Bethe Ansatz and Integrability in AdS/CFT correspondence

Thanks to:

Niklas Beisert Johan Engquist Gabriele Ferretti Rainer Heise Vladimir Kazakov Thomas Klose Andrey Marshakov Tristan McLoughlin Joe Minahan Radu Roiban Kazuhiro Sakai Sakura Schäfer-Nameki Matthias Staudacher Arkady Tseytlin Marija Zamaklar Konstantin Zarembo (Uppsala U.) “Constituents, Fundamental Forces and Symmetries of the Universe”, Napoli, 9.10.2006

AdS/CFT correspondence

Yang-Mills theory with N=4 supersymmetry

Exact equivalence Maldacena’97 Gubser,Klebanov,Polyakov’98 Witten’98

String theory on AdS

5

xS

5

background

Large-N limit: Planar diagrams and strings time

AdS/CFT correspondence Maldacena’97 Gubser,Klebanov,Polyakov’98 Witten’98

Spectrum of SYM = String spectrum but

λ<<1 Quantum strings Classical strings Strong coupling in SYM

1 + 0 Strong-weak coupling interpolation Gubser,Klebanov,Tseytlin’98; … λ SYM perturbation theory String perturbation theory + + … Circular Wilson loop (exact): Erickson,Semenoff,Zarembo’00 Drukker,Gross’00 Minimal area law in AdS 5

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 Gliozzi,Scherk,Olive’77 Field content: Action: Global symmetry: PSU(2,2|4)

Spectrum Basis of primary operators: Spectrum = {Δ n } Dilatation operator (mixing matrix):

Local operators and spin chains related by SU(2) R-symmetry subgroup i i j j

Tree level: Δ=L (huge degeneracy) One loop:

One loop planar dilatation generator: Heisenberg Hamiltonian Minahan,Z.’02

Lax operator: Monodromy matrix: Transfer “matrix”: Integrability Faddeev et al.’70-80s

Infinite tower of conserved charges: U – lattice translation generator: U=e iP

Spectrum: with eigenvalues Algebraic Bethe Ansatz are eigenstates of the Hamiltonian Provided (anomalous dimension) (total momentum) Bethe equations

Strings in AdS 5 xS 5 Green-Schwarz-type coset sigma model on SU(2,2|4)/SO(4,1)xSO(5).

Metsaev,Tseytlin’98 Conformal gauge is problematic: no kinetic term for fermions, no holomorphic factorization for currents, … Light-cone gauge is OK.

The action is complicated, but the model is integrable!

Bena,Polchinski,Roiban’03

Consistent truncation String on S 3 x R 1 :

Gauge condition : Equations of motion: Zero-curvature representation: equivalent Zakharov,Mikhaikov’78

Conserved charges Generating function (quasimomentum): time on equations of motion

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<

2→2 scattering in 2d

p 1 k 1 p 2 Energy and momentum conservation: k 2

Energy conservation k 2 II I k 1 Momentum conservation I: k 1 =p 1 , k 2 =p 2 (transition) II: k 1 =p 2 , k 2 =p 1 (reflection)

n→n scattering

p i k i 2 equations for n unknowns (n-2)-dimensional phase space

Unless there are extra conservation laws!

Integrability:

• No phase space:

• No particle production (all 2→many processes are kinematically forbidden)

2 1 Factorization: Consistency condition (Yang-Baxter equation): = 2 3 1 3

Integrability + Locality Bethe ansatz Strategy: • find the dispersion relation (solve the one-body problem): • find the S-matrix (solve the two-body problem): Bethe equations • find the true ground state full spectrum

What are the scattering states?

SYM: magnons Staudacher’04 String theory: “giant magnons” Hofman,Maldacena’06 Common dispersion relation: S-matrix is highly constrained by symmetries Beisert’05

Algebraic BA: one-loop su(2) sector Minahan,Z.’02 Rapidity: Zero momentum (trace cyclicity) condition: Anomalous dimension:

Algebraic BA: one loop, complete spectrum Beisert,Staudacher’03

Nested

BAE: - Cartan matrix of PSU(2,2|4) - highest weight of the field representation

bound states of magnons – Bethe “strings” 0 mode numbers u

Sutherland’95; Beisert,Minahan,Staudacher,Z.’03 Semiclassical states

Scaling limit: defined on a set of conoturs C k complex plane in the x 0

Classical Bethe equations Normalization: Momentum condition: Anomalous dimension:

Algebraic BA: classical string Bethe equation su(2) sector: Kazakov,Marshakov,Minahan,Z.’04 Normalization: Momentum condition: String energy: General classical BAE are known and have the nested structure consistent with the PSU(2,2|4) symmetry of AdS 5 xS 5 superstring Beisert,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 Beisert’05 • The Bethe equations are asymptotic: they describe infinitely long strings / spin chains.

Schäfer-Nameki,Zamaklar,Z.’06

Testing BA: semiclassical string in AdS 3 xS 1 global time radial coordinate in AdS angle in AdS angle on S 5

winds k times and rotates Rigid string solution AdS 5 Arutyunov,Russo,Tseytlin’03 S 5 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

classical energy one loop correction string fluctuation frequencies Explicitly, Park,Tirziu,Tseytlin’05

Quantum-corrected Bethe equations classical BE Kazakov,Z.’04 Anomaly Kazakov’04;Beisert,Kazakov,Sakai,Z.’05 Beisert,Tseytlin,Z.’05; Schäfer-Nameki,Zamaklar,Z.’05 Quantum correction to the scattering phase Hernandez,Lopez’06

Comparison

Large (long strings):

• String • BA BA misses exponential terms Schäfer-Nameki,Zamaklar,Z.’05

Conclusions • Large-N SYM / string sigma-model on AdS 5 xS 5 probably solvable by Bethe ansatz are • 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