The Calabi-Yau moduli and the Toda equations

Abstract

For a one-parameter family of Calabi-Yau d-fold embedded in C P d+1 we analyze the hermitian metrics on the moduli space of the non-linear sigma model by topological field techniques. The set of equations turns out to be the non-affine A-type Toda equation. It is contrasted with the C P d+1 case where the affine Toda equation appears. We obtain exact solutions of the hermitian metrics and associate their non-perturbative corrections to the holomorphic maps. In particular for the quintic case, the coefficients in the q-expansion are given by the number of rational curves. From these data, we construct the genus one partition function F 1 of the topological sigma model by the method of the holomorphic anomaly. In the asymmetrical limit of the Kähler parameter t → ∞ , the contribution from the A ∗- part is decoupled and we obtain the partition function of A-matter coupled with topological gravity at the stringy one-loop level. The coefficients of the series expansion are related with the integrals of the top Chem class of some vector bundle over the stable maps with definite degrees. We also comment on the behavior of the F 1 near the conifold locus.