GRAVITY DUALS OF 2D SUSY GAUGE THEORIES BASED ON: 0909.XXXX with E. - PowerPoint PPT Presentation

GRAVITY DUALS OF 2D SUSY GAUGE THEORIES BASED ON: ● 0909.XXXX with E. Conde and A.V. Ramallo (Santiago de Compostela) [See also 0810.1053 with C. Núñez, P. Merlatti and A.V. Ramallo] Daniel Areán Zürich, September 2009

OUTLINE ➣ INTRODUCTION. AdS/CFT and its generalisations ➣ GRAVITY DUAL OF 2d N=(1,1) from wrapped branes � � � ● Brane setup � � � � ● 10d SUGRA ansatz � � � ● Gauged SUGRA approach (7d) � � � ● Solution → Coulomb branch ➣ ADDING FLAVOR � � � ● Flavor D5s � � � ● Backreaction → smearing � � � ● Flavored solution ➣ GRAVITY DUAL OF 2d N=(2,2) from wrapped branes ➣ SUMMARY 1/12

AdS 5 × S 5 N D3-branes AdS / CFT ∼ Correspondence ( α � → 0) IIB ST 4d N = 4 SU ( N ) SYM 2/12

AdS 5 × S 5 N D3-branes AdS / CFT ∼ Correspondence ( α � → 0) IIB ST 4d N = 4 SU ( N ) SYM ★ d = 2 N = (2 , 2) ★ 2 (4) SUSYs GENERALISE 2 d SYM + N f flavors N = (1 , 1) ★ Conformal ★ Add Flavor (4d: Maldacena & Núñez, Gauntlett et al, Bigazzi et al) ★ USE WRAPPED BRANES (3d: Chamseddine & Volkov, Maldacena & Nastase, Schvellinger & Tran, Gomis & Russo, Gauntlett et al) 2/12

DUAL TO N=(1,1) SYM FROM WRAPPED D5s ★ BRANE SETUP G 2 D5s S 4 σ X R ( ρ ) S 2 R 1 , 1 G 2 � �� � R 1 , 1 S 4 N 3 R D5 − − � � � � N 3 : ( σ , θ , φ ) · · · · 3/12

DUAL TO N=(1,1) SYM FROM WRAPPED D5s ★ BRANE SETUP G 2 D5s S 4 σ X R ( ρ ) S 2 R 1 , 1 G 2 � �� � R 1 , 1 S 4 N 3 R D5 − − � � � � N 3 : ( σ , θ , φ ) · · · · ◆ G ➔ 1/8 SUSY 2 SUSYS 2 ◆ D5s (on a calibrated C ) ➔ 1/2 SUSY 4 3/12

G 2 � �� � ★ SUGRA ANSATZ R 1 , 1 S 4 N 3 R D5 − − � � � � · · · · R ( ρ ) N 3 : ( σ , θ , φ ) (Bryant, Salamon) 7 = ( d σ ) 2 + σ 2 4 + σ 2 1 − a 4 � � � ( E 1 ) 2 + ( E 2 ) 2 � ◆ (resolved) cone: ds 2 2 d Ω 2 G 2 (Gibbons, Page, Pope) 1 − a 4 4 σ 4 σ 4 G 2 S 4 σ S 2 4/12

G 2 � �� � ★ SUGRA ANSATZ R 1 , 1 S 4 N 3 R D5 − − � � � � · · · · R ( ρ ) N 3 : ( σ , θ , φ ) (Bryant, Salamon) 7 = ( d σ ) 2 + σ 2 4 + σ 2 1 − a 4 � � � ( E 1 ) 2 + ( E 2 ) 2 � ◆ (resolved) cone: ds 2 2 d Ω 2 G 2 (Gibbons, Page, Pope) 1 − a 4 4 σ 4 σ 4 G 2 d ξ 2 + ξ 2 � ( ω 1 ) 2 + ( ω 2 ) 2 + ( ω 3 ) 2 �� 4 ● : d Ω 2 S 4 � 4 = (1 + ξ 2 ) 2 4 S 4 σ ξ 2 S 2 E 1 = d θ + sin φ ω 1 − cos φ ω 2 � � 1 + ξ 2 ● fibered : S 2 ξ 2 ξ 2 � � E 2 = sin θ cos φ ω 1 + sin φ ω 2 � 1 + ξ 2 ω 3 � d φ − + 1 + ξ 2 cos θ 4/12

G 2 � �� � R 1 , 1 S 4 N 3 R D5 − − � � � � · · · · R ( ρ ) N 3 : ( σ , θ , φ ) + e − Φ + e − Φ z ◆ 10d metric ds 2 = e Φ � � d σ 2 + σ 2 � ( E 1 ) 2 + ( E 2 ) 2 �� dx 2 m 2 d Ω 2 m 2 ( d ρ ) 2 � 1 , 1 + 4 4 m 2 z 3 ◆ 3-form C 2 = g 1 E 1 ∧ E 2 + g 2 S ξ ∧ S 3 + S 1 ∧ S 2 � � F 3 = dC 2 , 5/12

