MIRROR MANIFOLDS IN HIGHER DIMENSION

We describe mirror manifolds in dimensions different from the familiar case of complex threefolds. We isolate certain simplifying features p resent only in dimension three, and supply alternative methods that do not rely on these special characteristics and hence can be generalize d to other dimensions. Although the moduli spaces for Calabi-Yau d-fol ds are not ''special Kahler manifolds'' when d > 3, they still have a restricted geometry, and we indicate the new geometrical structures wh ich arise. We formulate and apply procedures which allow for the const ruction of mirror maps and the calculation of order-by-order instanton corrections to Yukawa couplings. Mathematically, these corrections ar e expected to correspond to calculating Chern classes of various param eter spaces (Hilbert schemes) for rational curves on Calabi-Yau manifo lds. Our mirror-aided calculations agree with those Chern class calcul ations in the limited number of cases for which the latter can be carr ied out with current mathematical tools. Finally, we make explicit som e striking relations between instanton corrections for various Yukawa couplings, derived from the associativity of the operator product alge bra.