Brane webs in the presence of an O5−plane and 4d class S theories of type D
Accepted: June
plane and 4d class S theories of type
0 Haifa 32000 , Israel
1 Department of Physics, Technion, Israel Institute of Technology
In this article we conjecture a relationship between 5d SCFT's, that can be engineered by 5brane webs in the presence of an O5 plane, and 4d class S theories of type D. The speci c relation is that compacti cation on a circle of the former leads to the latter. We present evidence for this conjecture. One piece of evidence, which is also an interesting application of this, is that it suggests identi cations between di erent class S theories. This can in turn be tested by comparing their central charges.
Brane Dynamics in Gauge Theories; Field Theories in Higher Dimensions

Brane
O5
1 Introduction
2 Preliminaries
2.1
2.2
3.2 SCFT's with marginal deformations
4 Conclusions A Brane motions and free hypermultiplets 1 2
then study the 5d SCFT which can lead to interesting consequences also for the 4d SCFT,
see [11] for an example.
There are a large number of other 5d SCFT's, some of which can also be engineered by
generalizations of brane webs through the addition of orientifold planes [12{15]. In this
article we conjecture that compactifying a class of 5d SCFT's that can be engineered by brane
webs in the presence of an O5 plane leads to 4d type D class S theories. This can be
motivated by studying the dimension of the Higgs branch, and the global symmetries manifest
in the web description, and matching them against the properties of class S theories.
The major piece of evidence we provide, which is also an interesting application of
this relation, is by testing equivalences between di erent class S theories. Many of the 5d
SCFT's we consider can be mass deformed to a 5d gauge theory. In some cases this gauge
theory can also be realized by a di erent brane realization, either with the O5 plane, or
without it which leads to a web of the type considered in [4]. In both descriptions the
SCFT is realized when all mass parameters vanish. Thus, from 5d reasoning we conclude
that these are di erent string theory realizations of the same 5d SCFT. On one hand
compactifying this 5d SCFT should lead to one 4d SCFT. On the other hand, as the string
theory realizations are di erent, we get seemingly di erent class S theories. Consistency
now necessitates that these class S theories are in fact identical. This can be checked by
comparing the central charges of these class S theories. We indeed nd that they are equal
in all cases checked.
The structure of this article is as follows. Section 2 consists of a summary of properties
of D type class S theories and brane webs in the presence of an O5 plane that play a role
in this article. In section 3 we present our conjecture and provide supporting evidence for
it. We end with some conclusions. The appendix discusses the suspected creation of free
hypermultiplets accompanying certain 7brane manipulations.
2
Preliminaries
We begin by reviewing several aspects of class S technology and brane webs in the presence
of O5planes that play an important role in the proceeding discussion.
2.1
Class S technology
In this article we will be particularly interested in 4d class S SCFT's given by compactifying
the type D (2; 0) theory on a Riemann sphere with punctures. There are known methods
to compute various quantities of interest for this class of theories, and in this subsection
we shall summarize the ones used in this article. All of these, and much more, can be
found in [16, 17].
Like their A type cousins, punctures of D type class S theories are labeled by Young
diagrams. For a DN theory, this is given by a Young diagram with 2N boxes, subject
to the constraint that columns made of an even number of boxes must repeat with even
multiplicity. Furthermore if all the columns are made of an even number of boxes then
there are actually two di erent punctures associated with this Young diagram. These are
{ 2 {
HJEP07(216)35
usually called veryeven partitions, and the two punctures are usually distinguished by the
color of the diagram, red and blue being the common choice.
Each puncture has a global symmetry associated to it, where each group of nh columns
with the some height h contribute a USp(nh) factor if h is even, or an SO(nh) factor if h
is odd. Note that here the constraint that columns made of an even number of boxes must
repeat with even multiplicity is important. The contributions from all the punctures then
give part of the global symmetry of the SCFT, which may be further enhanced to a larger
global symmetry. In some cases we will also want to determine the full global symmetry.
In those cases we use the 4d superconformal index.
Since conserved currents are BPS operators they contribute to the index, and so
knowlcase in [21]. Occasionally a class S theory may contain free hypermultiplets in addition to,
or even without, a strongly interacting part. The 4d index then provides an excellent tool
for discovering if such a thing occurs, and to nd the number of such free hypers. A good
review for these applications of the index of class S theories can be found in [22].
