Exceptionally simple exceptional models
Received: September
Published for SISSA by Springer
0 Open Access , c The Authors
1 Haifa , 32000 , Israel
2 Physics Department , Technion
We discuss models with no dynamical vector elds in various dimensions which we claim might have exceptional symmetry on some loci of their parameter space. In particular we construct theories with four supercharges owing to theories with global symmetry enhancing to F4, E6, and E7. The main evidence for these claims is based on extracting information about the symmetry properties of the theories from their supersymmetric partition functions.
Conformal Field Theory; Field Theories in Lower Dimensions

Exceptionally simple exceptional models
1 Introduction 2
Model with E6 symmetry
The moduli space
2.2 3d supersymmetric index
Other partition functions
Model with F4 symmetry
Model with E7 symmetry
5 General properties 5.1
Example: G2
Some of the properties of the xed point, IR or UV, of a general quantum
eld theory are
not obvious from a given nonconformal description. For example, the global symmetry at
the xed point might be enhanced in dimension and/or rank. Such symmetry enhancements
are often encountered when discussing gauge theories in various dimensions. For example,
xed points with
avor Lie group corresponding to the Dynkin diagram [1]. In
xed points with
ENf +1 avor symmetry [2]. The enhancement of symmetry is due to instantons in the latter
case and monopoles in the former. Moreover in some cases the enhancement of symmetry
might only occur on a sub locus of the conformal manifold without obvious explanation
due to nonperturbative e ects. As an example we mention the enhancement to E7 of the
coupled through a quartic superpotential [3].
In this note we discuss certain models with four supercharges constructed from chiral
elds and no vector elds in various dimensions. We will present evidence that, choosing
the superpotentials in a careful way, these models ow to conformal theories, either in the
IR or (potentially) in the UV, with conformal manifolds with possible loci having
exceptional symmetries. The superpotential can be constructed in any theory allowing for four
superpotential will be either relevant or irrelevant leading to xed points with extended
supersymmetry either in the IR or (possibly) UV.
lines are bifundamental chiral elds. We have a superpotential term for each face of the tetrahedron.
The arguments in favor of enhancement of global symmetry are based on analysis
of partition functions. First we show that the partition functions are invariant under the
action of the Weyl group of that symmetry on the parameters, and in case these are indices
can be expanded in characters of the enhanced avor group. Moreover we will present an
argument in three dimensions, generalizing the four dimensional claim [4] that in a certain
order of the expansion of the index one can extract the number of marginal operators minus
the currents. This will let us identify the currents of the enhanced symmetry. The physical
interpretation of this result is that when the partition functions are consistent with the
enhanced symmetry there is a possibility of a locus of parameter space of the theory at
which the symmetry is actually enhanced.
The note is organized as follows. We will rst discuss a simple example of such a model
leading to SO(8)
U(1), and E6
avor symmetry in section two.
We will proceed to deforming the superpotential to obtain a theory with F4 symmetry in
section three. In section four we will consider a deformation of two copies of the theory
avor symmetry leading to a model with E7 symmetry. We will discuss several
general issues following from our construction in section ve.
Model with E6 symmetry
Let us consider a model built from 24 chiral elds. We will organize the elds into six
bifundamentals of SU(2)
SU(2). We will have four di erent SU(2)
avor groups and
denote the chiral elds by Q1, Qe1, Q2, Qe2, X, and Y . The superpotential is given by,
We can encode this superpotential in the tetrahedral quiver diagram of gure 1.
This model has the manifest symmetry SU(2)4
U(1)t. The four SU(2)
symmetries are manifest in the description above and under the U(1)t the Qi and Qei have
charge 12 while X and Y have charge
1. Under U(1)
Qi have charge 1 and Qei have
1 while X and Y have vanishing charge. All elds have R charge 23 . We have also
summarized the various charges, including those for elds we shall introduce later in the
paper, in table 1. The superpotential is irrelevant in four dimensions, and relevant in lower
under the various global symmetries.
dimensions. Thus we will think of the model as owing to an IR
xed point in two and
three dimensions while considering a possible ow to a UV completed xed point in four
First, we claim that the SU(2)4 symmetry here enhances to SO(8). We can check
this by studying di erent supersymmetric partition functions in various dimensions. For
example, in four dimensions the index is given by,1
The interpretation [4] of this result is that if there is a UV xed point for which this is the
index, it has an SO(8)
U(1) avor symmetry ( 28
1 terms in the order pq
of the index corresponding to the conserved currents), and it has a conformal manifold of
dimension 4 preserving this symmetry. Symmetry properties of the 4d index also give the
symmetry of the S3 partition function in three dimensions. In three dimensions this theory
ows to a CFT in the IR. In the next section we will generalize the arguments of [4] to
three dimensions, and show that also the index in three dimensions exhibits this symmetry.
