Quantum strings in AdS 5 S 5 and gauge-string duality

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1 Quantum strings in AdS 5 S 5 and gauge-string duality Arkady Tseytlin Review Quantum string corrections to dimension of short operators [ R. Roiban, AT, to appear]

2 Summary: planar N =4 SYM = g 2 N YM c cusp anomalous dimension f( 1) = [ 2π ( 73 f( 1) = π [ 1 3 ln 2 K ( ) 2...] (ζ(3))2 π 6 ) ] BES integral equation: any number of terms in expansions known anomalous dimension of Konishi operator γ( 1) = 12 [ (4π) (4π) (4π) 4 3 ] + [ ζ(3) 120ζ(5)] (4π) [ γ( 1) = b 1 ] + O( ( ) ) 2 b - leading correction to mass of lightest AdS 5 S 5 string state higher order terms? integral equation for γ?

3 AdS/CFT: progress largely using limited tools of supergravity + classical probe actions To go beyond: understand quantum string theory in AdS 5 S 5 Problems for string theory: find spectrum of states: energies/dimensions as functions of = g 2 N YM c construct vertex operators: closed and open (?) strings compute their correlation functions scattering amplitudes compute expectation values of Wilson loops generalizations to simplest less supersymmetric cases...

4 tree-level AdS 5 S 5 superstring = planar N = 4 SYM Recent remarkable progress in quantitative understanding interpolation from weak to strong t Hooft coupling based on/checked by perturbative gauge theory (4-loop in ) and perturbative string theory (2-loop in 1 ) data and (strong evidence of) exact integrability string energies = dimensions of local Tr(...) operators E(, C, m,...) = (, C, m,...) C - charges of SO(2, 4) SO(6): S 1, S 2 ; J 1, J 2, J 3 m - windings, folds, cusps, oscillation numbers,... Operators: Tr(Φ J 1 1 ΦJ 2 2 ΦJ 3 3 DS 1 + D S 2...F mn...ψ...) Solve supersymmetric 4-d CFT = Solve string in curved R-R background (2-d CFT): compute E = for any (and any C,m)

5 Problem: perturbative expansions are opposite 1 in perturbative string theory 1 in perturbative gauge theory weak-coupling expansion convergent defines () need to go beyond perturbation theory: integrability Last 7 years remarkable progress for subclass of states: semiclassical string states with large quantum numbers dual to long SYM operators (canonical dim. 0 1) [BMN 02, GKP 02, FT 03,...] E = same (in some cases!) dependence on C, m,... with coefficients = interpolating functions of Current status: 1. Long operators = strings with large quantum numbers: Asymptotic Bethe Ansatz (ABA) [Beisert, Eden, Staudacher 06] firmly established (including non-trivial phase factor) 2. Short operators = general quantum string states:

6 Partial progress based on improving ABA by Luscher corrections [Janik et al 08] Attempts to generalize ABA to TBA [Arutyunov, Frolov 08] Very recent (complete?) proposal for underlying Y-system [Gromov, Kazakov, Vieira 09] To justify from first principles need better understanding of quantum AdS 5 S 5 superstring theory: 1. Solve string theory on a plane R 1,1 relativistic 2d S-matrix asymptotic BA for the spectrum 2. Generalize to finite-energy closed strings the theory on R S 1 TBA (cf. integrable sigma models) Reformulation in terms of currents with Virasoro conditions solved ( Pohlmeyer reduction ) is promising approach [Grigoriev, AT 07; Roiban, AT 09]

7 Superstring theory in AdS 5 S 5 bosonic coset SO(2,4) SO(1,4) SO(6) SO(5) P SU(2,2 4) generalized to supercoset SO(1,4) SO(5) [Metsaev, AT 98] [ S = T d 2 σ G mn (x) x m x n + θ(d + F 5 )θ x + θθ θθ x x ] +... = 2π tension T = R2 2πα Conformal invariance: β mn = R mn (F 5 ) 2 mn = 0 Classical integrability of coset σ-model (Luscher-Pohlmeyer 76) true for AdS 5 S 5 superstring [Bena, Polchinski, Roiban 02] Progress in understanding of implications of (semi)classical integrability [Kazakov, Marshakov, Minahan, Zarembo 04,...]

