Bethe Ansatz in AdSCFT

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Bethe Ansatz in AdS/CFT
Marius de Leeuw
Introduction The AdS/CFT correspondence links
string theory on anti-de Sitter spaces to certain
gauge theories. This duality, 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 of strings
on the AdS side and to calculate the conformal
dimensions of operators on the gauge theory side
and compare the two. So far, this is even
unresolved in the basic example planar
N4 SYM ? single non-interacting superstring on
AdS5S5 An important recent development in this
area is the discovery of integrable structures. A
tool used to solve integrable models 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
? 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
, 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 The Bethe equations have been
formulated in the infinite volume limit. However,
to calculate the full spectrum, one also has to
take into account finite-size effects. How these
effects should be treated and calculated is still
unknown. Furthermore, the discussion in this
poster is based on the assumption of
integrability of the quantum string on AdS5xS5,
but this is still conjectural. It is important to
learn whether integrability is inherited by the
quantum string.
Symmetry Algebra The symmetry algebra of the
gauge fixed-light cone Hamiltonian consists of
two copies of centrally extended .
There is also a corresponding dynamic spin chain
with this algebra. The fundamental
representation of centrally extended
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 Hence, the spectrum can be found
by finding the set of momenta.

• Literature and References
• Hep-th/AFZ
• Hep-th/Beiser
• Hep-th/BS
• Hep-th/Leeuw

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