SciPost Phys. 14, 122 (2023)

Towards classification of 5d SCFTs: Single gauge node
Patrick Jefferson1,2 , Hee-Cheol Kim1,3 , Cumrun Vafa1 and Gabi Zafrir4
1 Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA
2 Center for Theoretical Physics, Department of Physics, Massachusetts Institute ofTechnology, Cambridge, MA 02139
3 Department of Physics, POSTECH, Pohang 37673, Korea
4 Kavli IPMU (WPI), UTIAS, the University of Tokyo, Kashiwa, Chiba 277-8583, Japan

Abstract
We propose a number of apparently equivalent criteria necessary for the consistency ofa 5d SCFT in its Coulomb phase and use these criteria to classify 5d SCFTs arising from agauge theory with simple gauge group. These criteria include the convergence of the 5sphere partition function; the positivity of particle masses and monopole string tensions;and the positive definiteness of the metric in some region in the Coulomb branch. We
find that for large rank classical groups simple classes of SCFTs emerge where the bounds
on the matter content and the Chern-Simons level grow linearly with rank. For classicalgroups of rank less than or equal to 8, our classification leads to additional cases which
do not fit in the large rank analysis. We also classify the allowed matter content for all
exceptional groups.
Copyright P. Jefferson et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 24-12-2022
Accepted 29-03-2023
Check forupdatesPublished 22-05-2023
doi:10.21468/SciPostPhys.14.5.122

Contents
1 Introduction

2

2 Effective Prepotentials on the Coulomb branch

4

3 New Criteria
3.1 Main conjecture
3.2 Additional conjectures88
14

4 Classification of 5d CFTs
4.1 Classification overview
4.2 Standard theories
4.2.1 SU(N ) gauge theories
4.2.2 S p(N ) gauge theories
4.2.3 SO(N ) gauge theories
4.3 Exceptional theories with rank ≤ 8
4.4 Exotic exceptional theories

16
16
18
18
21
23
24
28

5 6d Theories on a Circle

32
1

SciPost Phys. 14, 122 (2023)

5.1
5.2

Generic S p(N ) and SU(N ) with fundamental and antisymmetric matter
Generic SO(N ) and SU(N ) with symmetric matter

32
36

6 Discussion and Future Directions

39

A Mathematical Conventions
A.1 Root and coroot spaces
A.2 Changing from Dynkin to orthogonal basis
A.3 Lengths of weights, index of a representation

41
41
42
43

B Bound on Matter Representation Dimension

43

C Convergence of S 5 partition function
C.1 The 5d partition function

44
45

References

47

1

Introduction

One of the main achievements of string theory has been a deeper understanding of quantum
field theories. We have learned, in particular, of the existence of non-trivial superconformal
field theories, not only in dimensions d ≤ 4 but also in d = 5, 6. Studying the cases withd > 4 is interesting for a number of reasons: These theories turn out to involve, not onlyinteractions of massless particles, but also of tensionless strings, which are quite novel. Higher
dimensional SCFTs also seem to be simpler and in particular classifiable—in particular, the(2, 0) SCFTs in d = 6 are believed to be classified by ADE [1] and a classification for the d = 6,
(1, 0) SCFTs has been proposed [2–4]. Another motivation for studying cases with d > 4
is that higher dimensional theories can lead to lower dimensional ones by compactifications
and flowing to the IR. This relationship between lower dimensional theories and their higher
dimensional ‘parents’ also leads to the emergence of dualities in lower dimensional theories asa manifestation of the different possible ways of assembling the intermediate manifold fromelementary pieces.
It is thus natural to ask if we can also classify the d = 5 SCFTs. String theory suggeststhat this class of theories can be obtained geometrically as M-theory on non-compact CalabiYau 3-folds (CY3 ) where some compact 4-cycles have shrunk to a point. Unfortunately atpresent these singularities are not fully classified, although some progress has been made inthis regard [5]. Rather than attempting to classify these theories in terms of singularities, we
instead take a clue from the classification of 6d (1,0) theories, for which various geometricconditions were ultimately distilled into constraints on gauge theoretic data. It was found thatthe allowed 6d (1,0) theories can be viewed as generalized quiver theories. In that case, theclassification was split into two parts: first, a classification of the quiver node types (i.e. thesingle node case), followed by a specification of the rules according to which the quiver nodes
can be connected. There are a few exceptional cases where the quiver nodes had no gauge
theory associated with them, which were nevertheless reachable by Higgs branch deformationsof some gauge theory. One may hope that a similar story can play out in the case of 5d
SCFTs. An example of this structure is the class of SU(2) gauge theories coupled to matterstudied in [6–10], where it was found that SU(2) gauge theories with up to 7 fundamental

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SciPost Phys. 14, 122 (2023)

hypermultiplets can arise in a SCFT and are related to del Pezzo surfaces shrinking inside a
CY3 . The case of P2 is the only case which is not realizable directly in terms of gauge theory(formally, this theory could be interpreted as ‘SU(2) with -1 fundamental hypermultiplets’).
But even this case can be obtained from gauge theory by starting from SU(2) without matterand theta angle θ = π, and passing through a strong coupling point where some masslessmodes emerge, are given mass, and integrated out [7, 11]. In a sense, this is similar to the
exceptional cases in the 6d (1,0) classification.
Motivated by this analogy, we thus take seriously the idea of using gauge theoretic constructions as a window into the classification all 5d SCFTs. This approach was pioneered by
Intriligator, Morrison, and Seiberg (IMS) in [8]. The idea in that work is that a weakly-coupledgauge system in its Coulomb phase can emerge in the IR as a deformation of an SCFT, so aclassification of allowed IR theories would translate to a classification of UV fixed points, whichcorrespond (in gauge theoretic terms) to the regime where the gauge coupling is infinite and all
Coulomb branch parameters are set to zero. The assumption in [8] was that such an emergent
IR gauge theory should be physically sensible for all values of the Coulomb branch parameters.
More precisely, the condition that the vector multiplet scalars have a positive definite kinetic
term on the entire Coulomb branch was imposed. This condition is equivalent to requiring thatthe matrix of effective couplings on the Coulomb branch is positive semi-definite. Using thiscriterion, quiver gauge theories were ruled out and theories with a simple gauge group coupled to matter were fully classified. However, it was found using geometric constructions [10]as well as brane constructions in Type IIB [12] that not only are quiver gauge theories allowed, but also that these quiver theories are sometimes dual to gauge theories with a simple
gauge group. The loophole in the positivity condition of [8] ends up being the assumption thatthe gauge theoretic description should be valid for the entire Coulomb branch, since it could
nevertheless be possible that for some values of the Coulomb branch parameters massless nonperturbative states emerge that introduce boundaries where the effective description breaks
down. In some cases, these boundaries can be crossed at the expense of introducing a different
description of the physics. Indeed, by using brane constructions and instanton analysis, newexamples of gauge theories incompatible with the IMS criterion were found in [13–17], evenin cases involving simple gauge group.
We propose a modification of the IMS criterion that only imposes the condition of positive
definiteness of the metric to a subregion of Coulomb branch, rather than the entire Coulomb
branch. However, it turns out that there are additional conditions that need to be checked:
for example, 5d N = 1 gauge theories theories have monopole strings, and the regions on
the Coulomb branch where these strings become tensionless should be regarded as parts of aboundary where the effective gauge theory description breaks down. This is to say there may
exist regions for which the metric is positive definite but some string tensions are neverthelessnegative, which indicates that those regions are not physically sensible from the standpoint ofa given effective description.
Therefore, the most natural criteria necessary for the consistency of allowed gauge theories are positive semi-definiteness of the metric on some subregion of the Coulomb branch,
and furthermore positivity of masses and tensions of all physical degrees of freedom in that
region. Even though this seems to be the most natural set of criteria, we find that there arethree apparently simpler criteria, each of which we conjecture is equivalent to the previous
statement: 1) the metric is positive-definite somewhere on the Coulomb branch, even if thestring tensions are negative; 2) the tensions are positive in some region irrespective of metric positivity, as it seems that positivity of the metric in the same region automatically holds;
and 3) the perturbative prepotential is positive on the entire Coulomb branch. The motivation
for this last criterion is the convergence of the five-sphere partition function, which requires
positivity of the perturbative prepotential. However, the surprise is that to get an equivalent

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SciPost Phys. 14, 122 (2023)

condition to the other criteria, we need to check this positivity of the prepotential even in the
excluded regions where the naive tensions or metric eigenvalues may be negative. Why allthese criteria are equivalent to the existence of a region of Coulomb branch admitting positive
definite metric and positive mass (and tension) degrees of freedom is unclear, but we havefound no counterexamples to this assertion in the numerous examples we have explored.
The organization of this paper is as follows. In Section 2, we review some relevant facts
concerning 5d N = 1 supersymmetric gauge theories. In particular, we explain in detail ourassertion that the IMS criterion is incomplete. In Section 3, we propose our new (necessary)
criteria for the existence of 5d SCFTs. We present our criteria in the form of a series of conjectures we believe to be equivalent. In Section 4, we classify non-trivial SCFTs with simple gauge
group using our conjectures. We record the complete classification in a collection of several
tables indicating the gauge group, matter content, classical Chern-Simons level (if applicable),and global symmetry groups. In Section 5, we discuss the connection between 6d SCFTs compactified on a circle and a class of 5d gauge theories we have identified. Finally, in Section
6, we conclude with a summary of our results and prospects for future research associatedto this project. We summarize our conventions relevant to representation theoretic aspects ofthis paper in Appendix A; in Appendix B, we explain our rationale for excluding certain matter
representations from our classification; and in Appendix C we discuss the five-sphere partition
function along with its precise relation to the perturbative prepotential.

2

Effective Prepotentials on the Coulomb branch

We begin by presenting salient features of supersymmetric gauge theories in five-dimensions.
The miminal SUSY algebra in 5d is the N = 1 algebra, which contains eight real supersymmetries. A large class of 5d N = 1 gauge theories can be engineered by (p, q) five-branewebs in Type IIB string theory [12, 18, 19] and also by M-theory compactified on Calabi-Yau
threefolds [7–10].
A 5d gauge theory with a gauge group G has a Coulomb branch of moduli space where thereal scalar field φ in the vector multiplet Φ takes nonzero vacuum expectation values in the
Cartan subalgebra of the gauge group G. On a generic point of the Coulomb branch, therefore,
the gauge symmetry is broken to its maximal torus U(1) rG where rG = rank(G), and therefore the Coulomb branch C is isomorphic to a fundamental Weyl chamber, C ∼= R rG /W (G),
where W (G) is the Weyl group of G. The low-energy abelian theory is governed by a prepotential F (φi ), which is a cubic polynomial in φi . The perturbative quantum correction to the
prepotential is exactly determined by an explicit one-loop computation [20]. With matter hypermultiplets in generic representations R f , the exact prepotential of the low-energy abelian
theory is given by [6, 8]
!
X
X X
1
1
κ
F (φ) = 2 hi j φi φ j + di jk φi φ j φ k +|e · φ|3 −|w · φ + m f |3 ,(1)
612 e∈root
2g0
f w∈R
f

where g0 is the classical gauge coupling, hi j = Tr(Ti T j ), m f are the hypermultiplet masses, κ isthe classical Chern-Simons level, and di jk = 12 TrF (Ti {T j , Tk }), where F denotes the fundamental representation. Note that di jk is non-zero only for G = SU(N ≥ 3). Gauge invariance reP (3)(3)
quires the classical Chern-Simons level to be quantized as κ+ 21 f cR ∈ Z where cR is the cuff(3)

bic Dynkin index of the representation R f defined by the relation TrR f (Ti {T j , Tk }) = 2cR di jk .
f
The last two terms in parentheses are one-loop contributions coming from, respectively, themassive vector multiplets and massive hypermultiplets on the Coulomb branch. Due to the4SciPost Phys. 14, 122 (2023)

absolute values in the prepotential, the Coulomb branch is divided into distinct sub-chambers,
and the prepotential takes different expression in each sub-chamber.
The effective gauge coupling is given by a second derivative of the prepotential, and themetric on the Coulomb branch is set by the effecitve gauge coupling:−2= ∂i ∂ j F , ds2 = (τeff )i j dφi dφ j .
(2)(τeff )i j = geffij
The BPS spectrum for 5d N = 1 theories includes gauge instantons, electric particles andmagnetic monopole strings. The central charges of the electric particles and monopole strings
are, respectively
Ze =

rG
X

n(i)
e φi + m0 I ,

Zm =

i=1

rG
X

n(i)
m φ Di ,

(3)

i=1

(i)
where n(i)e , nm ∈ Z, I is the instanton number, and the dual coordinates φ Di are defined by
way of the prepotential:

φ Di = ∂i F (φ) .

(4)
( j)

Bearing in mind that not every choice of integer coefficients n(i)e , nm in (3) corresponds tothe central charge of a physical BPS state, we note that for any physical choice of coefficients
( j)
n(i)
e , nm the corresponding particle masses and monopole string tensions should be positive.
Geometrically, because the tension of a monopole string is proportional to the volume of a4-cycle, we can expect that there are at least as many independent monopole strings as there
are 4-cycles. The number of independent monopole strings is therefore bounded below by
rG . At low energies, we expect that the number of independent strings is equal to the numberof independent normalizable Kähler classes, implying that there are precisely rG independent
monopole tensions. We note that it is always possible to choose a coordinate basis for which
the independent physical string tensions are
Ti (φ) ≡ ∂i F (φ) = φ Di ,i = 1, . . . , rG .

(5)

In this paper, we will use the Dynkin basis (i.e. the basis of fundamental weights), in which we
expect the tensions of rG independent monopole strings are captured by the above expression.
The rationale for this choice is that a classical string tension in the Dynkin basis is a root
associated to gauge algebra and can thus be taken to have positive projection on the Weyl
chamber. (In the gauge theory description, these correspond to monopole solutions.) We
therefore expect that the quantum-corrected string tensions are also positive in this basis.
It was argued in [8] that when the quantum part of the metric is non-negative on entire
Coulomb branch, a gauge theory can have non-trivial renormalization group fixed point at
UV. Otherwise, the kinetic term may become negative at some point on the Coulomb branch,
thereby preventing the existence of a UV fixed point. This necessary condition for the existenceof 5d interacting quantum field theories is equivalent to the condition that F (φ) be convex
on all of C and, viewed in this fashion, imposes interesting constraints on the list of possible
gauge theories. For example, quiver gauge theories are ruled out, since for such theories there
is always a region on C for which the metric is negative. Thus, it was concluded in [8] that only
gauge theories with simple gauge groups can have non-trivial UV fixed points. The Coulomb
branch metrics of gauge theories with simple gauge groups were studied in relation to thiscondition and a full classfication was given in [8]. All the theories in this classification are
expected to have interacting fixed points in either 5d or 6d.
However, string dualities relate some theories predicted by the criterion in [8] to theories(such as quiver gauge theories) excluded by the same criterion. For example, an SU(3) gauge
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SciPost Phys. 14, 122 (2023)

theory with κ = 0 and NF = 2 is dual to an SU(2) × SU(2) gauge theory, as explained in [12].
The metric of the former theory is positive definite on the entire Coulomb branch, while thelatter theory has negative metric somewhere on the Coulomb branch. We discuss the detailsof this example below.
In fact, the existence of many interesting 5d field theories beyond the IMS bounds
was argued by studying the dynamics of branes in string theory [13, 14, 17, 21, 22], CY3 compactifications in M-theory [23–25], and also instanton analysis in 5d gauge theories[15, 16, 26]. For example, the authors in [8] predict that an SU(N ) gauge theory can couple to NF fundamental hypermultiplets within the bound NF ≤ 2(N − |κ|). However, recently,
the existence of the interacting fixed points for SU(N ) gauge theories with NF ≤ 2(N − |κ|) + 4
was argued in various literature [13–16]. These theories are obviously beyond the boundin [8]. See also [13–17, 21–26] for many other examples.
This tells us that the condition of the metric being positive semi-definite on the entire
Coulomb branch is too strong, and therefore we may be able to relax the resulting constraints
on the matter content and find a new classification which involves all known consistent 5dfield theories.
Indeed, we find that the previous constraint on the metric on the Coulomb branch is toostrong. This was already noticed for the simplest SU(2) × SU(2) quiver gauge theory, forinstance, in [12]. The SU(2) × SU(2) gauge theory has real two-dimensional Coulomb branchof the moduli space. The scalar fields φ1 , φ2 in the respective vector multiplets of each gauge
group parametrize the Coulomb branch of the moduli space. In the chamber with φ1 > φ2 > 0,
the quantum prepotential of this theory is given by

F (φ) =

1

φ2 +
2 1

2g1

1

φ2 +
2 2

2g2


1
8φ13 + 8φ23 − (φ1 + φ2 )3 − (φ1 − φ2 )3 ,6

(6)

where g1 , g2 are the respective gauge couplings for the two SU(2) gauge groups. Here wehave set the mass parameters of the bi-fundamental hypermultiplet to zero for convenience.
Let us examine if the metric is negative somewhere in the Weyl chamber. The matrix ofeffective couplings is
1/g12 + 6φ1−2φ2
τeff (φ) =
.
(7)
−2φ2
1/g22 − 2φ1 + 8φ2
One can easily check that one of the eigenvalues of the metric becomes negative near2
φ2 ∼ 0.255φ1 when the inverse couplings are small, 1/g1,2≪ 1. Naively, this quiver theory cannot have an interacting fixed point according to the criterion of [8]. However, it turns
out that this theory has a nice brane engineering in Type IIB string theory [12] as well as ageometric construction via compactification of M-theory on a CY3 [10]. The five-brane web in
Figure 1 will give rise to the 5d SU(2) × SU(2) gauge theory at low-energy.
It was pointed out in [12] that this theory has a singularity in the Coulomb branch owing
to the appearance of massless instanton particles and tensionless monopole strings. Thus theSU(2) × SU(2) gauge theory description breaks down beyond the singularity. This singularity
can be detected by computing the monopole string tensions explicitly. The tensions of themonopole strings are given by
T1 = ∂1 F = φ1 /g12 + 3φ12 − φ22 ,T2 = ∂2 F = φ2 /g22 − 2φ1 φ2 + 4φ22 .

