More Holographic M5 branes in AdS 7 × S 4

We study classical M5 brane solutions in the probe limit in the AdS 7 × S 4 spacetime geometry with worldvolume 3-form ﬂux. These solutions describe the holography of codimension-4 defects in the 6d boundary dual N = (0 , 2) supersymmetric gauge theories. Starting from some half-BPS solutions which follow stricter BPS conditions, we ﬁnd a general 116 -BPS solution. We show how some of the giant-like M5 brane solutions here may be related to the codimension-2 surface operators in the 4d gauge theory.


Introduction
The study of branes as probes in the anti-de Sitter spacetime provides valuable information about the non-local observables such as surface operators, Wilson lines, and other line operators in the boundary supersymmetric gauge theory [1][2][3], [4,5].In the M-theory framework, using the holographic AdS 7 /CF T 6 correspondence, M5 branes have been used as probes to know more about the non-local Wilson surface operators [6,7] which are codimension-4 in the 6d boundary gauge theory, and as well as the codimension-2 defects in the theory [8][9][10].Codimension-4 Wilson surface operators have also been analyzed due to the holography of probe M2 branes [11,12].The codimension-2 defects upon the dimensional reduction of the theory on a Torus T 2 become Gukov-Witten surface defects in the 4d gauge theory [13,14], [8,[15][16][17][18][19]. Whereas, in the literature, it has been discussed that the codimension-4 defects upon a suitable dimensional reduction may become Wilson lines in the large rank representations of the 5d SYM gauge theory [20], or they may become the codimension-2 surface operators in the 4d gauge theory [8,16].In [8] authors show how surface operators in the 4d gauge theory obtained from the codimension-2 and codimension-4 non-local operators are equivalent objects by showing the equivalence of the twisted-chiral superpotential functions that describe them in the IR regime.
In [9], we analyzed those M5 brane embedding solutions that are holographic duals of the codimension-2 surface defects and they end in the boundary of the global AdS 7 in an R × S 3 submanifold.We found the most general BPS M5 brane solutions that preserved just one spacetime supersymmetry.Out of these general 1  32 -BPS solutions, there were some solutions with at least 2 preserved supercharges that were shown to hit the AdS 7 boundary in 1 + 3 submanifold.For these 1  16 -BPS solutions, we also recovered the singularity profile of the complex scalar in the (2, 0) tensor multiplet theory when we took the large radius limit for the AdS 7 boundary.
In this article, the point of discussion is towards the codimension-4 surface operators in the 6d gauge theory.The holographic dual M5 branes hit the boundary of AdS 7 in a 1 + 1 dimensional submanifold.The M5 brane solutions that we analyze here in the probe limit can have self-dual 3-form flux turned on their world volume.Half-BPS solutions have the world volume of AdS 3 × S 3 , where the flux strength is proportional to the S 3 volume.There are two kinds of solutions possible here: the first kinds are the ones for which the S 3 part of the worldvolume is inscribed in the S 4 part of the background global AdS 7 × S 4 spacetime geometry; the second kinds are the ones for which all of the AdS 3 × S 3 worldvolume coordinates are identified with the AdS 7 coordinates.We will often refer to the first ones as the "giant-like" solutions and the second ones as the "dual-giant-like".
The "giant-like" solutions that we find here are similar to the ones discussed in [6,7].And like the "giant-like" solutions known from earlier, we are finding that the 3-form flux can be turned off to zero value for a certain instance, and the M5 world volume is still a stable supersymmetric solution.Another important result here is that we show these "giantlike" solutions are part of the most general 1 32 -BPS embedding conditions found in [9] when we analyzed the M5 brane solutions, which were the duals of codimension-2 defects in the boundary theory.We will give more comments on the relation with our previous results in subsection 3.1 and in the conclusion section 5.The "dual-giant-like" solutions that we find are also similar to the solutions that authors in [7] have found, and over there, the presence of the 3-form flux is indeed necessary to have a supersymmetric world volume.In section 2, we begin by describing the coordinate frame we choose for the AdS 7 × S 4 metric to find the desired solutions.In section 3, we give the details about the "giant-like" M5 solutions that we find and then show the possibility of combining them in ways so that their intersections preserve the common set of supercharges.The general "giant-like" M5 solutions preserve just 2 common supercharges which we derive in subsection 3.1 and App B and present in the equation (3.25).In section 4, we give the details of the "dual-giant-like" solutions that we find.In section 5, we end with an analysis of our results with some extra details and future outlooks from this work.

