Comments on the twisted punctures of Aeven class S theory
Accepted: June
Aeven class
Yuji Tachikawa 0 1 3
Yifan Wang 0 1 2
Gabi Zafrir 0 1 3
Global Symmetries, Supersymmetric Gauge Theory
0 Princeton , NJ 08544 , U.S.A
1 University of Tokyo , Kashiwa, Chiba 2778583 , Japan
2 Joseph Henry Laboratories, Princeton University
3 Kavli Institute for the Physics and Mathematics of the Universe
We point out that the USp symmetry associated to a full twisted puncture of a class S theory of type Aeven has the global anomaly associated to discuss manifestations of this fact in the context of the superconformal eld theory R2;2N introduced by Chacaltana, Distler and Trimm. For example, we nd that this theory can be thought of as a natural ultraviolet completion of an infraredfree SO(2N + 1) gauge theory with 2N avors, whose USp(4N ) symmetry clearly has the global anomaly.
Anomalies in Field and String Theories; Brane Dynamics in Gauge Theories

2.1
2.2
2.3
2.4
1 Introduction and summary 2
Class S constructions
Class S theories of type A2N with a twisted irregular puncture
R2;2N from twisted irregular punctures
Halfhypermultiplet from irregular twisted puncture
3 5d constructions
4 IIA constructions
class of 4d N =2 superconformal eld theory obtained by compactifying 6d N =(2; 0)
theory on a Riemann surface with punctures. This construction not only allowed a geometric
understanding of various Sdualities, but also provided a huge variety of new 4d N =2
theories. This variety comes from various sources: we have the choice of the initial 6d N =(2; 0)
theory, which comes with the ADE classi cation; then one can introduce punctures, which
come roughly in two varieties, called the regular ones and the irregular ones; then one can
further introduce twists by outer automorphisms.
The class S theories with regular punctures have been systematically explored by
Chacaltana, Distler and their collaborators [3{15], for almost all possible types of 6d N =(2; 0)
theories with outer automorphism twists. The remaining two cases are the twisted
punctures of type Aeven theories, and the case where Z2twisted and Z3twisted punctures of
type D4 are combined. The aim of this paper is to make a small comment on the former
case, namely the twisted punctures of type Aeven theories. More speci cally, we point out
that the avor symmetry USp(2N ) of the full twisted puncture of A2N theory has the global
anomaly associated to 4(USp(2N )) = Z2.
This point can be seen most succinctly as follows. Consider splitting a Riemann surface
on which the 6d N =(2; 0) theory of type A2N is compacti ed along a long tube around
which we have Z2 outerautomorphism twist. This results in two twisted full punctures
coupled by the 5d theory obtained by compactifying the A2N theory on S1 with Z2
outerautomorphism twist. This is the maximally supersymmetric 5d USp(2N ) theory with the
discrete theta angle
=
[16, 17]. The 4d class S theory with a twisted full puncture, in
this viewpoint, lives on a boundary of this 5d USp(2N ) theory.
{ 1 {
First, this determines that the current algebra central charge of USp(2N ) symmetry of
the twisted full puncture is k = 2N + 2 in the normalization where the halfhypermultiplet
in the fundamental has k = 1.1 More importantly for us, this means that the USp(2N )
symmetry of the twisted full puncture has the global anomaly. This is because of the
following. Note that the discrete theta angle is controlled by
4(USp(2N )) = Z2, which
also controls the global anomaly on the 4d USp(2N ) symmetry, as originally discussed by
Witten [19]. Therefore, there is an anomaly in ow from the bulk to the boundary, and the
twisted full puncture needs to carry the global anomaly. This is analogous to the fact that
if the bulk 5d theory has SU(n) gauge symmetry with the level k ChernSimons term, the
boundary 4d theory which is coupled to this bulk 5d theory should have 't Hooft anomaly
di erence here is that this puncture does not have the global anomaly.
