Surface operators in the 6d N = (2, 0) theory

Understanding the six dimensional $\mathcal{N}=\left(2,0\right)$ superconformal field theory is one of the most intriguing problems in theoretical physics. In this paper we revisit the most natural observables in this theory, surface operators [1]. If we define the theory as arising from N coincident M5-branes, the simplest surface operators correspond to the endpoints of M2-branes [2].

In some ways the surface operators in six dimensions are analogous to Wilson loops in lower dimensional gauge theories. Wilson loops are the boundaries of fundamental strings, which are the dimensional reduction of M2-branes, and indeed one obtains Wilson loops in compactifications of the 6d theory with surface operators. Wilson loops are not only interesting due to their physical importance, they are also accessible to many perturbative and non-perturbative calculational tools in supersymmetric field theories: Feynman diagrams, holographic descriptions [3–5], localization [6], the defect CFT framework and associated OPE techniques [7, 8], integrability [9, 10], duality to scattering amplitudes [11] and more. See for instance a recent survey of these techniques, as applied to supersymmetric Wilson loops in ABJM theory [12].

We do not expect all these techniques to extend to surface operators in six dimensions, but it is worthwhile to examine which of them may work, and we hope that some calculations may lead to exact results applicable for all N. Here we take the first step in such an examination, defining the notion of a 'locally BPS surface operator' and studying basic properties of their anomalies. This is mainly based on previous work [13–17], which we modify and refine in several ways.

As reviewed in the next section, the evaluation of generic surface operators leads to logarithmic divergences. The anomaly depends on the geometry of the surface, as well as intrinsic properties of the operator which are captured by three numbers, known as anomaly coefficients [18].

The 'locally BPS' operator couples to the scalar fields via a unit five-vector nⁱ . This can be viewed as a coupling to an R-symmetry background, and for non-constant nⁱ we find a new anomaly, proportional to (∂n)², with its own anomaly coefficient.

We perform explicit calculations of the three geometrical and one background coefficients in both the free theory at N = 1 and the holographic description valid at large N. An examination of our results reveals that the new anomaly coefficient matches (up to a sign) one of the geometric ones in both regimes. We present here a simple argument, relying on supersymmetry, why we expect this relation to hold for all N. A more rigorous proof of this relation based on the application of defect CFT techniques to surface operators will be presented in [19].

Beyond the study of $\mathcal{N}=\left(2,0\right)$ superconformal symmetry, surface operators in conformal field theories have drawn interest within a number of different contexts. Recent work on entangling surfaces in 4d [20–23] and theories with boundaries [24–26] uses some techniques which apply in our case as well. In particular, the classification of local conformal invariants of surfaces is independent of the codimension and translates to the 6d case [27].

Surface operators in the $\mathcal{N}=\left(2,0\right)$ theory have been studied both from a field theory perspective [14–17] and using holography [13, 28]. Corresponding soliton solutions of the M5-brane equations of motion have been discussed in the literature under the moniker of self-dual strings [1].

The resemblance to Wilson loops is evident in both the field theoretic and the holographic approach. In the former, for N = 1 as is studied in section 3, we define the surface operator in analogy to the Maldacena–Wilson loops [4] as

$$\begin{equation}{V}_{{\Sigma}}=\mathrm{exp}{\int }_{{\Sigma}}\left(i{B}^{+}-{n}^{i}{{\Phi}}_{i}\,{\text{vol}}_{{\Sigma}}\right),\end{equation} \tag{ 1.1 }$$

where B⁺ is the pullback of the chiral two-form to the surface Σ and Φⁱ are the scalar fields.

Since for N > 1 there is no realisation of the theory in terms of fundamental fields, we cannot give an analogous definition of the surface operator. However, by analogy with Wilson loops [3–5], in the large N limit, these operators in the fundamental representation have a nice holographic dual as M2-branes ending on the surface and extending into the AdS₇ × S⁴ bulk, as discussed in section 4. In the absence of a scalar coupling breaking the $\mathfrak{s}\mathfrak{o}\left(5\right)$ R-symmetry, these would be delocalised on the S⁴ [29, 30]. At leading order, we need only consider minimal three-volumes [4, 13] (similar to the minimal surfaces of interest in the Wilson loop case [3–5]), and to find the anomaly, which is a local quantity, it is enough to understand the volume close to the AdS boundary. High-rank (anti-)symmetric representations are dual to configurations involving M5 branes shrinking to the surface on the boundary of AdS₇ and have been considered in [31–34].

The definition in (1.1) includes BPS operators. Simple examples are the plane or sphere with constant unit nⁱ . Other examples are briefly discussed in section 3 and will be explored in more detail elsewhere [35]. We call operators with generic Σ and unit length nⁱ 'locally BPS', and show that they possess some nice properties, in particular that all power law divergences cancel.

In the next section we recall the structure of surface operator anomalies and introduce the anomaly coefficients. We evaluate these anomaly coefficients for the two known realisations of the $\mathcal{N}=\left(2,0\right)$ theory; first as the theory of a single M5-brane (N = 1) [36], for which the equations of motion are known [37], and second, using holography (for the large N limit) from M-theory on the AdS₇ × S⁴ background [38] found in [39]. The resulting anomaly coefficients are presented in equations (3.24) and (4.18). After performing the free field and holographic calculations, we address in section 5 surfaces with singularities. We discuss our results in section 6 and offer a simple argument for the relation between two of the anomaly coefficients. We collect some technical tools in appendices. Our conventions can be found in appendix A. Details of the geometry of submanifolds are compiled in appendix B. Appendix C contains an alternative, more geometric derivation of the field theory results in section 3.

The most natural quantities associated to surface operators in conformal field theories are their anomaly coefficients. To understand their origin, note that, unlike line operators, the expectation values of surface operators typically suffer from ultraviolet divergences, which cannot be removed by the addition of local counterterms. The regularised expectation value satisfies

$$\begin{equation}\mathrm{log}\left\langle {V}_{{\Sigma}}\right\rangle \sim \mathrm{log}\enspace {\epsilon}{\int }_{{\Sigma}}{\text{vol}}_{{\Sigma}}\,{\mathcal{A}}_{{\Sigma}}+\text{finite},\end{equation} \tag{ 2.1 }$$

where is a regulator, ${\mathcal{A}}_{{\Sigma}}$ is known as the anomaly density, and we suppressed possible power-law divergences.

${\mathcal{A}}_{{\Sigma}}$ is scheme independent and indicates an anomalous Weyl symmetry, since for a constant rescaling g → e^2ω g, the expectation value varies as

$$\begin{equation}\mathrm{log}{\left\langle {V}_{{\Sigma}}\right\rangle }_{{\text{e}}^{2\omega }g}-\mathrm{log}{\left\langle {V}_{{\Sigma}}\right\rangle }_{g}=\omega {\int }_{{\Sigma}}{\text{vol}}_{{\Sigma}}\,{\mathcal{A}}_{{\Sigma}},\end{equation} \tag{ 2.2 }$$

where the subscript ${\left\langle {\bullet}\right\rangle }_{g}$ denotes the background metric.

The anomaly is constrained by the Wess–Zumino consistency condition [18, 40] to be conformally invariant. In dimensions d ⩾ 3, the local geometric conformal invariants for a 2d submanifold, which have been classified in [27], are

• R^Σ: the Ricci scalar of the induced metric h_ab on Σ.
• H² + 4 tr  P: H^μ is the mean curvature, P_ab the pullback of the Schouten tensor (B.2).
• tr W: W_abcd is the pullback of the Weyl tensor.

Under conformal transformations, the first two change by a total derivative (type A anomalies) and the last is itself conformally invariant (type B).

As we allow for variable couplings to the scalars, parametrised by a unit five-vector nⁱ , we find an extra potential type B Weyl anomaly associated to it:

• (∂n)² ≡ ∂^a nⁱ ∂_a n_i .

This is (up to total derivatives) the only quantity of the correct dimension that can be constructed using only n.

The anomaly of a surface operator in any 6d $\mathcal{N}=\left(2,0\right)$ theory then takes the form

$$\begin{equation}{\mathcal{A}}_{{\Sigma}}=\frac{1}{4\pi }\left[{a}_{1}{\mathcal{R}}^{{\Sigma}}+{a}_{2}\left({H}^{2}+4\enspace \text{tr}\enspace P\right)+b\enspace \text{tr}\enspace W+c{\left(\partial n\right)}^{2}\right].\end{equation} \tag{ 2.3 }$$

The anomaly coefficients a₁, a₂, b and c depend on the theory (that is on N) and the type of surface operator (which, at least at large N, is specified by the representation of the A_N−1 algebra [41, 42]), but not on its geometry or n. They are the focus of this paper.

Let us mention that there exists another commonly used basis where

$$\begin{equation}{\mathcal{A}}_{{\Sigma}}=\frac{1}{4\pi }\left[a{R}^{{\Sigma}}+{b}_{1}\enspace \text{tr}\enspace {\tilde {\text{II}}}^{2}+{b}_{2}\enspace \text{tr}\enspace W+c{\left(\partial n\right)}^{2}\right],\end{equation} \tag{ 2.4 }$$

where ${\tilde {\text{II}}}_{ab}^{\mu }$ is the traceless part of the second fundamental form (see (B.8)). These bases are related through the Gauss–Codazzi equation (B.7). The relation between the coefficients is then

$$\begin{equation}\begin{aligned}\hfill {a}_{1}& =-{b}_{1}+a,\quad \hfill & \hfill 2{a}_{2}& ={b}_{1},\quad \hfill & \hfill b& ={b}_{2}+{b}_{1},\hfill \\ \hfill a& ={a}_{1}+2{a}_{2},\quad \hfill & \hfill {b}_{1}& =2{a}_{2},\quad \hfill & \hfill {b}_{2}& =b-2{a}_{2}.\hfill \end{aligned}\end{equation} \tag{ 2.5 }$$

Some results about these anomaly coefficients are known for surface defects in generic CFTs. The bound b₁ < 0 was derived in [22] by showing that b₁ captures the two-point function of the displacement operator, which is positive by unitarity. Similarly, it was shown in [22, 43] that b₂ is calculated by the one-point function of the stress tensor in the presence of the surface defect (this was also conjectured in [44]). Assuming that the average null energy condition holds in the presence of defects also leads to a bound b₂ > 0 [23].

For the surface operators at hand, these anomaly coefficients were also calculated previously. At large N, the first such result was a calculation of the 1/2-BPS sphere [28], with total anomaly −4N, implying ${a}_{1}^{\left(N\right)}+2{a}_{2}^{\left(N\right)}=-2N$, at leading order at large N. This was soon followed by the more detailed result ${a}_{2}^{\left(N\right)}=-N$ and ${a}_{1}^{\left(N\right)}={b}^{\left(N\right)}=0$ [13].

More recently, it was conjectured that $\mathcal{N}=\left(2,0\right)$ supersymmetry imposes b = 0 (or b₁ = −b₂) for any N [45]. a and b₂ were calculated at any N > 1 (and for any representation) by studying the holographic entanglement entropy in the presence of surface operators [23, 34, 46, 47]. This result is also supported by a recent calculation based on the superconformal index [48], which suggests that it is exact.

The anomaly coefficient c has previously not been discussed, to our knowledge.

In this section we study the anomaly coefficients of the surface operator in the abelian (2, 0) theory. This is the theory of a single M5-brane and the degrees of freedom form the tensor supermultiplet of the $\mathfrak{o}\mathfrak{s}\mathfrak{p}\left({8}^{{\ast}}\vert 4\right)$ symmetry algebra. It consists of three fields [49, 50] (see [51, 52] for an overview of superconformal multiplets in various dimensions)

• A real closed self-dual three-form H = dB⁺.
• A chiral spinor ${\psi }_{\alpha \check {\alpha }}$ subject to the symplectic Majorana condition $\overline{\psi }=-c{\Omega}\psi$ (A.14) where c and Ω are charge conjugation matrices, see (A.12).
• Five real scalar fields Φⁱ .

