Closed string exchanges on $C^2/Z_2$ in a background B-field

In an earlier work it was shown that the IR singularities arising in the nonplanar one loop two point function of a noncommutative ${\cal N}=2$ gauge theory can be reproduced exactly from the massless closed string exchanges. The noncommutative gauge theory is realised on a fractional $D_3$ brane localised at the fixed point of the $C^2/Z_2$ orbifold. In this paper we identify the contributions from each of the closed string modes. The sum of these adds upto the nonplanar two-point function.