N=1 N=1 QCD N=1 non-abelian QCD X 0
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- しおり かつま
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1 N=1 N=1 QCD N=1 non-abelian QCD X 0
2 moduli Witten thooft QCD < N c = N c = N c > N c N c N c < < 3N c N c N c QCD+ X electric
3 5.2 magnetic A 62 B Kähler 63 C Wess-Zumino 64 2
4 1 Higgs SU(3) SU(2) U(1) Higgs Higgs Higgs Higgs (naturalness) log log SU(5) SU(3) SU(2) U(1) Gev [1] 3
5 QCD Λ ) 1/N Schwinger-Dyson 1/N [2] QCD QCD 1994 Seiberg [3] Seiberg [7][4][5][8][9] N=2 Seiberg Witten [6][10] 0 thooft 4
6 N=1 non- Abelian [7] non-abelian non-abelian Seiberg N=1 non-abelian Wilson Wilson 5
7 thooft [11] thooft Seiberg N=1 QCD [3][7] SU(N c ) < N c = N c = N c + 1 N c N c magnetic 3 2 N c < < 3N c 3N c 3 2 N c < < 3N c non-abelian Kutasov Schwimmer QCD [12][13][14] N=1 QCD non-abelian QCD non-abelian non-abelian QCD N=1 non-abelian SO(n) Sp(n) [15][16][17] 6
8 2 2.1 Poincaré [18][19] Lie Weyl spinor Q A α, Q βb {Q A α, Q βb } = 2σ m α β P mδb A {Q A α, Q Ḃ } β = { Q A α, Q βb } = 0 [ ] Pm, Q A α = [ P m, Q ] αa = 0 (1) Weyl spinor [20] P m Weyl spinor (A,B) N N N=1 N=2 N Appendix (2.5) 7
9 [22] [21] [1] 2.2 P m i / x m Q, Q {x m } {θ α, θ α } 8
10 F (x, θ, θ) = f(x) + θϕ(x) + θ χ(x) + θθm(x) + θ θn(x) + θσ m θvm (x) +θθ θ λ(x) + θ θθψ(x) + θθ θ θd(x) (2) θ θ Lorentz F (x, θ, θ) Q, Q Q α = θ α iσm α α θ α (3) x m Q α = θ + α iθα σα m α (4) x m P m = +i / x m (1) f(x) ϕ(x) δ ξ F (x, θ, θ) (ξq + ξ Q)F = δ ξ f(x) + θδ ξ ϕ(x) + (5) δ ξ f(x) = (ξq+ ξ Q)f(x) ξ, ξ Q, Q (1) P m = i x m (1) (2) 9
11 2.3 D α = θ + α iσm α α θ α (6) x m D α = θ α iθα σα m α (7) x m D α Φ = 0 (8) y m = x m + iθσ m θ Φ(y, θ, θ) = A(y) + 2θΨ(y) + θθf (y) (9) D α y m = D α θ β = 0 (10) D D A Weyl Ψ F D, D Q, Q (5) δ ξ A = 2ξΨ (11) δ ξ Ψ = i 2σ m ξ m A + 2ξF (12) δ ξ F = i 2 ξ σ m m Ψ (13) n Φ i L = d 2 θd 2 θk(φ [ i, Φ j ) + ] d 2 θw (Φ i ) + h.c. (14) h.c. d 2 θ θθ = 1 K, W K Kähler 10
12 W Appendix K = Φ i Φ i,w Wess-Zumino [23] Yukawa Wess-Zumino V = V (15) V = V V (x, θ, θ) = C(x) + iθχ(x) i θ χ(x) + i 2 θθ [M(x) + in(x)] i 2 θ θ [M(x) in(x)] ] θσ m θvm i (x) + iθθ θ [ λ(x) + 2 σm m χ(x) [ i θ θθ λ(x) + i ] 2 σm m χ(x) + 1 [D(x) 2 θθ θ θ + 1 ] 2 C(x) (16) v m (v m V a V = T a V a Λ a e V = e iλ e V e iλ (17) (T a ) ij Λ = T a Λ a T a Hausdorff V = V Hausdorff Λ V = V + i(λ Λ ) + (18) 11
