Sfolds and 4d \( \mathcal{N} \) = 3 superconformal field theories
Accepted: May
0 University of Tokyo , Kashiwa, Chiba 2778583 , Japan
1 University of Tokyo , Bunkyoku, Tokyo 1130033 , Japan
2 Weizmann Institute of Science , Rehovot 7610001 , Israel
3 Department of Physics, Faculty of Science
Sfolds are generalizations of orientifolds in type IIB string theory, such that the geometric identi cations are accompanied by nontrivial Sduality transformations. They were recently used by Garc aEtxebarria and Regalado to provide the of four dimensional N =3 superconformal theories. In this note, we classify the di erent variants of these N =3preserving Sfolds, distinguished by an analog of discrete torsion, using both a direct analysis of the di erent torsion classes and the compacti cation of the Sfolds to three dimensional Mtheory backgrounds. Upon adding D3branes, these variants lead to di erent classes of N =3 superconformal eld theories. We also analyze the holographic duals of these theories, and in particular clarify the role of discrete gauge and global symmetries in holography.
Extended Supersymmetry; Brane Dynamics in Gauge Theories; FTheory

3 superconformal
eld theories
Dbranes
2.1
2.2
3.1
3.2
3.3
3.4
4.1
4.2
4.2.1
M2brane and D3brane charges
5 Special cases and N = 4 enhancement
1 Introduction and summary of results
2 Preliminary analysis
3 Holographic dual
D3branes on Sfolds and their discrete symmetries
1
Introduction and summary of results
Field theories with superconformal symmetry are useful laboratories for learning about
the behavior of quantum
eld theories in general, and strongly coupled
eld theories in
particular. This is because the superconformal symmetry allows many computations to
be performed in these theories, using methods such as localization, integrability, and the
superconformal bootstrap.
The
eld theories (above two dimensions) about which the most is known are N =
4 superconformal eld theories (SCFTs) in four dimensions, that have been called the
`harmonic oscillator of quantum
eld theories'. These theories have an exactly marginal
deformation, and it is believed that they are all gauge theories with some gauge group
G, such that the exactly marginal deformation is the gauge coupling constant. In these
theories many observables have already been computed as functions (trivial or nontrivial)
of the coupling constant, and there is a hope that they can be completely solved.
Four dimensional theories with N = 2 superconformal symmetry have also been
extensively studied. Some of these theories have exactly marginal deformations and
correobservables can be computed also in N = 2 SCFTs, at least if they have a weak coupling
limit, they are connected by renormalization group ows to theories that have such limits,
or they are one of the theories of class S.
Four dimensional N
= 3 theories should naively provide an intermediate class of
theories, that is more general than N = 4, but such that more computations can be made
than in general N = 2 theories. N = 3 SCFTs (that are not also N = 4 SCFTs) have no
exactly marginal deformations [5, 6] and thus no weak coupling limits that would aid in
classifying and performing computations in these theories. Until recently no N = 3 SCFTs
were known, but recently a class of such theories was constructed by Garc aEtxebarria and
Regalado in [7]. Their construction uses a generalization of orientifolds in string theory.
Orientifold 3planes (the generalization to other dimensions is straightforward) are
dened in type IIB string theory as planes in spacetime, such that in the transverse space
y 2 R6 to these orientifolds, there is an identi cation between the points y and ( y), but
with opposite orientations for strings (or, equivalently, with an opposite value for the B2
and C2 2form potentials of type IIB). This means that we identify con gurations related by
a Z2 symmetry that involves a spatial re ection in the transverse SO(6), and also a
transformation ( I) in the SL(2; Z) Sduality group of type IIB string theory. This breaks half
of the supersymmetry, preserving a four dimensional N = 4 supersymmetry. In particular,
putting N D3branes on the orientifold (these do not break any extra supersymmetries)
gives at low energies four dimensional N = 4 SCFTs.
In [7] this was generalized to identifying con gurations related by a Zk symmetry, that
acts both by a (2 =k) rotation in the three transverse coordinates in C
3 = R6, and by an
element of SL(2; Z) whose k'th power is the identity. We will call the xed planes of such
transformations Sfolds;1 for k = 2 they are the same as the usual orientifolds. Viewing
SL(2; Z) as the modular group of a torus, the Zk Sduality transformation may be viewed
as a rotation of the torus by an angle (2 =k); such a rotation maps the torus to itself if and
only if k = 3; 4; 6 and its modular parameter is
= e2i =k, so Sfolds of this type exist only
for these values of k and . It is natural to de ne such an identi cation using Ftheory [17],
in which the SL(2; Z) Sduality group is described as adding an extra zerosize torus whose
modular parameter is the coupling constant
of type IIB string theory; in this language
the Sfolds of [7] are the same as Ftheory on (C3
T 2)=Zk. This speci c Zk identi cation
preserves a four dimensional N = 3 supersymmetry. So, putting N D3branes on the Sfold
gives at low energies theories with N = 3 superconformal symmetry. Note that D3branes
sitting on the Sfold are invariant under all the transformations discussed above, and, in
particular, the D3brane charge of the Sfold is wellde ned.
In this note we analyze further the theories constructed in [7]. We focus on asking
what are the extra parameters associated with Sfolds, analogous to discrete torsion for
1This term, rst coined in [8], generalizes the term Tfolds that is used to describe identi cations by
elements of the Tduality group. Sfolds of string theory involving Sduality twists together with shifts
along a circle were studied, for instance, in [9{13], and similar Sfolds were studied in the 4d N = 4 SYM
theory in [14{16].
{ 2 {
orbifolds and orientifolds (namely, to nontrivial con gurations of the various pforms of
type IIB string theory in the presence of the Sfold). Sfolds with di erent parameters can
carry di erent D3brane charges. Upon putting N D3branes on the Sfold, these discrete
parameters label di erent N = 3 SCFTs. Our main result is a classi cation of these extra
parameters (following a preliminary discussion in [7]); we show that there are two variants
of Sfolds with k = 3; 4 and just a single variant with k = 6. Each variant leads to di erent
N = 3 SCFTs, with di erent central charges and chiral operators.
In some cases discrete global symmetries play an interesting role. In a speci c type
of orientifold (the O3
plane), the theory of N D3branes on the orientifold is an O(2N )
gauge theory, and this may be viewed as an SO(2N ) gauge theory in which a discrete
the e ects of discrete symmetries. In section 3 we discuss the holographic duals of the N = 3
SCFTs, and the realization of the discrete symmetries there. We show that holography
suggests that only some of the possibilities found in the naive analysis are consistent. The
compacti cation of Ftheory on (C3
T 2)=Zk on a circle gives Mtheory on (C3
T 2)=Zk,
with each Sfold splitting into several C4=Zl singularities in Mtheory. In section 4 we show
that the consistency of this reduction implies that indeed only the possibilities found in
section 3 are consistent. In section 5 we discuss the fact that for speci c values of k and
N some variants of the N = 3 theories have enhanced N = 4 supersymmetry, with gauge
groups SU(3), SO(5) and G2, and check the consistency of this. This provides new brane
constructions and new (strongly coupled) AdS duals for these speci c N = 4 SCFTs.
