11institutetext: Simons Center for Geometry and Physics, Stony Brook
University, Stony Brook, NY 11794-3840, USA22institutetext: C. N. Yang
Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY
11794-3840, USA33institutetext: Department of Mathematics, King’s College
London, The Strand, London WC2R 2LS, U.K.44institutetext: University of
Cincinnati, Physics Department, PO Box 210011, Cincinnati OH 45221, USA

# Needles in a haystack
An algorithmic approach to the classification of 4d ${\cal N}=2$ SCFTs

Justin Kaidi 1,2,3 Mario Martone 2 Leonardo Rastelli 4 Mitch Weaver

###### Abstract

There is a well-known map from 4d ${\cal N}=2$ superconformal field theories
(SCFTs) to 2d vertex operator algebras (VOAs). The 4d Schur index corresponds
to the VOA vacuum character, and must be a solution with integral coefficients
of a modular differential equation. This suggests a classification program for
4d ${\cal N}=2$ SCFTs that starts with modular differential equations and
proceeds by imposing all known constraints that follow from the 4d $\to$ 2d
map. This program becomes fully algorithmic once one specifies the order of
the modular differential equation and the rank (complex dimension of the
Coulomb branch) of the ${\cal N}=2$ theory. As a proof of concept, we apply
the algorithm to the study of rank-two ${\cal N}=2$ SCFTs whose Schur indices
satisfy a fourth-order untwisted modular differential equation. Scanning over
a large number of putative cases, only 15 satisfy all of the constraints
imposed by our algorithm, six of which correspond to known 4d SCFTs. More
sophisticated constraints can be used to argue against the existence of the
remaining nine cases. Altogether, this indicates that our knowledge of such
rank-two SCFTs is surprisingly complete.

YITP-SB-2022-02

##  1 Introduction

Superconformal field theories (SCFTs) are an ideal laboratory for the study of
quantum field theory. The space of superconformal field theories is much more
constrained than that of non-conformal, non-supersymmetric QFTs, so much so
that one might even envision a complete classification. While plausible
classification schemes have been proposed for SCFTs with maximal
supersymmetry, the case of half-maximal supersymmetry is still wide open. In
this paper we focus on four-dimensional ${\cal N}=2$ SCFTs, an extremely rich
subject with deep connections to mathematics and string theory.

Apart from rank-one SCFTs and Lagrangian gauge theories, which have been
completely classified in Bhardwaj:2013qia ; Argyres:2015ffa ; Argyres:2015gha
; Argyres:2016xmc ; Argyres:2016xua , there exists a bewildering and
constantly growing list of top-down constructions of 4d ${\cal N}=2$ SCFTs
that do not (yet?) admit a conventional Lagrangian description. Two broad
classes involve compactifications of six-dimensional SCFTs Gaiotto:2009we ;
Gaiotto:2009hg ; Chacaltana:2010ks ; Razamat:2016dpl ; Bah:2017gph ;
Kim:2017toz ; Kim:2018lfo ; Razamat:2018gro ; Ohmori:2018ona  and geometric
engineering on Calabi-Yau singularities in string theory Katz:1996fh ;
Shapere:1999xr ; Xie:2015rpa ; Wang:2016yha ; Chen:2016bzh ; Chen:2017wkw ;
Closset:2020scj .

In the spirit of the bootstrap, one would like to formulate a classification
program of 4d ${\cal N}=2$ SCFTs based only on general principles, such as
unitarity and superconformal invariance. In this paper we make some modest
progress in this direction. We leverage two classes of constraints:

  * $a.$

_VOAs and Higgs branch geometry_. It was shown in Beem:2013sza  that there
exists a map from (unitary) four-dimensional ${\cal N}=2$ SCFTs to a
restricted subset of (non-unitary) two-dimensional vertex operator algebras
(VOAs). This map identifies the Schur index of the ${\cal N}=2$ theory with
the vacuum character $\chi_{0}$ of the VOA, up to an overall factor,

$\displaystyle\chi_{0}(q)=q^{-{c_{\mathrm{2d}}\over 24}}Z_{\rm
Schur}(q)=q^{-{c_{\mathrm{2d}}\over
24}}(1+\sum_{n\in{{\mathbb{N}}/2}}a_{n}q^{n})~{},\hskip
28.90755pta_{n}\in{\mathbb{Z}}~{}.$ (1.1)

Here $c_{\mathrm{2d}}$ is the 2d central charge, related to the 4d Weyl
anomaly coefficient $c_{4d}$ by $c_{2d}=-12c_{4d}$. The vacuum character must
have integral Fourier expansion (since it counts operators in the theory), as
well as unit leading coefficient (signifying the presence of a unique vacuum).
Crucially, the VOAs associated to 4d SCFTs are believed to be of a special
type, known as “quasi-lisse,” which ensures that their vacuum characters
satisfy a monic modular differential equations (MDE).111A monic MDE is an MDE
with unit leading coefficient and holomorphic coefficient functions. See
Section 2.1 for explicit examples. The modular properties of the vacuum
character allow one to control its high temperature limit, from which one can
extract the other Weyl anomaly coefficient $a_{4d}$. These facts will provide
the starting point for our analysis. By looking for integral solutions to
monic MDEs of fixed order, we can identify Schur indices and central charges
of putative ${\cal N}=2$ SCFTs.

The Schur sector of an ${\cal N}=2$ SCFT comprises the Higgs branch chiral
ring. In particular, the Fourier coefficent $a_{1}$ counts the number of
moment maps of the 4d theory, i.e. the bottom components of the flavor current
supermultiplets. In fact, according to the central conjecture of Beem:2017ooy
, the entire Higgs branch (as a complex sympletic variety) is encoded in the
VOA.222However, in this work we will only discuss putative vacuum characters
and MDEs, which are often insufficient to reconstruct the full VOA.

  * $b.$

_Coulomb branch geometry._ Conjecturally, every 4d ${\cal N}=2$ theory has a
branch of the moduli space with the property that the low-energy theory at a
generic point is a free ${\cal N}=2$ supersymmetric $U(1)^{r}$ gauge theory
with no massless charged states. This is known as the Coulomb branch (CB). The
quantity $r$ is called the _rank_ of the theory and coincides with the complex
dimensionality of the CB. The CB is singular,333Purely on the basis of ${\cal
N}=2$ superconformal analysis, the CB can admit both _metric_ and _complex
structure_ singularities. While the former arise when BPS states charged under
the $U(1)^{r}$ become massless, the latter signal the fact that the CB chiral
ring is not freely generated Bourget:2018ond ; Argyres:2018wxu . Throughout
this manuscript we will restrict to ${\cal N}=2$ theories with freely
generated CB chiral ring and thus only consider metric singularities. and its
(complex codimension-one) singular loci encode many properties of the theory.
A related fact is that a globally-defined Lagrangian description of the low-
energy $U(1)^{r}$ theory is only possible by allowing electro-magnetically
dual changes of basis. This gives rise to non-trivial monodromies (lending the
CB a _Special Kähler_ structure Seiberg:1994rs ; Seiberg:1994aj ; Freed:1997dp
), which constrain the properties of CB operators.

The CB monodromies, which take value in the discrete electro-magnetic duality
group $Sp(2r,\mathbb{Z})$, can be used to restrict the scaling dimensions of
CB operators to a very small set of rational values, which depends solely on
the rank of the theory Argyres:2018urp ; Caorsi:2018zsq . Furthermore, the
stratification of the CB singular locus Argyres:2020wmq ; Martone:2020nsy ;
Cecotti:2021ouq  allows us to determine Weyl anomalies as well as flavor
symmetry data in terms of the scaling dimension of CB observables.

The structure of the CB geometry thus constrains some of the same SCFT
data—Weyl anomalies and flavor central charges—as are captured by the VOA. The
CB and VOA constraints are largely independent, making their compatibility a
highly non-trivial requirement.

A fundamental open question is how these two seemingly very different sets of
observables are related. Intricate examples of mutual compatibility will be
encountered throughout this work. But even more than just compatibility should
be true: conjecturally, the VOA and Coulomb branch geometry separately provide
complete characterizations of the 4d SCFT,444This statement concerning the
Coulomb branch requires some clarification. Since the early papers on the
subject, it has been known that for rank-1 theories where the Coulomb branch
stratification data is trivial, the scale-invariant limit of the Coulomb
branch is not enough to fully specify the SCFT Seiberg:1994aj . However, it is
still possible to enrich this geometric data with other purely Coulomb branch
quantities, _e.g._ the deformation pattern Argyres:2015ffa , to provide (up to
discrete gauging) a complete characterization. At higher ranks, the
stratification of singular loci becomes considerably more rich and, to the
authors’ knowledge, no inequivalent SCFTs share the same stratification data,
_i.e._ assignments of low-energy theories on sub-loci of all complex
codimensions.  in the sense that no two distinct SCFTs map to the same VOA or
to the same CB geometry. It should then in principle be possible to fully
reconstruct the CB geometry from the VOA, and vice versa. A tantalizing hint
in this direction is the experimental observation of Cordova:2015nma : in
several cases, one can precisely relate the Schur index with a certain wall-
crossing invariant computed from the massive BPS spectrum on the Coulomb
branch.

The current work has a more limited scope. We devise a procedure to identify
candidate ${\cal N}=2$ SCFTs by generating mutually compatible pairs of VOA
and Coulomb data. Our procedure becomes fully algorithmic once we specify, on
the VOA side, the order $d$ of the modular differential equation obeyed by the
Schur index (more precisely, by the vacuum character (1.1)), and, on the CB
side, the rank $r$ of the theory, i.e. the complex dimension of the CB.

$r,d,\chi_{0}$MDEIntegrality
(§2.1)$c_{\mathrm{4d}},a_{\mathrm{4d}},|\mathfrak{g}|$Shapere-
TachikawaHoffman-Maldacena
(§2.2)$\Delta_{u_{i}},\mathfrak{g}$$\begin{subarray}{c}\mathrm{CB\,\,strata}\\\
\mathrm{decomposition}\end{subarray}$CB constraints
(§2.3)$h_{\mathrm{ECB}},k_{\mathfrak{g}},\varkappa$${\begin{subarray}{c}\mathrm{Unitarity\,\,}(\S\ref{sec:unitsubsec})\\\
\varkappa\mathrm{\,\,constraints\,\,}(\S\ref{sec:chardim})\end{subarray}}$INOUT
Figure 1: A schematic depiction of our algorithm. As described in the main
text, we begin by fixing the rank $r$ and dimension $d$ of the MDE, and then
inputting the first few terms of the Fourier expansion of a candidate vacuum
character $\chi_{0}$. Various methods (given above the arrows) are then used
to extract increasingly refined sets of 4d data from these inputs. At each
step we impose constraints (given below the arrows) on the new data, which if
failed truncate the algorithm. The output of the algorithm is the data that
pass all of our constraints.

Concretely, the algorithm takes as input $r$, $d$, and the first few terms in
the Fourier expansion of a candidate vacuum character (the precise number of
terms is determined by $d$, as will be reviewed in Section 2.1). With this
input, we begin by using the MDE to verify that these first terms can be
“completed” to a fully integral solution $\chi_{0}(q)$. Assuming integrality
is satisfied, the MDE is then used to extract the possible central charges
$(a_{\mathrm{4d}},c_{\mathrm{4d}})$ as well as the dimension of the flavor
symmetry of the corresponding 4d ${\cal N}=2$ theory. This is discussed in
Section 2.2. This data must satisfy, among other things, the Hoffman-Maldacena
bounds. Assuming that it does, the Shapere-Tachikawa formula can be used to
extract the Coulomb branch scaling dimensions $\Delta_{u_{i}}$. The flavor
symmetry $\mathfrak{g}$ can also be restricted to a finite set of
possiblities. The data $\Delta_{u_{i}}$ and $\mathfrak{g}$ must satisfy a
number of stringent Coulomb branch consistency conditions, which we detail in
Section 2.3. As we will see, satisfying these conditions is roughly equivalent
to constructing a consistent CB geometry. Assuming that this is possible, the
stratification of the CB geometry can then be used to obtain more refined
quantities such as the flavor levels $k_{\mathfrak{g}_{i}}$, which are subject
to various unitarity constraints discussed in Section 2.4. Finally, a more
subtle constraint is discussed in Section 2.5. In the end, the algorithm
outputs the sets of data which satisfy all of these constraints. A pictorial
summary of this procedure is given in Figure 1.

Since imposing each of these steps by hand is rather laborious, the entire
process is automated on a computer. Automation also allows for the following
strategy. Instead of inputting a single vacuum character $\chi_{0}(q)$ and
checking for a consistent 4d interpretation, we can now scan over large
numbers of tentative vacuum characters (again fixing the rank $r$ and the
dimension $d$ of the MDE). Since the input for the algorithm is not actually
the vacuum character itself but rather just the first few terms of its Fourier
expansion, this amounts to a scan over tuples of integers. As long as the
integers scanned over take values in a sufficiently large range, we can be
fairly confident that our algorithm will discover all 4d ${\cal N}=2$ theories
with fixed $(r,d)$.

As a proof of concept, we implement such a scan in what is arguably the
simplest non-trivial case: namely rank-two theories whose Schur indices
satisfy a fourth-order untwisted MDE. As will be reviewed below, such MDEs—as
well as their solutions—are specified by three terms in the Fourier expansion
of $\chi_{0}(q)$. We must then scan over these three parameters, and for each
triplet apply our algorithm to check if the proposed vacuum character admits a
4d interpretation. Remarkably, of the order $O(10^{11})$ candidate triplets
that we examine, only 15 pass the full set of moduli space and unitarity
constraints imposed in our algorithm. The data of these 15 cases is summarized
in Table 3 and Figure 3. An even more careful analysis (using tools which are
difficult to automate on a computer) allows us to rule out nine of the 15
theories,555More precisely, seven candidates will be rigorously ruled out,
while two will be shown to be implausible. leaving only six candidates.
Comparing to the lists in e.g. Martone:2021ixp , all six candidates can be
seen to correspond to known physical ${\cal N}=2$ theory.

The fact that our scan does not produce any new candidate theories is perhaps
surprising, as we did not anticipate our current knowledge of ${\cal N}=2$
SCFTs to be so exhaustive. However this is likely an artifact of the
restriction to low order and low rank. The extension to higher orders and
higher ranks is conceptually straightforward, but quickly becomes
computationally intensive, and so we have not attempted to pursue it here.
With appropriate computer implementation, we hope that these difficulties can
be eventually overcome. We will comment on the feasibility of scaling up to
these richer setups in the concluding section.

##### Organization:

The organization of this paper is as follows. We begin in Section 2 by
introducing our general algorithm, which involves starting with a monic MDE of
fixed order, searching for solutions that are integral, and imposing various
four-dimensional Coulomb branch and unitarity constraints to determine whether
the solutions can correspond to legitimate 4d Schur indices. A visual summary
of this algorithm is given in Figure 1. In Section 3 we illustrate the
algorithm with a simple example—namely, we use our techniques to rediscover
the rank-two $D_{2}(SU(5))$ theory of Cecotti:2013lda . Then in Section 4 we
use the algorithm to do a computerized search for rank-two ${\cal N}=2$
theories with flavor symmetry having up to three simple factors, three $U(1)$
factors and with corresponding VOA satisfying a fourth-order untwisted MDE. We
rediscover six known 4d ${\cal N}=2$ theories, and identify nine candidate
novel theories. By using more refined Coulomb and Higgs branch constraints, we
are able to rigorously rule out seven of the candidates, and to argue that the
remaining two are also unlikely to be physical. Finally, in Section 5 we
conclude with a discussion of future directions.

We also include a number of appendices. Appendix A gives a review of some
relevant superconformal representation theory, and derives some constraints
which will be used in our analysis. In Appendix B, we rederive the list of
allowed pairs of CB scaling dimensions for rank-two theories, with results
differing slightly from those in the literature. Appendices C and D review
facts about rank-one IR-free theories and extended Coulomb branches, both of
which will be relevant for our scan. We also include Appendix E, which gives a
tangential but interesting discussion of one of the VOAs identified in our
scan.

##  2 General strategy

In this section we describe the algorithm used to identify sets of protected
data (i.e. Schur and Coulomb) that satisfy a variety of CB and unitarity
constraints. An explicit example will be given in Section 3, and the results
of a computer implementation will be given in Section 4.

###  2.1 Modularity and integrality constraints

Our program begins in two dimensions and relies crucially on the fact that the
Schur index of any 4d ${\cal N}=2$ superconformal field theory can be
interpreted as the vacuum character of a 2d VOA. As mentioned in the
introduction, for a generic VOA there is no reason to expect that the vacuum
character should have simple modular properties. We will take as a given the
central conjecture of Beem:2017ooy , which equates the Higgs branch of the 4d
${\cal N}=2$ theory with the associated variety of the corresponding VOA. A
corollary of this conjecture is that any VOA with a 4d avatar is “quasi-
lisse”,666We remind the reader that quasi-lisse VOAs Arakawa:2016hkg  are
those whose associated varieties are symplectic, or more generally symplectic
singularities brieskorn1970singular ; slodowy1980simple ;
beauville1999symplectic  with a finite number of symplectic leaves. and as
such its vacuum character must satisfy a monic modular differential equation
(MDE) Arakawa:2016hkg .777Explicit closed form expressions for Schur indices
in terms of quasi-modular forms were recently obtained in Pan:2021mrw ;
Beem:2021zvt  for a large class of SCFTs.

Depending on whether we require modular invariance under the full modular
group or just an index two subgroup Beem:2017ooy ; Bae:2020xzl ; Bae:2021mej ,
an MDE is called respectively untwisted or twisted. The space of possible MDEs
is labelled by a finite set of real parameters, the number of which depends
only on the degree of the differential equation and whether it is untwisted or
twisted. In this work we will focus mainly on untwisted MDEs, of which we will
give some examples below.888The twisted case is required if one allows for
Schur operators with half-integer chiral dimension, and as such is the most
general case. The untwisted case is a specialization of the twisted one
(allowing for Schur operators of only integer dimension), and thus at a given
order the untwisted case involves a smaller number of real parameters than the
twisted one.

An important property of any vacuum character is that it must have unit
leading coefficient, and that the remaining Fourier coefficients must be
integral. Not every choice of MDE, i.e. of the real parameters which determine
it, is compatible with such integrality. A first step in our classification
program is to identify the MDEs which do give rise to a sensible vacuum
character. In the context of 2d CFT, this idea goes back to the classic work
of Mathur, Mukhi, and Sen Mathur:1988rx ; Mathur:1988na ; Mathur:1988gt , see
e.g. Gaberdiel:2008pr ; Hampapura:2015cea ; Gaberdiel:2016zke ;
Hampapura:2016mmz ; Mukhi:2017ugw ; Chandra:2018pjq ; Mukhi:2019xjy ;
Mukhi:2020gnj ; Bae:2020xzl ; Das:2020wsi ; Mason:2021xfs ; Kaidi:2020ecu ;
Kaidi:2021ent ; Das:2021uvd ; Bae:2021mej ; Bae:2021jkc  for recent
developments. As scanning over the set of real numbers parameterizing the MDE
would clearly be impossible, our first step is to map the problem to a scan
over integers. This can be done as follows:

  * 1.

