The Candelas-de la Ossa-Green-Parkes formula

Abstract

In this note we discuss a recent proof of the formula for the worldsheet instanton prepotential predicted by Mirror Symmetry for the quintics in P 4. One of the key ingredients in the proof is the equivariant cohomology groups on the so-called linear sigma model moduli spaces. We introduce the notion of admissible data on the equivariant cohomology groups of the linear sigma model. An admissible data may be thought of as a sequence of equivariant classes satisfying certain algebraic conditions. It arises naturally from Kontsevich's stable map compactification of moduli spaces of maps from curves into projective manifolds. The structures of admissible data help reduce many counting problems to checking certain combinatorial structure of a compactification. The mirror transformation of Candelas et al turns out to be a transformation between two admissible data associated respectively to the linear and the non-linear sigma models. As an application, we prove the formula for the worldsheet instanton prepotential in terms of hypergeometric series. At the end we also interprete an infinite dimensional transformation group, called the mirror group, acting on admissible data, as a certain duality group of the linear sigma model.