gauge theory, superstrings and supermagnets volker schomerus sysy goettingen 2012
TRANSCRIPT
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