  Actions for noncommutative (NC) gauge field theories can be expanded
perturbatively in powers of the noncommutativity parameter $\theta$ using the
Seiberg-Witten map between ordinary classical fields and their NC counterparts.
The leading order term represents classical ($\theta=0$) action while
higher-order terms give us $\theta$-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 $\theta$-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 $\theta$.
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, $\theta$-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 $\theta$-dependent Pontryagin
density.
