Comments on the Mirror TBA

Transcription

1 Comments on the Mirror TBA Sergey Frolov Hamilton Mathematics Institute and School of Mathematics Trinity College Dublin Integrability and Gauge/String Duality, IAS at Hebrew University, April 3, 2012

2 Outline 1 Baby steps 2 Mirror TBA 3 Excited states 4 Results 5 Bound states 6 Summary

3 Mirror TBA approach to the AdS/CFT spectral problem Gauge-String Correspondence Matsubara Transform Thermodynamic Limit Gauge Theory String Sigma Model Mirror Theory TBA Spectrum Thermodynamics of the mirror theory determines the finite-size spectrum of the original model

4 Baby steps towards the goal 1 Green-Schwarz superstring on AdS 5 S 5 S = g dτdσ hh αβ α X M β X N G MN (X) + fermions, g = 2 2 L.c. string sigma model : E J = H = J 0 dσ H l.c. λ 2π

5 J J is the Noether charge corresponding to one of the Cartans of SO(6). J = # of Z s in a primary operator, e.g. ( J ) ( J ) sl(2) sector = Tr D n k Z or Tr X n k Z = su(2) sector k=1 k=1

6 Anti-de Sitter space Space of constant negative curvature E is the Noether charge corresponding to SO(2) of SO(4, 2). String energy E(g) = (λ) Scaling dimension

7 Baby steps towards the goal 1 Green-Schwarz superstring on AdS 5 S 5 S = g dτdσ hh αβ α X M β X N G MN (X) + fermions, g = 2 2 L.c. string sigma model : E J = H = J 0 dσ H l.c. 3 Space-time exchange = Mirror theory on a plane λ 2π

8 Mirror theory Al. Zamolodchikov 90 Partition function of the model Z (R, J) n ψ n e H R ψ n = n e En R = Dp Dx e R R 0 dτ R J 0 dσ(i pẋ H), where the integration is taken over x and p periodic in both τ and σ. R 0 dτ J dσ(i pẋ H) is Euclidean action. 0 R 0 dτ J dσ( pẋ H) is Minkowski action 0 Euclidean action: replace ẋ iẋ Wick rotation τ i σ. Take the Euclidean action Replace x ix Wick rotation of the σ-coordinate σ i τ. New action where τ is considered as the new time coordinate. Let H be the Hamiltonian with respect to τ H = R 0 d σ H

9 Mirror theory Al. Zamolodchikov 90 Partition function of the model Z (R, J) n ψ n e H R ψ n = n e En R = Dp Dx e R R 0 dτ R J 0 dσ(i pẋ H), where the integration is taken over x and p periodic in both τ and σ. R 0 dτ J dσ(i pẋ H) is Euclidean action. 0 R 0 dτ J dσ( pẋ H) is Minkowski action 0 Euclidean action: replace ẋ iẋ Wick rotation τ i σ. Take the Euclidean action Replace x ix Wick rotation of the σ-coordinate σ i τ. New action where τ is considered as the new time coordinate. Let H be the Hamiltonian with respect to τ H = R 0 d σ H

10 Mirror theory Al. Zamolodchikov 90 We refer to the model with the Hamiltonian H as to the mirror theory. No Lorentz inv H and H describe different Minkowski theories The partition function of the mirror model Z (J, R) n ψ n e e HJ ψ n = n e e E nj = D p Dx e R R 0 dτ R J 0 dσ(iepx e H) Integrating over p, we get the same Euclidean action Take the limit R Z (J, R) = Z (R, J). log Z (R, J) R E(J), where E(J) is the ground state energy. log Z (J, R) R J F(J), where F(J) is the bulk free energy of the mirror theory at temperature T = 1/J.

11 Mirror theory Al. Zamolodchikov 90 We refer to the model with the Hamiltonian H as to the mirror theory. No Lorentz inv H and H describe different Minkowski theories The partition function of the mirror model Z (J, R) n ψ n e e HJ ψ n = n e e E nj = D p Dx e R R 0 dτ R J 0 dσ(iepx e H) Integrating over p, we get the same Euclidean action Take the limit R Z (J, R) = Z (R, J). log Z (R, J) R E(J), where E(J) is the ground state energy. log Z (J, R) R J F(J), where F(J) is the bulk free energy of the mirror theory at temperature T = 1/J.

12 Mirror theory Al. Zamolodchikov 90 We refer to the model with the Hamiltonian H as to the mirror theory. No Lorentz inv H and H describe different Minkowski theories The partition function of the mirror model Z (J, R) n ψ n e e HJ ψ n = n e e E nj = D p Dx e R R 0 dτ R J 0 dσ(iepx e H) Integrating over p, we get the same Euclidean action Take the limit R Z (J, R) = Z (R, J). log Z (R, J) R E(J), where E(J) is the ground state energy. log Z (J, R) R J F(J), where F(J) is the bulk free energy of the mirror theory at temperature T = 1/J.

13 Mirror theory: Summary string of length L =J "mirror string" of length R One Euclidean theory two Minkowski theories. One is related to the other by the space-time exchange: σ = i τ, τ = i σ The Hamiltonian H w.r.t. τ defines the mirror theory. Ground state energy is related to the free energy of its mirror E(J) = J F(J) F is found from Bethe-Yang eqs for the mirror model as R

14 Baby steps towards the goal 1 Green-Schwarz superstring on AdS 5 S 5 2 L.c. string sigma model : E J = H = J 0 dσ H l.c. 3 Space-time exchange = Mirror theory on a plane: Massive integrable model in 2dim: 8 bosons + 8 fermions No 2-dim Lorentz invariance! 4 Symmetries of the mirror model fix the dispersion relation for mirror particles and the matrix form of the S-matrix 5 Crossing relations fix the dressing factor up to a CDD factor 6 Single out the right dressing factor 7 Use the S-matrix to identify the spectrum of mirror particles and to derive the mirror Bethe-Yang eqs

15 Baby steps towards the goal 1 Green-Schwarz superstring on AdS 5 S 5 2 L.c. string sigma model : E J = H = J 0 dσ H l.c. 3 Space-time exchange = Mirror theory on a plane: Massive integrable model in 2dim: 8 bosons + 8 fermions No 2-dim Lorentz invariance! 4 Symmetries of the mirror model fix the dispersion relation for mirror particles and the matrix form of the S-matrix 5 Crossing relations fix the dressing factor up to a CDD factor 6 Single out the right dressing factor 7 Use the S-matrix to identify the spectrum of mirror particles and to derive the mirror Bethe-Yang eqs

