SUSY DWs

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@ 2013 1 25 Supersymmetric Domain Walls Eric A. Bergshoeff, Axel Kleinschmidt, and Fabio Riccioni Phys. Rev. D86 (2012) 085043 (arxiv:1206.5697) ( )

Contents 1 2 SUSY Domain Walls Wess-Zumino Embedding Tensor Formalism

p-branes : D p p D 4 : standard branes p = D 3 : defect branes p = D 2 : Domain Walls T p (g s ) +α (l s = 1) α = 0 : fundamental α = 1 : Dirichlet α = 2 : solitonic S p-brane = T p ( ) + ( brane ) Dirac-Born-Infeld type Wess-Zumino type SUSY Domain Walls - 3 -

D = 10 IIA/IIB D = 10 p = 0 p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 p = 7 p = 8 p = 9 α = 0 F1 IIA/IIB α = 1 D0 IIA D1 IIB D2 IIA D3 IIB D4 IIA D5 IIB D6 IIA (D7) IIB (D8) IIA (D9) IIB α = 2 NS5 IIA/IIB SUSY Domain Walls - 4 -

D = 10 IIA/IIB D = 10 p = 0 p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 p = 7 p = 8 p = 9 α = 0 F1 IIA/IIB α = 1 D0 IIA D1 IIB D2 IIA D3 IIB D4 IIA D5 IIB D6 IIA (D7) IIB (D8) IIA (D9) IIB α = 2 NS5 IIA/IIB p 6 sources : F1 B (2), NS5 B (6), Dp C (p+1) (p 3), Dp C (p +1) (p > 4) db (2) = 10 db (6), dc (p+1) = 10 dc (7 p) 10 dc (p +1) Dp = standard branes (p 6) RR potentials C (p+1) D7 = defect branes scalar fields (+α) D8 = Domain Walls Romans mass ( ) D9 = spacetime-filling branes I SUSY Domain Walls - 5 -

Domain Walls (10 D8-brane Romans mass )

Motivation = 32 ( ) coset space G 0 /H D U-duality G 0 R- H dim(g 0 /H) T-duality 11 1 1 0 1 IIA R + 1 1 1 IIB SL(2, R) SO(2) 2 1 9 GL(2, R) SO(2) 3 SO(1, 1) 8 SL(3, R) SL(2, R) SO(3) SO(2) 7 SL(2, R) SL(2, R) 7 SL(5, R) Sp(2) 14 SL(4, R) 6 SO(5, 5) Sp(2) Sp(2) 25 SO(4, 4) 5 E 6(6) USp(8) 42 SO(5, 5) 4 E 7(7) SU(8) 70 SO(6, 6) 3 E 8(8) SO(16) 128 SO(7, 7) SUSY Domain Walls - 7 -

Motivation Domain Walls D8-brane in 10-dim. Ramond-Ramond potential C (9) 10 dc (9) = m ( ) IIA Romans massive IIA SUGRA SUSY Domain Walls - 8 -

Motivation Domain Walls D8-brane in 10-dim. Ramond-Ramond potential C (9) 10 dc (9) = m ( ) IIA Romans massive IIA SUGRA (D 2)-branes in D-dim. SUSY Domain Walls Domain Walls SUSY Domain Walls - 9 -

1 SUSY Domain Walls Wess-Zumino 7 Domain Walls

D8-brane in 10D D8-brane ds 2 10 = H 9 8 (y) dy 2 + H 1 8 (y) ds 2 9 e ϕ = H 5 4 (y) dilaton C 012 8 = ± 1 H(y), m = ± yh(y) RR potential (Romans mass) mass m 0 RR potentials δb 2 = dσ 1, δc 1 = mσ 1 F 2 dc 1 + mb 2 : Stückelberg pairing gauging C 1 B 2 C 3 C 5 B 6 C 7 m eaten massive massless massless eaten massive SUSY Domain Walls - 11 -

D8-brane in 10D D8-brane (D8 back reaction ) δc 9 = dλ 8 + H 3 λ 6 : RR tensor in bulk δb 2 = dσ 1 : NSNS tensor in bulk δx = 0 : transverse scalar δb µ = dσ 0 Σ 1 : D8-brane D8-brane SUSY {X, b µ ; ψ} : on-shell (8 boson + 8 fermion ) SUSY Wess-Zumino L WZ = C 9 + C 7 F 2 +... = ( C e F 2 ) 9 F 2 = db 1 + B 2, H 3 = db 2 ( ) m 0 SUSY Domain Walls - 12 -

D SUSY DWs D Wess-Zumino L WZ (A e F ) D 1 A : F : D ( ) Domain Walls A, F D U-duality G 0 1. (A, F) SUSY 2. SUSY Domain Walls SUSY Domain Walls - 13 -

