Short spinning strings, symmetries and the spectrum of the AdS 5 S 5 superstring

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1 Short spinning strings, symmetries and the spectrum of the AdS 5 S 5 superstring Radu Roiban PSU Based on M. Beccaria, S. Giombi, G. Macorini and A. Tseytlin

2 Integrability for N=4 sym and strings in AdS 5 S 5 Allows in principle to determine the spectrum of dimensions/energies at any value of the coupling constant Weak coupling integrability explicitly demonstrated though 4/5- loop order - - large quantum numbers provide guidance Strong coupling integrability explicitly demonstrated though 2- loop order The current strategy: assume all- order integrability and construct Bethe ansatze of appropriate type that explain exising data check predicions for other dimensions

3 Good understanding of operators with large quantum numbers AsymptoIc Bethe ansatz (ABA) (final form) Beisert, Eden, Staudacher Firmly established: tested through 4/2 loops at weak/strong coupling String theory tests: semi- classical expansion, states with large quantum numbers dual to operators with large classical dimension Most studied example of long operator: Tr[ S J : E = J +2+f( ) ln S +... h f( ) = 2 2 p J D S + ] i Both regimes captured by BES equaion; soluion by Basso, Korchemsky, Kotanski ` = f( ) = Long and fast string generalizaion h J p ln S fixed : E = J +2+ Captured by FRS equaion ln2 K i p ( p ) fw (, p `) f s ( p, `2) ln S +...

4 general short operators! general string quantum states Integrability- based results; improvements of ABA - Luscher correcions - ABA TBA Ambjorn, Kristjansen, Janik Bajnog, Janik, Lukowski; Bajnog, Hegedus, Janik, Lukowski! Arutyunov, Frolov - TBA in Y variables Gromov, Kazakov, Kozak, Vieira Bombardelli, FioravanI, Tateo Conjectured set of funcional equaions for the spectrum AddiIonal constraints needed for a physical soluion

5 general short operators! general string quantum states Integrability- based results; improvements of ABA - Luscher correcions - ABA TBA Ambjorn, Kristjansen, Janik Bajnog, Janik, Lukowski; Bajnog, Hegedus, Janik, Lukowski! Arutyunov, Frolov - TBA in Y variables Gromov, Kazakov, Kozak, Vieira Bombardelli, FioravanI, Tateo Conjectured set of funcional equaions for the spectrum AddiIonal constraints related to FiNLIE Z 4 automorphism of PSU(2, 2 4) Gromov, Kazakov, Leurent, Volin Main assumpions: - - quantum integrability in finite volume - - string model/gt spin chain obeys Hirota eq.

6 general short operators! general string quantum states Simplest example: the Konishi muliplet, containing Tr[ ( ) = (4 ) 2 h 4 (4 ) (4 ) 4 ( ) (4 ) 6 i i] 5- loop: IniIally predicted by from Luscher correcions; confirmed by direct calculaion Bajnok, Hegedus, Janik, Lukowski; Eden, Korchemsky, Smirnov, Sokatchev Strong coupling: Konishi muliplet + 8( ) Worldsheet analysis suggests that (more later) 3 st excited string level 4 i (4 ) ( ) = 2 4p + A 0 + A 4p + A 2 ( 4p ) RR, Tseytlin

7 general short operators! general string quantum states Simplest example: the Konishi muliplet, containing Tr[ Current strong coupling status: TBA/Y- system eqs. solved numerically up large 0 3 i i] TBA/Y- system eqs. solved numerically; fit reproduces QFT calculaion A 0 =0 A =.998 A 2 = 3. ( ) = 2 4p + A 0 + A 4p + A 2 ( 4p ) 3 + A 3 ( 4p ) and extrapolated to Gromov, Kazakov, Vieira Frolov worldsheet calculaion at - loop A 0 =0 A =2 RR, Tseytlin Gromov, Shenderovich, Serban, Volin using an all- order conjecture of Basso, the 2- loop coefficient is A 2 = ' Gromov, Valatka higher- order coefficients: A k 2k + lower transcendentality Beccaria, Giombi, Macorini, RR, Tseytlin e.g. A 3 = a 3,3 + a 3,2 3 + a 3, 5 with a 3, = 5 2

8 general short operators! general string quantum states Open quesions: IdenIfy general structures of the spectrum from the TBA/Y- system; what type of numbers can appear? AnalyIc calculaions for the TBA/Y- system approach? Other states? ( l, l, Li n (rational),...) A systemaic comparison with near- flat- space string spectrum?

