CERN-TH-2021-144

UPR-1315-T

Flavor Symmetries and Automatic Enhancement
in the 6d Supergravity Swampland

Mirjam Cvetič1,2,3,4, Ling Lin4, Andrew P. Turner1

1 Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, PA 19104, USA
2 Department of Mathematics, University of Pennsylvania,
Philadelphia, PA 19104, USA
3 Center for Applied Mathematics and Theoretical Physics,
University of Maribor, SI20000 Maribor, Slovenia
4CERN Theory Department, CH-1211 Geneva, Switzerland

We argue for the quantum-gravitational inconsistency of certain 6d
$\mathcal{N}=(1,0)$ supergravity theories, whose anomaly-free gauge algebra
$\mathfrak{g}$ and hypermultiplet spectrum $M$ were observed in [1] to be
realizable only as part of a larger gauge sector
$(\mathfrak{g}^{\prime}\supset\mathfrak{g},M^{\prime}\supset M)$ in F-theory.
To detach any reference to a string theoretic method of construction, we
utilize flavor symmetries to provide compelling reasons why the vast majority
of such $(\mathfrak{g},M)$ theories are not compatible with quantum gravity
constraints, and how the “automatic enhancement” to
$(\mathfrak{g}^{\prime},M^{\prime})$ remedies this. In the first class of
models, with $\mathfrak{g}^{\prime}=\mathfrak{g}\oplus\mathfrak{h}$, we show
that there exists an unbroken flavor symmetry $\mathfrak{h}$ acting on the
matter $M$, which, if ungauged, would violate the No-Global-Symmetries
Hypothesis. This argument also applies to 1-form center symmetries, which
govern the gauge group topology and massive states in representations
different than those of massless states. In a second class, we find that
$\mathfrak{g}$ is incompatible with the flavor symmetry of certain BPS strings
that must exist by the Completeness Hypothesis.

###### Contents

  1. 1 Introduction and Summary
  2. 2 Automatic enhancement as gauging of flavor symmetries
    1. 2.1 Review of 6d gauge anomalies
    2. 2.2 Cancellation of flavor anomalies
    3. 2.3 Examples
    4. 2.4 Gauge group topology from gauged 1-form symmetries
    5. 2.5 Comment on the Massless Charge Sufficiency Conjecture
  3. 3 Automatic enhancement enforced by BPS strings
    1. 3.1 Automatic enhancement enforced by the E-string
    2. 3.2 Automatic enhancement enforced by the M-string
    3. 3.3 Discussion
  4. A Anomaly cancellation conditions
  5. B E-string with gauged $\operatorname{SU}(9)/\mathbb{Z}_{3}$ flavor symmetry in F-theory
  6. C Automatic enhancement for F-theory on $\mathbb{F}_{2}$

##  1 Introduction and Summary

Physical features of string compactifications are highly constrained by
geometric restrictions on the underlying compactification space. One of the
main insights from the Swampland program [2] is the idea that, rather than
limitations of string model building, these restrictions should reflect
consistency conditions for coupling an effective quantum field theory to
gravity (see [3, 4, 5] for reviews). To draw lessons about generic features of
quantum gravity, it is therefore paramount that we understand the physical
principles corresponding to these geometric restrictions.

With gauge symmetries being a central building block of the effective field
theory framework, one important question to address is the landscape of
possible symmetry groups. In the context of effective supergravity theories,
recent progress in this direction has been made in spacetime dimensions $d>6$
[6, 7, 8, 9, 10, 11], by finding physical consistency conditions that lead to
the same restrictions as in known string compactifications, such as bounds on
the total gauge rank as well as the allowed gauge algebras and their global
structures.

In 6d, gauge dynamics of $\mathcal{N}=(1,0)$ supergravity theories is already
heavily constrained by the cancellation of chiral anomalies. Nevertheless, the
landscape of string-derived models, in particular F-theory constructions [12,
13, 14]111See [15, 16] for recent reviews on F-theory., appears to be much
smaller compared to the vast set of anomaly-free field theories [17, 18, 19].
Utilizing Swampland principles [20, 21, 22, 8, 23, 24, 25, 26, 27], we have
since learned that, indeed, most of such field theories are inconsistent when
coupled to quantum gravity. However, our knowledge about the boundary between
the Swampland and Landscape in the context of 6d supergravity theories is
still blurred when it comes to the precise set of consistent gauge symmetries.

A recent proposal [1], dubbed the “Automatic Enhancement Conjecture”, attempts
to sharpen this boundary considerably. This is based on the observation that,
in engineering certain gauge algebras $\mathfrak{g}$ with matter $M$ via
F-theory, the underlying geometry forces the appearance of a “larger” symmetry
$\mathfrak{g}^{\prime}\supset\mathfrak{g}$, whose matter spectrum
$M^{\prime}\supset M$ does not allow for a supersymmetric Higgsing of
$\mathfrak{g}^{\prime}$ to $\mathfrak{g}$. In other words, it appears that
F-theory can only realize $(\mathfrak{g},M)$ as a subsector of a larger gauge
symmetry, even though all gauge and gravitational anomalies are
cancelled.222More precisely, any 6d gauge symmetry has an associated anomaly
coefficient $b_{\mathfrak{g}}$, which is part of the defining data of the
supergravity model. See Section 2.1 for more details. This motivated the
conjecture [1] that, for such a pair of theories,
$(\mathfrak{g}^{\prime}\supset\mathfrak{g},M^{\prime}\supset M)$, where the
larger $(\mathfrak{g}^{\prime},M^{\prime})$ can be constructed in F-theory,
the smaller $(\mathfrak{g},M)$ theory belongs to the Swampland—the set of
apparently consistent effective gravitational theories which cannot be UV-
completed. This formulation makes explicit reference to an F-theoretic method
of construction, as the enhanced gauge group $\mathfrak{g}^{\prime}$ was found
through explicit geometric engineering. The geometric motivations obscure the
physical “origin” of $(\mathfrak{g}^{\prime},M^{\prime})$ apart from the non-
existence of a supersymmetric Higgs transition
$(\mathfrak{g}^{\prime},M^{\prime})\rightarrow(\mathfrak{g},M)$, and, in
particular, do not explain why the $(\mathfrak{g},M)$ theory is inconsistent
by itself.

The goal of this work is to shed light on the physical arguments underlying
the Automatic Enhancement Conjecture. While our analysis reveals no single,
unifying principle, two sets of arguments—related to flavor symmetries—emerge
that lend credence to the conjecture in different types of automatic
enhancement observed in [1].

The first type of enhancement we seek to explain are models where
$\mathfrak{g}^{\prime}=\mathfrak{g}\oplus\mathfrak{h}$, with the matter
$M^{\prime}=\bigoplus_{r}x^{\prime}_{r}\times(\bm{\mathrm{R}}_{\mathfrak{g}}^{(r)},\bm{\mathrm{R}}_{\mathfrak{h}}^{(r)})$
(for $\bm{\mathrm{R}}_{\mathfrak{f}}^{(r)}$ representations of
$\mathfrak{f}=\mathfrak{g},\mathfrak{h}$) being just a rearrangement of the
original matter
$M=\bigoplus_{r}(x^{\prime}_{r}\dim\bm{\mathrm{R}}_{\mathfrak{h}}^{(r)})\times\bm{\mathrm{R}}_{\mathfrak{g}}^{(r)}$
into bi-charged representations. In these instances, we show that
$\mathfrak{h}$ has the interpretation of the subalgebra of the classical
flavor symmetry $\hat{\mathfrak{h}}$ of $(\mathfrak{g},M)$ for which the
flavor–gauge, flavor–flavor, and flavor–gravitational anomalies can be
cancelled by a generalized Green–Schwarz (GS) mechanism [28, 29] with the
tensor multiplets present in the theory. The characterization of
$\mathfrak{h}$ as a subalgebra of the flavor symmetry implies in particular
that there is no Higgsing from $(\mathfrak{g}^{\prime},M^{\prime})$ to
$(\mathfrak{g},M)$, since an honest flavor symmetry of $\mathfrak{g}$ only
charges matter that is already charged under $\mathfrak{g}$, which would then
also break $\mathfrak{g}$ if given a vacuum expectation value (vev). The
automatic enhancement to
$(\mathfrak{g}^{\prime}=\mathfrak{g}\oplus\mathfrak{h},M^{\prime}\cong M)$ is
then the result of an equivalent formulation of the conjecture that needs no
reference to any string theory construction:

_If the flavor symmetry of $(\mathfrak{g},M)$ has a subalgebra $\mathfrak{h}$
whose anomalies can be cancelled by an appropriate GS term, then, in a
consistent 6d supergravity model, $\mathfrak{h}$ must also be gauged._

In particular, it implies that any potential sources of symmetry breaking in
the UV are entirely captured by anomalies in the low-energy effective gauge
theory. Therefore, the absence of these anomalies implies that the
$(\mathfrak{g},M)$ theory is inconsistent as an effective description of
quantum gravity, because it has an unbroken global symmetry. To conform with
the No-Global-Symmetries Hypothesis [30, 31, 32], this symmetry must hence be
gauged in quantum gravity. In Section 2, we will discuss this interpretation
in more detail, and provide detailed examples.

Moreover, we also relate the global structure of the gauge group in these
cases, which arises in F-theory models “automatically” in form of the
Mordell–Weil group [33, 34, 35], to an automatic enhancement of 1-form center
symmetries [36], which in 6d $\mathcal{N}=(1,0)$ theories have an anomaly [25]
that obstruct their gauging. As we point out in Section 2.4, the cases in [1,
37] where the gauge _group_ associated to the enhanced algebra
$\mathfrak{g}^{\prime}$ is automatically non-simply connected have no such
obstruction. Hence, the automatic gauging of the 1-form symmetry, leading to
the non-simply connected gauge group, can be understood equivalently as an
incarnation of the No-Global-Symmetries Hypothesis, now for generalized global
symmetries. In Section 2.5, we further comment on the relationship of the
aforementioned obstructions to the validity of the “Massless Charge
Sufficiency Conjecture” [37] in 6d supergravity theories.

While these arguments provide a consistent interpretation of the Automatic
Enhancement Conjecture in the larger web of quantum gravity constraints, this
bottom-up perspective does not provide the means to rule out UV-breaking
mechanisms not captured by anomalies; the observed enhancement is only a post-
factum confirmation of the absence of such a breaking. Hence, this
reformulation, while revealing the intimate connection to the No-Global-
Symmetries Hypothesis, does not constitute a definitive proof of the
conjecture. Moreover, there are enhancements
$(\mathfrak{g},M)\to(\mathfrak{g}^{\prime},M^{\prime})$ found in [1] that do
not have an interpretation as gauging a flavor symmetry of $(\mathfrak{g},M)$,
and, thus, would require a different set of arguments to explain.

As we will argue, the key objects to formulating such arguments are BPS
strings. However, because the Automatic Enhancement Conjecture places a
_lower_ bound on the allowed gauge symmetry, the methods of [8, 23, 27] to
utilize the so-called supergravity string—a type of BPS string present in all
6d $\mathcal{N}=(1,0)$ supergravity theories—are not immediately applicable,
as these place upper bounds. Instead, we find that one must inspect strings
specific to the model, whose worldsheet consistency then provides constraints
equivalent to the Automatic Enhancement Conjecture. In 6d $\mathcal{N}=(1,0)$
theories, such strings must exist by virtue of the Completeness Hypothesis
[32] as charged objects of 2-form fields in tensor multiplets, which
themselves are required to cancel gauge anomalies via the Green–Schwarz
mechanism. Then, the consistency of the worldsheet CFT poses conditions on the
spacetime gauge dynamics beyond effective field theory considerations. In
particular, from the worldsheet perspective, it is now the spacetime symmetry
$\mathfrak{g}$ that plays the role of a global, or flavor, symmetry. In this
case, the specific type of strings (with charge $b$) present in the
supergravity theory can constrain the symmetry algebra $\mathfrak{g}$, even if
it is consistent with gauge anomalies in the 6d bulk. From this perspective,
the enhancement $\mathfrak{g}\to\mathfrak{g}\oplus\mathfrak{h}$ observed in
F-theory constructions reflects the geometry’s “awareness” of these
constraints. It avoids the string of charge $b$ with an inconsistent flavor
symmetry $\mathfrak{g}$ by “pairing” it with a spacetime gauge symmetry
$\mathfrak{h}$, i.e., only allows instanton strings of $\mathfrak{h}$ with
compatible flavor symmetry to have charge $b$. In Section 3, we exemplify this
principle in two classes of models, where the constraining strings are strings
of 6d $\mathcal{N}=(1,0)$ superconformal field theories (SCFTs) with well-
known flavor symmetries. In Section 3.3, we will discuss some open questions
about the Automatic Enhancement Conjecture that will need further study of BPS
strings of 6d supergravity theories to elucidate.

