Brane webs, 5d gauge theories and 6d \( \mathcal{N}=\left(1,\;0\right) \) SCFT’s
HJE
Brane webs, 5d gauge theories and 6d
Gabi Zafrir 0 1
0 Haifa , 32000 , Israel
1 Department of Physics, Technion, Israel Institute of Technology
We study 5d gauge theories that go in the UV to 6d N = (1; 0) SCFT. We focus on these theories that can be engineered in string theory by brane webs. Given a theory in this class, we propose a method to determine the 6d SCFT it goes to. We also discuss the implication of this to the compacti cation of the resulting 6d SCFT on a torus to 4d. We test and demonstrate this method with a variety of examples.
Field Theories in Higher Dimensions; Brane Dynamics in Gauge Theories

SCFT's
1 Introduction
2 Preliminaries
2.1
2.2
Properties of the 6d theories
Class S technology
3.1
Generalizations
3.1.1
3.1.2
A simple example
Another example: the 5d TN theory with extra avors
4 Additional 5d theories
4.1
Quivers of SU groups
4.2 SU quivers with antisymmetric hypers
4.2.1
4.2.2
SU quivers with an antisymmetric hyper at one end
SU quivers with an antisymmetric hyper at both ends
4.3 Cases with completely broken groups
5 Conclusions
A Instanton counting for SU(N ) + 2AS + Nf F
3 The 5d (N + 2)F + SU0(N )k + (N + 2)F quiver and related theories
ries can also be realized in string theory using brane webs and geometric engineering [4{7].
The picture emerging from these methods is that 5d SCFT's exist and that they
sometimes posses mass deformations leading to 5d gauge theories, with the mass identi ed as
the inverse gauge coupling squared, g 2. These theories also posses some quite interesting
nonperturbative behavior. One such phenomenon is the occurrence of enhancement of
symmetry, in which the xed point has a larger global symmetry than that perturbatively
{ 1 {
exhibited in the gauge theory. An important ingredient in this is the existence of a
topological U(1) conserved current, jT =
Tr(F ^ F ), associated with every nonabelian gauge
group. The particles charged under this current are instantons.
These instantonic particles sometimes provide additional conserved currents leading to
an enhancement of the perturbative global symmetry. A simple example is SU(2) gauge
theory with Nf hypermultiplets in the doublet of SU(2). For Nf
7, this theory is known
to ow to a 5d
xed point, where the global symmetry is enhanced from U(1)
SO(2Nf )
to ENf +1 by instantonic particles [1]. This can be argued from string theory constructions,
and is further supported by the superconformal index [8, 9].
In many cases, a single 5d SCFT may have many di erent gauge theory deformations.
theory [5, 10].1 By now a great many examples of this are known, see [10{15].
String theory methods, such as brane constructions, also suggest the existence of
interacting 6d N = (1; 0) SCFT's [16{18]. These theories include massless tensor multiplets,
in addition to hyper and vector multiplets. The tensor multiplets contain a scalar leading
to a moduli space of vacua. In some cases, the low energy theory around a generic point in
this space is a 6d gauge theory, where g 2 is identi ed with the scalar vev [19]. By now, a
large number of such SCFT's are known. In fact, there exists a classi cation of N = (1; 0)
SCFT's using Ftheory [20, 21]. See also [22] for a classi cation of N = (1; 0) gauge theories.
There is an interesting relationship between 5d gauge theories and 6d N
theories, where, in some cases, a 5d gauge theory has a 6d N = (1; 0) UV completion. The
best known example is 5d maximally supersymmetric YangMills theory, which is believed
to lift to the 6d (2; 0) theory [23, 24]. Yet another notable example is the 5d gauge theory
with a USp(2N ) gauge group, a hypermultiplet in the antisymmetric representation, and
8 hypermultiplets in the fundamental representation, which is believed to lift to the 6d
rank N Estring theory [25]. Recently, another example was given in [26]. There the 6d
theory in question is known as the (DN+4; DN+4) conformal matter [27], which has a 6d
gauge theory description as USp(2N ) + (2N + 8)F . This theory is suspected to be the UV
completion of the 5d gauge theory SU0(N + 2) + (2N + 8)F .
The purpose of this paper is to extend these results to a large class of 5d gauge theories
with an expected 6d N = (1; 0) SCFT UV completion. We consider theories which can be
represented as ordinary 5brane webs. The starting point is to generalize the discussion
of [26] to the class of 5d gauge theories of the form (N + 2)F + SU0(N )k + (N + 2)F . These
were recently conjectured to lift to 6d SCFT [28]. Furthermore, in [29] a conjecture for
this 6d SCFT appeared. We start by generalizing the method of [26] to give evidence for
this conjecture.
1In 5d one can add a ChernSimons (CS) term to any SU(N ) gauge theory, for N > 2, and we use a
subscript under the gauge group to denote the CS level. For USp(2N ) groups, a CS term is not possible,
but there is a discrete Z2 parameter, called the
angle, which can be either 0 or
[2]. We again use a
subscript under the gauge group to denote it. Also, when denoting gauge theories we use F for matter
in the fundamental representation and AS for matter in the antisymmetric representation. When writing
hyper associated with every .
Using this result we then go on to propose a technique to determine the answer for
other 5d gauge theories, by thinking of them as a limit on the Higgs branch of a 5d gauge
theory (N + 2)F + SU0(N )k + (N + 2)F for some N and k. Then we can determine the 6d
SCFT by mapping the appropriate limit of the 5d Higgs branch to the corresponding one
of the 6d theory. We consider a variety of examples, exhibiting both the advantages and
limitations of this technique.
As an application of these results, we also consider the compacti cation on a torus
of the 6d SCFT's appearing as the lift of 5d gauge theories. For example, consider the
compacti cation of the rank 1 E string theory on a torus, where we take the limit of zero
area, keeping the 6d global symmetry unbroken. First compactifying to 5d, we get the 5d
theory SU(2) + 8F . We now want to compactify to 4d taking the limit of zero torus area,
but without breaking the E8 global symmetry. It turns out that the way to do this is by rst
integrating out a avor, owing to SU(2) + 7F .2 This leads to a 5d SCFT with E8 global
symmetry [1]. Compactifying this to 4d then leads to the rank 1 MinahanNemashansky E8
theory [30, 31]. For additional examples of the compacti cation of 6d N = (1; 0) SCFT's
on a torus, see [32{34].
We can now adopt a similar strategy to understand the result of compacti cation on
a torus of the 6d SCFT's we encounter. That is we rst compactify to 5d leading to the
5d gauge theory. Taking the R6 ! 0 limit, while keeping the 6d global symmetry, is
then implemented by integrating out a avor. This leads to a 5d SCFT with a brane web
description of the form of [35]. It is now straightforward to take the R5 ! 0 limit, leading
to a class S isolated SCFT, as shown in [35]. Thus, we conjecture that reducing the class
of 6d theories we consider on a torus leads to an isolated 4d SCFT. The main idea is
summarized graphically in gure 1.
We next seek to provide evidence for this relation. To this end we use the results
of [32], who found a way to calculate the central charges of a 4d theory resulting from
compacti cation of a 6d theory on a torus in terms of the anomaly polynomial of the 6d
theory. We can now compute the 4d central charges rst using class S technology (see [36]),
and second from the anomaly polynomial (using [37]), and compare the two. We indeed
nd that these match. This then provides evidence also for the original 5d
6d relation.
The structure of this article is as follows. Section 2 presents some preliminary
discussions about the computation of the anomaly polynomial for the 6d SCFT's considered
in this article, as well as the class S technology we use. In section 3 we consider the 5d
theory (N + 2)F + SU0(N )k + (N + 2)F . We rst generalize the methods of [26] to test the
conjecture of [29], and then go on to consider related theories. Section 4 deals with other
5d theories expected to lift to 6d, that are not of the form presented in section 3. We end
with some conclusions. The appendix discusses symmetry enhancement for a class of 5d
theories that play an important role in section 4, and which, to our knowledge, were not
previously studied.
like: g12
0
R6 g2 ! 0 limit.
2Note that this is a R6 ! 0 limit. This follows as one must keep the e ective coupling, which behaves
constant jmj, well de ned. Therefore, when taking the m ! 1 limit, one must also take the
{ 3 {
HJEP12(05)7
deform the 5d gauge theory, corresponding to taking the R6 ! 0 limit while keeping the 6d global
symmetry intact. This leads to a 5d SCFT. We then compactify this SCFT on a circle of radius
R5, and take R5 ! 0. This leads to a 4d class S SCFT, which can in turn be thought of as a result
of compactifying a 6d (2; 0) SCFT on a Riemann sphere with three punctures. We can use this
description as a consistency check by calculating the properties of this 4d SCFT when thought of as
a compacti cation of a 6d (2; 0) SCFT, known as class S technology, and comparing against what
is expected from the compacti cation of the 6d (1; 0) SCFT.
A word on notation.
Brane webs comprise an important part of our analysis and so
they appear abundantly in this article. In many cases only the external legs are needed and
not how they connect to one another. In these cases, for ease of presentation, we have only
depicted the external legs, using a large black oval for the internal part of the diagram.
