『共形場理論』

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へいぞうすみい
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1 T (z) SL(2, C) T (z)

2 SU(2)

3 S 1 /Z 2 SU(2) (ŜU(2) k ŜU(2) 1)/ŜU(2) k+1 ŜU(2)/Û(1) G H N =1 N =1

4 N =1 N =1 N =2 N =2 N =2 N =2 ĉ>1 N =2 N =2 N =4 N =4

5 1 2 2 z=x 1 +ix 2 z f(z) f(z) N =4

6 1 = = M 1 1 ϕ : x M x M ϕ 1 diffeomorphism M M N T μ 1,,μ N (x )=T ν1,,ν N (x) xν 1 x μ 1 xνn x μ N, 1.1 x, x 2 [14, 15, 16] [18]

7 1.2 T μ 1,,μ N (x ) dx μ 1 dx μ N = T ν1,,ν N (x) dx ν 1 dx ν N, T T ν1,,ν N (x) dx ν 1 dx ν N T ϕ 3 ϕ T (ϕ T ) μ1,,μ N (x) T ν1,,ν N (ϕ(x)) ϕ(x)ν 1 x μ ϕ(x)νn x μ N ϕ μ (x)=x μ +ɛv μ (x) ɛ v v μ (x) ɛ 2 ϕ T (x) =T (x)+ɛδ v T (x)+o(ɛ 2 ) 1.3 δ v T (ϕ T ) μν (x) T μν (x) ) = T αβ (x+ɛv) (δ αμ +ɛ )(δ vα x (x) βν +ɛ vβ μ x (x) T ν μν (x) ] [ = ɛ v ρ (x) x ρ T μν(x)+t αν (x) vα x μ (x)+t μβ(x) vβ x ν (x) +O(ɛ 2 ), δv T μν (x) =v ρ (x) x T μν(x)+t ρ αν (x) vα x (x)+t μβ(x) vβ (x), 1.4 μ xν N 3 pull-back ϕ(x) 1.3

8 1 δ v T μ1,,μ N (x)=v ν (x) x T ν μ 1,,μ N (x)+ N j=1 T μ1,,μ j 1,ν j,μ j+1,,μ N (x) vν j (x) x μ j 1.5, d M ds 2 =g μν (x)dx μ dx ν ϕ ρ(x) conformal transformation (ϕ g) μν (x) =e ρ(x) g μν (x). 1.6 isometry g μν (x) g μν (x) =e ρ(x) g μν (x) g(u, v) g(u, v) cos θ = =. 1.8 g(u, u) g(v, v) g(u, u) g(v, v) g(u, v) g(u, v)=g μν u μ v ν = 4 conformal 4 ρ(x)

9 1.2 ϕ(x) μ =x μ +ɛv μ (x) 1.6σ(x) δ v g μν (x) =σ(x)g μν (x), 1.4δ v g μν δ v g μν = v ρ ρ g μν +g αν μ v α +g μβ ν v β μ v ν + ν v μ, μ v ν (x)+ ν v μ (x) =σ(x)g μν (x), μ σ(x)= 2 d ρv ρ (x) ɛ μ (x) μ v ν (x)+ ν v μ (x) = 2 d ρv ρ (x)g μν (x), 1.11 v μ (x) g μν conformal Killing vector field 6 d g μν (x)=δ μν 1.11 μ v ν (x)+ ν v μ (x) = 2 d ρv ρ (x)δ μν, d d> δ v g μν ρ g μν =0 6 δ v g μν (x) μ v ν (x)+ ν v μ (x) =0

10 1.1 d a μ x μ =x μ +a μ d ω μν x ν d(d 1) (ω x μ =Λ μ ν xν, (Λ t Λ=I) μν = ω νμ ) 2 ɛx μ x μ =λx μ 1 x 2 b μ 2(b x)x μ x μ = xμ + x 2 b μ d 1+2b x+ b 2 x 2 d SO(d, 2) (d+1)(d+2) SO(d, 2) v 1 x 1 = v2 x 2, v 1 x 2 = v2 x v z v 1 +iv 2 z x 1 +ix 2 ( z=x 1 ix 2 ) 1.13 z z = 1 ( i 2 x 1 x 2 ), z z = 1 ( +i 2 x 1 ), x z vz =0, z z f(z), z z f(z), ( z f(z) = z f(z) =0) z, z 2

11 1.2 ds 2 = 1 2 dz 2, (z, z) (f(z, z), f(z, z)), f z f= z f=0 f ds 2 f (ds 2 )= 1 f f dz 2 z z dz 1 2 f 2 z ds 2, 1.16 f 2 z e ρ =1 U =C U\{0} z=0 z ϕ(z) z+ ɛ n z n n Z z l n z n+1, n Z z 7 =1 v(x) =v μ (x) x, μ

12 1 [l m,l n ]=(m n)l m+n, (m, n Z) CP 1 C { } z= n 1 v(z) = ɛ n z n+1 z. n= 1 z= w= 1 w=0 z v z (z) z =vw (w) w v w (w) =v z (z) w z = vz (z) ( 1z ) = ɛ 2 n z n 1 = ɛ n w n+1, n Z n Z w=0 n ɛ 1, ɛ 0, ɛ 1 {l 1,l 0,l 1 } 1.19 sl(2) [l 0,l ±1 ]= l ±1, [l 1,l 1 ]=2l sl(2) sl(2) SL(2, C) 1 z {( ) a b SL(2, C) SO(2, 2) = a, b, c, d C, ad bc =1} c d 1.21

13 1.2 SL(2, C) v z (z)=ɛ 1 z =z+a 2 v z (z)=ɛ 0 z, (ɛ 0 ir) z =cz, ( c =1) 1 v z (z)=ɛ 0 z, (ɛ 0 R) z =cz, (c>0) 1 v z (z)=ɛ 1 z 2 z = z 1 bz 2 z az+b cz+d c z z =cz, c C\{0} z z (h, h) [21] ( ) dz Φ(z, z) =Φ (z, z h ( d z ) h ) dz d z Φ(z, z)dz h d z h 1.24

14 1 h h h h ω ω(z)dz=ω (z )dz 2 Q Q(z)dz 2 =Q (z )dz 2 (h=1, h=0) (h=2, h =0) z z = e iθ z, z z = e iθ z, θ R, Φ(z, z) =Φ (z, z )e iθ(h h) 1.26 Φ s=h h SO(2)=U(1) s e isθ z z = λz, z z = λ z, λ > Φ(z, z) =Φ (z, z )λ (h+ h) 1.28 Φ Δ=h+ h h, h Δ s h+ h = Δ, h h = s h, h 0 h, h

l µ l µ l 0 (1, x r, y r, z r ) 1 r (1, x r, y r, z r ) l µ g µν η µν 2ml µ l ν 1 2m r 2mx r 2 2my r 2 2mz r 2 2mx r 2 1 2mx2 2mxy 2mxz 2my r 2mz 2 r

l µ l µ l 0 (1, x r, y r, z r ) 1 r (1, x r, y r, z r ) l µ g µν η µν 2ml µ l ν 1 2m r 2mx r 2 2my r 2 2mz r 2 2mx r 2 1 2mx2 2mxy 2mxz 2my r 2mz 2 r 2 1 (7a)(7b) λ i( w w ) + [ w + w ] 1 + w w l 2 0 Re(γ) α (7a)(7b) 2 γ 0, ( w) 2 1, w 1 γ (1) l µ, λ j γ l 2 0 Re(γ) α, λ w + w i( w w ) 1 + w w γ γ 1 w 1 r [x2 + y 2 + z 2 ] 1/2 ( w) 2 x2 + y 2 + z 2