Short spinning strings and AdS 5 S 5 spectrum

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1 Short spinning strings and AdS 5 S 5 spectrum Arkady Tseytlin R. Roiban, AT, arxiv: , arxiv: M. Beccaria, S. Giombi, G. Macorini, R. Roiban, AT, arxiv:

2 70-s: origin of string theory supersymmetric field theories in 2d, 4d, 0d [Lars Brink s talk] but also of conformal field theory in 4d [Polyakov, Ferrara et al,... ] integrability of 2d sigma models [Pohlmeyer, Luescher, Polyakov, Zamolodchikovs,...] and also of planar limit (N, = gym 2 N=fixed) [ t Hooft] N = 4 SYM: remarkable relation to all of these ideas Relation of SYM to string theory? known since 70 s / early 80 s: [Scherk et al; Brink, Green, Schwarz]

3 On-shell scattering amplitudes of N = 4 SYM may be found from α 0 limit of superstring amplitudes in flat 0d space R,3 T 6 But what about correlation functions of composite operators = natural observables in a CFT? can one compute 2-point (spectrum of dimensions) and 3-point (OPE coefficients) correlation functions of conformal primary operators [ e.g. TrF 2 mn(x )...TrF 2 pq(x n ) ] from 0d flat-space string theory?! not in an obvious or useful way (need an off-shell extension,...) the idea that one can use string theory to solve N = 4 SYM as a CFT using string theory would sound wild in 70 s especially if this string theory is in curved space and with finite tension...

4 Maximally symmetric case of gauge-string duality: [Maldacena, Polyakov, Klebanov, Witten,...] planar N = 4 super Yang-Mills free AdS 5 S 5 superstring closed string states on R S gauge-inv. SYM states on R S 3 marginal str. vertex ops on R 2 conf. primary SYM ops on R 4 Remarkably, correlators of AdS 5 S 5 string vertex operators analogs of S-matrix elements in flat 0d space are (expected to be) dual to correlators of conformal operators of planar N = 4 SYM In particular, relation of 2-point functions means that spectrum of AdS 5 S 5 string energies spectrum of dimensions of SYM primary operators Then the spectrum of N = 4 SYM conformal dimensions () should be possible to describe by 2d AdS 5 S 5 superstring sigma

5 Integrability: allows in principle to solve the problem of spectrum enormous progress in the last 0 years Some key inputs: [not a proper review!] SYM action + perturbation theory ( ) AdS 5 S 5 GS superstring action + α -expansion ( ) classical integrability of AdS 5 S 5 GS action [Bena, Polchinski, Roiban] perturbative integrability of SYM spectral problem: (-loop, 2-loop,...) dilatation operator = spin chain Hamiltonian [Minahan, Zarembo; Beisert, Staudacher,...] guidance from large-charge limits: BMN, GKP, FT Assume that integrability extends to all orders on both sides construct interpolating Bethe ansatz guided by general principles, symmetries and data from both weak+strong coupling check consistency of its predictions

6 I. Spectrum of long operators / semiclassical string states determined by Asymptotic Bethe Ansatz ( ) its final [Beisert-Eden-Staudacher] form found by intricate superposition of data from gauge theory (spin chain, BA,...) and perturbative string theory (classical and -loop phase, BMN), symmetries (S-matrix), assumption of exact integrability consequences checked against available gauge and string data Key example: cusp anomalous dimension of Tr(ΦD S Φ) = S f() ln S +..., S f = [ 2π ( 73 ] ζ2 3 π 6 ) [ f = 3 ln 2 K ] π ( )... 2 ζ k = ζ(k) = n= n k, K = β(2) = n=0 ( ) n (2n+) 2 = from 2-loop string sigma-model integrals [Roiban,Tirziu,AT] exact integral eq. [Basso, Korchemsky, Kotanski]: any order term