(8)

In the brane configuration in Figure 1 monopole strings correspond to D3-branes wrapping
two compact faces. Their tensions are proportional to the areas of two faces. We find thatT1 and T2 are the same as the areas of the right and the left compact face respectively. As
φ2 decreases, the monopole string coming from the D3-brane wrapping the left square of φ2
6

2/27

ExampleExampleSciPost Phys. 14, 122 (2023)

SU (2) ⇥ SU (2) gauge theory has two-dimensional Coulomb branchtheory has two-dimensional Coulomb branchSU(2) ⇥ SU (2)bygaugeparametrized
1 , 2 with 1 > 2 > 0 .

parametrized by

1,

2 with2 > 0.

1 >

1

1
2 2

2 1

2

1

Det(⌧ij ) > 0

Det(⌧ij ) > 0

As we decrease2

Det(⌧ij ) = 0 at2

2 ⇠ 0.25

Det(⌧ij ) = 0 at

2 ⇠2

, we meet massless instantons

Figure 1: Five-brane web for the SU(2) × SU(2) gauge theory.

As we decrease2, we meet massless1 instantons

2

1

2 2

2 1

massless instantons at1

2 ⇠ 0.5

massless instantons⇠at0.252

Det(⌧ij ) = 0 at22Det(⌧ij ) = 0 at2

2

Figure 2:
theory.

1
Appearance of massless
degrees of freedom in the SU(2) × SU(2) gauge

2tends to be lighter and eventually becomes massless near φ2 ∼ φ1 /2 with 1/g1,2≪ 1. It isclear that this point on the Coulomb branch where we meet the new light degrees of freedom
is far away from the region where we encounter the negative metric. The prepotential in (6)cannot describe the dynamics of the system beyond the singularity due to the new light objects.
The interior of the physical Coulomb branch of the SU(2) × SU(2) gauge theory is therefore
specified by the bounds φ1 > φ2 > φ1 /2. Notice, in particular, that this region is narrower
than the naive Coulomb branch φ1 > φ2 > 0. The SU(2) × SU(2) gauge theory is an effective
description of the system in Figure 1 only within the sub-region φ1 > φ2 > φ1 /2.
If we want to study the precise physics around φ2 = φ1 /2 we should use another effective
description which has the light particles in its perturbative spectrum. In this case, the relevant
description is the SU(3) gauge theory with NF = 2. The duality between the SU(2) × SU(2)gauge theory and the SU(3) gauge theory was studied in [12, 27]. Therefore, based on thisduality, since the SU(3) gauge theory is a non-tivial quantum theory with 5d CFT fixed point,
which is predicted already in [8], we can also expect that the SU(2) × SU(2) gauge theory is a
non-trivial quantum theory. These two gauge theories are good low-energy effective descriptions in different parameter regimes of the same theory.
In this example, by exploiting the S-duality of Type IIB, we can clearly understand what
happens when some non-perturbative degrees of freedom becomes massless. While theSU(2) × SU(2) gauge theory description breaks down near the point φ2 = φ1 /2 due to theemergence of light instanton particles, the perturbative description of the SU(3) gauge theory
is reliable at the point. The light instantons at φ2 = φ1 /2 give rise to non-abelian SU(2) gauge
symmetry enhancement in the dual SU(3) gauge theory. It follows that the Coulomb branchof the SU(3) gauge theory beyond the point φ2 = φ1 /2 is merely a Weyl copy of the enhanced
SU(2) gauge symmetry. This means that the correct prepotential on the chamber φ2 < 2φ1 after the transition is a copy of the original prepotential on the Weyl chamber φ2 > 2φ1 obtainedby exchanging φ1 ↔ φ2 − φ1 . Therefore we expect a discontinuity of the prepotential in theSU(2)×SU(2) gauge theory at the transition point φ2 = φ1 /2 detectable by a non-perturbative
quantum correction to the prepotential despite the fact that the perturbative prepotential cannot capture this. With the prepotential corrected by non-perturbative effects, the metric of the

7

2 ⇠

1

Therefore, around 2 ⇠ 0.5 1 , SU (2) ⇥ SU (2) gauge theory is not a goodeffective description for the theory. We should use another effective theory
to continue beyond 2 ⇠ 0.5 1 .
[Aharony, Hanany 97]

SciPost Phys. 14, 122 (2023)

In this case, we can use SU (3) gauge theory with Nf = 2 .

2 ⇠ 0.5 1

SU (2) ⇥ SU (2)SU (3) w/ Nf = 2

Figure 3: Duality of SU(2) × SU(2) theory and SU(3) theory with two fundamental
Note that the effective SU (2) ⇥ SU (2) theory breaks down before we
hypermultiplets, NF = 2.

encounter metric singularity on Coulomb branch.

SU(2) × SU(2) theory is always positive definite on the entire (physical) Coulomb branch.
The lesson from this simple example is that in general, the physical Coulomb branch may
be smaller than the naive Coulomb branch identified in the perturbative description. Stateddifferently, the actual physical Coulomb branch cannot necessarily be identified with a fundamental Weyl chamber.

3

New Criteria

The above example shows that the IMS criterion is too restrictive. In view of the above discussion, we will relax the condition that the metric be positive everywhere on the perturbative
Coulomb branch, since this condition imposes metric positivity even in unphysical regions.
Our reasoning is as follows: in some regions of the perturbative Coulomb branch, there may
exist boundaries where the non-perturbative degrees of freedom become massless or tensionless. Beyond these boundaries, these masses and tensions appear to become negative, which
signals that the perturbative computation can no longer be trusted. We therefore restrict ourattention to the physical regions in the Coulomb branch where all degrees of freedom havepositive masses or tensions.

3.1

Main conjecture

We claim that if the metric is positive definite on the physical Coulomb branch the gauge theory
has non-trivial interacting fixed point. For these theories we can reach their fixed points by
taking the infinite coupling limit g02 → ∞. It is therefore crucial to determine the exact
physical Coulomb branch that we will investigate. We claim that the physical Coulomb branch
is the subset Cphys ⊆ C bounded by the set of hyperplanes where some non-perturbative degrees
of freedom become massless and/or tensionless:

Cphys = {φ ∈ C | T (φ) > 0, m2I (φ) > 0} .

(9)Here m2I (φ) > 0, T (φ) > 0 indicates that the masses and tensions of all non-perturbative
degrees of freedom are positive at the point φ. Furthermore, the metric on the above physicalCoulomb branch must be positive-definite. Namely,τeff (φ) > 0 ,φ ∈ Cphys .

(10)This condition should be true for all physically-sensible 5d theories, including quiver theories.
It is crucial to our classification that we are able to identify the hyperplanes where massless
degrees of freedom arise on the naive Coulomb branch. However, from a purely field-theoretic
standpoint, the task of identifying such hyperplanes is somewhat subtle because it requires anon-perturbative analysis to identify exact instanton spectrum. On the other hand, from a8

SciPost Phys. 14, 122 (2023)

(a)

(b)

(c)

Figure 4: We illustrate the three types of geometric transitions occurring in CalabiYau threefolds using examples of (p, q) five brane webs. Transition (a) corresponds
to a 4-cycle shrinking to a 2-cycle, (b) corresponds to a 4-cycle shrinking to a point,and (c) is a flop transition obtained by first shrinking a 2-cycle to a point and blowing
up another 2-cycle.
geometric perspective, it is in principle a straightforward exercise to identify the hyperplanes
bounding the physical Coulomb branch Cphys —from a geometric point of view, these are regions where we encounter a geometric transition in the CY3 . Therefore, we embark on a shortdigression to explain how to characterize these hyperplanes, and moreover their relation to
the BPS spectrum.
We remark that there are three possible geometric transitions1 associated to particles ormonopole strings becoming massless: (a) a 4-cycle shrinking to a 2-cycle, (b) a 4-cycle shrinking to a point, and (c) a 2-cycle shrinking to a point. Examples of these three types of transitionsin the case of (p, q) five brane webs are depicted in Figure 4. We now describe these three
types of transitions in more detail, and explain their relevance to our classification.
The first type of transition results in a local ADE singularity, as was argued in [28], sothat there is always a gauge symmetry enhancement due to the appearance of a collectionof massless W-bosons arising from the shrinking 2-cycle. The low energy dynamics in thevicinity of the singularity can therefore be captured precisely by a gauge theory description
involving a simply-laced gauge group. The region beyond this wall should be regarded as animage generated by the Weyl group associated to the enhanced nonabelian gauge group, andis therefore unphysical.
The second type of transition should be distinguished from the first in the sense that a
4-cycle shrinking to a point always results in the appearance of an infinite number of massless particles associated to 2-cycles inside the shrinking 4-cycle. In this case, the low energy
dynamics is described by a CFT with tensionless monopole strings interacting with massless
electric particles [20]. The classic example of such a phenomenon is the local P2 shrinking
to a point. From a geometric point a view, continuing past this kind of wall is prohibited andtherefore corresponds to an unphysical region of the moduli space.
The third type of transition corresponds to a 2-cycle shrinking to a point, while all 4cycle volumes remain finite. This type of transition is typically described as a flop transition
in a geometric setting. When the volume of the 2-cycle corresponds to the mass of some
perturbative BPS particle, this transition can be captured by the one-loop prepotential (1) andNote that 5d N = 1 gauge theories can be constructed in string theory by compactifying M-theory on a CY3 inwhich M2-branes wrap compact 2-cycles and M5-branes wrap compact 4-cycles in the CY3 . This setup leads to a
5d description in which BPS particle masses are controlled by the volumes of the 2-cycles, while monopole string
tensions are controlled by the volumes of the 4-cycles.
1

9

SciPost Phys. 14, 122 (2023)

manifests itself as a discontinuity of the cubic coefficients. However, when the volume of the2-cycle corresponds to the mass of an instanton particle, the one-loop prepotential may not beable to detect this sort of transition at any point on the physical Coulomb branch. This sort oftransition can in principle be captured by a careful analysis of the non-perturbative physics,
such as a computation of the BPS partition function. Moreover, if a geometric descriptionof the Calabi-Yau geometry is known, one can detect the presence of such a transition bycomputing the prepotential in terms of geometric data and carefully studying how the volumesof various 2-cycles change in the moduli space. An example of this is the geometric transition
F1 → P2 [7, 11].
The relationship between the volumes of 2-cycles (4-cycles) and the central charges Ze(Zm ) described in (3) implies that some combination of central charges vanishes for each transition type discussed above. Moreover, restrictions on the “allowed” linear combinations of 2and 4-cycle classes in the geometric setting translate into charge quantization conditions in
the field theory setting that ensure the corresponding BPS states all lie within physical charge
lattices. Such constraints, for example,
P (i) imply that(i)the vanishing of irrational linear combinations of φi (i.e. combinations
ne φi where ne ∈ R\Q) cannot correspond to geometrictransitions. We will thus need to be careful to distinguish between physical and unphysicalboundary hyperplanes in our analysis.
In this paper, we conjecture that by turning off all of the mass parameters,2 we can identify
Cphys by studying, in addition to flops by W-bosons, the first two types of transitions describedabove—namely those for which a 4-cycle shrinks. We expect that the one-loop prepotential
can detect these types of transitions because the volumes of the string tensions are captured
by partial derivatives with respect to the scalars φi . The above conjecture also implies thatthe third type of transition can arise only as a pertubative flop transition by some massless
electric particle when the mass parameters are turned off. Therefore, we are led to the mainstatement of this paper, concerning the characterization of what we refer to as the ‘physical’
Coulomb branch:
Main Conjecture
The physical Coulomb branch Cphys is defined by

Cphys = C T ≥0 ,(11)

C T ≥0 = {φ ∈ C | T (φ) ≥ 0} and ∂ Cphys is rational.

(12)

where

Moreover, non-trivial 5d gauge theories have positive definite metric on the physicalCoulomb branch:τeff (φ) > 0 , φ ∈ Cphys .
(13)Note that τeff > 0 in the above equation indicates that τeff is a positive definite matrix,
and rational means that the boundary ∂ Cphys is specified by the setting
P some rational linear
combinations of the Coulomb branch parameters equal to zero, i.e.
i ni φi = 0, ni ∈ Z. In
other words, we require that the boundary ∂ Cphys is identified by the appearance of physicalmassless BPS particles, as discussed above. (Note that we find only five exceptional theories
2
Note that ‘mass parameters’ refers collectively to both hypermultiplet mass parameters m f and inverse gauge
couplings m0 = 1/g02 , which are treated on the same footing from the 5d perspective—they are mass parameters
for the global symmetries of the 5d theory.

10

SciPost Phys. 14, 122 (2023)

where our classification becomes subtle due to the rationality condition. These five exceptional
theories will be discussed in Section 4.4.) For the remainder of the paper, we use Cphys todenote the subset of the Coulomb branch defined in (11), while we use C to denote the (naive)
Coulomb branch isomorphic to a fundamental Weyl chamber. We will also use C T >0 to denotethe interior of the physical Coulomb branch Cphys .
These new criteria can be used to classify all candidate non-trivial 5d QFTs with simple
gauge groups. We emphasize that our main conjecture is a necessary (though perhaps not sufficient) set of criteria for the existence of a 5d interacting fixed point in the UV. Although braneconfigurations and geometric constructions can provide evidence confirming the existence of
these theories, we will leave explicit realizations of the theories in our classification to future
research. For the reason, it is important to bear in mind that our classification describes a
maximal set of theories that can possibly exist, and the set of theories that actually exist may
be a subset. For the remainder of this paper, we will be sloppy about the distinction between
theories that satisfy the necessary set of conditions described by our conjectures and theories
that have explicit realizations in string theory. In particular, we will refer to all candidates as‘non-trivial theories’.
We will classify these candidate non-trivial gauge theories below using our main conjecture
for lower rank cases rG ≤ 5. For higher rank theories, our main conjecture appears to be rather
inconvenient for the full classification of 5d QFTs. In the next sub-section, we will make three
additional conjectures, each of which we believe is equivalent to the main conjecture describedabove, which make the problem of classification significantly more tractable. These additional
conjectures impose constraints (involving at most one derivative of the prepotential), which
are easier to implement in practice and thus enable us to make a full classification of arbitrary
rank gauge theories with simple gauge groups. To our knowledge, our classification covers
all known gauge theories with non-trivial fixed points and furthermore predicts large classesof new non-trivial 5d gauge theories that may exist. However, we would like to emphasizethat only the main conjecture above is physically well-motivated at present. We do not havephysical or mathematical arguments for the three additional conjectures described in the next
sub-section.
Let us first demonstrate how to identify non-trivial QFTs and classify them using the Main
Conjecture with simple examples. In this paper, we shall focus on the N = 1 gauge theories
without adjoint hypermultiplets.3 We will also turn off all mass parameters except the gauge
coupling.
We begin with SU(3) gauge theories. An SU(3) gauge theory has a two-dimensional
Coulomb branch parametrized by scalar fields φ1 , φ2 . We find that an SU(3) gauge theory
can have hypermultiplets only in the fundamental or the symmetric representation. The dimensions of other SU(3) representations are so large that the prepotential becomes negativeat large gauge coupling when such hypermultiplets are included, and hence those representations must be excluded.
For some purposes, we will find it convenient to use Dynkin basis for roots and weights,
while in some other cases it will prove more convenient to use the orthogonal basis. In
the Dynkin basis, W-boson masses are given by the inner products φ · αi with simple roots{αi } = {(2, −1), (−1, 2), (1, 1)}. In the Weyl subchamber C (1) = {φ1 > φ2 > 0}, we find the3

Note that if we add a single adjoint hypermultiplet to a 5d theory with a simple gauge group, the theory will
be promoted to a N = 2 gauge theory. All such 5d N = 2 theories can be obtained from 6d N = (2, 0) SCFTs on acircle S 1 with or without outer automorphism twists [29]. Gauge theories with two or more adjoint hypermultiplets
have negative prepotentials, so they are not renormalizable.

