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We analyse the parity-odd ("type P") surface anomalies of the energy-momentumtensor correlators in conformal field theories, with an emphasis on d=4 and d=3dimensional spacetimes. Using cohomology analysis we construct the expressionfor the most general P-type surface trace anomaly on a singular 2-dimensionalsurface in 4-dimensional bulk spacetimes. As an important example, wespecialise to the case when the singular surface is a conical defect and showthat the bulk P-type Pontryagin trace anomaly induces such a surface traceanomaly. We show that this conical type P surface trace anomaly is given purelyby the outer curvature tensor. In addition, we analyse parity-odd surfacecontact terms in energy-momentum tensor correlators in the flat spacetimeinduced by the conical defect by studying two special cases in which thecontact terms are induced by, (1) type P trace anomaly in d=4 and, (2)gravitational Chern-Simons Lagrangian term in d=3 spacetime dimensions. In bothcases we show that the surface contact terms appear in correlators of the lowerrank than the corresponding bulk surface terms.

Parity-odd surface anomalies and correlation functions on conical defects