Gauge Theory, Superstrings and Supermagnets Volker Schomerus SYSY Goettingen 2012

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

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

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

Gauge-String Correspondence SUSY D-dimensional quantum gauge theory ↔ Superstring in AdS D+1 N=4 SYM ↔ String in AdS 5 x S 5 ([‘t Hooft] →).... [Maldacena]... λ = g 2 N c ↔ R/ℓ s quantum GT string geometry λ/N c ↔ g s Quantization of string geom solved by GT with finite N c ?

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

Plan:

N=4 SUSY YM Theory

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

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!

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

AdS Backgrounds

Maldacena’s AdS/CFT duality Conjecture: [Maldacena] N=4 SYM is dual to String theory on AdS 5 x S 5 x = (x 0,..x 3 ) line element on S 5 10 dimensional type IIB SUGRA possesses a solution with PSU(2,2|4) super-symmetry

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

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..but only describes R at λ = ∞ Kinematic data

Supermagnets

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

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

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

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