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SciPost Phys. 14, 122 (2023)

prepotential of SU(3)κ + NF F + NS S is given by

FSU(3) =

1
g02

 1

κ 2
φ1 φ2 − φ1 φ22 +8φ13 − 3φ1 φ22 − 3φ12 φ2 + 8φ2326
2φ13 − 3φ12 φ2 + 3φ1 φ22 .
(14)(φ12 − φ1 φ2 + φ22 ) +−NF − 9NS
12The prepotential in the second Weyl chamber C (2) = {φ2 > φ1 > 0} can be obtained byinterchanging φ1 and φ2 . The physical Coulomb branch is restricted to the sub-domain given
by
(2)
(1)
(15)
Cphys = {φ ∈ C (1) | T (φ) > 0} , Cphys = {φ ∈ C (2) | T (φ) > 0} ,in the first and the second Weyl subchamber, respectively. The condition for the gauge theory(2)(1)
being a non-trivial theory is that the metric on Cphys and Cphys is positive in the infinite coupling

limit g02 → ∞.
First consider the cases of κ = 0, NS = 0. Since κ is an integer, we require NF to beeven. The theories with NF ≤ 6 have a positive metric everywhere on the Coulomb branch,as discussed in [8], thus they are all non-trivial 5d QFTs. Therefore, we focus our attention
on the cases with NF > 6. The eigenvalues of the metric and string tensions are drawn in
Figure 5 for NF = 6, 8, 10. Unlike the case NF = 6 for which the metric always has positive
eigenvalues, theories with NF = 8, 10 have a negative eigenvalue of the metric in certain
sub-region of their Coulomb branches. However, one can easily check that their metrics are
positive in the physical Coulomb branches defined by T1 , T2 > 0. In other words, we encountera singularity at T2 = 0 in the naive Coulomb branch and the effective gauge theory description
breaks down before we reach at the point where the metric becomes negative. Within thephysical Coulomb branch Cphys , the SU(3) gauge theories with NF = 8, 10 have positive definite
metrics. Thus we claim that they are good theories with non-trivial UV fixed points. They aregood effective descriptions for the corresponding non-trivial theories at the fixed points inthe parameter space Cphys . We check that if we have NF > 10, the theories have no physicalCoulomb branch, i.e. Cphys = ;, and thus these theories cannot have interacting fixed points.
Indeed, it was conjectured in [14, 15] that the SU(3) gauge theories with NF ≤ 10 have nontrivial UV fixed points while the theories with NF > 10 may be inconsistent. Our computation
provides concrete physical evidence for this conjecture.
Let us now turn on the Chern-Simons term at level κ. SU(3)−1 + 8F can be engineered instring theory by a (p, q) 5-brane web [14]. Figure 6 is the 5-brane web diagram for this theory
at zero masses in the first Weyl chamber φ1 > φ2 > 0. The two 5-branes at the top of the diagram end on 7-branes. Since the length of the first D5-brane is given by 1/g02 −2φ1 , this brane
picture is a good description for the theory only within the parameter regime 1/g02 ≥ 2φ1 .
When 1/g02 < 2φ1 , two 7-branes at the top of the diagram need to be pulled inside the 5brane loop.
At small coupling 1/g02 ≫ φ1 , the brane description is reliable. The monopole strings
correspond to D3-branes wrapping the compact faces and the tensions of these strings T1 , T2
are given by areas of two compact faces. It is straightforward to read off the tensions from thediagram and the results are11
T1 = 2 (2φ1 − φ2 ) + 2φ2 (φ1 − φ2 ) , T2 =+ 2φ2 − φ1 (2φ2 − φ1 ) .
(16)
g0
g02

One can easily check that these string tensions agree with the tensions obtained from the first
derivatives of the prepotential FSU(3) in (14). Although the brane description breaks down atlarge coupling beyond 1/g02 = 2φ1 , the gauge theory description is still good. Thus we can
expect that the above tension formula for the monopole strings may give the correct tensions of12

SciPost Phys. 14, 122 (2023)

1.0

1.5

1.0

0.5

T1

0.8

T2

0.6

T2

M2 /100.4

M2 /100.2

M1 /10
0.6

0.7

0.8

0.9

T1

M1 /10
0.6

1.0

0.7

(a) NF = 6

0.8

0.9

1.0

(b) NF = 8

0.5
M2 /10

0.4
0.3
0.2

T1
T2

0.1
- 0.1

0.6

0.7

0.8

0.9

1.0

M1 /10- 0.2(c) NF = 10

Figure 5: Solid lines are eigenvalues M1 /10 and M2 /10 of the metric and dashed lines
are two string tensions T1 and T2 at g −2 = 0.1 for NF = 6, 8, 10. The horizontal axis
NF =6
is the ratio φ2 /φ1 . The physical Coulomb branches are Cphys= φ1 ≥ φ2 ≥ φ1 /2,
N =8

N =10

F
F
Cphys
= φ1 ≥ φ2 ≥ 0.7φ1 , and Cphys
φ1 ≥ 0.

= φ1 ≥ φ2 ≥ 0.95φ1 , respectively, with

1/g 2 2 1
1

T1

m=0
2

1

T2
2

1/g 2

Figure 6: 5-brane web diagram for the theory SU(3)−1 + 8NF at zero masses.
wrapped D3-branes even after 7-branes are moved inside the 5-brane loop at large coupling.
In Figure 6, the bottom face denoted by T2 shrinks to zero area when the last two internal
D5-branes are placed on top of each other, i.e. 2φ2 − φ1 = 0. Then the D3-brane wrapping
the bottom face contributes tensionless strings to the 5d gauge theory. The string tensions are
non-negative on the entire Coulomb branch in the first chamber φ1 > φ2 > φ1 /2 even in theinfinite coupling limit g02 → ∞. The non-negativity of the string tensions thus implies that theSU(3) gauge theory and the 5-brane web are both good descriptions for this theory in the firstchamber. This is consistent with the fact that the SU(3) gauge theory has a positive metric on
the first chamber as depicted in Figure 7a. In the second chamber, the metric becomes negativearound φ1 < 0.74φ2 . However, this is fine because the physical Coulomb branch is restricted
to φ2 > φ1 > 0.95φ2 due to the tensionless strings. Therefore, as expected, SU(3)−1 + 8F is a
non-trivial 5d QFT according to our new criteria.
Another interesting example is SU(3)−4 +6F. This is a new non-trivial 5d theory we predict
using our new criteria.4 This theory has the physical Coulomb branch only in the first chamber
4This theory is a marginal theory whose fixed point theory is not a 5d SCFT. We discuss marginal theories in

13

SciPost Phys. 14, 122 (2023)

T2

1.0

1.5

0.8

T2

1.0

0.6

M2 /10M2 /10

0.5

0.4
0.2

T1

T1
0.6

0.7

0.8

0.9

1.0

M1 /10

0.6

(1)

0.7

0.8

0.9

1.0

M1 /10(2)(a) Cphys : φ1 > φ2 > φ1 /2.

(b) Cphys : φ2 > φ1 > φ2 /2.

Figure 7: Eigenvalues of the metric (solid line) and string tensions (dashed lines) of
SU(3)−1 + 8F fundamentals at 1/g02 = 0.1. The horizontal axes are ratios of scalar
fields φ2 /φ1 and φ1 /φ2 respectively.
T2

3

3

T2

2
2

1

1
-1
0.6

0.7

0.8

0.9

0.7

0.8

0.9

1.0

T1

-2

1.0
T1

-1

0.6

-3

Det(τeff ) - 4(a) 1/g02 = 0.2.

Det(τeff )(b) 1/g02 = 0.

Figure 8: Det(τeff ) (solid line) and string tensions (dashes lines) of SU(3)−4 + 6F in
the first chamber.
and it becomes a one-dimensional real line Cphys (1) : 2φ = φ2 ≥ 0 at infinite coupling. In
the second chamber, there is no physical domain in the Coulomb branch. The metric on thephysical Coulomb branch with T ≥ 0 is non-negative as depicted in Figure 8.
Instanton operator analysis at 1-instanton level tells us that the global symmetry of thistheory will be enhanced to SO(12) × U(1) at the fixed point. Following [15,30], one can showthat there is a conserved current multiplet in the antisymmetric representation of the classicalSU(6) flavor symmetry due to a 1-instanton operator which enhances the global symmetry
U(6)×U(1) I to SO(12)×U(1), where U(1) I is the instanton symmetry. It is not clear, however,whether or not this theory has a 6d uplift in the UV because the enhanced global symmetry is
not of affine type.

3.2

Additional conjectures

In principle, we can classify all non-trivial 5d gauge theories for simple gauge groups of arbitrary rank using the criterion in (13). For low rank, this is a manageable problem and thus weclassify all gauge theories with rG ≤ 3 using (13) in the next section. However, our method ofclassification in this case involves an analysis on the entire Coulomb branch, and thus becomesa difficult task for higher rank gauge theories.
To circumvent this issue, we propose the following three additional conjectures, each ofwhich we have checked leads to the same classification of 5d theories obtained via our main
conjecture, up to rG ≤ 3 analytically and up to rG ≤ 5 numerically:
more detail in Section 4.

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SciPost Phys. 14, 122 (2023)

More Conjectures
1. If τeff (φ) > 0 somewhere in the naive Coulomb branch C (possibly including unphysical regions), then there exists a physical Coulomb branch Cphys .
2. If all T (φ) > 0 in some region C T >0 ⊆ C , then τeff (φ) > 0 in C T >0 .
3. If the prepotential F (φ) > 0 everywhere in the the Coulomb branch C (possibly
including unphysical regions), there exists a physical Coulomb branch Cphys .
4. Conjectures 1-3 above are equivalent to our main conjecture.

Physical or mathematical proofs for the additional conjectures above are currently lacking.
However, the checks we did up to rG ≤ 5 are very compelling evidences for the above conjectures. So we believe that they also hold for arbitrary higher-rank gauge theories. We leave theproof of the equivalence of these conjectures to future study.
Each of the above three conjectures has its own interesting implications, which we now
discuss briefly. Conjecture 1 can be viewed as an extension of the criterion previously discussed
in [8]. There, the authors imposed that the constraint of metric positivity everywhere on the
Weyl chamber C , whereas Conjecture 1 merely requires that there exist some regions in C forwhich the metric is positive definite. Stated differently, we do not assume that the physicalCoulomb branch Cphys = C .
Conjecture 2 is quite interesting because in all cases we have checked, the condition thatthe tensions be positive on some region Cphys guarantees that the metric is also positive definite on Cphys . Therefore, according to our main conjecture, the existence of such a region Cphys
is sufficient to imply the existence of a non-trivial fixed point. From a practical standpoint,
Conjecture 2 is much easier to utilize for a systematic classification of 5d theories because
the mathematical conditions we need to impose only involve the gradient of the prepotential,
whereas metric positivity requires diagonalization of the Hessian of the prepotential. Moreover, the constraint of positive tension is convenient because we only need to apply it to a local
region of the naive Coulomb branch (as opposed to the entire Coulomb branch) to identify alegitimate theory.
Conjecture 3 is essentially motivated by the convergence of the partition function on acompact five-manifold. More precisely, Conjecture 3 asserts that positivity of F on the entire
Coulomb C branch guarantees the convergence of the naive perturbative partition function.
However, it is not clear if Conjecture 3 can guarantee the convergence of the full partition function including non-perturabive contributions. We discuss the details of the partition function
on S 5 and its connection to Conjecture 3 in Appendix C. One practical advantage of Conjecture
3 is that it only involves checking the prepotential itself, rather than its first or second derivatives. However, this advantage comes at the expense of having to analyze the prepotential on
the entire Coulomb branch as opposed to a local region.
We remark that Conjectures 2 and 3 are complementary: Conjecture 2 can be used toargue the existence of non-trivial 5d gauge theories by locally identifying a physical Coulomb
branch Cphys , while Conjecture 3 can be used to exclude theories by again locally identifying a
region of C that does not admit a positive prepotential. Therefore, a combination of these two
conjectures gives us a powerful method to fully classify 5d SQFTs with simple gauge groups.
In the following section we will explicitly describe the classification we obtained by using
Conjectures 2 and 3.

15

SciPost Phys. 14, 122 (2023)

4Classification of 5d CFTs

4.1

Classification overview

Analysis on the Coulomb branch of the moduli space provides us a systematic way to classify non-trivial 5d gauge theories with interacting fixed points. For the first step toward thisdirection, we will classify all possible gauge theories with simple gauge group G.
An important and very useful assumption in our classification is the idea that the dimensionof the largest matter representation must be less than the dimension of the adjoint representation. The reason for this assumption is the fact that the matter hypermultiplets contributeterms that “reduce” the convexity of the perturbative prepotential (1). In order to ensure
the existence of a physical Coulomb branch for which the prepotential is convex, it is therefore necessary that the vector multiplet contributions to the prepotential “balance” against thehypermultiplet contributions. Building on an observation made in [8], we translate this requirement into the restriction that the lengths of the weights of a given representation be nolarger than those of the adjoint representation. If this were not the case, then even for a single
matter hypermultiplet it would be possible to locate a direction in the naive Coulomb branch
for which the prepotential is negative5 violating Conjecture 3.
In the following classification, we will distinguish between two types of theories—standard
and exceptional. Standard theories are classes of 5d gauge theories with classical gauge groups
for which the same specifications for the matter content (depending on the rank of the gauge
group) applies for all rank rG . The expressions bounding the matter content for standard
theories are simple linear expressions involving rG and the numbers of flavors. By contrast,
exceptional theories exist only for some sporadic lower rank cases, where rG ≤ 8. For these
theories, there is no simple formula constraining the matter content and therefore they must
be studied on a case-by-case basis.
In the first sub-section, we will present the classification of all standard theories. We will
first choose some particular subregions in the Weyl chamber (where the choices of subregions
are motivated by data from lower rank studies) and examine these subregions using Conjecture
3. This will set upper bounds on the matter content for non-trivial QFTs. Note that this does notguarantee the existences of such theories, but merely exclude the theories beyond the bounds.
Then we will apply Conjecture 2 along the same subregions to confirm the existence of thetheories below these bounds. This will allow us to finish the full classification of non-trivial
CFTs with single gauge nodes of higher ranks rG > 8.
In the second sub-section, we will deal with exceptional cases at lower ranks rG ≤ 8. Ourclassification program involves both analytic and numerical search methods. By ‘analytic’, wemean that we have used a symbolic computing tool to derive precise inequalities describing
the physical Coulomb branches with positive metric. By ‘numerical’, we mean that we have
implemented a numerical search on the lattice discretization of a fundamental Weyl chamber
to locate a region admitting physical Coulomb branch. Having determined such regions, wethen analytically test whether or not such regions admit non-trivial theories using Conjectures
2 and 3.
For rG ≤ 3, we employ only our main conjecture to analytically perform the classification,
without using the additional conjectures. For 3 < rG ≤ 5, we employ a numerical searchcombined with our main conjecture to identify suitable theories, and then use Conjectures 2
5

As described in Appendix A.3, the length l(e) of a root e is defined to be the sum of the coefficients of e in abasis of simple roots, and similarly for a weight w. We note that it is always possible to locate a region φ ∗ ∈ C for∗which e · φ ∗ =1/g02 = 0, the prepotential
is proportional
Pl(e) and w · φ P= l(w), and hence at infinite coupling PP
to 6F (φ ∗ ) = e∈roots |l(e)|3 − w∈R f |l(w)|3 , and the classical CS term w∈F (w · φ ∗ )3 = w∈F l(w)3 = 0. It is thenevident that F (φ ∗ ) will be negative if the quadratic Dynkin index of R f is larger than the quadratic Dynkin index
of the adjoint representation—see Appendix B for further discussion.

16

SciPost Phys. 14, 122 (2023)

and 3 to confirm the existence of these theories analytically. For 5 < rG ≤ 8, we numerically
perform the classification by using Conjectures 2 and 3.
Marginal theories vs. 5d CFTs One additional idea essential to our classification is thenotion of a marginal theory. Marginal theories are those theories for which there exists a
physical Coulomb branch Cphys with positive metric, but the strong coupling limit 1/g02 → 0
is not a 5d SCFT fixed point. One distinguished characteristic of marginal theories is thevanishing of the some of the eigenvalues of τeff at infinite coupling limit, i.e. the metric isdegenerate along Cphys . In addition, the physical Coulomb branch with positive string tensions
collapses to a lower dimensional space at infinite coupling limit:
dim Cphys g 2 →∞ < rG .

