SO(4, C) SL(2, C) SL(2, C) SL(2, C) positive chirality spinor : ψ a, negative chirality spinor : ψȧ. (1) SL(2,

Transcription

1 h = Cech h = +1/ h = N = 4 SYM hcs MHV MHV (disconnected prescription) ,,, MHV (connected prescription) i 1

2 SO(4, C) SL(2, C) SL(2, C) SL(2, C) positive chirality spinor : ψ a, negative chirality spinor : ψȧ. (1) SL(2, C) ɛ ψ a = ɛ ab ψ b, ψȧ = ɛȧḃ ψḃ. (2) ɛ 12 = ɛ 1 2 = ɛ 12 = ɛ 1 2 = 1 ɛ ab ɛ ab 4 (σ µ ) aḃ (σ µ ) aḃ = (1, iσ x, iσ y, iσ z ) (3) (σ µ )ḃa = (σ µ ) aḃ (σ µ )ȧa = ɛȧḃ (σµ )ḃb ɛ ba = (1, iσ x, iσ y, iσ z ) (4) ψ a χȧ 4 ( γ µ (σ µ ) aḃ ) =. (5) (σ µ )ȧb {γ µ, γ ν } = 2δ µν SO(4, C) Λ µν C T Λ = (1/4)Λ µν γ µν Λ µν SO(4)(= SO(4, R)) SU(2) SU(2) T Λ (T Λ ) = M 1 T Λ M, M = ( ɛ ɛ ). (6) ψ a χȧ 2

3 Λ 0m = Λ m0 m = x, y, z SO(3, 1) T ( ) (T Λ ) = M 1 ɛ T Λ M, M =. (7) ɛ ψ a χȧ Λ im = Λ mi m = x, z i = 0, y SO(2, 2) SL(2, R) SL(2, R) (T Λ ) = T Λ (8) ψ a χȧ 1: (+ + ++) (ψ a ) = (ψ ) a, (χȧ) = (χ )ȧ. (+ + + ) (ψ a ) = (ψ )ȧ, (χȧ) = (χ ) a. (+ + ) (ψ a ) = (ψ ) a, (χȧ) = (χ )ȧ. SO(3, 1) SO(3) S xy = 1 ( (σz ) a ) b (9) 2 ḃ (σ z )ȧ S yz S zx ψ, χ = ψ a χ a = ɛ ab ψ a χ b, [ ψ, χ] = ψȧ χȧ = ɛȧḃ ψȧ χḃ. (10) ɛ ab s ψ(x) = ζe ipx. (11) 3

4 p ψ ζ ζ ψ x e ipx ψ(p) 0 p 2 = 0 (12) (h = 0) ψ ζ ζ = 1 (h = ±1/2) s = 1/2 ψ a (x) = u a e ipx ψȧ(x) = vȧe ipx pȧa u a = 0, p aȧ vȧ = 0. (13) (13) z p 0 = p 3 = E pȧa = p µ σ µ ȧa = ( 2E ). (14) (13) (u 1, u 2 ) = (0, 1) (v 1, v 2) = (1, 0) ψ a h = 1/2 ψȧ h = +1/2 (h = ±1) s = 1 U(1) F µν F aȧb ḃ = F abɛȧḃ + F ȧḃɛ ab (15) F ab Fȧḃ F µν µ F µν = 0 [µ F νρ] = 0 F ab Fȧḃ pȧa F ab (p) = 0, p aȧ Fȧḃ (p) = 0. (16) z p (14) (16) F 1a = 0 F 2ȧ = 0 0 F 22 F 1 1 h = 1 h = +1 4

5 (h = ±3/2) ψµ a δψµ a = µ ɛ a ψµν a = 2 [µ ψν] a ψ c ab ψȧḃ c ψ c b ab γ- ψ ab 3 ψµν a : [(3, 1) (1, 3)] (2, 1) = (4, 1) (2, 1) (2, 3). (17) (σ µ ψ µν )ȧ = 0 (2, 2) (1, 2) = (2, 1) (2, 3) 0 (4, 1) ψ abc 3 [µ ψνρ] a = 0 ȧa ψ abc = 0. (18) z ψ 1ab = 0 0 ψ 222 h = 3/2 h = +3/2 (h = ±2) s = 2 R µνρσ 4 µ ν ρ σ F µν [(3, 1) (1, 3)] [(3, 1) (1, 3)] = (5, 1) (1, 5) (3, 3) (3, 1) (1, 3) (1, 1) (3, 3) (1, 1) (19) R [µνρ]σ = 0 (2, 2) (2, 2) = (3, 3) (3, 1) (1, 3) (1, 1) 0 (19) R µν = 0 (3, 3) (1, 1) 0 3 (5, 1) (1, 5) 4 R µνρσ (σ µ ) aȧ (σ ν ) b ḃ (σρ ) cċ (σ σ ) d d = ɛ ȧḃɛ ċ d R abcd + ɛ ab ɛ cd Rȧḃċ d (20) [λ R µν]ρσ = 0 R abcd Rȧḃċ d pȧa R abcd (p) = 0, p aȧ Rȧḃċ d (p) = 0. (21) z 0 R 2222 (p) R (p) h = 2 h = +2 5

6 h ȧa ψ ab c (x) = 0 (h 1/2), (22) aȧ ψȧḃ ċ (x) = 0 (h 1/2), (23) aȧ aȧ ψ(x) = 0 (h = 0). (24) ψ ab c (x) 2h = 2 h ψȧḃ ċ (x) 2h z p aȧ = λ a λȧ (25) λ, λ [ λ, λ] 0 (25) on-shell (12) p (12) (25) (25) (11) ζ ab c = λ a λ b λ c (h < 0), (26) ζȧḃ ċ = λȧ λḃ λċ (h > 0). (27) (25) p µ (λ, λ) η µν = diag( + ++) λ = λ η µν = diag( ++) λ = λ λ = λ tree (27) (26) ζ deg λ [ζ] deg λ[ζ] = 2h (28) deg X [F ] F X h = 0 ζ = (25) ζ 2h λ αλ, λ 1 α λ. (29) ζ F µν R µνρσ ɛ (28) 6

7 U(1) A µ (x) = ɛ µ e ipx Maxwell p µ ɛ µ = p a ḃ ɛaḃ = 0 (30) h = ±1 ɛ h λ λ ɛ aḃ +1 = µa λḃ µ, λ, ɛaḃ 1 = µḃλ a [ µ, λ] (31) µ µ 0 ɛ ±1 (28) µ µ µ µ ɛ +1 µ µ 2 µ, λ = 0 µ λ δµ = αµ + βλ α µ ɛ +1 β δɛ +1 p ɛ +1 µ ɛ µ ɛ µ ±1 ±1 (27) (26) F µν = p µ ɛ ν p ν ɛ ν F aȧb ḃ (p) = F µν(σ µ ) aȧ (σ ν ) b ḃ (p) = p [a {ȧ ɛ b] ḃ} + p {a [ȧ ɛ b} ḃ] (32) (25) (31) ɛ +1 p ɛ +1 λ 0 (32) F aȧb ḃ (p) = 1 λ, µ λ [aµ b] λȧ λḃ = 1 2 ɛ ab λȧ λḃ (33) Fȧḃ (p) = (1/2) λȧ λḃ h = +1 (27) µ h µν = g µν η µν ɛ aȧbḃ +2 = µa λȧµ b λḃ µ, λ, 2 ɛaȧbḃ 2 = µȧλ a µḃλ b [ µ, λ] 2 (34) µ µ 0 (28) 7

