#### Anomaly matching on the Higgs branch

HJE
Anomaly matching on the Higgs branch
Hiroyuki Shimizu 0
Yuji Tachikawa 0
Gabi Zafrir 0
0 Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo
We point out that we can almost always determine by the anomaly matching the full anomaly polynomial of a supersymmetric theory in 2d, 4d or 6d if we assume that its Higgs branch is the one-instanton moduli space of some group G. This method not only provides by far the simplest method to compute the central charges of known theories of this class, e.g. 4d E6,7,8 theories of Minahan and Nemeschansky or the 6d E-string theory, but also gives us new pieces of information about unknown theories of this class.
Anomalies in Field and String Theories; Supersymmetric Gauge Theory
1 Introduction
2 Basic idea
3 Summary of results
4 Six-dimensional theories
G is one of the exceptionals
G is SU(2)
G is SU(3)
Cases with pure gauge anomalies and/or gauge-R anomalies
Instantons of classical groups can be described in terms of the ADHM construction [1],
gauge groups, and if we probe them by lower-dimensional branes, we get supersymmetric
theories whose Higgs branch equals to the instanton moduli spaces of exceptional groups.
Among them we can count the 4d theories of Minahan and Nemeschansky [4, 5] for E6,7,8
instantons and the 6d E-string theory [6, 7].
The theories obtained this way do not usually have any conventional Lagrangian
descriptions, and were therefore rather difficult to study. Even their anomaly polynomials,
or equivalently the conformal central charges assuming that they become superconformal
in the infrared, needed to be computed first with stringy techniques [8, 9] and then with
rather lengthy field theoretical arguments on the Coulomb branch in 4d or on the tensor
always allows us to determine the full anomaly polynomial, when the theory is 6d N = (1, 0),
4d N = 2, or 2d N = (4, 0), and when the Higgs branch is assumed to be the one-instanton
moduli space of some group G. This is because on the generic point of the Higgs branch
the theory becomes free and the unbroken symmetry still knows the SU(2)R symmetry at
the origin.
This method provides the simplest way to compute the anomaly polynomials of 4d
theories of Minahan and Nemeschansky and the 6d E-string theory. But more importantly,
this method gives us new pieces of information about a theory whose Higgs branch is the
one-instanton moduli space of the group G, even when no string/M/F theory construction
is known. For example, in [13], the conformal bootstrap method was used to determine the
conformal central charges of the 4d theory whose Higgs branch is the one-instanton moduli
space of G2 or F4. Our method reproduces the values they obtained, and not only that,
we find a strong indication that the F4 theory does not exist because of a field theoretical
inconsistency. Similarly, we will see that there cannot be any 6d E6,7 theory.
The rest of the paper is organized as follows. In section 2, we describe in more detail
how the anomaly matching on the Higgs branch works if the Higgs branch is the
oneinstanton moduli space of some group G. In section 3, we summarize the results which we
obtained in this paper. Then in section 4, 5, 6, we study the 6d N = (1, 0) theories, the
4d N = 2 theories, and the 2d N = (4, 0) theories in turn. In appendix A, we collect the
formulas for characteristic classes used throughout in this paper.
2
Basic idea
We consider a theory with 6d N = (1, 0) or 4d N = 2 or 2d N = (4, 0) supersymmetry has a
Higgs branch given by the one-instanton moduli space MG of a group G.
Geometric data. Let us first recall some basic information on MG, whose detail can
be found e.g. in [14] and the references therein. The quaternionic dimension of MG is
h∨(G) − 1. We note that for G = Sp(n), the one-instanton moduli space is simply Hn/Z2,
where H is the space of quaternions.
Furthermore, the moduli space is smooth on a generic point, and the symmetry
SU(2)R × G acting on MG is broken to SU(2)D × G′, where SU(2)X × G′ ⊂ G is a particular
– 2 –
SU(n)
SO(n)
Sp(n)
E6
E7
E8
F4
G2
h
∨
n
n − 2
n + 1
12
18
30
9
4
SU(6)
SO(12)
E7
Sp(3)
SU(2)F
U(1)F × SU(n − 2)
(n − 2)−n ⊕ (n − 2)+n
SU(2)F × SO(n − 4)
Sp(n − 1)
2F ⊗ (n − 4)
2n − 2
14′
4
short comment
3-index antisym.
chiral spinor.
3-index antisym. traceless.
subgroup described in more detail below and SU(2)D is the diagonal subgroup of SU(2)R
and SU(2)X . The subgroup SU(2)X is the SU(2) subgroup associated to the highest root
of G and G′ is its commutant within G. The tangent space of MG at a generic point
transforms under SU(2)X × G′ as a neutral hypermultiplet and a charged half-hypermultiplet
in a representation R, with the rule
g = g′ ⊕ su(2) ⊕ R.