In this paper we discuss only the Sfolds which give rise to four dimensional N = 3
supersymmetric theories. It would be interesting to study Sfolds that preserve di erent
amounts of supersymmetry, and that have di erent dimensions. A particularly interesting
case, which should have many similarities to our discussion, is Sfolds preserving four
dimensional N = 2 supersymmetry. We leave the study of these theories to the future.
2
Preliminary analysis
In this paper we study the Sfolds introduced by Garc aEtxebarria and Regalado in [7],
which are equivalent to Ftheory on (C3
T 2)=Zk for k = 3; 4; 6, generalizing the standard
orientifold 3planes that arise for k = 2. Just as the orientifold 3planes have four variants,
O3 , Of3 , O3+ and Of3+, that di er by discrete uxes, we expect that these new Sfolds
could also have a few variants. In this section we perform a preliminary analysis of the
possible variants, from the viewpoint of the moduli space of the N = 3 superconformal eld
theories realized on N D3branes probing these Sfolds. In the following sections we will
discuss additional constraints on the possible variants.
{ 3 {
We start from the following three properties of theories of this type, that are generally
believed to hold:
The conformal anomalies (central charges) a and c are equal in any N = 3 SCFT [5]:
a = c:
The value of a = c determines also all anomalies of the SU(3)R
U(1)R symmetries
in these theories.
HJEP06(21)4
The geometry of the gravitational dual of N D3branes sitting on a singularity
involving a Zk identi cation of their transverse space, and carrying
units of D3brane
charge, is AdS5
S5=Zk, with (N + ) units of 5form
large N central charge of the corresponding CFTs is
ux. This implies that the
a
c
k(N + )2=4 + O(N 0):
In any N = 2 theory, the Coulomb branch operators (chiral operators whose
expectation values label the Coulomb branch of the moduli space, namely the space of
vacuum expectation values of the scalars in the vector multiplets) form a ring
generated by n operators, whose expectation values are all independent without any
relation. Furthermore, up to a caveat mentioned below, there is a relation
n
i=1
2a
c =
X(2 i
1)=4;
where
operators.
i are the scaling dimensions of the generators of the Coulomb branch
The property (2.2) can be derived easily following the analysis of [18{20]. Essentially,
the curvature of the AdS space is supported by the total energy of the veform
eld, and
the volume of the Zk quotient is 1=k of the volume before the quotient, changing the value
of Newton's constant on AdS5 by a factor of k.
The equation (2.3), originally conjectured in [21], was given a derivation that applies
to a large subclass of N = 2 theories in [22] (though it is not clear that this subclass includes
the theories we discuss here). More precisely, this relation applies only to gauge theories
whose gauge group has no disconnected parts. To illustrate this, consider the N = 4 super
YangMills theories with gauge groups U(1) and O(2). They have the same central charges
2a
c = 1=4, coming from a single free vector multiplet, but the gaugeinvariant Coulomb
branch operator has dimension 1 for the former and 2 for the latter, so that property 3
only holds for the former. This is because the scalars in the vector multiplet change sign
under the disconnected component of the O(2) gauge group. The O(2) theory is obtained
by gauging a Z2 global symmetry of the U(1) = SO(2) theory, where the Z2 symmetry is
generated by the quotient O(2)=SO(2); this operation does not change the central charges.
{ 4 {
(2.1)
(2.2)
(2.3)
This means that the O(2) theory includes a Z2 gauge theory, implying that it has a
nontrivial Z2 2form global symmetry in the sense of [23], as we further discuss below. We
assume in the following that the relation (2.3) holds whenever the theory does not have
any nontrivial 2form global symmetry, even if the theory does not have a gauge theory
description.
2.2
D3branes on Sfolds and their discrete symmetries
Let us now see what the three properties recalled above tell us about the properties of
Sfolds. Let us probe the Zk Sfold by N D3branes. The moduli space of N = 3 theories
is described by the expectation values of scalars in N = 3 vector multiplets, which are
identical to N = 4 vector multiplets; viewing an N = 3 theory as an N = 2 theory, these
decompose into a vector multiplet and a hypermultiplet. Let us choose a speci c N = 2
subgroup of the N = 3 symmetry, and consider the component of the moduli space which is
a Coulomb branch from the point of view of this N = 2 subgroup. This implies that all the
D3branes lie on a particular C=Zk within C3=Zk. Denote by zi (i = 1; : : : ; N ) the positions
on C of these D3branes. Since the D3branes are identical objects, the identi cations that
are imposed on the moduli space by the geometry are
erators are the symmetric polynomials of zik, and their generators are (PiN=1 zijk) for
j = 1;
; N , with dimensions
From (2.1) and (2.3), this naively implies that
k; 2k;
; N k:
4a = 4c = kN 2 + (k
1)N:
Applying (2.2), we nd that the D3brane charge of the Zk Sfold is
= (k
1)=(2k).
As discussed above, we know that even for k = 2 this is not the whole story, due to the
possibility of discrete torsion and discrete symmetries. Let us review in detail this k = 2
case, in a language that will be useful for our later analysis. Among the known O3planes,
the above analysis only applies to Of3 , O3+ and Of3+, for which
= (k
1)=(2k) = +1=4
is the correct value. When the orientifold is O3 , the O(2N ) gauge group arising on N
D3branes can be viewed as a Z2 gauging of the SO(2N ) gauge theory, and thus, to compute
a and c, we need to use the SO(2N ) theory instead of the O(2N ) theory. In the SO(2N )
theory the residual gauge symmetries on the Coulomb branch, acting on the eigenvalues of
a matrix in the adjoint representation of SO(2N ), are generated by
matrix of SO(2N ), z1z2 : : : zN . From this list we can compute a and c again using (2.1)
and (2.3) and nd
(2.8)
(2.9)
from which we nd
=
1=4 from (2.2). This is indeed the correct value for O3 .
The SO(2N ) theory has a discrete Z2 global symmetry, corresponding to gauge
transspace one of these transformations acts as z1 !
z1, and with this extra identi cation the
group generated by (2.7) becomes the same as the group (2.4) acting on the D3branes.
In particular, this identi cation projects out the Pfa an operator of dimension N , such
that after it we obtain the Coulomb branch operators with dimensions (2.5). So, for this
speci c orientifold plane, the theory on the D3branes is described by a Z2 quotient of
some `parent' theory which has a di erent group of identi cations (2.7), and
correspondingly di erent Coulomb branch operators. Only after gauging a Z2 global symmetry of this
`parent' theory do we get the theory of the D3branes on the O3
orientifold. We stress
here that this gauging of Z2 symmetry does not change the central charges.