8 1-loop quantum superstring corrections [Frolov, AT; Park, Tirziu, AT, 02-04,...] used as an input data to fix 1-loop term in strong-coupling expansion of the phase θ() in ABA [Beisert, AT 05; Hernandez, Lopez 06] 2-loop quantum superstring corrections [Roiban, Tirziu, AT; Roiban, AT 07] check of finiteness of the GS superstring implicit check of integrability of quantum string theory non-trivial confirmation of BES phase in ABA [Basso, Korchemsky, Kotansky 07]

9 Gauge states vs string states: principles of comparison 1. compare states with same global SO(2, 4) SO(6) charges e.g., (S, J) sl(2) sector Tr(D+Φ S J ) J=twist= spin-chain length 2. assume no level crosing while changing min/max energy (S, J) states should be in correspondence Gauge theory: E = S + J + γ(s, J, m, ), γ = k=1 k γ k (S, J, m) fix S, J,... and expand in ; then may expand in large/small S, J,... Semiclassical string theory: E = S + J + γ(s, J, m, ), γ = 1 k= 1 ( γ ) k(s, J, m) k fix semiclassical parameters S = S, J = J, m and expand in 1

10 To match in general will need to resum beyond ABA Various special limits studied: (i) Fast strings locally-bps long operators S GT: J 1, J =fixed ST: J 1, S J =fixed [BMN; Frolov, AT03; Beisert, Minahan, Staudacher, ] Zarembo03] E = S + J + J [h 1 ( SJ, m) + 1J h 2( SJ, m) (ii) Slow long strings long non-bps operators [GKP02] GT: ln S J 1 ST: ln S J, J = 0 or J =fixed E = S + f() ln S +... f( 1) = a , f( 1) = c (iii) Fast long strings GT: S J 1, j J ln S =fixed ST: S J 1, l J ln S =fixed = j [Belitsky, Gorsky, Korchemsky 06; Frolov, Tirziu, AT 06;...]

11 Key example of long operators Tr(ΦD+Φ) S dual to spinning string Folded spinning string in flat space: X 1 = ɛ sin σ cos τ, X 2 = ɛ sin σ sin τ ds 2 = dt 2 + dx i dx i = dt 2 + dρ 2 + ρ 2 dφ 2 t = ɛτ, ρ = ɛ sin σ, φ = τ tension T = 1 2πα 2π energy E = ɛ and spin S = ɛ2 2 Regge relation: E = 2 S Folded spinning string in AdS 5 : [Gubser, Klebanov, Polyakov 02] ds 2 = cosh 2 ρ dt 2 + dρ 2 + sinh 2 ρ dφ 2 t = κτ, φ = wτ, ρ = ρ(σ)

12 sinh ρ = ɛ sn(κɛ 1 σ, ɛ 2 ), 0 < ρ < ρ max coth ρ max = w 1 κ + 1 ɛ 2 ɛ measures length of the string κ = ɛ 2 F 1 ( 1 2, 1 2 ; 1; ɛ2 ) classical energy E 0 = E 0 and spin S = S E 0 = ɛ 2 F 1 ( 1 2, 1 2 ; 1; ɛ2 ), S = ɛ2 1 + ɛ 2 2 2F 1 ( 1 2, 3 2 ; 2; ɛ2 ) solve for ɛ analog of Regge relation E 0 = E 0 (S), E 0 = E 0 ( S )

13 short/long string flat space/ads interpolation: E 0 (S 1) = 2S +... E 0 (S 1) = S + 1 π ln S +... S : folds reach the boundary (ρ = ) solution drastically simplifies: length κ ln S t = κτ, φ κτ, ρ κσ, κ ɛ ln S E = S from massless end points at AdS boundary (null geodesic) E S π ln S from tension/stretching of the string quantum superstring corrections to E respect S + ln S form: Semiclassical string theory limit 1. 1, S = S = fixed; 2. S 1 E = S + f() ln S +..., [ a f( 1) = a 2 π ( ) +...] 2

