There and back again: a T-brane’s tale

Journal of High Energy Physics, Nov 2016

T-branes are supersymmetric configurations described by multiple Dp-branes with worldvolume flux and non-commuting vacuum expectation values for two of the worldvolume scalars. When these values are much larger than the string scale this description breaks down. We show that in this regime the correct description of T-branes is in terms of a single Dp-brane, whose worldvolume curvature encodes the T-brane data. We present the tale of the journey to reach this picture, which takes us through T-dualities and rugbyball-shaped brane configurations that no eye has gazed upon before.

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There and back again: a T-brane’s tale

Received: September There and back again: a T-brane's tale Iosif Bena 0 1 Johan Blaback 0 1 Ruben Minasian 0 1 Ra aele Savelli 0 1 Open Access 0 1 c The Authors. 0 1 0 Orme des Merisiers , F-91191 Gif-sur-Yvette , France 1 Institut de Physique Theorique, Universite Paris Saclay, CEA , CNRS T-branes are supersymmetric con gurations described by multiple Dp-branes with worldvolume ux and non-commuting vacuum expectation values for two of the worldvolume scalars. When these values are much larger than the string scale this description breaks down. We show that in this regime the correct description of T-branes is in terms of a single Dp-brane, whose worldvolume curvature encodes the T-brane data. We present the tale of the journey to reach this picture, which takes us through T-dualities and rugbyball-shaped brane con gurations that no eye has gazed upon before. D-branes; String Duality 1 Introduction T-branes The D6 picture The D8 picture 5 . . . and back again A Funnel shapes for the D8-branes Introduction T-branes were rst introduced [1] as special types of D7-brane vacuum con gurations, where the eigenvalues of the complex worldvolume scalar fail to capture the physics of the system (see also the earlier work [2]). As a consequence, the pro le of is always entangled with a non-trivial worldvolume ux, and the spectrum of low-energy uctuations typically features an interesting variety of unconventional brane-model-building phenomena. Tbranes are not really special to D7-branes, but also exist for other Dp-branes (as follows trivially from T-duality). Since their introduction, there has been a stream of e orts on uncovering the peculiarities of these supersymmetric vacua from multiple perspectives, and on investigating their potential applications to string phenomenology [3{13]. There is however an aspect of T-branes that so far has not been thoroughly investigated: the key feature of T-branes is the presence of a non-trivial worldvolume scalar commutator [ ; y], hinting to possible connections [1, 8] to other D-brane physics where non-Abelian e ects become important, such as the dielectric (Myers) e ect [14] or the non-Abelian realization [15] of the CallanMaldacena monopole [16]. However, it is unclear what this connection is. Non-Abelian elds on Dp-branes normally give rise to dielectric D(p + 2)-brane charges [14, 15], and this happens when three of their real worldvolume scalars have non-commuting expectation values. In contrast, for T-branes only two scalars are non-commuting, which makes the connection tenuous. Furthermore, for D7-branes the only possible dielectric dipole charge corresponds to D9branes. Since these branes are space- lling, they have in nite energy density from a D7 worldvolume perspective and therefore cannot be thought of as dielectric branes. There is another way to phrase this problem. The Myers e ect relates a non-Abelian Dp-brane description of a system to a description in terms of an Abelian D(p + 2)-brane that wraps a two-sphere and has worldvolume ux. The regimes of validity of the two descriptions are complementary. The non-Abelian description is valid when the commutators of the expectation values of the elds are small compared to the string scale, while the Abelian D(p + 2)-brane description is valid when these commutators are large (and hence the curvature of the two-sphere wrapped by the D(p + 2)-brane is lower than the string scale) [14, 15]. As we explained above, for T-branes the naive Abelian description in terms of dielectric branes is problematic. Therefore, we have no way to describe them when their worldvolume elds are large and the non-Abelian picture breaks down. The goal of this paper is to construct such a description. The nal result is that our Abelian description of the T-brane does not involve Dpbranes polarizing into D(p + 2)-branes, but rather a single curved Dp-brane, whose worldvolume curvature encodes the original ux (or equivalently the non-commutative) data of the T-brane. Focusing (without loss of generality) on D7-branes, we explicitly show that the easiest possible T-brane con guration preserving minimal supersymmetry in 5 + 1 dimensions corresponds to a single D7-brane with vanishing worldvolume ux, extended along a speci c holomorphic curve in the four remaining directions.1 To reach our result we make a detour through a T-dual type IIA system consisting of D6-branes ending on a D8-brane, for which the relation between the non-Abelian [15] and the Abelian [16] description is well understood. Our T-brane tale is schematically gure 1: we start with the traditional description of T-branes via Hitchin-like equations for a stack of D7-branes, which relate the non-commuting worldvolume scalars to a non-trivial worldvolume ux. We T-dualize this con guration along one of the D7 worldvolume directions aligned with the worldvolume ux; this transforms the T-brane into a smeared stack of D6-branes satisfying Nahm-like equations.2 We then construct the Abelian description of this system, in terms of a funnel-shaped D8-brane with a rugby-ball cross-section and non-trivial worldvolume ux (which can be thought of as a stretched Callan-Maldacena spike [16]). This description becomes more and more accurate as the number of D6-branes increases. Our nal step is to return to type IIB, by T-dualizing this Abelian D8-brane back along the same direction. This results in a single D7-brane extended along a two-dimensional holomorphic manifold, which we claim to be the correct description of T-branes away from the non-Abelian regime. As an immediate consistency check of our result, the fact that T-branes and D7-branes have compatible supersymmetries is encoded in the holomorphicity of this two-dimensional manifold. Along the way we nd a novel, two-parameter family of D8-brane solutions with funnel shape, whose cross-section is a generic tri-axial ellipsoid. For a particular limit of the parameters, this family of vacua degenerates to the well-known Callan-Maldacena solution 1Our result is reminiscent of brane recombination for T-branes [1, 9], but there is a crucial di erence: the e ect we nd is a large-N e ect, which cannot be captured by a gauge that only retains the holmorphic data (see section 6 for further remarks). 2The duality between Hitchin's and Nahm's systems was recently exploited in [8] to investigate the relationship between T-branes and D8/D6 systems with a Nahm pole. T-brane’s tale Polarization D7 D8 with spherical cross-section [16]. This class of solutions is interesting in its own right, and deserves further study. For example, it is likely that such general solutions will encode more general T-brane con gurations than the one we considered here, which corresponds to a one-parameter bi-axial ellipsoid (rugby ball) sub-class. Throughout the paper, we will consider branes in isolation and hence ignore their backreaction on the bulk elds (see section 6 for some considerations concerning the validity of this approximation). The organization of the paper follows the development of the tale. We start by brie y reviewing in section 2 the traditional presentation of T-branes, specifying the particular context and vacuum con gurations we will be focusing on. In section 3 we discuss the rst T-duality and describe the vacuum solution in the non-Abelian language of D6-branes. In section 4 we describe the same physical system using the Abelian Born-Infeld action of a funnel-shaped D8-brane with rugby-ball cross-section. Section 5 is the epilogue of the tale, where we T-dualize the system back to the original duality frame and obtain the description of the T-brane in terms of a curved D7-brane. This is the main result of this paper. In section 6 we make some concluding remarks and discuss our hopes for future adventures. In appendix A we construct the D8-brane solutions with generic tri-axial ellipsoidal cross-section. T-branes Our T-brane tale starts in this section, where we introduce its main character: a speci c supersymmetric vacuum con guration of type IIB string theory preserving a quarter of the original supercharges and six-dimensional Poincare invariance. We refer to [1, 7, 17] for background material relevant to the present discussion. Consider type IIB string theory on R5;1 C2, where C2 is parametrized by coordinates 3The 5 + 1 space-time dimensions play no role in our tale, and will hence not be discussed. are given by a set of complex equations and a set of real ones, given respectively by = 0 ; F + [ ; y] = 0 : is the Higgs eld made from the two worldvolume scalars of the D7, which transis diagonal, its eigenvalues give the transverse positions (in the z-plane) of the components of the D7 stack. The anti-holomorphic covariant derivative in the w coordinate