A question on generalization of partition functions of CY 3-folds in String Theory

TL;DR

This paper explores the generalization of topological string partition functions for CY 3-folds, connecting open Gromov-Witten theory, vertex operators, and Hilbert schemes, with a focus on extending existing frameworks.

Contribution

It introduces an extension of the partition function involving infinitely many Cassimir operators, linking open Gromov-Witten theory to Hilbert schemes of points on surfaces.

Findings

Extended the partition function with infinitely many twists.

Connected topological string theory to Hilbert schemes.

Provided insights into the generalization of Gromov-Witten invariants.

Abstract

This is an expositoray article on the topological string partition function promoting an extension of the partition function of open Gromov-Witten theory of CY 3-folds defined by the trace of vertex operators. We also give a brief survey of their connection to the theory of Hilbert scheme of points on surface. Specifically; we apply infinitely many Cassimir operators twisted to the vertex operator computing the amplitude. The case of finite number of twists has been well discussed in the mathematics and Physics literature.