We study BPS states in type IIA string compactification on a local
Calabi-Yau 3-fold which are related to the BPS states of the
E-string. Using Picard-Lefshetz transformations of the 3-cycles on
the mirror manifold we determine automorphisms of the K-theory of
the compact divisor of the Calabi-Yau which maps certain D-brane
configurations to a bound state of single D4-brane with multiple
D0-branes. This map allows us to write down the generating
functions for the multiplicity of these BPS states.