Twisted N = 4 Super Yang-Mills Theory in Background Katsushi Ito - PowerPoint PPT Presentation

Twisted N = 4 Super Yang-Mills Theory in Ω Background Katsushi Ito Tokyo Institute of Technology October 24, 2013@Todai/Riken Joint Workshop K.I., H. Nakajima, and S. Sasaki, arXiv:1209.2561, 1307.7565 . . . . . . Katsushi Ito (TokyoTech) omega background Hongo 1 / 33

. 1 Introduction . . 2 N = 2 Super Yang-Mills Theory in Ω -background . . 3 N = 4 super Yang-Mills theory in Ω -background . . 4 Off-Shell SUSY in Ω -background . . 5 Outlook . . . . . . Katsushi Ito (TokyoTech) omega background Hongo 2 / 33

Introduction: Ω -background x 5 compactified on S 1 5-dim flat metric: ˜ 4 ∑ ds 2 = x 2 x 2 u ∼ ˜ u + 2 πk ( k ∈ Z ) d ˜ i + d ˜ 5 , x 5 = R ˜ ˜ u, ˜ i =1 x 2 = ρ 1 e iθ 1 , ˜ x 4 = ρ 2 e iθ 2 cylindrical coordinates ˜ x 1 + i ˜ x 3 + i ˜ ds 2 = dρ 2 1 + ρ 2 1 dθ 2 1 + dρ 2 2 + ρ 2 2 dθ 2 2 + R 2 d ˜ u 2 identifications (twisted boundary condition) u ∼ ˜ ˜ u + 2 πk, θ 1 ∼ θ 1 − 2 πϵRk, θ 2 ∼ θ 2 + 2 πϵRk introduce 2 π -periodic coordinates u, ˜ ϕ 1 = θ 1 + ϵR ˜ u, ϕ 2 = θ 2 − ϵR ˜ u metric (Melvin background) ds 2 = dρ 2 u ) 2 + dρ 2 u ) 2 + R 2 d ˜ 1 + ρ 2 2 + ρ 2 u 2 1 ( dϕ 1 − ϵRd ˜ 2 ( dϕ 2 + ϵRd ˜ . . . . . . Katsushi Ito (TokyoTech) omega background Hongo 3 / 33

new Cartesian coordinates x 1 + ix 2 = ρ 1 e iϕ 1 , x 3 + ix 4 = ρ 2 e iϕ 2 ds 2 = x 5 ) 2 + ( dx 2 + ϵx 1 d ˜ x 5 ) 2 ( dx 1 − ϵx 2 d ˜ x 5 ) 2 + ( dx 4 − ϵx 3 d ˜ x 5 ) 2 + ( d ˜ x 5 ) 2 +( dx 3 + ϵx 4 d ˜   0 − ϵ 0 0   ϵ 0 0 0 = ( dx m + Ω mn x n d ˜ x 5 ) 2 + ( d ˜ ds 2 x 5 ) 2   Ω mn =   0 0 0 ϵ 0 0 − ϵ 0 self-dual: Ω mn = − 1 2 ϵ mnpq Ω pq x 1 = ρ 1 cos( ϕ 1 ) = ρ 1 cos( θ 1 + ϵR ˜ u ) x 2 = ρ 1 sin( ϕ 1 ) = ρ 1 sin( θ 1 + ϵR ˜ u ) rotating frame: angular velocities varies along ˜ u -axis U (1) vector field V = − ϵ ( x 1 ∂ 2 − x 2 ∂ 1 )+ ϵ ( x 2 ∂ 4 − x 4 ∂ 3 ) = Ω mn x n ∂ m . . . . . . Katsushi Ito (TokyoTech) omega background Hongo 4 / 33

