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arXiv:hep-th/0312022 v1 2 Dec 2003 - IHESpreprints.ihes.fr/P03/P03-65.pdf · x A _ W bc N K gON ac g H}z J N=] W t Z [ gP b P b [ WYP bB gOu gON W [a Z [Am [ PY[z t aVb z HI Jdt N

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Page 1: arXiv:hep-th/0312022 v1 2 Dec 2003 - IHESpreprints.ihes.fr/P03/P03-65.pdf · x A _ W bc N K gON ac g H}z J N=] W t Z [ gP b P b [ WYP bB gOu gON W [a Z [Am [ PY[z t aVb z HI Jdt N

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WYPM Jd Z Nu ur[ JdJd Z [ aVb W a_z [KgVN a W % N W[ a [Am \ Z Nu t uy[g_P b NAP£Z\[GM ]W !=bNZ Z Nu [AmºZ u WY[ a Z~[Am Æ [P[ H [ Jd_ W£u~Mg aVb P B W \ urg J ]AN g Jd5o © ª SW a uNZ W W£ZZ\g urW ac P tÆ [|Z\W W] << )?2.

a [ JybcJ! [IWPYPgOZ \J N ~ Z W bc NZ !B WPPxuy[ a Z J gOu N m N HI P Z W J Z ur[ a Z\W bcJ N,ur[ HI Nu ! N HI P [Am b W HIa Z\WY[ a [ a S a N HI P t WvZ$uN a M BJd NAPW % =bM t [B] N J W£NAM_P Z / g C 0 W \ [ a_ u N Jdz ] u [ J / 6g= 0 w ¦bca [ \$\ I¡N P J urPvNZdZ a! Nn] ' C

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Rm ! NAX / [M KgON ac W % b W a ©cª¦g a W ZS ©cª \a! uN a W bcac W¤m t_ Z N \ Z%[Am \WYPM Jd Z Nu W \ W aczJ$ [W ac Z 2 (g#2 C * SOZ\gM . u ! [

2 2 C / 0U[ B\ N 2 C 2 W HI PW Z \ N !! uN a W bac Wm tI\ WvZ$Z Nu W \ PvN \ Wvu ¢ [W ac Z%[ a\¢ [|Z\W W] PW a ZdN WvZLm t W az

2 a. N Jd WvurgPvN J$\ W£ZZ Nu NZ B [W ac ZSVZ z ur[ JdJd Z [ aOb W az I¡N P J urPvNZdZ o W£ZW bca W¤sNAMP ¢ W \ © ª W HI Z urg J ] N g Jd ur[ JdJd Z [ aVb W a_z [ PYW a Mg aOb P / 0$[A] J) W£ZºW£Z \ Mg aOb P [|Z [P[ H [ Jd WvuZ u WY[ a ZuN a M W bcac W¤ =b< W 5 [ H [ za [gOZ bzJd [P t|a [ H WvNAP£ZW a g C Z\[Pg WY[ a [Am / 0dS / 2 g#2 C 0dSur[ JdJd Z [ aOb Z [N H [ a [ H W£NAP C gOZ bzJd H [ a [ H W£NAPvZ¦N Jd~a N g J NAPYP t W bcac WsNAM_P W \ \ NAMO[A] [W a Z C ZG[g J:a N HI P ! ur[ a Z\W bJ N a [ a_ ur[ HI Nu :Ab W HIa Z\W[ a NAP$Z Nu WvZ WYPPur[ JdJd Z [ aOb< [ MP[ W az g C N N [W ac W£Z H N a Z !B NAX N [W ac [ a C N aOb5Jd PvNu W ¢ W \ N Iz [ HI\Jdta_ N JB\ W£Z W£Z /& 0 S/N aOb W a N Jd W£urgPvN J W W£Z a [ Nj NAPvNAMW NAg Z Nu NAX ¢\Jd ] N J W£NAM_P Z!N aOb< NAX [ a u N Jdz

