JHEP01(2018)124
projection on any of the gauge groups, and condition (2.3) simply imposes r
i
= n
0
i for
i = 1, . . . , N − 1 (with a flavor symmetry SU(r
N
= N n
0
) on the right). The (1, 0) SCFT
corresponding to this quiver is very similar to the so-called (rank-N) E-string theory [8],
with the crucial difference that it cannot be engineered in M-theory because of the D8’s
(sourcing a nonzero Romans mass F
0
=
n
0
2π
). For this reason it was dubbed (rank-N)
“massive E-string theory” in [52]. (In particular when n
0
= 1 we have an extra E
8
flavor
symmetry on the left, whose presence can be argued for by lifting the particular D8-O8
−
system to M-theory as in [64];
13
more generally, for 1 ≤ n
0
≤ 8 we have an E
1+(8−n
0
)
sym-
metry, using the definitions of [66].) The quiver was constructed in [52, figure 6] (which we
reproduce in figure 4c), the dual AdS
7
vacuum is given by [52, eq. (5.2)] and its a conformal
anomaly at large N by [52, eq. (5.13)].
14
Given that this case has already been discussed at
length in [52], in the remainder we will only treat the generic case where k 6= 0, i.e. r
0
6= 0.
In the second situation, the O8
±
sits on top of a half-NS5-brane stuck on the plane, at
x
6
= 0. The orientifold projects out the tensor multiplet contributed by that NS5, does not
act on the gauge group SU(r
0
), but acts on the bifundamental matter coming from strings
D6
0
-image D6
0
: we have a hypermultiplet in the (anti)symmetric of SU(r
0
= k). If we also
overlay n
0
pairs of image D8’s onto the O8
±
, all gauge groups will be SU(r
i
= k∓i(8±2n
0
))
for i = 1, . . . , N −1, and we have a flavor symmetry SU(r
N
= k∓N(8±2n
0
)) (again, this is
due to (3.10)). We can also add D8-branes as usual. The quivers are depicted in figure 4b.
The corresponding supergravity vacua will be defined in terms of a polynomial α(z)
that, in the first subinterval z ∈ [0, 1], is characterized by a nonzero constant term α
0
as
well as coefficient r
0
of the quadratic term, and in the last subinterval z ∈ [N − 1, N] by a
nonzero quadratic coefficient r
N
(this is a singular pole), but vanishing constant term α
N
,
as the right tail of the quiver is the same as for the pure SU(k) case. The correspondence
will be made more precise in section 2.
Let us now discuss the case with O6-planes overlaid onto D6-branes. The brane con-
figurations are depicted in figure 5. Given that the O6 charge changes sign whenever the
former crosses an NS5, the situation where an O8 is stuck on the latter at x
6
= 0, reflect-
ing an O6
±
into itself, is not consistent. Thus we only need to consider the first situation
(O8-plane between an NS5 at φ
1
> 0 and its image at −φ
1
). The combined O6
+
-O8
−
projection produces an SU(r
0
= k) gauge group (i.e. only half of the k D6 pairs under
the O6 projection count), followed by a sequence of gauge groups SO(p
i
) × USp(q
i
) for
i = 1, . . . , N −1, and an SO(p
N
) or USp(q
N
) flavor symmetry, according to the particular
theory at hand (i.e. q
N
= 0 or p
N
= 0 respectively). Notice that by simultaneously flip-
ping the O6 and O8 charge we can exchange the gauge factors as to produce a sequence
USp(q
i
) × SO(p
i
). Once again, we can add flavors of each gauge group by inserting D8-
branes across the finite-length D6 stacks (e.g. n
0
D8 pairs overlaid onto the O8
±
-plane will
produce a USp(2n
0
), respectively SO(2n
0
), flavor group) as long as conditions (3.10) are
satisfied at each node. The quivers are depicted in figure 5c.
13
The nonperturbative enhancement SO(2(8 − n
0
)) → E
1+(8−n
0
)
is due to D0-branes [65, 66], which
become tensionless as g
s
→ ∞ since T
D0
∼ g
−1
s
.
14
In that formula M is the number of D6-branes in the rightmost semi-infinite flavor stack, i.e. M := Nn
0
,
which also diverges as N → ∞.
– 10 –
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