Unquenched matter in the gauge/gravity correspondence A. V. - PowerPoint PPT Presentation

Unquenched matter in the gauge/gravity correspondence A. V. Ramallo Univ. Santiago Large-N gauge theories Florence, May, 2011

Plan of the talk Addition of flavor in AdS/CFT Quenched&Unquenched matter in AdS/CFT Smeared unquenched flavor in D3-D7 Holographic flavor in Chern-Simons-matter systems Smeared unquenched flavor in ABJM Flavor effects in Chern-Simons-matter theories

AdS/CFT correspondence Maldacena 97 closed string sector 5 AdS x S geometry 5 open string sector large N D3 stack at bottom of throat Sugra in AdS 5 × S 5 N c → ∞ , λ >> 1 , N = 4 SYM

flavor branes Addition of flavor D3/D7 setup quark mass 0 1 2 3 4 5 6 7 8 9 Karch&Katz 02 D3 X X X X Graña&Polchinski 01 D7 X X X X X X X X Bertolini et al 01 separation in 89 directions generalization of AdS/CFT N D3 0123 conventional open/closed string duality 4567 SYM AdS 5 89 3 − 3 quarks 3 − 7 7 − 7 flavour open/open string duality N probe D7 AdS f 4 R 5 brane λ D 7 = λ D 3 (2 πl s ) 4 N f λ D 7 → 0 when l s → 0 with λ D 3 fixed N c U ( N f ) flavor symmetry

Quenched approximation suppresed by factors N f Neglect quark loops N c Gravity side Small number of D7s treat D7s as probes Fluctuations of D7 dual to “mesons” D7 S 3 -exact mass formulae AdS 5 -matching fluctuations/operators r=L holography on the wv r (Kruczenski et al. 03)

Why going beyond the probe approximation? In real life N f ∼ N c In N = 4 SYM we want to capture the breaking of conformality due to the flavor Control of flavor dynamics QCD phase diagram Dualities in SUSY theories require N f ∼ N c

Including the backreaction Action of gravity+branes S = S IIB + S fl � � 1 � R − 1 2 ∂ M Φ ∂ M Φ − 1 (1) − 1 1 d 10 x √− g 10 2 e 2Φ F 2 5 ! F 2 S IIB = (5) 2 κ 2 2 10 �� � M 8 d 8 ξ e Φ √− g 8 − sources of gravity fields � � S fl = − T 7 M 8 C 8 N f Rewrite the WZ term as: � N f δ (2) � � S W Z = T D 7 M 10 Ω ∧ C 8 Ω = � M 8 ω 2 transverse volume element ω 2 charge distribution two-form Ω S W Z induces a violation of Bianchi identity of F 1 δ -function source term dF 1 = − Ω Einstein eqs. have also δ -function source terms

Localized embedding D3 D7 density distribution form with delta-functions

Smeared sources -no delta-function sources -still can preserve (less) SUSY -much simpler (analytic) solutions − flavor symmetry : U (1) N f what is the deformation of the metric due to smeared flavor?

S 5 as a U (1) bundle metric of C 3 ds 2 = | dZ 1 | 2 + | dZ 2 | 2 + | dZ 3 | 2 define r 2 = � | Z i | 2 Z i = rz i ds 2 = dr 2 + r 2 ds 2 S 5 | z 1 | 2 + | z 2 | 2 + | z 3 | 2 = 1 parametrize z 2 = cos χ 2 sin θ z 1 = cos χ 2 cos θ z 3 = sin χ i i 2 e iτ 2 (2 τ + ψ + ϕ ) 2 (2 τ + ψ − ϕ ) 2 e 2 e ds 2 S 5 = ds 2 CP 2 + ( dτ + A ) 2 CP 2 = 1 4 dχ 2 + 1 2 ( dθ 2 + sin 2 θdϕ 2 ) + 1 4 cos 2 χ 4 cos 2 χ 2 sin 2 χ ds 2 2 ( dψ + cos θdϕ ) 2 A = 1 2 cos 2 χ 2 ( dψ + cos θdϕ )

Deformation of AdS 5 × S 5 preserving four SUSYs (unflavored case) � dr 2 ds 2 = h − 1 CP 2 + r 2 F ( r ) ( dτ + A ) 2 � F ( r ) + r 2 ds 2 1 2 dx 2 1 , 3 + h 2 F ( r ) = 1 − b 6 r 6 The CP2 and the U(1) fiber are squashed differently Hint: this is the type of deformation induced by the smeared flavor branes

