"Lagrangian Disks"in M-theory

While the study of bordered (pseudo-)holomorphic curves with boundary on Lagrangian submanifolds has a long history, a similar problem that involves (special) Lagrangian submanifolds with boundary on complex surfaces appears to be largely overlooked in both physics and math literature. We relate this problem to geometry of coassociative submanifolds in $G_2$ holonomy spaces and to $Spin(7)$ metrics on 8-manifolds with $T^2$ fibrations. As an application to physics, we propose a large class of brane models in type IIA string theory that generalize brane brick models on the one hand and 2d theories $T[M_4]$ on the other.

Abstract: While the study of bordered (pseudo-)holomorphic curves with boundary on Lagrangian submanifolds has a long history, a similar problem that involves (special) Lagrangian submanifolds with boundary on complex surfaces appears to be largely overlooked in both physics and math literature. We relate this problem to geometry of coassociative submanifolds in G 2 holonomy spaces and to Spin(7) metrics on 8-manifolds with T 2 fibrations. As an application to physics, we propose a large class of brane models in type IIA string theory that generalize brane brick models on the one hand and 2d theories T [M 4 ] on the other.

Introduction and summary
A familiar setup, both in physics and in math, involves a (special) Lagrangian submanifold L in a Calabi-Yau 3-fold X together with bordered pseudoholomorphic curve Σ with boundary in L. Exploring all possible choices of Σ with fixed X and L is the main goal of "open Gromov-Witten theory" and plays an important role in counting disk instantons [1] which, in turn, are paramount for stability of string vacua.
In this paper we consider a much less explored version of this setup, which involves a similar triple, illustrated in Figure 1: X : Calabi-Yau 3-fold S : complex surface L : special Lagrangian X 7 : G 2 holonomy M 4 : coassociative X 8 : Spin(7) holonomy (1.1) Here, X, S, and L are in a similar mutual relation as ingredients of open Gromov-Witten theory; namely, S, L ⊂ X and L has boundary in S. As far as we know, moduli spaces of such special Lagrangians with boundary on S have not been studied in the previous literature. 1 In our models, we allow X and S to be compact or non-compact, whereas L is always assumed to be compact with non-trivial boundary Σ := ∂L ⊂ S (1. 2) In general, we allow L to have several connected components L i , i = 1, . . . , , possibly taken with multiplicity N i ∈ Z + . X S L Figure 1. Our geometric setup involves a holomorphic surface S in a Calabi-Yau 3-fold X, and a special Lagrangian submanifold L ⊂ X with boundary Σ := ∂L ⊂ S.
Examples of (X, S, L) include the following natural choices: • One can take X = C * × C * × C * = T * (T 3 ) and L = T 3 split into several "disks" L i ∼ = D 3 (each of which has topology of a 3-ball) by a holomorphic surface S ⊂ X. These are the so-called brane brick models (BBMs) [3][4][5].
• More generally, one can take X = T * L to be the total space of the cotangent bundle over a 3-dimensional base manifold L, with a simple choice of L such as R 3 , T 3 , S 3 , S 2 × S 1 , etc. Each of the resulting spaces T * L admits a complete Calabi-Yau metric.
• The previous class of examples contains the simplest Calabi-Yau 3-fold X = C 3 . Even with this choice of X, one can generate many interesting models by taking different hypersurfaces S defined by the zero locus of a polynomial P (z 1 , z 2 , z 3 ): • Note, same Σ can bound several Lagrangians which, moreover, can be special Lagrangian. In 2d N = (0, 2) theory, different choices of L bounded by the same Σ correspond to different (classical) vacua of the same 2d theory: For example, let X be the "deformed conifold" defined by a quadratic equation X : z 2 1 + z 2 2 + z 2 3 + z 2 4 = , (z 1 , z 2 , z 3 , z 4 ) ∈ C 4 (1.5) A hypersurface defined by imposing an additional equation z 4 = 0 has topology S ∼ = T * S 2 with a single non-trivial 2-cycle Σ ∼ = S 2 that bounds two special Lagrangian disks L 1 ∼ = L 2 ∼ = D 3 in X.
• Let I i ⊂ R 3 and C j ⊂ R 3 be a collection of intervals I i and 2-planes C j , such that all intervals are parallel to each other, with end-points on 2-planes, and such that C j 's are also mutually parallel and orthogonal to I i 's. Every such arrangement defines X = C 2 × T 2 , S = j (C j × T 2 ), and L = i (I i × T 2 ) which corresponds to 2d compactification of the models studied in [6].
• Let X be the famous quintic Calabi-Yau 3-fold: where f (z i ) is a homogeneous polynomial of degree 5. Imposing an extra linear relation 2 cuts out a degree-5 surface of general type, which is simply-connected and has b + 2 (S) = 9, b − 2 (S) = 44. Since b 2 (X) = 1, there are many primitive 3 homology classes in the 53-dimensional lattice H 2 (S; Z) that are trivial in X, thus providing many potential candidates for Σ that bound Lagrangian "disks" L ⊂ X.
While the input data for each of our models is the triple (X, S, L), the output can be summarized either as a pair (X 7 , M 4 ) of a G 2 holonomy 7-manifold X 7 with a coassociative submanifold M 4 , or as an 8-manifold X 8 of Spin(7) holonomy. They are produced from the input data via the map that we call "M-theory lift," because (X 7 , M 4 ) is the result of lifting to eleven dimensions the following type IIA brane configuration, space-time: R 4 × X ∪ ∪ NS5-branes: R 2 × S D4-branes: whereas X 8 is similarly produced [7,8] from D6-branes supported on R 3 × M 4 in type IIA space-time R 3 × X 7 . This M-theory lift of D6-branes supported on R 3 × M 4 is an auxiliary problem from the viewpoint of our main setup (1.7) and is modeled after [9,10], where analogous M-theory lift of D6-branes on special Lagrangian 3-manifolds was considered.
When the Calabi-Yau 3-fold X is compact and has irreducible SU (3) holonomy group, a compactification of type IIA string theory on X produces a 4d N = 2 effective supergravity theory coupled to matter. In this theory, a configuration of NS5 and D4-branes as in (1.7) engineers a 1 4 -BPS surface operator, which preserves N = (0, 2) supersymmetry on its twodimensional world-volume R 2 . Since the M-theory lift of this configuration has to preserve the same symmetry and supersymmetry, we quickly learn that it is given by M5-branes wrapped on a coassociative cycle in a G 2 holonomy manifold, When a certain anomaly discussed in section 3 vanishes, the topology of the G 2 manifold is simply X 7 ∼ = S 1 × X, and 4 When the anomaly of section 3 is non-trivial, the story involves additional ingredients and the geometry of X 7 and M 4 becomes more interesting. In particular, X 7 is topologically no longer a product S 1 × X, but rather a non-trivial circle fibration: Similarly, X 8 is a circle fibration over X 7 or, equivalently, a torus fibration over X: Upon the lift to (M 4 , X 7 ) or to X 8 , the moduli problem for Lagrangian "disks" L with boundary on S ⊂ X translates into a more tractable and better understood problem of moduli of Spin(7) metrics or moduli of coassociative submanifolds in G 2 holonomy spaces. We hope that families of new Spin(7) metrics on (1.11) labeled by (X, S, L) can be constructed using methods similar to those in [13,14], where many new G 2 analogues of Taub-NUT spaces were found. The paper is organized as follows. We start in section 2 with a simple example of the D4-NS5 brane model (1.7). In section 3 we describe a new chiral anomaly on 4d boundary of D4-branes. In section 4 we construct new coassociative submanifolds in R 3 × Taub-NUT which are ALF generalizations of the ALE coassociatives in R 7 constructed by Harvey and Lawson [15]. In section 5 we analyze in detail the physics of one of the D4-NS5 brane models (1.7) and make a proposal for the 2d N = (0, 2) effective theory. In particular, via lift to M-theory on a G 2 manifold, it gives us a concrete description of the moduli space (1.4) in that brane model: (1.12)

