GRAVITY DUALS OF 2D SUSY GAUGE THEORIES

GRAVITY DUALS OF 2D SUSY GAUGE THEORIES BASED ON: 0909.3106 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 Milos, 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 SUMMARY 1/11

N D3-branes AdS 5 S 5 AdS / CFT Correspondence d N = SU(N) SYM (α 0) IIB ST d = 2 GENERALISE 2 () SUSYs Conformal 2d N = (2, 2) N = (1, 1) SYM + N f flavors Add Flavor USE WRAPPED BRANES (d: Maldacena & Núñez, Gauntlett et al, Bigazzi et al) (3d: Chamseddine & Volkov, Maldacena & Nastase, Schvellinger & Tran, Gomis & Russo, Gauntlett et al) 2/11

DUAL TO N=(1,1) SYM FROM WRAPPED D5s BRANE SETUP D5s G 2 S S 2 σ X R (ρ) R 1,1 G 2 {}}{ R 1,1 S N 3 R D5 N 3 : (σ, θ, φ) 3/11

DUAL TO N=(1,1) SYM FROM WRAPPED D5s BRANE SETUP D5s G 2 S S 2 σ X R (ρ) R 1,1 G 2 {}}{ R 1,1 S N 3 R D5 N 3 : (σ, θ, φ) G 1/8 SUSY 2 D5s (on a calibrated C ) 1/2 SUSY 2 SUSYS 3/11

SUGRA ANSATZ G 2 {}}{ R 1,1 S N 3 R D5 N 3 : (σ, θ, φ) R (ρ) (resolved) G 2 cone: ds 2 7 = (dσ)2 1 a σ + σ2 2 dω2 + σ2 (1 a σ ) [(E 1 ) 2 + (E 2 ) 2] (Bryant, Salamon) (Gibbons, Page, Pope) G 2 S : dω 2 = (1 + ξ 2 ) 2 [dξ 2 + ξ2 ( (ω 1 ) 2 + (ω 2 ) 2 + (ω 3 ) 2)] S S 2 σ fibered : S 2 E 1 = dθ + ξ2 ( sin φ ω 1 1 + ξ 2 cos φ ω 2) ) E 2 = sin θ (dφ ξ2 1 + ξ 2 ω3 + ξ2 1 + ξ 2 cos θ ( cos φ ω 1 + sin φ ω 2) /11

G 2 {}}{ R 1,1 S N 3 R D5 N 3 : (σ, θ, φ) R (ρ) 10d metric ds 2 = e Φ [ dx 2 1,1 + z m 2 dω2 ] + e Φ m 2 z 3 [ dσ 2 + σ 2 ( (E 1 ) 2 + (E 2 ) 2 )] + e Φ m 2 (dρ)2 3-form F 3 = dc 2, C 2 = g 1 E 1 E 2 + g 2 ( S ξ S 3 + S 1 S 2) 5/11

G 2 {}}{ R 1,1 S N 3 R D5 N 3 : (σ, θ, φ) R (ρ) 10d metric ds 2 = e Φ [ dx 2 1,1 + z m 2 dω2 ] + e Φ m 2 z 3 [ dσ 2 + σ 2 ( (E 1 ) 2 + (E 2 ) 2 )] + e Φ m 2 (dρ)2 3-form F 3 = dc 2, C 2 = g 1 E 1 E 2 + g 2 ( S ξ S 3 + S 1 S 2) SUSY N=(1,1) BPSs z(ρ, σ) Φ(ρ, σ) g i (ρ, σ) SIZE OF C DILATON 3-FORM FLUX BPSs are PDEs, 7d Gauged SUGRA SOLUTION 5/11

GAUGED SUGRA APPROACH LINEAR DISTRIBUTION OF D5S Take 7d SO() Gauged SUGRA Domain wall problem z 1d problem BPSs easy Uplift 10d solution in terms of c ρ R (R 1,1, G 2 ) σ G 2 (z, ψ) S 6/11

