4d \( \mathcal{N}=1 \) from 6d \( \mathcal{N}=\left(1,0\right) \) on a torus with fluxes
Received: March
Published for SISSA by Springer
Ibrahima Bah 1 2 4 6 8 9 10 11
Amihay Hanany 1 2 4 7 9 10 11
Kazunobu Maruyoshi 1 2 3 4 9 10 11
Shlomo S. Razamat 1 2 4 5 9 10 11
Open Access 1 2 4 9 10 11
c The Authors. 1 2 4 9 10 11
0 IPMU, University of Tokyo
1 Prince Concert Road, South Kensington , London, SW7 2AZ, U.K
2 3400 North Charles Street, Baltimore, MD 21218 , U.S.A
3 Faculty of Science and Technology, Seikei University
4 San Diego , La Jolla, CA 92093 U.S.A
5 Department of Physics , Technion
6 Department of Physics and Astronomy, Johns Hopkins University
7 Imperial College London , Blackett Laboratory
8 Department of Physics, University of California
9 Kashiwa , Chiba 2778583 , Japan
10 Haifa , 32000 , Israel
11 331 KichijojiKitamachi , Musashinoshi, Tokyo, 1808633 , Japan
Compactifying N = (1; 0) theories on a torus, with additional uxes for global symmetries, we obtain N = 1 supersymmetric theories in four dimensions. It is shown that for many choices of ux these models are toric quiver gauge theories with singlet elds. In particular we compare the anomalies deduced from the description of the sixdimensional theory and the anomalies of the quiver gauge theories. We also give predictions for anomalies of fourdimensional theories corresponding to general compacti cations of M5branes probing C2=Zk singularities.
Field Theories in Higher Dimensions; Supersymmetric Gauge Theory

= 1 from
= (1; 0) on a torus with
1 Introduction
2 Anomalies from 6d 2.1 2.2 3.1
Anomaly polynomial of the 6d theory
Mapping the charges in 6d, 5d and 4d
2.3 Anomaly polynomial from 6d
3 4d theories from tori
Structure of the 4d quiver theories
Anomalies of the 4d quiver theories
4 Case studies N = k = 2
4.2 N = 3; k = 2
5.2 4d analysis
4.3 N = 2; k = 3, the orbifold C3=Z2
5 Compacti cation with discrete twists
A Anomalies of interacting trinions for general k; N
B Fluxes for u(1)t symmetry for k; N = 2
Introduction
Field theories in low dimensions can often be realized through compacti cations of higher
dimensional models. This point of view clari es some of the well known properties of
eld theories, and also predicts new properties and even new models; for
example, the appearance of theories which do not have a known semiclassical limit. Such
models are ubiquitous in compacti cations of sixdimensional supersymmetric theories to
four dimensions [1].
In this paper we mainly study some of the simpler compacti cations. We consider
theories living on the branes probing the singularity have in general some global symmetry,
which in our case, for general values of N and k, is su(k)
u(1). Upon compacti
cation we might turn on
uxes for abelian subalgebras of the global symmetry supported
on the torus (see [2] for early work on the subject). Without the uxes the theories have an
(see [4] for various ways to reduce on a torus without uxes). Turning the
uxes on we
turn on uxes only for subgroups of su(k)
su(k). In such a setup the compacti cations
give rise to theories with known Lagrangians. These turn out to be widely studied toric
quiver theories, albeit with additional singlet elds. We thus obtain a novel
parametrization of such theories labeling them with the number of M5branes N , the order of the
orbifold k, and the 2k
2 discrete numbers de ning the uxes through the torus.
The theories in four dimensions are constructed by studying renormalization group
(RG) ows of a quiver theory with su(N ) gauge nodes which together with the matter
elds triangulate the torus and has k gauge groups winding around one of the cycles of the
torus. The number of groups winding around the second cycle is related to the total ux
through the torus. Turning on vacuum expectation values for baryonic operators in the
setup one obtains theories which correspond to compacti cations on a torus with
dictionary between the compacti cations and the fourdimensional models was suggested
in [5]. For the dictionary to work one needs to introduce singlet
elds coupled through
superpotential terms to gauge invariant objects. These superpotential terms are in general
irrelevant giving rise to free elds in the IR. Thus, although with nontrivial uxes all the
gauge sectors are UV free, there are generally free chiral elds in the IR.
The dictionary is checked in two main ways. First by showing that the anomalies of
the compacti cation deduced by integrating the anomaly polynomial from six dimensions
to four are consistent with the fourdimensional construction. Next, the global symmetry
of the theory in four dimensions can be deduced from the compacti action details and we
give examples of how this works.
In addition to
uxes for continuous symmetries we can turn on uxes for discrete
symmetries of the sixdimensional model. The global structure of the avor symmetry is
class of models in four dimensions. These uxes can materialize in di erent ways. One way
is through fractional uxes whose quantization is consistent only for (SU(k)
SU(k))=Zk.
Another is by switching on almost commuting holonomies around the cycles of the torus,
in the sense that the holonomies commute in (SU(k)
SU(k))=Zk but do not in SU(k)
SU(k). In four dimensions this procedure corresponds to constructing the torus by gluing
a triangulated cylinder with a twist.
We also discuss the eld theories one obtains with uxes for all possible u(1) subgroups
in the special case of two M5branes on Z2 singularity where the eld theoretic construction
is known. Finally we give a prediction for anomalies of theories obtained from six
dimensions for general choices of Riemann surfaces. We have no eld theoretic constructions in
this case and this will serve as a prediction to be contrasted with future computations.
The paper is organized as follows. In section 2, we discuss the computation of the
anomalies from the sixdimensional vantage point. We consider the anomaly polynomial
presence of general values of uxes. We then derive the anomaly polynomial for the
fourdimensional models. The case of a torus is discussed in much detail. In section 3, we
consider the construction in four dimensions which should result in theories
corresponding to torus compacti cations. We compute the anomalies and see the agreement with
the sixdimensional predictions. In section 4, we detail several examples deriving
precise quiver diagrams and discussing symmetry properties which consistently enhance to
match expectations from six dimensions. In section 5, we discuss compacti cations with
SteifelWhitney classes and the fourdimensional theories related to these. We have two
appendices: in appendix A, we deduce some predictions from six dimensions for
anomalies of fourdimensional SCFTs. The appendix B details eld theoretic constructions of
strongly coupled models corresponding to compacti cations with general uxes.
Anomalies from 6d
We begin our discussion from the sixdimensional perspective. Consider taking N M5
six dimensions has su(k)b
u(1)s symmetry for general value of k and N ; here
Upon compacti cation we can choose an abelian subalgebra of this symmetry and turn
uxes supported on the torus (see for example [5, 6]). As the rst Chern classes of
uxes have to be properly quantized, the choice gives us models in four dimensions
which are labeled by discrete parameters. We can compute the 't Hooft anomalies of the
theories from the compacti cation setup by taking the anomaly eightform polynomial and
integrating this over the torus with the uxes turned on. This provides a prediction for
the fourdimensional models which we will now deduce.
Anomaly polynomial of the 6d theory
eightform polynomial I8, and can be computed using the methods developed in [7, 8],
using the fact that on the tensor branch this theory becomes a linear quiver gauge theory
with gauge group SU(k)N 1
. We use the normalization where the bifundamental
hypermultiplets in the quiver have charge
1 under u(1)s. The resulting anomaly polynomial is
I8 =
c2(R)(Tr Fb2 + Tr Fc2) +
c2(R)(4c2(R) + p1(T ))
(Ivec(b) + Ivec(c))
Tr Fb2 + Tr Fc2 c1(s)2 +
2 N N 3
c2(R)c1(s)2 +
the global symmetry.
where p1(T ) and p2(T ) are the rst and second Pontryagin classes of the tangent bundle,
c2(R) and c1(s) are the second and the rst Chern classes of the su(2)R and of the u(1)s
bundles of the 6d theory, respectively, Tr Fbn and Tr Fcn are parametrized below by Chern
Ivec(b) =
1)c2(R)p1(T )
Itensor =
Mapping the charges in 6d, 5d and 4d
We will match anomalies for various symmetries by performing computations in di erent
dimensions, thus we will start by matching the symmetries between di erent dimensions.
Let us map here the charges from 6d to lower dimensions.
We have su(k)b
u(1)s as the avor symmetry, in addition to the su(2)R
transform as a bifundamental representation of the global symmetry. Let us say that in
six dimensions the avor symmetry bundles split, and the Chern roots are given by
b1; : : : ; bk; c1; : : : ; ck; s
the line bundle with the Chern class ai
bj + s. The Chern class c1(s) used in (2.1) is
identi ed with this s. In our normalization,
Tr Fb2 =
c2(R) =
Trfund Fb4 =
TrfundFb3 =
for the su(2)R bundle with Chern roots (x; x).
