Surface defects in holographic 5d SCFTs

We use holography to study codimension-2 surface defects in 5d SCFTs engineered by (p, q) 5-brane webs. The three-dimensional defects are realized by D3-branes ending on the brane web. We identify the holographic representation of the defects in Type IIB AdS6 solutions as probe D3-branes, and study conformal and non-conformal defects which, respectively, preserve one half and one quarter of the supersymmetry. For a sample of 5d SCFTs, including the TN theories, we provide explicit solutions for conformal and non-conformal defects. For the conformal defects we obtain their contribution to the free energy on S5.


Introduction
Defects are important objects quantum field theory (QFT). They are sensitive to global aspects not captured by local operators, and can serve as non-local order parameters distinguishing different phases. A celebrated example is the area law for Wilson loops in confining gauge theories in four dimensions. Defects also lead to interesting mathematical structures such as generalized q-form symmetries [1] and higher group global symmetries [2].
In this paper we study codimension-2 surface defects in five-dimensional superconformal field theories (SCFTs). We will focus on 5d SCFTs realized by (p, q) 5-brane webs [3,4] and study codimension-2 defects, which generalize surface defects in 4d QFTs. Many 5d SCFTs can be understood as strongly-coupled UV fixed points of gauge theories [5][6][7], but the space of theories that can be realized by 5-brane webs and other string theory constructions is much broader. Aspects of higher symmetries in 5d were discussed recently in [8], and a classification program for defects in generic dimensions was initiated in [9]. Here we will use the holographic duals for 5d SCFTs engineered by (p, q) 5-brane webs, constructed in [10][11][12][13], as tool to study defects realized by D3-branes ending on the 5brane webs [14][15][16][17][18][19][20]. We will study conformal and non-conformal defects, and obtain the defect contribution to the SCFT free energy on S 5 for conformal defects. We hope this will be a starting point for fruitful interplay between AdS/CFT and field theory methods like supersymmetric localization in the study of these surface defects.
From a holographic perspective, defects with a small number of degrees of freedom compared to the degrees of freedom of the ambient SCFT are particularly accessible. They

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can be realized through probe branes [21], whose backreaction can be neglected at leading order in the planar limit. 1 A distinguished class of defects in conformal field theories are conformal defects, which preserve a conformal sub-algebra, SO(p + 1, 2) for a p + 1 dimensional defect, of the full conformal algebra of the ambient CFT. In superconformal field theories this may be enhanced to a superconformal sub-algebra. For conformal defects the dual probe branes wrap an AdS p subspace in the background geometry. The same probe brane embedding can then be used to describe planar defects in flat space and spherical defects in the ambient CFT on a sphere, depending on the coordinates chosen for the AdS factor in the background geometry. More general defects can be obtained by using operators localized on the defect to trigger "defect RG flows", leading to a breaking of the defect conformal symmetry to the isometries preserved by the defect.
We will identify D3-brane embeddings describing superconformal defects in 5d SCFTs engineered by (p, q) 5-brane webs. The D3-branes wrap AdS 4 in the AdS 6 part of the holographic duals and are localized at a distinguished point in the internal space. We will obtain the contribution of the defects to the SCFT free energy on S 5 from the brane on-shell action. We will also study more general non-conformal defects, realizing RG flows triggered by defect-localized operators for planar surface defects. The D3-branes still wrap an AdS 4 in the background geometry, but the position in the internal space now changes along the radial coordinate, thus breaking the AdS 4 isometries corresponding to defect conformal transformations.
The paper is organized as follows: in section 2 we review the construction of 5d SCFTs with surface defects using 5-brane webs with D3-branes. In section 3 we identify the holographic representation of the surface defects and find explicit solutions for supersymmetric conformal and non-conformal probe D3-brane embeddings. We also discuss the defect contribution to the free energy. We conclude in section 4. The probe brane BPS conditions are derived in appendix A.

