PHYSICAL REVIEW D 109, L021903 (2024)
LetterA new vista on the heterotic moduli space from six and three dimensions
Michele Del Zotto,1,2 Marco Fazzi ,2,3 and Suvendu Giri

2,4,5

1

Department of Mathematics, Uppsala University, SE-75120 Uppsala, Sweden
Department of Physics and Astronomy, Uppsala University, SE-75120 Uppsala, Sweden
3
NORDITA, Hannes Alfvéns väg 12, SE-10691 Stockholm, Sweden
4
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
5
Princeton Gravity Initiative, Princeton University, Princeton, New Jersey 08544, USA

2(Received 3 September 2023; accepted 19 December 2023; published 23 January 2024)
We settle a longstanding question about the hypermultiplet moduli spaces of the heterotic strings on
asymptotically locally Euclidean singularities. These heterotic backgrounds are specified by the singularity
type, an instanton number, and a (nontrivial) flat connection at infinity. Building on their interpretation as
six-dimensional theories, we determine a class of three-dimensional N ¼ 4 quiver gauge theories whose
quantum corrected Coulomb branch coincides with the exact heterotic hypermultiplet moduli space.
DOI: 10.1103/PhysRevD.109.L021903

Introduction and motivation. What is the hypermultiplet
moduli space of the heterotic string on an asymptotically
locally Euclidean (ALE) singularity? This question was
posed long ago by Witten, Sen and many others [1], andan answer was proposed for singularities of the form C2 =Zk
both in absence and presence of small instantons [2–4]. Upona further toroidal compactification, this moduli space should
be quaternionic Kähler [5] (i.e. it has holonony contained inUSpð2Þ × USpð2nÞ where n is its quaternionic dimension[6]) and receives α0 and world sheet instanton corrections [7].
However, in the limit of decoupled gravity it becomes
hyperkähler (with USpð2nÞ holonomy, and a unique
Ricci-flat metric) and most corrections disappear. The
corrections that remain are such that the resulting space isa smooth manifold, in all calculable examples. For instance,
for C2 =Z2 and in absence of small instantons this moduli
space is the celebrated Atiyah-Hitchin manifold MAH , ofunit quaternionic dimension [8]. Importantly, this manifold isalso the Coulomb branch (CB) of three-dimensional (3D)N ¼ 4 pure SUð2Þ gauge theory, which may be further
identified with the moduli space of two BPS ’t HooftPolyakov monopoles of SUð2Þ. It is then natural to conjecture that the heterotic moduli space on a C2 =Zk ALE spaceshould be the CB of pure SUðkÞ, which is the same as themoduli space [9] of k BPS monopoles of SUð2Þ [10–13].
This observation lends itself to a further natural generalization, which led [3] to propose that the hypermultipet moduli
space of the heterotic string on an ALE space locally modeled

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2470-0010=2024=109(2)=L021903(6)by a C2 =ΓG singularity (with ΓG ⊂ SUð2Þ finite) should be
the same as the CB of 3D N ¼ 4 pure G gauge theory, where
G is the McKay dual group [14] to ΓG [15]. This conjecture
has been verified using a variety of techniques [16–18]. For
instance, in the simplest case of G ¼ SUðkÞ, ΓSUðkÞ ¼ Zk
and the dimension of the heterotic moduli space is counted by
the number of resolution parameters of the C2 =Zk orbifold,i.e. k − 1. This is indeed the same as the (quaternionic)
dimension of the Coulomb branch of pure SUðkÞ.
The above setup requires no instantons of the heterotic
gauge group (i.e. no heterotic NS5-branes), that is we take
F ¼ 0 identically and thus the instanton number TrF2 isalso vanishing. The equation of motion for the dilaton in theheterotic string reads schematically Δϕ ¼ TrF2 − TrR2,
with R the Riemann tensor (regarded as a two-form valued
in the Lie algebra of the orthogonal group, i.e. the curvatureof the spin connection ω). With F ¼ 0 and R large we are
driven to weak string coupling, and we will not see
fluctuations around F ¼ 0 which may distinguish thetwo heterotic strings. The analysis of [3] then proceeds
by performing a calculation in the heterotic sigma model
CFT (valid at weak string coupling) to establish the correct
moduli space (the Atiyah-Hitchin manifold in the k ¼ 2
case of ΓG ¼ Zk , i.e. G ¼ SUð2Þ).
What happens if we add say M small (i.e. zero-size, orpointlike) instantons of the heterotic string gauge group?
On top of the instanton number M, these should be specified by the value of a (possibly nontrivial) flat connection
F ¼ 0 at the spatial infinity S3 =ΓG surrounding the orbifold point of the ALE space (since π 1 ðS3 =ΓG Þ ≠ ∅).
This flat connection is a representation (i.e. an injective
homomorphism, or embedding) of the orbifold group in
the gauge group, λ∶ ΓG → Spinð32Þ=Z2 or μ∶ ΓG → E8(one per E8 factor), for the two possible heterotic strings.

