Bethe Ansatz in the AdS/CFT correspondence
Konstantin Zarembo(Uppsala U.)
EuroStrings 2006Cambridge, 4/4/06
Thanks to:Niklas Beisert (Princeton)Johan Engquist (Utrecht)Gabriele Ferretti (Chalmers)Rainer Heise (Potsdam)Vladimir Kazakov (Paris)Thomas Klose (Uppsala)Andrey Marshakov (Moscow)Joe Minahan (Uppsala & Harvard)Kazuhiro Sakai (Paris)Sakura Schäfer-Nameki (Hamburg)Matthias Staudacher (Potsdam)Arkady Tseytlin (Imperial College)Marija Zamaklar (Potsdam)
AdS/CFT correspondence Maldacena’97
Gubser,Klebanov,Polyakov’98Witten’98
Strings in AdS5xS5
World-sheet theory is Green-Schwarz coset sigma model on SU(2,2|4)/SO(4,1)xSO(5).
Conformal gauge is problematic:no kinetic term for fermions, no holomorphic factorization for currents, …
RR flux requires manifest space-time supersymmetryGreen,Schwarz’84
Metsaev,Tseytlin’98
Integrability in string theory
But the model is integrable!Bena,Polchinski,Roiban’03
• infinite number of conserved charges• separation of variables in classical
equations of motion• quantum spectrum is determined by
Bethe equations
√
?
√Dorey,Vicedo’06
Integrability in N=4 SYM
• Bethe ansatz for planar anomalous dimensions:rigorous up to three loopsconjectured to all loop orders
• Infinite number of conserved charges associated with local operators (with unclear interpretation)
Minahan,Z.’02Beisert,Kristjansen,Staudacher’03Beisert,Staudacher’03
Beisert,Dippel,Staudacher’04Beisert,Staudacher’05; Rej,Serban,Staudacher’05
N=4 Supersymmetric Yang-Mills Theory
Field content:
The action:
Brink,Schwarz,Scherk’77Gliozzi,Scherk,Olive’77
Operator mixing
Renormalized operators:
Mixing matrix (dilatation operator):
Local operators and spin chains
related by SU(2) R-symmetry subgroup
a b
a b
One loop planar (N→∞) diagrams:
Permutation operator:
Integrable Hamiltonian! Remains such• at higher orders in λ• for all operators
Beisert,Kristjansen,Staudacher’03; Beisert’03; Beisert,Dippel,Staudacher’04; Rej,Serban,Staudacher’05
Beisert,Staudacher’03
Minahan,Z.’02
The spectrum
Ground state:
Excited states (magnons):
Zero momentum (trace cyclicity) condition:
Anomalous dimension:
Bethe’31
Exact spectrum
Rapidity:
Exact periodicity condition:
momentumscattering phase shifts
periodicity of wave function
u
0
bound states of magnons – Bethe “strings”
mode numbers
u
0
Sutherland’95; Beisert,Minahan,Staudacher,Z.’03
Macroscopic spin waves: long strings
defined on a set of conoturs Ck in the complex plane
Scaling limit:
x
0
Classical Bethe equations
Normalization:
Momentum condition:
Anomalous dimension:
Heisenberg model in Heisenberg representation
Heisenberg operators:
Hiesenberg equations:
Continuum + classical limit
Landau-Lifshitz equation
Consistent truncation
String on S3 x R1:
Conformal/temporal gauge:
Pohlmeyer’76Zakharov,Mikhailov’78Faddeev,Reshetikhin’86
2d principal chiral field – well-known intergable model
~energy
Equations of motion
Currents:
Virasoro constraints:
Light-cone currents and spins
Virasoro constraints:
Classical spins:
Equations of motion:
High-energy approximation
Approximate solution at :
The same Landau-Lifshitz equation.Kruczenski’03Kruczenski,Ryzhov,Tseytlin’03
Integrability
Zero-curvature representation:
Equations of motion:
equivalent
Conserved charges
time
on equations of motion
Generating function (quasimomentum):
Non-local charges:
Local charges:
Analyticity:
Classical string Bethe equation
Kazakov,Marshakov,Minahan,Z.’04
Normalization:
Momentum condition:
Anomalous dimension:
Quantum Bethe equations
(two particle factorization)
• find the dispersion relation (solve the one-body problem):
• find the S-matrix (solve the two-body problem):
Bethe equations full spectrum
• find the true ground state
Successfully used on the gauge-theory sideStaudacher’04; Beisert’05
WZ term:
Landau-Lifshitz model
Perturbation theory
in order to get canonical kinetic term Minahan,Tirziu,Tseytlin’04
= is not renormalized
= 0
…S 2→2 = Σp
p`
p
p`
Summation of bubble diagrams yields
Bethe equations
Heisenberg model: the same equations with
The difference disappears in the low-energy (u→∞) limit.
Klose,Z.’06
AAF model
SU(1|1) sector:
• in SYM:Callan,Heckman,McLoughlin,Swanson’04
• in string theory:
Alday,Arutyunov,Frolov’05
p0
p0
-μ
μ – chemical potential
μ→-∞ All poles are below the real axis.
-m
Empty Fermi sea:
E
+m
-m
E
+m
Physical vacuum:
Berezin,Sushko’65; Bergknoff,Thaker’79; Korepin’79
= 0
…S 2→2 = Σp
p`
p
p`
NO ANTIPARTICLES
Exact S-matrix
θ – rapidity:
Bethe ansatz
Im θj=0: positive-energy states
Im θj=π: negative-energy states
Klose,Z.’06
Ground state
θ
iπ
0
-k+iπ k+iπ
- UV cutoff
Mass renormalization:
This equation also determines the spectrum, the physical S-matrix, ...
• Weak-coupling (-1<g<1):Non-renormalizable (anomalous dimension of mass is complex)
• Strong attraction (g>1) Unstable
(Energy unbounded below)• Strong repulsion (g<-1):
Spectrum consists of fermions and anti-fermionswith non-trivial scattering
Questions
Continuous Discrete(string worldsheet) vs. (spin chain)
in Bethe ansatz:
Possible resolution:• extra hidden d.o.f. (“particles
on the world sheet that create the spin chain”)
• spin chain is thus dynamical
Choice of the reference state, the true vacuum, antiparticles, …
Mann,Polchinski’05Rej,Serban,Staudacher’05Gromov,Kazakov,Sakai,Vieira’06
Berenstein,Maldacena,Nastase’02 Beisert,Dippel,Staudacher’04