16 Baby steps towards the goal 1 Green-Schwarz superstring on AdS 5 S 5 2 L.c. string sigma model : E J = H = J 0 dσ H l.c. 3 Space-time exchange = Mirror theory on a plane: Massive integrable model in 2dim: 8 bosons + 8 fermions No 2-dim Lorentz invariance! 4 Symmetries of the mirror model fix the dispersion relation for mirror particles and the matrix form of the S-matrix 5 Crossing relations fix the dressing factor up to a CDD factor 6 Single out the right dressing factor 7 Use the S-matrix to identify the spectrum of mirror particles and to derive the mirror Bethe-Yang eqs

17 Baby steps towards the goal 1 Green-Schwarz superstring on AdS 5 S 5 2 L.c. string sigma model : E J = H = J 0 dσ H l.c. 3 Space-time exchange = Mirror theory on a plane: Massive integrable model in 2dim: 8 bosons + 8 fermions No 2-dim Lorentz invariance! 4 Symmetries of the mirror model fix the dispersion relation for mirror particles and the matrix form of the S-matrix 5 Crossing relations fix the dressing factor up to a CDD factor 6 Single out the right dressing factor 7 Use the S-matrix to identify the spectrum of mirror particles and to derive the mirror Bethe-Yang eqs

18 Baby steps towards the goal 1 Green-Schwarz superstring on AdS 5 S 5 2 L.c. string sigma model : E J = H = J 0 dσ H l.c. 3 Space-time exchange = Mirror theory on a plane: Massive integrable model in 2dim: 8 bosons + 8 fermions No 2-dim Lorentz invariance! 4 Symmetries of the mirror model fix the dispersion relation for mirror particles and the matrix form of the S-matrix 5 Crossing relations fix the dressing factor up to a CDD factor 6 Single out the right dressing factor 7 Use the S-matrix to identify the spectrum of mirror particles and to derive the mirror Bethe-Yang eqs

19 Toddlers Green-Schwarz superstring Metsaev, Tseytlin 98; Semi-classical strings Berenstein, Maldacena,Nastase 02; Gubser, Klebanov, Polyakov 02; Frolov, Tseytlin 02, 03; Bena, Polchinski, Roiban 03; Kazakov, Marshakov, Minahan, Zarembo 04; L.c. strings in AdS 5 S 5 Arutyunov, Frolov 04, 05; Frolov, Plefka, Zamaklar 06; First mentioning of TBA in AdS/CFT Ambjorn, Janik, Kristjansen 05; Symmetry algebra Beisert 05, 06; Arutyunov, Frolov, Plefka, Zamaklar 06; Dispersion relations Beisert, Dippel, Staudacher 04; Beisert 05; N.Dorey 06; S-matrix Staudacher 04; Beisert 05; Arutyunov, Frolov, Zamaklar 06; Crossing equations Janik 06; Dressing factor Arutyunov, Frolov, Staudacher 04; Beisert, Tseytlin 05; Hernandez, Lopez 06; Arutyunov, Frolov 06; Beisert, Hernandez, Lopez 06; Beisert, Eden, Staudacher 06; Bethe ansatz Minahan, Zarembo 02; Beisert, Staudacher 05;

20 Bethe-Yang equations for the mirror model Arutyunov, Frolov 07 1 = e iep k R 1 = 1 = j=1 N j=1 j k S 11 sl(2) (u k, u j ) α=± j=1 N (α) yɛ ɛ=± j=1 N N w (α) S yɛ,1 (v (α) k,y ɛ, u j ) S yɛ,w (v (α) k,y ɛ, w (α) j ) N w (α) l=1 l k S ww (w (α) k, w (α) j ) ɛ=± N (α) yɛ j=1 S 1,yɛ (u k, v (α) j,y ɛ ), S w,yɛ (w (α) k, v (α) j,y ɛ ) p k = p(u k ) is momentum of a fundamental mirror particle, while v (α) k,y ɛ and w (α) k with α = ± and ɛ = ± are rapidities of auxiliary y- and w-roots: < u <, < w <, 2 < v y < 2

21 Bound states of the mirror model Arutyunov, Frolov 07 The sl(2) S-matrix sl(2) (u, u ) = u u + 2i g u u 2i S 11 g 1 Σ 11 (u, u ) 2 exhibits a pole at u u = 2i/g implying the existence of Q-particles u u u u 1 2 u u 1 2 Q=6 Q=5 Bethe strings Q-particles with real rapidities

22 String hypothesis for the mirror model Arutyunov, Frolov 09(a) In the thermodynamic limit R with N/R, N y (α) ɛ /R, N w (α) /R fixed solutions of the BYE arrange themselves into nine different classes of Bethe strings ( ) 1 Q-particles with ẼQ = 2 arcsinh Q2 + p 2 and < u < 2 y (α) ɛ -particles with 2 < v < 2 3 M vw (α) -strings: M + 1 y (α) -roots, M 1 y (α) + -roots and M w (α) -roots with < v < 4 M w (α) -strings: M w (α) -roots with < w < 1 2g : y - : y + : w 1/g } y - y + 1 vw 2 vw 3 vw 1 w 2 w 3 w

23 Bethe-Yang eqs for Bethe strings Arutyunov, Frolov 09(a) N a strings with rapidities u a,k, k = 1,..., N a ( 1) ϕa = e iδaep a,k R b N b n=1 S ab (u a,k, u b,n ) Here δ a = 1 for strings carrying momentum and δ a = 0 otherwise, ϕ a are constants, S ab is scattering matrix of an a-string with a b-string. M w ( ) M vw ( ) y ( ) + y ( ) Q y (+) y (+) + M vw (+) M w (+) M w ( ) M vw ( ) y ( ) + y ( ) Q y (+) y (+) + M vw (+) M w (+)

24 Bethe strings of the Hubbard model Takahashi 72 Symmetry algebra psu(2 2) psu(2 2) R 3 For each α = ± one gets Bethe strings of the Hubbard model. Irreps of su(2) su(2) y ± (1/2, 1/2) M vw (M/2, 0) M w (0, M/2) M vw (+) (+) y - y + (+) + su(2) + su(2) M w (+)

25 Two copies of the Hubbard model M vw (-) M vw (+) (-) y 2 2 (+) + y M w (-) M w (+) y - (-) 1 - y - (+) +

26 String hypothesis for the mirror model Arutyunov, Frolov 09(a) M vw (-) Q M vw (+) (-) y (+) + y M w (-) M w (+) y - (-) y - (+) +

27 String hypothesis for the mirror model Arutyunov, Frolov 09(a) M vw (-) Q M vw (+) (-) y (+) + y M w (-) M w (+) y - (-) y - (+) +

