domingo, abril 17, 2016

Calabi-Yau



This image shows a local 2D cross-section of the real 6D manifold known in string theory as the Calabi-Yau quintic. This is an Einstein manifold and a popular candidate for the wrapped-up 6 hidden dimensions of 10-dimensional string theory at the scale of the Planck length. The 5 rings that form the outer boundaries shrink to points at infinity, so that a proper global embedding would be seen to have genus 6 (6 handles on a sphere, Euler characteristic -10). The underlying real 6D manifold (3D complex) has Euler characteristic -200, is embedded in CP4, and is described by this homogeneous equation in five complex variables: z05 + z15 + z25 + z35 + z45 = 0 The displayed surface is computed by assuming that some pair of complex inhomogenous variables, say z3/z0 andz4/z0, are constant (thus defining a 2-manifold slice of the 6-manifold), renormalizing the resulting inhomogeneous equations, and plotting the local Euclidean space solutions to the inhomogenous complex equation z15 + z25 = 1 This surface can be described as a family of 5x5 phase transformations on a fundamental domain, 1/25th of the surface, shown (slightly hidden) in blue. Each of the first set of phases mixes in a brighter red color to its patch, and the second set mixes in green. Thus the color alone shows the geometric parentage of each of the 25 patches. The resulting surface, which is embedded in 4D, is projected to 3D according to one's taste to produce the final rendering. Further details are given in Andrew J. Hanson, "A construction for computer visualization of certain complex curves," Notices of the Amer. Math. Soc. 41 (9): 1156-1163, (November/December 1994).

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