U-duality G 0 form fields D U-duality G 0 1-forms 2-forms 3-forms 4-forms 5-forms 6-forms 7-forms 8-forms 9-forms 10-forms IIA R + 1 1 1 1 1 1 1 1 1 1 IIB SL(2, R) 2 1 2 3 4 2 9 GL(2, R) 2 1 2 1 1 2 2 1 3 1 3 2 4 2 2 8 SL(3, R) SL(2, R) (3, 2) (3, 1) (1, 2) (3, 1) (3, 2) (8, 1) (1, 3) 7 SL(5, R) 10 5 5 10 24 40 15 6 SO(5, 5) 16 10 16 45 144 5 E 6(6) 27 27 78 351 4 E 7(7) 56 133 912 3 E 8(8) 248 3875 1 147250 3875 248 8645 133 1728 27 320 126 10 (6, 2) (3, 2) 70 45 5 (15, 1) (3, 3) (3, 1) (3, 1) F.Riccioni, D.Steele and P.West, arxiv:0906.1177 SUSY Domain Walls - 14 -

U-duality G 0 form fields D U-duality G 0 1-forms 2-forms 3-forms 4-forms 5-forms 6-forms 7-forms 8-forms 9-forms 10-forms IIA R + 1 1 1 1 1 1 1 1 1 1 IIB SL(2, R) 2 1 2 3 4 2 9 GL(2, R) 2 1 2 1 1 2 2 1 3 1 3 2 4 2 2 8 SL(3, R) SL(2, R) (3, 2) (3, 1) (1, 2) (3, 1) (3, 2) (8, 1) (1, 3) 7 SL(5, R) 10 5 5 10 24 40 15 6 SO(5, 5) 16 10 16 45 144 5 E 6(6) 27 27 78 351 4 E 7(7) 56 133 912 3 E 8(8) 248 3875 1 147250 3875 248 8645 133 1728 27 320 126 10 (6, 2) (3, 2) 70 45 5 (15, 1) (3, 3) (3, 1) (3, 1) Domain walls : (D 1)-forms F.Riccioni, D.Steele and P.West, arxiv:0906.1177 SUSY Domain Walls - 15 -

U-duality G 0 form fields D U-duality G 0 1-forms 2-forms 3-forms 4-forms 5-forms 6-forms 7-forms 8-forms 9-forms 10-forms IIA R + 1 1 1 1 1 1 1 1 1 1 IIB SL(2, R) 2 1 2 3 4 2 9 GL(2, R) 2 1 2 1 1 2 2 1 3 1 3 2 4 2 2 8 SL(3, R) SL(2, R) (3, 2) (3, 1) (1, 2) (3, 1) (3, 2) (8, 1) (1, 3) 7 SL(5) 10 5 5 10 24 40 15 6 SO(5, 5) 16 10 16 45 144 5 E 6(6) 27 27 78 351 4 E 7(7) 56 133 912 3 E 8(8) 248 3875 1 147250 3875 248 8645 133 1728 27 320 126 10 (6, 2) (3, 2) 70 45 5 (15, 1) (3, 3) (3, 1) (3, 1) 7 F.Riccioni, D.Steele and P.West, arxiv:0906.1177 SUSY Domain Walls - 16 -

7 SUSY DWs 7 Domain Walls = 5-branes 7D A G 0 = SL(5, R) A 1,[MN] 1-form 10 A M 2 2-form 5 A 3,M 3-form 5 A [MN] 4 4-form 10 A 5,M N 5-form 24 (adjoint) A 6,(MN) A [MN],P 6 6-forms 15 40 (M, N = 1,..., 5 of SL(5, R)) SUSY Domain Walls - 17 -

7 SUSY DWs 15 5-brane L 15 WZ A 6,(MN) + A 5,(M P F 1,N)P + A 3,(M F 3,N) +... F 3,N = db 2,N + A 3,N F 1,NP = db 0,NP + A 1,NP X y SUSY Domain Walls - 18 -

7 SUSY DWs 15 5-brane L 15 WZ A 6,(MN) + A 5,(M P F 1,N)P + A 3,(M F 3,N) +... F 3,N = db 2,N + A 3,N F 1,NP = db 0,NP + A 1,NP X y SUSY Domain Walls - 19 -

7 SUSY DWs 15 5-brane L 15 WZ A 6,(MN) + A 5,(M P F 1,N)P + A 3,(M F 3,N) +... F 3,N = db 2,N + A 3,N F 1,NP = db 0,NP + A 1,NP X y M = N(= 1) case : SUSY 5-branes b 2,N=1 : 4C 2 /2 = 3 b 0,[N=1,P ] : 1 4 = 4 X y : 1 8 boson + 8 fermion 1 2 -SUSY! M = 1,..., 5 SUSY Domain Walls - 20 -

