Quantum corrections in AdS/dCFT Matthias Wilhelm, Niels Bohr Institute . Nordic String Meeting 2017, Hannover February 10th, 2017 [1606.01886], [1611.04603] with I. Buhl-Mortensen, M. de Leeuw, A. C. Ipsen, C. Kristjansen Matthias Wilhelm Quantum corrections in AdS/dCFT
Table of contents Motivation 1 Defect theory & framework for quantum corrections 2 One-point functions 3 Conclusion and outlook 4 Matthias Wilhelm Quantum corrections in AdS/dCFT
Motivation Conformal field theories: Phenomenologically relevant Highly constrain the form of correlation functions Success of understanding standard AdS/CFT setup and N = 4 SYM theory, in particular due to integrability Matthias Wilhelm Quantum corrections in AdS/dCFT
Motivation Conformal field theories: Phenomenologically relevant Highly constrain the form of correlation functions Success of understanding standard AdS/CFT setup and N = 4 SYM theory, in particular due to integrability Defect CFTs: Equally relevant New features: Non-vanishing one-point functions Non-vanishing two-point functions between operators of different scaling dimensions New aspects of gauge gravity correspondence: AdS/dCFT Matthias Wilhelm Quantum corrections in AdS/dCFT
Table of contents Motivation 1 Defect theory & framework for quantum corrections 2 One-point functions 3 Conclusion and outlook 4 Matthias Wilhelm Quantum corrections in AdS/dCFT
String-theory construction D5-D3 probe brane set-up [Karch, Randall (2000)] N − k D3 N D3 D5 D3 brane ∼ R 1 , 3 D5 brane ∼ AdS 4 × S 2 with flux k through S 2 x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 D3 × × × × D5 × × × × × × � �� � defect Matthias Wilhelm Quantum corrections in AdS/dCFT
Gauge theory x 0 x 3 < 0: SU ( N − k ) x 3 > 0: (broken) SU ( N ) x 3 x 1 , 2 SU ( N ) broken by x 3 -dependent vacuum expectation values for scalars 3D fundamental hypermultiplet on defect S = S N =4 + S D =3 [DeWolfe, Freedman, Ooguri (2001)] , [Erdmenger, Guralnik, Kirsch (2002)] Matthias Wilhelm Quantum corrections in AdS/dCFT
Classical solution Classical fields ψ cl = ¯ ψ cl = 0 φ cl φ cl A cl 1 , 2 , 3 � = 0 = 0 = 0 4 , 5 , 6 µ Equations of motion [Constable, Myers, Tafjord (1999)] ∂ 2 φ cl i = [ φ cl j , [ φ cl j , φ cl i ]] ∂ x 2 3 x 3 : distance to defect Solution via k -dimensional irreducible representation of the SU (2) Lie algebra: � ( t i ) k × k � i = − 1 0 k × ( N − k ) φ cl 0 ( N − k ) × k 0 ( N − k ) × ( N − k ) x 3 t 1 , t 2 , t 3 with [ t i , t j ] = i ǫ ijk t k Also satisfies Nahm equation [Nahm (1979)] Matthias Wilhelm Quantum corrections in AdS/dCFT
Action Action of N = 4 SYM theory � � 2 − 1 4 F µν F µν − 1 2 D µ φ i D µ φ i + i ψ Γ µ D µ ψ d 4 x tr ¯ S N =4 = g 2 2 YM � + 1 ψ Γ i [ φ i , ψ ] + 1 ¯ 4[ φ i , φ j ][ φ i , φ j ] 2 Expand around classical solution φ i = φ cl i + ˜ φ i i = 1 , 2 , 3 2 tr( G 2 ), G = ∂ µ A µ + i [˜ Gauge fix with S gf = − 1 φ i , φ cl i ] S N =4 + S gf = S kin + S m + S cubic + S quartic [Buhl-Mortensen, de Leeuw, Ipsen, Kristjansen, MW (2016)] Matthias Wilhelm Quantum corrections in AdS/dCFT