In this paper, we discuss various manifestations of this fact, mainly using the 4d
N =2 superconformal
eld theory R2;2N introduced in [6, section 7.2] for N = 1 and
in [10] for general N . This theory is obtained by compactifying the 6d N =2 theory of
type A2N on a sphere with one simple puncture and two twisted full punctures. In [10],
it was shown that the symmetry USp(2N )2 apparent in this construction enhances to
USp(4N ), but Witten's anomaly could not be directly determined, since the diagonal
combination USp(2N )diag
USp(2N )2
USp(4N ) was gauged. Our main observation is
that by turning on the mass term associated to the simple puncture, the theory becomes an
SO(2N + 1) theory with 2N hypermultiplets in the vector representation, which is infrared
free; the USp(4N ) symmetry of this theory clearly has the global anomaly.
The rest of the paper consists of three sections, which can be read independently, and
are written using di erent techniques. Namely, the section 2 uses the class S
construction, the section 3 uses the dimensional reduction from 5d, and the section 4 uses a very
traditional orientifold construction.
In section 2, after brie y reviewing the original construction of R2;2N , we provide a
di erent construction of the same R2;2N theory as a sphere of A4N theory with a full twisted
puncture and an irregular puncture, generalizing the construction of [
20
]. This allows us
to perform a consistency check of the global anomaly.
In section 3, we point out that the mass deformation for the U(1) avor symmetry
associated to the simple puncture gives rise to the SO(2N + 1) theory coupled to 2N
avors. This will be done by constructing these theories by a twisted compacti cation of
5d N =1 theory, generalizing the work of [
21
].
In section 4, we revisit this mass deformation from the point of view of the oldfashioned
type IIA construction. We will see that the standard SeibergWitten curve of the SO(2N +
1) theory with 2N
avors, in the standard MQCD form, is literally equal to the
Seiberg1We emphasize that the
avor central charge here of the twisted full puncture is valid for class S
of the 4d theory and thus change the avor central charge which is related to the U(1)R anomaly [18].
theory is simply the mass parameter of the R2;2N theory.
Class S constructions
Review of R2;2N
Let us rst recall how the R2;2N nonLagrangian SCFT was constructed in [10]. We start
with a class S theory of A2N type with the UV curve given by a torus with minimal
untwisted puncture decorated by an Z2 twist line along a handle of the torus. The theory
has two Sdual frames with gauge theory descriptions:
S dual
SU(2N + 1)
USp(2N )
R2;2N
(2.1)
punctures, see gure 1.
SCFT to be
and the conformal central charges are
and the R2;2N SCFT emerges from the weak coupling limit of USp(2N ) gauge coupling.
Motivated by the decoupling picture, one can engineer this SCFT directly using 3 regular
A2N punctures on a sphere: one minimal untwisted puncture, and two maximal twisted
The decoupling picture above determines the Coulomb branch spectrum of R2;2N
2 f3; 5; 7; : : : ; 2N + 1g
a =
14N 2 + 19N + 1
24
; c =
8N 2 + 10N + 1
12
:
(2.2)
(2.3)
The theory has enhanced U(1)
USp(4N )2N+2 avor symmetry where the USp(4N ) factor
has the
avor central charge kUSp(4N) = 2N + 2. Consequently, the 2d chiral algebra of
the SCFT in the sense of [22] naturally contains the a ne KacMoody algebra of type
C2N with level k2d =
(N + 1) and a weight one current realizing the 4d U(1) symmetry.
Furthermore kUSp(4N) saturates a avor central charge unitary bound of [22], which means
that the stress tensor in the 2d chiral algebra is given by the Sugawara construction [10].2
2We emphasize here that the full chiral algebra of the R2;2N SCFT contains additional Virasoro primaries
regular
puncture
irregular
puncture
tent if the USp(4N ) symmetry of R2;2N also carries the global anomaly; then the USp(2N )
diagonal gauging is nonanomalous. Thus if we have an alternative construction R2;2N
that makes the enhanced avor symmetry manifest as a twisted puncture, this would o er
a nontrivial consistency check. We show this is indeed the case in the next section. Our
alternative construction involves irregular punctures.
2.2
Class S theories of type A2N with a twisted irregular puncture
Consider general 4d N = 2 SCFTs in class S of type A2N from one regular maximal twisted
puncture and one irregular twisted puncture, see gure 2.
