Observations on BPS observables in 6D

In the quest to ascend the precipice of the 6D $\mathcal{N}=\left(2,0\right)$ theory, we are in great need of edges or cracks to hold on to. Surface operators are ideal candidates, being the intrinsic, low dimensional, non-local observables of these theories [1–3]. Already, recent work combining the powerful techniques of the superconformal index [4, 5], holographic entanglement entropy [6–11] and defect CFT techniques [12–14] led to the exact calculation of the anomaly coefficients (see section 4), thus giving the leading property of surface operators: the logarithmic divergence in their expectation values, which is an analog of the anomalous dimensions of local operators [15–19].

A distinguished class of surface operators are BPS observables, i.e. those that preserve some of the supercharges of the theory. For these operators, supersymmetry affords many simplifications and, from the experience with BPS Wilson loops in lower dimensional theories [20–28], we expect BPS surface operators to be amenable to a variety of nonperturbative techniques, providing an ideal playground to learn more about the mysterious 6D $\mathcal{N}=\left(2,0\right)$ theories.

The obvious examples of BPS surface operators are the plane [29, 30] and the sphere [31], which preserve half of the supercharges and are analyzed in detail in [6–11, 13, 32–35]. There are no explicit expressions for the surface operators beyond the abelian theory and the holographic theory at large N, but they should be labeled by a representation of the ADE algebra of the theory. As far as the global symmetries, the 1/2 BPS operator is defined by a choice of plane in 6D, which breaks the 6D conformal symmetry $\mathfrak{s}\mathfrak{o}\left(2,6\right)$ to $\mathfrak{s}\mathfrak{o}\left(2,2\right){\times}\mathfrak{s}\mathfrak{o}\left(4\right)$, and a breaking of the $\mathfrak{s}\mathfrak{o}\left(5\right)$ R-symmetry to $\mathfrak{s}\mathfrak{o}\left(4\right)$, which can be realised by choosing a point on S⁴. Accounting for supercharges, the full preserved symmetry is an $\mathfrak{o}\mathfrak{s}\mathfrak{p}{\left({4}^{{\ast}}\vert 2\right)}^{2}$ subalgebra of the $\mathfrak{o}\mathfrak{s}\mathfrak{p}\left({8}^{{\ast}}\vert 4\right)$ symmetry algebra of the vacuum.

This analysis can be generalised to surface operators with an arbitrary geometry ${\Sigma}\to {\mathbb{R}}^{6}$ and a local breaking of R-symmetry n : Σ → S⁴ [19]. Pointwise these operators preserve half of the supersymmetries, but typically the supersymmetries are not compatible over the surface, thus can be called 'locally BPS'.

In this paper we present several large families of truly BPS surface operators, requiring the supercharges to be compatible over the full surface. As a simple illustration, consider two planes extended respectively along (x¹, x²) and (x¹, x³). For the first we take n^I = δ^I1, for the second, n^I = δ^I2. The two operators so defined share half of their supercharges, and as the planes intersect along a line, we can take half of each and form a corner (which is then 1/4 BPS). The operators presented below are massive generalisations of this example.

A word on notations: we use x^μ with μ = 1, ..., 6 as coordinates of 6D Euclidean space with metric δ_μν . Spacetime spinor indices are α and $\dot {\alpha }$ respectively for chiral and anti-chiral spinors, and for the fundamental of $\mathfrak{s}\mathfrak{p}\left(4\right)$ we use the indices A, all take four values. For the R-symmetry vectors (fundamental of $\mathfrak{s}\mathfrak{o}\left(5\right)$) we employ I, J = 1, ..., 5. On the surface Σ we use coordinates σ^a = (u, v) with a = 1, 2 and the metric induced from the embedding h_ab = ∂_a x^μ ∂_b x^μ with its inverse h^ab . We also use ɛ^ab to denote the Levi-Civita tensor density (including a factor $\mathrm{det}{\left({h}_{ab}\right)}^{-1/2}$).

Our construction relies on spinor geometry. To formulate the problem, recall that the supercharges of the $\mathcal{N}=\left(2,0\right)$ theory are parametrised by two 16-component constant spinors ɛ₀ and ${\bar{\varepsilon }}_{1}$. They are respectively chiral and anti-chiral in 6D and transform also under $\mathfrak{s}\mathfrak{p}\left(4\right)$. Since we work in Euclidean signature our spinors do not satisfy a reality condition. ɛ₀ parametrises the 16 super-Poincaré symmetries, Q, and ${\bar{\varepsilon }}_{1}$ the 16 superconformal ones, S. Together ${\varepsilon }_{0}+{\bar{\varepsilon }}_{1}\left(x\cdot \bar{\gamma }\right)$ are the conformal Killing spinors in flat space.

We use γ^μ and ${\bar{\gamma }}^{\mu }$ for the spatial gamma matrices in the chiral basis and ρ^I for the $\mathfrak{s}\mathfrak{o}\left(5\right)$ gamma matrices. They satisfy

$$\begin{equation}{\bar{\gamma }}_{\mu }{\gamma }_{\nu }+{\bar{\gamma }}_{\nu }{\gamma }_{\mu }=2{\delta }_{\mu \nu },\qquad {\gamma }_{\mu }{\bar{\gamma }}_{\nu }+{\gamma }_{\nu }{\bar{\gamma }}_{\mu }=2{\delta }_{\mu \nu },\qquad \left\{{\rho }_{I},{\rho }_{J}\right\}=2{\delta }_{IJ},\end{equation} \tag{ 1.1 }$$

and γ_μ and ρ_I commute. Explicitly the indices are ${{\left({\gamma }_{\mu }\right)}_{\alpha }}^{\dot {\alpha }}$, ${{\left({\bar{\gamma }}_{\mu }\right)}_{\dot {\alpha }}}^{\alpha }$, ${{\left({\rho }_{I}\right)}_{A}}^{B}$, ${\left({\varepsilon }_{0}\right)}^{\alpha A}$ and ${\left({\bar{\varepsilon }}_{1}\right)}^{\dot {\alpha }A}$, but we suppress them as there is no ambiguity. We also use the standard shorthand notation ${\gamma }_{\mu \nu }\equiv {\gamma }_{\left[\mu \right.}{\bar{\gamma }}_{\left.\nu \right]}$ for antisymmetrised products.

With these notations in hand we can be more precise about the conditions required to preserve supersymmetries. At any point along the surface, the supercharges preserved by the tangent plane to Σ and the choice of n in S⁴ are those satisfying the projector equation

$$\begin{equation}\left[{\varepsilon }_{0}+{\bar{\varepsilon }}_{1}\left({x}^{\rho }{\bar{\gamma }}_{\rho }\right)\right]\left[\frac{\mathrm{i}}{2}{\varepsilon }^{ab}{\partial }_{a}{x}^{\mu }{\partial }_{b}{x}^{\nu }{\gamma }_{\mu \nu }+{n}^{I}{\rho }_{I}\right]=0.\end{equation} \tag{ 1.2 }$$

This is a set of coupled equations depending on the embedding of the surface x^μ (u, v) and the scalar coupling along the surface n^I (u, v). If n is a unit vector, then this matrix is half-rank, so pointwise half the components of ${\varepsilon }_{0},{\bar{\varepsilon }}_{1}$ satisfy the equation, which is the 'locally BPS' notion already mentioned above.

To find global solutions to these equations with the same ɛ₀, ${\bar{\varepsilon }}_{1}$ along the surface, we do not attempt a full classification but present four main classes of solutions. These four classes have Σ in different subspaces of ${\mathbb{R}}^{6}$ and/or satisfying different constraints and with appropriate choices of n^I . Each of the four classes has many examples with extra properties. To avoid clutter, we give special names to only two of those subclasses, and all the other special examples are explained in the appropriate sections. The main examples are:

Type- $\mathbb{R}$: such surfaces are the product of a curve ${x}^{I}\left(u\right)\subset {\mathbb{R}}^{5}$ (I = 1, ..., 5) and the x⁶ = v direction. To guarantee that they are BPS we should choose the scalar coupling to be tangent to the curve

$$\begin{equation}{n}^{I}\left(u,v\right)=\frac{{\partial }_{u}{x}^{I}}{\vert {\partial }_{u}x\vert }.\end{equation} \tag{ 1.3 }$$

The notation x^I instead of x^μ is common in our examples, as in topological twisting, and is a crucial ingredient to get BPS observables. Examples of surfaces in this class were previously studied in [36].

This choice of identification allows for an immediate generalisation of multiplying the right-hand side by a fixed SO(5) matrix (or indeed replace x⁶ by any other line), so in total S⁵ × SO(5) of examples. Such modifications are also possible to all the classes below, but we do not worry about counting these generalisations and fix convenient representatives.

Type- $\mathbb{C}$: viewing ${\mathbb{R}}^{6}={\mathbb{C}}^{3}$ (with any standard complex structure), we can take the surface to be any holomorphic curve with a fixed scalar direction, say n¹.

Type- $\mathbb{H}$: restricting the surfaces to ${\mathbb{R}}^{4}$, we can take an arbitrary oriented surface and get a BPS observable, as long as

$$\begin{equation}{n}^{I}=\frac{1}{2}{\eta }_{\mu \nu }^{I}{\partial }_{a}{x}^{\mu }{\partial }_{b}{x}^{\nu }{\varepsilon }^{ab},\end{equation} \tag{ 1.4 }$$

where η is the 't Hooft chiral symbol (see (2.3)), and I = 1, 2, 3, so n^I are in S². This has a nice interpretation in terms of the Gauss map, which we explain in section 2. Particular special subclasses are surfaces with fixed n^I = (0, 0, 1) which are holomorphic with respect to an appropriate choice of complex structure, so they overlap with Type- $\mathbb{C}$ (except for the exchange of n¹ → n³). Another special subclass are Lagrangian submanifolds, for which n³ = 0. We denote those as Type-L. Lastly we should mention that for conical surfaces, the conformal transformation to surfaces in $\mathbb{R}{\times}{S}^{5}$ have been described and studied in great detail in [37].

Type-S: any surfaces within an ${S}^{3}\subset {\mathbb{R}}^{6}$ is BPS provided we choose

$$\begin{equation}{n}^{I}=\frac{1}{2}{\varepsilon }^{IJKL}{\varepsilon }^{ab}{\partial }_{a}{x}^{J}{\partial }_{b}{x}^{K}{x}^{L}.\end{equation} \tag{ 1.5 }$$

x^L appears explicitly and this expression assumes a sphere centered around the origin. We also assume a unit sphere, our results can be trivially extended to a sphere of arbitrary radius by dimensional analysis. These surfaces are different from the restriction of the Type- $\mathbb{H}$ ansatz to ${S}^{3}\subset {\mathbb{R}}^{4}$, but they do overlap at the vicinity of the north pole, where x^μ ∼ δ^μ4 and using the three-index antisymmetric tensor, both (1.4) and (1.5) reduce at leading order in $\left\vert {x}^{\mu }-{\delta }^{\mu 4}\right\vert$ to

$$\begin{equation}{n}^{I}=\frac{1}{2}{\varepsilon }^{IJK}{\partial }_{a}{x}^{J}{\partial }_{b}{x}^{K}{\varepsilon }^{ab}.\end{equation} \tag{ 1.6 }$$

We will refer to this subclass as Type-N.

