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Bethe ansatz in String Theory. Konstantin Zarembo (Uppsala U.). Integrable Models and Applications, Lyon, 13.09.2006. AdS/CFT correspondence. Maldacena’97. Gubser,Klebanov,Polyakov’98 Witten’98. Planar diagrams and strings. time. (kept finite). ‘t Hooft coupling: - PowerPoint PPT Presentation
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Bethe ansatz in String Theory
Konstantin Zarembo
(Uppsala U.)
Integrable Models and Applications, Lyon, 13.09.2006
The European Superstring Theory Network » Members
Co-ordinator Chalmers University of Technology (Sweden) Fundamental Physics Department of Physics* (Karlstad Univ.) Other Contractors
Uppsala University (Sweden) Theoretical Physics
Cosmology Particle Astrophysics and String theory* (Stockholm Univ.)
The Chancellor, Masters and Scholars of the University of Cambridge (UK)
Theoretical High Energy Particle Physics Group King's College (UK) Theoretical Physics Queen Mary and Westfield College (UK) String Theory Group Theory Group* (Imperial College) Centre National de la Reserche Scientifique (France) LTPENS*(École Normale Supérieure) LPTHE*(Univ. Pierre et Marie Curie) CRNS Universiteit van Amsterdam (The Netherlands) Institute for Theoretical Physics Institute for Theoretical Physics*(Utrecht Univ.) Theoretical Physics* (NIKHEF) Max-Planck-Gesellschaft (Germany) Quantum Gravity & Unified Theories (Max Planck Institute) Centre for Mathematical Physics (Univ. Hamburg & DESY) Universitá degli Studi di Roma "Tor Vergata" (I taly) String Theory Group Gruppo Teorico* (Univ. di Roma La Sapienza) Gruppo Teorico* (Univ. di Pisa) High Energy Group* (ICTP) University of Crete High Energy and Elementary Particle Physics Division * (Univ. of Athens) The Hebrew University of J erusalem (I srael) Racah Institute of Physics Dep. of Particle Physics*(Weizmann Inst. of Science) Dep. of Particle Physics*(Tel Aviv University) Masarykova univerzita v Brne (Czech Republic) Institute of Theoretical Physics and Astrophysics University of Cyprus (Cyprus) High Energy Physics Group
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
Circular Wilson loop (exact):Erickson,Semenoff,Zarembo’00
Drukker,Gross’00
0 λSYM perturbation
theory
1 + + …+
String perturbation
theory
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
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
a b
a b
Tree level: Δ=L (huge degeneracy)
One loop:
Minahan,Z.’02
Zero momentum (trace cyclicity) condition:
Anomalous dimensions:
Bethe’31
Bethe ansatz
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
Weak coupling:
Strong coupling:
Q: Is it exact at all λ?
Rej,Serban,Staudacher’05; Z.’05;
Feverati,Fiorovanti,Grinza,Rossi’06; Beccaria,DelDebbio’06
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
Bootstrap: SU(2|2)xSU(2|2) invariant S-matrix
asymptotic Bethe ansatz spectrum of an infinite spin chain
Ground state tr ZZZZ… breaks PSU(2,2|4) → P(SU(2|2)xSU(2|2))
Beisert’05
Beisert,Staudacher’05
STRINGS
String theory in AdS5S5Metsaev,Tseytlin’98
+ constant RR 4-form flux
Bena,Polchinski,Roiban’03
• Finite 2d field theory (¯-function=0)
• Sigma-model coupling constant:
• Classically integrable
Classical limit
is
AdS sigma-models as supercoset
S5 = SU(4)/SO(5)
AdS5 = SU(2,2)/SO(4,1)
Super(AdS5xS5) = PSU(2,2|4)/SO(5)xSO(4,1)
AdS superspace:
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)
= 8Fermions: - bifundamentals of su(2,2) x su(4)
4 x 4 x 2
= 32 real components
: 2 kappa-symmetry
: 2 (eqs. of motion are first order)
= 8
Quantization
• fix light-cone gauge and quantize:action is VERY complicated perturbation theory for the spectrum, S-matrix,…
• study classical equations of motion (gauge unfixed), then guess
• quantize near classical string solutions
Berenstein,Maldacena,Nastase’02
Callan,Lee,McLoughlin,Schwarz,
Swanson,Wu’03
Frolov,Plefka,Zamaklar’06
Callan,Lee,McLoughlin,Schwarz,Swanson,Wu’03; Klose,McLoughlin,Roiban,Z.’in progress
Kazakov,Marshakov,Minahan,Z.’04; Beisert,Kazakov,Sakai,Z.’05;
Arutyunov,Frolov,Staudacher’04; Beisert,Staudacher’05
Frolov,Tseytlin’03-04; Schäfer-Nameki,Zamaklar,Z.’05;
Beisert,Tseytlin’05; Hernandez,Lopez’06
Consistent truncation
String on S3 x R1:
Zero-curvature representation:
Equations of motion:
equivalent
Zakharov,Mikhaikov’78
Gauge condition:
Classical string Bethe equation
Kazakov,Marshakov,Minahan,Z.’04
Normalization:
Momentum condition:
Anomalous dimension:
Quantum string Bethe equations
extra phase
Beisert,Staudacher’05
Arutyunov,Frolov,Staudacher’04
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 and do not capture finite-size effects.
Beisert’05
Schäfer-Nameki,Zamaklar,Z.’06
• 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
• Algebraic structure: Algebraic Bethe ansatz?Yangian symmetries?Baxter equation?
Open problems
Teschner’s talk