Phy51c5 Letter5 8 302 ( 1993 ) 403-410
N0rth-H011and PHY51C5 LE77ER5 8
N0n-pertur6at1ve effect5 1n matr1x m0de15 and vacua 0f tw0 d1men510na1 9rav1ty
FranC015 Dav1d 1
5erv1ce de Phy514ue 7h~0r14ue 2, c E 5ac1ay, F-91191 61f/Yvette cedex, France Rece1ved 23 Decem6er 1992
7he m05t 9enera1 1ar9e N e19enva1ue5 d15tr16ut10n f0r the 0ne-matr1x m0de115 5h0wn t0 c0n515t 0f tree-11ke 5tructure5 1n the c0mp1ex p1ane. F0r the m = 2 cr1t1ca1 p01nt, 5uch a 501ut10n de5cr16e5 the 5tr0n9 c0up11n9 pha5e 0f 2D 4uantum 9rav1ty (c= 0 n0n- cr1t1ca1 5tr1n9). 1t 15 06ta1ned 6y tak1n9 c0m61nat10n5 0f c0mp1ex c0nt0ur5 1n the matr1x 1nte9ra1, and the re1at1ve we19ht 0f the c0nt0ur5 15 1dent1f1ed w1th the n0n-pertur6at1ve "0-parameter•• that f1xe5 un14ue1y the 501ut10n 0f the 5tr1n9 e4uat10n (Pa1n1ev6 1). 7h15 a110w5 t0 rec0ver 6y 1n5tant0n meth0d5 re5u1t5 0n the n0n-pertur6at1ve effect5 06ta1ned 6y the 150m0n0dr0m1c Def0r- mat10n Meth0d, and t0 c0n5truct f0r each 0-vacuum the 065erva61e5 (the 100p c0rre1at10n funct10n5) wh1ch 5at15fy the 100p e4uat10n5. 7he 6reakd0wn 0f ana1yt1c1ty 0f the 1ar9e N 501ut10n 15 re1ated t0 the ex15tence 0f p01e5 f0r the 100p 0perat0r5.
7 h e d15c0very 0 f the ••d0u61e 5ca11n9 501ut10n5•• 0f the matr1x m0de15 [ 1-3 ] 1ed t0 1mp0rtant Pr09re55 1n the under5tand1n9 0 f 5tr1n9 the0r1e5 1n d~< 2 6ack- 9r0und5 and 0 f 2D 9rav1ty (5ee ref5. [4,5] f0r re- v1ew5). H0wever, the 1mp0rtant 155ue 0 f the n0n-per- tur6at1ve 5tatu5 0f 50me 0f the5e the0r1e5 rema1n5 unc1ear, 1n part1cu1ar f0r 2D 9rav1ty c0up1ed t0 un1- tary matter f0r c~< 1.1n th151etter, we d15cu55 50me 0 f the5e 4ue5t10n5 1n the framew0rk 0 f the herm1t1an 0ne-matr1x m0de15. We 5haU 5h0w that a 51mp1e 9en- era112at10n 0f the c0mp1ex 1nte9rat10n c0nt0ur pre- 5cr1pt10n [ 6,7 ], wh1ch a110w5 t0 c0n5truct n0n-per- tur6at1ve - 6ut 1n 9enera1 c0mp1ex - 501ut10n5 0 f the 5tr1n9 e4uat10n5 and 0 f the c0nt1nu0u5 100p e4ua- t10n5,1ead5 t0 rea1 n0n-pertur6at1ve 501ut10n5 0f the5e e4uat10n5.7h15 9enera112at10n, wh1ch c0n515t51n tak- 1n9 c0m61nat10n5 0f 1ne4u1va1ent 1nte9rat10n c0n- t0ur5, ha5 6een a1ready d15cu55ed 6y F0ka5, 1t5 and K1taev [ 8 ] 1n the framew0rk 0 f the 150m0n0dr0m1c def0rmat10n m e t h 0 d (1DM) appr0ach t0 the 5tr1n9 e4uat10n5 [ 9 ], 6ut d0e5 n0t 5eem t0 have attracted m u c h attent10n. 0 u r treatment 15 6a5ed 0n the 8 1 P 2 501ut10n 0f the 0ne-matr1x m0de1 [ 10 ], and f0110w5
Phy514ue 7h6.0r14ue cNR5.
La60rat01re de 1a D1rect10n de5 5c1ence5 de 1a Mat1~re du c0mm155ar1at/t 1•Ener91e At0m14ue.
0Ur prev10U5 ana1y515 0fref. [ 11 ]. We 5ha115h0w that 1n the 11m1t N--,00, new 501Ut10n5 f0r the e19enVa1Ue5 (EV5) d15tr16Ut10n ex15t, wh1Ch have n0t 6een d15- cu55ed 6ef0re. 7 h e y c0rre5p0nd t0 a d15tr16ut10n 0 f EV5 a10n9 tree-11ke 5tructure5 1n the c0mp1ex p1ane.
M0re0ver, the5e 501ut10n5 depend n0n-ana1yt1ca11y 0n the c0up11n9 c0n5tant 0 f the matr1x m0de1, and w111 6e a550c1ated w1th the 5ect0r5 w1th an 1nf1n1te num- 6er 0 f p01e5 0 f the 5tr1n9 e4uat10n 501ut10n5.7he n0n- pertur6at1ve parameter wh1ch character12e5 the n0n- pertur6at1ve 501ut10n515 51mp1y re1ated t0 the d1ffer- ent we19ht5 ch05en f0r the c0nt0ur5, and 0ur treat- ment a110w5 t0 rec0ver ea511y 6y 1n5tant0n meth0d5 50me re5u1t5 0 f ref5. [ 7,8 ]. 1n add1t10n, we 5h0w that t0 each rea1 501ut10n 0f the 5tr1n9 e4uat10n 15 a550c1- ated a pre5cr1pt10n f0r the a5ympt0t1c5 0 f the 100p 0perat0r5 wh1ch def1ne5 un14ue1y 065erva61e5 (1.e.
macr05c0p1c 100p VEV) wh1ch 06ey the 100p e4ua- t10n5. F1na11y we 5ha11 5h0w that the5e new 501ut10n5 a110w t0 exp1a1n the pr0pert1e5 0 f t h e 501ut10n5 f0r the d0a61e we11 matr1x m0de15 recent1y d15cu55ed 6y 8r0wer, De0, Ja1n and 7 a n [ 12 ].
