ADS 4D/BPS 3D Correspondence John Terning with Csaba Csaki, Yuri - - PowerPoint PPT Presentation

ADS 4D/BPS 3D Correspondence

John Terning with Csaba Csaki, Yuri Shirman

A Brief History of Monopoles SUSY: 4D -> 3D x S1 N=2 SUSY in 4D Standard Model Conclusions

Outline

J.J. Thomson

• Philos. Mag. 8 (1904) 331

J = q g

• e

g

• R

J

q q g g q g

Dirac

• Proc. Roy. Soc. Lond. A133 (1931) 60

charge quantization

‘t Hooft-Polyakov

topological monopoles

• Nucl. Phys., B79 (1974) 276

JETP Lett., 20 (1974) 194

‘t Hooft-Polyakov

hedgehog gauge φa = ˆ rv h(vr) W a

i = ✏airˆ

rj f(vr) gr

‘t Hooft-Polyakov

U = 1 √ 2 ✓p 1 + ˆ r3I + i ˆ r2σ1 − ˆ r1σ2 √1 + ˆ r3 ◆ hedgehog gauge singular gauge φa = ˆ rv h(vr) W a

i = ✏airˆ

rj f(vr) gr U †τ aφaU = v h(vr)τ 3

‘t Hooft-Mandelstam

magnetic condensate confines electric charge

High Energy Physics Ed. Zichichi, (1976) 1225

• Phys. Rept. 23 (1976) 245

monopole solution

4D -> 3D x S1

SUSY SU(N) with F flavors

W a

µ → ~

W, a

4D -> 3D x S1

monopole solution Wick rotation

4D -> 3D x S1

monopole solution compactify

N-1 Embeddings of SU(2)

monopole solutions

. . .

N-1 diagonal generators

    

1 2

. . . − 1

2

. . . . . . . . . . . . . . . . . .           . . .

1 2

. . . − 1

2

. . . . . . . . . . . . . . .     

     . . . . . .

1 2

. . . . . . . . . . . . . . .     

. . .

H =

• T 3, T 8

Roots of SU(3)

 

1 2

− 1

2

  = β · H  

1 2

− 1

2

  = α · H α = (1, 0) β = (−1 2, √ 3 2 )

N-1 Embeddings of SU(2)

monopole charges

. . .

N-1 diagonal generators

. . . α1 · H α2 · H α3 · H α3 α2 α1

H =

• T 3, T 8

Roots of SU(3)

α = (1, 0) β = (−1 2, √ 3 2 ) hφi = a · H a = v1α1 + v2β

H =

• T 3, T 8

Roots of SU(3)

α = (1, 0) β = (−1 2, √ 3 2 ) hφi = a · H a = v1α1 + v2β α1 α2 α0

α1 α2 α0

Monopole Solutions

hφi = a · H φ = v1 α · H + ˆ raT a

β v2 h(v2r) ;

T 3

β = β · H

φ = v2 β · H + ˆ raT a

α v1 h(v1r) ;

T 3

α = α · H

a = v1 α + v2 β

4D -> 3D x S1

monopole solution Wick rotation

4D -> 3D x S1

KK monopole solution

3D x S1 -> 4D

N-1 monopole solutions + KK monopole

. . . + + + +

• > 4D instanton

as

R → ∞

Instanton Zero Modes

. . .

. . .

2N gauginos 2F quarks

Instanton Zero Modes

. . .

. . .

. . . + + + +

2N gauginos

. . .

→

Poppitz & Unsal hep-th/0812.2085 


Instanton Zero Modes

. . .

2N-2 F=N-1

= ∂W ∂Q∂Q

fermion mass

Instanton Superpotential

. . .

2N-2 F=N-1

W = Λ3N−F det Q∗Q

∗

|det QQ|2 = Λ3N−F det QQ

Affleck-Dine Seiberg Superpotential

F < N where does this come from?

WADS = (N − F) ✓ Λ3N−F det QQ ◆

1 N−F

Affleck-Harvey-Witten

• Nucl. Phys. B206 (1982) 413

W3D = X

i

1 Yi R → 0 φ = a · H Yi = ea·αi+iγi @mi = ✏mnpF np

i

Finite R

W = X

i

1 Yi + η YKK

Mixed Coulomb Branch SU(3) with F=1

Q = Q =   q  

SU(3)->U(1)xU(1) SU(3)->SU(2) SU(3)->U(1) monopoles are confined

φ = 1 2diag(v, 0, −v)

Mixed Coulomb Branch SU(3) with F=1

monopoles are confined superHiggs mechanism gives fermions masses

q ⌧ v

Mixed Coulomb Branch SU(3) with F=1

W = η Y1 Y2 + 1 Y1 Y2 QQ q ⌧ v W = 2 ✓ η det QQ ◆ 1

2

Mixed Coulomb Branch SU(3) with F=1

q 1 R, v

SU(3)->SU(2) in “4D”, F=0

W = ηLYL + 1 YL φ = a · H a = v(α + β) Λ8 = Λ6

Lq2

YL ∝ Y1Y2q2 ηL = η q2 matches, since

SU(N) with F < N-1

SU(N)->SU(F)xU(1)N-F SU(N)->SU(N-F) SU(N)->U(1)N-F-1 F+1 monopoles are confined 2F gauginos get masses 2(F+1)-2F= 2 2 gaugino legs => ADS super potential

φ has F zeros

Q, Q have F VEVs

Conclusions

Monopoles are still fascinating after all these years Confined monopoles relate 3D BPS monopoles to the 4D ADS superpotential