Topological string on elliptic CY 3-folds and the ring of Jacobi forms

Abstract

We give evidence that the all genus amplitudes of topological string theory on compact elliptically fibered Calabi-Yau manifolds can be written in terms of meromorphic Jacobi forms whose weight grows linearly and whose index grows quadratically with the base degree. The denominators of these forms have a simple universal form with the property that the poles of the meromorphic form lie only at torsion points. The modular parameter corresponds to the fibre class while the rôle of the string coupling is played by the elliptic parameter. As a consequence the topological string amplitudes are modular and quasi periodic in the string coupling. This leads to very strong all genus results on these geometries, which are checked against results from curve counting. The structure can be viewed as an indication that an N = 2 analog of the reciprocal of the Igusa cusp form exists that might govern the topological string theory on these Calabi-Yau manifolds completely.

Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds

“…We also show that similar features hold for the genus one amplitude, which is conjectured to be related to the gauge kinetic terms. As in the Calabi-Yau threefold case we find that these amplitudes are related via certain holomorphic anomaly equations, from which they can be reconstructed in simple situations [18][19][20][21][22].…”

“…It turns out that the intersection ring and the relevant topological invariants are completely determined by the base. Note that this geometric setup is completely analogous to the threefolds studied in [19,21].…”

“…As far as it carries over to the fourfold case, we follow the notation in [21] which we now quickly review. The generators of the Mori cone of the base B are given by…”

“…The definitions above are straightforward extensions of the corresponding threefold expressions introduced in [21]. For Calabi-Yau fourfolds a basis of middle-dimensional cycles has to be specified as well.…”

“…We choose a basis of branes 21) and calculate the monodromy for the large radius divisor corresponding to D j ,…”