is @A @ + [Aw; ], and Aw = (Aw)y is the anti-holomorphic part of an anti-Hermitian SU(N ) gauge connection with (self-adjoint) eld strength F = @Aw @Aw + [Aw; Aw]. Equations (2.1) are known as the Hitchin system [18].5 The vacuum solution we want to examine here is speci ed in terms of N matrix of the form6 = BB a = pa(N a) eCabfb=2 ; with Cab the Cartan matrix of SU(N ). The gauge eld A is diagonal, with anti-holomorphic part given by Aw = (Ca)ij = ia aj where Ca are Cartan generators for SU(N ). It is easy to verify that the above Ansatz automatically satis es the complex equations (2.1a). Since the gauge eld strength takes the form F = @@faCa, the real equations (2.1b) translate into a Toda-like system of second-order partial di erential equations on the w-plane for the N 1 functions fa [1]: @@fa = a(N This vacuum con guration does not describe a system of intersecting D7-branes, be cannot be diagonalized. It is, instead, the easiest instance of a T-brane, where the worldvolume ux and the non-commuting worldvolume scalars cannot be disentangled should be regarded as a (1; 0)-form on the w-plane, as re ected in equations (2.1), but in our non-compact setting, this is not essential. We also disregard the U(1) associated to the center of mass, 5It is possible to couple this system to defects of the worldvolume e ective theory, which show up as a triple of adjoint-valued moment maps in the right hand side of (2.1). The latter can be physically interpreted as vacuum expectation values of bifundamental matter elds localized on the points of C stack intersects other 7-branes [7]. In this paper we will just consider an isolated stack of D7-branes, and 2 where the D7 thus not allow for this possibility. 6We adopt the convention of summing over repeated indices and we work in units of 2 0. and permeate the entire brane worldvolume. Indeed, as explained in [1], the solution described above originates from the following constant nilpotent Higgs eld in the so-called \holomorphic" gauge7 h = BB ah = pa(N where the superscript h stands for holomorphic. In the language of nilpotent orbits, this solution corresponds to the principal orbit of the algebra sl(N; C), and hence completely breaks the gauge group. Although we will only be explicitly discussing this solution throughout the paper, the generalization to the other nilpotent orbits of sl(N; C) is straightforward. The eld h will in general have a Jordan block structure, which re ects the partition of N associated to the given nilpotent orbit. We can then repeat for each block a story analogous to the one we are about to tell for the single-block solution (2.5), by simply replacing N by the size of the block. To conclude this section, let us remark that, in our T-brane tale, the real BPS equations (2.1b) are the ones that play the key role. Therefore, the physics we discuss cannot be captured by working in the holomorphic gauge. The D6 picture In this section we would like to tell the rst episode of our T-brane tale, which consists in taking a T-duality in the w-plane of the con guration introduced in the previous section. We will obtain a type IIA con guration of N coincident D6-branes with non-commuting worldvolume scalars. To do that, we must rst compactify one of the real directions of the be a cylindrical coordinate,8 with s ' s + 2 the compact direction. Having an isometry along s amounts to reducing the Toda-like system (2.4) to an easier set of second-order ordinary di erential equations: where the symbol 0 denotes the derivative with respect to . Note that, if we write 1 equations (3.1) collapse to a single one for g( ): fa00( ) = 4a(N a) log g( ) ; 7In this gauge, all non-holomorphic data, like the (1; 1) component of the eld strength, are invisible, and moreover the gauge connection can be gauged away. Supersymmetric vacua are thus simply speci ed by a holomorphic Higgs eld modulo complexi ed gauge transformations. 8Such a product geometry spares us from getting an NS5-brane after T-duality, as opposed to situations with a non-trivially- bered circle. The gauge eld (2.3) is sent by T-duality to a D6 worldvolume scalar, which we call 3 and whose eigenvalues parametrize the position of the various components of the D6 stack along the direction dual to s. Denoting 1 = 2 = i( y ), we consider the following Ansatz 1( ) = g( ) 1 ; 2( ) = g( ) 2 ; 3( ) = hy, 2 = i( hy 3 = diag(N + 1 2m)m=1;:::;N . The latter determine a N N representation of the su(2) to (3.3), the worldvolume scalars (3.4) satisfy the so-called Nahm equations [19]: i( ) = which are indeed the T-dual version of the Hitchin equations (2.1b). As explained in [15], non-constant/non-commutative solutions of (3.5), just like those that this solution is static, one can immediately