The twisted punctures correspond to codimension2 defects in the (2,0) theory of A2N
type and are speci ed by a local singularity of the Higgs eld
at z = 0 on the UV curve.
For the Higgs eld to be wellde ned, we require
HJEP06(218)3
=
T
z2+ 4N+2
+
{ 4 {
(e2 iz) = g[o( (z))]g 1
as one circles the singularity. Here o denotes the Z2 outerautomorphism associated to the
A2N Dynkin diagram and g generates an inner automorphism of A2N .
Among the twisted punctures, regular punctures are associated with simple poles for
and are classi ed in [5], while the irregular punctures involve higher order poles and
a detailed classi cation will appear in [23]. For our purpose here, we list two
distinguished classes of twisted irregular punctures of A2N type and the relevant physical data
from [23] below.
Class I. has the Higgs eld of the form for odd, where
T = diag(1; !2; !4; : : : ; !4N );
!4N+2 = 1:
(2.4)
(2.5)
(2.6)
We have the conjectural avor central charge formula
Class II. has the Higgs eld of the form
kUSp(2N) = 2N + 2
1
4N + 2
2 4N + 2 +
:
=
T
z2+ 2N
+
kUSp(2N) = 2N + 2
1
2N
2 2N +
:
where
=
T
z2+ 1 4 N4N +
T = diag(0; 1; !; : : : ; !4N 1);
!4N = 1:
This is chosen such that under z ! ze2 i, the Higgs eld transform by an automorphism of
order 4N that is a product of the Z2 outerautomorphism o associated to the A4N Dynkin
diagram and an inner automorphism of A4N as in (2.4).
The singular SeibergWitten curve is where We have the conjectural avor central charge formula
T = diag(0; 1; !; : : : ; !2N 1);
!2N = 1:
The conformal central charges for the corresponding 4d SCFTs are determined by
for the above conjectures in [23].
2.3
R2;2N from twisted irregular punctures
Given the general construction of 4d N = 2 SCFTs from twisted irregular punctures in
the last section, we now construct R2;2N from 6d (2; 0) A4N type theory on a sphere with
two twisted punctures: one regular maximal puncture, and one irregular puncture of class
II with
= 1
4N in (2.8). The irregular twisted puncture is described by the following
local singularity for the Higgs eld ,
x4N+1 + xz1 4N = 0
{ 5 {
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
and the SeibergWitten 1form is
= xdz. The scaling dimensions of the coordinates are
Under the Z2 twist, the di erentials ` transform as
Therefore ` for ` even (odd) involves integral (halfintegral) powers of z. We can
immediately read o the spectrum of Coulomb branch operators to be
HJEP06(218)3
= f3; 5; 7; : : : ; 2N + 1g
which all come from the odddegree di erentials ` with ` = 2N + 3; 2N + 5; : : : ; 4N + 1.
The USp(4N ) avor symmetry comes from the regular twisted puncture and its avor
central charge is determined by (2.10) to be 2N + 2. The di erential 2N+1 contributes
the additional mass parameter responsible for the U(1) factor in the avor symmetry.
It is also easy to check that the central charges computed from (2.11) and (2.10) are
consistent with the results in the previous section. Moreover we can directly see that the
SeibergWitten curves from the two descriptions agree. For example let us look at the
N = 1 case which is a rank1 theory with a Coulomb branch operator u of dimension 3.
In this case [x] =
3 and [z] = 4. The full SeibergWitten curve in the A4 description is
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
Starting from the A4 curve, we perform a coordinate rede nition z !
which only changes the SeibergWitten di erential by an exact 1form, and then the A4
curve becomes identical to the A2 curve, after throwing out irrelevant overall factors.
