Gauge/String Duality and Integrable Systems Kostya Zarembo (Uppsala U.) Annual Theory Meeting, Durham, 18.12.2007

Yang-Mills fields = closed strings “rings of glue” or maybe ≈ Polyakov’80

string carries constant Coulomb field vs. Flux tube (QED) (QCD) q q string carries constant energy per unit length

More refined lattice simulations confirm that the string fluctuates. From Bali, Phys. Rep. 343 (2001) 1 More refined lattice simulations confirm that the string fluctuates. Caselle,Fiore,Gliozzi,Hasenbusch,Provero’97; Caselle,Pepe,Rago’04; …

“Index conservation law”: Large-N expansion ‘t Hooft’74 “Index conservation law”:

N N N N Two expansion parameters: ‘t Hooft coupling: string coupling:

Planar diagrams and strings Large-N limit: time

Conceptual problems: Closed strings describe gravity. What is graviton in YM? String theory is only consistent in ten dimensions. How does the string remember that it is made of gluons?

Large-N expansion of YM theory String theory Early examples: 2d QCD Matrix models ‘t Hooft’74 Brezin,Itzykson,Parisi,Zuber’78 4d gauge/string duality: AdS/CFT correspondence Maldacena’97

Bound states in QFT (mesons, glueballs) String states String states Local operators Resolves many puzzles of putative large-N string: graviton is not a massless glueball, but is the dual of the energy-momentum tensor Tμν extra dimensions are geometric images of the energy scale and of global symmetries sum rules are automatic

AdS/CFT correspondence Yang-Mills theory with N=4 supersymmetry Maldacena’97 Gubser,Klebanov,Polyakov’98 Witten’98 Exact equivalence String theory on AdS5xS5 background

Anti-de-Sitter space (AdS5) z 5D bulk strings gauge fields 4D boundary

N=4 Supersymmetric Yang-Mills Theory Gliozzi,Scherk,Olive’77 Brink,Schwarz,Scherk’77 Field content: Action: conformal field theory

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

Strong-weak coupling interpolation λ SYM perturbation theory String perturbation theory 1 + + + … Circular Wilson loop (exact): Erickson,Semenoff,Zarembo’00 Drukker,Gross’00 Minimal area law in AdS5

Correlation functions Dilatation operator: matrix of anomalous dimensions

Operators Protected Non-degenerate Degenerate (mixes with ) energy-momentum tensor Konishi operator (mixes with )

Local operators and spin chains j i j

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

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

The spectrum Ground state: Excited states (magnons): BMN (Berenstein,Maldacena,Natase’02) operators

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

Assumptions: Bethe equations R<<L particles can only exchange momenta no inelastic processes integrability!

Exact periodicity condition: momentum scattering phase shifts periodicity of wave function Exact periodicity condition:

Bethe equations for Heisenberg model Rapidity: Bethe’31 Anomalous dimension:

Sigma-model in AdS5xS5 AdS5 S5 Classical limit Metsaev,Tseytlin’98 AdS5 S5 Sigma-model coupling constant: Classical limit is

φ yi’ zi t S5 AdS5

Light-like geodesics: Gauge fixing Light-like geodesics: gauge condition: Berenstein,Maldacena,Nastase’02 Parnachev,Ryzhov’02 Callan,Lee,McLoughlin,Schwarz,Swanson,Wu’03 Arutyunov,Frolov,Plefka,Zamaklar’06

Bosonic Lagrangian in AdS5xS5 (up to quartic order in fields): Massive, integrable 2d field theory Lorentz-invariant kinetic terms Lorentz invariance is broken by interactions Gauge-dependent: a=1/2 (pure l.c. gauge) is the simplest case

Full asymptotic BA Beisert,Staudacher’05

Wrapping/finite size effects Ambjørn,Janik,Kristjansen’05 Weak coupling: wrapping order captured by ABA Beisert,Kristjansen,Staudacher’03

since length of the string = and mass gap = 1 in light-cone gauge Strong coupling: Schäfer-Nameki,Zamaklar,Z.’06 since length of the string = and mass gap = 1 in light-cone gauge asymptotic exact

Large-spin twist-2 operators where Beisert,Eden,Staudacher’06

Weak coupling: Strong coupling: Beisert,Eden,Staudacher’06 Confirmed by explicit four-loop calculations in SYM Bern,Czakon,Dixon,Kosower,Smirnov’06; Cachazo,Spradlin,Volovich’06 Strong coupling: Basso,Korchemsky,Kotański’07 Confirmed by explicit two-loop calculations in string theory Roiban,Tseytlin’07; Klose,Minahan,McLoughlin,Z.’07

Conclusions Large-N SYM is a solvable (and to certain degree solved) 4d theory. The solution is not simple! Finite-size/wrapping effects (Asymptotic Bethe ansatz => Thermodynamic Bethe Ansatz?) World-sheet sigma-model (continuous field theory) <=> spin chain (discrete system). How does it work? (Giant magnons? Hofman, Maldecena’06; Arutyunov,Frolov,Zamaklar’06) What can we learn from solving AdS/CFT exactly?