(τ τ ) τ, σ ( ) w = τ iσ, w = τ + iσ (w ) w, w ( ) τ, σ τ = (w + w), σ = i (w w) w, w w = τ w τ + σ w σ = τ + i σ w = τ w τ + σ w σ = τ i σ g ab w, w

S = 4π dτ dσ gg ij i X µ j X ν η µν η µν g ij g ij = g ij = ( 0 0 ) τ, σ (+, +) τ τ = iτ ds ds = dτ + dσ ds = dτ + dσ δ ij ( ) a =, a = τ b = σ g ij δ ab g g ( +, +,... ) S = 4π S = 4π ( i) = i 4π dτ dσ gg ij i X µ j X ν η µν dτ dτ dσ( τ Xµ dσ g (δ ab a X µ b X ν )η µν τ Xν + σ Xµ σ Xν )η µν τ ( ) e is e S E S E = 4π dτ dσδ ab a X µ b X ν η µν = 4π dτ dσ( τ Xµ τ Xν + σ Xµ σ Xν )η µν

(τ τ ) τ, σ ( ) w = τ iσ, w = τ + iσ (w ) w, w ( ) τ, σ τ = (w + w), σ = i (w w) w, w w = τ w τ + σ w σ = τ + i σ w = τ w τ + σ w σ = τ i σ g ab w, w ( g = g ww, g = g ww, g = g ww, g = g ww ) g µν = x x µ x β x ν g β g = x x w w δ + x x w w δ = τ τ w w + σ σ w w = 4 4 = 0 g = x x w w δ + x x w w δ = τ τ w w + σ σ w w = 4 + 4 = g = g, g = g g ab = ( 0 0 ), g ab = ( 0 0 ) gdτdσ ( ) gd D x = g d D x g dτdσ g dwdw dτdσ = 4 dwdw w, w

S = 4π 4π dτ dσ (δ ab a X µ b X ν )η µν = 4π = π dwdw ( gab a X µ b X ν )η µν dwdw ( wx µ w X ν + w X µ w X ν )η µν ( w = w, w = w ) dwdw w X µ (w, w) w X ν (w, w)η µν w =, w = δx µ S + δs = π = π dwdw((x µ + δx µ )X µ + X µ (X µ + δx µ )) dwdw(x µ X µ + δx µ X µ + X µ δx µ ) δs δs = π = π = π dwdw(δx µ X µ + X µ δx µ ) dwdw( δx µ X µ δx µ X µ ) dwdwδx µ (X µ ) δx µ w, w δs 0 X µ (w, w) = 0 X µ (w, w) = X µ (w) + X µ (w) X µ (w) (holomorphic function) X µ (w) (anti-holomorphic function) X µ (w) = X µ (τ iσ) right moving X µ (w) = X µ (τ + iσ) left moving ( τ ± σ i(τ ± iσ)) w w right left τ, σ w, w X µ w z z = e w = e τ iσ, z = e w = e τ+iσ 3

τ σ z σ σ = σ + π τ ( < τ < ) σ (σ π τ ) z σ π z z = e τ e τ τ (τ = ) τ = τ z e τ τ τ + a σ σ + b τ + a e a e τ iσ = e a z σ + b e ib e τ iσ = e ib z z σ 0 l z = e τ+iσ iσ X µ (τ, σ) X µ (τ, σ) = X µ L (τ + σ) + Xµ R (τ σ) X µ (τ, σ) = X µ (τ, σ + π) X µ L (τ + σ) = xµ 0 + pµ (τ + σ) + i n µ ne in(τ+σ) X µ R (τ σ) = xµ 0 + pµ (τ σ) + i n µ ne in(τ σ) w X µ L (w = τ + iσ) = xµ 0 + pµ ( iτ + σ) + i n µ ne in( iτ+σ) = xµ 0 i pµ (τ + iσ) + i = xµ 0 i pµ w + i n µ ne nw n µ ne n(τ+iσ) z = e w X µ L (z) = xµ 0 i pµ log z + i 4 n µ nz n

