Figure 1. Projection in the closed sector
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- Patience Carpenter
- 6 years ago
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2 " # FHGJIKGJLMN O5PQIKRJNMSM@T &%('+*-,%.0/21,365+%878%:;%<= >1?%#@A/;*=;%B,DC E5+%FG3+.H I;J3+*LKM/2*ONPA,QR S;TVUWTV7XTVUYT8Z\[/=];%3+1/^5+%FG3+.H I;J3+*_KM/;*OǸ A,QR K%QE5P/;aa:FG%<=/;*=<b^[<;%/;1,#%dce<= fhg ipipgajpj FG3+.H SAJDkmlon pgqrosutuv w=s yx {z # }x ~ By ` z h y È Oƒ Su 0 z# - #ˆƒ z u»šx#u ^ Qƒ} z u z # }x ~ zoœy z# ƒ`# }x ~ ˆ =x z m# } =x u 8ŽO yx # z }yx+ ux# z z (ƒ(ƒ z Su }x y z `-ƒ» y z z B+ S O -»-» z ` z y o S ẑ Œ š? œ 8 žÿ Š u S z K } = z u m e x+ }xo = m bx z V - #ˆƒ z? # `yxo u m Qƒ} z z # }x ~ z + y 'KyxH # (u  Yx X 8~ zm ^ ¾BGƒ z ou y = o Œ z zm" ƒ(ƒ+ x z ` O Œ Ú " + y B ;» u ª ª }x ~ ~yx yƒ " + x ƒ` z V O bx z m }x ~ Qƒ} z u {z # }x ~ z u m «œ ` u A O =x- V h ƒoƒ ƒš # ` ẑ Œ? z Š ƒ(ƒ} " x- ƒe - ƒ S }x # xo O }x+ S bx u A O bx V G yƒˆƒ ¾Š # ` z z u m z B z }x {z S z # z Oƒ S } =xèk# Jyx¾+ z =x x o } =x- O =xo ` # } bx z B? b = _ Sȃ( ` # x+ S z B. ƒ O bx V G yƒ x G yƒ} #O x z } z K m u? ¾ - ooƒp u # `yxo u G ƒ z u z # }x ~ zoœ z# ƒ` x~ƒ`b z uƒ # } =x z}" x m Syx+ œ V z # }x ~¾e - #Oƒ z [ S Qƒ` - z zm" z = m ƒ(ƒ} È SB? ; z }x ~± B ª `y + }xo? z # }x ~ x (A } =x ŒM² D z V x+ S«}x- z z u m S? yƒ³ƒêb z O =x z m bx z S # `u È z # }x ~ƒ + S } =x>by B " V? x 8~yx x œ Ju õ ˆ ƒ( }x yxo b y Œ u myxo? =x z x~x yƒ(» ` zm" }x z `u µ ƒ SB~ ˆ yx#.- ~yx ƒÿ S z Bƒ u yxb~= z# K~ y ª ` X ux+ 7y ª `y Ju œ }xx x P u + ˆ =xo # ` # } =x z B B z ; uƒ " z # }x ~ }x ~8xŠ " O bx+ m y ; #yx+ m V B±ƒ} =x~ Oƒ Oƒ( x~b+ S ƒ(ƒ z }x ~> y» z S z Èy u _ ª ( Êyx# h x ( z S } =x z M z x~ ƒ` x # #ƒ} ; }x ~>yxo# ` Œ ƒ + z X u J z yƒ z (ƒ(ƒ È z S u u b~ S y `u zbz Q S }x ~ŽO KŠx^ # z myx xo zb B z# - #ˆƒ z u `yx#u Qƒ} z z # }x ~ z ux+ " e X~yx yƒ³ƒ} " z ` S-ƒ»h O bx V G yƒoƒ y # ` ẑ Œ z8z S u }x ƒ SO }~=o# ` z ` + z ƒ u O Œ º¹u x z8z# Oƒ} ƒ ~b~ }x ~ y }x+ e O z» bx+ ª }x ~m V 7e - #Oƒ z # `yxo u Qƒ} z u z # }x ~ z}" + z Gx y ; uƒ h }xo e uxy ƒ}~b # `B S-ƒ» ¼ zbz - Q S }x ~' z# Qƒ} z u z u m e = ª `# } bx+ ƒ? yx z K m o `m Oƒ(ƒ #O+x S x z V ¾u }x ~ u. }xo Bx+ ƒ?b½?+ x bxÿ z oè À = {Œ«²z + ƒ(ƒ. ' # z }# `ƒ- z8z x~> O # }x ~>B y z `ȃ`e y z z S-ƒ»X u ª z -ƒ y ; }x ~ z Èœ u P ` zm Š ± zoœá u x y yƒ` # y (ƒ z h y ƒ V x+ X x ~u }x+ ź Á ÃÂ
3 3 3 3 Š Ú x+ }ẍ o V B+ O X }x ~' y ; `y # Qƒ` Œ8 z yx {z # }x ~' - ooƒ z±z ƒ`èyxobƒˆ z }x ~ ŽO # `yxo u Qƒ} z u ; " (ƒ`b u ª ª }x ~ z # }x ~ + z u # z }# `# } bx z V z Èe = ª ` bx+ ƒ Qƒ zz z u z O{~ ; ` Æe - #Oƒ zoœ8 u }yx# bx z # }x ~m yƒ³ `» zy ;»Š u G xy O bx# # ` } =x z z ¾ - }u ª }x ~ z±z ~=~ z z» zz»-ƒ» Sƒ( } =x z B > z B =x ~ ˆ ƒ( }x ~B y~u»èx z }x ~ e - #Oƒ zoœ z `-ƒ³ Q ` " ² z + yƒ³ƒá ˆ =x+x 7X h S yxo# } bxhb G ± Š y # ` z Á~yx +ze bx¾+ S# `# } =x x+ }# } =x z y Šx }x- z }~; Su }x z È # (ƒ Œ z `ȃ` z yx {z }x ~>e - #Oƒ # z myx+ xo z 5 # V²² z z x~ x+ G u B yx ª `yx z } =x+ yƒ = ; bx e - #Oƒ z Ž z }x# - mu }x O Œ u :ŒP² x8 # `# }x ~ BO ` = b Sȃ(» # z}"=²az + yƒ(ƒ+ X `[.e - ƒ SD }xo } =x z}" (ƒ`æ = ; x~ B> m O =x z ux# z V V e = } z ƒ`y ˆƒ =x 8 ¾ z yx# S bx zoœ z z `ȃ} HOƒ Su HB X =xµ x "+z }x+ m " # % ' ( % + ` B-,u # }x+ x+ m bx ".. 