1n the matr1x m0de1 f0rmu1at10n 0f 2D 9rav1ty, the part1t10n funct10n F (5um 0ver 0r1enta61e c0nnected 2D r1emann1an 5pace5) 15 d15cret12ed 1nt0 a 5um 0ver tr1an9u1at10n5, and 15 wr1tten a5 the 109ar1thm 0 f the part1t10n funct10n 2 f0r the herm1t1an 0ne-matr1x
m0de1 ( F = 1n 2 ) , wh1ch after d1a90na112at10n 0f the matr1x ~ can 6e wr1tten a5 an EV 1nte9ra1,
2 ~ = C N ~ dR~ exp [ - N V ( , L ) 1A~(2~) 2 ,
AN(21) = 1-1 ( 2 , - 2 j ) , ( 1 )
1<j
where CN 15 a n0rma112at10n fact0r, AN the Vander- m0nde determ1nant and Vthe matr1x p0tent1a1. 7he 1nte9ra1 ( 1 ) can 6e ca1cu1ated 1n term5 0f the matr1x e1ement5 0f the 0perat0r Q: ~t. (2) ~;t1t, (2), where the 1t. are 0rth0n0rma1 p01yn0m1a15 f0r the mea5ure dR exp [ - N V ( 2 ) ]. 1n the d0u61e 5ca11n911m1t, N-~ 00 and V+ Vcr1t1ca1 wh11e x = 1 - n / N 6ec0me5 a c0nt1nu- 0u5 parameter. 7hen Q 6ec0me5 a 5ec0nd 0rder d1f- ferent1a1 0perat0r 0fthe f0rm
d E
Q = - dx 2 + 2 u ( x ) , (2)
u 15 the 5tr1n9 5u5cept16111ty 02F
u ( t ) = - 0t 2 , (3)
where t 15 the ren0rma112ed c05m01091ca1 c0n5tant.
F0r the m = 2 cr1t1ca1 p01nt (pure c = 0 9rav1ty), t 5ca1e5 w1th N a5 t ~ N 4/5, and u 5at15f1e5 the Pa1n1ev6 1 5tr1n9 e4uat10n
1 02u u ~ x / ~ , t - + + m .
6 0t 2
+U2=t•
(4)1t 15 kn0wn that (4) f1xe5 un14ue1y the term5 0 f t h e a5ympt0t1c expan510n 0f u (1n p0wer5 0f t (1 --5k)/2) a5 t ~ +c0, 6ut that the c0rre5p0nd1n9 5er1e5 15 n0t 80re1 5umma61e, and that the 501ut10n5 0f (4) f0rm a 0ne-parameter fam11y 0f ••51mp1y truncated 501u- t10n5•• [ 13 ], wh1ch d1ffer 6y exp0nent1a11y 5ma11 term5 0f the f0rm
Ju0ct -1/5 exp( - ~.2x/~ t 514) . (5)
7 h e rea1 501ut10n5 0f (4) have an 1nf1n1te 5er1e5 0f d0u61e p01e5 (w1th re51due5 1 ) 0n the ne9at1ve rea1 ax15. 1f 0ne d1v1de5 the c0mp1ex p1ane 1nt0 5 5ect0r5 5=1, ..., V (wh1ch c0rre5p0nd t0 ( 5 - 1 ) . 2 7 t <
Ar9 (t) < 8.27t), the5e p01e5 extend t0 an 1nf1n1te net- w0rk 0f p01e5 1n the 5ect0r5 11, 111 and 1V, 50 that the a5ympt0t1c5 u ~ v / t h01d 0n1y 1n the tw0 p01e-free 5ec- t0r5 1 and V.
1t wa5 5u99e5ted 1n ref. [ 6 ], and 5h0wn m0re pre- c15e1y 1n ref5. [1 1,7,8], that, 1f 0ne c0n5truct5 the m = 2 the0ry 6y 5tart1n9 fr0m a cu61c p0tent1a1 0f the f0rm
V(2) = - 2 3 + . . . , (6)
and def1ne5 the 1nte9ra1 ( 1 ) 6y ch0051n9 a5 1nte9ra- t10n c0nt0ur f0r the 21~5 the c0mp1ex c0nt0ur ff+ (re- 5pect1ve1y c9•) wh1ch 90e5 fr0m - 0 0 t0 j00 ( j = e 1~/3) (re5pect1ve1y f ~ ) , 0ne 06ta1n5 the ••51mp1y truncated 501ut10n•• u+ (re5pect1ve1y u•) 0f (4) wh1ch ha5 p01e5 0n1y 1n the 5ect0r 11 (re5pect1ve1y 1V) and 5at15f1e5 the a5ympt0t1c5 u ~ x/~ 1n the rema1n1n9 f0ur p01e-free 5ect0r5. Fr0m the5e tw0 501ut10n5, wh1ch are ana1yt1c 0n the rea1 ax15, 0ne can c0n5truct w1th0ut am619u1ty the 0perat0r Q (2), wh1ch 15 n0t herm1t1an anym0re, and the 100p 0perat0r5 w ( p )
[ 14 ] ( w h e r e p 15 the 100p m 0 m e n t u m ), wh1ch 5at15fy the 100p e4uat10n5 [ 6,15,16 ].