14 a n Feynmann graphs of 2d CFT AdS 5 S 5 superstring a 1 = 3 ln 2: Frolov, AT 02 a 2 = K: Roiban, AT 07 K = k=0 ( 1) k (2k+1) 2 = (2-loop σ-model integrals) Gauge theory: dual operators Tr(ΦD+Φ), S S 2 = O() same ln S asymptotics of anomalous dimensions from symmetry argument [Alday, Maldacena 07] Perturbative gauge theory limit: 1. 1, S = fixed; 2. S 1 S 2 = f() ln S +... f( 1) = c 1 + c c c = 1 [ 2π ( 73 ] (ζ(3))2 π 6 )

15 c n are given by Feynmann graphs of 4d CFT N=4 SYM c 3 : Kotikov, Lipatov, et al 03; c 4 : Bern, Czakon, Dixon, Kosower, Smirnov 06; String and gauge limits are formally different but leading ln S term is universal single f() provides smooth interpolation Remarkably, both expansions are reproduced from single BES integral equation for f() [strong coupling expansion: numerical Benna, Benvenuti, Klebanov, Scardicchio 07; analytic Basso, Korchemsky, Kotansky 07] both expansions thus known in principle to any order exact expression for f() from BES equation? asymptotic nature of strong-coupling expansion: non-perturbative e 1 2 terms [BKK] fixed by AdS 5 S 5, symmetry?

16 Subleading terms in large S expansion string has large but finite length: does not reach boundary E 0 (S 1) = S + a 0 ln S + a S (a 2 ln S + a 3 ) + 1 S 2 (a 4 ln 2 S + a 5 ln S + a 6 ) + O( ln3 S S 3 ) a 0 = π, a 1 = π ln(8π) 1,... coeffs of lnk S happen to be related to coeff of ln S: S k a 2 = 1 2 a2 0, a 4 = 1 8 a3 0,... according to functional relation [Basso, Korchemsky 06] E S = f(e + S) = a 0 ln(s a 0 ln S +...) +... Why? In near-boundary limit for large S string end moves along nearly null line at the boundary: pp-wave limit cusp anomaly as pp-wave anomaly

17 pp-wave limit effectively establishes contact with collinear conformal group in the boundary theory [Kruczenski, AT 08; Ishizeki, Kruczenski, Titziu, AT 08] Some of coefficients in large S expansion are related due to reciprocity property in gauge theory true also at strong coupling [Basso, Korchemski 06; Beccaria, Forini, Tirziu, AT 08] Comparison of semiclassical string theory expansion to large S gauge theory expansion: E = S + f() ln S + h() + O( 1 S ) f and h are controlled by infinite length limit ABA but subleading coefficients require knowledge of wrapping / finite size corrections comparison in general will require resummation of the series

18 Dimensions of short operators = energies of quantum string states: progress in understanding spectrum of conformal dimensions of planar N = 4 SYM or spectrum of strings in AdS 5 S 5 based on quantum integrability Spectrum of states with large quantum numbers solution of ABA equations key example: cusp anomaly function Recent proposal of how to extend this to short states with any quantum numbers TBA or Y-system approach so far not checked/compared to direct quantum string results Aim: compute leading α 1 correction to dimension of lightest massive string state dual to Konishi operator in SYM theory data for checking future (numerical) prediction of Y-system

19 Konishi operator: operators (long multiplet) related to singlet [0, 0, 0] 2 (0,0) by susy = 0 + γ(), 0 = 2, 5 2, 3,..., 10 same anomalous dimension γ singlet eigen-state of anom. dim. matrix with lowest eigenvalue examples: Tr( Φ i Φ i ), i = 1, 2, 3, 0 = 2 Tr([Φ 1, Φ 2 ] 2 ) in su(2) sector 0 = 4 Tr(Φ 1 D+Φ 2 1 ) in sl(2) sector 0 = 4 Weak-coupling expansion of γ(): = g 2 N YM c [ γ() = 12 (4π) (4π) (4π) 6 4 ] + [ ζ(3) 120ζ(5)] (4π) [Fiamberti,Santambrogio,Sieg,Zanon; Bajnok,Janik; Velizhanin 08]

20 Finite radius of convergence (N c = ) if we could sum up and then re-expand at large what to expect? (cf. f()) AdS/CFT duality: Konishi operator dual to lightest among massive AdS 5 S 5 string states large = R2 α : small string at center of AdS 5 in nearly flat space 1 : ( 4) = 4 + a + O( 1 ) [ 2 = b ] 1 + O( ( ), b = 1 (a + 4) ) 2 8 a = first correction to mass of dual string state Evidence below: a = 4, b = 0