8? + S = m z + y b ª `# } =x ƒ ;» u ; ª }x ~ xh B =~b =x+ yƒá #u O + z `# } =x u [ z K m Œ² xb z x S } =x " B«}xo z }x ~ƒyx ª `yx z } =x+ yƒ e - #Oƒ z O z + =x+ ' } =x& x+ }# } =x z /' /' /'612 3< < 3 >«ƒ(ƒa B. yx ª `Èyx z bx+ ƒ - ooƒ z + y œƒ`o # }~=o z #È # z u m z# K z # }x ~ " z } =x& x+ }# } =x xb z x S bx u b z /'0017 : ; 3 = " }x+ Qƒ ; x~ # V²² = ¹ C M-DA FE žÿ HG + JI ž;gmki «DG?? x~æ+ }xo }x e ˆ =x z m# } =ẍ yx # z myx+ ux# z z L u }# } =xy u e Qƒ} z z u } X ( }x ~eƒ»ˆ ux+ Æ # }~=o u zoœ z z S }x z# -ƒ»yxo# }x ~ - }x # xo : ƒ} u»j +z + S# `# } bx& x+ m bx& ` Mœƒ`O }x + ƒ`e O =xo» # } =x Œ Figure 1. Projection in the closed sector ² x' z `ƒ` z œ u + zz `ƒ` z # }x ~ z}" ¾+ z =x z }x ~ " z K S } =x z + S }x' u Sƒ} y z # }x ~ y B ^ zxz y 0B &ƒ» = B ŽmO yx #u B YŽ} z } u 2+ S `B ON ` z B e - #Oƒ Œ6² xâ ŽmO Šx+ #u e - #Oƒ " ƒ(ƒ z }x "z " Gƒ( }~=o O =x ^ # z m» # } & y }xu = }# }x ~^ bx & b =»B + ƒ} -x X ƒ z }x B ux z z Gƒ`O e u ª }x ~ z Q zqpsrut ux+ " }x ux z z # }~=o e ; }x ~ z Q zwv PSTR " ` { SX ³ƒ( S N ` z m O bx z }xo8 =x ƒyxo# ` ˆ z + bx+ ª }x ~ B ƒ¾o Q } =x zoœ x Y G+ ux+ " }x BŽm z # m - #ˆƒ8 =xƒ} + ƒ} -x X ƒ z ` x = ª ` bx+ ƒ± ª z m z oè x+ #G }x# + ux ~ ± ƒ(ƒ B P R `B
4 PSR X yƒ(ƒ u Ju Œœ² x z x~ y B " ux h ƒ( ` z x H O ` G ux ( z S bxw }x B z O =xo ˆ " b #+ ƒ` zx Yy¹u q" ~ ux# B¾ # } b S# } bx V BMœƒ»ˆ x + #ƒ`b L u m " }x ~yx ƒ O =x- V h ƒ Oƒ Ây " O + z z ' # z m yx - ooƒ zm" O -»-» }x ~0 O ` x z u œ }xo Bx+ ƒ z oè h x+ 'ˆ» ƒ x S ƒ( ` }x ƒ³ƒ u ~u V Œ 8 V ƒ(ƒ} u z z ƒ`u = ; x~hb 'µ O =x z m# } bx u yx^ o z mšx+ xo z}" - ³ƒ`È O # }x ~ B>e - #Oƒ z # z }# `u xâb>y ª } +z u m } =x Œ ` z X ƒ(ƒ " =x z Y+ yƒ 'B> +z ˆ =x ` # } bx " ²Xz + ƒ(ƒ¾o y x B u x~ z 0ˆ ƒ( Q `#ƒ} Œ yx =x z b åb Mœƒ`O }x» ƒ» O bx## ` bx Œ z z Ž z e } Q yƒ z "z }x+ mg `e # X x z z oè B u B b S # } +z z m z BYŽ} z # mu Qƒ} z u z u } x #ƒb }xo Q+ ux ~h 1O `hžqƒ`o ux+ HŽm ~b# ẑ ŒM² xe~yx yƒ " Q u mœ u Mœƒ»ˆ x + #ƒ`¾ L u m# } bx z» x O b Á = (ƒ` " u x y yƒ³ƒ} " b S # } +z L u m# } bx z ª ( ÊM }x' z }~bx z V B = } z S ; u~= bx+ ƒ(ƒ} }x>b +z Sȃ( ` o " B+ Se o o }xè & ux# z oè Y B> z # m z u m ŒB ± }x z x m " # V²² >e - #Oƒ u Œ ˆ ƒ }xæ x Q `ƒ`x = oo ` V Ḿ ƒ`o }x + #ƒ`ƒb L u m } =x z}" O z» bx+ ª }x ~ ƒ(ƒ b y Q m z z }~bx z V B h b m z x W Œ > m ƒ(ƒ} " ± +z bx^ yƒ}~ }x# - m zxz bx ~ z }# } =x ẑ Œ b x m }x^~šx yƒ =xƒ} xo# V z #È # e O - x S# } bx z `ƒ`o x # }~=o N h e - ƒ` O `ƒ` ` e - ƒ S? }x b S xo +z Sȃ( ` # " ` zz `ȃ` x+ # z x+ u z È z }# } =x z S z >- ŽQ +z x~ + y» z u z S zoœh +z}" }x0b z y z È +z bx u # Y z = m # `È= z }x z ; `Oƒ z m " x+ 0B z u mb ƒ z X yƒ} y z z oè# u Œ ƒ` zz =x' S h # xe Qƒ} z }x z u }# } =xm - zz `ȃ`¾O z = } yƒ³ƒ} ` ~yx yƒ. y b (ƒ S-ƒ»e u } z O z + =x+ > B z z.bœ +z } =x