X µ R X µ R (z) = xµ 0 i pµ log z + i n µ nz n µ [x µ 0, pν ] = [xµ 0, ν 0] = iη µν [ µ m, ν n] = [ µ m, ν n] = mη µν δ m+n,0 η µν µ n, µ n n µ n = na µ n, µ n = na µ n µ n = n(a µ n), µ n = n(a µ n) n, µ µ n 0 µ n 0 = µ n 0 = 0, 0 µ n = 0 µ n = 0 (n ) p µ p µ 0 = 0 X µ ( )G X(τ, σ ), X(τ, σ ) τ, τ ( ) T (X µ (τ, σ )X ν (τ, σ )) = : X µ (τ, σ )X ν (τ, σ ) : +G(τ, σ ; τ, σ ) T : : G(τ, σ ; τ, σ ) = T (X µ (τ, σ )X ν (τ, σ )) : X µ (τ, σ )X ν (τ, σ ) : τ, σ z, z G(z, z ; z, z ) = T (X µ (z, z )X ν (z, z )) : X µ (z, z )X ν (z, z ) : () τ > τ z, z τ z > z c c τ > τ ( z > z ) 0 X µ (z, z )X ν (z, z ) 0 X µ X µ L + Xµ R 5

X µ (z, z )X ν (z, z ) = (X µ L (z ) + X µ R (z ))(X ν L(z ) + X ν R(z )) = (X µ L (z )X ν L(z ) + X µ R (z )X ν R(z ) + X µ L (z )X ν R(z ) + X µ R (z )X ν L(z )) () X µ L (z )X ν L (z ) 0 X µ L (z )X ν L(z ) 0 = 4 0 xµ 0 xν 0 0 4 log z log z 0 p µ p ν 0 i 4 log z 0 x µ 0 pν 0 + i i 4 log z 0 p µ x ν 0 0 + m 0 m 0 m, m z m 0 x µ 0 ν m 0 m z m 0 p µ ν m 0 + i n z n 0 µ nx ν 0 0 + i n z n 0 µ np ν 0 = 4 0 xµ 0 xν 0 0 m, mn z n z m 0 µ n ν m 0 mn z n z m 0 µ n ν m 0 i 4 log z 0 p µ x ν 0 0 (3) x 0, p µ n ( n n > 0 n < 0 ) 0 µ n ν m 0 m, mn z n z m 0 µ n ν m 0 = = = m= m= m= = η µν n= mn z n zm 0 µ n ν m 0 mn z n zm 0 ([ µ n, ν m] + ν m µ n) 0 mn z n zm nη µν δ n m,0 ( 0 0 = ) n z n zn n m = 0 n n 0 p µ x ν 0 0 0 p µ x ν 0 0 = 0 ([p µ, x ν 0] + x ν 0p µ ) 0 = 0 ( iη µν ) 0 = iη µν 0 X µ L (z )X ν L(z ) 0 = 4 0 xµ 0 xν 0 0 4 ηµν log z + ηµν 6 n= n (z z ) n (4)

X µ R (z )X ν R (z ) z z 0 X µ R (z )X ν R(z ) 0 = 4 0 xµ 0 xν 0 0 4 ηµν log z + ηµν n= n (z z ) n (5) X µ L (z )X ν R (z ) 0 µ n ν m 0 µ n ν m µ n ν m x µ 0 xν 0 xµ 0 pν 0 X µ L (z )X ν R(z ) 0 = 4 0 xµ 0 xν 0 0 i 4 log z 0 x µ 0 pν 0 = 4 0 xµ 0 xν 0 0 4 ηµν log z (6) X µ R (z )X ν L (z ) 0 X µ R (z )X ν L(z ) 0 = 4 0 xµ 0 xν 0 0 4 ηµν log z (7) () (4) (7) 0 X µ (z, z )X ν (z, z ) 0 = 0 x µ 0 xν 0 0 4 ηµν log z + ηµν 4 ηµν log z + ηµν n= 4 ηµν log z 4 ηµν log z n (z z ) n n= n (z z ) n = 0 x µ 0 xν 0 0 ηµν log z ηµν log z + ηµν n= n (z z ) n + ηµν n= n (z z ) n n= x n n = log[ x] n= n (z z ) n = log[ z z ] = log z [z z ] = log z log[z z ] 7

