E 8 orbits of IR dualities
HJE
Haifa 0
Israel 0
Theory, Supersymmetry and Duality
0 Kashiwa , Chiba 2778583 , Japan
1 Kavli IPMU (WPI), UTIAS, the University of Tokyo
2 Physics Department , Technion
We discuss USp(2n) supersymmetric models with eight fundamental fields and a field in the antisymmetric representation. Turning on the most generic superpotentials, coupling pairs of fundamental fields to powers of the antisymmetric field while preserving an R symmetry, we give evidence for the statement that the models are connected by a large network of dualities which can be organized into orbits of the Weyl group of E8. We make also several curious observations about such models. In particular, we argue that a USp(2m) model with the addition of singlet fields and even rank m flows in the IR to a CFT with E7 × U(1) symmetry. We also discuss an infinite number of duals for the USp(2) theory with eight fundamentals and no superpotential.
Duality in Gauge Field Theories; Global Symmetries; Supersymmetric Gauge

E
orbits of IR
1 Introduction 2 3 4
the moment.1 An interesting question one can ask to aid such an understanding is whether
there is any structure relating different theories in a certain universality class.
In this short note we will discuss such a structure in a very particular setup. We
consider USp(2m) gauge theories with eight fundamental chiral fields Qi, a field in the
antisymmetric representation X, and possibly a superpotential with gauge singlet fields.
It so happens that these models are interrelated by a large network of dualities and this
network has intriguing group theoretic structure. In particular there are dualities
relating models with fixed rank but different superpotentials and a non trivial map of
operators/symmetries between various sides of the duality. This duality web forms [9, 10] orbits
of the Weyl group of E7. Here we discuss yet another duality transformation which relates
theories with different rank. In particular a USp(2m) model with superpotential Q2Q1Xn
is in the same universality class as a USp(2n) model with superpotential Q2Q1Xm when
certain singlet fields and superpotentials involving them are added. This duality
transformation was considered implicitly in [9] as a property of integrals which imply equality of
the supersymmetric index of the two dual models. We will argue that turning on most
general superpotentials of the form above, breaking all the flavor symmetry of the gauge
models but preserving R symmetry, this duality transformation together with permutations
1There is though growing evidence that such dualities can be understood by constructing dual four
dimensional theories as geometrically equivalent but different looking compactifications of a six dimensional
model (see for examples [2–8]).
– 1 –
of the eight quarks generates orbits of the Weyl group of SO(16). Then together with the
dualities generating E7 orbits the full duality web is that of orbits of the Weyl group of E8.
The note is organized as follows. We start the discussion with a review of dualities
transforming USp(2m) models on an E7 orbit. We also make a curious observation about
a special property of the USp(2m) with even m. We argue that this model with a
particular superpotential flows to a SCFT with E7 × U(
1
) symmetry. In the particular case of
USp(4) this model sits on the same conformal manifold as the E7 surprise of Dimofte and
Gaiotto [11]. In section three we discuss a duality which relates USp(2m) and USp(2n)
models with superpotentials on both sides. In section four we finish by combining the two
types of dualities and explain how they build orbits of the Weyl group of E8.
are summarized in table 2.1.
ing manifestly the most symmetry is given by the same gauge theory but with a collection
of singlet fields coupled to gauge invariant operators through the superpotential [14],
(2.1)
(2.2)
(2.3)
(2.4)
The charges of the fields are in table 2.3.
The map between the operators is, W =
This is a generalization of IntriligatorPouliot duality [15] for n equals one case. We can
build many other duality frames which will have less symmetry manifest in a similar manner
– 2 –
with superpotential given here,
with the matter content given in table 2.9.
to the n = 1 case. The number of duals is 72 = W (E7)/W (A7). To construct these dualities
we split the eight fundamental fields into two groups of four. The matter content is then
written in table 2.5.
