Trace Identities for the Topological Vertex

TL;DR

This paper establishes new identities involving the topological vertex, connecting combinatorics, representation theory, and geometry, with implications for Donaldson-Thomas invariants of Calabi-Yau threefolds.

Contribution

It proves several new sum-to-product identities for the topological vertex using combinatorial and representation theoretic methods.

Findings

Derived closed-form identities for the topological vertex

Connected identities to Fourier expansions of Jacobi forms

Discussed applications to Donaldson-Thomas invariants

Abstract

The topological vertex is a universal series which can be regarded as an object in combinatorics, representation theory, geometry, or physics. It encodes the combinatorics of 3D partitions, the action of vertex operators on Fock space, the Donaldson-Thomas theory of toric Calabi-Yau threefolds, or the open string partition function of $\mathbb{C}^3$. We prove several identities in which a sum over terms involving the topological vertex is expressed as a closed formula, often a product of simple terms, closely related to Fourier expansions of Jacobi forms. We use purely combinatorial and representation theoretic methods to prove our formulas, but we discuss applications to the Donaldson-Thomas invariants of elliptically fibered Calabi-Yau threefolds at the end of the paper.