21 Flat space case: m 2 = 4(n 1) α, n = 1 2 (N + N) = 1, 2,..., N = N n = 1: massless IIB supergravity (BPS) level l.c. vacuum 0 >: (8 + 8) 2 = 256 states n = 2: first massive level (many states, highly degenerate) [(a i 1 + S a 1) 0 >] 2 = [(8 + 8) (8 + 8)] 2 in SO(9) reps: ([2, 0, 0, 0] + [0, 0, 1, 0] + [1, 0, 0, 1]) 2 = ( ) 2 e.g = = switching on AdS 5 S 5 background fields lifts degeneracy states with lightest mass at 1-st excited level should correspond to Konishi multiplet

22 string spectrum in AdS 5 S 5 : long multiplets A [k,p,q](j,j ) of P SU(2, 2 4) highest weight states: [k, p, q](j, j ) labels of SO(6) SO(4) Remarkably, flat-space string spectrum can be re-organized in multiplets of SO(2, 4) SO(6) P SU(2, 2 4) [Bianchi, Morales, Samtleben 03] SO(4) SO(5) SO(9) rep. lifted to SO(4) SO(6) rep. of SO(2, 4) SO(6) Konishi long multiplet T 1 = (1 + Q + Q Q +...)[0, 0, 0] (0,0) determines the KK floor of 1-st excited string level H 1 = J=0 [0, J, 0] (0,0) T 1

23 One expects for scalar massive state in AdS 5 ( 2 + m 2 )Φ +... = 0 ( 4) = (mr) 2 + O(α ) = 4(n 1) R2 α + O(α ) = 2 + (mr) O(α ) ( 1) = [Gubser, Klebanov, Polyakov 98] e.g., for first massive level: n = 2 : = (n 1) +... Subleading corrections?

24 Comparison between gauge and string theory states non-trivial: GT ( 1): operators built out of free fields, canonical dimension 0 determines states that can mix ST ( 1): near-flat-space string states built out of free oscillators, level n determines states that can mix meaning of 0 at strong coupling? meaning of n at weak coupling? 1. relate states with same global charges; 2. assume non-intersection principle [Polyakov 01]: no level crossing for states with same quantum numbers as changes from strong to weak coupling

25 Approaches to computation of corrections to string masses: (i) semiclassical approach: identify short string state as small-spin limit of semiclassical string state reproduce the structure of strong-coupling corrections to short operators [ Frolov, AT 03; Tirziu, AT 08] (ii) vertex operator approach: use AdS 5 S 5 string sigma model perturbation theory to find leading terms in anomalous dimension of corresponding vertex operator [Polyakov 01; AT 03]

26 (iii) space-time effective action approach: use near-flat-space expansion and NSR vertex operators to reconstruct α 1 corrections to corresponding massive string state equation of motion [Burrington, Liu 05] (iv) light-cone quantization approach: start with light-cone gauge AdS 5 S 5 string action and compute corrections to energy of corresponding flat-space oscillator string state [Metsaev, Thorn, AT 00 ]

27 Semiclassical expansion: spinning strings E = E( J, ) = E 0 (J ) + E 1 (J ) + 1 E 2 (J ) +... in short string limit J 1 E n = J (a 0n + a 1n J + a 2n J ) expansion valid for 1 and J = J fixed: J 1 but if knew all terms in this expansion could express J in terms of J, fix J to finite value and re-expand in J [ E = a 00 + a 10J + a 01 + a 20J 2 + a 11 J + a ] 02 ( +... ) 2 1 to trust the coeff of ( need coeff of up to n-loop terms ) n e.g. classical a 10 and 1-loop a 01 sufficient to fix 1 term [cf. fast string expansion J 1 [Frolov, AT 03]: for fixed J positive powers of need to resum]