Ansatz (massless quarks) � − 1 � 1 2 dx 2 ds 2 = � 1 , 3 + α � � 2 � e 2 f ( ρ ) dρ 2 + e 2 g ( ρ ) ds 2 dτ + A ) 2 � CP 2 + e 2 f ( ρ ) � h ( ρ ) h ( ρ ) F (5) = Q c (1 + ∗ ) ε ( S 5 ) F 1 = Q f ( dτ + A ) Φ = Φ( ρ ) Q c ≡ (2 π ) 4 g s N c Q f = V ol ( X 3 ) g s N f = g s N f = 16 πg s N c 4 V ol ( S 5 ) 2 π V ol ( S 5 ) BPS equations Φ = g s N f g = e 2 f − 2 g ˙ e Φ ˙ 2 π f = 3 − 2 e 2 f − 2 g − g s N f ˙ e Φ ˙ h = − Q c e − 4 g 4 π

Integration of the BPS equations Introduce a reference scale ρ = ρ ∗ → φ ∗ = φ ( ρ = ρ ∗ ) Deformation parameter � ∗ = g s N f 1 N f e φ ∗ � ∗ = 8 π 2 λ ∗ N c 2 π �� 1 1 � e φ − φ ∗ = � 1 6 √ e g = α � e ρ 1 + � ∗ 6 + ρ ∗ − ρ 1 + � ∗ ( ρ ∗ − ρ ) �� − 1 � � 1 3 e f = √ α � e ρ (1 + � ∗ ( ρ ∗ − ρ )) 1 1 + � ∗ 6 + ρ ∗ − ρ 2 �� − 2 � dh dρ = − Q c � 1 3 α � 2 e − 4 ρ 1 + � ∗ 6 + ρ ∗ − ρ

Properties of the solution (Landau pole) dilaton blows up at ρ = ρ LP = ρ ∗ + � − 1 ∗ metric singular at ρ = −∞ (IR) Good singularity that disappears when quarks are massive 1 regime of validity 1 << N c << N f << N c 3 Matching the field theory coupling constant ρ LP − ρ = log Λ L radius-energy relation Q 8 π 2 = N f log Λ L g 2 Y M = 4 π e Φ same running as in F. T. g 2 Q Y M

Perturbative solution expansion in powers of � ∗ h = R 4 change to a new radial coordinate such that: R 4 = 1 4 Q c = 4 π g s α � 2 N c r 4 r 4 � 2 r 4 r 8 � � 24(1 − 1 9 − 106 + 5 1 + � ∗ + 48 log( r e g = r � � + O ( � 3 ) + ∗ ) ∗ ) r 4 r 4 r 8 3 1152 9 9 r ∗ ∗ ∗ ∗ deviation from AdS 5 × S 5 r 4 � 2 r 4 r 8 � � 1 − � ∗ 24(1 + 1 17 − 94 + 5 − 48 log( r order by order in � ∗ e f = r � � + O ( � 3 ) + ∗ ) ∗ ) r 4 r 4 r 8 3 1152 9 9 r ∗ ∗ ∗ ∗ + � 2 1 − r 4 � � φ = φ ∗ + � ∗ log r + 12 log r + 36 log 2 r + O ( � 3 ∗ ∗ ) 72 r 4 r ∗ r ∗ r ∗ ∗ UV scale (in a Wilsonian sense) far below the Landau pole r ∗ << r LP 1 N f measures internal flavor loop contributions ∼ g 2 � ∗ = Y M ( r ∗ ) N f 8 π 2 λ ∗ N c In computing observables we should be sure that the UV pathological region is decoupled

One can study the effects of dynamical quarks in the screening of color charges (meson masses, quark potentials, screening lengths,..) (Biggazi et al., 0903.4747) Flavored black holes and hydrodynamics (Biggazi et al., 0909.2865, 0912.3256, 1101.3560) This method can be applied to add flavor to other backgrounds (Klebanov-Strassler, CVMN, ...) (Benini et al.,0706.1268, Casero, Nuñez&Paredes hep-th/0602027,...) For further results on this and other unquenched backgrounds, see the review 1002.1088