13 QED Coulomb (16) C, χ, M, N λ, v m, D Wess-Zumino Wess-Zumino Wess-Zumino Appendix v m λ 1/2 Weyl D (TΦ) a ij (Φ) i = Φ i Φ = e iλ Φ (19) Φ W α = 1 4 D De V D α e V (20) W W α = e iλ W α e iλ (21) L = 1 16kg 2 Tr ( ) d 2 θ W α W α + h.c. + d 2 θd 2 θφ e V Φ (22) Tr(T a T b ) = kδ ab V 2gV L = 1 4 va mnv a mn Da D a D m A D m A i Ψ σ m D m Ψ + F F +i 2g ( A T a ΦΨλ a λ a T a ΦA Ψ ) + gd a A T a ΦA (23) [24][25] D m A = m A + igv a mt a ΦA (24) 12
14 D m Ψ = m Ψ + igvmt a ΦΨ a (25) D m λ a = m λ a gt abc vmλ b c (26) vmn a = m vn a n vm a gt abc vmv b n c (27) [T a, T b ] = it abc T c (28) Yukawa Kähler [26] L F = d 2 θ W (Φ i ) + h.c. (29) W Φ i F 2.4 moduli Nambu-Goldstone Higgs flat directions 13
15 moduli moduli moduli flat directions, moduli moduli moduli moduli D,F V (A) = 1 D a2 + F 2g 2 i 2 (30) a i where D a = (A i ) l (Ti a ) l k (A i ) k (31) i,k,l F i = W (A) A i (32) D a F i T a i i W(A) D a = 0 for all a (33) F i = 0 for all i (34) (35) D-flatness flat directions 14
16 flat directions U(1) 1-1 Q Q D = Q Q Q Q = 0 (36) flat directions < Q > = < Q > = a (37) < Q >=< Q >= a (38) a 0 Higgs ( Higgs Weyl Dirac Higgs Higgs U(1) a a 2.5 Witten spinor Majorana Q i = Q i (39) i=1,2,3,4 H = 1 4 (Q i ) 2 (40) 4 i=1 15
17 < 0 H 0 > 0 (41) H H = 0 H > 0 H Q Q 0 >= 0 N > 1 H = 1 4 i1 ) 4 i=1(q 2 = (Q i2 ) 2 = (42) i=1 Q i1 H > 0 N=0 H > 0 Witten [27] 16
18 Witten Tr( 1) F (43) F Tr ( 1) F = exp(2πij z ) (44) J z z 90 Tr Majorana H = Q 2 1 = Q 2 2 = Q 2 3 = Q 2 4 Q i Q j + Q j Q i = 0, for i j (45) Q = Q 1 Q b > = H f > Q f > = H b > (46) ( 1) F Tr ( 1) F Witten Witten Tr( 1) F = n H=0 B n H=0 F (47) 17
19 n H=0 B (n H=0 F ) Witten Tr( 1) F 0 (48) (47) Witten Tr( 1) F 0 n H=0 B nh=0 F 0 Witten Witten Witten Euler [27] Witten dual Coxeter [27] SU(N) N Witten Witten Witten 18
20 3 3.1 BCS Ginzburg-Landau QCD QCD QCD 3.2 d 2 θ 19
21 Yukawa Λ [28] Wilson log Legendre β [29] Wilson β [30] Wilson Wess-Zumino W tree = mϕ 2 + gϕ 3 (49) 20
22 ϕ m g Yukawa U(1) U(1) U(1) R ϕ 1 1 m 2 0 g 3 1 U(1) U(1) R (θ, θ) (-1,1) d 2 θw tree θ W eff = mϕ 2 f ( ) gϕ m (50) gϕ m f g/m g 0, m 0 g,m f(t) = 1 + t (51) W tree = mϕ 2 + gϕ 3 = W eff (52) [31] Kähler Wess- Zumino Wess-Zumino 21
23 3.3 thooft Moduli thooft [11] Abel spectator Abel spectator Abel spectator spectator thooft thooft QCD 2 22
24 QCD 23
25 4 QCD N=1 QCD QCD SU(N c ) V a (a = 1, 2,, N c 2 1) SU(N c ) Q i r, Q s j (i, j = 1, 2,, ; r, s = 1, 2,, N c ) QCD D QCD 0 pure Yang-Mills QCD QCD Higgs [32] β β(g) = g3 16π 2 3N c + γ(g 2 ) 1 N c g 2 8π 2 γ(g 2 ) = g2 8π 2 N 2 c 1 N c + O(g 2 ) (53) [36] γ(g 2 ) 1-loop g 3 3N c 3N c < SU( ) L SU( ) R U(1) A U(1) B U(1) R (54) 0 QCD left-hand right-hand SU( ) U(1) 24