(2.1)
Here R is always of the form of the doublet of SU(2)X tensored with a representation R′
of G′. The subgroup G′ and the representation R′ are given in the table 1.
Strategy of the matching.
Now let us explain how the anomaly matching on the Higgs
branch works. At the origin, the theory has the symmetry SU(2)R × G where SU(2)R is
(part of) the R-symmetry.
On a generic point of the Higgs branch, we have a free theory whose unbroken symmetry
is SU(2)D × G′, where SU(2)D is the diagonal subgroup of SU(2)R and SU(2)X . The theory
is a collection of dH = h
∨ − 1 hypermultiplets. One, identified with changing the vev,
is neutral under the unbroken global symmetry. Considering that the scalars in a
halfhyper are doublets of SU(2)R, this hyper should just be a half-hyper in the 2 of SU(2)X .
Additionally we have the remaining dH − 1 hypers which transform as a doublet of SU(2)X
and in some representation R′ of G
′ given in table 1. Since SU(2)X and SU(2)R are
′
identified to be SU(2)D, this amounts to just dH − 1 free hypers in the representation R .
The anomaly of G of the original theory can be found from the anomaly of G′ of the
free theory in the infrared, if G′ is nonempty. This in turn determines the contribution
of SU(2)X to the anomaly of SU(2)D, which then fixes the anomaly of SU(2)R of the
original theory. Even if G′ is empty, this still constrains the anomaly of SU(2)R and G
of the original theory. Along the way, we might find that the anomaly matching cannot
be satisfied, in which case we conclude that such a theory cannot exist. There are cases
– 3 –
where the anomaly polynomials can be arranged to match but the global anomaly fails to
match.1 We call this the global anomaly matching test.
Finally, since the one-instanton moduli space of Sp(n) is Hn/Z2 as explained below, we
should always be able to match the anomaly in this case by n free hypermultiplets gauged
by Z2, or equivalently an O(1) − Sp(n) bifundamental gauged by O(1). This provides us a
simple way to check the computations.
Now that the strategy has been explained, we move on to the details. We first
summarize the results in the next section, and then look at the three cases in turn, in the order
6d, 4d and 2d.
3
Summary of results
In this section we summarize the results we obtain in each spacetime dimensions,
postponing the computational details in the following sections. We assume that there are just free
hypermultiplets on the generic point on the Higgs branch unless otherwise stated.
3.1
Six-dimensional theories
First we consider 6d N = (1, 0) theories. We find that the anomaly polynomials on the
Higgs branch can consistently be matched for
SU(2),
SU(3),
Sp(n),
E8,
and
G2.
(3.1)
• In the SU(2) case, we cannot completely determine the anomaly at the origin; we
find a three-parameter family of solutions (4.9). The result is consistent with one
known example, which is just a free hypermultiplet gauged by Z2.
• In the SU(3) case, we can unambiguously determine the anomaly as in (4.12). But
we do not know any example of 6d theories with this Higgs branch.
• The Sp(n) case reproduces the anomaly polynomial of n free hypermultiplets gauged
by Z2.
• The E8 case reproduces the anomaly of the rank-1 E-string theory.
• The G2 case does not pass the anomaly matching test of the global anomaly, as
detailed in section 4.7
3.2
Four-dimensional theories
Second we consider 4d N = 2 theories. We find that the anomaly polynomials on the Higgs
branch can be consistently matched only for
1The mismatch of the global anomaly associated to πd(G) cannot be canceled by adding topological
degrees of freedom. For details, see section 5 of [15].
– 4 –
HJEP12(07)
SU(2)
SU(3)
SO(8)
Sp(n)
E6
E7
E8
F4
G2
x + 1
k
3
4
1
6
8
12
5
10
3
nv
x
2
3
0
5
7
4
7
3
11
nh
x + 1
4
8
n
16
24
40
12
16
3
6x+1
3x+1
12
a
24
7
12
23
24
n
24
41
24
59
24
95
24
4
3
17
24
c
2
3
7
6
n
12
13
6
19
6
31
6
5
3
5
6
full symmetry.
only for
case suffers from the mismatch of the global anomaly.
The data is summarized in table 2, using the standard notations. The list of the groups
we found here is equal to the list of group compatible with the one-instanton moduli space
as Higgs branches, determined using the conformal bootstrap in [13, 16].2 Note however
that the F4 case does not pass the global anomaly matching test, as will be detailed in
section 5.4.
3.3
Two-dimensional theories
Third, we consider 2d N = (4, 0) theories. In two dimensions, the scalars always fluctuate
all over the moduli space, and the continuous symmetry never breaks. Therefore, it is not
technically correct to speak of the theory at the origin of the moduli space and compare
the anomaly computed at the generic point. Rather, what we do is to match the anomaly
polynomial as calculated using a semi-classical analysis at the generic point using the
unbroken symmetry at that point, with the anomaly polynomial written in terms of the
We find that the anomaly polynomials on the Higgs branch can consistently be matched
and
polynomial expanded as follows:
2Note that in table 4 of [13] Sp(n) is missing. This is because the authors of [13] assumed that the
SU(2)
Sp(n)
SU(3)
SO(8)
F4
E6
E7
E8
G2
n
3
4
5
6
8
12
nv
• The Sp(n) case gives us the anomaly of n free hypermultiplets gauged by Z2.