A simple instructive example
If one wants to construct the simplest model labeled by (X, S, L), for the choice of X nothing can be simpler than C 3 . For the choice of complex surface S, it is natural to take the zero locus (1.3) of a polynomial P (z 1 , z 2 , z 3 ). Since we want S to contain at least one non-trivial 2-cycle Σ, the degree of P (z 1 , z 2 , z 3 ) has to be at least 2. So, in our simple toy model we make a symmetric choice S : which has the added benefit of an extra symmetry SU (2) ∼ = SO (3). Without loss of generality, we can take the constant to be real and positive. Topologically, S is a line bundle L = O(−2) over CP 1 . In particular, S admits a non-contractible 2-cycle Σ ∼ = S 2 which, of course, is contractible in the ambient space X = C 3 . The 2-cycle Σ can be described explicitly by the same equation (2.1) with (z 1 , z 2 , z 3 ) restricted to the real slice: In the ambient space X = C 3 it can be "filled in" by a disk -or, rather, by a 3-dimensional ball -with boundary Σ: L : It is easy to check that L is, in fact, special Lagrangian with respect to the flat Calabi-Yau structure on X = C 3 : Once we introduced the key players, X, S, and L, they appear to define a perfectly sensible configuration of NS5 and D4-branes (1.7) with world-volumes R 2 × S and R 2 × L, respectively. This choice of (X, S, L), however, suffers from an important anomaly. As we explain in the next section, this anomaly can be easily cured. However, as it stands, the triple (X, S, L) introduced here does not define a consistent model and, in particular, does not admit a lift (1.1) to a coassociative submanifold M 4 ⊂ X 7 or to a Spin(7) manifold X 8 .

Euler number anomaly
The simple choice of (X, S, L) introduced in the previous section is a good example to illustrate an important anomaly in our class of models, which in general takes values in Specifically, for each connected component of Σ, there is a potential Z-valued anomaly given by the self-intersection of that component in the complex surface S: This anomaly can be seen both physically and geometrically. In this section we begin by illustrating the anomaly for the simple example in section 2, focusing on the M-theory geometric perspective. In section 3.2 we then present a general discussion of the anomaly in type IIA string theory, how this lifts to the M-theory picture, and how the anomaly can be cured (in a more or less canonical way). Finally, section 3.3 presents the QFT manifestation of the anomaly in our class of models.

Example
In our simple example (2.1)-(2.3), the lift to M-theory involves a 7-manifold of G 2 holonomy, X 7 , which topologically is simply a product of X and the M-theory circle S 1 . (This is a general feature of any type IIA background that does not involve D6-branes or RR 2-form fluxes.) Likewise, a lift of the Lagrangian disk (2.3) is also a product L×S 1 , which is automatically coassociative in X 7 = X × S 1 with respect to the natural G 2 structure (4.1). The 4-manifold L × S 1 , however, has boundary Σ × S 1 and needs something to end on. Indeed, both NS5 and D4-branes turn into M5-branes upon M-theory lift. And, just like in type IIA theory the NS5-brane (2.1) serves as a boundary condition for the D4-brane (2.3), in M-theory they join into one single 4-manifold since the M-theory lift of the NS5-brane is simply S ⊂ X 7 . Now we are starting to see the anomaly or, at least, its geometric manifestation. The boundary of L × S 1 is Σ × S 1 . The boundary of S \ Σ locally looks very similar; the normal sphere bundle to Σ ⊂ S is just a circle S 1 bundle. However, when the normal bundle is non-trivial, the boundary of S \ Σ is a non-trivial S 1 bundle over Σ of degree n given by the self-intersection (3.2). Therefore, when n = 0, we cannot simply glue S \ Σ and L × S 1 ⊂ X 7 into a single 4-manifold (1.9) along their boundaries since these boundaries are different. This is precisely the case in our simple example, where n = −Σ · Σ = 2.