26 U(1) R R U(1) R W α (θ) e λ W α (e λ θ) Nc Q j λ N r(θ) e f Q j r (e λ θ) Nc Q s N j(θ) e λ Q s f j(e λ θ) (55) Q, Q U(1) A U(1) B ( Q, 1, 1, 1, N ) f N c ( Q 1,, 1, 1, N ) f N c (56) U(1) R (-1) U(1) R (+1) U(1) R U(1) A Λ SU( ) L SU( ) R U(1) B U(1) R (57) N c = 2 Q Q N c 2 N c = 2 QCD moduli U(1) D a = (Q ) r i (T a ) i jq j r ( Q ) i s (T a ) j i Q s j = 0 (58) 25
27 [33] a 1 Q = Q a 2 = a Nf (59) < N c a i Q = a 1 a 2 a Nc (60) Q = ã 1 ã 2 ã Nc (61) a i 2 ã i 2 i (62) N c a i Q, Q moduli M i j = Q i Q j (63) B i 1...i N c = Q i1 Q i Nc B i1...i N c = Q i1 Q in c (64) 26
28 i, j SU(N c ) SU(N c ) ϵ i 1...i Nc ϵi1...i N Bose c < N c Bose N c 4.1 < N c W eff = C Nc, ( Λ 3N c det QQ ) 1 Nc (65) C Nc, subtraction scheme 0 < N c determinant 0 flat directions (59) a Nf Higgs SU(N c ) Nf SU(N c 1) Nf 1 SU(N c ) Nf N c SU(N c 1) Λ L SU(N c ) Λ a Nf 27
29 Λ 3(N c 1) ( 1) L = Λ3N c a 2 (66) DR scheme subtraction scheme DR dimensional reduction Siegel[48] (65) a Nf C Nc, = C Nc SU(N c 1) Nf 1 (65) W tree = mm Q Nf, Q Nf SU(N c ) Nf 1 Λ 3N c ( 1) L = mλ 3N c (67) W exact = ( Λ 3N c det M ) 1 Nc ( ) f t = mm Λ 3N c 1 Nc (68) det M m f(t) = C Nc, + mm t W exact = C Nc, ( Λ 3N c det QQ ) 1 Nc + mm (69) m C Nc, 1 = (N c + 1) ( CNc, N c ) Nc Nc +1 (70) C Nc, = C Nc C Nc, = (N c ) const. = N c 1 (65) 1-instanton = N c 1 3Nc Nc N Λ f = Λ 3Nc (71) 28
30 Higgs instanton N c = 2, = 1 instanton DR scheme W eff = (N c ) ( Λ 3N c det QQ ) 1 Nc (72) [34] = N c 1 instanton < N c 1 (72) 0 well-defind [33] W tree = T r mm W full = W eff +W tree M < Mj i >= ( ) det mλ 3N 1 ( c Nc 1 i m) j (73) N c N c, N c SU(N c ) pure Yang-Mills Witten N c 4.2 = N c M i j B = 1 N c! ϵ i 1,i 2,...,i Nf Q i 1 1 Q i 2 2 Q i 29
31 B = 1 N c! ϵi 1,i 2,...,i Nf Q 1 i 1 Q 2 i 2 Q i Nf (74) moduli M i j B, B det M BB = 0 (75) M j i B, B moduli, xy = 0 x = 0 y = 0 rank(m) N c 2 (76) SU(N c rank(m)) 0 moduli 0 W tree = Tr mm QCD, N c < N c < Mj i >= ( ) det mλ 3Nc N 1 ( f Nc 1 i m) j (77) instanton [2] < BB >= 0 (78) [2] 0 M i j 0 QCD = N c m det < M >= Λ 2N c (79) moduli moduli M i j B, B det < M > < BB >= Λ 2Nc (80) 30
32 [3] 0 1-instanton moduli xy = Λ moduli Λ moduli moduli Higgs SU( ) L SU( ) R U(1) B U(1) R < M >=< B >=< B >= 0 moduli moduli M i j = Λδ i j B = B = 0 (81) SU( ) L SU( ) R U(1) B U(1) R SU( ) V U(1) B U(1) R (82) U(1) R thooft Q ( ) 1, 1 31