7 II. Spectrum of short operators = quantum string states Thermodynamic Bethe Ansatz ( ) reconstructed from ABA using solely methods/intuition of 2-d integrable QFT, i.e. inspired by string-theory side highly non-trivial construction lack of 2-d Lorentz invariance in standard BMN-vacuum-adapted l.c. gauge in few cases ABA improved by Luscher corrections is enough: 4- and 5-loop Konishi dim, 4-loop dim. of twist 2 operator complicated set of integral equations in need of simplification; so far predictions extracted only numerically starting from weak coupling and interpolating to larger need more data to check predictions at and, i.e. against perturbative gauge-theory and string-theory data

8 Key example: dimension = 2 + γ() of Konishi operator Tr( Φ i Φ i ) γ( ) = 2 [ (4π) 2 4 (4π) (4π) 4 ( ) ζ ζ 5 (4π) ( ζ 3 54ζ3 2 ) 4 ] 90ζ ζ 7 (4π) loop result first found using integrability [Banjok,Janik] confirmed by more standard methods [Velizhanin; Eden et al 2] Suppose one can sum up (convergent) expansion and then re-expand at What one should expect to get for γ( )?

9 Duality to string theory predicts the structure of strong-coupling expansion: leading term near-flat-space expansion for fixed quant. numbers [Gubser, Klebanov, Polyakov 98] = 2N +... Subleading terms: α = expansion of 2d anom. dimensions of corresponding vertex operators [Roiban, AT 09] (N = 2) γ( ) = b 4 + b 2 ( 4 ) + b 3 3 ( ) 5/2 = 2 4 [ + b 2 + b 2 2( ) + b ] 3 2 2( ) Values of b k from string theory? From TBA?

10 Dimensions of short SYM operators = energies of quantum string states find leading α = corrections to energy of lightest massive string states on first massive string level dual to operators in Konishi multiplet in SYM theory compare with predictions of TBA approach important to check integrability-based approach which involves subtle assumptions directly against perturbative string sigma model [similar checks were done, e.g., in standard O(n) models]

11 TBA results: start at weak coupling for sl(2) Konishi descendant Tr(ΦD 2 Φ) use TBA to find () numerically; match to expected form of strong-coupling expansion to extract b k [Gromov, Kazakov, Vieira 09; Frolov 0, 2] b.988, b Compare to string theory: One can find b k using semiclassical short string expansion [Roiban, AT 09, ; Gromov, Serban, Shenderovich, Volin ] b = 2, b 2 = a 3ζ 3 rational a is found [Gromov, Valatka ] using also 2-loop coefficient in exact slope function (E 2 = h()s) [Basso ] b 2 = 2 3ζ Remarkable agreement with TBA - check of quantum integrability

12 Figure : Plot from Gromov, Kazakov, Vieira

13 Recent work on string side: [BGMRT 2] highest transcendentality terms in b k are ζ 2k and have -loop origin, e.g., b 3 = a + a 2 ζ 3 + a 3 ζ 5, a 3 = 5 2 rational a receives contribution from 3 loops; a 2 from 2-loops, etc.; b 4 ζ , etc. supermultiplet structure: universality of coefficients in E for string states with spins in different AdS 5 S 5 directions: dual operators from Konishi multiplet have same energy (up to constant shift depending on position in the multiplet) states on leading Regge trajectory: general structure of dependence of energy on string tension, string level (spin) and S 5 orbital momentum J

14 Some open questions: Analytic form of strong-coupling expansion from TBA? only ζ k coefficients in () in both weak and strong coupling expansions or other transcendental constants may also appear? (cf. cusp anomalous dimension) 2-loop string computation may shed light on this... Asymptotic form of strong coupling expansion: interpretation of possible e a corrections? Energies of other quantum states: similar behavior?