(17)

0

In order to preserve the full dimensionality of Cphys , it is necessary to keep the gauge coupling
finite, i.e. g02 < ∞. Finite gauge coupling introduces a scale to such theories, implying that
their UV completions cannot be 5d SCFTs. Some of these theories will have 6d UV completions instead of 5d SCFT completions. We will discuss such theories uplifted to 6d theories
in Section 5. For other marginal theories, we do not have a clear understanding of their UVcompletions, or if they even exist.
We use the term descendants to refer to theories that can be obtained from some othertheory with a larger number of matter hypermultiplets via RG flow after integrating out somemassive matter hypermultiplets. All the descendants of the marginal theories flow to 5d SCFT
fixed points at infinite coupling.
The reason for this, as explained in [8], is that the cubic deformations to the prepotential(coming from the terms |w·φ+m f |3 ) resulting from sending a mass parameter m f → ±∞ andintegrating out the matter make the prepotential “more convex”. The only other deformations
to the prepotential arising from integrating out NR f massive matter hypermultiplets in the(3)

representation R f with mass terms sgn(m f )|m f | are terms of the form −sgn(m f )NR f cR /2fcontributing to the classical CS level κ [8]. Therefore, for SU(N ≥ 3) theories, one can tradematter hypermultiplets for the CS level κ, and such theories will have interacting fixed points
as demonstrated by our classification. On the other hand, if one increases κ or adds morematter to these marginal theories described above, the resulting theories have no physicalCoulomb branch and consequently they are trivial.
Notational conventions

In the following discussion, we adopt the condensed notation
XGκ +NR f R f ,(18)
f

to refer to a gauge theory with simple gauge group G with CS level κ coupled to NR f matter hypermultiplets (or half-hypermultiplets) transforming in the representation R f . We encounterthe following representations:• F, fundamental• Sym, rank two symmetric• AS, rank-2 antisymmetric• TAS, rank-3 antisymmetric• S, spinor• C, conjugate spinor.
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SciPost Phys. 14, 122 (2023)

4.2

Standard theories

A class of gauge theories with gauge groups G = SU(N ), S p(N ), SO(N ) which exist even atlarge rank can be fully classified by employing our conjectures. We call such theories as standard theories as mentioned above. The results are summarized in Table 1. The theories inthese tables are marginal theories. We expect that all of the standard have 6d uplifts at strong
coupling, a point that we discuss further in Section 5. The descendants of these marginal
theories are expected to have non-trivial 5d SCFT fixed points at UV. Cphys is the subregion in
the Weyl chamber within which we have confirmed the metric and the string tensions of thecorresponding theory are both positive.
F is the global symmetry at the UV fixed point which we obtain by using 1-instanton analysis in [15, 16, 26, 30].6 Some lower rank theories have further symmetry enhancements which
are listed in Table 2. The full global symmetry could be in general different from those listed
in the table if the symmetry enhancement occurs at two or higher instanton level. In somecases, the global symmetry group is affinized by instanton operators. This may signal thatthe theory is uplifted to a 6d theory at the UV fixed point. Some of these global symmetries
have already been confirmed by identifying brane constructions or explicit 6d uplifts of these
theories [21, 22, 31].
In the following section we present an algorithm for the classification of the standard theories listed in Table 1.
4.2.1 SU(N ) gauge theories
Let us start with the classification of standard SU(N ) gauge theories with NF fundamental,
NSym symmetric, and NAS antisymmetric hypermultiplets. Note that other representations are
not allowed for higher rank theories rG > 8. For lower rank theories rG ≤ 8, we can also
add NTAS matters in the rank-3 antisymmetric representation. But theories with NTAS > 0 are
exceptional theories that we will discuss separately below. Matter hypermultiplets in otherrepresentations are not allowed.
In the following discussion, we express the roots and weights in the orthogonal basis. The
prepotential for SU(N )κ + NF F + NAS AS in the Weyl chamber a1 ≥ a2 ≥ · · · ≥ aN −1 ≥ 0 is givenby

FSU(N ) =N
1 X

2g02 i=1

ai2 +

N
N
N
N
NAS X
N X
κX 3 1Xai +(ai − a j )3 − F|ai |3 −|ai + a j |3 ,6 i=1
6 i< j
12 i=1
6 i< j(19)

P
with aN = − i ai . A symmetric matter hypermultiplet has the same contribution to the
prepotential as the contribution from (NAS = 1, NF = 8) hypermultiplets. It follows that
(NAS = 1, NF = 8) hypermultiplets can be traded with NSym = 1 at the level of the prepotential. So we will consider the cases NSym = 0 and retrive the cases with NSym > 0 by using the“equivalence” (NAS = 1, NF = 8) ∼ NSym = 1 as needed.
We will examine the following three regions of the Weyl chamber,(1)

SSU(N ) = (a, a, · · · , a, −(N − 1)a) ,(2)

SSU(N ) = (a, 0, · · · , 0, −a) ,(3)

SSU(N ) = (a, a, 0, · · · , 0, −2a) , (20)Firstly, Conjecture 3 imposes the condition for non-trivial SU(N ) theories that the prepotential
should be positive FSU(N ) > 0 along these three regions. Then each of three regions gives rise
In some cases, like S p(N ) with fundamental matter, the 1-instanton analysis was supplemented by additional
data. Also there are several subtleties and technical difficulties involved in the 1-instanton analysis, which in somecases prevent us from carrying the calculation. The cases of SU(N ) groups with symmetric matter hypermultiplets
are particularly subtle. The symmetries written in these cases are the minimal ones we are sure of, but there may
well be additional enhancements by 1-instanton states for low N cases.
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SciPost Phys. 14, 122 (2023)

Table 1: Classes of standard gauge theories. Note that all of these theories are
marginal and their descendants have 5d conformal fixed points in the UV. See (20),(35).
(a) Marginal SU(N ) theories with CS level κ, NSym symmetric, NAS antisymmetric, and NF fundamental
hypermultiplets. See (20) for an explicit description of the physical Coulomb branches described in the
far right column.

NSym

NAS

NF

|κ|

F

Cphys

1

1

0

0

A1 × U(1)SSU(N )

1

0

N −2

0

AN −2 × U(1)SSU(N )

1

0

0

N
2

U(1)2

SSU(N )

0

2

8

0

Even N : E7 ×A1 ×A1 ×SU(2)

(1)

(2)

(1)

(1)

(1)

(1)

(1)

(1)

(2)

(1)

(2)

SSU(N )

Odd N : D8 × A1 × SU(2)
0

2

(1)

3
2

7

(3)

SSU(N )

Even N : D8 × SU(2)(1)

Odd N : E7 × SU(2) × SU(2)
(1)

(2)

0

1

N +6

0

AN +6 × U(1)SSU(N )

0

1

8

N
2

SO(16) × U(1)2

SSU(N )

0

0

2N +4

0

D2N +4(1)(b) Marginal S p(N ) theories with NAS antisymmetric and NF fundamental hypermultiplets. See (35) for a description of the physicalCoulomb branches.

NAS

NF

F

Cphys

1

8

E8 × SU(2)SS p(N )0

2N + 6

D2N +6

(1)

(1)

(1)
(2)

SSU(N )(c) Marginal SO(N > 4) theories with NF fundamental hypermultiplets. See (42) for a description of the physical Coulomb branch.

NF

F

N −2

(2)
A2N −5Cphys
SSO(N )

SS p(N )Table 2: SU(N ) theories with further symmetry enhancement.
N

NAS

NF

|κ|

F

4

2

8

0

5

2

7

4

2

7

3
2
3
2

5

1

11

0

4

1

10

0

(1)
(1)
E7 × B3(1)
E 7 × G2
(1)
B9
(1)
(1)
A11 × A1
(1)
A11

6

1

8

3

SO(16) × SU(2) × U(1)

5

1

8

5
2

E8 × U(1)

4

1

8

2

E8 × U(1)

(1)
(1)

19

SciPost Phys. 14, 122 (2023)

to the following three inequalities,

SSU(N ) :(1)

NF (N 2 − 2N + 2) + 2|κ|N (N − 2) + NAS (N − 2)(N 2 − 4N + 8) ≤ 2N 3 ,SSU(N ) :(2)

NF + (N − 2)NAS ≤ 2N + 4 ,SSU(N ) :(3)

5NF + 5NAS (N − 2) + 6|κ| ≤ 10N + 24 .

(21)

For a given N , these inequalities set some upper bounds on the matter content of the theories
with non-trivial fixed points. We may be able to find other independent inequalities by examining other regions in the Weyl chamber. But we will show shortly that these inequalities areenough for higher rank theories. By analyzing these three inequalities, it can be shown thatfor N > 6 there are only five cases that saturate one of these inequalities while not violating
the others:(NAS , NF , |κ|) = (2, 8, 0) ,(2, 7, 32 ) ,(1, N + 6, 0) ,(1, 8, N2 ) ,(0, 2N + 4, 0) .

(22)

Other theories with N > 6 having more matter or higher Chern-Simons levels are all excluded
by Conjecture 3 since they have negative prepotential along one of the above three regions.
We will now confirm that the upper boundary theories in (22) are indeed non-tivial theories(i=1,2,3)
having interacting UV fixed points by testing Conjecture 2 along the three subregions SSU(N ) .
(1)

First, we expand the Coulomb branch parameters around SSU(N ) asai = a + δi ,δi ≪ 1 ,(23)

with δi > δi+1 . Then the string tensions can be approximated as
Ti = (∂ai − ∂ai+1 )FSU(N )

 

NF 3
1
− N NAS + 4NAS a (δi − δi+1 ) + O(δ2 ) ,=+ N +κ−
2
2
g02

(24)

TN −1 = ∂aN −1 FSU(N )

 2
NAS 3
N
N
a
= 2 a + N 3 −κN (N −2)− F (N 2 −2N +2)−
(N −6N 2 +16N −16)
+ O(δ) .
2
4
2
g0
It is obvious that all the string tensions are positive at weak coupling 1/g02 ≫ 1 since δi > δi+1 .
However, in the strong coupling limit 1/g02 → 0, all of the string tensions are positive, i.e.
T > 0, only if the matter content of the theory satisfies the following inequalities:
2N + 2κ − NF − NAS (3N − 8) ≥ 0 ,NAS 3N
N 3 −κN (N −2)− F (N 2 −2N +2)−
(N −6N 2 +16N −16) ≥ 0 .
2
4

(25)

When all the string tensions are positive, we identify the corresponding theory as a good theory
with an interacting UV fixed point by using Conjecture 2.
(2)
Similarly, it is straightforward to show that the string tension analysis around SSU(N ) leads
to the inequalities
2N + 4 − NF − NAS (N − 2) ≥ 0 , NAS ≤ 2 .
(26)

20

SciPost Phys. 14, 122 (2023)(3)

Lastly, the string tensions near SSU(N ) are approximated as


 
1
1
T1 =
+ N + k − NF − NAS (N − 2) a (δ1 − δ2 ) + O(δ2 ) ,
2
2
g02


1
1
a
T2 = 2 + N + 4 + k − NF − NAS (N + 2) a2 /2 + O(δ) ,
2
2
g
0

1
Ti =
+ (4 − 2NAS )a (δi − δi+1 ) + O(δ) ,
g02


1
2a
1
TN −1 = 2 + 2 N + 2 − k − NF − NAS (N − 3) a2 + O(δ) ,
2
2
g0
1



(27)

where a1 = a + δ1 , a2 = a + δ2 and ai = δi for 3 ≤ i ≤ N − 1 with infinitesimal parameters δi .
(3)
Therefore the analysis T > 0 around SSU(N ) gives the inequalities
NAS ≤ 2 , 2N + 2k − NF − NAS (N − 2) ≥ 0 ,2N + 4 − 2κ − NF − NAS (N − 3) ≥ 0 .

(28)

We now have three sets of inequalities (25), (26), (26). Conjecture 2 tells us that if an(i=1,2,3)
SU(N ) theory on one of the three subregions SSU(N ) satisfies the corresponding inequality, the theory is a non-trivial theory having interacting fixed point at UV. With this criterion,
one can easily show that all of the upper boundary theories in (22) are non-trivial theories.
We will call these theories, and their descendants obtained by integrating out massive matter, standard SU(N ) gauge theories. The standard SU(N ) theories are summarized in Table 1a. Note that we obtained results for the theories with symmetric hypermultiplets bytrading (NAS , NF ) = (1, 8) → NSym = 1. The theories (and their descendants) in Table 1a exist
at any N and they are the only non-trivial theories at N ≥ 9, which is proven above relying on
Conjecture 2 and 3. This completes the full classification of standard SU(N ) gauge theories.
In fact, all the theories listed in Table 1a lift to 6d SCFTs in the UV, and hence they are marginal
theories. We discuss these theories and their 6d uplifts in more detail in Section 5.
4.2.2 S p(N ) gauge theories
Let us turn to S p(N ) gauge theories. S p(N ) gauge theory can have NF fundamental, NASantisymmetric, and NTAS rank-3 antisymmetric representations of S p(N ). We claim that noother representations are permitted, and furthermore NTAS > 0 only for theories with N ≤ 4.
The theories with NTAS > 0 are exceptional theories, and will be discussed separately below.
In this section, we consider the standard theories S p(N ) + NF F + NAS AS. The S p(N ) theory
has a single Weyl subchamber a1 ≥ a2 ≥ · · · ≥ aN ≥ 0. The prepotential is given by
!
N
N
N
X
X
X

1
1
FS p(N ) = 2
ai2 +(ai − a j )3 + (ai + a j )3 (1 − NAS ) + (8 − NF )ai3 . (29)62g0 i=1i< ji=1Since the S p(N ) theory has a single Weyl chamber, in sharp contrast to the SU(N ) cases withmany distinct chambers, we can in fact examine every point on the Weyl chamber without
restricting to particular subregions and thereby obtain a full classification using only Conjecture 2.

21

SciPost Phys. 14, 122 (2023)

The string tensions are
Ti = (∂ai − ∂ai+1 )FS p(N )= (ai −ai+1 )

!
i
X
1
+
(2a j − ai − ai+1 )(1 − NAS ) + ((2N − 4)(1 − NAS ) + 8 − NF ) (ai + ai+1 ) ,2
g02 j=1
1

TN = ∂aN FS p(N ) = aN
where a T =

8 − NF+aN + 2(1 − NAS )a T
2
2
g0
1


,

(30)

PN −1

i=1 ai . The condition on the last string tension TN > 0 gives the inequality

NAS ≤8 − NF aN
+ 1.
4 aT

(31)

The right hand side of this equation is maximized near a1 = a2 = · · · = aN and the result isNAS ≤4N − NF + 4.
4(N − 1)(32)

For the theories with N ≥ 5, this condition allows only the cases with NAS = 0, 1.
When NAS = 1, the tension positivity conditions reduce to the single condition
NF ≤ 8(when NAS = 1) .

(33)

When NAS = 0, the string tensions can be written asi−1
X

2N + 8 − NF − 2i 2 2N + 4 − NF − 2i 2
Ti = (ai − ai+1 )
ai +
ai+1
+2
aj +
2
2
2
g0
j=1


8 − NF1
TN = aN+
aN + 2a T .
2
g02
1

!
,
(34)

The positive tension conditions T > 0 admit the maximum number of matter hypermultipletsnear the subregion

SSp(N ) = (a, 0, · · · , 0) .

(35)

Around SS p(N ) , the only non-trivial condition is T1 > 0 which gives rise to the bound
NF ≤ 2N + 6 .

(36)

The standard S p(N ) theories are those summarized in Table 1b along with their descendants.
The theories in Table 1b have 6d uplifts at the UV fixed point, so they are marginal theories.
Their descendants are expected to have 5d CFT fixed points in UV. Note that, when NF = 0, theS p(N ) gauge theory has a discrete theta angle θ = 0, π associated to π(S p(N )) ∼= Z2 . Thus
there are two more non-trivial S p(N ) theories with (NAS = 0, NF = 0) and (NAS = 1, NF = 0).
One can immediately show that, for these standard theories, the prepotential (29) is always
positive within the Weyl chamber. If we add more matter to these theories, their prepotential
will become negative somewhere in the Weyl chamber. This confirms Conjecture 3 for the
standard S p(N ) gauge theories.

22

SciPost Phys. 14, 122 (2023)

4.2.3 SO(N ) gauge theories
We consider the case of SO(N ) with NF fundamental flavors. Spinor hypermultiplets can also
be added when N ≤ 14, but because such theories are not standard, we discuss them in thenext subsection. In the case N = 2r + 1, there is only a single Weyl subchamber to consider,namely a1 ≥ a2 ≥ · · · ≥ a r ≥ 0. The prepotential for SO(2r + 1) + NF F isr

FSO(2r+1) =1 X
g02 i=1ai2 +rr
1 − NF X 3
1X[(ai + a j )3 + (ai − a j )3 ] +a .
6 i< j
6 i=1 i(37)

Using the above formula for the prepotential, we see that the tensions are given by:
Ti<r = ∂ai − ∂ai+1 FSO(2r+1)
=

2
g02

(ai − ai+1 ) + 2(ai − ai+1 )

i−1
X

aj

j=1


 

ai
1
2
+ ai+1
1 − (N − NF − 2i − 2) +(N − NF − 2i) − 2ai+1 ,
2
2
r−1
X
2
Tr = 2∂ar FSO(2r+1) = 2 a r + 4a r
a j + (1 − NF )a2r .
g0
j=1

(38)

(39)

Let us first focus on the expression for T1 :
T1 =

2
g02

1
(a1 − a2 ) + (a1 − a2 )((N − NF − 2)a1 + (N − NF − 6)a2 ) .
2(40)

It is possible to maximize the number of fundamental matters while keeping T1 positive by
setting a2 = 0 in the infinite coupling regime 1/g02 ≪ 1:
NF ≤ N − 2 .

(41)

Notice that when a2 = 0, the remaining Coulomb branch parameters a j = 0 for j > 2 according
to the definition of the fundamental Weyl chamber. Therefore, we see that we can maximize
NF while keeping all tensions positive along the locus

SSO(2n+1) = (a, 0, . . . , 0) .

