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