Rational Curves on Calabi-yau Threefolds

In the conformal field theory arising from the compactification of strings on a Calabi-Yau threefold X, there naturally arise fields correspond to harmonic forms of types (2,1) and (1,1) on X [4]. The uncorrected Yukawa couplings in H and H are cubic forms that can be constructed by techniques of algebraic geometry — [20] contains a nice survey of this in a general context in a language written for mathematicians. The cubic form on the space H of harmonic (p, 1) forms is given by the intersection product (ωi, ωj, ωk) 7→ ∫X ωi∧ωj∧ωk for p = 1, while for p = 2 there is a natural formulation in terms of infinitesimal variation of Hodge structure [20, 6]. The Yukawa couplings on H are exact, while those on H receive instanton corrections. In this context, the instantons are non-constant holomorphic maps f : CP → X. The image of such a map is a rational curve on X, which may or may not be smooth. If the rational curve C does not move inside X, then the contribution of the instantons which are generically 1-1 (i.e. birational) maps with image C can be written down explicitly — this contributions only depends on the topological type of C, or more or less equivalently, the integrals ∫ C f ∗J for (1files, a,1) forms J on X. If the conformal field theory could also be expressed in terms of a “mirror manifold” X ′, then the uncorrected Yukawa couplings on H(X ′) would be the same as the corrected Yukawa couplings on H(X). So if identifications could be made properly, the infinitesimal variation of Hodge structure on X ′ would give information on the rational curves on X. In a spectacular paper [5], Candelas et. al. do this when X is a quintic threefold. Calculating the Yukawa coupling on H(X ′) as the complex structure of X ′ varies gives the Yukawa coupling on H(X) as J varies. The coefficients of the resulting Fourier series are then directly related to the instanton corrections. To the mathematician, there are some unanswered questions in deducing the number of rational curves of degree d on X from this [20]. I merely cite one problem here, while I will state a little differently. It is not yet known how to carry out the calculation of the instanton