Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex

TL;DR

This paper calculates Donaldson-Thomas invariants for local elliptic surfaces with a new method combining motivic and toric techniques, expressing results via the topological vertex and deriving product formulas and connections to Jacobi forms.

Contribution

Introduces a novel computational approach for Donaldson-Thomas invariants of local elliptic surfaces using motivic and toric methods, linking to the topological vertex.

Findings

Derived product formulas for partition functions.

Expressed the connected partition function in terms of Jacobi forms.

Established a new derivation of the Katz-Klemm-Vafa formula for K3 surfaces.

Abstract

We compute the Donaldson-Thomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. This allows us to write the partition function for the invariants in terms of the topological vertex. Utilizing identities for the topological vertex proved in arXiv:1603.05271, we derive product formulas for the partition functions. The connected version of the partition function is written in terms of Jacobi forms. In the special case where the elliptic surface is a K3 surface, we get a derivation of the Katz-Klemm-Vafa formula for primitive curve classes which is independent of the computation of Kawai-Yoshioka.