Type IIA interpretation and M-theory lift
The geometric anomaly just illustrated may be seen from the type IIA brane perspective as follows. On the worldvolume of the NS5-brane propagates a periodic scalar field ϕ, where we normalize ϕ to have period 2π. This worldvolume theory arises from dimensional reduction of the M5-brane theory [16], where the NS5-brane is transverse to the M-theory circle direction, S 1 M . The scalar ϕ then corresponds to motion of the M5-brane in that direction. On the other hand, a D4-brane ending on an NS5-brane acts as a magnetic source for this scalar [6]. In our set-up, recall from (1.7) that the NS5-brane worldvolume is R 2 × S ⊂ R 4 × X, where the D4-brane wrapped on R 2 × L shares the R 2 directions with the NS5-brane. Suppressing the common R 2 directions, the end of the D4-brane is then ∂L = Σ ⊂ S. Since this has codimension 2 in S, it is linked by a circle S 1 . As one goes around such a transverse S 1 ⊂ S, the periodic scalar ϕ winds once around 2π, so that R 2 × Σ is effectively the locus of a "vortex" for the ϕ field.
We may describe this in more detail as follows. The scalar ϕ enters the worldvolume theory of the NS5-brane via its gauge-invariant curvature [16] where C 1 denotes the RR 1-form potential, and ι : R 2 ×S → R 4 ×X denotes the embedding of the NS5-brane into space-time, with ι * a pull-back to the worldvolume. Of course, we might a priori choose to turn off this RR potential, setting C 1 = 0, since neither the NS5brane nor D4-brane source it. To say that Σ ⊂ S is a magnetic source for ϕ then means that Here δ 2 is a delta-function representative of the Poincaré dual of Σ ⊂ S, i.e. δ 2 is a closed 2-form on S which restricts to a Dirac delta-function times the volume form in the normal R 2 directions of Σ. Integrating (3.4) along such a normal R 2 , with positive constant radius circle S 1 ⊂ R 2 , one obtains 2π = This says that ϕ winds once around 2π as it moves around an S 1 linking Σ. More generally for k D4-branes the winding is 2πk.
On the other hand, from (3.3) we have in general that where G 2 = dC 1 is the RR 2-form flux. Equating cohomology classes of the left and right hand sides of (3.6) then says In other words, when the end of the D4-brane Σ ⊂ S has a Poincaré dual δ 2 that defines a non-trivial cohomology class in H 2 (S, R), one must necessarily turn on a RR G 2 field obeying (3.7): to not do so leads to our anomaly. Let us examine this further. We denote the normal bundle of Σ inside S as N Σ, which is a complex line bundle. Such This generator is also known as the Thom class. There is a natural mapping from where the map simply forgets the compact support condition. This maps the Thom class 1 ∈ Z to the Euler number −n i ∈ Z. Putting all this together, we may integrate (3.6) over Σ i to obtain Recall that geometrically G 2 is the curvature of the M-theory circle bundle over spacetime. Thus (3.8) says that we must fiber the M-theory circle bundle in such a way that the Euler number of this fibration over Σ i , which is the left hand side of (3.8), is the same as the self-intersection number n i = −Σ i · Σ i , which is also minus the Euler number of the normal bundle N Σ i of Σ i in S. There are two ways to generate such a flux in our set-up, where recall that space-time is R 4 × X, with X a Calabi-Yau 3-fold: we may turn on RR 2-form flux on X, and/or introduce D6-branes into the space-time, which magnetically source such a flux. In order to preserve supersymmetry, the RR 2-form flux on X should be Hodge type (1,1), and D6-branes should wrap a special Lagrangian L D6 ⊂ X and are space-filling in the R 4 directions. The lift of X to M-theory is then not simply a product X × S 1 M , but rather the total space X 7 a non-trivial circle fibration over X, (1.10), which degenerates at D6-brane loci. Supersymmetry implies that X 7 is a G 2 holonomy manifold. Notice that in the simple case that X = C 3 , as in the example in section 2, since H 2 (X, Z) = 0 we must necessarily introduce D6-branes to source the required flux on the left hand side of (3.8).
We shall return to discuss this example at the end of the next subsection.
The condition (3.8) has a very elegant geometric interpretation, once we lift the NS5brane and D4-brane configuration to M-theory. Both objects descend from an M5-brane, where the NS5-brane is transverse to the M-theory circle while the D4-brane wraps the circle. Since the D4-brane ends on the NS5-brane, the configuration should lift to a single M5-brane wrapped on a 4-manifold M 4 in M-theory. The classical picture of the D4-brane ending on a definite submanifold Σ ⊂ S inside the NS5-brane is not quite accurate, due to the resulting distortion near to this locus. Let us examine the lift of both branes away from the intersection locus Σ. The NS5-brane wraps S \ Σ, which near to a connected component Σ i looks like N Σ i \ Σ i ∼ = I × M 3 , where I = (0, ] is an interval and M 3 is the total space of a degree n i circle bundle over Σ i On the other hand, the D4-brane wraps L \ Σ, which near to the ith end looks like I × Σ i . On lifting to M-theory, the M-theory circle fibres over this latter geometry to give the M5brane worldvolume, and (3.8) says that the total space of this S 1 M bundle over I × Σ i is also precisely I × M 3 . The condition (3.8) thus ensures that the boundaries of these two neck regions, around where the NS5-brane and D4-brane meet, lift to the same 3-manifold M 3 in M-theory, and as in [6] these are glued together to produce a single smooth M5-brane by connecting the necks via a small "tube" [− , ] × M 3 . The M5-brane is then wrapped on the smooth 4-manifold On the other hand, without the addition of the RR 2-form flux the boundaries are different 3-manifolds, so we cannot simply glue them together. We saw this for our explicit example in section 3.1. We conclude this subsection with a few more general remarks. Suppose one has a submanifold S in space-time over which there is a non-trivial RR 2-form flux, i.e. the pullback [ι * G 2 ] = 0 ∈ H 2 (S, R). Then one cannot wrap an NS5-brane over S. Geometrically, this is because the NS5-brane is an M5-brane transverse to the M-theory circle, and thus the M-theory circle bundle over S must have a global section. But this is true if and only if the S 1 M bundle is trivial over S, and hence [ι * G 2 ] = 0 ∈ H 2 (S, R). From the point of view of the NS5-brane worldvolume theory, one sees the same fact from the equation Since the worldvolume periodic scalar ϕ is also a section of the same M-theory circle bundle over S, this scalar field exists as a global function on S if and only if that bundle is trivial. In this case F 1 = dϕ − ι * C 1 is a global 1-form on S, and hence ι * G 2 is exact. If instead [ι * G 2 ] = 0 ∈ H 2 (S, R), the best one can do is to remove a submanifold Σ ⊂ S that is Poincaré dual to −[ι * G 2 /2π]. By construction, the M-theory circle bundle is then trivial over S \ Σ, and an NS5-brane can wrap this locus. Of course, physically we may then require a D4-brane to end on the locus Σ, thus satisfying (3.6), and this system then lifts to a single smooth M5-brane, without additional boundaries. In this case the worldvolume scalar ϕ is not defined at the locus Σ ⊂ S due to the winding by 2π in (3.5), just as the angular coordinate θ is not defined at the origin of R 2 in plane polar coordinates.
Similar remarks apply to a D2-brane with a fundamental string ending on it, which lifts to an M2-brane. In particular, the D2-brane worldvolume theory contains a periodic scalar corresponding to motion of the M2-brane in the M-theory circle direction. In three dimensions a periodic scalar is dual to an Abelian gauge field, which in this case is nothing but the usual U (1) gauge field on a D-brane. A fundamental string ending on a D-brane is of course an electric source for this gauge field, or equivalently for the D2-brane a magnetic source for the dual periodic scalar.

