Defect CFT techniques in the 6d $\mathcal{N} = (2,0)$ theory

Surface operators are among the most important observables of the 6d $\mathcal{N} = (2,0)$ theory. Here we apply the tools of defect CFT to study local operator insertions into the 1/2-BPS plane. We first relate the 2-point function of the displacement operator to the expectation value of the bulk stress tensor and translate this relation into a constraint on the anomaly coefficients associated with the defect. Secondly, we study the defect operator expansion of the stress tensor multiplet and identify several new operators of the defect CFT. Technical results derived along the way include the explicit supersymmetry tranformations of the stress tensor multiplet and the classification of unitary representations of the superconformal algebra preserved by the defect.


Introduction
A natural class of observables of the N = (2, 0) 6d SCFT is that of surface operators [1]. These operators share many properties with the much studied Wilson loops of gauge theories: they are extended objects which can enjoy superconformal symmetry [2], in some cases have a holographic description [3], and in the abelian N = (2, 0) theory admit a field realisation as the integral of the 2-form B field, akin to a gauge connection [4][5][6][7][8]. These similarities suggest that some of the methods which have proven successful in the study of Wilson loops can be applied to the N = (2, 0) theory as well, providing a window into its dynamics.
In this paper we apply the framework of defect CFT to the surface operators of the N = (2, 0) theory. We adopt the approach of the conformal bootstrap program [9][10][11][12][13][14] and use the symmetries preserved by the surface operators to constrain their correlators with other bulk operators as well as local operator insertions on the surface. One of the virtues of this description is that it does not rely on a field realisation and therefore is applicable to the nonabelian theory.
We focus on 1/2-BPS defects because they preserve the largest amount of symmetry. These are surface operators defined over a plane and expected to be labeled by a representation of the ADE group of the N = (2, 0) theory [15][16][17]. We consider local operator insertions into the defect, the simplest example encoding an infinitesimal geometric deformation of the plane itself. Because the plane preserves superconformal symmetry, the correlators of local operator insertions are constrained and obey the axioms of a dCFT-the 2-and 3-point functions are fixed up to a small set of numbers defining the dCFT, which make up the dCFT data.
Explicitly, consider a correlator involving such a surface operator V . While translating the plane along parallel directions leaves the correlator invariant, translations in directions transverse to the plane do not. Instead, the stress tensor receives a contribution from a contact term localised on the defect (at x = 0): (1.1) The index µ = 1, . . . , 6 runs over all spacetime coordinates, while m = 1, . . . , 4 are the coordinates transverse to the plane. We use the notation V [Ô(σ)] to denote the planar surface operator with a defect operatorÔ inserted at a point σ on the plane. Equation (1.1) is an operator equation, so it holds inside correlation functions. It defines D, known as the displacement operator. In addition, because V preserves some supersymmetries, the displacement operator sits in a multiplet containing also contact terms for the divergence of the broken super-and R-current, which we label Q and O, respectively.
It turns out that these defect operators enjoy a very favorable position: not only are they highly constrained by the residual symmetry (which includes the 2d rigid superconformal symmetry), but they also correspond to interesting physical quantities [18,19]. 1 Indeed it is easy to show that, as a consequence of (1.1), the insertion of a displacement operator D corresponds to small deformations of the plane, and thus captures the shape dependence of surface operators. This paper revolves around two correlators that capture physical properties of the defect. The first one is the 2-point function of displacement operators. Using the residual conformal symmetry of the plane and reading the conformal dimension ∆ D = 3 from (1.1), the 2-point function is constrained up to a single coefficient C D to be V [D m (σ)D n (0)] = C D δ mn π 2 |σ| 6 . (1.2) Notice however that, unlike most operators, the normalisation of D is already fixed by the normalisation of T µν from (1.1), so that C D is part of the data characterising the dCFT [21] (the factor π 2 is for convenience). The second is the stress tensor, which in the presence of the defect acquires an expectation value. Both the components of the tensor along the defect T ab and orthogonal to it T mn can have a nonzero 1-point function, and they are fixed by conformal invariance up to an arbitrary coefficient h T to be T (σ, x) is inserted at a distance x from the defect, and obviously the correlators do not depend on the coordinate σ by translation invariance along the plane. η ab = diag (−1, 1) is the Minkowski metric.
In theories with only conformal invariance the coefficients h T and C D are independent quantities [22], but in theories with enough supersymmetries one can use superconformal Ward identities to relate them [23]. For our surface operators we show in Section 3 that To derive this result, we obtain the transformations of the stress tensor multiplet under supersymmetry (3.7), which is also an important result of Section 3. Analogous relations between h T and C D were first derived using the same techniques for the 1/2-BPS Wilson loops of 4d N = 2 theories [23] and the 1/6-BPS bosonic loops of ABJM [24], proving the conjecture of [25,26]. A similar analysis was also applied recently to surface operators in 4d N = 1 theories [27]. All these different examples show how the language of dCFT is a powerful and universal tool to study superconformal defects.
More than simply equating different constants, the relation (1.4) has an important physical consequence. Recall that surface operators in CFTs typically have a conformal anomaly, which manifests itself as a divergence in the expectation value. The anomaly density A Σ is the sum of conformal invariants [28,29] and can be written as where the invariants R Σ , trĨI 2 , tr W, (∂n) 2 are local quantities depending on the embedding of the surface (see Appendix B for a review), while the coefficients a, b 1 , b 2 and c are known as anomaly coefficients and depend on the specifics of the theory and operator in question.
In Section 4 we relate the coefficients b 1 , b 2 , c to C D , h T and an additional constant C O to be introduced in (2.3). In the language of anomaly coefficients, the result (1.4) along with the relative normalisations (2.6) of the operators in the displacement multiplet can be stated as We emphasize that these identities are a consequence of supersymmetry and hold for any 1/2-BPS operator of the N = (2, 0) theory and for any ADE group. In particular, the second identity agrees with the explicit holographic calculations of [30,8,31] and was conjectured to come from supersymmetry in [27]. The two remaining anomaly coefficients a and b 1 were calculated at N = 1 in [8] and for N > 1 using holographic entanglement entropy in the presence of surface operators [32][33][34][35], and the superconformal index [36]. Finally, in Section 5 we expand our scope and consider the analog of the operator product expansion but for bulk operators in the presence of a defect-the defect operator expansion (dOE) [37,38]. This expansion gives a representation of bulk operators near the defect in terms of insertions of defect operators. To understand what these defect operators are more generally, we classify unitary multiplets of the algebra preserved by the defect. We then look at operators in the stress tensor multiplet and determine the short multiplets arising in their dOE. We find a new marginal defect operator, which we associate with the RG flow between the nonsupersymmetric and 1/2-BPS surface operator discussed in [31]. 2 In addition to this result, we find that the defect operator expansion provides a useful framework and makes the constraints imposed by the preserved symmetries manifest. In fact, in Section 5.4 we use the dOE and representation theory to give a different perspective on the relation (1.4). Unlike in Section 3, where (1.4) follows from a technical calculation, we are able to conclude directly that h T and C D must be related. This suggests a strategy for determining the minimal amount of supersymmetry required in order for the conjecture of [25], which relates these coefficients in the case of supersymmetric Wilson loops, to hold (see also [19] and references therein for a similar conjecture in the context of entanglement entropy).
Some auxiliary results are collected in appendices. Appendix A summarises our conventions and the gamma matrices used throughout the paper. Appendix B provides a short review of the Weyl anomaly for surface operators. In Appendix C we show how to constrain correlators containing both bulk and defect operators using conformal symmetry. Appendix D reviews the 2 algebras used in this paper: the osp(8 * |4) symmetry of the bulk theory and the osp(4 * |2) ⊕ osp(4 * |2) symmetry preserved by the defect.
Note added: In the last stages of preparation of this paper, the classification of unitary multiplets of osp(4 * |2) presented in Section 5.2 appeared in [41].

