Conference in honor of Kenzo Ishikawa and Noboru

Conference in honor of Kenzo Ishikawa and Noboru Kawamoto Sapporo, 8 -9 January 2009 Finite Volume Spectrum of 2 D Field Theories from Hirota Dynamics Vladimir Kazakov (ENS, Paris) with N. Gromov and P. Vieira, ar. Xiv: 0812. 5091

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 2 D 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 Tsuboi’ 00 V. K. , Sorin, Zabrodin’ 07, • A step towards the spectrum of anomalous dimensions of ALL operators of N=4 Super-Yang –Mills gauge theory, or its Ad. S/CFT dual superstring sigma model.

S-matrix for SU(2)x. SU(2) principal chiral field • S-matrix: Al. &A. Zamolodchikov’ 79 • Scalar (dressing) factor: Satisfies Yang-Baxter, unitarity, crossing and analyticity: • Footnote: Compare to Ad. S/CFT: SPSU(2, 2|4)(p 1, p 2) = S 02(p 1, p 2) SSU(2|2) (p 1, p 2) ×SSU(2|2) (p 1, p 2)

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

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

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

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

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

Numerical solution for ground state L Leading order L→∞ Our results From DDV-type eq. [Balog, Hegedus’ 04] 4 -0. 015513 0. 015625736 -0. 01562574(1) 2 -0. 153121 -0. 162028968 -0. 16202897(1) 1 -0. 555502 -0. 64377457 -0. 6437746(1) 1/2 -1. 364756 -1. 74046938 -1. 7404694(2) 1/10 -7. 494391 -11. 2733646 -11. 273364(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) mode numbers n=0, 1 L Ground state 2 -0. 16202897 One particle n=0 mass gap 0. 9923340596 One particle n=1 3. 24329692 0. 99233406(1) 1/2 -1. 74046938 0. 71072799 11. 49312617 0. 71072801(1) 1/10 -11. 2733646 -3. 00410986 -3. 0041089(1) From NLIE [Hegedus’ 04] 53. 97831155

Energy versus size for various states E 2 /L L

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

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: We get from Cauchy theorem Same for v-wing

Gauge equivalence of SU(2)L and SU(2)R wings • Wing exchange symmetry: • Gauge transformation relating two wings: • Can be recasted into a Destri-de. Vega 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 2 d integrable field theories. Useful for TBA and for quantum fusion • The method gives a rather systematic tool for study of 2 d 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 Ad. S 5 x. S 5 (wrapping). Nonstandard R-matrices, like Hubbard or su(2|2)ext S-matrix in Ad. S/CFT, are also described by Hirota eq. with different B. C. Hopefully the full Ad. S/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: X Fiamberti, Santambroglio, Sieg, Zanon’ 08, Velizhanin’ 08 Janik, Bajnok’ 08 Z Z X Janik, Lukowski’ 07 Frolov, Arutyunov’ 07 From TBA to finite size: double Wick rotation leads to “mirror” theory with spectrum: virtual particle S S Z-vacuum X X • TBA, with the full set of bound states should produce dimensions of all operators at any coupling λ