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

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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

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1 Introduction

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2 N = 2 Super Yang-Mills Theory in Ω-background

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3 N = 4 super Yang-Mills theory in Ω-background

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4 Off-Shell SUSY in Ω-background

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5 Outlook

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Introduction: Ω-background

5-dim flat metric: ˜ x5 compactified on S1 ds2 =

4

∑

i=1

d˜ x2

i + d˜

x2

5,

˜ x5 = R˜ u, ˜ u ∼ ˜ u + 2πk (k ∈ Z) cylindrical coordinates ˜ x1 + i˜ x2 = ρ1eiθ1, ˜ x3 + i˜ x4 = ρ2eiθ2 ds2 = dρ2

1 + ρ2 1dθ2 1 + dρ2 2 + ρ2 2dθ2 2 + R2d˜

u2

identifications (twisted boundary condition) ˜ u ∼ ˜ u + 2πk, θ1 ∼ θ1 − 2πϵRk, θ2 ∼ θ2 + 2πϵRk introduce 2π-periodic coordinates ˜ u, ϕ1 = θ1 + ϵR˜ u, ϕ2 = θ2 − ϵR˜ u

metric (Melvin background) ds2 = dρ2

1 + ρ2 1(dϕ1 − ϵRd˜

u)2 + dρ2

2 + ρ2 2(dϕ2 + ϵRd˜

u)2 + R2d˜ u2

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new Cartesian coordinates x1 + ix2 = ρ1eiϕ1, x3 + ix4 = ρ2eiϕ2 ds2 = (dx1 − ϵx2d˜ x5)2 + (dx2 + ϵx1d˜ x5)2 +(dx3 + ϵx4d˜ x5)2 + (dx4 − ϵx3d˜ x5)2 + (d˜ x5)2 ds2 = (dxm + Ωmnxnd˜ x5)2 + (d˜ x5)2 Ωmn =     −ϵ ϵ ϵ −ϵ     self-dual: Ωmn = − 1

2ϵmnpqΩpq

x1 = ρ1 cos(ϕ1) = ρ1 cos(θ1 + ϵR˜ u) x2 = ρ1 sin(ϕ1) = ρ1 sin(θ1 + ϵR˜ u) rotating frame: angular velocities varies along ˜ u-axis U(1) vector field V = −ϵ(x1∂2 −x2∂1)+ϵ(x2∂4 −x4∂3) = Ωmnxn∂m

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D = 4 + d-dimensional Omega-background

metric: ds2 = (dxm + Ωmn

a xndxa)2 + D

∑

a=5

dx2

a,

Ωa

mn =

    −ϵa

1

ϵa

1

ϵa

2

−ϵa

2

    commuting U(1)d-vector fields V a: [V a, V b] = 0 V a = −ϵa

1(x1∂2 − x2∂1) + ϵa 2(x3∂4 − x4∂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

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N = 2 Super Yang-Mills Theory in Ω-background

6-dim metric (xm, x5, x6) m = 1, 2, 3, 4 ds2

6 = 2dzd¯

z + (dxm + ¯ Ωmdz + Ωmd¯ z)2, z =

1 √ 2(x5 − ix6), ¯

z =

1 √ 2(x5 + ix6)

Ωm ≡ Ωmnxn, ¯ Ωm ≡ ¯ Ωmnxn : commuting U(1)2 vector fields Ωmn = −Ωnm, ¯ Ωmn = −¯ Ωnm

Ωmn = 1 2 √ 2     iϵ1 −iϵ1 −iϵ2 iϵ2     , ¯ Ωmn = 1 2 √ 2     −i¯ ϵ1 i¯ ϵ1 i¯ ϵ2 −i¯ ϵ2    

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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 AIJ ¯ AIJ DΛJ = [φ, ΛI] → [φ, ΛI] + AIJΛJ = ⇒ mass term for fermions

L(Ω,A) = 1 g2κTr [1 4FmnF mn + (Dmφ−FmnΩn)(Dm ¯ φ−F mp ¯ Ωp) + ΛIσmDm¯ ΛI − i √ 2ΛI[ ¯ φ, ΛI] + i √ 2 ¯ ΛI[φ, ¯ ΛI] + 1 √ 2 ¯ ΩmΛIDmΛI − 1 2 √ 2 ¯ ΩmnΛIσmnΛI − 1 √ 2Ωm¯ ΛIDm¯ ΛI + 1 2 √ 2Ωmn¯ ΛI ¯ σmn¯ ΛI + 1 2 ( [φ, ¯ φ]+iΩmDm ¯ φ − i¯ ΩmDmφ + i¯ ΩmΩnFmn )2 − 1 √ 2 ¯ AJ

