Witten Laplacian on pinned path group and its expected semiclassical behavior

Let (X, g) be a d-dimensional compact Riemannian manifold and let f be a Morse function on X. That is, the set of critical points of f is a finite set {c1, . . . , ck} and the Hessian there are nondegenerate. Let dλ = e−λf/2deλf/2. Here d denotes the exterior differential operator on X. Taking an adjoint of dλ on L2(∧T ∗X, dx) (dx is the Riemannian volume), we see dλ = e λf/2d∗e−λf/2 explicitly. dλ defines an elliptic complex which is called a Witten complex. Let ¤λ = dλdλ + dλdλ. Then we see that ¤λα = ¤α+ λ2 4 |∇f(x)|2α+ λ 2 d ∑