Exploring TBA in the mirror AdS 5 × S 5 Ryo Suzuki School of Mathematics, Trinity College Dublin Ryo Suzuki School of Mathematics, Trinity College Dublin Based on arXiv: , , Collaborators: Gleb Arutyunov (Utrecht), Sergey Frolov (TCD) Marius de Leeuw and Alessandro Torrilelli (Utrecht)

Motivation AdS/CFT correspondence predicts E string ( ) =  gauge ( ) for any string state and for any local gauge theory operator We want to ‘demonstrate’ this prediction by using the integrablity method called Bethe Ansatz The string/gauge theory becomes perturbatively integrable when they have large supersymmetry and global symmetry E string ( ) E Bethe ( )  gauge ( )

Motivation Consider maximally supersymmetric setup at large N limit Super Yang-Mills Superstring on AdS 5 × S 5 with N-units of RR flux Both theories are believed as perturbatively integrable Asymptotic Bethe Ansatz can demonstrate the relation [Bena Polchiski Roiban (2003)], [Minahan, Zarembo (2002)] and so on [Beisert Staudacher (2005)] E string ( ) E Bethe ( )  gauge ( ) for general ‘spinning’ strings and ‘long’ SYM operators, when wrapping corrections are negligible

Motivation The mirror TBA is expected to give the exact form of E Bethe ( ) for any string/SYM states It is far from trivial to see that the mirror TBA gives a ‘reasonable’ answer; one has to solve a set of nonlinear integral equations among infinitely many fields [Bombardelli, Fioravanti, Tateo (2009)] [Gromov, Kazakov, Kozak, Vieira (2009)] [Arutyunov, Frolov (2009)] I explain our findings on the mirror TBA for two-particle states, based on numerical study around the asymptotic limit All wrapping corrections can be summed up by Thermodynamic Bethe Ansatz (TBA) for the mirror AdS 5 × S 5 (N.B. no relationship between the mirror TBA and TBA for minimal surface)

Main Problem Compute two-point functions in D=4, N=4 SU(N) SYM Conformal symmetry constrains the two-point function [Fiamberti, Santambrogio, Sieg, Zanon (2007)] [Velizhanin (2008)]  can be computed at weak coupling by summng 131,015 diagrams at four loops Konishi state

Plan of Talk 1. Introduction Overview of AdS/CFT and Integrability 2. Results from Integrability Transfer-matrix formula T-system and Y-system 3. Excited-State TBA How to formulate TBA Contour deformation trick Konishi at five loops

1. Introduction

AdS/CFT setup Super Yang-Mills CFT side AdS side Superstring on AdS 5 × S 5 N-units of RR flux Global symmetry Theory Coupling constant ( λ is arbitrary, N is large)

Prediction 2. Conformal dimension of a local operator in SYM 1. Energy of a string state measured in AdS-time (t=τ) Given global charges ( S 1, S 2, J 1, J 2, J 3 ), compare

Difficulty Anomalous dimension in perturbative SYM String energy in semiclassical expansion

Correspondence as 2D models E(λ) = Worldsheet energy of a gauge-fixed Lagrangian for closed string on AdS 5 × S 5 Δ(λ) = Eigenvalue of anomalous dimension matrix Hamiltonian on 1-dim. lattice (with continuous time direction) Hamiltonian of 2-dim. field theory on cylinder

Anomalous dimension matrix [Minahan, Zarembo (2002)]

Hamiltonian of worldsheet theory Circumference of cylinder = L ? Hamiltonian of N=4 SYM spin chain Length of operator = L

Vacuum of 2D models Residual global symmetry is R × psu(2|2) 2 L = J 1 fields spin J 1 = L E = J 1 = Point particle moving along the geodesic of S 5 1/2 BPS operator (D, S1, S2, J1, J2, J3) Δ = J 1 = vacuum of GS action in the light-cone gauge,

Excitations of 2D models LeftRight singlet (vacuum) singlet Fundamental representation of PSU(2|2) 2 of PSU(2|2) 2 2D models have a mass gapfor fixed J 1

Infinite-size limit SYM operator of an infinite length Excitations on the decompactified worldsheet Periodicity condition Extra central charges

Magic of psu Closure of commutators constrains the dispersion relation [Beisert (2005)] Conjecture: for AdS 5 × S 5 /N=4 SYM Please remember that this dispersion is non-relativistic. 2D Hamiltonian is a part of

Multi-particle states Particles scatter (or create a bound state) Excitations on the decompactified worldsheet SYM operator of an infinite length