Displacement multiplet
As far as defect operators go, the displacement operator is pretty universal. As (1.1) suggests, any defect breaking translation symmetry contains that defect operator. For this reason, it has appeared in many contexts: the prototypical example is the 1/2-BPS Wilson line in N = 4 SYM, where the study of deformations and operator insertions was initiated in [42], but many other examples have been studied over the years and follow the general analysis of [22].
In the case of N = (2, 0), we are mostly interested in the multiplet which contains the displacement operator. Of the full superconformal algebra osp(8 * |4), the 1/2-BPS plane preserves a 2d conformal algebra so (2,2) in the directions parallel to the plane, along with rotations of the transverse directions so(4) ⊥ and an so(4) R R-symmetry. In addition, it also preserves half the supersymmetries Q + (andS + ) such that Q + V = 0. These are obtained by a half-rank projector Q + = Π + Q whose explicit definition can be found in (D.12). The preserved generators form an osp(4 * |2)⊕osp(4 * |2) subalgebra [43], detailed in Appendix D.2.
Importantly, in direct analogy to (1.1), the Ward identities associated to the remaining broken super-and R-symmetries also receive contributions localised on the defect, which give rise to defect operators Q and O i , encoding the nontrivial response of the defect to the broken generators. Explicitly, the conservation laws associated with the R-current j and the supercurrent J are broken as follows: As mentioned previously, the (nonabelian) theory does not have a known field realisation, so we cannot write these operators in terms of fundamental fields. We can however derive some of their properties purely from representation theory. The full multiplet as derived in Appendix D.2.1 reads (2.2) δ + = ε + Q + is a variation with respect to the preserved supercharges and ε + = ε + Π + .

Superconformal Ward identity
The 2-point functions of these operators is easy to find. Both D and O transform as scalars with respect to the 2d conformal symmetry, while Q is a spinor. Their conformal dimensions can also be read from (2.1) and are ∆ D = 3, ∆ Q = 5/2 and ∆ O = 2. Consequently, using the preserved bosonic symmetries, their 2-point functions are (up to some arbitrary coefficients As Q is a 2d spinor, its 2-point function should be written in terms of the corresponding 2d gamma matrices. In order to emphasize the relation between the respective symmetry algebras in 6d and 2d, we write these matrices as blocks of their 6d counterparts obtained by the projector Π − . We can now relate C O and C Q to C D using superconformal Ward identities associated to the preserved supersymmetries. Apply the supersymmetry transformations (2.2) to the vanishing correlator which serves to relate C D to C Q . Altogether, we find that the normalisations of the 2-point functions obey

Stress tensor correlators
Some of the most important operators in any theory are the stress tensor and its multiplet.
In the presence of the 1/2-BPS defect, their expectation values are highly constrained by the residual symmetry: typically the so(2, 2) ⊕ so(4) ⊥ ⊕ so(4) R bosonic subalgebra of preserved symmetries is powerful enough to fix them up a to a constant (see e.g. (1.3)).
In addition to the constraints imposed by conformal symmetry, supersymmetry relates correlators of different operators in the same multiplet. Adapting the strategy of [23,24,27], the key to deriving (1.4) is to focus on the correlator T µν (x)V [D m (σ)] , which is entirely fixed in terms of the constants C D and h T [22]. The kinematics of that correlator admit 2 independent tensor structures with their own coefficient. They are related to C D by taking the divergence and to h T by integrating the displacement operator over the surface, which simply translates the defect We stress that this does not provide in itself a relation between C D and h T , as can be checked using the explicit form of the correlators (see equation (6.2) of [19]). Instead, to derive the relation, we should use superconformal Ward identites to relate this correlator to O i5 V [O j ] , where O is the superconformal primary of the stress tensor multiplet. Because the latter admits only a single tensor structure, this would imply that C D and h T are related.
In order to derive this result, we need the explicit supersymmetry transformations of the stress tensor multiplet, which are summarised in (3.7). We also need the 1-point functions of the stress tensor appearing on the right-hand side of (3.2), which are derived in Section 3.2 (the 2-point functions of the displacement multiplet are given in (2.3)). Then, we use the supersymmetric Ward identities associated with correlators of the form OV [Ô] to derive (1.4).