8 1.3 h < 0 (22) ψ ab c (x) 2h 1.2 p (11) ζ (26) ψ ab c (x) = λ a λ b λ c f(x). (35) λ a 0 (35) (26) f(x) λ a aȧ f(x) = 0. (36) g f(x) = g(λ a x aȧ ) δ f (λ,µ) (x) = δ 2 (λ a x aȧ + µȧ) (37) δ 4 α-plane 2 α-plane (λ, µ) α-plane (λ, µ) (αλ, αµ) (38) CP 3 projective twistor space twistor T (37) x (λ, µ) λ a = 0 twistor T CP 3 CP 3 λ a = 0 V 1 = {z λ 1 0}, V 2 = {z λ 2 0}. (39) V 1 V 2 C 3 V 1 V 2 V 12 α-plane δ (37) twistor space ϕ(λ, µ) x ψ ab c (x) = Ωλ a λ b λ c δ 2 (λ a x aȧ + µȧ)ϕ(λ, µ) (40) T Ω twistor space 3 z I = (λ a, µȧ) Ω = 1 4! ɛ IJKLz I dz J dz K dz L (41) T projective space (λ, µ) 4 3 8

9 (40) twistor deg z [ϕ(z)] = 2h 2 (42) (37) δ x twistor T x D x D x T P 1 λ 1 λ 2 0 D x (39) V i U i = V i D x. (43) U 12 = U 1 U 2 (40) x x D x ϕ(z) (40) Ω = λ, dλ d 2 µ µ δ ψ ab c (x) = λ, dλ λ a λ b λ c ρ x ϕ(z) (44) C λ, dλ D x ρ x ϕ(z) ϕ(z) D x ρ x ϕ(z) = ϕ(λ a, λ a x aȧ ). (45) λ a (44) U 12 C U h = 0 subsection h 1/2 h 0 (42) twistor h = 0 twistor space 2 ϕ(z) (24) φ(x) φ(x) = 1 ρ x ϕ(z) λdλ (46) 2πi C measure λ, dλ 2 ϕ(z) 2 0 P 1 well-defined 9

10 (46) φ(x) φ(x) x aȧ ρ x ϕ(z) x (45) aȧ φ(x) = 1 2πi x aȧ ϕ(λa, x aȧ λ a ) λdλ = 1 λ a 2πi ϕ(λa, µȧ) λdλ (47) µȧ b ḃ aȧφ(x) = 1 λ b λ a 2πi ϕ(λa, µȧ) λdλ (48) µḃ µȧ a b ȧ ḃ 0 φ(x) (24) ϕ(z) (46) φ(x) φ(x) ϕ(z) φ(x) ϕ(z) ϕ (z) = ϕ(z) + h 1 (z) h 2 (z). (49) h i (z) V i ϕ(z) ϕ (z) ϕ(z) twistor space Cech co-homology Ȟ1 (T, O(n)) 1.5 Cech X Cech cohomology 1. U = {U i } 2. {f i } 0 co-chain 0 co-chain C 0 (U) twistor n C 0 (U, O(n)) 3. U i U j {f ij } f ij = f ji 1 co-chain C 1 (U) 4. U i0 U ip {f i0 i p } p co-chain C p (U) 5. ρ i f U f U i U C 0 (U) C 1 (U) δ f ij = ρ i f j ρ j f i (50) 10

11 6. C p 1 (U) C p (U) δ f i jk = (p + 1)ρ [k f i j]. (51) 7. δ co-cycle co-boundary δ co-homology Cech co-homology Ȟ p (U) 8. cover U Ȟ p (U) U cohomology Ȟp (X) P 1 Cech cohomology z 1 z 2 z i 0 U i U = {U 1, U 2 } U i f i f 1 (z 1, z 2 ) = a k z1 n k z2, k f 2 (z 1, z 2 ) = b k z1z k 2 n k. (52) k=0 co-chain {f 1, f 2 } C 0 (U) co-cycle U 1 U 2 f 12 = f 1 f 2 = 0 (52) n 1 a k = b k = 0 Ȟ 0 (U, O(n)) = 0 n 0 z 1 z 2 n f 1 (z 1, z 2 ) = f 2 (z 1, z 2 ) = a k z1 n k z2. k (53) k=0 Ȟ 0 (U, O(n)) (a 0,, a n ) C n+1 Ȟ 1 U 1 U 2 f 12 (z 1, z 2 ) f 12 (z 1, z 2 ) = a k z1 n k z2. k (54) k= 2 co-chain 1 co-cycle (52) f 1 f 2 co-homology 0 n 1 Ȟ 1 = 0 n 2 f 12 (z 1, z 2 ) = n 1 k=1 k=0 a k z n+k 1 z k 2. (55) z 1 z 2 f 1 f 2 Ȟ 1 (a n+1,..., a 1 ) C 1 n n Ȟ 0 (P 1, O(n)) C C 2 C 3 Ȟ 1 (P 1, O(n)) C 3 C 2 C (56) Cech cohomology Ȟ 1 (X, O(n)) = H 1 (X, O(n)) (57) 11

12 1.6 h = +1/2 h = +1/2 twistor space Ȟ1 (T, O( 1)) ϕ(λ, µ) D x g(λ, x) g(λ, x) = ρ x ϕ(z) = ϕ(λ, xλ). (58) H 1 (D x, O( 1)) = 0 g(λ, x) = g 1 (λ, x) g 2 (λ, x) (59) g 1 U 1 λ 1 0 g 2 U 2 λ 2 0 g 1 (λ, x) = 1 λ, dλ 2πi C 1 λ, λ g(x, λ ), g 2 (λ, x) = 1 λ, dλ 2πi C 2 λ, λ g(x, λ ) (60) C 1 C 2 )+*-,/."0213*-4(5 "!$#%&(' 6 * *-4(5 1: C 1 C 2 C 1 C 2 λ C 1 C 2 g(λ, x) g (58) chain rule λ a aȧ g(x, λ) = λ a λ a ρ x ϕ(z) = 0. (61) µȧ λ a aȧ g 1 (x, λ) = λ a aȧ g 2 (x, λ) g 1 g 2 D x 0 λ λ x x ψȧ ψȧ(x) = λ a aȧ g 1 (x, λ). (62) Dirac 12

13 ψȧ (62) (60) ψȧ ψȧ(x) = λ a aȧ g 1 (λ, x) = 1 λ, dλ 2πi C 1 λ, λ λa aȧ g(x, λ ) = 1 λ, dλ 2πi C 1 λ, λ λ, λ ρ x f(λ, µ) µȧ = 1 λ, dλ ρ x 2πi f(λ, µ) (63) C µȧ λ = λ pole C 1 C 2 b ḃ ψ a(x) = 1 2πi λ b f(z) λ, dλ (64) µḃ µȧ ȧ ḃ 0 Dirac aȧψȧ = 0 h 1/2 (23) (42) ϕ(z) ψȧ ḃċ (x) = 1 2πi C ρ x ϕ(z) λ, dλ (65) µȧ µḃ µċ 1.7 h = +1 F ab = 0 GL(N) A aȧ F ab = 0 A [28] anti-self dual GL(N) A(x) D x trivial P T N holomorphic vector bundle E (λ, µ) α-plane v aȧ = λ a ηȧ λ a λ b ɛȧḃ α-plane z 13

14 α-plane ψ N λ a D aȧ ψ = 0. (66) N z P 1 P 1 x P 1 D x D x x α-plane x ψ α-plane D x N (39) V 1 λ 1 0 (λ 1, λ 2, µ 1, µ 2 ) = (0, 1, 0, 0) p α-plane xȧ2 = 0 p V 1 z α-plane z xȧa λ a + µȧ = 0 p z (xȧ1, xȧ2 ) = ( µȧ/λ 1, 0) p z p α-plane p p 0 p 0 v i (i = 1,..., N) z x v i ( ) v i (x) = P exp v i (67) x p z p 0 A V 2 (λ 1, λ 2, µ 1, µ 2 ) = (1, 0, 0, 0) q α-plane q ( ) w i (x) = P exp A w i (68) x q z q 0 w i (x) = M ij (x)v i (x) (69) ( ) P exp w i = M ij (x)v i (70) p 0 p z x q z q 0 A M ij (x) z x z M ij V 12 holomorphic transition matrix U(λ, µ) GL(N) vector bundle E 14