/ 6g=6g 0 _aG\¢H [ HIac NZdN W£ZLm t

C h _ W£ZW bcac W¤ =b: W \5\ Z\g_MOZ Nu NAMV[ ] B W h SVN aOb:\a [ JdH NAPMg aOb P -/ 0[A] J WvZ W bcac W¤ =b5 W \ ] N Jdt W az h z NAW a Wm ! KgON ac W % !Æz m[ J¢\ NAPYP[ !=bW a\zJ NAP H [ HIac N uy[ aVb W W[ a \ N 2

N aVb

2 2 C 2 h U[ \ N WvZ$uN a M W bcac W¤ =bI W \5_ Z Nu [Am [|Z\W W] W aczJ NAP [W ac ZºW aG_ KgON bcJ N ac/ 2 g#2 C 0 W 5\ mg JdJJd Z \J Wvu W[ a \ N

2 h 2 2 C *

Page 15: arXiv:hep-th/0312022 v1 2 Dec 2003 - IHESpreprints.ihes.fr/P03/P03-65.pdf · x A _ W bc N K gON ac g H}z J N=] W t Z [ gP b P b [ WYP bB gOu gON W [a Z [Am [ PY[z t aVb z HI Jdt N

p

p2

1

( 8 ( & ! +;1 @ '!+17 # ^ 7 : ; +5 7GB#" !B#% _ & 53<N 5+#-3L17 #% & @):' 6<+ (;1 5;< D $&%

_ ¡N P J urPvNZdZ![A] J ur[ JdJd Z [ aVb Z [ _ / 0¦Mg aOb P ScNZ b WvZduygVZdZ b M m[ Jd W£ZWvZ$uNAPYP =b M_PY[ W az g N [W ac W a C M uNAgOZ !~ N=] ~JdH [A] =bI_ [ J W z W a N aOb:Jd PvNu =b W M tN S Wvu W£Z JdJd Z ac=b M t N a W ac\J ]ANAPOW a:\ P£N a_ U[ \ \ N ur[ HI N Jd=bÆ [5 C! Nn] bc P \=b N \J W£N az P [Am [W a Zm J [ H\ Z Nu SM t MP[ W az g I\ [W ac N !_ [ J W z W a a [ \J [ Jyb Z ! Nn] / 0 P ZdZ P HIac Z%W aG\ WYPM Jd Z Nu / W N a N g J NAP bc a W W[ a [Am(uy[g a W az&*|\ Z N \ Z[Am \ WPYM Jd Z Nu 0

e a uN a NAPvZ\[¢M_PY[ g : [,NZ\W a_z gP£N J Z Nu _[ J/ N HI P P gOZ!ur[ a Z\W bcJ NZ\W a_z gP£N J MPY[ g [AmE ur[ a Z\W bcJ!\Jd ] N J WvNAMP Z W [ a u N Jdz

/ (g=6g 0 _aG! Nn]

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Page 16: arXiv:hep-th/0312022 v1 2 Dec 2003 - IHESpreprints.ihes.fr/P03/P03-65.pdf · x A _ W bc N K gON ac g H}z J N=] W t Z [ gP b P b [ WYP bB gOu gON W [a Z [Am [ PY[z t aVb z HI Jdt N

_ ur[ JdJd Z [ aOb W az Z Nu NZ$N Z\W az gPvN J W t N (g C (g h f[IZ ~ WvZ a [ \ N \ Nu W[ a zaJ N \=b M t WvZ z NAg z |=b M tGz [W azI [< |=b ] NAPYg [Am %g WvZP Nn] ZN a1|\J N C N Jd [Am _ Nu WY[ a4$ W£u Z aOb Z / C g h 0 / C g h 0 U[ \ N z NAW a W a

\ W£Z N HI P W¤m ! WvZ N a W a\zJ NAPxZ\gMOZ Nu [Am¦Z N \ Z ! uN aG NAX ©cª$P N b W a_z, [ \ [|Z\W WY] W ac\zJ NAP H [ HIac Nur[ a Z \J NAW a_ b M t

