BPS M5-branes as Defects for the 3d-3d Correspondence

We study supersymmetric probe M5-branes in the AdS_4 solution that arises from M5-branes wrapped on a hyperbolic 3-manifold M_3. This amounts to introducing internal defects within the framework of the 3d-3d correspondence. The BPS condition for a probe M5-brane extending along all of AdS_4 requires it to wrap a surface embedded in an S^2-fibration over M_3. We find that the projection of this surface to M_3 can be either a geodesic or a tubular surface around a geodesic. These configurations preserve an extra U(1) symmetry, in addition to the one corresponding to the R-symmetry of the dual 3d N=2 gauge theory. BPS M2-branes can stretch between M5-branes wrapping geodesics. We interpret the addition of probe M5-branes on a closed geodesic in terms of conformal Dehn surgery.


Introduction
The 3d-3d correspondence associates a 3d N = 2 supersymmetric gauge theory to a 3manifold M 3 [1][2][3][4][5]. This theory can be thought of as the partially twisted compactification of the 6d (2, 0) superconformal field theory. It describes the low-energy effective theory on the worldvolume of a stack of N M5-branes wrapping R 1,2 × M 3 in the 11d spacetime R 1,4 × CY 3 , where M 3 is embedded in the Calabi-Yau 3-fold CY 3 as a special Lagrangian submanifold.
In this paper, we are interested in the large N limit of this setup with CY 3 taken to be the cotangent bundle T * M 3 . The stack of N M5-branes backreacts on the geometry and gives rise in the IR to an AdS 4 solution of 11d supergravity. We sketch this process as follows: (1.1) The radial coordinates of the factor R 2 and of the fibers R 3 in T * M 3 = (M 3 ← R 3 ) combine in the large N limit to give the radial coordinate of AdS 4 and a coordinate ρ on the interval I ρ . Accordingly, the cotangent bundle T * M 3 is replaced by the unit cotangent bundle T * 1 M 3 = (M 3 ← S 2 ). The circle S 1 R corresponds to the U (1) R-symmetry of the dual 3d N = 2 SCFT. This circle shrinks to a point at the origin of the interval I ρ , while the sphere S 2 shrinks at the other end, so that together they have the topology of an S 4 . This solution is in fact the M-theory uplift of a compactification of 7d gauged supergravity on AdS 4 × H 3 discovered by Pernici and Sezgin [6] (reviewed in section 2). Hyperbolic 3-space H 3 can be made compact by quotienting by a discrete subgroup Γ ⊂ P SL(2, C): In section 3, we consider adding supersymmetric probe M5-branes to the Pernici-Sezgin solution, in such a way that the superconformal symmetry of the dual theory is preserved. This corresponds to studying the space of 3d N = 2 SCFTs arising from M5-branes on 3-manifolds (some of our inspiration came from the interpretation of probe M5-branes as punctures on a Riemann surface associated with a 4d N = 2 SCFT [7] or with a 4d N = 1 SCFT [8]). We thus require that the probe M5-branes extend along all of AdS 4 and do not break the U (1) R-symmetry corresponding to S 1 R . The BPS condition coming from κ-symmetry then implies that a supersymmetric M5-brane wraps a surface embedded in the unit cotangent bundle T * 1 M 3 . The first type of BPS embedding that we find is an M5-brane wrapping a geodesic in M 3 as well as a great circle on the S 2 -fiber. This was anticipated, since line defects in 3-manifolds (such as the knot of a knot complement) are the analogues of punctures on a Riemann surface. Perhaps more unexpectedly, we also find that a BPS M5-brane can warp a tubular surface in M 3 , for instance a torus or an annulus.
These BPS configurations should be imagined as descending from a probe M5-brane wrapping a special Lagrangian submanifold of T * M 3 in the UV. It appears that this submanifold is the conormal bundle N * L of a submanifold L ⊂ M 3 , which can be a geodesic or a tube. Note that N * L is automatically Lagrangian in T * M 3 . Intersecting branes on conormal bundles have been used in various contexts to build knot complements (see for example [9][10][11]). In the IR, the probe M5-brane wraps the unit conormal bundle N * 1 L ⊂ T * 1 M 3 . The BPS embeddings that we found for a supersymmetric probe M5-brane are thus of the form BPS M5-brane : Interestingly, the unit conormal bundles N * 1 L that we obtain are flat (with the topology of a cylinder), which means that these configurations preserve an extra U (1) symmetry, in addition to the U (1) R-symmetry corresponding to S 1 R .
In section 4, we study BPS embeddings for supersymmetric probe M2-branes that correspond to BPS operators in the dual SCFT. We focus in particular on probe M2branes ending on probe M5-branes along two spacetime dimensions. We find BPS M2branes stretching between M5-branes on geodesics contained in the same geodesic surface, and M2-branes stretching between great circles wrapped by M5-branes on an S 2 -fiber. In contrast, we did not find M2-branes ending on M5-branes wrapping surfaces in M 3 . We take this as an indication that the two types of BPS M5-branes will play rather different roles in the dual 3d theories.
In the BPS calculations that we perform, we neglect the quotient by Γ that produces a closed 3-manifold M 3 = H 3 /Γ, and study embeddings in T * 1 H 3 . In section 5, we present a potential interpretation of our results in the actual manifold M 3 . We relate the addition of probe M5-branes on a closed geodesic to conformal Dehn surgery, which consists of excising a solid torus from M 3 , twisting it, and gluing it back in. The number of coincident M5branes should correspond to the amount of twisting, and taking it to be very large would produce a non-compact hyperbolic 3-manifold with a cusp, for example a knot complement.

