Finite Volume Spectrum of 2D Field Theories from Hirota Dynamics Vladimir Kazakov (ENS,Paris) Conference in honor of Kenzo Ishikawa and Noboru Kawamoto Sapporo, 8-9 January 2009 with N.Gromov and P.Vieira, arXiv:

Motivation and results Thermodynamical Bethe ansatz (TBA) is a powerful tool to get finite size solutions in relativistic sigma-models, including the spectrum of excited states. Al.Zamolodchikov’92,’00,… Bazhanov,Lukyanov,A.Zamolodchikov’94, Dorey,Tateo’94, Fendley’95, Ravanini,Hegedus‘95 Hagedus,Balog’98-’05……… TBA as a Y-system for finite size 2D field theories Al.Zamolodchikov’90 Subject of the talk: TBA as Hirota dynamics: Solution of finite size O(4) sigma model (equivalent to SU(2)×SU(2) Principle Chiral Field) for a general state. New and a very general method for such problems! Gromov,V.K.,Vieira’08 Hirota eq. and Y-system are examples of integrable discrete classical dynamics. We extensively use this fact. Krichever,Lipan,Wiegmann, Zabrodin’97 V.K.,Sorin,Zabrodin’07, Tsuboi’00 A step towards the spectrum of anomalous dimensions of ALL operators of N=4 Super-Yang –Mills gauge theory, or its AdS/CFT dual superstring sigma model.

S-matrix for SU(2)xSU(2) principal chiral field S-matrix: Al.&A.Zamolodchikov’79 Satisfies Yang-Baxter, unitarity, crossing and analyticity: Footnote: Compare to AdS/CFT: S PSU(2,2|4) (p 1,p 2 ) = S 0  (p 1,p 2 ) S SU(2|2) (p 1,p 2 ) ×S SU(2|2) (p 1,p 2 ) Scalar (dressing) factor:

Free energy – ground state I.e. from the asymptotic spectrum (R=∞) we can compute the ground state energy for ANY finite volume L! R=∞

Asymptotic Bethe Ansatz eqs. (L → ∞) Bethe equations from periodicity -variables describe U(1)-sector (main circle of S 3 in O(4) model), -“magnon” variables – the transverse excitations on S 3, or SU(2)xSU(2) Periodicity: Energy and momentum of a state:

Complex formation in (almost) infinite volume Magnon bound states for u-wing and v-wing, in full analogy with Heisenberg chain Thermodynamic equations for densities of bound states and their holes w.r.t. Minimization of the free energy at finite temperature T=1/L

SU(2) L SU(2)×SU(2) Principal Chiral Field in finite volume SU(2) R Yk(θ)Yk(θ) (densities of magnon holes/complexes) (densities of particles/holes) Thermodynamics of complexes → TBA → Y-system Gromov,V.K.,Vieira’08 Energy of vacuum Main Bethe eq. an exited state

a k Tk(θ)Tk(θ) SU(2) L Y-system and Hirota relation SU(2) R Parametrize: Hirota equation: Solution: linear Lax pair (discrete integrable dynamics!), Krichever, Lipan, Wiegmann, Zabrodin’97 Fateev,Onofri,Zamolodchikov’93 Fateev’96

Gauge transformation Deaterminant solution of Hirota eq. Wronskian relation Leaves Y’s and Lax pair invariant!

Analyticity and ground state solution Q=1 T0(x)T0(x) Solution in terms of T 0 (x), Φ(x )=T 0 (x+i/2+i0) and T -1 (x) (from Lax) - Baxter eq. relates T 0 and Φ to T -1 (x) through analyticity: TBA eq. for Y 0 is the final non-linear integral eq. for T -1 - “Jump” eq.

Numerical solution for ground state LLeading order L→∞ Our resultsFrom DDV-type eq. [Balog,Hegedus’04] (1) (1) (1) 1/ (2) 1/ (1) Solved by iterations on Mathematica

U(1)-states Particle rapidities – real zeroes Our solution generalizes to The same TBA eq. for Y 0 solves the problem

Numerical solution for one particle in U(1) LGround state One particle n=0 mass gap One particle n= (1) / (1) / (1) From NLIE [Hegedus’04] mode numbers n=0,1

E 2  /L L Energy versus size for various states

Strategy for general states with u,v magnons Solve T-system in terms ofor For each wing fix the gauge to make and polynomial Relate to by analyticity for each wing Find a gauge relating This closes the set of equations for a general state on (only one wing is analytical at a time)

Large Volume Limit L→∞ It is a spin chain limit: T-system splits into two wings with Y-system trivially gives Main BAE at large L: Auxiliary BAE – from polynomiality of(defined by Lax eq)

Analyticity (only for one wing at a time) From Lax: - Baxter eq. - “Jump” eq. Spectral representation relating with the spectral density from determinant solution of Hirota eq.

Calculating G(x) Choosing 3 different contours for 3 different positions of argument: Same for v-wing We get from Cauchy theorem

Gauge equivalence of SU(2) L and SU(2) R wings Gauge transformation relating two wings: Wing exchange symmetry: Can be recasted into a Destri-deVega type equation for

Bethe Ansatz Equations at finite L Main Bethe Ansatz equation (for rapidities of particles) Auxiliary Bethe equations for magnons (from regularity of on the physical strip): Our method works for all excited states and gives their unified description

Conclusions and Prospects Hirota discrete classical dynamics: A powerful tool for studying 2d integrable field theories. Useful for TBA and for quantum fusion The method gives a rather systematic tool for study of 2d integrable field theories at finite volume. We found Luscher corrections for arbitrary state. Y-system and TBA eqs. for gl(K|M) supersymmetric sigma-models are straightforward from Hirota eq. with “fat hook” boundary conditions. Our main motivation: dimensions of “short” operators (ex.: Konishi operator) in N=4 SYM using S-matrix for dual superstring on AdS 5 xS 5 (wrapping). Non- standard R-matrices, like Hubbard or su(2|2) ext S-matrix in AdS/CFT, are also described by Hirota eq. with different B.C. Hopefully the full AdS/CFT TBA as well. TBA should solve the problem.

Happy Birthday to Kawamoto-san and Ishikawa-san

Finite size operators and TBA ABA Does not work for “short” operators, like Konishi’s tr [Z,X] 2, due to wrapping problem. Finite size effects from S-matrix (Luscher correction) Four loop result found and checked directly from YM: Janik, Lukowski’07 Frolov,Arutyunov’07 X X Z-vacuum Z Janik,Bajnok’08 Fiamberti,Santambroglio, Sieg,Zanon’08,Velizhanin’08 XX S S virtual particle Z From TBA to finite size: double Wick rotation leads to “mirror” theory with spectrum: TBA, with the full set of bound states should produce dimensions of all operators at any coupling λ