Many of the diagrams also contain repeated parts shown by a sequence of black dots. This
should not be confused with 7branes.
In brane webs one can also add 7branes on which the 5branes can end. We have
in general suppressed the 7branes, with the exception of two cases. One, when several
5branes end on the same 7brane. In this case we depicted the 7brane as a black oval, the
type of which is understood by the type of 5branes ending on it. We in general also write
the number of 5branes ending on this 7brane. If no number is given then it is the number
visible in the picture. Any other numbers that appear stand for the number of 5branes.
The second case where we explicitly include 7branes is if no 5branes end on them. In
this case we denote a (1; 0) 7brane by an X and a (0; 1) 7brane by a square. Any other
7brane is denoted by a circle with the type written next to it.
We generically suppress the monodromy line of the 7branes. In the special cases when
we do draw it, we use a dashed line.
{ 4 {
This section discusses the type of 6d theories we encounter, the computation of the anomaly
polynomials for these theories, and the class S technology used in this article.
2.1
Properties of the 6d theories
We start by presenting the 6d gauge theories that we consider in this article. We rst
present them in their gauge theory description, namely at a generic point on the tensor
branch of the underlying 6d SCFT. In this description the gauge theory is made from a
quiver of SU(Ni) groups with one end being just fundamental hypers while the other end
being either a USp gauge group or an SU group with a hyper in the antisymmetric. The
freedom in the choice of the theory is given by the ranks of the groups Ni. The number of
avors for each group is uniquely determined by anomaly cancellation for each group. The
quiver diagrams for the theories we consider are shown in gure 2.
Next we wish to evaluate the anomaly polynomial that we use later. We concentrate
only on the terms in the anomaly polynomial that we need. By using the results of [37, 38],
we nd that the anomaly polynomial contains:
where Fi is the eld strength of the i'th group (we always denote the USp or SU with the
antisymmetric as i = 1), and a summation over repeated indices is implied. Also C2(R)
{ 5 {
stands for the second Chern class of the Rsymmetry bundle, and p1(T ); p2(T ) are the rst
and second Pontryagin classes of the tangent bundle, respectively. We use nh; nv and nt
for the number of hyper, vector and tensor multiplets respectively, and hGi for the dual
Coexter number of the i'th group. Finally:
.
.
.
2
tensor multiplet (see [19] for the details). For the case at hand this adds the following to
the anomaly polynomial:
X 1 tr(Fi2)
i
i
tr(Fi2+1) + C2(R) X hGj
j=1
p1(T )
4
32
5
2
collecting all the terms we nd:
where
C2(R)p1(T ) nv4d + dH
7p21(T )
4p2(T )
5760
(2.2)
(2.3)
(2.4)
(2.5)
where the sum i is over all the gauge groups.
The labels we used were chosen with the compacti cation to 4d in mind. When
compactifying to 4d on a torus we get some 4d SCFT in the IR. We can calculate the central
charges, particularly the a and c conformal anomalies, of this SCFT using the results
of [32].3 We nd that dH = 24(c
a) and nv4d = 4(2a
c). Thus, dH is the dimension of the
Higgs branch, and n4d the e ective number of vector multiplets of the 4d SCFT resulting
v
from the compacti cation of the 6d SCFT on a torus. In that light the equation for dH
has a rather nice interpretation as the classical dimension of the Higgs branch of the gauge
theory, nh
nv, plus the contribution of the tensor multiplets, each giving 29 dimensions,
like the rank 1 Estring theory.
Besides the a and c conformal anomalies, we also want to determine the central charges
for avor symmetries, kFi , associated to the avors under the i'th gauge group. From the
result of [32], this can be determined from the term 1k9F2i tr(Fg2lobali )p1(T ). Say we have a
avor symmetry, the elds charged under it being
avor of dimension
under the group
Gi. Then we nd that:
we're considering are of this type.
kFi = 12gi + 2d
(2.6)
3For these results to hold, the 6d SCFT must be veryHiggsable, as described in [32]. All the 6d SCFT's
{ 6 {
where gi = nG
representation .
i + 1, nG being the number of groups, and d is the dimension of the
Before continuing we note that some of the theories we consider also include gauging
the rank 1 Estring theory at one end of the quiver. This is a straightforward extension of
the quiver theories with a USp(2N1) end to the N1 = 0 case. This follows from the fact
that USp(2N1) + (2N1 + 8)F goes to a 6d SCFT known as the (DN1+4; DN1+4) conformal
matter [27], so this class of theories can be regarded as gauging a part of the SO(4N1 + 16)
global symmetry of (DN1+4; DN1+4) conformal matter. In this description we can also
consider the case of N1 = 0 relying on the fact that (D4; D4) conformal matter is the
rank 1 Estring theory. Going over the computation of the anomaly polynomial, we nd
that (2.5) is still valid, where we include the rank 1 Estring theory in the sum and take
hE string = 1.
kF1 = 12Q(nG + 1).
Generically when gauging a part of a rank Q Estring theory, some of the E8 global
symmetry remains unbroken and serves as a global symmetry. For these cases we nd
Finally, while we generally employ the gauge theory description of these (1; 0) SCFT's,
it is worthwhile to also specify their description as an Ftheory compacti cation. In this
language the theory is described as a long
1
2
2 curves and a USp or SU type group on the
For the details on the meaning of this notation we refer the reader to [21]. In a
nutshell, specifying a 6d SCFT requires enumerating its hyper, vector and tensor content.
The numbers represent the type of tensor multiplet, where a
2 curve represents a single
free N = (2; 0), tensor and a
1 curve the rank 1 Estring theory. The sequence of numbers
represents several tensor multiplets. For example,
2
An 1 theory where n is the number of
2 curves, and
2
1
2 : : :
2
2 gives the N = (2; 0)
2 : : :
2 gives the rank
n + 1 Estring theory.
One can add vector multiplets on these curves. When these are added, the theory on
the tensor branch acquires a gauge theory description. For a
2 curve, adding an SU(N )
type group, leads to an SU(N ) + 2N F gauge theory on the tensor branch. For a
1 curve,
adding a USp(2N ) type group leads to a USp(2N ) + (2N + 8)F gauge theory, while adding
an SU(N ) type group, leads to an SU(N ) + 1AS + (N + 8)F gauge theory at a generic
point on the tensor branch.4 It is now apparent that going to a generic point on the tensor
branch indeed gives the gauge theories we consider.
We can also consider the reverse process of removing vector multiplets from a curve.
This describes a Higgs branch limit of the 6d SCFT in which some of the vector multiplets
become massive and the theory
ows to a di erent IR SCFT. Note in particular, that
completely breaking a group, corresponding to removing all the vector multiplets from
that curve, still leaves the associated tensor multiplet. The resulting IR SCFT generically
has no complete Lagrangian description, but can still be described by a gauge theory
gauging part of the avor symmetry of a nonLagrangian part. We shall encounter several
examples of this later.
4For SU(
6
) there is an additional option giving an SU(
6
) + 12 20 + 15F gauge theory at a generic point
on the tensor branch. We brie y encounter this option later in this paper.
{ 7 {
Sometimes gauge theory physics is insu cient to fully determine the properties of the
SCFT. For example, in some cases the global symmetry naively exhibited by the gauge
theory, is larger than the one of the SCFT. We encounter some cases where this occurs,
and then it is useful to have an Ftheory description.
The results obtained from the 6d anomaly polynomial can be compared to the ones obtained
using class S technology. Speci cally, the theories we consider are all isolated SCFT's, that
can be represented as the compacti cation of an A type (2; 0) theory on a Riemann sphere
with 3 punctures. We also have a 5d brane web representation using [35]. It is known
how to calculate the central charges of such SCFT's from the form of the punctures. The
explicit formula used to calculate dH ; nv4d and kF can be found in [36, 39]. In practice, it
is usually simpler to calculate dH directly from the web.
We also want to determine the global symmetry of the SCFT. In general this can
be read of from the punctures, but in some cases the global symmetry can be larger than
is visible from the punctures [40]. One way to determine this is using the 5d description
either directly from the web, or using the gauge theory description.
A more intricate method is to use the 4d superconformal index. Since conserved
currents are BPS operators they contribute to the index, and so knowledge of the index
allows us to determine the global symmetry of the theory. In practice we do not need the
full superconformal index, just the rst few terms in a reduced form of the index called
the HallLittlewood index [41]. An expression for the 4d superconformal index for class S
theories was conjectured in [41{43], and one can use their results to determine the global
symmetry. For more on this application see [44].
In cases where the global symmetry is bigger than what is visible from the punctures,
we use the 4d superconformal index to show this. In cases where it is not di cult to argue
this also from the 5d description, we also use this as a consistency check.
3
The 5d (N + 2)F + SU0(N )k + (N + 2)F quiver and related theories
In this section we start analyzing the 6d lift of 5d theories. We start with the 5d quiver
theory (N + 2)F + SU0(N )k + (N + 2)F . Since a conjecture for this theory was already
given in [29], it is more convenient to start with the 6d theory. There are two slightly
di erent cases to consider. First, we have the 6d SCFT whose quiver description is shown
in gure 3. This theory can be realized in string theory by a system of D6branes crossing
an O8 plane and several NS5branes, shown in gure 4. Note, that this is a generalization
of the system in [26], by the addition of NS5branes. We can now repeat the analysis of [26].