We fix the order and the type of the MDE, i.e. untwisted or twisted. This
choice determines the number $n$ of real parameters that must be specified to
define the MDE.

  * 2.

Instead of specifying the $n$ real parameters directly, we specify an equal
number of integral Fourier coefficients of a putative vacuum character
$\chi_{0}$, and then fix the parameters of the MDE by demanding that the given
vacuum character is a solution up to the appropriate order in $q$. As will be
elaborated on below, certain parameters of $\chi_{0}$ have immediate physical
interpretation and thus are subject to additional constraints. For example,
the leading exponent of $\chi_{0}$ is related to the central charge
$c_{\mathrm{4d}}$ of the 4d ${\cal N}=2$ SCFT, while the $O(q)$ Fourier
coefficient gives the number of moment map operators, i.e. the dimension of
the flavor symmetry. The higher-order Fourier coefficients do not have such
straightforward physical interpretations, but they must be consistent with the
fact that they count operators of the 4d theory, and in particular they must
organize into representations of the flavor symmetry. This is discussed in
detail in Appendix A.

  * 3.

Since each coefficient of $\chi_{0}$ fixes one real parameter of the MDE, step
2 fixes the MDE completely. We now take the MDE constructed above as an input
and “complete” $\chi_{0}$ by assuming that it is a solution to all orders in
$q$. If all Fourier coefficients (at least up to some reasonable cutoff which
will depend on the order of the MDE) are integral, we will take $\chi_{0}$ to
be a candidate vacuum character and proceed. If the Fourier coefficients are
not integral, then $\chi_{0}$ is discarded.

Let us now briefly discuss the lowest-order monic untwisted MDEs, which will
be the focus of the rest of this work. Denoting the order of the MDE by $d$,
the most general MDE is given by

$\displaystyle d=2:$ $\displaystyle\hskip
36.135pt\left[D^{(2)}+\mu_{1}E_{4}\right]\chi=0~{},$ (2.2) $\displaystyle
d=3:$ $\displaystyle\hskip
36.135pt\left[D^{(3)}+\mu_{1}E_{4}D^{(1)}+\mu_{2}E_{6}\right]\chi=0~{},$
$\displaystyle d=4:$ $\displaystyle\hskip
36.135pt\left[D^{(4)}+\mu_{1}E_{4}D^{(2)}+\mu_{2}E_{6}D^{(1)}+\mu_{3}E_{4}^{2}\right]\chi=0~{},$

where $E_{i}$ are the holomorphic Eisenstein series, and the $D^{(k)}$ are the
$k$-th order modular covariant derivatives acting on weight-zero modular
forms,

$\displaystyle D^{(k)}:=\prod_{s=1}^{k}D_{2s-2}~{},\hskip
57.81621ptD_{w}:={1\over 2\pi i}{d\over d\tau}-{w\over 12}E_{2}(\tau)~{}.$
(2.3)

Clearly the equations (2.2) depend respectively on one, two, and three real
parameters, denoted by $\mu_{i}$.

Every solution of an untwisted MDE admits a Fourier expansion in integer
powers of $q:=e^{2\pi i\tau}$. In particular, the vacuum character will admit
an expansion

$\displaystyle\chi_{0}(q)=q^{-{c_{\mathrm{2d}}\over
24}}(1+\sum_{n>0}a_{n}q^{n})=q^{-{c_{\mathrm{2d}}\over
24}}(1+a_{1}q+a_{2}q^{2}+a_{3}q^{3}+\dots)~{},$ (2.4)

with $c_{\mathrm{2d}}$ the 2d central charge. As we have already mentioned, we
will not take the full vacuum character as an input, but rather only the first
few terms in its Fourier expansion. We then assume that these first few terms
provide a solution to one of the MDEs in (2.2). Since the $d=2,3,4$ MDEs are
specified by respectively $1,2,3$ free parameters, only this number of terms
in the expansion of $\chi_{0}(q)$ are needed to fix the MDE uniquely. In other
words, the input data will be taken to be

$\displaystyle d=2:$ $\displaystyle\hskip 36.135pt\\{c_{\mathrm{2d}}\\}~{},$
(2.5) $\displaystyle d=3:$ $\displaystyle\hskip
36.135pt\\{c_{\mathrm{2d}},a_{1}\\}~{},$ $\displaystyle d=4:$
$\displaystyle\hskip 36.135pt\\{c_{\mathrm{2d}},a_{1},a_{2}\\}~{}.$

Solving the MDE order-by-order in $q$, we may then fix the parameters
$\mu_{i}$ in terms of this input data as

$\displaystyle d=2:$ $\displaystyle\hskip 36.135pt\mu_{1}=-{1\over
576}c_{\mathrm{2d}}(c_{\mathrm{2d}}+4)~{},$ (2.6) $\displaystyle d=3:$
$\displaystyle\hskip
36.135pt\mu_{1}={c_{\mathrm{2d}}(21c_{\mathrm{2d}}^{2}+240\,c_{\mathrm{2d}}+704)-a_{1}(3c_{\mathrm{2d}}^{2}-48\,c_{\mathrm{2d}}+320)\over
576(a_{1}-31c_{\mathrm{2d}})}~{},$ $\displaystyle\hskip
36.135pt\mu_{2}=-{c_{\mathrm{2d}}^{2}(5c_{\mathrm{2d}}^{2}+66\,c_{\mathrm{2d}}+144)+a_{1}\,c_{\mathrm{2d}}(c_{\mathrm{2d}}^{2}-30\,c_{\mathrm{2d}}+144)\over
6912(a_{1}-31c_{\mathrm{2d}})}~{},$

and similarly for $d=4$.

Having fixed the MDE completely in terms of the data (2.5), we now solve the
MDE order-by-order to obtain the vacuum character to arbitrarily high order in
$q$. Assuming that it remains integral at all orders (in practice, up to some
reasonably high cut-off) we now ask whether there can exist a 4d ${\cal N}=2$
SCFT with that vacuum character as its Schur index.

###  2.2 Central charges

Assume that we have found an MDE compatible with integrality. This gives us a
candidate Schur index for a candidate 4$d$ ${\cal N}=2$ SCFT. In addition to
the vacuum character, an order-$d$ MDE will generically have $(d-1)$ non-
vacuum solutions, each with an associated chiral dimension $h_{i}$. These
dimensions are obtained as follows. Assume that the solutions to the MDE take
the form

$\displaystyle\chi_{i}(q)=q^{s_{i}}\sum_{n}a_{n}^{(i)}q^{n}~{},$ (2.7)

for $i=0,\dots,d-1$. The leading exponent of the vacuum character
$\chi_{0}(q)$ is $s_{0}=-{c_{2d}\over 24}$, while the leading exponents of the
remaining characters are $s_{i}:=h_{i}-{c_{2d}\over 24}$. These leading
exponents are given by the roots of the “indicial equation,” i.e. the order
$O(q^{s})$ portion of the MDE,

$\displaystyle d=2:$ $\displaystyle\hskip 36.135pts^{2}-{s\over
6}+\mu_{1}=0~{},$ (2.8) $\displaystyle d=3:$ $\displaystyle\hskip
36.135pts^{3}-{1\over 2}s^{2}+\left(\mu_{1}+{1\over 18}\right)s+\mu_{2}=0~{},$
$\displaystyle d=4:$ $\displaystyle\hskip
36.135pts^{4}-s^{3}+\left(\mu_{1}+{11\over 36}\right)s^{2}-\left({1\over
36}+{\mu_{1}\over 6}-\mu_{2}\right)s+\mu_{3}=0~{}.$

Hence the chiral dimensions can be identified by computing the indicial roots.

With the chiral dimensions in hand, it is now possible to restrict the central
charges of the putative 4d ${\cal N}=2$ theory to a discrete set of
possibilities. Indeed, the following standard formulae can be used
Beem:2013sza ; Cecotti:2015lab ; Beem:2017ooy ,999To be precise, in the
twisted case the $h_{*}$ appearing in (2.9) is one of the chiral dimensions of
the $S$-tranformed VOA—see e.g. Beem:2017ooy . Here we are focussing on the
untwisted case.

$\displaystyle c_{{\mathrm{4d}}}=-{c_{{\mathrm{2d}}}\over 12}~{},\hskip
36.135pta_{\mathrm{4d}}={1\over 48}(24h_{*}-5c_{\mathrm{2d}})~{}.$ (2.9)

When $c_{{\mathrm{4d}}}>a_{{\mathrm{4d}}}$, the $h_{*}$ appearing above is
required to be the minimal chiral dimension $h_{*}=\mathrm{min}\\{h_{i}\\}$
DiPietro:2014bca . When $c_{{\mathrm{4d}}}<a_{{\mathrm{4d}}}$ this need no
longer be the case, and it is in fact unclear if $h_{*}\in\\{h_{i}\\}$ at all.
But it is still seems reasonable to expect that $h_{*}$ takes values in
$\\{h_{i}\\}$ mod 1. The argument here relies on the study of the high-
temperature behavior of the index ArabiArdehali:2015ybk ; DiPietro:2014bca .
For Lagrangian ${\cal N}=2$ theories, it can be shown that there always exists
a contribution to the high-temperature behavior of the Schur index with
exponent proportional to $c_{\rm 4d}-a_{\rm 4d}$. Since the Schur index (or
really the vacuum character) forms a vector-valued modular function together
with the other solutions to the MDE, its $S$ transform is a linear combination
of these other solutions, and thus every exponent of the high-temperature
index is of the form $h_{i}-{c_{\mathrm{2d}}\over 24}+m$ for some
$h_{i}\in\\{h_{i}\\}$ and $m\in{\mathbb{N}}$. Together these imply that
$h_{*}=h_{i}+m$, and hence for Lagrangian theories we indeed have
$h_{*}\in\\{h_{i}\\}$ mod 1. All known examples suggest that this result holds
for non-Lagrangian theories as well, and we will assume it to be true here.

We are then left with a discrete but potentially infinite set of possibilities
for $(a_{\mathrm{4d}},c_{\mathrm{4d}})$. This set is made finite by imposing
the Hofman-Maldacena bounds Hofman:2008ar ,

$\displaystyle{1\over 2}\leq{a_{\mathrm{4d}}\over
c_{{\mathrm{4d}}}}\leq{5\over 4}~{}.$ (2.10)

Indeed, in the case that $c_{{\mathrm{4d}}}<a_{{\mathrm{4d}}}$, this will
restrict the quantity $m\in{\mathbb{N}}$ to a finite number of possible values
(since $a_{\mathrm{4d}}$ increases linearly with $m$ while $c_{\mathrm{4d}}$
is independent).

###  2.3 Coulomb geometry constraints

We now have a finite list of possible $(a_{\mathrm{4d}},c_{\mathrm{4d}})$ for
the candidate 4d ${\cal N}=2$ theory. We next consider constraints imposed by
consistency of the Coulomb branch (CB). Let’s start by reviewing some basic
facts about CBs of ${\cal N}=2$ SCFTs. If the CB is spanned by $r$ complex
coordinates ${\boldsymbol{u}}=\\{u_{i}\\}$, $i=1,...,r$, then the ${\cal N}=2$
theory is said to be a _rank_ -$r$ SCFT. Henceforth we will assume that the CB
chiral ring of the 4d ${\cal N}=2$ theory is freely-generated, in which case
it is possible to choose the coordinates of the CB to have globally defined
scaling dimensions $\Delta_{u_{i}}$ for $i=1,...,r$. There exist constraints
on the possible values of these scaling dimension. For example, at rank one it
turns out that there are only seven allowed values Argyres:2018urp ;
Caorsi:2018zsq ,

$\displaystyle\Delta\in\left\\{6,4,3,2,{3\over 2},{4\over 3},{6\over
5}\right\\}~{}.$ (2.11)

At rank two, there are again a finite number of possibilities for
$\Delta_{u_{1}}$ and $\Delta_{u_{2}}$. In fact, now there are constraints not
only on the individual scaling dimensions, but also on the allowed pairs. In
total, there are 79 allowed pairs, the full list of which is provided in Table
5 of Appendix B (note that the results here follow the revised analysis given
in Argyres:2022yet , differing slightly from those in Caorsi:2018zsq ). This
analysis can be extended, with enough effort and dedication, to any desired
rank, though the number of allowed _r_ -tuples increases considerably.

We now fix the rank of the putative theory to some particular $r$. The CB
scaling dimensions are known to be related to the central charges via the
Shapere-Tachikawa formula Shapere:2008zf ,101010There are known exceptions to
this formula in the case of theories obtained by discrete gauging of “more
fundamental” theories. We will assume that exceptions can only arise in this
way.

$\displaystyle\sum_{i=1}^{r}\Delta_{u_{i}}=2(2a_{\mathrm{4d}}-c_{\mathrm{4d}})+{r\over
2}~{}.$ (2.12)

Returning to our algorithm, we may now use the finite list of allowed
$(a_{\mathrm{4d}},c_{\mathrm{4d}})$ obtained from the MDE to compute a finite
list of candidate sums of CB scaling dimensions. Recall that the CB dimensions
themselves take values in a finite list, c.f. Appendix B, which means that
their sums do as well. For example, in the case of rank-two the sums
$\Delta_{u_{1}}+\Delta_{u_{2}}$ are constrained to take one of only 68
distinct values. If the value computed via the Shapere-Tachikawa formula is
not on the list of allowed rank-$r$ sums, then that particular pair
$(a_{\mathrm{4d}},c_{\mathrm{4d}})$ can be discarded. If every one of the
finite number of possible pairs $(a_{\mathrm{4d}},c_{\mathrm{4d}})$ is
discarded, then there is no 4d theory corresponding to the input data. If on
the other hand at least one pair $(a_{\mathrm{4d}},c_{\mathrm{4d}})$ gives
rise to a sum on the list, we proceed.

Our next step is to read off all possible ${\Delta}_{u_{i}}$ adding up to the
desired sum. For low enough rank there is usually a unique choice, though in
some cases there are multiple possibilities. As examples, at rank two one
might have

$\displaystyle\Delta_{u_{1}}+\Delta_{u_{2}}$ $\displaystyle=$
$\displaystyle{78\over 11}\hskip 36.135pt\Rightarrow\hskip
36.135pt(\Delta_{u_{1}},\Delta_{u_{2}})=\left({12\over 11},6\right)~{},$
$\displaystyle\Delta_{u_{1}}+\Delta_{u_{2}}$ $\displaystyle=$ $\displaystyle
18\hskip 36.135pt\Rightarrow\hskip
36.135pt(\Delta_{u_{1}},\Delta_{u_{2}})\in\left\\{(6,12),(8,10)\right\\}~{},$
(2.13)

as the reader may confirm by consulting Appendix B. We thus have identified a
relatively small (or empty) list of potential $\\{\Delta_{u_{i}}\\}$ for each
possible $(a_{\mathrm{4d}},c_{\mathrm{4d}})$.

Next we will need more detailed knowledge of the geometric structure of the
Coulomb branch. The Coulomb branch can have singular loci $\\{{\cal I}_{a}\\}$
falling into two broad classes Argyres:2018zay :

  * •

Unknotted: Locus defined by $u_{i}=0$ ,

  * •

Knotted: Locus defined by $P({\boldsymbol{u}})=0$, with $P$ an irreducible
homogeneous polyonomial in ${\boldsymbol{u}}$ .

We may associate to each singular locus ${\cal I}_{a}$ a value
$\Delta_{a}^{\mathrm{sing}}$ which is the degree of homogeneity of the
polynomial specifying the stratum, e.g. for unknotted strata ${\Delta}^{\rm
sing}\equiv{\Delta}_{u_{i}}$. Each singular locus can host a rank-one theory,
either conformal or IR-free. The full set of conformal rank-one theories is
known Argyres:2015ffa ; Argyres:2015gha ; Argyres:2016xua ; Argyres:2016xmc ,
and since we will make heavy use of them we include the relevant data in Table
1. On the other hand, the full set of IR-free rank-one theories is infinite,
but each such theory takes the form of $U(1)$ or $SU(2)$ gauge theory coupled
to sufficient numbers of hypermultiplets in arbitrary representations. In
practice, when we implement this algorithm on a computer it is necessary to
restrict to a finite subset of these IR-free theories. The particular subset
which we choose is discussed in Appendix C. We believe this to be a well-
motivated representative sample of the full space of IR-free theories.