1
z3x4; x ! x3z2
2.4
Halfhypermultiplet from irregular twisted puncture
Since we expect Witten's global anomaly to be a local property of the twisted full puncture,
which is eventually a boundary condition for the 5d super YangMills theory, it should not
depend on the types of other punctures that are involved in a given Class S setup. In
{ 6 {
x5 + x3 z22 + x2 m
On the other hand, the full SeibergWitten curve in the A2 description is
with [x] = 1 and [z] = 0. In both cases the SeibergWitten di erential is taken to be
= xdz. Here 2; 4 label Casimirs of the USp(4)
avor symmetries and
2; 20 label
Casimirs of the SU(2) SU(2) subgroup. The additional U(1) mass is labelled by m. For simplicity, let us look at the curves with the mass parameters turned o :
A4 : x5 +
x
z3 +
u
z 2
9 = 0;
A2 : x3
iu
5
z 2 (z
1)
= 0:
other words, if we can tune the choice of the irregular puncture in
gure 2 such that
the 4d theory has a free/weaklycoupled description that clearly exhibits the anomaly, it
strongly indicates that the twisted full puncture carries the global anomaly. Below we see
this is indeed the case for a free halfhyper multiplet in the fundamental representation
of USp(2N ).
The halfhyper can be constructed using A2N twisted punctures: a Class I twisted
irregular puncture with
and one twisted full regular puncture with USp(2N ) avor symmetry. The spectral curve is
x2N+1 + X z2i 2i + z 21 2N = 0:
Here the scaling dimensions are determined by [x] =
1
4N and [z] = 4N + 2. Thus 2i
with dimension 2i is a degree 2i Casimir for the USp(2N ) avor symmetry. The above is
also consistent with (2.7) which gives
as expected for a halfhyper.
3
5d constructions
In the previous section we relied on class S methods to argue that the R2;2N SCFT has
a global anomaly for the USp(4N ) group. Here we shall show further evidence for this
by using a di erent realization of the R2;2N SCFT. The realization we employ is the
compacti cation of 5d SCFTs with a global symmetry twist. In this manner, 4d N = 2
SCFTs can be engineered by the compacti cation of 5d SCFTs. This method can be used
to engineer many 4d N = 2 SCFTs including nonLagrangian theories appearing in class
S constructions [24].
Many 5d SCFTs possess discrete global symmetries. It is then possible to consider a
compacti cation with a twist under said discrete symmetry. In other words the
compacti cation is done such that upon traversing the circle one comes back to the theory acted
by the discrete symmetry element. To try to understand the results of such compacti
cations it is useful to consider 5d SCFTs with a string theory construction that exhibits the
global symmetry.
A convenient way to realize 5d SCFTs in string theory is using brane webs [25, 26].
Discrete symmetries of the SCFTs are then manifested by discrete symmetries of the brane
system. A particular interesting case is when the symmetry is manifested on the web as
a combination of spacetime re ections and an SL(2; Z) transformation. The cases where
the discrete symmetry is Z2 and Z3 were analyzed in [
21
] where it was argued that such a
construction can realize the R2;2N SCFT. We next review some aspects of this construction
that will be important for us.
{ 7 {
(2.21)
(2.22)
(2.23)
web after moving some of the 7branes. This corresponds to a mass deformation of the 5d SCFT.
As can be seen from the web, this mass deformation sends the 5d SCFT to an SU(2N + 1) gauge
theory with 4N
avors in the fundamental representation.
Consider the bane web shown in gure 3(a). This describes a 5d SCFT which lives at
the intersection of all the 5branes. This SCFT can be de ned in eld theory as the UV
xed point of an SU(2N + 1) gauge theory with 4N
avors in the fundamental
representation. This can be seen from
gure 3(b) which shows the web after a mass deformation
corresponding to the SU(2N + 1) coupling constant.
This 5d SCFT has an SU(4N )
U(1)2 global symmetry. Additionally, it also has a Z2
discrete symmetry, which in the brane web is given by a
rotation in the plane of the web
combined with
I in the SL(2; Z) transformation. In the lowenergy SU(2N + 1) gauge
theory it is manifested as charge conjugation which is a symmetry of the gauge theory.
We can now consider compactifying the 5d SCFT with a twist under this discrete
symmetry. It was argued in [
21
] that this should give the 4d R2;2N SCFT. There are
several reasons for this identi cation. One is that they have the same symmetries. Here
the SU(4N ) part is projected to USp(4N ) by the twist while one U(1) remains and another
is projected out. This is readily seen in the 5d gauge theory description by considering how
charge conjugation acts on these symmetries. Another reason is that one can argue from
the brane web that the resulting theory needs to participate in the same duality (2.1).