The paper is organised by themes. In section 2 we present the geometry underlying these constructions. The proof of supersymmetry for these classes of examples is contained in section 3. We discuss the Weyl anomaly in section 4, the reduction to line and surface operators in lower dimensional theories in section 5, and the holographic duals in section 6. The properties pertaining to each of the constructions above are scattered between these sections. To aid the reader we provided extensive cross referencing. Also, we summarize the main classification in table 1 and some of the most important examples in table 2, with links to the relevant pages where they are discussed.

Table 1. Main classes and subclasses of BPS surface operators. For each geometry we list the minimal number of preserved supercharges, which type it belongs to, and list the equation with its anomaly and the associated three-form defined on a subspace of AdS₇ × S⁴ that is compatible with supersymmetry.

Type	Geometry	nI in	SUSYs	Also in	Anomaly	3-form
$\mathbb{R}$	$\mathbb{R}{\times}\gamma$, $\gamma \subset {\mathbb{R}}^{5}$	S4	1Q		0	(6.7)
	$\mathbb{R}{\times}\gamma$, $\gamma \subset {\mathbb{R}}^{4}$	S3	1Q
	$\mathbb{R}{\times}\gamma$, $\gamma \subset {\mathbb{R}}^{3}$	S2	2Q	$\mathbb{H}$
	$\mathbb{R}{\times}\gamma$, $\gamma \subset {\mathbb{R}}^{2}$	S1	4Q	$\mathbb{H}$
$\mathbb{C}$	${\Sigma}\subset {\mathbb{C}}^{3}$ (holo.)	Point	2Q		(4.6)	(6.10)
	${\Sigma}\subset {\mathbb{C}}^{2}$ (holo.)	Point	4Q	$\mathbb{H}$, L
$\mathbb{H}$	${\Sigma}\subset {\mathbb{R}}^{4}$	S2	1Q		(4.11)	(6.12)
Subclass $\begin{cases}L\quad \hfill \\ N\quad \hfill \end{cases}$	Lagrangian	S1	2Q
	${\Sigma}\subset {\mathbb{R}}^{3}$	S2	2Q	(S)
S	Σ ⊂ S3	S3	2(Q + S)		(4.14)	(6.21)

Table 2. Examples of BPS operators with enhanced supersymmetry.

Name	Type	Geometry	nI in	SUSYs	See
Cones	$\mathbb{H}$	Over γ ⊂ S3	S2	Q, S	[37]
	$\mathbb{H}$, N	Over γ ⊂ S2	S2	2Q, 2S
Crease	$\mathbb{R}$, $\mathbb{H}$, N	Two half-planes	S0	$4\mathsf{Q},4\mathsf{S}\subset \mathfrak{o}\mathfrak{s}\mathfrak{p}\left({4}^{{\ast}}\vert 2\right)$	[38]
Tori	S	T2 ⊂ S3	S4	2(Q + S)
	$\mathbb{H}$, L	${T}^{2}\subset {\mathbb{R}}^{4}$	S1	2Q, 2S
Spheres	$\mathbb{H}$, N	${S}^{2}\subset {\mathbb{R}}^{3}$	S2	2Q, 2S
	S	Latitude S2 ⊂ S3	S3	4(Q + S)
	S	Large S2 ⊂ S3	Point	$16\left(\mathsf{Q}+\mathsf{S}\right)\subset \mathfrak{o}\mathfrak{s}\mathfrak{p}{\left({4}^{{\ast}}\vert 2\right)}^{2}$	[31]
Plane	$\mathbb{R}\enspace \mathbb{C}\enspace \mathbb{H}$ L N	${\mathbb{R}}^{2}$	Point	$8\mathsf{Q},8\mathsf{S}\subset \mathfrak{o}\mathfrak{s}\mathfrak{p}{\left({4}^{{\ast}}\vert 2\right)}^{2}$	[30]

In all our examples, we construct BPS observables by appropriately choosing the surface and a vector n over it. The details of these choices rely on varied notions of geometry of surfaces, which we review in turn for each type.

Type- $\mathbb{R}$: these surfaces are defined in terms of a curve in ${\mathbb{R}}^{5}$ and n^I (1.3) is its (normalised) tangent vector. Clearly for a general enough curve we can get any closed path n^I (u) on S⁴. If the curve is restricted to an ${\mathbb{R}}^{d}$ subspace, then n^I are constrained to an S^d−1. As we show in the next section, such loops preserve max(1, 2^4−d ) of the Poincaré supercharges Q.

Type-N: to ease into the other examples, let us start with this subclass of surfaces in ${\mathbb{R}}^{3}$ where n^I is the (unit) normal vector with a chosen orientation (1.6). This map from a surface to S² is known as the Gauss map and its degree is half of the Euler characteristic of the surface, which is important for the calculation of the anomaly below.

Type- $\mathbb{H}$: the Gauss map has an interesting generalisation to surfaces in ${\mathbb{R}}^{4}$ [39], which we can use to pick our vector n. At any point on the surface the tangent space is a plane in ${\mathbb{R}}^{4}$. The space of these planes is the Grassmanian ${G}_{2}\left({\mathbb{R}}^{4}\right)$, which is a homogenous space defined as the quotient of the orthogonal group SO(n)

$$\begin{equation}{G}_{k}\left({\mathbb{R}}^{d}\right)=\frac{SO\left(d\right)}{SO\left(k\right){\times}SO\left(d-k\right)}.\end{equation} \tag{ 2.1 }$$

For planes in 4D, the image of the Gauss map is thus a point on ${G}_{2}\left({\mathbb{R}}^{4}\right)={S}^{2}{\times}{S}^{2}$.

The ansatz (1.4) restricts n^I to S², which is just one of the two spheres of the Gauss map. To understand it, recall that two-forms in 4D can be decomposed into self-dual and anti-self-dual two-forms. ${\eta }_{\mu \nu }^{I}$ projects to the self-dual part, which are in the $\left(\mathbf{3},\mathbf{1}\right)$ representation of $\mathfrak{s}\mathfrak{u}{\left(2\right)}_{\mathrm{L}}{\times}\mathfrak{s}\mathfrak{u}{\left(2\right)}_{\mathrm{R}}\simeq \mathfrak{s}\mathfrak{o}\left(4\right)$. Another way to see it is that as a hyperkäler manifold, ${\mathbb{R}}^{4}$ has S² worth of Kähler forms ω^I (compatible with the orientation ɛ₁₂₃₄ = 1) and any surface is locally holomorphic with respect to one of them. We can write a basis for them in terms of the 't Hooft symbols

$$\begin{equation}{\omega }^{I}=\frac{1}{2}{\eta }_{\mu \nu }^{I}\enspace \mathrm{d}{x}^{\mu }\wedge \mathrm{d}{x}^{\nu }.\end{equation} \tag{ 2.2 }$$

Explicitly we use

$$\begin{equation}\begin{aligned}\hfill {\omega }^{3}=\mathrm{d}{x}^{1}\wedge \mathrm{d}{x}^{2}+\mathrm{d}{x}^{3}\wedge \mathrm{d}{x}^{4},\\ \hfill {\omega }^{2}=\mathrm{d}{x}^{3}\wedge \mathrm{d}{x}^{1}+\mathrm{d}{x}^{2}\wedge \mathrm{d}{x}^{4},\\ \hfill {\omega }^{1}=\mathrm{d}{x}^{1}\wedge \mathrm{d}{x}^{4}+\mathrm{d}{x}^{2}\wedge \mathrm{d}{x}^{3}.\end{aligned}\end{equation} \tag{ 2.3 }$$

If the map degenerates to a point n³ = 1, the self-dual projection of the tangent space is a linear combination of the (x¹, x²) and (x³, x⁴) planes, and in fact is holomorphic with respect to the complex structure associated to ω³, hence of Type- $\mathbb{C}$. The converse to this is a restriction to n³ = 0 everywhere, which means that the pullback to Σ of ω₃ vanishes. Surfaces satisfying this condition are Lagrangian submanifold, as in Type-L, and the half of the Gauss map represented by n^I reduces to an equator of S².

Type-S: these are also surfaces in ${\mathbb{R}}^{4}$, but now restricted to S³ with n^I as in (1.5). One way to understand this construction is that locally we have a surface in ${\mathbb{R}}^{3}$ with a version of the Type-N ansatz, so a local Gauss map, where the target S² is the one perpendicular to the vector x.

Note that restricting Σ to any great S² ⊂ S³ makes n^I a constant, so these are 1/2 BPS spheres, conformal to the plane. A more general subclass are surfaces foliated by arcs between the north and south pole, which describe a geometry reminiscent of a banana. In this case n^I belongs to an S², which is the sphere we get for the north pole Type-N surfaces. These 'banana' surfaces can be obtained by a conformal transformation (stereographic projection) of the cones in ${\mathbb{R}}^{3}$ and in fact the value of n^I exactly matches those of cones of Type-N also away from the north pole. These surfaces also have enhanced supersymmetry.

One more subclass of Type-S are non-maximal or 'latitude' spheres. They resemble in some ways the latitude Wilson loops of [26, 40] and their n^I image is also a latitude sphere there. One can think of those as interpolating between the 1/2 BPS sphere and the Type-N sphere. Similar interpolations are also possible between Type-L and Type- $\mathbb{C}$ within Type- $\mathbb{H}$, which are surfaces with fixed n³, so the Gauss map is into a latitude on S².

Type- $\mathbb{C}$: in this case we choose a complex structure on ${\mathbb{R}}^{6}$ and take any holomorphic curve. Regardless of how complicated the surface is, a constant n^I = δ^I1 yields a BPS surface. For a generic surface, as we show in the next section, it preserves two Q's. If it is located in a ${\mathbb{C}}^{2}$ subspace (so overlapping with Type- $\mathbb{H}$) it preserves four, and the plane $\mathbb{C}$ preserves eight.

The approach for proving that our examples preserve some supersymmetries, or in fact for constructing the examples relies on a generalisation of the idea already presented in the introduction. For each class of examples we can reduce the projector equation (1.2) over the whole surface to a set of independent constraints on ɛ₀ and ${\bar{\varepsilon }}_{1}$ associated with each independent tangent plane of the surface (and its vector n).

The number of supercharges preserved by a given surface operator depends on the number of constraints needed to satisfy (1.2). Since each independent constraint is half-rank (this can be seen pointwise from the projector equation) and ɛ has 32 components, in the most general case we could impose five independent conditions leaving a single Killing spinor. Each Killing spinor ɛ satisfying the constraints parametrises a supersymmetry transformation given by

$$\begin{equation}{\varepsilon }_{0}\mathsf{Q}+{\bar{\varepsilon }}_{1}\mathsf{S}.\end{equation} \tag{ 3.1 }$$

The analysis of supersymmetry captures many geometrical properties of the surfaces and the associated map n^I presented in section 2.