1n fact a 5tra19htf0rward 9enera112at10n 0f th15 pre- 5cr1pt10n 15 t0 c0n51der 11near c0m61nat10n5 0f the tw0 c0nt0ur5, 1.e. t0 rep1ace
1 c+1 +c ,7,
e9•+ ~+ ~¢~
(5ee f19. 1 ). 1ndeed, the part1t10n funct10n 2 w1115t111 6e rea1 (f0r rea1 V) 1f the we19ht c+ are c0mp1ex c0nju9ate
c•+ =••+10. (8)
, C1
F19. 1 . 7 h e three c0nt0ur5 f0r the cu61c p0tent1a1.
V01ume 302, num6er 4 PHY51C5 LE77ER5 8 1 Apr11 1993
W1th th15 pre5cr1pt10n, the 0rth090na1 p01yn0m1a1 meth0d 5t111 w0rk5, and the recurrence re1at10n5 (d15- crete 5tr1n9 e4uat10n5) 5t111 h01d. 7heref0re, 1f the d0u61e 5ca11n9 11m1t ex15t5, 0ne 5h0u1d 5t111 06ta1n 50me 501ut10n 0f the 5tr1n9 e4uat10n (4).
A5 a1ready ment10ned, 1t ha5 6een 5h0wn 1n re1". [ 8 ], w1th1n the 1DM appr0ach, that th15151ndeed the ca5e, and that there 15 a 0ne t0 0ne c0rre5p0ndence 6e- tween the we19ht rat10
c+/c•
and the 51mp1y trun- cated 501ut10n 0f (4) wh1ch 15 06ta1ned 1n the d0u61e 5ca11n9 11m1t. 7 0 rec0ver th15 re5u1t 1n the 81P2 ap- pr0ach, 1et u5 c0n51der the matr1x m0de11nte9ra1 ( 1 ) 1n the 1ar9e N 11m1t. 7 h e e19enva1ue pr06a6111ty den- 51ty ( 1 / N ) Y.~=~ ~(;t-;t~) 6ec0me5 the c1a551ca1 den- 51ty v(2). 1t 15 c0nven1ent t0 c0n51der the funct10n F ( 2 ) = = 1 1 m - - 7 - - . ( 9 )d/t 2 - / t ~r~00 N 2 - 4 5
1t can 6e 5h0wn, fr0m the 5add1e p01nt e4uat10n5 f0r the effect1ve act10n f0r v, 0r thr0u9h the 100p e4ua- t10n5 [6], that F m u 5 t 6e 0 f t h e f0rm
F ( 2 ) = ~ [ V~ (2) + x/-Q(-,~-) 1,
Q(2) = V• ( 2 ) 2 + 4 N ( 2 ) , (10) where N(2) 15 50me p01yn0m1a1 0fde9ree
de9ree N = d e 9 r e e V - 2 = m - 1 . ( 11 ) 6ener1ca11y, Q ha5 2m c0mp1ex 2er05, wh1ch c0rre- 5p0nd t0 54uare r00t cut5 f0r F. Fr0m (9) the EV den51ty v 15 pr0p0rt10na1 t0 the d15c0nt1nu1ty 0f F a10n9 the cut5, and can 6e rec0n5tructed fr0m F. 7he n0rma112at10n f d2 v= 1 1mp11e5 that F ~ 2 - ~ at 00 and th15 f1xe5 the c0eff1c1ent 0 f t h e 1ead1n9 term 0fN. Re- 4u1t1n9 that Fha5 0n1y 0ne cut (a5 d0ne f0r 1n5tance 1n ref. [ 10] ) 1mp11e5 that Q ha5 m - 1 d0u61e 2er05 and th15 f1xe5 un14ue1y N. H0wever, 1n 9enera1 F may have 5evera1 cut5.
Let u51a6e1 6y 0t the cut5 and 6y x,~ the fract10n 0f EV a10n9 each cut (x~ >1 0 and ~x~ = 1 ). 7 h e EV den- 51ty v mu5t m1n1m12e the act10n
5 = 1 d2 v ( 2 ) V ( 2 ) - 1 1 d2 d/2 v(2)v(/~)1n12--/t1
where F,~ are La9ran9e mu1t1p11er5.1n fact (10) 15 the
m05t 9enera1501ut10n when 0ne extrem12e5 5 w1th re- 5pect t0 var1at10n5 0f the den51ty wh1ch d0 n0t chan9e the x~5. A5 a c0n5e4uence, the effect1ve p0tent1a1 F ( 2 ) f0r 0ne EV 1n the 6ack9r0und created 6y the N - 1 0ther EV5,
F ( 2 ) = V(2) - 2 ~ d/t v(/t) 1 n ( 2 - #) (13) 15 c0n5tant a10n9 each cut ct, and e4ua1 t0 F ,
F ( 2 ) =F,~ 1 f 2 e a (14)
50 that the t0ta1 act10n 15
5=~(~d2v(2)V(2)+x, Fa).
(15)7 h e rema1n1n9 c0n5tra1nt5 wh1ch f1x N are the f0110w1n9:
(a) the EV fract10n5 x,~ mu5t 6e rea1,
(6) 0 n e mu5t m1n1m12e R e ( 5 ) w1th re5pect t0 var1at10n5 0f the
x~,~5,
5u6jected t0 the c0n5tra1nt5 that x~>~0 and that 2x,~= 1.51nce fr0m (12)85/0xa=F,~,
th15 1mp11e5 that the rea1 part 0f the effect1ve p0ten- t1a115 the 5ame a10n9 a11 the cut5:
Re(F~) =F0, un1e55 xa = 0 . (16)
1n fact the c0n5tra1nt5 (a) and (6) can 6e reca5t 1n the 5ame f0rm:
1 m ( • ) = 0 , ~ = 2-~nF(2), (17)
~a
where % 15 any c0nt0ur enc1rc11n9 a pa1r 0f 2er05 0f Q. 1ndeed, 1f ~9 enc1rc1e5 a cut a, J = x , ~ , wh11e 1f ~f enc1rc1e5 the end p01nt5 0ftw0 d1fferent cut5 0t and f1, 1t f0110w5 fr0m the fact that away fr0m the cut5, F• ( 2 ) = V• ( 2 ) - 2 F ( 2 ) (06ta1ned 6y tak1n9 the de- r1vat1ve 0f (13) ), that J = ( 1/1n) ( F p - F , ) ( 5ee f19.