USp(2n)
SU(4)1 SU(4)2 U(
1
)b U(
1
)
U(
1
)r
We have thirty five choices to perform such splitting. One set of thirty five duals is then
given by the following matter content,
USp(2n)
SU(4)1 SU(4)2 U(
1
)b U(
1
)
U(
1
)r
(Mˆ i,j/lqiqj Xe l−1 + Mˆ i′,j/lqj+4qi+4Xe l−1) ,
USp(2n)
SU(4)1 SU(4)2 U(
1
)b U(
1
)
U(
1
)r
1
2
1
2
0
1
2
1
2
0
1
1
2
1
2
0
1
1
The“baryons” map to the singlets and mesons to mesons. This is an analogue of the
duality [16] discussed by Csaki, Schmaltz, Skiba, and Terning. The last two sets of dual
descriptions were considered in [10].
It is convenient to encode the dualities as transformations on an ordered set of
fugacities for the different symmetries. We will parametrize the symmetries as follows using
(2.6)
(2.5)
(2.7)
(2.8)
(2.9)
Q1,...,4 2n
Q5,...,8 2n
X
– 3 –
1
−1
0
tributes to the index as Γe((qp) r2 h) [17, 18] with h being the fugacity of the U(
1
)
symmetry under which the field transforms. We will then denote by ui = (qp) 4 hit− n−41
1
the
weight for the ith quark. Here the fugacity t is for the U(
1
) symmetry, and fugacities
hj for SU(8) (then Q8
with one constraint coming from anomaly cancelation, t2n−2 Qj uj = (qp)2.
We define
an ordered set of fugacities parametrizing the gauge sector of the USp(2n) theory to be
(u1, u2, u3, u4, u5, u6, u7, u8, n, t). Denote u4+ = Q4
j=1 uj and u4− = Qj8=5 uj. Then the three
dualities we discussed imply the following transformations of this set,
k=1 hk = 1). We can take the parameters t and uk to be general
HJEP1(207)5
→
→
(u1, u2, u3, u4, u5, u6, u7, u8, n, t) →
(u1, u2, u3, u4, u5, u6, u7, u8, n, t) →
(u1, u2, u3, u4, u5, u6, u7, u8, n, t) →
u+u− , u+u− , u+u− , u+u− , u+u− , u+u− , u+u− , u+u− , n, t
u1
u2
u3
u4
u5
u6
u7
u8
u+ , u2+ , u2+ , u+ , u2− , u− , u2− , u− , n, t
2 2 2 2
u1, uu−+
u2, uu−+
u3, uu+−
u4, uu+−
u5, uu+−
u6, uu+−
u7, uu+−
u8, n, t
(2.10)
These transformations together with permutations of the first eight terms generate the
Weyl group of E7.
It has been observed by Dimofte and Gaiotto [11] that combining two copies of USp(
2
)
theories by coupling the gauge invariant operators as QjQiqjqi, the theory has a point
on the conformal manifold in the IR with, at least, E7 symmetry. This fact was related
in [8] to a statement that the two copies of USp(
2
) with that superpotential can be
obtained by compactification of the E string theory on a torus. We will now discuss here a
generalizations of the former fact to higher rank.
2.1
E7 × U(
1
) surprise
Consider T(0m) and assume m even. Turn on superpotential,
m/2
j=1
X QlQiXj−1Mil/j + X Xixi .
m
i=2
We claim this model is self dual under the dualities we have considered. Note that under all
the dualities it is either that QiQjXl−1 maps to itself or to Mij/n−l. The powers of X map
to the same powers on the dual side. This implies that the superpotential maps to itself
under the three dualities. The only effect of the duality is the non trivial identification of
symmetries. These imply that for example the protected spectrum is invariant under th
Weyl group of E7 and thus forms representations of E7. It is then plausible that on some
point on the conformal manifold of this model the group enhances to U(
1
) × E7. Let us
analyze one example in detail.