It is natural to ask how we could know directly from the theory of the D3branes
on the O3
plane that its central charges are not the same as the naive ones, because it
arises as a Z2 gauging of some other theory. As discussed in [23, 24], when we have a
discrete Zp global symmetry, we have local operators that transform under this symmetry,
as well as 3plane operators that describe domain walls separating vacua that di er by a Zp
transformation. When we gauge the Zp global symmetry, these local and 3plane operators
disappear from the spectrum. Instead we obtain new 2plane operators (that may be viewed
as worldvolumes of strings), characterized by having a Zp gauge transformation when we
go around them. These 2plane operators are charged under a 2form Zp global symmetry
in the language of [23]. So whenever we have a theory with a 2form Zp global symmetry,
it is natural to expect that it arises by gauging a Zp global symmetry of some `parent'
theory. And indeed, the analysis of [23] implies that this `parent' theory can be obtained
by gauging the 2form Zp global symmetry. Thus, whenever we have a theory with a Zp
2form global symmetry, we expect that its central charges would not be given by (2.3),
but rather by those of its `parent' theory.2
Suggested by this analysis for k = 2, we expect that also for k = 3; 4; 6, di erent
versions of Sfolds will be characterized by di erent 2form Zp
Zk global symmetries
for the corresponding theories on the D3branes, that will imply that these theories arise
from `parent' theories with Zp global symmetries. The analysis above suggests that the
identi cations on the moduli spaces of these `parent' theories should be subgroups of the
2Gauging a discrete Zp global symmetry does not change the dynamics on R4, but it does change the
spectrum of local and nonlocal operators.
{ 6 {
group generated by (2.4), such that the ring of invariants is polynomial without any relation,
and such that adding an extra Zp generator produces the group (2.4). Luckily such groups
are already classi ed and are known as ShephardTodd complex re ection groups; the fact
that the ring of invariants is polynomial without any relation if and only if the group acting
on CN is a complex re ection group is known as the theorem of Chevalley, Shephard and
Todd.3 For more on complex re ection groups, see [26].
In our case, the available groups are known as G(N; p; k), and are generated by the
following elements:
(zi; zj ) 7! ( zi;
zi 7!
pzi;
1zj );
where p is a divisor of k. By adding another Zp generator acting as zi 7!
zi, all these
groups become the groups (2.4). Denoting ` = k=p, we see that if there is a `parent' theory
with the identi cations (2.10) on its moduli space, then its gaugeinvariant Coulomb branch
operators would be generated by the symmetric polynomials of zik and by (z1z2
and therefore the dimensions of the Coulomb branch generators would be given by
k; 2k; : : : ; (N
1)k; N `:
From (2.1) and (2.3) we then nd that the central charges of the `parent' theory, and also
of its Zp quotient that would describe the theory of N D3branes on the corresponding
Sfold, are given by
4a = 4c = kN 2 + (2`
k
1)N:
Note that even though the Zpgaugings of the theories with di erent values of ` all have
the same moduli space, they are distinct theories with di erent central charges. When p
is not prime, one can also gauge subgroups of Zp, giving rise to additional theories, which
again are not equivalent to the theories for other values of ` (even though they may have
the same moduli space).
Therefore, it seems at this stage, that each Zk Sfold with k = 2; 3; 4; 6 can have
variants labeled by ` which is an integer dividing k. The D3brane charges k;` of these
Sfolds can be easily found using (2.2) and (2.12):
(2.10)
zN )`,
(2.11)
(2.12)
(2.13)
` = 1 ` = 2 ` = 3 ` = 4 ` = 6
Note that just from the analysis above it is far from clear that Sfolds and N = 3
theories corresponding to such identi cations actually exist. However, we expect any
Sfold to fall into one of these categories. In the next two sections, we will see that the
3See [25] for a recent use of ShephardTodd complex re ection groups in fourdimensional N = 2 theories.
{ 7 {
Sfold variants that really exist are only those shaded in the table (2.13) above, rst by
carefully studying the holographic dual and then by comparing with Mtheory. Note also
that there could be more than one theory with the same identi cations. For k = 2 there
are three orientifoldtypes with the same ` = 2 identi cations, though they all give rise to
the same theory on D3branes because they are all related by Sduality. We do not know
the corresponding situation in our case. We also cannot rule out the existence of N = 3
theories having Coulomb branch operators (2.11) also for the values of ` and k that do not
come from Sfolds. It would be interesting to shed further light on these questions, perhaps
by a conformal bootstrap analysis of N = 3 SCFTs, or by analyzing the two dimensional
chiral rings of the corresponding theories [27, 28].
3
3.1
Holographic dual
AdS duals of CFTs with discrete symmetries
Theories of quantum gravity are not expected to have any global symmetries (see, for
instance, [24]). There are very strong arguments that this is the case for continuous
symmetries, and it is believed to be true also for discrete symmetries. In the AdS/CFT
correspondence, this implies that any global symmetry in the CFT should come from a
gauge symmetry in antide Sitter (AdS) space.
For continuous global symmetries it is known that indeed they come from gauge elds
in the bulk, with a boundary condition that sets their eld strength to zero on the boundary.
In some cases (like AdS4 [29]) there is also another consistent boundary condition in which
the eld strength does not go to zero on the boundary, and its boundary value becomes a
gauge eld in the CFT. Changing the boundary condition may be interpreted as gauging
the global symmetry in the CFT (in cases where this gauging still gives a CFT).
In this section we discuss the analogous statements for discrete symmetries. We focus
on the AdS5 $ CFT4 case, but the generalization to other dimensions is straightforward.
The discussion here is a special case of the discussion in appendix B of [23], generalized to
ve dimensions, but as far as we know its implications for the AdS/CFT correspondence
were not explicitly written down before (see also [30]).
Consider a Zk gauge symmetry on AdS5 (thought of as a sector of a full theory of
quantum gravity on AdS5, that is dual to a 4d CFT). A universal way to describe such
a gauge symmetry in
ve spacetime dimensions is by a topological theory of a 1form A
and a 3form C, with an action
A can be thought of as the gauge eld for a Zk symmetry; for example, the action above
arises from a U(1) gauge symmetry that is spontaneously broken to Zk. The forms A
and C are both gauge elds, whose eld strengths dA and dC vanish by the equations of
motion. There are gauge transformations that shift the integrals of A and C over closed
cycles by one. A gauge symmetry is just a redundancy in our description, but an invariant
property of this theory is that it has line and 3surface operators, given by ei H A and ei H C ,
L =
ik
2
A ^ dC:
{ 8 {
(3.1)
multiplies the line operators ei H A by e2 i=k, and a similar 3form global symmetry.
with A and C integrated over closed cycles. And, it has a 1form Zk global symmetry that
When we put such a theory on AdS5, we need to choose boundary conditions; the
possible boundary conditions for such topological gauge theories were discussed in [23].
The variational principle implies that we need to set to zero A ^ C along the boundary. If
we set to zero C along the boundary, then the boundary value of A gives a gauge eld in
the dual CFT, corresponding to a Zk gauge symmetry in this CFT. With this boundary
condition line operators in the bulk are allowed to approach the boundary and to become
line operators in the CFT, while 3surface operators cannot approach the boundary, but
can end on the boundary, giving 2surface operators in the CFT. This is as expected for a
Zk gauge symmetry in four spacetime dimensions.
On the other hand, if we set A to zero on the boundary, we obtain a Zk global symmetry
in the dual CFT. The line operators ending on the boundary now give local operators in the
CFT, and the 1form global symmetry in the bulk becomes a standard Zk global symmetry
in the CFT, under which these local operators are charged. The 3surface operators going
to the boundary give 3surface operators in the CFT, which are domain walls between
di erent vacua related by Zk.
Thus, whenever we have a Zk gauge symmetry on AdS5, there are two natural boundary
conditions. One of them gives a conformal eld theory with a Zk global symmetry, and the
other gives a conformal eld theory with a Zk gauge symmetry (and a Zk 2form global
symmetry). The second theory is related to the rst one by gauging its Zk global symmetry,
and similarly the rst one arises from the second by gauging its Zk 2form global symmetry.
