Defect CFT in the 6d (2,0) theory from M2 brane dynamics in AdS$_7 \times$S$^4$

Surface operators in the 6d (2,0) theory at large $N$ have a holographic description in terms of M2 branes probing the AdS$_7 \times S^4$ M-theory background. The most symmetric, 1/2-BPS, operator is defined over a planar or spherical surface, and it preserves a 2d superconformal group. This includes, in particular, an $SO(2,2)$ subgroup of 2d conformal transformations, so that the surface operator may be viewed as a conformal defect in the 6d theory. The dual M2 brane has an AdS$_3$ induced geometry, reflecting the 2d conformal symmetry. Here we use the holographic description to extract the defect CFT data associated to the surface operator. The spectrum of transverse fluctuations of the M2 brane is found to be in one-to-one correspondence with a protected multiplet of operator insertions on the surface, which includes the displacement operator. We compute the one-loop determinants of fluctuations of the M2 brane, and extract the conformal anomaly coefficient of the spherical surface to order $N^0$. We also briefly discuss the RG flow from the non-supersymmetric to the 1/2-BPS defect operator, and its consistency with a"$b$-theorem"for the defect CFT. Starting with the M2 brane action, we then use AdS$_3$ Witten diagrams to compute the 4-point functions of the elementary bosonic insertions on the surface operator, and extract some of the defect CFT data from the OPE. The 4-point function is shown to satisfy superconformal Ward identities, and we discuss a related subsector of"twisted"scalar insertions, whose correlation functions are constrained by the residual superconformal symmetry.


Introduction and summary
Non-local operators are an important class of observables in conformal field theories in various dimensions. When they are defined over planar or spherical submanifolds, they may preserve a subgroup of the conformal symmetry of the "bulk" CFT, and are often referred to as conformal defects. Using the AdS/CFT duality, one may develop a strong-coupling perturbation theory approach to the computation of their expectation values and correlation functions of local operators inserted on them. The most familiar example is that of a fundamental string ending along a curve on the boundary of AdS 5 within type IIB string theory, dual to the Wilson loop operator in N = 4 SYM theory [1]. When the curve is a circle or infinite straight line, the Wilson loop is 1 2 -BPS and it preserves a 1d conformal symmetry, 1 corresponding to a string worldsheet with AdS 2 induced geometry [2]. The strong coupling expansion for the expectation value of the Wilson line and correlation functions of operators inserted along it is then controlled by the fluctuations [3] of the fundamental superstring near the static configuration (see, e.g., [4,5] and refs. therein).
One can generalize this to other branes in different AdS background ending along different dimensional submanifolds on the boundary (for example, D3-brane and D5-branes probes in AdS 5 describing line operators, surfaces and domain walls, see, e.g., [6,7]). In the most symmetric cases these branes have the world-volume metric of AdS p+1 × S k with appropriate p and k.
In this paper we study the simplest such example within M-theory: an M2-brane probe ending along a surface on the boundary of AdS 7 . The most symmetric configuration, which preserves half the supersymmetries of the bulk theory, is when the 3d world-volume of the M2-brane ends on a plane (or sphere) at the boundary. As the M-theory in the AdS 7 × S 4 vacuum is a dual description of the (2,0) conformal theory, this configuration should be representing a supersymmetric surface defect operator in this 6d CFT (for a recent discussion and refs. see [8]). Our aim is to study this system beyond the classical brane limit by calculating its one loop fluctuation determinant and performing the holographic computation of 4-point correlators of the simplest local insertions into the surface operator. This is the direct analog of the calculation of insertions into the Wilson loops, captured by string fluctuations [5].
Let us recall the case of insertions into Wilson loops in N = 4 SYM. The 1 2 -BPS Maldacena-Wilson line along the x 1 direction and coupling to the scalar Φ 6 is W = Tr Pe dx 1 (iA 1 +Φ 6 ) . We can insert any adjoint valued operators into the loop, the most natural being the remaining five scalars Φ a and the combination of field strength and scalar D i = F ti ≡ iF ti + D i Φ 6 . The latter, known as the displacement operator, represents small geometric deformations of the line. In the defect CFT (dCFT), the scalars Φ a have dimension one and the displacement dimension two. This translates in AdS to fluctuation modes of the AdS 2 worldsheet with m 2 = 0 and m 2 = 2. Their 4-point functions were studied in [5] by expanding the string action to quartic order and performing AdS 2 Witten-diagram calculations on the worldsheet. This allowed to deduce the spectrum of some of the operators appearing in their OPE, providing further details on the strong coupling Wilson loop dCFT. For instance, the scaling dimension of the singlet scalar insertion which is dual in AdS 2 to a two-particle "bound state" of string fluctuations along S 5 was found to be ∆ = 2 − 5 √ λ + . . .. Recently, this result was confirmed by integrability techniques in [9], which also obtained several more orders in the strong coupling expansion.
For the case at hand, the (2,0) supersymmetric 6d CFT describing multiple M5-branes may be thought of as a SU (N ) generalization of a free (2,0) tensor multiplet containing the B mn -field with self-dual strength H mnl , 5 real scalars Φ I and 4 symplectic Majorana fermions. In this abelian theory the locallysupersymmetric surface operator analogous to the Wilson loop operator of [1] may be defined as [8,10] where Φ 5 is one of the 5 scalars of the (2,0) tensor multiplet, X m (x) are the 6d coordinates describing the surface and we specified to the case when the defect is a plane in the (1, 2) directions. 3 The surface operator breaks the OSp(8 * |4) supersymmetry of the 6d theory to [OSp(4 * |2)] 2 with the bosonic subgroup SO(2, 2) × SO(4) × SO(4) = [SO(2, 1) × SU (2) × SU (2)] 2 . Here SO (2,2) corresponds to the 2d conformal symmetry, one SO(4) to rotations in the transverse directions to the surface, and the second SO(4) ⊂ SO (5) is the remaining R-symmetry that rotates the four scalars that do not couple to the operator. As natural in defect CFT, one can consider correlation functions of operators inserted on the defect surface: the basic short multiplet includes four transverse scalars Φ a (a = 1, · · · , 4) with dimension ∆ = 2, four displacement operators 4 D i = H 12i ≡ iH 12i + ∂ i Φ 5 (i = 1, · · · , 4) with ∆ = 3 and eight fermions with ∆ = 5/2. In the dual description this 1 2 -BPS surface operator is represented by a probe M2-brane with worldvolume ending on a plane at the R 6 boundary, stretched along z of AdS 7 and localized at a point in S 4 . The M2brane probe is described by a κ-symmetric generalization of the Dirac-Nambu action (see, e.g., [14,15]). The 2 The introduction of a surface operator with coupling to B-field [1,2,11] is natural by analogy with strings ending on D3branes case, i.e. in the picture where the dynamics of M5-branes is described in terms of M2-branes [12] ending on strings coupled to B-field. 3 Due to conformal invariance one can consider the defect with with either planar or spherical (S 2 ) geometry. 4 The displacement operator describes transverse deformations of the defect (see for instance [13] for a general discussion).
For a defect with co-dimension 6 − p, the displacement operator D i may be defined via ∂µT µi = δ (6−p) (x ⊥ )D i where T is the "bulk" stress tensor (the stress tensor of the 6d CFT), with ∆ = 6. For the surface defect (p = 2) the dimension of the displacement D i (i = 1, · · · , 4) is then ∆ = 3. In general, for CFT d with a co-dimension d − p defect, ∆(D i ) = p + 1.
induced 3-geometry in the static gauge is then AdS 3 and as in [3,16] one finds that the transverse fluctuations of the M2-brane surface are represented by: 4 scalars y a (S 4 fluctuations) with m 2 = 0, 4 scalars x i (AdS 7 fluctuations transverse to the 3-surface) with m 2 = 3 and 8 fermions with m 2 = 9 4 . The correlators of these "transverse" membrane fluctuations (and more generally their composites) should then define a 2d dCFT associated to the surface defect. Via AdS 3 /CFT 2 correspondence the dual boundary operators should have dimensions ∆ = 2, 3 and 5 2 matching those of the scalars, displacement operator and fermions on the defect. 5 Below we compute the correlators of the bosonic fluctuations X I = (x i , y a ) as defined by the M2-brane action in the inverse effective membrane tension T 2 = a 3 T 2 = 2 π N expansion (a is the radius of AdS 7 ). They should define the large N limit of the corresponding 6d correlators of the operators (

