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Covariant star product on semi-conformally flat noncommutative Calabi–Yau manifolds and noncommutative topological index theorem

Department of Applied Mathematics and Computer Science, Faculty of Mathematics and Statistics University of Isfahan, sfahan 81746-73441 Iran

School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran

E-mail Address: ab.varshovi@sci.ui.ac.ir

E-mail Address: amirabbassv@ipm.ir

E-mail Address: varshoviamirabbass@gmail.com

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https://doi.org/10.1142/S0219887823501682Cited by:0 (Source: Crossref)

Abstract

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 $X = Y \times Z$ $X = Y \times Z$ , wherein $Y$ $Y$ is a closed manifold, and $Z$ $Z$ is a flat Calabi–Yau $m$ $m$ -fold. Also, a semi-conformally flat metric is considered for $X$ $X$ which leads to a dynamical noncommutative spacetime from the viewpoint of noncommutative gravity. Based on the Kahler form of $Z,$ $Z,$ the noncommutative star product is defined covariantly on vector bundles over $X$ $X$ . This covariant star product leads to the celebrated Groenewold–Moyal product for trivial vector bundles and their flat connections, such as $C^{\infty} (X)$ $C^{\infty} (X)$ . Hereby, the noncommutative characteristic classes are defined properly and the noncommutative Chern–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.

Keywords:

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