Worldsheet Scattering in AdS5xS5

1
World-sheet Scattering in AdS5xS5

• Konstantin Zarembo
• (Uppsala U.)

T.Klose, T.McLoughlin, R.Roiban, K.Z.,
hep-th/0611169 T.Klose, K.Z., hep-th/0701240 T.Klo
se, T.McLoughlin, J.Minahan, K.Z., 0704.3891
Integrability in Gauge and String Theory, Paris,
21.06.07
2
AdS/CFT correspondence
Maldacena97
Gubser,Klebanov,Polyakov98 Witten98
3

• Spectral problem in large-N SYM
• can be reformulated in terms of an integrable
  spin chain
• with unknown Hamiltonian (beyond O(?))
• but with exactly known S-matrix!

Minahan,Z.02 Beisert,Kristjansen,Staudacher03 Be
isert,Staudacher03
Beisert05 Beisert,Eden,Staudacher06 Beisert,Hern
andez,Lopez06
4

• Sigma-model in AdS5xS5
• well-defined theory with known (although
  complicated) Lagrangian
• also integrabl
• How does the (discrete) spin chain arise from the
  (continuous) world-sheet?

Bena,Polchinski,Roiban03
Goal compute the S-matrix on the world-sheet
explicitly to few first orders in
perturbation theory
5
S5
AdS5
6
The metric
S5
AdS5
Transverse coordinates
RR flux
7
Sigma-model
Green-Schwarz-type coset PSU(2,24)/SO(4,1)xSO(5)
Metsaev,Tseytlin98
Sigma-model coupling constant
Classical limit is
8
RR flux requires manifest space-time supersymmetry
Fermion part of the sigma-model is of
Green-Schwarz type.

• Conformal gauge is problematic
• no kinetic term for fermions
• no holomorphic factorization for currents,

Standard CFT does not work
9
Kinetic term for fermions
If Xconst, ?a0 and the kinetic term
vanishes.
Need to fix the light-cone gauge Xt (then
?0G and the kin. term is not degenerate)
10
Gauge fixing
Light-like geodesics
angular momentum
Berenstein,Maldacen,Nastase02 Parnachev,Ryzhov02
Callan,Lee,McLoughlin,Schwarz,Swanson,Wu03 Aruty
unov,Frolov,Plefka,Zamaklar06
11
Unbroken bosonic subgroup
Taking into account supersymmetry
This also gets central extension with
Beisert05 Arutyunov,Frolov,Plefka,Zamaklar06
p world-sheet momentum
12
World-sheet fields
World-sheet fields (embedding coordinates in
AdS5xS5)
global time in AdS5 and angle on S5 eliminated
by gauge fixing
Fields form (22)x(22) multiplet of PSU(22)2
13
How to fix the gauge?
Step 1 T-duality
Arutyunov,Frolov,Zamaklar06
14
T-dual background
Metric
B-field
15
Step2 Nambu-Goto Lagrangian
Step3 Gauge condition
Step 4 Rescaling
16
Gauge-fixed Lagrangian (the bosonic part)
17
Gauge-fixed action
Loop counting parameter
Internal string length
Decompatification (infinite string) limit
18

• Static gauge-fixed T-dual Nambu-Goto string
• potenially problematic non-renormalizable field
  theory
• however in flat space perturbative calculcations
  in Nambu-Goto formalism (Lüscher,Symanzik,Weisz80
  Dietz,Filk83 Caselle,Hasenbusch,Panero05)
  agree, upon ?-function regularization, with
  conformal-gauge results (Olesen85)

(e.g. at two loops Roiban,Tirziu,Tseytlin07)
19
Bosonic Lagrangian in AdS5xS5 (up to quartic
order in fields)

• Massive, integrable 2d field theory
• Lorentz-invariant kinetic terms
• Lorentz invariance is broken by interactions
• Gauge-dependent a1/2 (pure l.c. gauge) is the
  simplest case

20
Spectrum
(88) massive states
Loop corrections should give
exact Beisert,Dippel,Staudacher04Beisert05
two-loop correction!
21
World-sheet scattering
S-matrix
Integrability (consistency with Yang-Baxter
equations) requires
SU(2)4 invariace
22
Beisert05
up to an overall phase
23
Symmetries of the S-matrix
Bosonic (su(4)4) generators
Supersymmetries
Spin chain explanation the chain is dynamical,
its length may decrease on increase in the
process of scattering
Beisert05
24
Hopf algebra
Co-product
promotes symmetries of the S-matrix to
non-trivial Hopf algebra
Gomez,Hernandez06 Plefka,Spill,Torrielli06
graded permutation
25
String explanation 1 non-locality of supercharges
Non-locality of supercurrents
local operator
Arutyunov,Frolov,Plefka,Zamaklar06
non-local
By contour manipulations
Klose,McLoughlin,Roiban,Z.06
non-trivial co-product
26
String explanation 2 fractional statistics
Hofman,Maldacena06
Fermion number
In principle, can be checked by summing over
known fermion modes Minahan06
Papathanasiou,Spradlin07 of the giant magnon
Jackiw,Rebbi76 Goldstone,Wilczek81
Niemi,Semenoff86
27
(Fermi statistics)
Bernard,Leclair90 Fendley,Intriligator91
and then
28
Tree level
Tree-level factorization
29
Scattering of bosons at tree level
Ym(p)Yn(p) ? Yk(p)Yl(p)
30
SO(4) ? SU(2)xSU(2)
31
Explicit calculation yields
32
Dressing phase
Beisert,Hernandez,Lopez06 Beisert,Eden,Staudacher
06
Dorey,Hofman,Maldacena07
To leading order in
agrees with tree-level scattering amplitudes in
the sigma-model
33
Near-flat limit
Maldacena,Swanson06
perturbative string modes
giant magnons
Hofman,Maldacena06
Left-moving sector
34
Reduced Sigma Model
Consistent truncation of the full sigma-model
Maldacena,Swanson06
- chiral SO(8) spinor
35
Dispersion relation
Exact dispersion relation
Beisert,Dippel,Staudacher04
At ??8
In the near-flat limit
(exact)
36
Tadpole cancellation
0

Yi (S5)
Zi (AdS5)
0
(fermions)
The same mechanism renders the model finite to
any order in perturbation theory
37
Mass renormalization

Mass-shell condition
38
S-matrix
One loop
Two loops
mass renormalization two-loop corrections to
Z-factors
39
Tree-level amplitude
One-loop correction
Klose,Z.07
Two loops
Klose,McLoughlin,Minahan,Z.07
Agrees!
40
Lorentz-invariant form of the S-matrix
All Lorentz non-invariance of amplitudes in the
reduced model can be hidden in the
momentum-dependent redefinition of the coupling
In the full s-model/spin chain, LI is possibly
q-deformed rather than violated
Gomez,Hernandez06 Young07
41
Conclusions

• At strong coupling, the SU(22)xSU(22) S-matrix
  describes the scattering of the string modes in
  the light-cone gauge
• Perturbative calculations of scattering
  amplitudes agree with the conjectured BHL/BES
  phase up to two loops at strong coupling (and to
  four loops at week coupling Bern,Czakon,Dixon,Koso
  wer,Smirnov06)
• Ultimately we want to quantize the s-model with
  periodic bounary conditions, when S-matrix is not
  well defined.
• S-matrix ? asymptotic Bethe ansatz
• Exact Bethe ansatz?