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

Abstract

A $\frac{1}{2}$-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$=12N- 9 - 3N^{-1}$. At the same time, one may expect that the probe M2 brane ending on a stack of $N$ M5 branes should represent a Wilson surface operator in the $U(N)$ rather than $SU(N)$ boundary 6d CFT. In this case one should get b$=12N- 9 $, i.e. the $N^{-1}$ term (that in the $SU(N)$ expression ensures that b vanishes for $N=1$) should be absent. By 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 of the value of the next 2-loop term in the M2 brane free energy, i.e. the coefficient of the $N^{-1}$ term in b. Remarkably, despite the general non-renormalizability of the standard BST M2 brane action we find that 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 both dimensional and $ζ$-function regularizations. This supports the expectation that the M2-brane probe computation captures the surface-defect anomaly of the $U(N)$ rather than the $SU(N)$ boundary 6d theory.