15 Konishi multiplet: long multiplet related to singlet [0, 0, 0] (0,0) by susy [J 2 J 3, J J 2, J 2 + J 3 ] (sl,s R ) s L,R = 2 (S ± S 2 ) SO(6) (J, J 2, J 3 ) and SO(4) (S, S 2 ) labels of SO(2, 4) SO(6) global symmetry [Andreanopoli,Ferrara 98; Bianchi,Morales,Samtleben 03] = 0 + γ(), 0 = 2, 5 2, 3,..., 0 same anomalous dimension γ for all members singlet eigen-state of anom. dim. matrix with lowest eigenvalue

16 Examples of gauge-theory operators in Konishi multiplet: [0, 0, 0] (0,0) : Tr( Φ i Φ i ), i =, 2, 3, 0 = 2 [2, 0, 2] (0,0) : Tr([Φ, Φ 2 ] 2 ) in su(2) sector, 0 = 4 [0, 2, 0] (,) : Tr(Φ D 2 Φ ) in sl(2) sector, 0 = 4

17 0 2 [0, 0, 0] (0,0) 5 [0, 0, ] 2 (0, 2 ) + [, 0, 0] ( 2,0) 3 [0, 0, 0] ( 2, 2 ) + [0, 0, 2] (0,0) + [0,, 0] (0,)+(,0) + [, 0, ] ( 2, 2 ) + [2, 0, 0] (0,0) 7 [0, 0, ] 2 ( 2,0)+( 2,)+( 3 2,0) + [0,, ] (0, 2 )+(, 2 ) + [, 0, 0] (0, 2 )+(0, 3 2 )+(, 2 ) + [, 0, 2] ( 2,0) +[,, 0] ( 2,0)+( 2,) + [2, 0, ] (0, 2 ) 4 [0, 0, 0] (0,0)+(0,2)+(,)+(2,0) + [0, 0, 2] ( 2, 2 )+( 3 2, 2 ) + [0,, 0] 2( 2, 2 )+( 2, 3 2 )+( 3 2, 2 ) + [2, 0, 2] (0,0) +[0,, 2] (,0) + [0, 2, 0] 2(0,0)+(,) + [, 0, ] (0,0)+2(0,)+2(,0)+(,) + [,, ] 2( 2, 2 ) + [2, 0, 0] ( 2 6 [0, 0, 0] 3(0,0)+3(,)+(2,2) + [0, 0, 2] 3( 2, 2 )+( 2, 3 2 )+( 3 2, 2 )+( 3 2, 3 2 ) + [0,, 0] 4( 2, 2 )+2( 2, 3 2 )+2( 3 2, 2 )+2 +[0,, 2] (0,0)+2(0,)+2(,0)+(,) + [0, 2, 0] 3(0,0)+(0,)+(0,2)+(,0)+3(,)+(2,0) + [0, 2, 2] ( 2, 2 ) +[0, 3, 0] 2( 2, 2 ) + [0, 4, 0] (0,0) + [, 0, ] (0,0)+3(0,)+3(,0)+4(,)+(,2)+(2,) + [, 0, 3] ( 2, 2 ) + [0, +[,, ] 4( 2, 2 )+2( 2, 3 2 )+2( 3 2, 2 ) + [, 2, ] (0,0)+(0,)+(,0) + [2, 0, 0] 3( 2, 2 )+( 2, 3 2 )+( 3 2, 2 )+( 3 2, 3 2 ) +[2, 0, 2] (0,0)+(,) + [2,, 0] (0,0)+2(0,)+2(,0)+(,) + [2, 2, 0] ( 2, 2 ) + [3, 0, ] ( 2, 2 ) + [4, 0, 0] (0,0 7 [0, 0, ] 2 (0, 2 )+(0, 3 2 )+(, 2 ) + [0,, ] ( 2,0)+( 2,) + [, 0, 0] ( 2,0)+( 2,)+( 3 2,0) + [, 0, 2] (0, 2 ) +[,, 0] (0, 2 )+(, 2 ) + [2, 0, ] ( 2,0) 9 [0, 0, 0] ( 2, 2 ) + [0, 0, 2] (0,0) + [0,, 0] (0,)+(,0) + [, 0, ] ( 2, 2 ) + [2, 0, 0] (0,0) 9 2 [0, 0, ] ( 2,0) + [, 0, 0] (0, 2 ) 0 [0, 0, 0] (0,0) Table : Long Konishi multiplet (a bit cut off)