15 D x D x C N x α-plane α-plane x y D x D y x y α-plane W (λ, x) U(z) D x W (λ, x) = U(λ, xλ). (71) U degree 0 W λ degree 0 W chain rule λ a aȧ W = 0. (72) D x triviality U 12 W = W 1 W2 1 (72) W1 1 (λ a aȧ )W 1 = W2 1 (λ a aȧ )W 2 λ 2 = 0 regular λ 1 = 0 regular regular 1 λ A aȧ (x) W 1 1 (λ a aȧ )W 1 = λ a A aȧ (λ, x). (73) A λ a D aȧ = λ a aȧ + λ a A aȧ (x) (74) (73) λ a λ b F ab ɛȧḃ = [λa D aȧ, λ b D b ḃ ] = 0 (75) F ab = 0 A anti-selfdual vector bundle E 3 bector bundle F G H F G injective linear map A G H surjective linear map B F A G B H (76) BA = 0 vector bundle E E = KerB/ Im A (77) F = k O( 1) H = k O(+1) G 2k + N trivial bundle A k (2k + N) B (2k + N) k z A a z a B a z a dim Im A = k dim KerB = k + N E N vector bundle E ADS ADHM [27] [28] 15

16 1.8 (25) d 4 pδ(p 2 ) = d3 p E = λ, dλ d2 λ (78) 4 3 (78) (29) λ 1 = c c p off-shell p 1ȧ = c λȧ, p 2ȧ = λ 2 λȧ + ɛκȧ. (79) κȧ [ λ, κ] 0 ɛ mass-shell p µ 4 (ɛ, λ 2, λ 1, λ 2 ) δ(p 2 ) d 4 p = c 2 [ λ, κ]dɛdλ 2 d 2 λ, δ(p 2 ) = δ(ɛ) c[ λ, κ] (80) ɛ (78) (79) δ 2 (ξ a p aȧ ) = δ(p 2 )δ( ξ, λ )(λ/ξ) (81) ξ 0 (λ/ξ) δ( ξ, λ ) λ ξ well-defined (78) δ E 1 δ 3 (p 1 p 2 ) = δ( λ 1, λ 2 )(λ 2 /λ 1 )δ 2 ( λȧ1 (λ 2 /λ 1 ) λȧ2) (82) p aȧ i = λ a i λȧi i = 1, 2 δ( λ 1, λ 2 ) λ, dλ δ( λ, λ )f(λ) = f(λ ) (83) well-defined f(λ) λ 1 16

17 1.9 twistor ϕ(z) ψ(p) ψ(x) h 0 ψ ab c (x) λ a λ b λ c δ 2 (λx + µ) ϕ(z) ψ ab c (x) e ipx ψ ab c (p) ψ ab c (p) λ λ ψ ab c (p) = λ a λ b λ c δ(p 2 )φ(λ, λ) (84) (deg λ deg λ)[φ(λ, λ)] = 2h (85) twistor ϕ(z) ψ ab c (p) φ(λ, λ) (27) (44) λ (26) (65) λȧ µȧ (86) λ µ (29) (38) (86) f(x) f(p) deg x [f] + deg p [ f] = 1 (87) (85) (42) ϕ(z) δ δ(p 2 )F (p ; λ, µ) = d 4 xe ip x δ 2 (λ a x aȧ + µȧ) (88) p p λ (λ, λ ) massless δ(p 2 ) F (p ; λ, µ) x x aȧ = λ a yȧ + ζ a zȧ, d 4 x = λ, ζ 2 d 2 yd 2 z. (89) 17

18 ζ a λ a δ(p 2 )F (p ; λ, µ) = d 2 yd 2 z λ, ζ 2 exp(iλ a p aȧ yȧ + iζ a p aȧ zȧ)δ 2 ( λ, ζ zȧ µȧ) ( = d 2 y exp iλ a p aȧ yȧ + i ζ ap aȧ ) µȧ λ, ζ ( = δ 2 (λ a p aȧ ) exp i ζ ap aȧ ) µȧ λ, ζ = δ(p 2 )δ( λ, λ )(λ /λ)e i(λ /λ)[ λ,µ]. (90) (81) λ λ p on-shell λ /λ δ( λ, λ ) λ λ well-defined F (p ; λ, µ) F (p ; λ, µ) F (λ, λ ; λ, µ) = δ( λ, λ )(λ /λ)e i(λ /λ)[ λ,µ]. (91) helicity h twistor ψ ab c (p ) = ΩF (λ, λ ; λ, µ)λ a λ b λ c ϕ(λ, µ) (92) φ(λ, λ ) (84) φ(λ, λ ) φ(λ, λ ) = ΩF (λ, λ ; λ, µ)(λ /λ) 2h ϕ(λ, µ) = d 2 µe i[ λ,µ] ϕ(λ, µ). (93) φ(λ, λ) ϕ(λ, µ) λ µ F (p; λ, µ) ΩF (λ λ ; λ, µ)f (λ λ ; λ, µ) = δ( λ, λ )(λ /λ )δ 2 ( λ ȧ (λ /λ ) λ ȧ ) (94) (82) δ (93) ϕ(λ, µ) = λ, dλ d 2 λ F (λ, λ ; λ, µ)(λ /λ) 2h φ(λ, λ ) (95) twistor p aȧ i = λ a i λȧi ϕ i(z) i ϕ i (λ, µ) = F (λ i, λ i ; λ, µ)(λ i /λ) 2h i = (λ i /λ) 1 2h i δ( λ, λ i ) exp(i(λ i /λ)[µ, λ i ]) (96) 18

19 1.10 twistor conformal spinor conformal generator twistor conformal λ a λȧ J ab = i ( ) λ a 2 λ + λ b b, λ a Jȧḃ = i ( ) λȧ + λḃ (97) 2 λḃ λȧ p p a ḃ = λ a λḃ (98) λ λ mass dimension 1/2 conformal boost mass dimension P 1 K a ḃ = λ a λḃ dilatation D λ λ mass dimension 1/2 D = i ( λ a 2 λ + ) λȧ a + 2 (100) λȧ K P amplitude (120) O (99) deg λ [O] deg λ[o] = 0 (101) twistor λ = i, µȧ = iµ ȧ. (102) λȧ deg λ[o] = deg µ [O] (101) deg λ [O] + deg µ [O] = 0 (103) O (λ, µ) J ab = i ( ) λ a 2 λ + λ b b, λ a Jȧḃ = i ( ) µȧ 2 + µ ḃ, µḃ µȧ p aȧ = iλ a, K aȧ = iµȧ µȧ λ, D = i ( λ a a 2 λ ) a µȧ. (104) µȧ 4 (λ, µ) SL(4) twistor conformal 19

