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

Outline A Brief History of Monopoles SUSY: 4D -> 3D x S 1 N=2 SUSY in 4D Standard Model Conclusions

J.J. Thomson q q g g J J = q g • • q g - g e R Philos. Mag. 8 (1904) 331

Dirac charge quantization Proc. Roy. Soc. Lond. A133 (1931) 60

‘t Hooft-Polyakov topological monopoles Nucl. Phys., B79 (1974) 276 JETP Lett., 20 (1974) 194

‘t Hooft-Polyakov hedgehog gauge φ a = ˆ rv h ( vr ) r j f ( vr ) W a i = ✏ air ˆ gr

‘t Hooft-Polyakov hedgehog gauge singular gauge φ a = ˆ U † τ a φ a U = v h ( vr ) τ 3 rv h ( vr ) r j f ( vr ) r 2 σ 1 − ˆ r 1 σ 2 ✓p ◆ 1 r 3 I + i ˆ W a i = ✏ air ˆ U = 1 + ˆ gr √ √ 1 + ˆ r 3 2

‘t Hooft-Mandelstam magnetic condensate confines electric charge High Energy Physics Ed. Zichichi, (1976) 1225 Phys. Rept. 23 (1976) 245

4D -> 3D x S 1 SUSY SU(N) with F flavors µ → ~ W a W, � a monopole solution

4D -> 3D x S 1 Wick rotation monopole solution

4D -> 3D x S 1 compactify monopole solution

N-1 Embeddings of SU(2) N-1 diagonal generators   1     0 0 0 0 0 0 0 0 . . . 2 . . . . . . 1 − 1 0 0 0 0 0 0 0       2 . . . 2 . . . . . .  − 1   1    0 0 0 0 . . . 0 0 0       2 2 . . . . . . . . .     . . . .  . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . monopole solutions

Roots of SU(3) T 3 , T 8 � � H =  1  0 0 2  = α · H − 1 0 0  2 0 0 0   0 0 0  = β · H 1 0 0  2 − 1 0 0 2 √ β = ( − 1 3 α = (1 , 0) 2 ) 2 ,

N-1 Embeddings of SU(2) N-1 diagonal generators α 1 · H α 2 · H α 3 · H . . . . . . monopole charges α 1 α 2 α 3

Roots of SU(3) T 3 , T 8 � � H = h φ i = a · H a = v 1 α 1 + v 2 β √ β = ( − 1 3 α = (1 , 0) 2 ) 2 ,

Roots of SU(3) α 2 T 3 , T 8 � � H = h φ i = a · H a = v 1 α 1 + v 2 β α 1 α 0 √ β = ( − 1 3 α = (1 , 0) 2 ) 2 ,

α 2 α 1 α 0

Monopole Solutions h φ i = a · H a = v 1 α + v 2 β T 3 φ = v 1 α · H + ˆ β v 2 h ( v 2 r ) ; β = β · H r a T a T 3 φ = v 2 β · H + ˆ α v 1 h ( v 1 r ) ; α = α · H r a T a

4D -> 3D x S 1 Wick rotation monopole solution

4D -> 3D x S 1 KK monopole solution

3D x S 1 -> 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 F=N-1 2N-2 . . . fermion ∂ W = mass ∂ Q ∂ Q

Instanton Superpotential F=N-1 2N-2 . . . W = Λ 3 N − F det Q ∗ Q ∗ = Λ 3 N − F | det QQ | 2 det QQ

Affleck-Dine Seiberg Superpotential ✓ Λ 3 N − F 1 ◆ N − F F < N W ADS = ( N − F ) det QQ where does this come from?

Affleck-Harvey-Witten 1 X W 3D = Y i i Y i = e a · α i + i γ i φ = a · H R → 0 @ m � i = ✏ mnp F np i Nucl. Phys. B206 (1982) 413

Finite R 1 X W = + η Y KK Y i i

Mixed Coulomb Branch SU(3) with F=1   0 φ = 1 q 2diag( v, 0 , − v ) Q = Q =   0 SU(3)->U(1)xU(1) SU(3)->SU(2) SU(3)->U(1) monopoles are confined

Mixed Coulomb Branch SU(3) with F=1 q ⌧ v monopoles are confined superHiggs mechanism gives fermions masses

Mixed Coulomb Branch SU(3) with F=1 q ⌧ v 1 W = η Y 1 Y 2 + Y 1 Y 2 QQ ◆ 1 ✓ 2 η W = 2 det QQ

Mixed Coulomb Branch SU(3) with F=1 q � 1 SU(3)->SU(2) in “4D”, F=0 R, v φ = a · H Λ 8 = Λ 6 L q 2 a = v ( α + β ) W = η L Y L + 1 Y L matches, since Y L ∝ Y 1 Y 2 q 2 η L = η q 2

SU(N) with F < N-1 Q, Q have F VEVs φ has F zeros 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

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