28 M w (+) String hypothesis for the mirror model Arutyunov, Frolov 09(a) M vw (-) Q M w (-) (-) (+) y y+ + + y - (-) M vw (+) y - (+)

29 String hypothesis for the mirror model Arutyunov, Frolov 09(a) M vw (-) Q M vw (+) (-) y (+) + y M w (-) M w (+) y - (-) y - (+) +

30 Thermodynamic limit ( 1) ϕa = e iδaep a,k R Y b N Y b n=1 Roots become dense as R ρ a(u) du: # roots for string of type a in du Integral eqs in the thermodynamic limit S ab (u a,k, u b,n ). ρ a + ρ a = R 2π δa dep a du + K ab ρ b In the AdS 5 S 5 case ep a does not vanish only for Q-particles. Star operation is defined as Z K ab ρ b (u) = du K ab (u, u )ρ b (u ), K ab (u, v) = 1 2πi The right action is defined as Z ρ b K ba (u) = du ρ b (u )K ba (u, u) d du log S ab(u, v)

31 Thermodynamic limit ( 1) ϕa = e iδaep a,k R Y b N Y b n=1 Roots become dense as R ρ a(u) du: # roots for string of type a in du Integral eqs in the thermodynamic limit S ab (u a,k, u b,n ). ρ a + ρ a = R 2π δa dep a du + K ab ρ b In the AdS 5 S 5 case ep a does not vanish only for Q-particles. Star operation is defined as Z K ab ρ b (u) = du K ab (u, u )ρ b (u ), K ab (u, v) = 1 2πi The right action is defined as Z ρ b K ba (u) = du ρ b (u )K ba (u, u) d du log S ab(u, v)

32 Thermodynamic limit ( 1) ϕa = e iδaep a,k R Y b N Y b n=1 Roots become dense as R ρ a(u) du: # roots for string of type a in du Integral eqs in the thermodynamic limit S ab (u a,k, u b,n ). ρ a + ρ a = R 2π δa dep a du + K ab ρ b In the AdS 5 S 5 case ep a does not vanish only for Q-particles. Star operation is defined as Z K ab ρ b (u) = du K ab (u, u )ρ b (u ), K ab (u, v) = 1 2πi The right action is defined as Z ρ b K ba (u) = du ρ b (u )K ba (u, u) d du log S ab(u, v)

33 Free energy and TBA equations Free energy: F µ (J) = R du P h i eea a ρ a + µ a ρ a 1 s(ρa), T = 1/J J Variations of the densities of particles and holes are subject to δρ a + δ ρ a = K ab δρ b. Using the extremum condition δf µ (J) = 0, one gets the canonical TBA eqs ɛ a = E e a + µ a 1 1 J log + e Jɛ b K ba F µ (J) = R J X a Z depa 2π log 1 + e Jɛa where P a P a δa and pseudo (or dressed) energies ɛa are ɛa = 1 ρa log J ρ a. The energy of the ground state of the l.c. string theory J E(J) = lim R R F(J) = X Z Q=1 depq 2π log (1 + Y Q), Y Q e Jɛ Q

34 Free energy and TBA equations Free energy: F µ (J) = R du P h i eea a ρ a + µ a ρ a 1 s(ρa), T = 1/J J Variations of the densities of particles and holes are subject to δρ a + δ ρ a = K ab δρ b. Using the extremum condition δf µ (J) = 0, one gets the canonical TBA eqs ɛ a = E e a + µ a 1 1 J log + e Jɛ b K ba F µ (J) = R J X a Z depa 2π log 1 + e Jɛa where P a P a δa and pseudo (or dressed) energies ɛa are ɛa = 1 ρa log J ρ a. The energy of the ground state of the l.c. string theory J E(J) = lim R R F(J) = X Z Q=1 depq 2π log (1 + Y Q), Y Q e Jɛ Q

35 Free energy and TBA equations Free energy: F µ (J) = R du P h i eea a ρ a + µ a ρ a 1 s(ρa), T = 1/J J Variations of the densities of particles and holes are subject to δρ a + δ ρ a = K ab δρ b. Using the extremum condition δf µ (J) = 0, one gets the canonical TBA eqs ɛ a = E e a + µ a 1 1 J log + e Jɛ b K ba F µ (J) = R J X a Z depa 2π log 1 + e Jɛa where P a P a δa and pseudo (or dressed) energies ɛa are ɛa = 1 ρa log J ρ a. The energy of the ground state of the l.c. string theory J E(J) = lim R R F(J) = X Z Q=1 depq 2π log (1 + Y Q), Y Q e Jɛ Q

36 Canonical TBA equations A Bethe string leads to a Y function Y a = e Jɛa = ρ a ρ a The canonical TBA eqs and the ground state energy log Y a = J(Ẽa + µ a ) + log (1 + Y b ) K ba E µ (J) = a d p a 2π log (1 + Y a) In the AdS 5 S 5 case (Q = 1, 2,...) Y Q = Y Q, Y (±) Q w = 1/Y(±) Q w, Y (±) Q vw = 1/Y(±) Q vw, Y (±) ± = 1/Y (±) ±

37 Simplified TBA equations Canonical TBA eqs Introduce log Y a = J( e E a + µ a) + log (1 + Y b ) K ba (K ab δ ab ) (S bc δ bc ) = δ ac Applying S bc δ bc to the canonical TBA eqs, one gets The simplified TBA equations log 1 «= J( E Y e b + µ b ) (S ba δ ba ) + log 1 + 1Yb S ba a They lead to Y-system and jump discontinuity conditions Hybrid TBA eqs: simplified for some Y-functions and partially simplified for others, most importantly for momentum carrying strings (Q-particles for AdS 5 S 5 ), that is the rhs of TBA equations contains terms of the form log (1 + Y) K where Y is a Y-function of a momentum carrying string.