D3-brane surface defects in 5-brane webs
We briefly review aspects of 5-brane webs [4] that will be relevant and discuss surface defects realized by D3-branes, to guide the holographic discussion of surface defects in the next section.
Supersymmetric configurations of (p, q) 5-branes, where we denote D5-branes by (1, 0) and NS5-branes by (0, 1), can be realized if all branes extend along the (01234) directions and are at an angle in the (56) plane such that p∆x 5 = q∆x 6 (with the axion-dilaton scalar τ = i). General planar junctions of (p, q) 5-branes at a point in the (56)-plane then define 5d SCFTs with 8 Poincaré and 8 superconformal supersymmetries. The SU(2) R-symmetry is realized by rotations in the remaining (789) directions.
The Coulomb branch of the SCFT is realized by resolving the 5-brane junction at a point into a 5-brane web, while mass deformations are realized by moving the external 5-branes relative to each other. This can often be used to obtain an effective description of the SCFT as a gauge theory. Our focus will be on the fixed-point SCFTs, described by JHEP04(2021)134 (d) Figure 1. From left to right: one-sided conformal defect, one-sided non-conformal defect, two-sided conformal defect, two-sided non-conformal defect. The black lines represent 5-branes in the (56) plane; the 5-brane junction has been resolved slightly for illustrative purposes. The vertical blue line represents a D3-brane, with the part above/below the (56) plane shown as solid/dashed line. 5-brane junctions at a point, but to highlight the features of the SCFTs we will nevertheless show resolved webs in the figures (e.g. in figures 2(a), 4(a), 5(a), which show the (56) plane).
More general SCFTs can be realized by including in addition 7-branes extending in the (01234) and (789) directions [26]. An example is shown in figure 7(a) below, and the brane orientations are summarized in the table above.
Our main interest are codimension-2 surface defects in the 5d SCFTs engineered by 5-brane junctions, which can be realized by adding D3-branes that end on the 5-brane web. The D3-branes extend along 3 of the 5 field theory directions, say (012), and one of the directions which none of the 5-branes extend into, say (7). Various examples of such configurations were studied e.g. in [3,[14][15][16][17][18][19][20]. From the field theory perspective these defects can be understood as extra degrees of freedom coupling to the 5d SCFT at the intersection point of the 5-brane junction and the D3-brane. They can also be obtained from RG flows with vortex configurations on the Higgs branch [14].
In this work we use AdS/CFT to study supersymmetric defects realized by D3-branes in 5d SCFTs. That is, the ambient QFT is at the conformal fixed point. If the position of the D3-brane in the (89) plane is fixed, the D3-brane preserves a U(1) subgroup of the R-symmetry corresponding to rotation in the (89) plane. All D3-branes studied below preserve this U(1) symmetry. We will distinguish between one-sided and two-sided defects: for one-sided defects the D3-brane ends on the 5-brane junction. For a conformal defect the D3-brane is located in the (56) plane at the point of the 5-brane junction ( figure 1(a)). Non-conformal defects can be realized by D3-branes ending on one of the external 5-branes away from the junction point, with the separation from the junction point introducing a JHEP04(2021)134 mass parameter, see figure 1(b). For two-sided conformal defects, two D3-branes are joined at the point of the 5-brane junction from opposite sides (figure 1(c)). More general mass deformations can in that case be obtained by arbitrarily moving the complete D3-brane away from the junction point in the (56) plane ( figure 1(d)). These features will be reflected in the discussion of probe D3-branes in the next section.
The AdS 6 solutions are defined in terms of locally holomorphic functions A ± on a Riemann surface Σ with boundary. The geometry is a warped product of AdS 6 and S 2 over Σ, with the S 2 collapsing on the boundary of Σ. The Einstein-frame metric, complex two-form C (2) , and axion-dilaton scalar B = (1 + iτ )/(1 − iτ ) are given by where w is a complex coordinate on Σ and ds 2 AdS 6 and ds 2 S 2 are the line elements for unit-radius AdS 6 and S 2 , respectively. The metric functions read and we have Explicit expressions for the functions A ± will be given below. The differentials ∂A ± generally have poles at isolated points r on the boundary of Σ, at which 5-branes with charge (p , q ) emerge, with (p , q ) given in terms of the residues of ∂A ± by This allows to identify the associated 5-brane junction. Solutions for 5-brane junctions with 7-branes in addition have punctures with SL(2, R) monodromy in the interior of Σ [13].

