The SL(2) sector of at strong coupling Ivan Kostov IPhT-Saclay - PowerPoint PPT Presentation

The SL(2) sector of at strong coupling Ivan Kostov IPhT-Saclay with Didina Serban and Dmytro Volin arXiv:hep-th/0703031, arXiv:0801.2542 GGI, Florence, 30 November 2008

The sl (2) sector of PSU (2,2|4) Excitations in the sl(2) sector: Lorentz spin Twist C lassical folded Gubser- strings propagating Klebanov- D M + Z L � � in AdS 3 x S 1 tr + . . . Polyakov’02 Bethe Ansatz equations: At one loop: [XXX] - ½ spin chain � 2 � � � � L M k − u + 1 − 1 /x + u − k x − � x + � Dressing phase j j e 2 i θ ( u k ,u j ) k = x − u + k − u − 1 − 1 /x − k x + j j k j � = k g 2 = g 2 YM N � ≡ 1 u ± = u ± i � , x ± = x ( u ± ) 16 π 2 . 4 g, u u ( x ) ≡ 1 � x + 1 � 2 x Large M limit: � � � 1 − 1 (BES) Beisert-Eden-Staudacher’06 ( L finite ) , x ( u ) = u 1 + u 2 Freyhult-Rej-Staudacher’07 (FRS) ( L~ Log M )

∆ = M + L + f ( g, L ) ln M + . . . Anomalous dimension for large M : universal scaling function Korchemsky’89; = cusp anomalous dimension GKP’02

∆ = M + L + f ( g, L ) ln M + . . . Anomalous dimension for large M : universal scaling function Korchemsky’89; = cusp anomalous dimension GKP’02 Provides a critical test of AdS/CFT: � 73 � f ( g ) = 8 g 2 − 8 3 π 2 g 4 + 88 630 π 6 + 4 ζ (3) 2 Weak coupling g 8 ± . . . . 45 π 4 g 6 − 16 expansion: From perturbative 3-loop guess SYM up to g 8 4-loop result [Moch, Vermasseren, Vogt’04; Lipatov at al’04] [Bern et al’06]

∆ = M + L + f ( g, L ) ln M + . . . Anomalous dimension for large M : universal scaling function Korchemsky’89; = cusp anomalous dimension GKP’02 Provides a critical test of AdS/CFT: � 73 � f ( g ) = 8 g 2 − 8 3 π 2 g 4 + 88 630 π 6 + 4 ζ (3) 2 Weak coupling g 8 ± . . . . 45 π 4 g 6 − 16 expansion: From perturbative 3-loop guess SYM up to g 8 4-loop result [Moch, Vermasseren, Vogt’04; Lipatov at al’04] [Bern et al’06] Strong coupling f ( g ) = 4 g − 3 log 2 − K 1 expansion: g + . . . 4 π 2 π From string perturbation theory [Gubser,Klebanov, Frolov,Tseytlin’02 Roiban,Tseytlin’07 Polyakov’02]

∆ = M + L + f ( g, L ) ln M + . . . Anomalous dimension for large M : universal scaling function Korchemsky’89; = cusp anomalous dimension GKP’02 Provides a critical test of AdS/CFT: � 73 � f ( g ) = 8 g 2 − 8 3 π 2 g 4 + 88 630 π 6 + 4 ζ (3) 2 Weak coupling g 8 ± . . . . 45 π 4 g 6 − 16 expansion: From perturbative 3-loop guess SYM up to g 8 4-loop result [Moch, Vermasseren, Vogt’04; Lipatov at al’04] [Bern et al’06] Strong coupling f ( g ) = 4 g − 3 log 2 − K 1 expansion: g + . . . 4 π 2 π From string perturbation theory [Gubser,Klebanov, Frolov,Tseytlin’02 Roiban,Tseytlin’07 Polyakov’02] [ Klebanov et al ’06, Both expansions Basso, Korchemsky, Casteill, Kotikov,Lipatov’06, Kotanski’07; should be reproduced Kristjansen’07; Alday et al ’07; IK, Serban, Volin’08 from BA equations Belitsky’07 I.K., Serban, Volin’07] (BES equation was taylored so that the weak coupling expans on is reproduced)

Functional Equation for resolvents at one loop ( x=2u , no dressing factor) M Baxter’s equation for : � Q ( u ) = ( u − u k ) k =1 T ( u ) = Q ( u + 2 i � ) ( u + i � ) L + Q ( u − 2 i � ) ( u − i � ) L Q ( u ) Q ( u ) For M → ∞ with u finite only one of => linear equations for the the terms of the Baxter equation survives magnon and hole resolvents j R m ( u ) ∼ d log Q (1 − D 2 ) R m + R h = ( � u > 0) ∼ du u + i � du j R h ( u ) ∼ d log T (1 − D − 2 ) R m + R h = ( � u < 0) u − i � du D = e i �∂ u : D is a shift operator: Df ( u ) = f ( u + i � ) j is related to L by , j = L/ log( M � ) → ∞ ± → ∓ R h → j ( u → ∞ ) . u 1 <<|u | << M ϵ : the density is => asymptotic R m → ∓ i ( u → ∞ ± i 0) conditions at infinity constant, of order Log(M ϵ ) �

