Witten-Helffer-Sjöstrand theory for $S^1$-equivariant cohomology

Given an $S^1$-invariant Morse function $f$ and an $S^1$-invariant Riemannian metric $g$, a family of finite dimensional subcomplexes $(\widetilde \Omega^*_{inv,sm}(M,t), D(t))$, $t\in [0,\infty)$, of the Witten deformation of the $S^1$-equivariant de Rham complex is constructed, by studying the asymptotic behavior of the spectrum of the corresponding Laplacian $\widetilde \Delta ^k(t)=D^*_k(t)D_k(t)+D_{k-1}(t)D^*_{k-1}(t)$ as $t\to \infty$. In fact the spectrum of $\widetilde \Delta^k(t)$ can be separated into the small eigenvalues, finite eigenvalues and the large eigenvalues. Then one obtains $( \widetilde \Omega^*_{inv,sm}(M,t),D(t))$ as the complex of eigenforms corresponding to the small eigenvalues of $\widetilde \Delta(t)$. This permits us to verify the $S^1$-equivariant Morse inequalities. Moreover suppose $f$ is self-indexing and $(f,g)$ satisfies the Morse-Smale condition, then it is shown that this family of subcomplexes converges as $t\to \infty$ to a geometric complex which is induced by $(f,g)$ and calculates the $S^1$-equivariant cohomology of $M$.