5d. Let us put the 6d theory on S1 with a nontrivial holonomy for the avor symmetry.
Then in the infrared, the 5d theory is dual, in the sense described in [9] as continuation
past in nite coupling, to the circular quiver su(N )k. Call Ii the Chern root for the
instanton number symmetry of su(N )i and t + Hi the Chern root for the baryon number
discussion in [10], we know that the oneinstanton operator of su(N )i, that becomes the
raising/lowering operators of the su(k)2 avor symmetry currents, couples to a line bundle
with the Chern class
This is to be identi ed with bi
cj 1. Therefore, we see
N Hi = bi
Next, to relate t and s, it is useful to consider the Higgs branch of the theory, when
we separate N M5branes. In the following, we will write down some key invariants on the
Higgs branch and specify some of the relations they satisfy. This will be su cient to derive
the quantum numbers. In the 5d description, one can use the Kronheimer construction for
su(N )i+1 as
i and ~ i, let diag(za) =
i ~ i, diag(xa) =
k, diag(ya) = ~
k. Therefore, u and v have the
u(1)tcharge
we have z = Tr
have u(1)s charge
In six dimensions, again when N M5branes are separated, the same Higgs branch can
be found as explained in [11]. Namely, when we denote the su(k)2 bifundamental by , ~ ,
1, thus x; y have charge
k, thus u; v have charge
can equate the u(1)s charge and u(1)t charge:
s = t:
The su(2)R symmetry in 5d and 6d can be naturally identi ed so the scalars in the
bifundamental hypermultiplets in the su(N )k quiver are su(2)R doublets.
Now we consider the situation in four dimensions. In the tube theory, most of the
analysis above can be directly applied. The su(2)R symmetry is broken to the Cartan.
We use the normalization where the supercharge has the charge
1 under the remaining
u(1)R0 symmetry. Here R0 emphasizes that this is a natural Rsymmetry coming from the
sixdimensional construction; this generically will not be the superconformal R symmetry
in the infrared, which needs to be determined by the amaximization [12].
In any case, the bifundamentals in the su(N )k quiver, before the supersymmetry is
broken by half, have the u(1)R0 charge 1 and the u(1)tcharge
1. Then, the surviving
chiral bifundamental in the su(N )k tube theory has the u(1)R0 charge 1 and the
u(1)tcharge 1. Together with (2.8), this data on the tube theory is enough to
nd the charge
assignment in the Lagrangian class Sk theory, as we will see in the next section.
Anomaly polynomial from 6d
We now compute the anomaly polynomial of the compacti ed theory from the 6d point
of view. Let Nbi , Nci , and Ns be the numbers of uxes of the u(1)bi , u(1)ci , and u(1)s
respectively. Let us also denote the rst Chern classes of line bundles in 4d as c1(R0), c1(t),
c1( i) and c1( i). The Chern roots introduced above are related as follows2
x = c1(R0)
bi = N c1( i)
s = c1(t) + Ns 2g
ci = N c1( i)
where c2(R) =
x2, c2(s) =
c1(s)2 =
Cg t = 2
proceeding equations hold also for this case.
By substituting these into the anomaly eightform and performing the integral over
the Riemann surface RCg I8, we get
I6 =
2)c1(R0)p1(T4) +
Nsc1(t)p1(T4)
1)(k2(N 2 + N
Nsc1(R0)2c1(t)
2)c1(R0)c1(t)2
1)c1(R0)2 + kN 2c1(t)2 X(Nbic1( i) + Ncic1( i))
kN X (Nbic1( i) + Ncic1( i)) p1(T4)
Nbi)c1(t)c1( i)2 + (N Ns + Nci)c1(t)c1( i)
Ns)c1( i)3 + (Nci + Ns)c1( i)3)
(X c1( i)2)(X Ncj c1( j)) + (X c1( i)2)(X Nbj c1( j)) ;
ed 4d theory.
the following anomaly polynomial:
N 2(N 1) (X c1( i)2)(X Nbj c1( j))+
N 2(N 1) (X c1( i)2)(X Ncj c1( j))
kN X (Nbic1( i)+Ncic1( i)) p1(T4)
(X c1( i)2)(X Ncj c1( j))+(X c1( i)2)(X Nbj c1( j)) :
I6 =
kN (N 1)c1(R0)2 +kN 2c1(t)2 X(Nbic1( i)+Ncic1( i))
Nbic1(t)c1( i)2 +Ncic1(t)c1( i)
where Pik=1 c1( i) = 0 and Pik=01 c1( i) = 0.
The triangle anomaly in the more traditional form Tr xyz can be read o
substituting c1( k) =
4d theories from tori
Structure of the 4d quiver theories
We consider now the eld theory construction corresponding to tori with
uxes with no
punctures [5]. The models are constructed by rst starting from a toric quiver built from
some number of free trinion theories ( gure 1), and then by higgsing some of the symmetries.
The free trinion theory corresponds to a sphere with three punctures: two are maximal
and one is minimal, and the u(1)s
maximal and the minimal punctures are associated with su(N )k and u(1) avor symmetries
respectively. The former is known to be labeled by the color c 2 Zk and the sign
have the same signs
other charges are denoted in gure 1.
Gluing of two maximal punctures corresponds to gauging of the su(N )k symmetry of
both punctures. Depending on the signs of the punctures we have two gluings [5, 13, 14]:
gluing: when the two punctures have the same sign, say
= +1, we add an
gauge factors cyclically, with superpotential coupling of the bifundamentals and the
mesonic operators associated to the punctures; the quiver in
gure 1 represents
multiplet with the superpotential coupling of two mesonic operators coming from the
These are associated to the theories on a tube without any ux. (See appendix for the tube
theory with u(1)s ux.) As already noticed in [6] one can see that the charge assignment is
consistent with the discussion in section 2.2 from 6d. If the two punctures have the same
color this preserves all internal symmetries. However if the two punctures have di erent
colors then all u(1) 's are broken.
We here only focus on the
gluings and construct a quiver theory associated to a torus
with only minimal punctures from a collection of kl free trinions, as in gure 1. When l is an
integer we can always glue two punctures that have the same color preserving all the internal
symmetries, so the global symmetry of this model consists of u(1)k 1
kl u(1) j symmetries which are associated to minimal punctures.
We obtain models with no punctures by giving vacuum expectation values to kl
baryonic operators charged under u(1) j symmetries and introducing certain gaugesinglet
chithese as bifundamentals of two copies of su(N )k. In the picture the circles are su(N ) groups and
one has k groups winding around a cylinder. The trinion is associated to a compacti cation on a
sphere with two maximal punctures (of di erent color) and a minimal puncture. On the right we
glue trinions together to triangulate a torus. We have lk trinions combined with every
introducing bifundamental elds which appear as vertical lines in the diagram.
ral multiplets ipping some of the other baryons,3 as sketched in gure 2. There are choices
to be made as to which baryons the vacuum expectation values are given and this choice
maps to a choice of uxes in six dimensions [5]. We will write down the exact
correspon. After higging all the u(1) j
do not in general have known regular Lagrangians (see for example the discussion in the
Let us make several general observations about these models. The quivers correspond
to tiling of the torus with triangular and square faces. The exact details depend on the
uxes and in fourdimensional language on the ways we close the minimal punctures. The
theory with the minimal punctures we utilize as a starting point of the construction
triangulates the torus. Importantly we supplement the quiver with singlet elds, some of which
might be free and some coupled to gauge singlet combinations of elds through additional
superpotential terms.
Such theories were widely studied in various contexts about ten years ago [18{20].
It is convenient to think about the theories in terms of zigzag paths on the torus. Each
symmetry factor u(1) (with exception of u(1)t) corresponds to a loop, zigzag path, winding
around the cycles of the torus. Let us call the cycle around which we have, in the theory in
3By ipping an operator O we mean the procedure of adding a chiral eld MO to the model with
superpotential W = MO O.
In every free trinion we give a vacuum expectation value to one of the baryons. The choice of the
baryons is related to the ux and in general di erent choices lead to di erent theories in the IR. The
baryons which do not receive a vev but are charged with same charge under the minimal puncture
symmetry as the baryon which does receive vev, are ipped. In the diagrams the elds with a cross
are the baryons which are ipped. In the picture the baryons which receive vacuum expectation
value are weighed as t 2 1N = N , t N2 2N
N
the UV, k gauge groups cycle A and the other one cycle B. The UV model then has loops
winding once around cycle A corresponding to puncture symmetries and k loops winding
l times around each one of the cycles A and B, and l loops winding
l times around cycle
A and l times around cycle B. The ow initiated by closing the punctures preserves the
symmetries not associated to the punctures and breaks symmetries associated to punctures.