Probe M5s in AdS 7 × S 4 geometry
The metric that we consider for the eleven-dimensional AdS 7 × S 4 geometry is in the global coordinates system 2) The background 3-form potential field is given by

Half-BPS M5 branes
We now spell out the type of half-BPS M5 brane embeddings that we are going to analyze.Such solutions will have the worldvolume of AdS 3 × S 3 metric.To find the desired supersymmetric M5 solutions, we follow the method employed in [7] which followed the classical equations obtained in the work [21] where the worldvolume superspace was considered to be embedded in the 11-dimensional target superspace.These field equations have been shown to be the same as Euler-Lagrange equations of motion [22], from the covariant action of Pasti-Sorokin-Tonin (PST) [23,24], which is a Dirac-Born-Infeld like covariant action.Here we only present the analysis of the κ-symmetry constraint equation but all the half-BPS solutions to be discussed solve all the remaining classical equations.The S 3 part of the induced metric has been understood to support the 3-form flux of the worldvolume.We will consider two kinds of solutions: i) giant-like: the ones which have the S 3 sphere expressed in terms of the coordinates that parametrize S 4 of the 11d spacetime.
ii) dual-giant-like: the second kinds are the ones where all the M5 coordinates comes from ⊂ AdS 7 coordinates of 11d.
The three-form flux here is self-dual and follows the constraint h = ⋆ g h1 .Hence we can derive a form for the field strength desired for our analysis throughout this note Worldvolume self-duality constraint allows us to write this in terms of the explicit expression where Ω 3 is the volume of the S 3 and the notation g is for the induced worldvolume metric.a is some common factor that turns out to be a constant in both the kinds of solutions that we discuss here.
We will begin by enumerating all the possible solutions that we found in the first category.
3 Solutions of the kind I : The giant-like M5 solutions which appeared in the literature [6,7,20] were understood to be related to the Wilson loop operators in the lower 5-and 4-dimensional gauge theories charged in the large rank anti-symmetric representations of the SU (N ) gauge group.The holographic duals of such Wilson loops in the 4d gauge theories were D5 branes, which were analyzed in detail in [25,26].The world volume metric of those D5 branes were of AdS 2 × S 4 type which hit the AdS boundary on the location of these Wilson lines.The authors in [6,7] discuss an upper bound on the value of fluxes due to 3-form field strength that can be allowed for probe M5 branes for such kind of embedding solutions in AdS 7 × S 4 spacetime geometry.In here, we have also found such a bound in the value of constant a in the 3-form field strength in (2.5), which will reflect that there is also an upper bound on the flux value on the world volume.
The first two solutions that we found had the metric in the static gauge with the following identification of the coordinates: The induced worldvolume metric is of this form The S 4 coordinate θ is a constant θ 0 that can take values between 0 and π and also controls the size of S 3 wrapped by the worldvolume.Here the κ symmetry constraint equation [21,22] is of this form γ τ 12345 is the product of six worldvolume Γ matrices and the indices m, n, p, q, r, s are the coordinate indices.ε is the Levi-Civita 6-tensor.Here matrices Γ here are the 11-dimensional matrices with the tangent space indices.