We will also need to evaluate various properties of the SCFT, notably, the spectrum
and dimension of Coulomb branch operators, dimension of the Higgs branch, and central
charges of the various global symmetries. Their evaluation from the Riemann surface is
known and given in [17]. One quantity that will play an important role is the dimension
of the Higgs branch for a SCFT arising from the compacti cation on a three punctured
sphere. The Higgs branch dimension is then given by:
where the compacti ed theory is DN , and the fi's are the contributions of the three
punctures. These can be evaluated directly from the Young diagram, and are given by:
dH = N + f1 + f2 + f3
f =
4
1 X r
j
2
j
2
1 X rj
j odd
(2.1)
(2.2)
where rj is the length of the j'th row, and the rst sum is over all rows while the second
is only over the odd numbered rows (j = 1; 3; 5 : : :).
2.2
Brane webs in the presence of O5planes
In this subsection we shall summarize some important results regarding brane webs in the
presence of O5planes. Such systems were
rst introduced in [12], and further studied
in [14, 15]. The particular system we are interested in consists of a group of D5branes,
an O5 plane parallel to the D5branes and several NS5branes stuck on the orientifold
5plane. Such a stuck NS5brane leads to a change in the orientifold type: an O5 plane
changes to an O5+plane and vice versa. We also take the number of such branes to be
even so that the asymptotic orientifold 5plane on both sides is an O5 plane. We can
further add 7branes on which the 5branes can end.
{ 3 {
The gauge symmetry living on N D5branes parallel to an O5 plane (O5+plane)
is SO(2N ) (USp(2N )). Thus, the resulting 5d SCFT has mass deformations leading to
a 5d quiver with alternating SO and USp gauge groups. The matter content includes
halfbifundamentals between each USp=SO pair, fundamental hypermultiplets for the USp
groups and vector matter for the SO groups. In some cases we can also incorporate spinor
matter for some SO groups. We refer the reader to [14] for the details.
Besides the USp=SO quiver, these SCFT's have an additional gauge theory description.
In the brane web this can be seen by performing Sduality on the con guration where the
O5 plane becomes a perturbative orbifold. Speci cally, an O5 plane with a full
D5brane on top of it is Sdual to an object called an ON 0 plane, which is a C2=I4(
1
)FL
orbifold [26, 27]. Studying the gauge theory living on the 5branes in this background
reveals that the 5d SCFT also possess mass deformations leading now to a 5d quiver of SU
groups in the shape of a D type Dynkin diagram.
We will also want to understand the global symmetry of the SCFT. This can be
determined by considering the gauge symmetry living on the 7branes, which is a global
symmetry from the point of view of the 5branes. For the system we will be considering,
there are three contributions given by the D7branes on each side of the orientifold as well
as the (0; 1) 7branes. At the
xed point the D7branes, on each side of the orientifold,
merge on the O5 plane while the (0; 1) 7branes merge outside the O5 plane.
We want to consider the classical gauge symmetry on the D7branes, which may
also have an arbitrary number of D5branes ending on them. To do this we envision
moving them through several NS5branes. Each such transition removes a D5brane via
the HananyWitten e ect, so as to arrive at a group of D7branes with no D5branes
ending on them.
Once this con guration is reached, the D7branes sit on top of an
O5+plane (O5 plane) if, in the initial con guration, the number of D5branes
ending on the D7branes is odd (even). The gauge symmetry living on a group of k stuck
D7branes intersecting an O5 plane (O5+plane) is USp(k) (SO(k)) where k must be
even for the O5 case.
For the (0; 1) 7branes each group of n such 7branes with the same number of
NS5branes ending on them contributes a U(n) global symmetry. Taking the direct product
of all three contributions then gives the classical global symmetry. This may be further
enhanced to a larger symmetry group.
We also wish to determine the dimension of the Higgs branch of the SCFT. This
can be done from the web by examining the number of motions of the 5branes parallel
to the 7branes. For the system we consider, these can be broken into four parts. First
we can separate the D7branes across the O5 plane, and pairwise separate D5branes,
stretched between neighboring D7branes, from the orientifold (see the discussion in [14]).
Doing this on either side of the orientifold comprises parts one and two, and doing so on
the central 5branes comprises part three. Finally we can break the NS5branes on the
(0; 1) 7branes and pairwise separate them from the O5 plane. This gives the fourth part.