The pattern of winding of the di erent lines can be translated to the uxes. For example,
the torus with no ux is formally mapped to con gurations with all windings vanishing.
Anomalies of the 4d quiver theories
The anomalies for these models can be rather easily derived. For the sake of computation
of the anomalies we do not need to
gure out the quiver diagram in the IR of the
triggered by vacuum expectation values turned on for baryonic operators when closing
minimal punctures. We can compute these in the UV, making sure to use the symmetries
surviving in the IR and decoupling the relevant Goldstone chiral multiplets. Let us give
the algorithm for computing the anomalies.
We will encode all anomalies in the trial
a conformal central charge and in the trace of a trial R symmetry where we will keep
dependence on possible mixing parameters with all the abelian symmetries. We denote the
R charge as
where qi , qi , qt and qu are the charges of u(1)k 1, u(1)k 1, u(1)t and u(1) j . We have the
constraint Pjk=1 sj = Pk
Rcharge z and of a vector multiplet of group of dimension h are given by
a (z) =
av(h) =
Then the anomaly of the free trinion is
at(N; k; s ; s ; st; s ) = N 2 X a
s i = 12 + 12 st
si and ip the baryons. The speci cation of the fugacities re ects
regarding u(1) i . We denote by Q
= Pik=1 Qi , Q
= Pk
u=1 Qu.
the fact that the operators receiving the vacuum expectation values have all their charges
vanishing in the IR. The function F (i) is an arbitrary function mapping (1;
; k) to itself.
We denote by Q
i the number of minimal punctures closed with si , and Qi is the same
We note that the color of the maximal punctures is not important in the computation of
the anomaly of the free trinion as it only determines the sequence in which the di erent
chiral elds are organized together.
The anomalies of each
gluing of maximal punctures of color c are easily seen to be
ag(N; k; c; s ; s ; st) = kav(N 2
3 X((su
where the indices are summed mod k. These elds are charged under both
symmetries and thus the color of the puncture we glue is important for the anomaly. The
anomaly of torus built from kl free trinions is then
N 2k(st +1)l(S2 +S2 )
The conformal anomaly a for the torus with no punctures but with uxes determined
by a choice of F is then
ator;F (N; k; Qi ; Qj ; s ; s ; st) = ator
i=1 u=1
X Qua (2
X Qua (2
i=1 u=1
where in the second line we have the contribution of the chiral elds which ip the baryons
and in the rst line we specialize the parameters of the torus with punctures to be consistent
the Goldstone bosons appearing in the ow as we break some symmetries. This evaluates to
vector whose components are (si ; )n.
i denotes the inner product of kdimensional vectors and (s ; )n stands for the
This expression has several nice features. If one shifts all Qi (or Qi ) by some integer
the above does not change. This corresponds to completely closing minimal punctures
exchanging all of Q
Q . This is consistent with the expectation that the
group here is enhanced to su(2k) as below these combinations are identi ed with the uxes.
The anomalies here are in agreement with the anomalies computed from six dimensions.
The map between the parameters is
Nbi = Q
Nci =
Ns = 0 :
For the free trinion we obtain that btrin: = kN 2(st
gauging we obtain bg =
N 2kst. This is independent of the puncture symmetries and the
close the minimal punctures we have to ip the baryons and the anomaly is
k2l. When we
b(Qi ; Qi ) =
k(Q + Q ) + k(Q + Q )
= N k(hQ ; s i + hQ ; s i) :
We observe that with our identi cation of uxes with multiplicities of the various choices
of closures of minimal punctures all anomalies agree between 4d and 6d.
We can use the trial aanomaly we have obtained to compute the conformal anomalies
of the theories. Here we have to be careful as in general the elds which ip the baryons
are coupled through irrelevant interactions and thus are free. One then needs to take this
into account in the computation of the conformal anomalies.
Case studies
We construct examples of various quiver gauge theories of class Sk type associated to a torus
for small values of k. We will study in greater detail some of their properties. Speci cally
we calculate the superconformal index and test the global symmetry of the 4d xed point
with that predicted from the 6d construction.
The superconformal index also allows us to compute the dimension of the conformal
manifold of the 4d theory. This can also be predicted based on the 6d construction as done
an exactly marginal operator for each complex structure modulus, and at connections for
the global symmetries. For the case of a torus, we always have a single complex structure
modulus. In addition we can also have at connections for the global symmetries with
nontrivial values around each of the two cycles of the torus. These must be abelian due to the
homotopy group relation of the torus. Thus we see that we get 2(2k
1) real parameters
1 complex marginal deformations. So to conclude, we expect:
dim(M) = 2k:
We can use the 4d superconformal index to check this prediction.
when considered as the AN 1 (2; 0) theory compacti ed on a torus. In that case, (4.1) gives
N = k = 2
Consider taking two free trinions and connecting maximal punctures of the same color
together. This results in a torus with two minimal punctures. Then by closing the
minimal punctures, we can get a theory corresponding only to a torus, as discussed in the
previous section.
First we begin with the theory corresponding to a torus with two minimal punctures
we get by connecting two free trinions. For the purpose of constructing these theories we
will leave N general, setting it to the desired value at the end. The quiver diagram of the
theory is shown in
gure 3. It has a cubic superpotential for any triangle. This is the
theory that lives on N D3branes probing a C3=Z2
in various contexts. See e.g. [21] and references therein.
Z2 singularity and has been studied
elds under all the non Rsymmetries: the internal u(1) ; u(1) ; u(1)t, and the minimal puncture
ones u(1) ; u(1) . Additionally there is a cubic superpotential for every triangle. Alternatively it
is given by the most general cubic superpotential that is gauge invariant and consistent with the
symmetry allocation in the table. All elds have the free Rcharge 23 .
Let us denote the uxes of the theory associated with the surface by (Nb; Nc; Ns). As
the free trinion has ux (0; 0; 12 ) [5, 6], this theory should correspond to ux (0; 0; 1).
Next we give a vev to the baryon made from Q1. This corresponds to closing a
minimal puncture. The resulting theory is associated to a toruAs with a 2minimal puncture
Further we can give a vev to another baryon, associated with1the o1tht2er2 puncture. This
will close the other puncture and leads us to a torus with no p2uncturte12s. We have three
distinct choices for the baryon. These will di er by the ux on t3he tortu11s.22
One choice is to close with the baryon made from Q~2. This4 will leta22d to a torus with
ux (0; 0; 0). This theory is somewhat singular and we shall refr5ain fr1omt2 1discussing it for
now. We can also close the puncture with the baryon made from6 Q~ 1t.2 1T1h2is will lead to a
ux (1; 0; 0). The quiver description of this theoryQi1s shtow1n1in2 gure 4. One
can see that it resembles an a ne A1 quiver with additional Qsi2ngletst c11oupled through a
superpotential. We shall refer to this theory as the a ne quiveQr.3
Next we can study some of its properties. Here we shall consider
N cases to the next subsection. We begin with studying its anomalies. In this case the
full superpotential, including the contribution from the ipping, is cubic, and performing
amaximization we indeed nd that under the correct u(1)R all chiral multiplets have the
Next to the elds are their charges summarized through fugacities. We use mostly standard notation
except for two points: lines from a group to itself represent N 2 hypermultiplets forming the adjoint
plus singlet representations of the group; we write an X over a
eld to represent the fact that
the baryon of that eld is ipped. The theory has a cubic superpotential for every triangle which
can also be derived by considering the most general cubic superpotential that is gauge invariant
and consistent with the symmetry allocation. There is also the superpotential term which is not
generally cubic coming from the ipping. All elds, save the ipping elds, have the free Rcharge 23 .
free eld Rcharge 23 . Further we nd that:
a =
c =
Note that these are the anomalies for the a ne A1 quiver in addition to two free
hypermultiplets.4 This fact and the values for the Rcharges suggest that this theory has
a subspace on its conformal manifold where it is indeed the a ne A1 quiver gauge theory,
where the supersymmetry enhances to N = 2.
We can also evaluate the index of this theory. Particularly we consider the a ne quiver
without the singlets as these are just free elds. The subgroup of so(7) which commutes
with the ux is u(1)
su(2)2 and so this is the expected global symmetry. Since without
with global symmetry u(1)
usp(4), the index in fact forms into characters of this group.
