Exact semiclassical strings

Transcription

1 Exact semiclassical strings Valentina Forini AEI Potsdam arxiv: in collaboration with M. Beccaria (Lecce U.), G. Dunne (Connecticut U.), M. Pawellek (Erlangen-Nuernberg U.) and A.A. Tseytlin (Imperial C.). Hannover, Nordic String Theory Meeting 2010

2 Motivation I. general aspects Suggestion of AdS/CFT (N=4 SYM in 4-dim Type II B strings in AdS 5 xs 5 ) > understanding quantum gauge theories at any coupling > understanding string theories in non-trivial backgrounds better address them together than separately. Fundamental insight: both theories might be integrable. Many efforts to translate into solvable. [Bena, Polchinski, Roiban, 02] [Minahan Zarembo 02] [Beisert, Kristjansen, Staudacher 03] [Beisert, Staudacher 05] Solvability even beyond the spectrum. [Beisert, Henn, McLaughlin, Plefka 10] [Alday, Maldacena, Sever, Vieira 10] Recent proposals to solve their full planar spectrum need explicit and independent checks. [Gromov, Kazakov, Vieira, 09] [Arutyunov, Frolov 09] [Bombardelli, Fioravanti, Tateo 09] Isometries of the two models coincide: PSU(2,2 4) Gauge theory operators & string states organized in reprs -(E; S 1,S 2 ; J 1,J 2,J 3 ) String energies = dimensions of dual gauge operators E( λ,c,...) (λ,c,...) C =(S 1,S 2 ; J 1,J 2,J 3 )

3 Motivation II. our aim Crucial role of semiclassical quantization of strings [Gubser, Klebanov, Polyakov 02] and crucial example: folded string rotating fastly (with spin S) in AdS 3 > Semiclassical expansion ( λ 1 & S = S/ λ finite ) & S 1 λ f(λ 1) = a 0 + a 1 + a 2 E = S + f(λ) lns +... π λ ( λ) [Frolov Tseytlin 02] [Roiban Tseytlin 07] exactly confirmed by solution of integrability-based cusp anomaly equation [Beisert, Eden, Staudacher 06] [Basso, Korchemsky, Kotansky] γ(s) = f(λ) log S + O(S 0 ) O S = Tr(ϕD S ϕ) S 1 checked at weak coupling by MHV 4-point gluon amplitudes. [Bern, Czakon, Dixon, Kosower, Smirnov, 06] In order to > go (safely) beyond the leading logarithmic behavior > extrapolate data for operators with finite quantum numbers one needs additional analytic tools on the string side, from which better understanding of quantum corrections for strings in AdS 5 xs 5.

4 Outlook The setup GS folded string The 1-loop exact calculation Gauge-related issues Solvable structures and solution Results useful in AdS/CFT Long strings Short strings Summary

5 Setup I: GS superstrings in AdS 5 xs 5 [Metsaev, Tseytlin 98] I = λ 2 π d 2 ξ L B (x, y)+l F (x, y, θ) λ 2π = R2 2πα = T string tension L B = 1 2 gg ab G (AdS 5) mn (x) a x m b x n + G (S5 ) m n (y) a y m b y n L F = i( gg ab δ IJ ab s IJ ) θ I ρ a D b θ J + O(θ 4 ) > 1/ 1/ λ semiclassical expansion in powers of λ E = λ E = λ E 0 + E 1 λ + E 2 ( λ) > invariant under world-sheet diffeo and local fermionic k-symmetry

6 Setup II: folded string in AdS 3 Ansatz for a solution rotating in AdS 3 ρ = ρ(σ) t = κ τ φ = ω τ > Sinh-Gordon EOM for ρ(σ) ρ = 1 2 (κ2 ω 2 )sinh(2ρ) > Conformal gauge constraint ρ 2 = κ 2 cosh 2 ρ ω 2 sinh 2 ρ 0 ρ(σ) < ρ max coth ρ max = ω 1+ κ 1 ~ length of the string 2 Exact solution κσ sinh ρ = sn, 2, 0 σ < π 2

