The double-trace spectrum of planar N = 4 SYM: an unexpected 10d conformal symmetry [arXiv: 1706.08456, 1802.06889] F. Aprile J. M. Drummond P. Heslop H. P. Tuesday Seminar, University of Southampton, 20/11/2018
Outline Motivation & General Setup Preliminaries Half-BPS operators Operator Product Expansion The double-trace spectrum Unexpected 10d conformal symmetry Conclusions
Motivation (i) Perturbative Quantum Gravity in AdS ◮ Want to study quantum corrections to supergravity: consider loops in AdS 5 ◮ Even tree level computations in AdS are hard: supergravity correlation functions only recently computed in full generality [Rastelli-Zhou’16, Arutyunov-Klabbers-Savin’18] ◮ Loops in AdS are even harder: so far only few examples of loop diagrams explicitly computed ( φ 3 , φ 4 -theory) [Aharony-Alday-Bissi-Perlmutter’16] [Yuan’17’18, Cardona’17, Ghosh’18, ...] ◮ Consistency checks of AdS/CFT correspondence
Motivation (ii) Spectrum of (S)CFT’s ◮ Study spectrum of operators and their three-point functions − → recent interest from ”bootstrap program” [Rattazzi-Rychkov-Tonni-Vichi’08 ...] ◮ Bootstrap approach to ”solving” N = 4 SYM in a 1 / N 2 expansion ◮ N = 4 SYM: connections to integrable systems away from the planar limit [Bargheer-Caetano-Fleury-Komatsu-Vieira’17]
General Setup AdS/CFT correspondence Supergravity on AdS 5 × S 5 N = 4 SYM with ⇐ ⇒ gauge group SU ( N ) ⇐ ⇒ 4pt correlation functions AdS amplitudes (Witten diagrams) 1 strong coupling ( λ → ∞ ) ⇐ ⇒ weak coupling → N 2 expansion ◮ Interested in loop corrections to supergravity in AdS 5 − → mixing problem of double-trace operators
General Setup single-trace operator O 2 ( x ) ⇐ ⇒ graviton multiplet higher charge operators O p ( x ) ⇐ ⇒ Kaluza-Klein modes H = + + + + + ... � �� � � �� � � �� � ↓ ↓ ↓ 1 1 N 0 N 2 N 4 free field classical supergravity quantum corrections theory � � to SUGRA
Preliminaries Simplest operators to consider: 1 2 -BPS single-trace operators � � y 2 = 0 O p ( x , t ) = y R 1 · · · y R p Tr ϕ R 1 ( x ) · · · ϕ R p ( x ) , ◮ 2pt- and 3pt-functions are protected by supersymmetry ◮ First non-trivial dynamics in 4pt-correlators: � p 1 p 2 p 3 p 4 � := �O p 1 ( x 1 , y 1 ) O p 2 ( x 2 , y 2 ) O p 3 ( x 3 , y 3 ) O p 4 ( x 4 , y 4 ) �
Preliminaries Simplest operators to consider: 1 2 -BPS single-trace operators � � y 2 = 0 O p ( x , t ) = y R 1 · · · y R p Tr ϕ R 1 ( x ) · · · ϕ R p ( x ) , ◮ 2pt- and 3pt-functions are protected by supersymmetry ◮ First non-trivial dynamics in 4pt-correlators: � p 1 p 2 p 3 p 4 � := �O p 1 ( x 1 , y 1 ) O p 2 ( x 2 , y 2 ) O p 3 ( x 3 , y 3 ) O p 4 ( x 4 , y 4 ) � Strategy: use OPE O p 1 ( x 1 ) O p 3 ( x 3 ) O ∆ ,ℓ O p 2 ( x 2 ) O p 4 ( x 4 ) ◮ Exchanged operators O ∆ ,ℓ can be unprotected
Preliminaries Analyse CFT data in large N expansion ∆ = ∆ (0) + 1 N 2 · η (1) + . . . ∆ ,ℓ + 1 C ∆ ,ℓ = C (0) N 2 · C (1) ∆ ,ℓ + . . .
