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arXiv 1803.04741gr-qchep-thmath-phmath.MP
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Chern-Weil定理，临界维数的Lovelock拉格朗日量和重力作用的边界项

Chern-Weil theorem, Lovelock Lagrangians in critical dimensions and boundary terms in gravity actions

Nathalie Deruelle, Nelson Merino, Rodrigo Olea

备注：
15 pages
发表信息：
Phys. Rev. D 98, 044031 (2018)
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摘要：

在本文中，我们展示了如何翻译成张量语言的陈-韦伊定理的洛伦兹对称性，这相当于两个流形的欧拉密度的差异的外导的海侵形式。为了做到这一点，我们需要引入一个辅助的，混合的，流形，我们明确地构造其几何。这使我们能够找到矢量密度，它只由时空量构成，其散度是海侵形式的外导数。因此，我们可以显示如何爱因斯坦-希尔伯特，高斯-邦纳，并在一般情况下，欧拉标量密度可以写为真正的矢量密度在临界尺寸$D=2,4$等的分歧洛夫洛克重力是一个维的延续欧拉密度，这些结果是相关的高斯-邦纳，并在一般情况下，洛夫洛克重力。事实上，这些向量，可称为广义Katz向量确保，特别是一个适定的狄利克雷变分原理。

英文摘要：

In this paper we show how to translate into tensorial language the Chern-Weil theorem for the Lorentz symmetry, which equates the difference of the Euler densities of two manifolds to the exterior derivative of a transgression form. For doing so we need to introduce an auxiliary, hybrid, manifold whose geometry we construct explicitely. This allows us to find the vector density, constructed out of spacetime quantities only, whose divergence is the exterior derivative of the transgression form. As a consequence we can show how the Einstein-Hilbert, Gauss-Bonnet and, in general, the Euler scalar densities can be written as the divergences of genuine vector densities in the critical dimensions $D=2,4$, etc. As Lovelock gravity is a dimensional continuation of Euler densities, these results are of relevance for Gauss-Bonnet and, in general, Lovelock gravity. Indeed, these vectors which can be called generalized Katz vectors ensure, in particular, a well-posed Dirichlet variational principle.

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