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- ゆきさ しげい
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1
2 Seiberg Witten 1994 N = 2 SU(2) Yang-Mills 1
3 N = 2 Yang-Mills SU(2) one-loop Duality BPS N =
4 1 SU(3) QCD (quantum chromodynamics) QCD QCD QCD QCD 3
5 1994 N = 2 N = 2 N = 2 SU(2) Yang-Mills 4
6 2 2.1 (Lorentz) (Poincare) 5
7 ϕ j ϕ i(x) = U j i (x)ϕ j(x) = [exp(igθ a (x)t a )] j i ϕ j(x) (2.1) T a θ a g charge ϕ i (x + dx) = ϕ i + ig(a µ ) j i ϕ j (x)dx µ (2.2) A µ = (A µ ) j i A µ T a A G dim G (A µ ) j i (x) = a=1 A a µ(x)(t a ) j i (2.3) ϕ µ φ ϕ D µ ϕ = ( iga µ )ϕ (2.4) D µ φ ( covariant derivative ) iga µ ϕ A µ ϕ g (gauge coupling) D µ ϕ D µ ϕ U(x)D µ ϕ(x) (2.5) (D µ ϕ) (D µ ϕ) D µ ϕ (D µ ϕ) (D µ ϕ) U 1 (2.6) (D µ ϕ) D µ ϕ (D µ ϕ) (U 1 U)D µ ϕ = (D µ ϕ) D µ ϕ (2.7) 6
8 ( ) 1 4 F a µνf aµν = 1 4 ( Aa µ A a ν + gf abc A b µa c ν)( A aµ A aν + gf ade A dµ A eν ) (2.8) 1 4 F a µνf aµν = 1 4 N 1 tr(f µν F µν ) (2.9) N T a tr(t a T b ) = Nδ ab F µν T a F µν Fµν a F µν F µν = F a µνt a (2.10) F µν U(x)F µν (x)u 1 (x) (2.11) tr(f µν F µν ) tr[(uf µν U 1 )(UF µν U 1)] = tr[u 1 UF µν F µν ] = tr[f µν F µν ] (2.12) 1F a 4 µνf aµν ϕ Lagrangian L = 1 4 F a µνf aµν + (D µ ϕ) D µ ϕ m 2 (ϕ ϕ) + L int (ϕ) (2.13) ϕ L int (ϕ) Yang-Mills Higgs QCD (Quantum chromodynamics) 7
9 QCD SU(3) 2.2 Wilson ( Wilsonian effective action ) Z = [dψ]e is(ψ) (2.14) [ ] Λ 0 Z = [dψ]e is(ψ) p Λ 0 (2.15) Z Wilsonian effective action S eff Λ Λ 0 Z = (2.16) p Λ 0 [dψ]e is eff (Ψ;Λ) Λ Λ 0 Wilsonian effective action Wilsonian effective action Wilsonian effective action Lagrangian Lagrangian Wilsonian effective action Lagrangian Wilsonian effective action Λ 0 8
10 Wilsonian effective action Wilsonian effective action ( Law energy effective action - LEEA ) N = 2 ( Wilsonian low energy effective theory) Lagrangian Wilsonian effecive theory N = 2 Wilsonian effective theory Lagrangian Seiberg [8] Seiberg-Witten [2] N = (assymptotic free) 9
11 N = 2 2 (supercharge - Noether charge ) N = 2 instanton [8] [9] 1 [2] N = 2 1 [9] [10] [11] 10
12 3 N = 2 Yang-Mills 3.1 N = 2 Yang-Mills ( Super Yang-Mills SYM ) Lagrangian adjoint L = 1 4 F aµν F a µν i λ a α σ µ αα D µ λ a α Da D a θ 32π F aµν a F µν (D µ φ a ) D µ φ a i ψ a α σ µ αα D µ ψ a α + f a f a (3.1) 2gf abc φ a λb α ψc α + 2gf abc φ a ψ bα λ c α + igf abc D a φ b φ c g, f abc (structure constant) Fµν a (field strength) φa (complex scalar) (ψ a λ a ) Weyl (spinor) D, f a, b, c µ, ν Lorentz α, α φ covariant derivative D µ φ a = µ φ a ig(ta b)a ca b µφ c (ψ a λ a ) (T A ) A (adjoint representation) field strength (T b A ) c a = if abc (3.2) F a µν = µ A a ν A a + gf abc A µ A ν (3.3) F µν a F aµν = 1 2 ɛµνρσ F a ρσ (3.4) 11