38 Excited states, CDT and EBE Arutyunov, Frolov, Suzuki 09 Inspired by P. Dorey, Tateo 96 TBA equations for excited states differ from each other only by a choice of integration contours = contour deformation trick Canonical TBA eqs and the energy of excited states log Y a = L TBA (Ẽa + µ a ) + log (1 + Y b ) Cb K ba E = a C a d p a 2π log (1 + Y a) An excited state is characterized by a set of rapidities z a,k which take values on Riemann surfaces of Y a, are between C a and the real mirror line, and satisfy (quantization cond.) Exact Bethe equations Bazhanov, Lukyanov, Zamolodchikov 96; P. Dorey, Tateo 96 Y a (z a,k ) = 1, k = 1, 2,..., N a

39 Excited states, CDT and EBE Arutyunov, Frolov, Suzuki 09 Inspired by P. Dorey, Tateo 96 TBA equations for excited states differ from each other only by a choice of integration contours = contour deformation trick Canonical TBA eqs and the energy of excited states log Y a = L TBA (Ẽa + µ a ) + log (1 + Y b ) Cb K ba E = a C a d p a 2π log (1 + Y a) An excited state is characterized by a set of rapidities z a,k which take values on Riemann surfaces of Y a, are between C a and the real mirror line, and satisfy (quantization cond.) Exact Bethe equations Bazhanov, Lukyanov, Zamolodchikov 96; P. Dorey, Tateo 96 Y a (z a,k ) = 1, k = 1, 2,..., N a

40 Excited states, CDT and EBE Arutyunov, Frolov, Suzuki 09 Inspired by P. Dorey, Tateo 96 TBA equations for excited states differ from each other only by a choice of integration contours = contour deformation trick Canonical TBA eqs and the energy of excited states log Y a = L TBA (Ẽa + µ a ) + log (1 + Y b ) Cb K ba E = a C a d p a 2π log (1 + Y a) An excited state is characterized by a set of rapidities z a,k which take values on Riemann surfaces of Y a, are between C a and the real mirror line, and satisfy (quantization cond.) Exact Bethe equations Bazhanov, Lukyanov, Zamolodchikov 96; P. Dorey, Tateo 96 Y a (z a,k ) = 1, k = 1, 2,..., N a

41 Driving terms 4 Integration contour Y 1 (z*k) = -1-1 If Y(z ) + 1 = 0 (or ) then taking the contour back to real mirror line produces driving term log S(z, z) from log(1 + Y) K ; K (w, z) = 1 d log S(w, z) 2πi dw log(1 + Y) K = 1 I log(1 + Y(w)) d log S(w, z) = 2πi z dw = 1 I d log(w z ) log S(w, z) = log S(z, z) 2πi z dw {z } new driving term Conjecture: Poles and zeroes of Y s are fixed by z a,k

42 Driving terms 4 Integration contour Y 1 (z*k) = -1-1 If Y(z ) + 1 = 0 (or ) then taking the contour back to real mirror line produces driving term log S(z, z) from log(1 + Y) K ; K (w, z) = 1 d log S(w, z) 2πi dw log(1 + Y) K = 1 I log(1 + Y(w)) d log S(w, z) = 2πi z dw = 1 I d log(w z ) log S(w, z) = log S(z, z) 2πi z dw {z } new driving term Conjecture: Poles and zeroes of Y s are fixed by z a,k

43 Excited states TBA with driving terms Let z a,k, k = 1, 2,..., N a obey Y a (z a,k ) = Excited states canonical TBA with driving terms E = a Y a = S ( ) ha exp L TBA (Ẽa + µ a ) + log (1 + Y b ) K ba S pa N a i p a (z a,k ) i p a (za,k ) ( N a k=1 k=1 d p ) a 2π log (1 + Y a) S pa (u) = b N b k=1 S ba (z b,k, u), S ha (u) = b Nb S ba (zb,k, u) k=1 Exact Bethe equations are derived by analytically continuing the excited state TBA equations for Y a to z a,k

44 Excited states TBA with driving terms Let z a,k, k = 1, 2,..., N a obey Y a (z a,k ) = Excited states canonical TBA with driving terms E = a Y a = S ( ) ha exp L TBA (Ẽa + µ a ) + log (1 + Y b ) K ba S pa N a i p a (z a,k ) i p a (za,k ) ( N a k=1 k=1 d p ) a 2π log (1 + Y a) S pa (u) = b N b k=1 S ba (z b,k, u), S ha (u) = b Nb S ba (zb,k, u) k=1 Exact Bethe equations are derived by analytically continuing the excited state TBA equations for Y a to z a,k

45 Spectrum of states with real momenta For the l.c. AdS 5 S 5 superstring in the case of states with real momenta p k one finds E J = N E(p k ) 1 d p log(1 + Y Q ) 2π k=1 Q=1 }{{}}{{} Bethe Yang finite size contribution Momenta p k (or rapidities u k ) are found from the exact Bethe equations (quantization cond.) Y 1 (p k ) = 1 One also finds Arutyunov, Frolov, Suzuki 09 L TBA = J + 2

46 CDT: General strategy To determine the analytic structure of Y-functions one uses 1 The asymptotic Bethe equations Beisert, Staudacher 05 to find a set of g-dependent momenta (or rapidities) for the state under consideration for small values of g 2 Solution of the asymptotic TBA equations Bajnok, Janik 08 Gromov, Kazakov, Vieira 09(a) obtained by dropping terms with log(1 + Y Q ) to find zeroes and poles of asymptotic 1 + Y and Y

47 CDT: General strategy To determine the analytic structure of Y-functions one uses 1 The asymptotic Bethe equations Beisert, Staudacher 05 to find a set of g-dependent momenta (or rapidities) for the state under consideration for small values of g 2 Solution of the asymptotic TBA equations Bajnok, Janik 08 Gromov, Kazakov, Vieira 09(a) obtained by dropping terms with log(1 + Y Q ) to find zeroes and poles of asymptotic 1 + Y and Y

48 BY equations Beisert, Staudacher 05 The main Bethe equations have the form K I K Y Y II 1 = e i J p x k k S sl(2) (u k, u l ) l k l=1 x + k y ( ) l y ( ) l v u t x + k x k K+Y II l=1 x k y (+) l x + k y (+) l v u t x + k x k They are supplied with auxiliary Bethe equations for the roots y (α) and w (α), α = ± v YK I y (α) x u k i t x + KαY III w (α) ν (α) i i i k g i=1 y (α) x + x = k i i i=1 w (α) ν (α), ν (α) y (α) i k k y (α) i k g k KαY II i=1 w (α) k w (α) k ν (α) + i i g ν (α) i i g KαY III = i=1 w (α) k w (α) k w (α) + 2i i g w (α) 2i i g Here K I is # of particles, Kα, II Kα III are weights of four SU(2) s of manifest symmetry of the l.c. string sigma model. SU(4) weights [q 1, p, q 2 ] and the spins [s 1, s 2 ] are q 1 = K II p = J 1 2 2K III + s 1 = K I K II 2K III, q 2 = K+ II (K II + K + II) + K III + K + III s 2 = K I K+ II Instead of weights of su(4) one can use the weights (J 1, J 2, J 3 ) of SO(6) J J 1 = 1 2 (q 1 + 2p + q 2 ), J 2 = 1 2 (q 1 + q 2 ), J 3 = 1 2 (q 2 q 1 )..