9 general short operators! general string quantum states Open quesions: IdenIfy general structures of the spectrum from the TBA/Y- system; what type of numbers can appear? AnalyIc calculaions for the TBA/Y- system approach? Other states? ( l, l, Li n (rational),...) A systemaic comparison with near- flat- space string spectrum? Here: understand the structure of the energy as a funcion of charges for some classes of string states evaluate / p correcions to energy of other states that should belong to the Konishi muliplet check symmetries (same E up to raional)/compare with TBA/Y- system

10 Outline OrganizaIon of flat space string state in AdS 5 S 5 muliplets Structure of energy correcions Short strings vs. the Konishi muliplet: vertex operator- improved semiclassical approach Summary

11 OrganizaIon of states In large volume limit AdS 5 S 5 is nearly flat Expect that some properies of the spectrum survive Flat space spectrum: best constructed in light- cone gauge n = 2 (N + Ñ) mass : p 2 = m 2 = 4(2 T )(n ) (NSR GSO) p 2 = m 2 = 4(2 T ) n (GS) ground state : 0i =(8 v +8 c ) 2 7! IIB supergravity multiplet 2 8 states st excited : (a i S a ) 2 0i ((8 v +8 c ) (8 v +8 c )) 2 7! 2 6 states - - Organize in representaions of SO(9) - - A single supersymmetry muliplet (44 b + 84 b + 28 f ) 2 higher levels : More oscillators; different indices, etc; representaions of SO(9), all of the same mass, several supersymmetry muliplets Some changes in the presence of a weakly curved AdS 5 S 5 background:

OrganizaIon of states Worldsheet perturbaion theory series in p Different flat space levels cannot belong to the same muliplet; level- by- level reorganizaion Assuming non- intersecion principle Same

12 OrganizaIon of states Worldsheet perturbaion theory series in p Different flat space levels cannot belong to the same muliplet; level- by- level reorganizaion Assuming non- intersecion principle Same should be true for all values of p Polyakov Flat space spectrum: reorganized in muliplets of SO(2, 4) SO(6) PSU(2, 2 4) Energy defined here Bianchi, Morales, Samtleben SO(9) 7! SO(4) SO(5)! SO(4) SO(6) SO(2, 4) SO(6) ; [k, p, q] (sl,s R ) X Example: st excited level [0,J,0] (0,0) ( + Q + Q ^ Q +...)[0, 0, 0] (0,0) J =0: Length of muliplets of 0d super- Poincare group Same content as the Konishi muliplet J=0 = Konishi muliplet Length of muliplets 4d superconformal group MulIplet structure is not severely reorganized; extra quantum number; Lij of flat space mass degeneracy (no useful noion of mass) st excited level (lowest KK)

13 TBA/Y

14 st excited level vs. higher levels st excited string level: minimal (2d RG) mixing of their vertex operators: - energies of states in the muliplet may differ only by (half- )integers - mild mixing may be possible between states in different KK mulip s Higher levels: lots of muliplets - each can acquire a different energy (i.e. different correcions) - nontrivial mixing problem of vertex operators (help from closed sectors) p Is there an integrable model for it? (w/ Hamiltonian ) L 0 + L 0 implicaions for correlaion funcions, worldsheet form factors, etc. No level mixing; what is the gauge theory analog of string level?

15 On vertex operators and the general structure of E(Q) Bosonic Lagrangian with manifest SO(2, 4) SO(6) + (Y Y + + (X X ) Vertex operators V = m...m s Z m...z @Z (Y 5 + Y 0e ) charges unrelated to level e.g. KK charge (n Z) J determine the level partly in terms of charges bulk- boundary prop. z K = z 2 +(x x) 2 analog of e iet Physical state condiion: 2d(Q) = 0 with Q fixed as! 2d(Q) from renormalizaion of vertex operators At L- loops: Feynman graphs with at most (L+) fields amached to V p L (Q L+ +lowerpowers) 2d Bosonic model: detailed expressions Wegner