13 1 Lagrangian field (A a µ ψ a λ a φ a ) adjoint A µ N = 2 (Pure) SYM Yang-Mills adjoint Lagrangian Yang-Mills adjoint (fundamental representation) Pure Yang-Mills η µν = diag( + ++) field Euler-Lagrange Lagrangian (D, f) Euler-Lagrange f = f = 0, D = igf abc φ b φ c (3.6) (φ a ψ a λ a A a µ) N = 2 multiplet( ) A µ (vector multiplet) N = 2 Super Yang-Mills vector multiplet 3.1 ψ φ A µ λ supercharge : Q 1 N = 2 supercharge : Q 2 A µ ψ λ φ (N = 2 hypermultiplet) Seiberg Witten (Non-abelian gauge group) Pure Yang-Mills 1 ef mn = 1 2 ɛ mnpqf pq (3.5) ( ) ( ) 12
14 3.1: N =2 vector multiplet Non-Abelian gauge theory SU(2) [2] SU(2) 3.2 SU(2) N = 2 SUSY U(1) R SU(2) R U(1) R SU(2) R R-symmetry (global symmetry) U(1) R φ a e i2α φ a λ a e iα λ a (3.7) ψ a e iα ψ a φ ψ λ 2 U(1) R charge 2 SU(2) R φ a φ a ( ) ψ a λ a e i τ i 2 θ i ( ψ a λ a ) (3.8) (i = 1, 2, 3) τ i Pauli Spin ψ a λ a (supercharge - Noether charge ) Q 1 Q 2 SU(2) R U(1) R U(1) R anomalous U(1 R ) anomaly Z 4Nc 2N f N f 13
15 (fundamental matter) (flavor) N f f N c color SU(N c ) N c SU(2) N c = 2 Pure Yang-Mills fundamental matter N f = 0 SU(2) R Z 8 SU(2) R Z 8 Z 8 φ e i 2π 4 α φ λ e i 2π 8 α λ (3.9) ψ e i 2π 8 α ψ φ U(1) R charge ψ λ charge 2 Z 8 ψ (λ U(1) R charge ) α ψ ψ e i π 4 ψ iψ e i 3 4 π ψ ψ e i 5π 4 ψ iψ e i 7π 4 ψ α = 4 φ φ ψ ψ λ λ (3.10) SU(2) R θ 1 = θ 2 = 0, θ 3 = 2π Z 2 SU(2) R Z 8 /Z 2 N = 2 SU(2) N = 2 (3.1) N = 2 SYM V (φ) = g 2 tr ( [φ, φ ] 2) (3.11) if abc = 2 tr ( T a[ T b, T c]) φ a T a φ 0 φ = 0 0 a 0 14
16 φ a SU(2) U(1) SU(2) Pauli-Matrix τ SU(2) τ 3 3 SU(2) 2 Pauli-Matrix τ (4-)Pauli-Matrix σ φ φ = a τ 3 2 (3.12) V (φ) = 0 (3.13) φ a C a a 0 SU(2) U(1) a = 0 SU(2) massless U(1) massless massless a (3.11) 0 a φ = a τ 3 (3.14) 2 SU(2) iτ 2 SU(2) Weyl group 15
17 φ adjoint φ φ (iτ 2 ) φ (iτ 2 ) 1 = (iτ 2 ) φ ( iτ 2 ) = a 2 τ 2 τ 3 τ 2 = a 2 τ 3 (3.15) = φ φ a u = trφ 2 a a 0 SU(2) U(1) Z 8 φ a Z 8 φ φ e i 2π 4 α φ, (N = 0, 1, 2 8) (3.16) α φ φ iφ φ iφ φ iφ φ iφ φ α = 0, 4 (3.17) Z 8 Z 2 φ Z 8 Z 2 SU(2) Weyl Weyl φ φ φ Z 8 α = 2, 6 (3.18) φ φ (3.19) < φ > Z 8 (α = 0, 4) (α = 2, 6) + ( Weyl ) Z 8 Z 2 Z 4 SU(2) R SU(2) R 16