ƒ}~ O `ƒ`g `ÂB> +z L u m# } bx Œ `B0 z B u ª z " =x y x z Šx ;»Èyx z } =x+ yƒ e - #Oƒ z u J Œ «ƒ(ƒ} S e z " # z Mœƒ`O }x + #ƒ`è L u m# } bx " B ~= Su ƒ(ƒ} b yƒ»šx# Œ z " e z 8x+ S yƒ u } " O z + =x+ z 'B x S ƒ + z z + S = m z V B z + = } ` Šx# ~B ± " x SÈOƒ} W " x z# ƒ` z }xæb¾ L m# } =x z #01012 # 617 ÿ : #612 - z u } z S z oè# u 8 (ƒ»&% % z u mb z S ux# z oè u Œ m z O z + =x+ B ƒb z z " B ƒb z z? ux+ + z z Œ? xƒ} Ž z Ḿ ƒ`o }x + #ƒ`¾ L m# } =x z ƒ(ƒ} u u X xè # o V²² z z # }x ~ " (ƒ`8ƒ z 1# " Qƒ`u Sƒ} H } b ƒ`yxo8 y S # ` ' x z V h S } =x " + x ~¾ Oƒ S } ˆ ~b# B&% % z u mb zm" x+ h S 1O V }x+ O `ƒ` ` B L u m } =xg u &' e - #Oƒ` z ˆƒ³ƒ Œ xæb + ux+ " x B &( e - #Oƒ =x z ƒ`u = ÆB BX b ª `# } bx+ ƒ~šx }xˆƒ G }x } b ƒ`yxo Mœƒ`O }x + #ƒ`è L u m# } bx z #* 62 ' ÿ : : # 62 + = ² z x z m B & Qƒ} z ƒ} ; SÈGƒ} u z G z ¾B &( o z mšx+ xo z z# ƒ`# }x ~ B J o z u B L u m bx Œ x S B # ` z u J yx ª `Èyx z bx+ ƒ Šx# ~B Yu Šx `ƒ} 0B+ S # 612 ƒ`u y z H = ; bx " &~ y ª `B =x x+ Y ª bx" }x0, - z m zm" z ˆƒ³ƒ z e z y z x+ ¾# e xo# z oè# yx z z }xƒb.%/% z m zm" x +z L m z ƒ(ƒ¼žy ` ƒ Qƒ} z u {z # }x ~ˆƒ zoœ x> œ?+ ux+ "0# 612 yƒ z ƒ`u y z 7 b Q- ; =x " ¹
5 ~ y ª `B =xe x ª bx0 }x * z u mb z}" ex u =BÆB % % z u m z O =xo y }x + B& z Oƒ( ƒ ux+ G ux xo# z ˆƒ³ yƒÿ V V "»B 7 z# ƒ» }x ~ L u m X Qƒ} z z u m -B+ S z ~2 }xex h Q-» ƒ ŒE Gƒ z L m# } =x " # " z = m ƒ(ƒ} BÆe z o }xo z x~^ =x Œ ² x+ #u " Y O z + =x+ ; x~ Qƒ} z u z K m z x G O bx# }xh b Q- ; =x " X =xƒ} H &~ y ;»B =xe ux+ ux xo# z #È # ' yx z X }x z u mb zm" ux+ z ` yƒ "z x m }x B % % z m z» O bx# }x z z Oƒ( ƒa V V " z Oƒ(ƒ` z x ux# z oè Wyx z x+ G z y yƒ S Œ > z z ƒ(ƒ z " h SG ` yƒ yx^ o z mšx+ xo z u &( - #ˆƒ? B yx - S m YyÂu Èu + x- z Oƒ( `o z ƒ(ƒ ~; ~m x+ ^~ y ª ` S# } bx+ ƒ x G yƒ(» zoœ0 x0e BX+ ux+ " b y O ª }x ~ m ƒ` " + B B &'& x B 0( - ooƒ z ƒ(ƒ} u =xƒ} Y =x Ḿ ƒ`o }x + #ƒ`>b L u m bx Œ V ƒ» = z ƒe - #Oƒ z 3 O Œ8 Ú " «ƒ`u b z o V² 8 ¹ z# z }x ~ Œ1 x- G J z ` SB O bx V G yƒqe - #Oƒ zz oè x #. }x# + ux ~ BO `ƒ»ˆ x # }~b# + S ẑ Œ. O Œ àO =xo y x z X ª z +zbz } =xy u yxy o z mšx+ xo z o }x `h ƒe - #Oƒ z `B ƒ(ƒ} z #È # Mœƒ`O }x + #ƒ`b L u m } =x " (ƒ` ˆ zoœ ˆ ; ux+ Y= O bx# }x z y yƒ # (ƒ z bx ±~yx ƒá y z 8 V. 8 e - #Oƒ ẑ Œ VERTICAL TIME HORIZONTAL TIME Figure 2. Different choices of "time" on the Klein bottle z b m yƒ(ƒ &x mg~š È Š ƒ z }x ~W V ƒbg b S +z + z m Q u m z }xâ Mœƒ»ˆ x + #ƒ` L u m# } bx zoœ² x+ #u " WMœƒ`O }x + #ƒ`æh y 0Æ m " z z ˆƒ³ }x# z u m x~ z # { b m " } Oƒ} " z > È o }x ~m S¾ m mb zz y S zoœ > mb zbz " 7u ƒ3 L m# } ƒ-ƒ ux " z z»èƒ`x x z ux+ mx u x =x»šx# Sƒ` z# { = m u.+ x+ S # ` z}" ux+ ÆG y ' y~2 S #u Y z G ª z# ƒ(ƒ ` z u + z»+ }x# z ƒ} ª }x ~h ƒ} bx ~> 7+ #yxo# (u Œ α β Figure 3. The crosscap } yxx» u ` # z x8+~ Œ?¹o" e u ª }x ~ =x B x m z K z BM x SÈyx yƒp~ G B z `ƒ} x+ m u z# { = m Œ > y O ª }x ~ { SX K # ` z u Qƒ} z u {z # }x ~ e - #Oƒ zm" x # m z B Æ yx ux z + # }x ~ bxæ x+ m m SB x+ B B P Œ u Š " z e >e u " 'Bƒ x+ m ¾G y 'X ª z ƒ = mu Æ» u }x#è ÈB~ B vbg B #yxo# ( + u xo Œ = }x+ }HB z e u z S z y ª } +z bx " bx z ƒ`o ` z È z }~=x z S m B¾KŠ+ y ª } u. b S # } +z + }xo x+ }# } =x zx + =xâ x z + e S x+ 0 m zz y S Œ e z }~=x b m yƒ(ƒ ÂB Æ O `ƒ` ` B +z } =xb ƒ` z}" x+ m O z + =x+ ' È z }~=x z }xæ +Mœƒ`O }x + ƒ`b L u m } =x ˆ ;? = :Œ ˆ V È x }x ~ B Æ G yx z u } " ƒ»ƒèg ª z +zz }x z # y (ƒ G BX y b (ƒ S-ƒ» mx u 8Žm ` V Mœƒ`O }x» ƒ» Œ ˆ x~hb +~ Œ#" By ª } +z Q u mg y BK Èu