Preliminaries Analyse CFT data in large N expansion ∆ = ∆ (0) + 1 N 2 · η (1) + . . . ∆ ,ℓ + 1 C ∆ ,ℓ = C (0) N 2 · C (1) ∆ ,ℓ + . . . Which operators contribute? Remember: N = 4 SYM at λ → ∞ ⇐ ⇒ supergravity limit ⇒ long single-trace operators (’string states’) decouple ⇒ remaining spectrum up to this order: double-trace operators ∼ O p � n ∂ ℓ O q
The double-trace spectrum Problem: double-trace operators are degenerate! → O i O t ,ℓ − t ,ℓ At twist t ≡ ∆ (0) − ℓ we have t 2 − 1 operators: � � t t O i 2 − 2 ∂ ℓ O 2 ) , ( O 3 � 2 − 3 ∂ ℓ O 3 ) , . . . , ( O t 2 � 0 ∂ ℓ O t t ,ℓ = ( O 2 � 2 )
The double-trace spectrum Problem: double-trace operators are degenerate! → O i O t ,ℓ − t ,ℓ At twist t ≡ ∆ (0) − ℓ we have t 2 − 1 operators: � � t t O i 2 − 2 ∂ ℓ O 2 ) , ( O 3 � 2 − 3 ∂ ℓ O 3 ) , . . . , ( O t 2 � 0 ∂ ℓ O t t ,ℓ = ( O 2 � 2 ) Unmixing equations take the form: � ◮ Disconnected free field theory: t ,ℓ ) 2 ( C i i � t ,ℓ ) 2 · η i ◮ Tree-level supergravity: ( C i t ,ℓ i
The system of equations Approach: use data from mixed correlators �O p O p O q O q � [Rastelli-Zhou’16] This gives us exactly as many equations as unknowns! → we can solve the mixing problem
The system of equations Approach: use data from mixed correlators �O p O p O q O q � [Rastelli-Zhou’16] This gives us exactly as many equations as unknowns! → we can solve the mixing problem The solution in the [0 , 0 , 0] channel takes the form t ,ℓ = − 2( t − 1) 4 ( t + ℓ ) 4 → − 1 η i ℓ 2 ( ℓ + 2 i − 1) 6 [arXiv:1706.08456]
The system of equations Approach: use data from mixed correlators �O p O p O q O q � [Rastelli-Zhou’16] This gives us exactly as many equations as unknowns! → we can solve the mixing problem Considering cases for different SU (4) channels [ a , b , a ] lead to a conjecture for the general solution: 2 M t M t + ℓ +1 η pq | [ a , b , a ] = − � � ℓ + 2 p − 2 − a − 1+( − 1) a + ℓ 2 6 M t = ( t − 1)( t + a )( t + a + b + 1)( t + 2 a + b + 2) [arXiv:1802.06889]
Last talk in February Open questions: ◮ Residual degeneracy for η pq | [ a , b , a ] : Is it lifted at higher orders? Or protected by some symmetry? UPDATE: (work in progress) setting up computation to study the first example of residual degeneracy: [0 , 2 , 0] channel at twist 8 � � 504( ℓ + 7)( ℓ + 8) 504 504 504( ℓ + 3)( ℓ + 4) , − , − , − − ( ℓ + 1)( ℓ + 2)( ℓ + 5)( ℓ + 6) ( ℓ + 5)( ℓ + 6) ( ℓ + 5)( ℓ + 6) ( ℓ + 5)( ℓ + 6)( ℓ + 9)( ℓ + 10)
Last talk in February Open questions: ◮ Residual degeneracy for η pq | [ a , b , a ] : Is it lifted at higher orders? Or protected by some symmetry? UPDATE: (work in progress) setting up computation to study the first example of residual degeneracy: [0 , 2 , 0] channel at twist 8 � � 504( ℓ + 7)( ℓ + 8) 504 504 504( ℓ + 3)( ℓ + 4) , − , − , − − ( ℓ + 1)( ℓ + 2)( ℓ + 5)( ℓ + 6) ( ℓ + 5)( ℓ + 6) ( ℓ + 5)( ℓ + 6) ( ℓ + 5)( ℓ + 6)( ℓ + 9)( ℓ + 10) Observation by [CaronHuot-Trinh’18]: Residual degeneracy explained by 10d conformal symmetry of tree level amplitudes! [arXiv:1809.09173]