IΛIΛJ − 1

√ 2AJ

I ¯

ΛI ¯ ΛJ ] , K.I., H. Nakajima, S. Saka and S. Sasaki, 1009.1212

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Deformed SUSY

Supersymmetry QI

α, ¯

QI

˙ 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 Qm = (¯ σm)IαQαI, ¯ Q = δ ˙

α I ¯

QI

˙ α,

¯ Qmn = −(¯ σmn) ˙

αI ¯

QI

˙ α

self-dual ϵ1 = −ϵ2: 4 anti-chiral SUSY ¯ QI

˙ α (ADS Qα I )

non self-dual case (generic) ϵ1 ̸= −ϵ2: no SUSY

▶ R-symmetry Wilson line gauge fields satisfy

AIJ = − 1

2Ωmn(¯

σmn)IJ, ¯ AIJ = − 1

2 ¯

Ωmn(¯ σmn)IJ anti-self-dual part of Ωmn=WL gauge field = ⇒ scalar supercharge ¯ Q is preserved

▶ ϵ1 = 0 (Nekrasov-Shatashvili limit): N = (2, 2)

super-Poincar´ e

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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´

• -Frau-Fucito-Lerda, INSS]

equivariant BRST charge ¯ QΩ ¯ Q2

Ω = 0 up to U(1)2 rotations and gauge transformation

instanton effective action SN=2

eff (M, ϵ) = ¯

QΩΞ = ⇒ Instanton partition function [Nekrasov] Nekrasov-Shatashvili limit: ϵ1 → 0 BPS-monopole solution [KI-Kamoshita-Sasaki]

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N = 4 SYM in Ω-background

10 dim metric xM = (xm, x4+a) (m = 1, 2, 3, 4; a = 1, · · · , 6) ds2 = ( dxm + Ωm

a dxa+4)2, +

( dxa+4)2 Ωm

a ≡ Ωmn a xn, Ωmn a

= −Ωnm

a

The U(1)6 vector fields Va = Ωma∂m commute each other. ΩmpaΩpnb − ΩmpbΩpna = 0 Ωmna =     ϵ1a −ϵ1a −ϵ2a ϵ2a     ,

• cf. T-dual description [Hellerman-Orlando-Reffert]

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10d N=1 SYM → 4d N=4 SYM ([Brink-Schwarz-Scherk, 1977]) gauge fields AM = (Am, φa) (a = 1, · · · , 6) Majorana-Weyl spinor Ψ → (ΛA, ¯ ΛA) (A = 1, 2, 3, 4)

LΩ = 1 κTr [ 1 4F mnFmn + iθg2 32π2 F mn Fmn + ΛAσmDm¯ ΛA + 1 2 ( Dmφa − gFmnΩn

a

)2 − g 2(Σa)AB ¯ ΛA[φa, ¯ ΛB] − g 2(¯ Σa)ABΛA[φa, ΛB] − g2 4 ( [φa, φb] + iΩm

a Dmφb − iΩm b Dmφa − igFmnΩm a Ωn b

)2 −ig 2 Ωm

a

( (Σa)AB ¯ ΛADm¯ ΛB + (¯ Σa)ABΛADmΛB) +ig 4 Ωmna ( (Σa)AB ¯ ΛA¯ σmn¯ ΛB + (¯ Σa)ABΛAσmnΛB)] ΣAB

a

, ¯ ΣaAB: SO(6) sigma matrices Σa ¯ Σb + Σb ¯ Σa = 2δab

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SU(4)I Wilson Line gauge fields

N = 4 SUSY: QA

α, ¯

Q ˙

αA

Ωmna: self-dual → anti-chiral SUSY ¯ Q ˙

αA [I-Nakajima-Saka-Sasaki]

Ωmna: non self-dual→ No SUSY We add the constant SU(4)I R-symmetry Wilson-line gauge field (Aa)AB to recover ( a part of ) SUSY spinor fields ((anti-)fundamental rep. of SU(4)I) Da+4ΛA = [ φa, ΛA] → [ φa, ΛA] + (Aa)ABΛB, Da+4¯ ΛA = [ φa, ¯ ΛA ] → [ φa, ¯ ΛA ] − ¯ ΛB(Aa)BA. scalar fields(antisymmetric rep of SU(4)R) [ φa, φb ] → [ φa, φb ] − 1 2 ( (Σb ¯ Σc)ABφc(Aa)BA − (Σa ¯ Σc)ABφc(Ab)BA ) .