16 In general: is a matrix the matrix representaion of 2d L 0 + L 0 d V i i i =(d i d ln j + ˆ2d j )V j 2d - - structure constrained by symmetries and charge conservaion - - eigenvalues are the anomalous dimensions of vertex operators Worldsheet theory is integrable in conformal gauge; seems plausible that it diagonalizes L 0 + L 0 ; eigenvectors related to physical vertex operators Ground state and st excited level are special: p no/minimal mixing! 2d dimension is a regular funcion of E(E 4) = 2 p X ai Q i + X i,j + p b ij Q i Q j + X c i Q i P ijk d ijkq i Q j Q k + P i,j e ijq i Q j + P f i Q i +... There exist singlet vertex operators which do not mix

17 In general: is a matrix the matrix representaion of 2d L 0 + L 0 d V i i i =(d i d ln j + ˆ2d j )V j 2d - - structure constrained by symmetries and charge conservaion - - eigenvalues are the anomalous dimensions of vertex operators Worldsheet theory is integrable in conformal gauge; seems plausible that it diagonalizes L 0 + L 0 ; eigenvectors related to physical vertex operators Ground state and st excited level are special: p no/minimal mixing! 2d dimension is a regular funcion of E(E 4) = 2 p X ai Q i Tree level + X i,j b ij Q i Q j + X c i Q i 2 loops loop + p P ijk d ijkq i Q j Q k + P i,j e ijq i Q j + P f i Q i +... There exist singlet vertex operators which do not mix

18 Simple examples: strings in flat space: E 2 / M 2 / J (- loop corr. is exact) supergravity states: E(E 4) = J(J + 4) i V = Y + h(@x Xk ) r +... =) (use NSR formulaion) 0= 2(r ) + 2 p ( 4) + 2r(r ) + p r(r )(r 7 2 ) Solve for - Inclusion of fermions may change some of the details Krastov, Lerner, Yudson; CasIlla, ChakravarI An excited string state: choose effecive level N (+ constant shijs)

19 Target space energy, in the presence of two charges; more subleading terms E 2 =2 p N + J 2 + n 02 N 2 + n N + p n 0 J 2 N + n 03 N 3 + n 2 N 2 + n 2 N + ( p ) 2 ñj 2 N +ñ 02 J 2 N 2 + n 04 N 4 + n 3 N 3 + n 22 N 2 + n 3 N + ( p ) 3 ñ0j 4 N +ñ 2 J 2 N +ñ 2 J 2 N 2 + n 05 N ( p ) 4 n J 4 N O( ( p ) 5 ) Other possible terms: E k,e l J m,j p : spurious sol s to marginality condiion in BMN limit; similar to higher powers of Laplacian in eom scheme dependent E k N p : treated perturbaively; slight redefiniions of Direct flat space interpretaion; N level coef s Valid for all states that have a vertex operator descripion, regardless of values of charges n

20 Target space energy, in the presence of two charges; more subleading terms E 2 =2 p N + J 2 + n 02 N 2 + n N + p n 0 J 2 N + n 03 N 3 + n 2 N 2 + n 2 N + ( p ) 2 ñj 2 N +ñ 02 J 2 N 2 + n 04 N 4 + n 3 N 3 + n 22 N 2 + n 3 N + ( p ) 3 ñ0j 4 N +ñ 2 J 2 N +ñ 2 J 2 N 2 + n 05 N ( p ) 4 n J 4 N O( ( p ) 5 ) the plan - - use universality of this structure to: - find the values for the coefficients from semiclasscial short string expansion - if several suitable expansions exist, check that they give same values n ij - jusify semiclassical approach to quantum string states - test realizaion of symmetries: classical soluions in the same muliplet have same energy correcions (up to integer shijs)

21 Expansions: p ) ConnecIon to quantum string states: large, fixed charges N and J E = q 2 p h N + A p + A 2 ( p ) + A i 3 2 ( p ) + O( 3 ( p ) ) 4 A = 4N J (n 02N + n ), A 2 = 2 A2 + 4 (n 0J 2 + n 03 N 2 + n 2 N + n 2 ) A 3 = h (n 3 8n n n 3 )+(3n 02 n 2 8n n 2 8n 02 n n 22 )N 28 i +(3n 2 02n 8n 03 n 8n 02 n n 3 )N Konishi muliplet states: N =2,J =2 Higher KK levels: tensor with [0,J,0]