derive its energy from the full non-Abelian DBI action of a stack of N D6-branes extended along . Remarkably, this energy density can be expressed as the square root of a sum of perfect squares [20]: where STr denotes the symmetrized trace [14, 21, 22]. Thus any solution of (3.3){(3.4) sets the rst square to zero and, as one expects for supersymmetric solutions, also satis es the full non-Abelian equations of motion [15]. Let us examine the pro le of the con guration given by (3.4). Note that, if we multiply both sides of equation (3.3) by 2(log g)0, we can easily realize that the combination p(g0=g)2 4g2 must be a constant, which we call C and whose physical meaning will soon be apparent. Equation (3.3) thus reduces to the rst-order equation g0( ) = g( ) = 2 sinh(C ) Fixing the integration constant in such a way that the domain of de nition of the solution > 0, we get This scalar con guration describes a fuzzy funnel, in which the N D6-branes, which are very close to each other at large , start expanding and eventually open up into a D8-brane located at = 0. The cross-section of this funnel at xed is a prolate ellipsoid of revolution (a rugby ball), which has two equal semi-axes ( ) and a third longer semi-axis (r), given by ( ) = r( ) = 2 = 2 sinh(C ) 2 tanh(C ) The rugby-ball ellipsoid becomes more and more like a basketball as we approach the D8of the funnel (for large ), the small semi-axes, , vanish exponentially, whereas the long = r = p semi-axis, r, goes to the constant C looking at the pro le of 3 in (3.4) for 1=2. This clari es the meaning of C: indeed, ! +1, we see that the N D6-branes are uniformly distributed along the direction 3 (the T-dual of s), with spacing given asymptotically by C. In the T-dual type IIB con guration of the previous section, a non-zero C corresponds to a non-trivial asymptotic value of the Wilson line along s for the stack of D7-branes. In the limit C ! 0, our ellipsoid solution becomes the sphere solution of [15], with radius Finally, we can also calculate the energy density of this D6 con guration, by plugging the solution (3.4) and (3.8) into (3.6). We nd: For large , the non-commutative picture just outlined is the accurate description of the physical system at hand, because the D6-branes are all very close to each other and the physics is non-Abelian. However, if C is larger than the string scale, the W-bosons are already massive at mode can condense to give rise to a T-brane. This is con rmed by the exponential fall-o of the nilpotent pro le of for large C. Nevertheless, for very small values of C, the o -diagonal degrees of freedom are approximately massless and can acquire a non-trivial vacuum expectation value. We will encounter again the smallness condition for the constant C when discussing validity regimes more systematically in section 6. The D8 picture When the number of D6-branes and the vevs of the non-Abelian elds are very large, the physics of the brane con guration considered in the previous section is captured by a di erent system, consisting of a single funnel-shaped D8-brane with non-trivial worldvolume gauge Hence, this can be thought of as a type of Myers e ect for the D6-branes, with the di erence that what keeps the D6-branes polarized are the boundary conditions, and not the bulk uxes as in [14]. When all the non-commutative corrections), and their regimes of validity are somewhat complementary, with an overlap whose size grows with N (for more details see section 6). matrices in (3.4) are equal, the corresponding Abelian D8-brane con guration is the CallanMaldacena solution [16], in which the boundary of the D6-branes ending on the D8-brane is viewed as a magnetic source for the gauge eld living on the D8. In this section we will present the analogous construction for the more general ellipsoidal solution (3.9), and this is the next episode of our T-brane tale. Given the absence of any Ramond-Ramond background elds, the physics of our D8brane is entirely determined by its DBI action 6 Z where jD8 denotes the pull-back of the at metric on the D8 worldvolume and the d6x is the R5;1 measure. We will choose the following embedding of the D8 into space-time, tailored to the rugby-ball ellipsoidal shape (3.9): where Xi are space-time coordinates, and are worldvolume coordinates. non-commutative coordinates ; 3 of the D6 picture have as commutative analogs X1; X2; X3 respectively. We also restrict to a worldvolume ux corresponding to uniformly distributed D6 charge With (4.2) and (4.3), we can compute the 3 3 matrix appearing in (4.1): jD8 + F2 = BBB The determinant of (4.4) splits into a sum of three perfect squares, and setting two of them to zero (as we did for (3.6)) gives the minimum-energy conditions for this system: X1 = ( ) sin cos ' ; X2 = ( ) sin sin ' ; X3 = r( ) cos ; X4 = F2 = 0 = r0 = ( ) = r( ) = 2 sinh(C ) 2 tanh(C ) These rst-order di erential equations are solved by: = 1, along X3, is CN . The shape of this D8-brane is sketched in gure 2. Likewise, one can compute the energy density of the system and nd agreement with the D6 picture, eq. (3.10), to the same level of approximation: where we recognize the D6 contribution in the term linear in N and the D8 contribution in the term cubic in N . In fact, the C-dependence of the second piece is illusory, as it disappears when integrating over with a very small cuto to avoid the divergence: In contrast to the D6 picture, the D8 picture is supposed to accurately describe the physics of this system in the region of space where both semi-axes are very large, and thus the curvature of the D8-brane is low in string units (see the discussion in section 6). Let us close this section by noting that the Ansatz (4.2) for the D8 embedding can be generalized further to a tri-axial ellipsoidal shape. To the best of our knowledge, all these supersymmetric solutions with ellipsoidal symmetry are new in the literature and interesting in their own right. More details about them, including their BPS equations and solutions thereof, are given in appendix A, which thus contains the derivation of (4.6) as a particular limit. 10More precisely O(1=N 2), and the same for r. . . . and back again Having derived the D8-brane shape in the previous section, we are approaching the epilogue of our T-brane tale: we return to type IIB by T-dualizing the Abelian D8-brane con guration back along the X3 direction. The result will be a single D7-brane with vanishing worldvolume ux, extended along a speci c holomorphic curve in C2. As emphasized in section 4, the D8 picture is reliable when the semi-axes (4.6) are very large in string units. Expanding the rugby-ball solution in this region of space (small ), we obtain N=2 . In order to see the fate of our funnel-shaped D8-brane after T-duality along X3, the rst step is to change coordinates on the D8 worldvolume, trading for X3. Remebering r, the matrix (4.4) becomes 1 (X3= )2 X3= 1 (X3= )2 X3= 1 (X3= )2 1 + 1 (X3= )2 Even though this D8 con guration displays no isometry along X3, it is not hard to see that the determinant of the above matrix, and thus the DBI dynamics of the system, does not depend on X3. In order to be able to T-dualize this solution, we zoom in on the equator of the rugby ball, thus is equivalent to approximating the rugby ball with a cylinder. The pro le = ( ) is una ected by this operation, because the equation of motion remains the same. Very roughly, this happens because, at each xed , the local D8 charge contributions on the X1;2-plane above the equator cancel out the corresponding ones below the equator. One may worry that zooming in on the equator discards information regarding the ends of the rugby ball. In fact, one could have raised a similar objection at the rst step of the tale, where we T-dualized the D7 stack but considered D6-branes spread over a interval of the T-duality direction.11 However, in the D6 picture, the type IIB T-brane data should be captured just by the D6-branes in the middle of the interval (where there is an approximate isometry along the T-duality direction), and not by the D6-branes near the boundaries. Zooming in on the equator implements this in the Abelian D8 picture. Therefore we are led to consider a D8-brane whose shape is sketched in gure 3 and whose de ning data are the following F' 3 = @'A3 = T-dualizing the above con guration along X3 is straightforward: we obtain a D7-brane with no worldvolume ux and with shape determined by the following pair of equations in 11Since we T-dualized at the level of the equations of motion, we did not discuss this issue at that stage. the four-dimensional ambient space parametrized by ; '; ; X~ 3: ' = Z = W = eiX~ 3= : ZW = R = where X~ 3, which the potential A3 is mapped to, denotes the coordinate T-dual to X3. In order to make supersymmetry manifest, we can translate the two real equations (5.3) into a single complex holomorphic one. Consider an ambient C 2 parametrized by the complex coordinates Note that the coordinate Z is identical to the z used in section 2, whereas W di ers from the w of section 2. However, for + iX~ 3 = w, where we identi ed X~ 3 Using (5.3) and (5.4), our D7-brane can then be seen to wrap the holomorphic curve A three-dimensional plot of this shape is displayed in gure 4. We have therefore reached an alternative description of our starting T-brane con guration (2.2), whereby the ux data of the latter is now encoded in the curvature of the worldvolume surface (5.5), whose Ricci scalar is given by plotted