 u111 \~M cquullon. hornOft'rieow hme-1pacc 
l\loth. S ubj. C ln1111.: 10515 

lt. i1:1 k11ow11ll tlmt onc of th c Oirnc'i;t2J co nfomml i;;pare .,.VI i!i' equh'tllc nt. to th c u 11 i-
tu ry group (2), which is cquivu[ cut. t.o thc om1>nc tcd spnCt' l\t OÍ all 2 X 2 Hcnui t.ion 
ml\I ri ces, nnd nn <"'.'<Jllicit glob nl !>olutlo u o f th Ynng-~lill~ ('Qunlion wns co nsLru cL in 

. 111 thb nrlick w'(' nt firsl co nstrucl o tlwr 'iOluti 11s írom difforcnt Lorcntz mc tri cs 
dt"fi11cd 011 .\lf 

1 Porlillll) UPf" rtf'Cf hy Nlllimml Nnlurnl &knC'(' h111ndA1K1n of China, Projt:<:t 
102!t\050/AOIOI09 



Qike11g: Lu 

\\ºe nrrnngc the coordi riatcs :1: = (:1;0, :i; 1, .1;2, x 3) of u poi11t in 1hc f\ li11kowski l'iJlllt'C 
f\ 1 into n 2 x 2 lforrnitinn nu1 tri x 

whcrc 

\Ve are to prove tlmt 

3 

HL = :1,Joj = L x1a1 , 
j=O 

is n su(2) gnuge potentinl(con11ecLio11 ), whcre the Creck indice; nm from 1 lo 3, ru1d 
Sl\li.'lfics thc Yang·. lills cqna t.ion 

F11;.1:!/1 (88~~1k+[A1 , Fj1,] - { ; , }Frk- { {1 }F1 r) = O, (2) 
where 

(3) 

nnd { ;, } is t he C hristoffel symbol of a 1netric ds2 = 9Jkd:i:Jdxl· wh ich i11 the pr~nt 
case we choosell J 

(4) 

\\'e shoulcl provc t hnt. AJ is ncl,unlly n au(2)·co nncction. As t.hc firs t stcp wc 
ron~truct n sl(2, C )·co1111 cct.ion(a 2-cotnponcnt. spino r conncction ) fron1 tbc tensor (4) 
ru1d rOOure it 10 11 t1u.(2)-comu.:cLio11. In fnct, M nnc\ d • .,2 nrc i1wnrilrnt m1rlcr thl' 
trru.i.-.for11mtio n 

1-1, = (A + 1-1,8)- ' (- 8+1-1, A). (5) 

whc rc A. B are 2 x 2 co111plcx urnlric~ 11ml satis fy the couditio n 

A 1 A + 131 IJ = 1, A 1 IJ - 8 1 A = O (G) 

with A' . B' d enotcd the co r11 ple x conj ugute n nd transpose: matrices of A,8 respcc· 
1iw ly 

Associnled lo the t.rn nsfo ri nnLion (5) thcrc is l.\ L(2. C ) matrix 

'll.,.(.") = dcL( A + H,8)1(A + 11, 8 )- 1 • (7) 

inl't' Ki .._,. crnmsltlvc undcr lhc group {i formcd from the 1r1\1ts formntions (5). 
lh•' rom':'ponding {IJlr(.r) }re.Q urc thc trnn~ition func tions o f thc ll{lt11ral principnl 
b11ndl1 ri l . L(2, )). \Ve opply t)I(' following thcorcm(c r IJJ Throrcm 2..t.2) 



Clobnl Sol11t.io11 s o í Ynng- Mills Ec1111u.io11 •19 

Thoor e m A. lf 9)1 l.'l o .{-rlm11•11sfo1"'l f~orr:u t.z ·"11¡,, mmufold. the n 

(fl~ = '72if,.17V i~ a !11(2. C )-t:o11ncc t1on 011 thc principal bmulle P(~Ul , SL(2 , C )), whcrc 

fl.111l 

d s 2 = .'Ji k1J:r.i tl:é = 1¡111,w" wb 
i.~ lh c !~ore.ni;. rnctrft· wit11 w" = e.Y' )rl!1;i tmd "fu) 1ml:isfyi,1g ~ .. )e~b) = ág. 

