5 Calabi-Yau web

Transcription

1 T NS5- T- 1 T Buscher T D- T Taub-NUT Kaluza-Klein monopole BPS NS5-branes T NS5-brane NS5-brane NS5-brane NS5-brane Geometric engineering NS5-D4 system Glueball superpotential NS5- Calabi-Yau Gukov-Vafa-Witten Elliptic model

2 5 Calabi-Yau web NS bipartite Brane tiling Zig-zag paths Elliptic model U(1) U(1) T- Calabi-Yau T- T- 1.1 T- S 1 Kaluza-Klein Kaluza-Klein modes Winding strings (1) Kaluza-Klein R S 1 ψ(x) ψ(x + 2πR) = ψ(x) p = i x p = 1 m, m Z. (2) R E = m R T str = 1 2πl 2 s (3) (4) 2

3 R S 1 w E = 1 2πls 2 2π w R = w R ls 2 (5) (3) (5) [ (m ) ( ) 2 2 1/2 Rw M = + R ls 2 + N] 2 ls 2 (6) N (1) T-duality Kaluza-Klein (3) (5) R R RR = l 2 s (7) S 1 (6) X(σ, τ) T- + X + X, X X. (8) σ ± = τ ± σ T-duality IIA IIB τ σ (8) τ σ τ X σ X, σ X τ X. (9) Dirichlet Neumann D-brane T-duality Wrapped Dp-branes Unwrapped D(p 1)-branes (10) D-brane II NS5-brane Wrapped NS5-branes Wrapped NS5-branes (11) Unrapped NS5-branes Kaluza-Klein monopole (12) N = 1 Calabi-Yau l s (string metric) S str = 1 2πl 2 s d 2 σ det G (13) 3

4 2πl s = 1. (14) T str = 1 2πl 2 s 1 = 2π, T Dp = (2π) p ls p+1 = 2π 1, T NS5 = g str g str (2π) 5 lsg 6 str 2 = 2π gstr 2. (15) (14) T- (7) RR = 1/(2π) 2 2π L = 2πR L = 2πR LL = 1. (16) 1.2 Buscher T-duality (9) α X η αβ ϵ βγ γ X (17) η ττ = η σσ = ϵ τσ = ϵ τσ = 1 (18) (17) T-duality [1] 10 X M = (X i, X 9 Y ) non-linear sigma model (NLSM) ( S[X i, α X i, α Y ] = d 2 σ T str 2 G MN α X M α X N + 2π ) 2 B MN ϵ αβ α X M β X N G MN B MN X i S Y Y G MN B MN Y Y 1 w y w y = σ Y dσ Z. (20) Y (19) S = S[X i, α X i, F α ] + d 2 σ2πϵ αβ α ZF β. (21) 1 (19) α Y F α 2 Z Y 1 (21) (19) Z ϵ αβ α F β = 0 F α = α Y (21) (19) Y (19) 4

5 Y Z (20) Y Z p z p z = dσ δs δ τ Z = 2π d 2 σ σ Y = 2πw y (22) p z = 2πm z m z Z (20) Y 1 Z F α S S[X i, α X i, α Z] = ( d 2 σ T str 2 G MN α X M α X N + 2π ) 2 B MN ϵ αβ α X M β X N X M 9 X M X 9 = Y X 9 = Z G MN B MN G ij = G ij G i9g j9 G 99 (23) + B i9b j9 G 99, G i9 = B i9 G 99, G 99 = 1 G 99, (24) B ij = B ij B i9g j9 G i9 B j9 G 99, B i9 = G i9 G 99. (25) T-dual Buscher [1] G 99 S 1 (16) C MN = G MN + B MN (26) C µν = C µν C µ9c 9ν C 99, C 9ν = C 9ν C 99, C µ9 = C µ9 C 99, C 99 = 1 C 99. (27) (27) C MN C MN ds 2 = ds G 99 (dy + V ) 2, B 2 = b 2 + W (dy + V ). (28) T-dual ds 2 = ds G 99 (dz + W ) 2, B 2 = b 2 + V dz. (29) B (28) B dy V (29) dz W b 2 b 2 + V W (28) B dy + V dy (29) B dz dz + W [1] worldsheet α sub-leading dilaton field 1 = G 99. (30) e 2ϕ e2ϕ NLSM R-R NS-NS R-R T-dual Green- Schwarz [2, 3, 4] pure spinor [5] 1.3 NS-NS R-R T-dual 5

6 1.3 T- 1.2 [6] IIA IIB NS-NS R-R IIA IIB R-R NS-NS [ R A + 4( µ ϕ A ) 2 1 L IIA NS = 2πeA e 2ϕA L IIB NS = 2πeB e 2ϕB 2 3! (HA µνρ) 2 [ R B + 4( µ ϕ B ) ! (HB µνρ) 2 ], (31) ]. (32) IIA IIB A B e A e B IIA IIB det e m µ H 3 dh 3 = 0 B 2 H 3 = db 2 (33) H 3 IIA IIB S 1 IIA S 1 y x 9 1 g99 A y IIB z = x 9 (28) ds 2 A = g (9) µν dx µ dx ν + e 2σ ( dy) 2, dy = dy + V1. (34) S 1 σ e 2σ = g99 A y 1 L A = (2πl s )e σ V 1 y U(1) y y = y + a(x µ ) V 1 = V 1 da. (35) (34) 1- dy 1- dy dy 0 (28) d( dy) = dv 1 = f 2. (36) H A 3 = h 3 + h 2 dy, B A 2 = b 2 + W 1 dy. (37) dy dy dy (37) (35) b 2 W 1 h 3 h 2 (37) (33) (36) h 3 = db 2 W 1 dv 1, h 2 = dw 1 (38) 9 ϕ A 9 φ ϕ A = φ σ (39) 6

7 (31) 9 L 9dim NS = 2πe(9) e 2φ [R (9) + 4( φ) 2 ( σ) 2 e2σ 4 f µν 2 e 2σ 4 h2 µν 1 ] 2 3! h2 µνρ (40) 9 IIB NS-NS (32) ds 2 B = g (9) µν dx µ dx ν + e 2σ ( dz) 2, dz = dx 9 + W 1. (41) H B 3 = h 3 + f 2 dz, B B 2 = b 2 + V 1 dz. (42) ϕ B = φ 1 σ. (43) 2 (29) (39) (43) (30) 1 e 2ϕB = ga 99 e 2ϕA (44) R-R IIB R-R 5- G B 5 IIA R-R IIB R-R G A even GB odd G A even = G A 0 + G A 2 + G A 4 + G A 6 + G A 8 + G A 10, G B odd = G B 1 + G B 3 + G B 5 + G B 7 + G B 9. (45) T 10 G A even + T G A even = 0, 10 G B odd + T G B odd = 0. (46) T (dx i1 dx in ) = dx in dx i1 (47) R-R IIA IIB dg A even = H A 3 G A even, dg B odd = H B 3 G B odd. (48) 9 IIA S 1 y x 9 R-R dy (34) G A even = g even + g odd dy. (49) IIA (46) (48) g even g odd G A even 10 Hodge 9 10 G A even = e σ 9 g odd + e σ 9 g even dy. (50) 7

8 σ (34) y G A even (46) 9 e σ 9 g odd = T g even, e σ 9 g even = T g odd. (51) 9 2 Hodge (48) G A even (49) dy dg even = h 3 g even + f 2 g odd, dg odd = h 3 g odd + h 2 g even. (52) IIA IIB R-R field G B odd G B odd = g odd + g even dz. (53) IIB (48) (46) (51) (52) RR- S 1 G A even GB odd 9 g even g odd (49) (53) R-R (48) G A even = e BA 2 dc A odd, G B odd = e BB 2 dc B even. (54) IIA Codd A IIB CB even 9 c odd c even Codd A = c odd + c even dy, Ceven B = c even + c odd dz. (55) dy dz dy dz (54) B (37) (42) (55) (49) (53) 9 g even = e b2 (dc odd dc even V 1 ), g odd = e b2 V1 W1 (dc even + dc odd W 1 ). (56) NS-NS R-R T-dual NS-NS field R-R 1.4 D- T- 1.3 T-duality D-brane D-brane IIA D(2n 2)- IIB D(2n 1)- NS-NS R-R D- NS-NS Born-Infeld SBI A 2π = d 2n 1 σ 1 e ϕa det[c A ij + F A ij ], S B BI 2π = 8 d 2n σ 1 e ϕb det[c B ij + F B ij ] (57)

9 C MN = g MN + B MN (26) C ij C MN NS-NS IIA IIB Born-Infeld IIA IIB R-R Chern-Simons SCS IIA = 2π Codde A IIA F2, SCS IIB = 2π Cevene B IIB F2. (58) 2n 1 C IIA IIB T- (10) T-duality D- D- IIA IIB 2n IIA IIB y y(x i ) z A 9 (x i ) 1: IIA D-brane T-duality IIB D-brane S 1 Wilson line IIA y = x 9 S 1 y IIA y IIB Wilson line A 9 T- A B µ = A A µ, A B 9 = y. (59) A B 1 = A A 1 + ydz. (60) IIB S 1 z Born-Infeld (57) IIA y Dp- Dp- x 9 y C MN C ij Cij y = C ij + C i9 x j + C y 9j x i + C y y 99 x i x j. (61) C MN = G MN + B MN y T- (27) (27) (61) (27) (57) Cij A + Fij A = Cij B + Fij A 1 C99 B (Ci9 B + i y)(c9j B j y) (62) 9

10 C µν C MN p + 1 p + 2 ( det(cij A + Fij A ) = 1 C99 B det C B ij + F A ij C B i9 + iy C B 9j jy C B 99 ). (63) (59) (60) IIB p + 1 F B 2 = F A 2 + dy dz. (64) F B ij = F A ij, F B i9 = F B 9i = i y. (65) (44) (63) 1 det(c A e ϕa ij + Fij A) = 1 det(c B e ϕb µν + Fµν) B (66) IIA D- Born-Infeld IIB D- Born-Infeld x 9 Kaluza-Klein T-dual Chern-Simons IIB (55) (64) S IIB CS 2π = = 2n 2n ( c even + c odd dz) e F A 2 dy dz (c even dy + c odd ) dz e F A 2 (67) IIB z dz dz IIA D-brane Chern-Simons (c even dy + c odd ) e F A 2 = Codd A e F A S IIA 2 = CS (68) 2π 2n 1 2n 1 Chern-Simons T-dual D-brane F 2 = F 2 + B 2 NS-NS 2-form (37) (42) (64) B B 2 = B A 2 W 1 dy + V 1 dz. (69) (64) dy dz F B 2 = F A 2 + dy dz. (70) 2 Taub-NUT Taub-NUT Taub-NUT 4 NS5-brane T-dual T-dual 10