Stress tensor multiplet
We begin by obtaining explicit supersymmetry transformations for the stress tensor multiplet, whose content is derived from representation theory and can be found in [44], where it is presented as a massless graviton multiplet (see also [45,46] for an overview of superconformal multiplets in various dimensions).
The primaries of any multiplet are labelled by their transformation under Lorentz symmetry [j 1 , j 2 , j 3 ] su(4) , R-symmetry (R 1 , R 2 ) sp(2) as well as their conformal dimension ∆. 3 In the notation of [45], the stress tensor multiplet is the D 1 [0, 0, 0]  Together with their descendants, these form an on-shell multiplet with 128 bosonic operators (and a matching number of fermionic operators).
In addition to the operator content, we need below the explicit supersymmetry transformations, which have not been calculated before to the best of our knowledge. These can be obtained in a variety of ways (e.g. oscillator constructions [44] and superspace transformations [47,48]), but here we simply list the terms allowed by Lorentz and R-symmetry and fix the coefficients by requiring closure of the algebra, i.e. imposing that on every operator {Q, Q} Φ = 2PΦ. Importantly, imposing this condition is made easy because we already know the operator content.
We start from the superprimary O IJ . Since Q transforms as [1,0,0] (1,0) 1/2 , we know from representation theory that the product QO can contain but as [1,0,0] (1,2) 9/2 does not appear in the multiplet, we remove it. The remaining term [1,0,0] (1,1) 9/2 can be constructed explicitly and is fixed up to a constant c 1 The transformation of χ is more complicated but the same analysis leads to It is easy to check that so the algebra closes provided c 1 d 1 = 1 (we identify P µ = ∂ µ , see (C.1)). We can proceed this way for the full multiplet and build the supersymmetry transformations. Checking for closure of the algebra becomes a tedious (if straightforward) task and is not very illuminating, so we omit the details. The end result is (with δ = ε αα Q αα ) There are still some arbitrary constants c i that remain unfixed and can be absorbed into the normalisations of O, χ and H. On the other hand, the normalisation of the conserved currents must match that of the algebra, so these operators cannot be rescaled. This can be seen by checking that the variation of the currents reproduces the corresponding commutator in (D.3). For example, the variation of j µ computed using (3.7) is which is indeed the correct normalisation for the commutator [Q αα , R IJ ] of (D.3).

Defect without insertions
Among the operators of the stress tensor multiplet, some can acquire an expectation value in the presence of V . For the stress tensor, this happens when h T = 0 in (1.3), and we can similarly constrain the 1-point functions of the other operators. This computation is done explicitly in Appendix C and the only nonvanishing correlators are where h O , h H , and h T are as yet undetermined constants. They are however related by the supersymmetry transformations (3.7) derived above. Specifically, consider the Ward identities associated with the preserved supersymmetries Q + = Π + Q (with the projector Π + defined in (D.12)) (3.10) These equations fix 11) and the correlators in (3.9) are fixed up to a single constant h T .

Defect with an insertion
We are now in a position to derive the result (1.
Explicitly, the Ward identity is Plugging in the explicit forms of these correlators (3.12), and demanding that the terms proportional to γ a σ a vanish, we obtain a linear relation 14) The terms proportional γ m x m give the same constraint. Next, recall that O and Q respectively encode the action of a broken infinitesimal Rsymmetry or supersymmetry variation. Therefore we can relate Using (3.9) and (3.12), we obtain A slightly more involved but entirely analogous calculation yields (3.17) Finally, C jO is related to the normalisation of the displacement operator multiplet by (2.1) Plugging the correlator of j µi5 and O j into the right hand side and integrating against a test function allows us to fix (3.19) Combining the above results into (3.14), we obtain which proves (1.4).

Relation to anomaly coefficients
In this section we explore the consequences of the relation between the coefficients C D and h T (1.4) for physical observables. These pieces of dCFT data appear in the Weyl anomaly of surface operators as defined in (1.5), and as we show below the relations (2.6) and (3.20) relate the anomaly coefficients as (1.6).
The relation between correlators and anomaly coefficients is not specific to 2d defects in the N = (2, 0) theory, but applies for any surface operator in a CFT. The anomaly coefficient b 1 was first shown to be related to C D in [19], while the relation between b 2 and h T was obtained in [18,19]. Here we review their derivation and apply it to surface operators in the (2,0) theory to prove In a slightly different direction, the anomaly coefficients also feature notably in entanglement entropy in 4d [49] and were discussed in the entanglement entropy literature, see [19] and references therein.

Displacement operator
In order to isolate the contribution of C D to the anomaly coefficients, we separately switch on each of the terms in (1.5). Since the displacement operator generates geometric deformations, one expects that inserting sufficiently many D m into the planar surface operator V leads to a logarithmic divergence in the expectation value, signalling a conformal anomaly associated to the curvature of the surface. Similarly, inserting O i to sufficient order will allow us to access the anomaly coefficient c associated with deformations in R-symmetry space.
To make this relation precise, we formally write deformations of the 1/2 BPS plane in terms of operator insertions Here P m = d 4 x∂ µ T µm generates translations transverse to the defect, while R-symmetry rotations are generated by R i5 = d 4 x∂ µ j µi5 . For constant parameters ξ, ω, the currents can be freely integrated and we recover the standard action of the charges P m and R i5 . Equation (4.1) is generally a complicated expression involving contact terms like (1.1), but also contact terms from P m acting on defect operators and possibly other operators from the OPE. We can calculate its expectation value to quadratic order by expanding the exponential and noting that the 1-point functions of defect operators vanish: We can discard log V since for the 1/2-BPS plane in a flat background, all anomaly terms vanish separately. Since the anomaly is quadratic in ξ and ω, it is related to the two point functions written here and we can safely ignore the higher order terms in the expansion.
To extract the anomaly coefficients, we study the UV divergence of the integrals in (4.2). The relevant correlators are found in (1.2) and (2.3). Fixing σ, the σ ′ integral can be evaluated explicitly by Taylor expanding ξ m (σ ′ ) and ω i (σ ′ ) around σ. Starting with the second integrand and using τ = σ ′ − σ, While this integral leads to power law singularities as well, a logarithmic divergence arises only from the term quadratic in τ . We adopt polar coordinates τ a = τ e a where e a are orthonormal vectors parametrised by an angle ϕ. Using the identities dϕ e a e b = πη ab , dϕ e a e b e c e d = π 4 (η ab η cd + η ac η bd + η ad η bc ) , (4.4) and dropping all but the logarithmic divergence, we obtain To leading order, the R-symmetry transformation in (4.1) takes the 1/2-BPS plane to a surface operator with ∂ a n i (σ) = ∂ a ω i , so we can read the anomaly coefficient as The logarithmic divergence of the first integrand in (4.2) can be evaluated in a similar way, and arises only from the fourth order in the Taylor expansion of ξ n (4.7) Performing the angular integral with (4.4) leads to This is the trace of the second fundamental form squared of the deformed surface (see (B.3)), which can be rewritten using the Gauss-Codazzi equation (B.4) as Since we are on flat space, the Weyl tensor vanishes. The volume form for the deformed surface gets corrected, but to leading order in ξ does not affect the calculation. Therefore the contribution of this term to the anomaly density is Note that the integral of R Σ vanishes for small deformations of the plane. It therefore does not contribute to the anomaly, and we find Using (2.6) along with (4.6) and (4.11) we find a relation for the anomaly coefficients