Since this is a simple generalization of their work we will be somewhat brief. We compactify
a direction shared by all the branes and preform Tduality. The O8
plane becomes two
O7
planes. Under strong coupling e ect, the O7
plane decomposes to a (1; 1) 7brane
and a (1; 1) 7brane [45]. We then end up with the web of gure 5. This web describes
the 5d gauge theory (N + 2)F + SU0(N )2l 1 + (N + 2)F , as shown in gure 6. Note that
the number of groups in 5d must be odd, owing to the even number of NS5branes.
{ 8 {
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part of the global symmetry of the shown 6d SCFT, in this case an SU(
8
) subgroup of E8.
gure 3. The horizontal lines represent
D6branes, and the number above the lines stand for the number of 6branes. The black circles
represent NS5branes, and their number is given below. Finally, the vertical line stands for the O8
plane. The con guration also include 2N + 4l D8branes, parallel to the O8
plane, on which the
asymptotic D6branes end. For clarity we have suppressed them in the gure.
This suggests that to get an even number of 5d SU(N ) groups, we need to take an odd
number of NS5branes, which we do by adding a stuck NS5brane on the O8
plane. The
brane and quiver description of the resulting 6d theory is shown in
gure 7. We can now
repeat the analysis. After Tduality we get again two O7 planes with the stuck NS5brane
stretching between the two. Decomposing the O7
planes with the stuck NS5brane, as
shown in [14], we arrive at the web of gure 8. As shown in
gure 9, this is the web of
(N + 2)F + SU0(N )2l + (N + 2)F . This agrees with the conjecture of [29], that this 6d
theory is the UV completion of the 5d gauge theory (N + 2)F + SU0(N )2l + (N + 2)F .
Note that in the 6d theories covered so far we have assumed that N > 2l
1. Naively,
this implies the same limitations on the 5d theories. However, it is not di cult to see that
performing Sduality on the web for (N +2)F +SU0(N )k 1 +(N +2)F results in the one for
(k + 2)F + SU0(k)N 1 + (k + 2)F . Thus, by doing an Sduality, one can map any 5d linear
{ 9 {
and resolving the O7 planes.
SU(N ) quiver to the required form. Also note that when k = N
1, both descriptions are of this form, and indeed the two 6d SCFT's are the same. We now wish to employ this relation to the compacti cation of the 6d SCFT on a torus,
2
preserving the global symmetry. Inspired by the Estring theory example, we are lead to
consider an in nite mass deformation limit of the related 5d theory. The natural candidate
is integrating out a fundamental avor. We have only one possibility, corresponding to the
5d theory (N + 2)F + SU0(N )k 2
SU 1 (N ) + (N + 1)F whose web is shown in gure 10.
This theory does give a 5d xed point shown in gure 10. We note that this web is of the
form of [35]. We can now employ class S technology to determine the global symmetry
of this theory, nding that its global symmetry is U(1)
SU(2)
SU(2N + 2k) when N = k.
SU(2N + 2k) when N 6= k and
Note that this is exactly the same as the global symmetry of the 6d theory. The
avors at the end give the SU(2N + 2k) part. The remaining U(1) is the anomalyfree
combination of the various baryonic and bifundamental U(1)'s. The case of N = k indeed
has an enhancement of symmetry to SU(2). For k = 2l + 1, this comes about because the
antisymmetric representation of SU(4) is real while for k = 2l, this comes about as the
gauging of SU(
8
)
E8 preserves an SU(2), since SU(
8
)
E7
E7
SU(2)
E8.
We now conjecture that compactifying this 5d theory to 4d should give the
compactication of the starting 6d theory on a torus. We know from the work of [35] that for the
theory of gure 10, this leads to a 4d isolated SCFT that can be described by a
compacti cation of the 6d (2; 0) theory of type A2N+2k 3 on the punctured sphere of gure 11.
on the left through the leftmost NS5brane leading to the web in (b). We can now push the (1; 1)
7brane and (1; 1) 7brane through the neighboring NS5brane. This changes the asymptotic
NS5brane to a D5branes, and is accompanied by a HananyWitten transition generating an additional
5brane ending on the 7brane. This gives the web in (c). Repeating this on the neighboring l
2
NS5branes, and also doing the same on the right hand side, we end up with the web in (d). This
is the web of gure 5 after pulling out the internal 7branes.
We next wish to test this conjecture by comparing the central charges of this 4d SCFT
with the ones expected from the compacti ed 6d (1; 0) SCFT which can be determined
through (2.5), (2.6).
From the 5d theory, using class S technology, we nd:
k, kSU(2) being relevant only for the N = k case. The results for
N < k can be generated from (3.1) by taking N $ k.
resolving the O7 planes.
SU 12 (N ) + (N + 1)F . The upper left shows
the web in its gauge theory description. Moving rst the two shown (0; 1) 7branes down to the
other side and then pulling out the two upper (1; 0) 7branes, doing HananyWitten transitions
when necessary, leads to the web on the upper right. Further pulling the remaining (1; 0) 7brane
to the right, doing all the HananyWitten transitions, leads to the web on the lower right. Finally,
exchanging the upper (0; 1) 7brane with N
1 NS5branes ending on it with the lower one with 2N + k 3 NS5branes ending on it, and also moving the left (1; 0) 7brane to the right leads us to the web in the lower left of the gure.
From the 6d theory we see that:
nv =
nh =
nt = l
8k(k
8k(k
1)(k
1)(k
3
3
2)
2)
for the case of k = 2l, and:
k
2
2
+ (N
k)(2N k + 2k2
2N
6k + 1);
+ 2k(N 2
k2 + 4k
8);
nv =
8(k
1)(k
2)(k
3)
3
k
1
2
+ 2(k
1)(N
k + 2)(N + k
4);
them to 4d on a torus leads to the isolated SCFT represented in the lower part.
nh =
nt = l
8(k
1)(k
2)(k
3)
3
+ 2kN 2 + 3N
2k3 + 16k2
43k + 30;
(3.3)
for the case of k = 2l + 1. Using these in (2.5) and (2.6) we indeed recover (3.1).
An interesting thing happens for k = 2. In that case the 6d theory becomes USp(2N
4) + (2N + 4)F , which is also known as (DN+2; DN+2) conforml matter [27]. The reduction
of this theory to 4d on a torus was recently studied in [32]. They found that it leads to
an isolated SCFT corresponding to compactifying the 6d (2; 0) theory of type DN+2 on
a Riemann sphere with three punctures shown in
gure 12 (b). If we are correct in our
description then these two SCFT's must be identical. Indeed, using the results of [39] we
can calculate the dimension of Coulomb branch operators and compare between the two
theories. We nd a perfect match.
on a torus should give the isolated 4d SCFT that is described by compactifying the 6d (2; 0) theory
of type A2N+1 on this punctured Riemann sphere. (b) A di erent analysis, done in [32], suggests
that the same theory compacti ed on a torus should give the isolated 4d SCFT that is described by
compactifying the 6d (2; 0) theory of type DN+2 on this punctured Riemann sphere. Our analysis
does imply that these two theories are in fact identical.
Before moving on to discuss other 5d theories, there is one more 6d SCFT, closely
related to the ones considered, that we would like to discuss. The quiver theory description
is given in
gure 13. We can repeat the previous analysis, now the di erence manifesting
in the 6d brane construction by adding a stuck 6brane. Upon performing Tduality this
becomes a stuck D5brane on one of the O7
planes. We can decompose the O7
planes
as done in [14], to get the nal web picture. The entire process is shown in gure 14. This
describes the 5d gauge theory of gure 15.
One can see that the Coulomb branch dimensions agree, and using the results of [29],
also the global symmetries agree, in particularly, we get an a ne A(21n)+8l. As a further test
we consider the compacti cation to 4d, where we expect to get the theory of gure 16.
Using class S technology we can indeed show that the 4d isolated theory in
gure 16 (b)
has the same global symmetry as the 6d quiver of gure 13. We can also calculate the
central charges nding:
dH = 2(n + 4l)2 + n + 6l;
Using (2.5), (2.6), this indeed matches what we expect from the theory of gure 13.
3.1
Generalizations
The next step is to consider generalizations to other 5d gauge theories with an expected
6d lift. Consider the 5d gauge theory given by a linear SU0(Ni) quiver with fundamental
Tdualize to the web system in the bottom of the gure.
matter, where each non edge group sees an e ective number of 2Ni avors. If in addition
the two edge groups see an e ective number of 2Ni + 2 avors, then it was argued in [29]
that this 5d theory should have an enhanced a ne A(1) symmetry. This strongly suggests
that these also lift to a 6d SCFT. Note that the previously considered theories are also of
this form.
Naturally, we would like to know to which 6d SCFT these theories lift. As there is an
in nite number of possibilities, a case by case study seems ine ective. Thus, we wish to
determine a procedure by which, given such a 5d quiver, the 6d SCFT can be determined.
To do this we can utilize the fact that any such quiver can be reached starting with the linear
the brane web, and doing several manipulations, we arrive at the web of gure 14.
