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Gauge Theory, Superstrings and Supermagnets

Gauge Theory, Superstrings and Supermagnets. Volker Schomerus SYSY Goettingen 2012. Prologue. Consider diffusion in a 1-dim system:. t > 0. t = 0. SO(2) symmetry. Rotational symmetry of 2-dim plane can appear in 1-dim physics!. Also SUSY can appear in non-SUSY world ….

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Gauge Theory, Superstrings and Supermagnets

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  1. Gauge Theory, Superstrings and Supermagnets Volker Schomerus SYSY Goettingen 2012

  2. Prologue Consider diffusion in a 1-dim system: t > 0 t = 0 SO(2) symmetry Rotational symmetry of 2-dim plane can appear in 1-dim physics! Also SUSY can appear in non-SUSY world …

  3. Quantum (Gauge) Field Theory Basic framework for the description of nature in hep & cond-mat Central challenge: Study of quantum effects ! → loops & legs, computer algebra, lattice gauge theory Fundamental DoF depend on scale/coupling quarks & gluons colorless hadrons QCD string ? failed

  4. Super-String Theory Candidate for a quantizable theory of gravity length scale in space-time GR is not quantizable Challenge 1: Space-time geom. for ℓs/R ~ 1? Stringy space-time differs from Einstein - Hilbert GR string length Possibly non-perturbative corrections in string length ℓs R Challenge 2: Quantize string geometry ↔ gs

  5. Gauge-String Correspondence SUSY D-dimensional quantum gauge theory ↔ Superstring in AdSD+1 N=4 SYM ↔ String in AdS5 x S5 ([‘t Hooft] →).... [Maldacena] ... λ = g2 Nc ↔ R/ℓs quantum GT string geometry λ/Nc ↔ gs Quantization of string geom solved by GT with finite Nc ?

  6. A Map of Physics [Polyakov, Maldacena] ls/R pert. String Theory pert. Gauge Theory Nc [‘t Hooft] gs λ

  7. Plan:

  8. N=4 SUSY YM Theory

  9. N=4 Super Yang-Mills Theory same on all length scales 6 matrix valued scalars → Conformal Quantum Field Theory Symmetries: U(4) ~ SO(6) R-symmetry and 4D conformal group SO(2,4) combine with 32 fermionic generators into Poincare, Dilations, Special Conformal Lie Supergroup PSU(2,2|4)

  10. 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!

  11. 6 gluon remainder function [Del Duca et al.] [Goncharov et al.] +  Spradlin’s talk xi± = xi±(ui) Lim - Poly-logs Lives on multi-sheeted cover of complex u-space Is there new (stringy) calculational scheme ?

  12. AdS Backgrounds

  13. Maldacena’sAdS/CFT duality 10 dimensional type IIB SUGRA possesses a solution with PSU(2,2|4) super-symmetry Conjecture:[Maldacena] N=4 SYM is dual to String theory on AdS5 x S5 x = (x0,..x3) line element on S5

  14. AdS/CFT duality Highly redshifted in center Strings soft  Mesons … Particle model at boundary Dictionary  Hubeny’s talk …... …...

  15. 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 only describes R at λ = ∞

  16. Supermagnets

  17. Magnets and Strings Particle Theory String Theory H = ∑ Li • Li+1 Δ = L • L Laplacian 1D Spin Chain 1D anisotropic spin ½ Heisenberg magnet: Continuum limit: Strings on S1 w. radius R SO(2) symmetry matches

  18. Supermagnets Magnets must have same symmetries as GT i.e. PSU(2,2|4) for N=4 SYM  Supermagnet Supermagnets known for many compact superspaces, But not for AdS yet [Mitev,Quella,VS,Saleur] Starting with work of Bethe, Onsager …... efficient techniques have been developed to compute quantities in (super-) magnets characterized by non-linear integral equations TBA

  19. Reformulation through TBA [Alday,Gaiotto,Maldacena] Area from nonlinear integral equations (NLIE): Kinematic data kernel fct K known ~ calculation of vacuum energy in 1D quantum systems not yet known Similar equations should determine R(λ,u) ~ determination of γcusp(λ) [Beisert, Eden, Staudacher]

  20. Summary Area of a bubble pulled by gravity SA ? Vacuum energy of supermagnet

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