6 V ŽO # y yƒ #" - ³ƒ` B z xy mbg y ÂÆ Èu HŽO # =xo yƒ Œ S B u -  { SX - > x z V } =x " ux+ 0 z u ˆ =x+ 0 # ± Sȃ(» # " ²µz + yƒ³ƒ #yx #0" O ` ` z BÈ + ~2 S# } bx" u B z K m yẍ >+ y ` «m zz y S ẑ ŒH x z ƒ Æ O V 8 og ux+ " }xæ z u. z z = } zm" O Q `yxo z }xb x z z Mœƒ`O }x + #ƒ`µ Sȃ(» # K1» z `# } Œ z z = m ƒ(ƒ} ux#u moƒ} È e y ª } +z ƒ` Œ +z}" V }x z x+ m " V # z myx+ xo z 5 7yx ª `Šx z } =x ƒ &( - ooƒ " # V 612 V # 612 V #  x " z xo# Q u " ƒ(ƒ¼ O Q `yxo }x+ # H+ z» } ± ( O u B B8x+ S yƒ+ z z z + b m # `È + S = m z}" Œe² z ƒ Q S z h O Q `yxo z + y h x- } - z y ƒ } =x F Š u ' - ƒ³ ¼.B8x G } =x z V ± bx + }x# x+ # } bx z 3 # `h S Gˆƒ z }x B =xo e mb zz y S Œ O ` = yƒ z S µ # X }xu B G ƒ S~O yxo - X mb zz y S ˆ =x z y }x# Y= YO ; = q" ux O }~yx- b ƒ S } =x O zz }x ~ b yƒ»šx+ m u J ª ( Ê Šx# z Š x~ z V ¾Be Sȃ( ` #h ~ Œe¹#" x W ƒ(ƒ} z V ƒ(ƒ 8 e - #Oƒ z x z # ` ẑ Œ ² x >yx ª `Šx z } =x ƒe - #Oƒ z}" B Mœƒ`O }x + #ƒ`0 L u }# } =x z # ux+ O z» bx+ ª }x ~ b = + x x OƒÁ Sȃ( ` # z # ¾ u ' 0h S # ( # X }xu  0B (ƒ( S'B ` z u x+ m# } bx zoœe m ƒ(ƒ + u z u ¾ Y }x b = + x x ` # z ẑ Œ  # z z È = ;» } =x+ yƒ8 O Èyxo z8z }x+ } " z z + yƒ³ƒ z " Š ¾ u > È z 8 3 yx {z }x ~&½?+ ux :¾ S bxy~b ẑ Œ y # O ` # }~u x x+ S yƒ z m `# } =xè #u yƒ `Be ª } ~yx#p u X ˆ =xo # ` # } =x z B B b = + ux xoƒ " - P b = ` # z S>O zz }xâ z u >e - ƒ( O `X ƒ`m O u ẑ Œ z z m# ` bx Á b ƒ(ƒ} u z 8 O bx yx-»šx#? ˆ È+ S z =x> Jyx O =xo» # } =x z [B z# # { = } z B ƒ`x = } # X x B z u } u KŠx :z x~ e - #Oƒ zm" Mœƒ`O }xæ+ #ƒ` " xx ƒ +z x+ ' - z z # ` " x+ G z } b ƒ`yxo #u yƒ( x~g ` b #+ ƒ` z ux+ ª } ~yx+ m z8 #ˆ :Œ " ¼ žÿ G I"*G u QG «P " G# «Ú X È - z u xo z o }xè ˆ =x z m# } =xh+ z 8 ` xš " Ž} z u #"-z u } Œ? z x- ; ƒ} z yx z # }x ~ z}" x+ m xy B L u m } =x }xu m = ; x~b e :+ yƒ» x x ƒ +z Sȃ( ` # +z O bx## ` bx Œ Ba Bb Ba Ba Figure. Projecion in the open sector
7 V z ƒ z 1# z # { b m z 8 b x- z }x ~ ƒ` + S = m ux+ Æ +z O =xo ` >B z SÈ # 5 S# } bxæ y B > " ª ( Ê Šx#ƒ e E +Mœƒ`O }x'+ ƒ` " + y z u m ˆƒ È» # ` z x bx¾+ x+ S x+ G =x } zbz Œ? + # ` z h y K¾ m G z # y xh- È Šx+ z u yx z # }x ~ zm" z ` u BO `œ½? x {¾ SB =x z À = :Œ? O V " ` z ƒ O È8 z x z# # z S z = ª ` bx+ ƒm O =xo» # } =x z S7+ uƒ -x o ƒ z ¼ #y~ ¾ e ux+ G =x8 }xhb¾ ƒ`# `ƒ( Q `# ` z ÊB b S } +z + SB~ z + = } zoœá z m u z + uƒ -x o ƒ z záz# L u }B z bx ~ O =x z }xo zm" z }x+ }± `«z X ˆ È+ S»-ƒ» `ÆB { = mb S# } bxy ª ` # ẑ Œ Ba Bb VERTICAL TIME Ba Bb HORIZONTAL TIME Figure 5. Different choices of "time" on the annulus y u "2z }x+ m8 XŽˆ =xo ƒ # `È«}x>B ux x ƒ ` # O ` ` z b = Q x xˆƒÿ Y Qƒ} z u z u m vb + z Jyx + `G u ƒ` zv+~ Œ " BY O Q `yxo z }x V z z V» z `# } ; `# h O bx z }xo z ` z K m µ K + z z u Á b m zm" }x z»x (ƒ S { z } =x' JX+ y z yx V #Œ8 V z e ˆ»Šx# yx #ƒ} & }x# u & z±z S Æ - ƒ³ 38x h ƒ( S bx z? bx + }x# x+ }# } =x z =xæb ª z#㜠² x ³ƒ ª }x ~ G ux x ƒ z Sȃ( ` o " ` z +z O ƒ B 'KŠ~ }x^ Be = b + ux xoƒ Œ z O z + =x+ z ÈhŽO # =xo yƒ.