See gure 1 for an example.
{ 4 {
D7branes on one side of the orientifold. In this case there are two such directions for each side of
the O5plane, where there is an arrow representing the pair. There are also two directions given
by pairwise breaking and separating the four D5branes stretching between the two groups of
D7branes on either side of the O5plane. Finally we can pairwise break and separate the NS5branes
along the (0; 1) 7branes. In this case there are two directions, one given by breaking the longer
part of the right NS5branes while the other by separating the remaining part of the NS5branes
from the O5plane.
3
Statement of the conjecture
In this section we present our conjecture that compactifying a class of 5d SCFT's on a
circle leads to 4d class S SCFT's given by compactifying the type D (2; 0) theory on a
Riemann sphere with punctures. We start with the cases related to compacti cation on a
three punctured sphere, and then move on to the general case.
3.1
Isolated 4d SCFT's
We start by presenting the class of 5d SCFT's we consider. These can be engineered by a
web with an O5plane, and a system of 5branes ending on (1; 0) or (0; 1) 7branes only.
More speci cally, asymptotically the orientifold plane is O5 in both directions, and there
are a total of 2N D5branes (in the covering space) ending on the (1; 0) 7branes whose
number may be less than 2N . The number of (0; 1) 7branes is at most two. If there is
only one all asymptotic NS5branes end on it, while if there are two then one NS5brane
ends on one of the (0; 1) 7branes, and the rest end on the other.
We claim that compactifying the 5d SCFT on a circle of radius R5 and taking the
R5 ! 0 limit, without mass deformations, leads to a 4d class S theory given by
compactifying the type DN (2; 0) theory on a Riemann sphere with three punctures. The three
punctures are associated with the 7branes, and the 5branes ending on them, where one
puncture is associated with the (0; 1) 7branes and two with the (1; 0) 7branes on either
side of the orientifold.
The punctures associated with the (1; 0) 7branes can be determined as follows: for
each such 7brane with n D5branes ending on it we associate a column with n boxes.
Starting from highest n to lowest we can combine them to form a Young diagram which
is the one associated with the punctures (see gure 2). It is not di cult to see that if
{ 5 {
ciated puncture. In all cases the 6d theory is the (2; 0) theory of type DN where N is determined
as explained in the text.
there are no stuck 7branes this generates a Young diagram of DN type on both sides.
This also straightforwardly works when there are an even number of stuck 7branes on
both sides of the O5plane. For an odd number of stuck 7branes, besides not giving a
D type Young diagram, in this case the number of D5branes on both sides is di erent.
Nevertheless, we can still associate a 4d class S theory with this case by moving one
of the stuck 7branes from the side with more D5branes to the other, doing
HananyWitten transitions when necessary. Note however that this may change the theory by the
generation of free hypermultiplets. We refer the reader to the appendix for details.
This leaves the question of the veryeven partitions which should give two di erent
punctures. We nd that we can only get one type of puncture. This appears to be related
to the question of changing the
angle for USp groups engineered by webs in the presence
of O5planes (see [14]). Presumably, if this problem can be resolved then one can also
realize the other choice of puncture in the 5d web.
{ 6 {
associated puncture. In all cases the 6d theory is the (2; 0) theory of type DN where N is determined
as explained in the text.
The punctures associated with the (0; 1) 7branes are shown in gure 3. Basically if
there is a single (0; 1) 7brane with 2n NS5branes ending on it then the corresponding
puncture is given by a Young diagram with two columns of length 2N
2n
1 and 2n + 1.
Alternatively, if there are two NS5branes, one with a single NS5brane ending on it and one
with 2n
1 NS5branes ending on it, then the Young diagram associated with this puncture
has four columns, one of length 2N
2n
1, another of length 2n
1 and two of length 1.
Next we want to provide evidence for this conjecture. First, note that the global
symmetry generated by the (1; 0) 7branes is in accordance with those given by the punctures.
As discussed in the previous section, a group of k D7branes with an even (odd) number of
5branes ending on it contributes a USp(k) (SO(k)) global symmetry. This indeed agrees
with the global symmetry associated with the punctures.
This usually also works for the (0; 1) 7branes. First note that one U(
1
) global
symmetry in the web, identi ed with translations along the O5plane, decouples from the SCFT.
