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1 70 80 90 50 2 3 3 84 first revolution 4 94 second revolution 5 6 2 1: 1

3 consistent 1-loop Feynman 1-loop Feynman loop loop loop Feynman 2 3 2: 1-loop Feynman loop 3 cycle 4 = 3: 4: 4 cycle loop Feynman Feynman consistent 4 Minkowski 5 Type I, Type IIA, Type 2

IIB, Heterotic SO(32), Heterotic E 8 E 8 1 open closed 5 Type I Type IIA Type IIB Het SO(32) Het E 8 E 8 O(32) U(1) SO(32) E 8 E 8 open + closed closed closed closed closed N = 1 N = 2 N = 2 N = 1 N = 1 10 10 10 10 10 1: open string closed string open string closed string 5: N = 1 N = 2 1 10 4 10 1 5 3 2 SU(3) SU(2) U(1) G W ± Z γ u, d e, ν e c, s µ, ν µ t, b τ, ν τ 2: SU(3) SU(2) U(1) 3

2 SU(3) SU(2) U(1) GUT 3 SU(5) SO(10) E 6 SU(5) GUT SO(10) GUT E 6 GUT SU(5) SO(10) E 6 (ψ 10,ψ 5,ψ 1 ) 3 ψ 16 3 ψ 27 3 3: 3 SO(10) GUT ψ 16 SO(10) 16 2 16 ψ 16 3 Heterotic E 8 E 8 4 10 6 M 6 6 M 6 4 4 4

N = 1 M 6 Calabi-Yau Calabi-Yau SU(3) 6 M 6 ψ SO(6) g gψ g SO(6) SU(3) 6 gψ ψ g SU(3) 6: SO(6) SU(4) SU(3) SO(6) SU(4) 4 SU(3) 4 N = 1 4 N = 1 6 M 6 Calabi- Yau E 6 GUT E 8 E 8 field strength F = 1F 2 µνdx µ dx ν 2 R = 1(R a 2 µν b )dxµ dx ν Σ tr(r R) = Σ tr(f F) (3.1) Σ M 6 4 M 6 R F minimal embedding M 6 SU(3) R M 6 SO(6) SU(3) F E 8 E 8 E 8 SU(3) E 8 SU(3) 1 N = 1, 2 10 10 N = 1 6 4 N = 0, 1, 2, 4 SO(6) Spin(6) Lie 5

E 8 SU(3) E 6 3 E 6 E 6 4 N = 1 6 SU(3) E 8 SU(3) E 6 minimal embedding Dirac E 6 27 ψ χ(m 6 )/2 χ(m 6 ) M 6 M 6 3 E 6 GUT Heterotic E 8 E 8 χ(m 6 )/2 3 consistent Heterotic 10 N = 1 consistent E 8 E 8 SO(32) consistent E 8 E 8 SO(32) GUT E 6 ψ 27 4 N = 1 GUT E 6 4 N = 1 E 6 4 GUT U(1) 496 E 8 U(1) 248 flat 6

10 4 Minkowski 6 M 6 6 M 6 6 Calabi-Yau M 6 Heterotic E 8 E 8 Hetero E 8 E 8 1 4 Second Revolution 94 D-brane M 7

4.1 (duality) (dual) T-duality Sduality T-duality Type IIA SR 1 ( R S1 ) Type IIB S1/R 1 ( 1/R S 1 ) Type IIA Type IIB 10 S 1 R 1/R 7 Type IIA SR 1 Type IIB S1/R 1 SR 1 1/R Type IIA on S 1 R R 7: T-duality Type IIB on S 1 1/R 1/R 1/R S1/R 1 2π/R ( 1/2π) 1/R R S-duality Type IIB Type IIB B µν, C µν µ, ν 0, 1,..., 9 µ ν B µν = B νµ g s g s 1/g s B µν C µν (4.1) C µν B µν g s 1/g s B µν C µν g s 1 1/g s 1 8