7 SUSY DWs 15 5-brane L 15 WZ A 6,(MN) + A 5,(M P F 1,N)P + A 3,(M F 3,N) +... F 3,N = db 2,N + A 3,N F 1,NP = db 0,NP + A 1,NP X y M = N(= 1) case : SUSY 5-branes 5 b 2,N=1 : 4C 2 /2 = 3 b 0,[N=1,P ] : 1 4 = 4 X y : 1 8 boson + 8 fermion 1 2 -SUSY! M = 1,..., 5 M N case : SUSY 5-branes boson 4 SUSY SUSY Domain Walls - 21 -

7 SUSY DWs 15 5-brane L 15 WZ A 6,(MN) + A 5,(M P F 1,N)P + A 3,(M F 3,N) +... F 3,N = db 2,N + A 3,N F 1,NP = db 0,NP + A 1,NP X y M = N(= 1) case : SUSY 5-branes 5 b 2,N=1 : 4C 2 /2 = 3 b 0,[N=1,P ] : 1 4 = 4 X y : 1 8 boson + 8 fermion 1 2 -SUSY! M = 1,..., 5 5 < 15 Elementary SUSY DWs M N case : SUSY 5-branes boson 4 SUSY SUSY Domain Walls - 22 -

7 SUSY DWs 40 L 40 WZ A [MN],P 6 + A 5,Q P F 1,RS ϵ MNQRS + A [MN] 4 F P 2 ( ) [MNP ] +... 5-brane F P 2 = db P 1 + A P 2 F 1,RS = db 0,RS + A 1,RS X y SUSY Domain Walls - 23 -

7 SUSY DWs 40 L 40 WZ A [MN],P 6 + A 5,Q P F 1,RS ϵ MNQRS + A [MN] 4 F P 2 ( ) [MNP ] +... 5-brane F P 2 = db P 1 + A P 2 F 1,RS = db 0,RS + A 1,RS X y SUSY Domain Walls - 24 -

7 SUSY DWs 40 L 40 WZ A [MN],P 6 + A 5,Q P F 1,RS ϵ MNQRS + A [MN] 4 F P 2 ( ) [MNP ] +... 5-brane F P 2 = db P 1 + A P 2 F 1,RS = db 0,RS + A 1,RS X y P = M(= 1 N) case : SUSY 5-branes 20 b P 1 : 4 b 0,[RS] : 1 3 = 3 X y : 1 8 boson + 8 fermion 1 2 -SUSY! M ( 5 C 2 = 20) P M N case : SUSY 5-branes boson 4 SUSY SUSY Domain Walls - 25 -

7 SUSY DWs 40 L 40 WZ A [MN],P 6 + A 5,Q P F 1,RS ϵ MNQRS + A [MN] 4 F P 2 ( ) [MNP ] +... 5-brane F P 2 = db P 1 + A P 2 F 1,RS = db 0,RS + A 1,RS X y P = M(= 1 N) case : SUSY 5-branes 20 b P 1 : 4 b 0,[RS] : 1 3 = 3 X y : 1 8 boson + 8 fermion 1 2 -SUSY! M ( 5 C 2 = 20) P M N case : SUSY 5-branes boson 4 SUSY SUSY Domain Walls - 26 -

7 SUSY DWs 40 L 40 WZ A [MN],P 6 + A 5,Q P F 1,RS ϵ MNQRS + A [MN] 4 F P 2 ( ) [MNP ] +... 5-brane F P 2 = db P 1 + A P 2 F 1,RS = db 0,RS + A 1,RS X y P = M(= 1 N) case : SUSY 5-branes 20 b P 1 : 4 b 0,[RS] : 1 3 = 3 X y : 1 8 boson + 8 fermion 1 2 -SUSY! M ( 5 C 2 = 20) 20 < 40 Elementary SUSY DWs P M N case : SUSY 5-branes boson 4 SUSY SUSY Domain Walls - 27 -