So a single (0; 1) 7brane has no associated global symmetry while two (0; 1) 7branes
have an associated U(
1
) global symmetry which is enhanced to SU(2) when the number
of 5branes ending on them is equal. This indeed agrees with the global symmetry
expected from the punctures with the exception of the case where the two longest columns
in the Young diagram are equal. Then there is an extra U(
1
) in the puncture, not
appearing in the web.
{ 7 {
We can also compare the Higgs branch dimension. As discussed in the previous section,
we can break the Higgs branch of the web to 4 contributions. Likewise we can break the
Higgs branch of the 4d SCFT to the 4 contributions appearing in equation (2.1). First
there are the two contributions coming from breaking the D5branes on the D7branes on
the two sides of the orientifold. The number of possibilities depends on the number of
D5branes ending on each 7brane and so on the choice of puncture. We identify these
with f1 and f2 in (2.1), where there are two for the two sides of the O5plane.
Next we need to show that these two quantities indeed agree, at least for the maximal
puncture. Using (2.2) we nd that:
HJEP07(216)35
Our mapping instructs us to map this to 2N (1; 0) 7branes with one D5brane ending
on each one. The number of possible separations is:
fMax = N (N
1)
2
N 1 !
X i = N (N
1)
i=1
(3.1)
(3.2)
agreeing with the class S result.
A second contribution comes from pairwise separating the 2N D5branes stretching
between the two stuck D7branes closest to the NS5branes. The number of possible
separations is always N , and is identi ed with the N in equation (2.1). The nal contribution
comes from breaking the NS5branes across the (0; 1) 7branes, and pairwise separating
them from the O5plane. The number of possibilities depends on the number of
NS5branes ending on each 7brane and so on the choice of the third puncture. It is thus
natural to identify this with f3. Using equation (2.2) we can evaluate f3 for the punctures
appearing in
gure 3, and we
nd that these indeed agree to what is expected from the
web. For example consider the minimal puncture, which is the left uppermost puncture in
gure 3. Using (2.2) we nd that f3min = 1. This indeed agrees with what is expected from
the web, corresponding to the 1 dimensional Higgs branch given by pairwise separating the
two NS5branes from the O5plane, along the (0; 1) 7brane. It is straightforward to carry
this over also to the other cases.
So far we gave some rather general arguments, but next we want to put more stringent
tests on this conjecture. Consider a 5d gauge theory which can be engineered by a web
of the previously given form. In some cases this gauge theory may also be constructed by
a brane web system without the orientifold. In that case the results of [4] suggest that
compactifying it to 4d should lead to a class S theory of type A. However, our conjecture
implies that compactifying the same theory, in the same limit, should yield a class S theory
of type D. Consistency now requires that these in fact be the same SCFT which we can
test using class S technology. Particularly, we can compute and compare their spectrum
and dimensions of Coulomb branch operators, Higgs branch dimension, global symmetry,
and central charges of the nonabelian global symmetries. If our conjecture is correct then
these must match.
{ 8 {
There are several types of theories where this can be done. One type are gauge theories
of the form USp(2N ) + Nf F . When Nf = 2N + 4; 2N + 5 the webs using an O5plane are
indeed of the previously considered form, and so are conjectured to lead to a 4d D type
class S theory (see
gure 4). These theories can also be engineered using an O7
plane,
which when decomposed leads to an ordinary brane web description [13]. Precisely when
Nf = 2N + 4; 2N + 5 these webs are of the form of [4], and so lead to an A type class S
theory (see gure 5). Note that the identi cation of the theories in gures 4(b) and 5(b)
was already done in [23] while the identi cation of the theories in gures 4(a) and 5(a) is
expected from the results of [24, 25].
As mentioned in the previous section, the Sdual of the webs considered here generally
leads to D shaped quivers of SU groups. In the case of D3 the D shape degenerates to a
linear quiver which can also be engineered using ordinary brane webs. This provides another
case where we can compare two di erent constructions of the same gauge theory, where one
is expected to give an A type class S theory, while the other a D type. Figures 6, 7 and 8
show two examples of theories in this class. Figure 6 shows the quiver diagram for the 5d
gauge theories we consider. Figure 7 then shows the realization of these gauge theories
using brane webs without orientifolds, which is possible since these are linear quivers of SU
groups. Finally gure 8 shows the realization of these gauge theories using brane webs in
the presence of an ON 0plan, which is possible as these are D3 shaped quivers. This can
be mapped to a con guration with an O5 plane using Sduality.