Mass terms Mass term � � 2 + 1 φ j ] + 1 i , ˜ i , ˜ j ][˜ φ i , ˜ d 4 x tr 2[ φ cl φ j ][ φ cl 2[ φ cl i , φ cl S m = φ j ] g 2 YM + 1 j ] + 1 i , ˜ φ j ][˜ i , ˜ j , ˜ 2[ φ cl φ i , φ cl 2[ φ cl φ i ][ φ cl φ j ] + 1 i ] + 2 i [ A µ , ˜ 2[ A µ , φ cl i ][ A µ , φ cl φ i ] ∂ µ φ cl i � + 1 ¯ ψ Γ i [ φ cl c [ φ cl i , [ φ cl i , ψ ] − ¯ i , c ]] 2 Properties: Non-diagonal in colour Mixing between the ˜ φ 1 , ˜ φ 2 , ˜ φ 3 and A 3 as well as between the fermion flavours Mass proportional to 1 / x 3 via φ cl i [Buhl-Mortensen, de Leeuw, Ipsen, Kristjansen, MW (2016)] Matthias Wilhelm Quantum corrections in AdS/dCFT
How to solve this? Matthias Wilhelm Quantum corrections in AdS/dCFT
Diagonalising the mass matrix � � A 0 , k × k A 0 , k × ( N − k ) Easy example: A 0 = A 0 , ( N − k ) × k A 0 , ( N − k ) × ( N − k ) Mass term: − 1 1 � � � � A 0 [ t i , [ t i , A 0 ]] A 0 , k × k [ t i , [ t i , A 0 tr = − tr k × k ]] 2 x 2 2 x 2 3 3 + 1 � � A 0 , k × ( N − k ) A 0 tr ( N − k ) × k t i t i x 2 ���� 3 k 2 − 1 4 L 2 = L i L i with L i = ad t i is the Laplacian on the fuzzy sphere: ⇒ Can be diagonalised by fuzzy spherical harmonics ˆ Y m ℓ Mass terms of { ˜ φ 1 , ˜ φ 2 , ˜ φ 3 , A 3 } and the fermions also contain σ i L i → Similar to spin-orbital interaction of the hydrogen atom! [Buhl-Mortensen, de Leeuw, Ipsen, Kristjansen, MW (2016)] Matthias Wilhelm Quantum corrections in AdS/dCFT
Spectrum of the mass matrix � m 2 + 1 Eigenvalues (for x 3 = 1) and multiplicities in terms of ν = 4 ν (˜ ν (˜ Multiplicity φ 4 , 5 , 6 , A 0 , 1 , 2 , c ) m ( ψ 1 , 2 , 3 , 4 ) φ 1 , 2 , 3 , A 3 ) ℓ + 1 ℓ + 3 ℓ = 1 , . . . , k − 1 ℓ + 1 2 2 ℓ + 1 ℓ − 1 ℓ + 1 ℓ 2 2 k k +1 k +2 ( k − 1)( N − k ) 2 2 2 k − 1 k − 2 k ( k + 1)( N − k ) 2 2 2 1 1 ( N − k )( N − k ) 0 2 2 [Buhl-Mortensen, de Leeuw, Ipsen, Kristjansen, MW (2016)] Matthias Wilhelm Quantum corrections in AdS/dCFT
Propagators Scalar propagator with x 3 -dependent mass term � � − ∂ µ ∂ µ + m 2 K ( x , y ) = g 2 YM 2 δ ( x − y ) ( x 3 ) 2 Standard scalar propagator K AdS ( x , y ) in AdS 4 with mass ˜ m m 2 ) K AdS ( x , y ) = δ ( x − y ) ( −∇ µ ∇ µ + ˜ √ g 1 with the metric of AdS 4 given as g µν = ( x 3 ) 2 η µν Scalar propagators K ( x , y ) = g 2 K AdS ( x , y ) YM 2 x 3 y 3 m 2 = m 2 − 2 upon identifying ˜ [Nagasaki, Tanida, Yamaguchi (2011)] , [Buhl-Mortensen, de Leeuw, Ipsen, Kristjansen, MW (2016)] Matthias Wilhelm Quantum corrections in AdS/dCFT
Table of contents Motivation 1 Defect theory & framework for quantum corrections 2 One-point functions 3 Conclusion and outlook 4 Matthias Wilhelm Quantum corrections in AdS/dCFT