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where µ = 0, 1, 2 are the field theory directions in which the defect should extend and (y 1 , y 2 ) is the plane transverse to the surface defect. From the field theory perspective there is a geometric SO(2) symmetry corresponding to rotations in the plane transverse to the surface defect. This symmetry should be preserved by the D3-brane, which thus has to be located at a fixed position in the (y 1 , y 2 ) plane. The worldvolume coordinates can be chosen as (r, x µ ).
The isometries of the S 2 in the background solution represent the SU(2) R-symmetry of the 5d SCFT, and we will consider only embeddings which preserve a U(1) subgroup of the R-symmetry. This forces the position of the D3-brane on the S 2 to be constant across the embedding. For a defect preserving 3d Poincaré symmetry, the only remaining freedom is then the position on Σ, and the entire embedding is characterized by one complex embedding function w(r). The induced metric for such an embedding is 6) and the action for the D3-brane becomes The equation of motion resulting from this action is a complex second-order non-linear ordinary differential equation for w(r). However, we are interested in supersymmetric defects and the BPS condition is more tractable. The derivation is spelled out in appendix A, and results in the condition As also shown in the appendix, this condition implies the equation of motion following from (3.7). As appropriate for a D3-brane, the BPS equation is invariant under Type IIB SL(2, R) transformations, since κ 2 and G are separately invariant. The solutions flow along the gradient of G in Σ, as dictated by (3.8), and generic embeddings of this form preserve 3d N = 2 supersymmetry. Embeddings preserving defect conformal symmetry, i.e. the SO(2, 3) isometries of the AdS 4 parametrized by (r, x µ ), have w = 0. The induced metric on the D3-brane is that of AdS 4 and the position on Σ is a constant. The BPS condition (3.8) reduces to The D3-brane thus is at an extremal point of G, which we denote as w = w c . Such points are also extrema of f 6 (and f 2 ), so that the D3-brane extremizes its action. 2 As shown in appendix A the embeddings preserve eight of the sixteen supersymmetries of the background, and realize the sub-superalgebra C(3) of F (4) [27] (see also table 1 of [28]). The on-shell action for these embeddings can be evaluated using that ∂ w G = 0 implies T = 1, such that f 4 6 = 6G. This leads to

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So far we have worked with Poincaré AdS 6 . However, for the embeddings with defect conformal symmetry one can do a conformal transformation to global AdS 6 coordinates, to describe an S 3 defect in the 5d SCFT on S 5 . The renormalized volume of global AdS 4 is given by Vol AdS 4 = 2 3 Vol S 3 with Vol S 3 = 2π 2 , and with these values the on-shell action in (3.10) yields the contribution of the defect described by the D3-brane to the free energy of the SCFT on S 5 .
An interpretation for the non-conformal embeddings described by (3.8) can be obtained by analyzing the behavior near the conformal boundary of AdS 6 . With a Fefferman-Graham radial coordinate z = − ln r, such that the boundary is at z = 0, the BPS equation becomes −κ 2 z∂ z w = ∂wG. At the boundary of AdS 6 the embedding approaches that of a conformal defect. The mass of fluctuations in the real and imaginary parts of w can be obtained by expanding (3.8) around the conformal embedding, using that (3.9) implies . This shows that the leading behavior of both fluctuations near the conformal boundary is linear in z. Fluctuations in the real and imaginary parts of w thus correspond to a pair of defect operators with scaling dimensions ∆ = 2 in standard quantization and ∆ = 1 in alternative quantization. The non-conformal embeddings we will discuss below describe defect RG flows triggered by combinations of these two relevant deformations.
The general BPS equation (3.8) can be integrated to obtain an implicit expression for the embedding function w(r). To that end, we note that (3.8) implies Integrating both sides leads to where m is a complex parameter encoding the mass parameters associated with the two relevant deformations discussed above. The conformal embedding corresponds to m = 0. The Type IIB SL(2, R) transformations are induced by SU(1, 1) ⊗ C transformations of A ± , spelled out in (5.12) of [10]. Under these transformations the combination A + − A − transforms by an overall factor, such that SL(2, R) transformations of the condition in (3.12) only transform the mass parameter.