##  2 Automatic enhancement as gauging of flavor symmetries

In this section, we focus on the bottom-up interpretation of the Automatic
Enhancement Conjecture as quantum gravity avoiding unbroken global symmetries.
First, we verify for enhancements $(\mathfrak{g},M)$ to
$(\mathfrak{g}^{\prime}=\mathfrak{g}\oplus\mathfrak{h},M^{\prime}\cong M)$,
with no hypermultiplets charged solely under $\mathfrak{h}$, that
$\mathfrak{h}$ is the subalgebra of the flavor symmetry $\hat{}\mathfrak{h}$
of $(\mathfrak{g},M)$ for which the flavor–gauge, flavor–flavor (or ’t Hooft),
and flavor–gravitational anomalies can be cancelled by a generalized
Green–Schwarz mechanism. We then use the same perspective to explain the
global gauge group topology, as quantum gravity avoiding an unbroken 1-form
global symmetry.

###  2.1 Review of 6d gauge anomalies

To begin with, we review gauge and gravitational anomalies of 6d
$\mathcal{N}=(1,0)$ gauge and supergravity theories, which also explains our
notation. For a pedagogical discussion, see, e.g., [38]. We are not concerned
here with global anomalies; for more on such anomalies, see [39, 40, 41].

Six-dimensional $\mathcal{N}=(1,0)$ supergravity theories famously have an
associated signature-$(1,T)$ unimodular lattice $\Gamma$ with pairing $\Omega$
that corresponds to the charge lattice of one self-dual and $T$ anti-self-dual
2-form fields (or, dually, with the BPS strings charged under these 2-forms)
in the supergravity and tensor multiplets, respectively. The vevs of the
scalars in these multiplets are parametrized by a vector
$\mathcal{J}\in\Gamma\otimes\mathbb{R}$ of unit length under $\Omega$, which
in turn defines a _positivity cone_ $\Gamma^{+}$, given by the requirement
$\Omega(v,\mathcal{J})\equiv v\cdot\mathcal{J}\geq 0$ for any
$v\in\Gamma^{+}$. There is a model-specific vector $a\in\Gamma$, called the
_gravitational anomaly coefficient_. To complete the data for a $(1,0)$
supergravity theory, we have to specify the gauge algebra,

$\mathfrak{g}=\bigoplus_{i}\mathfrak{g}_{i}\oplus\bigoplus_{\alpha}\mathfrak{u}(1)_{\alpha}\,,$
(2.1)

and hypermultiplet spectrum

$M=\bigoplus_{r}x_{r}\times\bm{\mathrm{R}}_{\mathfrak{g}}^{(r)}\equiv\bigoplus_{r}x_{r}\times(\bm{\mathrm{R}}_{\mathfrak{g}_{1}}^{(r)},\dotsc,\bm{\mathrm{R}}_{\mathfrak{g}_{i}}^{(r)},\dotsc)_{q_{1}^{(r)},\dotsc,q_{\alpha}^{(r)},\dotsc}\,.$
(2.2)

Here, the irreducible representations (irreps)
$\bm{\mathrm{R}}_{\mathfrak{g}}^{(r)}$ with multiplicity $x_{r}$ are
distinguished by irreps $\bm{\mathrm{R}}_{\mathfrak{g}_{i}}^{(r)}$ under each
non-abelian gauge factor $\mathfrak{g}_{i}$, and by charges $q_{\alpha}^{(r)}$
under each abelian $\mathfrak{u}(1)_{\alpha}$ factor.333Note that a
hypermultiplet we refer to as being in a representation $\bm{\mathrm{R}}$
actually contains field transforming in both $\bm{\mathrm{R}}$ and
$\overline{\bm{\mathrm{R}}}$, because the hypermultiplets live in a
quaternionic-Kähler manifold and thus can only be charged under quaternionic
representations of the gauge group. When the representation $\bm{\mathrm{R}}$
is itself quaternionic, we may form half-hypermultiplets that have half the
degrees of freedom of a full hypermultiplet.

We associate to each non-abelian factor $\mathfrak{g}_{i}$ an anomaly
coefficient $b_{i}\in\Gamma^{+}$, and to each _pair_ of abelian gauge factors
an anomaly coefficient $b_{\alpha\beta}=b_{\beta\alpha}$ that satisfies
$\frac{1}{2}b_{\alpha\alpha},b_{\alpha\beta}\in\Gamma$, and for which
$\mathcal{J}\cdot b_{\alpha\beta}$ is a positive definite matrix.444Note that
the integrality conditions, i.e., being elements of $\Gamma$ rather than
$\Gamma\otimes\mathbb{R}$, are related to a generalized Completeness
Hypothesis, see [21]. A tensor multiplet with charge $b$ that is an anomaly
coefficient of a gauge factor $\mathfrak{g}_{i}$ is said to be “paired” with
the corresponding vector multiplet [42]. Equivalently, this means that the
1-instanton configuration of $\mathfrak{g}_{i}$, which is string-like in 6d,
has charge $b$.

These anomaly coefficients encode the contributions to the total anomaly
polynomial $I_{8}^{\text{total}}=I_{8}^{\text{1-loop}}+I_{8}^{\text{GS}}$ from
a generalized version of the Green–Schwarz mechanism [28, 29] via the
inclusion of tree-level diagrams mediated by the (anti-)self-dual 2-form
fields:

$I_{8}^{\text{GS}}=\frac{1}{32}\Omega(X_{4},X_{4})\equiv\frac{1}{32}X_{4}\cdot
X_{4}\,,$ (2.3)

with

$X_{4}=\frac{1}{2}a\operatorname{tr}R^{2}+2\sum_{i}\frac{b_{i}}{\lambda_{i}}\operatorname{tr}F_{i}^{2}+2\sum_{\alpha,\beta}b_{\alpha\beta}F_{\alpha}F_{\beta}\,,$
(2.4)

where $F_{\alpha}$, $F_{i}$ are the field strengths of the (non-)abelian gauge
symmetries, $R$ is the spacetime curvature 2-form, and $\operatorname{tr}$
denotes trace in the fundamental representation. The $\lambda_{i}$ are
normalization constants associated with the simple non-abelian Lie algebras,
given by $\lambda_{i}=2c_{i}^{\vee}/A_{\mathfrak{g}_{i}}$ with $c_{i}^{\vee}$
the dual Coxeter number. Meanwhile, the 1-loop part $I_{8}^{\text{1-loop}}$
receives contributions from the gravity, tensor, vector, and hypermultiplets.
Their explicit expressions, as well as the resulting conditions for anomaly
cancellation, $I_{8}^{\text{total}}=0$, and the values for $\lambda_{i}$ are
summarized in Appendix A.

###  2.2 Cancellation of flavor anomalies

From the bottom-up perspective, it is natural to think about a supergravity
theory as coupling an effective 6d $\mathcal{N}=(1,0)$ field theory with gauge
algebra $\mathfrak{g}$ and massless matter $M$ to gravity in a specific way.
For the $(\mathfrak{g},M)$ gauge theory, there is an associated classical
flavor symmetry $\hat{\mathfrak{h}}$ that rotates $M$.555Associated to the
decomposition 2.2 of the hypermultiplets $M$, $\hat{\mathfrak{h}}$ contains a
$\mathfrak{u}(x_{r})$ factor rotating a complex irrep
$\bm{\mathrm{R}}_{\mathfrak{g}}^{(r)}$ with multiplicity $x_{r}$, an
$\mathfrak{sp}(x_{r})$ rotating a real irrep
$\bm{\mathrm{R}}_{\mathfrak{g}}^{(r)}$ (with
$\mathfrak{sp}(1)\cong\mathfrak{su}(2)$), and an $\mathfrak{so}(2x_{r})$
rotating a quaternionic irrep $\bm{\mathrm{R}}_{\mathfrak{g}}^{(r)}$; note
that in the last case, $x_{r}$ can be half-integer, see footnote 3. This re-
organizes the matter 2.2 as

$M=\bigoplus_{r}x_{r}\times\bm{\mathrm{R}}_{\mathfrak{g}}^{(r)}\cong\bigoplus_{r}\bm{\mathrm{R}}_{\mathfrak{g}\oplus\hat{\mathfrak{h}}}^{(r)}=\bigoplus_{r}(\bm{\mathrm{R}}_{\mathfrak{g}}^{(r)},\bm{\mathrm{R}}_{\hat{\mathfrak{h}}}^{(r)})\,,$
(2.5)

with $x_{r}=\dim\bm{\mathrm{R}}_{\hat{\mathfrak{h}}}^{(r)}$. We are interested
in the cancellation of flavor anomalies, which allows for gauging a part
$\mathfrak{h}\subset\hat{\mathfrak{h}}$ of the flavor symmetry, i.e., all
flavor–gauge, flavor–flavor, and flavor–gravitational anomalies associated
with $\mathfrak{h}$. To cancel the 1-loop contributions, one must introduce
appropriate Green–Schwarz terms, which structurally take the same form as in
Eq. 2.4, but now include the field strengths of $\mathfrak{h}$. In particular,
this requires the introduction of anomaly coefficients $b_{I}$,
$b_{\alpha\hat{\beta}}$, and $b_{\hat{\alpha}\hat{\beta}}$ for the flavor
factors
$\mathfrak{h}=\bigoplus_{I}\mathfrak{h}_{I}\oplus\bigoplus_{\hat{\alpha}}\mathfrak{u}(1)_{\hat{\alpha}}$;
in terms of the effective field theory action, this requires a coupling of the
2-form field dual to $b_{I}$ to the instanton density of $\mathfrak{h}$.

In the context of 6d superconformal field theories (SCFTs), such a gauging
procedure (of non-abelian flavor factors) can be used to construct new SCFTs
out of existing ones. However, this always introduces a new anomaly
coefficient $b_{I}$ that is linearly independent of those already present. In
other words, the lattice of string/tensor charges is extended in these
gaugings. On the other hand, the Automatic Enhancement Conjecture describes a
change of gauge symmetry without altering the tensors. In other words, the
required Green–Schwarz terms must be supplied by the existing tensors of the
supergravity model prior to enhancement. Moreover, we demand that this is
possible without the introduction of additional charged hypermultiplets that
are charged under $\mathfrak{h}$. This retains the interpretation of
$\mathfrak{h}$ as “just the flavor symmetry” of the gauge sector
$(\mathfrak{g},M)$, and, furthermore, prevents $\mathfrak{h}$ from being
Higgsed without also breaking $\mathfrak{g}$. In short, we would like to
determine the subalgebra $\mathfrak{h}\subset\hat{\mathfrak{h}}$ for which all
1-loop anomalies induced by the hypermultiplet spectrum 2.5 are cancelled by
the $\mathfrak{h}$-vector multiplets and Green–Schwarz terms built out of
tensors already present in the $(\mathfrak{g},M)$ theory.

If this is possible, then, from a bottom-up perspective, there are no
obstructions to turning on a “background gauge field” for the flavor symmetry
$\mathfrak{h}$—after all, the anomalies precisely measure such obstructions
that are detectable in the effective field theory description. However, in a
gravitational theory, there are no background gauge fields; in 6d
$\mathcal{N}=(1,0)$ theories, this would require the gauge coupling
$g_{I}^{2}\propto(\mathcal{J}\cdot b_{I})^{-1}$ to vanish, which gives an
infinite tension $\mathcal{J}\cdot b_{I}$ to the string dual to $b_{I}$.
Therefore, the background field for $\mathfrak{h}$ is allowed to fluctuate
(note that this requires also the cancellation of what would be ’t Hooft
anomalies for $\mathfrak{h}$ if treated as a global symmetry), and, in
particular, gauges $\mathfrak{h}$, thus avoiding an unobstructed global
symmetry. We have verified for all relevant examples discussed in [1] that the
observed automatic enhancement is precisely such an automatic gauging of an
unobstructed flavor symmetry $\mathfrak{h}$.

To perform this verification, one would, ideally, consider every possible
subalgebra embedding
$\mathfrak{h}=\bigoplus_{I}\mathfrak{h}_{I}\oplus\bigoplus_{\hat{\alpha}}\mathfrak{u}(1)_{\hat{\alpha}}\subset\hat{\mathfrak{h}}$
and consider, for each of these cases, if the associated flavor–gauge,
flavor–flavor, and flavor–gravitational anomalies can be cancelled by
assigning anomaly coefficients
$b_{I},b_{\alpha\hat{\beta}},b_{\hat{\alpha}\hat{\beta}}\in\Gamma$. However,
this approach can become quite computationally demanding when the flavor
symmetry is large and thus has a very large number of subalgebras. We propose
here an alternate approach that allows for a more direct computation of the
maximal gauge-able subalgebra of $\hat{\mathfrak{h}}$.

We first consider gauging an arbitrary $\mathfrak{u}(1)$ subalgebra of
$\hat{\mathfrak{h}}$, and then consider the constraints on the charges of the
resulting matter spectrum following from the 6d anomaly cancellation
conditions A.7. In practice, this is done by augmenting the gauge algebra
$\mathfrak{g}$ (with the anomaly coefficients $b_{i},b_{\alpha\beta}$) with an
additional $\mathfrak{u}(1)$ factor and considering all possible assignments
of $\mathfrak{u}(1)$ charges for irreps $\bm{\mathrm{R}}_{\mathfrak{g}}^{(r)}$
of the original matter $M$ and anomaly coefficient $b_{\mathfrak{u}(1)}$ that
solve the anomaly cancellation conditions.

Crucially, in solving for anomaly-consistent spectra, we allow for a
potentially _negative_ number of hypermultiplets charged only under the new
$\mathfrak{u}(1)$ gauge factor. The presence of a negative number of such
hypermultiplets signals that the newly-introduced $\mathfrak{u}(1)$ must be
embedded in a larger non-abelian subalgebra of $\hat{\mathfrak{h}}$ in order
to be consistent.

