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| Paper IPM / M / 17615 |
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| Abstract: | |
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A differential geometric statement of the noncommutative topological index theorem is
worked out for covariant star products on noncommutative vector bundles. To start,
a noncommutative manifold is considered as a product space X = Y Ã?? Z, wherein
Y is a closed manifold, and Z is a flat Calabiâ??Yau m-fold. Also, a semi-conformally
flat metric is considered for X which leads to a dynamical noncommutative spacetime
from the viewpoint of noncommutative gravity. Based on the Kahler form of Z, the
noncommutative star product is defined covariantly on vector bundles over X. This
covariant star product leads to the celebrated Groenewoldâ??Moyal product for trivial
vector bundles and their flat connections, such as Câ??(X). Hereby, the noncommutative
characteristic classes are defined properly and the noncommutative Chernâ??Weil theory is
established by considering the covariant star product and the superconnection formalism.
Finally, the index of the -noncommutative version of elliptic operators is studied and
the noncommutative topological index theorem is stated accordingly.
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