Discussion: In the equation (3.9) on the RHS, obtained after commuting through the matrix factor M , the parenthesis terms in the first two lines are contributions from the sixproduct γ τ 12345 , and the terms in the 3rd and 4th lines are from contributions due to the flux terms in the κ symmetry equation.Here the value of the 3-form field depends on the value of θ 0 where the S 4 coordinate θ is fixed.By choice from (3.7), h depends on the volume of S 3 , which varies if the fixed value of θ coordinate θ 0 is changed.And due to the condition (3.11), a is also θ 0 dependent.But when S 3 vanishes at θ equal to 0, and π (where the world volume is not AdS 3 × S 3 anymore), these are no longer M5 brane solutions.Whereas at θ equal to π 2 , although the size of the S 3 wrapped by the world volume is maximal, a becomes zero here.So in the limit when value of θ → π 2 , even with flux field h → 0, these are still stable supersymmetric AdS 3 × S 3 probe M5 solutions.
There are four more half-BPS solutions that contain such 3-form flux field.Identifications for the M5 worldvolume coordinates of these are different from (3.1).Γ κ can also be computed for all these four solutions separately and an equation similar to (3.9) can be found.The value of the constant a for all these are the same as in (3.11).Below we merely list these solutions with their embedding conditions and along with the respective projection conditions.We find that the three of them can be nicely written in terms of the complex AdS 7 coordinates defined in (A.1) In terms of the real coordinates the embedding conditions are: α = 0 and θ = θ 0 .When α = 0 the angular directions parametrized by the coordinates β, ϕ 2 and ϕ 3 shrink to zero sizes.The identification of the worldvolume coordinate in the static gauge is This solution is supersymmetric with κ-symmetry constraint true when In terms of the real coordinates the embedding conditions are: α = π 2 , β = 0 and, θ = θ 0 .When α = π 2 and β = 0 the angular directions parametrized by coordinates ϕ 1 and ϕ 3 shrink to zero size.The identification of the worldvolume coordinates is The solution is supersymmetric with κ-symmetry constraint true when Γ 025 ϵ 0 = ϵ 0 . (3.14) In terms of the real coordinates the embedding conditions are: α = π 2 , β = π 2 and, θ = θ 0 .When α = π 2 and β = π 2 the angular directions parametrized by coordinates ϕ 1 and ϕ 2 shrink to zero size.The identification of the worldvolume coordinates is The solution is supersymmetric when (3.15) 6 : For the last solution, the real embedding conditions are: α = π 2 , θ = θ 0 , ϕ 2 = 0 and ϕ 3 = 0.And the ϕ 1 direction shrinks to zero size.With the following worldvolume identification (τ, This solution is supersymmetric when

General giant-like solutions in kind I
It is also essential to identify the sets among these solutions that have common Supersymmetries th common supersymmetries: 1. Solution with projection (3.10) and the solution with projection (3.15).
Considering the above sets of projections this will give us the individual 1 4 th -BPS embedding solutions separately.
• 1 8 th common supersymmetries: There is only one combination that is possible with projections of (3.13), (3.14) and (3.15) which we also rewrite as follows along with the 1 8 -BPS projections in (3.17) if we also consider 11-Gamma product Γ 01234...10 = 1 then the following is also true In the remainder of the section, we aim to find a general expression solution that will also include the three half-BPS solutions that preserve the common supersymmetry due to (3.17).To find such an expression we further break the supersymmetry by introducing an extra projector The 1 16 -BPS projections above will also preserve the single supercharge that was determined by the 1 32 projections we wrote in [9] to calculate the most general 1 32 -BPS solution.θ equal to π 2 was another embedding condition for that solution.We rewrite the most general solution of [9] for the convenience below with the scaling condition on each F (I) : Since the 1  16 -BPS solution, for the giant-like M5 brane that we are after, will also preserve the single supersymmetry of (3.20).This will mean that it will be contained(as a subset conditions) within the (3.20) expression.But because it preserves at least one more supersymmetry than (3.20) the solution expression will be of a more constrained form.The form of our 1  16 -BPS solution here will be different from the ones in equation (4.1) of [9] where one of the Z coordinates of S 3 ⊂ S 4 was put to zero.We rewrite this particular equation as well Z 2 = 0 and F (ζΦ 0 , ζΦ 1 , ζΦ 2 , ζΦ 3 ) = 0 (3.22)where 2 .This equation is of further constrained form from (3.20) that goes on to describe the holographic duals of codimension-2 defects in the boundary 6d SCFT.This above equation of [9], now labeled (3.22), describes another 1  16 -BPS sector of M5 brane solutions different from our current 1  16 -BPS sector here due to (3.19).M5 worldvolume of (3.22) wraps an arbitrary 1d curve on the S 3 ⊂ S 4 and the remaining 1 + 4 directions of the 6d worldvolume lie completely in the AdS 7 part.While in the present case, the 3-dimensional space due to sphere S 3 ⊂ S 4 wrapped by a giant-like M5 solutions is always going to be the same, and hence the condition Z 2 = 0 will not enter here.