7 Setup III: folded string, classical results Integrals of motion: classical energy and spin E = P t = λ κ 2π 0 dσ 2π cosh2 ρ λ E S = P φ = λ ω 2π 0 dσ 2π sinh2 ρ λ S In parametric form E = E = 2 λ π E( 2 ), S = S = λ π E( 2 ) K( 2 ) > Long strings (large spin) + [Gubser, Klebanov, Polyakov 02] S λ E λ π ln S λ as twist operators at weak coupling > Short strings (small spin) 0 [Gubser, Klebanov, Polyakov 98] S λ E 2 λ S flat space λ 1 α

8 Semiclassical quantization of folded string Standard quantization of a soliton > Background field method L φ = φ cl + φ 4 λ Lfluct > Effective action Γ = ln Z = ln det fermions det bosons > 1-loop energy E 1 = Γ 1 κt, T dτ Stationary solution 1-dimensional determinants! det τ 2 σ 2 + M 2 (σ) = T dω 2π σ 2 + ω 2 + M 2 (σ)

9 Quantum fluctuations: Fermions [Drukker,Gross, Tseytlin 00] L GS 2F k-symmetry fix. θ 1 = θ 2 SO(1, 9) rotation L 2F =2 θ D F θ [Frolov, Tseytlin 02] D F = i(γ a a µ F Γ 234 ), µ F = ρ (σ) Determinant of the diagonalized laplacian ln det D F = 1 2 ln det(d F ) ln det F+ +4lndet F 2 > 8 species of Majorana fermions in 2 dimensions F± = a a +ˆµ 2 F ± ˆµ 2 F ± = ±ρ + ρ 2 Conformal gauge ghosts and k-symmetry ghosts decouple.

10 Quantum fluctuations: Bosons 1. Conformal gauge : nasty AdS 3 coupled sector L conf 2B = a t a t µ 2 t t 2 + a ρ a ρ + µ 2 ρ ρ 2 a + a φ φ + µ 2 φ φ2 + a βi a βi + µ 2 β i β2 i + +4 ρ k sinh ρ τ t cosh ρ τ φ µ 2 t =2ρ2 κ 2 etc. > highly non trivial masses! but regular > 3 massive and coupled AdS 3 fluctuations t, ρ, φ Q(σ) = 2 σ ω 2 ρ 2 + κ 2 2 ωκsinh ρ 0 2 ωκ sinh ρ 2 σ + ω 2 +2ρ 2 ω 2 p 2 ωω p cosh ρ 0 2 ωω p cosh ρ 2 σ + ω 2 + ρ 2 ω 2 p κ 2 Determinant of Q(σ) - hard to find in closed form expansion needed.

11 2. Static gauge ( ρ = t =0) : L st 2B = a φ a φ + m 2 φ + a βi a βi + µ 2 β i β2 i > all decoupled fluctuations! > but masses blow up at turning points > superficial UV divergence dτdσ gr (2)? m 2 φ =2ρ 2 +2 κ2 ω 2 ρ 2 Up to now conformal gauge preferred [Drukker Gross Tseytlin 00] [Frolov Tseytlin 02] [Roiban Tseytlin 07, 08, 09]

12 2. Static gauge ( ρ = t =0) : L st 2B = a φ a φ + m 2 φ + a βi a βi + µ 2 β i β2 i > all decoupled fluctuations! > but masses blow up at turning points > superficial UV divergence dτdσ gr (2)? m 2 φ =2ρ 2 +2 κ2 ω 2 ρ 2 Up to now conformal gauge preferred [Drukker Gross Tseytlin 00] [Frolov Tseytlin 02] [Roiban Tseytlin 07, 08, 09] HOWEVER > Equivalence of semiclassical (1-loop) partition fcs. for Nambu and Polyakov action. [Fradkin, Tseytlin 82] > gr (2) : total derivative, sensitive to topology of induced world-sheet (cylinder)

