C P -violating super-Weyl anomaly

In C P -violating conformal field theories in four dimensions, the Pontryagin density can appear in the Weyl anomaly. The Pontryagin density in the Weyl anomaly is consistent, but it has a peculiar feature that the parent three-point function of the energy-momentum tensor can violate C P only (semi)locally. In this paper, we study the supersymmetric completion of the Pontryagin density in the Weyl anomaly, where the central charge c effectively becomes a complex number. The supersymmetry suggests that it accompanies the graviphoton θ term associated with the R-symmetry gauging in the Weyl anomaly. It also accompanies new C P -violating terms in the R-current anomaly. While there are no conclusive perturbative examples of C P -violating super Weyl anomaly, we construct explicit supersymmetric dilaton effective action which generates these anomalies.