18 SU(2) R Z 4 Z 2 (3.20) N = 2 a Z 4 φ 2 a U(1) charge 2 u u u Z 2 3.2: u- Z 2 17
19 3.3: a- Z 4 4 Z
20 3.3 one-loop Seinberg [8] Wilsonian Lagrangian [ 1 4π Im d 4 θ F(A) A Ā + d 2 θ 1 ] 2 F(A) W α W 2 A 2 α (3.21) W W α (y, θ) = iλ α (y) + [δ β αd 1 2 (σµ σ ν ) β αf µν (y)]θ β + θθσ µ α α D µ λ α (y) (3.22) U(1) (field strength superfield ) A A(y, θ) = φ(y) + θψ(y) + θθf(y) (3.23) N = 2 vector multiplet N = 1 chiral multiplet superfield y θ θ y µ = x µ + iθ α σ µ α α θ α (3.24) SU(2) U(1) massless N = 2 U(1) vector multiplet : N =2 U(1) vector multiplet F(A) A Ā N = 2 Wilsoninan 19
21 F 0 A F one loop (non-perturbative) F non pert F 0 (A) + F one loop (A) + F non pert (A) (3.25) N = 2 vector multiplet A [8] F non pert (instanton) F non pert F instanton F 0 F 0 (A) = 1 2 τ cla 2 (3.26) τ cl = θ + i 4π 2π g 2 N = 2 1-loop 1-loop F one loop 1-loop g eff F one loop (A) = 1 4π (i )A 2 (3.27) 2 geff 2 g eff g 2 eff 4π = α (3.28) µ dα dµ = 1 2π bα2 (3.29) b β SU(N c ) 1-loop β b = 11 3 C 2(adj) + r n F r C 2(r) + r n B r 1 3 C 2(r) (3.30) n F r n B r : ( fermion ) : ( scalar boson ) (3.31) 20
22 C 2 2 adj adjoint (3.30) adjoint (3.30) r (representation) representation r fermion scalar boson adjoint C 2 (r) C 2 (adj) C 2 (F ) F fundamental representation SU(N c ) C 2 (adj) = N c C 2 (F ) = 1 2 (3.32) U(1) C 2 (adj) = 0 C 2 (F ) = 1 (3.33) U(1) β- N = 2 SU(2) Yang-Mills α N = 2 vector multiplet adjoint fermion 2 scalar boson 1 b = g 2 eff (A) 4π = 1 4 2π ln A Λ = 4 (3.34) (3.35) A φ µ F one loop (A) = i A2 A2 ln (3.36) 2π Λ 2 α(µ) 3.5 SU(2) U(1) 21
23 3.5: 1-loop SU(2) U(1) φ a β- 0 SU(2) 1-loop Λ a λ SU(2) U(1) a Λ a Λ a Λ a F F instanton (A) = ( 4k Λ F k A A) 2 (3.37) k=1 k 1 Seiberg Witten [2] 22
24 F 3.4 φ a 0 a u a u = a = 0 SU(2) : a = 0 1-loop a Λ 1-loop 3.7 a a ( 3.21) ( ) F(A) K = Im A Ā (3.38) 23
25 3.7: 1-loop a Λ Kähler A A(y, θ) = φ + θψ + θθf (3.39) a ā 2 Kähler Kähler (ds) 2 = Im 2 F(a) da dā (3.40) a 2 τ τ(a) = 2 F (3.41) a 2 τ(a) holomorphic( ) Im τ(a) da dā (3.42) Im τ(a) a a = 0 (positive definite) a 1-loop QCD scale strong coupling 24
26 1-loop a a a D a a D 25
27 4 4.1 Duality Low energy effective Lagrangian Kähler metric classical loop 3.7 Kähler metric (ds) 2 = Im τ(a) dadā (4.1) a τ(a) (3.41) τ = 2 F ā (4.2) a 2 a (holomorphic) (3.36) 1-loop τ(a) τ(a) i π a2 (ln( ) + 3) (4.3) Λ2 a a D = F a a D (ds) 2 = Imda D dā = i 2 (da Ddā dada D ) (4.4) a a D u u = 1 2 a2 (4.5) (ds) 2 = Im da D du dā dū du dū = i 2 (da D du da du 26 dā D )dudū (4.6) dū