Name | $12\,c_{\mathrm{4d}}$ | ${\Delta}_{u}$ | $h_{\mathrm{ECB}}$ | $\mathbf{R}$ | $T(\mathbf{R})$ | $b$ | $\mathfrak{g}$ | $k_{\mathfrak{g}}$ |
---|---|---|---|---|---|---|---|---|---
${\cal T}^{(1)}_{E_{8},1}$ | 62 | 6 | 0 | ${\mathbf{1}}$ | 0 | 10 | $\mathfrak{e}_{8}$ | 12 |
${\cal T}^{(1)}_{E_{7},1}$ | 38 | 4 | 0 | ${\mathbf{1}}$ | 0 | 9 | $\mathfrak{e}_{7}$ | 8 |
${\cal T}^{(1)}_{E_{6},1}/[{\cal T}^{(1)}_{E_{6},1}]_{{\mathbb{Z}}_{2}}$ | 26 | 3 | 0 | ${\mathbf{1}}$ | 0 | 8 | $\mathfrak{e}_{6}/\mathfrak{f}_{4}$ | 6 |
${\cal T}^{(1)}_{D_{4},1}/[{\cal T}^{(1)}_{D_{4},1}]_{{\mathbb{Z}}_{2}}/[{\cal T}^{(1)}_{D_{4},1}]_{{\mathbb{Z}}_{3}}$ | 14 | 2 | 0 | ${\mathbf{1}}$ | 0 | 6 | $\mathfrak{so}(8)/\mathfrak{so}(7)/\mathfrak{g}_{2}$ | 4 |
${\cal T}^{(1)}_{A_{2},1}/[{\cal T}^{(1)}_{A_{2},1}]_{{\mathbb{Z}}_{2}}/[{\cal T}^{(1)}_{A_{2},1}]_{{\mathbb{Z}}_{3}}$ | 8 | $\frac{3}{2}$ | 0 | ${\mathbf{1}}$ | 0 | 4 | $\mathfrak{su}(3)/\mathfrak{su}(3)/\mathfrak{su}(2)$ | 3 |
${\cal T}^{(1)}_{A_{1},1}/[{\cal T}^{(1)}_{A_{1},1}]_{{\mathbb{Z}}_{2}}$ | 6 | $\frac{4}{3}$ | 0 | ${\mathbf{1}}$ | 0 | 3 | $\mathfrak{su}(2)/\mathfrak{su}(2)$ | $\frac{8}{3}$ |
${\cal T}^{(1)}_{\varnothing,1}$ | $\frac{22}{5}$ | $\frac{6}{5}$ | 0 | ${\mathbf{1}}$ | 0 | 2 | $\varnothing$ | $\star$ |
1-9 ${\cal S}^{(1)}_{E_{6},2}$ | 49 | 6 | 5 | ${\mathbf{10}}$ | 1 | 7 | $\mathfrak{sp}(10)$ | 7 |
${\cal S}^{(1)}_{D_{4},2}$ | 29 | 4 | 3 | $({\mathbf{6,1}})$ | 1 | 6 | $\mathfrak{sp}(6){\times}\mathfrak{su}(2)$ | (5,8) |
${\cal S}^{(1)}_{A_{2},2}/[{\cal S}^{(1)}_{A_{2},2}]_{{\mathbb{Z}}_{2}}$ | 19 | 3 | 2 | ${\mathbf{4}}_{0}$ | 1 | 5 | $\mathfrak{sp}(4){\times}\mathfrak{u}(1)/\mathfrak{sp}(4)$ | $(4,\star)$ |
${\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.5}{0}{0}\pgfsys@color<EMAIL_ADDRESS>S}^{(1)}_{\varnothing,2}}$ | 9 | 2 | 1 | ${\mathbf{2}}$ | 1 | 3 | $\mathfrak{su}(2)$ | 3 |
1-9 ${\cal S}^{(1)}_{D_{4},3}$ | 42 | 6 | 4 | ${\mathbf{4}}\oplus\bar{\mathbf{4}}$ | 2 | 6 | $\mathfrak{su}(4)$ | 14 |
${\cal S}^{(1)}_{A_{1},3}$ | 24 | 4 | 3 | ${\mathbf{2}}_{+}\oplus{\mathbf{2}}_{-}$ | 2 | 5 | $\mathfrak{su}(2){\times}\mathfrak{u}(1)$ | $(10,\star)$ |
${\color[rgb]{0.4,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.4,1,0}\pgfsys@color@cmyk@stroke{0.6}{0}{1.}{0}\pgfsys@color<EMAIL_ADDRESS>S}^{(1)}_{\varnothing,3}}$ | 15 | 3 | 1 | ${\mathbf{1}}_{+}\oplus{\mathbf{1}}_{-}$ | 0 | 4 | $\mathfrak{u}(1)$ | $\star$ |
1-9 ${\cal S}^{(1)}_{A_{2},4}$ | 38 | 6 | 3 | ${\mathbf{3}}\oplus\bar{\mathbf{3}}$ | 2 | $\frac{11}{2}$ | $\mathfrak{su}(3)$ | 14 |
${\color[rgb]{0.4,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.4,1,0}\pgfsys@color@cmyk@stroke{0.6}{0}{1.}{0}\pgfsys@color<EMAIL_ADDRESS>S}^{(1)}_{\varnothing,4}}$ | 21 | 4 | 1 | ${\mathbf{1}}_{+}\oplus{\mathbf{1}}_{-}$ | 0 | $\frac{9}{2}$ | $\mathfrak{u}(1)$ | $\star$ |
Table 1: The list of rank-one SCFTs which can describe the low-energy physics
on codimension-one singular loci of the Coulomb branch. Additional information
can be found in Argyres:2016yzz . Theories in green are ${\cal N}=3$, while
the theory in blue is ${\cal N}=4$.

The singular loci are further constrained by the following three results
Argyres:2018urp ; Martone:2020nsy ; Martone:2021ixp :

###### Proposition 1

If a theory of rank $r>1$ is irreducible, i.e. not the product of lower rank
theories, then there must be at least one knotted stratum.

###### Proposition 2

An unknotted stratum ${u_{i}}=0$ is allowed only if the $r-1$ scaling
dimensions
$(\Delta_{u_{1}},...,\Delta_{u_{i-1}},\Delta_{u_{i+1}},...,\Delta_{u_{r}})$
give an allowed $(r-1)$-tuple of rank-$(r-1)$ Coulomb branch dimensions.

######  Proposition 3 (UV-IR Flavor constraint)

For a theory with flavor symmetry
$\mathfrak{g}=\mathfrak{u}(1)^{n}\bigoplus_{i}\mathfrak{g}_{i}$ with
$\mathfrak{g}_{i}$ simple, each $\mathfrak{g}_{i}$ must be realized on at
least one codimension-one stratum.

These results prove highly constraining. To see this, let us begin by
considering the flavor symmetry of our putative ${\cal N}=2$ theory. Recall
that the Fourier coefficient $a_{1}$ of $\chi_{0}$ counts the number of moment
map operators, and thus fixes the dimension of the flavor algebra of our
putative theory. This quantity will either be one of the inputs used to define
the MDE (for $d=4$) or will be an output obtained by solving the MDE at order
$q$ (for $d=2,3$); c.f. (2.5).

Clearly $a_{1}$ alone does not determine the flavor symmetry. Indeed, there
can be many Lie algebras with a given dimension $a_{1}$. For example, if
$a_{1}=248$, then we can have the following decompositions

$\displaystyle\\{248\\},\,\\{24,224\\},\,\\{80,168\\},\,\\{3,21,224\\},\,\dots$
(2.14)

which would be interpretable as the following Lie algebras,

$\displaystyle\mathfrak{e}_{8},\,\,\mathfrak{su}(5)\oplus\mathfrak{su}(15),\,\,\mathfrak{su}(9)\oplus\mathfrak{su}(13),\,\,\mathfrak{su}(2)\oplus\mathfrak{sp}(6)\oplus\mathfrak{su}(15),\,\,\dots$
(2.15)

If we allow for arbitrary numbers of $\mathfrak{u}(1)$ factors, then clearly
we should consider all integer partitions of $a_{1}$. In practice we will set
an upper bound on the number of $\mathfrak{u}(1)$ factors.

Given a choice of flavor symmetry, we may now attempt to reconstruct parts of
the CB geometry. To begin, Proposition 3 tells us that each simple flavor
factor must be realized on a codimension-one locus of the CB. Hence if we take
the flavor symmetry to be e.g. $\mathfrak{e}_{8}$, then we must identify a
rank-one theory which supports that flavor symmetry. Consulting Table 1, we
see that the only possibility is ${\cal T}_{E_{8},1}^{(1)}$. On the other
hand, if we take the flavor symmetry to be
$\mathfrak{su}(2)\oplus\mathfrak{sp}(6)\oplus\mathfrak{su}(15)$, then
$\mathfrak{su}(15)$ must be realized by a $[I_{15},\mathfrak{su}(15)]$
stratum, whereas $\mathfrak{su}(2)$ can be realized in three possible ways via
${\cal T}_{A_{1},1}^{(1)},{\cal S}_{0,2}^{(1)},$ and
$[I_{2},\mathfrak{su}(2)]$, and $\mathfrak{sp}(6)$ in two possible ways via
$[I_{6},\mathfrak{su}(6)]_{{\mathbb{Z}}_{2}}$ and
$[I_{2}^{*},\mathfrak{sp}(6)]$. In total then, we see that the $\\{248\\}$
decomposition is realizable on the Coulomb branch in only a single way,
whereas the $\\{3,21,224\\}$ decomposition is realizable in six ways. The
total number of Coulomb branch geometries that can capture a given $a_{1}$ is
given by summing over the different integer partitions of $a_{1}$, weighted by
the number of different ways of realizing a given decomposition on the CB
strata.

Note that simple flavor factors cannot enhance their ranks on any strata,
except perhaps for the appearance of additional $\mathfrak{u}(1)$ factors
Martone:2020nsy . This means, for example, that if we had a theory with
$\mathfrak{e}_{7}$ flavor symmetry, we would require a ${\cal
T}_{E_{7},1}^{(1)}$ stratum to satisfy Proposition 3, as opposed to a ${\cal
T}_{E_{8},1}^{(1)}$ stratum, even though the latter technically contains
$\mathfrak{e}_{7}\subset\mathfrak{e}_{8}$ flavor symmetry. On the other hand,
rank-preserving enhancements are technically allowed. For simplicity we will
assume that such enhancements do not take place.

With these data about the strata, together with the lists of possible
$\\{\Delta_{u_{i}}\\}$ identified before, we are now ready to proceed to the
next step. Let us denote the set of all Coulomb branch strata as ${\cal
I}=\\{{\cal I}_{a}\\}$, and the set of strata carrying the UV simple flavor
factor $\mathfrak{g}_{i}$ by ${\cal I}_{\mathfrak{g}_{i}}\subset{\cal I}$. We
may now make use of the following formulas relating the data of the theory at
the origin to the data of the theories living on each stratum (in the absence
of flavor symmetry enhancements) Martone:2020nsy ; Martone:2021ixp :

$\displaystyle h_{\mathrm{ECB}}$ $\displaystyle=$ $\displaystyle
12c_{\mathrm{4d}}-2r-\sum_{{\cal I}_{a}\in{\cal
I}}b_{a}\,\Delta_{a}^{\mathrm{sing}}~{},$ $\displaystyle k_{\mathfrak{g}_{i}}$
$\displaystyle=$ $\displaystyle\sum_{{\cal I}_{a}\in{\cal
I}_{\mathfrak{g}_{i}}}{\Delta_{a}^{\mathrm{sing}}\over\Delta_{a}}[k_{a}-T^{i}(\mathbf{R}_{a})]+T^{i}(\mathbf{R})~{},\,\,\,\,\,$
(2.16)

where $h_{\mathrm{ECB}}$ is the dimension of the extended Coulomb branch and
$\Delta_{a}^{\mathrm{sing}}$ was defined above Proposition 1.111111A theory
with a non-trivial extended Coulomb branch (ECB) is one such that a generic
point of the Coulomb branch contains free hypermultiplets. The quantity
$h_{\mathrm{ECB}}$ is the quaternionic dimension of the ECB, meaning that the
Coulomb branch of a theory with ECB of dimension $h_{\mathrm{ECB}}$ is
effectively enlarged to an $r+2h_{\rm ECB}$ complex-dimensional space, which
looks locally like a product CB$\times\mathbb{H}^{h_{\mathrm{ECB}}}$ with
$\mathbb{H}$ representing a free hypermultiplet. One must be careful in
distinguishing the case where the CB looks like
CB$\times\mathbb{H}^{h_{\mathrm{ECB}}}$ at all points (a free ECB), or only
away from singular loci (a coupled ECB)—see Appendix D for details. The
quantity $T^{i}(\mathbf{R})$ is somewhat more subtle, and will be described in
Appendix D. Finally, the quantities $b_{a},\,\Delta_{a},$ and
$T^{i}(\mathbf{R}_{a})$ depend only on the rank-one theories living on ${\cal
I}_{a}$, and are collected in Table 1.

For each choice of decomposition of $a_{1}$, and each choice of strata
realizing this decomposition, we can now compute $h_{\mathrm{ECB}}$ and
$k_{\mathfrak{g}_{i}}$. Note that $h_{\mathrm{ECB}}$ must be a non-negative
integer. If we compute $h_{\mathrm{ECB}}$ and find
$h_{\mathrm{ECB}}\notin{\mathbb{N}}$, we declare that case ruled out.
Otherwise, we take the values of $k_{\mathfrak{g}_{i}}$ and proceed to the
next step.

Before moving on, let us summarize what we have done so far. We began by
inputting the first few Fourier coefficients of the vacuum character and using
them to determine a finite list of possible
$(a_{\mathrm{4d}},c_{\mathrm{4d}})$ for the tentative 4d theory. For each
element of this list, we then used the Shapere-Tachikawa formula to obtain
$\sum_{i=1}^{r}\Delta_{u_{i}}$, from which we obtained a finite number of
possible $r$-tuples $(\Delta_{u_{1}},\dots,\Delta_{u_{r}})$. At the same time,
the quantity $a_{1}$ was used to obtain a finite number of possible flavor
symmetries $\mathfrak{g}=\mathfrak{u}(1)^{n}\bigoplus_{i}\mathfrak{g}_{i}$,
each of which could be realized on the CB strata in a finite number of ways.
We may then consider all possible combinations of $r$-tuples
$(\Delta_{u_{1}},\dots,\Delta_{u_{r}})$ and CB stratifications and compute
$h_{\mathrm{ECB}}$ and $k_{\mathfrak{g}_{i}}$ for each. Our goal is now to
impose enough constraints to whittle this large number of candidate cases down
to a small number of physical cases. Some first constraints are provided by
Propositions 1, 2, and 3, as well as by requiring that
$h_{\mathrm{ECB}}\in{\mathbb{N}}$. The rest of this section will describe
additional constraints.

###  2.4 Unitarity constraints

Four-dimensional unitarity can be used to obtain stringent constraints on the
data obtained thus far. As we now review, these constraints include lower
bounds on $k_{\mathfrak{g}}$, as well as upper and lower bounds on
$c_{{\mathrm{2d}}}$ Beem:2017ooy ; Beem:2018duj . Beginning with the former,
unitarity demands that

$\displaystyle k_{\mathfrak{g}}\geq\widetilde{k}_{\mathfrak{g}}~{},$ (2.17)

where $\widetilde{k}_{\mathfrak{g}}$ is a fixed value associated to each
$\mathfrak{g}$. The values of $\widetilde{k}_{\mathfrak{g}}$ are collected in
Table 2. If we have
$\mathfrak{g}=\mathfrak{u}(1)^{n}\bigoplus_{i}\mathfrak{g}_{i}$, these bounds
must be satisfied for each simple factor $\mathfrak{g}_{i}$.

$\mathfrak{g}$ | $\widetilde{k}_{\mathfrak{g}}$ | $h^{\vee}$
---|---|---
$\mathfrak{su}(N)$ |  $N\geq 3$, | $N$ | $N$
$\mathfrak{so}(N)$ | $N=4,\dots,8$ | $4$ |
$\mathfrak{so}(N)$ | $N\geq 8$ | $N-4$ | $N-2$
$\mathfrak{sp}(2N)$ | $N\geq 3$ | $N+2$ | $N+1$
$\mathfrak{g}_{2}$ | ${10\over 3}$ | 4
$\mathfrak{f}_{4}$ | $5$ | 9
$\mathfrak{e}_{6}$ | $6$ | 12
$\mathfrak{e}_{7}$ | $8$ | 18
$\mathfrak{e}_{8}$ | $12$ | 30
Table 2: Values of $\widetilde{k}_{\mathfrak{g}}$ and $h^{\vee}$ appearing in
the unitarity constraints. Algebras not appearing in this table do not have
any bounds on $k_{\mathfrak{g}}$.

As for the bounds on $c_{{\mathrm{2d}}}$, there is a general upper bound

$\displaystyle c_{{\mathrm{2d}}}\leq-{11\over 5}\left(1+\sqrt{1+{180\over
121}\sum_{i}{{k_{\mathfrak{g}_{i}}}\,{\mathrm{dim}}\mathfrak{g}_{i}\over
3k_{\mathfrak{g}_{i}}-h^{\vee}_{i}}}\right)~{}.$ (2.18)

Furthermore, when there is a sub-critical simple factor in the flavor group,
then one also has the lower bound121212Recall that a critical factor is one
such that $k_{\mathfrak{g}}=2h^{\vee}$, while a sub-critical factor is one
such that $k_{\mathfrak{g}}<2h^{\vee}$.

$\displaystyle c_{{\mathrm{2d}}}\geq c_{\mathrm{Sug}}~{},\hskip
36.135ptc_{\mathrm{Sug}}=\sum_{i}{k_{\mathfrak{g}_{i}}\,\mathrm{dim}\mathfrak{g}_{i}\over
k_{\mathfrak{g}_{i}}-2h^{\vee}_{i}}~{}.$ (2.19)

For convenience the dual Coxeter numbers $h^{\vee}_{i}$ of $\mathfrak{g}_{i}$
are tabulated in Table 2.

In addition to the bounds on $k_{\mathfrak{g}_{i}}$ and $c_{{\mathrm{2d}}}$,
unitarity also requires the following more subtle constraints Beem:2018duj :

  1. 1.

If the flavor symmetry has critical factors, there can be no sub-critical
factors.

  2. 2.

If the flavor symmetry has no critical factors, there can be at most one sub-
critical factor.

Imposing all of these constraints will reduce the number of candidate cases
drastically, as will be seen explicitly when we turn to our computerized scan.

###  2.5 Characteristic dimension constraints

There is one final constraint which we impose. To introduce it, we begin by
writing

$\displaystyle(\Delta_{u_{1}},\dots,\Delta_{u_{r}})=\lambda(d_{1},\dots,d_{r})$
(2.20)

where $\lambda\in{\mathbb{Q}}$ and $d_{i}$ are the unique integers with
$\mathrm{gcd}(d_{1},\dots,d_{r})=1$ satisfying the above. We then introduce a
quantity $\varkappa$ known as the “characteristic dimension,” defined in
Cecotti:2021ouq  to be:

$\varkappa:=\frac{1}{\\{\lambda^{-1}\\}}~{},$ (2.21)

where for any $x\in\mathbb{R}$, $\\{x\\}$ is the unique real number equal to
$x$ mod 1 such that $0<\\{x\\}\leq 1$. In terms of this quantity we have the
following result,

###### Proposition 4

The characteristic dimension can only take one of eight rational values
$\varkappa\in\\{1,6/5,4/3,3/2,2,3,4,6\\}$, and if $\varkappa\neq 1,2$ then
there can be no IR-free theories living on any Coulomb branch strata.

This proposition can be argued for as follows. First, when $\varkappa\neq 1,2$
the CB geometry is such that at a generic point the $U(1)_{r}$ symmetry is
only broken to a $\mathbb{Z}_{n}$ subgroup, with

$\displaystyle
n=\left\\{\begin{matrix}6&\,\,\,\mathrm{if}\,\,\,\varkappa={6\over 5},6~{},\\\
4&\,\,\,\mathrm{if}\,\,\,\varkappa={4\over 3},4~{},\\\
3&\,\,\,\mathrm{if}\,\,\,\varkappa={3\over 2},3~{}.\end{matrix}\right.$ (2.22)

This puts severe restriction on the allowed BPS spectra, which in particular
constrains the full CB geometry to be _isotrivial_ and _diagonal_.
Isotriviality means that the total space of the CB is locally a product
$A\times{\mathbb{C}}^{2}$ with $A$ a Jacobian variety, while diagonality means
that $A$ factorizes into a product of $r$ genus-one varieties
$A=E_{\tau}\times\dots\times E_{\tau}$ at the same coupling. Thus for a given
value of the rank $r$ and $\varkappa\neq 1,2$, the total structure of the
Coulomb branch can be one of only a handful possibilities—see Section 4.2 for
more detail.