– 4 –
Consider the USp(4) theory with the mesons and X2 flipped.2 That is the
superpotential is QlQmMml + X2x with M and x gauge singlet fields. The superconformal R
symmetry derived by a maximization [21] assigns R charge zero to X and R charge half
to the quarks. The superconformal (a, c) anomalies coincide with two copies of SU(
2
)
theories glued together with a superpotential, the E7 surprise model. This suggests that the
USp(4) model sits on the same conformal manifold. In particular as the superconformal
R charge of X is zero,3 giving it a vacuum expectation value takes us on the conformal
manifold of that model.4 Giving such an expectation value Higgses the USp(4) gauge group
to SU(
2
)2. This generates the E7 surprise model if the mesons are flipped. We thus expect
the USp(4) model to have E7 symmetry somewhere on its conformal manifold. The model
Here λ is the gaugino. The fields ψi are fermionic partners of Qi and the fields ψL are
fermionic partners of fields L with L being one of (M, X, x). The operator ψ¯iQj forms the
63 + 1 of SU(8) × U(
1
) and to construct the adjoint of E7 we also need the 70, fourth
completely antisymmetric power of the 8, in addition to the 63. The operators QiQlMkl
and ψ¯kMmMij cancel each other in the index computation as a consequence of the chiral
ring relation. The operator QiQj QkQnX is in the representation 378 + 336 of SU(8). In
particular this lacks the rank 4 antisymmetric representation of SU(8). This arises as to
get it one has to contract the Q’s antisymmetrically in the flavor index, but, since these are
bosonic fields, they must then be contracted antisymmetrically also in the gauge indices.
However the fourth totally antisymmetric product of the 4 of USp(4) is a singlet, and so the
product with X then cannot be made gauge invariant. The operator ψ¯lMk QiQj X is in the
28 × 28 = 336 + 378 + 70. The operator which gives us the required −70 is then obtained
2Flipping, that is introducing chiral fields φO and coupling them to a theory as φOO with O being an
operator to be removed in the IR, is a standard technique in CFT. For recent discussion of some aspects of
this procedure see [19, 20].
3
This will cease to be the case for higher rank, and in particular there will be no conformal manifold.
4More specifically, an expectation value for X is forbidden by both the Fterm and Dterm conditions.
from QlQmXψ¯iMk . The relevant operators in the 56 are constructed from Mij which form
the 28 of SU(8) and from XQiQj which forms the 28. Both operators have charge 21 under
the U(
1
). We note that the fact that at order qp we see −133 − 1, assuming that the theory
flows to interacting SCFT in the IR, is a proof, following from the superconformal
representation theory [22], that the symmetry of the fixed point enhances to at least U(
1
) × E7.
3
Rank changing duality
We consider a deformation of T(0n) by a superpotential term,
Wmn = Q1Q2Xm .
(3.1)
to T(m) with the additional superpotential and singlet fields,
n
The theory then will be denoted by T(mn).5 We claim that if m is bigger than n T(mn) is dual
ΔWnm =
xkXe k +
m
X
k=n+1
m−n
X
X
the same across the duality. The charges on the USp(2n) side are detailed in table 3.3.
The map of the operators is as follows,
Xj → Xe j ,
Q1,2Ql>2Xj−1 → q1,2ql>2Xe j−1 ,
j ≤ m − n Q1Q2Xj−1 → xm−j+1 ,
j ≤ n − 1
(Q4)ckXj−1 → Mck/n−j .
5Note that because of the anomaly condition, the superpotential Q3Q4Q5Q6Q7Q8X2n−2−m has R charge
4 minus the R charge of Wmn and opposite charges under other symmetries. This implies that at least one
of these operators is relevant if m < 2n − 2 and that the two terms are marginal in IR.
− 1 n
2
− 16 (2m − n − 2)
1
−y
− 31 (n − 2m + 3y − 1)
i, l 6= 1, 2
QiQlXj → qiqlXe m−n+j ,
j > m − n Q1Q2Xj−1 → q1q2Xe j−1−m+n ,
1
6
15
1
1
6
2
1
1
1
1
(q1, q2) 2m
q3,...,8
X
e
M/y
xy
2m
1
1
USp(2m)
SU(6) SU(
2
) U(
1
)
m(2m − 1) − 1 1
On the USp(2m) side we obtain the charges in table 3.4. – 6 –
The supersymmetric index of the two sides of the duality agrees as was shown by Rains [9].