Functional-integral equation at all orders (BES/FRS) R m ( u ) → − i � − j 2 u − 1 The universal scaling function can be extracted from 2 uf ( � , � ) + ... the behavior of the magnon resolvent at infinity: (1 − D 2 + K ) R m + R h = j Dd log x (UHP) du -- the kernel is given by the “magic formula” of � D 2 � K = D K − + K + + 2 K − 1 − D 2 K + D BES in terms of the even/odd kernels K ± K ± ( u, v ) = − 1 d � 1 − 1 � � 1 + 1 � � � ln ∓ ln 2 π i du xy xy x = x ( u + i 0) , y = y ( v − i 0) � K ± F ( u ) = dv K ± ( u, v ) F ( v ) = R − i 0 For functions F(u) analytic in UHP and the real axis and decaying faster than 1/u IK, Serban, Volin’08 1+ i 0 � v 2 − 1 � dv F ( v + i 0) ± F ( − v + i 0) K ± F ( u ) ≡ u 2 − 1 2 π i v − u − 1+ i 0 � 1 � a � 1 1 a � 1 � a 1 a � a 1 a

BES/FRS equation in the x-plane Express magnon resolvent R m ~ ∑ ( u-u i ) -1 in terms of resolvent in x -space S ~ ∑ ( x-x i ) -1 R m ( u ) = S ( x ) + S (1 /x ) and require that ( D - D -1 ) S ( x ) has at most a simple pole at x = ±1. Then the action of K + drastically simplifies: to any order in ϵ , K + DR m = ( D − D − 1 ) S (1 /x ) and the BES/FRS equation becomes ( D − 1 − D ) S ( x ) + K − D [ S ( x ) − S (1 /x )] + D − 1 R h = j ∂ u log x : (upper half plane u ) − − ( D − D − 1 ) S ( x ) − K − D − 1 [ S ( x ) − S (1 /x )] + DR h = j ∂ u log x . : (lower half plane u ) √ b 2 − x 2 − j � � S ( x ) = 1 Solution in the leading � , b = 1 + ( j � ) 2 − order (first obtained by x − 1 � x Casteil-Kristjansen’07) 2.3 2.3 2.2 2.2 2.1 2.1 j 2 + 16 g 2 log M � 2.0 2.0 ∆ = M + 1.8 1.6 1.4 1.2 1.2 1.4 1.6 1.8 √ Can be solved perturbatively in ϵ . The second order found by D. Volin’08 confirms the (formidable) calculation by N. Gromov’08.

The case j =0: BES equation The ϵ expansion is not uniform: two different strong coupling limits [IK, Serban, Volin’07] ϵ ➔ 0 with u fixed (Plane Waves/ Giant Magnons) ϵ ➔ 0 with z = ( u- 1 ) / ϵ fixed (Near Flat Space) u � ( ) NFS PW GM PW u � 1 1

BES equation ( j =0): Complete perturbative (in ϵ ) solution Basso, Korchemsky, Kotanski’07; IK, Serban, Volin’08 At j → 0 : homogeneous equation: ( D − D − 1 ) S (1 /x ) = K + D [ S ( x ) + S (1 /x )] (UHP) ( D − D − 1 ) S ( x ) = K − D [ S ( x ) − S (1 /x )] : 1 1 => S ( x + i 0) + S ( x − i 0) = 0 (valid perturbatively in ϵ ) S ( x ) + S ( − x ) = 0 S ( x ) → ∓ i � , ( x → ∞ ± i 0) � √ S ( x ) = 1 1 − x 2 Alday, Arutyunov, Benna, Solution in the leading order: x − 1 Eden, Klebanov’07 � x

1) Solution in the PW regime (| u | >1) General solution of the homogeneous equations: ∞ � 2 k c + � 2 k c − S = 1 x k [ � ] k [ � ] � (1 − x 2 ) 2 k + (1 − x 2 ) 2 k +1 . √ � 1 − x 2 k =0 The solution has 2 singular points: at x = ±1 or u = ±1 (NFS regime) . The coefficients can be fixed by comparing with the expansion near the singular points in the rescaled variable z = u − 1 �

From the homogeneous equations: G ± = 1 ± i -- analytic in ℂ / [- ∞ ,-1] ⋃ [1,+ ∞ ] ( D ∓ iD − 1 )[ S ( x ) ± iS (1 /x )] 2 ± ∓ 2 -- analytic in ℂ / [-1,1] g ± = ± i ( D − D − 1 )[ S ( x ) ± iS (1 /x )] g ± = 1 ± i D ∓ i ( D − 1) G ± h z = u − 1 Inverse Laplace w.r.t. 2 � Γ [ s s 2 Γ [ 1 2 π ± 1 2 π ] √ 4 ] 2 − ˜ 4 ]˜ g ± ( s ) = ± G ± ( s ) . s s Γ [ 1 2 π ∓ 1 Γ [1 − 2 π ] 2 + => no poles, only a branch analytic everywhere except the analytic everywhere except the cut [0, ∞] negative real axis positive real axis. expansion of rhs at s = ∞ coincides with expansion of lhs at s = 0 (known) the coefs c k ( ϵ )

3 different scaling regimes: Extend the method for the case when both S and L are large. L ~ log M Freyhult, Rej, Staudacher’ 07 Three different regimes: L /(g log M)~1 L /(g log M)~g -1/4 L /(g log M)~e -ag B. Basso, G. “Double scaling limit” Korchemsky’08 N. Gromov’08 Fioravanti, D. Volin, 08 Grinza,Rossi’08 Alday, O(6) Maldacena’07

Integral equation for the sl(2) sector (BES/FRS) u ± = u ± i � , x ± = x ( u ± ) � 2 � � � � L M k − u + 1 − 1 /x + u − k x − � x + � j j e 2 i σ ( u k ,u j ) k = u + k x + x − k − u − 1 − 1 /x − u ( x ) ≡ 1 � x + 1 � j j k j � = k 2 x � ≡ 1 u -- Repulsive interaction 4 g, � � � 1 − 1 => Bethe roots on the real axis , x ( u ) = u 1 + u 2 Take log, specify the root (mode number n k ) for each u k . In the limit M → ∞ n k => Integral equation for the magnon density ρ ( u ) = dk/du L k magnons holes magnons M+L