18 Comparison between gauge and string theory states: : gauge-theory operators built out of free fields, canonical dim. 0 determines operators that can mix : in near-flat-space expansion string states built out of free oscillators, level N determines states that can mix (i) relate states with same global charges (ii) assume direct interpolation (no level crossing ) for states with same quantum numbers as changes from small to large values

19 Gauge-string duality: Konishi operator dual to lightest among massive AdS 5 S 5 string states large = R2 α : short strings probe near-flat limit of AdS 5 S 5 members of supermultiplet: strings with spins/oscillators in different AdS 5 S 5 directions

20 Flat space case: m 2 = 2N α N = 0: 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 + S a ) 0 ] 2 = [(8 + 8) (8 + 8)] 2 in SO(9) reps: ([2, 0, 0, 0] + [0, 0,, 0] + [, 0, 0, ]) 2 = ( ) 2 e.g = = switching on AdS 5 S 5 background fields lifts degeneracy states with lightest mass at first excited string level should correspond to Konishi multiplet

21 String spectrum in AdS 5 S 5 : long multiplets of P SU(2, 2 4) highest weight states: [J 2 J 3, J J 2, J 2 + J 3 ] (sl,s R ), s L,R = 2 (S ± S 2 ) 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; Beisert et al 03] SO(4) SO(5) SO(9) rep. lifted to SO(4) SO(6) rep. of SO(2, 4) SO(6) Konishi long multiplet K = ( + Q + Q Q +...)[0, 0, 0] (0,0) determines the floor of -st excited string level J=0 [0, J, 0] (0,0) K

22 Spins: S, S 2 in AdS 5 ; (J, J 2 ) in S 5 orbital momentum J = J 3 in S 5 Examples: folded string with spin S and momentum J: S = J = 2 [0, 2, 0] (,), 0 = 4 folded string with spin J and momentum J: J = J = 2 [2, 0, 2] (0,0), 0 = 4 circular string with spins J = J 2 and momentum J: J = J 2 =, J = 2 [0,, 2] (0,0), 0 = 6 circular string with spins S = S 2 and momentum J: S = S 2 =, J = 2 [0,, 2] (0,0), 0 = 6 circular string with spins S = J and momentum J: S = J =, J = 2 [,, ] ( 2, 2 ), 0 = 6

23 Direct approaches to computation of quantum string energies: (not relying on integrability) (i) vertex operator approach: use AdS 5 S 5 string sigma model perturbation theory to find leading terms in 2d anomalous dimension of corresponding vertex operators and impose marginality condition [Polyakov 0; AT 03; pure spinors: Mazzucato,Vallilo ] (ii) light-cone gauge approach: start with AdS 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] both yet to be developed in detail; in practice, will be guided by vertex operator approach but use indirect but easier to implement semiclassical approach: short string limit of semiclassical expansion [Tirziu, AT 08; Roiban, AT 09, ]

24 Target space perspective: string in flat space: p 2 = m 2 = 2N α e.g. leading Regge trajectory ( x x) S/2 e ipx, N = S spectrum in (weakly) curved background: solve marginality (,) conditions on vertex operators e.g. 2d anomalous dimension operator γ on scalar T (x) differential operator in target space from β-function for the corresponding perturbation I = [ ] 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 + O(α 3 ), Ω... α n R... p Solving γ T + m 2 T = 0 (m 2 = 4 α ) same as diagonalizing γ

25 Massive string states in curved background: d D x [ ] g Φ... ( D 2 + m 2 + X)Φ m 2 = 2N α, X = R... + O(α ) case of AdS 5 S 5 background R mn 96 (F 5F 5 ) mn = 0, R = 0, F 2 5 = 0 Find leading-order term in X... leading α correction to scalar string state mass is 0 (?!) [ D 2 + m 2 + O( )]Φ = 0 = 2 + 2N O( ) N=2 = [ ] O( ( ) ) 2 Too naive: various subtleties (0d scalar vs singlet state, mixing) What is found for non-singlet (susy descendant) Konishi states?