Stress tensor
The relation between b 2 and h T is derived in a similar fashion, but instead of deforming the surface itself, we can relate the insertion of a stress tensor to a change in the background geometry. 5 The expectation value of the planar surface operator now receives a contribution from the metric variation: In this equation, the subscript • g means the expectation value is calculated on a curved background metric g.
Since the insertion of a stress tensor sources a metric perturbation of linear order δg, we can only reproduce the anomaly to that order, which, expanding (1.5), is (4.14) These two terms are respectively associated to T mn V and T ab V in (4.13), and the total derivative drops out of the integral over the plane. Using (3.9), we can evaluate the first term of (4.13). The logarithmic divergence arises as In the second step we expanded δg(x) in a Taylor series and dropped powers of x not contributing to the anomaly. We again switch to spherical coordinates x m = re m and take note of the 4d analogue of (4.4) The integral then becomes (4.17) Comparing against (4.14), we identify The calculation for T ab V is similar and gives the same result.
With expressions for b 1 , b 2 , c in terms of C D and h T in hand, we can finally translate the result of the previous section (3.20) into a constraint on the anomaly coefficients, and find as claimed.
A direct consequence of this relation (together with (4.12)) is that one only needs to calculate two nontrivial surface operators to calculate all the independent anomaly coefficients, for instance the sphere and cylinder.

Defect operator expansion
A useful tool in dCFT is the defect operator expansion (dOE), also known as the bulk-defect operator product expansion [37,38] (see [13] for a recent review of some dCFT techniques, including the dOE, in the context of the CFT bootstrap program). This is a convergent expansion representing bulk operators in terms of insertions of defect operators where the sum is over defect primaries. The differential operators C V ik (x, ∂ σ ) are fixed by conformal symmetry. Their exact form can be obtained from the corresponding bulk-defect 2-point function of O i andÔ k by equating where we denote by CÔ k the numerator of the 2-point function ofÔ k . Explicit expressions for C V ik can be found in [51,22], but are not needed in this paper. The list of defect primaries appearing on the right-hand side of (5.1) can include the defect operators of Section 3 (namely the defect identity and the displacement operator multiplet), but it certainly includes more defect operators. This can be viewed as a consequence of the associativity of the OPE: since (5.1) maps bulk operators to defect operators and is valid in any correlator, all the CFT data of the bulk operators must be encoded, in some way, in the OPE of defect operators. Hence there must be at least as many defect degrees of freedom as bulk degrees of freedom.
Here we initiate the study of these other defect operators. We first classify the unitary multiplets of defect operators in Sections 5.1 and 5.2. This allows us to find the decomposition of the stress tensor multiplet in multiplets of the preserved algebra, see Figures 1 and 2.
After this detour into representation theory, we write the leading terms in the dOE for some operators and discuss the appearance of a new marginal operator. We finally comment on constraints imposed by supersymmetry and show how the dOE sheds light on the derivation of Section 3.
As usual, we can label primaries by their representation under the bosonic subalgebra, which here is 3) The corresponding labels are [r 1 , r 2 ] h [r 1 ,r 2 ]h, with r 1 and r 2 the Dynkin labels for su(2) ⊥ and su(2) R , and h the conformal twist and labels representations of sl (2). The labelsr 1 , r 2 andh are similar, but for the second subalgebra. We note that while (5.3) is equivalent to so(2, 2) ⊕ so(4) ⊥ ⊕ so(4) R , the factorisation above in terms of 2 algebras is dictated by supersymmetry, see D.2 for more details. The joint representation has conformal dimension ∆ = h +h and spin s = h −h.
The simplest nontrivial example of a multiplet of osp(4 * |2) ⊕ osp(4 * |2) is the familiar displacement multiplet of section 2. Unlike our previous treatment however, here we label operators according to (5.3). In order to match that decomposition, we can express the superprimary O i ∼ (γ i ) α 2α2 O α 2α2 in spinor indices. In this notation, the indices α = 1, 2 are all su(2) indices. We use α 1 , β 1 , . . . for su(2) ⊥ and α 2 , β 2 , . . . for su(2) R ; similarly for the second set of su (2) The structure of the multiplet as a product of two representations of osp(4 * |2) is apparent in the diagram above. Under the action of Q, the operators transform as two multiplets of osp(4 * |2), for instance the lower diagonal is which is easily obtained from an ansatz as in Section 3.1 (the constant c is arbitrary). This is the simplest representation of osp(4 * |2) and it contains the weights [0, 1] 1 and [1, 0] 3/2 . Because it is ubiquitous, it is convenient to introduce some notation here and denote it B[0, 1], in anticipation of the results of Section 5.2.

