A note on the Atiyah-Singer index theorem for manifolds with totally antisymmetric H torsion
A note on the Atiyah-Singer index theorem for manifolds with totally antisymmetric H torsion
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关于完全反对称 H 挠率流形的 Atiyah-Singer 指数定理
DOI：
10.1088/0305-4470/21/10/008
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发表时间：
1988
期刊：
影响因子：
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通讯作者：
N. Mavromatos
中科院分区：
文献类型：
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作者：
N. Mavromatos
作者提出了狄拉克算子的 Atiyah-Singer 指数定理 (1969) 的两个证明，用于无边界的偶维可定向流形，具有消失旋度的完全反对称挠率（H 挠率），例如，可以由超弦启发的十维理论的超引力倍数中出现的反对称张量的场强提供，或者由 Kaluza-Klein 中的群流形的结构常数提供理论。其中一种方法被认为是正式的，它利用传统热核正则化的修改版本。另一个通过勒让德变换连接到第一个，使用超对称路径积分方法。当然，就指标而言，H部分形成了全局定义的精确形式，因此作者的结果与包含扭转情况下指标的不变性是一致的。作者的方法的明显优点是计算简单，与传统的正则化方法相比，尽管扭转对（物理）结果的影响很小，但实际上不可能处理超过四个维度的问题。
The author presents two proofs of the Atiyah-Singer index theorem (1969) for the Dirac operator for even-dimensional orientable manifolds without boundary with totally antisymmetric torsion with vanishing curl (H torsion) which, for example, may be provided by the field strength of the antisymmetric tensor occurring in the supergravity multiplet of superstring-inspired ten-dimensional theories, or by the structure constants of the group manifold in Kaluza-Klein theories. One of the methods, which is considered as formal, utilises a modified version of the conventional heat kernel regularisation. The other which is shown to be connected to the first one through a Legendre transformation uses the supersymmetric path integral approach. Of course, as far as the index is concerned, the H parts form a globally defined exact form and hence the author's result is consistent with the invariance of the index under the inclusion of torsion. The obvious advantage of the author's approach is the simplicity of the calculations in contrast to conventional regularisation methods which are practically impossible to handle beyond four dimensions, despite the trivial effects of the torsion on the (physical) results.