D = 4 + d -dimensional Omega-background metric:   − ϵ a 0 0 0 1 D ∑   ϵ a 0 0 0 ds 2 = ( dx m + Ω mn a x n dx a ) 2 +   dx 2 Ω a 1 a , mn =   ϵ a 0 0 0 2 a =5 − ϵ a 0 0 0 2 commuting U (1) d -vector fields V a : [ V a , V b ] = 0 V a = − ϵ a 1 ( x 1 ∂ 2 − x 2 ∂ 1 ) + ϵ a 2 ( x 3 ∂ 4 − x 4 ∂ 3 ) Supersymmetric gauge theories in D -dimensional Ω -background dimensional reduction to 4 dimensions D = 6 → N = 2 super Yang-Mills theory D = 10 → N = 4 super Yang-Mills theory . . . . . . Katsushi Ito (TokyoTech) omega background Hongo 5 / 33

N = 2 Super Yang-Mills Theory in Ω -background 6-dim metric ( x m , x 5 , x 6 ) m = 1 , 2 , 3 , 4 z + ( dx m + ¯ ds 2 z ) 2 , Ω m dz + Ω m d ¯ 6 = 2 dzd ¯ 2 ( x 5 − ix 6 ) , ¯ 2 ( x 5 + ix 6 ) 1 1 z = z = √ √ Ω m ≡ Ω mn x n , ¯ Ω m ≡ ¯ Ω mn x n : commuting U (1) 2 vector fields Ω mn = − Ω nm , ¯ Ω mn = − ¯ Ω nm     0 0 0 0 − i ¯ 0 0 iϵ 1 ϵ 1 1 − iϵ 1 0 0 0 1 i ¯ 0 0 0 ϵ 1 ¯     Ω mn = √  , Ω mn = √     0 0 0 − iϵ 2 0 0 0 i ¯ ϵ 2 2 2 2 2    0 0 0 0 0 − i ¯ 0 iϵ 2 ϵ 2 . . . . . . Katsushi Ito (TokyoTech) omega background Hongo 6 / 33

N = 2 SYM in Ω -background Dimensional reduction of 6d N = 1 SYM = ⇒ 4d N = 2 SYM add SU (2) I R-symmetry Wilson line gauge field A IJ ¯ A IJ D Λ J = [ φ, Λ I ] → [ φ, Λ I ] + A IJ Λ J = ⇒ mass term for fermions [ 1 1 4 F mn F mn + ( D m φ − F mn Ω n )( D m ¯ φ − F mp ¯ L (Ω , A ) = g 2 κ Tr Ω p ) i i + Λ I σ m D m ¯ Λ I [ φ, ¯ ¯ 2Λ I [ ¯ Λ I ] √ √ Λ I − φ, Λ I ] + 2 + 1 1 ¯ ¯ Ω m Λ I D m Λ I − Ω mn Λ I σ mn Λ I √ √ 2 2 2 − 1 1 Λ I + 2Ω m ¯ Λ I D m ¯ 2Ω mn ¯ σ mn ¯ Λ I √ √ Λ I ¯ 2 ( ) 2 + 1 φ − i ¯ Ω m D m φ + i ¯ φ ]+ i Ω m D m ¯ Ω m Ω n F mn [ φ, ¯ 2 ] − 1 I Λ I Λ J − 1 Λ I ¯ ¯ I ¯ A J 2 A J √ √ Λ J , 2 K.I., H. Nakajima, S. Saka and S. Sasaki, 1009.1212 . . . . . . Katsushi Ito (TokyoTech) omega background Hongo 7 / 33

