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

We consider a $\frac{1}{2}$-BPS solution for a D3 brane probe in AdS$_5 \times S^5$ that has world-volume geometry of AdS$_3 \times S^1$. 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 AdS$_3$ and may thus be interpreted as computing conformal anomaly of the supersymmetric $S^2$ 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 $\frac{1}{2}$-BPS M5 brane probe solution in AdS$_7 \times S^4$ with the world-volume geometry of AdS$_5 \times S^1$ that represents a dimension 4 defect in the boundary (2,0) 6d theory. Here the classical M5 brane action computes the leading order $N^2$ term in $a$-anomaly of the supersymmetric $S^4$ 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 $S^4$ defect $a$-anomaly coefficient.