33 Q ( ) 1, 1 W (1) 0,1 (N 2 f 1) (83) Q, Q, W ( ) 1, 1 SU( ) V U(1) B U(1) R (-1) M j i B, B (81) (80) TrM BB M (Nf 2 1) 0, 1 1 B (1) Nf, 1 1 B (1) Nf, 1 1 (84) thooft SU( ) 2 V U(1) R d (2) ( ) d (2) ( ) = d (2) (Nf 2 1) U(1) 3 R 2Nf 2 ( 1) 3 + (Nf 2 1) = (Nf 2 1)( 1) 3 2 U(1) 2 BU(1) R 2Nf 2 = 2Nf 2 U(1) R 2Nf 2 + (Nf 2 1) = (Nf 2 1) 2 (85) d (2) (r) SU( ) V r Casimir d (2) ( ) = d (2) ( ) = 1, 2 d(2) (Nf 2 1) = (80) moduli Mj i = 0 B = B = Λ Nc (86) 32
34 SU( ) L SU( ) R U(1) B U(1) R SU( ) L SU( ) R U(1) R (87) thooft (80) Lagrangian Lagrange A W = A(detM BB Λ 2N c ) (88) Lagrange A < N c (72) pure Yang-Mills 4.3 = N c + 1 = N c + 1 = N c M i j B i = ϵ i j1 j 2 j N c Qj 1 Q j2 Q j Nc B i = ϵ i j 1j 2 j N c Qj1 Q j2 Q jn c (89) moduli 1, 2,, N c ( ) 1 j det M B B j i = 0 M i M i jb i = M i j B j = 0 (90) 33
35 ( ) 1 j det M B B j i = Λ 2Nc 1 m j i (91) M i (77) 0 0 moduli moduli = N c = N c + 1 m moduli moduli moduli 0 Higgs 0 0 M = B = B = 0 (92) SU( ) L SU( ) R U(1) B U(1) R (93) 0 thooft Q Q (, 1) 1, Nc (1, ) 1, Nc N c N c W (1, 1) 0,1 (N 2 c 1) (94) 34
36 Q, Q, W (, 1) 1, Nc SU( ) L SU( ) R 1 U(1) B U(1) R ( Nc ) M i j B i, B j M thooft (, ) 0,1 2Nc B (, 1) Nf 1, B (1, ) Nf +1, 1 1 (95) SU( ) 3 ( 1) d (3) ( ) SU( ) 2 U(1) R ( 1) 2 d (2) ( ) U(1) 3 R Nf Nf 2 U(1) 2 BU(1) R 2( 1) 2 SU( ) 2 U(1) B ( 1) d (2) ( ) U(1) 2 RU(1) B 0 U(1) R N 2 f (96) SU( ) SU( ) L SU( ) R d (3) ( ) SU( ) 3 Casimir d (3) ( ) = d (3) ( ) M = B = B = 0 35
37 (90) M = B = B = 0 0 M = B = B = 0 < N c + 1 QCD W eff = 1 Λ 2N c 1 ( M i j B i B j det M ) (97) Λ Q, Q = N c + 1 QCD M = B = B = 0 [3] Higgs QCD Higgs 4.4 > N c + 1 > N c + 1 = N c + 1 moduli moduli 0 thooft 36
38 > N c N c, N c QCD screening Coulomb non-abelian free electric [7] N c < < 3N c, N c QCD QCD Λ QCD [35] QCD β β(g) = g3 16π 2 3N c + γ(g 2 ) 1 N c g 2 8π 2 37
39 γ(g 2 ) = g2 8π 2 N 2 c 1 N c + O(g 2 ) (98) γ(g 2 ) β 1-loop 2-loop > 3N 3 c 2N 2 c 1 (99) β 0 g g = 0 N c g 2 N c N c, = 3 ϵ N c g 2 = 8π2 3 ϵ + O(ϵ2 ) (100) β 0 N c ϵ = 3 N c N c < < 3N c, N c [7] 3 2 N c < < 3N c 0 electric Abelian Coulomb V 1 R (101) non-abelian Coulomb [7] θ, θ (1,-1) R dilatation D 3 R (102) 2 [37] D = 3R, D = 3 R N=
40 O 1 (x)o 2 (0) R R = R(O 1 )+R(O 2 ) D D(O 1 )+D(O 2 ) O 1 (x)o 2 (0) x = 0 x 0 0 D = D(O 1 ) + D(O 2 ) R QCD U(1) R R SU( ) L SU( ) R U(1) B R Q Q D( QQ) = 3 2 R( QQ) = 3 N c D(B) = D( B) = 3N c( N c ) 2 (103) (98) β γ = 3 Nc + 1 QQ 2 D = γ + 2 = 3 N c [37] D 1 (104) D = 1 [38] D = N c < < 3N c g QQ D( QQ) = 3 N c < 3 2 N c D < 1 = 3 2 N c D = 1 39
41 QQ M = QQ < 3N 2 c 3N 2 c QCD non-abelian Coulomb N=1 QCD magnetic non-abelian [7] non-abelian non-abelian SU(N c ) QCD electric SU( N c ) magnetic q i, q i i QCD Mj i 3N 2 c < < 3N c non-abelian Coulomb 3N 2 c < < 3N c 3(N 2 f N c ) < < 3( N c ) Mj i W = 1 µ M i jq i q j (105) D(q q) = 3N c D(W ) = 1+ 3N c < 3 relevant magnetic magnetic electric 40
