T-branes and matrix models

Journal of High Energy Physics, Jun 2017

We find that the equations describing T-branes with constant worldvolume fields are identical to the equations found by Banks, Seiberg and Shenker twenty years ago to describe longitudinal five-branes in the BFSS matrix model. Besides giving new ways to construct T-brane solutions, this connection also helps elucidate the physics of T-branes in the regime of parameters where their worldvolume fields are larger than the string scale. We construct explicit solutions to the Banks-Seiberg-Shenker equations and show that the corresponding T-branes admit an alternative description as Abelian branes at angles.

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T-branes and matrix models

Received: March T-branes and matrix models Iosif Bena 0 1 3 4 5 Johan Blaback 0 1 3 4 5 Ra aele Savelli 0 1 2 4 5 Open Access 0 1 4 5 c The Authors. 0 1 4 5 0 Cantoblanco , 28049 Madrid , Spain 1 Orme des Merisiers , F-91191 Gif-sur-Yvette , France 2 Instituto de F sica Teorica UAM-CSIC 3 Institut de Physique Theorique, Universite Paris Saclay, CEA , CNRS 4 [30] T.W. Grimm, R. Savelli and M. Weissenbacher , On 5 [29] N.R. Constable, R.C. Myers and O. Tafjord , The noncommutative bion core, Phys. Rev. D 6 corrections in N = 1 F-theory We nd that the equations describing T-branes with constant worldvolume elds are identical to the equations found by Banks, Seiberg and Shenker twenty years ago to describe longitudinal ve-branes in the BFSS matrix model. Besides giving new ways to construct T-brane solutions, this connection also helps elucidate the physics of T-branes in the regime of parameters where their worldvolume elds are larger than the string scale. We construct explicit solutions to the Banks-Seiberg-Shenker equations and show that the corresponding T-branes admit an alternative description as Abelian branes at angles. matrix; models; D-branes; String Duality 1 Introduction 2 3 4 From T-branes to matrix theory Finding a solution The \Abelian" picture Returning to the original frame T-branes are supersymmetric brane con gurations in which two scalars and the worldvol ux acquire non-commuting expectation values. They were rst introduced in [1], and have since received a fair bit of interest, with reasons ranging from their fundamental structure to the attractiveness of their low-energy features for model-building in string phenomenology [2{17]. Despite this, several aspects of T-branes have remained quite mysterious. In particular, the presence of non-Abelian scalar vevs seems to hint at a possible interpretation of T-branes in terms of higher-dimensional branes, similar in spirit to the Myers e ect [18]. In [19], the authors and Minasian have shown that for certain classes of T-branes such an interpretation is incorrect: in the regime of large worldvolume elds (in string units) these T-branes appear rather to be described by Abelian branes wrapping certain holomorphic surfaces, whose curvature encodes the original T-brane data. Roughly speaking, the non-Abelian vacuum pro les of T-branes give rise to brane bending, and not to brane The original purpose of the present investigation was to understand how universal the connection found in [19] is, by investigating other classes of T-brane solutions and, in particular, those with constant worldvolume elds. However, a surprise awaited us: we discovered that the Hitchin system describing this class of T-branes is exactly the same as the system of equations that was found by Banks, Seiberg and Shenker in [20] to describe longitudinal ve-branes in the BFSS matrix model [21] (see also [22{24]). Upon reduction to type IIA string theory, the Banks-Seiberg-Shenker equations describe a nonAbelian con guration of D0-branes that preserves eight supercharges and carries D2 and The fact that these equations are identical points to the existence of a more profound connection, which has to do with the fact that both the BFSS matrix model and the Hitchin system describing T-branes come from reductions of ten-dimensional super-YangMills theory to lower dimensions: the BFSS matrix model is the reduction of this theory to a particular one-dimensional matrix quantum mechanics, while the Hitchin system arises from an intermediate two-dimensional compacti cation of the self-duality equations of the super-Yang-Mills theory [25]. Armed with this connection, one can use the extensive technology developed in the good old matrix-model days to construct, rather straightforwardly, several solutions of Tbranes with constant elds. As we will show, to obtain such T-branes one has to consider in nite matrices, and we construct a map between these T-branes and their Abelian counterparts following a path similar to that of [19]: the system of equations we obtain in the T-brane frame is mapped to a dual system via two T-dualities along the worldvolume of the T-brane. The resulting dual system describes a particular D0-D2-D41 con guration from the perspective of D0-branes with non-Abelian worldvolume-scalar vacuum expectation values. The same system can be described as two or more D4-branes with Abelian worldvolume uxes, which, when T-dualized back to the original