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[K.I., H. Nakajima, S. Saka and S. Sasaki, arXiv:1111.6709]

L(Ω,A) = Tr [ 1 4F mnFmn + iθg2 32π2 F mn Fmn + 1 2 ( Dmφa−gFmnΩn

a

)2 + ΛAσmDm¯ ΛA − g 2(Σa)AB ¯ ΛA[φa, ¯ ΛB] − g 2(¯ Σa)ABΛA[φa, ΛB] − g2 4 ( [φa, φb] + iΩm

a Dmφb − iΩm b Dmφa − igFmnΩm a Ωn b

−1 2 ( (Σb ¯ Σc)A

Bφc(Aa)B A − (Σa ¯

Σc)A

Bφc(Ab)B A

))2 −ig 2 Ωm

a

( (Σa)AB ¯ ΛADm¯ ΛB + (¯ Σa)ABΛADmΛB) +ig 4 Ωmna ( (Σa)AB ¯ ΛA¯ σmn¯ ΛB + (¯ Σa)ABΛAσmnΛB) +g 2(Σa)AB ¯ ΛA¯ ΛD(Aa)D

B − g

2(¯ Σa)ABΛA(Aa)B

DΛD]

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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 SUSY conditions+gauge invariance in 4 dimensions = ⇒ constraints

• n the backgrounds

Ω-background case, solution to the constraints

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N = 1 SYM in 10 dimensions: flat spacetime

AM: gauge fields (M = 0, 1, · · · , 9) Ψ: Majorana-Weyl spinor 16-components Lagrangian in flat spacetime L = 1 g2 Tr [ −1 4FMNF MN − i 2 ¯ ΨΓMDMΨ ] FMN = ∂MAN − ∂NAM + i[AM, AN], DMΨ = ∂MΨ + i[AM, Ψ] ΓM: gamma matrices, {ΓM, ΓN} = 2ηMN

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N = 1 SUSY (on-shell) δAM = i¯ ξΓMΨ, δΨ = −1 2FMNΓ[MΓN]ξ

g2δL = Tr [ −FMNDMδAN + i 2 ¯ ΨΓM[iδAM, Ψ]+ i 2δ ¯ ΨDMδΨ + i 2 ¯ ΨDMδΨ ] = Tr [ ∂M ( i 2FMN ¯ ΨΓNξ − i 4FP Q ¯ ξΓP QMΨ ) + i 4 ¯ ΨΓP QMξDMFP Q + i 2 ¯ ΨΓM[¯ ξΓMΨ, Ψ] ]

red part: total derivative+Bianchi idenitity blue part: Fierz identity action: δS = ∫ d10xδL = 0

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10 dim SYM in curved background with torsion

eN

M: vielbein (M: curved, M: tangent)

• ωM,NP : spin connection with torsion

metric: GMN = eM

MeN N ηMN

torsion (Cartan’s 1st structure equation): T P

MN = ∂MeP N − ∂N eP M +

ωM, P QeQN − ωN , P QeQ

M

Ricci rotation coefficients: CMN P = ∂MeP

N − ∂N eP M

contorsion KM,NP = − 1

2

( TMN,P − TNP,M + TPM,N ) , spin connection=(vielbein part)+controsion

• ωM,NP = ωM,NP + KM,NP ,

ωM,NP = 1 2 ( CMN,P − CNP,M + CPM,N ) , covariant derivative:

• ∇MΨ = (∂M + 1

2 ωM,NP ΓNP )Ψ

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covariant derivative:

• ∇MAN = ∂MAN −

ΓMN PAP affine connection

• ΓMN P = ΓMN P + KM,PN ,

ΓMN P: Christoffel connection (symmetric) field strength:

• FMN =

∇MAN − ∇N AM + i[AM, AN ] = FMN − TMN PAP. Lagrangian

• L =

1 κg2 Tr [ −1 4e ( eM

M eN N

FMN )2 − i 2e ¯ ΨeM

M ΓM

∇(G)

M Ψ

] ,

• ∇(G)

M Ψ =

∇MΨ + i[AM, Ψ]