Type- $\mathbb{R}$: this case is particularly simple and closely resembles the beautiful construction of [41]. Choosing matching bases for the 2 unit $\mathfrak{s}\mathfrak{o}\left(5\right)$ vectors ∂_u x^I and n^I and identifying them via (1.3), the projector (1.2) factorises as

$$\begin{equation}\left[{\varepsilon }_{0}+{\bar{\varepsilon }}_{1}\left({x}^{\rho }{\bar{\gamma }}_{\rho }\right)\right]\left(\mathrm{i}{\gamma }_{I6}+{\rho }_{I}\right)\;{n}^{I}\left(u\right).\end{equation} \tag{ 3.2 }$$

For a generic curve x^I (u), all special supersymmetries are broken (${\bar{\varepsilon }}_{1}=0$) and we focus only on ɛ₀. At any point, the constraint imposed on the Killing spinor is a linear combination of the five projectors acting on ɛ₀

$$\begin{equation}\mathrm{i}{\gamma }_{I6}+{\rho }_{I},\quad I=1,\dots \hspace{-.3pt},5.\end{equation} \tag{ 3.3 }$$

To count the number of independent constraint we multiply them from the right by $\frac{\mathrm{i}}{2}{\gamma }_{56}$ and define

$$\begin{equation}{a}_{I}^{{\dagger}}=\frac{1}{2}\left({\gamma }_{I5}+\mathrm{i}{\gamma }_{56}{\rho }_{I}\right),\qquad {a}_{I}=\frac{1}{2}\left(-{\gamma }_{I5}+\mathrm{i}{\gamma }_{56}{\rho }_{I}\right),\quad I=1,\dots \hspace{-.3pt},4.\end{equation} \tag{ 3.4 }$$

These operators satisfy an oscillator algebra, which shows they are independent

$$\begin{equation}\left\{{a}_{I},{a}_{J}^{{\dagger}}\right\}={\delta }_{IJ},\qquad \left\{{a}_{I},{a}_{J}\right\}=\left\{{a}_{I}^{{\dagger}},{a}_{J}^{{\dagger}}\right\}=0.\end{equation} \tag{ 3.5 }$$

Note that since our gamma matrices are chiral, they satisfy ${\gamma }_{1}{\bar{\gamma }}_{2}{\gamma }_{3}{\bar{\gamma }}_{4}{\gamma }_{5}{\bar{\gamma }}_{6}=\mathrm{i}$, or equivalently ${\gamma }_{1}{\bar{\gamma }}_{2}{\gamma }_{3}{\bar{\gamma }}_{4}=-\mathrm{i}{\gamma }_{56}$. Likewise, ρ₁ ρ₂ ρ₃ ρ₄ = ρ₅, so the spinor annihilated by all four conditions ${a}_{I}^{{\dagger}}$ also satisfies ɛ₀(iγ_I5 + ρ₅) = 0.

Hence if the curve is a straight line, n is a constant vector and the projector equation imposes a single condition. This is the plane, which preserves eight Q supercharges (and exceptionally also eight S). For each extra dimension that the curve visits, the number of supercharges reduces by a half, except for the last dimension, that does not involve a new projector. Generically there are no solutions with nonzero ${\bar{\varepsilon }}_{1}$, so no preserved special supersymmetries S.

Type- $\mathbb{C}$: to make our discussion concrete it is convenient to choose an explicit complex structure for ${\mathbb{R}}^{6}$, represented here via the Kähler form

$$\begin{equation}\frac{1}{2}{\omega }_{\mu \nu }\enspace \mathrm{d}{x}^{\mu }\mathrm{d}{x}^{\nu }=\mathrm{d}{x}^{1}\wedge \mathrm{d}{x}^{2}+\mathrm{d}{x}^{3}\wedge \mathrm{d}{x}^{4}+\mathrm{d}{x}^{5}\wedge \mathrm{d}{x}^{6}.\end{equation} \tag{ 3.6 }$$

Holomorphic curves with respect to this complex structure are those satisfying the Cauchy–Riemann equations

$$\begin{equation}{\partial }_{a}{x}^{\mu }={{\omega }^{\mu }}_{\nu }{\varepsilon }_{ab}{h}^{bc}{\partial }_{c}{x}^{\nu }.\end{equation} \tag{ 3.7 }$$

The natural constraints we impose on ɛ₀ are then

$$\begin{equation}{\varepsilon }_{0}\left(\mathrm{i}{\gamma }_{\mu }+{{\omega }_{\mu }}^{\nu }{\gamma }_{\nu }{\rho }_{1}\right)=0.\end{equation} \tag{ 3.8 }$$

Of those six equations only the ones with μ = 1, 3, 5 are independent, as for example

$$\begin{align}\hfill {\varepsilon }_{0}\left(\mathrm{i}{\gamma }_{2}+{{\omega }_{2}}^{\nu }{\gamma }_{\nu }{\rho }_{1}\right)& ={\varepsilon }_{0}\left(\mathrm{i}{\gamma }_{2}-{\gamma }_{1}{\rho }_{1}\right)={\varepsilon }_{0}\left({\gamma }_{2}{\rho }_{1}+\mathrm{i}{\gamma }_{1}\right)\mathrm{i}{\rho }_{1}\hfill \\ \hfill & ={\varepsilon }_{0}\left(\mathrm{i}{\gamma }_{1}+{{\omega }_{1}}^{\nu }{\gamma }_{\nu }{\rho }_{1}\right)\mathrm{i}{\rho }_{1}=0.\hfill \end{align} \tag{ 3.9 }$$

Now for a generic holomorphic curve, we use the above conditions on ɛ₀ and then the Cauchy–Riemann equation to write

$$\begin{align}\hfill {\varepsilon }_{0}\mathrm{i}{\gamma }_{\mu \nu }{\partial }_{a}{x}^{\mu }{\partial }_{b}{x}^{\nu }{\varepsilon }^{ab}& =-{\varepsilon }_{0}\enspace {{\omega }_{\mu }}^{\sigma }{\gamma }_{\sigma }{\bar{\gamma }}_{\nu }{\rho }_{1}{\partial }_{a}{x}^{\mu }{\partial }_{b}{x}^{\nu }{\varepsilon }^{ab}\hfill \\ \hfill & =-{\varepsilon }_{0}{\gamma }_{\sigma }{\bar{\gamma }}_{\nu }{\rho }_{1}{\partial }_{a}{x}^{\sigma }{\partial }_{b}{x}^{\nu }{h}^{ab}.\hfill \end{align} \tag{ 3.10 }$$

The symmetry of the term on the right under σ ↔ ν reduces the right-hand side to −ɛ₀ ρ₁ (where we assume unit normalised tangent vectors), so the projector equation (1.2) is satisfied.

Again, each of the projectors (3.8) reduces the rank of ɛ₀ by half, giving eight, four and two Q supercharges for the plane, holomorphic curves in ${\mathbb{R}}^{4}$ and ${\mathbb{R}}^{6}$ respectively.

Type- $\mathbb{H}$: here we have arbitrary surfaces in ${\mathbb{R}}^{4}$. Choosing n according to (1.4), for each tangent plane we can assign a linear combination of the projectors (see (2.3))

$$\begin{equation}\begin{aligned}\hfill \frac{1}{2}\left(\mathrm{i}{\gamma }_{12}{\rho }_{3}+1\right),\qquad \frac{1}{2}\left(\mathrm{i}{\gamma }_{31}{\rho }_{2}+1\right),\qquad \frac{1}{2}\left(\mathrm{i}{\gamma }_{23}{\rho }_{1}+1\right),\\ \hfill \frac{1}{2}\left(\mathrm{i}{\gamma }_{34}{\rho }_{3}+1\right),\qquad \frac{1}{2}\left(\mathrm{i}{\gamma }_{24}{\rho }_{2}+1\right),\qquad \frac{1}{2}\left(\mathrm{i}{\gamma }_{14}{\rho }_{1}+1\right).\end{aligned}\end{equation} \tag{ 3.11 }$$

In contrast to the Type- $\mathbb{R}$ oscillators above, all these commute rather than anticommute.

Not all the projectors are independent. For a generic surface in ${\mathbb{R}}^{4}$, we need to satisfy all of them. Taking any projector of the first line and its complement on the second line imposes ɛ₀ = −ɛ₀ γ₁₂₃₄, which is a restriction to the supersymmetries antichiral with respect to 4D chirality. On these supersymmetries, all the projectors of the second line and equal to those of the first, so we are left with three independent conditions on the antichiral spinors. Hence generic surfaces preserve a single Q.

For a Lagrangian submanifold of Type-L one component of n is zero (e.g. n³ = 0). In that case we only need to take two columns of projectors out of (3.11), and this imposes again two conditions on antichiral spinors, so leaving two Q.

Conversely if n³ = 1 is constant (and all other components vanish), then the projector equation imposes only the first column of (3.11), and by the discussion above it preserves four Q. These are holomorphic curves, as in Type- $\mathbb{C}$ (except for the choice of n³ instead of n¹).

The Type-N surfaces have the projectors on the first line of (3.11), which are all independent, so a generic surface of this type preserves two Q supercharges. Note that the projectors on the second line are the same as (3.3) of Type- $\mathbb{R}$ if we replace x⁶ ↔ x⁴ in that construction, also preserving two Qs and manifesting another example of intersections between our examples.

Conical surfaces: the discussion so far focused only on the Poincaré supercharges Q, but if the surfaces are conical with the tip at the origin, they also preserve conformal supercharges S (for other locations of the tip, they would preserve other combinations of Q and S). We can express the conical surfaces by x^μ = uξ^μ (v) where ξ is a curve on an appropriate sphere.

The simplest example is the plane, where ξ is a circle centered around the origin. In addition to the eight Q's, it is annihilated by eight S's. The only other examples of Type- $\mathbb{R}$ are when the curve x^μ (u) is comprised of two rays meeting at any angle, which is a slight generalisation of the example mentioned in the introduction. The surface is then two half-planes glued along a line, so can be called crease. The base of the cone ξ^μ is comprised of two half-circles on S², so two longitude lines.

