✬ ✩ Deformations of the circular Wilson loop, defect-CFT data and spectral independence Nadav Drukker Based on: arXiv:1703.03812 - M. Cooke, A. Dekel and N.D. arXiv:18xx.xxxxx - M Cooke, A. Dekel, N.D., D. Trancanelli and E. Vescovi Workshop on Supersymmetric Localization and Holography: Black Hole Entropy and Wilson Loops ICTP, Trieste July 9, 2018 ✫ ✪ 1
✬ ✩ 1 / 2 BPS circular Wilson loop • The simplest and most symmetric Wilson loop in a CFT is a circle. • It preserves an SL (2 , R ) subgroup of the full conformal group. • In the case of N = 4 SYM it also preserves 1 / 2 of the supercharges and an OSp (4 ∗ | 4) supergroup which includes SL (2 , R ) × SO (3) × SO (5). � Erickson,Semeno ff � � drukker � � � • Its expectation value is well known Pestun Zarembo Gross � √ ⟨ W ⟩ 0 = 1 2 N − 1 ( λ / 4 N ) e λ / 8 N ∼ N L 1 � √ I 1 λ λ ✫ ✪ Nadav Drukker 2 deformed circle
✬ ✩ Deformations of the circle • We will view all closed loops as deformations of the circle. • Consider a Wilson loop in N = 4 SYM following a path in R 2 given by X ( θ ) = x 1 ( θ ) + ix 2 ( θ ) = e i θ + g ( θ ) . • It is convenient to write g ( θ ) in a Fourier decomposition ∞ � b n e in θ g ( θ ) = n = −∞ and without loss of generality g ( θ ) is real, so b − n = ¯ b n . • The expectation value of the Wilson loop can be written in an expansion in powers of b n ⟨ W ⟩ 2 n ∼ O ( b 2 n ) ⟨ W X ⟩ = ⟨ W ⟩ 0 + ⟨ W ⟩ 2 + ⟨ W ⟩ 4 + · · · , • At order b 0 we have the circular Wilson loop whose VEV I quoted alread. • As I will review, order b 2 is also known to all orders in the coupling. What I focus on is ⟨ W ⟩ 4 . ✫ ✪ Nadav Drukker 3 deformed circle
✬ ✩ Insertions into the circle � Drukker � Kawamoto • One can insert any number of adjoint valued operators into the Wilson loop O (1) ( x ( s 1 )) ... O ( n ) ( x ( s n )) � � � � = 1 � � ( i ˙ x | Φ 1 ( x ( s )) ) ds � O (1) ( x ( s 1 )) ... O ( n ) ( x ( s n )) e x µ A µ ( x ( s )) − | ˙ N tr P . • For example O can be a scalar field Φ I or the field strength F µ ν . • This is true for any Wilson loop. In the case of the circle we have Ward identities for conformal symmetry so for two insertiona a O (1) O (2) ( λ ) O (1) (0) O (2) ( θ ) � � � � = 2 ) ∆ O (1) + ∆ O (2) , (2 sin θ • For three scalar primary insertions c (123) ( λ ) O (1) ( θ 1 ) O (2) ( θ 2 ) O (3) ( θ 3 ) � � � � = | d 12 | ∆ 1 + ∆ 2 − ∆ 3 | d 13 | ∆ 1 − ∆ 2 + ∆ 3 | d 23 | − ∆ 1 + ∆ 2 + ∆ 3 , • In the case of the four point function we already have a single cross ratio. • These insertions have normalizations, dimensions and structure constants and should satisfy the OPE. • What can we say about those and how are they related to deformations of the circle? ✫ ✪ Nadav Drukker 4 deformed circle
✬ ✩ Outline • Introduction. • Deformed circle and defect CFT. • Integrability. • Perturbation theory. • dCFT data. • Bremsstrahlung function. • Results for ⟨ W ⟩ 4 . • Construction of string solution for near circular Wilson loop. • Spectral parameter (in)dependence. • Conclusions. ✫ ✪ Nadav Drukker 5 deformed circle
✬ ✩ Deformed circle and defect CFT • Deformations away from the circle can be represented by insertions of adjoint valued x ν . fields into the Wilson loop. Normally the first insertion is F µ ν ˙ • For a radial deformation of the circular Maldacena-Wilson loop this is replaced with F r φ = F r φ + iD r Φ 1 • All insertions can be classified by representations of OSp (4 ∗ | 4), and this first insertion, the displacement operator is in fact a protected operator of dimension 2. • The only insertions of classical dimension one are scalar fields Φ I . They decompose to the singlet Φ 1 and the 5 of SO (5). • The 5 is also protected, it’s a superpartner of the displacement operator. • The singlet is not protected. ✫ ✪ Nadav Drukker 6 deformed circle
✬ ✩ • Some operators of classical diemension two are D µ Φ 1 , D µ Φ a , Φ 1 Φ 1 , Φ 1 Φ a , Φ a Φ 1 , Φ a Φ b . iF i φ , , iF ir , iF ij , • They can be arranged in representations of the symmetry group of the Wilson loop including into supermultiplets (notations slight simpler when considering insertions the line instead of the circle). • There are also fermionic insertions, of course. • Details can be found in my paper... • Can calculate the dimensions and normalizations from – Perturbation theory. – AdS /CFT. – Integrability. – Localization. – Bootstrap. ✫ ✪ Nadav Drukker 7 deformed circle