42 non-abelian quantum equivalence [40] N=4 Yang-Mills [41] N=2 QCD [6] N=1 non-abelian magnetic electric non-abelian Coulomb 0 magnetic electric electric SU( ) L SU( ) R U(1) B U(1) R (106) magnetic electric magnetic SU( ) L SU( ) R U(1) B U(1) R q in (, 1, N c N c, Nc ) q in (1,, Nc N c, Nc ) ( N c ) ( N c ) M in (,, 0, 2 N c ) 1 (107) M electric QQ U(1) B magnetic q, q q, q 41
43 Q, Q q( q) Q(Q) SU( ) U(1) R U(1) R U(1) R 1 thooft 42
44 SU( ) 3 N c d (3) ( ) SU( ) 2 U(1) R U(1) 3 R U(1) 2 BU(1) R N 2 c d (2) ( ) N 2 c 1 2N 4 c N 2 f 2N 2 c SU( ) 2 U(1) B N c d (2) ( ) U(1) 2 RU(1) B 0 U(1) R Nc 2 1 (108) electric magnetic thooft (105) electric magnetic electric M i j = QQ B i 1,...,i N c = Q i1 Q i Nc B i1,...,i N c = Q i1 Q in c magnetic M j i M i j = QQ magnetic b i1,...,t Nc = q i1 q inc b i 1,...,i Nc = q i1 q i Nc (109) N c = N c magnetic electric B i 1,...,i Nc B i1,...,i Nc = C ϵ i 1,...,i Nc,j 1,...,j Nc = C ϵ i1,...,i Nc,j 1,...,j Nc bj1,...,j Nc b j 1,...,j Nc (110) 43
45 C = ( µ) N c Λ 3N c C magnetic electric magnetic magnetic q i q j (105) M (105) magnetic electric magnetic (105) 1 µ µ electric M i j M i j = Q i Q j (103) magnetic M m 1 µ M = µm m µ (105) M m µ magnetic Λ electric Λ M m M µ Λ 3Nc Λ 3( N c) = ( 1) N c µ (111) [39] scale matching relation scale matching relation ( 1) N c scale matching relation electric magnetic g 1 g 44
46 electric log Λ magnetic scale matching relation log Λ W 2 α = W 2 α (112) W α magnetic F 2 = F 2 E 2 B 2 = (Ẽ2 B 2 ) Wα 2 = W α 2 λλ = λ λ scale matching relation magnetic SU( ( N c )) = SU(N c ) M j i N j i = q i q j d i, d j scale matching relation Λ 3N c Nf Λ 3( N c ) = ( 1) ( N c) µ = ( 1) N c µ Λ = Λ µ = µ (113) W = 1 µ N j i d i d j + 1 µ M i jn j i = 1 µ N j i ( d i d j + M i j) (114) magnetic magnetic M, N N j i = 0 M i j = d i d j W = 0 d i, d j Q i, Q j electric N c N c non-abelian free electric 3N c N c 3 2 N c < < 3N c non- Abelian Coulomb N c N c magnetic 45
47 magnetic non-abelian Coulomb 3( N c ), electric non-abelian free electric magnetic magnetic irrelevant q, q, M electric electric N c + 2 3N 2 c magnetic magnetic electric non-abelian free magnetic = N c + 1 magnetic SU( N c ) = SU(1) flat directios electric W tree = mm SU(N c ) Nf 1 QCD Λ L SU(N c ) Nf Λ Λ 3N c ( 1) L = mλ 3N c SU(N c ) Nf 1 SU(N c ) Nf 46