More precisely, imagine the $\mathfrak{u}(1)$ arising from an adjoint Higgsing
of $\mathfrak{h}$, in which hypermultiplets charged solely under
$\mathfrak{h}$ generically decompose into charged singlets. These are counted
against those hypermultiplets that are “eaten” by the vectors in the Higgs
transition. A formally negative number of such charged singlets in the Higgsed
phase therefore signals that there were not enough hypermultiplets to begin
with to facilitate this Higgsing of $\mathfrak{h}$. The exact (negative)
number of charged singlets in the spectrum restricts the simple non-abelian
subalgebras $\mathfrak{h}$ of $\hat{\mathfrak{h}}$ into which the
$\mathfrak{u}(1)$ factor can consistently be embedded. One can then verify
whether this candidate algebra is anomaly-free with the anomaly coefficient
$b_{\mathfrak{h}}\propto b_{\mathfrak{u}(1)}$.

When solving the anomaly cancellation conditions for
$\mathfrak{g}\oplus\mathfrak{u}(1)$, we allow for the possibility that each
hypermultiplet in a given representation
$\bm{\mathrm{R}}_{\mathfrak{g}}^{(r)}$ has a different $\mathfrak{u}(1)$
charge, to accommodate any possible embedding into a larger algebra. However,
we only consider (non-positive multiplicities of) singlets of charge $1$ and
$2$ under the additional $\mathfrak{u}(1)$ factor. The reason for this is
that, although adjoint Higgsings of simple non-abelian factors can result in
singlets of other charges, higher-charge representations can always be
“exchanged” via anomaly equivalences until only charges $1$ and $2$ remain,
and the number of uncharged singlets will always increase in such an exchange.
Specifically, the following anomaly equivalence holds [43]:

$\bm{1}_{q}\longleftrightarrow-\frac{q^{2}\left(q^{2}-4\right)}{3}\times\bm{1}_{1}+\frac{q^{2}\left(q^{2}-1\right)}{12}\times\bm{1}_{2}+\left(\frac{q^{4}}{4}-\frac{5q^{2}}{4}+1\right)\times\bm{1}_{0}\,.$
(2.6)

We see that a single charge-$q$ singlet is equivalent under anomalies to a
negative number of charge-$1$ singlets along with positive numbers of
charge-$2$ and charge-$0$ singlets. As we are only using the charged singlets
as a computational tool to identify that the $\mathfrak{u}(1)$ factor is
embedded in a gauged non-abelian algebra, this allows us to reduce the scope
of the computation. Furthermore, we can reduce the list of possible charges of
other hypermultiplets by considering analogous anomaly equivalences for
$\mathfrak{u}(1)$-charged non-abelian representations other than singlets.
Including higher $\mathfrak{u}(1)$-charged representations will always reduce
the number of uncharged singlets; because each model $(\mathfrak{g},M)$ we
consider has only a finite number of uncharged singlets, there is a limit to
magnitude of the $\mathfrak{u}(1)$ charges under the new $\mathfrak{u}(1)$
factor before the number of uncharged singlets would be forced to be negative.

We stress that this procedure is merely a computational simplification to find
the gauge-able subalgebra $\mathfrak{h}$. We have not investigated potential
exceptions to this algorithm, since in all cases we utilize it, we find a
consistent subalgebra $\mathfrak{h}\subset\hat{\mathfrak{h}}$, which
furthermore agrees with the instances of automatic enhancement observed [1].
As we will see in Section 2.3.2, the algorithm might lead to an intermediate
$\mathfrak{u}(1)$ spectrum that does not embed into the full flavor
$\hat{\mathfrak{h}}$, but, in addition, also to a spectrum compatible with the
Cartan charges of a subalgebra $\mathfrak{h}\subset\hat{\mathfrak{h}}$ that
agrees with the automatic enhancement. Since the alternative spectrum has a
negative number of some singlets, it is inconsistent by itself anyways, so
does not pose a violation of the Automatic Enhancement Conjecture per se.666We
have also not found a consistent non-abelian algebra into which such spectra
embed, such that all negative number of singlets are “used up”, thus lending
further evidence to the efficacy of the algorithm.

In all examples we have examined, we only find one unique subalgebra
$\mathfrak{h}\subset\hat{\mathfrak{h}}$, which reflects the fact that in all
models studied in [1], there is a unique automatic enhancement. However, it is
in principle possible that there exist multiple (potentially identical)
subalgebras $\mathfrak{h}_{\ell}$ of $\hat{\mathfrak{h}}$ that can be
consistently gauged with different anomaly coefficients $b_{\ell}$, all of
which would be detectable by the algorithm above. By definition, there would
not be any supersymmetric Higgs transitions that flow from one case to
another, so, from our bottom-up perspective, the gauging of any
$\mathfrak{h}_{\ell}$ with $b_{\ell}$ would be a consistent model. If such
cases exist, then this must happen for $T\geq 1$, in which case it would be
interesting to investigate, from an F-theory perspective, how the multiple
allowed enhancements may be realized. Note that there can be additional
consistency constraints for $T\geq 1$ such as those that we discuss in Section
3, which could potentially exclude these cases.

###  2.3 Examples

####  2.3.1 Large $\operatorname{SU}(N)$ gauge group with $T=0$ and $b=1$

To illustrate the procedure, we will demonstrate how it is carried out for the
enhancements of large $\mathfrak{su}(N)$ gauge algebras with $T=0$ and $b=1$;
these appear in Section 4.3 of [1]. We will begin with the case $N=22$, where
the flavor symmetry is small enough that we can carry out the exhaustive
computation mentioned in Section 2.2, before addressing the same example and
all subsequent ones using the shortcut approach described there.

For the gauge symmetry $\mathfrak{su}(22)$ and $T=0$, we consider the
hypermultiplet spectrum

$x_{\bm{1}}=19\,,\quad x_{\bm{22}}=2\,,\quad x_{\bm{231}}=3\,,\quad
x_{\bm{483}}=0\,,$ (2.7)

which solves the anomaly cancellation conditions with anomaly coefficient
$b=1$. Here, $x_{\bm{\mathrm{R}}}$ denotes the multiplicity of the
representation $\bm{\mathrm{R}}$. The full flavor symmetry of this model is
$\hat{\mathfrak{h}}=\mathfrak{u}(2)\oplus\mathfrak{u}(3)$, associated with the
$2$ fundamental and $3$ anti-symmetric representations. We thus want to find
the subalgebras of $\mathfrak{u}(2)\oplus\mathfrak{u}(3)$ which can be gauged.

First, we note that the $\mathfrak{su}(2)\subset\mathfrak{u}(2)$ associated
with the fundamental flavors is non-anomalous; to see this, we look for
solutions to the anomaly cancellation conditions A.7 for an
$\mathfrak{su}(22)\oplus\mathfrak{su}(2)$ algebra with $T=0$ and
$b_{\mathfrak{su}(22)}=1$ that only have $\mathfrak{su}(22)$ fundamentals
charged under the additional $\mathfrak{su}(2)$ factor. The unique solution is

$x_{(\bm{1},\bm{1})}=22\,,\quad x_{(\bm{22},\bm{1})}=0\,,\quad
x_{(\bm{231},\bm{1})}=3\,,\quad x_{(\bm{483},\bm{1})}=0\,,\quad
x_{(\bm{22},\bm{2})}=1\,,$ (2.8)

with $b_{\mathfrak{su}(2)}=1$. Thus, the $\mathfrak{su}(2)$ fundamental flavor
symmetry can be gauged.

Next, we can consider gauging a subalgebra of the $\mathfrak{su}(3)$ anti-
symmetric flavor symmetry. Similarly to above, we can look for solutions to
the anomaly cancellation conditions in which only the $\mathfrak{su}(22)$
anti-symmetric representations are allowed to have non-trivial charge under
the additional gauge factor; we find that there are no valid solutions for any
subalgebra of the $\mathfrak{su}(3)$. To see how this occurs, consider the
case of $\mathfrak{su}(22)\oplus\mathfrak{su}(3)$. The solutions to the
anomaly cancellation equations take the form

$\displaystyle x_{(\bm{1},\bm{1})}$
$\displaystyle=5b_{\mathfrak{su}(3)}^{2}-\frac{687}{20}b_{\mathfrak{su}(3)}+19\,,$
(2.9) $\displaystyle x_{(\bm{22},\bm{1})}$ $\displaystyle=2\,,$ $\displaystyle
x_{(\bm{231},\bm{1})}$ $\displaystyle=3-\frac{3}{20}b_{\mathfrak{su}(3)}\,,$
$\displaystyle x_{(\bm{1},\bm{3})}$
$\displaystyle=b_{\mathfrak{su}(3)}\left(\frac{309}{20}-3b_{\mathfrak{su}(3)}\right)\,,$
$\displaystyle x_{(\bm{483},\bm{1})}$ $\displaystyle=0\,,$ $\displaystyle
x_{(\bm{1},\bm{8})}$
$\displaystyle=\frac{1}{2}(b_{\mathfrak{su}(3)}-1)(b_{\mathfrak{su}(3)}-1)\,,$
$\displaystyle x_{(\bm{231},\bm{3})}$
$\displaystyle=\frac{b_{\mathfrak{su}(3)}}{20}\,.$

Requiring that this $\mathfrak{su}(3)$ is subgroup of the flavor
$\mathfrak{u}(3)$ of the $\mathfrak{su}(22)$ anti-symmetrics sets
$x_{(\bm{231},\bm{3})}=1$ and $x_{(\bm{1},\bm{3})}=0$, which cannot be solved
simultaneously; thus, the $\mathfrak{su}(3)$ cannot be gauged. Similar results
hold for all subgroups of the $\mathfrak{su}(3)$. We then proceed to check
anomalies for the two $\mathfrak{u}(1)$ factors; the fundamental $\bm{22}$ is
charged only under the $\mathfrak{u}(1)\subset\mathfrak{u}(2)$, while the
anti-symmetric $\bm{231}$ is charged only under the second
$\mathfrak{u}(1)\subset\mathfrak{u}(3)$. Again, we find that the anomaly
conditions cannot be solved without the introduction of further charged
hypermultiplets. To rule out potential linear combinations of Cartan
$\mathfrak{u}(1)$s that can be gauged, we must then also consider linear
combinations of $\mathfrak{u}(1)$ charges for the bifundamental and anti-
symmetric hypermultiplets, which could result in an anomaly-free spectrum.
Since this procedure is precisely the starting point of the alternate approach
that we will discuss momentarily, we simply state that there are no such
$\mathfrak{u}(1)$s in
$\hat{\mathfrak{h}}=\mathfrak{u}(2)\oplus\mathfrak{u}(3)$, and so we conclude
that the $\mathfrak{su}(2)$ fundamental flavor symmetry is the unique
subalgebra of the full flavor symmetry that can be gauged without altering the
tensor or hypermultiplet spectrum; indeed, Eq. 2.8 is precisely the automatic
enhancement identified in [1].

Let us now see how we would reach this same conclusion by instead considering
arbitrary $\mathfrak{u}(1)$ subalgebras of the flavor symmetry. We solve the
system of equations A.7 for gauge algebra
$\mathfrak{su}(22)\oplus\mathfrak{u}(1)$ in terms of the multiplicities of the
representations $\bm{1}_{0}$, $\bm{22}_{n+22f_{1}}$, $\bm{22}_{n+22f_{2}}$,
$\bm{231}_{2n+22a_{1}}$, $\bm{231}_{2n+22a_{2}}$, $\bm{231}_{2n+22a_{3}}$,
$\bm{483}_{0}$, $\bm{1}_{22}$, $\bm{1}_{44}$, where we leave the integers
$-10<n\leq 11$, $f_{i}$, and $a_{i}$ unspecified. We have chosen to normalize
the $\mathfrak{u}(1)$ charges such that all charges will be integral
regardless of the presence or absence of quotients in the global structure of
the gauge group; as a result, two hypermultiplets transforming in the same
$\mathfrak{su}(22)$ representation must have $\mathfrak{u}(1)$ charges that
differ by a multiple of $22$. The variable $n$ allows for offsets related to
these possible quotients, while the variables $f_{i},a_{i}$ allow the various
hypermultiplets to have distinct charges. The offset of $2n$ for the anti-
symmetrics is required for consistency with the offset $n$ of the
fundamentals. We then look for solutions to the anomaly cancellation equations
for which $x_{\bm{1}_{22}},x_{\bm{1}_{44}}\leq 0$, all other multiplicities
are non-negative, and all multiplicities are integral. As discussed in Section
2.2, we can use the number of uncharged singlets in Eq. 2.7 to constrain the
magnitudes of $f_{i},a_{i}$ in our search for solutions. The resulting spectra
with non-trivial $\mathfrak{u}(1)$ are all of the form

$x_{\bm{1}_{0}}=22\,,\quad x_{\bm{22}_{11m}}=x_{\bm{22}_{-11m}}=1\,,\quad
x_{\bm{231}_{0}}=3\,,\quad x_{\bm{1}_{22m}}=-2$ (2.10)

for $m\in\mathbb{Z}$, with all other multiplicities zero. Here, we see the
negative number of charged singlets indicating that the $\mathfrak{u}(1)$
factor we have gauged must be embedded in a gauged non-abelian subalgebra of
$\hat{\mathfrak{h}}$ in order to be anomaly-free. Specifically, there are two
charged singlets in the branching rule of the $\mathfrak{su}(2)$ adjoint to
its Cartan $\mathfrak{u}(1)$ subalgebra, and so this is an indicator that the
appropriate anomaly-free subalgebra of $\hat{\mathfrak{h}}$ is in fact
$\mathfrak{su}(2)$; the fact that the $\mathfrak{su}(22)$ fundamentals are
charged under this $\mathfrak{u}(1)$ while the anti-symmetrics are not
indicates that this is the fundamental flavor $\mathfrak{su}(2)$, and the
appropriate anomaly-free enhancement is the one given in Eq. 2.8. As mentioned
above, this enhancement is precisely the one found in [1], and so we see that
the arguments here reproduce the results of the original Automatic Enhancement
Conjecture.