• The SU(3), SO(8), E6,7,8, F4 cases reproduce the anomaly on a single string in 6d
minimal gauge theories [17, 18]. These theories are also realized as compactifications
of the corresponding 4d N = 2 theories on S2 [19].
• For the SU(2) case, we cannot completely determine the anomalies.
• For the G2 case, we do not know any example of 2d theories with these values of
anomalies.
In two dimensions, we can slightly generalize the situation by allowing the massless
Fermi multiplets on the Higgs branch. The inclusion of Fermi multiplets opens the
possibility of matching the anomaly even for larger SU(n) and SO(n) groups. We analyze several
examples with relatively simple Fermi multiplet spectrum and reproduce the anomaly of
a single string in 6d non-anomalous gauge theory with various matter hypermultiplets, as
we will show in detail in section 6.
3.4
Cases with pure gauge anomalies and/or gauge-R anomalies
In two, four and six dimensions, we find that for larger SU(n) and SO(n) groups, we cannot
consistently match the anomaly polynomial. Still, if we ignore the matching of the terms
associated to U(1)F and SU(2)F (which are subgroups of unbroken flavor symmetries as
given in table 1), we find that our method somehow reproduces the values of anomalies
which one would naively associate to the ADHM gauge theories realizing the one-instanton
moduli spaces of these groups.
– 6 –
In four dimension, these theories are infrared free, and have a mixed gauge-gauge-R
anomaly. Moreover, for SO(odd), the gauge group has the global anomaly. In two and six
dimensions, these theories have a gauge anomaly.
We do not understand why the anomaly matching partially works for these cases. It
seems that the anomalies involving the gauge fields plays the role. We hope to come back
to study this case further.
4
Six-dimensional theories
In this section we perform the analysis for 6d N = (1, 0) theories.
First, let us specialize to the exceptional groups. Since there is no independent quartic
Casimir for exceptional groups, the anomaly polynomial at the origin can be written as
(4.1)
(4.2)
(4.3)
(4.4)
Iorigin = αc2(R)2 + βc2(R)p1(T ) + γp1(T )2 + δp2(T )
8
+
1
4
Tr FG2 κ4
Tr FG2 + λc2(R) + μp1(T ) ,
where c2(R) is the second Chern class of the SU(2) R-symmetry bundle and p1(T ), p2(T )
are the first and second Pontryagin classes of the tangent bundle respectively. We have
also introduced the unknown coefficients α, β, γ, δ, κ, λ, μ to be determined below. On the
generic point of the Higgs branch, using (A.2) we see that the anomaly polynomial (4.1)
On the other hand, the anomaly of free hypers is given as
In order to match (4.2) and (4.3), there should also be no independent quartic Casimir
′
invariant for G . This already excludes G = E6, E7, F4 and the remaining possibilities
When G = E8. Since tr56 FE47 = 23 (Tr FE27)2 and T E7(56) = 6, the anomaly (4.3)
Ihyper =
8
1
α =
which coincides with the anomaly of rank-1 E-string theory determined in [9].
α =
, β = −
, γ =
, δ = −
, κ =
, λ = −
, μ =
. (4.7)
41
108
5
72
4.2
Let us consider the SU(2) case, which is quite exceptional. In this case, the anomaly of the
polynomial of this theory is consistent with (4.9) with α = β = λ = 0.
4.3
The case of SU(3) is also exceptional since the fourth Casimir of SU(3) is zero and we can
take the SCFT anomaly polynomial to be of the form (4.1). Substituting the
decomposition (A.10) to (4.1), we obtain
Igeneric = (α + κ + λ)c2(D)2 + (β + μ)c2(D)p1(T ) + γp1(T )2 + δp2(T )
8
− 3c1(U(1)F )2 (λ + 2κ)c2(D) + μp1(T ) − 3κc1(U(1)F )2 .
(4.10)
– 8 –
In turn the anomaly polynomial of the free hypers is given by:
To our knowledge, a 6d SCFT with this Higgs branch is not known.
In this case, the anomaly of the free hypers is given by
Ihypers = 1
8
48 trfund FS4p(n−1) + 2c2(D)2 +
(2c2(D) + trfund FS2p(n−1))p1(T )
96
+ n 7p1(T )2 − 4p2(T )
Since the purely gravitational part of the anomaly can be reproduced from that of the
free hypers, we focus on the R-symmetry and the flavor symmetry part written as
Iorigin = αc2(R)2+βc2(R)p1(T )+x trfund FS4p(n)+y(trfund FS2p(n))2+trfund FS2p(n) κc2(R)+λp1(T ) .
