The $n$-Point Function of $t$-Core Partitions and Topological Vertex

TL;DR

This paper introduces a $q$-deformed $n$-point function for $t$-core partitions using the topological vertex, providing a closed formula and establishing quasimodularity of correlation functions.

Contribution

It generalizes the $n$-point function to $t$-core partitions via the topological vertex, deriving a closed-form expression and proving quasimodularity.

Findings

Derived a closed formula for the $n$-point function of $t$-core partitions.

Established that the correlation functions are quasimodular forms.

Unified the $n$-point functions of all partitions and $t$-core partitions.

Abstract

In this paper, we study the $n$-point function of $t$-core partitions. The main tool is the topological vertex, originally developed to study the topological string theory for toric Calabi--Yau 3-folds. By virtue of the topological vertex, we introduce the $q$-deformed $n$-point function that generalizes both the ordinary $n$-point function of all integer partitions studied by Bloch--Okounkov and $t$-core partition case treated here. As a consequence, we provide a closed formula for the $n$-point function of $t$-core partitions in terms of theta functions, and prove that the corresponding correlation functions are quasimodular forms.