Pernici-Sezgin AdS 4 solution
We start by reviewing the AdS 4 solution of M-theory that we will be considering in this paper. This solution is the 11d uplift of a 7d gauged supergravity solution originally found by Pernici and Sezgin (PS) [6], and was subsequently rediscovered in [12,13] from a study of wrapped M5-branes. It is of the form AdS 4 × Y 7 , where Y 7 is an S 4 -fibration over a hyperbolic 3-manifold M 3 .
This solution was shown in [14] to arise as a special case of a general class of N = 2 supersymmetric AdS 4 geometries describing the near-horizon limit of M5-branes wrapping a special Lagrangian 3-cycle in a Calabi-Yau 3-fold (the generality of this class was proven in [15]). The metric for this class of solutions takes the form ds 2 11 = λ −1 ds 2 (AdS 4 ) + ds 2 4 (M SU (2) ) +ŵ ⊗ŵ + λ 2 16 where λ is the warp factor, M SU (2) is a 4d space with SU (2)-structure,ŵ is a one-form, ρ is an interval coordinate, and ψ ∈ [0, 2π] is a coordinate on a circle S 1 R . The Killing vector field ∂/∂ψ is dual to the U (1) R-symmetry. The supersymmetry conditions reduce to a system of differential equations involving the standard two-forms {J 1 , J 2 , J 3 } defining the SU (2)-structure. The four-form flux is given by The PS solution was reproduced in section 9.5 of [14] by making the following ansatz: where Y a , a = 1, 2, 3, are constrained coordinates on an S 2 satisfying Y a Y a = 1, and e a are vielbeins for a 3-manifold M 3 . The covariant derivative is defined as with ω ab the spin connection of M 3 . The ansatz for the structure iŝ w = gY a e a , The supersymmetry conditions then determines the functions as and imply that M 3 is a hyperbolic 3-manifold and that the coordinates Y a together with ρ and ψ build up an S 4 . The hyperbolic 3-manifold M 3 can be expressed as a quotient of hyperbolic 3-space H 3 by an discrete subgroup Γ ⊂ P SL(2, C), that is M 3 = H 3 /Γ (see appendix A for a small review on hyperbolic 3-manifolds). The metric on M 3 is normalized such that the Ricci scalar is R = −3, so we take The spin connection on M 3 has the non-zero components ω 31 = dx/z and ω 32 = dy/z. The metric of the PS solution finally reads The holographic free energy can be easily calculated as 9) where G N 4 is the effective 4d Newton constant obtained by dimensional reduction (see [15] for more detail). The quantization of the four-form flux (2.2), with X 4 ⊂ Y 7 transverse to M 3 , then gives the expected N 3 -scaling: Note that the free energy corresponding to SCFTs on a squashed 3-sphere S 3 b with squashing parameter b is simply given by F b = (b + 1/b) 2 F/4 [16,17].

Supersymmetric probe M5-branes
In this section, we study a supersymmetric probe M5-brane that preserves the superconformal symmetries of the dual 3d N = 2 SCFT. This implies that it should extend along all of AdS 4 , and that the remaining two internal directions should wrap a 2d submanifold of Y 7 in such a way as to preserve the U (1) R-symmetry. There can be no three-form flux on the worldvolume of the M5-brane, since it would have to extend along at least one AdS 4 dimension, thus breaking the conformal symmetry. We will see that the BPS condition arising from κ-symmetry imposes that the probe M5-brane is located at the origin of the interval I ρ , where S 1 R shrinks to a point, and that it is calibrated by the two-form J 2 . The BPS configurations that we will find describe an M5-brane wrapping the unit conormal bundle N * 1 L ⊂ T * 1 M 3 of a submanifold L ⊂ M 3 . This submanifold L can be a geodesic curve, but also a surface that is equidistant from a point at infinity or from a geodesic curve. These BPS embeddings preserve an extra U (1) symmetry, in addition to the U (1) R-symmetry.