We nd it is given by:
IN=2;k=2
+pq 1 + [5]usp(4)
[10]usp(4) + : : : ;
where [4]usp(4) = t + t1 + t + t = [1; 0]usp(4).
4These are presumably the two chiral elds that accompany the adjoints plus the two chiral elds that
are introduced for the ipping.
5.2.1 and in particular equation (5.19). Furthermore it was found to be the closure of the
next to minimal orbit of usp(4) as in table 3 of [23] and tables 10 and 12 of [24], where
another description sets it as the Z2 orbifold of the closure of the minimal nilpotent orbit
of SL(4) (alternatively known as the reduced moduli space of 1 SU(4) instanton on C2).
This emphasizes that the global symmetry on this part is indeed usp(4). The unre ned
Hilbert Series takes the form
HNA=2n;ek=Q2uiver( ) =
and it admits the highest weight generating function [24, 25]
HW GN=2;k=2
A ne Quiver( ; 1; 2) =
with 1 and 2 the fugacities for the highest weights of usp(4). From this one deduces the
re ned Hilbert series that admits a character expansion
HNA=2n;ek=Q2uiver( ; t; ) =
n1=0 n2=0
In addition there are the 4 singlets, which in this construction, two are given the charge 2
and two the charge 14 . Therefore their presence does not interfere with the global
symmetry. Again it is reasonable to expect that at some sublocus of the conformal manifold these
are indeed free elds and so can be rotated separately leading to additional enhancement
of symmetry. We also note that this structure is common in the so called \ugly" class S
theories where the SCFT is accompanied by additional free hypers whose global symmetry
is identi ed with part of the global symmetry of the SCFT.
Next we can study the dimension of the conformal manifold for this theory. As the
singlets cannot add additional directions [26], it can be directly read from the index without
them (4.3). Particularly, we look at the pq order terms, which according to a result by [27],
are just the marginal operators minus the conserved currents. For the case at hand we nd
that there are 7 marginal operators where one is canceled against the u(1) conserved current
in (4.3). Applying the logic of [26] we nd a dimension 3 conformal manifold, reproducing
the result in section 3.2 of [28], along which the symmetry is broken to u(1)
su(2)2. This
similar to the story for k = 1.
by the compacti cation of the A1 (2; 0) theory on a torus with two maximal punctures.
Then the 6d analysis of [15] leads us to expect a threedimensional conformal manifold, two
directions of which preserve all the symmetries and correspond to the coupling constants
su(2)2 global symmetry. This agrees with our observation.
The class S and class S2 theories di er by the existence of the singlets. It is not
di cult to see one can build marginal operators uncharged under the 6d apparent global
elds are their charges summarized through fugacities. The theory has a
bifundamentals Qi and Rcharge 2
quartic superpotential involving the four bifundamentals as well as the superpotential coming from
the ipping. There is also an Rsymmetry where it is convenient to give Rcharge 12 to the four
symmetries. By the logic of [26] this should lead to exactly marginal operators. However,
this fails as these operators in fact become free leading to the appearance of accidental
symmetries invalidating the argument. Therefore we conclude that the 6d expectations
regarding the conformal manifold are too naive, and like the anomaly analysis, can be
modi ed due to the appearance of accidental symmetries.
The \KlebanovWitten" theory.
We can also close the puncture with the baryon
made from Q~4. This will lead to a torus with
ux ( 12 ; 12 ; 0). The quiver description of this
theory is shown in gure 5. One can see that it resembles the KlebanovWitten model [29],
but with additional singlets ; , coupled through superpotential terms. We shall refer to
this theory as the KW case.
Let us rst consider the theory without the singlets. In this case this model is known
to go to an interacting
xed point where the bifundamental elds have Rcharge 12 [29].
Now consider adding the free
elds and couple them through the superpotential. The
behavior of the resulting theory depends on the value of N . For N > 2, this superpotental
is irrelevant and the theory should
ow to the same
xed point, but with free singlets.
ow to a new
xed point. We shall now discuss the latter case in more detail. Note that for this special
case of N = 2, Q1 is a 2
2 matrix and the notation Q21 stands for det Q1, and similarly
for Q2. Each of these terms is invariant under a corresponding su(2) global symmetry and
the global su(4) symmetry which is present in the absence of these terms [30] is broken to
u(1) in the presence of these terms, where u(1) is the baryonic symmetry
which acts as +1 on Q1 and Q2 and as
1 on Q3 and Q4.
First we shall need to perform amaximization to determine the superconformal
Rsymmetry. It is straightforward to see that only the baryonic symmetry u(1) + u(1) can
mix with the naive u(1)R of the KW model. Thus, we de ne:
By performing amaximization we nd
0:027, so the Rcharges change
only slightly compared to their naive value. One can check that all gauge invariant elds
are above the unitary bound so this is consistent with the theory owing to an interacting
We can next evaluate the anomalies for this theory. Particularly, for the conformal
anomalies we nd:
Next we evaluate the index for this theory. First we note that the subgroup of so(7)
that commutes with the
ux is u(1)
usp(4) so this is the expected global symmetry,
enhancing the global symmetry of su(2)
u(1) found above. Indeed we nd that
the index naturally forms into characters of this symmetry, where it is given by:
INKW=2;k=2 = 1+p 21 q 12
u(1)0R = u(1)R +
u(1) + u(1)
a =
c =
+: : : (4.9)
so the true Rcharges of various operators should be shifted depending on their
It is interesting that the singlets are necessary to get the enhanced global symmetry
that is required from matching to 6d. They rst break su(4) to su(2) su(2) u(1) and then
enhance to u(1)
usp(4) which is evident from the index but not from the superpotential.
being relevant only for N = 2.
Another interesting computation is to check the moduli space for the conifold theory at
su(4) and the Hilbert series was shown in [30] and particularly in equation 4.16 to admit
a character expansion of the form
n1=0 n2=0
or alternatively, using the fugacities for the highest weights 1; 2, a highest weight
gener
HW G ( ; 1; 2)CNo=n2ifold =
These computations lead to a description of the moduli space as the set of all 4 by 4
complex symmetric matrices with rank at most 2. The natural guess after ipping and
symmetry enhancement, with highest weight fugacities 1 and 2 for usp(4), is given by
the highest weight generating function
)KNW=2 =
which leads to an unre ned Hilbert series
H ( )KNW=2 =
We can also calculate the dimension of the conformal manifold from the index. It is
again given by the pq order terms under the true Rsymmetry. For our case this translates to
the pq order operators which are uncharged under u(1) . Thus we nd an 11 dimensional
conformal manifold along which the usp(4) group is completely broken. This is greater
than the 4 dimensional one we expect from 6d reasoning. This does not contradict the
6d reasoning since there could be 4d marginal operators with no clear interpretation in
this with the 5 dimensional conformal manifold for the conifold theory which was found in
section 2.2 of [28].
Another interesting observation regarding the index (4.9) is the appearance of the 4
dimensional representation of usp(4). This implies that the global symmetry is USp(4) and
not SO(5). This group in 6d comes from breaking the SO(7) global symmetry of the 6d
SCFT. Naively this suggests that this group must be Spin(7) and not SO(7). However, [6]
found various 4d theories, matching 6d compacti cations with
uxes that are consistent
only with SO(7). These two observations suggest one of two scenarios. One, the 6d group
is in fact Spin(7) which naturally explains the appearance of the 4 in the index (4.9). Then
the 4d theories with nonstandard quantization should be viewed as 4d theories with no
valid 6d origin. An alternative explanation is that the 6d group is SO(7), which naturally
t the observations of [6]. However in that case one must view the spinors in the 4d
index (4.9) as 4d operators without a 6d origin, similarly to the excess marginal operators
we seem to nd for this theory. This interpretation then implies that the 4d theory in fact
undergoes an accidental discrete enhancement of symmetry SO(5) ! USp(4).
N = 3; k = 2
special features that are not present in the general case. We shall now discuss the
beindex calculations.
ne quiver" theory.
We start with the a ne A1 quiver case. The matter
content and charges are as in
gure 4. The models contain four singlet elds, two o
which come from the ipping and are coupled through a superpotential.
Without this
superpotential, all elds have free Rcharges and the theory is expected to sit on the
involving the ipped elds is irrelevant and so the theory with these terms is expected to
be additional symmetries rotating the free elds.
symmetry for k = 2; N > 2 is su(2)t
su(2) . This should be broken by the ux
which is the symmetry we expect in the 4d theory. We indeed
nd that the index forms characters of that symmetry. Ignoring the singlets, as these are
just free elds, we nd the index to be:
IN=3;k=2
A ne Quiver = 1 + p 23 q 23 2 4 +
Additionally there are the 4 singlets which in this construction two are given the charge
2 and two the charge 16 . Therefore their presence does not interfere with the symmetry.