8
Decomposing the characteristic classes for Sp(n) to those for Sp(n − 1) using (A.3)
and (A.4), we find that the anomaly becomes:
Igeneric = (α + 2x + 4y + 2κ)c2(D)2 + (β + 2λ)c2(D)p1(T )
8
+ x trfund FS4p(n−1) + 4yc2(D) trfund FS2p(n−1)
+ y trfund FS2p(n−1)
+ trfund FS2p(n−1) κc2(D) + λp1(T ) . (4.15)
Comparing (4.13) and (4.15), we find
α = 0,
β = 0,
x =
y = 0,
κ = 0,
λ =
(4.16)
1
96
which coincides with the anomaly of O(1) × Sp(n) half-hyper when we include the purely
gravitational part. This SCFT is the ADHM gauge theory for Sp(n).
1
48
,
2
– 9 –
In this case, the anomaly of the hypermultiplet is given by
8
Ihypers =
(n−4) trfund FF4 +6 trfund FF2 trfund FS2O(n−4)+2 trfund FS4O(n−4)+2c2(D)2
48
+
((n−4) trfund FF2 +2 trfund FS2O(n−4)+2c2(D))p1(T )
96
+(n−3)
7p1(T )2−4p2(T )
5760
−8xc1(U(1)F ) trfund FS3U(n−2)−4(6x+n(n−2)y)c1(U(1)F )2 trfund FS2U(n−2)
−2(n−2)(nκ+4yn+6(n−2)x)c1(U(1)F )2c2(D)+2n(n−2)(x(n2−6n+12)
+2yn(n−2))c1(U(1)F )4+(κ+4y)c2(D) trfund FS2U(n−2)
+λ trfund FS2U(n−2)p1(T )−4n(n−2)λc1(U(1)F )2p1(T ).
4.6
G is of type SU
8
Ihypers =
c2(D)2
24
+
We only need to consider n ≥ 4. Then, the anomaly of the hypermultiplets is given by
48
4
6
−
c2(D)p1(T )
n2(n − 2)c1(U(1)F )2p1(T )
48
−
−
n2c1(U(1)F )2 trfund FS2U(n−2) +
nc1(U(1)F ) trfund FS3U(n−2) + (n − 1)
n4(n − 2)c1(U(1)F )4
24
We take the flavor and R-symmetry part of the anomaly to be given by (4.14) with the
replacement Sp(n) → SU(n). Decomposing the characteristic classes of SU(n) into their
U(1)F × SU(n − 2) counterparts using (A.7) and (A.8), we find:
Igeneric = (α+2x+4y+2κ)c2(D)2+(β+2λ)c2(D)p1(T )+x trfund FS4U(n−2)+y(trfund FS2U(n−2))
8
2
Since the purely gravitational part reproduces the anomaly at the origin, we concentrate
on the part involving the R-symmetry and the flavor symmetry. We write the anomaly at
HJEP12(07)
the origin as
Iorigin = αc2(R)2+βc2(R)p1(T )+x trfund FS4O(n)+y(trfund FS2O(n))2+trfund FS2O(n)(κc2(R)+λp1(T )).
8
We can use equations (A.5) and (A.6) to get:
Igeneric = (α+4x+16y+4κ)c2(R)2+(β+4λ)c2(D)p1(T )+(x+4y)(trfund FF2 )2
8
+(12x+16y+2κ)c2(D) trfund FF2 +x trfund FS4O(n−4)+(16y+κ)c2(D) trfund FS2O(n−4)
+y(trfund FS2O(n−4))2+4y trfund FF2 trfund FS2O(n−4)+λp1(T )(trfund FS2O(n−4)+2 trfund FF2 ).
What we have to do is to match (4.17) and (4.20) and solve for α, β, x, y, κ, λ. We see
that the SU(2)F independent terms can be matched by setting α = − 81 , β = − 116 , x =
214 , λ = 418 , y = κ = 0. These are the values one get for an SU(2) gauge theory with n
half-hypermultiplets though it is anomalous in 6d. However it is not possible to much the
remaining SU(2)F dependent terms so there is no solution in this case.
Matching equations (4.20) and (4.21) we see that the U(1)F independent terms can be
matched by setting α = −x = − 214 , β = −λ = − 418 , y = κ = 0. These are the values one get
for a U(1) gauge theory with n hypermultiplets though it is anomalous in 6d. The U(1)F
dependent terms only match if n = 2 for which this analysis does not apply. Therefore we
conclude that there is no solution in this case.
Finally we consider anomalies under large gauge transformations. These exist only for
groups with π6(G) 6= 0 which are only SU(2), SU(3) and G2 for which π6(SU(2)) =
Z12, π6(SU(3)) = Z6 and π6(G2) = Z3. These anomalies are mapped to one another under
the embedding of SU(2) → SU(3) → G2. When embedded in groups with an independent
fourth Casimir, the global anomaly can match the standard square anomaly.
