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

Bethe Ansatz in AdS/CFT
Correspondence
Konstantin Zarembo
(Uppsala U.)
J. Minahan, K. Z., hep-th/0212208
N. Beisert, J. Minahan, M. Staudacher, K. Z., hep-th/0306139
V. Kazakov, A. Marshakov, J. Minahan, K. Z., hep-th/0402207
N. Beisert, V. Kazakov, K. Sakai, K. Z., hep-th/0503200
N. Beisert, A. Tseytlin, K. Z., hep-th/0502173
S. Schäfer-Nameki, M. Zamaklar, K.Z., hep-th/0507179
DGMTP, Tianjin, 23.08.05
Large-N expansion of gauge theory
String theory
Early examples:
• 2d QCD
• Matrix models
4d gauge/string duality:
• AdS/CFT correspondence
Macroscopic strings from planar diagrams
Large orders
of perturbation theory
or
Large number
of constituents
AdS/CFT correspondence
Maldacena’97
Gubser, Klebanov, Polyakov’98
Witten’98
λ<<1
Classical string
Quantum string
Strong coupling in SYM
Way out: consider states with large quantum numbers
= operators with large number of constituent fields
Price: highly degenerate operator mixing
Operator mixing
Renormalized operators:
Mixing matrix (dilatation operator):
Multiplicatively renormalizable operators
with definite scaling dimension:
anomalous dimension
N=4 Supersymmetric Yang-Mills Theory
Field content:
The action:
Local operators and spin chains
• Restrict to SU(2) sector
related by SU(2) R-symmetry subgroup
a
b
a
b
Operator basis:
• ≈ 2L degenerate operators
• The space of operators can be identified with the Hilbert
space of a spin chain of length L
with (L-M) ↑‘s and M ↓‘s
One loop planar (N→∞) diagrams:
Permutation operator:
Minahan, K.Z.’02
Integrable Hamiltonian! Remains such
Kristjansen, Staudacher’03
• at higher orders in λ Beisert,
Beisert, Dippel, Staudacher’04
• for all operators Beisert, Staudacher’03
Spectrum of Heisenberg ferromagnet
Ground state:
(SUSY protected)
Excited states:
flips one spin:
Non-interacting magnons
• good approximation if M<<L
Exact solution:
• exact eigenstates are still multi-magnon Fock states
• (**) stays the same
• but (*) changes!
Bethe ansatz
Rapidity:
Bethe’31
Zero momentum (trace cyclicity) condition:
Anomalous dimension:
bound states of magnons – Bethe “strings”
u
0
mode numbers
Macsoscopic spin waves: long strings
Sutherland’95;
Beisert, Minahan, Staudacher, K.Z.’03
Scaling limit:
defined on cuts Ck in the complex plane
x
0
Classical Bethe equations
Normalization:
Momentum condition:
Anomalous dimension:
Comparison to strings
• Need to know the spectrum of string states:
- eigenstates of Hamiltonian in light-cone gauge
or
- (1,1) vertex operators in conformal gauge
• Not known how to quantize strings in AdS5xS5
• But as long as λ>>1 semiclassical approximation is OK
Time-periodic classical solutions
Bohr-Sommerfeld
Quantum states
String theory in AdS5S5
Metsaev, Tseytlin’98
• Conformal 2d field theory (¯-function=0)
• Sigma-model coupling constant:
• Classically integrable
Bena, Polchinski, Roiban’03
Classical limit
is
Consistent truncation
Keep only
String on S3xR1
Conformal/temporal gauge:
2d principal chiral field – well-known intergable model
Pohlmeyer’76
Zakharov, Mikhailov’78
Faddeev, Reshetikhin’86
Integrability:
Time-periodic solutions of classical equations of motion
Spectral data (hyperelliptic curve + meromorphic differential)
AdS/CFT correspondence:
Noether charges in sigma-model
Quantum numbers of SYM operators (L, M, Δ)
Noether charges
Length of the chain:
Total spin:
Energy (scaling dimension):
Virasoro constraints:
BMN scaling
BMN coupling
Berenstein, Maldacena, Nastase’02
For any classical solution:
Frolov-Tseytlin limit:
If 1<<λ<<L2:
Which can be compared to perturbation theory even
though λ is large.
Frolov, Tseytlin’03
Integrability
Equations of motion:
Zero-curvature representation:
equivalent
on equations of motion
Infinte number of conservation laws
Auxiliary linear problem
quasimomentum
Noether charges are determined by asymptotic
behaviour of quasimomentum:
Analytic structure of quasimomentum
p(x) is meromorphic on complex plane with cuts along
forbidden zones of auxiliary linear problem and has poles
at x=+1,-1
Resolvent:
is analytic and therefore admits spectral representation:
and asymptotics at ∞
completely determine ρ(x).
Classical string Bethe equation
Kazakov, Marshakov, Minahan, K.Z.’04
Normalization:
Momentum condition:
Anomalous dimension:
Take
Normalization:
Momentum condition:
Anomalous dimension:
This is classical limit of Bethe equations for spin chain!
Q: Can we quantize string Bethe equations
(undo thermodynamic limit)?
A: Yes! Arutyunov, Frolov, Staudacher’04; Staudacher’04;Beisert, Staudacher’05
Quantum strings in AdS:
• BMN limit Berenstein, Maldacena, Nastase’02; Metsaev’02;…
• Near-BMN limit Callan, Lee,McLoughlin,Schwarz,Swanson,Wu’03;…
• Quantum corrections to classical string solutions
Frolov, Tseytlin’03
Frolov, Park, Tsetlin’04
Park, Tirziu, Tseytlin’05
Fuji, Satoh’05
Finite-size corrections to Bethe ansatz
Beisert, Tseytlin, Z.’05
Hernandez, Lopez, Perianez, Sierra’05
Schäfer-Nameki, Zamaklar, Z.’05
String on AdS3xS1:
radial coordinate in AdS
angle in AdS
angle on S5
Rigid string solution:
Arutyunov, Russo, Tseytlin’03
angular momentum on S5
AdS spin
One-loop quantum correction:
Park, Tirziu, Tseytlin’05
Bethe equations:
Even under L→-L
First correction is O(1/L2)
But singular if
simultaneously
Local anomaly
• cancels at leading order
• gives 1/L correction
Kazakov’03
Beisert, Kazakov, Sakai, Z.’05
Beisert, Tseytlin, Z.’05
Hernandez, Lopez, Perianez, Sierra’05
x
0
Locally:
Anomaly
local contribution
1/L correction to classical Bethe equations:
Beisert, Tseytlin, Z.’05
Re-expanding the integral:
Agrees with the string calculation.
Remarks:
• anomaly is universal: depends only on singular part
of Bethe equations, which is always the same
• finite-size correction to the energy can be always Beisert, Freyhult’05
expressed as sum over modes of small fluctuations