We can also compare the dimension of the conformal manifold with the 6d expectations.
Again we nd that the dimension of the conformal manifold is in fact greater than what is
expected from 6d.
The \KlebanovWitten" theory.
Next we consider the KW case. As we previously
discussed for N > 2 the superpotential coupling the singlets is irrelevant and the theory
should ow to the KW model with singlets.
the 4d theory to preserve an su(2)t
that the KW model shows an su(2)t
global symmetry. In fact we shall see
global symmetry which is broken
have a considerable enhancement of symmetry in the IR. As the singlets decouple in the
IR we concentrate only on the interacting part, for which we nd the index to be:
INKW=3;k=2 = 1 + p 21 q 12 [2]su(2)t
where [2]su(2)t = t + 1t and [2]su(2)
In addition there are two free singlets with charges 16 and 16 . These are inconsistent
with su(2) implying that it is broken to its Cartan by the superpotential only to return
in the deep IR. This again resembles some situations in class S theories where the global
symmetry of an interacting theory plus hypers is broken by mixing part of it with the
symmetry rotating the hypers.
The results of equation (4.15) agree with the computations in equation 3.83 of [31]
but are still missing two essential operators that transform as [2]su(2)t [1]su(2)
 the so called non factorizable baryons. It will be interesting to check
if higher order computations produce these two essential contributions.
Again we nd that the dimension of the conformal manifold is in fact greater than what is
expected from 6d.
We can in principle look at higher values of N and even the large N behavior. In fact
both of the theories considered here, without the singlets, have well known large N gravity
duals. As the number of singlets is order 1, it is reasonable that most of the properties of
these theories will be well described by the gravity duals. In this regard it is interesting
that both theories are reached by the compacti cation of the same 6d SCFT on the same
surface di ering by order 1
uxes. The models we consider here correspond to having
vanishing ux for u(1)t as in this case we have known Lagrangians. With the ux for u(1)t
the models are expected to be strongly coupled, see appendix. The properties of these
two types of models are qualitatively di erent. For example, the anomalies scale as N 2 for
the gauge theories we consider here (when the singlet elds are appropriately taken into
account), and are expected to scale as N 3 for the strongly coupled types as can be inferred
from the six dimensional analysis. As from the six dimensional perspective we cannot infer
existence of accidental symmetries this is just an expectation which can be invalidated in
N = 2; k = 3, the orbifold C3=Z2
consider taking three free trinions, connecting them together and closing three minimal
punctures. More speci cally we shall close two punctures with a vev to baryons charged
under 1 with the same charge.
Now we need to close the nal minimal puncture. We consider two di erent
possibilities. First we consider closing the last puncture also with a vev to a baryon charged under
1 with the same charge as the last two. This is similar to how we got the a ne quiver in
In this theory all gauge groups see 3N
avors and so are conformal. Thus, without the
ipping superpotential, all eld have the free Rcharge 23 . The ipping superpotential is
23 under the superconformal Rsymmetry. Again for N > 2 this entail an IR enhancement
of symmetry due to the ipping elds becoming free.
interacting part and ignore the singlets as these are free. The 6d global symmetry here
is enhanced to su(6), but our choice of ux breaks it to su(3)
Evaluating the index, we nd that it can be written as:
INAl=l2;1k=3 = 1 + p 32 q 23
with only 1
ux, while on the right is a table summarizing the charges of the various elds. Note
that several di erent elds have the same charges and so are represented with the same letter. This
theory has a rather large cubic superpotential involving the 12 triangles in the diagram. Again
these are most conveniently generated by taking all cubic terms consistent with the symmetries.
Additionally there are the superpotential terms coming from the ipping, which in general are
not cubic. It is again convenient to choose the Rsymmetry so that all non ipping elds have
where [2]su(2) =
and [3]su(3) =
12 + 22 + 121 22 . Additionally there are 6
singlet elds, 3 with charge 12 and 3 with charge 14 22. These can be written as 3 13 [2]su(2)
and so are consistent with the 6d global symmetry.
We can also consider the dimension of the conformal manifold. By the reasoning
of [26], the terms appearing in (4.16) do not contribute any exactly marginal operators
as one cannot form a
1 invariant from them. This leaves the marginal operators in the
adjoint of the full global symmetry G which must be present to cancel the contribution of
We can also consider closing the last puncture with a baryon charged under 2
ux leads to the symmetry breaking pattern su(6) ! su(4)
we expect to be the 4d global symmetry. In the eld theory the vev leads to a quiver with
an su(N ) group with N
avors. This group con nes in the IR leading to the identi cation
of the groups it's connected to and making the ipping
elds massive. After the dust
settles we end with the so called L222 [20] quiver theory in gure 7. This theory can also
be derived from 4 NS branes on the circle, with two types of orientation.
We next proceed to analyze this theory in detail. First we need to evaluate the
conformal Rsymmetry. By inspection one can see that there is only one u(1) that can mix with
ux and one of 2
ux. Next to the
elds are their charged summarized using fugacities.
This theory has a combination of cubic and quartic superpotential terms. Again these are most
conveniently generated by taking all terms consistent with the symmetries. Additionally there are
superpotential terms coming from the ipping. The theory also has an Rsymmetry, a convenient
choice for which is to give all the bifundamentals Rcharge 12 , and Rcharge 1 for the adjoints and
the Rsymmetry, which in our notation is u(1) 1 . The remaining u(1)'s can be grouped
into 4 baryonic u(1)'s, each rotating one of the four pairs of bifundamentals with opposite
charges while the adjoints and their associated singlets being neutral. Three of these u(1)'s
are combinations of u(1)t; u(1) 1 and u(1) 2 while the last is 2u(1) 2
Next we preform amaximization. We take the Rsymmetry to be:
u(1)0R = u(1)R + u(1) 1
where we take u(1)R to rotate the bifundamentals with charge 12 and the adjoints and their
associated singlets with charge 1. The ipping elds attached to the bifundamental then
also have R charge 1 while those attached to the adjoits have R charge 0. Performing the
amaximization we nd that:
= 3
6 5 . With this value we nd that a = 5 5
However with this Rcharge the adjoint ipping
elds are below the unitary bound.
Therefore the natural conjecture is that these elds become free at some point along the
ow leading to an accidental u(1) that mixes with the Rsymmetry. We can now repeat the
amaximization, but taking these to be free elds where we nd:
. We nd that
all elds have dimensions above the unitary bound and that the superpotential coupling
the ipping elds to the adjoints is irrelevant. All of these are consistent with our claim.
We can also preform amaximization considering all the ipping
elds as free, where we
indeed nd that, compared to that point, the superpotential coupling the ipping elds to
the adjoints is irrelevant while the one coupling the ipping elds to the bifundamentals
So to conclude we expect the theory in gure 7 to ow to an interacting xed point
consisting of the quiver theory, without the adjoint ipping, plus two free chiral elds. We
next want to evaluate the index of this xed point. Again for simplicity we shall rst ignore
the two free chiral elds. From 6d we expect an su(4)
u(1) global symmetry. We
indeed nd that the index can be written in characters of this symmetry, where it reads:
INM=ix2e;dk=3 = 1+p 21 q 21
4 2 +1+ 14 22 [5]su(2) + 14 22 [3]su(2) [6]su(4) + 14 22
[4]su(4) + [4]su(4) +2 [6]su(4) +2 [3]su(2) +: : :
where [2]su(2) =
22 + 13 and [4]su(4) = pt 2 ( 12 + 22 + 121 22 ) +
2
we have used the naive Rsymmetry, not the superconformal one, so the dimensions of
operators are shifted based on their 1 charge.
One can see that, as expected, the u(1) is given by u(1) 1 while 2u(1) 2
u(1) 1 forms
the nonabelian part. This is apparent as the fugacity of the u(1) is 12 2 so states charged
only under it are invariant under a 2u(1) 2
u(1) 1 transformation. Alternatively those
charged only under the nonabelian part are invariant under u(1) 1 transformations.
Finally we note that the dimension of the conformal manifold is again greater than
what is expected from 6d.
Compacti cation with discrete twists
For now we were satis ed with the discussion of the symmetries of the models at the level
of the algebra rather than the group. However, the global properties of the group in six
dimensions have far reaching implications as far as the choice of uxes goes. As we will
next discover the six dimensional constructions have a non trivial global structure which
allows for discrete uxes to be switched on which in particular break some of the continuous
global symmetry. This procedure has a four dimensional analogue. We turn our attention
to this next.