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PHYS. REV. D 109, L021903 (2024)

These instantons have to be interpreted appropriately instring theory. In the case of the Spinð32Þ=Z2 string, they
are dual to type I D5-branes [19], whereas in the case of theE8 × E8 string they are given by M5-branes in the dual
Hořava-Witten M-theory background on an interval [20,21].
(More precisely, we will have say N “full” instantons givenby the M5’s plus N μ “fractional” instantons produced by
the fractionalization of each of the two M9-walls against
the orbifold [4,22], so that M ¼ N þ N μ ). In the heterotic
string, they will act as sources in the Bianchi identity for thestrength
R of the B-field, dH ¼ TrF ∧ F − TrR ∧ R, with
M ¼ dH=ð4πÞ2 in appropriate units [23].
What is the role of these instantons in the 3D theory?
How do they modify its CB so as to capture the newheterotic moduli introduced by them? In the G ¼ SUðkÞ
case it was soon realized [16,18,24] that their addition in
the E8 × E8 heterotic string corresponds to adding M
flavors (i.e. fundamental hypermultiplets) to the 3D pureG gauge theory, when the flat connection at infinity istrivial, i.e. when the full E8 × E8 group is preserved.
Moreover [4] gave an interpretation of the 2 BPS monopoles of SUð2Þ in the heterotic setup (when k ¼ 2): theycorrespond to 2 half-NS5-branes stuck on an O8− -planeobtained by reducing each of the two E8 M9-walls of theHořava-Witten setup to Type IIA. (This can be generalized
to k NS5s.) The question as to what is the 3D theory
corresponding to the hypermultiplet moduli space inpresence of a nontrivial flat connection at infinity was
however left open in [4].
The main question we address in this work is how tocapture the quantum corrected hypermultiplet moduli
space in presence of a possibly nontrivial flat connection
at infinity in the heterotic string. Our main result is that we
can still construct 3D theories that capture this space via
their CB.
3D quivers from 6D. Consider the E8 × E8 heterotic string
compactified on a real four-dimensional ALE space,
namely a K3 surface with a singularity locally modeledby C2 =ΓG. The resulting 6D theories, dubbed T M ðμL ; μR Þ,
have recently been completely determined [25–27], building on [28–32], and have the structure of a fusion [33,34]
of two 6D orbi-instanton superconformal field theories
ΩN ðμÞ [35]. Schematically:
T M ðμL ; μR Þ ¼ ΩN L ðμL Þ − G − ΩN R ðμR Þ:

ð1Þ

The Higgs branch of this 6D theory is the quantum
corrected hypermultiplet moduli space of the heterotic
string MHet
G;M;μL ;μR . Since a fusion is the 6D generalizationof a gauging operation, the result is a simple hyperkähler
quotient [36]MHet
G;M;μL ;μR ¼ ðMΩN L ðμL Þ × MΩN R ðμR Þ Þ===Gð2Þ

of the Higgs branches of the known 6D theories ΩN L ðμL Þand ΩN R ðμR Þ. Here, as above, M ¼ N L þ N μL þ N R þ N μR .
This is the starting point for our paper: the 3D theories ofinterest can be determined starting from these 6D models.
We will outline the construction in full detail only in typeA, i.e. for G ¼ SUðkÞ; the other types are computationallymore challenging but not conceptually harder [37]. In
each case, the Higgs branch of interest is captured by a
3D N ¼ 4 quiver gauge theory, colloquially known as“magnetic quiver” [38], which flows in the infrared (IR) toa 3D N ¼ 4 superconformal field theory (SCFT). The
hypermultiplet moduli space of the heterotic string is
captured by the CB of the moduli space of this SCFT.
In the case where we already know the answer, i.e. the caseof M small instantons for G ¼ SUðkÞ with trivial flat
connection (i.e. the E8 × E8 case), this quiver ought to be
the one for SUðkÞ with M flavors, i.e.
ð3ÞThis quiver is related to the magnetic quiver constructionof [38] upon replacing SUðkÞ with UðkÞ and M with abouquet of M Uð1Þ gauge nodes. (The opposite operation is
the so-called hyperkähler implosion [39,40], [41] preserving the hyperkähler structure of the moduli space and theaction of (a maximal torus of) the flavor symmetry group.)
The SU group arises in 3D from a reduction of the 6D SU
gauge group (since the Uð1Þ centers of unitary groups are
massive in 6D, and hence decouple from the low-energy
dynamics); the (opposite of the) implosion is related to agauging of a SM discrete symmetry which exchanges the M
identical M5s=NS5s in 6D (see [42] for more details). The
result is:ð4Þwith an overall Uð1Þ decoupled from the IR dynamics.
In (3) and (4) (and henceforth) we have adopted standard
quiver notation, where a non-negative integer k denotes a
3D N ¼ 4 UðkÞ vector multiplet, an edge connecting two
such non-negative integers k1 and k2 denotes a bifundamental hypermultiplet [43], and p − k denotes p hypermultiplets transforming in the fundamental of the UðkÞgauge group.
Notice that, for this theory to be “good” in the sense of[44], i.e. to flow to a standard non-Gaussian (interacting)fixed point in the IR, we must have M ≥ 2k (or, moregenerally, N f ≥ 2N c ) [45]. In other words, a configuration
with M < 2k does not have a mirror dual [46]. For thisreason, one cannot simply take the M ¼ 0 limit and getback to Witten’s original configuration with no heterotic
instantons.

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A NEW VISTA ON THE HETEROTIC MODULI SPACE FROM SIX …
More generally, we will construct 3D quivers whose CB
captures the hypermultiplet moduli space of the E8 × E8
heterotic string on C2 =Zk for any k and choice of flat
connection at infinity. The latter (trivial and nontrivial
alike) are classified by group homomorphisms, i.e. embeddings, μL;R ∶ Zk → E8 (one per gauge group factor). Each
such embedding is specified by a so-called Kac label [47],i.e. an integer partition of k in terms of the Coxeter labels
1; …; 6; 40 ; 30 ; 20 of the affine E8 Dynkin diagram:
X

6
k¼
ini þ 4n40 þ 3n30 þ 2n20 ;ð5Þi¼1

which will be denoted in general μL;R ¼ ½1n1 ; …; 6n6 ; 4n40 ;3n30 ; 2n20  (and we will also say that the ni ; ni0 are themultiplicities of the parts of the Kac label). Each embedding preserves a subalgebra of E8 determined via a simple
algorithm: one simply “deletes” all nodes with nonzero
multiplicity ni ; ni0 in this partition, and reads off the Dynkinof the leftover algebra, which may be a sum of non-Abelian

