Transcript Bethe Ansatz in AdS/CFT: from local operators to classical
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< 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) p i k i 2 equations for n unknowns (n-2)-dimensional phase space Unless there are extra conservation laws! • No phase space: 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 • 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-vacuum2→2 scattering in 2d
n→n scattering
Integrability:
• No particle production (all 2→many processes are kinematically forbidden)
Large (long strings):