Fluctuations of BPS Wilson loop and AdS 2 /CFT 1 Arkady Tseytlin S. - PowerPoint PPT Presentation

Fluctuations of BPS Wilson loop and AdS 2 /CFT 1 Arkady Tseytlin S. Giombi, R. Roiban, AT arXiv:1706.00756

• novel sector of observables in AdS/CFT: gauge-invariant correlators of operators inserted on Wilson loop • described by an effective (“defect” ) CFT 1 ”induced” from N = 4 SYM • case of 1 2 -BPS straight-line WL: example of AdS 2 /CFT 1 quantum theory in AdS 2 defined by superstring action • surprisingly, in BPS WL “vacuum” AdS/CFT map: elementary SYM fields ( ⊥ to the line) ↔ string coordinates as fields in AdS 2 (in static gauge) [cf. Tr ( Φ n ... D m F k ... ) ↔ closed-string vertex operators] • aim: compute 4-point correlators at strong coupling • Witten diagrams for AdS/CFT correlators, CFT methods (OPE, etc.), localization, etc. should also have connections to integrability

Review: gauge theory side N = 4 SYM: special Maldacena-Wilson loop operator dt ( i ˙ x µ A µ + | ˙ x | θ I Φ I ) � W = tr Pe generic x µ ( t ) closed loop, θ I ( t ) unit 6-vector: “locally” susy • special choices preserve parts of global superconf symm [Zarembo:02; Drukker,Giombi,Ricci,Trancanelli:07] • max 16 susy – 1 2 BPS: infinite straight line (or circle), θ I =const x 0 = t ∈ ( − ∞ , ∞ ) , θ I Φ I = Φ 6 dt ( iA t + Φ 6 ) � W = tr Pe • local O i ( t i ) on WL: gauge inv correlator [Drukker, Kawamoto:06] �� O 1 ( t 1 ) O 2 ( t 2 ) · · · O n ( t n ) �� dt ( iA t + Φ 6 ) O 2 ( t 2 ) · · · O n ( t n ) e dt ( iA t + Φ 6 ) � � � � � ≡ � tr P O 1 ( t 1 ) e

�� 1 �� = � W � = 1 and similar normalization for circle • operator insertions are equivalent to deformations of WL [Drukker, Kawamoto:06; Cooke, Dekel, Drukker:17] complete knowledge of correlators ↔ expectation value of general Wilson loop – deformation of line or circle cf. “wavy line” WL [Mikhailov:03; Semenoff, Young:04] • symmetries preserved by 1 2 -BPS WL vacuum: SO ( 5 ) ⊂ SO ( 6 ) R -symmetry: rotates 5 scalars Φ a , a = 1, . . . , 5 SO ( 2, 1 ) × SO ( 3 ) ⊂ SO ( 2, 4 ) : SO ( 3 ) rotations around line SO ( 2, 1 ) – dilatations, transl and special conf along line d = 1 conformal group + 16 supercharges preserved by line: d = 1, N = 8 superconformal group OSp ( 4 ∗ | 4 ) • operator insertions O ( t ) classified by OSp ( 4 ∗ | 4 ) reps labelled by dim ∆ and rep of “internal” SO ( 3 ) × SO ( 5 )

• correlators define ”defect” CFT 1 on the line [Drukker et al:06; Sakaguchi, Yoshida:07; Cooke et al:17] determined by spectrum of dims and OPE coeffs • �� ... �� correlators satisfy all usual properties of CFT: call O ( t ) “operators in CFT 1 ” without reference to their (non-local) origin in SYM • “elementary excitations”: short rep of OSp ( 4 ∗ | 4 ) 8 bosonic (+ 8 fermionic) operators with protected ∆ : 5 scalars: Φ a ( ∆ = 1) that do not couple to WL; 3 “displacement operators”: F ti ≡ iF ti + D i Φ 6 ( i = 1, 2, 3) with protected ∆ = 2 (WI for breaking of ⊥ translations)

