Exact S-matrix for N=6 CS

1
Exact S-matrix for N6 CS
Changrim Ahn (Ewha Womans Univ.)
Work with Rafael Nepomechie (Miami) JHEP 0809,
010 a work in preparation
2
Summary of crazy June July
3
contents

• Introduction and motivation
• Perturbative integrability of N6 CS
• Non-perturbative integrability of N6 CS
• Conclusion

4
Why S-matrix?
Lessons from N4 SYM
5
2d integrable world

• Lattice model
• Spin chains
• YBE to define integrable model
• Bethe ansatz to find energy eigenvalues
• Difficult to get others
• (ex) XXX1/2

• Quantum field theory
• Symmetry
• Integrability from conserved charges
• Factorizable S-matrix
• YBE or symmetry to solve the model
• finite-size effects, correlation functions
  possible
• (ex) sine-Gordon model

6
Integrability in AdS/CFT

• Truly (quantum and quantitative), nonperturbative
  SYM in 4 dimensions
• Accurate agreements with 4-loop SYM perturbation
  theory and with 2-loop string world-sheet
  perturbation
• Rigorous realization and proof of the
  gauge/gravity duality
• Relations to other theoretical models and
  phenomena
• Hubbard model, O(6) sigma model, etc.
• Gluon amplitude, high energy scattering, etc

7
Still there is a problem left

• In addition to somewhat conceptual problems
• Hamiltonian for the all-loop BAE
• Relation to N4 SYM
• Quantization of the superstring theory
• After a lot of amazing successes, there is one
  most difficult problem left
• Wrapping
  problem
• When the length is shorter than the order of the
  perturbative expansion
• ? asymptotic Bethe ansatz nonperturbative
  results valid only when the length is infinite

8
S-matrix approach

• Symmetries spectrum determine S-matrix
• Staudacher, Beisert, Arutyunov-Frolov-Zamakl
  ar
• Crossing relation for overall dressing phase
  Janik
• Applications of S-matrix
• True meaning of asymptotic Bethe ansatz eq.
• Solution of Wrapping problem cf. Romuald
  Janiks talk
• Finite-size corrections from S-matrix
• Checked for large coupling limit (classical
  string computation)
• Checked for small coupling limit (perturbative
  SYM computation with wrapping interaction)

9
S-matrix program
10
N6 Chern-Simons Theory
Aharony, Bergman, Jafferis, Maldacena (ABJM)
11
Lagrangian and fields

• Fields
• AdS4/CFT3 N6 CS with SU(N) x SU(N) at level
  (k,-k) is dual to Type IIA superstring on AdS4 x
  CP3 as N,k ?8
• Parameter relations t Hooft coupling

12
Composite Fields

• BPS vacuum state (alternating form) (cf)
  near BPS Nishioka-Takayanagi
• Excited (Non-BPS) states
• Operator mixing under RG evolution

13
Perturbative Integrability
14
Leading two-loop (Minahan-Zarembo)

• Perturbative Feynman computation
  (cf) Dongsu Baks talk (w/ Rey)
• Given by SU(4) (alternating) integrable spin
  chain for scalar sector
• Exactly solvable by Bethe ansatz equations (BAE)

15
two-loop with fermions (Conjecture)

• BAE for full sector of Osp(2,26)
  Minahan-Zarembo

16
two-loop with fermions (Conjecture)

• Another (but equivalent) BAE
  Minahan-Zarembo v.3 only

17
Classical Integrability

• Type IIA superstring on AdS4 x CP3 as a coset
  sigma (integrable) model
• Arutyunov-Frolov, Stefanski
• Algebraic curve Gromov-Vieira

18
All-loop Conjecture (Gromov-Vieira)

• Combining the algebraic curve with two-loop BAE

19
Controversies over the conjecture

• Semi-classical quantization of spinning folded
  string on AdS3 x S1
• summing over quadratic fluctuations
  around classical configuration
• McLoughlin-Roiban, Alday-Arutyunov-Bykov,
  Krishnan
• All-loop BAE sl(2) sector
• Matter of summation of zero-point energy?
  Gromov-Mikhaylov

20
Non-Perturbative Integrabilitybased onS-matrix
21
S-matrix approach

1. Symmetries
2. Particle Spectrum
3. S-matrix from Symmetries
4. Dressing phase from Crossing and Unitarity
5. Minimality assumption

22
Factorizable S-matrix

• Integrability elastic S-matrix
• Quantum numbers are changed
• Factorizability Yang-Baxter Eq.

23
Excitation spectrum (Gaiotto-Giombi-Yin)

• BPS vacuum state
• Excited particle states
• SU(22) SU(22) symmetry for the particle
  states

24
S-matrix I. (Ahn-Nepomechie)

• A, B particles are charge conjugate of each other
• Reflectionless for AB scattering (assumption)
• Commutativity with SU(22) symmetry

25
SU(22) S-matrix (Beisert)
Yang-Baxter equation Arutyunov-Frolov-Zamaklar
26
Dressing phases

• Only difference is in the dressing phases
• Crossing relation
• Unitarity
• Solution upto CDD ambiguity

27
Derivation of all-loop BAE

• Periodic boundary condition
• Yang equation in terms of transfer matrix

28
Algebraic method (Martins-Melo)

• Diagonalization
• Match exactly with Gromov-Vieiras all-loop BAE
  conjecture

29
S-matrix II

• Most general case
• AB form a SU(2) doublet
• Symmetry SU(2) x SU(22)
• Yang-Baxter eq.

30
All-loop BAE II

• Yang equation
• Diagonalization of transfer matrices
• Similar structure of the second Minahan-Zarembo
  (two-loop) BAE!
• But we checked that they do not coincide in the
  limit of ? ? 0 limit

31
Concluding remarks

• Works to be done
• Finite-size effects in classical limit ? ? 8 and
  compare with classical string computation on AdS4
  x CP3
• Grignani-Harmark-Orselli-Semenoff,
  Bozhilov-CA, Astolfi-Puletti-Grignani-Harmark-Orse
  lli, Shenderovich, Lee-Panigrahi-Park,
  Bozhilov-Rashikov-CA
• Wrapping computation in perturbative ? ? 0 limit
  Bajnok-Janik
• Direct (string perturbation) computation of
  S-matrix
• Klose-McLoughlin-Minahan-Zarembo
• Open problems
• Can CDD ambiguity in S-matrix solve the
  discrepancy?
• gh(?) ?
• Is N6 CS theory really integrable in all-loop
  orders? (Higher order perturbative computations
  !)