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There is no known obstructions, but we have not been aware of any concreteexamples, either. The Wess-Zumino consistency condition for the conformalanomaly says that $a$ cannot change but does not say anything about $c$. Insupersymmetric models, both $a$ and $c$ are determined from the trianglet'Hooft anomalies and the unitarity demands that both must be fixed, so theunitary supersymmetric conformal field theories do not admit such apossibility. Given this field theory situation, we construct an effectiveAdS/CFT model without supersymmetry in which $c$ changes under exactly marginaldeformations. So, yes, we can.

Can we change $c$ in four-dimensional CFTs by exactly marginal deformations?