It is also easy to illustrate how the individual half-BPS solutions corresponding to (3.13), (3.14) and (3.15) will fit into the general form of (3.20).For example, the half-BPS solution due to (3.13) can also be expressed as Hence we can take the holomorphic functions in (3.20) to be Now, since the worldvolume wrap all of S 3 ⊂ S 4 , coordinates Z 1 , Z 2 will not appear as arguments.Therefore, it is natural to conclude that the answer for general 1  16 -BPS M5 solution of this "giant-like" kind can be expressed in terms of two arbitrary holomorphic conditions: along with the scaling condition with the coordinate θ fixed to any arbitrary value θ 0 between 0 and π.However, the most general solution (3.20) from [9] did not have the 3-form field strength of the M5 world volume turned on.
For completion, we give explicit proof for the general expression (3.25) in appendix B. It does not involve any new detail(method-wise) that we have not discussed so far here and in [9].

Boundary limit description
We can look at this general solution in the limit where the AdS radius is large.The complex coordinates Φ i ∈ C 1,3 chosen for the AdS 7 embedding take the form where r = 2l sinh ρ and ν 0 = e iϕ 0 , ν 1 = cos αe iϕ 1 , ν 2 = sin α cos βe iϕ 2 , ν 3 = sin α sin βe iϕ 3 .In this large r approximation(where the boundary theory lies), the induced metric is in the conformal class of R × S 5 (3.28) Now we need to find the locus of zeros of the functions in (3.25) as we near the boundary.It is important to note that the holomorphic functions F (I) given in (3.25) with a differential scaling condition given in (3.26) can be repackaged in the following way where H (I) are two arbitrary holomorphic functions.Near the boundary the functions become H (I) ν 1 ν 0 , ν 2 ν 0 , ν 3 ν 0 .So the worldvolume of the M5 brane intersects the boundary(at radial coordinate r = r c ) at the common intersection of zeros of the functions Next, we show how to connect this general solution with a class of solutions that appeared for Wilson surface operators in [28].It can be seen that if we shrink the size of the S 3 wrapped by the worldvolume(by moving to the θ position θ = 0) the resulting M2 branes are valid solutions preserving the same supersymmetry before the general M5 brane becomes degenerate, in this class of solutions.The M2 brane embedding will be described by the same conditions in (3.25) and will not wrap any directions contained in the S 4 part of the 11d background geometry.
First we do the Wick rotation τ E = −i τ , the functional conditions of (3.30) take the form Then we do the conformal coordinate transformation: (z 1 , z 2 , z 3 ) = e τ E cos α e iϕ 1 , sin α cos β e iϕ 2 , sin α sin β e iϕ 3 The metric transforms as follows It can now be seen that the functional conditions in (3.31) describe an arbitrary holomorphic curve in the C 3 space Therefore, we have shown that after we make our general 1 16 BPS M5 brane solution degenerate to M2 branes of the same supersymmetry(after shrinking the S 3 size to 0), in the large AdS radius boundary limit, our derived functional conditions in (3.31) describe an arbitrary holomorphic curve in the C 3 space, and which give the 'type-C' Wilson surfaces obtained in the reference [28].