1.2)
A novel feature of the present M2-brane in AdS 7 ×S 4 case compared to the string in AdS 5 × S 5 case in [5] is the presence of the WZ term in the action that contributes non-trivially to the 4-point correlator of the scalars y a . This term originates from the coupling of the M2-brane to the potential C 3 of the magnetic 4-form flux of the AdS 7 × S 4 background [17]. 6 Being intimately related to the underlying supersymmetry, the contribution of this term is important for the resulting 4-point function satisfying the constraints imposed by the residual superconformal symmetry. In contrast to the Wilson loop in N = 4 SYM case where one can also directly compute a weak-coupling limit of the corresponding correlators on the gauge theory side, it is not clear how to do this in the (2,0) 6d theory that currently lacks an intrinsic definition. 7 It would be interesting to make contact with the results of this paper by bootstrap methods, as was done in [20] for the case of the Wilson line dCFT. 8 The contents of this paper are as follows. Our starting point in Section 2 is the expansion of the M2brane action in AdS 7 × S 4 near the minimal 3-surface ending on a 2-plane or a 2-sphere at the boundary (and localized at a point in S 4 ). The value of the classical M2-brane action on this surface is proportional to the volume of AdS 3 . In the case of spherical boundary, the volume of AdS 3 is logarithmically divergent with the IR cutoff R. This is in contrast to the string in the AdS 5 case, where the classical value of the string action proportional to the volume of AdS 2 is finite (after subtraction), and matches the strong-coupling limit of the expectation value of the circular Wilson loop. Here instead the coefficient of log R term may be interpreted as one of the conformal anomaly coefficients in the defect CFT. 9 In Section 3 we compute the 1-loop correction to the logarithm of the partition function of the M2-brane ending on a spherical surface. This gives a correction of order N 0 to the leading result coming from the classical action of the surface, which is of order T 2 = 2 π N . In a choice of normalization that will be explained 5 In general, in AdSp+1/CFTp case we have ∆(∆ − p) = m 2 for scalars and ∆ = m + p/2 for the fermions. In the string (Wilson loop) case p = 1, while here p = 2. 6 A similar term is present, e.g., in the D3-brane probe action in AdS5 × S 5 [18]. 7 One can still mimic such computation by starting with the abelian 6d tensor multiplet theory and consider correlators of the fields with the defect (1.1). In particular, ref. [19] computed the 2-point function of the displacement operator by considering the second order in the "wavy surface" approximation. Its form D i ( x1)D j ( x2) ∼ 1 | x 12 | 6 is dictated by the associated dimension ∆ = 3.
8 Among possible generalizations one may consider a BPS configuration of a M5 brane probe intersecting M5 branes over a line and wrapped on S 3 ⊂ S 4 so that the resulting M5 brane world volume geometry is AdS3 × S 3 (cf. [21][22][23][24][25]). This should correspond to the case when the surface defect is in a large representation of SU (N ). 9 Similar logarithmic UV divergence appears in the log of expectation value of the surface operator (1.1) in the abelian (2,0) theory [8,10]. For the dual M2-probe discussion see also [26].
below, we find for the anomaly coefficient of the spherical surface b = 12N − 9 + O(N −1 ). These first two terms match the prediction b = 3(N − 1)(4 + N −1 ) following from [27,28]. In Section 3.2 we also comment on the holographic description of the non-supersymmetric surface defect operator which does not couple to the scalar fields, following the analogy with the standard Wilson loop case in [29][30][31][32]. In this case the M2-brane surface should be delocalized in S 4 , i.e. the scalars y a should satisfy the Neumann boundary condition. Adding a boundary perturbation to the M2-brane action leads to a 2d RG flow between the UV (non-supersymmetric) and IR (supersymmetric) fixed points with the resulting values of the boundary conformal anomaly coefficients consistent with the b-theorem for 2d defects [33][34][35].
In Section 4 we compute the 4-point correlation functions for the scalar fluctuations y a and x i near the BPS surface, in the leading tree-level approximation. We find the expressions following from the Dirac-Nambu part of the action for the general dimension p of the brane, with the p = 1 case reproducing the string-theory results of [5]. We observe that the yyyy 4-point function satisfies simple superconformal Ward identities that, as turns out, essentially determine its form. We also discuss the Mellin representation for the resulting AdS 3 correlators. In Section 4.3 we perform an OPE analysis of the correlator yyyy extracting the leading 1/N terms in the anomalous dimensions of composite y∂ n y operators appearing in different channels.
In Section 5 we discuss constraints on correlators imposed by a residual superconformal symmetry. In Section 5.1 we follow the analogy with the SYM case [36] and consider a special twisted combination Y = t a ( x) y a ( x) of scalar operator y a whose correlators are constrained by residual supersymmetry. The 4-point correlator of the twisted fields has a very simple form given in Section 5.2 and surprisingly has a very similar structure to that of the strong-coupling limit of the reduced correlator of 1 2 -BPS scalar operators in the N = 4 SYM theory. In Section 5.3 we show that the form of the yyyy correlator found in Section 4.  We are interested in studying the fluctuations of an M2-brane in the AdS 7 × S 4 background which is the near horizon geometry of N M5-branes Here a is the radius of AdS 7 , and ds AdS 7 and ds S 4 are the line elements on unit radius AdS 7 and S 4 . Ω 4 in (2.2) denotes the normalized volume form of S 4 , ℓ p is defined via 2κ 2 11 = (2π) 8 ℓ 9 p and F 4 = dC 3 . We use Euclidean signature throughout.
The (bosonic part of) the κ-symmetric action of an M2-brane probe [14] contains two terms: the standard Dirac-Nambu type term S 1 (the volume of the 3-surface in induced metric) and a WZ-type term S 2 of coupling to the 3-form C 3 S = S 1 + S 2 : 3) where the fundamental M2-brane tension T 2 is (2.5) The world-volume on an M2 brane ending on the plane (or sphere) at the 6-boundary of AdS 7 has the classical solution with AdS 3 induced metric. 10 In the case of the planar surface, this AdS 3 subspace is spanned by the plane and the z coordinate in (2.1). The effective membrane tension (i.e. the analog of the familiar fundamental string tension T 1 = a 2 2πα ′ = √ λ 2π in the AdS 5 × S 5 case) is then given by the product of the fundamental M2-brane tension T 2 and the cube of the AdS radius, i.e. (see, e.g., [38,39]) 11 We proceed now to expand the probe brane action about this classical solution. For generality, let us consider a p-brane in AdS d+1 with world volume ending along a p-plane at the boundary and also stretched along z of AdS d+1 . Following [5] where the case of p = 1 and d = 4 was discussed, let us choose the following "AdS p+1 -adapted" parametrization of AdS d+1 (with radius 1) where the indices of the boundary coordinates of AdS d+1 are split into v = 1, · · · , p and i = 1, · · · , d − p.
The minimal surface ending on a p-plane at the boundary is so that the corresponding induced metric is AdS p+1 . Choosing a static gauge in the p-brane action in AdS d+1 × S n we get for the S 1 part of its action in (2.3) where y a are coordinates of S n and r is its radius in units of the radius of AdS p+1 (which is absorbed into the dimensionless effective tension T p ). Expanding (2.9) in powers of the fluctuations x i and y a we get
(2.13) 10 The existence of such static M2 brane solution is related to the fact that M2 brane intersecting with a stack of M5 branes over a plane is a 1 2 -BPS configuration. This can be easily seen, e.g., from the absence of force on a static M2 brane in this case [37] (C3 here is purely magnetic). 11 In the notation of [38,39] The string in AdS 5 × S 5 case considered in [5] corresponds to p = 1, d = 4, n = 5, r = 1 while in the present M2-brane in AdS 7 × S 4 case we have p = 2, d = 6, n = 4, r = 1 2 . In the latter case we get 4 transverse AdS 7 fluctuation fields x i having m 2 = 3 and 4 massless S 4 fields y a propagating in induced AdS 3 geometry. One may also include the fermionic terms coming from the corresponding AdS 7 × S 4 supermembrane action as discussed in [16] getting (after fixing κ-symmetry gauge) eight 3d fermions with m = 3/2.
To find the explicit expression for the WZ term (2.4) in the M2-brane action let us note that the normalized volume form Ω 4 of a unit-radius S 4 in (2.2) may be expressed in terms of a unit 5-vector Y A as 12 (2.14) Using the expression (2.6) for the effective M2-brane tension the WZ term in (2.4) takes the form Like in the case of a similar WZ term in the D3-brane case [18] a manifestly SO(5) invariant form of the WZ term is non-local-given by an integral over a 4-surface that has the world-volume as its boundary. The normalization of (2.15) is checked by observing that if the 4-surface is S 4 the integral in (2.15) becomes −2πiN , i.e. e −S 2 = 1. 13 Setting where we rescaled y a by r to conform with (2.9), we find that the expansion of (2.15) in powers of y a starts with the y 4 term (as r = 1 2 we have 32πr 4 = 2π) The explicit normalization of the kinetic term for y a in (2.9) is (using (2.6))