Multi-particle states 1. S-matrix between two fundamental representations of psu(2|2) 2 is determined by symmetry up to a scalar factor psu(2|2) 2 is determined by symmetry up to a scalar factor 3. Yang-Baxter implies integrability; infinitely many charges are conserved during the scattering infinitely many charges are conserved during the scattering 4. Assuming Yang-Baxter relation, all S-matrices between two boundstates are determined two boundstates are determined [Arutyunov, de Leeuw, Torrielli (2009)] [Martins Melo (2007)] 2. The psu(2|2) S-matrix is equivalent to Shastry’s R- matrix of 1+1D Hubbard model, and satisfy Yang-Baxter relation of 1+1D Hubbard model, and satisfy Yang-Baxter relation [Beisert (2005)]

Evidences from Perturbative Studies Anomalous dimension in perturbative SYM String energy in semiclassical expansion [Minahan, Zarembo (2002)] and so on [Bena Polchiski Roiban (2003)] and so on and so on Integrable Integrable

Asymptotic Spectrum “Energy” (= Bethe Ansatz Equations) “Momentum” S0S0S0S0 [Beisert Staudacher (2005)] “dressing phase” [Beisert Hernández López (2006)] [Beisert Eden Staudacher (2006)] Assume AdS 5 × S 5 and N=4 SYM are integrable

Overview AdS/CFT correspondence and Integrability methods Finite-size corrections ? Asymptotic Bethe Ansatz Exact Spectrum Asymptotic Spectrum ?

Wrapping Problem No symmetry enhancement for the finite-length operator Dispersion relation receives wrapping corrections! Interactions wrap around at high loop orders Virtual particles travel around the world [Lüscher, Comm Math Phys 104 (1986)] [Janik Łukowski (2007)] [Bajnok Janik (2008)] [Fiamberti, Santambrogio, Sieg, Zanon (2007)] [Velizhanin (2008)]

Overview AdS/CFT correspondence and Integrability methods Finite-size corrections Thermodynamic Bethe Ansatz Asymptotic Bethe Ansatz Exact Spectrum Asymptotic Spectrum Generalized Lüscher formula

2. Results from Integrability

Two-particle States in sl(2) sector Asymptotic Bethe Ansatz [Beisert, Staudacher (2005)] [Beisert Hernández López (2006)] [Beisert Eden Staudacher (2006)] [Dorey Hofman Maldacena (2007)]

Konishi state [Beisert, Staudacher (2005)] Rapidity for Konishi (J=2, n=1) Asymptotic Bethe Ansatz equation and asymptotic energy This is correct up to three-loop, invalid at four-loop

TBA and Y/T-systems Mirror TBA for two-particle states Y-system ? Discrete Laplacian T-system Gauge-invariant combination of T Redefinition Transfer matrix for rectangular representaions Solution Need to know analytic structure of Y’s

Transfer matrix formulae [Bazhanov, Reshetikhin J Phys A23 (1990)] [Gromov, Kazakov, Vieira (2009)] Formula for general “rectangular representations” (a,s)

[Beisert (2006)] [Gromov, Kazakov, Vieira (2009)] [Arutyunov, de Leeuw, Torrielli, R.S. (2009)] Transfer matrix formulae

T-system Transfer matrices satisfy T-system equations T=1 along the bottom line of the fat hook of psu(2,2|4) T=0 outside the fat hook of psu(2,2|4) Nonzero gap on the complex v-plane [Arutyunov, Frolov (2009)] [Frolov, R.S. (2009)] Gauge symmetry can be used to fix boundary conditions

Fat-hook of psu(2,2|4) can split into two fat-hooks of psu(2|2) [Gromov, Kazakov, Vieira (2009)] [Gromov, Kazakov, Vieira (2009)] T=1 T=0T=0

Y-system Asymptotic Y-functions in terms of transfer matrix auxiliary momentum-carrying [Gromov, Kazakov, Vieira (2009)] Solution of the asymptotic Y-system (gauge invariant) (gauge ambiguity)

Y-system However, solution of the asymptotic Y- system is non-unique, because the discrete Laplacian has infinitely many solutions Non-canonical Y-system follows from the mirror TBA (depends on branch choice) Need to know the analytic structure of Y-functions consistent with TBA, to “integrate” the Y-system

3. Excited-State TBA

How to formulate TBA 1. Compute the partition function in the “mirror” theory in the “mirror” theory 3. Get TBA equations; a set of nonlinear integral equations nonlinear integral equations 2. Minimum of the partition function is related to ground state energy is related to ground state energy 4. Analytic continuation gives TBA equations for excited states TBA equations for excited states 5. Consistency with Lüscher formula