BPS condition
The requirement of κ-symmetry leads to a BPS bound on a supersymmetric probe M5brane [18] (see also [19]). A configuration that preserves some supersymmetry satisfies P − = 0, where is a Majorana spinor of 11d supergravity satisfying the Killing spinor equation, and P − is a κ-symmetry projector. Explicitly, we have where L M5 = √ g M5 is the Dirac-Born-Infeld Lagrangian on the M5-brane, and the subscript | M5 denotes the pullback to the worldvolume of the M5-brane. This leads to a BPS bound which is saturated if and only if the probe M5-brane is supersymmetric. We rewrite this bound as † L M5 vol 5 ≥ ν 5 | M5 , where vol 5 is the volume form on the spatial part of the worldvolume of the M5-brane, and the five-form ν 5 is defined as the bilinear with¯ = † Γ 0 and Γ (n) = 1 n! Γ N 1 ···Nn dX N 1 ∧ · · · ∧ dX Nn . We will now use the analysis of general AdS 4 solutions in [15] to obtain the BPS condition for a probe M5-brane extending along all of AdS 4 and wrapping a 2d submanifold of Y 7 . The 11d metric is written as the warped product g 11 = λ −1 (g AdS 4 + g 7 ) and the gamma matrices split accordingly as where α, β = 0, 1, 2, 3 and a, b = 1, . . . , 7 are orthonormal frame indices for AdS 4 and Y 7 respectively: {ρ α , ρ β } = 2η αβ , {γ a , γ b } = 2δ ab . We have defined the chirality matrix ρ 5 = iρ 0 ρ 1 ρ 2 ρ 3 . The 11d spinor splits into two spinors ψ + 1 , ψ + 2 on AdS 4 and two internal spinors χ 1 , χ 2 on Y 7 : where g 2 stands for the determinant of the metric induced on the internal submanifold. The right-hand side gives with r, α 1 , α 2 spatial coordinates on AdS 4 (see the AdS 4 metric in (B.1)). We see that the 2d submanifold wrapped by the M5-brane is calibrated by the following two-form: where we used the notation Note that the AdS 4 spinors appear in the same combination in the left-hand side (3.7) of the BPS condition and in front of V + in the right-hand side (3.9). This indicates that the M5-brane should be calibrated only by the term involving V + , since the other terms in (3.9) would generally lead to constraints on the AdS 4 spinors, thus breaking the N = 2 supersymmetry that we wish to preserve (we confirm this argument by explicit calculations in appendix B).
In the case of an AdS 4 solution arising from wrapped M5-branes (no electric flux), the calibration form reads This means that the probe M5-brane does not wrap the circle S 1 R parameterized by ψ and so, in order to preserve the corresponding U (1) R-symmetry, it must be located where this circle shrinks to a point, that is at We show in appendix B that this condition ensures that the pullbacks of the other twoforms appearing in (3.9) vanish: We remark that the condition that the probe M5-brane is calibrated essentially by J 2 at ρ = 0 is consistent with its origin from a special Lagrangian in the UV. Indeed, the SU (3)-structure of the Calabi-Yau 3-fold decomposes into the SU (2)-structure as withû a unit one-form on CY 3 , which reduces to λ −1/2 dr at ρ = 0 (see appendix C of [14]). It is then easy to see that, on the 3-cycle consisting of the internal 2d submanifold and the AdS 4 radial direction r, J and Im Ω restrict to zero, while Re Ω gives a volume form.
Focusing on the PS solution reviewed in section 2, we arrive at the following BPS condition for a supersymmetric probe M5-brane on AdS 4 : (3.14) Herevol 2 = √ḡ 2 dτ ∧ dσ is the volume form on the internal part of the M5-brane worldvolume induced from the metric 15) and the calibration two-form is given in terms of the vielbeinsē = {dx, dy, dz}/z bȳ