For the remaining examples, we will use only the shortcut approach in order to
reduce the necessary computations. Next we consider the model with gauge
symmetry $\mathfrak{su}(20)$, $T=0$, and hypermultiplet spectrum

$x_{\bm{1}}=23\,,\quad x_{\bm{20}}=4\,,\quad x_{\bm{190}}=3\,,\quad
x_{\bm{399}}=0\,,$ (2.11)

which solves the anomaly cancellation conditions with $b=1$. The flavor
symmetry of this model is $\mathfrak{u}(4)\oplus\mathfrak{u}(3)$. We add an
additional $\mathfrak{u}(1)$ gauge factor and solve the anomaly cancellation
conditions A.7 with $b_{\mathfrak{su}(20)}=1$, allowing for potentially
distinct $\mathfrak{u}(1)$ charges for each fundamental and anti-symmetric
hypermultiplet. The spectra with non-trivial $\mathfrak{u}(1)$ and satisfying
the requisite positivity conditions are all of the form

$x_{\bm{1}_{0}}=25\,,\quad x_{\bm{20}_{10m}}=x_{\bm{20}_{-10m}}=2\,,\quad
x_{\bm{190}_{0}}=3\,,\quad x_{\bm{1}_{20m}}=-2$ (2.12)

for $m\in\mathbb{Z}$. As before, the negative number of charged singlets along
with the charge assignments indicate that the appropriate anomaly-free
subalgebra of $\hat{\mathfrak{h}}$ is the $\mathfrak{su}(2)$ fundamental
flavor symmetry algebra, so the enhancement is to
$\mathfrak{su}(20)\oplus\mathfrak{su}(2)$ with spectrum

$x_{(\bm{1},\bm{1})}=25\,,\quad x_{(\bm{20},\bm{1})}=0\,,\quad
x_{(\bm{190},\bm{1})}=3\,,\quad x_{(\bm{399},\bm{1})}=0\,,\quad
x_{(\bm{20},\bm{2})}=2\,,$ (2.13)

This again matches the enhancement found in [1].

Continuing with the cases discussed in Section 4.3 of [1], we consider the
model with gauge symmetry $\mathfrak{su}(19)$, $T=0$, and hypermultiplet
spectrum

$x_{\bm{1}}=26\,,\quad x_{\bm{19}}=5\,,\quad x_{\bm{171}}=3\,,\quad
x_{\bm{360}}=0\,,$ (2.14)

which solves the anomaly cancellation conditions with $b=1$. The flavor
symmetry of this model is $\mathfrak{u}(5)\oplus\mathfrak{u}(3)$. Adding an
additional $\mathfrak{u}(1)$ gauge factor and solving the anomaly cancellation
conditions A.7 with $b_{\mathfrak{su}(19)}=1$, we find that all spectra with
non-trivial $\mathfrak{u}(1)$ and satisfying the requisite positivity
conditions are of the form

$x_{\bm{1}_{0}}=25\,,\quad x_{\bm{19}_{9m}}=5\,,\quad x_{\bm{171}_{-1m}}=3$
(2.15)

for $m\in\mathbb{Z}$. We see in this case that there is no negative number of
charged singlets, and so this $\mathfrak{u}(1)$ is not embedded in a non-
abelian anomaly-free subalgebra of the flavor symmetry. Furthermore, both the
fundamental and anti-symmetric representations are charged under this
$\mathfrak{u}(1)$; this $\mathfrak{u}(1)$ is apparently a combination of
$\mathfrak{u}(1)$s from within both factors of the
$\mathfrak{u}(5)\oplus\mathfrak{u}(3)$ flavor symmetry. We can carry out the
process again, adding another $\mathfrak{u}(1)$ factor to verify that this
enhancement is maximal, and we find that indeed it is. This reproduces the
enhancement of [1].

Finally, we consider the model with gauge symmetry $\mathfrak{su}(18)$, $T=0$,
and hypermultiplet spectrum

$x_{\bm{1}}=30\,,\quad x_{\bm{18}}=6\,,\quad x_{\bm{153}}=3\,,\quad
x_{\bm{323}}=0\,,$ (2.16)

which solves the anomaly cancellation conditions with $b=1$. The flavor
symmetry of this model is $\mathfrak{u}(6)\oplus\mathfrak{u}(3)$. We add an
additional $\mathfrak{u}(1)$ gauge factor and solve the anomaly cancellation
conditions A.7 with $b_{\mathfrak{su}(18)}=1$, the spectra with non-trivial
$\mathfrak{u}(1)$ and satisfying the requisite positivity conditions are all
of the form

$x_{\bm{1}_{0}}=30\,,\quad x_{\bm{18}_{9m}}=x_{\bm{18}_{-9m}}=3\,,\quad
x_{\bm{153}_{0}}=3$ (2.17)

for $m\in\mathbb{Z}$. We again see that the $\mathfrak{u}(1)$ is maximal in
this case, but this time it is a embedded solely in the $\mathfrak{u}(6)$
fundamental flavor symmetry. Adding another $\mathfrak{u}(1)$ factor and
carrying out the process again, we can verify that this enhancement is indeed
maximal, and it again matches the enhancement of [1].

####  2.3.2 $\mathfrak{su}(3)$ gauge algebra with $T=0$

In the case of $\mathfrak{g}=\mathfrak{su}(3)$, $T=0$, we encounter our first
example where the shortcut approach indicates that there are potentially
multiple distinct maximal anomaly-free subalgebras of $\hat{\mathfrak{h}}$.
For $b_{\mathfrak{su}(3)}=6$, there is an enhancement of the gauge algebra by
$\mathfrak{u}(1)^{2}$, as expected from [1]. The potential issue appears when
$b_{\mathfrak{su}(3)}=7$, with $M=42\times{\bm{3}}$, which was seen in [1] to
exhibit enhancement to gauge group
$G^{\prime}=[\operatorname{SU}(3)\times\operatorname{SU}(3)]/\mathbb{Z}_{3}$
(see Section 2.4 for further discussion of the global structure of the gauge
group). If we solve the anomaly cancellation conditions for
$\mathfrak{su}(3)\oplus\mathfrak{u}(1)$ and impose the relevant constraints,
we find four types of spectra with non-trivial $\mathfrak{u}(1)$ (after
removing cases that amount to rescaling the charges):

$\displaystyle b_{\mathfrak{u}(1)}=12\colon$ $\displaystyle\qquad
x_{\bm{1}_{0}}=40\,,\quad x_{\bm{3}_{1}}=28\,,\quad x_{\bm{3}_{-2}}=14\,,\quad
x_{\bm{8}_{0}}=15\,,\quad x_{\bm{1}_{3}}=-4\,,$ (2.18) $\displaystyle
b_{\mathfrak{u}(1)}=36\colon$
$\displaystyle\quad\left\\{\begin{gathered}x_{\bm{1}_{0}}=42\,,\quad
x_{\bm{3}_{0}}=x_{\bm{3}_{3}}=x_{\bm{3}_{-3}}=14\,,\quad
x_{\bm{8}_{0}}=15\,,\\\ x_{\bm{1}_{3}}=-4\,,\quad
x_{\bm{1}_{6}}=-2\,,\end{gathered}\right.$ $\displaystyle
b_{\mathfrak{u}(1)}=54\colon$ $\displaystyle\qquad x_{\bm{1}_{0}}=54\,,\quad
x_{\bm{3}_{3}}=x_{\bm{3}_{-3}}=21\,,\quad x_{\bm{8}_{0}}=15\,,\quad
x_{\bm{1}_{3}}=-18$ $\displaystyle b_{\mathfrak{u}(1)}=54\colon$
$\displaystyle\quad\left\\{\begin{gathered}x_{\bm{1}_{0}}=45\,,\quad
x_{\bm{3}_{6}}=1\,,\quad x_{\bm{3}_{3}}=18\,,\quad x_{\bm{3}_{0}}=3\,,\quad
x_{\bm{3}_{-3}}=20\,,\\\ x_{\bm{8}_{0}}=15\,,\quad x_{\bm{1}_{3}}=-6\,,\quad
x_{\bm{1}_{6}}=-3\,.\end{gathered}\right.$

The first two spectra correspond to different $\mathfrak{u}(1)$ subalgebras of
$\mathfrak{h}=\mathfrak{su}(3)$, and match with the expected
$\mathfrak{su}(3)\oplus\mathfrak{su}(3)$ multiplicities of

$x_{(\bm{1},\bm{1})}=43\,,\quad x_{(\bm{8},\bm{1})}=15\,,\quad
x_{(\bm{3},\bm{3})}=14\,,$ (2.19)

with anomaly coefficient $b_{\mathfrak{h}}=2$. The multiplicities and
$\mathfrak{u}(1)$ charges of the flavor-$\mathfrak{su}(3)$ fundamentals in
these cases correspond to the branching rules of the bifundamental
representation for these two embeddings, while the negative numbers of charged
singlets correspond exactly to the numbers expected from the branching rules
for the adjoint representation. The third and fourth spectra in Eq. 2.18,
however, cannot correspond to an embedding of the $\mathfrak{u}(1)$ factor
into an $\mathfrak{su}(3)$, and thus potentially indicate the existence of a
separate maximal anomaly-free subgroup of the flavor symmetry group. The final
spectrum can immediately be ruled out from the distribution of charges of its
$\mathfrak{su}(3)$ fundamental representations, while the third spectrum
requires more attention. In order to determine if this spectrum actually
admits an embedding into a larger algebra, we must explicitly check the other
possible non-abelian subalgebras of the $\mathfrak{u}(42)$ flavor symmetry.
Note that the enhancement cannot be by multiple non-abelian factors, as if
both factors had non-trivial anomaly coefficients $b$, then there would be a
non-zero number of hypermultiplets charged under the bifundamental
representation for these two factors, which cannot be the case for a
subalgebra of a flavor symmetry. Carrying out this explicit check, we find
that the third spectrum in Eq. 2.18 does not correspond to an actual maximal
subalgebra, and $\mathfrak{su}(3)$ is the unique maximal subalgebra of the
flavor symmetry that can be gauged.

This issue does not arise for $b_{\mathfrak{su}(3)}=8$; in this case, only the
expected spectra corresponding to the enhanced gauge group
$G^{\prime}=[\operatorname{SU}(3)\times\operatorname{SU}(3)]/\mathbb{Z}_{3}$
found in [1] appear.

####  2.3.3 Other cases

In addition to the examples worked out here, we have also checked all of the
relevant enhancements from Sections 4.2 and 4.4777For more on these examples,
see Section 3.1. of [1], and again find agreement with the enhancements
presented there. These include cases where $T=1$, which, as mentioned in
Section 2.2, could potentially exhibit multiple allowable enhancements.
Nevertheless, we again only find a unique gauge-able subalgebra
$\mathfrak{h}\subset\hat{\mathfrak{h}}$ in all of these cases.

###  2.4 Gauge group topology from gauged 1-form symmetries

Another observation based on F-theory models, named “Massless Charge
Sufficiency Conjecture” [37, 1], is that the absence of massless
hypermultiplets transforming non-trivially under $\mathcal{Z}$—a subgroup of
the center $Z(\widetilde{G})$ of the simply-connected cover $\widetilde{G}$
for the non-abelian gauge algebra $\mathfrak{g}$—automatically “enhances” the
gauge group to $G=\widetilde{G}/\mathcal{Z}$. The description as an
enhancement becomes literal, if one interprets the non-simply connected gauge
group as the result of having gauged a $\mathcal{Z}$ 1-form global symmetry of
the $\mathfrak{g}$ gauge theory [36]. Then, cases that satisfy the Massless
Charge Sufficiency Conjecture can be explained similarly as above, namely as
quantum gravity gauging an anomaly-free global symmetry.