✬ ✩ Integrability � � � � Correa Drukker Maldacena,Sever • One can use integrability to calculate the anomalous dimension of a cusped Wilson loop. • A Wilson loop is described by an open string in AdS , this translates to an open spin-chain (or other integrable model). • It is non-trivial, but true, that the boundary conditions appropriate for a cusp satisfy the boundary Yang-Baxter equation. • The same formalism allows to calculate a cusp with an operator insertion: – The insertion of Z L is the length L ground state of the system. – All other insertions can be viewed as excitations of this state. – We can find the anomalous dimensions of insertions into the circle by taking the cusp angle to be zero. • This procedure has not been applied in this case. ✫ ✪ Nadav Drukker 8 deformed circle
✬ ✩ Perturbation theory • One can consider the insertion of any operator and calculate Feynman diagrams. • We chose instead to look at smooth Wilson loops, for which the one loop VEV is I ( s 1 , s 2 ) = ˙ x 1 · ˙ x 2 + | ˙ x 1 || ˙ x 2 | λ � ⟨ W [ C ] ⟩ 1-loop = − ds 1 ds 2 I ( s 1 , s 2 ) , . x 2 16 π 2 12 • For curves in R 2 there is also a compact formula for two loop graphs � Bassetto,Griguolo � Pucci,Seminara λ 2 log x 2 ds 1 ds 2 ds 3 � ( s 1 , s 2 , s 3 ) I ( s 1 , s 3 ) x 32 · ˙ � x 2 21 ⟨ W [ C ] ⟩ 2-loop = − x 2 x 2 128 π 4 32 31 � 2 + λ 2 λ 2 � 1 � � ds 1 ds 2 I ( s 1 , s 2 ) − ds 1 ds 2 ds 3 ds 4 I ( s 1 , s 3 ) I ( s 2 , s 4 ) . 16 π 2 64 π 4 2 s 1 >s 2 >s 3 >s 4 • We have found e ffi cient algorithms to calculate these integrals for arbitrary curves, and then using the relation between deformations and insertions, extracted some CFT data. ✫ ✪ Nadav Drukker 9 deformed circle
✬ ✩ • Going back to a curve parametrized as X ( θ ) = x 1 ( θ ) + ix 2 ( θ ) = e i θ + g ( θ ) . • We can now expand the VEV of the WIlson loop in powers of g ( θ ), giving correlation functions in the defect-CFT, which are (schematically) � ⟨ W ⟩ = ⟨ W ⟩ 0 + g ( θ 1 ) g ( θ 2 ) ⟨ ⟨ F ( θ 1 ) F ( θ 2 ) ⟩ ⟩ d θ 1 d θ 2 � g 2 ( θ 1 ) g 2 ( θ 2 ) ⟨ + ⟨ D F ( θ 1 ) D F ( θ 2 ) ⟩ ⟩ d θ 1 d θ 2 � g 2 ( θ 1 ) g ( θ 2 ) g ( θ 3 ) ⟨ + ⟨ D F ( θ 1 ) F ( θ 2 ) F ( θ 3 ) ⟩ ⟩ d θ 1 d θ 2 d θ 3 � + g ( θ 1 ) g ( θ 2 ) g ( θ 3 ) g ( θ 4 ) ⟨ ⟨ F ( θ 1 ) F ( θ 2 ) F ( θ 3 ) F ( θ 4 ) ⟩ ⟩ d θ 1 d θ 2 d θ 3 d θ 4 + · · · • Expanding this further in powers of the coupling, we should find a match with the result of the Feynman diagram calculation. • Note that the one-loop graph sees only 2-point functions. ✫ ✪ Nadav Drukker 10 deformed circle
✬ ✩ The dCFT data • Matching the two expressions we find for the displacement operator: λ 2 a F = − 3 λ 32 π 2 + O ( λ 3 ) , 4 π 2 + γ F = 0 . • Some three point functions are c 0 Φ η ij i F i 3 ( s 1 ) i F j 3 ( s 2 ) Φ 1 ( s 3 ) � � � � = | s 12 | 3 | s 13 || s 23 | + O ( λ ) , = c 0 3 ( η ik η jm − η im η jk ) � � � � + O ( λ ) , i F i 3 ( s 1 ) i F j 3 ( s 2 ) iF km ( s 3 ) | s 12 | 2 | s 13 | 2 | s 23 | 2 = c 0 η ik η jm + η im η jk − 2 � � 3 η ij η km 5 � � � � i F i 3 ( s 1 ) i F j 3 ( s 2 ) iD { k F m } 3 ( s 3 ) + O ( λ ) , | s 12 | 2 | s 13 | 2 | s 23 | 2 and we found 1 1 5 c 0 c 0 c 0 Φ = − 32 π 4 , 3 = 16 π 4 , 5 = 16 π 4 . • For the unprotected singlet scalar we reproduced 1 1 a 0 Φ = 8 π 2 , γ Φ = 4 π 2 . • For the triplet and quintet states we calculated 3 = − 1 1 5 = 5 1 a 0 a 0 2 π 2 , γ 3 = 4 π 2 , π 2 , γ 5 = 4 π 2 . ✫ ✪ Nadav Drukker 11 deformed circle
✬ ✩ AdS /CFT � Giombi,Roiban � Tseytlin • The same story can be repeated using deformations propagating on a semiclassical string in AdS 5 . • One calculates Witten diagrams in the AdS 2 world-sheet. • For the singlet they find 5 √ + · · · γ Φ = 2 − λ • And some structure constant C ΦΦ ( Φ 2 ) = 2 43 5 − √ + · · · 30 λ • They used the OPE decomposition of the 4-point function to extract this and other structure constants. ✫ ✪ Nadav Drukker 12 deformed circle
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