String hypothesis and mirror TBA Excited states TBA: CDT Critical values of g Konishi at five loops Summary. The Mirror TBA

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1 The Mirror TBA Through the Looking-Glass, and, What Alice Found There Gleb Arutyunov Institute for Theoretical Physics, Utrecht University Lorentz Center, Leiden, 12 April 2010

2 Summary of the mirror TBA approach Gauge-String Correspondence Matsubara Transform Thermodynamic Limit Gauge Theory String Sigma Model Mirror Theory TBA Spectrum The mirror TBA approach as a tool to determine the spectrum of the AdS 5 S 5 superstring Frolov and G.A. 07

3 Alice falls down a rabbit hole... String Hypothesis Canonical TBA Equations (vacuum) Simplified TBA Equations (vacuum) TBA for Excited States

4 Outline 1 String hypothesis and mirror TBA 2 Excited states TBA: CDT 3 Critical values of g 4 Konishi at five loops 5 Summary

5 Mirror theory and TBA Inspired by Yang+Yang 69, Zamolodchikov 90 string of length L =J "mirror string" of length R Frolov and G.A. 07 Two Minkowski theories related by the double Wick rotation: σ = iτ, τ = iσ The Hamiltonian H with respect to τ defines the mirror theory Ground state energy is related to the free energy of its mirror L E(L) = lim F (L) = LF R R Free energy per unit length F is found from the Bethe ansatz for the mirror model

6 Define λ t Hooft coupling, g string tension λ = 4π 2 g 2, g = λ 2π p momentum of a string particle E energy of a string particle p momentum of a mirror particle Ẽ energy of a mirror particle J momentum carried by string along the equator of S 5, L length (will be related to J)

7 Mirror dispersion relation The dispersion relation and the S-matrix of the original string model are uniquely fixed by asymptotic symmetries, in particular by su(2 2) su(2 2) The pole of the Euclidean two-point function E 2 E + 4g 2 sin 2 p E In string theory: E E ie, p E p = E = 1 + 4g 2 sin 2 p 2 In mirror theory: E E p, p E iẽ = 1 + p 2 Ẽ = 2arcsinh 2g

8 BYE of the mirror model Construct the mirror Bethe-Yang equations for fundamental particles (α = 1, 2) I YK 1 = e iep k R 1 = 1 = K I Y l=1 K II (α) Y l=1 l=1 l k y (α) k K(α) 2Y Y II x k S( p k, p l ) α=1 l=1 x + k x l y (α) k x + l w (α) k w (α) k v u t x + l x l v (α) + i l g v (α) i l g K III (α) Y l=1 K III (α) Y l=1 l k v (α) k v (α) k w (α) k w (α) k y (α) l y (α) l v u t x + k x k w (α) i l g w (α) + i l g w (α) 2i l g w (α) + 2i l g Double Wick rotation (analytic continuation) of the string S-matrix. Subtleties related to unitarity and the choice of the reference state (vacuum) Frolov and G.A. 07

9 Bound states of the mirror theory The mirror S-matrix exhibits poles. For instance, S( p 1, p 2 ) u 1 u 2 + 2i g u 1 u 2 2i g The multi-particle S-matrix exhibits poles at which form a pattern of a Bethe string u j u j+1 2i g = 0, j = 1,..., Q 1 u j = u + (Q + 1 2j) i, j = 1,..., Q, u R g This string can be interpreted as the Q-particle bound state

10 Bethe strings u u v u v Q=6 Q=4 "Bethe strings" Quasi-particles with real rapidities u j = u + (Q + 1 2j) i, j = 1,..., Q, u R g Auxiliary roots v = y + 1/y and w participate in building up Bethe strings!