26 Vertex operator approach calculate 2d anomalous dimensions from first principles superstring theory in AdS 5 S 5 : [ I = d 2 σ Y p Y p + X k Xk + fermions 4π Y 2 0 Y Y Y 2 4 =, X X 2 6 = construct marginal (,) operators in terms of Y p and X k e.g. vertex operator for dilaton (in NSR framework) [ ] V J = (Y + ) (X x ) J Y p Y p + X k Xk + fermions Y + Y 0 + iy 5 = z + z x m x m e it X x X + ix 2 e iϕ 2 = [ ( 4) J(J + 4)] + O( ( ) 2 ) i.e. = 4 + J (BPS) ]

27 Vertex operators = eigenstates of 2d anomalous dimension matrix particular linear combinations like V = f k...k l m...m 2s X k...x kl X m Xm2... X m2s Xm2s their renormalization studied in O(n) sigma model [Wegner 90] simplest case: f k...k l X k...x kl with traceless f k...k l highest-weight rep V J = (X x ) J, γ = 2 2 J(J + 4) +... Higher massive states: Flat space: bosonic string state on leading Regge trajectory V S = e iet( x x ) S/2, x = x + ix 2, α E 2 = 2(S 2) AdS 5 S 5 : candidates for operators on leading Regge trajectory: V J = (Y + ) ( X x Xx ) J/2, Xx X + ix 2 V S = (Y + ) ( Y u Yu ) S/2, Yu Y + iy 2

28 + fermionic terms + α terms from diagonalization of anom. dim. op. mixing with operators with same charges and dimension. In general ( X x Xx ) J/2 mixes with singlets (X x ) 2p+2q ( X x ) 2 J 2p ( X x ) 2 J 2q ( X m X m ) p ( X k X k ) q Example of higher-level scalar/singlet operator: [ ] ( X k Xk ) r +..., N = 2(r ) Y + Marginality condition: [cf. Kravtsov, Lerner, Yudson 89; Castilla, Chakravarty 96] 0 = 2(r ) [ ] 2 ( 4) + 2r(r ) [ ] ( 2 ) 2 3 r(r )(r 7 2 ) + 4r +... r = : ground level fermions should make r = zero of γ

29 r = 2: excited level analog of singlet Konishi state 0 = 2 ( 4) = O( ), 0 = 2 4 [ ] O( ( ) ) 2 fermionic contributions change subleading coefficients Bosonic operators with momentum J and spin J J in S 5 : V J,J = Y + J /2 k,m=0 M km X J k m y c km M km Xx k+m ( X y ) k ( X x ) 2 J k ( X y ) m ( X x ) 2 J m highest and lowest eigen-values of -loop anom. dim. matrix γ min = 2 J + 2 γ max = 2 J + 2 [ ( 4) J(J + 4) 2 J (J + 0) 2JJ ] +... [ ] ( 4) J(J + 4) 2 J (J + 6) fermionic contributions change terms linear in J +...

30 How to take fermionic contributions into account? (i) compute energies of semiclassical string states in expansion using full AdS 5 S 5 Green-Schwarz action (ii) compare to structure of E = expected from marginality condition (iii) determine unknown coefficients in E( )

31 General structure of dimension/energy = E marginality condition condition on quantum numbers Q i Q = (E(), S, S 2 ; J, J 2, J 3 ;...); N = i a iq i = level 0 = 2N + ( + ( ) 2 ( i,j c ij Q i Q j + i i,j,k c ijk Q i Q j Q k + i,j c i Q i ) c ijq i Q j + i c iq i ) +... highest power of Q at n-loop order ( ) n Qn+2 dependence on momenta E and J = J 3 is special e.g., they do not enter N terms ( ) n Em are scheme-dependent (cf.(α D 2 ) n terms in γ) choose susy-preserving scheme where E = J is the only solution in BPS case only E 2 terms are then left (ignore const shifts) terms ( ) n E2 Q k can be traded for E-independent terms in perturbative solution for E

