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2010.31: Witten-Hodge theory for manifolds with boundary and equivariant cohomology

2010.31: Qusay Al-Zamil and James Montaldi (2010) Witten-Hodge theory for manifolds with boundary and equivariant cohomology.

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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 $\delta_{X_M}$. Witten \cite{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 $\delta_{X_M}$; the proof involves showing that certain boundary value problems are elliptic. 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 extension to an $X_M$-harmonic field on $M$, with corresponding boundary conditions.

Item Type:MIMS Preprint
Uncontrolled Keywords:Hodge theory, manifolds with boundary, equivariant cohomology, Killing vector fields
Subjects:MSC 2000 > 35 Partial differential equations
MSC 2000 > 53 Differential geometry
MSC 2000 > 55 Algebraic topology
MSC 2000 > 57 Manifolds and cell complexes
MIMS number:2010.31
Deposited By:Dr James Montaldi
Deposited On:10 September 2010

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