PROBE BRANES ON FLAVORED ABJM BACKGROUND Javier Mas Universidad de - PowerPoint PPT Presentation

PROBE BRANES ON FLAVORED ABJM BACKGROUND Javier Mas Universidad de Santiago de Compostela Heraklion June 2013 Niko Jokela, J. M., Alfonso V. Ramallo & Dimitrios Zoakos arXiv: 1211.0630 based on Eduardo Conde & Alfonso V. Ramallo 1105.6045

PLAN OF THE TALK • The ABJM theory • The flavored ABJM background • Probes on the flavored ABJM background • The flavored thermal ABJM background • Probes on the flavored thermal ABJM background • Conclusions

• The ABJM theory • The flavored ABJM background • Probes on the flavored ABJM background • The flavored thermal ABJM background • Probes on the flavored thermal ABJM background • Conclusions

The ABJM theory field theory Chern-Simons-matter theories in 2+1 dimensions gauge group: U ( N ) k × U ( N ) − k -Two gauge fields A µ , ˆ field content (bosonic) A µ -Four complex scalar fields: C I ( I = 1 , · · · , 4) bifundamentals ( N, ¯ N ) A ] − k D µ C I † D µ C I − V pot ( C ) S = k CS [ A ] − k CS [ ˆ action V pot ( C ) → sextic scalar potential Aharony, Bergman, Jafferis & Maldacena 0806.1218

The ABJM theory The ABJM model has N = 6 SUSY in 3d it has two parameters N → rank of the gauge groups ’t Hooft coupling λ ∼ N k k → CS level (1 /k ∼ gauge coupling) it is a CFT in 3d with very nice properties - partition function and Wilson loops can be obtained from localization Gaiotto&Jafferis 0903.2175 Drukker, Mariño & Putrov 1003.3837 - has many integrability properties (Bethe ansatz, Wilson loop/ T. Klose, 1012.3999 amplitude relation, ...) - connection to FQHE? Fujita, Li, Ryu & Takayanagi, 0901.0924 it is the 3d analogue of N=4 SYM

The ABJM theory sugra description in type IIA : 1 5 << k << N E ff ective description for N AdS 4 × CP 3 + fluxes CP 3 = C 4 / ( z i ∼ λ z i ) L 4 = 2 π 2 N k = 2 π 2 λ ds 2 = L 2 ds 2 AdS + L 2 ds 2 CP 3 F 4 = 3 π � 1 2 Ω AdS 4 � kN √ 2 J → Kahler form of CP 3 F 2 = 2 k J 1 � CP 1 F 2 = k 2 π � 2 N � 1 e φ = 2 L 4 = 2 √ π k 5 k

• The ABJM theory • The flavored ABJM background • Probes on the flavored ABJM background • The flavored thermal ABJM background • Probes on the flavored thermal ABJM background • Conclusions

Flavors in the ABJM background D6-branes extended in AdS 4 and wrapping RP 3 ⊂ CP 3 Hohenegger&Kirsch 0903.1730 Gaiotto&Jafferis 0903.2175 Introduce quarks in the ( N, 1) and (1 , N ) representation Q 2 → (1 , ¯ ˜ Q 1 → ( ¯ ˜ Q 1 → ( N, 1) N ) N, 1) Q 2 → (1 , N ) coupling to the vector multiplet 1 e − V Q 1 + Q † V Q 2 + antiquarks 2 e − ˆ Q † V, ˆ V vector supermultiplets for A , ˆ A C I = ( A 1 , A 2 , B † coupling to the bifundamentals 1 , B † 2 ) ˜ ˜ plus quartic terms in Q, ˜ Q 1 A i B i Q 1 , Q 2 B i A i Q 2 Q ’s

Flavors in the ABJM background ds 2 = L 2 ds 2 AdS + L 2 ds 2 CP 3 Write CP 3 as an S 2 -bundle over S 4 , ξ , ˆ θ , ˆ ( x 0 , x 1 , x 2 , r ψ , ˆ ) ϕ , θ , ϕ | {z } | {z } |{z} AdS 4 S 4 S 2 d ⇠ 2 + ⇠ 2  ✓ ( ! 1 ) 2 + ( ! 2 ) 2 + ( ! 3 ) 2 �◆ dx i + ✏ ijk A j x k ⌘ 2 � 4 ds 2 = L 2 ds 2 ⇣ AdS 4 + L 2 � + (1 + ⇠ 2 ) 2 4 where x 1 sin θ cos ϕ = cos ˆ ψ d ˆ θ + sin ˆ ψ sin ˆ ω 1 θ d ˆ ϕ = x 2 sin θ sin ϕ = sin ˆ ψ d ˆ θ − cos ˆ ψ sin ˆ ω 1 θ d ˆ ϕ = x 3 cos θ = d ˆ ψ + cos ˆ ω 3 θ d ˆ ϕ = ξ 2 A i = − 1 + ξ 2 ω i SU (2) instanton on S 4