2). 51nce there are 2 m - 2 1ndependent c0nt0ur5, (17) f1xe5 the m - 1 rema1n1n9 c0eff1c1ent5 0fN. Let u5 n0te that 1f Q ha5 a d0u61e 2er0, tw0 1ndependent c0n- 5tra1nt5 (17) are aut0mat1ca11y 5at15f1ed, 51nce f0r any c0nt0ur ~ wh1ch enc1rc1e5 the d0u61e 2er0 and at m05t 0ne 51n91e 2er0, J = 0.7heref0re we expect 1n 9enera1 t0 have 0ne 501ut10n 0f (17) w1th n0 d0u61e 2er0e5, 2 m - 1 501ut10n5 w1th 0ne d0u61e 2er0, etc. 50me 0f the5e d1fferent 501ut10n5 w111 6e exc1uded 6ecau5e:
(c) 50me x,~5 are <0.
f
(a)
/
(6)
F19. 2. 7he c0nt0ur5 1n (17 ) c0rre5p0nd1n9 t0 c0n5tra1nt5 (a) and (6).
( d ) 7 h e C0nt0Ur5 0 f 1nte9rat10n f0r the EV Cann0t pa55 thr0U9h 0ne 0 f the Cut5 a. F1na11y am0n9 the re- ma1n1n9 501ut10n5, 1t 15 the 0ne w1th m1n1ma1 R e ( 5 )
(rea1 part 0f the act10n ( 12 ) ) wh1ch 15 the phy51ca1 5add1e p01nt.
Let u5 d15cu55 exp11C1t1y the ca5e 0 f the m = 2 cr1t1- ca1 p01nt. We 5tart fr0m the p0tent1a1 V(2) = - ~23+92. 1n the cr1t1ca1 re91me we re5ca1e 9 c - 9 ~ - a 2 t , 2 - 2 ¢ ~ - a p [1 1 ]. t 15 the ren0rma112ed c05m01091ca1 c0n5tant and p the 100p m0mentum. 7 h e 5ca11n9 parameter (5h0rt-d15tance cut-0ff) a 15 de- f1ned 50 that the d0u61e 5ca11n9 11m1t 15 06ta1ned 6y tak1n9 N - , 0 0 , •-2 ~5tr1n5 =2~ , = 1.1n the p1anar 5ca11n9 ~72~5 11m1t ( N = 0 0 , then a - , 0 ) , the 9enera1 501ut10n (10) f0r F ( 2 ) 6ec0me5
F(~.)--* w ( p ) = a 5/2• 2 x / p 3 - 3 t p + c , ( 1 8 )
where ¢ 15 the 0n1y parameter t0 6e determ1ned 1n the p01yn0m1a1 N ( 2 ) wh1ch 15 re1evant 1n the 5ca11n911m1t.
1t c0rre5p0nd5 t0 the VEV 0 f the puncture 0perat0r [ 6 ]. 1n the weak c0up11n9 pha5e t > 0, the 5add1e p01nt 15 the 5tandard 0ne cut 501ut10n [ 10,6 ]
c = 2t 3/2 ,
w ( p ) = a 5/2" 23 ( t 1 / 2 - - p ) x / p + 2t1/2 . (19)
7 h e EV5 are 10Cated 0n the rea1 ax15 a10n9 the CUt ] --00, - - 2 P / 2 ] (5ee f19. 3a). 7 h e act10n f0r th15 50- 1Ut10n 5Ca1e5 a5
50c a5/2t 5 /2 . (20)
(a) (6)
(c) (d)
F19. 3.5chemat1c p1cture 0fthe EV d15tr16ut10n (61ack 11ne) and the Re(/•) > 0 d0ma1n5 (9rey) 1n the p c0mp1ex p1ane f0r the 9e- net1c 501ut10n 0fthe m=2 cr1t1ca1 p01nt: (a) rea1 t>0 (and 5ec- t0r5 1 and V); (6) rea1 t<0 (and 5ect0r 111); (c) t 1n 5ect0r 11;
(d) t 1n 5ect0r 1V.
7here are 0ther unphy51ca1 501ut10n5 wh1ch v101ate (c) 0r (d).
7h15 501ut10n can 6e ana1yt1ca11y c0nt1nued 1nt0 the 5tr0n9 c0up11n9 un5ta61e pha5e t < 0 . 7 h e tw0 c0m- p1ex c0nju9ate 501ut10n5 de5cr16e EV5 5t111 10cated a10n9 a 51n91e arc [ 11 ], and c0rre5p0nd t0 the tr1p1y truncated 501ut10n5 0f (4). F r 0 m (20) they have a pure1y 1ma91nary act10n. H0wever, f0r t < 0 the c0n- 5tra1nt5 (17) have an0ther 501ut10n, where w (p) ha5 n0w three 6ranch p01nt5. F0r th15 501ut10n c 15 91ven 6y
c = c ( - t ) 3/2, c > 0 (21)
and 1t c0rre5p0nd5 t0 the 6ranched d15tr16ut10n 0f the EV5 dep1cted 1n f19. 36. 7 h e den51ty 0f EV5 a10n9 the ne9at1ve rea1 ax15 van15he5 at the rea1 6ranch p01nt P0 a5 v/~0 - p , 6ut there are tw0 arc5 5tart1n9 fr0m P0 t0- ward the tw0 0ther c0mp1ex c0nju9ate 6ranch p01nt5 p x . 7heref0re e4ua1 fract10n5 x+ =x• 0f EV5 are 51t- t1n9 a10n9 the5e tw0 arc5. M0re0ver the act10n f0r th15 501ut10n 15 rea1 and ne9at1ve, and theref0re 1t 15 9e- ner1ca11y the d0m1nant 0ne. Away fr0m the ne9at1ve rea1 ax15, the 6ranched 501ut10n 5t111 ex15t5, 6ut w1th
V01ume 302, num6er 4 PHY51C5 LE77ER5 8 1 Apr11 1993 a5ymmetr1c 6ranche5 (x+ ~ x ~ ), pr0v1ded that 0ne
5tay5 1n the 5ect0r 111 ( - - ~ r t < A r 9 ( - t ) < 1 n ) , and 5t111 ha5 a 5ma11er act10n than the pertur6at1ve 0ne.
