AdS/CFT correspondence Addition of flavor Quenched and unquenched matter Our setup Summary D3-D5 theories with unquenched flavors Jos´ e Manuel Pen´ ın Ascariz November 16, 2016 Based on 1607.04998, in collaboration with E. Conde, H. Lin, A.V. Ramallo and D. Zoakos Jos´ e Manuel Pen´ ın Ascariz D3-D5 theories with unquenched flavors 1 / 18
AdS/CFT correspondence Addition of flavor Quenched and unquenched matter Our setup Summary AdS/CFT correspondence Addition of flavor Quenched and unquenched matter Our setup Brane intersection scheme Geometrical setup Smearing technique BPS system Integration of the BPS system Black hole Summary Jos´ e Manuel Pen´ ın Ascariz D3-D5 theories with unquenched flavors 2 / 18
AdS/CFT correspondence Addition of flavor Quenched and unquenched matter Our setup Summary AdS/CFT (Maldacena, 1997): relates gravity and field theory Observation: S.T. D-brane=SUGRA extremal p-brane ⇒ Two p.o.v. – D-BRANES � N = 4 gauge th. on the D3s open str. on D3 wv low E , α ′ → 0 � closed str. in the bulk − − − − − − − − − − − − → free gravity in the bulk interactions between them p-BRANES r → ∞ , E → 0, free gravity r → 0, also E → 0 observed at r → ∞ systems decoupled Conjecture: identify the low E system in the two p.o.v. { N =4 SYM, gauge SU(N), N >> 1 } = { SUGRA,r → 0, in D-brane bckg., α ′ → 0 � – ⇒ Duality: ( λ >> 1,grav.) & ( λ << 1,FT) - Jos´ e Manuel Pen´ ın Ascariz D3-D5 theories with unquenched flavors 3 / 18
AdS/CFT correspondence Addition of flavor Quenched and unquenched matter Our setup Summary Jos´ e Manuel Pen´ ın Ascariz D3-D5 theories with unquenched flavors 4 / 18
AdS/CFT correspondence Addition of flavor Quenched and unquenched matter Our setup Summary Motivation: d.o.f. of N = 4, SYM in adjoint rep. QCD: fundamental rep.! Fundamentals: how to...? N f << N c ⇒ probe Dp branes on D3 bckg. Correspondence now...? ⇒ two p.o.v. g s N c << 1 closed & open str. in flat space: - 3-3: N =4 SYM , [ g ] = 0 low E 1: closed str. in 10d flat & pp in Dp wv p-p: coupling ∝ E p − 3 2 sect. 2: N = 4 adj. SU(N c ) coupled 3-p: bifund. SU(N c ) × SU(N f ) − − − − → to 3-p in fund SU(N c ) × SU(N f ) g s N c >> 1: - � closed str. & open p-p in throat of AdS 5 xS 5 . Non interacting � closed str. & open p-p in asymptotically flat region. Interacting Conjecture: identify low E system: – { N =4 SYM, gauge SU(N), N >> 1, coupled to fundamentals } = { type IIB closed str. on AdS 5 × S 5 , coupled to open str. on wv of Dp-probes } Jos´ e Manuel Pen´ ın Ascariz D3-D5 theories with unquenched flavors 5 / 18
AdS/CFT correspondence Addition of flavor Quenched and unquenched matter Our setup Summary Jos´ e Manuel Pen´ ın Ascariz D3-D5 theories with unquenched flavors 6 / 18
AdS/CFT correspondence Addition of flavor Quenched and unquenched matter Our setup Summary Quenched gravity: ◮ N f → 0 ⇒ ’t Hooft limit ◮ No backreaction Quenched field theory: ◮ mass of fundamentals = ∞ ◮ quarks not running into loops ◮ not dynamical ... beyond the quenched approximation? - Real life N f ∼ N c - ⇒ Go to Veneziano limit: N c → ∞ , N f → ∞ , N c N f finite Captures more physics than ’t Hooft limit. Jos´ e Manuel Pen´ ın Ascariz D3-D5 theories with unquenched flavors 7 / 18
Brane intersection scheme AdS/CFT correspondence Geometrical setup Addition of flavor Smearing technique Quenched and unquenched matter BPS system Our setup Integration of the BPS system Summary Black hole D3-D5 system: 0 1 2 3 4 5 6 7 8 9 D3 x x x x - - - - - - D5 x x x - x x x - - - ◮ defect in ( x 0 , x 1 , x 2 ) where fundamentals live ◮ 2+1 dim. fundamental matter coupled to gauge theory in 3+1 dim. ◮ addition of massless hypermultiplete preserves conformality ⇒ dCFT Jos´ e Manuel Pen´ ın Ascariz D3-D5 theories with unquenched flavors 8 / 18
Brane intersection scheme AdS/CFT correspondence Geometrical setup Addition of flavor Smearing technique Quenched and unquenched matter BPS system Our setup Integration of the BPS system Summary Black hole D3 on the tip of a CY cone: CY = dr 2 + r 2 ds 2 ds 2 SE ds 2 SE = ds 2 KE + ( d τ + A ) 2 SE: 5d Sasaki-Einstein KE: 4d K¨ ahler-Einstein base fiber: ( d τ + A ) - Examples: S 5 , T 1 , 1 - - Ansatz: ds 2 = h − 1 KE + e 2 f ( d τ + A ) 2 ] 1 2 [ − ( dx 0 ) 2 +( dx 1 ) 2 +( dx 2 ) 2 + e 2 m ( dx 3 ) 2 ]+ h 2 [ dr 2 + e 2 g ds 2 Jos´ e Manuel Pen´ ın Ascariz D3-D5 theories with unquenched flavors 9 / 18
