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Witten-Hodge theory on manifolds with boundary and equivariant cohomology by Al-Zamil, Qusay S. A.; Montaldi, James is a scholarly article available to read on EtoBox.

What is Witten-Hodge theory on manifolds with boundary and equivariant cohomology about?

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 $X_M$-cohom

Author

Al-Zamil, Qusay S. A.; Montaldi, James

Published

2010

Language

EN