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Explict dependence on AM: 10 dimensional gauge symmetry T P

MN = 0

N = 1 SUSY δAM = ieM

M ¯

ζ ΓMΨ, δΨ = −1 2eM

M eN N

FMN Γ[MΓN]ζ. SUSY conditions: Bianchi identity

• ∇(G)

[M

FN P] = −(∂QA[M)TN P]

Q − (∂[MTN P] R + T[MN QTP]Q R)AR = 0

parallel spinor condition: → Ricci flatness

• ∇Mζ = 0 → ∇Mζ = 0

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dimensional reduction to 4 dimensions

xM = (xµ, xA) vielbein, torsion etc. are independent of xA gauge field AM = (Aµ, φA) MW fermion Ψ = (ΛA

α, ¯

Λ ˙

αA)

Γµ = −i ( 0 σµ ¯ σµ ) ⊗ 18, Γa = γ5 ⊗ ( 0 Σa ¯ Σa ) 4 dimensional gauge invariance T µ

MN = 0

4 dimensional SUSY Bianchi identity: ∂[MTNP]R + T[MN QTP]QR = 0 total derivativeness of δL: TMN M = 0 e ∇MV M = ∂M(eV M) − eTMN N V M modified parallel spinor condition ˆ ∇Mζ = 0

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10 dim Ω-background

ds2 = (dxm + Ωm

a dxa)2 + (dxa)2,

Ωm

a = Ωmn a xn

contorison ⇐ = R-symmetry WL gauge field KA,bc = −iδa

A(Aa)AB(Σbc)BA,

• ωA,mn

= δa

AΩmna,

• ωA,bc = KA,bc,
• ΓµAν

= ΩνµA,

• ΓABµ

= ΩµρAΩρB − δb

BΩµcKA,cb,

• ΓABC

= δb

BδC c KA,cb.

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4d gauge invariance T µ

MN = 0

= ⇒ (Ka,bc − Kb,ac)Ωmnc = 0 SUSY conditions:

• ∇Mζ = 0

∂[MTNP]

R + T[MN QTP]Q R = 0

= ⇒ T[ab

dTc]d e = 0

TMN M = 0 = ⇒ Taba = −Ka,ba = 0 parallel spinor condition:

• ∇Mζ = 0 =

⇒ {

• ∇µζ = ∂µζ = 0

ζ: constant

• ∇Aζ = 1

4δa A(ΩmnaΓmn + Ka,bcΓbc)ζ = 0

4D(vielbein) and 6D(controsion) constributions must be cancelled

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Topological twist of N = 4 SUSY

Lorentz group: SO(4) = SU(2)L × SU(2)R spinor indices: α, ˙ α = 1, 2 R-symmetry group SU(4)I vector indices: A = (A′, ˆ A) (A′ = 1, 2, ˆ A = 3, 4) ( SU(2)R′ SU(2)L′ ) , U(1)′ : ( 12 −12 ) embedding of SO(4) into SU(4)I = ⇒ new Lorentz group 3 types of N = 4 twists [Yamron 1988] half twist: SU(2)L × (SU(2)R × SU(2)R′)diag Vafa-Witten twist: SU(2)L × (SU(2)R × SU(2)L′ × SU(2)R′)diag Marcus twist, GL twist:[Marcus 1995, Kapustin-Witten 0604151] (SU(2)L × SU(2)L′)diag × (SU(2)R × SU(2)R′)diag

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Off-shell SUSY in Ω-background: half-twist

SU(2)R ∼ SU(2)R′ ζαA′ = (σm)αB′ϵA′B′ζm, ¯ ζ ˙

αA′ = δ ˙ αA′ ¯

ζ + (¯ σmn) ˙

αA′ ¯

ζmn φAB = i √ 2ΣAB

a

φa = ( φϵA′B′ φA′ ˆ

B

φ ˆ

AB′

− ¯ φϵ ˆ

A ˆ B

) , Ωµνa′ = ( ϵ1

a′iτ 2

−ϵ2

a′iτ2

) , (Aa′)AB = ( 1

4(ϵ1 a′ + ϵ2 a′)τ 3

ma′τ 3 ) , Ωµνˆ

a = (Aˆ a)AB = 0,

(a′ = 1, 2, ˆ a = 3, 4, 5, 6), N = 2∗ deformation: N = 2 vector (Am, ΛA′, ¯ ΛA′, φ, ¯ φ) +N = 2 adjoint hyper (Λ ˆ