28 Example: circular rotating string in S 5 with J 1 = J 2 = J: cf. Konishi descendant with J 1 = J 2 = 2: Tr([Φ 1, Φ 2 ] 2 ) try represent it by short classical string with same charges flat space R t R 4 : circular string solution x 1 + ix 2 = a e i(τ+σ), 4 E = α J, x 3 + ix 4 = a e i(τ σ) J = a2 α this solution can be directly embedded into R t S 5 in AdS 5 S 5 [Frolov, AT 03] : string on small sphere inside S 5 : X X 2 6 = 1 X 1 + ix 2 = a e i(τ+σ), X 3 + ix 4 = a e i(τ σ), X 5 + ix 6 = 1 2a 2, t = κτ J = J 1 = J 2 = a 2, E 2 = κ 2 = 4J Remarkably, exact E 0 is just as in flat space E 0 = E = 4 J, J = J

29 [cf. another (unstable) branch of J 1 = J 2 solution with J > 1 2 : E 0 = J 2 + = (1 + J ) ] 1-loop quantum string correction to the energy: sum of bosonic and fermionic fluctuation frequencies (n = 0, 1, 2,...) Bosons (2 massless + massive): AdS 5 : 4 ω 2 n = n 2 + 4J S 5 : 2 ω 2 n± = n 2 + 4(1 J ) ± 2 4(1 J )n 2 + 4J 2 Fermions: E 1 = 1 2κ 4 ωn 2 f ± = n J ± 4(1 J )n 2 + 4J ] n= [ 4ω n + 2(ω n+ + ω n ) 4(ω f n+ + ω f n ) expand in small J and do sums (UV divergences cancel) E 1 = 1 [ J [3 + ζ(3)]j 2 ] ] 1 J 4[ 5 + 6ζ(3) + 30ζ(5) J

30 J [ E = E 0 + E 1 = J ] 4( ) (1 + 2ζ(3)) if we could interpolate to fininte J = J 1 = J 2 = 2 that would suggest for Konishi state [ E = ] 2 + O( 1 ( ) ) 2 But: above still valid for large E and J; need to account for quantization of c.o.m. modes reinterpret as E(E 4) = 4 (J 1) 4 + O( 1 ) [ ] 1 J = 2 : E 2 = O( ( ) ) 2 same result will be found by different methods below

31 Spectrum of quantum string states from target space anomalous dimension operator Flat space: k 2 = m 2 = 4(n 1) α e.g. leading Regge trajectory ( x x) S/2 e ikx, n = S/2 spectrum in (weakly) curved background: solve marginality (1,1) conditions on vertex operators e.g. scalar anomalous dimension operator γ(g) on T (x) = c n...m x n...x m or on coefficients c n...m differential operator in target space found from β-function for the corresponding perturbation I = 1 4πα d 2 z[g mn (x) x m x n + T (x)] β T = 2T α 2 γ T + O(T 2 ) γ = Ω mn D m D n Ω m...k D m...d k +... Ω mn = G mn + p 1 α R mn + O(α 3 )

32 p 1 = 0,... in DR; Ω... α n R p...h q... Solve γ T + m 2 T = 0: diagonalize γ similarly for massless (graviton,...) and massive states e.g. β G mn = α R mn + O(α 3 ) gives Lichnerowitz operator as anomalous dimension operator ( γh) mn = D 2 h mn + 2R mknl h kl 2R k(m h k n) + O(α 3 ) Massive string states in curved background: d D x [ ] g Φ... ( D 2 + m 2 + X)Φ m 2 = 4 α (n 1), X = R... + O(α ) case of AdS 5 S 5 background R mn 1 96 (F 5F 5 ) mn = 0, R = 0, F 2 5 = 0 leading-order term in X should vanish for scalar state

33 leading α correction to scalar string state mass =0 (?!) [ D 2 + m 2 + O( 1 )]Φ = 0 = 2 + 4(n 1) O( 1 ) [ ] (n=2) = O( 1 ( ) ) 2 prediction (?) for leading term in strong-coupling expansion of singlet Konishi state dimension... but possible subtleties... 10d scalar vs singlet state... What about non-singlet Konishi descendant states?