49 Bajnok-Janik asymptotic solution ( large J or small g and finite J ) Generalized Lüscher formulae give the large J asymptotic solution Bajnok, Janik 08 Y o Q (v) = Υ Q(v) T Q, 1 (v u) T Q,1 (v u) Υ Q (v) = e J e E Q (v) NY i=1 S Q1 sl(2) (v, u i ) exp suppressed T Q,1 is an eigenvalue of a properly normalized su(2 2) C transfer matrix T Q,1 (v u) = str A S Q,1 A1 (v, u 1)S Q,1 A2 (v, u 2) S Q,1 AN (v, u N), conjectured by Beisert 06 and derived by Arutyunov, de Leeuw, Suzuki, Torrieli 09 The BY equations (in the sl(2) sector) follow from e E 1 (u k ) = i p k and T 1,±1(u k u) = 1 = 1 = e i J p k NY j=1 S 1 1 sl(2) (u k, u j ) All auxiliary Y o -functions are fixed by YQ o and (almost) agree with Arutyunov, Frolov 11 Gromov, Kazakov, Vieira 09(a)

50 Bajnok-Janik asymptotic solution ( large J or small g and finite J ) Generalized Lüscher formulae give the large J asymptotic solution Bajnok, Janik 08 Y o Q (v) = Υ Q(v) T Q, 1 (v u) T Q,1 (v u) Υ Q (v) = e J e E Q (v) NY i=1 S Q1 sl(2) (v, u i ) exp suppressed T Q,1 is an eigenvalue of a properly normalized su(2 2) C transfer matrix T Q,1 (v u) = str A S Q,1 A1 (v, u 1)S Q,1 A2 (v, u 2) S Q,1 AN (v, u N), conjectured by Beisert 06 and derived by Arutyunov, de Leeuw, Suzuki, Torrieli 09 The BY equations (in the sl(2) sector) follow from e E 1 (u k ) = i p k and T 1,±1(u k u) = 1 = 1 = e i J p k NY j=1 S 1 1 sl(2) (u k, u j ) All auxiliary Y o -functions are fixed by YQ o and (almost) agree with Arutyunov, Frolov 11 Gromov, Kazakov, Vieira 09(a)

51 Real vs Complex 1 If all momenta are real then no Y Q -function has poles inside the analyticity strip 1/g < Im < 1/g, and integration contours can be chosen so that in the regions bounded by the contours and real mirror line the structure of poles and zeroes of exact 1 + Y and Y is the same as that of asymptotic Y-functions if g is small enough 2 If some momenta are complex then some of Y Q -functions have poles inside the analyticity strip and the structure of poles and zeroes of exact 1 + Y and Y differs from that of asymptotic Y-functions no matter how small g is

52 Real vs Complex 1 If all momenta are real then no Y Q -function has poles inside the analyticity strip 1/g < Im < 1/g, and integration contours can be chosen so that in the regions bounded by the contours and real mirror line the structure of poles and zeroes of exact 1 + Y and Y is the same as that of asymptotic Y-functions if g is small enough 2 If some momenta are complex then some of Y Q -functions have poles inside the analyticity strip and the structure of poles and zeroes of exact 1 + Y and Y differs from that of asymptotic Y-functions no matter how small g is

53 Known excited state TBA equations The sl(2) sector ( J ) Tr D n k Z, k=1 J n k = N, n k 0 k=1 Canonical TBA eqs for two-particle Konishi-like states, λ < λ cr Gromov, Kazakov, Kozak, Vieira 09v3; Hybrid, simplified and canonical TBA eqs for arbitrary two-particle states Arutyunov, Frolov, Suzuki 09; Hybrid TBA eqs for twist-2 N-particle lightest state, λ < λ cr TBA eqs for a subsector of the sl(2) sector, λ < λ cr Balog, Hegedus 10; Balog, Hegedus 11; TBA eqs for two states not from the sl(2) sector which are degenerate asymptotically, λ < λ cr Sfondrini, van Tongeren 11; TBA eqs for some states from the su(2) sector with complex momenta Arutyunov, Frolov, van Tongeren 11

54 Spinning strings in the sl(2) sector Gromov 09 dual to operators ( J ) Tr D n k Z, k=1 J n k = N, n k 0 k=1 in the limit g, J/g and N/g are fixed Classical energy and a polynomial part of one-loop correction are captured by the BY eq The full one-loop correction requires TBA Precise agreement between the TBA result and the one-loop correction computed in perturbative string sigma model

55 Two-particle states in perturbation theory dual to spin-2 operators Tr(Z J 1 D 2 Z ), Tr(Z k 2 DZ Z J k DZ ) Log of the BY equation for two-particle states with p 1 + p 2 = 0 ip(j + 1) log x i θ(p, p) = 2πi n, n > 0 {z } x 2 dressing phase At g 0 the momentum is p o J,n = 2πn» J J + 1, n = 1,...,, 2 At large g the integer n coincides with the string level Arutyunov, Frolov, Staudacher 04 For Konishi J = 2 and n = 1

56 Two-particle states in perturbation theory dual to spin-2 operators Tr(Z J 1 D 2 Z ), Tr(Z k 2 DZ Z J k DZ ) Log of the BY equation for two-particle states with p 1 + p 2 = 0 ip(j + 1) log x i θ(p, p) = 2πi n, n > 0 {z } x 2 dressing phase At g 0 the momentum is p o J,n = 2πn» J J + 1, n = 1,...,, 2 At large g the integer n coincides with the string level Arutyunov, Frolov, Staudacher 04 For Konishi J = 2 and n = 1

57 Five-loop Konishi Direct field-theoretical computation of the four-loop contribution: E SYM = 4 + 3g 2 3g g ζ(3) 45 8 ζ(5) g Fiamberti, Santambrogio, Sieg, Zanon 07 ( 200 supergraphs!) Velizhanin 08 ( graphs!) It obviously agrees with TBA because one uses Bajnok-Janik asymptotic solution as an input Bajnok, Janik 08 At five-loops Lüscher s approach yields Bajnok, Hegedus, Janik, Lukowski 09 (10) = (10) asympt + g10n 81ζ(3)2 + 81ζ(3) 45ζ(5) + 945ζ(7) o 256 The same result follows from TBA Arutyunov, Frolov, Suzuki 10 Balog, Hegedus 10 and field theory considerations Eden, Heslop, Korchemsky, Smirnov, Sokatchev 12

