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Once we put a quantum field theory on a curved manifold, it is natural tofurther assume that coupling constants are position dependent. The positiondependent coupling constants then provide an extra contribution to the Weylanomaly so that we may attempt to cancel the entire Weyl anomaly on the curvedmanifold. We show that such a cancellation is possible for constant Weyltransformation or infinitesimal but generic Weyl transformation in two and fourdimensional conformal field theories with exactly marginal deformations. Whenthe Weyl scaling factor is annihilated by conformal powers of Laplacian (e.g.by Fradkin-Tseytlin-Riegert-Paneitz operator in four dimensions), thecancellation persists even at the finite order thanks to a nice mathematicalproperty of the $Q$-curvature under the Weyl transformation.

Cancelling Weyl anomaly from position dependent coupling