Bethe Ansatz in AdSCFT

1
Bethe Ansatz in AdS/CFT
Marius de Leeuw
Introduction The AdS/CFT correspondence links
string theory on anti-de Sitter spaces to gauge
theories. This connection, if correct, gives
valuable insights in gauge theories and string
theory. For example, the strong-coupling limit of
gauge theories could be computed with the help of
string theory. A proof of this duality would be
to calculate the full quantum spectrum (energy)
of strings and of the conformal dimensions of
operators on the gauge theory side and compare
the two. So far, even in the basic example
planar N4 SYM ? single non-interacting
superstring on AdS5S5 this is still unresolved
. An important recent development in this area
is the discovery of integrable structures. A tool
used to solve these systems is the Bethe ansatz.
This technique dates back to 1931 when H. Bethe
used it to solve the Heisenberg spin chain and it
turns out to be useful in this context as well.

• Integrability
• Integrability means that there is an infinite
  amount of conserved charges. This has a number of
  useful consequences. We consider the
  two-dimensional relativistic quantum integrable
  systems. That the superstring on AdS5 x S5 is
  such a system is an assumption and has not been
  proven, but there is evidence that points in this
  direction.
• Important features of these integrable models
• Scattering preserves particle number
• Scattering preserves set of on-shell particle
  momenta
• Scattering factorizes
• The factorization of scattering processes means
  that any S-matrix can be written as a product of
  two-body S-matrices. In other words, every
  scattering process is a repetition of two-body
  scattering processes. This implies a consistency
  condition, called the Yang-Baxter equation,
  schematically shown below
• Thus, two-dimensional relativistic integrable
  systems have the useful property that the
  two-body S-matrix contains all the scattering
  information.

Gauge Theory and Spin Chains Gauge theory is
linked to integrable systems in a remarkable way.
Operators from the N4 gauge theory are linked to
spin chains Tr(Fa...Fa)
? The scaling weights correspond to the
eigenvalues of a spin chain Hamiltonian. In other
words, to find the spectrum of conformal
dimensions one has to compute the energies of a
spin chain. The spin chains that we encounter
here are integrable systems.
Periodicity and Bethe Equations Periodicity puts
restrictions on the momenta of the excitations
present in the asymptotic states. The equations
that capture this are called the Bethe equations.
We impose periodicity by moving a particle around
the chain/string and by doing this, it scatters
with the other particles. This scattering is
described by the S-matrix. Periodicity just
means that this operation should leave our state
invariant. The equations that describe this are
the Bethe equations and are given
by Where x is a given function of the
momentum, y and w are auxiliary parameters and P
is the total momentum.
String Theory and S-Matrices We are working in
the uniform light-cone gauge. In this gauge the
length of the string world-sheet explicitly
depends on a charge J which corresponds to the
angular momentum of the string. By sending J-gt
inf, we decompactify our string and obtain a
two-dimensional, massive, quantum field theory,
which allows for scattering processes. We
restrict ourselves to asymptotic states, which
are the states that drop off rapidly.
The two-body S-matrix describing
scattering of asymptotic states for both the
string theory and the spin chain picture can be
determined up to a phase factor by imposing
compatibility with the symmetry algebra. This was
first done for spin chains (hep-th) and recently
for the AdS5 x S5 superstring (hep-th). The
remaining phase factor can be restricted by
requiring crossing symmetry. Finally, we assume
integrability for the strings.
Future Research Unresolved questions. Finite
size effects, check of integrability,
thermodynamic limit.
Symmetry Algebra The symmetry algebra of the
gauge fixed-light cone Hamiltonian consists of
two copies of centrally extended su(22). There
is also a corresponding dynamic spin chain with
this algebra. The fundamental representation of
centrally extended su(22) is four dimensional,
consisting of two bosonic vectors and two
fermionic vectors. This fundamental
representation depends on a parameter p, which
can be interpreted as momentum. One of the
central elements is given by the Hamiltonian and
it can be expressed in terms of the momentum as
follows Hsqrt(p). Hence, the energy can be
found by determining the set of allowed momenta.

• Literature
• He-th/78856789
• Eabae
• Abe
• Bae
• Ae
• aeb