Title: A note on the Atiyah-Singer index theorem for manifolds with totally antisymmetric H torsion
Abstract: 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.
Publication Year: 1988
Publication Date: 1988-05-21
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 41
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