QFT interpretation
Now let us discuss the QFT origin and interpretation of this "Euler number anomaly." Just like in its geometric manifestation, the crucial ingredient is the neck region along which S \ Σ and L × S 1 are supposed to be joined. Recall from the previous subsection that near the ith connected component of this neck region, S \ Σ looks like I × M 3 , where I = (0, ] and M 3 is the total space of a degree n i circle bundle over Σ i in (3.9). Let us look more closely at the physics of a five-brane near this neck region, i.e. a five-brane on R 2 × I × M 3 . More generally, we can consider N five-branes supported on R 2 × I × M 3 . Σ L Figure 2. An illustration of the anomaly inflow from the boundary into the bulk of the D4-brane world-volume theory.
Since I ×M 3 is part of the curved 4-manifold along which the five-brane is topologically twisted, a priori the interval I is not quite on the same footing as the R 2 part of the fivebrane world-volume. However, since I admits a flat metric, and since in flat space the full physical and topological theories are the same, we can treat R 2 × I as a space-time of the 3d physical theory obtained by compactifying a five-brane on M 3 . This physical 3d theory has N = 2 supersymmetry and is usually denoted T [M 3 ]. It is defined for any 3-manifold M 3 , but in our applications here we only need to know this theory for very special 3-manifolds of the form (3.9). For example, when Σ = S 2 , as in (2.2), we have M 3 = L(n, 1) and T [M 3 , G] is the following "Lens space theory" [12,17]: 3d N = 2 super-Chern-Simons with G n and an adjoint chiral (3.11) The important aspect of this theory is that it involves a Chern-Simons coupling for the gauge group G at level n. In our simple example (2.3), the interval I = (0, ] is parametrized by the radial coordinate r = | x|, and the 3d theory we described is a result of the reduction of the five-brane world-volume theory on the M-theory circle and Σ = S 2 (or, more precisely, on an S 1 bundle over Σ). At the end-points of the interval I we need to impose 2d boundary conditions, which must preserve N = (0, 2) supersymmetry because this is the amount of supersymmetry preserved by the brane configuration.
The 2d N = (0, 2) boundary conditions in question have a geometric origin: at one end of the interval the boundary condition is determined by the 4-manifold S \ Σ. This is where D4-brane runs into NS5-brane. At the other end of the interval, the boundary condition is determined by what happens at the limit r = 0. The resulting system looks like a 3d N = 2 theory (3.11) sandwiched by 2d N = (0, 2) boundary conditions, precisely of the type studied in [12] and illustrated in Figure 3.
2d boundary condition 2d boundary condition 3d theory Figure 3. An illustration of the anomaly inflow in 3d N = 2 theory with 2d N = (0, 2) boundary conditions. For example, if one boundary carries charged chiral fermions whose anomaly is compensated by Chern-Simons couplings of the 3d bulk theory, a similar mechanism must be at work at the other boundary.
In particular, when n = 0, there is a non-trivial anomaly inflow that also played an important role in [12]. And, what our above analysis shows is that the boundary condition where the D4-brane runs into the NS5-brane infuses n units of anomaly compensated by the Chern-Simons coupling of the 3d theory and also carried to the other boundary. Therefore, the other boundary condition (at r = 0 in our simple example) should also carry n units of anomaly, i.e. it should carry chiral degrees of freedom charged under G (= gauge group of D4-brane theory).
The original system of NS5 and D4-branes has no such degrees of freedom at r = 0. Therefore, it is anomalous. However, our analysis in the previous subsection also suggests what one should do in order to cancel this anomaly. The simplest possibility is to add n D6-branes that, geometrically, would replace a product with the M-theory circle by a non-trivial bundle and, physically, would produce n charged chiral fermions at the 2dimensional intersection of the D4 and D6-branes (the lowest modes of D4-D6 open strings). As mentioned earlier, in general another possibility is to turn on RR 2-form flux. In this paper, we mostly consider the first option.
The extra D6-branes that we need to add in order to cancel the Euler number anomaly (when n = 0) should have the following properties. First, their 7-dimensional worldvolume should intersect the world-volume R 2 × L of the D4-branes along R 2 (times a point in L, which in our simple example is best to be chosen at r = 0 in order to preserve SU (2) symmetry of the background). Second, the D6-branes must preserve the unbroken supersymmetry of the original brane system (1.7). These two conditions imply that D6branes must be supported on R 4 ×L D6 , where L D6 ⊂ X is a special Lagrangian submanifold calibrated by the 3-form Im(Ω), provided L is calibrated by Re(Ω): See Appendix A for details.
In our model example from section 2, a special Lagrangian L D6 ⊂ C 3 that meets (2.3) at x = 0 and is calibrated by Im(Ω) is a three-dimensional plane: In its presence, the corresponding M-theory lift of (3.12) is of the form (1.10): with a coassociative submanifold, cf. (1.9), where M 3 is the 3-manifold introduced in (3.9). Since in our example the complex surface S is itself asymptotically a cone on M 3 , and n = 2, it follows that Note, instead of D4-branes supported on Lagrangian disks (2.3), illustrated in Figure 2, one can consider non-compact D4-branes supported on the "disk complement," L : Although this option may seem less natural for the purpose of building a dynamical 2d gauge theory on D4-brane world-volume, it does have very interesting physics, as will be discussed in section 5. Brane models with such non-compact D4-branes also exhibit the familiar Euler number anomaly which, much like (3.2), can be expressed in terms of the D4brane boundary components Σ i ⊂ S. However, since the orientation of the boundary of a D4-brane on a "disk complement" is reversed compared to that supported on a Lagrangian disk, the Euler number anomaly in these two cases differs by sign, which is equivalent to n i → −n i . In our simple class of models with n D6-branes, replacing compact D4-branes on Lagrangian disks by non-compact D4-branes supported on disk complements (3.16) leads to coassociative submanifold in (3.14) that are double-ended cones on When the apex is smoothed out, these become topologically The next section offers an explicit construction of such coassociative submanifolds.

