Let $V$ be a chain complex, which is either $Z$ or $Z/2$ graded. A circle action on $V$ is
by definition an action of the dga $H_\ast(S^1)$. This consists of a map $D : V → V$ , which is of square zero, commutes with $d$, and in the $Z$ graded case is of degree 1.
the homotopy invariant is
$V^{hS^1}= V [[t]]$
with differential $d+tD$. That is if $vf(t)$ is an element $V [[t]]$, then
$d(vf(t)) = (dv)f(t) + (Dv)tf(t)$

Could anyone explain to me how this works? why homotopy invariant is that? I am not quite familiar with homotopy invariant.(but I know what it is)