# `È x& ~ Œ x & = Š O È - S z + ~2 S bx u J Gz u m " z - &x u x+ # O bx# uƒ V ~u Šx e - #Oƒ Œ> x z B z { b mu & ` mx u x z S Æ - ƒ³ ¼ ˆ u m bxy O Q `yxo z}" ux+ }x z µ y z zm" }x+ Qƒ ª }x ~ z»èƒ` =x #u ƒ( }x ~»B " B m z ƒ(ƒ} e # X x u ƒ z oè ŒP + z 7 ƒ`± zz `ȃ`ƒ Œ Šx& X ƒ z u z u m # z m# `u ' h z V ª ~= bx+ ƒ3+ S# `# } =x x+ }# } =xy u B¾# o / T T T e b m z T B+ S¾h u _ xy b b Q+ ux xoƒá µ x x ƒ +z S z Ú {ŒY z O z + =x+ z ' }xo ` e ou z S u }x ~ "Áz y " 'B }~=o± z ' x "P + =x u m bx " }x#b œ ` z O =x L 8 }x z yx z Ê ƒ z u :z # }x ~ B L u m bxÿ Œ² B + x SB z B z u } ~ } yxyo yx+ #» z #È " x bx # } ; yƒ u }# } =xg z + zz `ƒ`x bxƒ} ( 5 O bxl Syx z K ƒ z X O bxl `B> z u m. ¾ œ~u Šx z #È Œ >ƒ³ƒêb z z z z `ȃ` >B Q S B O # }x ~ ÈO Sȃ` zoœ +z}" }x B 0(G y z ' G z z }x ~ ƒ` z O z u } " ˆ z + bx+ ª }x ~>B " x " P ƒb u ƒ(ƒx G yƒ( S# } bx z Q z Šx V œƒ S ˆ =x- yx `yx+ m ŒW xæb œ+ ux+ " }x 7 e - #Oƒ =x-ƒ ux+ S7 ` u G `Bˆ» ˆ =xl ~2 S z - ȾB L u m bx 5 Œ " z }x+ }7+ ' z Oƒ( O =x L }xw 8 " x H O V V 617 " P = Â
8 V V T T 6 }x+ yƒ³ƒ} " xâb& &(e - #Oƒ«ƒ(ƒ = } z S>+ y»u ^ `ˆ»e ˆ =xl ~2 S z B Qƒ} z u {z # }x ~ B L u m } =x " ux+ HB x x ƒ +z = b Q+ ux xoƒá ˆ =xo }x z V u m } =xm O Q `yxo z V 612 P = Xx }x+ oyx+ #yxo 8 O Q `yxo z z G G `+ ux#µ S }x Be O bx z m } =x z }x+ } " z z + ƒ(ƒ z " ` o o }x z Bx K ` + SB~ z m z }xb - ooƒ zoœ. y u "bz }x+ } +zz # ` G y BK z yẍ z `B^ G uƒ» x+ m zz Š zm" X b b - z Q x xˆƒ z ƒ z # X x u > Ⱦ ` # z # V x V Œ? +zm" }xm - ooƒ zm" V W0102 V 617 P;01012 P V 612 P ' 62 8 = ² xæb z ¾O zz bx z 8 u yƒ(ƒá O Q `yx# z S~Š È Èu x z u B z ± }x # V ux+ V "»BY ux = ª `# } =x+ yƒ { = mb u J# > S¾ u m z e O # z u J +z = b Q x xˆƒ Œ x+ m e " z m h y 7 L +z (u ƒ- X O ƒ+ ux+ ƒ} z z u [ u z# " x+ e # y z ` z e } } =x ¾ z m `# } =xm.o # }x ~e ƒ(ƒ¼ Sȃ( ` # z B BO ` - ƒ`µ O u ẑ Œ x G y x u Bx V Sȃ( ` # z }xo m ª ` m Q x xˆƒ KŠx :z # ` bxw x+ m# } bx z +z x~0 h S ( 1'y ; +zez u m } =x ŒÂ xâb X+ x " z `o (ƒ } =x V È - ` # z ` z +z X G ª ( Ê Šx# h S# ( " "Oƒ S Æ >B È ± (ƒ( S h S m z x / / / / = y¹ S B+ xè # V X x~ƒ z x z V h S } =x z ª ` u }#ƒ} ";² ƒ oƒ( B 7 x # (ƒ} ª ` m + ux xoƒ " x # O -»-»Á xµ x z S V «½? x {¾ SB =xe + SB~ zbz }~=x yxo ẑ Œ? z m #y ^» z # } b S# } bx y ª } +z ½ S Ú= :Œ >«`X }x ~ S? e ª ` u }? # } =xe N #ƒ( }x+ # V ƒ Ú=¹ " + = h O =x z # B ux x ƒ ` # ` u ƒ+ h O =x+ ;» } =x z }xh &~yx h ª ~= bx+ ƒ S } =x+ yƒ O =x V h ƒˆƒ 0By B " B ; ª }x ~ &x u Oƒ? } =x 1 +z bx ƒ` O Q `yxo z y ^ o o }x'b> T ƒ ˆ =xo yxo» O z + =x+ ª }x ~ + x SB O bx+ ª `# } =x z x ŒY G z } =x 1 O Œ u z " ~u ŠxåBG bx =x & ˆ z + bx+ #yx+ } Šx ƒ ux+ + G z u } z ÊB z S bx+ ƒêe - #Oƒ zm" B x z S h y 7 x }x#b bx± V x x x ƒ ` # `B yxy½?