Triangle Seminars

Abstract:
We review some results concerning integrable systems on the moduli spaces of
holomorphic bundles. The interrelations between bundles of different
topological types are given by modifications (singular gauge
transformations). The modification provides canonical maps between
corresponding integrable systems. It also allows to describe the Backlund
transformations. We apply the general scheme to the elliptic Calogero-Moser
model and construct a map to an integrable Euler-Arnold top (the elliptic
SL(N)-rotator). The models correspond to the bundles of degree zero and one.
Explicit formulae represent a bosonization of the coadjoint orbit. The later
describes the phase space of the rotator while the Poisson structure of the
Calogero model is canonical. The construction can be generalized to more
complicated models. For example, field generalization (loop group or 1+1
theories) leads to relation between the field Calogero model and the
Landau-Lifshitz equation for XYZ model while the generalization to the
isomonodromic deformations leads to relation between Painleve VI equation
and Zhukovsky-Volterra gyrostat. In the end we note that the classification
of models is described by the center of the structure group which are shown
to be characteristic classes of the bundles. This remark allows to
generalize the construction to the case of an arbitrary simple Lie group.

Three-dimensional generalization of the construction naturally leads to the
Bogomolny equations. The modification in this case provides non-trivial
charge of the monopole solution. We find the Dirac monopole solution in the
case (R)x(Elliptic curve). This solution is a three-dimensional
generalization of the Kronecker series. We give two representations for this
solution and derive a functional equation for it.

Abstract:
The AdS/CFT correspondence establishes an equivalence of a four-dimensional supersymmetric gauge theory and string theory in Anti-de-Sitter space. Remarkably, the AdS/CFT system can be solved exactly with the help of integrability methods typical for two-dimensional models. I will review the general framework of the AdS/CFT duality, describe how integrability arises in AdS/CFT and how it can be used to compute the exact non-perturbative spectrum of supersymmetric Yang-Mills theory.

Abstract:
I will discuss a supergravity solution corresponding to
fivebranes wrapped on the S2 of the resolved conifold. By changing a
parameter the solution continuously interpolates between the deformed
conifold with flux and the resolved conifold with branes. Therefore, it
displays a geometric transition, purely in the supergravity context. The
solution is a simple example of torsional geometry and may be thought of
as a non-Kahler analog of the conifold. I will discuss how one can obtain
supersymmetric solutions of type IIB supergravity starting from simpler
non-Kahler geometries, by U-dualities or other methods. Applying these
transformations to the torsional conifold solution we obtain a solution
dual to the baryonic branch of the Klebanov-Strassler theory. Far along
the baryonic branch the solution resembles D5 branes wrapping a fuzzy
two-sphere in the resolved conifold and this can be matched to a weakly
coupled field theory analysis.

Abstract:
We review some results concerning integrable systems on the moduli spaces of
holomorphic bundles. The interrelations between bundles of different
topological types are given by modifications (singular gauge
transformations). The modification provides canonical maps between
corresponding integrable systems. It also allows to describe the Backlund
transformations. We apply the general scheme to the elliptic Calogero-Moser
model and construct a map to an integrable Euler-Arnold top (the elliptic
SL(N)-rotator). The models correspond to the bundles of degree zero and one.
Explicit formulae represent a bosonization of the coadjoint orbit. The later
describes the phase space of the rotator while the Poisson structure of the
Calogero model is canonical. The construction can be generalized to more
complicated models. For example, field generalization (loop group or 1+1
theories) leads to relation between the field Calogero model and the
Landau-Lifshitz equation for XYZ model while the generalization to the
isomonodromic deformations leads to relation between Painleve VI equation
and Zhukovsky-Volterra gyrostat. In the end we note that the classification
of models is described by the center of the structure group which are shown
to be characteristic classes of the bundles. This remark allows to
generalize the construction to the case of an arbitrary simple Lie group.

Three-dimensional generalization of the construction naturally leads to the
Bogomolny equations. The modification in this case provides non-trivial
charge of the monopole solution. We find the Dirac monopole solution in the
case (R)x(Elliptic curve). This solution is a three-dimensional
generalization of the Kronecker series. We give two representations for this
solution and derive a functional equation for it.