Witten–Hodge theory for manifolds with boundary and equivariant cohomology

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 of G   and corresponding vector field XMXM on M  , one defines Wittenʼs inhomogeneous coboundary operator dXM=d+ιXM:ΩG±→ΩG∓ (even/odd invariant forms on M  ) and its adjoint δXMδXM. Witten (1982) [18] showed that the resulting cohomology classes have XMXM-harmonic representatives (forms in the null space of ΔXM=(dXM+δXM)2ΔXM=(dXM+δXM)2), and the cohomology groups are isomorphic to the ordinary de Rham cohomology groups of the set N(XM)N(XM) of zeros of XMXM. 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 XMXM-cohomology and show the classes have representative XMXM-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 dXMdXM and δXMδXM. We also elucidate the connection between the XMXM-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(XM)N(XM) has a unique XMXM-harmonic field on M  , with corresponding boundary conditions. Finally, we define the XMXM-Poincaré duality angles between the interior subspaces of XMXM-harmonic fields on M with appropriate boundary conditions, following recent work of DeTurck and Gluck.