+ x bx z oè B " x+ SÈOƒ} T = O T > m ƒ(ƒ} " N ƒ³ }x+ o V ƒ Æh z B z O zbz } =x + S# #ƒ} ƒ( }x ~ "z x mg» `ȃ( ` z S V T x å O V Á O Q `yx# z }x S K # u m z S zm" z og ux+ #u z m B z O bx## ` bx Œ + y 8 #Š Oƒ} u ± V ; ƒ`b ; J - }u 7B e O =x z m B8 + SB~ zbz }~=xèyxo z }x ~Šx yƒê - #ˆƒ ź ˆ = Y= q" x+ }x+ #u }x> x =x ª ~b =x+ yƒ e - #Oƒ z Œ O - z x Sƒ} " "#&% '(*,+-%."/#01 *02#3 /657#&% *#8%+-#*/63. :;/6<2 "<2/=2 "/ /6?>@#+-202#(@#207#*/ACBD "6#E02#/. +,-3 "/5* <7#. :>@#GFH "<720+-I6#FA&JK LMA
9 V 3 V ~bhb z»èƒ`¾ - ooƒ z O bx z #u }x> z ƒ h y 8 ˆ =x z m u }x y e ª ` u }#ƒ} z# ~=~ z - z z B } =x Œ O V ª z ƒ y ; x~b ˆ z + bx+ ª }x ~e Sȃ( ` o zm" ƒ`.è«}xo - m8 ux B " u " O =x m Œœ - z + z G `ŽŠ O ȃ`O ½?+ x bx^ + SB~ z}" x z ~yx } =x o ƒ bx ~ V x u x + x+ S # ` z}" B+ SG }xe # `yxo 0 KŠx z # }x ~ z { " }xl m y z " }x `yx#u z m z u yx z u m ª Š S B u z B+ S b y O x# V BO ` # `yxo S } =x ŒX u x y yƒ³ƒ} " Žy ˆ ȃ`O» # ` z m + SB~ z = yƒ u }x& ƒ z yxo S } =x z u x &~ zm" u z Ž} u yƒ + x+ S # ` z Š S B Y + SB~ z b ƒ W }xw x SÈyxo ƒ3 z Šx# S# } bx z u z oèƒ`u m# h B =~b =x+ yƒ8~b zy b :Œ ² x B z He - #Oƒ z}" B z u ƒ( S `# z yxo z ` z Oƒ( yxy e Wx+ y } x z z Q+ ux ` # ƒ ª } + z `# } ª ` " z (ƒ(ƒ z yx8 xµ z `ƒ`jo Œ œ z # m# } bx z z B O =x z z Šx+ O h S " + z y ƒ(ƒ} " x # y ƒ` O bx z y xo z V J+ ` z Á O bxl 8 zoœ Ž ½? -ƒ»ˆ + SB~ z}" u y " ƒ z X yx z È8 Q S b m z Sx z Oƒ( ˆ =xl ~2 S u :Œ P Bx }x ~ O ƒ(» V ` # z}" ƒ» È - }u h O =x z # x~8 o z mšx ux# z u B # V²² Æe - #Oƒ ŒX z h S# o V² e - #Oƒ zm" ux+ `8+ z ƒ} bx ~'Kyx x x S8 =xƒ} =x X.B z ux h ƒ} & YyÂu :Œ² x& # B z 8 u 7 z # m bx z G ux ( z z Oƒ} zm" ƒ` ȱ ` ƒˆ ƒ( Q `#ƒ} m ` # z " V '8 '8 = yâ e u yƒ³ƒ z }~bx^ J +z z z O =x- yxo# } =x z8z ~ Œ > + z `# } ƒ & `ƒ} ẍ b~= bx+ ƒ ~; ~e~ " ~; ~È+ z bx zm" (ƒ`g Gxy~2 S ƒ & `-ƒ z oèƒ`u m#?~; ~~ " ` 3 3º 3 ~; ~J+ z bx zm" ux+ 7B. K? Q m z # o }xu - = #+ ƒ` O =x ª `# } =x Œ u Q z J } =xè yx b #+ ƒ` ˆ =x+ ª ` bx z x+ X ux h ƒ} e y ux+ moƒ(ƒ S } =x z z z.+ }xo u V. ok ²Áz# z # }x ~± }x>o Œ Yy¹u B x h ƒ( ` z S«ƒ( }x u = o» uƒ` z ˆŒ² xe z # }x ~8 `¼ zz `-ƒ»1 ¾ = ẽz# z S zm" ux+ «z =x }x z e - #Oƒ z `8 O z + =x z >B z u m x B b b Q+ ux xoƒ " ƒ ` h zz ƒ` zz % % z y ƒ SJ L u m o u [ Qƒ} z u z u m E Mœƒ`O }x + ƒ`± u Œ Œ # }x ~ e» z yƒ? =x + }xo x+ m ~Šx +z =x + yƒ³ " # o }xu ' ' zoœ  x+ yâ " ; `Oƒ z * b > S# } bx V " B+ S ˆ =x z m# } =x^ z ~ S xo m+ y m O u }x+ Q #yxoƒb ; z}" Y ~ ŒÂ 6 B z ~u }x+ S B #0" V H x+ V Œ > z xo# Q u " B x z ƒ } =x " ¹#" ƒ`u = z B G x- x h ƒ} 8 ¹ e - #Oƒ " ±ƒ} =x~ V x u x z# ƒ` yx x+ - e YyÂu :Œ n Figure 6. Tadpole condition for the Type-I superstring ( n 2 n Xx x >B &'Æ - ooƒ zm" y X x Mœƒ»ˆ x + #ƒ`' L u m# } bẍ x+ S +z z `ƒ` #u ƒ `B Œœ² x' z Š z µ ª ` u } Q+ ux xoƒá ux x ƒ +z + z h z m zm" x