58 Strong coupling expansion E (J,n) (λ) = c 4 1 n 2 λ+c 0 + c 1 4 n2 λ + c 2 n2 λ + c 3 (n 2 λ) + c 4 3/4 n 2 λ + c 5 + (n 2 λ) 5/4 Expansion is in 1/ 4 n 2 λ, n is the string level of a 2-particle state c 1 = 2 from the flat space string spectrum Gubser, Klebanov, Polyakov 98 and BYE Arutyunov, Frolov, Staudacher 04 c 0 = 0 from BYE and free fermion model Arutyunov, Frolov 05 Other coefficients are functions of J and n c 2 = 0??? due to supersymmetry Roiban, Tseytlin 09 c 2k = 0!?!?!? E (J,n) (λ) = 4 n 2 λ Gromov, Kazakov, Vieira 09(b) ( 2 + c 1 n2 λ + c ) 3 n 2 λ + c 5 (n 2 λ) + 3/2

59 Strong coupling expansion E (J,n) (λ) = c 4 1 n 2 λ+c 0 + c 1 4 n2 λ + c 2 n2 λ + c 3 (n 2 λ) + c 4 3/4 n 2 λ + c 5 + (n 2 λ) 5/4 Expansion is in 1/ 4 n 2 λ, n is the string level of a 2-particle state c 1 = 2 from the flat space string spectrum Gubser, Klebanov, Polyakov 98 and BYE Arutyunov, Frolov, Staudacher 04 c 0 = 0 from BYE and free fermion model Arutyunov, Frolov 05 Other coefficients are functions of J and n c 2 = 0??? due to supersymmetry Roiban, Tseytlin 09 c 2k = 0!?!?!? E (J,n) (λ) = 4 n 2 λ Gromov, Kazakov, Vieira 09(b) ( 2 + c 1 n2 λ + c ) 3 n 2 λ + c 5 (n 2 λ) + 3/2

60 Strong coupling expansion E (J,n) (λ) = c 4 1 n 2 λ+c 0 + c 1 4 n2 λ + c 2 n2 λ + c 3 (n 2 λ) + c 4 3/4 n 2 λ + c 5 + (n 2 λ) 5/4 Expansion is in 1/ 4 n 2 λ, n is the string level of a 2-particle state c 1 = 2 from the flat space string spectrum Gubser, Klebanov, Polyakov 98 and BYE Arutyunov, Frolov, Staudacher 04 c 0 = 0 from BYE and free fermion model Arutyunov, Frolov 05 Other coefficients are functions of J and n c 2 = 0??? due to supersymmetry Roiban, Tseytlin 09 c 2k = 0!?!?!? E (J,n) (λ) = 4 n 2 λ Gromov, Kazakov, Vieira 09(b) ( 2 + c 1 n2 λ + c ) 3 n 2 λ + c 5 (n 2 λ) + 3/2

61 Strong coupling expansion E (J,n) (λ) = c 4 1 n 2 λ+c 0 + c 1 4 n2 λ + c 2 n2 λ + c 3 (n 2 λ) + c 4 3/4 n 2 λ + c 5 + (n 2 λ) 5/4 Expansion is in 1/ 4 n 2 λ, n is the string level of a 2-particle state c 1 = 2 from the flat space string spectrum Gubser, Klebanov, Polyakov 98 and BYE Arutyunov, Frolov, Staudacher 04 c 0 = 0 from BYE and free fermion model Arutyunov, Frolov 05 Other coefficients are functions of J and n c 2 = 0??? due to supersymmetry Roiban, Tseytlin 09 c 2k = 0!?!?!? E (J,n) (λ) = 4 n 2 λ Gromov, Kazakov, Vieira 09(b) ( 2 + c 1 n2 λ + c ) 3 n 2 λ + c 5 (n 2 λ) + 3/2

62 Strong coupling expansion E (J,n) (λ) = c 4 1 n 2 λ+c 0 + c 1 4 n2 λ + c 2 n2 λ + c 3 (n 2 λ) + c 4 3/4 n 2 λ + c 5 + (n 2 λ) 5/4 Expansion is in 1/ 4 n 2 λ, n is the string level of a 2-particle state c 1 = 2 from the flat space string spectrum Gubser, Klebanov, Polyakov 98 and BYE Arutyunov, Frolov, Staudacher 04 c 0 = 0 from BYE and free fermion model Arutyunov, Frolov 05 Other coefficients are functions of J and n c 2 = 0??? due to supersymmetry Roiban, Tseytlin 09 c 2k = 0!?!?!? E (J,n) (λ) = 4 n 2 λ Gromov, Kazakov, Vieira 09(b) ( 2 + c 1 n2 λ + c ) 3 n 2 λ + c 5 (n 2 λ) + 3/2

63 Strong coupling expansion E (J,n) (λ) = c 4 1 n 2 λ+c 0 + c 1 4 n2 λ + c 2 n2 λ + c 3 (n 2 λ) + c 4 3/4 n 2 λ + c 5 + (n 2 λ) 5/4 Expansion is in 1/ 4 n 2 λ, n is the string level of a 2-particle state c 1 = 2 from the flat space string spectrum Gubser, Klebanov, Polyakov 98 and BYE Arutyunov, Frolov, Staudacher 04 c 0 = 0 from BYE and free fermion model Arutyunov, Frolov 05 Other coefficients are functions of J and n c 2 = 0??? due to supersymmetry Roiban, Tseytlin 09 c 2k = 0!?!?!? E (J,n) (λ) = 4 n 2 λ Gromov, Kazakov, Vieira 09(b) ( 2 + c 1 n2 λ + c ) 3 n 2 λ + c 5 (n 2 λ) + 3/2

64 c 1 = 2 was confirmed??? by various groups and methods Konishi dimension from TBA TBA eqs were solved numerically up to g = 7.2 (λ 2047) Frolov 10 E K Λ Λ dots numerics, red 2 4 λ, brown q E asym = g 2 sin 2 p asym 2 Up to g = 4.1 (λ 664) agreement with Setting c 0 = c 2 = c 4 = 0, one gets c 1 = 2 from GKV numerics E GKV K (λ) = 4 λ λ λ «λ 3/2 and from my numerics E K (λ) = 4 λ λ λ «λ 3/2 Gromov, Kazakov, Vieira 09(b)

65 Scaling dimensions from TBA TBA eqs were solved numerically for many other states Frolov 12 E J,n Λ J Λ

66 c 1 (J, n) =?, c 2 (J, n) = 0?, c 3 (J, n) =? E (J,n) (λ) = 2 4p n 2 λ + c 1(J, n) 4 n 2 λ + c 2(J, n) n 2 λ + c 3(J, n) (n 2 λ) 3/4 + From TBA c 2 (J, n) 0 and Frolov 12 c 1 (J, n) = J2 4 + n(3 n) 2 From semi-classical strings and Basso s conjecture Gromov, Valatka 11 c 3 (J, 1) = J J2 8 3ζ(3) 3 4 c 3 (J, 2) = J J2 8 24ζ(3) 11 4 c 3 (J, 3) = J J2 8 81ζ(3) 33 8 c 3 (J, 1) and c 3 (J, 2) agree with TBA. J-independent parts cannot be tested reliably.