28 a α = (a D, a), α = 1, 2 (4.7) (ds) 2 = i 2 ɛ da α αβ du dā β dudū (4.8) dū 2 2 U a α U α β aβ (4.9) ā α U α βāβ U U SL(2, R) (4.10) SL(2, R) v ( ) a D v = (4.11) a M SL(2, R) 2 2 v Mv + c (4.12) c 2 1 N = 2 (Pure) SYM M SL(2, R) SL(2, Z) c 0 SL(2, R) 1 SL(2, R) ( ) 1 b T b = 0 1 ( ) (4.13) 0 1 S =
29 b b R a α = (a D, a) (α = 1, 2) (4.14) T b a D a a D a D + ba a a (4.15) a D = F a (3.21) θ θ + 2πb b b b Z T b S SL(2, R) SL(2, Z) a D SL(2, Z) S U(1) ( QED ) F µν dual field strength F F µν = 1 2 ɛµνρσ F ρσ (4.16) F F 2 µν = ( F ) 2 µν (4.17) 1 32π Im τ(a) (F + i F ) 2 = 1 16π I τ(a) (F 2 + i F F ) (4.18) ( (4.17) F 2 = (F ) 2 ) F D F F F F integral out F F 28
30 F U(1) A F = da df = 0 F F 1 2 F µνdx µ dx ν (4.19) df = 1 2 ρf µν dx ρ dx µ dx ν (4.20) dx ρ, dx µ, dx ν df = 0 [ρ F µν] = 0 ( ) df = 0 ɛ µνρσ ν F ρσ = 0 (4.21) Lagrange V Dµ Lagrange 1 V Dµ ɛ µνρσ ν F ρσ = 1 F D F = 1 8π 8π 16π Re ( F D if D )(F + i F ) (4.22) F Dµν F Dµν = µ V Dν ν V Dµ V D field strength (4.18) (F + i F ) F D F D (4.18) (4.18) F D 1 32π Im 1 τ (F D + i F D ) 2 = 1 16π Im 1 τ (F 2 D + i F D F D ) (4.23) U(1) F D N = 1 U(1) N = 1 N = 1 field strength vector super field field strenth W α W field strength F µν (y) λ(y) D(y) W α = iλ α (y) + [δ β αd(y) i 2 (σµ σ ν ) β α F µν (y)]θ β + θθσ µ α β µ λ β(y) (4.24) y µ = x µ + iθσ µ θ θ chiral coodinate superspace Lagrangian 1 8π Im d 2 θτ(a)w 2 (4.25) 29
31 A N = 1 chiral superfield df = 0 N = 1 Im DW = 0 (4.26) D supercovariant derivative D α = + 2iɛ αβ σ µ β θ α θ α (4.27) α y µ Im DW = 0 chiral coordinate (y, θ) (x, θ, θ) ɛ µνρσ ν F ρσ = 0 Im DW = 0 df = 0 Lagrange vector superfield V D Lagrange 1 4π Im d 4 xd 4 θv D DW = 1 4π Im d 4 xd 2 θd 2 θ(dvd )W = 1 16π Im d 4 xd 2 θ D DDV D = 1 4π Im d 4 xd 2 θw D W (4.28) W Dα N = 1 U(1) vector superfield field strength W α = 1 4 D DD α V (4.29) D chiral coordinate D α = θ α (4.30) W Dα W Dα = 1 4 D DD α V D (4.31) W D W WD αw α [ 1 4π Im d 4 xd 2 θ 1 ] 2 τ(a)w W d 4 xd 2 θwd W (4.32) 30
32 W W 1 8π Im d 2 θ 1 τ(a) W DW D (4.33) F F D τ(a) 1 τ(a) field strength F µν F µν = 1 2 ɛµνρσ F ρσ (4.34) field strength N = 2 F(A) A = h(a) (4.35) Im d 4 θh(a)ā (4.36) Im A D = h(a) (4.37) d 4 θ h D (A D )ĀD (4.38) h D (h(a)) = A h (A) = τ(a) (4.39) D 1 τ(a) = 1 h (A) = h D(A D ) = τ D (A D ) (4.40) τ D (A D ) = 1 τ D (A) (4.41) 31
