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We consider a compact, oriented, smooth Riemannian manifold $M$ (with orwithout 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 $M$, onedefines 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 cohomologyclasses 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 areisomorphic to the ordinary de Rham cohomology groups of the set $N(X_M)$ ofzeros of $X_M$. Our principal purpose is to extend these results to manifoldswith boundary. In particular, we define relative (to the boundary) and absoluteversions of the $X_M$-cohomology and show the classes have representative$X_M$-harmonic fields with appropriate boundary conditions. To do this wepresent the relevant version of the Hodge-Morrey-Friedrichs decompositiontheorem for invariant forms in terms of the operators $d_{X_M}$ and$\delta_{X_M}$. We also elucidate the connection between the $X_M$-cohomologygroups and the relative and absolute equivariant cohomology, following work ofAtiyah and Bott. This connection is then exploited to show that every harmonicfield 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 subspacesof $X_M$-harmonic fields on $M$ with appropriate boundary conditions, followingrecent work of DeTurck and Gluck.

Witten-Hodge theory on manifolds with boundary and equivariant cohomology