PHYS. REV. D 109, L021903 (2024)

algebras plus a bunch of uð1Þ’s. As an example, the trivial
flat connection (embedding), which exists for any k, isgiven by the label k ¼ ½1k , and preserves the full E8 . In thiscase, as stated above, the moduli space is given by the CBof (4). As an example of nontrivial flat connection, consider
the case k even, which admits a partition k ¼ ½2m  andcorresponds to a nontrivial flat connection breaking theheterotic gauge group to E7 × SUð2Þ.
Can we see a more direct engineering of these 3D quivers
from the heterotic string? The answer is a resounding yes, andcomes from looking at the compactification of the heterotic
string on a K3 as a so-called 6D (1,0) little string theory (LST)[48]. In the E8 × E8 case in presence of M small instantons,
with nontrivial flat connection at infinity, such LSTs are
precisely the theories T M ðμL ; μR Þ in equation (1). The lattercan be given a quiver description via “geometric engineering” in terms of F-theory [28], which we reproduce below
in (15) for the case μL ¼ μR ¼ ½1k . Taking the T 3 compactification of this quiver produces a 3D theory which we willcall “electric quiver,” and which reads

ð6Þ

with an overall Uð1Þ decoupled from the IR dynamics [11].
The above quiver is mirror dual to the one in (3) [11,46]:
it clearly shows that we must have M − 2k þ 1 ≥ 1, i.e.
M ≥ 2k [which is the same as the s-rule constraint validin (4)], ensuring the existence of a mirror in the first place.
The rest of the paper showcases general 3D quivers(derived in the companion paper [49]), whose CB captures
the hypermultiplet moduli space of interest in the heterotic
string. As outlined above, when the latter is compactified
on a K3, and moreover gravity is decoupled, its dynamics is
captured by a 6D little string theory of type T M ðμL ; μR Þ,
which in turn is obtained by “gluing” together two 6D
SCFTs known as orbi-instantons as in Eq. (1). We willconsider LSTs specified by M small instantons, where
M ¼ M L þ M R ¼ ðN L þ N μL Þ þ ðN R þ N μR Þ:ð7ÞN ≡ N L þ N R is the total number of full heterotic NS5’sand N μL;R is the number of fractional instantons in the left
and right “half” (orbi-instanton) of the LST. This piece ofdata can be defined in terms of μL;R , generalizing appropriately the case N μL;R ¼ k of a trivial flat connection.

The “minimal” case (i.e. no full instantons) has N ¼ 0;
however, as is clear from the above formula, this does not
imply that M ¼ 0. This is because each M9-wall in theHořava-Witten setup fractionates once we introduce theorbifold C2 =Zk [22]. The number of fractions N μL;R
(corresponding to new NS5-branes in the dual Type I’
setup) depends on the specific choice of embedding μL;R .
For instance, in (4)–(6) we made the choice μL;R ¼ ½1k 
implying M ¼ 2k if N ¼ 0. In this case, we say themagnetic quiver is exactly balanced (since N f ¼ 2N c ) [44].
It may happen that for other choices of μL;R the quiver isoverbalanced (i.e. N f > 2N c ). Importantly, we will showthat the quiver is never underbalanced (i.e. bad, in the senseof [44]) for any choice of μL;R .
Coulomb branch and heterotic moduli spaces. The upshotof our companion paper [49] is the construction of new
3D N ¼ 4 QFTs flowing in the IR to SCFTs, whose CB
captures the hypermultiplet moduli space of the heterotic
string on the orbifold with a choice of flat connection at
infinity. Such QFTs are encoded in the following quiver
diagrams:

ð8Þ

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PHYS. REV. D 109, L021903 (2024)

where in the formula above the sums in the parentheses are
L;R
performed node-by-node and the ranks rL;Rof the Ui ; ri 0
gauge groups along the tails (some of which may be zero)are determined by the specific Kac label chosen to
determine the embeddings μL and μR . The explicit algorithm was obtained in [50] and is summarized below.
ð1ÞMoreover E8 stands for the quiver of affine E8 Dynkinshape, with the ranks of the U groups appearing therein
ð1Þ ∨being equal to the Coxeter labels, and E8 is the Dynkin
mirrored around the vertical axis, i.e. with the bifurcatedtail on the left:

dimH MHet
G;M;μL ;μR ¼ 30M þ k − 1 þ

6
X

ð9Þ

ð10Þ
It is easy to convince oneself that the above quiver isgenerically overbalanced, sometimes balanced, but neverunderbalanced, and hence it always flows to a well-defined
3D N ¼ 4 SCFT in the IR. We also have!
rLi

þ rL40 þ rL30 þ rL20 þ

i¼1

6−j
X

iniþj þ 2n20 þ 3n30 þ 4n40

ð12Þ

i¼1

for j ¼ 1; …; 6, andr20 ¼ n30 þ n40 ;ð13aÞ

r30 ¼ n20 þ n30 þ 2n40 ;ð13bÞ

r40 ¼ n20 þ 2n30 þ 2n40 :ð13cÞ

The dimension in (11) is obtained by summing thedimensions dimH CB3D of the CBs of the quivers in theparentheses in (8) and subtracting dimR SUðkÞ ¼ k2 − 1
because of the hyperkähler quotient by SUðkÞ performed toglue the magnetic quivers of left and right orbi-instanton
according to (2). It reduces to what we have already computed below (6) in the case μL ¼ μR ¼ ½1k , for which ri ¼
ri0 ¼ 0 for all i (left and right), according to the following
observation. In that case, (8) is nothing but the “infinitecoupling” magnetic quiver obtained from (4) performing a
total of M small E8 instanton transitions, which turn the
ð1Þ
bouquet of M 1’s into a sum of M E8 tails, or more
ð1Þ ∨
ð1Þ
precisely to M L E8 tails to the left, and M R E8 to theright of node k there. The dimension of its CB jumps by
29M, going from M þ k − 1 to 30M þ k − 1. More generalchecks can be found in [49].
∅ suð1Þ suð2Þ

!
rRi

þ rR40 þ rR30 þ rR20 ;

ð11Þ

i¼1

with
rj ¼ ð1 − δj6 Þ6Xsuðk−1Þ suðkÞ suðkÞ

As a simple consistency check of the above we canreproduce the case with no full small instantons (closer tothe original setup by Witten and Sen); it simply amounts totaking N ¼ 0 in (7) so that we are left only with the“inevitable” fractional instantons coming from the fractionalization of each of the M9’s (left and right) against
the orbifold. It is easy to determine explicitly N μL;R from
F-theory via a case-by-base analysis [25–27]. In the case of
singularities of type A, however, an explicit algorithm
determining N μL;R from μL;R was found in [51]. Once one
constructs the Type I’ setup dual to the heterotic string on
the orbifold one can easily read off, for each choice ofðμL ; μR Þ, the so-called largest linking number lL;R on
the left and right of the setup, i.e. the largest among the
L;R L;R
numbers ðlL;R
1 ; …; l8 ; l9 Þ, which are defined as thenumber of D6’s ending from the right on the ith D8,
minus the number of D6’s ending on the left, plus thenumber of NS5’s to the immediate left of it (where forconcreteness we have assumed the O8 sits on the left ofeach half, when considered individually). More precisely,
we have the relation
N μL;R ¼ lL;R ¼

6
X

ni þ pð14Þ

i¼1

with p ¼ min ðbðn30 þ n40 Þ=2c; bðn20 þ n30 þ 2n40 Þ=3cÞ. To
determine the Type I’ configuration dual to the heterotic
string, or equivalently to find the electric quiver in the
F-theory notation, one can apply the algorithm of [52]. For
instance, for μL ¼ μR ¼ ½1k  it producessuðkÞ suðkÞ suðk−1Þsuð2Þ suð1Þ ∅½E8 12 2  22
2  22
2    2 2 1½E8 :|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}½N f ¼1
½N f ¼1 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
N μL ¼k
N μR ¼k|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
N L þN R þ1¼Nþ1