Flavor in Chern-Simons-matter systems in 2+1 ABJM theory (Aharony et al. 0812.18) CS with gauge group U ( N ) k × U ( N ) − k + bifundamental fields k → CS level 1 k ∼ coupling constant M-theory description for large N → AdS 4 × S 7 / Z k AdS 4 × CP 3 + fluxes Sugra description in type IIA ds 2 = L 2 ds 2 AdS 4 + 4 L 2 ds 2 L 4 = 2 π 2 N CP 3 k F 4 = 3 π � 1 2 Ω AdS 4 � F 2 = 2 k J kN √ 2 � 2 N � 1 e φ = 2 L 4 = 2 √ π k 5 k 1 5 << k << N Effective description for N

Hohenegger&Kirsch 0903.1730 Flavor branes (massless quarks) Gaiotto&Jafferis 0903.2175 D6-branes extended in AdS 4 and wrapping RP 3 ⊂ CP 3 Introduce quarks in the ( N, 1) and (1 , N ) representation Backreaction N f � � � ˆ S W Z = T D 6 C 7 → T D 6 C 7 ∧ Ω M ( i ) M 10 i =1 7 Ω is a charge distribution 3-form Modified Bianchi identity dF 2 = 2 π Ω Localized solution in 11d for coincident massless flavors AdS 4 × M 7 with M 7 a hyperkahler 3-Sasakian manifold N = 3 with U ( N f ) flavor symmetry

Backreaction with smearing (E. Conde and AVR, to appear) Write CP 3 as an S 2 -bundle over S 4 ( x i ) 2 = 1 � CP 3 = 1 � dx i + � ijk A j x k � 2 � i ds 2 ds 2 � S 4 + 4 A i → SU (2) instanton on S 4 Fubini-Study metric The RR two-form F 2 can be written as: � 1 CP 1 F 2 = k F 2 = k � E 1 ∧ E 2 − S 4 ∧ S 3 + S 1 ∧ S 2 � � � 2 π 2 S i → (rotated) basis of one-forms along S 4 E i → one-forms along the S 2 fiber Some Killing spinors are constant in this basis deform to preserve them

Prescription: squash F 2 and the metric F 2 = k � E 1 ∧ E 2 − η S 4 ∧ S 3 + S 1 ∧ S 2 � � � 2 Induces violation of Bianchi identity Deformation parameter η ≡ 1 + 3 N f 4 k � ≡ N f = N f N λ k Flavored metric ds 2 = L 2 ds 2 AdS 4 + ds 2 6 6 = L 2 � dx i + � ijk A j x k � 2 � ds 2 q ds 2 � S 4 + b 2 q → C P 3 internal squashing b → relative AdS 4 / C P 3 squashing

N = 1 superconformal SUSY implies q 2 − 3(1 + η ) q + 5 η = 0 � q = 3 + 9 N f 1 + 3 N f � 3 � 4 � N f � 2 − 2 + k k k 8 4 4 Also q ( η + q ) b = 2( q + η q − η ) The new AdS 4 radius is: (2 − q ) b 4 L 4 = 2 π 2 N q ( q + ηq − η ) k

Dilaton and F 4 : e − φ = b η + q k F 4 = 3 kb η + q 2 − q L 2 Ω AdS 4 4 2 − q L 4 Regime of validity e φ << 1 L >> 1 , (same as in the unflavored case) 1 5 << k << N N If N f /k ∼ 1 When N f >> k � N � 1 L 4 ∼ N 4 1 e φ ∼ 5 << N f << N N N 5 N f f

Flavor effects Free energy on the 3-sphere (measures # dof’s) F ( S 3 ) = πL 2 1 1 e − 2 φ V ol ( M 6 ) F ( S 3 ) = − log | Z S 3 | = G N G 10 2 G N In flavored ABJM 5 2 ( η + q ) 4 √ ≡ 1 2 � N f q F ( S 3 ) = π � N f � � 1 3 2 N 2 ξ ξ k 1 7 16 2 ( q + ηq − η ) k 3 k (2 − q ) 2 For small N f /k � N f �� N f N f � 2 � 3 � ξ = 1 + 3 − 9 + O 4 k 64 k k √ √ √ N 2 2 2 λ − 3 π 2 F ( S 3 ) = π + π √ 3 2 + · · · N 2 N f N f λ √ 3 4 64 λ amazing field theory match by 3 unflavored term ∼ N 2 Drukker et al. (1007.3837) !