A hyper in the 7 of G2, one in the 3 of SU(3), and a half-hyper in the 2 of SU(2) both
contribute to the anomaly as the generator of π6(G) for their respective groups [20]. Under
the above mapping the 7 of G2 goes to the 3 + 3 of SU(3) and further to the 2 × 2 + singlets
of SU(2). Therefore the anomaly is consistently mapped across the groups.
The only non-excluded cases where the anomaly might be relevant are SU(2), SU(3)
and G2. For SU(2) and SU(3) the anomaly doesn’t exist on the Higgs branch which implies
that the anomaly vanishes in the SCFT. The situation for G2 is more involved as it is broken
to SU(2) on the Higgs branch where both groups have the discrete anomaly.
Let’s consider the 7 of G2. Under the SU(2)1 × SU(2)2 subgroup of G2, it decomposes
as: 7 → (2, 2) ⊕ (1, 3). As the anomaly must be preserved, and using the fact that the
3 of SU(2) contribute to the anomaly like 8 half-hyper doublets [20], we determine that
SU(2)1 has the same anomaly as G2 while SU(2)2 is non-anomalous. Therefore SU(2)2,
which is the remaining global symmetry on a generic point on the Higgs branch, must be
non-anomalous.
However on a generic point on the Higgs branch we have an half-hyper in the 4 of
SU(2)2 which does contribute to the anomaly. This can be readily seen by decomposing
the 14 of G2 under the SU(2)1 × SU(2)2 subgroup. Thus it is apparent that we cannot
match the SU(2) anomaly with the anomaly of G2. Therefore the G2 theory is inconsistent,
since global anomaly associated to π6(G) cannot be canceled by adding topological degrees
of freedom, as argued in e.g. section 5 of [15].
5
Four-dimensional theories
In this section we implement the strategy given in section 2 for 4d N = 2 theories. We
perform the analysis assuming that the theories in question are superconformal. The
analysis is slightly different depending on whether G is a group of type SU, SO or Sp and
the exceptional groups. We next discuss each in turn.
5.1
G is of type Sp or one of the exceptionals
When the group is of type Sp or the exceptionals then the symmetry G′ is a simple group.
We take the anomaly polynomial of the theory to be:
Iorigin = − dH
6
3 c1(R)3 +
dH
12 p1(T )c1(R) − nvc1(R)c2(R) +
kG
4 c1(R) Tr FG2
(5.1)
HJEP12(07)
T G(R)
14′
5
2
4
5
2n
1
2
n
1
n
1
2
of the anomaly polynomial is dictated by N = 2 SUSY [21]. The constants dH and nv are
related to the central charges a, c through: dH = 24(c − a), nv = 4(2a − c). The constant
kG is the central charge of the flavor symmetry G.
The anomaly polynomial of the free hypers is:
6
6
Next we need to decompose the G-characteristic classes to the SU(2)D ×G′ ones, where
the relation is given in (A.2). By matching (5.1) and (5.2) we find:
dH =
dR′ + 2
2
,
nv =
′
2T G (R′)
m
′
2T G (R′)
m
.
− 1,
kG =
(5.3)
The values of the Dynkin index are in table 4. The complete results are summarized in
For Sp(n) these are just the values of n free hypers. The SCFT consisting of an
O(1) × Sp(n) half-hyper indeed has this space as its Higgs branch.
5.2
G is of type SO
In this case the group G′ is SU(2)F × SO(n − 4) which is a semi-simple group. We again
take (5.1) as the anomaly polynomial of the SCFT and decompose the SO(n) characteristic
classes by (A.5), but now the half-hypers contribute:
6
Next we can proceed to match corresponding terms. Ignoring SU(2)F terms we
found that:
Interestingly these are exactly the values for an SU(2) gauge theory with n half-hypers
which classically has this space as its Higgs branch. This is despite the fact that this theory
has a global gauge anomaly for n odd and even for n even is not an SCFT unless n = 8.
Finally we need to match the last SU(2)F dependent term. This leads to the constraint
n − 4 = kSO(n) which is only obeyed if n = 8.
5.3
In this case the group G′ is U(1)F × SU(n − 2). We again take (5.1) as the anomaly
polynomial of the SCFT, but now the half-hypers contribute:
6
Assuming n > 3 and ignoring U(1)F terms we find that:
Interestingly these are exactly the values for a U(1) gauge theory with n hypers which
classically has this space as its Higgs branch. This is despite the fact that this theory is
not an SCFT. Indeed, to match the last U(1)F dependent term, we need the constraint
n2(n − 2) = 2n(n − 2) which has the solution n = 2. This is incompatible with n > 3.
When n = 3. For n = 3, we now only have U(1)F and so Tr(FS2U(n−2)) vanishes.