32 States on leading Regge trajectory : (max spin for given E in flat limit) Q = (E, J; N), N= spin component marginality condition: 0 = 2N + ( E 2 + J 2 + n 02 N 2 + n N + ( ) ( n 0 J 2 N + n 03 N 3 + n 2 N 2 + n 2 N ) 2 solution for E 2 takes form [Roiban, AT 09, ; BGMRT 2] E 2 = 2 N + J 2 + n 02 N 2 + n N + ( n0 J 2 N + n 03 N 3 + n 2 N 2 + n 2 N ) ) + O( ( ) ) 3 + ( (ñ J 2 N + ñ 02 J 2 N 2 + n 04 N 4 + n 3 N 3 + n 22 N 2 + n 3 N ) ) 2 + ( (ñ0 J 4 N + ñ 2 J 2 N + ñ 2 J 2 N 2 + n 05 N ) + O( ) 3 ( ) ) 4 J k N m terms: non-decoupling of c.o.m.

33 Expanding in large for fixed N, J E = 2 [ N + A + A 2 ( ) + O( ] 2 ( ) ) 3 A = 4N J (n 02N + n ) A 2 = 2 A2 + 4 (n 0J 2 + n 03 N 2 + n 2 N + n 2 ) Gives strong-coupling expansion of dimension of corresponding gauge-theory operator States on -st excited superstring level: N = 2 Konishi multiplet states: N = 2, J = 2 E = 4 [ 2 + b + b 2 ( ) + O( ] 2 ( ) ) 3 b = + n n b 2 = 4b 2 + 2n 0 + 2n 03 + n n 2

34 To find coefficients n km use semiclassical short string (small spin) expansion: start with solitonic string carrying same charges as vertex operator representing particular quantum string state perform semiclassical expansion: for fixed classical parameters (N, J ) N = N, J = J expand E in small values of N, J re-interpret the resulting E in terms of N, J and read off n km Key point: limit N = N 0, J = J 0 corresponds to for fixed values of quantum charges N, J

35 Rewrite E 2 in terms of N, J : ( E ) 2 = (2N + J 2 + n 0 J 2 N + n 02 N 2 + n 03 N 3 + ñ 0 J 4 N + ñ 02 J 2 N ) + (n N + ñ J 2 N + n 2 N 2 + ñ 2 J 2 N 2 + n 3 N ) + ( ) 2 (n 2N + ñ 2 J 2 N + n 22 N ) + O( ( ) 3 ), can now interpret n km as semiclassical k-loop contribution to N m term in E: -st line is classical energy, 2nd is -loop, etc. semiclassical loop expansion ( for fixed N, J ) different from first loop expansion in 2d anomalous dimension ( for fixed N, J) but they contain same coefficients each loop term in first expansion is polynomial in charges but in semiclassical expansion each term may contain infinite series in small J, N expansion

36 Semiclassical expansion of E 2 organized as expansion in small N formally looks like series in powers of N: E 2 = J 2 + h (, J) N + h 2 (, J) N 2 + h 3 (, J) N for fixed J and large h = 2 + n + n 2 + n 3 ( ) J 2( n 0 + h 2 = n 02 + n , h 3 = n exact result for slope h for AdS 5 folded spinning string state (N = S) from Bethe Ansatz [Basso ] ñ ( ) ) +... h (, J) = 2 d d ln I J( ) = 2 + J 2 + J 2 4 J ( + J 2 ) 5/2 = 2 + J 2 + J 2 ( 4 J 2 ) +... ( + J 2 ) 5/2

37 [Can one find h by direct summation of 4d or 2d Feynmann diagrams (or via localization)?] But one needs to know also coefficients in h 2, h 3,... (much more non-trivial, depend on wrapping corrections) Strategy: consider examples of small semiclassical string states corresponding to quantum string states with angular momentum J and few oscillator modes excited (carrying spin) start with classical string solutions in flat space representing states on leading Regge trajectory find the corresponding solutions in AdS 5 S 5 find -loop correction to their E expand E in N = N 0 interpolate result to finite N find the coefficients n km check universality of E for N = 2 (implied by susy)

38 Two basic classes of examples (N = spin, J= orbital momentum): circular string with 2 spins in two orthogonal planes folded string with spin in a plane Rigid circular string rotating in two planes of R 4 t = κτ, x x + ix 2 = a e i(τ+σ), x 2 x 3 + ix 4 = a e i(τ σ) E flat = κ 2 α = α N, N = J + J 2, J = J 2 = a2 α semiclassical counterpart of quantum string state created by e iet [ ( x x ) J 2 ( x2 x2 ) J ] Folded string rotating in a plane t = κτ, 2 E flat = α N, x x + ix 2 = a sin σ e iτ N = S = a2 2α, semiclassical counterpart [ of quantum string state ] e iet ( x x xx ) S/2 +...

