Vertical D4-D2-D0 bound states on K3 fibrations and modularity
  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. 
Vincent Bouchard, Thomas Creutzig, Duiliu-Emanuel Diaconescu, Charles   Doran, Callum Quigley, Artan Sheshmani
hep-th math.AG