Matching terms we find:
dH = 2,
nv = 2,
kG = 3 .
These are precisely the values of the AD SU(3) theory.
When n = 2. For SU(2) we have only SU(2)D as a remaining global symmetry and so
we only get the constraints:
dH = 1,
nv = kSU(2) − 1.
These are obeyed for both the O(1) × SU(2) half-hyper and the SU(2) AD theory, which
are the SCFTs known to have MSU(2) as their Higgs branch.3 Additionally it is obeyed for
a U(1) gauge theory with two charge +1 hypermultiplets, even though it is not an SCFT.
5.4
Global anomalies
So far we have used local anomalies to constraint properties of 4d N = 2 theories that have
the one-instanton moduli space MG as their Higgs branch. We can put one additional
constraint using anomalies under large gauge transformation of [23]. These exist only for
groups with π4(G) 6= 0, which are only Sp groups for which π4(Sp(n)) = Z2. When Sp
group is embedded in Sp group, the global anomaly should match the global anomaly.
When Sp group is embedded in SU group, the global anomaly can match the standard
triangle anomaly [24].
3Other known examples of the SCFT whose Higgs branch is MSU(2)=C2/Z2 include the superconformal
point of SO(4k + 2) SYM [22]. However, these theories do not fit within the class of theories considered in
this paper. At the generic point on the Higgs branch, the spectrum we obtain is not free hypermultplets
but the interacting SCFT without Higgs branch, i.e. the superconformal fixed point of SU(2k − 1) SYM.
This can be readily seen by the class S description of the SCFTs.
(5.7)
HJEP12(07)
(5.8)
(5.9)
In our case this implies a non-trivial constraint only for G = Sp(n), F4, G2. In the
first case, G = Sp(n), the unbroken group on the Higgs branch is Sp(n − 1), and as the
matter content is a single fundamental half-hyper, it suffers from this anomaly. This can
be accommodated in the SCFT if the original Sp(n) also has the same anomaly. This again
agrees with the expectation from the ADHM construction.
Both G2 and F4 cannot have an anomaly. However, they break on the Higgs branch
to groups that can, SU(2) for G2 and Sp(3) for F4. Therefore for these to be possible the
anomaly must vanish on the Higgs branch. This is true for G2 as the 4 of SU(2) does
not contribute to the anomaly. However, this is not true for F4 as the 14′ of Sp(3) does
contribute to the anomaly. Thus this excludes F4, since global anomaly associated to π4(G)
cannot be canceled by adding topological degrees of freedom; see e.g. section 5 of [15]. The
existence of the G2 theory is still an open question.
6
Two-dimensional theories
In this section we analyze the 2d N = (0, 4) theories.
We denote the R-symmetry as
SU(2)R × SU(2)I and the general form of the anomaly polynomial is written as
where nV , dH , nF and kG are the unknown coefficients determined below. Note that the
SU(2)I and the gravitational part of the anomaly can be matched directly on the Higgs
branch. We also note that there are no global gauge anomalies in 2d since π2(G) = 0 for
all Lie groups.
In 2d, we can consider the slightly generalized situation: we can also have Fermi
multiplets in addition to hypermultiplets on a generic point of the Higgs branch. Fermi
multiplet consists of a single left-moving Weyl fermion transforming some representation
RF under G.4 In this section, we also examine how the anomaly matching changes when
we allow the Fermi multiplets as the massless spectrum.5
6.1
G is of type Sp or one of the exceptionals
In this case, the unbroken subgroup G′ is simple. If we denote the representations of Fermi
multiplets under G′ as Pm Nm, then the anomaly polynomial of free multiplets is given as
Ifree = c2(D)+
4
2
2+dR′ c2(I)+
2+dR′ −Pm Nm
24
2T G (R′)−2 Pm T G (Nm)
′
4
Tr(FG2′ ).
On the other hand, by using (A.2) and (A.3), the anomaly (6.1) becomes
Igeneric = (kG − nv)c2(D) + dH c2(I) +
4
where m is 3 for G2 and 1 for other cases.
4Some references (e.g. [25]) define a pair of left-moving Weyl fermions transforming in conjugate
representations as a 2d N = (0, 4) Fermi multiplet. Here we choose not to use such a definition.
′
5In this note, we only consider Fermi multiplets transforming non-trivially under G . The effect of
neutral Fermi multiplets is to change the value of the gravitational anomaly.
(6.2)
(6.3)
branch.
the anomaly is given by
Without Fermi multiplets. If we assume that there are no Fermi multiplets, the
anomalies (6.2) and (6.3) can be matched by the data summarized in table 3.
The cases with G = E8, E7, E6, F4 reproduce the anomaly on a single self-dual string6
in minimal 6d N = (1, 0) theories for n = 12, 8, 7, 5:
Istring = −(n − 1)c2(R) + (3n − 7)c2(I) +
4
6d. To the best of our knowledge, we do not know an example of 2d N = (0, 4) SCFT with
Higgs branch MG2 and no Fermi multiplets.