20 1.11 twistor Yang-Mills super twistor space N = 4 N = R- SU(4) R δf ab = ξ Iċ Dċa χ Ib, δχ Ia = F ab ξ I b ɛijkl D aḃφ JK ξ L ḃ, δφ IJ = ɛ IJKL ξ K a χ al + ξ [Iȧ χȧj] δχ Iȧ = Fȧḃ ξḃ I + Dȧb φ IJ ξ Jb, δfȧḃ = ξ Ic D cȧ χ I ḃ. (105) I, J,... R- (F ab, χ Ia, φ IJ, χȧi, F ȧḃ) on-shell (G, χ I, φ IJ, χ I, A) F ab = λ a λ b G, χ Ia = λ a χ I, φ IJ = φ IJ, χȧi = λȧχ I, F ȧḃ = λȧ λḃa (106) on-shell λ λ δg = ξ Iȧ λȧχ I, δχ I = ξ Ia λ a G ɛijkl ξ Kȧ λȧφ IJ, δφ IJ = ɛ IJKL ξ K a λ a χ L + ξ [Iȧ λȧχ J], δχ I = ξȧ I λȧa + ξ Ja λ a φ IJ, δa = ξ Ia λ a χ I. (107) R- 4 ψ I ϕ(λ, λ, ψ) = A + ψ I χ I + ψ I ψ J φ IJ + ɛ IJKL ψ I ψ J ψ K χ L + ɛ IJKL ψ I ψ J ψ K ψ L G. (108) ( ) δϕ(λ, λ, ψ) = ξ Ia λ a ψ + ξiȧ λȧψ I ϕ(λ, λ, ψ). (109) I ψ I R- η I (108) G A ϕ(λ, λ, η) = G + η I χ I + ɛ IJKL η I η J φ KL + ɛ IJKL η I η J η K χ L + ɛ IJKL η I η J η K η L A. (110) 20

21 ϕ(λ, λ, ψ) ϕ(λ, λ, η) ϕ(λ, λ, ψ) = d 4 ηe iψi η I ϕ(λ, λ, η). (111) deg ψ [ f(ψ)] + deg η [f(η)] = 1. (112) (87) dη η 1 δ δ(η) = η 1 ϕ(z, η) ( δϕ(λ, λ, η) = ξ Ia λ a η I + ξ ) Iȧ λȧ ϕ(λ, λ, η). (113) η I λ µ (deg λ + deg µ + deg ψ )[ϕ(λ, µ, ψ)] = 0, (114) (deg λ + deg µ deg η )[ϕ(λ, µ, η)] = 4. (115) ψ µ ψ (λ, µ, ψ) 0 λ αλ, µ αµ, ψ A αψ A. (116) super twistor space 4 (λ, µ) 4 ψ A (116) P 3 4 Q Ia = λ a ψ, I Qİ a = ψ I. (117) µȧ R- L I J = ψ I ψ 1 J 4 δi Jψ K ψ. (118) K PSU(4 4) h i (96) super twistor space ϕ i (λ, µ, ψ A ) = (ψ A ) 2 2h i (λ i /λ) 1 2h i δ( λ, λ i ) exp(i(λ i /λ)[µ, λ i ]) (119) (ψ A ) 2 2h i 4 ψ A 2 2h i h i 4C 2 2hi vector multiplet helicity 21

22 2 2.1 n tree 1 n color ordering color tr(t a1 T a2 T an ) CPT h h p i ɛ i A = A(p i, ɛ i ) λ i λ i A = A(λ i, λ i ) A(p i, ɛ i ) ɛ i i λ i λ i (28) deg λi [A] deg λi [A] = 2h i (120) Amplitude A(λ i, λ i ) twistor A(λ i, µ i ) = n ( i=1 d 2 λ ) i e i[µ i, λi ] A(λ i, λ i ) (121) (87) (120) deg λi [A] + deg µi [A] = 2 2h i (122) (λ i, µ i ) A 2 2h i amplitude A(z 1, z 2,..., z n ) twistor ϕ i (z i ) M = ( ) Ω i ϕ i (z i ) A(z 1, z 2,..., z n ) (123) i Ω i z i (41) SL(4) well defined (38) ϕ A (42) (122) η A η deg η [A(λ i, λ i, η i,a )] = 2 2h i. (124) 22

23 η A component field amplitude (120) (124) deg λi [A] deg λi [A] deg η [A] = 2 (125) amplitude A(λ i, µ i, ψi A ) deg λi [A] + deg µi [A] + deg ψi [A] = 0 (126) ψ(λ, µ, ψ) supert wistor space P 3 4 measure (41) Ω = ɛ abcd z a dz b dz c dz d dψ 1 dψ 2 dψ 3 dψ 4 (127) (116) well-defined M = ( ) Ω i ϕ i (z i, ψ i ) A(z i, ψ i ) (128) i Yang-Mills on-shell 3 p 1 p 2 p 3 (3, 1) 3 (2, 2) (2, 2) 3 0 = λ a 1( p iaȧ ) λȧ2 = λ 1, λ 3 [ λ 3, λ 2 ] (129) i=1 λ 1, λ 3 = 0 [ λ 3, λ 2 ] = 0 i j λ i, λ j = 0 λ i (i = 1, 2, 3) p [µ 1 p ν] 2 ɛȧḃ[ λ 1, λ 2 ]λ {a 1 λ b} 2 (130) (null-plane) [ λ 3, λ 2 ] = 0 i j [ λ i, λ j ] = I 3 = I I I 312, I ijk = tr(f µν (i) A (j) µ A (k) ν ) (131) 23

24 λȧ2µ a I 123 = (ɛ ab λ 1ȧ λ 1 ḃ ) 2 λ 2, µ 2 λḃ3µ b 3 λ 3, µ 3 = [ λ 1, λ 2 ][ λ 1, λ 3 ] µ 2, µ 3 λ 2, µ 2 λ 3, µ 3 (132) µ i µ 1 = µ 2 = µ 3 (132) µ 2, µ 3 0 µ i (132) 0 3 I I I 231 = [ λ 1, λ 2 ]([ λ 1, λ 3 ] λ 1, µ 1 µ 2, µ 3 [ λ 2, λ 3 ] λ 2, µ 2 µ 3, µ 1 ) λ 1, µ 1 λ 2, µ 2 λ 3, µ 3 = [ λ 1, λ 2 ][ λ 1, λ 3 ]( λ 1, µ 1 µ 2, µ 3 λ 1, µ 2 µ 1, µ 3 ) λ 1, µ 1 λ 2, µ 2 λ 3, µ 3 = [ λ 1, λ 2 ][ λ 1, λ 3 ] λ 1, µ 3 µ 2, µ 1 λ 1, µ 1 λ 2, µ 2 λ 3, µ 3 = [ λ 3, λ 2 ][ λ 3, λ 1 ] µ 1, µ 2 λ 1, µ 1 λ 2, µ 2 = I 312 (133) tree level gluon amplitude 1 F µν ± h = 1 A = F a1 b 1 (x 1 )F a2 b 2 (x 2 ) F an b n (x n ) (134) N = 1 {Q a, λ b } = F ab, [Q a, F bc ] = 0. (135) A = {Q a1, λ b1 (x 1 )F a2 b 2 (x 2 ) F an b n (x n )} (136) Q 0 = 0 Q =

