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Of Mesons and Metals –
Bethe & the 5th Dimension
Heidelberg, June 2008
Volker Schomerus
- DESY Hamburg -
Introduction: Mesons
Can meson physics be modeled by strings ?
flux tubes
But: String amplitudes are too soft
~ strings
(↔ extended nature) ↔experiment!
QCD → description of perturbative
regime through gauge theory (GT).
Non-perturbative QCD physics remains to be
understood → confinement, sQG plasma ...
Even perturbative physics difficult Loops & legs
Similar issues for SM of many body systems
The 5th Dimension
● Large Nc exp. of GT↔String theory [‘t Hooft]
● Flux tube extension ↔ 5th dim. ? [Polyakov]
Maldacena ’97: Many strongly coupled 4D GTs
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
Bethe, Metals and Strings
Particle Theory
String Theory
Δ=L•L
H = ∑ Li • Li+1
Laplacian
1D Spin Chain
e.g. spherically symmetric problem in ED
continuum limit
non-linear σ model
Solution method: (thermodynamic) Bethe Ansatz
Spectrum from solutions of non-linear integral equations
Plan:
Part I
Part II
Part III
Strings & Gauge Theory
Closed strings and SUGRA
String
length
Closed string theory in background X
Infinite tower of vibrational modes
M2
String interaction through 3-vertex
gs
~
2
n/ℓs
String
coupling
-1
At low energies (E « ℓs ): massless closed str.
modes behave like gravitons
10D SUGRA
Solitonic & Dirichlet p-branes
10-dim SUGRA has
solutions describing
massive & charged
objects localized on
p+1 - dim. surfaces.
D-branes are objects on
[black branes]
which open strings can
Branes in string theory ?
end [Polchinski]
Open strings & gauge theory
-1
Aab
At low energies (E«ℓs ):
massless modes of the
open string behave like
gauge bosons + matter
on the world-volume of
D-branes.
a .. b
N
p+1- dimensional
gauge theory
Gauge-String theory dualities
Depending on Δy
one side simpler
Δy << ℓs: gauge th
↔ closed string th
← Δy →
Holography
• Closed string in curved 10-dimensional space
• Gauge theory loop  classical string theory !
High redshift ↔ soft strings ↔ meson resonances
Example: AdS5 / CFT4 duality
Gauge theory on stack
Strings in their near
of Nc D3 - branes
horizon geometry
N=4 4D S Yang-Mills
Strings in AdS5 x S5
U(Nc) gauge field A; 6 scalars Φ
par.: λ=g2YM Nc ; Nc
R4 / ℓs4=λ ; gs=λ/Nc
Gauge inv. operators
Closed string states
Anomalous dimensions
Mass of string mode
…...
…...
Map of Physics
ls/R
Nc
pert.
[‘t Hooft] Gauge
Theory
[Polyakov, Maldacena]
pert. String Theory
Lattice
Gauge Theory
gs
Quantum Gravity
λ
Strings & Spin Chains
Particle vs. String Geometry
Particle in S1 with radius R:
String on
S1:
q = e-β
momentum
winding
oscillations
Does Z(q) encode spectrum of operator ?
String Geometry & Magnets
1D anisotropic spin ½ Heisenberg magnet:
Pauli matrices
L ∞
HXXZ ↔ String spectrum on S1 with radius R
Luttinger liquid
Spectrum of many 1D magnets is known !
Factorized scattering → Bethe Ansatz [Bethe 31] ...
String geometry & σ-models
↔ Spectrum of Hamiltonian of 1D Quantum
Field Theory defined by the action: σ-model
Σ
Xμ
e.g. S1
Strings ~ MagnetsL→∞ ~ Sigma Models
1D Systems for Gauge Theory
Magnets must have same symmetries as GT
Anisotropic Heisenberg chain has SO(2) sym
Supermagnets
L→∞
V rep of PSU(2,2|4)
OR
Sym of N=4 SYM
Supercoset σ-Models
σ-Model on PSU(2,2|4)/SO(1,4) x SO(5) …
Conclusion: Toolkit of String Geometry
~ systems of
2d stat mech
.. is toolkit of 1d quantum
systems
Successful recent applications to N=4 SYM theory:
anomalous dimensions and
all loops
gluon amplitudes
0,1,∞ loops
Scattering in N=4 SYM theory
p2
s
p3
(MHV, color ordered,planar)
t
p4
p1
n-gluon Scattering Amplitude
n-gluon SA depends
cutoff
known
coupling
Finite Remainder
on 3n-10 variables: BDS conjecture: [Bern et al.]
s = (p1+p2)2
t = (p2+p3)2
Holds for n = 4 !
known from ST!
Gluon Scattering in AdS gravity
Gluon SA at strong coupling:
Kinematic data
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 surface very hard to find
Reformulation through TBA
Thermodynamik Bubble Ansatz [Alday,Gaiotto,Maldacena]
Area from nonlinear integral equations (NLIE):
Kinematic data
kernel fct K known
~ calculation of vacuum energy in 1D quantum systems
m, .. - parameters Y - density of part./holes A - energy
If you want to see more come to …
Amplitudes 2012
Hamburg, Mar 5-9 2012
Organizers:
R. Boels, G. Heinrich,
J. Henn, P. Mastrolia
J. Plefka, V. Schomerus
Summary
Strings in 5D
Strings from
for mesons
magnets
Gluon scattering from TBA