The direct one-to-one map to the solutions of [28] also makes sense with the understanding that the 'scalar couplings' n I , which are defined in [28], were constant vectors along the whole of Wilson surfaces only for the 'type-C' class surfaces.The n I vectors were unit vectors that took values in R 5 , the same space in which the 5 scalar fields of the boundary theory take their values.For our general solutions from (3.25), the associated n I vectors for the boundary scalars are going to be constants by construction, since the holographic dual M5 solutions always wrap all of the S 3 directions inside S 4 (which are the orthogonal directions to the AdS 7 boundary) and are located at fixed θ coordinate values.And in the degenerate limit to get M2 branes θ = 0, where S 3 size is zero.Some comments: It is also worthwhile to note that because S 4 coordinates: Z 1 and Z 2 are not the arguments of these holomorphic conditions(as they are all identified with the worldvolume coordinates that describe the S 3 part wrapped by the general M5 solution), they do not appear as arguments in the general embedding conditions we have presented in (3.25).Due to this, we cannot do the analysis that we did for the profile of the scalar fields singularities associated with the codimension-2 defects in the later part of section 4 in [9], where the coordinate Z 1 was still in the argument (refer (3.22)).And we had taken the boundary limit for the AdS radial direction and then found the singularity profile of the complex scalar in the boundary (2, 0) tensor multiplet theory.
Therefore, for these giant-like M5 branes, it appears that unlike the previous solutions in [9], taking the large radius limit will not reflect any singular behaviour for the boundary gauge theory scalar fields(when the codimension-4 defects are present).This is very much consistent with what is known about surface operators and also discussed in [8,16] where, upon dimensional reduction, the codimension-4 defects were associated with the coupled 2d/4d description of surface operators in the 4-dimensional gauge theory.

Solutions of the kind II: S3 ⊂ AdS 7
The dual-giant-like solutions that have been known are also understood to be related to those Wilson operators which are charged in the large rank symmetric representations of the SU (N ) gauge group of the 4-and 5-dimensional gauge theories.The holographic dual probe D3 branes in the AdS 5 × S 5 spacetime were analyzed in [3,26] where world volume metric is of AdS 2 × S 2 type and hit the AdS boundary on the location of such Wilson lines.For the dual-giant-like kind that we present, there are three half-BPS solutions preserving supersymmetries in the different sectors.Each has a respective projection condition on the constant spinor ϵ 0 required to satisfy the κ-symmetry constraint.They all have the following identification for the worldvolume coordinates The three M5 brane solutions that we discuss here directly couple to the dual background 6-form potential field The embedding condition: sinh ρ sin α = 1 is also the same for all three.The induced worldvolume metric of AdS 3 × S 3 topology has the following expression What is different are the following embedding conditions • Solution 1: the point chosen is θ = π 2 , χ = 0 and ξ 1 = 0 where the circle parametrized by ξ 2 shrinks.
• Solution 3: the point θ = 0 is chosen where the size of S 3 ⊂ S 4 shrinks to zero.
We demonstrate our calculation a bit for Solution 1 here and later state the final answer for all of them.For convenience we write the Γ κ matrix in its general form again: For the chosen solution the matrix and the flux-dependent part is equal to Here the non-zero components of the self-dual field strength are After commuting all the six-product Γ matrices in (4.5) and all the 3-product Γ matrices in (4.6) through the matrix factor M in the killing spinor ϵ (A.5) we find that the κ-symmetry constraint is satisfied.
The projection required on the constant spinor ϵ 0 is the following Along with this, we find that the value of the constant factor a in the h needs to be Similarly, the answers for the remaining two solutions are Here the contributions from both γ τ 12345 and the flux-dependent part in the Γ κ matrix (4.4) are needed in the κ-symmetry equation for the solutions to be supersymmetric.Therefore, we find that for these half-BPS solutions the value of 3-form flux field h cannot vanish in order to have a supersymmetric worldvolume of AdS 3 × S 3 .The same conclusion was also found for the "dual-giant-like" solutions in [7].
This also means that it will not be possible for these solutions to be combined with those that we saw in the previous section.Hence they will not belong to the same BPS sector and they will not belong to the embedding conditions in (3.20) as done by those giant-like kinds and also by the solutions in [9].