11 2.1 Kaluza-Klein monopole D S 1 D 1 U(1) W ds 2 D = ds 2 2 D 1 + g zz dz, dz = dz + W. (71) IIB S 1 S 1 z U(1) W (35) S 1 W L = 1 gg 2 µν µ ϕ ν ϕ (72) ϕ y 1 ϕ = ϕ n e 2πiny (73) n Z ϕ n y D 1 (73) (72) z D 1 L = 1 gzz g 2 n [ g ij D i ϕ n D j ϕ n + 1 ] q 2 g nϕ n ϕ n zz g ij (71) D 1 ds 2 D 1 g ij g g ij D i q n Kaluza-Klein p z (74) D i ϕ n = i ϕ n iw i q n ϕ n (75) q n = 2πn (76) (75) Kaluza-Klein ϕ n W i q n W 1 W 1 U(1) Kaluza-Klein Q = h 2. (77) S 2 h 2 h 2 = dw 1 (78) U(1) S 2 S

12 10 S 1 9 Kaluza-Klein 5-brane D 1 D D 1 D x 6 x 7 x 8 z x 9 10 (r, θ, ϕ) (x 6, x 7, x 8 ) = (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ). (79) Kaluza-Klein W = Q (1 cos θ)dϕ. (80) 4π r = 0 (80) θ = π 678 (35) W = W Q dϕ (81) 2π z z = z + Q 2π ϕ (82) (35) IIA IIB z 1 ϕ 2π (82) Q Q Z. (83) Kaluza-Klein (76) 2π ρ ρ h 2 dh 2 = 3 ρ (84) Maxwell B = ρ Hodge 3 3 ds 2 = dx dx dx 2 8 r I ρ ρ = δ 3 (r r I ) (85) I 678 r Q Q r I Q 12

13 D (71) W 1 D 0 R MN = 0 ds 2 D = ds 2 D 4 + ds 2 TN (86) ds 2 D 4 D 4 Kaluza-Klein ds 2 TN Kaluza-Klein 3 S1 4 Taub-NUT 4 ds 2 TN = Hdr 2 + H 1 dz 2, dz = dz + W. (87) H 3 3 H = ρ (88) 3 (85) H(r) = 1 L TN 4π r r i i L TN (84) (88) H h 2 (89) h 2 = 3 dh. (90) 2.2 (87) Taub-NUT r H R 3 S 1 S 1 L TN r = r I center center r I center 678 (79) (r, θ, ϕ) H = 1 4πr r (89) (90) W 1 = 1 cos θdϕ. (92) 4π S 1 z ψ (91) ψ = 4πz (93) 4π (87) 4πds 2 T N = 1 r (dr2 + r 2 (dθ 2 + sin 2 θdϕ 2 )) + r(dψ cos θdϕ) 2 (94) 13

14 r = ρ 2 /2 (94) dω S3 4πds 2 = dρ 2 + ρ 2 dω 2 3. (95) dω 2 3 = 1 4 [dθ2 + sin 2 θdϕ 2 + (dψ cos θdϕ) 2 ]. (96) (95) R 4 center (87) Taub-NUT center 4 center (96) 1 S u C 2 ( ) ( iϕ u = exp 2 τ z exp iθ ) ( ) ( ) iψ 2 τ y exp 2 τ 1 z 0 τ m u (97) u u = 1 (98) 1 S 3 S 3 S 3 (96) 0 θ π, 0 ϕ < 2π, 0 ψ < 4π. (99) ds 2 = du du = dω 2 3. (100) (97) S 3 S 2 S 1 S 3 S 2 f : u n = u τu = (sin θ cos ϕ, sin θ sin ϕ, cos θ). (101) τ = (τ x, τ y, τ z ) ψ u u e iα u S 1 Hopf center 678 N center H W 1 N ψ (93) ψ = 4π N z (102) N (94) 4π N ds2 = dρ 2 + ρ 2 dω 2 3. (103) ρ r = ρ 2 /2 1/N (94) 4 (102) ψ 4π 4π/N R 4 orbifold R 4 /Z N R 4 (z 1, z 2 ) = (ρu 1, ρu 2) u i u = (u 1, u 2 ) (97) C 2 14

15 ρ z i ds 2 = (N/4π)dz i dzi ψ 4π/N ψ ψ + 4πk/N (k Z) z i (z 1, z 2 ) (e 2πik/N z 1, e 2πik/N z 2 ), k Z. (104) Taub-NUT N center C 2 /Z N orbifold A N 1 orbifold holomorphic 2-form ω (2,0) dz 1 dz 2 (105) (104) orbifold z i u i Z N holomorphic 2-form 2.3 Taub-NUT hyper Kähler 3 (87) Kähler form k 6 = dx 6 dx 9 H(r)dx 7 dx 8, k 7 = dx 7 dx 9 H(r)dx 8 dx 6, k 8 = dx 8 dx 9 H(r)dx 6 dx 7. (106) z x 9 dx 9 = dx 9 + W volume form Ω Ω = 1 2 k 6 k 6 = 1 2 k 7 k 7 = 1 2 k 8 k 8 = H(r)dx 6 dx 7 dx 8 dx 9 (107) k m closed (90) dk 6 = ( h H)dx 6 dx 7 dx 8 (108) h 78 h 2 (90) 0 Taub-NUT (87) Hodge dual k-form A k 4 A k = ( ) k H 1 k 3 A k dx 9, 4 A k dx 9 = H 2 k 3 A k. (109) 4 k m = k m (110) k m k m closed form Taub-NUT 2- center x m I x m 0 = k2 m (111) S I 15

16 S I 2 I center 2- x m 0 center xm I x m 0 S I center S 2 S I S IJ = S I S J S IJ center x m I x m J = k2 m (112) S IJ x 8 S 2 x 7 S 12 center x 6 2: Taub-NUT 2- (106) 3 4 SU(2) Spin(4) = SU(2) L SU(2) R SU(2) SU(2) L γ 5 SU(2) R γ 5 SU(2) (110) γ 5 ϵ γ mnpq = ϵ mnpq γ 5. (113) SU(2) L SU(2) R (3, 1) (3, 1) SU(2) R SU(2) R η1 a (ηa 1) η1 a = 1 η η2 a = ϵ ab (η1) b ηm a = ϵ mn ϵ ab (ηn) b, ηmϵ a ab ηn b = ϵ mn, (ηm) a ηn a = δn m, ηm(η a m) b = δb a. (114) η 3 kµν(σ A A ) m n = iη m γ µν η n (115) σ A η (113) kµ 1 κ kκν 1 = g µν, kµ 1 κ kκν 2 = kµ 2 κ kκν 1 = kµν, 3 etc. (116) 16

17 η 1 η 2 (115) k A (106) Taub-NUT manifold w = x 6 + ix 7. (117) center w w I = x 6 I + ix7 I w 2 x 8 R x 9 x 8 S 1 w w I S 1 S 1 R x, y C C 2 xy = c (118) c 0 (118) x y 0 x 0 y (118) 3: xy = c 0 1 xy = 0 w w I S 1 w I xy = 0 (119) x = 0 y = 0 x = 0 y y y = 0 x x x = y = 0 x y (119) Taub-NUT xy = I (w w I ). (120) x y (120) x 8 x 9 17

18 3 Kähler form (117) k 8 2-form ω (2,0) = k 6 + ik 7 = idw ( Hdx 8 + i dx 9 ) (121) dw 2-form (2, 0)-form Taub-NUT 2 ρ (121) F ω (2,0) = idw dρ (122) Hdx 8 + i dx 9 = dρ + F dw (123) (121) F (122) F (123) dh dx 8 + ih 2 = df dw (124) (90) w w H = ih 8w, w H = ih 8w, 8 H = 2ih ww. (125) (124) dw dx 8 dw dx 8 dw dw 8 F = 2ih 8w, w F = ih w w. (126) h 2 = dv 1-form V ϕ I w w I W = I dϕ I = 1 2i f I ( w w I, x 8 ) dϕ I 2π ( dw w w I dw w w I ) (127) (128) f I (127) f I w = x 8 = 0 W = c cos θ dϕ. (129) 4π c (127) f I I (126) F dw = 2iW w dw (127) w W w = w W w ρ dρ = Hdx 8 + idx 9 + iw w dw iw w dw = Hdx 8 + i dx 9 2iW w dw. (130) 18

19 [7] (87) (106) k 8 (130) (1, 0)- dρ ds 2 TN = 1 H dρ dρ + Hdwdw. (131) k 8 = 1 2Hi dρ dρ + H 2i dw dw (132) dρ = dρ + 2iW w dw = Hdx 8 + i dx 9. (133) ρ (130) W W (127) x 8 center x 8 x 8 W + x 8 x 8 W W = W + W = dϕ 2π = 1 4πi i ( dw w w i dw w w i ) (134) (130) dx 9 ρ+ x 8 x 8 w w ρ 4: ρ + ρ W ± ρ ± ρ + ρ = 1 2π log(w w I ) (135) x 9 1 S 1 ρ ± i x y I x = e 2πρ+, y = e 2πρ (136) x x 8 x y x 8 y W 0 ρ + ρ (135) x y Taub-NUT (120) 19

20 ρ ± center center X ρ ρ X ρ X = ρ + 1 log(w w I ) = ρ + 1 log(w w I ) (137) 2π 2π I X I / X Taub-NUT center x i I Taub- NUT (120) 2 (138) w I w I x 8 I L TN (2, 0) (122) ω (2,0) = idw dρ = 1 2πi dw dx x = i 2π dx dy w I (w w I) (138) Taub-NUT F (x, y, w) xy (w w I ) = 0. (139) 2.2 center singular center 3 (w I, x 8 I ) (139) w I x 8 I manifold singular w I (139) singular x 8 center (139) singular C 2 x, y, w regular singular (x, y) regular x y (139) w F/ w 0 2 singular F = F x = F y = F = 0. (140) w (139) F = 0 (140) F x = y = (w w I ) = d (w wi ) = 0. (141) dw w w I regular w I w singular 20