To see the extended supersymmetry, specializing to a conical surface over ξ^μ , (1.2) becomes ($\dot {\xi }=\mathrm{d}\xi /\mathrm{d}v$)

$$\begin{equation}\left[{\varepsilon }_{0}+{\bar{\varepsilon }}_{1}\left(u\enspace \xi \cdot \bar{\gamma }\right)\right]\left[\mathrm{i}{\varepsilon }^{ab}{\gamma }_{\mu \nu }{\xi }^{\mu }{\dot {\xi }}^{\nu }+{n}_{I}{\rho }_{I}\right]=0.\end{equation} \tag{ 3.12 }$$

Of course ${\xi }^{\rho }{\bar{\gamma }}_{\rho }{\gamma }_{\mu \nu }{\xi }^{\mu }{\dot {\xi }}^{\nu }={\xi }^{\nu }{\bar{\gamma }}_{\nu }$, but more importantly, we can also permute the first gamma matrix through, such that the equation becomes

$$\begin{equation}{\varepsilon }_{0}\left[\mathrm{i}{\varepsilon }^{ab}{\gamma }_{\mu \nu }{\xi }^{\mu }{\dot {\xi }}^{\nu }+{n}_{I}{\rho }_{I}\right]+{\bar{\varepsilon }}_{1}\left[-\mathrm{i}{\varepsilon }^{ab}{\bar{\gamma }}_{\mu \nu }{\xi }^{\mu }{\dot {\xi }}^{\nu }+{n}_{I}{\rho }_{I}\right]\left(u\enspace \xi \cdot \bar{\gamma }\right)=0.\end{equation} \tag{ 3.13 }$$

So the equations for ${\bar{\varepsilon }}_{1}$ involve the complementary projectors to ɛ₀, and certainly the dimension of the space of solutions for the two are the same.

Tori: Within the Type-L surfaces in ${\mathbb{R}}^{4}$, the simplest representatives are tori which are the product of circles in the (x¹, x²) and (x³, x⁴) planes. In addition to the two Q supercharges that any of the Lagrangian surfaces preserve, these also preserve a pair of S generators. The derivation is similar to (3.13) and uses that for these surfaces x² = 1.

Type-S: to understand this class, recall that it includes 1/2 BPS spheres, which are not annihilated by any one Q or S, but by 16 linear combinations of the form Q + S. We can get this by a stereographic map of the plane, and likewise we can understand the surfaces foliated by arcs between the north and south pole by their stereographic projection to cones of Type-N, which preserve two Q and two S. So the analogs on S³ preserve four independent combinations Q + S. As a special case, the conformal map of the crease give surfaces comprised of two hemispheres of different great S² and preserves eight combinations Q + S (to visualise this, think of two longitudinal lines, or hemicircles, meeting at the north and south pole, and replace them with hemispheres).

The most general surface on S³ preserves two supercharges. To see this, write the projector equation (1.2) as

$$\begin{equation}\frac{1}{2}{\varepsilon }^{ab}{\partial }_{a}{x}^{\mu }{\partial }_{b}{x}^{\nu }\left[{\varepsilon }_{0}+{\bar{\varepsilon }}_{1}\left(x\cdot \bar{\gamma }\right)\right]\left[\mathrm{i}{\gamma }_{\mu \nu }+{\varepsilon }_{\mu \nu \sigma I}{x}^{\sigma }{\rho }_{I}\right]=0.\end{equation} \tag{ 3.14 }$$

We want this equation to hold for all orientations, meaning all μ, ν, so expanding and reorganising (assuming orthogonality) we find

$$\begin{equation}\left[{\varepsilon }_{0}\mathrm{i}{\gamma }_{\mu \nu }+{\bar{\varepsilon }}_{1}{\bar{\gamma }}^{\sigma }{\varepsilon }_{\mu \nu \sigma I}{\rho }_{I}\right]+\left[{\varepsilon }_{0}{\varepsilon }_{\mu \nu \sigma I}{\rho }_{I}+\mathrm{i}{\bar{\varepsilon }}_{1}{\bar{\gamma }}_{\mu \nu \sigma }\right]{x}^{\sigma }=0.\end{equation} \tag{ 3.15 }$$

Since ${\varepsilon }_{0},{\bar{\varepsilon }}_{1}$ are constant the two terms in brackets need to vanish separately, and in fact imposing one implies the other.

To count the number of independent constraints, note that referring to the ansatz (1.5) every tangent plane on S³ is associated to a vector n on S³. For the latter there is obviously four possible independent vectors, so for a generic surface we impose four independent conditions on ${\varepsilon }_{0},{\bar{\varepsilon }}_{1}$. This leaves two preserved supercharges.

Recall that surface operators have Weyl anomalies whose form is constrained by the Wess–Zumino consistency conditions to be a sum of conformal invariants [15–17, 19]. They can be expressed as the integral of the density

$$\begin{equation}{\mathcal{A}}_{{\Sigma}}=\frac{1}{4\pi }\left[{a}_{1}{\mathcal{R}}^{{\Sigma}}+{a}_{2}\left({H}^{2}+4\enspace \text{tr}\enspace P\right)+b\enspace \text{tr}\enspace W+c{\left(\partial n\right)}^{2}\right].\end{equation} \tag{ 4.1 }$$

The first invariant is ${\mathcal{R}}^{{\Sigma}}$, which is the Ricci scalar of the induced metric h_ab . The second piece includes H^μ , the mean curvature vector and tr P = h^ab P_ab , the trace of the pullback of the Schouten tensor. W is the pullback of the Weyl tensor and (∂n)² = h^ab ∂_a n^I ∂_b n^I is the norm of the variation of n over the surface.

We work on flat Euclidean space so P = W = 0. It is useful in the following to have an explicit expression for the other invariants, and they can be written nicely in terms of the second fundamental form $\text{II}$ as

$$\begin{equation}{\text{II}}_{ab}^{\mu }={\partial }_{a}{\partial }_{b}{x}^{\nu }\left({\delta }_{\nu }^{\mu }-{h}^{cd}{\partial }_{c}{x}_{\nu }{\partial }_{d}{x}^{\mu }\right),\qquad {H}^{\mu }={h}^{ab}{\text{II}}_{ab}^{\mu },\qquad {\mathcal{R}}^{{\Sigma}}={\text{II}}_{ab}^{\mu }{\text{II}}_{cd}^{\mu }{\varepsilon }^{ac}{\varepsilon }^{bd}.\end{equation} \tag{ 4.2 }$$

The term $\left({\delta }_{\nu }^{\mu }-{h}^{cd}{\partial }_{c}{x}_{\nu }{\partial }_{d}{x}^{\mu }\right)$ is a projector to the components normal to the surface.

Another simplification occurs in our case. While the Wess–Zumino consistency condition allow the four independent coefficients a₁, a₂, b and c above, supersymmetry imposes a₂ = −c and b = 0 for locally BPS surface operators [13, 42]. The anomaly density then reduces to

$$\begin{equation}{\mathcal{A}}_{{\Sigma}}=\frac{1}{4\pi }\left[{a}_{1}{\mathcal{R}}^{{\Sigma}}+c\left(\partial {n}^{2}-{H}^{2}\right)\right].\end{equation} \tag{ 4.3 }$$

The constants a₁ and c are observables that depend on the theory (the choice of ADE algebra underlying the $\mathcal{N}=\left(2,0\right)$ theory), and the representation of the surface operator (at least for the A_N theories they are classified by representations of that algebra). According to [5, 8–11, 14] they are given by

$$\begin{equation}{a}_{1}=\frac{1}{2}\left({\Lambda},{\Lambda}\right),\qquad c={C}_{2}\left({\Lambda}\right)-{a}_{1},\end{equation} \tag{ 4.4 }$$

where Λ is the highest weight defining the representation and C₂(Λ) is the quadratic Casimir of the representation. For the fundamental representation of $\mathfrak{s}\mathfrak{u}\left(N\right)$ this is

$$\begin{equation}{a}_{1}^{\left(N\right)}=\frac{1}{2}-\frac{1}{2N},\qquad {c}^{\left(N\right)}=N-\frac{1}{2}-\frac{1}{2N},\end{equation} \tag{ 4.5 }$$

but we leave them as a₁ and c in the following.

The above expressions give a precise expression for the anomaly of all the surfaces studied in this paper, requiring just the evaluation of the integrals of the expressions in (4.3). But as in all our examples the vector n depends in varying ways on x^μ , we can evaluate (∂n)² relying on the geometry as studied in section 2 and simplify the expression for the anomaly even further.

Type- $\mathbb{R}$: for BPS surfaces satisfying (1.3), the anomaly vanishes since (∂n)² = H² and the Ricci scalar is zero. In the abelian theory one can make a stronger statement, and using the field theory propagators given in [19, 43–45] one can check that the expectation value also vanishes identically. It is tempting to conjecture that this is true in general.

Type- $\mathbb{C}$: in this case n^I is a constant, so (∂n)² vanishes. H² also vanishes, which can be seen by a direct computation, or by noting that the Kähler form ω (3.6) act as a calibration for these surfaces, so that any holomorphic curve is a minimal surface. The only contribution to the anomaly then comes from the Ricci scalar, so it is proportional to the Euler characteristic of the curve

$$\begin{equation}{\int }_{{\Sigma}}{\mathcal{A}}_{{\Sigma}}^{\mathbb{C}}\enspace {\text{vol}}_{{\Sigma}}={a}_{1}\chi \left({\Sigma}\right).\end{equation} \tag{ 4.6 }$$

Note that for non-compact surfaces the genus is defined by this integral, and may not match that expected for the analogous curve in $\mathbb{C}{\mathbb{P}}^{3}$.

Type- $\mathbb{H}$: here the anomaly is more interesting. Using (1.4) and the definition of the geometric invariants (4.2), we can write

$$\begin{equation}{\left(\partial n\right)}^{2}-{H}^{2}={\eta }_{\mu \nu }^{I}{\eta }_{\rho \sigma }^{I}\left({\text{II}}_{ab}^{\mu }{\partial }_{e}{x}^{\nu }\right)\left({\text{II}}_{cd}^{\rho }{\partial }_{f}{x}^{\sigma }\right)\left({\varepsilon }^{be}{\varepsilon }^{df}{h}^{ac}-\frac{1}{2}{h}^{ab}{h}^{cd}{h}^{ef}\right).\end{equation} \tag{ 4.7 }$$

The 't Hooft symbol (2.3) satisfies

$$\begin{equation}{\eta }_{\mu \nu }^{I}{\eta }_{\rho \sigma }^{I}={\delta }_{\mu \rho }{\delta }_{\nu \sigma }-{\delta }_{\mu \sigma }{\delta }_{\nu \rho }+{\varepsilon }_{\mu \nu \rho \sigma }.\end{equation} \tag{ 4.8 }$$

If we focus on the δ contractions, only the first is nonzero. We identify δ_νσ ∂_e x^ν ∂_f x^σ = h_ef , and this reduces the last term to −ɛ^ac ɛ^bd , so we immediately recognize $-{\mathcal{R}}^{{\Sigma}}$ from (4.2).

The contraction involving ɛ_μνρσ gives us the Hodge dual of the two-form $\left({\text{II}}_{cd}^{\rho }{\partial }_{f}{x}^{\sigma }\right)\mathrm{d}{x}^{\rho }\enspace \mathrm{d}{x}^{\sigma }$. This interchanges the tangent and normal space, and one can verify that it calculates the curvature of the normal bundle, which has a coordinate expression

$$\begin{equation}{\mathcal{R}}_{\perp }^{{\Sigma}}=-\frac{1}{2}{\varepsilon }_{\mu \nu \rho \sigma }\left({\text{II}}_{ab}^{\mu }{\text{II}}_{cd}^{\nu }{h}^{bc}{\varepsilon }^{ad}\right){\varepsilon }^{ef}{\partial }_{e}{x}^{\rho }{\partial }_{f}{x}^{\sigma }.\end{equation} \tag{ 4.9 }$$

Its integral, the Euler class of the normal bundle, is also equal in 4D to the self-intersection number (denoted [Σ] ⋅ [Σ]), see e.g. [46].

