Recent Developments in Topological String Theory

TL;DR

This review highlights recent progress in topological string theory, focusing on computational methods for elliptic Calabi-Yau manifolds and links to quantum systems, advancing understanding of non-perturbative effects.

Contribution

The paper introduces Jacobi forms as an ansatz for partition functions and explores their connection to quantum non-perturbative effects, providing new computational approaches.

Findings

Complete solutions for non-compact models' partition functions.

Higher genus computations for compact models.

Exact quantization conditions linked to topological strings.

Abstract

This review summarizes the recent developments in topological string theory from the author's perspective, mostly focused on aspects of research in which the author is involved. After a brief overview of the theory, we discuss two aspects of these developments. First, we discuss the computational progress in the topological string partition functions on a class of elliptic Calabi-Yau manifolds. We propose to use Jacobi forms as an ansatz for the partition function. For non-compact models, the techniques often provide complete solutions, while for compact models, though it is still not completely solvable, we compute to higher genus than previous works. Second, we explore a remarkable connection of refined topological strings on a class of non-compact toric Calabi-Yau threefolds with non-perturbative effects in quantum-mechanical systems. The connections provide rarely available exact quantization conditions for quantum systems and new insights on non-perturbative formulations of topological string theory.