21 w I N w I = 0 w w = 0 (139) xy = w N. (142) w = 0 O(w N+1 ) 2.2 N center A N 1 (104) C 2 /Z N z 1 z 2 orbifold C 2 /Z N covering space C 2 orbifold (z 1, z 2 ) (104) (z 1, z 2 ) Z N C 2 /Z N (z 1, z 2 ) orbifold orbifold (104) x = z1 N, y = z2 N, w = z 1 z 2. (143) (142) C 2 /Z 2 (120) 67 w singular center x 8 singular (142) singular x = y = w = 0 3 w = 0 center x 8 I = 1,..., N 5 ρ + W + w = 0 W + =W 0 W 1 W 2 W =W 3 5: x 8 x 8 1 ρ + x 0 w = 0 x 8 x 8 N ρ y = 0 x 8 N x8 x 8 1 x = y = 0 w = 0 x, y, w w = 0 center N center N + 1 x 8 x 8 S 0, S 1,..., S N S k w = 0 x 8 k x8 x 8 k 1 x8 0 = x 8 N+1 = S1 S 0 S N S 2 (137) S k W k ρ k S k 21

22 (137) t k = e 2πρ k t k = e 2πρ k t k = w k x, t k = w k N y. (144) t k S k 0 t k t k = 1 (144) (142) (t k, t k ) S k stereographic coordinates (144) (142) yt k = w N k, xt k = w k. (145) A N k 1 A k 1 N center A N 1 S k A k 1 A N k 1 S k (k = 1,..., N 1) N 1 (144) A N 1 N 1 A 0 S 2 A N 1 N 1 S 2 A N 1 2- H 2 A N 1 A N Taub-NUT non-compact 2-cycle Poincare dual center 2-form zero-mode [8, 9] zero mode center 2-2- Taub-NUT SU(2) U(1) 2-form η = fh 2 + df dx 9 (146) center closed 4 fh 2 = fh 1 dh dx 9 self-dual anti-self dual dh/h = ±df/f anti-self-dual 4 η = η 4 η = η f = ch ±1 center center f = 0 f = ch 1 η η η = 1 L 2 (H 1 h 2 + dh 1 dx 9 ) B [ ] i 1 = 8πr 3 L 2 B H 2 (w dw dρ wdw dρ) + x8 H dw dw x8 H dρ 3 dρ. (147) η (1, 1)-form c = L 2 A = 1/L2 B η 1 22

23 η 1 gηµνη µν = η η = 2 2 L 4 H 1 h 2 dh 1 dx 9 = 1 B L 4 h 2 dh 2 dx 9 (148) B x 9 1 h 2 S gηµνη µν = 1 r= dh 2 = 1. (149) 2 L 4 B 2- S 1 η 1 η = 1 S 1 L 2 dh 1 dx 9 = 1 r=0 B S 1 L 2 dh 1 = 1. (150) B r= Taub-NUT r 1 = (x 6 1, x 7 1, x 8 1) k 8 x 8 1 ( ) k8 = η. (151) x 8 1 ρ,w (132) x 8 w ( ) k 8 ik x 8 dz i dz j = L 2 BHη. (152) 1 x 8,w (ρ, w) (130) x 8 x 8 1 center x 8 x 8 1 r=0 ρ ( ) k8 = 1 ( ) k8 L 2 B H (151) x 8 1 ρ,w x 8 1 x 8,w = η (153) ω (2,0) ω (2,0) + δw 1 η 2 (2, 0) center center harmonic 2-form η I η J = η I η J = δ IJ, 4 η I = η I. (154) S I η I (151) center x 8 I 2.7 BPS Taub-NUT 2- A det A = da = Gαβ d 2 σ (155) 23

24 G αβ Σ G αβ G αβ = α x m β x n g mn (156) det G αβ = 1 2 ϵαγ ϵ βδ G αβ G γδ = 1 2 S mn S pq g mp g nq (157) S mn S mn = ϵ αβ α x m β x n (158) (157) (158) ϵ αβ ϵ αβ / G S mn S mn S mn S mn = S mn G (159) S Smn S pq g mp g nq = 1. (160) X mn X 2 S mn S mn da = da dx m dx n (161) da K K mp K p n = g mn (162) K S mn 2 Wirtinger s inequality 1 2 K mns mn 1 (163) A = da 1 K mn S mn da = K mn dx m dx n = K 2 (164) K (162) K k k (164) k Calibration form (164) Z Σ Z = k 2 (165) Z 3.4 (164) K k k Z A A Z. (166) 24

25 (166) manifold calibrated submanifold (164) (163) 1 2 k mns mn = ±1 (167) k 2 Σ Σ k 2 Σ = ±ω 2. (168) Σ z x i x i (168) i x i x j g ij i x i x j g ij = det G = ±ig zz (169) x i x j g ij = 0 x i = 0 x i z Taub-NUT 4 4 k (m) k (n) = ϵ mnp k (p) δ mn k (m). (170) 3 k (m) Z m = k2 m (171) 3 n m n m k2 m (166) A Zm 2 (172) m Taub-NUT (111) S I center Z m [S I ] = x m I x m 0 = k m. S I (173) Z m = A 678 S NS5-branes T- NS5-brane Kaluza-Klein T-dual NS5-brane Taub-NUT 2-form 3.1 NS5-brane NS-NS 2-form B 2 Q H 3 = db 2 Q = H 3 S 3 (174) 25

26 3 3 Q S 3 S 3 S NS-NS 2-form B 2 5- NS5- H 3 J 4 = dh 3 (175) (174) J 4 Q[M] = J 4 (176) M (174) S 3 4 world volume 4 C (176) M C J 4 C 0 δ 4-form δ 4 (C) δ(c) = M, C. (177) M M C M δ(c) (177) NS5-brane J 4 = ρ(x i )dx 6 dx 7 dx 8 dx 9. (178) I 6789 x i I ρ(xi ) ρ(x i ) = δ 4 (x i x i I). (179) I M x i I ρ(x i ) 6789 H 4 H = ρ(r). (180) (180) = c H = c + I 1 4π 2 r r I 2 (181) ds 2 = η µν dx µ dx ν + Hδ ij dx i dx j. (182) 26

27 e ϕ = g str H 1/2 (183) g str H 1 g str e ϕ H 3 (175) H (180) d 4 dh = 4 ρ(r) J 4 = 4 ρ H 3 = 4 dh. (184) δ 3.2 NS5-brane NS5-brane S 1 T-duality IIA (182) (183) (184) NS5-brane x 5 L A S 1 1 Buscher s rule x 5 x 5 x g 55 = L 2 A IIA ds 2 = ds L 2 A(dx 5 ) 2 + Hδ ij dx i dx j (185) B (34) (37) 9 U(1) V W 0 B IIB ds 2 = ds L 2 (dx 5 ) 2 + Hδ ij dx i dx j (186) A B (184) (44) (183) g B str = ga str L A (187) S 1 NS5-brane T-dual dual S 1 NS5-brane T-duality 9 h 3 NS5-brane T-dual NS5-brane 3.3 NS5-brane NS5-brane 1 Buscher s rule x x 9 1 S 1 H (182) g 99 H L 2 L 27

28 Buscher s rule x 9 x 9 NS5-brane x 9 NS5-brane x 9 smear (179) x 9 ρ = I δ 3 (r r I ) (188) Taub-NUT (85) r I I NS5-brane (180) 3 (88) (89) H(r) = L 2 A π r r I I (189) L 2 A IIA S1 L A (34) ds 2 9 = η µν dx µ dx ν + Hdr 2, e 2σ = H, v 1 = 0. (190) v 1 = 0 dx 9 = dx 9 NS5-brane x 9 H 3 dx 9 (37) H A 3 = h 2 dx 9, B A 2 = b 1 dx 9. (191) (175) h 2 dh 2 = ρ(r)dx 6 dx 7 dx 8. (192) h 2 ρ(r) 3 U(1) (90) h 2 = 3 dh (184) Buscher s rule T-dual (41) ds 2 = ds e 2σ (dx 9 + b 1 ) 2 = η µν dx µ dx ν + ds 2 TN. (193) (190) ds 2 TN (87) Taub-NUT NS-brane T-dual Taub-NUT center (190) v 1 = 0 T-dual B 2 = 0 ϕ = ϕ σ = g str (194) NS5-brane T-dual NS5-brane NS5-brane x 9 T-duality B

29 x 8 L A C 2 C 12 NS5 w IIA x 8 S 12 IIB S 2 center w 6: S 1 NS5-brane T-duality Taub-NUT center Kaluza- Klein monopole I NS5-brane C I J NS5-brane I NS5-brane C IJ = C I C J Taub-NUT 2- S I S IJ S I S IJ S cycle Σ k A Z A Σ IIB B 2 C even ξ I ξ I δ δ (33) H 2 (54) G odd [δ, δ]b 2 = dλ NS 1, [δ, δ]c even = e B 2 dλ odd. (195) H 3 G odd 0 0 Λ ξ I ξ I Λ NS 1 = 1 4 (σ z) IJ (ξ Iγ [1] ξ J ), Λ 2n 1 = 1 4e ϕ (σ xσ n+1 z ) IJ (ξ Iγ [2n 1] ξ J ). (196) γ [n] n γ [n] = γ i1 γ in dx i1 dx in. (197) Σ B Z F1 S brane B (198) B 2 (x) B B 29

30 3 k A δb 2 = ϵ A k A 3 ZF A 1 = 1 2π ϵ A S brane[c even, B 2 + ϵ A k A ] (171) S F 1 = S NG + 2π B 2. (200) S NG - B ϵa =0 Taub-NUT S I (199) Z m F 1[F1 wrapped on S I ] = x m I x m 0. (201) (173) S I D1-brane D- D- RR- C Ĉ Ĉ A odd = e BA 2 C A odd, Ĉ B even = e BB 2 C B even. (202) (195) B Λ NS 1 Z F1 D Λ odd (195) C B (202) Ĉ Ĉ2 ZD1 A ZD1 A = 1 2π ϵ A S brane[ĉ2 + ϵ A k A, B 2 ] (203) 2- D1-brane (171) D- Chern- ϵa =0 Simons (58) Ĉ = 2π Ĉodde A IIA F2, SCS IIB = 2π S IIA CS 2n 1 2n Ĉevene B IIB F2. (204) (204) C Z D1 D1- (203) Z D1 0 Z F 1 D- D-brane Z F 1 (199) S brane D-brane RR- (199) C B Ĉ C Λ NS 1 C Ĉ Taub-NUT S I (203) Z m D1[D1 wrapped on S I ] = x m I x m 0. (205) 30