These fields transform into each other under superconformal transformations as [36]

$$\begin{equation}\begin{aligned}\hfill {\delta }_{\varepsilon }{B}_{\mu \nu }^{+}& =\varepsilon \left(x\right){\gamma }_{\mu \nu }\psi ,\hfill \\ \hfill {\delta }_{\varepsilon }\psi & =-{\gamma }^{\mu }{\partial }_{\mu }{{\Phi}}^{i}{\check {\gamma }}_{i}\bar{\varepsilon }\left(x\right)+\frac{1}{12}{\gamma }^{\mu \nu \rho }{H}_{\mu \nu \rho }\bar{\varepsilon }\left(x\right)+4{{\Phi}}^{i}{\check {\gamma }}_{i}{\varepsilon }^{1},\hfill \\ \hfill {\delta }_{\varepsilon }{{\Phi}}^{i}& =-\varepsilon \left(x\right){\check {\gamma }}^{i}\psi .\hfill \end{aligned}\end{equation} \tag{ 3.1 }$$

The parameter $\overline{\varepsilon }\left(x\right)$ is an antichiral spinor of the form ${\overline{\varepsilon }}_{\dot {\alpha }\check {\alpha }}\left(x\right)={\overline{\varepsilon }}_{\dot {\alpha }\check {\alpha }}^{0}+{\left({\overline{\gamma }}_{\mu }\right)}_{\dot {\alpha }}^{\beta }{x}^{\mu }{\varepsilon }_{\beta \check {\alpha }}^{1}$, where ${\bar{\varepsilon }}^{0}$ and ɛ¹ are constant spinors parametrising, respectively, the supersymmetry and special supersymmetry transformations. Our spinor conventions are summarised in appendix A.

3.1. Surface operators and BPS condition

We define the surface operators V_Σ of the abelian theory as in (1.1). We restrict to space-like surfaces in flat 6d Minkowski space (with mostly positive signature). Null surfaces could be interesting by analogy with null polygonal Wilson loops, which are dual to scattering amplitudes in $\mathcal{N}=4$ SYM [11], but lie beyond the scope of this work (see however [53]).

A surface operator is BPS provided that its variation under the supersymmetry transformations (3.1) vanishes

$$\begin{equation}{\delta }_{\varepsilon }{V}_{{\Sigma}}=-\int \varepsilon \left(x\right)\left[\frac{i}{2}{\gamma }_{\mu \nu }{\partial }_{a}{x}^{\mu }{\partial }_{b}{x}^{\nu }{{\epsilon}}^{ab}-{n}^{i}{\check {\gamma }}_{i}\right]\psi \,{\text{vol}}_{{\Sigma}}\,{V}_{{\Sigma}}=0.\end{equation} \tag{ 3.2 }$$

Since this is an integral over the insertion of an operator ψ along the surface, this is satisfied only when the integrand vanishes at every point along the surface, leading to the projector equation

$$\begin{equation}\varepsilon {{\Pi}}_{-}=0,\quad {{\Pi}}_{-}=\frac{1}{2}-\frac{i}{4}{\partial }_{a}{x}^{\mu }{\partial }_{b}{x}^{\nu }{{\epsilon}}^{ab}\frac{{n}^{i}}{{n}^{2}}{\gamma }_{\mu \nu }{\check {\gamma }}_{i}.\end{equation} \tag{ 3.3 }$$

If we impose that n² ≡ nⁱ nⁱ = 1, then Π₋ is a half rank projector and otherwise it is a full rank matrix. In the case of a planar surface with constant unit nⁱ , this is a single condition, so the surface preserves 16 supercharges, i.e. is 1/2-BPS ² .

In analogy to Wilson loops in 4d theories, it is natural to discuss 'locally BPS operators' [5], where the equations (3.3) are satisfied at every point along the surface, but without a global solution. This amounts to the requirement n² = 1, and as shown below, leads to the cancelation of all power-like divergences in the evaluation of the surface operator.

One can also look for surfaces, other than planes, that preserve some smaller fraction of the supersymmetry by relating nⁱ (σ) to x^μ (σ) and its derivatives. One simple way to realise this is for surfaces with the geometry $\mathbb{R}{\times}S$, for some curve $S\subset {\mathbb{R}}^{1,4}$. Upon dimensional reduction this becomes a Wilson loop in 5d maximally supersymmetric Yang–Mills (or 4d upon further dimension reduction). Then one can choose nⁱ to follow the construction of globally BPS Wilson loops of [54] or [55] to find globally BPS surface operators. Indeed this was realised recently in [56] (see also [57]).

There are further examples of globally BPS surface operators, which do not follow this construction. The simplest is the spherical surface, but there are several other classes of such operators, which will be explored elsewhere [35].

3.2. Propagators

Since the abelian theory is non-interacting, the expectation value of V_Σ reduces to

$$\begin{equation}\mathrm{log}\left\langle {V}_{{\Sigma}}\right\rangle =\frac{1}{2}\int \left[-\left\langle {B}^{+}\left(\sigma \right){B}^{+}\left(\tau \right)\right\rangle +\left\langle {{\Phi}}_{i}\left(\sigma \right){{\Phi}}_{j}\left(\tau \right)\right\rangle {n}^{i}\left(\sigma \right){n}^{j}\left(\tau \right)\sqrt{h\left(\sigma \right)h\left(\tau \right)}{d}^{2}\sigma \enspace {d}^{2}\tau \right],\end{equation} \tag{ 3.4 }$$

where h is the determinant of the induced metric on Σ. Evaluating this requires expressions for the propagators of the tensor and scalar fields.

While one would preferably derive the propagators from an action, none is readily available. Many actions for the abelian $\mathcal{N}=\left(2,0\right)$ theory have been proposed over the years, but they all suffer from some pathologies regarding the self-dual two-form (see [36, 58–61] for examples of available actions, and [62–64] and references therein for recent accounts of the various approaches in the abelian theory). In any case, gauge fixing and inverting the kinetic operator is not straightforward.

3.2.1. Tensor structure

We sidestep these obstacles by determining the propagators in other ways. The scalar propagator in flat 6d is fixed by conformal symmetry to be

$$\begin{equation}\left\langle {{\Phi}}_{i}\left(x\right){{\Phi}}_{j}\left(y\right)\right\rangle =\frac{{C}_{{\Phi}}{\delta }_{ij}}{{\left\vert x-y\right\vert }^{4}}.\end{equation} \tag{ 3.5 }$$

The proportionality constant depends on the normalisation of the fields. It could be determined from an action, but in its absence it is fixed by supersymmetry below.

The more complicated question is the self-dual two-form propagator. Let us start by considering an unconstrained two-form field B with a free Maxwell type action

$$\begin{equation}{S}_{\text{tot}}\propto \int {\mathrm{d}}^{6}x\enspace {B}^{\mu \nu }\left(-\left({\delta }_{\mu }^{\rho }{\delta }_{\nu }^{\sigma }-{\delta }_{\nu }^{\rho }{\delta }_{\mu }^{\sigma }\right){\partial }^{2}+4\left(1-\alpha \right){\partial }_{\mu }{\partial }^{\rho }{\delta }_{\nu }^{\sigma }\right){B}_{\rho \sigma },\end{equation} \tag{ 3.6 }$$

were α is a gauge fixing parameter. In Feynman gauge α = 1, this gives the propagator

$$\begin{equation}\left\langle {B}^{\mu \nu }\left(x\right){B}_{\rho \sigma }\left(y\right)\right\rangle =\frac{{C}_{B}\left({\delta }_{\mu }^{\rho }{\delta }_{\nu }^{\sigma }-{\delta }_{\nu }^{\rho }{\delta }_{\mu }^{\sigma }\right)}{{\left\vert x-y\right\vert }^{4}}.\end{equation} \tag{ 3.7 }$$

Now we decompose the field into its self-dual and anti-self-dual parts ${B}_{\mu \nu }={B}_{\mu \nu }^{+}+{B}_{\mu \nu }^{-}$ and try to deduce the propagators for each component.

Since there is no covariant four-tensor satisfying the self-duality properties of a mixed correlator $\left\langle {B}^{+}{B}^{-}\right\rangle$, we can decompose

$$\begin{equation}\left\langle BB\right\rangle =\langle {B}^{+}{B}^{+}\rangle +\langle {B}^{-}{B}^{-}\rangle .\end{equation} \tag{ 3.8 }$$

The two terms on the right-hand side need not be identical, but the difference between them should be parity-odd ³ . The only such term of the right scaling dimension which we can write down is

$$\begin{equation}\langle {B}^{+}{B}^{+}\rangle -\langle {B}^{-}{B}^{-}\rangle \propto {{\epsilon}}_{\rho \sigma \kappa \lambda }^{\mu \nu }\frac{{x}^{\kappa }{y}^{\lambda }}{{\left\vert x-y\right\vert }^{6}}.\end{equation} \tag{ 3.9 }$$

However, terms of this type do not contribute to (3.4), since the integration is symmetric in x and y. Therefore, for the purpose of our calculation we can take $\left\langle {B}^{+}{B}^{+}\right\rangle =\left\langle BB\right\rangle /2$. Note that in curved space we can add to the right-hand side a term proportional to the Weyl tensor with all the required symmetries.

3.2.2. Normalisation

The normalisation of the tensor field propagator is fixed by the assumption that the surface operator defined in (1.1) corresponds to a single unit of quantised charge. First, for any closed surface Σ, we can rewrite the surface operator (without scalars) in terms of the field strength as

$$\begin{equation}\mathrm{exp}{\int }_{{\Sigma}}i{B}^{+}=\mathrm{exp}{\int }_{V}iH,\end{equation} \tag{ 3.10 }$$

where ∂V = Σ. In order for this to be well-defined, any two such V with the same boundary must yield the same result. Equivalently, for every closed three-manifold V

$$\begin{equation}{\int }_{V}H\in 2\pi \mathbb{Z},\end{equation} \tag{ 3.11 }$$

and similarly for *H.

Now consider a flat surface operator in the (x¹, x²) plane, which we view as a source for the self-dual B field. The solution to the equations of motion would be given by convoluting the propagator with this source. Using the expression in (3.7) and adding the factor 1/2 to account for restricting to the self-dual sector, we get

$$\begin{equation}{B}_{\mu \nu }\left(x\right)={\int }_{{\mathbb{R}}^{2}}\frac{1}{2}\frac{{C}_{B}\left({\delta }_{\mu }^{1}{\delta }_{\nu }^{2}-{\delta }_{\nu }^{1}{\delta }_{\mu }^{2}\right)}{\vert x-y{\vert }^{4}}\enspace \mathrm{d}{y}^{1}\enspace \mathrm{d}{y}^{2}.\end{equation} \tag{ 3.12 }$$

Again, because we do not know the self-dual propagator, the field strength we obtain is not self-dual, but the quantisation condition should still be satisfied. Imposing that the charge enclosed in a transverse sphere is quantised leads to

$$\begin{equation}{\int }_{{S}^{3}}{\ast}H=2{\pi }^{3}{C}_{B}=2\pi \quad {\Rightarrow}\quad {C}_{B}=\frac{1}{{\pi }^{2}}.\end{equation} \tag{ 3.13 }$$

The normalisation of the scalar propagator is then fixed by supersymmetry. A simple way to implement that is to compare with the classical BPS solution of the self-dual string [1] which gives ⁴ 2C_Φ = C_B . Overall, we are left with

$$\begin{equation}\left\langle {{\Phi}}_{i}\left(x\right){{\Phi}}_{j}\left(y\right)\right\rangle =\frac{{\delta }_{ij}}{2{\pi }^{2}{\left\vert x-y\right\vert }^{4}},\end{equation} \tag{ 3.14a }$$

$$\begin{equation}\left\langle {B}_{\mu \nu }^{+}\left(x\right){B}_{\rho \sigma }^{+}\left(y\right)\right\rangle =\frac{{\delta }_{\mu \rho }{\delta }_{\nu \sigma }-{\delta }_{\mu \sigma }{\delta }_{\nu \rho }}{2{\pi }^{2}{\left\vert x-y\right\vert }^{4}}.\end{equation} \tag{ 3.14b }$$

We emphasise that this normalisation is obtained by imposing a quantisation condition on the self-dual sector of an unconstrained B-field. This follows the discussion in [58], however some caution is warranted. A proper treatment of the quantisation of a self-dual two-form could add a factor of 2 on the right-hand side of (3.12), changing the resulting anomaly coefficients.

With the flat space propagators we are able to determine the anomaly coefficients a₁, a₂, and c. The calculation of b, however, requires the curved space propagator, where the right-hand side of (3.9) could pick up contributions whose integral does not vanish. Since we do not know how to fix these terms, we cannot determine b.