Deformed SUSY α , ¯ Supersymmetry Q I Q I α : SU (2) L × SU (2) R × SU (2) I ( α , ˙ α , I ) ˙ topological twist[Witten]: SU (2) L × ( SU (2) I × SU (2) R ) diag ¯ I ¯ ¯ αI ¯ σ m ) Iα Q αI , Q = δ ˙ α Q I σ mn ) ˙ Q I Q m = (¯ Q mn = − (¯ α , ˙ α ˙ self-dual ϵ 1 = − ϵ 2 : 4 anti-chiral SUSY ¯ Q I α (ADS Q α I ) ˙ non self-dual case (generic) ϵ 1 ̸ = − ϵ 2 : no SUSY ▶ R-symmetry Wilson line gauge fields satisfy ¯ 2 ¯ A IJ = − 1 A IJ = − 1 σ mn ) IJ , σ mn ) IJ 2 Ω mn (¯ Ω mn (¯ anti-self-dual part of Ω mn =WL gauge field ⇒ scalar supercharge ¯ = Q is preserved ▶ ϵ 1 = 0 (Nekrasov-Shatashvili limit): N = (2 , 2) super-Poincar´ e . . . . . . Katsushi Ito (TokyoTech) omega background Hongo 8 / 33

N = 2 Ω -background and instantons ADHM construction of instanton in Ω -deformed N = 2 super Yang-Mills theory [INSS, 1009.1212] D(-1)-D3 system: Ω -background+WL gauge fields ↔ R-R 3-form field strength background [Bill´ o-Frau-Fucito-Lerda, INSS] equivariant BRST charge ¯ Q Ω Ω = 0 up to U (1) 2 rotations and gauge transformation ¯ Q 2 eff ( M, ϵ ) = ¯ instanton effective action S N =2 Q Ω Ξ = ⇒ Instanton partition function [Nekrasov] Nekrasov-Shatashvili limit: ϵ 1 → 0 BPS-monopole solution [KI-Kamoshita-Sasaki] . . . . . . Katsushi Ito (TokyoTech) omega background Hongo 9 / 33

N = 4 SYM in Ω -background 10 dim metric x M = ( x m , x 4+ a ) ( m = 1 , 2 , 3 , 4 ; a = 1 , · · · , 6 ) ( a d x a +4 ) 2 , + ( d x a +4 ) 2 d s 2 = d x m + Ω m Ω m a ≡ Ω mn a x n , Ω mn = − Ω nm a a The U (1) 6 vector fields V a = Ω ma ∂ m commute each other. Ω mpa Ω pnb − Ω mpb Ω pna = 0   0 0 0 ϵ 1 a   − ϵ 1 a 0 0 0   Ω mna =  ,  0 0 0 − ϵ 2 a 0 0 ϵ 2 a 0 cf. T-dual description [Hellerman-Orlando-Reffert] . . . . . . Katsushi Ito (TokyoTech) omega background Hongo 10 / 33

10d N=1 SYM → 4d N=4 SYM ([Brink-Schwarz-Scherk, 1977]) gauge fields A M = ( A m , φ a ) ( a = 1 , · · · , 6 ) Majorana-Weyl spinor Ψ → (Λ A , ¯ Λ A ) ( A = 1 , 2 , 3 , 4 ) [ 1 4 F mn F mn + iθg 2 L Ω = 1 32 π 2 F mn � F mn + Λ A σ m D m ¯ κ Tr Λ A ( ) 2 + 1 D m φ a − gF mn Ω n a 2 − g Λ B ] − g 2(Σ a ) AB ¯ Λ A [ φ a , ¯ 2(¯ Σ a ) AB Λ A [ φ a , Λ B ] ( ) 2 − g 2 [ φ a , φ b ] + i Ω m a D m φ b − i Ω m b D m φ a − igF mn Ω m a Ω n b 4 ( Σ a ) AB Λ A D m Λ B ) − ig (Σ a ) AB ¯ 2 Ω m Λ A D m ¯ Λ B + (¯ a Σ a ) AB Λ A σ mn Λ B )] ( + ig (Σ a ) AB ¯ σ mn ¯ Λ B + (¯ 4 Ω mna Λ A ¯ , ¯ Σ aAB : SO(6) sigma matrices Σ a ¯ Σ b + Σ b ¯ Σ AB Σ a = 2 δ ab a . . . . . . Katsushi Ito (TokyoTech) omega background Hongo 11 / 33