We can test these identi cations by comparing the spectrum and dimensions of
Coulomb branch operators, Higgs branch dimension, global symmetry, and central charges
of the nonabelian global symmetries of the two SCFT's.1 For the A type class S
theory in gure 7(a) we nd using class S technology that it has a Higgs branch dimension of
dH = 4N 2 + 10N + 15 and global symmetry SO(4N + 6)8N+4
SU(4)12 where the subscript
denote the central charge. We also nd the spectrum of Coulomb branch operators to be:
d2 = d3 = 0; d4 = 1
di =
di =
1
0
( 2 i even )
( 1 i even )
i odd
i odd
for 4 < i
2N + 1
for 2N + 1 < i
4N + 2
where di stands for the number of such operators with dimension i, and we assumed N > 1
(for N = 1 the matching reduces to the N = 1 case of gures 4(b) and 5(b)). We can now
compare these values with the ones for the D type class S theory in gure 8(a). Using class
S technology we carried out this calculation nding complete agreement.
For the A type class S theory in gure 7(b) we nd using class S technology that it has
a Higgs branch dimension of dH = 4N 2 + 4N + 7 and global symmetry SO(4N + 4)8N
1We can also compare the a and c conformal anomalies of the SCFT's. However, for class S theories
these are completely determined in terms of the spectrum and dimensions of Coulomb branch operators
and the dimension of the Higgs branch [31], and so do not provide an independent check.
{ 9 {
(b) using an O5plane. The upper part shows the webs describing the 5d SCFT's. These webs are
of the form we consider and so we conjecture that their compacti cation on a circle results in the
D type class S theories shown below them.
(b) using a resolved O7 plane. The left part shows the webs one gets after resolving the O7
plane. By performing a series of 7brane motions these can be recast into the the form of [4]. The
resulting webs are shown in the middle part of the
gure. Upon compacti cation to 4d, these are
expected to give the type A class S theories shown on the right.
and 8. All groups are of type SU with the CS level written above the group.
gure 6(b). The left part shows the webs in a form where the
gauge theory description is manifest. Note that we use black X's for (1; 0) 7branes. By performing
a series of 7brane motions, these can be recast into the the form of [4]. The resulting webs are
shown in the middle part of the gure. Upon compacti cation to 4d, these are expected to give the
type A class S theories shown on the right.
SU(2)28
U(
1
). We also nd the spectrum of Coulomb branch operators to be:
d2 = 0
di =
di =
(
(
1
0
2 i even
1 i even
i odd
i odd
)
)
for 2 < i
2N
for 2N < i
4N
We can now compare these values with the ones for the D type class S theory in
gure 8(b). Using class S technology we carried out this calculation
nding complete
agreement.
gure 6(a), now using an ON
0plane. (b) The brane web for the 5d gauge theory shown in gure 6(b), now using an ON 0plane.
The left part shows the webs using the ON 0plane, which makes the D3 shaped quiver nature of
the gauge theories manifest. Again we use black X's for (1; 0) 7branes. Performing Sduality leads
a web using an O5plane where for ease of presentation we have suppressed the bending due to the
change in O5plane type. By performing a series of 7brane motions these can be recast into the
form we consider. Therefore we conjecture that their compacti cation on a circle results in the D
type class S theories shown on the right.
We can also consider SO(N ) gauge theories with spinor matter for N
6. In that
range the theory can also be engineered using brane webs without the orientifold. Selected
examples are shown in
gure 9, for cases with a D4 class S theory, and gure 10 for cases
with a D5 class S theory. All the class S theories of type D4 were analyzed in [17], and
their results indeed agree with our expectation from the 5d SCFT.
It is straightforward to also carry this analysis for the theories in gure 10. For the D
type class S theory in
gure 10(a) we nd it has an SU(4)10
SU(8)12 global symmetry,
a 33 dimensional Higgs branch and Coulomb branch operators of dimensions 4; 5; 6. This
indeed matches the properties of the A type class S theory in gure 10(a) evaluated using
class S technology.