Global form of the avor symmetry.
There is no doubt that the 6d N =(1; 0) theory
we have been using in this paper has the avor symmetry su(k)
u(1)t at the Lie
algebra level. What is exactly the
avor symmetry group? Let us for now concentrate
our attention to the subgroup connected to the identity, neglecting the u(1)t part. On the
generic point of the tensor branch, the theory becomes a linear quiver gauge theory. It
contains a gaugeinvariant operator which is a bifundamental of su(k)
su(k), obtained by
multiplying all the bifundamentals of the quiver. This suggests that the group is
where the quotient is with respect to the diagonal combination of the two centers.
If this is the case, on a compacti cation on T 2, we should be able to turn on the
(generalized) StiefelWhitney class w2 2 Zk. In the next subsection, we will nd the
corresponding operation in the 4d
eld theory language, and will check that the anomaly
computed in 4d agrees with the expectation from 6d.
Before doing that, let us remind ourselves the basics of the (generalized)
StiefelWhitney class w2 of bundles of nonsimplyconnected groups on T 2. (A detailed account
readable for physicists can be found in [32, 33]. For complete generality, the reader should
consult [34].) Let us say we construct a G bundle on T 2 by rst having a Gbundle on a
rectangle by identifying two sets of parallel edges. This identi cation involves specifying
the gauge transformation used in the gluing process along the boundary of the rectangle.
This becomes a closed path within the group manifold, which is topologically classi ed by
Zk. This is the (generalized) StiefelWhitney class w2. When the StiefelWhitney class of
Flat bundles with nontrivial w2.
First are the at bundles. In this case we have
the holonomy gA along the A cycle and the holonomy gB along the B cycle. They should
We can easily nd such a pair: take
gA = diag(1; !; !2; : : : ; !k 1);
! = e2 i=k
gB = BB
for physicists these matrices were familiar from the work of 't Hooft [35]. There is no
unbroken symmetry.
!`gBgA such that the unbroken symmetry is su(m). This is done by rstly recalling that
regarding the resulting gA;B as matrices in SU(k). As the SU(m) part is untouched, clearly
the avor symmetry is su(m).
Abelian bundles with nontrivial w2. Another are Abelian bundles. Let us denote by
An Abelian SU(k) bundle is speci ed by a map U(1) ! T . In particular, specifying
one on T 2 corresponds to specifying a point in a lattice
of rank k 1 that can be naturally
identi ed with the root lattice of SU(k). They can be thought of as k integers summing
by a point in a lattice
again of rank k
1, but now identi ed with the weight lattice
of SU(k). If we use the Chern classes normalized to the subgroup of SU(k), they can now
look rational, with denominator k. The StiefelWhitney class can be easily read o by
consider (SU(k)
StiefelWhitney class. For example, one can choose at bundles for both, Abelian bundles
for both, or a at bundle for one and an Abelian bundle for the other. The computation of
the 4d anomaly is straightforward: one just has to plug in the Chern classes of the Abelian
parts in the formulas we have been using.
StiefelWhitney class and the symmetry of the quiver graph.
Now, to bridge
our discussion here to the 4d analysis below, consider rst the compacti cation to 5d.
Let us put the SU(k) holonomy gA (5.2) around S1. Then we have a circular SU(N )k
quiver with the same gauge coupling for all groups. Now, the operation gB (5.3) naturally
corresponds to the symmetry of the circular quiver shifting the node by one. Therefore, by
compactifying the 5d theory on an additional S1 with a twist rotating the circular quiver,
we can realize the compacti cation of the 6d theory with a nontrivial StiefelWhitney class.
We have constructed tori theories by combining free trinions in multiples of k. This way
we always glued punctures of the same color and preserve all the internal symmetries. It
is however possible to glue punctures of di erent colors at the price of breaking some of
the internal symmetry. For example, constructing a loop out of l free trinions the group is
su(gcd(k; l)) . We can then close the punctures and try to identify the six
dimensional compacti cation leading to such theories.
Gluing two punctures of a single trinion. Let us consider taking a free trinion
and gluing the two maximal punctures to each other. The symmetry preserved here is
u(1)t and the puncture symmetry u(1) . The theory is the a ne quiver with
k nodes and with a singlet
eld associated to every node coupling to charged
same manner as the adjoint chirals. The theory is superconformal with the coupling of the
singlet elds being marginally irrelevant leading to them decoupling in the IR as free elds
and the symmetry enhancing in the IR to su(k)
rotating the free elds. The adjoint chiral elds are charged t 2
u(k) with the last factor
1, and the bifundmental
elds between i
1 and i node have charges t 21 i
1 and ptq i. We can close the minimal
puncture giving a vacuum expectation value to one of the baryons weighed t 2 j
Closure of the minimal puncture also entails ipping the rest of the baryons charged under
bifundamental chirals are charged i 1 j, ptq i. The singlets are k
1 nodes and with singlets. The
1 having charges t j
as the chiral adjoint elds, and k
1 chiral elds which ip the baryons and have charges
pq iN j N . The chiral elds couple through either irrelevant or marginally irrelevant terms
the minimal one.
and thus ow to free elds in the IR. The theory in the IR is then a collection of 2k
free chiral elds and the a ne quiver with k
1 nodes (see gure 8).
The anomalies of this model match the anomalies of six dimensional compacti cation
k1 for the u(1)i with i not equal to j. Such
of uxes to
ows. This picture also predicts that the a ne quiver should have loci on
conformal manifold with the
avor group being su(k
u(1)t. This is not
trivial from the Lagrangian of the theory which for general number of branes exhibits
u(1)k symmetry. The index will be consistent with this claim because the a ne quiver has
dualities interchanging minimal punctures. This implies that the index will be invariant
for permutations of i. This permutation symmetry implies that the index can be written
in terms of characters of su(k
1)u(1) j parametrized by i
. Moreover at order pq of the
index (when one takes the free superconformal R charges to chiral elds) the index has
the conformal manifold are such. Thus, we expect the index to be consistent with the
enhanced symmetry.
Gluing maximal punctures with a shift. Let us consider another operation we can
legally do on the eld theory side. Gluing together two maximal punctures of a given theory
we can twist them relatively to each other. The twist corresponds to matching i to i+u
and i with u+i. This breaks the global symmetry to su(gcd(k; u))
These operations are considered for toric quivers in [36]. Performing both operation of
gluing punctures with color di ering by l units and twisting with u we obtain the group,
su(gcd(k; l; u))
We can also consider gluing punctures of same
orientation which will exchange roles of
For the anomaly polynomials that we have computed the e ect of the twist and the
gluing is simple. We only have to remove the parameters which correspond to symmetries
Thus we conclude that the anomaly polynomial is still given by the same expressions. This
also means that the matching between the 6d and 4d analysis persists also for these cases.
Matching with 6d.
We are now in a position to merge the 4d and 6d observations in
this section to a consistent picture. On the 4d side we observed that we can form a torus
by gluing together l free trinions and closing the minimal punctures. When gluing to form
the torus we may also twist the gluing by say u units so that in the last gluing we match
i to i+u. Either of these breaks the global symmetry unless l; n are a multiplet of k.
For simplicity let us consider each of these separately. First we have seen that if we take
polynomial calculations in 4d and 6d agree. The matching, speci cally equation (3.9),
implies that closing a minimal puncture shifts the ux in a U(1) embedded inside one of
the SU(k)'s so that its commutant is SU(k
normalization the
ux for (SU(k)
describes an SU(k)
SU(k) consistent
ux. However if the
uxes of both SU(k)'s are
fractional then we have a nontrivial StiefelWhitney class on the torus that is materialized
through abelian
uxes for both SU(k)'s. This in general breaks the global symmetry to
it's Cartan subalgebra.
case the global symmetry is broken down at least to su(gcd(k; u))
and may be further broken to the Cartan due to the uxes. In this case we identify n with
a nontrivial StiefelWhitney class on the torus that is manifested using at connections for
both SU(k)'s. This naturally accounts for the global symmetry breaking pattern. Also it is
quite natural from the 5d viewpoint. It is also consistent with the matching of the anomaly
polynomial between 6d and 4d as besides the breaking of symmetries this does not e ect
either of the calculations. Note that again if the uxes of both SU(k)'s are fractional then
we have an additional nontrivial StiefelWhitney class so that the total StiefelWhitney
class, which may be trivial, is materialized partially by at connections and partially by
abelian uxes.
broken down at least to su(gcd(k; l)). We identify l with a nontrivial StiefelWhitney class
on the torus that is manifested using
at connections for one SU(k), the broken one, and
uxes for the other. This correctly accounts for the symmetry breaking pattern,
and also agrees with the matching of the anomaly polynomial since the two match where
we have fractional uxes for the unbroken symmetry. Note that the at connections are
necessary so that the total con guration be consistent with (SU(k)
SU(k))=Zk.