48 magnetic W tree = mm W = 1 µ M j i q j q i + mm (115) M, M j, M i, q Nf, q q Nf q = µm, q i q = q Nf q i = 0 (116) M = M j = M i = 0 (117) (116) magnetic Higgs SU( N c 1) Nf 1 W = 1 M j i i ˆq jˆ q µ (118) M, ˆq, ˆ q SU(Nf N c 1) Nf 1 1 i, ĩ = 1,..., 1 electric SU(N c ) Nf 1 SU( N c 1) Nf 1 = SU(( 1) N c ) Nf 1 magnetic Λ 3(( N c ) 1) ( 1) L = Λ (3 Nc) <q Nf q Nf > = Λ (3Nf Nc) Nf µm magnetic scale matching relations = N c + 2 magnetic Higgs instanton W inst = Λ 6 N c+2 L det( 1 µ M) q Nf q = det M Λ 3N c (N c +1) L (119) 47
49 W = 1 Λ 2Nc 1 L ( M i jq i q j det M) (120) q, q electric = N c + 1 B i = q i B j = q j (120) = N c + 2 electric magnetic magnetic electric QCD (120) electric Λ L magnetic loop instanton electric flat directions < Q >=< Q Nf > < Q N c >=< Q Nc > electric SU(N c ) Nf Higgs SU(N c 1) Nf 1 Λ 3(Nc 1) ( 1) L = Λ3N c <Q QNf > electric magnetic < M > q Nf, q magnetic SU( N c ) Nf 1 electric Λ 3( N c) ( 1) L = 1 µ < M > Λ 3( N c) magnetic 48
50 Λ 3(Nc 1) ( 1) L Λ 3( N c) ( 1) L = 1 µ Λ3Nc N Λ f 3( N c) = ( 1) N c µ 1 scale matching relations flat directions electric magnetic electric magnetic moduli electric rank < M > N c magnetic M magnetic M N j i = q i q j < N j i >= 0 rank < M > magnetic rank < M >< N c rank <M > < N c < N j i >= 0 magnetic rank < M > N c rank < M >= N c magnetic 0 rank < M >= N c = Ñc detn b b = Λ 2Ñ c L (121) Λ L Λ 2Ñ c L = det < 1 µ M > Λ 3Ñ c det 0 (121) b, b B, B < N j i >= 0 scale matching relations < B B >= det < M > (122) electric 49
51 non-abelian, N c QCD 3N c non Abelianfree electric electric magnetic N c + 2 3N 2 c 3 2 c < < 3N c non AbelianCoulomb electric magnetic N c + 2 3N 2 c non Abelianfree magnetic magnetic electric 3N c = N c + 1 Higgs moduli = moduli = N c Higgs moduli moduli < N c instanton W eff = 0 N c pure Y amg Mills = 2N c (123) N=2 QCD N=1 non-abelian N=2 QCD [42] 50
52 5 QCD+ X QCD X j i (TrX = 0) > 0 QX Q N=2 N=2 Seiberg Witten N=1 Kähler N=1 W = s 0 k + 1 TrXk+1 (124) Kutasov Schwimmer [12][13] QCD 5.1 electric electric SU(N c ) Q i α, Q β j ; α, β = 1,, N c ; i, j = 1,, X β α; α, β = 1,, N c ; TrX = 0 W = s 0 k + 1 TrXk+1 (125) k > 2 irrelevant dangerously irrelevant relevant 51
53 SU( ) L SU( ) R U(1) B U(1) R (126) U(1) R R Q (, 1, 1, 1 2 N c ) k + 1 Q (1,, 1, 1 2 N c ) k X (1, 1, 0, k + 1 ) (127) 1-loop β (2N c ) 2N c < (M j ) ĩ i = QĩX j 1 Q i ; j = 1, 2,, k (128) B (n 1,n 2,,n k ) = Q n 1 (1) Qn 2 (2) Qn k (k) k n l = N c (129) l=1 ϵ 1,2,,N c Q (l) Q (l) = X l 1 Q; l = 1,, k (130) TrX j ; 2 j k (131) j (125) 52