25 h 1 = h 2 = +1 h 3 = 1 I ijk λȧ2µ a I 123 = (ɛ ab λ 1ȧ λ 1 ḃ ) 2 λ b 3 µḃ3 λ 2, µ 2 [ λ 3, µ 3 ] = [ λ 1, λ 2 ] [ λ 1, µ 3 ] λ 3, µ 2 [ λ 3, µ 3 ] λ 2, µ 2, (137) λḃ1µ b 1 I 231 = (ɛ ab λ 2ȧ λ 2 ḃ ) λa 3 µȧ3 [ λ 3, µ 3 ] λ 1, µ 1 = [ λ 1, λ 2 ] [ λ 2, µ 3 ] λ 3, µ 1 [ λ 3, µ 3 ] λ 1, µ 1, (138) λȧ1µ a 1 I 312 = (ɛȧḃ λ 3aλ 3b ) λ 1, µ 1 λḃ2µ b 2 λ 2, µ 2 = [ λ 1, λ 2 ] λ 3, µ 1 λ 3, µ 2 λ 1, µ 1 λ 2, µ 2 [ λ 1, λ 2 ] 0 λ i λ 3, µ 1 / λ 1, µ 1 λ 3 /λ 1 (139) I I 231 = [ λ 1, λ ( 2 ] [ λ 1, µ 3 ] λ 1λ 3 + [ λ 2, µ 3 ] λ ) 2λ 3 = [ λ 1, λ 2 ] λ 3λ 3 = I 312 (140) [ λ 3, µ 3 ] λ 1 λ 2 λ 1 λ 2 λ 1 λ 2 I I I 312 = 2I 312 λ 3 = λ 3 [ λ 3, λ 2 ] λ 1 λ 1 [ λ 3, λ 2 ] = λ 1 [ λ 1, λ 2 ] λ 1 [ λ 3, λ 2 ] = [ λ 1, λ 2 ] [ λ 3, λ (141) 2 ] 1 2 [ λ 1, λ 2 ] 3 I I I 312 = 2 [ λ 2, λ 3 ][ λ 3, λ 1 ] (142) 0 λ 1, λ 2 3 I( +) = 2 λ 2, λ 3 λ 3, λ 1 (143) x µ P 1 D x 4 D x (+, +, )

26 2: null plane λ D x (+, +, ) (+ + ) (+ + ) Yang-Mills ( ) S = d 4 x tr F ab G ab + g2 2 G2 ab (144) F ab F = da + A 2 G ab F ab G ab + + Yang-Mills G ab G ab = g 2 F ab S = 1 2gYM 2 d 4 x tr(f +2 ) = 1 4gYM 2 d 4 x[tr(f +2 + F 2 ) + tr(f +2 F 2 )] (145) F F (144) GdA GAA g 2 G 2 3 g (144) Yang-Mills 3: F -G ( + +) ( +) 26

27 G ab F ab G ab 1 GdA F ab = 0 A +1 ( + +) (144) GAA ( +) F -G GAA GG ( + +) g ( +) g 2 g 0 G g 1 G A ga g = 0 ( +) S = d 4 x tr(f ab G ab ) (146) A G ϕ A ϕ G (42) ϕ A 0 ϕ G 4 O O( 4) I int = Ω 3 ϕ A ϕ A ϕ G. (147) 0 well-defined (96) h i I int = Ω 3 ϕ h 1 ϕ h 2 ϕ h 3. (148) 0 h 1 + h 2 + h 3 = 1. (149) (147) h 1 = h 2 = +1 h 3 = 1 (96) M = 3 Ω 3 i=1 [ (λi /λ) 1 2h i δ( λ, λ i ) exp(i(λ i /λ)[µ, λ i ]) ] (150) Ω 3 = λ, dλ d 2 µ µ δ ( ) 3 A = λ, dλ (λ i /λ) 1 2h i δ( λ, λ i ) δ 2 ( (λ i /λ) λȧi ) (151) i i=1 27

28 λ 0 λ 2 = 1 ( 3 ) A = dλ 1 3 (λ 2 i ) 1 2h i δ(λ 1 i λ 1 λ 2 i ) δ 2 ( λ 2 i λȧi ) (152) i=1 λ 1 δ(λ 1 3 λ 1 λ 2 3) amplitude ( ) λ 2 1 2h1 ( ) A = 1 λ 2 1 2h2 2 3 (λ 2 λ 2 3 λ 3) 2 δ( λ 2 3, λ 1 )δ( λ 3, λ 2 )δ 2 ( λ 2 i λȧi ) (153) 3 i=1 δ 3 λ i δ 3i=1 λ a i λȧi = 0 δ δ 4 ( 3 i=1 λ a i λȧi ) (153) δ i=1 2 λȧi λ 2 i = λȧ3λ 2 3 (154) i=1 λ 2 1 λ2 2 λ 2 1 = [2, 3] [1, 2] λ2 3, λ 2 2 = [3, 1] [1, 2] λ2 3. (155) [1, 2]2 δ( 3, 1 )δ( 3, 2 ) = (λ 2 3) 2 δ(λ1 1[1, 2] + λ 1 3[3, 2])δ(λ 1 2[2, 1] + λ 1 3[3, 1]). (156) [1, 1] = [2, 2] = 0 [1, 2]2 3 3 δ( 3, 1 )δ( 3, 2 ) = (λ 2 3) δ( λ 1 2 i [i, 2])δ( λ 1 i [i, 1]). (157) i=1 i=1 δ argument δ( 3, 1 )δ( 3, 2 ) = [1, 2] 3 (λ 2 3) 2 δ2 ( λ 1 i λȧi ) (158) i=1 (153) (155) amplitude 3 A = [2, 3] 1 2h 1 [3, 1] 1 2h 2 [1, 2] 1 2h 3 δ 4 ( λ a i λȧi ) (159) i=1 h 1 = h 2 = +1 h 3 = 1 (142) 4 (146) (147) Cech cohomology ϕ A ϕ G (0, 1)- A G 28

29 H 1 Cech Ȟ 1 [4] appendix A(z, z) = ϕ A (z) i θ(z, z)dz i, G(z, z) = ϕ G (z) i θ(z, z)dz i. (160) A G H 1(CP 3, O) H 1(CP 3, O( 4)) (147) 3 I int = Ω 3 tr(a A G) (161) Ω 3 CP 3 holomorphic 3- (160) (147) I = Ω 3 tr[(da + A A) G] (162) δa = dλ A + [A, λ A ], δg = [G, λ A ] + dλ G + [A, λ G ]. (163) λ A λ G N = 4 SYM hcs Witten B-model D1-brane N = 4 Yang-Mills + + N = 4 holomorphic CS R- SU(4) ξ Ia ξȧ I F ab, χ Ia, φ IJ, χ Iȧ, Fȧḃ. (164) + ++ (F ab ) = Fȧḃ, (χia ) = χȧi, (φ IJ ) = 1 2 ɛijkl φ KL (165) ++ R- SL(4, R) ξ Ia ξȧ I (164) ξ ξ 29

30 {Q, Q} = P ξ I g 1/2 ξ I, ξ I g 1/2 ξ I. (166) g 0 (105) F ab Fȧḃ (166) Fȧḃ F ȧḃ, χ Iȧ g 1/2 χ Iȧ, φ IJ gφ IJ, χ I a g 3/2 χ I a, F ab g 2 F ab (167) F ab Fȧḃ F ab = G ab G ab F µν F µν, G ab g 2 G ab. (168) N = 4 G ab L = 1 g 2 tr ( F ab G ab G2 ab + ɛ IJKL D µ φ IJ D µ φ KL + ɛɛφφφφ +χ Ia D aȧ χȧi + χ Ia φ IJ χ J a ɛijkl χȧiφ JK χ Lȧ ) (169) (167) (168) g g 2 L = tr ( ) F ab G ab + ɛ IJKL D µ φ IJ D µ φ KL + χ Ia D aȧ χȧi + ɛ IJKL χȧiφ JK χ Lȧ ( ) 1 +g 2 tr 2 G2 ab + ɛɛφφφφ + χ Ia φ IJ χ J a (170) A G (144) massless on-shell (170) 0 g 0 L = tr ( ) F ab G ab + ɛ IJKL D µ φ IJ D µ φ KL + χ Ia D aȧ χȧi + ɛ IJKL χȧiφ JK χ Lȧ (171) 4 tree 0 3 (171) GAA φ IJ (p) = 1, χ Ia (p) = λ a, χȧi(p) = λȧ, A aȧ (p) = µ a λȧ λ, µ. (172) 30

