35 th summer institute @ ENS Perturbative Integrability of large Matrix Theories August 9 th, 2005 Thomas Klose Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute), Potsdam, Germany Thanks for collaborations / discussions to Thomas Fischbacher, Nakwoo Kim and Jan Plefka Abishek Agarwal, Niklas Beisert and Matthias Staudacher [hep-th/0306054: SYM to PWMT] [hep-th/0310232: Integrability] [hep-th/0412331: 4-loop] [hep-th/0507217: SO(6)] The exploration of the AdS/CFT correspondence received a fresh impetus after J. Minahan and K. Zarembo found integrability properties in N=4 supersymmetric Yang-Mills theory three years ago. This integrability is described most conveniently, when the planar dilatation operator of this conformal theory is considered as a spin-chain Hamiltonian. Even though an enormous progress has been made in understanding and utilizing the uncovered integrable structures, their fundamental origin and the precise mechanism of symmetry enhancement in the planar limit is still unclear. In this talk, we review recent studies of quantum mechanical matrix theories and their contribution to a better insight into perturbative integrability. We derive two matrix models from super Yang-Mills theory and recall how the associated spin-chain Hamiltonians are determined. We investigate these spin-chain systems with respect to integrability and present new results.
Perturbative Integrability of large Matrix Theories 4-dim susy Yang-Mills truncation toy model Matrix Theory gauge invariant QM, mass planar limit energy shifts anomalous dimensions planar limit long-range integrable spin-chains Strings on AdS/CFT Comparison Integrable Structures
Perturbative Integrability of large Matrix Theories From SYM to Matrix Models ❶ (PWMT & SO(6)-model) 4-dim susy Yang-Mills truncation toy model Matrix Theory gauge invariant QM, mass energy shifts planar limit ❷ From Matrix Models to long-range Spin-Chains long-range integrable spin-chains ❹ Wrapping in Matrix Model language Compare Integrability ❸ of different Models
❶ From SYM to Matrix Models (Plane-wave matrix model & SO(6)-model)
❶ From SYM to Matrix Models Motivation from AdS/CFT 4d SYM AdS/CFT corresp. Strings on BMN limit Penrose limit BMN sector of 4d SYM BMN corresp. Strings on plane-wave Holographic principle wants: 1d theory 1d boundary M-Theory 10d SYM 4d SYM BFSS matrix model Plane-wave matrix model (parameter: radius of sphere, discrete spectrum, pert. theory)
❶ SYM on 4d SYM is conformally invariant Harmonic field-expansion on : are harmonic functions on Label harmonic functions by representations scalars fermions gauge field
❶ Truncations of SYM Mass 3 heavy bosons 8 fermions 6 light bosons SYM PWMT 17 SO(6)-model 6
❶ Classical Equivalence Original Action derive eq. of motion consistent truncation truncate Action of reduced theory find reduced eq. of motion SO(6)-model Plane-wave Matrix Theory (PWMT) M-Theory on plane-wave [Berenstein, Maldacena, Nastase 2002] DLCQ of supermembrane or ensemble of N D0-branes [Dasgupta, Sheikh-Jabbari, van Raamsdonk 2002]
❶ Quantization of PWMT Introduce modes creation/annihilation oscillators Notation 6 light bosons 6 1 1/2 (, ) 1/2 3 heavy bosons 1 3 1 (, ) 1 17 8 fermions 4 2 3/4 (, ) 3/4 Super-oscillator: Hamiltonian: Gauge invariant composite states:
❶ Quantum Equivalence!?! 4d SYM is conformally invariant Dilatations on gen. by SYM-Dilatation operator restricted to SYM-Dilataton operator MM Spectrum of conf. dim s?? Time-translations on gen. by SYM-Hamiltonian truncated to Matrix Model Hamiltonian Spectrum of energies Free theories: Scalars Field strength Fermions Light bosons Heavy bosons Fermions
❷ From Matrix Models to long-range Spin-Chains large N = large N
❷ Indication / Meaning of Integrability Look at energy spectrum Energy [Beisert, Kristjansen, Staudacher 2003] Degeneracies in planar limit between parity pairs Conserved charge: Bethe equation: Planar limit
❷ Why Perturbation Theory? Ideally: Don t know how to take the planar limit if there are: pure annihilation operators pure creation operators Get rid of them in perturbation theory Construct with i.e. preserves free energy no pure annihilators / pure creators Actually this is yet more like dilatation operator:
