  We consider a compact, oriented, smooth Riemannian manifold $M$ (with or
without boundary) and we suppose $G$ is a torus acting by isometries on $M$.
Given $X$ in the Lie algebra and corresponding vector field $X_M$ on $M$, one
defines Witten's inhomogeneous coboundary operator $d_{X_M} = d+\iota_{X_M}:
\Omega_G^\pm \to\Omega_G^\mp$ (even/odd invariant forms on $M$) and its adjoint
$\delta_{X_M}$. In the 1980s Witten showed that the resulting cohomology
classes have $X_M$-harmonic representatives (forms in the null space of
$\Delta_{X_M} = (d_{X_M}+\delta_{X_M})^2$), and the cohomology groups are
isomorphic to the ordinary de Rham cohomology groups of the set $N(X_M)$ of
zeros of $X_M$. Our principal purpose is to extend these results to manifolds
with boundary. In particular, we define relative (to the boundary) and absolute
versions of the $X_M$-cohomology and show the classes have representative
$X_M$-harmonic fields with appropriate boundary conditions. To do this we
present the relevant version of the Hodge-Morrey-Friedrichs decomposition
theorem for invariant forms in terms of the operators $d_{X_M}$ and
$\delta_{X_M}$. We also elucidate the connection between the $X_M$-cohomology
groups and the relative and absolute equivariant cohomology, following work of
Atiyah and Bott. This connection is then exploited to show that every harmonic
field with appropriate boundary conditions on $N(X_M)$ has a unique
$X_M$-harmonic field on $M$, with corresponding boundary conditions. Finally,
we define the $X_M$-Poincar\'{e} duality angles between the interior subspaces
of $X_M$-harmonic fields on $M$ with appropriate boundary conditions, following
recent work of DeTurck and Gluck.