It is a non-trivial result that isotriviality implies that all of the
monodromies around singular loci have to be of finite order Cecotti:2021ouq .
Since the hallmark of IR-free theories is monodromy of infinite order (i.e. of
parabolic-type), isotriviality also implies that no IR-free theory can live on
any CB strata. This gives the desired statement.

In practice, this constraint is implemented in our algorithm as follows. If
the steps outlined in the previous subsections produce a candidate theory with
a set of $\\{{\Delta}_{u_{i}}\\}$ for which $\varkappa\neq 1,2$, and the only
compatible Coulomb branch stratification includes IR-free theories, we may
discard that case.

##  3 An Example: Rediscovering $D_{2}(SU(5))$

We now illustrate the algorithm outlined above by means of a simple rank-two
example. As we shall see, even in this simple example the analysis is lengthy,
and one is well-advised to implement the algorithm via a computer.

When restricting to rank two, the CB analysis outlined in Section 2.3
simplifies significantly. For example, Proposition 2 reduces to the statement
that an unknotted stratum on $u_{1}=0$ can appear only if $\Delta_{u_{2}}$
takes values in (2.11), and vice versa. A further simplifying feature is that
at rank-two, knotted singularities are specified by a single type of
irreducible homogeneous polynomial, taking the generic form

$P(u_{1},u_{2})=u_{1}^{p}+\alpha\,u_{2}^{q}~{},$ (3.23)

where $\alpha$ is a free parameter and $p,q$ are chosen such that the left-
hand side is homogeneous under a ${\mathbb{C}}^{*}$-action, _i.e._
$p/q:=\Delta_{u_{2}}/\Delta_{u_{1}}$ such that $(p,q)=1$. Knotted
singularities have $\Delta^{\mathrm{sing}}=p\Delta_{u_{1}}$.

Let us consider the following input data,

$\displaystyle(c_{\mathrm{2d}},a_{1},a_{2})=\left(-24,\,24,\,300\right)\hskip
14.45377pt\Rightarrow\hskip
14.45377pt\chi_{0}(q)=q+24q^{2}+300q^{3}+O(q^{4})~{},$ (3.24)

and assume that the vacuum character solves an order-4 untwisted MDE. We now
ask if there exists any rank-two unitary ${\cal N}=2$ SCFT compatible with
this data. First, we solve the corresponding MDE order-by-order and check if
the corresponding vacuum character has integer Fourier expansion and unit
leading coefficient. Indeed, one finds the closed form solution

$\displaystyle\chi_{0}(q)=\left(\eta(q^{2})\over\eta(q)\right)^{24}=q\,(1+24q+300q^{2}+2624q^{3}+18126q^{4}+105504q^{5}+\dots)\,$
(3.25)

which is integral to all orders in $q$.131313This can be seen, for example, by
relating it to the McKay-Thompson series for Co0 Duncan:2014eha . We next note
that the indicial equation (2.8) can be used to obtain the chiral dimensions
$\\{h_{i}\\}=\left\\{0,\,-{1\over 2},\,-1,\,-{3\over 2}\right\\}$. These may
be used to compute the 4d central charges via (2.9). We start with the choice
$h_{*}=-{3\over 2}$, giving

$\displaystyle(a_{\mathrm{4d}},c_{\mathrm{4d}})=\left({7\over
4},\,2\right)~{}.$ (3.26)

Since $c_{\mathrm{4d}}>a_{\mathrm{4d}}$, this is only legitimate if $h_{*}$ is
the minimal chiral dimension, which is indeed the case. These values also
satisfy the Hoffman-Maldacena bounds. If we choose other values for $h_{*}$ we
will get other candidates for $(a_{\mathrm{4d}},c_{\mathrm{4d}})$, and all of
them must be checked. For the moment let us simply focus on (3.26).

We next use the Shapere-Tachikawa formula (2.12) to obtain the sum of the CB
dimensions, which gives $\Delta_{u_{1}}+\Delta_{u_{2}}=4$. We then decompose
the sum into allowed pairs of dimensions, giving

$\displaystyle\left(\Delta_{u_{1}},\Delta_{u_{2}}\right)\in\left\\{\left({4\over
3},{8\over 3}\right),\left({3\over 2},{5\over 2}\right),\left({8\over
5},{12\over 5}\right),\left(2,2\right)\right\\}~{},$ (3.27)

_c.f._ Appendix B. We also note that $a_{1}=24$ admits the following
decompositions into dimensions of simple Lie algebras,

$\displaystyle\\{24\\}:$ $\displaystyle\hskip 36.135pt\mathfrak{su}(5)$ (3.28)
$\displaystyle\\{3,21\\}:$ $\displaystyle\hskip
36.135pt\mathfrak{su}(2)\oplus\mathfrak{so}(7)\,\,\,\mathrm{or}\,\,\,\mathfrak{su}(2)\oplus\mathfrak{sp}(6)$
$\displaystyle\\{10,14\\}:$ $\displaystyle\hskip
36.135pt\mathfrak{so}(5)\oplus\mathfrak{g}_{2}$ $\displaystyle\\{8,8,8\\}:$
$\displaystyle\hskip
36.135pt\mathfrak{su}(3)\oplus\mathfrak{su}(3)\oplus\mathfrak{su}(3)$

among others. If we allow for $\mathfrak{u}(1)$ factors, then we must allow
for arbitrary integer partitions of 24, of which there are 1575.

With these decompositions in hand, we now proceed by constructing possible CB
stratifications. Let us begin by assuming flavor symmetry $\mathfrak{su}(5)$
and scaling dimensions
$\left(\Delta_{u_{1}},\Delta_{u_{2}}\right)=\left({4\over 3},{8\over
3}\right)$. From Proposition 3, we know that the full flavor symmetry must be
realized on at least one codimension-one stratum, which according to Table 1
can only be a $[I_{5},\mathfrak{su}(5)]$ stratum. We may still allow for
unflavored strata ${\cal T}_{\varnothing,1}^{(1)}$ and
$[I_{1},\varnothing]$.141414The careful reader might notice that
$[I_{1},\varnothing]$ has a $U(1)$ flavor symmetry, and thus might object to
use of the word “unflavored.” However, here we are only interested in flavor
factors which act on the CB by “splitting” the singular locus. Because the
$U(1)$ mass deformation does not split the $I_{1}$ singularity (see _e.g._
Argyres:2015ffa ) the $[I_{1},\varnothing]$ stratum is effectively unflavored.
This is to be contrasted with the cases of ${\cal S}_{\varnothing,3}^{(1)}$
and ${\cal S}_{\varnothing,4}^{(1)}$, which also have $U(1)$ flavor symmetry,
but for which the $U(1)$ splits the singular locus. From Proposition 2, we
know that unknotted strata can only be of the form $u_{2}=0$, since
$\Delta_{u_{2}}={8\over 3}$ is not on the list of allowed rank-one Coulomb
branch dimensions. Thus both knotted and unknotted strata must have
$\Delta^{\mathrm{sing}}_{i}={8\over 3}$.

Using (2.3) to compute $k_{\mathfrak{su}(5)}$, we find that
$k_{\mathfrak{su}(5)}={16\over 3}$ regardless of whether the flavored stratum
is unknotted or knotted. This value of the level satisfies the unitarity bound
(2.17). It also satisfies the bounds in (2.18) and (2.19), which read
$-27\lesssim c_{\mathrm{2d}}\lesssim-11.6$. However, note that the
characteristic dimension $\varkappa$ in this case is $\varkappa={4\over 3}$.
Thus by Proposition 4, there can be no IR-free theories living on any Coulomb
branch strata. Since $[I_{5},\mathfrak{su}(5)]$ is IR-free, it is not allowed,
and so we conclude that there is no consistent Coulomb branch geometry with
$\mathfrak{su}(5)$ and
$\left(\Delta_{u_{1}},\Delta_{u_{2}}\right)=\left({4\over 3},{8\over
3}\right)$.

Next consider the choice of flavor symmetry $\mathfrak{su}(5)$ and
$\left(\Delta_{u_{1}},\Delta_{u_{2}}\right)=\left({3\over 2},{5\over
2}\right)$. In this case the unknotted strata can again only be of the form
$u_{2}=0$. We see that unknotted strata have
$\Delta^{\mathrm{sing}}_{i}={5\over 2}$, while knotted strata have
$\Delta^{\mathrm{sing}}_{i}={15\over 2}$. Turning again to the levels, we find
that $k_{\mathfrak{su}(5)}={5}$ (resp. ${15}$) if the flavored stratum is
unknotted (resp. knotted). Both of these satisfy the unitarity bound (2.17),
but only one satisfies the bounds (2.18) and (2.19). Indeed, for the case
$k_{\mathfrak{su}(5)}={5}$ the bounds read $-24\leq
c_{\mathrm{2d}}\lesssim-11.7$ and are satisfied (in fact, we see that the
lower bound is saturated, signaling that the theory is Sugawara). So for
consistency we must have the flavored stratum be unknotted.

To complete the construction of the Coulomb branch geometry, we must now
identify the relevant unflavored strata. First we note that the characteristic
dimension is $\varkappa=1$, so by Proposition 4 IR-free strata are allowed.
Since the unknotted $u_{2}=0$ stratum is already “occupied” by the
$[I_{5},\mathfrak{su}(5)]$ theory, the unflavored strata must be knotted. The
most general scenario is to assume $n_{1}$ knotted $[I_{1},\varnothing]$
strata, and $n_{2}$ knotted ${\cal T}_{\varnothing,1}^{(1)}$ strata. Then from
(2.3) we compute the dimension of the extended Coulomb branch to be

$\displaystyle h_{\mathrm{ECB}}={15\over 2}-{15\over 2}(n_{1}+2n_{2})~{}.$
(3.29)

This is required to be a non-negative integer. Furthermore, from Proposition 1
we see that we must have at least one knotted stratum, and hence
$n_{1}+2n_{2}\geq 1$. There is a single non-trivial solution to these
equations, namely $(n_{1},n_{2})=(1,0)$ and $h_{\mathrm{ECB}}=0$. The fact
that there exists a solution at all is an extremely non-trivial consistency
check. Finally, the two unitarity bounds mentioned after (2.19) are trivially
satisfied since the flavor symmetry is simple.

In summary, based on the input data
$(c_{\mathrm{2d}},a_{1},a_{2})=\left(-24,\,24,\,300\right)$, we have used two-
and four-dimensional consistency conditions to gain evidence for the existence
of a rank-two ${\cal N}=2$ theory with
$(a_{\mathrm{4d}},c_{\mathrm{4d}})=({7\over 4},2)$, flavor symmetry
$\mathfrak{su}(5)_{5}$, Coulomb branch dimensions
$(\Delta_{u_{1}},\Delta_{u_{2}})=({3\over 2},{5\over 2})$, and extended
Coulomb branch dimension $h_{\mathrm{ECB}}=0$. Furthermore, the Coulomb branch
of the tentative theory is required to have the particular form given in
Figure 2. Such a theory would satisfy all consistency conditions of which we
are aware.

$\mathfrak{su}(5)_{5}$$[I_{1},\varnothing]$
$[I_{5},\mathfrak{su}(5)]$$\big{[}u_{1}^{5}+u_{2}^{3}=0\big{]}$$\big{[}u_{2}=0\big{]}$
Figure 2: Hasse diagram for the CB of the theory with $\mathfrak{su}(5)_{5}$
flavor symmetry.

In fact, precisely such a theory is known—it is the $D_{2}(SU(5))$ theory of
Cecotti:2013lda . What we have seen is that, rather remarkably, the data of
this theory is fixed almost entirely in terms of three parameters (3.24). Of
course, we have not yet shown that this is the only consistent ${\cal N}=2$
theory that can follow from this 2d data. In other words, though it is
conjecturally believed that the map from ${\cal N}=2$ theories to VOAs is one-
to-one, this is something which must be checked. In order to check it, we must
show that the remaining combinations of $(a_{\mathrm{4d}},c_{\mathrm{4d}})$,
of the flavor symmetry ($\mathfrak{su}(2)\oplus\mathfrak{so}(7)$ etc.), and of
the Coulomb branch scaling dimensions do not give rise to any other seemingly
legitimate theories. By repeating the steps above for all of the remaining
case (with the help of a computer), one can indeed verify that this is the
case.

##  4 Rank-two scan

In the previous sections we introduced an algorithm for identifying mutually
consistent sets of VOA and CB data which could correspond to legitimate 4d
${\cal N}=2$ theories. In particular, the algorithm allowed us to determine
the central charges and Coulomb branch data of the ${\cal N}=2$ theories,
including scaling dimensions and possible stratifications. In this section we
will apply these techniques to search for rank-two theories with Schur indices
satisfying an order-four untwisted MDE. In particular, by scanning over a wide
range of values for $(c_{\mathrm{2d}},a_{1},a_{2})$ and applying our algorithm
in each case, we aim to obtain a chart of the landscape of such 4d theories.

One obvious drawback of this strategy is that it is quite computationally
intensive. Indeed, our goal is to scan over a large enough range of values
$c_{\mathrm{2d}}=-{p\over q}$, $a_{1}$, and $a_{2}$, with $p,q,a_{1},a_{2}$
positive integers, such that we are confident that all 4d ${\cal N}=2$
theories would fall into that range. While there is no guarantee that any
finite scan will be exhaustive, we find it reasonable to restrict to $p,q\leq
200$, $a_{1}\leq 248$, and $a_{2}\leq 1000\,a_{1}$—the latter in particular
seems reasonable since the coefficient $a_{2}$ must be decomposable into
representations of a flavor symmetry of dimension $a_{1}$, which for
$a_{1}\leq 248$ will not be exorbitantly larger than $a_{1}$ itself, unless
rather baroque representations make an appearance. Restricting to
$\mathrm{gcd}(p,q)=1$, the total number of cases which we will scan over is
then of order $O(10^{11})$. Furthermore, for each case one must consider all
ways of partitioning $a_{1}$ into dimensions of simple Lie algebras. If we
truly allow for all possible decompositions, the rough number of cases which
must be checked is of order $O(10^{25})$. Scanning over such a large number of
cases is clearly impractical, even with a computer.

This number may be reduced drastically by restricting the allowed partitions
of $a_{1}$. One simplification is to restrict to flavor symmetries with small
numbers of $U(1)$ factors—for example, only up to two $U(1)$ factors. Another
is to restrict to flavor symmetries with only a certain number of simple
factors—for example, up to 3 simple factors. Since these two conditions are
satisfied by all known rank-two theories Martone:2021ixp , we find it
reasonable to impose them both, though it is certainly possible for there to
exist unknown theories violating them. These conditions cut the number of
cases down to something of order $O(10^{12})$—a large number to be sure, but
one which is now accessible with advanced enough computing.

###  4.1 Results

We now report the results of the scan described above. Running a number of
Mathematica tasks in parallel, we carried out a scan over $O(10^{11})$
triplets $(c_{\mathrm{2d}},a_{1},a_{2})$. In each case we applied our
algorithm to search for data consistent with a 4d ${\cal N}=2$ interpretation.
Remarkably, amongst all of these cases we identify only 15 candidate theories,
of which six have appeared in the literature Martone:2021ixp . We give the
data of all of these candidate theories in Table 3, as well as their CB Hasse
diagrams in Figure 3.

Before providing information about the putative new theories, let us briefly
summarize the known theories that our algorithm rediscovers. These theories
are usually denoted by $D_{1}^{20}(E_{8})$, $(A_{1},A_{4})$,
$\mathrm{USp}(4)+6F$, $D_{2}(SU(5))$, $(A_{1},D_{5})$, and ${\cal
T}_{\varnothing,1}^{(1)}\times{\cal T}_{\varnothing,1}^{(1)}$. Their data,
including Coulomb and Higgs branch geometries, can be found in e.g. Table 1 of
Martone:2021ixp . In Section 3, we showed explicitly how our algorithm
reconstructs the $D_{2}(SU(5))$ theory. The other cases proceed similarly.

Clearly, we have only reproduced a small fraction of the known rank-two
theories catalogued in Martone:2021ixp . This is of course to be expected,
since we searched only for theories whose Schur indices satisfy untwisted
order-four MDEs. A preliminary analysis shows that the overwhelming majority
of known rank-two theories satisfy twisted fourth-order MDEs.151515It remains
an open question whether there is a universal upper bound on the order of the
MDE for a fixed rank SCFT. For this reason, we anticipate that extending the
current scan to the order-four twisted case would give even more interesting
results.