SU (4) I Wilson Line gauge fields α , ¯ N = 4 SUSY: Q A Q ˙ αA Ω mna : self-dual → anti-chiral SUSY ¯ Q ˙ αA [I-Nakajima-Saka-Sasaki] Ω mna : non self-dual → No SUSY We add the constant SU (4) I R-symmetry Wilson-line gauge field ( A a ) AB to recover ( a part of ) SUSY spinor fields ((anti-)fundamental rep. of SU (4) I ) [ φ a , Λ A ] [ φ a , Λ A ] D a +4 Λ A = + ( A a ) AB Λ B , → [ ] [ ] D a +4 ¯ φ a , ¯ φ a , ¯ − ¯ Λ B ( A a ) BA . → Λ A = Λ A Λ A scalar fields(antisymmetric rep of SU (4) R ) [ ] [ ] ( ) − 1 (Σ b ¯ Σ c ) AB φ c ( A a ) BA − (Σ a ¯ Σ c ) AB φ c ( A b ) BA φ a , φ b → φ a , φ b . 2 . . . . . . Katsushi Ito (TokyoTech) omega background Hongo 12 / 33

[K.I., H. Nakajima, S. Saka and S. Sasaki, arXiv:1111.6709] [ 1 4 F mn F mn + iθg 2 ( ) 2 F mn + 1 32 π 2 F mn � D m φ a − gF mn Ω n L (Ω , A ) = Tr a 2 Λ A − g Λ B ] − g 2(Σ a ) AB ¯ + Λ A σ m D m ¯ Λ A [ φ a , ¯ 2(¯ Σ a ) AB Λ A [ φ a , Λ B ] ( − g 2 [ φ a , φ b ] + i Ω m a D m φ b − i Ω m b D m φ a − igF mn Ω m a Ω n b 4 )) 2 ( − 1 (Σ b ¯ A − (Σ a ¯ Σ c ) A B φ c ( A a ) B Σ c ) A B φ c ( A b ) B A 2 ( Σ a ) AB Λ A D m Λ B ) − ig (Σ a ) AB ¯ Λ A D m ¯ Λ B + (¯ 2 Ω m a ( Σ a ) AB Λ A σ mn Λ B ) + ig (Σ a ) AB ¯ σ mn ¯ Λ B + (¯ 4 Ω mna Λ A ¯ D Λ D ] + g B − g 2(Σ a ) AB ¯ Λ A ¯ 2(¯ Λ D ( A a ) D Σ a ) AB Λ A ( A a ) B . . . . . . Katsushi Ito (TokyoTech) omega background Hongo 13 / 33

Torsion and SUSY in Ω -background [K.I., H. Nakajima, and S. Sasaki, arXiv:1209.2561] Construction of Deformed supersymmetry in 4 dimensions: very complicated, classification? in 10 dimensions: relatively easy geometrical origin of R-symmetry WL gauge fields? setup: general curved background + torsion dimensional reduction to 4 dimensions ⇒ constraints SUSY conditions+gauge invariance in 4 dimensions = on the backgrounds Ω -background case, solution to the constraints . . . . . . Katsushi Ito (TokyoTech) omega background Hongo 14 / 33

N = 1 SYM in 10 dimensions: flat spacetime A M : gauge fields ( M = 0 , 1 , · · · , 9 ) Ψ : Majorana-Weyl spinor 16-components Lagrangian in flat spacetime [ ] L = 1 − 1 4 F MN F MN − i ΨΓ M D M Ψ ¯ g 2 Tr 2 F MN = ∂ M A N − ∂ N A M + i [ A M , A N ] , D M Ψ = ∂ M Ψ + i [ A M , Ψ] Γ M : gamma matrices, { Γ M , Γ N } = 2 η MN . . . . . . Katsushi Ito (TokyoTech) omega background Hongo 15 / 33