GAUGED SUGRA APPROACH LINEAR DISTRIBUTION OF D5S Take 7d SO() Gauged SUGRA Domain wall problem z 1d problem BPSs easy Uplift 10d solution in terms of c ρ R (R 1,1, G 2 ) σ G 2 (z, ψ) S UV (z ): ds 2 D5s along R 1,1 S [ Linear dilaton ] IR (for c<-1): Singularity (good) at z = z 0 Linear distribution (ψ) > ψ z = z 0 6/11

Changing vbles. (z, ψ) (ρ, σ) Analytic (implicit) sol. for z(ρ, σ) G 2 {}}{ R 1,1 S N 3 R D5 N 3 : (σ, θ, φ) R (ρ) z 10 8 z 10 8 e 2 $ 5 6 3 z 0 6 z 0-2 -2!" 0 c " 2 c!" 0 c " 0 2 " 1 " 0 2 3 1 # 2 3 # c = 3 2 2 1 0-2!" c 0 " c 2 " 0 1 2 3 # Figure 3: Plots of z(ρ, σ) (left) and e 2Φ (right) obtained from (3.21) and (3.23). In both cases we have taken c = 1.5. On the left plot we have represented by a line the segment z = z 0, ρ c ρ ρ c in the σ = 0 plane. Linear Distribution of D5s COULOMB BRANCH One can also verify that e Φ vanishes along this constant z = z 0 segment. Thus our solution has a linear distribution of(z singularities. = z 0, ψ) One ( ρ can < check ρ c, σ that = 0) these singularities and good in the sense of [3] (see appendix A). Therefore, our solution can be naturally interpreted as 7/11

ADDING FLAVOR Add an open string sector FLAVOR BRANES Flavor Color Brane setup Flavor D5s Non-compact C G 2 At fixed ρ = ρ Q Global Sym: flavor m Q ρ Q Same SUSY 8/11

ADDING FLAVOR Add an open string sector FLAVOR BRANES Flavor Color Brane setup Flavor D5s Non-compact C G 2 At fixed ρ = ρ Q Global Sym: flavor m Q ρ Q Same SUSY Probe approximation N f N c, N c (Karch & Randall, Karch & Katz) Flavor D5 C σ Quenched flavor in the large N limit. c D5s ρ Q 8/11

Computing the backreaction is difficult S = S IIB + S flavor DBI + S flavor W Z Smearing (Bigazzi et al, Casero et al) D5s φ D5s φ ρ = ρ Q ρ = ρ Q U(N f ) U(1) N f S flavor W Z = T 5 N f (i) M6 Ĉ 6 = T 5 M 10 Ω C 6 df 3 = 2κ 2 10 T 5 Ω Bianchi identity Ω + metric Flavored BPSs D5 embeddings (κ-symmetry) Ω, this is hard!! D5-branes at ρ = ρ Q Same SUSY (2) No new deformations of g ab generic Ω / Consistent BPSs ( EoM) Color Flavor = 9/11

Particular charge distribution / homogeneous charge distribution along R 3 Numerical solution with z, φ, g i continuous at ρ = ρ Q Coincides with the unflavored for ρ < ρ Q x 18π n ρ = ρ Q f = 0.2 x = 1 N c z 8 6 2.8 1.6 " z 8 6 2.8 1.6 " 0 1 2 3 0. 0 1 2!! 3 0. Flavor contributes as expected [ 1/gY 2 M z 2 (ρ, σ = 0) ] Figure : Plot of z(ρ, σ) obtained by the numerical integration of (3.61) for ρ Q = 3 and x 18πn f N c = 0.2 (left) and x = 1 (right). In order to integrate (3.61) we assume that z(ρ, σ) is given by the unflavored solution (3.21) for ρ ρ Q, while at ρ = ρ Q the derivative of z with respect to ρ jumps in the form dictated 10/11

SUMMARY / TO TRY Gravity duals of 2d N=(1,1) & (2,2) SUSY theories from wrapped D5s Large number of flavors via backreacting flavor D5s Explore the F.T. (a little) color probe brane (E-r relation missing) Higgs branch Color & flavor branes recombining Alternative setup D3s on a 2-cycle of a CY3. Better UV. Non-singular background? Less SUSY D5s on a -cycle of a Spin(7) 11/11

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