frame, give rise to several intersecting D2 branes. In \black-hole" language, the map between the D0 and the D4 descriptions that we construct is not a microscopic map, but a macroscopic one. To see this, it is important to recall that the D0-D4 system has a very large number of states, of order e2 p2N0N4 , and each of these states can be in principle described either from a D0-brane perspective, as a vacuum con guration where the scalars of the D0-brane worldvolume have non-commutative vacuum expectation values, or from the D4 perspective, as an instanton con guration on the D4-brane worldvolume. The precise map between individual microstates is only known for a few very speci c microstates, and requires in general pretty complicated technology. Our purpose is not to construct this detailed microscopic map, but rather to identify ensemble representatives that have the same overall D4, D2, and D0 charges. The Abelian system that we nd is then brought back to the original T-brane frame by reversing the two T-dualities. At the end of this last step, we recover a D2-brane system, which gives the Abelian description of the original non-Abelian T-brane system. Thus, we nd the same underlying physics as in [19]: T-brane con gurations of stacks of Dp-branes can be mapped to Abelian systems of Dp-branes. Our map can clearly be made more precise, both on the lower side of gure 1 (by nding for example relations between three-point functions in the matrix-model description and D0 density modes in the D4 worldvolume description) and on the upper side of gure 1 (by relating the T-brane data to the precise shape of the holomorphic curves wrapped by D2-branes), and we leave this investigation for future work. The paper is organized as follows. In section 2 we present our T-brane system and map it to the Banks-Seiberg-Shenker system in Matrix Theory through two T-dualities. In the language of gure 1 we start in the upper left corner, and move downwards. In the lower left corner we construct an explicit solution, which is presented in section 3. We 1We will mostly refer to the T-brane as made of D2-branes in this paper, for historical matrix-model reasons. This is however done without loss of generality; all the same conclusions can be drawn for any Dp-brane stack for p = 2; : : : ; 7. work out a map between the lower left and right corners in section 4, and present the resulting D4-brane solution. In section 5, we move to the upper right corner of gure 1, where we construct the Abelian intersecting-brane con guration that corresponds to our original T-brane. The paper is concluded with some observations in section 6. From T-branes to matrix theory T-branes preserving eight supercharges are non-trivial solutions of the so-called Hitchin = 0 ; F + [ ; y] = 0 : This system is de ned on Cw Cz, parametrized by the complex coordinates w and z, that are parallel and transveral to the D-brane directions, respectively. The anti-holomorphic @Aw + [Aw; Aw], where @A @ + [Aw; ]. Moreover, , usually called the \Higgs eld", is the complex combination of two of the worldvolume scalars of the D-brane stack, and is a holomorphic (1,0)-form valued in the adjoint representation of the gauge group. Before beginning we would like to make some preliminary observations on these equations. T-brane con gurations are characterized by a non-trivial commutator [ ; y] and, because of the cyclicity of the trace, have a traceless worldvolume ux. The eld ever, is not necessarily traceless. In this paper we are interested in T-branes that have constant worldvolume elds, for which the equations above are written solely in terms of [Aw; ] = 0 ; [Aw; Aw] + [ ; y] = 0 : Since, as we will reiterate below, these equations can only be non-trivially solved for in nite matrices, all commutators can in principle admit a non-trivial trace. However, since we nite-N T-branes in mind, we will still keep the commutators appearing in (2.2b) traceless, whereas we will allow for non-trivial traces in (2.2a) (as we will see, these just give rise to additional harmless brane charges, without spoiling supersymmetry). Upon expressing the complexi ed elds Aw and in terms of their Hermitian components2 the system (2.2) becomes where the rst two equations come from the anti-Hermitian and Hermitian parts of (2.1a) respectively, and the last comes from (2.2b). Following a train of logic similar to that of [19], we now T-dualize the T-brane equations (2.4) twice along the worldvolume directions 3 and 4 (see table 1). This maps the gauge potentials A3;4 into worldvolume scalars 3;4, and the T-brane equations become: or more concisely 2From now on we only consider the matrix-valued coe cients of the di erential forms, but refrain from introducing a new notation. Aw = Aw = (A3 + iA4) ; [ 1; A4] = [ 2; A3] ; [ 1; A3] = [A4; 2] ; [ 1; 2] = [A3; A4] ; [ 1; 4] = [ 2; 3] ; [ 1; 3] = [ 4; 2] ; [ 1; 2] = [ 3; 4] ; ijkl[ i; j ] = [ k; l]: The rst surprise in our investigation is that this system is exactly the same as the Banks-Seiberg-Shenker system of equations [20] that describes longitudinal ve-branes in the BFSS matrix model [21]. Upon compactifying to type IIA string theory, these equations describe multiple D0-branes