Unitary multiplets of osp(4 * |2)
Since the algebra preserved by the defect factorises, we now turn our focus to general multiplets of a single copy of osp(4 * |2). Importantly, we can classify allowed multiplets by working out the constraints imposed by unitarity. 6 This follows the method described in [52] used to classifiy multiplets in superconformal theories for d ≥ 3.
The idea is the following. In radial quantisation, any operator O defines a corresponding state |O . While |O has positive norm (by assumption), there is no guarantee that the norm of all the other states of the multiplet is also positive, as required by unitarity. Demanding that negative norm states are absent from the multiplet leads to a lower bound on the conformal dimension of the superprimary h ≥ h A . In particular, as we show below, at h = h A (5.7) some states become null, and the corresponding multiplets are the short multiplets A. In addition, we find yet shorter multiplets B with superprimary of conformal dimension h B (5.8).
Consider the state |O of a superprimary operator in the representation [r 1 , r 2 ] h . Unitarity constrains the states Q|O to satisfy where we use Q † α 1 α 2 = S α 1 α 2 and the anticommutator (D.11), written in terms of su(2) ⊥ and su(2) R generators T i (1,2) . We suppress the indices of Q and |O , but the constraint should hold for any choice of Q, |O , and linear combinations thereof.
The matrix elements s|σ i T i |s are bounded by the eigenvalues of σ i T i . Since σ i is the fundamental representation, the product σ i T i can be decomposed as [1] , for both r 1 and r 2 . The eigenvalues are expressed in terms of the quadratic Casimirs C 2 (j) = j(j + 2)/4 (using e.g. equation (2.38) of [52]), so that (5.5) takes the form with j 1 and j 2 taking any values in r 1 ± 1 and r 2 ± 1. This assumes that both r 1 > 0 and r 2 > 0, otherwise the tensor product decomposition is simply [1] ⊗ [0] = [1] and j = 1. For r 1 > 0, we then find that the strongest bound on the scaling dimension implied by (5.6) is For r 1 = 0, we should instead take j 1 = 1 and we obtain If these bounds are saturated, a subset of states become null and may be consistently removed from the multiplet. While (5.7) and (5.8) are necessary conditions for unitarity, there could be, in principle, additional states whose norm becomes null (or negative), imposing further restrictions on h. It would be tedious to perform the above calculation for all states, but fortunately the conditions under which a representation is reducible (but not necessarily unitary) are listed by Kac in [53] (see also [54]). These match precisely the values obtained for the 4 choices of j 1 and j 2 in (5.6), which indicates that there are no further constraints.
We therefore conclude that for multiplets satisfying h ≥ h A , with h A given in (5.7), there are no stronger constraints from requiring unitarity at higher levels. Generically, these are long multiplets, and they thus contain 2 4 (r 1 + 1)(r 2 + 1) operators. Multiplets saturating the bound h = h A have a null state at level one, [r 1 − 1, r 2 + 1] h+1/2 , and their dimension is reduced. The special case r 1 = 0 still leads to a unitary multiplet, but in this case the first null state is at level 2.
In the case h A > h ≥ h B (5.8) however, since h is below the unitarity bound h A , some states in the multiplet would have a negative norm unless h = h B exactly: this is an isolated multiplet. It has a null state at level one, [1, r 2 + 1] h+1/2 . These short multiplets A and B are important to our discussion. For example, the B[0, 1] multiplet of Section 5.1 contains only 2 + 2 operators, so it is indeed a short multiplet. From the argumentation above, the conformal dimension of its superprimary is thus fixed by unitarity to h = h B = 1, in accordance with (2.1).
The broader question of determining the content of all short multiplets is interesting but lies beyond the scope of this work. However, specific short multiplets play a role in Section 5.3, and it is useful to know their content explicitly. It is sufficient for our present purposes to construct some representations heuristically by taking the tensor product decomposition of known multiplets.

The stress tensor dOE
Having gained some understanding of representations of the preserved algebra, we turn now to the main goal of this section: constructing the dOE (5.1) for the bulk operators of our theory. We focus on operators of the stress tensor multiplet (which should exist in any local quantum field theory), but the same analysis could be applied to other multiplets. A naive way of thinking about (5.1) is as branching rules for the breaking of symmetry due to the presence of the defect. Indeed, it is natural to decompose, for example, the While this expression merely amounts to a rewriting of the bulk degrees of freedom, the dOE becomes much more interesting if we consider a defect V which interacts with the bulk nontrivially.
A first sign that the dOE for general V contains additional terms is that the bulk operators couple to the defect identity 1 V and the displacement multiplet (cf. for instance (3.9) and (3.12)). It is clear that these operators do not appear in the branching rules and encode additional interactions between bulk and defect degrees of freedom.
The second way in which the dOE is interesting is more subtle. The decomposition of operators in terms of the preserved algebra can be performed, as above, for all the operators in the stress tensor multiplet. The resulting representations can be organised in the multiplets of Figures 1 and 2 and the displacement multiplet, leading to the branching rules under the breaking of symmetry osp(8 * |4) → osp(4 * |2) ⊕ osp(4 * |2). The superprimaries of the multiplets in Figure 1 are easily identified as the defect counterparts of the operators O 55 and O ij by their representation, and with a bit of work this correspondence between bulk and defect operators can be also established for all the other operators.   Observe that the conformal dimension of these defect operators is, in some cases, lower than that of the corresponding bulk operators, leading to singular terms in the dOE. For instance, the dimension ofÔ 55  takes the form In that case, the dOE of the stress tensor superprimary includes a defect operator of dimension 1, which can be understood as the insertion of Φ 6 in the line. Here, we do not have a field realisation of the N = (2, 0) theory butÔ 55 plays an analogous role. Consider then the dOE for O 55 . From Figures 1 and 2 we know some of the defect operators that can appear on the right-hand side of (5.1). This leads to The list of defect operators that may appear in this expansion is constrained by supersymmetry and can be treated systematically, but we do not pursue this direction further. Equation 14) The coefficient of the displacement operator can be found without computing O 55 V [D m ] explicitly, using the fact that the displacement operator is related to the broken translation symmetry. Integrating over the position of D m , we can replace it by a derivative: The left hand side is easily computed from (5.13) and related to C D and C V OD , while the right hand side is given in terms of h O . Matching coefficients, we find By contrast, the coefficient C V OÔ is not obviously related to the remaining coefficients, and thus an independent piece of dCFT data.