We have found that in all cases, for a given set of legitimate
$(c_{\mathrm{2d}},a_{1},a_{2})$, there is always a unique possibility for the
data of the corresponding 4d ${\cal N}=2$ theory. This gives non-trivial
evidence for the injective nature of the map from 4d ${\cal N}=2$ theories to
chiral algebras.

| $(c_{\mathrm{2d}},a_{1},a_{2})$ | $c_{\mathrm{4d}}$ | $a_{\mathrm{4d}}$ | $(\Delta_{u_{1}},\Delta_{u_{2}})$ | $h_{\mathrm{ECB}}$ | $\varkappa$ | $\mathfrak{g}_{k}$ | Status
---|---|---|---|---|---|---|---|---
1. | $(-124,\,248,\,31124)$ | ${31\over 3}$ | ${101\over 12}$ | $(4,10)$ | 0 | 2 | $(\mathfrak{e}_{8})_{20}$ | $D_{1}^{20}(E_{8})$
2. | $(-{68\over 7},\,0,\,1)$ | ${17\over 21}$ | ${67\over 84}$ | $({8\over 7},{10\over 7})$ | 0 | 2 | $\varnothing$ | $(A_{1},A_{4})$
3. | $(-44,\,66,\,2200)$ | $11\over 3$ | $37\over 12$ | $(2,4)$ | 0 | 2 | $\mathfrak{so}(12)_{8}$ | $\mathrm{USp}(4)+6F$
4. | $(-{24},\,24,\,300)$ | $2$ | $7\over 4$ | $({3\over 2},{5\over 2})$ | 0 | 1 | $\mathfrak{su}(5)_{5}$ | $D_{2}(SU(5))$
5. | $(-{12},\,3,\,9)$ | $1$ | $19\over 20$ | $({6\over 5},{8\over 5})$ | 0 | 2 | $\mathfrak{su}(2)_{16\over 5}$ | $(A_{1},D_{5})$
6. | $(-{44\over 5},\,0,\,2)$ | $11\over 15$ | $43\over 60$ | $({6\over 5},{6\over 5})$ | 0 | $6\over 5$ | $\varnothing$ | ${\cal T}_{\varnothing,1}^{(1)}\times{\cal T}_{\varnothing,1}^{(1)}$
7. | $(-64,\,64,\,2080)$ | $16\over 3$ | $14\over 3$ | $(4,5)$ | 0 | 1 | $\mathfrak{su}(8)_{10}\times\mathfrak{u}(1)$ | ?
8. | $(-{44\over 5},\,2,\,2)$ | $11\over 15$ | $43\over 60$ | $({6\over 5},{6\over 5})$ | 0 | $6\over 5$ | $\mathfrak{u}(1)^{2}$ | ?
9. | $(-{44\over 3},\,0,\,3)$ | $11\over 9$ | $49\over 36$ | $({4\over 3},{8\over 3})$ | 0 | $4\over 3$ | $\varnothing$ | ✗
10. | $(-{36},\,18,\,189)$ | $3$ | $11\over 4$ | $(3,3)$ | 5 | 3 | $\mathfrak{su}(2)_{4}\times\mathfrak{su}(4)_{8}$ | ✗
11. | $(-{28},\,8,\,38)$ | $7\over 3$ | $29\over 12$ | $(3,3)$ | 0 | 3 | $\mathfrak{su}(3)_{6}$ | ✗
12. | $(-{24},\,8,\,44)$ | $2$ | $15\over 8$ | $({3\over 2},3)$ | 2 | $3\over 2$ | $\mathfrak{su}(2)_{4}\times\mathfrak{su}(2)_{4}\times\mathfrak{u}(1)^{2}$ | ✗
13. | $(-{20},\,4,\,17)$ | $5\over 3$ | $19\over 12$ | $({3\over 2},{5\over 2})$ | 1 | 1 | $\mathfrak{su}(2)_{7/2}\times\mathfrak{u}(1)$ | ✗
14. | $(-{10},\,2,\,7)$ | $5\over 6$ | $11\over 12$ | $({3\over 2},{3\over 2})$ | 0 | $3\over 2$ | $\mathfrak{u}(1)^{2}$ | ✗
15. | $(-{84\over 5},\,2,\,7)$ | $27\over 20$ | $7\over 5$ | $({6\over 5},{12\over 5})$ | 2 | $6\over 5$ | $\mathfrak{u}(1)^{2}$ | ✗
Table 3: The 15 tentative theories identified in our scan. The first six are
already known, c.f. Table 1 of Martone:2021ixp . The remaining nine theories
are discussed in the rest of this work. Those marked with an ✗ will be shown
not to exist, while those marked with a ? have not yet been ruled out but seem
unlikely to exist.

###  4.2 New rank-two theories?

We now discuss the nine candidate 4d ${\cal N}=2$ SCFTs which our scan
identifies but which have not appeared in the literature. We will denote these
tentative theories by ${\cal T}_{i}$, with the index $i=7,\dots,15$
corresponding to the label in Table 3. It is useful to split them into two
categories: those with characteristic dimension $\varkappa\neq 1,2$ and those
with $\varkappa=1,2$.

####  4.2.1 Cases with $\varkappa\neq 1,2$

We start with the six candidate theories that have characteristic dimension
$\varkappa\neq 1,2$. As discussed in in Section 2.5, theories of this kind
satisfy stronger CB constraints, which allow us to make more sophisticated
statements about the proposed CB stratifications. We warn the reader that the
arguments in this section are somewhat technical, and can be skipped on a
first reading.

To begin, theories with $\varkappa\neq 1,2$ cannot support any IR-free strata
on their CBs, and this is indeed reflected in the Hasse diagrams shown in
Figure 3. This property stemmed from the more fundamental fact that the CBs of
such theories must be isotrivial and diagonal. The condition of isotriviality
means that the total space of the CB is locally a product
$A\times{\mathbb{C}}^{2}$, while the diagonality means that $A$ factors into a
product of two genus-one varieties $A=E_{\tau}\times E_{\tau}$. Furthermore,
the value of $\tau$ is fixed by the value of $\varkappa$: in particular, we
have (locally)

$\displaystyle\varkappa$ $\displaystyle\in$ $\displaystyle\left\\{4,{4\over
3}\right\\}:\hskip 59.26192pt\tau=i~{},$ $\displaystyle\varkappa$
$\displaystyle\in$ $\displaystyle\left\\{3,{3\over 2},6,{6\over
5}\right\\}:\hskip 36.135pt\tau=\rho~{},$ (4.30)

with $\rho:=e^{\pi i/3}$. The global structure of the Coulomb geometry $X$ is
then given by choosing a discrete group $G$ and a group homomorphism

$\displaystyle\sigma:\,G\rightarrow\mathrm{Aut}(A)$ (4.31)

and defining

$\displaystyle
X=A\times{\mathbb{C}}^{2}/[(a,x)\sim(\sigma(g)a,\,gx)~{},~{}~{}g\in G]~{}.$
(4.32)

As mentioned before, we assume that the CB chiral ring is freely-generated (or
alternatively that the CB has no complex singularities Argyres:2017tmj ;
Argyres:2018wxu ), and thus $G$ is restricted to be a complex reflection group
by the Chevalley-Shephard-Todd theorem Shephard:1954 ; Chevalley:1955 . We
will also for the moment assume that it is indecomposable, i.e. that the rank-
two Coulomb branch is not simply a product of two rank-one Coulomb branches.
We will lift this assumption later.

${\cal T}_{7}$$[I_{8},\mathfrak{su}(8)]$
$[I_{1},\varnothing]$$\Delta^{\mathrm{sing}}={5}$$\Delta^{\mathrm{sing}}={20}$${\cal
T}_{8}$${\cal T}_{\varnothing,1}^{(1)}$ ${\cal
T}_{\varnothing,1}^{(1)}$$\Delta^{\mathrm{sing}}={6\over
5}$$\Delta^{\mathrm{sing}}={6\over 5}$${\cal T}_{9}$${\cal
T}_{\varnothing,1}^{(1)}$ ${\cal
T}_{\varnothing,1}^{(1)}$$\Delta^{\mathrm{sing}}={8\over
3}$$\Delta^{\mathrm{sing}}={8\over 3}$
${\cal T}_{10}$${\cal S}_{\varnothing,2}^{(1)}$ ${\cal
S}_{D_{4},3}^{(1)}$$\Delta^{\mathrm{sing}}={3}$$\Delta^{\mathrm{sing}}={3}$${\cal
T}_{11}$${\cal T}_{\varnothing,1}^{(1)}$ ${\cal T}_{A_{2},1}^{(1)}$${\cal
T}_{\varnothing,1}^{(1)}$
$\Delta^{\mathrm{sing}}={3}$$\Delta^{\mathrm{sing}}={3}$${\cal T}_{12}$${\cal
S}_{\varnothing,2}^{(1)}$ ${\cal
S}_{\varnothing,2}^{(1)}$$\Delta^{\mathrm{sing}}={3}$$\Delta^{\mathrm{sing}}={3}$
${\cal T}_{13}$${\cal S}_{\varnothing,2}^{(1)}$
$[I_{1},\varnothing]$$\Delta^{\mathrm{sing}}={5\over
2}$$\Delta^{\mathrm{sing}}={15\over 2}$${\cal T}_{14}$${\cal
T}_{\varnothing,1}^{(1)}$ ${\cal
T}_{\varnothing,1}^{(1)}$$\Delta^{\mathrm{sing}}={3\over
2}$$\Delta^{\mathrm{sing}}={3\over 2}$${\cal T}_{15}$${\cal
S}_{\varnothing,4}^{(1)}$ $\Delta^{\mathrm{sing}}={12\over 5}$ Figure 3: Hasse
diagrams for the CB of the candidate new theories ${\cal T}_{i}$ for
$i=7,\dots,15$, as output by the algorithm.

It is useful to introduce notation $G^{\prime}$ for the image of the map
$\sigma$,

$\displaystyle G^{\prime}:=\sigma(G)\subset\mathrm{Aut}(A)~{}.$ (4.33)

The unbroken R-symmetry generated by $e^{2\pi i/n}$ (with $n$ as given in
(2.22)) must be contained in $G^{\prime}$, so we have the chain of inclusions

$\displaystyle\langle e^{2\pi i/n}\rangle\subset
G^{\prime}\subset\mathrm{Aut}(A)~{}.$ (4.34)

It is also clear that $G^{\prime}$ must be crystallographic, since it should
leave the lattice of $A$ invariant. Noting that the maximal automorphism
groups of $E_{\tau}\times E_{\tau}$ at the points $\tau=i,\rho$ are given by
Fujiki:1988

$\displaystyle A$ $\displaystyle=$ $\displaystyle E_{i}\times E_{i}:\hskip
38.30301pt\mathrm{Aut}(A)\in\left\\{G_{8},\,G(4,1,2)\right\\}~{},$
$\displaystyle A$ $\displaystyle=$ $\displaystyle E_{\rho}\times
E_{\rho}:\hskip
36.135pt\mathrm{Aut}(A)\in\left\\{G_{5},\,G(6,1,2),\,G(6,2,2)\right\\}~{},$
(4.35)

it is then possible to determine all possible groups $G^{\prime}$. Indeed, one
finds

$\displaystyle n$ $\displaystyle=$ $\displaystyle 4:\hskip
46.97505ptG^{\prime}\in\left\\{G_{8},\,G(2,1,2),\,G(4,1,2),\,G(4,2,2)\right\\}~{},$
$\displaystyle n$ $\displaystyle=$ $\displaystyle 3,6:\hskip
36.135ptG^{\prime}\in\left\\{G_{4},\,G_{5},\,G(3,1,2),\,G(6,1,2),\,G(6,2,2)\right\\}~{}.$
(4.36)

Note that the full group $G$ is potentially a non-trivial extension of
$G^{\prime}$ by a group $H$, with $H$ such that the extension preserves the
rank of $G^{\prime}$. It is actually possible to list all valid $G$ and $H$
for a given $G^{\prime}$, as will be done in upcoming work Cecotti:2021yet .
Here we will only quote the portion of those results relevant for the current
analysis.

We begin with the case of $n=4$, which in particular means that
$\varkappa\in\\{4,{4\over 3}\\}$. The only such case in our list of tentative
theories is ${\cal T}_{9}$, which has
$(\Delta_{u_{1}},\Delta_{u_{2}})=({4\over 3},{8\over 3})$ and two ${\cal
T}_{\varnothing,1}^{(1)}$ Coulomb branch strata. From the above discussion, we
conclude that the group $G$ must be a (potentially trivial) extension of
$G^{\prime}=G_{8},\,G(2,1,2),\,G(4,1,2),$ or $G(4,2,2)$. It is an important
fact that the pair of Coulomb branch dimensions
$(\Delta_{u_{1}},\Delta_{u_{2}})$ is proportional to the pair of degrees
$(d_{1},d_{2})$ of invariant polynomials of $G$ (see the discussion around
(4.40) for more on this). Since we would like
$(\Delta_{u_{1}},\Delta_{u_{2}})=({4\over 3},{8\over 3})$, it is clear that we
must have $d_{2}=2d_{1}$. Not every complex reflection group satisfies this
property—indeed, the only ones which do are

$\displaystyle
G\in\left\\{G(m,1,2),\,G_{5},\,G_{10},\,G_{15},\,G_{18}\right\\}~{}.$ (4.37)

If $G^{\prime}=G(2,1,2)$ or $G(4,1,2)$, then it is consistent to have
$G=G^{\prime}$ since these appear in the list (4.37). In fact, these cases do
not admit any extensions to alternative entries in (4.37), and thus having
$G=G^{\prime}$ is the only possibility for them. On the other hand, $G_{8}$
and $G(4,2,2)$ do not appear in the list, nor can they be extended to elements
of the list, and thus they must be ruled out.

We are thus left with two choices for $G$. We would now like to see if either
of these choices can reproduce the desired Coulomb branch stratification.
Recall that we want two ${\cal T}_{\varnothing,1}^{(1)}$ strata, which are of
Kodaira-type $II$. In other words, the monodromy around these singular loci
must be of order 6. For a quotient of ${\mathbb{C}}^{2}$ by $G$, the singular
loci are in one-to-one correspondence with conjugacy classes of maximal cyclic
subgroups of $G$ generated by reflections. If we consider the fixed locus of a
generator $g$ of one of the maximal cyclic subgroups, then the monodromy
around that locus is given by $\sigma(g)$. Noting that neither $G(2,1,2)$ nor
$G(4,1,2)$ have any elements of order 6 and recalling that $\sigma$ is a group
homomorphism, we conclude that there cannot be any monodromy of order 6, and
hence that the proposed structure of the Coulomb branch is inconsistent.

Before moving on, it is worth noting that though a 4d interpretation for this
case can be ruled out, this does not mean that the corresponding VOA is ill-
defined. Indeed, in the current case the VOA can be identified with a simple-
current extension of a ${\mathbb{Z}}_{3}$ gauging of the $(4,9)$ ${\cal
W}_{3}$ minimal model, as shown in Appendix E. What we have shown here is
simply that this VOA does not lie in the image of the 4d $\to$ 2d map.

Next consider the case of $n=3,6$, which means that
$\varkappa\in\left\\{3,{3\over 2},6,{6\over 5}\right\\}$. Looking at Table 3,
we see that there are two subcases here: those with
$\Delta_{u_{2}}=2\Delta_{u_{1}}$ (namely ${\cal T}_{12}$ and ${\cal T}_{15}$)
and those with $\Delta_{u_{2}}=\Delta_{u_{1}}$ (namely ${\cal T}_{8},{\cal
T}_{10},{\cal T}_{11},$ and ${\cal T}_{14}$). Let us begin with the former. In
this case we again require $G$ to be an element of the list (4.37). If
$G^{\prime}=G_{5},\,G(3,1,2),$ or $G(6,1,2)$, then we can have $G=G^{\prime}$,
and indeed this is the only choice since there does not exist any non-trivial
extension to other elements in the list. On the other hand, for $G_{4}$ and
$G(6,2,2)$ the extension must be non-trivial. For $G_{4}$ there is in fact no
valid extension, so this case must be ruled out, whereas for $G(6,2,2)$ there
does exist an extension by the quaternion group $H={\cal Q}_{8}$ to $G_{15}$.
There are then a total of four options for $G$, namely
$G=G_{5},\,G(3,1,2),G(6,1,2),$ or $G_{15}$.

Next note that the candidate theory ${\cal T}_{12}$ has two ${\cal
S}_{\varnothing,2}^{(1)}$ strata, which are of Kodaira-type $I_{0}^{*}$. The
monodromy around each must be of order 2. For $G_{5}$ the only cyclic
subgroups are of order 3 (see e.g. Table D.1 of lehrer2009unitary ) and hence
this cannot give the desired structure of the Coulomb branch. Likewise for
$G(3,1,2)$ or $G(6,1,2)$, each has only a single cyclic subgroup of order 2,
so these cannot work either. Finally for $G_{15}$, there do in fact exist two
cyclic subgroups of order 2, but there is also a subgroup of order 3. In this
case we would obtain three singular loci, which is incompatible with the
proposed Coulomb branch stratification.

As for the candidate theory ${\cal T}_{15}$, this has a single ${\cal
S}_{\varnothing,4}^{(1)}$ stratum of Kodaira type $III^{*}$, and hence must be
of order 4. By the comments in the previous paragraph, there is again no way
to realize the desired Coulomb branch. We may thus rule this case out as well.

We now return to the case of $\Delta_{u_{2}}=\Delta_{u_{1}}$. In this case we
must consider complex reflection groups such that the degrees are equal,
$d_{1}=d_{2}$. The only complex reflection groups for which this holds are

$\displaystyle G(m,2,2),\,\,G_{7},\,\,G_{11},\,\,G_{19}~{}.$ (4.38)

For $G^{\prime}=G_{4},\,G_{5},\,G(3,1,2)$ or $G(6,1,2)$, we conclude that we
must have a non-trivial extension to a group in this list. For
$G^{\prime}=G_{4},\,G_{5}$ no such extension is possible, but for
$G^{\prime}=G(3,1,2),\,G(6,1,2)$ we can do a ${\mathbb{Z}}_{2}$ extension to
obtain $G=G(6,2,2),\,G(12,2,2)$ respectively. On the other hand, for
$G^{\prime}=G(6,2,2)$ we can take $G=G^{\prime}$, and indeed there exists no
extension giving an alternative element of (4.38). Hence we have a total of
two possibilities for $G$, namely $G=G(6,2,2),G(12,2,2)$ (and for the case of
$G=G(6,2,2)$ we have seen that there are two choices for $G^{\prime}$).

We now ask whether any of these cases can give rise to the desired Coulomb
branch stratifications. The Kodaira type of the strata of the various theories
are as follows:

$\displaystyle{\cal T}_{8}:$ $\displaystyle\hskip 21.68121pt(II,II)\hskip
61.42993pt{\cal T}_{10}:\hskip 21.68121pt(I_{0}^{*},II^{*})$ (4.39)
$\displaystyle{\cal T}_{11}:$ $\displaystyle\hskip 21.68121pt(II,II,IV)\hskip
41.19443pt{\cal T}_{14}:\hskip 21.68121pt(II,II)$

The strata of type $I_{0}^{*}$ have monodromy of order 2, while those of type
$II/II^{*}$ have monodromy of order 6, and those of type $IV$ have monodromy
of order 3. Note that both $G(3,1,2)$ and $G(6,2,2)$ have two conjugacy
classes of maximal cyclic subgroups generated by reflections, of order 2 and 3
respectively, while $G(12,2,2)$ has subgroups of order 2 and 6 instead. It is
thus clear that the Coulomb branch stratifications proposed for the ${\cal
T}_{8},{\cal T}_{11}$, and ${\cal T}_{14}$ theories can never be realized in
this way. These theories may thus be ruled out as indecomposable rank-two
theories (we will discuss the possibility of them being decomposable below).