S i11cc u¿ = oo n11d o~ = - 0 0 (0· = l , 2, 3), Tl1c !1l(2, C )-conncction in Thcornm A 
cnri b u wril,tcn into 

(8) 

w! wru t.hu Grcck iud icCii run fr0t 11 1t.o31.111d {a 0 ,'i.a 0 } 0 ::::i1 , 2 ,:i is l\ bnsiil oí t he Lic 
ulgubrn lil(2, C ) oí S J..,(2, C ) nud {'ia0 }c,.=1,2,:J is t.lmt. o í t.he Lie nlgcbrn s u(2) oí SU(2). 
Siuce {Uauu- 1 } 0 .1,2.3 íor nny U E SU(2) is sWl n bns is oí lhc vcclor spnce geuern t.ed 
by {u0 } 0 = 1•2 ,3 , nccordin g l,o Lhe red uct.io 11 t.heore m o í con necLio ns, 

(Y) 

is/\ o·n(2 )-co1111cc io n on t hc rcd 11ced p rit icitnil bumlle P 1 (9Jt. SU(2)) or P (9Jt, S /_,(2 , C}) . 
In cnsc t !Lnt. 9Jl = Af nnd d.~2 is dofincd by (4 ), AJ is cxnctly cxprcssed by ( 1). !!~ 

ru111 uit1s l:o provc t hot. such A 1 sllt.isflcs tfo,: Y11 11g-ivlilh; CQ\UlLion (2). In fnct., lll:corcl-
i11t1. t.o ( ! ) t ite •lc mcnt.s o í t.hc nial.rices AJ(j = O, l , 2, 3) m e llll odd íunct.ions o f ~,) . 
Ohvio 11s ly [AJ(.t))r=o =O. Hc11 ec itll olmucuts oí F J.1: nrc cvcu fonclions oí :JJ. T hcrc-
foru 11 11 d 1::mcuts o í F, .1::/ nrc odd fu11ct io11s o f :r,J und con$1.-quc11Lly jF (r)Jk:il.r.=O = O. 
S i11ce Mis Lntns.ili,·efl l uuder 1.lie g ro up Q, for uuy poi nl :r0 of :1 t here is t\I. JcosL n 
Lrnnsformnt.ion (5) which cmries t lie poi11 t. :t = xo to t.he poi nl y= O. Sincc bot.h !}jl· 
011d F jA·:I nre C0\'8.til\nL ullder t.he Lhe Lro.ns fo rm otio n (5), 

O= [ 
f}.r,P {J:r,q fJ:r.rl 

(y),. ,,J •• o = ~lr(")F('"),,.,,..'llr{r)- 1 8yo 8y' 8y' z~z,' 

whi uh im pl k-s thnt jF (.:r)1111,r].r.m:r.u = O. Siucu :ro c1m be 1111 l\rbitrnry point o f M, wc 
lnwu F(:i.:)~1 =O nnd obvious ly iL s1.1l. ii! liCs t.hc Yung-~ l ills cquotion gt·l p jl·:I = O. 

Since M is Lhe compncted Minkowski spucc M nnd Lhc Yru1g·~lills eq uut.ion is 
confo r111 nl inV8.Tinnt, the i:.11 (2)-co1111ect.io 11 AJ dcfüicd by{! ) nlso satis fles 1.hc Yn 11g-
J\i!il ls cqu ntion 

"( O F F ) '1 f).r.1 ' 11;+ A 1 · Jk- F ,.1,. A 1 = O 

i11 l1hu Mi ukow"Ski s pucc M . 



Q ikcug: Lu 

Ano thc r :.0lu1io n o f 11bc Y1111g-t-.11i lls eq1 mt.io11 on M cnu be d<.'(lucc<I from 1mo1hl·r 
Lo rt•nt z mc tric. M is d iffoomorphic1'11 to S 1 X S 3 1md on l hc Jnt cr spncc thcrc i._.¡ 
nnturnlly o Lorcntz metric 

(11) 

which cnn be regnrd cd ns t.he metric of M. By T lu .. 'Orcm A nnd for11111 ln (8), 

is n su(2) ·ronncction. Since S 1 x S 3 1md d.~2 dcti ncd by ( 11 ) is invnrinni. mtdt'r 
S U(2)); (·1) which 11ct.s O l l S 1 X s:J ns t.hc t.rnnsfonnnt iou group of 5 1 X Sª nnd 1hc 
d cmcnt.s of t hc mnt.rices A ; nre odd f111 1ctio11s of xi, it. cn n be proved t lrnt A J sntiNfil~ 
the nng-~l ills cquot.iou by t.bc sume 11rgmnc11 t. 1L'> in cuse thut. ds2 is d e f111ccl by {11 ). 
This is n tat ic solut ion bec:lu1se A ; tu·e noL dep e nd 011 x 0 . 