Constraints from supersymmetry
We conclude this section by sketching an alternative derivation of the results of Section 3. It turns out that the dOE provides a simple and elegant way to understand the origin of the linear relations (3.10) and (3.14) without doing explicit calculations, by reframing them in terms of coefficients of displacement primaries in the stress tensor dOE. Indeed, the method we use can in principle be applied far more generally to obtain analogous constraints for the remaining dOE coefficients.
To reproduce these results, consider the dOE of χ 5 . Following the analysis of Section 5.3, we decompose χ 5 into representations of the preserved algebra We only need the dOE of χ 5 α 1α2 , which takes the form Again, there are other terms that could be included in this expansion, but they don't play a role in what follows so we ignore them. We also emphasise that (5.18) is related to the dOE of the stress tensor superprimary by supersymmetry.
We can now proceed as in Section 3 and find the constraints imposed by the preserved supersymmetries. Consider first acting with Q on the bulk operator χ 5 α 1α2 to find with some coefficients. (The exact expression can be obtained by restricting (3.7) to the relevant representations of the preserved algebra.) Using the dOE on the right-hand side and focusing on the defect identity component gives Note that C j1 = 0 and C ∂O1 can be obtained from (5.15). We call this the "bulk" channel, since we calculate the action of Q on χ before taking the dOE.
The expression (5.20) is to be contrasted with the "defect" channel, where we first use (5.18) and then apply Q. Clearly, since 1 V is not the variation of anything 1 V = Q(. . . ), the result does not have an identity component. Consequently, the identity component of (5.20) must vanish as well, giving a linear constraint equivalent to (3.10) relating the normalisations of the stress tensor 1-point functions.
Similarly, (3.14) can be reproduced by focusing on the scalar displacement component of the same equation. The bulk channel gives schematically For the defect channel, we act on (5.18) with Q. From (5.4), we see that the variation only leads to descendants like ∂O, and no primary. Since equality between defect and bulk channel must hold at the level of each defect operator, we conclude that the contribution of the displacement superprimary O to the bulk channel must vanish, and we obtain a linear constraint on the dOE coefficients C V jO , C V HO , C V ∂OO , which is equivalent to (3.14). These two relations are only the simplest examples of a much larger set of constraints obeyed by the dOE coefficients. Indeed, equating the bulk and defect channel of any supercharge acting on any primary dOE at the level of each defect operator, it is straightforward to derive further such linear relations. These conditions greatly reduce the number of independent coefficients of stress tensor dOE coefficients, until we are left with what we could call a super-dOE, i.e. a set of dOEs which is fully consistent under the preserved supersymmetry.

Conclusion
In this paper, we initiate the application of defect CFT techniques to describe surface operators of the 6d N = (2, 0) theory, that is, we apply the ideas and tools of CFT to study local operator insertions into the 1/2-BPS plane. An important insertion is the displacement operator (1.1) which literally deforms the plane, but there are also other defect operators corresponding to inserting bulk operators near the defect-they are captured by the dOE (5.1).
One of our results is the classification of unitary multiplets of osp(4 * |2) ⊕ osp(4 * |2), the algebra preserved by a 1/2-BPS defect, in Section 5.2. These multiplets were not classified before to the best of our knowledge 7 and are the building blocks for discussing other aspects of the dCFT, like its spectrum, the OPE of defect operators and the dOE. In this work we focus on the dOE, but it would also be interesting to pursue these other directions, for instance using the tools of conformal bootstrap [13].
There are two important applications of the dOE (5.1) in our analysis: in Section 5.3 we use it to find new defect operators and in Section 5.4 we sketch how it makes the preserved symmetries manifest.
First, we use it to give the example of how the bulk stress tensor multiplet decomposes into defect multiplets. There are of course the operators D, Q and O of the displacement multiplet, but also other defect multiplets whose operator content is shown in Figure 1 and 2. Although we focus on the stress tensor multiplet, this analysis could also be applied to any other multiplet of the N = (2, 0) theory. In addition to the multiplets presented above, the dOE can include additional terms, and it would be interesting to obtain the selection rules as was done for 4d N = 4 SYM [55], by treating systematically all the superconformal Ward identities.
The important aspect of this decomposition of bulk operators is that it is convergent. In particular, it encodes all the information of the bulk OPE, which opens the possiblity of studying the N = (2, 0) theory from the point of view of a 2d defect CFT. This direction could lead to additional constraints on the bulk theory, since the defect operators are not a trivial rewriting of those in the bulk. This is manifested for instance by the appearance of divergences in the dOE of O 55 (5.13).
Instead, the dOE captures some important reorganisation of degrees of freedom in the dCFT. For instance, in the expansion of the bulk operator O 55 (5.13) we find a defect operator which is of dimension 2 and therefore marginal (we expect it to be marginally irrelevant). The analogous expansion of the superprimary of the stress tensor multiplet is well understood in the context of Wilson loops in 4d N = 4 SYM: using the definition of the 1/2-BPS Wilson loop (5.12) the marginal operator there corresponds to inserting Φ 6 into the line defect [39]. Here the interpretation is similar: inserting the analog ofÔ 55 in the non-supersymmetric surface operator triggers an RG flow which comes to a stop whenÔ 55 becomes marginal at the conformal fixed point, which is the 1/2-BPS surface operator. This flow is verified in holography [31] and should hold more generally for all N = (2, 0) theories.
A second use of the dOE is to make the preserved symmetries manifest. As we sketch in Section 5.4, we can explain the origin of the relation between h T and C D (1.4) simply by looking at the structure of multiplets of defect operators. This is to be contrasted with the derivation of Section 3, where the relation is the result of a calculation and not obvious from the outset. We believe this approach could shed light on determining the minimal amount of supersymmetry required to prove (1.4), that is whether it also holds for defects of the N = (1, 0) theory, and more generally what are the necessary conditions to prove the conjecture of [25].
In addition to the methods, the result (1.4) and the corresponding relation between the anomaly coefficients (1.6) are themselves interesting. In the context of Wilson loops, C D was shown to appear in the Bremsstrahlung function [21] and h T both in the radiation emitted by a quark undergoing constant acceleration and the entanglement entropy associated with Wilson lines [25]. While these calculations can be generalised to the case of surface operators, they do not give a finite answer: as shown in Section 4 respectively inserting displacement operators in the defect or introducing a stress tensor in its vicinity leads to a logarithmic divergence caused by an anomaly. To obtain a finite result, one can define a renormalised surface operator so that V ǫ effectively acquires a dimension (A Σ is defined in (1.5)). The interpretation of C D and h T are then as the anomaly coefficients b 1 and b 2 , which are the relations (4.11), (4.18) (also (4.6) between c and C O ). The net result of the relations between the anomaly coefficients (1.6) is that the anomaly depends on the geometry only through the combination (H 2 +4 tr P )−(∂n) 2 (defined in (B.4)), while the anomaly term R Σ integrates to a topological invariant, the Euler characteristic of the surface Σ. We note that for some classes of BPS operators, H 2 and (∂n) 2 are related and either cancel out or give interesting quantities. A simple example is the uplift of the BPS Wilson loops of [56] for which the anomaly vanishes, but more examples will be presented elsewhere [57]. Finally, there are other interesting directions which we haven't explored in this paper. For the Wilson line, a point of confluence between different techniques is the cusp, whose anomalous dimension at small angles is related to the Bremsstrahlung function [21] and can be calculated using integrability [58][59][60] and supersymmetric localization [61]. Its analog here are conical singularities which exhibit a peculiar log 2 ǫ divergence, see [62][63][64][65]8]. The coefficient of the divergence is entirely fixed by the behavior of the surface near the singularity, so it is natural to consider an operator inserting a conical singularity and to try and find its interpretation in the dCFT.
Another possibility is to study further the OPE for BPS operators. The N = (2, 0) theory contains a sector isomorphic to a chiral algebra [66] which can be used to calculate for instance the 3-point functions of 1/4-BPS local operators. For 4d N = 2 SCFTs, it was shown in [67] that the supercharges defining the cohomology are compatible with N = (2, 2) surface defects, and it would be interesting to extend their construction to the N = (2, 0) theory with 1/2-BPS surface defects. This could lead to exact results for a sector of the dOE and defect OPE.
It would also be interesting to study BPS operators in the context of the AGT correspondence. At large N one can use holography to calculate the expectation values, in the presence of the defect, of operators in the traceless symmetric representation of so(5) R [68], which contains in particular O IJ in the stress tensor multiplet. Since the the AGT correpondence can be used to calculate the expectation value of the stress tensor [36], it might also calculate expectation values for this larger class of operators at finite N.
acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). Cette recherche aété financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG).

