Dirac operators on cobordisms: degenerations and surgery. (English) Zbl 1275.58015
Summary: 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 \( C_0\).
MSC:
| 58J20 | Index theory and related fixed-point theorems on manifolds |
| 58J28 | Eta-invariants, Chern-Simons invariants |
| 58J30 | Spectral flows |
| 58J32 | Boundary value problems on manifolds |
| 53B20 | Local Riemannian geometry |
| 35B25 | Singular perturbations in context of PDEs |
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