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Of Mesons and Metals – Bethe & the 5th Dimension Heidelberg, June 2008 Volker Schomerus - DESY Hamburg - Introduction: Mesons Can meson physics be modeled by strings ? flux tubes But: String amplitudes are too soft ~ strings (↔ extended nature) ↔experiment! QCD → description of perturbative regime through gauge theory (GT). Non-perturbative QCD physics remains to be understood → confinement, sQG plasma ... Even perturbative physics difficult Loops & legs Similar issues for SM of many body systems The 5th Dimension ● Large Nc exp. of GT↔String theory [‘t Hooft] ● Flux tube extension ↔ 5th dim. ? [Polyakov] Maldacena ’97: Many strongly coupled 4D GTs possess a weakly coupled description through strings in a curved 5D geometry. compare high-T low-T dualities Geometrization of Quantum Physics String theoretic descriptions for non-relativistic 3D systems are being investigated crit cold atoms strange metals Bethe, Metals and Strings Particle Theory String Theory Δ=L•L H = ∑ Li • Li+1 Laplacian 1D Spin Chain e.g. spherically symmetric problem in ED continuum limit non-linear σ model Solution method: (thermodynamic) Bethe Ansatz Spectrum from solutions of non-linear integral equations Plan: Part I Part II Part III Strings & Gauge Theory Closed strings and SUGRA String length Closed string theory in background X Infinite tower of vibrational modes M2 String interaction through 3-vertex gs ~ 2 n/ℓs String coupling -1 At low energies (E « ℓs ): massless closed str. modes behave like gravitons 10D SUGRA Solitonic & Dirichlet p-branes 10-dim SUGRA has solutions describing massive & charged objects localized on p+1 - dim. surfaces. D-branes are objects on [black branes] which open strings can Branes in string theory ? end [Polchinski] Open strings & gauge theory -1 Aab At low energies (E«ℓs ): massless modes of the open string behave like gauge bosons + matter on the world-volume of D-branes. a .. b N p+1- dimensional gauge theory Gauge-String theory dualities Depending on Δy one side simpler Δy << ℓs: gauge th ↔ closed string th ← Δy → Holography • Closed string in curved 10-dimensional space • Gauge theory loop classical string theory ! High redshift ↔ soft strings ↔ meson resonances Example: AdS5 / CFT4 duality Gauge theory on stack Strings in their near of Nc D3 - branes horizon geometry N=4 4D S Yang-Mills Strings in AdS5 x S5 U(Nc) gauge field A; 6 scalars Φ par.: λ=g2YM Nc ; Nc R4 / ℓs4=λ ; gs=λ/Nc Gauge inv. operators Closed string states Anomalous dimensions Mass of string mode …... …... Map of Physics ls/R Nc pert. [‘t Hooft] Gauge Theory [Polyakov, Maldacena] pert. String Theory Lattice Gauge Theory gs Quantum Gravity λ Strings & Spin Chains Particle vs. String Geometry Particle in S1 with radius R: String on S1: q = e-β momentum winding oscillations Does Z(q) encode spectrum of operator ? String Geometry & Magnets 1D anisotropic spin ½ Heisenberg magnet: Pauli matrices L ∞ HXXZ ↔ String spectrum on S1 with radius R Luttinger liquid Spectrum of many 1D magnets is known ! Factorized scattering → Bethe Ansatz [Bethe 31] ... String geometry & σ-models ↔ Spectrum of Hamiltonian of 1D Quantum Field Theory defined by the action: σ-model Σ Xμ e.g. S1 Strings ~ MagnetsL→∞ ~ Sigma Models 1D Systems for Gauge Theory Magnets must have same symmetries as GT Anisotropic Heisenberg chain has SO(2) sym Supermagnets L→∞ V rep of PSU(2,2|4) OR Sym of N=4 SYM Supercoset σ-Models σ-Model on PSU(2,2|4)/SO(1,4) x SO(5) … Conclusion: Toolkit of String Geometry ~ systems of 2d stat mech .. is toolkit of 1d quantum systems Successful recent applications to N=4 SYM theory: anomalous dimensions and all loops gluon amplitudes 0,1,∞ loops Scattering in N=4 SYM theory p2 s p3 (MHV, color ordered,planar) t p4 p1 n-gluon Scattering Amplitude n-gluon SA depends cutoff known coupling Finite Remainder on 3n-10 variables: BDS conjecture: [Bern et al.] s = (p1+p2)2 t = (p2+p3)2 Holds for n = 4 ! known from ST! Gluon Scattering in AdS gravity Gluon SA at strong coupling: Kinematic data Given by area of a 2D surface ending on the polygon P{pj} & pulled by gravity into AdS [Alday, Maldacena] Confirms n = 4 gluon BDS amplitude & new prediction for SA with n > 5 gluons ..but surface very hard to find Reformulation through TBA Thermodynamik Bubble Ansatz [Alday,Gaiotto,Maldacena] Area from nonlinear integral equations (NLIE): Kinematic data kernel fct K known ~ calculation of vacuum energy in 1D quantum systems m, .. - parameters Y - density of part./holes A - energy If you want to see more come to … Amplitudes 2012 Hamburg, Mar 5-9 2012 Organizers: R. Boels, G. Heinrich, J. Henn, P. Mastrolia J. Plefka, V. Schomerus Summary Strings in 5D Strings from for mesons magnets Gluon scattering from TBA