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Non-projective twistor space is extended by fermionic coordinates where is the number of supersymmetries so that a twistor is now given by (, ′,), =, …, with anticommuting.
In mathematics and theoretical physics (especially twistor theory), twistor space is the complex vector space of solutions of the twistor equation ′ =. It was described in the 1960s by Roger Penrose and Malcolm MacCallum. [ 1 ]
The "twistor space" Z is complex projective 3-space CP 3, which is also the Grassmannian Gr 1 (C 4) of lines in 4-dimensional complex space. X = Gr 2 (C 4), the Grassmannian of 2-planes in 4-dimensional complex space. This is a compactification of complex Minkowski space. Y is the flag manifold whose elements correspond to a line in a plane of C 4.
The twistor approach simplifies calculations of particle interactions. In a conventional perturbative approach to quantum field theory, such interactions may require the calculation of thousands of Feynman diagrams , most describing off-shell "virtual" particles which have no directly observable existence.
A spin network, immersed into a manifold, can be used to define a functional on the space of connections on this manifold. One computes holonomies of the connection along every link (closed path) of the graph, determines representation matrices corresponding to every link, multiplies all matrices and intertwiners together, and contracts indices in a prescribed way.
By the more narrow definition, commonly used in mathematics, the term Killing spinor indicates those twistor spinors which are also eigenspinors of the Dirac operator. [1] [2] [3] The term is named after Wilhelm Killing. Another equivalent definition is that Killing spinors are the solutions to the Killing equation for a so-called Killing number.
The first Penrose tiling (tiling P1 below) is an aperiodic set of six prototiles, introduced by Roger Penrose in a 1974 paper, [16] based on pentagons rather than squares. Any attempt to tile the plane with regular pentagons necessarily leaves gaps, but Johannes Kepler showed, in his 1619 work Harmonices Mundi , that these gaps can be filled ...
In mathematics, a torsion sheaf is a sheaf of abelian groups on a site for which, for every object U, the space of sections (,) is a torsion abelian group.Similarly, for a prime number p, we say a sheaf is p-torsion if every section over any object is killed by a power of p.