SU(N ) quiver considered before, for some N and k, and going on the Higgs branch. Also,
for theories with 8 supercharges, the Higgs branch does not receive quantum corrections,
and so the 5d and 6d Higgs branches must agree. Therefore, one possible strategy is to
start from one of the previous cases, where we know the 6d SCFT, and determine the Higgs
branch limit needed to get the required 5d quiver. Then, by mapping this to the 6d SCFT,
we can determine the 6d lift of the 5d quiver.
To understand the mapping, we can again rely on the brane description. Starting with
the 5d case, the Higgs branch limits we are interested in are represented, in the brane web,
by forcing a group of 5branes to end on the same 7brane. For example consider a group
of N parallel 5branes, crossing some NS5branes, each ending on a di erent 7brane, see
gure 17 (a). This describes a quiver tail of the form N F + SU0(N )
SU0(N ) : : :. If we
force two 5brane to end on the same 7brane then, because of the Srule, one Coulomb
modulus of the edge SU(N ) group is lost. Thus, this describes the Higgs branch breaking
N F + SU0(N )
SU0(N ) to (N
2)F + SU0(N
1)
SU0(N ) + 1F (see gure 17 (b)).
is integrated out. Opening out the web we get to the presentation of (b). We could have also
integrated out any other avor, and obtained the same theory.
We can of course repeat this and force two other 5branes to end on the same 7brane.
This leads to a similar breaking on the new quiver (see
gure 17 (c)). However, we can
also consider forcing an additional 5brane to end on the same 7brane, so as to have three
5branes ending on it (see
gure 17 (d)). Now the Srule not only eliminates a Coulomb
moduli of the edge SU(N ) group, but also one from the adjacent group. This describes the
Higgs branch breaking associated with giving a vev to the gauge invariant made from a
avor of the edge group, the bifundamental, and the avor from the adjacent group. The
quiver left after this breaking is shown in (see gure 17 (d)).
It is now straightforward to generalize to an arbitrary con guration. Before moving
to the corresponding limits in the 6d theory, we note that this correspondence may not
hold when completely breaking a gauge group. In general, the topological symmetry of
the broken group survives the breaking and remains in the resulting theory, sometimes
manifesting as extra avors. In these cases, perturbative reasoning alone may be inadequate
to determine the answer. For our purposes, this can always be avoided. Also note, that
this can be related to the classi cation of 4d quiver tails of [40] by using the results of [35].
This is an alternative way to argue this mapping.
Next, we consider the implications of this on the 6d theory. Under Tduality, the
D5branes are mapped to D6branes and the D7branes to D8branes, so the analogous
breaking on the 6d side is represented in the brane con guration by forcing a group of
D6branes to end on the same D8brane. If the breaking is not too extreme, this translates
to a limit on the perturbative Higgs branch of the 6d SCFT. In fact, as the Srule is the
same as in the 5d case, we nd that this induces exactly the same e ect on the quiver tail.
The only di erence is that now there is only one quiver tail. Each action performed on any
of the two tails of the 5d quiver is mapped to the corresponding action done on the single
6d tail.
Nevertheless, complications can arise in some instances, for example, when the 6d
SCFT has a tensor multiplet without an associated gauge theory. For example, consider
the 6d quiver of gure 3, for N = 2l. In that case the 6d SCFT has a nonLagrangian
part, the rank 1 Estring theory, possessing a 29 dimensional Higgs branch. Some of the
breaking we consider may be mapped to the Higgs branch of the Estring theory, where we
have no perturbative description. This can happen even in cases where the initial theory
has a complete Lagrangian description, but on the Higgs branch limit the gauge group
is completely broken leaving its associated tensor multiplet.5 Note that this method can
still be used to determine the 6d SCFT, but is somewhat complicated as the Higgs branch
limits may not be perturbatively realized. Thus, determining the resulting 6d SCFT will
probably require string theoretic methods like the ones in [46].
3.1.1
A simple example
We next wish to illustrate this with a simple example. First, consider the 5d theories shown
in gure 18. We can get these theories from the one in
gure 6 (a) by going on the Higgs
branch. On the web system this is manifested by breaking two pairs of 5branes so that
each of them end on the same 7brane, the di erence between them being whether the pair
are on the same side or opposite sides. In the 6d theory these are mapped to the same
breaking, indicating that these two quivers are dual, in the sense of both lifting to the same
6d SCFT.
Taking the corresponding limit in 6d, we get to the quiver of gure 19, which is the
desired 6d SCFT. By construction, we are now assured that doing the Tduality on the
brane system of this 6d quiver leads to the webs in
gure 18. We can also consider
compactifcation of the 6d theory to 4d. As the Higgs branch limit and dimensional reduction
should commute, we again expect the resulting 4d theory to be given by the class S theory
whose 5d analogue is given by integrating out a avor from the theories of 18. Naively, we
have several di erent choices of which avor to integrate out, but we nd these all lead to
the same class S theory, shown in gure 20 (c).
5This is manifested in the web when one is forced to coalesce 2 NS5branes or an NS5brane and the O8
plane due to the constraints of the Srule.
some of the 7 branes, we arrive at the spiraling con guration shown on the bottom left. (b) Starting
with the same con guration, doing some 7brane gymnastics, we get to the web on the right. One
can note that this is a Higgs branch limit of a theory of the form of gure 6 (a).
presented in section 3. Besides supporting the claim that this theory lifts to 6d, we can,
by taking the required Higgs branch limit on the 6d lifts given in section 3, also argue that
the quivers given in gure 33 are indeed the required 6d lifts. This is shown for the k > N
case in gure 34, and for the k < N case in gures 35.
We can again consider the reduction to 4d on a torus. We expect the 4d theory to be
described by the case with one less avor shown in
gure 36. The punctures suggests a
global symmetry of SU(2N + 2k + 1)
U(1)3 except in some special cases, for
example, when k = N or k = N
1 where the symmetry enhances to SU(2N + 2k + 1)
SU(3)
SU(2)
enhancement of U(1)
when k = N or k = N
U(1)2. From the 4d superconformal index we see that there is a further
SU(2)2 ! SU(4), which becomes U(1)
SU(2)
SU(3) ! SU(5)
1. This enhancements, including the special cases with enhanced
symmetry, exactly matches the ones expected from the 6d SCFT of gure 33. We can also
this brane web. One can see that it is in the form of [35] so compacti cation to 4d will yield the
isolated SCFT of (b).
theory of gure 32.
Higgs branch limit of the theories in gure 6 (a) given by the Sdual of the web on the right.
;
;
(4.3)
(4.4)
Higgs branch limit of the theories in gure 8.
calculate the central charges of this theory nding:
dH = 2N 2 + 2k2 + 4kN + 17N + 9k;
(2N
1)(6k2 + 4N
4N 2 + 9k + 18kN )
kSU(2N+2k+1) = 2(2N + 2k + 3);
kSU(4) = 4k + 8N
for k
N where for k = N SU(4) ! SU(5), and
dH = 2N 2 + 2k2 + 4kN + 17k + 9N + 4;
k(12N 2 + 24N
13
18k + 36kN )
3
8k2
3
kSU(2N+2k+1) = 2(2N + 2k + 3);
kSU(4) = 4N + 8k + 2
for N > k. This indeed matches the results we get from (2.5) and (2.6).
4.2.2
SU quivers with an antisymmetric hyper at both ends
We can next consider the case where both ends are SU groups with an antisymmetric so
we have the quiver theory of gure 37. We conjecture the 6d lift to be the one shown
in
gure 38.
We can repeat the same steps as before, rst deform the web to give a
Higgs branch limit of a theory of gure 6 (a). This gives the web shown in gure 39. By
series of HW transitions, we get to the brane web in the bottom left. One can see that it is in the
form of [35] so compacti cation to 4d will yield the appropriate isolated SCFT. This is the form
most suited to the k
N case. For the N > k case, the one in (b), gotten from (a) by shu ing
some of the 7branes, is more adequate.
implementing the required breaking on the 6d SCFT of gure 3 we indeed get the quiver
of gure 38.
As an additional test, we can again consider the reduction to 4d on a torus. We
expect the 4d theory to be described by the case with one less avor shown in gure 40.
Higgs branch limit of the web in gure 6 (a).
The global symmetry visible from the punctures is SU(2k + 2)
SU(4)
which is further enhanced when k = 0 or N = 2. When k 6= 0, we can show from the
superconformal index that there is an enhancement of SU(4)
U(1) ! SO(12).
This, including the enhancement when N = 2, exactly matches what is expected from the
6d global symmetry. However, the k = 0 case, the 5d SCFT of which corresponds to the 5d gauge theory
2
SU 1 (2N )+2AS+7F , has some puzzling features. First, let's start with the global symmetry
for the SCFT of gure 40. As argued in the appendix, instanton counting methods suggests
this theory has an E7
SU(2)3 global symmetry which is further enhanced to E7
SO(7)
for N = 2. This is further supported by the 4d superconformal index. Note that the N = 2
case discussed here is identical to the N = 2 case for the theory in gure 27 (b), which
provides a dual gauge theory description for the same xed point.