The fact that the index agrees guarantees in particular that the anomalies agree and that
the protected operators map to each other. Nevertheles, let us detail the anomalies here.
We can encode anomalies involving abelian symmetries in the trial c and a anomalies.
Defining R = R′ + sq with R′ and q the R symmetry and the U(
1
) charge in the tables
above, with s a parameter, the conformal anomalies are,
a(s) =
c(s) =
1
32
1
32
s3 −3 m2 +4m−2 n2 +6(m+2)n3 −(m−1)2(4m+5)n−4n4 −9
(3.6)
−3 s2 2m2 +8m−1 n+(2−8m)n2 +8n3 −9 +12s 2mn+n2 +n−2 +4n+6
HJEP1(207)5
s3 −3 m2 +4m−2 n2 +6(m+2)n3 −(m−1)2(4m+5)n−4n4 −9
−3 s2 2m2 +8m−1 n+(2−8m)n2 +8n3 −9 +2s 3(4m+1)n+8n2 −11 +16n+4 .
A more symmetric way to think about the duality is to define the model T(0n) with the
superpotential given by the following.
n
k=2
n
We denote this theory as I¯(0n) and the model with Q1Q2Xm as I¯(mn). We then claim that
Let us define the index of a theory I¯(n) to be Inm(u1, u2, · · · , u8, t) with the condition
m
coming from anomalies t2n−2 Q8
i=1 ui = p2q2. Here again ui are the weights of the quarks
as defined in the previous section and t is the fugacity for the U(
1
) under which the
antisymmetric chiral field is charged. Turning on the superpotential Q1Q2Xm we identify
2 = pq ut2−un1 , the duality implies that the index satisfies the
Inm(u1, u2, u3, u4, u5, u6, u7, u8, t) = Imn u1u, u2u, u3 , u4 , u5 , u6 , u7 , u8 , t .
(3.8)
u u u u
u u
This can be derived from the map of symmetries between the two dualities.
3.1
Duals of SU(2) SQCD with four flavors
Consider an example of the rank changing duality with n = 1 and general m. The theory
on one side is always USp(
2
) = SU(
2
) SQCD with eight fundamental chiral fields and no
superpotential. On the dual side we have a USp(2m) model with superpotential W = q2q1Xe
and other terms involving singlet fields we have discussed. We then have an infinite number
of duals for the SU(
2
) theory with eight fundamental chiral fields. The fact that this model
has an infinite number of duals is not surprising [4], but the surprising point is that the
duals are rather simple.
Let us work out a simple example. We consider m to be two. At the fixed point of
the gauge theory with no superpotential the operator q2q1Xe has R charge 1.15749 and
thus is relevant. The operator Xe2 violates the unitarity bound and needs to be decoupled
by introducing a flip x appearing in the superpotential as xXe2. After decoupling the Xe2
– 7 –
operator we get that the R charge of Xe q1q2 is 1.15331. We turn it on and flow to a new
fixed point. At that point the operators (i, j 6= 1, 2) qiqj violate the unitarity bound and
need to be decoupled by introducing flippers. After this there are no operators violating
unitarity bounds and we get precisely the superpotential we obtain from the duality. Thus
USp(4) theory with the superpotenial flipping Xe 2 and qiqj (i, j 6= 1, 2), and superpotential
term q2q1Xe flows to USp(
2
) with no superpotential. For low values of m we can repeat
such an analysis though for higher values it becomes rather intricate.
4
The duality orbit
We consider the more general superpotential for a USp(2a9) theory,
HJEP1(207)5
Wa9;a1,··· ,a8 =
X QiQj Xai+aj .
i6=j
This superpotential breaks all the flavor symmetry but preserves the R symmetry.6 The
parameters aj are either all integer or all half integer. The R charges are,
rQi = 1 − airX ,
rX =
4
2 − 2a9 + P8
j=1 aj
.