One-loop partition function: defect conformal anomaly
In this section we calculate the fluctuation determinants about the AdS 3 classical M2-brane solution. The more complicated problem of deriving the 1-loop quadratic fluctuation for 2 parallel planes was discussed in [16]. The fluctuation spectrum presented in the preceding section indeed matches their spectrum in the limit of large separation. The discussion is parallel to the one in the string case in [3,42]. 12 In general, one has . Ω d appears in the expression for the 13 Being topological this WZ term should not be renormalized and should be derivable as in [18,40,41] from the 1-loop fermionic determinant in the dual 6d theory in the presence of a defect represented by the surface operator.
To recall, our spectrum has 4 bosons with m 2 = 3 plus 4 bosons with m = 0 and 8 fermions with m = 3/2. 14 The resulting partition function Z is then given by To evaluate the determinants we may follow the approach used in the AdS 5 × S 5 string case in [3,43], i.e. use the results for heat kernels of operators in AdS space from [44].
Cubic UV divergence cancels out due to the equal number of bosons and fermions. The linear divergence of log det and thus, as in string case, is proportional to the Euler number (assuming boundary terms are taken into account, cf. footnote 30 in [3]). In any case, such divergences are absent in an analytic regularization like ζ-function one and may be ignored.
There is no bulk logarithmic UV divergence in 3d but the resulting UV finite F contains a logarithmic IR divergence. It can be found using the results in Section 3 of [45] (see also [46]), which studied the determinant of higher spin theory on AdS 3 . For a scalar of mass m (with m 2 = ∆(∆ − 2)), the contribution to F is 15 F For spin 1/2 fermion with ∆ = p/2 + m = 1 + m we get Introducing R as the AdS 3 IR cutoff regularizing the AdS 3 volume (e.g. the radius of the boundary S 2 ) we have (for the unit-radius As a result, we get for (3.1)