Conormal bundles
We have derived that a supersymmetric probe M5-brane that extends along all of AdS 4 must be at ρ = 0, where S 1 R shrinks, and wrap a surface calibrated by the two-formJ 2 in the 5d space with metricḡ given in (3.15). In the next two subsections we will present some natural classes of solutions to the BPS condition (3.14). They all share the interesting feature that they appear to descend from conormal bundles in the cotangent bundle T * M 3 .
Recall that in the UV the M5-branes should be thought of as wrapping special Lagrangian submanifolds of T * M 3 . The original stack of N M5-branes is wrapping the 3manifold M 3 itself, while there might be additional M5-branes that wrap other intersecting Lagrangians. A simple example of a Lagrangian submanifold of T * M 3 is the conormal bundle N * L = Ann(T L) of a submanifold L ⊂ M 3 . If L is a knot in M 3 , a well-known construction of the knot complement M 3 \L consists in intersecting branes wrapped on M 3 and on N * L inside T * M 3 (see [9] and some recent applications in [10,11]).
In the AdS 4 geometries that we study, what remains of the UV cotangent bundle T * M 3 is the unit cotangent bundle T * 1 M 3 described by the metricḡ in (3.15) -the radial direction in the cotangent fibers has been absorbed in the radial direction of AdS 4 and in ρ (see section 5 in [14]). Similarly, a Lagrangian conormal bundle N * L ⊂ T * M 3 descends to a Legendrian unit conormal bundle N * 1 L ⊂ T * 1 M 3 . The embedding of a probe M5-brane is therefore fully specified by the submanifold L that it wraps in M 3 . The remaining position of the M5-brane on S 2 is then simply given by the fiber of N * 1 L. Since the M5-brane is calibrated by the two-form there are essentially two options for the submanifold L ⊂ M 3 : it can be a curve or a surface. We will analyze these two cases in turn in the next subsections.
For the moment, we forget about the quotient by the discrete subgroup Γ ⊂ P SL(2, C) that produces the closed 3-manifold M 3 = H 3 /Γ, and study submanifolds L ⊂ H 3 of hyperbolic 3-space itself. We will come back to the interpretation of our results in terms of the closed 3-manifold M 3 in section 5.
Before proceeding we remark that the BPS condition (3.14) is invariant under any isom- . This invariance will allow us to focus first on solutions involving submanifolds L ⊂ H 3 that take particularly simple forms in the upper half-space model of hyperbolic 3-space, and then to obtain entire classes of solutions by acting with elements of P SL(2, C).

Line defects in H 3
We now present a class of BPS probe M5-branes that extend along all of AdS 4 and wrap the unit conormal bundle N * 1 L ⊂ T * 1 H 3 , with L a geodesic curve in H 3 . In the upper half-space model of hyperbolic 3-space, geodesics can be either vertical straight lines, or semicircles orthogonal to the boundary at z = 0 (see figure 1). In the case where L is a straight geodesic along the z-axis, the conormal bundle N * L intersects the S 2 -fibers of the unit cotangent bundle T * 1 H 3 along the equator (see figure 2). Denoting the internal worldvolume coordinates by τ ∈ [0, ∞) and σ ∈ [0, 2π], we can write the embedding for this solution as We can then generate solutions for any geodesic in H 3 by acting with transformations γ ∈ P SL(2, C). To obtain a semicircle geodesic we act on the straight geodesic with the transformation Applying the P SL(2, C) action (A.4) we get which describes a semicircle centered at {x, y, z} = {R, 0, 0} with radius R (see figure 1): On the S 2 -fibers, the M5-brane is now wrapping a great circle whose inclination depends on the base point (see figure 3). We can also shift the geodesic by acting with a parabolic element of P SL(2, C), and rotate it with an elliptic element. We will discuss in section 5 how such probe M5-branes on geodesics in H 3 can be interpreted as wrapping closed geodesics in M = H 3 /Γ. If we were to wrap an increasing number of M5-branes on a closed geodesic L, we would eventually produce a knot complement M \L.

Surface defects in H 3
We now consider a supersymmetric probe M5-brane wrapping AdS 4 × N * 1 L, where L is a surface in H 3 . We found two classes of solutions, namely surfaces that are equidistant from a point at infinity (horospheres) or from a geodesic in H 3 (tubes).

Horospheres
Horospheres are surfaces that are equidistant from a point p on the boundary ∂H 3 = S 2 ∞ of hyperbolic 3-space. In the upper-half plane model of H 3 , the point p can be either on the plane z = 0, in which case the horosphere is a Euclidean sphere tangent to p, or it can be at ∞, in which case the horosphere is a horizontal plane (see figure 4). The embedding corresponding to a horizontal plane is where {σ 1 , σ 2 } ∈ R 2 are worldvolume coordinates and z 0 is a constant. On the S 2 -fibers the M5-brane can be at the north and south poles. Just like for the geodesic solution, we can then generate any horosphere by acting on this horizontal plane with some transformation γ ∈ P SL(2, C). We can parameterize the horosphere with center p = {x 0 , y 0 , 0} and radius H as where now σ 1 ∈ [0, π] and σ 2 ∈ [0, 2π]. The conormal fiber over a point of this horosphere intersects the S 2 -fiber of T * 1 H 3 at the corresponding point (or at the antipode):