G 2 � �� � R 1 , 1 S 4 N 3 R D5 − − � � � � · · · · R ( ρ ) N 3 : ( σ , θ , φ ) + e − Φ + e − Φ z ◆ 10d metric ds 2 = e Φ � � d σ 2 + σ 2 � ( E 1 ) 2 + ( E 2 ) 2 �� dx 2 m 2 d Ω 2 m 2 ( d ρ ) 2 � 1 , 1 + 4 4 m 2 z 3 ◆ 3-form C 2 = g 1 E 1 ∧ E 2 + g 2 S ξ ∧ S 3 + S 1 ∧ S 2 � � F 3 = dC 2 , z ( ρ , σ ) SIZE OF C 4 N=(1,1) BPSs SUSY Φ ( ρ , σ ) DILATON g i ( ρ , σ ) 3-FORM FLUX 5/12

G 2 � �� � R 1 , 1 S 4 N 3 R D5 − − � � � � · · · · R ( ρ ) N 3 : ( σ , θ , φ ) + e − Φ + e − Φ z ◆ 10d metric ds 2 = e Φ � � d σ 2 + σ 2 � ( E 1 ) 2 + ( E 2 ) 2 �� dx 2 m 2 d Ω 2 m 2 ( d ρ ) 2 � 1 , 1 + 4 4 m 2 z 3 ◆ 3-form C 2 = g 1 E 1 ∧ E 2 + g 2 S ξ ∧ S 3 + S 1 ∧ S 2 � � F 3 = dC 2 , z ( ρ , σ ) SIZE OF C 4 N=(1,1) BPSs SUSY Φ ( ρ , σ ) DILATON g i ( ρ , σ ) 3-FORM FLUX ● BPSs are PDEs ☹ , 7d Gauged SUGRA ➞ SOLUTION ☺ 5/12

★ GAUGED SUGRA APPROACH LINEAR DISTRIBUTION OF D5S ◆ Take 7d SO(4) Gauged SUGRA Domain wall problem Uplift ● 1d problem → BPSs easy 10d solution in terms of c z ρ → R ⊥ ( R 1 , 1 , G 2 ) ( z, ψ ) σ → G 2 S 4 6/12

★ GAUGED SUGRA APPROACH LINEAR DISTRIBUTION OF D5S ◆ Take 7d SO(4) Gauged SUGRA Domain wall problem Uplift ● 1d problem → BPSs easy 10d solution in terms of c z ρ → R ⊥ ( R 1 , 1 , G 2 ) ( z, ψ ) σ → G 2 S 4 ◆ UV (z → ∞ ): ds 2 → D5s along R 1 , 1 × S 4 [ ➡ Linear dilaton ] ● Singularity (good) at z = z 0 ◆ IR (for c<-1): > z = z 0 ● Linear distribution ( ψ ) ψ 6/12

� � � � � � � � G 2 � �� � R 1 , 1 S 4 N 3 R ● Changing vbles. D5 ( z, ψ ) → ( ρ , σ ) − − � � � � · · · · R ( ρ ) N 3 : ( σ , θ , φ ) ➥ Analytic (implicit) sol. for z ( ρ , σ ) e 2 � 5 z 10 4 8 3 6 2 3 � c = − 3 3 � 4 1 2 z 0 2 2 0 -2 -2 1 � � � � 1 0 0 c c � � c 2 2 c 0 0 � � 7/12

� � � � � � � � G 2 � �� � R 1 , 1 S 4 N 3 R ● Changing vbles. D5 ( z, ψ ) → ( ρ , σ ) − − � � � � · · · · R ( ρ ) N 3 : ( σ , θ , φ ) ➥ Analytic (implicit) sol. for z ( ρ , σ ) e 2 � 5 z 10 4 8 3 6 2 3 � c = − 3 3 � 4 1 2 z 0 2 2 0 -2 -2 1 � � � � 1 0 0 c c � � c 2 2 c 0 0 � � Linear Distribution of D5s COULOMB BRANCH ( z = z 0 , ψ ) → ( | ρ | < ρ c , σ = 0) 7/12

★ ADDING FLAVOR Flavor ◆ Add an open string sector ➔ FLAVOR BRANES Color 8/12

★ ADDING FLAVOR Flavor ◆ Add an open string sector ➔ FLAVOR BRANES Color Flavor D5s ★ Global Sym: flavor ● Non-compact C 4 ⊂ G 2 ● Brane setup ★ m Q ∼ ρ Q ● At fixed ρ = ρ Q ★ Same SUSY 8/12

★ ADDING FLAVOR Flavor ◆ Add an open string sector ➔ FLAVOR BRANES Color Flavor D5s ★ Global Sym: flavor ● Non-compact C 4 ⊂ G 2 ● Brane setup ★ m Q ∼ ρ Q ● At fixed ρ = ρ Q ★ Same SUSY ● Probe approximation N f ≪ N c , N c → ∞ (Karch & Randall, Karch & Katz) Flavor D5 C 4 σ ρ Q Quenched flavor in the large N limit. c D5s 8/12