39 3 ways to embed circular solutions into AdS 5 S 5 : (i) the two 2-planes in S 5 : J = J 2 small string (ii) the two 2-planes in AdS 5 : S = S 2 small string (iii) one plane in AdS 5 and the other in S 5 : S = J small string 2 choices AdS 5 or S 5 for folded string for N = 2 all 5 cases represent states on st string level; for N = J = 2 they are particular members of Konishi multiplet can be used to check universality of -dependent part of = E for different states in supermultiplet

40 Spins: S, S 2 in AdS 5 ; (J, J 2 ) in S 5 orbital momentum J = J 3 in S 5 Examples: folded string with spin S and momentum J: S = J = 2 [0, 2, 0] (,), 0 = 4 folded string with spin J and momentum J: J = J = 2 [2, 0, 2] (0,0), 0 = 4 circular string with spins J = J 2 and momentum J: J = J 2 =, J = 2 [0,, 2] (0,0), 0 = 6 circular string with spins S = S 2 and momentum J: S = S 2 =, J = 2 [0,, 2] (0,0), 0 = 6 circular string with spins S = J and momentum J: S = J =, J = 2 [,, ] ( 2, 2 ), 0 = 6

41 Results: for several states on leading Regge trajectory E 2 = 2 N + J 2 + n 02 N 2 + n N + ( n0 J 2 N + n 03 N 3 + n 2 N 2 + n 2 N ) + ( (ñ J 2 N + ñ 02 J 2 N 2 + n 04 N 4 + n 3 N 3 + n 22 N 2 + n 3 N ) ) 2 + ( (ñ0 J 4 N + ñ 2 J 2 N + ñ 2 J 2 N 2 + n 05 N ) + O( ) 3 ( ) ) 4 n 0 =, ñ 0 = 4,... from near-bmn expansion (J ) E 2 = J 2 + 2N + J = J 2 + N(2 + J ) tree-level coeffs n 02, n 03, n 04,... are all rational leading -loop n is rational [Roiban, AT 09; Gromov et al ] ñ = n, i.e in general [BGMRT 2] h = 2 + J 2 + n + J 2 + [ n2 + ñ 2 J 2 + O(J 4 ) ] + O( ( ) 2 ) h 2 = n 02 + J 2 + J 2 + [ n2 + ñ 2 J 2 + O(J 4 ) ] + O( ( ) 2 )

42 n 2 = n 2 3ζ 3, n 2 = 3 8 2n 03 is rational [Tirziu, AT 08; Roiban, AT 09; Gromov-Valatka ] ζ 3 term is universal for states on leading Regge trajectory ñ 2 = ñ 2 + 3ζ ζ 5, ñ 2 rational n k contains universal ζ 2k (universal UV n asymptotics) e.g. n 3 = ñ 2 + ñ k ζ ζ 5 leading 2-loop coefficient n 2 is universal: n 2 = 4 for folded string state [Basso]; evidence from universality [BGMRT] of the Konishi state energy (J = N = 2) E N=J=2 = 4 [ 2 + b + b 2 ( ) + b ] 3 2 ( ) b = + n n = 2 b 2 = 4 b2 + 2n 0 + 2n 03 + n n 2 = 2 3ζ 3 b 3 = a + a 2 ζ ζ 5,... b, b 2 : match TBA predictions interpolated to