On the other hand, it is seemingly possible for ${\cal T}_{10}$ to be realized
by $G=G(12,2,2)$. To rule this out, we will need additional information about
the form of the Seiberg-Witten differential. First, note that there are two
possibilities for $\sigma$ in (4.31), namely $(i)$ the natural map
$G(12,2,2)\rightarrow G(6,1,2)$ and $(ii)$ the conjugate to the natural map
Cecotti:2021yet . Denoting the coordinates on $A\times{\mathbb{C}}^{2}$ as
$(a,x)$, the Seiberg-Witten differential may then be written as

$\displaystyle(i):$ $\displaystyle\qquad\lambda=x^{1}da_{1}+x^{2}da_{2}~{},$
(4.40) $\displaystyle(ii):$ $\displaystyle\qquad\lambda={\partial
p_{1}\over\partial x^{1}}da_{1}+{\partial p_{2}\over\partial x^{1}}da_{2}~{},$

where $da_{i}$ are a basis of holomorphic differentials on $A$ and the
coefficient functions were chosen such that $\lambda$ is well-defined on the
orbifold space. This in particular means that the $p_{i}(x)$ are built from
invariant polynomials of $G(12,2,2)$. Let us say that these $p_{i}(x)$ are
degree-$k$ polynomials in $x$, and since we are restricting to complex
reflection groups whose invariant polynomial have the same dimensions
$d_{1}=d_{2}=d$, we can assume without loss of generality that $k=n\cdot d$
for some $n\in{\mathbb{N}}$. Since the integrals of $\lambda$ along the cycles
of $A$ give rise to masses, we expect that $[\frac{\partial p_{1}(x)}{\partial
x}]=[p_{1}(x)]-[x]=1$ and hence that $[x]={1\over k-1}$. The Coulomb branch
scaling dimensions are obtained by the resulting dimension of the invariant
polynomials in $x$:

$\displaystyle(i):$
$\displaystyle\qquad(\Delta_{u_{1}},\Delta_{u_{2}})=(d,d)~{},$ (4.41)
$\displaystyle(ii):$
$\displaystyle\qquad(\Delta_{u_{1}},\Delta_{u_{2}})={1\over n\,d-1}(d,d)~{}.$

In our case $d=12$, since the degree of the invariant polynomials of
$G(12,2,2)$ are $(12,12)$, and it is immediate to see that no $n$ exists for
which $(\Delta_{u_{1}},\Delta_{u_{2}})=(3,3)$. Hence we arrive at a
contradiction and ${\cal T}_{10}$ must be ruled out as well.

We have thus seemingly ruled out all of our candidate theories with
$\varkappa\neq 1,2$. However, it should be noted that the previous analysis
assumed that the Coulomb branch is indecomposable. We may now consider lifting
this assumption. For this to be possible, the four-dimensional central charges
of the rank-two SCFT should be decomposable into the sum of two rank-one
central charges. By referring to Table 1, it is easy to check that this is
only possible for the theory ${\cal T}_{8}$. Indeed, in this case
$c_{\mathrm{4d}}=-{44\over 5}$ is simply twice the central charge of the
${\cal T}_{\varnothing,1}^{(1)}$ SCFT. This suggests that the Coulomb branch
is simply the product of two copies of the Coulomb branch of ${\cal
T}_{\varnothing,1}^{(1)}$. We will discuss this case in more detail now.

#####  ${\cal T}_{8}$:

The candidate theory ${\cal T}_{8}$ is specified by 2d data
$(c_{\mathrm{2d}},a_{1},a_{2})=(-{44\over 5},2,2)$. As shown in Table 3, one
finds that this 2d data is consistent with a 4d theory with central charges
$(a_{\mathrm{4d}},c_{\mathrm{4d}})=({43\over 60},{11\over 15})$, Coulomb
branch scaling dimensions $(\Delta_{u_{1}},\Delta_{u_{2}})=({6\over 5},{6\over
5})$, and $U(1)^{2}$ flavor symmetry. This looks nearly identical to the data
of the ${\cal T}_{6}={\cal T}_{\varnothing,1}^{(1)}\times{\cal
T}_{\varnothing,1}^{(1)}$ theory, except for the presence of two spin-one
currents. The Schur index of the theory is given by

$\displaystyle Z_{{\cal T}_{8}}$ $\displaystyle=$ $\displaystyle
1+2q+2q^{2}+4q^{3}+5q^{4}+8q^{5}+11q^{6}+16q^{7}+21q^{8}+30q^{9}+39q^{10}+\dots~{},$

to be contrasted with that of ${\cal T}_{6}$,

$\displaystyle Z_{{\cal T}_{6}}$ $\displaystyle=$
$\displaystyle\mathrm{PE}\left[{2(q^{2}-q^{4})\over(1-q)(1-q^{5})}\right]$
(4.43) $\displaystyle=$ $\displaystyle
1+2q^{2}+2q^{3}+3q^{4}+4q^{5}+7q^{6}+8q^{7}+13q^{8}+16q^{9}+23q^{10}+\dots~{}.$

In the current case the minimal MDE satisfied by the vacuum character is of
order four, with the tentative chiral dimensions of the corresponding VOA
being

$\displaystyle\\{h_{i}\\}=\left\\{0,\,-{1\over 3},\,-{2\over 5},\,{4\over
15}\right\\}~{}.$ (4.44)

The general lore that no two SCFTs should have the same CB geometry seems to
suggest that this candidate theory is invalid, since it has the same geometry
as ${\cal T}_{6}$, which is known to exist. What’s more, we could not come up
with a natural ansatz for the strong generators of the VOA associated to the
putative index (4.2.1). Our hunch is that this candidate theory is
inconsistent, but unfortunately we we will not be able to say anything more
conclusive.

####  4.2.2 Cases with $\varkappa=1,2$

We next consider the theories with $\varkappa=1,2$. Amongst the candidate new
theories, only two satisfy this property, namely ${\cal T}_{7}$ and ${\cal
T}_{13}$. Because in these cases the CB is no longer required to be isotrivial
and diagonal, the powerful tools used in the previous subsection are no longer
applicable.

#####  ${\cal T}_{13}$ :

The candidate theory ${\cal T}_{13}$ is defined by 2d data
$(c_{\mathrm{2d}},a_{1},a_{2})=(-20,4,17)$. As shown in Table 3, one finds
that this 2d data is consistent with a 4d theory with central charges
$(a_{\mathrm{4d}},c_{\mathrm{4d}})=({19\over 12},{5\over 3})$, Coulomb branch
scaling dimensions $(\Delta_{u_{1}},\Delta_{u_{2}})=({3\over 2},{5\over 2})$,
and flavor symmetry $\mathfrak{su}(2)_{7/2}\times\mathfrak{u}(1)$. The Schur
index of the tentative theory is obtained by solving the MDE, giving

$\displaystyle Z_{{\cal T}_{13}}$ $\displaystyle=$ $\displaystyle
1+4q+17q^{2}+56q^{3}+163q^{4}+428q^{5}+1063q^{6}$ (4.45) $\displaystyle\hskip
72.26999pt+2472q^{7}+5515q^{8}+11792q^{9}+24404q^{10}+\dots~{}$

Let us check whether this putative Schur index is compatible with the sum rule
(A.54). First note that there are no Kac-Moody nulls at level two, so
$a_{2}^{\rm KM}=14$. It is simple to check that the Sugawara condition is
obeyed, so $n^{\rm T}=0$. Additional $h=2$ states must be affine Kac-Moody
(AKM) primaries. An AKM primary in the spin-$j$ representation of
$\mathfrak{su}(2)_{-7/4}$ and with charge $q$ under $\mathfrak{u}(1)$ has
holomorphic dimension

$h(j,q)=\frac{j(j+1)}{-\frac{7}{4}+2}-q^{2}=4j(j+1)-q^{2}\,,$ (4.46)

and thus in order to have $h=2$ we need both a non-trivial $\mathfrak{su}(2)$
representation and a non-zero $\mathfrak{u}(1)$ charge (for example, $j=1/2$
and $q=\pm 1$). CPT invariance implies that both signs of the
$\mathfrak{u}(1)$ charge must be present. We then conclude that $\\#\hat{\cal
B}_{2}$ must be even, and thus from (A.54) that $a_{2}$ itself is even, in
contradiction with the value $17$ in (4.45). We conclude that our candidate
character does not have a consistent 4d interpretation.

It is intriguing that an actual ${\cal N}=2$ SCFT with the same flavor
symmetry (including the level) and central charges does exist, but at rank
three. This is the $(A_{1},D_{8})$ theory with CB scaling dimensions
$(\frac{3}{2},\frac{5}{4},\frac{7}{4})$ and Schur index Buican:2015ina

$\displaystyle Z_{(A_{1},D_{8})}$ $\displaystyle=$ $\displaystyle
1+4q+18q^{2}+56q^{3}+167q^{4}+436q^{5}+1086q^{6}$ (4.47) $\displaystyle\hskip
72.26999pt+2520q^{7}+5631q^{8}+12024q^{9}+24906q^{10}+\dots~{}$

Comparing with (4.45), the two indices look remarkably similar. Indeed the
$O(q^{2})$ term is off by only one (allowing the coefficient to circumvent the
issue that led to the previous inconsistency) and the $O(q^{3})$ term matches.
It is unclear whether this coincidence has any meaning.

#####  ${\cal T}_{7}$ :

The candidate theory ${\cal T}_{7}$ is defined by 2d data
$(c_{\mathrm{2d}},a_{1},a_{2})=(-64,64,2080)$. As shown in Table 3, one finds
that this 2d data is consistent with a 4d theory with central charges
$(a_{\mathrm{4d}},c_{\mathrm{4d}})=({14\over 3},{16\over 3})$, Coulomb branch
scaling dimensions $(\Delta_{u_{1}},\Delta_{u_{2}})=(4,5)$, and flavor
symmetry $\mathfrak{su}(8)_{10}\times\mathfrak{u}(1)$. The Schur index of the
tentative theory is obtained by solving the corresponding MDE, giving

$\displaystyle Z_{{\cal T}_{7}}$ $\displaystyle=$
$\displaystyle\mathrm{PE}\left[64(q+q^{3})\over
1-q^{4}\right]=1+64q+2080q^{2}+45824q^{3}+770576q^{4}\dots$ (4.48)

We have found it difficult to obtain a compelling guess for the Higgs branch
of this theory. Indeed, the simplest guess would be that the first stratum is
located on the minimal nilpotent orbit of $\mathfrak{a}_{7}$, and that the
theory eventually Higgses to free hypermultiplets. This simplest guess turns
out to be inconsistent.

To see this, begin by noting the following. Say that upon moving onto the
Higgs branch, we spontaneously break a $\mathfrak{g}$ flavor symmetry, thereby
taking a putative theory ${\cal T}$ to a new SCFT ${\cal T}_{\rm Higgs}$.
Denote the sublocus (i.e. symplectic leaf) of the HB where this specific
Higgsing takes place $\mathfrak{S}$. Using anomaly matching, the central
charges of the two theories can be related by Giacomelli:2020jel ; CCLMW2021

$12c_{{\cal T}}=12c_{{\cal T}_{\rm
Higgs}}+2\left(\frac{3}{2}k_{\mathfrak{g}}-1\right)+{\rm
dim}_{\mathbb{H}}\mathfrak{S}-1~{}.$ (4.49)

In the current case, we have $12c_{{\cal T}_{7}}=64$,
$k_{\mathfrak{su}(8)}={10}$, and with the assumption that the theory Higgses
to free hypers, we also conclude that $d_{\mathrm{HB}}^{{\cal
T}_{7}}=24(c_{\mathrm{4d}}-a_{\mathrm{4d}})=16$. Since ${\rm
dim}_{\mathbb{H}}\mathfrak{S}=d_{\mathrm{HB}}^{{\cal
T}_{7}}-d_{\mathrm{HB}}^{\mathrm{Higgs}}$, where
$d_{\mathrm{HB}}^{\mathrm{Higgs}}$ is the dimension of the HB of ${\cal
T}_{\rm Higgs}$, we then conclude that the theory on $\mathfrak{S}$ must have

$\displaystyle 12c_{{\cal
T}_{\mathrm{Higgs}}}-d_{\mathrm{HB}}^{\mathrm{Higgs}}={21}~{}.$ (4.50)

Furthermore, by definition $h_{\mathrm{ECB}}=0$ implies that Higgsing ${\cal
T}_{7}$ should necessarily lead to theories of rank lower than two. Under the
assumption that the only rank-zero theories are free hypermultiplets and
discrete gauging thereof—which are inconsistent with the analysis above—the
theory on the lowest stratum should have rank one. We are thus led to search
for a rank-one theory satisfying (4.50). The only such theory is ${\cal
S}_{A_{1},3}^{(1)}$, with flavor level $k_{IR}$ given in Table 1. Noting that
the flavor level $k_{\mathfrak{su}(8)}$ of the UV theory must be related to
that on the minimal nilpotent orbit by Giacomelli:2020jel ; Beem:2019tfp ;
Beem:2019snk

$\displaystyle
k_{\mathfrak{su}(8)}={2+k_{IR}I_{\mathfrak{f}_{\mathrm{IR}}\hookrightarrow\mathfrak{a}_{7}}\over
I_{\mathfrak{a}_{5}\hookrightarrow\mathfrak{a}_{7}}}~{},$ (4.51)

where $I_{\mathfrak{h}\hookrightarrow\mathfrak{g}}$ is the embedding index of
$\mathfrak{h}$ in $\mathfrak{g}$, we see that in no case does this combination
reproduce the correct value of $k_{\mathfrak{su}(8)}={10}$.

Of course this alone does not rule this out as a consistent theory—it simply
rules out the simplest guess for the structure of the Higgs branch. We may for
example consider more complicated guesses in which the lowest stratum is not
located on the minimal nilpotent orbit of $\mathfrak{a}_{7}$, or cases in
which the theory does not Higgs completely down to free hypermultiplets. It
would be interesting to use more sophisticated Higgs branch technology to
reach a definitive conclusion about the status of this putative theory. The
candidate data also look somewhat implausible from the point of view of the
sum rule (A.54). Substituting $a_{2}=2080$, $a_{2}^{\rm KM}=2144$ and $n^{\rm
T}=1$ we find

$\\#\hat{\cal B}_{2}-2\,\\#{\cal D}_{1(0,0)}+2\,\\#{\cal
D}_{\frac{1}{2}(0,\frac{1}{2})}=-65\,.$ (4.52)

There are of course many ways to assign flavor representations to the
additional generators so that this sum rule is satisfied, but there appears to
be no economical or natural choice.

##  5 Outlook

In this paper we have introduced an algorithm that takes as input the first
few terms of a putative Schur index and gives as output a (potentially empty)
set of mutually consistent Schur and Coulomb data. The algorithm uses
constraints encoding the general principles of unitarity and superconformal
invariance. These constraints turn out to be extremely powerful, vastly
winnowing down the number of consistent possibilities. As a proof of concept,
we performed a search for rank-two ${\cal N}=2$ superconformal field theories
with Schur indices satisfying an untwisted modular differential equation of
order four. Our analysis reproduces a variety of known examples, but perhaps
surprisingly it does not conclusively produce any new candidate theories. This
may be evidence that our understanding of rank-two theories is more complete
than expected, or just an artifact of our restriction to low order and low
rank.

Only six of the nearly seventy known rank-two theories Martone:2021ixp  have a
vacuum character satisfying an untwisted order-four MDE; most of them have
characters satisfying a twisted order-four MDE. So even the simplest extension
of our scan to twisted fourth-order is expected to have a large payoff. This
extension is conceptually straightforward, but rather challenging from a
computational viewpoint. A solution of a fourth-order twisted MDE is specified
by eight parameters, and thus the number of cases that serve as input would be
too large for a scan as thorough as the one given here. One might hope to
implement a more limited scan (e.g. searching for cases with no or small-
dimensional flavor symmetry) or parallelize on a cluster. More speculatively,
since we now have a method for generating large data sets of consistent versus
inconsistent inputs (in the form of tuples of integers), one might envision
the use of machine learning techniques.

The other obvious extension is to perform searches for 4d SCFTs of higher
rank. Unfortunately, it is not in general understood how the rank is related
to the order of the MDE. Both are in some sense measures of the complexity of
the SCFT, and experimentally one notices that the order grows at least
linearly with the rank. An order-four untwisted MDE scan is thus likely to
produce sparser and sparser candidate theories at higher rank, and it will be
necessary to consider higher-order MDEs. Apart from the increased
computational cost, there are also some conceptual obstacles that must be
overcome starting at rank three. For example, the scaling dimension
${\Delta}^{\rm sing}$ for a knotted stratum is no longer set solely by the
scaling dimension of the unknotted ones. This difficulty arises from the known
fact that the set of irreducible homogenous polynomial in three complex
variables is infinite. To explore theories of rank three or higher, one would
thus need to find an efficient way to get around this limitation.

Altogether, we have seen that the space of consistent 4d ${\cal N}=2$ theories
is subject to a stringent set of constraints, which, when imposed in their
entirety, admit a surprisingly small set of solutions. The approach of this
paper is similar to that used in the conformal bootstrap to identify candidate
theories from general principles, though in the current case the complete set
of principles remains unknown. Furthermore, the principles which are known are
likely still not phrased in their most economical form. Once the full set of
compatibility conditions between CB and VOA data is uncovered, one could
imagine carrying out an even more stringent scan over candidate theories. If
some of the new constraints are imposable already at the level of the inputs
to the algorithm, scans could be done over smaller numbers of input
parameters, thereby allowing for significant computational improvement. We
hope that this work provides a first step in a new, computational approach to
the classification of 4d ${\cal N}=2$ theories.

## Acknowledgements

JK would like to thank Kavli IPMU for their hospitality—a significant portion
of this work was completed there. MM is supported in part by the NSF grant
PHY-1915093, by the Simons Foundation grant 815892 and STFC grant
ST/T000759/1. LR is supported in part by the NSF grant PHY-1915093. MW is
supported in part by DOE grant SC0011784.

##  Appendix A Superconformal representation theory constraints

In order for a vacuum character to be identifiable with a 4d Schur index, one
must be able to interpret its Fourier coefficients as counting Schur
operators. Superconformal representation theory imposes certain constraints.
In Table 4 we list the 4d superconformal multiplets that contain Schur
operators (one for each multiplet). It will be useful to recall the
interpretation of a few of them (see e.g. Dolan:2002zh ; Beem:2013sza  for
more details):

  * •

$\hat{{\cal B}}_{R}$: Higgs branch multiplets. Their bottom components are
Higgs chiral ring operators, with conformal dimension $\Delta=2R$. The
corresponding VOA operators have chiral dimension $h=R$. The cases
$R=\frac{1}{2}$ and $R=1$ are special: $\hat{{\cal B}}_{\frac{1}{2}}$ are free
hypermultiplets, whereas $\hat{{\cal B}}_{1}$ are moment-map multiplets, which
contain the conserved flavor currents of the theory. The VOA counterparts of
the latter are affine Kac-Moody currents.

  * •

${\cal D}_{0(0,j_{2})}$ and $\overline{{\cal D}}_{0(j_{1},0)}$ are free field
multiplets. For $j_{1}=j_{2}=0$ they correspond to the standard free vector
multiplet, while for higher $j_{1}$, $j_{2}$ they are (exotic) higher-spin
free fields.

  * •

$\hat{{\cal C}}_{0(j_{1},j_{2})}$ contain conserved currents of spin
$2+j_{1}+j_{2}$. In particular, $\hat{{\cal C}}_{0(0,0)}$ is the standard
stress tensor multiplet.