13 A posteriori proof conformal gauge = static gauge UV finiteness of static gauge action Static gauge results reproduce conformal gauge ones

14 A posteriori proof conformal gauge = static gauge UV finiteness of static gauge action Static gauge results reproduce conformal gauge ones Equivalence (numerical) of determinants! Γ CG 1 = T 4π R dω det 8 O ψ det 2 O β det Q det 3 ( 2 ) Γ SG 1 = T 4π R dω det 8 O ψ det 2 O β det O φ det 5 ( 2 ) det Q =deto φ det 2 ( 2 ) 60 product coupled 40 determinants iω

15 How one could proceed... (non exactly ) Non trivial masses m 2 (σ) ρ 2 = κ 2 cn 2 κσ, 2 Expanding, leading order is sigma independent: ok! ρ 2 κ 2 0 k 0 = 1 π ln[16 2 ] Γ (0) = T dω 2π ln det( ω 2 + κ 2 0) 8 det( ω2 +4κ 2 0 )det( ω2 +2κ 2 0 )2 det( ω2 ) 5 > Leading contribution (constant mass relativistic fields on the cylinder!) E (0) = 1 2 κ n= Euler-MacLaurin E (0) 2 n 2 +2κ 20 + n 2 +4κ n 2 8 n 2 + κ = Γ(0) 1 κ T = 1 κ [Frolov, Tseytlin 02] 3 ln 2 κ O(e 2πκ 0 ), κ 0 1-loop correction to cusp anomaly ln 2 ln S [Frolov, Tseytlin 02] [Schaefer-Nameki, Zamaklar 05]

16 How one could proceed... (non exactly ) Non trivial masses m 2 (σ) ρ 2 = κ 2 cn 2 κσ, 2 Expanding, leading order is sigma independent: ok! ρ 2 κ 2 0 k 0 = 1 π ln[16 2 ] Γ (0) = T dω 2π ln det( ω 2 + κ 2 0) 8 det( ω2 +4κ 2 0 )det( ω2 +2κ 2 0 )2 det( ω2 ) 5 > Leading contribution (constant mass relativistic fields on the cylinder!) E (0) = 1 2 κ n= Euler-MacLaurin E (0) 2 n 2 +2κ 20 + n 2 +4κ n 2 8 n 2 + κ = Γ(0) 1 κ T = 1 κ [Frolov, Tseytlin 02] 3 ln 2 κ O(e 2πκ 0 ), κ 0 1-loop correction to cusp anomaly ln 2 ln S [Frolov, Tseytlin 02] [Schaefer-Nameki, Zamaklar 05] BUT further expansion breaks down!! ρ 2 = κ κ 0 [πκ 0 cosh(2κ 0 σ) 2] at turning points 2 e 2σ 1 0 π ln 162

17 The exact way Eigenvalue fluctuation equation, eg. two fluctuations β i ( m 2 β i =2ρ 2 ) σ 2 + ω 2 +2ρ 2 β i (x) =λβ i (x)

18 The exact way Eigenvalue fluctuation equation, eg. two fluctuations β i ( m 2 β i =2ρ 2 ) σ 2 + ω 2 +2ρ 2 β i (x) =λβ i (x) k 2 = 2 x = 2K 1+ 2 π σ β i (x +4K) =β i (x) x 2 +2k 2 sn 2 [x + K,k 2 ]+Ω 2 β i (x) =λβ i (x)

19 The exact way Eigenvalue fluctuation equation, eg. two fluctuations β i ( m 2 β i =2ρ 2 ) σ 2 + ω 2 +2ρ 2 β i (x) =λβ i (x) k 2 = 2 x = 2K 1+ 2 π σ β i (x +4K) =β i (x) x 2 +2k 2 sn 2 [x + K,k 2 ]+Ω 2 β i (x) =λβ i (x) Lamé equation with periodic b.c. Case j=1 of the Lamé equation in Jacobian form x 2 +2j(j + 1) k 2 sn 2 [x, k 2 ] h Ψ =0