Interpretation of the result
Equation (3.5) is the 1-loop correction to the tree level contribution given by the value of the M2-brane action which is just the M2-brane tension (2.6) times the regularized volume of the induced AdS 3 metric (cf. [2]) 14 The bosonic and fermionic operator are essentially universal for the straight line or two parallel line configurations: what changes is just the induced geometry. The same was in the case of a string in AdS5 × S 5 : there one had [3,42]: 2 bosons with m 2 = 2/a 2 ; 1 boson with m 2 = 4/a 2 + R (2) ; 5 massless S 5 bosons; 8 fermions with m = 1/a or squared operator −∇ 2 + 1 4 R (2) + 1/a 2 . In the straight-line or circular line surface the induced geometry was AdS2 so R (2) = −2/a 2 . It is remarkable that the structure of the partition function in the string and M2 cases is very similar. This has to do, in particular, with the universal form of the Nambu-type term in the p-brane action and also the fact that in a natural κ-symmetry gauge the fermionic kinetic term comes from the supergravity covariant derivative projected to the world volume that contains the F -flux term that gets contribution from the sphere magnetic part that is not sensitive to the details of surface in the AdS space. 15 The general formulas for the AdS d+1 spectral density for bosonic totally symmetric rank-s and fermionic in the [s, 1/2, · · · 1/2] representation in general boundary dimension d (see [44]) are presented in [46] (see eqs. (3.20) and (3.22)). For d = 2, they happen to coincide as a function of s. 16 In general, the regularized volume of global AdSp+1 space with S p as its boundary for even p is, discarding power-law divergences (see, e.g., [47] The coefficient of log R in equation (3.6) and in (3.5) has the interpretation of a conformal anomaly coefficient in the defect CFT. Surface operators have three anomaly coefficients, 17 each multiplying a particular conformally invariant integral on the surface related to its topology, extrinsic curvature and background Weyl tensor (see [49] for details). Since our calculation is focused on the single surface geometry of the sphere (the plane has trivial anomaly), our result captures one particular combination of the anomaly coefficients which we denote as b. Thus F can be expressed as where dots stand for possible higher-loop 1/T 2 ∼ 1/N terms.
The leading order at large N of the other two anomaly coefficients were calculated holographically by Graham and Witten [50] by considering M2-branes ending on arbitrary surfaces. They can also be inferred in other ways: the coefficient related to extrinsic curvature is proportional to the normalization of the displacement operator, as mentioned in Section 4 below eq. (4.6). The remaining one was conjectured to also be fixed by the same normalization constant in theories with enough supersymmetry [51] (based on [52,53]) as is indeed verified in [54].
Going back to our expression for the coefficient b in (3.7), we observe that it happens to be consistent with the result for the corresponding anomaly coefficient found in [27,28] from the entanglement entropy for the "bubbling" M5-M2 geometry with M2-branes corresponding to a 1 2 -BPS surface defect operator in (2,0) theory in a su(N ) representation with the Young tableau with a large number of boxes. 18 In the notation of [27] where ρ is the Weyl vector of su(N ) and λ is the highest weight of a particular su(N ) representation. If we formally assume that this expression should be valid not just for large representations but also for the ones with finite number of boxes then in the present case of a single M2-brane corresponding to the surface operator in the fundamental representation one finds (ρ, λ) Remarkably, this is in agreement with (3.7) and suggests that the perturbative expansion in (3.7) may terminate after the 2-loop 1 N term. 19 It would be very interesting to compute this term directly from the 2-loop supermembrane Witten diagrams in AdS 3 . 20

Non-supersymmetric surface defect and 2d RG flow
The above discussion applied to the case of the dual description of the 1 2 -BPS surface operator which (at least in the abelian case) should be represented by an analog of the Wilson-Maldacena exponent (1.1) with 17 A fourth can be defined for nontrivial coupling to the scalar fields, see [8]. By adjusting the scalar coupling, the total anomaly of some BPS surface operators, different from the sphere studied here, vanishes [48]. 18 An exact expression for another anomaly coefficient is derived in [55] from the computation of the associated superconformal index. 19 If one assumes that the series in (3.7) terminates at 1/N order then the coefficient of this term can be of course fixed by requiring that the full expression should vanish for N = 1. 20 For comparison, let us recall the expressions for the conformal anomaly coefficients of the "bulk" theory-the (2, 0) theory describing N coincident M5 branes (see, e.g., [56] for a review): The leading N 3 terms follow [57] from the classical supergravity action, the order N terms originate from the R 4 corrections in 11d action [39] and order N 0 terms are from the 1-loop 11d supergravity corrections [58,59]. The exact expressions follow also from non-perturbative approaches based on supersymmetry constraints [60,61].
one of the 5 scalars of the (2,0) tensor multiplet coupled to the induced metric as in [8,10]. This breaks SO(5) R-symmetry to SO(4) and corresponds, in the M2-theory in AdS 7 × S 4 , to an expansion near a point of S 4 with 4 S 4 massless scalars subject to the Dirichlet boundary condition in AdS 3 .
By analogy with the Wilson loop case [29][30][31][32] we may also consider the dual description of nonsupersymmetric surface operator without scalar coupling [11] preserving SO(5) symmetry. In this case the classical M2-brane minimal surface is the same AdS 3 but one is to impose the Neumann boundary condition on S 4 fluctuations (and average over an expansion point in the sphere) to preserve the SO(5) symmetry.
As in the case of the surface for the circular Wilson loop in AdS 5 × S 5 the 1-loop contributions of the 4 massive AdS 7 scalars and 8 fermions in (3.1) remain the same as in (3.5) while to find the contribution of the 4 massless S 4 fluctuations with alternative b.c. (i.e. with ∆ = ∆ − = 0 + O( 1 N )) we may use, e.g., the relation [47,62,63] between the AdS d+1 bulk field and S d boundary conformal field partition functions: In the present case d = 2, ∆ + = 2, ∆ − = 0, ν = 1 so that the induced boundary CFT 2 has the standard kinetic operator −∇ 2 on S 2 , i.e. the difference of the scalar free energies is where R plays the role of a UV cutoff in 2d. The positivity of the difference is in agreement with the expected "defect b-theorem" [33,34], viewing, by analogy with the circular Wilson loop case [30,31], the non-BPS surface operator as the UV limit deformed by the relevant operator Y 5 ∼ Φ 5 to flow to the BPS surface operator, i.e. b UV − b IR = +1 > 0. As a result, taking into account the multiplicity 4 of S 4 scalars in (3.1), we conclude that in the nonsupersymmetric (non-BPS) defect case we should get instead of the b = b susy in (3.7) 21 b non−susy = b susy + 4 = 12N − 5 + . . . . (3.11) One may attempt to understand the RG flow between the non-supersymmetric and supersymmetric cases by using the same approach as in the string theory description of the circular Wilson loop [30,31]. Starting with the (super)membrane action 22 (3.12) Here √ g 2 = 1 z 2 z→0 and κ is a new coupling which will run between the UV and IR fixed points (see Section 4.2 in [31] for details). κY 5 term should correspond to a similar scalar Φ 5 term in the exponent in the surface operator (cf. (1.1)) with coefficient running between 0 and 1. The variation of (3.12) implies that y a should satisfy the massless wave equation in AdS 3 with the metric ds 2 = 1 z 2 (dz 2 + d xd x) subject to the Robin boundary condition (∂ n = n µ ∂ µ , n µ = (−z, 0, 0)) (3.13) 21 Compared to the non-supersymmetric circular Wilson loop case in [31] here the S 4 zero mode contribution ∼ 4 log N appears only in the finite part of F , i.e. is not relevant for the 1-loop conformal anomaly. 22 One may expect that the standard first-derivative supermembrane action in AdS7 × S 4 is not renormalized (i.e. tension is not renormalized): it contains fermionic and bosonic WZ terms that can not be renormalized, and they are related by κsymmetry to the rest of the terms (this is analogous, e.g., to non-renormalization of 11d supergravity action). Loop corrections may induce higher-derivative terms but presumably they should not be relevant for the discussion below.
The parameter 0 ≤ κ ≤ ∞ thus interpolates between the Neumann (κ = 0) and Dirichlet (κ → ∞) boundary conditions corresponding to y a = z ∆ − v a + O(z 2 ) = v a + O(z 2 ) and y a = z ∆ + u a + O(z 2 ) = z 2 u a + O(z 2 ) respectively. κ will be running with UV scale Λ of the 3d theory. Integrating y a out we get at leading 1-loop order the following boundary divergence (ignoring power divergent terms) where A 3 is the relevant Seeley coefficient (see, e.g., eq. (5.32) in [64] where the Robin parameter S is equal to 4κ here). Adding (3.14) to the bare action (3.12) and taking into account that κ has canonical dimension 2 we conclude that the renormalized κ should run according to (cf. [31]) Here we used (2.6). As a result, we get the UV fixed point at κ = 0 and also a possible IR fixed point at κ = − 1 2 N . Assuming the latter can be trusted in the large N expansion it should represent the Dirichlet limit of the Robin b.c. (3.13). Since the derivative of the β-function gives anomalous dimension at a fixed point the total dimension (2 + β ′ ) of the perturbing operator Y 5 = 1 − 2y a y a + . . . in (3.12) is then Since y a corresponds to ∆ = 2, the value of ∆ + = 4 + . . . is consistent with the leading-order dimension of the composite y a y a operator. To go beyond the leading order one would need to include higher order terms in the 3d action (2.13).

