Topological partition function and string-string duality

Evidence for string/string-duality can be given not only by matching the vector couplings but also by matching the gravitational couplings. In this note this is shown in the rank three example, the closest stringy analog of the Seiberg/Witten-setup, which is related to the Calabi-Yau WP1,1,2,2,64(12). I provide an exact analytical verification of a relation checked by coefficient comparison to fourth order by Kaplunovsky, Louis and Theisen.