Alberto Richtsfeld: Index theory for geometric elliptic differential operators of first order

Abstract:

This talk explores novel advances in index theory for first-order geometric elliptic operators, extending beyond the classical Dirac operator framework. We introduce chiral geometric operators, a class—including both Dirac and Rarita-Schwinger operators—characterized by a natural \(\mathbb{Z}_2\)-grading induced by manifold orientation. Employing invariance theory methods, we establish a Local Index Theorem showing that the supertrace of the heat kernel converges to a characteristic form as dictated by the Atiyah-Singer Index Theorem. Furthermore, we demonstrate the equivalence between the Atiyah-Patodi-Singer index as formulated by Bär and Bandara and the APS index in the b-calculus framework.