34 How to find γ: Effective action approach derive equation of motion for a massive string field in curved background from quadratic effective action S reconstructed from flat-space NSR S-matrix Example: totally symmetric NS-NS 10-d tensor state on leading Regge trajectory in flat space symmetric tensor Φ µ1...µ 2n in metric+rr background (m 2 = 4(n 1) α ) L = R 1 2 5! F O(α 3 ) 1 2 (D µφd µ Φ + m 2 Φ 2 ) + k 1(α ) k 1 ΦX k (R, F 5, D)Φ +... assumption: α nr 1, i.e. n : small massive string in the middle of AdS 5 : near-flat-space expansion should be applicable

35 then eq. for Φ to leading α order [Burrington, Liu 05] [ D 2 + m 2 + X 1 + O(α )]Φ µ1 µ 2n = 0 ΦX 1 Φ = c 1 Φ µ1 µ 2 µ 2n R µ 1ν 1 µ 2 ν 2 Φ ν1 ν 2 µ 3 µ 2n +c 2 Φ µ1 µ 2n F µ 1ν 1 α 3 α 5 F µ 2ν 2 α3 α 5 Φ ν1 ν 2 µ 3 µ 2n +c 3 Φ µ1 µ 2 µ 2n F µ 1α 2 α 5 F ν 1 α2 α 5 Φ ν1 µ 2 µ 2n c 1 = n 2, c 2 = 1 4!, c 3 = 1 4 4! check: reproduces eq for graviton perturbation around R µν 1 4 4! (F 5F 5 ) µν = 0 AdS 5 S 5 background: R ab = 4 R 2 g ab, R mn = 4 R 2 g mn µ, ν,... = 0, 1,...9; a, b,... in AdS 5 and m, n,... in S 5 L = 1 2 Φ µ 1 µ 2n ( D 2 + m 2 )Φ µ 1 µ 2n + n2 R 2 (Φ a 1 a 2 µ 3 µ 2n Φ a 1a 2 µ 3 µ 2n Φ m1 m 2 µ 3 µ 2n Φ m 1m 2 µ 3 µ 2n ) +... background is direct product can consider particular tensor with S indices in AdS 5 and K indices in S 5 :

36 end up with anomalous dimension operator [ D 2 + (m 2 + K2 S 2 2R )]Φ = 0, D 2 = D 2 2 AdS 5 + DS 2 5 m 2 = 4 α (n 1) = 2 α (S + K 2), 2n = S + K symmetric transverse traceless tensor highest-weight state Young table labels (, S, 0; J, K, 0), J K extract AdS 5 radius R and set = R2 α ( D 2 AdS 5 + M 2 )Φ = 0 M 2 = 2 (S + K 2) (K2 S 2 ) + J(J + 4) K For symmetric traceless rank S tensor in AdS 5 : 2 = M 2 + S + 4 = 2 (S + K 2) (S + K 2)(K S) + J(J + 4) O( )

37 To summarize: condition of marginality of (1,1) vertex operator for (, S 1, S 2 ; J 1, J 2, J 3 ) = (, S, 0; J, K, 0) state 0 = (S + K 2) [ ( 4) S(S 2) 1 K(K 2) J(J + 4)] + O( 1 2 ) BPS level n = 1 2 (S + K) = 1: J = K + J, J = 0, 1, 2,... S = 2, K = 0: = 4 + J ; K = 2, S = 0: = 6 + J ; etc First massive level: n = 1 2 (S + K) = 1 case of minimal dimension shift S = 4, K = J = 0: dual to 0 = 6 Konishi state [0, 0, 0] (2,2) [ ] 0 = 2 + O( 1 1 ) = O( ( ) ) 2 what about other states in Konishi multiplet?

38 Vertex operator approach [Polyakov 01; AT 03] I = 1 4πα d 2 ξ G mn (x) x m x n +... perturbed by V (f) = f m1...m s (x) k 1 x m 1... k h x m s compute the renormalization of f m1...m j and set β f = γf +...=0 γf = [2 J α D 2 + c k α k (R...) n...d p ]f = 0 diagonalize anomalous dimension operator but γ for generic f and G not known even to α order calculate anomalous dimensions from first principles superstring theory in AdS 5 S 5 : I = 4π [ d 2 σ N p N p + n k nk + fermions N + N N u N u N v N v = 1, n x n x + n y n y + n z n z = 1 N ± = N 0 ± in 5, N u = N 1 + in 2,..., n x = n 1 + in 2,... construct marginal (1,1) operators in terms of N p and n k ]

39 e.g. vertex operator for dilaton sugra mode [ ] V J (ξ) = (N + ) (n x ) J N p N p + n k nk + fermions N + N 0 + in 5 = 1 z (z2 + x m x m ) e it n x n 1 + in 2 e iϕ 0 = [ ( 4) J(J + 4)] + O( 1 ( ) 2 ) i.e. = 4 + J (BPS) candidate operators for states on leading Regge trajectory: V J (ξ) = (N + ) ( n x nx ) J/2, nx n 1 + in 2 V S (ξ) = (N + ) ( N u Nu ) S/2, Nu N 1 + in 2 + fermionic terms + α 1 terms from diagonalization of anom. dim. op. how they mix with ops with same charges and dimension?