31 R- tr( φ[a, φ]) : (p µ 1 p µ 2)A µ (p 3 ) = λ a µ a λ 1 λȧ1 3ȧ λ 3, µ [23][31] (1 2) =, [12] tr(χ[a, χ]) : λ a µ a λ 3ȧ 1 λȧ2 = [23]2 λ 3, µ [12], tr(χ[φ, χ]) : λ a 1λ 2a = [12] (173) (148) (159) A G χ I χ I φ IJ ϕ A ϕ G ϕ I ϕ I ϕ IJ I int = Ω 3 (ϕ G ϕ A ϕ A + ɛ IJKL ϕ IJ ϕ KL ϕ A + ϕ I ϕ I ϕ A + ϕ I ϕ J ϕ IJ ) (174) ϕ(z, ψ) = ϕ A (z) + ϕ I (z)ψ I + ϕ IJ (z)ψ I ψ J + ɛ IJKL ϕ I (z)ψ J ψ K ψ L + ϕ G (z)ψ 1 ψ 2 ψ 3 ψ 4 (175) I int = Ω 3 dψ 1 dψ 2 dψ 3 dψ 4 ϕ(z, ψ) 3 (176) H 1 Ȟ1 ϕ(z, ψ) (0, 1) A A(Z, Z, ψ) = dz I( A I + ψ A χ IA ψa ψ B φ IAB + 1 3! ɛ ABCDψ A ψ B ψ C χ D I + 1 4! ɛ ABCDψ A ψ B ψ C ψ D G I ) (177) (0, 1)- H 1 Ȟ1 ϕ(z, ψ) A I int = Ω 3 dψ 1 dψ 2 dψ 3 dψ 4 A A A (178) H 1 holomorphic-chern-simons I = 1 Ω 3 dψ 1 dψ 2 dψ 3 dψ 4 tr (A A + 2 ) 2 3 A A A. (179) truncated N = 4 holomorphic CS in super twistor space (180) 31

32 2.5 MHV n tree n +1 1 helicity helicity 2 n 2 n 2 + n n 2 4 n n gluon n n 2 tree level ±n ±(n 2) 4 0 [1, 2] n = 4 gluon scattering gluon 1 helicity 4 ±(n 4) maximally helicity violating (MHV) n 2 + MHV n 2 + MHV MHV googly amplitude MHV amplitude [3] A(λ i, λ i ) = δ 4 ( i λ a i λḃi) λ r, λ s 4 ni=1 λ i, λ i+1 r s n = 3 (143) δ λ MHV MHV λ λ A(λ i, λ i ) = δ 4 ( i λ a i λḃi) [ λ r, λ s ] 4 ni=1 [ λ i, λ i+1 ] n = 3 (142) (120) N = 4 MHV amplitude scattering amplitude MHV amplitude η I A(λ i, λ i, η i ) = δ 4 ( i λ a i λḃi)δ 8 ( i η i,a λ a 1 i ) ni=1 λ i, λ i+1 (181) (182) (183) MHV ψ I A(λ i, λ i, ψ i ) = δ 4 ( i λ a i λḃi)δ 8 ( i ψi A 1 λȧi ) ni=1 [ λ i, λ i+1 ] (184) 1 Peskin Prob.17.3(b) 32

33 η ψ 3 (184) ψ δ 8 ( i ψi A λȧi ) 8 ψ 3 A(λ i, λ i, ψ i ) = δ 4 ( i λ a i λḃi) h 1 +h 2 +h 3 =1 (ψ 1 λ 1 ) 2+2h 1 (ψ 2 λ 2 ) 2+2h 2 (ψ 3 λ 3 ) 2+2h 3 ni=1 [ λ i, λ i+1 ] (185) ψ (108) λ i A(λ i, λ i, ψ i ) = δ 4 ( i λ a i λḃi) h 1 +h 2 +h 3 =1 (ψ 1 ) 2+2h 1 (ψ 2 ) 2+2h 2 (ψ 3 ) 2+2h 3 [ λ 2, λ 3 ] 2 2h 1 [ λ 3, λ 1 ] 2 2h 2 [ λ 1, λ 2 ] 2 2h 3 ni=1 [ λ i, λ i+1 ] (186) (142) (173) MHV amplitude (183) twistor space amplitude A(λ i, µ i, ψ i ) MHV amplitude δ λ A(λ i, λ i, η i,a ) = δ 4 ( i λ a i λḃi)δ 8 ( i η i,a λ a i )F (λ i ) (187) F λ F (λ i ) = 1 ni=1 λ i, λ i+1 (188) (187) (λ, µ, ψ) A(λ i, µ i, ψi A ) = d 2n λe i i [µ i, λi ] d 4n ηe i i η i,aψi A δ 4 ( λ a λḃi)δ i 8 ( η i,a λ a i )F (λ i ) i i = = d 2n λ d 4n η d 4 x d 4 x exp ( i i ( λȧi (µ i,ȧ + x aȧ λ a i ) d 8 θ exp i η i,a (ψi A + θa A λ a i ) F (λ i ) i [ ] d 8 θ δ 2 (µ i,ȧ + x aȧ λ a i )δ 4 (ψi A + θa A λ a i ) F (λ i ) (189) i δ n (λ i, µ i, ψi A ) µ i,ȧ = x aȧ λ a, ψi A = θa A λ a. (190) 33 ) )

34 super twistor space holomorphic P 1 ϕ i (λ i, µ i, ψi A ) M µ ψ M = d 4 x d 8 θ i [ ] λ i, dλ i ϕ i (λ i, x aȧ λ a i, θa A λ a i ) F (λ i ) (191) twistor f(λ) = 1 ɛ abcd z a dz b dz c dz d = λ, dλ d 2 µ λ 1 = 1 (191) P 1 λ λ µ λ µ P 1 (σ 1, σ 2 ) λ λ a = u a pσ p u a p (190) σ twistor σ λ a (σ i ) = u a pσ p i, µȧ(σ i ) = x pȧσ p i, ψ A (σ i ) = θ A p σ p i. (192) x pȧ = x aȧ u a p, θ A p = θ A a u a p. (193) F (λ i ) = (det u) n 1 i σ i σ i+1, d 4 xd 8 θ = (det u) 2 d 4 x d 8 θ, λ, dλ = (det u)σ dσ. (194) M M = (det u) 2 d 4 x d 8 θ i [ ] σ i dσ i ϕ i (λ a (σ i ), µȧ(σ i ), ψ A 1 (σ i )) (195) i σ i σ i+1 (195) u x θ λ µ (det u) 2 d d 4 x d 8 θ 4 ud 4 x d 8 θ = vol GL(2) µ λ 1/ vol GL(2) P 1 SL(2) (λ, µ, ψ) GL(2) u a p u FP-determinant (det u) 2 (196) ϕ i (λ a (σ i ), µȧ(σ i ), ψ A (σ i )) ϕ i (σ i ) (195) d 4 ud 4 x d 8 θ [ ] σi dσ i M = ϕ i (σ i ) (197) vol GL(2) σ i σ i+1 i 34 (196)

35 2.6 moduli D1-brane A i D1-brane D5-brane J A D1-brane J J D1-brane D1-brane I D1 = dzα( + A)β (199) P 1 α β 1-5 string 1/(σ i σ j ) 1 1/(σ i σ j ) i σ i σ i+1 contraction contraction multi-trace amplitude conformal gravity 2.7 MHV (disconnected prescription) truncated N = 4 super twistor holomorphic CS truncate N = 4 IIB D3- brane super twistor holomorphic CS super twistor B-model B-model Calabi-Yau N = 4 CP 3 4 Calabi-Yau B-model U(N) N D5-brane D(5 8)-brane [4] D5-brane (0, 1) A (0, 1)- BRS ψ D5- (z, z, ψ) ψ ψ = A Witten B-model truncate full N = 4 MHV (197) Witten twistor space holomorphic D1-brane [4] D-instanton D5-brane A amplitude moduli A 1(Z(σ 1 ))A 2 (Z(σ 2 )) A n (Z(σ n )) J(σ 1 )J(σ 2 ) J(σ n ) (198) D C instanton connected instanton disconnected instanton twistor 35

