Abstract
We consider the semiclassical asymptotic expansion of the heat kernel coming from Witten’s perturbation of the de Rham complex by a given function. For the index, one obtains a time-dependent integral formula which is evaluated by the method of stationary phase to derive the Poincaré–Hopf theorem. We show how this method is related to approaches using the Thom form of Mathai and Quillen. Afterwards, we use a more general version of the stationary phase approximation in the case that the perturbing function has critical submanifolds to derive a degenerate version of the Poincaré–Hopf theorem.
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Ludewig, M. A semiclassical heat kernel proof of the Poincaré–Hopf theorem. manuscripta math. 148, 29–58 (2015). https://doi.org/10.1007/s00229-015-0741-y
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DOI: https://doi.org/10.1007/s00229-015-0741-y