Vacuum charge Q N

Transcription

1 Vacuum charge Q N

2 introduction 2 2 free bosonic string old-covariant approach BRST quantization g-loop amplitude 3 3. free bosonic field with vacuum charge Q V N;0 for free bosonic theory V N;g for free bosonic theory A 52 B Cµν, Cµ, 2 C 3, N 56 B. Cµν B.2 Cµ B.3 C B.4 N C Canonical form 62 64

3 introduction bosonic string Tachyon bosonic string bosonic string string string string field theory [9, 0, ] open bosonic string conformal field theory [3, ] string string comformal field theory physical state) vertex operator 2 bosonic string comformal field theory BRST formalism 2

4 (World sheet) [7] g-loop moduli g Dual Resonance Model [2] tree g-loop sewing procedure [8] Vecchia conformal field theory [6] 3 g g-loop N 3

5 2 free bosonic string classical action D Minkowski D x µ (τ) (µ = 0, D) τ τ dτ dx µ = xµ τ dτ dx µ dx µ τ action S = dτ x µ (τ) x µ (τ) (2.) action τ τ D x µ (τ) x µ (τ ) action D (worldsheet) σ, τ X µ (τ, σ) τ σ τ σ X τ X dτ σ dσ 2 Xµ τ dτ X µ σ dσ 2 4

6 * T 2 τ σ action S = T = T dσdτ ( X τ )2 ( X σ )2 + (( X X )( τ σ ))2 dσdτ det( α X β X) (2.2) α β τ σ 0 (2.2) action action S = T dσdτ hh αβ α X β X (2.3) 2 h αβ (σ, τ) world sheet h αβ X h det(hαβ ) det(h αβ ) > 0 (2.3) h αβ h αβ h αβ h αβ h det(h) δ det(h) δh αβ = 2 det(h) δ det(h) δh αβ = 2 h ( det(h)h αβ) = 2 hhαβ (2.4) δs δh αβ = T htαβ 2 T αβ = α X β X 2 h αβh α β α X β X (2.5) T αβ h T αβ = 0 (2.6) (2.5) h αβ = 2 α X β X h α β α X β X (2.7) * D µ 5

7 (2.3) h S = T det( α X β X) dσdτ2 2 h α β α X β X hαβ α X β X = T det( α X β X) (2.8) (2.2) (2.2) (2.3) (2.3) (2.2) local symmetry σ α ˆ σ α (σ) h αβ (σ) ρ(σ)h αβ (σ) world sheet world sheet r (r =, 2 ) (2.3) α β 0 r S = T d r σ hh αβ α X β X 2 action r = 2 T αβ =0 Conformal field theory [] local symmetry h αβ h αβ = ( 0 0 ) (2.9) (2.3),(2.5) S = T 2 = T 2 dσdτη αβ α X β X dσdτ[ ( τ X) 2 + ( σ X) 2 ] T αβ = α X β X 2 η αβη α β α X β X (2.0) 6

8 action (2.0) + coordinate σ + = τ + σ (2.) σ = τ σ (2.3) S = T 2 = T 2 = T 2 dτdσ[ ( X τ )2 + ( X σ )2 ] dσ + dσ [ 2( X X )( σ + σ )] dσ + dσ [η αβ α X β X] (2.2) η + = η + = σ + = σ + (σ + ) σ = σ (σ ) X X ( σ+ σ + ) +X X dσ + dσ ( σ + σ + ) dσ + dσ ((2.2)) ((2.3)) action (2.3) ((2.3)) ( σ + σ ) = e + S = T 2 ( d 2 σ) ( h) (h αβ ) ( α X β X) = T (e d 2 σ)(e h)(e h αβ )(e α X β X) 2 = T d 2 σ(e h)(e h αβ ) α X β X 2 7

9 ((2.3)) 2..2 action (2.0) X X (τ, σ = 0) = (τ, σ = π) (2.3) σ σ X(τ, σ = 0) = X(τ, σ = 2π) (2.4) (2.0) [X σ X] 0 2π [X τ X] τ f τ i D 2 X µ τ 2 2 X µ σ 2 = 0 (2.5) X µ (τ, σ) = x µ + p µ τ + i n αµ n e inτ cos nσ (2.6) n 0 X µ (τ, σ) = 2 xµ + 2 pµ (τ σ) + i 2 n αµ n e 2in(τ σ) n xµ + 2 pµ (τ + σ) + i α n µ e 2in(τ+σ) 2 n n 0 X R + X L (2.7) 8

10 α n α n 2.2 (2.0) X µ lightcone gauge quantization Lorentz covariance QED old covariant quantization QED Gupta-Bleuler BRST quantization FP ghost BRST charge 2.3 old-covariant approach X µ T = π (2.0) P µ (τ, σ) = δs δ X X = µ (τ, σ) π (2.8) 9

11 [X µ (τ, σ), X ν (τ, σ )] = 0 [X µ (τ, σ), P ν (τ, σ )] = iη µν δ(σ σ ) (2.9) (2.6) (2.9) [x µ, p ν = iη µν [ µ α n, α ν ] m = iη µν δ m+n (2.20) X Conformal field theory free String D X µ (τ, σ) τ, σ (τ, σ) X L X R (2.6) X µ (τ, σ) = 2 xµ + 2 pµ (τ + σ) + i 2 n xµ + 2 pµ (τ σ) + i 2 n 0 n αµ n e in(τ+σ) n αµ n e in(τ σ) X(σ + ) + X(σ ) (2.2) action ((2.2)) τ iτ e τ+iσ = e ρ z e τ iσ = e ρ z (2.22) ((2.2)) S = 4π T zz T(z) dzd z z X z X = 2 z X z X T z z T( z) = 2 z X z X (2.23) 0

12 X ((2.2)) X(z, z) = X(z) + X( z) = x µ ip µ log z + i n αµ n z n n 0 + x µ ip µ log z + i n αµ n z n (2.24) (2.23) conformal invariance * 2 action σ+ f (σ + ), σ f (σ ) f (σ + ) = e (τ+iσ) Energy momentum Tensor T T well difined ((2.6)) n 0 (2.25) state physical state T(z) T(z) = T ψ> = 0 (2.25) + n= L n z 2 z n (2.26) L n (2.23) L n = T(z)z n+ z=0 = z X z Xz n+ (2.27) 2 z=0 (2.27) (2.24) L n = 2 m= α n m α m (2.28) α 0 p L n well difined n 0 L 0 * 2 (2.23) z ρ (2.23)

13 L 0 = 2 m= α m α m (2.29) α m normal order α m (m > 0) 2 m= α m α m = α m α m + α m α m + α 0 α m=<0 m=>0 = α m α m + 2 p2 + m m= L 0 L 0 normal ordered product physical condition (2.25) L n m= L n ψ> = 0 (2.30) Gupta-Bleuler L 0 L n ψ> = 0 (n > 0) (2.3) (L n a) ψ> = 0 (2.32) a Operator X(z)X(z ) z = e τ+iσ t > t z > z T Radial ordered product X z > z X(z; +) ip log z + i X(z; ) x i n= n= n α nz n n α nz n (2.33) R(X(z)X(z )) = [X(z; +) + X(z; )][X(z ; +) + X(z ; )] = X(z; +)X(z ; +) + X(z; )X(z ; +) + X(z; )X(z ; ) + X(z; +)X(z ; ) (2.34) 2