+2 2 C 2 h uN a NAP£Z\[ÆMP[ g H [ Jd \ N a [ a [W ac [ a C [ J N HI P ur[ a Z\W bJ m[g J ] N J WvNAMP W \ [:u N Jdz Z / 6g=6g 6gd 0 g

*C / 6gd(g=6g 0 WY]W az

C h g h $ C

W£ZZ Nu ur[ JdJd Z [ aOb Z [ \ Z\gMOZ Nu [Am _¢ [|Z\W W] / g C 0!KgON bcJ N ac ZdN WvZLm t W az C g

) C C z NAW a SOW¤m ! W£Z 5 [ [M NAW a W aczJ NAP [W a Z ! uN a ur[ a Z\W bcJ KgON ac W %W a_z g C W a g a W Z

[Am©ª!P N b W a_z, [2 2 C 2 h g

+2 2 C 2 $ C z # j P N J P t5! uN a ur[ a Z\W bcJ H [ Jd MP[ g Z a W£Z N t5! uN a [M NAW a N J MW \J N Jdtur[ a ] Z\gMVZ [Am/W aczJ NAP [W ac Zº[Amc C X h [ J [g J¦a¦ N HI P ur[ a Z\W bJ h MP[ a g N N [W ac W£Z%ur[ JdJd Z [ aOb Z [ JdH [A]|W azN [W ac W a h N aOb Jd PvNurW az W G W N 4C H M b_b W a_z [Am 4C W a Z Nu z W] Z-/ 0 C U[ \ N N z NAW aI\ W£Z%WvZ a [ N j NAP£NAMW NAg z [ H\Jdt /NZ _ a [ JdH NAPcMg aOb P W£Za [ -/ 0 0 WvZ$uN a M BJd NAPYW % =b M t ur[ a Z\W bJ W az ] N J W£NAM_P Z W \ [ a u N Jdz / 6g 6g=6g 0 gP N b W azÆ [

C h $ 2g$ W£u uN a M W bac W b W G_¢ [|Z\W W] [u N a / g C g h 0 W 5\ uy[ Jda_J!JdH [A] b

$ C h *r

Page 17: arXiv:hep-th/0312022 v1 2 Dec 2003 - IHESpreprints.ihes.fr/P03/P03-65.pdf · x A _ W bc N K gON ac g H}z J N=] W t Z [ gP b P b [ WYP bB gOu gON W [a Z [Am [ PY[z t aVb z HI Jdt N

p

p2

1

&9 + & *# ! +;1 @' ^ 7 5):: 7 #% _ D $&% 76 ^ D $ %

);1 !17G@ #% )0 17A l 5): )7 *#%45 178 #% & 5):7&' #- 3 l 764: ;517LB# !B#% #-07 ;1+

_ ur[ JdaJ WvZ Jd PvNu =b M tI\¢\J WvN az P . C h

$ W£u ur[ JdJd Z [ aOb Z [ C [AmZ\W % NZ$Z [ a W a z " #

Page 18: arXiv:hep-th/0312022 v1 2 Dec 2003 - IHESpreprints.ihes.fr/P03/P03-65.pdf · x A _ W bc N K gON ac g H}z J N=] W t Z [ gP b P b [ WYP bB gOu gON W [a Z [Am [ PY[z t aVb z HI Jdt N

p1

p2

< + !#% +;1 !17 @ !+17 # 17 l 76 B#B*: ;5*7 # !B#% ^ ' & 53<N 5+#-3 5):: 7 "#%2#- + (;1'5;<: D Ê $&%

z NAW a Wm ! uy[ a Z\W bcJ $ [GM KgON ac W % bk! Nn] m[ J \ NAPYP[ !=b KgON ac g H Z N Z W \2

c

2 2 C$ 2 h 2 $ j [ HI N Jd=b [ h !G Nn] bc P \=b / 0 / 0 ". [W ac Z /Z N \ Z10¢m J [ H WYPM JdZ Nu /Z z 0 Z NAPYP HI~ WvZ N HI P M P[ SW a \ ur[ ac| [Am a [ a ur[ HIH g N W] W a Z N ac [ a Z