10d conformal symmetry [CaronHuot-Trinh’18] made the following observations: ◮ AdS 5 × S 5 metric is conformally equivalent to flat space ◮ The four-point tree amplitude of identical complex axi-dilatons in IIB supergravity is conformally invariant: 10 = 8 π G N δ 16 ( Q ) , with K µ · 1 A tree stu = 0 stu ◮ After identifying 8 π G N = π 5 L 8 (where c = N 2 − 1 ), our 4 c conjecture for η t ,ℓ looks like partial wave coefficients of A tree 10 : ( L √ s / 2) 8 A ℓ ( s ) = 1 + i π A tree ⇒ 10 c ( ℓ + 1) 6 ∆ 8 e − i πη t ,ℓ = 1 + i π compared to c ( ℓ eff + 1) 6
10d conformal symmetry Taking this coincidence seriously, the 10d conformal group SO (10 , 2) should relate different supergravity correlators. ◮ SO (10 , 2) ⊃ SO (4 , 2) × SO (6) ↔ isometries of AdS 5 × S 5 . ◮ expect natural action of SO (10 , 2) on 12-vectors w i : w i ≡ ( X i , y i ) X i ↔ x µ where i space-time point of SO (4 , 2) y i ↔ parametrisation of SO (6) R-symmetry Conjecture: all tree-level supergravity four-point correlators arise from a single SO (10 , 2)-invariant object G 10 ( u 10 , v 10 ) � φ ( w 1 ) φ ( w 2 ) φ ( w 3 ) φ ( w 4 ) � 10 ≡ � 34 ) 2 � 2 ( x 2 12 − y 2 12 ) 2 ( x 2 34 − y 2
10d conformal symmetry Consequences for tree-level supergravity amplitudes: F (1) p 1 p 2 p 3 p 4 ( u , v ) = D p 1 p 2 p 3 p 4 G 10 ( u , v ) , where D p 1 p 2 p 3 p 4 is a differential operator obtained by Taylor- expanding � φ ( w 1 ) φ ( w 2 ) φ ( w 3 ) φ ( w 4 ) � 10 in the y i ’s of SO (6) Conjecture was checked against ◮ old results in the literature [Arutyunov, Eden, Frolov, Petkou, Sokatchev,...] ◮ result for all supergravity correlators in Mellin-space [Rastelli-Zhou’16] Implications for 4d anomalous dimensions: Double-trace operators O p � n ∂ ℓ O q stem from bilinears φ∂ ℓ φ of a single 10d field φ ( w ). Degeneracy occurs when different 4d operators come from the same 10d primary!
Summary and Outlook ◮ Conjectured anomalous dimensions of double-trace operators 2 M t M t + ℓ +1 η pq | [ a , b , a ] = − � � ℓ + 2 p − 2 − a − 1+( − 1) a + ℓ 2 6 has a residual degeneracy in some SU (4) channels. ◮ This residual degeneracy is explained by an unexpected 10d conformal symmetry of supergravity tree-level amplitudes. Outlook: ◮ Study first example of degeneracy ([0 , 2 , 0] channel): is it lifted at the next order? i.e. is the 10d tree-level conformal symmetry broken by quantum effects? ◮ Study implications of 10d symmetry for higher-loop correlators
Structure of 4-point functions Consider the correlator ( p ≤ q ) � � ppqq � = g p 12 g q H R i ( u , v ) , 34 i where R i ∈ [0 , p , 0] × [0 , p , 0] u = g 13 g 24 x and v = g 13 g 24 g 12 g 34 ≡ x ¯ g 14 g 23 ≡ (1 − x )(1 − ¯ x ).
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