14 (2.34) normal ordering X(z; +)X(z ; ) = [ ip log z + i n= n α nz n ][x + i = X(z ; )X(z; +) log z + (2.34) z > z n= m= n ( z n ) z m α nz m ] = X(z ; )X(z; +) log z log( z z ) = X(z ; )X(z; +) log(z z ) (2.35) R(X(z)X(z )) = log(z z )+ : X(z)X(z ) : (2.36) z > z log(z z) Wick (2.36) X(z) X(z ) Wick contraction X(z) X(z ) normal ordered z = z X Wick contraction X R R( f (X)g(X)) α (2.20) (2.36) α n L n [α n, α m ] = i 2 = i 2 = i 2 z=0 z=0 z=0 z =0 z =0 z =0 α n = i dzz n z X(z) (2.37) z=0 dzdz z n z m [ z X(z), z X(z )] dzdz z n z m z X(z) z X(z ) z=0 dzdz z n z m R( z X(z) z X(z )) z=0 z =0 (2.38) dzdz z n z m z X(z ) z X(z) (2.39) dzdz z n z m R( z X(z) z X(z )) z z z > z z > z R dz dz f (z z ) = dz dz f (z z ) + dz dz f (z z ) z>z z >z z 0 3 z =0

15 f (z z ) z = z pole [α n, α m ] = i 2 dz dzz z =0 z=z n z m R( z X(z) z X(z )) = i 2 dz dzz z =0 z=z n z m z z[ log(z z ) + (regular at z = z )] = dz nz n z =0 = nδ n+m,0 (2.20) energy momentum tensor T(z) X(z) T αβ (2.3) h αβ generator T αβ comformal gauge σ + = f (σ + ) σ = g(σ ) T(z) [ dz ϵ(z )T(z ), X(z)] = ϵ(z) z X(z) (2.40) ϵ(z) z = 0 (2.38) (2.40) (2.23) z dz R(T(z )X(z))ϵ(z ) R(T(z )X(z)) = 2 z X z z( log(z z)) 2 = z X z z (2.4) T normal ordered T contraction [ dz ϵ(z )T(z ), X(z)] = = z dz z X(z ) z z ϵ(z ) dz z X(z) z z z ϵ(z) = ϵ(z) z X(z) (2.42) ϵ X z z z = z ϵ(z ) = ϵ(0) + ϵ (0)z + + ϵ (n) (0) n! z n z n 4

16 [ dz z n+ T(z ), X(z)] = [L n, X(z)] = z n+ z X(z) L n δz = z n+ generator [L n, L m ] α (2.27) [L n, L m ] = dz dz R(T(z)T(z )) (2.43) z 0 T contraction 2 contraction contraction 2 2 z X z X = 2 (z z ) 4 (2.44) 4 4 z X z X : z X z X : = : z X z X : (z z ) 2 = : ( z X + 2 z X(z z ) + ) z X : (z z ) 2 = : z X z X : (z z ) 2 : 2 z z X : (z z ) = 2T(z ) (z z ) 2 + z T(z ) (z z ) (2.45) R(T(z)T(z )) = D 2 (z z ) 4 + 2T(z ) (z z ) 2 + z T(z ) (z z ) + (regular atz = z ) (2.46) comformal ϕ(z) R(T(z)ϕ(z )) = + hϕ(z ) (z z ) 2 + z ϕ(z ) (z z + (reg) (2.47) ) ϕ comformal weight h z = z X(z) c.w 0,T(z) c.w 2 (2.4) 5

17 ,(2.46) (2.42) [ dz ϵ(z )T(z ), ϕ(z)] = dz ϵ(z )R(T(z )ϕ(z)) z = z + ϵ(z) z = ϵ(z) ϕ(z) + h ϵ(z)ϕ(z) (2.48) ϕ(z) ( z z )h ϕ(z ) (2.49) comfomal dimension h naive conformal gauge h action conformal gauge confomal z = z + ϵ(z) h (2.49) f (z, z) z z z z conformal T L n (2.43) [L n, L m ] = (n m)l n+m + D(m3 m) δ m+n,0 (2.50) 2 Virasoro Lie string L n conformal generator (2.50) anomaly α n poisson braket T normal ordering anomaly (2.46) comfomal (2.50) anomaly α (2.50) 6

18 2.3. free string spectrum vertex operator free spectrum Hamiltonian H = dσ : ( xp L) : = dσ : ( x 2 + X 2 ) : 2 = : α n α n : 2 = 2 p2 + α n α n = L 0 (2.5) (2.32) L 0 ( 2 p2 + α n α n a) ψ> = 0 (2.52) p 2 = M 2 p 2 mass operator L 0 on shell M 2 ψ> = (2a α n α n ) ψ> (2.53) 0; 0 > α n 0; 0 > = 0 (n > 0) p 0; 0 > = 0 p 0; k> = e ik x 0; 0 > Hamiltonian α n (n < 0) Fock (α µ i )ni (α µ j 2 )n j p; 0> (2.54) physical state L 0 a = 0 state L n = 0 7

19 (2.54) (2.54) QED D metric 0 α n 0; 0 0 > L n = 0 Physical State α 0; k µ > physical state condition (L 0 a) ψ> = 0 L 0 2 p2 + α n α n L n ψ> = 0 (n > 0) L n α n m α m (2.55) 2 m= k 2 = 2a 0; p µ > physical state 0; p µ > (k 2 = 2a) ζ α 0; k> state L 0 L physical state condition k 2 = 2a 2 k ζ = 0 (2.56) a = ζ polarization vector massless ζ ζ ζ ζ k ζ k k 2 k 2 k 2 > 0 (a > ) k µ = (0, h, 0, 0 ) k 2 = 0 (a = ) k µ = (h, h, 0, 0 ) k 2 < 0 (a < ) k µ = (h, 0, 0, 0 ) k ζ D a > nagative norm,positive norm (ζ = (, 0, )) a = zero norm,positive norm (ζ = (,, 0, )) 8

20 a < positive norm (ζ = (0,, 0 )) negative norm a a = ζ µ = ck µ a = a = 0; k> k 2 = 2 = M 2 a = superstring m 2 < 0 string m 2 < 0 X(z) m 2 < 0 D massless X µ (τ, α) (conformal field theory) m 2 < 0 Targrt spase (D dim Minkowshki space) mass X(z) mass vertex operator physical state physical state physical state confomal field confomal weight h primary fileld (2.47) R(T(z)ϕ(z )) = hϕ(z ) (z z ) 2 + z ϕ(z ) (z z + (reg) (2.57) ) L n ϕ(z) (2.42) [L n, ϕ(z)] = (n + )z n hϕ(z) + z n+ z ϕ(z) (2.58) 9

21 ϕ(z) physical state ϕ(z) 0; 0 > physical state [L n, ϕ(z)] = 0, n > 0 (2.59) [L 0, ϕ(z)] = ϕ(z) L normal ordered L n 0; 0 >= 0 (2.58) ϕ(0) [L n, ϕ(0)] = 0, n > 0 (2.60) [L 0, ϕ(0)] = hϕ(0) comfomal weight V(z) V(0) 0; 0 > physical state (2.60) h = vertex operator V(z) 0; k> vertex opreator V 0 (k, z) =: e ik X(z) := e ik X (z) e ik X+ (z) (2.6) T R((X(z)) n : e X(z ) :) = [R(X(z)X(z )] n : e X(z ) : (2.62) R(T(z)V 0 (k, z )) = k 2 2 V 0(k, z ) (z z ) 2 + z V 0(k, z ) (z z ) (2.63) k 2 = 2 V 0 comformal weight V 0 (k, 0) 0; 0 > physical state 0; k> = e ik x 0; 0 > V 0 (k, 0) (2.6) V 0 (k, z) = e ik x e k p log z e ik α n n z n e ik αn n z n (2.64) n z = 0 0; 0 > α p z = 0 x V 0 (k, 0) 0; 0 >= e ik x 0; 0 >= 0; k> (2.65) vertex operator 0; k> 20