Some operators violate the unitarity bounds for general choices of ai and need to be
decoupled. We define the parameters ui and t to be as in previous sections,
– 8 –
1
Note also that now t = (qp) 2 rX . The index is given by
ui = (pq) 12 t−ai .
Iaa91+a2 ((pq) 12 t−a1 , · · · , (pq) 12 t−a8 , t) .
The duality of the previous section then implies that this index is equal to,
Iaa19+a2 (pq) 12 t− 21 (a1−a2+a9), (pq) 12 t− 21 (a2−a1+a9), (pq) 12 t− 21 (2a3+a2+a1−a9), · · · , t
We parametrize again the theory by the nine numbers, which here are integers (or half
integers), (a9, a1, a2, a3, a4, a5, a6, a7, a8). The duality transforms
(a9, a1, a2, a3, a4, a5, a6, a7, a8) →
→
a1 + a2, 21 (a1 − a2 + a9), 1 (a2 − a1 + a9), (2a3 + a1 + a2 − a9), · · · .
1
2 2
The transformations with permutations of the last eight numbers generate the Weyl group
of SO(16).
8 a
6Here we assume that a9 6= 1 + Pj=1 2j . If this is not true then there is no R symmetry, but instead
there is an anomaly free U(
1
) global symmetry. We shall not discuss this case in any detail.
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
We can also act with the duality generating the E7 orbit. Denote a
i=5 ai, a = 12 (a− + a+), a′ = 21 (a− − a+). Then the three dualities generating the
E7 orbit imply the following transformations of this set,
(a9, a1, a2, a3, a4, a5, a6, a7, a8) →
→ (a9, a+ − a1, a+ − a2, a+ − a3, a+ − a4, a− − a5, a− − a6, a− − a7, a− − a8) →
→ (a9, a − a1, a − a2, a − a3, a − a4, a − a5, a − a6, a − a7, a − a8) →
→ (a9, a′ + a1, a′ + a2, a′ + a3, a′ + a4, −a′ + a5, −a′ + a6, −a′ + a7, −a′ + a8) . (4.7)
These transformations and permutations of last eight elements generate the action of the
invariant under all the transformations.
The transformations can generate a full orbit of the Weyl group of E8 for general values
of the parameters. For special values they might not act faithfully and generate smaller
orbits. For example take all ai for i = 1 . . . 8 to be 21 a9. Then the SO(16) transformations
become self dualities.
There is an issue of whether or not the theories defined via the superpotential (4.1) are
indeed all unique and define a nontrivial SCFT. The basic problem is that the
superpotentials seem irrelevant in the IR so one may fear that the RG flow they generate is trivial.
A related issue is that for generic values of ai there will be operators with R charge below
the unitarity bound. To deal with the latter problem we can add, to all sides of the duality,
flipping fields that remove these operators. The exact number of operators required may
depend on the value of 2−2a9 +P8
i=1 ai, but one can still perform this action so as to result
in an orbit where all operators are above the unitarity bound. Here it is important that
the R symmetry is fixed so this does not lead to any flow that may cause more operators
to go below the unitarity bound. Therefore this statement can be phrased as an identity
between a collection of theories where all operators are above the unitarity bound, and
so there is no contradiction with them flowing to an SCFT. Of course we cannot rule out
the possibility that in some cases the flow may be trivial and the superpotential are truly
irrelevant rather then dangerously irrelevant.
It will be interesting to understand whether the fact that there are theories residing
on an orbit of the Weyl group of E8 implies that there is a model with E8 symmetry, and
in which particular way this fact is related to compactifications of six dimensional (
1, 0
)
models.
Acknowledgments
We would like to thank O. Aharony, D. Gaiotto, K. Intriligator, H. C. Kim, Z. Komargodski,
and C. Vafa for useful comments and discussions. GZ is supported in part by World Premier
International Research Center Initiative (WPI), 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.
– 9 –
Open Access.
