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Bethe ansatz in String Theory

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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

time

(kept finite)

‘t Hooft coupling:

String coupling constant =

(goes to zero)

λ

0

SYM perturbation

theory

String perturbation

theory

1

+

+

+ …

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.

Field content:

Gliozzi,Scherk,Olive’77

Action:

Global symmetry: PSU(2,2|4)

Basis of primary operators:

Spectrum = {Δn}

Dilatation operator (mixing matrix):

related by SU(2) R-symmetry subgroup

b

a

b

a

Tree level: Δ=L (huge degeneracy)

One loop:

Minahan,Z.’02

Bethe’31

Zero momentum (trace cyclicity) condition:

Anomalous dimensions:

Requirments ofintegrability 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

Beisert,Dippel,Staudacher’04

In Hubbard model, these equations are approximate

with O(e-f(λ)L) corrections at L→∞

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?

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

asymptotic Bethe ansatz

spectrum of an infinite spin chain

Beisert’05

Beisert,Staudacher’05

STRINGS

Metsaev,Tseytlin’98

+ constant RR 4-form flux

- Finite 2d field theory (¯-function=0)
- Sigma-model coupling constant:
- Classically integrable

Classical limit

is

Bena,Polchinski,Roiban’03

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!

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)

4 x 4 x 2

= 32 real components

: 2 kappa-symmetry

: 2 (eqs. of motion are first order)

= 8

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

String on S3 x R1:

Gauge condition:

Equations of motion:

Zero-curvature representation:

equivalent

Zakharov,Mikhaikov’78

Kazakov,Marshakov,Minahan,Z.’04

Normalization:

Momentum condition:

Anomalous dimension:

Arutyunov,Frolov,Staudacher’04

extra phase

Beisert,Staudacher’05

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?

Teschner’s talk