Gauge Theory, Superstrings Gauge Theory, Superstrings

and Supermagnetsand SupermagnetsVolker Schomerus

SYSY

Goettingen 2012

ProloguePrologue

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 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 TheorySuper-String Theory

Candidate for a quantizable theory of gravity GR is not quantizable

Challenge 1: Space-time geom. for ℓs/R ~ 1?

Stringy space-time differs

from Einstein - Hilbert GR

string length

length scale in space-time

Challenge 2: Quantize string geometry ↔ gs

R

Possibly non-perturbative

corrections in string length ℓs

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 ?

pert. Gauge Theory

pert. String TheoryNc

λ

ls/R

gs

[‘t Hooft]

[Polyakov, Maldacena]

A Map of PhysicsA Map of Physics

Plan:Plan:

N=4 SUSY YM TheoryN=4 SUSY YM Theory

N=4 Super Yang-Mills 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 theoryScattering in N=4 SYM theory

n-gluon Scattering Amplitude

(MHV, color ordered, planar)

p1

p2 p3

p4

n-gluon SA depends

on 3n-10 variables:

s = (p1+p2)2

t = (p2+p3)2

s tcutoff coupling

known Finite Remainder

BDS conjecture: [Bern et al.]

Holds for n = 4 ! known from ST!

6 gluon remainder function 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

xi± = xi

±(ui) Lim - Poly-logs Spradlin’s talk

AdS AdS BackgroundsBackgrounds

Maldacena’s AdS/CFT dualityMaldacena’s AdS/CFT duality

Conjecture: [Maldacena] N=4 SYM is dual to

String theory on AdS5 x S5

x = (x0,..x3)

line element on S5

10 dimensional type IIB SUGRA possesses

a solution with PSU(2,2|4) super-symmetry

AdS/CFT dualityAdS/CFT duality

par.: par.: λλ=g=g22YM YM NNc c ; N; Ncc R4 / ℓs

4=λ ; gs=λ/Nc

Gauge inv. operatorsGauge inv. operators Closed string states

Anomalous dimensionsAnomalous dimensions Mass of string mode

…... …...

Highly redshifted in center

Strings soft Mesons …

Particle model at boundary

Dictionary Hubeny’s talk

Gluon Scattering in AdS gravityGluon 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[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 Supermagnets

Particle Theory String Theory

Δ = L • L

Laplacian

H = ∑ Li • Li+1

1D Spin Chain

Magnets and StringsMagnets and Strings

Continuum limit: Strings on S1 w. radius R

1D anisotropic spin ½ Heisenberg magnet:

SO(2) symmetry matches

Supermagnets 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 TBAReformulation 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 γcusp(λ) [Beisert, Eden, Staudacher]

SummarySummary

SA ? Area of a bubble pulled by gravity

Vacuum energy of supermagnet