Tubes
A BPS M5-brane can also wrap a surface that is equidistant from a geodesic in H 3 , or in other words a tube. In the case of a vertical geodesic, such an equidistant surface is simply given by a vertical cone with its apex on the plane z = 0 (see figure 5). The cone with its apex at {x 0 , y 0 , 0} and with aperture 2α 0 can be parameterized as with τ ∈ [0, ∞) and σ ∈ [0, 2π]. The conormal fiber over a point of the cone intersects S 2 at a point that lies on a horizontal circle of radius cos α 0 : To obtain a surface equidistant from a semicircular geodesic, we again apply a transformation γ ∈ P SL(2, C). The resulting surface looks like a banana 1 (see figure 6). Note that this banana solution contains the other solutions that we found as special limits. Indeed, if we send the aperture α 0 of a banana to zero, we obtain a geodesic as in section 3.3, while if we bring the two endpoints of a banana together, we obtain a horosphere as in section 3.4.1 (see figure 7). Thus the space of BPS probe M5-branes is connected.

Extra U (1) symmetry
For all the solutions that we found (geodesic, horosphere, tube) the metric induced on the 2d submanifold wrapped by the M5-brane turns out to be flat. Interestingly, a theorem of Sasaki says that any complete flat surface in H 3 is either a horosphere or a surface equidistant from a geodesic [20]. These are precisely the two classes of solutions that we found for an M5-brane on a surface in H 3 .
Taking the limit α 0 → 0 of the vertical cone gives the straight geodesic, while for α 0 → π/2 the cone coincides with the horizontal plane with z 0 → 0. We can then produce general geodesics, horospheres, or bananas by acting with elements of P SL(2, C) in H 3 and with the corresponding SO(3) transformations on the coordinates Y a .

M2-branes ending on M5-branes
Probe M2-branes wrapping 2d internal submanifolds correspond to BPS operators in the dual 3d N = 2 gauge theory. In the presence of probe M5-branes, we can consider a probe M2-brane on submanifolds with a 1d boundary on probe M5-branes. The BPS condition implies that M2-branes are located at ρ = 0 and are calibrated by the two-form J 3 . We will describe M2-branes wrapping the S 2 -fiber over a point in H 3 , as well as M2-branes wrapping a hemisphere in H 3 . Such embeddings can end on M5-branes wrapping geodesics, but not on M5-branes wrapping surfaces in H 3 .

BPS condition
Similar arguments to the ones reviewed in section 3.1 lead to a BPS bound for a supersymmetric probe M2-brane: Using the decomposed gamma matrices (3.5) and spinor (3.6), we find This leads to the BPS condition If the probe M2-brane is calibrated by the first term, the condition on the AdS 4 spinors is which imposes that the M2-brane is at the center of AdS 4 , that is at r = 0, as can be seen from the explicit expressions of the spinor bilinears given in (B.6) and (B.9). On the other hand, if the probe M2-brane is calibrated by the second term in (4.3), the AdS 4 condition cannot be solved for finite r, and we therefore exclude this case. Since the M2-brane is essentially calibrated by the two-form J 3 given in (2.5), it does not wrap S 1 R , and so, in order to preserve the corresponding U (1) R-symmetry, it must be located at ρ = 0, where S 1 R shrinks. This is also the location of the probe M5-branes on which we want the M2-brane to end.
The BPS condition for a supersymmetric probe M2-brane in the PS solution is then wherevol 2 = √ḡ 2 dσ 1 ∧ dσ 2 is the volume form on the internal part of the M2-brane worldvolume induced from the metric (3.15), and the calibration two-form is We see that there are in principle three types of embeddings to consider: the M2-brane could wrap a point, a line, or a surface in H 3 . There are no solutions to the BPS condition for an M2-brane on a line, but we found BPS M2-branes at points and on surfaces in H 3 .

Spherical M2-branes
If the M2-brane sits at a constant point in H 3 , the BPS condition does not impose any constraints on the coordinates Y a on the S 2 -fiber in T * 1 H 3 : Without any additional probe M5-brane, the M2-brane would simply wrap the whole S 2 . However, there is a new possibility in the presence of an M5-brane that wraps a geodesic in H 3 and a great circle on the S 2 : the M2-brane can sit at a constant point on the geodesic, and wrap an hemisphere of the S 2 -fiber that ends on the great circle. We can also consider a probe M2-brane sitting at the intersection of two geodesics wrapped by M5-branes. On the S 2 -fiber over the intersection point, the M2-brane stretches between the two great circles wrapped by the M5-branes (see figure 8). In contrast, an M2-brane at a point in H 3 cannot end along a 1d boundary on an M5-brane wrapping a surface in H 3 (horosphere or tube) because such an M5-brane is point-like on the S 2 -fiber.