67 c 1 (J, n) =?, c 2 (J, n) = 0?, c 3 (J, n) =? E (J,n) (λ) = 2 4p n 2 λ + c 1(J, n) 4 n 2 λ + c 2(J, n) n 2 λ + c 3(J, n) (n 2 λ) 3/4 + From TBA c 2 (J, n) 0 and Frolov 12 c 1 (J, n) = J2 4 + n(3 n) 2 From semi-classical strings and Basso s conjecture Gromov, Valatka 11 c 3 (J, 1) = J J2 8 3ζ(3) 3 4 c 3 (J, 2) = J J2 8 24ζ(3) 11 4 c 3 (J, 3) = J J2 8 81ζ(3) 33 8 c 3 (J, 1) and c 3 (J, 2) agree with TBA. J-independent parts cannot be tested reliably.

68 c 1 (J, n) =?, c 2 (J, n) = 0?, c 3 (J, n) =? E (J,n) (λ) = 2 4p n 2 λ + c 1(J, n) 4 n 2 λ + c 2(J, n) n 2 λ + c 3(J, n) (n 2 λ) 3/4 + From TBA c 2 (J, n) 0 and Frolov 12 c 1 (J, n) = J2 4 + n(3 n) 2 From semi-classical strings and Basso s conjecture Gromov, Valatka 11 c 3 (J, 1) = J J2 8 3ζ(3) 3 4 c 3 (J, 2) = J J2 8 24ζ(3) 11 4 c 3 (J, 3) = J J2 8 81ζ(3) 33 8 c 3 (J, 1) and c 3 (J, 2) agree with TBA. J-independent parts cannot be tested reliably.

69 c 1 (J, n) =?, c 2 (J, n) = 0?, c 3 (J, n) =? E (J,n) (λ) = 2 4p n 2 λ + c 1(J, n) 4 n 2 λ + c 2(J, n) n 2 λ + c 3(J, n) (n 2 λ) 3/4 + From TBA c 2 (J, n) 0 and Frolov 12 c 1 (J, n) = J2 4 + n(3 n) 2 From semi-classical strings and Basso s conjecture Gromov, Valatka 11 c 3 (J, 1) = J J2 8 3ζ(3) 3 4 c 3 (J, 2) = J J2 8 24ζ(3) 11 4 c 3 (J, 3) = J J2 8 81ζ(3) 33 8 c 3 (J, 1) and c 3 (J, 2) agree with TBA. J-independent parts cannot be tested reliably.

70 Excitations with complex momenta Arutyunov, Frolov, van Tongeren 11 su(2) sector contains states composed of particles with complex momenta or rapidities Simplest state is composed of one particle with real rapidity u 1 and two complex conjugate rapidities u 2,3 If u 3 u 2 2i/g and J is large enough the state can be thought of as a scattering state of a fundamental particle and a two-particle bound state There are many three-particle solutions with (k 1)/g < Im(u) < k/g, k = 2, 3,... Explicitly consider the state J = 4, L = 7

71 Solution to the asymptotic Bethe eqs for u 2 with L = 7 Reu g Img u g The imaginary part of the rapidity is rescaled by a factor of g The rapidity u 2 asymptotes to the branch point 2 i/g before breakdown of the BYE at g 0.53 u 2 and u 3 are outside the analyticity strip 1/g < Im(u) < 1/g

72 Ω 1 Ω Ω 2 Ω 2 u 2 u u 3 3 u 3 u 1 Ω 2 2 u 3 u 2 u 1 Ω 2 2 u 3 u 1 u 2 u 2 u 2 u 1 0 u 3 u 2 u 1 0 u u 3 3 u 3 u 1 Ω 2 2 Ω 2 2 Ω Ω 1 2

73 Relevant roots/poles of the asymptotic Y s on the mirror u-plane u 3 u 2 Y 1 Y 3 ig u 1 u 3 u 2 Y 2 Y 2 r 0... r u 1 Y 1 vw Y 1 vw r 00 u 3 Y 1 Y 3 u 2 r 0... r Y M vw r 1 M Y r 1 0 ig u 1 Y Y Q with Q 3 have poles at u 2 + i g (Q 1), u 3 i g (Q 1)

74 Analytic structure of the exact solution The functions Y 1, Y 2 and Y 3 have poles inside the analyticity strip! Around a pole Y (u) = y(u) u u Within the analyticity strip y(u) is small and of the order g 2L 1 This implies that there is a point u 1 close to u such that Y (u 1 ) = 1 Indeed Y (u 1 ) = 1 = u 1 u + y(u 1 ) = 0, and expanding around u, one finds u 1 u y(u ) = u Res Y (u ) u 1 is also inside the analyticity strip!

75 Analytic structure of the exact solution The functions Y 1, Y 2 and Y 3 have poles inside the analyticity strip! Around a pole Y (u) = y(u) u u Within the analyticity strip y(u) is small and of the order g 2L 1 This implies that there is a point u 1 close to u such that Y (u 1 ) = 1 Indeed Y (u 1 ) = 1 = u 1 u + y(u 1 ) = 0, and expanding around u, one finds u 1 u y(u ) = u Res Y (u ) u 1 is also inside the analyticity strip!

76 Y Q = Υ Q T Q, 1 T Q,1 T Q 1,0 T Q+1,0, 1 + Y Q = T + Q,0 T Q,0 T Q 1,0 T Q+1,0 Prefactor Υ Q has poles at u 2 + i (Q 1) and u g 2 + i (Q + 1) g Asymptotically T Q,0 = 1 For an exact solution T Q,0 1 and T Q,0 (u 2 + i g Q) =, T Q,0(u (Q) 2 + i g Q) = 0 This implies Y Q u (Q) 2 + i g (Q 1) «Y Q u (Q±1) 2 + i g (Q ± 1) «= 1, = In addition 1 + Y 1 has zero at real u 1 which is in the string region