22 Expansions: p ) ConnecIon to quantum string states: large, fixed charges N and J Konishi muliplet states: N =2,J =2 Higher KK levels: tensor with [0,J,0] E N=2 = Symmetry consequences (uniqueness of muliplet) are not universal; however - - n ij q 2 p N (A ) N=2 = 8 J (2n 02 + n ) 2n 02 + n =fixed h + (A ) N=2 p + (A 2) N=2 ( p ) 2 + (A 3) N=2 ( p ) 3 + O( ( p ) 4 ) i (A 2 ) N=2 = 2 (A ) 2 N=2 + 4 n 0J (4n 03 +2n 2 + n 2 ) (A 3 ) N=2 = (A A 2 ) N=2 + 4 (2ñ 02 +ñ )J 2 + i 4 (8n 04 +4n 3 +2n 22 + n 3 ) 2ñ 02 +ñ =fixed n 0 =fixed 4n 03 +2n 2 + n 2 =fixed 8n 04 +4n 3 +2n 22 + n 3 =fixed

23 Expansions: 2) Semiclassical expansion: fixed p large, then small and E p = J + h N J ( + 2 n 0J 2 + 2ñ0J ) + p + N 2 2J 3 N 2 2J 3 Q = Q/ p N J +(n 0 n 02 )J 2 +(ñ 0 ñ n2 0)J h N 2J (n +ñ J 2 + n J ) i +... n +(n 2 2 n 0n ñ )J 2 +(ñ 2 n 2 n 0ñ 2ñ0n )J N 3 4J 5 3n +[3ñ 2n 2 +(3n 0 n 02 )n ]J 2 + 2(n 3 ñ 2 ) n 0 n 2 +3 n +(3ñ 0 ñ n2 0)n +(3n 0 n 02 )ñ J h N i ( p ) 2 2J n 2 +ñ 2 J O( ( p ) ) 3 Each loop order is a nontrivial funcion of charges E 2 is a much more economical presentaion of the spectrum i +...

Expansions: ResummaIon of small J expansion: contact with the slope funcion(s) E 2 = J 2 + h (,J)N + h 2 (,J)N 2 + h 3 (,J)N 3 +... h =2 p + n + n 2 p + n 3 ( p ) +.

24 Expansions: ResummaIon of small J expansion: contact with the slope funcion(s) E 2 = J 2 + h (,J)N + h 2 (,J)N 2 + h 3 (,J)N h =2 p + n + n 2 p + n 3 ( p ) J 2 n 0 p + ñ 2 ( p ) + ñ2 2 ( p ) 3 h 2 = n 02 + n 2 p + n 22 ( p ) J 2 ñ 02 2 ( p ) + ñ2 2 ( p ) 3 h 3 = n 03 p + n 3 ( p ) , h 4 = n 03 ( p ) E at small N and fixed J : E = J + 2J h (,J) N Conjectured by Basso in sl(2) sector based on ABA - Proven by Basso and Gromov su(2)sector is more intricate - Conjectured by Gromov and Beccaria and Tseytlin informaion on the 2- loop coefficient n why raional?

25 h =2 p + n + n 2 p + n 3 ( p ) J 2 n 0 p + ñ 2 ( p ) + ñ2 2 ( p ) 3 h sl(2) =2 p d ln I J ( p ) d p =2 p 4 J 2 p =2 p p +J 2 +J 2 h su(2) ( p, J )= h su(2) ( h su(2) = 2 p d ln K J ( p ) d p =2 p + =2 p p +J J 2 p + 4 J 2 ( p ) 2 p, J ) +J 2 4 J 2 ( p ) J J 4 ( p +... ) 3 4 J 2 4 p ( + J 2 ) 5/2 h su(2) ( p,j)= h su(2) ( J 2 + J 4 ( p ) 2 ( + J 2 ) J J 4 ( p ) Basso p, J) Gromov Becaria, Tseytlin 4 J 2 5 p ( + J 2 ) J 2 + J 4 5/2 ( p ) 2 ( + J 2 ) informaion on the 2- loop coefficient n why raional?