We can do the same for the theory in gure 10(b) now nding an
SU(2)9
(E7)16 global symmetry, a 35 dimensional Higgs branch and Coulomb branch
operators of dimensions 4; 8. This indeed matches the properties of the rank 2 E7 theory
as can be computed using class S technology from the construction of this theory with a
free hyper given in [19].
We do note that sometimes the class S theory contains free hypers in excess of those
expected from 7brane motions (as explained in the appendix). This seems common for
SO(3) and SO(4), and an example for this is given by the rst case in gure 9. Nevertheless,
the Higgs branch of the class S theory with the free hypers agrees with that of the web.
Thus, the natural interpretation is that the SCFT described by the web in these cases also
contains free hypers. As mentioned in [14], the spinors are thought to come from instantons
associated class S theories being a compacti cation of the D4 theory where we have used that
SO(3) = SU(2), SO(4) = SU(2)2 and SO(5) = USp(4). Up to the free hypers in the rst case, these
match what is expected from the gauge theory description, see [17].
of an SU(2) gauge theory broken by a motion on the Higgs branch. Thus, it is tempting to
identify these free hypers as directions on that Higgs branch that somehow remain in the
lowenergy theory.
A further consistency check is to match the global symmetry expected from the 5d
description to the one of the 4d SCFT calculated using class S technology. As an example,
consider an SO(N ) gauge theory with spinor matter. In [28] an analysis of the 1
instanton operators was done for these theories, which in turn suggests that enhancement of
symmetries should occur in some cases. The class S theories which we conjecture result
from compactifying these theories do not show that symmetry, and in many cases not even
the classically visible symmetry. Consistency now requires that the 4d global symmetry is
actually enhanced to a larger global symmetry than is visible from the punctures, which we
associated D5 theory and the construction using ordinary brane webs leading to an A5 theory. (b)
Shows another example now using the rank 2 E7 theory.
can test in turn using class S technology. Several examples are shown in gures 11 and 12,
where in all cases complete agreement with the results of [28] is found, and further that
the Coulomb branch dimension of the class S theory agrees with that expected from the
web. Also class S technology calculations are consistent with the two theories in gure 11
being the same theory up to a di ering number of free hypers.
The webs are shown on the left, and upon moving the rightmost stuck D7brane to the leftmost side
of the web, they can be cast in the form we consider. Note that this 7brane motion results in the
creation of free hypers. We conjecture that compacti cation on a circle to 4d results in the D type
class S theories shown on the right. Above each class S theory we have written its global symmetry
evaluated using the superconformal index. These agree with the expected global symmetry of the
5d SCFT determined from the 1 instanton analysis of [28].
3.2
SCFT's with marginal deformations
Next we want to discuss theories related to the compacti cation of the D type (2; 0) theory
on a sphere with more than three punctures. Unlike the previous case the reduction now
involves several scaling limits where in addition to taking the R5 ! 0 limit one also takes
the limit mi ! 1, for several mass parameters mi, while keeping miR5 xed. These then
become the marginal deformations that exist in compacti cation of a (2; 0) theory on a
sphere with more than three punctures. These mass parameters can be chosen as mi = g152d
corresponding to the couplings of gauge groups. These then become marginal gauge group
couplings in 4d which are indeed related to their 5d counterparts by g142d
gR525d .
The actual correspondence is a straightforward generalization of the previous case.
The theories in this class are of the same form as the previous ones except that the number
of (0; 1) 7branes is unconstrained save for the demand that they can be partitioned into
groups each one in the form of gure 3. We can then do the reduction in the limit where
(upper part) and SO(8) + 4V + 2S + 1C (lower part). The webs are shown on the left, where
the lower one is already in the form we consider. The upper web can also be cast in this form
by moving the rightmost stuck D7brane to the leftmost side of the web, though this leads to the
creation of free hypers. We conjecture that compacti cation on a circle to 4d results in the D type
class S theories shown on the right. Above each class S theory we have written its global symmetry
evaluated using the superconformal index. These agree with the expected global symmetry of the
5d SCFT determined from the 1 instanton analysis of [28].
each group of (0; 1) 7branes is in nitely separated from the other. This corresponds to
a weakly coupled limit for the gauge theory living on the D5branes stretched between
neighboring groups of (0; 1) 7branes.
It is thus straightforward to conjecture that the resulting theory is given by a collection
of isolated SCFT's connected by the gauge groups whose weak coupling limit was taken.
