Remarks on partition functions of topological string theory on generalized conifolds

The notion of topological vertex and the construction of topological string partition functions on local toric Calabi-Yau 3-folds are reviewed. Implications of an explicit formula of partition functions for the generalized conifolds are considered. Generating functions of part of the partition functions are shown to be tau functions of the KP hierarchy. The associated Baker-Akhiezer functions play the role of wave functions, and satisfy $q$-difference equations. These $q$-difference equations represent the quantum mirror curves conjectured by Gukov and Su\lkowski.