26 Classical soluions in the same muliplet: flat space Circular strings: Folded strings: t = apple, x = x + ix 2 = ae i( + ), x 2 = x 3 + ix 4 = ae i( ) E 2 flat = 2 0 N, N= J + J 2, J = J 2 = a2 - - Vertex operator of the h corresponding quantum state: i V = e iet ) J/2 (@x ) J2/ t = apple, x + ix 2 = a sin e i Eflat 2 = 2 2a2 N, N= S, S= Vertex operator of h the corresponding quantum state: i V = e iet ) S/ Other orientaions related by Lorentz transformaions - Add translaional momentum

27 Classical soluions in the same muliplet: Circular strings: three inequivalent embeddings AdS 5 S st excited level: N =2 - - member of Konishi muliplet: add orbital angular momentum: J =2 0 =6 - - two planes in : S two planes in AdS 5 : - - one plane in each: J = J 2 =,J=2! [0,, 2] (0,0) S = S 2 =,J=2! [0, 2, 0] (,0) S = J =,J=2! [,, ] ( 2, 2 ) Folded strings: two inequivalent embeddings 0 =4 - - one planes in : S one planes in AdS 5 : J = J =2! [2, 0, 2] (,) S = J =2! [0, 2, 0] (,) - Evaluate loop correcions to their energies in the semiclassical regime (no longer in the same muliplet) - Extract n coefficients - Find (same) correcion in short string regime; predict other n coefficients

28 RegularizaIon and symmetries Lorentz- invariant field theories in infinite volume - - limited choices of regulator: dimensional, higher- derivaive, etc QFT in finite volume: - - more choices as space/ime different: e.g. mode number cutoff * for diagonal propagators argued to preserve supersymmetry Rebhan, van Nieuvenhuizen, et al Off- diagonal propagators, finite volume: - - more choices due to absence of natural i prescripion * expand each field in modes, each mode may have its own independent i prescripion Z dp0 2 ln(p 0!(n) is i,n )= s i,n 2!(n)! E 2d = 4 X i X s b i,n! i b (n) s f i,n!f i (n) - Choice of signs realizaion of target space symmetry algebra - transcendental part of E 2d controlled by n ; limle sensiivity to summaion prescripion =

29 ProperIes of the coefficients: Near- BMN expansion is universal E 2 =2 p N + J 2 + n 02 N 2 + n N + p n 0 J 2 N + n 03 N 3 + n 2 N 2 + n 2 N + E 2 = J 2 +2N p + J = J 2 + N 2 p + p J 2 4( p ) J ) n 0 = ñ 0 = 4 Other classical coefficients: raional + state- dependent From analysis of folded strings and circular strings: 2n 02 + n =2 Universality of ( p ) 2 ñj 2 N +ñ 02 J 2 N 2 + n 04 N 4 + n 3 N 3 + n 22 N 2 + n 3 N + ( p ) ñ0j 4 N +ñ 2 J 2 N +ñ 2 J 2 N 2 + n 05 N ( p ) n J 4 N O( 4 ( p ) ) 5 RR, Tseytlin Gromov, Serban, Shenderovich, Volin Beccaria, Giombi, Macorini, RR, Tseytlin - dependence in terms in slope funcion(s) ñ = n, n = n, 2ñ 02 +ñ =0 J h =2 p p +J 2 + n +J 2 + p (n 2 +ñ 2 J 2 + O(J ) 4 )) + O( ( p ) 2 ) h 2 = n 02 + J 2 +J 2 + O( p )

30 E 2 =2 p N + J 2 + n 02 N 2 + n N + p n 0 J 2 N + n 03 N 3 + n 2 N 2 + n 2 N More properies and predicions: n 2 = n universal + ( p ) 2 ñj 2 N +ñ 02 J 2 N 2 + n 04 N 4 + n 3 N 3 + n 22 N 2 + n 3 N + ( p ) 3 ñ0j 4 N +ñ 2 J 2 N +ñ 2 J 2 N 2 + n 05 N ( p ) 4 n J 4 N O( ( p ) 5 ) Tirziu, Tseytlin; RR, Tseytlin; Gromov, Valatka Folded string slope funcion: n 2 = 4 4n 03 +2n n 2 = n 0 2 = 3 8 2n 03 should be universal Basso Remarkably val s of n and imply that n 2 = 03 n should be universal from a 2- loop sigma model calculaion; why raional? More universal behavior: n k 2k e.g. ñ 2 =ñ n 3 = n n Top transcendental component: high modes, local ws structure

32 An example: J 0 J = J 2 6= J 3 J in S 5 The soluion in embedding coordinates: X X 2 6 = X = ae i(w + ), X 2 = ae i(w ), X 3 = p 2a 2 e i E 2 0 = 2 =4a = 2 + 4J 0 + 2, w2 =+ 2 J J = J 2 = a 2 w, J J 3 = p 2a 2, = J q 2J Expansions: E 0 =2q p J 0 h + p J 2 8J 0 ( p ) 2 J 4 28J i E 0 = J + 2 J E 0 =2 p + 2 q p J 0 p +J 2 J 0 2( + 2J 2 ) J 3 ( + J 2 ) J 02 + O(J 03 ) apple + p 2 J 0 ( + 2 ) + ( p ) 2 ( )J 02 2( + 2 ) 2 + O(( p ) 3 )

