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Heidelberg, June 2008 Volker Schomerus - DESY Hamburg -

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 ~ strings. But: String amplitudes are too soft (↔ extended nature) ↔experiment!

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Heidelberg, June 2008 Volker Schomerus - DESY Hamburg -

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  1. Of Mesons and Metals – Bethe & the 5th Dimension Heidelberg, June 2008 Volker Schomerus - DESY Hamburg -

  2. Introduction: Mesons Can meson physics be modeled by strings ? flux tubes ~ strings But: String amplitudes are too soft (↔ 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

  3. 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

  4. Bethe, Metals and Strings Particle Theory String Theory H = ∑ Li • Li+1 Δ = L • L Laplacian 1D Spin Chain e.g. spherically sym- metric problem in ED continuum limit non-linear σ model Solution method: (thermodynamic) Bethe Ansatz Spectrum from solutions of non-linear integral equations

  5. Plan: Part I Part II Part III

  6. Strings & Gauge Theory

  7. Closed strings and SUGRA Closed string theory in background X Infinite tower of vibrational modes M2 ~ n/ℓs String interaction through 3-vertex gs String length 2 String coupling -1 At low energies (E « ℓs ): massless closed str. modes behave like gravitons 10D SUGRA

  8. Solitonic & Dirichlet p-branes 10-dim SUGRA has solutions describing massive & charged objects localized on p+1 - dim. surfaces. D-branes are objects on which open strings can end [Polchinski] [black branes] Branes in string theory ?

  9. Open strings & gauge theory -1 At low energies (E«ℓs ): massless modes of the open string behave like gauge bosons + matter on the world-volume of D-branes. Aab p+1- dimensional gauge theory a .. b N

  10. Gauge-String theory dualities Depending on Δy one side simpler Δy << ℓs: gauge th ↔ closed string th Holography ←Δy → • Closed string in curved 10-dimensional space • Gauge theory loop  classical string theory ! High redshift ↔ soft strings ↔ meson resonances

  11. N=4 4D S Yang-Mills Strings in AdS5 x S5 Example: AdS5 / CFT4 duality Gauge theory on stack Strings in their near of Nc D3 - branes horizon geometry U(Nc) gauge field A; 6 scalars Φ …... …...

  12. Map of Physics [Polyakov, Maldacena] ls/R pert. String Theory pert. Gauge Theory Nc Lattice Gauge Theory [‘t Hooft] gs Quantum Gravity λ

  13. Strings & Spin Chains

  14. Particle vs. String Geometry Particle in S1 with radius R: momentum winding q = e-β String on S1: oscillations Does Z(q) encode spectrum of operator ?

  15. String Geometry & Magnets 1D anisotropic spin ½ Heisenberg magnet: HXXZ ↔String spectrum on S1 with radius R Pauli matrices L ∞ Spectrum of many 1D magnets is known ! Luttinger liquid [Bethe 31] ... Factorized scattering → Bethe Ansatz

  16. 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

  17. 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) Sym of N=4 SYM OR Supercoset σ-Models σ-Model on PSU(2,2|4)/SO(1,4) x SO(5) …

  18. Conclusion: Toolkit of String Geometry ~ systems of 2d stat mech .. is toolkit of 1d quantum systems Sigma Models Integrability Spin Chains affine algebras numerical studies non-linear integral equations Bethe Ansatz Yangians Yang-Baxter equation Successful recent applications to N=4 SYM theory: anomalous dimensions and gluon amplitudes all loops 0,1,∞ loops

  19. Scattering in N=4 SYM theory n-gluon Scattering Amplitude (MHV, color ordered,planar) p2 p3 t s cutoff coupling p4 p1 known Finite Remainder n-gluon SA depends on 3n-10 variables: s = (p1+p2)2 t = (p2+p3)2 BDS conjecture: [Bern et al.] Holds for n = 4 ! known from ST!

  20. Gluon Scattering in AdS gravity Gluon SA at strong coupling: Given by area of a 2D surface ending on the polygon P{pj} & pulled by gravity into AdS Kinematic data [Alday, Maldacena] Confirms n = 4 gluon BDS amplitude & new prediction for SA with n > 5 gluons ..but surface very hard to find

  21. 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

  22. Amplitudes 2012 If you want to see more come to … Organizers: R. Boels, G. Heinrich, J. Henn, P. Mastrolia J. Plefka, V. Schomerus Hamburg, Mar 5-9 2012

  23. Summary Strings in 5D for mesons Strings from magnets Gluon scattering from TBA

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