_[ J [g J a| N H P ur[ a Z\W bcJ [:MP[ g Z¨[AmX J Z M_PY[ N [W ac g [ CN aObB\a MPY[ g N [W ac [ aÆ_ MPY[ $a g C% [N WvZ/uN a M $bc Zdu J WM b M t# ] N J WvNAMP Z W \ [:Z [Am¦u N Jdz Z

/ 6g=6g 6g 6gd 0 g C / 6gd(g¬(g=6g 0 g

"

Page 19: arXiv:hep-th/0312022 v1 2 Dec 2003 - IHESpreprints.ihes.fr/P03/P03-65.pdf · x A _ W bc N K gON ac g H}z J N=] W t Z [ gP b P b [ WYP bB gOu gON W [a Z [Am [ PY[z t aVb z HI Jdt N

p

p

p1

2

3

p = 01

p = 02

p = 03

p = 05

p = 04

&2 !2#0 ! ;1 @* l ;13)17 #% ^ +7 1< )3@)5 @ %+5 _

$ W£u P N b Z [ C h $ g $ 4 C

W£Z z [ HI\Jdt W£Z b W£u \ b W a z Rm C W£ZBur[ JdJd Z [ aOb Z [ \z [ H\Jdt [Am$N ur[ aa u \ bk [5NG$W J % M J gOu Z\g J moNu SW a Z\W bc\ MP[ a g h z NAW a W¤m ! ur[ a Z\W bJ\ W aczJ NAPZ\gMOZ Nu \ Z N \ Z ! X uN a M B NZ\WP t b b gOu b j P N J P t [ a uN a ur[ ac W a g ,\ WvZ [ H [ Jd ur[ HI PW£uN \=b MP[ g Z e a_ Z\gVu N HI P W£ZZ [ $a W a z

-/ 0 -/ 0 uN a NAP£Z\[uy[ a Z\W bcJxb [ J Wvu z [ H\J W Z $ Wvu N JdÂa [ [M NAW a=bM t MPY[ W az g h _[ J! N HI P ~! uN a uy[ a Z\W bcJ m[g J ] N J WvNAMP Z W [ a u N Jdz / 6g 6g 6g= 0 g

n

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)4"537Q<; 25)7( ( 9): (7<;#%) 5 ; @ l P N b W azÆ [

C h $ W£ZuN a M ]W !=b NZ _ Z\gMVZ Nu [Am \ [u N ac / g C g h 0 W \G mg Jd\J ur[ a Z \J NAW ac

$ C h W£Z z [ HIJdt WvZ b W£u \ b W a z U[ \¢\ N h W aG WvZZ Nu P N b Z [ \

J [. u WY[ a [Am [Am¦Z\W % S N J N HIJ W % =b M t / C g $ 0¦W a \BH [ Hac g H Z Nu WvZ!W a moNu ur[ JdJd Z [ aOb Z [ / 0 /& 0 z [ HIJdt S $ Wvu W£Z!N j NAPvNAMW NAgZ Nu uN a< NAX ©cªN aOb [M NAW aG [W ac Z%[Am \ $WPM Jd Z Nu ~ W 2

* ZdN WvZLm t W az

2 2 C 2 h 2 $ uN a NAPvZ\[GMPY[ g [W ac ZW a\ W£ZBZ Nu x[ b [ Z\[ ! W a\J [ b gOu [ a,H [ Jd ] N J WvNAMP

/ a g J NAPg aObcJ!_¢Jd ]WY[gOZ!u N Jdz 0¦N aOb uy[ a Z\W bcJ!\ N bb W W[ a NAPu N Jdz/ 6g¬(g=6g=6g 0

$ W£u M_PY[ Z~g 1\: [W ac h $ e J! uy[g_P b NAPvZ\[5MP[ g NAPY[ a_z ) [ J N HI P ! ur[ a Z\W bcJ W a Z \ N b N bb W WY[ a NAPu N Jdz/ (g¬(g=6g=6g 0