We can now use the brane construction to study various properties of the R2;2N SCFT.
The particular properties, that are of interest to us here, are mass deformations. We have
already encountered one such mass deformation, the one leading to the SU(2N + 1) gauge
theory. This deformation is invariant under the Z2 discrete symmetry as can be seen from
the lowenergy gauge theory. We thus expect it to remain also in the R2;2N SCFT where
it should correspond to a deformation in the U(1) global symmetry.
We can infer the result of this deformation from the 5d construction, where it should
just be the twisted compacti cation of the 5d gauge theory. As the twist acts on it by
charge conjugation, we expect the SU(2N + 1) to be projected to SO(2N + 1) while the 4N
avors should be split to two groups of 2N each mapping to the other. The end result is a
4d SO(2N + 1) gauge theory with 2N hypermultiplets in the vector representation. This
theory indeed has a USp(4N ) global symmetry as required. Furthermore it is easy to see
{ 8 {
same web after moving some of the 7branes. This corresponds to a mass deformation of the 5d
SCFT. As can be seen from the web, this mass deformation sends this 5d SCFT to the 5d SCFT
in gure 3(a).
that it has a global Witten's anomaly. By anomaly matching arguments this implies that
the starting theory, the R2;2N SCFT, must also have the same anomaly.
We can also consider mass deformations leading to the R2;2N SCFT. For instances
consider the 5d SCFT shown in gure 4(a). This is a 5d SCFT that has the same discrete
symmetry and so we can also consider its compacti cation to 4d with a twist. The result of
such a compacti cation was considered in [
21
] where it was argued that it leads to a known
4d SCFT with SU(2)
USp(4N + 2) global symmetry. An interesting property of this 4d
SCFT is that it is dual to an SO(2N + 3) gauge theory with 2N + 1 hypermultiplets in the
vector representation upon gauging its SU(2) global symmetry. This duality in particular
implies that its USp(4N + 2) global symmetry has a global anomaly. This theory can also
be engineered by a Class S of A4N+2 type involving one twisted irregular puncture and
one twisted regular full puncture that realizes the USp(4N + 2) avor symmetry with the
global anomaly [23].
As shown in
gure 4(b) we can ow from this SCFT to the R2;2N SCFT via a mass
deformation. Then anomaly matching arguments again suggest that the USp(4N ) global
symmetry of the R2;2N SCFT has a global anomaly.
4
IIA constructions
In this section we provide another way to see that the R2;2N SCFT can be mass deformed to
be the SO(2N +1) gauge theory with 2N
avors, using a traditional brane construction [
27
]
using orientifolds [16, 28]. Let us rst recall the brane construction of 4d SO(2N + 1) gauge
theories with hypermultiplets in the vector representation using O4, D4 and NS5 branes,
see gure 5(a).
f
+
As shown in the gure, this requires the use of Of4 plane to realize the SO(2N + 1)
gauge group. Then, across a halfNS5brane, the type of the orientifold plane switches to
O4 plane, and we have a halfhypermultiplet in the bifundamental of USp and SO(2N +1)
at the intersection of the D4branes and the halfNS5brane. Note that this bifundamental
halfhypermultiplet has the global anomaly for USp symmetry. From the anomaly in ow
{ 9 {
+ n D4
USp(2n)
+ N D4
SO(2N+1)
(a)
½NS5
½NS5
+ nʹ D4
USp(2nʹ)
+ N D4
USp(2N)
+ N D4
USp(2N)
full NS5
(b)
+
requires the use of two halfNS5branes. (b) When we collapse the two halfNS5branes, we obtain
the IIA reduction for the R2;2N theory.
= xdz=z. This becomes, for n = n0 = N ,
argument, this implies that the 5d USp gauge symmetry on the Of4 plane has to have the
discrete theta angle
= . This is a much simpler argument for this theta angle than the
one given in [17].