New coassociative submanifolds
When X 7 = S 1 × X, the associative and coassociative forms on X are where ψ is a coordinate on S 1 . In particular, it is clear that, given a special Lagrangian L ⊂ X calibrated by Re(Ω) and a complex surface S ⊂ X, both S 1 × L and S are individually coassociative in X 7 . We are interested in a coassociative 4-manifold M 4 ⊂ X defined by "smoothing" of S 1 × L and S near the "neck" region S ∩ (L × S 1 ) = S 1 × Σ. As we saw in section 3, such smoothing is possible only if the self-intersection of Σ inside S vanishes. If that is the case, M 4 is given by the (deformation of) eq. (1.9). When the self-intersection of Σ in S is non-zero, we need to replace X 7 = S 1 × X by a non-trivial circle bundle (1.10), such that Motivated by our discussion in sections 2 and 3, let us analyze more carefully how this works in the class of examples with X = C 3 and one component (3.13) of the codimension-4 fixed point set supported on R 3 ⊂ C 3 . For a circle bundle of degree n, the total space (3.14) carries a G 2 structure determined by the first relation, where ω i are self-dual 2-forms on the Taub-NUT space TN n of "charge" n. The standard way to write a hyper-Kähler metric on the Taub-NUT space, which is ideally suited for our application to (3.14) with coordinates ( x, y, ψ), is Equivalently, dχ = * 3 dH, where * 3 is the Hodge dual with respect to the flat metric of R 3 parametrized by x. In these conventions, the triplet of the self-dual 2-forms on TN n that appear in (4.4) can be explicitly written as In order to preserve the SO (3)  and Σ ∼ = S 2 was already described in (2.2) as a sphere in x-plane of radius . The Lagrangian L is a 3-ball ("disk") within this plane, whereas the Lagrangian L D6 ∼ = R 3 is a copy of y-plane at x = 0, cf. (3.13).
In order to construct new coassociative 4-manifolds M 4 in (4.3) with the SO(3) symmetry, we need to fix the angular dependence as in (4.9) and allow more general radial dependence that will be encoded in a single function g of the radial variable r = | x|. In particular, separating radial and angular variables, the equation for complex surface (4.9) can be written as x = (cosh ρ) m , y = (sinh ρ) n , | m| = | n| = 1 , m · n = 0 . (4.10) Note, the locus of small constant ρ defines a circle bundle over Σ = S 2 .
Then, inspired by (4.9) and (4.10), we look for a coassociative 4-manifold M 4 of the form y = g(r) n(θ, φ, ψ) , n · n = 1 , m · n = 0 . where m = (θ, φ, ψ) denote the standard angle coordinates on S 3 . In these coordinates, the explicit expression for 1-form χ compatible with our choice of the harmonic function (4.8) is χ = n cos θdφ, and the Taub-NUT space is parametrized by the "polar" coordinates (r, θ, φ, ψ) instead of the original ones ( x, ψ). The angular coordinate ψ here is related to ψ in (4.5) by ψ = nψ . From the last two equations in (4.11) it follows that, for each given m, the unit vector n takes values in a circle. In fact, here we are interested in a solution such that n winds exactly once around this circle as ψ runs from 0 to 2π. Implementing this in our ansatz (4.11) and requiring that M 4 is calibrated with respect to (4.4)-(4.8), we obtain a single ordinary differential equation (ODE) for the function g(r): In fact, as we explain momentarily, the specific value of B is irrelevant, as long as B = 0. It can be changed to any other value by rescaling r and g. Therefore, as far as the dependence on B is concerned, there are essentially two cases to consider: B = 0 and B = 0. The ordinary differential equation The substitution z = (A − 1 + Br)dr brings it to the canonical form 6 where Φ(z) is defined parametrically, by eliminating r in the relations 5 We thank Robert Bryant for pointing this out to us. 6 Also note that, via a change of variable F = f −1 , we can bring (4.15) into the form of the Abel equation of the first kind: In analyzing the solutions of (4.13) and (4.14) it will be convenient to consider even a larger family of ODEs: with three parameters, A, B and C. When parameters A and C are related as in (4.13), i.e. A = 2n = 4C, this more general ODE has a peculiar property that all parameters can be scaled away completely 7 by rescaling g and r, In particular, when B = 0, without loss of generality we can bring it to a form (4.14) with A = 4 and B = 1. Now, let us explore solutions to the above Abel equation, starting with the special ones that correspond to setting either A or B to zero. When n = 0, i.e. A = C = 0 in (4.18), a simple solution is NS5-brane : It describes the original NS5-brane (4.9) without D6 or D4-branes. Another special case corresponds to setting B = 0 in (4.14) and leads to a famous asymptotically conical coassociative submanifold in R 7 constructed by Harvey and Lawson [15]. Indeed, after a change of variable r = ρ 2 /8 our ODE with B = 0 has the property that both the numerator and denominator on the right-hand side are homogeneous (of degree 2) in g and ρ. As usual in such cases, the standard substitution u = g ρ gives and allows to write the explicit solution: The case A = 4 is precisely the solution of Harvey and Lawson: Topologically, this coassociative 4-manifold is a spin bundle over S 2 . Therefore, as expected, our coassociative submanifolds defined by the ansatz (4.11) generalize the Harvey-Lawson construction by replacing R 7 with a more general 7-manifold X 7 = R 3 × TN n .

Solving the Abel equation
Now, let us study solutions to (4.13) and (4.14) more systematically. Motivated by the special solutions (4.20) and (4.23) we first consider the behavior of g in the limits of small and large r. 7 Specifically, under (4.19) this equation becomes

Asymptotic behavior
Small r: In the limit r → 0, the equation (4.14) can be approximated by , (4.24) whose solution is where c 1 is an integration constant determined by the boundary conditions. For concreteness, let us consider a separatrix solution that corresponds to the initial condition g(0) = 0, i.e. c 1 = 0. We thus get g = (2 + 2A)r. (4.26) Large r: Assuming that g grows faster than r 1/2 as r → ∞, an assumption that we shortly verify to be self-consistent, (4.14) becomes: where we set B = 1 by scaling symmetries (4.19). The solution to this approximate equation is g = r 2 + ζ. If we are interested in the leading behavior at large r, the integration constant ζ can be neglected.

Full solution: the interpolating function
Let us introduce the following function which reproduces the small and large r asymptotic behavior of g. Figure 4 presents a comparison between the numerical solution for g and g 0 for A = 1. The two functions rapidly become indistinguishable for larger values of r. We will later discuss the accuracy of this approximation in further detail and its dependence on A.
It is convenient to decompose g into a product as follows where g 0 takes care of the asymptotic behavior as r → 0 and ∞. We refer to f as the interpolating function. It is clearly a function with the boundary values f (0) = f (∞) = 1. Plugging (4.30) into (4.14), we obtain the following differential equation for f   The A = 1 case Let us first consider the case of A = 1. Figure 5 shows the function f , obtained by numerically solving (4.31). It is interesting to note that the maximum value of f is around 1.04, which means that in this case g 0 is already a reasonably good approximation to g, differing from it by at most 4%. This function becomes remarkably simple and suggestive when plotted in log-log scale, as shown in Figure 6.
An approximate solution. Motivated by the simplicity of Figure 6, we propose an analytical approximation f app to f . This is given by the 4-parameter function (4.32) Let us briefly motivate this expression. First of all, at α = 1, it gives a Gaussian function in log-log scale, which is a natural ansatz in view of Figure 6. To achieve a better fit, we introduced an additional parameter α. The function (4.32) provides an excellent, albeit not perfect, fit to f . Indeed, it is possible to verify that there is no choice of parameters such that f app is a solution of (4.31). It would be interesting to investigate whether a small modification of it yields an exact solution.   Figure 9. Numerical solution for f for A = 1 (blue) to A = 10 (red).