M0re0ver, 51nce the c0n5tra1nt5 (17) are n0n-ana- 1yt1c, th15 501ut10n d0e5 n0t depend ana1yt1ca11y 0n t (1n 0ther w0rd5, the num6er c 1n (21) depend5 0n Ar9(t) ).
F1na11y, 1n the 5ect0r5 11 and 1V, an0ther k1nd 0f 501ut10n5, w1th tw0 cut5, 15 d0m1nant (5ee f195. 3c, 3d). 7he5e 501ut10n5 are 5t111 n0n-ana1yt1c 1n t. 1n the 5ect0r5 1 and V, the pertur6at1ve ana1yt1c 501ut10n (19) 15 the phy51ca1 0ne.
A5 15 c1ear fr0m f19. 3, f0r the 9enera1 1nte9rat10n pre5cr1pt10n (7), 0ne can 06ta1n the5e new, n0n-an- a1yt1c, 501ut10n5 1n the 5ect0r5 11, 111 and 1V. 51nce they have a 5ma11er act10n than the ana1yt1c 501ut10n, they w111 d0m1nate the 1ar9e N 11m1t, un1e55 c+ 0r c•
15 2er0.
7h15 new 501ut10n f0r the EV d15tr16ut10n 15 a550- c1ated w1th the 51mp1y truncated 501ut10n5 0f (4). 7h15 can 6e 5een a5 f0110w5. F1r5t, we have 5een that the5e 501ut10n5 are n0n-ana1yt1c 1n t 1n the 5ect0r5 where 51mp1y truncated 501ut10n5 have p01e5. 51nce d0u61e p01e5 0f u(t) 5h0u1d c0rre5p0nd t0 51mp1e 2er05 0f the 2 funct10n, wh1ch c0rre5p0nd5 t0 the part1t10n funct10n 2 f0r the matr1x m0de1, and 51nce 1n the 1ar9e N11m1t 2 15 06ta1ned fr0m the act10n 5 f0r the 5add1e p01nt EV c0nf19urat10n 6y 2=exp(N25), u51n9 the Cauchy f0rmu1a f0r the der1vat1ve 0f 1n(2) and 5t0ke•5 f0rmu1a we 06ta1n the e5t1mate f0r the num- 6er 0f2er05 1n a d0ma1n D:
# 0f2er051n d0ma1n D = ~ d9 0 1 n ( 2 ) 21~2 09
0D
47t 0 9 ~ t 9 9 ) . (22)
D
7hu5 the new 501ut10n de5cr16e5 a part1t10n funct10n w1th a p051t1ve den51ty 0f 2er05 p0c 005 1n the three 5ect0r5 11, 111, 1V, wh1ch 5ca1e5 a5 p0cN2a5/21t[1/2
× f ( 0 ) , where 0 = Ar9 (t).
0 n e can make the 1dent1f1cat10n m0re prec15e 6y re1at1n9 the c0n5tra1nt5 (17) t0 the a5ympt0t1c5 0fthe 51mp1y truncated 501ut10n5 0f (4). F0110w1n9 ref. [ 13 ] (5ee ref5. [17,18] f0r m0re recent d15cu5510n5) we make the chan9e 0f var1a61e
u=t1/2U, 7=~t 5/4 (23)
1n (4), wh1ch 6ec0me5 U• 4 U
U " - 6 U 2 + 6 = - - ~ +2-5 7 - 5 " (24) A5 1 71--,~, U 15 a5ympt0t1c t0 a We1er5tra55 e111pt1c
funct10n U0 ( 7, E0), 501ut10n 0f
(U~)2= 4U 3 - 12U0 + E 0 , (25)
wh1ch 15 d0u61y per10d1c (w1th a 1att1ce 0f d0u61e p01e5) w1th per10d5
~1,2 = ~ d U 0 ( 4 U ~ - 12U0+E0) -1/2 ,
~1,2
(26)
where ¢¢1,2 are tw0 c0nt0ur5 enc1rc11n9 pa1r5 0f 2er05 0f the RH5 0f (25). 1n a ne19h60rh00d 0f 50me 7 = 70, 0ne can treat E0 a5 a 510w1y varY1n9 var1a61e E 0 ( 7 ) . Fr0m (24), 1n the 10ca1 per10d5 c00rd1nate5
7 - 70 =121y 1 +922y 2, E0 var1e5 a5
020 2
0 y 1 , 2 - ~ : , , 2 ~ , ~ 0 j ,
1
~ , 2 ( E 0 ) = ~ dU0(4U30-12U0+E0) 1/2. (27)
:~1,2
501v1n9 the f10w e4uat10n5 (27) 0ne can check that a5ympt0t1ca11y, E0 depend5 0n1y 0n the ar9ument 0f 7, 0 = A r 9 ( 7 ) , 6ut n0t 0n 1t5 m0du1u5, and 15 501u- t10n 0f the tw0 c0n5tra1nt5
Re[e1°~.2(E0) 1 = 0 . (28)
8ut the5e c0n5tra1nt5 are exact1y e4u1va1ent t0 (17) f0r ( 18 ), 0ne we 1dent1fy E0= t-3/2c and u5e the fact that 0 = ~Ar9(t).