When k is not prime, there are also additional possible boundary conditions, corresponding
to gauging subgroups of Zk.
In the context of our discussion in the previous section, this implies that the `parent'
theories and their Zkgaugings should arise from the same holographic dual, just with
di erent boundary conditions for the Zk gauge elds. We will see in section 3.4 how the
Zk gauge elds of (3.1) arise in Ftheory on AdS5
(S5
T 2)=Zk from integrals of the
type IIB
veform
eld F5 on discrete cycles. Note that all this applies already to the
AdS5
S5=Z2 case discussed in [31]; typically in that case only the option of having a
global Z2 symmetry, leading to the SO(2N ) gauge theory, is discussed.
Finally, note that a very similar story occurs already in type IIB string theory on
AdS5
S5, which includes a topological sector in the bulk corresponding to a 1form ZN
gauge symmetry, with an action i2N B2 ^ dC2 (where B2 and C2 are the 2form
elds of type
IIB string theory) [32, 33]; this topological theory was discussed in section 6 of [23]. In this
case both simple boundary conditions give rise to a 1form ZN global symmetry in the dual
CFT, and the resulting theories are SU(N ) and SU(N )=ZN gauge theories [23, 34, 35]. In
this speci c case there is also an option of coupling these theories to a continuous U(1)
gauge symmetry, leading to a U(N ) theory [36, 37]. One can also quantize the topological
theory in ways that do not lead to local eld theories [38].
{ 9 {
In [31], Witten showed how to characterize the variants of O3planes by studying the
discrete torsion on S5=Z2. In this section we generalize this to S5=Zk Sfolds, for k = 3; 4; 6.
For k = 2, the discrete torsion of the NSNS and of the RR threeform
eld strengths
takes values in H3(S5=Z2; Z~), where the tilde over Z means that the coe cient system is
multiplied by ( 1) when we go around the Z2 torsion 3cycle of S5=Z2. For k = 3; 4; 6, we
have a Zk torsion 3cycle in S5=Zk, and we have an action of an element
NSNS and RR
eld strengths in Z
Z when we go around the Zk torsion cycle of S5=Zk.
2 SL(2; Z) on the
We can choose a speci c form for this , given, say, by
=
for k = 3,
=
these matrices are
and
1. The discrete torsion is then given by H3(S5=Zk; (Z
Z) ),
and its computation is standard in mathematics.4
In general, H (S2n 1=Zk; A) where A is a Zkmodule is given by the cohomology of
the complex
C0
1 t
!
C1
1+t+ +tk 1
!
C2
1 t
!
1+t+ +tk 1
!
C2n 2
1 t
!
C2n 1 (3.2)
where all Ci ' A, t is the generator of Zk, and the di erential d is alternately given
by the multiplication by 1
t or by 1 + t +
(1
t)(1 + t +
+ tk 1) = 1
+ tk 1. It is easy to see that d
When k = 2, t is just the multiplication by
1. Then 1 + t = 0 and 1
t = 2, from
which we conclude H3(S5=Z2; (Z
Z) ) = Z2
Z2, reproducing four types of O3planes.
When k = 3; 4; 6, the action
of the generator of Zk on Z
Z obeys 1+ +
+ k 1 = 0,
and det(1
) = 3; 2; 1 for k = 3; 4; 6, respectively, and therefore
8
>>Z3 (k = 3);
Z2 (k = 4);
>
>:Z1 (k = 6):
This gives the discrete torsion groups arising from the 3form
elds of type IIB string
theory on these Sfolds.
For k = 3, we have three di erent possibilities, but two nontrivial elements of Z3 are
related by conjugation in SL(2; Z), so up to Sduality transformations there are just two
types of Sfolds with k = 3 (in the same sense that up to Sduality there are just two
types of O3planes that give di erent theories for the D3branes on them). For k = 4, the
cohomology is Z2, and therefore we expect two types of Sfolds. Similarly, for k = 6, there
is only one type of Sfold.
3.3
Generalized Pfa ans
In the holographic duals of the Sfolds, the discrete torsion described above corresponds to
discrete 3form
uxes on the S3=Zk discrete 3cycle in S5=Zk, and it a ects the spectrum
4The method to compute cohomologies with twisted coe cients is explained in e.g. Hatcher [39] chapter
example 2.43. Section 5.2.1 of DavisKirk [40] was also quite helpful.
of wrapped branes on this 3cycle. We can use this to match the discussion of the previous
subsection to our analysis of section 2.
For k = 2 this was analyzed in [31]. There is a Z2torsion threecycle of the form S3=Z2
within S5=Z2, and only when the discrete torsion is zero, i.e. when the 3plane is O3 , we
can wrap a D3brane on this cycle. By analyzing the properties of this wrapped D3brane
we
nd that it corresponds to a dimension N operator in the dual CFT, which can be
naturally identi ed with the Pfa an operator, that only exists in the SO(2N ) theory but
not in SO(2N + 1) or Sp(N ) theories. Note that, according to the discussions above, this
operator exists in the `parent' SO(2N ) theory, but not after we gauge the Z2 to get the
O(2N ) theory. So in the AdS dual it exists when we choose the boundary condition for the
Z2 gauge theory in the bulk that gives a Z2 global symmetry, corresponding to the `parent'
theory, but not for the other boundary condition, that corresponds to the theory on the
D3branes.
The obstruction to wrap D3branes on S3=Z2 can be understood as follows. The NSNS
3form
ux G can in general be in a nontrivial cohomological class. But when pulledback
to the worldvolume of a single D3brane, G is the exterior derivative of a gaugeinvariant
object B
F , where F is the gauge eld on the D3brane, and therefore the cohomology
class [G] should be trivial. The argument for the RR
ux is the Sdual of this.
For all k = 3, 4, 6, there is a Zktorsion threecycle in S5=Zk of the form S3=Zk. When
the discrete torsion is zero, there is no obstruction to wrapping a D3brane on this cycle.
The scaling dimension of this wrapped D3brane can be easily found to be kN=k = N . This
matches the scaling dimensions we found for the ` = 1 variants of section 2. So we identify
the Sfold with no discrete torsion with the ` = 1 case (either the `parent' theory, or its Zk
gauging that gives the theory on the D3branes, depending on the boundary conditions).
For k = 6, there is nothing more to discuss, since H3(S5=Zk; (Z
Z) ) itself is trivial,
so we do not nd any variant except ` = 1.
For k = 3, when the discrete torsion in H3(S5=Zk; (Z
Z) ) = Z3 is nontrivial, we
cannot wrap a D3brane on this cycle. So this should correspond to the ` = 3 variant (with
no discrete symmetries).
The most subtle is the k = 4 case, when the discrete torsion is given by an element in
Z) ) = Z2. We claim that the nontrivial element of this discrete torsion
gives the ` = 4 variant of section 2, not the ` = 2 variant.
To see this, it is instructive to recall why in the case k = 2 we can wrap two D3branes
on S3=Z2 even with the discrete torsion. Note that it is not just that for N D3branes
wrapping on the same cycle, the triviality of N [G] su ces. For one thing, if we have two
D3branes wrapping on the same locus with at least U(1)
U(1) unbroken, then B
F1
and B
F2 are both gaugeinvariant (where F1;2 are the U(1) gauge elds coming from the
rst and the second ChanPaton factor), and therefore [G] still needs to be zero.