31 (173) (201) D1-brane Z D1 Ĉ2 (106) δĉb 2 = ϵ m k m 2 = ϵ m dx m dx 9 H 2 ϵ mnpϕ m dx n dx p (206) m, n, p = 6, 7, 8 Σ x 9 S I (206) 2 δĉb m9 = ϵ m Z m D1 = 1 2π S D1 Ĉm9. (207) T-dual (207) type IIA Ĉ T-dual Ĉ A odd = ĉ odd + ĉ even dy, Ĉ B even = ĉ even + ĉ odd dz (208) 9 ĉ odd = e b 2 (c odd c even V ), ĉ even = e b 2 V W (c even + c odd W ). (209) ĈA m = ĈB mz (207) ZD1 m = 1 SD1 B = 1 SD0 A = Z 2π ĈB 2π m9 ĈA D0 m (210) m 2- D1-brane D0- D1-brane S I D0-brane 6 C I NS5-brane Z m D0[D0 along C I ] = x m I x m 0. (211) (210) NS5-brane 678 T-dual Taub-NUT center NS5-brane x 9 NS5-brane x 9 D0-brane Z 9 D0 ĈA 9 T-dual ĈA 9 = ĈB ZD0 9 = 1 SD0 A = 1 SD( 1) B = Z m 2π ĈA 2π 9 ĈB D( 1) (212) S I D1-brane x 9 I x 9 0 = Z 9 D0[D0 along C I ] = Z m D( 1) [D1 wrapped on S I] (213) D1- (204) x 9 I x 9 0 = F 2 = S I B 2 S I A 1. S I (214) 31

32 (214) T-dual (70) F2 B = S I (F2 A + dy dz) = S I dy dz = S I dy C I (215) C I y (214) (214) 2 S I I A 9 T-dual x 9 0 x9 x 9 0 x 9 I x 9 0 = B2 B. S I (216) 678 (173) 9 (216) Taub-NUT T-dual NS5-brane NS5-brane center w = x 6 + ix 7 (138) w I w 0 = ω (2,0) S I (217) x 9 x NS5-brane (x 8 I x 8 0) + i(x 9 I x 9 0) = J. S i (218) J k 8 B J = k 8 + ib (219) 3.5 NS5-brane NS5-brane 6- N = 2 IIA NS5-brane NS5-brane N = (2, 0) c D0-0-form NS5-brane D4-brane NS5-brane D4-brane NS5-brane S 1 c dc D4-brane N D4 = dc. (220) c 1 S 1 M- NS5-brane x 11 M5-brane c M5-brane x 11 32

33 NS5-brane 2 c 2 h 3 = dc 2 6 h 3 = h 3. (221) NS5-brane x 6,..., x 9 c c 2 T-dual Kaluza-Klein monopole x m T-dual Taub-NUT center 678 NS5-brane center (151) 3 δk A = I δx A I (x µ )η I (222) center η I (154) harmonic 2-form IIB (222) NS5-brane IIB S = d 10 x 2π g (10) e 2ϕB (R (10) + 4( M ϕ) 2 ) (223) ds 2 = g µν (x µ )dx µ dx ν + g mn (x µ, x m )dx m dx n g mn = g mn (t A (x µ ), x m ) (224) t A x µ (223) S = d 6 x 2π ( g e 2ϕ 4V ( ϕ) 2 4( µ ϕ)( µ V ) 1 ) 2 G AB( µ t A )( µ t B ) (225) x m 4 V V = d 4 y g (4) (226) x m V = t A G AB = d 4 x ( 1 g (4) 2 gmp g nq (g mn,a )(g pq,b ) 1 ) 2 (gmn g mn,a )(g pq g pq,b ) (227) k 8 (t, y) = k 8 + t I η I (228) δg ij = it I η I ij δg mn g mn δg mn = k mn η I mn k mn η I mn 0 δg mn (227) 0 V (225) 0 t I t A G IJ = δ IJ (229) 33

34 t I S = 2π d 6 x( µ t I )( µ t I ) = 2(g B str) 2 2π 2(g A str) 2 d 6 x( µ t I )( µ t I )L 2 A (230) 2π/(gstr) A 2 5-brane IIA NS5-brane L 2 A g 88 NS5-brane 9 NS5-brane 9 T-dual Taub-NUT manifold x 9 B- x 9 Taub-NUT manifold center B B B B 2 = f I (x µ )η I (231) I f I x µ (µ = ) (150) f I = S I B 2 (216) f NS5-brane x 9 f B B (231) S IIB = 1 2π d 10 x g 12 (10) H B2 e 2ϕB 3 = 1 2π d 10 x g 4 (10) µ f I µ f J η Iµν η µν e 2ϕB J = 1 2π d 6 x g 2 (6) µ f I µ f I (232) e 2ϕB e 2ϕB = e 2ϕA /L 2 A S IIB = L2 A 2 2π e 2ϕA d 6 x g (6) ( µ f I ) 2 (233) 2πe 2ϕ IIA NS5- L 2 A x 9 g A 99 = L 2 A self-dual 4-form zero-mode C 4 = I c I 2(x µ ) η I (234) c I 2 I NS5-brane 2- C 4 10 G 5 = G 5 (154) η I h 3 (221) IIB S IIB = 2π d 10 x g 2 5! (10) G 2 5 = 2π d 6 x g 12 (6) (h I 3) 2 (235) NS5-brane c 2 RR 2-form zero mode C 2 C 2 = I c I (x µ ) η I (236) 34

35 NS5-brane 0-form c S IIB = 2π 12 d 10 x g (10) (G B 3 ) 2 = 2π 2(g A str) 2 d 6 x g (6) µ f I µ f I (g A str) 2. (237) (g A str) 2 NS5-brane M5-brane T NS5 = T M5 = 2π(g A str) 2 g 11,11 = (g A str) 2 M5-brane x 11 (236) S I D1-brane S D1 = 2π C 2 = 2πc I 2 CI (238) S I C I C I C I NS5-brane IIA D0-brane NS5-brane c I (173) (216) NS5-brane c M5-brane x 11 x 11 I x 11 0 = c I c 0 = C 2. S I (239) C 2 (236) c 0 center NS5-brane NS5-brane IIB NS5-brane N U(1) U(N) T-dual Taub-NUT N center cycle D2-brane U(1) N U(N) Taub-NUT IIA 2- D3-brane string 0 little string theory 4 Geometric engineering 3 NS5-brane Taub-NUT N = 1 NS5-D4 T-dual [10] T-dual Geometric engineering NS5-brane [11] N = 1 NS5-brane NS5-brane NS5-brane 35

36 1: N = NS5 1,2 D4 N 4.1 NS5-D4 system IIA NS5-brane NS5 1 NS5 2 N D4-brane 1 IIA 1/4 4 N = 2 Hanany Witten[12] 3 Hanany-Witten D4-brane NS5-brane Witten[13] 4 N = 2 NS5-brane D4-brane 4 N = 4 A µ (x µ, x 9 ), A 9 (x µ, x 9 ), ϕ i (x µ, x 9 ), µ = 0, 1, 2, 3, i = 4, 5, 6, 7, 8. (240) N U(N) NS5-brane N = 2 (240) N = 2 vector mult. : A µ (x µ, x 9 ), ϕ 4 (x µ, x 9 ), ϕ 5 (x µ, x 9 ), (241) hyper mult. : ϕ 6 (x µ, x 9 ), ϕ 7 (x µ, x 9 ), ϕ 8 (x µ, x 9 ), A 9 (x µ, x 9 ). (242) D4-brane NS5-brane x 9 D4-brane N = 2 pure U(N) N = 2 N = 1 A µ V ϕ = ϕ 4 + iϕ 5 Φ L = 1 2gYM 2 tr( µ ϕ µ ϕ + F F ) (243) F D-term V tr[ϕ, ϕ ] 2. (244) [ϕ, ϕ ] = 0 ϕ N D4-brane u N N ϕ ϕ = T str diag(u 1, u 2,..., u N ). (245) NS5-brane N = 1 ϕ W = tr W (ϕ) L = 1 2 tr(f W (ϕ) + F W (ϕ) ) (246) 36

37 F ϕ V = g2 YM 2 tr(w (ϕ)w (ϕ) ). (247) ϕ W = 0 N 0 W (u) = 0 u u k 0 D4-brane N k U(N) U(N) k U(N k ). (248) U(1) decouple U(1) NS5-brane x 9 x 9 1,2 NS5-brane M5-brane x 11 c 1,2 x 9 = x 9 2 x 9 1, c = c 2 c 1 = x11 2 x 11 1 (249) L 11 1 x 11 L 11 x 6 1,2 = x 7 1,2 = x 8 1,2 = 0 (250) D4-brane D4-brane U(1) U(N) D4-brane Born-Infeld 4 effective action S = d 4 x [ x9 x 9 (2π) 4 lsg 5 + ( 14 str (2π) 2 l s g F µνf µν 12 ) ] µϕ µ ϕ + str D4-brane NS5-brane c S = 2π cf F (252) 2 D4 (251) (251) 1 2 A F ϕ D4-45 D4-brane ϕ = 1 2πls 2 (x 4 + ix 5 ). (253) (252) 2π g 2 YM = x9 2πl s g str, (253) (254) u = x 4 + ix 5, w = x 6 + ix 7, s = x11 + ix 9 37 θ = c. (254) 2π L 11. (255)

38 w N = 1 x 9 1 τ gauge = θ 2π + 2πi gym 2 = x11 + i x 9 = s s 2 s 1. (256) L 11 l s 0 N = 1 NS5-brane w = x 6 + ix 7 NS5-brane u NS5 1,2 w = w 1,2 (u) w = w 2 w 1 uw 4567 NS5-brane (w w 1 (u))(w w 2 (u)) = 0 (257) singular w L 9 NS52 D4 NS5 1 u u w 1 =w 2 7: NS5-brane D4-brane u w 2 (u) w 1 (u) 0 NS5-brane D4-brane x 9 L = x 29 + w(u) 2 = x 9 + w 2 2 x 9 + (258) (251) S pot = [ d 4 x 1 w 2 ] (2π) 4 lsg 5 str 2 x 9 (247) W (ϕ) w 1,2 (u) W (ϕ) = (259) w(u) (2πl s ) 3 g str (260) l s 0 W w(u) x 9 x 9 L (254) (260) x 9 x 9 (254) (260) S = d 4 x 1 [ gym F µνf µν 1 ] 2 µϕ µ ϕ g4 YM 2 W (ϕ) 2 (261) 38