Comparing with the 6d side, we naively encounter a contradiction. When k = 0 we have
a long quiver of SU(2) groups leading to an enhancement of the U(1) bifundamental global
symmetries to SU(2)'s. More importantly the mixed anomalies leading to the breaking of
most of these U(1)'s now vanish so we naively expect to have an SU(2)N+1 global symmetry
contradicting the global symmetry suggested by the 5d description. The issue appears to
be the discrepancies between the global symmetry suggested from the gauge theory and
the one that actually exists in the SCFT mentioned in section 2. To truly understand the
6d SCFT we should consider a string theory realization of it.
Fortunately, the 6d SCFT at hand was considered in [21]. They considered a class of
theories engineered in string theory by a group of M5branes probing a C2=Z2k+2 orbifold
and an M9plane. One of the theories in this class is the theory with gauge theory
description given in
gure 38. This is no coincidence as the original 5d gauge theory, shown in
gure 37, can be engineered by a group of D4branes probing a C2=Z2k+2 orbifold and an
O8 plane [47] so it is natural to expect the 6d lift to be of this form.
According to the analysis of [21], the nonabelian global symmetry of this 6d SCFT
is indeed SO(12)
SU(2k + 2)
SU(2). The case k = 0 is special: the nonabelian
global symmetry is actually E7
SU(2)3. The extra SU(2) is there since the orbifold
C2=Z2 preserves the full SO(4) symmetry, while C2=Z2k+2 breaks one of the SU(2)'s.7
So this appears to agree with what we see from the instanton counting analysis done in
the appendix.
The case k = 0; N = 2 is more special. Then the 6d theory is known as the (E7; SO(7))
conformal matter [27]. Again the gauge theory shows an SO(
8
) global symmetry, while it
is known the SCFT only has SO(7). This indeed agrees with the results from instanton
counting done in the appendix.
We can also to calculate the central charges of this theory nding:
dH = 2k2 + 30N + 19k + 3;
This indeed matches the results we get from (2.5) and (2.6), supporting the claim
that compactifying the 6d SCFT of gure 38 on a torus leads to the isolated 4d SCFT of
gure 40. Incidentally, the compacti cation of the (E7; SO(7)) conformal matter on a torus
was already considered in [32]. They conjectured that the resulting theory is given in terms
of a compacti cation of the E6 (2; 0) theory on a Riemann sphere with three punctures
labeled: 0; 2A1; E6(a1) (see [44], for a discussion on the meaning of the notation and for
properties of this SCFT). They further compared the central charges of this theory to the
ones expected from the compacti cation, nding an exact match.
Consistency of these two approaches then suggests that these two theories are in fact
the same theory. Indeed, we calculated the central charges and spectrum of Coulomb
branch operators of the theory in gure 41, nding exact matching to the previously
mentioned SCFT from compactifying the E6 (2; 0) theory.
We can also consider the even rank case shown in gure 42. While this can be gured
out from the previous case by going on the Higgs branch, we will mention this case. We
expect the 6d theory to be the one shown in gure 43. This can be argued by manipulating
the brane web into a form, shown in gure 44, as a Higgs branch limit of the theory of
gure 8.
We can again consider the reduction to 4d on a torus. We expect the 4d theory
to be described by the case with one less
avor shown in
gure 45. The discussion is
quite similar to the odd rank case. The global symmetry visible from the punctures is
SU(2)
SU(
6
)
SU(2k + 2)
U(1)2 which gets further enhanced when N = 1 or k = 1; 0.
7I am grateful for J. J. Heckman for making his work known to me and for discussing this point.
this brane web. One can see that it is in the form of [35] so compacti cation to 4d will yield the
appropriate isolated SCFT.
2
sentation of its associated 4d SCFT as a compacti cation of an A type (2; 0) theory on a three
punctured sphere.
theory of gure 42.
limit of the web in gure 8. This is given by the Sdual of the rightmost web.
From the 4d superconformal index we nd an enhancement of SU(2) SU(
6
) U(1) ! SU(
8
)
which is further enhanced to SU(2k + 10) for N = 1. This agrees with what is seen from
the gauge theory description of gure 43 except for the case of k = 0. In this case the
2
gauge theory is SU 1 (5) + 2AS + 7F and as discussed in the appendix, we expect to have an
SO(16) SU(2)2 global symmetry. This is also con rmed from the 4d superconformal index.
In the 6d theory we again encounter a series of SU(2) groups and we naively have a
problem with matching the global symmetry. However, this theory was also considered
in [21], as expected since the 5d theory is related to the previous one by adding D4branes
stuck on the orbifold and so should lift to a 6d SCFT of this type. The analysis of [21]
suggests the nonabelian global symmetry of this theory is indeed SU(
8
)
SU(2k + 2). The
k = 0 case is again special, and then the nonabelian global symmetry should indeed be
SO(16)
SU(2)2.
We can also calculate the central charges of this theory nding:
dH = 2k2 + 30N + 19k + 3;
kSU(2k+2) = 4k + 16;
kSU(
8
) = 12N + 4k + 4
This indeed matches the results we get from (2.5) and (2.6).
Cases with completely broken groups
Finally, we wish to consider several additional cases. The common thread in all of them
is that they involve completely breaking a 6d gauge group leaving a tensor multiplet. As
our
rst example we consider the case of USp(2N ) + AS + 8F . As mentioned in the
introduction, this theory is known to lift to the rank N Estring theory. It also has a brane
web description given in
gure 46 (a) [14]. We can now recast this web as a Higgs branch
limit of the theory in
gure 6 (a). Carrying out this breaking on the 6d SCFT, one nds
that this completely breaks the gauge symmetry leaving only the tensor multiplets. Indeed,
as mentioned in section 2, the theory described by such a structure of tensor multiplets is
the rank N Estring theory.
Next we consider a case in which only part of the gauge theory is broken. Take the 5d
gauge theory Nf F + USp(2N + 4)
USp0(2N ) whose web is shown in gure 47 (a). First,
this brane web. One can see that it is in the form of [35] so compacti cation to 4d will yield the
appropriate isolated SCFT.
limit of the theory in gure 6 (a).
let us analyze the global symmetry of this theory. Instanton counting methods suggest that
the (0; 1) instantons should lead to an enhancement of the USp0(2N ) topological U(1) to
SU(2) [14]. In addition we expect an enhancement of U(1)
SO(2Nf ) to ENf +1. This is
most notable from the gauge symmetry on the 7branes using the results of [48, 49]. Thus,
we conclude that this theory has an ENf +1
SU(2)2 global symmetry. The case of N = 1
is exceptional as then there is an additional enhancement of SU(2)2 ! G2 [12] so in that
case the global symmetry is ENf +1
G2.
USp0(2). The generalization to USp(2N + 4)
USp0(2N ) is apparent and we only show the shape of the external legs, shown on the right. The
generalization to Nf F + USp(2N + 4)
USp0(2N ) is also straightforward and is done by adding
7branes. For example consider the web of (b) describing 8F + USp(2N + 4)
USp0(2N ). By
manipulating the 7branes we can get to the web on the right which is in the form as a Higgs
branch limit of the web in gure 6 (a).
E8
gauge theory.
In the case of Nf = 8 we get an E(1) global symmetry and the theory is expected to
8
lift to 6d. Indeed, as shown in gure 47 (b), the web for this theory can be cast into a form
as a Higgs branch limit of the web in gure 6 (a). We can now implement this breaking on
the 6d theory. Doing this one can see that we are left with the two free tensor multiplets
of type
1
2. This gives the rank 2 Estring theory. The remaining quiver connects to
this theory by gauging the SU(2) subgroup of the SU(2)
E8 global symmetry of this 6d
SCFT. This leaves an E8 global symmetry, as expected from the 5d theory.
The explicit 6d theory we get is shown in
gure 48. Like in previous cases, we expect
most of the SU(2) global symmetries to be anomalous even though this is not visible in
the gauge theory. The case of N = 1 is known as the (E8; G2) conformal matter [27] and
there it is known that the global symmetry of the SCFT is actually E8
G2 and not the
SO(7) visible from the gauge theory. This indeed matches what is expected from the
number stands for an odd number of halfhypers, possible since the group is SU(2).
We can also consider compacti cation to 4d on a torus. For simplicity, we only consider
the N = 1 case. We expect the resulting 4d theory to be the one described by reducing the
SU0(2), shown in gure 49, on a circle. This indeed preserves
the 6d global symmetry. We can further test this by matching the central charges of the
4d SCFT with the one expected from the 6d theory. Using class S technology, we nd that
this theory has Coulomb branch operators of dimensions: 6; 8; 12; 18. We further nd:
dH = 92; nv4d = 84; kE8 = 36; kG2 = 16
(4.7)
Using the methods of [37], we nd that this indeed matches the result we expect from
(E8; G2) conformal matter.
Like the previous case, the compacti cation of the (E8; G2) conformal matter on a
torus was already considered in [33]. They conjectured that the resulting theory is given in
terms of a compacti cation of a speci c E8 (2; 0) theory on a Riemann sphere with three
punctures. Consistency of these two approaches then suggests that these two theories are
in fact the same theory. Since the class S analysis for compacti cation of E8 (2; 0) theory
is not yet available we cannot compare the two theories. It will be interesting to check this
if the classi cation becomes available.