40 in general ( n x nx ) J/2 mixes with singlets (n x ) 2p+2q ( n x ) J/2 2p ( n x ) J/2 2q ( n m n m ) p ( n k n k ) q S 5 = SO(6)/SO(5) sigma model S = d 2 ξ n m nm, n m n m = 1 4π ġ = ɛg + 4g 2 + 4g , g 1 = α R 2, ɛ = d 2 running is cancelled if embedded into AdS 5 S 5 string σ-model ops. for states on leading Regge trajectory O l,s = f k1...k l m 1...m 2s n k1...n kl n m1 nm2... n m2s 1 nm2s their renormalization studied before [Wegner 90]

41 renormalization of composite operators to leading order in 1 use pairing rules (and ignore on-shell operators): < AB >=< A > B + A < B > + < A, B > < A, B >= d 2 ξd 2 ξ < n k (ξ), n m (ξ ) > δa < A(n) >= 1 2 d 2 ξd 2 ξ < n k (ξ), n m (ξ ) > δn k (ξ) δb δn m (ξ ) δ 2 A δn k (ξ)δn m (ξ ), etc. < n k >= 5 2 In k, < n k, n l >= I(n k n l δ kl ), I = 1 2πɛ < n k, n l >= I n k n l, < n k, n l >= I n k n l, < n k, n l >= In k n l n m n m, < n k, n l >= In k n l nm nm, < n k, n l >= I( n k n l δ kl n m nm ) < ( n k nk ) >= 0, < ( n k n k ) >= 4I n k n k, < ( n k nk ) >= 4I n k nk

42 simplest case: f k1...k l n k1...n kl with traceless f k1...k l same anom. dim. γ as its highest-weight rep V J = (n x ) J γ = [5J + J(J 1)] +... = 2 2 J(J + 4) +... scalar spherical harmonic that solves Laplace eq. on S 5 similarly for AdS 5 or SO(2, 4) model: replacing n J x and n m nm with N+ and N p Np, with J = and g = 1 1 e.g. dimension of n J x n m nm : γ = 1 2 J(J + 4) + O( 1 ( ) ) 2 dimension of N + N p Np : γ = 1 2 ( 4) + O( 1 ( ) 2 )

43 example of scalar higher-level operator: N + [( n k nk ) r +...], r = 1, 2,... [Kravtsov, Lerner, Yudson 89; Castilla, Chakravarty 96] 0 = 2(r 1) [ ( 4) + 2r(r 1)] + 1 ( ) [ r(r 1)(r 7 2 ) + 4r] +... r = 1: ground level fermionic contributions should make r = 1 exact zero of γ r = 2: first excited level candidate for singlet Konishi state 0 = 2 ( 4) = O( 1 ), [ ] 1 0 = O( ( ) ) 2 same as for (S = 4, K = 0) Konishi state with 0 = 6

44 Operators with two spins J, K in S 5 : V K,J = N + M uv n J u v y K/2 u,v=0 n u+v x c uv M uv ( n y ) u ( n x ) K/2 u ( n y ) v ( n x ) K/2 v highest and lowest eigen-values of 1-loop anom. dim. matrix γ min = 2 K [ ( 4) 1 2 K(K + 10) J(J + 4) 2JK] + O( 1 ( ) 2 ) γ max = 2 K [ ( 4) 1 2 K(K + 6) J(J + 4)] + O( 1 ( ) 2 ) fermions may alter terms linear in K K = 4: first massive level Konishi state identify operators with right representations more evidence for b = 0 [R.Roiban, AT, in progress]

45 Conclusions Beginning of understanding quantum string spectrum in AdS 5 S 5 = spectrum of short SYM operators more progress expected soon aiding/checking integrability approach