Discussion of the results and conclusion
In this article, we presented some probe M5 branes in the AdS 7 ×S 4 spacetime with half-BPS solutions having AdS 3 × S 3 world volume.These solutions have the world volume 3-form flux field turned on.Two kinds of solutions appeared in our discussion i) giant-like solutions and ii) dual-giant-like solutions.Both the kinds of M5 solutions will hit the boundary region of AdS 7 at the locations of 1 + 1 dimensional Wilson surface operators in the dual gauge theory.From supersymmetry analysis, we found these solutions to be consistent with the ones discussed earlier in the literature [6,7,20].For the giant-like solutions, we see that the worldvolume 3-form flux field value becomes maximum when the M5 brane is located at a fixed point of θ 0 , which is somewhere between 0 and π 2 .An increment of θ 0 from this 'maximum point' decreases the value of the flux field before it vanish at θ 0 = π 2 .
After this, we found that 3 of the half-BPS giant-like solutions can be combined such that a general solution can be constructed that preserves the common four supersymmetries of these.On further breaking the supersymmetry by an extra 1 2 factor, we found that the common supersymmetries due to the 4 independent projections in equation (3.19) were among the 1 32 -BPS projections we found in [9].In equation (3.25), we also determine the expression for a more general 1  16 -BPS solution that can have the 3-form flux field turned on.By AdS/CFT holographic principle, this general solution will describe the probe M5 branes that are dual to those 1  16 -BPS codimension-4 surface defects in the 6d boundary gauge theory which would be charged in the higher rank representation of the gauge symmetry.Towards the end of section 3, we further show how the general solution in (3.25) is mapped to the 'type-C' Wilson surfaces of [28] (refer to [29] for surfaces of other general shapes and classes).In [9], we also found a solution for the M5 duals of 1  16 -BPS codimension-2 defects in the gauge theory, which we write in (3.22).In this concluding section, we would like to end with some comments on the relation between these two different 1  16 -BPS solutions.In the AdS 7 × S 4 background spacetime, the brane construction of these can be understood from the table below 0 1 2 3 4 5 6 7 8 9 10 The construction of the probe M5 duals of codimension-2 and codimension-4 defects in the background AdS 7 × S 4 spacetime geometry can be understood as above.
In [8], authors discussed this relationship between these two kinds of probe M5 branes in terms of deformation, such that M5 codim-2 brane that intersects with the boundary region of AdS 7 in four directions, can be deformed to an M5 codim-4 brane which has just two common directions with the AdS boundary (also see fig. 1).For us this kind of deformation doesn't change the number of supersymmetry preserved, but the deformed version, the M5 codim-4 , now belong to a different 1  16 -BPS sector of the solutions.However, for the construction of Frenkel, Gukov and Teschner in [8], the BPS sector also remained the same after such worldvolume deformation.For our case, the equation below captures this deformation process in a precise way The authors in [8] also discussed Hanany-Witten-like phenomenon [30] when such deformation of M5 brane was considered.They show the emergence of M2 branes when the deformed M5 codim-4 brane is moved in the direction that is orthogonal to all the three kinds of M5s shown in table 1.But here in the current scenario, such M2 branes, if present, should be stretched along the radial AdS 7 direction and also intersect the boundary at the same 1 + 1 dimensional location where M5 codim-4 brane is hitting.But the presence of these M2 branes is obscured at the moment.Perhaps, a close analysis of the movement of M5 codim-4 brane along the θ ⊂ S 4 coordinate will help us in recovering them.We will look forward to addressing this shortly.Another direction that one can pursue is to also turn on the flux field h on the worldvolume of M5 codim-2 branes from reference [9].Our analysis for M5 codim-4 branes in this article suggests that changing the position of M5 codim-2 from θ = π 2 to an arbitrary θ 0 may allow for such a possibility.Doing this may also shine more light on the intricacies in the relation between the two descriptions of surface defects(which are of codimension-2 and codimension-4) in the six-dimensional boundary gauge theories.