The anomaly in this case is a mixed anomaly between the 1-form center symmetry
and the large gauge transformations of the (anti-)self-dual tensors
originating in the Green–Schwarz term in the effective action [25], which can
be quantified as follows. Let the gauge algebra
$\mathfrak{g}=\bigoplus_{i}\mathfrak{g}_{i}$ have the simply-connected cover
$\widetilde{G}=\prod_{i}\widetilde{G}_{i}$ with center
$Z(\widetilde{G})=\prod_{i}Z(\widetilde{G}_{i})\cong\prod_{i}\mathbb{Z}_{n_{i}}$.888We
are ignoring here the more subtle case
$Z(\text{Spin}(4n))=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, see [25] for details.
Then, there is an obstruction to turning on a background field for—and,
therefore, also to gauging—a subgroup $\mathcal{Z}=\mathbb{Z}_{m}\subset
Z(\widetilde{G})$ generated by $(k_{i})\in\prod_{i}\mathbb{Z}_{n_{i}}$, if
there exists a tensor with charge vector $b$ for which

$\sum_{i}b\cdot b_{i}\,\alpha_{i}\,k_{i}^{2}\notin\mathbb{Z}\,.$ (2.20)

This is a non-trivial constraint, since the numbers $\alpha_{i}$ associated to
each gauge factor $\mathfrak{g}_{i}$ with anomaly coefficient $b_{i}$ are in
general fractional; for $\mathfrak{g}_{i}=\mathfrak{su}(n_{i})$, we have
$\alpha_{i}=\frac{n_{i}-1}{2n_{i}}$ [44] (see [45] for values for other Lie
algebras).

As an example, consider a $T=0$ theory containing a
$\mathfrak{g}=\mathfrak{su}(2)$ gauge sector with anomaly coefficient
$b_{\mathfrak{g}}=12$. This is free of (0-form) gauge symmetries with only
adjoint hypermultiplets, which are invariant under the center
$\mathcal{Z}=\mathbb{Z}_{2}$. For any other tensor with charge $b$, the
anomaly above, $b\cdot
b_{\mathfrak{g}}\,\alpha_{\mathfrak{g}}=3b\in\mathbb{Z}$, is always trivial.
Assuming that there are no other obstructions to turning on a $\mathcal{Z}$
1-form symmetry background, it would constitute a global symmetry, which must
be gauged in a gravitational theory. This gauging leads to an
$\operatorname{SO}(3)$ gauge group, which is also reflected in the geometry of
the F-theory realization [37, 1]. Another $T=0$ example is a
$\mathfrak{g}=\mathfrak{su}(24)$ gauge algebra with $b_{\mathfrak{g}}=1$,
which has 2-index anti-symmetric hypermultiplets. This preserves a
$\mathbb{Z}_{2}\subset\mathbb{Z}_{24}=Z(\widetilde{G})$, which is generated by
$k=12\mod 24\in\mathbb{Z}_{24}$. Since
$\alpha_{\mathfrak{g}}\,k^{2}=\frac{23}{48}\,12^{2}=69\in\mathbb{Z}$, this
center subgroup is also anomaly-free; again, the corresponding 1-form symmetry
is automatically gauged in F-theory [46] in the gauge group
$G=\operatorname{SU}(24)/\mathbb{Z}_{2}$.

Moreover, the logic also applies to automatic enhancements
$\mathfrak{g}^{\prime}=\mathfrak{g}\oplus\mathfrak{h}$ discussed above, where
there is a resulting non-trivial global group structure
$G^{\prime}=[\widetilde{G}\times\widetilde{H}]/\mathcal{Z}$ involving the
enhanced flavor symmetry $\widetilde{H}$. From the perspective of the original
$\mathfrak{g}$ gauge theory, this non-trivial global structure arises from a
$\mathcal{Z}$-twisted gauge bundle that is compensated by a corresponding
$\mathcal{Z}$-twist in the flavor bundle (see, e.g., [47, 48, 49, 50]). In
particular, it means that there is the option to turn on a background 1-form
$\mathcal{Z}\subset Z(\widetilde{G})$ gauge field that is “locked” to an
analogous background for the flavor symmetry. In 6d $\mathcal{N}=(1,0)$
theories, such a background can have a similar obstruction as above [25];
since $\mathfrak{h}$ is gauged through automatic enhancement, this anomaly
boils down to including the anomaly coefficient $b_{\mathfrak{h}}$ in the sum
in Eq. 2.20. The model discussed in Section 2.3.2 provides an example of this
sort. There, the enhanced gauge algebra is
$\mathfrak{g}^{\prime}=\mathfrak{su}(3)_{\mathfrak{g}}\oplus\mathfrak{su}(3)_{\mathfrak{h}}$,
with $b_{\mathfrak{g}}=7$ and $b_{\mathfrak{h}}=2$, and the hypermultiplet
spectrum contains only bifundamental matter, so a diagonal
$\mathcal{Z}=\mathbb{Z}_{3}$ generated by
$(k_{\mathfrak{g}},k_{\mathfrak{h}})=(1,1)\in\mathbb{Z}_{3}\times\mathbb{Z}_{3}=Z(\operatorname{SU}(3)\times\operatorname{SU}(3))$
center is left unbroken. Since this is a $T=0$ model, the only constraint
comes from the self-dual tensor:
$\alpha_{\mathfrak{su}(3)}(b_{\mathfrak{g}}k_{\mathfrak{g}}^{2}+b_{\mathfrak{h}}k_{\mathfrak{h}}^{2})=\frac{1}{3}(7+2)\in\mathbb{Z}$.
So there is indeed no obstruction for the $\mathcal{Z}$ 1-form symmetry, whose
gauging leads to the gauge group
$G^{\prime}=[\operatorname{SU}(3)\times\operatorname{SU}(3)]/\mathbb{Z}_{3}$.

###  2.5 Comment on the Massless Charge Sufficiency Conjecture

Lastly, let us briefly comment on the exceptions to the Massless Charge
Sufficiency Conjecture discussed in [37], that is, cases where a simple gauge
sector $\mathfrak{g}$ has no massless non-adjoint representations, yet there
is no non-trivial global gauge group, i.e., $G=\widetilde{G}$. These fall into
two categories, namely either when $\mathfrak{g}$ is non-Higgsable, or when it
is an $\mathcal{N}=(1,1)$ gauge sector (i.e., its anomaly coefficient
satisfies $b_{\mathfrak{g}}\cdot b_{\mathfrak{g}}=b_{\mathfrak{g}}\cdot a=0$),
which is coupled to an $\mathcal{N}=(1,0)$ supergravity system.

As standalone gauge theories, i.e., when decoupled from the rest, these indeed
do not have the anomaly 2.20 that would obstruct the center 1-form symmetry
from being gauged; for the non-Higgsable clusters this was discussed in [25],
and for the $\mathcal{N}=(1,1)$ sectors, it follows from the absence of a
Green–Schwarz term due to the properties of $b_{\mathfrak{g}}$. In F-theory,
it turns out that the _local_ geometries describing such gauge theories do
exhibit Mordell–Weil torsion, reflecting an unbroken 1-form center symmetry
[33, 34, 51]. For the non-Higgsable theories, whose 1-form symmetries were
first analyzed in [52], we refer again to [25] for the detailed relationship
between Mordell–Weil torsion and absence of the anomaly 2.20. For the $(1,1)$
sectors, the local base geometry is simply $T^{2}\times\mathbb{C}$, where the
torus is the curve associated to the anomaly coefficient $b_{\mathfrak{g}}$.
Since the axio-dilaton is constant along the $T^{2}$, the elliptic fibration
only changes non-trivially in the $\mathbb{C}$ direction, where there is a
(non-compact) K3-surface with one singular fiber corresponding to the gauge
algebra $\mathfrak{g}$; equivalently, this is a trivial $T^{2}$ reduction of
an 8d $\mathcal{N}=1$ F-theory model with a $\mathfrak{g}$ gauge symmetry. The
8d gauge theory has an unbroken $Z(\widetilde{G})$ 1-form center symmetry [51]
(see also [52, 53]), which therefore also survives after a trivial $T^{2}$
reduction. Geometrically, it is reflected in the Mordell–Weil torsion
preserved locally around the singular fiber in $\mathbb{C}$, which extends
trivially to Mordell–Weil torsion over $\mathbb{C}\times T^{2}$.

We therefore see that the Massless Charge Sufficiency Conjecture does—if
extended to non-gravitational theories—apply to the pure non-Higgsable
clusters and $(1,1)$ gauge sectors: in the absence of massless hypermultiplets
that break the 1-form center symmetry, the latter constitutes a global
symmetry of the gauge theory. From this perspective, it is clear that the
violation of the conjecture, at least in the models considered in [37], must
arise from coupling to gravity. Indeed, this can be easily understood from the
mixed 1-form anomaly. Namely, in these examples, there always exists at least
one other dynamical tensor with charge $b$, such that Eq. 2.20 is not an
integer. For the simplest gravitational models containing a non-Higgsable
cluster, corresponding to F-theory on Hirzebruch surfaces $\mathbb{F}_{n}$,
the tensor dual to the fibers of the ruling on $\mathbb{F}_{n}$ induces the
anomaly. Likewise, the simplest models with $(1,1)$ sectors have $T=9$,
corresponding to F-theory on a rational elliptic dP9 surface. While the
$(1,1)$ sectors live on the generic elliptic fibers of this surface, they are
intersected by the sections of the fibration, which give rise to independent
tensors whose large gauge transformations have a non-trivial anomaly 2.20. One
may wonder how these exceptions to the Massless Charge Sufficiency Conjecture
satisfy the Completeness Hypothesis. Clearly, since the gauge group is
$\widetilde{G}$ rather than $\widetilde{G}/\mathcal{Z}$, there must be massive
states transforming non-trivially under $\mathcal{Z}$. As elaborated on in
[25], these states are precisely provided by the excitations of the BPS
strings that are dual to the tensor multiplets with a non-trivial anomaly
2.20.

We have not checked other examples apart from these “simplest” gravitational
models that violate the Massless Charge Sufficiency Conjecture. However, it is
suggestive that such models must always contain tensor multiplets that give
rise to the anomaly 2.20, by virtue of the necessary unimodularity of the
tensor charge lattice. It would be interesting to attempt such a proof, at
least geometrically for models constructable in F-theory.

##  3 Automatic enhancement enforced by BPS strings

We have seen that a large class of automatic enhancements is consistent with
the gauging of an anomaly-free subalgebra $\mathfrak{h}$ of the flavor
symmetry of a $\mathfrak{g}$ gauge theory. However, the absence of anomalies
is a necessary, but not sufficient condition for the appearance of the
additional gauge sector $\mathfrak{h}$. That is, these arguments do not
explain why the absence of additional gauge sectors would be _inconsistent_.
Moreover, there are cases with hypermultiplets charged just under
$\mathfrak{h}$, although not enough to allow for a supersymmetric Higgsing;
these cases do not allow for an interpretation of $\mathfrak{h}$ as a flavor
symmetry.

In this section, we analyze two classes of $T=1$ models without non-Higgsable
clusters that exhibit automatic enhancements of this sort. In these models, we
can argue physically why certain gauge algebras $\mathfrak{g}$ with anomaly
coefficient $b_{\mathfrak{g}}$ satisfying $b_{\mathfrak{g}}\cdot b_{1}=1$ are
inconsistent, where $b_{1}$ is the charge of the (unpaired) anti-self-dual
tensor that necessarily has $b_{1}\cdot b_{1}=-1$ or $-2$. The key ingredient
is—by the Completeness Hypothesis—the existence of a dynamical string charged
under $b_{1}$. Under some mild assumptions, these strings can be identified
with the E-string and the M-string (the string of an $\mathcal{N}=(2,0)$
$A_{1}$ SCFT) for $b_{1}\cdot b_{1}=-1$ and $b_{1}\cdot b_{1}=-2$,
respectively. Then, the inconsistency of models that are observed to
automatically enhance is an incompatibility of the gauge symmetry
$\mathfrak{g}$ with the flavor symmetry of these SCFT strings.

###  3.1 Automatic enhancement enforced by the E-string

Let us first consider $T=1$ models with tensor pairing

$\Omega=\begin{pmatrix}0&1\\\ 1&-1\end{pmatrix}\,,$ (3.1)

where the basis is chosen such that the positivity cone is spanned by $(1,0)$
and $(0,1)$. The gravitational anomaly coefficient is $a=(-3,-2)$. In
F-theory, such models are realized with the Hirzebruch surface
$\mathbb{F}_{1}$ as the base manifold.