46 Happy Birthday Misha! Thank you for your inspiration, insight and help!

47 Long Konishi multiplet 0 min = 2, [m, n, k](s, s ) = [0, 0, 0](0, 0) SO(6) and SO(4) labels [Andreanopoli,Ferrara 98; Bianchi,Morales,Samtleben 03]

48 0 2 [0, 0, 0] (0,0) 5 [0, 0, 1] 2 (0, 1 2 ) + [1, 0, 0] ( 1 2,0) 3 [0, 0, 0] ( 1 2, 2 1 ) + [0, 0, 2] (0,0) + [0, 1, 0] (0,1)+(1,0) + [1, 0, 1] ( 1 2, 1 2 ) + [2, 0, 0] (0,0) 7 [0, 0, 1] 2 ( 1 2,0)+( 1 2,1)+( 3 2,0) + [0, 1, 1] (0, 2 1 )+(1, 1 2 ) + [1, 0, 0] (0, 2 1 )+(0, 3 2 )+(1, 1 2 ) + [1, 0, 2] ( 2 1,0) +[1, 1, 0] ( 1 2,0)+( 1 2,1) + [2, 0, 1] (0, 2 1 ) 4 [0, 0, 0] (0,0)+(0,2)+(1,1)+(2,0) + [0, 0, 2] ( 1 2, 1 2 )+( 3 2, 1 2 ) + [0, 1, 0] 2( 1 2, 1 2 )+( 1 2, 3 2 )+( 3 2, 1 2 ) + [2, 0, 2] (0,0) +[0, 1, 2] (1,0) + [0, 2, 0] 2(0,0)+(1,1) + [1, 0, 1] (0,0)+2(0,1)+2(1,0)+(1,1) + [1, 1, 1] 2( 1 2, 1 2 ) + [2, 0, 0] ( [0, 0, 0] 3(0,0)+3(1,1)+(2,2) + [0, 0, 2] 3( 1 2, 1 2 )+( 1 2, 3 2 )+( 3 2, 1 2 )+( 3 2, 3 2 ) + [0, 1, 0] 4( 1 2, 1 2 )+2( 1 2, 3 2 )+2( 3 2, 1 2 )+2 +[0, 1, 2] (0,0)+2(0,1)+2(1,0)+(1,1) + [0, 2, 0] 3(0,0)+(0,1)+(0,2)+(1,0)+3(1,1)+(2,0) + [0, 2, 2] ( 1 2, 1 2 ) +[0, 3, 0] 2( 1 2, 1 2 ) + [0, 4, 0] (0,0) + [1, 0, 1] (0,0)+3(0,1)+3(1,0)+4(1,1)+(1,2)+(2,1) + [1, 0, 3] ( 1 2, 1 2 ) + [0, +[1, 1, 1] 4( 1 2, 1 2 )+2( 1 2, 3 2 )+2( 3 2, 1 2 ) + [1, 2, 1] (0,0)+(0,1)+(1,0) + [2, 0, 0] 3( 1 2, 1 2 )+( 1 2, 3 2 )+( 3 2, 1 2 )+( 3 2, 3 2 ) +[2, 0, 2] (0,0)+(1,1) + [2, 1, 0] (0,0)+2(0,1)+2(1,0)+(1,1) + [2, 2, 0] ( 1 2, 1 2 ) + [3, 0, 1] ( 1 2, 1 2 ) + [4, 0, 0] (0,0 17 [0, 0, 1] 2 (0, 1 2 )+(0, 3 2 )+(1, 1 2 ) + [0, 1, 1] ( 1 2,0)+( 1 2,1) + [1, 0, 0] ( 2 1,0)+( 2 1,1)+( 3 2,0) + [1, 0, 2] (0, 1 2 ) +[1, 1, 0] (0, 1 2 )+(1, 1 2 ) + [2, 0, 1] ( 2 1,0) 9 [0, 0, 0] ( 1 2, 1 2 ) + [0, 0, 2] (0,0) + [0, 1, 0] (0,1)+(1,0) + [1, 0, 1] ( 1 2, 1 2 ) + [2, 0, 0] (0,0) 19 2 [0, 0, 1] ( 1 2,0) + [1, 0, 0] (0, 2 1 ) 10 [0, 0, 0] (0,0)