39 N = 2 U(1) W (ϕ) N = 1 Φ N = 1 super Yang-Mills ϕ w i (u) w 1 (u) = 0 w 2 (u) = µu µ N = 1 super Yang-Mills NS5 1 u NS5 2 w NS5-brane [14] N = 1 2: NS5 1 NS5 2 D4 N Seiberg duality ([14] [15] N = 1 Yang-Mills D4- NS5-brane N = 2 [13] N = 1 [15] D4-brane NS5-brane D4-brane NS5-brane w 1 (u) = w 0 (u), w 2 (u) = w 0 (u). (262) W 2w 0 = (2πl s ) 3 gstr A (263) uw 4567 NS5-brane w 2 = w 0 (u) 2 (264) ϕ n + 1 w 0 (u) u n NS5-brane x 9 (264) x 9 = x 9 (u) u NS5-brane NS5-brane D4-brane D4-brane NS5-brane (264) w 2 = w 0 (u) 2 + f n 1 (u). (265) D4-brane NS5-brane w = ±w 0 + O(u 1 ) f n 1 n 1 u (265) ( w = ± w0 2 + f n 1 = ± w 0 + f ) n 1 + 2w 0 (266) 39

40 u w ±w 0 f w 0 f n 1 a k n 1 f n 1 (u) = a k u k. (267) f n 1 (264) D4- k=0 w L 9 NS52 u NS5 1 u 8: D4-brane u 0 D4-brane NS5-brane uw x 9 x 9 (u) x 9 (u) β i β i dx 9 = x 9 (268) B i i cut NS5-brane x 9 β i D4-brane NS5-brane D4-0-form c c(u) α i dc = N i. (269) α i i cut α i NS5-brane D4- N i D4-brane (255) s ds = N i, ds = τ + n i. (270) α i β i (270) 2 (268) β i τ s x 11 n i n i β i α i n i N i n i = 0,..., N i 1 N i N i D4-brane unbroken SU(N i ) N i (270) differential ds (265) f n 1 40

41 (270) ds = N (271) ds β (270) 2 w mod = w 0 + un+1 M n+1 (272) u n (271) ds 1 β i i 9 B i B i A i A i (a) (b) 9: (a) B i Λ cutoff (b) cut cutoff (270) ds genus n holomorphic 1-form n η k = uk du, k = 0,..., n 1. (273) w f n 1 (270) α i ds n f n 1 f n 1 1 h α i β i A i = h, B i = h (274) α i β i 1 τ ij B i = τ ij A j (275) τ ij f n 1 (270) ds (270) (275) f n 1 τ + n i = τ ij (a k )N j. (276) 41

42 f n 1 τ ij f n 1 a k 4.2 Glueball superpotential NS5-brane (276) curve (270) 1-form ds (276) f n 1 τ ij (276) S i = wdu, α i Π i = wdu. β i (277) f n 1 = 0 w = ±w 0 dw = W 2π dϕ = (2πls)g 5 str A wdu (278) Si cl = 0, Π cl i = ga str 2π [W (u ) W (u i )]. (279) S i f n i Π i i (277) 1-form wdu f n 1 a k (265) f n 1 a k ak w = u k /(2w) a k wdu = uk du 2w. (280) holomorphic 1-form α i β i Π i a k = τ ij S i a k (281) a k S i Π i n (276) τ ij = Π i S j (282) [Π j N j (τ + n j )S j ] = 0. (283) S i j F W eff (S i ) = 2π gstr A [(τ + n i )S i Π i N i ]. (284) i (283) F-term (279) tree level superpotential W (284) (279) W eff (S i ) classical = i W (u i )N i = tr W (Φ). (285) 42

43 (284) S i SU(N i ) glueball super field S i tr(w i W i ) = T D4 S i (286) S i W W + S i 2π (284) n i W eff (S i ) = 2π g A str (277) (270) [15] W eff (S i ) = 2π gstr A wdu ds = 2π Σ gstr A [τs i Π i N i ]. (287) i B dw du ds (288) Σ u-w-s M5-brane B 3 Σ B = Σ B Σ Σ 0 B = Σ Σ 0 B Σ Σ 0 curve B Domain Wall B M5-brane Domain Wall (288) M5-brane (288) M5-brane central charge BPS bound Witten [15] domain wall (288) 4.3 NS5- Calabi-Yau NS5-brane x 9 L 9 T-dual 1 (182) (189) NS5 IIA ds 2 A = L 2 A(dx dx dx dx 2 9) (289) Taub-NUT IIB (193) ds 2 B = L 2 A(dx dx dx 2 8) + L 2 Bdx 2 9, L B = 1 L A (290) M5-brane x m new = L A x m old, x 9 new = L A x 9 old. (291) M5-brane IIB 2-form x m I x m 0 = L A k2 S m, x 9 I x 9 0 = L A B 2, x 11 I x 11 0 = L 11 C 2. (292) I S I S I w Taub-NUT 2 wi new w0 new = ω(2,0) new. (293) D I 43

44 2 ω new (2,0) = L A(k ik 2 2) = L A ω old (2,0) = 1 2πi dw new dx x. (294) ω (2,0) L A s IIB 2- s I s 0 = x11 + ix 9 = C2 B + il A B2 B = (C2 B + τ str B2 B ). (295) L 11 S I L 11 S I S I τ str = il 11 /L A = i/gstr B 0 x 8 + ix 9 x 8 x 11 B RR C 2 w s u T-duality N = 1 u w s (255) NS5-brane w-s w = w I (u), s = s I (u) (296) I NS5-brane u NS5-brane f n 1 D4-brane NS5-brane T-dual NS5-brane Taub-NUT geometry (120) xy = P (u, w) I (w w I (u)). (297) 4 Taub-NUT 2 u 3 w-u-x 8 S 1 P (u, w) = x 8 = 0 3 x 9 w L 9 C 21 C 2 NS52 w S 21 S 2 shrinking cycles NS5 1 u u 10: NS5-brane T-dual 2-3- ω (3,0) k CY Calabi-Yau 3- ω (3,0) = du ω (2,0) = 1 du dw dx dy = 1 2πi d(xy P (u, w)) 2πi du dw dx x. (298) ds 2 CY = dudu + ds 2 TN k CY = k 8 i 2 du du. (299) 44

45 k 8 w I (u) Taub-NUT k 8 u (299) closed (299) u IIA NS5-brane D4-brane D4-brane NS5-brane I J C IJ C IJ D4-brane T-dual S IJ D5-brane IIB D5-brane IIA NS5-D4 D4- (256) (256) s NS5-brane I J u s = s I (u) s J (u) (295) τ gauge = s I s J = (C 2 + τ str B 2 ) S IJ (300) S IJ = S I S J S 2 S 2 D5-brane (300) T-duality D5-brane (57) worldvolume R 4 S 2 R 4 S 2 B S = 2π d 4 x det(g µν + F µν ) da det(g ij + B ij ) (301) g str R 4 S 2 S 2 decoupling S 2 0 da det(g ij + B ij ) = B 2 = b (302) S 2 S 2 F (302) ( S = d 4 x 1 ) 2πb Fµν 2 R 4 g 4 str (300) B Chern-Simons (58) S = 2π C 2 F 2 F 2 (304) 2 C 2 S 2 (303) c = C 2 (305) S 2 4 S = 2πc F 2 F 2, (306) 2 (300) C 2 D4-brane NS5-brane NS5-brane u-w Calabi-Yau (297) w I (u) w 1 (u) w 2 (u) u = u 0 w 0 = w 1 (u 0 ) = w 2 (u 0 ) u = u 0 w = w w 0 u = u u 0 (297) xy = c(w au )(w bu ) (307) 45

46 a, b, c U V xy UV = 0 (308) (140) x = y = U = V = 0 singular conifold singularity NS5-brane D4-brane NS5-brane (297) (265) f n 1 conifold (308) xy UV = ϵ (309) deformation resolution 4.1 NS5-brane α β deform Calabi-Yau Calabi-Yau α- β- A- B- 3- A-cycle NS5-brane C 1 α i -cycle C I 11 (a) T-dual C I S I 3 S 3 3-cycle A i α i i u α α α S I α S 1 S 2 A i α S (b) A u u α α' 11: Calabi-Yau A i B i C I β i β i C 1 C 2 β i C (a) T-dual C I S I S B i β i β i α i β i S I β i S 21 = S 2 S 1 B i β i S (b) 46

47 B u u β β' 12: Calabi-Yau B i A i B i α i β i A i, B j = δ ij. (310) NS5-brane x 8 0 u Taub- NUT 2-cycle k 8 0 Lagrangian u Taub-NUT 2- A i B i (299) 0 (299) 4.4 Gukov-Vafa-Witten (287) Calabi-Yau (270) (277) Calabi-Yau (277) w (293) 2 S I wdu α i S i = wdu = α i α i ( S I ω (2,0) ) du (311) S I α i Calabi-Yau A i Calabi-Yau holomorphic 3-form (298) S i ω (3,0) A i Π i S i = ω (3,0), A i Π i = ω (3,0) B i (312) (270) differential ds u N i τ u s 2-form (295) (270) s α i β i A i B i G 3 N i = G C 3, τ + n i = G C 3. (313) A i B i 47

48 3- G C 3 G C 3 = G B 3 + τ str H B 3. (314) (313) A i N i N i D5-brane A i S 2 τ (300) S 2 D5-brane (312) (313) (287) W eff = 2π [ ] gstr A ω (3,0) G 3 ω (3,0) G 3 A i B i B i A i gstr A T-dual IIA IIB gstr A = L A gstr B 3 (315) Ω (3,0) = 1 L A ω (3,0) (316) ω (3,0) = du ω (2,0) (294) L A Ω (3,0) Ω (3,0) Ω (3,0) = 4i 3 k CY k CY k CY, Ω (3,0) Ω mnpω mnp = 8. (317) Ω (3,0) (315) W eff = T D5 Ω (3,0) G 3 [ A i B i B i Ω (3,0) A i G 3 T D5 = 2π/gstr B D5-brane Gukov-Vafa-Witten [16] IIB ] (318) S i tr(w i W i ) = 2π gstr A S i = T D5 Ω (3,0) (319) A i (318) Calabi-Yau A i n D5- domain-wall G 3 B i G 3 = n. (320) B B i A i S 3 W eff = nt D5 Ω (3,0) A = nt D5V (321) i A i ω (3,0) = V (322) A i Ω (3,0) BPS (321) D5-brane 48