To wrap two D3branes on S3=Z2 with discrete torsion consistently, one needs to require
that the ChanPaton indices 1,2 are interchanged when we go around the Z2 cycle. This
means that in fact there is a single connected D3brane of the shape S3, which is wrapped
on S3=Z2 using a 2:1 quotient map. In this particular setting, we know how to judge the
consistency of wrapping: on S3, (B
F ) is a gaugeinvariant object, so [G] should vanish
there. What needs to be checked is then to pull back the spacetime G using the map
S
3
! S3=Z2 ! S5=Z2;
where the rst is the 2:1 projection and the second is the embedding. The result is zero,
from the trivial fact that H3(S3; Z) = Z does not have nonzero twotorsion elements. Thus
we see that we can wrap two D3branes in this way.
Now, let us come back to the k = 4 case, and try to wrap two D3branes on S3=Z4
with discrete torsion, which should naively be possible since the discrete torsion lies in
Z2. Again, if the ChanPaton structure is trivial, we cannot wrap them. If we try to do
the analogue of the k = 2 case above, we can try wrapping two D3branes such that the
ChanPaton indices 1 and 2 are exchanged. Just as above, this is equivalent to wrapping
one D3brane on S3=Z2, covering S3=Z4 with a 2:1 quotient map. To test the consistency
of the wrapping, one needs to pull back the spacetime obstruction on S5=Z4 via
S3=Z2 ! S3=Z4 ! S5=Z4:
(3.4)
(3.5)
One nds that the pullback is nontrivial, showing that this embedding is inconsistent.5
It is clear that we can wrap four D3branes on the discrete cycle, so this implies that the
theory with this discrete torsion should be the ` = 4 variant discussed in section 2, and
that we cannot have Sfolds with the ` = 2 variant.
This does not prove, we admit, that there are no other nonAbelian con gurations on
two D3branes on S3=Z4 that still allow the wrapping. But we will see that the choice
` = 4 for this discrete torsion variant matches with the Mtheory computation below.
3.4
Realization of the Zk gauge theory in Ftheory
After these discussions we can understand how the Zk gauge theory of section 3.1 arises in
Ftheory on S5=Zk, when the discrete torsion in H3(S5=Zk; (Z
Z) ) is zero. Recall that
Hn(S5=Zk; Z) is Zk for n = 1 and 3, and its generator is S1=Zk and S3=Zk, respectively.
We can wrap a D3brane on these cycles to obtain a world 3cycle and a worldline on
AdS5. (Note that the latter is the generalized Pfa an particle discussed above.) Clearly,
they both carry Zk charges. We are then naturally led to identify the worldline as coupling
to the 1form A and the world 3cycle to the 3form C in (3.1). For this identi cation to
make sense, it should be the case that if we rotate the generalized Pfa an around an S1
that has a unit linking number with the world 3cycle of the D3brane wrapped on S1=Zk,
there should be a nontrivial Zk holonomy of unit strength.
This can be seen as follows. A D3brane wrapped on S1=Zk creates an F5 ux given
by the Poincare dual of its worldvolume. In this case this is given by s
c, where s is
the generator of H1(S1; Z) where this S1 is linked in AdS5 around the worldvolume of the
D3brane, and c is the generator of H4(S5=Zk; Z) = Zk.
Now, we use the cohomology long exact sequence associated to the short exact sequence
0 ! Z ! R ! U(1) ! 0 to conclude that there is a natural isomorphism
H3(S5=Zk; U(1)) ' H4(S5=Zk; Z) = Zk:
(3.6)
5In general, the pullback map from Sn=Zab to Sn=Zb is a multiplication map by 1 + t + t2 + : : : + ta 1,
where t is the generator of the Za action that divides Sn=Zb to Sn=Zab.
In physics terms, this means that the discrete Zk eld strength (which is Zvalued) with
four legs along S5=Zk can be naturally identi ed with the discrete Zk holonomy (which is
U(1)valued) with three legs along S5=Zk.
From this we see that we have a holonomy of the type IIB C4 eld given by the element
s
c 2 H1(S1; Z)
H3(S5=Zk; U(1));
(3.7)
which can be naturally integrated on the cycle S1
S3=Zk to give exp(2 i=k). This means
that the generalized Pfa an particle wrapped on S3=Zk, when carried around the S1
linking the worldvolume of the D3brane wrapped on S1=Zk, experiences this holonomy.
This is indeed the behavior we expect for the objects charged under the Zk 1form A and
In this section, we consider Mtheory con gurations on (C3
T 2)=Zk. We know that we
obtain such con gurations from any of our Sfolds on a circle, using the standard relation
between type IIB theory on a circle and Mtheory on a torus. However, the opposite is
not true, since when we translate some con guration of discrete
uxes in Mtheory on
(C3
T 2)=Zk to Ftheory on a circle, we could also have some nontrivial action of the
shift around the Ftheory circle, corresponding to a `shiftSfold' where there is a rotation
on the transverse C
3 when we go around the compacti ed S1. By analyzing all possible
discrete charges in Mtheory and translating them to Ftheory, we learn about all possible
variants of Sfolds.
4.1
Discrete
uxes in Mtheory on (C3
Let us start by analyzing the possible discrete uxes in Mtheory, which come from 4form
uxes. For a given k, (C3
T 2)=Zk has several xed points of the form C4=Z`i , each of
which has an associated H4(S7=Z`i ; Z) = Z`i in Mtheory [41]. From this viewpoint, the
possible discrete charges are given by the orbifold actions on the xed points,
M H3(C4=Z`i ; Z) = <>Z3
8
>>Z2
>
>
>
>
>>Z4
>:Z6
Z2
Z3
Z4
Z3
Z2
Z3
Z2
Z2
We can alternatively measure the same charges by considering H4 of the `asymptotic
in nity' of (C3
T 2)=Zk, which has the form (S5
T 2)=Zk. This is a T 2 bundle over
S5=Zk, and as such, one can apply the LeraySerre spectral sequence,6 that says that it
has the ltration
H4((S5
T 2)=Zk; Z) = F 2;2
F 3;1
F 4;0;
6For an introduction on the LeraySerre spectral sequence, see e.g. [42]. In our case, the computation
goes as follows. The second page E2p;q of the spectral sequence is given by E2p;q = Hp(S5=Zk; Hq(T 2; Z)),
with the di erentials d2p;q : E2p;q p+2;q 1. The third page is given by E3p;q = Ker d3p;q=Im dp 2;q+1, and
has the di erentials d3p;q : E3p;q !!EE3p+23;q 2. The second page and the part relevant for us of the third page
(4.1)
(4.2)
where
F 2;2=F 3;1 = H2(S5=Zk; H2(T 2; Z)) = H2(S5=Zk; Z);
F 3;1=F 4;0 = H3(S5=Zk; H1(T 2; Z)) = H3(S5=Zk; (Z
Z) );
F 4;0 = H4(S5=Zk; H0(T 2; Z)) = H4(S5=Zk; Z):
(4.3)
(4.4)
(4.5)
The di erent subgroups here correspond in a sense to the `number of legs along the base
S5=Zk and the
ber T 2'. As we are taking the Zvalued cohomology, we only have a
ltration and it is not guaranteed that the group (4.2) is a direct sum of (4.3){(4.5).
