Transcription

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2

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4

5 D D = 2, 3, 4, 5, 6

6 D D = 2, 3, 4, 5, 6

7

8 N =(2, 0)

9 N =(2, 0)

10 N =(2, 0)

11 N =2 (2) l (2) r (2) R (2) r (2) R (2) r N =2 S 4 N =2 S 4

12 N =2 (2) l (2) r (2) R (2) r (2) R (2) r N =2 S 4 N =2 S 4

13 T µν g µν T µν g µν S µα ψ µα Jµ R AR µ X AB M AB T µν g µν A R µ M AB

14 Q M ξ δ ξ O = 0 O Q M ϵ δ ϵ O = 0 O

15 S S + t d d xδ ϵ O, δ ϵ 2 O = 0, δ ϵ O ψ δψ 2. t Z = d d x δ ϵ O = 0. t δψ = 0 M = M i Z = Z Z i M i

16 Ω R d S 2 RP 2 S 3 S 3 /Z k S 2 S 1 S 4 S 3 S 1 S 3 /Z k S 1 S 5 S 4 S 1 δϵ 2

17 S 3

18 N =1 N =2 N =2 N =4 δ ϵ

19 F F E Nf +1 m = 1/g 2 (2) N f (2N f ) N =(2, 0) S 1 m KK = 1/g 2

20 S 4 S 1 S 5

21 E 6 (2) Z(S 1 S 4 )

22 N =(2, 0) S 4 C C C S 4

23 N =(2, 0) S 4 C (S 4 /S 1 ) C C C S 4

24 N =(2, 0) S 1 S 3 C q C C S 1 S 3

25 N =(2, 0) S 1 S 3 C S 3 C q C C S 1 S 3

26 N =(2, 0) S 3 X X X S 3

27 N =(2, 0) S 3 X S 2 X X X S 3

28 n S n n S d (n d) N =(2, 0) S 1

29 N =(2, 0)

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31 N =(2, 0) C :

32 a i u a i u T N (N) 3 T N C N =2 N =2 C N =1 N =1 N =2

33 T N a i u (N) 1 a= 1,..., N (N) 2 i = 1,..., N (N) 3 u= 1,..., N N =2 (N) 3 T 2 Q aiu T 3 E 6 (3) 3 Q aiu Q aiu µ a b µi j ˆµu v T N

34 q 2 µ a b, µi j, ˆµu v 1(N 1) Q aiu 2(N 2) Q [ab][ij][uv] k(n k) Q [a1 a k ][i 1 i k ][u 1 u k ] (N 1)1 Q [a1 a N 1 ][i 1 i N 1 ][u 1 u N 1 ] = Q aiu

35 T N k (µ a b )k = ( µ i j )k = (ˆµ u v )k N =1 (N) a i u β N c = N f ( µ i j )N = (ˆµ u v )N + Λ 2N. N = 2 (2)

36 M j i ˆM v u µ = ˆµ = 0 W = M i j µj i + ˆM u v ˆµv u, ( µ i j )N = (ˆµ u v )N + Λ 2N. N = 2 (2) = (1) (N) (N) T N

37 T N T N (Γ, Γ ) D p (G) N =2 N =1 N =1

38 Supersymmetric theories Lagrangian theories

39 Supersymmetric theories Lagrangian theories Holographic theories

40 Supersymmetric theories Lagrangian theories 6d constructions Holographic theories

41 patching two disks x 2 + y 2 + z 2 =1 S 2 dr 2 + r 2 sin 2 d 2 {(z,w) (cz, cw)} each can give complementary info no one thing privileged

42 Lagrangian gauge theory description 1 Lagrangian gauge theory description 2 construction using 6d theory holographic construction a QFT Q each can give complementary info no one thing privileged

43 N =(2, 0)

44 L A su(n) L B su(n) su(n) τ = il B /L A su(n) su(n) τ = il A /L B

45 su(n) Z 2 M M T L A su(2n) L B su(2n) usp(2n) τ = 2iL B /L A

46 su(n) Z 2 M M T L A su(2n) L B su(2n) usp(2n) τ = 2iL B /L A so(2n+1) so(2n+1) τ = il A /L B

47 N =(2, 0) su(2n) Z 2 su(2n) S 1 Z 2 su(2n) S 1 Z 2 so(2n + 1) so(2n + 1) su(2n)

48 N =(2, 0) su(2n) Z 2 su(2n) S 1 Z 2 su(2n) S 1 Z 2 so(2n + 1) so(2n + 1) su(2n) N =(2, 0) su(2n) S 1 Z 2 so(2n + 1)

49 L A su(n) L B su(n) (N)/Z N τ = il B /L A su(n) (N) τ = il A /L B

50 N =(2, 0) su(n)

51 su(n) X Z(X) a a H 2 (X, Z N ) N =(2, 0) su(n) M C a H 3 (M, Z N ) a Z N C C C C C 0

52 H 3 (M, Z N ) = A B a a = 0 a, a A M b b = 0 b, b B M a A b B {Z(M) a a A} {Z(M) b b B} Z a e i M a b Z b. b

53 Z Z a = Z a, Z b = Z b, { a ; a A} { b ; b B} a b = e i M a b

54 su(n) T 2 M = T 2 Y T 2 = SA 1 S1 B H 3 (M, Z N ) H 2 (Y, Z N ) A H 2 (Y, Z N ) B, H 2 (Y, Z N ) A (N)/Z k θ

55 g C 2g T N 3g su(n) su(n) 3g

56 su(n) M = S 3 S 1 C su(n) S 3 S 1 H 3 (M) = H 3 (S 3 ) H 3 (S 1 C). {Z a a H 3 (S 3 ) = Z N } {Z b b H 3 (S 1 C) = Z N } Z a = b e i2πab/n Z b. a b

57 T [C] Z N Z a = Ha ( 1) F e βh. Z N a T [C] S 3 S 1 = q su(n) C Z b = a e i2πab/n Z a q su(n) b C

58 N T N S 1 N N N T N (N)/Z N T N Z N (N) S 3 S 2 S 1

59 N =(2, 0)

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