Elementary SUSY DWs fundamental Dirichlet solitonic (brane s tension) (g s ) +α D U T # of EDWs α = 0 α = 1 α = 2 α = 3 α = 4 α = 5 IIA R + 1 1 1 9 GL(2, R) SO(1, 1) 2 3 U 1 1 8 SL(3, R) SL(2, R) SL(2, R) SL(2, R) 6 (6, 2) U (1, 2) T 4 (3, 2) T 7 SL(5, R) SL(4, R) 20 40 U 4 T 4 10 T 12 20 T 5 15 U 4 10 T 1 T 6 SO(5, 5) SO(4, 4) 80 144 U 8 S T 32 56 C T 32 56 S T 8 C T 5 E 6(6) SO(5, 5) 216 351 U 16 T 80 120 T 80 144 T 40 45 T 4 E 7(7) SO(6, 6) 576 912 U 32 T 160 220 T 192 352 T 160 220 T 32 T 3 E 8(8) SO(7, 7) 2160 3875 U 1 T 64 T 280 364 T 448 832 T 560 1001 T 14 104 T 448 832 T (α 6) 280 364 T, 6 64 T, 7 1 T, 8 D = 3, 4, 6 S-dual branes α = α 4(D 1) D 2 D = 3, 4, 6 S-dual branes by D-dim. S-duality (g µν) S ((g s ) α d D 2 x [NG(g µν )] = (g s ) α = e 8ϕ/(D 2) (g µν ) S ) d D 2 x [NG(g µν)] SUSY Domain Walls - 28 -

String theory origin of Domain Walls in D-dim. fundamental Dirichlet solitonic α = α 4(D 1) D 2 via S-duality D α = 0 α = 1 α = 2 α = 3 α = 4 α = 5 α = 6 α = 7 α = 8 IIA C 9 [D8] 9 C 8 [D7] E 9,1,1 [7 (0,1) 3 ] 8 C 7 [D6] E 9,2,1 [6 (1,1) 3 ] 7 C 6 [D5] D 6 [NS5] D 7,1 [KK5] D 8,2 [5 2 2] E 9,3,1 [5 (2,1) 3 ] F 9,3 [5 3 4] 6 C 5 [D4] E 9,4,1 [4 (3,1) 3 ] F 9,4,1 [4 (3,1) 4 ] 5 C 4 [D3] E 9,5,1 [3 (4,1) 3 ] F 9,5,2 [3 (3,2) 4 ] 4 C 3 [D2] E 9,6,1 [2 (5,1) 3 ] F 9,6,3 [2 (3,3) 4 ] 3 B 2 [F1] C 2 [D1] E 9,7,1 [1 (6,1) 3 ] F 9,7,4 [1 (3,4) 4 ] F 9,7,1,1 [1 (6,0,1) 4 ] G 9,6,2m G 9,6,2m+1 G 9,7,2m,1 G 9,7,2m+1,1 H 9,7,4+n,n (S 3 (C 2 )) (S 3 (B 2 )) A D T,I1 +I 2,I 2 -forms : mixed-symmetry tensors p (I 1,I 2 ) α -branes T + p + i I i = D 1 with T = 1 : transverse, p : spatial, I i : isometry directions E.A. Bergshoeff et al, arxiv:1108.5067, arxiv:1210.1422 SUSY Domain Walls - 29 -

Elementary SUSY DWs Z (a) : a-form central charge D R- H Z (1) Z (2) # of EDWs 9 SO(2) 1 2 2 8 SO(3) SO(2) (1, 2) 6 3 7 Sp(2) 5 + 1 20 + 5 4 (V), 5 (T) 6 Sp(2) Sp(2) (4, 4) 80 5 5 U Sp(8) 36 216 6 4 SU(8) 36 + + 36 576 8 3 SO(16) 135 2160 16 # of EDWs = {(10 D) + 1} (# of Z (2) ) 5 D 9 8 D = 4 16 D = 3 ( ) standard branes central charges 1 1 SUSY Domain Walls - 30 -

Elementary SUSY DWs D # of (D 1)-forms # of EDWs # of non-edws 9 3 2 2 + 0 1 + 2 8 (6, 2) (3, 2) 6 + 0 6 + 6 7 40 15 20 + 5 20 + 10 6 144 80 64 5 351 216 135 4 912 576 336 3 3875 1 2160 + 0 1715 + 1 Elementary SUSY DWs (EDWs) (D 1)-forms EDWs (D 2)-branes ( ) EDWs 1 2-SUSY threshold bound states of EDWs EDWs 1 2-SUSY non-threshold bound states of EDWs SUSY Domain Walls - 31 -

2 ( ) Embedding Tensor Formalism

coset space G 0 /H D U-duality G 0 R- H dim(g 0 /H) T-duality 11 1 1 0 1 IIA R + 1 1 1 IIB SL(2, R) SO(2) 2 1 9 GL(2, R) SO(2) 3 SO(1, 1) 8 SL(3, R) SL(2, R) SO(3) SO(2) 7 SL(2, R) SL(2, R) 7 SL(5, R) Sp(2) 14 SL(4, R) 6 SO(5, 5) Sp(2) Sp(2) 25 SO(4, 4) 5 E 6(6) USp(8) 42 SO(5, 5) 4 E 7(7) SU(8) 70 SO(6, 6) 3 E 8(8) SO(16) 128 SO(7, 7) SUSY Domain Walls - 33 -

