Geometric Aspects of Mirror Symmetry
The past twenty years have seen a number of fruitful exchanges of ideas between pure mathematics and theoretical physics. On the one hand, deep results in mathematics arising out of studies in geometry and topology have had unexpected and powerful applica
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R.
MORRISON *
Introduction The past twenty years have seen a number of fruitful exchanges of ideas between pure mathematics and theoretical physics. On the one hand, deep results in mathematics arising out of studies in geometry and topology have had unexpected and powerful applications to theoretical physics. On the other hand, insights gleaned from physical models have led to a number of conceptual revolutions in mathematics, particularly in geometry and topology. One of the most exciting of these conceptual revolutions, one which is still in the process of unfolding, goes by the name "mirror symmetry." Mirror symmetry predicts a completely unexpected relationship between certain pairs of Calabi-Yau manifolds.) From a mathematical point of view, the connections between the two "mirror" manifolds are extremely indirect, and the effort to explain them has led to a number of mathematical extensions of the original mirror phenomenon. Some of these extensions have proven useful in physics as well, and physicists have also found more general contexts in which mirror symmetry can be observed. Mirror symmetry has become a large field, and there have been a number of expository surveys [10,22,27,32,48,54,58,64, 77] of portions of the theory, and at least two books devoted to the subject [20,75]. In this paper, we will focus on the geometric connections between mirror pairs, using the example of the quintic threefold and its mirror as a guide. We have not attempted to be complete, but have instead concentrated on a few selected developments which are not covered in other recent surveys, and which hold much promise for future interesting work. In the first part of the paper, we review in some detail the known geometric connections (in the context of toric geometry). In the second part of the paper, we outline some new directions in which the geometric understanding of mirror symmetry has been moving recently, and indicate some of the important problems remaining to be solved. Although the issues discussed in this paper are mathematical in nature, the reader may wish to acquire some knowledge of physics in order to fully appreciate their context. A good place to start is Deligne et al. [21] (although
* Research
partially supported by National Science Foundation grant DMS-9401447 and by the Institute for Advanced Study, Princeton, New Jersey, USA, whom the author also thanks for hospitality. ) A Calabi-Yau manifold is a Kahler manifold with a nowhere-vanishing holomorphic form of top degree.
B. Engquist et al. (eds.), Mathematics Unlimited — 2001 and Beyond © Springer-Verlag Berlin Heidelberg 2001
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D. R. MORRISON
this assumes some knowledge of quantum mechanics, which can be obtained, for example, in [72]). A general introduction to string theory can be found in the IeM address of Witten [76], and the standard physics textbooks on string theory by Green, Schwarz, and Witten [34] and by Po1chinski [65] can be profitably studied. The more modem aspects of string theory are discussed in detail in [28].
1. Duality in
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