We can write down other partition functions in other dimensions exhibiting the symmetry
(elliptic genus, spheres, indices). The details of the physics will depend on the dimension
but the symmetry will remain.
We can slightly complicate the model by adding more elds. For example, we can add
the elds Z
which are singlets under SU(2)4, have U(1)t charge
1, U(1) charges
and R charge 23 . We couple these elds as,
WSO(10) U(1) = WSO(8) U(1) U(1) + Z+(
1For notations and de nitions of supersymmetric partition functions the reader can consult [5].
This theory has symmetry SO(10)
U(1)t, where SO(8)
U(1) enhances to SO(10).
Giving the example of the index in four dimensions we obtain,
1 1
1 + (10t 1 + 16t 2 )(pq) 3 +
1 + 1050 +
Here we deduce that we have SO(10)
U(1) symmetry (the
45 giving the currents)
and that there is no marginal operator preserving this symmetry.
We can add another eld to enhance the symmetry farther. We add a single eld
which is charged under U(1)t with charge 2 and has R charge 23 , and is a singlet under all
the other symmetries. The superpotential is,
This theory has an E6 avor symmetry. Again in four dimensions the index is,
WE6 = WSO(10) U(1) + (X2 + Y 2 + Z Z+) :
1 + 27(pq) 3 + 3510(pq) 3 + ( 78 + 3003)pq +
Like in the previous cases, this result suggests that if there is a UV
xed point, for which
this is the index, it has an E6 global symmetry. One can also consider the analogue theory
in three or two spacetime dimensions. Particularly, in section 2:2 we shall examine the 3d
index of the analogous three dimensional theory and argue that we can derive a similar
result also for this case. However now the theory is expected to ow to an IR
and the index can be readily interpreted as its supercoformal index.
The fact that the superpotential 2.5 gives rise to E6 symmetry is not surprising. One
can realize this symmetry as the group of transformations xing the determinant of a three
by three hermitian matrix built from octonions. This determinant gives the polynomial
WE6 with very speci c numerical coe cients. Since the supersymmetric partition functions
are insensitive to such parameters we allow ourselves to be agnostic about them in our
The moduli space
The model we presented has a moduli space spanned by the vacuum expectation values
of the scalars in the chiral elds modulo the superpotential constraints. In this subsection
we try to identify this space. The physics data describes it as an algebraic variety of
C27 de ned by 27 quadratic equations. As a rst step we note that the equations are
homogeneous so the moduli space must be a complex cone over another space B.
This structure of the superpotential leads to two interesting features. First, there
should be a conical singularity at the origin. This is expected as there are massless elds
there. Second, there is a natural U(1) action on the cone which we identify as the U(1)R
symmetry of the theory. Indeed all elds have the same Rcharge which agrees with the
U(1) action on the cone.
So now we need to identify the space B which the equations de ne as an algebraic
variety of CP26. We propose that this space is the complex Cayley plane which is a 16
dimensional complex manifold. This space can indeed be de ned as an algebraic variety
of CP26 via 27 quadratic equations [6]. Alternatively it can be de ned as the symmetric
space E6=(SO(10)
U(1)). This de nition manifests its E6 isometry.
We can also provide additional evidence for this identi cation. First the Hilbert series
for the complex Cayley plane was calculated in [7]. The
rst few terms of their results
suggest the space is spanned by functions in the 27 of E6 subject to the condition that the
27 does not appear in their symmetric product. This agrees with the result we observe
from the index.
We can also try to infer the dimension of the manifold from the equations. Say we
choose a nonsingular point on the manifold and expand the equations around this point.