Flavors in the ABJM background D6-branes extended in AdS 4 and wrapping RP 3 ⊂ CP 3 , ξ , ˆ θ = 0 , ˆ ( x 0 , x 1 , x 2 , r ψ , ˆ ϕ = 0 , θ = θ ( r ) , ϕ ) | {z } | {z } | {z } AdS 4 S 4 S 2 Z Z d 7 ζ e − φ p ˆ S = S DBI + S W Z = − T D 6 − det ˆ g 7 + T D 6 C 7 θ ( r ) m 0 θ ( r ) = π / 2 m 0 = 0 ⇒

The ABJM flavored background the idea is now to smear over positions and orientations E. Conde & A. V. Ramallo 1105.6045 ⇣ ⌘ P N f M ( i ) d 7 ζ e − φ √− det ˆ M ( i ) ˆ Backreaction R R S flav − T D 6 g 7 + T D 6 C 7 = i =1 ✓ ◆ Z Z 1 d 10 xe 3 φ / 4 p d 10 x C 7 ∧ Ω − det g 10 | Ω | + → − κ 2 10 Ω is a charge distribution 3-form C 7 = e − φ K is the calibration form preserve N=1 SUSY - no delta-function sources m 0 - much simpler (analytic) solutions − flavor symmetry : U (1) N f

The ABJM flavored background modified Bianchi identity dF 2 = 2 π Ω

The ABJM flavored background modified Bianchi identity dF 2 = 2 π Ω θ ( r ) = π / 2 solution for massless flavors

The ABJM flavored background modified Bianchi identity dF 2 = 2 π Ω θ ( r ) = π / 2 solution for massless flavors go to vielbein basis S 4 S i = ( S 1 , S 2 , S 3 , S 4 ) along the base ( d ξ , ω 1 , ω 2 , ω 3 , d θ , d ϕ ) → E a = ( E 1 , E 2 ) S 2 along the fiber

The ABJM flavored background modified Bianchi identity dF 2 = 2 π Ω θ ( r ) = π / 2 solution for massless flavors go to vielbein basis S 4 S i = ( S 1 , S 2 , S 3 , S 4 ) along the base ( d ξ , ω 1 , ω 2 , ω 3 , d θ , d ϕ ) → E a = ( E 1 , E 2 ) S 2 along the fiber 2 E 1 d θ + ξ S 1 S ξ = 1 + ξ 2 d ξ = E 2 sin d ϕ − ξ S 2 ξ = � sin ϕ ω 1 − cos ϕ ω 2 � S 1 = 1 + ξ 2 ξ cos ϕ ω 1 + sin ϕ ω 2 �� sin θ ω 3 − cos θ S 2 � � = 1 + ξ 2 ξ cos ϕ ω 1 + sin ϕ ω 2 �� − cos θ ω 3 − sin θ � � S 3 = 1 + ξ 2

The ABJM flavored background ! 4 2 ds 2 = L 2 ds 2 ( S i ) 2 + X X AdS + L 2 ( E a ) 2 a =1 i =1 F 2 = k ( S 4 ∧ S 3 + S 1 ∧ S 2 ) h i E 1 ∧ E 2 − 2 F 4 = 3 k 2 L 2 Ω AdS 4

The ABJM flavored background ! 4 2 Flavor backreaction ds 2 = L 2 ds 2 ( S i ) 2 + X X AdS + L 2 ( E a ) 2 a =1 i =1 F 2 = k ( S 4 ∧ S 3 + S 1 ∧ S 2 ) h i E 1 ∧ E 2 − η 2 η = 1 + 3 N f F 4 = 3 k 2 L 2 Ω AdS 4 k 4

The ABJM flavored background ! 4 2 1 Flavor backreaction ds 2 = L 2 ds 2 ( S i ) 2 + X X AdS + L 2 ( E a ) 2 q b 2 a =1 i =1 F 2 = k ( S 4 ∧ S 3 + S 1 ∧ S 2 ) h i E 1 ∧ E 2 − η 2 η = 1 + 3 N f F 4 = 3 k 2 L 2 Ω AdS 4 k 4

