= Atiyah-Singer index theorem
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{title2=$\operatorname{ind}D_W^+=\int_M\widehat A(TM)\operatorname{ch}(W)$}
{wiki=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.
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