Nested One-point Functions in AdS/dCFT Georgios Linardopoulos NCSR - PowerPoint PPT Presentation

One-point functions in the D3-D5 system One-point functions in the D3-D7 system Summary Nested One-point Functions in AdS/dCFT Georgios Linardopoulos NCSR ”Demokritos” and National & Kapodistrian University of Athens Workshop on higher-point correlation functions and integrable AdS/CFT Hamilton Mathematics Institute – Trinity College Dublin, April 16th 2018 based on Phys.Lett. B781 (2018) 238 [arXiv:1802.01598] and J.Phys. A: Math.Theor. 50 (2017) 254001 [arXiv:1612.06236] with Charlotte Kristjansen and Marius de Leeuw 1 / 46

One-point functions in the D3-D5 system One-point functions in the D3-D7 system Summary Table of Contents 1 One-point Functions in the D3-D5 System Introducing the D3-D5 system Nested one-point functions at tree-level su (2) k representations Determinant formulas 2 One-point Functions in the D3-D7 System Introducing the D3-D7 system Nested one-point functions at tree-level Determinant formulas 3 Summary 2 / 46

Introducing the D3-D5 system One-point functions in the D3-D5 system Nested one-point functions at tree-level One-point functions in the D3-D7 system su (2) k representations Summary Determinant formulas Section 1 One-point Functions in the D3-D5 System 3 / 46

Introducing the D3-D5 system One-point functions in the D3-D5 system Nested one-point functions at tree-level One-point functions in the D3-D7 system su (2) k representations Summary Determinant formulas The D3-D5 system: description In the bulk, the D3-D5 system describes IIB Superstring theory on AdS 5 × S 5 bisected by D5 branes with worldvolume geometry AdS 4 × S 2 . The dual field theory is still SU ( N ), N = 4 SYM in 3 + 1 dimensions, that now interacts with a SCFT that lives on the 2+1 dimensional defect. Due to the presence of the defect, the total bosonic symmetry of the system is reduced from SO (4 , 2) × SO (6) to SO (3 , 2) × SO (3) × SO (3). The corresponding superalgebra psu (2 , 2 | 4) becomes osp (4 | 4). 4 / 46

Introducing the D3-D5 system One-point functions in the D3-D5 system Nested one-point functions at tree-level One-point functions in the D3-D7 system su (2) k representations Summary Determinant formulas The (D3-D5) k system Add k units of background U (1) flux on the S 2 component of the AdS 4 × S 2 D5-brane. Then k of the N D3-branes ( N ≫ k ) will end on the D5-brane. On the dual SCFT side, the gauge group SU ( N ) × SU ( N ) breaks to SU ( N − k ) × SU ( N ). Equivalently, the fields of N = 4 SYM develop nonzero vevs... (Karch-Randall, 2001b) 5 / 46

Introducing the D3-D5 system One-point functions in the D3-D5 system Nested one-point functions at tree-level One-point functions in the D3-D7 system su (2) k representations Summary Determinant formulas Subsection 2 Nested one-point functions at tree-level 6 / 46

Introducing the D3-D5 system One-point functions in the D3-D5 system Nested one-point functions at tree-level One-point functions in the D3-D7 system su (2) k representations Summary Determinant formulas The dCFT interface of D3-D5 An interface is a wall between two (different/same) QFTs It can be described by means of classical solutions that are known as ”fuzzy-funnel” solutions (Constable-Myers-Tafjord, 1999 & 2001) Here, we need an interface to separate the SU ( N ) and SU ( N − k ) regions of the (D3-D5) k dCFT... For no vectors/fermions, we want to solve the equations of motion for the scalar fields of N = 4 SYM: d 2 Φ i A µ = ψ a = 0 , dz 2 = [Φ j , [Φ j , Φ i ]] , i , j = 1 , . . . , 6 . A manifestly SO (3) ≃ SU (2) symmetric solution is given by ( z > 0): � � Φ 2 i − 1 ( z ) = 1 ( t i ) k × k 0 k × ( N − k ) & Φ 2 i = 0 , 0 ( N − k ) × k 0 ( N − k ) × ( N − k ) z Nagasaki-Yamaguchi, 2012 where the matrices t i furnish a k-dimensional representation of su (2): [ t i , t j ] = i ǫ ijk t k . 7 / 46

Introducing the D3-D5 system One-point functions in the D3-D5 system Nested one-point functions at tree-level One-point functions in the D3-D7 system su (2) k representations Summary Determinant formulas k -dimensional Representation of su (2) We use the following k × k dimensional representation of su (2): k − 1 k − 1 k � � � c k , i E i c k , i E i +1 d k , i E i t + = i +1 , t − = , t 3 = i i i =1 i =1 i =1 t 1 = t + + t − t 2 = t + − t − , 2 2 i d k , i = 1 � c k , i = i ( k − i ) , 2 ( k − 2 i + 1) , where E i j are the standard matrix unities that are zero everywhere except ( i , j ) where they’re 1. 8 / 46