20 The spectral problem of Lamé potential λ Band structure determined by properties of Floquet exponent (quasi-momentum) λ 2 = Ω 2 +1+k 2 λ 1 = Ω 2 +1 forbidden β ± (x +4K) =e ±if β ± (x) λ 0 = Ω 2 + k 2

21 The spectral problem of Lamé potential λ Band structure determined by properties of Floquet exponent (quasi-momentum) λ 2 = Ω 2 +1+k 2 λ 1 = Ω 2 +1 forbidden β ± (x +4K) =e ±if β ± (x) λ 0 = Ω 2 + k 2 Periodic boundary conditions: spectrum discrete λ n = 2 k2 3 P(iy n ) F (iy n )=2K i ζ(iy n )+2y n ζ(k) =2π n n =1, 2,...

22 The spectral problem of Lamé potential λ Band structure determined by properties of Floquet exponent (quasi-momentum) λ 2 = Ω 2 +1+k 2 λ 1 = Ω 2 +1 forbidden β ± (x +4K) =e ±if β ± (x) λ 0 = Ω 2 + k 2 Periodic boundary conditions: spectrum discrete λ n = 2 k2 3 P(iy n ) F (iy n )=2K i ζ(iy n )+2y n ζ(k) =2π n n =1, 2,... The spectral way to functional determinants is feasible [only need a finite subset of eigenvalues: the 3 band edges!] but there is a powerful short-cut

23 Gel fand-yaglom Theorem (1960) Consider K g (x) = 2 for x [0,L] and g (0, 1) x + gv(x) K g (x) φ(x) =λφ(x) with Dirichlet bc φ(0) = φ(l) =0 To compute the determinant, solve the initial value problem K g (x) φ(x) =0 φ(0) = 0 φ (0) = 1 det K g = φ(l) K 0 L

24 Gel fand-yaglom Theorem (1960) Consider K g (x) = 2 for x [0,L] and g (0, 1) x + gv(x) K g (x) φ(x) =λφ(x) with Dirichlet bc φ(0) = φ(l) =0 To compute the determinant, solve the initial value problem K g (x) φ(x) =0 φ(0) = 0 φ (0) = 1 det K g = φ(l) K 0 L Example: Helmoltz operator [ 2 x + m 2 ] with Dirichlet b.c. > Dirichlet spectrum det[ 2 x + m 2 ] det[ 2 x] = n=1 m 2 +( n π L )2 ( n π L )2 = sinh ml ml > Gel fand-yaglom det[ 2 x + m 2 ] = det[ 2 x] φ(x) φ 0 (x) = sinh(ml) ml φ(x) =sinh(mx) φ 0 (x) =x

25 GY at work for spinning string Gel fand-yaglom for periodic b.c. x [0,P] given y 1 (0) = 1 y 2 (0) = 0 y 1(0) = 0 y 2(0) = 1 det O = y 1 (P )+y 2(P ) 2

26 GY at work for spinning string Gel fand-yaglom for periodic b.c. x [0,P] given y 1 (0) = 1 y 2 (0) = 0 y 1(0) = 0 y 2(0) = 1 det O = y 1 (P )+y 2(P ) 2 Solutions of the associated homogeneous equation [Hermite 1872] β ± (x) = sn(α,k 2 ) = H(x ± α) Θ(x) e Z(α)x 1+ 1 k 2 1+ π2 ω 2 4 K 2 (k 2 ) Z(u) = π 2K θ 4 πu 2K,q θ 4 πu 2K,q Θ(u) =θ 4 πu 2K,q H(u) =θ 1 πu 2K,q

