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Non-Integrability of Some Higher-Order Painlev\'e Equations in the Sense of Liouville
In this paper we study the equation which is one of the higher-order
Painlev\'e equations (i.e., equations in the polynomial class having the
Painlev\'e property). Like the classical Painlev\'e equations, this equation
admits a Hamiltonian formulation, B\"acklund transformations and families of
rational and special functions. We prove that this equation considered as a
Hamiltonian system with parameters , , , is not integrable in Liouville sense by means of
rational first integrals. To do that we use the Ziglin-Morales-Ruiz-Ramis
approach. Then we study the integrability of the second and third members of
the -hierarchy. Again as in the previous case it
turns out that the normal variational equations are particular cases of the
generalized confluent hypergeometric equations whose differential Galois groups
are non-commutative and hence, they are obstructions to integrability
Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals
Lattice statistical mechanics, often provides a natural (holonomic) framework
to perform singularity analysis with several complex variables that would, in a
general mathematical framework, be too complex, or could not be defined.
Considering several Picard-Fuchs systems of two-variables "above" Calabi-Yau
ODEs, associated with double hypergeometric series, we show that holonomic
functions are actually a good framework for actually finding the singular
manifolds. We, then, analyse the singular algebraic varieties of the n-fold
integrals , corresponding to the decomposition of the magnetic
susceptibility of the anisotropic square Ising model. We revisit a set of
Nickelian singularities that turns out to be a two-parameter family of elliptic
curves. We then find a first set of non-Nickelian singularities for and , that also turns out to be rational or ellipic
curves. We underline the fact that these singular curves depend on the
anisotropy of the Ising model. We address, from a birational viewpoint, the
emergence of families of elliptic curves, and of Calabi-Yau manifolds on such
problems. We discuss the accumulation of these singular curves for the
non-holonomic anisotropic full susceptibility.Comment: 36 page
Lectures on quantization of gauge systems
A gauge system is a classical field theory where among the fields there are
connections in a principal G-bundle over the space-time manifold and the
classical action is either invariant or transforms appropriately with respect
to the action of the gauge group. The lectures are focused on the path integral
quantization of such systems. Here two main examples of gauge systems are
Yang-Mills and Chern-Simons.Comment: 63 pages, 22 figures. Based on lectures given at the Summer School
"New paths towards quantum gravity", Holbaek, Denmark, 200
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