Note though that we can calculate the contribution of the scalars to the anomaly coefficient b. The propagator of a conformal scalar in a curved background can be expanded in powers of the geodesic distance [17], and the contribution to the anomaly coefficient b is read off as −1/3.

If we give up the requirement of self-duality, we can use the short-distance expansion of an unconstrained two-form propagator on curved space, which has been computed in [14, 16], and again, the Weyl tensor of the background explicitly contributes to the curvature corrections. Halving that to try to account for self-duality and adding to it the contribution from the scalars, one obtains b = −4/3 [17]. This is in disagreement with the conjecture b = 0 [45] and therefore one may not trust it.

3.3. Evaluation of the anomaly

With the propagators at hand, we can compute the expectation value of the surface operator by evaluating the integrals in (3.4). Generically, these integrals are divergent and must be regularised.

In this section we take a rather naive approach of placing a hard UV cutoff on the double integral (3.4), so as to restrict |σ − τ| > (where the distance is measured with the induced metric), the same regularisation that is used in [14]. A different regularisation is employed in [17], where the surface is assumed to be contained within a 5d linear subspace of ${\mathbb{R}}^{6}$ and the two copies of the surface are displaced by a distance in the 6th direction. This restriction to ${\mathbb{R}}^{5}$ must still yield the correct answer, since even for surfaces in 4d the geometric invariants in the anomaly (2.3) are independent of each other. Still, in appendix C we redo the calculation removing this assumption by displacing the two copies of the surface along geodesics in the direction of an arbitrary normal vector field. That approach could be important for the calculation of surface operators in four dimensions, where the restriction to a 3d linear subspace does not allow to resolve all the anomaly coefficients.

To find the anomalies we only need the short-distance behaviour of the propagators, so we use normal coordinates η^a about a point σ on Σ. The notations and required geometry are presented in appendix B.

Starting from the scalar contribution to (3.4), the integrand is

$$\begin{equation}\frac{1}{4{\pi }^{2}}\frac{{n}^{i}\left(\sigma \right){n}^{i}\left(\tau \right)}{\vert x\left(\sigma \right)-x\left(\tau \right){\vert }^{4}}\sqrt{h\left(\sigma \right)}\sqrt{h\left(\tau \right)}.\end{equation} \tag{ 3.15 }$$

Using nⁱ nⁱ = 1 and (B.14), (B.12) we have

$$\begin{equation}\begin{aligned}\hfill {n}^{i}\left(\sigma \right){n}^{i}\left(\tau \right)& =1-\frac{1}{2}\left({\partial }_{a}{n}^{i}{\partial }_{b}{n}^{i}\right){\eta }^{a}{\eta }^{b}+\mathcal{O}\left({\eta }^{3}\right),\hfill \\ \hfill \sqrt{h\left(\tau \right)}& =1-\frac{1}{6}{R}_{ab}^{{\Sigma}}{\eta }^{a}{\eta }^{b}+\mathcal{O}\left({\eta }^{3}\right),\hfill \\ \hfill {\left\vert x\left(\sigma \right)-x\left(\tau \right)\right\vert }^{2}& ={\eta }^{a}{\eta }_{a}-\frac{1}{12}{\text{II}}_{ab}\cdot {\text{II}}_{cd}{\eta }^{a}{\eta }^{b}{\eta }^{c}{\eta }^{d}+\mathcal{O}\left({\eta }^{5}\right).\hfill \end{aligned}\end{equation} \tag{ 3.16 }$$

The integral computing the density of the scalar contribution to $\mathrm{log}\left\langle {V}_{{\Sigma}}\right\rangle$ is then

$$\begin{equation}\frac{1}{4{\pi }^{2}}\int \frac{{\mathrm{d}}^{2}\eta }{\vert \eta {\vert }^{4}}\left[1-\left(\frac{1}{6}{R}_{ab}^{{\Sigma}}+\frac{1}{2}{\partial }_{a}{n}^{i}{\partial }_{b}{n}^{i}\right){\eta }^{a}{\eta }^{b}+\frac{1}{6\vert \eta {\vert }^{2}}{\text{II}}_{ab}\cdot {\text{II}}_{cd}{\eta }^{a}{\eta }^{b}{\eta }^{c}{\eta }^{d}+\mathcal{O}\left({\eta }^{3}\right)\right].\end{equation} \tag{ 3.17 }$$

Using polar coordinates η^a = ηe^a (φ), where e is a 2d unit vector, and the identities

$$\begin{equation}{\int }_{0}^{2\pi }\mathrm{d}\varphi \enspace {e}^{a}{e}^{b}=\pi {\delta }^{ab},\quad {\int }_{0}^{2\pi }\mathrm{d}\varphi \enspace {e}^{a}{e}^{b}{e}^{c}{e}^{d}=\frac{\pi }{4}\left({\delta }^{ab}{\delta }^{cd}+{\delta }^{ac}{\delta }^{bd}+{\delta }^{ad}{\delta }^{bc}\right),\end{equation} \tag{ 3.18 }$$

we are left with the radial integral, for which we introduce the cutoff

$$\begin{align}\hfill & \frac{1}{2\pi }{\int }_{{\epsilon}}\frac{\mathrm{d}\eta }{{\eta }^{3}}\left(1-\frac{{\eta }^{2}}{48}\left(4{R}^{{\Sigma}}+12{\left(\partial n\right)}^{2}-{H}^{2}-2{\text{II}}^{ab}\cdot {\text{II}}_{ab}\right)+\mathcal{O}\left({\eta }^{3}\right)\right)\hfill \\ \hfill & \quad =\frac{1}{4\pi {{\epsilon}}^{2}}+\frac{1}{32\pi }\left(2{R}^{{\Sigma}}-{H}^{2}+4{\left(\partial n\right)}^{2}\right)\mathrm{log}\enspace {\epsilon}+\text{finite}.\hfill \end{align} \tag{ 3.19 }$$

To get the expression in the second line we also used the Gauss–Codazzi equation (B.7).

The calculation of the contribution of the two-form field is very similar. Expanding the tensor structure, we have

$$\begin{equation}\frac{1}{2}\left\langle {B}^{+}\left(\sigma \right){B}^{+}\left(\tau \right)\right\rangle =\frac{1}{8{\pi }^{2}}\frac{{\delta }_{\mu \rho }{\delta }_{\nu \sigma }}{\vert x\left(\sigma \right)-x\left(\tau \right){\vert }^{4}}\enspace \mathrm{d}{x}^{\mu }\left(\sigma \right)\wedge \mathrm{d}{x}^{\nu }\left(\sigma \right)\otimes \mathrm{d}{x}^{\rho }\left(\tau \right)\wedge \mathrm{d}{x}^{\sigma }\left(\tau \right).\end{equation} \tag{ 3.20 }$$

In terms of η^a , the differential forms read (see (B.9))

$$\begin{equation}\begin{aligned}\hfill {\left.\mathrm{d}{x}^{\mu }\wedge \mathrm{d}{x}^{\nu }\right\vert }_{\sigma }& ={\varepsilon }^{ab}{v}_{a}^{\mu }{v}_{b}^{\nu }{\mathrm{d}}^{2}\eta ,\hfill \\ \hfill {\left.\mathrm{d}{x}^{\rho }\wedge \mathrm{d}{x}^{\sigma }\right\vert }_{\tau }& ={\varepsilon }^{cd}\left({v}_{c}^{\left[\right. \rho }{v}_{d}^{\sigma \left.\right]}+2{v}_{c}^{\left[\right.\rho }{v}_{de}^{\sigma \left.\right]}{\eta }^{e}+\left({v}_{ce}^{\left[\right. \rho }{v}_{df}^{\sigma \left.\right]}+{v}_{c}^{\left[\right. \rho }{v}_{def}^{\sigma \left.\right]}\right){\eta }^{e}{\eta }^{f}+\mathcal{O}\left({\eta }^{3}\right)\right){\mathrm{d}}^{2}\eta .\hfill \end{aligned}\end{equation} \tag{ 3.21 }$$

Collecting terms and introducing a radial cutoff as above, we find the contribution

$$\begin{equation}-\frac{1}{4\pi {{\epsilon}}^{2}}-\frac{1}{32\pi }\left(-2{R}^{{\Sigma}}+3{H}^{2}\right)\mathrm{log}\enspace {\epsilon}+\text{finite}.\end{equation} \tag{ 3.22 }$$

Finally, combining (3.19) and (3.22) we find that the quadratic divergences cancel and we are left with

$$\begin{equation}\mathrm{log}\left\langle {V}_{{\Sigma}}\right\rangle =\frac{1}{8\pi }\enspace \mathrm{log}\enspace {\epsilon}{\int }_{{\Sigma}}{\text{vol}}_{{\Sigma}}\left[{R}^{{\Sigma}}-{H}^{2}+{\left(\partial n\right)}^{2}\right]+\text{finite}.\end{equation} \tag{ 3.23 }$$

Comparing to (2.3), we can read off the anomaly coefficients

$$\begin{equation}{a}_{1}^{\left(1\right)}=+\frac{1}{2},\quad {a}_{2}^{\left(1\right)}=-\frac{1}{2},\quad {c}^{\left(1\right)}=+\frac{1}{2}.\end{equation} \tag{ 3.24 }$$

As discussed above, since we do not know the contribution of the Weyl tensor to the B-field propagator, we cannot determine b⁽¹⁾. According to the conjecture of [45] however, it should vanish. This relation is the subject of work in progress [19].

Equation (3.23) differs from (2.3) by the absence of the tr  P term, which also vanishes in flat space. Since H² does not vanish in flat space, it determines a₂ unambiguously and in curved space H² is necessarily accompanied by 4  tr  P, based on the general argument for the form of the anomaly reviewed in section 2.

Finally, we reiterate that, depending on the form of the quantisation condition, the result for the anomaly coefficients may be changed by a factor of 2, see the discussion following (3.14). In any case, the abelian theory should have surface operators with an integer multiple of iB⁺ − nⁱ Φ_i in (1.1), and for all of them it is still true that ${a}_{1}^{\left(1\right)}=-{a}_{2}^{\left(1\right)}={c}^{\left(1\right)}$.

3.4. Generalising the scalar coupling

Note that the preceding calculation is applicable regardless of whether the operator is locally BPS or not, so we may relax the condition n² = 1. In that case the result for the anomaly coefficients is

$$\begin{equation}{a}_{1}^{\left(1\right)}=\frac{{n}^{2}+1}{4}.\quad {a}_{2}^{\left(1\right)}=-\frac{{n}^{2}+3}{8},\quad {c}^{\left(1\right)}=\frac{1}{2}.\end{equation} \tag{ 3.25 }$$

If we replace nⁱ → inⁱ , we recover the surface operator studied in [17]. An operator with n² = 0 was also studied in [14], but assuming a non-self-dual two-form. The anomaly coefficients computed in [14, 17] are respectively twice and four times the ones we obtain by substituting the values of n in (3.25), due to a difference in the overall normalisation of the propagator.

It would be interesting to study this system in the large n² limit. This is similar to the 'ladder' limit of the cusped Wilson loop in $\mathcal{N}=4$ SYM in 4d first suggested in [65] which is related to a special scaling limit of that theory, dubbed the 'fishnet' model, which also has a 6d version [66].

The holographic calculation of the Weyl anomaly for surface operators was pioneered by Graham and Witten in [13]. Here we present a rewriting of their argument, which we also generalise slightly to include operators extended on the S⁴.

4.1. Surface operators

The $\mathcal{N}=\left(2,0\right)$ theory is described at large N by 11d supergravity on an asymptotically AdS₇ × S⁴ geometry [38]

$$\begin{equation}\mathrm{d}{s}^{2}=\frac{{L}^{2}}{{y}^{2}}\left(\mathrm{d}{y}^{2}+{g}^{\left(0\right)}+{g}^{\left(1\right)}{y}^{2}\right)+\frac{{L}^{2}}{4}{g}_{{S}^{4}}^{\left(0\right)}+\mathcal{O}\left({y}^{2}\right),\quad L={\left(8\pi N\right)}^{1/3}{l}_{P},\end{equation} \tag{ 4.1 }$$

such that g⁽⁰⁾ is the metric of the dual field theory ⁵ and ${g}_{{S}^{4}}^{\left(0\right)}$ is the metric of S⁴.