Defect 4-point correlation functions at large N from M2-brane action
Here we follow the same strategy as in the case of the correlators on the BPS Wilson line in [5] and compute the tree-level (large N ) 4-point functions of the bosonic fields X I = (x i , y a ) (representing the displacement operator and the 4 scalars other than the one coupled to the surface operator in the 6d theory) directly from the M2-brane action (2.9)-(2.13). Let us first discuss the normalization of the two-point functions. Given a scalar action in AdS p+1 with the metric ds 2 = 1 z 2 (dz 2 + d xd x) (cf. (2.8)) the bulk-to-boundary propagator will be normalized as in [65] (here x = (z, x)), i.e. 23 and the two-point function of the corresponding boundary operator O( x) will be defined as 3) 23 Note that in [5,32] a different normalization was used: .
In the case of the scalars x i and y a in (2.10) with masses m 2 x = p + 1 and m 2 = 0 corresponding to ∆ x = p + 1 and ∆ y = p respectively we thus find .

(4.4)
In particular, in the case of the string in AdS 5 ×S 5 (T 1 = √ λ 2π ) we find C x = 6C y , C y = √ λ 2π 2 (in agreement with direct identification of x i with displacement operator and y a with "transverse" scalars and strong-coupling limit of exact prediction in [66,67]). In the present case of p = 2, T 2 = 2N π we find 24 Since the x i fluctuations are dual to the displacement operator D i , from the above results we can read off the normalization C D of the D i two-point function on the surface defect in the (2,0) theory The normalization constant C D also determines the anomaly coefficient associated to extrinsic curvature of the surface [68].

The yyyy correlator
Let us first compute the 4-point boundary correlator of the four S 4 fluctuations where we have multiplied each y a ( x i ) with an auxiliary constant 4-vector t a i to remove the SO(4) R-symmetry indices y( x; t) ≡ t a y a ( x) . Here x stands for 2 boundary coordinates of AdS 3 . It is convenient to extract a kinematic factor so that the correlation function can be expressed in terms of the cross ratios where t ij ≡ t i · t j , and (4.10) The holographic computation of the correlator leads to a 1/N expansion G = G disc + G tree + . . . , (4.11) 24 Note that here we start with (2.10), i.e. assume that the scalars y a are normalized in the same way as x i (after the rescaling y a by r = 1 2 , cf. (2.1)). If the corresponding scalar operator on the defect is identified with unrescaled y a we get Cy → r 2 Cy = N 2π 2 while Cx is the same. See also [54].
where the leading order contribution is given by the disconnected diagrams Note that this scales as (C y ) 2 in agreement with (4.4),(4.5). The order N tree-level contributions to the 4-point function can be divided into two parts: G 1 from the Dirac-Nambu type action (2.9),(2.13) which is parity-even, and G 2 from WZ type action (2.15),(2.17) which is parity odd The first contribution can be straightforwardly computed from the action, and after some simplification reads (after using (2.6)) (4.14) HereD ∆ 1 ∆ 2 ∆ 3 ∆ 4 are functions of cross-ratios, and are related to the D-functions defined by The three-derivative contact Witten diagram corresponding to the WZ vertex (2.17) has already been computed in [69], and with the normalization of the WZ term in (2.17) we have Note that the combination (4.13) of the two contributions (4.14) and (4.17) is very special, in that the resulting correlator satisfies differential relations which resemble superconformal Ward identities (see, e.g., [70] for examples in SCFT d with d > 2) Here it is understood that we first act with the differential operators on the correlator, and then set the R-symmetry cross ratios to specific values. One can easily check that the disconnected correlator G disc in (4.12) also satisfies the identities above. We will show in Section 5 below that (4.18) are indeed Ward identities following from the superconformal symmetry [OSp(4 * |2)] 2 . We note that the same differential identities are satisfied by the correlators on the 1 2 -BPS Wilson loop in the string theory case (see Appendix B). The major difference, however, is that in the 1d Wilson loop case there is only a single conformal cross ratio χ, and both α andᾱ can be twisted with respect to χ. Here the parity odd term (4.17) has broken the symmetry of interchanging χ andχ in the correlator: only the simultaneous interchange of χ ↔χ, α ↔ᾱ remains a symmetry of the 4-point function. Therefore, we have two differential identities which separately pair up χ with α andχ withᾱ, and the chirality of these relations parallels the factorized structure of the superconformal group OSp(4 * |2) × OSp(4 * |2). The same structure of the superconformal Ward identities was observed in [69] for P SU (1, 1|2) × P SU (1, 1|2).

