- last post: 01.01.0001 12:00 AM PDT
Calabi-Yau spaces are important in string theory, where one model posits the geometry of the universe to consist of a ten-dimensional space of the form , where M is a four dimensional manifold (space-time) and V is a six dimensional compact Calabi-Yau space. They are related to Kummer surfaces. Although the main application of Calabi-Yau spaces is in theoretical physics, they are also interesting from a purely mathematical standpoint. Consequently, they go by slightly different names, depending mostly on context, such as Calabi-Yau manifolds or Calabi-Yau varieties.
Although the definition can be generalized to any dimension, they are usually considered to have three complex dimensions. Since their complex structure may vary, it is convenient to think of them as having six real dimensions and a fixed smooth structure.
A Calabi-Yau space is characterized by the existence of a nonvanishing harmonic spinor . This condition implies that its canonical bundle is trivial.
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[Edited on 6/22/2004 4:18:29 PM]