 Atiyah-Singer index theorem

For a Twisted Dirac operator on a closed even-dimensional spin manifold, the theorem identifies its analytic Fredholm index with the integral of the top-degree part of the A-hat form times the Chern character. This is one geometric specialization of the general theorem for elliptic differential operators, which expresses the analytic index in terms of the topological class of the principal symbol. The heat proof pairs all positive eigenvalues by the odd Dirac operator, leaving only the index in the supertrace; Getzler rescaling computes its local limit from curvature.