Prepotentials for local mirror symmetry via Calabi-Yau fourfolds

TL;DR

This paper develops a new intrinsic approach to compute prepotentials for local mirror symmetry involving Calabi-Yau fourfolds and threefolds, avoiding instanton expansions and fixing prepotentials up to quadratic terms.

Contribution

It introduces an intrinsic definition of intersection numbers for toric surfaces and extends the formalism to local fourfolds, providing a novel method to determine prepotentials.

Findings

Derived an intrinsic triple intersection number for toric surfaces.

Computed extended Picard-Fuchs systems without instanton expansion.

Fixed prepotentials of local Calabi-Yau threefolds up to quadratic terms.

Abstract

In this paper, we first derive an intrinsic definition of classical triple intersection numbers of K_S, where S is a complex toric surface, and use this to compute the extended Picard-Fuchs system of K_S of our previous paper, without making use of the instanton expansion. We then extend this formalism to local fourfolds K_X, where X is a complex 3-fold. As a result, we are able to fix the prepotential of local Calabi-Yau threefolds K_S up to polynomial terms of degree 2. We then outline methods of extending the procedure to non canonical bundle cases.