Supersymmetry and the Atiyah-Singer Index Theorem II: The Scalar Curvature Factor in the Schr{\" o}dinger Equation

The quantization of the superclassical system used in the proof of the index theorem results in a factor of $\hbar^{2}R/8 $ in the Hamiltonian. The path integral expression of the kernel is analyzed up to and including 2-loop order. The existence of the scalar curvature term is confirmed by comparing the linear term in the heat kernel expansion with the 2-loop order terms in the loop expansion.