Mirror transformation Exact ground state energy Asymptotic free energy [Zamolodchikov, Nucl Phys B342 (1990)] Wrapping corrections No wrapping

AdS 5 × S 5 Mirror model for AdS 5 × S 5 [Arutyunov, Frolov, Zamaklar (2006)] [Klose, McLoughlin, Roiban, Zarembo (2006)] (Bosonic part of) the gauge-fixed sigma-model on AdS 5 × S 5 Mirror model is defined by the rotation

AdS 5 × S 5 Mirror model for AdS 5 × S 5 After the double Wick-rotation on worldsheet Different real section of complexified psu(2|2) 2 Residual symmetry is again psu(2|2) 2 Residual symmetry is again psu(2|2) 2 Same S-matrix, except for the dressing phase Same S-matrix, except for the dressing phase Particles (giant magnons) live in AdS 5, not in S 5 Particles (giant magnons) live in AdS 5, not in S 5 Dispersion relation are doubly Wick-rotated Dispersion relation are doubly Wick-rotated Residual symmetry is again psu(2|2) 2 Residual symmetry is again psu(2|2) 2 Same S-matrix, except for the dressing phase Same S-matrix, except for the dressing phase Particles (giant magnons) live in AdS 5, not in S 5 Particles (giant magnons) live in AdS 5, not in S 5 Dispersion relation are doubly Wick-rotated Dispersion relation are doubly Wick-rotated [Arutyunov Frolov (2009)] [Volin (2009)] [Gromov Kazakov Vieira (2009)]

String theory and Mirror theory String Mirror Asymptotic dispersion for Q-particle boundstate (Q=1,2,... ∞) Related by analytic continuation

x and u variables (rapidity) Physical Regions

Counting Mirror Spectrum Assume that the dominant contribution to the mirror free energy comes from thermodynamical states 1. Classify mirror boundstates using string hypothesis 2. Take thermodynamic limit of ABA for mirror boundstates ⇒ the constraint equations 3. Extremize free energy by varying the density of particles (or holes) [Takahashi, Prog Theor Phys 47 (1972)] [Zamolodchikov, Nucl Phys B342 (1990)]

Labelling mirror boundstates by Bethe roots (momentum-carrying + auxiliary) Dynkin diagram for su(3|2) Four types of boundstates Q-particlesy-particlesM|w-stringsM|vw-strings momentum carrying boundstates fermionic auxiliary particles bosonic auxiliary boundstates bosonic boundstate of y, y, M|w (Q=1,2,... and M=1,2,...)

Q-particle M|vw-string M|w-string y-particle [Bombardelli, Fioravanti, Tateo (2009)] [Gromov, Kazakov, Kozak, Vieira (2009)] [Arutyunov, Frolov (2009)] Ground State TBA

Exact Groundstate Energy [Frolov, R.S. (2009)] Extremum of mirror free energy Y Q is meromorphic on the physical region of z-torus Small h expansion (Witten index at h=0)

Relation to Y-system [Gromov, Kazakov, Vieira (2009)] [Arutyunov, Frolov (2009)] ⇔ Y-system can be derived by applying discrete Laplacian to the TBA equations This Y-system is same as the Y-system from transfer matrix (up to branch choice)

Excited-State TBA Analytic continuation of coupling constant brings TBA from the ground state to excited states [Dorey Tateo (1996,1997)] Deformed contour Original contour

Excited-State TBA Analytic continuation of coupling constant brings TBA from the ground state to excited states [Dorey Tateo (1997)] Stokes Phenomena Analytic continuation Asymptotic limit (large L) Analytic continuation Asymptotic limit

Contour Deformation Trick TBA equations and the exact energy are universal. Only the integration contours depend on the state under consideration. The large L (= small g) solution must be consistent with the generalized Lüscher formula (= transfer matrix formula). The contours are deformed smoothly as g increases. TBA equations and the exact energy are universal. Only the integration contours depend on the state under consideration. The large L (= small g) solution must be consistent with the generalized Lüscher formula (= transfer matrix formula). The contours are deformed smoothly as g increases. Analytic continuation of g Deformation of integration contours Contour in z-torus

Contour Deformation Trick Contour in z-torus Our contour Naive guess How to choose? Contours in u-plane

Contour Deformation Trick How to determine the deformed contour All modification is explained by our choice of contour Excited-state TBA equations at large L must be solved by asymptotic Y-functions given by transfer-matrix formula Consider 2-particle states in the sl (2) sector with general L, What are the integral equations they satisfy? Naive choice of contour is inconsistent when L > 6 Extra terms are needed to get consistent TBA