The background also includes N units of F₄ flux

$$\begin{equation}\frac{1}{{\left(2\pi \right)}^{2}{l}_{P}^{3}}{\int }_{{S}^{4}}{F}_{4}=2\pi N.\end{equation} \tag{ 4.2 }$$

The full form of the metric is determined by the supergravity equations of motion in the presence of fluxes and by requiring the geometry to close smoothly in the interior. While the latter requires nonlocal information, the near-boundary expansion is fixed to the required order by local information about the boundary. Following [67, 68], the first term in this expansion was found in [13] as

$$\begin{equation}{g}_{\mu \nu }^{\left(1\right)}=-{P}_{\mu \nu }^{\left(0\right)}\equiv -{P}_{\mu \nu }{\vert }_{g={g}^{\left(0\right)}}.\end{equation} \tag{ 4.3 }$$

At this order the S⁴ is round, so to leading order the solution to (4.2) is simply

$$\begin{equation}{F}_{4}=\frac{3}{8}{L}^{3}\,{\text{vol}}_{{S}^{4}}.\end{equation} \tag{ 4.4 }$$

The holographic description of the surface operators (1.1) is by M2-branes anchored along Σ on the boundary of AdS [4]. Using $\hat{{\Sigma}}$ for the world-volume of the M2-brane, it has a boundary at y = 0 with $\partial \hat{{\Sigma}}={\Sigma}$. The expectation value of the surface operators is then given by the minimum of the M2-brane action, reading (in Euclidean signature and with all fermionic terms suppressed) [69]

$$\begin{equation}\mathrm{log}\left\langle {V}_{{\Sigma}}\right\rangle \simeq -{S}_{\text{M}2}=-{T}_{\text{M}2}{\int }_{\hat{{\Sigma}}}\left({\text{vol}}_{\hat{{\Sigma}}}+i{A}_{3}\right),\quad {T}_{\text{M}2}=\frac{1}{4{\pi }^{2}{l}_{P}^{3}}=\frac{2N}{\pi {L}^{3}},\end{equation} \tag{ 4.5 }$$

where T_M2 is the tension of the brane, proportional to N. ${\text{vol}}_{\hat{{\Sigma}}}$ is the volume form calculated from the induced metric and A₃ is the pullback of the three-form potential.

4.2. Local supersymmetry

Before studying the M2-brane embeddings, let us note that the M2-brane minimizing (4.5) is also locally supersymmetric. The supergravity fields appearing there sit in the supergravity multiplet, which transform as

$$\begin{equation}\begin{aligned}\hfill \delta {A}_{MNP}& =-3\overline{\varepsilon }{{\Gamma}}_{\left[\right. MN}{{\Psi}}_{P\left.\right]},\hfill \\ \hfill \delta {{\Psi}}_{M}& ={D}_{M}\varepsilon +\frac{1}{288}\left({{\Gamma}}_{M}^{PQRS}-8{{\Gamma}}^{QRS}{\delta }_{M}^{P}\right){F}_{PQRS}\varepsilon ,\hfill \\ \hfill \delta {E}_{M}^{\overline{M}}& =\overline{\varepsilon }{{\Gamma}}^{\overline{M}}{{\Psi}}_{M},\hfill \end{aligned}\end{equation} \tag{ 4.6 }$$

where ${E}_{M}^{\overline{M}}$, Ψ_M and A₃ are respectively the vielbein, gravitino and three-form potential of F₄ ($\overline{M}=1,\enspace \dots ,\enspace 11$ is the frame index). Using these transformations, the variation of (4.5) is

$$\begin{equation}{\delta }_{\varepsilon }S={T}_{\text{M}2}{\int }_{\hat{{\Sigma}}}\bar{\varepsilon }\left({{\Gamma}}^{\hat{a}}-\frac{i}{2}{\varepsilon }^{\hat{a}\hat{b}\hat{c}}{{\Gamma}}_{\hat{b}\hat{c}}\right){{\Psi}}_{\hat{a}}\,{\text{vol}}_{\hat{{\Sigma}}}=0.\end{equation} \tag{ 4.7 }$$

We here denote the coordinates on the world-volume by ${\hat{\sigma }}^{\hat{a}}$. The projector equation is then

$$\begin{equation}\bar{\varepsilon }{{\Pi}}_{-}=0,\quad {{\Pi}}_{-}=\frac{1}{2}\left[1-\frac{i}{6}{\varepsilon }^{\hat{a}\hat{b}\hat{c}}{{\Gamma}}_{\hat{a}\hat{b}\hat{c}}\right].\end{equation} \tag{ 4.8 }$$

The projector is again half-rank, so that the M2-brane locally preserves half of the supersymmetries (16 supercharges). These supercharges can be shown to agree with the field theory BPS condition (3.3) on Σ once we decompose x^M into coordinates on the boundary of AdS, x^μ , and the S⁴ coordinates nⁱ .

4.3. Holographic calculation

To find the saddle points of the action (4.5), we parametrise the M2-brane by y, σ^a where σ^a are coordinates for Σ. We then use the static gauge to describe the embedding by {u^a '(y, σ), nⁱ (y, σ)}, where u^a ' are the normal directions to the surface Σ at y = 0. In this setup, the boundary conditions are u^a '(y = 0, σ) = 0 and nⁱ (y = 0, σ) = nⁱ (σ) (where the right-hand side has the nⁱ from (1.1)).

Because the metric (4.1) diverges at the boundary of AdS, the volume element on the M2-brane diverges as y⁻³, which leads to divergences in the action. Finding the shape of the embedding requires knowledge of the full surface and is generally a hard problem. But since we are only interested in the logarithmically divergent part of the action, it is sufficient to solve the equations of motion for small y. We do this perturbatively following [13], mirroring the solution of the background supergravity equations above.

Using (4.3), the lowest order terms in the metric for our coordinates normal and tangent to the surface, are

$$\begin{equation}\begin{aligned}\hfill {g}_{ab}\left(y,\sigma ,u\right)& ={\left.{h}_{ab}-{P}_{ab}^{\left(0\right)}{y}^{2}+{\partial }_{{a}^{\prime }}{g}_{ab}^{\left(0\right)}\right\vert }_{u=0}{u}^{{a}^{\prime }}+\mathcal{O}\left({y}^{4},{u}^{2}\right),\hfill \\ \hfill {g}_{a{a}^{\prime }}\left(y,\sigma ,u\right)& =\mathcal{O}\left({y}^{2},u\right),\hfill \\ \hfill {g}_{{a}^{\prime }{b}^{\prime }}\left(y,\sigma ,u\right)& ={\left.{g}_{{a}^{\prime }{b}^{\prime }}^{\left(0\right)}\right\vert }_{u=0}+\mathcal{O}\left({y}^{2},u\right).\hfill \end{aligned}\end{equation} \tag{ 4.9 }$$

Here ${h}_{ab}={g}_{ab}^{\left(0\right)}{\vert }_{u=0}$ is the metric on Σ. Note that away from y = 0, this metric depends on u^a ' (for y ≠ 0, generically u^a ' ≠ 0), as in the first line.

To write down the M2-brane action we need the induced metric _ab = ∂_a X^M ∂_b X^N g_MN (including also the S⁴ directions). We expand the embedding coordinates as

$$\begin{equation}\begin{aligned}\hfill {u}^{{a}^{\prime }}\left(y,\sigma \right)& =\mathcal{O}\left({y}^{2}\right),\hfill \\ \hfill {n}^{i}\left(y,\sigma \right)& ={n}^{i}\left(\sigma \right)+\mathcal{O}\left({y}^{2}\right).\hfill \end{aligned}\end{equation} \tag{ 4.10 }$$

It is easy to check that higher order terms are not required. Then the S⁴ metric can be replaced with ${g}_{{S}^{4}}^{\left(0\right)}={\delta }_{ij}\mathrm{d}{n}^{i}\mathrm{d}{n}^{j}$ and the second fundamental form is ${\text{II}}_{ab}^{{a}^{\prime }}=-\frac{1}{2}{g}^{{a}^{\prime }{b}^{\prime }}{\partial }_{{b}^{\prime }}{g}_{ab}$.

Dropping the explicit $\mathcal{O}\left({y}^{\star }\right)$ as well as the subscript |_u=0 along with the superscript ⁽⁰⁾, since all the quantities are evaluated on the surface, we find

$$\begin{equation}\begin{aligned}\hfill {\hat{h}}_{yy}& \simeq \frac{{L}^{2}}{{y}^{2}}\left[1+{\partial }_{y}{u}^{{a}^{\prime }}{\partial }_{y}{u}^{{b}^{\prime }}{g}_{{a}^{\prime }{b}^{\prime }}\right],\hfill \\ \hfill {\hat{h}}_{ay}& \simeq 0,\hfill \\ \hfill {\hat{h}}_{ab}& \simeq \frac{{L}^{2}}{{y}^{2}}\left[{h}_{ab}+\left(-{P}_{ab}+\frac{1}{4}{\partial }_{a}{n}^{i}{\partial }_{b}{n}^{j}{\delta }_{ij}\right){y}^{2}-2{\text{II}}_{ab}^{{a}^{\prime }}{u}^{{b}^{\prime }}{g}_{{a}^{\prime }{b}^{\prime }}\right].\hfill \end{aligned}\end{equation} \tag{ 4.11 }$$

The determinant of the metric is then

$$\begin{equation}\mathrm{det}\enspace \hat{h}\simeq \frac{{L}^{6}}{{y}^{6}}\left(1+{\partial }_{y}{u}^{{a}^{\prime }}{\partial }_{y}{u}^{{b}^{\prime }}{g}_{{a}^{\prime }{b}^{\prime }}-2{H}^{{a}^{\prime }}{u}^{{b}^{\prime }}{g}_{{a}^{\prime }{b}^{\prime }}+\left(-\text{tr}\enspace P+\frac{1}{4}{\left(\partial n\right)}^{2}\right){y}^{2}\right)\mathrm{det}\enspace h,\end{equation} \tag{ 4.12 }$$

while the pullback of the three-form

$$\begin{equation}{A}_{3}=\frac{1}{3!}{A}_{ijk}\enspace \mathrm{d}{n}^{i}\wedge \mathrm{d}{n}^{j}\wedge \mathrm{d}{n}^{k}\sim \mathcal{O}\left(y\right),\end{equation} \tag{ 4.13 }$$

does not contribute to the divergences. We thus find the action

$$\begin{equation}{S}_{\text{M}2}\simeq \frac{{L}^{3}}{{\left(2\pi \right)}^{2}{l}_{P}^{3}}{\int }_{{\Sigma}}{\text{vol}}_{{\Sigma}}\underset{y{\geqslant}{\epsilon}}{\int }\frac{\mathrm{d}y}{{y}^{3}}\left[1+\frac{1}{2}{\left({\partial }_{y}{u}^{{a}^{\prime }}\right)}^{2}-H\cdot u+\left(-4\enspace \text{tr}\enspace P+{\left(\partial n\right)}^{2}\right)\frac{{y}^{2}}{8}\right].\end{equation} \tag{ 4.14 }$$

At order $\mathcal{O}\left({y}^{2}\right)$, we need only solve for u^a '(y), which has the equation of motion

$$\begin{equation}{y}^{3}{\partial }_{y}\left({y}^{-3}{\partial }_{y}{u}^{{a}^{\prime }}\right)+{H}_{{a}^{\prime }}\simeq 0\quad {\Rightarrow}\quad {u}^{{a}^{\prime }}\simeq \frac{1}{4}{H}^{{a}^{\prime }}{y}^{2}.\end{equation} \tag{ 4.15 }$$

The action evaluated at the classical solution is then

$$\begin{equation}{S}_{\text{M}2}\simeq \frac{{L}^{3}}{{\left(2\pi \right)}^{2}{l}_{P}^{3}}{\int }_{{\Sigma}}{\text{vol}}_{{\Sigma}}\underset{y{\geqslant}{\epsilon}}{\int }\frac{\mathrm{d}y}{{y}^{3}}\left[1-\frac{{y}^{2}}{8}\left({H}^{2}+4\enspace \text{tr}\enspace P\right)+\frac{{y}^{2}}{8}{\left(\partial n\right)}^{2}\right]\end{equation} \tag{ 4.16 }$$

where we see that the anomaly indeed takes the form (2.3). The result is

$$\begin{equation}\mathrm{log}\left\langle {V}_{{\Sigma}}\right\rangle =\frac{N}{4\pi }\enspace \mathrm{log}\enspace {\epsilon}{\int }_{{\Sigma}}{\text{vol}}_{{\Sigma}}\left[-\left({H}^{2}+4\enspace \text{tr}\enspace P\right)+{\left(\partial n\right)}^{2}\right]\mathrm{log}\enspace {\epsilon}+\text{finite},\end{equation} \tag{ 4.17 }$$

where we discarded an irrelevant term proportional to ⁻² (see the discussion below).