22 vertex operator vertex operator V (k, z) =: ζ z X(z)e ik X(z) : (2.66) R(T(z)V (k, z )) = ik ζ : eik X(z) : (z z ) 3 + k (z z ) 2 V (k, z ) + z V (k, z ) (z z ) (2.67) k 2 = 0, k ζ = 0 V vertex operator V (k, 0) 0; 0 > normal ordiring z X(0) α V (k, 0) 0; 0 >= ζ α 0; k> 2.4 BRST quantization 2.4. b,c ghost system action h X µ S[h, X] = d 2 σ hh αβ α X β X (2.68) Z = DhDXe is[h,x] (2.69) h Dh = Dh ++ Dh Dh +, ((2.)) + coordinate action h ++ = h = 0 h h World sheet ξ +, δh αβ = α ξ β + β ξ α (2.70) δh ++ = 2 + ξ + δh = 2 ξ (2.7) 2

23 h αβ = h α β + δh α β Z Z = = Dh DXe is[h,x] DXDh ++Dh Dh + DXe is[h,x] S h ++, h h ++, h h 0 ++, h 0 (2.70) h ++, h ξ +, ξ Dh ++ = det( δh ++ δξ )Dξ + (2.72) = det( δ +ξ + (σ) δξ + (σ ) )Dξ + = det(2 + δ(σ σ ))Dξ + det( + )Dξ + Z Z = DXDh + Dξ + Dξ det( + )det( )e is[h +,h++ 0,h0,X] (2.73) ξ ++, h++ 0 = h 0 = 0 ξ h + = e ϕ Z = DXDh + det( + )det( )e is[h +,X] (2.74) = DXDϕdet( + )det( )e is[x] S[X] Flat Worldsheet action (2.0) ϕ Dϕ factorize [5] det( + ) = det( ) = Dc(σ )Db(σ )e i d 2 σc(σ ) + b(σ ) Dc(σ + )Db(σ + )e i d 2 σc(σ + ) b(σ + ) 22 (2.75)

24 Z = S[X, c, b] = = DXDcDbe is[x,c,b] (2.76) d 2 σ[ + X X + c(σ + ) b(σ + ) + c(σ ) + b(σ )] d 2 σ[ + X X + c(σ + ) b(σ + ) + c(σ ) + b(σ )] + coodinate z, z S[X, c, b] = d 2 z[ 2 z X z X + c(z) z b(z) + c( z) z b( z)] (2.77) (2.77) X /2 b, c normalization /2 classical (2.77) S c b c conformal weight λ b conformal weight ( λ) λ b c conformal weight c b c(z) = c n z n λ (2.78) b(z) = n= n= b n z n +λ {b(z), c(w)} = δ(z w) (2.78) {b n, c m } = δ n+m,0 (2.79) operator product expansion (2) R[b(z)c(w)] = c[b(z)c(w)]+ : b(z)c(w) : (2.80) c[b(z)c(w)] R z = w normal ordering c[b(z)c(w)] 23

25 normal ordering normal ordering b, c c.w λ ϕ ϕ(z) = a n z n λ (2.8) n= lim z=0 ϕ(z) 0>, t = (in-state) (2.8) z 0 a n 0> = 0, (n λ + ) (2.82) 0> <0 lim z= <0ϕ(z) t (out-state) z <0 a n = 0, (n λ ) (2.83) <0 a λ [4] [3] a n 0> = 0 <0 a n = 0 a n 0> 0 <0 a n 0 (2.82) { an 0> = 0, n λ + <0 a n = 0 n λ (2.84) (2.84) b, c 24

26 { cn 0> = 0, n λ + <0 c n = 0 n λ { bn 0> = 0, n λ <0 b n = 0 n λ (2.85) normal ordered product z > w R(b(z)c(w)) = b n z n ( λ) c m w m λ (2.86) n m = (normal ordered pert) + = : b(z)c(w) : + n λ m λ n λ m λ = : b(z)c(w) : + z n ( λ) w n λ n λ = : b(z)c(w) : + ( w z )n ( z w )λ z n λ = : b(z)c(w) : + ( w z )n z n 0 = : b(z)c(w) : + z w b c contractionc[b(z)c(w)] b n c m z n ( λ) w m λ (2.87) δ n+m,0 z n ( λ) w m λ (2.88) c[b(z)c(w)] = z w (2.89) b, c OPE(Operator Product Expansion) b, c local (2.89) b, c T bc (z) T bc (z) =: ( λ)( z c(z))b(z) λc(z)( z b(z)) : (2.90) (2.90) T bc b, c T bc 25

27 (2.89) R ( λ)b(w) R[T bc (z)b(w)] = (z w) 2 + wb(w) z w + (2.9) R[T bc (z)b(w)] = λc(w) (z w) 2 + wc(w) z w + (2.92) (2.47) (2.9) b, c conformal weight λ, λ primary field T bc R[T bc (z)t bc (w)] = A (z w) 4 + 2T bc(w) (z w) 2 + wt(w) z w + (2.93) A A = 6λ( λ) (2.94) (2.93) X (2.46) T(z) X b, c T X (z) + T bc (z) conformal (2.94) λ = T X (z) + T bc (z) D/2 + A D = 26, λ = 0 λ = BRST bosonic string BRST symmetry 2.4. BRST action (2.77) BRST [5] X(z) BRST c(z) δ X(z, z) = ηc(z) z X + ηc( z) z X (2.95) η (2.95) 26

28 δ δx(z, z) = c(z) z X + c( z) z X (2.96) δ Grassman Odd c delta δ 2 = 0 δ 2 X = 0 δc(z) = c(z) z c(z) (2.97) c( z) (2.97) z z b(z) δs[x, b, c] = 0 (2.77) BRST [ ] δs[x, b, c] = ( z c(z)) 2 z X z X (c(z) z b(z) + 2 z c(z)b(z)) + δb(z) [ ] + ( z c( z)) 2 z X z X (c( z) z b( z) + 2 z c( z)b( z)) + δb( z) action BRST (2.98) δb(z) = T X (z) + T bc (z) T(z) (2.99) δb( z) = T X ( z) + T bc ( z) T( z) (2.99) T X (z) (2.23) T zz T bc (2.90) λ = (2.96) c z z + ϵ(z) ϵ(z) = ηc(z) z conformal weight - λ = BEST BRST current J B (2.96),(2.97),(2.99) J B J B (z) = : c(z)[t X (z) + 2 T bc(z)] : (2.00) J B ( z) (2.00) z z BRST R (2.00) normal oreder (2.86),(2.36) R[J B (z)x(w)] = c(w) w X(w) + z w R[J B (z)c(w)] = c(w) wc(w) + z w R[J B (z)b(w)] = T(w) z w + c(w)b(w) (z w) 2 + (2.0) 27