The xxxx and xxyy correlators and Mellin representation
The calculation for the xxxx and xxyy correlators is almost identical to that of the the parity-even part of the yyyy correlator as they are not affected by the WZ term at tree level, and only the Dirac-Nambu term (2.9) contributes. We first present the results for generic defect dimension p and then specify to p = 2. For the string (Wilson loop) case of p = 1 the expressions below agree with the results of [5].
Using the action in (2.9)-(2.13) we find for the 4-point function of the transverse AdS d+1 fluctuations where we have already used the relation ∆ x = p + 1, and the D-function identities (summarized in, e.g., Appendix D of [71]) to simplify the expression. For our present case, p = 2, T 2 = 2N π , this correlator may be written explicitly in terms of theD-functions of cross ratios as (cf. (4.14)) (4.20) Similarly, the xxyy correlator for generic p reads where we have used that ∆ x = p + 1 and ∆ y = p. For the p = 2 the correlator may be written in terms of functions of cross ratios as (after using again the D-function identities) For comparison, let us also record the parity-even part of the yyyy correlator with general p and r following from (2.13) (generalizing the p = 2, r = 1 2 expression in (4.9),(4.13),(4.14))  p+1,p+1,p+1 , G 1 in (4.23) is related to G 1 defined in (4.13) as in (4.9), i.e.
Note that the WZ term in (2.15),(2.17) is specific to p = 2 case so that its contribution to the 4-point correlator (4.17) does not admit a generalization to an arbitrary p.
We can develop a better intuition about the above expressions for the correlators using the Mellin representation [72,73]. It is straightforward to translate the D-functions into the Mellin space (see, e.g., Appendix A of [74] for explicit expressions), and we find (s + t + u = r ∆ r ) where the Mellin amplitudes are given by Similarly, the parity-even part of the yyyy correlator (4.23) admits the following Mellin representation As expected, the contact interactions with up to four derivatives (cf. (2.10)-(2.13)) give amplitudes that are quadratic polynomials in the Mellin-Mandelstam variables.
Let us also point out that the parity-odd part (4.17) of the yyyy correlator, which only exists for p = 2 as it derives from the WZ action (2.15), does not admit a Mellin representation. Indeed, the parity-odd contribution (4.17) gains a minus sign under χ ↔χ while the variables U and V (that appear in the standard definition of the Mellin representation) are invariant under this transformation (cf. their definition in (4.10)).

OPE analysis
In this subsection, we perform a preliminary OPE analysis to extract the CFT data for low-lying operators from the 4-point correlator yyyy . Because of the chiral nature of the correlator, we have to use 2d conformal blocks which are not symmetrized with respect to χ andχ This conformal block corresponds to an operator with holomorphic dimension h and anti-holomorphic dimensionh. 25 For the same reason, we should use SU (2) R-symmetry polynomials for SU (2) L ×SU (2) R = SO(4) where m = 0, 1, · · · corresponds to the spin-m 2 representation of SU (2). We can analyze the 4-point function in small χ (orχ) expansion, 26 which is dominated by operators with small h (orh). By going to higher orders in the expansion, we can systematically read off CFT data for operators with increasing conformal twists. However, at higher conformal twists the mixing effect of operators become important, and analyzing the yyyy correlator alone gives only "averages" of anomalous dimensions over the operators appearing in the mixing (weighted by the corresponding OPE coefficients).
We will postpone the unmixing analysis (which involves xxxx and xxyy as well) for a future study and focus on the leading double-trace operators with h = 4 where the mixing is absent. Since we are only interested in the leading operators in the OPE, we do not need superconformal blocks and bosonic conformal blocks are sufficient.
It is useful to decompose the correlator (4.11) into different R-symmetry channels where R = (1, 1), . . . in G R (χ,χ) labels the representation of SU (2) L × SU (2) R that is exchanged. Each G R (χ,χ) can be decomposed into the conformal blocks The (1, 1) channel: From the χ 2 coefficient in the small χ expansion of the disconnected part of the correlator in (4.12) comparing to (4.33) we find that From the χ 2 log(χχ) coefficient of the tree-level correlator (4.13) we can read off γ . We find contributions only fromh = 4, 8, which correspond to spin 0 and spin 2 operators respectively. As a result, The fact that the anomalous dimensions have a finite support on spins is expected because the tree-level correlator is computed as a finite sum of contact diagrams, and each contact diagram has a finite support on spins [75].
The (1, 2) channel: From the χ 2 expansion coefficient in the disconnected part (4.12) we find From the χ 2 log(χχ) coefficient in the tree-level correlator (4.13) we can extract the corresponding anomalous dimension. We find that there is only one operator withh = 6 contributing and its anomalous dimension is The (2, 1) channel: From the small χ expansion of the disconnected correlator we find again However, here there is no χ 2 log(χχ) term in the tree-level correlator. The operator with h = 4 andh = 6 receives only a correction to the OPE coefficient The first logarithmic singularity arises at χ 3 order which corresponds to h = 6. There are two operators responsible for this logarithmic singularity, with the anti-holomorphic dimensionsh = 4, 8.
The (2, 2) channel: Here the zeroth order OPE coefficients read As in the (2, 1) channel here we find no χ 2 log(χχ) term in the tree-level correlator. This is consistent with the fact that the (2, 2) channel receives contribution from the 1 2 -BPS operator with h =h = 4 that has no anomalous dimension. The tree-level correlator leads only to a correction to the OPE coefficient of this operator