Finally we can consider an arbitrary con guration with any value of l and n. This
should correspond to the 6d theory on torus with uxes determined by l through (3.9) and
with an n StiefelWhitney class manifested using at connections. From the discussion
so far it is apparent that the symmetry breaking pattern as well as the matching of the
anomaly polynomial are consistent with this. Note that in the generic case the total
StiefelWhitney class is manifested using both at connections and abelian uxes and may vanish
specify the 6d con guration we must enumerate the StiefelWhitney class in addition to
the uxes.
Acknowledgments
We are grateful to K. Intriligator, H. C. Kim, Z. Komargodski, and C. Vafa for useful
conversations. The work of Y.T. is supported in part by JSPS GrantinAid for Scienti c
Research No. 25870159. The work of Y.T. and G.Z. is supported by World Premier
International Research Center Initiative (WPI Initiative), MEXT, Japan. SSR is a Jacques
Lewiner Career Advancement Chair fellow. The research of SSR was also supported by
Israel Science Foundation under grant no. 1696/15 and by ICORE Program of the Planning
and Budgeting Committee. The Work of I.B. is supported by UC president's postdoctoral
fellowship and in part by DOE grant DESC0009919. A. H. is supported by STFC
Consolidated Grant ST/J0003533/1, and by EPSRC Programme Grant EP/K034456/1.
Anomalies of interacting trinions for general k; N
In section 2, we have obtain the anomaly coe cients of class Sk theories from the anomaly
polynomial of the sixdimensional theory. By using these, we now predict the anomalies of
the trinion models in this section.
genus g Riemann surface
First of all, let us reproduce here the anomalies of the class Sk theory associated to
Tr R0 =
Tr R03 =
Tr R0t2 =
1)(k2(N 2 + N
Tr t =
Tr R02t =
Tr t3 =
We omitted the anomalies involving i and i which depend on the su(k)
for simplicity. By subtracting the contribution of the 3g
3 tubes from these, one would
get the anomalies of 2g
2 trinions from which one can deduce the single contribution.
g 1. There is a duality frame where the fourdimensional theory consists of 2g
= +1, combined
gluing corresponding to the
SU(N )k vector multiplets and the bifundamental chiral multiplets between them. Thus
one tube contributes to the anomalies as
Tr R0 = Tr R03 = k(N 2
1); Tr t = Tr t3 =
Tr R02t = Tr R0t2 = 0:
Tr R0 =
Tr R03 =
Tr R0t2 =
Tr R0 =
Tr R03 =
Tr R0t2 =
Tr R0 =
Tr t = 8;
Tr R03 = 2;
Tr R02t = 4;
Tr R0t2 =
Tr t3 =
Interacting trinions for this case were constructed and studied in [6]. Particularly there are
three theories appearing there that have this u(1)s ux, with di erent su(k)
su(k) uxes.
These were dubbed TA, TB and the so(5) trinion that has a Lagrangian description. In all
three cases the anomalies agree with the result above.
With these anomalies one can obtain those of the trinion with arbitrarily ux Ns
(k2 2)(N 1)+3k(N 2 1)
(N 1)(k2(N 2 +N 1)+2) 3k(N 2 1)
Tr R02t =
Tr t3 =
k2N 3n 3kN 2
(k2 2)(N 1)+3k(N 2 1)
(N 1)(k2(N 2 +N 1)+2) 3k(N 2 1)
Tr R02t =
Tr t3 =
By subtracting these tube contributions from (A.1) and dividing by (2g
the anomalies of the trinion theory with Ns = 12 :
Note that these anomalies are independent of the su(k) su(k) uxes. For example when
N = k = 2, these are
1 can be easily obtained. The answer is simply (A.3) where the signs
of the anomalies involving odd power of t are changed.
We can now subject this construction to the following consistency conditions. Take
1) trinions, each with some value of the u(1)s ux ni, and glue them together. In this
manner we get a genus g Riemann surface with ux Pi ni. Consistency now demands that
1) trinions and 3(g
It is straightforward to show that this is indeed true.
gluings we should recover (A.1) with Ns = Pi ni.
We can further complicate by adding the conjugate trinion. Consider taking 2g 2 a
surface with
uxes ni and a conjugate trinions with
uxes nj to build a genus g Riemann
Pj nj. Punctures of opposite sign are glued together by S
gluing. Now when constructing the Riemann surface we use some combination of
of punctures with a positive sign, which contribute the anomalies in (A.2),
punctures with a negative sign, which contribute the anomalies in (A.2) but with t !
and S gluing which only contributes to the R symmetry anomalies where it gives the same
contribution as in (A.2). In fact we can construct the same theory in di erent ways using
di erent combinations of the above. Particularly say we use b
gluings of punctures with
a negative sign then we must use 3a
punctures with a positive sign. It is straightforward to show that due to the structure of
the contributions b will drop out as required and further that summing all contribution we
indeed recover (A.1).
It is straightforward to generalize this to more complicated cases. First we can
consider trinions with punctures of di erent signs. We can also consider anomalies involving
the u(1) i and u(1) i symmetries. These will be sensitive also to the
uxes under these
symmetries, and to the colors of the punctures. These generalizations should be messy,
but straightforward and we shall not carry them out here.
of certain trinions under the assumption that these trinion have certain symmetries. It
might happen that the puncture symmetries in certain situations are inconsistent with
preserving some of the u(1)k 1
u(1)k 1 symmetries. Evidence for this was found
in [6] where certain trinions were possible to construct only under assumptions that some
of the symmetries are broken. This should be related to issues with discrete
have studied in the previous section and we do not study this question in the current
Let us here mention a caveat of the construction. We try to predict the anomalies
Fluxes for u(1)t symmetry for k; N = 2
In the case of two M5 branes and Z2 singularity we can construct eld theories associated
to non vanishing
ux of u(1)t. These theories do not have a regular Lagrangian
description, rather are described by Lagrangians with parameters
ne tuned in strong coupling
domain. To construct a general model we rst build a theory corresponding to sphere with
two maximal punctures of same color and having a
ux corresponding to u(1)t. The
construction is based on singular Lagrangians one can obtain for models in this class derived
in [6] following [37]. To obtain such a tube model we start with the TA trinion of [6] which
has uxes ( 14 ; 14 ; 1) for (u(1) ; u(1) ; u(1)t) associated to the surface. Then we ip the sign
of one of the punctures by ipping the mesons corresponding to it [5] (see [16, 17, 27, 38, 39]
uxes. Then we close the ipped maximal puncture rst to minimal with vacuum
expectation value for a meson shifting uxes with ( 14 ; 14 ; 12 ). Finally we close the minimal
puncture with vacuum expectation value for a baryon shifting
theory in the end has two maximal punctures of the same color and
ux (0; 0; 2). We can
then insert this theory in our construction of torus models together with free trinions and
close the minimal punctures to obtain theories with
ux for all three symmetries.
We now construct the tube model in more detail. We refer the interested readers for
details to [6] and here we just summarize the construction of the TA trinion. The TA trinion
can be built by taking a sphere with two minimal and two maximal punctures and
(0; 0; 1), which is constructed by combining together two free trinions, and tuning to the
point on the conformal manifold of the model where the abelian symmetries coming from
minimal punctures enhance to su(2)
u(1)c and gauging the su(2) with special choice of
matter. With that choice of matter the u(1)c symmetry enhances also to su(2) and one
obtains additiona su(2) factor rotating the additional matter elds. We thus obtain model
with three factors of su(2)
su(2) symmetry associated to a triplet of maximal punctures.
We can thus begin our procedure of ipping and closing maximal puncture by closing the
puncture of the sphere with two minimal and two maximal punctures to obtain a sphere
with two minimal and one maximal punctures and then perform the gauging needed to
obtain the interacting trinion.
First we ip the sign of one of the maximal punctures. This is done by ipping the
mesons associated to that puncture [5]. We add singlet elds mi and couple them through
a superpotential to the mesons, W = m
M . Next we close the puncture by giving vacuum
expectation values to a particular combinations of m. As the mesons in the sphere with
two minimal and two maximal punctures are built from QQe combinations of chiral elds,
the vacuum expectation values induce mass terms for some of the avors. The sphere with
one maximal and two minimal punctures we obtain thus has Lagrangian in terms of two
su(2) gauge groups each having ve avors and a bunch of singlets.