O ur mc thod cn n nlso be appl ierl to cons truct. ::;olutions of t hc Yung- Mills L'flllB· 
tion on 1hc d c-Sit.t.c r s p11ccs d eli11orl by Oirnc15J. Thc d c-Sittc r spnc of cosmology 
con.:;tntll ;\ is demote<l by r/S{A ). Somc 1u1 t.ho rs cn ll it nmi-d e-Sit.tcr s p1lcc wlwn 
.\ < O ruHI de not e it by J\dS. In foct, tlS(A) is a dou min (con ncctcd open set ) of lhr 
rcnl p rojl'C't i\"C s pncr RP'1. ! u Uic locnl c:oordinntc ( uou-homogcncous coordinnt4') 
r = (r0.r1 .r~,r3) of RP'1 1J 1c do11 111 i11 dS(A) is dc ñ ned by lite incqu11li1y l6l 

(13) 

ru1d 1hc rc ¡_., n l...orl'nt:-.. 111 ctric 

Tht.> t-1IDC'e el ( A) b, iuvmiu nL 11wlor Lhu trnnsforumtion 

' = a{u)~~/Y. 
11 l - A111,,,fJl'a'' k ' 

(15) 

bnou.o¡Jy .. whcn a = O, {15) is 11 1,orcntz trnnsformntio n. The me1ric ds~ is in\lnrinnt 
u ndrr tll<' 1mn. .. for111uLio11 {15) 11ud dS( A) is trnm1lti\'C und r thc g·rou p of 1111 s11ch 
trul."fonnntmlL'l. In fncL t his is thc group 50(2, 3)(in ca.«e A < O) o r t.hc group 
~Ot l. -l)(m C".(L.'<' A > O) 1lm1, uc1s · o11 r/S(A) . ~loroovcr 1hc mc lril' el.¡~ undc.r llw 
coordillftt c 1nuu.fo r111nt io11 

(16) 



.~pl . lolml 'e>lu tio us o í Y1 11 1g-~ l il\~ E:q untion 51 

i ~ d1u11 gcd to bl• 

( 17) 

wlii cl 1 is n co u~ rmal flnt. me t ric. 
Th nt th e d e-Sitt r s pnc nre s pi n urn nifolds is imp lied in tht> Dirnc' oonstru c Ll on 

or th c '¡Ji,,4 wnve l"ql1Ation15J. App ly ing T heorcm A nncl the theorem of recl uct io11 
of coiincc ti o ns, wc obl ai n t. hc !lll(2)-co n ncc ti on 

A ,= -~1\ ( I - t\~111u1 11 1'11.'1 ) - 1 (&~ 11 ti - 6:u°')6!}'1o1 , ( 18) 
wh k h im ti s fi th '{nng-1\ lills cqu nt io n bcc nu se t. hc el mcnts oí , 8ít' odd fun ct.i oua 

of " ' · 
W(• co 11 cl11 cil' in gc nr rnl 1 l111t. 
T li coro m B. /f 9)1 1.~ fl 4·tfimtmsimwl .~pi 1 l maiujoltl anti pDA"l('.<JS{'fi r1 /.,rmml.z 

m c t1 fr wlw ·I• '" 11111tlr'1rm t 1m tle1· " l ie !JfOll/J \B llrnt r1c ls 011 !)JI tr011-"l11·dy, rmrl l h 1' 1Y' 
1.~ rw tui mi.mble local coorvliiw le .1J(j = O, 1, 2 , 3) s utlt tlial the su(u) -oo n11cc t·io 11 AJ 
dt·1fo ce1l f mm 1'hcortm A nre utld fuu ct.ions o/ x' . thc11 A, ,,otr.5ji~.s the Ya 11!J· Ali/l:j 
e1¡11 utio 11 . 

Received: Ma r c h 2005 . Revised: Apr il 2005. 

Rc f r e n ces 

[I J Q IKENG Lu, JI global .o;olulion uf th c Ei r1,,tcw- l'ang-M1lls equal.iu n on tlw 
c0t1/ on11al spoce, Scicuc:o i11 Cld 110 (Scrics A). ~1 5 (2002). ~2-355. 

[2] P . A. M . Om,, c, Wr1:11c N¡un.lio1l iii thc ronformal .poct', J\nn. l\fat.h., 
37(1936), 17 1-20 1. 

[3 ) Q 1KE:-.iC Lu , D1ffe,.e111iol Gco metry cm d 1ts Ap11f1rol1011 ta Phy:jfr.:s{i 11 Chi-
uc e). ience P ress, 13cij ii1 g:, l!J83. 

1 ~1 1 Q1 KESC Ltl, lll KllN W ANC 1 1 { 1~ W L'. Global .i•olut1011$ o/ El7lst.r:in-Din1c 
Of11nt1ot1. A .. -.iau .J. f\ l11t h., 8 (2004 ), 1-26. 

[5] P. 1 1 DIRA , '!'h e f' fcrl ro11 wt1ue eq11atro11 m tht de· rttcr spocc, A nn . 
Mnth .. 36 {1935), 657-609 . 

IGI H. K LooK(=Q 11<0NC: L u), L. Tsou(=Z ll ESLONc Z11 ou), H. Y . 
K 1\ 0(=HANYINC C 1\ ), 'J'lw kwcmaltc rffcct m th c d~ reo/ dom11 ifls wul 
lht red· 'h1/I phrno111r1 w of cJtlll·!Jlllahc ob;cc( (111 Chmc.sr). Act.n P hys icn 
'inica.. ... 23 (197-1), 225-23 . 


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