• protected dims: exact 2-point functions in planar SYM �� Φ a ( t 1 ) Φ b ( t 2 ) �� = δ ab C Φ ( λ ) t 12 = t 1 − t 2 , t 2 12 C F ( λ ) �� F ti ( t 1 ) F tj ( t 2 ) �� = δ ij t 4 12 √ √ λ I 2 ( λ ) √ C Φ ( λ ) = 2 B ( λ ) , C F ( λ ) = 12 B ( λ ) , B ( λ ) = 4 π 2 I 1 ( λ ) B ( λ ) – Bremsstrahlung function [Correa, Henn, Maldacena, Sever:12] • 3-point functions of these elementary bosonic operators vanish by SO ( 3 ) × SO ( 5 ) symmetry • 4-point functions: depend on t 1 , ..., t 2 and λ constrained by 1d conf symm; only leading O ( λ 2 ) term in 4-point F ti known [Cooke et al:17]

String theory side 4-point functions at strong coupling ( N = ∞ , λ ≫ 1) Aim: from string theory in AdS 5 × S 5 • WL → open string minimal surfaces in AdS 5 ending on contour defining WL operator at the boundary • 1 2 -BPS Wilson line (or circle): minimal surface – AdS 2 embedded in AdS 5 (at fixed point on S 5 ) • fundamental open string stretched in AdS 5 : preserves same OSp ( 4 ∗ | 4 ) as 1 2 -BPS WL 1d conf group SO ( 2, 1 ) realized as isometry of AdS 2 • expanding string action around AdS 2 surface: AdS 2 multiplet of fluctuations transverse to string – 5 ( m 2 = 0) scalars y a in S 5 ; 3 ( m 2 = 2) scalars x i in AdS 5 ; 8 ( m 2 = 1) fermions [Drukker, Gross, AT:00]

• identify 8+8 fields in AdS 2 with elementary CFT 1 insertions [Sakaguchi, Yoshida:07; Faraggi, Pando Zayas:11; Fiol et al:13] • m 2 = ∆ ( ∆ − d ) for AdS d + 1 scalar masses and CFT d dims: massless S 5 fields y a should be dual to Φ a in CFT 1 with ∆ = 1 massive AdS 5 fields x i should be dual to F ti with ∆ = 2 • same spectrum as in “non-relat. limit” of AdS 5 × S 5 string [Gomis et al:05] and in OSp ( 4 ∗ | 4 ) invariant N = 8 superconformal QM [Belucci et al:03] • AdS/CFT: closed superstring vertex operators → single-trace gauge inv local operators in SYM; add open-string sector (strings ending at bndry) → gauge-inv operators = WL with insertions of local operators • here: only to insertions of ops with protected dims dual to “light” fields on AdS 2 string world-sheet

• open problem: description of unprotected insertions with large anom dim at λ = ∞ , e.g. insertion of Φ 6 [Alday, Maldacena:07] √ expect duals of “heavy” insertions to have m 2 ∼ 1 α ′ ∼ λ corresponding to massive states of open string; CFT 1 spectrum of ops on WL will have large ∆ gap ∼ λ 1/4 as for closed string states • other gauge-invariant correlators: (i) WL with single-trace ops e.g. � W tr Z J � open-closed string sector: closed string from worldsheet to bdry point away from line ( tr 2 : subleading at large N ) [Berenstein et al:98; Semenoff, Zarembo:01; Pestun, Zarembo:02] (ii) mixed correlators of ops on line and ops away from line