A, ¯

Λ ˆ

A, φA′ ˆ B) with mass ma′

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N = (0, 2) on-shell SUSY ¯ Q, ¯ Q12; ¯ QAµ = Λµ, ¯ QΛµ = −2 √ 2(Dµφ − FµνΩν), ¯ Qφ = ΩµΛµ, ¯ Q ¯ φ = − √ 2¯ Λ + ¯ ΩµΛµ, ¯ Q¯ Λ = −2i ( [φ, ¯ φ] + iΩµDµ ¯ φ − i¯ ΩµDµφ + i¯ ΩµΩνFµν ) , ¯ Q¯ Λµν = −2F −

µν − i(¯

σµν)

˙ β ˙ α[φ ˙ α ˆ A, ¯

φ ˆ

A ˙ β],

¯ Qφ ˙

α ˆ A = −

√ 2¯ Λ ˙

α ˆ A,

¯ Q¯ Λ ˙

α ˆ A = −2i

( [φ, φ ˙

α ˆ A] + iΩµDµφ ˙ α ˆ A + M ˆ A ˆ Bφ ˙ α ˆ B)

− Ωµν(¯ σµν) ˙

α ˙ βφ ˙ β ˆ A,

¯ QΛ

ˆ A α =

√ 2(σµ)α ˙

αDµφ ˙ α ˆ A.

¯ Q2¯ Λµν = ¯ Q2Λ ˆ

A α = 0 up to e.o.m.

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• ff-shell SUSY

auxiliary fields Dµν, K ˆ

A ˙ α

L′ = L + 1 κg2 Tr [ −1 2(Dµν)2 + 1 2K

ˆ A α Kα ˆ A

] . modify the SUSY transformations: ¯ Q¯ Λµν = 2Dµν − 2F −

µν − i(¯

σµν)

˙ β ˙ α[φ ˙ α ˆ A, ¯

φ ˆ

A ˙ β],

¯ QΛ

ˆ A α = 2K ˆ A α +

√ 2(σµ)α ˙

αDµφ ˙ α ˆ A.

¯ QL + ¯ QLaux = 0= ⇒ ¯ QDµν = · · · , ¯ QK ˆ

A ˙ α = · · ·

¯ Q2Ψ = 2 √ 2 ( δgauge(φ) + δLorentz(Ω) + δflavor(M) ) Ψ. L′ = ¯ QΞ + 1 κg2 Tr [1 4Fµν ˜ F µν ]

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Vafa-Witten twist

SU(2)R ∼ [SU(2)L′ × SU(2)R′]diag ζα

A′ = (σm)αB′ϵA′B′ζm,

¯ ζ ˙

α A′ = δ ˙ α A′ ¯

ζ + (¯ σmn) ˙

α A′ ¯

ζmn, ζα

ˆ A = (σm)α ˆ Bϵ ˆ A ˆ B ˆ

ζm, ¯ ζ ˙

α ˆ A = δ ˙ α ˆ Aˆ

¯ ζ + (¯ σmn) ˙

α ˆ Aˆ

¯ ζmn. Ωµνa = ( ϵ1

aiτ2

−ϵ2

aiτ2

) , (a = 1, 2, 5, 6) (Aa)A

B =

(

1 4(ϵ1 a + ϵ2 a)τ 3 1 4(ϵ1 a + ϵ2 a)τ 3

) , Ωµν3 = Ωµν4 = (A3)A

B = (A4)A B = 0.

• n-shell SUSY ¯

Q, ¯ Q12, ˆ ¯ Q, ˆ ¯ Q12;

• ff-shell SUSY: ¯

Q, ˆ ¯ Q: L′ = ¯ Q ˆ ¯ QF + 1 κg2 Tr [1 4Fµν ˜ F µν ]

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Marcus twist

SU(2)L ∼ SU(2)L′, SU(2)R ∼ SU(2)R′ ζαA′ = (σm)αB′ϵA′B′ζm, ¯ ζ ˙

αA′ = δ ˙ αA′ ¯

ζ + (¯ σmn) ˙

αA′ ¯

ζmn, ζα

ˆ A = δα ˆ Aζ + (σmn)α ˆ Aζmn,

¯ ζ ˙

α ˆ A = (¯

σm) ˙

α ˆ Bϵ ˆ A ˆ B ¯

ζm.