Page 21: arXiv:hep-th/0312022 v1 2 Dec 2003 - IHESpreprints.ihes.fr/P03/P03-65.pdf · x A _ W bc N K gON ac g H}z J N=] W t Z [ gP b P b [ WYP bB gOu gON W [a Z [Am [ PY[z t aVb z HI Jdt N

p1

p2 = 0

p3 = 0

= 0

= 0p4

!)! &" !"#% &"#%)015 )*+#--,Q H >G0 È%C H >G0 ÈC 2 _

) ( ! +;1 @ >G0 È%C >G0 È%C 2 _ 3; 7 +17 #B'##- .<# 597 _ <;1)7 #% &'3)

uN a NAPvZ\[ ur[ a Z\W bcJ MV[ b W HIa Z\WY[ a NAP¦MP[ g ZSÂW S%MP[ g Z,NAP[ az N aOb4b W HIa Z\WY[ a NAPM_PY[ g [AmºN [W a uN a NAP£Z\[:ur[ a Z\W bcJ ] N J WY[gVZ H gP W MP[ g Z NZ N a N HI P z <4<;4" X.

I WPYP!NAP£Z\[5M W acJd Z =b W a wfN zJ N az W£N a1 M J N a Z $ Wvu uN a M bc Zdu J WM =b M t [ J W£u

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ap

p

1

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( 9 ! 7 6:7 #-3 , < &32; 17&8 ? (15 6 5+#B#%9 6<; ; 17 17 #% & #-)015 <+#B'#% +;13 , _ ^ +7 L#% &: ' "17(13; 5 !B#% :3 27 17 )7 #% ! )+7 #% &'51:5;1*5)*B:) #%

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< <;[@HZ#/)F6.,. a5\ PW a N J Z\W zH N H [ bc P/uy[ a Z J gOu WY[ a [Am [ J Wvu j NAPvNAMW NAg H [ a uN a W a\J [ b gOu

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S 6g 'g Z NAPYP ba [ B_H M t

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Z t|HI P u Wvu N aOb W£Z zaJ N =b M t Z\[ HI N H WYP [ a W£N a

o g

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u N az¦\ Z _JdÂz [ HI\J W£uNAPP t SAMg Jd ] J Z Z _ [ J W ac N W[ a [Am _!J [ N WY[ a\ ] u [ J P b

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z 1 =0

z 2 =0

! ! &&(3! 7 # ; 5)@+' ^ ' & <; ; 51:5;1" & #% 3 +17 #!17A#- )9+#%-,

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g C

0 _ uy[[ Jyb W a N Z[AmX C a\ WvZuNZ \] u [ J P b WvZ z W] a M t

C C

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/ g C 0 / g C

0

gOZc C NZ[ a P t [ a =b [W ac/z W] a M t / g C 0 / (g¬ 0 Jd N Jd NAPvZ\[ [ |=bÆ P£N a_ Zg aObJ z WY] a M t / g C 0 / (g C 0!N aOb / g C 0 / g¬ 0 gOZ%[ aG\ / C g C C 0 PvN a ! uN aGJdJd Z ac C NZZ [ a W a z _ PYW a N J Z\W zH N H [ bc Pxu N Jdz Zim[ J$\ WvZ z [ HI\Jdt N Jd / 6g= 0 W£Z HI N a Z N ! uN aJdJd Z a NZ Kg_[ W a [Am C M tI\ m[PYP[ W a_z Nu W[ ax/ g C 0 / g C C 0 g

_ ] u [ J P b C bc Zdu aOb Z [:NI] u [ J P b I [ a a5\ ur[|[ Jyb W a N \ N u / C 8 0 _ PY[cuNAP/ur[|[ Jyb W a N \ [ a WvZ + + / C 0

N aOb<_ ] u [ J P b I WvZ z W] a M tI

g (g

I g C

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Ω

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