The last case we wish to consider involves a 2 type tensor multiplet. Consider the
2
2
5d theories SU 3 (2N ) + 2AS + 7F and SU 3 (2N + 1) + 2AS + 7F . The instanton analysis
calculation, done in the appendix, suggests these have an enhanced a ne global symmetry
and so may lift to 6d. For simplicity, we concentrate on the N = 2 case, the generalization
to other N being straightforward.
Figure 50 shows the brane webs for these theories, and how they can be cast as a Higgs
branch limit of the theories of gure 14. Implementing this breaking on the appropriate
6d SCFT yields the theories described in
gure 51 which are the appropriate 6d lifts. One
can see that indeed the theory of gure 51 (a) has the SO(19) symmetry expected from the
5d description. However, the one of gure 51 (b) shows an E7
SO(7), the E7 agreeing
with the gauge theory expectations. We expect the SCFT to not posses the SO(7) global
symmetry, but only have the G2 subgroup, like the (E8; G2) conformal matter case. It
would be interesting to test this using the Ftheory description.
Figure 49. The brane web for 7F + USp(
6
)
SU0(2). From this one can arrive at the repre
sentation of its associated 4d SCFT as a compacti cation of an A type (2; 0) theory on a three
punctured sphere.
5
Conclusions
In this article we studied 5d gauge theories that are expected to lift to 6d SCFT's. Given
such a 5d gauge theory, we are interested in determining its 6d lift. We have proposed a
method to do this for 5d gauge theories with an ordinary brane web description. We have
provided several examples of these, showcasing its usefulness as well as its limitations.
One such limitation is that to properly utilize it, one must be able to cast the web as a
Higgs branch limit of a known theory. It is not immediately clear if this can be done for an
arbitrary theory. However, we have checked a number of examples in which this appears
to be true. This leads us to conjecture that all 5d gauge theories with an ordinary brane
web description that lift to 6d, lift to the family of theories discussed in section 2. It will
be interesting to further explore this.
Another direction is to nd further evidence for the relations proposed in this article.
One possible direction is to compute a quantity in the 5d theory and compare it against the
expected result from the 6d SCFT. Such a thing was done, for example, in the case of the
rank 1 Estring case in [28, 50], the quantity in question being the 5d superconformal index.
It is interesting if this can also be carried out for some of the examples presented here.
limit of the web in 14.
2
2
of the web in 14. (b) The brane web for SU 3 (5) + 2AS + 7F converted to a form as a Higgs branch
It is also interesting to consider other 5d gauge theories. While it is not yet completely
clear what gauge and matter content are allowed for the theory to posses 5d or 6d
xed
points, there are several cases that can be engineered in string theory and thus are known
to exist. In particular one can generalize brane webs by adding O7 planes [14] or O5
planes [51] leading to additional possibilities. Some theories in these classes are known to
have an enhancement to an a ne symmetry and so are expected to lift to 6d [29, 52]. It
will be interesting to also determine the 6d SCFT's in these cases.
Acknowledgments
I would like to thank Oren Bergman, Soek Kim, Kimyeong Lee, HeeChoel Kim, Kazuya
Yonekura, Shlomo S. Razamat and Jonathan J. Heckman 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
{ 41 {
lift of the 5d gauge theory SU 3 (5) + 2AS + 7F . The rightmost circle in both quivers, corresponds
to a single N = (2; 0) tensor multiplet where an SU(2) subgroup of the USp(4) (2; 0) Rsymmetry
2
is gauged.
A
Instanton counting for SU(N ) + 2AS + Nf F
In this appendix we consider symmetry enhancement in theories of the form SU(N ) +
2AS + Nf F . The method we employ borrows signi cantly from [53]. The essential idea is
to identify the states, coming from 1 instanton con gurations, that are conserved currents.
This sometimes allows one to determine what the enhanced symmetry is. The methods
relies on the following observations of [53]:
1. The 1 instanton of SU(2), when properly quantizing the zero modes coming from the
gaugino, forms a multiplet which is exactly the one associated to a broken current
supermultiplet.
2. Any 1 instanton of some Lie group G can be embedded in an SU(2) subgroup of G.
Therefore, to determine the spectrum of 1 instanton con gurations of arbitrary G it
is su cient to decompose it to SU(2) representations.
Particularly, for our case we consider gauge group SU(N ) with matter in the
fundamental or antisymmetric. The case of SU(N ) with matter in the fundamental was studied
already in [53] and later in [29], which also discussed antisymmetric matter. Yet, to our
knowledge, a complete analysis of the case of SU(N ) + 2AS + Nf F was not done, even
though the building blocks are in essence already known.
Consider a 1 instanton of SU(N ) + 2AS + Nf F . It breaks the SU(N ) gauge symmetry
to U(1)
SU(N
2). We can decompose all fermionic matter under the reduced gauge
symmetry and determine the zero modes provided by them. Particularly, there is only one
state in the adjoint of SU(2) whose quantization provides the broken current supermultiplet.
The remaining elds are all in the fundamental of SU(2) and so provide one raising operator
B
C
A
N
2
4)
N
N
2
2
2
2
1
1
Nf
SU(N ) + 2AS + Nf F . The B operators come from the gaugino, the C from the fundamentals and
A from the antisymmetrics.
per fermion. By either doing the decomposition, or simply burrowing the results of [53],
we nd the zero modes spectrum given in table 1.
The full spectrum is now given by acting with these operators on the ground state, j0i,
whose charges are: QUG(1) = (N
2)(
Nf
2
4), QUB(1) =
N2f and QUAS(1) = N
2,
where
is the CS level. Furthermore, recall that the ground state is a broken current
supermultiplet. Thus, to get a conserved current we need to enforce two conditions:
1. The state must be gauge invariant under the unbroken UG(1)
SU(N
2) gauge
symmetry.
2. The state must remain a broken current supermultiplet, particularly, it must have as
the lowest component, a triplet of scalar operators under SUR(2).
The implications of these two conditions is that we must look at all operators made
from the
elds in table 1 that are SU(N
2) and SUR(2) singlets. The application of
any combination of these on the ground state gives an SU(N
2) invariant broken current
supermultiplet. Next, one must enforce UG(1) invariance.
Going over table 1 we see that the only SU(N
2) and SUR(2) singlets are:
B2(N 2),
AN 2
, C and ( AlBN 2 l)2 for l = 1; 2 : : : ; N
1, where the SU(N
2) indices are
contracted with the epsilon symbol.
Before looking at all these operators, we should discuss under what conditions we
expect a
xed point.
We answer this question by analyzing brane webs.
We
nd two
cases with a spiral tau type diagram, or alternatively, a web description as a Higgs branch
limit of a 6d lifting theory. These suggest that these theories lift to 6d. The cases are
2
SU0(N ) + 2AS + 8F (see gure 39 for the web in the N even case and
gure 44 in the
N odd case) and SU 3 (N ) + 2AS + 7F (see gure 50 for the web in the N = 4; 5 cases).
Integrating out avors from these theories gives well de ned webs leading us to believe that
this class of theories indeed go to a 5d
xed point.
Next, we want to determine what conserved currents are provided by the 1 instanton
con guration in these cases. First, let's look at all gauge invariant states made by applying
A and B on the ground state. These are:
j0i ; B2(N 2) j0i ; AN 2 j0i ; ( AN 2)2 j0i ; AN 2
B2(N 2) j0i ; ( AN 2)2
B2(N 2) j0i ;
( AlBN 2 l)2 j0i
(A.1)
where in the last term l = 1; 2 : : : ; N
1. We can also act on each of these states with
k C operators for k = 0; 1 : : : ; Nf . Next, we need to determine when each of these states
j0i ; ( AN 2)2 B2(N 2) j0i can contribute if 2j j + Nf
the only contribution can come from ( AlBN 2 l)2 j0i.
is UG(1) invariant and thus give a conserved current. We only consider theories in the
previously discussed class. We also assume N > 3 as the other choices reduce to known
N = 4, AN 2 j0i and AN 2 B2(N 2) j0i can contribute if 2j j + Nf
cases.8 We nd that B2(N 2) j0i and ( AN 2)2 j0i can only contribute if 2j j + Nf
8 and N = 4; 5 and
8 and
8. Thus, as long as 2j j + Nf < 8
The behavior of these changes depending on whether N is even or odd. If N is even
then we can
nd a conserved current from the l = N2 2 case, ( A N2 2 B N2 2 )2 j0i. This
contribute conserved currents when
= 0; Nf = 0. When
avors are added then we can
also conserved currents from the l = N 2
1 case.
still get conserved currents by acting with C operators. If 2j j + Nf
2
If N is odd then we can nd a conserved current from the l = N 1 and l = N 3 cases.
The rst contribute when
= 2; Nf = 0 while the second when
when avors are added then we can still get conserved currents by acting with C operators.
We next need to go over all cases, and see what conserved currents we get. This tells
us whether symmetry enhancement occurs in the theory, and if so, helps us determine
the enhanced symmetry. Since we only see contributions from the 1 instanton, there can
sometimes be further enhancements coming from higher instantons. In fact, the need to
complete a Lie group sometimes necessitates the existence of conserved currents from higher
order instantons. In the following, when writing the global symmetry of a theory, we write
the minimal one consistent with the conserved currents we observe.