One-point functions in defect CFTs New feature of dCFTs: operators O can have nonvanishing one-point functions [Cardy (1984)] �O� = C x ∆ 3 ∆: scaling dimension of O , x 3 : distance to defect, C : constant Studied in this dCFT at tree level for BPS operators [Nagasaki, Tanida, Yamaguchi (2011)] and operators in the SU (2) sector [de Leeuw, Kristjansen, Zarembo (2015)] , [Buhl-Mortensen, de Leeuw, Kristjansen, Zarembo (2015)] , where integrability was found. Study loop corrections → Start with simplest operator: O ( x ) = tr( Z L )( x ) , Z ( x ) = φ 3 ( x ) + i φ 6 ( x ) BPS → corrections to C but not to ∆ Matthias Wilhelm Quantum corrections in AdS/dCFT
One-point functions at tree level Tree-level one-point function of O = tr( Z L ) [Nagasaki, Tanida, Yamaguchi (2011)] [de Leeuw, Kristjansen, Zarembo (2015)] 3 ) L ) = ( − 1) L = tr(( Z cl ) L ) = tr(( φ cl tr( t L �O� tree-level = t 3 ) x L 3 k � k − 2 i + 1 � L = ( − 1) L � x L 2 3 i =1 � 0 , L odd = � 1 − k � 2 − , L even 3 ( L +1) B L +1 x L 2 B L +1 ( u ): Bernoulli polynomial Matthias Wilhelm Quantum corrections in AdS/dCFT
One-loop corrections to one-point functions One-loop correction: two diagrams 1. Two quantum fields in O : 2. One quantum field in O , tadpole diagram one cubic vertex: lollipop diagram �O� 1-loop , tad = �O� 1-loop , lol = t t [Buhl-Mortensen, de Leeuw, Ipsen, Kristjansen, MW (2016)] Matthias Wilhelm Quantum corrections in AdS/dCFT
Tadpole diagram Tadpole diagram � tr( Z cl . . . ˜ Z . . . ˜ Z . . . Z cl ) �O� 1-loop , tad = = t Planar limit → quantum fields need to be adjacent Regulate scalar loop K ( x , x ) in dimensional regularisation in the d = 3 − 2 ε dimensions parallel to the defect Result: � 1 − k � λ 2 L �O� 1-loop , tad = − 3 ( L − 1) B L − 1 16 π 2 x L 2 [Buhl-Mortensen, de Leeuw, Ipsen, Kristjansen, MW (2016)] Matthias Wilhelm Quantum corrections in AdS/dCFT
Lollipop diagram Lollipop diagram � tr( Z cl . . . � ˜ Z � 1-loop . . . Z cl ) �O� 1-loop , lol = = t where � � � ˜ Z � 1-loop ( x ) = ˜ d 4 y Z ( x ) V 3 (Φ 1 , Φ 2 , Φ 3 )( y ) Φ 1 , Φ 2 , Φ 3 Result: � ˜ Z � 1-loop = 0 ⇒ �O� 1-loop , lol = 0 Crucially depends on the use of a supersymmetry-preserving regularisation scheme ` a la dimensional reduction! [Buhl-Mortensen, de Leeuw, Ipsen, Kristjansen, MW (2016)] Matthias Wilhelm Quantum corrections in AdS/dCFT
String-theory calculation Double-scaling limit suggested in [Nagasaki, Tanida, Yamaguchi (2011)] to compare gauge-theory and string-theory results and thus test AdS/dCFT: λ N → ∞ k → ∞ k ≪ N k 2 ≪ 1 Dual description of one-point function of O [Nagasaki, Yamaguchi (2012)] : y x 3 x 0 , 1 , 2 point-like string stretching from boundary of AdS 5 to D5 brane, calculable in supergravity approximation λ Suggests perturbative expansion in k 2 Matthias Wilhelm Quantum corrections in AdS/dCFT
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