27 GY at work for spinning string Gel fand-yaglom for periodic b.c. x [0,P] given y 1 (0) = 1 y 2 (0) = 0 y 1(0) = 0 y 2(0) = 1 det O = y 1 (P )+y 2(P ) 2 Solutions of the associated homogeneous equation [Hermite 1872] β ± (x) = sn(α,k 2 ) = H(x ± α) Θ(x) e Z(α)x 1+ 1 k 2 1+ π2 ω 2 4 K 2 (k 2 ) Z(u) = π 2K θ 4 πu 2K,q θ 4 πu 2K,q Θ(u) =θ 4 πu 2K,q H(u) =θ 1 πu 2K,q The result Using GY for P =4K and exploiting β ± (x +4K) =e 4Z(α)K β ± (x) det O β =4sinh 2 2K Z(α)

28 The exact way II: ubiquitous Lamé! Bosonic mode φ, m 2 φ =2ρ 2 + 2κ2 ω 2 2 x +2k 2 sn 2 [x, k 2 ]+ ρ 2 2 sn 2 [x, k 2 ] + Ω2 φ(x) =0 modular transformations 2 2 x +2 k 2 sn 2 x, k 2 + Ω 2 φ(x) =0 Lamé equation 2 Fermions ˆµ 2 F ± = ±ρ + ρ 2 2 x + k 2 sn 2 [x, k 2 ] ± k 2 cn 2 [x, k 2 ]dn 2 [x, k 2 ]+Ω 2 ψ(x) =0 modular transformation 3 2 x + ˆk 2 sn 2 [x, ˆk 2 ]+ˆΩ 2 ψ(x) =0 Lamé equation 3

29 The exact way III: Exact expression for 1-loop semiclassical energy Given the determinants deto β =sinh 2 [2 K(k 2 ) Z(α)] sn(α,k 2 ) = 1+ 1 k 2 1+ π2 ω 2 4 K 2 (k 2 ) deto φ =sinh 2 K deto ψ = cosh 2 K 4k Z( α) (1 + k) 2 4k Z(ˆα) (1 + k) 2 sn α, 4k (1 + k) 2 sn ˆα, 4k (1 + k) 2 = = (1 + k) 2 2+ π2 ω 2 8k K 2 4k ( (1 + k) 2 1+ π2 ω 2 4k K 2 4k ( (1+k) 2 ) (1+k) 2 ) the one-loop effective action reads Γ 1 = T dω ln 4π from which the one-loop energy R E 1 = Γ 1 κt, det 8 O ψ det 2 O β det O φ det 5 ( 2 ) T dτ 5 trivial fluctuations in S 5

30 UV-finiteness The behavior of the integrand for ω ln det O i = r 0 ω + r 1,i ω + O(ω 3 ), is determined by the fluctuations potentials V β, φ, ζi, ψ r 0 =2π r 1,i = π V i e.g. V β = 1 4K 4K 0 sn 2 (x k 2 ) Collecting altogether ln det 8 O ψ det 2 O β det O φ det 5 ζ 2K π (K E) =0

31 1-loop energy: exact vs. expanded E short exact k long Beautiful approximations! 3 ln 2 ln 16k2 1 k 2 logarithmic divergence cusp-anomaly

32 Long strings - Large Spin Expansion Leading behavior. Constant potential fluctuations ρ κ 0, cusp anomaly. Expanding further Γ NLO 1 = T π κ 0 (π + 6 ln 2), κ 0 recover a first correction missing in previous analysis! (turning point contribution) [Gromov unpublished 09] [Freyhult, Zieme 09]

33 Long strings - Large Spin Expansion Leading behavior. Constant potential fluctuations ρ κ 0, cusp anomaly. Expanding further Γ NLO 1 = T π κ 0 (π + 6 ln 2), κ 0 recover a first correction missing in previous analysis! Going to many subleading orders in 1/ 6, namely in 1/S 3 (turning point contribution) [Gromov unpublished 09] [Freyhult, Zieme 09] E 1 = κ 0 κ 1 c 01 κ 0 + c 00 + d 01 π κ c 10 + d κ 0 4 c 21 κ 0 + c 20 + d κ 0 6 c 31 κ 0 + c 30 + d κ 0 c 01 = 3π log 2, c 00 = π + 6 log 2, d 01 = 5 π 12, c 11 = 0, c 10 = 3 log 2, d 11 = ln 2 π c 21 = π π log 2, c 20 = π 16 c 31 = π π log 2, c 30 = 3π log 2 +, d 21 = log 2, d 30 = log 2 32π, + 85 log 2 64π Expansion compatible with reciprocity? At weak coupling, reciprocity is an observed regularity in the large spin expansion of the anomalous dimension for twist operators.