29 (2.0) BRST J B BRST charge Q B Q B ϕ(w) d 2 z(j(z) + J( z)) (2.02) z > w z > w ϕ(w) X, c.b (2.0) Q B ϕ(w) = δϕ(w) (2.03) BRST δ 2 = 0 BRST (2.03) δ 2 = 0 (2.03) δ 2 = 0 Q 2 B = 0 J B (z) R D = 26 Q 2 B = 0 [4] Q B phys> = 0 BRST bosonization 2.4. b(z), c(z) bosonization [5] ϕ(z) ϕ(z) = x + N log z + n 0 α n n z n (2.04) [α n, α m ] = nδ n+m,0, [x, N] = (2.05) ϕ(z) b(z), c(z) c(z) =: e ϕ(z) :, b(z) =: e ϕ(z) : (2.06) (2.06) b(z), c(z) [b(z), c(w)] = δ(z w) R (2.86) (2.06) vertex operator (2.6) (2.62) R[b(z)c(w)] = R[: e ϕ(z) :: e ϕ(w) :] = z w + (2.07) 28

30 T bc (2.90) λ = T bc (z) =: c(z)( z b(z)) + 2( z c(z))b(z) : (2.08) (2.08) (2.06) (2.08) : c(z)( z b(z)) : = lim w z : c(w)( z b(z)) : (2.09) = lim w z [R[c(w) z b(z)] C[c(w)][ z b(z)]] C[c(w)][ z b(z)] c(w) z b(z) contraction w > z R (2.08) (2.09) (2.06) T bc =: 2 ( zϕ)( z ϕ) z ϕ : (2.0) λ = λ T bc =: 2 ( zϕ)( z ϕ) + 2λ z 2 ϕ : (2.) 2 j(z) =: b(z)c(z) : T bc j(z) = lim z w b(w)c(z) dz j(z) = z=0 j(z) = z ϕ(z) (2.2) z=0 dz ϕ(z) = N (2.3) N (2.04) ϕ(z) fermion ghost (2.90) (2.) T f ermion bc T boson bc = : 2 c( b) + 2λ ( c)b + ( bc) : (2.4) 2 2 = : 2λ ( ϕ)( ϕ) + ( ϕ) :

31 (2.4) ϕ bc b(z) c(z) ϕ(z) ϕ(z) 3 action S[b, c] S[b, c] d 2 z [c(z) z b(z) + c( z) z b( z)] (2.5) (2.06) (2.5) (2.) action S[ϕ] = d 2 σ [ h h αβ α ϕ β ϕ + 2λ ] R (2) ϕ 4 (2.6) action (2.6) h (2.) 30

32 3 g-loop amplitude [6] g-loop object boson ϕ vacuum charge Q b, c bosonization b, c g-loop amplitudes Q = 0 X loop vacuum charge Q operator product expansion Q (2) fermion number g-loop amplitude tree vertex [8, 7] tree amplitude physical state g-loop amplitude tree vertex prime form,g abelian differential,vector Riemann constant,period matrix Schottky 3

33 3. free bosonic field with vacuum charge Q vacuum charge Q S[ϕ] = d 2 σ h[h αβ α ϕ β ϕ 2π Σ 4 QR(2) ϕ] (3.) h metric R ( 2) R (2) = hαµ h βν (h αν,βµ h αµ,βν + h βµ,αν h βν,αµ ) (3.2) 2 Σ R 2 ϕ Q (2.0) T zz (++ ) ghost number current p R T(z) =: 2 [ zϕ(z) z ϕ(z) Q z z ϕ(z)] : (3.3) j(z) = z ϕ(z) (3.4) R[ϕ(z)ϕ(w)] = log(z w)+ : ϕ(z)ϕ(w) : (3.5) (3.5) OPE T(z)T(w) = [ 3Q2 ]/2 (z w) T(w) (z w) 2 + wt(w) + (nomal ordered) (3.6) z w Q T(z)j(w) = (z w) 3 + j(w) (z w) 2 + w j(w) + (nomal ordered) z w j(z)j(w) = + (nomal ordered) (z w) 2 Q T central charge /2 [ 3Q 2 ]/2 j primary field 32

34 ϕ ϕ(z) = x + N log z + bosonic string n 0 α n n z n (3.7) [α n, α m ] = nδ n+m,0, [x, N] = (3.8) (3.6),(3.7),(3.8) L n j n L n,j n T(z),j(z) conformal weight z (2) L n = : α m α n m : 2 2 Q(n + )α n (3.9) m j n = α n [L n, L m ] = (n m)l n+m + 2 ( 3Q2 )n(n 2 )δ n+m,0 [L n, j m ] = mj n+m 2 Qn(n + )δ n+m,0 [j n, j m ] = nj n+m δ n+m,0 Q = 2λ b(z)(conformal weight λ) c(z) ϕ b, c (2.06) j(z) j(z) = : b(z)c(z) : (3.0) (3.7) N N = dz j(z) = j 0 (3.) boson ϕ b, c boson N (2.85) N L 0 (3.9) L 0 = α m α m + 2 α 0(α 0 Q) (3.2) m 0 33

35 L 0 = L 0 α 0 = α 0 α 0 = α 0 + Q α 0 = N N α 0 = α 0 + Q (3.3) N + N + Q = 0 (3.4) N N q> q > < q q >= δ(q + q + Q) (3.5) q> vertex oparator q> = lim z 0 : e qϕ(z) : 0> (3.6) N q T(z) OPE T(z) : e qϕ(w) := 2 q(q + Q): eqϕ(w) : (z w) 2 + w : e qϕ(w) : + (3.7) (z w) : e qϕ(z) : conformal weight 2 q(q + Q) primary field (3.3) L 0 q> = q(q + Q) q> (3.8) 2 L n q> = 0 n > V N;0 for free bosonic theory (3.) tree level ( world-sheet N V qi (z i ) : e q i ϕ(z i ): N T(z ; q z N ; q N ) = <q = 0 R[ V qi (z i )] q = 0> (3.9) i= 34

36 N g-loop N (3.9) vertex function vertex function DS vertex [7, 8] W i = <n i, O a : exp[ dzϕ( z) z ϕ i (z)] : (3.20) n i 0 ϕ i ϕ (auxilialy field) <n i, O a N i n i (3.6) q = n i n i W i primary field α> i auxilialy field z = W i lim z 0 V i (z) 0> i = V() (3.2) V i (z) = V[ϕ i (z)], V(z) = V[ϕ(z)] (3.2) i auxilialy space vertex operator q i > (3.2) W i q i > i = W i lim z 0 : e q i ϕ(z) i : 0> i (3.22) = : e q i ϕ() : W i ( ) n dzϕ( z) z ϕ i (z) : = dz( n ϕ()z n )( 0 0 n! n=0 ( ) n = n ϕ()αn i n! n=0 + m= α i mz m ) (3.23) q i > N i = α i 0 n > 0 αi n q i > = 0 35

37 W i q i > = <n i, O a : e n=0 ( ) n n! nϕ() : q i > (3.24) n i = <n i, O a : e q i ϕ() : q i > n i = : e q i ϕ() : primary field : e qϕ(z) : Q = 0 vertex operetor vertex operator (3.2) Q primary field W i z = primary field z = z i primary field conformal V i (z) V i (0) = z i conformal mapping( V i (z) z = 0 conformal weight λ Φ V i V i Φ(V i (z)) = ( V i z ) λ Φ(z) (3.25) z = V i (z) Φ(z V ) = ( i (z ) z ) λ Φ(V i (z )) (3.26) (3.24) W i q i > = Φ(), q i > = lim z 0 Φ(z) 0> (3.27) conformal z γ i (z) γ i z i γ i (z) = V i ( z) γ i, γ i (3.27) γ i W i γi V i (z i ) q i > = ( z ) λ Φ(z i ) (3.28) γ i W i γi V i ( z ) λ q i > = Φ(z i ) V i W i ( z ) λ q i > = Φ(z i ) 36