❷ Perturbation Theory Task: Computation of interacting energy spectrum Ordinary quantum mechanical perturbation theory for small Textbook-approach: Concentrate on some free energy Pick basis for states of free energy and orthogonal basis Find and from matrix elements of in basis
❷ Perturbation Theory Task: Computation of interacting energy spectrum Ordinary quantum mechanical perturbation theory for small Here: Pre-diagonalisation: Define such that = eigenvalues of, = eigenvectors of This procedure decouples mixing of states within degenerate subspace admixture of states from outside this subspace Definition of Energy Operator without reference to state
❷ Matrix Theory Energy Operator For Hamiltonian of the form, we have Perturbative expansion: where we say: with Projector onto subspace of states with Propagator
❷ Matrix Theory Energy Operator Feynman-Graphs in SU(N)-space Computation of Application of Part of out-state Attach to in-state
❷ Planar Limit Planar limit after pre-diagonalization and normal-ordering Discard most pieces, for instance: Only keep these: Define planar operators: Examples: identity permutation trace
❷ Energy Operator Spin-Chain Ham. One-loop results These operators define integrable spin-chains systems! Include higher orders obtain long-range spin-chain Eigenvalues Energy operator Spin-chain Hamiltonian
❸ Compare Integrability of different Models PWMT SYM Inozemtsev SO(6)-model AFS BDS
❸ Integrability Results SYM Don t know any of PWMT SO(6) Limited order in perturbation theory su(3 2) su(2) Limited to specific subsectors Out -Space Interactions In -Space
❸ Integrability Results SYM Model Order Integrability PWMT SO(6) su(3 2) su(2) SYM PWMT SO(6)-model 1-loop YBE [Beisert, Staucher 2003] 3-loop 1-loop YBE [Kim, TK, Plefka 2003] 3-loop [TK, Plefka 2003] 4-loop [Fischbacher, TK, Plefka 2004] 2-loop 1-loop 3-loop 2-loop YBE [TK, Plefka 2003] 4-loop [TK 2005] [Beisert 2003] [Eden, Jarczak, Sokachev 2004] No invariant long-range integrable spin-chain from Matrix Model with at most 4-valent vertices
❸ su(2) Bethe ansätze SYM Inozemtsev BDS AFS PWMT SO(6)-model know to 3-loop map-able to SYM, no BMN scaling at 4-loop all-loop proposal for SYM, to account for deviations in near-bmn limit computed to 4-loop, as BDS as AFS but 4-loop violation of BMN scaling up to 3-loop qualitatively similar no BMN scaling [Beisert 2003] [Serban, Staudacher 2004] [Beisert, Dippel, Staudacher, 2004] [Arutyunov, Frolov, Staudacher 2004] [Fischbacher, TK, Plefka 2003] [TK 2005]
❹ Wrapping in Matrix Model language growing interaction range loop order fixed spin-chain length
❹ The Wrapping Problem or How to take the planar limit of a matrix model operator? Planar limit at the level of eigenvalues is no problem: but at the level of operators: = sample 4-loop interaction generically non-planar but sometimes planar length of states Workaround: 2 3 4 5 6 planar asymptotic operator + extra data for short states loop order < length of state loop order 1 2 3 4 5 0 0 0 0 0 0 0
❹ The Wrapping Problem Determine full planar operator from its asymptotic piece Naive ideas: 1 2 3 4 5 5 1 2 3 sample 4-loop interaction discard wrap around 4 Periodization à la Inozemtsev spin-chain asymptotic: L-periodic: with So far unsuccessful, but data available to test further ideas
❹ Wrapping and Integrability Asymptotic regime: Existence of local very non-trivial: Freedom in << dimension of Wrapping regime: is non-local, commuting operators exists always
❹ Wrapping in Matrix Model language Wrapping interactions do not act locally, but replace the entire state. Recall planar operators: with Define wrapping operators: Full planar (bosonic) algebra: [Lee, Rajeev 1999]
Summary and Conclusions Matrix Models Super Yang-Mills long-range integrable spin-chains Long-range integrable Spin-chains from planar Matrix Models E.g. Plane-wave Matrix Theory, being closely related to SYM agree * at planar level * in subsector * up to 3-loop Toy models for integrability in AdS/CFT Possibility to explicitly study wrapping