8ef0re d15cu551n9 the a5ympt0t1c5 f0r the 100p 0p- erat0r5, 1et u5 5h0w h0w the ch01ce 0f we19ht c0n- t0ur5 c+ 1n (7) f1xe5 un14ue1y the n0n-pertur6at1ve part 0 f t h e 501ut10n 0f (4). Let u5 den0te 6y 2~¢(0) (re5pect1ve1y FN(0) the part1t10n funct10n (1) (re- 5pect1ve1y 1t5 L09ar1thm) f0r N EV5 w1th the c0nt0ur c0eff1c1ent5 (8). 7ak1n9 the der1vat1ve 0f F w1th re- 5pect t0 0 51n91e5 0ut 0ne 0f the EV5
d E N . N
×f d2°fN(-1~d213~v(21'2exP(-N~V(21))
(29) where ~ 15 the c0nt0ur 901n9 fr0m f00 t0 j00. 0 n e e5t1mate5 th151nte9ra16y f1r5t 1nte9rat1n9 0ut the N - 1 1a5t EV5 (6y u51n9 the 81P2 meth0d), 1n the effect1ve p0tent131
9(2) = V(2) + ~ [ V(2) - 2 1n(20 - 2 ) ] 1
m0d1f1ed 6y the f1r5t EV. 7he re5u1t1n9 effect1ve p0- tent1a1 f0r the f1r5t EV 15 1n 9enera1 c0mp11cated, 51nce 1t take5 1nt0 acc0unt the 6ackreact10n 0f th15 EV 0n the 6u1k N - 1 0ther EV5. H0wever 1t take5 a 51mp1e f0rm 1f20 15 c105e t0 the end-p01nt 2e 0fthe EV d15- tr16ut10n, 51nce we 06ta1n
dFu 1
d0 -- 8-n / d2° ).0 1-- J.e
×exp{--N[F(20)--F(2~)
]}[1 + 0 ( 1 ) ] , (30, where F(2) 15 the effect1ve p0tent1a1 (13). F0r the m = 2 Cr1t1ca1 p01nt, 1n the 5ca11n9 re91me F• ( p ) = --2w(p), w1thw(p)
91ven 6y (19). At 1ar9e N the 1nte9ra1 (30) 15 d0m1nated 6y the 1n5tant0n c0nf19u- rat10n 0fref. [ 11 ], where the EV 51t5 at the t0p 0fthe p0tent1a1 p = t ~/2.7he re5u1t, 1nc1ud1n9 the c0ntr16u- t10n 0f f1uctuat10n5 ar0und the 5add1e p01nt, 153 - 3 / 4
dE 91/2 " t-5/8
d 0 - -
× e x p ( - 1 ~ 2 x / ~ t 5/~)[1 + 0 ( 1 ) ] , (31,
where 9, = N - ~a- 5/2 15 the 5tr1n9 c0up11n9 c0n5tant.
1n the d0u61e 5ca11n9 11m1t 95 = 1, and (31 ) 91ve5 the n0n-pertur6at1ve 0-dependence 0f the 5t1n9 5u5cept1- 6111ty u = - F " . U51n9 the fact that the tr1p1y trun- cated 501ut10n5 u+• c0rre5p0nd t0 0= -7- •1, 0ne thu5 rec0ver5 the re5u1t5 0f ref5. [7,8 ] f0r the n0n-pertur- 6at1ve part 0f the 51mp1y truncated 501ut10n5 0f Pa1n- 1ev6 1.
F1na11y, 1et u5 return t0 the c0n5truct10n 0fthe 100p 0perat0r5. 1t 15 very ea5y t0 check that w1th the 9e- net1c ch01ce 0f c0nt0ur5 (7), the f1n1te-N 100p e4ua- t10n5 0f the matr1x m0de1 are 5t1115at15f1ed 6y the 100p 0perat0r5. 1t rema1n5 t0 under5tand what 15 the c0n- t1nuum 11m1t 0f the5e 0perat0r5. 1n 5tr1n9 pertur6a- t10n the0ry, the 0perat0r wh1ch create5 a macr0- 5c0p1c 100p w1th m0mentum p (c0nju9ate t0 the 100p 1en9th 1),
w(p)
= ~f d1e-P1w(1), 0can 6e expre55ed 1n term5 0f matr1x e1ement5 0f the 0perat0r Q 91ven 6y (2) [ 14]. F0r 1n5tance the 0ne- 100p c0rre1at0r 15
~ 1
( w ( p ) ) = d x < x 1 n ~ 1 x ) , (32)
t
and the pr061em 15 t0 def1ne the re501vent
6(x;
p) = (x1 ( p - Q)-• 1x)f0r the 9ener1c rea1501ut10n5 0f (4), w1th p01e5 0n the ne9at1ve rea1 ax15. 6 mu5t 5at15fy the 6e1fand-D1k11 e4uat10n
-266" +6•2 +4[p+ 2u(x) ]62=
1, (33)and 9ener1ca11y 6 ha5 a150 d0u61e p01e5 at the p01e5 0f u. 1n fact there 15 a un14ue a5ympt0t1c pre5cr1pt10n f0r 6 (x) 1n the 5tr0n9 c0up11n9 re91me x--, - ~ , wh1ch 15 c0n515tent w1th the c0nt0ur pre5cr1pt10n (7) (de- f1ned 6y the 0-parameter), and the 5pec1f1c u. 1f we perf0rm the re5ca11n9 (51m11ar t0 (23) )
X = 4 X 5/4, 6 : X - 1 / 4 H , P = X - 1 / 2 p , (34) 1n the 11m1t 1x1 --,~ (33) 6ec0me5
- 2 H H "
+H•2+4[P+2U0(x)
] H E=1 + 0 ( 1 ) , (35,
Where
U0(X)
15 the e111pt1C fUnCt10n 91Ven 6y (23), Wh1Ch 15 501Ut10n 0f (25). 7here 15 a Un14Ue 501Ut10n 0f (35) Wh1Ch 15 d0U61y per10d1C W1th the 5ame pe- r10d5 921,2(E0) than U0. 1t 15 91ven exp11c1t1y 6yP-U0( X)
H(x; P ) = x / 4 p 3 • 1 2 P + E 0 " (36)