11 The spectrum of TBA particles String hypothesis suggests the existence of nine types of TBA vacuum particles (α = 1, 2): Frolov and G.A. 09(a) Q-particles (Q-particle bound states) carrying momentum p Q y ±(α) -particles corresponding to fermionic Bethe roots M vw (α) -strings M w (α) -strings Derivation of the canonical TBA equations follows the textbook root Essler, Frahm, Göhmann, Klümper, Korepin, The One-Dimensional Hubbard Model

12 TBA equations Thermodynamic limit: R, N i with a ratio N i /R kept fixed ρ i (u) + ρ i (u) = K ij ρ j (u) K ij (u, v) = 1 d 2πi du log S ij (u, v), where ρ i and ρ i are the densities of the particles/holes Mirror free energy F(L) = R du P k h eek ρ k 1 i {z } L s(ρ k ) {z } energy T entropy The TBA equations δf δρ k = 0 = ɛ k = L e E k log `1 + e ɛ j K jk = canonical TBA The pseudo-energies ɛ k are e ɛ k = ρ k ρ k Y k E(L) = 1 2π X Z Q=1 du depq du log(1 + Y Q)

13 Simplified TBA equations Frolov and G.A. 09(b) 1 1 M w-strings: log Y (α) (α) (α) = log(1 + Y )(1 + Y M w M 1 w M+1 w ) s + δ Y (α) M1 log ˆ s 1 1 Y (α) + M vw-strings: log Y (α) M vw (α) (α) (1 + Y )(1 + Y M 1 vw M+1 vw = log ) (α) 1 Y (α) Y M+1 s + δ M1 log 1 Y ˆ s y-particles: log Y (α) + = log(1 + Y Y (α) Q ) K Qy, log Y (α) + Y (α) = log (1 + Y Q) K Q + 2 log Q-particles for Q 2: log Y Q = log Q = 1-particle: log Y 1 = log Y2 s + log 1+ 1 Y (α) M vw 1+ 1 Y (α) M w 1+ 1 Y (1) 1+ 1 Y (2) Q 1 vw Q 1 vw (1+ 1 Y Q 1 )(1+ 1 Y Q+1 ) 1 1 Y (1) 1 1 Y (2) s K M ˆ s (L) ˇ s

14 z-torus x(u) = 1 2 (u ip 4 u 2 ); Imx(u) < 0 u Mirror u String x s (u) = u 2 ³ 1 + q1 4u 2 jx s (u)j 1 Mirror and string regions on the z-torus. The boundaries are mapped onto the cuts on the u- and u -planes, respectively

15 Alice gets excited P. Dorey, Tateo 96; Bazhanov, Lukyanov, Zamolodchikov 96; NLIE 4 Integration contour Y 1 (z*k) = -1-1 TBA equations for excited states differ from each other only by a choice of the integration contour Taking the contour back to the real mirror line produces extra contributions log S(z, z) from log(1 + Y 1 ) K, where K (w, z) = 1 d log S(w, z) 2πi dw

16 Alice s adventures in the sl(2)-sector The Bethe-Yang equations have a rank one sector called sl(2) States are dual to the gauge theory operators Tr(D N Z J ) +... Y-functions Y (α) M vw, Y (α) M w, Y (α) ± satisfy Y (1) = Y (2) = Y Two-particle states are dual to the gauge theory operators Tr(D 2 Z J ) +...

17 Bethe-Yang equations BY equations for two-particle states in the sl(2) sector with p 1 + p 2 = x s ip(j + 1) log i θ(p, p) = 2πi n, n > 0 {z } xs 2 dress phase At small g the momentum is pj,n o = 2πn» J + 1 J + 1, n = 1,...,, 2 The corresponding rapidity u J,n 1 g uo J,n, uo J,n = cot πn J + 1 At large g the integer n will coincide with the string level

18 How to choose the contour? Generalized Lüscher formulae give the large J asymptotic solution Y o Q (v) = e J e E Q (v) T Q,1 (v u) 2 N Y i=1 S Q1 sl(2) (v, u i ) exp suppressed Bajnok and Janik 08 {u 1,..., u N } is the set of rapidities of string theory particles (solutions of BYE) The BY equations then follow from the fact that e E 1 (u k ) = ip k and T 1,1(u k u) = 1 = 1 = e ijp k NY i=1 S 1 1 sl(2) (u k, u i ) The remaining Y-functions are given in terms of T a,s-transfer matrices Y o = T 2,1 T 1,2, Y o + = T 2,3T 2,1 T 3,2 T 1,2, Y o Q vw = T Q+2,1T Q,1 T Q+1,2, Y o Q w = T 1,Q+2T 1,Q T 2,Q+1 T 0,Q+1 Kuniba,Nakanishi,Suzuku 93; Tsuboi 97; Gromov, Kazakov and Vieira 09