Instead, consider reducing the threepunctured sphere de ning the R2;2N theory given
+
in gure 1 from Mtheory to type IIA. A twisted full puncture corresponds to a semiin nite
segment of N D4branes on the Of4 plane, and the untwisted simple puncture corresponds
to a full NS5 brane, see gure 5(b). Clearly, the two setups shown in gure 5(a) and 5(b)
are related by the motion of the two halfNS5branes, when n = n0 = N . Since the
distance between the two halfNS5branes specify the squared inverse gauge coupling of
the SO(2N + 1) gauge theory at the string scale, this clearly means that the R2;2N theory
is at the in nite coupling limit of the SO(2N + 1) gauge theory with 2N
avors.
What remains is to show that how the gauge coupling looks like from the point of view
of the R2;2N theory. Here we encounter a mild surprise: the SeibergWitten curve of the
SO(2N + 1) theory with 2N
avors is the same with the SeibergWitten curve of the R2;2N
theory, without making any approximation.
To see this, recall the SeibergWitten curve of SO(2N + 1) theory with n + n0 avors,
as determined by lifting the brane setup shown in gure 5(a) to Mtheory:
x
"
2N 1 2n 1
z1=2
n
Y(x2
i=1
mi2) + 2N 1 2n0z1=2 Y(x2
m~ i20)
#
= x2N + u2x2N 2 +
+ u2N :
(4.1)
#
n0
i=1
m~i20) = x2N + u2x2N 2 +
+ u2N :
(4.2)
We now compare this to the SeibergWitten curve of the R2;2N theory. We put two
full twisted punctures at z = 0 and z = 1, and the untwisted simple puncture at z = 1.
The SeibergWitten curve is given by
x2N+1 + 1(z)x2N + 2(z)x2N 1 +
+ 2N+1(z) = 0;
(4.3)
= xdz=z. Note that we allow
1(z) to be nonzero
to simplify the description of the massdeformed untwisted simple puncture, where the
conditions just become that for all k, k(z) have at most a simple pole at z = 1. Combining
with the conditions from the twisted full punctures, we see that the SeibergWitten curve
of the R2;2N theory to be
x2N+1 +
x2N +
mz1=2
z
1
z
1
2z + 02 x2N 1
+
u^3z1=2
z
1
x2N 2 +
+
2N z + 0
z
1
2N x +
u^2N+1z1=2
z
1
where m is the mass parameter for the untwisted simple puncture,
0 ;
2
; 02N are the mass parameters for the two twisted full punctures, and u^3;
; u^2N+1
are the Coulomb branch parameters of the R2;2N theory.
We see that the curve (4.2) for the SO(2N + 1) theory with 2N
avors and the
curve (4.4) for the R2;2N theory are the same up to a minor relabeling of the
parameters as one can see by dividing (4.2) by (z1=2 + z 1=2)=
to make the coe cient of x2N+1
to be one. In particular, we see that the scale
of the Landau pole of the infrared gauge
theory can simply be equated to the mass deformation parameter m of the R2;2N theory.
It might be interesting to look for similar phenomena in other 4d N =2 gauge theories with
infraredfree matter content, such that its SeibergWitten curve is the same as that of some
= 0;
(4.4)
2; : : : ; 2N and
Acknowledgments
YT is partially supported by JSPS KAKENHI GrantinAid (WakateA), No.17H04837
and JSPS KAKENHI GrantinAid (KibanS), No.16H06335, and also by WPI Initiative,
MEXT, Japan at IPMU, the University of Tokyo. YW is supported in part by the US
NSF under Grant No. PHY1620059 and by the Simons Foundation Grant No. 488653.
GZ is supported in part by World Premier International Research Center Initiative (WPI),
MEXT, Japan.
Open Access.
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[arXiv:1008.5203] [INSPIRE].
[arXiv:1106.5410] [INSPIRE].
[1] D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
[2] D. Gaiotto, G.W. Moore and A. Neitzke, Wallcrossing, Hitchin systems and the WKB
approximation, arXiv:0907.3987 [INSPIRE].