For the S 3 ⊂ S 4 we define the complex cooordinates describing it embedded in C 2 space We choose the same frame vielbein that we had taken in [9] where the AdS 7 part can be written as a U (1) Hopf fibration over a Kähler manifold CP

3
(also refer to [27]).Here CP 3 is the hyperbolic version of the complex projective space and it is defined as the set of rays in the complex space C 1,3 ( in place of the C 4 space).Similarly for the S 3 ⊂ S 4 part, the frame vielbein are chosen so that U (1) Hopf fibration over a Kähler manifold CP 1 becomes manifest.The frame vielbein that we use are the following e 0 = 2l cosh 2 ρ dϕ 0 − sinh 2 ρ cos 2 α dϕ 1 + sin 2 α cos 2 β dϕ 2 + sin 2 α sin 2 β dϕ 3 The solution for the Killing spinor in the above coordinate is given by : ϵ = e where γ denotes the four-product Γ 78910 .
B Proof of the general giant-like solution in (3.25)In this appendix, we give the proof of our result in (3.25).The M5 brane solution that we seek has a worldvolume 3-form field strength turned on.The value of the field strength will remain proportional to the size of the S 3 sphere wrapped by the world volume (3.7).The κ-symmetry equation is given by The world volume coordinates σ 3 , σ 4 , σ 5 are always going to be χ, ξ 1 , and ξ 2 , respectively, which are on S 3 ⊂ S 4 of AdS 7 × S 4 .Therefore, the factors h χξ 1 ξ 2 and γ 345 are fixed constants here, proportional to the volume of the S 3 .And γ 345 is simply equal to Ω 3 Γ 8910 .So Γ κ can also be written as where a is the same parameter we saw in section 3 which is equal to 1−sin θ cos θ .The worldvolume coordinates τ , σ 1 and σ 2 are chosen from the AdS (here we have introduced the notation e a i for pull-backs of the 11d frame vielbein as e a i ≡ e a µ ∂ i X µ ).
In the next step of this proof, we will show what the factor e a τ e b 1 e c 2 Γ abc yields in the l.h.s. of the κ-symmetry equation.For the four independent projection conditions from (3.19) the killing spinor reduces to this simplified form We put the coefficients of every Γ factor to zero in the above equation.And with the help of the following complex 1-form definitions we write down every constraint in the parenthesis brackets in (B.8) in these compact forms For the 1  16 BPS embedding solution that we are after, has the S 4 polar coordinate θ fixed to an arbitrary value.And the coordinates of S 3 wrapped by the world volume are identified with χ, ξ 1 , ξ 2 of S 4 .So we just need 4 real conditions to describe the embedding of this general M5 brane solution.Therefore, we consider two arbitrary complex functional conditions where P denotes pullback onto the world volume.We rewrite this in terms of the complex oneforms defined in (B.9) using the frame vielbeins in (A.3) and (A.4) (given in the appendix) As the next step, we solve for e 0 from above (B.13)and substitute in the complex conjugate of one of the equations in (B.10) which is On account of the remaining 3-form constraints which are  On solving these in (B.15), we find that the functional conditions F (I) have holomorphic dependence on AdS 7 complex coordinates (defined in (A.1)) for I equal to 1 and 2.
In addition to (B.15) the functional constraints F I also satisfy the scaling condition This can be seen when we pick a 1-form E a and solve for it from (B.13) in terms of other 1-forms and substitute in the 3-form constraint For a = 1−sin θ cos θ this is equal to ϵ and we confirm the supersymmetry of the solution in (3.25).

Figure 1 :
Figure 1: This figure depicts the probe M5 brane dual to codimension-2 defects in the background spacetime geometry of AdS 7 × S 4 getting deformed to become the M5 which is dual to codimension-4 defects.

10 )
These are 3 + 3 + 1 = 7 equations and here ω is a real 2-form defined as ω ≡ e 14 + e 25 + e 36 .This exercise with the BPS constraints (B.10) suggests to us that the volume form e a τ e b 1 e c 2 ε abc in the 3-directions τ , σ 1 , σ 2 (on the M5 world volume) is equal to e 014 + e 025 + e 036 .
14)the l.h.s. e 0 ∧ E 1 ∧ E a vanishes, if we also consider the following differential equations which are the coefficients of E