We can easily compute (4.3){(4.5) to obtain
HJEP06(21)4
F 2;2=F 3;1 = Zk;
F 3;1=F 4;0 = <>Z3
;
F 4;0 = Zk:
(4.6)
8
>>Z2
>
>
>
>
>>Z2
>:Z1
Z2 (k = 2)
(k = 3)
(k = 4)
(k = 6)
From the standard Ftheory / Mtheory mapping, we see that each piece (4.3){(4.5)
has the following e ect in the Ftheory language:
The piece F 2;2=F 3;1 becomes a component of the metric of Ftheory on S1.
Concretely, it speci es the amount of the rotation on the transverse space while we go
around S1. Therefore, the system is a plain S1 compacti cation if the ux in this
piece is zero, whereas it is a shift Sfold if it is nonzero.
The piece F 3;1=F 4;0 becomes the discrete RR and NSNS 3form
uxes around the
Sfold. Indeed, the coe cient system H1(T 2; Z) of the ber is exactly the one (Z
Z)
that we discussed in the previous section, and there is a natural isomorphism of
F 3;1=F 4;0 with the discrete torsion (3.3).
The piece F 4;0 becomes an F5
ux having one leg along the S1. As discussed in
section 3.4, we have a natural identi cation H4(S5=Zk; Z) = H3(S5=Zk; U(1)). So,
this piece can also be regarded as specifying a C4 holonomy having one leg along
the S1.
Note that
are then given by
(F 2;2=F 3;1)
(F 3;1=F 4;0)
F 4;0 = <>Z3
2 Z 0 Zk 0 Zk Z
1 0 X 0 X 0 X
0 Z 0 Zk 0 Zk Z
0 1 2 3 4 5
;
8
>>Z2
>
>
>
>
>>Z4
>:Z6
2
1
0
Z2
Z3
Z4
Z3
0 Zk
X
Z2
Z3
Z2
Z2
Zk Z
Z2; Z3; Z2; Z1 for k = 2; 3; 4; 6, respectively. From this we conclude that in fact
F p;q=F p+1;q 1 = E1p;q ' E2p;q for p + q = 4.
this case the map
that
is the `sum' and that
The cases we use are
(M2brane charge) =
1
24
1
k
+
`(k
`)
:
2k
` = 0 ` = 1 ` = 2 ` = 3 ` = 4 ` = 5
is the diagonal embedding. Note that a b is indeed a zero map.
We assume that a similar relation holds also in the other cases. Namely, we assume
a : M H3(C4=Z`i ) ! F 2;2=F 3;1 = Zk
b : F 4;0 = Zk !
M H3(C4=Z`i )
is the `diagonal embedding'. For k = 4 and k = 6, we need to use natural maps such as
Z2 ! Z6 and Z6 ! Z2. We just choose a multiplication by 3 in the former, and the mod2
map in the latter, and similarly for k = 4. We see that the composition a b is indeed a
zero map, which gives a small check of our assumptions.
4.2
M2brane and D3brane charges
Using the discussions above, let us test our identi cation by working out the M2brane and
D3brane charges of our various con gurations. We use the formula of [43] for the charge
of C4=Zk orbifolds with discrete torsion,
and we see that (4.7) and (4.1) are the same as abstract groups. What remains is to gure
out the precise mapping between the two. This is a purely geometrical question since it is
just the relation of the H4 of (C3
T 2)=Zk computed at the origin and at the asymptotic
in nity. As we have not yet done this computation, we will use guesswork, and then verify
that it leads to consistent results for the M2brane and D3brane charges of the various
The k = 2 case can be worked out using the known properties of the orientifolds. In
is given by the sum of the four Z2's, and the map
a : M H3(C4=Z`i ) = (Z2)4 ! F 2;2=F 3;1 = Z2
b : F 4;0 = Z2 !
M H3(C4=Z`i ) = (Z2)4
1
4
1
3
3
8
5
12
A long list of tables analyzing all possible cases follows. The end result is that every
possible Mtheory con guration can be interpreted as a plain S1 compacti cation or a
k = 2 :
k = 3 :
k = 4 :
k = 6 :
shiftSfold on S1 of precisely the Sfolds that we discussed in the previous section (the
types shaded in (2.13)), namely the Sfolds with
(k; `) = (2; 1); (2; 2); (3; 1); (3; 3); (4; 1); (4; 4); and (6; 1):
(4.14)
In all cases we compute the D3brane charge using the sum of the M2brane charges of the
orbifold singularities (4.12) and successfully compare it with our expectation (2.13).
Let us denote the discrete charges of the four xed points as elements in
piece F4;0 = Z2 is generated by ( 00 00 ) and ( 11 11 ). The piece F 2;2=F 3;1 is given by the sum
of the four entries. In the interpretations below, we consider the columns of the matrices
to correspond to O2planes, when we interpret our Mtheory con guration in type IIA by
shrinking one cycle of the torus (as we can do for k = 2). Of course everything should
work out correctly in this k = 2 case, and has been already worked out in [44, 45]. We
reproduce the analysis here since it is a useful warmup for k = 3; 4; 6.
ZZ22 ZZ22 . The
Shift = 0=2 2 F 2;2=F 3;1:
O2+ + O2+
+ 1
4
4
label
IIA
#D3
label
IIA
#D3
label
IIA
#D3
label
IIA
#D3
label
IIA
#D3
label
IIA
#D3
4
( 10 01 )
4
( 10 00 )
f
O2+ + O2
0
O2 + Of2
The rst two come from O3 , but the latter of the two has one additional mobile
D3brane stuck at the origin, due to a nontrivial Wilson line around S1 in the component
of O(2N ) disconnected from the identity. The rest all come from the other three O3planes
wrapped on S1.
Shift = 1=2 2 F 2;2=F 3;1:
O2+ + Of2
f
( 10 11 )
bration
over S1 such that when we go around S1, we have a multiplication by ( 1) on C3. The ones
with D3brane charge 0 are an empty shiftorientifold, and the ones with charge 0 + 1=2
have one D3brane wrapped around S1. Note that the charge 1=2 we are seeing here re ects
the fact that the T 2 of Mtheory is bered over C3=Z2. In the type IIB frame, a ber over
a particular point on S1 is C3, and therefore the D3brane charge is 1 if we integrate over
the asymptotic in nity of this C3.
Let us denote the discrete charges of the three xed points as elements in (Z3; Z3; Z3). The
piece F4;0 = Z3 consists of (0; 0; 0), (1; 1; 1), (2; 2; 2) =
(1; 1; 1). The piece F 2;2=F 3;1 = Z3
corresponding to the shift is given by the sum of the entries. There are 27 choices in total.