With Fermi multiplets. Next we consider the cases with Fermi multiplets on the Higgs
As examples, let us consider nf fundamental Fermi multiplets of G′. For the G = E7,
Ifull = −(7 − nf )c2(R) + 17c2(I) +
4
17 − 3nf
12
where we included the SO(nf ) symmetry acting on Fermi multiplets. This anomaly
precisely agrees with that of a single string in 6d E7 gauge theory with nf /2 hypermultplets.
Similarly, G = E6, F4 cases reproduce the anomaly of a single string in 6d G = E6, F4
gauge theory with nf fundamental hypermultiplets.
Finally, we consider G = G2. The anomaly can be matched by
Ifull = −
4
7 − nf
3
c2(R) + 3c2(I) +
where we included the SU(nf ) flavor symmetry acting on the Fermi multiplets. For nf =
1, 4, 7, (6.6) reproduces the anomaly of a string in the 6d G2 gauge theory with nf = 1, 4, 7
fundamental hypermultiplets.
6.2
G is of type SO
In this case, the unbroken group is SU(2)F × SO(n − 4). If we denote the representation
of the Fermi multiplets by Pm(nm, Nm), the anomaly of the free multiplets is given by
Ifree = c2(D)+(n−3)c2(I)+
4
2n−6−Pm nmNm
24
p1(T )
+
n−4−2 Pm NmT SU(2)F (nm)
4
Tr(FF2 )+
2−Pm nmT SO(n−4)(Nm)
2
Tr(FS2O(n−4)).
(6.7)
On the other hand, by using (A.5), the anomaly (6.1) becomes
Igeneric = (kG−nv)c2(D)+dH c2(I)+
4
2dH − nF
24
4
p1(T )+ kG Tr(FF2 )+ kG Tr(FS2O(n−4)). (6.8)
4
6We have subtracted the anomaly of the center-of-mass mode from the result presented in [18].
Without Fermi multiplets.
Comparing (6.7) and (6.8) in the case of Fermi multiplets,
the anomaly can be solved by
if we ignore the SU(2)F part. This precisely agrees with the values of the SU(2) gauge
theory with n half-hypers, though it is anomalous in 2d. If we include the matching of
SU(2)F , the solution exists only for G = SO(8) and we obtain the anomaly of (6.4) for
n = 4. Indeed, the worldsheet theory on a single string in minimal 6d N = (1, 0) SCFT for
n = 4 has the Higgs branch MSO(8).
With Fermi multiplets. Let us consider the cases with Fermi multiplets. The matching
of SU(2)F puts a constraint
An example of solution of these constraints is obtained by setting 1 ≤ m ≤ n − 8, Nm = 1
and nm = 2 for all m. The anomaly polynomial is
Ifull = −3c2(R) + (n − 3)c2(I) +
4
5
12 p1(T ) + Tr(FS2O(n)) +
1
4
Tr(FS2p(n−8)),
(6.11)
where we have included the global symmetry acting on (n−8) free Fermi multiplets. This is
precisely the anomaly of a single string in 6d SO(n) gauge theory with (n − 8) fundamental
hypermultiplets.
6.3
G is of type SU
If we denote the representation of the Fermi multiplets as ⊕m(Nm)nm under SU(n − 2) ×
U(1)F , the anomaly of the free multiplets is
(6.12)
(6.13)
(6.14)
Ifree = c2(D)+(n−1)c2(I)+
4
− n2(n−2)−
2 m
Igeneric = (kSU(n) − nv)c2(D) + dH c2(I) +
4
− kSU(n)n(n − 2)c1(U(1)F )2.
On the other hand, by using the decomposition (A.7), we have the anomaly
Without Fermi multiplets. Let us first consider the case n ≥ 4. If we ignore the U(1)F
part, the matching between (6.12) and (6.13) can be solved by
which precisely agrees with the values of the U(1) gauge theory with n hypermultiplets,
though it is anomalous in 2d. The matching of U(1)F forces us to set n = 2, which
contradicts with our assumption.
When G = SU(2), the matching can be solved by
Ifull = −(kSU(2) − 1)c2(R) + c2(I) +
4
1
12 p1(T ) +
4
kSU(2) Tr(FS2U(2)),
where kSU(2) is an undetermined coefficient. If we set kSU(2) = 1, we reproduce the anomaly
of the O(1) × SU(2) half-hyper as in 4d and 6d.
When G = SU(3), the matching can be solved by
which coincides with the anomaly (6.4) for n = 3. Indeed, the worldsheet theory of a single
string in minimal 6d N = (1, 0) SCFT for n = 3 is the SCFT with Higgs branch MSU(3).