and rotated on the diagonal. The size of the minimal circle is proportional to N . Remarkably, even if our derivation of (5.5) was performed in the regime of small , where the Ricci curvature of the D8-brane is small, our nal D7-brane con guration never su ers from a large-curvature problem. Indeed, as manifest from (5.6), working at large N is enough to guarantee that our D7-brane has a small curvature for all values of . Thus 0 corrections are negligible, which suggests that our Abelian picture of the T-brane could give a correct description of the physics even for small vevs of the non-Abelian elds. We plan to investigate this exciting possibility in the future. In this paper we have considered a speci c class of T-brane solutions with 5+1-dimensional Poincare invariance, realized as a particular set of eight-supercharge vacua on a stack of D7-branes with non-commuting worldvolume scalars and non-trivial worldvolume ux. We have worked in a local patch of the internal part of the D7-brane worldvolume and of its transverse space, thus decoupling gravitational e ects and neglecting all issues related to compacti cation. The class of T-branes we have focused on is speci ed by an integer partition of N , the number of D7-branes composing the stack. When N is large, we have found a novel description of the T-brane corresponding to the maximal partition, consisting of a single D7-brane with no worldvolume When all the ni are large, our new description will consist of k uxless D7-branes wrapping gauge group when two or more ni coincide, which is consistent with the original T-brane description. The maximal partition that we have analyzed corresponds to a vacuum which breaks the SU(N ) gauge symmetry completely. We have achieved our result by making a detour through type IIA string theory, and relating the T-branes to certain supersymmetric systems of Dp-branes ending on D(p + 2)branes that give rise to rugby-ball-shaped bion cores. Throughout our T-brane tale, we have disregarded the backreaction of all D-branes on the bulk geometry, by tuning the string coupling constant gs such that gsN 1. This weak coupling limit has no e ect on the funnel shapes we have discussed, because they are determined by the interplay between the D6 and D8 tensions, and hence are independent of gs [16]. Nevertheless, since our focus was mainly on D7-branes, which strictly speaking cannot be treated as probes, it is worth specifying what this approximation is supposed to a stack of N D7-branes, one can readily see that (gsN ) 1 mean for them. By solving the monodromy problem inside a disk of radius R1 surrounding log (jzj=R1), with z the local transverse coordinate. As expected, this means that the string coupling diverges a distance away from the stack, but can be kept in a perturbative regime within a region of order e 1=N R1. We have argued that our Abelian description of the T-brane is the correct one when vevs of the non-Abelian elds and ux densities are large in string units, and hence the nonAbelian description [1], based on equation (2.1b), breaks down. Indeed, in the T-dual type IIA description, the Abelian and the non-Abelian pictures have complementary regimes of validity. However, as pointed out in [15], these two regimes are expected to overlap, in a region whose size grows as N increases. To see this for our rugby-ball D6-D8 funnels, we have to analyze when higher-order variations of the worldvolume elds are much smaller than rst-order variations, which are the only ones captured by the DBI action. For the non-Abelian physics of the D6-branes, this condition is 8 i = 1; 2; 3 : Using the explicit solution (3.4) and (3.8), it is easy to express this as an upper bound for the larger semi-axis: r N . Since, at = 1, r approaches a non-zero minimal value, CN , the non-Abelian description is consistent when C 1. We thus recover a condition that we guessed on physical grounds at the end of section 3. The Abelian D8 picture is valid when (in string units) d By inspecting the solution (4.6), we easily see that both these conditions are satis ed so CN , which gives a lower bound for the smaller semi-axis. Therefore the quantity CN should not be too large, in order not to spoil the D8 picture. Thus, when overlap is proportional to N .12 Based on this analysis, one would expect that our Abelian description of the T-brane is not only valid for very large non-Abelian vevs and ux densities (when the non-Abelian description breaks down), but also when these vevs are small and the non-Abelian description is reliable. In fact, the regime of validity of the Abelian T-brane description appears to extend over the whole non-Abelian regime of validity: this is because for large N the 12This is similar to the analysis