Ωµνa′ = ( ϵ1

a′iτ2

−ϵ2

a′iτ2

) , (Aa′)A

B =

(

1 4(ϵ1 a′ + ϵ2 a′)τ 3 1 4(ϵ1 a′ − ϵ2 a′)τ 3

) Ωµνˆ

a = (Aˆ a)A B = 0,

(a′ = 1, 2, ˆ a = 3, 4, 5, 6).

N = (2, 2) SUSY ¯ Q, ¯ Q12, Q, Q12

• ff-shell SUSY Q = uQ + v ¯

Q (u, v ∈ C) L′ = QΞ + 1 κg2 Tr [ (u2 − v2) 4(u2 + v2)Fµν ˜ F µν ]

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Deformed N = 4: Summary

half-twist

= ⇒ ma′→ 1

4 (ϵ1a′−ϵ2a′)

Marcus twist N = (0, 2) SUSY N = (2, 2) SUSY Q uQ + v ¯ Q ↓ ma′ → 1

4(ϵ1a′ + ϵ2a′)

N = 2Ω+ adj hyper

⇐ = ϵ1,2ˆ

a→0

Vafa-Witten twist with mass 1

4(ϵ1a′ + ϵ2a′)

N = (0, 4) SUSY ¯ Q, ˆ ¯ Q

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Nekrasov-Shatashvili limit

ϵ1a → 0 (ϵ2a → 0) Poincar´ e invariance in (x1, x2)-plane ((x3, x4)-plane) is recovered enhancement of SUSY

▶ half-twist N = (0, 2) =

⇒ N = (2, 2)

▶ Vafa-Witten twist N = (0, 4) =

⇒ N = (4, 4)

▶ Marcus twist N = (2, 2) =

⇒ N = (4, 4)

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Superstrings and Instantons: N = 4

Microscopic approach type IIB Superstrings N=4 U(N) SYM N D3-branes ↓ ↓ k-instanton k D(-1)-branes within D3-branes ADHM moduli

• pen string zero modes [Douglas,Witten]

↓ ↓ α′ → 0 (field theory limit) instanton effective action D(-1)-low energy effective actions Gauge theory IIB Superstrings N = 4 U(N) SYM N D3-branes k instantons k D(-1)-branes within N D3 branes ADHM moduli

• pen string connecting D-branes

FI parameters NS-NS B-fields Ω-backgrounds (S,A)-type R-R 3-form R-symmetry Wilson lines (A,S)-type R-R 3-form

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R-R background Decomposition of R-R field strength: bispinor of SO(10), decompose into SO(4) × SO(6) (α, ˙ α = 1, 2, A, B = 1, · · · , 4) FαβAB = F[αβ][AB] + F[αβ](AB) + F(αβ)[AB] + F(αβ)(AB) = Faϵαβ(Σa)AB + Fabcϵαβ(Σabc)AB +Fµνa(σµν)αβ(Σa)AB + Fµνabc(σµν)αβ(Σabc)AB (Σabc)AB ≡ (Σ[a ¯ ΣbΣc])AB symmetric in A, B (σµν)αβ = 1

4(σµ¯

σν − σν¯ σµ)α

γεγβ: symmetric in α, β

Ω-background Ωmna: F(αβ)[AB], F( ˙

α ˙ β)[AB] (S,A)-type

R-symmetry Wilson line: mAB: F[αβ](AB), F[ ˙

α ˙ β](AB) (A,S)-type

mAB = 1 8 ( (¯ Σa)AC(Aa)CB + (¯ Σa)BC(Aa)CA ) .

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Outlook

classification of deformed SUSY on the curved 4-manifold (parallell spinor→ Killing spinor) [Dumitrescu-Festuccia-Seiberg, Hama-Hosomichi] N = 1∗ deformation A = diag(0, m1, m2, m3), quiver (orbifolding),

• ther gauge groups(orientifold)

S-duality, instanton partition function large N, AdS/CFT correspondence (distribution of a′

m in the large N)

Ω-background and D-brane system in various dimensions (vortex, monopole, domain wall etc.) [I-Kamoshita-Sasaki, Bulycheva-Chen-Gorsky-Koroteev] String theory interpretation [Hellerman-Orlando-Reffert, Nakayama-Ooguri, Bill´

• -Frau-Fucito-Lerda, INSS,

Antoniadis-Flokakis-Hohenegger-Narain-Zein Assi]

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