  We present BPS indices of the supergroup WZW models that live on intersecting
M2-M5 brane systems. They can encode data of the stretched M2-branes between
M5-branes and count the BPS states. They are generally expressed in terms of
mock theta functions via the Kac-Wakimoto character formula of the affine Lie
superalgebra. We give an explicit expression of the index for the
$PSL(2|2)_{k=1}$ WZW model in terms of the second order multi-variable
Appell-Lerch sum. It indicates that wall-crossing occurs in the BPS state
counting due to the $C$-field on the M5-branes.
