Partition functions and Jacobi fields in the Morse theory

We study the semiclassical partition function in the frame work of the Morse theory, to clarify the phase factor of the partition function and to relate it to the eta invariant of Atiyah. Converting physical system with potential into a curved manifold, we exploit the Jacobi fields and their corresponding eigenvalue equations to be associated with geodesics on the curved manifold and the Hamilton-Jacobi theory.