13,118 research outputs found
Quantum deformations of projective three-space
We describe the possible noncommutative deformations of complex projective
three-space by exhibiting the Calabi--Yau algebras that serve as their
homogeneous coordinate rings. We prove that the space parametrizing such
deformations has exactly six irreducible components, and we give explicit
presentations for the generic members of each family in terms of generators and
relations. The proof uses deformation quantization to reduce the problem to a
similar classification of unimodular quadratic Poisson structures in four
dimensions, which we extract from Cerveau and Lins Neto's classification of
degree-two foliations on projective space. Corresponding to the ``exceptional''
component in their classification is a quantization of the third symmetric
power of the projective line that supports bimodule quantizations of the
classical Schwarzenberger bundles.Comment: 27 pages, 1 figure, 1 tabl
Fuzzy Toric Geometries
We describe a construction of fuzzy spaces which approximate projective toric
varieties. The construction uses the canonical embedding of such varieties into
a complex projective space: The algebra of fuzzy functions on a toric variety
is obtained by a restriction of the fuzzy algebra of functions on the complex
projective space appearing in the embedding. We give several explicit examples
for this construction; in particular, we present fuzzy weighted projective
spaces as well as fuzzy Hirzebruch and del Pezzo surfaces. As our construction
is actually suited for arbitrary subvarieties of complex projective spaces, one
can easily obtain large classes of fuzzy Calabi-Yau manifolds and we comment on
fuzzy K3 surfaces and fuzzy quintic three-folds. Besides enlarging the number
of available fuzzy spaces significantly, we show that the fuzzification of a
projective toric variety amounts to a quantization of its toric base.Comment: 1+25 pages, extended version, to appear in JHE
On the spectra of the quantized action-variables of the compactified Ruijsenaars-Schneider system
A simple derivation of the spectra of the action-variables of the quantized
compactified Ruijsenaars-Schneider system is presented. The spectra are
obtained by combining Kahler quantization with the identification of the
classical action-variables as a standard toric moment map on the complex
projective space. The result is consistent with the Schrodinger quantization of
the system worked out previously by van Diejen and Vinet.Comment: Based on talk at the workshop CQIS-2011 (Protvino, Russia, January
2011), 12 page
Fuzzy toric geometries
We describe a construction of fuzzy spaces which approximate projective toric varieties. The construction uses the canonical embedding of such varieties into a complex projective space: The algebra of fuzzy functions on a toric variety is obtained by a restriction of the fuzzy algebra of functions on the complex projective space appearing in the embedding. We give several explicit examples for this construction; in particular, we present fuzzy weighted projective spaces as well as fuzzy Hirzebruch and del Pezzo surfaces. As our construction is actually suited for arbitrary subvarieties of complex projective spaces, one can easily obtain large classes of fuzzy Calabi-Yau manifolds and we comment on fuzzy K3 surfaces and fuzzy quintic three-folds. Besides enlarging the number of available fuzzy spaces significantly, we find evidence for the conjecture that the fuzzification of a projective toric variety amounts to a quantization of its toric base
Quantum Mechanics Model on K\"ahler conifold
We propose an exactly-solvable model of the quantum oscillator on the class
of K\"ahler spaces (with conic singularities), connected with two-dimensional
complex projective spaces. Its energy spectrum is nondegenerate in the orbital
quantum number, when the space has non-constant curvature. We reduce the model
to a three-dimensional system interacting with the Dirac monopole. Owing to
noncommutativity of the reduction and quantization procedures, the Hamiltonian
of the reduced system gets non-trivial quantum corrections. We transform the
reduced system into a MIC-Kepler-like one and find that quantum corrections
arise only in its energy and coupling constant. We present the exact spectrum
of the generalized MIC-Kepler system. The one-(complex) dimensional analog of
the suggested model is formulated on the Riemann surface over the complex
projective plane and could be interpreted as a system with fractional spin.Comment: 5 pages, RevTeX format, some misprints heve been correcte
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