General A
We now consider the case of general A, including the value A = 4 of interest in (4.13).
Since the analysis is identical to the one for A = 1, our presentation will be more succinct. Figure 9 shows f for various values of A ranging from 1 to 10, obtained by numerically solving (4.31). Once again, f becomes extremely simple in log-log scale, as shown in Figure 10.
Parameter fit. As in the case A = 1, f app provides an excellent approximation to f for all A. Figures 11 and 12 compare f for values of A in the range 1, . . . , 10 to their fits by f app . The parameters have been fitted using the same method as before. Below we present them for reference.   Figure 11. Comparison between f , for A = 1 (blue) to A = 10 (red), to the fitted f app (dotted).   The full function We are now ready to put together the asymptotic function g 0 with the analytic approximation to the interpolating function f app and compare it to g.
Comparison to g 0 . In order to appreciate how things improve by introducing f app , it is convenient to first compare g to g 0 , as shown in Figure 13. We show two ranges of small r, that is where the deviations are most noticeable. We see that g 0 becomes a better approximation to g as A increases. We will revisit this observation shortly.  Figure 13. Comparison between g, for A = 1 (blue) to A = 10 (red), to g 0 (dotted). We consider the ranges: a) 0 ≤ r ≤ 5 and b) 0 ≤ r ≤ 20.
Comparison to g 0 f app . Figure 14 compares g to the analytic approximation given by g 0 f app for A = 1, . . . , 10. The functions become indistinguishable to a naked eye.
The A → ∞ limit. Figures 9 and 10 show that the maximum value of f decreases as A grows. In other words, g 0 becomes a better approximation to g as A is increased. The maximum discrepancy between the two functions goes from 4% for A = 1 to 0.7% for A = 10. For A = 40 this number reduces to 0.2%. We can discuss this behavior in terms of f app , since it gives a good approximation to f . The maximum of f app is f app (r 0 ) = 1 + α(e B − 1). Computing the fits up to A = 40,  Figure 14. Comparison between g, for A = 1 (blue) to A = 10 (red), to g 0 f app (dotted). We consider the ranges: a) 0 ≤ r ≤ 5 and b) 0 ≤ r ≤ 20.
we observe that B seems to converge to a finite value while α appears to decrease to zero. This leads us to conjecture that f app → 1 as A → ∞ or, equivalently, that g 0 becomes the exact solution in the A → ∞ limit. Notice that this is not just the small r solution (4.26), since for sufficiently large r the behavior of g is controlled by B.
We can offer more details on how this limit is approached. The interpolating function f accounts for the transition between the small and large r regimes of g. The value of r 0 is a natural indicator of where this transition occurs. Interestingly, r 0 depends linearly on A, as shown in Figure 15. As A grows, not only f app approaches 1, signaling that g 0 becomes a better approximation to g, but also r 0 grows, indicating that the small r approximation is valid up to larger values of r, as expected.

L 2 norms
For the Harvey-Lawson solution (4.23), it was shown in [8] that the deformation mode is not L 2 normalizable: For B = 0 and more general (integer) values of A, from (4.22) we find This exhibits a faster than ρ −2 decay when A < 2, which is the condition for the deformation mode to be L 2 normalizable. If A > 2, the -mode is not L 2 normalizable and usually this means that it should be interpreted as a parameter (coupling constant) of the IR theory, rather than a dynamical field with finite kinetic term. This is not the full story, however, in our present context. First, our IR theory is a two-dimensional N = (0, 2) theory that "lives" on the R 2 part of the fivebrane worldvolume (1.8). As is well known, in two space-time dimensions, long range quantum fluctuations do not allow fixing vacuum expectation values, and so all moduli tend to be dynamical. A natural question, then, is whether a singularity = 0 occurs at finite distance in the moduli space metric, see e.g. [18] for a close cousin of our problem where "compactification" on a non-compact 4-manifold is replaced by a "compactification" on a non-compact Calabi-Yau 4-fold.
Another obvious modification of the estimate (4.35) is due to the fact that it was deduced under the assumption B = 0. Indeed, as we noted several times earlier, the large-r behavior of the solution is controlled entirely by B, not A. Thanks to the detailed analysis in this section, however, it is easy to see that incorporating B = 0 still leads to the same conclusion as (4.35). Namely, from (4.29)-(4.30) and (4.32) we quickly find the large-r behavior of g ∼ r and δg δζ ∼ r −1 , which means that the deformation mode is not L 2 normalizable.

Reversing the degree
The differential equation (4.13) and its solutions described above correspond to D4-NS5 brane models with non-compact D4-branes of the form (3.16). Moreover, as explained in the end of section 3, brane models with compactly supported D4-branes, as in (2.3), differ by the orientation reversal on the fibers of X 7 or, equivalently, n → −n.
In our parametrization of the coassociative submanifolds (4.11), this corresponds to simultaneously changing the sign of g and r. Therefore, the corresponding version of the ODE (4.13) reads (n > 0) dg dr = (−n + |n| + Br)g g 2 +nr = Brg g 2 +nr . (4.36) The term |n| is not affected by the sign flip, cancels against (−n), and leads to the main qualitative difference between (4.13) and (4.36). The |n| term is needed to maintain the components of Taub-NUT metric positive. One important consequence is that, although the resulting configuration is free of the winding number anomaly, it breaks supersymmetry and hence is likely to be unstable. In Figure 16, we plot a numerical solution to the differential equation (4.36) with B = 1. Note, just like SO(3)-rotation (4.10)-(4.11) of the graph of function g(r) discussed earlier in this section produces coassociative submanifolds that correspond to NS5-D4 brane systems with non-compact D4-branes on "disk complements" (3.16), a similar SO(3)-rotation of the graph of function g(r) presented in Figure 16 clearly has only one asymptotic region ∼ = R + × S 3 /Z n that corresponds to M-theory lift of NS5-branes and, therefore, D4-branes in this brane model are compactly supported on Lagrangian disks (2.3).  Figure 16. A numerical solution to the differential equation (4.36).

