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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 most recent proposal in this direction by Penrose in 2015 was based on noncommutative geometry on twistor space and referred to as palatial twistor theory. [46] The theory is named after Buckingham Palace, where Michael Atiyah [47] suggested to Penrose the use of a type of "noncommutative algebra", an important component of the theory.
In the converse direction, a result of LeBrun [13] shows that any Fano manifold that admits both a Kähler–Einstein metric and a holomorphic contact structure is actually the twistor space of a quaternion-Kähler manifold of positive scalar curvature, which is moreover unique up to isometries and rescalings.
Twistor string theory is an equivalence between N = 4 supersymmetric Yang–Mills theory and the perturbative topological B model string theory in twistor space. [1] It was initially proposed by Edward Witten in 2003. Twistor theory was introduced by Roger Penrose from the 1960s as a new approach to the unification of quantum theory with gravity.
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.
Given a quaternionic -manifold , the unit 2-sphere subbundle = corresponding to the pure unit imaginary quaternions (or almost complex structures) is called the twistor space of . It turns out that, when n ≥ 2 {\displaystyle n\geq 2} , there exists a natural complex structure on Z {\displaystyle Z} such that the fibers of the projection Z → ...
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The scattering amplitude can thus be thought of as the volume of a certain polytope, the positive Grassmannian, in momentum twistor space. [1] When the volume of the amplituhedron is calculated in the planar limit of N = 4 D = 4 supersymmetric Yang–Mills theory, it describes the scattering amplitudes of particles described by this theory. [1]