54 5.2 magnetic ppp 2N c > > 3 N c non-abelian Coulomb 2 k non-abelian magnetic magnetic SU( N c ); Nc = k N c q α i, q j β ; α, β = 1,, N c ; i, j = 1,, Y β α ; α, β = 1,, N c ; TrY = 0 (M j ) ĩ i = QĩX j 1 Q i ; j = 1, 2,, k (132) electric magnetic electric (126) q Y q M j ( (, 1, N c k N c, 1 2 k + 1 ( N c 1,,, 1 2 k N c k + 1 ( ) 1, 1, 0, 2 k + 1,, 0, 2 4 ) k N c k N c ) N c + 2 ) (j 1) k + 1 k + 1 (133) electric magnetic thooft SU( ) 3 N c d (3) ( ) SU( ) 2 U(1) R 2 Nc 2 d (2) ( ) k + 1 SU( ) 2 U(1) B N c d (2) ( ) U(1) R 2 k + 1 (N c 2 1) ( ) U(1) 3 2 R ( k + 1 1)3 + 1 (Nc 2 1) 16 Nc 4 (k + 1) 3 Nf 2 53
55 U(1) 2 BU(1) R 4 k + 1 N 2 c (134) magnetic W mag = s 0 k + 1 TrY k+1 + s 0 µ 2 k M j qy k j q (135) j=1 s 0 Y s 0 = s 0 [14] electric magnetic TrY j = TrX j ; j = 2,, k 1 TrY k = N c N c TrX k (M j ) ĩ i = QĩX j 1 Q i ; j = 1, 2,, k B (n 1,n 2,,n k ) el B (m1,m2,,m k) mag ; m l = n k+1 i ; l = 1, 2,, k (136) B mag magnetic electric (125) k l=1 n l = N c, kl=1 m l = N c Q (l) 0 n l, 0 m l k l=1 m l = N c k l=1 n l = N c k k k m l = ( n k+1 i ) = k n k+1 i = k N c = N c (137) l=1 l=1 l=1 TrY j [14] 54
56 k=1 electric X QCD QCD k electric electric W el = s 0 Tr X k+1 + m Q Nf Q (138) (N c, ) (N c, 1) (k N c k, 1) magnetic W mag = s 0 k + 1 TrY k+1 + s 0 µ 2 k j=1 M j qy k j q + m(m 1 ) (139) q Nf Y l 1 q = δ l,k m; l = 1, 2,, k (140) q α = δ α,1 q α = δ α,k Y α β = δ α β+1 β = 1,, k 1 0 (141) Higgs (k N c, ) (k N c k, 1) electric 55
57 W el = k i=0 s i k + 1 i Tr Xk+1 i (142) s k TrX s k TrX = 0 k=2 s i ( 1 W el = Tr 3 X3 + m ) 2 X2 + λx (143) X X x 2 + mx + λ = 0 (144) V = W (X) 2 x +, x X x + r = 0, 1,, N c N c + 1 x N c r r N c r Λ TrX = rx + (Λ) + (N c r)x (Λ) = 0 r Higgs SU(N c ) SU(r) SU(N c r) U(1) (145) X QCD magnetic electric (2 N c ) + 1 Higgs SU(2 N c ) SU(l) SU(2 N c r) U(1) (146) electric magnetic QCD N c > l, 2 N c l 56
58 l = N c, N c 1,, (147) magnetic electric QCD, l = r magnetic M 1, M 2 QCD k > 2 k=2 k W (X) = 0 X k x l ; l = 1,, k X x i i l k l=1 i l = N c SU(N c ) SU(i 1 ) SU(i 2 ) SU(i k ) U(1) k 1 (148) magnetic k l=1 j l = k N c k j l ; l = 1,, k SU(k N c ) SU(i 1 ) SU(i 2 ) SU(i k ) U(1) k 1 (149) k=2 QCD j l = i l < N c k X W = s o k+1 TrXk+1 < N c k k > 2 s l W (X) = 0 W (x) = i (x a i ) n i ; n i = k (150) 57
59 n i 1 r i X a i SU(N c ) i SU(r i ) U(1) k 1 ; ri = N c (151) SU(r i ) X i 0 W L = i W (n i+1) (a i )TrX (n i+1) i + X i (152) W (n i+1) (a i ) W(x) n i x=a i magnetic SU(k N c ) i SU( r i ) U(1) k 1 ; ri = k N c (153) W = s o k+1 TrXk+1 < N c k r i = n i r i SU(N c ) SO(N), Sp(N) X QCD non-abelian [43][44] 58