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A NEW VISTA ON THE HETEROTIC MODULI SPACE FROM SIX …
(All more general choices have been analyzed in detailin [25–27,32].) The T 3 compactification of the above
electric quiver is nothing but (6), which is mirror to (4).
The dimension of the HB of the above quiver equals
M þ k − 1, as expected. To obtain 30M þ k − 1 we firstmove to infinite coupling in each of the two orbi-instanton
constituents, compute the dimension of their HB, andsubtract the (real) dimension of the diagonal SUðkÞ flavor
group from the two which we are gauging, producing one
exra suðkÞ algebra [53].

PHYS. REV. D 109, L021903 (2024)

strata of the so-called affine Grassmannian of E8 . It would
be extremely interesting to prove this conjecture. The
minimum symmetry on this CB in the IR can also beeasily computed, which may help with the proof.
As a final remark, the hyperkähler/quaternionic Kähler
correspondence [57,58] suggests that one can also determine the gravitational quantum moduli space of theheterotic string on these backgrounds building from ourresults.

Conclusions. In this paper we have identified the hypermultiplet moduli space of the E8 × E8 heterotic string on an
A-type ALE space with the exact quantum corrected CB ofa 3D quiver gauge theory [the magnetic quiver (8)] flowing
in the IR to an SCFT. The magnetic quiver is the mirror of
the T 3 compactification of a 6D quiver gauge theory (plus
tensor multiplets) that gives the dynamics of a 6D (1,0) LSTwith a Higgs branch that coincides with the heterotic
moduli space (in the limit of decoupled gravity). To obtain
this magnetic quiver, one also has to perform M small E8
instanton transitions in 6D.
The dimension of the moduli space in question can becomputed both in absence and in presence of N full small
instantons, where one is also required to specify a flat
connection at infinity for the heterotic gauge group. The
various choices of flat connection give rise to different
magnetic quivers, with different CBs. One is moreover led
to conjecture (due the same string theory arguments usedby [4]) that this CB (which is a hyperkähler space) issmooth, as was the case in absence of small instantons[2,3]. It remains to be understood what is the underlying
geometry of the CB. In the companion paper [49], based on
observations about 6D (1,0) SCFTs made in [51,54,55], wehave proposed that this CB is given by the holomorphic
symplectic quotient construction of [56] applied to two

Acknowledgments. M. D. Z. would like to thank Timo
Weigand for an interesting conversation at the Strings
and Geometry 2023 conference held at the University ofPennsylvania which sparked our interest into this subject,
and Kantaro Ohmori for several enlightening discussions.
We would like to thank Sav Sethi for interesting correspondence. M. F. would like to thank the University of
Milano–Bicocca, the Pollica Physics Center, the Universityof Lancaster for hospitality during the completion of thiswork. M. F. and S. G. gratefully acknowledge support fromthe Simons Center for Geometry and Physics, Stony Brook
University (2023 Simons Physics Summer Workshop) at
which some of the research for this paper was performed.
The work of M. D. Z. and M. F. has received funding
from the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovation
program (Grant agreement No. 851931). M. D. Z. also
acknowledges support from the Simons Foundation
Grant No. 888984 (Simons Collaboration on Global
Categorical Symmetries). The work of M. F. is also
supported in part by the Knut and Alice Wallenberg
Foundation under Grant No. KAW 2021.0170, theSwedish Research Council Grant No. VR 2018-04438,
the Olle Engkvists Stiftelse grant No. 2180108. The workof S. G. was conducted with funding awarded by the
Swedish Research Council grant No. VR 2022-06157.