38 V i SL(2, C) V i (z) = A iz + B i C i z + D i, A i D i B i C i =, A i, B i, C i, D i C (3.29) W i [ W i = <n i, O a exp ] dz ϕ i (z)(α0 i 2 Q) log[v i (z)] n i 0 [ ] : exp ϕ[v i (z)]ϕ i (z) : (3.30) 0 V conformal ( z ) (3.28) (3.30) q i > q i > W i [ ] W i q i > = <n i ; O a exp 2 N i(n i + Q) log[v i (0)] : exp[n i ϕ(z i )] : q i > (3.3) n i [ ] = exp 2 q i(q i + Q) log[v i (0)] : exp[q i ϕ(z i )] : ( V ) (z i ) 2 q i (q i +Q) = : exp[q i ϕ(z i )] : z i W i N tree vertex function V N;0 = <q = 0 N W i q = 0> (3.32) i= q = 0>, <q = 0 auxiliary field V N; 0 (i = N) normal orderd N N <q = 0 : e A i : q = 0> = <q = 0 : e A i :: e A j : q = 0> (3.33) i= i<j <q = 0 : e A i :: e A j : q = 0> = e <q=0 A i A j q=0> 37

39 (3.32) V N;0 = [ N [<n i, O a ]] exp i= [ 2 N i= dz ϕ i (z)(α i 0 Q) log[v i (z)] ] N <q = 0 : dzϕ[vi (z)]ϕ e i (z) :: dyϕ[vj (y)]ϕ e j (y) : q = 0> i<j (3.34) auxilialy field (3.33) = N <q = 0 : dzϕ[vi (z)]ϕ e i (z) :: dyϕ[vj (y)]ϕ e j (y) : q = 0> (3.35) i<j [ = exp i<j dz ] N dy log[v i (z) V j (y)]ϕ i (z)ϕ j (y) δ( N i + Q) 0 z > y auxilialy field N i α i vertex V N;0 = [ N <n i, O a ]e 2 i= [ exp i<j dz Ni= dz ϕ i (z)(α0 i Q) log[v i (z)] (3.36) i= ] N dy log[v i (z) V j (y)]ϕ i (z)ϕ j (y) δ( N i + Q) (3.29) (3.36) [ N <n i, O a ] exp 2 i= N i, j=,i j n,m=0 i= and i nm (U i V j )am j N δ( N i + Q) (3.37) i= (3.37) a n α n normalization a 0 = α 0 N, α n = na n, n > 0 (3.38) U i (z) V i (z) Γ(z) = z U i(z) Γ Vi (z) 38

40 D nm (V(z)) D nm (V) = lim z=0 m! m n m z [V(z)] n n m (3.39) D n0 (V) = lim n [V(z)] n n 0 z=0 D 0m (V) = lim z=0 D 00 (V) = lim z=0 2 log[v (z)] m 2m! m z log[v (z)] m 0 (3.36) (3.37) D nm (V V 2 ) = D nl (V )D lm (V 2 ) + D n0 (V )δ mo + D 0m (V 2 )δ no (3.40) l= D nm (ΓV Γ) = D mn (V) (3.4) (3.37) D nm (U i V j ) ((3.40)) D(ΓV i ) D(V j ) D (3.36) ((3.40)) n, m 0 D nm (V) SL(2, C) tree-level N V N;0 α α N N <q = 0 N N : e q i ϕ(z i ) : q = 0> = V N;0 q i > i (3.42) i= i= N δ( q i + Q) (z i z j ) q i q j (3.43) i= (3.3) q i > i z = z i primary field q i > i lim [ V z zi i = lim [ V z zi i i<j (z)] q i(q i +Q) : exp q i ϕ i [V i (z)] : 0> i (3.44) (z)] q i(q i +Q) : exp[q i ϕ i (0)] : 0> i exp (3.36) Q : exp[ϕ(0)] : 0> i N i, α n : exp[ϕ(0)] : 0> i = 0 (3.43) 39

41 3.3 V N;g for free bosonic theory 3.2 N vertexv n;0 g-loop N vertexv n;g tree vertex vertex (N + 2g) g g g N V N;g tree vertex V N+2g N 2g i = N 2g 2µ 2µ µ = g 2g V N+2g;0 (2µ) 2µ <n 2µ n 2µ >, N 2µ N 2µ, a2µ n a n 2µ (3.45) vertex V N+2g;0 2µ 2µ sewing operatorp(x µ ); (µ = g) x u P(x µ ) 2µ an 2µ z P(x µ )[z] P(x µ ) =: exp [ n,m=0 a 2µ n D nm (P(x µ ))a 2µ m + n= a 2µ n a 2µ n ] : (3.46) V N+2g;0 V V i V i P g-loop N g V N;g Tr (2µ,2µ) V N+2g;0 µ= g P(x µ ) µ= (3.47) (3.47) (α n, n 0) [a, a ] = 40

42 ψ> ψ> exp(ψa 0>) = n=0 λ n n! n> (3.48) ψ n> a a ψ> a (3.48) I = d 2 z ψ>e ψ 2 <ψ (3.49) O Tr(O) = d 2 ψ<ψ O ψ>e ψ 2 (3.50) Tr (2µ,2µ) (O) = d 2 ψ n n= < ψ n O ψ n >e n= ψ n 2 (3.5) r µ > Tr2µ,2µ 0 (O) = < r µ Q O r µ > (3.52) r µ (3.5) < r µ Q r µ >= r µ (3.5),(3.52) V N;g V N+2g;0 n= 4

43 N V N+2g;0 = [ <n i, O a ] exp N a i= n i 2 nd i nm (U i V j )a j m (3.53) i, j=,i j n,m=0 g [ <n 2µ, O a ]δ N g N i + [N 2µ + N 2µ ] + Q µ= n 2µ i= µ= N g exp an[d i nm (U i V 2µ )am 2µ + D nm (U i V 2µ )a 2µ m ] i= µ= n,m=0 g exp an 2µ D nm (U 2µ V 2ν )am 2ν + a n 2µ D nm (U 2µ V 2ν )a 2ν m µ,ν= n,m=0 g exp a n 2µ D nm (U 2µ V 2ν )a 2ν g m [ n 2µ, O a >] µ,ν= n,m=0 µ= n 2µ (3.53) D nm D nm (U I V J ) = D mn (U J V I ) (3.54) (3.53) µ, ν µ = ν tree vertex (3.37) µ = ν D nm (U 2µ V 2µ ) = D nm (U 2µ V 2µ ) 0 (3.55) P(x µ ) V N;g * V 2µ Ṽ 2µ V 2µ P(x µ ) (3.56) U 2µ Ũ 2µ Γ[V 2µ P(x µ )] (3.47) g V N;g Tr (2µ,2µ) V N+2g;0 µ= g P(x µ ) µ= = g µ= ] Tr (2µ,2µ) [Ṽ N+2g;0 (3.57) * C 42