49 NS5-brane D4-brane N = 2 NS5-brane w N = T-dual Calabi-Yau Calabi-Yau Calabi-Yau Calabi-Yau D4-brane w NS5-brane NS5-brane w D4-brane NS5-brane NS5-brane w x 8 = x 9 = x 11 = 0 N = 2 SU(2) R (x 8, x 9, x 11 ) NS5-brane NS5-brane D2-brane D2-brane N = 2 deformation N = 2 N = 1 V i S i (x 8, x 9, x 11 ) NS5-brane SU(2) R 3 N = 2 FI s = (x 11 + ix 9 )/L 11 s = τ x 8 ζ = δx 8 W K W = τs, K = ζv (323) (287) N i = 0 (287) 2 N i D4-brane NS5-brane FI N = 2 N = 2 Fayet-Illiopoulos S i S N = 2 S NS5-brane deformation w NS5-brane w S = 2π 2(g A str) 2 d 6 x µ w µ w (324) (15) NS5-brane w z i w = w(u, z i ). (325) (324) z i S = 2πi 4(g A str) 2 d 4 x du du w w z i z µz i µ z j = j 2 K d 4 x z i z µz i µ z j (326) j 49

50 K K = 2πi 4(gstr) A wdu w du (327) 2 u-w NS5-brane (293) Calabi-Yau K = 2πi 4(gstr) A ω 2 (3,0) ω(0,3) = 2πi 4(gstr) B Ω 2 (3,0) Ω (0,3) (328) CY IIB Calabi-Yau IIB Calabi-Yau N = 2 IIB (32) CY δg mn = g i,mn δz i = 1 Ω 2 Ω i,mpqω m pq δz i (329) g 10 g ij g kl g µν µ g ik ν g jl (330) z 4 S = d 4 x g 2 K z i z µz i µ z j (331) j V 6 K = 4πV ( ) 6 (gstr) B log i Ω 2 (3,0) Ω (0,3) Calabi-Yau V 6 Calabi-Yau ( ) i Ω Ω = V 6 Ω 2 + iδ Ω (3,0) Ω (0,3) CY CY CY (332) (333) (332) V 6 4πi K = (gstr) B 2 Ω 2 CY Ω (3,0) Ω (0,3) (334) Ω 2 = 8 NS5-brane (328) 4.6 NS5-brane NS5-brane Calabi-Yau NS5-brane Bucher 50

51 [17] Calabi-Yau Calabi-Yau NS5-brane NS5-brane w = w(u), s = s(u), (335) x M (255) s M- x 11 M5-brane T-dual ds 2 = du 2 + ds 2 TN(u) (336) ds 2 TN (u) center u Taub-NUT ds TN u G 3 H 3 (314) 3-form field strength G C 3 T-dual (295) NS5-brane G C 3 = I ds I η I (337) η I Taub-NUT ds 2 TN (u) η I Taub-NUT self-dual 2 du = idu 6 G C 3 = ig C 3 (338) imaginary self-dual (336) (337) (338) Calabi-Yau dg C 3 = 0 G C 3 k 8 η I 0 k 8 η I = 0 G C 3 k CY = 0 (339) k CY (299) G C 3 primitive (339) (339) [18, 19] G C 3 SU(3) 3-form (2, 1)-form 6 (0, 3)-form (337) (0, 3) G C 3 (0, 3) Gukov-Vafa-Witten (318) 0 51

52 3: 3- form (3, 0) (2, 1) (1, 2) (0, 3) SU(3) [333] = 1 [33]3 = [33] = [333] = 1 self-dual primitive N = 1 Yang-Mills GVW NS5-brane (337) Calabi-Yau 4 W = 0 (0, 3) 3-form flux 5-form H 3 G 3 0 RR 5- H 3 G 3 0 (48) G 5 dg 5 = H 3 G 3. (340) G 5 = G 5 G 5 (340) H 3 G 3 effective D3-brane charge ds 2 CY = g mndx m dx n Calabi-Yau D3-brane D3-brane ρ g mn Calabi-Yau volume form ω 6 dg 5 = ρω 6 (341) D3-brane ds 2 = h 1/2 η µν dx µ dx ν + h 1/2 ds 2 CY (342) h Calabi-Yau ( g) h = g str ρ. (343) RR 4-form potential 0123 C 0123 = 1 g str h (344) D3-brane 3-form D3-brane H 3 G 3 = ρ D3 ω 6. (345) Gukov-Vafa-Witten 52

53 3- D5-brane D5-brane (330) 4 g µν Calabi-Yau g mn S (g µν ) 1 (g mn ) 3 h 1 (328) [20] K = 2πi 4(gstr) B hω 2 (3,0) Ω (0,3) (346) CY h h Calabi-Yau h 3-form g G C 3 2 h 1 4 K ij W,i W,j (346) D-term NS5-brane M5-brane D-term F-term 4.7 Elliptic model 4.1 NS5-brane D4-brane NS5-brane tree level x 9 L 9 (1) P NS5-brane x 6 = x 7 = x 8 = 0 x 9 NS5-brane I w u x 9 Y Φ X D4 NS5 1 NS5 2 NS5 3 13: N = 2 NS5-brane I NS5-brane I + 1 NS5-brane N I D4-brane I = 1 I = P + 1 P NS5-brane 1 NS5-brane D4-brane D4-brane x 9 U(N I ) N = 2 P I=1 U(N I ) U(N I ) N = 2 N = 1 V I Φ I D4- u SU(N I ) Φ I 53

54 SU(N I ) τ I NS5 (256) τ I = s I+1 s I. (347) NS5-brane D4-brane X I Y I X : (N I, N I 1 ), Y : (N I, N I 1 ). (348) N = 2 Φ I W (X, Y, Φ) = Φ I (X I Y I Y I+1 X I+1 ) (349) I Φ I Φ I 1 X I Y I D4-brane quiver 15 N I Y 1 Φ 1 Φ 2 Φ 3 Y 2 Y 3 X 1 X 2 X 3 14: N = 2 2N I β 0 N = 2 N = 1 I NS5-brane w w = w I (u) Φ I (260) W (Φ I ) = w I+1(u) w I (u) (2πl s ) 3 gstr A. (350) NS5-brane w I (u) w I (u) = µq I u. (351) µ q I I q I = 0 q I = 1 q I = 1 NS5-brane u w 15 µ NS5-brane w W = µ (q I+1 q I )Φ 2 I (352) 2 I Φ I q I+1 = q I 0 q I+1 q I ±µ Φ I decouple 16 Φ I 54

55 w u x 9 D4 NS5 1 NS5 2 NS5 3 15: N = 1 Y 1 Φ 1 Y 2 Y 3 X 1 X 2 X 3 16: SPP quiver diagram F-term Φ I = q I+1 q I (X I Y I Y I+1 X I+1 ) (353) µ Φ I Φ I decouple W = 1 2µ (q I+1 q I )(X I Y I Y I+1 X I+1 ) 2 (354) W = Φ I (X I Y I Y I+1 X I+1 ) 1 (q I+1 q I )(X I Y I Y I+1 X I+1 ) 2 (355) 2 q I =q I+1 q I q I+1 µ F -term Φ I = Φ I + 2q I 1 (X I Y I + Y I+1 X I+1 ) (356) 2 (X I Y I ) 2 W = Φ I (X I Y I Y I+1 X I+1 ) + (q I+1 q I )(X I Y I Y I+1 X I+1 ) (357) q I =q I+1 q I q I+1 Φ I Φ I N I Φ I decouple SU(N I ) N = 2 β = 0 Φ I decouple SU(N I ) 2N I NS5-brane 4.3 T-duality NS5-brane q I = 0 w = 0 m q I = 1 u = 0 n NS5 world volume w m u n = 0 x 9 T-dual Calabi-Yau xy = w m u n (358) 55

56 m = n = 1 conifold m, n generalized conifold m = 2, n = 1 suspended pinch point SPP moduli space U(1) Φ I, x X P X 2 X 1, y Y 1 Y 2 Y P, X I Y I. (359) (q 1, q 2, q 3 ) = (0, 0, 1) quiver 16 W = Φ 1 (X 1 Y 1 Y 2 X 2 ) + (X 2 Y 2 Y 3 X 3 ) (X 3 Y 3 Y 1 X 1 ). (360) F -term 0 F X1 = (Φ 1 X 3 Y 3 )Y 1, F Y1 = (Φ 1 X 3 Y 3 )X 1, F X2 = (Φ 1 X 3 Y 3 )Y 2, F Y2 = (Φ 1 X 3 Y 3 )X 2 (361) F -term F = 0 Φ 1 X 3 Y 3 3 F Φ1 = X 1 Y 1 X 2 Y 2, F X3 = (X 1 Y 1 X 2 Y 2 )Y 3, F Y3 = (X 1 Y 1 X 2 Y 2 )X 3. (362) X 1 Y 1 X 2 Y 2 w = X 1 Y 1 = X 2 Y 2, u = X 3 Y 3 = Φ 1, x, y, (363) 4 xy = w 2 u. (364) SPP (363) u w S 1 D4-brane u w NS5 1 NS5 2 D4-brane u Φ u- D4-brane NS5 3 D4 NS5-brane X 3 Y 3 w D4-brane S 1 T-dual D3-brane D3-brane T-dual Calabi-Yau N = 2 curve cut Coulomb moduli D4-brane N = 1 cut i N i NS5- N i cut monopole condensation [21] 56

57 NS5-brane q I = 0, 1 F -term w = X I Y I (q I = 0) = Φ I (q I = 1), u = X I Y I (q I = 1) = Φ I (q I = 0). (365) xy = w m u n. (366) T-dual generalized conifold (358) 5 Calabi-Yau 5.1 2n n n n B T n T n M (367) B B T n R T n R ~ B 17: ϕ a (a = 1,..., n) 2π 1 n ϕ ϕ a n T n covering space R v R ϕ ϕ = ϕ + av a R (368) U(1) U(1) n e a v = v a e a e a ϕ a 57

58 e a R Γ T n = R/Γ e a n U(1) µ a n R n R v = v a e a U(1) U(1) n v a µ a R R e a R ẽ a Γ Γ Γ ẽ a dϕ a B R R I I s I R s I R R Γ Γ B R s I µ ξ I I. (369) M resolution I s I 18 s I isometry s I B z n 18: B I s I z I s I s I = iz I izi z I z I (370) s I I {s I } Γ b n,0 = 1 (n, 0) Ω Ω (368) L si Ω = iq I Ω (371) 58