dissolved into D4-branes (with extra possible D2 charges) from the perspective of the worldvolume non-Abelian Born-Infeld action of the D0-branes. As noted in [20], this system of equations admits no non-trivial solutions in terms of matrices, and hence to proceed we will henceforth use in nite matrices. We will further discuss the relevance of this construction for nite-N T-branes in section 6. To demonstrate that indeed this system contains D2-branes as well as D4-branes, we can derive an expression for their charge densities from the Wess-Zumino part of the nonAbelian Born-Infeld action of N D0-branes [18]: SWD0Z = 0 = 2 `s2 = 2 that the volume3 of the D2-branes is L2 and the volume of the D4-branes is L4. The induced numbers of Dp-branes, Np, are given by the electric couplings between the D0-brane elds and Cp+1 SWD0Z = : : : + pNp quantities ~ and to express them in terms of matrices it is convenient to de ne the dimensionless . The D2 and D4 numbers are then L= 2 N2ij = N4 = where the ij superscript on the number denoting D2-branes signify their orientation, according to the left hand side of table 2. From now throughout the rest of this paper we will exclusively use the dimensionless elds ~ i, but proceed to drop the tilde in order to un-clutter the formulae. Note that K can be thought of as the dimensionless size of the box in which our D0-branes are distributed, and, like N , must be taken to in nity. Equations (2.9) and the cyclicity of the trace make it clear that to be able to induce non-trivial D2 charges one has to use in nite matrices i. As explained at the beginning of this secthe tracelessness of equation (2.1b) for nite matrices. We will impose this condition in order not to introduce new features unrelated to T-branes. However, at the same time, we which are allowed for nite matrices. 3We are here quite liberal with the use of the phrase volume, as L is derived from the topological Wess-Zumino term: it does not strictly give a volume but rather gives information about the boundaries. However, for the at branes we are considering here, these two agree and we will keep on slightly abusing is the resulting branes after reversing the T-dualities depicted in table 1, e.g. the T-brane frame. The branes colored in red and underlined are not present in a T-brane solution. Finding a solution The goal of this section is to nd an explicit solution to the system (2.5). The building blocks for constructing solutions to this system of equations are two in nite Hermitian traceless matrices D and X, analogous to momentum and position operators, satisfying [D; X] = iIM ; where the size of the matrices, M , is actually in nity, but we keep track of it for the purpose of making the normalizations clear. Explicitly, these matrices can be constructed from the creation and annihilation operators of Quantum Mechanics via ay = BBBBB 0... Di = Xi = ij )IM + ij D) ; ij )IM + ij X) ; Our dynamics takes place in four dimensions and we can construct four-dimensional momentum and position operators of size M 4 M 4 = N 4This particular choice is not compulsory, there exist other types of in nite matrices that can represent a and ay, but this choice makes the calculations more straightforward. [Di; Xj ] = i ij IM = i ij IN : We can now construct Ansatze for the matrices i in terms of Di and Xi. As already mentioned, the goal is to nd a solution that has non-vanishing charge for all the D2; 4]. This can be achieved for example by the following three-parameter family of solutions 1 = D1 2 = D2 + A13X4 3 = D3 ; 4 = D4 + p (X2X4 + X1X3) ; where A13; A14; are constants whose physical meaning will be clear shortly. The matrices i in (3.6) have the commutators and hence equation (2.9) implies that the D2-brane charges are [ 1; 2] = [ 3; 4] = i p [ 1; 3] = [ 4; 2] = iA13IN + i p [ 1; 4] = [ 2; 3] = iA14IN ; N (12) = N (34) = 0 N4 = These charges do not depend on , because the Xi are traceless. However, the D4-brane charge does depend on : These dependences highlight the crucial role played by the parameter of our family of solutions. If a solution allows the following decomposition of the trace N4 = ijklTr l k j i = then it features supersymmetry enhancement and preserves 16 supercharges. As one can see from (3.9), this condition is broken by , and therefore only the solutions with non-zero will preserve just 8 supercharges. Hence, it is that gives to our solution a T-brane character, because it is the only parameter appearing in equation (2.1b). On the other hand, the parameters A13 and A14 are only there to \dress" the T-brane with additional D2 charges, without spoiling its features. Let us now work out the nite physical quantities of our family of solutions. We have a number N of D0-branes which we are implicitly sending to in nity. These branes are distributed over an in nite four-dimensional space of volume K4, and the appropriate nite quantity in our solution is the average density of D0-branes: The same can be said for D2-branes: they are distributed in a subspace of volume K2 so their number is in nite but their density is nite: 0 = i2j = K22 = 0Aij < 1 : 4 0 = Tr [Di; Xj ] = i ij 0 : Since the D4-branes wrap the whole four-dimensional space, the