Strategy: string action → interaction vertices for ”light” AdS 2 fields → tree-level Witten diagrams in AdS 2 → prediction for 4-point functions of protected ops on WL: √ √ 1 d 2 σ λ � λ (action S = h ∂ x ∂ x + ...) √ expansion parameter d 5 x √ gR + ...) N 2 in 4-points in AdS 5 sugra: S = N 2 � 1 (cf. • AdS 2 QFT: superstring action UV finite – √ λ AdS 2 /CFT 1 duality should hold for any T = 2 π : • AdS 2 Witten diagrams with loops should be well defined e.g. 1-loop correction to boundary-to-boundary propagator protected 2-point function: subleading term in √ λ 8 π 2 + O ( 1 3 4 π 2 − B ( λ ) = λ ) √ checked earlier [Buchbinder, AT:13]

AdS 2 AdS 5 O(t 4 ) O(t 3 ) R 4 O(t 2 ) t O(t 1 ) • Plan: (i) compute tree-level 4-point functions (ii) use OPE to extract strong coupling corrections to dims of “2-particle” ops built of 2 of protected insertions: Φ ∂ n t Φ , etc. (iii) compare with localization to YM 2

• e.g. for singlet “2-particle” operator O = Φ a Φ a ��O ( t 1 ) O ( t 2 ) �� = C OO 5 ∆ O = 2 − λ + . . . , √ 2 ∆ O t 12 • correlators on circle WL: get by large conf transf; corr. of class of S 5 ops on circle WL captured by localization [Drukker, Giombi, Ricci, Trancanelli:07, Giombi,Pestun:09,12] compare result of Witten diagram calculation in AdS 2 to prediction of localization to solvable YM 2 on S 2 [Migdal:75; Witten:91] • open question: connection to integrability approach? [cf. TBA of Drukker:12; Correa, Maldacena, Sever:12 ] relation of AdS 2 Witten diagrams to factorization of 2d S-matrix in flat space?

AdS 5 × S 5 string in static gauge as AdS 2 bulk theory √ bosonic part of superstring action in AdS 5 × S 5 ( T = λ 2 π ) h h µν � 1 √ � + ∂ µ y a ∂ ν y a � ∂ µ x r ∂ ν x r + ∂ µ z ∂ ν z � S B = 1 d 2 σ � 2 T z 2 ( 1 + 1 4 y 2 ) 2 σ µ = ( t , s ) , r = ( 0, i ) = ( 0, 1, 2, 3 ) , a = 1, ..., 5 minimal surface for straight Wilson line at Euclidean boundary x 0 = t , x i = 0 , y a = 0 z = s , g µν d σ µ d σ ν = 1 s 2 ( dt 2 + ds 2 ) . induced metric is AdS 2 : • Aim: study correlators of small fluctuations of “transverse” coordinates ( x i , y a ) near AdS 2 minimal surface • global symmetry of action SO ( 2, 1 ) × [ SO ( 3 ) × SO ( 6 )]

• make SO ( 2, 1 ) symmetry manifest: choose AdS 2 adapted coordinates and fix static gauge: z = s and x 0 = t AdS5 = ( 1 + 1 4 x 2 ) 2 dx i dx i AdS2 = 1 ds 2 4 x 2 ) 2 ds 2 ds 2 z 2 ( dx 2 0 + dz 2 ) AdS2 + 4 x 2 ) 2 , ( 1 − 1 ( 1 − 1 • Nambu action in static gauge √ d 2 σ d 2 σ L B S B = T � h = T � h µν = ( 1 + 1 4 x 2 ) 2 g µν ( σ ) + ∂ µ x i ∂ ν x i 4 x 2 ) 2 4 x 2 ) 2 + ∂ µ y a ∂ ν y a g µν = 1 4 y 2 ) 2 , s 2 δ µν ( 1 − 1 ( 1 − 1 ( 1 + 1 • action of straight fundamental string in AdS 5 × S 5 along z : 2d theory of 3+5 scalars in AdS 2 with SO ( 2, 1 ) × [ SO ( 3 ) × SO ( 6 )] • bulk AdS 2 theory ↔ CFT 1 at z = s = 0 bndry: CFT 1 defined by operator insertions on straight WL