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
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
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 AdS/CFT dual superstring sigma model.
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)
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 S3 in O(4) model),
-“magnon” variables – the transverse excitations on S3, or SU(2)xSU(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 T0(x),
Φ(x )=T0(x+i/2+i0) and T-1(x) (from Lax)
- Baxter eq.
- “Jump” eq.
relates T0 and Φ to T-1(x) through analyticity:
T0(x)
• TBA eq. for Y0 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 Y0 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-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). Nonstandard 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:
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