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

$\frac{1}{2}$-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.

💡 Research Summary

This paper investigates the holographic computation of the conformal anomaly coefficient (b) for a 1/2-BPS surface defect in the six-dimensional (2,0) conformal field theory. The defect is described by a probe M2-brane wrapping an AdS3 subspace within the AdS7×S4 M-theory background, which is dual to the state of N coincident M5-branes. The effective M2-brane tension is T2 = (2/π)N, making the large tension expansion equivalent to a 1/N expansion in the boundary theory.

Previous work established that the expected anomaly coefficient for a surface defect in the fundamental representation of SU(N) (2,0) theory is b = 12N - 9 - 3N^{-1}. The first two terms (12N and -9) were successfully reproduced by the classical action and the 1-loop correction to the M2-brane partition function, respectively. The central question addressed here is whether the 2-loop correction to the M2-brane free energy reproduces the subleading -3N^{-1} term.

The authors expand the BST M2-brane action around the AdS3 minimal surface up to quartic order in fluctuations. A suitable κ-symmetry gauge is chosen that eliminates cubic interaction terms. The fluctuation spectrum consists of 4 massive bosons (m^2=3) from AdS7 transverse directions, 4 massless bosons from the S4, and 8 massive Majorana fermions (m=3/2). Consequently, the relevant 2-loop diagrams are simple “bubble” diagrams involving products of two propagators evaluated at coincident points in AdS3.

Despite the general non-renormalizability of the supermembrane action, the calculation reveals a remarkable finiteness property for this specific setup: the 2-loop correction is UV finite modulo power divergences, which are removed using analytic regularization schemes. The core technical result is expressed in Eq. (1.14), where the 2-loop coefficient f2 is given as a combination of coincident-point limits of the massive scalar (G_x) and fermion (G_θ) propagators in a generalized AdSd+1 space with d=2-2ε (dimensional reduction regularization). Crucially, this combination is proportional to (d-2), leading to a vanishing result in the limit ε→0 (i.e., back to three dimensions). The same vanishing result is confirmed using ζ-function regularization.

This finding, b2 = 0, is in clear disagreement with the predicted value of b2 = -3 from the field theory expression b = 12N - 9 - 3N^{-1}. The paper discusses several potential resolutions to this puzzle: 1) The possible contribution of higher-derivative counterterms required for the non-renormalizable M2-brane action. 2) More fundamental issues in how the AdS/CFT dictionary is applied to brane partition functions. 3) The most straightforward explanation, suggested to the authors, is that the M2-brane probe computation might actually capture the surface defect anomaly in the U(N) version of the (2,0) theory, rather than the SU(N) theory. For the U(N) group, the analogous anomaly formula, with an appropriate generalization of the group-theoretic data, yields b = 12N - 9, which lacks the 1/N term and is thus consistent with a vanishing 2-loop contribution. This implies that the defect operator described by the M2-brane ending on the boundary may couple to the U(1) sector which does not decouple in this context, unlike for bulk observables.

In conclusion, the paper presents a precise and robust 2-loop calculation showing a vanishing result, which challenges the initial field theory expectation and opens up interesting questions about the precise holographic dictionary for defect anomalies and the role of the global structure (SU(N) vs. U(N)) of the dual CFT.