This in turn is equal to the compacti cation of a D type (2; 0) theory on a Riemann sphere
with punctures where each group of (0; 1) 7branes and the (1; 0) 7branes on the two sides
of the O5plane gives a puncture of the type explained in the previous subsection.
As an example we consider SO gauge theories with spinors and vector matter, for
example the SO(8) and SO(7) theories shown in gure 13. With the chosen matter content,
and SO(8) + 4V + 2S (in (c)). These are in a form that we conjecture that when reduced to 4d,
in the in nite weakly coupled limit, lead to a compact cation of the D4 theory on the shown four
punctured sphere. Indeed the analogue 4d gauge theories are conformal, and as was found in [17]
have the shown representation as a compacti cation of the D4 (2; 0) theory. Also note that the 4d
theory in (a) also contains a free hyper that from the 5d point of view is generated when moving
the right stuck 7brane to the left side.
the webs are of the form expected to lead to a four punctured sphere when reduced to 4d.
Furthermore it is clear from the web that the reduction is done in the limit where the
SO group becomes weakly coupled, and we expect it to descend to the corresponding SO
group with marginal coupling. In fact the matter content is exactly the one required for the
analogous 4d theory to be conformal. So we expect the resulting theory to have a completely
perturbative description given by an SO gauge group with vector and spinor matter.
In fact such 4d theories were constructed, using compacti cation of D type (2; 0)
theories, in [17], and we can compare the class S construction they give with the one
generated from the web by our prescription,
nding complete agreement (see
gure 13).
This naturally leads to one frame of the many possible frames obtained by taking di erent
pair of pants decompositions of the Riemann surface. One may ask whether the other are
also manifest in the 5d description. Some of them, the ones corresponding to exchanging
the punctures associated with a group of (0; 1) 7branes, are manifest as we can exchange
them also in the web. This leads to 5d lifts of Gaiotto dualities, similar to the cases studied
in [9] for the A type theories.
In the 4d theory we can exchange any of the punctures while in the 5d theories there
does not appear to be a way to exchange the punctures associated with the (0; 1) 7branes
with the ones associated with the (1; 0) 7branes. This is already apparent from the
mapping, since we can essentially get any puncture from the (1; 0) 7branes, but only a subset
from the (0; 1) 7branes. It would be interesting to nd a generalization of these mappings
allowing a description of all types of punctures also with the (0; 1) 7branes.
Finally we wish to discuss the issue of veryeven partitions. To do this we draw your
attention to
gure 13(c), in which there is an example of a class S theory with two
veryeven partitions. In these cases there are two possible theories depending on whether the
two veryeven partitions are of the same or di ering colors. In both cases the theory is
an SO(8) gauge theory with four hypermultiplets in the vector representation and two in
the spinor. The spinors have the same chirality if the colors are the same, but di erent
otherwise. So one can see that the choice of veryeven partition is related to the chirality
of the spinors. It is currently unclear how one can change the chirality of the spinors in
the web, and so how the other choice of veryeven partition can be realized. See [14] for a
discussion of this point.
4
Conclusions
In this article we have proposed a connection between a class of 5d SCFT's engineered
by brane webs in the presence of O5planes and 4d class S theories of type D. It will be
interesting to further explore this, and see if further evidence for this proposal can be found.
This has several interesting implications for the class S theories involved. First, the
5d theory may have one or more gauge theory descriptions. This implies that the
corresponding class S theory has mass deformations leading to the analogous 4d gauge theory.
We have also seen that this proposal motivates identi cations of apparently distinct class S
theories. Using the work of [29], this then also implies an identi cation of the dual mirror
theories, which may lead to interesting 3d dualities.
Also there are several other intriguing questions that warrant further exploration. We
have seen that there are class S theories of type D for which we do not know the 5d lift,
notably the DN theory. It is interesting if there is some generalization of these constructions
that may allow the realization of the 5d lifts of these theories as well. On the other hand,
there are other brane webs system with an O5plane that are not of the desired form. It
will be interesting to also study their 4d limit.
Acknowledgments
I would like to thank Oren Bergman and Shlomo S. Razamat for useful comments and
discussions. G.Z. is supported in part by the Israel Science Foundation under grant no.