With such a tensor pairing, any 6d supergravity theory with gauge algebra
$\mathfrak{g}=\mathfrak{su}(N>3)$ and anomaly coefficient
$b_{\mathfrak{g}}=(2,1)$ (with $b_{\mathfrak{g}}\cdot b_{\mathfrak{g}}=3$ and
$b_{\mathfrak{g}}\cdot a=-5$) is anomaly-free with hypermultiplet spectrum
$M=(40-3N)\times\bm{\mathrm{F}}\oplus 5\times\bm{\mathrm{AS}}$, where
$\bm{\mathrm{F}}$ (resp. $\bm{\mathrm{AS}}$) denotes the fundamental (resp.
anti-symmetric) representation. However, F-theory models with $N>8$
automatically enhance to $\mathfrak{g}^{\prime}\supset\mathfrak{su}(N)$ [1],
indicating the inconsistency of a bare $\mathfrak{su}(N)$ gauge algebra in
these cases. Note that the enhancement patterns
$\mathfrak{g}^{\prime}=\overline{\mathfrak{g}}\oplus\mathfrak{h}$ (where
$\overline{\mathfrak{g}}$ has anomaly coefficient
$b_{\overline{\mathfrak{g}}}=b_{\mathfrak{g}}$) do not fit into the framework
discussed previously, in that the additional gauge symmetry that appears in
the F-theory constructions are not flavor symmetries of $(\mathfrak{g},M)$
(they have additional charged matter999 For the $N=9$ case, the hypermultiplet
spectrum of the enhanced model is
$1\times(\bm{1},\bm{2})+5\times(\bm{36},\bm{1})+11\times(\bm{9},\bm{1})+1\times(\bm{9},\bm{2})$.
For the $N=10$ case, it is
$5\times(\bm{45},\bm{1},\bm{1})+4\times(\bm{10},\bm{1},\bm{2})+1\times(\bm{10},\bm{2},\bm{1})$.
For the $N=11,12$ cases, it is
$5\times(\bm{66},\bm{1})+1\times(\bm{12},\bm{4})$. In the $N=10$ case, the
anomaly coefficient of the additional $\mathfrak{su}(2)$ is $(2,2)$. , which
however does not allow for a supersymmetric Higgsing by itself):

${\begin{array}[]{*{3}{c}}\hline\cr\hline\cr
N&\overline{\mathfrak{g}}&\mathfrak{h}=\bigoplus_{I}\mathfrak{h}_{I}\\\
\hline\cr 9&\mathfrak{su}(9)&\mathfrak{su}(2)\\\
10&\mathfrak{su}(10)&\mathfrak{su}(2)\oplus\mathfrak{su}(2)\\\
11&\mathfrak{su}(12)&\mathfrak{sp}(2)\\\
12&\mathfrak{su}(12)&\mathfrak{sp}(2)\\\ \hline\cr\hline\cr\end{array}}$ (3.2)

The common feature of all these enhanced algebras is that the additional gauge
factor $\mathfrak{h}_{1}$ has anomaly coefficient $b_{1}=(0,1)$. This
corresponds to an anti-self-dual tensor with $b_{1}\cdot b_{1}=-1$. The
validity of the Automatic Enhancement Conjecture would therefore imply that,
for $N>8$, an unpaired tensor $b_{1}$, i.e., no gauge algebra with anomaly-
coefficient $b_{1}$, is inconsistent.

To give a physical argument to this, observe that, by the Completeness
Hypothesis, there must be a dual string charged under the tensor with charge
$b_{1}$. If $b_{1}$ is unpaired, there are compelling arguments [54] based on
anomaly considerations on the worldsheet of such a string that there must be
an $E_{8}$ flavor symmetry in 6d; since it agrees with that of the E-string,
it is generally believed that any such string must be an E-string. When
coupled to gravity, the worldsheet $\operatorname{E}_{8}$ flavor symmetry is
generically broken in the bulk. A subalgebra
$\mathfrak{g}\subset\mathfrak{e}_{8}$ may be gauged if one can assign an
anomaly coefficient $b_{\mathfrak{g}}$, with $b_{\mathfrak{g}}\cdot b_{1}=1$,
to $\mathfrak{g}$, provided all gauge and gravitational anomalies are
cancelled (which may require hypermultiplets charged under $\mathfrak{g}$).
However, it is clear that any such gauging can not “exceed”
$\operatorname{E}_{8}$, as it would be otherwise inconsistent with the
worldsheet dynamics of the string.

This “$\operatorname{E}_{8}$-rule” [55, 56, 57, 58] immediately explains why
$\mathfrak{g}=\mathfrak{su}(N)$ with $N>9$ are inconsistent, as these
$\mathfrak{su}$ algebras are not contained in $\mathfrak{e}_{8}$. By forcing
such a gauge algebra with anomaly coefficient $b_{\mathfrak{g}}=(2,1)$ in an
F-theory construction [1], the observed automatic enhancement can be
interpreted as the geometry attempting to rectify this inconsistency by
pairing the string with charge $b_{1}=(0,1)$ with a gauge symmetry
$\mathfrak{h}$. In this case, the string is no longer an E-string, but rather
an instanton string of $\mathfrak{h}$, which is not subject to the
“$\operatorname{E}_{8}$-rule”. Note that this does not explain why, for
$N=11$, the $\mathfrak{su}(11)$ has to enhance to
$\overline{\mathfrak{g}}=\mathfrak{su}(12)$, but it does rule out a bare
$\mathfrak{su}(11)$.

At first sight, the “$\operatorname{E}_{8}$-rule” does not seem to explain the
inconsistency of the model with $N=9$, since
$\mathfrak{su}(9)\subset\mathfrak{e}_{8}$. However, it is important to realize
that, at the group level, $\operatorname{E}_{8}$ does not have an
$\operatorname{SU}(9)$-subgroup, but rather
$\operatorname{SU}(9)/\mathbb{Z}_{3}$. To retain this non-trivial global
structure, it means that any gauging of an
$\mathfrak{su}(9)\subset\mathfrak{e}_{8}$ subalgebra must have an unbroken
$\mathbb{Z}_{3}\subset\mathbb{Z}_{9}=Z(\operatorname{SU}(9))$ 1-form symmetry.
In particular, this requirement demands the absence of any fundamental and
2-index anti-symmetric hypermultiplets. Within the 6d SCFT landscape, the only
possible gauging of this kind is $\mathfrak{g}=\mathfrak{su}(9)$ with anomaly
coefficient $b_{\mathfrak{g}}$ satisfying $b_{\mathfrak{g}}\cdot
b_{\mathfrak{g}}=-2$ and $b_{\mathfrak{g}}\cdot b_{1}=1$ (where $b_{1}$ is the
tensor dual to the E-string), which has no massless hypermultiplets. A
supergravity model where an $\mathfrak{su}(9)\subset\mathfrak{e}_{8}$
subalgebra of the E-string flavor symmetry is gauged can be constructed in
F-theory on a dP9 surface, i.e., a model with $T=9$, see Appendix B; in this
geometric construction, the $\mathbb{Z}_{3}$ 1-form symmetry is manifestly
gauged by the presence of Mordell–Weil torsion.101010In fact, while in both
the SCFT and the supergravity model, one would naively expect the full
$\mathbb{Z}_{9}$ center to be preserved, the excitations of the E-string break
it to an $\mathbb{Z}_{3}$ [59], which can also be seen from the 1-form
anomalies [25]. This is another exception to the Massless Charge Sufficiency
Conjecture, as the $\mathfrak{g}=\mathfrak{su}(9)$ by itself forms an
$\mathcal{N}=(1,1)$ gauge sector, see Section 2.5.

Returning to the $T=1$ supergravity models with tensor pairing 3.1 and gauge
symmetry $\mathfrak{g}=\mathfrak{su}(N)$ with anomaly coefficient
$b_{\mathfrak{g}}=(2,1)$, we now see that the inconsistency of $N=9$ model
stems from the presence of the fundamental and anti-symmetric hypermultiplets,
which are needed to cancel the gauge anomalies and which explicitly break the
$\mathbb{Z}_{3}$ center symmetry.

###  3.2 Automatic enhancement enforced by the M-string

The simplest supergravity models that contain an M-string SCFT sector have a
tensor pairing

$\Omega=\begin{pmatrix}0&1\\\ 1&-2\end{pmatrix}\,,$ (3.3)

where we have chosen the basis in which the positivity cone is spanned by
$(1,0)$ and $(0,1)$, with the latter being the anti-self-dual tensor $b_{1}$
that has self-pairing $-2$. The gravitational anomaly coefficient is
$a=(-4,-2)$. In F-theory, such models are realized with the Hirzebruch surface
$\mathbb{F}_{2}$ as the base manifold.

We can easily verify that, just based on gauge and gravitational anomalies, a
$\mathfrak{g}=\mathfrak{su}(N)$ model with $M=(48-4N)\times{\bf F}\oplus
6\times{\bf AS}$ and anomaly coefficient $b_{\mathfrak{g}}=(3,1)$ (satisfying
$b_{\mathfrak{g}}\cdot b_{\mathfrak{g}}=4,b\cdot a=-6,b_{\mathfrak{g}}\cdot
b_{1}=1$) would be consistent for $4\leq N\leq 9$; for $N=2,3$, the anomaly-
free hypermultiplet spectrum would be $40\times{\bf 2}$ and $42\times{\bf 3}$,
respectively. However, if we attempt to engineer such models in F-theory, we
see that there is automatic enhancement
$\mathfrak{g}^{\prime}=\overline{\mathfrak{g}}\oplus\mathfrak{h}$ for $N\geq
3$, with an additional gauge factor $\mathfrak{h}_{1}\subset\mathfrak{h}$
having anomaly coefficient $b_{1}$; see Appendix C for details.

Again, a physical argument for the Automatic Enhancement Conjecture in this
instance—i.e., why an unpaired tensor $b_{1}$ would be inconsistent for
$N>2$—comes from the flavor symmetry of the string with charge $b_{1}$, which
exists by the Completeness Hypothesis. By a similar argument as for the
E-string, one can show that the spacetime flavor symmetry is limited via
worldsheet anomaly arguments to be $\mathfrak{su}(2)$, which agrees with the
flavor symmetry of what is known in the literature as the M-string
[60].111111The name “M-string” originates from the M-theory realization as
M2-branes ending on M5-branes, which have 6d $\mathcal{N}=(2,0)$ worldvolume
dynamics. Alternatively, such a string arises as the charged object of a 6d
$\mathcal{N}=(2,0)$ tensor multiplet (which also furnishes the tensor branch
description of a $(2,0)$ $A_{1}$ SCFT). When coupled to an $\mathcal{N}=(1,0)$
background, an $\mathfrak{su}(2)$ subalgebra of the $(2,0)$ R-symmetry may be
interpreted as the flavor symmetry of the SCFT.121212The $(2,0)$ R-symmetry is
$\mathfrak{sp}(2)\cong\mathfrak{so}(5)$, which in particular acts on the
fermions in the $(2,0)$ tensor multiplet in the spinor representation $\bm{4}$
of $\mathfrak{so}(5)$. Under the branching
$\mathfrak{sp}(2)\supset\mathfrak{su}(2)\oplus\mathfrak{su}(2)$, where one of
the factors is identified with the flavor symmetry in the $\mathcal{N}=(1,0)$
background, the decomposition
$\bm{4}\rightarrow(\bm{2},\bm{1})\oplus(\bm{1},\bm{2})$ yields fundamental
degrees of freedom, thus indicating that the flavor group is
$\operatorname{SU}(2)$. This is confirmed in F-theory models, see Appendix C.
Therefore, any model with an $\mathfrak{su}(N>2)$ symmetry on
$b_{\mathfrak{g}}$ with $b_{\mathfrak{g}}\cdot b_{1}=1$ can only give a
globally consistent supergravity model if the tensor $b_{1}$ is paired with a
non-trivial spacetime gauge symmetry itself.

###  3.3 Discussion

It would be desirable to explain all instances of automatic enhancement using
the consistency of BPS strings, as these provide, at least in the examples
above, a definitive of proof why the lack of enhancement would be
inconsistent. In the case where these are strings of anti-self-dual tensors
$b_{1}$, the distinction between consistent and inconsistent gauge symmetries
$\mathfrak{g}$ with anomaly coefficient $b_{\mathfrak{g}}$ satisfying
$b_{\mathfrak{g}}\cdot b_{1}=1$ is analogous to the question of whether a
$\mathfrak{g}$ flavor symmetry of a 6d SCFT containing the tensor $b_{1}$ can
be gauged (see [61] and references therein). For supergravity models, the
constraints are relaxed compared to SCFTs in the sense that the anomaly
coefficient $b_{\mathfrak{g}}$ of $\mathfrak{g}$ can be a self-dual tensor,
and $\mathfrak{g}$ is allowed to have adjoint hypermultiplets (i.e., the genus
of $b_{\mathfrak{g}}$ can be non-zero). As we have seen above in simple
models, this still poses strong restrictions. Moreover, even though we have
not discussed them in detail, it is rather obvious that, in the presence of
multiple anti-self-dual tensors $b_{A}$, there can be independent constraints
on $\mathfrak{g}$ with anomaly-coefficient $b_{\mathfrak{g}}$ from each
$b_{A}$ with $b_{A}\cdot b_{\mathfrak{g}}\neq 0$. By forcing the existence of
the gauge algebra $\mathfrak{g}$ in the corresponding F-theory model, one
might then find a large, automatically enhanced algebra
$\mathfrak{g}^{\prime}=\overline{\mathfrak{g}}\oplus\bigoplus_{A}\mathfrak{h}_{A}\oplus\dotsb$.
Examples of this type appear, e.g., in F-theory models on blow-ups of
Hirzebruch surfaces, see [1].

The main challenge to extend this method to other cases of automatic
enhancement, such as those discussed in Section 2, is that the strings in
question are not SCFT strings, i.e., they are associated to tensors with
$b_{1}\cdot b_{1}\geq 0$. In the case $b_{1}\cdot b_{1}=0$ and $b_{1}\cdot
a=-2$ (with $a$ the gravitational anomaly coefficient), the corresponding
string may be identified with the heterotic string, which has either
$\operatorname{E}_{8}\times\operatorname{E}_{8}$ or
$\operatorname{Spin}(32)/\mathbb{Z}_{2}$ flavor symmetry. The latter contains
an $\operatorname{SU}(N)$ subgroup for $N\leq 15$, yet one observes automatic
enhancements for $12\leq N\leq 15$ in $\mathfrak{su}(N)$ gauge symmetries with
$b_{\mathfrak{su}}$ such that $b_{\mathfrak{su}}\cdot b_{1}=1$ [1]. Moreover,
to the best of our knowledge, the maximal flavor symmetry of strings with
$b_{1}\cdot b_{1}=1$ and $b_{1}\cdot a=-2$, present in all $T=0$ supergravity
models, has not been studied.