10 V =x + z B O =x z z Šx# z u? zz ~bxèyx# z 617 W617 V " W617 V S # X }xƒbx O Q `yxo z x Œ Œ8 z8z `-ƒ»h Qƒ zbz J - ooƒ z z }xo z }x ~ "z }x+ }» O ` ` z x y Šx Èyx bx " ½?+ x bx z oè B u }x ~ Œ Figure 7. Chan-Paton symmetry breaking O }x ~ÆB ~ Œ& bxg Š x Su Q S> S ƒ(ƒ} u x~ "z y " " z BÈO Êu m u # ; }x ~ ±½?+ ux {¾ SB =xm Q S~h S # m z u [ œ~2 ~ u mb z (ƒ»1 u ; x~ƒ O ` z B 8xy z u mb z B z u m Œ y u " B z z ƒ(ƒ¾h x- ³ z #ƒ} O È+ S `ƒ` ` { b m } =x Œ +zm" V }x z x+ } " ( x+ S+» z `# } È e S ~2 ~+ z =x z ˆ z + bx+ ª }x ~ÆB ÿ }x = L }x# z yxo S } =x& u 8 8 " = ; bx z O z + =x+ ª }x ~ ÿ }xæb z #È # yx z? x z y ƒ S5 z Šx# S# } bx z ÊB " x L x+ «X bx z }x z yxo S } =x Œ zgz u m \ z h x ( z #ƒ} x =x Q-» ƒ " + }x BY Qƒ} z u z u m L u } u = Š O ª }x ~> Œ ux+ }x' ± yx z u mb Œ ½? z + bx+ ª }x ~ ƒ} " B =x-ƒ 'h zbz ƒ` zz b #+ ƒ`± }xb b = Q+ ux xoƒ ˆ z + bx+ z B B z u m " W z y yƒ bx " ux+ # o }x z B ; =xè b #+ ƒ` S~yx +z = #+ ƒ` O bx+ ª `# } =x " B+ S ƒ X >B ƒ z u [ : =~= bx+ ƒv ~; ~ ~ åb ¹#" z Oƒ Su m~; ~m x ~ ; } =x+ yƒ8 ux h ƒ( ` z z }xâ # V²² Š z Œ? " B B ƒ`8 5 z 8 = ª `# } =x+ yƒ = #+ uƒ»µ O =x ª `# } =x z z x x+ o z ; - HB X u x z S# z { = m bx Œ ƒè x u - x0b ÆBQ &( e - #Oƒ z}" y~u xx }x ~ `BåB +z# yƒ x z S " }xâ z y z = z 612 " " W612 V 612 W = u =B& KŠxH # z myx+ xo z S mx =x ` yƒ }x O `G Qƒ} z u z u m Q ` ƒ x ˆ» yx z u mb 8 ux+ +zm" x z# # z x~ ƒ} " È = o» uƒ` O =x ª `# } =x z z e ƒ z B Œ  Oƒ S
11 H Ž x - z y ƒ ^z y ƒ S z O z + =x+ ; x~0 ux+ Œ Æ z# ƒ` x~ O =x ª `# } =x z}" ux+ " ; z + z e «ƒ(ƒ?~2 ~e x G yƒ³ ` ẑ ŒX b #+ ƒ`o z u~; y x - z Š ƒ? bx " O z + =x+ ; x~ B ª (ƒ S bx Œ }x ~».B B ƒ µ Âu B yƒ z ÊB~; ~~B Œ ² x z Qƒ zz 1e - #Oƒ z h S # Æe G } =x z V Mœƒ`O }x + ƒ`æb L u m bx " x +z =x G y 8 x- z # B O z + =x+ ª }x ~X yx z m zoœ²áz ƒ(ƒ ª z +zz }x z # (ƒa =xƒ} È e - #Oƒ z O z» bx+ ª }x ~' # + J Œ "Pz x mƒby m S ± }xo z }x ~ Œ² xy~yx ƒ " Šx& =x e z B bx B L u m bx> ux B " O ȃ`O h + SB~ z Œ xu V J z z u z ³ƒ} z yx - x # Q }x }x' z y z bxƒ} y ux u }xhb +z Sȃ( ` # Œ z + z x `È+ xo O bx z yx m " ' ±~2 ~X u mb z O z + =x+ ÆB x-» SB ~ zm" x+ ' +z Š S B >+ y» z M O =xl ~; SX + SB~ ẑ Œ 8 ; z ƒ y ` # z}"ÿz x m¾ z ³ƒ(ƒ.h 1X z Qƒ`u S V V " ; ; ; = ² z ƒ Q Su e O Q `yx# z u x+ W +z + B b ux z "z }x+ }O ` z }~=x z xu } zz S # (ƒ} ^ }x+ O u } :> z ux# Q `+ Su " u Jy " B z O bx+ ª `# } =x z ;»u mƒ»èƒ( `u - 0ˆ» } =xâb Æ e ª `yx z } =x z u Bo x+ SÈyxo yƒ z } =x z x ~B ẑ Œ ½? È+ S }x ~ ` Œ  z u z SƒBo z x = ª ` bx+ ƒ1 b #+ ƒ`y ˆ =x+ ª ` bx^ V &Ž x - z y ƒ z y e - #«}xè z u m ŒÁ z ;»ˆƒ z Oƒ S# } bx> yx ;»Èyx z } =x z [B x ` SB ~ zm" Âu # z# Q `yxo¾ 'yx z Bh y x+ } =x u ƒ(ƒ~; ~h x ~ ; } =x+ yƒ ux h ƒ( ` zm" z h y Æ z yx h x~ z u. ƒ z# ƒ` z J O Œ Ú = {Œ² xb z Š z =xµ z x + y bx z 8 ª (ƒ S bxæ = #+ ƒ` " ux+ B z œ~2 ~±~ G z O V x # o }xu Œ # } bx u ˆƒ y }x ~ 8 ; =x = #+ uƒ»e ˆ =x+ ª ` bx " O =x z # }x' O Œ Ú " h y ƒ`u = > X }x# z # }x ~ =~ zbz }x yx {z # }x ~ z # }x ~e - #OƒÊ (ƒ ª }x ~ Œ œƒ z Qƒ zz ` yxm # z myx+ xo z G y O =x z m o }x z `o (ƒ S { z } =x " x+ o z ƒ`o z x>ˆ Q z V «}xo z u u = o Œ, y (ƒ z u O bx z m } =x ` yxh # z myx+ ux# z V Qƒ zbz z ` O bx V G yƒê - #ˆƒ z}" x Qƒ ª }x ~ƒb { = } S bx>b ` z M ª z# Sȃ( ` # z V X }x `G yƒ? ux+ 8 e - #Oƒ zm" G y > V x+ }xæo zoœ àYO = Yu :Œ u J ª `Šx z } =x ƒ¾e - #Oƒ z G y 0Y ˆ =x z m u ^ }xh z `o (ƒ SG { z } =x Œ0² x z ( ;»Èyx z } =x &ƒ SB~ Qƒ zz z u e ` yƒ z# z oè# Æ - ooƒ z `B0 }xo z x~"~2 ~ ~ z B+ S O ` `Y ^x u Oƒ u S z Š yƒ xo# z oè# Æ Šx z & ; z + z =~ X~2 ~^ ux+ H~ y } =x ƒ x G yƒ³ ` z &~yx yƒ³ yx - S u + ux z " u Š " ` ƒ V ª `yx z } =x+ yƒ¾e - #Oƒ z S x x+ o z ; - åb X u x z { b m# } bx " x S + Syx#ƒ} ƒ`u = 0 x+ ux# b #+ ƒ`e O bx+ ª `# } =x z `X» O `œ~2 ~~B z B G z h ƒ(ƒ z Œ ² ẍ O bx+ Qƒ +z bx "² + y e # z m»u Y x z Èh # y (ƒ Bh O =x z m# } bx^ yx # z myx xo z V z z `ȃ` Qƒ} z u»šx# Sƒ` e - #Oƒ zoœ ; È?~yx ƒ; S z u z - ` SB ƒ`o }~=o z #È # O =x- V h ƒ e - #Oƒ z}" x+ 0yx u B»B bx e m Qƒ zz z u KŠx # z myx+ xo zoœ O ƒ(ƒ} B z O ƒ( Q `X # # } b S# } bx" (ƒ(ƒ Oƒ` }x# z u = # z &~ = y xo ` z X u +z }x ~'B # ` z yx :z # }x ~B - ooƒ z B+ S "Pz e { S " X+ y ƒšxy S-ƒ»È x- O (ƒ Œ
12 Œ Ež #œ«a «¼? ² O ƒ5 B B~; x- z V ˆ» x }x ; } =x " x+ ' zz ` x+ " x ux+ O ¾ b ª z ` x+ zbz yx> xš ± V ¼ xèyxl u - Sƒ` O ƒ(ƒ S+ S =x bxh x- 8 [BM zbz# z ª z +zz u }x z ƒ ÃŒ? z zz#» u ± }x+ S X½ Œ Œ8Œ Œ`" ux+ m " x ƒ Œ Œ ½ Œ xo½ ½ =¹@ ¹
13 ¹ ¹ Œ Œ Œ * 8*,.*7; *76 > Œ " ½ Œ "  ½ z Œ ½ M Œ Œ ", Œ ½ ", Œ, > Œ " Ž ¾ ¾ " z ¾ z Œ Œ " z Œ " z Œ > Œ Œ > Œ > " > Œ Œ > Œ > Œ " ½ Œ z " > Œ Œ Œ z " > Œ Œ Œ Œ " z z ¾ Œ, " z Œ Œ z z Œ Œ ½ ˆ /Œ,«( bx " +Œ S Š ŒDN y { ux+ ` yx ~ l Š i} #" = ²OŒ >œxo =x- b ª zm" Œ+ b Q x W½ M xx+ z}"q~ l Š Žim ²OŒ >œxo =x- b ª zm" Œ+ b Q zm" ~ l Š i} bâ S y }x "Q ^Œ Œ - SG b ª ¼ x+ Y½ ŒDN y { "~ l Š Žim # Œ M8 y y Œ y Oƒ(ƒ`yxm x ŒÚ Œ " ~ l Š i} _Œ Œ +z 8 x Q> Œ - Q Oƒ(ƒ» yx zm" ~ l Š i} # ;Œ u Œ ƒ( } " +Œ ª Q x+ ŒQ ƒ³ } "~ l Š i} " U b¹œ º¹u - u~=x =x :¾ } # ux# `Oƒ y B o" u zoœ Œ b ƒ? B~; Se =x zbzm"[ Œo=Œ b +Œ Œ bxy x Œ ^Œ ½?+ ux "~ l Š i} % = #ŠÂ +Œ Œ - " }x Žu½ yxo ¾ ƒ` }x S Qƒ` y #" ¾B - Œ +Œ }x ½? =x- Œ ƒ} yx+ m "Q +z ux+ Q> - ~bx # " i} a@] ] & "&' -Œ u ^Œ x+ Á ux+ Q> - ~bx # "& i} a@] ] ~ l Š i} ( # #Œ o ºÂu ^Œ x+ Œ+¾ = ª ¼ x Q> - ~bx # "& Žim a@] ] ( # ¹Œ Š ^Œ x+ Œ+¾ = ª ¼ x Q> - ~bx # "[~ lešq Žim ( ¹bÂ-Œ u - u~=x "* + Žim a ] ] =Â-Œ u Œ ˆ»B~G ux+ ` yx "~ l Š + Žim Œ- Oƒ y ; x Œ ^Œ¾ ƒ} - u ux+ Q> Œd Œ, Se ƒ} - u "Ê~ l Š + Žim & u ¹b¹=¹Œ O ` yx "- d B. Z]/ [ Žim NXŒ M x x Q" uƒ - Q u "~ l Š Žim =¹Œ Œ ² z bxm x+ +Œd Œ,+ K "- d_è_b. Z]/ [ Žim &'( =Œ Š¹ +Œ+¾ ƒ }x z# Á x+ ½ y "Q~ l Š + Žim o=œ ˆ = Œ+¾ = ª - u~=x # ¼ x xš " i} a@] ] (*0 Œ Œ+¾ = ª - u~=x # ¼ x xš " i} a@] ] (*0 =¹ ŠÂ ^Œd Œ yxm ux+ +Œ Œ - - S " i} a@] ] &' 2Œ y ^Œ Œ Šx " +Œ Œ - S x+ ½ S # ~21œx } ` zzm"[ ` yx Ž@ }x ~ y Œ } b ux# Œ+¾ = ;? ux+ Q> - ~bx # " i} a@] ] (' ¹ Œ ^Œd Œ yxm ux+ +Œ Œ - - S " i} a@] ] &0' o=œ ^Œ x+ Á ux+ Q> - ~bx # "& i} a@] ] ' =Œ oo ^Œ8Œ,œ ~ ƒ x+ Œ # }x O }x "& i} a@]] &'( ˆ xo~ " Žim a@]] &' ;Œ +Œ Œ ½ S "~ l Š Žim ( #bœ b¹ ŒDN ƒ( x # "~ lešq Žim (%% ¹b u = Œ >«ƒ} 3 x ` Šx "~ l Š i} ( =ÂŒ Œ+ = + z}" # }x#
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