v01ume 302, num6er 4 PHY51C5 LE77ER5 8 1 Apr11 1993 7h15, t09ether w1th (32) and the c0n5tra1nt5 (28)
wh1ch f1x E0, 91ve5 the 5ame a5ympt0t1c expre5510n f0r the 0ne-100p c0rre1at0r
w(p)
1n the n0n-pertur- 6at1ve pha5e t < 0 than the expre5510n (18) that we have 06ta1ned prev10u51y thr0u9h the 81P2 ap- pr0ach. 7h15 ach1eve5 the 1dent1f1cat10n 0f 0ur 1ar9e N n0n-pertur6at1ve 501ut10n 0f the matr1x m0de1 w1th the rea1 501ut10n5 0f the 5tr1n9 e4uat10n (4). 7he5e 100p 0perat0r5 w111 5at15fy the 100p e4uat10n5 [6,15,16], at var1ance w1th the 0perat0r5 c0n- 5tructed 0n1y 1n the pertur6at1ve pha5e w1th the pre- 5cr1pt10n 0f ref5. [2,3,14 ]. Each 100p 0perat0r w111 have a51n91ep01e
1n t wherever the 5tr1n9 5u5cept1611- 1ty ha5 a d0u61e p01e. 7h15151n fact natura1, 51nce the 5tr1n9 5u5cept16111ty 15 the VEV 0f tw0 m1cr05c0p1c 100p5 (puncture 0perat0r).Let u5 5ummar12e and d15cu55 0ur re5u1t5 f0r the ca5e 0f pure 9rav1ty.
7he pr0p05a10f ref. [ 8 ] t0 take a rea1 c0m61nat10n 0f c0mp1ex 1nte9rat10n c0nt0ur5 1n the 0ne-matr1x m0de1 t0 06ta1n rea1501ut10n5 0 f t h e Pa1n1ev6 15tr1n9 e4uat10n f0r pure 2D 9rav1ty ( c = 0 ) ha5 6een f0r- mu1ated here 1n the framew0rk 0f the 1ar9e N 501u- t10n 0f the matr1x m0de1 ~ 1a 81P2, 1.e. 1n term5 0f d15tr16ut10n 0f e19enva1ue5. We have 5h0wn that the 5tr0n9 c0up11n9 pha5e, wh1ch c0rre5p0nd5 t0 ne9at1ve va1ue5 0f the ren0rma112ed c05m01091ca1 c0n5tant t, and 1n wh1ch the 5tr1n9 5u5cept16111ty ha5 p01e5, can 6e de5cr16ed 51mp1y 1n term5 0f 5p11tt1n9 0f the mean- f1e1d d15tr16ut10n 0f the e19enva1ue51nt0 tw0 6ranche5 at the end 0f the EV d15tr16ut10n.
7he5e tw0 d1fferent 6ranche5 can 6e v1ewed a5 tw0 d1fferent t0p01091ca1 5ect0r5 1n the 1nte9ra1 0ver the EV5 and the n0n-pertur6at1ve 0-parameter wh1ch d15t1n9u15he5 the d1fferent 501ut10n5 0f the 5tr1n9 e4uat10n 15 51mp1y the pha5e d1fference 6etween the5e tw0 5ect0r5, wh1ch ha5 t0 6e 5pec1f1ed 1n the func- t10na1 1nte9ra1. 7heref0re, at a f0rma1 1eve1, each 0 def1ne5 a 0-vacuum 0f 2D 9rav1ty, a51n f1e1d the0r1e5 w1th t0p01091ca15ect0r5, 5uch a5 4D n0n-a6e11an 9au9e the0r1e5 0r 50me 2D a-m0de15. 7he n0n-pertur6at1ve effect5 a550c1ated w1th th15 0 parameter can 6e e5t1- mated 6y 51mp1e 1n5tant0n meth0d5.
F1na11y, we have 5h0wn that, f0r each rea1 501ut10n 0f the 5tr1n9 e4uat10n (0-vacuum), 1t 15 p055161e t0 c0n5truct 1n a c0n515tent way 065erva61e5 (100p 0p- erat0r5), 1n 5uch a way that the 100p e4uat10n5 5h0u1d 6e 5at15f1ed n0n-pertur6at1ve1y.
0 f c0ur5e, many 1ntere5t1n9 4ue5t10n5 are 5t1110pen.
1t 15 c1ear that 0ne can def1ne n0n-pertur6at1ve1y the 0ne-matr1x m0de1 f0r 9enera1 p0tent1a1, and pr06a61y rec0n5truct 6y ade4uate ch01ce 0f c0nt0ur5 the rea1 n0n-pertur6at1ve 501ut10n5 0f the un5ta61e even rn 5tr1n9 e4uat10n5. We 5ha11 d15cu55 6e10w the ca5e 0f the d0u61e we11 p0tent1a1. 51m11ar1y, the 5ame rec1pe can 6e app11ed t0 the mu1t1-matr1x m0de15, and u5ed t0 5tudy the 9enera1 (p, 4) 5tr1n9 e4uat10n5 (a1- th0u9h f0r the mu1t1-matr1x m0de15 there 15 n0 51m- p1e p1cture 0f the 1ar9e N 11m1t 1n term5 0f EV d15tr16ut10n).
7he fact that the 100p e4uat10n5 are 5t111 va11d n0n- pertur6at1ve1y 1n the framew0rk d15cu55ed here 15 4u1te appea11n9. 7he5e e4uat10n5 can 6e der1ved fr0m var10u5 p01nt5 0f v1ew: Dy50n-5chw1n9er e4uat10n5 f0r the matr1x m0de15, V1ra50r0 c0n5tra1nt5 f0r the KdV h1erarchy, recur510n re1at10n51n 2D t0p01091ca1 9rav1ty. 7h1515 at var1ance w1th 0ther 5cheme5 wh1ch have 6een pr0p05ed f0r def1n1n9 n0n-pertur6at1ve1y 2D 9rav1ty [ 19,20].