HJEP06(21)4
Shift = 0=3 2 F 2;2=F 3;1:
They are the Z3 shiftSfolds, with the rotation angle 2 =3, of the at background on
S1, with 0, 1, 2 D3brane(s) wrapped around at the origin.
label (0; 0; 0) (1; 1; 1) (2; 2; 2)
label (0; 1; 2) (1; 2; 0) (2; 0; 1)
label (0; 2; 1) (1; 0; 2) (2; 1; 0)
13 + 1
13 + 1
label (1; 0; 0) (2; 1; 1) (0; 2; 2)
label (1; 1; 2) (2; 2; 0) (0; 0; 1)
label (1; 2; 1) (2; 0; 2) (0; 1; 0)
Among the rst three, the rst entry has the right D3brane charge to be the (k =
3; ` = 1) Sfold, see (2.13). The other two have one more mobile D3brane, stuck at the
origin through a nontrivial Z3 background holonomy around S1.
The others all have the right D3charge to be the (k = 3; ` = 3) Sfold, see (2.13)
again. The second three and the third three have opposite Z3 charge characterizing the
type of the Sfold.
Shift = 1=3 2 F 2;2=F 3;1:
#D3
#D3
#D3
#D3
#D3
#D3
1
3
3
3
0
2
3
2
3
3
3
2
3
1
3
1
3
They are again the Z3 shiftSfolds, but with the rotation angle 4 =3, of the at
HJEP06(21)4
background on S1, with 0, 1, 2 mobile D3brane(s) wrapped around at the origin.
We denote the charges at the xed points as elements in (Z4; Z4; Z2). The piece F4;0 = Z4
consists of (0; 0; 0), (1; 1; 1), (2; 2; 0), (3; 3; 1).
Shift = 0=4 2 F 2;2=F 3;1:
label (0; 0; 0) (1; 1; 1) (2; 2; 0) (3; 3; 1)
label (0; 2; 1) (1; 3; 0) (2; 0; 1) (3; 1; 0)
38 + 1
38 + 1
+ 3
8
4
+ 1
2
These are Z4 shiftSfolds of at space, with rotation angle =2, and with 0, 1, 2 or 3 mobile
D3brane(s) around S1.
Shift = 2=4 2 F 2;2=F 3;1:
#D3
#D3
#D3
#D3
3
8
38 + 1
18 + 12
+ 18 + 12
Among the rst four, the rst has the right D3brane charge to be the (k = 4; ` = 1)
Sfold, see (2.13). The other three have one more mobile D3brane, stuck at the origin
through nontrivial Z4 background holonomy around S1. All the second four have the
right D3charge to be the (k = 4; ` = 4) Sfold, see (2.13).
Shift = 1=4 2 F 2;2=F 3;1:
These are the Z2 shiftSfolds on S1 of the O3planes.
#D3
0
#D3
label (0; 3; 1) (1; 0; 0) (2; 1; 1) (3; 2; 0)
label (0; 0; 1) (1; 1; 0) (2; 2; 1) (3; 3; 0)
label (0; 2; 0) (1; 3; 1) (2; 0; 0) (3; 1; 1)
18 + 1
+ 18 + 12
charge of O3 is
is 1=2 ( 1=4) =
1=8.
trapped at the origin.
by this Z2 operation.
Shift = 3=4 2 F 2;2=F 3;1:
Among the rst four, the rst has the right D3brane charge to be the Z2
shiftSfold of O3 . Recall that the base of the Mtheory con guration is C3=Z4, but the in
the type IIB description, the
ber at a particular point on S1 is C3=Z2. The D3brane
1=4, and therefore the charge as seen from the Mtheory con guration
The others have one or two additional mobile D3brane(s) trapped at the origin.
Among the second four, the rst and the third have the right D3brane charge to be the
Z2 shiftSfold of Of3+. The second and the fourth have one additional mobile D3brane
Note that we cannot take the Z2 shift Sfold of Of3 or O3+, since they are exchanged
4
0
2
+ 1
4
4
+ 1
2
0
4
These are Z4 shiftSfolds of at space, with rotation angle 3 =2, and with 0, 1, 2 or 3
mobile D3brane(s) around S1.
4.2.4
k = 6
We denote the charges at the xed points as elements in (Z6; Z3; Z2). The piece F4;0 = Z6
consists of (0; 0; 0), (1; 1; 1), (2; 2; 0), (3; 0; 1), (4; 1; 0), (5; 2; 1).
Shift = 0=6 2 F 2;2=F 3;1:
label (0; 0; 0)
(1; 1; 1)
(2; 2; 0)
(3; 0; 1)
(4; 1; 0)
(5; 2; 1)
#D3
5
12
152 + 1
152 + 1
152 + 1
152 + 1
152 + 1
The rst has the right D3brane charge to be the (k = 6; ` = 1) Sfold, see (2.13). The
other
ve have one more mobile D3brane, stuck at the origin through a nontrivial Z6
background holonomy around S1.
Shift = 1=6 2 F 2;2=F 3;1:
label (1; 0; 0) (2; 1; 1) (3; 2; 0) (4; 0; 1) (5; 1; 0) (0; 2; 1)
#D3
0
5
6
2
3
1
2
1
3
1
6
This is the Z6 shiftSfold of at space, with zero to ve D3branes stuck at the origin. The
rotation angle is 2 =6.
label
#D3
(2; 0; 0)
112 + 13
(3; 1; 1)
112 + 1
(4; 2; 0)
112 + 23
(5; 0; 1)
112 + 13
(0; 1; 0)
1
12
(1; 2; 1)
112 + 23
The fth has the correct charge to be the Z3 shiftSfold of the standard O3 plane. Since
the O3 plane itself has an identi cation by the angle
, its Z3 quotient involves the
rotation by
=3 of the transverse space. Note also that the O3 plane has the D3brane
charge
1=4, therefore we see 1=3 ( 1=4) =
1=12 in Mtheory. The others have one or
two additional D3brane(s) at the origin.
Shift = 3=6 2 F 2;2=F 3;1:
label (3; 0; 0) (4; 1; 1) (5; 2; 0) (0; 0; 1) (1; 1; 0) (2; 2; 1)
#D3
16 + 12
16 + 1
16 + 12
1
6
16 + 12
16 + 1
The fourth has the correct charge to be the Z2 shiftSfold of the (k = 3; ` = 1) Sfold. The
others have one or two additional mobile D3brane(s) on top.
Shift = 4=6 2 F 2;2=F 3;1:
label
#D3
(4; 0; 0)
112 + 13
(5; 1; 1)
112 + 23
(0; 2; 0)
1
12
(1; 0; 1)
112 + 13
(2; 1; 0)
112 + 23
(3; 2; 1)
112 + 1
The third has the correct charge to be the Z3 shiftSfold, of rotation angle 2 =3, of
the standard O3 plane. The others have one or two additional D3brane(s) at the origin.
Shift = 5=6 2 F 2;2=F 3;1:
label (5; 0; 0) (0; 1; 1) (1; 2; 0) (2; 0; 1) (3; 1; 0) (4; 2; 1)
#D3
0
1
6
1
3
1
2
2
3
5
6
This is the Z6 shiftSfold of at space, with zero to ve D3branes stuck at the origin.
The rotation angle is 2 5=6.
Comments: note that in the cases with shift 2; 4 2 Z6, we only nd Z3 shiftSfolds of
O3 , but we do not have Z3 shiftSfolds of Of3 , O3+ and Of3+. This is as it should be,
because these three types of O3planes are permuted by the Z3 action.
Similarly, there is no Z2 shiftorientifold of the (k = 3; ` = 3) orientifold, since the Z2
action exchanges the two subtly di erent versions that we denoted by (1; 1; 1) and (2,2,2)
in section 4.2.2.