Embedding tensor formalism embedding tensor Θ M α T M Θ M α t α t α Lie G 0 global T M Lie G local µ D µ µ ga M µ T M SUSY Domain Walls - 34 -

Embedding tensor formalism embedding tensor Θ M α T M Θ M α t α t α Lie G 0 global T M Lie G local µ D µ µ ga M µ T M [T M, T N ] = T MN P T P, T MN P Θ M α (t α ) N P [D µ, D ν ] gf M µν T M F M µν µ A M ν ν A M µ + gt [NP ] M A N µ A P ν : 0 = f αβ γ Θ M α Θ N β + (t α ) N P Θ M α Θ P γ SUSY Domain Walls - 35 -

Embedding tensor formalism T (MN) P Θ P α = 0 [T M, T N ] = T MN P T P T (MN) P = 0 δf M µν = 2D [µ δa M ν] 2g T (P Q) M A P [µ δaq ν] δa M µ = D µ Λ M tensor gauge fields B (NP ) µν Stückelberg pairing H M µν F M µν + g T (NP ) M B (NP ) µν SUSY Domain Walls - 36 -

Embedding tensor formalism : Θ M α dim G dim G 0 ( Dµ = µ ga M µ Θ M α t α ) M G 0 SUSY Domain Walls - 37 -

Embedding tensor formalism : Θ M α dim G dim G 0 ( Dµ = µ ga M µ Θ M α t α ) M G 0 D U-duality G 0 constraints on R(M) R(α) 9 GL(2, R) (2 1) (3 1) = 1 2 2 3 4 8 SL(3, R) SL(2, R) (3, 2) [(1, 3) (8, 1)] = (3, 2) (3, 2) (3, 4) (6, 2) (15, 2) 7 SL(5, R) 10 24 = 10 15 40 175 6 SO(5, 5) 16 45 = 16 144 560 5 E 6(6) 27 78 = 27 351 1728 4 E 7(7) 56 133 = 56 912 6480 3 E 8(8) 248 248 = 1 248 3875 27000 30380 F.Riccioni, D.Steele and P.West, arxiv:0906.1177 SUSY Domain Walls - 38 -

Θ M α D (D 1)-form SUSY Domain Walls - 39 -

U-duality G 0 form fields D U-duality G 0 1-forms 2-forms 3-forms 4-forms 5-forms 6-forms 7-forms 8-forms 9-forms 10-forms IIA R + 1 1 1 1 1 1 1 1 1 1 IIB SL(2, R) 2 1 2 3 4 2 9 GL(2, R) 2 1 2 1 1 2 2 1 3 1 3 2 4 2 2 8 SL(3, R) SL(2, R) (3, 2) (3, 1) (1, 2) (3, 1) (3, 2) (8, 1) (1, 3) 7 SL(5, R) 10 5 5 10 24 40 15 6 SO(5, 5) 16 10 16 45 144 5 E 6(6) 27 27 78 351 4 E 7(7) 56 133 912 3 E 8(8) 248 3875 1 147250 3875 248 8645 133 1728 27 320 126 10 (6, 2) (3, 2) 70 45 5 (15, 1) (3, 3) (3, 1) (3, 1) (D 1)-forms Embedding Tensors F.Riccioni, D.Steele and P.West, arxiv:0906.1177 SUSY Domain Walls - 40 -

Θ α M D (D 1)-form (D 1)-form DWs Elementary SUSY DWs Θ α M SUSY Domain Walls - 41 -

Θ α M D (D 1)-form (D 1)-form DWs Elementary SUSY DWs Θ α M SUSY Domain Walls - 42 -

9 A 1, A 1,a, A 2,a, A 3 (a = 1, 2 of GL(2, R)) SUSY Domain Walls - 43 -

9 A 1, A 1,a, A 2,a, A 3 (a = 1, 2 of GL(2, R)) embedding tensors Θ a in 2, Θ ab in 3 ; with constraints Θ a Θ bc ϵ ab = 0, Θ (a Θ bc) = 0 SUSY Domain Walls - 44 -