T N and Y N theories
We start with the T N and Y N theories. The 5d T N theories introduced in [29] are realized by junctions of N D5, N NS5 and N (1, 1) 5-branes (figure 2(a)). The Y N theories of [30] correspond to a junction of 2N NS5 branes, N (1, 1) 5-branes and N (1, −1) 5-branes ( figure 4(a)). The functions A ± and G realizing holographic duals for these theories were given explicitly in [31], as where w is a complex coordinate on the upper half plane and 2πα = 1. The poles on ∂Σ are at w ∈ {0, ±1}, and encode the charges of the 5-brane junction via (3.4). The two supergravity solutions are related by a combination of SL(2, R) transformation and rescaling of the charges, as discussed in [30]. The free energies obtained holographically were matched to field theory computations in [32,33].
Since the equation governing the D3-brane embedding is invariant under SL(2, R) transformations, the solutions for the D3-brane embeddings in T N and Y N can be discussed together. The functions G for T N and Y N/ √ 2 are given by where D is the Bloch-Wigner function defined by Since G depends on N only through the overall coefficient in (3.14), the D3-brane BPS equation in (3.8) is independent of N . We start with the embedding describing conformal defects. With constant w, (3.8) has one solution. The solution and the on-shell action obtained from (3.10) are given by One can do a conformal transformation to global AdS 6 coordinates, with the D3-brane embedding unchanged, to describe an S 3 defect in the 5d SCFT on S 5 . The action in (3.16) with Vol AdS 4 as below (3.10) then gives the contribution of an S 3 defect to the free energy on S 5 for the T N and Y N/ Non-conformal embeddings can be obtained by inverting (3.12). Using the explicit expression for A ± for the T N theory in (3.13), the real and imaginary parts of (3.12) lead to where m 1 and m 2 are, respectively, related to the real and imaginary part of m in (3.12) by rescaling. The general solution in Σ, with Fefferman-Graham radial coordinate z, is given by This is a family of curves which connect the conformal point w c to points on the boundary of Σ, reaching the boundary when the square root vanishes. Embeddings with a simple form in the upper half plane, which follow a circle or straight line connecting the point w c = i/ √ 3 to one of the poles, can be found from the fact that G is invariant under reflection across the imaginary axis: if the initial departure from the conformal point w c has vanishing real part this is preserved along the flow, leading to the solutions (3.18) with m 1 = 0, The supergravity solution further has a Z 3 group of SL(2, R) transformations of the upper half plane which can be combined with Type IIB SL(2, R) transformations to form symmetries of the solution. Since the D3-brane BPS condition is invariant under Type IIB SL(2, R) transformations, this leads to two related branches of solutions connecting w c to the other two poles, These D3-brane embeddings resemble the string embeddings discussed in [30]. The form of the embeddings is not symmetric in m 2 → −m 2 : for m 2 < 0 the D3-brane reaches all the way into the IR region of AdS 6 , where z → ∞, without reaching the pole on the boundary of Σ. For m 2 > 0, on the other hand, the D3-brane reaches a regular point on the boundary of Σ at m 2 z = ln 4.
As the D3-brane reaches the boundary of Σ, it can not just end at the corresponding value of the AdS 6 radial coordinate z: since the D3-brane does not wrap any part of the internal space, one can not obtain a smooth worldvolume without boundary by shrinking an internal cycle. However, since the S 2 in which the D3-brane is localized at a point collapses on the boundary of Σ, one can connect the embedding to a second D3-brane, described by the same profile w(r) but located at the antipodal point of form a smooth D3-brane worldvolume, with the branches at the two antipodal points of S 2 corresponding to the two branches of the two-sided defects in figure 1. In the UV region of AdS 6 , where z is small, the two D3-branes are close to the conformal embedding on Σ. As one moves into the IR region of AdS 6 both branches approach the boundary of Σ, where they meet for zm 2 = ln 4. At that point they disappear from the AdS 6 perspective. In summary, mass deformations with m 2 < 0 can be realized for one-sided defects in the sense of figure 1. They can also be realized for two-sided defects, which simply corresponds to adding a second embedding at the antipodal point of S 2 . Mass deformations with m 2 > 0 can only be realized for two-sided defects in the sense of figure 1 with two branches of D3-brane embeddings, and the mass deformations completely gap the defect in the IR. This realizes brane configurations of the type shown in figure 1(d).
The embeddings are illustrated in figures 2(b) and 2(c) for the T N solution. Figure 2(b) illustrates the supergravity solution with Σ mapped to the unit disc with coordinate u via This maps the Z 3 -symmetric point w = i/ √ 3 to the center of the disc. The poles of ∂A ± , where the 5-brane stacks are located, are indicated as solid circles on the boundary, and the phase of u in (3.21) is chosen such that the locations of the poles roughly line up with the external 5-brane stacks in the brane web in figure 2(a). 3 As shown in [31] by considering Wilson loops represented by different D3-brane embeddings, each point of Σ can be identified with a face of the associated 5-brane web. Namely, each point on Σ can be assigned coordinates (N D1 , N F 1 ) given by  These coordinates translate to coordinates on the brane web, with N D1 labeling the faces in the horizontal direction and N F1 labeling the faces in the vertical direction. The space of charges carved out when moving along Σ is shown by the blue closed curves in figure 2(c), which correspond to curves of constant radius on the disc. The points in the triangle correspond to the closed faces of the brane web in figure 2(a), and the shape represents the associated Newton polygon.
In figure 2(b) curves of constant N D1 /N F1 are shown in red/blue. The defect embeddings are the straight lines connecting the center of the disc to boundary points, with each branch shown in a different color. The lines ending at the poles describe non-conformal D3-brane defects that reach into the IR region of AdS 6 -these embeddings never actually reach the poles. The lines ending at regular boundary points represent a pair of embeddings at antipodal points of the S 2 which are joined at the boundary of Σ, and describe two-sided defects. These embeddings describe defects that are gapped in the IR. Figure 2(c) shows how the D3-brane embeddings w 1 , w 2 , w 3 trace through the (N D1 , N F1 ) coordinates. The quiver gauge theory deformation of the T N theory is given by (3.23) The discrete symmetries of the supergravity solution and brane web do not translate to simple symmetries of the quiver gauge theory. This will be different for the Y N theory. More general embeddings (3.18) are shown in figure 3. The general picture is that, for generic choices of the phase of m 1 + im 2 , the D3-brane embedding reaches the boundary of Σ at a regular point, so that it has to be combined with a second branch to form a two-sided defect. There are three distinguished choices of the phase for which the defect approaches the boundary of Σ at a pole, so that a one-sided embedding can be realized and extends all the way into the IR. This mirrors the discussion of section 2, where the one-sided embeddings can only be moved along specific directions in the (56) plane in which the external 5-branes of the junction extend, while two-sided D3-brane defects can be moved arbitrarily into the (56) plane.