S-duality S-duality (4.1) Type IIA Type IIB Type I Heterotic 8 Type I Heterotic SO(32) S-dual Type IIA T Type IIB S Type I S Het SO(32) T Het E 8 E 8 8: Duality Web Type I Heterotic SO(32) 4.2 D-brane Type IIB Type IIB 2 B µν A µ 4.1 Type IIB S-duality C µν C µν D-string D1-brane T-duality Dp-brane (1 + p) ( 1 p ) p Type IIA Type IIB Dp-brane Dp-brane Polchinski 9 open string (1 + p) Dp-brane D-brane D open string Dirichlet boundary condition Dp-brane 9

open string (1 + p) 9: Dp-brane second revolution 4.3 M Type IIA Type IIA g µν φ C µ 2 B µν 3 C µνρ µ, ν, ρ 0, 1,..., 9 φ g 1010 C µ g µ10 B µν C µν10 11 3 g MN C MNP (M, N, P = 0, 1,..., 10) 10 11 11 x 10 10 Type IIA Witten 11 Type IIA 4.2 D-brane D0-brane 11 11 M Membrane M M 3 C MNP (1 + 2) M M2-brane (1 + 2) x 10 S 1 x 10 Type IIA M Membrane x 10 Type IIA D2-brane 10

Mother M M M 11 S 1 Type IIA 4 Type IIB M 11 M 0 M on S 1 Type IIA M on S 1 S 1 Type IIB M on I Het E 8 E 8 M on I S 1 Het SO(32) M on S 1 I Type I 4: M S 1 S 1 S 1 M S 1 S 1 9 Type IIA S 1 7 T-duality Type IIB S1/R 1 10 4 I (I = S 1 /Z 2 ) M x 10 I E 8 Heterotic E 8 E 8 Miracle, Magic, Mystery M M S-duality Type IIB 4 M S 1 S 1 S 1 Type IIB S-duality S 1 Type IIB g s S 1 g s 1/g s Heterotic SO(32) Type I S-duality 4 Heterotic SO(32) Type I M I S 1 S 1 I x 10 I S 1 Heterotic SO(32) Type I I S 1 M S 1 I I S 1 Matrix M M M(atrix)-theory 11

M looks like W. 1999 Strings 99 Townsend W Witten W M 1 M 10 5 M S T Type IIB Type I Type IIA 11 M T S Het SO(32) Het E 8 E 8 10: 1995 11 M 4.4 D-brane M D-brane D-brane Seiberg N = 1 Seiberg-Witten N = 2 9 D-brane open string Dp-brane (1 + p) 12

M M 4 Seiberg-Witten brane QCD D-brane D-brane Bekenstein-Hawking S = log W S = A/4 D-brane 11: D-brane D-brane Type IIA D0-brane D0-brane S m 1 (ẋ k ) 2 dt (4.2) 13

x k (t) (k = 1, 2,..., 9) D0-brane open string D0-brane N open string 12 D0-brane N N x k (t) open string t open string D0-brane 12: D0-brane N N N D0-brane x k (t) D0-brane D0-brane N x k N N N N 4.3 M x k D-brane Type IIA 1 Type IIA closed string open string (1 + 9) D-brane D9-brane open string 4.2 Dp-brane p Type IIA p D-brane non-bps D-brane non-bps D9-brane N open string N N U(N) 14

Type I Type IIB open string 1 5 N Type I Type IIA Type IIB O(N + 32) O(N) U(N) U(1) U(N) U(N) open + closed open + closed open + closed (, ) adjoint (, ) 5: N open string 13 V (T) T non-bps D-brane V (T) 13: Sen 13 T non-bps D-brane 1 13 T D-brane 4.2 open string 5 D-brane open string Type I Type II T 1 5 15 T

5 10 3 3 3 Heterotic E 8 E 8 5 4.3 M M 5 M 3 4 3 4 D-brane M 4 80 second revolution D-brane 3 second revolution 16

4 M 4 revolution 6 17

4.4 10 QCD QED 18

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