0 n e 1mp0rtant 155ue ha5 t0 6e pr0per1y under- 5t00d. 1n the 5tr0n9 c0up11n9 pha5e (t < 0) the part1- t10n funct10n 2 = e x p ( F ) ha5 2er05, and the 100p 0p- erat0r5 have p01e5. 7h15 1mp11e5 that the n0n- pertur6at1ve rea1501ut10n5 0f 2D 9rav1ty that we have d15cu55ed here 5h0u1d 5uffer fr0m n0n-pertur6at1ve v101at10n 0f p051t1v1ty, even 1n the weak c0up11n9 re- 91me t> 0. 1t rema1n5 t0 under5tand what th15 rea11y mean5 when 0ne f0rmu1ate5 the5e 501ut10n51n term5 0f 5tr1n9 f1e1d the0r1e5 1n 10w d1men510na1 6ack- 9r0und5, 1n part1cu1ar f0r p051t1v1ty and un1tar1ty.
7he c = 1 matr1x m0de1501ut10n 5tud1ed 1n ref. [21 ] d0e5 n0t 5uffer fr0m the k1nd 0f 1n5ta6111ty 0fthe c = 0 m0de1, 51nce 1t c0rre5p0nd5 t0 free ferm10n51n an 1n- verted harm0n1c p0tent1a1, w1th the tw0 we115 0f the p0tent1a1 f111ed at the 5ame Ferm1 1eve1. C0n5e- 4uent1y, a1th0u9h the 5tr1n9 pertur6at10n the0ry f0r the c-- 1 m0de115 n0t 80re15umma61e, there 15 a we11 def1ned 5ummat10n pre5cr1pt10n wh1ch a110w5 t0 re- c0n5truct th15 n0n-pertur6at1ve 501ut10n. D0e5 the k1nd 0f 1dea5 d15cu55ed here a110w t0 c0n5truct 0ther n0n-pertur6at1ve 501ut10n5 0 f t h e c = 1 m0de1•
F1na11y 1et u5 6r1ef1y d15cu55 the ca5e 0f the d0u61e we11 p0tent1a1,
V ( ; t ) - 1 - ~ p 2 +~;t . 2 1 4 (37) F0r p < 0 1ar9e en0u9h, the EV5 are d15tr16uted a10n9
tw0 cut5 (5ymmetr1c u n d e r 2 . ~ , - 2 ) . At the cr1t1ca1 p 0 1 n t / ~ , the tw0 cut5 fu5e ( a t ) . = 0 ) 1nt0 0ne 5e9- ment. 1n the d0u61e 5ca11n9 11m1t the 5tr1n9 e4uat10n f0r th15 cr1t1ca1 p01nt 15 the Pa1n1ev6 11 e4uat10n [ 22 ].
Recent1y, 8 r 0 w e r et a1. 5h0wed that 6y re1ax1n9 the par1ty c0nd1t10n ~tn(2)= ( - 1 ) n 7 t ~ ( - 2 ) 0n the 0r- th0n0rma1 p01yn0m1a1 a n d 0n the a550c1ated 501u- t10n5 0 f the recurrence e4uat10n5, new 5ymmetry 6reak1n9 501ut10n5 0 f the m0de1 c0u1d 6e 06ta1ned [ 12 ]. 7h15 can 6e ea511y under5t00d 6y c0n51der1n9 the three 1ndependent path5 0 f 1nte9rat10n f0r the p0- tent1a1 ( 3 7 ) . 1n add1t10n t0 the rea1 ax15 ¢fr we can a150 1nte9rate 0ver the path5 cf+ 901n9 f r 0 m - ~ t0
•+ 100.7he m05t 9enera1 we19ht fact0r5 f0r the5e path5 wh1ch 91ve a rea1 part1t10n funct10n are
cr=1-2x, c+=x+•10. ( 3 8 ) 1f c+ # 0, 1n the 5tr0n9 c0up11n9 pha5e/2>/~c the EV d15tr16ut10n 15 n0 m 0 r e the 0ne cut 501ut10n 6ut a cr055-5haped d15tr16ut10n w1th f0ur cut5 meet1n9 at the 0r191n. 5ett1n9 x ~ 0 6reak5 exp11c1t1y the 5ymme- try 2~-~-2 a n d 5h0u1d a110w t0 rec0ver the 5ymmetry 6reak1n9 501ut10n5 0 f ref. [ 12] (wh1ch d1ffer fr0m the 5tandard 501ut10n 6y 5u6d0m1nant term5 0 f 0 r d e r
1/N 2). 5ett1n9 x = 0 6ut 0 # 0 91ve5 501ut10n5 wh1ch d1ffer n0n-pertur6at1ve1y f r 0 m the 5tandard 0ne, a n d wh1ch c0rre5p0nd t0 501ut10n5 0 f the Pa1n1e+6 11 e4uat10n w1th (51mp1e) p01e5 0n the ne9at1ve rea1 ax15.
7he5e c0n51derat10n5 can 6e e x t e n d e d ea511y t0 the mu1t1cut matr1x m0de15 5tud1ed 1n ref. [23 ].
1 a m v e r y 1nde6ted t0 5. Ja1n f0r exp1a1n1n9 t0 m e the re5u1t5 0 f ref. [ 12 ] pr10r t0 pu611cat10n. 1 t h a n k R. C0nte, P. D1 France5c0, J. 21nn-Ju5t1n a n d J.-8.
2 u 6 e r f0r very u5efu1 d15cu5510n5, a n d P. D1 F r a n - ce5c0 f0r a carefu1 read1n9 0 f the manu5cr1pt.
R e f e r e n c e 5
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