The term proportional to c in the anomaly density is then

$$\begin{equation}{\left(\partial n\right)}^{2}-{H}^{2}=-{\mathcal{R}}^{{\Sigma}}+{\mathcal{R}}_{\perp }^{{\Sigma}}.\end{equation} \tag{ 4.10 }$$

Including the a₁ anomaly and integrating over the surface, we obtain a topological quantity

$$\begin{equation}{\int }_{{\Sigma}}{\mathcal{A}}_{{\Sigma}}^{\mathbb{H}}\enspace {\text{vol}}_{{\Sigma}}=\left({a}_{1}-c\right)\chi \left({\Sigma}\right)+c\left[{\Sigma}\right]\cdot \left[{\Sigma}\right].\end{equation} \tag{ 4.11 }$$

This has a natural interpretation in terms of the Gauss map. Recall the image of the Gauss map is two S²'s and n^I is in one of these spheres. The topological invariant calculated by integrating $-{\mathcal{R}}^{{\Sigma}}+{\mathcal{R}}^{\perp }$ is (2 times) the degree of this map Σ → S².

It is interesting to compare this result with our discussion for holomorphic surfaces in ${\mathbb{C}}^{2}$. As argued around (4.6), for holomorphic surfaces (∂n)² − H² = 0. Indeed in this case the Gauss map gives a constant n and certainly the degree of the map vanishes, and indeed the self-intersection is equal to the Euler characteristic. The same is true also for Lagrangian submanifolds, as the image is on the equator of S², so again the degree vanishes.

Type-S: here we simply expand (∂n)² using (1.5) to obtain

$$\begin{equation}{\left(\partial n\right)}^{2}={\partial }_{a}{\partial }_{b}{x}^{\mu }\left({\delta }^{\mu \nu }-{h}^{cd}{\partial }_{c}{x}^{\mu }{\partial }_{d}{x}^{\nu }\right){\partial }_{e}{\partial }_{f}{x}^{\nu }{h}^{ae}{h}^{bf}-2.\end{equation} \tag{ 4.12 }$$

The factor two seems surprising at first, as it leads to a term in the anomaly proportional to the area of the surface Σ. This in fact is not in contradiction to conformal invariance, as this construction has a dimensionful parameter, the radius of the sphere (which we set to one). As for the first term, we can identify the second fundamental form from (4.2) to rewrite this as

$$\begin{equation}{\left(\partial n\right)}^{2}-{H}^{2}=-{\mathcal{R}}^{{\Sigma}}-2.\end{equation} \tag{ 4.13 }$$

The anomaly is then

$$\begin{equation}\int {\mathcal{A}}_{{\Sigma}}^{S}\enspace {\text{vol}}_{{\Sigma}}=\left({a}_{1}-c\right)\chi \left({\Sigma}\right)-c\frac{\text{vol}\left({\Sigma}\right)}{2\pi }.\end{equation} \tag{ 4.14 }$$

For infinitesimal surfaces on S² we should recover the anomaly of Type-N. In that limit the area of the surface vanishes giving (4.11), since in 3D the self-intersection number (4.9) is always zero.

4.1. Conical singularities

Surfaces with conical singularities have two sources of anomalies. First there are the usual anomalies discussed above but whose density may have a singular contribution at the apex of the cone. Second, the tips are distinguished points, so we may find new anomalies when integrating finite quantities over a scale-invariant cone. In fact, for non-BPS surfaces these two sources of divergences can conflate to log²  divergences [19, 47–50]. For BPS surfaces they split into two independent contributions. We discuss them in turn.

We start with a cone over a curve $\mathcal{K}\subset {S}^{2}$ (parametrised by ξ^μ ) of Type-N, which is a subclass of Type- $\mathbb{H}$. The self-intersection number (4.9) vanishes in three dimensions, so the anomaly (4.11) is determined solely by the Euler characteristic. The Ricci scalar vanishes along the cone, but diverges at the tip. It can be regularised by smoothing the tip at distances u ⩽ to form the regularised surface ${S}_{\mathcal{K}}$, and the contribution to the anomaly is evaluated using the Gauss–Bonnet theorem

$$\begin{equation}\frac{1}{4\pi }{\int }_{{S}_{\mathcal{K}}}{\mathcal{R}}^{{\Sigma}}\enspace {\text{vol}}_{{\Sigma}}=\chi \left({S}_{\mathcal{K}}\right)-\frac{1}{2\pi }{\int }_{\mathcal{K}}{\kappa }_{g}\enspace {\text{vol}}_{\mathcal{K}}.\end{equation} \tag{ 4.15 }$$

Note that we defined here the anomaly for the non-compact cone without a boundary term at infinity, which would be on the left-hand side, so instead it is on the right. If we choose the disc topology for ${S}_{\mathcal{K}}$, the first term gives 1.

To evaluate the boundary contribution we can take our parametrisation ξ^μ to be unit speed, so that the geodesic curvature κ_g is constant and equal to 1/ when evaluated on the sphere of radius . The line integral then calculates the arc length of the curve $\mathcal{K}$, which is ${\epsilon}{L}_{\mathcal{K}}$ (${L}_{\mathcal{K}}$ being the arc length on a unit sphere).

Therefore the anomaly is

$$\begin{equation}{\int }_{{\Sigma}}{\mathcal{A}}_{{\Sigma}}\enspace {\text{vol}}_{{\Sigma}}=\left(1-\frac{{L}_{\mathcal{K}}}{2\pi }\right)\left({a}_{1}-c\right).\end{equation} \tag{ 4.16 }$$

The story for general cones of Type- $\mathbb{H}$ is more interesting, since the curve $\mathcal{K}$ can have the topology of a nontrivial knot on S³. In the case of the unknot, we could choose the disc topology for our regularisation, but more generally this is a Seifert surface (see e.g. [51]). Given a knot there are many different possible such surfaces of different topology. This is easy to see already in the case of the disk to which we could add any number of handles, which would modify the anomaly above to

$$\begin{equation}\left(1-2n-\frac{{L}_{\mathcal{K}}}{2\pi }\right)\left({a}_{1}-c\right),\quad n\in \mathbb{N}.\end{equation} \tag{ 4.17 }$$

For a general knot there would be Seifert surfaces of lowest genus, whose Euler characteristic would replace the 1 in this equation, and other regularisations would still allow for the integer n. A slight generalisation to this result is to allow for multiple curves ${\mathcal{K}}_{\mathit{i}}$ defining a link. There is again a Seifert surface of lowest genus whose boundary is the link, and in addition one needs to replace ${L}_{\mathcal{K}}$ by ${\sum }_{i}{L}_{{\mathcal{K}}_{i}}$.

The discussion thus far focused on regular surface anomalies. They are localized at the apex of the cone and can be calculated by choosing an appropriate regularisation of the conical singularity. The second type of divergences arising from cones come from terms that would be finite upon regularisation. Consider a cone with no anomaly density (apart from the apex) and restrict to an open subset u ∈ (u_min, u_max). While the contribution to the expectation value is finite for any such open subset, it would naturally scale like log(u_min/u_max) and, after taking the limit u_min → , contribute a new term to the anomaly.

An example of such surfaces are the cones of Type- $\mathbb{H}$. They are related to BPS Wilson loops in 5D, see the discussion in section 5.1 below.

We can consider also surfaces that are not globally conical, but have conical singularities. Examples are the surfaces foliated by hemicircles of Type-S, conformal to Type-N cones. Those have the topology of the sphere, so we expect that the anomaly does not require special regularization.

4.2. Anomaless surfaces

Surfaces that are not anomalous can have finite expectation values, which one may hope to be able to calculate exactly, as is done for BPS line operators in 3D and 4D (as well as other observables).

In the examples above we found that those of Type- $\mathbb{R}$ are indeed anomaless, though we also expect their expectation value to vanish identically. This is certainly true in the free abelian theory, where using the propagators of [19] and the ansatz (1.3) one can check that the propagator for the two-form B_μν exactly cancels the scalar Φ^I propagator.

Both the anomaly of Type- $\mathbb{C}$ (4.6) and Type- $\mathbb{H}$ (4.11) are purely topological, so any surface with vanishing χ(Σ) and self-intersection number is anomaless. This includes all (topological) tori of Type- $\mathbb{H}$, with examples in Type-L and Type-N.

The formula for the anomaly of Type- $\mathbb{H}$ cones, (4.17) vanishes only for n = 0 and ${L}_{\mathcal{K}}=2\pi$. This means it is a cone over the unknot with zero deficit angle and includes the crease of arbitrary opening angle (and in particular the plane). These surfaces cannot have finite expectation values, as they are non-compact. Note that by conformally transforming to a compact manifold like the sphere (or the crease generalisation of it discussed above equation (3.14)) changes the topology, so those examples are in fact anomalous.

Finally, we can look for cases where the surfaces are anomaless only for particular values of a₁ and c (4.4). For example, if we restrict to surfaces in the fundamental representation, ${c}^{\left(N\right)}=\left(2N+1\right){a}_{1}^{\left(N\right)}$ (4.5). Now looking at the Type-S anomaly (4.14), this vanishes for

$$\begin{equation}\frac{\text{vol}\left({\Sigma}\right)}{4\pi }=-\chi \left({\Sigma}\right)\frac{N}{2N+1}.\end{equation} \tag{ 4.18 }$$

For example for χ = −2, or genus two surfaces, the resulting area would be close to that of a maximal S². For higher genus, the surfaces would have to be even larger.

For Type- $\mathbb{H}$ surfaces the analogous equation (4.11) vanished for

$$\begin{equation}\left[{\Sigma}\right]\cdot \left[{\Sigma}\right]=\chi \left({\Sigma}\right)\frac{2N}{2N+1}.\end{equation} \tag{ 4.19 }$$

The simplest possible solution would require Euler characteristic χ = −4N − 2 and self-intersection number [Σ] ⋅ [Σ] = −4N. We do not know whether there are solutions to these conditions. One can of course also go to other representations using the expressions in (4.4).

The expressions in (4.4) do not extend to the abelian theory, where a₁ = c = 1/2. In this case any surface of Type- $\mathbb{H}$ with vanishing self-intersection number is anomaless. In the large N limit a₁ ≪ c, so the classical supergravity calculation [19, 52] is only sensitive to the c anomaly. In this limit all Type- $\mathbb{H}$ surfaces of equal Euler characteristic and self-intersection number as well as all Type- $\mathbb{C}$ surfaces are effectively anomaless.