Hyperbolic M2-branes
A BPS M2-brane can wrap a geodesic plane in H 3 , which is either a vertical plane, or a hemisphere ending on the boundary z = 0. Note that geodesic planes are copies of H 2 inside H 3 .
In the case of a vertical plane, the M2-brane is at a constant point on the equator of S 2 depending on the orientation β 0 of the plane: with σ 1 ∈ R + and σ 2 ∈ R. For a hemisphere, in each S 2 -fiber over a point of the hemisphere the M2-brane is located at the corresponding point (or at the antipode): {x, y, z} = {x 0 + R 0 sin σ 1 cos σ 2 , y 0 + R 0 sin σ 1 sin σ 2 , R 0 cos σ 1 } , with now σ 1 ∈ [0, π/2] and σ 2 ∈ [0, 2π]. These two configurations are shown in figure 9.
It is amusing to note that, just like for the M5-branes, the submanifolds wrapped by the M2-branes are unit conormal bundles in T * 1 H 3 . If a geodesic curve is embedded inside a geodesic plane (vertical plane or hemisphere), the conormal fiber of the curve automatically contains the conormal fiber of the plane. We can thus consider M2-branes stretching between geodesic M5-branes in the same geodesic plane (see figure 10). Although it would naively appear that a hemispheric M2-brane can have a 1d boundary on a horosphere or on a tube wrapped by an M5-brane, closer inspection reveals that the M2-brane and the M5-brane are located at different points on the S 2 -fibers, and hence do not genuinely meet.

From geodesics to knot complements
In this section we propose a potential interpretation of our results in terms of the geometry of hyperbolic 3-manifolds (see appendix A for a small review).
We started with the Pernici-Sezgin AdS 4 solution, which involves a closed hyperbolic 3-manifold M 3 = H 3 /Γ. However, in the analysis of sections 3 and 4 we neglected the quotient by Γ and obtained BPS embeddings for probe M5-branes and M2-branes in T * 1 H 3 . We found in particular that a BPS M5-brane can wrap a geodesic in H 3 , that is a vertical line or semicircle with its endpoints on S 2 ∞ . We would like to understand how to think about this geodesic after the quotient to M 3 .
In the closed 3-manifold M 3 = H 3 /Γ, we expect that probe M5-branes wrap closed geodesics, and these correspond to loxodromic elements γ ∈ Γ. To see this, recall that a loxodromic transformation acts as a screw motion (rotation plus translation) around its axis, which is the unique geodesic in H 3 between its two fixed points on S 2 ∞ . A point p on this axis is simply translated along the axis by γ, and so the segment [p, γ(p)] projects to a closed geodesic in H 3 / γ (see figure 11). It is thus natural to associate geodesic M5-branes to loxodromic elements in Γ. Figure 11. Solid torus H 3 / γ , where γ is a loxodromic transformation (here with real eigenvalue). The corresponding closed geodesic is shown in blue. A meridian on the boundary of the solid torus is shown in red.
One of our motivations for studying the addition of a probe M5-brane on a closed geodesic in M 3 is that this can be seen as the first step towards the creation of the complement M 3 \K of a knot K ⊂ M 3 . The knot K that has been removed from M 3 should be thought of as being located at a fixed point on S 2 ∞ that is shared by two parabolic elements γ 1 , γ 2 ∈ Γ. This parabolic fixed point, or cusp, is in a sense "stretched" into the knot. The parabolic group γ 1 , γ 2 turns the horospheres around the fixed point into nested tori, so that the cusp neighborhood looks like T 2 × [0, ∞) (see figure 12). 2 A knot complement M 3 \K is thus a non-compact manifold with a cusp torus surrounding the knot at infinity. To produce such a drastic deformation of M 3 we would certainly need to wrap a very large number of coincident M5-branes on a closed geodesic.
It turns out that there is a well-known procedure in hyperbolic geometry, called conformal Dehn surgery, which precisely generates a cusp torus from a sequence of loxodromic Figure 12. A pair of parabolic elements γ 1 , γ 2 ∈ Γ with common fixed point (here at ∞) turn the surrounding horospheres (horizontal planes) into nested cusp tori. elements [21] (see also the video Not Knot [22]). 3 It consists in drilling out a tubular neighborhood of a closed geodesic in M 3 , and then filling it in with increasingly twisted solid tori. In the limit of infinite twisting, the solid torus "fractures" and creates a cusp torus on S 2 ∞ . It is then natural to interpret the addition of coincident M5-branes on a closed geodesic as the operation of conformal Dehn surgery, with the twist number related to the number of M5-branes.