So far, we saw that N D3branes probing various variants of the Sfolds give rise to N = 3
superconformal eld theories characterized by
(k; `) = (3; 1); (3; 3); (4; 1); (4; 4); (6; 1):
(5.1)
When ` < k we expect to have both `parent' theories and their discrete gaugings. In
this section we discuss some interesting special cases, including cases where the N = 3
supersymmetry is enhanced to N = 4.
As discussed in [5], an enhancement of supersymmetry to N = 4 occurs if and only
if there is a Coulomb branch operator of dimension 1 or 2, since N = 3 supersymmetry
then dictates the presence of extra supercharges. The dimensions of the Coulomb branch
operators of our `parent' N = 3 theories were given in (2.11). For k = 2 we always have
such an enhancement, but for k = 3; 4; 6 we see that it happens just for ` = 1 and N = 1; 2.
Note that the theory with the lowest central charge which does not have any enhancement
is the N = 1, k = ` = 3 theory, whose only Coulomb branch operator has dimension 3.
The central charges of this theory are the same as those of ve vector multiplets. Since
the Coulomb branch operators of N = 3 theories must be integers as shown in section 3.1
of [28], then this must be the `minimal' N = 3 SCFT, assuming the general validity of the
formula (2.3). It would be interesting to test this by a superconformal bootstrap analysis,
generalizing the ones in [46, 47].
Going back to theories with N = 4 supersymmetry, the case N = 1 is rather trivial:
we just have a Coulomb branch operator of dimension one, so the moduli space is just C3,
and we get the N = 4 super YangMills theory with gauge group U(1). So let us discuss
the N = 2 cases.
The spectrum of the Coulomb branch operators of the `parent' theory is given by
>
<
8>2; 3 (k = 3);
2; 4 (k = 4);
>>:2; 6 (k = 6):
(5.2)
These spectra agree with those of an N = 4 super YangMills theory with gauge group
SU(3), SO(5) and G2, respectively. Below we give evidence that indeed, the `parents' of
these Sfold con gurations give rise to these N = 4 super YangMills theories, realized in
a somewhat unusual manner. Note that N = 4 theories always have an exactly marginal
deformation, sitting in the same multiplet as the dimension two Coulomb branch operator,
and our conjectured relation implies that for our N = 2 theories this is the gauge coupling
of these N = 4 gauge theories. Our discussion in the previous sections implies that in the
AdS dual of these `parent' theories, this marginal deformation corresponds to a scalar eld
coming from a D3brane wrapped on the torsion 3cycle; for this speci c case this wrapped
D3brane gives rise to a massless eld.
Again let us limit ourselves to the points on the moduli space corresponding to a
Coulomb branch from the point of view of an N
= 2 description of our SCFTs. In
the N = 2; ` = 1 theories, this subspace is parameterized by z1;2 2 C, with the gauge
symmetry (2.10)
(z1; z2) 7! ( nz2;
nz1);
where
= e2 i=k and n is any integer. The charges in a basis that is natural from this
point of view can be written as (e1; m1; e2; m2).
Naively, one would expect the electric charges e1; e2 to correspond to electric charges
of the corresponding N = 4 theory, but this cannot be the case because of the nontrivial
Sp(4; Z) action on these charges, induced by the SL(2; Z) transformation that accompanies
the identi cation (5.3). So instead we consider the rank2 sublattice containing charges of
the form (Q; Q) where we regard (e; m) 2 Z
Z as a complex number Q = e + m .
Two charges from this sublattice are local with respect to each other. To see this, note
that given Q = e + m
and Q0 = e0 + m0 , their Dirac pairing is
em0
e0m = (QQ0
Q0Q)=(
):
Then the Dirac pairing between the two charges (Q; Q) and (Q0; Q0) is clearly zero. This
means that we can take these charges to be the \electric charges" in the N = 4 description.
We provide some consistency checks for this below.
Now, note that the SL(2; Z) element associated to the Zk orbifold then acts on Q just
by multiplication by . Then the gauge transformation (5.3) acts on this variable Q as
Q 7!
nQ;
For k = 3, Q = !n and Q = (1 + !)!n =
!n 1 where ! = e2 i=3,
For k = 4, Q = in and Q = (1 + i)in,
For k = 6, Q =
n and Q = (1 + ) n.
Clearly they can be identi ed with the roots of SU(3), SO(5), and G2, respectively.
(5.3)
(5.4)
(5.5)
(5.6)
and nz2 for n = 0; : : : ; k
of halfBPS particles are given by
which is a re ection of the complex plane along the line e in=kR. This makes it clear that
the group generated by (5.3) for k = 3; 4; 6 is the Weyl group of SU(3), SO(5) and G2,
respectively.
Let us test our identi cation by looking at halfBPS particles. There's no string
connecting z1 and
nz1, since we know nothing happens when z1 = 0 for ` = 1. So there are
only strings connecting
nz1 and
mz2. Using (5.3) we can always restrict z1 to have a
phase between 0 and 2 =k. Then we just have to consider all (p; q)strings connecting z1
1. Since the IIB coupling constant is
= , the central charges
(p + q )(z1
nz2):
We conjectured that \electric" states have the charge (Q; Q), and then their central
charges are given by Qz1
Qz2. Comparing with (5.6), we see that \electric" objects have
the following Q:
The metric on the moduli space is also correctly mapped to that on the Cartan
subalgebra of these groups. The original metric is dz1dz1 + dz2dz2 on C2, and we choose a
real subspace R2 of the form Qz1
Qz2. Then, two vectors Qz1
Qz2 and Q0z1
0
Q z2 in
R2 have the induced inner product (2Re(QQ0)). Using this, we can easily check that the
vectors listed above have the same inner products as the root vectors of SU(3), SO(5), and
G2, in the normalization that the short roots have length squared 2.
Finally, recall that the dyons of N = 4 SYM have central charges of the form
(p + q Y M )( s
p + q Y M
r
( l
);
(5.7)
where s;l are short and long roots, and r is the length squared of the long roots divided by
that of the short roots. We can check that the spectrum (5.6) can be matched with (5.7)
with the identi cation of the roots given above, if we take Y M =
1=(1 + ), uniformly
for k = 3; 4; 6. As a further check, note that for k = 6 this is exactly the value of Y M for
which the G2 N = 4 theory has a discrete Z6 symmetry [48].
Acknowledgments
The authors would like to thank I. Garc aEtxebarria, D. Harlow, Z. Komargodski, T.
Nishinaka, H. Ooguri and N. Seiberg for useful discussions. The work of OA is supported
in part by an Israel Science Foundation center for excellence grant, by the ICORE program
of the Planning and Budgeting Committee and the Israel Science Foundation (grant number
1937/12), by the Minerva foundation with funding from the Federal German Ministry for
Education and Research, by a Henri Gutwirth award from the Henri Gutwirth Fund for
the Promotion of Research, and by the ISF within the ISFUGC joint research program
framework (grant no. 1200/14). OA is the Samuel Sebba Professorial Chair of Pure and
Applied Physics. The work of YT is partially supported in part by JSPS GrantinAid
for Scienti c Research No. 25870159, and by WPI Initiative, MEXT, Japan at IPMU, the
University of Tokyo.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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