[1] For compactifications on tori, a detailed analysis was carriedout in [59].
[2] A. Sen, Adv. Theor. Math. Phys. 1, 115 (1998).
[3] E. Witten, J. High Energy Phys. 02 (2000) 025.
[4] A. Hanany and A. Zaffaroni, J. High Energy Phys. 12(1999) 014.
[5] J. Bagger and E. Witten, Nucl. Phys. B222, 1 (1983).
[6] A. L. Besse, Einstein Manifolds (Springer-Verlag, Berlin,
Heidelberg, New York, 1987).
[7] E. Witten, J. High Energy Phys. 02 (2000) 030.
[8] M. F. Atiyah and N. J. Hitchin, Phil. Trans. R. Soc. A 315,459 (1985).
[9] A more precise statement is the following. The reduced
moduli space of k monopoles of SUð2Þ (with thecenter-of-mass modulus factored out) is the same as the
CB of pure SUðkÞ, i.e. the Uð1Þ center of UðkÞ decouples
from the dynamics as it is free in the infrared. Alternatively,
the moduli space of k SUð2Þ monopoles is the CB of SUðkÞ
times the moduli space of a free vector.
[10] N. Seiberg and E. Witten, in Conference on the Mathematical Beauty of Physics (In Memory of C. Itzykson) (1996),
pp. 333–366, arXiv:hep-th/9607163.
[11] A. Hanany and E. Witten, Nucl. Phys. B492, 152 (1997).
[12] G. Chalmers and A. Hanany, Nucl. Phys. B489, 223(1997).
[13] N. Dorey, V. V. Khoze, M. P. Mattis, D. Tong, and S.
Vandoren, Nucl. Phys. B502, 59 (1997).
[14] J. McKay, Proc. Symp. Pure Math. 37 (1980).

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PHYS. REV. D 109, L021903 (2024)[15] The pure G theory with standard Yang–Mills kinetic term
for the gauge field has 4r massless real scalars if G issemisimple of rank r. Namely, the quaternionic dimensionof its CB is dimH CBðGÞ ¼ r. The physics is free in theinfrared for large vacuum expectation values of the scalars[10]. This can be extended to quivers with ADE shape,Uðni Þ gauge groups, and (only)P
bifundamental matter. The
CB has dimension dimH CB ¼ i ni and is interpreted as
the moduli space of BPS monopoles of SUðr þ 1Þ withr ≥ 1, if r is the rank of the ADE group [60]. In type A, thismatches what found in [11] via type IIB brane constructions.
[16] M. Rozali, J. High Energy Phys. 12 (1999) 013.
[17] P. Mayr, J. High Energy Phys. 08 (2000) 042.
[18] P. S. Aspinwall and M. R. Plesser, J. High Energy Phys. 04(2000) 025.
[19] E. Witten, Nucl. Phys. B460, 541 (1996).
[20] P. Horava and E. Witten, Nucl. Phys. B460, 506 (1996).
[21] P. Horava and E. Witten, Nucl. Phys. B475, 94 (1996).
[22] M. Del Zotto, J. J. Heckman, A. Tomasiello, and C. Vafa,
J. High Energy Phys. 02 (2015) 054.
[23] Equivalently, H ¼ dB þ CSðAÞ − CSðωÞ, where CS is the
Chern–Simons invariant, A the gauge connection, and ω thespin connection. E.g. CSðAÞ ¼ TrðA ∧ dA þ 23 A ∧ A ∧ AÞ.
[24] A. Hanany and A. Zaffaroni, Nucl. Phys. B529, 180 (1998).
[25] M. Del Zotto, M. Liu, and P.-K. Oehlmann, J. High Energy
Phys. 01 (2023) 176.
[26] M. Del Zotto, M. Liu, and P.-K. Oehlmann, arXiv:2212.05311.
[27] M. Del Zotto, M. Liu, and P.-K. Oehlmann (to be published).
[28] P. S. Aspinwall and D. R. Morrison, Nucl. Phys. B503, 533(1997).
[29] J. D. Blum and K. A. Intriligator, Nucl. Phys. B506, 199(1997).
[30] J. D. Blum and K. A. Intriligator, Nucl. Phys. B506, 223(1997).
[31] K. A. Intriligator, Adv. Theor. Math. Phys. 1, 271 (1998).
[32] L. Bhardwaj, M. Del Zotto, J. J. Heckman, D. R. Morrison,
T. Rudelius, and C. Vafa, Phys. Rev. D 93, 086002 (2016);
100, 029901(E) (2019).
[33] J. J. Heckman, T. Rudelius, and A. Tomasiello, J. HighEnergy Phys. 02 (2019) 167.
[34] M. Del Zotto and G. Lockhart, J. High Energy Phys. 08(2018) 173.
[35] These are specific 6D theories denoted ΩN ðμÞ, whereμ∶ ΓG → E8 is a group homomorphism and N is a nonnegative integer (for their classification see e.g. Heckman:
2015bfa,Frey:2018vpw).