Conformal Dehn surgery
We illustrate conformal Dehn surgery with an explicit example [21] (see section 4.9 in [23] and also chapter 9 in [24]). Consider first a non-compact cusped hyperbolic 3-manifold H 3 /Γ, where Γ is a parabolic group of rank 2: with w = x + iy and Im τ > 0. There is a cusp torus T 2 = C/Γ at the boundary at infinity. We choose a pair of simple loops {α, β} on T 2 that intersect once, so that every simple loop can be expressed in the form pα + qβ, with p and q relatively prime integers. Performing (p, q)-Dehn surgery means gluing a solid torus S 1 × D 2 to T 2 such that the curve pα + qβ matches the meridian on boundary of the solid torus. In particular, (1, 0)-Dehn surgery applied to a knot complement M 3 \K just gives the closed manifold M 3 . We will now perform (1, q)-Dehn surgery and see how the cusp torus is recreated in the limit q → ∞. The suitable solid torus (see figure 11) is where L q ⊂ P SL(2, C) is the loxodromic transformation The boundary of the solid torus, T 2 q = C/ L q , is identified with T 2 via the conformal map Indeed, we see that the generators γ 1 and γ 2 producing T 2 can be expressed in terms of the generator L q on T 2 q : Note that if we change the basis on T 2 via γ 1 (w) → γ 1,q (w) = w + (1 + qτ ), which corresponds to replacing α by α + qβ, we get This means that the image of α + qβ is a meridian on the solid torus. A straight line with tangent vector 1 + qτ is mapped by u q to a circle centered at 0, which then projects to a meridian on T 2 q (see figure 11). However, given that λ q → 1 for q → ∞, the sequence of loxodromic transformations {L q } converges to the identity. In order to obtain a sequence that converges geometrically to Γ = γ 1 , γ 2 , we conjugate L q such that the fixed points are at p 0 = τ /(1 − λ q ) and ∞: We now have lim q→∞Lq (u) = u + τ . Note also that the fixed point p 0 tends to the other fixed point at ∞ as q → ∞, and the axis ofL q contracts to a point (see figure 13). The conformal map is taken to beũ which also satisfies properties such as (5.5) and (5.6). Finally, from the fact that limũ q (w) = w it follows that The sequence of loxodromic groups is said to converge geometrically to the rank-2 parabolic group Γ = γ 1 , γ 2 (see figure 14). In conclusion, the sequence of solid tori H 3 / L q converges for q → ∞ to the non-compact 3-manifold H 3 /Γ with a cusp at its core. Figure 14. Left: Acting with the loxodromic transformationsL q orL −q q on a point p translates it in different directions on a cone around the axis ofL q . Right: in the limit q → ∞, the fixed point p 0 recedes to ∞ as in figure 13, and the cone becomes a horizontal plane (the horosphere around ∞). We are left with two parabolic elements γ 1 = limL −q q and γ 2 = limL q .

