8 J. Rasmussen / Nuclear Physics B 950 (2020) 114865
such that all t ∈ Q
+
are covered. Accordingly, we refer to levels of the form (3.1)simply as
fractional.
For fractional le
vel, following (3.2), the Verma module characters are invariant under simul-
taneous translations of r, s by multiples of p, p
such that (r + p)(s + p
) > 0or r + p >
0, s + p
= 0:
χ
j
r+p ,s +p
(q, z) =
χ
j
r,s
(q, z), ∈ Z. (3.5)
Contrarily, the affine Kac characters (2.42)are all distinct.
3.2. Ve r m a modules
For t ∈ Q
+
and j
r,s
as in (2.37), the submodule structure of the Verma module V
j
r,s
depends
critically on whether or not r is a multiple of p. Indeed, for r not a multiple of p, the submodule
structure is described by a Loewy diagram of the form [46,44,47,3]
r/∈ pZ:
•
•
→
•
→
•
→
•
→
•
→
•
→
•
→
•
→
•
→
•
→
•
→
•
→
α
0
β
0
α
−1
β
−1
α
−2
β
−2
α
−3
β
1
α
1
β
2
α
2
β
3
α
3
...
...
(3.6)
whereas for r a nonzero multiple of p, it is described by a Loewy diagram of the form
r ∈ pZ
×
: •
→
•
→
•
→
•
→
•
→
•
→
γ
1
γ
2
γ
3
γ
4
γ
5
γ
6
...
(3.7)
Each node in these diagrams represents a Verma module of highest weight α
i
, β
i
or γ
i
, where
an arrow connecting two such modules, V
μ
→ V
μ
, indicates that the target module, V
μ
, is a
submodule of the initial module, V
μ
. Moreover , the submodules in (3.6)or (3.7)are all generated
by singular vectors, where the maximal dimension of the space of singular vectors of any given
weight is one.
Every Ve
rma module of the form V
j
r,s
appears as a submodule of exactly one of the (p + 1)p
distinct Verma modules
V
j
ρ,s
0
, 0 ≤ ρ ≤ p, 0 ≤ s
0
≤ p
− 1, (3.8)
where it is noted that j
0,s
0
= j
−p,−(p
−s
0
)
= j
p,p
+s
0
and j
p,s
0
are indeed of the form (2.37). The
submodule structure of V
j
r,s
thus follows from the Loewy diagram, (3.6)or (3.7), of the corre-
sponding ambient Verma module (3.8); one merely has to identify the ambient Loewy diagram
and locate the position of the Verma module V
j
r,s
in that diagram. To see this in action, we first
recall the submodule structures of the Verma modules (3.8).
For ρ = r
0
, where as usual 1 ≤ r
0
≤ p − 1, the Loewy diagram is of the form (3.6) with
α
i
(r
0
,s
0
) = j
r
0
+2ip,s
0
= j
r
0
,s
0
+ ip,
β
i
(r
0
,s
0
) = j
−r
0
+2ip,s
0
= j
r
0
,s
0
− r
0
+ ip, i ∈ Z.
(3.9)
For ρ = δp, where δ = 0, 1, the Loewy diagram can be obtained from a Loewy diagram of the
form (3.6)by imposing
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