19 Zeroes of Y k vw ^r 2 u ^r 2 r 2 r 2 u u r1 r 1 r 1 ^r 1 ^r 1 r 2 r 1 ^r 1 ^r 2 r 2 ^r 1 ^r 1 For finite values of g any function Y k vw has four zeros It turns out that ˆr (k 1) j = r (k+1), i.e. Y j k vw has zeroes {r (k), r (k+2) } j j Positions of zeroes of Y k vw govern the analytic structure of other Y-functions: (k+1) Y k vw `r ± i j g = 1, Yk+1`r (k+1) j = 0, k = 1,...,, Y±`r (2) j = 0

20 Types of states At g 0 the following classification of two-particle states in the sl(2)-sector takes place Type of a state Y-functions Number of zeros I Y 1 vw 2 II Y 1 vw, Y 2 vw 2+2 III Y 1 vw, Y 2 vw, Y 3 vw IV Y 1 vw, Y 2 vw, Y 3 vw, Y 4 vw k Y 1 vw, Y 2 vw, Type of a state depends on how many Y-functions have zeroes in the rescaled physical strip Im(u) < 1

21 Evolution of a state Initial cond. Y 1 vw, Y 2 vw 2+2 Y 1 vw, Y 2 vw, Y 3 vw g Y 1 vw, Y 2 vw, Y 3 vw, Y 4 vw Y 1 vw, Y 2 vw, Y 3 vw, Y 4 vw, Y 5 vw Y 1 vw, Y 2 vw,

22 Critical values of g u physical 0 strip 2 i g 2 i g The (simplified) TBA equations must be modified each time one crosses a critical value of g Y r+1 vw (± i, g) = 0, r = 1, 2,..., g For the Konishi state J = 2, n = 1 there are 7 critical values for g < 100 g asympt crit {4.4, 11.5, 21.6, 34.8, 51.2, 70.7, 93.3}

23 Singularities evolve and call to modify the TBA Zeros of Y k+1jvw for g < g cr Zeros of 1 + Y kjvw u u r r log(1 + Y kjvw )? s cancel S(r i g u)s(r + i g u) = 1 Zeros of Y k+1jvw for g cr < g < ¹g cr u u r r log(1 + Y kjvw )? s do not cancel S( r i g u)s(r i g u) 6= 1

24 Five-loop Konishi from Lüscher s approach The CDT provides the spectrum of excited states NX E = J + E(p i ) 1 X Z 2π i=1 Q=1 {z } Bethe Yang The perturbative Lüscher approach yields du depq du log(1 + Y Q) {z } finite size corr. (10) = (10) asympt + g10n 81ζ(3)2 + 81ζ(3) 45ζ(5) + 945ζ(7) o 256 Bajnok, Hegedus, Janik and Lukowski 09 The five-loop correction comes from two sources 1 Position of Bethe roots shifts from their asymptotic value determined by BYE BYE(u k ) = 0 Y 1 (u k ) = 1 2 Finite-size correction to the BY energy (the same for Lüscher s and TBA)

25 Finite-size correction to BYE from Lüscher q As E(p) = 1 + 4g 2 sin 2 p 2 it is enough to compute δu k u k = u o k + g8 δu k from N=2 X j=1 δbye(u k ) δu j δu j + Φ (8) k = 0. Here Φ (8) is the leading finite-size correction to the BYE of order g 8 for which the Lüscher approach yields Φ (8) k = X Z M=1 du 2π u 1 Y o M (u) u1 +u 2 =0, Here Y o M is an asymptotic Y M function taken at leading order g 8