This result agrees with the original calculation of [13] and adds to it the coupling to (∂n)². It is also consistent with the explicit calculation of the 1/2-BPS sphere [28], for which the anomaly is −4N. The anomaly coefficients at leading order in N are then

$$\begin{equation}\begin{aligned}\hfill {a}_{1}^{\left(N\right)}& =\mathcal{O}\left({N}^{0}\right),\hfill & \hfill \quad {b}^{\left(N\right)}& =\mathcal{O}\left({N}^{0}\right),\hfill \\ \hfill {a}_{2}^{\left(N\right)}& =-N+\mathcal{O}\left({N}^{0}\right),\hfill & \hfill \quad {c}^{\left(N\right)}& =+N+\mathcal{O}\left({N}^{0}\right).\hfill \end{aligned}\end{equation} \tag{ 4.18 }$$

As in the case of Wilson loops in $\mathcal{N}=4$ SYM in 4d, we expect this holographic description to be correct in the locally BPS case when the scalar couplings satisfy n² = 1. Following [29, 30], the case of n² = 0 should be described by the same surface inside AdS₇, but completely smeared over the S⁴. In this case we find the same result for the geometric anomaly coefficients as above, and, since the corresponding anomaly term vanishes, c^(N) does not apply.

4.3.1. Power-law divergence

Note that in addition to the log divergence in (4.17), (4.16) produces also a power-law divergence

$$\begin{equation}\frac{{L}^{3}}{{\left(2\pi \right)}^{2}{l}_{P}^{3}}\frac{\text{Area}\left({\Sigma}\right)}{2{{\epsilon}}^{2}}.\end{equation} \tag{ 4.19 }$$

While such divergences can be removed by the addition of a local counter-terms, in the field theory result (3.23), they cancelled without extra counter-terms (for the locally BPS operator).

A more elegant way of eliminating the power law divergences also in this holographic calculation follows the example of the locally BPS Wilson loops [5]. A careful treatment of the boundary conditions suggests that the natural action is a Legendre transform of (4.5), which differs from the action we used by a total derivative. This modification does not change the equations of motion, but gives a contribution on the boundary, where it precisely cancels the divergence above.

By looking at the M5-brane metric before the decoupling limit, we can identify the coordinate to use in the transform as rⁱ = L³ nⁱ /2y². Defining its conjugate momentum by differentiating with respect to the boundary value of the coordinate (where y = )

$$\begin{equation}{p}_{i}\left(\sigma \right)=\frac{\delta S\left[{x}^{\mu },{r}^{i}\right]}{\delta {r}^{i}}=-\frac{{{\epsilon}}^{3}{n}^{i}}{{L}^{3}}\frac{\delta S\left[{x}^{\mu },{n}^{i},{\epsilon}\right]}{\delta {\epsilon}}=\frac{{{\epsilon}}^{3}{n}^{i}}{{L}^{3}}\frac{{L}^{3}}{{\left(2\pi \right)}^{2}{l}_{P}^{3}}\left(\frac{1}{{{\epsilon}}^{3}}+\mathcal{O}\left(\frac{1}{{\epsilon}}\right)\right).\end{equation} \tag{ 4.20 }$$

In the last equality we used the value of the classical action (4.16), undoing the integration, so the classical Lagrangian density.

The Legendre transformed action is then

$$\begin{equation}\tilde {S}\left[{x}^{\mu },{p}^{i}\right]=S\left[{x}^{\mu },{r}^{i}\right]-{\int }_{{\Sigma}}{p}_{i}{r}^{i}{\text{vol}}_{{\Sigma}}=S\left[{x}^{\mu },{n}^{i},{\epsilon}\right]-\frac{{L}^{3}}{2{\left(2\pi \right)}^{2}{l}_{P}^{3}{{\epsilon}}^{2}}{\int }_{{\Sigma}}{\text{vol}}_{{\Sigma}}.\end{equation} \tag{ 4.21 }$$

The last term exactly cancels the power law divergence in (4.19).

An interesting class of surface operators that has received some attention recently is surfaces with conical singularities. For these surfaces, it was found that the regularised expectation value typically diverges as [21, 70–72]

$$\begin{equation}\mathrm{log}\left\langle {V}_{{{\Sigma}}_{c}}\right\rangle \sim A\enspace {\mathrm{log}}^{2}\enspace {\epsilon}+\mathcal{O}\left(\mathrm{log}\enspace {\epsilon}\right).\end{equation} \tag{ 5.1 }$$

Let us consider a conical defect (on flat space) of the form

$$\begin{equation}{x}^{\mu }\left(r,s\right)=r{\gamma }^{\mu }\left(s\right),\quad {\gamma }^{2}=1,\quad {n}^{i}\left(r,s\right)={\nu }^{i}\left(s\right).\end{equation} \tag{ 5.2 }$$

We allow here also a 'conical singularity' in the scalar couplings, which has s dependence even as r → 0. It is possible to also allow x^μ and nⁱ to have higher order terms in r, but since those lead to subleading divergences, they are unimportant.

We can try to use the usual formula for the anomaly (2.3) by plugging in the geometric invariants

$$\begin{equation}{R}^{{\Sigma}}={\Omega}\delta \left(r\right),\quad {H}^{2}=\frac{{\kappa }^{2}-1}{{r}^{2}},\quad {\left(\partial n\right)}^{2}=\frac{{\left({\partial }_{s}\nu \right)}^{2}}{{r}^{2}},\end{equation} \tag{ 5.3 }$$

where Ω is the deficit angle, $\kappa ={\ddot {\gamma }}^{2}/\vert \dot {\gamma }{\vert }^{2}$ is the curvature of γ. Plugging into (2.3), the Ricci scalar gives a finite contribution, but H² and (∂n)² diverge as r → 0. Introducing a cutoff $\hat{{\epsilon}}$ on the r integration, this gives

$$\begin{equation}\frac{1}{4\pi }\enspace \mathrm{log}\enspace {\epsilon}\enspace \mathrm{log}\enspace \hat{{\epsilon}}{\int }_{\gamma }{a}_{2}\left(1-{\kappa }^{2}\left(s\right)\right)-c{\left({\partial }_{s}\nu \right)}^{2}\mathrm{d}s+\mathcal{O}\left(\mathrm{log}\enspace {\epsilon}\right).\end{equation} \tag{ 5.4 }$$

This expression is a bit naive, as we should treat all divergences on the same footing and should identify $\hat{{\epsilon}}={\epsilon}$. But then we should not use (2.3), rather go back one step and regularise the divergences that gave rise to the original log divergence while also applying it to the r integration. As we show below, this leads to the expression in (5.4) with $\mathrm{log}\enspace {\epsilon}\enspace \mathrm{log}\enspace \hat{{\epsilon}}\to \frac{1}{2}\enspace {\mathrm{log}}^{2}\enspace {\epsilon}$. In both the free field case and the holographic realisation this factor of 1/2 is a simple consequence of the usual coefficient of the quadratic term in the Taylor expansion, or in other words of an integral of the form ∫log  r d  log  r.

This factor of 1/2 was noticed already in the calculations of [21, 70] and justified in [72] by a careful treatment of the holographic calculation, which is repeated below. It was also studied in the context of defect CFT in [22]. We think that the comparison of this to the free-field calculation and the universal nature of our result further elucidates this mismatch from the naive expectation. Our calculation is also more generic, for allowing arbitrary conical singularities and incorporating the scalar singularities too.

Beside this factor 1/2, it is interesting to compare the log²  divergence of surface operators to the log divergence of cusped Wilson loops. In $\mathcal{N}=4$ SYM, the cusp anomalous dimension is a complicated function of the opening angle ϕ. At small angles, it is related to the Bremsstrahlung function, which encodes the radiation emitted by heavy probe particles. It is therefore an interesting quantity to compute, and the exact Bremsstrahlung function has been obtained using supersymmetric localization in [73].

In contrast, the expression (5.7) is not an approximation for small angles, but the exact result. The relation to physical quantities is unclear as well. It would be interesting to interpret it as a Bremsstrahlung function, but computing the radiation emitted by a probe string in 6d would require a more careful treatment of the self-dual field strength.

We should also note, as already noticed in [21], that surfaces with 'creases', i.e. co-dimension one singularities, do not lead to additional log²  divergences and the expression (2.3) can be immediately applied to them.

5.1. Field theory

Here we do not rely on (3.23), but go further back to where the log  arises from an integral of the form (3.19)

$$\begin{equation}{\int }_{{\epsilon}}^{\rho }\frac{\mathrm{d}\eta }{\eta }=-\mathrm{log}\enspace {\epsilon}+\text{finite},\end{equation} \tag{ 5.5 }$$

where η is a radial coordinate around the point x, and ρ is an IR cutoff related to the overall size of the surface, or at least a large smooth patch where we defined our local coordinate. Near the cone the smooth patch is bounded by the distance from x to the apex, which we denote by r. The integral instead gives

$$\begin{equation}{\int }_{{\epsilon}}^{r}\frac{\mathrm{d}\eta }{\eta }=-\mathrm{log}\enspace \frac{{\epsilon}}{r}.\end{equation} \tag{ 5.6 }$$

With this careful treatment of the log, we can go back to (2.3), plug in the expressions from (5.3) and integrate over r and with the same UV cutoff to find

$$\begin{align}\hfill \mathrm{log}\left\langle {V}_{{\Sigma}}\right\rangle & =-\frac{1}{4\pi }{\int }_{\gamma }\mathrm{d}s\underset{{\epsilon}}{\int }\frac{\mathrm{d}r}{r}\left[{a}_{2}\left(1-{\kappa }^{2}\right)-c{\left({\partial }_{s}n\right)}^{2}\right]\mathrm{log}\enspace \frac{{\epsilon}}{r}+\text{finite}\hfill \\ \hfill & =\frac{1}{8\pi }\enspace {\mathrm{log}}^{2}\enspace {\epsilon}{\int }_{\gamma }\left[{a}_{2}\left(1-{\kappa }^{2}\left(s\right)\right)-c{\left({\partial }_{s}\nu \right)}^{2}\right]\mathrm{d}s+\mathcal{O}\left(\mathrm{log}\enspace {\epsilon}\right).\hfill \end{align} \tag{ 5.7 }$$

5.2. Holography

The derivation in holography is similar. We first note that conformal symmetry fixes the form of the solution as

$$\begin{equation}y\left(r,s\right)=ru\left(s\right)\end{equation} \tag{ 5.8 }$$

To get to (4.17), we integrate over y, but the conformal ansatz suggests to impose the range ⩽ y ⩽ ru_max. Plugging the curvatures from (5.3) into equation (4.17) we arrive at

$$\begin{equation}\mathrm{log}\left\langle {V}_{{\Sigma}}\right\rangle =-\frac{1}{4\pi }{\int }_{\gamma }\mathrm{d}s\underset{{\epsilon}}{\int }\frac{\mathrm{d}r}{r}\left[{a}_{2}\left(1-{\kappa }^{2}\right)-c{\left({\partial }_{s}n\right)}^{2}\right]\mathrm{log}\enspace \frac{{\epsilon}}{r{u}_{\mathrm{max}}\left(s\right)}+\text{finite}.\end{equation} \tag{ 5.9 }$$

which again gives the log²  divergence with the same 1/2 prefactor, as in the field theory (5.7).