In $T=0$ models, one observes an additional type of enhancement pattern,
$\mathfrak{g}\to\mathfrak{g}^{\prime}\supset\mathfrak{g}$ with a simple
algebra $\mathfrak{g}^{\prime}$, discussed in [1], which does not fall into
the classes discussed above. Another instance of this type is the $N=11$ case
in Eq. 3.2, where $\mathfrak{g}=\mathfrak{su}(11)$ enhances to
$\overline{\mathfrak{g}}=\mathfrak{su}(12)$, in addition to the appearance of
$\mathfrak{h}_{1}=\mathfrak{sp}(2)$. Presumably, this particular enhancement
is related to the supergravity string present in this model, rather than the
E-string. Hence, a better understanding of the flavor symmetry of strings with
$b_{1}\cdot b_{1}\geq 0$ will be necessary to scrutinize the physical
explanation of the Automatic Enhancement Conjecture for these and more general
cases.

## Acknowledgments

We thank Craig Lawrie for many valuable discussions and comments on the draft.
The work of MC and APT is supported in part by the DOE (HEP) Award DE-
SC0013528. MC further acknowledges support by the Simons Foundation
Collaboration grant #724069 on “Special Holonomy in Geometry, Analysis and
Physics”, the Fay R. and Eugene L. Langberg Endowed Chair, and the Slovenian
Research Agency (ARRS No. P1-0306).

##  Appendix A Anomaly cancellation conditions

In this appendix, we collect the explicit anomaly cancellation conditions for
a 6d $\mathcal{N}=(1,0)$ supergravity theory with tensor lattice $\Gamma$,
pairing $\Omega(b_{1},b_{2})\equiv b_{1}\cdot b_{2}$ for
$b_{1},b_{2}\in\Gamma$, gauge algebra
$\mathfrak{g}=\bigoplus_{i}\mathfrak{g}_{i}\oplus\bigoplus_{\alpha}\mathfrak{u}(1)_{\alpha}$,
and matter 2.2 [62, 28, 29, 63]. The Green–Schwarz contributions cancel the
1-loop contributions from the various multiplets, as encoded in their
respective anomaly polynomials [64]:

$\displaystyle\begin{split}I^{\text{grav}}_{8}&=-\frac{273}{5760}\left(\operatorname{tr}R^{4}\frac{5}{4}+(\operatorname{tr}R^{2})^{2}\right)+\frac{9}{128}(\operatorname{tr}R^{2})^{2}\,,\\\
I^{\text{tensor}}_{8}&=\frac{29}{5760}\left(\operatorname{tr}R^{4}+\frac{5}{4}(\operatorname{tr}R^{2})^{2}\right)-\frac{1}{128}(\operatorname{tr}R^{2})^{2}\,,\\\
I_{8}^{\text{vector}}(\mathfrak{g}_{i})&=-\frac{\dim(\mathfrak{g}_{i})}{5760}\left(\operatorname{tr}R^{4}+\frac{5}{4}(\operatorname{tr}R^{2})^{2}\right)-\frac{1}{24}\operatorname{tr}_{\bf
adj}F_{i}^{4}+\frac{1}{96}\operatorname{tr}_{\bf
adj}F_{i}^{2}\operatorname{tr}R^{2}\,,\\\
I_{8}^{\text{vector}}(\mathfrak{u}(1)_{\alpha})&=-\frac{1}{5760}\left(\operatorname{tr}R^{4}+\frac{5}{4}(\operatorname{tr}R^{2})^{2}\right)\,,\end{split}$
(A.1) $\displaystyle I^{\text{hyper}}_{8}(\bm{\mathrm{R}})$
$\displaystyle=\frac{\dim(\bm{\mathrm{R}})}{5760}\left(\operatorname{tr}R^{4}+\frac{5}{4}(\operatorname{tr}R^{2})^{2}\right)+\frac{1}{24}\operatorname{tr}_{\bm{\mathrm{R}}}F^{4}-\frac{1}{96}\operatorname{tr}_{\bm{\mathrm{R}}}F^{2}\operatorname{tr}R^{2}\,,$
(A.2)

where $\operatorname{tr}_{\bm{\mathrm{R}}}$ denotes the trace in the
representation $\bm{\mathrm{R}}$. In the hypermultiplet contribution, the
traces over the combined field strength $F$ of the full gauge symmetry
$\mathfrak{g}$ can be disentangled into products of field strengths of the
individual (non-)abelian factors, by applying recursively the formulae

$\displaystyle\operatorname{tr}_{\bm{\mathrm{R}}}(F_{\mathfrak{f}}^{2})$
$\displaystyle=\dim(\bm{\mathrm{R}}_{1})\operatorname{tr}_{\bm{\mathrm{R}}_{2}}(F_{2}^{2})+\dim(\bm{\mathrm{R}}_{2})\operatorname{tr}_{\bm{\mathrm{R}}_{1}}(F_{1}^{2})\,,$
(A.3) $\displaystyle\operatorname{tr}_{\bm{\mathrm{R}}}(F_{\mathfrak{f}}^{4})$
$\displaystyle=\dim(\bm{\mathrm{R}}_{1})\operatorname{tr}_{\bm{\mathrm{R}}_{2}}(F_{2}^{4})+6\operatorname{tr}_{\bm{\mathrm{R}}_{1}}(F_{1}^{2})\operatorname{tr}_{\bm{\mathrm{R}}_{2}}(F_{2}^{2})+\dim(\bm{\mathrm{R}}_{2})\operatorname{tr}_{\bm{\mathrm{R}}_{1}}(F_{1}^{4})\,,$
(A.4)

for $\mathfrak{f}=\mathfrak{f}_{1}\oplus\mathfrak{f}_{2}$ with
$\mathfrak{f}_{i}$ simple non-abelian and
$\bm{\mathrm{R}}=(\bm{\mathrm{R}}_{1},\bm{\mathrm{R}}_{2})$, and

$\displaystyle\operatorname{tr}_{\bm{\mathrm{R}}}(F_{\mathfrak{f}}^{2})$
$\displaystyle=\operatorname{tr}_{\bm{\mathrm{R}}_{\text{n}}}(F_{\text{n}}^{2})+q_{\alpha}q_{\beta}F_{\alpha}F_{\beta}\,,$
(A.5) $\displaystyle\operatorname{tr}_{\bm{\mathrm{R}}}(F_{\mathfrak{f}}^{4})$
$\displaystyle=\operatorname{tr}_{\bm{\mathrm{R}}_{\text{n}}}(F_{\text{n}}^{4})+4\operatorname{tr}_{\bm{\mathrm{R}}_{\text{n}}}(F_{\text{n}}^{3})q_{\alpha}F_{\alpha}+6\operatorname{tr}_{\bm{\mathrm{R}}_{\text{n}}}(F_{\text{n}}^{2})q_{\alpha}q_{\beta}F_{\alpha}F_{\beta}+q_{\alpha}q_{\beta}q_{\gamma}q_{\delta}F_{\alpha}F_{\beta}F_{\gamma}F_{\delta}\,,$
(A.6)

for
$\mathfrak{f}=\mathfrak{f}_{\text{n}}\oplus\bigoplus_{\alpha}\mathfrak{u}(1)_{\alpha}$
with $\mathfrak{f}_{\text{n}}$ and
$\bm{\mathrm{R}}=(\bm{\mathrm{R}}_{\text{n}})_{(q_{\alpha})}$. Here, summation
over repeated Greek indices is implied. Then, the vanishing of the anomaly
polynomial $I_{8}^{\text{GS}}+I_{8}^{\text{1-loop}}$ is ensured term-wise if:

$\displaystyle\operatorname{tr}(R^{4})\colon$ $\displaystyle 0$
$\displaystyle=H-V+29T-273\,,$ (A.7a)
$\displaystyle\operatorname{tr}(R^{2})^{2}\colon$ $\displaystyle a\cdot a$
$\displaystyle=9-T\,,$ (A.7b)
$\displaystyle\operatorname{tr}(F_{i}^{2})\operatorname{tr}(R^{2})\colon$
$\displaystyle a\cdot b_{i}$
$\displaystyle=-\frac{1}{6}\lambda_{i}\left(\sum_{r}d_{i}^{(r)}A_{\bm{\mathrm{R}}_{\mathfrak{g}_{i}}^{(r)}}-A_{\mathfrak{g}_{i}}\right)\,,$
(A.7c) $\displaystyle\operatorname{tr}(F_{i}^{4})\colon$ $\displaystyle 0$
$\displaystyle=\sum_{r}d_{i}^{(r)}B_{\bm{\mathrm{R}}_{\mathfrak{g}_{i}}^{(r)}}-B_{\mathfrak{g}_{i}}\,,$
(A.7d) $\displaystyle\operatorname{tr}(F_{i}^{2})^{2}\colon$ $\displaystyle
b_{i}\cdot b_{i}$
$\displaystyle=\frac{1}{3}\lambda_{i}^{2}\left(\sum_{r}d_{i}^{(r)}C_{\bm{\mathrm{R}}_{\mathfrak{g}_{i}}^{(r)}}-C_{\mathfrak{g}_{i}}\right)\,,$
(A.7e)
$\displaystyle\operatorname{tr}(F_{i}^{2})\operatorname{tr}(F_{j}^{2})\colon$
$\displaystyle b_{i}\cdot b_{j}$
$\displaystyle=\lambda_{i}\lambda_{j}\sum_{r}d_{i,j}^{(r)}A_{\bm{\mathrm{R}}_{\mathfrak{g}_{i}}^{(r)}}A_{\bm{\mathrm{R}}_{\mathfrak{g}_{j}}^{(r)}}\,,\quad
i\neq j\,,$ (A.7f)
$\displaystyle\operatorname{tr}(R^{2})F_{\alpha}F_{\beta}\colon$
$\displaystyle a\cdot b_{\alpha\beta}$
$\displaystyle=-\frac{1}{6}\sum_{r}d^{(r)}q_{\alpha}^{(r)}q_{\beta}^{(r)}\,,$
(A.7g) $\displaystyle\operatorname{tr}(F_{i}^{3})F_{\alpha}\colon$
$\displaystyle 0$
$\displaystyle=\sum_{r}d_{i}^{(r)}E_{\bm{\mathrm{R}}_{\mathfrak{g}_{i}}^{(r)}}q_{\alpha}^{(r)}\,,$
(A.7h) $\displaystyle\operatorname{tr}(F_{i}^{2})F_{\alpha}F_{\beta}\colon$
$\displaystyle b_{i}\cdot b_{\alpha\beta}$
$\displaystyle=\lambda_{i}\sum_{r}d_{i}^{(r)}A_{\bm{\mathrm{R}}_{i}^{(r)}}q_{\alpha}^{(r)}q_{\beta}^{(r)}\,,$
(A.7i) $\displaystyle F_{\alpha}F_{\beta}F_{\gamma}F_{\delta}\colon$
$\displaystyle\quad b_{\alpha\beta}\cdot
b_{\gamma\delta}+b_{\alpha\gamma}\cdot b_{\beta\delta}+b_{\alpha\delta}\cdot
b_{\beta\gamma}$
$\displaystyle=\sum_{r}d^{(r)}q_{\alpha}^{(r)}q_{\beta}^{(r)}q_{\gamma}^{(r)}q_{\delta}^{(r)}\,,$
(A.7j)

where $d^{(r)}=x_{r}\prod_{i}\dim(\bm{\mathrm{R}}_{\mathfrak{g}_{j}}^{(r)})$,
$d_{i}^{(r)}=x_{r}\prod_{j\neq
i}\dim(\bm{\mathrm{R}}_{\mathfrak{g}_{j}}^{(r)})$, and
$d_{i,j}^{(r)}=x_{r}\prod_{k\neq
i,j}\dim(\bm{\mathrm{R}}_{\mathfrak{g}_{k}}^{(r)})$. Here, $H$, $V$, and $T$
are, respectively, the numbers of hypermultiplets, vector multiplets, and
tensor multiplets in the theory. The quantities
$A_{\bm{\mathrm{R}}},B_{\bm{\mathrm{R}}},C_{\bm{\mathrm{R}}},E_{\bm{\mathrm{R}}}$
are group theoretic coefficients defined by

$\operatorname{tr}_{\bm{\mathrm{R}}}F^{2}=A_{\bm{\mathrm{R}}}\operatorname{tr}F^{2}\,,\quad\operatorname{tr}_{\bm{\mathrm{R}}}F^{4}=B_{\bm{\mathrm{R}}}\operatorname{tr}F^{4}+C_{\bm{\mathrm{R}}}\left(\operatorname{tr}F^{2}\right)^{2}\,,\quad\operatorname{tr}_{\bm{\mathrm{R}}}F^{3}=E_{\bm{\mathrm{R}}}\operatorname{tr}F^{3}\,,$
(A.8)

with $\operatorname{tr}_{\bm{\mathrm{R}}}$ the trace in representation
$\bm{\mathrm{R}}$ and $\operatorname{tr}$ the trace in the fundamental (or
defining) representation. We use
$A_{\mathfrak{g}_{i}},B_{\mathfrak{g}_{i}},C_{\mathfrak{g}_{i}},E_{\mathfrak{g}_{i}}$
to denote these coefficients for the adjoint representation of
$\mathfrak{g}_{i}$. The values of the coefficients $\lambda_{i}$ are as
follows:

$\begin{array}[]{*{10}{c}}\hline\cr\hline\cr\mathfrak{g}&\mathfrak{su}(N)&\mathfrak{so}(N)&\mathfrak{sp}(N)&\mathfrak{e}_{6}&\mathfrak{e}_{7}&\mathfrak{e}_{8}&\mathfrak{f}_{4}&\mathfrak{g}_{2}\\\
\hline\cr\lambda&1&2&1&6&12&60&6&2\\\ \hline\cr\hline\cr\end{array}$ (A.9)

##  Appendix B E-string with gauged $\operatorname{SU}(9)/\mathbb{Z}_{3}$
flavor symmetry in F-theory

In this appendix, we give a concrete F-theory example where an
$\operatorname{SU}(9)/\mathbb{Z}_{3}$ gauge group is located next to a
$(-1)$-curve. In particular, this provides a simple example of a “non-Tate”
tuning of the $\mathfrak{su}(9)$ subalgebra of the $\mathfrak{e}_{8}$ flavor
symmetry of an E-string, which was not possible with Tate’s algorithm [58].