The inspiration for this project comes from the rich history of BPS Wilson loops in $\mathcal{N}=4$ SYM in 4D [20–22, 26, 41]. It is therefore natural to examine whether any of the families of surfaces uncovered here arise as the dimensional uplift of BPS loops in 4D or 5D. Likewise we would like to address the relation to BPS surfaces in lower dimensions.

5.1. Wilson and 't Hooft loops

5.2. Surfaces in 4D

The other way to dimensionally reduce a surface is if it is at a fixed point along a compact direction. In that case we would get a surface operator in the lower dimensional theory. This can give surface operators in any dimension, but let us focus on 4D, either $\mathcal{N}=4$ theory or theories of class-$\mathcal{S}$. Theories with $\mathcal{N}=1$ supersymmetry in 4D also have BPS surface operators, but we will not touch upon those. As the construction of surfaces of Type- $\mathbb{H}$ and Type-S is inherently four dimensional, we expect them to be realised in 4D as well.

A first comment is that when restricting to ${\mathbb{R}}^{4}$ both Type- $\mathbb{C}$ and Type- $\mathbb{R}$ become subclasses of Type- $\mathbb{H}$, and Type- $\mathbb{R}$ is essentially the same as Type-N. This was discussed already below equations (2.3) and (3.11) respectively. So we need only discuss Type- $\mathbb{H}$ (and its subclasses: Type- $\mathbb{C}$, Type- $\mathbb{R}$, Type-L and Type-N) and separately Type-S.

A second comment is that within the AGT correspondence [61] it is very natural to distinguish the origin of a surface operator in 4D as arising from a codimension-4 operator in 6D, or from ann a codimension-2 defect wrapping the Riemann surface. While our discussion is explicitly relevant only for the former case, it is only based on supersymmetry analysis, so is very likely to carry over, as to any other operator involving the coupling of an appropriate 2D superconformal field theory to the 4D theory.

Let us separate the discussion to theories of class-$\mathcal{S}$ and $\mathcal{N}=4$ SYM.

$\mathcal{N}=2$ theories:

There is an immense literature on surface operators in 4D class-$\mathcal{S}$ theories focusing mostly on operators breaking $\mathfrak{s}\mathfrak{u}\left(2,2\vert 2\right)\to \mathfrak{s}\mathfrak{u}{\left(2\vert 1\right)}^{2}$ (see, e.g. [62–65]). This is indeed the symmetry preserved by the 1/2 BPS surface operator of the 6D theory appropriately compactified to $\mathcal{N}=2$ in 4D. In 2D language, these are $\mathcal{N}=\left(2,2\right)$ defects.

To make the relation between 6D and 4D explicit, it involves replacing the x⁵, x⁶ directions with a compact Riemann surface. In order to preserve supersymmetry away from flat space, one performs a topological twist which preserves the supercharges satisfying

$$\begin{equation}\varepsilon \left({\gamma }_{56}+{\rho }_{45}\right)=0.\end{equation} \tag{ 5.1 }$$

The resulting supersymmetries are the Q and $\bar{\mathsf{Q}}$ of the 4D theory. It is a simple check that (5.1) is automatically satisfied if we impose all the projectors (3.11).

Viewing things from the 4D point of view, the equations of BPS surface operators are very similar to (1.2), with only three ρ's and the four dimensional Killing spinors. As mentioned above, all but Type-S are subclasses of Type- $\mathbb{H}$, so we analyze the latter first in turn.

Type- $\mathbb{H}$: for the most general surface of this type n^I ∈ S², and we need to solve all the equations of (3.11). The projectors in the first column give the 4D chirality projector γ₁₂₃₄, eliminating the $\bar{\mathsf{Q}}$ supercharges. This is not surprising as the Type- $\mathbb{H}$ ansatz (1.4) is chiral. In 4D the three projectors on the first line of (3.11) are not independent, as the gamma matrices of $\mathfrak{s}\mathfrak{u}{\left(2\right)}_{\mathrm{R}}$ symmetry satisfy ρ₁ ρ₂ ρ₃ = −i. We find that the generic surface of this type preserves a single Q.

Type- $\mathbb{R}$: here we have surfaces extended along x⁴ and an arbitrary surface in ${\mathbb{R}}^{3}$ and generically n^I ∈ S². As already mentioned, these surfaces are now a subclass of Type- $\mathbb{H}$ and one should impose the projector equations on the second line of (3.11). Given that ɛ₀ is comprised of one four-component chiral and one four-component antichiral spinors, the two independent equations in the second line reduce to one Q and one $\bar{\mathsf{Q}}$.

Type-N: this case is equivalent to Type- $\mathbb{R}$ in four dimensions, except that we need to impose the conditions on the first line of (3.11).

Type-L: in this case we need to impose the equations associated to two of the columns in (3.11). As mentioned above, these automatically imply that the third column is also satisfied, so there is no supersymmetry enhancement for Lagrangian surfaces.

Type- $\mathbb{C}$: for holomorphic surfaces which are not a plane, we need to impose both equations in one of the columns in (3.11), so we find that the surfaces preserve a pair of Q supercharges. Indeed holomorphic surface operators in $\mathcal{N}=4$ SYM were constructed in [66] and the analysis here suggests that this should extend to any class-$\mathcal{S}$ theory.

Type-S: in this case the ansatz (1.5) requires n^I ∈ S³, so cannot be fully realised for $\mathcal{N}=2$ theories. Recall though that surfaces of Type-N are conformal to 'banana' surfaces on S³ and for those n⁴ = 0. We can implement this conformal transformation for $\mathcal{N}=2$ theories in 4D as well, so they too preserve a pair of superchages: one Q + S and one $\bar{\mathsf{Q}}+\bar{\mathsf{S}}$.

$\mathcal{N}=4$ SYM:

The dimensional reduction on a torus to $\mathcal{N}=4$ theories does not break any supersymmetries, thus guaranteeing that our analysis carries over. The most studied surface operators are those of Gukov and Witten [67], and though they arise from codimension-2 defects in six dimensions, the 1/2 BPS plane preserves the same superalgebra as arising from 6D surface operators, $\mathfrak{s}\mathfrak{u}{\left(2\vert 2\right)}^{2}$, or in two dimension language $\mathcal{N}=\left(4,4\right)$.

The projector equations in 4D take a slightly different form from (1.2), as is clear, since the R-symmetry breaking is $\mathfrak{s}\mathfrak{o}\left(6\right)\to \mathfrak{s}\mathfrak{o}\left(4\right)$, which requires two ρ matrices. Also, the structure of the Killing spinors is slightly different. Yet, the dimensional reduction guarantees that solutions to all our classes exist. The number of preserved supercharges is also slightly different from 6D, and is double the number listed for the appropriate subclasses of Type- $\mathbb{H}$ for $\mathcal{N}=2$ theories above.

Type-S: In this case the we can realise the full ansatz (1.5) and the supersymmetry analysis in section 3 is indeed consistent in 4D. The most general surface preserves two supercharges.

We should mention that this supersymmetry analysis does not provide a construction of these operators, but suggests which symmetries to try to realise. It would be interesting to extend the constructions of [67, 68] as defect operators and those of [62, 63] as coupled 2D–4D systems to arbitrary SUSY preserving geometries, generalising the Type- $\mathbb{H}$ case of [66].

The large N limit of the $\mathcal{N}=\left(2,0\right)$ theory has a holographic representation in terms of M-theory on AdS₇ × S⁴ [69]. Therein, the surface operators are described by M2-branes ending along the surface at the boundary of space [30]. Globally BPS surface operators should be described by M2-brane embeddings which realise the same supersymmetries.

Here we take the first few steps to realise the different classes of BPS surfaces. The simplest examples of M2-branes embeddings are those whose boundary is a plane or a sphere, they were presented respectively in [30, 31].

We take again inspiration from the holographic realisation of the large N limit of BPS Wilson loops in $\mathcal{N}=4$ SYM. For each of the construction of [26, 41], the BPS conditions on the field theory could be extended to an almost complex structure on a subspace of AdS₅ × S⁵. In both cases the dual classical string surfaces are pseudo-holomorphic with respect to the almost complex structure. Furthermore, the action of the string can be evaluated by the pullback of an appropriate form in the bulk, which manifests as a (generalised) calibration. In the following we present some analogous structures for the four main classes of BPS surface operators. We refer to the original papers [26, 57] for more details on the string constructions.

Our starting point is the metric on AdS₇ × S⁴,

$$\begin{equation}\mathrm{d}{s}^{2}=\frac{y}{L}\enspace \mathrm{d}{x}^{\mu }\enspace \mathrm{d}{x}^{\mu }+{\left(\frac{L}{y}\right)}^{2}\enspace \mathrm{d}{y}^{I}\enspace \mathrm{d}{y}^{I},\end{equation} \tag{ 6.1 }$$

where y ≡ |y^I |. To get the regular Poincaré patch metric redefine y = 4L³/z² and parametrise y^I /y as angular coordinates on S⁴. Note that here L is the radius of S⁴ and the curvature radius of AdS₇ is 2L. The background also has a four-form field strength proportional to the volume form on S⁴.

The Killing spinors in this coordinate system can be constructed from the same pair of constant spinors ɛ₀ and ${\bar{\varepsilon }}_{1}$ of section 3. Using the 11-dimensional curved space Γ-matrices satisfying

$$\begin{equation}\left\{{{\Gamma}}_{\mu },{{\Gamma}}_{\nu }\right\}=\frac{2y}{L}{\delta }_{\mu \nu },\qquad \left\{{{\Gamma}}_{\mu },{{\Gamma}}_{I}\right\}=0,\qquad \left\{{{\Gamma}}_{I},{{\Gamma}}_{J}\right\}=2{\left(\frac{L}{y}\right)}^{2}{\delta }_{IJ},\end{equation} \tag{ 6.2 }$$

they are given by [70]

$$\begin{equation}\varepsilon ={\left(\frac{y}{L}\right)}^{1/4}{\varepsilon }_{0}+{\left(\frac{L}{y}\right)}^{1/4}{\bar{\varepsilon }}_{1}\left({x}^{\mu }{{\Gamma}}_{\mu }-2{y}^{I}{{\Gamma}}_{I}\right).\end{equation} \tag{ 6.3 }$$

In the limit of y → ∞, the last term drops out and we recover the conformal Killing spinors of flat space appearing in (1.2), ${\left(y/L\right)}^{1/4}\left({\varepsilon }_{0}+{\bar{\varepsilon }}_{1}{x}^{\mu }{\gamma }_{\mu }\right)$. This enables a very direct map between the BPS conditions that we found for the different types of surfaces in section 3 to the bulk.

In the following we use the notation X^M for coordinates of AdS₇ × S⁴ and index m = 1, 2, 3 for the M2-brane. G_MN is the metric (6.1) and g_mn is the induced metric on the M2-brane.