Singularity of Y M|vw, 1+Y M|vw Y M|vw functions are auxiliary Y-functions coupled to the Y Q functions which carry the energy

Singularity of Y M|vw, 1+Y M|vw All asymptotic Y M|vw functions have four zeroes, either on the real axis or on the imaginary axis of the mirror All asymptotic Y M|vw functions have four zeroes, either on the real axis or on the imaginary axis of the mirror Zeroes of Y M-1|vw are related to those of (1+Y M|vw ), Y M+1|vw, Zeroes of Y M-1|vw are related to those of (1+Y M|vw ), Y M+1|vw, exhibit the universal structure as in the table exhibit the universal structure as in the table Type I Y 1|vw 2 Type II Y 1|vw,Y 2|vw 2+2 Type III Y 1|vw,Y 2|vw, Y 3|vw Type IV Y 1|vw,Y 2|vw, Y 3|vw, Y 4|vw Type V...

Evolution of Zeroes of Y M|vw Zeroes of move as g increases Zeroes of Y M|vw move as g increases Principal-value prescription is needed for singularities on the real axis Subcritical values

Evolution of Zeroes of 1+Y M|vw Zeroes of 1+ move as g increases Zeroes of 1+Y M|vw move as g increases Critical values Two of them pinches the real axis, a new residue appears, an old disappears

Evolution of Zeroes of 1+Y M|vw Zeroes of 1+ move as g increases Zeroes of 1+Y M|vw move as g increases Critical values Two of them pinches the real axis, a new residue appears, an old disappears

Zeroes of asymptotic Y M|vw All asymptotic Y M|vw functions have four zeroes, either on the real axis or on the imaginary axis of the mirror All asymptotic Y M|vw functions have four zeroes, either on the real axis or on the imaginary axis of the mirror Konishi at g=0, k>2 Four extra log S terms cancel Zeroes of Y M-1|vw are related to those of (1+Y M|vw ), Y M+1|vw Zeroes of Y M-1|vw are related to those of (1+Y M|vw ), Y M+1|vw

Critical points Three extra log S terms do not cancel Zeroes of asymptotic Y k|vw move as g increases. Zeroes of asymptotic Y k|vw move as g increases. The first critical value is approximately g=4.429 or λ=774 The first critical value is approximately g=4.429 or λ=774 Konishi (J=2,n=1) at g>4.429 or Konishi-like (J>4, n=1) at g=0 for k=2 Asymptotic Bethe Ansatz : Asymptotic Bethe Ansatz :

Subcritical Points Three extra log S terms do not cancel Two pure imaginary zeroes collide, split into two real zeroes Two pure imaginary zeroes collide, split into two real zeroes Extra terms in Canonical TBA equations change. Extra terms in Canonical TBA equations change. The first subcritical value is approximately g=4.495 or λ=798 The first subcritical value is approximately g=4.495 or λ=798 [Arutyunov, Frolov, R.S. (2009)]; see also [Dorey, Tateo (1997)]

Exact Konishi Spectrum? [Gromov, Kazakov, Vieira (2009)] Numerical solution of TBA up to λ=700, extrapolate to λ=∞ Numerical solution of TBA up to λ=700, extrapolate to λ=∞ Disagree slightly with semiclassical string results Disagree slightly with semiclassical string results [Roiban, Tseytlin (2009)] Existence of infinitely many critical points may explain the mismatch

Konishi at weak coupling The exact energy of TBA for Konishi state is expanded as Four-loop part agrees with the known results in SYM There is only a conjecture at five loops

Konishi at five loops Need to know the correction to the asymptotic Bethe roots A refined generalized Lüscher formula was conjectured [Bajnok, Hegedűs, Janik, Łukowski (2009)] Five-loop anomalous dimension from Lüscher

Konishi at five loops The correction from TBA looks very different Solve the TBA, linearized around asymptotic solution We find TBA=Lüscher numerically [Arutyunov, Frolov, R.S. (2010)] [Balog, Hegedűs (2010)] there is an analytic proof

Conclusion and Outlook TBA equations for excited states are dynamical; depend on coupling constant as well as the state under consideration Contour deformation trick explains modification Five-loop Konishi agrees with the Lüscher formula TBA equations for excited states are dynamical; depend on coupling constant as well as the state under consideration Contour deformation trick explains modification Five-loop Konishi agrees with the Lüscher formula agrees with BFKL equation, The Lüscher formula agrees with BFKL equation, for five-loop twist-two operators -- how about TBA? [Łukowski, Rej, Velizhanin (2009)] Deeper understanding on analytic structure of TBA

Thank you for attention