(42)

For NF = N − 2, the only way to maintain positive tension is to keep the gauge coupling finite,
g02 < ∞. The resulting theory is a marginal theory with a 6d uplift at the UV fixed point.
When the number of fundamental matters NF > N − 2, there is no physical Coulomb branch at
infinite coupling 1/g02 = 0 with positive string tensions, and therefore such theories are trivial.
For SO(2r) gauge theories with fundamental matter, the proof works identically to theabove case, as we will now show. The prepotential for the SO(2r) theory isrFSO(2r) =1 X
g02 i=1

ai2 +

r
r
N X 3
1X
[(ai + a j )3 + (ai − a j )3 ] − Fa .
6 i< j
6 i=1 i(43)

We again parametrize the fundamental Weyl chamber by the condition a1 ≥ a2 ≥ · · · ≥ a r ≥ 0.
By computing derivatives of the above formula with respect to φi , we find that the monopole

23

SciPost Phys. 14, 122 (2023)

tensions are given by
Ti<r = ∂ai − ∂ai+1 FSO(2r)
=

2
g02

(ai − ai+1 ) + 2(ai − ai+1 )

i−1
X

aj

j=1

(44)


 

ai
1
2
+ ai+1 1 − (N − NF − 2i − 2) +(N − NF − 2i) − 2ai+1 ,22
Tr = (∂ar−1 + ∂ar )FSO(2r)
=

2
g02

(a r−1 + a r ) + 2(a r−1 + a r )

r−2
X

aj

j=1

(45)





a r−1
1
+ a r−1 2a r +(N − NF − 2r + 2) + a2r 1 − NF .
2
2
From the above expressions, we see that the only difference with the tensions Ti associated to
the SO(2r + 1) gauge theory is in the expression for Tr . By arguments identical to the abovecase G = SO(2r + 1), we find again that the bound on the number of fundamental flavors is
NF ≤ N − 2, with the case NF = N − 2 corresponding to a marginal theory.

4.3

Exceptional theories with rank ≤ 8

Rank 2In the previous section, we confirmed that some 5d theories which go beyond the constraints
introduced in [8] are in fact consistent theories with interacting fixed points. We now turn ourattention to the case of simple gauge groups G with rG = 2, namely G = SU(3), S p(2), G2 .
We begin by classifying non-trivial SU(3) gauge theories. As discussed in the previous
section, SU(3)κ + NF F + NSym Sym has two Weyl subchambers, assuming the presence of fundamental hypermultiplets. Applying our main conjecture to the Coulomb branch, we find thelist of marginal SU(3) gauge theories displayed in Table 3. On the other hand, all the descendants of these theories obtained by integrating out matter hypermultiplets have non-trivial 5d
conformal fixed points. This full list includes all known non-trivial 5d SU(3) gauge theories,
SU(3)0 + 1F + 1Sym, and SU(3)0 + 10F, and their descendants. Other theories in the list are
new non-trivial SU(3) theories we predict based on new criteria.
We now turn to S p(2) gauge theories. S p(2) gauge theories can only have fundamental
and antisymmetric matter, again because the dimensions of other representations are too large
to admit a positive semi-definite metric on the physical Coulomb branch. There is a single Weyl
chamber φ2 ≥ φ1 ≥ φ2 /2 ≥ 0 in the Dynkin basis. We find the marginal S p(2) gauge theories
listed in the Table 4. The global symmetry algebras of these theories are all affine type [26].
(2)Here AN denote the affine algebras with outer automorphism twists discussed in [29]. Thismeans that all of them have 6d uplifts at their UV fixed points and their descendants have
5d CFT fixed points. This list completely agrees with matter bounds for S p(2) gauge theories
given in [26]. S p(2) + 1AS + 8F has a two-dimensional Coulomb branch at infinite coupling.
This implies that this theory has a two-dimensional tensor branch in 6d. Indeed, this theory is
the 6d rank-2 E-string theory on a circle.
Lastly, the G2 gauge theory can contain only fundamental hypermultiplets. This theory
has a single Weyl subchamber 2φ1 ≥ φ2 ≥ 3φ1 /2 ≥ 0 in the Dynkin basis. We find that these
theories have a non-trivial physical Coulomb branch when NF ≤ 6. The theory at NF = 6
has an affine type global symmetry algebra and thus it will be uplifted to 6d at the UV fixed
point [26].

24

SciPost Phys. 14, 122 (2023)

Table 3: Marginal theories SU(3) with CS level κ, NSym symmetric and NF fundamental hypermultiplets. The theories with κ < 0 can be obtained by the interchange
φ1 ↔ φ2 .
NSym

NF

κ

F

1

0

3
2

SU(2) × U(1)2φ1 = φ2 ≥ 0
φ1 = φ2 ≥ 0
2φ1 = φ2 ≥ 0

Cphys

1

1

0

0

10

0

0

9

3
2

(1)
A1 × U(1)
(1)
D10
(1)
E8 × SU(2)

0

6

4

SO(12) × U(1)2φ1 = φ2 ≥ 0

0

3

13
2

U(3) × U(1)2φ1 = φ2 ≥ 0

0

0

9

U(1)

2φ1 = φ2 ≥ 0

φ1 = φ2 ≥ 0

Table 4: Marginal S p(2) gauge theories with NAS antisymmetric, NF fundamental
matters.
NAS

NF

F

3

0

A5 at θ =0,

(2)

Cphys

2φ1 = φ2 ≥ 0

S p(3)×U(1) at θ =π
2

4

1

8

0

10

A11

(2)

2φ1 = φ2 ≥ 0(1)
E8 × SU(2)(1)
D10

φ1 , φ 2 ≥ 0
φ 1 = φ2 ≥ 0

Rank 3Let us start with the SU(4) gauge theories. An SU(4) gauge theory can have matter hypermultiplets in the fundamental, the antisymmetric, and the symmetric representations. Whenthe theory contains symmetric or antisymmetric hypermultiplets, there are four distinguished
Weyl chambers. We examine all these chambers and find the list of marginal SU(4) theories
in Table 6. Again, this list covers all previously known SU(4) gauge theories in [8, 13–16,
21] and additionally it predicts many new interacting 5d gauge theories. The theories with
NAS = 4, NSym = NF = 0 and κ = 0, 1, 2, 3, 4 are all marginal theories which cannot be obtained
from other theories by integrating out massive matter.
Similarly, we can classify all non-trivial S p(3) gauge theories. The list of marginal S p(3)theories are given in Table 7. We can also have half-hypermultiplets in the rank-3 antisymmetric representation. However, due to the global anomaly described by Witten in [32], consistent S p(3) gauge theories with rank-3 antisymmetric matter must also have an odd numberof half-hypermultiplets in the fundamental representation. The new criteria confirms that all
previously known S p(3) gauge theories in [8, 26] are physical, with one exception. In [26],S p(3) + 12 TAS + 72 F + AS may have a fixed point of 6d SCFT on a circle. However, we findthat this theory has no physical Coulomb branch. So our classification rules out this theory. It
would be interesting to confirm this result using other arguments.
Finally, we come to the case of 5d SO(7) gauge theories, which can have matter hypermul25

SciPost Phys. 14, 122 (2023)

Table 5: A marginal G2 gauge theory with NF fundamental matters.
NF

F

6

A11

(2)

Cphys

2φ1 = φ2 ≥ 0

tiplets in fundamental and spinor representations. The marginal theories SO(7) + NF F + NS S
are listed in Table 8.
Rank 4In this section, we discuss exceptional 5d gauge theories, with gauge algebras SO(8), SO(9),F4 , and SU(5). To begin, we consider SO(8) gauge theories. An SO(8) gauge theory can have
NF fundamental, NS spinor, and NC conjugate spinor representations. When there is matter inthese three representations, the Weyl chamber splits into six distinct subchambers. We study
each of these chambers and find the marginal theories summarized in Table 9d. Due to thetriality of the D4 Dynkin diagram, given such an SO(8) gauge theory, there exists anotherSO(8) gauge theory obtained by permuting NF , NS , NC . However, we stress that the physicalCoulomb branches associated to different permutations are in general not identical. These
results cover all known cases of SO(8) gauge theories [26].
For SO(9) gauge theories, there can be NF fundamental and NS spinors. When spinor
matter is included, the fundamental Weyl chamber splits into three subchambers. We study
each of these chambers and find the marginal theories summarized in Table 9e.
For SU(5) gauge theories, we can consider hypermultiplets in the antisymmetric and thefundamental representations. We find four marignal theories as summarized in Table 9a. We
would like to remark that our criteria for SU(5) 3 + 3AS + 2F is more involved. We find that2
this theory has a single sub-chamber which has a region C T >0 with positive string tensions
and positive definite metric at infinite coupling. Therefore, naively, this theory is not marginal
and it may have a 5d SCFT fixed point at UV. However, a careful analysis shows us that theboundary ∂ C T >0 of the region is irrational. This indicates that this sub-chamber should not
be a physical Coulomb branch. We expect that there exists a flop transition point where some
non-perturbative instantons become massless so that we cannot reach the irrational boundary.
We expect that, after taking into account the flop transition, this theory becomes marginal.
The enhanced affine-type global symmetry of this theory in Table 9a supports this conjecture.
The Coulomb branch with irrational boundaries and the geometric consideration associatedto it will be discussed in Section 4.4.
Finally, we come to the case of G = F4 . Since the only representation with dimension
smaller than the adjoint is the fundamental representation, an F4 gauge theory can only havefundamental hypermultiplets. In this case, the Weyl chamber only consists of a single component, and therefore the task of determining a bound on the number NF of fundamental
half-hypers is straightforward. We find that NF ≤ 3 and all these theories have 5d conformal
fixed points at UV. So all the F4 theories are non-marginal. The physical Coulomb branch of
the theory with NF = 3 is given by

Cphys = {2φ1 > φ2 , φ1 + 2φ3 > 2φ2 , 2φ4 ≥ φ3 , 2φ3 > φ2 + φ4 }.

(46)

Rank 5,6,7,8
For SU(6) theories, we can have NTAS rank-3 antisymmetric hypermultiplets. Since TAS is
pseudo-real, we can add half-hypermultiplets which we donote by NTAS = 1/2. The marginal
26

SciPost Phys. 14, 122 (2023)

Table 6: Marginal SU(4) theories with CS level κ, NSym symmetric, NAS antisymmetric, NF fundamental matters. The theories with negative CS level are obtained by
φ1 ↔ φ 3 .
NSym

NAS

NF

κ

1

1

0

0

1

0

2

0

1

0

0

F
(1)
A1 × U(1)(1)
A2 × U(1)Cphys

2φ1 ≥ φ2 ≥ φ1 = φ3 ≥ 0
φ1 = φ 2 = φ 3 ≥ 0

2

U(1)

φ1 = φ2 /2 = φ3 /3 ≥ 0
3φ1 ≥ φ3 ≥ φ1 = φ2 /2 ≥ 0

2

0

4

0

4

(2)
E6

0

4

0

1, 2, 3

S p(4) × U(1)φ1 = φ2 /2 = φ3 ≥ 0

0

4

0

0

A7

(2)

φ1 = φ2 /2 = φ3 ≥ 0

0

3

4

0

A11 × U(1)φ1 = φ2 /2 = φ3 ≥ 0

0

3

4

1

S p(7) × U(1)φ1 = φ2 /2 = φ3 ≥ 0

0

3

4

2

E6 × SU(4)φ1 = φ2 /2 = φ3 ≥ 0

0

3

0

5

S p(3) × U(1)φ1 = φ2 /2 = φ3 /3 ≥ 0
2φ1 ≥ φ2 ≥ φ1 = φ3 ≥ 0

0

2

(2)

(2)

8

0

(1)
(1)
E7 × B3
(1)
B9

0

2

7

3
2

0

2

0

6

S p(2) × U(1)φ1 = φ2 /2 = φ3 /3 ≥ 0

0

1

10

0

A11

(1)

φ1 = φ 2 = φ 3 ≥ 0

0

1

8

2

E8 × U(1)φ1 = φ2 /2 = φ3 /3 ≥ 0

0

1

0

7

SU(2) × U(1)φ1 = φ2 /2 = φ3 /3 ≥ 0

0

0

12

0

D12

(1)

φ1 = φ 2 = φ 3 ≥ 0

0

0

8

3

E8 × U(1)φ1 = φ2 /2 = φ3 /3 ≥ 0

0

0

0

8

U(1)

φ1 = φ2 /2 = φ3 /3 ≥ 0(1)

φ1 = φ2 /2 = φ3 /2 ≥ 0

SU(6) theories are summarized in Table 10a. For SU(7) theories, we can also have rank-3
antisymmetric hypermultiplets, but only a single full-hypermultiplet is allowed. All the exceptional cases have NTAS = 1. The SU(7) theories are summarized in Table 11a. The first twoof them are marginal. However, due to computational limits, we could not clearly identify
the last theory whether it is non-marginal or exotic, which will be discussed in Section 4.4.
For SU(8)κ + NTAS TAS we find that there are no non-trivial exceptional SU(8) theories for any
value of NTAS .
For SO(10) theories, we can have NS spinor and NF fundamental hypermultiplets. The
marginal SO(10) theories are summarized in Table 10b. For SO(11) theories, we can have 2NS
spinor half-hypermultiplets and NF vector hypermultiplets. The marginal SO(11) theories are
summarized in Table 10c. For SO(12) theories, we can have NF fundamental hypermultiplets,
along with 2NS spinor and 2NC conjugate spinor half-hypermultiplets. So we have a large
number of exceptional theories as summarized in Table 11b. Note that the last theory in thistable could either be non-marginal or exotic, as we are presently unable to verify its status
due to computational limitations. All the other theories are marginal. For SO(13) theories,

27

SciPost Phys. 14, 122 (2023)

Table 7: Marginal S p(3) gauge theories with NTAS rank-3 antisymmetric, NAS antisymmetric, NF fundamental matters.

Cphys

NTAS NAS NF

F

1

0

5

E6

(1)

3φ1 ≥ φ3 ≥ 2φ1 = φ2 ≥ 0

1
2
1
2

1

S p(4)

φ1 = φ2 /2 = φ3 /3 ≥ 0

0

5
2
19
2

SO(19) × U(1)2φ1 ≥ φ2 = φ3 ≥ φ1 ≥ 0

0

2

0

C2 at θ =0,(1)

φ1 = φ2 /2 = φ3 /3 ≥ 0

S p(2)×U(1) at θ =π
0

1

(1)

{2φ1 ≥φ3 ≥φ1 ∩ φ3 ≥φ2 ≥(φ1 +φ3 )/2} ∪E8 × SU(2)8{3φ1 ≥φ3 ≥2φ1 ∩ 2φ1 ≥φ2 ≥(φ1 +φ3 )/2}, φ1 ≥ 0
0

0

(1)

12

φ 1 = φ 2 = φ3 ≥ 0

D12Table 8: Marginal SO(7) gauge theories with NS spinor and NF fundamental matters.

Cphys

NS

NF

F

7

0

S p(7) × U(1)61

A11 × SU(2)φ1 = φ2 /2 = φ3 ≥ 0

5

2

C7

(1)

φ1 = φ2 /2 = φ3 ≥ 0(2)(2)
E6 × A5(2)
A12

φ1 = φ2 ≥ φ3 ≥ φ1 /2 ≥ 0

4

3

2

4

φ1 = φ2 /2 = φ3 ≥ 0(2)

φ1 = φ2 = 2φ3 ≥ 0

we can have 2NS spinor half-hypermultiplets and the marginal theories are summarized inTable 11c. SO(14) case has merely one exceptional theory with a NS = 1 spinor (or NC = 1
conjugate spinor) and NF = 6 fundamental hypermultiplets as listed in Table 11d. An E6 gauge
theory can have NF ≤ 4 fundamental hypermultiplets and an E7 gauge theory can have 2NF ≤ 6fundamental half-hypermultilets. However, we are not sure whether or not the theories E6 +4F
and E7 + 3F are exotic. See Section 4.4 for more discussions about this. The N = 1 E8 gauge
theory cannot have any hypermultiplets, and the E8 theory without matter flows to a 5d SCFT
at the UV fixed point.

4.4

Exotic exceptional theories

In our main conjecture, we require that a physical Coulomb branch should have rationalboundary. In geometry this means that a physical Coulomb branch must be determined asa subregion of the Weyl chamber bounded by hyperplanes where some of 4-cycles shrink to
2-cycles or contract to a point. In addition, it is necessary that when a 4-cycle shrinks to a 2cycle, there must exist an associated shrinking 2-cycle. Such hyperplanes can be characterized
by the condition that some of string tensions vanish,i.e. Ti (φ) = 0 for some i. The rationalPboundary condition can therefore be recast as j n j φ j = 0 with n j ∈ Z for each hyperplane.
However, we notice
P that there are some cases where sub-chambers are bounded by irrational boundaries at j n j φ j = 0 with n j ∈ R\Q. Such irrational boundaries are not allowed

28

SciPost Phys. 14, 122 (2023)

Table 9: Rank 4 exceptional gauge theories.
(d) Marginal SO(8) gauge theories with NS
spinor, NC conjugate spinor, NF fundamental
matters. Due to triality of the D4 Dynkin diagram, for each of the above theories there
exists an entire family of theories obtained bypermutation of the values (NF , NS , NC ).

(a) Exceptional SU(5)κ gauge theories with
NAS antisymmetric, NF fundamental matters.
The first three are marginal theoriese and thelast one is an exotic theory.