Brane intersection scheme AdS/CFT correspondence Geometrical setup Addition of flavor Smearing technique Quenched and unquenched matter BPS system Our setup Integration of the BPS system Summary Black hole Problem to solve? S = S IIB + S BRANES S BRANES = S DBI + S WZ D3 ⇒ coupled to RR F (5) F 5 = Q c (1 + ∗ ) ǫ ( M 5 ) Qc? ⇒ charge quantization: Q c = (2 π ) 4 g s α ′ 2 N c Vol ( M 5 ) D5 ⇒ coupled to RR F (3) � � ˆ S WZ = T 5 C 6 M 6 N f 1 � d 10 x √ g 10 � R − 1 2( ∂φ ) 2 − 1 1 (3) − 1 1 � 3! e φ F 2 5! e 2 φ F 2 ⇒ S IIB = (5) 2 κ 2 2 2 10 ⇒ violation of Bianchi id. for F (3) : dF (3) ∼ δ (2) ( M 6 ). (Challenging to solve) We use a different approach!!! Jos´ e Manuel Pen´ ın Ascariz D3-D5 theories with unquenched flavors 10 / 18
Brane intersection scheme AdS/CFT correspondence Geometrical setup Addition of flavor Smearing technique Quenched and unquenched matter BPS system Our setup Integration of the BPS system Summary Black hole Smeared embedding (massive case) Localized embedding � � � ˆ Ξ ∧ C (6) T 5 C 6 → T 5 M 6 M 10 N f ⇒ dF (3) = 2 κ 2 10 T 5 Ξ Ξ ∼ charge distribution Features of smearing: ◮ no δ -function sources ◮ still SUSY ◮ U ( N f ) → U (1) N f - Jos´ e Manuel Pen´ ın Ascariz D3-D5 theories with unquenched flavors 11 / 18
Brane intersection scheme AdS/CFT correspondence Geometrical setup Addition of flavor Smearing technique Quenched and unquenched matter BPS system Our setup Integration of the BPS system Summary Black hole Ansatz for F (3) (massless flavor) - K¨ ahler form of the KE manifold: J KE = e 1 ∧ e 2 + e 3 ∧ e 4 ⇒ J KE = dA 2 Define: Ω 2 = e i 3 τ ( e 1 + ie 2 ) ∧ ( e 3 + ie 4 ) ˆ Ansatz: F (3) = Q F dx 3 ∧ Im (ˆ Ω 2 ) ⇒ dF (3) = − 3 Q f dx 3 ∧ Re ( ˆ Ω 2 ) ∧ ( d τ + A ) = 2 κ 2 (10) T 5 Ξ ⇒ Dictates smearing form Jos´ e Manuel Pen´ ın Ascariz D3-D5 theories with unquenched flavors 12 / 18
Brane intersection scheme AdS/CFT correspondence Geometrical setup Addition of flavor Smearing technique Quenched and unquenched matter BPS system Our setup Integration of the BPS system Summary Black hole Bosonic SUSY background in type IIB SUGRA? - � δλ = 0 ( dilatino ) � ⇒ BPS equations: δψ µ = 0 ( gravitino ) - Leads to: h ′ = − Q c e − 4 g − f − Q f e φ 2 − m − 2 g h φ ′ = Q f e φ 2 − m − 2 g g ′ = e f − 2 g f ′ = 3 e − f − 2 e f − 2 g + Q f φ 2 − m − 2 g 2 e Jos´ e Manuel Pen´ ın Ascariz D3-D5 theories with unquenched flavors 13 / 18
Brane intersection scheme AdS/CFT correspondence Geometrical setup Addition of flavor Smearing technique Quenched and unquenched matter BPS system Our setup Integration of the BPS system Summary Black hole Unflavored solution: � d ξ 2 � unflav = h − 1 1 ⇒ ds 2 2 dx 2 F ( ξ ) + ξ 2 ds 2 KE + ξ 2 F ( ξ )( d τ + A ) 2 1 , 3 + h 2 ξ 6 , b 6 = g 3 where F ( ξ ) = 1 − b 6 Case g 3 = 0 ⇒ Conformal AdS 5 × S 5 4 - Flavored scaling solution: ds 2 scaling = ds 2 s 2 5 + d ˆ 5 � 4 3 ( dx 3 ) 2 5 = r 2 + R 2 dr 2 � � 4 Q f � ds 2 dx 2 1 , 2 + r 2 , R 2 3 4 r 3 � KE + 9 � s 2 5 = ¯ R 2 ds 2 8( d τ + A ) 2 d ˆ - ⇒ Anisotropic scale transf. invariance: r → r x 0 , 1 , 2 → λ x 0 , 1 , 2 , x 3 → λ e φ → λ − 2 1 3 x 3 , 3 e φ λ, - - Jos´ e Manuel Pen´ ın Ascariz D3-D5 theories with unquenched flavors 14 / 18
Brane intersection scheme AdS/CFT correspondence Geometrical setup Addition of flavor Smearing technique Quenched and unquenched matter BPS system Our setup Integration of the BPS system Summary Black hole Massive flavors: Cavity r < r q without charge. r q ∼ m q - Modified ansatz: Q f → Q f p ( r ) - � p ( r → ∞ ) = 1 p ( r ) profile p ( r < r q ) = 0 Solution for step function: p ( r ) = Θ( r − r q ) � r < r q ⇒ unflavored � Interpolation r > r q ⇒ massless flavored - Jos´ e Manuel Pen´ ın Ascariz D3-D5 theories with unquenched flavors 15 / 18
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