44 Ṽ N+2g;0 V N+2g;0 (3.56) (3.5),(3.52) < ψ µ 2µ, ψ µ > 2µ, < r µ Q 2µ, r µ > 2µ an 2µ, a n 2µ, a 2µ 0, a 2µ 0 Ṽ N+2g;0 an 2µ ψ n µ (3.58) a n 2µ ψn µ a 2µ 0 r µ a 2µ 0 r µ + Q ψ n V N;g = [ N N <n i, O a ]δ( i= n= µ= i= g [d 2 ψ n µ ] exp N i + Q) [ (B, B 2 ) exp(a) (3.59) r µ ( ) ψ ψ ( )] ψ 2 (ψ, ψ)( H) ψ (3.59) (X, X 2 ) ( Y Y 2 ) g µ= n= ( (X ) µ n(y ) µ n + (X 2 ) µ n(y 2 ) µ n) H ( ) H nm µν Dnm (U = 2µ Ṽ 2ν ) D nm (U 2µ V 2ν ) D nm (Ũ 2µ Ṽ 2ν ) D nm (Ũ 2µ V 2ν ) (3.60) (3.6) 43

45 A A = N N i, j=,i j n,m=0 g i= ν= n=0 g µ,ν=,µ ν g µ,ν=,µ ν a i nd nm (U i V j )a j m (3.62) an[d i n0 (U i Ṽ 2ν )r ν D n0 (U i V 2ν )(r ν + Q)] r µ D 00 (Ũ 2µ Ṽ 2ν )r ν + g µ,ν= (r µ + Q)D 00 (U 2µ V 2ν )(r ν + Q) r µ D 00 (Ũ 2µ V 2ν )(r ν + Q) B r µ g (B I ) ν m = ( ) I r ν [D 0m (Ũ 2ν V I ) D 0m (U 2ν V I )] + ( ) I N i= n=0 ν= and i nm (U i V I ) + ( ) I Q I =, 2 V = Ṽ 2µ, V 2 = V 2µ (3.63) g D 0m (U 2ν V I ) (3.64) (3.59) ψ n ( ) ψ (B, B 2 ) + ( ) ψ 2 2 (ψ ψ 2, ψ )K ψ 2 = ψ 2 [[(ψ 2, ψ ) + (B, B 2 )K ][( /2 ) ( ) (B, B 2 )K B ] B 2 ψ 2 ν= ( ) + K /2 B ] B 2 (3.65) (3.66) ( ψ ) ψ = K /2 ( ψ ) 2 ψ 2, K = H ψ ψ [det(k)] /2 (3.59) 44

46 N N V N;g = [det( H)] /2 [ <n i, O a ]δ[ N i (g )Q] (3.67) i= n i i= [ exp A + ( )] 2 (B, B 2 )( H) B2 B r µ (3.67) ( H) H H H l (3.68) (3.68) (3.67) [ exp A + ( ) ] 2 (B, B 2 ) H l B2 (3.69) B H l (A) ( (H l ) µν Dnm (U 2µ Σ l nm = (, )Ṽ2ν ) D nm (U 2µ Σ l Σ l=0 l=0 D nm (Ũ 2µ Σ l (+, )Ṽ2ν ) (+, ) D nm ( Σ l (+, ) ) = α:n α =l (,+) V 2ν) D nm (Ũ 2µ Σ l (+,+) V 2ν) ) (3.70) D nm ( T α ) (3.7) T α S µ = Ṽ 2µ U 2µ, (µ = g) A (3.7) order l S µ S ν (3.55) (3.64) B (B I ) µ m = ( ) I ( ) I s= ν= l,s= i= δ I, QD 0m (V I ) g r ν [D 0s (Ũ 2ν ) D 0s (U 2ν )]D sm (V I ) r µ D 0m (V I ) (3.72) N a i l D ls(u i )D sm (V I ) + ( ) I Q 45 s= ν= g D 0s (U 2ν )D sm (V I )

47 (B I ) µ m = ( ) I ( ) I s= ν= l,s= j= δ I, QD m0 (U I ) g D ms (U I )[D s0 (Ṽ 2ν ) D 0s (V 2ν )]r ν r µ D m0 (U I ) (3.73) N D ms (U i )D sl (V I )a j l + ( )I Q s= ν= g D ms (U I )D s0 (V 2ν ) I 2 V = Ṽ 2µ, V 2 = V 2µ, U = Ũ 2µ, U 2 = U 2µ (3.72),(3.73) (3.55) (3.59) B B 2 (3.72) B B 2 (3.73) (3.59) (3.59) (3.70),(3.72),(3.73) r µ V N;g N N V N;g = [ <n i, O a ]δ[ N i (g )Q] (3.74) i= n i i= = N exp[ g g r µ C µν () r ν + r µ C µ (2) + C (3) ] 2 r µ µ,ν= (3.74) N, C () µν, C (2) µ, C (3) * 2 C() µν, C µ (2), C (3), N B µ= * 2 A 46

48 C () µν = 2πiτ µν (3.75) C µ (2) = 2πi[ N Vi dz ϕ (i) (z) (z)[ ω µ ] Q( z 0 µ + 2πi 0 z 0 2 ) C (3) = N i= N i, j=,i<j 0 0 N i, j= 0 + N 2 Q i= i= dz ϕ (i) (z) log[v i (z)]α(i) 0 dz dy ϕ (i) (z) log[v i (z) V j (y)] ϕ (j) (y) 0 dz dy ϕ (i) (z) log E(V i(z), V j (y)) 0 V i (z) V j (y) ϕ(j) (y) 0 dz ϕ (i) (z)(log[v i (z)] + 2 log σ[v i(z)]) N = (det( H)) /2 = α ( kα) n 2 (3.76) (3.75) τ µν, ω µ, z 0 µ, E(z, y), σ(z) Σ g A (3.75) (3.74) V N;g N N V N;g = N [ <n i, O a ]δ[ N i (g )Q] (3.77) i= n i i= ( exp N ) dz ϕ (i) (z)[α0 i Q] log[v i 2 (z)] i= exp N 2 i, j= exp N n= dz dy ϕ (i) (z) log E(V i(z), V j (y)) 0 V i (z) V j (y) ϕ(j) (y) dz dy ϕ (i) (z) log[v i (z) V j (y)] ϕ (j) (y) 0 i, j=,i<j ( Θ [ N dz 2πi 0 i= ( N ) exp Q dz ϕ (i) (z) log σ[v i (z)] i= 0 ) Vi ϕ (i) (z) (z)[ ω µ ] Q( z 0 µ + z 0 2 )] τ 47

49 Θ Θ(z τ) = exp 2πni n µ g µ,ν= 2 n µτ µν n ν + g µ= n µ z µ (3.78) n µ tree vertex V N;0 (3.77) N V N;g = N ˆV N;0 δ[ N i (g )Q] (3.79) i= exp N 2 i, j= 0 ( Θ [ N dz 2πi 0 i= ( N ) exp Q dz ϕ (i) (z) log σ[v i (z)] i= 0 dz dy ϕ (i) (z) log E(V i(z), V j (y)) 0 V i (z) V j (y) ϕ(j) (y) ) Vi ϕ (i) (z) (z)[ ω µ ] Q( z 0 µ + z 0 2 )] τ ˆV N;0 (3.36) V N;0 g-loop vertex primary field N tree vertex (3.42) <q = 0 N N : e q i ϕ(z i ) : q = 0> g V N;g q i > i (3.80) i= i= N = δ[ N i (g )Q]N [E(z i, z j )] q i q j [ Θ i= ( [ 2πi 0 dz i<j N ϕ (i) (z)[ i= Vi (z) N [σ(z i )] q i Q i= z 0 ω µ ] Q( z 0 µ + 2 )] τ )] Q Q b, c Q = 3 c N = q = > b N = q = > N 48