26 Finite-size correction to BYE from the mirror TBA Frolov, Suzuki and G.A Linearize TBA equations Y = Y o (1 + Y ), 2 Linearize the exact Bethe equation Y 1 (u k ) = 1 where δr k = + 2X π(2n k + 1) = J p k + i log S sl(2) (u j, u k ) +R k j=1 {z } BYE X Z du Y o M (u) 1 u u k M=1 π (M + 1) 2 + (u u k ) 2 Z 1 + du A 1 vw (u)y 1 vw (u) 2 sinh π 2 (u u k ) X Z du Y o M+1 (u) 1 h 4(u u k ) i M=1 4πi M 2 + (u u k ) 2 + ψ` M i (u u ) k ψ` M 4 4 i (u u ) k ψ` M i 4 (u u k ) + ψ` M+2 4 i 4 (u u k ) i and A M vw = Y M vw o 1+Y M vw o. Looks puzzling, but...

27 Finite-size correction to BYE from the mirror TBA Frolov, Suzuki and G.A Linearize TBA equations Y = Y o (1 + Y ), 2 Linearize the exact Bethe equation Y 1 (u k ) = 1 where δr k = + 2X π(2n k + 1) = J p k + i log S sl(2) (u j, u k ) +R k j=1 {z } BYE X Z du Y o M (u) 1 u u k M=1 π (M + 1) 2 + (u u k ) 2 Z 1 + du A 1 vw (u)y 1 vw (u) 2 sinh π 2 (u u k ) X Z du Y o M+1 (u) 1 h 4(u u k ) i M=1 4πi M 2 + (u u k ) 2 + ψ` M i (u u ) k ψ` M 4 4 i (u u ) k ψ` M i 4 (u u k ) + ψ` M+2 4 i 4 (u u k ) i and A M vw = Y M vw o 1+Y M vw o. Looks puzzling, but...

28 Five-loop Konishi from the mirror TBA Equation for Y Q vw : Y Q vw Ω QM = Y o M+1 s, s(u) = where the fluctuation operator is 1 4 cosh πu 2 Ω QM = δ QM δ Q,M 1 A M 1 vw s δ Q,M+1 A M+1 vw s. This equation is solved by iterations. Formal solution is Y M vw = Y o Q+1 s Ω 1 QM The solution Y 1 vw is plugged into δr k

29 Five-loop Konishi from the mirror TBA Comparison of the numerical results M TBA Lüscher TBA-Lüscher TBA-Lüscher Lüscher The contributions to δr 1 and Φ (8) (u 1 ) produced by individual Y M s for M = 2,..., 6 The perfect numerical agreement δr k + Φ (8) k = 0 Frolov, Suzuki and G.A. 10 Very recently this agreement has been also confirmed analytically Balog, Hegedus 10

30 In progress: Four magnons at 5-loops Each of Y M vw has four real zeroes (Y 1 vw has six) already at g 0 Such an analytic structure leads to extra driving terms 1 in the TBA equations already at weak coupling. Two-particle states are a bit special in this respect As, in the two-particle case, we find a universal relation between length and charge L = J + 2 Recall that semi-classically L = J 1 That is to those which are not induced by log(1 + Y 1 )

31 Four magnons at weak coupling Example: J = 2, N = 4, n 1 = n 2 = 1 Four Bethe roots u 1 = u 3, u 2 = u Roots Y M vw (r (M) i ) = 0 log Y M vw (u) = 2+2δ X M1 i=1 log S(r (M) i i 2X g u) + i=1 log S(r (M+2) i i g u) + log(1 + Y M 1 vw )(1 + Y M+1 vw ) s + δ M1 log 1 Y ˆ s 1 Y + log(1 + Y M+1 ) s

32 ... and What Alice Found The mirror TBA: from string hypothesis to the exact spectrum 1 The TBA equations for the vacuum follow from the string hypothesis 2 The TBA equations for excited states are determined by the TBA equations for the vacuum in conjunction with the contour deformation trick 3 With the integration contour tightened back to the real mirror line, the TBA equations for different states have different driving terms, even in the two-particle case! 4 The form of the equations does depend on the coupling constant. For any state there are critical values of g, crossing which the TBA equations must be modified 5 Relation length charge is independent of a state (a conjecture explicitly checked for 2-,4-, and 6-particle states) L = J + 2