5.3. Example: circular cone

As a simple example of a singular surface we compute explicitly the anomaly of a cone. Denoting the deficit angle by ϕ (see figure 1) and including an internal angle θ for the scalar coupling nⁱ , we parametrise the cone as follows

$$\begin{equation}{\gamma }^{\mu }\left(s\right)=\begin{pmatrix}\hfill \mathrm{cos}\enspace \phi \enspace \mathrm{sin}\enspace s\hfill \\ \hfill \mathrm{cos}\enspace \phi \enspace \mathrm{cos}\enspace s\hfill \\ \mathrm{sin}\enspace \phi \hfill \end{pmatrix},\quad {n}^{i}\left(s\right)=\begin{pmatrix}\hfill \mathrm{sin}\enspace \theta \enspace \mathrm{sin}\enspace s\hfill \\ \hfill \mathrm{sin}\enspace \theta \enspace \mathrm{cos}\enspace s\hfill \\ \mathrm{cos}\enspace \theta \hfill \end{pmatrix},\quad 0{\leqslant}r,\enspace 0{\leqslant}s{< }2\pi .\end{equation} \tag{ 5.10 }$$

The conformal invariants are explicitly

$$\begin{equation}{\kappa }^{2}=\frac{1}{{\mathrm{cos}}^{2}\phi },\quad {\left({\partial }_{s}n\right)}^{2}=\frac{{\mathrm{sin}}^{2}\theta }{{\mathrm{cos}}^{2}\phi }.\end{equation} \tag{ 5.11 }$$

The divergence is then

$$\begin{equation}\mathrm{log}\left\langle {V}_{{\Sigma}}\right\rangle =-\frac{{a}_{2}{\mathrm{sin}}^{2}\phi +c\enspace {\mathrm{sin}}^{2}\theta }{4\enspace \mathrm{cos}\enspace \phi }{\mathrm{log}}^{2}\enspace {\epsilon}+\mathcal{O}\left(\mathrm{log}\enspace {\epsilon}\right).\end{equation} \tag{ 5.12 }$$

Notice that as long as the anomaly coefficients satisfy the relation a₂ = −c, which we have shown to hold in the abelian and large N case, the anomaly vanishes for configurations θ = ±ϕ, which correspond generically to 1/8-BPS configurations.

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In this paper we calculated the anomaly coefficients of locally supersymmetric surface operators in the $\mathcal{N}=\left(2,0\right)$ theory in 6d, refining and generalising the calculations of [13, 17]. We first introduced a new anomaly coefficient c (2.3) arising from non-constant dependence on the internal R-symmetry directions. These are explicit scalar couplings in the abelian theory and motion on S⁴ in the holographic realisation.

We then presented an explicit calculation for the abelian theory and for the large N limit (using holography). The results are in equations (3.24) and (4.18). Although we are not able to compute the anomaly coefficient b at N = 1 because we do not know the general curved space propagator for the self-dual two-form, we found the others in both cases.

Making all N conjectures based on the asymptotics is a fool's errand, which we carefully tread. This is especially true given that the abelian theory is not the same as the A_N−1 theory at N = 1, since the latter is the empty theory. Nevertheless, in both cases we see that a₂ = −c, and we expect this to hold generally. The argument is based on the BPS Wilson loops of [54], where nⁱ is parallel to ^μ and which have trivial expectation values. If we uplift them to the 6d theory we expect to find surface operators with no anomaly (and no finite part as well). These operators satisfy H² = (∂n)² and indeed they do not contribute to the anomaly ⁶ for a₂ = −c. A proof of this relation as well as properties of b, based on defect CFT techniques, will be presented elsewhere [19].

Two more results are the formalism for regularising surface operators presented in appendix C and the expression for the divergences due to conical singularities over arbitrary curves in section 5.

All our calculations are for a surface operator in the fundamental representation. It is expected that 1/2-BPS surface operators are classified by representations of the A_N−1 algebra of the theory. At large N this is proven, since the asymptotically AdS₇ × S⁴ solutions of 11d supergravity preserving the symmetry algebra of 1/2-BPS surface operators can be classified in terms of Young diagrams [41, 42].

A calculation of anomalies of surface operators in arbitrary representation, based on the bubbling geometries and holographic entanglement entropy was undertaken in [47]. If we assume b = 0, then for a the fundamental representation, their result reads

$$\begin{equation}{a}_{1}^{\left(N\right)}=\frac{1}{2}-\frac{1}{2N},\quad {a}_{2}^{\left(N\right)}=-N+\frac{1}{2}+\frac{1}{2N}.\end{equation} \tag{ 6.1 }$$

This is supported by an independent calculation using the superconformal index [48]. In the large N limit, our result [13] indeed agrees with theirs. These calculations do not determine the remaining anomaly coefficients in generic representations. But if we believe the b = 0 conjecture of [45] and our argumentation above for c = −a₂, this fixes the remaining ones.

It would be interesting to reproduce these finite N corrections using other methods as well as do direct holographic calculations for higher-dimensional surface operators.

The anomalies studied here are the most basic properties of surface operators, but finding them is only a first step in understanding these observables and the mysterious theory they belong to. Planar/spherical surface operators preserve part of the conformal group (and with the scalar coupling also half the supersymmetries) and their deformations behave like operators in a defect CFT. A natural next step is to study the defect CFT data: spectrum and structure constants.

Another natural question is the classification of globally BPS surface operators (and local operators within the surface operators associated with their deformations) beyond the case of the plane/sphere.

We hope to report progress on these questions in the near future.

We would like to thank Lorenzo Bianchi, Robin Graham, Nikolay Gromov, Chris Herzog, Elias Kiritsis, Neil Lambert, Ki-Myeong Lee, Andy O'Bannon, Ronnie Rodgers and Itamar Shamir for interesting discussions. ND Would like to thank the University of Modena and Reggio Emilia, the Pollica Summer Workshop, the Hebrew University, CERN and EPFL Lausanne for hospitality in the course of this work. His work is supported by an STFC Grant Number ST/P000258/1. The Pollica Summer workshop was supported in part by the Simons Foundation (Simons Collaboration on the Non-perturbative Bootstrap) and in part by the INFN. The author is grateful for this support. MT acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG).

In this paper we work in Minkowski space with mostly positive signature. We make use of the following indices:

$$\begin{equation*}\begin{matrix}\hfill \text{Index}\hfill & \hfill \text{Usage}\hfill \\ \hfill M=1,\dots ,11\hfill & \hfill 11\mathrm{d}\;\text{spacetime}\;\text{vector}\;{X}^{M}\hfill \\ \hfill A=1,\dots ,32\hfill & 11\mathrm{d}\;\text{spinors}\hfill \\ \hfill \mu =1,\dots ,6\hfill & \hfill 6\mathrm{d}\;\text{spacetime}\;\text{vectors}\;{x}^{\mu }\hfill \\ \hfill \alpha \enspace \left(\dot {\alpha }\right)=1,\dots ,4\hfill & \hfill 6\mathrm{d}\;\text{chiral}\;\left(\text{antichiral}\right)\;\text{spinors}\hfill \\ \hfill i=1,\dots ,5\hfill & \hfill \text{R-}\text{symmetry}\;\text{vectors}\hfill \\ \hfill \check {\alpha }=1,\dots ,4\hfill & \hfill \text{R-}\text{symmetry}\;\text{spinors}\hfill \\ \hfill {a}^{\prime }=1,\dots ,4\hfill & \hfill \text{spacetime}\;\text{vectors}\;\text{orthogonal}\;\text{to}\;\text{the}\;\text{surface}\hfill \\ \hfill a=1,2\hfill & \hfill \;\text{worldsheet}\;\text{coordinates}\;{\sigma }^{a}\hfill \\ \hfill \hat{a}=1,2,3\hfill & \hfill \text{worldvolume}\;\text{coordinates}\;{\hat{\sigma }}^{\hat{a}}\hfill \end{matrix}\end{equation*}$$

Our usage of spinors is restricted to the supersymmetry transformations (3.1) and (4.6) but we include our conventions for completeness. In general we follow the NW-SE convention for indices summation

$$\begin{equation}\overline{{\Phi}}{\Psi}\equiv {\overline{{\Phi}}}^{A}{{\Psi}}_{A},\end{equation} \tag{ A.1 }$$

The conjugate and transpose act as

$$\begin{equation}{\left({{\Psi}}_{A}\right)}^{{\ast}}={\left({{\Psi}}^{{\ast}}\right)}^{A},\quad {\left({\mathcal{C}}^{AB}\right)}^{T}={\mathcal{C}}^{BA}.\end{equation} \tag{ A.2 }$$

Below we detail the properties of gamma matrices in d = 11 and d = 6, and we state the reality condition on spinors. More details can be found in [36] and references therein.

A.1.  d = 11 Clifford algebra

The 11d Clifford algebra is generated by the set of matrices ${\left({{\Gamma}}_{M}\right)}_{A}^{B}$ satisfying

$$\begin{equation}\left\{{{\Gamma}}_{M},{{\Gamma}}_{N}\right\}=2{\eta }_{MN}.\end{equation} \tag{ A.3 }$$

Here for readability M is used for flat spacetime, unlike (4.6) where it denotes curved spacetime.

The matrices may be chosen such that ${{\Gamma}}_{0}^{{\dagger}}=-{{\Gamma}}_{0}$ is antihermitian while the others are Hermitian ${{\Gamma}}_{M}^{{\dagger}}={{\Gamma}}_{M}\enspace \left(M\ne 0\right)$. In addition, there is an orthogonal, real anti-symmetric matrix ${\mathcal{C}}_{AB}$ such that ${{\Gamma}}_{M}\mathcal{C}=-{\left({{\Gamma}}_{M}\mathcal{C}\right)}^{T}$. $\mathcal{C}$ naturally defines a real structure by relating Ψ and Ψ^† as

$$\begin{equation}\overline{{\Psi}}\equiv -i{{\Gamma}}_{0}{{\Psi}}^{{\dagger}}={\mathcal{C}}^{{\dagger}}{\Psi}.\end{equation} \tag{ A.4 }$$

This is the Majorana condition.

A.2.  d = 6 Clifford algebra

An easy way to construct the 6d Clifford algebra is to decompose ${{\Gamma}}_{M}=\left\{{{\Gamma}}_{\mu },{{\Gamma}}_{i}\right\}$ by introducing a chirality matrix Γ_* = Γ₀Γ₁Γ₂Γ₃Γ₄Γ₅. The matrices are then (in the chiral basis)

$$\begin{equation}{{\Gamma}}_{\mu }=\begin{pmatrix}\hfill 0\hfill & \hfill {\overline{\gamma }}_{\mu }\hfill \\ \hfill {\gamma }_{\mu }\hfill & \hfill 0\hfill \end{pmatrix}\otimes {I}_{4},\quad {{\Gamma}}_{i}=\begin{pmatrix}\hfill -{I}_{4}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {I}_{4}\hfill \end{pmatrix}\otimes {\check {\gamma }}_{i},\quad {{\Gamma}}_{{\ast}}=\begin{pmatrix}\hfill -{I}_{4}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {I}_{4}\hfill \end{pmatrix}\otimes {I}_{4},\end{equation} \tag{ A.5 }$$

where the algebra is

$$\begin{equation}{\overline{\gamma }}_{\mu }{\gamma }_{\nu }+{\overline{\gamma }}_{\nu }{\gamma }_{\mu }=2{\eta }_{\mu \nu },\quad {\gamma }_{\mu }{\overline{\gamma }}_{\nu }+{\gamma }_{\nu }{\overline{\gamma }}_{\mu }=2{\eta }_{\mu \nu },\quad \left\{{\check {\gamma }}_{i},{\check {\gamma }}_{j}\right\}=2{\delta }_{ij}.\end{equation} \tag{ A.6 }$$

Since γ_μ and ${\check {\gamma }}_{i}$ commute, they define independent spinor representations. Explicitly, we decompose $A=\left(\dot {\alpha }\oplus \alpha \right)\otimes \check {\alpha }$, so that the indices are ${\left({\gamma }_{\mu }\right)}_{\alpha }^{\dot {\beta }}$, ${\left({\overline{\gamma }}_{\mu }\right)}_{\dot {\alpha }}^{\beta }$ and ${\left({\check {\gamma }}_{i}\right)}_{\check {\alpha }}^{\check {\beta }}$. The chiral and antichiral representations are related through

$$\begin{equation}{\overline{\gamma }}_{\mu }^{{\dagger}}={\gamma }_{0}{\overline{\gamma }}_{\mu }{\gamma }_{0}{\Rightarrow}\begin{cases}{\overline{\gamma }}_{0}^{{\dagger}}=-{\gamma }_{0},\quad \hfill & \hfill \\ {\overline{\gamma }}_{\mu }^{{\dagger}}={\gamma }_{\mu },\quad \hfill & \mu \ne 0.\hfill \end{cases}\end{equation} \tag{ A.7 }$$

