Calabi-Yau Manifolds and Mirror Symmetries

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-duality can give rise to new dualities. Specifically, Calabi-Yau manifolds are very special kinds of manifolds, each possessing a dual or “mirror” manifold that has a different complex topology. Although these two Calabi-Yau manifolds would be classified as distinct from a mathematics standpoint, that’s not necessarily the case in string theory where manifolds of different topology can nevertheless give rise to the same physics. This generalization of -duality is called mirror symmetry .

Dualities turn questions into dual questions, meaning that for every question you can ask in one framework, there is a dual question that can be asked in a dual framework. What is an example of an interesting question in one setting that gets reformulated as a dual question in another ? Well, in order to compute physical interactions in string theory, we start with 10 dimensions and reduce the theory to 4 dimensions by assuming that the remaining dimensions are curled up and hidden on a tiny, 6-dimensional space, a typical example of it is known as a “Calabi-Yau manifold.” We then need to to calculate the number of minimal area spheres that can be placed inside that manifold, which can be a very difficult task that is, in some cases, beyond us. Because of the duality, however, we can answer the same question by computing some simple integrals on the mirror Calabi-Yau–in this way replacing a vexing problem with a much easier one. Using this method, physicists computed the number of minimal area spheres for spheres of different degrees–each degree corresponding to the winding number of the sphere, or the number of ways a sphere can wrap around the space. Mathematicians had previously worked out this problem for degrees and , and their answer agreed with the number that physicists had obtained. However, by using mirror symmetry, physicists were able to determine the answer not just for degrees and but for any degree.

Mathematicians had originally tried to get these numbers using traditional methods and, after some hard work, they obtained a solution to the degree 3 problem. Their number, however, disagreed with that arrived at by physicists through an approach involving mirror symmetry. Many people assumed the string theorists had gotten it wrong, but the mathematicians later discovered an error in their own work. After redoing their calculations, they confirmed the physicists’ computations. This gave us further confidence in using dualities to solve difficult problems in physics and math, because this approach yielded reliable predictions that could not be obtained by other known means.