This article is distributed under the terms of the Creative Commons Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
HJEP1(207)5
JHEP 06 (2012) 005 [arXiv:1203.0303] [INSPIRE].
JHEP 10 (2015) 035 [arXiv:1505.00255] [INSPIRE].
[4] P. Agarwal, K. Intriligator and J. Song, Infinitely many N = 1 dualities from m + 1 − m = 1,
[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
[15] K.A. Intriligator and P. Pouliot, Exact superpotentials, quantum vacua and duality in
supersymmetric Sp(Nc) gauge theories, Phys. Lett. B 353 (1995) 471 [hepth/9505006]
[16] C. Csa´ki, M. Schmaltz, W. Skiba and J. Terning, Selfdual N = 1 SUSY gauge theories,
Phys. Rev. D 56 (1997) 1228 [hepth/9701191] [INSPIRE].
[17] J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An index for 4 dimensional super
conformal theories, Commun. Math. Phys. 275 (2007) 209 [hepth/0510251] [INSPIRE].
[math.QA/0309252].
6d N = (
1, 0
) on a torus with fluxes, JHEP 06 (2017) 022 [arXiv:1702.04740] [INSPIRE].
theories with an antisymmetric tensor, Nucl. Phys. B 487 (1997) 128 [hepth/9607210]
[arXiv:0801.4947] [INSPIRE].
chiral ring stability, arXiv:1706.02225 [INSPIRE].
dimensions, arXiv:1706.04949 [INSPIRE].
[1] N. Seiberg , Electricmagnetic duality in supersymmetric nonAbelian gauge theories , Nucl. Phys. B 435 ( 1995 ) 129 [ hep th/9411149] [INSPIRE].
[2] D. Gaiotto , N = 2 dualities , JHEP 08 ( 2012 ) 034 [arXiv: 0904 .2715] [INSPIRE].
[3] I. Bah , C. Beem , N. Bobev and B. Wecht , Fourdimensional SCFTs from M 5branes , [7] I. Bah , A. Hanany , K. Maruyoshi , S.S. Razamat , Y. Tachikawa and G. Zafrir , 4d N = 1 from [8] H.C. Kim , S.S. Razamat , C. Vafa and G. Zafrir , Estring theory on Riemann surfaces , [9] E.M. Rains , Transformations of elliptic hypergometric integrals , Ann. Math. 171 ( 2010 ) 169 [10] V.P. Spiridonov and G.S. Vartanov , Superconformal indices for N = 1 theories with multiple duals , Nucl. Phys. B 824 ( 2010 ) 192 [arXiv: 0811 . 1909 ] [INSPIRE].
[11] T. Dimofte and D. Gaiotto , An E7 surprise , JHEP 10 ( 2012 ) 129 [arXiv: 1209 .1404] [12] K.A. Intriligator , R.G. Leigh and M.J. Strassler , New examples of duality in chiral and nonchiral supersymmetric gauge theories , Nucl. Phys. B 456 ( 1995 ) 567 [ hep th/9506148] [13] K.A. Intriligator , New RG fixed points and duality in supersymmetric Sp(Nc) and SO(Nc) gauge theories , Nucl. Phys. B 448 ( 1995 ) 187 [ hep th/9505051] [INSPIRE].
[14] C. Csa ´ki, W. Skiba and M. Schmaltz , Exact results and duality for Sp(2N ) SUSY gauge [18] F.A. Dolan and H. Osborn , Applications of the superconformal index for protected operators and qhypergeometric identities to N = 1 dual theories , Nucl. Phys. B 818 ( 2009 ) 137 [19] S. Benvenuti and S. Giacomelli , Supersymmetric gauge theories with decoupled operators and [20] S. Benvenuti and S. Giacomelli , Abelianization and sequential confinement in 2 + 1 [21] K.A. Intriligator and B. Wecht , The exact superconformal R symmetry maximizes a, Nucl . Phys. B 667 ( 2003 ) 183 [ hep th/0304128] [INSPIRE].
[22] C. Beem and A. Gadde , The N = 1 superconformal index for class S fixed points ,