43 need 2-loop string sigma model computation to confirm universality of n 2, fix n 22 determine b 3 Some details: Circular rotating string in S 5 with J = J 2 J : flat space R t R 4 : circular string solution x + ix 2 = a e i(τ+σ), 4 E = α J, x 3 + ix 4 = a e i(τ σ) J = a2 α 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 = X + ix 2 = a e i(τ+σ), X 3 + ix 4 = a e i(τ σ), X 5 + ix 6 = 2a 2, t = κτ J = J = J 2 = a 2, E0 2 = κ 2 = 4J

44 E 0 is just as in flat space E 0 = E = 4 J, J = J -loop correction to energy: closed string (R S ) sum of bosonic and fermionic fluctuation frequencies (n = 0,, 2,...) Bosons (2 massless + massive): AdS 5 : 4 ω 2 n = n 2 + 4J S 5 : 2 ω 2 n± = n 2 + 4( J ) ± 2 4( J )n 2 + 4J 2 Fermions: E = 2κ 4 ωn 2 f ± = n J ± 4( 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 (ζ k come from n ) E = [ J (3 + ζ J 3 )J 2 ( ) ] ζ3 + 30ζ 5 J

45 J [ E = E 0 + E = ( + 2ζ 3) J ] ( ) To get a state on the first excited string level (N = 2J ) should choose J =, i.e. J = J 2 = for minimal non-trivial value of J = J 3 = 2 there is unique corresponding state in Konishi multiplet table: [0,, 2] (0,0) at level 0 = 6 and thus ( J 2 b = 2 8J + ) 2 J=2,J = = 2 Small circular spinning string with S = S 2 rigid circular string with two equal spins in AdS 5 and orbital momentum J = J 3 in S 5 Y 0 + iy 5 = + 2r 2 e iκt, Y + iy 2 = r e i(wτ+σ), Y 3 + iy 4 = r e i(wτ σ) X 5 + ix 6 = e iντ, w 2 = κ 2 +, κ 2 ( + 2r 2 ) = 2r 2 ( + w 2 ) + ν 2

46 E 0 = ( + 2r 2 )κ = κ + 2κS + κ 2, S = S = S 2 = r 2 w, J = ν short string expansion of the classical energy E 0 = E 0, E 0 = 2 S ( + S + J 2 8S +... ) including -loop correction: S [ E 0 + E = 2 + ( S + J 2 8S ) 2 ] + O( ( ) ) 2 state on the first excited level (N = 2S) has two excited oscillators, i.e. should have S = S = S 2 = for J = 2 the dual state in representation [0, 2, 0] (,0) there is just one state in Konishi table with 0 = 6 b = 2 (S + J 2 8S ) 2 S=,J=2 = 2

47 Small circular spinning string with S = J and J = J 2 0 rigid circular solution with one spin in AdS 5 and one spin in S 5 and orbital momentum J in S 5 Y 0 + iy 5 = + r 2 e iκt, Y + iy 2 = r e i(wτ+σ), w 2 = κ 2 +, X + ix 2 = a e i(w τ σ), X 3 + ix 4 = a 2 e iντ, w 2 = ν 2 +, E 0 = 2 S ( + 2 S + J 2 ) 8S +... The leading -loop correction to the energy vanishes (cancellation of AdS and sphere contributions) S [ E 0 + E = 2 + ( 2 S + J 2 2 ) ] + O( 8S ( ) ) 2 state on the first excited level: S = J = for J = 2 get state [,, ] ( 2, at 2 ) 0 = 6 level ( b = 2 2 S + J 2 ) = 2 8S S=,J=2

48 Conclusions beginning of understanding of AdS 5 S 5 string spectrum = spectrum of conformal N = 4 SYM operators agreement with numerical results from TBA: non-trivial check of quantum integrability prediction of transcendental structure of leading coefficients: reproduce them by an analytic solution of TBA at strong coupling? evidence of universality of some coefficients in strong coupling expansion of dimensions of states on leading Regge trajectory need systematic study of quantum string theory in AdS 5 S 5 in near-flat-space expansion and of course we still need first-principles solution for the spectrum of AdS 5 S 5 superstring = spectrum of N = 4 SYM based on integrability it is now appearing to be within reach...