9 A 1, A 1,a, A 2,a, A 3 (a = 1, 2 of GL(2, R)) embedding tensors Θ a in 2, Θ ab in 3 ; with constraints Stückelberg pairing Θ a Θ bc ϵ ab = 0, Θ (a Θ bc) = 0 δa 1 = dλ 0 Θ a λ 1,a δa 1,a = dλ 0,a ϵ ab Θ bc λ 1,c δa 2,a = dλ 1,a ϵ ab Θ b λ 2 δa 3 = dλ 2 F 2 = da 1 + Θ a A 2,a F 2,a = da 1,a + ϵ ab Θ bc A 2,c F 3,a = da 2,a + ϵ ab Θ b A 3 F 4 = da 3 SUSY Domain Walls - 45 -

9 A 1, A 1,a, A 2,a, A 3 (a = 1, 2 of GL(2, R)) embedding tensors Θ a in 2, Θ ab in 3 ; with constraints Stückelberg pairing Θ a Θ bc ϵ ab = 0, Θ (a Θ bc) = 0 δa 1 = dλ 0 Θ a λ 1,a δa 1,a = dλ 0,a ϵ ab Θ bc λ 1,c δa 2,a = dλ 1,a ϵ ab Θ b λ 2 δa 3 = dλ 2 F 2 = da 1 + Θ a A 2,a F 2,a = da 1,a + ϵ ab Θ bc A 2,c F 3,a = da 2,a + ϵ ab Θ b A 3 F 4 = da 3 Minimal Gauging (Θ a, Θ ab ) Gauging A 1 A 1,a=1 A 1,a=2 A 2,a=1 A 2,a=2 A 3 Θ 1 = 1, Θ 2 = 0, Θ ab = 0 eaten massless massless massive eaten massive Θ a = 0, Θ 11 = 1, Θ 22 = ±1 massive eaten eaten massive massive massless Θ a = 0, Θ 11 = 1, Θ 22 = 0 massive massless eaten massive massless massless 2 SUSY Domain Walls - 46 -

8 A 1,Ma, A M 2, A 3,a (M = 1, 2, 3 of SL(3, R), a = 1, 2 of SL(2, R)) SUSY Domain Walls - 47 -

8 A 1,Ma, A M 2, A 3,a Minimal gauging (M = 1, 2, 3 of SL(3, R), a = 1, 2 of SL(2, R)) (Θ Ma in (3, 2) EDWs ) Θ MN a = {Θ 11 1, Θ 11 2, Θ 22 1, Θ 22 2, Θ 33 1, Θ 33 2 } in (6, 2) 6 SUSY Domain Walls - 48 -

8 A 1,Ma, A M 2, A 3,a Minimal gauging (M = 1, 2, 3 of SL(3, R), a = 1, 2 of SL(2, R)) (Θ Ma in (3, 2) EDWs ) Θ MN a = {Θ 11 1, Θ 11 2, Θ 22 1, Θ 22 2, Θ 33 1, Θ 33 2 } in (6, 2) 6 Θ MN 1 Θ 1 MN = diag(1 p, 1 q, 0 r ) with p + q + r = 3 CSO(p, q, r) with f MN P = ϵ MNQ Θ P Q [T 1, T 2 ] = Θ 1 33 T 3, [T 2, T 3 ] = Θ 1 11 T 1, [T 3, T 1 ] = Θ 1 22 T 2 (Θ MN 2 = 0) Minimal Θ 22 1 = Θ 33 1 = 0 CSO(1, 0, 2) = Heisenberg algebra (i = 2, 3) Gauging A 1,11 A 1,12 A 1,i1 A 1,i2 A 1 2 A i 2 A 3,a Θ 11 1 = 1, others = 0 massless eaten massive massless massive massless massless (i = 2, 3) SUSY Domain Walls - 49 -

7 A 1,MN, A M 2 (M = 1, 2,..., 5 of SL(5, R)) SUSY Domain Walls - 50 -

7 A 1,MN, A M 2 (M = 1, 2,..., 5 of SL(5, R)) embedding tensors Θ [MN],P v [M w N]P in 40 20 w NP = diag(1 p, 1 q, 0 r ) with p + q + r = 4 minimal gauging = CSO(1, 0, 3) Gauging A 1,ij A 1,12 A 1,1i A 1,2i A 1 2 A 2 2 A i 2 Θ 12,1 = 1, others = 0 massive eaten massless massless massive massless massless (i = 3, 4, 5) embedding tensors Θ (MN) in 15 5 Θ MN = diag(1 p, 1 q, 0 r ) with p + q + r = 5 minimal gauging = CSO(1, 0, 4) Gauging A 1,1i A 1,ij A 1 2 A i 2 A 3,1 Θ 11 = 1, others = 0 massive massless eaten massless massive (i = 2, 3, 4, 5) SUSY Domain Walls - 51 -

minimal gauging elementary SUSY Domain Walls non-minimal gauging (non)-threshold bound states of EDWs SUSY Domain Walls - 52 -