36 [12] MHV amplitude p 1 λ λ (25) p on-shell [12] p p 2 λ a = p aȧ ηȧ (200) ηȧ η p on-shell (200) (25) λ a [ λ, η] 2.8,,,+ + (a) 4:,,,+ I (a) = λ 2, λ q 3 1 λ 3, λ 4 3 λ 1, λ 2 λ q, λ 1 q 2 λ 4, λ q λ q, λ 3 (201) λ q λ q = qη = (p 1 + p 2 )η = λ 1 [ λ 1, η] λ 2 [ λ 2, η] = λ 3 [ λ 3, η] + λ 4 [ λ 4, η] (202) 36

37 I (a) I (a) = φ3 1 φ 2 φ 3 φ 4 1 q 2 λ 1λ 2 λ 3 λ 4 (203) q 2 = (p 1 + p 2 ) 2 = 2p 1 p 2 = λ 1, λ 2 [ λ 1, λ 2 ] I (a) = φ3 1 φ 2 φ 3 φ 4 1 q 2 λ 3, λ 4 [ λ 1, λ 2 ] (204) I (b) I (a) 2 4 I (a) I (b) φ 3 ( 1 1 λ3, λ 4 I (a) + I (b) = φ 2 φ 3 φ 4 q 2 [ λ 1, λ 2 ] + λ ) 3, λ 2 [ λ 1, λ 4 ] = φ3 1 1 λ 3, λ 4 [ λ 4, λ 1 ] + λ 3, λ 2 [ λ 2, λ 1 ] φ 2 φ 3 φ 4 q 2 [ λ 1, λ 2 ][ λ 1, λ 4 ] = φ i=1 λ 3, λ i [ λ i, λ 1 ] φ 2 φ 3 φ 4 q 2 [ λ 1, λ 2 ][ λ 1, λ 4 ] (205) 4i=1 λ i λ i 0 amplitude 0 amplitude [16] MHV vertex googly amplitude vertex [13] 2.9 amplitude η 3 amplitude 5: 3 r s t I Γ = λ r, λ s 4 1 λ, λ t 4 λi, λ i+1 P 2 λj, λ j+1 = 1 g(λ) (206) P 2 37

38 MHV λ a = P aḃηḃ 1/P 2 g(λ) λ a 0 (200) λ g(λ) λ amplitude η I Γ η 1 I Γ = λ, dλ [ λ, d λ] g(λ) (207) (P λ λ) 2 0 meromorphic 2-form CP 2 CP 2 2-cycle λ [ λ, d λ] (P λ λ) = 1 2 (P λη) d λ ( ) [ λ, η] λ (P λ λ) (208) λ (207) I Γ = λ, dλ g(λ) (P λη) d λ ( ) [ λ, η] λ (P λ λ) 2-cycle λ λ 2-cycle P timelike P λ λ 0 timelike P I Γ = λ, dλ g(λ) ( (P λη) d λ λ + ) ( ) [ λ, η] (210) λ (P λ λ) λ λ 0 z (1/z) δ(z) pole δ P λ λ 0 / λ + / λ λ = λ 2-cycle 2-cycle λ λ / λ + / λ / λ (209) I Γ = λ, dλ [ λ, η] (P λ λ) d λ λ 38 g(λ) (P λη) (211)

39 dλ 1 λ ζ, λ = 2πδ( ζ, λ ) (212) I Γ = λ, dλ [ λ, η] δ(p λη)g(λ) + (213) (P λ λ) g(λ) pole Γ 0 λ, dλ δ( ζ, λ )B(λ) = B(ζ) (214) I Γ = [ λ, η] 1 P λ, g(p η) = g(p η) (215) P η P 2 (206) 2.10 twistor space holomorphic CS [4] A A λ 1 = λ 1 = 1 G = 1 (2π) δ(λ 2 λ 2 )δ(µ 1 µ 1 )δ(µ 2 µ 2 ) (216) G = 1 (2π) 2 δ(λ 2 λ 2 )δ(µ 1 µ 1 1 ) µ 2 µ 2 (217) δ 4 (ψ A ψ A ) 3 f = 0 G line (µ 1, µ 2 ) = const λ 2 (λ 2, µ 2 ) = const µ 1 line λ 2 µ 1 (217) δ line 1/(µ 2 µ 2 ) 39

40 G line line pole line µȧ = x aȧ λ a, µ ȧ = x aȧ λ a. (218) (217) δ λ 2 λ a = λ a dλ 2 G = δ(µ 1 µ 1 1 ) µ 2 = δ(y1 1 λ 2 y ) µ 2 y 1 2 λ 2 y 2 2 (219) y = x x λ 2 dλ 2 dλ 2 G = 1 1 y 1 1 y 1 2 (y 2 1 /y 1 1 ) y = y aḃy a ḃ (220) exp(i[µ, λ i ]) (190) exp(ixp ix P ) y = x x G(P ) = d 4 y y 2 eiyp (221) P y curve moduli y d 4 y/y 2 pole G(P ) = tdt λ, dλ [ λ, d λ]e iyp (222) pole y 2 = 0 y = tλ λ SL(2) y = tλ λ t λ λ scaling weight 1 t 0 (207) 1 G(P ) = λ, dλ [ λ, d λ] (223) (P λ λ) MHV (connected prescription) Witten 1 MHV amplitude MHV [4] M = moduli n i=1 [ ] σi dσ i ρ C ϕ i (σ i ) σ i σ i+1 40 (224)

41 i = 1,..., n n ρ C curve λ a (σ), µȧ(σ), ψ A (σ). (225) (225) 3 σ = (σ 1, σ 2 ) d P 1 d super twistor P 3 4 d = 1 MHV amplitude moduli (225) σ d + 1 (225) P 1 SL(2) 3 moduli parameter d λ a (σ) = P a (σ) = Pk a u k (σ), k=0 d µȧ(σ) = Qȧ(σ) = Qȧku k (σ), k=0 d ψ A (σ i ) = R A (σ) = Rk A u k (σ). (226) k=0 Pk a Qȧk RA k d = 1 ua p x pȧ θ p A d + 1 u k (σ) (k = 0,..., k) P 1 d u k (σ) = (σ 1 ) k (σ 2 ) d k u k (224) measure d 2(d+1) P d 2(d+1) Qd 4(d+1) R d amplitude Rk A n ψ 4(d + 1) h i ψ 2 2h i n (1 h i ) = 2(d + 1) (227) i=1 d = 1 MHV bosonic δ (119) n δ d 2d+2 Qȧk exp 2d + 2 δ 2d n δ moduli d 2d+2 P d n σ 4 4 δ 41

42 h i = ±1 h i = 1 d p 1 q p + q = n, q = d + 1. (228) 1. Witten [4] amplitude A(λ i, µ i ) 0 n (λ i, µ i ) q 1 curce (p, q) MHV [4] curve 0 F (λ i, µ i ) F (λ i, / λ i )A(λ, λ) = 0 (229) (p, 2) MHV p (p, q) = (2, 3), (3, 3) (3, 3) 2. [10, 11] (p, q) = (2, q) MHVamplitude (224) amplitude moduli A(λ, µ) A(λ, λ) moduli (p, q) = (2, 3) [10] [11] q 3. witten [15] (224) (p, q) amplitude p q MHV (224) MHV (224) [10, 11] MHV + 1 p = 2 q = 1 (224) q = 0 0 small d = 0 λ a (σ) = P a, µȧ(σ) = Qȧ, ψ A (σ) = R A, (230) σ (224) R ( ) 1 3 σ i dσ i M = Ω 3 ϕ 1 (z)ϕ 2 (z)ϕ 3 (z). (231) vol SL(2) σ i σ i+1 i=1 42

