921 research outputs found
Toric K\"ahler metrics seen from infinity, quantization and compact tropical amoebas
We consider the metric space of all toric K\"ahler metrics on a compact toric
manifold; when "looking at it from infinity" (following Gromov), we obtain the
tangent cone at infinity, which is parametrized by equivalence classes of
complete geodesics. In the present paper, we study the associated limit for the
family of metrics on the toric variety, its quantization, and degeneration of
generic divisors.
The limits of the corresponding K\"ahler polarizations become degenerate
along the Lagrangian fibration defined by the moment map. This allows us to
interpolate continuously between geometric quantizations in the holomorphic and
real polarizations and show that the monomial holomorphic sections of the
prequantum bundle converge to Dirac delta distributions supported on
Bohr-Sommerfeld fibers.
In the second part, we use these families of toric metric degenerations to
study the limit of compact hypersurface amoebas and show that in Legendre
transformed variables they are described by tropical amoebas. We believe that
our approach gives a different, complementary, perspective on the relation
between complex algebraic geometry and tropical geometry.Comment: v1: 32 pages, 5 figures; v2: 1 figure added; v3: 1 reference added;
v4: some reorganization, 1 theorem (now 1.1) added; v5: final version, to
appear in JD
Laughlin states change under large geometry deformations and imaginary time Hamiltonian dynamics
We study the change of the Laughlin states under large deformations of the
geometry of the sphere and the plane, associated with Mabuchi geodesics on the
space of metrics with Hamiltonian -symmetry.
For geodesics associated with the square of the symmetry generator, as the
geodesic time goes to infinity, the geometry of the sphere becomes that of a
thin cigar collapsing to a line and the Laughlin states become concentrated on
a discrete set of --orbits, corresponding to Bohr-Sommerfeld orbits of
geometric quantization.
The lifting of the Mabuchi geodesics to the bundle of quantum states, to
which the Laughlin states belong, is achieved via generalized coherent state
transforms, which correspond to the KZ parallel transport of Chern-Simons
theory
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