NAS

NF

|κ|

F

3

3

0

(1)
C3 × SU(3)

NS

NC

NF

F

3

1

3

SU(4) × SU(2) × U(1)

5

2

0

0

5

11
2

A14

SO(10) × U(1)

5

1

1

A13

3

2

3
2

E6 × U(1)440

E6 × E6

4

3

1

F4 × S p(4)

4

2

2

A7 × D73

3

2

C6 × S p(2)(2)(b) Marginal S p(4) gauge theories with NTAS
rank-3 antisymmetric, NF fundamental matters.

NTAS

NF

F

1
2

4

SO(8) × U(1)

(2)
(2)
(2)

(2)

(1)

(2)

(2)

(1)

(e) Marginal SO(9) gauge theories with NS
spinor, NF fundamental matters.

(c) Exceptional F4 gauge theories with NF fundamental matters.

NS

NF

F

4

1

E6 × SU(2)

(2)

(1)

NF

F

3

3

C3 × S p(3)3S p(3) × SU(2)25A9 × A4

1

6

A14

(2)

(2)

(2)

due to geometric considerations, as the class of the would-be 2-cycle that vanishes on such aboundary would have irrational coefficients and therefore be ill-defined. We also remark thatthe dictionary between geometry and field theory teaches us that these boundaries can equivalently be characterized by the vanishing of some BPS particle with irrational central charge,
and therefore we find we must also dismiss such boundaries on physical grounds. This may
imply that either we cannot enter such sub-chambers or that we will encounter a geometrictransition such as a flop transition before we meet the irrational hyperplane, which cannot
be captured by our perturbative analysis. When this happens, we admit that our perturbative
analysis using the condition Ti (φ) = 0 cannot determine the physical Coulomb branch in the
sub-chamber exactly. If a theory has another sub-chamber with rational boundaries, we canapply our criteria for the sub-chamber and classify the theory accordingly. However, when all
the sub-chambers are bounded by irrational hyperplanes, it turns out to be rather difficult todetermine whether or not the theory has a 5d CFT fixed point. We find a handful of such exotic
theories, which we describe in more detail below.
For example, 5F + SU(2) × SU(2) + 2F has a single physical sub-chamber with irrationalboundaries. The sub-chamber is given by2
1
φ1 < φ2 < p φ1 ,
3
2

(47)

where φ1 , φ2 are scalar vevs of two SU(2) gauge groups respectively. In the above subchamber, string tensions are positive and moreover the metric is positive definite at infinite
29

SciPost Phys. 14, 122 (2023)

Table 10: Rank 5 exceptional gauge theories.
(a) Marginal SU(6)κ gauge theories with NTAS rank-3 antisymmetric, NSym symmetric, NAS antisymmetric, NF fundamental matters.

NTAS NSym NAS NF(b) Marginal SO(10) gauge theories
with NS spinor, NC conjugate spinor,
NF fundamental matters.

|κ|

F

4

2

NS NF

F

2

0

0

0

0

(2)
(2)
A2 × A2

3
2
3
2
3
2

0

0

5

0

SO(3)×SU(5)×U(1)2

3

4

0

0

3

2

SO(3)×SU(3)×U(1)2

2

6

0

0

0

9
2

SO(3) × U(1)2

1

7

1

0

1

4

0

D4 × A1 × U(1)

1

0

1

3

3
2

A2 × A2 × SU(4)

1

0

1

0

4

SU(2) × U(1)2

1

0

0

10

0

D10

1

0

0

9

3
2

E8 × A2

NS

NF

F

1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2

1

0

1

0

?

5
2

0

S p(2) × U(1)

?

2

3

A5 ×A2 ×A2

D4 × SU(2) × U(1)32

5

A2 × S p(5) × U(1)(2)

SU(5) × U(1)1

7

A13 × A1 × U(1)(2)

SU(3) × U(1)12

8

S p(9)

(1)

(1)

(2)

(2)

(1)

(1)

(1)

1

0

0

0

2

2

0

2

2

0

2

0

3
2
3
2
1
2
7
2

0

1

9

0

SU(11) × U(1)

0

1

8

3
2

SO(16)×SU(2)×U(1)

0

0

13

0

SU(13) × U(1)2

0

0

9

3

SU(9) × U(1)2

0

0

3

0

3

D4 × SU(2)

0

0

3

0

1

Sp(3) × U(1)

0

0

3

0

0, 2

SU(3) × U(1)2

0

0

0

0

9

U(1)

(3)

(1)
B3 × SU(4)(1)
C3 × S p(4)
(1)
(2)
(2)
A11 ×A2 ×A1(2)
A15(c) Marginal SO(11) gauge theories with NS spinor, NF fundamental matters. We use U(1)(2) for aU(1) compactified with an outer automorphism twist.

(2)

(2)

(2)

(2)

(2)

(1)

(3)

coupling. However, it is clear from the above expression that one of the boundaries is definedby an irrational linear combination of Coulomb branch parameters,p
φ1 + 2φ2 = 0 .
(48)
From geometric considerations, we believe this boundary cannot correspond to the vanishingof a 4-cycle with rational divisor class, and hence the correspondence between BPS data andgeometric data is invalid in this description. This is an indication that a flop transition ofsome sort—possibly due to the appearance of massless instantons—must occur before we meetthis wall on the Coulomb branch. The presence of spurious boundaries of the above type isindicative of our inability to compute instanton masses and other non-perturbative data using
only the perturbative expression for the prepotential.
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SciPost Phys. 14, 122 (2023)

Table 11: Rank 6,7,8 exceptional gauge theories.
(a) Exceptional SU(7)κ gauge theories with
NTAS rank-3 antisymmetric and NF fundamental matters. The first two are marginal, while
the last one could be non-marginal or exotic.

NTAS NF

|κ|

F

1

6

0

D6

1

5

3
2

SO(10)×SU(2)×U(1)10(c) Marginal SO(13) gauge theories with NS
spinor, NF fundamental matters.

NS

(1)

U(1)

(b) Exceptional SO(12) gauge theories with
NS spinor, NC conjugate spinor, NF fundamental matters. The last theory could be nonmarginal or exotic and the other theories are
all marginal.

NS

NC

NF

F

5
2

0

0

S p(2) × U(1)

2

0

4

E6 × A2 × A2

3
2
3
2
3
2

1

(2)

5

1
2

9

NS
1

NF

F

6

(2)
(1)
A11 × A1(e) Exceptional E6 gauge theories with NF fundamental matters. This theory can be nonmarginal or exotic.

(2)

(2)

1

F
(1)
(2)
A9 × A1(2)
A17(d) Marginal SO(14) gauge theories with Ns
spinor, NF fundamental matters.

2

5

NF

NF

F

4

SU(4)×SU(2)×U(1)1

SU(4) × SU(2)

1
2

4

(2)
A2 × S p(4)06

S p(6)×SU(2)×U(1)

1

1

4

A7 × A1 × A1 × U(1)(2)

NF

F

1

1
2

6

S p(6) × SU(2)2

3

?

1

0

8

A15 × A1

1
2
1
2

1
2

8

(2)
A15 × U(1)(2)09

S p(9) × U(1)

2

1
2

0

SU(4)

(2)

(1)

(2)

(f) Exceptional E7 gauge theories with NF fundamental matters. This theory can be nonmarginal or exotic.

(1)

(1)

(g) E8 gauge theory has no matters.

NF

F

0

;

It turns out that the 5F + SU(2) × SU(2) + 2F theory is dual to SU(3) 3 + 9F. These two2dual theories admit the same brane construction in Type IIB string theory. As we discussed
in Section 4.3, the dual SU(3) gauge theory has non-trivial 6d completion in the UV fixed
point having a zero eigenvalue of the metric at infinite coupling. In fact, we can obtain theSU(2) × SU(2) gauge theory description from the SU(3) gauge theory by turning on mass parameters such as a1 > m1,2,3,4,5,6,7 > a2 , a2 > m8,9 > a3 where a1,2,3 are Coulomb parametersof SU(3) and mi is the mass of the i-th hypermultiplet. This corresponds to two consecutive
flop transitions in the corresponding geometry. This supports our expectation that the geometry undergoes flop transitions before we meet the irrational boundary in the Coulomb branch.
This example also shows that while we may not be able to reach the UV fixed point with theSU(2) × SU(2) gauge theory description, the 6d fixed point can nevertheless be attained by
way of the SU(3) gauge theory description after the flop transitions. Therefore, when we have
only physical Coulomb branches bounded by irrational curves, then the naive application of
our criteria cannot tell us whether or not the theory is marginal. We call such theory as exotic.
31

SciPost Phys. 14, 122 (2023)

In our classification, we have encountered five possible cases of exotic theories, only oneof which we have explicitly confirmed as such. They are:
1. SU(5) 3 + 3AS + 2F (confirmed)
2

2. SU(7)5 + 1TAS (undetermined)
3. SO(12) + 2S + 21 C (undetermined)
4. E6 + 4F (undetermined)
5. E7 + 3F (undetermined)
For the first case above, the SU(5) theory, Cphys consists of a single component with irrationalboundaries. Within this chamber, as expected, the string tensions are positive and the metricis positive definite. However, we expect that this theory has a 6d fixed point based on the(2)affinization of the flavor symmetry E6 × U(1) according to a 1-instanton analysis. However,if this is case, it remains unclear what dual physical description we must introduce (as in thecase of 5F + SU(2) × SU(2) + 2F) in order to reach the 6d fixed point through the Coulomb
branch.
We have also numerically confirmed the existence of physical Coulomb branches Cphys forthe other theories in the above list. Unfortunately, due to computational limitations, we havebeen unable to precisely test whether or not the boundaries of these regions are rational or
irrational. For this reason, it is unclear whether or not these theories are exotic in the sense
we have described above. However, we note the two exceptional theories in the above list
each have two distinct candidates for their 6d lifts, namely 6d SCFTs corresponding to a single
curve with self-intersection −n, where n = 1, 2. In the case of E6 , the n = 1 theory has NF = 5,
while the n = 2 theory has NF = 4. In the case of E7 , the n = 1 theory has NF = 7/2, while then = 2 theory has NF = 3.

56d Theories on a Circle

In this section we present some 6d SCFTs that are the lifts of some of the marginal 5d theories
we found. We note that most of these theories have already been discussed elsewhere in theliterature. We discuss the new theories in detail, while for the known theories we merely quote
the results.

5.1

Generic Sp(N ) and SU(N ) with fundamental and antisymmetric matter

We start with the cases S p(N ) and SU(N ), for generic N , with matter in the fundamental andantisymmetric representations. These theories can be constructed with ordinary brane web
systems, where the S p(N ) and SU(N ) theories with antisymmetric matter can be brought tothis state by using an O7− -plane and then decomposing it into a pair of 7-branes [17]. The 6dlifts of these types of theories were studied extensively in [14, 21, 31, 33], and we shall quote
their results for the theories of interest.
The 6d SCFTs appearing in these cases can all be engineered by a brane system involving
NS5-branes, D6-branes and D8-branes in the presence of an O8− -plane. When compactified
and T-dualized the system becomes a brane configuration of NS5-branes, D5-branes and D7branes in the presence of two O7− -planes. Upon decomposing the two O7− -planes to a pairof 7-branes, one obtains the brane webs of the associated 5d gauge theories.

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SciPost Phys. 14, 122 (2023)

SU(N)0 + (2N + 4)F and Sp(N − 1) + (2N + 4)F
The 5d gauge theory SU(N )0 + (2N + 4)F is known to lift to 6d [34]. The 6d lift was worked
out in [15, 31], and is the so-called (DN +4 , DN +4 ) conformal matter [35]. This theory has alow-energy description on the tensor branch as the 6d gauge theory S p(N − 2) + (2N + 4)F.
In F-theoretic notation, this theory is denoted
spN −2

1[SO(4N + 8)] .

It is straightforward to see that the dimensions of the Coulomb branch in both descriptions(1)
agree. Also the 1-instanton spectrum of the 5d theory is consistent with an enhanced D2N +4 ,agreeing with the spectrum expected from the compactification of the 6d theory on a circle.
The 5d gauge theory S p(N − 1) + (2N + 4)F also lifts to 6d and is in fact dual toSU(N )0 + (2N + 4)F on account of both theories lifting to the same 6d SCFT [31].
SU(N)0 + 1AS + (N + 6)F
The 5d gauge theory SU(N )0 + 1AS + (N + 6)F lifts to the 6d SCFT which has a low-energy
description on the tensor branch as the 6d gauge theory SU(N − 1) + 1AS + (N + 7)F [21, 31].
In F-theoretic notation this 6d SCFT is denotedsuN −11[SU(N + 7)] .

Again it is easy to see that the Coulomb branch dimensions agree. The 6d and 5d theories bothhave a Higgs branch, associated with giving a vev to the antisymmetric matter, upon which we
reduce to the previous case.
We can also compare the global symmetries of these theories, which are summarized inTables 1a and 2. For N > 4 The 1-instanton analysis suggests that the current spectrum of the(1)
5d theory is consistent with AN +6 which again agrees with the 6d theory on a circle. There
are however additional currents when N ≤ 5. For N = 3 an antisymmetric hypermultiplet is
identical to an antifundamental hypermultiplet so in 5d this case is identical to the previous
case. This feature is correctly captured by the 6d description, as for SU(2) the antisymmetricrepresentation is a singlet. For N = 4 the antisymmetric represenation is real and the U(1)associated with this representation in the general case is perturbatively enhanced to SU(2).
There are then additional 1-instanton currents, and the spectrum of the 5d theory is consistent(1)with a larger enhancement to A11 . This again agrees with the perturbative enhancement in 6d.
Finally when N = 5 we also get additional currents, and the spectrum is consistent with(1)
(1)
an enhancement to A11 × A1 . This again agrees with the perturbative enhancement in 6d.
SU(N)0 + 2AS + 8F
The 5d gauge theory SU(N )0 + 2AS + 8F has different 6d lifts depending on whether N is evenor odd [21]. The 6d lift can be described on the tensor branch as a low-energy 6d semi-gauge
theory. The description for these two cases is shown in Figure 9, where (a) is the N = 2n case,and (b) is the N = 2n + 1 case. Alternatively, we also provide an F-theoretic description ofboth cases,
[E7 ]

1

[SO(16)]su2

2[SU(2)] ,
sp1

1

su2

2

su2

···

2

su2

2

···
33

su2

su2

su2

su2

2

2

2

2

[SU(2)][SU(2)] ,SciPost Phys. 14, 122 (2023)

n−1

𝑟𝑎𝑛𝑘 1
𝐸 𝑠𝑡𝑟𝑖𝑛𝑔

SU

SU

2

2

...

SU

SU

2

2

2

SU

SU

2

2

2

(𝑎)
n

8

SU

SU

SU

2

2

2

...

2

(𝑏)

Figure 9: The low-energy description on the tensor branch of the 6d SCFTs which are
the lifts of the 5d SU(N )0 + 2AS + 8F gauge theories. Quiver (a) describes the caseN = 2n, while quiver (b) the case N = 2n + 1. In quiver (a) the arrow into the rank
1 E-string theory represents gauging a subgroup SU(2) ⊂ E8 of the global symmetrygroup of the E-string theory.
where the upper diagram is the N = 2n case, the lower diagram is the N = 2n + 1 case. Notethat there are n − 1 2 curves in both diagrams above.
One can see that the Coulomb branch dimensions of the 5d theories and 6d theories ona circle agree. The global symmetries also agree, where instantons in the 5d gauge theory
lead to additional conserved currents consistent with what is expected from the 6d theory on
a circle. The results for the various cases are summarized in Tables 1a and 2.
SU(N)± 3 + 2AS + 7F
2

The 5d gauge theory SU(N )± 3 +2AS+7F is also known to lift to 6d, and again goes to different
2
6d theories depending on whether N is even or odd. The cases N = 3, 4, 5 were explicitly
discussed in [21] and it is straightforward to generalize the construction to generic N . The 6d
SCFTs can be again conveniently represented using the low-energy description on the tensor
branch. Descriptions of the two cases are shown in Figure 9, where (a) is the N = 2n case,and (b) is the N = 2n + 1 case. We also provide F-theoretic descriptions,[SO(16)]

[E7 ]

1

sp1

1

su2

su2

2

2
[SU(2)]

su2

2

su2

2

···

···

su2

su2

2

2
[SU(2)] ,

su2

su2

2

2
[SU(2)] ,

II

2
II

2

where the upper diagram is the N = 2n case and the lower diagram is the N = 2n + 1 case.
Note that there are n − 1 2 curves in the upper diagram and n 2 curves in the lower diagram.
One can see that the Coulomb branch dimensions of the 5d theories and 6d theories on acircle agree. The global symmetries also appear to agree, at least as far as can be seen fromthe 1-instanton analysis. Again the results for the various cases are summarized in Tables 1aand 2.
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SciPost Phys. 14, 122 (2023)

n−1

8

SU

SU

SU

2

2

2

...

SU

SU

2

2

(𝑎)

N= (2,0)𝑡𝑒𝑛𝑠𝑜𝑟32
n−1

𝑟𝑎𝑛𝑘 1
𝐸 𝑠𝑡𝑟𝑖𝑛𝑔

SU

SU

2

2

...