77 Integration contours for Y Q enclose all zeroes of 1 + Y Q which are in the analyticity strip of the string region, and all zeroes and poles which are below the real line in the mirror region The contours never go to the anti-mirror region of the z-torus or equivalently outside the analyticity strip of the string region, and thus they do not enclose u 3 For the L = 7 state this means that the contours enclose the poles of Y 1 at u (2) 3, of Y 2 at u (1)+ 2, u (3) 3 3i g, and of Y Q, Q 3 at u (Q±1) 3 i (Q ± 1), and g 3 3i, and of then the zeroes of 1 + Y 1 at u (1) 3, u (1) 2, of 1 + Y 2 at u (2)+ 2, u (2) g 1 + Y Q, Q 3 at u (Q) 3 i (Q ± 1) in the mirror u-plane, and finally the zero of g 1 + Y 1 at u 1 in the string u-plane With this choice, one gets the driving terms from log(1 + Y Q ) CQ K Q log S 1 (u 1, v) log S 1(u (1) 2, v) S 1 (u (1) 3, v) + log The choice of integration contours is not unique S 2 (u (1)+ 2, v) S 2 (u (2)+ 2, v) S 2 (u (2) 3, v) S 2 (u (1) 3, v) Integration contours for Y aux are similar to the ones for the real rapidities cases The integration contours discussed above are for the canonical TBA equations, and they are different from the contours for the simplified equations

78 Simplified equations for Y M w and Y M vw functions log Y M w = log(1 + Y M 1 w )(1 + Y M+1 w ) s + δ M1 log 1 1 Y 1 1 Y + ˆ s log S(r M 1 v)s(r M+1 v) log Y M vw = log(1 + Y M+1 ) s + log(1 + Y M 1 vw )(1 + Y M+1 vw ) s + δ M1 log 1 Y ˆ s 1 Y + +δ M1 log S(u(2)+ 2 v) S(u (2) 3 v) log S(u 1 v)s(r 0 v)

79 Simplified equations for Y ± functions log Y+ Y = log(1 + Y Q ) K Qy 3X log S 1 y(u (1) i, v) + log S 2y(u (1)+ 2, v) S 2y (u (2)+ 2, v) i=1 S 2y (u (2) 3, v) S 2y (u (1) 3, v) log Y +Y = 2 log 1 + Y 1 vw s log (1 + Y Q ) K Q + 2 log(1 + Y Q ) Kxv Q1 s 1 + Y 1 w +2 log S(r 1 v) 2 log S1 1 xv (u 1, v) s + log S 2 (u 1 v) s 2 log S11 xv (u (1) 2, v) Sxv 11 (u (1) 3, v) s + log S (1) 1(u2 v) S 1 (u (1) 3 v) log S 2(u (1)+ 2 v) S 2 (u (2) 3 v) S 2 (u (2)+ 2 v) S 2 (u (1) 3 v) + 2 log S 21 xv (u (1)+ 2, v)sxv 21 (u (2) 3, v) S 21 xv (u (2)+ 2, v)s 21 xv (u (1) 3, v) s

80 Compatibility of quantization conditions Y 1 (u (1) 3 ) = 1 Y 1(u (1) 3 ) = 1 Y 1 (u (1) 3 ) = 1 The exact Bethe equations representing these quantization conditions are compatible in a non-trivial manner which involves crossing symmetry There are similar quantization conditions involving Y 2 Y 2 (u (2) 3 ) = 1 Y 2 (u (2) 3 ) = 1

81 The fact that 1 + Y Q functions have zeroes and poles in the analyticity strip and the choice for the integration contours lead to the energy formula E = 3X i=1 E(u (1) i ) 1 2π and the total momentum formula P = X i X Q=1 Z du d p Q du log(1 + Y Q) i p 2 (u (1)+ 2 ) + i p 2 (u (2)+ 2 ) i p 2 (u (2) 3 ) + i p 2 (u (1) 3 ) p i 1 2π Z du dẽ Q du log(1 + Y Q) iẽ 2 (u (1)+ 2 ) + iẽ 2 (u (2)+ 2 ) iẽ 2 (u (2) 3 ) + iẽ 2 (u (1) 3 ) The g 0 and J finite limit provides the leading wrapping correction Z E wrap = 1 X 2π Q=1 h d p2 i Res du d p Q du Y Q du (u+ 2 )Y 2 (u + 2 ) Res d p2 du (u 3 )Y 2(u 3 ) i

82 Three-particle state with u 2 and u 3 lying in the 3rd strip Reu g Img u g The imaginary part of the rapidity is rescaled by a factor of g The rapidity u 2 crosses the line Im u = 2/g and enters the 2nd strip.

83 A generalization of our construction to a three-particle state with u 2 and u 3 lying in the kth strip Four functions Y k 2,..., Y k+1 will have poles in the analyticity strip, with the poles of Y k 2 and Y k being closest to the real line The driving terms in the corresponding TBA equations will depend on u (k 1) 2,3 and u (k) 2,3 whose locations are determined by the corresponding exact Bethe equations for Y k 1 and Y k The energy E = 3X i=1 E(u (k 1) i ) 1 2π X Q=1 Z i p k u (k 1) 2 + (k 1) i g du d p Q du log(1 + Y Q) + i p k u (k) i p k u (k) 3 (k 1) i g 2 + (k 1) i g + i p k u (k 1) 3 (k 1) i g

84 Assumptions 1 Quantum integrability of l.c. string theory and its mirror 2 Symmetry algebra of l.c. string theory and its mirror in the decompactification limit Beisert 05; Arutyunov, Frolov, Plefka, Zamaklar 06 3 BES dressing factor Beisert, Eden, Staudacher 06 4 Bajnok-Janik asymptotic solution Bajnok, Janik 08 5 String hypothesis for the mirror model Arutyunov, Frolov 09 6 Universality of the contour deformation trick 7 Universality of the exact Bethe equations Y a (z a,k ) = 1

85 The mirror TBA: from string hypothesis to the exact spectrum 1 TBA eqs are written for the whole superconformal multiplet. PSU(2, 2 4) invariance is built-in Arutyunov, Frolov 11 2 TBA eqs for any regular state can be obtained 3 Singular states which at g = 0 contain particles with rapidities u = ±i still represent a challenge Arutyunov, Frolov, Sfondrini (work in progress) 4 Driving terms depend on λ. For infinitely many states there are critical values of λ, crossing which the driving terms must be changed. 5 For a given operator, is the number of critical values infinite or finite (or even 0)? 6 We found no evidence (for n 2) that up to the overall factor 2 4 n 2 λ the large λ expansion is in powers of 1/ λ. Is the expansion in powers of 1/ 4 λ? 7 The numerics we performed is not sufficient to give definite answers to many questions. Analytical methods are necessary. NLIE??? Gromov 10; Suzuki 11; Gromov, Kazakov, Leurent, Volin 11; Balog, Hegedus 12; 8 TBA for orbifold and γ-deformed models Ahn, Bajnok, Bombardelli, Nepomechie 11; Arutyunov, de Leeuw, van Tongeren 10; de Leeuw, van Tongeren 12; 9 TBA from Y-system and jump discontinuity conditions Cavaglia, Fioravanti, Tateo 10; Balog,Hegedus 11;