D4 charge N4 is the same To summarize, we may formulate our non-Abelian picture solely in terms of nite quantities as follows. We start by xing the quantity 0, which is the analogue of the size of nite-dimensional matrices. By rescaling our in nite matrices, we can make it appear in the fundamental commutation relation (3.1), so that Now, the three-parameter family of explicit solutions is formally given by (3.6), from which, by computing the relevant traces and using (3.14), we can extract the D2-brane densities (3.12) and the D4-brane charge (3.13). The \Abelian" picture In the previous section we constructed a family of eight-supercharge con gurations with D4, D2 and D0 charges, from the non-Abelian D0 perspective. Following the same logic as in [19], we now want to work out the corresponding D4-brane picture for these con gurations. As we explained in the Introduction, we will only construct a macroscopic map between these pictures, by building a D4 con guration that has the same D0, D2 and D4 charges as that of the previous section. A system of N4 at D4-branes with non-trivial worldvolume ux can carry D2 and D0 charges [26], given by the electric couplings to C3 and C1 SWD4Z = 4 in the conventions of [18]. Just as in section 2, we prefer to use dimensionless quantities and de ne F~2 F2. From here on we will exclusively use F~2 but drop the tilde, and all integrals are now over boxes with sides of (dimensionless) size K. In these conventions, the brane numbers are given by N2ij = N0 = Much like in the D0 picture, this system of branes displays an enhancement of supersymmetry if the trace can be split according to N0 = TrN4 f?F2 ^ F2g and our interest here is to prevent this enhancement. numbers according to5 The macroscopic map between the D4 and the D0 descriptions preserves the brane N4N0 = N2ij = N (ij) = N0N4 = A three-parameter family of D4 con gurations with these charges can be obtained using a constant worldvolume ux of the form F12 = F34 = 0 ; F13 = F42 = F14 = F23 = is any traceless N4 above con gurations contain the same amounts of D0, D2 and D4 charges as in (3.11), (3.12) and (3.13) respectively. However, this constant Ansatz is clearly only applicable if N4 > 1. From the D0 point of view discussed in the previous section, nothing appears to prevent us from considering a solution to the T-brane equations whose scalar pro le gives not be chosen, as it would correspond to a 16-supercharge con guration. One is therefore bound to rely on non-constant ux pro les. If the number of D0-branes were would have been impossible to describe them from the perspective of a single D4-brane Here, however, the number of D0-branes must be in nite, which allows to 5A similar map, as we use here to identify the D4 picture of the D0-D2-D4 state, can be found in [27], where they use such a map as a technique to nd solutions, and also in [22, 28], in which they perform four T-dualities along a D0-D2-D4 system. 6This is due to the well-known fact that there are no Abelian instantons on R4. relax the nite-action requirement when trying to solve the self-duality equation for the D4-brane ux. Nevertheless we still believe that there exists no description of our system from the D4relax the condition of nite action, we still need to demand that the density of D0-branes is nite. This means that either the integral determining the D0-brane number scales as K4 | the same as if the integrand were a constant, or equivalently, as the volume of R the expression for the worldvolume ux must contain explicit K dependence. Any explicit K dependence is ruled out since it would, as we will point out later, fail to produce and non-vanishing T-brane dynamics in the K ! 1 limit. Hence we conclude that the D0brane number must make the integral scale as K4 to be a solution of interest. This is in turn not possible since a component of the worldvolume gauge potential must satisfy the Laplace equation if its eld-strength is to satisfy the Bianchi identity and the self-duality condition. This indicates that this eld-strength and its derivatives in Euclidean coordinates, have to obey the \maximum principle", which states that these functions cannot have local extrema. This in turn implies that the function cannot be bounded at in nity (and be regular at nite distances at the same time), and hence must have an integral that scales as K>4. Although this constitutes no formal proof, this argument is for us convincing of view. It would be interesting to look into these discrepancies between the D0-brane and D4-brane pictures further. We hope to provide more insight into this in future work. Returning to the original frame In this section we start from the Abelian D4-brane perspective of the previous section and perform two T-dualities in order to return to the original T-brane frame. We will reverse the T-duality performed in section 2 along the directions x3;4, and the resulting system will be a set of intersecting D2-branes. The latter will be extended along non-compact two-dimensional planes parameterized by the coordinates x1 and x2. Performing the Tduality along the directions of the worldvolume ux of the previous section (equation (4.5)) produces the following