Phase transitions and Spin(7) manifolds
In the previous section, we constructed new coassociative submanifolds M 4 ⊂ X 7 = Λ 2,+ (TN n ) of topology L(n, 1) × R, or TN n , or O(n) → CP 1 , or any combinations thereof. These are particular examples of a more general construction (3.15), which can be carried out in detail following the steps of section 4. As explained in the Introduction, every such pair (M 4 , X 7 ) determines a manifold X 8 of Spin(7) holonomy that, in physics, arises from an (auxiliary) problem of D6-branes supported on R 3 × M 4 . The topology of X 8 is determined by the topology of M 4 and X 7 . Specifically, X 8 is the total space of a circle bundle over X 7 with a codimension-4 singular locus M 4 ⊂ X 7 ∼ = X 8 /S 1 . (Note, codimension of the singular locus is always even, and must be equal to 4 in order for the fibration to be smooth.) In our class of examples, this means that the space of Spin(7) holonomy metrics consists of several branches (or phases).
Indeed, consider, for concreteness, the case of n = 1. Then, one choice of the coassociative submanifold M 4 is a disjoint union of a Taub-NUT space TN 1 and O(+1) → CP 1 : In this phase, the corresponding S 1 -fibration has topology 8 X (A) 8 ∼ = R 4 × CP 2 and depends on one real deformation parameter that we can choose to be Vol(CP 2 ). As Vol(CP 2 ) → 8 Sometimes we write non-trivial vector bundles simply as products (or, more precisely, our use of " ∼ =" means homotopy eqivalence). For example, X here is the total space of the universal quotient bundle over CP 2 . 0, the Spin(7) holonomy metric on X (A) 8 develops an isolated conical singularity which, aside from X (A) 8 , admits another resolution ("desingularization") by a family of Spin (7) holonomy metrics on X (B) 8 ∼ = R 3 × S 5 . This family of Ricci-flat metrics, conjectured in [8] and only recently constructed in [19], can also be obtained as the total space of a circle bundle over the same X 7 , but with a different locus of singular S 1 fibers Phase B: The topology changing transition between these two coassociative submanifolds -or, between the corresponding Spin(7) manifolds X  A natural question, then, is: What is the physics of this transition? From the M-theory viewpoint, it requires studying the effective 3d N = 1 theory obtained by compactification on X 8 . From the perspective of type IIA theory on X 7 , it requires understanding worldvolume theory of the D6-brane supported on R 3 ×M 4 . Either way, the result of this analysis is a simple 3d N = 1 theory [8]: U (1) gauge theory with Chern-Simons coupling at level k = ± 1 2 and a N = 1 matter multiplet of charge +1 such that what we call phase A and phase B correspond to Higgs and Coulomb branches of this effective theory. A notable feature of this low-energy dynamics is the half-integer Chern-Simons term k = ± 1 2 that on D6-brane world-volume originates from the flux quantization condition [20]: and the fact that O(+1) → CP 1 is not Spin. In particular, because of this, the parity symmetry of the three-dimensional theory is spontaneously broken on the Higgs branch, which therefore in turn splits into two components labeled by k = + 1 2 and k = − 1 2 .
Two geometric phases of Spin (7) holonomy manifolds X 8 correspond to Higgs and Coulomb phases of 3d N = 1 effective field theory.
2d phase transitions for n = 1 Naively, one might expect that the effective 2d N = (0, 2) theory on the fivebrane worldvolume in our setup (1.8) is similar to the effective theory of a D6-brane compactified on the same coassociative 4-manifold. In particular, for the two choices (5.1)-(5.2) of M 4 , with n = 1, one might expect two phases, which we still continue calling phase A and phase B.
However, looking into the analysis a little more closely one quickly runs into questions, which indicate that the physics of M5-branes can be rather different from the physics of D-branes, even when they wrap the same coassociative submanifolds. For example, one question about the fivebrane system (1.8) is: What is the analogue of the Freed-Witten anomaly (5.3)? And, does it lead to two vacua in phase (5.1)?
Since the world-volume theory of a single fivebrane has a Lagrangian description, one might hope to deduce the effective 2d N = (0, 2) theory simply by following the standard rules of the Kaluza-Klein reduction. However, by looking at the KK spectrum in [21], it is not immediately clear what the answer should be for M 4 = S 3 × R. This manifold has no homology in degrees up to 3, whereas all fields of 6d (0, 2) theory on the fivebrane worldvolume are represented by differential forms of degree less than 3, even after the partial topological twist along M 4 . So, naively, the resulting 2d N = (0, 2) theory looks completely empty in phase (5.2). Another issue is that M 4 = S 3 × R is non-compact, whereas the Kaluza-Klein spectrum summarized in Table 1 of [21] assumes compactness of M 4 . So, how shall we even think of this compactification from 6d to 2d on a non-compact M 4 ?
Another puzzle has to do with the fact that, in M-theory compactification on Spin(7) manifold X (B) 8 ∼ = R 3 × S 5 a key role is played by the charged particle coming from M5brane wrapped on the 5-sphere S 5 . In the corresponding type IIA setup with a D6-brane supported on a coassociative 4-manifold M 4 = S 3 × R, this charged particle comes from a D4-brane with world-volume R × D 4 , such that ∂D 4 = S 3 . But, what is the corresponding analogue of this charged state in a setup with M5-branes on the same coassociative manifold M 4 = S 3 × R? The only candidate could be an M2-brane ending on the M5-brane, but the dimension of M2-brane world-volume is too small for this. On the other hand, in a different phase, where M 4 ∼ = R 2 × S 2 , M2-branes supported on D 3 and ending on M5-brane along the S 2 = ∂D 3 produce instantons (local operators) in 2d (0, 2) theory. What is their role?
Luckily, all these questions conveniently resolve one another. For example, in the case of 3d N = 1 phases, the spontaneous symmetry breaking in phase A that leads to two vacua with k = + 1 2 and k = − 1 2 is accompanied by the existence of a domain wall that interpolates between these vacua. In M-theory on R 3 × X (A) 8 , this half-BPS domain wall is a M5-brane supported on R 2 × CP 2 , such that R 2 ⊂ R 3 and CP 2 ⊂ X (A) 8 is a topologically non-trivial calibrated (Cayley) 4-cycle. In the corresponding type IIA setup with a D6brane supported on R 3 × M 4 , this half-BPS domain wall is a D4-brane with world-volume R 2 × D 3 , such that R 2 ⊂ D 3 as before, and ∂D 3 = S 2 ⊂ M 4 is a topologically non-trivial 2-cycle.
Since, upon reduction on S 1 , the 2-form field on fivebrane world-volume gives rise to a gauge field 1-form on D4-brane world-volume, one might expect that anomalous quantization condition (5.3) on D4-brane world-volume comes from a similar anomaly for the 2-form. Moreover, one would also expect that, in case of M5-branes, this anomaly is responsible for spontaneous breaking of 1-form symmetry, since after compactification on S 1 it becomes the origin of a spontaneous breaking of ordinary symmetry on D4-brane theory. Both of these expectations are, in fact, correct, cf. [12,22].
Indeed, just like ordinary symmetry breaking is accompanied by the existence of domain walls that interpolate between vacua (in our case, labeled by k = + 1 2 and k = − 1 2 ), 1-form symmetry breaking is accompanied by codimension-2 vortices. In our fivebrane setup (1.8), these half-BPS vortices are precisely M2-branes supported on {pt} × D 3 that were mentioned in one of the above questions. Note, that these M2-brane vortices only exist in phase (5.1), same geometric phase that hosts half-BPS domain walls in 3d N = 1 theory and has w 2 (M 4 ) = 0. Now let us address non-compactness of M 4 , which also quickly turns on its head from an ugly problem into a nice feature. Indeed, since both choices of M 4 in phases (5.1) and (5.2) are non-compact, with two asymptotic regions ≈ R × S 3 , we should really think of these 4-manifolds as cobordisms from M −  ] on both sides of the interface. In fact, we already encountered this 3d N = 2 theory for general value of n in our previous discussion, cf. (3.11). Here we need its version with n = 1 and G = U (1), that is a U (1) super-Chern-Simons theory at level 1 with an additional 3d 9 equivalently, a non-dynamical domain wall N = 2 free chiral multiplet. Therefore, it remains to identify two 2d N = (0, 2) interfaces in this theory, which correspond to (5.1) and (5.2), respectively (or, more precisely, two phases of the same interface related by a 2d phase transition).
In one of these cases, the answer is simple: namely, in phase (5.2) the interface is "trivial" or "fully transparent." Concretely, this means that it identifies the U (1) + and U (1) − gauge multiplets of theories T [M + 3 ] and T [M − 3 ], as well as free chiral multiplets on both sides. Note, in the 2d N = (0, 2) theory on the interface, U (1) + and U (1) − appear as global symmetries.
The phase A of our 2d interface is more interesting. (Recall, that the bulk 3d theory does not change, and it is only the 2d interface that undergoes a phase transition.) In this phase, our 2d interface is geometrically "engineered" by a compactification of 6d fivebrane theory on a 4-manifold with two connected componets, where, according to (5.1), written here, for convenience, 10 with general value of n. Each of these components has one asymptotic region of the form R × S 3 , which corresponds to either left or right side of Figure 18: To summarize so far, our 2d interface I has two branches, geometrically engineered by topological reduction on (5.1) and (5.2). These branches are parametrized by Vol(S 2 ) and Vol(S 3 ), respectively. And, asymptotically, in the large volume limits, the interface becomes either fully transmissive or totally reflective, cf. ] are known, it only remains to couple them in a consistent way, so that the combined 2d theory has two branches with the desired asymptotic behavior (5.7). 10 For the arguments to follow, it will be important to pay careful attention to orientations, which is easier if we restore general n for a moment. As explained in [12], the manifolds An−1 and O(−n) → CP 1 have oppositely oriented boundaries and unique Spin structures. Therefore, they can be glued to form a closed 4-manifold which, however, is not Spin. In our case, M + 4 and M − 4 should be oriented in a way that allows a topology changing transition to M4 = R × L(n, 1). This means if M + 4 = An−1, then M − 4 must be O(+n) → CP 1 . chiral multiplets Φ − and P − : There is an obvious candidate for the supersymmetric interaction of these fields, namely a J-term superpotential, that leads to the desired vacuum structure (5.7): In phase B, when a complex scalar in Φ 0 gets a vev, this superpotential acts as a mass term for P − and Ψ + , effectively setting these fields to zero. The variation with respect to so that either Φ + − Φ − = 0 with Φ 0 = 0 (phase A) or vice versa (phase B). We believe the two-dimensional phase transition described here is in the same universality class as the phase transition in a simple type IIB brane model that we describe next. It would be nice to understand the relation between these brane models better, e.g. find a precise duality that relates the two.