The chirality operator gives 2 additional constraints

$$\begin{equation}{\gamma }_{012345}=I,\quad {\overline{\gamma }}_{012345}=-I,\end{equation} \tag{ A.8 }$$

with ${\gamma }_{\mu \nu \dots \rho }\equiv {\gamma }_{\left[\right. \mu }{\overline{\gamma }}_{\nu }\dots {\gamma }_{\rho \left.\right]}$ the antisymmetrised product of γ-matrices ⁷ . The charge conjugation matrix takes the form

$$\begin{equation}{\mathcal{C}}_{AB}=\begin{pmatrix}\hfill 0\hfill & \hfill {c}_{\dot {\alpha }\beta }\hfill \\ \hfill {c}_{\alpha \dot {\beta }}\hfill & \hfill 0\hfill \end{pmatrix}\otimes {{\Omega}}_{\check {\alpha }\check {\beta }},\quad c\equiv {c}_{\dot {\alpha }\beta },\end{equation} \tag{ A.9 }$$

and is used to lower (or raise) spinor indices. The matrix ${{\Omega}}_{\check {\alpha }\check {\beta }}$ is the real, antisymmetric symplectic metric of $\mathfrak{s}\mathfrak{p}\left(4\right)$ and c is unitary:

$$\begin{equation}{c}^{{\dagger}}c={c}^{\alpha \dot {\alpha }}{c}_{\dot {\alpha }\beta }={\delta }_{\alpha }^{\beta },\quad {c}^{{\ast}}{c}^{T}={c}^{\dot {\alpha }\alpha }{c}_{\alpha \dot {\beta }}={\delta }_{\dot {\alpha }}^{\dot {\beta }},\quad {{\Omega}}^{{\dagger}}{\Omega}={{\Omega}}^{\check {\alpha }\check {\beta }}{{\Omega}}_{\check {\beta }\check {\gamma }}={\delta }_{\check {\gamma }}^{\check {\alpha }}.\end{equation} \tag{ A.10 }$$

They satisfy

$$\begin{equation}\left({\gamma }_{\mu }c\right)=-{\left({\gamma }_{\mu }c\right)}^{T},\quad \left({\overline{\gamma }}_{\mu }{c}^{T}\right)=-{\left({\overline{\gamma }}_{\mu }{c}^{T}\right)}^{T},\quad \left({\check {\gamma }}_{i}{\Omega}\right)=-{\left({\check {\gamma }}_{i}{\Omega}\right)}^{T}.\end{equation} \tag{ A.11 }$$

A representation of this algebra is given by

$$\begin{align}\hfill {\gamma }_{0}& ={\overline{\gamma }}_{0}=i{I}_{2}\otimes {I}_{2},\hfill & \hfill \quad {\gamma }_{1}& =-{\overline{\gamma }}_{1}=-i{\sigma }_{1}\otimes {I}_{2},\hfill & \hfill \quad {\gamma }_{2}& =-{\overline{\gamma }}_{2}=-i{\sigma }_{2}\otimes {I}_{2},\hfill \\ \hfill {\gamma }_{3}& =-{\overline{\gamma }}_{3}=i{\sigma }_{3}\otimes {\sigma }_{1},\hfill & \hfill \quad {\gamma }_{4}& =-{\overline{\gamma }}_{4}=i{\sigma }_{3}\otimes {\sigma }_{2},\hfill & \hfill \quad {\gamma }_{5}& =-{\overline{\gamma }}_{5}=-i{\sigma }_{3}\otimes {\sigma }_{3},\hfill \\ \hfill {\check {\gamma }}_{1}& ={\sigma }_{1}\otimes {\sigma }_{2},\quad {\check {\gamma }}_{2}={\sigma }_{2}\otimes {\sigma }_{2},\enspace \hfill & \hfill {\check {\gamma }}_{3}& ={\sigma }_{3}\otimes {\sigma }_{2},\quad {\check {\gamma }}_{4}={I}_{2}\otimes {\sigma }_{1},\enspace \hfill & \hfill {\check {\gamma }}_{5}& ={I}_{2}\otimes {\sigma }_{3},\hfill \\ \hfill c& =-{c}^{T}={\sigma }_{1}\otimes i{\sigma }_{2},\hfill & \hfill \quad {\Omega}& =i{\sigma }_{2}\otimes {I}_{2}.\enspace \hfill & \hfill \end{align} \tag{ A.12 }$$

A.3. Symplectic Majorana condition

In 6d the spinor Ψ decomposes into a chiral and an antichiral 6d spinor as

$$\begin{equation}{{\Psi}}_{A}=\begin{pmatrix}\hfill {\overline{\chi }}_{\dot {\alpha }\check {\alpha }}\hfill \\ \hfill {\psi }_{\alpha \check {\alpha }}\hfill \end{pmatrix},\quad {\overline{{\Psi}}}^{A}\equiv \begin{pmatrix}\hfill -i{\left({\psi }^{{\dagger}}\right)}^{\alpha \check {\alpha }}{\left({\gamma }_{0}\right)}_{\alpha }^{\dot {\alpha }}\hfill & \hfill \quad -i{\left({\overline{\chi }}^{{\dagger}}\right)}^{\dot {\alpha }\check {\alpha }}{\left({\overline{\gamma }}_{0}\right)}_{\dot {\alpha }}^{\alpha }\hfill \end{pmatrix}\equiv \begin{pmatrix}\hfill {\overline{\psi }}^{\dot {\alpha }\check {\alpha }}\hfill & \hfill {\chi }^{\alpha \check {\alpha }}\hfill \end{pmatrix}.\end{equation} \tag{ A.13 }$$

The Majorana condition on Ψ then translates to

$$\begin{equation}{\chi }^{\alpha \check {\alpha }}={\left({c}^{{\dagger}}{{\Omega}}^{{\dagger}}\overline{\chi }\right)}^{\alpha \check {\alpha }}={\left(c{\Omega}\overline{\chi }\right)}^{\alpha \check {\alpha }},\quad {\overline{\psi }}^{\dot {\alpha }\check {\alpha }}={\left({c}^{{\ast}}{{\Omega}}^{{\dagger}}\psi \right)}^{\dot {\alpha }\check {\alpha }}=-{\left(c{\Omega}\psi \right)}^{\dot {\alpha }\check {\alpha }},\end{equation} \tag{ A.14 }$$

where in the second equality we use the properties of our representation. The inclusion of the symplectic form Ω in (A.14) is the reason these equations are known as the symplectic Majorana condition. The spinors ${\overline{\varepsilon }}^{0}$, ɛ¹, and ψ in (3.1) are of this type.

In this appendix we assemble the geometry results used throughout the main text and in appendix C. Appendices B.1 and B.2 contain our conventions for Riemann curvature and the definition of the second fundamental form of an embedded submanifold as well as some standard results relating the two. In appendix B.3 the second fundamental form is related to the coefficients of the normal coordinate expansion of the embedding.

B.1. Riemann curvature

We adopt the convention where the Riemann tensor is defined as

$$\begin{equation}{{R}^{\mu }}_{\nu \rho \sigma }={\partial }_{\rho }{{\Gamma}}_{\nu \sigma }^{\mu }-{\partial }_{\sigma }{{\Gamma}}_{\nu \rho }^{\mu }+{{\Gamma}}_{\rho \lambda }^{\mu }{{\Gamma}}_{\nu \sigma }^{\lambda }-{{\Gamma}}_{\sigma \lambda }^{\mu }{{\Gamma}}_{\nu \rho }^{\lambda }.\end{equation} \tag{ B.1 }$$

It is convenient to split it into a conformally invariant Weyl tensor W_μνρσ and the Schouten tensor P_μν ,

$$\begin{equation}{P}_{\mu \nu }=\frac{1}{d-2}\left({R}_{\mu \nu }-\frac{R}{2\left(d-1\right)}{g}_{\mu \nu }\right),\end{equation} \tag{ B.2 }$$

$$\begin{equation}{W}_{\mu \nu \rho \sigma }={R}_{\mu \nu \rho \sigma }-{g}_{\mu \rho }{P}_{\nu \sigma }+{g}_{\mu \sigma }{P}_{\nu \rho }+{g}_{\nu \rho }{P}_{\mu \sigma }-{g}_{\nu \sigma }{P}_{\mu \rho }.\end{equation} \tag{ B.3 }$$

B.2. Extrinsic curvature

We define the second fundamental form to be

$$\begin{equation}{\text{II}}_{ab}^{\mu }=\left({\partial }_{a}{\partial }_{b}{x}^{\lambda }+{\partial }_{a}{x}^{\rho }{\partial }_{b}{x}^{\sigma }{{\Gamma}}_{\rho \sigma }^{\lambda }\right)\left({\delta }_{\lambda }^{\mu }-{g}_{\kappa \lambda }{\partial }^{c}{x}^{\kappa }{\partial }_{c}{x}^{\mu }\right).\end{equation} \tag{ B.4 }$$

The second part is the projector to the components orthogonal to the surface (defined by its embedding x^μ (σ)), while the first part is the action of the covariant derivative on the (pullback) of x^λ (σ). The mean curvature vector is then

$$\begin{equation}{H}^{\mu }={h}^{ab}{\text{II}}_{ab}^{\mu }.\end{equation} \tag{ B.5 }$$

These invariants are related to the intrinsic curvature of Σ and M by the Gauss–Codazzi equation

$$\begin{equation}{R}_{abcd}^{{\Sigma}}={R}_{abcd}^{M}+2{\text{II}}_{a\left[\right. b}^{\mu }{\text{II}}_{c\left.\right]d}^{\nu }{g}_{\mu \nu }.\end{equation} \tag{ B.6 }$$

Contracting twice with h⁻¹ and expanding the Riemann tensor in terms of the Weyl and Schouten tensors, we obtain

$$\begin{equation}\left({H}^{2}+4\enspace \text{tr}\enspace P\right)=2{R}^{{\Sigma}}+2\enspace \text{tr}\enspace {\tilde {\text{II}}}^{2}-2\enspace \text{tr}\enspace W,\end{equation} \tag{ B.7 }$$

where ${\tilde {\text{II}}}_{ab}^{\mu }$ is the traceless part of the second fundamental form

$$\begin{equation}{\tilde {\text{II}}}_{ab}^{\mu }={\text{II}}_{ab}^{\mu }-\frac{{H}^{\mu }}{2}{h}_{ab}.\end{equation} \tag{ B.8 }$$

B.3. Embedding in normal coordinates

Using these standard geometry results, we now derive the expressions needed for (3.16) and (3.21). Unlike in section 3, we state here the result for a generic curved spacetime M. This allows us to perform the calculation in appendix C on curved space.