33 S a a D τ 1 S a a D S (duality transformation) dual a a D τ D 4.2 BPS a a D BPS BPS saturated states (BPS ) Bogomolny [4] Prasad-Sommerfield [5] Bogomolny BPS Prasad-Sommerfield dyon dyon monopole monopole ( ) U(1) U(1) U(1) t Hooft-Polyakov monopole dyon Witten-Olive [6] N 2 BPS 32
34 monopole dyon a charge Z( ) Z Z = an e + a D n m (4.42) BPS BPS (BPS saturated state) charge Z M 2 = 2 Z 2 (4.43) [6] SU(2) U(1) monopole, dyon a D charge Z cl = a(n e + τ cl n m ) (4.44) τ cl = θ + i 4π n 2π g 2 e n m BPS M 2 2 Z 2 (4.45) a D = F (4.46) a a D Z = a n e + a D n m (4.47) τ cl a D a D a, a D n e, n m BPS N = 2 hypermultiplet massive 2 2N = = 16 massless N = 2 2 N = 2 2 = 4 BPS massive N = 2 massless hypermultiplet 33
35 5 5.1 a a D u a D = F a (3.21) a a D a a D u u a a D u-plane u (a(u), a D (u)) u e i2π u a a D a a D u-plane u e i2πu a a D M ( ) ( ) a D a D M (5.2) a a (5.1) M a a D SL(2, Z) SL(2, Z) M (a D, a) v Kähler v Mv + c (5.3) 34
36 c BPS states BPS states charge Z = n e a + n m a D (5.4) Z charge (n e, n m ) Z (a, a D ) c (n e, n m ) [3] Z c N = 2 u a a D u u 1 2 u Z 2 1 a u 0 u u Z 2 u 3 35
37 1+2n(n 1; n Z) U(1) massless vector superfield massless massless massless massless singular ( ) 2 massless U(1) SU(2) massless massless massless u = tr(φ 2 ) = 0 2 massless BPS state massless monopole massless 36
38 5.3 a D a D = F a 2ia ( a ) π ln + ia (5.5) Λ π a u 1 2 a2 (5.6) u e 2πi u (5.7) ln a ln a + iπ (5.8) a D a D + 2aa a (5.9) ( ) M = P T = (5.10) 0 1 P ( P = ) (5.11) a D = F (a) a = τ cl a (5.12) u e 2πi u (5.13) a a (5.14) a D a D 37
39 P T T = ( ) (5.15) P T M 1 massless monopole monopole 2 nm a D (5.16) a D (u 0 ) = 0 (5.17) u 0 dual τ τ D a magnetic photon dual electron N = 2 U(1) electron electron 2 2 U(1) hypermultiplet U(1) vectormultiplet (3.30) b = 2 (5.18) τ D i π ln a D (5.19) u 0 a D a D c 0 (u u 0 ) ( c 0 ) (5.20) τ D = da da D (5.21) 38
40 a a(u) a 0 + i π a D ln a D a 0 + i π c 0(u u 0 ) ln(u u 0 ) (a 0 ) (5.22) u u 0 e 2πi (u u 0 ) a a D a D a D a a 2a D (5.23) ( ) 1 0 M 1 = 2 1 (5.24) M 1 u u 0 u 0 = 1 3 u = M 1 u = 1 Z 2 u = 1 u M 1 M 1 = M (5.25) (5.10) (5.24) M, M 1 M 1 ( ) M 1 = (T S)T 2 (T S) = (5.26) 2 3 A ( ) 1 1 A = T M 1 = 2 1 (5.27) 39
41 5.1: u- M 1 M 1 M 1 = AM 1 A 1 (5.28) M 1 M 1 q = (n m, n e ) (5.29) v = ( a D a ) (5.30) M v Mv (5.31) q qm 1 40
42 A v Av (5.32) q qa 1 u = 1 q = (1, 0) u = 1 ( ) ( ) qa = 1, 0 = (1, 1) (5.33) 2 1 massless dyon M q qm = q (5.34) M 1 q 1 = (1, 0) (5.35) q 1 M 1 = q 1 (5.36) M 1 q 1 = (1, 1) (5.37) dyon massless 41
