Dirac operators on cobordisms: degenerations and surgery

We investigate the Dolbeault operator on a pair of pants, i.e., an elementary cobordism between a circle and the disjoint union of two circles. This operator induces a canonical selfadjoint Dirac operator $D_t$ on each regular level set $C_t$ of a fixed Morse function defining this cobordism. We show that as we approach the critical level set $C_0$ from above and from below these operators converge in the gap topology to (different) selfadjoint operators $D_\pm$ that we describe explicitly. We also relate the Atiyah-Patodi-Singer index of the Dolbeault operator on the cobordism to the spectral flows of the operators $D_t$ on the complement of $C_0$ and the Kashiwara-Wall index of a triplet of finite dimensional lagrangian spaces canonically determined by $C_0$.