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Noncommutative $D=5$ Chern-Simons Gravity: Kaluza-Klein Reduction and Chiral Gravitational Anomaly

Actions for noncommutative (NC) gauge field theories can be expanded perturbatively in powers of the noncommutativity parameter $θ$ using the Seiberg-Witten map between ordinary classical fields and their NC counterparts. The leading order term represents classical ($θ=0$) action while higher-order terms give us $θ$-dependent NC corrections that ought to capture some aspects of quantum gravity. Building on previous work of Aschieri and Castellani on NC Chern-Simons (CS) gauge and gravity theories, showing that non-trivial $θ$-dependence exists only for spacetime dimensions $D\geq 5$, we investigate a correlated effect of these extra spatial dimensions and noncommutativity on four-dimensional physics, up to first-order in $θ$. Assuming that one spatial dimension is compactified into a circle, we apply the Kaluza-Klein reduction procedure on the NC $D=5$ CS theory for the conformal gauge group $SO(4,2)$, to obtain an effective, $θ$-dependent four-dimensional theory of gravity that has Einstein-Hilbert gravity with negative cosmological constant as its commutative limit. We derive field equations for this modified theory of gravity and study the effect of NC interactions on some classical geometries, such as the AdS-Schwarzschild black hole. We find that this NC background spacetime gives rise to chiral gravitational anomaly due to the nonvanishing $θ$-dependent Pontryagin density.