#### 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].
We would like to thank Ulf Danielsson, Giuseppe Dibitetto, Mariana Gran~a, Fernando
Marchesano, Washington Taylor, and Angel Uranga for interesting discussions. The work
of I.B. and J.B. was supported by the John Templeton Foundation Grant 48222 and by the
ANR grant Black-dS-String. The work of J.B. was also supported by the CEA Eurotalents
program. The work of R.S. was supported by the ERC Advanced Grant SPLE under
contract ERC-2012-ADG-20120216-320421. In our calculations we have used SymPy [33]
and we would like to thank the developers.
Open Access.
This article is distributed under the terms of the Creative Commons
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any medium, provided the original author(s) and source are credited.
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