Adiabatic Limit and Analytic Torsion of Vector Bundles

For a vector bundle $E^{n+k}$ over a closed manifold $M^n$ with $k$ even and $n$ odd, we equip the metric with an adiabatic parameter, and prove that the index of $E$ is the same as the index of $M$. We also introduce an analog of analytic torsion on $E$ using the Witten Laplacian. Moreover, we prove that the Quillen metric associated with this analytic torsion coincides with that of $M$.

💡 Research Summary

This paper studies the adiabatic limit and analytic torsion for a vector bundle (E^{n+k}\to M^{n}) where the fiber dimension (k) is even and the base dimension (n) is odd. The authors equip the total space with a family of metrics \