21,072 research outputs found
On the geometry of the energy operator in quantum mechanics
We analyze the different ways to define the energy operator in geometric
theories of quantum mechanics. In some formulations the operator contains the
scalar curvature as a multiplicative term. We show that such term can be
canceled or added with an arbitrary constant factor, both in the mainstream
Geometric Quantization and in the Covariant Quantum Mechanics, developed by
Jadczyk and Modugno with several contributions from many authors.Comment: 18 pages; paper in honour of the 70th birthday of Luigi Mangiarotti
and Marco Modugn
On the hyperkaehler/quaternion Kaehler correspondence
A hyperkaehler manifold with a circle action fixing just one complex
structure admits a natural a hyperholomorphic line bundle. This forms the basis
for the construction of a corresponding quaternionic Kaehler manifold in the
work of A.Haydys. We construct in this paper the corresponding holomorphic line
bundle on twistor space and compute many examples, including monopole and Higgs
bundle moduli spaces. We also show that the bundle on twistor space has a
natural meromorphic connection which realizes it as the quantum line bundle for
the hyperkaehler family of holomorphic symplectic structures. Finally we give a
twistor version of the HK/QK correspondence.Comment: 35 page
Integral geometry of complex space forms
We show how Alesker's theory of valuations on manifolds gives rise to an
algebraic picture of the integral geometry of any Riemannian isotropic space.
We then apply this method to give a thorough account of the integral geometry
of the complex space forms, i.e. complex projective space, complex hyperbolic
space and complex euclidean space. In particular, we compute the family of
kinematic formulas for invariant valuations and invariant curvature measures in
these spaces. In addition to new and more efficient framings of the tube
formulas of Gray and the kinematic formulas of Shifrin, this approach yields a
new formula expressing the volumes of the tubes about a totally real
submanifold in terms of its intrinsic Riemannian structure. We also show by
direct calculation that the Lipschitz-Killing valuations stabilize the subspace
of invariant angular curvature measures, suggesting the possibility that a
similar phenomenon holds for all Riemannian manifolds. We conclude with a
number of open questions and conjectures.Comment: 68 pages; minor change
Geometric Aspects of Mirror Symmetry (with SYZ for Rigid CY manifolds)
In this article we discuss the geometry of moduli spaces of (1) flat bundles
over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills
bundles over complex submanifolds in Calabi-Yau manifolds.
These moduli spaces reflect the geometry of the Calabi-Yau itself like a
mirror. Strominger, Yau and Zaslow conjecture that the mirror Calabi-Yau
manifold is such a moduli space and they argue that the mirror symmetry duality
is a Fourier-Mukai transformation. We review various aspects of the mirror
symmetry conjecture and discuss a geometric approach in proving it.
The existence of rigid Calabi-Yau manifolds poses a serious challenge to the
conjecture. The proposed mirror partners for them are higher dimensional
generalized Calabi-Yau manifolds. For example, the mirror partner for a certain
K3 surface is a cubic fourfold and its Fano variety of lines is birational to
the Hilbert scheme of two points on the K3. This hyperkahler manifold can be
interpreted as the SYZ mirror of the K3 by considering singular special
Lagrangian tori.
We also compare the geometries between a CY and its associated generalized
CY. In particular we present a new construction of Lagrangian submanifolds.Comment: To appear in the proceedings of International Congress of Chinese
Mathematicians 2001, 47 page
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