Polynomial Structure of the (Open) Topological String Partition Function

TL;DR

The paper addresses how to determine the full topological string partition function Z = exp(sum_g lambda^{2g-2} F^{(g)}) for general Calabi-Yau manifolds with multiple moduli, including open strings. It introduces a generator-based polynomial framework that encodes non-holomorphic data, derives a degree-bound recursion, and fixes holomorphic ambiguities via boundary conditions and gap structures. The authors prove that F^{(g,h)} admit polynomial expressions in a finite set of generators and demonstrate the method on the real quintic, reproducing Walchers open-string results and yielding higher-genus/open amplitudes and Ooguri-Vafa invariants. This work broadens the applicability of polynomial integration techniques in topological string theory and provides a practical computational tool across Calabi-Yau moduli spaces.

Abstract

In this paper we show that the polynomial structure of the topological string partition function found by Yamaguchi and Yau for the quintic holds for an arbitrary Calabi-Yau manifold with any number of moduli. Furthermore, we generalize these results to the open topological string partition function as discussed recently by Walcher and reproduce his results for the real quintic.