正博 昆

Bio: 正博 昆 is an academic researcher. The author has contributed to research in topics: Invariant (mathematics). The author has an hindex of 3, co-authored 3 publications receiving 862 citations.

Hyperkahler Metrics and Supersymmetry

Abstract: We describe two constructions of hyperkahler manifolds, one based on a Legendre transform, and one on a sympletic quotient. These constructions arose in the context of supersymmetric nonlinear σ-models, but can be described entirely geometrically. In this general setting, we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry, along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience.

Geometrical Formulation of Quantum Mechanics

Abstract: States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a Uhler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics which, although equivalent to the standard algebraic formulation, has a very different appearance. In particular, states are now represented by points of a symplectic manifold (which happens to have in addition a compatible Riemannian metric), observables are represented by certain real-valued functions on this space, and the Schrodinger evolution is captured by the symplectic flow generated by a Hamiltonian function. There is thus a remarkable similarity with the standard symplectic formulation of classical mechanics. Features such as uncertainties and state vector reductions—which are specific to quantum mechanics can also be formulated geometrically but now refer to the Riemannian metric—a structure which is absent in classical mechanics. The geometrical formulation sheds considerable light on a number of issues such as the second quantization procedure, the role of coherent states in semiclassical considerations, and the WKB approximation. More importantly, it suggests generalizations of quantum mechanics. The simplest among these are equivalent to the dynamical generalizations that have appeared in the literature. The geometrical reformulation provides a unified framework to discuss these and to correct a misconception. Finally, it also suggests directions in which more radical generalizations may be found.

Vanishing theorems and string backgrounds

Abstract: We show various vanishing theorems for the cohomology groups of compact Hermitian manifolds for which the Bismut connection has a (restricted) holonomy contained in SU(n) and classify all such manifolds of dimension four. In this way we provide necessary conditions for the existence of such structures on compact Hermitian manifolds with vanishing first Chern class of non-Kahler type. Then we apply our results to solutions of the string equations and show that such solutions admit various cohomological restrictions such as, for example, that under certain natural assumptions the plurigenera vanish. We also find that under some assumptions the string equations are equivalent to the condition that a certain vector is parallel with respect to the Bismut connection.