Finite Volume Spectrum of 2D Field Theories from Hirota Dynamics

1. Conference in honor of Kenzo Ishikawa and Noboru Kawamoto Sapporo, 8-9 January 2009 Finite Volume Spectrum of 2D Field Theories from Hirota Dynamics Vladimir Kazakov (ENS,Paris) with N.Gromov and P.Vieira, arXiv:0812.5091

2. 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’97V.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.

3. S-matrix for SU(2)xSU(2) principal chiral field • S-matrix: Al.&A.Zamolodchikov’79 • Scalar (dressing) factor: Satisfies Yang-Baxter, unitarity, crossing and analyticity: • Footnote: Compare to AdS/CFT: SPSU(2,2|4)(p1,p2) = S02(p1,p2) SSU(2|2) (p1,p2)×SSU(2|2) (p1,p2)

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

5. AsymptoticBethe Ansatz eqs. (L → ∞) • Periodicity: • Bethe equations from periodicity • -variables describe U(1)-sector (main circle of S3 in O(4) model), • -“magnon” variables – the transverse excitations on S3, or SU(2)xSU(2) • Energy and momentum of a state:

6. 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

7. 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 vacuum an exited state • Main Bethe eq.

8. 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

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

10. Analyticity and ground state solution Q=1 • Solution in terms of T0(x), Φ(x)=T0(x+i/2+i0) and T-1(x) (from Lax) - Baxter eq. - “Jump” eq. relates T0andΦto T-1(x) through analyticity: T0(x) • TBA eq. for Y0 is the final non-linear integral eq. for T-1

11. Numerical solution for ground state • Solved by iterations on Mathematica

12. U(1)-states • Particle rapidities – real zeroes Our solution generalizes to • The same TBA eq. for Y0 solves the problem

13. Numerical solution for one particle in U(1) mode numbers n=0,1 From NLIE [Hegedus’04]

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

15. 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 by analyticity for each wing • For each wing fix the gauge to make and polynomial • Find a gauge relating • This closes the set of equations for a general state on

16. 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)

17. 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.

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

19. 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-deVega type equation for

20. 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

21. 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 AdS5xS5 (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.

22. Happy BirthdaytoKawamoto-sanandIshikawa-san

23. Finite size operators and TBA • ABA Does not work for “short” operators, like Konishi’s tr [Z,X]2, due towrapping problem. • Finite size effects from S-matrix (Luscher correction) Four loop result found and checked directly from YM: X Z Z X Fiamberti,Santambroglio, Sieg,Zanon’08,Velizhanin’08 Janik,Bajnok’08 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