Bethe Ansatz and Integrability in AdS/CFT correspondence
Konstantin Zarembo
(Uppsala U.)
“Constituents, Fundamental Forces and Symmetries of the Universe”,Napoli, 9.10.2006
Thanks to:Niklas BeisertJohan EngquistGabriele FerrettiRainer HeiseVladimir KazakovThomas KloseAndrey MarshakovTristan McLoughlinJoe MinahanRadu RoibanKazuhiro SakaiSakura Schäfer-NamekiMatthias StaudacherArkady TseytlinMarija Zamaklar
AdS/CFT correspondence
Yang-Mills theory with
N=4 supersymmetry
String theory on
AdS5xS5 background
Maldacena’97
Gubser,Klebanov,Polyakov’98
Witten’98
Exact equivalence
Planar diagrams and strings
time
Large-N limit:
AdS/CFT correspondence Maldacena’97
Gubser,Klebanov,Polyakov’98
Witten’98
λ<<1 Quantum strings
Classical strings Strong coupling in SYM
Spectrum of SYM = String spectrum
but
Strong-weak coupling interpolation
Circular Wilson loop (exact):Erickson,Semenoff,Zarembo’00
Drukker,Gross’00
0 λSYM perturbation
theory
1 + + …+
String perturbation
theory
Minimal area law in AdS5
Gubser,Klebanov,Tseytlin’98; …
SYM is weakly coupled if
String theory is weakly coupled if
There is an overlap!
Q:HOW TO COMAPARE
SYM AND STRINGS?
A(?): SOLVE EACH
WITH THE HELP OF BETHE ANSATZ
Plan
1. Integrability in SYM
2. Integrability in AdS string theory
3. Integrability and Bethe ansatz
4. Bethe ansatz in AdS/CFT
5. Testing Bethe ansatz against string quantum corrections
N=4 Supersymmetric Yang-Mills Theory
Field content:
Action:
Gliozzi,Scherk,Olive’77
Global symmetry: PSU(2,2|4)
Spectrum
Basis of primary operators:
Dilatation operator (mixing matrix):
Spectrum = {Δn}
Local operators and spin chains
related by SU(2) R-symmetry subgroup
i j
i j
One loop:
Tree level: Δ=L (huge degeneracy)
One loop planar dilatation generator:
Minahan,Z.’02
Heisenberg Hamiltonian
Integrability
Lax operator:
Monodromy matrix:
Faddeev et al.’70-80s
Transfer “matrix”:
Infinite tower of conserved charges:
U – lattice translation generator: U=eiP
Algebraic Bethe Ansatz
Spectrum:
are eigenstates of the Hamiltonian
with eigenvalues
(anomalous dimension)
(total momentum)
ProvidedBethe equations
Strings in AdS5xS5
Green-Schwarz-type 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, …
Light-cone gauge is OK.
Metsaev,Tseytlin’98
The action is complicated, but the model is integrable!Bena,Polchinski,Roiban’03
Consistent truncation
String on S3 x R1:
Zero-curvature representation:
Equations of motion:
equivalent
Zakharov,Mikhaikov’78
Gauge condition:
Conserved charges
time
on equations of motion
Generating function (quasimomentum):
Non-local charges:
Local charges:
Bethe ansatz
• Algebraic Bethe ansatz: quantum Lax operator + Yang-Baxter equations → spectrum
• Coordinate Bethe ansatz: direct construction of the wave functions in the Schrödinger representation
• Asymptotic Bethe ansatz: S-matrix ↔ spectrum (infinite L) ? (finite L)
Spectrum and scattering phase shifts
periodic short-range potential
• exact only for V(x) = g δ(x)
Continuity of periodized wave function
where
is (eigenvalue of) the S-matrix
• correct up to O(e-L/R)• works even for bound states via analytic
continuation to complex momenta
Multy-particle states
Bethe equations
Assumptions:• R<<L• particles can only exchange momenta• no inelastic processes
2→2 scattering in 2d
p1
p2
k1
k2
Energy and momentum conservation:
I
II
Momentum conservation
Energy conservation
k1
k2
I: k1=p1, k2=p2 (transition)
II: k1=p2, k2=p1 (reflection)
n→n scattering
2 equations for n unknowns
(n-2)-dimensional phase space
pi ki
Unless there are extra conservation laws!
Integrability:
• No phase space:
• No particle production (all 2→many processes are kinematically forbidden)
Factorization:
Consistency condition (Yang-Baxter equation):
1
2
3
1
2
3=
Strategy:
• 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
Integrability + Locality Bethe ansatz
What are the scattering states?
SYM: magnons
String theory: “giant magnons”
Staudacher’04
Hofman,Maldacena’06
Common dispersion relation:
S-matrix is highly constrained by symmetriesBeisert’05
Zero momentum (trace cyclicity) condition:
Anomalous dimension:
Algebraic BA: one-loop su(2) sector
Rapidity:Minahan,Z.’02
Algebraic BA: one loop, complete spectrumBeisert,Staudacher’03
Nested BAE:
- Cartan matrix of PSU(2,2|4)
- highest weight of the field representation
bound states of magnons – Bethe “strings”
mode numbers
u
0
Sutherland’95;
Beisert,Minahan,Staudacher,Z.’03
Semiclassical states
defined on a set of conoturs Ck in the complex plane
Scaling limit:
x
0
Classical Bethe equations
Normalization:
Momentum condition:
Anomalous dimension:
Algebraic BA: classical string Bethe equation
Kazakov,Marshakov,Minahan,Z.’04
Normalization:
Momentum condition:
String energy:
su(2) sector:
General classical BAE are known and have the nested structure
consistent with the PSU(2,2|4) symmetry of AdS5xS5 superstringBeisert,Kazakov,Sakai,Z.’05
Asymptotic BA: SYM
Beisert,Staudacher’05
Asymptotic BA: string
extra phase
Arutyunov,Frolov,Staudacher’04
Hernandez,Lopez’06
• Algebraic structure is fixed by symmetries
• The Bethe equations are asymptotic: they describe infinitely long strings / spin chains.
Beisert’05
Schäfer-Nameki,Zamaklar,Z.’06
Testing BA: semiclassical string in AdS3xS1
- radial coordinate in AdS
- angle in AdS
- angle on S5
- global time
Rigid string solution
Arutyunov,Russo,Tseytlin’03
AdS5S5
winds k times
and rotates
winds m times
and rotates
Internal length of the string is
Perturbative SYM regime:
(string is very long)
For simplicity, I will consider
(large-winding limit) Schäfer-Nameki,Zamaklar,Z.’05
string fluctuation frequencies
Explicitly,
Park,Tirziu,Tseytlin’05
classical energy one loop correction
Quantum-corrected Bethe equations
classical BEKazakov,Z.’04
AnomalyKazakov’04;Beisert,Kazakov,Sakai,Z.’05
Beisert,Tseytlin,Z.’05; Schäfer-Nameki,Zamaklar,Z.’05
Quantum correction to the scattering phaseHernandez,Lopez’06
Large (long strings):
Comparison
• String
• BA
BA misses exponential termsSchäfer-Nameki,Zamaklar,Z.’05
Conclusions
• Large-N SYM / string sigma-model on AdS5xS5 are probably solvable by Bethe ansatz
• Open problems: Interpolation from weak to strong coupling Finite-size effects Appropriate reference state / ground state Algebraic formulation:
– Transfer matrix
– Yang-Baxter equation
– Pseudo-vacuum
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