Vertical D4-D2-D0 bound states on K3 fibrations and modularity

TL;DR

This paper derives an explicit generating function for vertical D4-D2-D0 bound states on K3 fibered Calabi-Yau threefolds, demonstrating its modularity and introducing a new construction of vector valued modular forms.

Contribution

It provides a generalized explicit formula for bound state generating functions and reveals their strong modularity properties, connecting string theory predictions with mathematical structures.

Findings

Derived explicit formula for generating functions

Proved strong modularity properties of the formula

Constructed new vector valued modular forms with generalized Hecke features

Abstract

An explicit formula is derived for the generating function of vertical D4-D2-D0 bound states on smooth K3 fibered Calabi-Yau threefolds, generalizing previous results of Gholampour and Sheshmani. It is also shown that this formula satisfies strong modularity properties, as predicted by string theory. This leads to a new construction of vector valued modular forms which exhibits some of the features of a generalized Hecke transform.