Download Bethe Ansatz in AdS/CFT: from local operators to classical strings

Bethe Ansatz in the AdS/CFT
correspondence
Thanks to:
Konstantin Zarembo
(Uppsala U.)
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)
EuroStrings 2006
Cambridge, 4/4/06
AdS/CFT correspondence
Maldacena’97
Gubser,Klebanov,Polyakov’98
Witten’98
Strings in AdS5xS5
RR flux requires manifest space-time supersymmetry
Green,Schwarz’84
World-sheet theory is Green-Schwarz coset sigma model
on SU(2,2|4)/SO(4,1)xSO(5). Metsaev,Tseytlin’98
Conformal gauge is problematic:
no kinetic term for fermions, no holomorphic
factorization for currents, …
Integrability in string theory
But the model is integrable!
Bena,Polchinski,Roiban’03
• infinite number of conserved charges
• separation of variables in classical
Dorey,Vicedo’06
equations of motion
• quantum spectrum is determined by
Bethe equations
√
√
?
Integrability in N=4 SYM
• Bethe ansatz for planar anomalous
Minahan,Z.’02
dimensions:
Beisert,Kristjansen,Staudacher’03
rigorous up to three loops Beisert,Staudacher’03
conjectured to all loop orders
Beisert,Dippel,Staudacher’04
Beisert,Staudacher’05; Rej,Serban,Staudacher’05
• Infinite number of conserved charges
associated with local operators (with
unclear interpretation)
N=4 Supersymmetric Yang-Mills Theory
Brink,Schwarz,Scherk’77
Gliozzi,Scherk,Olive’77
Field content:
The action:
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:
Minahan,Z.’02
Integrable Hamiltonian! Remains such
Beisert’03;
• at higher orders in λ Beisert,Kristjansen,Staudacher’03;
Beisert,Dippel,Staudacher’04; Rej,Serban,Staudacher’05
• for all operators Beisert,Staudacher’03
The spectrum
Ground state:
Excited states (magnons):
Exact spectrum
Rapidity:
Bethe’31
Zero momentum (trace cyclicity) condition:
Anomalous dimension:
scattering phase shifts
momentum
Exact periodicity condition:
periodicity of wave function
u
0
bound states of magnons – Bethe “strings”
u
0
mode numbers
Macroscopic spin waves: long strings
Sutherland’95;
Beisert,Minahan,Staudacher,Z.’03
Scaling limit:
defined on a set of conoturs Ck in the
complex plane
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:
~energy
2d principal chiral field – well-known intergable model
Pohlmeyer’76
Zakharov,Mikhailov’78
Faddeev,Reshetikhin’86
Equations of motion
Currents:
Virasoro constraints:
Light-cone currents and spins
Classical spins:
Virasoro constraints:
Equations of motion:
High-energy approximation
Approximate solution at
:
The same Landau-Lifshitz equation.
Kruczenski’03
Kruczenski,Ryzhov,Tseytlin’03
Integrability
Equations of motion:
Zero-curvature representation:
equivalent
Conserved charges
Generating function (quasimomentum):
time
on equations of motion
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 side
Staudacher’04; Beisert’05
Landau-Lifshitz model
WZ term:
Perturbation theory
in order to get canonical kinetic term
Minahan,Tirziu,Tseytlin’04
=
is not renormalized
=
S 2→2 =
Σ
p
p`
0
…
p
p`
Summation of bubble diagrams yields
Bethe equations
Klose,Z.’06
Heisenberg model: the same equations with
The difference disappears
in the low-energy (u→∞) limit.
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.
Empty Fermi sea:
Physical vacuum:
E
E
+m
+m
-m
-m
Berezin,Sushko’65; Bergknoff,Thaker’79; Korepin’79
NO ANTIPARTICLES
=
S 2→2 =
Σ
p
p`
0
…
p
p`
Exact S-matrix
θ – rapidity:
Bethe ansatz
Klose,Z.’06
Im θj=0: positive-energy states
Im θj=π: negative-energy states
Ground state
θ
-k+iπ
iπ
0
- UV cutoff
k+iπ
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-fermions
with non-trivial scattering
Questions
Continuous
(string worldsheet)
vs.
Discrete
(spin chain)
in Bethe ansatz:
Berenstein,Maldacena,Nastase’02
Possible resolution:
• extra hidden d.o.f. (“particles
on the world sheet
that create the spin chain”)
• spin chain is thus dynamical
Beisert,Dippel,Staudacher’04
Mann,Polchinski’05
Rej,Serban,Staudacher’05
Gromov,Kazakov,Sakai,Vieira’06
Choice of the reference state, the true vacuum, antiparticles, …