We can then linearize the equations and solve the resulting linear system. The dimension
of the solution space is then the dimension of the manifold. Of course this only works if
we choose a nonsingular point. As the Cayley plane is a symmetric space, if the moduli
space is as we proposed, any point save the origin will do. Say we take all elds to be
; (Q1)22; (Q2)22; (Qe1)22 and (Qe2)22 (we regard the bifundamental chirals as
matrices and use the subscript as an entry in the matrix). It is easy to check that this is
a solution. We then expanded around this solution and found that there is indeed a 17
dimensional solution space in accordance with our picture of the moduli space.2
3d supersymmetric index
We can also look at partition functions in other dimensions, notably the 3d supersymmetric
index and 2d elliptic genus. We shall show that they also can be expanded in characters
of E6. We shall start with the 3d supersymmetric index as for the 3d case the E6 model
leads to a conformal xed point in the IR. For the 3d theory the supersymmetric index is
1 + 27x 32 + 3510x 34 + ( 78 + 3003)x2 +
This is very similar in structure to the 4d index. Again all states fall in characters of E6.
An interesting question is whether we can identify the superconformal multiplets
contributing to the index similarly to the results [4] we stated for the 4d index. For this we
consider the possible short multiplets and their contribution to the index. The shortening
conditions for various 3d superconformal algebras were extensively discussed in [8]. A more
concise summary can be found in [9], and we shall employ their notations for the various
denoted as Q and Q. The superconformal multiplet is then generated by acting with them,
and with the translation generators P , on a superconformal primary.3 Short
representations are those for which the superconformal primary is annihilated by some combination
of Qs and Qs. Due to the superconformal algebra this necessarily
the superconformal primary in term of its Rcharge and angular momentum.
xes the dimension of
2Note that this requires some tuning of the constants appearing in the superpotential.
3These are states annihilated by the generators of special conformal transformations K , and their
fermionic partners S and S.
Shortening conditions
ab QajSCP ib = 0,
(Q)2jSCP i = 0,
abQajSCP ib = 0,
(Q)2jSCP i = 0,
QajSCP i = 0,
QajSCP i = 0,
= j
= r + 1, j = 0, r < 0
= j + r + 1, j
= r + 1, j = 0, r > 0
= r, j = 0, r <
= r, j = 0, r > 12
abQajSCP ib = 0 and abQajSCP ib = 0,
= j +1, j 12 , r = 0 I(2j + 2; 2j + 1)
(Q)2jSCP i = 0 and (Q)2jSCP i = 0,
QajSCP i = 0 and (Q)2jSCP i = 0,
QajSCP i = 0 and (Q)2jSCP i = 0,
= 1, j = r = 0
= 12 , j = 0, r =
= 12 , j = 0, r = 12
I(2 + r + 2j; 2j + 1)
we have adopted the notations of [9] in the naming of the various short multiplets. We have also used
jSCP i for the state associated with the superconformal primary,
for its Rcharge and j for its angular momentum.
for its conformal dimension, r
In table 2 we have summarized the various short representations, shortening conditions
and their contribution to the index. For the index contribution we have de ned,
I(l; s) = ( 1)s
We now study what multiplets can contribute to the 3d index at order xl for l
From table 2 we see that the only multiplets that can contribute are: A2B1, B1A2, LB1 and
A2A2. The multiplets A2B1 and B1A2 are free elds and indeed their combination is the
free chiral multiplet. The LB1 type multiplets are chiral elds and thus their contributions
conserved current multiplet.
From this we see that, similarly to 4d, the x2 order is the marginal operators minus the
conserved currents. Particularly for the E6 model we indeed see a negative contribution,
at order x2, in the adjoint of E6. This supports our claim that this model has an IR
point with E6 global symmetry somewhere on its conformal manifold.
Other partition functions
We can also evaluate the 2d elliptic genus, which is given by,
where P E stands for plethystic exponential. The structure again has some similarities
with the 4d and 3d indices though it contains more terms. Particularly it can be cast in
characters of E6.