50 b N 2 c N N 2 V N +N 2 ;g i= j= = N N q i = > q j = > (3.8) i, j=;i<j E(z i, z j ) N h,k=;h<k E(y h, y k ) N i= N2 h= E(z i, y h ) δ(n N 2 + 3(g ))Θ ( Ω + Ω 2 3( z 0 µ ) τ ) N2 h= σ(y h) 3 N i= σ(z i) 3 Ω k 2πi 0 N k dz i= zi ϕ (i) (z)[ ω µ ] (3.82) z 0 (3.8) N = N 2 3(g ) X µ (z) (µ = 0,, 25) vertex Q = 0 N (2.24) X g-loop vertex V X N;g = O26 exp [ N [ i= µ= 2 i, j= N X µ (z) = x µ iα µ 0 log z + i ] d D p i <p i ; O a δ 0 [d D k µ ] exp g ˆV N;0 tree vertex N V N;0 ˆ = exp [ exp 2 i ( i= N n 0 α µ n n z n (3.83) p i ) ˆ V N;0 (3.84) dz dy X i (z) log E(V ] i(z), V j (y)) 0 V i (z) V j (y) X j (y) g N g k µ (2πiτ) µν k ν + i dz X i (z) 2 µ= i= µ= i, j=,i<j 0 dz 0 ( Vi (z) dy X i (z) log[v i (z) V j (y)] X j (y) dz X i (z)α i 0 log V i (z) ] 49 z 0 ω µ ) k µ (3.85)

51 3.4 2 bosonic string conformal field theory conformal vertex operator conformal couple Di Vecchia [6] 3 2 Vacuum charge Q g-loop vertex tree vertex g-loop vertex Schottky tree vertex g-loop vertex moduli g-loop vertex N X [6] -loop [2] [3, 2] technical [2] 50

52 5

53 A g-loop vertex (3.79) S(2, C) S µ (z) µ V N;g S µ (z) = a µz + b µ c µ z + d µ, a µ d µ b µ c µ =, a, b, c, d C (A.) (A.) (A.2) S µ (z) η µ S µ (z) ξ µ = k µ z η µ z ξ µ, k µ < (A.2) a µ = η µ k µ ξ µ kµ η µ ξ µ c µ = k µ kµ η µ ξ µ b µ = η µ ξ µ ( k µ ) kµ η µ ξ µ d µ = k µη µ ξ µ kµ η µ ξ µ (A.3) (A.2) z lim n Sn µ(z) = η µ ; lim S n n µ (z) = ξ µ (A.4) η µ, ξ µ S µ, S µ S µ z B µ, B µ B µ, B µ ds µ dz /2 = c µ z + d µ = ; 52 dsµ dz /2 = c µ z a µ = (A.5)

54 z R µ, R µ J µ, J µ R µ = R µ = k µ ξ µ η µ k µ (A.6) J µ = d µ c µ = ξ µ k µ η µ k µ ; J µ = a µ c µ = η µ k µ ξ µ k µ ; (A.7) (A.6),(A.7),(A.4) S µ B µ B µ S µ B µ B µ η µ B ξ B µ B µ, B µ F S µ, S µ B µ, B µ B B S µ, S µ F, B B F F S g S µ (µ = g) T α = S n µ S n 2 µ 2 S n r µ r, r =, 2, µ i =, 2, g (A.8) n i (i =, r) T α G g T α S S T α n α i= n α = n i (A.9) r G g T α SL(2, C) (A.2) η, ξ, k A T = AT A T G g G g G g η, ξ A(η), A(ξ) k G g G g A 2g S µ η µ, ξ µ 2g 3 S µ F B B T α T α B, B 2,, B g B, B 2,, B g H 53

55 S µ B µ B µ B µ B µ F * H g Bµ, B µ g g B µ B µ a µ B µ z B µ S µ(z) B µ * 2 F + (3.75) Cµ 2 ω µ ω µ = (,µ) T α ( ) z T α (η µ ) z T α (ξ µ ) (A.0) (,µ) T α S µ (A.0) ω µ a µ ω µ = ω µ = 2πiδ µν a ν B ν µ ν (A.0) T α S n ν (A.) (n > 0) T α (η µ ), T α (ξ µ ) a µ (A.0) (A.0) µ = ν T α (A.) ω µ b ν (3.75) C µν τ µν (2πi)τ µν = b ν ω µ (A.2) b µ B µ z 0 B µ S µ(z 0 ) * C * 2 b µ F 54

56 (2πi)τ µν = ω µ = b ν = (,µ) Sν (z 0 ) z 0 ω µ (A.3) log S ν(z 0 ) T α (η µ ) S T ν (z 0 ) T α (ξ µ ) α (ν,µ) = δ µν T α z 0 T α (ξ µ ) z 0 T α (η µ ) log η ν T α (η µ ) ξ ν T α (η µ ) η ν T α (η µ ) ξ ν T α (ξ µ ) (A.4) (A.3) (ν,µ) T α S ± ν S ± µ µ = ν (A.3) τ µν z 0 C 3 E(z, w) prime form E(z, w) = (z w) α z T α (w) z T α (z) w T α (z) w T α (w) (A.5) α T α, Tα * 3 (A.5) a µ b µ E(S µ (z), w) = exp z 0 µ σ(z) [ 2πi( w ] 2 τ µµ + ω µ ) E(z, w) z (A.6) g > z 0 µ = 2πi 2 log K µ πi + g (µ,ν) ν= α log ξ ν T α (η µ ) η ν T α (η µ ) z 0 T α (ξ µ ) z 0 T α (η µ ) (A.7) log σ(z) = g = 2(g ) + g (µ,ν) µ,ν= α I log ξ µ T α (ξ ν ) z T α (ξ ν ) log σ(z) = log[(z ξ)(ξ z)] 2 (A.9) z T α (z) ξ µ T α (a) + ξ µ ξ ν log (z ξ µ ν ν )(ξ µ z) (A.8) * 3 T α Tα 55

57 B C µν, C 2 µ, C 3, N (3.74) C µν, C 2 µ, C 3 N (3.75) (3.75) B. C µν C µν (3.59) r µ (3.72),(3.73),(3.68) r µ D (3.40) U V S µ, S µ 56

58 Cµν = (D 00 (S µ ) + D 00 (Sµ ))δ µν (B.) [ ] + (D 0n (S µ ) D 0n (Sµ )) D nm (Σ l ) (D m0 (S ν ) D m0 (Sν )) l= (D 0n (Ũ 2µ ) D 0n (U 2µ )) [ ] δ nm + D nm (Σ) + [D nm (Σ l+ ) 2D nm (Σ l ) + D nm (Σ l )] l= (D m0 (Ṽ 2ν ) D m0 (V 2µ )) (D 0n (Ũ 2µ ) D 0n (U 2µ )) [ ] δ nm + [D nm (Σ l ) D nm (Σ l )] l= (D m0 (S ν ) D m0 (Sν )) + (D 0n (S µ ) D 0n (Sµ )) [ ] δ nm + [D nm (Σ l ) D nm (Σ l )] l= (D m0 (Ṽ 2ν ) D m0 (V 2µ )) (B.) n, m (3.70) l l= lim N N l= (B.2) (B.2) (B.) U, V S, S (3.70) Cµν = (D 00 (S µ ) + D 00 (Sµ ))δ µν (B.3) (±µ,±ν) + [(D 0n (S µ ) D 0n (Sµ ))D nm (T α )(D m0 (S ν ) D m0 (Sν ))] l=0 α;n α =l (±µ,±ν) α;n α (3.40) D(S) D(T =l α ) S µ ± T α S ν ± T α 57

