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Hyperkähler Quotient
In mathematics, the hyperkähler quotient of a hyperkähler manifold acted on by a Lie group ''G'' is the quotient of a fiber of a hyperkähler moment map M \to \mathfrak \otimes \mathbb^3 over a ''G''-fixed point by the action of ''G''. It was introduced by Nigel Hitchin, Anders Karlhede, Ulf Lindström Ulf Lindström (born 12 November 1947) is a Swedish theoretical physicist working in the fields of string theory, supersymmetry, and general relativity. He earned his fil. kand. university degree at Stockholm University in 1972 and continued un ..., and Martin Roček in 1987. It is a hyperkähler analogue of the Kähler quotient. References * Differential geometry {{differential-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperkähler Manifold
In differential geometry, a hyperkähler manifold is a Riemannian manifold (M, g) endowed with three integrable almost complex structures I, J, K that are Kähler with respect to the Riemannian metric g and satisfy the quaternionic relations I^2=J^2=K^2=IJK=-1. In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds. Hyperkähler manifolds were defined by Eugenio Calabi in 1979. Equivalent definition in terms of holonomy Equivalently, a hyperkähler manifold is a Riemannian manifold (M, g) of dimension 4n whose holonomy group is contained in the compact symplectic group . Indeed, if (M, g, I, J, K) is a hyperkähler manifold, then the tangent space is a quaternionic vector space for each point of , i.e. it is isomorphic to \mathbb^n for some integer n, where \mathbb is the algebra of quaternions. The compact symplectic group can be considered as the group of orthogonal transformations of \mathbb^n whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nigel Hitchin
Nigel James Hitchin FRS (born 2 August 1946) is a British mathematician working in the fields of differential geometry, gauge theory, algebraic geometry, and mathematical physics. He is a Professor Emeritus of Mathematics at the University of Oxford. Academic career Hitchin attended Ecclesbourne School, Duffield, and earned his BA in mathematics from Jesus College, Oxford, in 1968.''Fellows' News'', Jesus College Record (1998/9) (p.12) After moving to Wolfson College, he received his D.Phil. in 1972. From 1971 to 1973 he visited the Institute for Advanced Study and 1973/74 the Courant Institute of Mathematical Sciences of New York University. He then was a research fellow in Oxford and starting in 1979 tutor, lecturer and fellow of St Catherine's College. In 1990 he became a professor at the University of Warwick and in 1994 the Rouse Ball Professor of Mathematics at the University of Cambridge. In 1997 he was appointed to the Savilian Chair of Geometry at the University of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ulf Lindström
Ulf Lindström (born 12 November 1947) is a Swedish theoretical physicist working in the fields of string theory, supersymmetry, and general relativity. He earned his fil. kand. university degree at Stockholm University in 1972 and continued under the supervision of Bertel Laurent with doctoral studies. The title of his PhD thesis was "Extensions of general relativity: scalar tensor theory, topology of space-time and supergravity." He spent the year 1978-1979 at Brandeis University as a post-doc working with Stanley Deser, and after getting his doctoral degree, became a docent in Stockholm. During 1986-1987, he spent time as a research fellow at Stony Brook University. In 2002 he moved from Stockholm University to Uppsala University where he is now the chairman of the theoretical physics department. His most well-known contributions to theoretical physics are in the field of supersymmetry where he was one of the first people to discuss the hyper-Kähler quotient construction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Roček
Martin Roček is a professor of theoretical physics at the State University of New York at Stony Brook and a member of the C. N. Yang Institute for Theoretical Physics. He received A.B. and Ph.D. degrees from Harvard University in 1975 and 1979. He did post-doctoral research at the University of Cambridge and Caltech before becoming a professor at Stony Brook University. He was one of the co-inventors of hyperkähler quotients, a hyperkahler analogue of Marsden–Weinstein reduction and the structure of Bihermitian manifolds. His research interests include supersymmetry, string theory and applications of generalized complex geometry In the field of mathematics known as differential geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic structure. Generalized complex structures were ..., and with S. J. Gates, M. T. Grisaru, and W. Siegel, Rocek coauthored ''Superspace, or One thou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kähler Quotient
In mathematics, specifically in complex geometry, the Kähler quotient of a Kähler manifold X by a Lie group G acting on X by preserving the Kähler structure and with moment map \mu : X \to \mathfrak^* (with respect to the Kähler form) is the quotient :\mu^(0)/G. If G acts freely and properly, then \mu^(0)/G is a new Kähler manifold whose Kähler form is given by the symplectic quotient construction. By the Kempf-Ness theorem, a Kähler quotient by a compact Lie group G is closely related to a geometric invariant theory quotient by the complexification of G.* See also *Hyperkähler quotient In mathematics, the hyperkähler quotient of a hyperkähler manifold acted on by a Lie group ''G'' is the quotient of a fiber of a hyperkähler moment map M \to \mathfrak \otimes \mathbb^3 over a ''G''-fixed point by the action of ''G''. It was intr ... References {{DEFAULTSORT:Kahler quotient Complex manifolds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