We can also calculate other 4d partition functions. For instance, the lens space
inT 2 partition function. The latter is hindered by the fact that it (see
refs. [10{12]) requires integer Rcharges. We can try to correct this by mixing the U(1)R
symmetry with U(1)t which in the S2
T 2 partition function formalism is associated with
adding magnetic
ux on S2. Unfortunately adding the magnetic
ux breaks E6 down to
U(1) subgroup. The resulting partition function depends on the choices of the
magnetic uxes but can be expressed in characters of SO(10)
The lens space index is quite similar to the 4d index, but with more terms, and it in
we consider without holonomies, the lens space index reads,
This di ers from the 4d index by the factor of Fk(p; q) whose exact form is given in [13]. The
expression is inherently written in characters of E6. Adding holonomies under a collection
of U(1)'s will change the factor Fk(p; q) for each chiral eld based on its charges under
these symmetries. Naturally this will break E6. Still we retain the action of the Weyl
group which implies that di erent holonomies, related by the action of the Weyl group,
should have the same index. This again is manifest in the expression as the chiral elds sit
in characters of E6 which ensures they are properly transformed under the action of the
Model with F4 symmetry
Let's return to the SO(10)
U(1) model, and consider reducing the symmetry by enlarging
the superpotential. Speci cally, we consider breaking U(1)t and U(1)
while preserving
U(1)R and the four SU(2) groups. Adding all terms compatible with these requirements
gives the superpotential,
WF4 = WSO(10) U(1) + (Z
This theory has an F4 avor symmetry. Speci cally, we consider the three dimensional
model for which the 3d index is,
1 + 26x 23 + (324 + 1)x 3 + ( 52 + 2652)x2 +
4
We interpret this as the IR
xed point of this model having a conformal manifold with
a point with enhanced symmetry which is F4 in this case. The four dimensional index,
relevant for the 4d model, also has a similar structure but with x2 replaced by pq. Since the
superpotential is irrelevant in 4d, this model is only interesting if there is a UV completed
We can again inquire about the moduli space. The structure of the equations is quite
similar so we again expect the moduli space to be a complex cone over another space B.
The space B can be described as an algebraic variety of CP25 by 26 quadratic equations.
We also expect B to have an F4 isometry. A natural guess is that B is a symmetric space
similarly to the E6 case. This is reasonable as given a solution to the equations we can
generate more solutions by acting with the F4 global symmetry. Assuming this covers all
solutions, the resulting space is a symmetric space given by F4 moded by the symmetry
keeping the solution
plane) and F4=(SU(2)
USp(6)). The rst is 8 complex dimensional space and the second
is 14 complex dimensional space. We next analyze the equations linearized around the
solution where the only nonvanishing
elds are: (Q1)22; (Q2)22; (Qe1)22 and (Qe2)22. We
nd a 15 dimensional solution space.4 This is consistent with the moduli space being a
Model with E7 symmetry
We can use the E6 model to generate a model with E7 global symmetry. To do this we take
56 chiral multiplets and split them into two copies of the 27 chiral elds in the E6 model
and two additional chiral elds P+ and P . The elds interact through the superpotential,
WE7 = P+WE16 + P WE26 + Wint ;
where we use WE16 and WE26 for the superpotential of the E6 model involving chiral elds
from just one of the two copies, these being the rst or second copy respectively. We use
Wint for the most general quartic superpotential involving only the combinations P+P
and products of elds in the rst copy with its image in the second copy.
The classical avor symmetry is SU(2)4
one copy of the 27 transform as before under SU(2)4
U(1)p. Fields belonging to
while the other copy
transforms as the complex conjugate. Under U(1)p copy one has charge
has charge 1. The elds P+ and P
charge 3 and
3 under U(1)p, respectively.
are singlets under SU(2)4
The theory also has a U(1)R symmetry where now the Rcharge of all the elds is 12 .
The superpotential is now quartic so it is irrelevant in four dimensions, marginally irrelevant
in three dimensions and relevant in two dimensions.
We claim that this theory has E7 global symmetry. Again in four dimensions the
1 while copy two
U(1) and carry
1 + 56(pq) 4 + (1463 + 133)(pq) 2 + (24320 + 6480)(pq) 4
+ ( 133 + 293930 + 150822 + 7371)pq +
The three dimensional index also has a similar structure but with pq replaced by x2. If
either the 3d or the 4d models possess a UV completed xed point, then the indices suggest
it should have an E7 global symmetry.
The analogue two dimensional model is expected to ow to an IR
xed point, the
elliptic genus of which, can be cast in characters of E7. Therefore one may also expect this
xed point to have an E7 global symmetry at a point on its conformal manifold.
4This requires some tuning of the constants in the superpotential.
General properties
Finally, we wish to discuss some general properties that emerge from our construction.