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For the Y N solution the D3-brane embeddings take the same form in Σ, though the brane web and gauge theories are different. The brane web and supergravity solution are shown in figure 4. The Y N theories have two S-dual gauge theory deformations that differ in their form. The one corresponding to the brane web in figure 4(a) is given by (3.24) Along the first/second ellipsis the rank of the gauge groups increases/decreases in steps of one, and the central node has a Chern-Simons term. The quiver is symmetric under reflection across the central node, and this symmetry is realized in the supergravity solution as reflection across the vertical diameter of the disc in figure 4(b). The form of the D3brane embeddings suggests that the mass deformation described by w 1 is symmetric under reflection of the quiver across the central node. The Chern-Simons term explains the lack of symmetry under m 2 → −m 2 .

+ N,M and X N,M theories
The + N,M theories are realized by intersections of N D5 and M NS5-branes (figure 5(a)), and were studied already in [4]. The X N,M theories of [30] correspond to intersections of N (1, −1) 5-branes and M (1, 1) 5-branes. Similar to the relation between the T N and Y N theories, the supergravity solutions are related by a combination of Type IIB SL(2, R) transformations and charge rescaling. The functions A ± are The free energy obtained from the supergravity solution was matched to field theory computations in [32,33] for the + N,M theory and in [31] for X N,N . Since N and M only appear as combined overall factor in (3.26), the D3-brane BPS equation Non-conformal defects are obtained by inverting (3.12). With the expressions for A ± for the + N,M theory, the real and imaginary parts of (3.12) lead to These embeddings connect the point w c to the two D5-brane poles. In the IR the embeddings w 1 approach the D5-brane poles on Σ but never reach them, so that the D3-branes extend all the way into the IR region of AdS 6 . In figure 5 These embeddings correspond to mass deformations of the conformal defect that are S-dual to the mass deformations leading to the embeddings w 1 ; they connect the point w c to the NS5-brane poles. In figure 5(b) they correspond to the green horizontal line. Two more simple solutions correspond to the diagonal lines in figure 5(b). They can be obtained by performing SL(2, R) transformations of the upper half plane that map the poles to locations that are symmetric with respect to reflection across the imaginary axis, and then making an ansatz for imaginary embeddings. This corresponds to m 1 = ±m 2 in (3.29), leading to (3.32) In figure 5(b) the purple line shows w 3 and the cyan line w 4 . The solutions w 3 and w 4 reach the boundary at 2|m 1 |z = ln(3 + 2 √ 2). They do not extend all the way into the IR region of AdS 6 and have to be combined with a second branch of the same embedding on the antipodal point on S 2 , to form a two-sided defect in the sense of section 2.
The general picture is similar to the one for the T N and Y N theories, and illustrated in figure 6. For generic choices of m 1 + im 2 , the embedding reaches the boundary of Σ at finite AdS 6 radial coordinate, and has to be combined with a second branch on the antipodal point of S 2 to form a regular embedding. The defect degrees of freedom are gapped in the IR. For discrete choices of the phase of m 1 + im 2 , in this case four, the nonconformal embeddings approach the poles without ever reaching them, and can describe one-sided defects. The embedding then reaches all the way into the IR of AdS 6 and the defect contains light degrees of freedom.
The quiver gauge theory deformations of the + N,M junction reflect the Z 2 symmetries in the supergravity solution. The quiver is given by with a total of M − 1 gauge nodes. The S-dual quiver deformation, corresponding to a ninety degree rotation of the brane web, has the same form but with N and M exchanged. The plots in figure 5(c) suggest that the defects described by the embedding w 2 correspond to mass deformations that are symmetric under reflection of the quiver (3.33) across the central node. Analogous comments hold for the defects described by the embedding w 1 in the S-dual quiver. The analog of figure 5(c) for the X N,M theory follows from a 45 degree rotation, similarly to the relation between the T N and Y N theories before (the horizontal and vertical axes in figure 5(c) are scaled differently for N = M , and the rotation is to be performed with unrescaled axes).