59 s 4 s 3 s 1 s 2 19: q I s I Ω µ R q I = s I µ (372) f v L v f = i(v q)f (373) q R µ Ω f Γ (n, 0) Ω R- Ω µ B I s I z I z I n 1 y 1,..., y n 1 I C n (n, 0) Ω dz I dy 1 dy n 1 (374) Ω 0 0 s I U(1) Ω 1 I s I µ = 1. (375) s I Γ n 1 n web Γ web s I web µ R (375) (375) 59

60 µ R B (369) a µ + aµ B B M R µ B µ B R = R/(µ 0) web µ R ~ R 20: web Γ SL(n, Z) µ = ẽ n v R µ R v µ µ R µ µ (376) 1 s I = (s I, 1) (377) (376) I J s I µ = s I µ + µ n = k I, s J µ = s J µ + µ n = k J. (378) µ n R n 2 (s I s J ) µ = k I k J. (379) s I s J n 2 k I web Calabi-Yau 3-fold web-diagram NS5 web R µ v µ = 0. (380) U(1) n 1 M n + 1 T n 1 B 60

61 B S 1 S 1 R- S 1 B shrink B T n 1 (n, 0) Ω B 1 η dζ = Ω. (381) T n 1 1 dζ B ζ ζ B 0 B B = R C ζ (382) C ζ ζ B C ρ ρ = 0 web R codimension 1 B codimension 3 S 2 I K S IJ J ~ R 21: web S 2 T n s I s J shrink T n 1 s I s J shrink NUT-singularity NUT charge ds 2 = ds 2 B + n 1 a,b=1 g ab (dϕ a A a )(dϕ b A b ) (383) ds 2 B B g ab T n 1 g ab B I s I = (s I, 1) shrink C ρ A a = s a I (384) S 2 NUT S 2 F a = s a I s a J (385) 5.3 µ a B 2n M n U(1) n B T n n U(1) a = 61 ϕ a (386)

62 ϕ a a = 1,..., n ϕ a 2π 0 = L a k = (i a d + di a )k = di a k (387) i a a (387) µ a dµ a = i a k. (388) µ a M 2n {µ a, ϕ a } (387) d(i a dµ b ) = di a i b k = L a i b k = [L a, i b ]k + i b L a k = 0 (389) [L X, i Y ] = i [X,Y ] (390) (389) i a dµ b 0 µ a µ a R s I shrink 0 s I = s a I a bdr = 0 (391) i si k bdr = d(s a I µ a ) bdr = 0 (392) s a I µ a (388) k = f ab (µ)dµ a dµ b + dµ a dϕ a (393) 1 1 f ab (µ)dµ a dµ b = da = d(a a dµ a ) = dµ a da a (394) k = dµ a d(ϕ a A a ) (395) ϕ a new = ϕ a A a (396) ϕ a k = dµ a dϕ a. (397) ζ a dζ a = dy a + idϕ a (398) 62

63 y a ϕ a µ a k = id h d a K (399) d h ζ d a ζ d h = dζ a ζ a = 1 ( 2 dζa y a i ) µ a, d a = dζ a ζ a = 1 ( 2 dζ a y a + i ) µ a. (400) µ a y a (402) G(µ a ) k = 1 ( ) K 2 d y a dϕ a (401) µ a = 1 K 2 y a. (402) G = µ a y a 1 2 K (403) y a = G µ a (404) ds 2 = 1 2 ( K 2 y a y b (dya dy b + dϕ a dϕ b ) = 2 G 2 ) 1 G dµ a dµ b + dϕ a dϕ b (405) µ a µ b µ a µ b 5.4 C 3 (z 1, z 2, z 3 ) 3 U(1) z a = e ya +iϕ a y a ϕ a r a = e ya = z a B shrink B r a R shrinking cycle s 1 = ϕ 1, s 2 = ϕ 2, s 3 = ϕ 3. (406) 2n 1 1 B shrink (406) C 3 Calabi-Yau 3 63

64 s 3 s 2 f 2 s 1 s 2 s 1 s 3 f 1 22: C 3 web U(1) 3 U(1) 2 (3, 0) Ω C 3 Ω Ω = dz 1 dz 2 dz 3 = z 1 z 2 z 3 (dy 1 + idϕ 1 ) (dy 2 + idϕ 2 ) (dy 3 + idϕ 3 ). (407) z 1 z 2 z 3 e i(ϕ1 +ϕ 2 +ϕ 3) (406) 3 Ω z 1 z 2 z 3 e iφ3 ϕ 1 = φ 1, ϕ 2 = φ 2, ϕ 3 = φ 3 φ 1 φ 2 (408) Ω φ 1 φ 2 f 1 := φ 1 = ϕ 1 ϕ 3 = s 1 s 3, f 2 := φ 2 = ϕ 2 ϕ 3 = s 2 s 3. (409) 2 Ω φ 1 φ 2 1 dζ = Ω = (2π) 2 d(z 1 z 2 z 3 ) (410) T 2 ζ ζ = (2π) 2 z 1 z 2 z 3 (411) 3 1 µ C 3 ds 2 = (dz a ) dz a (402) K = 1 2 ( z1 2 + z z 3 2 ) = 1 2 (e2y1 + e 2y2 + e 2y3 ) (412) (403) G = µ a = 1 K 2 y a = 1 a 2 e2y. (413) 3 µ a y a 1 2 K = 1 2 a=1 ds 2 = 3 a=1 3 (µ a log(2µ a ) µ a ) (414) a=1 ( ) 1 (dµ a ) 2 + 2µ a (dϕ a ) 2. (415) 2µ a 64

65 conifold C n n C n gaiged linear sigma model (GLSM) F-term D-term conifold C 4 z i z 1 z 2 = z 3 z 4 (416) conifold 3 R R 3 C 4 (416) z i ϕ i = arg z i 4 R R z i 0 (416) ϕ i ϕ 1 + ϕ 2 = ϕ 3 + ϕ 4 ϕ F = dϕ 1 + dϕ 2 dϕ 3 dϕ 4 R (417) conifold 3 R R R = { v R v F = 0} (418) (416) z 1 z 2 z 3 z 4 2 s 1 = ϕ 1 + ϕ 3, s 2 = ϕ 1 + ϕ 4, s 3 = ϕ 2 + ϕ 3, s 4 = ϕ 2 + ϕ 4 R R (419) (417) F 3 R q 1 + q 4 = q 2 + q 3 µ = dϕ 1 + dϕ 2 R (420) v µ = (416) 3 φ a ϕ 1 = φ 2 + φ 3, ϕ 2 = φ 2, ϕ 3 = φ 1 + φ 3, ϕ 4 = φ 1. (421) R 3 e 1 = φ1 = ϕ3 + ϕ4, e 2 = φ2 = ϕ1 + ϕ2, e 3 = φ3 = ϕ1 + ϕ3 R R (422) s 1 = e 3, s 2 = e 3 + e 1, s 3 = e 3 + e 2, s 4 = e 3 + e 1 + e 2 R R (423)

66 s 3 s 1 s 4 s 2 s 3 e 2 s 1 e 1 s 4 s 2 23: web (3, 0) Ω = dz 1 dz 2 dz 3 z 3 (424) e i(ϕ 1+ϕ 2 ) = e iφ 3 φ 1 φ 2 e 1 e 2 Ω φ 1 φ 2 T 2 dζ = Ω = (2π) 2 d(z 1 z 2 ) (425) T 2 ζ ζ = (2π) 2 z 1 z 2 (426) GLSM conifold 4 ρ i i = 1, 2, 3, 4 C 4 ϕ i = arg ρ i R C 4 4 R R R 1 U(1) C (ρ 1, ρ 2, ρ 3, ρ 4 ) (e iα ρ 1, e iα ρ 2, e iα ρ 3, e iα ρ 4 ) (427) R R g = ϕ1 + ϕ2 ϕ3 ϕ4 R (428) D-term g µ = 0 (429) R R = { µ R g µ = 0} (430) M 1 ρ i C 4 1 ρ i R s 1 = ϕ1, s 2 = ϕ2, s 3 = ϕ3, s 4 = ϕ4 R. (431) GLSM 1 ρ i ρ i ρ i = 0 C n 1 66

67 1 R R (431) R 4 R f 1 = dϕ 2 + dϕ 4, f2 = dϕ 3 + dϕ 4, f3 = dϕ 1 + dϕ 2 + dϕ 3 + dϕ 4 R R. (432) (431) s 1 = (0, 0, 1), s 2 = (1, 0, 1), s 3 = (0, 1, 1), s 4 = (1, 1, 1) R (433) 23 GLSM R R C 4 ρ i O[n i ] = i ρ ni i (434) n i R n i (434) n i 0 (427) O[n i ] g n = 0 (435) D-term (429) R 4 z i z 1 = ρ 1 ρ 3, z 2 = ρ 2 ρ 4, z 3 = ρ 1 ρ 4, z 4 = ρ 2 ρ 3. (436) 3 (416) R q R s I s I B I C n s I (370) z I n 1 y 1,..., y n 1 f f z I f s I f I f q s I q 0 I. (437) q Γ (437) Γ (369) B (437) {s I } 67

68 (437) 0 B 2 C 2 (x, y) O = x m y n, m, n 0 (438) (m, n) Z 2 Z 2 + C2 x = r 1 e iϕ1, y = r 2 e iϕ 2 r a ϕ a C 2 R 2 + T2 Z 2 + R2 + 1 C z z n n Z + z = re iϕ r R + z 0 S 1 C z n n Z C R S 1 f ψ = fe K/2 (439) K e K/2 f ψ 2 3 (464) z a (465) z a O = 3 a=1 z n a a (440) z a = y a + iϕ a y a ϕ a ψ 2 = e 2 a naya K (441) (402) µ a = n a (442) f ψ 2 68