Superconformal symmetry of holographic correlators
In Section 4.1 we observed that the yyyy correlator satisfies intricate differential relations (4.18) which can be interpreted as superconformal Ward identities following from the global symmetry [D(2, 1| − 1 2 )] 2 = [OSp(4 * |2)] 2 . We give a derivation of these identities in Section 5.2 by studying the supersymmetry of "twisted" operators defined below. In fact, these superconformal Ward identities are so restrictive to essentially fix the form of the 4-point function. This is shown in Section 5.3, where we "bootstrap" the yyyy tree-level correlator from the Ward identities and crossing symmetry without inputting the precise form of the AdS 3 "bulk" vertices in the M2 brane action, thus shortcutting the computation in Section 4.1.
The inner product is now simply We define the half-twisted operator by setting a = v We can also define a half-twist in the opposite chirallity by settingā =v, or twist both. As y a is a massless (∆ = 2) field whose 2-point function is given by (4.4), eq.(5.3) implies that Y behaves like This operator Y twists the left-moving SU (2) of SO(4) ≃ SU (2) × SU (2) R-symmetry into the left-moving SO(2, 1) of the SO(2, 2) ≃ SO(2, 1) × SO(2, 1) conformal group. To see that, note that the doublets a α are in the fundamental of the left-moving SU (2). We can implement then the R-symmetry transformations of y a in terms of the following differential operators acting on y(v,v; a,ā) The action of the conformal group on a field is implemented, as usual by Note that the dimension δ for the field y is equal to 1 (the total conformal dimension is ∆ = δ +δ = 2). 27 This should not be confused with the chiral algebra twist of [60]. The construction of [60] requires twisting a subalgebra psu(1, 1|2) which does not fit into D(2, 1|− 1 2 ). Moreover, it will be clear that our twisted correlator has both χ andχ dependence, rather than being chiral. (5.8) Hence Y transforms covariantly under the twisted generators Let us now examine the supersymmetry constraints on the twisted operators. The surface defect breaks the OSp(8 * |4) of the N = (2, 0) 6d theory to [OSp(4 * |2)] 2 . The OSp(8 * |4) transformations can be parameterized by the spinor ǫ(x) = ǫ 0 +ǭ 1 x µ γ µ , where ǫ 0 corresponds to the super-Poincaré transformations andǭ 1 to the superconformal ones. They are chiral and antichiral respectively, are in a spinor representation of the Sp(4) R-symmetry group and satisfy the symplectic Majorana conditionǭ = −cΩǫ, where c is the charge conjugation matrix and Ω the symplectic form.
Using γ µ and ρ a for space-time and R-symmetry gamma matrices, the surface in the (x 1 , x 2 ) plane (v = x 1 + ix 2 ) imposes the condition ǫ(x)(1 + iγ 12 ρ 5 ) = 0 . (5.10) Defining γ v = 1 2 (γ 1 − iγ 2 ) and likewise γv, we have iγ 12 = γ v γv − γvγ v . Then the above equation splits into where ǫ v = ǫγ v and ǫv = ǫγv are the generators of the two D(2, 1; − 1 2 ) superalgebras. In addition to the generators in (5.6) and (5.7), each algebra has another SU (2) acting on the x i fields (whose generators we denote by T ) and supercharges Q and S. The algebra (see e.g. [76,77]) is where ε = iσ 2 . The supersymmetry transformations of the y fields are For the twisted field at z = 0 we need to also include the S transformations Let us define a fermionic field which is the action of one of the supersymmetry generators on Y We now examine the Ward identity for the supercharge Q m v for the correlation function of this fermion with any number of Y fields. Since Q m v annihilates Y, we get where in the last step we have used again that Q m v annihilates Y, leaving the anticommutator of Q m v and Q 2n v . Using (5.12), this anticommutator evaluates to (5.17) and its action on Y(v,v;ā) can be expressed as the differential operators (5.6),(5.7) Let us apply this relation to the 4-point function (4.7),(4.9) of the Y(v,v;ā) operators. Examining the definitions of conformal cross-ratios in (4.10), we conclude that in this case α = 1/χ, i.e.
Noting that the differential operator (5.18) annihilates the prefactor in the second line above, we can translate the Ward identity (5.16), (5.18) to act on the function of cross ratios G. Moreover, if we fix v 1 = 0, v 3 = 1 and v 4 → ∞, we find from (4.10) that χ = v 2 , and since the twisting (5.4) fixes a 2 = v 2 , then α = 1/a 2 , and we reproduce precisely the first relation in (4.18).
Note that our construction has only used the left-moving copy of D(2, 1; − 1 2 ). We can repeat the entire analysis by twisting instead the right-moving D(2, 1; − 1 2 ), which leads to the second identity in (4.18). Thanks to the factorization property, our derivation above implies that the superconformal Ward identity 20) applies as well to the 4-point functions of 1 2 -BPS insertions on a line defect in 4d N = 2 theories preserving half of the supersymmetry. This system has the same D(2, 1; − 1 2 ) = OSp(4 * |2) symmetry [78]. Here χ is the conformal cross ratio on a straight line, and α is the cross ratio for the SO(3) R-symmetry.

Twisted 4-point correlator and a curious relation to special 4-point function in N = 4 SYM
We can twist both D(2, 1; − 1 2 ) algebras with a = v andā =v. This gives a dimension-one non-chiral scalar operator Now take the 4-point function of such double-twisted operators Using (4.14),(4.17) we find that the correlator (5.19) has a remarkably simple expression proportional to just one D-function where Φ I (I = 1, · · · , 6) are the 6 real scalars of N = 4 SYM. 28 This operator has protected conformal dimension ∆ = 2, and transforms in the rank-2 symmetric traceless representation of SO(6) R-symmetry. We contracted the indices with a null vector t I satisfying t 2 = 0, which automatically performs the projection to the symmetric traceless representation. Thanks to superconformal symmetry, the 4-point function has a "partially nonrenormalized" structure [80,81] Here G free ( x i ; t i ) is the correlator in the free SYM theory where we assumed the canonical normalization O 2 ( x 1 ; t 1 )O 2 ( x 2 ; t 2 ) = and H( x i ) is the reduced correlator which encodes all the dynamical information. We can compute H( x i ) in 1/c expansion at strong coupling, using the dual bulk description of IIB supergravity on AdS 5 × S 5 The tree-level reduced correlator reads [82] H tree ( Comparing this to the twisted correlator in (5.23) 29 we conclude that they match up to an overall constant 28 We will be brief in the following about the superconformal kinematics of 1 2 -BPS 4-point functions in N = 4 SYM, and refer the interested reader to Section 2 of [79] for a more detailed review. 29 The reader might be concerned that we are comparing results in different spacetime dimensions. However, a nice feature of D-functions defined for AdS d+1 is that the d-dependence only appears in the overall normalization. The functional dependence on x 2 ij is the same for all d. Moreover, for four points we can always use a conformal transformation to restrict them on a two-dimensional plane. and a factor that can be interpreted as a "tetrahedron" contraction of generalized free fields 30 (5.30) The observed relation 31 between the twisted correlator on BPS surface defect in 6d (2, 0) theory, and the BPS correlator in strongly coupled N = 4 SYM theory is quite curious and we hope to shed light on its meaning in the future.

