On co-dimension 2 defect anomalies in N=4 SYM and (2,0) theory via brane probes in AdS/CFT

We consider a 12-BPS solution for a D3 brane probe in AdS₅ S⁵ that has world-volume geometry of AdS₃ S¹. It intersects the boundary over a surface that represents a dimension 2 defect in the boundary N=4 SYM theory. The effective action of the probe brane is proportional to the logarithmically divergent volume of AdS₃ and may thus be interpreted as computing conformal anomaly of the supersymmetric S² defect. The classical action scales as N. We compute the 1-loop correction to it due to quantum fluctuations of the D3 brane world-volume fields and compare the result to an earlier suggested expression for the defect anomaly. We also perform a similar analysis of a 12-BPS M5 brane probe solution in AdS₇ S⁴ with the world-volume geometry of AdS₅ S¹ that represents a dimension 4 defect in the boundary (2, 0) 6d theory. Here the classical M5 brane action computes the leading order N² term in a-anomaly of the supersymmetric S⁴ defect. We perform a detailed computation of the 1-loop correction to the M5 brane effective action and thus provide a prediction for the subleading constant in the S⁴ defect a-anomaly coefficient.