The Mirror Formula for Quintic Threefolds

has been intriguing algebraic and symplectic geometers since the beginning of the decade. The first proof of this formula was given two years ago in the extensive paper [9] among a number of other theorems on equivariant Gromov – Witten theory. Several authors managed to adjust the approach of that paper to complete intersections in homogeneous Kähler spaces [13, 2, 14], in toric manifolds [11, 12] and to symmetric products of Riemann surfaces [4]. We present here a shortcut to our original proof in the case of quintic threefolds. Several variants of the proof can be found in [9, 11, 12, 17, 5, 20] as particular cases of more general theorems. Yet it seems useful to illustrate all ingredients of the proof in the simplest nontrivial example. We will assume that the reader is familiar with generalities on orbifolds and orbibundles, equivariant cohomology and localization formulas, Kontsevich’s moduli spaces of stable maps [15, 3] and with the formulation of the conjecture. We will concentrate therefore only on the issues relevant for the proof of the mirror formulas.