Tuesday, January 29, 2013

What, exactly, is a Calabi-Yau manifold?

String theory describes one of the smallest things you can possibly imagine — six-dimensional geometric spaces that may be more than a trillion times smaller than an electron — that could be one of the defining features of our universe. The story of these spaces, can be explained by what physicists call “Calabi-Yau manifolds,”

Superstring theory is a unified theory for all the forces of nature including quantum gravity. In superstring theory, the fundamental building block is an extended object, namely a string, whose vibrations would give rise to the particles encountered in nature. The constraints for the consistency of such a theory are extremely stringent. They require in particular that the theory takes place in a 10-dimensional space-time. To make contact with our 4-dimensional world, it is expected that the 10-dimensional space-time of string theory is locally the product M 4 ×X of a 4-dimensional Minkowski space M 3,1 with a 6-dimensional space X . The 6-dimensional space X would be tiny, which would explain why it has not been detected so far at the existing experimental energy levels. Each choice of the internal space X leads to a different effective theory on the 4-dimensional Minkowski space M 3,1 , which should be the theory describing our world.

It has long been argued that, in order to solve certain classic problems of unified gauge theories such as the gauge hierarchy problem, the 4-dimensional effective theory should admit an N=1 supersymmetry. In a fundamental paper, Candelas, Horowitz, Strominger and Witten (1985) analyzed what the constraint of that N=1 supersymmetry would mean for the geometry of the internal space X . They found that, for the most basic product models with N=1 supersymmetry, the space X must be a Calabi-Yau manifold of complex dimension 3. Shortly afterwards, Strominger (1986) considered slightly more general models, allowing warped products. For these models, the N=1 supersymmetry constraint results in a modification of the Ricci-flat equation of the earlier model.

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