43 6 6.1 ( ) 1 2 M = 0 ( 1 ) M 1 = 2 ( ) (6.1) M 1 = a a D Im (τ) > 0 (6.2) a, a D a = 2 2π dx x u 2 γ 2 x2 1 = π 2 u a D = 2π dx x u x2 1 dx x u x2 1 (6.3) (6.4) 42
44 6.2 2 a = π 2 a D = π 1 1 u 1 dx x u x 2 1 dx x u x2 1 u 2u 1 dx a π 1 x = 2u (6.6) 2 1 (6.5) a D x = uz 2u 1 dz z 1 a D = (6.7) π z 2 u 2 1/u u log 2u ln u a D i (6.8) π a a D u = 1 a D 2u 1 dz z 1 a D = π z 2 u 1 1 dz z 1 = i 2 π z u 1 2 (1 1 i(u 1) ) (6.9) u 2 1/u 1/u a u = 1 u = 1 a u = 1 a(u = 1) = da 2 du = 2π π 1 1 dx x + 1 = 4 π (6.10) dx (x + 1)(x 1)(x u) (6.11) a a = 4 (u 1) ln(u 1) (6.12) π 2π u = 1 43
45 7 F µν (self-dual equation) F µν = ɛµνρσ F ρσ (7.1) Seiberg-Witten [2] N = 1 D-term N = 1 N = trivial N = 4 running 44
46 N = 1 F instanton 2002 N.Nekrasov k F k [9] Okounkov Nekrasov [10] 2 45
47 46
48 [1] J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton University Press, Princeton, (1983), (second edition: 1992). [2] N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory, Nucl.Phys.B426:19-52,1994 (arxiv:hep-th/ ). [3] N. Seiberg and E. Witten, Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD, Nucl.Phys.B431: ,1994 (arxiv: hepth/ ) [4] E.B.Bogomolny, The stability of classical solutions,sov.j.nucl.phys.24:449,1976. [5] M.K.Prasad and Charles M. Sommerfield, Exact Classical Solution for the t Hooft Monopole and the Julia-Zee Dyon, Phys.Rev.Lett.35: ,1975. [6] E. Witten and D. Olive, Supersymmetry algebras that include topological charges, Phys.Lett.B78:97,1978. [7] E. Witten, Dyons of Charge eθ 2π, Phys.Lett.B86: ,1979. [8] N. Seiberg, Phys.Lett. 206B (1988) 75. [9] N.A.Nekrasov, SEIBERG-WITTEN PREPOTENTIAL FROM INSTANTON COUNTING, Adv.Theor.Math.Phys.7: ,2004 (arxiv: hep-th/ ). [10] Nikita Nekrasov and Andrei Okounkov, SEIBERG-WITTEN THEORY AND RANDOM PARTITIONS, (arxiv hep-th/ ). [11] Hiraku Nakajima and Kota Yoshioka, INSTANTON COUNTING ON BLOWUP. 1., (arxiv: math.ag/ ). 47
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18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α
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24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x
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H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [
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数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published
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