The analog of the projector equation in supergravity is a consequence of the κ-invariance of the M2-brane action [71]. It reads

$$\begin{equation}-\frac{\mathrm{i}}{6}\varepsilon {{\Gamma}}_{MNP}{\partial }_{m}{X}^{M}{\partial }_{n}{X}^{N}{\partial }_{p}{X}^{P}{\varepsilon }^{mnp}=\varepsilon ,\end{equation} \tag{ 6.4 }$$

where ɛ^mnp is the Levi-Civita tensor density and includes $1/\sqrt{g}$. On the boundary this equation reduces to (1.2), so the supercharges preserved by the M2-brane are constructed from the same ɛ₀ and ${\bar{\varepsilon }}_{1}$ as in field theory.

Given a preserved supercharge represented by ɛ, we can construct a three-form [37]

$$\begin{equation}\phi =-\mathrm{i}\frac{\varepsilon {{\Gamma}}_{MNP}{\varepsilon }^{{\dagger}}}{\varepsilon {\varepsilon }^{{\dagger}}}\enspace \mathrm{d}{X}^{M}\wedge \mathrm{d}{X}^{N}\wedge \mathrm{d}{X}^{P}.\end{equation} \tag{ 6.5 }$$

By construction, the pullback of this three-form to an M2-brane satisfying (6.4) is its volume form. ϕ then serves a role analogous to the almost complex structure in string theory.

If ϕ is closed dϕ = 0, then the form is a calibration [57, 72, 73]; its integral is the same on all three-volumes of the same homology class and is equal to the minimal volume, so the classical M2-brane action.

If ϕ is exact then the action comes only from the boundary at large y and is simply proportional to y vol(Σ). This divergent term is removed by the Legendre transform of [9, 19, 74] or equivalently by renormalisation [75], and the expectation value of the surface operator vanishes.

In each of our main examples we construct the appropriate ϕ and comment on its properties. It is particularly useful when one can write an equation of the form

$$\begin{equation}{\partial }_{m}{X}^{M}=\frac{1}{2}{g}_{ml}{\varepsilon }^{lnp}{G}^{ML}{\phi }_{LNP}{\partial }_{n}{X}^{N}{\partial }_{p}{X}^{P}.\end{equation} \tag{ 6.6 }$$

This replaces the equations of motion with a set of first order nonlinear equations and is the generalisation of the pseudo-holomorphicity condition in string theory. The exact form of this equation varies between the examples presented below.

Type- $\mathbb{R}$: using ɛ with ɛ₀ annihilated by all the a_J oscillators in (3.5), equation (6.5) gives the three-form

$$\begin{equation}{\phi }^{\mathbb{R}}=-\mathrm{d}{x}^{6}\wedge {\omega }^{\mathbb{R}},\quad {\omega }^{\mathbb{R}}=\sum\limits _{I=1}^{5}\left(\mathrm{d}{x}^{I}\wedge \mathrm{d}{y}^{I}\right).\end{equation} \tag{ 6.7 }$$

${\phi }^{\mathbb{R}}=-\mathrm{d}\left({x}^{6}{\omega }^{\mathbb{R}}\right)$ is clearly exact, so the expectation value of the Type- $\mathbb{R}$ surface operators vanishes at large N.

By translation symmetry, the three-volume should extend in the x⁶ direction and identifying σ³ = x⁶, we expect the remaining coordinates to satisfy the pseudo-holomorphicity condition

$$\begin{equation}{\partial }_{m}{X}^{M}={g}_{ml}{\varepsilon }^{ln3}{G}^{ML}{\omega }_{LN}^{\mathbb{R}}{\partial }_{n}{X}^{N}.\end{equation} \tag{ 6.8 }$$

Type- $\mathbb{C}$: in this case there are two preserved supercharges, so (6.5) gives two choices of three-form. The components shared by both are

$$\begin{equation}{\varphi }^{\mathbb{C}}=\left(\mathrm{d}{x}^{1}\wedge \mathrm{d}{x}^{2}+\mathrm{d}{x}^{3}\wedge \mathrm{d}{x}^{4}+\mathrm{d}{x}^{5}\wedge \mathrm{d}{x}^{6}\right)\wedge \mathrm{d}{y}^{1}.\end{equation} \tag{ 6.9 }$$

This is obviously the uplift of the Kähler form ${\omega }^{\mathbb{C}}$ of ${\mathbb{R}}^{6}$ (3.6) to AdS₇: ${\varphi }^{\mathbb{C}}={\omega }^{\mathbb{C}}\wedge \mathrm{d}{y}^{1}$. It is natural to augment it to

$$\begin{equation}{\phi }^{\mathbb{C}}=\left(\mathrm{d}{x}^{12}+\mathrm{d}{x}^{34}+\mathrm{d}{x}^{56}\right)\wedge \mathrm{d}{y}^{1}+{\left(\frac{y}{L}\right)}^{3/2}\left(\mathrm{d}{x}^{136}+\mathrm{d}{x}^{145}+\mathrm{d}{x}^{235}-\mathrm{d}{x}^{246}\right),\end{equation} \tag{ 6.10 }$$

which has the form of a G₂-structure (the shorthand notation is dx^μν... = dx^μ ∧ dx^ν ∧...).

In this case the first order equation (6.6) is indeed satisfied. This is a consequence of properties of G₂-structures as described below, around (6.14), but this equation is also satisfied with the restricted version of ${\varphi }^{\mathbb{C}}$ with just the three components (6.9). That three-form is closed, which means that minimal volumes are parallel to it. Choosing a gauge where X⁷ = y¹ = σ³, this is the requirement that ∂₃ x^μ = 0, so the pullback of ${\varphi }^{\mathbb{C}}$ is

$$\begin{equation}{\varphi }_{\mu \nu 7}^{\mathbb{C}}{\partial }_{m}{x}^{\mu }{\partial }_{n}{x}^{\nu }\enspace \mathrm{d}{\sigma }^{m}\wedge \mathrm{d}{\sigma }^{n}\wedge \mathrm{d}{\sigma }^{3}.\end{equation} \tag{ 6.11 }$$

This is the pullback of the Kähler form ${\omega }^{\mathbb{C}}$ to the surface at fixed y¹ times dσ³. We can immediately integrate over σ³ and, using (3.7), simply find a linear divergence times the area element of the two-surface Σ. The expectation value of the surface operator therefore vanishes, which is also clear, since this version of the three-form is exact. This is consistent with (4.6), as this calculation captures only $\mathcal{O}\left(N\right)$ terms in the large N limit [19, 52] and the anomaly coefficient ${a}_{1}^{\left(N\right)}\to 1/2$ (4.5).

Type- $\mathbb{H}$: the natural three-form extending (3.11) to AdS₅ × S² is

$$\begin{align}\hfill {\phi }^{\mathbb{H}}& ={\eta }_{\mu \nu }^{I}\enspace \mathrm{d}{x}^{\mu }\wedge \mathrm{d}{x}^{\nu }\wedge \mathrm{d}{y}^{I}-\frac{{L}^{3}}{{y}^{3}}\mathrm{d}{y}^{123}\hfill \\ \hfill & =\left(\mathrm{d}{x}^{12}+\mathrm{d}{x}^{34}\right)\wedge \mathrm{d}{y}^{3}+\left(\mathrm{d}{x}^{31}+\mathrm{d}{x}^{24}\right)\wedge \mathrm{d}{y}^{2}+\left(\mathrm{d}{x}^{23}+\mathrm{d}{x}^{14}\right)\wedge \mathrm{d}{y}^{1}\hfill \\ \hfill & \quad -\frac{{L}^{3}}{{y}^{3}}\enspace \mathrm{d}{y}^{123}.\hfill \end{align} \tag{ 6.12 }$$

This is indeed the result of (6.5) with the supercharge preserving all Type- $\mathbb{H}$ surfaces (3.11). This form is closed, so defines a calibration, and as in the Type- $\mathbb{C}$ above (6.10), this is a G₂-structure [76–79] (see also [80] for a gentle introduction).

Let us prove this implies that the first order equation (6.6) is satisfied. Following [57], consider the quantity

$$\begin{equation}{a}_{m}^{M}={\partial }_{m}{X}^{M}-\frac{1}{2}{g}_{ml}{\varepsilon }^{lnp}{G}^{ML}{\phi }_{LNP}{\partial }_{n}{X}^{N}{\partial }_{p}{X}^{P}.\end{equation} \tag{ 6.13 }$$

Squaring this we get

$$\begin{align}\hfill {g}^{mn}{a}_{m}^{M}{a}_{n}^{N}{G}_{MN}& ={g}^{mn}{\partial }_{m}{X}^{M}{\partial }_{n}{X}^{N}{G}_{MN}-{\varepsilon }^{mnp}{\partial }_{m}{X}^{M}{\partial }_{n}{X}^{N}{\partial }_{p}{X}^{P}{\phi }_{MNP}\hfill \\ \hfill & \quad +\frac{1}{2}{g}^{mn}{g}^{pq}{\partial }_{m}{X}^{M}{\partial }_{n}{X}^{N}{\partial }_{p}{X}^{P}{\partial }_{q}{X}^{Q}{G}^{RS}{\phi }_{RMP}{\phi }_{SNQ}.\hfill \end{align} \tag{ 6.14 }$$

The first term gives the trace of the induced metric and the second using (6.5), the contraction of two Levi-Civita tensors. Their sum is −3. The terms of the second line can be evaluated using properties of the three-form ϕ. In particular G₂-structures satisfy

$$\begin{equation}{G}^{RS}{\phi }_{RMP}{\phi }_{SNQ}={G}_{MN}{G}_{PQ}-{G}_{MQ}{G}_{PN}-{\left({\ast}\phi \right)}_{MPNQ}.\end{equation} \tag{ 6.15 }$$

The asymmetric term does not contribute and the contractions with the first two reduce everything to traces of the induced metric and the second line evaluates to 3, showing that ${a}_{m}^{M}=0$, thus proving (6.6).

Surfaces analogous to cones of Type- $\mathbb{H}$ were studied in [37] (see the discussion in section 5.1), including their holographic realisation. There it was shown that the M2-branes are calibrated with respect to a three-form, which should be the same as (6.12) in a different coordinate system. A reduction of (6.6) to conical surfaces (which looks similar to (6.8)) was also derived there.

