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

2. Introduction: Mesons Even perturbative physics difficult flux tubes ~ strings Can meson physics be modeled by strings ? Non-perturbative QCD physics remains to be understood → confinement, sQG plasma... But: String amplitudes are too soft (↔ extended nature) ↔experiment! QCD → description of perturbative regime through gauge theory (GT). Similar issues for SM of many body systems Loops & legs

3. The 5 th Dimension The 5 th Dimension ● Large N c exp. of GT↔String theory [‘t Hooft] ● Flux tube extension ↔ 5 th dim. ? [Polyakov] Maldacena ’97: Many strongly coupled 4D GT s 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. Particle Theory String Theory Δ = L L Laplacian H = ∑ L i L i+1 1D Spin Chain Bethe, Metals and Strings non-linear σ model continuum limit Solution method: (thermodynamic) Bethe Ansatz e.g. spherically sym- metric problem in ED 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 M 2 ~ n/ℓ s String interaction through 3-vertex g s String length At low energies (E « ℓ s ): massless closed str. modes behave like gravitons 10D SUGRA String coupling 2

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

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

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

11. Example: AdS 5 / CFT 4 duality N=4 4D S Yang-MillsStrings in AdS 5 x S 5 U(N c ) gauge field A; 6 scalars Φ Gauge theory on stack Strings in their near of N c D3 - branes horizon geometry par.: λ=g 2 YM N c ; N c par.: λ=g 2 YM N c ; N c R 4 / ℓ s 4 =λ ; g s =λ/N c Gauge inv. operators Gauge inv. operators Closed string states Anomalous dimensions Anomalous dimensions Mass of string mode …...

12. pert. Gauge Theory pert. String Theory NcNc λ l s /R gsgs Lattice Gauge Theory [‘t Hooft] [Polyakov, Maldacena] Map of Physics Quantum Gravity

13. Strings & Spin Chains

14. Particle vs. String Geometry Particle in S 1 with radius R: String on S 1: momentum winding oscillations q = e -β Does Z (q) encode spectrum of operator ?

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

16. String geometry & σ-models Σ XμXμ e.g. S 1 ↔ Spectrum of Hamiltonian of 1D Quantum Field Theory defined by the action: σ-model Strings ~ Magnets L→∞ ~ Sigma Models

17. 1D Systems for Gauge Theory Supermagnets Supercoset σ-Models Magnets must have same symmetries as GT Anisotropic Heisenberg chain has SO(2) sym OR σ-Model on PSU(2,2|4)/SO(1,4) x SO(5) … V rep of PSU(2,2|4) L → ∞ Sym of N=4 SYM

18. Conclusion: Toolkit of String Geometry.. is toolkit of 1d quantum systems ~ systems of 2d stat mech Sigma Models Bethe Ansatz Integrability Yangians Spin Chains affine algebras non-linear integral equations Yang-Baxter equation numerical studies 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) p1p1 p2p2 p3p3 p4p4 n-gluon SA depends on 3n-10 variables: s = (p 1 +p 2 ) 2 t = (p 2 +p 3 ) 2 s t cutoff coupling known Finite Remainder 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{p j } & pulled by gravity into AdS [Alday, Maldacena] Confirms n = 4 gluon BDS amplitude & new prediction for SA with n > 5 gluons Kinematic data..but surface very hard to find

21. Reformulation through TBA Thermodynamik Bubble Ansatz [Alday,Gaiotto,Maldacena] ~ calculation of vacuum energy in 1D quantum systems m,.. - parameters Y - density of part./holes A - energy Kinematic data kernel fct K known Area from nonlinear integral equations (NLIE):

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

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