The Geometry Underlying Mirror Symmetry

The recent result of Strominger, Yau and Zaslow relating mirror symmetry to the quantum field theory notion of T-duality is reinterpreted as providing a way of geometrically characterizing which Calabi–Yau manifolds have mirror partners. The geometric description—that one Calabi–Yau manifold should serve as a compactified, complexified moduli space for special Lagrangian tori on the other Calabi–Yau manifold—is rather surprising. We formulate some precise mathematical conjectures concerning how these moduli spaces are to be compactified and complexified, as well as a definition of geometric mirror pairs (in arbitrary dimension) which is independent of those conjectures. We investigate how this new geometric description ought to be related to the mathematical statements which have previously been extracted from mirror symmetry. In particular, we discuss how the moduli spaces of the ‘mirror’ Calabi–Yau manifolds should be related to one another, and how appropriate subspaces of the homology groups of those manifolds could be related. We treat the case of K3 surfaces in some detail. Precise mathematical formulations of the string theory phenomenon known as “mirror symmetry” [21, 33, 19, 27] have proved elusive up until now, largely due to one of the more mysterious aspects of that symmetry: as traditionally formulated, mirror symmetry predicts an equivalence between physical theories associated to certain pairs of Calabi–Yau manifolds, but does not specify any geometric relationship between those manifolds. However, such a geometric relationship has recently been discovered in a beautiful paper of Strominger, Yau and Zaslow [53]. Briefly put, these authors find that the mirror partner X of a given Calabi–Yau threefold Y should be realized as the (compactified and complexified) moduli space for special Lagrangian tori on Y . This relationship was derived in [53] from the assumption that the physical theories associated to the pair of Calabi–Yau threefolds satisfy a strong property called “quantum mirror symmetry” [52, 6, 12, 42]. In the present paper, we will invert the logic, and use this geometric relationship as a characterization of mirror pairs, which we formulate in arbitrary dimension. On the one hand, this characterization can be stated in purely Our definition appears to produce valid mirror pairs of conformal field theories in any dimension, even though the string-theoretic arguments of [53] cannot be directly