★ ADDING FLAVOR Flavor ◆ Add an open string sector ➔ FLAVOR BRANES Color Flavor D5s ★ Global Sym: flavor ● Non-compact C 4 ⊂ G 2 ● Brane setup ★ m Q ∼ ρ Q ● At fixed ρ = ρ Q ★ Same SUSY ● Probe approximation N f ≪ N c , N c → ∞ (Karch & Randall, Karch & Katz) Flavor D5 C 4 σ ρ Q Quenched flavor in the large N limit. c D5s Veneziano limit N f , N c → ∞ ● Backreaction N f ∼ N c Quarks loops included N f /N c fixed 8/12

S = S IIB + S flavor DBI + S flavor ◆ Computing the backreaction is difficult W Z D5s D5s ➥ Smearing φ φ (Bigazzi et al, Casero et al) ρ = ρ Q ρ = ρ Q → U (1) N f U ( N f ) − 9/12

S = S IIB + S flavor DBI + S flavor ◆ Computing the backreaction is difficult W Z D5s D5s ➥ Smearing φ φ (Bigazzi et al, Casero et al) ρ = ρ Q ρ = ρ Q → U (1) N f U ( N f ) − N f � � � Bianchi identity dF 3 = 2 κ 2 S flavor ˆ 10 T 5 Ω = T 5 C 6 = ⇒ − T 5 Ω ∧ C 6 W Z M ( i ) M 10 6 ➥ Ω + metric → Flavored BPSs 9/12

S = S IIB + S flavor DBI + S flavor ◆ Computing the backreaction is difficult W Z D5s D5s ➥ Smearing φ φ (Bigazzi et al, Casero et al) ρ = ρ Q ρ = ρ Q → U (1) N f U ( N f ) − N f � � � Bianchi identity dF 3 = 2 κ 2 S flavor ˆ 10 T 5 Ω = T 5 C 6 = ⇒ − T 5 Ω ∧ C 6 W Z M ( i ) M 10 6 ➥ Ω + metric → Flavored BPSs ◆ D5 embeddings ( κ -symmetry) ➞ Ω , this is hard!! ● D5-branes at ρ = ρ Q ● Consistent BPSs ( ➡ EoM) generic Ω / ● Same SUSY (2) ● Color ∩ Flavor = ∅ ● No new deformations of g ab 9/12

◆ Particular charge distribution / homogeneous charge distribution along ⊥ R 3 ● Numerical solution with continuous at z, φ , g i ρ = ρ Q ● Coincides with the unflavored for ρ < ρ Q ρ = ρ Q x ≡ 18 π n f = 0 . 2 x = 1 N c 8 8 z z 4 4 6 6 2 2.8 2 2.8 4 4 4 4 � � 1.6 1 1.6 1 0 0 1 1 0.4 0.4 0.4 0.4 2 2 3 3 4 4 � � ● Flavor contributes as expected [ ] 1 /g 2 Y M ∼ z 2 ( ρ , σ = 0) 10/12

★ SUGRA DUALS OF 2D THEORIES WITH N=(2,2) SUSY CY 3 R 1 , 1 ● D5s on a 4-cycle of a CY3 ~ 2d N = (2,2) S 2 × S 2 R 2 ( ρ , χ ) σ X ◆ CY3 ➔ 1/ 4 SUSY ψ 2 SUSYS ◆ D5s ➔ 1/2 SUSY 11/12

★ SUGRA DUALS OF 2D THEORIES WITH N=(2,2) SUSY CY 3 R 1 , 1 ● D5s on a 4-cycle of a CY3 ~ 2d N = (2,2) S 2 × S 2 R 2 ( ρ , χ ) σ X ◆ CY3 ➔ 1/ 4 SUSY ψ 2 SUSYS ◆ D5s ➔ 1/2 SUSY ■ 10d Ansatz ■ Metric → z ( ρ , σ ) & ϕ ( ρ , σ ) ■ Analyt. sol ✓ BPSs ■ 3-form → g( ρ , σ ) ■ EoM ✓ [ 7d Gauged SUGRA ] 11/12

★ SUGRA DUALS OF 2D THEORIES WITH N=(2,2) SUSY CY 3 R 1 , 1 ● D5s on a 4-cycle of a CY3 ~ 2d N = (2,2) S 2 × S 2 R 2 ( ρ , χ ) σ X ◆ CY3 ➔ 1/ 4 SUSY ψ 2 SUSYS ◆ D5s ➔ 1/2 SUSY ■ 10d Ansatz ■ Metric → z ( ρ , σ ) & ϕ ( ρ , σ ) ■ Analyt. sol ✓ BPSs ■ 3-form → g( ρ , σ ) ■ EoM ✓ [ 7d Gauged SUGRA ] ■ Flavoring → D5s on a non-compact 4-cycle → Embeddings found ➥ Ω constructed → new BPSs → (Numeric) Flavored background 11/12