Speci cally we seek to summarize the salient features of our construction in a way that
facilitates generalizations to other systems. In general we have a collection of chiral elds
that we choose to form a representation R of a chosen group G, where for simplicity we
consider only a single representation of G. The chiral elds carry charges under the classical
symmetry so that they correctly form the representation R of G.
This generally requires a superpotential to force all elds to carry the desired charges
and eliminates additional symmetries.
We shall limit ourselves to theories with an
Rsymmetry as in these cases we can preform more stringent tests using the superconformal
index. Furthermore, as by assumption all chiral elds form a single representation R of
G, they must have the same Rcharge. The results of these two conditions is that the
superpotential must be a polynomial in the elds of degree r. One obstruction for this
construction is that one must be able to nd the desired superpotential. We can formulate
some necessary conditions using group theory.
First, group theory gives a limitation on the possible values of r. The chiral ring of
the theory is made from the symmetric products of the chiral elds and so is in G
representations appearing in such products. The superotential constraints eliminate chiral ring
elements made from the r
1 symmetric product of the basic chiral elds, and carry charges
in the conjugate representation to R. Thus consistency necessitates that the representation
rSym1R. This in turn constrains r.
R must appear in
For example, for E6 we have
r is 3. Likewise for F4,
again 3. However for E7 we have
2
Sym27 = 3510 + 27 so the minimal possible value of
2
Sym26 = 324 + 26 + 1 so the minimal nontrivial value of r is
2
Sym56 = 1463 + 133 so a cubic superpotential is not
56 so the minimal nontrivial value of r in this case is 4.
An additional condition can then be given using the 4d supersymmetric index (or
as we have seen the three dimensional one). This receives contributions from the chiral
elds modulo the superpotential constraints. A nice feature of the 4d index is that the pq
order receives contributions only from marginal operators, which contribute positively, and
conserved currents, which contribute negatively. Therefore we can look at the negative
terms in the pq order and see whether or note we indeed get the adjoint, and only the
adjoint representation of G. In fact it is straightforward to write the contribution for the
pq order to be:
Example: G2
R. This essentially reduces the problem to group theory:
which representation appearing in the direct product R
R do not appear in
For example, in the E6, F4 and E7 theories the answer to this is indeed only the adjoint
As an illustrating example let's consider the exceptional group G2 and its 7 dimensional
representation. It is convenient to form the chiral elds in representations of the SU(3)
maximal subgroup of G2. Under it the 7 of G2 decomposes as 1 + 3 + 3 so we shall use
3 chiral elds F in the 3 of the classical SU(3), 3 chiral elds F in the 3 and a singlet X.
Next we need to nd a superpotential that limits the elds to these charges. However we
shall now argue that this is not possible.
The superpotential must be SU(3) invariant and so must be made from the meson F F .
Note that baryonic products vanish as the elds are bosonic. This implies that the minimal
nontrivial order for the superpotential is 4, and also that there is an additional U(1) under
which F and F carry opposite charges that we cannot eliminate. The superpotential that
we can add has the form,
W = (F F )2 + F F X2 + X4 :
This leads to a classical U(1)
SU(3) global symmetry under which the elds are charged
. These in fact form the 7 of SO(7) under its U(1)
SU(3) subgroup.
So we conclude that we cannot build a G2 model. Attempting to build one leads to model
with SO(7) global symmetry.
We can also see all these statements materialize just from group theory analysis. First
3
Sym7 = 77 + 7 so indeed the minimal nontrivial order
of the superpotential is 4. Next we look at the conserved currents given by the terms in
the product 7
7 that are not contained in
4
Sym7. Doing the group theory we nd these
to be the adjoint 14 of G2 and the 7. Thus we see that there are additional conserved
currents, which in fact form the adjoint of SO(7) signaling that such a theory must have
a larger global symmetry. So the group theory analysis supports the previous claim that
there is no analogous model wth G2 as its global symmetry.
We would like to thank Nathan Seiberg, Brian Willett and Amos Yarom for useful
comments and discussions. GZ is supported in part by the Israel Science Foundation under
grant no. 352/13, and by the GermanIsraeli Foundation for Scienti c Research and
Development under grant no. 1156124.7/2011. SSR is a Jacques Lewiner Career Advancement
Chair fellow. This research was also supported by Israel Science Foundation under grant
no. 1696/15 and by ICORE Program of the Planning and Budgeting Committee.
Open Access.
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