Type-S: these surfaces are restricted to ${S}^{3}\subset {\mathbb{R}}^{4}$ and n^I ∈ S³, so the M2-brane duals are located within an AdS₄ × S³ ⊂ AdS₇ × S⁴. Using the metric (6.1), we take the eight coordinates x^I and y^I with I = 1, 2, 3, 4, set x⁵ = x⁶ = y⁵ = 0 and impose the further constraint

$$\begin{equation}\vert x{\vert }^{2}+\frac{4{L}^{3}}{y}=1.\end{equation} \tag{ 6.16 }$$

Using the two supercharges that preserve the Type-S surfaces (3.14) gives according to (6.5) two three-forms. The terms appearing in both are

$$\begin{align}\hfill {\phi }^{S}& =\frac{1}{6}{\varepsilon }_{IJKL}\left[\mathrm{d}{x}^{I}\enspace \mathrm{d}{x}^{J}\left(3{x}^{K}\enspace \mathrm{d}{y}^{L}-2{y}^{K}\enspace \mathrm{d}{x}^{L}\right)\right.\hfill \\ \hfill & \quad -\left.{\left(\frac{L}{y}\right)}^{3}\enspace \mathrm{d}{y}^{I}\enspace \mathrm{d}{y}^{J}\left({x}^{K}\enspace \mathrm{d}{y}^{L}-6{y}^{K}\enspace \mathrm{d}{x}^{L}\right)\right].\hfill \end{align} \tag{ 6.17 }$$

To simplify this, we define the complex vielbeins ζ^I

$$\begin{equation}{\zeta }^{I}=\sqrt{\frac{y}{L}}\enspace \mathrm{d}{x}^{I}+\mathrm{i}\frac{L}{y}\mathrm{d}{y}^{I},\qquad {\bar{\zeta }}^{I}=\sqrt{\frac{y}{L}}\enspace \mathrm{d}{x}^{I}-\mathrm{i}\frac{L}{y}\enspace \mathrm{d}{y}^{I}.\end{equation} \tag{ 6.18 }$$

They are null with respect to the metric (6.1), and $\frac{\mathrm{i}}{2}{\sum }_{I}{\zeta }^{I}\wedge {\bar{\zeta }}^{I}$ defines an almost complex structure on AdS₅ × S³. Taking Ξ to be the unit vector normal to (6.16), its contraction with ζ^I gives

$$\begin{equation}{\Xi}\cdot {\zeta }^{I}=\sqrt{\frac{L}{y}}\left({x}^{J}{\partial }_{{x}^{J}}-2{y}^{J}{\partial }_{{y}^{J}}\right)\cdot {\zeta }^{I}={x}^{I}-2\mathrm{i}{\left(\frac{L}{y}\right)}^{3/2}{y}^{I},\end{equation} \tag{ 6.19 }$$

and the complex conjugate for ${\Xi}\cdot {\bar{\zeta }}^{I}$.

Now taking the (anti)holomorphic four-forms on AdS₅ × S³

$$\begin{equation}{\Omega}={\zeta }^{1}\wedge {\zeta }^{2}\wedge {\zeta }^{3}\wedge {\zeta }^{4},\qquad \bar{{\Omega}}={\bar{\zeta }}^{1}\wedge {\bar{\zeta }}^{2}\wedge {\bar{\zeta }}^{3}\wedge {\bar{\zeta }}^{4},\end{equation} \tag{ 6.20 }$$

ϕ^S is the projection of their difference onto the hypersurface (6.16)

$$\begin{equation}{\phi }^{S}=\frac{1}{2\mathrm{i}}{\Xi}\cdot \left({\Omega}-\bar{{\Omega}}\right)=\frac{\mathrm{i}}{12}{\varepsilon }_{IJKL}\left({\zeta }^{I}\wedge {\zeta }^{J}\wedge {\zeta }^{K}\left({\Xi}\cdot {\zeta }^{L}\right)-{\bar{\zeta }}^{I}\wedge {\bar{\zeta }}^{J}\wedge {\bar{\zeta }}^{K}\left({\Xi}\cdot {\bar{\zeta }}^{L}\right)\right).\end{equation} \tag{ 6.21 }$$

This representation makes it manifest that though we write ϕ in terms of eight coordinates, it is contained within (the cotangent to) the AdS₄ × S³ subspace defined by (6.16). Perhaps it is consistent with some notion of generalised calibrations as in [26, 81, 82].

Minimal volumes: the discussion above allowed us to reduce the BPS equations to first order equations for Type- $\mathbb{R}$, Type- $\mathbb{C}$ and Type- $\mathbb{H}$. The structure of three-form for Type-S in (6.17) is more complicated, and it is not clear what should be the form of (6.6) in this case.

The equations for Type- $\mathbb{R}$ and Type- $\mathbb{C}$ can be written as pseudo-holomorphicity equations (6.8), (6.11) and in the latter case the solutions are simply ${\Sigma}{\times}{\mathbb{R}}_{+}$ where Σ is the original surface and ${y}^{1}\in {\mathbb{R}}_{+}$. For Type- $\mathbb{C}$ the equation can also be written in a form identical to (6.6) and this is also the form of the equation for Type- $\mathbb{H}$ surfaces.

We leave it to future work to find explicit solutions to those equations.

This paper is meant to reveal some cracks in the steep granite face of the $\mathcal{N}=\left(2,0\right)$ theory. They are unlikely to offer an easy path to the summit, but the anchors we placed should provide a good starting route for further exploration. At the very least, they promise to be a fun playground for those interested in this mysterious theory.

In the absence of a fully satisfying Lagrangian (or other traditional) formulation, we relied here on symmetry, geometry and algebra, the knowledge that these theories have planar surface operator observables and the large symmetry that they preserve. It is a very natural (and mild) assumption that such operators can have arbitrary geometries and an associated vector field n representing the local R-symmetry breaking. Indeed this assumption can easily be checked and realised in both the abelian theory and the holographic realisation.

This philosophy leads to trying to find geometries and vector fields for which the projector equation (1.2) has global solutions. To our surprise we found a very intricate structure of solutions to this set of equations which we split into several main classes, subclasses and examples preserving different numbers (and types) of supersymmetries, see tables 1 and 2.

The most obvious question is whether the examples we found are exhaustive. In one sense the answer is clearly no: acting with a global symmetry gives more examples and we chose to focus on single representatives of those. While changing x⁶ to x⁵ in Type- $\mathbb{R}$ or the three components of n^I that are turned on by the Type- $\mathbb{H}$ ansatz are rather trivial, acting with conformal transformations is less so. For example, they map ${\mathbb{R}}^{4}$ of Type- $\mathbb{H}$ to S⁴ and for surfaces that extend to infinity, they change the topology, so can affect the anomaly, or in the absence of an anomaly the finite expectation value. A simple illustration of this is the relation between the non-anomalous plane and its anomalous counterpart, the sphere. Likewise for Type- $\mathbb{C}$ surfaces, their compact versions could also be interesting.

Another generalisation is to allow n to be complex, while keeping (n)² = 1 (which is required in order to satisfy (1.2)). In this case the analytic continuation to Lorentzian signature would lead to non-unitary operators, but the analog for those in the case of Wilson loops have been instrumental for localisation [22], the ladder limit [83] and the related fishnet theories [84–86].

Complexifying n is somewhat akin to smearing chiral operators over the surface, but in fact one can consider combinations of surface operators and local chiral operators. Those can either be operators constrained to the surface such that they do not exist in its absence, or be away from it. The former case is studied extensively for maximally symmetric setups as part of the defect CFT program [13, 42, 87–91]. This could be extended to general surfaces and indeed a natural approach to try to prove that Type- $\mathbb{R}$ surfaces have trivial expectation value is to view all surfaces of this type as deformations of one another by the insertion of appropriate combinations of geometric and R-symmetry-changing local operators. In the latter case the question is which combination of surfaces and local operators are mutually BPS along the lines of [92–97].

Our approach here was to try auspicious ansätze and we were fortunate to uncover some examples, but it should be possible to formalise the problem and treat it systematically. This could lead to a classification of BPS surface operators, as was done for Wilson loops in 4D [53].

Another generalisation would be to study surfaces when the theory itself is on curved space. We touch on that when relating to surface operators in class-$\mathcal{S}$ theories in section 5.2. One can likewise reduce the theory to 3D [98] and 2D [99, 100], and one can look for other geometries it can be placed on and find compatible surface operators.

Lower dimensional line operator versions of Type- $\mathbb{R}$ and conical Type- $\mathbb{H}$ Wilson loops have been previously described in [26, 37, 41, 54] as were 4D surface operators similar to Type- $\mathbb{C}$ in [66]. It would be interesting to realise the surface operators of Type- $\mathbb{H}$ and Type-S there as well.

In four dimensional field theories entangling surfaces are also two dimensional and have been studied extensively in recent years [10, 18, 48, 49]. They share many properties with surface operators, in particular the anomaly structure. Even more relevant for our discussion here is the supersymmetric Rényi entropy [101, 102], and it would be interesting to see whether more results on entropy can be gleaned from entangling surfaces inspired by our different classes.

In the cases where we can define the (2, 0) theory, namely the abelian theory and the large N holographic realisation, we can proceed to try to evaluate expectation values of surface operators explicitly. The anomaly has already been reproduced in both cases in [19, 43, 45, 52]. To go beyond that, one can try to calculate finite expectation values for anomaless surfaces in the abelian theory employing the free-field propagators [19]. In the holographic setup it would be interesting to utilise the first order equations in section 6 to find explicit examples, at least for highly symmetric surfaces. In addition to providing finite expectation values for anomaless surfaces, it would allow to calculate the correlation function with local operators as in [31].

One may hope to be able to realise the BPS surface operators within some old and new approaches to the $\mathcal{N}=\left(2,0\right)$ theory [103–108]. Many of those formulations address the presence of planar surfaces and it would be interesting to look for the richer spectrum of BPS observables found here.

Even limited to the abelian theory, we can use it to get clues to the nonabelian theory. Following [109] we could look for an effective theory calculating the BPS surface operators. An example is the proposal of [37] that the (compactified) cones of Type- $\mathbb{H}$ are calculated by a 3D Chern–Simons theory. Attempting this for Type-S, we can write down an effective propagator which includes the contribution from the two-form B and the scalars Φ^I that couples to n^I in (1.5) [19] to get an effective two-form propagator

$$\begin{equation}\left\langle {B}^{\mu \nu }\left(x\right){B}_{\rho \sigma }\left(y\right)\right\rangle =\frac{1}{4{\pi }^{2}{\left\vert x-y\right\vert }^{2}}[\frac{1}{2}{\delta }_{\mu \nu }^{\rho \sigma }-\frac{4{\left(x-y\right)}_{[\mu }{\left(x-y\right)}^{\left[\rho \right.}{\delta }_{\left.\nu \right]}^{\sigma]}}{{\vert x-y\vert }^{2}}].\end{equation} \tag{ 7.1 }$$

This behaves somewhat like the propagator of a two-form in 4D, and can also be dualised to a compact scalar. Integrating this over the surface easily reproduces the Type-S anomaly in section 4 including the area term (4.14). Finding an appropriate nonabelian generalisation could help in guessing an expression for the surface operators a generic (2, 0) theory.

Finally, the $\mathcal{N}=\left(2,0\right)$ theory also contains co-dimension two defects, i.e. observables defined over four-manifolds [110–114] (see also [115]). It would be interesting to study their supersymmetric embeddings in 6D, the corresponding anomalies and the compatibility with the surface operators presented here.

It is a pleasure to thank Dima Panov and Simon Salamon for explaining to us a lot of the geometry underlying these constructions and to Malte Probst for continued discussion, related collaborations, and just for being an awesome guy.

ND's research is supported by the Science Technology & Facilities council under the Grants ST/T000759/1 and ST/P000258/1. 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).

No new data were created or analysed in this study.