Let x^μ and η^a be Riemann normal coordinates on M and Σ about the same point. In terms of these, the embedding Σ ↪ M may be expanded as

$$\begin{equation}{x}^{\mu }\left(\eta \right)={x}^{\mu }\left(0\right)+{\eta }^{a}{v}_{a}^{\mu }+\frac{1}{2}{\eta }^{a}{\eta }^{b}{v}_{ab}^{\mu }+\frac{1}{6}{\eta }^{a}{\eta }^{b}{\eta }^{c}{v}_{abc}^{\mu }+\mathcal{O}\left({\eta }^{4}\right).\end{equation} \tag{ B.9 }$$

These coefficients are constrained by the condition that straight lines in normal coordinates correspond to geodesics. In particular, a curve on Σ given by a straight line in η has constant speed and its curvature in M is normal to Σ at every point, which gives the constraints

$$\begin{equation}\begin{aligned}\hfill {\delta }_{ab}& ={v}_{a}\cdot {v}_{b},\hfill \\ \hfill 0& ={v}_{ab}\cdot {v}_{c},\hfill \\ \hfill 0& =3\enspace {v}_{d}\cdot {v}_{abc}+{v}_{ab}\cdot {v}_{cd}+{v}_{ac}\cdot {v}_{bd}+{v}_{ad}\cdot {v}_{bc}.\hfill \end{aligned}\end{equation} \tag{ B.10 }$$

Using (B.4) one easily checks that the second order coefficient equals the second fundamental form

$$\begin{equation}{\text{II}}_{ab}^{\mu }{\vert }_{\eta =0}={v}_{ab}^{\mu }.\end{equation} \tag{ B.11 }$$

The geodesic distance between ξ(η) and the origin of the normal frame is found from (B.9)

$$\begin{equation}{\left\vert x\left(\eta \right)-x\left(0\right)\right\vert }^{2}={\eta }^{a}{\eta }_{a}-\frac{1}{12}{\text{II}}_{ab}\cdot {\text{II}}_{cd}{\eta }^{a}{\eta }^{b}{\eta }^{c}{\eta }^{d}+\mathcal{O}\left({\eta }^{5}\right).\end{equation} \tag{ B.12 }$$

Furthermore, in normal coordinates, the metrics take the form

$$\begin{equation}\begin{aligned}\hfill {g}_{\mu \nu }& ={\delta }_{\mu \nu }-\frac{1}{3}{R}_{\mu \rho \nu \sigma }^{M}{\xi }^{\rho }{\xi }^{\sigma }+\mathcal{O}\left({\xi }^{3}\right),\hfill \\ \hfill {h}_{ab}& ={\delta }_{ab}-\frac{1}{3}{R}_{acbd}^{{\Sigma}}{\eta }^{c}{\eta }^{d}+\mathcal{O}\left({\eta }^{3}\right),\hfill \end{aligned}\end{equation} \tag{ B.13 }$$

which yields an expansion for the volume factor

$$\begin{equation}\sqrt{h\left(\eta \right)}=1-\frac{1}{6}{R}_{ab}^{{\Sigma}}{\eta }^{a}{\eta }^{b}+\mathcal{O}\left({\eta }^{3}\right).\end{equation} \tag{ B.14 }$$

In this appendix we present an alternative regularisation of (3.4), essentially point splitting, displacing one copy of the surface operator by a distance in an arbitrary normal direction ν. This regularisation is used in [16, 17], but there the vector ν is taken to be a constant, and therefore the method is only applicable if the operators are restricted to a codimension-one subspace.

The technology used to define this regularisation scheme applies for generic smooth embedded surfaces in a Riemannian manifold, and we present here a curved space calculation, as opposed to section 3.3, where for brevity we restricted ourselves to flat space. However, we still have to restrict to conformally flat backgrounds, since otherwise we do not have a short-distance expansion for the propagator and therefore still cannot infer the anomaly coefficient b.

As expected, we recover the result (3.23) exactly, and thus verify scheme-independence.

C.1. Displacement map

We can regularise the integral (3.4) by displacing a copy of the surface a distance along a unit normal vector field ν. Under that map, which we denote by $\mathcal{T}$, the geodesic distance admits an expansion of the form

$$\begin{equation}{\left\vert \mathcal{T}\left({x}^{\mu }\left(\sigma \right)\right)-{x}^{\mu }\left(\sigma +\eta \right)\right\vert }^{2}={{\epsilon}}^{2}+{\eta }^{2}+\sum _{k=3}^{\infty }\sum _{l=0}^{k}{f}_{l}^{\left(k\right)}{\eta }^{l}{{\epsilon}}^{k-l}.\end{equation} \tag{ C.1 }$$

We can combine the terms of fixed k in terms of degree k polynomials f^(k)

$$\begin{equation}\sum _{l=0}^{k}{f}_{l}^{\left(k\right)}{\eta }^{l}{{\epsilon}}^{k-l}={{\epsilon}}^{k}{f}^{\left(k\right)}\left(\eta /{\epsilon}\right).\end{equation} \tag{ C.2 }$$

We calculate the higher order terms in (C.1) explicitly in (C.7), but first we note that the only terms contributing to the divergent part are f⁽³⁾ and f⁽⁴⁾. To see that, the integrals computing the expectation value take the form

$$\begin{equation}\underset{0}{\overset{\rho }{\int }}\frac{{\eta }^{m+1}\mathrm{d}\eta }{{\left\vert \mathcal{T}\left({x}^{\mu }\left(\sigma \right)\right)-{x}^{\mu }\left(\sigma +\eta \right)\right\vert }^{4}},\end{equation} \tag{ C.3 }$$

where ρ is an arbitrary but fixed IR cutoff. We can evaluate (C.3) by expanding the integrand in . Writing s ≡ η/, we obtain

$$\begin{equation}{{\epsilon}}^{m-2}\underset{0}{\overset{\rho /{\epsilon}}{\int }}\frac{{s}^{m+1}}{{\left(1+{s}^{2}\right)}^{2}}\left[1-\frac{2{f}^{\left(3\right)}\left(s\right)}{1+{s}^{2}}{\epsilon}+\left(\frac{3{\left({f}^{\left(3\right)}\left(s\right)\right)}^{2}}{{\left(1+{s}^{2}\right)}^{2}}-\frac{2{f}^{\left(4\right)}\left(s\right)}{1+{s}^{2}}\right){{\epsilon}}^{2}+\mathcal{O}\left({{\epsilon}}^{3}\right)\right]\mathrm{d}s.\end{equation} \tag{ C.4 }$$

By application of Faà di Bruno's formula one checks that the terms in brackets of order ⁿ contribute to the divergence only if m + n ⩽ 2. We can therefore safely ignore higher orders in . Only a finite number of terms remains to be computed and we find that the only divergent integrals (C.3) are:

$$\begin{equation}m=0:\quad \frac{1}{2{{\epsilon}}^{2}}-\frac{1}{8{\epsilon}}\left(4{f}_{0}^{\left(3\right)}+\pi {f}_{1}^{\left(3\right)}+4{f}_{2}^{\left(3\right)}+3\pi {f}_{3}^{\left(3\right)}\right)+\left(-3{\left({f}_{3}^{\left(3\right)}\right)}^{2}+2{f}_{4}^{\left(4\right)}\right)\mathrm{log}\enspace {\epsilon},\end{equation} \tag{ C.5a }$$

$$\begin{equation}m=1:\quad \frac{\pi }{4{\epsilon}}+2{f}_{3}^{\left(3\right)}\mathrm{log}\enspace {\epsilon},\end{equation} \tag{ C.5b }$$

$$\begin{equation}m=2:\quad -\mathrm{log}\enspace {\epsilon}.\end{equation} \tag{ C.5c }$$

The relevant coefficients can be read off of the expansion of the geodesic distance up to combined order of 4 in η and . The second term on the left-hand side of (C.1) can be expanded simply using the embedding (B.9). For the first term, we solve the geodesic equation order by order in the displacement to obtain

$$\begin{equation}\mathcal{T}\left({x}^{\mu }\right)={x}^{\mu }+{\epsilon}{\nu }^{\mu }-\frac{{{\epsilon}}^{2}}{2}{{\Gamma}}_{\kappa \lambda }^{\mu }{\nu }^{\kappa }{\nu }^{\lambda }+\frac{{{\epsilon}}^{3}}{6}\left(-{\partial }_{\nu }{{\Gamma}}_{\rho \sigma }^{\mu }+2{{\Gamma}}_{\nu \lambda }^{\mu }{{\Gamma}}_{\rho \sigma }^{\lambda }\right){\nu }^{\nu }{\nu }^{\rho }{\nu }^{\sigma }+\mathcal{O}\left({{\epsilon}}^{4}\right).\end{equation} \tag{ C.6 }$$

Combining these expressions, and writing η^a = ηe^a (φ) as in (3.18) and onwards, the only two non-vanishing relevant coefficients read

$$\begin{equation}\begin{aligned}\hfill {f}_{2}^{\left(3\right)}& =-{e}^{a}{e}^{b}{\text{II}}_{ab}\cdot \nu ,\hfill \\ \hfill {f}_{4}^{\left(4\right)}& =-\frac{1}{12}{e}^{a}{e}^{b}{e}^{c}{e}^{d}{\text{II}}_{ab}\cdot {\text{II}}_{cd}.\hfill \end{aligned}\end{equation} \tag{ C.7 }$$

The first contributes to a scheme-dependent divergence ⁻¹, while the second contributes to the anomaly.

C.2. Evaluation of the anomaly

With the displacement map (C.6) in hand, we can evaluate (3.4). The propagators on a conformally flat background can be obtained by considering curved space actions for a conformal scalar and a Maxwell-type two-form and inverting the kinetic operators order by order, following [14, 17]. We find:

$$\begin{equation}\left\langle {{\Phi}}_{i}\left(x\right){{\Phi}}_{j}\left(x+\xi \right)\right\rangle =\frac{{\delta }_{ij}}{2{\pi }^{2}{\left\vert \xi \right\vert }^{4}}\left[1+\frac{1}{3}{P}_{\mu \nu }{\xi }^{\mu }{\xi }^{\nu }+\mathcal{O}\left({\xi }^{3}\right)\right],\end{equation} \tag{ C.8 }$$

$$\begin{align}\hfill \left\langle {B}^{+\mu \nu }\left(x\right){B}_{\rho \sigma }^{+}\left(x+\xi \right)\right\rangle & =\frac{1}{2{\pi }^{2}{\left\vert \xi \right\vert }^{4}}\left[{\delta }_{\mu }^{\rho }{\delta }_{\nu }^{\sigma }-{\delta }_{\nu }^{\rho }{\delta }_{\mu }^{\sigma }\right.\hfill \\ \hfill & \quad \left.-\frac{4}{3}\left(4{P}_{\left[\right.\rho }^{\left[\right. \mu }{\delta }_{\sigma \left.\right]}^{\nu \left.\right]}{\delta }_{\lambda \tau }+{P}_{\lambda \left[\right.\rho }{\delta }_{\sigma \left.\right]}^{\left[\right. \mu }{\delta }_{\tau }^{\nu \left.\right]}+{\delta }_{\lambda \left[\right.\rho }{P}_{\sigma \left.\right]}^{\left[\right. \mu }{\delta }_{\tau }^{\nu \left.\right]}\right){\xi }^{\lambda }{\xi }^{\tau }+\mathcal{O}\left({\xi }^{3}\right)\right].\hfill \end{align} \tag{ C.9 }$$

To apply our regularisation, we should replace ξ by (C.1) in the denominator of the propagators before performing the integral over η. A priori, we should also perform the displacement in the numerator, since a term of order $\mathcal{O}\left({\epsilon}\right)$ can contribute to the ⁻¹ divergence by multiplying (C.5a). However, one easily checks that the only terms of that order are accompanied by nonzero powers of η, and therefore do not contribute to the divergence of (3.4). We therefore drop the in the numerators of the propagators.

The expansion of the numerators is then assembled, as before, from (3.16) and (3.21), but in addition, since we are working on curved space, we obtain an additional term at $\mathcal{O}\left({\eta }^{2}\right)$ explicitly involving tr  P from the propagators (C.8). Collecting terms in analogy to section 3.3, and integrating out the angular coordinate using (3.18), we obtain the scalar contribution

$$\begin{equation}\frac{1}{4\pi {{\epsilon}}^{2}}+\frac{H\cdot \nu }{8\pi {\epsilon}}+\frac{1}{32\pi }\left(2{R}^{{\Sigma}}-\left({H}^{2}+4\enspace \text{tr}\enspace P\right)+4{\left(\partial n\right)}^{2}\right)\mathrm{log}\enspace {\epsilon}+\text{finite},\end{equation} \tag{ C.10 }$$

while the tensor field yields

$$\begin{equation}-\frac{1}{4\pi {{\epsilon}}^{2}}-\frac{H\cdot \nu }{8\pi {\epsilon}}-\frac{1}{32\pi }\left(-2{R}^{{\Sigma}}+3\left({H}^{2}+4\enspace \text{tr}\enspace P\right)\right)\mathrm{log}\enspace {\epsilon}+\text{finite}.\end{equation} \tag{ C.11 }$$

Combining these terms, we find

$$\begin{equation}\mathrm{log}\left\langle {V}_{{\Sigma}}\right\rangle =\frac{1}{8\pi }\enspace \mathrm{log}\enspace {\epsilon}{\int }_{{\Sigma}}{\text{vol}}_{{\Sigma}}\left[{R}^{{\Sigma}}-\left({H}^{2}+4\enspace \text{tr}\enspace P\right)+{\left(\partial n\right)}^{2}\right]+\text{finite},\end{equation} \tag{ C.12 }$$

which agrees exactly with (3.23). Note that the scheme dependence, which is present in the simple pole of both (C.10) and (C.11), cancels in the final result, and the terms H² and tr  P combine to an anomaly term as in (2.3), as required.