Such an F-theory geometry is an elliptic fibration over a
$B=\operatorname{dP}_{9}$ surface, given by the Weierstrass model
$y^{2}=x^{3}+fx+g$ with

$f=\frac{1}{2}a_{1}b_{1}^{3}-\frac{1}{48}a_{1}^{4}\,,\quad
g=\frac{1}{4}b_{1}^{6}+\frac{1}{864}a_{1}^{6}-\frac{1}{24}a_{1}^{3}b_{1}^{3}\,,$
(B.1)

which is just the generic Weierstrass model with $\mathbb{Z}_{3}$ Mordell–Weil
torsion [33], specialized to $a_{3}=b_{1}^{3}$. From the discriminant,
$\Delta=4f^{3}+27g^{2}=\frac{1}{16}b_{1}^{9}(3b_{1}-a_{1})(a_{1}^{2}+3a_{1}b_{1}+9b_{1}^{2})$,
we see that there is an $I^{\text{s}}_{9}$ fiber over $\\{b_{1}=0\\}$
(splitness follows from $\left.9g/2f\right|_{\\{b_{1}=0\\}}=-a_{1}^{2}/4$
being a perfect square), which corresponds to an $\mathfrak{su}(9)$ gauge
algebra.

Note that the divisor classes $[\\{a_{1}=0\\}]=[\\{b_{1}=0\\}]$ are just the
anti-canonical class $-K$ of $B=\operatorname{dP}_{9}$, which coincides with
the fiber class of the rational elliptic surface
$\operatorname{dP}_{9}\rightarrow\mathbb{P}^{1}$. Then, $\\{a_{1}=0\\}$ and
$\\{b_{1}=0\\}$, for generic choices of sections of $\mathcal{O}_{B}(-K)$,
correspond to two distinct, smooth, and irreducible elliptic fibers of this
fibration. This means that, apart from the $\mathfrak{su}(9)$ on
$\\{b_{1}=0\\}$, the F-theory model described by the Weierstrass model above
has no other non-abelian gauge factor. Moreover, since generic fibers do not
intersect, there are no codimension-two singularity enhancements, i.e., no
massless matter other than the single adjoint hypermultiplet, associated to
the genus of the elliptic curve $\\{b_{1}=0\\}$ on which the
$\mathfrak{su}(9)$ lives.

So far, we have only made manifest the $\operatorname{SU}(9)/\mathbb{Z}_{3}$
gauge group on $\\{b_{1}=0\\}$, with the $\mathbb{Z}_{3}$ quotient coming from
the Mordell–Weil group. To see that this gauges part of the
$\operatorname{E}_{8}$ flavor symmetry of an E-string, recall that a
$\operatorname{dP}_{9}$ surface also has sections of the elliptic fibration
$B\rightarrow\mathbb{P}^{1}$, which themselves are genus 0 curves with self-
intersection $(-1)$, i.e., give rise to E-strings via wrapped D3-branes. Since
a section $S$ intersects any smooth fiber exactly once, in particular also
$\\{b_{1}=0\\}$, we see that the $\operatorname{SU}(9)/\mathbb{Z}_{3}$ indeed
can be identified with the gauged flavor symmetry of the E-string on $S$. Note
that the $\operatorname{dP}_{9}$ surface actually has 9 independent (in
homology) sections, so that the $\operatorname{SU}(9)/\mathbb{Z}_{3}$ on
$\\{b_{1}=0\\}$ is in fact the “diagonal” flavor symmetry of all 9
(independent) E-string sectors.

##  Appendix C Automatic enhancement for F-theory on $\mathbb{F}_{2}$

In this appendix, we describe the F-theory models on a $B=\mathbb{F}_{2}$ base
that undergo automatic enhancement. To begin with, we recall that a Hirzebruch
surface is a $\mathbb{P}^{1}$-bundle over a base $\mathbb{P}^{1}$. We
introduce homogeneous coordinates $[s:t]$ for the fiber $\mathbb{P}^{1}$, and
$[u:v]$ for the base $\mathbb{P}^{1}$, and denote by $[x]$ the divisor class
of the codimension-one variety $\\{x=0\\}$. Then, the second homology lattice
of $\mathbb{F}_{2}$, which is identified with the tensor lattice, is generated
by $[s]$ and $[v]$, with linear relations $[s]=[t]$, $[u]=[v]+2[s]$, and
intersection numbers $[s]^{2}=0,[s]\cdot[v]=1,[v]^{2}=-2$. The canonical class
$K$, which is identified with the gravitational anomaly coefficient $a$, is
$K=-(4[s]+2[v])$.

The geometric origin of automatic enhancement can be attributed to generic
representatives of certain homology classes becoming reducible [1]. Such a
representative can be expressed in terms of the vanishing locus $\\{p=0\\}$ of
a homogeneous polynomial of the appropriate multi-degree, which becomes
reducible if any such polynomial factorizes. For a representative of an
effective divisor class $n[s]+m[v]$, with $n,m\geq 0$, the homogeneous
polynomials are

$p=v^{m}\,q_{n}+v^{m-1}u\,q_{n-2}+...+v^{m-k}u^{k}\,q_{n-2k}+...+u^{m}q_{n-2m}\,,$
(C.1)

where $q_{d}$ are homogeneous polynomials of degree $d$ in $(s,t)$. If $d<0$,
then $q_{d}\equiv 0$, which means that a generic polynomial of multi-degree
$n[s]+m[v]$ factorizes,

$p=v^{m-\lfloor n/2\rfloor}\,\tilde{p}\,,\quad\text{if and only if}\quad
n<2m\,.$ (C.2)

Note that a generic representative of any multiple of the anti-canonical
divisor class $-K=4[s]+2[v]$ is irreducible.

The relevant supergravity models considered in Section 3 have
$\mathfrak{su}(N)$ gauge symmetry with anomaly coefficient
$b_{\mathfrak{g}}=(3,1)=3[s]+[v]$, which corresponds to a curve class $[C]$
with $[C]^{2}=4,[C]\cdot K=-6$, and, hence, a generic representative is an
irreducible rational curve. In F-theory, engineering an $\mathfrak{su}(N)$ on
a generic curve $C=\\{\sigma=0\\}$ in this homology class requires a
Weierstrass model $y^{2}=x^{3}+f\,x+g$ to take the following forms [46, 65]

$\displaystyle N$ $\displaystyle=2k\colon$ $\displaystyle f$
$\displaystyle=-\frac{1}{3}\Phi^{2}+f_{k}\,\sigma^{k}\,,\quad
g=\frac{2}{27}\Phi^{3}-\frac{1}{3}\Phi\,f_{k}\,\sigma^{k}+g_{N}\,\sigma^{N}\,,$
(C.3) $\displaystyle N$ $\displaystyle=2k+1\colon$ $\displaystyle f$
$\displaystyle=-\frac{1}{3}\Phi^{2}+\frac{1}{2}\phi_{0}\psi_{k}\sigma^{k}+f_{k+1}\sigma^{k+1}\,,$
$\displaystyle g$
$\displaystyle=\frac{2}{27}\Phi^{3}-\frac{\Phi}{6}(\phi_{0}\psi_{k}+2f_{k+1}\sigma)\sigma^{k}+\frac{1}{4}\psi_{k}^{2}\sigma^{2k}+g_{N}\sigma^{N}\,,$

with $\Phi=\frac{1}{4}\phi_{0}^{2}+\phi_{1}\sigma$.

Focusing on $N=2k$ first, observe that the polynomials $f_{k}$ and $g_{N}$
have divisor classes

$\begin{split}[f_{k}]&=[f]-k[\sigma]=-4K-k[\sigma]=(16-3k)[s]+(8-k)[v]\,,\\\
[g_{N}]&=[g]-N[\sigma]=-6K-N[\sigma]=(24-6k)[s]+(12-2k)[v]\,.\end{split}$
(C.4)

By Eq. C.2, we have $f_{k}=v^{\lceil k/2\rceil}\tilde{f}_{k}$ and
$g_{N}=v^{k}\tilde{g}_{N}$ for $k\geq 1$, where $\tilde{f}_{k}$ and
$\tilde{g}_{N}$ are irreducible. Note that, since $[\phi_{0}]=-K$ does not
factorize, $\Phi^{2}$ and $\Phi^{3}$ do not contain any non-trivial power of
$v$ as common factor. This means that $f$ and $g$ remain generically
irreducible (as long as $[f_{k}]$ and $[g_{N}]$ are effective). Then, it is
straightforward to check that the discriminant of the Weierstrass model
factorizes as

$4f^{3}+27g^{2}=\sigma^{N}\left(4f_{k}^{3}\sigma^{k}+27g_{N}^{2}\sigma^{N}-18f_{k}g_{N}\Phi\sigma^{k}-f_{k}^{2}\Phi^{2}+4g_{N}\Phi^{3}\right)=\sigma^{N}v^{k}(\dotsb)\,.$
(C.5)

Note that for $k=1$, i.e., an $\mathfrak{su}(2)$ on $\\{\sigma=0\\}$, this
factorization gives rise to an $I_{1}$ singularity over $\\{v=0\\}$, which
does not give rise to any gauge symmetry. For larger $k$, we find an
$I^{\text{s}}_{k}$ fiber, which gives rise to an $\mathfrak{su}(k)$ gauge
symmetry over $\\{v=0\\}$. Note that the splitness here comes about because
$[\Phi]=-2K$ has trivial intersection number with $[v]$, so $\Phi$ does not
vanish anywhere on $\\{v=0\\}$, hence $\left.9g/2f\right|_{\\{v=0\\}}$ is a
perfect square.

In the odd ($N=2k+1$) cases, the relevant divisor classes are

$\begin{split}[\psi_{k}]&=[f]-[\phi_{0}]-k[\sigma]=-3K-k[\sigma]=(12-3k)[s]+(6-k)[v]\,,\\\
[f_{k+1}]&=[f]-(k+1)[\sigma]=-4K-(k+1)[\sigma]=(13-3k)[s]+(7-k)[v]\,,\\\
[g_{N}]&=[g]-N[\sigma]=-6K-N[\sigma]=(21-6k)[s]+(11-2k)[v]\,,\end{split}$
(C.6)

which again all factorize by Eq. C.2:

$\psi_{k}=v^{\lceil k/2\rceil}\tilde{\psi}_{k}\,,\quad
f_{k+1}=v^{\lfloor\frac{k+2}{2}\rfloor}\tilde{f}_{k+1}\,,\quad
g_{N}=v^{k+1}\tilde{g}_{N}\,.$ (C.7)

By carefully distinguishing between even and odd $k$, we find a factorization
of the discriminant:

$4f^{3}+27g^{2}=\begin{cases}\sigma^{N}v^{k}\,(\dotsb)&k\,\text{ even}\,,\\\
\sigma^{N}v^{k+1}\,(\dotsb)&k\,\text{ odd}\,.\end{cases}$ (C.8)

This gives rise to either an $\mathfrak{su}(k)$ (for even $k$) or an
$\mathfrak{su}(k+1)$ (for odd $k$) gauge symmetry on $\\{v=0\\}$.

In summary, we see that F-theory constructions on $\mathbb{F}_{2}$ of
supergravity models with $\mathfrak{g}=\mathfrak{su}(N)$ and anomaly
coefficient $b_{\mathfrak{g}}=(3,1)=3[s]+[v]$ always force a non-trivial gauge
symmetry $\mathfrak{h}$ along the $(-2)$-curve $\\{v=0\\}$ when $N>2$,
indicating an automatic enhancement with anomaly coefficient $[v]\equiv
b_{\mathfrak{h}}=(0,1)$. For $N=2$, where there is no enhancement, the
presence of 40 fundamental hypermultiplets confirms that the gauge group is
$G=\operatorname{SU}(2)$.

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