A Conventions
We make use of the following indices: We work in Minkowski space with mostly positive signature. Below we detail the properties of gamma matrices arising in the osp(8 * |4) algebra. More details can be found in [69] and references therein.

A.1 Spinors and γ-matrices
We suppress contracted spinor indices in the main text. We use the NW-SE spinor index convention, so that where α is the index of a chiral 6d spinor (α for antichiral) andα that of an R-symmetry spinor. These indices are raised and lowered by the charge conjugation matrices c, Ω, which obey We also make use of two types of γ-matrices: 6d chiral (γ µ )β α (antichiral (γ µ ) β α ) and 5d (γ I )β α associated to R-symmetry. Their algebra is The chiral and antichiral representations are related through with γ µν...ρ ≡ γ [µγν . . . γ ρ] the antisymmetrised product of γ-matrices.
A representation of this algebra is given by

B Weyl anomaly for surface operators
Surface operators in CFT typically suffer from UV divergences which make their expectation value ill-defined. Up to power-law divergences (which can be removed by appropriate counterterms) their expectation value takes the form where ǫ is a regulator, Σ is the surface (in this paper we take the plane) and A Σ is known as the anomaly density. This conformal anomaly is constrained by the Wess-Zumino condition to take the form The conformal invariants of this expression are R Σ : The Ricci scalar on Σ.
The square of the traceless part of the second fundamental form.
tr W : W is the pullback of the Weyl tensor.
(∂n) 2 : The norm of the variation of the coupling to R-symmetry.
The exact definition of these invariants in our conventions can be found in Appendix B of [8].
Here we use the definition of the second fundamental form in (4.9) which for Γ λ ρσ = 0 and x m = ξ m reduces to (4.9). It can be decomposed into its trace, H µ , and its traceless part,ĨI

C Conformal Ward identities for defect correlators
In this appendix, we derive explicit expressions for the structure of the expectation values of stress tensor primaries in the presence of a flat conformal surface defect. Up to overall normalisation constants, which we further constrain in Section 3 using supersymmetry, these correlators are completely fixed by the bosonic symmetries (conformal and R-symmetry) preserved by the defect. We consider both defects with an insertion of a single primary of the displacement operator multiplet, and defects without such insertions. For brevity, we do not give an exhaustive list of such correlators and instead focus on those we require in the main text. More specifically, we compute only the expectation values of the primaries in the stress tensor multiplet, and some 2-point functions involving low-level primaries, namely O IJ , χ I αα , H I λµν in the stress tensor, and O i , Q αα in the displacement multiplet. The remaining correlators can of course be calculated using the same method.
We proceed in two steps. First, we fix the dependence on σ and x by implementing the Ward identities associated with the conformal symmetry preserved by the defect as well as transverse rotational symmetry. For clarity, in this calculation we suppress the R-symmetry indices of the operators and leave the scaling dimensions general. Indeed, as much of the kinematics is easily generalised to defects of dimension p in arbitrary spacetime dimension d = p + q, we state the more general result wherever we can do so without obscuring the results we presently need. Secondly, we fix the R-symmetry tensor structure of these correlators by demanding invariance under the residual so(4) R symmetry. Throughout, we denote generic operators in the bulk O and on the defectÔ.
Many of the kinematical results have been obtained by different methods in the past. In particular, the embedding space formalism allows for the efficient computation of bosonic correlators [22]. However, it is not straightforwardly applicable to correlators involving fermions.

