Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians

Abstract

We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -expansion principle for functions on the moduli space of Calabi-Yau manifolds, and the “mirror symmetry” phenomenon recently observed by string theorists.