34 Reciprocity at weak coupling reviewed in [Beccaria, Forini, Macorini, 10] Operators O = Tr{D k 1 X...D k J X} k k J = S Rephrase the large S expansion of γ γ =f(s γ 1 2 β) in terms of another function f In QCD f the evidence is that has a (large S) parity invariant C C expansion f S = n a n (ln C) C 2n [Basso, Korchemsky 06] [Dokshitzer, Marchesini 06] C 2 = (S + J )(S + J 1) Casimir of SL(2,R) SO(4,2) J : twist : ϕ λ A

35 Reciprocity at weak coupling reviewed in [Beccaria, Forini, Macorini, 10] Operators O = Tr{D k 1 X...D k J X} k k J = S Rephrase the large S expansion of γ γ =f(s γ 1 2 β) in terms of another function f In QCD f the evidence is that has a (large S) parity invariant C C expansion f S = n a n (ln C) C 2n [Basso, Korchemsky 06] [Dokshitzer, Marchesini 06] C 2 = (S + J )(S + J 1) Casimir of SL(2,R) SO(4,2) J : twist : ϕ λ A In Mellin space, parity invariance becomes 1 F(x) = x F where x f(x) = 1 0 dx x S 1 F(x) or a generalized (Gribov-Lipatov) reciprocity. [Gribov, Lipatov, 72]

36 Evidence at weak coupling All twist-2 anomalous dimensions in QCD (3 loops) [Basso, Korchemsky 06] Twist 2-3 in various sectors of N=4 SYM also with wrapping O # loops wrapping reciprocity ϕϕ, ψψ, AA 5 yes ϕϕϕ 5 yes ψψψ 5 yes AAA 4 no (ABA) [Beccaria, Marchesini, Dokshitzer 07] [Beccaria 07] [Beccaria Forini 08]

37 Evidence at weak coupling All twist-2 anomalous dimensions in QCD (3 loops) [Basso, Korchemsky 06] Twist 2-3 in various sectors of N=4 SYM also with wrapping O # loops wrapping reciprocity ϕϕ, ψψ, AA 5 yes ϕϕϕ 5 yes ψψψ 5 yes AAA 4 no (ABA) [Beccaria, Marchesini, Dokshitzer 07] [Beccaria 07] [Beccaria Forini 08] Reciprocity has been even assumed to simplify multiloop calculations Ex.1 Twist three at 5 loops Tr (D s 1 Z D s 2 Z D s 3 Z) with S = s 1 + s 2 + s 3 Reciprocity-respecting ansatz for the anomalous dimension. [Beccaria,Forini, Lukowski, Zieme 09] Verified with Y-system! [Gromov, Kazakov, Vieira 09] Verified field-theoretically! [Fiamberti, Santambrogio, Sieg 09] Ex.2 Twist two at 5 loops Tr (D s 1 Z D s 2 Z) with S = s 1 + s 2 [Rej, Lukowski, Velizhanin 09]

38 Reciprocity at strong coupling From anomalous dimension 0 (S) =E 0 S 1 (S) =E 1 define F via (S) =F S (S) (S) = 0 (S)+ 1 λ 1 (S)+ F(S) =F 0 (S)+ 1 λ F 1 (S)+ and expand at large S. Re-express in terms of the semiclassical Casimir C S C 2 = S(S + 1) 1 ( λ) 2 C 2 = S S + 1 λ Coefficients of odd terms under S S vanish! c 10 = 1 π c 01, d 11 = 1 2π c 00, c 30 = c π c π c 21 c 31 = c 21, d 31 = d π 2 c π c π c 20. Reciprocity holds up to 1/S 3