made in [15] for the spherical solution. curvature (5.6) is low for the whole range of and there is no worldvolume ux. We believe that this remarkable phenomenon deserves a deeper understanding.13 As mentioned in the Introduction, our Abelian description of T-branes is reminiscent of brane recombination for T-branes associated to \reconstructible" Higgs elds14 [1]. Our Higgs eld (2.2) is non-reconstructible, and if one tries to carry over the brane recombination analysis of [1], one nds a highly singular con guration [9]. The e ect that we nd in this paper is a large-N e ect, which cannot be captured in a gauge where one keeps only the holomorphic data, as done in [1, 9]. It would be interesting to understand if there is any connection between the e ect we nd and brane recombination, and whether our solution can be thought of as the smoothing out of the singular \brane-recombination" shape of [9] by non-holomorphic physics. The main result of this paper raises the obvious question of whether there is a direct way of deriving our Abelian description of T-branes without the need of following an indirect path, like in gure 1. A positive answer would open up a plethora of new research directions. For example, it would allow one to explore more complicated T-brane solutions, such as those involving non-constant holomorphic functions in the (holomorphic) Higgs eld (2.5). It would also be very exciting, especially in light of possible phenomenological applications, to extend our derivation to four-dimensional T-brane vacua with four supercharges, possibly containing non-trivial monodromies. Acknowledgments We would like to thank In~aki Garc a-Etxebarria, Alessandro Tomasiello and Gianluca Zoccarato for helpful discussions. The work of I.B. and J.B. was supported by the John Templeton Foundation Grant 48222. The work of J.B. was also supported by the CEA Eurotalents program. The work of R.M. was supported by the ANR grant 12-BS05-003-01. The work of R.S. was supported by the ERC Starting Grant 259133 - ObservableString. R.M. and R.S. would like to thank the Institut Henri Poincare for hospitality during this work. Funnel shapes for the D8-branes funnel-shaped D8-branes whose cross-section is a generic tri-axial ellipsoid. They naturally arise as solutions of the Abelian DBI dynamics of a single D8-brane with magnetic To the best of our knowledge they are new in the literature, and reduce to the long-known spherical solution of [16] for a particular choice of parameters. The more general bi-axial ellipsoid solutions (rugby balls), relevant for this paper and presented in section 4, can be recovered as a limit of these most general solutions. Let us start with the following tri-axial Ansatz for the embedding of a D8-brane with a worldvolume parametrized by ; '; , into a at Euclidean ambient space with coordinates 13For instance, equation (2.1b) is know to receive 0 corrections [23], which our Abelian picture should automatically encode. 14A Higgs eld is called reconstructible if its spectral surface is non-singular [1]. X1 = r1( ) sin cos ' ; X2 = r2( ) sin sin ' ; X3 = r3( ) cos ; X4 = jD8 + F2 = perfect squares: The worldvolume ux and the pull-back of the metric give the 3 This results in a DBI Lagrangian density, whose square can be written as a sum of four + (sin2 cos ')2 + (sin cos ) N r30 + r1r2 N r10 + r2r3 + (sin2 sin ')2 N r20 + r1r3 which makes it easy to extract the BPS equations as minimum-energy conditions ri0+1 = These equations can be disentangled to give the following system: r1 = = 0 ; = 0 ; = 0 ; where C2 and C3 are integration constants. The only di erential equation left in (A.4) has where \sn" is the Jacobi sine elliptic function and C1 is the remaining integration constant [25]. This function has poles at 2m1K + (2m2 + 1)iK0 ; 15Solutions with non-uniform F2 can in principle also be constructed. See [24] for some examples involving r1 = r1 = Cl2im!0 r1 = lim r1 = 2 sinh(C~3 ) 2 tanh(C3 ) This expression is generally periodic, and describes periodically reoccurring D8-branes. However, there are two limits in which the period becomes in nite: where C~32 = C2. The rst limit requires C3 to be purely imaginary, while in the second 3 the tri-axial ellipsoid is reduced to the rugby ball discussed in section 4. The other limit is equivalent, but with r1 r2 = r3. and r3 = r. Its small- behavior is given in (A.7), while for large where mi are integers, K = K(k) and K0 = K(1 k), where K is the Elliptic K function 4m1K + 2m2iK0, and it is odd under shifts of the argument by 2m1K. 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Iosif Bena, Johan Blåbäck, Ruben Minasian. There and back again: a T-brane’s tale, Journal of High Energy Physics, 2016, 179, DOI: 10.1007/JHEP11(2016)179