2-loop free energy of M2 brane in AdS$_7 \times$ S$^4$ and surface defect anomaly in (2,0) theory

Abstract: $\frac{1}{2}$-BPS surface operator viewed as a conformal defect in rank $N$ 6d (2,0) theory is expected to have a holographic description in terms of a probe M2 brane wrapped on AdS$_3$ in the AdS$_7\times S^4$ M-theory background. The M2 brane has an effective tension T$_2= \frac{2}{ π} N$ so that the large tension expansion corresponds to the $1/N$ expansion. The value of the defect conformal anomaly coefficient in $SU(N)$ (2,0) theory was previously argued to be b$=12N- 9 - 3N^{-1}$. Semiclassically quantizing M2 brane it was found in arXiv:2004.04562 that the first two terms in b are indeed reproduced by the classical and 1-loop corrections to the M2 free energy. Here we address the question if the 2-loop term in the M2 brane free energy reproduces the $N^{-1}$ term in b. Despite the general non-renormalizability of the standard BST supermembrane brane action we find that, remarkably, the 2-loop correction to the free energy of the AdS$_3$ M2 brane in AdS$_7\times S^4$ is UV finite (modulo power divergences that can be removed by an analytic regularization). Moreover, the 2-loop correction vanishes in the dimensional and $ζ$-function regularizations. This result appears to be in disagreement with the non-vanishing of the coefficient of the $N^{-1}$ term in the expected expression for the anomaly coefficient b. We discuss possible resolutions of this puzzle, including the one that the M2 brane probe computation may be capturing the surface defect anomaly in the $U(N)$ rather than the $SU(N)$ boundary 6d CFT.