set of di erential equations This system can be easily integrated and shown to describe the embedding @1X3 = @2X4 = @1X4 = @2X3 = X3 = X4 = X1 = x1IN4 ; X2 = x2IN4 : derived from the constant C, and k from . It should be noted that these matrix-valued coordinates describe the embedding of N4 D2-branes at once. Furthermore, even though this embedding has a matrix structure, the solution is still Abelian, in the sense that any commutator between the coordinates is zero, embedding in a holomorphic way7 W = CZ + ; where C and are given by C = = 2 + i 1 : According to equation (5.3), the surface over which each D2-brane extends is a at complex plane embedded in the C2 parameterized by Z and W . Under a certain projection onto a R2 subspace of C2, the embedding for one of these branes can be described by gure 2. We see that Aij and describe the angle the branes make and the integration constants describe shifts of the branes. Note that in the absence of the term proportional to , the branes would all be parallel and the supersymmetry would be enhanced to 16 supercharges. It is only the parameter that ensures that the branes are not parallel and hence that the system has only 8 supercharges. The at shape of these D2-branes is just a consequence of the constant- ux \ensemble representative" solution we chose to focus on in the previous section. A generic member of the ensemble will have non-constant uxes on the system of D4-branes, which in turn would give rise to curved D2-brane embeddings after the T-dualities. Hence, the conclusion of our investigation is that the non-Abelian T-brane con guration we started with, made in a number of (generally curved) D2-branes intersecting in the Z; W plane. As we will explain further below, since all the quantities in equation (5.4) are independent of K and N , our result is nite and hence applies to T-branes with large but nite N our coordinates are matrices. 7With some care, W and Z can be compared to the coordinates with the same labels in [19], although Our result con rms the claim made in [19] that T-branes admit an alternative Abelian description in terms of branes wrapping holomorphic cycles. We focused on solutions preserving eight supercharges and we restricted to T-branes characterized by constant pro les of the worldvolume scalars, which forced us to consider stacks made of an in nite number of D-branes. The non-commutative scalar pro le encodes important physical information, which we extracted and connected to the number of D-branes (of the same dimensionality) needed to describe the system from the Abelian perspective.8 The detour we took to link the two pictures allowed us to discover an intriguing connection between the BPS equations governing T-branes and the twenty-year-old BanksSeiberg-Shenker equation that describe longitudinal ve-branes in Matrix Theory. found a three-parameter family of explicit solutions to these equations and discussed their brane interpretation. As we have pointed out several times throughout this paper, our matrix-model-inspired construction of T-branes uses in nite matrices, and one may ask whether similar conclusions apply to T-branes made from a nite number of branes. Departing from the in nite-N sources of such corrections. The rst originates from higher-derivative terms in the nonAbelian Born-Infeld action [29].9 The second comes from assuming that the physics of the lower part of gure 1 takes place on an R4 space. However, for a T-brane in a compacti cathe compacti cation. When the D0-density, 0, is nite, N and K are related, and therefore related to the rst) is going from the D0 to the D4 description. The exact relation between nite- and in nite-N map is still not concrete, and left for future study. However, we believe that the map from the Hitchin system to an Abelian system is valid in general. In this paper we have limited ourselves to matching the macroscopic charges between the D0-brane and the D4-brane descriptions, and our purpose has not been to precise con guration of D4-branes with worldvolume ux providing the alternative Abelian description of the particular D0-brane solution in (3.6). To construct such a microscopic map, one would need to nd for example the precise distribution of D0-branes in the noncompact four-dimensional space, encoded by the details of the scalars i. In analogy with nite-dimensional systems studied in [18, 29], one should be able to reconstruct the \fuzzy" distribution from traces of powers of the scalars i, something which we did not attempt here. For this reason we focused on the easiest possible ux pro le reproducing the same macroscopic charges from the D4 perspective, namely the constant- ux solution, which leads to a uniform distribution of D0-branes. We hope to provide a more re ned analysis in a future work. 8This quantity can be roughly seen as the analogue of the number of Jordan blocks characterizing 9For D7-branes one could try to extract some of these corrections from known 0 corrections in nite-N T-branes [19]. 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Iosif Bena, Johan Blåbäck, Raffaele Savelli. T-branes and matrix models, Journal of High Energy Physics, 2017, 1-15, DOI: 10.1007/JHEP06(2017)009