ηµν log z ηµν log z + ηµν n= n (z z ) n + ηµν n= n (z z ) n = ηµν log z ηµν log z + ηµν (log z log[z z ]) + ηµν (log z log[z z ]) = ηµν log[z z ] ηµν log[z z ] 0 X µ X ν 0 0 X µ (z, z )X ν (z, z ) 0 = 0 x µ 0 xν 0 0 ηµν log[(z z )(z z )] (8) z, z (τ, τ ) (µ, ν z, z ) 0 X µ (z, z )X ν (z, z ) 0 = 0 X ν (z, z )X µ (z, z ) 0 (8) () () : p µ x ν 0 := x ν 0p µ, : m n := n m, : m n := n m (m, n ) (3) 0 x µ 0 xν 0 0 (8) 0 : X µ (z, z )X ν (z, z ) : 0 = 0 x µ 0 xν 0 0 0 X µ (z, z )X ν (z, z ) 0 0 : X µ (z, z )X ν (z, z ) : 0 = ηµν log[(z z )(z z )] = ηµν log z z G(z, z ; z, z ) = ηµν log[(z z )(z z )] = ηµν log z z (9) z z (9) z, z z z G(z, z ; z, z ) = z X µ (z, z ) z X ν (z, z ) = ηµν z z z = ηµν (z z ) 8

X(z, z )X(z, z ) z X(z, z ) z X(z, z ) ((8) ) ( ) z z λz ϕ(z, z) ϕ(z, z) ϕ (z, z) = λ h λ h ϕ(λz, λz) h, h (conformal dimension) (h h ) A = ab a λa, b λb A λλa z f(z) ϕ(z, z) ϕ (z, z) = ( f z )h ( f z )h ϕ(f(z), f(z)) (0) primary field ϕ primary field quari-primary field z = z + n ϵ n ( z n+ ), z = z + n ϵ n ( z n+ ) l n = z n+ z, l n = z n+ z z n+ z = 0 z = n ( ) n z = 0 z = z = /v z = v z v = z v = v v l n = z n+ z = ( v )n+ ( v ) v = ( v )n v z = v 0 n + l n l, l 0, l + l, l 0, l +, l, l 0, l + (globally) primary field quasi-primary field primary fieldϕ (u), ϕ (v) (u, v ) ϕ (u)ϕ (v) n = u = u + ϵ = u + a, v = v + a ϕ (u)ϕ (v) ϕ (u + a)ϕ (v + a) ϕ (u)ϕ (v) 9

D(u, v) = ϕ (u)ϕ (v) D(u, v) = D(u v) n = 0 u = u + ϵ 0 ( u) = λu, v = λv primary field ϕ (u)ϕ (v) λ h ϕ (λu)λ h ϕ (λv) = λ h +h ϕ (λu)ϕ (λv) λ h +h ϕ (λu)ϕ (λv) = λ h +h D(λ(u v)) λ h+h D(λ(u v)) = D(u v) D(u v) d D(u v) = () (u v) h+h ( (λu λv) h +h = λ (h +h ) ) (u v) h +h d n = + u = u + ϵ + ( u ) = ( cu)u, v = ( cv)v (x /x) u = u, v = v D(u v) = ϕ (u)ϕ (v) D( u v ) = ϕ ( u )ϕ ( v ) = ϕ ( u u)ϕ ( v v) = u h v h ϕ ( u )ϕ ( v ) () 0

u h v h ϕ ( u )ϕ ( v ) = u h v h d ( u + v )h+h D(u v) = D( u v ) d (u v) h +h = u h v h d ( u + v )h +h h = h = h d u h v h ( u + = d v )h u h v h (u v/uv) h = d (u v) h n =, 0, + u = u + a, v = v + a u = λu, v = λv u = ( cu)u, v = ( cv)v ϕ (u)ϕ (v) ϕ (u)ϕ (v) = d (u v) h l, l 0, l + quasi-primary field 3 (Möbius transformation) z z = az + b cz + d (ad cb =, ( cz)z z cz + ) G(z, z ; z, z ) = X µ (z, z )X ν (z, z = ηµν log[(z z )(z z )] X µ (z, z) (quasi)-primary field (0) z X µ z X µ (z, z ) z X ν (z, z ) = ηµν (z z ) z X µ (z, z) primary field (0)