C.1 Defect without insertions
Treating separately the coordinates along the plane σ a and tranverse x m , translation invariance on the plane implies that O(σ, x)V is a function of x m only. The other Ward identities can be cast into the form: These constraints are now straightforwardly solved. We focus on scalars O, vectors j µ , selfdual 3-forms H λµν and traceless symmetric 2-tensors T µν , as operators of those types make up the bosonic degrees of freedom of the stress tensor multiplet, while the correlators of fermionic operators with a scalar defect vanish identically.
For a Lorentz scalar O, all S µν vanish and the conformal Ward identities (C.2) are immediately solved to give with h O an as yet undetermined constant. The transformation law for a vector reads which, plugged into (C.2) eventually leads to 8 For higher spin bosonic operators, each Lorentz index separately transforms as (C.4). For a 3-form H λµν , the Ward identities (C.2) imply that the only components with nonvanishing expectation value in the presence of V are H abm and H lmn , and furthermore restricts the available terms for their one-point functions to In this work, we are concerned with 3-forms which come with a selfduality condition, which serves to relate the proportionality constants in (C.6). We are left with Lastly, we repeat the same analysis for a symmetric traceless 2-tensor. Exactly the same line of argument as above yields We are now in a position to construct the correlator of V with any bosonic primary in the stress tensor multiplet. To that end, recall that, under the unbroken so(5) R , O IJ and H I λµν transform as a symmetric traceless 2-tensor and a vector, respectively, while the stress tensor T µν is an R-symmetry singlet. 9 Without explicitly applying the Ward identities associated with the preserved so(4) R , we can fix the R-symmetry structure of the 1-point functions by writing down the available terms and, for O IJ , implementing tracelessness. Plugging in the correct scaling dimensions ∆ O = 4, ∆ H = 5, and ∆ T = 6, we find the only nonvanishing 1-point functions of stress tensor primaries in the presence of V are (3.9).

C.2 Defect with an insertion
We now repeat the above discussion for correlators O(σ, x)V [Ô(σ ′ )] involving a defect with an insertion of a displacement multiplet primary. The kinematical analysis is more involved than, but technically very similar to, the previous subsection. We use translation invariance to centerÔ at σ ′ = 0 and suppress the arguments of O(σ, x). The conformal Ward identities may be cast into the form: (C.9) 9 The R-symmetry current j IJ µ transforms as an antisymmetric tensor, but as seen above, its 1-point function vanishes identically regardless of the R-symmetry structure.
For the simplest case of a scalar O on the defect and a scalar O in the bulk, (C.9) become particularly simple, and imply 10 with C OO some normalisation constant. For a defect scalar O and a bulk vector j µ we obtain: (C.11) Indeed, these correlators are exactly the same for defects of generic dimension and codimension. It is easily checked that (C.11) is compatible with conservation of j in the bulk if and only if ∆ = d − 1 and∆ = p, which is indeed satisfied by the displacement superprimary O i and the bulk R-symmetry current j IJ µ . The conservation equation In order to apply the Ward identities (C.9), we expand χ α V [Q β ] in terms of antisymmetrised products of gamma matrices. The only such matrices with the appropriate chirality properties are γ µ and γ µνρ (we can omit γ µνρστ since it is related to γ µ by duality): Writing out and simplifying the conformal Ward identities explicitly then leads to Having completed the kinematic analysis, we can now restore the R-symmetry structure in order to construct the full bulk-defect 2-point functions. The Ward identities associated with the generators of so(4) R decouple from the kinematics, and therefore take a purely algebraic form (with R,R the representations of O,Ô) (C.17) Among the bosonic 2-point functions we consider, the only nonvanishing ones are (we again suppress coordinate dependence and Lorentz indices): To restore the correct R-symmetry structure of the fermionic 2-point function, recall that χ I αα transforms in the tensor product of the vector and spinor representation of so(5) R and is subject to a constraintγ I χ I = 0, while Q αα transforms as an ordinary R-symmetry spinor but obeys a constraint Π + Q = 0 mixing Lorentz and R-symmetry. Since we only need the correlator involving χ 5 α , we make the ansatz which is indeed compatible with (C.17).
With the kinematical data and R-symmetry structure in hand, we can now assemble the full 2-point functions. Plugging in the correct defect operator scaling dimensions ∆ O = 2 and ∆ Q = 5/2, we obtain (3.12).
The precise embedding of these supercharges inside Q αα is obtained by restricting to the preserved supercharges Π + Q = Q, where the projector is [8] (Π ± ) ββ αα = which gives, respectively for the two terms, two anticommuting superchargesQα 1α2 and Q α 1 α 2 . Their chirality is derived from the projector: (1 + γ 01 ) projects onto the positive chirality component, which is correlated with the positive chirality under so(4) ⊥ since γ 01 = γ 2345 .

D.2.1 Subalgebra as an embedding inside osp(8 * |4)
Lastly, in Section 2 and 3 it is convenient to discuss the subalgebra directly within the larger osp(8 * |4). Here we decompose some of the commutators of osp(8 * |4) into preserved and broken generators directly with the projector. We make use of the following identities Π † ± = Π ± , (Π ± C) T = −Π ± C T , [Π ± , Γ a ] = [Π ± ,γ 5 ] = 0 , (D.14) Note that here we don't differentiate between the action of Q andQ for simplicity. Using these properties, one can easily derive the induced subalgebra and its representation by acting with Π ± . The only nontrivial part of the preserved algebra is for the supercharges, which now obey Q + αα , Q + ββ = 2 (γ a Π + cΩ) αβαβ P a , S + αα ,S + ββ = 2 γ a Π + c T Ω αβαβ K a , The broken generators satisfy (D. 16) These transformations are related to (2.2) using (2.1) to write the displacement operator as contact terms in the presence of the defect: We can recover the full representation by acting with Q + , e.g.,