43 σ (147) 2.12 Yang-Mills MHV amplitude d = 1 curve MHV amplitude d = n 3 curve MHV amplitude (λ, λ) amplitude (λ, µ) d = 1 [4] moduli A(λ, λ) A(λ, µ) (232) MHV amplitude λ amplitude curve moduli moduli MHV amplitude moduli A(λ, λ) (233) moduli (p, q) = (2, 3) [10] [11] q amplitude MHV amplitude [15] Parity (224) (119) (λ i, λ i, h i ) amplitude A(λ i, λ i, h i ) (224) u k moduli u k q u k h = index I u k k I moduli moduli u k (σ i ) = δ k,i, k, i I. (234) σ i σ i curve u k q 1 q u k Pk a σ k (k I ) P a (σ k ) Qȧk Rk A R A (σ) ϕ i (i I ) 4 R A k 43

44 (119) R (224) A(λ i, λ d 2q P d 2q ( ) Q 1 2hi σi dσ i, h i ) = i λi δ( P (σ i ), λ i ) exp(i(λ i /P (σ i ))[Q(σ i ), λ i ]) vol GL(2) i n σ i σ i+1 P (σ i ) (235) σ i σ i dσ i δ( P (σ i ), λ i ) = d 2 σ i δ 2 (P (σ i ) λ i ) (236) σ i λ i rescale λ i αλ i σ i α 1/(q 1) σ i q > 1 λ(σ i )/λ i 1 (235) A(λ i, λ i, h i ) = d 2q P d 2q Q vol GL(2) i n [ d 2 σ i σ i σ i+1 δ 2 (P (σ i ) λ i ) exp(i[q(σ i ), λ i ]) ] (237) + P Q measure (234) Q Q d 2d+2 Q exp(i[q(σ i ), λ i ]) (238) i n 2q u k (σ i ) λȧi = 0. (239) i λȧi i n k q (239) p Witten p 1 P [15] 2 λȧi = P ȧ(σ i ) k i(σ k σ i ) (240) P p p 1 P ȧ(σ) = P ȧk ũk(σ) (241) + p λ(σ) ũ k ũ k (σ i ) = δ k,i, k, i I +. (242) 2 P [4] T 44

45 (239) (240) f(σ i ) Q (238) d 2q Q i exp(i[q(σ i ), λ i ]) = f(σ i ) d 2p P δ ( λȧi 2 i n P ȧ(σ i ) k i(σ k σ i ) f(σ i ) 2q λ i (243) d λ = d 2q Q exp(i[µ(σ i ), λ i ]) δ 2 (µȧ(σ i )) = 1 (244) i p i q (243) ) (243) ( 2p P d λ = f(σ i ) d P ȧ(σ ) δ 2 i ) λȧi = f(σ i ) ( ) 2 1 i I + k i(σ k σ i ) i p k i(σ k σ i ) (245) (244) (245) f(σ i ) f(σ i ) = k i(σ k σ i ) 2 (246) i I + 1 (243) (237) A(λ i, λ i, h i ) = d 2q 2p P d P vol GL(2) i n i n [ d 2 σ i σ i σ i+1 ] [ δ 2 (λ i P (σ i ))δ 2 ( λȧi 1 i I + k i(σ k σ i ) 2 P ȧ(σ ) i ) ] (247) k i(σ k σ i ) σ i I σ i (σ k σ i ) 1/(q 1), σ i I+ σ i, k i P P, P P (σ k σ i ) 1/(q 1). (248) j I,k j A(λ i, λ i, h i ) = d 2q 2p P d P [ d 2 σ ] i 1 vol GL(2) i σ i σ i+1 l k(σ l σ k ) 2 ( [ δ 2 λ i P (σ ) i) δ 2 ( λȧi P ȧ(σ i ) ) ] i I + M i ( [ P ȧ(σ ) δ 2 (λ i P (σ i ))δ 2 i ) ] λȧi (249) i I M i 45

46 I + I λ λ (240) (239) P σ 1 q σ i 1 pole holomorphic differential 3 p 1 P P ȧ σ dσ (σ) = ni=1 P ȧ(σ) (250) (σ i σ) P ȧ (σ) 0 0 q 1 u 0 i u(σ i ) Res σ i P ȧ (σ) = 0 (251) (239) λ i = Res P σi (239) (240) 2.13 amplitude [17] MHV vertex line pole [17] connected instanton instanton line pole amplitude pole degree 2 curve curve C 2 : z A = β A 0 + β A 1 σ + β A 2 σ 2 (252) β A 0 0, β A 2 0. (253) βk A σ SL(2, R) P [4] P 46

47 curve Z 4 = 0 curve SL(2, R) β0 4 = β2 4 = 0 β 1 4 = 1 (253) Z A (σ = 0) = β0 A, Z A (σ = ) = β2 A, (up to rescaling) (254) Z 4 = 0 σ = 0 σ = C 0 : Z A = β A 0 σ 1 + β A 1, C : Z A = β A 1 + β A 2 σ. (255) C 0 σ = 0 curve C σ = curve β1 A βa 0 β A 2 6: β1 A, n A 0 = βa 0, n A β0 3 2 = βa 2. (256) β2 3 β0 3 = β2 3 τ β A k β A 1 = (β 1 1, β 2 1, β 3 1, 1 β 1 1, β 2 1, β 3 1, β 4 1 ), (257) β A 0 = τ(n 1 0, n 2 0, 1, 0 n 1 0, n 2 0, n 3 0, n 4 0 ), (258) β A 2 = τ(n 1 2, n 2 2, 1, 0 n 1 2, n 2 2, n 3 2, n 4 2 ). (259) 47

48 J dβk A δ(β 4 0/β )δ(β0)δ(β )δ(β2) 4 = τ 3 dτ dβ1 A k,a A =4 A 3,4 dn A 0 dn A 2. (260) curve ω(σ 1,, σ n ) = n i=1 dσ i σ i σ i+1 (261) curve τ 0 σ 1,, σ m σ m+1,, σ n σ i τ 0 Z A = β1 A σ i {1,2,...,m} = σ i /τ, σ i {m+1,...,n} = τ/ σ i. (262) up to rescaling z A i {1,...,m} = τ 2 n A 0 σ 1 i + β A 1 + n A 2 σ i, z A i {m+1,...,n} = n A 0 σ i + β A 1 + τ 2 n A 2 σ 1 i (263) τ 0 C 0 C (262) (261) dσ i {1,...,m 1} d σ i =, σ i σ i+1 σ i σ i+1 (264) dσ m d σ m =, σ m σ m+1 σ m τ 2 / σ m+1 (265) dσ {i m+1,...,n 1} σ i σ i+1 = d σ i σ i σ m+1 σ i+1, (266) σ i dσ n = τ 2 σ n σ 1 d σ n σ 1 σ n 2 τ 2 σ n. (267) ( m 1 ω(σ 1,..., σ n ) = τ 2 i=1 ) n 1 d σ i d σm d σ i d σ n σ i σ i+1 σ 1 σ m i=m+1 σ i σ i+1 σ m+1 σ n = τ 2 ω (0, σ 1,..., σ m )ω (0, σ m+1,..., σ n ) (268) (260) τ τ = 0 disconnected prescription [1] Parke, Taylor, PLB157(1985)81 48

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