Fixing the yyyy correlator from the superconformal Ward identities
In this subsubsection, we provide an alternative perspective on the tree-level holographic 4-point function yyyy . We show that the holographic correlator (4.13) can be "bootstrapped" by imposing superconformal constraints captured by the the relations (4.18), without using any precise information about the coefficients of the bulk vertices. Similar techniques have already been implemented in a number of maximally supersymmetric AdS backgrounds, and lead to unique answers for tree level 4-point functions in theories with no defects [84][85][86][87].
We start from an ansatz for a local AdS 3 bulk action which consists of all possible contact interactions of y a fields with up to four derivatives. The structure of the corresponding Witten diagrams translates into the ansatz for the following tree-level 4-point function where G 1 is the parity-even part given by a linear combination of contributions of all possible 0-, 2-, and 4-derivative contact diagrams and G 2 is the parity odd part coming from the 3-derivative contact interaction. The precise combination of these Witten diagrams can be fixed as in Section 4.1 using the explicit form of the M2 brane Lagrangian (2.13), but here we will leave them arbitrary and to be determined by symmetries. A convenient parameterization for G 1 is given by the linear combination of all possible D-functions that can show up at this order with the coefficients that are any possible parity-even R-symmetry structures We also require that the parity-even part G 1 (as coming from a local bulk action) should be crossing symmetric. 30 Here we do not want to absorb T into the definition of the reduced correlator Htree, because it is important that Htree has conformal dimension 4 to exhibit the ten-dimensional hidden conformal symmetry [83]. Replacing the argument x 2 ij in Htree( x 2 ij ) with x 2 ij + tij gives a generating function for the reduced correlators of higher Kaluza-Klein modes (see Section 2.3 of [79] for a discussion of this point). 31 It might also be instructive to view the relation from the Mellin perspective. For the twisted correlator (5.23), we have a factorized polynomial Mellin amplitude and it has an interpretation of contact interactions with up to six derivatives. For the supergravity case (5.29), the Mellin amplitude is the inverse of the above expression The only parity-odd 4-vertex allowed by symmetries is the one in (2.17), i.e. ∼ ǫ abcd ǫ µνλ y a ∂ µ y b ∂ ν y c ∂ λ y d and thus G 2 is proportional to the 3-derivative contact Witten diagram in AdS 3 discussed in [69] (cf. (4.2)) It is easy to check by integration by parts that this contribution is antisymmetric with respect to all four points x i . In order for G 2 to be crossing symmetric, the R-symmetry factor has to be anti-symmetric and can only be Using the result of [69], G 2 can be written as (cf. (4.17)) where λ is an undetermined coefficient. Imposing the superconformal Ward identities (4.18), we find that all the coefficients µ k in the ansatz (5.31) can be fixed except for an overall scaling factor. 32 The overall normalization is not fixed by (4.18) because these relations are homogenous. The solution is proportional to G tree we obtained above (4.13) (4.14),(4.17) by the direct computation from the M2-brane action.
Thus the superconformal symmetry is effectively determining the relative coefficients in the underlying bulk action. Note that even though we included the zero-derivative contact interactions in the ansatz for G 1 in (5.32), such contributions are absent in the final solution fixed by superconformal symmetry. This is consistent with the fact that the are no such terms in the M2-brane action for y a in (2.13).
We can also apply a similar bootstrap approach to the case of 1 2 -BPS Wilson loop. The corresponding 4-point function for the dimension 1 operators can be uniquely fixed by the superconformal Ward identity, up to an overall constant which can be determined using supersymmetric localization [5]. We will give the details of this calculation in Appendix B.

AD-functions
For reader's convenience, we collect some useful properties of theD-functions in (4.15),(4.16)) (see, e.g., [71]), which can be used to obtain the explicit form of correlators as functions of cross ratios U and V or χ and χ in (4.10).

B 4-point correlator on 1 2 -BPS Wilson line from superconformal invariance
In this Appendix we implement the techniques of Section 5.3 to determine the tree-level contribution to the the 4-point function of insertions in the 1 2 -BPS Wilson loop from the superconformal Ward identity and crossing. This reproduces the expression derived in [5] from the fundamental string action in AdS 5 × S 5 .
Recall that the Wilson loop has an OSp(4 * |4) superconformal symmetry. The insertions Φ a with a = 1, · · · , 5 have conformal dimension 1, and transforms as a vector under the SO(5) = Sp(4) R-symmetry. Holographically, they corresponds to the S 5 fluctuation y a . As in (4.8), we contract the R-symmetry index of Φ a with a constant auxiliary vector t a Φ(w; t) = t a Φ a (w) , (B.1) so that its correlators depend on the coordinates w i ∈ R 1 parametrizing the straight Wilson line and on the internal "coordinates" t i .
For simplicity, we will fix the normalization of the two-point function so that ⟪Φ(w 1 ; t 1 )Φ(w 2 ; t 2 )⟫ = t 12 |w 12 | 2 , (B.2) where t ij = t i · t j , w ij = w i − w j . The 4-point function can be written in terms of a function F of cross ratios as A = ⟪Φ(w 1 ; t 1 )Φ(w 2 ; t 2 )Φ(w 3 ; t 3 )Φ(w 4 ; t 4 )⟫ = t 12 t 34 |w 12 | 2 |w 34 | 2 F(χ; α,ᾱ) , Note that unlike the SO(4) case of the M2-brane theory in the main text, here the R-symmetry cross ratios α,ᾱ should appear symmetrically in F(χ; α,ᾱ) (we cannot have det(t ij ) ∝ t 12 t 34 (α −ᾱ)). In other words, F(χ; α,ᾱ) is a linear function of σ and τ . The 4-point function also needs to satisfy the superconformal Ward identities [20] analogous to (4.18) The ansatz for A, the analog of (5.32), now has only coefficients that are linear in σ and τ A = t 12 t 34 i (µ 1,i + µ 2,i σ + µ 3,i τ ) W i . (B.8) Note that the dimension d = p dependence of the D-functions only comes as an overall factor and does not affect the dependence on the cross ratios (as is clear, for example, from the Mellin representation). Therefore, we can compute them for generic d, and set χ =χ, d = 1 in the end.