59 (B.3) D 0n (S µ )D nm (T α )D m0 (S ν ) (B.4) T α T α = S µ S ν (B.2) (B.3) (B.4) (B.4) T α S µ, Sν order l T α ( µ,+ν)(l) T ( µ,+ν)(l) α = (n,m);n+m l;n,m 0 (S µ ) n T (±µ,±ν)(l n m) α (S ν ) m (B.5) T α (±µ,±ν)(l n m) S µ, Sµ, S ν, Sν order (l n m) (3.40),(B.5),(B.2) (B.3) S C µν = lim N (µ,ν) α + [(D 00 (S N ν ) D 00 (S N ν (D 0n (S N µ ) D 0n (S N µ ))D nm (T α )(D m0 (S N ν ) D m0 (S N ν )) (B.6) )) (D 00 (Sν N ) D 00 (Sν N ))]δ µν T α S µ ±n S±m ν D nm (S) (3.39) η µ, ξ µ k µ lim D 0n(Sµ N ) = ( ) n, lim N n ξ D 0n(Sµ N ) = ( ) n, µ N n η µ lim D m0(sν N ) = (η µ ) m, lim D m0(sν N ) = (ξ µ ) m, N m N m (B.7) lim (D 00(Sν N ) D 00 (Sν N )) = N 2 log k µ (B.8) (B.7) (B.8) (B.6) (µ,ν) Cµν = α = 2πiτ µν log η µ T α (ξ µ ) ξ µ T α (η ν ) η µ T α (η ν ) ξ µ T α (ξ ν ) (B.9) 58

60 C µν τ µν* B.2 C 2 µ C 2 µ B. C 2 µ = [ l=0 α;n α =l n,k= N i= m=0 = lim N [(D 0n (S µ ) D 0n (Sµ ))D nk (T α )] (B.0) D km (V i )am i Q N i= lim N Q α µ [ g µ,µ ν= g ν= D k0 (S ν ) ] + QD 00 (S µ ) (D 0n (Sµ N ) D 0n (Sµ N ))D nk (T α )D km (V i )am i n,k= α ] (D 0n (Sµ N ) D 0n (Sµ N ))D nk (T α )D k0 (Sν N ) n,k= + lim Q[D 00(Sµ N ) D 00 (S N N N = i= α (µ) + Q[ ν µ (µ) α m=0 α i m m! m z µ ) ( log ξ µ T α (V i (z)) η µ T α (V i (z)) log ξ µ T α (ξ µ ) η µ T α (z 0 ) η µ T α (η µ ) ξ µ T α (z 0 ) ] (B.0) A (3.75) ) η µ T α (z 0 ) z=0 ξ µ T α (z 0 ) B.3 C 3 C 3 B.2 * A 59

61 C 3 = N and i nk (U i )D kh (T α )D hm (V j )am j l=0 α;n α =l i, j= n,m=0 k,h=0 N i= m=0 2 Q [(a0 i Q)D 0m(V i )am i + amd i 0m (U i )(a0 i Q)] N [D 0n (S ν )D nk (T α )D km (V i )am i l=0 α;n α =l i= n,k=0 m=0 + amd i mk (V i )D kn (T α )D n0 (Sν )] g 2 Q2 [D 0n (S µ )D nk (T α )D km (Sν )] l=0 α;n α =l µ,ν= n,m= (B.) (B.0) D nm (S) lim N D nm (S N ) C 3 (3.75) B.4 N g-loop vertex (3.77) N = (det( H)) /2 log det( H) = n= n Tr[Hn ] = n= T α T α = U α D α Uα ( ) kα 0 D α = 0 n α:n α =n D mm (T α ) (B.2) D α (B.3) K α T α D mm (T α ) m > 0 D U α, U α D mm (T α ) = m= D mm (D α ) = m= kα m = m= k α k α (B.4) k α order n conjugacy class (B.2) log det( H) = n= 60 n α;n α =n k α k α r α (B.5)

62 α conjugacy class r α class condjugacy class conjugacy class primary class Tα p primary class (Tα p ) m primary class conjugacy class k p (Tα ) m = k m, n Tα p p (Tα ) m = m n Tα p, r (Tα p ) m = r Tα p =, n Tα p (B.6) (B.6) (B.5) log det( H) = α m 0 k (T p α ) m k (T p α ) m r (T p α ) m n (T p α ) m (B.7) primary class (B.6),(B.7) det( H) = α ( kα) n 2 (B.8) α primary class T α N N = α n= n= ( k n α) (B.9) 6

63 C Canonical form (3.47) P(x µ ) (3.56) canonical form canonical form O O exp( a, A) : exp( a, (C )a) : exp( B, a) exp( ϕ) (C.) ϕ a, a (a, A) = a n A n, (B, a) = B n a n, n= n= (a, (C )a) = a n(c nm δ nm )a m n,m= (C.2) canonical form (C.) canonical form O O 2 = O 3 (C.3) A 3 = A + C A 2, B 3 = B 2 + B C 2 (C.4) C 3 = C C 2 ϕ 3 = ϕ + ϕ 2 + B A 2 62

64 vertex canonical form (3.46) [ ] P(x µ ) =: exp a n 2µ D nm (P(x µ ))am 2µ + a n 2µ an 2µ n,m=0 n= : (C.5) canonical form C nm = D nm (P(x µ )), ϕ = a 0 D 00(P(x µ )), A n = D n0 (P(x µ ))a 0,B n = a 0 D 0n(P(x µ )), (C.6) (3.37) V N;0 2µ a 2µ canonical form V N;0 = V exp 2 N k=,k 2µ m=0 a k n[d nm (U k V 2µ ) + D mn (U 2µ V k )]a 2µ m (C.7) (C.7) (C.) B m = 2 ak n[d nm (U k V 2µ ) + D mn (U 2µ V k )] (C.8) ϕ = 2 ak n[d n0 (U k V 2µ ) + D 0n (U 2µ V k )]a 2µ 0 C, A (C.5) (C.7) canonical form P(x µ ) Ã, B, C, ϕ V N;0 B, ϕ V N;0 P(x µ ) = V exp( a 2µ, Ã) : exp( a 2µ, ( C )a 2µ ) : exp( ( B + B C, a 2µ ) exp( ( ϕ + ϕ + (B, Ã))) (C.9) (C.9) exp vertex <n 2µ, O a (C.8),(C.6) (C.9) V N;0 P(x µ ) = V exp 2 N k=,k 2µ m=0 a k n[d nm (U k Ṽ 2µ ) + D mn (Ũ 2µ V k )]a 2µ m (C.0) Ṽ 2µ = V 2µ P(x µ ), Ũ 2µ = ΓP(x µ ) ΓU 2µ (C.0) (C.7) U 2µ Ũ 2µ, V 2µ Ṽ 2µ 63

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