With Fermi multiplets. Let us consider the case with Fermi multiplets for n ≥ 4. We
consider two cases. When Nm≥1 = 1, the matching can be solved by
as long as the U(1)F charges satisfy
nV = 1,
An example of solutions of (6.18) is obtained by setting 1 ≤ m ≤ 2n and nm = (n − 2) for
all m. The full anomaly is
Ifull = −c2(R) + (n − 1)c2(I) −
4
1
12 p1(T ) +
2
4
1
4
Tr(FS2U(n)) +
Tr(FS2U(2n)F ),
where we have included the contribution of the flavor symmetry SU(2n)F , acting on the
Fermi multiplets of the same U(1)F charges. This is precisely the anomaly of a single string
in 6d SU(n) gauge theory with 2n fundamental hypermultiplets.
When N1 = (n − 2), Nm≥2 = 1, the matching can be solved by
as long as the U(1)F charges satisfy
nv = 0,
An example of solutions of (6.21) is obtained by setting 1 ≤ m ≤ n + 9, n1 = (n − 4) and
nm = (n − 2) for all m ≥ 2. The total anomaly is given by
Ifull = (n − 1)c2(I) −
4
1
3 p1(T ) +
1
4
1
4
Tr(FS2U(n)) +
Tr(FS2U(n+8)F ),
where we have included the global symmetry SU(n + 8) acting on the Fermi multiplets of
the same U(1)F charge. This precisely agrees with the anomaly of a single string in 6d
SU(n) gauge theory with Nf = n + 8, NΛ2 = 1 hypermultiplets.
(6.15)
(6.16)
(6.17)
(6.18)
(6.19)
(6.20)
(6.21)
(6.22)
Acknowledgments
The authors would like to thank Kantaro Ohmori and Kazuya Yonekura for useful
discussions. HS is partially supported by the Programs for Leading Graduate Schools, MEXT,
Japan, via the Leading Graduate Course for Frontiers of Mathematical Sciences and
Physics. HS is also supported by JSPS Research Fellowship for Young Scientists. The
work of YT is partially supported in part by JSPS Grant-in-Aid for Scientific Research
No. 25870159. The work of YT and GZ are partially supported by WPI Initiative, MEXT,
Japan at IPMU, the University of Tokyo.
A
Decomposition of characteristic classes
′
In this appendix, we collect the formulas relating the characteristic classes for G and G ,
used in the main body of the paper. We define the Tr by the trace in the adjoint
representation, divided by the dual Coxeter number of G. The Dynkin index of the representation
R of gauge group G relates the Tr FG2 via
trR FG2 = T G(R) Tr FG2 ,
where trR is the trace in the representation R. We list the values of T G(R) relevant in this
paper in table 4.
When G is one of the exceptionals.
Since there are no independent quartic Casimir
invariants in this case, we only have to consider Tr FG2 . The unbroken subgroup G′ is
simple. The formula is
Tr(FG2 ) = 4c2(D) + m Tr FG2′ ,
where m = 3 for G2 and 1 for any other group.
When G is of type Sp. The unbroken subgroup is Sp(n − 1) in this case. The Tr FS2p(n)
is related that of Sp(n − 1) via
The trfund FS4p(n) is related by
Tr(FS2p(n)) = 4c2(D) + Tr FS2p(n−1).
trfund FS4p(n) = 2c2(D)2 + trfund FS4p(n−1).
When G is of type SO. The unbroken subgroup is SU(2)F × SO(n − 4) in this case.
The Tr FS2O(n) is related via
Tr FS2O(n) = 4c2(D) + Tr FF2 + Tr FS2O(n−4).
The trfund FS4O(n) is related by
trfund FS4O(n) = 4c2(D)2 + 2 trfund FF4 + 12c2(D) trfund FS2O(n−4) + trfund FS4O(n−4). (A.6)
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
HJEP12(07)
When G is of type SU. First, we assume n ≥ 4. The Tr FS2U(n) becomes
Tr FS2U(n) = 4c2(D) − 4n(n − 2)c1(U(1)F )2 + Tr(FS2U(n−2)),
and trfund FS4U(n) becomes
trfund FS4U(n) = trfund FS4U(n−2)−8c1(U(1)F ) trfund FS3U(n−2)−24c1(U(1)F )2 trfund FS2U(n−2)
+2c2(D)2−12(n−2)2c1(U(1)F )2c2(D)+2n(n−2)(n2−6n+12)c1(U(1)F )4.
The cases of SU(2) and SU(3) are quite exceptional since these groups have no
indeHJEP12(07)
pendent quartic Casimir and we only have to consider Tr F 2. For G = SU(2), SU(2)R is
identified with the original G and we simply take
For the case of SU(3), we use
Tr FS2U(2) = 4c2(D) = 4c2(R).
Tr FS2U(3) = 4c2(D) − 12c1(U(1)F )2.
(A.7)
(A.8)
(A.9)
(A.10)
Open Access.
Attribution License (CC-BY 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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