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

Understand

There is no known obstructions, but we have not been aware of any concrete examples, either.

• The Wess-Zumino consistency condition for the conformal anomaly says that $a$ cannot change but does not say anything about $c$.
• In supersymmetric models, both $a$ and $c$ are determined from the triangle t'Hooft anomalies and the unitarity demands that both must be fixed, so the unitary supersymmetric conformal field theories do not admit such a possibility.
• Given this field theory situation, we construct an effective AdS/CFT model without supersymmetry in which $c$ changes under exactly marginal deformations.