arXiv 1004.26872010-04math.DGmath-phmath.ATmath.MP

具有边界和等变上同调的流形上的Witten-Hodge理论

Witten-Hodge theory on manifolds with boundary and equivariant cohomology

Qusay S. A. Al-Zamil, James Montaldi

备注：: 17 pages

摘要：

考虑一个紧致的、有向的、光滑的黎曼流形$M$（有边界或无边界），并假设$G$是一个等距作用在$M$上的环面.给定李代数中的$X$和$M$上相应的向量场$X_M$，定义了Witten非齐次共边界算子$d_{X_M} = d+ι_{X_M}: Ω_G^\pm \toΩ_G^\mp$（$M$上的偶/奇不变形式）及其伴随$δ_{X_M}$ .在20世纪80年代威滕表明，由此产生的上同调类有$X_M$ -调和代表（形式在零空间的$Δ_{X_M} = (d_{X_M}+δ_{X_M})^2$），和上同调群是同构的普通德拉姆上同调群的集合$N(X_M)$的零$X_M$。我们的主要目的是将这些结果推广到带边界的流形。特别是，我们定义的相对（边界）和绝对版本的$X_M$ -上同调，并显示类有代表性的$X_M$ -调和字段与适当的边界条件。为了做到这一点，我们提出了相关版本的Hodge-Morrey-Friedrichs分解定理的不变形式的运营商$d_{X_M}$和$δ_{X_M}$。我们还阐明了$X_M$ -上同调群和相对和绝对等变上同调之间的联系，以下工作的Atiyah和博特。然后利用这种联系表明，每个谐波场与适当的边界条件在$N(X_M)$上有一个唯一的$X_M$ -谐波场在$M$，相应的边界条件。最后，我们定义了$M$上的$X_M$ -调和场的内部子空间之间的$X_M$ -Poincaré对偶角，并给出了相应的边界条件。

英文摘要：

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 $d_{X_M} = d+ι_{X_M}: Ω_G^\pm \toΩ_G^\mp$ (even/odd invariant forms on $M$) and its adjoint $δ_{X_M}$. In the 1980s Witten showed that the resulting cohomology classes have $X_M$-harmonic representatives (forms in the null space of $Δ_{X_M} = (d_{X_M}+δ_{X_M})^2$), 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 $δ_{X_M}$. 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 $X_M$-Poincaré duality angles between the interior subspaces of $X_M$-harmonic fields on $M$ with appropriate boundary conditions, following recent work of DeTurck and Gluck.

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