60 6 Seiberg QCD Kutasov-Schwimmer non-abelian Coulomb non- Abelian Moduli N=1 thooft N=1 QCD QCD 3N 2 c < < 3N c non-abelian QCD Kutasov-Schwimmer non-abelian Kutasov-Schwimmer 59
61 non-abelian non-abelian non-abelian [45] N=2 non-abelian electric magnetic electric magnetic [46] N=1 non-abelian 60
62 7 61
63 { Q L α, QᾱM } = 2σ m αᾱ p m δ L M A Poincarè [P m, P n ] = 0 [ Pm, Qα] L = [ ] P m, QᾱL = 0 [P m, B l ] = [ P m, X ] LM = 0 { Q L α, Q βm } = ϵαβ X LM { QᾱL, Q βm } = ϵᾱ βx LM [ X LM, QᾱK ] = [ X LM, Q K α ] = 0 [ X LM, X KN ] = [ X LM, B l ] = 0 [B l, B m ] = ic k lmb k [ Q L α, B l ] [ QᾱL, B l ] = S L l MQ M α = S l M L QᾱM X LM = a l,lm B l (154) X LM a l,lm L, M B l c k lm XLM Sl L M Jacobi B l Jacobi S M l Ka k,kl = a k,mk S l L K (155) a S ( S ) intertwiner Weyl spinorq Q 1 N=1 ) Lorentz 62
64 B Kähler n Φ i L = d 2 θd 2 θk(φ [ i, Φ j ) + ] d 2 θw (Φ i ) + h.c. (156) A i χ i L = g ij m A i m A j ig ij χ j σ m D m χ i R ij kl χi χ k χ j χ l 1 2 D id j W χ i χ j 1 2 D i D j W χ i χ j g ij D i W D j W (157) D i W = A i W D i D j W = 2 A i A W j Γk ij A W (158) k g ij = K(A) A i A j g ij,k = A g k ij = g mj Γm ik g ij,k = A g k ij = g im Γm i k (159) g ij K Kähler Kähler Γ m ik Kähler [47] 63
65 C Wess-Zumino Wess-Zumino v m v m + m χ Wess-Zumino Wess-Zumino Wess-Zumino δ ξ A = 2ξΨ δ ξ Ψ = i 2σ m ξdm A + 2ξF δ ξ F = i 2 ξ σ m D m Ψ + i2gt (a) A ξ λ (a) δ ξ v (a) m = i λ (a) σ m ξ + i ξ σ m λ (a) δ ξ λ (a) = σ mn ξv (a) mn + iξd (a) δ ξ D (a) = ξσ m D m λ(a) D m λ (a) σ m ξ (160) 64
66 [1] S.Dimopoulos,S.Raby and F.Wilczek, Phys. Rev. D24 (1981) [2] D.Amati, K.Konishi, Y.Meurice, G.C.Rossi and G.Veneziano, Phys. Rep. 162 (1988) 169 and references therein. [3] N.Seiberg, Phys. Rev. D49 (1994)6857. [4] K.Intriligator and N.Seiberg, Nucl. Phys. B431 (1994) 551. [5] K.Intriligator and N.Seiberg, Nucl. Phys. B444 (1995) 125. [6] N.Seiberg and E.Witten, Nucl. Phys. B431 (1994) 484. [7] N.Seiberg, Nucl. Phys. B435 (1995) 129. [8] K.Intriligator and R.G.Leigh and N.Seiberg, Phys. Rev. D50 (1994) [9] K.Intriligator, Phys. Lett. 336B (1994) 409. [10] N.Seiberg and E.Witten, Nucl. Phys. B426 (1994) 19. [11] G. thooft: in Recentdevelopments in Gauge Theoriees,eds. G. thooft et al.(plenum Press, New York,1980). [12] D.Kutasov, Phys. Lett. 351B (1995) 230. [13] D.Kutasov and A.Schwimmer, Phys. Lett. 354B (1995) 315. [14] D.Kutasov and A.Schwimmer and N.Seiberg, hep-th/ [15] K.Intriligator and R.G.Leigh and M.Strassler, hep-th/ [16] M.Berkooz, Nucl.Phys. B452 (1995)
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