  * •

${\cal D}_{\frac{1}{2}(0,0)}$, $\overline{{\cal D}}_{\frac{1}{2}(0,0)}$ are
“additional” supercurrent multiplets, on top of the supercurrent contained in
the stress tensor multiplet. Their presence signals supersymmetry enhancement.
In the VOA, they correspond to fermionic $h=\frac{3}{2}$ operators (i.e. 2d
supercurrents).161616 4d ${\cal N}=3$ SCFTs map to VOAs containing an ${\cal
N}=2$ super-Virasoro algebra, while 4d ${\cal N}=4$ SCFTs map to VOAs
containing a (small) ${\cal N}=4$ super-Virasoro algebra.

We wish to assume that the 4d SCFT is fully interacting, i.e. that it doesn’t
contain any free field subsector. It follows that:

  1. (i)

No free field supermultiplets are allowed. This forbids $\hat{{\cal
B}}_{\frac{1}{2}}$, ${\cal D}_{0(0,j_{2})}$, and $\overline{{\cal
D}}_{0(j_{1},0)}$ multiplets.

  2. (ii)

No supermultiplets containing conserved currents of spin greater than two are
allowed. Indeed, such higher spin currents would imply the existence of a free
subsector Bhardwaj:2013qia . This forbids $\hat{{\cal C}}_{0(j_{1},j_{2})}$
with $j_{1}+j_{2}>0$.

It is also natural to assume:

  1. (iii)

The theory contains a single spin-two conserved current, which is identified
with the stress tensor operator. This demands the presence of precisely one
$\hat{{\cal C}}_{0(0,0)}$ multiplet.

This last assumption is meant to capture locality (the existence of a local
stress tensor) and indecomposibility (a direct product of SCFTs would have
multiplet stress tensors).171717Strictly speaking, we are not aware of a
rigorous argument showing the converse, i.e. that a theory with multiple spin
two conserved currents is necessarily decomposable. Nevertheless it is a
standard assumption.

Multiplet | $h$ | $\bf{r}$
---|---|---
$\hat{{\cal B}}_{R}$ | $R$ | $0$
${\cal D}_{R(0,j_{2})}$ | $R+j_{2}+1$ | $j_{2}+\frac{1}{2}$
$\overline{{\cal D}}_{R(j_{1},0)}$ | $R+j_{1}+1$ | $-j_{1}-\frac{1}{2}$
$\hat{{\cal C}}_{R(j_{1},j_{2})}$ | $R+j_{1}+j_{2}+2$ | $j_{2}-j_{1}$
Table 4: List of 4d ${\cal N}=2$ superconformal multiplets that contain Schur
operators (one for each multiplet). Here $R$ and $(j_{1},j_{2})$ denote the
Cartan of the $SU(2)_{R}$ symmetry and the Lorentz quantum numbers of the
superconformal primary of the multiplet, while $h$ is the chiral dimension and
$\bf{r}$ the $U(1)_{\bf r}$ charge of the Schur operator. In our conventions,
$R$, $j_{1}$, and $j_{2}$ take non-negative half-integer values. Even and odd
values of $2\bf{r}$ correspond to bosonic and fermionic Schur operators,
respectively.

The Schur index only gives us information about the chiral dimension $h$, and
we see from Table 4 that for fixed $h$ there is a certain ambiguity in
identifying the corresponding 4d supermultiplet, which gets worse for
increasing $h$. Nevertheless, leveraging the physical assumptions above, some
useful statements can be made for low $h$. A general Schur index has a series
expansion of the form181818In most of the paper we focus on untwisted modular
equations, which yield characters with an integer power expansion, but here we
discuss the general case.

$Z_{\rm
Schur}(q)=\sum_{k\in\frac{\mathbb{N}}{2}}a_{k}q^{k}=1+a_{\frac{1}{2}}q^{\frac{1}{2}}+a_{1}q+a_{\frac{3}{2}}q^{\frac{3}{2}}+a_{2}q^{2}+\dots\,,\quad
a_{k}\in\mathbb{Z}\,.$ (A.53)

Let us interpret the first few coefficient:

  * •

We see at once that $a_{\frac{1}{2}}=0$, because the only supermultiplet that
could yield $h=\frac{1}{2}$ states is the hypermultiplet $\hat{{\cal
B}}_{\frac{1}{2}}$, violating assumption (i).

  * •

$a_{1}$ counts the number of $\hat{{\cal B}}_{1}$ multiplets, which are the
only non-free multiplets with $h=1$. This basic fact is used throughout the
paper.

  * •

$a_{\frac{3}{2}}$ gets a positive contribution from the number of $\hat{{\cal
B}}_{\frac{3}{2}}$ multiplets and a negative contribution from the number of
additional supercurrent multiplets (if any). One often has an _a priori_
opinion about the number of supersymmetries of the theory, and thus can read
off the number of Higgs operators with $R=\frac{3}{2}$. (They are necessarily
generators of the chiral ring since they cannot be composites of operators
with lower $R$).

  * •

$a_{2}$ gets a variety of contributions. There are first of all the affine
Kac-Moody algebra states
$\\{J^{a}_{-2}|0\rangle,J_{-1}^{a}J_{-1}^{b}|0\rangle\\}$, where $a,b$ are
adjoint indices of the flavor symmetry algebra. We denote their number by
$a_{2}^{\rm KM}$.191919To determine $a_{2}^{\rm KM}$, one must of course take
care of subtracting possible null states, which are determined if one knows
the levels of the (simple factors of the) affine current algebra. In the
absence of null states, $a_{2}^{\rm KM}=a_{1}+a_{1}(a_{1}+1)/2$. If the
central charge satisfies the Sugawara condition (2.19), the 4d stress tensor
multiplet $\hat{{\cal C}}_{0(0,0)}$ is already accounted for by the 2d
Sugawara construction Beem:2013sza ; Beem:2018duj . If the Sugawara condition
is not obeyed, there must be an independent generator with $h=2$ to account
for the 4d stress tensor. Finally, there can be additional generators of type
$\hat{\cal B}_{2}$ (bosonic), ${\cal D}_{1(0,0)}$, $\overline{{\cal
D}}_{1(0,0)}$ (fermionic), or ${\cal D}_{\frac{1}{2}(0,\frac{1}{2})}$,
$\overline{{\cal D}}_{\frac{1}{2}(\frac{1}{2},0)}$ (bosonic). It is important
to note that CPT invariance of the 4d theory implies that ${\cal D}_{R(0,j)}$
and $\overline{{\cal D}}_{R(j,0)}$ multiplets must come in pairs. All in all
then, we have202020Here $\\#\hat{\cal B}_{2}$ denote the number of additional
$\hat{\cal B}_{2}$ multiplets, on top of the ones that correspond to the
composites $J_{-1}^{a}J_{-1}^{b}|0\rangle$. These additional $\hat{\cal
B}_{2}$ operators are generators of the Higgs chiral ring.

$a_{2}=a_{2}^{\rm KM}+n^{\rm T}+\\#\hat{\cal B}_{2}-2\,\\#{\cal
D}_{1(0,0)}+2\,\\#{\cal D}_{\frac{1}{2}(0,\frac{1}{2})}\,,$ (A.54)

where $n^{\rm T}\equiv 0$ if the Sugawara condition is obeyed and $n^{\rm
T}\equiv 1$ if it is not obeyed.

The sum rule (A.54) can sometimes be used to rigorously rule out putative
Schur indices. In the context of this paper, our algorithm yields the
requisite information to compute $a_{2}^{\rm KM}$ and $n^{\rm T}$. If we
happen to also have an opinion about the Higgs branch of the theory, in
particular on the number of $\hat{\cal B}_{2}$ generators, we get a sharp
constraint on the parity (even or odd) of $a_{2}$. A more heuristic use of the
sum rule is as a plausibility check. The additional level-two generators must
form representations of the flavor algebra. There are usually ways to satisfy
the sum rule, but if they require baroque choices of flavor decomposition, we
may conclude that the character is unlikely to have a 4d interpretation.

##  Appendix B Allowed rank-two Coulomb branch dimensions

In this appendix we rederive the allowed pairs of Coulomb branch dimensions
for rank-two ${\cal N}=2$ SCFTs, which will be necessary data for our rank-two
scan. We begin by noting that, based solely on the fact that CB monodromies
must take values in the discrete group $Sp(2r,{\mathbb{Z}})$, the allowed
scaling dimensions of Coulomb branch operators are constrained to take values
in Argyres:2018urp ; Caorsi:2018zsq :

${\Delta}\in\left\\{\frac{n}{m}\,\Big{|}\,\,n,m\in\mathbb{N},\,0<m\leq
n,\,{\rm gcd}(n,m)=1,\,\varphi(n)\leq 2r\right\\}~{},$ (B.55)

where $\varphi(n)$ is Euler’s totient function. This set depends only on the
rank $r$ of the SCFT. The maximum allowed value scales as ${\Delta}_{\rm
max}\sim r\ln\ln r$, while the total numbers of allowed scaling dimensions
$\boldsymbol{N}$ and integer scaling dimensions $\boldsymbol{N}_{\rm int}$
scale as Caorsi:2018zsq

$\displaystyle\boldsymbol{N}\sim
2r^{2}\frac{\zeta(2)\zeta(3)}{\zeta(6)}~{},\hskip 36.135pt\boldsymbol{N}_{\rm
int}\sim 2r\frac{\zeta(2)\zeta(3)}{\zeta(6)}~{},$ (B.56)

with $\zeta(s)$ the Riemann zeta function. Naively, the number of allowed
$r$-tuples would be

${\rm
Number\,of\,\emph{r-}tuples}=\left(\begin{array}[]{c}\boldsymbol{N}+r-1\\\
\boldsymbol{N}-1\end{array}\right)~{}.$ (B.57)

However, we will now see that this number can be cut down significantly
Argyres:2022yet .

Genuinely rank-two pairs
---
$P_{i}(z)$ | $\\{{\Delta}_{1},{\Delta}_{2}\\}$
$\Phi_{5}(z)$ | $\\{\frac{4}{3},\frac{5}{3}\\}\ \\{\frac{5}{4},\frac{3}{2}\\}\ \\{\frac{3}{2},\frac{5}{2}\\}\ \\{\frac{5}{4},3\\}\ \\{\frac{5}{3},3\\}\ \\{\frac{5}{2},3\\}\ \\{3,5\\}\ \\{\frac{5}{4},4\\}$
$\\{\frac{5}{3},4\\}\ \\{\frac{5}{2},4\\}\ \\{4,5\\}\ \\{\frac{5}{4},8\\}\
\\{\frac{5}{3},8\\}\ \\{\frac{5}{2},8\\}\ \\{5,8\\}$
$\Phi_{8}(z)$ | $\\{\frac{6}{5},\frac{8}{5}\\}\ \\{\frac{4}{3},\frac{8}{3}\\}\ \\{\frac{8}{7},\frac{10}{7}\\}\ \\{\frac{8}{7},\frac{12}{7}\\}\ \\{\frac{8}{5},\frac{12}{5}\\}\ \\{\frac{8}{3},\frac{10}{3}\\}\ \\{\frac{8}{7},4\\}\ \\{\frac{8}{5},4\\}$
$\\{\frac{8}{3},4\\}\ \\{4,8\\}\ \\{\frac{8}{7},6\\}\ \\{\frac{8}{5},6\\}\
\\{\frac{8}{3},6\\}\ \\{6,8\\}\ \\{\frac{8}{7},12\\}\ \\{\frac{8}{5},12\\}$
$\\{\frac{8}{3},12\\}\ \\{8,12\\}$
$\Phi_{10}(z)$ | $\\{\frac{10}{9},\frac{4}{3}\\}\ \\{\frac{4}{3},\frac{10}{3}\\}\ \\{\frac{10}{9},4\\}\ \\{\frac{10}{7},4\\}\ \\{\frac{10}{3},4\\}\ \\{4,10\\}\ \\{\frac{10}{9},8\\}\ \\{\frac{10}{7},8\\}$
$\\{\frac{10}{3},8\\}\ \\{8,10\\}$
$\Phi_{12}(z)$ | $\\{\frac{6}{5},\frac{12}{5}\\}\ \\{\frac{12}{11},6\\}\ \\{\frac{12}{7},6\\}\ \\{\frac{12}{5},6\\}\ \\{6,12\\}\ \\{\frac{12}{11},8\\}\ \\{\frac{12}{7},8\\}\ \\{\frac{12}{5},8\\}$
Non-Genuine Rank-Two Pairs
$\\{{\Delta}_{1},{\Delta}_{2}\\}$
$\\{\frac{6}{5},\frac{6}{5}\\}\ \\{\frac{6}{5},\frac{4}{3}\\}\
\\{\frac{6}{5},\frac{3}{2}\\}\ \\{\frac{6}{5},2\\}\ \\{\frac{6}{5},3\\}\
\\{\frac{6}{5},4\\}\ \\{\frac{6}{5},6\\}\ \\{\frac{4}{3},\frac{4}{3}\\}\
\\{\frac{4}{3},\frac{3}{2}\\}\ \\{\frac{4}{3},2\\}$
$\\{\frac{4}{3},3\\}\ \\{\frac{4}{3},4\\}\ \\{\frac{4}{3},6\\}\
\\{\frac{3}{2},\frac{3}{2}\\}\ \\{\frac{3}{2},2\\}\ \\{\frac{3}{2},3\\}\
\\{\frac{3}{2},4\\}\ \\{\frac{3}{2},6\\}\ \\{2,2\\}\ \\{2,3\\}$
$\\{2,4\\}\ \\{2,6\\}\ \\{3,3\\}\ \\{3,4\\}\ \\{3,6\\}\ \\{4,4\\}\ \\{4,6\\}\
\\{6,6\\}$
Table 5: Full list of allowed pairs of scaling dimensions at rank-two. The
_genuinely rank-two_ pairs are those which include at least one entry which is
not allowed at rank-one. In the first column we report the form of the
characteristic polynomial where $\Phi_{d}(z)$ denotes a degree-$d$ cyclotomic
polynomial.

First observe that the product of a rank-$n$ SCFT and a rank-$(r-n)$ SCFT (for
$n<r$) gives a consistent rank-$r$ ${\cal N}=2$ theory. This implies that the
union of any allowed $(r-n)$-tuple with an allowed $n$-tuple gives an allowed
$r$-tuple. Thus, without loss of generality, we can focus on determining the
_genuine_ $r$-tuples, i.e. those which contain at least one Coulomb branch
scaling dimension which is not allowed at rank $r-1$. Now consider the
following locus of the Coulomb branch,

${\cal I}_{i}:=\\{u_{1}=0\,,\dots,\,u_{i-1}=0,\,u_{i}\neq
0,\,u_{i+1}=0,\dots,\,u_{r}=0\\}$ (B.58)

and assume that this locus is regular, i.e. ${\cal I}_{i}$ is not part of the
singular locus. In particular, by Proposition 2, if we choose $i$ such that
${\Delta}_{i}$ is not an allowed scaling dimension at rank-$(r-1)$, then
${\cal I}_{i}$ is necessarily non-singular.

Since ${\cal I}_{i}$ is non-singular, the fiber $X^{i}_{u}$ is a rank-$r$
Abelian variety for all $u\in{\cal I}_{i}$. That is,

$X^{i}_{u}={\mathbb{C}}^{r}/\Lambda^{i}_{u}~{},\qquad\forall u\in{\cal I}_{i}$
(B.59)

with $\Lambda^{i}_{u}$ a rank-$2r$ lattice. The following $U(1)_{r}$
transformation

$\xi^{i}:\ u_{k}\to\exp\left(2\pi
i{\Delta}_{k}/{\Delta}_{i}\right)u_{k}~{},\hskip 36.135pt\xi^{i}\in U(1)_{r}$
(B.60)

acts trivially on ${\cal I}_{i}$ and thus generates an automorphism of
$X^{i}_{u}$, i.e. an element $\boldsymbol{\xi}^{i}\in GL(r,{\mathbb{C}})$ such
that $\boldsymbol{\xi}^{i}(\Lambda^{i}_{u})=\Lambda^{i}_{u}$. Using the fact
that the holomorphic symplectic form on $X\to{\cal C}$ is non singular
Donagi:1995cf , given (B.60) it is possible to derive an explicit
representation of $\boldsymbol{\xi}^{i}$:

$\displaystyle\boldsymbol{\xi}^{i}$ $\displaystyle:=\rho_{a}(\xi^{i})\in
GL(r,{\mathbb{C}})$ $\displaystyle:={\rm diag}\left(e^{2\pi
i({\Delta}_{1}-1)/{\Delta}_{i}},...,e^{2\pi
i({\Delta}_{i-1}-1)/{\Delta}_{i}},e^{-2\pi i/{\Delta}_{i}},e^{2\pi
i({\Delta}_{i+1}-1)/{\Delta}_{i}},...,e^{2\pi
i({\Delta}_{r}-1)/{\Delta}_{i}}\right)$

which explicitly depends on the $r$-tuple of Coulomb branch scaling dimensions
$\\{{\Delta}_{1},...,{\Delta}_{r}\\}$. We now observe that there is a general
relation between $\boldsymbol{\xi}^{i}$, which is also called the _analytic
representation_ of $\xi^{i}$, and the induced action $\xi^{i}_{H}\in
Sp(2r,\mathbb{Z})$ on the homology of $X^{i}_{u}$, i.e. the _homological
representation_ of $\xi^{i}$. In particular, the following isomorphism of
complex representations holds:

$\xi^{i}_{H}\otimes{\mathbb{C}}\cong\boldsymbol{\xi}^{i}\oplus(\boldsymbol{\xi}^{i})^{*}$
(B.62)

where ∗ denotes complex conjugation. Given (B.62) and (B), we obtain the
restriction that the following quantities

$\exp\left(2\pi i\frac{{\Delta}_{j}-1}{{\Delta}_{i}}\right),\quad i=1,2,...,r$
(B.63)

must be roots of a degree-$2r$ polynomial $P_{i}(z)$ (the characteristic
polynomial of $\xi_{H}^{i}$) which is a product of degree-$d$ cyclotomic
polynomials $\Phi_{d}(z)$:

$P_{i}(z)=\prod_{d}\Phi_{d}(z)^{n(d)}~{},\hskip
36.135pt\sum_{d}\varphi(d)n(d)=2r~{},$ (B.64)

where again $\varphi(d)$ is Euler’s totient function. Imposing that
${\Delta}_{i}$ is not a rank-$(r-1)$ dimension further constraints the form of
$P_{i}(z)$, giving

$P^{\rm new}_{i}(z)=\Phi_{d}(z)~{},\hskip 36.135pt\varphi(d)=2r~{}.$ (B.65)

Carrying out the analysis for $r=2$, we then obtain 51 genuinely rank-2 pairs
of CB scaling dimensions. Together with 28 pairs having both entries being
allowed at rank-one, this gives a total of 79 allowed pairs of scaling
dimensions at rank-two. These are tabulated in Table 5. As promised, this
provides a great improvement on the naive counting in (B.57), which would have
predicted a total of 276 allowed pairs.