I M,j theories
As a last example we consider a 5-brane junction with 7-branes. It is composed of two stacks of M NS5-branes on a system of 2j D7-branes. The D7-branes are split into two groups of j D7-branes which have their branch cuts oriented in opposite directions ( figure 7(a)). This is a special case of the + N,M,j,k theories discussed in [34]. Orientifolds, dubbed I ± M,j , were discussed in [35]. The functions A ± can be written as (see (3.4

) of [35])
where v is a complex coordinate on the unit disc. The two poles on the boundary of Σ corresponding to the external NS5-brane stacks are at v = ±i, and there are 2j D7-branes at the center of the disc. Half of the D7-branes have their branch cut along the positive real axis, the other half has their branch cut along the negative real axis. The solutions have two Z 2 symmetries, corresponding to reflection across the real and imaginary axes. The quiver gauge theory is given by (4.17) of [34] with k = j and 2N = M j. The free energy was matched to a field theory computation in [35]. The conformal defect is thus localized at the center of the disc, on top of the D7-brane punctures.
Non-conformal embeddings can be obtained from (3.12). With A ± in (3.34) the real and imaginary parts of (3.12) lead to The form of the embeddings is symmetric in m 1 → −m 1 and m 2 → −m 2 . The background solution has two Z 2 symmetries corresponding to reflection across the real and imaginary axes, so there are two distinguished families of embeddings along these fixed lines. For embeddings along the imaginary axis the BPS equations simplify since the dilogarithm terms drop out. They correspond to m 2 = 0 and can be given explicitly as (3.37) The embeddings connect the center of the disc to the NS5-brane poles. They describe onesided defects and extend all the way into the IR region of AdS 6 . For embeddings along the real axis, with v 2 (r) =v 2 (r), the first equation in (3.36) is trivial, and the second equation simplifies to where D is the Bloch-Wigner function defined in (3.15). Since D 2 (±i) = ±C, where C is Catalan's constant, the D3-brane reaches the boundary of Σ, with v 2 reaching ±1, when |m 2 |e −r = 2C. These embeddings have to be combined with a second branch on the antipodal point of S 2 to form regular embeddings describing two-sided defects.
More general solutions are shown in figure 8. As before, generic embeddings reach regular points on the boundary of Σ at finite radial coordinate in AdS 6 , and have to be combined with a second branch on the antipodal point of S 2 to form a smooth worldvolume. Only the embeddings v 1 and v 2 approach the poles and describe one-sided defects.