352/13, and by the GermanIsraeli Foundation for Scienti c Research and Development
under grant no. 1156124.7/2011.
after moving the two D7branes past each other. (c) The brane con guration one gets after
performing two Tdualities in the directions parallel to both the D7branes and the D5branes, followed
by an Sduality, on the system of (a). The vertical wide black lines are NS5branes and the thin
horiozontal lines are D3branes. (d) The resulting brane con guration after the same transformations
are done on the system in (b).
A
Brane motions and free hypermultiplets
In this appendix we wish to discuss the creation of free hypermultiplets as a consequence
of brane motions. Consider the two systems shown in
gures 14(a) and (b), related by a
7brane motion. The system in (a) has a one dimensional Higgs branch, corresponding to
breaking the extended D5brane on the 7brane. However, in (b) there is now a two
dimensional Higgs branch so it appears that this motion has a ected the theory. Furthermore, as
the Coulomb branch moduli hasn't changed, the most logical explanation is that a single
free hyper was created in this motion.
Next, we wish to show that this leads to known relations in other dimensions. Consider
taking the systems in gure 14(a)+(b), performing two Tdualities and an Sduality, so that
the 7branes become NS5branes and the D5branes become D3branes. This leads to the
brane con guration of gures 14(c) and (d) which are again related by brane motion, now
of the NS5branes. These motions and the dualities they imply for the theories on the
D3branes, where considered in [30] who found that indeed the theory in (d) is the one in
(c) with the addition of a free twisted hypermultiplet.
As an additional example consider the theory shown in
gure 15(a). This describes
the 5d T4 theory. Figure 15(b) shows the system after several brane motions which also
involves one motion of the type considered before. Thus, we conclude that the theory
shown in
gure 15(b) is the 5d T4 theory with a single free hyper. This theory is also in
the form of [4] so it leads to 4d class S theory. This theory is indeed known to be T4 with
a single free hyper [31].
at the con guration in (b). Note that these include a transition of the type shown in
gure 14.
(c) and (d) are the mirror duals of the 3d analogue class S theory for the theories in (a) and (b)
respectively.
Another interesting test is given by employing the relation of [29] between the 4d class S
theory and the 3d mirror dual. These are shown in gure 15(c) and (d) for the two theories
involved. The preceding imply that these two are identical up to a free twisted hyper, and
indeed this is just an application of the previous duality of [30] to the central node.
directions parallel to both the D7branes and the D5branes, and an Sduality.
We can ask what happens in the more general case shown in gure 16(a). Counting
the Higgs branch moduli we nd that it changes by jn
kj so the natural expectation is
that jn
kj free hypers are generated. We can again consider the related 3d system shown
in gure 16(b). If we are correct then these must be dual up to jn
kj free twisted hypers.
This has indeed been argued in [32] from 3d reasoning.
We can also consider the analogue application for class S theories. Figure 17 shows
similar motions as the ones done for the T4 theory now done for TN . This leads to identifying
the two resulting class S theories up to N
3 free hypermultiplets. For N > 4 these theories
are bad in the sense of [19] so index analysis is unknown for them. Also application of
class S technology leads to a negative number of Coulomb branch operators and so is
not applicable in this case. Yet, the preceding analysis suggests that these theories exist
and are given by the TN theory with free hypermultiplets. One supporting evidence is to
consider the 3d mirror duals shown in gure 17, which are indeed related by the duality
of [32] on the central node.
Finally we wish to discuss the implications of these on the systems considered in
this article. This phenomenon still occurs with little changes due to the presence of the
orientifold plane. For example consider the systems shown in gure 18(a) and (b). Counting
the Higgs branch dimensions we see that in the case of (a) we get one free hyper while in
the case of (b) we get two. It is straightforward to generalize this to other cases.
We next ask whether we can test this also in this case against known phenomena in
4d or 3d. The discussion in section 3 implies that this should lead to similar relationships
between di erent class S theories of type D, but it appears at least one of them is bad
(again in the sense of [19]) making explicit comparison di cult. One interesting implication
of this is a generalization of the dualities in [32] also to SO or USp gauge theories. To our
knowledge these have not been previously discussed from purely 3d reasoning. It will be
interesting to further explore this.
leads to the web in (b). (c) and (d) are the mirror duals of the 3d analogue class S theories for the
theories in (a) and (b) respectively.
case where one of the 7branes is stuck while (b) shows the case where both are full.
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