We write our results for N > 5 odd in table 2, and for N > 4 even in table 3. As is clear
already from the analysis of the currents the N = 4; 5 cases are special. In the N = 4 case
this is manifested already at the perturbative level as the antisymmetric representation
is real and the SUAS(2)
UAS(1) symmetry is enhanced to USp(4). Then there are also
further conserved currents completing the SUAS(2)
UAS(1) representations to USp(4)
2
2
2; Nf = 0. Again,
ones. We write our ndings for this case in table 4.
In the N = 5 case, the di erence only arises when Nf + 2j j = 10. In this case
we nd that there is a further enhancement of SU(2)
SU(2) ! G2. This is related to
the enhancement to G2 in the USp(
6
)
SU(2) theories mentioned in section 4.3 as, by
manipulating brane webs, we nd that the theories SU 9 Nf (2n + 1) + 2AS + (Nf + 1)F and
Nf F + USp (2n + 2)
USp(2n
2) + 1F are dual (the
angle for USp(2n + 2) is relevant
only in the Nf = 0 case). One implication of this is that, besides the enhancement revealed
from the 1 instanton analysis, there should be an additional enhancement of U(1) ! SU(2)
coming from higher instantons. This is also apparent in the N = 2 case as this is necessary
2
8 then there can
to complete the Lie group G2.
For general N , this can be argued from the Nf F + USp (2n + 2)
USp(2n
2) + 1F
description. According to the results of [14], as the USp(2n 2) group e ectively sees 2n+3
8For N = 2 the antisymmetric completely decouples and we just get the rank 1 ENf +1 theories. For
N = 3 the antisymmetric is just the antifundamental so the problem reduces to analyzing SU(3) with
fundamentals where this analysis was done in [15, 26, 29] expect the case of Nf + 2j j = 12. However,
the brane webs describing these theories are identical to the rank 2 E~Nf theory so these are just dual
descriptions of known xed points.
= 0 U(1)2
U(1)3
U(1)2
U(1)2
U(1) SU(2)4
U(1) SU(2)2
SU(3)
U(1)2
SU(2) SU(5)
U(1) SU(2)2
U(1) SU(2)3
SU(4)
U(1) SU(2) SU(
8
)a
SO(16)b
SU(2)
SU(3) U(1) SU(2)2
SU(2) SU(4)
U(1)
SU(
6
)
SO(10)
SU(2)2 U(1) SU(2)2
SU(4) SU(2)2
SO(12)d
SU(2)2 SU(2)3
SU(4)
of SU(2)2 ! G2. Also note that for Nf + 2j j = 10 one of the SU(2) results from contributions of
higher instantons and is inferred from a dual description of the
xed point. (a) To get this global
symmetry requires also two conserved currents that are
avor singlet with instanton number
(b) To get this global symmetry requires also two conserved currents with instanton number
2
that are in the 7 of SU(7). (c) To get this global symmetry requires also two conserved currents
with instanton number
2 that are SU(4) singlets. (d) To get this global symmetry requires also
two conserved currents with instanton number
2 that are in the 5 of SU(5). (e) To get this global
symmetry requires also several conserved currents with instanton number
2 that are in the 1 and
15 of SU(
6
), and another two with instanton number
3 that are in the 6 of SU(6).
avors, the (0; 2) instanton should provide two conserved currents with charges
1 under
SOF (2). These lead to an enhancement of at least U(1)2
! SU(2)2. Furthermore, as argued
in section 4, when Nf > 0 we expect a further enhancement of at least SO(2Nf )
ENf +1, where the U(1) containing the USp(2n + 2) topological symmetry. The minimal
implication of these on the SU description is that a further enhancement of U(1) ! SU(2)
should occur in this theory. Note that this argument does not hold for the pure case,
SU5(2n + 1) + 2AS. Nevertheless, since this enhancement appears to be una ected by
integrating out avors, as long as Nf + 2j j = 10, we conjecture that it should occur also
U(1) !
for this case, and have included it in table 2.
Finally, we want to discuss the cases where we expect a 6d
xed point.
First
we have SU0(2n + 1) + 2AS + 8F , where we
nd several conserved currents with
the charges: (1; 28; 1; 2); (1; 28; 1; 2); (1; 1; N
2; 4) and (1; 1; (N
2); 4), under
SU(
8
)
UAS (1)
UB(1). All these currents cannot form a nite Lie group. The
rst two seem to suggest that U(1)2
SU(
8
) is enhanced to the a ne D8(1). The last two
then imply that the remaining U(1) should also form an a ne group. SUAS (2) does not
appear to be a nized at least at this level.
For SU0(2n) + 2AS + 8F , the conserved currents are a bit di erent. First there is one
current in the 70 of SU(
8
). This cannot lead to any nite Lie group, but can form an a ne
= 0 U(1)
U(1)2
U(1)2
U(1)3
U(1)
U(1)
U(1)2
U(1)2
SU(2)4
U(1)
SU(2)
SU(2)
SU(4)
SO(
8
)
SU(5)
SU(2)
SO(10)e
U(1)2
SU(2)
Ea
6
SU(2)
SO(10) SU(2)3
Eb
7
SU(
6
)
SO(12)f
(a) To get this global symmetry requires also two conserved currents that are
avor singlets with
instanton number
instanton number
2. (b) To get this global symmetry requires also two conserved currents with
2 that are in the 7 of SU(7). (c) To get this global symmetry requires also
two conserved currents with instanton number
2 that are SUF (2) singlets. (d) To get this global
symmetry requires also two conserved currents with instanton number
2 that are in the 3 of SU(3).
(e) To get this global symmetry requires also two conserved currents with instanton number
2
that are in the 6 of SU(4). (f ) To get this global symmetry requires also two conserved currents
with instanton number
2 that are in the 10 of SU(5). (g) To get this global symmetry requires
also several conserved currents with instanton number
2 that are in the 1; 1 and 15 of SU(6), and
another two with instanton number
3 that are in the 6 of SU(6).
one E7(1). If n 6= 2 then we also have 4 additional currents, which are singlets of SU(2)
SU(8), with charges (4; N
2); ( 4; (N
2)); (4; 2) and ( 4; 2) under UB(1)
UAS (1).
In light of the enhancement of SU(
8
) to an a ne group, we also expect these currents to
enhance U(1)2 to an a ne group. If n = 2 then we get two conserved currents in the
UB(1). These indeed cannot t in a nite Lie group, but
Next, we consider the case of SU 3 (2n) + 2AS + 7F . First, we nd a conserved current
2
The last case we consider is SU 3 (2n+1)+2AS +7F . We nd conserved currents in the
(1; 35; 1; 12 ), (1; 7; (N
2); 52 ), and (1; 1; 1; 72 ) under SUAS (2) SU(7) UAS(1) UB(1).
The rst two cannot t in a nite group, rather forming the a ne E7(1). Like in the other
case, we expect the last current to a nize the remaining U(1). In the N = 5 case, there is
(5; 1; 4) of USpAS(4)
can form an a ne one, B3(1).
SU(
8
)
32 ), under SUAS (2)
2 additional currents in the (1; 7; 2n
(5; 7; 52 ) of USpAS (4)
can form an a ne one, B9(1).
SU(7)
to the a ne group D8(1). SUAS (2) does not appear to be a nized at least at this level. If
n = 2 then these two currents merge with additional currents to form one current in the
UB(1). These indeed cannot t in a nite Lie group, but
2
SU(7)
2; 52 ) and (1; 7; 2; 52 ). These suggest an enhancement
UAS (1)
UB(1). If n 6= 2 then we also have
U(1)
SU(3)
USp(4) U(1)
USp(4) SU(2)
SO(7)
U(1)
USp(4)
USp(4)
SU(4)
SO(13)d
SO(
8
)
SU(5)
SO(7)
U(1)
U(1)
SO(7)
SO(19)f
USp(4)
Ea
6
USp(4)
USp(4)
SU(4)
USp(4)
SO(10) SO(7)
Eb
7
USp(4) SU(2)
USp(4)
U(3) SO(7)
USp(4) U(1)
SO(7)
SU(2)
SU(
6
)
SO(15)e
= 0 SU(2)
USp(4) U(1)
= 5 U(1)
USp(4)
= 1 U(1)
= 2 U(1)
= 3 U(1)
= 4 SO(7)
U(1)
U(1)2
U(1)
U(1)
U(1)
U(1)
USp(4)
global symmetry requires also two conserved currents that are avor singlets with instanton number
2. (b) To get this global symmetry requires also two conserved currents with instanton number
2 that are in the 7 of SU(7). (c) To get this global symmetry requires also two conserved currents
with instanton number
2 that are in the 3 of SU(3). (d) To get this global symmetry requires
also two conserved currents with instanton number
2 that are in the 6 of SU(4). (e) To get this
global symmetry requires also two conserved currents with instanton number
2 that are in the 10
of SU(5). (f ) To get this global symmetry requires also several conserved currents with instanton
number
2 that are in the (1; 5) and (15; 1) of SU(
6
)
USp(4), and another two with instanton
number 3 that are in the 6 of SU(
6
).
an additional current in the (4; 1; 0; 72 ) which lead to the enhancement to G2. In light of
the enhancement to E7(1), we also expect the G2 to be a nized though whether this indeed
happens is not visible from this method.