The ABJM flavored background ! 4 2 1 Flavor backreaction ds 2 = L 2 ds 2 ( S i ) 2 + X X AdS + L 2 ( E a ) 2 q b 2 a =1 i =1 F 2 = k ( S 4 ∧ S 3 + S 1 ∧ S 2 ) h i E 1 ∧ E 2 − η 2 η = 1 + 3 N f F 4 = 3 k 2 L 2 Ω AdS 4 k 4 s  N f 1 + 3 ◆ 4 ✓ N f ◆ 2 ✓ 3 q = 3 + 9 N f 1 + 3 N f k − ...  8 k − 2 k + k 4 4 4 ( N f q → 5 k → ∞ )  3 r ⌘ 2 ⇣ N f N f N f 4 + 39 1 + 3 9  N f k + 1 + 3 k − ... k − 16 4 16 k  16 b = N f ( N f 3 + 3 q → 5 k → ∞ )  2 k 4

The ABJM flavored background ! 4 2 1 Flavor backreaction ds 2 = L 2 ds 2 ( S i ) 2 + X X AdS + L 2 ( E a ) 2 q b 2 a =1 i =1 F 2 = k ( S 4 ∧ S 3 + S 1 ∧ S 2 ) h i E 1 ∧ E 2 − η 2 η = 1 + 3 N f F 4 = 3 k 2 L 2 Ω AdS 4 k 4 s  N f 1 + 3 ◆ 4 ✓ N f ◆ 2 ✓ 3 q = 3 + 9 N f 1 + 3 N f k − ...  8 k − 2 k + k 4 4 4 ( N f q → 5 k → ∞ )  3 r ⌘ 2 ⇣ N f N f N f 4 + 39 1 + 3 9  N f k + 1 + 3 k − ... k − 16 4 16 k  16 b = N f ( N f 3 + 3 q → 5 k → ∞ )  2 k 4 L 2 = π √ where is related to the quark-antiquark potential screening 2 λ σ σ

The ABJM flavored background 4 q q 2 b b Σ σ � Ε N f 0 5 10 15 20 25 30 k √ 2 π 3 q = − Q Q = 4 √ potential screening V q ¯ ; λ σ → l Γ (1 / 4) 4 N f 8 1 − 3 q 3 / 2 ( η + q ) 2 (2 − q ) 1 / 2 σ = 1 k − ... 8 < q ( N f ( q + η q − η ) 5 / 2 4 k k → ∞ ) → : N f

The ABJM flavored background 4 q q 2 b b Σ σ � Ε N f 0 5 10 15 20 25 30 k √ 2 π 3 q = − Q Q = 4 √ potential screening V q ¯ ; λ σ → l Γ (1 / 4) 4 N f 8 1 − 3 q 3 / 2 ( η + q ) 2 (2 − q ) 1 / 2 σ = 1 k − ... 8 < q ( N f ( q + η q − η ) 5 / 2 4 k k → ∞ ) → : N f ◆ 1 / 4 (2 − q ) 5 / 4 ✓ 2 N e φ = 4 √ π dilaton shifts ( η + q )[ q ( q + η q − η )] 1 / 4 k 5

The ABJM flavored background 4 q q 2 b b Σ σ � Ε N f 0 5 10 15 20 25 30 k √ 2 π 3 q = − Q Q = 4 √ potential screening V q ¯ ; λ σ → l Γ (1 / 4) 4 N f 8 1 − 3 q 3 / 2 ( η + q ) 2 (2 − q ) 1 / 2 σ = 1 k − ... 8 < q ( N f ( q + η q − η ) 5 / 2 4 k k → ∞ ) → : N f ◆ 1 / 4 (2 − q ) 5 / 4 ✓ 2 N e φ = 4 √ π dilaton shifts ( η + q )[ q ( q + η q − η )] 1 / 4 k 5 regime of validity N 1 / 5 ⌧ N f ⌧ N

• The ABJM theory • The flavored ABJM background • Probes on the flavored ABJM background • The flavored thermal ABJM background • Probes on the flavored thermal ABJM background • Conclusions

Probe Branes D6-branes extended in AdS 4 and wrapping RP 3 ⊂ CP 3 , ξ , ˆ θ = 0 , ˆ ( x 0 , x 1 , x 2 , r ψ , ˆ ϕ = 0 , θ ( r ) , ϕ ) | {z } | {z } | {z } AdS 4 S 4 S 2 u = r b new cartesian-like coordinates ✓ dr 2 r 2 + d θ 2 ◆ L 2 R = u cos θ ρ = u sin θ ; b 2 R L 2 d ρ 2 + dR 2 � � b 2 ( ρ 2 + R 2 ) θ ( r ) ρ R = R ( ρ ) profile