D Domain Walls (DWs) 1 2-SUSY DWs (EDWs) U-duality non-edws (EDWs ) Central charges ( ) EDWs minimal gauging Non-EDWs non-minimal gauging SUSY Domain Walls - 54 -

Elementary SUSY DWs D U T # of EDWs α = 0 α = 1 α = 2 α = 3 α = 4 α = 5 IIA R + 1 1 1 9 GL(2, R) SO(1, 1) 2 3 U 1 1 8 SL(3, R) SL(2, R) SL(2, R) SL(2, R) 6 (6, 2) U (1, 2) T 4 (3, 2) T 7 SL(5, R) SL(4, R) 20 40 U 4 T 4 10 T 12 20 T 5 15 U 4 10 T 1 T 6 SO(5, 5) SO(4, 4) 80 144 U 8 S T 32 56 C T 32 56 S T 8 C T 5 E 6(6) SO(5, 5) 216 351 U 16 T 80 120 T 80 144 T 40 45 T 4 E 7(7) SO(6, 6) 576 912 U 32 T 160 220 T 192 352 T 160 220 T 32 T 3 E 8(8) SO(7, 7) 2160 3875 U 1 T 64 T 280 364 T 448 832 T 560 1001 T 14 104 T 448 832 T (α 6) 280 364 T, 6 64 T, 7 1 T, 8 D R- H Z (1) Z (2) # of EDWs 9 SO(2) 1 2 2 8 SO(3) SO(2) (1, 2) 6 3 7 Sp(2) 5 + 1 20 + 5 4 (V), 5 (T) 6 Sp(2) Sp(2) (4, 4) 80 5 5 U Sp(8) 36 216 6 4 SU(8) 36 + + 36 576 8 3 SO(16) 135 2160 16 SUSY Domain Walls - 55 -

Embedding tensor D 32-SUSY 16-SUSY 8-SUSY 9 arxiv:1105.1760 (unknown) 8 arxiv:1203.6562 (unknown) 7 hep-th/0506237 (unknown) 6 arxiv:0712.4277 (unknown) arxiv:1012.1818 5 hep-th/0412173 hep-th/0702084 (unknown) 4 arxiv:0705.2101 hep-th/0602024 arxiv:1107.3305 3 hep-th/0103032 arxiv:0806.2584 arxiv:0807.2841 SUSY Domain Walls - 56 -

Defect branes

U-duality G 0 form fields D U-duality G 0 1-forms 2-forms 3-forms 4-forms 5-forms 6-forms 7-forms 8-forms 9-forms 10-forms IIA R + 1 1 1 1 1 1 1 1 1 1 IIB SL(2, R) 2 1 2 3 4 2 9 GL(2, R) 2 1 2 1 1 2 2 1 3 1 3 2 4 2 2 8 SL(3, R) SL(2, R) (3, 2) (3, 1) (1, 2) (3, 1) (3, 2) (8, 1) (1, 3) 7 SL(5, R) 10 5 5 10 24 40 15 6 SO(5, 5) 16 10 16 45 144 5 E 6(6) 27 27 78 351 4 E 7(7) 56 133 912 3 E 8(8) 248 3875 1 147250 3875 248 8645 133 1728 27 320 126 10 (6, 2) (3, 2) 70 45 5 (15, 1) (3, 3) (3, 1) (3, 1) (D 2)-forms U-duality group G 0 F.Riccioni, D.Steele and P.West, arxiv:0906.1177 SUSY Domain Walls - 59 -

Wess-Zumino terms for 7-branes in 10D L WZ i A 8,i + F 2 Γ i A 6 +... A 8,i : 8-forms in bulk Γ i : SO(2, 1) SL(2, R) (i = +,, 3) F 2 = (db 1, S(db 1 )) : curvatures of DBI vector and its S-dual / spinor repr. of SO(2, 1) A 6 = (B (6), C (6) ) : 6-forms in bulk / spinor repr. of SO(2, 1) i = + i = 7-brane S-dual project-out i = 3 SUSY Domain Walls - 60 -

Wess-Zumino terms for defect branes in D-dim. Defect branes (D 2)-form potentials scalar fields U-duality group G 0 T-duality group ( R + ) d = 10 D fundamental Dirichlet solitonic U T α = 0 α = 1 α = 2 α = 3 α = 4 D 5 E d+1(d+1) SO(d, d) Adj U spinor T (Adj + singlet) T conj. spinor T D = 4 E 7(7) SO(6, 6) Adj U singlet T spinor T (Adj + singlet) T conj. spinor T singlet T D = 3 E 8(8) SO(7, 7) Adj U vector T spinor T (Adj + singlet) T conj. spinor T vector T α = α 4 by D-dim. S-duality (g µν) S ((g s ) α d D 2 x [NG(g µν )] = (g s ) α = e 8ϕ/(D 2) (g µν ) S ) d D 2 x [NG(g µν)] E.A. Bergshoeff et al, arxiv:1009.4657, arxiv:1102.0934, arxiv:1108.5067 SUSY Domain Walls - 61 -

