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An explicit formula is derived for the generating function of verticalD4-D2-D0 bound states on smooth K3 fibered Calabi-Yau threefolds, generalizingprevious results of Gholampour and Sheshmani. It is also shown that thisformula satisfies strong modularity properties, as predicted by string theory.This leads to a new construction of vector valued modular forms which exhibitssome of the features of a generalized Hecke transform.

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