SU

SU

2

2

N= (2,0)𝑡𝑒𝑛𝑠𝑜𝑟3
2

2(𝑏)Figure 10: The low-energy description on the tensor branch of the 6d SCFTs which
are the lifts of the 5d SU(N )± 3 +2AS+7F gauge theory. Quiver (a) describes the case2
N = 2n while quiver (b) the case N = 2n + 1. The arrow into the N = (2, 0) tensor,
which appears on the rightmost in both quivers, represents gauging the SU(2) global
symmetry of the tensor (this is part of its S p(2) R-symmetry that is seen as a global
symmetry from the N = (1, 0) point of view).
Sp(N − 1) + 1AS + 8F and SU(N)± N + 1AS + 8F
2

The case of S p(N − 1) + 1AS + 8F is well known to lift to the rank N − 1 E-string theory [36].
On the other hand, to our knowledge SU(N )± N + 1AS + 8F, for N > 3, has not been discussed
2
in the literature. From brane webs we can argue that the 5d gauge theory SU(N )± N +1AS+8F
2should have a 6d lift.
To argue this we consider the brane web description of this theory, which can be engineeredusing an O7− -plane and resolving it. The brane web looks slightly different depending onwhether N is even or odd. For simplicity, we shall show here only the N even case, thoughmost of the steps and results apply also to the N odd case.
Figure 11 shows the web for an SU(N ) N −NF +8 + 1AS + NF F gauge theory. For simplicity
2
we have only shown the external legs and not how these connect inside the web, as that isirrelevant for this discussion. By manipulating the web we arrive at the web on the bottom
left of Figure 11. When N > 10 one of the legs collides with another leg and further transitions ensue. We can calculate the monodromy “felt” by this leg when going around the entire
web, and we find it to be the S L(2, Z) transformation T 8−NF . Thus we see that if NF < 8 andthe transition does not terminate before going full circle, the brane will come back less divergent. Therefore this process will terminate after a finite number of rotations. In particular, for
NF = 0, this means that a fixed point exists for every N . For NF = 8 the brane returns back toitself while for NF > 8 the brane returns more divergent.
This suggests then that for NF < 8 a 5d fixed point exists, while for NF > 8 neither a 5d nor
a 6d fixed point exists. The case NF = 8 seems to be a 6d lift. For N ≤ 10, the process alreadyterminates at the step shown in the bottom left of Figure 11, and we can show this explicitly.
Furthermore we can argue that this theory is dual to S p(N − 1) + 1AS + NF F.
This follows from 7-brane manipulations. We have shown the relevant steps in Figure 12
for the simpler case of N = 6. The manipulations for the other cases are similar, and we omit
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SciPost Phys. 14, 122 (2023)

3(3, −1)n−1(3, −1)n−1(5, −1)(2n + 3, −1)n(2n + 3, −1)

n
𝑁𝐹

𝑁𝐹

(8n − 13, −(2n − 3))n+57n − 5(2n − 1)(5, −1)
2n − 13n − 43𝑁𝐹

n(2n − 5, −1)

𝑁𝐹(2n − 1)(5, −1)

Figure 11: Brane web manipulation for SU(N ) N −NF +8 +1AS+NF F, in the case N = 2n.
2For ease of drawing, only the external legs are shown. We use a black ‘×’ to represent
the NF 7-branes that give the flavors. The picture on the top left is for the initial web.
Moving the (3, −1) 7-brane through the single (1, −1) 5-brane leads to the picture in
the top right. Further moving the (3, −1) 7-brane through the n − 1 (1, −1) 5-branes
leads to the picture in the bottom right, where we have also moved the (5, −1) 7brane through the (2n + 3, −1) 5-brane. Finally, we move the (8n − 13, −(2n − 3)) 7brane through all the (1, −1) and (1, 1) 5-branes leading to the picture at the bottom
left. If n ≤ 5 then no further transitions are required.
them for brevity. Therefore, in that range at least, the SU(N ) N −NF +8 +1AS+NF F class of theories
2

go to a 5d fixed point for NF < 8, and the SU(N )± N + 1AS + 8F theory should lift to the 6d2
rank N − 1 E-string theory. We suspect this to be true for all N .

5.2

Generic SO(N ) and SU(N ) with symmetric matter

Next we examine the cases of SO(N ) and SU(N ) theories with matter in the symmetric representation for generic N . This class of theories can be engineered in string theory by branewebs in the presence of O7+ -planes [17]. The 6d lifts for SO(N ) with fundamental matter, andSU(N ) with symmetric and fundamental matter were studied in [22], and it is straightforward
to extend these results to some of the other cases.
The 6d SCFTs in these cases can be engineered in string theory by a brane system involving
NS5-branes, D6-branes and D8-branes, but the compactification is done with a Z2 twist in a
discrete symmetry. In the brane system this discrete symmetry is manifested as worldsheet
parity combined with a reflection in the direction orthogonal to the D8-branes. Performing
T-duality, the authors of [22] mapped this system to a configuration involving NS5-branes andD5-branes in the presence of an O7+ and O7− -planes. Decomposing the O7− -plane then leads
to the brane webs for the 5d gauge theories.
SO(N) + (N − 2)F
The case of 5d SO(N ) + (N − 2)F gauge theory is known to lift to a 6d SCFT. The 6d SCFT was
worked out in [22] to be the one living on 2 M5-branes probing a C2 /ZN −2 singularity. The
compactification is done with a Z2 twist in a discrete symmetry of the 6d SCFT. Particularly,

36

SciPost Phys. 14, 122 (2023)

5(3,1)
16

5(3,1)
5(5, −1)

5

5(5, −1)
5

𝑁𝐹

4

𝑁𝐹
𝑇 2 transformation

5
5

5(3, −1)

4

𝑁𝐹

Figure 12: Brane web manipulation for SU(6) 14−NF + 1AS + NF F. The picture on
2

the top left is just the web on the bottom left of Figure 11 for the case of n = 3.
Moving the (1, −1) 7-brane from top to bottom, we obtain the web on the top right.
Performing an S L(2, Z) transformation T 2 , we end up with the web on the bottom
right—this web is a known description of S p(5) + 1AS + NF F, suggesting that these
two theories are dual up to free hypermultiplets.
the 6d SCFT has a 1-dimensional tensor branch along which the theory has a low-energy
description as an SU(N − 2) + (2N − 4)F gauge theory. In F-theory it is described by:suN −22[SU(2N − 4)] .

The discrete symmetry in question acts on the gauge theory as charge conjugation. It can also
be thought of as the orbifold generalization of the outer automorphism twist compactificationsof the A type (2, 0) theory.
(2)
The global symmetry of the 6d theory is consistent with the A2N −5 found for SO(N )+(N −2)F
using 1-instanton analysis. Additionally, the Coulomb branch dimension agrees between thetwo, after accounting for the fact that the twist projects out the odd dimension Coulomb branch
generators that would have come from the SU(N − 2) vector upon reduction to 5d.
SU(N)0 + 1Sym + (N − 2)FThis case was also covered in [22]. The 6d lift is the SCFT living on 3 M5-branes probing a
C2 /ZN −1 singularity, which has the F-theoretic description:[SU(N − 1)]

suN −1

2

suN −12[SU(N − 1)] .

The compactification is again done with a twist, generalizing the outer automorphism twist
now to the case of the A2 (2, 0) theory, so the twist also acts non-trivially on the tensors. In
the low-energy gauge theory description, (N − 1)F + SU(N − 1) × SU(N − 1) + (N − 1)F, thisdiscrete symmetry is manifested as a combination of quiver reflection and charge conjugation.
As consistency checks, we note that the global symmetries and Coulomb branch dimensions
agree.

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SciPost Phys. 14, 122 (2023)

SU(N)0 + 1Sym + 1AS
We can also consider the case of 5d SU(N )0 + 1Sym + 1AS. This theory also lifts to 6d, andbehaves differently depending on whether N is even or odd. For odd N we can find the 6d
SCFT by generalizing the results of [22]. We find the 6d SCFT to be the one living on N
M5-branes probing a C2 /Z2 singularity, where again the compactification is done with a Z2
twist generalizing the outer automorphism twist for the AN −1 (2, 0) theory. The 6d SCFT has
a low-energy gauge theory description as 2F + SU(2) × SU(2) · · · SU(2) × SU(2) + 2F, where
there are N − 1 SU(2) groups. This theory also has the following F-theory description:[SU(2)]

su2

2

su2

2

···

su2

2

su2

2

[SU(2)] ,where there are N − 1 2 curves in the above diagram. The Z2 discrete symmetry by which we
twist acts on the gauge theory as quiver reflection. The web for the even N case has a different
form, and appears to go to a different type of theory.
We can again perform several consistency checks for the odd N case. As indicated in(1)
Table 1a, the symmetry of the 5d theory is at least A1 × U(1). This is consistent with theglobal symmetry of the twisted theory. We also note that the Coulomb branch dimension
agrees.
It is instructive to also consider Higgs branch flows. For instance, the 5d theory has a flowfrom SU(N )0 + 1Sym + 1AS to SO(N ) + 1AS triggered by giving a vev to the symmetric matter,and one to S p( N 2−1 ) + 1S triggered by giving a vev to the antisymmetric matter—these are justthe maximally supersymmetric SO and S p gauge theories. These flows should then also exist
in the 6d theory. Indeed we can go on the Higgs branch of the 6d SCFT and flow to either the
AN −1 or AN −2 (2, 0) theories. Compactification with a twist of the latter yields maximally supersymmetric SO(N ) while the same for the former yields maximally supersymmetric S p( N 2−1 )(with theta angle θ = π) [29].
This leaves the case of even N , which seems to be of a different nature than the theories
discussed so far. This is also apparent from the Higgs branch discussion. We again can flow
to either the maximally supersymmetric SO(N ) or S p( N2 ) gauge theory. However the former
when N is even lifts to the D N type (2, 0) theory with untwisted compactification. The latter,2
alternatively, lifts to twisted compactifications of either the D N +1 type (2, 0) theory or the AN
2type (2, 0) theory depending on the theta angle of the 5d theory [29]. It seems quite non-trivial
to find a theory with this behavior, and therefore we reserve a determination of this case for
future work.
SU(N)± N + 1Sym
2

To our knowledge this class of theories has not been discussed in the literature. We can argue
that this class should exist, at least as a 6d lift gauge theory, as follows. We can try to engineer
this theory by using brane webs in the presence of an O7+ -plane. The existence of the theory isentirely determined by the asymptotic behavior of the 5-branes. Consider changing the O7+ plane to an O7− -plane with 8 D7-branes. This does not change the monodromy of the object
and so does not change the asymptotic behavior of the 5-branes. However, this changes thegauge theory to the previously discussed SU(N )± N +1AS+8F, which we have argued has a 6d
2lift. Therefore, it is reasonable to assume that SU(N )± N +1Sym also has a 6d lift. We note that2
the equivalence in monodromy between an O7+ -plane and an O7− -plane with 8 D7-branes isthe brane web manifestation of the invariance of the prepotential under exchanging a single
symmetric hypermultiplet with a single antisymmetric and 8 fundamental hypermultiplets.
We can then ask what theory is the 6d lift. By examining the brane web in the case N = 3,
we can reformulate the problem in a manner where we are in a position to apply the results
38

SciPost Phys. 14, 122 (2023)

of [22]. We then find that the 6d theory is the A4 (2, 0) theory compactified with an outerautomorphism twist. This implies that this theory is dual to maximally supersymmetric S p(2)π ,and should thus have an enhancement of SUSY in the UV. Similarly, for N > 3, we suspect thatthe 6d theory is dual to maximally supersymmetric S p(N − 1)π and goes to the A2N −2 (2, 0)
theory compactified with an outer automorphism twist.

6

Discussion and Future Directions

In this paper, we have given a clear physical explanation for why the naive Coulomb branch (i.e.
the fundamental Weyl chamber) is not necessarily physical, and why it is therefore necessary to
restrict a weakly-coupled gauge theory description to a subset of the Coulomb branch satisfying
certain necessary physical criteria. It is necessary that the BPS spectrum consists of degrees of
freedom with positive masses and tensions, and the metric be positive definite on this physicalregion of the Coulomb branch. By extending these ideas, we have proposed a set of conjecturesin Section 3.1, 3.2 which we can use for the classification of 5d N = 1 gauge theories withinteracting fixed points.
By using these conjectures, we have given an exhaustive classification of 5d N = 1 gauge
theories with simple gauge group, which to our knowledge includes all known examples in
the literature as a subset—for example, the theories described in [13–17, 21, 22, 26]. Ourclassification consists of two types of theories, namely standard theories (see Table 1) and
exceptional theories (see Tables 3-11). Our strategy has been to identify extremal theories
saturating the matter bounds imposed by our physical criteria and then use the fact that alldescendants of such extremal theories will have non-trivial UV fixed points.
In practice, we used our main conjecture to test a number of low rank cases explicitly using
a symbolic computing tool. Furthermore, we combined Conjectures 2 and 3 to circumvent the
problem that a complete analysis of the naive Coulomb branch rapidly becomes intractable
as the rank of the gauge group increases; the combination of our conjectures enabled us tomake a full classification using a mixture of symbolic and numerical searches. The standard
theories are particularly tractable and thus we were able to perform the classification by hand.
Computational expense has only hindered our understanding of a few isolated cases of possible
exotic theories, as discussed in Section 4.4.
There are a plethora of open questions that remain unresolved. The first question concerns
our main conjecture, which asserts that the positivity of monopole string tensions can correctly
identify the physical Coulomb branch when all of the mass parameters are switched off. As
we explained in Section 3.1, the criterion of positive string tensions can in general miss floptransitions associated to the emergence of massless instantons without tensionless strings (notethat the perturbative flop transitions are already captured by the gauge theory interpretation of
the prepotential.) However, we remark that the massless instanton spectrum can be capturedby an explicit computation of the BPS partition function as described in [37–42], and thus it
would be interesting to see if we could test our main conjecture against an explicit computationof the instanton spectrum.
Another puzzle is why, as asserted by Conjecture 2, the positivity of the monopole string
tensions guarantees the positivity of the metric. It is unclear from a mathematical perspective why this property should hold true for any gauge theoretic prepotential evaluated on
the naive Coulomb branch, and moreover what properties of this function are essential forthis phenomenon. Futhermore, Conjectures 1 and 3 seem to lack both mathematical proofs
and physical explanations, given that these conjectures involve testing regions of the naive
Coulomb branch that are obviously unphysical.
As explained in Section 4.4, there are four potentially exotic theories that we were not able

39

SciPost Phys. 14, 122 (2023)

to confirm due to computational limitations. With a better algorithm or more computational
resources, it is possible to check explicitly whether or not these theories are indeed exotic.
However, there is a more striking puzzle presented by the notion of exotic theories. From geometric considerations, we expect that truly exotic theories (such as SU(5) 3 +3AS+2F) undergo2
some number of flop transitions before reaching the irrational boundaries of the putative physical Coulomb branch. If these theories could be constructed using geometry, this notion could
tested explicitly. Alternatively, dual descriptions (as in the case of 5F + SU(2) × SU(2) + 2F,
which has a dual description as SU(3) 3 +9F) that are valid in the vicinity of the irrational com2
ponents of the boundary could provide a clear indication that such theories are well-definedand have an interacting fixed point.
Lingering puzzles aside, there are also many future directions to be explored. For instance, many of the 5d theories appearing in our classification are still lacking a realization
either in terms of brane configurations or local Calabi-Yau singularities, and it would be desirable to work out such descriptions in order to provide further evidence for the existence of
these theories, as well as to capture the subtleties of the non-perturbative physics. It would
be worthwhile to explore the possibility that all of the marginal theories we have identified
can be obtained by compactifying some 6d SCFT; developing an understanding of the 6d compactifications, including a possible classification of the intermediary Kaluza-Klein theories, iscritical for this purpose.
It would also be quite interesting to further develop a precise geometric understandingof the vanishing monopole string tensions and instanton masses that signal the breakdown
of an effective gauge theory description. The field theoretic expression for F clearly encodes
the appearance of massless electric particles in a collection of hyperplanes w · φ + m f = 0
subdividing the fundamental Weyl chamber, across which the Chern-Simons levels jump discontinuously. Geometrically, these “jumps” are interpreted as the result of flops, which can beassociated with the interchange of two nonisomorphic resolutions of the same singular CalabiYau geometry sharing a common blowdown. Both field theoretically and geometrically thesetransitions are rather transparent. However it is much more difficult to analyze the extremaltransitions associated with the vanishing of monopole tensions or instanton masses away fromthe conformal point. This point is especially relevant for the distinction between rational and‘irrational’ classes characterizing the exotic theories in our classification.
Finally, we would like to extend our classification of simple gauge groups to a full classification of 5d N = 1 SCFTs, including quiver gauge theories and non-Lagrangian theories. It
is obvious that a physically sensible theory should have a positive definite metric on its physical Coulomb branch having only degrees of freedom with positive messes and tensions. In
principle, we can classify general theories with gauge theory description by applying the same
techniques we used for the present classification though it may be practically a tremendous task
at the moment. A classification of non-Lagrangian theories in the same spirit as our current
classification will require a better understanding of how geometric constraints translate into
constraints on field theory, much as the case of the geometric description P2 in CY3 translates
to the (formal) field theory description SU(2) with NF = −1.