Let us conclude by noting that for cases in which one of the Coulomb branch
scaling dimensions is $\Delta_{u_{i}}=2$, the theory has an exactly marginal
operator parameterizing a conformal manifold. In Beem:2014zpa ;
Perlmutter:2020buo  it was conjectured that any ${\cal N}=2$ $n$-dimensional
conformal manifold arises by gauging $n$ simple factors in the global symmetry
of isolated ${\cal N}=2$ SCFTs. This means that if $\Delta_{u_{i}}=2$ the
corresponding theory is either Lagrangian, or an $SU(2)$ conformal gauging of
a lower rank theory. Since all Lagrangian theories have already been
classified, we will for the most part exclude such cases from our scan.

##  Appendix C Rank-one IR-free theories

In this appendix we discuss the set of rank-one IR-free theories, which will
serve as necessary data for our rank-two scan. The full set of rank-one IR-
free is obtained by considering $U(1)$ or $SU(2)$ gauge theories coupled to
sufficient numbers of hypermultiplets (or discrete gaugings thereof). Let us
begin with the case of $U(1)$ gauge theory. The most general matter content
consists of $n_{i}$ hypermultiplets of charge $q_{i}$ for $i=1,\dots,N$. The
beta-function for this theory is given by
$\beta\propto\sum_{i}q_{i}^{2}n_{i}$. The relevant Kodaira fiber is then of
type $I_{\sum_{i}q_{i}^{2}n_{i}}$, and the corresponding flavor symmetry is
$\bigoplus_{i}\mathfrak{su}(n_{i})$. By nature of being a $U(1)$ theory we
have $\Delta_{u}=1$, and it is easy to see that
$12c_{\mathrm{4d}}=2+\sum_{i}n_{i}$ and $h_{\mathrm{ECB}}=0$. This is enough
data to compute $b=\sum_{i}n_{i}$ and $k_{i}=2$. A summary of the CFT data for
$U(1)$ IR-free theories is given in Table 6.

$\left[I_{\sum\limits_{i}q_{i}^{2}n_{i}},\bigoplus\limits_{i}\mathfrak{su}(n_{i})\right]$
---
$12c_{\mathrm{4d}}$ | $\Delta_{u}$ | $h_{\mathrm{ECB}}$ | $\mathbf{R}$ | $T(\mathbf{R})$ | $b$ | $\mathfrak{g}$ | $k_{\mathfrak{g}}$
$2+\sum_{i}n_{i}$ | 1 | 0 | $(\mathbf{1},\dots,\mathbf{1})$ | 0 | $\sum_{i}n_{i}$ | $\bigoplus_{i}\mathfrak{su}(n_{i})$ | $(2,\dots,2)$
Table 6: CFT data of ${\cal N}=2$ IR-free theories with $U(1)$ gauge group and
$n_{i}$ hypermultiplets of charge $q_{i}$.

One may also consider discrete gaugings of this class of theories. In
particular, when one of the $n_{i}$ is even a ${\mathbb{Z}}_{2}$ gauging can
be done. This results in a theory with the same data as above, but with the
corresponding $\mathfrak{su}(2n_{i})$ flavor factor replaced by
$\mathfrak{sp}(2n_{i})$.

Next consider IR-free $SU(2)$ gauge theories. The most general matter content
consists of $n_{i}$ hypermultiplets in real representations $\mathbf{R_{i}}$
for $i=1,\dots,N_{R}$, and $n_{i}$ hypermultiplets in pseudoreal
representations $\mathbf{R_{i}}$ for $i=N_{R}+1,\dots,N_{R}+N_{PR}$. The beta-
function is given by $\beta\propto\sum_{i}n_{i}T(\mathbf{R_{i}})-4$. The
relevant Kodaira fiber is of type $I^{*}_{\sum_{i}n_{i}T(\mathbf{R_{i}})-4}$,
and the corresponding flavor symmetry is
$\bigoplus_{i=1}^{N_{R}}\mathfrak{sp}(2n_{i})\bigoplus_{i=N_{R}+1}^{N_{R}+N_{PR}}\mathfrak{so}(2n_{i})$.
Since it is a $SU(2)$ gauge theory, we must have $\Delta_{u}=2$. Note that
each $\mathfrak{sp}(2n_{i})$ flavor factor contributes a factor of $n_{i}$ to
the dimension of the extended Coulomb branch, i.e.
$h_{\mathrm{ECB}}=\sum_{i=1}^{N_{R}}n_{i}$, and the central charge receives
contributions from all matter fields
$12c_{\mathrm{4d}}=6+\sum_{i}n_{i}\,\mathrm{dim}\,\mathbf{R_{i}}$. From this
information we may then compute the quantity $b$ as well as the flavor levels.
A summary of the CFT data for $SU(2)$ theories is reported in Table 7.

Note that in the $SU(2)$ case there is an additional subtlety which is that
changing the normalization of the generators can lead to different
monodromies—see e.g. Section 4.2 of Argyres:2015ffa . However, this change in
normalization does not affect any of the data in Table 7, and hence will be
irrelevant for the purposes of this paper. One could also consider discrete
gaugings of this class of theories, but we will not do so here.

$\left[I^{*}_{\sum_{i}n_{i}T(\mathbf{R_{i}})-4},\,\bigoplus\limits_{i=1}^{N_{R}}\mathfrak{sp}(2n_{i})\bigoplus\limits_{i=N_{R}+1}^{N_{R}+N_{PR}}\mathfrak{so}(2n_{i})\right]$
$12c_{\mathrm{4d}}$ $\Delta_{u}$ $h_{\mathrm{ECB}}$ $\mathbf{R}$
$T(\mathbf{R})$ $b$ $\mathfrak{g}$ $k_{\mathfrak{g}}$
$6\text{+}\sum\limits_{i=1}^{\scaleto{N_{R}+N_{PR}}{4pt}}n_{i}\,d_{\mathbf{R_{i}}}$
2 $\sum\limits_{i=1}^{N_{R}}n_{i}$
$(\mathbf{2n_{1}},...,\mathbf{2n}_{N_{R}}\underbrace{\mathbf{1},...,\mathbf{1}}_{N_{PR}\
{\rm times}})$ $N_{R}$
$2\text{+}\sum\limits_{i=1}^{N_{R}}\frac{n_{i}(d_{\mathbf{R}_{i}}-1)}{2}\text{+}\sum\limits_{\scaleto{i=N_{R}+1}{4pt}}^{\scaleto{N_{R}+N_{PR}}{4pt}}\frac{n_{i}\,d_{\mathbf{R}_{i}}}{2}$
$\bigoplus\limits_{i=1}^{N_{R}}\mathfrak{sp}(2n_{i})\bigoplus\limits_{\scaleto{i=N_{R}+1}{4pt}}^{\scaleto{N_{R}+N_{PR}}{4pt}}\mathfrak{so}(2n_{i})$
$(\underbrace{3,\dots,3}_{N_{R}\ {\rm times}},\underbrace{4,\dots,4}_{N_{PR}\
{\rm times}})$

Table 7: CFT data of ${\cal N}=2$ IR-free theories with a $SU(2)$ gauge group
and with $N_{R}$ real and $N_{PR}$ pseudoreal representations.
$d_{\mathbf{R}_{i}}$ indicates the dimension of the representation
$\mathbf{R}_{i}$.

For the purposes of our computerized scan, it will be necessary to restrict to
a finite subset of rank-one IR-free theories. As will be discussed in the main
text, our scan will restrict to theories with at most three simple factors in
their flavor symmetry, and hence we will impose the same on the allowed IR-
free theories. In the case of $U(1)$ gauge theories and their discrete
gaugings, this gives rise to infinite families of theories labelled by charges
$(q_{1},q_{2},q_{3})$ and degeneracies $(n_{1},n_{2},n_{3})$. Since the
charges do not appear anywhere in the data in Table 6, and since we do not
discuss the monodromy of IR-free theories anywhere in this paper, we can
neglect them for the purposes of the scan. As for the degeneracies, we simply
restrict them to a finite range—in practice $0\leq n_{i}\leq 10$.

Turning next to the case of $SU(2)$ gauge theories, these are labelled by
representations $(\mathbf{R}_{1},\mathbf{R}_{2},\mathbf{R}_{3})$ and
degeneracies $(n_{1},n_{2},n_{3})$. As before, we restrict the degeneracies to
take values in a finite range $0\leq n_{i}\leq 10$. However, in the current
case the dependence on the representations
$(\mathbf{R}_{1},\mathbf{R}_{2},\mathbf{R}_{3})$ does not drop out, and hence
we must place restrictions on these as well. We find it well-motivated to
restrict $\mathbf{R}_{i}$ to be either the fundamental or adjoint
representation. This gives the finite set of theories which is fed into the
algorithm.

##  Appendix D General structure of the extended Coulomb branch

In this appendix we collect a number of observations on the generic structure
of the extended Coulomb branch (ECB) of ${\cal N}=2$ SCFTs and the way it
transforms under the flavor symmetry $\mathfrak{g}$. These observations are
necessary to properly run our scan and, to the authors’ knowledge, have not
yet appeared in the literature. After introducing the general idea, we will
provide two concrete Lagrangian examples where these features arise. For
simplicity we focus on rank-two theories, though the entire discussion
straightforwardly generalizes to higher rank (and non-Lagrangian) SCFTs.

### General discussion

We begin with some general remarks. At rank-one there is only one way in which
the ECB arises, namely as a series of free hypermultiplets with perhaps global
quotienting structure, thus allowing for $\mathfrak{g}\neq\mathfrak{sp}(2n)$.
At rank-two there are instead two conceptually distinct ways:

  * •

The ECB arises from free hypermultiplets tensored over every point of the CB
(up to global twists). This means that the CB has locally a
CB$\times{\mathbb{H}}^{n}$ structure over _all_ points. This is the way in
which the ECB of only one of the two examples below (${\cal N}=4$
$\mathfrak{su}(3)$ theory) behaves as well as all ECBs of rank-one theories.

  * •

The ECB might arise from rank-one theories supported on singular strata
carrying a non-trivial ECB themselves. In this case, the local
CB$\times{\mathbb{H}}^{n}$ structure breaks down on the complex codimension-
one locus where the rank-one theory is supported, and the $n$ free hypers in
question become strongly-coupled there. This is a distinctively higher rank
feature and in fact applies to both rank-two cases discussed below.

We will call the former possibility a _free ECB_ and the latter a _coupled
ECB_. There is a major difference among these two possibilities:

  * (1)

A _free ECB_ carries a flavor symmetry which _does not act_ on BPS states
which are both massless and charged under the low-energy
$\mathfrak{u}(1)^{2}$. As we will soon argue, a free ECB necessarily implies a
coupled ECB, and that the ECB does not transform irreducibly under
$\mathfrak{g}$.

  * (2)

A _coupled ECB_ carries a flavor symmetry which instead _acts_ on charged
states which become massless where the rank-one theory is supported.

The UV-IR simple flavor condition Martone:2020nsy  forces any simple flavor
factor to act on massless charged states somewhere on the CB, and thus free
ECBs seem to contradict this condition. However there are situations in which
a free ECB can be realized: namely, when one of the rank-one theories on the
codimension-one strata carries the same flavor symmetry which acts on the free
ECB (either $\mathfrak{sp}(2n)$ or $\mathfrak{su}(n)/\mathfrak{u}(n)$) and the
flavor symmetry of the rank-two SCFT is the diagonal combination of the two.
Thus a free ECB _necessarily requires_ a coupled one with the same flavor
symmetry factor. This implies that the complex scalars of the free
hypermultiplets of the ECB transform in a reducible representation of the
flavor symmetry:

$\mathbf{R}=\underbrace{\oplus_{a}\mathbf{R}_{a}}_{\text{Coupled\
ECB}}\overbrace{\oplus\mathbf{R}_{\mathbb{H}}}^{\text{Free\ ECB}}~{},$ (D.66)

where the sum is over all the irreducible singular complex codimension-one
strata which realize the flavor symmetry $\mathfrak{g}$. Here
$\mathbf{R}_{\mathbb{H}}$ is the contribution to the ECB which is decoupled on
the whole CB$\setminus\\{0\\}$.

How is this discussion relevant for the algorithm outlined in the main text?
Recall that the way in which we compute the flavor central charge for a given
simple flavor symmetry factor $k_{\mathfrak{g}_{i}}$ was to use (2.3), which
we reproduce here for convenience:

$k_{\mathfrak{g}_{i}}=\sum_{{\cal I}_{a}\in{\cal
I}_{\mathfrak{g}_{i}}}{\Delta_{a}^{\mathrm{sing}}\over\Delta_{a}}[k_{a}-T^{i}(\mathbf{R}_{a})]+T^{i}(\mathbf{R})~{}.$
(D.67)

This sum is over all the irreducible singular complex co-dimension one strata
which realize the flavor symmetry $\mathfrak{g}_{i}$ and we have introduced an
indexed Dynkin index $T^{i}(\cdot)$, which indicates the Dynkin index of the
corresponding irreducible representaion with respect to the flavor factor
$\mathfrak{g}_{i}$, i.e. acting on a representation
$\mathbf{R}=(\mathbf{R}_{1},\mathbf{R}_{2})$ of
$\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$, we have

$\displaystyle T^{1}(\mathbf{R})$ $\displaystyle:={\rm
dim}\mathbf{R}_{2}\,T(\mathbf{R}_{1})~{},$ (D.68) $\displaystyle
T^{2}(\mathbf{R})$ $\displaystyle:={\rm
dim}\mathbf{R}_{1}\,T(\mathbf{R}_{2})~{}.$ (D.69)

The Dynkin index also satisfies
$T^{i}(\mathbf{R}_{1}\oplus\mathbf{R}_{2})=T^{i}(\mathbf{R}_{1})+T^{i}(\mathbf{R}_{2})$
and thus, using (D.66) and assuming that each $\mathfrak{g}_{i}$ is realized
on a single irreducible stratum, we find:

$T^{i}(\mathbf{R})=\left\\{\begin{array}[]{cl}T^{i}(\mathbf{R}_{a})&\hskip
14.45377pt\text{Coupled\ ECB}~{},\\\ 2\,T^{i}(\mathbf{R}_{a})&\hskip
14.45377pt\text{Free\ ECB}~{}.\end{array}\right.$ (D.70)

The upshot is then that a free versus coupled ECB affects the computation of
the quantities $k_{\mathfrak{g}_{i}}$ appearing in our algorithm. In cases in
which one cannot rigorously exclude a free ECB based on flavor symmetries, our
algorithm checks both the free and coupled cases. In the scan carried out in
Section 4, the only case that we identify which has a free ECB is ${\cal
T}_{15}$.

The discussion above generalizes straightforwardly to the ECB of rank-$r$
SCFTs, but with caveat that there are $r-1$ different types of free ECB
depending on the co-dimensionality, strictly larger than one, of the locus
where the free hyper is decoupled.

### Examples

Let us now see how these features arise in two rank-two Lagrangian examples,
namely ${\cal N}=4$ $\mathfrak{su}(3)$ SYM and ${\cal N}=2$ $\mathfrak{sp}(4)$
gauge theory with four hypers in the ${\bf 4}$ and one in the ${\bf 5}$. Both
of these theories have a non-trivial ECB, but the former has a _free ECB_
while the latter has a _coupled_ one.

##### $\boldsymbol{{\cal N}=4\ \mathfrak{su}(3)}:$

From the ${\cal N}=2$ point of view, this theory is an $\mathfrak{su}(3)$
vector multiplet with a single hypermultiplet transforming in the adjoint
representation. The globally well-defined Coulomb branch coordinates are
$u_{1}=\left\langle{\rm Tr}\left(\Phi^{2}\right)\right\rangle$ and
$u_{2}=\left\langle{\rm Tr}\left(\Phi^{3}\right)\right\rangle$, where $\Phi$
is the complex scalar of the ${\cal N}=2$ vector multiplet. It follows that
${\Delta}_{u_{1}}=2$ and ${\Delta}_{u_{2}}=3$.

A generic point on the Coulomb branch is parametrized, up to Weyl
transformations, by:

$\langle\Phi\rangle=\left(\begin{array}[]{ccc}a_{1}&0&0\\\ 0&a_{2}&0\\\
0&0&-a_{1}-a_{2}\end{array}\right)~{}.$ (D.71)

As expected, this breaks $\mathfrak{su}(3)\to\mathfrak{u}(1)^{2}$. Notice that
on the generic point of the CB not all components of the adjoint
hypermultiplet are massive. In fact, the massless components are those which
are uncharged under both $\mathfrak{u}(1)$ factors. It is then a
straightforward exercise (involving decomposing the adjoint representation) to
show that at a generic point of the CB there are two free hypermultiplets
giving, locally, a CB$\times\mathbb{H}^{2}$ structure and $h_{\rm ECB}=2$.
This matches our expectation that any rank-$r$ ${\cal N}\geq 3$ theory has an
$r$-(quaternionic-)dimensional ECB.

Now let’s extend the analysis to the singular strata. Extra massless states
appear on special loci where either there is an enhancement of the unbroken
gauge group, or some component of the hypermultiplet become massless. The only
relevant enhancement in this case is when $a_{1}=a_{2}$ (and their Weyl
equivalents) which corresponds to an enhancement
$\mathfrak{u}(1)^{2}\to\mathfrak{su}(2)\oplus\mathfrak{u}(1)$. This locus can
be written in terms of the CB coordinates as

$u_{1}^{3}+u_{2}^{2}=0\qquad\Rightarrow\qquad{\Delta}^{\rm sing}=6~{},$ (D.72)

and the ${\bf 8}$ decomposes as:

${\bf 8}\to{\bf 3}_{0}\oplus{\bf 2}_{\pm 2}\oplus{\bf 1}_{0}~{},$ (D.73)

where the subscript denotes the $\mathfrak{u}(1)$ charge. To understand which
component charged under the $\mathfrak{su}(2)$ is massless, we need to look at
its charge under the commutant of $\mathfrak{su}(2)$ inside
$\mathfrak{su}(3)$. This is simply $\mathfrak{u}(1)$. All in all we find that

  * $a$)

${\bf 1}_{0}$ denotes a massless component of the adjoint hyper which is a
singlet under both the $\mathfrak{su}(2)$ and the $\mathfrak{u}(1)$ and which
is therefore massless not just over the generic point of the CB but over the
entire CB$\setminus\\{0\\}$ \- the aforementioned hallmark of a free ECB.

  * $b$)

The ${\bf 3}_{0}$ instead gives rise to ${\cal N}=4$ $\mathfrak{su}(2)$ SYM,
denoted ${{\cal S}^{(1)}_{\varnothing,2}}$ in what follows, which is then
identified as the rank-one theory describing the low-energy effective theory
along the singularity. Notice that ${{\cal S}^{(1)}_{\varnothing,2}}$ itself