Discussion
We have presented supersymmetric probe M5-branes in the Pernici-Sezgin AdS 4 solution that preserve the superconformal symmetries of the dual N = 2 SCFT. They extend along all of AdS 4 and are located where the S 1 R dual to the R-symmetry shrinks to a point. We have shown that the BPS condition for M5-branes (but also for M2-branes ending on them) then boils down to the calibration of a surface in the unit cotangent bundle T * 1 M 3 (with S 2 -fibers) of the hyperbolic 3-manifold M 3 = H 3 /Γ. We have found solutions corresponding to all the natural objects appearing in the geometry of hyperbolic 3-manifolds. BPS M5-branes can wrap geodesic curves in M 3 , on which BPS M2-branes wrapping geodesic surfaces can end. M5-branes can also wrap invariant surfaces which are equidistant from the axis of a loxodromic or elliptic transformation (tubes), or from a parabolic fixed point (cusp torus or annulus).
In all these cases, the calibrated surface turns out to be simply the unit conormal bundle N * 1 L ⊂ T * 1 M 3 of a submanifold L ⊂ M 3 . For example, an M5-brane on a geodesic in M 3 also wraps a rotating great circle in the S 2 -fibers. This suggests that in the UV regime the M5-branes are wrapping special Lagrangian submanifolds given by the conormal bundles N * L ⊂ T * M 3 . It would be very interesting to make this perspective more precise and to study the flow from the UV to the IR.
If we were to wrap a large number of supersymmetric M5-branes on N * 1 L, they would ultimately backreact on the Pernici-Sezgin geometry and produce a new AdS 4 solution, arising from intersecting stacks of M5-branes (this would be an AdS 4 analogue of the general AdS 5 solution found in [25]). Just like in conformal Dehn surgery, we expect that the original closed 3-manifold M 3 will develop cusps along the way, and could then be a knot (or link) complement. Since the BPS M5-branes that we found preserve an extra S 1 in addition to S 1 R , we predict that this general AdS 4 solution will have a U (1) 2 isometry. An important open problem is to match our results to the dual 3d N = 2 SCFTs. Gauge theories associated with 3-manifolds have been constructed in various ways, starting with [1,2], and for higher rank in [3]. Theories associated with closed 3-manifolds have been presented in [26,27]. The volume of the hyperbolic 3-manifold M 3 will give the free energy of the 3d theory on an ellipsoid [16] (see [17] for calculations for knot complements).
The probe M5-branes on geodesics are expected to correspond to flavor symmetries. In particular, SCFTs associated with knot complements should have an SU (N ) flavor symmetry [3]. Probe M2-branes ending on M5-branes correspond to BPS operators, and the volumes of the surfaces they wrap will give their conformal dimensions. We can anticipate that M5-branes on geodesics and M5-branes on surfaces will play very different roles in the 3d theory, since M2-branes can end on the former but not on the latter. We have found two types of M2-branes, which will correspond to two types of BPS operators: those stretching between geodesics on a hyperbolic surface, or those stretching between great circles on an S 2 -fiber. On the other hand, M5-branes on tubes in M 3 might find an interpretation in terms of domain walls and couplings to 4d N = 2 theories, as for example in [28].
We have seen that the worldvolume of a BPS M5-brane is of the form AdS 4 × R × S 1 . The Kaluza-Klein reduction on this S 1 of the two-form potential on the worldvolume of a probe M5-brane produces a U (1) gauge field in AdS 4 , corresponding to a global U (1) symmetry in the dual 3d theory. This extra U (1) symmetry, which comes in addition to the U (1) corresponding to the R-symmetry, was recently shown in [27] to play a key role in the 3d-3d correspondence.
There are several commonly used models of hyperbolic 3-space H 3 . We mostly use the upper half-space model with the metric We also refer occasionally to the ball model { x ∈ R 3 : | x| < 1} with the metric We denote the boundary at infinity of H 3 by S 2 ∞ , which in the case of the upper-half space model has to be understood as the plane z = 0 together with the point at infinity, C ∪ {∞}.
In the upper half-space model, geodesics are vertical lines and semicircles orthogonal to S 2 ∞ , while they are diameters and arcs orthogonal to S 2 ∞ in the ball model (see figure 15). Geodesic surfaces are vertical half-planes and hemispheres orthogonal to S 2 ∞ . The group of orientation-preserving isometries of H 3 is Isom + (H 3 ) ∼ = P SL(2, C). The action of P SL(2, C) on H 3 can be expressed as fractional linear transformation on a quaternion q = x + iy + jz: In terms of the complex coordinate w = x + iy this gives w → (aw + b)(cw +d) + acz 2 |cw + d| 2 + |c| 2 z 2 , z → z |cw + d| 2 + |c| 2 z 2 . (A.4) Any element of P SL(2, C) is conjugate to one of the following three standard matrices: Parabolic elements in Γ generate cusps, which make M 3 non-compact. If a fixed point is shared by a pair of parabolic elements, its cusp neighborhood is isometric to T 2 × [0, ∞), where T 2 is a torus generated by the pair of translations on a horosphere around the fixed point (as in figure 12). A parabolic fixed point that is not shared is associated with a cusp annulus that relates two punctures on the boundary of M 3 .
To perform explicit calculations, we choose a basis for the Clifford matrices: Expanding the BPS condition (3.3) in powers of r, we see that the M5-brane must be calibrated by the term involving V + as claimed in section 3.1, and that the pullback of U to its worldvolume must vanish. We saw that the first condition requires the M5-brane to be at ρ = 0 in order to preserve the U (1) R-symmetry, which then implies the second condition U | M5 = 0 since we have U = S * Re [χ c + γ (2) χ − ] + L * ∧ P = λ 2 e −i(ψ−τ ) −λ 1/2 ρJ 1 + 1 4 Note that the other two-form appearing in (3.9) is given by which also vanishes at ρ = 0. We also need the following spinor bilinear that appears in the BPS condition (4.3) for a probe M2-brane:ψ + 1 (ψ + 2 ) c = 1 + i sinh r sin α 1 cos α 2 . (B.9)