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: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 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:

SPSU(2,2|4)(p1,p2) = S0(p1,p2) SSU(2|2) (p1,p2) ×SSU(2|2) (p1,p2)

• 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 S3 in O(4) model),

-“magnon” variables – the transverse excitations on S3, 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)RYk(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)

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’93Fateev’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)

• Solution in terms of T0(x), Φ(x )=T0(x+i/2+i0) and T-1(x) (from Lax)

- Baxter eq.

relates T0 and Φ to T-1(x) through analyticity:

• TBA eq. for Y0 is the final non-linear integral eq. for T-1

- “Jump” eq.

Numerical solution for ground state

L Leading orderL→∞

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 Y0 solves the problem

Numerical solution for one particle in U(1)

L Ground state One particle n=0 mass gap

One particle n=1

2 -0.16202897 0.9923340596

0.99233406(1)

3.24329692

1/2 -1.74046938 0.71072799

0.71072801(1)

11.49312617

1/10 -11.2733646 -3.00410986

-3.0041089(1)

53.97831155

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 of or

• 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 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.

Happy Birthday

toKawamoto-san

andIshikawa-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’07Frolov,Arutyunov’07

X

X

Z-vacuum

Z

Janik,Bajnok’08 Fiamberti,Santambroglio,Sieg,Zanon’08,Velizhanin’08

XX

SSvirtual 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