Defect branes (co-dim. 2) solitonic defect brane (α = 2) supersymmetric fundamental Dirichlet solitonic (brane s tension) (g s ) +α D # of SUSY defect branes α = 0 α = 1 α = 2 α = 3 α = 4 IIB 2 3 1 1 9 2 3 3 1 1 8 6 (8, 1) (2, 1) 2 (3, 1) (2, 1) 2 (1, 3) 2 (1, 3) 7 20 24 4 12 15 4 6 40 45 8 V 24 28 8 V 5 72 78 16 40 45 16 4 126 133 1 32 60 66 32 1 3 240 248 14 64 84 91 64 14 E.A. Bergshoeff et al, arxiv:1009.4657, arxiv:1102.0934, arxiv:1108.5067 SUSY Domain Walls - 62 -

String theory origin of defect branes in D-dim. fundamental Dirichlet solitonic S D -dual of (Dirichlet) S D -dual of (fundamental) D α = 0 α = 1 α = 2 α = 3 α = 4 IIB C 8 [D7] E 8 = S 10 (C 8 ) [7 3 ] 9 C 7 [D6] E 8,1 = S 9 (C 7 ) [6 1 3] 8 C 6 [D5] D 6 [NS5] D 7,1 [KK5 = 5 1 2] D 8,2 [5 2 2] E 8,2 = S 8 (C 7 ) [5 2 3] 7 C 5 [D4] E 8,3 = S 7 (C 5 ) [4 3 3] 6 C 4 [D3] E 8,4 = S 6 (C 4 ) [3 4 3] 5 C 3 [D2] E 8,5 = S 5 (C 3 ) [2 5 3] 4 B 2 [F1] C 2 [D1] E 8,6 = S 4 (C 2 ) [1 6 3] F 8,6 = S 4 (B 2 ) [1 6 4] 3 [P] C 1 [D0] E 8,7 = S 3 (C 1 ) [0 7 3] F 8,7,1 [0 (6,1) 4 ] p (I 1,I 2 ) α -brane A D T,I1 +I 2,I 2 (T, p, I 1, I 2 ) α with T + p + i I i = D 1 Mass (T,p,I1,I 2 ) α = R 1 R p (R p+1 R p+i1 ) 2 (R p+i1 +1 R p+i1 +I 2 ) 3 (g s ) α SUSY Domain Walls - 63 -

Defect branes (co-dim. 2) D G 0 /H n P n D n S IIB SL(2, R)/SO(2) 3 2 2 9 SL(2, R)/SO(2) R + 3 + 1 2 + 0 2 + 1 8 SL(3, R)/SO(3) SL(2, R)/SO(2) 8 + 3 6 + 2 5 + 2 7 SL(5, R)/SO(5) 24 20 14 6 SO(5, 5)/[SO(5) SO(5)] 45 40 24 5 E 6(6) /Sp(8) 78 72 42 4 E 7(7) /SU(8) 133 126 70 3 E 8(8) /SO(16) 248 240 128 n P = dim G 0 : # of (D 2)-form potentials n D = dim G 0 rank G 0 : # of SUSY defect branes (rank G 0 = rank T + 1) n S = dim G 0 dim H : # of coset scalars in D-dim. maximal SUGRA SUSY Domain Walls - 64 -

Defect branes (co-dim. 2) Z (a) : a-form central charge D R- H Z (0) Z (1) Z (2) Z (3) n D IIB SO(2) 1 2 2 9 SO(2) 1 2 2 8 SO(3) SO(2) 3 + 1 6 + 2 2 7 Sp(2) 10 20 2 6 Sp(2) Sp(2) (10, 1) + + (1, 10) 40 2 5 U Sp(8) 36 72 2 4 SU(8) 63 126 2 3 SO(16) 120 240 2 SUSY Domain Walls - 65 -

CSO(p, q, r) CSO(p, q, r) jump CSO(p, q, 0) = SO(p, q) CSO(p, q, 1) = ISO(p, q) CSO(p, q, r) SO(p, q) U(1) r(r 1) 2 for r 2 C.M. Hull, PL 142B (1984) 39, PL 148B (1984) 297, NPB 253 (1985) 650 L. Andrianopoli et al, hep-th/0009048, etc. SUSY Domain Walls - 66 -

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