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

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.