Acknowledgements
We would like to thank Thomas T. Dumitrescu, Hirotaka Hayashi, Sheldon Katz, Seok Kim,
Sung-Soo Kim, Kimyeong Lee, Kantaro Ohmori, Yuji Tachikawa, Masato Taki, Itamar Yaakov,
Futoshi Yagi and Dan Xie for useful comments and discussions. The research of P.J. and H.K.
and C.V. is supported in part by NSF grant PHY-1067976. H.K. would like to thank the Aspen
Center for Physics Winter Conference on Superconformal Field Theories in d ≥ 4, which is sup40

SciPost Phys. 14, 122 (2023)

ported by NSF grant PHY-1066293, and also the Erwin Schrödinger Institute for its workshop
on Elliptic hypergeometric functions in Combinatorics, Integrable systems and Physics for hospitality during the final stages of this work. G.Z. is supported by World Premier International
Research Center Initiative (WPI), MEXT, Japan.

A

Mathematical Conventions

Much of the notation and content of the following section is adapted from [43].

A.1

Root and coroot spaces

Let R ⊂ VR be a set of roots in the root space VR , and let VR∨ be the coroot space dual to VR . A
subset S ⊂ R is called a system of simple roots if all elements e ∈ S are linearly independentand every root e′ ∈ R belongs to the nonnegative or nonpositive integer linear span of S. A
positive root is a positive linear combination of elements of S, and similarly for negative roots.
As suggested by this definition, there exists a partition R = R+ ∪ R− ∪ R0 , where R+ is the set of
positive roots, R− = −R+ and R0 is the set of zero roots. To every choice of S, one can associatea distinguished Weyl chamber C (S), where

C (S) = {φ ∈ VR∨ | 〈φ, e〉 ≥ 0 , e ∈ S} .

(A.1)

We define simple coroots byα∨i = 2αi,
〈αi , αi 〉(A.2)

where above 〈, 〉 is the Euclidean inner product. The Cartan matrix is defined as:
Ai j = 〈αi , α∨j 〉 = 2〈αi , α j 〉〈α j , α j 〉≡ D−1
j 〈αi , α j 〉.

(A.3)

In practice, we expand the real vector multiplet scalar vev φ in a basis of simple coroots:Xφ=φi α∨i .
(A.4)
i

For some computations, we will find it convenient to expand the simple roots in the Dynkinbasis,7 or the basis of fundamental weights, as fundamental weights are canonically dual tosimple coroots. To illustrate this point, consider the inner product
X
X
X〈φ, αi 〉 =φ j Aik 〈α∨j , w k 〉 =φ j Aik δ jk =Aik φk .
(A.5)
j,k

j,k

k

We can apply the same logic above to the scalar product as applied to two coroots:
XX
X〈φ, φ〉 =φi φ j 〈α∨i , α∨j 〉 =φ j D−1
φ j h ji φi ,j A ji φi ≡
i, j

i, j

(A.6)

i, j

where in the above equation we have defined the metric tensor(h−1 )i j = (A−1 )i j D j .

(A.7)

Therefore, the vector multiplet scalars for the classical kinetic term can be said to be in the(canonical dual of the) Dynkin basis when the quadratic form is given by the inverse of the Liealgebra metric tensor (A.7).
7

Note that the rows of the Cartan matrix A are the simple roots in the Dynkin basis.

41

SciPost Phys. 14, 122 (2023)

A.2

Changing from Dynkin to orthogonal basis

In some cases, we find it more convenient to work in the orthogonal basis. Therefore, for reference we derive the linear transformation necessary to express the coroot φ in the orthogonalbasis, given an expansion of φ in terms of simple coroots α∨i .
To begin, recall that the Dynkin basis is canonically dual to the basis of simple coroots inthe following sense〈α∨i , w j 〉 = δi j ,(A.8)

In matrix form, the above equation readsAΩ t = I ,(A.9)

where we define Ω through the following relation:Ω = (AAt )−1 A.

(A.10)

Note that the rows of the above matrix Ω are the fundamental weights, where Ω can be viewed
as a change of basis matrix from the orthogonal basis to the Dynkin basis:XΩi j ê j .
(A.11)wi =j

The matrix Ω will be useful in the following derivation.
In order to determine the correct change of basis for φ, we will first convert the simple
roots αi to the orthogonal basis, using the Dynkin basis as an intermediate step. Expanding
αi , we findXX
αi =Ai j w j =Ai j Ω jk ek ,(A.12)
jj,kwhere êk is a standard basis vector. By inverting Ω and exploiting the canonical pairing between simple coroots and fundamental weights, we notice that we may write:XX(Ω−1 )ki φi ,(A.13)〈φ, êk 〉 =(Ω−1 )kl φi α∨i , w l =ii,lwhere Ω−1 is the pseudo-inverse of Ω. Therefore, the inner product between φ and a simple
root must satisfy the following equation:
!

X X
X
X
−1
〈φ, αi 〉 =Ai j Ω jk(Ω )kl φl =Ai j φ j .
(A.14)kjlj

In the above equation, we identify
eik =
A

X

Ai j Ω jk ,(A.15)

jas the components of the matrix whose rows are the simple roots in the orthogonal basis.
Therefore, it must be the case that the coroot is expressed in an orthogonal basis as
X
ej .
φi =
Ωi j φ(A.16)
j

In practice, we adopt the notationei ,ai ≡ φto be consistent with commonly used notation in the literature.
42(A.17)

SciPost Phys. 14, 122 (2023)

A.3

Lengths of weights, index of a representation

In this section, we expand roots and weights in a basis of simple roots αi=1,...,rG (this basis isP
referred to as the α-basis in [43]). The length8 l(e) of a root e = i ei αi is defined as follows:Xl(e) =
ei .
(A.18)
i

By linearity, we can extend this notion to weights w =Xl(w) =wi .

Pi w i αi :(A.19)i

The lengths of the weights of an irreducible representation R(w+ ) with highest weight w+(2)
are related to the quadratic Dynkin index cR(w+ ) (also referred to as simply ‘the index’ of therepresentation R(w+ )) [44]:(2)cR(w+ ) =X
dim(R(w+ )) + +〈w , w + 2δ〉 = C(G)l(w)2 ,
dim(G)+(A.20)w∈R(w )

where in the above expression
δ=

1 X
e.
2 e∈R

(A.21)+For classical Lie algebras G, the normalization constant C(G) is chosen so that the index of thefundamental representation is equal to 1:
C(G) = P

1.
2w∈F l(w)(A.22)

However, a different normalization is chosen for the exceptional Lie algebras.

B

Bound on Matter Representation Dimension

In our classification, we assume that the dimension of the largest matter representation (orhalf-dimension, in the case of pseudoreal representations) is no larger than the dimensionof the adjoint representation. In this section, we explain why this assumption is implied by
Conjecture 3, which asserts that the prepotential must be positive on the entire fundamental
Weyl chamber.
There exists a point φ ∗ in the fundamental Weyl chamber where the inner products
e ·φ, w·φ are equal to the lengths of the corresponding root or weight. This can be understoodas follows. In a basis of simple roots, the inner product e · φ may be expressed as:
X
X
e·φ =
ei φ j 〈αi , α∨j 〉 =ei φ j Ai j ,
(B.1)
i, j

i, j

and a similar expression holds for w · φ. The conditions placed on φ parametrizing the fundamental Weyl chamber C (S) thus translate to the following inequalities:X
Ai j φ j ≥ 0 .
(B.2)
j
8

In [43], the ‘length’ of a root or weight is referred to as the ‘height’.

43

SciPost Phys. 14, 122 (2023)

Evidently, it is always possible to find a solution φ ∗ to the equation
X
Ai j φ j = 1 , for all i ,

(B.3)

j

P
since the only constraint on the coefficients j Ai j φ j is that they be non-negative, and a Cartan
matrix is by definition nonsingular. At this point φ ∗ , we findXXe · φ∗ =e j = l(e) ,w · φ∗ =w j = l(w) .
(B.4)
j

j

Therefore, at infinite coupling 1/g02 = 0, the prepotential evaluated at this point φ ∗ is proportional toXX
6F (φ ∗ ) =|l(e)|3 −|l(w)|3 ,(B.5)
e∈rootsw∈R f

and the classical CS termX

(w · φ ∗ )3 =w∈FXl(w)3 = 0 .

(B.6)

w∈F

We claim that the difference of sums (B.5) is negative if
(2)

(2)

cadj < cR .

(B.7)

f

(2)

(2)

f

f

(In the case of pseudoreal representations, we replace cR with 12 cR .) To see this, assume
that the above inequality holds, and observe that l(w) ∈ 12 Z for any weight w, and therefore
up to an overall constant, the quadratic indices
X
X
(2)
(2)
(B.8)
cadj =
n2e ,
n2w ,
cR =
f

w∈R f

e∈adj

where nw ∈ Z≥0 . We therefore find that the quadratic indices are proportional to sums ofsquares of positive integers. If the above sums satisfy the inequality (B.7), then it must also
be true that the sums of cubes n3w , n3e satisfy the same inequality. But the sums of the cubes ofthese positive integers are proportional to the sums of the cubes of the lengths. Therefore, itfollows that (B.5) must be negative, which violates the condition imposed by Conjecture 3 atthe point φ ∗ .
(2)(2)
One can verify that dim(R f ) > dim(adj) implies that cR > cadj , which justifies the exf
clusion of matter representations with dimension greater than that of the adjoint from ourclassification.

C

Convergence of S 5 partition function

In this appendix we collect several formulas for the expression of the S 5 partition function
evaluated using localization. The main result here is that the convergence of the perturbative
contribution to the S 5 partition function is equal to the demand that the perturbative prepotential be positive on the entire Coulomb branch. We can use this to give another criterion for
the UV completeness of 5d gauge theories, which appears to coincide with the other criteriapresented in this article, at least in all examples we have checked. This is summarized in the
conjectures in section 3.2, particularly conjecture 3.
44

SciPost Phys. 14, 122 (2023)

C.1

The 5d partition function

The partition function can be represented by the path integral on S 5 , which in turn can beevaluated using localization; this partition function receives contributions from a perturbative
part and a non-perturbative part, where some aspects regarding the latter are still conjectural[45–52]. The general form is suspected to be:
Z
5

ZS =

3
Zpert Zinst
,

(C.1)

where Zpert is the perturbative part and Zinst is the non-perturbative contribution. The latter
arises from contact instantons [46, 52], which fill the S 1 in the Hopf-fibration realization of S 5
as an S 1 fibered over CP2 . They are localized at three points on CP2 hence the power in the3

integral. In general they can be expanded in a power series in q = eZinst = 1 + O e

3

r
− 16π
2
g

YM

r
− 16π
2
g

YM

:


,

(C.2)

2
is the ratio of the radius of S 5 to the coupling constant.
where r/gYM
Let us first ignore the instanton contribution, and consider only the perturbative part. We
shall use the expression for it given in [46, 47]. The full expression is then:

Z
3
3


− 4π2 r trF σ2 + πκ
1
1
3 trF σgZ=
dσe YM
detadj sin(iπσ)e 2 f (iσ)|W | Cartan
Y


1
1
1
1
1
detRI (cos(iπσ)) 4 e− 4 f ( 2 −iσ)− 4 f ( 2 +iσ) + instanton contributions ,
×

(C.3)

I

where |W | is the size of the Weyl group. Also, κ is the Chern-Simons level (if present), and f
is a function defined by:f ( y) ≡iπ y 3iy1
ζ(3)+ y 2 log(1 − e−2πi y ) + Li2 (e−2πi y ) +Li3 (e−2πi y ) −
.
2
3
π
2π
2π2

(C.4)

The integral is done over the Cartan subalgebra and the notation trR , detR stands for the
trace and determinant in the representation R, where F denote the fundamental represetnation
and adj denotes the adjoint representation.
Ignoring the instanton contributions, the above integral can be recast in the following
form [53]:Z1Z=dσe−F (σ) ,(C.5)|W | Cartan
where:F (σ) =

4π3 r
2
gYM

trF σ2 +Xπk
trF σ3 + tradj FV (σ) +trRI FH (σ) .
3
I

(C.6)

We will be concerned with the conditions needed for the convergence of the integral (C.5), forwhich we will need the asymptotic expansion of F (σ), which is:
π 3|σ| − π|σ| ,
6

(C.7)

π
π
FH (σ) ≈ − |σ|3 − |σ| .
6
8

(C.8)

FV (σ) ≈45

SciPost Phys. 14, 122 (2023)

One can see that the leading contributions are of order σ3 , where the contribution of thegauge multiplets generally makes the integral converge while the ones from matter and ChernSimons terms make it diverge. Thus, the convergence of the integral constrains the number
and representations of matter fields and the Chern-Simons level. Hence, it is tempting toconjecture that convergence of the integral may be related to the existence of a non-trivial
fixed point.
In the σ → ∞ limit the resulting expression for F (σ) can be identified with the gauge
theory prepotential [54]. This allows us to recast the condition as follows:
σ → ∞ ⇒ F (σ) → ∞ ,(C.9)

where F is now the prepotential.
Thus we see that the convergence of the perturbative part of the partition function is equivalent to the demand that the perturbative prepotential be everywhere positive. We stress herethat the above (piecewise) expression F for the prepotential is naive because in general adifferent prepotential is needed for each effective description of the theory valid in a given
subset of the naive Coulomb branch. In the absence of instanton corrections, we expect that F
will not have a well-defined global expression valid on all of C . In fact, we have encounterednumerous examples in our classification program where the prepotential associated to somegauge theory description is only valid on a particular subset of the naive Coulomb branch.
Nevertheless, this criterion (i.e. Conjecture 3) leads to physically sensible results consistent
with the predictions of our other conjectures.
An interesting case is when the prepotential vanishes along one direction but is positive
along all others. In that case the perturbative part of the partition function may diverges by
the linear term when gYM → ∞, but should converge for every other value. These cases arethen marginal, and in all examples we have checked correspond to 6d lifting theories.
Thus we see that examining the perturbative part of the partition function, we are naturally
led to a criterion for the existence of a UV fixed point for a 5d gauge theories. In all theexamples we have checked, this criterion gives identical results as the other criteria introducedin this article.
Before ending we wish to discuss the physical aspects of this criterion as well as possible
generalizations. A physical argument for this criterion is that the partition function is a characteristic of the theory and must be a finite number so it seems strange for it to diverge. It is
known that in some cases partition functions diverges. For instance the index of some 3d theories, the so called ‘bad’ by [55], diverges. In that case it is due to monopole operators going
below the unitary bound, and is believed to signal the appearance of singlets. When the S 5partition function diverges it is expected that likewise there should be a physical explanation
for this.
A natural physical explanation then is that the theory needs a UV completion and thus
additional degrees of freedom that could curve the divergence. This is also supported as thisdivergence is of UV type, that is in the limit of infinite Coulomb vev.
There are however several arguments one can raise against this. One such argument is that
we only look at part of the expression since we ignored the instanton corrections. So, first, it
is possible that even if the perturbative part converges, the instanton contribution diverges.
This just means that this condition is not sufficient, but that might be so even when the fullpartition function converges. It is still reasonable that this is a necessary condition.
However, it is possible that both the perturbative part and some of the instanton contributions diverges, but their sum converges. This seems extremely unlikely since the perturbative
part and instanton contributions have different dependence on the coupling constant, which
is a mass parameter in 5d controlling the instanton masses. So even if such a cancellation occurs changing the value of the coupling constant should destroy it and leads to a divergence.
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SciPost Phys. 14, 122 (2023)

Since the partition function must converge for every value of gYM , it still seems to be a goodnecessary condition.
Special attention should be given to the marginal case when the coefficient of the σ3 term
vanishes exactly along some direction. In that case the partition function only diverges whengYM → ∞ so one cannot rule out that the full partition function also converges for gYM → ∞where the divergence is canceled by instanton contributions.
Another issue that can be raised is how well can we trust the result for the S 5 partition
function particularly in the gYM → ∞ limit. In other words, how well does the gauge theory
partition function captures properties of the underlying SCFT when it exists. It is natural toexpect that the 5d SCFT should have a physical object that can be identified as its S 5 partition
functions. This will be a function of various mass deformation, identified with flavor masses
and coupling constants in low-energy gauge theory descriptions. The masses in turn can be
defined as central charges in the SUSY algebra. Physically we expect this object to be finite
for any value of the masses. If we identify this object with the gauge theory partition function
then a physical argument for its convergence naturally arises. In that lights, the agreement of
this criterion with the others seem to imply that the localized partition function of the gauge
theory captures some aspects of the partition function of the underlying SCFT.
Finally we comment on possible generalizations of this criterion. The most obvious one
is to generalize to the complete partition function by including the non-perturbative part. As
we explained when we discussed the exotic cases, it is expected that there will be some nonperturbative corrections to the perturbative prepotential. Given the identity of the large value
behavior of the exponent of the integrand with the perturbative prepotential, and the existenceof the non-perturbative instanton corrections, it is tempting to suspect that the latter may giveas a window into non-perturbative corrections to the perturbative prepotential. We also only
considered the round sphere, while there are generalizations of all the expressions also to thesquashed sphere. It might be interesting to explore the generalization to the squashed sphere.
It may also be interesting to better understand the relation between the gauge theory partition
function and that of the underlying SCFT. We reserve further study of all these issues to future
work.

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