Alberto Richtsfeld (UP)
This thesis investigates the Local Index Theorem and the Atiyah-Patodi-Singer Index Theorem for geometric elliptic operators of first order beyond the classical Dirac-type operators.
Chiral geometric operators are introduced as first-order elliptic differential operators whose principal symbols are determined by universal elliptic symbols. The chirality condition ensures that the orientation of a manifold induces a Z2 -grading
on the bundle upon which the operator acts, rendering the operator odd with respect to this grading. The class of chiral geometric operators includes both the Dirac operator and the Rarita-Schwinger operator.
The Local Index Theorem for chiral geometric operators is then established,
demonstrating that the supertrace of the heat kernel converges to a characteristic form as time approaches zero. This characteristic form is determined by the Atiyah-Singer Index Theorem, and the result is achieved by adapting invariance theory methods developed by Gilkey and by Atiyah, Bott, and Patodi.
Furthermore, the thesis demonstrates the equality between the APS index as formulated by Bär and Bandara and the APS index in the b-calculus setting. Specifically, it is shown that the APS index, in the sense of Bär and Bandara, coincides with the index of the operator extended to a manifold with a cylindrical end attached to each boundary component. The index of these extended operators can then be calculated using index formulas from the b-calculus, providing an APS index formula for general first-order elliptic differential operators that are of product type
near the boundary.
Additionally, for chiral geometric operators, the product structure assumption near the boundary can be removed by introducing a boundary integral involving the transgression form. An algorithm is presented for calculating transgression forms of characteristic classes, expressing them in terms of the second fundamental form and the Riemannian curvature tensor.