Hanany-Witten type brane models
Here we propose another brane system that realizes the field theory setup of the 3d N = 2 gauge theory (3.11) with a half-BPS interface (5.12), thus making a full circle back to type IIB string theory and closer to the brane brick models [3][4][5] that were our original inspiration.
Following [25], we can consider a stack of N D3-branes suspended between two fivebranes in type IIB string theory. We choose one of the fivebranes to be a NS5-brane, with world-volume along directions 012345, and the second fivebrane to be of type (p, q) and oriented along directions 01234[59] θ , where [59] θ indicates a rotation in (x 5 , x 9 ) plane by angle θ. It is well known [26][27][28] that such brane configuration carries 3d N = 2 gauge theory (3.11) on the D3-brane world-volume, provided that is an integer (equal to the value of the Chern-Simons coefficient). We assume the worldvolume of D3-branes to be along the directions (x 0 , x 1 , x 2 ) and where in the last line we also included a D5-brane localized along the x 2 direction. This D5-brane breaks half of the supersymmetries of the 3d N = 2 theory on D3-brane, and therefore creates a codimension-1 half-BPS defect. 13 Also indicated in this Table, and illustrated in Figure 19, is the possibility of D3-brane(s) splitting into two pieces along the D5-brane. Since the only directions shared by all three fivebranes in our IIB system are (x 0 , x 1 ) and (x 3 , x 4 ), the D3-branes can only separate along (x 3 , x 4 ), Here we identified this separation with the parameters in (5.9) since the physics on the branch parametrized by = 0 matches that of phase A in the above discussion. Similarly, phase B of the 2d N = (0, 2) theory at the D3-D5 intersection corresponds to moving the D5-brane along the (x 7 , x 8 ) directions, as illustrated on the second panel of Figure 19. It should not be too difficult to generalize this analysis of brane configurations -in M-theory as well as in type IIB string theory -to arbitrary values of n and N . More generally, it is natural to ask: What kinds of topology changing transitions can coassociative 4-manifolds have? We leave these interesting problems to future work.

A Supersymmetry conditions for D6-branes
Here we show that brane configuration (1.7) in type IIA string theory allows adding D6branes supported on special Lagrangian submanifolds in X, without breaking supersymmetry further. Moreover, and especially important for our applications, the orientation of extra D6-branes needs to be strictly correlated with the orientation of D4-branes; namely, if D4-branes are calibrated by Re(e iθ Ω), then D6-branes must be calibrated by Im(e iθ Ω).
For purposes of analyzing supersymmetry conditions, we can replace X by C 3 , with its flat Calabi-Yau structure (2.4), and imitate S by a triple of complex hyperplanes supported at z 1 = 0, z 2 = 0, and z 3 = 0, cf. (1.3). In these conventions, D4-branes are calibrated by the 3-form Re(Ω), and the brane configuration looks as follows: The supersymmetry condition for a NS5-brane with world-volume along the directions 012345 is given by L,R = −Γ 012345 L,R , and similarly for the other two branes NS5 and N S5 . Here, L and R are 10d spinors of left and right chirality: Combining the supersymmetry condition for a D4-brane, L = Γ 01246 R , with the above mentioned condition for a NS5-brane, after simple gamma-matrix algebra we obtain L = Γ 0124789 R . This is precisely the supersymmetry condition for a D6-brane with worldvolume along the directions 0124789. Similarly, combining the D4-brane supersymmetry condition with those for NS5 and NS5 branes, we learn that any of the following D6-branes can be introduced into the brane configuration (A.1) without breaking supersymmetry further [30]: 0 1 2 3 4 5 6 7 8 9 Note, all of these D6-branes meet the original D4-branes along two directions inside X, precisely as special Lagrangian submanifolds calibrated by the 3-form Im(Ω) (with our convention that D4-branes are calibrated by Im(Ω)). The other set of D6-branes, with world-volume along directions 0135789 is also calibrated by Im(Ω) and, as can be seen directly via gamma-matrix algebra, does not break supersymmetry further.

B Calibration condition
Here we show how to derive the ODE (4.13) from the calibration condition Φ| M 4 = 0, Ψ| M 4 = d(vol) M 4 and the SO(3)-invariant ansatz introduced in section 4. For concreteness, we focus on the case n = 2, but generalization to other n is straightforward.
Embedding ansatz We embed M 4 into R 3 × TN 2 by expressing y as functions of ( x, ψ).