JHEP04(2021)134 4 Discussion
We have investigated planar three-dimensional surface defects in 5d SCFTs engineered by the UV limit of (p, q) 5-brane webs. The surface defects are realized either by D3branes ending on the 5-brane junction arising as UV limit of the brane web, or by D3branes intersecting the plane in which (p, q) 5-branes are at an angle dictated by their charges. Conformal defects are realized by D3-branes at the point where the 5-branes form a junction. For non-conformal defects the D3-branes are displaced from the junction point. We have identified the holographic representation of the defects as probe D3-branes in the AdS 6 × S 2 × Σ supergravity duals of the 5d SCFTs and studied conformal and nonconformal defects.
The conformal defects preserve half of the supersymmetries of the ambient SCFT. The embeddings wrap an AdS 4 in AdS 6 and are localized on S 2 and on the Riemann surface Σ. On Σ they are located at points where the AdS 6 and S 2 radii are extremal. For the solutions considered here there is exactly one such extremal point on Σ, and we found one half-BPS conformal defect described by a probe D3-brane. The planar defects can be mapped to S 3 defects in the 5d SCFTs on S 5 by a conformal transformation, and we obtained the defect contribution to the free energy on S 5 . It would be interesting to study these defects from the field theory perspective and e.g. reproduce their contribution to the free energy on S 5 using supersymmetric localization. It would also be interesting to consider the limit where the number of D3-branes ending on the 5-brane web becomes large. Such configurations should be described by AdS 4 solutions incorporating the backreaction of the D3-branes on the 5-brane web. At least certain defects may be described by uplifting AdS 4 solutions of 6d gauged supergravity to Type IIB, and it would be interesting to further explore this for the theories studied here. Related discussions based on the Brandhuber/Oz solution of massive Type IIA can be found in [36][37][38][39].
We also considered non-conformal defects which preserve half of the Poincaré supercharges of the ambient SCFT and a U(1) R-symmetry, while breaking the conformal supersymmetries. The defects are described by D3-brane embeddings which wrap (part of) AdS 4 in AdS 6 but where the location on Σ depends on the AdS 4 radial coordinate in Poincare coordinates. Fluctuations of the D3-brane away from the conformal point on Σ are dual to a pair of relevant operators, and the non-conformal embeddings describe defect RG flows triggered by combinations of these operators.
The features of the non-conformal embeddings suggest a natural brane-web interpretation: for special values of the phase of the complex number formed out of the two mass parameters triggering the flow, the embeddings approach, without ever reaching, one of the poles on the boundary of Σ, which represent the external 5-branes of the associated 5-brane junction. These embeddings seem to describe D3-branes displaced in the 5-brane web along one of the external 5-branes. Embeddings for more general values of the phase can be realized if there are two D3-branes ending from opposite sides on the brane web, corresponding to two D3-branes on antipodal points of the S 2 in the supergravity solution. These two embeddings reach a generic point on the boundary of Σ at a finite value of the AdS 4 radial coordinate. Since the S 2 collapses on the boundary of Σ, the two D3-brane JHEP04(2021)134 embeddings can combine to form a smooth worldvolume without boundary. In the brane web these embeddings seem to correspond to more general displacements of the entire D3brane away from the 5-brane junction, which do not need to be along one of the external 5-branes.
An interesting subject in the context of defect RG flows are RG monotones. It would be interesting to study the contribution of the non-conformal defects to the entanglement entropy, which can be done without computing the backreaction of the branes using the method of [40,41].

A.1 Conformal defect
To find the projection conditions on the Killing spinors it is convenient to start with conformal defects. For the D3-brane wrapping AdS 4 , localized at a fixed point in S 2 and Σ, and using complex notation for the Killing spinors, The condition we aim to implement is with λ 2 = 1. Note that χ −η 1 ,−η 2 = (γ (1) ⊗ γ (2) )χ η 1 η 2 . For x 3 = x 4 = 0 the γ 3 , γ 4 terms drop out in the AdS 6 Killing spinor in (A.6). Moreover, the matrices multiplying ± S,0 are constant on the brane embedding. The relation (A.8) can be realized by constraining the constant spinor to satisfy where R = exp i 2 θ 2 σ 2 exp − i 2 θ 1 σ 3 . This is a constant projection condition which squares to one, and implies (A. These conditions can be further evaluated using the expressions for the Killing spinor components in (4.9) of [10]. This will be covered by the discussion for more general nonconformal embeddings below. Embeddings at points satisfying (A.12) preserve half the background supersymmetries, and realize the sub-superalgebra C(3) of F (4) [27] (see also table 1 of [28]).
We have yet to check the equation of motion resulting from the action in (3.7). Using the expressions for the metric functions one may write the Lagrangian as

JHEP04(2021)134
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