Let us gure this out in complete detail. The discussion is most easily performed at
the level of the index as it captures all the relevant information. The index of the sphere
with two maximal and two minimal punctures is
Iz;v;a;b = (p; p)2 (q; q)2
We ip the z puncture,
Then we give vacuum expectation values for the meson weighed ptq (z1z2) 1. In the index
. We also need to introduce
new chiral elds coupling them through superpotential [5]. In the index this amounts
to multiplying with
2) ee(( ppttqq 22 2)2)e(ep(tqptq 2 2) . Then we need to give a vacuum
2 2)
expectation value to a speci c baryonic operator which amounts to setting
We also need to ip the second baryon charged under u(1) by multiplying the index with
e(pq 4). After all the above manipulations index of the sphere with two minimal and one
= ( ptq ) 12
maximal puncture we obtain is,
Iv;a;b = e
Ifz1= ;z2= ptq g;v;a;b ;
form mass term and decouple. Thus the index becomes,
Ifz1= ;z2= ptq g;v;a;b = (p; p)2 (q; q)2
Which is the index of two copies of su(2) SQCD with ve avors coupled together through
superpotential terms and bifundamental elds.
Finally the index of the tube theory with ux for u(1)t and two maximal punctures of
the same color is obtained by gauging the su(2) enhances group of u(1)a and u(1)b with
appropriate chiral elds,
Iv;c = e t
(p; p) (q; q)
z 1v1 1 Ic;pzv2;pv2=z :
smoothly becomes
to be one the above index
Ifz1! ;z2! ptq g;c;pzv2;pv2=z j !1 :
Smooth deformation of this sort breaking symmetry might correspond to marginal
deformations. Indeed we do expect to have marginal deformations of this sort which break
puncture symmetries down to the Cartan [6]. We thus have evidence that the tube theory
corresponds to sphere with four minimal punctures and
ux two for u(1)t when we tune
the couplings to point where the abelian symmetries enhance.
The anomalies of this theory are easily computed from Lagrangian implied by this
index. Let us write the trial a anomaly for the model one obtains gluing together Q + Q
Q =2 units of
ux, and 2Qt units of t ux. The anomalies are given,
a =
2Qtst 18s 2 + 18s 2 + 6st2
from which we determine the cubic anomalies which depend on ux of u(1)t,
kttt =
We can compute the dependence of T rR on Qt and get that it is equal to
8stQt which
tells us that Tr u(1)t =
8Qt. All other anomalies do not depend on Qt. The anomalies
here coincide with the ones deduced from (2.11).
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and class S theories: part II, JHEP 12 (2015) 131 [arXiv:1508.00915] [INSPIRE].
and grid diagrams, JHEP 01 (1998) 002 [hepth/9710116] [INSPIRE].
class S theories: Part I, JHEP 07 (2015) 014 [arXiv:1503.06217] [INSPIRE].
[17] E. Nardoni, 4d SCFTs from negativedegree line bundles, arXiv:1611.01229 [INSPIRE].
[18] A. Hanany and K.D. Kennaway, Dimer models and toric diagrams, hepth/0503149
gauge theories, JHEP 01 (2006) 096 [hepth/0504110] [INSPIRE].
toric geometry and brane tilings, JHEP 01 (2006) 128 [hepth/0505211] [INSPIRE].
series, JHEP 01 (2015) 150 [arXiv:1410.1548] [INSPIRE].
10 (2014) 152 [arXiv:1408.4690] [INSPIRE].
theories, JHEP 08 (2005) 024 [hepth/0502043] [INSPIRE].
m = 1,
[2] C.S. Chan , O.J. Ganor and M. Krogh , Chiral compacti cations of 6D conformal theories , [3] K. Ohmori , H. Shimizu , Y. Tachikawa and K. Yonekura , 6d N = ( 1 ; 0) theories on S1 /T2 [4] M. Del Zotto , C. Vafa and D. Xie , Geometric engineering, mirror symmetry and [5] D. Gaiotto and S.S. Razamat , N = 1 theories of class Sk , JHEP 07 ( 2015 ) 073 [6] S.S. Razamat , C. Vafa and G. Zafrir , 4d N = 1 from 6d ( 1 , 0), JHEP 04 ( 2017 ) 064 [7] K. Ohmori , H. Shimizu , Y. Tachikawa and K. Yonekura , Anomaly polynomial of general 6d [8] K. Intriligator , 6d , N = ( 1 ; 0) Coulomb branch anomaly matching , JHEP 10 ( 2014 ) 162 [9] O. Aharony , A. Hanany and B. Kol , Webs of (p,q) vebranes, vedimensional eld theories [10] Y. Tachikawa , Instanton operators and symmetry enhancement in 5d supersymmetric gauge [11] K. Ohmori , H. Shimizu , Y. Tachikawa and K. Yonekura , 6d N = ( 1 ; 0) theories on T 2 and [12] K.A. Intriligator and B. Wecht , The exact superconformal R symmetry maximizes a , Nucl.
[13] A. Hanany and K. Maruyoshi , Chiral theories of class S, JHEP 12 ( 2015 ) 080 [14] S. Franco , H. Hayashi and A. Uranga , Charting Class Sk Territory, Phys. Rev. D 92 ( 2015 ) [15] I. Bah , C. Beem , N. Bobev and B. Wecht , FourDimensional SCFTs from M5Branes , JHEP [16] M. Fazzi and S. Giacomelli , N = 1 superconformal theories with DN blocks , Phys. Rev. D 95 [19] S. Franco , A. Hanany , K.D. Kennaway , D. Vegh and B. Wecht , Brane dimers and quiver [20] S. Franco , A. Hanany , D. Martelli , J. Sparks , D. Vegh and B. Wecht , Gauge theories from [21] B. Feng , A. Hanany and Y.H. He , Dbrane gauge theories from toric singularities and toric duality, Nucl . Phys . B 595 ( 2001 ) 165 [hepth /0003085] [INSPIRE].
[22] A. Hanany and N. Mekareeya , Trivertices and SU ( 2 )'s, JHEP 02 ( 2011 ) 069 [23] S. Cremonesi , A. Hanany , N. Mekareeya and A. Za aroni , T (G) theories and their Hilbert [24] A. Hanany and R. Kalveks , Quiver Theories for Moduli Spaces of Classical Group Nilpotent Orbits , JHEP 06 ( 2016 ) 130 [arXiv:1601.04020] [INSPIRE].
[25] A. Hanany and R. Kalveks , Highest Weight Generating Functions for Hilbert Series, JHEP [26] D. Green , Z. Komargodski , N. Seiberg , Y. Tachikawa and B. Wecht , Exactly Marginal Deformations and Global Symmetries , JHEP 06 ( 2010 ) 106 [arXiv:1005.3546] [INSPIRE].
[27] C. Beem and A. Gadde , The N = 1 superconformal index for class S xed points , JHEP 04 [28] S. Benvenuti and A. Hanany , Conformal manifolds for the conifold and other toric eld [29] I.R. Klebanov and E. Witten , Superconformal eld theory on threebranes at a CalabiYau singularity, Nucl . Phys . B 536 ( 1998 ) 199 [hepth /9807080] [INSPIRE].
[30] D. Forcella , A. Hanany , Y.H. He and A. Za aroni, Mastering the Master Space , Lett. Math.
Phys. 85 ( 2008 ) 163 [arXiv:0801.3477] [INSPIRE].
[31] D. Forcella , A. Hanany and A. Za aroni , Baryonic Generating Functions , JHEP 12 ( 2007 ) [32] E. Witten , Supersymmetric index of threedimensional gauge theory , hepth/ 9903005 [33] E. Witten , Supersymmetric index in fourdimensional gauge theories , Adv. Theor. Math.
[35] G. 't Hooft, A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories, Nucl.
[36] A. Hanany and A.M. Uranga , Brane boxes and branes on singularities , JHEP 05 ( 1998 ) 013 [34] A. Borel , R. Friedman and J.W. Morgan , Almost commuting elements in compact Lie groups , [37] A. Gadde , S.S. Razamat and B. Willett , \ Lagrangian" for a NonLagrangian Field Theory with N = 2 Supersymmetry , Phys. Rev. Lett . 115 ( 2015 ) 171604 [arXiv:1505.05834] [38] D. Gaiotto , L. Rastelli and S.S. Razamat , Bootstrapping the superconformal index with surface defects , JHEP 01 ( 2013 ) 022 [arXiv:1207.3577] [INSPIRE].
[39] P. Agarwal , K. Intriligator and J. Song , In nitely many N = 1 dualities from m + 1