[36] N. J. Hitchin, A. Karlhede, U. Lindstrom, and M. Rocek,
Commun. Math. Phys. 108, 535 (1987).
[37] Importantly, they may be approached with techniques
already available in the literature.
[38] S. Cabrera, A. Hanany, and M. Sperling, J. High Energy
Phys. 06 (2019) 071; 07 (2019) 137(E).
[39] A. Dancer, F. Kirwan, and A. Swann, arXiv:1209.1578.
[40] A. Dancer, A. Hanany, and F. Kirwan, Adv. Theor. Math.
Phys. 25, 1367 (2021).
[41] In physics, it simply means we can ungauge a Uð1Þ bygauging the topological Uð1Þ symmetry that comes with it.
[42] A. Hanany and G. Zafrir, J. High Energy Phys. 07 (2018)168.
[43] I.e., a hypermultiplet transforming in the ðk1 ⊗ k2 Þ ⊕ðk1 ⊗ k2 Þ representation of Uðk1 Þ × Uðk2 Þ.
[44] D. Gaiotto and E. Witten, Adv. Theor. Math. Phys. 13, 721(2009).
[45] This is a consequence of the “s-rule” (ensuring thatsupersymmetry is preserved) in a type IIB engineering
via D3-D5-NS5-branes of this quiver [11,44].
[46] K. A. Intriligator and N. Seiberg, Phys. Lett. B 387, 513(1996).
[47] V. Kac, Infinite-Dimensional Lie Algebras (Cambridge
University Press, Cambridge, England, 1990).
[48] N. Seiberg, Phys. Lett. B 408, 98 (1997).
[49] M. Del Zotto, M. Fazzi, and S. Giri, arXiv:2307.11087.
[50] S. Cabrera, A. Hanany, and M. Sperling, J. High Energy
Phys. 02 (2020) 184.
[51] M. Fazzi, S. Giri, and P. Levy, arXiv:2312.17178.
[52] N. Mekareeya, K. Ohmori, Y. Tachikawa, and G. Zafrir,
J. High Energy Phys. 09 (2017) 144.
[53] Remember that the quaternionic dimension of the HB in atheory with eight supercharges in any spacetime dimension
is found by subtracting the total number of vector multipletsfrom that of hypermultiplets.
[54] M. Fazzi and S. Giri, J. High Energy Phys. 12 (2022) 076.
[55] M. Fazzi, S. Giacomelli, and S. Giri, J. High Energy Phys.
03 (2023) 089.
[56] G. W. Moore and Y. Tachikawa, Proc. Symp. Pure Math. 85,
191 (2012).
[57] S. Cecotti, S. Ferrara, and L. Girardello, Int. J. Mod. Phys. A
04, 2475 (1989).
[58] N. Hitchin, Commun. Math. Phys. 324, 77 (2013).
[59] J. de Boer, R. Dijkgraaf, K. Hori, A. Keurentjes, J. Morgan,
D. R. Morrison, and S. Sethi, Adv. Theor. Math. Phys. 4,995 (2002).
[60] D. Tong, Phys. Lett. B 448, 33 (1999).

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