33 - loop correcion to energy: from 2d pariion funcion - find frequencies of small oscillaions about the classical soluion - sum them up (signs for fermions) = 2 q =2J 0 / p + 2 solve perturbaively in

34 - loop energy in terms of the parameters of the soluion: E = apple f 0( )+f ( ) a 2 + f 2 ( ) a f 0 ( ) =0 f ( ) =2 2 + O( 4 ) f 2 ( ) = O( 2 ) semiclassical expansion E = 2 J 4 2J +2J 3 + O(J 5 ) J + J J O(J ) J pp J 0 E =2p + 2 = J /(2 p J 0 ) fixed expansion ( ) h 2 p + J ( + 2 ) O( p 3 ) i

35 E = 2 J 4 2J +2J 3 + O(J 5 ) J + J J E = N 2J (n +ñ J 2 + n J ) Vs. + N 2 2J 3 n +(n 2 2 n 0n ñ )J 2 +(ñ 2 n N 3 + O J (N =2J ) O(J ) J n 0ñ 2ñ0n )J Read off n ij coeficients: n 0 = n 02 =0 n 03 =0 RR, Tseytlin Beccaria, Giombi, RR, Tseytlin n =2, ñ = 2, n 2 = n = coefficients and relaions as menioned n j 3 contain higher zeta numbers in expected pamern Konishi dimension: (assume susy restricion for n 2 ) E =2 4p apple + b 2 p + b 2 2( p ) 2 + b 3 2( p ) b =2 b 2 = b 3 = a 3,3 + a 3,

36 S = S = S 2 in AdS 5 ; J in S 5 soluion in embedding coordinates: Y 2 5 Y Y Y 2 4 = Y 0 + iy 5 = p +2r 2 e i,y + iy 2 = re i(w + ),Y 3 + iy 4 = re i(w E 0 =(+2r 2 ) =2 S +S + J 2 8S +... ),X + ix 2 = e i Slightly more involved technical details q h E = E 0 + E =2 S + level state for minimal nontrivial S = J =2 S + J 2! b =2 8S 2 + O( i ( ) ) 2 unique state in representaion [0, 2, 0] (,0) n 2 = n = ; n sill needed at 2 loops b 0

37 S = J with J 2 6=0 soluion in embedding coordinates: Y 0 + iy 5 = p +r 2 e i, Y + iy 2 = re i(w + ) X + ix 2 = ae i(w0 ), X 3 + ix 4 = p a 2 e i E 0 =2 S + 2 S + J 2 2 8S +... Slightly more involved technical details; q E = E 0 + E =2 level state for minimal nontrivial S = state in representaion S J 2 =2 h + 2 S + J 2 2 8S! b =2 [,, ] (/2,/2) n vanishes + O( i ( ) ) 2 + b 0 not unique; choose dim=6 n 2 = n = 5 ; sill needed at 2 loops 8 (?) 3 3 n 2

38 Summary general structure of energy as funcion of conserved charges - based on the structure of vertex operators for general states - only some states in the muliplet should be visible semiclasically discussed a number of examples of members of Konishi muliplet E =2 4p + b 0 + b 4p + b 2 ( 4p ) b =2 b 2 = 2 - direct calculaion of - b b and of transcendental part of agrees with TBA/Y- system conjecture - consistent with supersymmetry - transcendental part of is universal - predicion for transcendental structure of higher coefficients - susy: 2- loop contrib. to Test of finite- size integrability b 2 b 2 Most open quesion are sill open 3 3 b 2 Gromov, Kazakov, Vieira Frolov is raional (surprising) and universal Special simplificaions in the 2- loop sigma model calculaion?

39 Extra slides

Progress in understanding quantum spectrum of AdS 5 S 5 superstring

Progress in understanding quantum spectrum of AdS 5 S 5 superstring Arkady Tseytlin R. Roiban, AT, arxiv:0906.494, arxiv:0.09 M. Beccaria, S. Giombi, G. Macorini, R. Roiban, AT, arxiv:03.570 M. Beccaria,