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

Bethe ansatz in String Theory
Konstantin Zarembo
(Uppsala U.)
Integrable Models and Applications, Lyon, 13.09.2006
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AdS/CFT correspondence
Maldacena’97
Gubser,Klebanov,Polyakov’98
Witten’98
Planar diagrams and strings
time
‘t Hooft coupling:
String coupling constant =
(kept finite)
(goes to zero)
Strong-weak coupling interpolation
λ
0
SYM perturbation
theory
1+
+
String perturbation
theory
+…
Circular Wilson loop (exact):
Erickson,Semenoff,Zarembo’00
Drukker,Gross’00
Minimal area law in AdS5
Weakly coupled SYM is reliable if
Weakly coupled string is reliable if
Can expect an overlap.
N=4 Supersymmetric Yang-Mills Theory
Gliozzi,Scherk,Olive’77
Field content:
Action:
Global symmetry: PSU(2,2|4)
Spectrum
Basis of primary operators:
Spectrum = {Δn}
Dilatation operator (mixing matrix):
Local operators and spin chains
related by SU(2) R-symmetry subgroup
a
b
a
b
Tree level: Δ=L
(huge degeneracy)
One loop:
Minahan,Z.’02
Bethe ansatz
Bethe’31
Zero momentum (trace cyclicity) condition:
Anomalous dimensions:
Higher loops
Requirments of integrability and BMN scaling
uniquely define perturbative scheme to construct
dilatation operator through order λL-1:
Beisert,Kristjansen,Staudacher’03
The perturbative Hamiltonian turns out to coincide
with strong-coupling expansion of Hubbard model
at half-filling:
Rej,Serban,Staudacher’05
Asymptotic Bethe ansatz
Beisert,Dippel,Staudacher’04
In Hubbard model, these equations are approximate
with O(e-f(λ)L) corrections at L→∞
Anti-ferromagnetic state
Rej,Serban,Staudacher’05; Z.’05;
Feverati,Fiorovanti,Grinza,Rossi’06; Beccaria,DelDebbio’06
Weak coupling:
Strong coupling:
Q: Is it exact at all λ?
Arbitrary operators
Bookkeeping:
“letters”:
“words”:
“sentences”:
Spin chain:
infinite-dimensional
representation of
PSU(2,2|4)
• Length fluctuations:
operators (states of the spin chain) of different length mix
• Hamiltonian is a part of non-abelian symmetry group:
conformal group SO(4,2)~SU(2,2) is part of PSU(2,2|4)
so(4,2):
Mμν - rotations
Pμ - translations
Kμ - special conformal transformations
D
- dilatation
Ground state tr ZZZZ… breaks PSU(2,2|4) → P(SU(2|2)xSU(2|2))
Bootstrap: SU(2|2)xSU(2|2) invariant S-matrix
Beisert’05
spectrum of an infinite spin chain
asymptotic Bethe ansatz
Beisert,Staudacher’05
STRINGS
String theory in AdS5S5
Metsaev,Tseytlin’98
+ constant RR 4-form flux
• Finite 2d field theory (¯-function=0)
• Sigma-model coupling constant:
• Classically integrable
Bena,Polchinski,Roiban’03
Classical limit
is
AdS sigma-models as supercoset
S5 = SU(4)/SO(5)
AdS5 = SU(2,2)/SO(4,1)
AdS superspace:
Super(AdS5xS5) = PSU(2,2|4)/SO(5)xSO(4,1)
Z4 grading:
Coset representative: g(σ)
Currents: j = g-1dg = j0 + j1 + j2 + j3
Action:
Metsaev,Tseytlin’98
In flat space:
Green,Schwarz’84
no kinetic term for fermions!
Degrees of freedom
Bosons:
15 (dim. of SU(2,2)) + 15 (dim. of SU(4))
- 10 (dim. of SO(4,1)) - 10 (dim. of SO(5))
= 10 (5 in AdS5 + 5 in S5)
- 2 (reparameterizations)
= 8
Fermions:
- bifundamentals of su(2,2) x su(4)
4x4x2
= 32
real components
:
2
kappa-symmetry
:
2
(eqs. of motion are first order)
= 8
Quantization
• fix light-cone gauge and quantize:
Berenstein,Maldacena,Nastase’02
Callan,Lee,McLoughlin,Schwarz,
Swanson,Wu’03
Frolov,Plefka,Zamaklar’06
action is VERY complicated
perturbation theory for the spectrum, S-matrix,…
Callan,Lee,McLoughlin,Schwarz,Swanson,Wu’03; Klose,McLoughlin,Roiban,Z.’in progress
• study classical equations of motion (gauge unfixed),
then guess
Kazakov,Marshakov,Minahan,Z.’04; Beisert,Kazakov,Sakai,Z.’05;
Arutyunov,Frolov,Staudacher’04; Beisert,Staudacher’05
• quantize near classical string solutions
Frolov,Tseytlin’03-04; Schäfer-Nameki,Zamaklar,Z.’05;
Beisert,Tseytlin’05; Hernandez,Lopez’06
Consistent truncation
String on S3 x R1:
Gauge condition:
Equations of motion:
Zero-curvature representation:
equivalent
Zakharov,Mikhaikov’78
Classical string Bethe equation
Kazakov,Marshakov,Minahan,Z.’04
Normalization:
Momentum condition:
Anomalous dimension:
Quantum string Bethe equations
Arutyunov,Frolov,Staudacher’04
extra phase
Beisert,Staudacher’05
Arutyunov,Frolov,Staudacher’04
Hernandez,Lopez’06
• Algebraic structure is fixed by symmetries Beisert’05
• The Bethe equations are asymptotic: they describe
infinitely long strings / spin chains and do not capture
finite-size effects. Schäfer-Nameki,Zamaklar,Z.’06
Open problems
• Interpolation from weak to strong coupling in the
dressing phase
• How accurate is the asymptotic BA? (Probably up to
e-f(λ)L)
• Eventually want to know closed string/periodic chain
spectrum
need to understand finite-size effects
Teschner’s talk
• Algebraic structure:
Algebraic Bethe ansatz?
Yangian symmetries?
Baxter equation?