[3] O. Chacaltana and J. Distler, Tinkertoys for Gaiotto Duality, JHEP 11 (2010) 099
[4] O. Chacaltana and J. Distler, Tinkertoys for the DN series, JHEP 02 (2013) 110
[5] O. Chacaltana, J. Distler and Y. Tachikawa, Nilpotent orbits and codimensiontwo defects of
6d N = (2; 0) theories, Int. J. Mod. Phys. A 28 (2013) 1340006 [arXiv:1203.2930]
[7] O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the twisted Dseries, JHEP 04
(2015) 173 [arXiv:1309.2299] [INSPIRE].
[arXiv:1403.4604] [INSPIRE].
(2015) 027 [arXiv:1404.3736] [INSPIRE].
HJEP06(218)3
[10] O. Chacaltana, J. Distler and A. Trimm, A Family of 4D N = 2 Interacting SCFTs from the
Twisted A2N Series, arXiv:1412.8129 [INSPIRE].
[11] O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the twisted E6 theory,
arXiv:1501.00357 [INSPIRE].
arXiv:1601.02077 [INSPIRE].
arXiv:1704.07890 [INSPIRE].
arXiv:1802.09626 [INSPIRE].
[12] O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the Z3twisted D4 theory,
[13] O. Chacaltana, J. Distler, A. Trimm and Y. Zhu, Tinkertoys for the E7 theory,
[14] J. Distler, B. Ergun and F. Yan, Product SCFTs in classS, arXiv:1711.04727 [INSPIRE].
[15] O. Chacaltana, J. Distler, A. Trimm and Y. Zhu, Tinkertoys for the E8 theory,
[16] K. Hori, Consistency condition for vebrane in Mtheory on R5=Z2 orbifold, Nucl. Phys. B
539 (1999) 35 [hepth/9805141] [INSPIRE].
[arXiv:1110.0531] [INSPIRE].
[17] Y. Tachikawa, On Sduality of 5d super YangMills on S1, JHEP 11 (2011) 123
[18] A.D. Shapere and Y. Tachikawa, Central charges of N = 2 superconformal eld theories in
four dimensions, JHEP 09 (2008) 109 [arXiv:0804.1957] [INSPIRE].
D 94 (2016) 065012 [arXiv:1509.00847] [INSPIRE].
[arXiv:1605.08337] [INSPIRE].
(2015) 1359 [arXiv:1312.5344] [INSPIRE].
[22] C. Beem et al., In nite chiral symmetry in four dimensions, Commun. Math. Phys. 336
[23] Y. Wang and D. Xie, Codimensiontwo defects and ArgyresDouglas theories from
outerautomorphism twist in 6d (2; 0) theories, arXiv:1805.08839 [INSPIRE].
[24] F. Benini, S. Benvenuti and Y. Tachikawa, Webs of vebranes and N = 2 superconformal
eld theories, JHEP 09 (2009) 052 [arXiv:0906.0359] [INSPIRE].
[25] O. Aharony and A. Hanany, Branes, superpotentials and superconformal xed points, Nucl.
Phys. B 504 (1997) 239 [hepth/9704170] [INSPIRE].
[8] O. Chacaltana , J. Distler and A. Trimm , Tinkertoys for the E6 theory , JHEP 09 ( 2015 ) 007 [9] O. Chacaltana , J. Distler and A. Trimm , SeibergWitten for Spin(n) with spinors , JHEP 08 [19] E. Witten , An SU(2) anomaly , Phys. Lett. B 117 ( 1982 ) 324 [INSPIRE].
[20] Y. Wang and D. Xie , Classi cation of ArgyresDouglas theories from M 5 branes , Phys. Rev.
[21] G. Zafrir , Compacti cations of 5d SCFTs with a twist , JHEP 01 ( 2017 ) 097 [26] O. Aharony , A. Hanany and B. Kol , Webs of (p; q) vebranes, vedimensional eld theories and grid diagrams , JHEP 01 ( 1998 ) 002 [ hep th/9710116] [INSPIRE].
[27] E. Witten , Solutions of fourdimensional eld theories via Mtheory , Nucl. Phys. B 500 [28] E.G. Gimon , On the Mtheory interpretation of orientifold planes , hepth/9806226