39 Short strings Realized sending 0, k 0 det O f=β,φ,ψ = D (0) f (ω)+2 D (1) f (ω)+4 D (2) f (ω)+, Isolating lowest eigenvalues E 1 =1 1 4πκ dω ln (det O ψ ) 8 (det O β ) 2 det O φ The 1-loop correction in the short string limit reads E1 an = 3 2 S 2 4 ln ln ζ(3) S 3/ ln ζ(3) 480 ζ(5) S 5/2 + O(S 7/2 ) E1 nan = 1+O(S) Disagreement with semiclassical evaluation based on algebraic curve approach. [Gromov, unpublished]

40 Work in progress: QFT vs geometry [Beccaria, Dunne, Forini, Pawellek,Tseytlin] + [Gromov] QFT: fluctuations governed by the single-gap operators Riemann surface and elliptic curve interpretation [Belokos, Bobenko, Enolskii, Its, Matseev, 94] Algebraic curve approach to integrability > Classical string sols map to Riemann surfaces with several sheets and cuts. Classical energy is a contour integral. [Kazakov, Marshakov, Minahan, Zarembo 04] > Semiclassical quantization: pinching the surface by adding extra cuts δe 1 loop = 1 ( 1) F ij Ω ij n 2 n,ij [Beisert, Kazakov, Sakai, Zarembo 05] i,j: 8+8 bos. and ferm. polarizations F ij = ±1 [Gromov, Vieira 07]

41 Work in progress: QFT vs geometry [Beccaria, Dunne, Forini, Pawellek,Tseytlin] + [Gromov] QFT: fluctuations governed by the single-gap operators Riemann surface and elliptic curve interpretation [Belokos, Bobenko, Enolskii, Its, Matseev, 94] Algebraic curve approach to integrability > Classical string sols map to Riemann surfaces with several sheets and cuts. Classical energy is a contour integral. [Kazakov, Marshakov, Minahan, Zarembo 04] > Semiclassical quantization: pinching the surface by adding extra cuts δe 1 loop = 1 ( 1) F ij Ω ij n 2 n,ij 1. Non trivial dictionary to be constructed! [Beisert, Kazakov, Sakai, Zarembo 05] i,j: 8+8 bos. and ferm. polarizations F ij = ±1 > From alg.curve quasi-momentum and eigenfrequencies (x: spectral paramer) [Gromov, Vieira 07] dp dx dx dω dp dω ω 2 + f(k 2 ) (ω ω 2 )(ω ω2 )(ω ω2 ) our single gap problem 2. Interesting also to explain disagreements between the two approaches

42 Concluding remarks & perspectives Exact starting point for 1-loop corrections to the energy of folded string: fluctuations with ubiquitous, diagonalizable, Lamé operators Generalization to (S,J) solution. Still finite gap expected! caveat: fluctuations coupled even in static gauge

43 Concluding remarks & perspectives Exact starting point for 1-loop corrections to the energy of folded string: fluctuations with ubiquitous, diagonalizable, Lamé operators AdS/CFT: short string limit (not confirming previous results), QCD-like properties for large spin structure Generalization to (S,J) solution. Still finite gap expected! caveat: fluctuations coupled even in static gauge (Detailed) comparison with algebraic curve approach

44 Concluding remarks & perspectives Exact starting point for 1-loop corrections to the energy of folded string: fluctuations with ubiquitous, diagonalizable, Lamé operators AdS/CFT: short string limit (not confirming previous results), QCD-like properties for large spin structure Integrability: inherited from classical solution, redescovered with the integrable Lamé equation. Generalization to (S,J) solution. Still finite gap expected! caveat: fluctuations coupled even in static gauge (Detailed) comparison with algebraic curve approach Classify integrable matrix differential operators corresponding to sigma model classical solutions.