69 6 T 2 IIB NS5-brane T 2 T-dual IIB Calabi-Yau NS5-brane brane tiling N = 1 Brane tiling [22] [23] [24] brane tiling [25] Calabi-Yau T- Calabi-Yau moduli space Calabi-Yau brane tiling T-duality Calabi-Yau moduli space [26] Brane tiling [27], [28] Calabi-Yau Calabi-Yau U(1) NS5-5.2 Calabi-Yau n-fold R n+1 T n 1 singular codimension 3 n = 3 T-duality NS5-brane n = 3 B 4 web 1-brane string CY R 4 R 4 string T 2 string Buscher T-duality (383) ds 2 = ds g kl (dϕ k + V k )(dϕ l + V l ) (443) k,l=1,2 ds 2 4 B g kl T 2 V k B U(1) (385) V k F k = dv k F k s k I s k J = s k IJ. (444) S IJ web string s k IJ T 2 T-duality NS-NS 3-form Buscher T-dual 3-form H 3 = F 1 dx 5 + F 2 dx 7. (445) ϕ 1 ϕ 2 x 5 x 7 69

70 web-diagram (444) H 3 = s 1 IJ, H 3 = s 2 IJ. (446) S IJ S 1 (x 5 ) S IJ S 1 (x 7 ) S 1 (x i ) x i S 1 (446) T-dual H 3 NS5-brane NS5-brane x 5 -x 7 T 2 (446) J = dh 3 J = s 1 IJ, J = s 2 IJ. (447) D IJ S 1 (x 5 ) D IJ S 1 (x 7 ) D IJ S IJ B 3 x 5 NS5-brane s 1 IJ x7 s 2 IJ NS5-brane (s 2 IJ, s1 IJ ) web R 4 T 2 NS5-brane Σ Buscher Busher H 3 NS5-brane 0 NS5-brane 0 web-diagram worldvolume Web-diagram NS5-brane Σ B s I µ k I, I (448) B µ 3 = max(k I s 1 Iµ 1 s 2 Iµ 2 ) (449) e µ 3 = e k I +ib I x s1 I y s 2 I I (450) x y x = e µ 1+i ϕ 1, y = e µ 1+i ϕ 2 (451) b I k I (449) Web-diagram (449) (450) web-diagram B P (x, y) c I x s1 I y s 2 I = 0, ci = e k I +ib I (452) I P (x, y) c I (448) 70

71 k I c I (452) 0 web web-diagram 24(b) (a) (b) (c) 24: (p, q)-web amoeba 6.2 bipartite toric Calabi-Yau T-dual NS5-brane C 2 NS5-brane T 2 D-brane NS5-brane T 2 AdS/CFT Calabi-Yau cone k I = 0 web µ 1 -µ 2 (s 2 IJ, s1 IJ ) l k = (m k, n k ) m k n k T-dual NS5-brane l k 46 NS5-brane NS5-brane T 2 NS5-brane T 2 C 3 C 3 (p, q)-web junction 3 (1, 0), (0, 1), ( 1, 1). (453) NS5-brane 25(a) 3 NS5-brane NS5-brane 71

72 (a) (b) 25: NS5-brane charge NS5-brane NS5-brane 25(b) 25(b) NS5-brane charge 0 NS5-brane +1 1 NS5-brane NS5-brane charge +1 NS5-brane charge 1 25 (a) (b) (b) δ- 1- (a) (b) 0 (m k, n k ) 0 (m k, n k ) (m k, n k ) = (s 2 I+1, s 1 I+1) (s 2 I, s 1 I) (454) web 0, +1, 1 NS5-brane charge +1 NS5-brane charge 1 charge ± (a) (b) 26: bipartite graph bipartite graph 72

73 0, ±1 bipartite graph 0, ±1 NS5-brane 0, ±1 2 A Q A C C Q A Q 1 Q 2 A C Q 1 Q Q 2 Q 1 27: NS5-brane charge A A Q 1 Q 3 Q ±1 0, ±1 Calabi-Yau Seiberg-dual 6.3 Brane tiling NS5-brane Calabi-Yau D-brane Calabi-Yau cone D3-brane N T-dual web-diagram T 2 D5-brane N D5-brane 5-brane diagram NS5-brane charge ±1 N D5-brane NS5-brane (N, ±1)-brane NS5-brane charge 0 N D-brane (N, ±1-brane SL(2, Z) 1 D5-brane U(1) SU(N) NS5-brane charge 0 Calabi-Yau string coupling g str T-duality NS5-brane string coupling g str = A g str A fivebrane T 2 Calabi-Yau cone r T 2 dual 73

74 A A 0 fivenrane g str 0 T NS5 T D5 (455) D5-brane NS5-brane NS5-brane D5-brane worldvolume NS5-brane bound state NS5-brane charge 0 ±1 D5-brane NS5-brane 28 (a) NS5-brane 28: NS5-brane D5-brane NS5-brane g str 1 g str D5-brane NS5-brane NS5-brane NS5-brane charge 2 NS5-brane (b) D5-brane worldvolume NS5-brane charge ±1 NS5-brane charge g str NS5-brane D5-brane worldvolume D5-brane N bipartite graph U(N) U(1) [30] SU(N) SU(N) 29: 74

75 bipartite graph (N, N) (N, N) bipartite graph N N brane tiling quiver diagram quiver diagram SU(N) quiver diagram brane tiling brane tiling superpotential bipartite graph 30: k I I Φ I k O k = tr Φ I, (456) I k W = k h k O k (457) h k Φ I h k 0 h k 0 k:white β = ( h k) k:black h k (458) β-deformation B [31] 75

76 β = 1 5- T-dual Calabi-Yau [26] β = 1 h k = 1 h k = 1 W = O k O k (459) k:black k:white 6.4 Zig-zag paths bipartite graph NS5-brane zig-zag path zig-zag path bipartite graph zig-zag path bipartite zig-zag path 31: zig-zag path zig-zag path NS5-brane boundary bipartite graph NS5-brane zig-zag path web 6.5 Elliptic model 4.7 elliptic model brane tiling T-dual 2 NS5-brane NS5-brane NS5a NS5b D4-brane 4: T-duality brane tiling Hanany Witten NS5 a NS5 b D4 N x 9 S 1 76

77 T-duality brane tiling 4567 R 3 S 1 R 3 r ( ) r = u x 4 + ix 5 σu, u = (460) x 6 + ix 7 σ S 1 ψ 5 NS5-brane S 1 fiber 5: ± s shrinking cycle ψ r 1 r 2 r NS5 a + NS5 b shrinking cycle s D4 N r 3 (NS5a) (NS5b) S 1 r = 0 x 8 -r 3 NS5-brane shrinking cycle Figure 32) x 9 T-dual NS5-brane shrinking cycle Calabi-Yau NS5a r 3 x 8 NS5b shrinking cycle 32: IIA NS5-brane generalized conifold D3-brane ψ T-dual NS5-brane NS5-brane shrinking cycle NS5-brane D4-brane D5-brane brane tiling 5-brane web brane tiling Figure 33 3 (N, 0) SU(N) 3 elliptic model W = ΦX 1 Y 1 ΦY 2 X 2 X 1 X 2 Y 2 Y 1 + Y 2 Y 3 X 3 X 2. (461) (360) 77

78 X 3 X 3 X 2 X 2 Φ Y 2 Y 3 Y 1 Φ Y 2 Y 3 Y 1 X 1 X 1 33: T- Toric O = tr(φ 1 Φ 2 ) (462) 33 O = tr(φy 2 X 2 ) 34 X 2 Φ Y 2 34: U(1) D3-brane U(1) O = I Φ n I I, n I 0. (463) F -term 78

79 I Φ I F -term F I = 0 Φ I I Φ I I Φ I Φ I Φ I F I = 0 F -term C Φ 2 I C 1 35: Φ I F -term C 1 C 2 3 α β γ O α, O β, O γ (464) γ β α 36: 3 SU(N) U(1) 3 3 (463) O O O n α α O n β β On γ γ, n α, n β, n γ Z (465) 79

80 F -term U(1) U(1) U(1) Φ I U(1) charge Q I Φ I Q I I U(1) P f f P (466) P 37 P [O] cycle flow 37: O f[u(1)] U(1) flow (O U(1) charge) = f[u(1)] P [O] (467) U(1) F -term R- R regular flow regular flow U(1) regular flow U(1) U(1) SU(N) U(1) +1 U(1) 1 80

81 U(1) 38: U(1) 1 SU(N) det Φ I U(1) U(1) 1- U(1) 6.8 U(1) [32] U(1)SU(N) 2 U(1) bi-fundamental field bi-fundamental field U(1) C I Q 2 aic I = 0. (468) I C I 0 zig-zag path zig-zag path (468) C I zig-zag path C I = sign(µ, ν)(c µ C ν ) (469) µ ν I zig-zag path C µ zig-zag path C I C µ C µ C I = zig-zag path 1 (470) U(1) 81

82 RR X 5 Sasaki-Einstein H 3 (X 5 ) ω i C 4 = A i 1 ω i (471) b 2 A i 1 AdS H 3 (X 5 ) 2- N N 3 X 5 T 3 fibration dual cone T 3 2 junction 39: Sasaki-Einstein 2- junction string string charge charge shrink string junction string charge 3 2-cycle n 3 R- Q 2 air I = 2 (472) I R I zig-zag path πr I = sign(µ, ν)(ϕ µ ϕ ν ) (473) ϕ µ zig-zag path π π sign(µν) R-charge 0 1 R I (472) 2π ϕ µ zig-zag path zig-zag path S 1 U(1) R 82

83 6.9 U(1) G = U(1) R U(1) 2 F U(1) n F 1 B (474) G = U(1) R U(1) 2 F U(1) b B (475) b zigzag path 3 U(1) regular flow regular flow bipartite graph SPP 40 1 (a) (b) (c) (d) (e) (f) 40: SPP regular flow regular flow regular flow 40 (a) (b) (c) + (d) = 0 (476) F -term µ = 1, 2,... regular flow f µ 83

84 Φ I P I = P [Φ I ] I F -term Φ I Φ I = µ ρ f µ P I µ (477) ρ µ F -term O γ (477) O γ = ρ µ (478) µ F -term F -term (477) f µ U(1) ρ µ 1 U(1) 6.10 U(1) U(1) U(1) s I µ s µ s µ 3 R- 3 s µ (464) (465) f µ 3 3 f µ α f µ s µ = f µ β (479) f µ γ 3 γ 1 SPP

85 (0,0) (1,0) (1,0) (2,0) (0,1) (1,1) 41: 42: α β α 42 1 µ ν s µ s ν f µ f ν (1, 1) (2, 0) 43 (a) (1, 1) 2 (2, 0) (0, 0) (b) D5-brane (a) (b) 43: NS5-brane (a) 43 (b) web- NS5-brane web- 85

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