19,519 research outputs found
Sasaki-Einstein Manifolds
This article is an overview of some of the remarkable progress that has been
made in Sasaki-Einstein geometry over the last decade, which includes a number
of new methods of constructing Sasaki-Einstein manifolds and obstructions.Comment: 58 pages. Invited contribution to Surveys in Differential Geometry.
v2: references and discussion adde
Symplectic Techniques for Semiclassical Completely Integrable Systems
This article is a survey of classical and quantum completely integrable
systems from the viewpoint of local ``phase space'' analysis. It advocates the
use of normal forms and shows how to get global information from glueing local
pieces. Many crucial phenomena such as monodromy or eigenvalue concentration
are shown to arise from the presence of non-degenerate critical points.Comment: 32 pages, 7 figures. Review articl
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
On minimal eigenvalues of Schrodinger operators on manifolds
We consider the problem of minimizing the eigenvalues of the Schr\"{o}dinger
operator H=-\Delta+\alpha F(\ka) () on a compact manifold
subject to the restriction that \ka has a given fixed average \ka_{0}.
In the one-dimensional case our results imply in particular that for
F(\ka)=\ka^{2} the constant potential fails to minimize the principal
eigenvalue for \alpha>\alpha_{c}=\mu_{1}/(4\ka_{0}^{2}), where is
the first nonzero eigenvalue of . This complements a result by Exner,
Harrell and Loss (math-ph/9901022), showing that the critical value where the
circle stops being a minimizer for a class of Schr\"{o}dinger operators
penalized by curvature is given by . Furthermore, we show that the
value of remains the infimum for all . Using
these results, we obtain a sharp lower bound for the principal eigenvalue for a
general potential.
In higher dimensions we prove a (weak) local version of these results for a
general class of potentials F(\ka), and then show that globally the infimum
for the first and also for higher eigenvalues is actually given by the
corresponding eigenvalues of the Laplace-Beltrami operator and is never
attained.Comment: 7 page
Compact Einstein Spaces based on Quaternionic K\"ahler Manifolds
We investigate the Einstein equation with a positive cosmological constant
for -dimensional metrics on bundles over Quaternionic K\"ahler base
manifolds whose fibers are 4-dimensional Bianchi IX manifolds. The Einstein
equations are reduced to a set of non-linear ordinary differential equations.
We numerically find inhomogeneous compact Einstein spaces with orbifold
singularity.Comment: LaTeX 28 pages, 5 eps figure
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