Introducing the D3-D5 system One-point functions in the D3-D5 system Nested one-point functions at tree-level One-point functions in the D3-D7 system su (2) k representations Summary Determinant formulas 1-point functions Following Nagasaki & Yamaguchi (2012), the 1-point functions of local gauge-invariant scalar operators �O ( z , x ) � = C z ∆ , z > 0 , can be calculated within the D3-D5 dCFT from the corresponding fuzzy-funnel solution, for example: 1 SU (2) O ( z , x ) = Ψ i 1 ... i L Tr [Φ 2 i 1 − 1 . . . Φ 2 i L − 1 ] z L · Ψ i 1 ... i L Tr [ t i 1 . . . t i L ] − − − − − → interface where Ψ i 1 ... i L is an so (6)-symmetric tensor and the constant C is given by (MPS= matrix product state ) � � MPS | Ψ � ≡ Ψ i 1 ... i L Tr [ t i 1 . . . t i L ] � � 8 π 2 � L / 2 (”overlap”) · � MPS | Ψ � 1 √ C = , , 1 λ L � Ψ | Ψ � ≡ Ψ i 1 ... i L Ψ i 1 ... i L � Ψ | Ψ � 2 which ensures that the 2-point function will be normalized to unity ( O → (2 π ) L · O / ( λ L / 2 √ L ) 1 �O ( x 1 ) O ( x 2 ) � = | x 1 − x 2 | 2∆ within SU ( N ), N = 4 SYM (i.e. without the defect). 9 / 46

Introducing the D3-D5 system One-point functions in the D3-D5 system Nested one-point functions at tree-level One-point functions in the D3-D7 system su (2) k representations Summary Determinant formulas Example: chiral primary operators The one-point functions of the chiral primary operators � 8 π 2 � L / 2 1 · C i 1 ... i L Tr [Φ i 1 . . . Φ i L ] , √ O CPO ( x ) = λ L where C i 1 ... i L are symmetric & traceless tensors satisfying 6 9 � � i = cos 2 ψ, i = sin 2 ψ C i 1 ... i L C i 1 ... i L = 1 Y L = C i 1 ... i L ˆ x 2 x 2 & x i 1 . . . ˆ x i L , ˆ ˆ i =4 i =7 and Y L ( ψ ) is the SO (3) × SO (3) ⊆ SO (6) spherical harmonic, have been calculated at weak coupling: � 2 π 2 � L / 2 � L / 2 Y L ( π/ 2) � 1 k 2 − 1 �O CPO ( x ) � = √ k ≪ N → ∞ . k , z L λ L Nagasaki-Yamaguchi, 2012 The large- k limit agrees with the supergravity calculation at tree-level: � L / 2 Y L ( π/ 2) � 2 π 2 � � �O CPO ( x ) � = k L +1 1 + λ I 1 I 1 ≡ 3 2 + ( L − 2) ( L − 3) √ · π 2 k 2 + . . . , . z L λ 4 ( L − 1) L 10 / 46

Introducing the D3-D5 system One-point functions in the D3-D5 system Nested one-point functions at tree-level One-point functions in the D3-D7 system su (2) k representations Summary Determinant formulas Dilatation operator The mixing of single-trace operators O ( x ) is generally described by the integrable so (6) spin chain: ∞ L � � λ I j , j +1 − P j , j +1 + 1 � λ n · D n , � λ = g 2 D = L · I + 8 π 2 · H + H = 2 K j , j +1 , YM N , n =2 j =1 Minahan-Zarembo, 2002 up to one loop in N = 4 SYM, where I · | . . . Φ a Φ b . . . � = | . . . Φ a Φ b . . . � P · | . . . Φ a Φ b . . . � = | . . . Φ b Φ a . . . � 6 � K · | . . . Φ a Φ b . . . � = δ ab | . . . Φ c Φ c . . . � . c =1 The above result is unaffected by the presence of a defect in the SCFT (DeWolfe-Mann, 2004). 11 / 46

Introducing the D3-D5 system One-point functions in the D3-D5 system Nested one-point functions at tree-level One-point functions in the D3-D7 system su (2) k representations Summary Determinant formulas Bethe eigenstates In the following we will examine eigenstates of the so (6) spin chain which can be written as: � | Ψ � ≡ ψ i ( u 1 , u 2 , u 3 ) · | • . . . • ↑ • . . . • ↓ • . . . • ⇑ • . . . • ⇓ • . . . � , x 1 x 2 x 3 x 4 x i where u 1 , 2 , 3 are the rapidities of the excitations at x i . The corresponding single-trace operator is � � Z x 1 − 1 WZ x 2 − x 1 − 1 YZ x 3 − x 2 − 1 WZ x 4 − x 3 − 1 Y . . . | • . . . • ↑ • . . . • ↓ • . . . • ⇑ • . . . • ⇓ . . . � ∼ Tr , x 1 x 2 x 3 x 4 where Z (ground state field), W , Y (excitations) are the following three complex scalars: W = Φ 1 + i Φ 2 ∼ ↑ Y = Φ 3 + i Φ 4 ∼ ↓ Z = Φ 5 + i Φ 6 ∼ • W = Φ 1 − i Φ 2 ∼ ⇑ Y = Φ 3 − i Φ 4 ∼ ⇓ Z = Φ 5 − i Φ 6 ∼ ◦ The wavefunction ψ ( u 1 , u 2 , u 3 ) can be constructed with the (nested) coordinate Bethe ansatz... 12 / 46

Introducing the D3-D5 system One-point functions in the D3-D5 system Nested one-point functions at tree-level One-point functions in the D3-D7 system su (2) k representations Summary Determinant formulas Nesting Let us first construct the kets | • . . . • ↑ • . . . • ↓ • . . . • ⇑ • . . . • ⇓ • . . . � . x 1 x 2 x 3 x 4 13 / 46

Introducing the D3-D5 system One-point functions in the D3-D5 system Nested one-point functions at tree-level One-point functions in the D3-D7 system su (2) k representations Summary Determinant formulas Nesting Let us first construct the kets | • . . . • ↑ • . . . • ↓ • . . . • ⇑ • . . . • ⇓ • . . . � . x 1 x 2 x 3 x 4 Because the excitations can have 5 different polarizations, we apply a procedure called ”nesting”. 13 / 46