Ketov, Conformal Field Theory.pdf

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% *, *=:8 )R $N 2I$ *= ' ) !) 2 $ :! * 6: " - $ / 2 ! A *931KL ( * * -/ (! : iS3] 3 ` ] 3 .S`O$N 2 NL G 3 ),!RQ / ) NL G 3 ),!RQ / D8N $N 2 $ ( :! * A ! 8 ] `$! ] -) ` A ! $ h_ *JA '' 3>KEZ6: *98 ! <&) 2$ " ' ! 6 * f =( * 8 ! 6 - .A: (' $#&) 2$ C$ "d :' $? * # $ BA * ' 2 6?$ (' N * $6 *- $k #-5 6:O6 A: " $/- .A: (' M31 ( :XW * A ! R d ! " * P ! ! R $< * d : .f O HsN $ * *=C * * x 3j ' *= 6 *) Jv ) i ' A * * * : 2 : 3 -5BA: 8 ' : !' &) T](3 63=8 l$ :' A * * $ : :8Q!Z! .31K <3 :`G8>$ ( :! * *= ! ] -) ` A ! A '' X2 D .C " P *+&) 2$8-/ . R ?$7&) !T :! W d A * 8 2/* . &/N A:*=:3

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. 2 8 ; w w $ ( : $ " GF !! k $ ! $ ! d! ! 8 -/ ' 6mi;3O] :3/*H2`D $ :! * 6: " ] :3/*HF`G3 >i;3O] :3/*H2` " A: $*= !$ @ m$ ( :! 3 ] 3 ?` >@>@> B G 3 ] &B G 3 `] G B`] &B<` / G 9 3 ] 3 ?` >@>@> B G 3 ] &B G 3 ` Bm] &B8` D) ] -. F` -/ 7$ =t * : 2 : ] :` B G 3 ] G / " ] "" + &` B ` ] " &B` G 3 ] -. ` 3 N " ! ' ] :`G3 >i;35] 3 F` " (' $ 6: * x $ :! *, 2 (' !" $3 zk - - !C - $ (' * * ( :! * A ! E- g & ! $ $ WEd : :! *, 2 (' gh_#3daD $ W 8 + W . WEd : (' 8;$B -5 * . @! .3JK)3 3#KE ' *, 8di;3>] :3 .7,:` " <6: * x $ WEd : :! *, 2 (' " ' 6$ ( :! * l !Z / ( * * -/ ] G `] ` 3 ] 3 ?` >@>@> Bm] B` 3 ] " ` G 3 ] #` ] ` 3 ] 3 ?` >@>@> Bm] B` B ] /{] " ` G 3 ] " + ` ` " ] ` G 3 9 ] ` 3 ] 3 ?` >@>@> Bm] B8` D . ] -. H2` " " 3

3 X2 D / X2 D / 2H8M-5N .A:8 ' *, 8 ( : / % / 2q] % *, *= ( : 3 ` B 9 3 ] 3 ?` >@>@> BC] B8` D&/ 2 ) 3 B ] -. ` ] ` 9 3 ] 3 ?` >@>@> BC] B8` D&/ 2 ) 3 B ` 9 3 ] 3 ?` >@>@> BC] B8` D&/ 2 ) ] ) N 6 !*, 2 N *98 3 *, 2) 2 N *J * ( :! * A ! 8 3 6q$ R -/\ ( :! 2 ! : $ :! *, 2 : ' $! ] -) ` ( :! 2 M] ` /y] B`] I 3` G 3G8 B 9 3 }] M] 3 `!?` >@>@> Bm}] M] B<`!8` D&/ J ] I 3` ) 9 3 ] 3 ?` >@>@> Bm] B8` D ) ] -. :` +8

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. $ :! *, 2 (' 9 3 >@>@> B D " C- ! !$j (' " + !] -) ` W A ! 2 @](* R / 3 ) 3 ) ) / " 8 ( : ( :'Cd : $ (` <' *= * k" m$ WL : -/ T ' $# ( :! 2 C*, .- ] 3 :` 9 3 ] 3 `?>@>@> Bm] B`8D&/ J G %1] ` / 1)...) " + . ] -. H+` % i;3] 3 :`G8H$ ! :' * $ 2 k$ *, 2 ] -. .S` / ) / &. KE ' *, 82 V$ W 83+ W . WEd : (' 1 8:$ BA :' ' *=$ 6 A: " miS3] :3 .0`G8I] :3 .7*:` ] :3 . `G8 3=8d ! ' 3 w $ . ! 6 mi;3] 3 -,:`5 7& % B ! 8N $B : D- $ $7 ' ! * : C$ + WEd : (' ! : ! msN* 8 9 3 ) ] ) ` 0D/ 9 3 ] ` ) ] ) ` 2]'28` D / 3 ) : G < G ! : G ! < G ! . ] -. 3,:` KL I$4 $ : # %S$N 2$ w $ ( : $ " 5GF !! m$ ! $ ! ( :! ! 8 G G ! / G G ! ) ]( ' A: ) ` ) ] -. H0` -/ O$ f 1 Q$ w $ ( : *- -1sN* 8: 5] V `I O : ('

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Fig. 5. For the calculation of the integral on the r.h.s. of eq. (2.27).

i ' 2 M8 $ ' * * ' *, 2 " : 5A: $ :' 1 21 6:<* BA:* /](! 3 # 8 + % ( :#% : `G3 0 BA: $* !8$6: *Q&) 2$ !$ l" i;35] 3 H0` " * . ( :' &) 6k T * * ' *, 2 -5* *93 % * = $D ) 2 BA: % : 8$ $ RS 2 * WE $ 'd j$ ) ' !! 3oh $ : 2 :! ' j$Z*93 M3 3 i;3V] 3 -, " ` *=k$ : 2 : P] ` 3 3 ] ` ] ` ]'2` . ] -. 3*:` N] ) ] ` 3 ]'2`!` / I 3 )

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: 2 : ( * * -/8H-/ ' 6$ :! * 6: " H3 z * ' 5$N 21$/h_ :% * B$*=P sN " $ * N 6: 8$ ( :XW *+-5 6:$%] : * :' )` ) $ :! 6:& W -5 6:#&) 2$# $ w $Z] :' * 6:)` " BE-5l$ ! @sN* 3Zk*, 2!$ % sHFH j" ( :! * A ! 8> $BzN .A:C @" B$ !% z(! : N % $ ' *= 6 *I&&$ 3 7 $v %(+ ( K!K W o] `%h $ ( :! * l ! Eb] :3 0 ` n$@* % + &Bk8 ! A:i;3Q] 3 F` (! : .3 # ?GF i;3Q] :3 0 ` d -5 ] " &B`G8+ ' ! iS3] :3/,H0` ] :3/,3*:`G3 K!K W + mh5 )R i;31] 3 F` $*=:8 " :%<' 6 G $P* T$P*93 M3 3 # ) R &B / 2 mi;3] 3 F` Z' !Pi;3V] 3 `G3 K!K W . k R P$P! / p! A:7$N 2$ /* . $ (* qY ) # + N8 " 8;-/ i ' ! 6$ :! *, 2 (' " P 6:* W A * ' 3OKE/$ % *, & ! m$7 * 6 0

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Fig. 6. The crossing (duality) symmetry of a 4-point function.

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G G ! ! G G ! ]'2 ) 2` 2XD .

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Fig. 9. The graphical representation of a general conformal block.

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