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[1] N. Seiberg, Fivedimensional SUSY
eld theories, nontrivial xed points and string
dynamics, Phys. Lett. B 388 (1996) 753 [hepth/9608111] [INSPIRE].
[2] D.R. Morrison and N. Seiberg, Extremal transitions and vedimensional supersymmetric
eld theories, Nucl. Phys. B 483 (1997) 229 [hepth/9609070] [INSPIRE].
[3] K.A. Intriligator, D.R. Morrison and N. Seiberg, Fivedimensional supersymmetric gauge
theories and degenerations of CalabiYau spaces, Nucl. Phys. B 497 (1997) 56
[hepth/9702198] [INSPIRE].
[4] M.R. Douglas, S.H. Katz and C. Vafa, Small instantons, Del Pezzo surfaces and typeIprime
theory, Nucl. Phys. B 497 (1997) 155 [hepth/9609071] [INSPIRE].
[5] O. Aharony and A. Hanany, Branes, superpotentials and superconformal xed points, Nucl.
Phys. B 504 (1997) 239 [hepth/9704170] [INSPIRE].
and grid diagrams, JHEP 01 (1998) 002 [hepth/9710116] [INSPIRE].
Symmetry, JHEP 10 (2012) 142 [arXiv:1206.6781] [INSPIRE].
[9] C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5d SCFT, JHEP 07
(2015) 063 [arXiv:1406.6793] [INSPIRE].
[10] O. Bergman, D. Rodr guezGomez and G. Zafrir, 5d superconformal indices at largeN and
holography, JHEP 08 (2013) 081 [arXiv:1305.6870] [INSPIRE].
Duality, JHEP 04 (2012) 105 [arXiv:1112.5228] [INSPIRE].
[11] L. Bao, E. Pomoni, M. Taki and F. Yagi, M5Branes, Toric Diagrams and Gauge Theory
[12] G. Zafrir, Duality and enhancement of symmetry in 5d gauge theories, JHEP 12 (2014) 116
[arXiv:1408.4040] [INSPIRE].
[arXiv:1410.2806] [INSPIRE].
arXiv:1507.03860 [INSPIRE].
arXiv:1506.03871 [INSPIRE].
109 [hepth/9705022] [INSPIRE].
[13] O. Bergman and G. Zafrir, Lifting 4d dualities to 5d, JHEP 04 (2015) 141
[14] O. Bergman and G. Zafrir, 5d xed points from brane webs and O7planes,
[15] D. Gaiotto and H.C. Kim, Duality walls and defects in 5d N = 1 theories,
[16] I. Brunner and A. Karch, Branes and sixdimensional xed points, Phys. Lett. B 409 (1997)
JHEP 03 (1998) 003 [hepth/9712143] [INSPIRE].
Phys. B 529 (1998) 180 [hepth/9712145] [INSPIRE].
B 390 (1997) 169 [hepth/9609161] [INSPIRE].
[17] I. Brunner and A. Karch, Branes at orbifolds versus Hanany Witten in sixdimensions,
[18] A. Hanany and A. Za aroni, Branes and sixdimensional supersymmetric theories, Nucl.
[19] N. Seiberg, Nontrivial xed points of the renormalization group in sixdimensions, Phys. Lett.
[20] J.J. Heckman, D.R. Morrison and C. Vafa, On the Classi cation of 6D SCFTs and
Generalized ADE Orbifolds, JHEP 05 (2014) 028 [Erratum ibid. 1506 (2015) 017]
[arXiv:1312.5746] [INSPIRE].
Fortsch. Phys. 63 (2015) 468 [arXiv:1502.05405] [INSPIRE].
[21] J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, Atomic Classi cation of 6D SCFTs,
[22] L. Bhardwaj, Classi cation of 6d N = (1; 0) gauge theories, JHEP 11 (2015) 002
[arXiv:1502.06594] [INSPIRE].
[arXiv:1012.2880] [INSPIRE].
[23] M.R. Douglas, On D = 5 super YangMills theory and (2,0) theory, JHEP 02 (2011) 011
[24] N. Lambert, C. Papageorgakis and M. SchmidtSommerfeld, M5Branes, D4branes and
Quantum 5D superYangMills, JHEP 01 (2011) 083 [arXiv:1012.2882] [INSPIRE].
083B02 [arXiv:1504.03672] [INSPIRE].
[25] O.J. Ganor, D.R. Morrison and N. Seiberg, Branes, CalabiYau spaces and toroidal
[hepth/9610251] [INSPIRE].
minimal conformal matter, JHEP 08 (2015) 097 [arXiv:1505.04439] [INSPIRE].
[29] K. Yonekura, Instanton operators and symmetry enhancement in 5d supersymmetric quiver
gauge theories, JHEP 07 (2015) 167 [arXiv:1505.04743] [INSPIRE].
[30] J.A. Minahan and D. Nemeschansky, An N = 2 superconformal xed point with E6 global
symmetry, Nucl. Phys. B 482 (1996) 142 [hepth/9608047] [INSPIRE].
[31] J.A. Minahan and D. Nemeschansky, Superconformal xed points with En global symmetry,
Nucl. Phys. B 489 (1997) 24 [hepth/9610076] [INSPIRE].
[32] K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6d N = (1; 0) theories on T 2 and
class S theories: Part I, JHEP 07 (2015) 014 [arXiv:1503.06217] [INSPIRE].
[33] M. Del Zotto, C. Vafa and D. Xie, Geometric engineering, mirror symmetry and
6d(1;0) ! 4d(N =2), JHEP 11 (2015) 123 [arXiv:1504.08348] [INSPIRE].
[34] K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6d N =(1; 0) theories on S1=T 2 and
class S theories: part II, arXiv:1508.00915 [INSPIRE].
[35] F. Benini, S. Benvenuti and Y. Tachikawa, Webs of vebranes and N = 2 superconformal
eld theories, JHEP 09 (2009) 052 [arXiv:0906.0359] [INSPIRE].
[36] O. Chacaltana and J. Distler, Tinkertoys for Gaiotto Duality, JHEP 11 (2010) 099
[arXiv:1008.5203] [INSPIRE].
[37] K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, Anomaly polynomial of general 6d
SCFTs, PTEP 2014 (2014) 103B07 [arXiv:1408.5572] [INSPIRE].
[38] J. Erler, Anomaly cancellation in sixdimensions, J. Math. Phys. 35 (1994) 1819
[hepth/9304104] [INSPIRE].
[arXiv:1106.5410] [INSPIRE].
[39] O. Chacaltana and J. Distler, Tinkertoys for the DN series, JHEP 02 (2013) 110
[40] D. Gaiotto, N=2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
[41] A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge Theories and Macdonald
Polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].
[42] D. Gaiotto and S.S. Razamat, Exceptional Indices, JHEP 05 (2012) 145 [arXiv:1203.5517]
[43] D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with
surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].
[44] O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the E6 theory, JHEP 09 (2015) 007
[arXiv:1403.4604] [INSPIRE].
09 (2015) 052 [arXiv:1505.00009] [INSPIRE].
171 [arXiv:1206.3503] [INSPIRE].
7branes: KacMoody algebras and beyond, Adv. Theor. Math. Phys. 3 (1999) 1835
and exceptional gauge theories, JHEP 07 (2015) 087 [arXiv:1503.08136] [INSPIRE].
theories, PTEP 2015 (2015) 043B06 [arXiv:1501.01031] [INSPIRE].
[6] O. Aharony , A. Hanany and B. Kol , Webs of (p,q) vebranes, vedimensional eld theories [7 ] O. DeWolfe , A. Hanany , A. Iqbal and E. Katz , Fivebranes, sevenbranes and vedimensional E(n) eld theories , JHEP 03 ( 1999 ) 006 [ hep th/9902179] [INSPIRE].
[8] H.C. Kim , S.S. Kim and K. Lee , 5 dim Superconformal Index with Enhanced En Global compacti cation of the N = 1 sixdimensional E8 theory, Nucl . Phys. B 487 ( 1997 ) 93 [26] H. Hayashi , S.S. Kim , K. Lee , M. Taki and F. Yagi , A new 5d description of 6d Dtype [27] M. Del Zotto , J.J. Heckman , A. Tomasiello and C. Vafa , 6d Conformal Matter, JHEP 02 [28] S.S. Kim , M. Taki and F. Yagi , Tao Probing the End of the World , PTEP 2015 ( 2015 ) [46] J.J. Heckman , D.R. Morrison , T. Rudelius and C. Vafa , Geometry of 6D RG Flows , JHEP [47] O. Bergman and D. RodriguezGomez , 5d quivers and their AdS6 duals , JHEP 07 ( 2012 ) [48] O. DeWolfe , T. Hauer, A. Iqbal and B. Zwiebach , Uncovering the symmetries on [p; q] sevenbranes: Beyond the Kodaira classi cation , Adv. Theor. Math. Phys. 3 ( 1999 ) 1785 [49] O. DeWolfe , T. Hauer, A. Iqbal and B. Zwiebach , Uncovering in nite symmetries on [p; q]